LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 Class 
 
FIELD 
 
 OOK 
 
 FOR 
 
 CIVIL ENGINEERS. 
 
 BY 
 
 DANIEL C^RHART, C.E., 
 
 DEAN AND PROFKSSOK OK CIVIL ENGINEERING IN THE WESTERN 
 UNIVERSITY OF PENNSYLVANIA. 
 
 OF THE 
 
 UNIVERSITY 
 
 BOSTON, U.S.A.: 
 GINN AND COMPANY. 
 
 1903. 
 

 COPYRIGHT, 1893, 1903, BY 
 DANIEL CAJRHART 
 
 ALL BIGHTS RESERVED 
 
- 
 
 PREFACE. 
 
 THE work of the Civil Engineer is varied and extensive. 
 He may be called upon to survey a tract of land ; to lay out a 
 town ; plan a system of water supply, and sewerage ; to locate 
 and construct a road, canal or railroad ; to design and erect a 
 bridge, or roof ; to build a light-house ; improve the regimen 
 of a water course ; maintain a highway in good condition ; in 
 fact, several of these pages would be necessary to even enu- 
 merate the requirements which the modern engineer is ex- 
 pected to meet, and almost every topic would furnish material 
 for a manual. In this book it is proposed to treat that part of 
 the Civil Engineer's work which will enable him to locate the 
 center line of a railroad ; to set the stakes incident to its con- 
 struction ; to compute the quantities ; and to solve the prob- 
 lems pertaining to track laying. 
 
 The book is written for students of civil engineering, and 
 to satisfy a demand, often expressed by field engineers, for a 
 manual convenient in size, containing the desired information, 
 systematically arranged, fully illustrated, and easy of reference 
 
 It abounds in problems, such as are likely to arise in ordi- 
 nary field practice, accompanied by full explanations, and with 
 illustrative examples wherever it seems at all necessary. 
 
 The arrangement of the matter is the natural one ; in other 
 words, the various subjects are introduced and treated in the 
 order in which they present themselves in actual work. Ac- 
 cordingly reconnoissance, preliminary survey and location, 
 with the required instruments, occupy the first two chapters. 
 In Chapters III. and IV. there are numerous formulas derived, 
 
(V PKEFACE. 
 
 practical problems proposed, and solutions indicated for them 
 in connection with running simple and compound curves, 
 which are supplemented in Chapter Y. by a set of miscella- 
 neous questions, among which several forms of the Y problem 
 are discussed. 
 
 Chapter VI. treats of construction, consequently here, among 
 other things, are introduced methods of setting out the work 
 and computing quantities. The difficulties which the young 
 engineer experiences in setting slope stakes, were kept vividly 
 in mind when writing this chapter. It is believed that the 
 subject is presented in such a simple manner, and so fully 
 illustrated, that he can easily comprehend it, and make ready 
 use of the methods explained. 
 
 Chapter VII., on Frogs and Switches, is from the pen of my 
 esteemed friend and former student, Lewis C. Weldin, C. E., 
 Assistant Engineer Pennsylvania Railroad. Mr. Weldin's 
 twenty years' experience in the engineering department of this 
 famous highway, qualifies him to present, with the authority 
 of an expert, this subject in a most practical manner. By 
 adopting a notation slightly different from that of other writers 
 under this head, he has, in many cases, obtained results much 
 simpler than any hitherto published, and he has increased the 
 value of his work by the introduction of numerous formulas 
 and examples, selected from his extensive practice. I there- 
 fore desire to express here my acknowledgments to Mr. Weldin 
 for this valuable chapter. 
 
 The book contains all the necessary tables for field use. 
 Among them are included tables of natural trigonometric func- 
 tions, sines, cosines, secants, cosecants, tangents, cotangents, 
 versed sines, and exsecants, as well as tables of radii, long 
 chords, squares, cubes, functions of a one-degree curve, and 
 numerous others. The author believes that time is saved by 
 
PREFACE. V 
 
 using, in the field, the natural instead of the logarithmic trigo- 
 nometric functions ; accordingly he has omitted the long 
 tables, usually found in field books, of logarithmic sines, tan- 
 gents, etc. A table of logarithms of numbers, however, is 
 inserted. The sines, cosines, secants and cosecants, are on 
 tinted paper, and placed near the middle of the set of tables. 
 All others are on white paper ; tangents, versed sines, and ex- 
 secants, near the end, and logarithms of numbers, radii, long 
 chords, etc., near the beginning of the set. Thus the tinted 
 pages indicate plainly where four important tables may be 
 found, as well as favor the eye while rea ling thereon, and 
 with a little familiarity, soon acquired, as to the relative 
 position of others, a person can quickly turn to any one which 
 he may need. 
 
 Besides the introduction of tinted paper, there will be 
 noticed, for the first time in a field book, the absence of 
 vertical lines in some of the tables, and the consequent 
 facility and ease with which these pages can be consulted, 
 will doubtless be remarked. 
 
 In submitting this work to the profession, the author trusts 
 that it will prove helpful to the student of civil engineering in 
 acquiring a knowledge of field practice, and useful to the 
 young practitioner in pursuing his profession, as well as a con- 
 venient and reliable work of reference to all who use engineer- 
 ing instruments in the field. With such results accomplished, 
 
 his aim will be attained. 
 
 D. C. 
 
 SEPTEMBER, 1893. 
 
NOTE. 
 
 AT the earnest solicitation of those who make use of the 
 book both in office and field work, Tables XX and XXI of the 
 logarithms of trigonometric functions have been added. These 
 tables were printed from electrotypes from Nagle's "Field 
 Manual for Railroad Engineers." Also, a growing demand 
 for some method, easily applied, to pass from tangent to curve 
 or vice versa will be met, it is believed, by an Appendix on the 
 Transition Curve. My acknowledgments are due and are 
 hereby made to W. C. Armstrong, Bridge Engineer of the 
 Chicago and Northwestern Railway, for permission to use 
 matter from his publications for this Appendix. His method 
 will be found extremely simple as regards both elucidation 
 
 and application. 
 
 D. C. 
 JANUARY 3, 1903. 
 
COISTTEOTS. 
 
 CHAPTER I. 
 RECONNOISSANCE. 
 
 PAGE 
 
 The Instruments 1 
 
 General Directions 2, 3 
 
 CHAPTER II. 
 
 PRELIMINARY SURVEY. 
 A. THE TRANSIT PARTY. 
 
 Adjustment of transit 4,5 
 
 The stadia 6-9 
 
 The gradienter 10 
 
 The vernier 11,12 
 
 Duties of members of the corps 13-15 
 
 B. THE LEVEL PARTY. 
 
 Adjustment of level 15, 16 
 
 Leveling 17,22 
 
 Duties of the leveler and the rodman 23 
 
 C. THE TOPOGRAPHIC PARTY. 
 
 Contouring : 24, 26 
 
 Remarks on locating the line 27,28 
 
 CHAPTER III. 
 
 SIMPLE CURVES. 
 
 A. DEFINITIONS AND FUNDAMENTAL FORMULAS. 
 
 Definitions , 29,30 
 
 To find the radius R in terms of the degree of curve D 31, 32 
 
 The deflection angle denned 33 
 
 The long chord C and external distance E denned 34 
 
Vlll CONTENTS. 
 
 PAGE 
 
 To find the length of a curve L 34 
 
 Observations and examples 35 
 
 Remarks on field practice 36 
 
 To find the tangent T, given E, or D, and central angle a 37 
 
 To find R, given Tand a 37 
 
 To find C, given R and a 38 
 
 To find the mid-ordinate M, given R and a 38 
 
 To find M, given E and a 38 
 
 To find Jf, given R and C 38 
 
 To find any ordinate, given R, C and d 39, 40 
 
 To find E, given R and a. 40 
 
 To find E, given T and a 41 
 
 To find T, given E and a 41 
 
 To find E, given M and a 41 
 
 Remarks on the application of formulas 41 
 
 Application of Tables 42, 43 
 
 Formulas grouped for convenience 43,44 
 
 B. LOCATING SIMPLE CURVES. 
 
 To locate a curve, given D 44 
 
 To find direction of tangent at a given point 45 
 
 Remarks on methods of procedure 46 
 
 Methods of keeping notes 47 
 
 Remarks on the notes 48 
 
 To locate a curve by offsets from tangent 48 
 
 To locate a curve by offsets from chords produced 49 
 
 C. OBSTACLES. 
 
 To pass an obstacle on a curve 51 
 
 To locate a curve when the P.I. is inaccessible 52 
 
 To locate a curve when the P.C. is inaccessible 52 
 
 To locate a curve when the P.C. and P.I. are inaccessible 53 
 
 To pass from curve to tangent when the P.T. is inaccessible. 54, 55 
 To extend a curve across a stream , 56 
 
 D. PROBLEMS IN CHANGE OF LOCATION. 
 
 To find the change in R and E for a given change in T. 56, 57 
 
 To find the change in R and T for a given change in E 58 
 
 To find the change in T and E for a given change in R 58 
 
CONTENTS. IX 
 
 PAGE 
 
 To find new R for curve to connect P.C. and a parallel tangent 59 
 
 To find new P.C. to connect by same R with parallel tangent 60 
 To find new R and P.C. for curve ending in parallel tangent 
 
 at a point on same radial line 62 
 
 To find new P.C. for new tangent from same P.T 02 
 
 To find new R and P.C. for new tangent from same P.T 63 
 
 CHAPTER IV. 
 
 COMPOUND CURVES. 
 
 A. PROBLEMS IN LOCATION. 
 
 To find JR', given R, T, T' and a 64 
 
 To find R', given JJ, d, and angles between the chord and tan- 
 gents 65 
 
 To find T, T', d, and the angles between them, given JJ, jR', 
 
 a' and a" 66 
 
 To find R and R' for tangent parallel to d, given d and the 
 angles between it and the tangents 67 
 
 B. OBSTACLES. 
 
 To locate a compound curve when the P.C.C. is inaccessible.... 68 
 Various conditions and methods 68, 69 
 
 C. PROBLEMS IN CHANGE OF LOCATION-^ 
 
 To find the change in P.C.C. for parallel tangent outside the 
 
 terminal and last R the longer 70 
 
 To find the change in P. C. C. for parallel tangent inside the 
 
 terminal and last R the longer 72 
 
 To find the change in P.C.C. for parallel tangent outside the 
 
 terminal and last R the shorter 73 
 
 To find the change in P.C.C. for parallel tangent inside the 
 
 terminal and last R the shorter 74 
 
 To find a P.C.C. from which a curve of known R may run 
 
 and end in a parallel tangent 75 
 
 To find a P.C.C. and last R for curve ending in parallel tan- 
 gent at a point on same radial line. Four cases 76-79 
 
 To find P. C. C. and change last R to end curve at some other 
 point in terminal tangent. Two cases 79-81 
 
 To substitute a three-centered compound curve for a simple 
 
 curve . 81 
 
X CONTENTS. 
 
 CHAPTER V. 
 
 MISCELLANEOUS PROBLEMS. 
 
 PAGE 
 
 Given a simple curve intersected by a line, to find a point on 
 the curve whence to run a curve of given R to end in the 
 
 line as tangent. Two cases 83, 84 
 
 Given a tangent and a curve to connect by another curve 
 
 forming a Y 84 
 
 Given a curve, and radii of two others, to connect, forming a Y. 
 
 Two cases 85,86 
 
 To lay out a track of given length having circular ends 87 
 
 To substitute a simple curve for a tangent between two curves, 88 
 
 To locate a tangent to a given curve from a point without 89 
 
 To locate a definite point in a curve from some point in the 
 
 tangent....; 90 
 
 To run a curve, with a given .R, from a tangent which shall 
 
 pass through a given point 90 
 
 To prolong a line beyond an obstacle. Several methods 91, 92 
 
 To find the radius of a railroad track 93 
 
 To locate a curve parallel to a given curve 93 
 
 To connect two parallel tangents by a reversed curve. Two 
 
 94 
 
 CHAPTEE VI. 
 
 CONSTRUCTION. 
 A. GENERAL DIRECTIONS, DEFINITIONS AND PROBLEMS. 
 
 Definitions 96,97 
 
 To find where two grades will meet 97 
 
 To find where a grade will pass from cut to fill and vice versa 98 
 
 Vertical curves 99 
 
 Difference in elevation of the rails on curves 100-102 
 
 B. SETTING SLOPE STAKES. 
 
 When the ground is level transversely. Formula 103 
 
 When the ground slopes transversely. Formula 104, 105 
 
 Side-hill work 106, 107 
 
 Compound section 108 
 
 The common practice of setting slope stakes 108-110 
 
 Shrinkage Ill 
 
CONTENTS. XI 
 
 C. CALCULATING THE EARTH WORK. 
 
 PAGE 
 
 Volume of prism, pyramid, and wedge 112 
 
 The prismoidal formula 113 
 
 Sectional areas 114, 115 
 
 The volume by prismoidal formula, and by averaging end 
 
 areas 116-118 
 
 Excavation on curves 118-122 
 
 Overhaul 123 
 
 D. CULVERTS, BRIDGES AND TUNNELS. 
 
 To stake out culverts 124 
 
 To stake out bridges and trestles 125-127 
 
 Tunnels 127-132 
 
 Ballast stakes 133 
 
 CHAPTER VII. 
 FROGS' AND SWITCHES. 
 
 The gauge g^ frog angle F, lead L, point of switch, etc. , defined 
 
 134-136 
 
 Turnout from straight track. 
 
 To find L and R, given F and g 136 
 
 To find L and F, given R and g 137 
 
 Turnout from outside of curved track. 
 
 To find L and R', given R and F 138, 139 
 
 To find L and F, given R and R' 140 
 
 Turnout from inside of curved track. 
 
 To find L and R', given J^and R 141, 142 
 
 To find F and L, given R and R' 143 
 
 Crossovers. 
 
 Between straight -tracks 144 
 
 Between curved tracks 145, 146 
 
 Crossing frogs. 
 
 In straight tracks 147 
 
 In curved tracks ....147-152 
 
Xll CONTENTS. 
 
 Crossing slips. 
 
 PAGE 
 
 Between straight tracks 152, 153 
 
 Between curved tracks 153-157 
 
 Trigonometric formulas 158 
 
 Solution of right and oblique triangles 159 
 
 General formulas 100, 161 
 
 Miscellaneous formulas , 162 
 
 TABLP:S. 
 
 I. Kadii of curves 165 
 
 II. Tangents and externals to a 1 curve 168 
 
 III. Tangential offset at 100 feet 172 
 
 IV. Mid-ordinates to 100-foot chords 172 
 
 V. Long chords 173 
 
 VI. Mid-ordinates to long chords 175 
 
 VII. Minutes in decimals of a degree 177 
 
 VIII. Squares, cubes, square and cube roots 178 
 
 IX. Logarithms of numbers 195 
 
 X. Natural sines and cosines r 213 
 
 XL Natural secants and cosecants 222 
 
 XII. Natural tangents and cotangents 235 
 
 XIII. Natural versines and exsecants 247 
 
 XIV. Cubic yards per 100 feet, in terms of center-height .... 270 
 XV. Cubic yards per 100 feet, in terms of sectional area.. 276 
 
 XVI. Mutual conversion of feet and inches into meters 
 
 and centimeters 280 
 
 XVII. Mutual conversion of miles and kilometers 281 
 
 XVIII. Length of V arc of latitude and longitude 281 
 
 XIX. Stadia measurements 282 
 
 XX. Logarithmic sines and cosines 290 
 
 XXI. Logarithmic tangents and cotangents 305 
 
CONTENTS OF APPENDIX. 
 
 PAGE 
 
 Transition curve, or spiral 321 
 
 Angle turned by the spiral curve 321 
 
 Properties of the spiral..: ... 322 
 
 Comparison of the transition curve with the cubic parabola.... 322 
 Deflection from any point on curve to any other point on 
 
 curve 323 
 
 General rule regarding deflections from point to point on 
 
 curve : 324 
 
 Formulas for semi-tangent T and external secant E 324 
 
 Diagram showing parts in true proportion 325 
 
 How to lay out the curve 326 
 
 Diagram and illustrative example 327 
 
 Modification of above example 328 
 
 Additional examples 329-330 
 
 Table I. Elements of a No. 1 spiral 331 
 
 Table II. Deflections of a No. 1 spiral 332-333 
 
 xiii 
 
ERRATA. 
 
 PAGE 
 
 57. In first part of example at bottom, cot 17 was used instead 
 of 16, causing an error of one minute in degree of curve. 
 65. Tenth line from bottom, for o^Fread of. 
 
 68. The letter A is wanting in the figure at end of line DP. 
 
 69. Fifth line from top, E should be E\. 
 74. In the figure the upper T should be 2". 
 
 81. In eighth line from bottom, for radii read radius. 
 
 81. a in ninth line from bottom should be a'. 
 
 83. In last line, for PD read PP'. 
 
 85. In Fig. 49 the at the right-hand angle should be (X. 
 
 93. Right-hand member of eq. (90) should read 2 B' sin $ EOD. 
 
 94. Second line from bottom, the first DT should be TT'. 
 
 cc" 
 
 95. First equation should read E' = := * 
 
 96. Eleventh line from bottom, for affected read effected. 
 98. Seventeenth line from top should have before -J- x. 
 
 12 12 
 
 116. Third line from bottom, for read 
 
 119. Fourth line from bottom, for POP read POP'. 
 
 120. Middle term in tenth line from top should read 
 
 120. Fifth line from bottom, bGE should be CGE. 
 
 121. Second line from top, for r read -^. 
 121. Third line from top, for T ff read - 1 /. 
 121. Fourth line from top, for T 2 read -y-. 
 
 121. Sixth line from top should read, between equality signs, 
 
 J 7 (746 + 2180 + 370) <M>. 
 
 144. Third line from bottom, for sin F read sin F'. 
 
 147. Eleventh line from bottom, for C read B. 
 
 148. First member of eq. (145) should be ED. 
 
 152. In Fig. 93 there should be a B at intersection of lines KD 
 
 and AE. 
 
 153. In eqs. (160) and (161), for 360 read 180. 
 
 155. In eqs. (165) and (166), for 360 read 180. 
 
 156. In eqs. (171) and (172), for 360 read 180. 
 
 157. In eq. (173), for O'O read O"0. 
 
 216. Opposite the cosine 18 57', for 93580 read 94580. 
 
FIELD BOOK. 
 
 CHAPTER I. 
 
 BECONNOISSANCE. 
 
 1. In determining the best location for any highway, es- 
 pecially if it is to be of considerable length, it is customary 
 for the engineer to make a hasty examination of the country 
 lying in the general direction of the proposed route ; gathering 
 facts regarding streams, mountains, valleys, soil, etc., and 
 other data bearing upon the construction of the highway, the 
 business it may command, and the cost of operating it, seeking 
 thereby to limit and minimize the detailed work which follows. 
 Such examination is called a Reconnaissance. 
 
 2. The Instruments which may be used advantageously on 
 such surveys, are the pocket compass, locke level, tape measure, 
 aneroid barometer, and field-glass. 
 
 a. The Pocket Compass may be used for roughly deter- 
 mining the direction of any line. 
 
 b. The Locke Level for obtaining approximately the differ- 
 ence in height of two points by direct measurement, where the 
 points are not far apart, either vertically or horizontally. 
 
 c. The Aneroid Barometer may be used to determine quite 
 closely the difference in elevation of two places whose distance 
 in any direction may be considerable, in the following manner : 
 
 Take the barometric reading at each station ; denote the 
 reading at the lower and upper station respectively by b and 
 &', and the required difference in feet by d. 
 
 Then d = 60300 (log b log I'}. 
 
RECONNOISSANCE. 
 
 EXAMPLE. 
 
 If the reading of the barometer at the foot of a mountain 
 be 28.5 inches, and at the top, 26 inches, the height of the 
 mountain will be 
 
 d = 60300 (log 28.5 log 26) = 2404 feet. 
 
 3. In preparing to make a reconnoissance, the engineer 
 should also provide himself, if he can, with a good map of the 
 locality to be traversed, and, as far as possible, obtain from 
 persons acquainted with the country information pertaining 
 to the case in hand. From a map he will perceive the direc- 
 tion, length, and location of the water courses and their 
 tributaries, the position of mountains, valleys, etc. He will 
 ascertain upon inquiry something concerning " high " and 
 " low " water, the behavior of streams during floods, ice 
 gorges, maximum snow fall, etc. 
 
 4. Ordinarily in making a railroad along a river, bridges 
 have to be built to span tributaries. If the divide is kept, 
 
 few, if any, bridges will 
 be necessary, as may be 
 seen in the figure oppo- 
 site, where AB represents 
 the line near bank of 
 stream, and A'B' that 
 on the dividing ridge 
 between two valleys or 
 streams. To pass from 
 the valley of one water- 
 course to another it is 
 sometimes practicable to 
 do so with easy grades, 
 where their sources are quite near together, by following up 
 one and crossing the ridge by cut or tunnel, gaining the other 
 as CED. To go directly across from C to D would generally 
 involve considerable earth work, or heavier grades, though 
 there would be a saving in distance. The items of bridges, 
 
GENERAL DIRECTIONS. 6 
 
 however, on tributaries, grades, cost of earth work, and busi- 
 ness to be acquired on the different lines, as well as the main- 
 tenance of the permanent way and other operating expenses, 
 must have great weight in deciding the route. 
 
 FIG. 2. 
 
 5. Having studied carefully the maps, and obtained all 
 information possible from other available sources, the engineer 
 proceeds to traverse the country in both directions, observing 
 its topography, nature of the soil, banks, beds, and accessibility 
 of rivers; and objective points, such as passes in mountains to 
 be crossed, and depressions in ridges through which the best 
 grade may be secured, adding to his knowledge already 
 acquired, and thereby qualifying himself for an intelligent 
 decision in the matter of location. 
 
 The young engineer will gain much general information 
 bearing on this subject by careful examination of some exist- 
 ing lines traversing our country, especially will he be benefited 
 by a study of the history and location of our trans-continental 
 lines. 
 
CHAPTER II. 
 
 PRELIMINARY SURVEY. 
 
 6. A Preliminary Survey is an examination in detail of the 
 belt of country somewhere in which the location of the line is 
 likely to be made; the data for determining its limits having 
 been obtained on the reconnoissance. The field corps required 
 to make it may be organized into three parties as follows : 
 
 THE TRANSIT PARTY. 
 
 THE LEVEL PARTY. 
 
 THE TOPOGRAPHIC PARTY. 
 
 A. THE TRANSIT PARTY. 
 
 7. The Transit Party is composed of one chief engineer or 
 senior assistant, a transitman, two chainmen, two or more axe- 
 men, a stakeman, and a flagman. 
 
 The instruments needed by this party are the transit, two 
 sight-poles, a hundred-foot chain or tape, and axes for the 
 axemen. 
 
 THE TRANSIT. 
 
 The principal adjustments are 
 
 1. The Levels. 
 
 2. The Line of Collimation. 
 
 3. The Standards. 
 
 8. To adjust the levels, that is, to make the level-tubes parallel 
 to the vernier plate, or perpendicular to the axis of the instrument. 
 Set up the instrument firmly, and by the leveling screws 
 
 bring each bubble to the center of its run. Then turn the 
 plate half-way round. The bubbles should remain centered; if 
 they do not, then with the adjusting pin turn the small screws 
 at the end of the level until the bubbles are moved over half 
 the error. Then bring the bubbles again to the center by the 
 leveling screws, and repeat the operation, if necessary, until 
 the bubbles will remain centered during a complete revolution 
 of the plate. 
 
ADJUSTMENT OF TRANSIT. 5 
 
 9. To adjust the line of collimation, or, to make the line of 
 collimation perpendicular to the horizontal axis of the telescope. 
 Set up the instrument on tolerably level ground, level it, and 
 bring the intersection of the cross-hairs on a definite point two 
 or three hundred feet away. Clamp the plate to prevent 
 horizontal motion, plunge the telescope, and locate a point 
 now covered by the intersection of the cross-hairs, opposite the 
 first and at about the same distance from the instrument. Now 
 unclamp the limb, revolve it horizontally half-way round, and 
 set the intersection of the cross-hairs on the point first observed. 
 Clamp as before, and plunge the telescope again ; the intersec- 
 tion of the cross-hairs should cover the second point set; if it 
 do not, then with the adjusting screws at the side of the tele- 
 scope, screwing one in and the other out simultaneously, move 
 the vertical hair until it covers a point one-fourth the distance 
 between the last two. Repeat the operation to check the work. 
 
 10. To adjust the standards, that is, to make the horizontal 
 axis of the telescope parallel to the vernier plate;, or the locus of the 
 line of sight a vertical plane. 
 
 Set up the instrument firmly and level as before, set the in- 
 tersection of the cross-hairs on some high, well-defined point, 
 as a church spire, or some projection of a high chimney. Clamp 
 the plates, to prevent horizontal motion, depress the object end 
 of the telescope, and direct the intersection of the cross-hairs 
 to a point on the ground a few feet from the instrument. Now 
 unclamp and turn the instrument horizontally half-way round, 
 sight the first point, clamp as before, and note if in depressing 
 the telescope the intersection of the cross-hairs covers the 
 lower point. If it do not, then with the adjusting screws in 
 one of the standards,, raise or lower that end of the horizontal 
 axis until the motion of the line of sight in a vertical plane is 
 assured. Check as before. 
 
PRELIMINARY SURVEY. 
 
 AUXILIARIES. 
 
 1. The Stadia. 
 
 11. The Stadia is a compound cross-wire ring or diaphragm, 
 shown in Fig. 3, having three horizontal wires, of which 
 the middle one is cemented to the ring as usual, while the 
 others, lb and cc, are fastened to small slides, held apart by a 
 slender brass spring hoop, and actuated by independent screws 
 dd, by which the distance between the two movable wires can 
 be adjusted to include a given space; as, 1 foot on a rod 100 
 feet distant. These wires will in the same manner include 
 2 feet on a rod 200 feet distant, or half a foot at a distance of 
 50 feet, and so on in the same proportion; thus furnishing a 
 means of measuring distances especially over broken ground 
 much more easily, and even more accurately, than with a 
 tape or chain. 
 
 FIG. 3. 
 
 12. Its principles may be explained more fully as follows: 
 
 Let Fig. 3a represent a section of a common telescope with 
 but two lenses, between which the diaphragm with the stadia 
 wire is placed, and assume that 
 
 f =the focal distance of the object glass; 
 
 p = the distance of the stadia wires a and b from each other; 
 
 d = the horizontal distance of the object glass to the stadia; 
 
THE STADIA. 
 
 a = the stadia reading' (BA ); 
 
 D = the horizontal distance from middle of instrument to 
 stadia. 
 
 The Telescope is leveled and sighted 
 to a leveling or stadia rod, which is 
 held vertically, hence at right angles 
 with the line of sight. According to 
 a principle of optics, rays parallel to 
 the axis of the lens meet, after being 
 refracted, in the focus of the lens. 
 Suppose the two stadia wires are the 
 sources of those rays, w r e have, from 
 the similarity of the two triangles 
 a'lt'F and FAB, the proportion 
 d-f:a=f:p. 
 
 The quotient / : p is, or at least can 
 be made, constant, and may be desig- 
 nated by k, hence we may write 
 df=FC = ka. 
 
 To get the distance from the center 
 JV of the instrument there must be 
 added to FC the value 
 
 c = OF + ON. 
 
 ON is mostly equal to half the focal 
 length of the object glass ; hence, 
 
 c = 1.5/. 
 
 Therefore the formula for the dis- 
 tance of the stadia from the center 
 of the instrument, when that stadia is at right angles to the 
 level line of sight, is 
 
 D = ka + c. (I) 
 
 13. When the line of sight is not level it is impracticable, 
 especially in long distances, to hold the rod in a vertical plane, 
 and at the same time perpendicular to the line of sight ; hence 
 it is customary to hold the rod vertical, as in the preceding 
 case, and obtain the true distance by applying a correction 
 depending upon the angle of inclination of the sight. 
 
PRELIMINARY SURVEY. 
 
 Mt 
 
 This correction is deduced as follows : 
 Let AGB 2m; 
 
 n = the angle of inclination ; 
 
 CD must be expressed by AB ; 
 MP = the horizontal distance = J/ cos n = D ; 
 AB = a. 
 
 _J -rj 
 
 FIG. 4. 
 Now the angle 
 
 BA G = 90 + (n m) 
 ... ABG = 90 - (n + m) ; 
 AF sin 7?i 
 
 Hence 
 
 or, 
 
 and 
 
 or. 
 
 GF sin [90 + (n m)] ' 
 AF = GFsmm 
 
 cos(w m)' 
 BF sin 7/1 
 
 GF sin [90 - (71 + m)] 
 GF sin 771 
 
 cos (n + m) 
 
 .-, AF+ BF=GFs'mm[ 
 But AF+BF=a, 
 and C 
 
 Lcos(7i m) 
 CD cos 7/1 
 
 cos (71 + m)_ 
 
 2 tan tyi 2 sin m 
 
THE STADIA. V 
 
 Substituting this value of GF in the equation above, we 
 obtain 
 
 CD cos m [cos (n + m) + cos (n m)] . 
 
 * -^ 
 
 2 [cos (n + m) cos (n m)] 
 cos 2 n cos 2 m sin 2 n sin 2 m 
 
 cos n cos 2 ??7 
 cos 2 n cos 2 m sin 2 n sin 2 m 
 
 and IX = c + ka 
 
 cosn cos 2 m 
 
 Whence D = c cosn + fca cos 2 w ka sin' 2 n tan 2 m. 
 
 The third term of the second member of this equation may 
 be safely neglected, as it is very small, even for long distances 
 and large angles of elevation (for 1500', n = 45 and k = 100, 
 it is but 0.07') ; therefore, the final formula for distances, with 
 a stadia kept vertical, and with wires equidistant from the 
 center wire, is the following : 
 
 D = c cos n + ak cos %. (2) 
 
 The value of c cos n is usually neglected, as it amounts to 
 but 1 or 1.5 feet ; it is exact enough to add always 1.25' to the 
 distance as derived from the formula, 
 
 D = ak cos 2 n. (2a)* 
 
 14. The focal length of the object-glass may be found by 
 focussing the instrument upon some distant object, say a heav- 
 enly body, and measuring then the distance between the 
 plane of the cross-wires and that of the objective. ON, being 
 equal to the distance between the objective and the intersec- 
 tion of a plumb-line with the horizontal axis of the telescope, 
 may be obtained by direct measurement. The distance p, 
 between the stadia wires, may be determined as follows : 
 
 Set up the instrument on level ground, and measure forward 
 from the pLumb-line a distance equal to c, and mark the point ; 
 measure onward from the mark any convenient distance d, 400 
 or 500 feet as a base. The telescope being level, observe care- 
 fully the space a intercepted by the stadia wires on a leveling 
 
 * The above explanation of the stadia is substantially that given by 
 Mr. G. J. Specht, published by Van Nostrand, 1884, though corrected 
 and simplified. See Table XIX for reducing stadia measurements. 
 
10 PRELIMINARY SURVEY. 
 
 rod held vertically at the farther extremity of the base. Then 
 from the proportion d f : a ==f : p, the required distance p 
 may be obtained. 
 
 EXAMPLES. 
 
 1. Given f= 8 inches, base = 500 feet, and a = 5.25 feet. 
 Find p = .084 inches. 
 
 2. At what fractional part of the focal length must the 
 stadia wires be separated so that one foot on the rod will 
 correspond to 100 feet base V 
 
 2. The Gradienter. 
 
 15. The gradienter is an attachment to the transit which 
 may be used for running grades, determining distances, etc. 
 In its construction, a clamping arm extends downwards from 
 the axle upon which the telescope revolves, and is forked at 
 lower extremity to embrace a micrometer headed nut. This 
 nut moves along a screw accurately cut, making a certain 
 number of revolutions to the hundredth of a foot. The head 
 of the screw is graduated into one hundred parts, and attached 
 is a zero edge for reading the graduations. 
 
 As the proportion of the screw is such that a complete 
 revolution gives one foot vertical in a distance of 100 feet 
 horizontal, when the motion of the telescope measures this 
 foot, it necessarily follows that the rod must be 100 feet 
 distant ; or if telescope measures 1.5 feet the rod must be 150 
 feet distant. 
 
 Hence, to run a certain gradient, bring the telescope level by 
 means of the milled head screw, and note the reading ; then continue 
 the motion of the milled head one revolution and part thereof for 
 each foot and part thereof, of foot per hundred of the desired 
 gradient. 
 
 Thus, to set off a gradient of 0.5 foot per 100 feet, move 
 micrometer milled head 50 graduations from the level. 
 
 To set off 1.25 foot per 100 feet, move one revolution and 
 twenty-five graduations. 
 
THE VERNIER. 
 
 11 
 
 To measure distances, note the space on rod passed over by 
 one revolution of the micrometer head. Thus, if one revolution 
 of micrometer head passes from 4.2 to 5.6, the difference, 1.4, 
 indicates the distance of rod from instrument of 140 feet. 
 
 3. The Vernier. 
 
 16. The Vernier. Though, perhaps, it cannot be con- 
 sidered an auxiliary to the transit in the same sense as the 
 preceding, it is thought best to give in this place some 
 explanation of it, more especially for the benefit of the 
 young engineer. 
 
 It is an auxiliary scale for measuring smaller divisions than 
 those into which a graduated scale or limb is divided. The 
 smallest reading of the vernier, or least count, is the difference 
 in length between one division on the graduated scale or limb, 
 and one on the vernier. If the divisions on the vernier are 
 smaller than those on the limb, the vernier is direct; if the 
 reverse, retrograde. 
 
 10 
 
 11 
 
 12 13 
 
 15 
 
 16 
 
 17 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 R 
 
 0123 456788 10 
 
 FIG. 5. 
 
 Let LM represent any scale divided into tenths, and we wish 
 to measure or read to tenths of these divisions, i. e., to T ^. 
 
 Using a direct vernier, we should have 10 spaces on it equal 
 to 9 on the scale, and each one of them equal to -^ of T ^, or 
 y^, of the scale of graduation, giving a least count of -^^ 
 - Tim = Ttj<y> as desired. 
 
 To read to twentieths of the divisions on the scale, w^e 
 should have 20 divisions on the vernier corresponding to 19 
 on the scale, or ^each space on the vernier equal to $ X -^ 
 = *W and S ivin g a least count of inT <y = - 
 
12 PRELIMINARY SURVEY. 
 
 In general, if s = the smallest division of the scale or limb, 
 v = the smallest division of the vernier, 
 n the number of divisions on the vernier, 
 
 we shall have least count = s v = ^. 
 
 Or, the least count of a vernier is equal to the smallest 
 division of the scale or limb divided by the number on the 
 vernier. * 
 
 If s = \ degree, and n = 30, as ordinarily found on transit 
 plates, the least count will be ^ 4- 30 = ^ of a degree = one 
 minute. 
 
 If s = ^ degree, and n = 40, as sometimes found on vertical 
 arcs to solar attachments, the smallest reading = 1 -^- ? 1 ^= T ^ T j 
 of a degree = \ minute. 
 
 To space a vernier for a given least count, say 10" on a 
 
 limb graduated to 10', we must have n = = 10 -j- i = 60 
 
 sv 
 
 spaces, covering 59 spaces on the limb. 
 
 17. To read an instrument having a vernier consists in 
 determining the number of units and fractional parts thereof, 
 into which its scale or limb may be divided, from the zero 
 point on the limb, where the graduation begins, to the zero 
 point of the vernier. 
 
 It is accomplished as follows: Take the reading of the scale 
 or limb, as shown by the last graduation preceding the zero of 
 the vernier; then find a line on the vernier which coincides 
 with a line on the scale or limb. The number of this line, as 
 indicated by the graduation on the vernier, shows how many 
 units of the least count are to be added to the first reading. 
 
 EXERCISES. 
 
 1. An arc is graduated into quarter-degrees, and a vernier 
 of 30 parts covers 29 parts of the arc; find the least count. 
 
 2. Design a vernier, which, when applied to a limb gradu- 
 ated into 20', will give a least count of 20". f 
 
 * It is evidently immaterial whether LM be straight or curved. 
 t The foregoing description of the vernier is taken from the author's 
 Treatise on Plane Surveying. 
 
THE TRANSIT PARTY. 13 
 
 18. In running a long tangent, or prolonging a straight 
 line with the transit, the instrument should be in good adjust- 
 ment ; it should be properly centered, that is, set precisely over 
 the center of the station from which the observation is to be 
 made, especially if the point to be sighted back sight is 
 near the observer. The error arising from an eccentric setting 
 is inversely as the distance of the object sighted ; an eccentric 
 setting of one inch producing an error of nearly three (3') 
 minutes of arc in sighting 100 feet, while the error arising 
 from a sight of 900 feet is less than one-third (') of a minute. 
 The instrument should be level, especially across the line of 
 sight. The sight-pole should be held plumb, and exactly on the 
 proper point. The observation should be made as near the 
 bottom of the sight-pole as possible ; the line of sight as 
 nearly horizontal, and the range in both directions as nearly 
 equal, as practicable. 
 
 When a well-defined distant object can be sighted ahead, it 
 is better to set the instrument by it than to trust to a back- 
 sight. 
 
 When great accuracy is required, errors of adjustment may 
 be lessened by reversing the instrument in altitude and azimuth, 
 making two observations at each station, and taking the mean 
 of their readings. 
 
 The transit notes are written from the bottom of the page 
 upwards, analogous to the " column form " generally used in 
 surveying land. The left-hand page of the transit note-book 
 is usually prepared for this purpose, while the right-hand page 
 is suitably ruled for recording some more details of the work 
 as it may be necessary. 
 
 19. The chief of the field corps directs the operations of 
 the party, provides accommodations and subsistence, and 
 pays the necessary expenses. He indicates the direction of 
 the line, establishes the deflection points, selects suitable 
 sites for the crossing of streams, being careful always to 
 run the line as nearly as his judgment dictates, and with 
 the minimum grade in view, over ground likely to be chosen 
 for location. 
 
14 PRELIMINARY SURVEY. 
 
 On preliminary work, especially in settled districts, the 
 clearing should be kept at a minimum, that growing crops and 
 forests be injured as little as possible. 
 
 The chief verifies and supplements the work of reconnoissance, 
 he observes the quality of material to be moved, and the 
 timber and rock, with a view of using them in the construction. 
 He should be considerate of the rights of landholders and not 
 do or allow anything to be done by any member of the corps 
 which would tend to arouse active opposition to the project, 
 but, on the contrary, by due regard endeavor to secure their 
 aid. 
 
 20. The transitman keeps his instrument in adjustment, 
 observes the direction of the line either by needle or plates, 
 keeps the axemen in line when clearing, and the chainmen 
 when measuring ; he notes the directions and names of the 
 principal highways and streams intersected, and, when practi- 
 cable, property lines with the names of the owners. He records 
 also the lengths of the lines run. 
 
 21. The head chainman when measuring advances with the 
 chain and a sight pole, and being put in line, usually at a 
 chain's length, by the transitman, directs the stakeman to 
 drive a stake there. In a wooded district or brush land where 
 much clearing is to be done, the head chainman should aid 
 the transitman in giving line to the axemen, by going ahead 
 and ranging it out. 
 
 22. The rear chainman should see that his end of the chain 
 is held at the proper point, and that the chain is horizontal and 
 taut when the head chainman is setting the next succeeding 
 stake. If, for any reason, a portion of the line is run without 
 stakes, and pins are used, he should keep the tally. He should 
 see that the numbers placed upon the stakes by the stakeman 
 indicating the stations are correct, and with this in view he 
 should call out the numbers of the stations as he approaches 
 them. 
 
 Should a change be made in the direction of the line, he 
 should note mentally the plus if any, and be careful that the 
 
THE LEVEL PARTY. 15 
 
 next stake is set in its proper place, that the uniform distance 
 between stations may be preserved. He should be provided 
 with a book in which to record turning points, the intersection 
 of streams, highways, and, when practicable, property lines. 
 
 23. The axemen do the necessary chopping and clearing the 
 way, so that the transitman and leveler may have, if practicable, 
 unobstructed sight, consistent however with the directions to 
 the chief of party, and for economic reasons keeping the 
 width of the cutting a minimum. 
 
 24. The stakeman marks or numbers the stakes and drives 
 them vertically at points indicated by the head chainman, with 
 the numbers so that they can be read. by the rodman and 
 topographer as they advance along the line. If stakes are not 
 provided the party, it is the duty of the stakeman to keep 
 himself supplied, using the proper means at hand. If, however, 
 but little or no clearing is to be done, an axeman should be 
 detailed to keep up the supply. 
 
 If the deflections of the lines are determined by the plates, a 
 back flagman is required to give position of last transit point, 
 but his services will not be needed if, as is sometimes the case, 
 the bearings are taken directly by needle. 
 
 B. THE LEVEL PARTY. 
 
 25. The level party consists of a leveler and a rodman, and 
 the instruments needed are a level, a rod, and a small axe. 
 
 THE Y LEVEL. 
 The principal adjustments are 
 
 1. The Line of Collimation. 
 
 2. The Level Bubble. 
 
 3. The Wyes. 
 
 26. To adjust the line of collimation, or to make- it coincide 
 with the optical axis of the telescope. 
 
 Set up the instrument firmly, remove the pins from the 
 clips, clamp to spindle, and bring the intersection of the cross- 
 
16 PRELIMINARY SURVEY. 
 
 hairs upon a well-defined point a few hundred feet distant. 
 Then carefully turn the telescope half-way round in the wyes ; 
 the bubble-tube will then be above the telescope. Observe 
 again the point, and see if the intersection of the cross-hairs is 
 still on it. If it be not, then bring either or both cross-hairs, as 
 may be required, half-way back, using the capstan-head screws 
 perpendicular to the one which it is desired to move. 
 
 27. To adjust the level bubble, or to make the axis of the 
 bubble-tube parallel to the longitudinal axis of the telescope. 
 
 a. Clamp the telescope over either .pair of leveling screws, 
 and bring the bubble to the center of its run. Turn the tele- 
 scope in the wye's, so as to bring the bubble-tube a little to 
 either side of the center of the bar. If the bubble runs towards 
 either end, bring it back to the center by the capstan-head 
 screws, which are set in either side of the tube-holder. 
 
 Again bring the bubble-tube to the center of the bar and 
 the bubble to the center of its run ; turn the tube to either 
 side, and repeat the correction if necessary until the bubble 
 will keep its position when its tube is turned half an inch or 
 more on either side of the bar. 
 
 The necessity for this operation arises from the fact that, 
 when the telescope is reversed end for end in the wyes, in the 
 other and principal adjustments of the bubble, we are not 
 certain of placing the bubble-tube in the same vertical plane ; 
 and, therefore, it would be almost impossible to effect the 
 adjustment without a lateral correction. 
 
 b. Now bring the bubble to the centers of its run, and with- 
 out jarring the instrument take the telescope out of the wyes 
 and reverse it end for end. Should the bubble run to either 
 end, bring it half-way back by lowering that end or raising 
 the other, using the capstan-head screws at the end of the 
 tube. Verify the adjustment. 
 
 28. To Adjust the Wyes, or to make the axis of the bubble- 
 tube perpendicular to the vertical axis of the instrument. 
 
 Clamp the telescope in the wyes ; release from spindle, place 
 the telescope over one pair of leveling screws and bring the 
 bubble to the center of its run ; then turn the telescope hori- 
 
LEVELING. 17 
 
 zontally half-way round. If the bubble runs towards either 
 end bring it half-way back by the adjusting screws at the end 
 of the bar, and one-half by the leveling screws. Proceed in 
 the same manner with the telescope for the other pair of level- 
 ing screws. Repeat the operation. 
 
 29. A surface like that of still water may be called a level 
 surface. The curve formed by the intersection with such a 
 surface of a vertical plane is a line of true level; a line tangent 
 to the latter is a line of apparent level. 
 
 Leveling is the art of determining the differences of eleva- 
 tion of two or more points, or of determining how much one 
 point is above or below a line of true -level passing through the 
 other point. 
 
 30. From the foregoing it is evident that, on account of the 
 curvature of the earth, a horizontal line is not really through- 
 out its length a level line; that of two points in the same level 
 line each will have its own horizon. Hence in leveling the 
 effect of the curvature of the earth upon the comparative ele- 
 vations of different points must be taken into consideration. 
 The effect of the curvature is to make objects appear lower 
 than they really are. The air nearer the surface of the earth 
 is denser than that farther removed from the surface. This 
 difference in density, causing refraction of light, will affect the 
 elevation of a point as observed through the telescope of a 
 level, so that it also must be taken into consideration. Its 
 effect is to make objects appear higher than they really are. 
 The error caused by refraction is one-seventh as great as that 
 caused by curvature. 
 
 Let us first find an expression for the correction due to the 
 curvature of the earth. That is 
 
 31. To find the deviation from its tangent of a line of true 
 level. 
 
 Let represent the center of the earth, PN a line of 
 true level, and PN' its tangent, or a line of apparent level. 
 The distance NN' corresponding to the length of sight PN 
 is required. 
 
18 
 
 PRELIMINARY SURVEY. 
 
 O 
 
 From Geometry, 
 
 PN' 2 = NN' (2 ON + NN'); 
 
 or, 
 
 , PN 
 
 2 ON + NN' 
 
 FIG. 6. 
 
 For ordinary distances, the length 
 of the arc may be regarded as that of 
 the tangent, and NN 7 as inconsider- 
 able in comparison with 2 ON, the 
 diameter of the earth. Therefore, call- 
 ing the length of sight rf, the correc- 
 tion c, and the radius of the earth r, 
 we have 
 
 and the correction for refraction 
 
 = - 
 
 2r 14r' 
 
 (4) 
 
 then the correction due to curvature and refraction, w r hich we 
 will call C, is 
 
 or, 
 
 7 2r Ur' 
 
 Ir 
 
 (4a) 
 
 This correction must be added to the height of the object as 
 found by the level. 
 
 In practice, the necessity for using the above formula is 
 avoided whenever it is possible to set the level at equal 
 distances from the points whose difference of height is 
 required. 
 
 EXERCISES. 
 
 1. Assuming the diameter of the earth 7,926 miles, show 
 that for a mile sight c = about 8 inches. Find the value of C 
 for the same distance. 
 
LEVELING. 
 
 19 
 
 2. What is the correction due to curvature for half a mile ? 
 Two miles ? 
 
 3. What is the length of sight when C equals one-tenth of a 
 foot? 
 
 4. Show that, practically, the correction for curvature in 
 feet is equal to two-thirds the square of the distance in miles. 
 
 32. If two points M, N, whose difference of elevation is 
 required, can be observed upon from some point P about equi- 
 distant* from them, not necessarily in their line, set up the level 
 at P, and note the reading of a rod held vertically over each 
 point. The difference of the two readings will indicate the 
 difference of level required. 
 
 FIG. 8. 
 
 33. If the above method is impracticable, set up the instru- 
 ment at some point P either in or out of the line, no matter 
 which from which a rod maybe observed on the first station 
 M, and also on another point 0, in the direction of JV, about 
 equidistant with M from the instrument. Remove the level 
 to a new position P', whence observe again the rod on O, also 
 the rod reading at N. 
 
 The difference between the readings of the rod at M and 
 shows how much higher the latter is than the former, and in 
 like manner the difference of the readings at and N gives 
 
 * Placing the instrument in this position lessens the effects of inaccu- 
 rate adjustment and renders unnecessary the corrections indicated in 
 Article 31. 
 
20 PRELIMINARY SURVEY. 
 
 the difference in elevation of these points, and so on, no matter 
 what the number of stations. The difference in height of M 
 and TV 
 
 = M m Oo + Oo' Nn ; 
 or, Mm + Oo' Oo Nn 
 
 = Mm + Oo' - (Oo + Nn). 
 
 Calling Mm and Oo' back-sights, and the other two, fore- 
 sights, we perceive that the difference of level of two points is 
 shown by subtracting the sum of the fore-sights from the sum 
 of the back-sights. 
 
 34. Again, in leveling, we measure, by means of the rod, 
 how much lower than the line of sight (height of instrument) 
 certain points are. Thus we may determine the relative eleva- 
 tions of the points. Suppose, for example, it be required to 
 determine the difference in elevation of any two points. For 
 reasons already given, set the level equally distant from the 
 points. If this cannot be done, and both observations have to 
 be taken from one of the stations, especially if the distance 
 between them is considerable, correction as previously de- 
 scribed must be made. But in this case suppose it is possible ; 
 and suppose that when held on one point, the rod reads 7.255 ; 
 that is, this point may be considered 7.255 below the line of 
 sight, and 4.755 when held on the other ; then the first may 
 be considered 7.255 4.755, or 2.500 farther than the second 
 below the line of sight, or lower than the second. 
 
 35. Suppose it be required to determine the difference in 
 elevation between two points, of which one is so much higher 
 than the other that the rod is too short to give a reading 011 
 both points for one position of the instrument. In such a case 
 one or more auxiliary points, called turning points (T. P.), must 
 be used, and their relative elevations determined. Suppose the 
 reading on the first point is 0.824, and on a turning point is 
 10.432 ; the latter is then 9.608 below the former. Now the 
 instrument must be moved and set up so as to obtain a reading 
 on the turning-point, and (we will suppose) on the other of 
 the given points. 
 
LEVELING. 21 
 
 Suppose that on the former it is 1.302, and on the latter 
 8.634 ; the latter is then 7.332 below the turning-point, or 
 9.608 + 7.332, or 16.940 below the first of the two given 
 points. 
 
 The first sight taken after setting up the level is called a 
 back-sight, or plus sight ; those taken after this, and before 
 the instrument is moved, are called fore-sights, or minus sights. 
 As the difference of the readings of the rod on two points 
 gives their difference of elevation, so the difference of the sum 
 of the plus sights, and the sum of the minus sights on T. P.'s 
 and the last point will give the difference in elevation of the 
 extreme points. In the above example. 
 
 0.824 10.432 
 
 1.302 8.634 
 
 2.126 19.066 
 
 19.066 - 2.126 = 16.940, as before. 
 
 This is used as a check on level-notes. 
 
 In extended leveling, permanent elevations, fixed during the 
 progress of the work for future reference, are called bench 
 marks, or benches (B. M.). 
 
 36. In leveling, it is customary to refer all elevations to an 
 assumed level plane, called the plane of reference, the datum 
 plane, or simply the datum. Points are then said to be so 
 much above or below the datum. As this plane may be 
 assumed at pleasure, it is generally so taken as to be lower 
 than any point whose elevation is to be determined. 
 
 If the beginning of a survey is in the vicinity of tide-water, 
 this plane is assumed at the height of mean low water, which 
 elevation may be called zero. Then a point which has the 
 elevation 125.37 will be 125.37 above low water. 
 
 If two points have the elevations 125.375 and 105.213, re- 
 spectively, the former is 125.375 105.213, or 20.162 higher 
 than the latter. 
 
 The datum having once been determined, its elevation, or 
 that of a point a known distance above it, should be perma- j 
 nently fixed for future reference and comparison. 
 
PRELIMINARY SURVEY. 
 
 37. There are various forms employed for recording level 
 notes. The following is simple and convenient : 
 
 T.P 
 
 T.P. 
 
 STA. 
 
 + s. 
 
 H.I. 
 
 -S. 
 
 ELEV. 
 
 KKMARKS. 
 
 B.M. 
 
 4.725 
 
 
 
 100.000 
 
 On S. end, lower 
 
 
 
 
 
 
 step, University, 
 
 
 
 
 104.725 
 
 2.44 
 
 102.285 
 
 Main Bldg. 
 
 1 
 
 
 
 1.25 
 
 103.475 
 
 
 +60 
 
 8.417 
 
 
 .50 
 
 104.225 
 
 Pt. on sidewalk, 
 
 
 
 
 
 
 S. of Main Bldg. 
 
 2 
 
 
 112.642 
 
 7.80 
 
 104.842 
 
 
 3 
 
 
 
 6.50 
 
 106.142 
 
 
 4 
 
 
 
 4.28 
 
 108.362 
 
 
 5 
 
 3.365 
 
 
 1.36 
 
 111.282 
 
 
 
 
 
 114.647 
 
 1.25 
 
 113.397 
 
 
 7 
 
 
 
 5.45 
 
 109.197 
 
 
 The bench mark is assumed to be 100 feet above the datum. 
 The first plus sight is 4.725, which added to 100 gives 104.725, 
 the height of the instrument (H. T.) above the datum. The 
 first minus sight is taken on station 0, and is 2.44, which sub- 
 tracted from 104.725 gives 102.285, the height of this station 
 above the datum. Similarly, the height of station 1 and plus 
 60 are obtained, the latter being a turning point. The instru- 
 ment is then carried forward, set up again in a convenient 
 place to sight the T.P. and other points in the line, and thus 
 the work proceeds. 
 
 At one setting of the instrument the elevations of points, 
 besides the turning points, which are not too high or too low 
 to be reached, may be ascertained. It is evident that if any 
 error be made at a T. P., all the following elevations will 
 thereby be affected ; but if made at one of the other points, 
 only the elevation of that point will be affected. Hence the 
 importance of careful observations at the T. P.'s. 
 
 38. Wind and sunshine affect the accuracy of leveling as 
 of work with the transit. For very good work it is desirable 
 to have a calm day on which the sun is obscured by clouds. 
 In addition to a proper manipulation of the instrument, the 
 sights should not exceed 300 feet, the rod should be held ver- 
 
THE TOPOGRAPHIC PAKTY. 23 
 
 tical, and the rodman should select for turning-points good and 
 firm points on stones, pegs, etc., on which the rod may be 
 freely turned or spun around. Test the arithmetical work in 
 the foregoing table as indicated in Art. 35. 
 
 39. The leveler keeps his instrument in adjustment, sights 
 the rod when held on stations, turning points, or benches of 
 the rodman, and records his readings in a book provided for 
 the purpose. He ordinarily takes a bench reading every 1500 
 or 2000 feet, and oftener in a hilly district. If the benches 
 are judiciously chosen they may frequently serve as turning- 
 points, arid a saving of time be thereby effected. 
 
 The rod should be read to tenths of a foot on intermediates 
 and to thousandths of a foot on benches and turning points. 
 
 The leveler should observe the surface, and when practicable 
 the high water mark, of creeks and rivers. He should use a 
 Locke level to take the heights when crossing deep gulches or 
 narrow ravines, and thus save the time required to peg in the 
 usual way. The profile should be made up daily. 
 
 40. The rodman holds his rod vertically for observation by 
 the leveler at every stake, the number of which he calls out to 
 him, and wherever there is an observation to be taken on a 
 plus, for instance on bank or in bed of stream, he should make 
 known to the leveler its amount. He should be quick to 
 perceive a singular point, and prompt to decide whether a plus 
 observation is required on it. He should note the position of 
 benches and turning points, so as to find them readily, should 
 it be desired to re-level the line. He should assist the leveler 
 when required to make up the profile. 
 
 C. THE TOPOGRAPHIC PARTY. , 
 
 41. It is the duty of the Topographic party to indicate the 
 position of the prominent features of a belt of country extend- 
 ing both ways from the center line of the survey, including 
 woodland, streams, roads, buildings, etc., and to obtain the 
 necessary data to make a contour map of it. 
 
 The party usually consists of a topographer and two 
 assistants. They need a tape, rod, and an instrument for 
 
24 PRELIMINARY SURVEY. 
 
 measuring slopes, which may be a clinometer, cross-section 
 pole (slope board), or a Locke level. A small prismatic 
 compass is sometimes carried to observe the direction of 
 objects. Buildings, roads, streams and other objects are 
 usually located by offsets from the center line. 
 
 42. The data for the contour map is obtained by observing 
 from every station the slope of the ground at right angles to 
 the proposed line, and on both sides of it.* The distance 
 between points of successive contours being taken by the 
 assistants, and the total distance out from the center varying 
 from about 50 feet to several hundred feet, as may be needed 
 by the chief engineer in determining the final location. 
 
 It will be perceived that the topographic work is connected 
 with that of the level-party, and since the elevations of the 
 stations are given by the latter, the data for a contour map 
 can be easily obtained. 
 
 For instance, if a station in the center line, as shown by the 
 level notes, has an elevation of 852.4 feet, and five-foot con- 
 tours are being taken, it is evident that the 850-foot contour 
 passes 2'.4 below this point. Suppose the Locke level is used ; 
 then if the observer, whose eye we shall assume is five feet 
 high, stands at the station, and his assistant holds a rod at a 
 point where the line of sight intersects it at 7'.4, the foot of the 
 rod will be on the 850-foot contour. Measure from the station 
 to this point, and from the point observe a reading of ten feet 
 on the rod, which will show the 845-foot contour. Measure to 
 this point, and continue the observations until the limit of 
 measurement in this direction is attained. Proceed in a 
 similar manner to take the slope of the upper side, and so pass 
 along the entire line. Otherwise the observer places himself 
 where the rod reading on the station is 2.6 feet, then his line 
 of sight will indicate the 855-foot contour; he evidently stands 
 on the 850-foot contour, and without changing place he dis- 
 covers the 845-foot contour where the rod reads 10 feet. 
 
 * Sometimes on preliminary work, and especially where the ground 
 is gently undulating, the contours are taken from stations, 200 or 300 
 feet apart. 
 
THE TOPOGRAPHIC PARTY. 25 
 
 Measure from the station and locate the points as above. In 
 a similar manner the Topographer will stand on the 840-foot 
 contour when sighting the* bottom of rod held on the 845- 
 foot contour, whence a rod placed so as to give a reading of 
 10 feet will indicate the 835-foot contour. In a correspond- 
 ing manner the contours on the upper side of the line may be 
 shown. If the height of the observer's eye is not 5 feet pro- 
 ceed in general as though using the wye level. The record 
 may be kept in fractional form, the numerator indicating the 
 contour, and the denominator the distance of the observed 
 point in it from the center line. 
 
 In a similar manner a simple cross-section at any station can 
 be obtained j" the height of the eye giving the difference in 
 elevations at the successive points of observation, the distances 
 between these being measured by the tape. If a cross-section 
 pole (called also a slope board) and leveling rod are used, the 
 pole is held level and the rod indicates the difference in height 
 of its ends, while the length of the pole, usually 10 or 12 feet, 
 gives the distance between the successive readings. Cross- 
 sections may be taken very rapidly by either of these methods. 
 
 In the office the contour map is made by connecting the 
 points of equal height, and writing the elevation on the 
 several lines. 
 
 Let Fig. 9 represent a portion of a contour map drawn to a 
 scale of 400 feet to the inch. We will suppose the line nearest 
 the top of page to be 100 feet above datum or the 100-foot 
 contour, and the farthest one the 130-foot contour. The differ- 
 ence between the successive contours being 5 feet. The dotted 
 line represents the contour of the grade rising one foot per 
 station, or a one per cent, grade, from L to N. If this line be 
 adopted for the location, there would be neither center-cut nor 
 center-fill. If the straight line LN is adopted, the plan shows 
 that there would be a center-cut at b of about 11 feet, one at 
 g of 3 feet ; and a center-fill at e of 14 feet, but at d and/ 
 there would be neither cut nor fill, hence these are grade points. 
 The cuts and fills at the center of the line being shown by the 
 number of spaces and fractions thereof between the adopted 
 line and the grade-contour, and hence a tolerably close approxi- 
 
26 
 
 PRELIMINARY SURVEY. 
 
 mation may be made as to quantities. Table XIV will be 
 found useful in this connection. While, therefore, the line 
 LN has the advantage of being shorter than the dotted line, it 
 has the disadvantage of a heavier grade, besides the cuts and 
 fills named. 
 
 Various conditions and circumstances, it will be readily 
 perceived, present themselves to the engineer in deciding the 
 precise location of a center line of any great extent. 
 
 FIG. 9. 
 
 43. The small plan above simply serves as an illustration, 
 but to more fully comprehend the difficulties surrounding this 
 important matter of location, one should imagine that the 
 engineer has before him a contour map of a belt of country 
 several miles in extent, that he is endeavoring to decide upon 
 the best location of a line in that belt ; and consider what are 
 the questions which present themselves to him for solution, 
 and what are his limitations. He desires to make the line as 
 near straight as practicable between certain points ; the grade 
 the best possible, in no case to exceed a certain amount ; the 
 cost of construction a minimum. It is evident also that the 
 expense of operating and maintaining the road is involved in 
 this decision. The cost of the earth work will in general be 
 
LOCATING THE LINE. 27 
 
 lessened by equalizing as far as practicable the material in 
 cuts and fills. This, however, is not always possible, because 
 of the necessity of crossing other public highways or streams 
 at fixed grade. To lessen solid rock cutting he may consider 
 the question of change of grade or direction from what other- 
 wise he would deem the proper location. This change, if 
 made, may necessitate borrowing material elsewhere. To secure 
 proper drainage in a flat and wet section, the question of 
 waste of material or its transportation a distance to fill must 
 be settled. 
 
 These, and numerous other questions of a similar character, 
 present themselves in the decision of this matter ; and to 
 answer them properly, and thus make the best location 
 possible under all the conditions and circumstances, an op- 
 portunity is offered the engineer for the display of his best 
 judgment. 
 
 In passing upon the question of equalizing cuts and fills, it 
 must be remembered that rock measured in excavation will, 
 when broken and thrown in embankment, show an increase in 
 volume of about two-thirds ; that is to say, 3 cubic yards of 
 solid rock in cut will make about 5 cubic yards in fill. The 
 shrinkage of earth soils, which is about one-tenth, is generally 
 ignored in this connection, though it too should be considered 
 when setting out the work for the contractor. See Art. 113. 
 
 The student may sketch a contour map showing a descend- 
 ing grade of H feet per station, and determine the depth of 
 cuts and fills. 
 
 44. The engineer is thus enabled to make a paper location, 
 establish grades, and estimate approximately the quantity of 
 material to be removed. The center line must then be staked 
 out, curves run in to connect the tangents, stakes being set 
 carefully and firmly every 100 feet, at the beginning and end- 
 ing of every curve, on the banks of creeks and edges of ravines 
 intersecting the line, and at all other points where, in the 
 judgment of the engineer, the work of construction will be 
 expedited thereby ; the more important points in the line 
 being also referred to other points at known distances and 
 
28 PRELIMINARY SURVEY. 
 
 directions. These reference points (R.P.'s) should be located 
 sufficiently far from the field of operations that they will not 
 likely be disturbed during the progress of the work. The 
 levels must be carefully taken on all points in the line set by 
 transit, and plusses taken and the rod read on other points 
 in the line where there is a noticeable change in its direction 
 vertically, that a correct profile may be made, and the work of 
 cross-sectioning be facilitated. 
 
 The leveler should select his benches so far from the line 
 that they will be undisturbed during the construction, and 
 when practicable make them nearly at grade for convenient 
 future reference.. The rod should be read to thousandths on 
 benches and turning points, but only to tenths on intermediates. 
 
 The width of strip, or right of way, required for railroad 
 purposes is a variable quantity depending upon the width of 
 the roadbed and the depth of cuts and fills. A general rule 
 adopted by some roads is to have 30 feet besides the slopes for 
 single track and 60 feet and slopes for double tracks. That is 
 to say, where there is a 16-foot cut with slopes 1 : 1 the 
 required width would be 
 
 30 + 2 (1| X 16) == 78 feet. 
 
 At grade 30 feet would be the width required for single 
 track, and 60 feet for double track. It will readily be per- 
 ceived that the right of way cannot always be figured from the 
 center cut or fill, since there may be a cut or fill at the side 
 where the center is at grade. 
 
CHAPTER III. 
 
 p.c.c. 
 
 SIMPLE CURVES. 
 
 A. DEFINITIONS AND FUNDAMENTAL FORMULAS. 
 
 45. The center line of a railroad is composed of straight 
 lines and curves. The straight lines are called tangents ; the 
 curves are usually arcs of cir- 
 cles, and are simple, com- 
 pound or reversed. * 
 
 a. A simple curve is the 
 arc of a circle. 
 
 b. A compound curve is 
 composed of two simple curves, 
 or branches, of different radii, 
 both lying on the same side 
 of a common tangent drawn 
 
 at their point of union, as BE, FlG> 10- 
 
 Fig. 10. 
 
 c. A reversed curve is composed of two simple curves, or 
 branches, of the same or different radii, lying on opposite sides 
 of a common tangent drawn 
 
 at their point of union, as 
 BE, Fig. 11. 
 
 (L The point of curve, or 
 the P.O., is the point at 
 which the tangent AB ends, 
 and the curve BNE begins, 
 as B, Fig. 12. 
 
 e. The point of tangent, or 
 the P.T., is the point at which the curve BNE ends, and the 
 tangent EF begins, as E, Fig. 12. 
 
 * As a rule reversed curves are not admissible on main line, but they 
 are properly used in connection with cross-overs, sidings, and in yai-ds. 
 
 FIG. 11. 
 
30 
 
 SIMPLE CURVES. 
 
 /. The point of intersection, or the P.T., is the point of 
 intersection of the tangents drawn through the P.C. and the 
 P.T., as I. 
 
 g. The radius BO, or EO, is denoted by R. 
 
 h. The point of compound curve, or the P.C.C., is the point 
 of common tangent of its two branches. See Fig. 10. 
 
 i. The point of reversed curve, or the P.R.C., is the point 
 of common tangent of its two branches. See Fig. 11. 
 
 j. The angle of intersection, or a-, indicates the amount of 
 divergence of the tangents BI and IE. Fig. 12. 
 
 k. The tangent distance, or T, is the distance BI or El 
 from the P.C. or P.T. to the point of intersection. 
 
 46. Chords of one hundred feet are generally used, not the 
 actual arc, in running a curve, and the amount of curvature is 
 designated by the degree, though sometimes by the radius of 
 the curve. 
 
 47. The degree of the curve D, as usually defined is the 
 angle A OB, Fig. 13, at the center, subtended by a chord as 
 AB of 100 feet length. If it were practicable to measure 100 
 feet along the arc AB instead of the chord, and substitute 
 hence, in the definition for degree of curve, the word arc for 
 chord, a precise and convenient ratio would be always avail- 
 able between the radius and degree of curve; for we should 
 then have 
 
FUNDAMENTAL FORMULAS. 
 
 31 
 
 For a 1 curve, 
 
 = 100 X 360; 
 
 = 100 X 180; .-. 7? 2 = m^ = 2864.79. 
 2it 
 
 = 100 X 120; .-. R 3 = 1^P-P_9 = 1909.86. 
 
 2lt 
 
 u u 
 
 360 
 
 100 .360 
 
 2*JR. = 100x; ,.R H = ^.x'. 
 
 n 
 
 The inverse ratio existing between the degree of curve and 
 the radius being apparent. But since the arc measurement is 
 impracticable the chord being substituted therefor, and since the 
 difference in length between an arc and its chord increases with 
 the degree of curve the radii obtained as functions of the chords, 
 will not agree exactly 
 with those computed 
 as above. The ques- 
 tion therefore arises to 
 whal extent is it con- 
 sistent with good prac- 
 tice to assume the 
 equality of arc and 
 chord. In this connec- 
 tion consider the fol- 
 lowing problem. 
 
 48. To find the ra- 
 dius R, in terms of 
 the degree of curve 
 J). Draw OM perpen- 
 dicular to the 100-foot chord AB. Denote OA by R, and 
 the angle AOB by D. Then in, the right triangle AMO we 
 have 
 
 50 
 
 R sin-i-D = AM, whence R = 
 
 = 50cosec|D, 
 
 sin % D 
 and conversely the degree of curve in terms of the radius. 
 
 (6) 
 
32 SIMPLE CURVES. 
 
 Applying this formula to compute R for a one degree curve 
 according to the definition of the degree of curve we obtain, 
 
 1 curve, E = 5729.65 8 curve, R = 717.78 
 
 2 " R = 2864.93 10 " R = 573.69 
 
 3 " 12 = 1910.08 14 " 12 = 410.28 
 
 5 " 12 = 1146.28 15 " 12 = 383.06 
 
 7 " 12= 819.02 20 " 12 = 287.94 
 
 Comparing the corresponding values of 12 in the two sets 
 given above we perceive the difference. 
 
 For a 1 curve = 5729.65 - 5729.58 = 0.07 
 
 " .2 " 2864.93-2864.79 = 0.14 
 
 " 3 " 1910.08-1909.86=0.22 
 
 " 5 " 1146.28 1145.92=0.36 
 
 " 7 " 819.02- 818.51=0.51 
 
 " 8 " 716.78- 716.20 = 0.58 
 
 " 10 " 573.69- 572.96 = 0.73 
 
 " 14 " 410.28- 409.26 = 1.02 
 
 " 15 " 383.06- 381.97 = 1.09 
 
 " 20 " 287.94- 286.48 = 1.46 
 
 The difference is about a half a foot in a 7 curve and about 
 a foot in a 14 curve ; so that for ordinary work it is permis- 
 sible to stake out curves from 1 to 7 inclusive with chords of 
 100 feet, using formula (5) to obtain 12. From 8 to 14 inclu- 
 sive use chords of 50 feet, whence 
 
 12 = ^ = 25 cosec i D. (6a) 
 
 sin D 
 
 From 15 to 28 inclusive use chords of 25 feet, whence 
 
 12.5 = 12 .5 cosec D. (66) 
 
 sin^D 
 
 For a greater degree, stakes should be set in the curve every 
 10 feet, and the value of the corresponding radius 
 
 12 = - T -^ = 5 cosec & D. (6c) 
 
 sin ^L D 
 
 The practical effect of the application of the above formulas 
 is that when the degree of curve is assumed, the radius can 
 be determined at once by simple division, and vice versa, if the 
 
FUNDAMENTAL FORMULAS. 33 
 
 radius is known, the degree of curve Z), can be found. Thus, 
 suppose D is 2. 
 
 Then E = 572 t 9 ' 65 = 2864.83 feet. 
 
 Or, as is frequently done, taking the radius of a 1 curve as 
 5730 feet, 
 
 = 5730 = 2865 feet. 
 
 The assumption of 5730 feet as the length of the radius of a 
 1 curve lessens the slight error committed in assuming the 
 equality of arc and chord for the same central angle and 
 radius when the angle is 2 and over ; and the difference is 
 inconsiderable for angles even less than 2. That is to say, 
 
 the radius of a 2 30' curve is more nearly ' = 2292, than 
 
 5799 65 
 it is- riJ_L = 2291.86, and the radius of a 3 curve more 
 
 ^2 
 
 nearly one-third of 5730 = 1910 than it is one-third of 5729.65 
 = 1909.88. 
 
 Tlie student may verify these results, and explain why the 
 above assumption lessens the error. 
 
 EXAMPLES. 
 
 1. Find the length of R when D = 6 38'. 
 
 2. Find D if 7* = 5000 feet. 
 
 3. Find R it D = 30 minutes. 
 
 4. Find R if D = 30 degrees. 
 
 5. Find the degree of curve when R is 250 feet, and state 
 how far apart the stakes should be set. 
 
 49. Stations are usually placed 100 feet apart and num- 
 bered, beginning at zero (0). Short chords, called sub-chords, 
 may, therefore, occur at the ends of a curve. 
 
 a. The deflection angle is the angle IB G or GBH, Fig. 14, 
 at any point in the curve subtended by a chord of 100 feet, 
 and = i D. 
 
 b. The middle ordinate M is the perpendicular MN from 
 the middle of a chord to the arc, or it is that part of the radius 
 
34 
 
 SIMPLE CURVES. 
 
 intercepted between the middle of the chord and its arc, corre- 
 sponding to the versed sine of half the arc. 
 
 c. The long chord C, as usually denned, is the line BE 
 joining the beginning and ending points of the curve, though 
 the term long chord is often used in practice to designate any 
 chord of the curve greater than 100 feet in length, as BH. 
 
 d. The external distance E is the line Nf joining the 
 middle of the curve with the intersection point 7, and is, there- 
 fore, the prolongation of the radius from N to /. 
 
 o 
 
 FIG. 14. 
 
 50. The length of a curve L is given by the number of 
 applications and fractions thereof, if any, of the chord used in 
 laying out the curve, or the number of stations and fractions 
 thereof, if any, which compose the curve. 
 
 In general = A. 
 
 The actual length of arc = 
 
 300 
 
FUNDAMENTAL FORMULAS. 35 
 
 The distance from station 44 to station 55 = 11 stations or 
 1100 feet. From station 44 to a point 60 feet beyond station 
 55 = 1160 feet, and such point would be marked and called 
 55 _j_ 60, or may be said to be 11.6 stations from 44. 
 
 51. From the properties of the circle it will readily appear 
 
 (a) That the radii OB and OE are respectively perpen- 
 dicular to the tangents IB and IE. 
 
 (&) That the tangents IB and IE are equal. 
 
 (c-) That the angle of intersection a is equal to the angle 
 BOE at the center, called also the central angle. 
 
 (d) That the angle I BE = IEB = $ BOE = $ a. 
 
 (e) That the angle IBN = NEI = NBE = NEB = \ a. 
 (/) That the angle IBG = BOG = %D when BG =* 100 
 
 feet. 
 
 When the length of a circular arc as BGE = the length of 
 the radius BO or OE it is called the unit arc, and the angle 
 BOE = 57.o ; hence if one degree is subtended by a chord of 
 
 100 feet 
 
 * r R = 57. 3 X 100 = 5730 feet. 
 
 Since the chords and sines of small arcs or angles practi- 
 cally coincide, the value of either for 1 being .01745, the 
 divergence per station, or 100 feet, will be nearly 1.75 feet, 
 for D = 1. For a 2 curve, 3.5 feet ; and so on in arith- 
 metical ratio, approximately correct for 6 or 7, and may be 
 used for roughly setting out a curve, or as a help and check 
 on the instrumental operations in locating points in a curve. 
 
 EXAMPLES. 
 
 1 . Given the angle of intersection a = 22 40' and D = 3 20', 
 to find the length of the curve : 
 
 3 20'= 3i=y> 
 
 <y -j- -y> = fi A x -^ = 6.8 stations = 680 feet. Ans. 
 2. Given a = 31 51' and D = 2 46', to find the length of 
 the curve : 
 
 31 51' = 1911 minutes 
 2 46' = 166 " 
 
 = 1 1.51 stations = 1151 feet. Ans, 
 
36 SIMPLE CURVES. 
 
 REMARK 1. If, as in Example 1, the minutes can be 
 readily turned into convenient fractional parts of a degree, 
 it is best so to reduce them, and then divide out. If, 
 however, as in Example 2, the fractional parts would not 
 be so convenient, it is best to reduce both to minutes before 
 dividing. 
 
 REMARK 2. In ordinary preliminary work, it will be suffi- 
 cient to measure lengths to the nearest foot, and to take the 
 needle-bearings of the tangents. This practically requires the 
 reading of the tape to be made to half a foot : that is, a line 
 which, if read to tenths, would equal 348.7 feet, \vould be 
 recorded 349 feet ; if 348.4, would be recorded 348. On loca- 
 tion, however, the angles should be read off the plates by the 
 vernier to the minute, and the measurements should be made 
 ordinarily to within two or three tenths. If there be con- 
 ditions requiring greater accuracy, as in determining the area 
 of valuable property in connection with the right of way, etc., 
 then of course the measurements should be correspondingly 
 close, say to a tenth of a foot. Xo strict rule can be laid 
 down ; the engineer must use his judgment in this matter as 
 in many others, and make the degree of precision consistent 
 with the interests involved. 
 
 REMARK 3. For all ordinary calculations in the field use 
 the natural functions, sines, tangents, secants, etc., true to four 
 places of decimals, tables of which are found in this book. 
 Waste no time with logarithms, for even Without wind, rain, or 
 bright sun to contend with, the seeking of the logarithmic 
 quantities in the different tables in all field books the figures 
 in the tables are necessarily small and then after addition 
 finding the corresponding number, will consume so much time 
 that the Napierian follower will generally be found far behind 
 one who adopts the more primitive mode of computation. 
 Table II, explained on page 42, containing tangents and ex- 
 ternal distances of a 1 curve, will also be found useful, easily 
 applied, and sufficiently accurate for common field practice, 
 and by consulting it for these functions the transitman will 
 often save himself much calculation. 
 
FUNDAMENTAL FORMULAS. 
 
 37 
 
 3. Given the length of a curve 510 feet, and D = 2 40' to 
 find a. Ans. 13 36'. 
 
 4. A 3 curve begins at station 22 + 36, and ends at 29 + 84, 
 find the length of the curve, 
 
 the radius, and central angle. 
 
 5. Given the angle of inter- 
 section 53 57', and the degree 
 5 43' to find the length of 
 the curve. 
 
 6. How much longer is the 
 arc than its 100-foot chord, 
 the radius being 500 feet ? If 
 the radius is 1000 feet ? 2000 
 feet? 
 
 52. Given the angle of 
 intersection a, and the radius 
 R, or degree D, to find the 
 tangent distance. 
 
 Ir* the right triangle BOI the angle BOI= a. 
 
 Therefore r=/?tan|a, (7) 
 
 and substituting for R its value from (5) 
 
 This formula is useful when the radius OB or degree of curve 
 is assumed, the tangent distance IB must then be calculated. 
 If, on the other hand, T be assumed, then we may find R, and 
 hence Z), from the problem 
 
 53. Given the tangent distance T and the central angle a, 
 to find the radius. 
 
 From (7) 
 
 R=Tcota. (9) 
 
 EXAMPLES. 
 
 1. Given the tangent distance 450 feet, and the angle of 
 intersection 23 42' to find the radius. 
 
38 SIMPLE CURVES. 
 
 2. Given the angle of intersection 17 56', and the degree of 
 curve 2 40', to find the tangent distance. 
 
 54. Given the radius R and central angle a, to find the 
 long chord C. 
 
 In the right triangle BOm Fig. 16 
 
 Bm = BOsmBOm 
 
 C 
 or, = Rsm$a. 
 
 Hence C = 2Rsm$a. (10) 
 
 The student may show that, 
 
 (11) 
 
 and that C = 2 M cot* or. (12) 
 
 Example Given a 2 50' curve, having a central angle of 
 28 30', to find the long chord. 
 
 55. Given the radius R and central angle a, to find the 
 middle ordinate M. 
 
 mN= BO versine BOm-, 
 or, M = R versine a. (13) 
 
 Example Given a 3 40' curve having a central angle of 
 32 50' to find the middle ordinate. 
 
 If a and the external distance E are given, to find M, 
 substitute in (13) the value of R from the proportion, 
 
 cosine : R = versine : exsec, or JE", and we obtain, 
 
 M=Ecos$a. (14) 
 
 The student may verify the last equation by another 
 method. 
 
 56. Given the radius R and chord C to find the middle 
 ordinate M. 
 
 mN = ON Om. 
 
 But Om = *J OE" mE 2 > 
 
 and substituting values, there results, 
 
 ? ( 15 > 
 
FUNDAMENTAL FORMULAS. 
 
 39 
 
 57. Given the radius and chord, to find any ordinate of a 
 curve, its distance from the center of chord being known. 
 
 In Fig*. 16 let ap be the ordinate whose length is required. 
 Extend pa to Q, draw OQ, parallel to the chord BE, and join 
 Op. Denote the distance am by f/, and the other notation as 
 above; then in the right triangle p 0Q, 
 
 But 
 
 and 
 
 substituting these values, there results, 
 
 (16) 
 
 If d = 0, (16) reduces to (15), as it evidently should. 
 
 For most practical purposes a modification of formula (15) 
 may be employed to locate points in a curve. 
 
 By expanding the binomial of the right-hand member in a 
 series to three terms, there results, 
 
40 SIMPLE CURVES. 
 
 The last term of this series will not affect the result .03 of a 
 foot for M of a chord 100 feet and radius 300 feet, and may 
 be, therefore, safely rejected; we have then, 
 
 *=g; (IT) 
 
 And hence for any other middle ordinate M l and chord C l 
 in the same curve, 
 
 . (17a) 
 
 SB 
 
 or, M:M l =C*: 
 
 If Ci = iC Mi = . (18) 
 
 Assuming BN % BE (which may often be done with suf- 
 ficient accuracy), its middle ordinate m'n' = M, and the mid- 
 dle ordinate of En' = m'n', and so on, numerous points in a 
 curve may be established. 
 
 Practically these points may be located by measuring off the 
 computed distance, as m'n', from the middle point of a tape 
 stretched between the extremities of an arc, as BN. 
 
 EXAMPLES. 
 
 1. Find the middle ordinate, and the length of one 25 feet 
 therefrom, of a 100-foot chord, the radius being 1000 feet. 
 
 2. Given the radius of a curve 500' and chord 100' to locate 
 in the arc points whose projections on the chord shall be 12^ 
 feet apart. Compare results by different methods. 
 
 58. Given the radius R, and central angle a, to find the 
 external distance E. As Bin is the sine and mN the versed 
 sine to radius R of the arc BON, Fig. 17, so is NI the external 
 secant of the same arc and radius. 
 
 Hence given the external distance E and central angle a 
 to find the radius R, 
 
 R = (20) 
 
 exsec^a 
 
FUNDAMENTAL FORMULAS. 
 
 41 
 
 The student may show that 
 
 E= T cottar. exsec-Jar, 
 and that E = M sec a. 
 
 (21) 
 (22) 
 
 59. Given the tangent distance T, and the central angle 
 a, to find the external distance E. 
 
 In Fig. 17 extend the radius OB until it meets at K, the 
 tangent drawn from the 
 point N of the 
 Also extend ON 
 
 middle 
 
 curve. 
 
 until it meets, at /, the 
 
 tangent drawn from the 
 
 extremity of the curve at 
 
 E. Join IK, and BN. 
 
 By this construction it 
 
 is evident that IK and 
 
 BN are parallel ; that 
 
 NK = BI; that NT is 
 
 the external secant of 
 
 the arc BON, and the 
 
 angle IKN = EBN = $ 
 
 BON = \ a, and therefore in the right triangle NIK 
 
 NI = NKtsuiNKI, 
 
 or, E=Tta,na. (23) 
 
 Hence given the external distance E and central angle a 
 to find T, we obtain from (23) 
 
 (24) 
 
 The student may show that 
 
 E 
 
 M 
 
 = M seeder. 
 
 (25) 
 
 REMARK. Equations (20) and (24) will aid us in deter- 
 mining the elements and retracing the curved track of a rail- 
 road when all notes concerning it are defaced, and the location 
 of the P.C. and P.T. are unknown. Extend the centre lines 
 of the tangents to the curve, to their intersection at /, Fig. 17 ; 
 observe the angle supplementary to a, bisect it, and measure 
 
42 SIMPLE CURVES. 
 
 on the bisector from I to the center of the track, and thereby 
 get E. The values of E and a, being thus discovered and 
 substituted in the equations named, make known R and T. 
 
 EXAMPLES. 
 
 1. Given the central angle 28 48', and degree of curve 4 40', 
 to find the external distance. 
 
 2. Given the angle of intersection 44 56', and the tangent 
 distance 560 feet, to find how far from / to the nearest point 
 on the curve. 
 
 3. The angle of intersection is 32 20', and the curve 
 passes within 60 feet of /. Find the distance from the P.I. 
 to the P.O. 
 
 4. The angle of intersection is 26 40', and the nearest 
 point of the curve is to be 56 feet from /. Find the radius 
 and degree of curve. 
 
 Tangents and external distances for a one-degree curve for 
 every ten minutes of central angle are arranged in table II. 
 In this table, therefore, one may find the length of tangent or 
 external distance at once corresponding to a radius of 5730 
 feet, and a central angle varying by ten minutes of arc, and 
 by interpolation he may obtain these lengths for any minute 
 whatever. 
 
 For the required length of a corresponding tangent, or 
 external distance of any other radius or degree of curve, take 
 the proportional part thus: To find the length of a tangent 
 corresponding to a 3 curve and central angle 24, look in 
 table II under 24, and take out of the column of tangents, 
 opposite 24, the number 1218. This is the length of the tan- 
 gent of a 1 curve having a central angle of 24. 
 
 Now the tangent of a 3 curve, and the same central angle, 
 is only one-third as long; hence, 1218 -f 3 =406 = length of 
 chord required. 
 
 The student may verify this result by either formula (7) 
 or (8). 
 
FUNDAMENTAL FORMULAS. 43 
 
 EXAMPLES. 
 
 [To be solved by aid of Table II.] 
 
 1. Given the intersection angle a = 48 26', and D = 5, to 
 find the tangent distance. 
 
 2. Given a = 28 20', and the length of the tangent = 361.6 
 feet, to find the radius. Ans. 1432.5 feet. 
 
 3. Given the angle a == 36 40', to find the external distance 
 and tangent of a 3 40' curve. 
 
 4. Given the tangent distance T = 559 feet, and D = 4, to 
 find a. Ans. 42 38'. 
 
 5. Given the external distance 126.1 feet, and D = 2 40', 
 to find a. 
 
 SOLUTION. Find the product of 126.1 by 2f = 336.3 in 
 the table in the E column, and note the degrees and minutes 
 corresponding thereto = 38 20. Ans. 
 
 The following formulas derived on preceding pages are 
 grouped here for convenience of reference. 
 
 E = Radius. C = Long chord. 
 
 L = Length of curve. M = Middle ordinate. 
 
 T = Tangent distance. E = External distance. 
 
 c = Any chord. 
 
 sin^D 2 
 
 R = - - T = 
 
 = 
 
 M = 
 
 exsec^a: 
 C 
 
 versine 4- a 
 L = 100 - 
 
SIMPLE CURVES. 
 
 =- (nearly) 
 
 E= Ttan^o: 
 
 E = M sec \ a 
 
 O 
 
 FIG. 18. 
 
 B. LOCATING SIMPLE CURVES. 
 
 60. Given the degree D to locate a curve from a known 
 point in a given tangent. 
 
 Let B in the above figure be the known point in the tangent 
 AL Set up the transit at 5, level, and for convenience make 
 the zeros of the plates coincide. Without disturbing the rela- 
 tive positions of the zeros, observe some point, as / or J in the 
 tangent for direction; then turn off the angle IBC = ^D, and 
 
LOCATING SIMPLE CURVES. 45 
 
 measure in the direction of the line of sight 100 feet and there 
 set C. Deflect again CBF= Z>, i.e., make the reading of the 
 plates now =Z), and measure from C onward 100 feet, to a 
 point F in the line of sight, thus locating F. Deflect again 
 the angle FBG = D, making the reading of the instrument 
 | D, and measure 100 feet from F to G, and so on as far as 
 may be necessary. If the curve is to be extended farther than 
 it can be seen from the point B, the direction of the tangent 
 at the point 011 the curve on which it is desired to place the 
 instrument must be ascertained; hence the problem. 
 
 61. To find the direction of a tangent at a given point in 
 a curve. Let G be the point, and GP the tangent. The read- 
 ing of the plates when sighting G was |Z>; clamp at that read- 
 ing, and transfer the instrument to G, see that the index is not 
 disturbed, and sighl to B, then turn off an angle BGK= GBK, 
 i.e., = -|Z),* or make the reading of the instrument 3Z), and the 
 telescope will point in the direction of the required tangent 
 GK, whence inverting the telescope and making proper de- 
 flections as before, other points in the curve may be found. 
 Having run the curve from this new tangent point G, two 
 more stations to E, let us suppose it is desired to turn into 
 tangent at E. Clamp the index at its last reading, that is 4D, 
 set up at E, and with the index undisturbed observe G, then 
 turn off an angle GEP = EGP, the angle deflected from the 
 tangent at the last station, and the telescope will point in the 
 direction of the required tangent. 
 
 The following is a general rule for this common operation : 
 
 For direction of tangent at any point in a circular curve : 
 From twice the reading of the instrument when locating the point, 
 subtract its reading at the last tangent, or as we sometimes say 
 for shortness, double the index minus the last tangent. 
 
 To illustrate further : 
 
 Suppose the reading of the instrument at G, when the tele- 
 scope pointed in the direction of GK, to be 12, the degree of 
 curve being 4 ; then when sighting E, the reading or index 
 will be 16. Now after making at E the observation on G as 
 
 * The student will perceive that the triangle BGK is isosceles. 
 
46 SIMPLE CURVES. 
 
 directed above, turn into tangent by setting the index at 2 x 
 16 12 = 20, and having started at B with the index at 
 zero, the reading 20 indicates the magnitude of the central 
 angle BOE, or the angle of intersection a. 
 
 The student may verify by summing the deflections. 
 
 REMARK!. It will be perceived that at any tangent point 
 when the telescope points in the direction of the tangent the 
 reading of the vernier gives exactly the amount of the central 
 angle consumed. 
 
 Hence, when the angle of intersection, or, of two tangents is 
 measured, the P.C. and P.T. located, and then a curve traced 
 uniting these points, a check on the work is secured, for at the 
 P.T. when the telescope points in the direction of the located 
 tangent, the reading of the plates should be a. 
 
 REMARK 2. Some engineers before taking the back sight 
 from the new tangent point, turn back the zero of the vernier 
 past the zero of the limb, just as far as it was on the other 
 side when the new tangent point was sighted. Then after 
 sighting the previous tangent point, move the vernier plate 
 back to zero, thus bringing the telescope to point in the direc- 
 tion of the new tangent. But this is objectionable, since it 
 always requires two changes and two readings of the plates, 
 takes more time, it is obviously no more accurate, and more- 
 over renders impossible the convenient check which the more 
 expeditious method introduces. 
 
 The following are the field notes of a 3 curve, central angle 
 25 30', P.C. at station 24, curve turned to the right. 
 
 The student should calculate the tangent distance, or find its 
 length 432 feet from Table IT. He should also compute the 
 length of the curve = 8 stations 850 feet; determine the 
 deflection angle, and the amount to deflect at the P.T. for the 
 sub chord ; the reading of the vernier at the tangent points 
 when telescope is pointing in direction of tangent ; and in fact 
 he should verify the work in every particular. 
 
LOCATING SIMPLE CURVES. 
 
 47 
 
 
 Sta- 
 tion. 
 
 De- 
 flect. 
 
 Read- 
 ing. 
 
 Tan- 
 gent. 
 
 C'mputed 
 Course. 
 
 Magnetic 
 Course. 
 
 Remarks. 
 
 P.T. 
 
 32+50 
 
 045' 
 
 21 45' 
 
 25 30' 
 
 N7520'E 
 
 N7530'E 
 
 
 
 32 
 
 
 21 00' 
 
 
 
 
 
 
 31 
 
 
 19 30' 
 
 
 
 
 
 T.P. 
 
 30 
 
 
 13 30' 
 
 18 00 
 
 
 
 
 
 29 
 
 
 12 00' 
 
 
 
 
 
 
 28 
 
 
 10 30' 
 
 
 
 
 
 T.P. 
 
 27 
 
 
 4 30' 
 
 9 00' 
 
 
 
 
 
 26 
 
 
 3 00' 
 
 
 
 
 
 
 25 
 
 130' 
 
 130' 
 
 
 
 
 
 P.C. 
 
 24 
 
 
 
 
 
 
 
 
 3 curve, turning right, 
 
 
 
 
 
 
 
 
 a = 25 30' T- 432.2 feet. 
 
 Make a complete table, as above, for the following 
 problems 
 
 1. A3 40' curve, turning right, central angle 48 40', P.C. 
 at station 51 -f- 60. Make three tangent points between the 
 P.C. and the P.T. 
 
 2. A 9 30' curve to the right, a = 70 P.C. at station 84, 
 make two T.P.'s. 
 
 Another method, in which the deflections for all proposed 
 stations may be calculated before going on the field. The 
 record is shown in the table below. 
 
 P.T. 
 
 T.P. 
 T.P. 
 
 P.C. 
 
 Sta- 
 tion. 
 
 Deflec- 
 tion. 
 
 Read- 
 ing. 
 
 C'mputed 
 Course. 
 
 Magnetic 
 Course. 
 
 Remarks. 
 
 + 50 
 
 040' 
 
 10 00' 
 
 
 
 
 17 
 
 
 9 20' 
 
 
 
 
 16 
 
 
 8 00' 
 
 
 
 
 15 
 
 
 6 40' 
 
 
 
 
 14 
 
 
 5 20' 
 
 
 
 
 13 
 
 
 4 00' 
 
 
 
 
 12 
 
 
 2 40' 
 
 
 
 
 11 
 
 I 8 20' 
 
 120' 
 
 
 
 
 10 
 
 
 
 
 
 
 
 2 40 curve, turn'g right, 
 a = 20 T= 378.9 
 
 Set up the transit at station 10, the P.C., and proceed as in 
 the previous case to locate 11, 12, and 13. Carry the transit 
 to 13, clamp the zeros together, and then observe the P.C. 
 Then, if the vernier -plate is moved over 4 in the direction of 
 the curve, the telescope will point in the line of tangent at 
 station 13 ; and, with the index at 5 20' (the reading oppo- 
 
48 SIMPLE CURVES. 
 
 site 14), station 14 may be set, and 6 40' will locate 15. 
 Then transfer the instrument to 15, make the reading 4 (the 
 deflection corresponding to 13, the last T.P.), and observe 13 ; 
 clamp the lower plate. Now, when the vernier reads 6 40', 
 the telescope will point in the direction of the tangent at 
 station 15, and readings of 8, 9 20', and 10, respectively, will 
 set the remaining stations and sight the P.T. If now the 
 instrument be set up at the P.T. and an observation made 
 on station 15 with the vernier at 6 40', the lower plate then 
 clamped, and the reading made 10, the telescope will point in 
 the direction of the tangent, and this reading is evidently one 
 half the central angle. 
 
 REMARK. Observing T.P. 13 from T.P. 15 with the index 
 at 4, turning into tangent at 6 40', and sighting station 16 
 with the vernier at 8 is evidently the same as though the P.O. 
 were sighted from 15 with the plates at zero, the tangent turned 
 at 6 40' and station 16 set with the index at 8. Before 
 making an observation at any station set the vernier at the 
 reading opposite the point on which the observation is to be 
 made, observe the point, clamp the lower plate, then any station 
 in the curve, either way from the instrument, may be located 
 by setting the vernier at the reading opposite that station, 
 and the telescope will point the direction of tangent at the 
 station when the index points to the reading opposite the 
 station. 
 
 The advantages which this method possesses over all others 
 are that the deflections required for all proposed stations or 
 known chord lengths may be tabulated in advance, and the 
 deflections may be used to run the curve in from either end, or 
 from any intermediate point, working either way from the 
 instrument. Of course any odd plus employed to locate a 
 point not predetermined will have to be calculated on the 
 field. 
 
 62. To locate points in a curve of given radius by off- 
 sets from a given tangent. 
 
 Let B represent the beginning point ; c', e', f, points in the 
 tangent to be found in order to establish c, e,f, points 100 feet 
 
LOCATING SIMPLE CURVES. 
 
 49 
 
 apart in the curve. The distance from B along the tangent 
 to the points, and the lengths of the offsets are required. From 
 the known radius the angle 
 BOc = D is found, and it 
 it is evident that 
 
 FIG. 19. 
 
 Be' = m'e = R sin 2 D ; 
 and similarly for other 
 points. 
 Also 
 
 c'c = Bin = E versine D ; 
 
 e'e = Bin' = R versine 2 D ; 
 and so on for other points. 
 From these equations 
 the distances and offsets 
 may be computed, and 
 then measured off, or they may be taken at once from 
 tables of middle ordinates and long chords, since it will 
 be perceived that the offsets Bm, Bm', etc., correspond to middle 
 ordinates of double the arcs Be, Be, etc., and me, m'e, etc., cor- 
 respond to one-half the long chords of double the arcs. If 
 there is a sub-chord at B the above equations will still give 
 the proper distances by substituting for D the value of the 
 angle, say 8, which the sub-chord subtends, or 
 
 * R sin 8, and for the next point 
 R sin (8 + D), and so on. 
 
 The same substitutions will be required in the equations 
 containing the versines. 
 
 EXAMPLE. Find the distances along the tangent and the 
 lengths of the offsets for a curve of 2000 feet radius. 
 
 63. To locate a curve of given radius by offsets from 
 chords produced. 
 
 Produce the chord ac to e', Fig. 20, draw bch tangent to the 
 curve at c, and draw ee' through h perpendicular to ch, and 
 suppose a, c, e, and y points in the curve. Then since the angle 
 e'ch = angle ech, and the side ch perpendicular to ee', these tri- 
 
50 
 
 SIMPLE CURVES. 
 
 angles are equal ; the side ce' = ce and the triangle cee' is 
 isosceles. Moreover, since the angle ece f = cOe and both tri- 
 angles are isosceles, they are similar and the corresponding- 
 sides are proportional ; hence, 
 
 cO : ce = ce: ee', 
 or E : C = C : ee' ; 
 
 .-. ee' = <-. (26) 
 
 If C = 100, as it does usually, then 
 
 100 2 
 R 
 
 
 
 ee' = 
 
 (2Ca) 
 
 FIG. 20. 
 
 Otherwise from the equality of the triangles ceh and ce'h, it 
 will be readily perceived that the chord offset ee' = twice the 
 tangenf offset he, and therefore the equations for tangent 
 offsets, last articles are applicable. Practically each chord as 
 ac must be extended through c to e' a distance of 100 feet, and 
 then, while preserving the point e', draw the tape over from e' 
 past h keeping it pivoted at c until the point e is found, which 
 shall be 100 feet from c and the computed distance from e'. 
 
 If there is a sub-chord at the beginning, its complement, or 
 what it lacks of 100 feet, may be laid off in the opposite direc- 
 tion by offsets from the tangent, and then having the extremi- 
 
OBSTACLES. 
 
 51 
 
 ties of the chord proceed as above. In either case if the chord 
 is not given in this direction find it by offset from tangent, as 
 in article 62. 
 
 EXAMPLE. Given the sub-chord of 30 feet, located in a 
 curve whose radius = 1910 feet, to find the necessary deflections 
 and lay out the curve. 
 
 C. OBSTACLES. 
 64. To pass an obstacle on a curve : 
 
 Let be the center of the curve y x, which we may suppose 
 is laid out in the usual manner by deflection angles, until the 
 point p is reached, when obstacles shown in the figure inter- 
 vene, requiring evidently some modification or change in the 
 method being employed. 
 
 FIG. 21. 
 
 I The best way ordinarily is to turn off an angle equal to a 
 certain number of deflection angles to just pass the obstacle. 
 In the present case four times one deflection will do ; compute 
 the long chord, as pt, or take its length from table Y. Meas- 
 ure out this distance to t, set up there ; turn into tangent, make 
 the proper deflection, and if possible set some of the back 
 stations, as s and r, and then before lifting the instrument 
 locate other points if possible between t and x. Points q and 
 r may be located by offsets from the tangent pv, Art. 62. 
 
52 
 
 SIMPLE CURVES. 
 
 65. To locate a curve when the point of intersection is 
 inaccessible. 
 
 When the P.T. is inaccessible neither the tangent distance 
 nor the angle a can be measured directly, hence, some equiva- 
 lent must be discovered. 
 
 The tangents being lo- 
 cated as near as practicable 
 to the P. I., set up at some 
 point I in the one, and ob- 
 serve some point n in the 
 other; note the angle nil, 
 measure In, and also the 
 angle Inl. The sum of the 
 two angles measured = cr. 
 From the data obtained com- 
 pute the distance II ; meas- 
 ure from / towards p, a dis- 
 tance = to the difference be- 
 tween the assumed tangent distance and 77, thus locating the 
 P.C., whence the curve may be traced as usual. 
 
 If it is impossible to run a straight line, survey a broken 
 line, and, as in working a traverse, determine the direction 
 and length of In ; then proceed, as before, to find the P.C. 
 
 66. To locate a curve when the point of curve is inac- 
 cessible. 
 
 Let I be a numbered station at the point of intersection of 
 two tangents al and It, Fig. 23; 73, the P.C, the distance to which 
 from / being known. Then if there are two or more stations in 
 the vicinity of a and either of them numbered, the location of 
 any point as c' is discovered; a tangent offset therefrom will fix 
 a point c in the curve extended back of the P.C. Set up at c, 
 and deflect from cc' a right angle less B Oc, the telescope will 
 then point in the direction of tangent at c. Then deflect suf- 
 ficiently to clear the obstacle, usually to some station, measure 
 the long chord eg, and obtain thereby a point in the curve 
 beyond 73, whence the curve may be traced in the usual man- 
 ner. If the preceding is impracticable, set up at some point 
 
OBSTACLES. 
 
 53 
 
 /'; its position may be known by the stakes in tangent or its 
 distance from / may be measured. Compute, and observe the 
 
 FIG. 23. 
 
 angle a:', and measure the distance I'n = FB. Move to n, sight 
 /', the telescope will then point in the direction of the tangent, 
 and deflections may be made either way to set points in the 
 
 67. To locate a curve when both the point of curve 
 and point of intersection are inaccessible. 
 
 o 
 FIG. 24. 
 
 Let a I and It represent the tangents. Set up at some point 
 I in the tangent al ; run a line In, joining the tangents, as in 
 
54 
 
 SIMPLE CURVES. 
 
 Article 65 ; measure the angles at I and n, and compute the 
 distance II. Then, knowing the radius and central angle, 
 compute IB, and thence Bl becomes known. Now, by tan- 
 gent offset, as in the previous article, a point c in the curve 
 may be located ; place the transit there, make proper deflec- 
 tion to clear the obstacle, obtain the corresponding long chord 
 ce, and proceed to completely trace the curve, as in the pre- 
 ceding article. 
 
 68. To pass from curve to tangent when the point of 
 tangent is inaccessible. 
 
 Let tt' be the tangent sought 
 and t, the inaccessible P.T. 
 Locate as many stations in 
 the curve as practicable; then 
 deflect at some point n, so as 
 to clear the obstacle and set 
 a point c in the extension of 
 the curve beyond the P.T. 
 ^ by the long chord as in 
 previous articles. Compute 
 the tangent offset cc' = R 
 versine cOt ; measure it off, 
 making the angle ncc' = a 
 right angle plus the angle 
 which the long chord makes 
 with the tangent, that is 90 
 -f ncrc', (n'c being drawn 
 parallel to the tangent). A 
 
 right angle turned at c' with c'c will give the direction of 
 tangent, and the distance c't = R sin cOt. 
 
 If c is farther than n from t, the angle c'cn will equal 90 
 minus the angle which the chord makes with the tangent. 
 In general c'cn = 90 ncn'. 
 
 If the deflection for long chord at n is made for a certain 
 number of stations, the length of the chord may be taken 
 direct from Table V. 
 
 Or compute the length of the tangent offset at the known 
 
 FIG. 25. 
 
OBSTACLES. 
 
 55 
 
 point n, and sight from n and measure direct to some point 
 c" on the tangent, its distance being 
 
 nc" = 
 
 nn 
 
 sin nc"n" 
 
 Then in the triangle nc"n" calculate c"n" and substract from 
 it tn" = 11 sin. n0f, thus obtaining c'% whence the numbering 
 of the stations may be properly continued. If an angle 
 tnc" be turned at n and made one-half ntn" = one-fourth nOt, 
 ic" will equal tn", and no calculation, except for nc", will be 
 necessary. 
 
 If the angle at c" is very acute this method will not be very 
 reliable. 
 
 Otherwise, if the obstruction does not prevent alignment. 
 
 FIG. 26. 
 
 From a known point n, in the curve, Fig. 26, compute the 
 tangent distance ni ; measure it oif, locate i and set the instru- 
 ment there. Deflect a' from ni, plunge the telescope and set a 
 point e in tangent ; sight another point c along the bank of 
 the stream, measure the angle eic and the length of ec. Then 
 in the triangle cie compute ie, subtract ti = ni from it and 
 thereby obtain te. 
 
56 
 
 SIMPLE CURVES. 
 
 69. To extend a curve across a pond or stream. 
 
 Let pt be the tangent to a curve pv at the point p. Set a 
 point c in the tangent, and estimate the distance to a point 
 n across the water, using a certain number of stations, and 
 deflect accordingly the angle tpn. Move to n, measure the 
 angle pnc and the distance nc. Then by the triangle pnc com- 
 pute pn, compare the computed and estimated distances, and 
 
 FIG. 27. 
 
 discover thereby the precise position of n. If it is not on the 
 curve (it probably will not be), measure from n to n', the 
 difference between the computed and estimated distances of_/m, 
 either forward or backward, as the case may require. Place 
 the transit at n', turn into tangent, and proceed in the usual 
 manner to set other points in the curve. 
 
 D. PROBLEMS IN CHANGE OF LOCATION. 
 
 70. Having located a curve between two tangents, it is 
 required to determine the necessary change in the radius, 
 and the external distance for any desired change in the tan- 
 gent distance. 
 
 Denote BI = IN by T. 
 
 B'I = IN'\>y T'. 
 OB = ON by R. 
 O'B' = &N' by R'. 
 " Jm by E. 
 * Im' by E'. 
 
PROBLEMS IN CHANGE OF LOCATION. 57 
 
 Then BE' =T T' = the given change in tangent distance. 
 Draw O'D perpendicular to OB, it will equal T T, and OD 
 
 Then in the right triangle ODO', 
 
 R R' = (T T) cot. i a. (27) 
 
 From (23) E = T tang. a. 
 
 E' = T tang. \ a. 
 .:E-E' = (T- T) tang. a. (27a) 
 
 EXAMPLES. 
 
 1. Two tangents which intersect at an angle of 32 are 
 united by a 3 30' curve. It is desired to lengthen the tan- 
 gent distance 72 feet; find the necessary change in R and E. 
 
 By (27) E E' = 72 cot 16, 
 
 JJ-E / =72 X 3.2709 = 237.5; 
 .-. R = 1637.28 + 237.5 = 1874. 78, 
 or a curve of about 3 3'. 
 
 By (27a) E-E'=72 tan 8, 
 
 or E-E' = 72 X 0.1405 = 10.12. 
 
58 SIMPLE CURVES. 
 
 2. Find the change in the degree of curve in Example 1, on 
 the assumption that T is to be shortened 60 feet. 
 
 71. Having located a curve between two tangents it is 
 required to determine the necessary change in the radius 
 and tangent distance, for any desired change in the external 
 distance. (Fig. 28.) 
 
 By (19) mI=E = R exsec | a-, 
 
 m'l = E' = R' exsec a; 
 
 or, R-R' = E ~ E/ (28) 
 
 exsec i- a 
 
 By (27a) T- T' = (E- E') cot* a. (29) 
 
 EXAMPLE. Two tangents which intersect at an angle of 
 40 are united by a 3 40' curve. It is desired to bring the 
 middle point of the curve 30 feet nearer the P.I. Find the 
 length of Ef and T. 
 
 30 
 
 By (28) R-R' = 
 
 exsec 20 
 
 jl-jl' = _ = 407.29. 
 .0642 
 
 ... E' = 1562.88 - 467.29 = 1095.59, 
 or a curve of 5 14' 
 
 By (29) T T = 30 cot 10, 
 
 T - T = 30 X 5.6713 = 170.14. 
 
 ... T = 568.84 - 170.14 = 398.7. 
 
 72. Having located a curve between two tangents it is 
 required to determine the necessary change in the tangent 
 distance and external distance for any desired change in the 
 radius. 
 
 From (27) we obtain 
 
 T T = (R R') tan a. (30) 
 
 From (28) E-E' = (R- R') exsec | a. (31 ) 
 
 EXAMPLE. Two tangents which intersect at an angle of 
 28 40' are united by a 4 20' curve. Find what change would 
 be made in T and E, by the substitution of a 5 curve. 
 
PROBLEMS IN CHANGE OF LOCATION. 
 
 59 
 
 73. Having located a curve between two tangents it is 
 required to determine the radius of a curve which from 
 the same point of curve will terminate in a defined parallel 
 tangent. 
 
 Let BE represent 
 the located curve and 
 BE' the curve re- 
 quired. 
 
 Since the tangents 
 El and ET are paral- 
 lel the angles QBE, 
 OEB and O'E'B are 
 equal, the central angle 
 a remains unchanged, 
 and EE' is a prolong- 
 ation of the chord BE 
 
 FIG. 29. 
 
 Denote AE' the per- 
 pendicular distance be- 
 tween the tangents by d, and draw OD perpendicular to the 
 radii OE = R and O'E' = R'] 
 
 Then 
 
 and 
 whence 
 
 (R' R) cosa + R + d = R', 
 R' (1 cosa) = R (1 cosa) -f- 
 
 ,_ d 
 
 versine a 
 
 (32) 
 
 In practice, before removing the instrument from E, the dis- 
 tance and direction to E' should be computed and a stake 
 driven there to serve as a check on the location of the curve 
 BE'. 
 
 The student will observe that 
 
 EE' = d . sec 
 
 180 or 
 2 ' 
 
 or, 
 
 EE' = d . cosec 
 
 (33) 
 
 and the angle which EE' makes with AE is 
 
60 SIMPLE CUKVES. 
 
 If BE' represents the located curve and BE the curve re- 
 quired, then R' will be known and R required. 
 
 or, R = R' ? (34) 
 
 versine a 
 
 If it is determined on the ground through what point in the 
 prolongation of the chord BE, as E', the tangent must pass ; 
 then measure d' the distance from E to E', and proceed other- 
 wise, as before 
 
 E' = (fl' - P) cos a + R + d' sin| 
 
 ,, . (t 
 d sin 
 
 or, R = R + 
 
 I cos a 
 
 whence R' = E + - (since 1 cos a = 2 sin 2 (35) 
 2 sin i a \_ 2J 
 
 If the parallel tangent is on the center side of the terminal 
 tangent. 
 
 (36) 
 
 The student -may find an expression for R f in terms of the 
 chords, BE, BE' and R. 
 
 EXAMPLE. A3 30' curve having a central angle of 34 40 
 ends in a tangent IE. It is required to substitute a curve 
 having the same P,C. but to terminate in a parallel 16 feet 
 from IE. 
 
 74. Having located a curve between two tangents, it 
 is required to determine the change in the point of curve 
 so that, with the same radius, the curve may end in a 
 given parallel tangent. 
 
 The required curve as to size and shape, is the given curve in 
 another position ; its elements are precisely the same. Imagin- 
 ing the one to merge into the other, it will be perceived that as 
 B Fig. 29a, approaches B', will approach 0', and N and / re- 
 spectively N' and /'. Hence the lines joining these like-lettered 
 points, are equal and parallel. Therefore turn oft' from the ter- 
 
PROBLEMS IN CHANGE OF LOCATION. 01 
 
 minal tangent at N, an angle = , and measure in the direction 
 of the telescope to a point N' in the given parallel tangent. 
 This distance laid off from .Z> will give the position of the 
 new P.C. 
 
 If the perpendicular distance NK between the parallel tan- 
 gents is given, then in the triangle NKN', observing that the 
 angle NN'K = a and putting NK = <L 
 
 NN' = BE' = d cosec a. (37) 
 
 As in the preceding case the direction of B' from B is depen- 
 dent upon the relative position of the terminal and parallel 
 tangents. 
 
 EXAMPLE. A 4 20' curve having a central angle of 32 40' 
 ends in the tangent IN. Compute the distance the P.C. must 
 be advanced along Bl so that with the same radius the curve 
 may be run ending in a tangent parallel to IN, and 16 feet 
 farther from the centre. 
 
 BB' = 16 . cosec 32 40'. 
 BB' = 16 X 1.8527 = 29.64. 
 
62 
 
 SIMPLE CURVES. 
 
 75. Having located a curve between two tangents, it is 
 required to change the radius and the point of curve, so that 
 the curve may terminate in a given parallel tangent at a 
 point on the same radial line as the first. 
 
 Let /JVbe the located tangent, 
 and I'N' the given parallel 
 tangent, and 0" the cor- 
 responding centers of curvature, 
 and R and R' the radii. Drop 
 a perpendicular from 0' on OB, 
 prolong the tangent BI and 
 radius ON to their intersecting 
 point P, and denote the distance 
 NN' on the radial line by d. 
 
 Then PN-PN' = d, 
 
 or (R R') exsec a = d ; 
 d 
 
 FIG. 30. 
 
 whence (R-R')=. 
 
 exsec a 
 
 and 
 
 exsec a 
 
 BB' = DO' = (R R') tan a, 
 or substituting the value of R R' from (38) 
 
 (38) 
 
 (39) 
 (40) 
 
 dtan a 
 
 But 
 
 (41) 
 
 exsec a 
 
 tang a = exsec a cot \ a, 
 .-. BB' = deotgla. 
 
 76. Having located a curve between two tangents, it is 
 required to find the change in the point of curve con- 
 sequent upon a given change in the direction of the ter- 
 minal tangent at the point of intersection, the radius 
 remaining the same. 
 
 Denote BI and B'l Fig. 31 by T and T' respectively, and 
 the angles NOB and N'O'B' by a and a' respectively. 
 
 Then BB' = T - T' = R (tan \ a - tan i a'). (42) 
 
PROBLEMS IN CHANGE OF LOCATION. 
 
 63 
 
 If the position of the P.O. remains fixed and R change, 
 the other condition as above in this article, find the new 
 radius. 
 
 Calling a and a' the angles 
 of intersection, R and R' radii, 
 and T the length of the tan- 
 gent. We shall have 
 
 T=R tan i or, 
 and T=R / tan|a / , 
 whence 
 
 R' = R tan $ a cot a'. 
 
 The student may supply the 
 figure. 
 
 FIG. 31. 
 
 EXAMPLES. 
 
 1. Given a 3 40' curve, the angle of intersection 28 20', to 
 find the position of the P.C., if the terminal tangent IN' makes 
 an angle of 6 with IN; the radius remaining the same length. 
 
 2. Given the same degree of curve, and a as in the last 
 example, to find the change in the radius, so that the position 
 of the P.C. may remain unchanged if ex. be diminished 6. 
 
 77. Having located a curve between two tangents, it is 
 required to change the radius and the point of curve so 
 that the terminal tangent may be changed in its di- 
 rection at the point of tangent. 
 
 Let IN represent the lo- 
 cated tangent ; I'N the ter- 
 minal tangent ; INF the given 
 change in the angle at N. 
 The student may show that 
 a' = a -f INI' ; that 
 
 E' = 
 
 R versine a 
 versine a' 
 
 and BR = Rsma-R' sin a'. (45) 
 
CHAPTER IV. 
 
 COMPOUND CURVES. 
 A. PROBLEMS IN LOCATION. 
 
 78. Given two tangents of unequal length, their angle 
 of intersection, and one radius, to find the length of the 
 radius of the other branch of a compound curve which 
 will unite the tangents. 
 
 
 FIG. 33. 
 
 Let BPE represent the curve. 
 
 Denote the radii PO and PO' by B and R' respectively. 
 " tangents BI and IE by T and T " 
 ' angles B OP and P O'E by a' and a" ' ' 
 " angle of intersection by a. 
 
PROBLEMS IN LOCATION, 65 
 
 Extend the first branch of the curve to D, where its terminal 
 tangent is parallel to the tangent IE, and draw DF parallel to 
 BL Draw the chord PE, it will pass through D. Draw also 
 01' and BD. In the triangle DEF, DF is equal to //' by con- 
 struction, and, therefore, 
 
 DF= T R t&n$a, 
 
 FE= T' - Btanio-, 
 and angle F = 180 - a, 
 
 whence the angle FED and side ED may be found. Now 
 2 FED a", and hence a' = a a" becomes known. 
 
 Again 2R sinFED = PD, 
 
 2R'sinFED = PE, 
 Therefore 2 R ' sin FED = 2 R sin FED + DE, 
 
 whence ir = B + __. (40) 
 
 The point D may be located by turning off half a from the 
 P.C. on tangent BI, and measuring the corresponding long 
 chord. Then, having computed the angle FED, set up the 
 instrument at D, turn off from the parallel tangent through 
 D the angle FED, and thereby locate P, the P.C.C., and E, the 
 P.T., after which in the usual manner the curve may be staked 
 out. 
 
 EXAMPLES. 
 
 1. Given BI and El 500 and 600 feet respectively, their 
 angle of intersection 21 40', and the radius of the first branch 
 2500 feet. Find the radius of the second branch, and number 
 the P.C.C. and P.T., assuming the P.C. to be 24 -f 60. 
 
 2. The student may reason out by aid off a diagram the 
 case where the length BI is greater than El, and the given 
 radius of the first branch is the longer ; and verify his deduc- 
 tions by numerical example and construction. 
 
 79. Given the length of the straight line between the P.C. 
 and the P.T., the angles which it makes with the tangents, 
 and the radius of the first branch, to find the radius of the 
 second branch of a compound curve uniting the tangents. 
 
 In Fig. 33 let d denote the distance from B to E, B and E 
 the angles, at the respective points, between the chord and the 
 
66 
 
 COMPOUND CURVES. 
 
 tangents, R and R' the radii of the first and second branches 
 respectively, and a' and a" as shown in the figure. The angle 
 IBD = $a = (B + E). In the triangle BDE, BE is given, ED 
 = 2R sin i (B + E) and the angle DEE = E \(E + E}= 
 (B E). Therefore the angle BED and side DE can be 
 computed, and E BED= the angle which the chord PDE 
 makes with the tangent IE, and thus a." and of become known. 
 Then, as before, using the relation between an angle, its sub- 
 tended chord, and radius, the value may be found of 
 
 DE 
 
 2 sin PEI 
 
 (47) 
 
 which agrees with (46). 
 
 80. Given the radii and the central angles of a compound 
 curve uniting two tangents, to find the lengths of the tan- 
 gents, the line connecting the P.O. and P.T., and the angles 
 which this line makes with the tangents. 
 
 In Fig. 34 let 
 B and E represent 
 the P.C. and P.T. 
 respectively, P the 
 P.C.C.jOandO'the 
 centers, IB and IE 
 tangents, d the con- 
 necting line, and 
 other notation as 
 shown in the fig- 
 ure. With the 
 given central an- 
 gles and radii com- 
 pute the lengths 
 of the chords BP 
 and PE ; find also 
 the angles OPE = 
 OBP and 0'PE = 0'EP. Then in the triangle BEP find 
 the angles PEE and PEE and the side BE = d. The un- 
 known angles may now be found as follows : 
 
PROBLEMS IN LOCATION. 
 
 6T 
 
 The angle EBI = 9Q PBO + PEE. 
 The angle BE I = 9Q PEO' + PEE. 
 
 Then in the triangle BE I having all the angles and the side 
 d, compute the tangents El and IE. 
 
 EXAMPLE. Given R, 1432.7 feet, R', 2148.8 feet a', 20 
 and a", 45 ; to find the lengths of the tangents, the line BE, 
 and the angles which the tangents make with BE. 
 
 81. Given the length of the straight line between the 
 P.O. and the P.T., and the angles which it makes with the 
 tangents, it is required to find the radii of a compound 
 curve having the common tangent parallel to the straight 
 line. 
 
 In the figure let 
 DF represent the 
 tangent parallel to 
 the given line BE. 
 PtheP.C.C., ZXBE 
 and FEE the given 
 angles, and d the 
 length of BE. 
 
 It is evident that 
 
 = FP and the an- 
 gles DBF and FEP 
 = % B and E re- 
 spectively. 
 
 In the triangle 
 BEP we have the 
 proportion 
 
 BP: BE = sin BEP: sin [180 - \ (B + E)], 
 dsinlE 
 
 E) 
 
 and 
 
 PE = - 
 
 E) 
 
68 
 
 COMPOUND CURVES. 
 
 Having found the chords of the branches the radii may be 
 calculated by the well known relation between the radius, 
 central angle, and chord, which gives 
 
 sin i B sin \ B . sin (13 + E)' 
 
 and 
 
 (48) 
 
 (49) 
 
 sin i#. sin 
 
 EXAMPLE. Given the distance between the P.C. and P.T. 
 1000 feet, angle B 12 30', E 14 40'. Find the radii R 
 and R'. 
 
 B. OBSTACLES. 
 
 82. To locate a compound curve when the P.C.C. is inac- 
 cessible ; the central angles of and a", and the degree of each 
 branch of the curve being given. 
 
 If the P.C. and P.T. are determined, the curve may be staked 
 out from these points. 
 
 Or, if the obstructions do not extend far from the P.C.C., as 
 indicated in Fig. 36, set as many stakes on the first branch as 
 
OBSTACLES. 69 
 
 possible ; calculate the chord B^m, parallel to the common 
 tangent through P, and the length of the middle ordinate mP 
 for double the arc B^P. Then in the second branch find 
 the angle ft" having the same middle ordinate mP, for 
 double its arc, whence the length of the chord mE, may be 
 computed. 
 
 Then deducting a! -f ft' from a the angle E^O'E will be 
 obtained, and thence the distance to the P.T. 
 
 If the curves are much obstructed, so that few, or no stakes, 
 can be set in them, run direct from the P.O. to some point on 
 the second branch obtaining the necessary data, as indicated 
 in the foregoing. Or, if it is more practicable, run the com- 
 mon tangent APD. 
 
 The length of this line can readily be found, since AP = AB 
 and PD = DE are tangent distances, by applying the well- 
 known rule : Radius multiplied by tangent of one-half the 
 central angle is equal to the tangent distance. 
 
 EXAMPLES. 
 
 1. Given a' = 12, a" = 10 30', and the degrees respectively 
 3 and 5 to locate the P.T., the P.C.C. being obstructed by a 
 building : 
 
 BA = AP = 1910 tan 6 = 201.6 feet, 
 PD = DE = 1146 tan 5 15' = 105.3 feet. 
 
 With the instrument at B, set as many stakes as possible on 
 the curve BP, then sight along the tangent and measure BA 
 201.6 feet ; remove to A, turn off the angle a', and measure 
 AD 306.9 feet, offsetting at P. With the instrument at 1), 
 turn off the angle ot" as indicated in the figure, and measure 
 DE 105.3 feet to the P.T. Having now located the P.T., if 
 practicable, set one or two stakes in the curve EP. 
 
 The student may determine where to place these last 
 stakes and how to number them, that they may appear 
 consecutive, the P.O. being at number 42. State also the plus 
 of the P.T. 
 
 2. Given a 2 and a 4 curve, of and a" respectively 9 and 
 13, conditions favorable to setting the first three stakes on 
 
70 
 
 COMPOUND CURVES. 
 
 first branch, and from the third station run a parallel to the 
 common tangent past the obstacle. Show how to continue 
 the work, find the point where the parallel line intersects the 
 second branch, the reading of the instrument at this point 
 when telescope sights the tangent, supposing the index was at 
 zero at the P.C., and find the plus of the P.T., calling the P.C. 
 17 + 40. 
 
 C. PROBLEMS IN CHANGE OF LOCATION. 
 
 83. Having located a compound curve terminating in a 
 tangent, it is required to change the location, so that, with 
 the same radii, the curve may end in a parallel tangent at 
 a given perpendicular distance from the terminal tangent. 
 
 FIRST CASE. The parallel tangent farther out than the 
 terminal tangent, and the second branch of the curve having 
 the longer radius. 
 
 The figure opposite 
 represents the case. 
 The first curve with 
 center having been 
 run from A, com- 
 pounded at P into 
 one with radius O'P, 
 and terminating in a 
 tangent at T. It is 
 desired to locate the 
 tangent further out 
 at a given perpendic- 
 ular distance from the 
 present one. It is 
 
 evident that the P.C.C. must be moved back on the sharper 
 curve to some point P'. The practical question is to find this 
 point. 
 
 Let 0"P' represent the required position of the longer 
 radius, and draw OMN perpendicular to the radii O'T and 
 O"T. It is evident that the angle POP' is equal to a' a = 
 OO"N OO'M, and as the latter is known we may find the 
 former as follows : 
 
PROBLEMS IN CHANGE OF LOCATION. 71 
 
 Let R denote the radius of the first branch of the curve. 
 " R' " " " " second " " " 
 " a " " angle " OO'M. 
 a' " " " " 00"N. 
 
 " d " " perpendicular distance between the tangents. 
 Then (R' - R) cos a = O'M, 
 
 and ( R' - R) cos a' = 0"N. 
 
 But O'M O"N = d substituting we obtain 
 
 (R' R) cos a (R' R) cos a' = d, 
 
 or, cos a.' = cos a (50) 
 
 R' R 
 
 and a' a = POP'. 
 
 Divide a' a by the degree of the first branch and thus 
 ascertain the distance from P to P'. 
 
 To find the distance and direction of T' from T; connect 
 the points O'O", and draw a perpendicular from O to the line 
 O'O"] also draw the perpendicular O'K on the radius 0"T' 
 prolonged. Then, evidently, 
 
 TT = O'O" = 2(R'- R) sin^a' - a). (51) 
 
 Now considering the angles about the point 0', we perceive 
 that 
 
 90 + a = KO'O" + 90-| (a' - a), 
 or, KO' O" = a + \ (a' - a) ; 
 
 .-. angle STT = KO'O" = ^(a' + a). (52) 
 
 With the instrument then at T", turn off from the tangent 
 an angle equal to the arithmetical mean of the given and com- 
 puted angles; the telescope will then point in the direction of 
 the new P.T. Measure off the distance computed by (51) ; set, 
 and center a stake there before running in the new curve. 
 
 REMARK. This method, based on the arithmetical mean, 
 for obtaining direction between the tangent points, holds true 
 for all four of the cases coming under this head. It was sug- 
 gested by Mr. Edward Godfrey, Class of '93, W.U.P. 
 
 EXAMPLE. Given a 5 curve compounding into a 2 40' for 
 12 50', and terminating in a tangent; it is desired to move the 
 tangent out 10 feet. Ascertain the change necessary in the 
 P.C.C. Ans. 47 feet towards P.O. 
 
72 
 
 COMPOUND CURVES. 
 
 SECOND CASE. The parallel tangent farther in than the 
 terminal tangent and the second branch of the curve having 
 the longer radius. 
 
 O' 
 
 FIG. 38. 
 
 Adopting the notation of the first case (R' = radius of second 
 branch) we have, 
 
 (R' R) cos of (R' R) cos a = d ; 
 d 
 
 whence 
 
 and 
 
 cos a = cos a 
 
 IF-JB' 
 
 (53) 
 
 Divide as before this angle by the degree of the first branch, 
 and thereby ascertain the distance to P' ', or how much the 
 sharper branch must be lengthened. 
 
 Again connect CfO" and the equidistant points TT' and 
 draw perpendiculars from 0' to 0"T' and from O to O'O" and 
 as before we shall have 
 
 TT = (y 0" = 2 (R - R) sin \ (a - a') , (54) 
 
 and K&O" = i (a + a'}. (55) 
 
 EXAMPLE. Given a 5 curve compounding into a 3 for 
 12 and terminating in a tangent. It is desired to move the 
 tangent in 8 feet. Show that the new P.C.C. is 67 feet farther 
 from the P.C., and locate the required P.T, 
 
PROBLEMS IN CHANGE OF LOCATION. 
 
 73 
 
 THIRD CASE. The parallel tangent farther out than 
 the terminal tangent, and the second branch of the 
 curve having the shorter radius. 
 
 It is evident that the flat- 
 ter branch must be length- 
 ened at the expense of the 
 sharper one, and, as before, 
 the angle POP' must be de- 
 termined in order to locate 
 P f . 
 
 Denote the angle O'OM 
 by a, 0"ON by ', and R 
 and 7t', the radii of the first 
 and second branches respec- 
 tively. 
 
 Then 
 
 or, 
 and 
 
 FIG. 39. 
 
 (R R') cos a' = (R- R') cos a + d, 
 d 
 
 cos of = cos a 
 
 a a' = POP'. 
 
 R 
 
 (56) 
 
 Whence the distance PP' may be found as before, and the 
 first branch extended. 
 
 The distance between the tangent points may be determined 
 as in the second case thus, 
 
 TT' = O'O" = 2R R' 
 
 a a'). 
 
 (57) 
 
 The student may show that the direction of T' from T is 
 obtained, as in the other cases, by deflecting from the tangent 
 at !T, Ka + > 
 
 EXAMPLE. Given a 3 20' curve compounding into a 5 40' 
 for 22, and terminating in a tangent. It is desired to 
 have the tangent 16 feet farther out. Locate the new 
 P.C.C., and the new P.T., and state the length of the curve 
 required. 
 
74 
 
 COMPOUND CURVES. 
 
 FOURTH CASE. The parallel tangent farther in than the 
 terminal, and the second branch of the curve having the 
 shorter radius. 
 
 FIG. 40. 
 
 Adopting the notation of the third case (R 
 second branch) we have 
 
 (R R') cos a^ = (R R') cos a d, 
 d 
 
 or, 
 and 
 
 cos a' = cos a 
 
 R- R'' 
 
 = radius of 
 
 (58) 
 
 Whence the distance from P to P', the new P.C.C., is readily 
 obtained. 
 
 The distance between the tangent points is shown by the 
 equation 
 
 TT' = (yO" = 2(R- R') sin (a' - a). (59) 
 
 and the angle to be turned off from the tangent at T = 
 i (X + "> 
 
 EXAMPLE. Given a 2 curve compounding into a 3 for 
 10, and terminating in a tangent. It is desired to move the 
 tangent in 12 feet. Show that the P.C.C. has to be moved 
 176.5 feet, and give the direction and distance of the new P.T. 
 from T. 
 
PROBLEMS IN CHANGE OF LOCATION. 
 
 75 
 
 84. A method of solution, similar to the preceding, may 
 be applied to the following problem: 
 
 Given a simple curve AT terminating in a tangent at 
 T\ it is required to find the point P, at which, by 
 compounding with a curve of known radius, the curve 
 may end in a given parallel tangent. 
 
 A 
 
 Then 
 or, 
 
 whence 
 
 FIG. 41. 
 
 Denote the unknown angle POT by a, 
 
 " perp. dist. between tangs, by d, 
 
 " first radius OTby J?, 
 
 " second radius (XT' by R'. 
 
 00 / cosnr= O'JW, 
 (fi' - R) cosa = R' -(R + d); 
 d 
 
 cos a = 1 
 
 R' R 
 
 (60) 
 
 EXAMPLES. 
 
 1. Given a 5 20' curve terminating in a tangent at T\ it is 
 required to locate a point P, whence a 3 30' curve may be 
 run which shall end in a parallel tangent 18' farther out. 
 Ans. a = 14 32', and distance from T to the P.C.C. = 272.5 ft. 
 
 Solve the following, making such modifications of the pre- 
 ceding formula as may be necessary. 
 
 2. Given a 2 40' curve terminating in a tangent; it is 
 required to locate a point P, whence a 4 20' curve may be run 
 which shall end in a parallel tangent 16' farther in. 
 
76 
 
 COMPOUND CURVES. 
 
 85. Having located a compound curve terminating in 
 a tangent, it is required to change the P.C.C. and the 
 radius of the second branch, so that the curve may end 
 in a parallel tangent at a given point on the same radial 
 line. 
 
 A 
 
 FIG. 42 
 
 FIRST CASE. The parallel tangent outside the terminal 
 tangent, and the second branch of the curve having the 
 longer radius. 
 
 Let APT be the located curve, and AP'T' the required 
 curve, having the centers and 0' respectively, and P' the 
 new point of compound curvature. 
 
 Let 7? = OP, the radius of the nrst branch, 
 R' = O'P, " " second " 
 
 " R" = 0"P' " " required " 
 
 " <1 = the given distance between tangents, 
 and the angles as indicated in the figure. 
 
 Extend the first branch of the curve from P till it termi- 
 nates at E in a tangent parallel to the terminal tangent TF. 
 Draw the radius OE, and with the radius 0"P' describe the 
 arc P'T. The angle POE being equal to PO'T, the chord 
 PT will pass through E ; and the angle P'OE being equal to 
 
PROBLEMS IN CHANGE OF LOCATION. 77 
 
 P'O"T', the chord P'T will also pass through E. The point 
 P f may, therefore, be constructed by extending the chord from 
 T' through E till it meets the first branch. 
 
 There are two principal steps in the solution. 
 
 First to find a?, and with it, second, to find R". 
 
 To find a' the figure shows that the angle P'T'K is equal to 
 a', hence, 
 
 But = tan i- a and T'K --= TF=OM= (R' R) sin a. 
 
 TF 
 
 Substituting, there results, 
 
 (61) 
 
 In the triangle 00' 0", by the law of sines 00" may be 
 found, which added to OP', will give the length of R". 
 Or, in the triangles OO"M and OO'M find at once 
 
 (R" - R) sin a' = (R' - R) sin a; 
 
 whence E" = (R' - R) > + A (62) 
 
 SECOND CASE. The parallel tangent inside of the terminal 
 tangent and the second branch having the longer radius. 
 
 Using the same figure and notation as in the preceding, a and 
 R' become the unknown quantities, and we have 
 
 tanl^-* 
 T'K ' 
 
 or, tan \ a = tan | a' _ . ; (63) 
 
 sin a.' 
 and B'^K" _*)__+. (64) 
 
 EXAMPLES. 
 
 1. Given a 5 20' curve compounded at P, Fig. 42, into a 
 2 40' and terminating in a tangent FT at 7 1 , making a: 16 30'; 
 it is required to end in a parallel tangent KT' intersecting the 
 prolongation of O'T 18 feet farther out. Find P', and O"P', 
 the new P.C.C. and radius respectively. 
 
78 
 
 COMPOUND CURVES. 
 
 2. Given a 5 40' curve compounded into a 2 30', and ter- 
 minating in a tangent making a' 32 40'; it is required to end 
 the curve in a parallel tangent intersecting the same radius 
 12 feet farther in. Locate the new P.C.C., and determine the 
 radius of the last branch. 
 
 THIRD CASE. The parallel tangent outside the termi- 
 nal tangent and the second branch of the curve having 
 the shorter radius. 
 
 FIG 43. 
 
 Let R = OP, the radius of the first branch, 
 R' O'P, " " second " 
 
 R" 0"P', " " required " 
 
 d = distance between tangents, and the angles as 
 shown in the figure, a' and R" are to be determined. 
 = EF = EK d 
 
 - r F ~ TK ' 
 
 and 
 
 whence 
 
 (R R") sin a' = (R 
 
 R" = R (R R'} 
 
 sin a: 
 
 (65) 
 
 (66) 
 
PROBLEMS IN CHANGE OF LOCATION. 79 
 
 FOURTH CASE. The parallel tangent inside the termi- 
 nal tangent and the second branch having the shorter 
 radius. 
 
 Using the same notation as in the preceding case, a and R 
 become the unknown quantities, and we have 
 
 or, g tan^ = tani' + (fi _ fi)s . na/ , (67) 
 
 and (R R") sin a' = (R R') sin a 
 
 sin cc f 
 whence R' = R-(R- R") - (68) 
 
 EXAMPLES. 
 
 1. Given a 2 50' curve, compounded at P, Fig. 43, into a 
 4 50', and terminating in a tangent at T, making a 30 ; it is 
 required to end the curve in a parallel tangent intersecting 
 the prolongation of O'T at T', 24 feet distant from T. Find 
 P' and the length of 0"P'. 
 
 2. Given a 2 40' curve compounded into a 4 40', and ter- 
 minating in a tangent, making a' 24 ; it is required to end in 
 a parallel tangent, intersecting the same radius 20 feet farther 
 in. Locate the new P.C.C., and determine the radius of the 
 last branch. 
 
 86. Having located a compound curve between two 
 tangents, it is required to shift the P.C.C. and change the 
 radius of the last branch so that the curve may end at some 
 other point in the terminal tangent. 
 
 FIRST CASE. When the second branch of the curve has 
 the longer radius and the point in the tangent is given 
 as at E'. 
 
 In the figure there are given the central angles at 0, and O', 
 the radii drawn from these points, and the distance d between 
 E and E'. It is required to find the angle at O" and the 
 radius 0"E'. 
 
80 
 
 COMPOUND CURVES. 
 
 Extend the first branch of the curve to D, where its terminal 
 tangent becomes parallel to IE ; draw the chord PE ; it will 
 pass through D, and the line drawn from E' through D 
 prolonged to the curve will indicate at P' the new P.C.C. 
 (See Art. 73.) Finally draw from 0, the line OMN perpen- 
 dicular to R' and R", and with center 0' describe the arc OQ. 
 
 By construction DF= MQ. 
 
 .-. DF= (R' -R)(l- cos a') 
 = (R' R) versine a'. 
 
 In the triangles DEF and DE'F, 
 
 <=- 
 
 whence, by substitution, 
 
 2 
 
 or, 
 
 , 
 
 DF 1 
 
 cot = cot + 
 
 9 9. ' I ft" 
 
 (70) 
 
 (R' R) versine a' 
 Divide a' a'' by the degree of curve of first branch, and 
 the result will show how far to extend the first branch to 
 reach the new P.C.C. 
 
PROBLEMS IN CHANGE OF LOCATION. 81 
 
 In the triangles OO'M and 00"N, 
 
 (R / R) sin a' = OM , 
 (R"- R)sina"=ON, 
 or, by subtraction, 
 
 (R" R) sin a?' (R R) sin a' = MN. = d ; 
 
 (R' R) sin a' + d ,--, v 
 
 whence, JJ" = B + L si ' n ^ < 71 > 
 
 Assuming the terminus of the curve at E', write the equa- 
 tions for a' and R' so that the P.T. may be at E, d distance 
 from E'. 
 
 SECOND CASE. Conditions the same as in the preceding, 
 except the point is not definitely located ; it must be, how- 
 ever, somewhere on the terminal tangent. 
 
 Here we may assume new central angles, that is, a new 
 P.C.C., and calculate Jf2"; or, we may assume R" ', and compute 
 the change in the angles. The student may show that, assuming 
 the P.C.C., 
 
 R" = R+ ?^, (72) 
 
 versme a'' 
 
 and, assuming R", 
 
 veisinea" = -^-. (73) 
 
 xt _tt 
 
 Show also how to find the distance d in each of the exam- 
 ples of this case, so that in practice a check may be had on 
 the work. 
 
 EXAMPLE. Having located a compound curve terminating 
 in a tangent, the radii of the first and second branches respec- 
 tively, 1600 and 2500 feet, and the angle a = 32; it is required 
 to move the P.C.C. back 150 feet. Find the radii of a curve 
 which shall end somewhere in the terminal tangent, and locate 
 the new P.T. 
 
 87. To substitute a three-centred compound curve for a 
 simple curve. 
 
 In the figure let a denote the central angle at 0, a' the 
 angle at 0', R, R', R", R", the radii, the last two being equal, 
 and sweeping equal arcs P'E and BP. It is evident that 
 
82 
 
 COMPOUND CURVES. 
 
 the intersection 0' of the radii n"R" will be on the 
 line bisecting the simple curve, and (7 is, therefore, the cen- 
 ter of the middle part of the 
 compound curve, 0"O"' being 
 centers of the other parts, 
 and a = the sum of the angles 
 at the centres O/ 0" and 0'". 
 In practice we assume the 
 radii R' and R", or, R' and 
 the equal angles at 0" and 
 0"' ', and compute what is re- 
 quired. 
 
 Proceeding under the first 
 assumption, and using the 
 triangle 0, 0', 0" (0, 0', 0'" 
 would answer as well), we 
 have the proportion, 
 
 R" R : R" R = sin^ : sin^- ; 
 
 whence, 
 
 and the angle at 
 
 O" = 0'" = 
 
 (74) 
 
 (75) 
 
 Under the second assumption we find from the above pro- 
 portion, 
 
 (R" - R) sin^ = (R" R'} sin^ ; 
 
 whence, 
 
 JR" = 
 
 sin - sin 
 
 2 2 
 
 (76) 
 
 a:' being known, since all the other angles are given. 
 EXAMPLE. Given, R =1910, a = 36, 
 
 #' = 1508, " = 5730. 
 Find EP' = PB. Ans. 177ft. 
 
CHAPTER V. 
 
 MISCELLANEOUS PROBLEMS. 
 
 88. Given a simple curve intersected by a straight line ; 
 it is required to find a point on the curve from which to run 
 a curve of given radius that shall terminate in the straight 
 line as a tangent. 
 
 FIRST CASE. The P.T. on the straight line inside of the 
 given curve. Let BE represent the curve, TN the straight 
 line, and N the point of 
 intersection. Measure for- 
 ward and backward from 
 N to points equidistant on 
 the curve; bisect the line 
 connecting these points, 
 and thus obtain the direc- 
 tion of the radius ON. 
 Or, more accurately, with 
 the instrument at N and 
 the reading zero, direct 
 the telescope to a point in 
 the curve 100 feet distant, and turn off from the curve an 
 angle equal to one-half the degree of the curve, then the 
 telescope will point in the direction of the tangent NG. 
 Observe the angle ONT. Suppose the curve produced to E' 
 where its terminal tangent becomes parallel to the given line 
 NTj draw O'M perpendicular to OE'. Denote the radii of the 
 given and required curves by R and R' respectively, the known 
 angle NO Thy a, and the required angle PO'P' by p. P being 
 the point sought. 
 
 It is evident from the figure that, 
 
 R cos a = (R R) cos /? + JB' ; 
 
 Rcosa R' 
 or cos/3= R K ' ( 77 ) 
 
 and yg a = angle PON, 
 
 whence P may be located, and the curve PD set out. 
 
84 MISCELLANEOUS PROBLEMS. 
 
 SECOND CASE. The P.T. on the straight line outside of 
 the given curve. 
 D 
 
 In Fig. 47, a is found as before, and ft is to be determined. 
 Similarly as in the preceding case, we have 
 
 and (R + R') cos ft + R = OT, 
 
 or, R cos a = (R + R') cos (3 + R', 
 
 Rcosa R' 
 
 whence 
 
 , 
 
 and ft OL = the angle PON, with which P may be located, 
 and the curve PD set out. 
 
 89. Given a tangent TT', and a curve TV, it is required to 
 connect these by a curve VT' of known radius forming a Y 
 at some point V. The tangent points T', V, and the angles 
 a and a' are to be determined. 
 
 In Fig. 48, suppose P V drawn from the middle of TT' tan- 
 gent to the curves at V. This construction makes P T, P V, and 
 PT' equal. Connect the centers and 0'. Now since the sup- 
 plementary angles TP V and T'P V are bisected respectively by 
 OP and O'P the angle OPO f is a right angle and PV is there- 
 fore a mean proportional between the radii R and R'. 
 
 Or denoting PV by x and the angles at and 0' by a 
 and of 
 
TO LOCATE A Y. 
 
 85 
 
 hence 
 and 
 
 tania = -= A /^ 
 
 (79) 
 (80) 
 
 a 7 
 
 180 - a. 
 P 
 
 \ - 
 
 FIG. 48. 
 
 90. Given a curve TT located, and the radii of two 
 other curves, it is required to connect the system forming a 
 Y, as indicated in figures 49 and 50. 
 
 FIG. 49. 
 
 FIRST CASE. The curves being convex to each other, as 
 in figure 49. Connect the centers of the curves O,0',O", thus 
 
86 
 
 MISCELLANEOUS PROBLEMS. 
 
 forming a triangle in which the three sides are known, and 
 compute the angles and thence the common tangent distance. 
 Practically find one of the central angles as <*, then 
 
 Tangent distance = x = R tan i a = PT=PT=PV 
 
 tan 
 
 n ' x - 
 a -R'~ 
 
 and 
 
 tan i 
 
 (81) 
 
 (82) 
 
 R 
 
 whence the limiting points of the curves are determined. 
 
 EXAMPLE. Given TT', a 3 20' curve, and TV, a 4 40' 
 curve, it is required to connect them by VT' a 6 curve. Com- 
 pute the central angle and the common tangent distance. 
 
 SECOND CASE. The curves being convex in the same 
 general direction. Fig. 50. 
 
 OT=R 
 
 0'T=R f 
 
 O"T = R" 
 
 Connect the centers 
 O O' O" of the curves 
 thus forming a tri- 
 angle and compute the 
 central angles and 
 common tangent dis- 
 tance as in the first 
 case. 
 
 The student may 
 show, by construction, 
 how to locate P, the 
 intersecting point of 
 the tangents. 
 
 EXAMPLE. Given 
 TT a 1 20' curve, 
 and TV a 7 curve, to 
 
 connect by an 8 30' curve VT'. Compute the central angle 
 
 and the common tangent distance. 
 
A TRACK WITH CIRCULAR ENDS. 
 
 87 
 
 91. To lay out a track of a given length, having cir- 
 cular ends connected by two tangents of known direc- 
 tions, and of given distance apart at either end. 
 
 FIG. 51. 
 
 In the figure denote the line AB by &, the known angles 
 FAG and AFG, as found from the direction of the tangents, 
 by a and ft respectively. Call DB, y; DP, x ; DC, c; BF, R ; 
 and DE, r. 
 
 Then the distance round, or length of track, 
 
 (83 > 
 
 Now, 
 
 E = " sec a, 
 
 r = - sec a, 
 
 x = - cosec a, 
 y + x = - cosec a, 
 
 V=(-. -) cosec a. 
 Substituting in (83), we obtain 
 
 - cosec a 
 
 (84) 
 
 in which c is the only unknown quantity. 
 
MISCELLANEOUS PROBLEMS. 
 
 EXAMPLE. If AB = 50Q, the distance round mile and 
 ft = 80. 
 Then, 2640 = 2 [- TT . 250 sec 10 + f 250 - -\ cosec 10 
 
 From which c may be found, thence the arcs DMC, ANB, and 
 the tangent DB. 
 
 92. Given two curves united by a tangent, to substitute 
 for the tangent a simple curve of known radius, com- 
 pounded with the others. 
 
 Denote by R the radius OT, 
 
 R' " O"T, 
 
 R" 0"P = 0"P', 
 
 " d the distance between tangent points TT', 
 
 a the angle TOP, 
 
 " DOO'. 
 
 Draw O'D perpendicular to OT. 
 
 Then 
 
 and OO' = (R-R')secj3. 
 
 Now in the triangle 00' 0", the sides being known, find the 
 angle 0, hence the angle <*, and thereby locate the point P. 
 In a similar manner the point P' may be located. 
 
TANGENT TO CURVE FROM POINT. 
 
 89 
 
 Can a radius as short as i (R -f- R' -f 0(7) be employed? 
 If R = R', show that 
 
 EXAMPLE. Given two curves, TP and T'P', of 4 40' and 
 5 50' respectively, connected by a tangent of 500 feet in 
 length, to replace by a simple curve of 1 30'. 
 
 93. To locate a tangent to a given curve from a fixed 
 point without. 
 
 Let QTR be the given curve and P the point. If the ground 
 is clear and the point not over 200 feet distant, proceed as 
 follows : 
 
 Measure in the direction of the curve to Q, and onward to 
 R. Then by geometry 
 
 x PQ. (85) 
 
 With one end of a tape pivoted at P, observe where the 
 length PT cuts the curve. This will be the point of tangency. 
 If the distance is greater, measure as before to Q, and observe 
 the angle which the chord QR makes with the tangent Q7, at 
 Q. Thus the central angle becomes known, OM and QM may 
 be calculated, and thence 
 
 But 
 
 and the angle 
 
 tan OPT = , 
 MPT = OPT- 0PM. 
 
90 MISCELLANEOUS PROBLEMS. 
 
 Hence deflecting at P the angle MPT, the direction of the 
 tangent is indicated. Its length is given in Eq. (85) the chord 
 being either measured or computed. 
 
 EXAMPLE. Given QTR a 4 40' curve, P a point 1000 feet 
 from Q, and the angle 1QR 18 30', to find the angle MPT 
 and the length of the tangent PT. 
 
 94. To locate a definite point in a given curve from some 
 point in the tangent. 
 
 Let JK be a tangent to an 8 curve, T the point of tangency. 
 It is required to locate from some point in the tangent, a point 
 P in the curve, two stations from T. 
 
 From the known degree of curve and the number of stations 
 it will be perceived that a = l 6. 
 
 Calculate TP = 2 R sin 8, and establish P' at the same 
 distance from T. Then in the isosceles triangle PTP' find 
 P'P = 2 PT cos TPM = 2 PT cos 4. 
 
 Set up the instrument at P' and deflect from the tangent, in 
 the direction of the curve, one fourth the central angle or 4, 
 and measure off the distance PP' to the point P. 
 
 95. Given the perpendicular distance of a point from 
 a tangent, it is required to find the point on the tangent 
 whence a curve with a given radius may be run which 
 shall pass through the given point. 
 
 Let BT be the tangent, TP = d the perpendicular to the 
 given point P, OP = H the given radius ; it is required to 
 find x, the distance from T to the point of curve fi. 
 
TO PASS A CURVE THROUGH A POINT. 
 
 91 
 
 Draw PE parallel to HT ; then,*in the right triangle POE, 
 we have 
 
 x\ 
 
 or 
 
 (86) 
 
 Given 7J7' and 7V, 
 the student may write 
 an expression for 7i. 
 
 Show how either of 
 the problems in this 
 article may be solved by 
 trigonometry. 
 
 96 To prolong a straight line, as LN, beyond a tree, a 
 building, or any obstacle. 
 
 FIRST METHOD. Set up the instrument at any point of 
 the line, as TV, and deflect sufficient to pass the obstacle to any 
 point 1\ Measure NP, remove to 7 J , deflect to 0, making the 
 angle QJ'O double the angle at TV. 
 
 FIG. 56. 
 
 Measure PO = PN, place the instrument at 0, observe 
 7', plunge the telescope and deflect to 72, so that SOU = },- 
 OPQ, the telescope will then be in the prolongation of LN, 
 and 
 
 (87) 
 
 SKCOND METHOD. Deflect 00 from the direction of the 
 line at TV, measure to P a distance sufficient that 7*0, making 
 an angle of 60 with PN, will clear the obstacle. Measure 
 
92 
 
 MISCELLANEOUS PROBLEMS. 
 
 PO = PN, and turn the felescope in the direction of O/t, the 
 prolongation of LN, by deflecting 60 from the direction of 
 PO. NO is evidently equal to PO = PN. 
 
 THIRD METHOD. Erect a perpendicular, NK, of sufficient 
 length that a line passing through A' parallel to LA 7 " will clear 
 
 the obstacle ; run KM ; lay off MO = N K, and a right angle 
 turned from MO will indicate the direction of LN, or its 
 prolongation OR. 
 
 FIG 59. 
 
 OTHERWISE, if a stream or pond, measure a base line 
 LP, and the angles at L and J > ; then, by the law of sines, 
 
 ~ ~ X sin P 
 
 LN = 
 ~ 
 
 sin (L + P) 
 
 (88) 
 
TO FIND THE RADIUS OF A TRACK. 
 
 93 
 
 97. Given a railroad track on a curve to find the radius. 
 On the curved track 
 
 M 
 
 ^\1V 
 
 XY take any point L, 
 and measure a straight 
 line LN, and from its 
 middle point Q, measure 
 the perpendicular QM 
 to the track. Then de- 
 noting the radius MO 
 by R, the chord LN by 
 2 c, and the middle ordi- 
 nate MQ by m, we have, 
 from a well-known prop- 
 osition in Geometry, 
 
 Q 
 
 FIG. 60 
 
 whence 
 
 m : c = c : 2 R m, 
 c? + m* 
 
 R = 
 
 2m 
 
 (89) 
 
 98. To locate a curve parallel to a given curve and at a 
 definite distance from it. 
 
 Let BCDE be the given 
 curve, B'B", E'E", etc., points 
 on radial lines through B, E, 
 etc. To locate B'E' parallel, 
 and at a given distance from 
 the first, use the same deflec- 
 tion angle and find the length 
 of chord E'D' from the pro- 
 portion 
 
 (90) 
 
 : E'V, 
 
 or, denoting R + EE' by jR', 
 
 a well-known formula will give 
 
 E'D' =2R' sin E OD ; 
 and similarly for the curve B"E". 
 
 EXAMPLE. Given BE, a 4 curve of three stations, to find 
 the length of chord required to run in the curve B'E' 60 feet 
 distant. 
 
94 
 
 MISCELLANEOUS PROBLEMS. 
 
 99. To connect two parallel tangents by a reversed curve. 
 
 FIRST CASE. Given the length of the straight line con- 
 necting the tangent points, and the perpendicular distance 
 between the tangents, to find one of the equal radii which 
 shall unite the tangents by a reversed curve. 
 
 FIG. 62. 
 
 Denote OD = DO' by R. 
 TT by c. 
 Q:r = perp. dist. by d. 
 
 Draw OE to the middle of TD, then the triangles TOE and 
 TQT are similar, and the homologous sides give the proportion. 
 
 or R== f~' < J1 ) 
 
 The student may show, what is assumed in the foregoing, 
 namely, that the point of reversed curvature D, is at the mid- 
 dle of TT. 
 
 SECOND CASE. The radii unequal, the same lines TT = c, 
 and QT d given as before, and the length of one of the 
 radii OT = R. 
 
 The student may find the unequal chord lengths DT and 
 DT, and show that the radius 0'T = R' is equal to the 
 product of DT and DT divided by 2 QT, or denoting DT by c' 
 and DT by c", show that 
 
REVERSED CURVES. 
 
 95 
 
 Again, since 
 
 and 
 
 7?' = 
 ~ 2d' 
 
 R _ c' 
 ~R'~ c c' 
 
 (92) 
 
 R' = ^-.-R. (93) 
 
 Given a 3 curve AB, a straight line TT' intersecting it at 
 7 1 , making an angle of 40 54' with the tangent TN ; it is 
 required to find the reversing point P whence an 8 curve may 
 be run terminating in the given straight line TT'. 
 
 fo' 
 
 Let E indicate the P. T. of the required curve, and draw 
 TC and ED perpendicular, and OB parallel to the given line 
 TT. Now R and R' are known, the angle CTO = NTT' = 
 40 54' whence the angle COT and side CT= ED can be found, 
 hence the angles TOP and 0' and the point P determined. 
 
CHAPTER VI. 
 
 CONSTRUCTION. 
 
 A. GENERAL DIRECTIONS, DEFINITIONS AND 
 PROBLEMS. 
 
 100. The position of the center line of the road being finally 
 determined upon; its place indicated by stakes, their elevations 
 taken, the profile made, and grades established, the next thing 
 in order is to build the road. The work must be carefully set 
 out; for example, stakes must be set for excavations and em- 
 bankments, and for culverts, trestles, etc. The precise location 
 and elevation of bridges and tunnels, if any, must be marked 
 out; the amount of cutting and filling necessary to reduce the 
 inequalities of the ground to conform to the grades must be 
 ascertained; the kind, quality, and quantity of materials to be 
 used in the construction, their most economic transportation, 
 where, when, and how to be delivered. These and innumer- 
 able other problems and questions, present themselves to 
 the engineer during the progress of the work, for solution or 
 answer. 
 
 To proceed advantageously with this work, especially if the 
 line is of considerable length, a re-organization of the engineer 
 department is usually affected, the chief engineer having now 
 division and resident engineers to assist him. 
 
 The chief, as before, has charge of the work, gives general 
 directions, passes upon bids, estimates, etc., and decides numer- 
 ous questionable matters referred to him by the division 
 engineer. 
 
 A division engineer is placed in charge of several miles 
 of the line in which there are a number of residencies, and 
 to him the resident engineers report. From these reports 
 monthly estimates are made, and forwarded to the chief for 
 examination and approval. 
 
DEFINITIONS AND PROBLEMS. 
 
 97 
 
 A resident engineer has charge of the construction of a 
 few miles of the road, and it is his duty to personally superin- 
 tend it. With his assistant or rodman, he will show grade, 
 or line, perform the necessary measurements and computations 
 for the monthly estimates, and make the required report to his 
 superior officer. 
 
 101. A cross-section is a vertical section taken at right 
 angles to the vertical plane which embraces the center line ; 
 its extreme limits to the right and left depend upon the width 
 of the road-bed, the transverse slope of ground, the side slope 
 of cut or fill, and the cut at the center. The difference in the 
 elevations of the surface and grade at the center gives the cut 
 at that point. 
 
 A cross-section should be taken at every regular station, and 
 at every other point where the surface of the ground changes 
 its slope perceptibly, whether at the center or near the place of 
 the side stakes, so that data may be had to calculate closely 
 the amount of material removed. 
 
 102. A grade point, or a point where the natural surface of 
 the ground intersects the grade, is neither in cut nor fill ; such 
 a point is discovered by setting the target of a leveling rod 
 equal to the difference between the height of instrument and 
 elevation of grade, and having the rod moved around until a 
 place is found where the target is equal in height to the line 
 of sight. A stake marked 0,0, should be set at such a point, 
 and its position noted in the cross-section book. 
 
 FIG. &t. 
 
 103. Given the elevations of two points A and B, their 
 distance apart, and the gradients, to find some point P, 
 where the grades will meet. 
 
98 CONSTRUCTION. 
 
 Denote the horizontal distance of AP by x. 
 " " " PB " y. 
 
 " " " AB " d. 
 
 " gradient of AP by . 
 
 " " PB " 6. 
 
 " difference in elevation of .1 and B by e. 
 
 Then x -f ?/ = J, and a: + by = e, using the plus sign when 
 the grade rises from P to B and the minus sign when the grade 
 falls from P to B. 
 
 From the last two equations the values of x and y may be 
 found, and the point P thereby located. 
 
 For example, suppose the gradient from A rises .5 foot per 
 station and from P to B it falls .8 foot per station, the differ- 
 ence in elevation of A and B 2.8 feet, B being lower than A, 
 and from A to B there are 10 stations. 
 
 Then x + y = 10 
 
 i+|y= 2.8, 
 
 whence x = 4 and ?/ = 6. 
 
 Showing the point P to be four stations from A. 
 
 104. To find where a grade will pass from cut to fill or 
 vice versa, the slope of the ground being uniform between 
 stations. 
 
 r 
 
 .41 
 
 FIG. 65. 
 
 1. Given the cut at station 40 equal to a, and the fill at 41 
 equal to b, find the grade point P. By similar triangles we 
 have 
 
 a : x = b : 100 x, 
 
 whence x = . (94) 
 
 a + b 
 
 If a = 10, and b = 4, x will be 71.4'. 
 
VERTICAL CURVES. 99 
 
 2. Given cut at station 40 equal to a, rise of grade per 
 station equal to m, and the slope of ground n to 1, find x, the 
 distance to the grade point P. 
 
 whence x= 100yia . (95) 
 
 mn + 100 
 
 If a = 8, slope of ground 10 to 1, and rise of grade per 
 station 1.5', x will = G9.G'. 
 
 3. Find an expression for , assuming the grade falls b ft. 
 per station; the other conditions as in example 2. 
 
 105. Vertical Curves. Where two grades meet an angle is 
 formed, and it is necessary to lessen the grade at the point of 
 meeting, and for a short distance both ways therefrom, say 
 from 100' to 200'. 
 
 FIRST METHOD. A very practical and generally sufficiently 
 accurate method to round off two grades is to make, on profile 
 paper, a drawing of the grades to a large scale, say 2 feet or 
 2 feet to the inch vertically, and 40 or 50 feet to the inch 
 horizontally; then fit a proper curve to them and scale off every 
 25' or 50' the distances to points therein. 
 
 SECOND METHOD -- Let MN and NO represent the grades 
 to be rounded off. Measure equal distances in both directions 
 
 from the point where the grades meet, as NP = NQ equal say 
 150' or 200', and connect the points P and Q by a straight 
 line. If the angle of intersection of the grades is very small, 
 this line PQ, with a little rounding up at its extremities, may 
 be taken for the grade required. Then the ground must be cut 
 away or filled up to it, depending upon the work whether it is 
 in cut or fill. By repeating the operation, that is to say 
 
100 CONSTRUCTION. 
 
 connecting R and S, points 100" each way from Q, and also R' 
 and S', equidistant from P, a nearer approximation to a curve 
 will be attained, and in surfacing the road any slight angulari- 
 ties at the points of meeting, as at R and S, can be removed. 
 
 106. Difference in elevation of the rails on curves. The 
 
 centrifugal force F, of a car moving in a curved track of 
 radius R, and velocity v, is given by the well-known general 
 
 ,,2 
 
 formula in Mechanics F = . To oppose this force the outside 
 
 rail on curves is set higher and the inside rail lower than the 
 grade of the center line. 
 
 The problem is to find the difference in elevation of the 
 rails, so that as a car moves round a curve on the inclined 
 plane thus formed by the unequal elevation of the track, the 
 action of gravity to draw it down the plane will just equal the 
 centrifugal force. We have given R, v, and the distance 
 between the centers of the rails d, to find the difference in 
 height h, the linear dimensions being in feet, and v in feet per 
 second. The action of gravity upon a body lying on an 
 inclined plane varies with the sine of the angle of the plane. 
 The component of gravity, therefore, that opposes the cen- 
 trifugal force = 32| sin a, a being the angle of the plane. 
 
 Substituting the value of sine a, and equating the forces we 
 obtain 
 
 I 6d' 
 whence = 1|. (96) 
 
 If instead of v = feet per second, we write F = miles per 
 
 3600 
 hour = rQ v there results 
 
 22\ 2 6dF 2 _. 06687 dF 2 
 
 Substituting 4'.9 for d, which is about its proper value for 
 the standard gauge of 4' 8", and writing for the radius its 
 equivalent in terms of the degree of curve, there results 
 
ELEVATION OF OUTER RAIL ON CURVES. 101 
 
 Which is the formula employed to calculate the following 
 table : 
 
 V 
 
 DEGREE OF CURVE. 
 
 1 
 
 2 
 
 . 3 
 
 4 
 .09 
 
 5 
 .11 
 
 6 
 .14 
 
 7 
 .16 
 
 8 
 .18 
 
 10 
 .23 
 
 20 
 
 .02 
 
 .05 
 
 .07 
 
 30 
 
 .05 
 
 .10 
 
 .15 
 
 .21 
 
 .26 
 
 .31 
 
 .36 
 
 .41 
 
 .51 
 
 40 
 
 .09 
 
 .18 
 
 .27 
 
 .37 
 
 .46 
 
 .55 
 
 .64 
 
 .73 
 
 .91 
 
 50 
 60 
 
 .14 
 
 .29 
 
 .43 
 
 .57 
 
 .71 
 
 .87 
 
 1.00 
 
 
 
 .21 
 
 .41 
 
 .62 
 
 .82 
 
 1.03 
 
 1.23 
 
 
 
 
 In calculating the height in any given case, the velocity 
 assumed should be that of the train of the highest speed which 
 will regularly pass around the curve, since if the centrifugal 
 force developed be not thus counteracted, an accident might 
 occur from the excessive pressure of the flanges of the wheels 
 against the outside rail. On the other hand, the flanges of 
 wheels on cars running at a lower rate of speed around the 
 same curve would be forced by gravity against the inside rail. 
 The effect of this would be to wear off the inside surface of 
 that rail, but it would not jeopardize life and property to the 
 extent of the former pressure, if the outside rail were not prop- 
 erly elevated ; and as it is obviously impracticable to guard 
 against both, the value of V indicated above should be used. 
 It is better, however, all things considered, to reduce the speed 
 of fast trains when running round sharp curves than to elevate 
 the rail unduly. 
 
 With regard to the elevation of the outer rail, the practice 
 among engineers is not precisely uniform. Some think inch 
 elevation per degree sufficient for speeds up to 50 miles per 
 hour. Others give f of an inch per degree up to 5, and ^ inch 
 per degree thence up to 10, 
 
102 CONSTRUCTION. 
 
 Another rule is to elevate the outer rail 
 
 1 inch for a 1 degree curve, 
 
 2 inches " 2 " 
 
 3 " "3 " " 
 31 4 
 
 4 "5 " 
 
 41. g u 
 
 and slacken speed for greater curvature rather than elevate the 
 rail above the last height. It is evidently unwise to lay down 
 a specific rule, based upon the degree of curve simply, since 
 the location of the curve will in practice enter the problem as 
 a factor. When the curve is in a low place approached by a 
 heavy grade, more elevation should be given than when the 
 curve is on a summit, or at or near an important station, or 
 wherever the conditions favor an easy approach. 
 
 The table shows the theoretical requirements, and the 
 engineer must exercise good judgment in its application as in 
 all things else. 
 
 The difference in elevation of the rails on a curve, if 
 transition curves are not used, is the same from the P.C. to the 
 P.T. From each of these points it is diminished gradually 
 along the tangent until it becomes zero. The range for this 
 distance is from 50 to 200 feet, depending upon the curvature. 
 For a 3 or 4 curve the distance should be about 100 feet; for 
 an 8 or 10 curve about 200 feet. For a compound curve the 
 average of the elevations due to its branches gives the proper 
 difference in height at the P.C.C. 
 
 It is the practice on most roads to increase the gauge on 
 curves varying from \" to ". 
 
 107. Inasmuch as it requires more force on a curved track 
 than on a tangent to overcome the resistance of motion, it 
 is customary and proper to make the grade somewhat less on 
 curves than on tangents. This is called easing, lessening, or 
 reducing grades on curves. The rules adopted are based upon 
 the assumption that the resistance increases with the curva- 
 ture ; that is, the resistance offered by a curve of 2 is twice 
 that of a 1 curve. 
 
SETTING SLOPE STAKES. 103 
 
 On the Central Pacific Railroad, in the Sierra Xevada Moun- 
 tains, grades were lessened for curvature from 2 to 2 feet per 
 mile per degree, on curves from 2 up. Or on an 8 curve, 
 the grade was made from 16 to 20 feet per mile less than on 
 the tangents. On the location through Weber Canon, on the 
 Union Pacific Railroad, grades were reduced T ^ per degree 
 per hundred feet ; that is, if a maximum grade on tangent 
 was one foot per hundred feet on an 8 curve, it was made 
 1 T ^ X 8 = /^ of a foot, or a reduction of about 13 feet 
 per mile. Other roads use y^ per degree per hundred feet, 
 which, for an 8 curve, makes a reduction of about 21 feet 
 per mile. 
 
 B. SETTING SLOPE STAKES. 
 
 108. The nature of the ground to be excavated being 
 known, or assumed to be known, the ratio of the side slopes 
 base to height may be fixed upon ; the depth of cut at 
 center being computed, and the width of road-bed decided, 
 sufficient data are furnished to set the dope stakes, which indi- 
 cate the limit of cut or fill. 
 
 H 
 
 -4 D 
 
 FIG. 67. 
 
 Suppose the ground is level transversely, and that ABHI 
 represents the cross-section. 
 
 Denote the width of road-bed AB by ?, 
 
 " " cut CD, at center, " c, 
 
 " CI or CH, the distance from center 
 to place of slope stake, distance out, " d, 
 
 and the ratio of base EM to height MH " r : s. 
 
 Then d = + c 
 
104 
 
 CONSTRUCTION. 
 
 ordinarily for clay cuts r : s = 3 : 2, 
 or, d = - + '- 
 
 (100) 
 
 If the ground slopes transversely, let ABFCE represent 
 the section. The line EF may be straight, or it may be 
 broken at C. The distance out on the right-hand side will 
 evidently be greater than that for the level section, while that 
 on the left side will be less. In all cases, if the inclination 
 CF of the ground be ascertained, the distance out to F can be 
 readily calculated as follows : 
 
 Obtain directly from the given slope of ground the depth of 
 cut at foot of slope, as at B. 
 
 Denote this cut, the line BO by c', 
 
 OP 8, 
 
 PF x, 
 
 " the slope of ground, horizontal to vertical " m : n, 
 " " side slopes as before " r : . 
 
 Then g = x, 
 
 8 c> 
 
 and - o = c + # 
 
 r 
 
 whence, 
 and 
 
 w . 
 a = H 
 
 2 
 
 ms nr 
 
 mrc' 
 
 (101) 
 (102) 
 
SETTING SLOPE STAKES. 105 
 
 or assuming and inserting values as in the preceding, namely 
 r : a = 3 : 2, there results, 
 
 Measure out the distance d, sight a rod on the point thus 
 reached, and see if the observed and computed heights agree 
 quite closely, say within a tenth of a foot. If they do not, 
 probably the ground does not slope uniformly as was assumed, 
 and the work must be revised. 
 
 A stake must be driven at the point F, marked with figures 
 indicating the depth of cut there, preceded by a C for cut, facing 
 the center line of the road, and a record made in the cross 
 section book of the distance out and depth of side cut. In case 
 of a fill, the letter F should be substituted for C. Some 
 engineers prefer the signs -f- and for cut and fill respectively, 
 but the letters are preferable. 
 
 Evidently the same general formula (102) for upper side 
 stake in cut, will answer for the lower side stake in fill, and 
 if the ratio of side slopes is the same in both, 3 : 2, the particu- 
 lar formula (103) may be used. On account of drainage, the 
 width of road-bed must be from 4 to 6 feet wider in cut than 
 in fill. 
 
 REMARK. The ratio for ordinary earths is f to 1; for 
 solid rock, \ to 1 ; and for loose rock, and sand in embank- 
 ment, 1 to 1. 
 
 For distance out, lower side stake in cut, we use the 
 
 following notation : 
 
 Denote ET\)y&'. 
 
 AQ " c", 
 
 ' QT a:', 
 
 " distance out " d'. 
 
 And the other notation as above. 
 
 Then - g' = x, 
 
 m 
 
 and 
 
CONSTRUCTION. 
 
 
 
 whence ' 
 
 and < r = + . (105) 
 
 2 7NS 4- nr 
 
 which formula will answer for distance out, to down-hill 
 stake in cut, or up-hill stake in fill, and may be modified as 
 (102) when the ratio of the side slopes is determined. If r : s 
 = 2:1, 
 
 (106) 
 
 EXAMPLES. 
 
 1. Given width of road bed 20 feet, depth of center cut 13', 
 side slopes r : s = 3 : 2 ; slope of ground m : n = 10 : 1 . Find d. 
 
 and substituting in (103) 
 
 The result shows that the rod reading at the side stake, if 
 the ground slopes uniformly from center out 10 : 1, should be 
 3'.47 less than that at the center. 
 
 2. Find in Example 1 the distance out to lower side stake, 
 and depth of side cut. State what the rod reading should 
 be. 
 
 3. Given, width of road-bed, 26 feet ; side slopes, to 1 ; 
 surface slope, 14 to 1 ; center cut, 16 feet. Find the distance 
 out both ways from center, and the rod readings. 
 
 109. When the surface of the ground cuts the road- 
 bed, part of the work is in excavation and part in embank- 
 ment ; this is called side-hill work. 
 
 In this case, the upper side can evidently be determined 
 as before. For the low r er side, a formula can be readily 
 deduced. 
 
SETTING SLOPE STAKES. 
 
 107 
 
 The distance DG, and hence the fill A 0, can be determined 
 from the slope of the ground. It remains to find the distance 
 OQ, which shows the position of E. 
 
 FIG. 69. 
 
 Denote A O by c", 
 OQ 8', 
 " QE x, 
 and the other notation as before. 
 
 Then 
 
 V x 
 
 m 6 ' 
 
 and 
 
 (107) 
 
 whence 8' = 
 
 ?ns nr 
 
 which, added to half width of road-bed, gives the distance out, 
 and it will be perceived that (107) is analogous to (101), as 
 might have been inferred. 
 
 EXAMPLES. 
 
 1. Given center cut 2 feet, slope of ground 4 to 1, slope of 
 sides 3: 2, width of road-bed 20 feet. Find distance out both 
 ways and the corresponding heights. Locate also the grade 
 point. 
 
 2. The problem shown in figure 70 may be solved like the 
 preceding. For from the slopes the value of c" can be found, 
 
108 
 
 CONSTRUCTION. 
 
 and the preceding formula will give the proper result. Given 
 the center cut CD = 4 feet, width of road-bed 20 feet, and data 
 as shown in the figure, locate the grade point, and side stakes, 
 and state rod readings thereat. 
 
 FIG. 70. 
 
 110. A compound section, or one in which different mate- 
 rials are found in the same section, as rock with earth super- 
 imposed, may be staked out according to the principles already 
 established if the center depths and slopes of the materials be 
 known. In widening an old cut, or making new excavations 
 in the vicinity of old workings, such information may be sup- 
 plied. With these exceptions, however, in the majority of 
 cases it is expedient to make approximate settings of stakes 
 for the loose earth, and when this is removed, rectify and 
 complete the work. 
 
 111. With a little practice in setting slope stakes, especially 
 if the method above given be used, in which the judgment of 
 the inexperienced may be improved, the young engineer learns 
 to make a very close estimate regarding the position of a side 
 stake, and when he is confident of his ability to do this he 
 should abandon the use of formulas and proceed as follows : 
 Estimate the rise of the ground from the center stake to the 
 place where the side stake should be placed; set up the level so 
 as to take a rod reading on both these points. Observe the 
 center first, and with the known center cut, width of road bed, 
 
SETTING SLOPE STAKES. 
 
 109 
 
 ratio of side slopes, and assumed rise from center, calculate the 
 distance out. Observe the rod on this point and if it agrees 
 within the prescribed limits, say a tenth of a foot, drive a stake 
 there; if not, estimate again, and profiting by the knowledge 
 obtained during the preceding effort, one or two more trials 
 should be sufficient, and less time consumed than by the 
 formula. 
 
 FIG. 71. 
 
 For example suppose the road bed 20 feet; center cut 8 feet; 
 side slopes 3 to 2, and we estimate the rise from the center 
 stake to the place where the side stake should be set equal to 
 4 feet. Then the distance out should be 10 + (8 + 4) =28 
 feet, and if the rod reading at center was 7 feet, that at the 
 side should be 3 feet. Now suppose the reading should be 
 found only 2 feet, at 28 feet distance from the center. Such a 
 result would show the ground to be higher at P than it was 
 estimated, and therefore the position of the stake farther out. 
 
 AVe perceive that an additional distance of three halves of a 
 foot will take up the rise of one foot, which is the difference 
 between the estimated and observed rise, but since the ground 
 rises, the rod reading at this distance, 29.5 feet, will be less 
 than 2 feet, and therefore a little greater length must be taken. 
 Try (7 = 30' which gives a rise from road bed of f (30 10) 
 = 13.3', and a corresponding rod equal to 15' 13.3'= 1.7'. 
 Measure out this distance, sight the rod and see if it does not 
 agree within a tenth. 
 
 Proceeding in a similar manner to set the down-hill stake, 
 \ve may estimate that the ground falls 2 feet from the center 
 
110 
 
 CONSTRUCTION. 
 
 to its place, making the distance out, 19 feet, and supposing 
 the rod reading at center 2', that at 19' should be 4'. Sup- 
 pose it is found to be 3.5' ; this shows that the slope of^the 
 ground is not as great as it was estimated, and that the posi- 
 tion of the stake is a little farther out. Now we perceive that 
 three-fourths of a foot more horizontally will bring the side 
 slope half a foot higher, which is the difference between the 
 estimated and observed rod reading, but since the ground falls 
 in the direction of the measurement, we do not require quite 
 a half-foot rise, perhaps a tenth less ; that is, instead of a 
 reading of 3.5' feet, we should expect a reading of about 3.6'. 
 For this we must go out 10 -f- f (8 -f 2 3.6) = 19.6' Measure 
 out this distance, observe the rod, and see if the agreement is 
 not sufficiently close. 
 
 If the rod reading at 19' had been greater than four feet, 
 it would show that the stake should not be as far as 19' from 
 the center. 
 
 From the foregoing considerations respecting slope staking 
 we can write the following 
 
 RULE. 
 
 In excavation, if the observed rod reading is J I 
 
 L greater j 
 
 than the computed, for the supposed site of stake, it indicates 
 that the true position of the stake is farther J t . 
 
 In embankment, if the observed rod reading is J I 
 
 L greater j 
 
 than the computed, for the supposed site of stake, it indicates that 
 the true position of the stake is farther -< > 
 
 L out J 
 
 Sta. 
 
 Eleva- 
 tions. 
 
 Grade. 
 
 Left. 
 
 Center. 
 
 Right. 
 
 Area. 
 
 C. Yds. 
 
 Remarks. 
 
 43 
 
 44 
 
 490.8 
 499.7 
 
 473.2 
 474.3 
 
 + 16.5 
 
 + 17.6 
 + 25.4 
 
 + 20.5 
 29.5 
 
 + 31.7 
 
 650.5 
 1129 
 
 3260 
 
 Width of road- 
 bed, 18'. 
 Grade rises I'.l 
 per station. 
 Slopes, j|: 1. 
 
 25.5 
 + 20.7 
 
 29.7 
 
 40.7 
 
SETTING SLOPE STAKES. Ill 
 
 112. The above table exhibits a form of recording the 
 notes. The last two columns will be explained farther on. 
 The first two columns are taken from the level notes ; the 
 third and fifth calculated from the elevations and adopted 
 grade ; and the fourth and sixth supplied on the field during 
 the operation of setting slope stakes. The plus sign indicates 
 a cut at station 43 of 17'.6, and that the right and left side 
 stakes for that station are respectively 20'. 5 and 16'. 5 above 
 the plane of the road bed. A minus sign correspondingly 
 placed would indicate a fill. The denominators of the frac- 
 tional expressions show how far these stakes are placed hori- 
 zontally from the center. In general, the numerators of the 
 fractions, found in the notes, exhibit the cuts or fills and the 
 denominators the distances out. 
 
 If the surface of the ground transversely is such that it can- 
 not be considered level, or as having a uniform slope to the 
 side stake, the cross-section party must ascertain the inequal- 
 ities and make record of them in the cross-section book. 
 
 113. In staking out the work allowance should be made 
 for the change in volume of the material to be removed from 
 excavation to embankment. If the nature of this material is 
 earthy it will occupy less space in embankment than in exca- 
 vation, and vice versa if rocky. 
 
 In regard to the shrinkage, however, much depends upon the 
 condition as well as the composition of the material, and also 
 the manner in which it is placed in the fill. If it is wet, or 
 frozen, and simply shovelled or even dumped from scrapers a 
 greater allowance should be made than if it is dry; or if the 
 conditions be the same, and carts or wagons be used to trans- 
 fer the material to a long, large fill, a less allowance should be 
 made because the earth becomes solidified by the impact of 
 the horses and loads. Moreover sandy soils will not shrink as 
 much as those in which clay preponderates. 
 
 A fair average for the shrinkage is taken at one-tenth, T a ^. 
 That means that a fill which is to be 13.5 feet at grade must 
 be made 15 feet high at first so as to allow for a settlement of 
 1.5 feet ; or a fill finished at 10 feet will settle to one of 9 feet. 
 
112 CONSTRUCTION. 
 
 Rock, on the contrary, when broken increases in bulk ; the 
 increase depending upon the size of the pieces, being greater 
 for small pieces than for large ones. A fair average increase 
 may be taken at two-thirds, f , or 3 cubic yards of rock in cut 
 will make 5 cubic yards in fill. 
 
 C. CALCULATING THE EARTH WORK. 
 
 114. From the cross section book we now obtain data 
 sufficient to determine the quantity of earth to be removed, or 
 the amount of cutting and filling. 
 
 The cross sections being parallel,* and having been taken 
 sufficiently near each other that the lines connecting the cor- 
 responding points of any two consecutive cross-sections may be 
 considered straight, and the sides of the figure planes, the 
 prismoidal formula will give the exact quantity of earth in the 
 solid. Or, if one end-section vanishes, being at grade as at the 
 end of a cut or side hill w r ork, there will be a wedge-shaped mass, 
 or a pyramid formed; but since this formula is applicable to 
 the wedge or pyramid, it may also be used in these cases to 
 determine accurately the cubic contents. For illustration, 
 conceive a prism, a wedge, and a pyramid having equal bases 
 and altitudes; and let b denote the area of each base and h 
 the common altitude. 
 
 Then the volume of the prism = Hi, 
 
 45 + &). 
 The vol ume of the wedge = bh t 
 
 The volume of the pyramid = 
 
 It will be perceived that 6, 6, and * b, represent respectively 
 the middle area of the prism, wedge, and pyramid, and therefore 
 
 * Except on curves where a correction is made. See Art. 118. 
 
CALCULATING THE EARTH WORK. 113 
 
 the volume of either of these solids, or any combination of them 
 may be expressed by the following equation, known as the 
 
 PRISMOIDAL FORMULA. 
 
 r*=(A,+ 4M+B)\ (108) 
 
 In which V denotes the volume, 
 
 h " " distance between the ends, 
 A and B " " end areas, 
 
 and M " " area of section midway between 
 
 the ends A and B. 
 
 REMARK. The term prismoid usually suggests a body com- 
 posed of the solids just named, that is, one having six plane 
 surfaces of which only two are parallel, yet the " prismoidal 
 formula " has a much wider application as was first shown by 
 Ellwood Morris, C.E., in the Journal of the Franklin Institute 
 in 1840. Trautwine says:* "It embraces all parallelepipeds, 
 pyramids, prisms, cylinders, cones, wedges, etc., whether regular 
 or irregular, right or oblique, together with their frustums, 
 when cut by planes parallel to their bases; in a word, any solid 
 whatever, which has two parallel ends, connected together by either 
 plane or by longitudinally unwarped surfaces." 
 
 Gillespie f shows that if the surface is warped, being " gene- 
 rated by a straight line resting on the two straight lines which 
 join the extremities of the tw r o end sections, and moving 
 parallel to their planes or perpendicular to the axis of the 
 road," the prismoidal formula will give the correct result. 
 
 Or if the natural surface is generated by " a straight line 
 which rests on the two end sections and moving on them in 
 such a way as always to divide them proportionally " the formula 
 is applicable. 
 
 If a ridge or hollow runs obliquely in one direction across 
 the solid, from end to end, and its position determined with 
 sufficient accuracy that the area of the mid-section as well as 
 
 * Trautwine on Excavations and Embankments, page 5. 
 t Gillespie, Roads and Railroads, pages 3G8-9. 
 
114 
 
 CONSTRUCTION. 
 
 that of the end may be computed, the prismoidal formula will 
 still hold. 
 
 115. Sectional Areas. To calculate the contents, we must 
 compute the end areas and the area of the mid-section. 
 
 D 
 FIG. 72. 
 
 If the ground is level transversely, the area is evidently 
 that of a trapezoid, having for bases the width of road-bed 
 and the sum of the distances out, and for height the center 
 cut. 
 
 Let d and d' denote distance out to right and left respec- 
 tively, w the width of road-bed, and c the center cut. 
 
 Then 
 
 A = ^ (w + d + d') . 
 
 (109) 
 
 If the ground slopes uniformly from F to E through C, 
 or if it has a uniform slope from either C' or C" to E and F, 
 CD being the center cut in the first case, and C'D and C"D 
 center cuts in the last cases. 
 
 F 
 
 In either case two triangles may be formed, having for bases 
 AD and DB, and FG and EH their respective heights, and 
 two more triangles having for their common base the center 
 
CALCULATING THE EARTH WORK. 115 
 
 cut, and for heights the distances out. Calling h and li' the 
 right and left side heights respectively, and using the notation 
 above, the equation for the area may be written, 
 
 A == H? (h + h') + c - (d + d') . (110) 
 
 4 2 
 
 The area of a section, as ABECF, Fig. 73, may be found 
 without using the center cut, simply by subtracting from the 
 area of the trapezoid EFGH the area of the triangles DEH 
 and AFG. 
 
 In case the ground is irregular, sufficient measurements 
 should be made, so that the section may be divided up into 
 triangles and trapezoids and its area thereby computed. 
 
 If the inequalities of the surface are numerous, it is gen- 
 erally sufficiently accurate to plot the section, average the 
 inequalities by stretching a silk thread over them, scale off 
 the necessary distances and heights, and compute the area by 
 some of the preceding methods. 
 
 The middle area is found by first averaging the correspond- 
 ing lines of the end sections, thereby obtaining the mid-section, 
 and then in the usual manner, by formula (110) compute its 
 area. 
 
 For example, if the center cuts at the ends A and B are 
 12' and 8' respectively, 
 
 and the corresponding distance 
 
 out to the right 34' and 28', 
 
 and the distance out to the left 23.5' " 17.5', 
 the side heights on the right 16' " 12', 
 the side heights on the left 9' " 5' ; 
 
 then the mid-section would have for center cut 10', 
 
 " " " distance out to right 31', 
 
 " " left 20.5', 
 
 " side height to right 14', 
 
 " ' " left 7', 
 
 and its area 
 
 = ?2 (14 + 7) + ~ (31 + 20.5) = 362.5 sq. ft. 
 
116 CONSTRUCTION. 
 
 116. The volume of the section referred to in the last para- 
 graph, supposing the distance between the ends .1 and B to be 
 100 feet, and the width of road-bed 20 feet, will be computed 
 as follows : 
 
 Area of end A 
 
 16 9 
 
 = ?2 (16 + 9) + 1? (34 + 23.5) = 470 sq. ft. 
 
 Area of end B 
 
 f)f\ 
 
 = (12 + 5) + 5 (28 + 17.5) =267 sq.ft. 
 4 2 
 
 Area of mid-section from last paragraph 362.5 sq. ft. 
 
 ... V = (470 + 4 X 362.5 + 267) = 36450 cu. ft. 
 6 
 
 or, 36450 + 27 = 1350 cu. yds. 
 
 The area of the cross-section at each station, when computed, 
 is placed in the column of areas in the table on page 110, and 
 the quantity of material between two cross-sections is placed, 
 as shown, in the column of cubic yards. 
 
 The student may verify the areas and cubic yards in the 
 table on page 110. 
 
 117. Instead of the prismoidal formula the method of aver- 
 aging end-areas is very frequently employed. It consists sim- 
 ply in computing the areas of the ends of the sections, taking 
 their arithmetical mean, and multiplying it by the length of 
 the section. For example, suppose we take a mass of earth, 
 level on top, center cut at one end 12', distance out each way 
 28', the road-bed being 20' and sides slopes to 1 ; the cut 
 at the other end 14', distance out each way 31', the road-bed 
 and side slope as before, and the distance between ends 100' ; 
 then, by averaging end areas, we obtain, 
 
 Area first end = (20 + 56) 1^ = 456 sq. ft. 
 " second " =(20 + 62)11= 574 " ' 
 
 Average = = 515 sq. ft, 
 
CALCULATING THE EARTH WORK. 117 
 
 and the volume - 1907 cu. yds. 
 
 By the prisrnoidal formula the correct volume is found to 
 be 1904 cubic yards, but more labor is involved in the calcula- 
 tion. 
 
 Again, suppose the larger end section to remain as before, 
 but the smaller to have only a 4' cut at center, the width of 
 road-bed, side slopes, and length of section remaining un- 
 changed, then we shall find by averaging end-areas F=1255-f 
 cubic yards, but by the prismoidal formula the correct 
 quantity is found to be 1163 cubic yards, so that in this case, 
 by averaging end-areas, we get nearly 8 per cent, too much. 
 Furthermore, if the smaller end-area vanishes, the surface 
 being at grade there, the other dimensions and conditions the 
 same as before, and the volume be computed by averaging 
 end-areas and also by the prismoidal formula, we shall find 
 an excess by the former method over the exact amount, com- 
 puted by the latter, of more than 20 per cent. And in general 
 it will be found that in a prismoid the greater the difference 
 between the end-areas, the greater will be the departure from 
 the true volume when the method of averaging end areas is 
 used. The reason becomes obvious upon analyzing the figures 
 found in the various cases. In the first and second, conceive 
 a plane drawn parallel to the road-bed, and at a distance from 
 it equal to the center cut at the smaller end. Also conceive 
 two other planes extending throughout the section longitudi- 
 nally and perpendicularly to the road-bed, one through each 
 side stake at the smaller end-section. 
 
 These planes, with the surface of the ground, faces of the 
 slopes, and the end sections divide the solid into a prism, a 
 wedge, and two pyramids. In the first example the pyramids 
 have very small bases, and therefore multiplying as we did, 
 when averaging end-areas, by one-half the altitude instead of 
 one-third, we increased the volume but little. In the second 
 example, however, the bases of the pyramids are larger and 
 the difference in the product between one-half and one-third is 
 considerable. 
 
118 
 
 CONSTRUCTION. 
 
 In the third or last example, computed above, the mass may 
 be divided into a wedge and two pyramids, the bases of the 
 latter forming a still larger part of the area of the cross-section 
 than in either of the other examples, an increased percentage 
 in quantity obtained by end-areas over the correct result might 
 be expected. While, therefore, in many cases quantities may 
 be computed with sufficient accuracy by averaging end-areas, 
 it will be perceived from the foregoing what conditions are 
 favorable, and what unfavorable to an approximate result by 
 this method, and that the engineer must exercise judgment in 
 determining when to use the prismoidal formula instead. 
 When the ground is very much broken and irregular, or in the 
 case of borrow pits, the mass may be divided into small regular 
 solids, by the level and tape, and their volumes determined by 
 well-known rules of geometry. 
 
 When the precise quantities are required, as in piers, abut- 
 ments, etc., the prismoidal formula should be used. 
 
 118. Excavation on Curves. In calculating the amount 
 of excavation, we have thus far assumed that the cross-sections 
 are perpendicular to a straight center line. The assumption is j 
 not theoretically correct on curves, though where the curve is j 
 
CALCULATING THE EARTH WORK. 
 
 119 
 
 not very sharp, the error arising is generally slight. Moreover, 
 when the method of averaging end areas is employed instead 
 of the prismoidal formula, in computing quantities, any attempt 
 to correct errors arising under the above assumption would 
 be a needless refinement ; still, where greater accuracy is de- 
 manded, and the prismoidal formula, therefore, freely used to 
 obtain the volumes, it may be well not to ignore the effect of 
 curvature, especially where the depth of cut is great, the radius 
 small, and the transverse slope steep. 
 
 E 
 
 Let Fig. 74 x represent the horizontal projection of a por- 
 tion of a road-bed in excavation on a curve, the center of 
 which curve is at 0, LCL' the center line, MEN the outside, 
 and UTV the inside line of slope stakes. Now, in calculating 
 the volume between the stations L and C, in the ordinary way 
 v/e obtain the contents of the solid lying between the planes 
 PCQ and M'L V, drawn at right angles to the straight line 
 7,6', thereby getting too much by the volumes of the wedge- 
 shaped masses QTC and LW, and too little by the wedge- 
 shaped masses MM'L and PEC. If the distances out were 
 equal, as shown in cross-section ABGCF Fig. 74y, the over- 
 Lipping on one side of the center line would counterbalance 
 the gap on the other, and no correction would be necessary. 
 Iti other cases, the overlap and gap may in general be repre- 
 sented by solids similar to QCQ' and POP respectively. 
 
 Suppose the case in question has a cross-section as A BE Fin 
 74y. We may proceed as follows, employing one of the theorems 
 of Pappus, namely : If a plane area revolve round any axis in 
 
120 CONSTRUCTION. 
 
 % 
 
 its plane, the volume generated is equal to the area of the 
 revolving figure multiplied by the length of the path described 
 by its center of gravity. 
 
 Kow, the center of gravity of a triangle is on a line joining 
 the vertex with the middle of the base, and at two-thirds the 
 distance from the vertex. 
 
 Let H indicate the middle of GE. Draw CE' horizontally, 
 and project // and E on it at H' and E' respectively. Calling 
 the distance out to E, d, and to F, d', \ve perceive that 
 
 - \CH' = \ j.l(d-d') + d' = l(d + d') 
 o o 2 o 
 
 or the sum of the distances out. 
 
 This added to the known radius, will give the radius 7t', by 
 which we may determine the distance traversed by the center 
 of gravity of the plane in question.* 
 
 Let RS in (x) represent the required distance traversed, 
 ES = G'S - G'R = G'S - 100. 
 TtR'D 
 
 But G'S = 
 
 180 
 
 180 
 
 Find then in the ordinary way, for straight center-line by 
 the prismoidal formula, the volume of the mass between the 
 stations L and C. From the mid-section find the area of the 
 triangle CGE, by subtracting from the whole area, already 
 computed, the part ABGCF, which we may assume to be the 
 plane area that generates the solid, or the wedge-shaped body 
 sought. I 
 
 Then the product of the length of RS, and the area of bGE 
 will give C the correction required. 
 
 The correction must be added when the highest ground is on 
 the convex side, and substracted when the highest ground is on 
 the concave side. 
 
 * Practically correct, but not precisely so theoretically. 
 
CALCULATING THE EARTH WORK. 121 
 
 EXAMPLE. 
 
 Given a 10 curve, or 72 = 573.7. 
 
 Cross section at A, . . . . T 4 F , T , f , and its area = 746 
 "A . . . . A, T %,|, " " =370 
 " M, .... A, A, If, =545 
 Vol. between the planes 
 
 PCQ and M'LV' = f (746 + 2180 + 370) 100 = 2035 cu. yds. 
 Area of triangle CGE in 74y = 335 
 
 CH' = i (15 + 50) =.21.67 . 
 
 E' = 573.7 + 21.67 = 595.4. 
 
 3.1416X595.4X10 
 
 = 
 
 180 
 
 Therefore we obtain for the entire volume between MV and 
 ET, 2035 + 48 = 2083 cu. yds. 
 
 119. The following is Henck's method of making the cor- 
 rection at each station.* Adopting the notation employed in 
 the preceding articles, namely, 
 c = the cut at the center, 
 
 d and </', the greater and less distances out, respectively, 
 h and /*', the corresponding side heights, 
 w = the width of road bed. 
 Then the area of the triangle 
 
 CGE = c (d d') + $ w (h hy 
 
 The wedge-shaped mass horizontally projected at PCP' is 
 considered a truncated prism, its edges PP f and 7t.S, always 
 short, are taken as straight lines, and at C the height of the 
 .solid vanishes. 
 
 Now PP' = 2 d sin $ D, 
 
 and J2,S' = 2 d' sin D. 
 
 Then since the volume of the prism is equivalent to the 
 product of the base, and one-third the sum of the edges, the 
 
 * Henck's Field Book, page 112. 
 
122 CONSTRUCTION. 
 
 formula for the volume and hence the correction sought is 
 
 C = [$c(d - d') + w(h - h')]$(d + d') sin \D (111) 
 
 or, writing for sin ^ D, its value in terms of the radius R, 
 
 i on 
 
 C = [i c (d - d') + i w (h - h')] (d + d'). (112) 
 
 In side-hill work with such a cross-section as sBE, let I 
 denote the base of the cut rB, Fig. 74 (#), then the height of 
 the solid at E is the same as before, or 2 d sin D, the height 
 at B is w sin $ D, and at any point between D and B as s, the 
 distance of which from the center is w ft, the height is 
 o (i w ft) sin D = (w 2 ft) sin $ D. Now calling b the 
 base of the cutting, the area of the cross-section sBE = \ ft/, 
 and hence the correction 
 
 = i M i (9 j + w+ ,,,_o &) S i n i 7), 
 
 or C = i ft/i (W + ?c ft) sin D. (113) 
 
 When the excavation lies on both sides of the center lint 
 having a cross-section rBE, its area= ft/i, the height of the 
 solid at E and B will be as in the preceding paragrapl 
 respecting 2 </ sin D and w sin D; but at a point betweei 
 A and D as at r, the distance of which from the center i; 
 ft ^ iv, the height will be 
 
 2 (ft i M?) sin i 79 = (2 ft w) sin D. 
 
 I 
 
 This last height being on the opposite side of the center lin< 
 from the others is considered negative in the product for tin 
 volume, hence the correction 
 
 = i bh (2 d + w 2 ft + w) sin 7)"; 
 or C = bh (V + -a? ft) sin D, (114) 
 
 precisely the same value of C given in (113). 
 
 Substituting for sin D its value in terms of the radius w 
 obtain 
 
 C = lbh(,l+w-b-). (115) 
 
 6 K 
 
 Add or subtract as indicated above. 
 
CALCULATING THE EARTH WORK. 
 
 123 
 
 EXAMPLE. Given the road bed 30 feet, the radius of the 
 curve 800, the base of a side-hill cut 26, distance out to highest 
 stake 60, and its corresponding height 28, all dimensions in 
 feet, to find the correction. 
 
 C = i X 26 X 28 x 2Yoo ( G0 + 30 26 ) = 971 cubic feet - 
 
 120. Overhaul. When contracting for the removal of ma- 
 terial from excavation to embankment, it is sometimes stipu- 
 lated that extra pay shall be had for the transportation of 
 material through a distance greater than that specified, rang- 
 ing usually from 300 to 500 feet. It may be necessary, there- 
 fore, to ascertain what is this extra distance known as overhaul, 
 and on how much material the contractor is entitled to extra 
 pay. 
 
 FIG. 75. 
 
 In the figure let the straight line SPOR represent the grade 
 line, and the curved line NOQR the profile of a proposed rail- 
 road, where OQR is to be cut away and OTS filled up. 
 
 With a little computation of quantities, based upon the 
 cross-section notes, and a few measurements and trials on the 
 profile, the points P and M can be located, so that the cubic 
 yards in MOQ and PON will be approximately equal --near 
 enough for practical purposes while the limit of free haul 
 will be indicated by the distance MP. 
 
 Now, the contractor is entitled to additional compensation 
 for the transport of all material from MQR to NPST, that is 
 to say, if the center of gravity of MQR be at g and that of 
 NPST at (f, then the extra distance which this material is 
 hauled = gtf PM, cr eg -f c'y'. With a few trials, and a 
 little figuring as before, a tolerably close approximation as to 
 quantity and distance can be made. If greater accuracy is 
 required, the center of gravity of each mass should be deter- 
 
124 
 
 CONSTRUCTION. 
 
 mined by the principles of Mechanics. This distance, usually 
 reckoned in stations of 100 feet as the unit, multiplied by the 
 quantity of material transported and by the price agreed upon 
 for the overhaul, will give the amount due the contractor. 
 
 EXAMPLE. If the limit of free haul be 400 feet, the dis- 
 tance between the centers of gravity of cut and fill be 1000 
 feet, the price one and a quarter cents per cubic yard per sta- 
 tion of 100 feet, and the quantity of material, exceeding that 
 of free haul, transported from cut to fill = 8000 cubic yards ; 
 then the extra pay 
 
 P = (10 - 4) 80CO X = $600. 
 
 D. CULVERTS, BRIDGES, AND TUNNELS. 
 
 121. Culverts are used for the passage of water from one 
 side of the road to the other under the track. If practicable, 
 they should be constructed perpendicularly to the center line 
 of the roadway. They are set out by driving stakes on the 
 center line, and at the corners or angles indicating the limits 
 of the foundations ; and on each stake should be marked the 
 depth required to be dug. In the note-book there should be 
 made a sketch of the culvert, accompanied by a record of its 
 dimensions, the amount of cutting at each stake, location of 
 reference points, etc. 
 
 FIG. 76. 
 
 The length of a box culvert cd placed at right angles to the 
 roadway may be found as follows : 
 
 Let w = the width of the roadway, 
 a = the altitude of embankment, 
 
 " : 1, the ratio of side slopes, 
 n 
 
 and h = height of culvert ; 
 
CULVERTS AND BRIDGES. 125 
 
 then its length l = w + '2-(a h). (1 1(5) 
 
 n 
 
 If TO : n = 3 : 2, 
 
 then l = w + 3(a h). (117) 
 
 For a 20-foot fill, 16-foot roadway, and a culvert 6 feet high, 
 slopes | : 1, 
 
 I = 10 + 3 (20 ) = 58 feet. 
 
 Searles * says that in box culverts the span varies from 2 to 
 5 feet, the height in the clear from 2 to 6 feet, the thickness of 
 walls from 3 to 4 feet, the thickness of cover from 12 to 18 
 inches, and its length at least 2 feet greater than the span. 
 Furthermore, when the span required is more than 5 feet, and 
 the embankment too high to warrant carrying the walls up to 
 grade as an open culvert, an arch culvert should be used. The 
 span varies from 6 to 20 feet ; the arch is a semicircle, the 
 thickness varying from 10 or 12 inches to 18 or 20 inches. 
 The height of abutments to the springing lines varies from 2 
 to 10 feet, the thickness at the springing line from 3 to 5 feet, 
 and at the base from 3 to 6 feet, the back of the abutment 
 receiving the batter. The wing-walls stand at an angle to 
 30 with the axis of the culvert. 
 
 122. To set out bridge abutments when the bridge is on 
 a tangent, proceed in a manner similar to that indicated for 
 culverts, working from the axis of the roadway to locate the 
 center line of the foundation for the main part of the abut- 
 ment and its outside limits, and then the direction and extent 
 of the pit for the wing-walls. All governing points to be 
 referenced, care being taken that these stakes be placed where 
 they will not likely be disturbed during construction. 
 
 When the bridge is on a curve, and especially if the span is 
 considerable, the center of the abutment or pier should not be 
 at the intersection of the axis of the bridge with either 
 of these axis, but on a line LN, Fig. 77, called the bridge.- 
 chord. 
 
 * Field Engineering, page 209. 
 
126 
 
 CONSTRUCTION. 
 
 FIG. 77. 
 
 The bridge-chord is a 
 line midway between 
 the tangent T at the 
 mid-span, and the chord 
 M whose arc is limited 
 by its intersection with 
 the axis of the masonry 
 represented by ah and 
 cd. The line LN is 
 then used as the basis 
 of measurement, whence 
 the limits of the work 
 b are determined. 
 
 If L is inaccessible it 
 may be located as fol- 
 lows: From some point 
 P in the center line set 
 off PQ perpendicular 
 
 to the bridge-chord ; make 
 
 PQ^RTLb 
 
 = R (vers POT vers iOT). 
 
 and then will 
 
 L = R sin POT 
 
 LN 
 
 (118) 
 
 The point L may then be found by measuring direct with a 
 tape, or if this is impracticable, it may be indicated by the 
 intersection of transit lines. 
 
 123. Trestles may be staked out by locating the position of 
 the center of each bent, and then measuring -the proper 
 distance right and left, to fix the limit of the foundation for 
 the sill. If pile bents are used stakes should be driven, if 
 practicable, to indicate the position of the piles. If a swamp 
 or body of water is to be trestled, and the line is straight, 
 stakes or poles a few feet high may be set in the line on firm 
 ground on each side of the water, and the piles ranged in with 
 sufficient accuracy by them. The location of each additional 
 bent being ascertained by measurement from the one immedi- 
 ately preceding. 
 
CULVERTS, BRIDGES AND TUNNELS. 127 
 
 If the line curves, two transits may be employed to indicate 
 by the intersection of their lines of sight the place for each 
 bent. 
 
 The bents may be placed 12 or 15 feet apart, and for single 
 track are usually composed of a sill, cap, two vertical posts, two 
 batter posts, and two braces, running diagonally from sill to 
 cap; all of 12 X 12 timber except the batter posts which may 
 be 10 X 12, and the braces 3 or 4 inches by 10 or 12. For 
 double track roadway some modification of the preceding 
 is necessary, more bracing is introduced, and for greater 
 height than 28 or 30 feet some form of built-up post is 
 employed, the posts being fastened together, thus forming 
 a bent throughout the series, or the bents may be braced 
 in pairs, making a pier, and the space between the piers 
 spanned by a truss. 
 
 High wooden trestles are not as common in this country as 
 formerly, having been largely superseded by iron structures. 
 
 AC D B 
 
 FiG. 78. 
 
 124. Tunnels. Great care should be exercised in setting 
 out tunnels. A first-rate instrument in good adjustment 
 should be employed for observing direction of the line, and 
 the best steel tape used for measuring the distance. A spring 
 balance should be used with the tape so that a constant ten- 
 sion may be had, the measurements being made between plugs 
 driven specially for this purpose, and the readings should be 
 corrected for temperature. 
 
 If possible, some point should be selected in the line on the 
 summit of the mountain as at M, whence an unobstructed 
 view each way down the mountain may be obtained. Here a 
 
128 CONSTRUCTION. 
 
 monument should be erected, whence by numerous and careful 
 observations made at different times, and using the mean re- 
 sult, stations A and B in the vicinity of the ends of the tunnel 
 should be established precisely in a line which includes a point 
 on the monument at M, from which the direction of the head- 
 ings at D and C may be given. 
 
 It was discovered at the Musconetcong Tunnel that the best 
 time during daylight to make an observation was just after 
 sunrise. In summer it was quite impossible to do accurate 
 work during the middle of the day. A round iron pole of one- 
 half inch diameter, painted white and red alternately, answered 
 well for a sight pole. The plummet lamp, however, sighted 
 on a calm clear night gave the best results. 
 
 It is sometimes necessary to establish two or more stations 
 on the mountain. In the alignment of the Hoosac Tunnel 
 there were four permanent stations used. In such cases the 
 difficulty of obtaining accurately the direction to drive the 
 headings is considerably increased. In the Mont Cenis Tun- 
 nel an extended system of triangulation was resorted to in 
 order to secure precision in the location of the axis of the 
 tunnel. 
 
 In locating the Musconetcong Tunnel the first method above 
 exhibited was employed with remarkable results ; the difference 
 in the alignment of the east and west headings was only four 
 hundredths of a foot. 
 
 If in addition to driving a tunnel from its ends, work is to 
 be conducted from the foot of a shaft, carefully constructed 
 apparatus must be provided, and extraordinary care observed 
 in its manipulation to transfer accurately the direction of the 
 line from the surface to the foot of the shaft. Various devices 
 have been employed by engineers in solving this difficult prob- 
 lem. The principle thing is to suspend from two points pre- 
 cisely in the line, but on opposite sides of the shaft, at the 
 surface, two plumb lines reaching to the place of the tunnel 
 and obtaining thence the proper direction of the underground 
 working. 
 
 In the Hoosac Tunnel the line was transferred 1000 feet 
 down a shaft with such precision that when the heading, driven 
 
TUNNELS. 129 
 
 over two miles from the west end, met the one produced over 
 2050 feet from the plumb lines, the error in the alignment was 
 found to be only nine-sixteenths of an inch, and the heading 
 driven from the shaft and plumb lines 1560 feet in the oppo- 
 site direction met the corresponding one from the east with an 
 error of only Jive-sixteenths of an inch. 
 
 125. In running the levels over the surface corresponding 
 care must be ^exercised. The instrument should be kept in 
 good adjustment, all the observations made on benches, and 
 the readings taken to thousandths of a foot. The levels should 
 be repeatedly tested so as to reduce the error to a minimum. 
 
 The difference of levels between points at the top and bottom 
 of a shaft may be obtained by measuring with a rod the 
 distance between lines made on a number of bolts, driven from 
 10 to 12 feet apart, in a vertical line down the shaft. 
 
 The grade of a tunnel should be at least sufficient for 
 drainage, or about 0.15 of a foot per station, the least width in 
 the clear for double track should be 28 feet, and the least 
 height in the clear above the outside rail 16 feet. The center 
 of the tunnel will of course be somewhat higher, depending 
 upon the form of its cross-section. 
 
 Tunneling, like bridge building, has become an engineering 
 specialty, and one who desires more knowledge on the subject 
 should consult Drinker and Siinms. 
 
 The information contained in the following extract from 
 Mr. II. 8. Drinker's paper on the Musconetcong Tunnel is as 
 important to the engineer studying tunneling to-day as when 
 it was written ; it is thought to be proper, therefore, to give it 
 place here.* 
 
 The approach to the tunnel on the west begins on a 5 curve, 
 the P.T. of which is about 800 feet from the entrance, and the 
 tunnel itself located on a tangent throughout its length, the 
 said tangent terminating in a curve, having its P.C. some 
 1850 feet beyond the east portal. The grade ran to a summit 
 
 **Read before the American Institute of Mining Engineers at New 
 Haven, Coun., Feb. 25, 1875. 
 
130 CONSTRUCTION. 
 
 in the middle of the tunnel, the same being the summit for the 
 road. It was reached by a rise of two-tenths (0.2) to the 
 hundred feet on the west side, or 10.56 feet to the mile, falling 
 on the east at fifteen-hundredths (0.15) to the hundred feet, or 
 7.92 feet to the mile. 
 
 To determine the line after its preliminary location, an 
 observatory was erected on the summit of the mountain, about 
 12 feet high, with an eight-foot square base, battering on the 
 four sides about 1^ inches to the foot. T\vo solid stone founda- 
 tions were also built on line, one on a hill about half a mile 
 from the west entrance, the other on the grading, at the east 
 end, and about half a mile from the eastern portal. As the 
 observatory was located about midway over the tunnel, this 
 gave, approximately, equidistant sights of about, say, a mile 
 and a quarter each, at the farthest. This, however, was done 
 after the tunnel had been started from points established on 
 both sides by repeated and carefully checked runnings. The 
 tower being subsequently built, two points were established, 
 one each on the foundations, on either side, from the lines by 
 which the work at either end had been so far run, and then 
 assuming these end points as correct, by a series of repeated 
 and careful trials, the center point on the tower, or permanent 
 back-sight for both ends was determined by setting up, approx- 
 imately, over it, and then reversing and sighting repeatedly, 
 moving the instrument to and fro sideways, within a variable 
 distance of about fifteen-hundredths (0.15) of a foot, in which 
 the sights all came, and finally taking their mean. This was 
 at first done, as soon as the observatory could be located and 
 built, with sufficient accuracy to test the preliminary lines. 
 Subsequently this center point was tested, and retested, and 
 determined with extreme accuracy, by the mean of very many 
 trials made both by sighting by day and by night, and in 
 winter and summer. Different objects were used for sighting 
 on in day work. Both the ordinary red and white round pole, 
 also a flat 2x1 inch black pole, with a white center streak. 
 This latter, from its shape, was found difficult to keep plumb, 
 either when held or fastened. Also a pole of one-half inch 
 round iron, painted white, was tried, and found to answer well, 
 
TUNNELS. 
 
 better than either of the others. But far better and more 
 accurate than any daylight back-sight, whether permanent or 
 movable, was found the simple expedient of using plummet- 
 lamps on clear calm nights. They worked admirably outside, 
 a flame f inch high, by 5 inch in diameter, being distinctly 
 seen in the long sights ; and with a fine hair, the sights were 
 found, finally, to repeatedly test within practically such exact 
 limits (two or three hundredths), that, the point being once 
 fixed, it was not subsequently found advisable to move it. 
 Now, these three reference points being located, at the west 
 end a center was set at the mouth of the slope, and from it 
 another at the bottom. This gave a back-sight of 276 feet to 
 run from into the heading. At each shaft a center was first 
 set, with great care, about twenty-five feet off, and from this 
 the line prolonged to two staples driven into the timbers on 
 each side. On the mean of many sights being determined, the 
 points on both staples were notched, the notches tested, and 
 fine plummet lines dropped from them, the weights being 
 steadied at the bottom, in water. Then the line was continued 
 from these, as in ordinary mine surveying, in running from a 
 shaft, the instrument being first approximately set up in line, 
 and then moved sideways, until the hair exactly bisected the 
 mean of the slight oscillations observable in the lines. Though 
 the distance to be run from the shafts was not great, this care 
 was necessary from the shortness of the back-sight, the distance 
 between staples being only some 7 feet, and from the fact 
 that the headings were through earth, it being very necessary 
 i;i enlarging through earth to be able to have the crown bars 
 closely located at equidistant spaces from center. On the 
 headings between the shafts and slope meeting, the various 
 runnings all tested closely ; but it was the long line between 
 the main east and west headings that required, of course, the 
 most care, and caused the most anxiety. This line, at the east 
 end, was simply continued on the grading, up into the heading, 
 at first with one, and, subsequently, as the headings advanced, 
 with two intermediate centers. At the west end, the line was 
 at first run into the main heading (Xo. 1) down the slope, but 
 as the enlargement in soft ground proceeded between the slope 
 
132 CONSTRUCTION. 
 
 and west end, in time a clear sight was obtained from the 
 mouth of the tunnel to the slope, and thence into the heading, 
 making two intermediate centers, as at the east end. It was 
 always necessary to have a station where the slope came down, 
 since the latter was driven, after meeting rock, sixteen feet 
 wide thirteen on the left and three on the right of center 
 line, leaving at its foot about ten feet of space for passage on 
 the right, as the line ran, and, of course, cutting off center line. 
 The three feet on the right, however, were dressed off, sub- 
 sequently, at the level of the heading, so as to give a clear 
 back-sight to the mouth. 
 
 These east and west lines were repeatedly run and tested 
 as the headings advanced, and, besides the work continually 
 spent on them by the division and resident engineers, they 
 were frequently checked by the principal assistant engineer. 
 They finally tested within four hundredths (0.04) of a foot, or 
 less than one half an inch. The levels were carried over the 
 mountain by a series of test benches run until succeeding 
 benches tested within 0.005 of a foot. On meeting, the face 
 benches on either side were found to test within 0.015 of a 
 foot, or less than one-fifth of an inch. Owing to the system 
 of center cuts, used in blowing the rock, in which ten feet at 
 a time were brought out, it was especially necessary that the 
 chaining should be accurate, so that the distance apart of 
 the headings might be safely determined. To measure over the 
 mountain, two stout frames were made, steadied by weights 
 on the legs. They each simply consisted of a vertical shaft 
 with three legs, one movable. From a board nailed on the 
 top of the shaft a fine plummet was hung. The two were put 
 in line, the plummets centered by the transit, and a point at 
 the top of one line leveled with a point near the bottom of 
 the other, and the measurement thereon taken between the 
 two with steel tapes. The hind frame was then moved on, 
 and the chaining so carried up or down hill in successive steps. 
 This method was found to be satisfactory ; for, on the head- 
 ings coming together, the distance apart, predicted and marked, 
 was found to agree with the measured distance within fifty-two 
 hundredths of a foot (0.52), or about six inches out in a total 
 
TUNNELS. 133 
 
 chaining of about eight thousand feet, four thousand through 
 headings, and four thousand over the mountain, the test 
 measurement being brought down the slope on angle instead 
 of in at the west entrance. 
 
 126. Ballast Stakes are set every 50 feet at the proper dis- 
 tance transversely, to indicate the width of the base of the 
 ballast and by their tops the upper surface. The depth of the 
 ballast is about 18 or 20 inches, or it is 12 or 14 inches below 
 the lower surface of the tie. The center line of the track must 
 again be retraced, stakes driven and centered ; on curves every 
 50 or 25 feet, depending upon the curvature ; on tangents 
 every 200 feet. 
 
 
CHAPTER VII. 
 
 FROGS AND SWITCHES. 
 DEFINITIONS. 
 
 127. The gauge of a track is the distance between the rails 
 of the track. It is measured from the inside of the rails as 
 from A to B, Fig. 79. 
 
 a. The gauge line is the line from which the gauge is 
 measured. It is used instead of the rail in these calculations. 
 
 b. The distance between tracks is the perpendicular dis- 
 tance between the gauge lines of the tracks, as BA'> 
 
 c. A turnout is used to connect one track with another, as 
 AB, Fig. 80. 
 
 FIG. 80. 
 
 d. A crossover is used to pass from one to another of two 
 parallel tracks, as AB, Fig. 81. 
 
 FIG. 81. 
 
DEFINITIONS. 
 
 e. The point of switch is the point at \vhich a turnout or 
 crossover begins, as the point A, Fig. 80. 
 
 f. The point of frog is the point at which the gauge lines 
 of two rails intersect, as at C, Fig. 80. 
 
 FIG. 82. 
 
 g. The frog angle is the angle formed by th3 gauge lines 
 at the point of frog, as F at 5, Fig. 82. 
 
 li. The number of a frog is found by constructing an 
 isosceles triangle 
 upon the lines en- 
 closing the frog 
 
 ir 
 
 angle and dividing 
 its altitude by its A 
 base. 
 
 If A, in Fig. 83, 
 
 be the frog angle 
 
 , , D FIG. 83. 
 
 make AB = AC, 
 
 and draw AD perpendicular to BC, then 
 
 the number of frog = n 
 If AD = 8 and BC == 1, then n = 8. 
 
 i. A crossing frog is formed by the intersection of two 
 rails which are on the same sides of their respective tracks, 
 as at A and D, Fig. 90. The frogs at A, D, B and E, taken 
 collectively, are sometimes called a set of crossing frogs. 
 
 . 
 
 BC 
 
136 
 
 FROGS AND SWITCHES. 
 
 k. The lead L is the distance from the point of switch to the 
 point of frog, measured on the chord of that rail of the turn- 
 out which passes through the frog, as AB, Fig. 82. 
 
 L The radius of a turnout is the radius of the gauge 
 line of the rail which passes through the frog as OA, OB,* 
 Fig. 82. 
 
 m. The radius of the main track on a curve is the radius 
 of the gauge line of the rail which passes through the frog.* 
 
 n. A crossing slip is an arrangement of two sets of switch 
 rails in connection with a set of crossing frogs by which two 
 tracks, which cross each other, are connected, as AB, ML, in 
 Fig. 94. 
 
 PROBLEMS. 
 
 128. Given the angle of the frog F, and the gauge g, of 
 a turnout from a straight track, Fig. 84, to find the lead 
 L and the radius R of the turnout. 
 
 In the right triangle ABE. The angle A = F, since the 
 angle C at the center = F, and the angle BAE at the circum- 
 ference subtends the same chord AB, 
 
 * This definition we prefer to that usually given, as it enables us to 
 simplify the formulas. 
 
TURNOUT FROM A STRAIGHT TRACK. 137 
 
 hence A B = -5fu = BE cosec ^1 , 
 
 sm^l 
 
 or L = g = gcosec$F. (119) 
 
 The isosceles triangle A CB gives 
 
 and 72 = - = - cosec | F. (120) 
 
 2 
 
 EXAMPLE. Given the frog angle = 7 10', and the gauge 
 4.75 feet ; required the lead and the radius of the turnout. 
 
 Ans. Z = 76/; 72 = 608.' 
 
 129. Given the radius R, and the gauge g, of a turnout 
 from a straight track, to find the lead L and the frog 
 angle F, Pig. 84. 
 
 The right triangle *HCB gives 
 
 or cosF = - ; (121) 
 
 R 
 
 and the isosceles triangle A CB gives 
 
 ' 
 
 (122) 
 
 EXAMPLES. 
 
 1. Given the radius 771.85, and the gauge 4.75 ; required 
 the lead and the frog angle. 
 
 2. Given the radius 1151.92, and the gauge 4.75 ; required 
 the lead and the frog angle. 
 
138 
 
 FROGS AND SWITCHES. 
 
 TABLE FOR TURNOUTS FROM A STRAIGHT TRACK.* 
 
 No. of 
 Frog. 
 
 Angle 
 of Frog. 
 
 Lead 
 of 
 Turnout. 
 
 Radius 
 of 
 Turnout.t 
 
 Degree of 
 Curve of 
 Turnouts.! 
 
 4 
 
 14 15' 0" 
 
 38.15 
 
 154.45 
 
 ,370 47 / 
 
 5 
 
 11 25' 16" 
 
 47.74 
 
 239.9 
 
 24 04' 
 
 6 
 
 9 31' 38" 
 
 57.20 
 
 344.4 
 
 16 42' 
 
 7 
 
 8 10' 16" 
 
 66.67 
 
 467.9 
 
 12 17' 
 
 8 
 
 7 9' 10" 
 
 76.15 
 
 610.4 
 
 9 24' 
 
 9 
 
 6 21' 35" 
 
 85.67 
 
 771.85 
 
 7 26' 
 
 10 
 
 5 43' 29" 
 
 95.12 
 
 952.4 
 
 6 01' 
 
 11 
 
 5 12' 18" 
 
 104.61 
 
 1151.9 
 
 4 59' 
 
 12 
 
 4 46' 19" 
 
 114.10 
 
 1370.4 
 
 4 11' 
 
 13 
 
 4 24' 19" 
 
 123.58 
 
 1607.8 
 
 3 34' 
 
 14 
 
 4 5' 28" 
 
 133.07 
 
 1864.4 
 
 3 04' 
 
 15 
 
 3 49' 06" 
 
 142.58 
 
 2139.9 
 
 2 41' 
 
 * g = 4.75. 
 
 t These refer to the rail running through the frog. For approximate 
 degree of curve of turnout from a curved track, use degree of curve of 
 turnout = degree of curve in table degree of curve of main track. 
 
 130. Given, in Pig. 85, the radius CB of the^ main 
 track = R, and the frog angle F, to find the lead L and 
 the radius R' of the turnout from the outside of the main 
 track. 
 
 In the triangle CAB, A B = F. For A + C'AB = 180, 
 and the isosceles triangle C'AB gives C'A B = C'BA and 
 C'BA + B + CBE = 1 80. Therefore, since 
 
 CBE = F, A + C'AB = C'BA + B + F, or A - B == F. 
 Again 
 
 CB-CA : CB + CA = tanl(A-B):tzn(A + B), 
 
 or, substituting values, there results, 
 
 (2R- 
 
 9 
 
 (123) 
 
TURNOUT PROM A CURVED TRACK. 
 
 139 
 
 The half sum of A and B being thus found, and (A B) 
 
 p 
 being equal to , A and B are readily determined. 
 
 
 
138 
 
 FROGS AND SWITCHES. 
 
 TABLE FOR TURNOUTS FROM A STRAIGHT TRACK.* 
 
 No. of 
 Frog. 
 
 Angle 
 of Frog. 
 
 Lead 
 of 
 Turnout. 
 
 Radius 
 of 
 Turnout.! 
 
 Degree of 
 Curve of 
 Turnouts. t 
 
 4 
 
 14 15' 0" 
 
 38.15 
 
 154.45 
 
 37 47' 
 
 5 
 
 11 25' 16" 
 
 47.74 
 
 239.9 
 
 24 04' 
 
 6 
 
 9 31' 38" 
 
 57.20 
 
 344.4 
 
 16 42' 
 
 7 
 
 8 10' 16" 
 
 66.67 
 
 467.9 
 
 12 17' 
 
 8 
 
 7 9' 10" 
 
 76.15 
 
 610.4 
 
 9 24' 
 
 9 
 
 6 21' 35" 
 
 85.67 
 
 771.85 
 
 7 26' 
 
 10 
 
 5 43' 29" 
 
 95.12 
 
 952.4 
 
 6 01' 
 
 11 
 
 5 12' 18" 
 
 104.61 
 
 1151.9 
 
 4 59' 
 
 12 
 
 4 46' 19" 
 
 114.10 
 
 1370.4 
 
 4 11' 
 
 13 
 
 4 24' 19" 
 
 123.58 
 
 1607.8 
 
 3 34' 
 
 14 
 
 4 5' 28" 
 
 133.07 
 
 1864.4 
 
 3 04' 
 
 15 
 
 3 49' 06" 
 
 142.58 
 
 2139.9 
 
 2 41' 
 
 TABLE FOR TURNOUTS FROM A STRAIGHT TRACK. 
 
 g = 4' 8** 
 
 No. of 
 Frog. 
 
 Angle 
 of Frog. 
 
 Lead 
 of 
 
 Turnout. 
 
 Radius 
 of 
 Turnout. 
 
 Degree of 
 Curve of 
 Turnout. 
 
 4 
 
 14 15' 0" 
 
 37.96 
 
 153.0 
 
 38 09' 
 
 5 
 
 11 25' 16" 
 
 47.32 
 
 237.8 
 
 24 17' 
 
 6 
 
 9 31' 38" 
 
 56.70 
 
 341.4 
 
 1(5 51' 
 
 7 
 
 8 10' 16" 
 
 66.08 
 
 463.8 
 
 12 23' 
 
 8 
 
 7 9' 10" 
 
 75.48 
 
 605.0 
 
 9 29' 
 
 9 
 
 6 21' 35" 
 
 84.88 
 
 765.1 
 
 7 30' 
 
 10 
 
 5 43' 29" 
 
 94.27 
 
 943.7 
 
 6 04' 
 
 11 
 
 5 12' 18" 
 
 103.69 
 
 1141.8 
 
 5 01' 
 
 12 
 
 4 46' 19" 
 
 113.10 
 
 1358.4 
 
 4 13' 
 
 13 
 
 4 24' 19" 
 
 122.51 
 
 1593.9 
 
 3 36' 
 
 14 
 
 4 5' 28" 
 
 131.91 
 
 1847.8 
 
 3 06' 
 
 15 
 
 3 49' 06" 
 
 141.33 
 
 2121.1 
 
 2 42' 
 
TURNOUT FROM A CURVED TRACK. 
 
 139 
 
 The half sum of A and B being thus found, and (A 
 
 p 
 being equal to , A and B are readily determined. 
 
 Then C = 180 (A + B), 
 
 and the exterior angle 
 
 CBE=C + C" ^F, 
 or C' = FC. 
 
 In the triangle 
 
 (124) 
 
 smA 
 
 sin A 
 
 (125) 
 
140 
 
 FROGS AND SWITCHES. 
 
 In the triangle ABC', 
 C'B = 
 
 or R' 
 
 AB 
 
 2 sin | C" 
 L 
 
 (126) 
 
 EXAMPLE. In a turnout from the outside of a 6 curve 
 with a number 10 frog, find the lead, and radius of the 
 turnout. 
 
 131. Given, in Fig. 85, the radius CB of the main track 
 = /?, and the radius C'B of the turnout = R' t to find the 
 lead AB = L and the frog angle F. 
 
TURNOUT FROM A CURVED TRACK. 141 
 
 Draw ED and C'G perpendicular to CH and AB respec- 
 tively. CAC/ is a straight line, for the curves AM and AL 
 are tangent to each other at A. In the triangle CBC', CC' 
 = R + R' g, hence all the sides are known, and we have 
 the proportion 
 
 CC' : BC + EC' = EC EC' : CD - C'D, 
 or R + R'-g: R + R' = R-R': CD C'D. 
 
 R + R'-g 
 
 The difference between CD and C'D being thus found, and 
 having their sum = R + R' g, CD and C'D are readily 
 determined. 
 
 In the triangle BCD, 
 
 and in the triangle C'BD, 
 
 m- c -w-' ** 
 
 now the angle CBE, which is = F = the sum of the angles at 
 C and C", 
 
 or, F= C + C'. (129) 
 
 The isosceles triangle AC'E gives 
 
 = 2E'siniC". (130) 
 
 EXAMPLE. The radius of a turnout from the outside of 
 a 4 curve = 1060.22. Find the lead and frog angle.* 
 
 Ans. Z = 76.27 feet; F=79 / . 
 
 132. Given the frog angle F, the radius CH=CB = R 
 of the main track, Fig. 86, to find the lead AB = L, and 
 the radius C'A = C'E = R' of the turnout from the inside 
 of a curved track. 
 
 Let AM represent the outside rail of the main track. Then 
 C, C', and A are in the same straight line, since the arcs A M 
 and AL are tangent at A. 
 
 * When g is not given use 4.75. 
 
142 
 
 FROGS AND SWITCHES. 
 
 In the triangle A CB, B A = F. For F being the angle 
 between the tangents at B, and C'E and CB being radii, it 
 follows, therefore, that 
 
 Then 
 
 C'BC = F=B- C'BA = B-A. 
 F=B- A. 
 
 FIG. 86. 
 
 Now AC BC : AC + BC = 
 hence, tan*<4 + B} = 
 
 or, substituting, there results 
 
 - B) : t&nt(A + J5); 
 - B 
 
 AC BC 
 
 tanJ-F 
 9 
 
 With this half sum of A and B, and the half difference, 
 A and B may be found. 
 
 Again, C = 180 - (A + J5), 
 
 BC . sin C 
 
 and 
 
 AB = 
 L = 
 
 sin A 
 E . sin C 
 
 (131) 
 
TURNOUT FROM A CURVED TRACK. 143 
 
 The exterior angle at C" = C + F, and the isosceles triangle 
 A C'B, gives 
 
 AC' = AB , 
 2 sin 4- C' 
 
 or, ', *' = _J, = co S eciC'. (132) 
 
 133. In Fig. 86, given the radii CB = R, C'B = R', to 
 find the frog angle and the lead. 
 
 In the triangle CC'B, CC' = CA C'A = R + g R'. 
 The three sides of the triangle are therefore known, and 
 drawing the perpendicular C'D, we have from a well-known 
 proposition, 
 
 BD-CD = (BC'+CC')(BC'-CC>) 
 
 BC 
 or, substituting, there results 
 
 BD CD = (2R' R g). 
 
 R 
 
 With this difference of BD and CD, and CB = R as their 
 sum, find BD and CD. Then, since the angle C'BC = F, 
 
 ^, (133) 
 
 -K 
 
 and cos C'CB = CD CD 
 
 C'C R + g R' 
 In the isosceles triangle A C'B, the angle 
 
 AC'S = C'BC + BCC'-, 
 
 hence A B = 2 C'B sin 1 A C'B, 
 
 or, L = 2 R' sin 1 A C'B. (134) 
 
 EXAMPLES. 
 
 1. Given the frog angle = 5 43' 29", the radius of the main 
 track = 1436. 69, to find the lead and the radius of the turnout. 
 
 2. Given the radius of the main track = 955.37, and that of 
 the turnout = 477.8, to find the frog angle and the lead. 
 
 Ans. F= 5 43' 29"; L = 94.83. 
 
144 
 
 FROGS AND SWITCHES. 
 
 134. Given the angle of the frogs F=F / , the gauge g, 
 and the distance between the tracks b, Fig. 87, of a cross- 
 over on straight tracks, to find the distance F'K. 
 
 H 
 
 F'K 
 
 FIG. 87. 
 In the triangle HFG, the angle HFG = F, and 
 
 cosF 
 In the triangle HF / K^ the angle at F f = F, and 
 
 therefore F'K = (b g sec F) cot F. (135) 
 
 135. Given the frog angles F and F', the gauge g, and 
 the distance between tracks b, Fig. 88, of a crossover on 
 straight tracks, to find the distance CE and the radius 
 
 Make DE and AB perpendicular to BE, and A G parallel to 
 it. Make AC = g, and draw A K parallel to the tangent of 
 
 the frog at C. The 
 L D/^ angle LA G = F'. 
 
 In the triangle 
 KLD, the angle LDK 
 = F, and the exterior 
 - angle KLM = F', 
 hence the angle LKD 
 
 = A OD = F' F. 
 
 In the triangle A CB, 
 AC = g, A=F', AB 
 = AC cos A =g cos F', 
 and BC g sin F . 
 In the triangle DGA, DG = b GE = b g cos F', and 
 DA G = LA G [LA D = (F' F)] = (F' + F), 
 
CROSSOVER ON CURVED TRACKS. 
 
 145 
 
 AG = DG cotDAG = (b - g cosF') cot | (F' + F), 
 and CE = BE-BC = AG BC, 
 or CE = (b-gcosF')cot\(F' + F)-gs'mF'. (136) 
 
 and the radius 
 
 DP 
 
 sinDOP sinKF'-F) 
 
 substituting the value of AD found above, we obtain 
 R== _ b-gcosF' _ . 
 
 ' 
 
 . 
 
 or 
 
 R = b 9 cosF ' cosec 
 
 cosec 
 
 F). (137a) 
 
 EXAMPLES. 
 
 F' == 7 9', the gauge 
 
 4.75, and 
 4.75, and 
 
 1. Given, in Fig. 87, F 
 6 = 7.417, to find F'K. 
 
 2. Given, in Fig. 88, F=-.79 / , F' = 9 32', 
 I) = 7.42, to find CE and 72. 
 
 136. Given the radius FO = R of one rail, and distance 
 b between two concentric tracks, and the angles F and F' 
 of two frogs in a crossover between them, Fig. 89, to find 
 the distance FD measured on a chord of the rail BFD 
 and CO' = radius of the outer rail of the crossover. 
 
 Let MC and BD represent the 
 gauge lines of the rails which 
 pass through the frogs and A C, 
 one rail of the crossover. Draw 
 the radii AO, CO, AO', and CO'; 
 also FO, A C, and FD. 
 
 In the triangle A OF the ex- 
 terior angle OFO' = F, A F=y, 
 and FO = 7i, so that we have 
 two sides and the included angle 
 given, whence 
 
 R + g : R g = tan * F : tan -J (OAF A OF), 
 
146 FHOGS AND SWITCHES. 
 
 or 
 
 g 
 
 With this half difference, and % F as the half sum, the 
 angles A and O are readily found. 
 Then, by the law of sines, 
 
 sinFAO : smAFO = FO : OA, 
 
 or A = RsmF = R sin p cosec FA 0. (138) 
 
 sinFAO 
 
 In the triangle AOC we now know OA, OC = R -j- b, and 
 the difference of the angles A and C= (9J.jF-f- .F', for the 
 angle OA C = OA F + O'A C, and CA = O'A C0 CO'. Now 
 ' = F r , but (yCA OCA=F'\ hence 
 
 Then, having the two sides AO and CO, and the difference 
 of the angles opposite them, we obtain, by the law of tangents, 
 
 (70+ AO : CO-AO = tanl(OAC + OCA) : tan$(OAC OCA); 
 
 .-.tan (04 C+ OCA) = CO + AO tanl(OAC- OCA), 
 CO AO 
 
 or tani(OyK?+ OCA)= R + b + A0 tznl(F' + OAF). 
 
 R ~\~ b A 
 
 With this half sum and as the half difference, we 
 
 find OA C = their sum and OCA their difference. Then, by 
 the law of sines, 
 
 sin OAC 
 
 In the triangle AO'C, / AC=0 / CA = OCA + O'CO and 
 .40'C^ISO 2 O'CA, there are then known all the angles 
 and the side AC, so that 
 
 / (140) 
 
 sin^O'C 
 In the isosceles triangle FOB, FOD = BOD EOF, and 
 
 or FD = 2RsmiFOD. (141) 
 
CROSSING FROGS. 
 
 147 
 
 It is evident that F F' must be a small angle, since, if it 
 were not, R' would be too small for practical purposes. 
 
 EXAMPLE. Given F= 7 9' 10", F' = 5 43' 29", the gauge 
 4.75, and b = 7.417, to find R' and FD. 
 
 137. Given the angle of the crossing frogs = F, and the 
 gauges ff and g' of two straight tracks, Fig. 90. to find 
 the distances EA = DB and AB 
 
 FIG. 90. 
 
 Draw HA and EC perpendicular respectively to ED and 
 AC. On account of the parallelism of the lines ED and AB, 
 and of DB and A C, the angles at E, D, A , and C are equal 
 to F. In the right triangle ABC, BC = g, and BAC=F, 
 
 hence, 
 
 AB= = q cosec F. 
 sinF 
 
 Similarly, the right triangle EAH gives 
 
 EA = -3 = 0' cosec F. 
 sin ,P 
 
 (142) 
 
 (143) 
 
 138. Given the radius EC, of a rail of a curved track, 
 Pig. 91, and one angle F, at E, made by a straight track 
 crossing it. Required the angles F', F", F'" situated at 
 D, A, and B respectively, and the distances EA DB ED 
 and AB 
 
148 
 
 FROGS AND SWITCHES. 
 
 In the triangle EDO the angle E = 90 + F, the angle at 
 = 9Q F'EC = Ra,ndDC = R . Then 
 
 . n _ 
 
 sinJ0 X 
 
 DC 
 
 sin (90 + F)R 
 R + g 
 
 or 
 and 
 
 R 
 
 cosF, 
 
 F' = 90 JD. 
 The angle ECD = F' F, and 
 
 D sin ECD X EC R sin (ff x F) 
 
 sinEDC 
 
 (144) 
 
 (145) 
 
 FIG. 91. 
 The right triangle EHG gives 
 
 ~~ cosGEII cosF' 
 
 In the triangle CGA, CA = R, CG = R EG and angle 
 CGA = 90 + F. Then 
 
 sin CGA X CG 
 
 J2 
 
 Now 
 
 (146) 
 
CROSSING FROGS. 149 
 
 The angle EC A = F" F. 
 The isosceles triangle A CE, gives 
 
 EA = EC X 2sin$ACE, 
 or EA = 2 R sin i (F" F). (147) 
 
 The triangle A CB gives 
 
 sin ABC = * CABX AC = _R_ cosj ^ 
 CB R + g 
 
 and F'" = 90-^4 BC. (148) 
 
 The angle A CB = F" f F", 
 
 A T, sinACB X BC 
 
 AB = -, (149) 
 
 cosjF 1 " 
 
 In the triangle BCD the angle C = F"' F', and 
 
 or DjB = 2 (R + 0) sin ('" F'). (150) 
 
 EXAMPLES. 
 
 1. Given, in Fig. 90, the angle F = 9 31' 38" (No. 6 frog) 
 to find EA and ED. 
 
 2. Given, in Fig. 91, a 4 curve, and a No. 6 frog at E, 
 required F', F", F"', and distances EA, DB, ED, and AB. 
 
 139. Given the radii A C = EC = R, and A C' = BC' = 
 R', of two curved tracks crossing each other, Fig. 92, and 
 forming the angle F at the point E. Required the angles 
 F'j F", and F'" , formed at A, D, and B, respectively, and 
 the distances EA, ED, AB, and DB. 
 
 In the triangle CEC' we have given CE = R, C'E = R' 
 + g', and the included angle CEC' = F. Then 
 
 tan* (ECC- - EC'C) = (EC' - BC)i^(EQC' + EC'C) 
 
 EC' + EC 
 
 R' + g' + R 
 
 
150 
 
 FROGS AND SWITCHES. 
 
 With this half difference and 
 
 180 
 
 -, as the half sum, 
 
 the angles ECC' and EC'C are readily found. 
 This triangle also gives 
 
 ECsinCEC' _ RsinF 
 
 Now, in the triangle A CC', CC' is given by the last equa- 
 tion, the side A C = R, AC' = R', hence the angles may be 
 computed. 
 
 Draw A H perpendicular to CC', then 
 
 C'H CH : G'A CA = C'A + CA : CC", 
 
 - C H = 
 
 CC' CC' 
 
 This half difference of the segments of the base added to 
 their half sum will give the longer segment C'H, and being 
 subtracted from the half sum will give the shorter CH. 
 Then 
 
 CA C' A 
 
 and 1?" = 180 (^CC' + AC'C). ' (151) 
 
 Now, in the isosceles triangle A CE, ECA = ECC' ACC', 
 and CA = CE = R, and the included angle C are known, 
 hence 
 
 EA=2Rsin$ACE. (152) 
 
CROSSING FROGS. 151 
 
 In the triangle CDC', CD = R + //, C'D = A' + </', and 
 CC" is known, hence 
 
 nw r>' it' (K + 9) 2 ~~ 
 u/z Lf jl = * 
 
 and 
 
 and CDC' = F" = 180 - (DO//' + DC' II'). (153) 
 Again, the angle DC'C EC'C=EC'D. 
 In the isosceles triangle C'DE, 
 
 C'E = C'D = R f + <j, and ED = 2(R' + g') sin | #C'D. (1 54) 
 
 In the triangle CZJC", CB = R + y, C'B = A J/ , and CC" is 
 known, hence 
 
 Proceeding as above, with the half difference and half sum 
 we obtain C'H" and CH", then 
 
 
 C B+flr 
 
 cos BC' C = C/H// = G ' H " , 
 C',8 R' 
 
 and F'" = C7^C 7 = 180 - (BCC' + BC'C). (155) 
 
 In the isosceles triangle AC'S, AC" = BC" R', and the 
 angle A C'B = BC'C A C'C, hence 
 
 ^4 7J = 2 7i" sin i ^. C'7^. ( 1 56) 
 
 In the isosceles triangle DCR, 
 
 DCB = DCC' BCC', 
 and DB = 2(R + g)smiDCB. (157) 
 
 EXAMPLE. Given, in Fig. 92, a 1 curved track, crossing 
 another of 4, and a number 6 frog at E. Find F', F", F'", 
 and the chords EA, ED, AB, and DB. 
 
152 
 
 FROGS AND SWITCHES. 
 
 140. If the tracks cross as in Fig. 93 then the solution 
 is the same, except that CEC' = 180 F, and F=ECC' 
 + EC'C. 
 
 n 
 
 The half sum of ECC' and EC'C = , and (151) becomes 
 
 F' = A CC' + A C'C ; (153) becomes F" = Z>C7/' + DC'H' ; 
 (155) becomes F x// = CC" -f C'C. 
 
 EXAMPLE. Given, Fig. 93, a 1 curved track, crossing an- 
 other of 2, and a No. 8 frog at E, to find JF', F", F'", and 
 the chords ED, EA, ED, and EA. 
 
 141. Given F, the angle of intersection of two straight 
 tracks, Fig. 94, to find the radii A and MO, and the 
 lengths of the curved rails AD and ML, of a crossing slip 
 connecting the tracks. 
 
 Draw the radii AO and BO, and connect C and .0. By 
 Article 137, Fig. 90, find the distances GC, GK, and C7/, 
 HK, and assume 6V1 as small as the construction of the frog 
 at G will permit. 
 
 Then, since the arc ADD must be tangent to GC and CH 
 at J. and D respectively, AC= CB = GC GA. F is the 
 
CROSSING SLIPS. 
 
 153 
 
 angle at the vertex, and A C the tangent distance of the curve 
 
 A DB. Hence 
 
 AO = ACcotF. (158) 
 
 MO = AO g. (159) 
 
 j 
 
 The length of the arc 
 
 and the arc 
 
 AB = AO x 3.1410 X -- 
 
 ML = MO X 3.1410 X ~ 
 oOu 
 
 (160) 
 
 (161) 
 
 EXAMPLE. Given, in Fig. 94, F= 9 31' 38", to find the 
 radii and lengths of the curved rails AB and ML, GA being 
 5 feet. 
 
 142. In a crossing of a curved track by a straight 
 track, Fig. 95. Having given the radii of the rails of the 
 curved track, and the angle of the frog at E, to find A 0', 
 MO', the radii, and the lengths A B and MN of the curved 
 rails of a crossing slip connecting the tracks. 
 
 By Article 138, Fig. 91, find the angles F', F", and F'", of 
 the frogs at C, K, and //respectively, and the distances EC, 
 EK, CH, and HK. 
 
154 
 
 FROGS AND SWITCHES. 
 
 Draw the radii EO, CO, BO, and HO. Assume BH as short 
 as the construction of the frog at H will admit of. Draw DB 
 tangent to CBH, and AO* at right angles to ED, making 
 
 FIG. 95. 
 
 In the isosceles triangle BOH, 
 
 BH 
 
 the angle 
 
 COB = COH - BOH = F" -F' - BOH, 
 
 and the chord 
 
 CB = 2 sin i COB X CO. (162) 
 
 In the triangle DCB, CBD = $ COB, DCB = F' + $ COB, 
 and the exterior angle FDB = their sum = F'-\- COB* Then, 
 by the law of sines, CD and DB are readily found. It will 
 now be seen that FDB is the angle at the vertex, and DE 
 the tangent distance of the curve AB. Then 
 
 = BO' = BDcotiFDB. (163) 
 
 = BO'-g. (164) 
 
 * When the slip rails are on the outside of the curve FOB =F' l COB, 
 and FDB = F' COB. 
 
CROSSING SLIPS. 
 
 155 
 
 The length of the arc 
 
 AO' x 3.1416 
 
 360 
 The length of the arc 
 
 MO' X 3.1416 
 360 
 
 X the angle FDB in degrees. (165) 
 
 X the angle FDB in degrees. (166) 
 
 EXAMPLE. Given, Fig. 95, the curved track on a 2 curve, 
 and the frog at E, a No. 6 frog, to find the radii and lengths 
 of the curved rails AB and MN, BH being 4| feet. 
 
 143. Given the radii of the rails of two curved tracks 
 which cross each other, as in Fig. 96, the angle of the 
 frog at G' = F, and AG, to find the radii and lengths of 
 the curved rails AB and EF, of a -crossing slip connecting 
 the tracks. 
 
 and 0' being the centers of the rails GC and GF respec- 
 tively, and the figure completed as shown, 0" will be the 
 center of the arcs AB and EF. 
 
 Let the radius GO = R and GO' = R', then BO = R + g. 
 
156 F11OGS AND SWITCHES. 
 
 By Article 139, Fig. 92, find the angles F', F", and F" f of 
 the frogs at C, D, and // respectively ; also the chords GC, 
 GD, CH, and the side OO', and angle at 0' of the triangle 
 GOO'. 
 
 In the triangle GO' A, 
 
 GA 
 
 2GO f 
 In the triangle OO"O', the angle 
 
 0'=OO'G+ GO' A, 
 and 0"0' 0"0 E' E g (167) 
 
 for O"0 = /I 0' A 0", and 0"0 = OB 0"B, also <9" 
 = A O". 
 
 Lay off 0"P = 0"0 on CT0', then 
 
 0'F=R'-n g. (168) 
 
 In the triangle OPO', 
 
 tan J (P - 0) = 00 , ~ Q/p x tan ^90 - ~Y 
 
 With this value of the half difference, and ^90 J as 
 
 the value of the half sum, the angles P and are readily 
 found. Then 
 
 The exterior angle OPO" =0+0'. 
 The isosceles triangle 00"P gives 
 
 0"0= OP 
 
 2 cos 0" OP 
 
 Then the radius 0"B of the arc AB=OB 00", 
 or 0" = R + c) - OO". (169) 
 
 The radius FO" of the arc EF= 0"B </, 
 or FO" =R- OO". (170) 
 
 The length of the arc 
 
 AB= 0"B X 3.1416 X 4^' (171) 
 
 and the length of the arc 
 
 EF = 0"F X .3.1416 X 4r' ( 172 ) 
 
CROSSING SLIPS. 
 
 157 
 
 Tf the tracks cross each other, as in Fig. 97, the solution 
 will be the same as above, except that (167) becomes 
 
 O"(y + (JO = R + g + R', (173) 
 
 0"0' = AO' + A 0" and 0"0 = OB + 0"B, and (168) becomes 
 OT^R + cj + R'. (174) 
 
 EXAMPLES. 
 
 1. Given, in Fig. 96, a 2 curved track crossed by a 2 30' 
 curved track making 
 
 a No. 8 frog at G, 
 to find the radii and 
 lengths of the curved 
 rails AB and EF, AG 
 being 4' 10". 
 
 2. Given, in Fig. 97, 
 a 1 curved track 
 crossed by a 1 30' 
 curved track making 
 a No. 8 frog at G, 
 to find the radii and 
 lengths of the curved 
 rails AB and EF, AG 
 being 5 feet. 
 
158 
 
 TRIGONOMETRIC FORMULAS. 
 
 TRIGONOMETRIC FORMULAS. 
 
 FIG. 98. 
 
 In Fig 99, let DCE be the arc of a quadrant, ABC a right 
 triangle, the angle BAC subtended by the arc CE = A, and 
 consider the radius A C = unity. Then 
 
 AF= cosecvl. 
 BE = versin^l . 
 DI =coversin 
 CH=exsecA. 
 
 CF = 
 
 BC =si 
 AB = cos A. 
 HE = tan A. 
 DF = cot^l. 
 AH = sec A. 
 
 Using the small letters a, &, c, to represent the sides of a 
 right triangle in Fig. 98 or 99, we may write 
 
 sin A = - cosecvl 
 6 
 
 c b 
 
 = -; secA=-; 
 
 sin A = 
 
 cos A 
 
 = -; .-. tan^l = 
 
 cosec A 
 
 I 
 
 secJ. 
 
 I 
 cot 4* 
 
SOLUTION OF TRIANGLES. 
 
 159 
 
 SOLUTION OF RIGHT TRIANGLES. 
 
 Required. 
 
 Given. 
 
 A, C,c 
 
 a, 6 
 
 A, C, b 
 
 a, c 
 
 C, 6, c 
 C, a, c 
 C, a, 6 
 
 A, a 
 A,b 
 
 A,c 
 
 Formulas. 
 smA=cosC = -; c = /(&+ a) (6 a), 
 
 tan J. = cot J5 = - ; b== *a? + c' 2 . 
 c 
 
 (7 = 90 ^4; c = 
 C = 90 A; a= 
 C = 90 ^1 a = 
 
 Required. 
 
 Given. 
 
 6 
 
 A, B, a 
 
 * 
 
 A, a, 6 
 
 K^- - ^) 
 
 Ufc,CJ 
 
 ! 1 
 
 J 
 
 j I 
 i r 
 
 
 a, 6, c 
 
 Area 
 Area 
 
 A, 6, c 
 
 Area 
 
 -4, #, c 
 
 SOLUTION OF OBLIQUE TRIANGLES. 
 
 Formulas. 
 asinJ5 
 
 sin^l 
 
 bsinA 
 
 sin B = 
 
 ta,ni(A - B) = 
 
 C) 
 
 a ~ 
 
 a + b 
 B) + \(A B} 
 B)-i(A- B) 
 
 + B) 
 
 If 
 
 c), 
 
 tan-M = 
 
 =A F 
 
 V" 
 
 -V 1 
 
 sin A = 2x/ s(s-a)(s-6)(.s-c) 
 6c 
 
 Area = *s (s a) (s b)(s c) 
 Area = | be sin J. 
 
 C ' sin ^ sin B 
 
 Area = 
 
 2sin( +.B) 
 
160 GENERAL FORMULAS. 
 
 GENERAL FORMULAS. 
 
 =-\l cos: 2 A = 
 sin A = 
 
 sin A = - - - = >/-Hl cos A). 
 cosec A 
 
 cos A = = = xfi sin 2 .4 = 
 sec A 
 
 cos A =12 sin 2 -|-yl = 1 versvl. 
 
 cos A = * 4- $cos2A cos-jrA sin 2 
 
 tan ^4 =j = 
 cos A 
 
 cosyl 1 + cos2A 
 
 cot^l sin 2^1 
 
 t&nA s'mA 
 
 sin 2 ^4 = 1_ 
 1 cos 2 A sin 2 A 
 
 sec A = = the reciprocal of any expression for cos A 
 
 cos^l 
 
 cosec A = = the reciprocal of any expression for sin ^4. 
 
 sin A 
 
 \ersA = 1 cos A = 2 sin 2 \A. 
 
 .. versA 
 
 cos A 
 
 V 1 
 
 sin|,4=. *--4 
 
GENERAL FORMULAS. 161 
 
 sin A 1 cos A 
 
 sin 2^1 = 28*11^4 cos A. 
 cos 2 A cos' 2 yl sin 2 A = 2 cos 2 A 1. 
 
 1 tan 2 A 
 cot 2 A 1 
 
 2cot^l 
 
 sin ( A B) = sin A cos B cos A sinB. 
 cos (A B) = cosA cosB q: sin A sinB. 
 tan (A B) = tan X tan 7* 
 
 sin J. + sin B = 2 sin \ (A + 7;?) cos| (yl 5). 
 
 sin ^4 - sin B = 2cosl(A + 7^) sini(4 - B). 
 
 cos A + coB=2coslr(A + B) cos(A B). 
 
 cosB cosA=2s'mi(A + B) sm$(A B). 
 
 sin* A sin 2 B = cos 2 B cos- A = sin (A + 7?) sin (A B), 
 
 cos' 2 A s'm' 2 B = cos(A + B) cos(A B). 
 
 cos A cosB 
 
 sin A sinB 
 
162 
 
 MISCELLANEOUS FORMULAS. 
 
 MISCELLANEOUS FORMULAS. 
 
 Required. 
 
 Given. 
 
 Formulas. 
 
 Area of 
 
 Parallel sides = m and n 
 
 
 Trapezoid 
 
 Perp. dist. bet. them =p 
 
 - (m + ?i) 
 
 Regular Polygon 
 
 Length of side = I 
 Number of sides = n 
 
 nl' 2 ,180 
 cot 
 4 n 
 
 Circle 
 
 Radius = r 
 
 Ttr 2 [X = 3. 1410 
 
 Ellipse 
 
 Semi-axes = a and 6 
 
 Ttab 
 
 Parabola 
 
 Base = b, height = h 
 
 \bh 
 
 Surface of 
 
 Radius of base = r 
 
 
 Cone 
 
 Slant height = s 
 
 Ttrs 
 
 Cylinder 
 
 Radius = r, height = h 
 
 Z-rtrh 
 
 Sphere 
 
 Radius = r 
 
 4rtr 2 
 
 Zone 
 
 Height = h 
 
 Zrtrh 
 
 
 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 = b and &' 
 Height = h 
 
 3 
 
 Sphere 
 
 Radius = r 
 
 irtr 3 
 
TABLES. 
 
 The plates for Table IX and for I and II 
 in the Appendix were prepared by Messrs. 
 J. S. Gushing & Co., Norwood, Mass. All 
 other tables except Table XI are printed 
 from electrotypes furnished by Messrs. 
 John Wiley and Sons, New York. 
 
 163 
 
TABLE I. RADII. 
 
 Deg. 
 
 ladius. 
 
 Deg. 
 
 ladius. 
 
 Deg. 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 1 0' 
 
 nflnite 
 
 Q/ 
 
 5729.65 
 
 ()/ 
 
 2864.93 
 
 3 0' 
 
 1910.08 
 
 4 0' 
 
 1432.69 
 
 U 1 
 
 343775. 
 
 1 
 
 5635.72 
 
 1 
 
 2841.26 
 
 1 
 
 1899.53 
 
 1 
 
 1426.74 
 
 2 
 
 171887. 
 
 2 
 
 5544.83 
 
 2 
 
 2817.97 
 
 2 
 
 1889.09 
 
 2 
 
 1420.85 
 
 3 
 
 114592. 
 
 3 
 
 5456.82 
 
 3 
 
 2795.06 
 
 3 
 
 1878.77 
 
 3 
 
 1415.01 
 
 4 
 
 85943.7 
 
 4 
 
 5371.56 
 
 4 
 
 2772.53 
 
 4 
 
 1868.56 
 
 4 
 
 1409.21 
 
 5 
 
 68754.9 
 
 5 
 
 5288.92 
 
 5 
 
 2750.35 
 
 5 
 
 1858.47 
 
 5 
 
 1403.46 
 
 6 
 
 57295.8 
 
 6 
 
 5208.79 
 
 6 
 
 2728.52 
 
 6 
 
 1848.48 
 
 6 
 
 1397.76 
 
 7 
 
 49110.7 
 
 7 
 
 5131.05 
 
 7 
 
 2707.04 
 
 7 
 
 1838.59 
 
 7 
 
 1392.10 
 
 8 
 
 42971.8 
 
 8 
 
 5055.59 
 
 8 
 
 2685.89 
 
 8 
 
 1828.82 
 
 8 
 
 1386.49 
 
 9 
 
 38197.2 
 
 9 
 
 4982.33 
 
 9 
 
 2665.08 
 
 9 
 
 1819.14 
 
 9 
 
 1380.92 
 
 10 
 
 34377.5 
 
 10 
 
 4911.15 
 
 10 
 
 2644.58 
 
 10 
 
 1809.57 
 
 10 
 
 1375.40 
 
 11 
 
 31252.3 
 
 11 
 
 4841.98 
 
 11 
 
 2624.39 
 
 11 
 
 1800.10 
 
 11 
 
 1369.92 
 
 12 
 
 28647.8 
 
 12 
 
 4774.74 
 
 12 
 
 2604.51 
 
 12 
 
 1790.73 
 
 12 
 
 1364.49 
 
 13 
 
 26444.2 
 
 13 
 
 4709.33 
 
 13 
 
 2584.93 
 
 13 
 
 1781.45 
 
 13 
 
 1359.10 
 
 14 
 
 24555.4 
 
 14 
 
 4645.69 
 
 14 
 
 2565.65 
 
 14 
 
 1772.27 
 
 14 
 
 1353.75 
 
 15 
 
 22918.3 
 
 15 
 
 4583.75 
 
 15 
 
 2546.64 
 
 15 
 
 1763.18 
 
 15 
 
 1348.45 
 
 16 
 
 21485.9 
 
 16 
 
 4523.44 
 
 16 
 
 2527.92 
 
 16 
 
 1754.19 
 
 16 
 
 K343.15 
 
 17 
 
 20222.1 
 
 17 
 
 4464. 10 
 
 17 
 
 2509.47 
 
 17 
 
 1745.26 
 
 17 
 
 1337.65 
 
 18 
 
 19098.6 
 
 18 
 
 4407.46 
 
 18 
 
 2491.29 
 
 18 
 
 1736.48 
 
 18 
 
 1332.77 
 
 19 
 
 18093.4 
 
 19 
 
 4351.67 
 
 19 
 
 2473.37 
 
 19 
 
 1727.75 
 
 19 
 
 1327.63 
 
 20 
 
 17188.8 
 
 20 
 
 4297.28 
 
 20 
 
 2455.70 
 
 20 
 
 1719.12 
 
 20 
 
 1322.53 
 
 21 
 
 16370.2 
 
 21 
 
 4244.23 
 
 21 
 
 2438.29 
 
 21 
 
 1710.56 
 
 21 
 
 1317.46 
 
 22 
 
 15626.1 
 
 22 
 
 4192.47 
 
 22 
 
 2421.12 
 
 22 
 
 1702.10 
 
 22 
 
 1312.43 
 
 23 
 
 14946.7 
 
 23 
 
 4141.96 
 
 23 
 
 2404.19 
 
 23 
 
 1693.72 
 
 23 
 
 1307.45 
 
 24 
 
 14323.6 
 
 24 
 
 4092.66 
 
 24 
 
 2387.50 
 
 24 
 
 1685.42 
 
 24 
 
 1302.50 
 
 25 
 
 13751.0 
 
 25 
 
 4044.51 
 
 25 
 
 2371 .04 
 
 25 
 
 1677.20 
 
 25 
 
 1297.58 
 
 26 
 
 13222.1 
 
 26 
 
 3997.49 
 
 26 
 
 2354.80 
 
 26 
 
 1669.06 
 
 26 
 
 1292.71 
 
 27 
 
 12732.4 
 
 27 
 
 3951.54 
 
 27 
 
 2338.78 
 
 27 
 
 1661.00 
 
 27 
 
 1287.87 
 
 28 
 
 12277.7 
 
 28 
 
 3906.54 
 
 28 
 
 2322.98 
 
 28 
 
 1653.01 
 
 28 
 
 1283.07 
 
 29 
 
 11854.3 
 
 29 
 
 3862.74 
 
 29 
 
 2307.39 
 
 29 
 
 1645.11 
 
 29 
 
 1278.30 
 
 30 
 
 11459.2 
 
 30 
 
 3819.83 
 
 30 
 
 2292.01 
 
 30 
 
 1637.28 
 
 30 
 
 1273.57 
 
 /31 
 
 11089.6 
 
 31 
 
 3777.85 
 
 31 
 
 2276.84 
 
 31 
 
 1629.52 
 
 31 
 
 1268.87 
 
 32 
 
 1*0743 
 
 32 
 
 3736.79 
 
 32 
 
 2261.86 
 
 32 
 
 1621.84 
 
 32 
 
 1264.21 
 
 33 
 
 10417.5 
 
 33 
 
 3696.61 
 
 33 
 
 2247.08 
 
 33 
 
 1614.22 
 
 33 
 
 1259.58 
 
 34 
 
 10111.1 
 
 34 
 
 3657.29 
 
 34 
 
 2232.49 
 
 34 
 
 1606.68 
 
 34 
 
 1254.98 
 
 35 
 
 9822.18 
 
 35 
 
 3618.80 
 
 35 
 
 2218.09 
 
 35 
 
 1599.21 
 
 35 
 
 1250.42 
 
 36 
 
 9549.34 
 
 36 
 
 3581.10 
 
 36 
 
 2203.87 
 
 36 
 
 1591.81 
 
 36 
 
 1245.89 
 
 37 
 
 9291.29 
 
 37 
 
 3544.19 
 
 37 
 
 2189.84 
 
 37 
 
 1584.48 
 
 37 
 
 1241.40 
 
 38 
 
 9046.75 
 
 38 
 
 3508.02 
 
 38 
 
 2175.98 
 
 38 
 
 1577.21 
 
 38 
 
 1236.94 
 
 39 
 
 8814.78 
 
 39 
 
 3472.59 
 
 39 
 
 2162.30 
 
 39 
 
 1570.01 
 
 39 
 
 1232.51 
 
 40 
 
 8594.42 
 
 40 
 
 3437.87 
 
 40 
 
 2148.79 
 
 40 
 
 1562.88 
 
 40 
 
 1228.11 
 
 41 
 
 8384.80 
 
 41 
 
 3403.83 
 
 41 
 
 2135.44 
 
 41 
 
 1555.81 
 
 41 
 
 1223.74 
 
 4-4 
 
 8185.16 
 
 42 
 
 3370.46 
 
 42 
 
 2122.26 
 
 42 
 
 1548.80 
 
 42 
 
 1219.40 
 
 43 
 
 7994.81 
 
 43 
 
 3337.74 
 
 43 
 
 2109.24 
 
 43 
 
 1541.86 
 
 43 
 
 1215.30 
 
 44 
 
 7813.11 
 
 44 
 
 3305.65 
 
 44 
 
 2096.39 
 
 44 
 
 1534.98 
 
 44 
 
 1210.82 
 
 45 
 
 7639.49 
 
 45 
 
 3274.17 
 
 45 
 
 2083.68 
 
 45 
 
 1528.16 
 
 45 
 
 1206.57 
 
 46 
 
 7473.42 
 
 46 
 
 3243.29 
 
 46 
 
 2071.13 
 
 46 
 
 1521.40 
 
 46 
 
 1202.36 
 
 47 
 
 7314.41 
 
 47 
 
 3212.98 
 
 47 
 
 2058.73 
 
 47 
 
 1514.70 
 
 47 
 
 1198.17 
 
 48 
 
 7162.03 
 
 48 
 
 3183.23 
 
 48 
 
 20-16.48 
 
 48 
 
 1508.06 
 
 48 
 
 1194.01 
 
 49 
 
 7015.87 
 
 49 
 
 3154.03 
 
 49 
 
 20:M.37 
 
 49 
 
 1501.48 
 
 49 
 
 1189.88 
 
 50 
 
 6875.55 
 
 50 
 
 3125.36 
 
 50 
 
 2022.41 
 
 50 
 
 1494.95 
 
 50 
 
 1185.78 
 
 51 
 
 6740.74 
 
 51 
 
 3097.20 
 
 51 
 
 2010.59 
 
 51 
 
 1488.48 
 
 51 
 
 1181.71 
 
 52 
 
 6611.12 
 
 52 
 
 3069.55 
 
 52 
 
 1998.90 
 
 52 
 
 1482.07 
 
 52 
 
 1177.66 
 
 53 
 
 6486.38 
 
 53 
 
 3042.39 
 
 53 
 
 1987.35 
 
 53 
 
 1475.71 
 
 53 
 
 1173.65 
 
 54 
 
 6366.2 
 
 54 
 
 3015.71 
 
 54 
 
 1975.93 
 
 54 
 
 1469.41 
 
 54 
 
 1169.66 
 
 55 
 
 6250.5 
 
 55 
 
 2989.48 
 
 55 
 
 1964.64 
 
 55 
 
 1463.16 
 
 55 
 
 1165.70 
 
 56 
 
 6138.90 
 
 56 
 
 2963.71 
 
 56 
 
 1953.48 
 
 56 
 
 1456.96 
 
 56 
 
 1161.76 
 
 57 
 
 6031.2 
 
 57 
 
 2938.39 
 
 57 
 
 1942.44 
 
 57 
 
 1450.81 
 
 57 
 
 1157.85 
 
 58 
 
 5927.2 
 
 58 
 
 2913.49 
 
 58 
 
 1931.53 
 
 58 
 
 1444.72 
 
 58 
 
 1153.97 
 
 59 
 
 5826.7 
 
 59 
 
 2889.01 
 
 59 
 
 1920.75 
 
 59 
 
 1438.68 
 
 K 
 
 1150.11 
 
 60 
 
 5729.6 
 
 60 
 
 2864 -98 
 
 60 
 
 1910.08 
 
 60 
 
 1432.69 
 
 60 
 
 1146.28 
 
 165 
 
TABLE I. RADII. 
 
 Deg 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 50 
 
 1146 28 
 
 60 
 
 955.37 
 
 70 
 
 818 G4 
 
 80' 
 
 716.34 
 
 90' 
 
 636.78 
 
 I 
 
 1142.47 
 
 1 
 
 952.72 
 
 1 
 
 816.70 
 
 1 
 
 714.85 
 
 1 
 
 635.61 
 
 2 
 
 1138 69 
 
 2 
 
 950.09 
 
 2 
 
 814.76 
 
 2 
 
 713.37 
 
 2 
 
 634.44 
 
 3 
 
 1134.94 
 
 3 
 
 947.48 
 
 3 
 
 812.83 
 
 3 
 
 711.90 
 
 3 
 
 633.27 
 
 4 
 
 1181.21 
 
 4 
 
 944.88 
 
 4 
 
 810.92 
 
 4 
 
 710.43 
 
 4 
 
 632.10 
 
 5 
 
 1127 50 
 
 5 
 
 942.29 
 
 5 
 
 809.01 
 
 5 
 
 708.96 
 
 5 
 
 630.94 
 
 6 
 
 1123.82 
 
 6 
 
 939.72 
 
 6 
 
 807.11 
 
 6 
 
 707,51 
 
 6 
 
 629.79 
 
 7 
 
 1120.16 
 
 7 
 
 937.16 
 
 7 
 
 805.22 
 
 7 
 
 706.05 
 
 7 
 
 6-28.64 
 
 8 
 
 1116.52 
 
 8 
 
 934.62 
 
 8 
 
 803.34 
 
 8 
 
 704.60 
 
 8 
 
 6:27.49 
 
 9 
 
 1112.91 
 
 9 
 
 932.09 
 
 9 
 
 801.47 
 
 9 
 
 703 16 
 
 9 
 
 626.35 
 
 10 
 
 1109.33 
 
 10 
 
 929.57 
 
 10 
 
 799.61 
 
 10 
 
 701.73 
 
 10 
 
 6:25.21 
 
 11 
 
 1105 76 
 
 11 
 
 927.07 
 
 11 
 
 797 75 
 
 11 
 
 700.30 
 
 31 
 
 624.08 
 
 12 
 
 1102.22 
 
 12 
 
 924.58 
 
 12 
 
 795.91 
 
 1-2 
 
 698.88 
 
 12 
 
 622.95 
 
 13 
 
 1098.70 
 
 13 
 
 922.10 
 
 13 
 
 794.07 
 
 13 
 
 697 46 
 
 13 
 
 621.82 
 
 14 
 
 1095.20 
 
 - 14 
 
 919.64 
 
 14 
 
 792.24 
 
 14 
 
 696.05 
 
 14 
 
 620.70 
 
 15 
 
 1091.73 
 
 15 
 
 917.19 
 
 15 
 
 790.42 
 
 15 
 
 694.65 
 
 15 
 
 619 58 
 
 16 
 
 1088.28 
 
 16 
 
 914.75 
 
 16 
 
 788.61 
 
 16 
 
 693. -24 
 
 16 
 
 618.47 
 
 17 
 
 1084.85 
 
 17 
 
 912.33 
 
 17 
 
 786.80 
 
 17 
 
 691.85 
 
 17 
 
 617.36 
 
 18 
 
 1081.44 
 
 18 
 
 909.92 
 
 18 
 
 785.01 
 
 18 
 
 690.46 
 
 18 
 
 616 25 
 
 19 
 
 1078.05 
 
 19 
 
 907.52 
 
 19 
 
 783.22 
 
 19 
 
 689.08 
 
 19 
 
 615 15 
 
 20 
 
 1074.68 
 
 20 
 
 905.13 
 
 20 
 
 781.44 
 
 20 
 
 687.70 
 
 20 
 
 614.05 
 
 21 
 
 1071.34 
 
 21 
 
 902.76 
 
 21 
 
 779.67 
 
 21 
 
 686 33 
 
 21 
 
 612.96 
 
 2:2 
 
 1068.01 
 
 22 
 
 900.40 
 
 22 
 
 777 91 
 
 22 
 
 684 96 
 
 22 
 
 611 87 
 
 23 
 
 1064.71 
 
 23 
 
 898.05 
 
 23 
 
 776.15 
 
 23 
 
 683 60 
 
 23 
 
 610.78 
 
 24 
 
 1061 .43 
 
 24 
 
 895.71 
 
 24 
 
 774.40 
 
 24 
 
 682 25 
 
 24 
 
 609.70 
 
 25 
 
 1058.16 
 
 25 
 
 893.39 
 
 25 
 
 772.66 
 
 25 
 
 680.89 
 
 25 
 
 608.62 
 
 26 
 
 1054.92 
 
 26 
 
 891 .08 
 
 26 
 
 770.93 
 
 26 
 
 679.55 
 
 26 
 
 607.55 
 
 27 
 
 1051.70 
 
 27 
 
 888.78 
 
 27 
 
 769.21 
 
 27 
 
 678.21 
 
 27 
 
 606.48 
 
 28 
 
 1048.48 
 
 28 
 
 886.49 
 
 28 
 
 767.49 
 
 28 
 
 676.88 
 
 28 
 
 605.41 
 
 29 
 
 1045.311 
 
 29 
 
 884.21 
 
 29 
 
 765.78 
 
 29 
 
 675.54 
 
 29 
 
 604.35 
 
 30 
 
 1042.14 
 
 30 
 
 881.95 
 
 30 
 
 764.08 
 
 30 
 
 674 22 
 
 30 
 
 603.29 
 
 31 
 
 1039.00 
 
 31 
 
 879.69 
 
 31 
 
 760.39 
 
 31 
 
 672.90 
 
 31 
 
 602.23 
 
 32 
 
 1035.87 
 
 32 
 
 877.45 
 
 32 
 
 700.70 
 
 32 
 
 671.59 
 
 32 
 
 601.18 
 
 33 
 
 1032.76 
 
 33 
 
 875.22 
 
 33 
 
 759.02 
 
 33 
 
 670.28 
 
 33 
 
 600.13 
 
 34 
 
 1029.67 
 
 34 
 
 873.00 
 
 34 
 
 757.35 
 
 34 
 
 668.98 
 
 34 
 
 599.09 
 
 35 
 
 10sJ6 60 
 
 35 
 
 870.80 
 
 35 
 
 755.69 
 
 35 
 
 667.68 
 
 35 
 
 598.04 
 
 36 
 
 1023.55 
 
 36 
 
 868.60 
 
 36 
 
 754.03 
 
 36 
 
 666.39 
 
 36 
 
 597.01 
 
 37 
 
 1020.51 
 
 37 
 
 866.41 
 
 37 
 
 752.38 
 
 37 
 
 665.10 
 
 37 
 
 595.97 
 
 38 
 
 1017.49 
 
 38 
 
 864.24 
 
 38 
 
 750.74 
 
 38 
 
 603.82 
 
 38 
 
 594.94 
 
 39 
 
 1014.50 
 
 39 
 
 862.08 
 
 39 
 
 749.10 
 
 39 
 
 662.54 
 
 . 39 
 
 593.91 
 
 40 
 
 1011.51 
 
 40 
 
 859.92 
 
 40 
 
 747.48 
 
 40 
 
 661.S6 
 
 40 
 
 592.89 
 
 41 
 
 1008.55 
 
 41 
 
 857.78 
 
 41 
 
 745.86 
 
 41 
 
 659.99 
 
 41 
 
 591.87 
 
 4-2 
 
 1005.60 
 
 4-2 
 
 855.65 
 
 42 
 
 744.24 
 
 42 
 
 658.73 
 
 42 
 
 590.85 
 
 43 
 
 100-2.67 
 
 43 
 
 853.53 
 
 43 
 
 742.63 
 
 43 
 
 657.47 
 
 43 
 
 589.84 
 
 44 
 
 999.76 
 
 44 
 
 851.42 
 
 44 
 
 741.03 
 
 44 
 
 656.22 
 
 44 
 
 588.83 
 
 45 
 
 996.87 
 
 45 
 
 849.32 
 
 45 
 
 739.44 
 
 45 
 
 654.97 
 
 45 
 
 587.83 
 
 46 
 
 993.99 
 
 46 
 
 847.23 
 
 46 
 
 737.86 
 
 46 
 
 653.72 
 
 46 
 
 586.82 
 
 47 
 
 991.13 
 
 47 
 
 845.15 
 
 47 
 
 736.28 
 
 47 
 
 65-2.48 
 
 47 
 
 585.83 
 
 48 
 
 988.28 
 
 48 
 
 843.08 
 
 48 
 
 7:34.70 
 
 48 
 
 651.25 
 
 48 
 
 584.83 
 
 49 
 
 985.45 
 
 49 
 
 841.02 
 
 49 
 
 733.14 
 
 49 
 
 650.02 
 
 49 
 
 583.84 
 
 50 
 
 982.64 
 
 50 
 
 838.97 
 
 50 
 
 731.53 
 
 50 
 
 648.79 
 
 50 
 
 582.85 
 
 51 
 
 979.84 
 
 51 
 
 836.93 
 
 51 
 
 730.03 
 
 51 
 
 647.57 
 
 51 
 
 581.86 
 
 52 
 
 977.06 
 
 52 
 
 834.90 
 
 52 
 
 728.48 
 
 52 
 
 646.35 
 
 52 
 
 580.88 
 
 53 
 
 974.29 
 
 53 
 
 832.89 
 
 53 
 
 726.94 
 
 53 
 
 645.14 
 
 53 
 
 579.90 
 
 54 
 
 971.54 
 
 54 
 
 830 88 
 
 54 
 
 7-25.41 
 
 54 
 
 643.94 
 
 54 
 
 578.92 
 
 55 
 
 968.81 
 
 55 
 
 828.88 
 
 55 
 
 723.88 
 
 55 
 
 642.73 
 
 55 
 
 577.95 
 
 56 
 
 966.09 
 
 56 
 
 826.89 ' 
 
 56 
 
 72-2.36 
 
 56 
 
 641.53 
 
 56 
 
 576 98 
 
 57 
 
 963.39 
 
 57 
 
 824.91 
 
 57 
 
 720 85 
 
 57 
 
 640.34 
 
 57 
 
 576.02 
 
 58 
 
 960.70 
 
 58 
 
 822.93 
 
 58 
 
 719.34 
 
 58 
 
 639.15 
 
 58 
 
 575.06 
 
 59 
 
 958.03 
 
 59 
 
 8-.J0.97 
 
 59 
 
 717.84 
 
 59 
 
 637.96 
 
 59 
 
 574.10 
 
 , eo 
 
 955.37 
 
 60 
 
 819.03 
 
 60 
 
 716.34 
 
 60 
 
 636.78 
 
 60 
 
 573.14 
 
 166 
 
TABLE I. RADII. 
 
 Dee. 
 
 Radius. Deg. 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 Deg. 
 
 Radius. 
 
 Deg. 
 
 Katiiur.. 
 
 10 0' 
 
 573.14 
 
 12 0' 
 
 477.68 
 
 14 0' 
 
 409.32 
 
 16 0' 
 
 358.17 
 
 18 0' 
 
 818.31 
 
 2 
 
 571.24 
 
 2 
 
 476.36 
 
 2 
 
 408.35 
 
 2 
 
 357,43 
 
 2 
 
 317.80 
 
 4 
 
 5G9.35 
 
 4 
 
 475.05 
 
 4 
 
 407.38 
 
 4 
 
 356.69 
 
 4 
 
 317.2& 
 
 6 
 
 567 47 
 
 6 
 
 473.74 
 
 6 
 
 406.42 
 
 6 
 
 355.95 
 
 6 
 
 316.63 
 
 8 
 
 565 60 
 
 8 
 
 472.44 
 
 8 
 
 405.40 
 
 8 
 
 355.21 
 
 8 
 
 316 05 
 
 10 
 
 563.75 
 
 10 
 
 471.15 
 
 10 
 
 404.51 
 
 10 
 
 354.48 
 
 10 
 
 315.47 
 
 12 
 
 561 91 
 
 12 
 
 469 86 
 
 12 
 
 403.56 
 
 12 
 
 353.75 
 
 12 
 
 314.89 
 
 14 
 
 560.08 
 
 14 
 
 468.58 
 
 14 
 
 402.61 
 
 14 
 
 353 03 
 
 14 
 
 314 32 
 
 16 
 
 558.26 
 
 16 
 
 467.31 
 
 16 
 
 401. G7 
 
 16 
 
 352.30 
 
 16 
 
 313 75 
 
 18 
 
 556.45 
 
 18 
 
 466.04 
 
 18 
 
 400.74 
 
 18 
 
 351.58 
 
 18 
 
 313.18 
 
 20 
 
 554.66 
 
 20 
 
 464.78 
 
 20 
 
 399.80 
 
 20 
 
 350.86 
 
 20 
 
 312.61 
 
 22 
 
 552.88 
 
 22 
 
 463.53 
 
 22 
 
 398.88 
 
 22 
 
 350.15 
 
 22 
 
 312.04 
 
 24 
 
 551.11 
 
 24 
 
 462 29 
 
 24 
 
 397.95 
 
 24 
 
 349.44 
 
 24 
 
 311.47 
 
 26 
 
 549.35 
 
 26 
 
 461.05 
 
 ' 26 
 
 397.03 
 
 26 
 
 348.72 
 
 26 
 
 310 91 
 
 28 
 
 547.60 
 
 28 
 
 459.82 
 
 28 
 
 396.13 
 
 28 
 
 348.02 
 
 28 
 
 310 35 
 
 30 
 
 545.87 
 
 30 
 
 458.59 
 
 30 
 
 395.21 
 
 30 
 
 347.32 
 
 30 
 
 309.79 
 
 32 
 
 544.14 
 
 32 
 
 457.38 
 
 32 
 
 394.30 
 
 32 
 
 346.62 
 
 32 
 
 309.23 
 
 34 
 
 542.42 
 
 34 
 
 456.16 
 
 34 
 
 393.40 
 
 34 
 
 345.93 
 
 34 
 
 308.68 
 
 36 
 
 540.72 
 
 36 
 
 454.96 
 
 36 
 
 392.50 
 
 36 
 
 345.23 
 
 36 
 
 308/13 
 
 38 
 
 539.03 
 
 38 
 
 453.76 
 
 38 
 
 391.61 
 
 38 
 
 344.54 
 
 38 
 
 307.58 
 
 40 
 
 537.34 
 
 40 
 
 452.57 
 
 40 
 
 390.72 
 
 40 
 
 343.85 
 
 40 
 
 307.03 
 
 42 
 
 535.67 
 
 42 
 
 451.38 
 
 42 
 
 389.83 
 
 42 
 
 343.16 
 
 42 
 
 306.48 
 
 44 
 
 534.01 
 
 44 
 
 450.20 
 
 44 
 
 388.95 
 
 44 
 
 342.48 
 
 44 
 
 305.93 
 
 46 
 
 532.36 
 
 46 
 
 449.02 
 
 46 
 
 388.07 
 
 46 
 
 341.80 
 
 46 
 
 305.39 
 
 48 
 
 530.71 
 
 48 
 
 447.86 
 
 48 
 
 387.20 
 
 48 
 
 341.12 
 
 48 
 
 304.85 
 
 50 
 
 529.08 
 
 50 
 
 446 69 
 
 50 
 
 386.33 
 
 50 
 
 340.45 
 
 50 
 
 304.31 
 
 52 
 
 5:27. 46 
 
 52 
 
 445 54 
 
 52 
 
 385.47 
 
 52 
 
 339.78 
 
 52 
 
 303.77 
 
 54 
 
 525.85 
 
 54 
 
 444.39 
 
 54 
 
 384.60 
 
 54 
 
 339.11 
 
 54 
 
 303.24 
 
 56 
 
 524.25 
 
 56 
 
 443.54 
 
 56 
 
 383.75 
 
 56 
 
 338.44 
 
 56 
 
 302.70 
 
 58 
 
 522.65 
 
 58 
 
 442.11 
 
 58 
 
 882.89 
 
 58 
 
 337.77 
 
 58 
 
 302.17 
 
 1100' 
 
 521 .07 
 
 13 0' 
 
 440.97 
 
 16 0' 
 
 38-2.04 
 
 170' 
 
 337.11 
 
 19 0' 
 
 301.64 
 
 2 
 
 519.50 
 
 2 
 
 439.85 
 
 2 
 
 381.19 
 
 2 
 
 336.45 
 
 2 
 
 301.13 
 
 4 
 
 517.93 
 
 4 
 
 438.73 
 
 4 
 
 380.35 
 
 4 
 
 335.80 
 
 4 
 
 300.59 
 
 6 
 
 516.38 
 
 6 
 
 437.61 
 
 6 
 
 379.51 
 
 6 
 
 335.14 
 
 6 
 
 300.07 
 
 8 
 
 514.84 
 
 8 
 
 436.50 
 
 8 
 
 378.68 
 
 8 
 
 334.49 
 
 8 
 
 299. E4 
 
 10 
 
 513.30 
 
 10 
 
 435.40 
 
 10 
 
 377.84 
 
 10 
 
 333.84 
 
 10 
 
 299.0-2 
 
 12 
 
 511.77 
 
 12 
 
 434.30 
 
 12 
 
 377.02 
 
 12 
 
 333.19 
 
 12 
 
 298.50 
 
 14 
 
 510.26 
 
 14 
 
 433.21 
 
 14 
 
 376.19 
 
 14 
 
 332.55 
 
 14 
 
 297.99 
 
 16 
 
 503.75 
 
 16 
 
 432.12 
 
 16 
 
 375.37 
 
 16 
 
 331.91 
 
 16 
 
 297.47 
 
 18 
 
 507.25 
 
 18 
 
 431.04 
 
 18 
 
 374.55 
 
 18 
 
 331. 27 
 
 18 
 
 296.96 
 
 20 
 
 505.76 
 
 20 
 
 429.96 
 
 20 
 
 373.74 
 
 20 
 
 330.63 
 
 20 
 
 296.45 
 
 22 
 
 504.28 
 
 22 
 
 428.98 
 
 22 
 
 372.93 
 
 22 
 
 330.00 
 
 22 
 
 295.94 
 
 24 
 
 502.80 
 
 24 
 
 4-27.82 
 
 24 
 
 372.12 
 
 24 
 
 329.37 
 
 24 
 
 295.43 
 
 26 
 
 501.34 
 
 26 
 
 426.76 
 
 26 
 
 371.32 
 
 26 
 
 328.74 
 
 26 
 
 294.92 
 
 28 
 
 499.88 
 
 28 
 
 425.71 
 
 28 
 
 370.52 
 
 28 
 
 328.11 
 
 28 
 
 294.4:2 
 
 30 
 
 498.43 
 
 30 
 
 424.66 
 
 30 
 
 369.72 
 
 30 
 
 327.48 
 
 30 
 
 293.91 
 
 32 
 
 496.99 
 
 3:2 
 
 423.61 
 
 32 
 
 368.93 
 
 32 
 
 326.86 
 
 32 
 
 293.41 
 
 34 
 
 495.56 
 
 34 
 
 422.57 
 
 34 
 
 368.14 
 
 34 
 
 326.24 
 
 34 
 
 292.91 
 
 36 
 
 494.14 
 
 36 
 
 421.54 
 
 36 
 
 367.35 
 
 36 
 
 325.62 
 
 36 
 
 292.41 
 
 38 
 
 492.73 
 
 38 
 
 420.51 
 
 38 
 
 366.57 
 
 38 
 
 325.01 
 
 38 
 
 291.92 
 
 40 
 
 491.32 
 
 40 
 
 419.49 
 
 40 
 
 365.79 
 
 40 
 
 324.40 
 
 40 
 
 291.42 
 
 42 
 
 489.92 
 
 42 
 
 418.47 
 
 42 
 
 365.01 
 
 42 
 
 323.79 
 
 42 
 
 290.93 
 
 44 
 
 488.53 
 
 44 
 
 417.45 
 
 44 
 
 364.24 
 
 44 
 
 323.18 
 
 44 
 
 290.44 
 
 46 
 
 487.15 
 
 46 
 
 416.44 
 
 46 
 
 363.47 
 
 46 
 
 322.57 
 
 46 
 
 289.95 
 
 48 
 
 485.77 
 
 48 
 
 415.44 
 
 48 
 
 362.70 
 
 48 
 
 321.97 
 
 48 
 
 289.46 
 
 50 
 
 484.40 
 
 50 
 
 414.44 
 
 50 
 
 361.94 
 
 50 
 
 321.37 
 
 50 
 
 288.98 
 
 tt) 
 
 483.05 
 
 52 
 
 413.44 
 
 52 
 
 361.18 
 
 52 
 
 320.77 
 
 52 
 
 288.49 
 
 54 
 
 481.69 
 
 54 
 
 412.45 
 
 54 
 
 360.42 
 
 54 
 
 320.17 
 
 54 
 
 288.01 
 
 56 
 
 480.35 
 
 56 
 
 411.47 
 
 56 
 
 359.67 
 
 56 
 
 319.57 
 
 56 
 
 287.53 
 
 58 
 
 479.01 
 
 58 
 
 410.49 
 
 58 
 
 358.92 
 
 58 
 
 318.98 
 
 58 
 
 287.05 
 
 60 
 
 477.68 
 
 60 
 
 409.51 
 
 60 
 
 358.17 
 
 60 
 
 318.39 
 
 60 
 
 286.57 
 
 
 
 
 
 I 
 
 
 
 
 
TABLE II. TANGENTS AND EXTERNALS TO A 1 CURVE. 
 
 Angle. 
 
 Tan- 
 gent. 
 
 Exter- 
 nal. 
 
 Angle. 
 
 Tan- 
 gent. 
 
 Exter- 
 nal. 
 
 Angle. 
 
 Tan- 
 gent. 
 
 Exter- 
 nal. 
 
 oc 
 
 T. 
 
 E. 
 
 oc 
 
 T. 
 
 E. 
 
 oc 
 
 T. 
 
 E. 
 
 1 
 
 50.00 
 
 .218 
 
 11 
 
 551.70 
 
 26.500 
 
 21 
 
 1061.9 
 
 97.577 
 
 10' 
 
 58.34 
 
 .297 
 
 10' 
 
 560.11 
 
 27.313 
 
 !(/ 
 
 1070.6 
 
 99.155 
 
 20 
 
 66.67 
 
 .388 
 
 20 
 
 568.53 
 
 28.137 
 
 - 20 
 
 1079.2 
 
 100.75 
 
 30 
 
 75.01 
 
 .491 
 
 30 
 
 576.95 
 
 28.974 
 
 30 
 
 1087.8 
 
 102.35 
 
 40 
 
 as. 34 
 
 .606 
 
 40 
 
 585.36 
 
 29.824 
 
 40 
 
 1096.4 
 
 103.97 
 
 50 
 
 91.68 
 
 .733 
 
 50 
 
 593.79 
 
 30.686 
 
 50 
 
 1105.1 
 
 105.60 
 
 2 
 
 100.01 
 
 .873 
 
 12 
 
 602.21 
 
 31.561 
 
 22 
 
 1113.7 
 
 107.24 
 
 10 
 
 108.35 
 
 1.024 
 
 10 
 
 610.64 
 
 32.447 
 
 10 
 
 1122.4 
 
 108.90 
 
 20 
 
 116.68 
 
 1.188 
 
 20 
 
 619.07 
 
 33.347 
 
 20 
 
 1131.0 
 
 110.57 
 
 30 
 
 125.02 
 
 1.364 
 
 30 
 
 627.50 
 
 34.259 
 
 30 
 
 1139.7 ! 112.25 
 
 40 
 
 133.36 
 
 1.552 
 
 40 
 
 635.93 '35.183 
 
 40 
 
 1148.4 113.95 
 
 50 
 
 141.70 
 
 1.752 
 
 50 
 
 644.37 
 
 36.120 
 
 50 
 
 1157.0 115.66 
 
 3 
 
 150.04 
 
 1.964 
 
 13 
 
 652.81 
 
 37.070 
 
 23 
 
 1165.7 
 
 117.38 
 
 10 
 
 158.38 
 
 2.188 
 
 10 
 
 661.25 38.031 
 
 10 
 
 1174.4 
 
 119.12 
 
 20 
 
 166.72 
 
 2.425 
 
 20 
 
 669.70 39.006 
 
 20 
 
 1183.1 
 
 120.87 
 
 30 
 
 175.06 
 
 2.674 
 
 30 
 
 678.15 
 
 39.993 
 
 30 
 
 1191.8 
 
 122.63 
 
 40 
 
 183.40 
 
 2.934 
 
 40 
 
 686.6.) 
 
 40.992 
 
 40 
 
 1200.5 
 
 124.41 
 
 50 
 
 191.74 
 
 3.207 
 
 50 
 
 695.06 
 
 42.004 
 
 50 
 
 1209.2 
 
 12C.20 
 
 4 
 
 200.08 
 
 3.492 
 
 14 
 
 703.51 
 
 43.029 
 
 24 
 
 1217.9 
 
 128.00 
 
 10 
 
 208.43 
 
 3.790 
 
 10 
 
 711.97 
 
 44.066 
 
 10 
 
 1226.6 
 
 129.82 
 
 20 
 
 216.77 
 
 4.099 
 
 20 
 
 720.44 
 
 45.116 
 
 20 
 
 1235.3 
 
 131.65 
 
 30 
 
 225.12 
 
 4.421 
 
 30 
 
 728.90 
 
 46.178 
 
 30 
 
 1244.0 1 133.50 
 
 40 
 
 233.47 
 
 4.755 
 
 40 
 
 737.37 
 
 47.253 
 
 40 
 
 1252.8 135.35 
 
 50 
 
 241.81 
 
 5.100 
 
 50 
 
 745.85 
 
 48.341 
 
 50 
 
 1261.5 
 
 137.23 
 
 5 
 
 250.16 
 
 5.459 
 
 15 
 
 754.32 
 
 49.441 
 
 25 
 
 1270.2 
 
 139.11 
 
 10 
 
 258.51 
 
 5.829 
 
 10 
 
 762.80 
 
 50.554 
 
 10 
 
 1279.0 
 
 141.01 
 
 20 
 
 266.86 
 
 6.211 
 
 20 
 
 771.99 
 
 51.679 
 
 20 
 
 1287.7 
 
 142.93 
 
 30 
 
 275.21 
 
 6.606 
 
 30 
 
 779.77 
 
 52.818 
 
 30 
 
 1296.5 
 
 144.85 
 
 40 
 
 283.57 
 
 7.013 
 
 40 
 
 788.26 
 
 53.969 
 
 40 
 
 1305.3 
 
 146.79 
 
 50 
 
 291.92 
 
 7.432 
 
 50 
 
 796.75 
 
 55.132 
 
 50 
 
 1314.0 
 
 148.75 
 
 6 
 
 300.28 
 
 7.863 
 
 16 
 
 805.25 
 
 56.309 
 
 26 
 
 1322.8 
 
 150.71 
 
 10 
 
 308.64 
 
 8.307 
 
 10 
 
 813.75 57.498 
 
 10 
 
 1331.6 
 
 152.69 
 
 20 
 
 316.99 
 
 8.762 
 
 20 
 
 822.25 58.699 
 
 20 
 
 1340.4 
 
 154.69 
 
 30 
 
 325.35 
 
 9.230 
 
 30 
 
 830.76 59.914 
 
 30 
 
 1349.2 
 
 156.70 
 
 40 
 
 333.71 
 
 9.710 
 
 40 
 
 839.27 
 
 61.141 
 
 40 
 
 1358.0 
 
 158.72 
 
 50 
 
 342.08 
 
 10.202 
 
 50 
 
 847 ..78 
 
 62.381 
 
 50 
 
 1366.8 
 
 160.76 
 
 7 
 
 350.44 
 
 10.707 
 
 17 
 
 856.30 
 
 63.634 
 
 27 
 
 1375.6 
 
 162.81 
 
 10 
 
 358.81 
 
 11.224 
 
 10 
 
 864.82 i 64.900 
 
 10 
 
 1384.4 
 
 164.86 
 
 20 
 
 367.17 
 
 11.753 
 
 20 
 
 873.35 | 66.178 
 
 20 
 
 1393.2 
 
 166.95 
 
 30 
 
 375.54 
 
 12.294 
 
 30 
 
 881.88 67.470 
 
 30 
 
 1402.0 
 
 169.04 
 
 40 
 
 383.91 
 
 12.847 
 
 40 
 
 890.41 
 
 68.774 
 
 40 
 
 1410.9 
 
 171.15 
 
 50 
 
 392.28 
 
 13.413 
 
 50 
 
 898.95 
 
 70.091 
 
 50 
 
 1419.7 
 
 173.27 
 
 8 
 
 400.66 
 
 13.991 
 
 18 
 
 907.49 
 
 71.421 
 
 28 
 
 14:28.6 
 
 175.41 
 
 10 
 
 409.03 
 
 14.582 
 
 10 
 
 916.03 
 
 72.764 
 
 10 
 
 1437.4 
 
 177 . 55 
 
 20 
 
 417.41 
 
 15.184 
 
 20 
 
 924.58 
 
 74.119 
 
 20 
 
 1446.3 
 
 179.72 
 
 30 
 
 425.79 
 
 15.799 
 
 30 
 
 933.13 
 
 75.488 
 
 30 
 
 1455.1 
 
 181.89 
 
 40 
 
 434.17 
 
 16.426 
 
 40 
 
 941.69 
 
 76.869 
 
 40 
 
 1484.0 
 
 184.08 
 
 50 
 
 442.55 
 
 17.065 
 
 50 
 
 950.25 
 
 78.264 
 
 50 
 
 1472.9 
 
 186.29 
 
 9 
 
 450.93 
 
 17.717 
 
 19 
 
 958.81 
 
 79.671 
 
 29 
 
 1481.8 
 
 188.51 
 
 10 
 
 459.32 
 
 18.381 
 
 10 
 
 967.38 
 
 81.092 
 
 10 
 
 1490.7 
 
 190.74 
 
 20 
 
 467.71 
 
 19.058 
 
 20 
 
 975.96 
 
 82.525 
 
 20 
 
 1499.6 
 
 192.99 
 
 30 
 
 476.10 
 
 19.746 
 
 30 
 
 984.53 
 
 83.972 
 
 30 
 
 1508.5 
 
 195.25 
 
 40 
 
 484.49 
 
 20.447 
 
 40 
 
 993.12 
 
 85.431 
 
 40 
 
 1517.4 
 
 197.53 
 
 50 
 
 492.88 
 
 21.161 
 
 50 
 
 1001.7 
 
 86.904 
 
 50 
 
 1526.3 
 
 199.82 
 
 10 
 
 501.28 
 
 21.887 
 
 20 
 
 1010.3 
 
 88.389 
 
 30 
 
 1535.3 
 
 202.12 
 
 10 
 
 509.68 
 
 22.624 
 
 10 
 
 1018.9 
 
 89.888 
 
 10 
 
 1544.2 
 
 204.44 
 
 20 
 
 518.08 
 
 23.375 
 
 20 
 
 1027.5 
 
 91.399 
 
 20 
 
 1553.1 
 
 206.77 
 
 30 
 
 526.48 
 
 24.138 
 
 30 
 
 1036.1 
 
 92.924 
 
 30 
 
 1562.1 
 
 209.12 
 
 40 
 
 534.89 
 
 24.913 
 
 40 
 
 1044.7 
 
 94.462 
 
 40 
 
 1571.0 
 
 211.48 
 
 50 
 
 543.29 
 
 25.700 
 
 50 
 
 1053.3 
 
 96.013 
 
 50 
 
 1580.0 
 
 213.80 
 
 168 
 
TABLE II. TANGENTS AND EXTERNALS TO A 1 CURVE. 
 
 Angle. 
 <x 
 
 Tan- 
 gent. 
 
 T. 
 
 Exter- 
 nal. 
 
 E. 
 
 Angle. 
 
 oc 
 
 Tan- 
 gent. 
 
 T. 
 
 Exter- 
 nal. 
 
 E. 
 
 Angle, 
 oc 
 
 Tan- 
 gent. 
 
 T. 
 
 1 
 
 Exter- 
 nal. 
 
 E. 
 
 31 
 
 1589.0 
 
 216.25 
 
 41 
 
 2142.2 
 
 387.38 
 
 51 
 
 2732.9 
 
 618.39 
 
 10 
 
 1598.0 
 
 218.66 
 
 10' 
 
 2151.7 
 
 390.71 
 
 10' 
 
 2743.1 
 
 622.81 
 
 20 
 
 1606.9 
 
 221.08 
 
 20 
 
 2161.2 
 
 394.06 
 
 20 
 
 2753.4 
 
 627.24 
 
 30 
 
 1615.9 
 
 223.51 
 
 30 
 
 2170.8 
 
 397.43 
 
 30 
 
 2763.7 
 
 631.69 
 
 40 
 
 1624.9 
 
 225.96 
 
 40 
 
 2180.3 
 
 400.82 
 
 40 
 
 2773.9 
 
 636.17 
 
 50 
 
 1633.9 
 
 228.42 
 
 50 
 
 2189.9 
 
 404.22 
 
 - 50 
 
 2784.2 
 
 640.66 
 
 82 
 
 1643.0 
 
 230.90 
 
 42 
 
 2199.4 
 
 407.64 
 
 52 
 
 2794.5 
 
 645.17 
 
 10 
 
 1652.0 
 
 233.39 
 
 10 
 
 2209.0 
 
 411.07 
 
 10 
 
 2804.9 
 
 649.70 
 
 20 
 
 1661.0 
 
 235.90 
 
 20 
 
 2218.6 
 
 414.52 
 
 20 
 
 2815.2 
 
 654.25 
 
 30 
 
 16TO.O 
 
 238.43 
 
 30 
 
 2228.1 
 
 417.99 
 
 30 
 
 2825.6 
 
 658.83 
 
 40 
 
 1679.1 
 
 240.96 
 
 ,40 
 
 2237.7 
 
 421.48 
 
 40 
 
 2835.9 
 
 663.42 
 
 50 
 
 1688.1 
 
 243.52 
 
 (60 
 
 2247.3 
 
 424.98 
 
 50 
 
 2846.3 
 
 668.03 
 
 83 
 
 1697.2 
 
 246.08 
 
 43 
 
 2257.0 
 
 428.50 
 
 53 
 
 2856.7 
 
 672.66 
 
 10 
 
 1706.3 
 
 248.66 
 
 10 
 
 2266.6 
 
 432.04 
 
 10 
 
 2867.1 
 
 677.32 
 
 20 
 
 1715.3 
 
 251.26 
 
 20 
 
 2276.2 
 
 435.59 
 
 20 
 
 2877.5 
 
 681.99 
 
 30 
 
 1724.4 
 
 253.87 
 
 30 
 
 2285.9 
 
 439.16 
 
 30 
 
 2888.0 
 
 686.68 
 
 40 
 
 1733.5 
 
 256.50 
 
 40 
 
 2295.6 
 
 422.75 
 
 40 
 
 2898.4 
 
 691.40 
 
 50 
 
 1742.6 
 
 259.14 
 
 50 
 
 2305.2 
 
 446.35 
 
 50 
 
 2908.9 
 
 693.13 
 
 84 
 
 1751.7 
 
 261.80 
 
 44 
 
 2314.9 
 
 449.98 
 
 54 
 
 2919.4 
 
 700.89 
 
 10 
 
 1760.8 
 
 264.47 
 
 10 
 
 2324.6 
 
 453.62 
 
 10 
 
 2929.9 
 
 705.66 
 
 20 
 
 1770.0 
 
 267.16 
 
 20 
 
 2334.3 
 
 457.27 
 
 20 
 
 2940.4 
 
 710.46 
 
 30 
 
 1779.1 
 
 269.86 
 
 30 
 
 2344.1 
 
 460.95 
 
 30 
 
 2951.0 
 
 715.28 
 
 40 
 
 1788.2 
 
 272.58 
 
 40 
 
 2353.8 
 
 464.64 
 
 . 40 
 
 2961.5 
 
 720.11 
 
 50 
 
 1797.4 
 
 275.31 
 
 50 
 
 2363.5 
 
 468.35 
 
 50 
 
 2972.1 
 
 724.97 
 
 35 
 
 1806.6 
 
 278.05 
 
 45 
 
 2373.3 
 
 472.08 
 
 55 
 
 2982.7 
 
 729.85 
 
 10 
 
 1815.7 
 
 280.82 
 
 10 
 
 2383.1 
 
 475.82 
 
 10 
 
 2993.3 
 
 7'34.76 
 
 20 
 
 1824.9 
 
 283.60 
 
 20 
 
 2392.8 
 
 479.59 
 
 20 
 
 3003.9 
 
 739.68 
 
 30 
 
 1834.1 286.39 
 
 30 
 
 2402.6 
 
 483.37 
 
 30 
 
 3014.5 
 
 744.62 
 
 40 
 
 1843.3 I 289.20 
 
 40 
 
 2412.4 
 
 487.17 
 
 40 
 
 3025.2 
 
 749.59 
 
 50 
 
 1852.5 
 
 292.02 
 
 50 
 
 2422.3 
 
 490.98 
 
 50 
 
 3035.8 
 
 754.57 
 
 36 
 
 1861.7 
 
 294.86 
 
 46 
 
 2432.1 
 
 494.82 
 
 56 
 
 3046.5 
 
 759.58 
 
 10 
 
 1870.9 
 
 297.72 
 
 10 
 
 2441.9 
 
 498.67 
 
 10 
 
 3057.2 
 
 764.61 
 
 20 
 
 1880.1 
 
 300.59 
 
 20 
 
 2451.8 
 
 502.54 
 
 20 
 
 3067.9 
 
 769.66 
 
 30 
 
 1889.4 
 
 303.47 
 
 30 
 
 2461.7 
 
 506.42 
 
 30 
 
 3078.7 
 
 774.73 
 
 40 
 
 1898.6 
 
 306.37 
 
 40 
 
 2471.5 
 
 510.33 
 
 40 
 
 3089.4 
 
 779.83 
 
 50 
 
 1907.9 
 
 309.29 
 
 50 
 
 2481.4 
 
 514.25 
 
 50 
 
 3100.2 
 
 784.94 
 
 87 
 
 1917.1 
 
 312.22 
 
 47 
 
 2491.3 518.20 
 
 57 
 
 3110.9 
 
 790.08 
 
 10 
 
 1926.4 
 
 315.17 
 
 10 
 
 2501.2 
 
 522.16 
 
 10 
 
 3121.7 
 
 795.24 
 
 20 
 
 1935.7 
 
 318.13 
 
 20 
 
 2511.2 
 
 526.13 
 
 20 
 
 3132.6 
 
 800.42 
 
 30 
 
 1945.0 
 
 321.11 
 
 30 
 
 2521.1 
 
 530.13 
 
 30 
 
 3143.4 
 
 805.62 
 
 40 
 
 1954.3 
 
 324.11 
 
 40 
 
 2531.1 
 
 534.15 
 
 40 
 
 3154.2 
 
 810.85 
 
 50 
 
 1963.6 
 
 327.12 
 
 50 
 
 2541.0 
 
 538.18 
 
 50 
 
 3165.1 
 
 816.10 
 
 88 
 
 1972.9 
 
 330.15 
 
 43 
 
 8561.0 
 
 542.23 
 
 58 
 
 3176.0 
 
 821.37 
 
 10 
 
 1982.2 
 
 333.19 
 
 10 
 
 2561.0 
 
 546.30 
 
 10 
 
 8186. 9 
 
 826.66 
 
 20 
 
 1991.5 
 
 a36.25 
 
 20 
 
 2571.0 
 
 550.39 
 
 20 
 
 3197.8 
 
 831.98 
 
 30 
 
 2000.9 
 
 339.32 
 
 30 
 
 2581.0 
 
 554.50 
 
 30 
 
 3208.8 
 
 837.31 
 
 40 
 
 2010.2 
 
 342.41 
 
 40 
 
 2591.1 
 
 558.63 
 
 40 
 
 3219.7 
 
 842.67 
 
 50 
 
 2019.6 
 
 345.52 
 
 50 
 
 2601.1 
 
 562.77 
 
 50 
 
 3230.7 
 
 848.06 
 
 39 
 
 2029.0 
 
 348.64 
 
 49 
 
 2611.2 
 
 566.94 
 
 59 
 
 3241.7 
 
 853.46 
 
 10 
 
 2038.4 
 
 351.78 
 
 10 2021.2 
 
 571.12 
 
 10 
 
 3252.7 
 
 858.89 
 
 20 
 
 2047.8 
 
 354.94 
 
 20 2631.3 
 
 575.32 
 
 20 
 
 3263.7 
 
 864.34 
 
 30 
 
 2057.2 
 
 358.11 
 
 30 2641.4 
 
 579.54 
 
 30 
 
 3274.8 
 
 869.82 
 
 40 
 
 2066.6 361.29 
 
 40 2651.5 
 
 583.78 I 
 
 40 
 
 3285.8 
 
 875.32 
 
 50 
 
 2076.0 
 
 3&4.50 
 
 50 
 
 2661.6 
 
 588.04 
 
 50 
 
 3296.9 
 
 880.84 
 
 40 
 
 2085.4 
 
 367.72 
 
 50 
 
 2671.8 
 
 592.32 i 
 
 60 
 
 3308.0 
 
 886.38 
 
 10 
 
 2094.9 
 
 370.95 
 
 10 
 
 2681.9 
 
 596.62 j 
 
 10 
 
 3319.1 
 
 891.95 
 
 20 
 
 2104.3 374.20 
 
 20 
 
 2692.1 
 
 600.93 j 
 
 20 
 
 3330.3 
 
 897.64 
 
 30 
 
 2113.8 377.47 
 
 30 
 
 2702.3 
 
 605.27 1 
 
 30 
 
 3341.4 
 
 903.15 
 
 40 
 
 2123.3 
 
 380.76 
 
 40 
 
 2712.5 
 
 609.62 i 
 
 40 
 
 3352.6 
 
 908.79 
 
 50 
 
 2132.7 
 
 384.06 
 
 50 
 
 2722.7 
 
 614.00 ! 
 
 50 
 
 8363.8 
 
 914.45 
 
TABLE II. TANGENTS AND EXTERNALS TO A 1 CURVE. 
 
 Angle. 
 
 oc 
 
 Tan- 
 gent. 
 
 T. 
 
 Exter- 
 nal. 
 
 E. 
 
 Angle, 
 cc 
 
 Tan- 
 gent. 
 
 T. 
 
 Exter- 
 nal. 
 
 E. 
 
 Angle. 
 
 oc 
 
 Tan- 
 gent. 
 
 T. 
 
 Exter- 
 nal. 
 
 E. 
 
 61 
 
 3375.0 
 
 920.14 
 
 71 
 
 4086.9 
 
 1308.2 
 
 81 
 
 4893.6 
 
 1805.3 
 
 10' 
 
 3386.3 
 
 925.85 
 
 10' 
 
 4099.5 
 
 1315.6 
 
 10' 
 
 4908.0 
 
 1814.7 
 
 20 
 
 3397.5 
 
 931.58 
 
 20 
 
 4112.1 
 
 1322.9 
 
 20 
 
 4922.5 
 
 1824.1 
 
 30 
 
 '3408.8 
 
 937.34 
 
 30 
 
 4124.8 
 
 1330.3 
 
 30 
 
 4937.0 
 
 1833 6 
 
 40 
 
 3420.1 
 
 943.12 
 
 40 
 
 4137.4 
 
 1337.7 
 
 40 
 
 4951.5 
 
 1843.1 
 
 50 
 
 3431.4 
 
 948.92 
 
 50 
 
 4150.1 
 
 1345.1 
 
 50 
 
 4966.1 
 
 1852.6 
 
 62 
 
 3442.7 
 
 954.75 
 
 72 
 
 4162.8 
 
 1352.6 
 
 82 
 
 4980.7 
 
 1862.2 
 
 10 
 
 3454.1 
 
 960.60 
 
 10 
 
 4175.6 
 
 1360.1 
 
 10 
 
 4995.4 
 
 1871.8 
 
 20 
 
 3465.4 
 
 966.48 
 
 20 
 
 4188.5 
 
 1367.6 
 
 20 
 
 5010.0 
 
 1881.5 
 
 30 
 
 3476.8 
 
 972.38 
 
 30 
 
 4201.2 
 
 1375.2 
 
 30 
 
 5024.8 
 
 1891.2 
 
 40 
 
 3488.3 
 
 978.31 
 
 40 
 
 4214.0 
 
 1382.8 
 
 40 
 
 5039.5 
 
 1900.9 
 
 50 
 
 3499.7 
 
 984.27 
 
 50 
 
 4226.8 
 
 1390.4 
 
 50 
 
 5054.3 
 
 1910.7 
 
 63 
 
 3511.1 
 
 990.24 
 
 73 
 
 4239.7 
 
 1398.0 
 
 83 
 
 5069.2 
 
 1920.5 
 
 10 
 
 3522.6 
 
 996.24 
 
 10 
 
 4252.6 
 
 1405.7 
 
 10 
 
 5084.0 
 
 1930.4 
 
 20 
 
 3534.1 
 
 1002.3 
 
 20 
 
 4265.6 
 
 1413.5 
 
 20 
 
 5099.0 1940.3 
 
 30 
 
 3545.6 
 
 1008.3 
 
 30 
 
 4278.5 
 
 1421.2 
 
 30 
 
 5113.9 1950.3 
 
 40 
 
 3557.2 
 
 1014.4 
 
 40 
 
 4291.5 
 
 1429.0 
 
 40 | 5128.9 
 
 1960.2 
 
 50 
 
 3568.7 
 
 1020.5 
 
 50 
 
 4304.6 
 
 1436.8 
 
 50 i 5143.9 
 
 1970.3 
 
 64 
 
 3580.3 
 
 1026.6 
 
 74 
 
 4317.6 
 
 1444.6 
 
 84 1 5159.0 
 
 1980.4 
 
 10 
 
 3591.9 
 
 1032.8 
 
 10 
 
 4330.7 
 
 1452.5 
 
 10 5174.1 
 
 1990.5 
 
 20 
 
 3603.5 
 
 1039.0 
 
 20 
 
 4343.8 
 
 1460.4 
 
 20 5189.3 
 
 2000.6 
 
 30 
 
 3615.1 
 
 1045.2 
 
 30 
 
 4356.9 
 
 1468.4 
 
 30 
 
 5204.4 
 
 2010.8 
 
 40 
 
 3626.8 
 
 1051.4 
 
 40 
 
 4370.1 
 
 1476.4 
 
 40 
 
 5219.7 
 
 2021.1 
 
 50 
 
 3638.5 
 
 1057.7 
 
 50 
 
 4383.3 
 
 1484.4 
 
 50 
 
 5234.9 
 
 2031.4 
 
 65 
 
 3650.2 
 
 1063.9 
 
 75 
 
 4396.5 
 
 1492.4 
 
 85 
 
 5250.3 ! 2041.7 
 
 10 
 
 3661.9 
 
 1070 2 
 
 10 
 
 4409.8 
 
 1500.5 
 
 10 
 
 5265.6 2052.1 
 
 20 
 
 3673.7 
 
 1076.6 
 
 20 
 
 4423.1 
 
 1508.6 
 
 20 1 5281.0 2062.5 
 
 30 
 
 3685.4 
 
 1082.9 
 
 30 
 
 4436.4 
 
 1516.7 
 
 30 1 5296.4 2073.0 
 
 40 
 
 3697.2 
 
 1089.3 
 
 40 
 
 4449.7 
 
 1524.9 
 
 40 i 5311.9 2083 5 
 
 50 
 
 3709.0 
 
 1095.7 
 
 50 
 
 4463.1 
 
 1533.1 
 
 50 5327.4 
 
 2094.1 
 
 66 
 
 3720.9 
 
 1102.2 
 
 76 
 
 4476.5 
 
 1541.4 
 
 86 
 
 5343.0 
 
 2104.7 
 
 10 
 
 3732.7 
 
 1108.6 
 
 10 
 
 4489.9 
 
 1549.7 
 
 10 
 
 5358.6 
 
 2115.3 
 
 20 
 
 3744.6 
 
 1115.1 
 
 20 
 
 4503.4 
 
 1558.0 
 
 20 5374.2 
 
 2126.0 
 
 30 
 
 3756.5 
 
 1121.7 
 
 30 
 
 4516.9 
 
 1566.3 
 
 30 
 
 5389.9 
 
 2136.7 
 
 40 
 
 3768.5 
 
 1128.2 
 
 40 
 
 4530.4 
 
 1574.7 
 
 40 
 
 5405.6 
 
 2147.5 
 
 50 
 
 3780.4 
 
 1134.8 
 
 50 
 
 4544.0 
 
 1583.1 
 
 50 
 
 5421.4 
 
 2158.4 
 
 67 
 
 3792.4 
 
 1141.4 
 
 77 
 
 4557.6 
 
 1591.6 
 
 87 
 
 5437.2 
 
 2169.2 
 
 , 10 
 
 3804.4 
 
 1148.0 
 
 10 
 
 4571.2 
 
 1600.1 
 
 10 
 
 5453.1 
 
 2180.2 
 
 20 
 
 3816.4 
 
 1154.7 
 
 20 
 
 4584.8 
 
 1608.6 
 
 20 
 
 5469.0 2191.1 
 
 30 
 
 3828.4 
 
 1161.3 
 
 30 
 
 4598.5 
 
 1617.1 
 
 30 
 
 5484.9 
 
 2202.2 
 
 40 
 
 3840.5 
 
 1168.1 
 
 40 
 
 4612.2 
 
 1625.7 
 
 40 
 
 5500.9 
 
 2213.2 
 
 50 
 
 3852.6 
 
 1174.8 
 
 50 
 
 4626.0 
 
 1634.4 
 
 50 
 
 5517.0 
 
 2224.3 
 
 68 
 
 3864.7 
 
 1181.6 
 
 78 
 
 4639.8 
 
 1643.0 
 
 88 
 
 5533.1 
 
 2235.5 
 
 10 
 
 3875.8 
 
 1188.4 
 
 10 
 
 4653.6 
 
 1651.7 
 
 10 
 
 5549.2 
 
 2246.7 
 
 20 
 
 3889.0 
 
 1195.2 
 
 20 
 
 4667.4 
 
 1660.5 
 
 20 
 
 5565.4 
 
 2258.0 
 
 30 
 
 3901.2 
 
 1202.0 
 
 30 
 
 4681.3 
 
 1669.2 
 
 30 
 
 5581.6 
 
 2269.3 
 
 40 
 
 3913.4 
 
 1208.9 
 
 40 
 
 4695.2 
 
 1678.1 
 
 40 
 
 5597.8 
 
 2280.6 
 
 50 
 
 3925.6 
 
 1215.8 
 
 50 
 
 4709.2 
 
 1686.9 
 
 50 
 
 5614.2 
 
 2292.0 
 
 39 
 
 3937.9 
 
 1222.7 
 
 79 
 
 4723.2 
 
 1695.8 
 
 89 
 
 5630.5 
 
 2303 5 
 
 10 
 
 3950.2 
 
 1229.7 
 
 10 
 
 4737.2 
 
 1704.7 
 
 10 
 
 5646.9 
 
 2315.0 
 
 20 
 
 3962.5 
 
 1236.7 
 
 20 
 
 4751.2 
 
 1713.7 
 
 20 
 
 5663.4 
 
 2326.6 
 
 30 
 
 3974.8 
 
 1243.7 
 
 30 
 
 4765.3 
 
 1722.7 
 
 30 
 
 5679.9 
 
 2338.2 
 
 40 
 
 3987.2 
 
 1250.8 
 
 40 
 
 4779.4 
 
 1731.7 
 
 40 
 
 5696 4 
 
 2349.8 
 
 50 
 
 3999.5 
 
 1257.9 
 
 50 
 
 4793.6 
 
 1740.8 
 
 50 
 
 5713.0 
 
 2361.5 
 
 70 
 
 4011.9 
 
 1265.0 
 
 80 
 
 4807.7 
 
 1749.9 
 
 90 
 
 5729.7 
 
 2373.3 
 
 10 
 
 4024.4 
 
 1272.1 
 
 10 
 
 4822.0 
 
 1759.0 
 
 10 
 
 5746.3 
 
 2385.1 
 
 20 
 
 4036.8 
 
 1279.3 
 
 
 
 4836.2 
 
 1768.2 
 
 20 
 
 5763.1 
 
 2397.0 
 
 30 
 
 4049.3 
 
 1286.5 
 
 30 
 
 4850.5 
 
 1777.4 
 
 30 
 
 5779.9 
 
 2408.9 
 
 40 
 
 4061.8 
 
 1293.6 
 
 40 
 
 4864.8 
 
 1786.7 
 
 40 
 
 5796.7 
 
 2420.9 
 
 50 
 
 4074.4 
 
 1300.9 
 
 50 
 
 4879.2 
 
 1796.0 
 
 50 
 
 5813.6 
 
 2432.9 
 
 170 
 
'TABLE H. TANGENTS AND EXTERNALS TO A 1 CURVE. 
 
 Angle. 
 
 Tan- 
 gent. 
 
 Ex- 
 ternal. 
 
 Angle. 
 
 Tan- 
 gent. 
 
 Ex- 
 ternal. 
 
 Angle. 
 
 Tan- 
 gent. 
 
 Ex- 
 ternal 
 
 cc 
 
 
 
 T. 
 
 E. 
 
 
 oc 
 
 
 T. 
 
 E. 
 
 
 
 oc 
 
 T. 
 
 
 E. 
 
 91 
 
 
 5830.5 
 
 2444 
 
 9 
 
 97 
 
 
 6476.2 
 
 2917.3 
 
 103 
 
 7203. 
 
 2 
 
 34" 
 
 '4.4 
 
 10' 
 20 
 
 5847.5 
 5864.6 
 
 2457.1 
 
 2469.3 
 
 10 
 20 
 
 6495.2 
 6514.3 
 
 2931.6 
 2945.9 
 
 
 10 
 20 
 
 7224.7 
 7246.3 
 
 3491.3 
 3508.2 * 
 
 30 
 
 5881.7 
 
 2481 
 
 .5 
 
 30 
 
 65? 
 
 J3.4 
 
 2960.3 
 
 
 30 
 
 7268. 
 
 
 
 3525.2 
 
 i 
 
 
 
 5 
 
 98.8 
 
 2493 
 
 8 
 
 i 
 
 ) 
 
 65, 
 
 52.6 
 
 2974. 
 
 r 
 
 
 40 
 
 7289. 
 
 8 
 
 35^ 
 
 12.4 
 
 50 
 
 5916.0 
 
 2506.1 
 
 50 
 
 65 
 
 "1.9 
 
 2989.2 
 
 
 50 
 
 7311. 
 
 7 
 
 3559.6 
 
 92 
 
 10 
 20 
 
 5933.2 
 5950.5 
 5967.9 
 
 2518 
 2531 
 2543 
 
 5 
 .0 
 .5 
 
 98 
 
 10 
 20 
 
 6591.2 
 6610.6 
 6630.1 
 
 3003 8 
 3018.4 
 3033.1 
 
 104 
 
 10 
 20 
 
 7333.6 
 7355.6 
 7377.8 
 
 3576.8 
 3594.2 
 3611.7 
 
 30 
 
 5985.3 
 
 2556 
 
 .0 
 
 30 
 
 6649.6 
 
 3047. 
 
 3 
 
 
 30 
 
 7399 
 
 9 
 
 36 
 
 29.2 
 
 4 
 
 
 
 6( 
 
 )02.7 
 
 2568 
 
 .6 
 
 4! 
 
 3 
 
 66 
 
 39.2 
 
 3062. 
 
 3 
 
 
 40 
 
 7422 
 
 2 
 
 86 
 
 46.8 
 
 50 
 
 6020.2 
 
 2581 
 
 .8 
 
 50 
 
 6688.8 
 
 3077.7 
 
 
 50 
 
 7444 
 
 6 
 
 3664.5 
 
 93 
 
 10 
 
 6037.8 
 6055.4 
 
 2594.0 
 2606.8 
 
 99 
 
 10 
 
 6708.6 
 6728.4 
 
 3092.7 
 3107.7 
 
 108 10 
 
 7467.0 
 7489.6 
 
 3682.3 
 3700.2 
 
 1 
 
 JO 
 
 1 
 
 )73.1 
 
 2619 
 
 .7 
 
 2 
 
 
 
 67 
 
 48.2 
 
 3122. 
 
 9 
 
 
 20 
 
 7512 
 
 2 
 
 37 
 
 ia.2 
 
 ; 
 
 JO 
 
 6C 
 
 )90.8 
 
 2G32 
 
 .6 
 
 3 
 
 a 
 
 67 
 
 38.1 
 
 3138. 
 
 1 
 
 
 30 
 
 7534 
 
 9 
 
 37 
 
 36.2 
 
 40 
 
 6108.6 
 
 2645 
 
 .5 ! 
 
 40 
 
 6788.1 
 
 3153.3 
 
 
 40 
 
 7557 
 
 7 
 
 3754.4 
 
 
 )0 
 
 6 
 
 L26.4 
 
 2058 
 
 .5 
 
 5 
 
 
 
 68 
 
 38.2 
 
 3168. 
 
 7 
 
 
 50 
 
 7580 
 
 5 
 
 sr 
 
 72.6 
 
 94 
 
 
 6 
 
 144.3 
 
 2671 
 
 .6 
 
 100 
 
 
 68 
 
 28.3 
 
 3184. 
 
 1 
 
 1 
 
 96 
 
 7603 
 
 5 
 
 37 
 
 91.0 
 
 10 
 
 6162.2 
 
 2684 
 
 .7 
 
 10 
 
 6848.5 
 
 3199.6 
 
 
 10 
 
 7626.6 
 
 3809.4 
 
 1 
 
 JO 
 
 6 
 
 180.2 
 
 2697 
 
 .9 
 
 2 
 
 
 
 68 
 
 S8.8 
 
 3215. 
 
 1 
 
 
 20 
 
 7649 
 
 7 
 
 38 
 
 27.9 
 
 30 
 
 6198.3 
 
 2711 
 
 .2 
 
 3 
 
 
 
 6889.2 
 
 3230.8 
 
 
 30 
 
 7672.9 
 
 a 
 
 46.5 
 
 
 to 
 
 6 
 
 216.4 
 
 2724 
 
 .5 
 
 4 
 
 
 
 69< 
 
 (39.6 
 
 3246. 
 
 5 
 
 
 40 
 
 7696 
 
 .3 
 
 3i* 
 
 65.2 
 
 50 
 
 6234.6 
 
 2737 
 
 .9 
 
 50 
 
 69 
 
 30.1 
 
 3262. 
 
 3 
 
 
 50 
 
 7719 
 
 .7 
 
 3884.0 
 
 95 
 
 
 6252.8 
 
 2751 
 
 .3 
 
 101 
 
 
 6950.6 
 
 3278. 
 
 1 
 
 107 
 
 7743 
 
 .2 
 
 3902.9 
 
 
 10 
 
 6 
 
 271.1 
 
 2764 
 
 .8 
 
 1 
 
 0' 
 
 69 
 
 71.3 
 
 3294. 
 
 1 
 
 
 10 
 
 7766 
 
 .8 
 
 3 
 
 21.9 
 
 20 
 
 6289.4 
 
 2778.3 
 
 20 
 
 6992.0 
 
 3310. 
 
 
 
 20 
 
 7790 
 
 .5 
 
 3940.9 
 
 
 30 
 
 6. 
 
 307.9 
 
 2792 
 
 .0 
 
 3 
 
 
 
 70 
 
 12.7 
 
 3326. 
 
 1 
 
 
 30 
 
 7814 
 
 .3 
 
 3 
 
 60.1 
 
 
 40 
 
 6 
 
 326.3 
 
 2805 
 
 .6 
 
 4 
 
 
 
 n 
 
 33.6 
 
 3342. 
 
 3 
 
 
 40 
 
 7838 
 
 .1 
 
 at 
 
 79.4 
 
 
 50 
 
 & 
 
 344.8 
 
 2819 
 
 .4 
 
 1 
 
 
 
 70 
 
 54.5 
 
 3358. 
 
 5 
 
 
 50 
 
 78G2 
 
 .1 
 
 31 
 
 >98.7 
 
 96 
 
 
 6363.4 
 
 2833.2 
 
 102 
 
 
 7075.5 
 
 3374. 
 
 9 
 
 108 
 
 7886 
 
 .2 
 
 4018.2 
 
 
 10 
 
 6, 
 
 382.1 
 
 2847 
 
 .0 
 
 1 
 
 
 
 70 
 
 96.6 
 
 3391. 
 
 2 
 
 
 10 
 
 7910 
 
 .4 
 
 4C 
 
 )37.8 
 
 
 20 
 
 6400.8 
 
 2861.0 
 
 20 
 
 7117.8 
 
 3407. 
 
 7 
 
 
 20 
 
 7934 
 
 .6 
 
 4057.4 
 
 
 30 
 
 6 
 
 419.5 
 
 287E 
 
 .0 
 
 $ 
 
 
 
 71 
 
 39.0 
 
 3424. 
 
 3 
 
 
 30 
 
 7959 
 
 .0 
 
 4C 
 
 yr7.2 
 
 
 40 
 
 6 
 
 438.4 
 
 288 
 
 
 4 
 
 
 
 71 
 
 60.3 
 
 3440. 
 
 9 
 
 
 40 
 
 7983 
 
 .5 
 
 4( 
 
 KW.l 
 
 50 
 
 6457.3 
 
 290J 
 
 .1 
 
 50 
 
 7181.7 
 
 3457. 
 
 6 
 
 
 50 
 
 8008.0 
 
 4117.0 
 
 CORRECTIONS FOR TANGENTS AND EXTERNALS. 
 
 
 FOB TANGENTS, ADD 
 
 
 
 FOR 
 
 EXTERNALS, 
 
 ADD 
 
 
 Ang 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 Ang 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 oc 
 
 Cur. 
 
 Cur. 
 
 Cur. 
 
 Cur. Cur. 
 
 Cur. 
 
 GC 
 
 Cur. 
 
 Cur. 
 
 Cur. 
 
 Cur. 
 
 Cur. 
 
 Cur. 
 
 10 
 
 .03 
 
 .06 
 
 09~ 
 
 .1 
 
 3 .16 
 
 .19 
 
 10 
 
 001 
 
 .003 
 
 .004 
 
 .006 
 
 .007 
 
 .008 
 
 20 
 
 .06 
 
 .13 
 
 '.19 
 
 .2 
 
 3 .32 
 
 .39 
 
 20 
 
 .006 
 
 .011 
 
 .017 
 
 .022 
 
 .028 
 
 .034 
 
 30 
 
 .10 
 
 .19 
 
 .29 
 
 .3 
 
 3 .49 
 
 .59 
 
 30 
 
 .013 
 
 .025 
 
 .038 
 
 .051 
 
 .065 
 
 .078 
 
 40 
 
 .13 
 
 .26 
 
 .40 
 
 .5. 
 
 3 .67 
 
 80 
 
 1 40 
 
 .023 
 
 .046 
 
 .070 
 
 .093 
 
 .117 
 
 .141 
 
 50 
 
 .17 
 
 .34 
 
 .51 
 
 .6 
 
 3 .85 
 
 1 02 
 
 j 50 
 
 .037 
 
 .075 
 
 .116 
 
 .151 
 
 .189 
 
 .227 
 
 60 
 
 .21 
 
 .42 
 
 .63 
 
 .8 
 
 4 1.05 
 
 1.27 
 
 60 
 
 .056 
 
 .112 
 
 .168 
 
 .225 
 
 .283 
 
 .340 
 
 70 
 
 .25 
 
 .51 
 
 .76 
 
 1.0 
 
 2 1.28 
 
 1.54 
 
 i 70 
 
 .080 
 
 .159 
 
 .240 
 
 .321 
 
 .403 
 
 ,.485 
 
 80 
 
 .30 
 
 .61 
 
 .91 
 
 1.22 1.53 
 
 1.84 
 
 80 
 
 .110 
 
 .220 
 
 .332 
 
 .445 
 
 .558 
 
 .671 
 
 90 
 
 .36 
 
 .72 
 
 1.09 
 
 1.4 
 
 5 1.83 
 
 2.20 
 
 90 
 
 .149 
 
 .299 
 
 .450 .603 
 
 .756 
 
 .910 
 
 100 
 
 .43 
 
 .86 
 
 1.30 
 
 1.7 
 
 4 2 18 
 
 2.62 
 
 100 
 
 .200 
 
 .401 
 
 .604 .809 
 
 1.015 
 
 1.221 
 
 110 
 
 .51 
 
 1.03 1.56 
 
 2 
 
 8 2.61 
 
 3.14 
 
 110 
 
 .268 
 
 .536 
 
 .806 1.082 
 
 1.355 
 
 1.633 
 
 120 
 
 .62 
 
 1.25 
 
 1.93 
 
 2.5 
 
 2 3 16 
 
 3.81 
 
 120 
 
 ,360 
 
 .721 
 
 1.086 1.456 
 
 1.825 
 
 2.197 
 
 
 
TABLE III. ^TANGENTIAL OFFSETS 100 FT. ALONG THE CURVE. 
 
 Deg. of 
 Curve. 
 
 0' 
 
 10' 
 
 20' 
 
 30' 
 
 40' 
 
 50' 
 
 
 
 
 
 000 
 
 0.145 
 
 
 0.291 
 
 0< 
 
 136 
 
 0.582 
 
 
 0.727 
 
 1 
 
 
 
 873 
 
 1.01 
 
 8 
 
 
 1.164 
 
 i!i 
 
 J09 
 
 1.454 
 
 
 1 
 
 .600 
 
 2 
 
 1 
 
 745 
 
 1.891 
 
 
 2.036 
 
 2.181 
 
 2.327 
 
 2.472 
 
 3 
 
 2 
 
 618 
 
 2.76 
 
 I 
 
 
 2.908 
 
 3 ( 
 
 154 
 
 3.199 
 
 
 a 
 
 .345 
 
 40 
 
 3 
 
 490 
 
 3.63 
 
 5 
 
 
 3.781 
 
 3. 
 
 
 4.071 
 
 
 4 
 
 .217 
 
 5 
 
 4 
 
 362 
 
 4.507 
 
 
 4 ."653 
 
 4.798 
 
 4.943 
 
 5.088 
 
 6 
 
 5 
 
 234 
 
 5.37 
 
 D 
 
 
 5.524 
 
 5. 
 
 369 
 
 5.814 
 
 
 E 
 
 .960 
 
 r-o 
 
 6 
 
 105 
 
 6.250 
 
 
 6.395 
 
 6.540 
 
 6.685 
 
 6 
 
 .831 
 
 8 
 
 6 
 
 976 
 
 7.1* 
 
 1 
 
 
 7.206 
 
 7 
 
 111 
 
 7.556 
 
 
 
 .701 
 
 9 
 
 7 
 
 846 
 
 7.99 
 
 1 
 
 
 8.136 
 
 8. 
 
 281 
 
 8.426 
 
 
 I 
 
 .57i 
 
 10 
 
 8 
 
 716 
 
 8.860 
 
 
 9.005 
 
 9.150 
 
 9.295 
 
 1 
 
 .440 
 
 11 
 
 9 
 
 585 
 
 9.72 
 
 9 
 
 
 9.874 
 
 10. 
 
 )19 
 
 10.164 
 
 
 1C 
 
 .308 
 
 12 
 
 10 
 
 453 
 
 10.59 
 
 
 1 
 
 0.742 
 
 10. 
 
 387 
 
 11.031 
 
 
 11 
 
 176 
 
 13 
 
 11 
 
 320 
 
 11.465 
 
 11.609 
 
 11.754 
 
 11.898 
 
 12.043 
 
 14 
 
 12 
 
 187 
 
 12.33 
 
 1 
 
 1 
 
 2.476 
 
 12. 
 
 120 
 
 12.764 
 
 
 12 
 
 .908 
 
 15 
 
 13 
 
 053 
 
 13.197 
 
 13.341 
 
 13.485 
 
 13.629 
 
 
 13.773 
 
 16 
 
 13 
 
 917 
 
 14.06 
 
 1 
 
 1 
 
 4 205 
 
 14. 
 
 349 
 
 14.493 
 
 
 14 
 
 .637 
 
 17 
 
 14 
 
 781 
 
 14.92 
 
 5 
 
 1 
 
 5.069 
 
 15. 
 
 212 
 
 15.356 
 
 
 15 
 
 .500 
 
 18 
 
 15.643 
 
 15.787 
 
 15.931 
 
 16.074 
 
 16.218 
 
 16.301 
 
 19 
 
 16 
 
 505 
 
 16.64 
 
 8 
 
 16.792 
 
 16.935 
 
 17.078 
 
 17 
 
 .224 
 
 20 
 
 17 
 
 365 
 
 17.50 
 
 8 
 
 1 
 
 7.651 
 
 17. 
 
 '94 
 
 17.937 
 
 
 if 
 
 .081 
 
 21 
 
 18 
 
 224 
 
 18.36 
 
 
 1 
 
 8.509 
 
 18. 
 
 352 
 
 18.795 
 
 
 1 
 
 .938 
 
 22 
 
 19 
 
 081 
 
 19.224 
 
 19.366 
 
 19.509 
 
 19.652 
 
 19.794 
 
 23 
 
 19 
 
 937 
 
 20.07 
 
 i) 
 
 I 
 
 0.222 
 
 20. 
 
 364 
 
 20.507 
 
 
 20 
 
 .649 
 
 24 
 
 20 
 
 791 
 
 20.933 
 
 21.076 
 
 21.218 
 
 21.360 
 
 21 
 
 .502 
 
 TABLE IV. MID-ORDINATES TO A 100-FT. CHORD. 
 
 D of 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 7 
 
 8' 
 
 9 
 
 Curve. 
 
 
 
 
 
 
 
 
 
 
 
 
 0.000 
 
 0.21* 
 
 0.436 
 
 0.655 
 
 0.873 
 
 1 091 
 
 1.309 1.528 
 
 1.746 
 
 1.965 
 
 10 
 
 2.183 
 
 2.40x 
 
 2.620 
 
 2.839 
 
 3.058 
 
 3.277 
 
 3.496 3.716 
 
 3.935 
 
 4.155 
 
 20 
 
 4.374 
 
 4,594 
 
 4.814 
 
 5.035 
 
 5.255 
 
 5.476 
 
 5.697 5.918 
 
 6.139 
 
 6.360 
 
 Note. As an example illustrating the use of Table IT, suppose we 
 require the value of T for a 5 curve, where / = 40 20'. Then 
 
 2104.3 
 
 + .13 = 420.99. 
 
TABLE V.-LONG CHORDS. 
 
 Degree 
 of 
 Curve. 
 
 Actual 
 Arc, 
 One 
 Station. 
 
 LONG CHORDS. 
 
 2 
 
 Stations. 
 
 3 
 
 Stations. 
 
 4 
 
 Stations. 
 
 5 
 
 Stations. 
 
 6 
 
 Stations. 
 
 QW 
 
 100.000 
 
 200.000 
 
 299.999 
 
 399.998 
 
 499.996 
 
 599.993 
 
 20 
 
 .000 
 
 199.999 
 
 299.997 
 
 399.992 
 
 499.983 
 
 599.970 
 
 30 
 
 .000 
 
 199.998 
 
 299.992 
 
 399.981 
 
 499.962 
 
 599.933 
 
 40 
 
 .001 
 
 199.997 
 
 299.986 
 
 399.966 
 
 499.932 
 
 599.882 
 
 50 
 
 .001 
 
 199.995 
 
 299.979 
 
 399.947 
 
 499.894 
 
 599.815 
 
 1 
 
 100.001 
 
 199.992 
 
 299.970 
 
 399.924 
 
 499.848 
 
 599.733 
 
 10 
 
 .002 
 
 199.990 
 
 299.959 
 
 399.896 
 
 499.793 
 
 599.637 
 
 20 
 
 .002 
 
 199.986 
 
 299.946 
 
 399.865 
 
 499.729 
 
 599.526 
 
 SO 
 
 .003 
 
 199.983 
 
 299.932 
 
 399.829 
 
 499.657 
 
 599.401 
 
 40 
 
 .003 
 
 199.979 
 
 299.915 
 
 399.789 
 
 499.577 
 
 599.260 
 
 50 
 
 .004 
 
 199.974 
 
 299.898 
 
 399.744 
 
 499.488 
 
 599.105 
 
 2 
 
 100.005 
 
 199.970 
 
 299.878 
 
 399.695 
 
 499.391 
 
 598.934 
 
 10 
 
 .006 
 
 199.964 
 
 299.857 
 
 399.643 
 
 499.285 
 
 598.750 
 
 20 
 
 .007 
 
 199.959 
 
 299.83-1 
 
 399.586 
 
 499.171 
 
 598.550 
 
 30 
 
 .008 
 
 199.952 
 
 299.810 
 
 399.524 
 
 499.049 
 
 598.336 
 
 40 
 
 .009 
 
 199.946 
 
 299.783 
 
 399.459 
 
 498.918 
 
 598.106 
 
 50 
 
 .010 
 
 199.939 
 
 299.756 
 
 399.389 
 
 498.778 
 
 597.862 
 
 3 
 
 100.011 
 
 199.931 
 
 299.726 
 
 399.315 
 
 498.630 
 
 597.604 
 
 10 
 
 .013 
 
 199.924 
 
 299.695 
 
 399.237 
 
 498.474 
 
 597.331 
 
 20 
 
 .014 
 
 199.915 
 
 299.662 
 
 399.154 
 
 498.309 
 
 597.043 
 
 30 
 
 .015 
 
 199.907 
 
 299.627 
 
 399.068 
 
 498.136 
 
 596.740 
 
 40 
 
 017 
 
 199.898 
 
 299.591 
 
 398.977 
 
 497.955 
 
 596.423 
 
 50 
 
 .019 
 
 199.888 
 
 299.553 
 
 398.882 
 
 497.765 
 
 596.091 
 
 4 
 
 100.020 
 
 199.878 
 
 299.513 
 
 398.782 
 
 497.566 
 
 595.744 
 
 10 
 
 .022 
 
 199.868 
 
 299.471 
 
 398.679 
 
 497.360 
 
 595.383 
 
 20 
 
 .024 
 
 199.857 
 
 299.428 
 
 398.571 
 
 497.145 
 
 595.007 
 
 30 
 
 .026 
 
 199.846 
 
 299.383 
 
 398.459 
 
 496.921 
 
 594.617 
 
 40 
 
 .028 
 
 199.834 
 
 299.337 
 
 398.343 
 
 496.689 
 
 594.212 
 
 50 
 
 .030 
 
 199.822 
 
 299.289 
 
 398.223 
 
 496.449 
 
 593.792 
 
 5 
 
 100.032 
 
 199.810 
 
 299.239 
 
 398.099 
 
 496.201 
 
 593.358 
 
 10 
 
 .034 
 
 199.797 
 
 299.187 
 
 397.970 
 
 495.944 
 
 592.909 
 
 20 
 
 .036 
 
 199.783 
 
 299.134 
 
 397.837 
 
 495.678 
 
 592.446 
 
 30 
 
 .038 
 
 199.770 
 
 299.079 
 
 397.700 
 
 495.405 
 
 591.968 
 
 40 
 
 .041 
 
 199.756 
 
 299.023 
 
 397.559 
 
 495.123 
 
 591.476 
 
 50 
 
 .043 
 
 199.741 
 
 298.964 
 
 397.413 
 
 494.832 
 
 590.970 
 
 6 
 
 100.046 
 
 199.726 
 
 298.904 
 
 397.264 
 
 494.534 
 
 590.449 
 
 10 
 
 .048 
 
 199.710 
 
 298.843 
 
 397.110 
 
 494.227 
 
 589.913 
 
 20 
 
 .051 
 
 199.695 
 
 298.779 
 
 396.952 
 
 493.912 
 
 589.364 
 
 30 
 
 .054 
 
 199.678 
 
 298.714 
 
 396.790 
 
 493.588 
 
 588.800 
 
 40 
 
 .056 
 
 199.662 
 
 298.648 
 
 396.623 
 
 493.257 
 
 588.221 
 
 50 
 
 .059 
 
 199.644 
 
 298.579 
 
 396.453 
 
 492.917 
 
 587.628 
 
 7 
 
 100.062 
 
 199.627 
 
 298.509 
 
 396.273 
 
 492.568 
 
 587.021 
 
 10 
 
 .065 
 
 199.609 
 
 298.438 
 
 396.099 
 
 492.212 
 
 586.400 
 
 20 
 
 .068 
 
 199.591 
 
 298.364 
 
 395.916 
 
 491.847 
 
 585.765 
 
 30 
 
 .071 
 
 199.572 
 
 298.289 
 
 395.729 
 
 491.474 
 
 585.115 
 
 40 
 
 .075 
 
 199.553 
 
 298.212 
 
 395.538 
 
 491.093 
 
 584.451 
 
 50 
 
 .078 
 
 199.533 
 
 298.134 
 
 395.342 
 
 490.704 
 
 583.773 
 
 8 
 
 100.081 
 
 199.513 
 
 298.054 
 
 395.142 
 
 490.306 
 
 583.081 
 
 10 
 
 .085 
 
 199.492 
 
 297.972 
 
 394.938 
 
 489.900 
 
 582.375 
 
 20 
 
 .088 
 
 199.471 
 
 297.888 
 
 394.731 
 
 489.486 
 
 581.654 
 
 30 
 
 .092 
 
 199.450 
 
 297.803 
 
 394.518 
 
 489.064 
 
 580.920 
 
 40 
 
 .095 
 
 199.428 
 
 297.716 
 
 394.302 
 
 488.634 
 
 580.172 
 
 50 
 
 .099 
 
 199.406 
 
 297.628 
 
 394.082 
 
 488.196 
 
 579.409 
 
 9 
 
 100.103 
 
 199.383 
 
 297.538 
 
 393.857 
 
 487.749 
 
 578.633 
 
 10 
 
 .107 
 
 199.360 
 
 297.446 
 
 393.629 
 
 487.294 
 
 577.843 
 
 20 
 
 .111 
 
 199.337 
 
 297.352 
 
 393.396 
 
 486.832 
 
 577.039 
 
 30 
 
 .115 
 
 199.313 
 
 297.257 
 
 393.159 
 
 486.361 
 
 576.222 
 
 40 
 
 .119 
 
 199.289 
 
 297.160 
 
 392.918 
 
 485.882 
 
 575.390 
 
 50 
 
 .123 
 
 199.264 
 
 297.062 
 
 392.673 
 
 485.395 
 
 574.545 
 
 10 
 
 100.127 
 
 199.239 
 
 296.962 
 
 392.424 
 
 484.900 | 573.686 
 
 173 
 
TABLE V. LONG CHORDS. 
 
 Degree 
 of 
 Curve. 
 
 Actual 
 Arc, 
 One 
 Station. 
 
 LONG CHORDS. 
 
 2 
 
 Stations. 
 
 3 
 
 Stations. 
 
 4 
 Stations. 
 
 5 
 
 Stations. 
 
 6 
 
 Stations. 
 
 10 10' 
 
 100.131 
 
 199.213 
 
 296.860 
 
 392.171 
 
 484.397 
 
 572.813 
 
 20 
 
 .136 
 
 199.187 
 
 396.756 
 
 391.914 
 
 483.886 
 
 571.926 
 
 30 
 
 140 
 
 199.161 
 
 296.651 
 
 391.652 
 
 483.367 
 
 571.027 
 
 40 
 
 .145 
 
 199.134 
 
 296.544 
 
 391.387 
 
 482.840 
 
 570.113 
 
 50 
 
 .149 
 
 199.107 
 
 296.436 
 
 391.117 
 
 482.305 
 
 569.186 
 
 11 
 
 100.154 
 
 199.079 
 
 296.325 
 
 390.843 
 
 481.762 
 
 568.245 
 
 10 
 
 .158 
 
 199.051 
 
 296.214 
 
 390.565 
 
 481.211 
 
 567.292 
 
 20 
 
 .163 
 
 199.023 
 
 296.100 
 
 390.284 
 
 480.653 
 
 566.324 
 
 30 
 
 .168 
 
 198.994 
 
 295.985 
 
 389.998 
 
 480.086 
 
 565.343 
 
 40 
 
 .173 
 
 198.964 
 
 295.868 
 
 389.708 
 
 479.511 
 
 564.349 
 
 50 
 
 .178 
 
 198.935 
 
 295.750 
 
 389.414 
 
 478.929 
 
 563.341 
 
 12 
 
 100.183 
 
 198.904 
 
 295.629 
 
 389.116 
 
 478.338 
 
 562.321 
 
 10 
 
 .188 
 
 198.874 
 
 295.508 
 
 388.814 
 
 477.740 
 
 561.287 
 
 20 
 
 .193 
 
 198.843 
 
 295.384 
 
 388.508 
 
 477.135 
 
 560.240 
 
 30 
 
 .199 
 
 198.811 
 
 295.259 
 
 388.197 
 
 476.521 
 
 559.180 
 
 40 
 
 .204 
 
 198.779 
 
 295.132 
 
 387.883 
 
 475.899 
 
 558.107 
 
 50 
 
 .209 
 
 198.747 
 
 295.004 
 
 387.565 
 
 475.270 
 
 557.020 
 
 13 
 
 100.215 
 
 198.714 
 
 294.874 
 
 387.243 
 
 474.633 
 
 555.921 
 
 10 
 
 220 
 
 198.681 
 
 294.742 
 
 386.916 
 
 473.988 
 
 554.809 
 
 20 
 
 226 
 
 198.648 
 
 294.609 
 
 386.586 
 
 473.336 
 
 553.684 
 
 30 
 
 .232 
 
 198.614 
 
 294.474 
 
 386.252 
 
 472.675 
 
 552.546 
 
 40 
 
 .237 
 
 198.579 
 
 294.337 
 
 385.914 
 
 472.007 
 
 551.395 
 
 50 
 
 .243 
 
 198.544 
 
 294.199 
 
 385.572 
 
 471.332 
 
 550.232 
 
 14 
 
 100.249 
 
 198.509 
 
 294.059 
 
 385.225 
 
 470.649 
 
 549.056 
 
 10 
 
 255 
 
 198.474 
 
 293.918 
 
 384.875 
 
 469.958 
 
 547.867 
 
 20 
 
 .261 
 
 198.437 
 
 293.774 
 
 384.521 
 
 469.260 
 
 546.666 
 
 30 
 
 .267 
 
 198.401 
 
 293.629 
 
 384.163 
 
 468.554 
 
 545.452 
 
 40 
 
 .274 
 
 198.364 
 
 293.483 
 
 383.801 
 
 467.840 
 
 544.226 
 
 50 
 
 .280 
 
 198.327 
 
 293.335 
 
 383.435 
 
 467.119 
 
 542.987 
 
 15 
 
 100.286 
 
 198.289 
 
 293.185 
 
 383.065 
 
 466.390 
 
 541.736 
 
 10 
 
 292 
 
 198.251 
 
 293.034 
 
 382.691 
 
 465.654 
 
 540.472 
 
 20 
 
 299 
 
 198.212 
 
 292.881 
 
 382.313 
 
 464.911 
 
 539.196 
 
 30 
 
 306 
 
 198.173 
 
 292.726 
 
 381.931 
 
 464.160 
 
 537.908 
 
 40 
 
 312 
 
 198.134 
 
 292.570 
 
 381.546 
 
 463.401 
 
 536.608 
 
 50 
 
 .319 
 
 198.094 
 
 292.412 
 
 381.156 
 
 462.635 
 
 535.296 
 
 IG; 
 
 100.326 
 
 198.054 
 
 292.252 
 
 380.763 
 
 461.862 
 
 533.972 
 
 10 
 
 .333 
 
 198.013 
 
 292.091 
 
 380.365 
 
 461.081 
 
 532.635 
 
 20 
 
 .339 
 
 197.972 
 
 291.928 
 
 379.964 
 
 460.293 
 
 531.287 
 
 30 
 
 .346 
 
 197.930 
 
 291.764 
 
 379.559 
 
 459.498 
 
 529.927 
 
 40 
 
 .353 
 
 197.888 
 
 291.598 
 
 379.150 
 
 458.695 
 
 528.555 
 
 50 
 
 .361 
 
 197.846 
 
 291.430 
 
 378.737 
 
 457.886 
 
 527.171 
 
 17 
 
 100.368 
 
 197.803 
 
 291.261 
 
 378.320 
 
 457.069 
 
 525.778 
 
 10 
 
 .375 
 
 197.760 
 
 291.090 
 
 377.900 
 
 456.244 
 
 524.369 
 
 20 
 
 .382 
 
 197.716 
 
 290.918 
 
 377.475 
 
 455.413 
 
 522.950 
 
 30 
 
 .390 
 
 197.672 
 
 290.743 
 
 377.047 
 
 454.574 
 
 521.519 
 
 40 
 
 .397 
 
 197.628 
 
 290.568 
 
 376.615 
 
 453.728 
 
 520.073 
 
 50 
 
 .405 
 
 197.583 
 
 290.390 
 
 376.179 
 
 452.875 
 
 518.625 
 
 18 
 
 100.412 
 
 197.538 
 
 290.211 
 
 375.739 
 
 452.015 
 
 517.160 
 
 10 
 
 .420 
 
 197.492 
 
 290.031 
 
 375.295 
 
 451.147 
 
 515.685 
 
 20 
 
 .428 
 
 197.446 
 
 289.849 
 
 374.848 
 
 450.373 
 
 514.198 
 
 30 
 
 .436 
 
 197.399 
 
 89.665 
 
 374.397 
 
 449.392 
 
 512.699 
 
 40 
 
 .444 
 
 197.352 
 
 289.479 
 
 373.942 
 
 448.504 
 
 511.190 
 
 50 
 
 A52 
 
 197.305 
 
 289.292 
 
 373.483 
 
 447.608 
 
 509.670 
 
 19 
 
 100.4GO 
 
 197.256 
 
 289.104 
 
 373.021 
 
 446.706 
 
 508.139 
 
 10 
 
 .468 
 
 197.209 
 
 288.913 
 
 372.554 
 
 445.797 
 
 506.597 
 
 20 
 
 .476 
 
 197.160 
 
 288.722 
 
 372.084 
 
 444.881 
 
 505.043 
 
 30 
 
 .484 
 
 197.111 
 
 288.528 
 
 371.610 
 
 443.957 
 
 503.479 
 
 40 
 
 .493 
 
 197.062 
 
 288.333 
 
 371.133 
 
 443.028 
 
 501.905 
 
 50 
 
 .501 
 
 197.012 
 
 288.137 
 
 370.652 
 
 442.091 
 
 500.320 
 
 20 
 
 100.510 
 
 196.963 
 
 287.939 
 
 370.167 
 
 441.147 
 
 498.724 
 
 174 
 
TABLE VI.-MID-ORDINATES TO LONG CHORDS. 
 
 Degree 
 of 
 Curve. 
 
 1 
 
 Station. 
 
 a 
 
 Stations. 
 
 3 
 
 Stations. 
 
 4 
 
 Stations. 
 
 5 
 
 Stations. 
 
 6 
 
 Stations. 
 
 l</ 
 
 .036 
 
 .145 
 
 .327 
 
 .582 
 
 .909 
 
 1.309 
 
 20 
 
 .073 
 
 .291 
 
 .654 
 
 1.164 
 
 1.818 
 
 2.618 
 
 80 
 
 .109 
 
 .436 
 
 .982 
 
 1.745 
 
 2.727 
 
 3.926 
 
 40 
 
 .145 
 
 .582 
 
 1.309 
 
 2.327 
 
 3.636 
 
 5.235 
 
 50 
 
 183 
 
 .727 
 
 1.636 
 
 2.909 
 
 4.545 
 
 6.544 
 
 1 
 
 218 
 
 .873 
 
 1.963 
 
 3.490 
 
 5.453 
 
 7.852 
 
 10 
 
 255 
 
 1.018 
 
 2.291 
 
 4.072 
 
 6.362 
 
 9.160 
 
 20 
 
 231 
 
 1.164 
 
 2.618 
 
 4.654 
 
 7.270 
 
 10.468 
 
 30 
 
 .327 
 
 1.309 
 
 2.945 
 
 5.235 
 
 8.179 
 
 11.775 
 
 40 
 
 364 
 
 1.454 
 
 3.272 
 
 5.816 
 
 9.087 
 
 13.082 
 
 50 
 
 .400 
 
 1.600 
 
 3.599 
 
 6.398 
 
 9.994 
 
 14.389 
 
 2 
 
 436 
 
 1.745 
 
 3.926 
 
 6.979 
 
 10.902 
 
 15.694 
 
 10 
 
 473 
 
 1.891 
 
 4.253 
 
 7.560 
 
 11.809 
 
 17.000 
 
 20 
 
 .509 
 
 2.036 
 
 4.580 
 
 8.141 
 
 12.716 
 
 18.304 
 
 30 
 
 .545 
 
 2.181 
 
 4.907 
 
 8.722 
 
 13.623 
 
 19.608 
 
 40 
 
 .582 
 
 2.327 
 
 5.234 
 
 9.303 
 
 14.529 
 
 20.912 
 
 50 
 
 .618 
 
 2.472 
 
 5.561 
 
 9.883 
 
 15.435 
 
 22.214 
 
 3 
 
 .654 
 
 2.618 
 
 5.888 
 
 10.464 
 
 16.341 
 
 23.516 
 
 10 
 
 .691 
 
 2.763 
 
 6.215 
 
 11.044 
 
 17.246 
 
 24.817 
 
 20 
 
 .727 
 
 2.908 
 
 6.542 
 
 11.624 
 
 18.151 
 
 26.117 
 
 30 
 
 .763 
 
 3.054 
 
 6.868 
 
 12.204 
 
 19.055 
 
 27.416 
 
 40 
 
 .800 
 
 3.199 
 
 7.195 
 
 12.784 
 
 19.959 
 
 28.714 
 
 50 
 
 .836 
 
 3.345 
 
 7.522 
 
 13.363 
 
 20.863 
 
 30.C12 
 
 4 
 
 .872 
 
 3.430 
 
 7.848 
 
 13.943 
 
 21.766 
 
 31.308 
 
 10 
 
 .009 
 
 3.635 
 
 8.175 
 
 14.522 
 
 22.668 
 
 32.6(3 
 
 20 
 
 .945 
 
 3.781 
 
 8.501 
 
 15.101 
 
 23.570 
 
 83.893 
 
 30 
 
 .983 
 
 3. 920 
 
 8.828 
 
 15.680 
 
 24.471 
 
 35.189 
 
 40 
 
 .018 
 
 4.071 
 
 9.154 
 
 16.258 
 
 25.372 
 
 36.480 
 
 50 
 
 .054 
 
 4.217 
 
 9.480 
 
 16.837 
 
 26.272 
 
 37.770 
 
 5 
 
 .091 
 
 4.362 
 
 9.807 
 
 17.415 
 
 27.171 
 
 39.053 
 
 10 
 
 .127 
 
 4.507 
 
 10.133 
 
 17.992 
 
 28.070 
 
 40.346 
 
 20 
 
 .164 
 
 4.653 
 
 10.459 
 
 18.570 
 
 28.968 
 
 41.631 
 
 30 
 
 .200 
 
 4.798 
 
 10.785 
 
 19.147 
 
 29.866 
 
 42.916 
 
 40 
 
 .237 
 
 4.943 
 
 11.111 
 
 19.724 
 
 30.762 
 
 44.198 
 
 50 
 
 .273 
 
 5.088 
 
 11.436 
 
 20.301 
 
 31.658 
 
 45.479 
 
 6 
 
 .309 
 
 5.234 
 
 11.762 
 
 20.877 
 
 32.553 
 
 46.759 
 
 10 
 
 .346 
 
 5.379 
 
 12.088 
 
 21.453 
 
 33.448 
 
 48.037 
 
 20 
 
 .382 
 
 5.524 
 
 12.413 
 
 22.029 
 
 34.341 
 
 "49.313 
 
 30 
 
 1.418 
 
 5.663 
 
 12.739 
 
 22.604 
 
 35.234 
 
 50.587 
 
 40 
 
 1.453 
 
 5.814 
 
 13.064 
 
 23.179 
 
 36.126 
 
 51.860 
 
 50 
 
 1.491 
 
 5.960 
 
 13.389 
 
 23.754 
 
 37.017 
 
 53.130 
 
 7 
 
 1.523 
 
 6.105 
 
 13.715 
 
 24.328 
 
 37.907 
 
 54.399 
 
 10 
 
 1.564 
 
 6.250 
 
 14.040 
 
 24.902 
 
 38.796 
 
 55.660 
 
 20 
 
 1.600 
 
 6.395 
 
 14.365 
 
 25.476 
 
 39.684 
 
 56.931 
 
 30 
 
 1.637 
 
 6.540 
 
 14.689 
 
 26.049 
 
 40.571 
 
 58.193 
 
 40 
 
 1.673 
 
 6.683 
 
 15.014 
 
 26.622 
 
 41.458 
 
 59.451 
 
 50 
 
 1.710 
 
 6.831 
 
 15.339 
 
 27.195 
 
 42.343 
 
 60.712 
 
 8 
 
 1.746 
 
 6.976 
 
 15.663 
 
 27.767 
 
 43.227 
 
 61.969 
 
 10 
 
 1.782 
 
 7.121 
 
 15.988 
 
 28.338 
 
 44.110 
 
 63.223 
 
 20 
 
 1.819 
 
 7.266 
 
 16.312 
 
 28.910 
 
 44.992 
 
 64.475 
 
 30 
 
 1.855 
 
 7.411 
 
 16.636 
 
 29.481 
 
 45.873 
 
 65.724 
 
 40 
 
 1.892 
 
 7.556 
 
 16.960 
 
 30.051 
 
 46.753 
 
 66.972 
 
 50 
 
 1.928 
 
 7.701 
 
 17.284 
 
 30.621 
 
 47.632 
 
 68.216 
 
 9 
 
 1.965 
 
 7.846 
 
 17.608 
 
 31.190 
 
 48.510 
 
 69.459 
 
 10 
 
 2.001 
 
 7.991 
 
 17.932 
 
 31.759 
 
 49.386 
 
 70.699 
 
 20 
 
 2.037 
 
 8.136 
 
 18.255 
 
 32.328 
 
 50.261 
 
 71.936 
 
 30 
 
 2.074 
 
 8.281 
 
 18.578 
 
 32.896 
 
 51.135 
 
 73.171 
 
 40 
 
 2.110 
 
 8.426 
 
 18.902 
 
 33.464 
 
 52.008 
 
 74.403 
 
 50 
 
 2.147 
 
 8.571 
 
 19.225 
 
 34.031 
 
 52.880 
 
 75.632 
 
 10 
 
 2.183 
 
 8.716 
 
 19.548 
 
 34.597 
 
 53.750 
 
 76.859 
 
TABLE VI. MID-ORDINATES TO LONG CHORDS. 
 
 Degree 
 of 
 Curve. 
 
 1 
 
 Station. 
 
 
 
 Stations. 
 
 3 
 
 Stations. 
 
 4 
 
 Stations. 
 
 5 
 
 Stations. 
 
 6 
 
 Stations. 
 
 10 10' 
 
 2.219 
 
 8.860 
 
 19.870 
 
 35.164 
 
 54.619 
 
 78.083 
 
 20 
 
 2.256 
 
 9.005 
 
 20.193 
 
 35.729 
 
 55.486 
 
 79.305 
 
 30 
 
 2.293 
 
 9.150 
 
 20.516 
 
 36.294 
 
 56.353 
 
 80.523 
 
 40 
 
 2.329 
 
 9.295 
 
 20.838 
 
 36.859 
 
 57.218 
 
 81.739 
 
 50 
 
 2.365 
 
 9.440 
 
 21.160 
 
 37.423 
 
 58.081 
 
 82.951 
 
 11 
 
 2.402 
 
 9.585 
 
 21.483 
 
 37.986 
 
 58.943 
 
 84.161 
 
 10 
 
 2.438 
 
 9.729 
 
 21.804 
 
 38.549 
 
 59.804 
 
 85.368 
 
 20 
 
 2.475 
 
 9.874 
 
 22.126 
 
 39.111 
 
 60.663 
 
 86.571 
 
 80 
 
 2.511 
 
 10.019 
 
 22.448 
 
 39.673 
 
 61.521 
 
 87.772 
 
 40 
 
 2.547 
 
 10.164 
 
 22.7'69 
 
 40.234 
 
 62.377 
 
 88.969 
 
 50 
 
 2.584 
 
 10.308 
 
 23.090 
 
 40.795 
 
 63.232 
 
 90.164 
 
 12 
 
 2.620 
 
 10.453 
 
 23.412 
 
 41.355 
 
 64.085 
 
 91.355 
 
 10 
 
 '2.657 
 
 10.597 
 
 23.732 
 
 41.914 
 
 64.937 
 
 92.542 
 
 20 
 
 2.693 
 
 10.742 
 
 24.053 
 
 42.473 
 
 65.787 
 
 93.727 
 
 30 
 
 2.730 
 
 10.887 
 
 24.374 
 
 43.031 
 
 66.636 
 
 94.908 
 
 40 
 
 2.766 
 
 11.031 
 
 24.694 
 
 43.588 
 
 67.482 
 
 96.086 
 
 50 
 
 2.803 
 
 11.176 
 
 25.014 
 
 44.145 
 
 68.328 
 
 97.260 
 
 13 
 
 2.839 
 
 11.320 
 
 25.334 
 
 44.701 
 
 69.171 
 
 98.431 
 
 10 
 
 2.876 
 
 11.465 
 
 25.654 
 
 45.256 
 
 70.013 
 
 99.598 
 
 20 
 
 2.912 
 
 11.609 
 
 25.974 
 
 45.811 
 
 70.854 
 
 100.762 
 
 30 
 
 2.949 
 
 11.754 
 
 26.293 
 
 46.365 
 
 71.692 
 
 101 . 922 
 
 40 
 
 2.985 
 
 11.898 
 
 26.612 
 
 46.919 
 
 72.529 
 
 103. O r , 9 
 
 50 
 
 3.022 
 
 12.043 
 
 26.931 
 
 47.472 
 
 73.364 
 
 104.232 
 
 14 
 
 3.058 
 
 12.187 
 
 27.250 
 
 48.024 
 
 74.197 
 
 105. S81 
 
 10 
 
 3.095 
 
 12.331 
 
 27.569 
 
 48.575 
 
 75.029 
 
 106.527 
 
 20 
 
 3.131 
 
 12.476 
 
 27.887 
 
 49.126 
 
 75.859 
 
 107. 6C9 
 
 30 
 
 3.168 
 
 12.620 
 
 28.206 
 
 49.676 
 
 76.687 
 
 108.807 
 
 40 
 
 3.204 
 
 12.764 
 
 28.524 
 
 50.225 
 
 77.513 
 
 109.941 
 
 50 
 
 3.241 
 
 12.908 
 
 28.841 
 
 50.773 
 
 78.337 
 
 111.071 
 
 15 
 
 3.277 
 
 13.053 
 
 29.159 
 
 51.321 
 
 79.159 
 
 112.197 
 
 10 
 
 3.314 
 
 13.197 
 
 29.476 
 
 51.868 
 
 79.979 
 
 113.319 
 
 20 
 
 3.350 
 
 13.341 
 
 29.794 
 
 52.414 
 
 80.798 
 
 114.4^8 
 
 30 
 
 3.387 
 
 13.485 
 
 30.111 
 
 52.959 
 
 81.614 
 
 115.552 
 
 40 
 
 3.423 
 
 13.629 
 
 30.427 
 
 53.504 
 
 82.429 
 
 116.662 
 
 50 
 
 3.460 
 
 13.773 
 
 30.744 
 
 54.048 
 
 83.241 
 
 117.768 
 
 16 
 
 3.496 
 
 13.917 
 
 31.060 
 
 54.591 
 
 84.052 
 
 118.870 
 
 10 
 
 3.533 
 
 14.061 
 
 31.376 
 
 55.133 
 
 84.861 
 
 119.967 
 
 20 
 
 3.569 
 
 14.205 
 
 31.692 
 
 55.675 
 
 85.667 
 
 121.061 
 
 30 
 
 3.606 
 
 14.349 
 
 32.008 
 
 56.215 
 
 86.471 
 
 122.150 
 
 40 
 
 3.643 
 
 14.493 
 
 32.323 
 
 56.755 
 
 87.274 
 
 123. 5 
 
 50 
 
 3.679 
 
 14.637 
 
 32.638 
 
 57.294 
 
 88.074 
 
 124.315 
 
 17 
 
 3.716 
 
 14.781 
 
 32.953 
 
 57.832 
 
 88.872 
 
 125. S91 
 
 10 
 
 3.752 
 
 14.925 
 
 33.267 
 
 58.369 
 
 89.C68 
 
 126.463 
 
 20 
 
 8.789 
 
 15.069 
 
 33.582 
 
 58.906 
 
 90.462 
 
 127. E30 
 
 30 
 
 3.825 
 
 15.212 
 
 33.896 
 
 59.441 
 
 91.254 
 
 128.593 
 
 40 
 
 3.862 
 
 15.356 
 
 34.210 
 
 59.976 
 
 92.043 
 
 129. GC1 
 
 50 
 
 .899 
 
 15.500 
 
 34.523 
 
 60.510 
 
 92.830 
 
 130.704 
 
 18 
 
 3.935 
 
 15.643 
 
 34.837 
 
 61.042 
 
 93.616 
 
 131.753 
 
 10 
 
 3.972 
 
 15.787 
 
 35.150 
 
 61.574 
 
 94.398 
 
 132.797 
 
 20 
 
 4.008 
 
 15.931 
 
 35.463 
 
 62.106 
 
 95.179 
 
 133.837 
 
 30 
 
 4.045 
 
 16.074 
 
 35.775 
 
 62.636 
 
 95.957 
 
 134.872 
 
 40 
 
 4.081 
 
 16.218 
 
 36.088 
 
 63.165 
 
 96.733 
 
 135.902 
 
 50 
 
 4.118 
 
 16.361 
 
 36.400 
 
 63.693 
 
 97.506 
 
 136.928 
 
 19 
 
 4.155 
 
 16.505 
 
 36.712 
 
 64.221 
 
 98.278 
 
 137.948 
 
 10 
 
 4.191 
 
 16.648 
 
 37.023 
 
 64.747 
 
 99.047 
 
 138.964 
 
 20 
 
 4.228 
 
 16.792 
 
 37. M 
 
 65.273 
 
 99.813 
 
 139.975 
 
 30 
 
 4.265 
 
 16 935 
 
 37.645 
 
 65.797 
 
 100.577 
 
 140.981 
 
 40 
 
 4.301 
 
 17.078 
 
 37.956 
 
 66.321 
 
 101.339 
 
 141.982 
 
 50 
 
 4.338 
 
 17.222 
 
 38.266 
 
 66.843 
 
 102.098 
 
 142.978 
 
 20 
 
 4.374 
 
 17.365 38.576 
 
 67.365 
 
 102.855 
 
 143.969 
 
TABLE VII. -MINUTES IN DECIMALS OF A DEGREE. 
 
 t 
 
 0" 
 
 10" 
 
 15' 
 
 20" 
 
 30" 
 
 40" 
 
 45' 
 
 50' 
 
 / 
 
 
 
 .00000 
 
 00278 
 
 .00417 
 
 .00556 
 
 .00833 
 
 .01111 
 
 .01250 
 
 .01389 
 
 
 
 1 
 
 .01667 
 
 .01944 
 
 .02083 
 
 .02222 
 
 .02500 
 
 .02778 
 
 .02917 
 
 .03055 
 
 1 
 
 2 
 
 .03333 
 
 .03611 
 
 .03750 
 
 .03889 
 
 .04167 
 
 .04444 
 
 .04583 
 
 .04722 
 
 2 
 
 3 
 
 .05000 
 
 .05278 
 
 .05417 
 
 .05556 
 
 .05833 
 
 .06111 
 
 .06250 
 
 .06389 
 
 3 
 
 4 
 
 .06667 
 
 .06944 
 
 .07083 
 
 .07222 
 
 .07500 
 
 .07778 
 
 .07917 
 
 .08056 
 
 4 
 
 5 
 
 .08333 
 
 .08611 
 
 .08750 
 
 .08889 
 
 .09167 
 
 .09444 
 
 .09583 
 
 .09722 
 
 5 
 
 6 
 
 .10000 
 
 .10278 
 
 .10417 
 
 .10556 
 
 .10833 
 
 .11111 
 
 .11250 
 
 .11389 
 
 6 
 
 7 
 
 .11667 
 
 .11944 
 
 .12083 
 
 .12222 
 
 .12500 
 
 .12778 
 
 .12917 
 
 .13056 
 
 7 
 
 8 
 
 .13333 
 
 .13611 
 
 .13750 
 
 .13889 
 
 .14167 
 
 .14444 
 
 .14583 
 
 .14722 
 
 8 
 
 9 
 
 .15000 
 
 .15278 
 
 .15417 
 
 .15556 
 
 .15833 
 
 .16111 
 
 .16250 
 
 .16389 
 
 9 
 
 10 
 
 .16667 
 
 .16944 
 
 .17083 
 
 .17222 
 
 .17500 
 
 .17778 
 
 .17917 
 
 .18056 
 
 10 
 
 11 
 
 .18333 
 
 ..18611 
 
 .18750 
 
 .18889 
 
 .19167 
 
 .19444 
 
 .19583 
 
 .19722 
 
 11 
 
 12 
 
 .20000 
 
 .20278 
 
 .20417 
 
 .20556 
 
 .20833 
 
 .21111 
 
 .21250 
 
 .21389 
 
 12 
 
 13 
 
 .21667 
 
 .21944 
 
 .22083 
 
 .22222 
 
 .22500 
 
 .22778 
 
 .22917 
 
 .23056 
 
 13 
 
 14 
 
 .23333 
 
 .23611 
 
 .23750 
 
 .23889 
 
 .24167 
 
 .24444 
 
 .24583 
 
 .24722 
 
 14 
 
 15 
 
 .25000 
 
 .25278 
 
 .25417 
 
 .25556 
 
 .25833 
 
 .26111 
 
 .26250 
 
 .26389 
 
 15 
 
 16 
 
 .26067 
 
 .26944 
 
 .27083 
 
 .27222 
 
 .27500 
 
 .27778 
 
 .27917 
 
 .28056 
 
 16 
 
 17 
 
 .28333 
 
 .28611 
 
 .28750 
 
 .28889 
 
 .29167 
 
 .29444 
 
 .29583 
 
 .29722 
 
 17 
 
 18 
 
 .30000 
 
 .30278 
 
 .30-117 
 
 .30556 
 
 .30833 
 
 .31111 
 
 .31250 
 
 .31389 
 
 18 
 
 19 
 
 .31667 
 
 .31944 
 
 .32083 
 
 .32222 
 
 .32500 
 
 .32778 
 
 .32917 
 
 .33056 
 
 19 
 
 20 
 
 .33333 
 
 .33611 
 
 .33750 
 
 .33889 
 
 .34167 
 
 .34444 
 
 .34583 
 
 .34722 
 
 20 
 
 21 
 
 .35000 
 
 .35278 
 
 .35417 
 
 .35556 
 
 .35833 
 
 .36111 
 
 .36250 
 
 .36389 
 
 21 
 
 22 
 
 .36667 
 
 .36944 
 
 .37033 
 
 .37222 
 
 .37500 
 
 .37778 
 
 .37917 
 
 .38056 
 
 22 
 
 23 
 
 .38333 
 
 .38611 
 
 .38750 
 
 .38389 
 
 .39167 
 
 .39444 
 
 .39583 
 
 39722 
 
 23 
 
 24 
 
 .40000 
 
 .40278 
 
 .40417 
 
 .40556 
 
 .40833 
 
 .41111 
 
 .41250 
 
 .41389 
 
 24 
 
 25 
 
 .41667 
 
 .41944 
 
 .42083 
 
 .42222 
 
 .42500 
 
 .42778 
 
 .42917 
 
 .43056 
 
 25 
 
 26 
 
 .43333 
 
 .43611 
 
 .43150 
 
 .43389 
 
 .44167 
 
 4441'! 
 
 .44583 
 
 .44722 
 
 26 
 
 27 
 
 .45000 
 
 .45278 
 
 .45417 
 
 .45556 
 
 .45833 
 
 146111 
 
 .46250 
 
 .46389 
 
 27 
 
 28 
 
 .46667 
 
 .46944 
 
 .47083 
 
 .47222 
 
 .47500 
 
 .47778 
 
 .47917 
 
 .48056 
 
 28 
 
 29 
 
 .483.33 
 
 .48611 
 
 .48750 
 
 .48889 
 
 .49167 
 
 .49444 
 
 .49583 
 
 .49722 
 
 29 
 
 30 
 
 .50000 
 
 .50278 
 
 .50417 
 
 .50556 
 
 .50833 
 
 .51111 
 
 .51250 
 
 .51389 
 
 30 
 
 31 
 
 .51667 
 
 .51944 
 
 .52083 
 
 .52222 
 
 .52500 
 
 .52778 
 
 .E2917 
 
 .53056 
 
 31 
 
 32 
 
 .53333 
 
 .53611 
 
 .53750 
 
 .53839 
 
 .54167 
 
 .54444 
 
 .54583 
 
 .54722 
 
 32 
 
 33 
 
 .55000 
 
 .55278 
 
 .55417 
 
 .55556 
 
 .55a33 
 
 .56111 
 
 .56250 
 
 .56389 
 
 33 
 
 34 
 
 .56667 
 
 .56944 
 
 .57083 
 
 .57222 
 
 .57500 
 
 .57778 
 
 .57917 
 
 .58056 
 
 34 
 
 35 
 
 .58333 
 
 .58611 
 
 .58750 
 
 .53889 
 
 .59167 
 
 .59444 
 
 .59583 
 
 .59722 
 
 35 
 
 36 
 
 .60000 
 
 .60278 
 
 .60417 
 
 .60556 
 
 .60833 
 
 .61111 
 
 .61250 
 
 .61389 
 
 36 
 
 37 
 
 .61667 
 
 .61944 
 
 .62083 
 
 .62222 
 
 .62500 
 
 .62778 
 
 .62917 
 
 .63056 
 
 37 
 
 38 
 
 .63333 
 
 .63611 
 
 .637'50 
 
 .63889 
 
 .64167 
 
 .64444 
 
 .64583 
 
 .64722 
 
 38 
 
 39 
 
 .65000 
 
 .65278 
 
 .65417 
 
 .65556 
 
 .65833 
 
 .66111 
 
 .66250 
 
 .66389 
 
 39 
 
 40 
 
 .66667 
 
 .66944 
 
 .67083 
 
 .67222 
 
 .67500 
 
 .67778 
 
 .67917 
 
 .68056 
 
 40 
 
 41 
 
 .68333 
 
 .68611 
 
 .68750 
 
 .68889 
 
 .69167 
 
 .69444 
 
 .69583 
 
 .69722 
 
 41 
 
 42 
 
 .70000 
 
 .70278 
 
 .70417 
 
 .70556 
 
 .70833 
 
 .71111 
 
 .71250 
 
 .71389 
 
 42 
 
 43 
 
 .71667 
 
 .71944 
 
 .72083 
 
 72222 
 
 .72500 
 
 .72778 
 
 .72917 
 
 .73056 
 
 43 
 
 44 
 
 .73333 
 
 .73611 
 
 .73750 
 
 !73889 
 
 .74167 
 
 .74444 
 
 .74583 
 
 .74722 
 
 44 
 
 45 
 
 .75000 
 
 .75278 
 
 .75417 
 
 .75556 
 
 .75833 
 
 .76111 
 
 .76250 
 
 .76389 
 
 45 
 
 46 
 
 .76667 
 
 .76944 
 
 .77083 
 
 .77222 
 
 .77500 
 
 .77778 
 
 .77917 
 
 .78056 
 
 46 
 
 47 
 
 .78333 
 
 .78011 
 
 .78750 
 
 .78889 
 
 .79167 
 
 .79444 
 
 .79583 
 
 .79722 
 
 47 
 
 48 
 
 .8COOO 
 
 .80278 
 
 .80417 
 
 .80556 
 
 .80833 
 
 .81111 
 
 .81250 
 
 .81389 
 
 48 
 
 49 
 
 .81667 
 
 .81944 
 
 .82083 
 
 .82222 
 
 .82500 
 
 .82778 
 
 .82917 
 
 .83056 
 
 49 
 
 50 
 
 .83333 
 
 .83611 
 
 .83750 
 
 .83889 
 
 .84167 
 
 .84444 
 
 .84583 
 
 .84722 
 
 50 
 
 51 
 
 .85000 
 
 .85278 
 
 .85417 
 
 .85556 
 
 .85833 
 
 86111 
 
 .86250 
 
 .86389 
 
 51 
 
 52 
 
 .86667 
 
 .86944 
 
 .87083 
 
 .87222 
 
 .87500 
 
 .87778 
 
 .87917 
 
 .88056 
 
 52 
 
 53 
 
 .88333 
 
 .88611 
 
 .88750 
 
 .88889 
 
 .89167 
 
 .89444 
 
 .89583 
 
 .89722 
 
 53 
 
 54 
 
 .90000 
 
 .90278 
 
 .90417 
 
 .90556 
 
 .90833 
 
 .91111 
 
 .91250 
 
 .91389 
 
 54 
 
 55 
 
 .91667 
 
 .91944 
 
 .92083 
 
 .92222 
 
 .92300 
 
 .92778 
 
 .92917 
 
 .93056 
 
 55 
 
 56 
 
 .93333 
 
 .93611 
 
 .93750 
 
 .93889 
 
 .94167 
 
 .94444 
 
 .94583 
 
 .94722 
 
 56 
 
 57 
 
 .95000 
 
 .95278 
 
 .95417 
 
 .95556 
 
 .95833 
 
 .96111 
 
 .96250 
 
 .96389 
 
 57 
 
 58 
 
 .90667 
 
 .96944 
 
 .97083 
 
 .97222 
 
 .97500 
 
 .97778 
 
 .97917 
 
 .98056 
 
 58 
 
 59 
 
 .98333 
 
 .98611 
 
 .98750 
 
 .98889 
 
 .99167 
 
 .99444 
 
 .99583 
 
 .99722 
 
 59 
 
 / 
 
 0" 
 
 10" 
 
 15' 
 
 20" 
 
 30" 
 
 40" 
 
 45" 
 
 50" 
 
 / 
 
TABLE VIII. SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOT 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 X a | CubeB*. 
 
 Reciprocals. 
 
 1 
 
 1 
 
 1 
 
 1.0000000 
 
 1.0000000 
 
 1.000000000 
 
 2 
 
 4 
 
 8 
 
 1.4142136 
 
 1.2599210 
 
 .500000000 
 
 3 
 
 9 
 
 27 
 
 1.7320508 
 
 1.4422496 
 
 .333333333 
 
 4 
 
 16 
 
 64 
 
 2.0000000 
 
 1.5874011 
 
 .250000000 
 
 5 
 
 25 
 
 125 
 
 2.2360680 
 
 1.7099759 
 
 .200000000 
 
 6 
 
 36 
 
 216 
 
 2 4494897 
 
 1.8171206 
 
 .166666667 
 
 7 
 
 49 
 
 343 
 
 2.6457513 
 
 1.9129312 
 
 .142857143 
 
 8 
 
 64 
 
 512 
 
 2.8284271 
 
 2.0000000 
 
 .125000000 
 
 9 
 
 81 
 
 729 
 
 3.0000000 
 
 2.0800837 
 
 .111111111 
 
 10 
 
 100 
 
 1000 
 
 3.1622777 
 
 2.1544347 
 
 .100000000 
 
 11 
 
 121 
 
 1331 
 
 3.3166248 
 
 2.2239801 
 
 .090909091 
 
 12 
 
 144 
 
 1728 
 
 3 4641016 
 
 2.2894286 
 
 .083333333 
 
 13 
 
 169 
 
 2197 
 
 3.6055513 
 
 2.3513347 
 
 .076923077 
 
 14 
 
 196 
 
 2744 
 
 3.7416574 
 
 2.4101422 
 
 .071428571 
 
 15 
 
 225 
 
 3375 
 
 3.8729833 
 
 2.4662121 
 
 .066666667 
 
 16 
 
 256 
 
 4096 
 
 4.0000000 
 
 2.5198421 
 
 .062500000 
 
 17 
 
 289 
 
 4913 
 
 4.1231056 
 
 2.5712816 
 
 .058823529 
 
 18 
 
 324 
 
 5832 
 
 4.2426407 
 
 2.6207414 
 
 .055555556 
 
 19 
 
 361 
 
 6859 
 
 4.3588989 
 
 2.6684016 
 
 .052631579 
 
 20 
 
 400 
 
 8000 
 
 4.4721360 
 
 2.7144177 
 
 .050000000 
 
 21 
 
 441 
 
 9261 
 
 4.5825757 
 
 2.7589243 
 
 .047619048 
 
 22 
 
 484 
 
 10648 
 
 4.6904158 
 
 2.8020393 
 
 .045454545 
 
 23 
 
 529 
 
 12167 
 
 4.7958315 
 
 2.8438670 
 
 .043478261 
 
 24 
 
 576 
 
 13824 
 
 4.8989795 
 
 2.8844991 
 
 .041666667 
 
 25 
 
 625 
 
 15625 
 
 5.0000000 
 
 2.9240177 
 
 .040000000 
 
 26 
 
 676 
 
 17576 
 
 5.0990195 
 
 2.9624960 
 
 .038461538 
 
 27 
 
 729 
 
 19683 
 
 5.1961524 
 
 3.0000000 
 
 .037037037 
 
 28 
 
 784 
 
 21952 
 
 5.2915026 
 
 3.0365889 
 
 .035714286 
 
 29 
 
 841 
 
 24389 
 
 5.3851648 
 
 3.0723168 
 
 .034482759 
 
 30 
 
 900 
 
 27000 
 
 5.4772256 
 
 3.1072325 
 
 .033333333 
 
 31 
 
 961 
 
 29791 
 
 5.5677644 
 
 3.1413806 
 
 .032258065 
 
 32 
 
 1024 
 
 32768 
 
 5.6568542 
 
 5.1748021 
 
 .031250000 
 
 33 
 
 1089 
 
 35937 
 
 5.7445626 
 
 8.2075343 
 
 .030303030 
 
 34 
 
 1156 
 
 39304 
 
 5.8309519 
 
 3.2396118 
 
 .029411765 
 
 35 
 
 1225 
 
 42875 
 
 5.9160798 
 
 3.2710663 
 
 .028571429 
 
 36 
 
 1296 
 
 46656 
 
 6.0000000 
 
 3.3019272 
 
 .027777778 
 
 37 
 
 1369 
 
 50653 
 
 6.0827625 
 
 3.3322218 
 
 .027027027 
 
 38 
 
 1444 
 
 54872 
 
 6.1644140 
 
 3.3619754 
 
 .026315789 
 
 39 
 
 1521 
 
 59319 
 
 6.2449980 
 
 3.3912114 
 
 .025641026 
 
 40 
 
 1600 
 
 64000 
 
 6.3245553 
 
 3.4199519 
 
 .025000000 
 
 41 
 
 1681 
 
 68921 
 
 6.4031242 
 
 3.4482172 
 
 .024390244 
 
 42 
 
 1764 
 
 74088 
 
 6.4807407 
 
 3.4760266 
 
 .023809524 
 
 43 
 
 1849 
 
 79507 
 
 6.5574385 
 
 3.5033981 
 
 .023255814 
 
 44 
 
 1936 
 
 85184 
 
 6.6332496 
 
 3.5303483 
 
 .022727273 
 
 45 
 
 2025 
 
 91125 
 
 6.7082039 
 
 3.5568933 
 
 .022222222 
 
 46 
 
 2116 
 
 97336 
 
 6.7823300 
 
 3.5830479 
 
 .021739130 
 
 47 
 
 2209 
 
 103823 
 
 6.8556546 
 
 3.G088261 
 
 .021276600 
 
 48 
 
 2304 
 
 110592 
 
 6.9282032 
 
 3.6342411 
 
 .020833333 
 
 49 
 
 2401 
 
 117649 
 
 7.0000000 
 
 3.6593057 
 
 .020408163 
 
 50 
 
 2500 
 
 125000 
 
 7.0710678 
 
 3.6840314 
 
 .020000000 
 
 51 
 
 2601 
 
 132651 
 
 7.1414284 
 
 3.7084298 
 
 .019607843 
 
 52 
 
 2704 
 
 140608 
 
 7.2111026 
 
 3.7325111 
 
 .019230769 
 
 53 
 
 2809 
 
 148877 
 
 7.2801099 
 
 3.7562858 
 
 .018867925 
 
 54 
 
 2916 
 
 157464 
 
 7.3484692 
 
 3.7797631 
 
 ,0185ia519 
 
 55 
 
 3025 
 
 166375 
 
 7.4161985 
 
 3.8029525 
 
 .018181818 
 
 56 
 
 3136 
 
 175616 
 
 7.4833148 
 
 3.8258624 
 
 .017857143 
 
 57 
 
 3249 
 
 185193 
 
 7.5498344 
 
 3.8485011 
 
 .017543860 
 
 58 
 
 3364 
 
 195112 
 
 7.6157731 
 
 3.8708766 
 
 .017241379 
 
 59 
 
 3481 
 
 205379 
 
 7.6811457 
 
 3.8929965 
 
 .016949153 
 
 60 
 
 3600 
 
 216000 
 
 7.7459667 
 
 3.9148676 
 
 .016666667 
 
 61 
 
 3721 
 
 ' 226981 
 
 7.8102497 
 
 3.9364972 
 
 .016393443 
 
 62 
 
 3844 
 
 238328 
 
 7.8740079 
 
 3.9578915 
 
 .016129032 
 
TABLE VIII.- Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Roots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 63 
 
 3969 
 
 250047 
 
 7.9372539 
 
 3.9790571 
 
 .015873016 
 
 64 
 
 4096 
 
 262144 
 
 8.0000000 
 
 4.0000000 
 
 .015625000 
 
 65 
 
 4225 
 
 274025 
 
 8.0622577 
 
 4.0207256 
 
 .015384615 
 
 68 
 
 4356 
 
 287496 
 
 8.1240384 
 
 4.0412401 
 
 . .015151515 
 
 67 1 
 
 4489 
 
 300763 
 
 8.1853528 
 
 4.0615480 
 
 .014925373 
 
 68 
 
 46.24 
 
 314432 
 
 8.2462113 
 
 4.0816551 
 
 .014705882 
 
 69 
 
 4761 
 
 328509 
 
 8.3066239 
 
 4.1015661 
 
 .014492754 
 
 70 
 
 4900 
 
 343000 
 
 8.3666003 
 
 4.1212853 
 
 .014285714 
 
 71 
 
 5041 
 
 357911 
 
 8.4261498 
 
 4.1408178 
 
 .014084507 
 
 72 
 
 5184 
 
 373248 
 
 8.4852814 
 
 4.1601676 
 
 .013888889 
 
 73 
 
 5329 
 
 389017 
 
 8.5440037 
 
 4.1793390 
 
 .013698630 
 
 74 
 
 5476 
 
 405224 
 
 8.6023253 
 
 4.1983364 
 
 .013513514 
 
 75 
 
 5625 
 
 421875 
 
 8.6602540 
 
 4.2171633 
 
 .013333333 
 
 76 
 
 5776 
 
 438976 
 
 8,7177979 
 
 4.2358236 
 
 .013157895 
 
 77 
 
 5929 
 
 456533 
 
 8* 7749644 
 
 4.2543210 
 
 .012987013 
 
 78 
 
 6084 
 
 474552 
 
 8.8317609 
 
 4.2726586 
 
 .012820513 
 
 79 
 
 6241 
 
 493039 
 
 8.8881944 
 
 4.2908404 
 
 .012658228 
 
 80 
 
 6400 
 
 512000 
 
 8.9442719 
 
 4.3088695 
 
 .012500000 
 
 81 
 
 G5G1 
 
 531441 
 
 9.0000000 
 
 4.3267487 
 
 .012345679 
 
 82 
 
 6724 
 
 551368 
 
 9.0553851 
 
 4.3444815 
 
 .012195122 
 
 83 
 
 6889 
 
 571787 
 
 9.1104336 
 
 4.3620707 
 
 .012048193 
 
 84 
 
 7056 
 
 592704 
 
 9.1651514 
 
 4.3795191 
 
 .011904762 
 
 85 
 
 7225 
 
 614125 
 
 9.2195445 
 
 4.3968296 
 
 .011764706 
 
 86 
 
 7396 
 
 636056 
 
 9.2736185 
 
 4.4140049 
 
 .011627907 
 
 87 
 
 7569 
 
 658503 
 
 9.327'3791 
 
 4.4310476 
 
 .011494253 
 
 88 
 
 7744 
 
 681472 
 
 9.3808315 
 
 4.4479602 
 
 .011363636 
 
 89 
 
 7921 
 
 704969 
 
 9.4339811 
 
 '4.4&47451 
 
 .011235955 
 
 90 
 
 8100 
 
 729000 
 
 9.4868330 
 
 4.4814047 
 
 .011111111 
 
 91 
 
 8281 
 
 753571 
 
 9.5393920 
 
 4.4979414 
 
 .010989011 
 
 92 
 
 8464 
 
 778683 
 
 9.5916630 
 
 4.5143574 
 
 .010869565 
 
 93 
 
 8649 
 
 804357 
 
 9.6436508 
 
 4.5306549 
 
 .010752688 
 
 94 
 
 8836 
 
 830584 
 
 9.6953597 
 
 4.5468359 
 
 .010638298 
 
 95 
 
 9025 
 
 857375 
 
 9.74679-43 
 
 4.5629026 
 
 .010526316 
 
 96 
 
 9216 
 
 884736 
 
 9.7979590 
 
 4.5788570 
 
 .010416667 
 
 97 
 
 9409 
 
 912673 
 
 9.8488578 
 
 4.5947009 
 
 .010309278 
 
 J98 
 
 9604 
 
 941192 
 
 9.8904949 
 
 4.6104363 
 
 .010204082 
 
 99 
 
 9801 
 
 970299 
 
 9.9498744 
 
 4.0260G50 
 
 .010101010 
 
 100 
 
 10000 
 
 1000000 
 
 10.0000000 
 
 4.6415888 
 
 .010000000 
 
 101 
 
 10201 
 
 1030301 
 
 10.0498756 
 
 4.6570095 
 
 .009900990 
 
 102 
 
 10404 
 
 1061208 
 
 10.0995049 
 
 4.6723287 
 
 .009803922 
 
 103 
 
 10609 
 
 1092727 
 
 10.1488916 
 
 4.6875482 
 
 .009708738 
 
 104 
 
 10816 
 
 1124864 
 
 10.1980390 
 
 4.7026694 
 
 .009615385 
 
 105 
 
 11025 
 
 1157625 
 
 10.2469508 
 
 4.7176940 
 
 .009523810 
 
 106 
 
 11236 
 
 1191016 
 
 10.2956301 
 
 4.7326235 
 
 .009433962 
 
 107 
 
 11449 
 
 1225043 
 
 10.3440804 
 
 4.7474594 
 
 .009345794 
 
 108 
 
 11G64 
 
 1259712 
 
 10.3923048 
 
 4.7622032 
 
 .003259259 
 
 109 
 
 11881 
 
 1295029 
 
 10.4403065 
 
 4.7768562 
 
 .009174312 
 
 110 
 
 12100 
 
 1331000 
 
 10.4880885 
 
 4.7914199 
 
 .009090909 
 
 111 
 
 12321 
 
 1367631 
 
 10.5356538 
 
 4.8058955 
 
 .009009009 
 
 112 
 
 12544 
 
 1404928 
 
 10.5830052 
 
 4.8202845 
 
 .008928571 
 
 113 
 
 12769 
 
 1442897 
 
 10.6301458 
 
 4.8345881 
 
 .008849558 
 
 114 
 
 12996 
 
 1481544 
 
 10. 770783 
 
 4.848S076 
 
 .008771930 
 
 115 
 
 13225 
 
 1520875 
 
 10.7238053 
 
 4.8629442 
 
 .008695652 
 
 116 
 
 13456 
 
 1560896 
 
 10.7703296 
 
 4.8769990 
 
 .008620690 
 
 117 
 
 13689 
 
 1601613 
 
 10.8166538 
 
 4.8909732 
 
 .008547009 
 
 118 
 
 13924 
 
 1643032 
 
 10.8627805 
 
 4.9048681 
 
 .008474576 
 
 119 
 
 14161 
 
 1685159 
 
 10.9087131 
 
 4.9186847 
 
 .008403361 
 
 120 
 
 14400 
 
 1728000 
 
 10.9544513 
 
 4.9324242 
 
 .008333333 
 
 121 
 
 14641 
 
 1771561 
 
 1 1.00. -0000 
 
 4.9460874 
 
 .008264463 
 
 132 
 
 14884 
 
 1815848 
 
 11.0453610 
 
 4.9596757 
 
 .008196721 
 
 123 
 
 15129 
 
 1860867 
 
 11.0905365 
 
 4.9731898 
 
 .008130081 
 
 124 
 
 15376 
 
 1906624 
 
 11.1355287 
 
 4.9866310 
 
 .008064516 
 
 179 
 
TABLE VIII. Continued. 
 
 ~ NO. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Roots. 
 
 . 
 Reciprocals^ 
 
 125 
 
 15625 
 
 1953125 
 
 11.1803399 
 
 5.0000000 
 
 .008000000 
 
 126 
 
 15876 
 
 2000376 
 
 11.2249722 
 
 5.0132979 
 
 .007936508 
 
 127 
 
 16129 
 
 2048383 
 
 11.2694277 
 
 5.0265257 
 
 .007874016 
 
 128 
 
 16384 
 
 2097152 
 
 11.3137085 
 
 5.0396842 
 
 .007812500 
 
 129 
 
 16641 
 
 2146689 
 
 11.3578167 
 
 5,0527743 
 
 .007751938 
 
 130 
 
 16900 
 
 2197000 
 
 11.4017543 
 
 5.0657970 
 
 .007692308 
 
 131 
 
 17161 
 
 2248091 
 
 11.4455231 
 
 5.0787531 
 
 .007633588 
 
 132 
 
 17424 
 
 2299968 
 
 11.4891253 
 
 5.0916434 
 
 .007575758 
 
 133 
 
 17689 
 
 2352637 
 
 11.5325626 
 
 5.1044687 
 
 .007518797 
 
 134 
 
 17956 
 
 2406104 
 
 11.5758369 
 
 5.1172299 
 
 .007462687 
 
 135 
 
 18225 
 
 2460375 
 
 11.6189500 
 
 5.1299278 
 
 .007407407 
 
 136 
 
 18496 
 
 2515456 
 
 11.6619038 
 
 5.1425632 
 
 .007352941 
 
 137 
 
 18769 
 
 2571353 
 
 11.7046999 
 
 5.1551367 
 
 .007299270 
 
 138 
 
 19044 
 
 2628072 
 
 11.7473401 
 
 5.1676493 
 
 .007246377 
 
 139 
 
 19321 
 
 2685619 
 
 11.7898261 
 
 5.1801015 
 
 .007194245 
 
 140 
 
 19600 
 
 2744000 
 
 11.8321596 
 
 5.1924941 
 
 .007142857 
 
 141 
 
 19881 
 
 2803221 
 
 11.8743421 
 
 5.2048279 
 
 .007092199 
 
 142 
 
 20164 
 
 2863288 
 
 11. 91637'53 
 
 5.2171034 
 
 .007042254 
 
 143 
 
 20449 
 
 2924207 
 
 11.9582607 
 
 5.2293215 
 
 .006993007 
 
 144 
 
 20736 
 
 2985984 
 
 12.0000000 
 
 5.2414828 
 
 .006944444 
 
 145 
 
 21025 
 
 3018625 
 
 12.0415946 
 
 5.2535879 
 
 .006896552 
 
 146 
 
 21316 
 
 3112136 
 
 12.0830460 
 
 6.2656374 
 
 .006849315 
 
 147 
 
 21609 
 
 3176523 
 
 12.1243557 
 
 5.2776321 
 
 .006802721 
 
 148 
 
 21904 
 
 3241792 
 
 12.1655251 
 
 5.2895725 
 
 .006756757 
 
 149 
 
 22201 
 
 3307949 
 
 12.2065556 
 
 5.3014593 
 
 .006711409 
 
 150 
 
 22500 
 
 3375000 
 
 12.2474487 
 
 5.3132928 
 
 .006666667 
 
 151 
 
 22801 
 
 3442951 
 
 12.2882057 
 
 5.3250740 
 
 .006622517 
 
 152 
 
 23104 
 
 3511808 
 
 12.3288280 
 
 5.3368033 
 
 .006578947 
 
 153 
 
 23409 
 
 3581577 
 
 12.3693169 
 
 5.3484812 
 
 .006535948 
 
 , 154 
 
 23716 
 
 3652264 
 
 12.4096736 
 
 5.3601084 
 
 .006493506 
 
 155 
 
 24025 
 
 3723875 
 
 12.4498996 
 
 5.3716854 
 
 .006451613 
 
 156 
 
 24336 
 
 3796416 
 
 12.48999CO 
 
 5.3832126 
 
 .006410256 
 
 157 
 
 24649 
 
 3869893 
 
 12.5299641 
 
 5 3946907 
 
 .006369427 
 
 158 
 
 24964 
 
 3944312 
 
 12.5698051 
 
 6.4061202 
 
 .006329114 
 
 159 
 
 25281 
 
 4019679 
 
 12.6095203 
 
 5.4175015 
 
 .006289308 
 
 160 
 
 25600 
 
 4096000 
 
 12.6491106 
 
 5.4288353 
 
 .006250000 
 
 161 
 
 25921 
 
 4173281 
 
 12.G885775 
 
 5.4401218 
 
 .COG211180 
 
 162 
 
 26244 
 
 4251528 
 
 12.7279221 
 
 5.4513618 
 
 .006172840 
 
 1G3 
 
 26569 
 
 4330747 
 
 12.7671453 
 
 5.4625556 
 
 .006134969 
 
 164 
 
 26896 
 
 4410944 
 
 12.80G2485 
 
 5.4737037 
 
 .006097561 
 
 165 
 
 27225 
 
 4492125 
 
 12.8452326 
 
 5.4848066 
 
 .006060606 
 
 166 
 
 27556 
 
 4574296 
 
 12.8840987 
 
 5.4958647 
 
 .006024096 
 
 167 
 
 27889 
 
 4657463 
 
 12.9228480 
 
 6.50G8784 
 
 .005988024 
 
 168 
 
 28224 
 
 4741632 
 
 12.9G14814 
 
 5.5178484 
 
 .005952381 
 
 169 
 
 28561 
 
 4826809 
 
 13.0000000 
 
 5.5287748 
 
 .005917160 
 
 170 
 
 28900 
 
 4913000 
 
 13.0384048 
 
 5.5396583 
 
 .005882353 
 
 171 
 
 29241 
 
 5000211 
 
 13.07G6968 
 
 5.5504991 
 
 .005847953 
 
 172 
 
 29584 
 
 5088448 
 
 13.1148770 
 
 5.5612978 
 
 .005813953 
 
 173 
 
 29929 
 
 5177717 
 
 13.1529464 
 
 5.5720546 
 
 .005780347 
 
 174 
 
 30276 
 
 5268024 
 
 13.1909060 
 
 5.5827702 
 
 .005747126 
 
 17'5 
 
 30625 
 
 5359375 
 
 13.2287566 
 
 5.5934447 
 
 .005714286 
 
 176 
 
 30976 
 
 5451776 
 
 13.2664992 
 
 5.G040787 
 
 .005681818 
 
 177 
 
 31329 
 
 5545233 
 
 13.3041347 
 
 5.6146724 
 
 .005649718 
 
 178 
 
 31684 
 
 5639752 
 
 13.3416641 
 
 6.6252263 
 
 .005617978 
 
 179 
 
 32041 
 
 5735339 . 
 
 13.3790882 
 
 5.6357408 
 
 .005586592 
 
 180 
 
 32400 
 
 5832000 
 
 13.4164079 
 
 5.6462162 
 
 .005555556 
 
 181 
 
 32761 
 
 5929741 
 
 13.4536240 
 
 5.6566528 
 
 .005524862 
 
 182 
 
 33124 
 
 6028568 
 
 13.4907376 
 
 5.G670511 
 
 .005494505 
 
 183 
 
 33489 
 
 6128487 
 
 13.5277493 
 
 5.6774114 
 
 .005464481 
 
 184 
 
 33856 
 
 6229504 
 
 13.5646600 
 
 5.6877340 
 
 .005434783 
 
 " y J85 
 
 34225 
 
 6331625 
 
 13.6014705 
 
 5.6980192 
 
 .005405405 
 
 186 
 
 34596 
 
 6434856 
 
 13.6381817 
 
 6.7082675 
 
 .005376344 
 
 180 
 
TABLE VlU.-Continued. 
 
 No. 1 Squares. 
 
 Cubes. 
 
 Square 
 Roots. 
 
 Cube Boots. 
 
 Reciprocals. 
 
 187 
 
 34969 
 
 6539203 
 
 13.6747943 
 
 5.7184791 
 
 .005347594 
 
 188 
 
 35344 
 
 6644672 
 
 13.7113092 
 
 5.7286543 
 
 .005319149 
 
 189 
 
 35721 
 
 6751269 
 
 13.7477271 
 
 5.7387936 
 
 .005291005 
 
 190 
 
 36100 
 
 6859000 
 
 13.7840488 
 
 5.7488971 
 
 .005263158 
 
 191 
 
 36481 
 
 6967871 
 
 13.8202750 
 
 5.7589652 
 
 .005235602 
 
 192 
 
 36864 
 
 7077888 
 
 13.8564065 
 
 5.7689982 
 
 .005208333 
 
 193 
 
 37249 
 
 7189057 
 
 13.8924440 
 
 5.7789966 
 
 .005181347 
 
 194 
 
 37636 
 
 7301384 
 
 13.9283883 
 
 5.7889604 
 
 .005154639 
 
 195 
 
 38025 
 
 7414875 
 
 13.9642400 
 
 5.7988900 
 
 .005128205 
 
 196 
 
 38416 
 
 7529536 
 
 14.0000000 
 
 5.8087857 
 
 .005102041 
 
 197 
 
 38809 
 
 7645373 
 
 14.0356688 
 
 5.8186479 
 
 .005076142 
 
 198 
 
 39204 
 
 7762392 
 
 14.0712473 
 
 5.8284767 
 
 .005050505 
 
 199 
 
 39601 
 
 7880599 
 
 14.1067360 
 
 5.8382725 
 
 .005025126 
 
 200 
 
 40000 
 
 8000000 
 
 14.1421356 
 
 5.8480355 
 
 .005000000 
 
 201 
 
 40401 
 
 8120601 
 
 14.1774469 
 
 5.8577660 
 
 .004975124 
 
 202 
 
 40804 
 
 8242408 
 
 14.2126704 
 
 5.8674643 
 
 .004950495 
 
 203 
 
 41209 
 
 8365427 
 
 14.2478068 
 
 5.8771307 
 
 .004926108 
 
 204 
 
 41616 
 
 8489664 
 
 14.2828569 
 
 5.8867653 
 
 .004901961 
 
 205 
 
 42025 
 
 8615125 
 
 14.3178211 
 
 5.8963685 
 
 .004878049 
 
 206 
 
 42436 
 
 8741816 
 
 14.3527001 
 
 5.9059406 
 
 .004854369 
 
 207 
 
 42849 
 
 8869743 
 
 14.3874946 
 
 5.9154817 
 
 .004880918 
 
 203 
 
 43264 
 
 8998912 
 
 14.4222051 
 
 5.9249921 
 
 .004807692 
 
 209 
 
 43681 
 
 9129329 
 
 14.4568323 
 
 6.9344721 
 
 .004784689 
 
 210 
 
 44100 
 
 9261000 
 
 14.4913767 
 
 5.9439220 
 
 .004761905 
 
 211 
 
 44521 
 
 S393931 
 
 14.5258390 
 
 5.95S3418 
 
 .004739336 
 
 212 
 
 44944 
 
 9528128 
 
 14.5602198 
 
 5.9627320 
 
 .004716981 
 
 213 
 
 45369 
 
 9663597 
 
 14.5945195 
 
 5.9720926 
 
 .004694836 
 
 214 
 
 45796 
 
 9800344 
 
 14.6287388 
 
 5.9814240 
 
 .004672897 
 
 215 
 
 46225 
 
 9938875 
 
 14.6628783 
 
 5.9907264 
 
 .004651163 
 
 216 
 
 46656 
 
 10077696 
 
 14.6969385 
 
 6.0000000 
 
 .004629630 
 
 217 
 
 47089 
 
 10218313 
 
 14.7309199 
 
 6.0092450 
 
 .004608295 
 
 218 
 
 47524 
 
 10360232 
 
 14.7648231 
 
 6.0184617 
 
 .004587156 
 
 219 
 
 47961 
 
 10503459 
 
 14.7986486 
 
 6.0276502 
 
 .004566210 
 
 220 
 
 48400 
 
 10648000 
 
 14.8323970 
 
 6.0368107 
 
 .004545455 
 
 221 
 
 48841 
 
 10793861 
 
 14.8660687 
 
 6.0459435 
 
 .004524887 
 
 222 
 
 49284 
 
 10941048 
 
 14.8996644 
 
 6.0550489 
 
 .004504505 
 
 223 
 
 49729 
 
 11089567 
 
 14.9331845 
 
 6.0641270 
 
 .004484305 
 
 224 
 
 50176 
 
 11239424 
 
 14.9666295 
 
 6.0731779 
 
 .004464286 
 
 225 
 
 50625 
 
 11390625 
 
 15.0000000 
 
 6.C822020 
 
 .004444444 
 
 226 
 
 51076 
 
 11543176 
 
 15.0332964 
 
 6.0911994 
 
 : 004424779 
 
 227 
 
 51529 
 
 11697083 
 
 15.0665192 
 
 6.1001702 
 
 .004405S86 
 
 228 
 
 51984 
 
 11852352 
 
 15.0996689 
 
 6.1091147 
 
 .004385965 
 
 229 
 
 52441 
 
 12008989 
 
 15.1327460 
 
 6.1180332 
 
 .0043G6812 
 
 230 
 
 52900 
 
 12167000 
 
 15.1657509 
 
 6.1269257 
 
 .004347826 
 
 231 
 
 53361 
 
 12326391 
 
 15.1986842 
 
 6.1 357 924 
 
 .004329004 
 
 232 
 
 53824 
 
 12487168 
 
 15.2315462 
 
 6.1446337 
 
 .004310345 
 
 233 
 
 54289 
 
 12649337 
 
 15.2643375 
 
 6.1534495 
 
 .004291845 
 
 234 
 
 54756 
 
 12812904 
 
 15.2970585 
 
 6.1622401 
 
 . 00427 a504 
 
 235 
 
 55225 
 
 12977875 
 
 15.3297097 
 
 6.1710058 
 
 .004255319 
 
 236 
 
 55G96 
 
 13144256 
 
 15.3622915 
 
 6.1797466 
 
 .004237288 
 
 237 
 
 561G9 
 
 13312053 
 
 15.3948043 
 
 6.1884628 
 
 .004219409 
 
 238 
 
 56644 
 
 13481272 
 
 15.4272486 
 
 6.1971544 
 
 .004201681 
 
 239 
 
 57121 
 
 13651919 
 
 15.4596248 
 
 6.2058218 
 
 .004184100 
 
 240 
 
 57600 
 
 13824000 
 
 35.4919334 
 
 6.2144C50 
 
 .004166667 
 
 241 
 
 58081 
 
 13997521 
 
 15.5841747 
 
 6.2230843 
 
 .004149378 
 
 242 
 
 58564 
 
 14172488 
 
 15.5563492 
 
 6.2316797 
 
 .004132231 
 
 243 
 
 9049 
 
 14348907 
 
 15.5884573 
 
 6.2402515 
 
 .004115226 
 
 244 
 
 595:36 
 
 14526784 
 
 15.6204994 
 
 6.2487998 
 
 .004098361 
 
 245 
 
 60025 
 
 14706125 
 
 15.6524758 
 
 6.2573248 
 
 .004081633 
 
 246 
 
 60516 
 
 14886936 
 
 15.6843871 
 
 6.2658266 
 
 .004065041 
 
 247 
 
 61009 
 
 15069223 
 
 15.7162336 
 
 6.2743054 
 
 .004048583 
 
 248 
 
 61504 
 
 15252992 
 
 15.7480157 
 
 6.2827613 
 
 .004032258 
 
 181 
 
TABLE VIII. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Boots. 
 
 Reciprocals. 
 
 249 
 
 62001 
 
 15438249 
 
 15.7797338 
 
 6.2911946 
 
 .004016064 
 
 250 
 
 62500 
 
 15625000 
 
 15.8113883 
 
 6.2996053 
 
 .004000000 
 
 251 
 
 63001 
 
 15813251 
 
 15.8429795 
 
 6.3079935 
 
 .003984064 
 
 252 
 
 63504 
 
 16003008 
 
 15.8745079 
 
 6.3163596 
 
 .003968254 
 
 253 
 
 64009 
 
 16194277 
 
 15.9059737 
 
 6.3247035 
 
 .003952509 
 
 254 
 
 64516 
 
 16387064 
 
 15.9373775 
 
 6.3330256 
 
 .003937008 
 
 255 
 
 65025 
 
 16581375 
 
 15.9687194 
 
 6.3413257 
 
 .003921509 
 
 256 
 
 65536 
 
 16777216 
 
 16.0000000 
 
 6.3496042 
 
 .003906250 
 
 257 
 
 615049 
 
 16974593 
 
 16.0312195 
 
 6.3578611 
 
 .003891051 
 
 258 
 
 66564 
 
 17173512 
 
 16.0623784 
 
 6.3600908 
 
 . 00387590 J 
 
 259 
 
 67081 
 
 17373979 
 
 16.09347'G9 
 
 6.3743111 
 
 .00:3861004 
 
 260 
 
 67600 
 
 17576000 
 
 16.1245155 
 
 6.3825043 
 
 .003846154 
 
 261 
 
 68121 
 
 17779581 
 
 16.1554944 
 
 6.3906765 
 
 .003831418 
 
 262 
 
 68644 
 
 17984728 
 
 16.1864141 
 
 6.3988279 
 
 .003816794 
 
 263 
 
 69169 
 
 18191447 
 
 16.2172747 
 
 6.4069585 
 
 .003802281 
 
 264 
 
 69696 
 
 18399744 
 
 16.2480768 
 
 6.4150G87 
 
 .003787879 
 
 265 
 
 70225 
 
 18609625 
 
 16.2788206 
 
 6.4231583 
 
 .003773585 
 
 266 
 
 70756 
 
 18821096 
 
 16.3095064 
 
 6.4312276 
 
 .003759398 
 
 267 
 
 71289 
 
 190:34163 
 
 16.3401346 
 
 6.4392767 
 
 .003745318 
 
 268 
 
 718.24 
 
 19248832 
 
 16.3707055 
 
 6.4473057 
 
 .003731343 
 
 269 
 
 72361 
 
 19465109 
 
 16.4012195 
 
 C. 4553148 
 
 .003717472 
 
 270 
 
 72900 
 
 19683000 
 
 16.4316767 
 
 6.4633041 
 
 .003703704 
 
 271 
 
 73441 
 
 19902511 
 
 16.4620776 
 
 6.4712736 
 
 .003090037 
 
 272 
 
 73984 
 
 20123648 
 
 16.4924225 
 
 6.4792236 
 
 .003676471 
 
 273 
 
 74523 
 
 20346417 
 
 16.5227116 
 
 6.4871541 
 
 .003663004 
 
 274 
 
 75076 
 
 20570824 
 
 16.5529154 
 
 6.4950653 
 
 .003649035 
 
 275 
 
 75625 
 
 20796875 
 
 16.5831240 
 
 6.5029572 
 
 .003636304 
 
 276 
 
 76176 
 
 21024576 
 
 16.6132477 
 
 6.5108300 
 
 .003623183 
 
 277 
 
 76729 
 
 21253933 
 
 16.6433170 
 
 6.5186839 
 
 .003610108 
 
 278 
 
 77284 
 
 21484952 
 
 16.6733320 
 
 6.5265189 
 
 .003597122 
 
 279 
 
 77841 
 
 21717639 
 
 16.7032931 
 
 6.5343351 
 
 .003584229 
 
 280 
 
 78400 
 
 21952000 
 
 16.7332005 
 
 6.5421326 
 
 .003571429 
 
 281 
 
 78961 
 
 22188041 
 
 16.7630546 
 
 6.5499116 
 
 .003558719 
 
 282 
 
 79524 
 
 22425703 
 
 16.7928556 
 
 6.5576722 
 
 .003546099 
 
 283 
 
 80089 
 
 22G65187 
 
 16.8226038 
 
 6.5654144 
 
 .003533509 
 
 284 
 
 80656 
 
 22906304 
 
 16.8522995 
 
 6.5731385 
 
 .003521127 
 
 285 
 
 81225 
 
 23149125 
 
 16.8819430 
 
 6.5808443 
 
 .003508772 
 
 286 
 
 81796 
 
 23393656 
 
 16.9115345 
 
 6.5885323 
 
 .003496503 
 
 287 
 
 82369 
 
 23639903 
 
 16.9410743 
 
 6.5962023 
 
 .003484321 
 
 288 
 
 82944 
 
 23887872 
 
 16.9705G27 
 
 6.G038545 
 
 .00347222-3 
 
 289 
 
 83521 
 
 241375(39 
 
 17.0000000 
 
 6.0114890 
 
 .003460208 
 
 290 
 
 84100 
 
 24389000 
 
 17.0293864 
 
 6.6191060 
 
 .003448276 
 
 291 
 
 84081 
 
 24642171 
 
 17.0587221 
 
 6.G267054 
 
 .003436426 
 
 292 
 
 85264 
 
 24897088 
 
 17.0880075 
 
 6.6342874 
 
 .003424658 
 
 293 
 
 85S49 
 
 25153757 
 
 17.1172428 
 
 6.6418522 
 
 .003412909 
 
 294 
 
 86436 
 
 25412184 
 
 17.1464282 
 
 6.6493998 
 
 .003401301 
 
 295 
 
 87025 
 
 25672375 
 
 17.1755640 
 
 6.6569302 
 
 .003389831 
 
 296 
 
 87616 
 
 25934336 
 
 17.2046505 
 
 6.G644437 
 
 .003378378 
 
 297 
 
 88209 
 
 20198073 
 
 17.2336879 
 
 6.G719403 
 
 .003367003 
 
 298 
 
 88804 
 
 26403592 
 
 17.2626765 
 
 6.G794200 
 
 .003355705 
 
 299 
 
 89401 
 
 26730899 
 
 17.2916165 
 
 6.6868831 
 
 .003344482 
 
 800 
 
 90000 
 
 27000000 
 
 17.3205081 
 
 6.6943295 
 
 .00333.3333 
 
 301 
 
 90601 
 
 27270901 
 
 17.3493510 
 
 6.7017593 
 
 .00:3322259 
 
 302 
 
 91204 
 
 27543608 
 
 17.3781472 
 
 6.7091729 
 
 .003311258 
 
 303 
 
 91809 
 
 27818127 
 
 17.4068952 
 
 6.7165700 
 
 .003300330 
 
 304 
 
 92416 
 
 28094464 
 
 17.4355958 
 
 6.7239508 
 
 .003289474 
 
 305 
 
 93025 
 
 28372625 
 
 17.4642492 
 
 6.7313155 
 
 .003278089 
 
 306 
 
 93636 
 
 28652616 
 
 17.4928557 
 
 6.7386641 
 
 .003267974 
 
 307 
 
 94249 
 
 28934443 
 
 17.5214155 
 
 6.7459967 
 
 .003257329 
 
 308 
 
 94864 
 
 29218112 
 
 17.5499288 
 
 6.7533134 
 
 .003246753 
 
 309 
 
 95481 
 
 29503G29 
 
 17 5783958 
 
 6.7606143 
 
 .003236246 
 
 310 
 
 96100 
 
 29791000 
 
 17! 6068169 
 
 6.7G78995 
 
 .003225806 
 
 182 
 
TABLE VIII. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Koots. 
 
 Cube Boots. 
 
 Reciprocals. 
 
 311 
 
 96721 
 
 30080231 
 
 17.6351921 
 
 6.7751690 
 
 .003215434 
 
 313 
 
 97344 
 
 30371328 
 
 17.6635217 
 
 6.7824229 
 
 .003205128 
 
 313 
 
 97969 
 
 30664297 
 
 17.6918060 
 
 6.7896613 
 
 .003194888 
 
 314 
 
 98596 
 
 30959144 
 
 17.7200451 
 
 6.7968844 
 
 .003184713 
 
 315 
 
 99225 
 
 31255875 
 
 17.7482393 
 
 6.8040921 
 
 .003174603 
 
 316 
 
 99856 
 
 31554496 
 
 17.7763888 
 
 6.8112847 
 
 .003164557 
 
 317 
 
 100489 
 
 31855013 
 
 17.8044938 
 
 6.8184620 
 
 .003154574 
 
 318 
 
 101124 
 
 32157432 
 
 17.8325545 
 
 6.8256242 
 
 .003144654 
 
 319 
 
 101761 
 
 32461759 
 
 17.8605711 
 
 6.8327714 
 
 .003134796 
 
 320 
 
 102400 
 
 32768000 
 
 17.8885438 
 
 6.8399037 
 
 .003125000 
 
 321 
 
 103U41 
 
 33076161 
 
 17.9164729 
 
 6.8470213 
 
 .003115265 
 
 322 
 
 103684 
 
 33386248 
 
 17.9443584 
 
 6.8541240 
 
 .003105590 
 
 323 
 
 104329 
 
 33698267 
 
 17.9722008 
 
 6.8612120 
 
 .003095975 
 
 324 
 
 104976 
 
 34012224 
 
 18.0000000 
 
 6.8682855 
 
 .003086420 
 
 325 
 
 105625 
 
 34328125 
 
 18.0277564 
 
 6.8753443 
 
 .003076923 
 
 326 
 
 106276 
 
 34645976 
 
 18.0554701 
 
 6.8823888 
 
 .003067485 
 
 327 
 
 106929 
 
 34965783 
 
 18.0831413 
 
 6.8894188 
 
 .003058104 
 
 328 
 
 107584 
 
 35287552 
 
 18.1107703 
 
 6.8964345 
 
 .003048780 
 
 329 
 
 108241 
 
 35011289 
 
 18.1383571 
 
 6.9034359 
 
 .003039514 
 
 330 
 
 108900 
 
 35937000 
 
 18.1659021 
 
 6.9104232 
 
 .003030303 
 
 331 
 
 1005(51 
 
 36264691 
 
 18.1934054 
 
 6.917'3964 
 
 .003021148 
 
 332 
 
 110224 
 
 36594368 
 
 18.2208672 
 
 6.9243556 
 
 .003012048 
 
 333 
 
 110889 
 
 361)26037 
 
 18.2482876 
 
 6.9313008 
 
 .003003003 
 
 334 
 
 111556 
 
 37259704 
 
 18.2756669 
 
 6.9382321 
 
 .002994012 
 
 335 
 
 112225 
 
 37595375 
 
 18.3030052 
 
 6.9451496 
 
 .002985075 
 
 336 
 
 112896 
 
 37933056 
 
 18.3303028 
 
 6.9520533 
 
 .002976190 
 
 337 
 
 113569 
 
 38272753 
 
 18.3575598 
 
 6.9589434 
 
 .002967359 
 
 338 
 
 114244 
 
 38614472 
 
 18.3847763 
 
 6.9658198 
 
 .002958580 
 
 339 
 
 114921 
 
 38958219 
 
 18.4119526 
 
 6.9726826 
 
 .002949853 
 
 340 
 
 115600 
 
 39304000 
 
 18.4390889 
 
 6.9795321 
 
 .002941176 
 
 341 
 
 116281 
 
 39651821 
 
 18.4661853 
 
 6.9863681 
 
 .002932551 
 
 342 
 
 116964 
 
 40001688 
 
 18.4932420 
 
 6.9931906 
 
 .002923977 
 
 343 
 
 117649 
 
 403:53607 
 
 18.5202592 
 
 7.0000000 
 
 .002915452 
 
 344 
 
 118336 
 
 40707584 
 
 18.5472370 
 
 7.0067962 
 
 .002906977 
 
 345 
 
 119025 
 
 41063G25 
 
 18.5741756 
 
 7.0135791 
 
 .002898551 
 
 346 
 
 119716 
 
 41421736 
 
 18.6010752 
 
 7.0203490 
 
 .002890173 
 
 347 
 
 120409 
 
 41781923 
 
 18.6279360 
 
 7.0271058 
 
 .002881844 
 
 348 
 
 121104 
 
 42144192 
 
 18.6547581 
 
 7.0338497 
 
 .002873563 
 
 349 
 
 121801 
 
 42508549 
 
 18.6815417 
 
 7.0405806 
 
 .002865330 
 
 350 
 
 122500 
 
 42875000 
 
 18.7082869 
 
 7.0472987 
 
 .002857143 
 
 351 
 
 123201 
 
 43243551 
 
 18.7349940 
 
 7.0540041 
 
 .002849003 
 
 352 
 
 123904 
 
 43614208 
 
 18.7616630 
 
 7.0006967 
 
 .002840909 
 
 353 
 
 124G09 
 
 43986977 
 
 18.7882942 
 
 7.0673767 
 
 .002832861 
 
 354 
 
 125316 
 
 44361864 
 
 18.8148877 
 
 7.0740440 
 
 .002824859 
 
 355 
 
 126025 
 
 44738875 
 
 18.8414437 
 
 7.0806988 
 
 .002816901 
 
 356 
 
 126736 
 
 45118016 
 
 18.8679623 
 
 7.0873411 
 
 .002808989 
 
 357 
 
 127449 
 
 45499293 
 
 18.8944436 
 
 7.0939709 
 
 .002801120 
 
 358 
 
 1281 G4 
 
 45882712 
 
 18.9208879 
 
 7.1005885 
 
 .002793296 
 
 359 
 
 128881 
 
 46268279 
 
 18.9472953 
 
 7.1071937 
 
 .002785515 
 
 SCO 
 
 129600 
 
 46656000 
 
 18.9736660 
 
 7.1137866 
 
 .002777778 
 
 361 
 
 130321 
 
 47045881 
 
 19.0000000 
 
 7.1203674 
 
 .002770083 
 
 362 
 
 131044 
 
 47437928 
 
 19.0262976 
 
 7.1269360 
 
 .002762431 
 
 363 
 
 131769 
 
 47832147 
 
 19.0525589 
 
 7.1334925 
 
 .002754821 
 
 364 
 
 132496 
 
 48228544 
 
 19.0787840 
 
 7.1400370 
 
 .002747253 
 
 365 
 
 133225 
 
 48627125 
 
 19.1049732 
 
 7.1465695 
 
 .002739726 
 
 366 
 
 133956 
 
 49027896 
 
 19.1311265 
 
 7.1530901 
 
 .002732240 
 
 367 
 
 134689 
 
 49430863 
 
 19.1572441 
 
 7.1595988 
 
 .002724796 
 
 368 
 
 135424 
 
 49836032 
 
 19.1833261 
 
 7.1660957 
 
 .002717391 
 
 369 
 
 136161 
 
 50243409 
 
 19.2093727 
 
 7.1725809 
 
 .002710027 
 
 370 
 
 136900 
 
 60653000 
 
 19.2353841 
 
 7.1790544 
 
 .002702703 
 
 37} 
 
 137641 
 
 51064811 
 
 19.2613003 
 
 7.1855162 
 
 .002695418 
 
 372 
 
 138384 
 
 51478848 
 
 19.2873015 
 
 7.1919663 
 
 .002688172 
 
 183 
 
TABLE VIII. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 373 
 
 139129 
 
 51895117 
 
 19.3132079 
 
 7.1984050 
 
 .002680965 
 
 374 
 
 139876 
 
 52313G24 
 
 19.3390796 
 
 7.2048322 
 
 .002673797 
 
 375 
 
 140625 
 
 52734375 
 
 19.3649167 
 
 7.2112479 
 
 .002666667 
 
 376 
 
 141376 
 
 53157376 
 
 19.3907194 
 
 7.2176522 
 
 .002659574 
 
 377 
 
 142129 
 
 53582633 
 
 19.4104878 
 
 7.2240450 
 
 .002652520 
 
 378 
 
 142884 
 
 54010152 
 
 19.4422221 
 
 7.2304268 
 
 .002645503 
 
 379 
 
 143641 
 
 54439939 
 
 19.4679223 
 
 7.2367972 
 
 .002638522 
 
 380 
 
 144400 
 
 54872000 
 
 19.4935887 
 
 7.2431565 
 
 .002631579 
 
 381 
 
 145161 
 
 55306341 
 
 19.5192213 
 
 7.2495045 
 
 .002624672 
 
 382 
 
 145924 
 
 55742968 
 
 19.5448203 
 
 7.2558415 
 
 .002617801 
 
 383 
 
 146689 
 
 56181887 
 
 19.5703858 
 
 7.2621675 
 
 .002610966 
 
 884 
 
 147456 
 
 56623104 
 
 19.5959179 
 
 7.2684824 
 
 .002604167 
 
 385 
 
 148225 
 
 57066625 
 
 19.6214169 
 
 7.2747864 
 
 .002597403 
 
 386 
 
 148996 
 
 57512456 
 
 19.6468827 
 
 7.2810794 
 
 .002590674 
 
 387 
 
 149769 
 
 57960603 
 
 19.6723156 
 
 7.2873617 
 
 .002583979 
 
 388 
 
 150544 
 
 58411072 
 
 19.6977156 
 
 7.2936330 
 
 .002577320 
 
 389 
 
 151321 
 
 58863869 
 
 19.7230829 
 
 7.2998936 
 
 .002570694 
 
 390 
 
 152100 
 
 59319000 
 
 19.7484177 
 
 7.3061436 
 
 .002564103 
 
 391 
 
 152881 
 
 59776471 
 
 19.7737199 
 
 7.3123828 
 
 .002557545 
 
 392 
 
 153664 
 
 60236288 
 
 19.7989899 
 
 7.3186114 
 
 .002551020 
 
 393 
 
 154449 
 
 60698457 
 
 19.8242276 
 
 7.3248295 
 
 .002544529 
 
 394 
 
 155.236 
 
 61162984 
 
 19.8494332 
 
 7.3310369 
 
 .002538071 
 
 395 
 
 156025 
 
 61629875 
 
 19.8746069 
 
 7.3372339 
 
 .002531646 
 
 396 
 
 156816 
 
 62099136 
 
 19.8997487 
 
 7.3434205 
 
 .002525253 
 
 397 
 
 157609 
 
 62570773 
 
 19.9248588 
 
 7.3495966 
 
 .002518892 
 
 398 
 
 158404 
 
 63044792 
 
 19.9499373 
 
 7.3557624 
 
 .002512563 
 
 399 
 
 159201 
 
 63521199 
 
 19.9749844 
 
 7.3619178 
 
 .002506266 
 
 400 
 
 160000 
 
 64000000 
 
 20.0000000 
 
 7.3680630 
 
 .002500000 
 
 401 
 
 160801 
 
 64481201 
 
 20.0249844 
 
 7.3741979 
 
 .002493766 
 
 402 
 
 161604 
 
 64964808 
 
 20.0499377 
 
 7.3803227 
 
 .002487562 
 
 403 
 
 162409 
 
 65450827 
 
 20.0748599 
 
 7.S864373 
 
 .002481390 
 
 404 
 
 163216 
 
 65939264 
 
 20.0997512 
 
 7.3925418 
 
 .002475248 
 
 405 
 
 164025 
 
 66430125 
 
 20.1246118 
 
 f. 3986363 
 
 .002469136 
 
 406 
 
 164836 
 
 66923416 
 
 20.1494417 
 
 7.4047206 
 
 .002463054 
 
 407 
 
 165649 
 
 67419143 
 
 20.1742410 
 
 7.4107950 
 
 .002457002 
 
 408 
 
 166464 
 
 67917312 
 
 20.1990099 
 
 7.4168595 
 
 .002450980 
 
 409 
 
 167281 
 
 68417929 
 
 20.2237484 
 
 7.4229142 
 
 .002444988 
 
 410 
 
 168100 
 
 68921000 
 
 20.2484567 
 
 7.4289589 
 
 .002439024 
 
 411 
 
 168921 
 
 69426531 
 
 20.2731349 
 
 7.4349938 
 
 .002433090 
 
 412 
 
 169744 
 
 69934528 
 
 20.2977831 
 
 7.4410189 
 
 .002427184 
 
 413 
 
 170569 
 
 70444997 
 
 20.3224014 
 
 7.4470342 
 
 .002421308 
 
 414 
 
 171396 
 
 70957944 
 
 20.3469899 
 
 7.4530399 
 
 .002415459 
 
 415 
 
 172225 
 
 71473375 
 
 20.3715488 
 
 7.4590359 
 
 .002409639 
 
 416 
 
 173056 
 
 71991296 
 
 20.3960781 
 
 7.4650223 
 
 .002403846 
 
 417. 
 
 173889 
 
 72511713 
 
 20.4205779 
 
 7.4709991 
 
 .002398082 
 
 418 
 
 174724 
 
 73034632 
 
 20.4450483 
 
 7.4769664 
 
 .002392344 
 
 419 
 
 17'5561 
 
 73560059 
 
 20.4694895 
 
 7.4829242 
 
 .002386635 
 
 420 
 
 176400 
 
 74088000 
 
 20.4939015 
 
 7.4888724 
 
 .002380952 
 
 421 
 
 177241 
 
 74618461 
 
 20.5182845 
 
 7.4948113 
 
 .002375297 
 
 422 
 
 178084 
 
 75151448 
 
 20.5426386 
 
 7.5007406 
 
 .002369668 
 
 423 
 
 178929 
 
 75686967 
 
 20.5669638 
 
 7.5066607 
 
 .002364066 
 
 424 
 
 179776 
 
 76225024 
 
 20 5912603 
 
 7.5125715 
 
 .002358491 
 
 425 
 
 180625 
 
 76765625 
 
 20.6155281 
 
 7.5184730 
 
 .002352941 
 
 426 
 
 181476 
 
 77308776 
 
 20.6397674 
 
 7.5243652 
 
 .002347418 
 
 427 
 
 182329 
 
 77854483 
 
 20.6639783 
 
 7.5302482 
 
 .002341920 
 
 428 
 
 183184 
 
 78402752 
 
 20.6881609 
 
 7.5361221 
 
 .002336449 
 
 429 
 
 184041 
 
 78953589 
 
 20.7123152 
 
 7.5419867 
 
 .002331002 
 
 430 
 
 184900 
 
 79507000 
 
 20.7364414 
 
 7.5478423 
 
 .002325581 
 
 431 
 
 185761 
 
 800G2991 
 
 80:7805895 
 
 7.5536888 
 
 .002320186 
 
 432 
 
 18G624 
 
 80621568 
 
 20.7846097 
 
 7.5595263 
 
 .002314815 
 
 433 
 
 187489 
 
 81182737 
 
 20.8086520 
 
 7.5653548 
 
 .002309469 
 
 434 
 
 188356 
 
 81746504 
 
 20.8326667 
 
 7.5711743 
 
 .002304147 
 
 184 
 
TABLE VIII. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Koots. 
 
 Reciprocals. 
 
 435 
 
 189225 
 
 82312875 
 
 20.8566536 
 
 7.5769849 
 
 .002298851 
 
 436 
 
 190096 
 
 82881856 
 
 20.88061:30 
 
 7.5827865 
 
 .002293578 
 
 437 
 
 190969 
 
 83453453 
 
 20.9045450 
 
 7.5885793 
 
 .002288330 
 
 438 
 
 191844 
 
 84027672 
 
 20.9284495 
 
 7.5943033 
 
 .002283105 
 
 439 
 
 192721 
 
 84604519 
 
 20.9523268 
 
 7.6001385 
 
 .002277904 
 
 440 
 
 193600 
 
 85184000 
 
 20.9761770 
 
 7.6059049 
 
 .002272727 
 
 441 
 
 194481 
 
 85760121 
 
 21.0000000 
 
 7.6116020 
 
 .002267574 
 
 442 
 
 195304 
 
 86350888 
 
 21.0237900 
 
 7.6174116 
 
 .002262443 
 
 443 
 
 196249 
 
 86938307 
 
 21.0475052 
 
 7.6231519 
 
 .002257336 
 
 444 
 
 197136 
 
 87528384 
 
 21.0713075 
 
 7.6288837 
 
 .002252252 
 
 445 
 
 198025 
 
 88121125 
 
 21.0950231 
 
 7.6346067 
 
 .002247191 
 
 446 
 
 198916 
 
 88716536 
 
 21.1187121 
 
 7.6403213 
 
 .002242152 
 
 447 
 
 199809 
 
 89314023 
 
 21.1423745 
 
 7.6460272 
 
 .002237136 
 
 448 
 
 200704 
 
 89915392 
 
 21.1600105 
 
 7.6517247 
 
 .002232143 
 
 449 
 
 201601 
 
 90518849 
 
 21.1896201 
 
 7.6574133 
 
 .002227171 
 
 450 
 
 202500 
 
 91125000 
 
 21.2132034 
 
 7.6630943 
 
 .002222222 
 
 451 
 
 203401 
 
 91733851 
 
 21.2307600 
 
 7.0687665 
 
 .002217295 
 
 452 
 
 204304 
 
 92345408 
 
 21.2602916 
 
 7.6744303 
 
 .002212389 
 
 453 
 
 205209 
 
 92959677 
 
 21.2337967 
 
 7.0800857 
 
 .002207506 
 
 454 
 
 206116 
 
 93576664 
 
 21.3072758 
 
 7.6857328 
 
 .002202643 
 
 455 
 
 207025 
 
 94196375 
 
 21.3307290 
 
 7.6913717 
 
 .002197802 
 
 456 
 
 207936 
 
 94818816 
 
 21.a>41565 
 
 7.6970023 
 
 .002192982 
 
 457 
 
 208849 
 
 95443993 
 
 21.3775583 
 
 7.7026246 
 
 .002188184 
 
 458 
 
 209764 
 
 96071912 
 
 21.4009346 
 
 7.7082388 
 
 .C02183406 
 
 459 
 
 210081 
 
 96702579 
 
 21.4242853 
 
 7.7138448 
 
 .002178649 
 
 460 
 
 211600 
 
 97336000 
 
 21.4476106 
 
 7.7194426 
 
 .002173913 
 
 461 
 
 212521 
 
 97972181 
 
 21.4709106 
 
 7.7250325 
 
 .002169197 
 
 402 
 
 213444 
 
 98611128 
 
 21.4941853 
 
 7.7306141 
 
 .002164502 
 
 463 
 
 214369 
 
 99252847 
 
 21.5174348 
 
 7.7361877 
 
 .002159827 
 
 464 
 
 215296 
 
 99897344 
 
 21.5400592 
 
 7.7417532 
 
 .002155172 
 
 465 
 
 216225 
 
 100544625 
 
 21.5638587 
 
 7.747'3109 
 
 .002150538 
 
 466 
 
 217156 
 
 101194696 
 
 21.5870331 
 
 7.7528606 
 
 .002145923 
 
 467 
 
 218089 
 
 101847563 
 
 21.6101828 
 
 7.7584023 
 
 .002141328 
 
 468 
 
 219024 
 
 102503232 
 
 21.6333077 
 
 7.7639361 
 
 .002136752 
 
 469 
 
 219961 
 
 103161709 
 
 21.6564078 
 
 7.7694620 
 
 .002132196 
 
 470 
 
 220900 
 
 103823000 
 
 21.6794834 
 
 7.7749801 
 
 .002127660 
 
 471 
 
 221841 
 
 104487111 
 
 21.7025344 
 
 7.7804904 
 
 .032123142 
 
 472 
 
 222784 
 
 105154048 
 
 21.7255610 
 
 7.7859928 
 
 .C021 18644 
 
 473 
 
 223729 
 
 105823817 
 
 21.7485632 
 
 7.7914875 
 
 .002114165 
 
 474 
 
 224676 
 
 106496424 
 
 21.7715411 
 
 7.7969745 
 
 .002109705 
 
 475 
 
 225025 
 
 107171875 
 
 21.7944947 
 
 7.80.24538 
 
 .002105263 
 
 476 
 
 220576 
 
 107850176 
 
 21.8174242 
 
 7.8079254 
 
 .002100840 
 
 477 
 
 227529 
 
 108531333 
 
 21.8403297 
 
 7.8ia3892 
 
 .002096436 
 
 478 
 
 223484 
 
 109215352 
 
 21 8632111 
 
 7. C 188456 
 
 .002092050 
 
 479 
 
 229441 
 
 109902239 
 
 21.8860086 
 
 7.8242942 
 
 .002087683 
 
 480 
 
 230400 
 
 110592000 
 
 21.9089023 
 
 7.8297353 
 
 .002088333 
 
 481 
 
 231301 
 
 111284641 
 
 21.9317122 
 
 7.8:351088 
 
 .002079002 
 
 482 
 
 232:324 
 
 111980168 
 
 21.9544984 
 
 7.8405949 
 
 .002074689 
 
 483 
 
 233289 
 
 112678587 
 
 21.9772610 
 
 7.8460134 
 
 .002070393 
 
 484 
 
 234256 
 
 113379904 
 
 22.0000000 
 
 7.8514244 
 
 .002066116 
 
 485 
 
 235225 
 
 114084125 
 
 22.0227155 
 
 7.85682S1 
 
 .002061856 
 
 486 
 
 236196 
 
 114791256 
 
 22.0454077 
 
 7.8622242 
 
 .002057613 
 
 487 
 
 237169 
 
 115501303 
 
 22.0680765 
 
 7.8670130 
 
 .002053388 
 
 488 
 
 238144 
 
 116214272 
 
 22.0907220 
 
 7.8729944 
 
 .002049180 
 
 489 
 
 239121 
 
 116930169 
 
 22.1133444 
 
 7.8783684 
 
 .002044990 
 
 490 
 
 240100 
 
 117649000 
 
 22.1359436 
 
 7.8837352 
 
 .002040816 
 
 491 
 
 241081 
 
 Iia370771 
 
 22.1585198 
 
 7.8890916 
 
 .002036660 
 
 492 
 
 242064 
 
 119095488 
 
 22.1810730 
 
 7.8944-463 
 
 .002032520 
 
 493 
 
 243049 
 
 119823157 
 
 22.2036033 
 
 7.8997917 
 
 .002028398 
 
 494 
 
 244036 
 
 120553784 
 
 22.2261108 
 
 7.9051294 
 
 .002024291 
 
 495 
 
 245025 
 
 121287375 
 
 22.24a5955 
 
 7.9104599 
 
 .002020202 
 
 496 | 240016 
 
 122023936 
 
 22.2710575 | 7.9157832 
 
 .002016129 
 
 185 
 
TABLE VITL-Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Boots. 
 
 Reciprocals. 
 
 497 
 
 ' 247009 
 
 122763473 
 
 22.2934968 
 
 7.9210994 
 
 .002012072 
 
 493 
 
 248004 
 
 123505992 
 
 22.3159136 
 
 7.9264085 
 
 .002008032 
 
 499 
 
 249001 
 
 13U51499 
 
 22.3383079 
 
 7.9317104 
 
 . 003004008 
 
 500 
 
 250000 
 
 125000000 
 
 22.3606798 
 
 7.9370053 
 
 .002000000 
 
 501 
 
 251001 
 
 125751501 
 
 22.3830293 
 
 7.9422931 
 
 .001996003 
 
 502 
 
 252. .04 
 
 126506008 
 
 22.4053565 
 
 7.9475739 
 
 .001992032 
 
 503 
 
 253009 
 
 12?'26352r 
 
 22.4276615 
 
 7.9528477 
 
 .001988072 
 
 504 
 
 254016 
 
 128024064 
 
 22.4499443 
 
 7.9581144 
 
 .001984127 
 
 505 
 
 255025 
 
 128787625 
 
 22.4722051 
 
 7.9633743 
 
 .001980198 
 
 506 
 
 256036 
 
 129554216 
 
 22.4944438 
 
 7.9686271 
 
 .001976285 
 
 507 
 
 257049 
 
 130323843 
 
 22.5166605 
 
 7.9738731 
 
 .001972387 
 
 508 
 
 258064 
 
 131096512 
 
 22.5388553 
 
 7.9791122 
 
 .001968504 
 
 509 
 
 259081 
 
 131872229 
 
 22.5610283 
 
 7.9843444 
 
 .001964037 
 
 510 
 
 260100 
 
 132651000 
 
 22.5&31796 
 
 7.9895697 
 
 .001900781 
 
 511 
 
 261121 
 
 133432831 
 
 22.6053091 
 
 7.9947883 
 
 .001950947 
 
 513 
 
 262144 
 
 134217728 
 
 22.6274170 
 
 8.0000000 
 
 .001953125 
 
 513 
 
 263169 
 
 135005697 
 
 22.6495033 
 
 8.0052049 
 
 .001949318 
 
 514 
 
 261196 
 
 135796744 
 
 22.6715681 
 
 8.0104032 
 
 .001945525 
 
 515 
 
 265225 
 
 136590875 
 
 22.6936114 
 
 8.0155946 
 
 .0019417-33 
 
 516 
 
 266256 
 
 137388096 
 
 22.7156334 
 
 8.0207794 
 
 .001937984 
 
 517 
 
 267289 
 
 138188413 
 
 22.73763-10 
 
 8.0259574 
 
 .001934236 
 
 518 
 
 268324 
 
 138991832 
 
 22.7596134 
 
 8.0311287 
 
 .001930502 
 
 510 
 
 269361 
 
 139798359 
 
 22.7815715 
 
 8.0362935 
 
 .001926782 
 
 520 
 
 270400 
 
 140608000 
 
 22.8035085 
 
 8.0414515 
 
 .001923077 
 
 521 
 
 271441 
 
 141420761 
 
 22.8254244 
 
 8.04G6030 
 
 .001919336 
 
 522 
 
 272484 
 
 142236648 
 
 22.8473193 
 
 8.0517479 
 
 .001915709 
 
 523 
 
 273529 
 
 143055667 
 
 22.8691933 
 
 8.0368862 
 
 .001912046 
 
 524 
 
 274576 
 
 143877824 
 
 22.8910463 
 
 8.0620180 
 
 .001908397 
 
 525 
 
 275625 
 
 144703125 
 
 22.9128785 
 
 8.0671432 
 
 .001904763 
 
 526 
 
 276676 
 
 145531576 
 
 22.9346899 
 
 8.0722620 
 
 .001901141 
 
 527 
 
 277729 
 
 146363183 
 
 22.9564806 
 
 8.0773743 
 
 .001897533 
 
 528 
 
 278784 
 
 147197952 
 
 22.9782506 
 
 8.0824800 
 
 .001893939 
 
 529 
 
 279841 
 
 148035889 
 
 23.0000000 
 
 8.0875794 
 
 .001890359 
 
 530 
 
 280900 
 
 148877000 
 
 23.0217289 
 
 8.0926723 
 
 .001886792 
 
 531 
 
 281961 
 
 149721231 
 
 23.043437'2 
 
 8.0977589 
 
 .001883239 
 
 532 
 
 283024 
 
 150568763 
 
 23.0051252 
 
 8.1028390 
 
 .001879699 
 
 533 
 
 284089 
 
 151419437 
 
 23.0867928 
 
 8.1079128 
 
 .001876173 
 
 534 
 
 285156 
 
 152273304 
 
 23.1084400 
 
 8.1129803 
 
 .001872659 
 
 535 
 
 286225 
 
 153130375 
 
 23.1300670 
 
 8.1180414 
 
 .001869159 
 
 536 
 
 2S7296 
 
 153990656 
 
 23.1516738 
 
 8.1230962 
 
 .001865672 
 
 537 
 
 288369 
 
 154854153 
 
 23.1732605 
 
 8.1281447 
 
 .C01862197 
 
 538 
 
 280444 
 
 155720872 
 
 23.1948270 
 
 8.1331870 
 
 .001858736 
 
 539 
 
 290521 
 
 156590819 
 
 23.2163735 
 
 8.1382230 
 
 .001855288 
 
 540 
 
 291000 
 
 157464000 
 
 23.2379001 
 
 8.1432529 
 
 .001851852 
 
 541 
 
 292681 
 
 158340421 
 
 23.2594067 
 
 8.1482765 
 
 .001848423 
 
 542 
 
 293764 
 
 159220088 
 
 23.2808935 
 
 8.1532939 
 
 .001845018 
 
 543 
 
 294S49 
 
 16:103007 
 
 23.3023604 
 
 8.1583051 
 
 .001841621 
 
 544 
 
 295936 
 
 160989184 
 
 23.3238076 
 
 8.1633102 
 
 .001838235 
 
 545 
 
 297035 
 
 161878625 
 
 23.3452351 
 
 8.1683092 
 
 .001834862 
 
 546 
 
 298116 
 
 162771336 
 
 23.3666429 
 
 8.1733020 
 
 .001831502 
 
 547 
 
 299209 
 
 163667323 
 
 23.3880311 
 
 8.1782888 
 
 .001828154 
 
 518 
 
 300304 
 
 164566592 
 
 23.4093998 
 
 8.1832695 
 
 .001824818 
 
 549 
 
 301401 
 
 165469149 
 
 23.4307490 
 
 8.1882441 
 
 .001821494 
 
 550 
 
 302500 
 
 166375000 
 
 23.4520788 
 
 8.1932127 
 
 .001818182 
 
 551 
 
 303601 
 
 167284151 
 
 23.4733892 
 
 8.1981753 
 
 .001814882 
 
 552 
 
 304704 
 
 168196608 
 
 23.4946802 
 
 8.2031319 
 
 .001811594 
 
 553 
 
 305809 
 
 169112377 
 
 23.5159520 
 
 8.2080825 
 
 .001808318 
 
 554 
 
 306916 
 
 170031464 
 
 23.5372046 
 
 8.2130271 
 
 .001805054 
 
 555 
 
 308025 
 
 170953875 
 
 23.5584380 
 
 8.2179657 
 
 .001801803 
 
 556 
 
 309136 
 
 171879616 
 
 23.5796522 
 
 8.2228985 
 
 .001798561 
 
 557 
 
 310249 
 
 172808693 
 
 23.600r>i74 
 
 8.2278254 
 
 .001795332 
 
 558 
 
 311364 
 
 173741112 
 
 83.0230336 
 
 8.2327463 .0017:2115 
 
 186 
 
TABLE VIII. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 559 
 
 312481 
 
 174676879 
 
 23.6431808 
 
 8.2376614 
 
 ..001788909 
 
 560 
 
 313600 
 
 175616000 
 
 23.6643191 
 
 8.2425706 
 
 .001785714 
 
 561 
 
 314721 
 
 176558481 
 
 23.6854386 
 
 8.247'47'40 
 
 .001782531 
 
 562 
 
 315844 
 
 177504328 
 
 23.7065392 
 
 8.2523715 
 
 .001779359 
 
 563 
 
 316969 
 
 178453547 
 
 23.7'276210 
 
 8.2572C33 
 
 .001776199 
 
 564 
 
 81K096 
 
 179406144 
 
 23.7486842 
 
 8.2621492 
 
 .001773050 
 
 565 
 
 819225 
 
 180362125 
 
 23.7697286 
 
 8.2670294 
 
 .001769912 
 
 566 
 
 320356 
 
 181321496 
 
 23.7907545 
 
 8.2719039 
 
 .001766784 
 
 567 
 
 321489 
 
 182284263 
 
 23.8117618 
 
 8.2767726 
 
 .0017'63668 
 
 568 
 
 322624 
 
 183250432 
 
 23.8327506 
 
 8.2816355 
 
 .0017'60563 
 
 509 
 
 323761 
 
 184220000 
 
 #o. 6567 209 
 
 8.J&64928 
 
 .001757469 
 
 170 
 
 324900 
 
 185193000 
 
 23.8746728 
 
 8.2913444 
 
 .001754386 
 
 571 
 
 326041 
 
 186169411 
 
 23.8956063 
 
 8. 961903 
 
 .001751313 
 
 572 
 
 327184 
 
 187149248 
 
 23.9165215 
 
 8.3010304 
 
 .001748252 
 
 573 
 
 3281323 
 
 188132517 
 
 23. 9374184 
 
 8.3058651 
 
 .001745201 
 
 574 
 
 329476 
 
 189119224 
 
 23.9582971 
 
 8.3106941 
 
 .001742160 
 
 575 
 
 330625 
 
 190109375 
 
 23.9791576 
 
 8.3155175 
 
 .001739130 
 
 576 
 
 331776 
 
 191102976 
 
 24.0000000 
 
 8. 203353 
 
 .001736111 
 
 577 
 
 332929 
 
 1921000:33 
 
 24.0208243 
 
 8. 25147 5 
 
 .001733102 
 
 578 
 
 334084 
 
 1931C0552 
 
 24.0416306 
 
 8.8299542 
 
 .001730104 
 
 579 
 
 3352-11 
 
 1D4104529 
 
 24.0624188 
 
 8.3347553 
 
 .001727116 
 
 580 
 
 336400 
 
 1D5112000 
 
 24.C831891 
 
 8.3395509 
 
 .001724138 
 
 C81 
 
 837561 
 
 106122941 
 
 24.1039416 
 
 8.3443410 
 
 .00172117'0 
 
 582 
 
 338724 
 
 197137368 
 
 24.1246762 
 
 8.3491256 
 
 .001718213 
 
 583 
 
 839889 
 
 198155287 
 
 24.1453929 
 
 8. 539047 
 
 .001715266 
 
 584 
 
 341056 
 
 199176704 
 
 4.1660919 
 
 8.3586784 
 
 .001712329 
 
 585 
 
 342225 
 
 200201625 
 
 24.1867732 
 
 8. 634466 
 
 .001709402 
 
 586 
 
 843396 
 
 201230056 
 
 24.2074369 
 
 8.3682095 
 
 .001706485 
 
 587 
 
 344569 
 
 202262003 
 
 24.2280829 
 
 8.87'29668 
 
 .001703578 
 
 588 
 
 345744 
 
 203297472 
 
 4.2487113 
 
 8.3777188 
 
 .001700680 
 
 589 
 
 346921 
 
 204336469 
 
 4.693222 
 
 8.3824653 
 
 .001697793 
 
 590 
 
 348100 
 
 205379000 
 
 24.2899156 
 
 8.S872065 
 
 .001694915 
 
 591 
 
 349281 
 
 206425071 
 
 4.3104916 
 
 8.3919423 
 
 .001692047 
 
 592 
 
 350464 
 
 207474688 
 
 24.3310501 
 
 8. 966729 
 
 .001689189 
 
 593 
 
 351649 
 
 208527857 
 
 24.3515913 
 
 8.4013981 
 
 .001686341 
 
 594 
 
 352836 
 
 209584584 
 
 24.3721152 
 
 8.4061180 
 
 .001683502 
 
 595 
 
 354025 
 
 210644875 
 
 24.3926218 
 
 8.4108326 
 
 .001680672 
 
 596 
 
 355216 
 
 211708736 
 
 24.4131112 
 
 8.4155419 
 
 .001677852 
 
 597 
 
 356409 
 
 212776173 
 
 24.4335834 
 
 8.4202460 
 
 .001675042 
 
 598 
 
 357604 
 
 213847192 
 
 24.4540385 
 
 8.4249448 
 
 .001672241 
 
 599 
 
 358801 
 
 214921799 
 
 24.4744765 
 
 8.4296383 
 
 .001669449 
 
 600 
 
 360000 
 
 216000000 
 
 24.4948974 
 
 8.4348267 
 
 .001666667 
 
 C01 
 
 361201 
 
 17081801 
 
 24.5153013 
 
 8.4390098 
 
 .001668894 
 
 C02 
 
 362404 
 
 218167208 
 
 24.5356883 
 
 8.4436877 
 
 .0016611.30 
 
 C03 
 
 363609 
 
 219256227 
 
 24.5560583 
 
 8.4483605 
 
 .001658375 
 
 C04 
 
 364816 
 
 220348864 
 
 24.5764115 
 
 8.4530281 
 
 .001655629 
 
 05 
 
 366025 
 
 221445125 
 
 24.5967478 
 
 8.4576906 
 
 .001652893 
 
 606 
 
 367236 
 
 222545016 
 
 24.6170673 
 
 8.4623479 
 
 .001650165 
 
 607 
 
 368449 
 
 223648543 
 
 24.6373,00 
 
 8.4670001 
 
 .001647446 
 
 08 
 
 369664 
 
 224755712 
 
 24.6576560 
 
 8.4716471 
 
 .001644737 
 
 609 
 
 370881 
 
 225866529 
 
 24.6779254 
 
 8.4762892 
 
 .00164203li 
 
 610 
 
 372100 
 
 226981000 
 
 24.6981781 
 
 8.4809261 
 
 .001639344 
 
 611 
 
 873321 
 
 228099131 
 
 24.7184142 
 
 8.4855579 
 
 .001636661 
 
 12 
 
 74544 
 
 229220928 
 
 24.7386,338 
 
 8.4901848 
 
 .001633987 
 
 613 
 
 375769 
 
 230346397 
 
 24.7588368 
 
 8.4948065 
 
 .001631321 
 
 614 
 
 376996 
 
 231475544 
 
 24.7790234 
 
 8.4994233 
 
 .001628664 
 
 615 
 
 378225 
 
 232608375 
 
 24.7991935 
 
 8.5040350 
 
 .001626016 
 
 616 
 
 379456 
 
 233744896 
 
 24.8193473 
 
 8.5086417 
 
 .001623377 
 
 617 
 
 380689 
 
 234885113 
 
 24.8394847 
 
 8.5132435 
 
 .001620746 
 
 618 
 
 381924 
 
 236029032 24.85960.58 
 
 8.5178403 
 
 .001618123 
 
 619 
 
 883161 237176659 24.8797106 
 
 8.5224321 
 
 .001615509 
 
 'teo 
 
 384400 238328000 i 24.8997992 
 
 8.5270189 
 
 .001612903 
 
 187 
 
TABLE VIII. Continued. 
 
 No. 
 
 Square 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Roots 
 
 Reciprocals. 
 
 621 
 
 385641 
 
 239483061 
 
 24.9198716 
 
 8.5316009 
 
 .001610306 
 
 622 
 
 386884 
 
 240041848 
 
 24.9399278 
 
 8.5361780 
 
 .001607717 
 
 623 
 
 388129 
 
 241804867 
 
 24.9599679 
 
 8.5407501 
 
 .001605136 
 
 624 
 
 389376 
 
 242970624 
 
 24.9799920 
 
 8. .5453173 
 
 .001602564 
 
 625 
 
 390625 
 
 244140625 
 
 25.0000000 
 
 8.5498797 
 
 .001600000 
 
 626 
 
 391876 
 
 245314376 
 
 25.0199920 
 
 8.:'j544372 
 
 .001597444 
 
 627 
 
 393129 
 
 246491883 
 
 25.0399681 
 
 8.5589899 
 
 .001594896 
 
 628 
 
 394384 
 
 247673152 
 
 25.0599282 
 
 8.5635377 
 
 .001592357 
 
 629 
 
 395641 
 
 248858189 
 
 25.0798724 
 
 8.5080807 
 
 .001589825 
 
 630 
 
 396900 
 
 250047000 
 
 25.0998008 
 
 8.5720189 
 
 .001587302 
 
 631 
 
 398161 
 
 251239591 
 
 25.1197134 
 
 8.5771523 
 
 .001584780 
 
 632 
 
 399424 
 
 252435968 
 
 25.1396102 
 
 8.5816809 
 
 .001582278 
 
 633 
 
 400689 
 
 253636137 
 
 25.1594913 
 
 8.5862047 
 
 .001579779 
 
 634 
 
 401956 
 
 254840104 
 
 25.1793566 
 
 8.5907238 
 
 .001577287 
 
 635 
 
 403225 
 
 256047875 
 
 25.1992063 
 
 8.5952380 
 
 .001574803 
 
 636 
 
 404496 
 
 257259456 
 
 25.2190404 
 
 8.5997476 
 
 .001572327 
 
 637 
 
 405769 
 
 258474853 
 
 25.2388589 
 
 8.6042525 
 
 .001569859 
 
 638 
 
 407044 
 
 259694072 
 
 25.2586619 
 
 8.6087526 
 
 .001567398 
 
 639 
 
 408321 
 
 200917119 
 
 25.2784493 
 
 8.6132480 
 
 .001564945 
 
 640 
 
 409600 
 
 262144000 
 
 25.2982213 
 
 8.6177388 
 
 .001562500 
 
 641 
 
 410881 
 
 263374721 
 
 25.3179778 
 
 8.6222248 
 
 .001560002 
 
 642 
 
 412164 
 
 264609288 
 
 25.3377189 
 
 8.6267063 
 
 .001557632 
 
 643 
 
 413449 
 
 265847707 
 
 25.3574447 
 
 8.6311830 
 
 .001555210 
 
 644 
 
 414736 
 
 267089984 
 
 25.3771551 
 
 8.6356551 
 
 .001552795 
 
 645 
 
 416025 
 
 268336125 
 
 25.3968502 
 
 8.6401226 
 
 .001550388 
 
 646 
 
 417316 
 
 269586136 
 
 25.4165301 
 
 8.6445855 
 
 .001547988 
 
 647 
 
 418609 
 
 270840023 
 
 25.4361947 
 
 8.6490437 
 
 .001545595 
 
 648 
 
 419904 
 
 272097792 
 
 25.4558441 
 
 8.6534974 
 
 .001543210 
 
 649 
 
 421201 
 
 273359449 
 
 25.4754784 
 
 8.6579465 
 
 .001540832 
 
 650 
 
 422500 
 
 274625000 
 
 25.4950976 
 
 8.6623911 
 
 .001538462 
 
 651 
 
 423801 
 
 275894451 
 
 25.5147016 
 
 8.6668310 
 
 .001536098 
 
 652 
 
 425104 
 
 277167808 
 
 25.5342907 
 
 8.6712665 
 
 .001533742 
 
 653 
 
 426409 
 
 278445077 
 
 25.5538647 
 
 8.6756974 
 
 .001531394 
 
 654 
 
 427716 
 
 279726264 
 
 25.5734237 
 
 8.6801237 
 
 .001529052 
 
 655 
 
 429025 
 
 281011375 
 
 25.5929078 
 
 8.6845456 
 
 .001526718 
 
 656 
 
 430336 
 
 282800416 
 
 25.6124969 
 
 8.6889630 
 
 .001521390 
 
 657 
 
 431649 
 
 283593393 
 
 25.6320112 
 
 8.6933759 
 
 .001522070 
 
 658 
 
 432964 
 
 284890312 
 
 25.6515107 
 
 8.6977843 
 
 .001519757 
 
 659 
 
 434281 
 
 286191179 
 
 25.6709963 
 
 8.7021882 
 
 .001517451 
 
 660 
 
 435600 
 
 287496000 
 
 25.6904652 
 
 8.7065877 
 
 .001515152 
 
 661 
 
 436921 
 
 288804781 
 
 25.7099203 
 
 8.7109827 
 
 .001512859 
 
 662 
 
 438244 
 
 290117528 
 
 25.7293607 
 
 8.7153734 
 
 .001510574 
 
 663 
 
 439569 
 
 291434247 
 
 25.7487864 
 
 8.7197596 
 
 .001508296 
 
 664 
 
 440896 
 
 292754944 
 
 25.7681975 
 
 8.7241414 
 
 .001506024 
 
 65 
 
 442225 
 
 294079625 
 
 25.7875939 
 
 8.7285187 
 
 .001503759 
 
 66 
 
 443556 
 
 295408296 
 
 25.8069758 
 
 8.7328918 
 
 .001501502 
 
 667 
 
 444889 
 
 296740963 
 
 25.8263431 
 
 8.7372604 
 
 .001499250 
 
 668 
 
 446224 
 
 298077632 
 
 25.8456960 
 
 8.741C246 
 
 .001497006 
 
 669 
 
 447561 
 
 299418309 
 
 25.8650343 
 
 8.7459846 
 
 .001494708 
 
 670 
 
 448900 
 
 300763000 
 
 25.8843582 
 
 8.7503401 
 
 .001492537 
 
 71 
 
 450241 
 
 302111711 
 
 25.9036677 
 
 8.7546913 
 
 .001490313 
 
 72 
 
 451584 
 
 303464448 
 
 25.9229628 
 
 8.7590383 
 
 .001488095 
 
 673 
 
 452929 
 
 304821217 
 
 25.9422435 
 
 8.7633809 
 
 .001485884 
 
 74 
 
 454276 
 
 306182024 
 
 25.9615100 
 
 8.7677192 
 
 .001483680 
 
 675 
 
 455625 
 
 307546875 
 
 25.9807621 
 
 8.7720532 
 
 .001481481 
 
 76 
 
 456976 
 
 30891*5776 
 
 26.0000000 
 
 8.7763830 
 
 .001479290 
 
 677 
 
 458329 
 
 310288733 
 
 26.0192237 
 
 8.7807084 
 
 .001477105 
 
 678 
 
 459684 
 
 311665752 
 
 26.0384331 
 
 8.7850296 
 
 .001474926 
 
 679 
 
 461041 
 
 313046839 
 
 26.0576284 
 
 8.7893466 
 
 .004472754 
 
 680 
 
 462400 
 
 314432000 
 
 26.0768096 
 
 8.7936593 
 
 .001470588 
 
 681 
 
 463761 
 
 315821241 
 
 26.09597'67 
 
 8.7979679 
 
 .001468429 
 
 682 
 
 465124 
 
 317214568 
 
 26.1151297 
 
 8.8022721 
 
 .001466276 
 
 188 
 
TABLE Vm. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Roots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 683 
 
 466489 
 
 318611987 
 
 26.1342687 
 
 8.8065722 
 
 .001464129 
 
 684 
 
 467856 
 
 320013504 26.1533937 
 
 8.8108681 
 
 .001461988 
 
 685 
 
 469225 
 
 321419125 26.1725047 
 
 8.8151598 
 
 .001459854 
 
 686 
 
 470596 
 
 322828856 
 
 26.1916017 
 
 8.8194474 
 
 .001457726 
 
 687 
 
 471969 
 
 324242703 
 
 26.2106848 
 
 8.8237307 
 
 .001455604 
 
 688 
 
 47&344 
 
 325660672 
 
 26.2297541 
 
 8.8280099 
 
 .001453488 
 
 689 
 
 474721 
 
 327082769 
 
 26.2488095 
 
 8.8322850 
 
 .001451379 
 
 690 
 
 476100 
 
 328509000 
 
 26.2678511 
 
 8.8365559 
 
 .001449275 
 
 691 
 
 477481 
 
 329939371 
 
 26.2868789 
 
 8.8408227 
 
 .001447178 
 
 692 
 
 478864 
 
 331373888 
 
 26.3058929 
 
 8.8450854 
 
 .001445087 
 
 693 
 
 480249 
 
 332812557 
 
 26.3248932 
 
 8.8493440 
 
 .001443001 
 
 694 
 
 481636 
 
 334255384, 
 
 26.3438797 
 
 8.8535985 
 
 .001440922 
 
 695 
 
 483025 
 
 35702375 
 
 26.3628527 
 
 8.8578489 
 
 .001438849 
 
 696 
 
 484416 
 
 337153536 
 
 26.3818119 
 
 8.8620952 
 
 .001436782 
 
 697 
 
 485809 
 
 338608873 
 
 26.4007576 
 
 8.8663375 
 
 .001434720 
 
 698 
 
 487204 
 
 340068392 
 
 26.4196896 
 
 8.8705757 
 
 .001432665 
 
 699 
 
 488601 
 
 341532099 
 
 26.4386081 
 
 8.8748099 
 
 .001430615 
 
 700 
 
 490000 
 
 343000000 
 
 26.4575131 
 
 8.8790400 
 
 .001428571 
 
 701 
 
 491401 
 
 344472101 
 
 26.4764046 
 
 8.8832661 
 
 .001426534 
 
 702 
 
 492804 
 
 345948408 
 
 26.4952826 
 
 8.8874882 
 
 .001424501 
 
 703 
 
 494209 
 
 347428927 
 
 26.5141472 
 
 8.8917063 
 
 .001422475 
 
 704 
 
 495616 
 
 348913664 
 
 26.5329983 
 
 8.8959204 
 
 .001420455 
 
 705 
 
 497025 
 
 350402625 
 
 26.5518361 
 
 8.9001304 
 
 .001418440 
 
 706 
 
 498436 
 
 351895816 
 
 26.5706605 
 
 8.9043366 
 
 .001416431 
 
 707 
 
 499849 
 
 353393243 
 
 26.5894716 
 
 8.9085387 
 
 .001414427 
 
 708 
 
 501264 
 
 354894912 
 
 26.6082694 
 
 8.9127369 
 
 .001412429 
 
 709 
 
 502681 
 
 356400829 
 
 26.6270539 
 
 8.9169311 
 
 .001410437 
 
 '710 
 
 504100 
 
 357911000 
 
 26.6458252 
 
 8.9211214 
 
 .001408451 
 
 711 
 
 505521 
 
 359425431 
 
 26.6645833 
 
 8.9253078 
 
 .001406470 
 
 712 
 
 506944 
 
 360944128 
 
 26.6833281 
 
 8.9294902 
 
 .001404494 
 
 713 
 
 508369 
 
 362467097 
 
 26.7020598 
 
 8.9336687 
 
 .001402525 
 
 714 
 
 509796 
 
 363994344 
 
 26.7207784 
 
 8.9378433 
 
 .001400560 
 
 715 
 
 511225 
 
 365525875 
 
 26.7394839 
 
 8.9420140 
 
 .001398601 
 
 716 
 
 512656 
 
 367061696 
 
 26.7581763 
 
 8.9461809 
 
 .001396648 
 
 717 
 
 514089 
 
 368601813 
 
 26.7768557 
 
 8.9503438 
 
 .001394700 
 
 718 
 
 515524 
 
 370146232 
 
 26.7955220 
 
 8.9545029 
 
 .001392758 
 
 719 
 
 516961 
 
 371694959 
 
 26.8141754 
 
 8.9586581 
 
 .001390821 
 
 720 
 
 518400 
 
 373248000 
 
 26.8328157 
 
 8.9628095 
 
 .001388889 
 
 721 
 
 519841 
 
 374805361 
 
 26.8514432 
 
 8.9639570 
 
 .(,01386963 
 
 722 
 
 521284 
 
 376367048 
 
 26.8700577 
 
 8.9711007 
 
 .001385042 
 
 723 
 
 522729 
 
 377933067 
 
 26.8886593 
 
 8.9752406 
 
 .001383126 
 
 724 
 
 524176 
 
 379503424 
 
 26.9072481 
 
 8.9793766 
 
 .001381215 
 
 725 
 
 525625 
 
 381078125 
 
 26.9258240 
 
 8.9835089 
 
 .001379310 
 
 726 
 
 527076 
 
 382657176 
 
 26.9443872 
 
 8.9876373 
 
 .001377410 
 
 727 
 
 528529 
 
 384240583 
 
 26.9629375 
 
 8.9917620 
 
 001375516 
 
 728 
 
 529984 
 
 385828352 
 
 26.9814751 
 
 8.9958829 
 
 .001373626 
 
 729 
 
 531441 
 
 387420489 
 
 27.0000000 
 
 9.0000000 
 
 .001371742 
 
 730 
 
 532900 
 
 389017000 
 
 27.0185122 
 
 9.0041134 
 
 .001369863 
 
 731 
 
 534361 
 
 390617891 
 
 27.0370117 
 
 9.0082229 
 
 .001367989 
 
 732 
 
 535824 
 
 392223168 
 
 27.0554985 
 
 9.0123288 
 
 .001366120 
 
 733 
 
 537289 
 
 393832837 
 
 27.0739727 
 
 9.0164309 
 
 .001364256 
 
 734 
 
 538756 
 
 395446904 
 
 27.0924344 
 
 9.0205293 
 
 .001362:398 
 
 735 
 
 540225 
 
 397065375 
 
 27.1108834 
 
 9.0246239 
 
 .001360544 
 
 736 
 
 541696 
 
 398688256 
 
 27.1293199 
 
 9.0287149 
 
 .00ia58696 
 
 737 
 
 543169 
 
 400315553 
 
 27.1477439 
 
 9.0328021 
 
 .001356852 
 
 738 
 
 544644 
 
 401947272 
 
 27.1661554 
 
 9.0368857 
 
 .001355014 
 
 739 
 
 546121 
 
 403583419 
 
 27.1845544 
 
 9.0409G55 
 
 .001353180 
 
 740 
 
 547600 
 
 405224000 
 
 27.2029410 
 
 9.0450419 
 
 .001351351 
 
 741 
 
 549081 
 
 406869021 
 
 27.2213152 
 
 9.0491142 
 
 .001349528 
 
 742 
 
 550564 
 
 408518488 
 
 27.2396769 
 
 9.0531831 
 
 .001347709 
 
 ,743 
 
 552049 
 
 410172407 
 
 27.2580263 
 
 9.0572482 
 
 .001345895 
 
 744 
 
 553536 
 
 411*30784 | 27.2763634 
 
 9.0613098 
 
 .001344086 
 
TABLE VIIL Continued. 
 
 ''NO. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 745 
 
 555025 
 
 413493625 
 
 27.2946881 
 
 9.0653677 
 
 .0013:2282 
 
 746 
 
 556516 
 
 415160936 
 
 27.3130006 
 
 9.0094220 
 
 .001340483 
 
 747 
 
 558009 
 
 4168327'23 
 
 27.3313007 
 
 9.07'34726 
 
 .001338088 
 
 748 
 
 559504 
 
 4185081)92 
 
 27.3495887 
 
 .9.0775197 
 
 .001336898 
 
 749 
 
 561001 
 
 420189749 
 
 27.3678044 
 
 5). 0815031 
 
 .001335113 
 
 750 
 
 562500 
 
 421875000 
 
 27.3861379 
 
 9.085G030 
 
 .001333333 
 
 751 
 
 5(54001 
 
 428564751 
 
 27.4043 r <U2 
 
 9. 0890392 
 
 .001331558 
 
 752 
 
 565504 
 
 425259008 
 
 27.4220184 
 
 9.0930719 
 
 ,00132978? 
 
 753 
 
 567009 
 
 426957777 
 
 27.4408455 
 
 9.0977010 
 
 .001328021 
 
 754 
 
 568516 
 
 428001004 
 
 27.4590004 
 
 9.1017205 
 
 .001320200 
 
 755 
 
 570025 
 
 430308875 
 
 27.4772033 
 
 9.1057'485 
 
 .001324503 
 
 756 
 
 571536 
 
 432081216 
 
 27.4954542 
 
 9.1(/J7'G69 
 
 .CC1122751 
 
 757 
 
 573049 
 
 483798093 
 
 27.5130330' 
 
 9.11^7818 
 
 .GG1321004 
 
 758 
 
 574564 
 
 435519512 
 
 27.5317998 
 
 9.1177931 
 
 .001319201 
 
 759 
 
 576081 
 
 437245479 
 
 27.5499546 
 
 9.1218010 
 
 .G01i>17523 
 
 760 
 
 577600 
 
 438976000 
 
 27.5680975 
 
 9.1258053 
 
 .001815789 
 
 761 
 
 579121 
 
 440711081 
 
 27.5862284 
 
 9.1X96001 
 
 .101314000 
 
 762 
 
 580644 
 
 442450728 
 
 27.604347'5 
 
 9.13o8034 
 
 .001312336 
 
 763 
 
 582169 
 
 444194947 
 
 27.62^4540 
 
 9.1377971 
 
 .001310010 
 
 764 
 
 583696 
 
 445943744 
 
 27.6405499 
 
 9.1417874 
 
 .001308901 
 
 765 
 
 585225 
 
 447'697125 
 
 27.0586334 
 
 9.1457742 
 
 .001 307190 
 
 766 
 
 586756 
 
 449455096 
 
 27.6767050 
 
 9.1497576 
 
 .001305483 
 
 767 
 
 588289 
 
 451217663 
 
 27.0947'G48 
 
 9.1537'375 
 
 .001:^03781 
 
 768 
 
 589824 
 
 452984832 
 
 27.7128129 
 
 <J.15771S9 
 
 .001308088 
 
 769 
 
 591361 
 
 454750009 
 
 27.7308492 
 
 ( J.101G80'J 
 
 .001300390 
 
 770 
 
 592900 
 
 456533000 
 
 27.7488739 
 
 9.1656565 
 
 .001298701 
 
 771 
 
 594441 
 
 458314011 
 
 27.7008868 
 
 9.1096225 
 
 .001*97017 
 
 772 
 
 595984 
 
 460099048 
 
 27.7848880 
 
 9.1735852 
 
 .001*95337' 
 
 773 
 
 597529 
 
 461889917 
 
 27.8028770 
 
 9.1775445 
 
 .001293001 
 
 '774 
 
 599076 
 
 463684824 
 
 27.8208555 
 
 9.1815003 
 
 .001291990 
 
 775 
 
 600625 
 
 465484375 
 
 27.8388218 
 
 9.1854527 
 
 .0012U0323 
 
 776 
 
 602176 
 
 467288570 
 
 27.b567760 
 
 9.1b94018 
 
 .001288060 
 
 777 
 
 603729 
 
 469097433 
 
 27.8747197 
 
 9.1933474 
 
 .001287001 
 
 778 
 
 605284 
 
 470910952 
 
 27.8926514 
 
 9.1972897 
 
 .001*85847 
 
 779 
 
 606841 
 
 472729139 
 
 27.9105715 
 
 9.2012286 
 
 .001283097 
 
 780 
 
 608400 
 
 474552000 
 
 27.9284801 
 
 9.2051641 
 
 .001282051 
 
 781 
 
 609961 
 
 476379541 
 
 27.9463772 
 
 9. 2010962 
 
 .001280410 
 
 782 
 
 611524 
 
 478211708 
 
 27.9642029 
 
 9.*lcO*50 
 
 .001278772 
 
 783 
 
 613089 
 
 480048687 
 
 27.9821372 
 
 9.2109505 
 
 .001*77139 
 
 784 
 
 614656 
 
 481890304 
 
 28.0000000 
 
 9.2208720 
 
 .001275510 
 
 785 
 
 616225 
 
 483730025 
 
 28.0178515 
 
 9.2247914 
 
 .001*7U885 
 
 786 
 
 617796 
 
 485587056 
 
 28.0356915 
 
 9.2267008 
 
 .001272205 
 
 787 
 
 619369 
 
 4874434C3 
 
 28.0535203 
 
 9.23*Glb9 
 
 .001*70048 
 
 788 
 
 620944 
 
 489303872 
 
 28.0713377 
 
 9.2305277 
 
 .G012WJ030 
 
 789 
 
 622521 
 
 491109009 
 
 28.0891438 
 
 9.2404333 
 
 .001*07427 
 
 790 
 
 624100 
 
 493039000 
 
 28.1069386 
 
 9.2443355 
 
 .001205823 
 
 791 
 
 625681 
 
 494913071 
 
 28.1247'222 
 
 9.2482344 
 
 .001204223 
 
 792 
 
 627264 
 
 496793088 
 
 28.1424946 
 
 9.2521300 
 
 .001202620 
 
 793 
 
 628849 
 
 498677257 
 
 28.1602557 
 
 9.2500224 
 
 .001201034 
 
 794 
 
 630436 
 
 500566184 
 
 28.1780056 
 
 9.2599114 
 
 .001*59440 
 
 795 
 
 632025 
 
 50245987'5 
 
 28.1957444 
 
 9.2037973 
 
 .001257862 
 
 796 
 
 633616 
 
 504358336 
 
 28.2134720 
 
 9.2076798 
 
 .001250281 
 
 797 
 
 635209 
 
 506201573 
 
 28.2311884 
 
 9.2715592 
 
 .0015:54705 
 
 798 
 
 636804 
 
 508169592 
 
 28.2488938 
 
 9.2754352 
 
 .001253133 
 
 799 
 
 638401 
 
 510082399 
 
 28.2665881 
 
 9.2793081 
 
 .001251504 
 
 800 
 
 640000 
 
 512000000 
 
 28.2842712 
 
 9.2831777 
 
 .001250000 
 
 801 
 
 641601 
 
 513922401 
 
 28.8019434 
 
 9.2870440 
 
 .001248439 
 
 802 
 
 643204 
 
 515849608 
 
 28.3196045 
 
 9.2909072 
 
 .001246883 
 
 803 
 
 644809 
 
 517781627 
 
 28.3372546 
 
 9.2947071 
 
 .001245330 
 
 804 
 
 646416 
 
 5197184&4 
 
 28.3548938 
 
 9.2986239 
 
 .001243781 
 
 805 
 
 648025 
 
 521660125 
 
 28.3725219 
 
 9.3024775 
 
 .001242236 
 
 806 
 
 649636 
 
 523606616 
 
 28.3901391 
 
 9.3003278 
 
 .001240695 
 
 190 
 
TABLE VIII. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Boots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 807 
 
 6512-19 
 
 525557943 
 
 28.4077454 
 
 9.3101750 
 
 .001239157 
 
 803 
 
 652864 
 
 527514112 
 
 28.4253408 
 
 9.3140190 
 
 .001237624 
 
 809 
 
 654481 
 
 529475129 
 
 28.4429253 
 
 9.3178599 
 
 .001236094 
 
 810 
 
 656100 
 
 531441000 
 
 28.4604989 
 
 9.3216975 
 
 .001234568 
 
 811 
 
 657721 
 
 533411731 
 
 28.4780617 
 
 9.3255320 
 
 .001233046 
 
 812 
 
 659344 
 
 535387328 
 
 28.4956137 
 
 9.3293634 
 
 .001231527 
 
 813 
 
 660969 
 
 537367797 
 
 28.5131549 
 
 9'. 3331816 
 
 .001230012 
 
 814 
 
 662596 
 
 539353144 
 
 28.5306852 
 
 9.3370167 
 
 .001228501 
 
 815 
 
 664225 
 
 541343375 
 
 28.5482048 
 
 9.3408386 
 
 .001226994 
 
 816 
 
 665856 
 
 543338496 
 
 28.5657137 
 
 9.3446575 
 
 .001225490 
 
 817 
 
 667489 
 
 545338513 
 
 28.5832119 
 
 9.3484731 
 
 .001223990 
 
 818 
 
 669124 
 
 547343432 
 
 28.6006993 
 
 a. 3522857 
 
 .001222494 
 
 819 
 
 670761 
 
 549353259 
 
 28.6181760 
 
 9.3560952 
 
 .001221001 
 
 820 
 
 672400 
 
 551368000 
 
 28.6356421 
 
 9.3599016 
 
 .001219512 
 
 821 
 
 674041 
 
 553:387661 
 
 28.6530976 
 
 9.3637'049 
 
 .001218027 
 
 822 
 
 675684 
 
 555412248 
 
 28.6705424 
 
 9.3675051 
 
 .001216545 
 
 823 
 
 677329 
 
 557441767 
 
 28. 68797 G6 
 
 9.3713022 
 
 .001215067 
 
 824 
 
 678976 
 
 559476224 
 
 28.7054002 
 
 9.3750963 
 
 .001213592 
 
 825 
 
 680625 
 
 561515625 
 
 28.7228132 
 
 9.3788873 
 
 .001212121 
 
 828 
 
 682276 
 
 563559976 
 
 28.7402157 
 
 9.3826752 
 
 .001210654 
 
 837 
 
 683929 
 
 565609283 
 
 28.7576077 
 
 9.3864600 
 
 .001209190 
 
 838 
 
 685584 
 
 567663552 
 
 28.7749891 
 
 9.3902419 
 
 .001207729 
 
 829 
 
 687241 
 
 569722789 
 
 28.7923601 
 
 9.3940206 
 
 .001206273 
 
 830 
 
 688900 
 
 571787000 
 
 28.8097206 
 
 9.3977964 
 
 .001204819 
 
 831 
 
 690561 
 
 573856191 
 
 28.82707'06 
 
 9.4015691 
 
 .001203369 
 
 832 
 
 692224 
 
 575930368 
 
 28.8444102 
 
 9:4053387 
 
 .001201923 
 
 8-33 
 
 693889 
 
 578009537 
 
 28.8617394 
 
 9.4091054 
 
 .001200480 
 
 834 
 
 695556 
 
 580093704 
 
 28.8790582 
 
 9.4128690 
 
 .001199041 
 
 835 
 
 697225 
 
 582182875 
 
 28.8963666 
 
 9.4166297 
 
 .001197605 
 
 836 
 
 698890 
 
 584277056 
 
 28.9136646 
 
 9.4203873 
 
 .001196172 
 
 637 
 
 700569 
 
 586376253 
 
 28.9309523 
 
 9.4241420 
 
 .001194743 "" 
 
 833 
 
 702244 
 
 588480472 
 
 28.9482237 
 
 9.4278936 
 
 .001193317 
 
 839 
 
 703921 
 
 590589719 
 
 28.9654967 
 
 9.4316423 
 
 .001191895 
 
 840 
 
 705600 
 
 592704000 
 
 28.9827535 
 
 9.4353880 
 
 .001190470 
 
 841 
 
 707281 
 
 59482:3321 
 
 29.0000000 
 
 9.4391307 
 
 .001189061 
 
 842 
 
 708964 
 
 596947688 
 
 29.0172363 
 
 9.4428704 
 
 .001187648 
 
 843 
 
 710649 
 
 599077107 
 
 29.0344023 
 
 9.4466072 
 
 .001186240 
 
 844 
 
 712336 
 
 601211584 
 
 29.05167'81 
 
 9.450ailO 
 
 .001184834 
 
 845 
 
 714025 
 
 603351125 
 
 29.0688837 
 
 9.4540719 
 
 .001183432 
 
 846 
 
 715716 
 
 605495736 
 
 29.08607D1 
 
 9.4577999 
 
 .001182033 
 
 847 
 
 717409 
 
 607645423 
 
 29.1032644 
 
 9.4615249 
 
 .001180633 
 
 848 
 
 719104 
 
 609800192 
 
 29.1204396 
 
 9.4652470 
 
 .001179245 
 
 849 
 
 720801 
 
 611960049 
 
 29.1376046 
 
 9.4689661 
 
 .001177856 
 
 850 
 
 722500 
 
 614125000 
 
 29.1547595 
 
 9.4726824 
 
 .001176471 
 
 851 
 
 724201 
 
 616295051 
 
 29.1719043 
 
 9.4763957 
 
 .001175088 
 
 852 
 
 725904 
 
 618470208 
 
 29.1890390 
 
 9.4801061 
 
 .001173709 
 
 853 
 
 727609 
 
 620650477 
 
 29.2061637 
 
 9.4838136 
 
 .001172333 
 
 854 
 
 729316 
 
 622835864 
 
 29.2232784 
 
 9.4875182 
 
 .001170960 
 
 855 
 
 731025 
 
 625026375 
 
 29.2403830 
 
 9.4912200 
 
 .001169591 
 
 853 
 
 732736 
 
 627222016 
 
 29.2574777 
 
 9.4949188 
 
 .001168224 
 
 857 
 
 734449 
 
 629422793 
 
 29.2745623 
 
 9.4986147 
 
 .001166861 
 
 858 
 
 73()1G4 
 
 631628712 
 
 29.2916370 
 
 9.5023078 
 
 .001165501 
 
 859 
 
 737881 
 
 633839719 
 
 29.3087018 
 
 9.5059980 
 
 .001164144 
 
 860 
 
 739600 
 
 636056000 
 
 29.3257566 
 
 9.5096854 
 
 .001162791 
 
 861 
 
 741321 
 
 638277381 
 
 29.3428015 
 
 9.5133699 
 
 .001161440 
 
 862 
 
 743044 
 
 640503928 
 
 29. 598365 
 
 9.5170515 
 
 .001160093 
 
 863 
 
 744769 
 
 642735647 
 
 29.3768616 
 
 9.5207303 
 
 .001158749 
 
 864 
 
 746496 
 
 644972544 
 
 29.3938769 
 
 9.5244063 
 
 .001157407 
 
 865 
 
 748225 
 
 64721 4625 
 
 29.4108823 
 
 9.5280794 
 
 .001156069 
 
 866 
 
 749956 
 
 649461896 
 
 29.4278779 
 
 9.5317407 
 
 .001154734 
 
 867 
 
 751689 
 
 651714363 
 
 29.4448ffl7 
 
 9.5354172 
 
 .001153403 
 
 68 
 
 753434 
 
 653972032 29.4618397 9.5390818 001152074 
 
 191 
 
TABLE V1IL. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Hoots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 869 
 
 755161 
 
 656234909 
 
 29.4788059 
 
 9.5427437 
 
 .001150748 
 
 870 
 
 756900 
 
 658503000 
 
 29.4957624 
 
 9.5464027 
 
 .001149425 
 
 871 
 
 758641 
 
 660776311 
 
 29.5127091 
 
 9.5500589 
 
 .001148106 
 
 872 
 
 760384 
 
 663054848 
 
 29.5296461 
 
 9.5587123 
 
 .001146789 
 
 873 
 
 762129 
 
 665338617 
 
 .5466784 
 
 9.5573030 
 
 .001145475 
 
 874 
 
 763876 
 
 667627624 
 
 29.5634910 
 
 9.5610108 
 
 .001144165 
 
 875 
 
 765625 
 
 669921875 
 
 29.5803989 
 
 9.5646559 
 
 .001142857 
 
 876 
 
 767376 
 
 67'2221376 
 
 29.5972972 
 
 9.5682982 
 
 .001141553 
 
 877 
 
 769129 
 
 674526133 
 
 29.6141858 
 
 9.5719377 
 
 .001140251 
 
 878 
 
 770884 
 
 676836152 
 
 29.6310648 
 
 9.5755745 
 
 .001138952 
 
 879 
 
 772641 
 
 679151439 
 
 29.6479342 
 
 9.5792085 
 
 .001137656 
 
 880 
 
 774400 
 
 681472000 
 
 29.6647939 
 
 9.5828397 
 
 .001136364 
 
 881 
 
 776161 
 
 683797841 
 
 29.6816442 
 
 9.5864682 
 
 .001135074 
 
 882 
 
 777924 
 
 686128968 
 
 29.6984848 
 
 9.5900939 
 
 .001133787 
 
 883 
 
 779689 
 
 688465387 
 
 29.7153159 
 
 9.5937169 
 
 .001132503 
 
 884 
 
 781456 
 
 690807104 
 
 29.7321375 
 
 9.5973373 
 
 .001131222 
 
 885 
 
 783225 
 
 693154125 
 
 29.7489496 
 
 9.6009548 
 
 .001129944 
 
 886 
 
 784996 
 
 695506456 
 
 29.7657'521 
 
 9.6045096 
 
 .001128668 
 
 887 
 
 786769 
 
 697864103 
 
 29.7825452 
 
 9.0081817 
 
 .001127396 
 
 888 
 
 788544 
 
 700227072 
 
 29.7993289 
 
 9.6117911 
 
 .001126126 
 
 889 
 
 790321 
 
 702595369 
 
 29.8101030 
 
 9.G153977 
 
 .001124859 
 
 890 
 
 792100 
 
 704989000 
 
 29.8328678 
 
 9.6190017 
 
 .001123596 
 
 891 
 
 793881 
 
 707347971 
 
 9. 8496231 
 
 9.0220030 
 
 .001122334 
 
 892 
 
 795664 
 
 709732288 
 
 29.8663G90 
 
 9.6202016 
 
 .001121076 
 
 893 
 
 797449 
 
 712121957 
 
 29.8831056 
 
 9.6297975 
 
 .001119821 
 
 894 
 
 799236 
 
 714516984 
 
 29.8998328 
 
 9.0333907 
 
 .001118568 
 
 895 
 
 801025 
 
 716917375 
 
 29.9165506 
 
 9.G369812 
 
 .001117318 
 
 896 
 
 802816 
 
 719323136 
 
 29.9332591 
 
 9.6405090 
 
 .001116071 
 
 897 
 
 804609 
 
 721734273 
 
 29.9499583 
 
 9.6441542 
 
 .001114827 
 
 898 
 
 806404 
 
 724150792 
 
 29.9666481 
 
 9.6477367 
 
 .001113586 
 
 .899 
 
 808201 
 
 726572699 
 
 29.9833287 
 
 9.6513166 
 
 .001112347, 
 
 900 
 
 810000 
 
 729000000 
 
 30.0000000 
 
 9.6548938 
 
 .001111111 
 
 901 
 
 811801 
 
 731432701 
 
 30.0166620 
 
 9.0584084 
 
 .001109873 
 
 902 
 
 813604 
 
 733870808 
 
 30.0333148 
 
 9.GG20403 
 
 .001108647 
 
 903 
 
 815409 
 
 736314327 
 
 30.0499584 
 
 9.6650096 
 
 .001107420 
 
 904 
 
 817'216 
 
 738763264 
 
 30.06G5928 
 
 9.6691762 
 
 .0011C6195 
 
 905 
 
 819025 
 
 741217625 
 
 30.0832179 
 
 9.6727403 
 
 .001104972 
 
 906 
 
 820836 
 
 743677416 
 
 30.0998339 
 
 9.6763017 
 
 .001108753 
 
 907 
 
 822649 
 
 746142643 
 
 30.1164407 
 
 9.6798604 
 
 .001102^3(5 
 
 908 
 
 824464 
 
 748613312 
 
 30.1330383 
 
 9.6834166 
 
 .001101322 
 
 909 
 
 826281 
 
 751089429 
 
 30.1496269 
 
 9.6869701 
 
 .001100110 
 
 910 
 
 828100 
 
 753571000 
 
 30.1662063 
 
 9.0905211 
 
 .001098901 
 
 911 
 
 829921 
 
 756058031 
 
 30.1827765 
 
 9.0940094 
 
 .001097095 
 
 912 
 
 831744 
 
 758550528 
 
 30.1993377 
 
 9.6976151 
 
 .001096491 
 
 913 
 
 833569 
 
 761048497 
 
 30.2158899 
 
 9.7011583 
 
 .001095290 
 
 914 
 
 835396 
 
 763551944 
 
 30.2324329 
 
 9.7046989 
 
 .001094002 
 
 915 
 
 837225 
 
 766060875 
 
 30.2489669 
 
 9.7082369 
 
 .001092896 
 
 916 
 
 839056 
 
 768575296 
 
 30.2654919 
 
 9.7117723 
 
 .001091703 
 
 917 
 
 840889 
 
 771095213 
 
 30.2820079 
 
 9.7153051 
 
 .001090513 
 
 918 
 
 842724 
 
 773620632 
 
 30.2985148 
 
 9.7188354 
 
 .001089335 
 
 919 
 
 844561 
 
 776151559 
 
 30.3150128 
 
 9.7223631 
 
 .001088139 
 
 920 
 
 846400 
 
 778688000 
 
 30.&315018 
 
 9.7258883 
 
 .001086957 
 
 921 
 
 848241 
 
 781229961 
 
 30.3479818 
 
 9.7294109 
 
 .00108577'6 
 
 922 
 
 850084 
 
 783777448 
 
 30.3644529 
 
 9.7329309 
 
 .001084599 
 
 923 
 
 851929 
 
 786330467 
 
 30.3809151 
 
 9.7304484 
 
 .001083423 
 
 924 
 
 853776 
 
 788889024 
 
 30.3973683 
 
 9.7399634 
 
 .001082251 
 
 925 
 
 855625 
 
 791453125 
 
 30.4138127 
 
 9.7434758 
 
 .001081081 
 
 926 
 
 857476 
 
 794022776 
 
 30.4302481 
 
 9.7469857 
 
 .001079914 
 
 927 
 
 859329 
 
 796597983 
 
 30.4466747 
 
 9.7504930 
 
 .001078749 
 
 928 
 
 861184 
 
 799178752 
 
 30.4630924 
 
 9.75S9979 
 
 .001077580 
 
 9S9 
 
 863041 
 
 801765089 
 
 30.4795013 
 
 9.7575002 
 
 .001076426 
 
 930 
 
 864900 
 
 804357000 
 
 30.4959014 9.7610001 
 
 .001075269 
 
TABLE VTH. Continued. 
 
 No. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Roots. 
 
 Cube Roots, j Reciprocals. 
 
 931 
 
 866761 j 806954491 
 
 30.5122926 
 
 9.7644974 
 
 .001074114 
 
 932 
 
 868624 J 809557568 
 
 30.52S6750 
 
 9.7679922 
 
 .001072961 
 
 933 
 
 870489 
 
 812166237 
 
 30.5450487 
 
 9.7714845 
 
 .001071811 
 
 934 
 
 872356 
 
 814780504 
 
 30.5614136 
 
 9.77'49743 
 
 .001070664 
 
 935 
 
 874225 
 
 817400375 
 
 30.5777697 
 
 9.7784616 
 
 .001069519 
 
 936 
 
 876096 
 
 820025856 
 
 30.5941171 
 
 9.7819466 
 
 .001068376 
 
 937 
 
 877969 
 
 822656953 
 
 30.6104557 
 
 9.7854288 
 
 .001067236 
 
 938 
 
 79844 
 
 825293672 
 
 30.6267857 
 
 9.7889087 
 
 .001066098 
 
 939 
 
 881721 
 
 827936019 
 
 30.6431069 
 
 9.7923861 
 
 .001064963 
 
 940 
 
 883600 
 
 830584000 
 
 30.6594194 
 
 9.7958611 
 
 .001063830 
 
 941 
 
 885481 
 
 833237621 
 
 30.6757233 
 
 9.7993336 
 
 .001062699 
 
 942 
 
 887364 
 
 835896888 
 
 30.6920185 
 
 9.8028036 
 
 .001061571 
 
 943 
 
 889249 
 
 838561807 
 
 30.7083051 
 
 9.8062711 
 
 .001060445 
 
 944 
 
 891136 
 
 841232384 
 
 :.0. 7245830 
 
 9.8097362 
 
 .001059322 
 
 945 
 
 893025 
 
 843908625 
 
 30.7408523 
 
 9.8131989 
 
 .001058201 
 
 946 
 
 894916 
 
 846590536 
 
 30.7571130 
 
 9.8166591 
 
 .001057082 
 
 947 
 
 896809 
 
 849278123 
 
 30.7733651 
 
 9.8201169 
 
 .001055966 
 
 948 
 
 898704 
 
 851971392 
 
 30.78C6G86 
 
 9.8235723 
 
 .001054852 
 
 949 
 
 900601 
 
 854670349 
 
 30.8058436 
 
 9.8270252 
 
 .001053741 
 
 950 
 
 902500 
 
 857375000 
 
 30.8220700 
 
 9.8304757 
 
 001052632 
 
 951 
 
 904401 
 
 860085351 
 
 30.8382879 
 
 9.8338238 
 
 .001051525 
 
 952 
 
 906304 
 
 862801408 
 
 30.8544972 
 
 9.8373695 
 
 .001050420 
 
 953 
 
 908209 
 
 865523177 
 
 30.8706981 
 
 9.8408127 
 
 .001049318 
 
 954 
 
 910116 
 
 868250664 
 
 30.8868904 
 
 9.8442536 
 
 .001048218 
 
 955 
 
 912025 
 
 870983875 
 
 30.9030743 
 
 9.8476920 
 
 .001047120 
 
 956 
 
 913936 
 
 873722816 
 
 30.9192497 
 
 9.8511280 
 
 .001046025 
 
 957 
 
 915849 
 
 876467493 
 
 30.9354166 
 
 9.8545617 
 
 .001044932 
 
 958 
 
 917764 
 
 879217912 
 
 30.9515751 
 
 9.8579929 
 
 .001043841 
 
 959 
 
 919681 
 
 881974079 
 
 30.9677251 
 
 9.&614218 
 
 .001042753,, 
 
 ,960 
 
 921600 
 
 884736000 
 
 SO. 9838668 
 
 9.8648483 
 
 .00104166A 
 
 '061 
 
 923521 
 
 887503681 
 
 31.0000000 
 
 9.8682724 
 
 .001040583 
 
 962 
 
 925444 
 
 90277128 
 
 31.0161248 
 
 9.8716941 
 
 .001039501 
 
 063 
 
 927369 
 
 893056347 
 
 31.0322413 
 
 9.87'51135 
 
 .001038422 
 
 964 
 
 929296 
 
 95841344 
 
 31.0483494 
 
 9.8785305 
 
 .001037344 
 
 965 
 
 931225 
 
 198632125 
 
 31.0644491 
 
 9.8819451 
 
 .001036269 
 
 966 
 
 933156 
 
 901428696 
 
 31.0805405 
 
 9.885357'4 
 
 .001035197 
 
 967 
 
 935089 
 
 904231063 
 
 31.0966236 
 
 9.8887673 
 
 .001134126 
 
 968 
 
 937024 
 
 007039232 
 
 31.1126984 
 
 9.8921749 
 
 .00103EC58 
 
 969 
 
 938961 
 
 909853209 
 
 31.1287648 
 
 9.8955801 
 
 .(J01031092 
 
 070 
 
 940900 
 
 912673000 
 
 31.1448230 
 
 9.8989830 
 
 .C0103C928 
 
 971 
 
 942841 
 
 915498611 
 
 31.16C6729 
 
 9.9C28835 
 
 .00102C8G6 
 
 972 
 
 944784 
 
 918330048 
 
 31.1769145 
 
 9.0057817 
 
 .001C2fc807 
 
 973 
 
 946729 
 
 921167317 
 
 31.1929479 
 
 9.9091776 
 
 .001027749 
 
 974 
 
 948676 
 
 924010424 
 
 31.2089731 
 
 9.9125712 
 
 .C01 026694 
 
 975 
 
 950625 
 
 926859375 
 
 31.2249900 
 
 9.9150624 
 
 .C01025641 
 
 976 
 
 952576 
 
 929714176 
 
 31.2409987 
 
 9.9108513 
 
 .001024590 
 
 977 
 
 954529 
 
 932574833 
 
 31.2569992 
 
 9.0827379 
 
 .001023541 
 
 978 
 
 956484 
 
 935441352 
 
 31.2729915 
 
 9.0261222 
 
 .001022495 
 
 979 
 
 958441 
 
 038313739 
 
 31.2889757 
 
 9.0295042 
 
 .001021450 
 
 980 
 
 960400 
 
 941192000 
 
 31.3049517 
 
 9.9328839 
 
 .001020408 
 
 981 
 
 '962361 
 
 944076141 
 
 31.3209195 
 
 9.9362613 
 
 .001010368 
 
 082 
 
 964324 
 
 946966168 
 
 31.3368792 
 
 9.9396363 
 
 .001018330 
 
 9a3 
 
 966289 
 
 9498G2087 
 
 31.3528308 
 
 9.9430092 
 
 .001017294 
 
 984 
 
 968256 
 
 952763904 
 
 31.3687743 
 
 9.9463797 
 
 .001016260 
 
 985 
 
 970225 
 
 955671 C25 
 
 31.3847097 
 
 9.9497479 
 
 .001015228 
 
 986 
 
 972196 
 
 958585256 
 
 31.4006369 
 
 9.9531138 
 
 .001014199 
 
 987 
 
 974169 
 
 961504803 
 
 31.4165561 
 
 9.9564775 
 
 .001013171 
 
 988 
 
 976144 
 
 964430272 
 
 31.4324673 
 
 9.9598389 
 
 .001012146 
 
 989 
 
 978121 
 
 967361669 
 
 31.4483704 
 
 9.9631981 
 
 .001011122 
 
 990 
 
 980100 
 
 970299000 
 
 31.4642654 
 
 9.9665549 
 
 .001010101 
 
 \801 
 
 982081 
 
 973242271 
 
 31.4801525 
 
 9.9699095 
 
 .001009082f' 
 
 992 
 
 984064 
 
 976191488 
 
 31.4960315 
 
 9.9732619 
 
 .001008066 
 
 193 
 
TABLE VIII. Continued. 
 
 >. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 Roots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 993 
 
 986049 
 
 979146657 
 
 31.5119025 
 
 9.9766120 
 
 .001007049 
 
 994 
 
 988036 
 
 982107784 
 
 31.5277655 
 
 9.9799599 
 
 .001006036 
 
 995 . 
 
 990025 
 
 085074875 
 
 31.5436206 
 
 9.9833055 
 
 .001005025 
 
 996 
 
 992010 
 
 988047936 
 
 31.5594677 
 
 9.9866488 
 
 .001004016 
 
 997 
 
 994003 
 
 991026973 
 
 31.5753068 
 
 9.9899900 
 
 .001003009 
 
 993 
 
 99G004 
 
 994011992 
 
 31.5911380 
 
 9.9933289 
 
 .001002004 
 
 999 
 
 998301 
 
 997002999 
 
 31.6069613 
 
 9.99GG056 
 
 . .001001001 
 
 1000 
 
 1000000 
 
 1000000000 
 
 31.6227766 
 
 10.0000000 
 
 .001000000 
 
 1001 
 
 1002001 
 
 1003003001 
 
 31.6385840 
 
 10.0033322 
 
 .0009990010 
 
 100 2 
 
 1034004 
 
 1006012008 
 
 31.6543836 
 
 10.0066622 
 
 .0009980040 
 
 1003 
 
 1006009 
 
 1009027027 
 
 31.6701752 
 
 10.0099899 
 
 .0009970090 
 
 1004 
 
 1008016 
 
 1012J48064 
 
 31.6859590 
 
 10.0133155 
 
 .0009960159 
 
 1005 
 
 1010025 
 
 1015075125 
 
 31.7017349 
 
 10.0166389 
 
 .0009950249 
 
 1008 
 
 1012036 
 
 1018108216 
 
 31.7175030 
 
 10.0199601 
 
 .0009940358 
 
 1007 
 
 1014049 
 
 1021147'343 
 
 31.7332633 
 
 10.0232791 
 
 .0009930487 
 
 1003 
 
 1016064 
 
 1024192512 31.7490157 
 
 10.0265958 
 
 .0009920635 
 
 1009 
 
 1018081 
 
 1027243729 31.7647603 
 
 10.0299104 
 
 .0009910803 
 
 1010 
 
 1020100 
 
 1030301000 j 31.7804972 
 
 10.0332228 
 
 .0009900990 
 
 1011 
 
 1022121 
 
 1033364331 ] 31.79622G2 
 
 10.0365330 
 
 .0009891197 
 
 1012 
 
 1024144 
 
 103S433723 
 
 31.8119474 
 
 10.0398410 
 
 .0009881423 
 
 1013 
 
 1026169 
 
 1033509197 
 
 31.8276609 
 
 10.0431469 
 
 .0009871668 
 
 1014 
 
 1028196 
 
 1042593744 
 
 31.8433666 
 
 10.0464506 
 
 .0009861933 
 
 1015 
 
 1030225 
 
 1045678375 
 
 31.8590646 
 
 10.0497521 
 
 .0009852217 
 
 1016 
 
 103-2256 
 
 1048772096 
 
 31.8747549 
 
 10.0530514 
 
 .0009842520 
 
 ioir 
 
 1034289 
 
 1051871913 
 
 31.8904374 
 
 10.0563485 
 
 .0009832842 
 
 1018 
 
 103G324 
 
 1054977832 
 
 31.9061123 
 
 10.0596435 
 
 .0009823183 
 
 1019 
 
 1038361 
 
 1053089859 
 
 31.9217794 
 
 10.0629364 
 
 .0009813543 
 
 .1020 
 
 1040400 
 
 1061208000 
 
 31.9374388 
 
 10.0662271 
 
 .0009803922 
 
 \1021 
 
 1042441 
 
 1064332261 
 
 31.9530906 
 
 10.0695156 
 
 .0009794319 
 
 1022 
 
 1044484 
 
 1067462648 
 
 31.9637347 
 
 10.0728020 
 
 .0009784736 
 
 1023 
 
 1046529 
 
 1070599167 
 
 31.9843712 
 
 10.0760863 
 
 .0009775171 
 
 1024 
 
 1048576 
 
 1073741824 
 
 32.1.000000 
 
 10.0793G84 
 
 .0009765625 
 
 1025 
 
 1050625 
 
 1076890625 
 
 32.0156212 
 
 10.0826484 
 
 .0009756098 
 
 1026 
 
 1052676 
 
 1080045576 
 
 32.0312348 
 
 10.0859262 
 
 .0009746589 
 
 1027 
 
 1054729 
 
 1083206683 
 
 32.0468407 
 
 10.0892019 
 
 .0009737098 
 
 1028 
 
 1056784 
 
 1086373952 
 
 32.0624391 
 
 10.0924755 
 
 .0009727626 
 
 1029 
 
 1058841 
 
 1039547389 
 
 32.0780298 
 
 10.095746D 
 
 .0009718173 
 
 1030 
 
 1060900 
 
 1092727000 
 
 32.0936131 
 
 10.0990163 
 
 .0009708733 
 
 1031 
 
 1062961 
 
 1095912791 
 
 32.1091887 
 
 10.1022835 
 
 .0009699321 
 
 1032 
 
 1065024 
 
 1099104768 
 
 32.1247568 
 
 10.1055487 
 
 .00 9689922 
 
 1033 
 
 1067089 
 
 1102302937 
 
 32.1403173 
 
 10.1088117 
 
 .0009680542 
 
 1034 
 
 1069156 
 
 1105507304 
 
 32.1558704 
 
 10.1120726 
 
 .0009671180 
 
 1035 
 
 1071225 
 
 1108717875 
 
 32.1714159 
 
 10.1153314 
 
 .0009661836 
 
 1036 
 
 1073296 
 
 1111934656 
 
 32.1869539 
 
 10.1185882 
 
 .0009652510 
 
 1037 
 
 1075369 
 
 1115157653 
 
 32.2024844 
 
 10.1218428 
 
 .0009643202 
 
 1038 
 
 1077444 
 
 1118386872 
 
 32.2180074 
 
 10.1250953 
 
 .0009633911 
 
 1039 
 
 1079521 
 
 1121622319 
 
 32.2335229 
 
 10.1283457 
 
 .0009624639 
 
 1040 
 
 1081600 
 
 1124864000 
 
 32.2490310 
 
 10.1315941 
 
 .0009615385 
 
 1041 
 
 1083681 
 
 1128111921 
 
 32.2645316 
 
 10.1348403 
 
 .0009606148 
 
 1042 
 
 1085764 
 
 1131366088 
 
 32.2800248 
 
 10.1380845 
 
 . .0009596929 
 
 1043 
 
 1087849 
 
 1134626507 
 
 32.2955105 
 
 10.1413266 
 
 .0009587733 
 
 1044 
 
 1089936 
 
 1137893184 
 
 32.3109888 
 
 10.1445667 
 
 .0009578544 
 
 1045 
 
 1092025 
 
 1141166125 
 
 32.3264598 
 
 10.1478047 
 
 .0009569378 
 
 1046 
 
 1094116 
 
 1144445336 
 
 32.3419233 
 
 10.1510406 
 
 .0009560229 
 
 1047 
 
 1096209 
 
 1147730823 
 
 32.a573794 
 
 10.1542744 
 
 .0009551098 
 
 1048 
 
 1098304 
 
 1151022592 
 
 32.3728281 
 
 10.1575062 
 
 .0009541985 
 
 1049 
 
 1100401 
 
 1154320649 
 
 32.3882695 
 
 10.1607359 
 
 .0009532888 
 
 1050 
 
 1102500 
 
 1157625000 
 
 32.4037035 
 
 10.1639636 
 
 .0009523810 
 
 1051 
 
 1104601 
 
 1160935651 
 
 32.4191301 
 
 10.1671893 
 
 .0009514748 
 
 1052 
 
 1106704 
 
 1164252608 
 
 32.4345495 
 
 10.1704129 
 
 - .0009505703 
 
 Viass 
 
 1108809 
 
 1167575877 
 
 32.4499615 
 
 10.1736344 
 
 .0009496676 
 
 1054 
 
 1110916 
 
 1170905464 
 
 32.4653662 
 
 10.1768539 
 
 .0009487666 
 
 194 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 NO123456789 
 
 100 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 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 
 
 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 311 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 
 
 5308 5346 5385 5423 5461 5500 5538 5576 5614 5(552 
 
 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 
 
 3672 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 5806 5897 5927 5957 5987 6017 6047 6077 6107 
 
 5 6137 6167 6197 6227^ 6256 6286 6316 6346 6376 6406 
 
 6 6435 6465 6495 6524 6554 6584 6613 6643 6673 6702 
 
 7 6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 
 
 8 7026 7056 7085 7114 7143 7173 7202 7231 7260 7289 
 
 9 7319 7348 7377 7406 7435 7464 7493 7522 7551 7580 
 
 >0 ! 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 NO1234567 ~~8 
 
 150 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 
 
 1 7898 7926 7955 7984 8013 8041 8070 8099 8127 8156 
 
 2 8184 8213 8241 8270 8298 8327 8355 8384 8412 8441 
 
 3 8469 8498 8526 8554 8583 8611 8639 8667 8696 8724 
 
 4 8752 8780 8808 8837 8865 8893 8921 8949 8977 9005 
 
 5 9033 9061 9089 9117 9145 9173 9201 9229 9257 9285 
 
 6 9312 9340 9368 9396 9424 9451 9479 9507 9535 9562 
 
 7 9590 9618 9(545 9673 9700 9728 9756 9783 9811 9838 
 
 8 9866 9893 9921 9948 99762000320030200582008520112 
 
 9 20140 20167 20194 20222 20249 0276 0303 0330 0358 0385 
 
 160 20412 20439 20466 20493 20520 20548 20575 20602 20629 20656 
 
 1 0683 0710 0737 0763 0790 0817 0844 0871 0898 0925 
 
 2 0952 0978 1005 1032 1059 1085 1112 1139 1165 1192 
 
 3 1219 1245 1272 1299 1325 1352 1378 1405 1431 1458 
 
 4 1484 1511 1537 1564 1590 1617 1643 1669 1696 1722 
 
 5 1748 1775 1801 1827 1854 1880 1906 1932 1958 1985 
 
 6 2011 2037 2063 2089 2115 2141 2167 2194 2220 2246 
 
 7 2272 2298 2324 2350 2376 2401 2427 2453 2479 2505 
 
 8 2531 2557 2583 2608 2634 2660 2686 2712 2737 2763 
 
 9 2789 2814 2840 2866 2891 2917 2943 2968 2994 3019 
 
 170 23045 2307JO 23096 23121 23147 23172 23198 23223 23249 23274 
 
 1 3300 3325 3350 3376 3401 3426 3452 3477 3502 3528 
 
 2 3553 3578 3603 3629 3654 3679 3704 3729 3754 3779 
 
 3 3805 3830 3855 3880 3905 3930 3955 3980 4005 4030 
 
 4 4055 4080 4105 4130 4155 4180 4204 4229 4254 4279 
 
 5 4304 4329 4353 4378,4403 4428 4452 4477 4502 4527 
 
 6 4551 4576 4601 4625 4650 4674 4699 4724 4748 4773 
 
 7 4797 4822 4846 4871 4895 4920 4944 4969 4993 5018 
 
 8 5042 5066 5091 5115 5139 5164 5188 5212 5237 5261 
 
 9 5285 5310 5334 5358 5382 5406 5431 5455 5479 5503 
 
 180 25527 25551 25575 25600 25624 25648 25672 25696 25720 25744 
 
 1 5768 5792 5816 5840 5864 5888 5912 5935 5959 5983 
 
 2 6007 6031 6055 6079 6102 6126 6150 6174 6198 6221 
 
 3 6245 6269 6293 6316 6340 6364 6387 6411 6435 6458 
 
 4 6482 6505 6529 6553 6576 6600 6623 6647 6670 6694 
 
 5 6717 6741 6764 6788 6811 6834 6858 6881 6905 6928 
 
 6 6951 6975 6998 7021 7045 7068 7091 7114 7138 7161 
 
 7 7184 7207 7231 7254 7277 7300 7323 7346 7370 7393 
 
 8 7416 7439 7462 7485 7508 7531 7554 7577 7600 7623 
 
 9 7646 7669 7692 7715 7738 7761 7784 7807 7830 7852 
 
 190 27875 27898 27921 27944 27967 27989 28012 28035 28058 28081 
 
 1 8103 8126 8149 8171 8194 8217 8240 8262 8285 8307 
 
 2 8330 8353 8375 8398 8421 8443 8466 8488 8511 8533 
 
 3 8556 8578 8601 8623 8646 8668 8691 8713 8735 8758 
 
 4 8780 8803 8825 8847 8870 8892 8914 8937 8959 8981 
 
 5 9003 9026 9048 9070 9092 9115 9137 9159 9181 9203 
 
 6 9226 9248 9270 9292 9314 933(5 9358 9380 9403 9425 
 
 7 9447 9469 9491 9513 9535 9557 9579 9601 9623 9645 
 
 8 9667 9688 9710 9732 9754 9776 9798 9820 9842 9863 
 
 9 9885 9907 9929 9951 9973 9994 30016 30038 30060 30081 
 
 200 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 
 
 196 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 3456789 
 
 200 
 
 1 
 2 
 3 
 4 
 5 
 
 9 
 
 210 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 230 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 
 
 0320 0341 0363 0384 0406 0428 0449 0471 0492 0514 
 
 0535 0557 0578 0600 0621 0643 0664 0685 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 1534 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 
 
 250 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 
 
 197 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 260 
 1 
 2 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 270 
 1 
 
 2 
 3 
 
 4 
 5 
 
 280 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 678 
 
 39794 39811 39829 39846 39803 39881 39898 39915 39933 39950 
 
 9967 9985 40002 40019 40037 40054 40071 40088 40106 41)123 
 
 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 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 N 
 
 0123456789 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 310 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 320 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 
 9 
 
 340 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 
 
 7857 7871 7885 7900 7914 7929 7943 7958 7972 7986 
 
 8001 8015 8029 8044 8058 8073 8087 8101 8116 8130 
 
 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 970,7 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 50569 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 4:532 4345 4357 4370 4382 4394 
 
 54407 54419 54432 54444 54456 54469 54481 54494 54506 54518 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 56789 
 
 54407 54419 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 5400 5413 5425 5437 5449 5461 5473 5485 5497 
 
 5509 5522 5534 5546 5558 5570 5582 5594 5606 5618 
 
 55630 55642 55654 55666 55678 55691 55703 55715 55727 55739 
 
 5751 5763 5775 5787 5799 5811 5823 5835 5847 5859 
 
 5871 5883 5895 5907 5919 5931 5943 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 
 
 7237 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 9632 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 
 
 9 | 60097 60108 0119 0130 0141 0152 0163 0173 0184 0195 
 
 400 i 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 N0123456789 
 
 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 
 
 4 0638 0649 0660 0670 0681 0692 0703 0713 0724 0735 
 
 5 0746 0756 0767 0778 0788 0799 0810 0821 0831 0842 
 0853 0863 0874 0885 0895 0906 0917 0927 0938 0949 
 
 7 0959 0970 0981 0991 1002 1013 1023 1034 1045 1055 
 
 1066 1077 1087 1098 1109 1119 1130 1140 1151 1162 
 
 9 1172 1183 1194 1204 1215 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 
 
 3 1595 1606 1616 1627 1637 1648 1658 1669 1679 1690 
 
 4 1700 1711 1721 1731 1742 1752 1763 1773 1784 1794 
 
 5 1805 1815 1826 1836 1847 1857 1868 1878 1888 1899 
 
 6 1909 1920 1930 1941 1951 1962 1972 1982 1993 2003 
 
 7 2014 2024 2034 2045 2055 2066 2076 2086 2097 2107 
 
 8 2118 2128 2138 2149 2159 2170 2180 2190 2201 2211 
 
 9 2221 2232 2242 2252 2263 2273 2284 2294 2304 2315 
 
 420 62325 62335 62316 62356 62366 62377 62387 62397 62408 62418 
 
 1 2428 2439 2449 2459 2469 2480 2490 2500 2511 2521 
 
 2 2531 2542 2552 2562 2572 2583 2593 2603 2613 2624 
 
 3 2634 2644 2655 2665 2675 2685 2696 2706 2716 2726 
 
 4 2737 2747 2757 2767 2778 2788 2798 2808 2818 2829 
 
 5 2839 2849 2859 2870 2880 2890 2900 2910 2921 2931 
 
 6 2941 2951 2961 2972 2982 2992 3002 3012 3022 3033 
 
 7 3043 3053 3063 3073 3083 3094 3104 3114 3124 3134 
 
 8 3144 3155 3165 3175 3185 3195 3205 3215 3225 3236 
 
 9 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 
 
 4 3749 3759 3769 3779 3789 3799 3809 3819 3829 3839 
 
 5 3849 3859 3809 3879 3889 3899 3909 3919 3929 3939 
 
 6 3949 3959 3969 3979 3988 3998 4008 4018 4028 4038 
 
 7 4048 4058 4068 4078 4088 4098 4108 4118 4128 4137 
 
 8 4147 4157 4167 4177 4187 4197 4207 4217 4227 4237 
 
 9 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 
 
 2 4542 4552 4562 4572 4582 4591 4601 4611 4621 4631 
 
 3 4640 4650 4660 4670 4680 4689 4699 4709 4719 4729 
 
 4 4738 4748 4758 4768 4777 4787 4797 4807 4816 4826 
 
 5 4836 4846 4856 4865 4875 4885 4895 4904 4914 4924 
 
 6 4933 4943 4953 4963 4972 4982 4992 5002 5011 5021 
 
 7 5031 5040 5050 5060 5070 5079 5089 5099 5108 5118 
 
 8 5128 5137 5147 5157 5167 5176 5186 5196 5205 5215 
 
 9 5225 5234 5244 5254 5263 5273 5283 5292 5302 5312 
 
 450 i 65321 65331 65341 65350 65360 65369 65379 65389.65398 65408 
 
 201 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 6789 
 
 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 5772 5782 5792 
 
 5801 5811 5820 5830 5839 5849 5858 5868 5877 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 7330 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 8511 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 
 
 500 69897 69906 69914 69923 69932 69940 69949 69958 69966 69975 
 
 202 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 N0123456789 
 
 500 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 520 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 
 9 
 
 7 
 8 
 9 
 
 540 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 69897 69906 69914 69923 699:52 69940 69949 69958 69966 69975 
 
 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 026;) 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 
 
 2263 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 
 
 2509 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 3^23 3231 
 
 73239 73247 73255 73263 73272 73280 73288 73296 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 3656 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 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 NO123456789 
 
 550 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 
 9 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 570 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 580 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 
 
 4115 4123 4131 4139 4147 4155 4162 4170 4178 4186 
 
 4194 4202 4210 4218 4225 4233 4241 4249 4257 4265 
 
 4273 4280 4288 4296 4304 4312 4320 4327 4335 4343 
 
 4351 4359 4367 4374 4382 4390 4398 4406 4414 4421 
 
 4429 4437 4445 4453 4461 4468 4476 4484 4492 4500 
 
 4507 4515 4523 4531 4539 4547 4554 4562 4570 4578 
 
 4586 4593 4601 4609 4617 4624 4632 4640 4648 4656 
 
 4663 4671 4679 4687 4695 4702 4710 4718 4726 4733 
 
 4741 4749 4757 4764 4772 4780 4788 4796 4803 4811 
 
 74819 74827 74834 74842 74850 74858 74865 74873 74881 74889 
 
 4896 4904 4912 4920 4927 4935 4943 4950 4958 4966 
 
 4974 4981 4989 4997 5005 5012 5020 5028 5035 5043 
 
 5051 5059 5066 5074 5082 5089 5097 5105 5113 5120 
 
 5128 5136 5143 5151 5159 5166 5174 5182 5189 5197 
 
 5205 5213 5220 5228 5236 5243 5251 5259 5266 5274 
 
 5282 5289 5297 5305 5312 5320 5328 5335 5343 5351 
 
 5358 5366 5374 5381 5389 5397 5404 5412 5420 5427 
 
 5435 5442 5450 5458 5465 5473 5481 5488 5496 5504 
 
 5511 5519 5526 5534 5542 5549 5557 5565 5572 5580 
 
 75587 75595 75603 75610 75618 75626 75633 75641 75648 75656 
 
 5664 5671 5679 5686 5694 5702 5709 5717 5724 5732 
 
 5740 5747 5755 5762 5770 5778 5785 5793 5800 5808 
 
 5815 5823 5831 5838 5846 5853 5861 5868 5876 5884 
 
 5891 5899 5906 5914 5921 5929 5937 5944 5952 5959 
 
 5967 5974 5982 5989 5997 6005 6012 6020 6027 6035 
 
 6042 6050 6057 6065 6072 6080 6087 6095 6103 6110 
 
 6118 6125 6133 6140 6148 6155 6163 6170 6178 6185 
 
 6193 6200 6208 6215 6223 6230 6238 6245 6253 6260 
 
 6268 6275 6283 6290 6298 6305 6313 6320 6328 6335 
 
 76343 76350 76358 76365 76373 76380 76388 76395 76403 76410 
 
 6418 6425 6433 6440 6448 6455 6462 6470 6477 6485 
 
 6492 6500 6507 6515 6522 6530 6537 6545 6552 6559 
 
 6567 6574 6582 6589 6597 6604 6612 6619 6626 6634 
 
 6641 6649 6656 6664 6671 6678 6686 6693 6701 6708 
 
 6716 6723 6730 6738 6745 6753 6760 6768 6775 6782 
 
 6790 6797 6805 6812 6819 6827 6834 6842 6849 6856 
 
 6864 6871 6879 6886 6893 6901 6908 6916 6923 6930 
 
 6938 6945 6953 6960 6967 6975 6982 6989 6997 7004 
 
 7012 7019 7026 7034 7041 7048 7056 7063 7070 7078 
 
 77085 77093 77100 77107 77115 77122 77129 77137 77144 77151 
 
 7159 7166 7173 7181 7188 7195 7203 7210 7217 7225 
 
 7232 7240 7247 7254 7262 7269 7276 7283 7291 7298 
 
 7305 7313 7320 7327 7335 7342 7349 7357 7364 7371 
 
 7379 7386 7393 7401 7408 7415 7422 7430 7437 7444 
 
 7452 7459 7466 7474 7481 7488 7495 7503 7510 7517 
 
 7525 7532 7539 7546 7554 7561 7568 7576 7583 7590 
 
 7597 7605 7612 7619 7627 7634 7641 7648 7656 7663 
 
 7670 7677 7685 7692 7699 7706 7714 7721 7728 7735 
 
 7743 7750 7757 7764 7772 7779 7786 7793 7801 7808 
 
 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 
 2(H 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 6789 
 
 600 
 
 1 
 
 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 
 7887 7895 7902 7909 7916 7924 7931 7938 7945 7952 
 
 2 7960 7967 7974 7981 7988 7996 8003 8010 8017 8025 
 
 3 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 
 
 1 
 2 
 
 3 \ 8746 8753 8760 8767 8774 8781 8789 8796 8803 8810 
 
 8817 8824 8831 8838 8845 8852 8859 8866 8873 8880 
 
 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 9341 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 95(50 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 80638 80645 80652 80659 80665 80672 80679 
 
 0686 06D3 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 
 
 0895 0002 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 
 
 050 81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 
 
 iius 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 0123456789 
 
 (>50 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 
 680 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 
 9 
 
 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 3503 
 
 3569 3575 3582 3588 3594 3601 3607 3613 3620 3626 
 
 3632 3639 3645 3651 3658 3664 3670 3677 3683 3689 
 
 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 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 NO123456789 
 
 00 
 
 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 
 8 
 9 
 
 730 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 740 
 
 1 
 2 
 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 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 5467 5473 5479 5485 
 
 5491 5497 5503 5509 5516 5522 5528 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 5908 
 
 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 
 207 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 STO12345678 
 
 790 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 
 
 7564 7570 7576 7581 7587 7593 7599 7604 7610 7616 
 
 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 
 
 8988 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 9f,<)2 
 
 9597 9603 9609 9614 9620 9625 9631 9636 9642 9047 
 
 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 
 
 208 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 56789 
 
 810 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 820 
 1 
 2 
 3 
 4 
 5 
 
 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 
 
 6 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 
 
 I960 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 
 
 "209 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 NO123456789 
 
 850 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 
 860 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 
 9 
 
 92942 92947 92952 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 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 0123456789 
 
 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 
 
 5472 5477 5482 5487 5492 5497 5501 5506 5511 5516 
 
 5521 5525 5530 5535 5540 5545 5550 5554 5559 5564 
 
 5569 6574 5578 5583 5588 5593 5598 5602 5607 6612 
 
 5617 5622 5626 5631 5636 5641 5646 5650 5655 5660 
 
 5665 5670 5674 5679 5684 5689 5694 5698 5703 5708 
 
 5713 5718 5722 5727 5732 5737 5742 5746 5751 5756 
 
 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 95938 95942 95947 
 
 5952 5957 5961 5966 5971 5976 5980 5985 5990 5995 
 
 5999 6004 6009 6014 6019 6023 6028 6033 6038 6042 
 
 6047 6052 6057 6061 6066 6071 6076 6080 6085 6090 
 
 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 
 
 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 7085 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 
 
TABLE IX. LOGARITHMS OF NUMBERS. 
 
 0123456789 
 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 960 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 980 
 
 1 
 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 
 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 , 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 1000: 00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 
 
 "212 
 
 9695 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 
 
TABLE X. SINES AND COSINES. 
 
 
 
 
 1 
 
 
 
 2 
 
 3 
 
 4 
 
 / 
 
 Sine Cosin 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 
 o 
 
 00000 
 
 One. 
 
 01745 
 
 99985 
 
 03490 
 
 99939 
 
 05234 
 
 99863 
 
 06976 
 
 99756 
 
 60 
 
 1 
 
 U0029 
 
 One. 
 
 01774 
 
 99984 
 
 03519 
 
 99938 
 
 05263 
 
 99861 
 
 07005 
 
 99754 
 
 59 
 
 2 
 
 00058 
 
 One. 
 
 01803 
 
 99984 
 
 03548 
 
 99937 
 
 05292 
 
 99860 
 
 07034 
 
 99752 
 
 58 
 
 3 
 
 00087 
 
 One. 
 
 01832 
 
 99983 
 
 03577 
 
 99936 
 
 05321 
 
 99858 
 
 07063 
 
 99750 
 
 57 
 
 4 
 
 00116 
 
 One. 
 
 01862 
 
 99983 
 
 03606 
 
 99935 
 
 05350 
 
 99857 
 
 07092 
 
 99748 
 
 56 
 
 5 
 
 00145 
 
 One. 
 
 01891 
 
 99982 
 
 03635 
 
 99934 
 
 05379 
 
 99855 
 
 07121 
 
 99746 
 
 55 
 
 6 
 
 00175 
 
 One. 
 
 01920 
 
 99982 
 
 03664 
 
 99933 
 
 05408 
 
 99854 
 
 07150 
 
 99744 
 
 54 
 
 7 
 
 00204 
 
 One. 
 
 01949 
 
 99981 
 
 03693 
 
 99932 
 
 05437 
 
 99852 
 
 07179 
 
 99742 
 
 53 
 
 8 
 
 00233 
 
 One. 
 
 01978 
 
 99980 
 
 03723 
 
 99931 
 
 05466 
 
 99851 
 
 07208 
 
 99740 
 
 52 
 
 9 
 
 00262 
 
 One. 
 
 02007 
 
 99980 
 
 03752 
 
 99930 
 
 05495 
 
 99849 
 
 07237 
 
 99738 
 
 51 
 
 iO 
 
 00291 
 
 One. 
 
 02036 
 
 99979 
 
 03781 
 
 99929 
 
 05524 
 
 99847 
 
 07266 
 
 99736 
 
 50 
 
 11 
 
 00320 
 
 99999 
 
 02065 
 
 99979 
 
 OS810 
 
 99927 
 
 05553 
 
 99846 
 
 07295 
 
 99734 
 
 49 
 
 12 
 
 00349 
 
 99999 
 
 02094 
 
 99978 
 
 03839 
 
 99926 
 
 05582 
 
 99844 
 
 07324 
 
 99731 
 
 48 
 
 13 
 
 00378 
 
 99999 
 
 02123 
 
 99977 
 
 03868 
 
 99925 
 
 05611 
 
 99842 
 
 07353 
 
 99729 
 
 47 
 
 14 
 
 00407 
 
 99999 
 
 02152 
 
 99977 
 
 03897 
 
 99924 
 
 05640 
 
 99841 
 
 07382 
 
 99727 
 
 46 
 
 15 
 
 00436 
 
 99999 
 
 02181 
 
 99976 
 
 '03926 
 
 99923 
 
 05669 
 
 99839 
 
 07411 
 
 99725 
 
 45 
 
 16 
 
 00465 
 
 99999 
 
 02211 
 
 99976 
 
 03955 
 
 99922 
 
 05698 
 
 99838 
 
 07440 
 
 99723 
 
 44 
 
 17 
 
 00495 
 
 99999 
 
 02240 
 
 99975 
 
 03984 
 
 99921 
 
 05727 
 
 99836 
 
 07469 
 
 99721 
 
 43 
 
 18 
 
 00524 
 
 99999 
 
 02269 
 
 99974 
 
 04013 
 
 99919 
 
 05756 
 
 99834 
 
 07498 
 
 99719 
 
 42 
 
 19 
 
 00553 
 
 99998 
 
 02298 
 
 99974 
 
 04042 
 
 99918 
 
 05785 
 
 99833 
 
 07527 
 
 99716 
 
 41 
 
 20 
 
 00582 
 
 99998 
 
 02327 
 
 99973 
 
 04071 
 
 99917 
 
 05814 
 
 99831 
 
 07556 
 
 99714 
 
 40 
 
 21 
 
 00611 
 
 99998 
 
 02356 
 
 99972 
 
 04100 
 
 99916 
 
 05S44 
 
 99829 
 
 07585 
 
 99712 
 
 39 
 
 22 
 
 00640 
 
 99998 
 
 02385 
 
 99972 
 
 04129 
 
 99915 
 
 05873 
 
 99827 
 
 07614 
 
 99710 
 
 38 
 
 23 
 
 00669 
 
 99998 
 
 02414 
 
 99971 
 
 04159 
 
 99913 
 
 05902 
 
 99826 
 
 07643 
 
 99708 
 
 37 
 
 24 
 
 00698 
 
 99998 
 
 02443 
 
 99970 
 
 04188 
 
 99912 
 
 05931 
 
 99824 
 
 07672 
 
 99705 
 
 36 
 
 25 
 
 00727 
 
 99997 
 
 02472 
 
 99969 
 
 04217 
 
 99911 
 
 05960 
 
 99822 
 
 07701 
 
 99703 
 
 35 
 
 26 
 
 00756 
 
 99997 
 
 02501 
 
 99969 
 
 04246 
 
 99910 
 
 05989 
 
 99821 
 
 07730 
 
 99701 
 
 34 
 
 27 
 
 00785 
 
 99997 
 
 02530 
 
 99968 
 
 04275 
 
 99909 
 
 06018 
 
 99819 
 
 07759 
 
 99699 
 
 33 
 
 28 
 
 00814 
 
 99997 
 
 02560 
 
 99967 
 
 04304 
 
 99907 
 
 06047 
 
 99817 
 
 07788 
 
 99696 
 
 32 
 
 29 
 
 00844 
 
 99996 
 
 02589 
 
 99966 
 
 04333 
 
 99906 
 
 06076 
 
 99815 
 
 07817 
 
 99694 
 
 31 
 
 30 
 
 00873 
 
 99996 
 
 02618 
 
 99966 
 
 04362 
 
 99905 
 
 06105 
 
 99813 
 
 07846 
 
 99692 
 
 30 
 
 31 
 
 00902 
 
 99996 
 
 02647 
 
 99965 
 
 04391 
 
 99904 
 
 06134 
 
 99812 
 
 07875 
 
 99689 
 
 29 
 
 32 
 
 00931 
 
 99996 
 
 02676 
 
 99964 
 
 04420 
 
 99902 
 
 06163 
 
 99810 
 
 07904 
 
 99687 
 
 28 
 
 33 
 
 00960 
 
 99995 
 
 02705 
 
 99963 
 
 04449 
 
 99901 
 
 06192 
 
 99808 
 
 07933 
 
 99685 
 
 27 
 
 34 
 
 00989 
 
 99995 
 
 02734 
 
 99963 
 
 04478 
 
 99900 
 
 06221 
 
 99806 
 
 07962 
 
 99683 
 
 26 
 
 35 
 
 01018 
 
 99995 
 
 02763 
 
 99962 
 
 04507 
 
 99898 
 
 06250 
 
 99804 
 
 07991 
 
 99680 
 
 85 
 
 36 
 
 01047 
 
 99995 
 
 02792 
 
 99961 
 
 04536 
 
 99897 
 
 06279 
 
 99803 
 
 08020 
 
 99678 
 
 24 
 
 37 
 
 01076 
 
 99994 
 
 02821 
 
 99960 
 
 04565 
 
 99896 
 
 06308 
 
 99801 
 
 08049 
 
 99676 
 
 23 
 
 38 
 
 01105 
 
 99994 
 
 02850 
 
 99959 
 
 04594 
 
 99894 
 
 06337 
 
 99799 
 
 08078 
 
 99673 
 
 22 
 
 39 
 
 01134 
 
 99994 
 
 02879 
 
 99959 
 
 04623 
 
 99893 
 
 06366 
 
 99797 
 
 08107 
 
 99671 
 
 21 
 
 40 
 
 01164 
 
 99993 
 
 02908 
 
 99958 
 
 04653 
 
 99892 
 
 06395 
 
 99795 
 
 08136 
 
 99668 
 
 20 
 
 41 
 
 01193 
 
 99993 
 
 02938 
 
 99957 
 
 04682 
 
 99890 
 
 06424 
 
 99793 
 
 08165 
 
 99666 
 
 19 
 
 42 
 
 01222 
 
 99993 
 
 02967 
 
 99956 
 
 04711 
 
 99889 
 
 06453 
 
 99792 
 
 08194 
 
 99664 
 
 18 
 
 43 
 
 01251 
 
 99992 
 
 02996 
 
 99955 
 
 04740 
 
 99888 
 
 06482 
 
 99790 
 
 08223 
 
 99661 
 
 17 
 
 44 
 
 01280 
 
 99992 
 
 03025 
 
 99954 
 
 04769 
 
 99886 
 
 06511 
 
 99788 
 
 08252 
 
 99659 
 
 16 
 
 45 
 
 01309 
 
 99991 
 
 03054 
 
 99953 
 
 04798 
 
 99885 
 
 06540 
 
 99786 
 
 08281 
 
 99657 
 
 15 
 
 46 
 
 01338 
 
 99991 
 
 03083 
 
 99952 
 
 04827 
 
 99883 
 
 06569 
 
 99784 
 
 08310 
 
 99654 
 
 14 
 
 47 
 
 01367 
 
 99991 
 
 03112 
 
 99952 
 
 04856 
 
 99882 
 
 06598 
 
 99782 
 
 08339 
 
 99652 
 
 13 
 
 48 
 
 01396 
 
 99990 
 
 03141 
 
 99951 
 
 04885 
 
 99881 
 
 06627 
 
 99780 
 
 08368 
 
 99649 
 
 12 
 
 49 
 
 01425 
 
 99990 
 
 03170 
 
 99950 
 
 04914 
 
 99879 
 
 06656 
 
 99778 
 
 08397 
 
 99647 
 
 11 
 
 50 
 
 01454 
 
 99989 
 
 03199 
 
 99949 
 
 04943 
 
 99878 
 
 06685 
 
 99776 
 
 08426 
 
 99644 
 
 10 
 
 51 
 
 01483 
 
 99989 
 
 03228 
 
 99948 
 
 04972 
 
 99876 
 
 06714 
 
 99774 
 
 08455 
 
 99642 
 
 9 
 
 52 
 
 01513 
 
 99989 
 
 
 99947 
 
 05001 
 
 99875 
 
 06743 
 
 99772 
 
 08484 
 
 99639 
 
 8 
 
 53 
 
 01542 
 
 99988 
 
 03286 
 
 99946 
 
 05030 
 
 39873 
 
 06773 
 
 99770 
 
 08513 
 
 99637 
 
 7 
 
 54 
 
 01571 
 
 99988 
 
 03316 
 
 99945 
 
 05059 
 
 99872 
 
 06802 
 
 99768 
 
 08542 
 
 99635 
 
 6 
 
 55 
 
 0160G 
 
 99987 
 
 03345 
 
 99944 
 
 05088 
 
 99870 
 
 06831 
 
 99766 
 
 08571 
 
 99632 
 
 5 
 
 56 
 
 01629 
 
 99987 
 
 03374 
 
 99943 
 
 05117 
 
 99869 
 
 06860 
 
 99764 
 
 08600 
 
 99630 
 
 4 
 
 57 
 
 01658 
 
 99986 
 
 03403 
 
 99942 
 
 05146 
 
 99867 
 
 06889 
 
 99762 
 
 08629 
 
 99627 
 
 3 
 
 58 
 
 01687 
 
 99986 
 
 03432 
 
 99941 
 
 05175 
 
 99866 
 
 06918 
 
 99760 
 
 08658 
 
 99625 
 
 2 
 
 59 
 
 01716 
 
 99985 
 
 03461 
 
 99940 
 
 05205 
 
 99864 
 
 06947 
 
 99758 
 
 0868V 
 
 99622 
 
 1 
 
 60 
 
 01745 
 
 99985 
 
 03490 
 
 99939 
 
 05234 
 
 99863 
 
 06976 
 
 99756 
 
 08716 
 
 99619 
 
 J) 
 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 89 88 
 
 87 
 
 86' 
 
 85 
 
 
 213 
 
TABLE X. SINES AND COSINES. 
 
 37 
 
 Sine 
 0871 
 08745 
 08774 
 
 13975 
 14004 
 14033 
 14061 
 14090 
 14119 
 14148 
 14177 
 14205 
 
 99607 
 99604 
 99602 
 99599 
 
 10771 
 10800 
 10829 
 10858 
 10887 
 10916 
 10945 
 10973 
 11002 
 11031 
 
 12504 
 12533 
 12562 
 12591 
 
 15959 
 15988 
 16017 
 16046 
 16074 
 16103 
 16132 
 16160 
 16189 
 16218 
 
 99586 
 99583 
 99580 
 9957S 
 99575 
 99572 
 99570 
 99567 
 
 12649 
 
 12678 
 12706 
 12735 
 12764 
 
 99178 
 99175 
 99171 
 99167 
 99163 
 99160 
 99156 
 99152 
 99148 
 99144 
 
 14522 
 14551 
 14580 
 14608 
 14637 
 14666 
 
 99380 
 99377 
 99374 
 99370 
 99367 
 99364 
 99360 
 99357 
 
 16361 
 16390 
 16419 
 16447 
 16476 
 16505 
 
 99551 
 99548 
 99545 
 99542 
 99540 
 
 11349 
 11378 
 11407 
 11436 
 11465 
 11494 
 11523 
 11552 
 11580 
 11609 
 
 99354 
 99351 
 99347 
 99344 
 99341 
 99337 
 99334 
 
 13081 
 13110 
 13139 
 13168 
 13197 
 
 16533 
 16562 
 16591 
 16620 
 16648 
 16677 
 16706 
 16734 
 16763 
 16792 
 
 l 1985 
 14954 
 14982 
 15011 
 15040 
 15069 
 
 13254 
 13283 
 13312 
 13341 
 
 13370 
 13399 
 13427 
 1345G 
 13485 
 13514 
 13543 
 13572 
 13600 
 13629 
 
 16820 
 
 16849 
 16878 
 16906 
 
 15097 
 15126 
 15155 
 15184 
 15212 
 15241 
 15270 
 15299 
 15327 
 15350 
 
 99506 
 99503 
 99500 
 99497 
 99494 
 99491 
 99488 
 99485 
 
 11927 
 11956 
 11985 
 12014 
 12043 
 12071 
 12100 
 12129 
 12158 
 12187 
 Cosin 
 
 13658 
 136S7 
 13716 
 13744 
 13773 
 13802 
 13831 
 13860 
 
 15385 
 15414 
 15442 
 15471 
 15500 
 15529 
 15557 
 15586 
 15615 
 15643 
 Cosin 
 
 17107 
 17136 
 17164 
 17193 
 17222 
 17250 
 17279 
 17308 
 17336 
 17365 
 
 99063 
 99059 
 99055 
 99051 
 99047 
 99043 
 
 98511 
 98506 
 98501 
 98496 
 98491 
 
 84 
 
 214 
 
TABLE X.-SINES AND COSINES. 
 
 1 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 Sine Cosin 
 
 
 
 
 17365 
 
 98481 
 
 '19081 
 
 98163 
 
 "20791 
 
 97815 
 
 22495 
 
 97437 
 
 24192 
 
 97030 
 
 60 
 
 1 
 
 17393 
 
 98476 
 
 19109 
 
 98157 
 
 20820 
 
 97809 
 
 22523 
 
 97430 
 
 24220 
 
 97023 
 
 59 
 
 2 
 
 17422 
 
 98471 
 
 19138 
 
 98152 
 
 20848 
 
 97803 
 
 22552 
 
 97424 
 
 2424'J 
 
 97015 
 
 58 
 
 3 
 
 17451 
 
 98466 
 
 19167 
 
 98146 
 
 20877 
 
 97797 
 
 22580 
 
 97417 
 
 24277 
 
 97008 
 
 57 
 
 4 
 
 17479 
 
 98461 
 
 19195 
 
 98140 
 
 20905 
 
 97791 
 
 22608 
 
 97411 
 
 24305 
 
 97001 
 
 56 
 
 5 
 
 17508 
 
 98455 
 
 19224 
 
 9S135 
 
 20933 
 
 97784 
 
 22637 
 
 97404 
 
 24:333 
 
 96994 
 
 55 
 
 6 
 
 17537 
 
 98450 
 
 19252 
 
 98129 
 
 20962 
 
 97778 
 
 22665 
 
 07398 
 
 24362 
 
 96987 
 
 54 
 
 7 
 
 17565 
 
 98445 
 
 19281 
 
 98124 
 
 20990 
 
 97772 
 
 22693 
 
 97391 
 
 24390 
 
 96980 53 
 
 8 
 
 17594 
 
 98440 
 
 19309 
 
 98118 
 
 21019 
 
 97766 
 
 22722 
 
 97384 
 
 24418 
 
 96973) 52 
 
 9 
 
 17623 
 
 98435 
 
 19338 
 
 98112 
 
 21047 
 
 97760 
 
 22750 
 
 97378 
 
 24446 
 
 96966 51 
 
 10 
 
 17651 
 
 98430 
 
 19366 
 
 98107 
 
 21076 
 
 97754 
 
 22778 
 
 97371 
 
 24474 
 
 96959 
 
 50 
 
 11 
 
 17680 
 
 98425 
 
 19395 
 
 98101 
 
 21104 
 
 97748 
 
 22807 
 
 97365 
 
 24503 
 
 96952 
 
 49 
 
 12 
 
 17708 
 
 98420 
 
 19423 
 
 98096 
 
 21132 
 
 97742 
 
 22835 
 
 97358 
 
 24531 
 
 96945, 48 
 
 13 
 
 17737 
 
 98414 
 
 19452 
 
 98C90 
 
 21161 
 
 97735 
 
 22863 
 
 97351 
 
 24559 
 
 96937 
 
 47 
 
 14 
 
 17766 
 
 98409 
 
 19481 
 
 98084 
 
 21189 
 
 97729 
 
 22892 
 
 97345 
 
 24587 
 
 96930 
 
 46 
 
 15 17794 
 
 98404 
 
 19509 
 
 98079 
 
 21218 
 
 97723 
 
 22020 
 
 97338 
 
 24615 
 
 96923! 45 
 
 16 17823 
 
 98399 
 
 19538 
 
 98073 
 
 21246 
 
 97717 
 
 22948 
 
 97331 
 
 24644 
 
 96916 44 
 
 17 
 
 17852 
 
 98394 
 
 19566 
 
 98067 
 
 21275 
 
 97711 
 
 22977 
 
 97325 
 
 24672 
 
 96909 43 
 
 18 
 
 17880 
 
 98389 
 
 19595 
 
 98061 
 
 21303 
 
 97705 
 
 23005 
 
 97318 
 
 24700 
 
 96902 42 
 
 19 
 
 17909 
 
 98383 
 
 19623 
 
 98056 
 
 21331 
 
 97698 
 
 23033 
 
 97311 
 
 24728 
 
 968941 41 
 
 20 
 
 17937 
 
 98378 
 
 19652 
 
 98050 
 
 21360 
 
 97692 
 
 23062 
 
 97304 
 
 24756 
 
 968871 40 
 
 21 
 
 17966 
 
 98373 
 
 19680 
 
 98044 
 
 21388 
 
 97686 
 
 23090 
 
 97298 
 
 24784 
 
 96880!39 
 
 22 
 
 17995 
 
 98368 
 
 19709 
 
 98039 
 
 21417 
 
 97680 
 
 23118 
 
 97291 
 
 24813 
 
 968731 38 
 
 23 
 
 18023 
 
 98362 
 
 19737 
 
 93033 
 
 21445 
 
 97673 
 
 23146 
 
 97284 
 
 24841 
 
 9686(5 ! 37 
 
 24 
 
 18052 
 
 98357 
 
 19766 
 
 98027 
 
 21474 
 
 97667 
 
 23175 
 
 97278 
 
 24869 
 
 96858 36 
 
 25 
 
 18081 
 
 98352 
 
 19794 
 
 98021 
 
 21502 
 
 97661 
 
 23203 
 
 97271 
 
 24897 
 
 96851 i 35 
 
 26 
 
 18109 
 
 98347 
 
 19823 
 
 98016 
 
 21530 
 
 97655 
 
 23231 
 
 97264 
 
 2-4925 
 
 96844134 
 
 27 
 
 18138 
 
 98341 
 
 19851 
 
 98010 
 
 21559 
 
 97648 
 
 23260 
 
 97257 
 
 24954 
 
 96837 33 
 
 28 
 
 18166 
 
 98336 
 
 19880 
 
 93004 
 
 21587 
 
 97642 
 
 23288 
 
 97251 
 
 24982 
 
 96829 
 
 32 
 
 29 
 
 18195 
 
 98331 
 
 19908 
 
 97998 
 
 21616 
 
 97636 
 
 23316 
 
 97244 
 
 25010 
 
 96822 
 
 31 
 
 30 
 
 18224 
 
 98325 
 
 19937 
 
 97992 
 
 21644 
 
 97630 
 
 23345 
 
 97237 
 
 25038 
 
 96815 
 
 30 
 
 31 
 
 18252 
 
 98320 
 
 19965 
 
 97987 
 
 21672 
 
 97623 
 
 23373 
 
 97230 
 
 25066 
 
 96807 
 
 29 
 
 32 
 
 18281 
 
 98315 
 
 19994 
 
 97981 
 
 21701 
 
 97617 
 
 23401 
 
 97223 
 
 25094 
 
 96800 
 
 28 
 
 33 
 
 18309 
 
 98310 
 
 20022 
 
 97975 
 
 21729 
 
 97611 
 
 23423 
 
 97217 
 
 25122 
 
 96793 
 
 27 
 
 34 
 
 18338 
 
 98304 
 
 20051 
 
 97969 
 
 21758 
 
 97604 
 
 23458 
 
 97210 
 
 25151 
 
 96786 26 
 
 35 
 
 18367 
 
 98299 
 
 20079 
 
 97963 
 
 21786 
 
 97598 
 
 23486 
 
 97203 
 
 25179 
 
 96778' 25 
 
 36 
 
 18395 
 
 98294 
 
 20108 
 
 97958 
 
 21814 
 
 97592 
 
 23514 
 
 97196 
 
 25207 
 
 96771 
 
 24 
 
 37 
 
 18424 
 
 98288 
 
 20136 
 
 97952 
 
 21843 
 
 97585 
 
 23542 
 
 97189 
 
 25235 
 
 96764 
 
 23 
 
 38 
 
 18452 
 
 98283 
 
 20165 
 
 97946 
 
 21871 
 
 97'579 
 
 23571 
 
 97182 
 
 25263 
 
 96756 
 
 22 
 
 39 
 
 18481 
 
 98277 
 
 20193 
 
 97940 
 
 21899 
 
 97573 
 
 23599 
 
 97176 
 
 25291 
 
 96749 
 
 21 
 
 40 
 
 18509 
 
 98272 
 
 20222 
 
 97934 
 
 21928 
 
 97566 
 
 23627 
 
 97169 
 
 25320 
 
 96742 
 
 20 
 
 41 
 
 18538 
 
 98267 
 
 20250 
 
 97928 
 
 21956 
 
 97560 
 
 23656 
 
 97162 
 
 25348 
 
 96734 
 
 19 
 
 42 
 
 18567 
 
 98261 
 
 20279 
 
 97922 
 
 
 97553 
 
 23684 
 
 97155 
 
 25376 
 
 96727 
 
 18 
 
 43 
 
 18595 
 
 98256 
 
 20307 
 
 97916 
 
 22013 
 
 97547 
 
 23712 
 
 97148 
 
 25404 
 
 96719 
 
 17 
 
 44 
 
 18624 
 
 98250 
 
 20336 
 
 97910 
 
 22041 
 
 97541 
 
 23740 
 
 97141 
 
 25432 
 
 96712! 16 
 
 45 
 
 18652 
 
 98245 
 
 20364 
 
 97905 
 
 22070 
 
 97534 
 
 23769 
 
 97134 
 
 25460 
 
 9G705 15 
 
 46 
 
 18681 
 
 98240 
 
 20393 
 
 97899 
 
 22098 
 
 97528 
 
 23797 
 
 97127 
 
 25488 
 
 96697 14 
 
 47 
 
 18710 
 
 98234 
 
 20421 
 
 97893 
 
 22126 
 
 97521 
 
 23825 
 
 97120 
 
 25516 
 
 96690 13 
 
 48 
 
 18738 
 
 98229 
 
 20450 
 
 97887 
 
 22155 
 
 97515 
 
 23853 
 
 97113 
 
 25545 
 
 96682 12 
 
 49 
 
 18767 
 
 98223 
 
 20478 
 
 97881 
 
 22183 
 
 97508 
 
 23882 
 
 97106 
 
 25573 
 
 96675 11 
 
 50 
 
 18795 
 
 98218 
 
 20507 
 
 97875 
 
 22212 
 
 97502 
 
 23910 
 
 97100 
 
 25601 
 
 96667 10 
 
 51 
 
 18824 
 
 98212 
 
 20535 
 
 97869 
 
 22240 
 
 97496 
 
 23938 
 
 97093 
 
 25629 
 
 96660 
 
 9 
 
 52 
 
 18852 
 
 98207 
 
 20563 
 
 97863 
 
 22268 
 
 97489 
 
 23966 
 
 97086 
 
 25657 
 
 96653, 8 
 
 53 
 
 18881 
 
 98201 
 
 20592 
 
 97857 
 
 22297 
 
 97483 
 
 23995 
 
 97079 
 
 25685 
 
 96645 7 
 
 54 
 
 18910 
 
 98196 
 
 20620 
 
 97851 
 
 22325 
 
 97476 
 
 24023 
 
 97072 
 
 25713 
 
 966:38 6 
 
 55 
 
 18938 
 
 98190 
 
 20649 
 
 97845 
 
 22353 
 
 97470 
 
 24051 
 
 97065 
 
 25741 
 
 96630 5 
 
 56 
 
 18967 
 
 98185 
 
 20677 
 
 97839 
 
 22382 
 
 97463 
 
 24079 
 
 97058 
 
 25769 
 
 96623 4 
 
 57 
 
 18995 
 
 98179 
 
 20706 
 
 97833 
 
 22410 
 
 37457 
 
 24108 
 
 97051 
 
 25798 
 
 96615 3 
 
 58 
 
 19024 
 
 98174 
 
 207:34 
 
 97827 
 
 22438 
 
 97450 
 
 24136 
 
 97044 
 
 25826 
 
 96608 2 
 
 59 
 
 19052 
 
 98168 
 
 20763 
 
 9782^ 
 
 22467 
 
 97444 
 
 24164 
 
 97037 
 
 25854 
 
 96600 1 
 
 GO 
 
 19081 
 
 98163 
 
 20791 
 
 9781? 
 
 22495 
 
 97437 
 
 24192 
 
 97030 
 
 25882 
 
 96593 
 
 t 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 f 
 
 
 79 
 
 78 
 
 . 
 
 76 
 
 
 
 75 
 
 
 
 
 215 
 
TABLE X.-SINES AND COSINES. 
 
 
 15 16 17 18 19 
 
 
 
 Sine Cosin Sine Cosin Sine 
 
 Cosin Sine 
 
 Cosin Sine Cosin 
 
 
 
 
 25882 96593 27564 96126 29237 
 
 95630 30902 
 
 95106 32557 .94552 
 
 60 
 
 1 
 
 25910 96585 27592 96118 29265 
 
 95622 30929 
 
 95O7 32584 94542 
 
 59 
 
 2 
 
 25938 96578 27620 96110 29293 
 
 95613 30957 
 
 95088 32612 94533 
 
 58 
 
 3 
 
 25966 96570 27648 96102 29321 
 
 95605 30985 
 
 95079 32639 94523 
 
 57 
 
 4 
 
 25994 96562 27676 96094 29348 
 
 95596 31012 
 
 95070 32667 94514 
 
 56 
 
 5 
 
 26022 96555 27704 96086 29376 
 
 95588 31040 
 
 95061 32694 94504 
 
 55 
 
 6 
 
 26050 96547 27731 96078 29404 
 
 95579 31068 
 
 95052 32722 94495 
 
 54 
 
 7 
 
 26079 96540 27759 96070 29432 
 
 95571 31095 
 
 95043 32749 94485 
 
 53 
 
 8 
 
 26107 96532 27787 96062 29460 
 
 95562 31123 
 
 95033 32777 94476 
 
 52 
 
 9 
 
 26135 96524 27815 96054 29487 
 
 95554 31151 
 
 95024 32804 94466 
 
 51 
 
 10 
 
 26163 96517 27843 96046 29515 
 
 95545 31178 
 
 95015 32832 94457 
 
 50 
 
 11 
 
 26191 96509 27871 96037 29543 
 
 95536 31206 
 
 95006 32859 94447 
 
 49 
 
 12 
 
 26219 96502 27899 96029 29571 
 
 95528 31233 
 
 94997 32887 94438 
 
 48 
 
 13 
 
 26247 96494 27927 96021 29599 
 
 95519 31261 
 
 94988 32914 94428 
 
 47 
 
 14 
 
 26275 96486 27955 96013 29626 
 
 95511 31289 
 
 94979 32942 94418 
 
 46 
 
 15 
 
 26303 96479. 27983 96005 29654 
 
 95502 31316 
 
 94970 32969 94409 
 
 45 
 
 16 
 
 26331 96471 28011 95997 29682 
 
 95493 31344 
 
 94961 32997 94399 
 
 44 
 
 17 
 
 26359 96463 28039 95989 29710 
 
 95485 31372 
 
 94952 33024 94390 
 
 43 
 
 18 
 
 2638? 96456 28067 95981 29737 
 
 95476 31399 
 
 94943 33051 94380 
 
 42 
 
 19 
 
 26415 96448 28095 95972 29765 
 
 95467 31427 
 
 94933 33079 94370 
 
 41 
 
 20 
 
 26443 96440 28123 95964 29793 
 
 95459 31454 
 
 94924 33106 94361 
 
 40 
 
 21 
 
 26471 96433 28150 95956 29821 
 
 95450 31482 
 
 94915 33134 94351 
 
 39 
 
 22 
 
 26500 96425 28178 95948 29849 
 
 95441, 31510 
 
 94906 33161 94342 
 
 38 
 
 23 
 
 26528 96417 28206 95940 29876 
 
 95433, 31537 
 
 94897 33189 94332 
 
 37 
 
 24 
 
 26556 96410 28234 95931 29904 
 
 95424 31565 
 
 94888 33216 94322 
 
 36 
 
 25 
 
 26584 96402 28262 95923 29932 
 
 95415 31593 
 
 94878 33244 94313 
 
 35 
 
 26 
 
 26612 96394 28290 95915 29960 
 
 95407 31620 
 
 94869 33271 94303 
 
 34 
 
 27 
 
 26640 96386 28318 95907 29987 
 
 95398 31648 
 
 94860 33298 94293 
 
 33 
 
 28 
 
 26668 96379 28346 95898 30015 
 
 95389 31675 
 
 94851 33326 94284 
 
 32 
 
 29 
 
 26696 96371 28374 95890 30043 
 
 95380 31703 
 
 94842 33353 94274 
 
 31 
 
 30 
 
 26724 96363 28402 95882 30071 
 
 95372 31730 
 
 94832 33381 94264 
 
 30 
 
 31 
 
 26752 96355 28429 95874 30098 
 
 95363 31758 
 
 94823 33408 94254 
 
 29 
 
 32 
 
 26780 9634? 28457 95865 30126 
 
 95354 31786 
 
 94814 33436 94245 
 
 28 
 
 33 
 
 26808 96340 28485 95857 30154 
 
 95345 31813 
 
 94805 33463 94235 
 
 27 
 
 34 
 
 26836 96332 28513 95849 30182 
 
 95337 31841 
 
 94795 33490 94225 
 
 26 
 
 35 
 
 26864 96324 28541 95841 30209 
 
 95328 31868 
 
 94786 33518 94215 
 
 25 
 
 36 
 
 26892 96316 28569 95832 30237 
 
 95319 31896 
 
 94777 33545 94206 
 
 24 
 
 37 
 
 26920 96308 28597 95824 30265 
 
 95310 31923 
 
 94768 33573 94196 
 
 23 
 
 38 
 
 26948 96301 28625 95816 30292 
 
 95301 31951 
 
 94758 33600 94186 
 
 22 
 
 39 
 
 26976 96293 28652 95807 '30320 
 
 95293 31979 
 
 94749 33627 94176 
 
 21 
 
 40 
 
 27004 96285 28680 95799 30348 
 
 95284 32006 
 
 94740 33655 94167 
 
 20 
 
 41 
 
 27032 96277 28708 95791 30376 
 
 95275 32034 
 
 94730 33682 94157 
 
 19 
 
 42 
 
 27060 96269 28736 95782 30403 
 
 95266 32061 
 
 94721 33710 94147 
 
 18 
 
 43 
 
 27088 96261 28764 95774 30131 
 
 95257 32089 
 
 94712 38737 94137 
 
 17 
 
 44 
 
 27116 96253 28792 95766 30459 
 
 95248 32116 
 
 94702 33764 94127 
 
 16 
 
 45 
 
 27144 96246 28820 95757 30486 
 
 95240 32144 
 
 94693 33792 94118 
 
 15 
 
 46 
 
 27172 96238 28847 95749 30514 
 
 95231 32171 
 
 94684 33819 94108 
 
 14 
 
 47 
 
 27200 96230 28875 95740 30542 
 
 95222 32199 
 
 94674 33846 94098 
 
 13 
 
 48 
 
 27228 96222 28903 95732 30570 
 
 95213 32227 
 
 94665 33874 94088 
 
 12 
 
 49 
 
 27256 96214 28931 95724 30597 
 
 95204 32254 
 
 94656 33901 94078 
 
 11 
 
 50 
 
 27284 96206 28959 95715 30625 
 
 95195- 32283 
 
 94641) 33929 94068 
 
 10 
 
 51 
 
 27312 96198 28987 95707 30653 
 
 95186 32309 
 
 94637 33956 94058 
 
 9 
 
 52 
 
 27340 96190 29015 95698 30680 
 
 9517? 32337 
 
 94G27 33983 94049 
 
 8 
 
 53 
 
 27368 96182 29042 95690 30708 
 
 95168 32364 
 
 94618 34011 94039 
 
 7 
 
 54 
 
 27396 96174 29070 95681 30736 
 
 95159 32392 
 
 94609 34038 94029 
 
 6 
 
 55 
 
 27424 96166 29098 95673 30763 
 
 95150 32419 
 
 94599 34065 94019 
 
 5 
 
 56 
 
 27452 96158 29126 95664 30791 
 
 95142 32447 
 
 94590 34093 94009 
 
 4 
 
 57 
 
 27480 96150 29154 95656 30819 
 
 95133 32474 
 
 93580 34120 93999 
 
 3 
 
 58 
 
 27508 96142 29182 95647 30846 
 
 95124 32502 
 
 94571 34147 93989 
 
 2 
 
 59 
 
 27536 96134 29209 95639 30874 
 
 95115 32529 
 
 94561 34175 93979 
 
 1 
 
 60 
 
 27564 96126 29237 95630 30902 
 
 95106 . 32557 , 
 
 94552 34202 93969 
 
 
 
 / 
 
 Cosin Sine Cosin Sine Cosin 
 
 Sine Cosin 
 
 Sine Cosin Sine 
 
 / 
 
 
 74 73 72 71 70 
 
 
 216 
 
TABLE X. SINES AND COSINES. 
 
 
 20' 
 
 21 
 
 o 
 
 a 
 
 5< 
 
 23 
 
 
 
 24 
 
 
 
 ' 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 Sine Cosin 
 
 
 ~o 
 
 34202 
 
 93969 
 
 35837 
 
 93358 
 
 37461 
 
 92718 
 
 39073 
 
 92050 
 
 40674 
 
 91355 
 
 60 
 
 1 
 
 34229 
 
 93959 
 
 35864 
 
 93348 
 
 37488 
 
 92707 
 
 39100 
 
 92039 
 
 40700 
 
 91343 
 
 59 
 
 2 
 
 34257 
 
 93949 
 
 35891 
 
 93337 
 
 37515 
 
 92697 
 
 39127 
 
 92028 
 
 40727 
 
 91331 1 58 
 
 3 
 
 34284 
 
 93939 
 
 35918 
 
 93327 
 
 37542 
 
 92686 
 
 39153 
 
 92016 
 
 40753 
 
 91319' 57 
 
 4 
 
 34311 
 
 93929 
 
 35945 
 
 93316 
 
 37569 
 
 92675 
 
 39180 
 
 92005 
 
 40780 
 
 91307 
 
 56 
 
 5 
 
 34339 
 
 93919 
 
 35973 
 
 93306 
 
 37595 
 
 92664 
 
 39207 
 
 91994 
 
 40806 
 
 91295 
 
 55 
 
 6 
 
 34366 
 
 93909 
 
 36000 
 
 93295 
 
 37622 
 
 92653 
 
 39234 
 
 91982 
 
 40833 
 
 91283 
 
 54 
 
 7 
 
 34393 
 
 93899 
 
 36027 
 
 93285 
 
 37649 
 
 92642 
 
 39260 
 
 91971 
 
 40860 
 
 91272 
 
 53 
 
 8 
 
 34421 
 
 93889 
 
 36054 
 
 93274 
 
 37676 
 
 92631 
 
 3928 1 ; 
 
 91959 
 
 40886 
 
 91260 
 
 52 
 
 9 
 
 34448 
 
 93879 
 
 36081 
 
 93264 
 
 37703 
 
 92620 1 
 
 39314 
 
 91948 
 
 40913 
 
 91248 
 
 51 
 
 10 
 
 34475 
 
 93869 
 
 36108 
 
 93253 
 
 37730 
 
 92609 
 
 393-41 
 
 91936 
 
 40939 
 
 91236 
 
 50 
 
 11 
 
 34503 
 
 93859 
 
 36135 
 
 93243 
 
 37757 
 
 92598 
 
 39367 
 
 91925 
 
 40966 
 
 91224 
 
 49 
 
 12 
 
 34530 
 
 93840 
 
 361G2 
 
 93232 
 
 37784 
 
 92587 
 
 39394 
 
 91914 
 
 40992 
 
 91212 
 
 48 
 
 13 
 
 34557 
 
 93839 
 
 36190 
 
 93222 
 
 37811 
 
 92576 
 
 39421 
 
 91902 
 
 41019 
 
 91200 
 
 47 
 
 14 
 
 34584 
 
 93829 
 
 36217 
 
 93211 
 
 37838 
 
 92565 
 
 39448 
 
 91891 
 
 41045 
 
 91188 
 
 46 
 
 15 
 
 34612 
 
 93819 
 
 36244 
 
 93201 
 
 37865 
 
 92554 
 
 39474 
 
 91879 
 
 41072 
 
 91176 
 
 45 
 
 16 
 
 34G39 
 
 93809 
 
 36271 
 
 93100 
 
 37892 
 
 92543 
 
 39501 
 
 91868 
 
 41098 
 
 91164 
 
 44 
 
 17 
 
 34G66 
 
 93799 
 
 36298 
 
 93180 
 
 37919 
 
 92532 
 
 39528 
 
 91856 
 
 41125 
 
 91152 43 
 
 18 
 
 84094 
 
 93789 
 
 36325 
 
 931G3 
 
 37946 
 
 92521 
 
 39555 
 
 91845 
 
 41151 
 
 91140 J 
 
 19 
 
 34721 
 
 93779 
 
 36352 
 
 93159 
 
 37973 
 
 92510 
 
 39581 
 
 91833 
 
 41178 
 
 91128 
 
 41 
 
 20 
 
 34748 
 
 93769 
 
 36379 
 
 93148 
 
 37999 
 
 .92499 
 
 39608 
 
 91822 
 
 41204 
 
 91116 
 
 40 
 
 21 
 
 34775 
 
 93750 
 
 36406 
 
 93137 
 
 38026 
 
 92488 
 
 39635 
 
 91810 
 
 41231 
 
 91104 
 
 39 
 
 22 
 
 34803 
 
 93748 
 
 36434 
 
 93127 
 
 38053 
 
 92477 
 
 39661 
 
 91799 
 
 41257 
 
 91092 
 
 38 
 
 23 
 
 34830 
 
 93738 
 
 36461 
 
 9311G 
 
 38080 
 
 92466 
 
 39688 
 
 91787 
 
 41284 
 
 91080 
 
 37 
 
 24 
 
 34857 
 
 93728 
 
 36488 
 
 93106 
 
 38107 
 
 92455 
 
 39715 
 
 91775 
 
 41310 
 
 91068 
 
 36 
 
 25 
 
 34884 
 
 93718 
 
 36515 
 
 93095 
 
 38134 
 
 92444 
 
 39741 
 
 91764 
 
 41337 
 
 91056 
 
 35 
 
 26 
 
 34912 
 
 93708 
 
 36542 
 
 93084 
 
 38161 
 
 92432 
 
 39768 
 
 91752 
 
 41363 
 
 91044 
 
 34 
 
 27 
 
 34939 
 
 93698 
 
 3G5G9 
 
 93074 
 
 38188 
 
 92421 
 
 39795 
 
 91741 
 
 41390 
 
 91032 
 
 33 
 
 23 
 
 34966 
 
 93688 
 
 36596 
 
 93063 
 
 38215 
 
 92410 
 
 39822 
 
 91729 
 
 41416 
 
 91020 
 
 32 
 
 29 
 
 34993 
 
 93677 
 
 3GG23 
 
 93052 
 
 8241 
 
 92399 
 
 39846 
 
 91718 
 
 41443 
 
 91008 
 
 31 
 
 30 
 
 35021 
 
 93667 
 
 36650 
 
 93042 
 
 38268 
 
 92388 
 
 39875 
 
 91706 
 
 41469 
 
 90996 
 
 30 
 
 31 
 
 35048 
 
 93657 
 
 36677 
 
 93031 
 
 38295 
 
 92377 
 
 39902 
 
 91694 
 
 41496 
 
 90984 
 
 29 
 
 32 
 
 35075 
 
 93647 
 
 36704 
 
 93020 
 
 38322 
 
 92366 
 
 39928 
 
 91C88 
 
 41522 
 
 9097'2 
 
 28 
 
 33 
 
 35102 
 
 93637 
 
 36731 
 
 93010 
 
 38349 
 
 92355 
 
 39955 
 
 91671 
 
 41549 
 
 90960 
 
 27 
 
 34 
 
 35130 
 
 93626 
 
 36758 
 
 92999 
 
 38376 
 
 92343 
 
 39982 
 
 91660 
 
 41575 
 
 90948 
 
 26 
 
 35 
 
 35157 
 
 93616 
 
 36785 
 
 92988 
 
 38403 
 
 92332 
 
 40008 
 
 91648 
 
 41602 
 
 90936 
 
 25 
 
 36 
 
 35184 
 
 93606 
 
 36812 
 
 92978 
 
 38430 
 
 92321 
 
 40035 
 
 91636 
 
 41628 
 
 90924 
 
 24 
 
 37 
 
 35211 
 
 93596 
 
 36839 
 
 92967 
 
 38456 
 
 92310 
 
 40062 
 
 91625 
 
 41655 
 
 90911 
 
 23 
 
 38 
 
 35239 
 
 93585 
 
 36867 
 
 92956 
 
 38483 
 
 92299 
 
 40088 
 
 91613 
 
 41681 
 
 90899 
 
 22 
 
 39 
 
 35266 
 
 93575 
 
 36894, 
 
 92945 
 
 38510 
 
 92287 
 
 40115 
 
 91G01' 
 
 41707 
 
 90887 
 
 21 
 
 40 
 
 35293 
 
 93565 
 
 36921 
 
 92935 
 
 38537 
 
 92276 
 
 40141 
 
 91590 
 
 41734 
 
 90875 
 
 20 
 
 41 
 
 35320 
 
 93555 
 
 36948 
 
 92924 
 
 38564 
 
 92265 
 
 40168 
 
 91578 
 
 41760 
 
 90863 
 
 19 
 
 42 
 
 35347 
 
 93544 
 
 36975 
 
 92913 
 
 38591 
 
 92254 
 
 40195 
 
 91566 
 
 41787 
 
 90851 
 
 18 
 
 43 
 
 35375 
 
 93534 
 
 37002 
 
 92902 
 
 38617 
 
 92243 
 
 40221 
 
 91555 
 
 41813 
 
 90839 
 
 17 
 
 44 
 
 35402 
 
 93524 
 
 37029 
 
 92892 
 
 38644 
 
 92231 
 
 40248 
 
 91543 
 
 41840 
 
 90826 
 
 16 
 
 45 
 
 35429 
 
 93514 
 
 37056 
 
 92881 
 
 38671 
 
 92220 
 
 40275 
 
 91531 
 
 41866 
 
 90814 
 
 15 
 
 46 
 
 35456 
 
 93503 
 
 37083 
 
 92870 
 
 38698 
 
 92209 
 
 40301 
 
 91519 
 
 41892 
 
 90802 14 
 
 47 
 
 35484 
 
 93493 
 
 37110 
 
 92859 
 
 38725 
 
 92198 
 
 40328 
 
 91508 
 
 41919 
 
 90790 13 
 
 48 
 
 35511 
 
 93483 
 
 37137 
 
 92849 
 
 38752 
 
 92186 
 
 40355 
 
 91496 
 
 41945 
 
 90778 12 
 
 49 
 
 35538 
 
 93472 
 
 37164 
 
 92838 
 
 38778 
 
 92175 
 
 40381 
 
 91484 
 
 41972 
 
 90766 11 
 
 50 
 
 35565 
 
 93462 
 
 37191 
 
 92827 
 
 38805 
 
 92164 
 
 40408 
 
 91472 
 
 41998 
 
 90753 
 
 10 
 
 51 
 
 35592 
 
 93452 
 
 37218 
 
 92816 
 
 38832 
 
 92152 
 
 40434 
 
 91461 
 
 42024 
 
 90741 
 
 9 
 
 52 
 
 35619 
 
 93441 
 
 37245 
 
 92805 
 
 38859 
 
 92141 
 
 40461 
 
 91449 
 
 42051 
 
 90729 
 
 8 
 
 53 
 
 35647 
 
 93431 
 
 37272 
 
 92794 
 
 38886 
 
 92130 
 
 40488 
 
 91437 
 
 42077 
 
 90717 
 
 7 
 
 54 
 
 35674 
 
 93420 
 
 37299 
 
 92784 
 
 38912 
 
 92119 
 
 40514 
 
 91425 
 
 42104 
 
 90704 
 
 6 
 
 55 
 
 35701 
 
 93410 
 
 37326 
 
 92773 
 
 38939 
 
 92107 
 
 40541 
 
 91414 
 
 42130 
 
 90692 
 
 5 
 
 56 
 
 35728 
 
 93400 
 
 37353 
 
 92762 
 
 38966 
 
 92096 
 
 40567 
 
 91402 
 
 42156 
 
 90680 
 
 4 
 
 57 
 
 35755 
 
 93389 
 
 37380 
 
 92751 
 
 38993 
 
 92085 
 
 40594 
 
 91390 
 
 42183 
 
 90668 
 
 3 
 
 58 
 
 35782 
 
 9.3379 
 
 37407 
 
 92740 
 
 39020 
 
 92073 
 
 40621 
 
 91378 
 
 42209 
 
 90655 
 
 2 
 
 59 
 
 35810 
 
 93368 
 
 37434 
 
 92729 
 
 39046 
 
 92062 
 
 40647 
 
 91366 
 
 42235 
 
 90643 
 
 1 
 
 60 
 
 35837 
 
 93358 
 
 37461 
 
 92718 
 
 39073 
 
 92050 
 
 40674 
 
 91355 
 
 42262 
 
 90631 
 
 
 
 9 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 f 
 
 
 69 68 
 
 67 
 
 66 
 
 65 
 
 o 
 
 
 217 
 
TABLE X. SINES AND COSINES. 
 
 
 Sine Cosin 
 
 48481 "87462 
 48506 87448 
 48532 87434 
 48557 87421 
 48583 8740( 
 
 44124 
 44151 
 44177 
 44203 
 44229 
 44255 
 44281 
 44307 
 44333 
 44359 
 
 45684 
 45710 
 45736 
 45762 
 45787 
 45813 
 
 48761 
 
 48786 
 48811 
 48837 
 
 90483 
 90470 
 90458 
 90446 
 90433 
 90421 
 90408 
 
 42762 
 
 42788 
 
 42815 
 42841 
 
 44385 
 44411 
 44437 
 44464 
 44490 
 44516 
 44542 
 44568 
 44594 
 44620 
 
 43077 
 43104 
 43130 
 43156 
 
 44646 
 44672 
 44698 
 44724 
 44750 
 44776 
 44802 
 
 47741 
 47767 
 47793 
 47818 
 47844 
 47869 
 47895 
 47920 
 47946 
 47971 
 
 43418 
 43445 
 43471 
 43497 
 
 45166 
 45192 
 45218 
 45243 
 45269 
 45295 
 45321 
 45347 
 45373 
 
 89219 
 89206 
 89193 
 89180 
 89167 
 8915S 
 89140 
 89127 
 89114 
 89101 
 Sine 
 
 43706 
 43733 
 43759 
 43785 
 43811 
 43837 
 
 89943 
 89930 
 89918 
 89905 
 89892 
 89879 
 
 49924 
 49950 
 49975 
 50000 
 
 46921 88308 
 46947 88295 
 Cosin Sine 
 
 218 
 
TABLE X.-SINES AND COSINES. 
 
 30 
 
 31 
 
 o 
 
 32 33 
 
 34 
 
 f 
 
 / 
 
 Sine Cosin ' 
 
 Sine Cosin 
 
 Sine Cosin 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 
 ~o 
 
 ~50000 
 
 86G03 
 
 ^51504' 
 
 85717 
 
 52992 
 
 84805 
 
 54464 
 
 83867 
 
 '55919 82904 
 
 60 
 
 1 
 
 50025 
 
 86588 
 
 51529 
 
 85702 
 
 53017 
 
 84789 
 
 54488 
 
 83851 
 
 55943 
 
 82887 
 
 5S 
 
 2 
 
 50050 
 
 86573 
 
 51554 
 
 85687 
 
 53041 
 
 84774 
 
 54513 
 
 83835 
 
 55968 
 
 82871 
 
 58 
 
 3 
 
 50076 
 
 86559 
 
 51579 
 
 85672 
 
 53066 
 
 84759 
 
 54537 
 
 83819 
 
 55992 
 
 82855 
 
 57 
 
 4 
 
 50101 
 
 86544 
 
 51604 
 
 85657 
 
 53091 
 
 84743 
 
 54561 
 
 83804 
 
 56016 
 
 82839 
 
 56 
 
 5 
 
 50126 
 
 86530 
 
 51628 
 
 85G42 
 
 53115 
 
 84728 
 
 54586 
 
 83788 
 
 56040 
 
 82822 
 
 55 
 
 6 
 
 50151 
 
 86515 
 
 51653 
 
 85627 
 
 53140 
 
 84712 
 
 54610 
 
 83772 
 
 56064 
 
 82806 
 
 54 
 
 7 
 
 50176 
 
 86501 
 
 51678 
 
 85612 
 
 53164 
 
 84697 
 
 54635 
 
 83756 
 
 56088 
 
 82790 
 
 53 
 
 8 
 
 50201 
 
 86486 
 
 51703 
 
 85597 
 
 53189 
 
 84681 
 
 54659 
 
 83740 
 
 56112 
 
 82773 
 
 52 
 
 9 
 
 50227 
 
 86471 
 
 51728, 
 
 85582 
 
 53214 
 
 84666 
 
 54683 
 
 83724 
 
 56136 
 
 82757 
 
 51 
 
 10 
 
 50252 
 
 86457 
 
 51753 
 
 85567 
 
 53238 
 
 84650 
 
 54708 
 
 837U8 
 
 56160 
 
 82741 
 
 50 
 
 11 
 
 50277 
 
 86442 
 
 51778 
 
 85551 
 
 53263 
 
 84635 
 
 54732 
 
 83692 
 
 56184 
 
 82724 
 
 4C 
 
 12 
 
 50302 
 
 86427 
 
 51803 
 
 85536 
 
 53288 
 
 84619 
 
 54756 
 
 83676 
 
 56208 
 
 82708 
 
 48 
 
 13 
 
 50327 
 
 86413 
 
 51828 
 
 85521 
 
 53312 
 
 84604 
 
 54781 
 
 83660 
 
 56232 
 
 82692 
 
 47 
 
 14 
 
 50352 
 
 86398 
 
 51852 
 
 85506 
 
 53337 
 
 84588 
 
 54805 
 
 83645 
 
 56256 
 
 82675 
 
 46 
 
 15 
 
 50377 
 
 86384 
 
 51877 
 
 85491 
 
 53361 
 
 84573 
 
 54829 
 
 83629 
 
 56280 
 
 82659 
 
 45 
 
 16 
 
 50403 
 
 86369 
 
 51902 
 
 85476 
 
 53386 
 
 84557 
 
 54854 
 
 83613 
 
 56305 
 
 82643 
 
 44 
 
 17 
 
 50428 
 
 86354 
 
 51927 
 
 85461 
 
 53411 
 
 84542 
 
 54878 
 
 83597 
 
 56329 
 
 82626 
 
 43 
 
 18 
 
 50453 
 
 86340 
 
 51952 
 
 85446 
 
 53435 
 
 84526 
 
 54902 
 
 83581 
 
 56353 
 
 82610 
 
 42 
 
 19 
 
 50478 
 
 86325 
 
 51977 
 
 85431 
 
 534.60 
 
 84511 
 
 54927 
 
 83565 
 
 56377 
 
 82593 
 
 41 
 
 20 
 
 50503 
 
 86310 
 
 52002 
 
 85416 
 
 53484 
 
 84495 
 
 54951 
 
 83549 
 
 56401 
 
 82577 
 
 40 
 
 21 
 
 50528 
 
 86295 
 
 52026 
 
 85401 
 
 53509 
 
 84480 
 
 54975 
 
 83533 
 
 56425 
 
 82561 
 
 89 
 
 22 
 
 50553 
 
 86281 
 
 52051 
 
 85385 
 
 53534 
 
 84464 
 
 54999 
 
 83517 
 
 56449 
 
 82544 
 
 88 
 
 23 
 
 50578 
 
 86266 
 
 52076 
 
 85370 
 
 53558 
 
 84448 
 
 55024 
 
 83501 
 
 56473 
 
 82528 37 
 
 24 
 
 50C03 
 
 86251 
 
 52101 
 
 85355 
 
 53583 
 
 84433 
 
 55048 
 
 83485 
 
 56497 
 
 S2511 
 
 36 
 
 25 
 
 50628 
 
 86237 
 
 52126 
 
 85340 
 
 53607 
 
 84417 
 
 55072 
 
 83469 
 
 56521 
 
 82495 
 
 35 
 
 26 
 
 50654 
 
 86222 
 
 52151 
 
 85325 
 
 53632 
 
 84402 
 
 55097 
 
 83453 
 
 56545 
 
 82478 
 
 34 
 
 27 
 
 50679 
 
 86207 
 
 52175 
 
 85310 
 
 53656 
 
 84386 
 
 55121 
 
 83437 
 
 56569 
 
 82462 
 
 33 
 
 28 
 
 50704 
 
 86192 
 
 52200 
 
 85294 
 
 53681 
 
 84370 
 
 55145 
 
 83421 
 
 56593 
 
 82446 
 
 32 
 
 29 
 
 50729 
 
 86178 
 
 52225 
 
 85279 
 
 53705 
 
 84355 
 
 55169 
 
 83405 
 
 56617 
 
 82429 
 
 31 
 
 30 
 
 50754 
 
 86163 
 
 52250 
 
 85264 
 
 53730 
 
 84339 
 
 55194 
 
 83389 
 
 56641 
 
 82413 
 
 30 
 
 31 
 
 50779 
 
 86148 
 
 52275 
 
 85249 
 
 53754 
 
 84324 
 
 55218 
 
 83373 
 
 56665 
 
 82396 
 
 29 
 
 32 
 
 50804 
 
 86133 
 
 52299 
 
 85234 
 
 53779 
 
 84308 
 
 55242 
 
 83356 
 
 56689 
 
 82380 
 
 28 
 
 33 
 
 50829 
 
 86119 
 
 52324 
 
 85218 
 
 53804 
 
 84292 
 
 55266 
 
 83340 
 
 56713 
 
 82363 
 
 27 
 
 34 
 
 50854 
 
 86104 
 
 52349 
 
 85203 
 
 53828 
 
 84277 
 
 55291 
 
 83324 
 
 56736 
 
 82347 
 
 26 
 
 35 
 
 50879 
 
 86089 
 
 52374 
 
 85188 
 
 53853 
 
 84261 
 
 55315 
 
 83308 
 
 56760 
 
 82330 
 
 25 
 
 36 
 
 50904 
 
 86074 
 
 52399 
 
 85173 
 
 53877 
 
 84245 
 
 55339 
 
 83292 
 
 56784 
 
 82314 
 
 24 
 
 37 
 
 50929 
 
 86059 
 
 52423 
 
 85157 
 
 53902 
 
 84230 
 
 55363 
 
 83276 
 
 56808 
 
 82297 
 
 23 
 
 38 
 
 50954 
 
 86045 
 
 52448 
 
 85142 
 
 53926 
 
 84214 
 
 55388 
 
 83260 
 
 56832 
 
 82281 
 
 22 
 
 39 
 
 50979 
 
 86030 
 
 52473 
 
 85127 
 
 53951 
 
 84198 
 
 55412 
 
 83244 
 
 56856 
 
 82264 
 
 21 
 
 40 
 
 51004 
 
 86015 
 
 52498 
 
 85112 
 
 53975 
 
 84182 
 
 55436 
 
 83228 
 
 56880 
 
 82248 
 
 SO 
 
 41 
 
 51029 
 
 86000 
 
 52522 
 
 85096 
 
 54000 
 
 84167 
 
 55460 
 
 83212 
 
 56904 
 
 82231 
 
 19 
 
 42 
 
 51054 
 
 85985 
 
 52547 
 
 85081 
 
 54024 
 
 84151 
 
 55484 
 
 83195 
 
 56928 
 
 82214 
 
 18 
 
 43 
 
 51079 
 
 85970 
 
 52572 
 
 85066 
 
 54049 
 
 84135 
 
 55509 
 
 83179 
 
 56952 
 
 82198 
 
 17 
 
 44 
 
 51104 
 
 85956 
 
 52597 
 
 85051 
 
 54073 
 
 84120 
 
 55533 
 
 83163 
 
 56976 
 
 82181 
 
 16 
 
 45 
 
 51129 
 
 85941 
 
 52621 
 
 85035 
 
 54097 
 
 84104 
 
 55557 
 
 83147 
 
 57000 
 
 82165 
 
 15 
 
 46 
 
 51154 
 
 85926 
 
 52646 
 
 85020 
 
 54122 
 
 84088 
 
 55581 
 
 83131 
 
 57024 
 
 82148 
 
 14 
 
 47 
 
 51179 
 
 85911 
 
 52671 
 
 85005 
 
 54146 
 
 84072 
 
 55605 
 
 83115 
 
 57047 
 
 82132 
 
 13 
 
 48 
 
 51204 
 
 85896 
 
 52696 
 
 '84989 
 
 54171 
 
 84057 
 
 55630 
 
 83098 
 
 57071 
 
 82115 
 
 12 
 
 49 
 
 51229 
 
 85881 
 
 52720 
 
 84974 
 
 54195 
 
 84041 
 
 55654 
 
 83082 
 
 57095 
 
 82098 
 
 11 
 
 50 
 
 51254 
 
 85866 
 
 52745 
 
 84959 
 
 54220 
 
 84025 
 
 55678 
 
 83066 
 
 57119 
 
 82082 
 
 10 
 
 51 
 
 51279 
 
 85851 
 
 52770 
 
 84943 
 
 54244 
 
 34009 
 
 55702 
 
 83050 
 
 57143 
 
 32065 
 
 9 
 
 52 
 
 51304 
 
 85836 
 
 52794 
 
 84928 
 
 54269 
 
 83994 
 
 55726 
 
 83034 
 
 57167 
 
 82048 
 
 8 
 
 53 
 
 51329 
 
 85821 
 
 52819 
 
 84913 
 
 54293 
 
 83978 
 
 55750 
 
 83017 
 
 57191 
 
 82032 
 
 7 
 
 54 
 
 51354 
 
 85806 
 
 52844 
 
 84897 
 
 54317 
 
 83962 
 
 55775 
 
 83001 
 
 57215 
 
 82015 
 
 6 
 
 55 
 
 51379 
 
 85792 
 
 52869 
 
 84882 
 
 54342 
 
 83946 
 
 55799 
 
 82985 
 
 57238 
 
 81999 
 
 5 
 
 56 
 
 51404 
 
 85777 
 
 52893 
 
 84866 
 
 54366 
 
 83930 
 
 55823 
 
 82969 
 
 57262 
 
 81982 
 
 4 
 
 57 
 
 51429 
 
 85762 
 
 52918 
 
 84851 
 
 54391 
 
 83915 
 
 55847 
 
 82953 
 
 57286 
 
 81965 
 
 3 
 
 58 
 
 51454 
 
 85747 
 
 52943 
 
 84836 
 
 54415 
 
 83899 
 
 55871 
 
 82936 
 
 57310 
 
 81949 
 
 2 
 
 59 
 
 51479 
 
 85732 
 
 52967 
 
 84820 
 
 54440 
 
 83883 
 
 55895 
 
 82920 
 
 57334 
 
 81932 
 
 1 
 
 60 
 
 51504 
 
 85717 
 
 52992 
 
 84805 
 
 54464 
 
 83867 
 
 55919 
 
 82904 
 
 57358 
 
 81915 
 
 
 
 / 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin | Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 / 
 
 
 59 
 
 58 
 
 57 
 
 56 55 
 
 
 219 
 
TABLE X.-SINES AND COSINES. 
 
 
 35 
 
 
 
 36 
 
 37 
 
 38 
 
 39 
 
 
 
 Sine Cosin 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 f 
 
 9 
 
 57358 
 
 81915 
 
 ^8779 
 
 80902 
 
 60182 
 
 79864 
 
 61566 
 
 78801 
 
 62932 
 
 77715 
 
 tiO 
 
 u. 
 
 57381 
 
 81899 
 
 58802 
 
 80885 
 
 60205 
 
 79846 
 
 61589 
 
 78783 
 
 62955 
 
 77696 
 
 59 
 
 2 
 
 57405 
 
 81882 
 
 58826 
 
 80867 
 
 60228 
 
 79829 
 
 61612 
 
 78765 
 
 62977 
 
 77678 
 
 58 
 
 3 
 
 57429 
 
 81865 
 
 58849 
 
 80850 
 
 60251 
 
 79811 
 
 61635 
 
 78747 
 
 63000 
 
 77660 
 
 57 
 
 4 
 
 57453 
 
 81848 
 
 58873 
 
 80833 
 
 60274 
 
 79793 
 
 61658 
 
 78729 
 
 63022 
 
 77641 
 
 56 
 
 5 
 
 57477 
 
 81832 
 
 58896 
 
 80816 
 
 60298 
 
 79776 
 
 61681 
 
 78711 
 
 63045 
 
 77623 
 
 55 
 
 6 
 
 57501 
 
 81815 
 
 58920 
 
 80799 
 
 60321 
 
 79758 
 
 61704 
 
 78694 
 
 63068 
 
 77605 
 
 54 
 
 7 
 
 57524 
 
 81798 
 
 58943 
 
 80782 
 
 60344 
 
 79741 
 
 61726 
 
 78676 
 
 63090 
 
 77586 
 
 53 
 
 8 
 
 57548 
 
 81782 
 
 58967 
 
 80765 
 
 60367 
 
 79723 
 
 61749 
 
 78658 
 
 63113 
 
 77568 
 
 52 
 
 9 
 
 57572 
 
 81765 
 
 58990 
 
 80748 
 
 60390 
 
 79706 
 
 61772 
 
 78640 
 
 63135 
 
 77550 
 
 51 
 
 10 
 
 57596 
 
 81748 
 
 59014 
 
 80730 
 
 60414 
 
 79688 
 
 61795 
 
 78622 
 
 63158 
 
 77531 
 
 50 
 
 11 
 
 57619 
 
 81731 
 
 59037 
 
 80713 
 
 60437 
 
 79671 
 
 61818 
 
 78604 
 
 63180 
 
 '77513 
 
 40 
 
 12 
 
 57643 
 
 81714 
 
 59061 
 
 80696 
 
 60460 
 
 79653 
 
 61841 
 
 78586 
 
 63203 
 
 77494 
 
 48 
 
 13 
 
 57667 
 
 81698 
 
 59084 
 
 80679 
 
 60483 
 
 79635 
 
 C1864 
 
 78568 
 
 63225 
 
 77476 
 
 47 
 
 14 
 
 57691 
 
 81681 
 
 59108 
 
 80662 
 
 60506 
 
 79618 
 
 61887 
 
 78550 
 
 63248 
 
 77458 
 
 46 
 
 15 
 
 57715 
 
 81664 
 
 59131 
 
 80644 
 
 60529 
 
 79600 
 
 61909 
 
 78532 
 
 63271 
 
 77439 
 
 45 
 
 16 
 
 57738 
 
 81647 
 
 59154 
 
 80627 
 
 60553 
 
 79583 
 
 619S2 
 
 78514 
 
 63293 
 
 77421 
 
 44 
 
 17 
 
 57762 
 
 81631 
 
 59178 
 
 80610 
 
 60576 
 
 79565 
 
 61955 
 
 78496 
 
 63316 
 
 77402 
 
 43 
 
 18 
 
 57786 
 
 81614 
 
 59201 
 
 80593 
 
 60599 
 
 79547 
 
 61978 
 
 78478 
 
 63338 
 
 77384 
 
 42* 
 
 19 
 
 57'810 
 
 81597 
 
 59225 
 
 80576 
 
 60622 
 
 79530 
 
 62001 
 
 78460 
 
 63361 
 
 77366 
 
 41 
 
 20 
 
 57833 
 
 81580 
 
 59248 
 
 80558 
 
 ,.60645 
 
 79512 
 
 62024 
 
 78442 
 
 63383 
 
 77347 
 
 40 
 
 21 
 
 57857 
 
 81563 
 
 59272 
 
 80541 
 
 60668 
 
 79494 
 
 62046 
 
 78424 
 
 63406 
 
 77329 
 
 39 
 
 22 
 
 57881 
 
 81546 
 
 59295 
 
 80524 
 
 60691 
 
 79477 
 
 62069 
 
 78405 
 
 63428 
 
 77310 
 
 38 
 
 23 
 
 57904 
 
 81530 
 
 59318 
 
 80507 
 
 60714 
 
 79459 
 
 62092 
 
 78387 
 
 63451 
 
 77292 
 
 37 
 
 24 
 
 57928 
 
 81513 
 
 59342 
 
 80489 
 
 60738 
 
 79441 
 
 62115 
 
 78369 
 
 63473 
 
 77273 
 
 36 
 
 25 
 
 57952 
 
 81496 
 
 59365 
 
 80472 
 
 60761 
 
 79424 
 
 62138 
 
 78351 
 
 63496 
 
 77255 
 
 35 
 
 26 
 
 57976 
 
 81479 
 
 59389 
 
 80455 
 
 60784 
 
 79406 
 
 62160 
 
 78333 
 
 63518 
 
 77236 
 
 34 
 
 27 
 
 57999 
 
 81462 
 
 59412 
 
 80438 
 
 60807 
 
 79388 
 
 62183 
 
 78315 
 
 63540 
 
 77218 
 
 33 
 
 28 
 
 58023 
 
 81445 
 
 59436 
 
 80420 
 
 60830 
 
 79371 
 
 62206 
 
 78297 
 
 63563 
 
 77199 
 
 32 
 
 29 
 
 68047 
 
 81428 
 
 59459 
 
 80403 
 
 60853 
 
 79353 
 
 62229 
 
 78279 
 
 63585 
 
 77181 
 
 31 
 
 30 
 
 58070 
 
 81412 
 
 59482 
 
 80386 
 
 60876 
 
 79335 
 
 62251 
 
 T8261 
 
 63608 
 
 77162 
 
 30 
 
 31 
 
 58094 
 
 81395 
 
 59506 
 
 80368 
 
 60899 
 
 79318 
 
 62274 
 
 r8243 
 
 63630 
 
 77144 
 
 29 
 
 32 
 
 58118 
 
 81378 
 
 59529 
 
 80351 
 
 60922 
 
 79300 
 
 62297 
 
 78225 
 
 63653 
 
 77125 
 
 28 
 
 33 
 
 58141 
 
 81361 
 
 59552 
 
 80334 
 
 60945 
 
 79282 
 
 62320 
 
 78206 
 
 63675 
 
 77107 
 
 27 
 
 34 
 
 58165 
 
 81344 
 
 59576 
 
 80316 
 
 60968 
 
 79264 
 
 62342 
 
 78188 
 
 63698 
 
 77088 
 
 20 
 
 35 
 
 58189 
 
 81327 
 
 59599 
 
 80299 
 
 60991 
 
 79247 
 
 62365 
 
 78170 
 
 63720 
 
 77070 
 
 25 
 
 36 
 
 58212 
 
 81310 
 
 59622 
 
 80282 
 
 61015 
 
 79229 
 
 62388 
 
 78152 
 
 63742 
 
 77051 
 
 24 
 
 37 
 
 58236 
 
 81293 
 
 59646 
 
 80264 
 
 61038 
 
 79211 
 
 62411 
 
 78134 
 
 63765 
 
 77033 
 
 23 
 
 38 
 
 58260 
 
 81276 
 
 59669 
 
 80247 
 
 61061 
 
 79193 
 
 62433 
 
 78116 
 
 63787 
 
 77014 
 
 22 
 
 39 
 
 58283 
 
 81259 
 
 59693 
 
 80230 
 
 61084 
 
 79176 
 
 62456 
 
 78098 
 
 63810 
 
 76996 
 
 21 
 
 40 
 
 58307 
 
 81242 
 
 59716 
 
 80212 
 
 61107 
 
 79158 
 
 62479 
 
 78079 
 
 63832 
 
 76977 
 
 20 
 
 41 
 
 58330 
 
 81225 
 
 59739 
 
 80195 
 
 61130 
 
 79140 
 
 62502 
 
 78061 
 
 63854 
 
 76959 
 
 19 
 
 42 
 
 58354 
 
 81208 
 
 59763 
 
 80178 
 
 61153 
 
 79122 
 
 62524 
 
 78043 
 
 63877 
 
 76940 
 
 18 
 
 43 
 
 58378 
 
 81191 
 
 59786 
 
 80160 
 
 61176 
 
 79105 
 
 62547 
 
 78025 
 
 63899 
 
 76921 
 
 17 
 
 44 
 
 58401 
 
 81174 
 
 59809 
 
 80143 
 
 61199 
 
 79C87 
 
 62570 
 
 78007 
 
 63922 
 
 76903 
 
 16 
 
 45 
 
 58425 
 
 81157 
 
 59832 
 
 80125 
 
 61222 
 
 79069 
 
 62592 
 
 77988 
 
 63944 
 
 76884 
 
 15 
 
 46 
 
 58449 
 
 81140 
 
 59856 
 
 80108 
 
 61245 
 
 79051 
 
 62615 
 
 77970 
 
 63966 
 
 76866 
 
 14 
 
 47 
 
 58472 
 
 81123 
 
 59879 
 
 80091 
 
 61268 
 
 79033 
 
 62638 
 
 77952 
 
 63989 
 
 76847 
 
 13 
 
 48 
 
 58496 
 
 81106 
 
 59902 
 
 80073 
 
 61291 
 
 79016 
 
 62660 
 
 77934 
 
 64011 
 
 76828 
 
 12 
 
 49 
 
 58519 
 
 81089 
 
 59926 
 
 80056 
 
 61314 
 
 78998 
 
 62683 
 
 77916 
 
 64033 
 
 76810 
 
 11 
 
 50 
 
 58543 
 
 81072 
 
 59949 
 
 80038 
 
 61337 
 
 78980 
 
 62706 
 
 77897 
 
 64056 
 
 76791 
 
 10 
 
 51 
 
 58567 
 
 81055 
 
 59972 
 
 80021 
 
 C1360 
 
 78962 
 
 62728 
 
 77879 
 
 64078 
 
 76772 
 
 9 
 
 52 
 
 58590 
 
 81038 
 
 59995 
 
 800Q3 
 
 61383 
 
 78944 
 
 62751 
 
 77861 
 
 64100 
 
 76754 
 
 8 
 
 53 
 
 58614 
 
 81021 
 
 60019 
 
 79986 
 
 61406 
 
 78926 
 
 62774 
 
 77843 
 
 64123 
 
 76735 
 
 7 
 
 54 
 
 58637 
 
 81004 
 
 60042 
 
 79968 
 
 61429 
 
 78908 
 
 62796 
 
 77824 
 
 64145 
 
 70717 
 
 6 
 
 55 
 
 58661 
 
 80987 
 
 60065 
 
 79951 
 
 61451 
 
 78891 
 
 62819 
 
 77806 
 
 64167 
 
 76698 
 
 5 
 
 56 
 
 58684 
 
 80970 
 
 60089 
 
 79934 
 
 61474 
 
 78873 
 
 62842 
 
 77788 
 
 64190 
 
 76679 
 
 4 
 
 57 
 
 58708 
 
 80953 
 
 60112 
 
 79916 
 
 61497 
 
 78855 
 
 62864 
 
 77769 
 
 64212 
 
 76661 
 
 3 
 
 58 
 
 58731 
 
 80936 
 
 60135 
 
 79899 
 
 61520 
 
 78837 
 
 62887 
 
 77751 
 
 64234 
 
 76642 
 
 2 
 
 59 
 
 58755 
 
 80919 
 
 6C158 
 
 79881 
 
 61543 
 
 78819 
 
 62909 
 
 77733 
 
 64256 
 
 76623 
 
 1 
 
 60 
 
 58779 
 
 80902 
 
 60182 
 
 79864 
 
 61566 
 
 78801 
 
 62932 
 
 77715 
 
 64279 
 
 76604 
 
 
 
 
 Cosin 
 
 Sine" 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 f 
 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 
 220 
 
TABLE X. SINES AND COSINES. 
 
 40 
 
 41 
 
 o 
 
 42 
 
 43 
 
 e 
 
 440 
 
 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 Sine 
 
 Cosin 
 
 Sine Cosin 
 
 Sine Cosin 
 
 
 ~o 
 
 64279 
 
 76604 
 
 65606 
 
 75471 
 
 6G913 
 
 74314 
 
 68200 
 
 73135 
 
 69466 
 
 71934 
 
 60 
 
 1 
 
 64301 
 
 76586 
 
 65628 
 
 75452 
 
 6C935 
 
 74295 
 
 68221 
 
 73116 
 
 69487 
 
 71914 
 
 59 
 
 2 
 
 64323 
 
 7G567 
 
 65650 
 
 75433 
 
 66956 
 
 74276 
 
 68242 
 
 73096 
 
 69508 
 
 71894 
 
 58 
 
 3 
 
 64346 
 
 76548 
 
 65672 
 
 75414 
 
 66978 
 
 74256 
 
 68264 
 
 73076 
 
 69529! 
 
 71873 
 
 57 
 
 4 
 
 64368 
 
 76530 
 
 65694 
 
 75395 
 
 CG999 
 
 74237 
 
 68285 
 
 73056 
 
 69549 
 
 71853 
 
 56 
 
 5 
 
 64390 
 
 76511 
 
 65716 
 
 75375 
 
 C7021 
 
 74217 
 
 68306 
 
 73036 
 
 69570 
 
 71833 
 
 55 
 
 6 
 
 64412 
 
 76492 
 
 65738 
 
 75356 
 
 67043 
 
 74198 
 
 68327 
 
 73016 
 
 69591 
 
 71813 
 
 54 
 
 7 
 
 64435 
 
 76473 
 
 65759 
 
 75337 
 
 67064 
 
 74178 
 
 68349 
 
 72996 
 
 69612 
 
 71792 
 
 53 
 
 8 
 
 61457 
 
 7G455 
 
 65781 
 
 75318 
 
 C708G 
 
 74159 
 
 68370 
 
 72976 
 
 69633 
 
 71772 
 
 52 
 
 9 
 
 64479 
 
 76436 
 
 65803 
 
 75239 
 
 67107 
 
 74139 
 
 68391 
 
 72957 
 
 69654 
 
 71752 
 
 51 
 
 10 
 
 64501 
 
 76417 
 
 65825 
 
 75280 
 
 67129 
 
 74120 
 
 68412 
 
 72937 
 
 69675 
 
 71732 
 
 50 
 
 11 
 
 64524 
 
 76398 
 
 65847 
 
 75261 
 
 67151 
 
 74100 
 
 68434 
 
 72917 
 
 69696 
 
 71711 
 
 49 
 
 12 
 
 64546 
 
 76330 
 
 658G9 
 
 75241 
 
 67172 
 
 74080 
 
 68455 
 
 72897 
 
 69717 
 
 71G91 
 
 43 
 
 13 
 
 64568 
 
 76361 
 
 65391 
 
 752.22 
 
 67194 
 
 74061 
 
 68476! 
 
 72877 
 
 69737 
 
 71671 
 
 47 
 
 14 
 
 64590 
 
 76342 
 
 65913 
 
 75203 
 
 67215 
 
 74041 
 
 68497 
 
 72857 
 
 69758 
 
 71650 
 
 46 
 
 15 
 
 64612 
 
 76323 
 
 65935 
 
 75184 
 
 67237 
 
 74022 
 
 68518 
 
 72837 
 
 69779 
 
 71G30 
 
 45 
 
 16 
 
 64635 
 
 76304 
 
 65956 
 
 75165 
 
 67258 
 
 74002 
 
 68539 
 
 72817 
 
 69800 
 
 71610 
 
 44 
 
 17 
 
 64657 
 
 76286 
 
 65978 
 
 75146 
 
 67280 
 
 73983 
 
 68561 
 
 72797 
 
 69821 
 
 71590 
 
 43 
 
 18 
 
 64679 
 
 76267 
 
 66000 
 
 75126 
 
 67301 
 
 739G3 
 
 68582 
 
 72777 
 
 69842 
 
 71569 
 
 42 
 
 19 
 
 64701 
 
 76248 
 
 6G022 
 
 75107 
 
 67323 
 
 73944 
 
 68603 
 
 72757 
 
 69862 
 
 71549 
 
 41 
 
 20 
 
 64723 
 
 76229 
 
 66044 
 
 75088 
 
 67344 
 
 73924 
 
 68624 
 
 72737 
 
 69883 
 
 71529 
 
 40 
 
 21 
 
 64746 
 
 76210 
 
 66066 
 
 75069 
 
 67366 
 
 73904 
 
 68645 
 
 72717 
 
 69904 1 
 
 71508 
 
 39 
 
 22 
 
 64768 
 
 76192 
 
 6GOS8 
 
 75050 
 
 67387 
 
 738S5 
 
 68GG6 
 
 72G97 
 
 69925 
 
 71488 
 
 38 
 
 23 
 
 64790 
 
 76173 
 
 66109 
 
 75030 
 
 67409 
 
 738G5 
 
 68G88 
 
 72677 
 
 69946 
 
 71468 
 
 37 
 
 24 
 
 64812 
 
 76154 
 
 66131 
 
 75011 
 
 67430 
 
 73846 
 
 68709 
 
 72G57 
 
 69966 
 
 71447 
 
 36 
 
 25 
 
 64834 
 
 76135 
 
 66153 
 
 74392 
 
 67452 
 
 73820 
 
 68730 
 
 72G37 
 
 69987 
 
 71427 
 
 35 
 
 26 
 
 64856 
 
 76116 
 
 66175 
 
 74973 
 
 67473 
 
 73806 
 
 68751 
 
 72617 
 
 70008 
 
 71407 
 
 34 
 
 27 
 
 64878 
 
 76097 
 
 66197 
 
 74953 
 
 67495 
 
 73787 
 
 68772 
 
 72597 
 
 70029 
 
 71386 
 
 33 
 
 28 
 
 64901 
 
 76078 
 
 66218 
 
 74931 
 
 67516 
 
 7376? 
 
 68793 
 
 72577 
 
 70049 
 
 713G6 
 
 32 
 
 29 
 
 64923 
 
 76059 
 
 66240 
 
 74915 
 
 67538 
 
 73747 
 
 68814 
 
 72557 
 
 70070 
 
 71345 
 
 31 
 
 30 
 
 64945 
 
 76041 
 
 66262 
 
 74896 
 
 67559 
 
 73728 
 
 68835 
 
 72537 
 
 70091 
 
 ri325 
 
 30 
 
 31 
 
 64967 
 
 76022 
 
 66284 
 
 74876 
 
 67580 
 
 73708 
 
 68857 
 
 72517 
 
 70112 
 
 71305 
 
 29 
 
 32 
 
 64989 
 
 76003 
 
 66306 
 
 74857 
 
 67602 
 
 73688 
 
 68878 
 
 72497 
 
 70132 
 
 71284 
 
 28 
 
 33 
 
 65011 
 
 75984 
 
 66327 
 
 74833 
 
 67623 
 
 73669 
 
 68899 
 
 72477 
 
 70153 
 
 71264 
 
 27 
 
 34 
 
 65033 
 
 75965 
 
 66349 
 
 74818 
 
 67645 
 
 73649 
 
 68920 
 
 72457 
 
 70174 
 
 71243 
 
 26 
 
 35 
 
 65055 
 
 75946 
 
 66371 
 
 74709 
 
 67666 
 
 73629 
 
 68941 
 
 72437 
 
 70195 
 
 71223 
 
 35 
 
 36 
 
 65077 
 
 75927 
 
 66393 
 
 74780 
 
 67688 
 
 73610 
 
 68962 
 
 72417 
 
 7C215 
 
 71203 
 
 24 
 
 37 
 
 65100 
 
 75908 
 
 66414 
 
 74700 
 
 67709 
 
 73590 
 
 68983 
 
 72397 
 
 70236 
 
 71182 
 
 23 
 
 38 
 
 65122 
 
 75889 
 
 66436 
 
 74741 
 
 67730 
 
 73570 
 
 69004 
 
 72377 
 
 70257 
 
 71162 
 
 22 
 
 39 
 
 65144 
 
 75870 
 
 6G45S 
 
 74722 
 
 67752 
 
 73551 
 
 69025 
 
 72357 
 
 70277 
 
 71141 
 
 21 
 
 40 
 
 65166 
 
 75851 
 
 66480 
 
 74703 
 
 67773 
 
 73531 
 
 69046 
 
 72337 
 
 70298 
 
 71121 
 
 20 
 
 41 
 
 65188 
 
 75832 
 
 66501 
 
 74683 
 
 67795 
 
 73511 
 
 69067 
 
 72317 
 
 70319 
 
 71100 
 
 19 
 
 42 
 
 '65210 
 
 75813 
 
 66523 
 
 74664 
 
 67816 
 
 73491 
 
 69088 
 
 72297 
 
 70339 
 
 71080 
 
 18 
 
 43 
 
 65232 
 
 75794 
 
 6G545 
 
 74844 
 
 67837 
 
 73475 
 
 69109 
 
 70077 
 
 70360 
 
 71059 
 
 17 
 
 44 
 
 65254 
 
 75775 
 
 66566 
 
 74625 
 
 67859 
 
 73452 
 
 69130 
 
 72257 
 
 70381 
 
 71039 
 
 16 
 
 45 
 
 65276 
 
 75756 
 
 66588 
 
 74606 
 
 67880 
 
 73432 
 
 69151 
 
 72236 
 
 70401 
 
 71019 
 
 15 
 
 46 
 
 65298 
 
 75738 
 
 66610 
 
 74586 
 
 67901 
 
 73413 
 
 69172 
 
 72216 
 
 70422 
 
 70998 
 
 14 
 
 47 
 
 65320 
 
 75719 
 
 66632 
 
 74567 
 
 67923 
 
 73393 
 
 69193 
 
 72196 
 
 70443 
 
 70978 
 
 13 
 
 48 
 
 G5342 
 
 75700 
 
 66653 
 
 74548 
 
 67944 
 
 73373 
 
 69214 
 
 72176 
 
 70463 
 
 70957 
 
 12 
 
 49 
 
 G5364 
 
 75680 
 
 66675 
 
 74528 
 
 67965 
 
 73353 
 
 69235 
 
 72156 
 
 70484 
 
 70937 
 
 11 
 
 50 
 
 65386 
 
 75661 
 
 66697 
 
 74509 
 
 67987 
 
 73333 
 
 69256 
 
 72136 
 
 70505 
 
 70916 
 
 10 
 
 51 
 
 65408 
 
 75642 
 
 66718 
 
 74489 
 
 68008 
 
 73314 
 
 69277 
 
 72116 
 
 70525 
 
 70896 
 
 9 
 
 52 
 
 65430 
 
 75623 
 
 66740 
 
 74470 
 
 68029 
 
 73294 
 
 69298 
 
 72095 
 
 70546 
 
 70875! 8 
 
 53 
 
 65452 
 
 75604 
 
 66762 
 
 74451 
 
 68051 
 
 73274 
 
 69319 
 
 72075 
 
 70567 
 
 70855 i 7 
 
 54 
 
 65474 
 
 75585 
 
 66783 
 
 74431 
 
 68072 
 
 73254 
 
 69340 
 
 72055 
 
 70587 
 
 70834 6 
 
 55 
 
 65496 
 
 75566 
 
 66805 
 
 74412 
 
 68093 
 
 73234 
 
 69361 
 
 72035 
 
 70608 
 
 70813| 5 
 
 56 
 
 65518 
 
 75547 
 
 66827 
 
 74392 
 
 68115 
 
 73215 
 
 69382 
 
 72015 
 
 70628 
 
 707931 4 
 
 57 
 
 65540 
 
 75528 
 
 66848 
 
 74373 
 
 68136 
 
 73195 
 
 69403 
 
 71995 
 
 70649 
 
 70772 
 
 3 
 
 58 
 
 65562 
 
 75509 
 
 66870 
 
 74353 
 
 68157 
 
 73175 
 
 69424 
 
 71974 
 
 70670 
 
 70752 
 
 2 
 
 59 
 
 65584 
 
 75490 
 
 66891 
 
 74334 
 
 68179 
 
 73155 
 
 69445 
 
 71954 
 
 70690 
 
 70731 
 
 1 
 
 60 
 
 65606 
 
 75471 
 
 66913 
 
 74314 
 
 68200 
 
 73135 
 
 69466 
 
 71934 
 
 70711 
 
 70711 
 
 
 
 / 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 Cosin 
 
 Sine 
 
 / 
 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 
 
 
 221 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 Natural Secants and Co -secants. 
 
 o it 1 
 
 2 
 
 3 
 
 SECANT. 
 
 CO-SECANT. 1 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-BEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 
 Infinite. 
 
 I.OOOI 
 
 57-299 
 
 i. 0006 
 
 28.654 
 
 1.0014 
 
 19. 107 
 
 
 3437-7 
 
 .0001 
 
 6-359 
 
 .0006 
 
 8.417 
 
 .0014 
 
 9.002 
 
 
 1718.9 
 
 .0002 
 
 5-45 
 
 .0006 
 
 8.184 
 
 .0014 
 
 8.897 
 
 
 145-9 
 
 .0002 
 
 4-57 
 
 .0006 
 
 7-955 
 
 .0014 
 
 8-794 
 
 
 859.44 
 
 .0002 
 
 3-7i8 
 
 .0006 
 
 7-73 
 
 .0014 
 
 8.692 
 
 
 687.55 
 
 1.0002 
 
 52.891 
 
 1.0007 
 
 27.508 
 
 1.0014 
 
 18.591 
 
 
 572.96 
 
 .OOO2 
 
 2.09 
 
 .0007 
 
 7.29 
 
 .0015 
 
 8.491 
 
 
 491.11 
 
 OOO2 
 
 1-313 
 
 .0007 
 
 7-075 
 
 .0015 
 
 8-393 
 
 
 29.72 
 
 .0002 
 
 0-558 
 
 .0007 
 
 6.864 
 
 .0015 
 
 8.295 
 
 
 381.97 
 
 .OOO2 
 
 49.826 
 
 .0007 
 
 6.655 
 
 .0015 
 
 8.198 
 
 
 343-77 
 
 1.0002 
 
 49.114 
 
 1.0007 
 
 26.45 
 
 1.0015 
 
 18.103 
 
 
 12.52 
 
 .OOO2 
 
 8.422 
 
 .0007 
 
 6.249 
 
 .0015 
 
 8.008 
 
 
 286.48 
 
 .0002 
 
 7-75 
 
 .0007 
 
 6.05 
 
 .0016 
 
 7.914 
 
 
 64.44 
 
 .OOO2 
 
 7.096 
 
 .0007 
 
 5-854 
 
 .0016 
 
 7.821 
 
 
 45-55 
 
 .0002 
 
 6.46 
 
 .0008 
 
 5-66i 
 
 .0016 
 
 7-73 
 
 
 229.18 
 
 I.OOO2 
 
 45.84 
 
 1.0008 
 
 25-47 1 
 
 i. 0016 
 
 17-639 
 
 
 14-86 
 
 .0002 
 
 5-237 
 
 .0008 
 
 5-284 
 
 .0016 
 
 7-549 
 
 
 02.22 
 
 .OOO2 
 
 4-65 
 
 .0008 
 
 5-i 
 
 .0016 
 
 7-46 
 
 
 190.99 
 
 .0002 
 
 4-077 
 
 .0008 
 
 4.918 
 
 .0017 
 
 7-372 
 
 
 80.73 
 
 .OOO3 
 
 3-52 
 
 .0008 
 
 4-739 
 
 .0017 
 
 7.285 
 
 
 171.89 
 
 1.0003 
 
 42.976 
 
 1.0008 
 
 24.562 
 
 1.0017 
 
 17.198 
 
 
 63-7 
 
 .0003 
 
 2-445 
 
 .0008 
 
 4-358 
 
 .0017 
 
 7-"3 
 
 
 56.26 
 
 .0003 
 
 1.928 
 
 .0008 
 
 4.216 
 
 .0017 
 
 7.028 
 
 
 49-47 
 
 .0003 
 
 1.423 
 
 .0009 
 
 4.047 
 
 .0017 
 
 6-944 
 
 
 43-24 
 
 .0003 
 
 40-93 
 
 .0009 
 
 3.88 
 
 .0018 
 
 6.861 
 
 
 i37-5i 
 
 I.OOO3 
 
 40.448 
 
 1.0009 
 
 23.716 
 
 1.0018 
 
 16.779 
 
 
 32.22 
 
 .0003 
 
 39.978 
 
 .0009 
 
 3-553 
 
 .0018 
 
 6.698 
 
 
 27.32 
 
 .0003 
 
 9.518 
 
 .0009 
 
 3-393 
 
 .0018 
 
 6.617 
 
 
 22.78 
 
 .0003 
 
 9.069 
 
 .0009 
 
 3-235 
 
 .0018 
 
 6.538 
 
 
 18.54 
 
 .0003 
 
 8.631 
 
 .0009 
 
 3-079 
 
 .0018 
 
 6-459 
 
 
 "4-59 
 
 1.0003 
 
 38.201 
 
 1.0009 
 
 22.925 
 
 1.0019 
 
 16.38 
 
 
 10.9 
 
 .0003 
 
 7.782 
 
 .001 
 
 2.774 
 
 .0019 
 
 6.303 
 
 
 07-43 
 
 .0003 
 
 7-371 
 
 .001 
 
 2.624 
 
 .0019 
 
 6.226 
 
 
 04.17 
 
 .0004 
 
 6.969 
 
 .001 
 
 2-476 
 
 .0019 
 
 6.15 
 
 
 01. II 
 
 .0004 
 
 6.576 
 
 .001 
 
 2-33 
 
 .0019 
 
 6.075 
 
 
 98.223 
 
 I.OOO4 
 
 36.191 
 
 1. 001 
 
 22.186 
 
 1.0019 
 
 16 
 
 
 5-495 
 
 .0004 
 
 5.814 
 
 .001 
 
 2.044 
 
 .002 
 
 5-926 
 
 
 2.914 
 
 .0004 
 
 5-445 
 
 .001 
 
 1.904 
 
 .002 
 
 5-853 
 
 .0001 
 
 2.469 
 
 .0004 
 
 5-084 
 
 .001 
 
 1-765 
 
 ,OO2 
 
 5-78 
 
 .0001 
 
 88. 149 
 
 .0004 
 
 4.729 
 
 .0011 
 
 1.629 
 
 .002 
 
 5.708 
 
 I.OOOI 
 
 85.946 
 
 1.0004 
 
 34-382 
 
 I.OOII 
 
 21.494 
 
 I.OO2 
 
 J 5-637 
 
 .0001 
 
 3-849 
 
 .0004 
 
 4.042 
 
 .0011 
 
 1.36 
 
 .0021 
 
 5-566 
 
 .0001 
 
 1-853 
 
 .0004 
 
 3.708 
 
 .OOII 
 
 1.228 
 
 .OO2I 
 
 5-496 
 
 .0001 
 
 79-95 
 
 .0004 
 
 3-38i 
 
 .0011 
 
 1.098 
 
 .0021 
 
 5-427 
 
 .0001 
 
 8-133 
 
 .0004 
 
 3.06 
 
 .OOII 
 
 20.97 
 
 .OO2I 
 
 5-358 
 
 I.OOOI 
 
 76.396 
 
 1.0005 
 
 32-745 
 
 I.OOII 
 
 20.843 
 
 1. 0021 
 
 15-29 
 
 .0001 
 
 4.736 
 
 .0005 
 
 2-437 
 
 .0012 
 
 0.717 
 
 .OO22 
 
 5-222 
 
 .0001 
 
 3.146 
 
 .0005 
 
 2.134 
 
 .0012 
 
 0-593 
 
 .0022 
 
 5-155 
 
 .0001 
 
 1.622 
 
 .0005 
 
 1-836 
 
 .0012 
 
 0.471 
 
 .OO22 
 
 5-089 
 
 .0001 
 
 1.16 
 
 .0005 
 
 1-544 
 
 .0012 
 
 0-35 
 
 .0022 
 
 5.023 
 
 I.OOOI 
 
 68-757 
 
 I.OOO5 
 
 3I-257 
 
 I.OOI2 
 
 20.23 
 
 I.OO22 
 
 14.958 
 
 .0001 
 
 7.409 
 
 .0005 
 
 30.976 
 
 .0012 
 
 O. 112 
 
 .0023 
 
 4-893 
 
 .0001 
 
 6.113 
 
 .0005 
 
 0.699 
 
 .0012 
 
 19-995 
 
 .0023 
 
 4.829 
 
 .0001 
 
 4.866 
 
 .0005 
 
 0.428 
 
 .0013 
 
 9.88 
 
 .OO23 
 
 4-765 
 
 .0001 
 
 3-664 
 
 .0005 
 
 o. 161 
 
 0013 
 
 9 . 7 66 
 
 .0023 
 
 4.702 
 
 I.OOOI 
 
 62.507 
 
 1.0005 
 
 29.899 
 
 1.0013 
 
 I9-653 
 
 I.OO23 
 
 14.64 
 
 .0001 
 
 J-39 1 
 
 .0006 
 
 9.641 
 
 .0013 
 
 9-541 
 
 .0024 
 
 4.578 
 
 .0001 
 
 I-3M 
 
 .0006 
 
 9-388 
 
 .0013 
 
 9-43 1 
 
 .0024 
 
 4-5*7 
 
 .0001 
 
 59- 2 74 
 
 .0006 
 
 9- r 39 
 
 .0013 
 
 9.322 
 
 .0024 
 
 4-456 
 
 .0001 
 
 8.27 
 
 .0006 
 
 8.894 
 
 .0013 
 
 9.214 
 
 .OO24 
 
 4-395 
 
 I.OOOI 
 
 57-299 
 
 I. OOo6 
 
 28.654 
 
 1.0014 
 
 19. 107 
 
 1.0024 
 
 14-335 
 
 Co-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 89 
 
 88 
 
 87 
 
 860 
 
 From Haswell's "Engineering." Copyright, 1884, by Harper & Brothers 
 
 222 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 40 
 
 50 
 
 60 
 
 70 
 
 SECANT. 
 
 CO-SKC'T. i 
 
 SECANT. 
 
 CO-SKC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SKCANT. 
 
 CO-SKC'T. 
 
 1.0024 
 
 14-335 
 
 1.0038 
 
 11.474 
 
 1.0055 
 
 9.5668 
 
 1.0075 
 
 8.2055 
 
 .0025 
 
 4.276 
 
 .0038 
 
 1.436 
 
 0055 
 
 544 
 
 .0075 
 
 .1861 
 
 .0025 
 
 4.217 
 
 .0039 
 
 1.398 
 
 .0056 
 
 SMI 
 
 .0076 
 
 .1668 
 
 .0025 
 .0025 
 
 1.0025 
 
 4-159 
 4.101 
 
 .0039 
 .0039 
 1.0039 
 
 1-36 
 1-323 
 11.286 
 
 .0056 
 .0056 
 1.0057 
 
 .488 
 .462 
 9-4362 
 
 .0076 
 .0076 
 1.0077 
 
 .1476 
 .1285 
 8.1094 
 
 .0026 
 
 '3-986 1 
 
 .004 
 
 1.249 
 
 .0057 
 
 .4105 
 
 .0077 
 
 -0905 
 
 .0026 
 
 3-93 
 
 .004 
 
 1.213 
 
 .0057 
 
 385 
 
 .0078 
 
 .0717 
 
 .0026 
 
 
 .004 
 
 1.176 
 
 .0057 
 
 3596 
 
 .0078 
 
 .0529 
 
 .0026 
 
 3'8i8 
 
 .004 
 
 1.14 
 
 .0058 
 
 3343 
 
 .0078 
 
 .0342 
 
 1.0026 
 
 13-763 
 
 1.0041 
 
 ii. 104 
 
 1.0058 
 
 9.3092 
 
 1.0079 
 
 8.0156 
 
 .0027 
 
 3.708 
 
 .0041 
 
 1.069 
 
 .0058 
 
 .2842 
 
 .0079 
 
 7.9971 
 
 .0027 
 
 3-654 
 
 .0041 
 
 1-033 
 
 .0059 
 
 2593 
 
 .0079 
 
 .9787 
 
 .0027 
 
 3-6 
 
 .0041 
 
 0.988 
 
 .0059 
 
 .2346 
 
 .008 
 
 .9604 
 
 .0027 
 
 3-547 
 
 .0042 
 
 0.963 
 
 0059 
 
 .21 
 
 .008 
 
 .9421 
 
 1.0027 
 
 13-494 
 
 1.0042 
 
 10.929 
 
 i. 006 
 
 9- l8 55 
 
 1.008 
 
 7.924 
 
 .0028 
 
 3-44 1 
 
 .0042 
 
 0.894 
 
 .006 
 
 .1612 
 
 .0081 
 
 9059 
 
 .0028 
 
 3-389 
 
 0043 
 
 o 86 
 
 .006 
 
 137 
 
 .0081 
 
 .8879 
 
 .0028 
 
 3-337 
 
 .0043 
 
 0^826 
 
 .0061 
 
 .1129 
 
 ,0082 
 
 .87 
 
 .0028 
 
 3.286 
 
 .0043 
 
 0.792 
 
 .0061 
 
 .089 
 
 0082 
 
 .8522 
 
 1.0029 
 
 I3-235 
 
 1.0043 
 
 10.758 
 
 i. 0061 
 
 9.0651 
 
 1.0082 
 
 7.8344 
 
 .0029 
 
 3.184 
 
 .0044 
 
 0.725 
 
 .0062 
 
 .0414 
 
 .0083 
 
 .8168 
 
 .0029 
 .0029 
 
 3-134 
 3.084 
 
 .0044 
 .0044 
 
 0.692 
 0.659 
 
 .0062 
 .0062 
 
 .0179 
 8-9944 
 
 .0083 
 .0084 
 
 7992 
 .7817 
 
 .0029 
 
 3.034 
 
 .0044 
 
 0.626 
 
 .0063 
 
 97" 
 
 .0084 
 
 .7642 
 
 1.003 
 
 12.985 
 
 1.0045 
 
 10.593 
 
 1.0063 
 
 8-9479 
 
 1.0084 
 
 7.7469 
 
 .003 
 
 2.937 
 
 0045 
 
 0.561 
 
 .0063 
 
 .9248 
 
 .0085 
 
 .7296 
 
 .003 
 003 
 
 2.84 
 
 .0045 
 .0046 
 
 0.529 
 
 0-497 
 
 .0064 
 .0064 
 
 .9018 
 .879 
 
 .0085 
 .0085 
 
 .7124 
 6953 
 
 .0031 
 
 2-793 
 
 .0046 
 
 0.465 
 
 .0064 
 
 8563 
 
 .0086 
 
 -6783 
 
 1.0031 
 
 12.745 
 
 1.0046 
 
 10-433 
 
 1.0065 
 
 8-8337 
 
 1.0086 
 
 7-6613 
 
 .0031 
 
 2.698 
 
 .0046 
 
 0.402 
 
 .0065 
 
 .8112 
 
 .0087 
 
 6444 
 
 0031 
 
 2.652 
 
 .0047 
 
 0.371 
 
 .0065 
 
 .7888 
 
 .0087 
 
 .6276 
 
 .0032 
 
 2.606 
 
 .0047 
 
 o-34 
 
 .0066 
 
 .7665 
 
 .0087 
 
 .6108 
 
 .0032 
 
 2.56 
 
 .0047 
 
 0.309 
 
 .0066 
 
 7444 
 
 .0088 
 
 5942 
 
 1.0032 
 
 12.514 
 
 1.0048 
 
 10.278 
 
 1.0066 
 
 8.7223 
 
 1.0088 
 
 7-5776 
 
 .0032 
 
 2.469 
 
 .0048 
 
 0.248 
 
 .0067 
 
 .7004 
 
 .0089 
 
 .5611 
 
 .0032 
 
 2.424 
 
 .0048 
 
 0.217 
 
 .0067 
 
 .6786 
 
 .0089 
 
 5446 
 
 -0033 
 
 2-379 
 
 .0048 
 
 0.187 
 
 .0067 
 
 .6569 
 
 .0089 
 
 .5282 
 
 0033 
 
 2-335 
 
 .0049 
 
 0-157 
 
 .0068 
 
 6 353 
 
 .009 
 
 5"9 
 
 1.0033 
 
 12.291 
 
 1.0049 
 
 10.127 
 
 i. 0068 
 
 8.6138 
 
 1.009 
 
 7-4957 
 
 .0033 
 
 2.248 
 
 .0049 
 
 0.098 
 
 .0068 
 
 59 2 4 
 
 .009 
 
 4795 
 
 .0034 
 
 2.204 
 
 .005 
 
 0.068 
 
 .0069 
 
 57" 
 
 .0091 
 
 4634 
 
 .0034 
 
 2.161 
 
 .005 
 
 0.039 
 
 .0069 
 
 5499 
 
 .0091 
 
 4474 
 
 .0034 
 
 2.118 
 
 .005 
 
 O.OI 
 
 .0069 
 
 5289 
 
 .0092 
 
 4315 
 
 1.0034 
 
 12.076 
 
 1.005 
 
 9.9812 
 
 1.007 
 
 8.5079 
 
 1.0092 
 
 7-4*56 
 
 0035 
 
 2.034 
 
 0051 
 
 9525 
 
 .007 
 
 .4871 
 
 .0092 
 
 .3998 
 
 .0035 
 
 1.992 
 
 0051 
 
 9239 
 
 .007 
 
 .4663 
 
 .0093 
 
 384 
 
 .0035 
 
 i-95 
 
 .0051 
 
 .8955 
 
 .0071 
 
 4457 
 
 .0093 
 
 3683 
 
 .0035 
 
 1.909 
 
 .0052 
 
 .8672 
 
 .0071 
 
 .4251 
 
 .0094 
 
 3527 
 
 1.0036 
 
 11.868 
 
 1.0052 
 
 9.8391 
 
 1.0071 
 
 8.4046 
 
 1.0094 
 
 7-3372 
 
 .0036 
 
 1.828 
 
 .0052 
 
 .8112 
 
 .0072 
 
 3843 
 
 .0094 
 
 3217 
 
 .0036 
 
 1.787 
 
 .0053 
 
 7834 
 
 .0072 
 
 .3640 
 
 .0095 
 
 -3063 
 
 .0036 
 
 i-747 
 
 0053 
 
 7558 
 
 .0073 
 
 3439 
 
 .0095 
 
 .2909 
 
 .0037 
 
 1.707 
 
 53 
 
 .7283 
 
 .0073 
 
 3238 
 
 .0096 
 
 2757 
 
 1.0037 
 
 11.668 
 
 1-0053 
 
 9.701 
 
 1.0073 
 
 8-3039 
 
 1.0096 
 
 7.2604 
 
 .0037 
 
 !.6 2 8 
 
 0054 
 
 6739 
 
 .0074 
 
 .2840 
 
 .0097 
 
 2453 
 
 .0037 
 
 1.589 
 
 -0054 
 
 .6469 
 
 .0074 
 
 .2642 
 
 0097 
 
 .2302 
 
 .0038 
 
 
 .0054 
 
 .62 
 
 .0074 
 
 .2446 
 
 .0097 
 
 .2152 
 
 .0038 
 
 1-512 
 
 0055 
 
 5933 
 
 .0075 
 
 225 
 
 .0098 
 
 .2002 
 
 1.0038 
 
 11.474 
 
 1-0055 
 
 9. 5668 
 
 1.0075 
 
 8.2055 
 
 .0098 
 
 7-1853 
 
 CO-SKC'T.; SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T 
 
 SKCANT. 
 
 850 
 
 840 
 
 83 
 
 820 
 
 From Haswell's " Engineering." Copyright, 1884, hy Harper & Brothers 
 
 223 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 
 8 
 
 90 
 
 10 11 11 
 
 t 
 
 SECANT. 
 
 CO-SEC'T 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-8KC'T. 
 
 o 
 
 1.0098 
 
 7-I853 
 
 1.0125 
 
 6.3924 
 
 1.0154 
 
 5.7588 
 
 1.0187 
 
 5.2 4 08 
 
 I 
 
 .0099 
 
 .1704 
 
 .0125 
 
 .3807 
 
 0155 
 
 7493 
 
 .0188 
 
 2 33 
 
 2 
 
 .0099 
 
 1557 
 
 .0125 
 
 369 
 
 0155 
 
 7398 
 
 .0188 
 
 .2252 
 
 3 
 
 .0099 
 
 .1409 
 
 .0126 
 
 3574 
 
 .0156 
 
 7304 
 
 .0189 
 
 2174 
 
 4 
 
 C 
 
 .01 
 
 1263 
 
 .0126 
 
 3458 
 6 IIAI 
 
 .0156 
 
 I OI C7 
 
 .721 
 
 571 17 
 
 .0189 
 
 I.OJO 
 
 .2097 
 C 2OIQ 
 
 6 
 
 .0101 
 
 .0972 
 
 .0127 
 
 U * JJ*T J 
 
 .3228 
 
 * U1 D/ 
 0157 
 
 . / J.1/ 
 
 .7023 
 
 .0191 
 
 j' *"Hy 
 
 .1942 
 
 7 
 
 .OIOI 
 
 .0827 
 
 .0128 
 
 3"3 
 
 .0158 
 
 693 
 
 .0191 
 
 .1865 
 
 8 
 
 .0102 
 
 .0683 
 
 .0128 
 
 .2999 
 
 .0158 
 
 .6838 
 
 .0192 
 
 .1788 
 
 9 
 
 .0102 
 
 539 
 
 .0129 
 
 .2885 
 
 .0159 
 
 6745 
 
 .0192 
 
 .1712 
 
 10 
 
 I.OIO2 
 
 7.0396 
 
 1.0129 
 
 6. 2772 
 
 I.OI59 
 
 5-6653 
 
 1.0193 
 
 5.1636 
 
 ii 
 
 .0103 
 
 .0254 
 
 .013 
 
 .2659 
 
 .Ol6 
 
 .6561 
 
 .0193 
 
 .156 
 
 12 
 
 .0103 
 
 .0112 
 
 .013 
 
 .2546 
 
 .Ol6 
 
 .647 
 
 .0194 
 
 .1484 
 
 13 
 
 .0104 
 
 6.9971 
 
 .0131 
 
 2434 
 
 .Ol6l 
 
 6379 
 
 .0195 
 
 .1409 
 
 14 
 
 .0104 
 
 983 
 
 .0131 
 
 .2322 
 
 .Ol62 
 
 .6288 
 
 .0195 
 
 I 333 
 
 15 
 
 I.OI04 
 
 6.969 
 
 1.0132 
 
 6.22II 
 
 I.0l62 
 
 5-6197 
 
 1.0196 
 
 5.1258 
 
 16 
 
 .OIO5 
 
 955 
 
 .0132 
 
 .21 
 
 .0163 
 
 .6107 
 
 .0196 
 
 .1183 
 
 i? 
 
 .0105 
 
 .9411 
 
 .0133 
 
 .199 
 
 .0163 
 
 .6017 
 
 .0197 
 
 .1109 
 
 18 
 
 ,OIo6 
 
 9 2 73 
 
 0133 
 
 .188 
 
 .0164 
 
 .5928 
 
 .0198 
 
 .1034 
 
 *9 
 
 .0106 
 
 9*35 
 
 .0134 
 
 .177 
 
 .0164 
 
 .5838 
 
 .0198 
 
 .096 
 
 20 
 
 I.OIOJ 
 
 6.8998 
 
 1.0134 
 
 6. 1661 
 
 I.0l65 
 
 5-5749 
 
 1.0199 
 
 5.0886 
 
 21 
 
 .0107 
 
 .8861 
 
 0135 
 
 1552 
 
 .0165 
 
 .566 
 
 .0199 
 
 .0812 
 
 22 
 
 .0107 
 
 .8725 
 
 0135 
 
 J 443 
 
 .Ol66 
 
 557 2 
 
 .02 
 
 0739 
 
 2 3 
 
 .0108 
 
 -8589 
 
 .0136 
 
 I 335 
 
 .0166 
 
 5484 
 
 .0201 
 
 .0666 
 
 24 
 
 .OI08 
 
 8454 
 
 .0136 
 
 .1227 
 
 .0167 
 
 5396 
 
 .O2OI 
 
 593 
 
 2 5 
 
 I.OlOg 
 
 6.832 
 
 1.0136 
 
 6. 1 12 
 
 I.0l67 
 
 5-5308 
 
 1.0202 
 
 5-052 
 
 26 
 
 .0109 
 
 8185 
 
 0137 
 
 .1013 
 
 .Ol68 
 
 .5221 
 
 .O2O2 
 
 .0447 
 
 2 7 
 
 .Oil 
 
 .8052 
 
 OI 37 
 
 .0906 
 
 .0169 
 
 5134 
 
 .0203 
 
 0375 
 
 28 
 
 .Oil 
 
 .7919 
 
 .0138 
 
 .08 
 
 .0169 
 
 5047 
 
 .0204 
 
 .0302 
 
 29 
 
 .OIII 
 
 .7787 
 
 .0138 
 
 .0694 
 
 .017 
 
 496 
 
 .0204 
 
 .023 
 
 30 
 
 I.OIII 
 
 6 -7 6 55 
 
 1.0139 
 
 6.0588 
 
 I.OI7 
 
 5-4874 
 
 1.0205 
 
 5.0158 
 
 31 
 
 .OIII 
 
 7523 
 
 .0139 
 
 .0483 
 
 .0171 
 
 .4788 
 
 .0205 
 
 .0087 
 
 32 
 
 .0112 
 
 7392 
 
 .014 
 
 0379 
 
 .0171 
 
 .4702 
 
 .O2O6 
 
 .0015 
 
 33 
 
 .OII2 
 
 .7262 
 
 .014 
 
 .0274 
 
 .0172 
 
 .4617 
 
 .0207 
 
 4-9944 
 
 34 
 
 .0113 
 
 7 I 3 2 
 
 .0141 
 
 .017 
 
 .0172 
 
 4532 
 
 .0207 
 
 9873 
 
 35 
 
 I.OII3 
 
 6.7003 
 
 1.0141 
 
 6.0066 
 
 I.OI73 
 
 5-4447 
 
 1. 0208 
 
 4.9802 
 
 36 
 
 .0114 
 
 .6874 
 
 .0142 
 
 5-9963 
 
 .0174 
 
 4362 
 
 .0208 
 
 9732 
 
 37 
 
 .OII4 
 
 .6745 
 
 .0142 
 
 .986 
 
 .0174 
 
 .4278 
 
 .0209 
 
 .9661 
 
 38 
 
 .OII5 
 
 .6617 
 
 .0143 
 
 9758 
 
 OI 75 
 
 .4194 
 
 .O2I 
 
 959 1 
 
 39 
 
 .0115 
 
 .649 
 
 .0143 
 
 9655 
 
 0175 
 
 .411 
 
 .021 
 
 952i 
 
 40 
 
 I.OII5 
 
 6.6363 
 
 1.0144 
 
 5-9554 
 
 1.0176 
 
 5.4026 
 
 I.02II 
 
 4-9452 
 
 4i 
 
 .OIl6 
 
 .6237 
 
 .0144 
 
 9452 
 
 .0176 
 
 3943 
 
 .0211 
 
 .9382 
 
 42 
 
 .OIl6 
 
 .6111 
 
 .0145 
 
 9351 
 
 .0177 
 
 386 
 
 .0212 
 
 93i3 
 
 43 
 
 .0117 
 
 5985 
 
 .0145 
 
 925 
 
 0177 
 
 3777 
 
 .0213 
 
 9243 
 
 44 
 45 
 
 .OII7 
 I.OIlS 
 
 .586 
 6-5736 
 
 .0146 
 1.0146 
 
 9'S 
 5.9049 
 
 .0178 
 1.0179 
 
 3695 
 5-3612 
 
 0213 
 I.02I4 
 
 9*75 
 4.9106 
 
 46 
 
 .OIl8 
 
 .5612 
 
 0147 
 
 
 .0179 
 
 353 
 
 .0215 
 
 937 
 
 47 
 
 .0119 
 
 .5488 
 
 .0147 
 
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 .018 
 
 3449 
 
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 .8969 
 
 48 
 
 .Olig 
 
 5365 
 
 .0148 
 
 8751 
 
 .018 
 
 3367 
 
 .02l6 
 
 .8901 
 
 49 
 
 .0119 
 
 5243 
 
 .0148 
 
 .8652 
 
 .0181 
 
 .3286 
 
 ,O2l6 
 
 8833 
 
 5 
 
 I.OI2 
 
 6.5121 
 
 1.0149 
 
 5-8554 
 
 1.0181 
 
 5'3205 ! 
 
 1.0217 
 
 4.8765 
 
 5i 
 
 .012 
 
 4999 
 
 015 
 
 .8456 
 
 .0182 
 
 3124 ; 
 
 .02l8 
 
 .8697 
 
 52 
 
 .OI2I 
 
 .4878 
 
 .015 
 
 8358 
 
 .0182 
 
 344 j 
 
 .02l8 
 
 .863 
 
 53 
 
 .0121 
 
 4757 
 
 .0151 
 
 .8261 
 
 .0183 
 
 .2963 
 
 .0219 
 
 .8563 
 
 54 
 
 .0122 
 
 4637 
 
 .0151 
 
 .8163 
 
 .0184 
 
 .2883 
 
 .022 
 
 .8496 
 
 55 
 
 I OI22 
 
 6.4517 
 
 1.0152 
 
 5-8067 
 
 1.0184 
 
 5.2803 1 
 
 1.022 
 
 4.8429 
 
 56 
 
 .0123 
 
 439 
 
 .0152 
 
 797 
 
 .0185 
 
 .2724 
 
 .0221 
 
 .8362 
 
 57 
 58 
 
 .OI23 
 .0124 
 
 4279 
 .416 
 
 0153 
 0153 
 
 7874 
 .7778 
 
 .0185 
 .0186 
 
 .2645 
 .2566 
 
 .O22I 
 .0222 
 
 .8296 
 .8229 
 
 59 
 
 .OI24 
 
 .4042 
 
 .0154 
 
 .7683 
 
 .0186 
 
 .2487 
 
 .0223 
 
 ,8163 
 
 60 
 
 I.OI25 
 
 6.3924 
 
 1.0154 
 
 5-7588 
 
 1.0187 
 
 5-2408 
 
 I.O223 
 
 4.8097 
 
 / 
 
 CO-SKC'T. 
 
 SECANT. 
 
 CO-SEC'T. SECANT. 
 
 CO-SBC'T. 
 
 SECANT. 
 
 CO-SKC'T. : SECANT. 
 
 | 81 
 
 80 
 
 790 
 
 78 
 
 From Haswell's " Engineering." Copyright, 1884, by Harper & Brothers. 
 
 224 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 120 
 
 13 
 
 140 
 
 150 
 
 SECANT. 
 
 CO-SC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SlC A NT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 1.0223 
 
 4.8097 
 
 1.0263 
 
 4- 4454 
 
 1.0306 
 
 4-I336 
 
 1-0353 
 
 3-8637 
 
 .0224 
 
 .8032 
 
 .0264 
 
 4398 
 
 .0307 
 
 . 287 
 
 353 
 
 8595 
 
 .0225 
 
 .7966 
 
 .0264 
 
 4342 
 
 .0308 
 
 239 
 
 0354 
 
 8553 
 
 0225 
 
 .7901 
 
 .0265 
 
 .4287 
 
 .0308 
 
 . 191 
 
 0355 
 
 .8512 
 
 .0226 
 
 7835 
 
 .0266 
 
 .4231 
 
 .0309 
 
 144 
 
 .0356 
 
 .847 
 
 1.0226 
 
 4-777 
 
 1.0266 
 
 4.4176 
 
 1.031 
 
 4. 096 
 
 1-0357 
 
 3.8428 
 
 .0227 
 
 .7706 
 
 .0267 
 
 .4121 
 
 .0311 
 
 . 048 
 
 .0358 
 
 .8387 
 
 .0228 
 
 .7641 
 
 .0268 
 
 4065 
 
 .0311 
 
 . 001 
 
 0358 
 
 8346 
 
 .0-28 
 .0229 
 
 7576 
 7512 
 
 .0268 
 .0269 
 
 .4011 
 3956 
 
 .0312 
 0313 
 
 0953 
 
 0359 
 .036 
 
 .8304 
 .8263 
 
 1.023 
 
 4.7448 
 
 1.027 
 
 4.3901 
 
 1.0314 
 
 4.0859 
 
 1.0361 
 
 3.8222 
 
 023 
 
 7384 
 
 .0271 
 
 3847 
 
 .0314 
 
 .0812 
 
 .0362 
 
 .8181 
 
 .0231 
 
 732 
 
 .0271 
 
 379 2 
 
 0315 
 
 .0765 
 
 .0362 
 
 .814 
 
 .0232 
 
 7257 
 
 .0272 
 
 3738 
 
 .0316 
 
 .0718 
 
 0363 
 
 .81 
 
 .0232 
 
 7*93 
 
 .0273 
 
 3684 
 
 3 I 7 
 
 .0672 
 
 .0364 
 
 .8059 
 
 1.0233 
 
 4-7I3 
 
 1.0273 
 
 
 1.0317 
 
 4-0625 
 
 1.0365 
 
 3.8018 
 
 .0234 
 
 .7067 
 
 .0274 
 
 3576 
 
 .0318 
 
 0579 
 
 .0366 .7978 
 
 .0234 
 
 
 .0275 
 
 3522 
 
 .0319 
 
 0532 
 
 0367 -7937 
 
 0235 
 
 
 .0276 
 
 3469 
 
 032 
 
 .0486 
 
 .0367 | .7897 
 
 0235 
 
 ,J 
 
 .0276 
 
 3415 
 
 .032 
 
 .044 
 
 .0368 
 
 7857 
 
 1.0236 
 
 4.6817 
 
 1.0277 
 
 
 1.0321 
 
 4.0394 
 
 1.0369 
 
 3-7816 
 
 .0237 
 
 6754 
 
 .0278 
 
 3309 
 
 .0322 
 
 .0348 
 
 037 
 
 7776 
 
 .0237 
 
 .6692 
 
 .0278 
 
 3256 
 
 0323 
 
 .0302 
 
 .0371 
 
 7736 
 
 .0238 
 
 .6631 
 
 .0279 
 
 3203 
 
 0323 
 
 .0256 
 
 037 1 
 
 7697 
 
 .0239 
 
 .6569 
 
 .028 
 
 
 .0324 
 
 .0211 
 
 .0372 
 
 7657 
 
 1.0239 
 
 4.6507 
 
 1.028 
 
 4.3098 
 
 1.0325 
 
 4.0165 
 
 1-0373 
 
 
 .024 
 
 .6446 
 
 .0281 
 
 345 
 
 .0326 
 
 .012 
 
 0374 
 
 7577 
 
 .0241 
 
 6385 
 
 .0282 
 
 -2993 
 
 .0327 
 
 .0074 
 
 375 
 
 7538 
 
 .0241 
 
 .6324 
 
 .0283 
 
 .2941 
 
 .0327 
 
 .0029 
 
 .0376 
 
 .7498 
 
 .0242 
 
 .6263 
 
 .0283 
 
 .2888 
 
 .0328 
 
 3.9984 
 
 .0376 
 
 7459 
 
 1-0213 
 
 4.6202 
 
 1.0284 
 
 4.2836 
 
 1.0329 
 
 3-9939 
 
 1-0377 
 
 3-742 
 
 .0243 
 
 6142 
 
 0285 
 
 .2785 
 
 033 
 
 .9894 
 
 .0378 
 
 .738 
 
 .0244 
 
 6081 
 
 .0285 
 
 2733 
 
 33 
 
 985 
 
 0379 
 
 7341 
 
 .0245 
 
 .6021 
 
 .0286 
 
 .2681 
 
 0331 
 
 .9805 
 
 .038 
 
 .7302 
 
 -0245 
 
 .5961 
 
 .0287 
 
 .263 
 
 0332 
 
 .976 
 
 -0381 
 
 7263 
 
 1.0246 
 
 4.5901 
 
 1.0288 
 
 4-2579 
 
 1-0333 
 
 3-9716 
 
 1.0382 
 
 3-7224 
 
 .0247 
 
 .5841 
 
 .0288 
 
 .2527 
 
 334 
 
 .9672 
 
 .0382 
 
 .7186 
 
 .0247 
 
 .5782 
 
 .0289 
 
 .2476 
 
 0334 
 
 .9627 
 
 0383 
 
 7'47 
 
 .0248 
 
 .5722 
 
 .029 
 
 2425 
 
 0335 
 
 9583 
 
 .0384 
 
 .7108 
 
 .0249 
 
 5663 
 
 .0291 
 
 2375 
 
 0336 
 
 9539 
 
 -0385 
 
 .707 
 
 1.0249 
 
 4.5604 
 
 1.0291 
 
 4.2324 
 
 1-0337 
 
 3-9495 
 
 1.0386 
 
 3-7031 
 
 .025 
 
 5545 
 
 .0292 
 
 .2273 
 
 0338 
 
 9451 
 
 .0387 
 
 6993 
 
 .0251 
 
 -5486 
 
 .0293 
 
 2223 
 
 0338 
 
 .9408 
 
 .0387 
 
 6955 
 
 .0251 
 
 5428 
 
 .0293 
 
 2173 
 
 339 
 
 9364 
 
 .0388 
 
 .6917 
 
 .0252 
 
 5369 
 
 .0294 
 
 .2122 
 
 034 
 
 932 
 
 .0389 
 
 .6878 
 
 1.0253 
 
 4-53" 
 
 1.0295 
 
 4.2072 
 
 1.0341 
 
 
 1.039 
 
 3.684 
 
 0253 
 
 5253 
 
 .0296 
 
 .2022 
 
 .0341 
 
 9234 
 
 .0391 
 
 .6802 
 
 .0254 
 
 
 .0296 
 
 .1972 
 
 .0342 
 
 .919 
 
 .0392 
 
 .6765 
 
 0255 
 
 .5137 
 
 .0297 
 
 .1923 
 
 343 
 
 .9147 
 
 393 
 
 .6727 
 
 0255 
 
 1.0256 
 
 5079 
 4.5021 
 
 .0298 
 1.0299 
 
 .1873 
 4.1824 
 
 0344 
 1-0345 
 
 .9104 
 3.9061 
 
 0393 
 1.0394 
 
 .6689 
 3-6651 
 
 .0257 
 
 4964 
 
 .0299 
 
 1774 
 
 0345 
 
 .9018 
 
 0395 
 
 .6614 
 
 .0257 
 
 
 
 1725 
 
 .0346 
 
 .8976 
 
 0396 
 
 6576 
 
 .0258 
 
 .485 
 
 0301 
 
 .1676 
 
 0347 
 
 8933 
 
 0397 
 
 6539 
 
 .0259 
 
 4793 
 
 .0302 
 
 . 1627 
 
 .0348 
 
 .899 
 
 .0398 
 
 .6502 
 
 1.026 
 
 4-4736 
 
 1.0302 
 
 4-I578 
 
 1.0349 
 
 3.8848 
 
 1.0399 
 
 3.6464 
 
 .026 
 
 4679 
 
 0303 
 
 .1529 
 
 0349 
 
 .8805 
 
 0399 
 
 6427 
 
 .0261 
 .0262 
 
 .4623 
 .4566 
 
 .0304 
 .0305 
 
 .1481 
 .1432 
 
 035 
 0351 
 
 .8763 
 .8721 
 
 .04 
 .0401 
 
 639 
 6353 
 
 .0262 
 
 451 
 
 0305 
 
 .1384 
 
 0352 
 
 .8679 
 
 .0402 
 
 .6316 
 
 1.0263 
 
 4-4454 
 
 1.0306 
 
 4-I336 
 
 1-0353 
 
 3-8637 
 
 1.0403 
 
 3.6279 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 770 
 
 760 
 
 75 
 
 74 
 
 From Bagwell's "Engineering." Copyright, 18S4J by Harper * Brother 
 
 225 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 16 
 
 170 
 
 18 
 
 19 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T 
 
 SECANT. | CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 1.0403 
 
 3.6279 
 
 1-0457 
 
 3-4203 
 
 1-0515 
 
 3.2361 
 
 1.0576 
 
 3-7 I 5 
 
 .0404 
 
 .6243 
 
 .0458 
 
 .417 
 
 .0516 
 
 2332 
 
 577 
 
 .069 
 
 .0405 
 
 .6206 
 
 0459 
 
 .4138 
 
 OS 1 / 
 
 2303 
 
 0578 
 
 .0664 
 
 .0406 
 
 .6169 
 
 .046 
 
 .4106 
 
 0518 
 
 .2274 
 
 579 
 
 .0638 
 
 .0406 
 
 6i33 
 
 .0461 
 
 4073 
 
 .0519 
 
 .2245 
 
 .058 
 
 .0612 
 
 1.0407 
 
 3.6096 
 
 1.0461 
 
 3.4041 
 
 1.052 
 
 3.2216 
 
 1.0581 
 
 3.0586 
 
 .0408 
 
 .606 
 
 .0462 
 
 .4009 
 
 .0521 
 
 .2188 
 
 .0582 
 
 .0561 
 
 .0409 
 
 .6024 
 
 .0463 
 
 3977 
 
 .0522 
 
 .2159 
 
 .0584 
 
 535 
 
 .041 
 
 5987 
 
 .0464 
 
 3945 
 
 .0523 
 
 .2131 
 
 0585 
 
 .0509 
 
 .0411 
 
 595 1 
 
 0465 
 
 39i3 
 
 .0524 
 
 .2102 
 
 .0586 
 
 .0484 
 
 1.0412 
 
 3-59*5 
 
 1.0466 
 
 3-388i 
 
 1-0525 
 
 3.2074 
 
 1.0587 
 
 3-0458 
 
 .0413 
 
 5879 
 
 .0467 
 
 -3849 
 
 .0526 
 
 .2045 
 
 .0588 
 
 433 
 
 .0413 
 
 5843 
 
 .0468 
 
 -3817 
 
 .0527 
 
 .2017 
 
 0589 
 
 .0407 
 
 .0414 
 
 5807 
 
 .0469 
 
 3785 
 
 .0528 
 
 .1989 
 
 59 
 
 .0382 
 
 .0415 
 
 5772 
 
 .047 
 
 3754 
 
 0529 
 
 .196 
 
 .0591 
 
 357 
 
 1.0416 
 
 3-573<5 
 
 1.0471 
 
 3-3722 
 
 1-053 
 
 3-I932 
 
 1.0592 
 
 3-033I 
 
 .0417 
 
 57 
 
 .0472 
 
 369 
 
 0531 
 
 .1904 
 
 593 
 
 .0306 
 
 .0418 
 
 5665 
 
 473 
 
 3659 
 
 .0532 
 
 .1876 
 
 594 
 
 .0281 
 
 .0419 
 
 .5629' 
 
 .0474 
 
 3627 
 
 0533 
 
 .1848 
 
 595 
 
 .0256 
 
 .042 
 
 5594 
 
 475 
 
 3596 
 
 534 
 
 .182 
 
 .0596 
 
 .0231 
 
 1.042 
 
 3-5559 
 
 1.0476 
 
 3-3565 
 
 1-0535 
 
 3- I 79 2 
 
 1.0598 
 
 3.0206 
 
 .0421 
 
 5523 
 
 .0477 
 
 3534 
 
 .0536 
 
 .1764 
 
 599 
 
 .0181 
 
 .0422 
 
 -5488 
 
 .0478 
 
 3502 
 
 0537 
 
 1736 
 
 .06 
 
 .0156 
 
 .0423 
 
 5453 
 
 .0478 
 
 347 1 
 
 0538 
 
 .1708 
 
 .0601 
 
 .0131 
 
 .0424 
 
 .5418 
 
 .0479 
 
 344 
 
 539 
 
 .1681 
 
 .0602 
 
 .0106 
 
 1.0425 
 
 3-5383 
 
 1.048 
 
 3- 349 
 
 1.054 
 
 3-i653 
 
 1.0603 
 
 3.0081 
 
 .0426 
 
 5348 
 
 .0481 
 
 3378 
 
 .0541 
 
 .1625 
 
 .0604 
 
 .0056 
 
 .0427 
 
 5313 
 
 .0482 
 
 3347 
 
 .0542 
 
 .1598 
 
 .0605 
 
 0031 
 
 .0428 
 
 5279 
 
 .0483 
 
 33i6 
 
 543 
 
 1 57 
 
 .0606 
 
 .0007 
 
 .0428 
 
 5244 
 
 .0484 
 
 .3286 
 
 .0544 
 
 1543 
 
 .0607 
 
 2.9982 
 
 1.0429 
 
 3- 5209 
 
 1.0485 
 
 3-3255 
 
 1-0545 
 
 3- I 5 I 5 
 
 1. 0608 
 
 2 -9957 
 
 043 
 
 5175 
 
 .0486 
 
 3224 
 
 0546 
 
 .1488 
 
 .0609 
 
 9933 
 
 .0431 
 
 5i4 
 
 .0487 
 
 3*94 
 
 547 
 
 .1461 
 
 .0611 
 
 .9908 
 
 .0432 
 
 .5106 
 
 .0488 
 
 3163 
 
 .0548 
 
 1433 
 
 .0612 
 
 .9884 
 
 433 
 
 .5072 
 
 .0489 
 
 3i33 
 
 0549 
 
 .1406 
 
 .0613 
 
 9 8 59 
 
 1.0434 
 
 3- 5037 
 
 1.049 
 
 3-3102 
 
 1-055 
 
 3-1379 
 
 1.0614 
 
 2.9835 
 
 0435 
 
 .5003 
 
 .0491 
 
 .3072 
 
 0551 
 
 1352 
 
 .0615 
 
 .981 
 
 .0436 
 
 .4969 
 
 .0492 
 
 .3042 
 
 0552 
 
 1325 
 
 .0616 
 
 .9786 
 
 437 
 
 4935 
 
 0493 
 
 .301 
 
 553 
 
 .1298 
 
 .0617 
 
 .9762 
 
 .0438 
 
 .4901 
 
 494 
 
 .298 
 
 554 
 
 .1271 
 
 .0618 
 
 9738 
 
 1.0438 
 
 3.4867 
 
 1.0495 
 
 3-295 
 
 J-oSSS 
 
 3.1244 
 
 1.0619 
 
 2.9713 
 
 439 
 
 4833 
 
 .0496 
 
 .292 
 
 55 6 
 
 .1217 
 
 .062 
 
 .9689 
 
 .044 
 
 4799 
 
 .0497 
 
 .289 
 
 0557 
 
 
 .0622 
 
 -9665 
 
 .0441 
 
 .4766 
 
 .0498 
 
 .286 
 
 0558 
 
 .1163 
 
 .0623 
 
 .9641 
 
 .0442 
 
 4732 
 
 .0499 
 
 -283 
 
 -0559 
 
 "37 
 
 .0624 
 
 .9617 
 
 1.0443 
 
 3-4698 
 
 1.05 
 
 3.280 
 
 1.056 
 
 3.111 
 
 1.0625 
 
 2-9593 
 
 .0444 
 
 4665 
 
 .0501 
 
 .277 
 
 .0561 
 
 .1083 
 
 .0626 
 
 95 6 9 
 
 445 
 
 4632 
 
 .0502 
 
 274 
 
 .0562 
 
 i57 
 
 .0627 
 
 9545 / 
 
 .0446 
 
 4598 
 
 0503 
 
 .271 
 
 .0563 
 
 .103 
 
 .0628 
 
 .9521 
 
 .0447 
 
 4505 
 
 .0504 
 
 .2683 
 
 0565 
 
 . 1004 
 
 .0629 
 
 9497 
 
 1.0448 
 
 3-4532 
 
 1-0505 
 
 3-2653 
 
 1.0566 
 
 3-0977 
 
 1.063 
 
 2-9474 
 
 .0448 
 
 .4498 
 
 .0506 
 
 .2624 
 
 .0567 
 
 .0951 
 
 .0632 
 
 945 
 
 .0449 
 
 4465 
 
 .0507 
 
 2594 
 
 .0568 
 
 
 0633 
 
 .9426 
 
 045 
 
 4432 
 
 .0508 
 
 2565 
 
 .0569 
 
 .0898 
 
 .0634 
 
 .9402 
 
 .0451 
 
 
 0509 
 
 2535 
 
 057 
 
 .0872 
 
 0035 
 
 9379 
 
 1.0452 
 
 3-4366 
 
 1.051 
 
 3.2506 
 
 1.0571 
 
 3.0846 
 
 1.0636 
 
 2-9355 
 
 453 
 
 4334 
 
 0511 
 
 .2477 
 
 .0572 
 
 .082 
 
 .0637 
 
 9332 
 
 454 
 
 .4301 
 
 .0512 
 
 .2448 
 
 0573 
 
 0793 
 
 .0638 
 
 .9308 
 
 0455 
 
 .4268 
 
 0513 
 
 .2419 
 
 574 
 
 .0767 
 
 .0639 
 
 -9285 
 
 .0456 
 
 4236 
 
 .0514 
 
 239 
 
 575 
 
 .0741 
 
 .0641 
 
 .9261 
 
 1-0457 
 
 3-4203 
 
 1-0515 
 
 3.2361 
 
 1.0576 
 
 3-0715 
 
 1.0642 
 
 2.9238 
 
 CO-SEC'T. SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. SECANT, j CO-SEC'T. 
 
 SECANT. 
 
 730 
 
 720 | 
 
 71 Jl 70 
 
 bruin Haswell's " .Engineering." Copyright, 1884, by Harper & Brothers. 
 
 226 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 20 
 
 SHCANT. | CO-SEC 'T 
 
 2 
 
 SBC ANT. 
 
 L 
 
 CO-SKC'T 
 
 2 
 
 SECANT. 
 
 2 
 CO-SEC'T. 
 
 2 
 
 SECANT. 
 
 3 
 CO-SKC'T. 
 
 1.0642 
 
 2.9238 
 
 1.0711 
 
 2.7904 
 
 1.0785 
 
 2.6695 
 
 1.0864 
 
 2-5593 
 
 .0643 
 
 .9215 
 
 0713 
 
 .7883 
 
 .0787 
 
 .6675 
 
 .0865 
 
 5575 
 
 .0644 
 
 .9191 
 
 .0714 
 
 .7862 
 
 .0788 
 
 .6656 
 
 .0866 
 
 5558 
 
 .0645 
 
 .9168 
 
 0715 
 
 .7841 
 
 .0789 
 
 .6637 
 
 .0868 
 
 554 
 
 .0646 
 1.0647 
 
 .9145 
 2.9122 
 
 .0716 
 1.0717 
 
 .782 
 2.7799 
 
 .079 
 1.0792 
 
 .6618 
 2.6599 
 
 .0869 
 1.087 
 
 5523 
 2.5506 
 
 .0648 
 
 .9098 
 
 .0719 
 
 .7778 
 
 0793 
 
 .658 
 
 .0872 
 
 .5488 
 
 .065 
 
 9075 
 
 .072 
 
 7757 
 
 .0794 
 
 .6561 
 
 .0873 
 
 547 1 
 
 .0651 
 
 .9052 
 
 .0721 
 
 7736 
 
 795 
 
 6542 
 
 .0874 
 
 5453 
 
 .0652 
 
 .9029 
 
 .0722 
 
 77 I 5 
 
 .0797 
 
 .6523 
 
 .0876 
 
 543 6 
 
 1.0653 
 
 2.9006 
 
 1.0723 
 
 2.7694 
 
 1.0798 
 
 2.6504 
 
 1.0877 
 
 2.5419 
 
 .0654 
 
 .8983 
 
 .0725 
 
 .7674 
 
 .0799 
 
 6485 
 
 .0878 
 
 .5402 
 
 0655 
 
 .896 
 
 .0726 
 
 7653 
 
 .0801 
 
 .6466 
 
 .088 
 
 5384 
 
 .0656 
 
 8937 
 
 .0727 
 
 .7632 
 
 .0802 
 
 .6447 
 
 .0881 
 
 5367 
 
 .0658 
 
 
 .0728 
 
 .7611 
 
 .0803 
 
 .6428 
 
 .0882 
 
 535 
 
 1.0659 
 
 2.8892 
 
 1.0729 
 
 2-7591 
 
 1.0804 
 
 2.641 
 
 1.0884 
 
 2-5333 
 
 .066 
 
 .8869 
 
 0731 
 
 757 
 
 .0806 
 
 .6391 
 
 .0885 
 
 53i6 
 
 .0661 
 
 8846 
 
 .0732 
 
 755 
 
 .0807 
 
 6372 
 
 .0886 
 
 5299 
 
 .0662 
 
 .8824 
 
 0733 
 
 7529 
 
 .0808 
 
 6353 
 
 .0888 
 
 .5281 
 
 .0663 
 1.0664 
 
 .8801 
 2.8778 
 
 0734 
 1.0736 
 
 7509 
 2.7488 
 
 .081 
 1.0811 
 
 6335 
 2.6316 
 
 .0889 
 1.0891 
 
 .5264 
 2-5247 
 
 .0666 
 
 .8756 
 
 0737 
 
 .7468 
 
 .0812 
 
 .6297 
 
 .0892 
 
 523 
 
 .0667 
 
 8733 
 
 .0738 
 
 7447 
 
 0813 
 
 .6279 
 
 .0893 
 
 5213 
 
 .0668 
 
 .8711 
 
 0739 
 
 .7427 
 
 -0815 
 
 .626 
 
 .0895 
 
 .5196 
 
 .0669 
 
 .8688 
 
 .074 
 
 .7406 
 
 .0816 
 
 .6242 
 
 .0896 
 
 5i79 
 
 1.067 
 
 2.8666 
 
 1.0742 
 
 2-7386 
 
 1.0817 
 
 2.6223 
 
 1.0897 
 
 2.5163 
 
 .0671 
 
 .8644 
 
 743 
 
 .7366 
 
 .0819 
 
 .6205 
 
 .0899 
 
 .5146 
 
 .0673 
 
 .8621 
 
 .0744 
 
 734 6 
 
 .082 
 
 .6186 
 
 .09 
 
 .5129 
 
 .0674 
 
 8599 
 
 745 
 
 7325 
 
 .0821 
 
 .6168 
 
 .0902 
 
 5"2 
 
 .0675 
 
 8577 
 
 .0747 
 
 735 
 
 .0823 
 
 .615 
 
 0903 
 
 5095 
 
 1.0676 
 
 2-8554 
 
 1.0748 
 
 2.7285 
 
 1.0824 
 
 2.6131 
 
 1.0904 
 
 2.5078 
 
 .0677 
 .0678 
 
 .8532 
 .851 
 
 .0749 
 075 
 
 .7265 
 7245 
 
 .0825 
 .0826 
 
 .6113 
 6095 
 
 .0906 
 .0907 
 
 .5062 
 5045 
 
 .0679 
 
 .8488 
 
 0751 
 
 7225 
 
 .0828 
 
 .6076 
 
 .0908 
 
 .5028 
 
 .0681 
 
 .8466 
 
 753 
 
 .7205 
 
 .0829 
 
 .6058 
 
 .091 
 
 .5011 
 
 1.0682 
 
 2.8444 
 
 I -754 
 
 2.7185 
 
 1.083 
 
 2.604 
 
 1.0911 
 
 2.4995 
 
 .0683 
 
 .8422 
 
 0755 
 
 7165 
 
 .0832 
 
 .6022 
 
 .0913 
 
 .4978 
 
 .0684 
 
 .84 
 
 .0756 
 
 7 J 45 
 
 0833 
 
 .6003 
 
 .0914 
 
 .4961 
 
 .0685 
 
 .8378 
 
 .0758 
 
 7125 
 
 .0834 
 
 5985 
 
 .0915 
 
 4945 
 
 .0686 
 
 8356 
 
 759 
 
 7 I0 5 
 
 .0836 
 
 59 6 7 
 
 .0917 
 
 .4928 
 
 1.0688 
 
 2-8334 
 
 1.076 
 
 2.7085 
 
 1.0837 
 
 2-5949 
 
 1.0918 
 
 2.4912 
 
 .0689 
 
 .8312 
 
 .0761 
 
 .7065 
 
 0838 
 
 5931 
 
 .092 
 
 4895 
 
 .069 
 
 .829 
 
 .0763 
 
 745 
 
 .084 
 
 59 J 3 
 
 .0921 
 
 .4879 
 
 .0691 
 
 .8269 
 
 .0764 
 
 .7026 
 
 .0841 
 
 5895 
 
 .0922 
 
 .4862 
 
 .0692 
 
 .8247 
 
 .0765 
 
 .7006 
 
 .0842 
 
 5877 
 
 .0924 
 
 .4846 
 
 1.0694 
 
 2.8225 
 
 1.0766 
 
 2.6986 
 
 1.0844 
 
 2-5859 
 
 1.0925 
 
 2.4829 
 
 .0695 
 
 .8204 
 
 .0768 
 
 .6967 
 
 .0845 
 
 .5841 
 
 .0927 
 
 .4813 
 
 .0696 
 
 .8182 
 
 .0769 
 
 .6947 
 
 .0846 
 
 .5823 
 
 .0928 
 
 4797 
 
 .0697 
 
 .816 
 
 .077 
 
 .6927 
 
 .0847 
 
 5805 
 
 .0929 
 
 .478 
 
 .0698 
 
 8139 
 
 .0771 
 
 .6908 
 
 .0849 
 
 5787 
 
 .0931 
 
 .4764 
 
 1.0699 
 
 2.8117 
 
 I -773 
 
 2.6888 
 
 1.085 
 
 2.577 
 
 1.0932 
 
 2.4748 
 
 .0701 
 
 .8096 
 
 .0774 
 
 .6869 
 
 .0851 
 
 5752 
 
 0934 
 
 4731 
 
 .0702 
 
 .8074 
 
 775 
 
 .6849 
 
 .0853 
 
 5734 
 
 0935 
 
 47i5 
 
 .0703 
 
 8053 
 
 .0776 
 
 683 
 
 .0854 
 
 57i6 
 
 .0936 
 
 .4699 
 
 .0704 
 
 .8032 
 
 .0778 
 
 .681 
 
 0855 
 
 5 6 99 
 
 .0938 
 
 4683 
 
 1.0705 
 
 2.801 
 
 1.0779 
 
 2.6791 
 
 1.0857 
 
 2.5681 
 
 1.0939 
 
 2.4666 
 
 .0707 
 
 .7989 
 
 .078 
 
 .6772 
 
 .0858 
 
 .5663 
 
 .0941 
 
 465 
 
 .0708 
 
 .7968 
 
 .0781 
 
 .6752 
 
 .0859 
 
 .5646 
 
 .0942 
 
 4^34 
 
 .0709 
 
 7947 
 
 .0783 
 
 6733 
 
 .0861 
 
 .5628 
 
 943 
 
 .4618 
 
 .071 
 
 79 2 5 
 
 .0784 
 
 .6714 
 
 .0862 
 
 .561 
 
 945 
 
 .4602 
 
 1.0711 
 
 2.7904 
 
 1.0785 
 
 2.6695 
 
 1.0864 
 
 2-5593 
 
 .0946 
 
 2.4586 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SBCANT. 
 
 CO-SEC'T. ! SECANT. 
 
 JO-SKC'T. 
 
 SECANT. 
 
 69 
 
 68 
 
 67 
 
 66 
 
 rom Haswell's Engineering." Copyright, 1884, by Harper & Brother*. 
 
 227 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 24 
 
 25 
 
 26 I! 270 
 
 SECANT. 
 
 CO-SEC'T 
 
 SECANT. 
 
 CO-SEC'T 
 
 SECANT. CO-SEC'T. 1 1 SECANT. | CO-SKC'T. 
 
 1.0946 
 
 2.4586 
 
 1.1034 
 
 2.3662 
 
 1.1126 
 
 2.2812 
 
 1.1223 
 
 2.2027 
 
 .0948 
 
 457 
 
 i35 
 
 .3647 
 
 .1127 
 
 .2798 
 
 .1225 
 
 .2014 
 
 .0949 
 
 4554 
 
 .1037 
 
 3632 
 
 .1129 
 
 .2784 
 
 .1226 
 
 .2002 
 
 .0951 
 
 4538 
 
 .1038 
 
 3618 
 
 1131 
 
 .2771 
 
 .1228 
 
 .1989 
 
 .0952 
 
 .4522 
 
 .104 
 
 36 o 
 
 .1132 
 
 2757 
 
 .123 
 
 1977 
 
 1-0953 
 
 2.4506 
 
 1. 1041 
 
 2.3588 
 
 1.1134 
 
 2.2744 
 
 1.1231 
 
 2.1964 
 
 0955 
 
 449 
 
 -1043 
 
 3574 
 
 "35 
 
 273 
 
 I2 33 
 
 .1952 
 
 .0956 
 
 4474 
 
 .1044 
 
 3559 
 
 "37 
 
 .2717 
 
 1235 
 
 1939 
 
 .0958 
 
 4458 
 
 .1046 
 
 3544 
 
 "39 
 
 2703 
 
 1237 
 
 .1927 
 
 0959 
 
 .4442 
 
 .1047 
 
 353 
 
 .114 
 
 .269 
 
 .1238 
 
 .1914 
 
 1.0961 
 
 2.4426 
 
 1.1049 
 
 2-35I5 
 
 1.1142 
 
 2.2676 
 
 1.124 
 
 2. 1902 
 
 .0962 
 
 44" 
 
 .105 
 
 35i 
 
 "43 
 
 .2663 
 
 .1242 
 
 .1889 
 
 .0963 
 
 4395 
 
 .1052 
 
 -3486 
 
 "45 
 
 .265 
 
 .1243 
 
 .1877 
 
 0965 
 
 4379 
 
 1053 
 
 3472 
 
 "47 
 
 .2636 
 
 .1245 
 
 .1865 
 
 .0966 
 
 4363 
 
 1055 
 
 3457 
 
 .!I 4 8 
 
 .2623 
 
 .1247 
 
 .1852 
 
 1.0968 
 
 2-4347 
 
 1.1056 
 
 2-3443 
 
 i."5 
 
 2.261 
 
 1.1248 
 
 2.184 
 
 .0969 
 
 4332 
 
 1058 
 
 .3428 
 
 ."5i 
 
 2596 
 
 .125 
 
 .1828 
 
 .0971 
 
 .4316 
 
 .1059 
 
 34*4 
 
 "53 
 
 2583 
 
 .1252 
 
 .1815 
 
 .0972 
 
 43 
 
 .1061 
 
 3399 
 
 "55 
 
 257 
 
 1253 
 
 .1803 
 
 0973 
 
 -4285 
 
 .1062 
 
 3385 
 
 ."56 
 
 2556 
 
 1255 
 
 .1791 
 
 *-975 
 
 2.4269 
 
 1.1064 
 
 2-3371 
 
 1-1158 
 
 2-2543 
 
 1.1257 
 
 2.1778 
 
 .0976 
 
 4254 
 
 .1065 
 
 -3356 
 
 "59 
 
 253 
 
 .1258 
 
 .1766 
 
 .0978 
 
 .4238 
 
 .1067 
 
 3342 
 
 .1161 
 
 .2517 
 
 .126 
 
 I 754 
 
 .0979 
 
 .4222 
 
 .1068 
 
 3328 
 
 .1163 
 
 2503 
 
 .1262 
 
 .1742 
 
 .0981 
 
 .4207 
 
 .107 
 
 33i3 
 
 .1164 
 
 .249 
 
 .1264 
 
 !73 
 
 1.0982 
 
 2.4191 
 
 1.1072 
 
 2.3299 
 
 1.1x66 
 
 2.2477 
 
 1.1265 
 
 2.1717 
 
 .0984 
 
 .4176 
 
 I0 73 
 
 3285 
 
 .1167 
 
 .2464 
 
 .1267 
 
 1705 
 
 .0985 
 
 .416 
 
 1075 
 
 3271 
 
 .1169 
 
 .2451 
 
 .1269 
 
 .1693 
 
 .0986 
 
 4145 
 
 .1076 
 
 3256 
 
 .1171 
 
 .2438 
 
 .127 
 
 .1681 
 
 .0988 
 
 4*3 
 
 .1078 
 
 .3242 
 
 .1172 
 
 2425 
 
 .1272 
 
 .1669 
 
 1.0989 
 
 2.4114 
 
 1.1079 
 
 2.3228 
 
 1.1174 
 
 2.2411 
 
 1.1274 
 
 2.1657 
 
 0991 
 
 .4099 
 
 .1081 
 
 3214 
 
 .1176 
 
 2398 
 
 I2 75 
 
 .1645 
 
 0992 
 
 .4083 
 
 .1082 
 
 32 
 
 "77 
 
 2385 
 
 .1277 
 
 l6 33 
 
 .0994 
 
 .4068 
 
 .1084 
 
 3186 
 
 "79 
 
 .2372 
 
 .1279 
 
 .162 
 
 0995 
 
 4053 
 
 .1085 
 
 3172 
 
 .118 
 
 2359 
 
 .1281 
 
 .1608 
 
 i 0997 
 
 2.4037 
 
 1.1087 
 
 2-3158 
 
 1.1182 
 
 2.2346 
 
 1.1282 
 
 2.1596 
 
 .0998 
 
 .4022 
 
 .1088 
 
 3i43 
 
 .1184 
 
 2333 
 
 .1284 
 
 .1584 
 
 .1 
 
 .4007 
 
 .109 
 
 .3129 
 
 "85 
 
 .232 
 
 ,1286 
 
 1572 
 
 .1001 
 
 399 2 
 
 .1092 
 
 3"5 
 
 .1187 
 
 .2307 
 
 .1287 
 
 156 
 
 .1003 
 
 397 6 
 
 .1093 
 
 3101 
 
 . .1189 
 
 -2294 
 
 .1289 
 
 .1548 
 
 1.1004 
 
 2.3961 
 
 1.1095 
 
 2.3087 
 
 1. 119 
 
 2.2282 
 
 1.1291 
 
 2.1536 
 
 1005 
 
 3946 
 
 .1096 
 
 3073 
 
 .1192 
 
 .2269 
 
 .1293 
 
 1525 
 
 .1007 
 
 3931 
 
 1098 
 
 359 
 
 "93 
 
 2256 
 
 ,1294 
 
 1513 
 
 .1008 
 
 39 J 6 j 
 
 .1099 
 
 3046 
 
 "95 
 
 2243 
 
 .1296 
 
 .1501 
 
 .101 
 
 .3901 
 
 IIOI 
 
 3032 
 
 "97 
 
 223 
 
 .1298 
 
 .1489 
 
 i. ion 
 
 2.3886 
 
 I. IIO2 
 
 2.3018 
 
 1.1198 
 
 2.2217 
 
 1.1299 
 
 2.1477 
 
 .1013 
 
 3871 
 
 .1104 
 
 3004 
 
 .12 
 
 .2204 
 
 .1301 
 
 .1465 
 
 .1014 
 
 3856 
 
 .1106 
 
 299 
 
 .1202 
 
 2192 
 
 TSOS 
 
 I 453 
 
 .1016 
 
 .3841 
 
 .1107 
 
 .2976 
 
 .1203 
 
 .2179 
 
 1305 
 
 .1441 
 
 .1017 
 
 3826 
 
 . IIO9 
 
 .2962 
 
 1205 
 
 .2166 
 
 .1306 
 
 H3 
 
 i.. 1019 
 
 2.3811 
 
 I. Ill 
 
 2.2949 
 
 I.I2O7 
 
 2.2153 
 
 1.1308 
 
 2.1418 
 
 .102 
 
 379 6 
 
 .1112 
 
 2935 
 
 .1208 
 
 .2141 
 
 131 
 
 .1406 
 
 .1022 
 
 378i 
 
 III3 
 
 .2921 
 
 .121 
 
 .2128 
 
 1312 
 
 !394 
 
 .IO23 
 
 .3766 
 
 HIS 
 
 .2907 
 
 .1212 
 
 -2115 
 
 1313 
 
 .1382 
 
 .1025 
 
 375 1 
 
 .IIl6 
 
 .2894 
 
 .1213 
 
 2103 
 
 1315 
 
 I37 1 
 
 I.IO26 
 
 2.3736 
 
 1.1118 
 
 2.288 
 
 I.I2I5 
 
 2.209 
 
 1-1317 
 
 2-1359 
 
 .1028 
 
 3721 
 
 .112 
 
 .2866 
 
 .!2I 7 
 
 .2077 
 
 I3I9 
 
 1347 
 
 .1029 
 
 .3706 
 
 .1121 
 
 2853 
 
 .I2l8 
 
 .2065 
 
 .132 
 
 !335 
 
 .1031 
 
 .3691 
 
 .1123 
 
 .2839 
 
 .122 
 
 .2052 
 
 .1322 
 
 .1324 
 
 .1032 
 
 
 .1124 
 
 .2825 
 
 .1222 
 
 2039 
 
 .1324 
 
 .1312 
 
 I.I034 
 
 2.3662 
 
 I.II26 
 
 2.2812 
 
 I.I223 
 
 2.2027 
 
 1.1326 
 
 2.13 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 65 
 
 64 
 
 63 | 
 
 620 
 
 From Haswell's " Engiueering." Copyright, 1=>84, by Harper & Brothers. 
 
 228 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 28 
 
 290 
 
 30 
 
 310 
 
 SECANT. 
 
 CO-SKC'T. 
 
 SECANT. 
 
 CO-SBC'T. 
 
 SECANT. 
 
 CO-SKC'T. 
 
 SECANT. 
 
 CO-SKC'T. 
 
 1.1326 
 
 2-13 
 
 I-I433 
 
 2.0627 
 
 I-I547 
 
 2 
 
 1. 1666 
 
 1.9416 
 
 1327 
 
 .1289 
 
 1435 
 
 .0616 
 
 1549 
 
 1.999 
 
 .1668 
 
 9407 
 
 .1329 
 
 .1277 
 
 1437 
 
 .0605 
 
 1551 
 
 .998 
 
 .167 
 
 9397 
 
 133 1 
 
 .1266 
 
 1439 
 
 0594 
 
 553 
 
 997 
 
 .1672 
 
 -9388 
 
 !333 
 
 1254 
 
 .1441 
 
 0583 
 
 1555 
 
 .996 
 
 .1674 
 
 9378 
 
 i-i334 
 
 2. 1242 
 
 I-I443 
 
 2-0573 
 
 i-i557 
 
 1-995 
 
 i 1676 
 
 i 9369 
 
 .1336 
 
 -I23I 
 
 1445 
 
 .0562 
 
 1559 
 
 994 
 
 .1678 
 
 93 6 
 
 1338 
 
 .1219 
 
 .1446 
 
 0551 
 
 .1561 
 
 993 
 
 .1681 
 
 935 
 
 134 
 
 .1208 
 
 .1448 
 
 054 
 
 .1562 
 
 .992 
 
 .1683 
 
 9341 
 
 I34 1 
 
 . 1196 
 
 145 
 
 053 
 
 .1564 
 
 .991 
 
 .1685 
 
 9332 
 
 I-I343 
 
 2.1185 
 
 1.1452 
 
 2.0519 
 
 1.1566 
 
 1.99 
 
 1.1687 
 
 1.9322 
 
 '1345 
 
 -"73 
 
 1454 
 
 .0508 
 
 .1568 
 
 .989 
 
 .1689 
 
 9313 
 
 I 347 
 
 .1162 
 
 .1456 
 
 .0498 
 
 !57 
 
 .988 
 
 1691 
 
 9304 
 
 !349 
 
 "5 
 
 .1458 
 
 .0487 
 
 1572 
 
 .987 
 
 .1693 
 
 9 2 95 
 
 J 35 
 
 "39 
 
 1459 
 
 .0476 
 
 1574 
 
 .986 
 
 .1695 
 
 .9285 
 
 1-1352 
 
 2.1127 
 
 1.1461 
 
 2.0466 
 
 1.1576 
 
 1.985 
 
 1.1697 
 
 1.9276 
 
 1354 
 
 .1116 
 
 .1463 
 
 0455 
 
 1578 
 
 .984 
 
 .1699 
 
 .9267 
 
 i35 6 
 
 .1104 
 
 .1465 
 
 .0444 
 
 .158 
 
 983 
 
 .1701 
 
 .9258 
 
 1357 
 
 .1093 
 
 .1467 
 
 434 
 
 .1582 
 
 .982 
 
 !703 
 
 .9248 
 
 1359 
 
 .1082 
 
 .1469 
 
 0423 
 
 1584 
 
 .9811 
 
 .1705 
 
 9239 
 
 1.1361 
 
 2.107 
 
 1.1471 
 
 2.0413 
 
 1.1586 
 
 1.9801 
 
 1.1707 
 
 1.923 
 
 1363 
 
 .1059 
 
 1473 
 
 .0402 
 
 .1588 
 
 .9791 
 
 .'1709 
 
 .9221 
 
 1365 
 
 .1048 
 
 .1474 
 
 .0392 
 
 159 
 
 .9781 
 
 .1712 
 
 .9212 
 
 .1366 
 
 .1036 
 
 .1476 
 
 .0381 
 
 .1592 
 
 .9771 
 
 i7 J 4 
 
 .9203 
 
 .1368 
 
 1025 
 
 .1478 
 
 37 
 
 1594 
 
 .9761 
 
 .1716 
 
 9'93 
 
 i-i37 
 
 2.1014 
 
 1.148 
 
 2.036 
 
 1. 159 6 
 
 1-9752 
 
 1.1718 
 
 1.9184 
 
 1372 
 
 .1002 
 
 .1482 
 
 349 
 
 .1598 
 
 .9742 
 
 .172 
 
 9'75 
 
 J 373 
 
 .0991 
 
 .1484 
 
 0339 
 
 .16 
 
 9732 
 
 .1722 
 
 .9166 
 
 1375 
 *377 
 
 .098 
 .0969 
 
 .1486 
 .1488 
 
 .0329 
 .0318 
 
 .1602 
 . 1604 
 
 .9722 
 97i3 
 
 .1724 
 .1726 
 
 9*57 
 .9148 
 
 I-I379 
 
 2.0957 
 
 1.1489 
 
 2.0308 
 
 1. 1606 
 
 I-9703 
 
 1.1728 
 
 i-9'39 
 
 .1381 
 
 .0946 
 
 .1491 
 
 0297 
 
 .1608 
 
 .9693 
 
 173 
 
 9*3 
 
 .1382 
 
 0935 
 
 I 493 
 
 .0287 
 
 .161 
 
 .9683 
 
 1732 
 
 .9121 
 
 .1384 
 
 .0924 
 
 *495 
 
 .0276 
 
 .1612 
 
 .9674 
 
 1734 
 
 .9112 
 
 -1386 
 1.1388 
 
 .0912 
 2.0901 
 
 .1497 
 1.1499 
 
 .0266 
 2.0256 
 
 .1614 
 1. 1616 
 
 .9664 
 1.9654 
 
 1737 
 i-i739 
 
 .9102 
 1.9093 
 
 139 
 
 .089 
 
 .1501 
 
 .0245 
 
 .1618 
 
 9 6 45 
 
 .1741 
 
 .9084 
 
 J 39 T 
 
 .0879 
 
 1503 
 
 0235 
 
 .162 
 
 9 6 35 
 
 1 743 
 
 975 
 
 J 393 
 
 .0868 
 
 i55 
 
 .0224 
 
 .1622 
 
 -9625 
 
 1745 
 
 .9066 
 
 1395 
 
 .0857 
 
 1507 
 
 .0214. 
 
 . 1624 
 
 .9616 
 
 J 747 
 
 957 
 
 I -. I 397 
 
 2.0846 
 
 1.1508 
 
 2.0204 
 
 1.1626 
 
 1.9606 
 
 1.1749 
 
 1.9048 
 
 1399 
 
 0835 
 
 151 
 
 0194 
 
 .1628 
 
 959 6 
 
 I75i 
 
 939 
 
 .1401 
 
 .0824 
 
 .1512 
 
 0183 
 
 .163 
 
 9587 
 
 1753 
 
 93 
 
 . 1402 
 
 .0812 
 
 1514 
 
 0173 
 
 .1632 
 
 9577 
 
 1756 
 
 .9021 
 
 .1404 
 
 .0801 
 
 1516 
 
 .0163 
 
 .1634 
 
 .9568 
 
 1758 
 
 .9013 
 
 1. 1406 
 
 2.079 
 
 1.1518 
 
 2.0152 
 
 1.1636 
 
 1-9558 
 
 1.176 
 
 
 .1408 
 
 .0779 
 
 .152 
 
 .0142 
 
 1638 
 
 9549 
 
 .1762 
 
 .8995 
 
 .141 
 
 .0768 
 
 .1522 
 
 .0132 
 
 .164 
 
 9539 
 
 .1764 
 
 .8986 
 
 .1411 
 
 0757 
 
 .1524 
 
 .0122 
 
 .1642 
 
 953 
 
 .1766 
 
 .8977 
 
 HI3 
 1.1415 
 
 .0746 
 2-0735 
 
 .1526 
 1-1528 
 
 .OIII 
 2.OIOI 
 
 l6 >4 
 1. 1646 
 
 952 
 i-95i 
 
 .1768 
 1.177 
 
 .8968 
 1.8959 
 
 .1417 
 
 .0725 
 
 153 
 
 .0091 
 
 .1648 
 
 .9501 
 
 .1772 
 
 895 
 
 .1419 
 
 .0714 
 
 I53 1 
 
 .Oo8l 
 
 .165 
 
 .9491 
 
 1775 
 
 .8941 
 
 .1421 
 
 .0703 
 
 1533 
 
 .0071 
 
 .1652 
 
 .9482 
 
 .1777 
 
 8932 
 
 .1422 
 
 .0692 
 
 1535 
 
 .Oo6l 
 
 .1654 
 
 9473 
 
 .1779 
 
 .8924 
 
 1.1424 
 
 2.0681 
 
 i-i537 
 
 2.005 
 
 1.1656 
 
 1.9463 
 
 1.1781 
 
 1-8915 
 
 .1426 
 
 .067 
 
 1539 
 
 .004 
 
 .1658 
 
 9454 
 
 1783 
 
 .8906 
 
 .1428 
 
 .0659 
 
 i54i 
 
 .003 
 
 .166 
 
 9444 
 
 1785 
 
 .8897 
 
 143 
 
 .0648 
 
 1543 
 
 .OO2 
 
 .1662 
 
 9435 
 
 .1787 
 
 .8888 
 
 .1432 
 
 .0637 
 
 1545 
 
 .OOI 
 
 .1664 
 
 94 2 5 
 
 .179 
 
 .8879 
 
 J-I433 
 
 2.0627 
 
 i-i547 
 
 2 
 
 1. 1666 
 
 1.9416 
 
 1.1792 
 
 1.8871 
 
 CO-SEC'T 
 
 SECANT. 
 
 CO-SKC'T 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 61 
 
 60 
 
 59 
 
 58 
 
 From Haswell's " Engineering." Copyright, 1884, by Harper & Brother* 
 
 229 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 31 
 
 jo 
 
 3: 
 
 $o 
 
 3' 
 
 1 
 
 3d 
 
 o 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SKCANT. 
 
 CO-SKC'T. 
 
 SECANT. 
 
 CO-SKC'T. 
 
 SKCANT. 
 
 CO-SKC'T. 
 
 1.1792 
 
 1.8871 
 
 1.1924 
 
 1.8361 
 
 1.2062 
 
 1.7883 
 
 1.2208 
 
 *-7434 
 
 .1794 
 
 .8862 
 
 . 1926 
 
 .8352 
 
 .2064 
 
 7875 
 
 .221 
 
 .7427 
 
 .1796 
 
 8853 
 
 .1928 
 
 8344 
 
 .2067 
 
 .7867 
 
 .2213 
 
 742 
 
 .1798 
 
 .8844 
 
 193 
 
 8336 
 
 .2069 
 
 .786 
 
 .2215 
 
 74i3 
 
 .18 
 
 .8836 
 
 1933 
 
 .8328 
 
 .2072 
 
 .7852 
 
 .2218 
 
 7405 
 
 1. 1802 
 
 1.8827 
 
 I-I935 
 
 1.832 
 
 1.2074 
 
 1.7844 
 
 1.222 
 
 I-7398 
 
 -1805 
 
 .8818 
 
 1937 
 
 .8311 
 
 2076 
 
 7837 
 
 . 22 3 
 
 739 1 
 
 .1807 
 
 .8809 
 
 *939 
 
 8303 
 
 .2079 
 
 .7829 
 
 . 225 
 
 7384 
 
 .1809 
 
 .8801 
 
 .1942 
 
 .8295 
 
 .2081 
 
 .7821 
 
 . 228 
 
 7377 
 
 .1811 
 
 .8792 
 
 .1944 
 
 .8287 
 
 .2083 
 
 .7814 
 
 23 
 
 7369 
 
 1.1813 
 
 1.8783 
 
 1.1946 
 
 1.8279 
 
 1.2086 
 
 1.7806 
 
 I- 233 
 
 1.7362 
 
 .1815 
 
 .8785 
 
 .1948 
 
 .8271 
 
 .2088 
 
 .7798 
 
 235 
 
 7355 
 
 .1818 
 
 .8766 
 
 I95 1 
 
 8263 
 
 .2091 
 
 .7791 
 
 2 3 8 
 
 7348 
 
 .182 
 .1822 
 
 8757 
 .8749 
 
 1953 
 1955 
 
 8255 
 .8246 
 
 2093 
 2095 
 
 7783 
 .7776 
 
 . 24 
 243 
 
 7341 
 7334 
 
 1.1824 
 
 1.874 
 
 1.1958 
 
 1.8238 
 
 1.2098 
 
 1.7768 
 
 I. 245 
 
 I-7327 
 
 .1826 
 
 8731 
 
 .196 
 
 .823 
 
 '21 
 
 .776 
 
 . 248 
 
 
 .1828 
 
 .8723 
 
 . 1962 
 
 .8222 
 
 .2103 
 
 7753 
 
 25 
 
 7312 
 
 .1831 
 
 .8714 
 
 .1964 
 
 .8214 
 
 2105 
 
 7745 
 
 253 
 
 735 
 
 1833 
 
 .8706 
 
 .1967 
 
 8206 
 
 .2107 
 
 7738 
 
 255 
 
 .7298 
 
 1-1835 
 
 1.8697 
 
 1.1969 
 
 1.8198 
 
 1. 211 
 
 J -773 
 
 I- 2 5 8 
 
 1.7291 
 
 1837 
 
 .8688 
 
 .1971 
 
 .819 
 
 .2112 
 
 7723 
 
 .226 
 
 .7284 
 
 .1839 
 
 .868 
 
 .1974 
 
 .8182 
 
 .2115 
 
 77 J 5 
 
 .2263 
 
 7277 
 
 .1841 
 
 .8671 
 
 .1976 
 
 .8174 
 
 .2II 7 
 
 .7708 
 
 2265 
 
 727 
 
 .1844 
 
 .8663 
 
 .1978 
 
 .8166 
 
 .2119 
 
 77 
 
 .2268 
 
 .7263 
 
 1.1846 
 
 1.8654 
 
 1.198 
 
 1.8158 
 
 I. 2122 
 
 1-7693 
 
 1.227 
 
 1.7256 
 
 .1848 
 
 .8646 
 
 .1983 
 
 -815 
 
 .2124 
 
 .7685 
 
 .2273 
 
 7249 
 
 .185 
 
 -8637 
 
 .1985 
 
 .8142 
 
 .2127 
 
 .7678 
 
 .2276 
 
 .7242 
 
 .1852 
 
 .8629 
 
 .1987 
 
 -8i34 
 
 .2129 
 
 767 
 
 .2278 
 
 7234 
 
 1855 
 
 .862 
 
 .199 
 
 .8126 
 
 .2132 
 
 .7663 
 
 .228l 
 
 7227 
 
 1.1857 
 
 1.8611 
 
 1.1992 
 
 1.8118 
 
 I-2I34 
 
 1-7655 
 
 1.2283 
 
 1.722 
 
 .1859 
 
 .8603 
 
 .1994 
 
 .811 
 
 .2136 
 
 .7648 
 
 .2286 
 
 7213 
 
 .1861 
 
 8595 
 
 .1997 
 
 .8102 
 
 .2139 
 
 .764 
 
 .2288 
 
 .7206 
 
 .1863 
 
 .8586 
 
 .1999 
 
 .8094 
 
 .2141 
 
 7633 
 
 .2291 
 
 .7199 
 
 .1866 
 
 .8578 
 
 .2001 
 
 .8086 
 
 .2144 
 
 .7625 
 
 .2293 
 
 .7192 
 
 1. 1868 
 
 1.8569 
 
 1 . 2004 
 
 1.8078 
 
 1.2146 
 
 1.7618 
 
 1.2296 
 
 1.7185 
 
 .187 
 
 8561 
 
 .2OO6 
 
 .807 
 
 .2149 
 
 .761 
 
 .2298 
 
 .7178 
 
 .1872 
 
 -8552 
 
 .2008 
 
 .8062 
 
 .2151 
 
 .7603 
 
 .23OI 
 
 .7171 
 
 .1874 
 
 8544 
 
 .201 
 
 8054 
 
 2153 
 
 
 .2304 
 
 .7164 
 
 .1877 
 
 8535 
 
 .2013 
 
 .8047 
 
 2156 
 
 .7588 
 
 .2306 
 
 7*57 
 
 1.1879 
 
 1.8527 
 
 I.2OI5 
 
 1.8039 
 
 1.2158 
 
 1.7581 
 
 1.2309 
 
 
 .1881 
 
 .8519 
 
 .2017 
 
 .8031 
 
 .2l6l 
 
 7573 
 
 .2311 
 
 .7144 
 
 .1883 
 
 .851 
 
 .202 
 
 .8023 
 
 .2163 
 
 .7566 
 
 .2314 
 
 7137 
 
 .1886 
 
 .8502 
 
 .2022 
 
 .8015 
 
 .2166 
 
 7559 
 
 .2316 
 
 
 .1888 
 
 8493 
 
 .2024 
 
 .8007 
 
 .2168 
 
 7551 
 
 .2319 
 
 .7123 
 
 1.189 
 
 1.8485 
 
 I. 2027 
 
 1.7999 
 
 I.2I7I 
 
 J -7544 
 
 1.2322 
 
 1.7116 
 
 .1892 
 
 .8477 
 
 .2029 
 
 .7992 
 
 2173 
 
 7537 
 
 .2324 
 
 .7109 
 
 .1894 
 
 .8468 
 
 .2031 
 
 .7984 
 
 2175 
 
 7529 
 
 .2327 
 
 .7102 
 
 .1897 
 
 .846 
 
 .2034 
 
 .7976 
 
 .2178 
 
 7522 
 
 .2329 
 
 7095 
 
 .1899 
 
 8452 
 
 .2036 
 
 .7968 
 
 .218 
 
 7514 
 
 2332 
 
 .7088 
 
 1. 1901 
 
 1.8443 
 
 1.2039 
 
 1.796 
 
 1.2183 
 
 I-7507 
 
 1-2335 
 
 1.7081 
 
 .1903 
 
 8435 
 
 .2041 
 
 7953 
 
 .2185 
 
 75 
 
 2337 
 
 775 
 
 .1906 
 
 .8427 
 
 .2043 
 
 7945 
 
 .2188 
 
 7493 
 
 234 
 
 .7068 
 
 .1908 
 
 .8418 
 
 .2046 
 
 7937 
 
 .219 
 
 7485 
 
 .2342 
 
 .7061 
 
 .191 
 
 .841 
 
 .2048 
 
 .7929 
 
 .2193 
 
 .7478 
 
 2345 
 
 7054 
 
 1.1912 
 
 1.8402 
 
 1.205 
 
 1.7921 
 
 1.2195 
 
 1.7471 
 
 1.2348 
 
 1.7047 
 
 19'S 
 
 8394 
 
 2053 
 
 .7914 
 
 .2198 
 
 7463 
 
 235 
 
 .704 
 
 .1917 
 
 .8385 
 
 2055 
 
 .7906 
 
 .22 
 
 7456 
 
 2353 
 
 7033 
 
 .1919 
 
 8377 
 
 .2057 
 
 .7898 
 
 .2203 
 
 7449 
 
 2355 
 
 .7027 
 
 .1921 
 
 -8369 
 
 .206 
 
 .7891 
 
 .2205 
 
 .7442 
 
 2358 
 
 .702 
 
 1.1922 
 
 1.8361 
 
 1.2062 
 
 1.7883 
 
 1. 2208 
 
 J -7434 
 
 1.2361 
 
 1.7013 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SKC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SKC'T. 
 
 SECANT. 
 
 5' 
 
 r 
 
 5C 
 
 o 
 
 5v 
 
 r 
 
 ) 
 
 64 
 
 L 
 
 i'rojn Haswell's " Engineering." 
 
 Copyright, 1884, by Harper <5t Brothers. 
 
 230 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 36 
 
 370 
 
 38 
 
 39 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SKCANT. CO-SEC'T. 
 
 SECANT. | CO-SEC'T. 
 
 SECANT. CO-SEC'T. 
 
 1.2361 
 
 1.7013 
 
 1.2521 
 
 1.6616 
 
 1.269 
 
 1.6243 
 
 1.2867 
 
 1.589 
 
 2363 
 
 .7006 
 
 .2524 
 
 .661 
 
 .2693 
 
 6237 
 
 .2871 
 
 -5884 
 
 .2366 
 
 6999 
 
 .2527 
 
 .6603 
 
 .2696 
 
 6231 
 
 .2874 
 
 5879 
 
 .2368 
 
 6993 
 
 253 
 
 6597 
 
 .2699 
 
 .6224 
 
 .2877 
 
 5873 
 
 237 1 
 
 .6986 
 
 
 .6591 
 
 .2702 
 
 .6218 
 
 .288 
 
 .5867 
 
 1-2374 
 
 1.6979 
 
 1-2535 
 
 1.6584 
 
 1.2705 
 
 1.6212 
 
 1.2883 
 
 1.5862 
 
 .2376 
 
 .6972 
 
 .2538 
 
 .6578 
 
 .2707 
 
 .6206 
 
 .2886 
 
 5856 
 
 2379 
 
 .6965 
 
 .2541 
 
 6572 
 
 .271 
 
 .62 
 
 .2889 
 
 585 
 
 .2382 
 
 6959 
 
 2543 
 
 6565 
 
 2713 
 
 .6194 
 
 .2892 
 
 5845 
 
 .2384 
 
 .6952 
 
 .2546 
 
 6559 
 
 .2716 
 
 .6188 
 
 .2895 
 
 5839 
 
 1.2387 
 
 1.6945 
 
 1-2549 
 
 1-6552 
 
 1.2719 
 
 1.6182 
 
 1.2898 
 
 1-5833 
 
 .2389 
 
 .6938 
 
 2552 
 
 .6546 
 
 .2722 
 
 .6176 
 
 .2901 
 
 .5828 
 
 2392 
 
 .6932 
 
 2554 
 
 654 
 
 .2725 
 
 .617 
 
 .2904 
 
 .5822 
 
 2395 
 
 6925 
 
 2557 
 
 6533 
 
 .2728 
 
 .6164 
 
 .2907 
 
 .5816 
 
 2397 
 
 .6918 
 
 .256 
 
 .6527 
 
 2731 
 
 .6159 
 
 .291 
 
 .5811 
 
 1.24 
 
 1.6912 
 
 1.2563 
 
 1.6521 
 
 1-2734 
 
 1.6153 
 
 1.2913 1.5805 
 
 .2403 
 
 .6905 
 
 2565 
 
 .6514 
 
 2737 
 
 .6147 
 
 .2916 .5799 
 
 2405 
 
 .6898 
 
 .2568 
 
 .6508 
 
 2739 
 
 .6141 
 
 .2919 .5794 
 
 .2408 
 
 .6891 
 
 257 1 
 
 .6502 
 
 .2742 
 
 6i35 
 
 .29* .5788 
 
 .2411 
 
 .6885 
 
 2574 
 
 .6496 
 
 2745 
 
 .6129 
 
 .2926 i .5783 
 
 1.2413 
 
 1.6878 
 
 1-2577 
 
 1.6489 
 
 1.2748 
 
 1.6123 
 
 1.2929 
 
 1-5777 
 
 .2416 
 
 .6871 
 
 2579 
 
 .6483 
 
 2751 
 
 .6117 
 
 .2932 
 
 .5771 
 
 .2419 
 
 .6865 
 
 .2582 
 
 6477 
 
 2754 
 
 .6111 
 
 2Q35 
 
 .5766 
 
 .2421 
 
 .6858 
 
 2585 
 
 .647 
 
 2 757 
 
 6105 
 
 .2938 
 
 .576 
 
 .2424 
 
 .6851 
 
 .2588 
 
 .6464 
 
 .276 
 
 .6099 
 
 .2941 
 
 5755 
 
 1.2427 
 
 1-6845 
 
 1.2591 
 
 1.6458 
 
 1.2763 
 
 1.6093 
 
 1.2944 
 
 J -5749 
 
 .2429 
 
 .6838 
 
 2593 
 
 6452 
 
 .2766 
 
 .6087 ' 
 
 2947 
 
 5743 
 
 .2432 
 
 .6831 
 
 .2596 
 
 6445 
 
 .2769 
 
 .6081 
 
 295 
 
 5738 
 
 2435 
 
 .6825 
 
 2599 
 
 6439 
 
 .2772 
 
 .6077 
 
 2953 
 
 5732 
 
 2437 
 
 .6818 
 
 .2602 
 
 6433 
 
 2775 
 
 .607 
 
 .2956 
 
 5727 
 
 1.244 
 
 1.6812 
 
 1.2605 
 
 1.6427 
 
 1.2778 
 
 1.6064 
 
 1.296 
 
 1.5721 
 
 2443 
 
 .6805 
 
 .2607 
 
 .642 
 
 .2781 
 
 .6058 
 
 .2963 
 
 .5716 
 
 2445 
 
 .6798 
 
 .261 
 
 .6414 
 
 .2784 
 
 .6052 
 
 .2966 
 
 571 
 
 .2448 
 
 .6792 
 
 .2613 
 
 ' .6408 
 
 .2787 
 
 .6046 
 
 .2969 
 
 575 
 
 .2451 
 
 .6785 
 
 .2616 
 
 .6402 
 
 .279 
 
 .604 
 
 .2972 
 
 5699 
 
 1-2453 
 
 1.6779 
 
 1.2619 
 
 1.6396 
 
 1-2793 
 
 1.6034 
 
 1-2975 
 
 1.5694 
 
 .2456 
 
 .6772 
 
 .2622 
 
 .6389 
 
 2795 
 
 .6029 
 
 .2978 
 
 .5688 
 
 2459 
 
 .6766 
 
 .2624 
 
 6383 
 
 .2798 
 
 .6023 
 
 .2981 
 
 -5683 
 
 .2461 
 
 6759 
 
 .2627 
 
 6377 
 
 .2801 
 
 .6017 
 
 .2985 
 
 5677 
 
 .2464 
 
 6752 
 
 .263 
 
 6371 
 
 .2804 
 
 .6011 
 
 .2988 
 
 .5672 
 
 1.2467 
 
 1.6746 
 
 1.2633 
 
 1-6365 
 
 1.2807 
 
 1.6005 
 
 1.2991 
 
 1.5666 
 
 .247 
 
 6739 
 
 .2636 
 
 6359 
 
 .281 
 
 .6 
 
 2994 
 
 5661 
 
 .2472 
 
 
 .2639 
 
 6352 
 
 .2813 
 
 5994 
 
 .2997 
 
 5655 
 
 2475 
 
 .6726 
 
 .2641 
 
 6346 
 
 .2816 
 
 .5988 
 
 3 
 
 -565 
 
 .2 47 3 
 
 .672 
 
 .2644 
 
 634 
 
 .2819 
 
 .5982 
 
 3003 
 
 5644 
 
 1.248 
 
 1.6713 
 
 1.2647 
 
 I.6334 
 
 1.2822 
 
 I-597 6 
 
 1.3006 
 
 1-5639 
 
 .2483 
 
 .6707 
 
 .265 
 
 .6328 
 
 .2825 
 
 
 .301 
 
 5633 
 
 .2486 
 
 .67 
 
 2653 
 
 .6322 
 
 .2828 
 
 59 6 5 
 
 3 OI 3 
 
 .5628 
 
 .2488 
 
 .6694 
 
 .2656 
 
 .6316 
 
 .2831 
 
 5959 
 
 .3016 
 
 .5622 
 
 .249 
 
 .6687 
 
 .2659 
 
 .6309 
 
 .2834 
 
 5953 
 
 .3019 
 
 5617 
 
 1.2494 
 
 i. 6681 
 
 1.2661 
 
 1.6303 
 
 1.2837 
 
 1-5947 
 
 1.3022 
 
 1.5611 
 
 .2497 
 
 .6674 
 
 .2664 
 
 .6297 
 
 .284 
 
 5942 
 
 .3025 
 
 .5606 
 
 2499 
 
 .6668 
 
 .2667 
 
 .6291 
 
 .2843 
 
 5936 
 
 3029 
 
 -56 
 
 .2502 
 
 .6661 
 
 .267 
 
 .6285 
 
 .2846 
 
 593 
 
 3 32 
 
 5595 
 
 2505 
 
 6655 
 
 .2673 
 
 .6279 
 
 .2849 
 
 5924 
 
 335 
 
 559 
 
 1.2508 
 
 1.6648 
 
 1.2676 
 
 1.6273 
 
 1.2852 
 
 
 1-3038 
 
 1-5584 
 
 251 
 
 .6642 
 
 .2679 
 
 .6267 
 
 2855 
 
 59 J 3 
 
 .3041 
 
 5579 
 
 2513 
 
 .6636 
 
 .2681 
 
 .6261 
 
 .2858 
 
 597 
 
 344 
 
 5573 
 
 2516 
 
 .6629 
 
 .2684 
 
 6255 
 
 .2861 
 
 
 .3048 
 
 5568 
 
 2519 
 
 .6623 
 
 .2687 
 
 .6249 
 
 .2864 
 
 .5896 
 
 3051 
 
 5563 
 
 1.2521 
 
 i. 6616 
 
 1.269 
 
 1.6243 
 
 1.2867 
 
 1.589 
 
 I -354 
 
 I -5557 
 
 CO-SEC'T 
 
 SECANT. 
 
 CO-SEC'T 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 530 
 
 520 
 
 510 
 
 500 
 
 From Haswell's " Engineering." Copyright, 1884, by Harper & Brothera 
 
 231 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 40 
 
 41 
 
 42 
 
 43 
 
 SECANT. 
 
 CO-SKC'T. 
 
 SECANT. 
 
 CO-SHC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 I- 354 
 
 1-5557 
 
 L325 
 
 1.5242 
 
 I-3456 
 
 1-4945 
 
 I-3673 
 
 1.4663 
 
 :p 7 
 
 5552 
 5546 
 
 3253 
 
 5237 
 5232 
 
 .346 
 3463 
 
 494 
 4935 
 
 3677 
 3681 
 
 .4658 
 4654 
 
 .3064 
 
 5541 
 
 326 
 
 5227 
 
 3467 
 
 493 
 
 .3684 
 
 4649 
 
 .3067 
 
 5536 
 
 3263 
 
 .5222 
 
 347 
 
 4925 
 
 .3688 
 
 4644 
 
 1.307 
 
 !-553 
 
 1.3267 
 
 1.5217 
 
 1-3474 
 
 1.4921 
 
 1.3692 
 
 1.464 
 
 373 
 
 5525 
 
 327 
 
 5212 
 
 3477 
 
 .4916 
 
 3695 
 
 4635 
 
 .3076 
 
 552 
 
 3274 
 
 .5207 
 
 .3481 
 
 49" 
 
 3699 
 
 .4631 
 
 .308 
 
 5514 
 
 .3277 
 
 .5202 
 
 3485 
 
 .4906 
 
 3703 
 
 .4626 
 
 3083 
 
 5509 
 
 328 
 
 5197 
 
 .3488 
 
 .4901 
 
 377 
 
 .4622 
 
 1.3086 
 
 I-5503 
 
 1.3284 
 
 1.5192 
 
 1.3492 
 
 1.4897 
 
 I-37I 
 
 1.4617 
 
 3089 
 
 5498 
 
 .3287 
 
 5187 
 
 3495 
 
 .4892 
 
 37M 
 
 .4613 
 
 3092 
 
 5493 
 
 329 
 
 .5182 
 
 3499 
 
 .4887 
 
 
 .4608 
 
 3096 
 
 5487 
 
 3294 
 
 5177 
 
 3502 
 
 .4882 
 
 .3722 
 
 .4604 
 
 3099 
 
 5482 
 
 3297 
 
 5171 
 
 35o6 
 
 .4877 
 
 3725 
 
 4599 
 
 1.3102 
 
 1-5477 
 
 i-33 01 
 
 1.5166 
 
 1-3509 
 
 1-4873 
 
 I-3729 
 
 1-4595 
 
 3105 
 
 5471 
 
 3304 
 
 .5161 
 
 3513 
 
 .4868 
 
 3733 
 
 459 
 
 .3109 
 
 .5466 
 
 3307 
 
 5156 
 
 3517 
 
 .4863 
 
 3737 
 
 4586 
 
 .3112 
 
 .5461 
 
 33" 
 
 .5151 
 
 352 
 
 4858 
 
 374 
 
 .4581 
 
 3"5 
 
 5456 
 
 
 .5146 
 
 3524 
 
 4854 
 
 3744 
 
 4577 
 
 1.3118 
 
 1-545 
 
 1.3318 
 
 1.5141 
 
 1-3527 
 
 1.4849 
 
 I-3748 
 
 I-457 2 
 
 .3121 
 
 5445 
 
 3321 
 
 5136 
 
 3531 
 
 4844 
 
 3752 
 
 4568 
 
 3125 
 
 544 
 
 3324 
 
 
 3534 
 
 4839 
 
 3756 
 
 4563 
 
 .3128 
 
 5434 
 
 3328 
 
 .5126 
 
 3538 
 
 4835 
 
 3759 
 
 4559 
 
 3 I 3 I 
 
 5429 
 
 333 1 
 
 5121 
 
 3542 
 
 483 
 
 3763 
 
 4554 
 
 I-3I34 
 
 1-5424 
 
 1-3335 
 
 1.5116 
 
 1-3545 
 
 1.4825 
 
 1-3767 
 
 1-455 
 
 3138 
 
 5419 
 
 3338 
 
 .5111 
 
 3549 
 
 .4821 
 
 3771 
 
 4545 
 
 3 1 4 I 
 
 54 I 3 
 
 3342 
 
 .5106 
 
 3552 
 
 .4816 
 
 3774 
 
 454 1 
 
 3 J 44 
 
 5408 
 
 3345 
 
 .5101 
 
 3556 
 
 .4811 
 
 3778 
 
 4536 
 
 .3148 
 
 543 
 
 3348 
 
 5096 
 
 356 
 
 .4806 
 
 3782 
 
 453 2 
 
 
 I-5398 
 
 1-3352 
 
 1.5092 
 
 1-3563 
 
 1.4802 
 
 1-3786 
 
 14527 
 
 3154 
 
 539 2 
 
 3355 
 
 .5087 
 
 3567 
 
 4797 
 
 379 
 
 4523 
 
 
 5387 
 
 3359 
 
 .5082 
 
 3571 
 
 .4792 
 
 3794 
 
 .4518 
 
 .3161 
 
 5382 
 
 3362 
 
 5077 
 
 3574 
 
 .4788 
 
 3797 
 
 45H 
 
 .3164 
 
 5377 
 
 3366 
 
 .5072 
 
 3578 
 
 4783 
 
 3801 
 
 451 
 
 1.3167 
 
 
 1-3369 
 
 1.5067 
 
 i-358i 
 
 1.4778 
 
 1-3805 
 
 I-4505 
 
 3 J 7 
 
 '5366 
 
 3372 
 
 .5062 
 
 3585 
 
 4774 
 
 .3809 
 
 .4501 
 
 
 -536i 
 
 3376 
 
 5057 
 
 3589 
 
 .4769 
 
 -3813 
 
 4496 
 
 3177 
 
 5356 
 
 3379 
 
 5052 
 
 3592 
 
 4764 
 
 -3816 
 
 .4492 
 
 318 
 
 5351 
 
 3383 
 
 547 
 
 
 -476 
 
 382 
 
 4487 
 
 1-3184 
 
 1-5345 
 
 1-3386 
 
 1.5042 
 
 
 1-4755 
 
 1.3824 
 
 1.4483 
 
 3187 
 
 534 
 
 339 
 
 5037 
 
 3603 
 
 475 
 
 .3828 
 
 4479 
 
 3 I 9 
 
 5335 
 
 3393 
 
 .5032 
 
 3607 
 
 .4746 
 
 3832 
 
 4474 
 
 3J93 
 
 533 
 
 3397 
 
 .5027 
 
 .36" 
 
 .4741 
 
 -3836 
 
 447 
 
 
 5325 
 
 34 
 
 .5022 
 
 .3614 
 
 4736 
 
 3839 
 
 4465 
 
 1.32 
 
 
 1.3404 
 
 1.5018 
 
 1.3618 
 
 1-4732 
 
 1-3843 
 
 1.4461 
 
 3203 
 
 53 J 4 
 
 3407 
 
 5013 
 
 .3622 
 
 4727 
 
 3847 
 
 4457 
 
 .3207 
 
 5309 
 
 34" 
 
 .5008 
 
 3625 
 
 4723 
 
 3851 
 
 4452 
 
 .321 
 
 534 
 
 34H 
 
 5003 
 
 3629 
 
 .4718 
 
 3855 
 
 .4448 
 
 3213 
 
 5299 
 
 .3418 
 
 .4998 
 
 3633 
 
 
 3859 
 
 4443 
 
 1.3217 
 
 1.5294 
 
 1.3421 
 
 1-4993 
 
 1-3636 
 
 1.4709 
 
 1-3863 
 
 J -4439 
 
 322 
 
 .5289 
 
 3425 
 
 .4988 
 
 364 
 
 474 
 
 .3867 
 
 4435 
 
 3223 
 
 5283 
 
 3428 
 
 4983 
 
 3644 
 
 4699 
 
 -387 
 
 443 
 
 .3227 
 
 .5278 
 
 3432 
 
 4979 
 
 3647 
 
 .4695 
 
 3874 
 
 .4426 
 
 323 
 
 5273 
 
 3435 
 
 4974 
 
 3651 
 
 .469 
 
 .3878 
 
 44E2 
 
 I-3233 
 
 1.5268 
 
 J-3439 
 
 1.4969 
 
 1-3655 
 
 1.4686 
 
 1.3882 
 
 1.4417 
 
 3237 
 324 
 
 5263 
 5258 
 
 3442 
 3446 
 
 4964 
 4959 
 
 3658 
 
 .4681 
 .4676 
 
 .3886 
 389 
 
 .4408 
 
 3243 
 
 5253 
 
 3449 
 
 4954 
 
 .'3666 
 
 .4672 
 
 
 .4404 
 
 3247 
 
 5248 
 
 3453 
 
 4949 
 
 -3669 
 
 .3667 
 
 .3898 
 
 44 
 
 1-325 
 
 1.5242 
 
 I-3456 
 
 r -4945 
 
 1-3673 
 
 1.4663 
 
 1.3902 
 
 1-4395 
 
 CO-SEC'T. 
 
 SKCANT. 
 
 CO-SEC'T. 
 
 SKCANT. 
 
 CO-SBC'T. 
 
 SECANT. 
 
 CO-SEC'T. 
 
 SECANT. 
 
 49 
 
 48 
 
 470 
 
 46 
 
 From Haswell's " Engineering." Copyright, 1884, by Harper k Brother* 
 
NATURAL SECANTS AND CO-SECANTS. 
 
 
 44 
 
 to 
 
 
 
 44 
 
 
 
 
 
 44 
 
 to 
 
 
 ; 
 
 SECANT. 
 
 CO-SEC'T. 
 
 f 
 
 i 
 
 SECANT. 
 
 CO-SEC'T. 
 
 / 
 
 1 
 
 SECANT. 
 
 CO-SKC'T. 
 
 ' 
 
 
 
 1.3902 
 
 1-4395 
 
 60 
 
 21 
 
 1.3984" 
 
 I-4305 
 
 39 
 
 41 
 
 1.4065 
 
 1.4221 
 
 19 
 
 I 
 
 395 
 
 4391 
 
 59 
 
 22 
 
 .3988 
 
 .4301 
 
 38 
 
 42 
 
 .4069 
 
 .4217 
 
 18 
 
 c 
 
 399 
 
 4387 
 
 58 
 
 23 
 
 3992 
 
 4297 
 
 37 
 
 43 
 
 4073 
 
 .4212 
 
 17 
 
 3 
 
 39 X 3 
 
 .4382 
 
 57 
 
 24 
 
 .3996 
 
 .4292 
 
 36 
 
 44 
 
 .4077 
 
 .4208 
 
 16 
 
 4 
 
 39*7 
 
 4378 
 
 56 
 
 25 
 
 1.4 
 
 1.4288 
 
 35 
 
 45 
 
 1.4081 
 
 1.4204 
 
 15 
 
 5 
 
 1.3921 
 
 *-4374 
 
 55 
 
 26 
 
 .4004 
 
 .4284 
 
 34 
 
 46 
 
 .4085 
 
 .42 
 
 M 
 
 6 
 
 39 2 5 
 
 437 
 
 54 
 
 2 7 
 
 .4008 
 
 .428 
 
 33 
 
 47 
 
 .4089 
 
 .4196 
 
 J3 
 
 7 
 
 39 2 9 
 
 43 6 5 
 
 53 
 
 28 
 
 .4012 
 
 4276 
 
 S 2 
 
 48 
 
 493 
 
 .4192 
 
 12 
 
 8 
 
 3933 
 
 on"37 
 
 .4361 
 
 A OC7 
 
 5 2 
 
 CT 
 
 29 
 
 1O 
 
 .4016 
 
 I J.O2 
 
 .4271 
 
 3i 
 
 49 
 
 .4097 
 
 .4188 
 
 II 
 
 IO 
 
 9 
 o 
 
 oVO/ 
 
 1-3941 
 
 43j/ 
 
 1-4352 
 
 J X 
 
 5 
 
 J^ 
 
 31 
 
 **^M 
 
 .4024 
 
 I. 42t>7 
 
 .4263 
 
 3O 
 
 29 
 
 5^ 
 
 51 
 
 .4105 
 
 .4179 
 
 9 
 
 ii 
 
 3945 
 
 4348 
 
 49 
 
 32 
 
 .4028 
 
 4259 
 
 28 
 
 52 
 
 .4109 
 
 4!75 
 
 8 
 
 12 
 
 13 
 
 3949 
 3953 
 
 4344 
 4339 
 
 48 
 47 
 
 33 
 34 
 
 .4032 
 .4036 
 
 4254 
 425 
 
 27 
 26 
 
 53 
 54 
 
 4H3 
 .4117 
 
 .4171 
 .4167 
 
 I 
 
 14 
 
 3957 
 
 4335 
 
 46 
 
 35 
 
 1.404 
 
 1.4246 
 
 2 5 
 
 55 
 
 1.4122 
 
 1.4163 
 
 5 
 
 5 
 
 1.396 
 
 I -433 I 
 
 45 
 
 36 
 
 .4044 
 
 .4242 
 
 24 
 
 56 
 
 .4126 
 
 4*59 
 
 4 
 
 6 
 
 39 6 4 
 
 4327 
 
 44 
 
 37 
 
 .40 4 8 
 
 .4238 
 
 23 
 
 57 
 
 4*3 
 
 4154 
 
 3 
 
 7 
 
 .3968 
 
 .4322 
 
 43 
 
 38 
 
 .4052 
 
 4333 
 
 22 
 
 58 
 
 4!34 
 
 4*5 
 
 2 
 
 8 
 
 397 2 
 
 .4318 
 
 42 
 
 39 
 
 .4056 
 
 .4229 
 
 21 
 
 59 
 
 .4138 
 
 .4146 
 
 I 
 
 9 
 
 3976 
 
 43H 
 
 4i 
 
 40 
 
 1.406 
 
 1.4225 
 
 20 
 
 60 
 
 1.4142 
 
 1.4142 
 
 
 
 20 
 
 1.398 
 
 i-43i 
 
 40 
 
 
 
 
 
 
 
 
 
 / 
 
 CO-SEC 'T. 
 
 SECANT. 
 
 > 
 
 / 
 
 CO-SKC'T. 
 
 SECANT. 
 
 ; 
 
 / 
 
 CO-SKC'T. 
 
 SECANT. 
 
 / 
 
 
 4t 
 
 > 
 
 
 
 4 
 
 ) 
 
 
 
 4 
 
 50 
 
 
 Preceding Table contains Natural Secants and Co-secants for every 
 minute of the Quadrant to Radius i. 
 
 If Degrees are taken at head of column, Minutes, Secant, and Co-secant 
 must be taken from head also; and if they are taken at foot of column, 
 Minutes, etc., must be taken from foot also. 
 
 ILLUSTRATION. 1.05 is secant of 17 45' and co-secant of 72 15'. 
 
 To Compute Secant or Co-secant of any Angle. 
 RULE. Divide i by Cosine of angle for Secant, and by Sine for Co-secant. 
 EXAMPLE i. What is secant of 25 25'? 
 
 Cosine of angle = .903 21. Then i -r- .903 21 = 1. 1072, Secant. 
 2. What is co-secant of 64 35'? 
 
 Sine of angle = .903 21. Then 1-^.903213=1.1072, Co-secant. 
 
 From Haswell's "Engineering. 
 
 Copyright, 1884, by Harper & Brothers. 
 
 233 
 
TABLE XII. TANGENTS AND COTANGENTS. 
 
 
 0* 
 
 1 
 
 o 
 
 2 
 
 
 
 3 
 
 o 
 
 1 
 
 / 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang, 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 
 
 00000 
 
 Infinite. 
 
 01746 
 
 57.2900 
 
 03492 
 
 28.6363 
 
 05241 
 
 19.0811 
 
 GO 
 
 1 
 
 00029 
 
 3437.75 
 
 01775 
 
 56.3506 
 
 03521 
 
 28.3994 
 
 05270 
 
 18.9755 
 
 59 
 
 2 
 
 00058 
 
 1718.87 
 
 01804 
 
 55.4415 
 
 03550 
 
 28.1664 
 
 05299 
 
 18.8711 
 
 58 
 
 3 
 
 00087 
 
 1145.92 
 
 01833 
 
 54.5613 
 
 03579 
 
 27.9372 
 
 05328 
 
 18.7678 
 
 57 
 
 4 
 
 00116 
 
 859.436 
 
 01862 
 
 53.7086 
 
 03609 
 
 27.7117 
 
 05357 
 
 18.6656 
 
 56 
 
 5 
 
 00145 
 
 687.549 
 
 01891 
 
 52.8821 
 
 03638 
 
 27.4899 
 
 05387 
 
 18.5645 
 
 55 
 
 G 
 
 00175 
 
 572.957 
 
 01920 
 
 52.0807 
 
 03667 
 
 27.2715 
 
 05416 
 
 18.4645 
 
 54 
 
 7 
 
 00204 
 
 491.106 
 
 01949 
 
 51.3032 
 
 03696 
 
 27.0566 
 
 05445 
 
 18.3655 
 
 53 
 
 8 
 
 00233 
 
 429.718 
 
 01978 
 
 50.5485 
 
 03725 
 
 26.8-150 
 
 05474 
 
 18.2677 
 
 52 
 
 9 
 
 00262 
 
 381.971 
 
 02007 
 
 49.8157 
 
 03754 
 
 26.6367 
 
 05503 
 
 18.1708 51 
 
 10 
 
 00291 
 
 343.774 
 
 02036 
 
 49.1039 
 
 03783 
 
 26.4316 
 
 05533 
 
 18.0750 
 
 50 
 
 11 
 
 00320 
 
 312.521 
 
 02066 
 
 48.4121 
 
 03812 
 
 26.2296 
 
 05562 
 
 17.9802 
 
 49 
 
 12 
 
 00349 
 
 286.478 
 
 02095 
 
 47.7395 
 
 03842 
 
 26.0307 
 
 05591 
 
 17.8863 
 
 48 
 
 13 
 
 00373 
 
 2G4.441 
 
 02124 
 
 47.0353 
 
 03871 
 
 25.8348 
 
 05620 
 
 17.7934 
 
 47 
 
 14 
 
 00407 
 
 245.552 
 
 02153 
 
 46.4489 
 
 03900 
 
 25.6418 
 
 05649 
 
 17.7015 
 
 46 
 
 15 
 
 00436 
 
 229.182 
 
 02182 
 
 45.8294 
 
 03929 
 
 25.4517 
 
 05678 
 
 17.6106 
 
 45 
 
 16 
 
 004G5 
 
 214.858 
 
 02211 
 
 45.2261 
 
 03958 
 
 25.2644 
 
 05708 
 
 17.5205 
 
 44 
 
 IT 
 
 00495 
 
 202.219 
 
 02240 
 
 44.6386 
 
 03987 
 
 25.0798 
 
 05737 
 
 17.4314 
 
 43 
 
 18 
 
 00524 
 
 190.984 
 
 02269 
 
 44.0661 
 
 04016 
 
 24.8978 
 
 05766 
 
 17.3432 
 
 42 
 
 19 
 
 00553 
 
 180.932 
 
 02298 
 
 43.5081 
 
 04046 
 
 24.7185 
 
 05795 
 
 17.2558 
 
 41 
 
 20 
 
 00583 
 
 171.885 
 
 02328 
 
 42.9641 
 
 04075 
 
 24.5418 
 
 05824 
 
 17.1693 
 
 40 
 
 21 
 
 00611 
 
 163.700 
 
 02357 
 
 42.4335 
 
 04104 
 
 24.3675 
 
 05854 
 
 17.0837 
 
 39 
 
 22 
 
 00640 
 
 156.259 
 
 02386 
 
 41.9158 
 
 04133 
 
 24.1957 
 
 05S83 
 
 16.9990 
 
 38 
 
 23 
 
 006G9 
 
 149.465 
 
 02415 
 
 41.4106 
 
 04162 
 
 24.0263 
 
 05912 
 
 16.9150 
 
 37 
 
 24 
 
 00698 
 
 143.237 
 
 02444 
 
 40.9174 
 
 04191 
 
 23.8593 
 
 05941 
 
 16.8319 
 
 36 
 
 25 
 
 00727 
 
 137.507 
 
 02473 
 
 40.4358 
 
 04220 
 
 23.6945 
 
 05970 
 
 16.7496 
 
 35 
 
 20 
 
 00756 
 
 132.219 
 
 02502 
 
 39.9655 
 
 04250 
 
 23.5321 
 
 05999 
 
 16.6681 
 
 34 
 
 27 
 
 00785 
 
 127.321 
 
 02531 
 
 39.5059 
 
 04279 
 
 23.3718 
 
 06029 
 
 16.5874 
 
 33 
 
 28 
 
 00815 
 
 122.774 
 
 02560 
 
 39.0568 
 
 04308 
 
 23.2137 
 
 06058 
 
 16.5075 
 
 32 
 
 29 
 
 008*^4 
 
 118.540 
 
 02589 
 
 38.6177 
 
 04337 
 
 23.0577 
 
 06087 
 
 16.4283 
 
 31 
 
 30 
 
 00873 
 
 114.589 
 
 02619 
 
 38.1885 
 
 04366 
 
 22.9038 
 
 06116 
 
 16.3499 
 
 30 
 
 31 
 
 00902 
 
 110.892 
 
 02648 
 
 37.7686 
 
 04395 
 
 22.7519 
 
 06145 
 
 16.2722 
 
 29 
 
 oo 
 
 00931 
 
 107.426 
 
 02677 
 
 37.3579 
 
 04424 
 
 22.6020 
 
 06175 
 
 16.1952 
 
 23 
 
 88 
 
 00960 
 
 101.171 
 
 02706 
 
 36.9560 
 
 04454 
 
 22.4541 
 
 06204 
 
 16.1190 
 
 27 
 
 34 
 
 00989 
 
 101.107 
 
 02735 
 
 36.5G27 
 
 04483 
 
 22.3081 
 
 06233 
 
 16.0435 
 
 26 
 
 35 
 
 01018 
 
 98.2179 
 
 02764 
 
 36.1776 
 
 04512 
 
 22.1640 
 
 06262 
 
 15.9687 
 
 25 
 
 36 
 
 01047 
 
 95.4895 
 
 02793 
 
 35.8006 
 
 04541 
 
 22.0217 
 
 06291 
 
 15.8945 
 
 24 
 
 37 
 
 01076 
 
 92.9085 
 
 02822 
 
 35.4313 
 
 04570 
 
 21.8813 
 
 06321 
 
 15.8211 
 
 23 
 
 88 
 
 01105 
 
 90.4633 
 
 02351 
 
 35.0G95 
 
 04599 
 
 21.7426 
 
 06350 
 
 15.7483 
 
 22 
 
 39 
 
 01135 
 
 83.1436 
 
 02881 
 
 34.7151 t 
 
 04628 
 
 21.6056 
 
 06379 
 
 15.6762 
 
 21 
 
 40 
 
 011G4 
 
 85.9398 
 
 02910 
 
 34.3678 
 
 04658 
 
 21.4704 
 
 06408 
 
 15.6048 
 
 20 
 
 41 
 
 01193 
 
 83.8435 
 
 02939 
 
 34.0273 
 
 04687 
 
 21.3369 
 
 06437 
 
 15.5340 
 
 19 
 
 42 
 
 01222 
 
 81.8470 
 
 02963 
 
 33.6935 
 
 04716 
 
 21.2049 
 
 06467 
 
 15.4638 
 
 18 
 
 43 
 
 01251 
 
 79.9434 
 
 02997 
 
 33.3662 
 
 04745 
 
 21.0747 
 
 06496 
 
 15.3943 
 
 17 
 
 44 
 
 01280 
 
 78.1263 
 
 03026 
 
 83.0452 
 
 04774 
 
 20.9460 
 
 06525 
 
 15.3254 
 
 16 
 
 45 
 
 01309 
 
 76.3900 
 
 03055 
 
 32.7303 
 
 04803 
 
 20.8188 
 
 06554 
 
 15.2571 
 
 15 
 
 46 
 
 C1338 
 
 74.7292 
 
 03084 
 
 32.4213 
 
 04833 
 
 20.6932 
 
 06584 
 
 15.1893 
 
 14 
 
 47 
 
 01367 
 
 73.1390 
 
 03114 
 
 32.1181 
 
 04862 
 
 20.5691 
 
 06613 
 
 15.1222 
 
 13 
 
 48 
 
 01396 
 
 71.6151 
 
 03143 
 
 31.8205 
 
 04891 
 
 20.4465 
 
 06642 
 
 15.0557 
 
 12 
 
 49 
 
 01425 
 
 70.15,33 
 
 03172 
 
 31.5284 
 
 04920 
 
 20.3253 
 
 06671 
 
 14.9898 
 
 11 
 
 50 
 
 01455 
 
 68.7501 
 
 03201 
 
 31.2416 
 
 04949 
 
 20.2056 
 
 06700 
 
 14.9244 
 
 10 
 
 51 
 
 01484 
 
 67.4019 
 
 03230 
 
 30.9599 
 
 04978 
 
 20.0872 
 
 06730 
 
 14.8596 
 
 9 
 
 52 
 
 01513 
 
 66.1055 
 
 03259 
 
 30.6833 
 
 05907 
 
 19.9702 
 
 06759 
 
 14.7954 
 
 8 
 
 53 
 
 01542 
 
 64.8580 
 
 03288 
 
 30.4116 
 
 05037 
 
 19.8546 
 
 06788 
 
 14.7317 
 
 7 
 
 54 
 
 01571 
 
 63.G567 
 
 03317 
 
 30.1446 
 
 05066 
 
 19.7403 
 
 06817 
 
 14.6685 
 
 6 
 
 55 
 
 01600 
 
 62.4992 
 
 03346 
 
 29.8823 
 
 05095 
 
 19.6273 
 
 06847 
 
 14.6059 
 
 5 
 
 56 
 
 01629 
 
 61.3829 
 
 03376 
 
 29.6245 
 
 05124 
 
 19.5156 
 
 06876 
 
 14.5438 
 
 4 
 
 57 
 
 01658 
 
 60.3058 
 
 03405 
 
 29.3711 
 
 05153 
 
 19.4051 
 
 06905 
 
 14.4823 
 
 3 
 
 58 
 
 01687 
 
 59.2659 
 
 03434 
 
 29.1220 
 
 05182 
 
 19.2959 
 
 06934 
 
 14.4212 
 
 2 
 
 59 
 
 01716 
 
 58.2612 
 
 03463 
 
 28.8771 
 
 05212 
 
 19.1879 
 
 06963 
 
 14.3607 
 
 1 
 
 60 
 
 01746 
 
 57.2900 
 
 03492 
 
 28.6363 
 
 05241^ 
 
 19.0811 
 
 06993 
 
 14.3007 
 
 
 
 / 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 / 
 
 
 89" 
 
 88 
 
 87' 
 
 86 
 
 
 '235 
 
TABLE XII.-TANGENTS AND COTANGENTS. 
 
 
 
 4 
 
 5 
 
 6 
 
 
 7 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 ' 
 
 
 
 .06993 
 
 14.3007 
 
 08749 
 
 11.4301 
 
 10510 
 
 9.51436 
 
 ' 12278 
 
 8.14435 
 
 00 
 
 1 
 
 07022 
 
 14.2411 
 
 08778 
 
 11.3919 
 
 10540 
 
 9.48781 
 
 12308 
 
 8.12481 
 
 59 
 
 2 
 
 07051 
 
 14.1821 
 
 08807 
 
 11.3540 
 
 10569 
 
 9.46141 
 
 12338 
 
 8.10530 
 
 58 
 
 3 
 
 07080 
 
 14.1235 
 
 08837 
 
 11.3163 
 
 10599 
 
 9.43515 
 
 12367 
 
 8.08600 
 
 57 
 
 4 
 
 07110 
 
 14.0655 
 
 08866 
 
 11.2789 
 
 10628 
 
 9.40904 
 
 12397 
 
 8.06674 
 
 56 
 
 5 
 
 07139 
 
 14.0079 
 
 08895 
 
 11.2417 
 
 10657 
 
 9.38307 
 
 12426 
 
 8.04756 
 
 55 
 
 6 
 
 07168 
 
 13.9507 
 
 08925 
 
 11.2048 
 
 10687 
 
 9.35724 
 
 12456 
 
 8.02848 
 
 54 
 
 7 
 
 07197 
 
 13.8940 
 
 08954 
 
 11.1681 
 
 10716 
 
 9.33155 
 
 12485 
 
 8.00948 
 
 53 
 
 8 
 
 07227 
 
 13.8378 
 
 08983 
 
 11.1316 
 
 10746 
 
 9.30599 
 
 12515 
 
 7.99058 
 
 52 
 
 9 
 
 07256 
 
 13.7821 
 
 09013 
 
 11.0954 
 
 10775 
 
 9.28058 
 
 12544 
 
 7.97176 
 
 51 
 
 10 
 
 07285 
 
 13.7267 
 
 09042 
 
 11.0594 
 
 10805 
 
 9.25530 
 
 12574 
 
 7.95302 
 
 50 
 
 11 
 
 07314 
 
 13.6719 
 
 09071 
 
 11.0237 
 
 10834 
 
 9.23016 
 
 12603 
 
 7.93438 
 
 49 
 
 12 
 
 07344 
 
 13.6174 
 
 09101 
 
 10.9882 
 
 108C3 
 
 9.20516 
 
 12633 
 
 7.91582 
 
 48 
 
 13 
 
 07373 
 
 13.5634 
 
 09130 
 
 10.9529 
 
 10893 
 
 9.18028 
 
 12662 
 
 7 89734 
 
 47 
 
 14 
 
 07402 
 
 13.5098 
 
 09159 
 
 10.9178 
 
 10922 
 
 9.15554 
 
 12692 
 
 7.87895 
 
 40 
 
 15 
 
 07431 
 
 13.4566 
 
 09189 
 
 10.8829 
 
 10952 
 
 9.13093 
 
 12722 
 
 7.86064 
 
 45 
 
 16 
 
 07461 
 
 13.4039 
 
 09218 
 
 10.8483 
 
 10981 
 
 9.10646 
 
 12751 
 
 7.84242 
 
 44 
 
 17 
 
 07490 
 
 13.3515 
 
 09247 
 
 10.8139 
 
 11011 
 
 9.08211 
 
 12781 
 
 7.82428 
 
 4,3 
 
 18 
 
 07519 
 
 13.2996 
 
 09277 
 
 10.7797 
 
 11040 
 
 9.05789 
 
 12810 
 
 7 80622 
 
 42 
 
 19 
 
 07548 
 
 13.2480 
 
 09306 
 
 10.7457 
 
 11070 
 
 9.03379 
 
 12840 
 
 7.78825 
 
 41 
 
 20 
 
 07578 
 
 13.1969 
 
 09335 
 
 10.7119 
 
 11099 
 
 9.00983 
 
 12869 
 
 7.77035 
 
 40 
 
 21 
 
 07607 
 
 13.1461 
 
 09365 
 
 10.6783 
 
 11128 
 
 8.98598 
 
 12899 
 
 7.75254 
 
 39 
 
 22 
 
 07636 
 
 13.0958 
 
 09394 
 
 10.6450 
 
 11158 
 
 8.96227 
 
 12929 
 
 7.73480 
 
 3H 
 
 23 
 
 07665 
 
 13.0458 
 
 09423 
 
 10.6118 
 
 11187 
 
 8.93867 
 
 12958 
 
 7.71715 
 
 37 
 
 24 
 
 07695 
 
 12.9962 
 
 09453 
 
 10.5789 
 
 11217 
 
 8.91520 
 
 12988 
 
 7.69957 
 
 30 
 
 25 
 
 07724 
 
 12.9469 
 
 09482 
 
 10.5402 
 
 11246 
 
 8.89185 
 
 13017 
 
 7.68208 
 
 35 
 
 20 
 
 07753 
 
 12.8981 
 
 09511 
 
 10.5136 
 
 11276 
 
 8.86802 
 
 13047 
 
 7.66466 
 
 34 
 
 27 
 
 07783 
 
 12.8496 
 
 09541 
 
 10.4813 
 
 11305 
 
 8.84551 
 
 13076 
 
 7.64732 
 
 38 
 
 ys 
 
 07812 
 
 12.8014 
 
 09570 
 
 10.4491 
 
 11335 
 
 8.82252 
 
 13106 
 
 7.63005 
 
 32 
 
 29 
 
 07841 
 
 12.7536 
 
 09600 
 
 10.4172 
 
 11364 
 
 8.79964 
 
 13136 
 
 7.61287 
 
 31 
 
 30 
 
 07870 
 
 12.7062 
 
 09029 
 
 10.3854 
 
 11394 
 
 8.77689 
 
 13165 
 
 7.59575 
 
 30 
 
 31 
 
 07899 
 
 12.6591 
 
 09658 
 
 10.3538 
 
 11423 
 
 8.75425 
 
 13195 
 
 7.57872 
 
 29 
 
 32 
 
 07929 
 
 12.6124 
 
 09088 
 
 10.3224 
 
 11452 
 
 8.73172 
 
 13224 
 
 7.56176 
 
 js 
 
 33 
 
 07958 
 
 12.5660 
 
 09717 
 
 10.2913 
 
 11482 
 
 8.70931 
 
 13254 
 
 7.54487 
 
 27 
 
 34 
 
 07987 
 
 12.5199 
 
 09746 
 
 10.2602 
 
 11511 
 
 8.68701 
 
 13284 
 
 7.52806 
 
 20 
 
 35 
 
 08017 
 
 12.4742 
 
 09776 
 
 10.2294 
 
 11541 
 
 8.66482 
 
 13313 
 
 7.51132 
 
 25 
 
 30 
 
 08046 
 
 12.4288 
 
 09805 
 
 10.1988 
 
 11570 
 
 8.64275 
 
 13343 
 
 7.49465 24 
 
 37 
 
 08075 
 
 12.3838 
 
 09834 
 
 10.1683 
 
 11600 
 
 8.62078 
 
 13372 
 
 7.47806 
 
 23 
 
 38 
 
 08104 
 
 12.3390 
 
 09864 
 
 10.1381 
 
 11629 
 
 8.59893 
 
 13402 
 
 7.46154 
 
 .,'.> 
 
 39 
 
 08134 
 
 12.2946 
 
 09893 
 
 10.1080 
 
 IK > 
 
 8.57718 
 
 13432 
 
 7.44509 
 
 21 
 
 40 
 
 08163 
 
 12.2505 
 
 09923 
 
 10.0780 
 
 11688 
 
 8.55555 
 
 13461 
 
 7.42871 
 
 20 
 
 41 
 
 08192 
 
 12.2067 
 
 09952 
 
 10.0483 
 
 11718 
 
 P. 53402 
 
 13491 
 
 7.41240 
 
 19 
 
 42 
 
 08221 
 
 12.1632 
 
 09981 
 
 10.0187 
 
 11747 
 
 8.51259 
 
 13521 
 
 7.39616 
 
 18 
 
 43 
 
 08251 
 
 12.1201 
 
 10011 
 
 9.98931 
 
 11777 
 
 8.49128 
 
 13550 
 
 7.37999 
 
 17 
 
 44 
 
 08280 
 
 12.0772 
 
 10040 
 
 9.96007 
 
 11806 
 
 8.47007 
 
 13580 
 
 7.30389 
 
 16 
 
 45 
 
 08309 
 
 12.0346 
 
 10069 
 
 9.93101 
 
 11836 
 
 8.44896 
 
 13609 
 
 7.34786 
 
 15 
 
 40 
 
 08339 
 
 11.9923 
 
 10099 
 
 9.90211 
 
 11865 
 
 8.42795 
 
 18639 
 
 7.33190 
 
 14 
 
 47 
 
 08368 
 
 11.9504 
 
 10128 
 
 9.87338 
 
 11895 
 
 8.40705 
 
 13669 
 
 7.31600 
 
 13 
 
 48 
 
 08397 
 
 11.9087 
 
 10158 
 
 9.84482 
 
 11924 
 
 8.38625 
 
 13698 
 
 7.30018 
 
 12 
 
 49 
 
 08427 
 
 11.8673 
 
 10187 
 
 9.81641 
 
 11954 
 
 8.36555 
 
 13728 
 
 7.28442 
 
 11 
 
 50 
 
 08456 
 
 11.8262 
 
 10216 
 
 9.78817 
 
 11983 
 
 8.34496 
 
 13758 
 
 7.26873 
 
 10 
 
 51 
 
 08485 
 
 11.7853 
 
 10246 
 
 8.76009 
 
 12013 
 
 8.32446 
 
 13787 
 
 7.25310 
 
 9 
 
 52 
 
 08514 
 
 11.7448 
 
 10275 
 
 9.73217 
 
 12042 
 
 8.30406 
 
 13817 
 
 7.23754 
 
 8 
 
 53 
 
 08544 
 
 11.7045 
 
 10305 
 
 9.70441 
 
 12072 
 
 8.28376 
 
 13846 
 
 7.22204 
 
 7 
 
 54 
 
 08573 
 
 11.6645 
 
 10334 
 
 9.67680 
 
 12101 
 
 8.26355 
 
 13876 
 
 7.20661 
 
 6 
 
 55 
 
 08602 
 
 11.6248 
 
 10363 
 
 9.64935 
 
 12131 
 
 8.24345 
 
 13906 
 
 7.19125 
 
 5 
 
 50 
 
 08632 
 
 11.5853 
 
 10393 
 
 9.62205 
 
 12160 
 
 8.22344 
 
 13935 
 
 7.17594 
 
 4 
 
 57 
 
 08661 
 
 11.5461 
 
 10422 
 
 9.59490 
 
 12190 
 
 8.20352 
 
 13965 
 
 7.16071 
 
 3 
 
 58 
 
 08690 
 
 11.5072 
 
 10452 
 
 9.56791 
 
 12219 
 
 8.18370 
 
 13995 
 
 7.14553 
 
 2 
 
 59 
 
 08720 
 
 11.4685 
 
 10481 
 
 9.54106 
 
 12249 
 
 8.16398 
 
 14024 
 
 7.13042 
 
 1 
 
 CO 
 
 08749 
 
 11.4301 
 
 10510 
 
 9.51436 
 
 12278 
 
 8.14435 
 
 14054 
 
 7.11537 
 
 
 
 / 
 
 Cotang 
 
 Tang 
 
 Cotang | 
 
 Tang 1 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 / 
 
 
 85 
 
 84 
 
 83 82 
 
 
 236 
 
TABLE XII.-rANUENTS AND COTANGENTS. . 
 
 
 
 8 
 
 9 10 
 
 11 
 
 f 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 ~o 
 
 14054 
 
 7.11537 
 
 15838 
 
 6.31375 
 
 17633 
 
 5.67128 
 
 19438 
 
 5.14455 
 
 60 
 
 1 
 
 14084 
 
 7.10038 
 
 15S68 
 
 6.30189 
 
 17663 
 
 5.66165 
 
 19468 
 
 5.13658 
 
 59 
 
 2 
 
 14113 
 
 7.08546 
 
 15898 
 
 6.29007 
 
 17693 
 
 5.65205 
 
 19498 
 
 5.12862 
 
 58 
 
 3 
 
 14143 
 
 7.07059 
 
 15928 
 
 6.27829 
 
 17723 
 
 5.64248 
 
 19529 
 
 5.12069 
 
 57 
 
 4 
 
 14173 
 
 7.05579 
 
 15958 
 
 6.26655 
 
 17753 
 
 5.63295 
 
 19559 
 
 5.11279 
 
 56 
 
 5 
 
 14202 
 
 7.04105 
 
 15988 
 
 6.25486 
 
 17783 
 
 5.62344 
 
 19589 
 
 5.10490 
 
 55 
 
 6 
 
 14232 
 
 7.02637 
 
 16017 
 
 6.24321 
 
 17813 
 
 5.61397 
 
 19619 
 
 5.09704 
 
 54 
 
 7 
 
 14262 
 
 6.91174 
 
 16047 
 
 6.23160 
 
 17843 
 
 5.60452 
 
 19649 
 
 5.08921 
 
 53 
 
 8 
 
 14291 
 
 6.99718 
 
 16077 
 
 6.22003 
 
 17873 
 
 5.59511 
 
 19680 
 
 5.08139 
 
 52 
 
 9 
 
 14321 
 
 6.98268 
 
 16107 
 
 6.20851 
 
 17903 
 
 5.58573 
 
 19710 
 
 5.07360 
 
 51 
 
 10 
 
 14351 
 
 6.96823 
 
 16137 
 
 6.19703 
 
 17933 
 
 5.57638 
 
 19740 
 
 5.06584 
 
 50 
 
 11 
 
 14381 
 
 6.95385 
 
 16167 
 
 6.18559 
 
 17963 
 
 5.56706 
 
 19770 
 
 5.05809 
 
 49 
 
 12 
 
 14410 
 
 6.93952 
 
 16196 
 
 6.17419 
 
 17993 
 
 5.55777 
 
 19801 
 
 5.05037 
 
 48 
 
 13 
 
 14440 
 
 6.92525 
 
 16226 
 
 6.16283 
 
 18023 
 
 5.54851 
 
 19831 
 
 5.04267 
 
 47 
 
 14 
 
 14470 
 
 6.91104 
 
 16256 
 
 6.15151 
 
 18053 
 
 5.53927 
 
 19861 
 
 5.03499 
 
 46 
 
 15 
 
 14499 
 
 6.89688 
 
 16286 
 
 6.14023 
 
 18083 
 
 5.53007 
 
 19891 
 
 5.02734 
 
 45 
 
 16 
 
 14529 
 
 6.88278 
 
 16316 
 
 6.12899 
 
 18113 
 
 5.52090 
 
 19921 
 
 5.01971 
 
 44 
 
 17 
 
 14559 
 
 6.86874 
 
 16346 
 
 6.11779 
 
 18143 
 
 5.51176 
 
 19952 
 
 5.01210 
 
 43 
 
 18 
 
 14588 
 
 6.85475 
 
 16376 
 
 6.10664 
 
 18173 
 
 5.50264 
 
 
 5.00451 
 
 42 
 
 19 
 
 14618 
 
 6.84082 
 
 16405 
 
 6.09552 
 
 18203 
 
 5.49356 
 
 20012 
 
 4.99695 
 
 41 
 
 20 
 
 14648 
 
 6.82694 
 
 16435 
 
 6.08444 
 
 18233 
 
 5.48451 
 
 20042 
 
 4.98940 
 
 40 
 
 21 
 
 14678 
 
 6.81312 
 
 16465 
 
 6.07340 
 
 18263 
 
 5.47548 
 
 20073 
 
 4.98188 
 
 39 
 
 
 14707 
 
 6.79936 
 
 16495 
 
 6.06240 
 
 18293 
 
 5.46648 
 
 20103 
 
 4.97438 
 
 38 
 
 -'] 
 
 14737 
 
 6.78564 
 
 16525 
 
 6.05143 
 
 18323 
 
 5.45751 
 
 20133 
 
 4.96690 
 
 87 
 
 24 
 
 14767 
 
 6.77199 
 
 16555 
 
 6.04051 
 
 
 5.44857 
 
 20164 
 
 4.95945 
 
 36 
 
 25 
 
 14796 
 
 6.75838 
 
 16585 
 
 6.02962 
 
 18384 
 
 5.43966 
 
 20194 
 
 4.95201 
 
 35 
 
 26 
 
 14826 
 
 6.74483 
 
 16615 
 
 6.01878 
 
 18414 
 
 5.45077 
 
 20224 
 
 4.94460 
 
 34 
 
 27 
 
 14856 
 
 6.73133 
 
 16645 
 
 6.00797 
 
 18444 
 
 5.42192 
 
 20254 
 
 4.93721 
 
 33 
 
 28 
 
 14886 
 
 6.71789 
 
 16674 
 
 5.99720 
 
 18474 
 
 5.41309 
 
 20285 
 
 4.92984 
 
 32 
 
 29 
 
 11915 
 
 6.70450 
 
 16704 
 
 5.98646 
 
 18504 
 
 5.40429 
 
 20315 
 
 4.92249 
 
 31 
 
 30 
 
 14945 
 
 6.69116 
 
 16734 
 
 5.97576 
 
 18534 
 
 5.39552 
 
 20345 
 
 4.91516 
 
 30 
 
 31 
 
 14975 
 
 6.67787 
 
 16764 
 
 5.96510 
 
 18564 
 
 5.38677 
 
 20376 
 
 4.90785 
 
 29 
 
 32 
 
 15005 
 
 6.66463 
 
 16794 
 
 5.95448 
 
 18594 
 
 5.37805 
 
 20406 
 
 4.90056 
 
 28 
 
 33 
 
 15034 
 
 6.65144 
 
 16824 
 
 5.94390 
 
 18624 
 
 5.36936 
 
 20436 
 
 4.89330 
 
 27 
 
 34 
 
 15064 
 
 6.63831 
 
 16854 
 
 5.93335 
 
 18654 
 
 5.36070 
 
 20466 
 
 4.88605 
 
 26 
 
 35 
 
 15094 
 
 6.62523 
 
 16884 
 
 5.92283 
 
 18684 
 
 5.35206 
 
 20497 
 
 4.87882 
 
 25 
 
 36 
 
 15124 
 
 6.61219 
 
 16914 
 
 5.91236 
 
 18714 
 
 5.34345 
 
 20527 
 
 4.87162 
 
 24 
 
 37 
 
 15153 
 
 6.59921 
 
 16944 
 
 5.90191 
 
 18745 
 
 5.33487 
 
 20557 
 
 4.86444 
 
 23 
 
 88 
 
 15183 
 
 6.58627 
 
 16974 
 
 5.89151 
 
 18775 
 
 5.32631 
 
 20588 
 
 4.85727 
 
 22 
 
 39 
 
 15213 
 
 6.57&39 
 
 17004 
 
 5.88114 
 
 18805 
 
 5.31778 
 
 20618 
 
 4.85013 
 
 21 
 
 40 
 
 15243 
 
 6.56055 
 
 17033 
 
 5.87080 
 
 18835 
 
 5.30928 
 
 20648 
 
 4.84300 
 
 20 
 
 41 
 
 15272 
 
 6.54777 
 
 1 17063 
 
 5.86051 
 
 18865 
 
 5.30080 
 
 20679 
 
 4.83590 
 
 19 
 
 42 
 
 15302 
 
 6.53503 
 
 17093 
 
 5.85024 
 
 18895 
 
 5.29235 
 
 20709 
 
 4.82882 
 
 18 
 
 43 
 
 15332 
 
 6.52234 
 
 17123 
 
 5.84001 
 
 18925 
 
 5.28393 
 
 20739 
 
 4.82175 
 
 17 
 
 44 
 
 15362 
 
 6.50970 
 
 17153 
 
 5.82982 
 
 18955 
 
 5.27553 
 
 20770 
 
 4.81471 
 
 16 
 
 45 
 
 15391 
 
 6.49710 
 
 17183 
 
 5.81966 
 
 18986 
 
 5.26715 
 
 20800 
 
 4.80769 
 
 15 
 
 46 
 
 15421 
 
 6.48456 
 
 17213 
 
 5.80953 
 
 19016 
 
 5.258SO 
 
 20830 
 
 4.80068 
 
 14 
 
 47 
 
 15451 
 
 6.47206 
 
 17243 
 
 5.79944 
 
 19046 
 
 5.25048 
 
 208Q1 
 
 4.79370 
 
 13 
 
 48 
 
 15481 
 
 6.45961 
 
 17273 
 
 5.78938 
 
 19076 
 
 5.24218 
 
 20891 
 
 4.78673 
 
 12 
 
 49 
 
 15511 
 
 6.44720 
 
 17303 
 
 5.77936 
 
 19106 
 
 5.2-3391 
 
 20921 
 
 4.77978 
 
 11 
 
 50 
 
 15540 
 
 6.43484 
 
 17333 
 
 5.76937 
 
 19136 
 
 5.22566 
 
 20952 
 
 4.77286 
 
 10 
 
 51 
 
 15570 
 
 6.42253 
 
 17363 
 
 5.75941 
 
 19166 
 
 5.21744 
 
 20982 
 
 4.76595 
 
 9 
 
 52 
 
 15600 
 
 6.41026 
 
 17393 
 
 5.74949 
 
 19197 
 
 5.20925 
 
 21013 
 
 4.75906 
 
 8 
 
 53 
 
 15630 
 
 6.39804 
 
 17423 
 
 5.73960 
 
 19227 
 
 5.20107 
 
 21043 
 
 4.75219 
 
 7 
 
 54 
 
 15660 
 
 6.38587 
 
 17453 
 
 5.72974 
 
 19257 
 
 5.19293 
 
 21073 
 
 4.74534 
 
 6 
 
 55 
 
 15689 
 
 6.37374 
 
 17483 
 
 5.71992 
 
 19287 
 
 5.18480 
 
 21104 
 
 4.73851 
 
 5 
 
 56 
 
 15719 
 
 6.36165 
 
 17513 
 
 5.71013 
 
 19317 
 
 5.17671 
 
 21134 
 
 4.73170 
 
 4 
 
 57 
 
 15749 
 
 6.34961 
 
 17543 
 
 5.70037 
 
 19347 
 
 5.16863 
 
 21164 
 
 4.72490 
 
 3 
 
 58 
 
 15779 
 
 6.33761 
 
 17573 
 
 5.69064- 
 
 19378 
 
 5.16058 
 
 21195 
 
 4.71813 
 
 2 
 
 59 
 
 15809 
 
 6.32566 
 
 17603 
 
 5.68094 
 
 19408 
 
 5.15256 
 
 21225 
 
 4.71137 
 
 1 
 
 no 
 
 15&38 
 
 6.31375 
 
 17633 
 
 5.67128 
 
 19438 
 
 5.14455 
 
 21256 
 
 4.70463 
 
 
 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 f 
 
 
 81 
 
 80 
 
 79 
 
 78 
 
 
 237 
 
^ w .** 
 
 V OF THE 
 
 TABLE 'XIL TANGENTS AND COTANGENTS. 
 
 
 12 
 
 13 
 
 14 
 
 15 
 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 
 
 21256 
 
 4.70463 
 
 ^23087 
 
 4.33148 
 
 24933 
 
 4.01078 
 
 26795 
 
 3.73205 
 
 60 
 
 1 
 
 21286 
 
 4.69791 
 
 23117 
 
 4.32573 
 
 24964 
 
 4.00582 
 
 26826 
 
 3.72771 
 
 59 
 
 2 
 
 21316 
 
 4.G9121 
 
 23148 
 
 4.32001 
 
 24995 
 
 4.00086 
 
 26857 
 
 3.72338 
 
 53 
 
 3 
 
 21347 
 
 4.68452 
 
 23179 
 
 4.31430 
 
 25026 
 
 3.99592 
 
 26888 
 
 3.71907 
 
 57 
 
 4 
 
 21377 
 
 4.67786 
 
 23209 
 
 4.30860 
 
 25056 
 
 3.99099 
 
 26920 
 
 3.71476 
 
 56 
 
 5 
 
 21408 
 
 4.67121 
 
 23240 
 
 4.30291 
 
 25087 
 
 3.98607 
 
 26951 
 
 3.71046 
 
 55 
 
 6 
 
 21438 
 
 4.66458 
 
 23271 
 
 4.29724 
 
 25118 
 
 3.98117 
 
 26982 
 
 3.70616 
 
 54 
 
 7 
 
 21469 
 
 4.65797 
 
 23301 
 
 4.29159 
 
 25149 
 
 3.97627 
 
 27013 
 
 3.70188 
 
 53 
 
 8 
 
 21499 
 
 4.65138 
 
 23332 
 
 4.28595 
 
 25180 
 
 3.97139 
 
 27'044 
 
 3.69761 
 
 52 
 
 9 
 
 215S9 
 
 4.G4480 
 
 23363 
 
 4.28032 
 
 25211 
 
 3.9G651 
 
 27076 
 
 3.69335 
 
 51 
 
 10 
 
 21560 
 
 4.63825 
 
 23393 
 
 4.27471 
 
 25242 
 
 3.96165 
 
 27107 
 
 3.68909 
 
 50 
 
 11 
 
 21590 
 
 4.63171 
 
 23424 
 
 4.26911 
 
 25273 
 
 3.95680 
 
 27138 
 
 3.68485 
 
 49 
 
 13 
 
 21621 
 
 4.62518 
 
 23455 
 
 4.26352 
 
 25304 
 
 3.95196 
 
 27169 
 
 3.680G1 
 
 48 
 
 13 
 
 21651 
 
 4.61868 
 
 23485 
 
 4.25795 
 
 25335 
 
 3.94713 
 
 27201 
 
 3.67638 
 
 47 
 
 14 
 
 21683 
 
 4.61219 
 
 23516 
 
 4.25.239 
 
 253G6 
 
 3.94232 
 
 27232 
 
 3.G7217 
 
 4G 
 
 15 
 
 21712 
 
 4.60572 
 
 23547 
 
 4.24685 
 
 25397 
 
 3.93751 
 
 27263 
 
 3.66796 
 
 45 
 
 16 
 
 21743 
 
 4.59927 
 
 23578 
 
 4.24132 
 
 25428 
 
 3.93271 
 
 27294 
 
 3.66376 
 
 44 
 
 17 
 
 21773 
 
 4.59283 
 
 28606 
 
 4.23580 
 
 25459 
 
 3.92793 
 
 27326 
 
 3.65957 
 
 48 
 
 is 
 
 21804 
 
 4.58G41 
 
 23639 
 
 4.23030 
 
 25490 
 
 3.92316 
 
 27357 
 
 3.G5538 
 
 42 
 
 19 
 
 21834 
 
 4.58001 
 
 23670 
 
 4.22481 
 
 25521 
 
 3.91839 
 
 27388 
 
 3.65121 
 
 11 
 
 20 
 
 21864 
 
 4.57363 
 
 23700 
 
 4.21933 
 
 25552 
 
 3.91364 
 
 27419 
 
 3.64705 
 
 40 
 
 21 
 
 21895 
 
 4.56726 
 
 23731 
 
 4.21387 
 
 25583 
 
 3.90890 
 
 27451 
 
 3.64289 
 
 39 
 
 23 
 
 21925 
 
 4.56091 
 
 23762 
 
 4.20842 
 
 25614 
 
 3,. 90417 
 
 27482 
 
 3.63874 
 
 38 
 
 23 
 
 21956 
 
 4.55458 
 
 23793 
 
 4.20298 
 
 25645 
 
 3\ 89945 
 
 27513 
 
 3.63461 
 
 37 
 
 24 
 
 21986 
 
 4.54826 
 
 23823 
 
 4.19756 
 
 25G76 
 
 3.89474 
 
 27545 
 
 3.63048 
 
 86 
 
 23 
 
 22017 
 
 4.54196 
 
 23S54 
 
 4.19215 
 
 25707 
 
 3.89004 
 
 27576 
 
 3.C2G36 
 
 35 
 
 2G 
 
 22047 
 
 4.53568 
 
 23885 
 
 4.18675 
 
 25738 
 
 3.88536 
 
 27607 
 
 3.62224 
 
 34 
 
 27 
 
 22078 
 
 4.52941 
 
 23916 
 
 4.18137 
 
 25769 
 
 3.88068 
 
 27638 
 
 3.61814 
 
 33 
 
 28 
 
 22108 
 
 4.52316 
 
 23946 
 
 4.17600 
 
 25800 
 
 3.87601 
 
 27G70 
 
 3.61405 
 
 32 
 
 29 
 
 22139 
 
 4.51693 
 
 23977 
 
 4.17064 
 
 25831 
 
 3.87136 
 
 27701 
 
 3.G0996 
 
 81 
 
 30 
 
 22169 
 
 4.51071 
 
 24008 
 
 4.16530 
 
 25862 
 
 3.86671 
 
 27732 
 
 3 60588 
 
 30 
 
 31 
 
 22200 
 
 4.50451 
 
 24039 
 
 4.15997 
 
 25893 
 
 3.86208 
 
 27764 
 
 3.60181 
 
 29 
 
 32 
 
 22231 
 
 4.49832 
 
 24069 
 
 4.15465 
 
 25924 
 
 3.85745 
 
 27795 
 
 3.59775 
 
 28 
 
 33 
 
 22261 
 
 4.49215 
 
 24100 
 
 4.14934 
 
 25955 
 
 3.85284 
 
 27826 
 
 3.59370 
 
 27 
 
 34 
 
 22292 
 
 4.48600 
 
 24131 
 
 4.14405 
 
 25986 
 
 3.84S24 
 
 27858 
 
 3.58966 
 
 26 
 
 35 
 
 22322 
 
 4.47986 
 
 24162 
 
 4.13877 
 
 26017 
 
 3.84364 
 
 27889 
 
 3.58562 
 
 ;:.-> 
 
 3G 
 
 22353 
 
 4.47374 
 
 24193 
 
 4.13350 
 
 26048 
 
 3.83906 
 
 27921 
 
 3.58160 
 
 24 
 
 37 
 
 22383 
 
 4.46764 
 
 24223 
 
 4.12825 
 
 26079 
 
 3.83449 
 
 27952 
 
 3.57758 
 
 23 
 
 38 
 
 22414 
 
 4.46155 
 
 24254 
 
 4.12301 
 
 26110 
 
 3.82992 
 
 27983 
 
 3.57357 
 
 2-> 
 
 39 
 
 22444 
 
 4.45548 
 
 24285 
 
 4.11778 
 
 26141 
 
 3.82537 
 
 28015 
 
 3.56957 
 
 21 
 
 40 
 
 22475 
 
 4.44942 
 
 24316 
 
 4.11256 
 
 26172 
 
 3.82083 
 
 28046 
 
 3.56557 
 
 20 
 
 41 
 
 22505 
 
 4.44338 
 
 24347 
 
 4.10736 
 
 26203 
 
 3.81630 
 
 28077 
 
 3.56159 
 
 19 
 
 42 
 
 22536 
 
 4.43735 
 
 24377 
 
 4.10216 
 
 26235 
 
 3.81177 
 
 28109 
 
 3.55761 
 
 18 
 
 43 
 
 22567 
 
 4.43134 
 
 24408 
 
 4.09699 
 
 26266 
 
 3.807'26 
 
 28140 
 
 3.55364 
 
 17 
 
 44 
 
 22597 
 
 4.42534 
 
 24439 
 
 4.09182 
 
 26297 
 
 3.80276 
 
 28172 
 
 3.54068 
 
 16 
 
 45 
 
 22628 
 
 4.41936 
 
 24470 
 
 4.08666 
 
 26328 
 
 3.79827 
 
 28203 
 
 3.54573 
 
 15 
 
 46 
 
 22658 
 
 4.41340 
 
 24501 
 
 4.08152 
 
 26359 
 
 3.79378 
 
 28234 
 
 3.54179 
 
 14 
 
 47 
 
 22G89 
 
 4.40745 
 
 24532 
 
 4.07639 
 
 26390 
 
 3.78931 
 
 28266 
 
 3.53785 
 
 13 
 
 48 
 
 22719 
 
 4.40152 
 
 24562 
 
 4.07127 
 
 26421 
 
 3.78485 
 
 28297 
 
 3.53393 
 
 12 
 
 49 
 
 22750 
 
 4.30560 
 
 24593 
 
 4.06G16 
 
 2G452 
 
 3.78040 
 
 28329 
 
 3.53001 
 
 11 
 
 50 
 
 22781 
 
 4.38969 
 
 24624 
 
 4.06107 
 
 26483 
 
 3.77595 
 
 28360 
 
 3.52609 
 
 10 
 
 51 
 
 22811 
 
 4.38381 
 
 24655 
 
 4.05599 
 
 26515 
 
 3.77152 
 
 28391 
 
 3.52219 
 
 G 
 
 52 
 
 22S42 
 
 4.37793 
 
 24G86 
 
 4.05092 
 
 26546 
 
 3.76709 
 
 28423 
 
 3.51829 
 
 8 
 
 53 
 
 22872 
 
 4.37207 
 
 24717 
 
 4.04586 
 
 26577 
 
 3.7G2G8 
 
 28454 
 
 3.51441 
 
 7 
 
 54 
 
 22903 
 
 4.3GG23 
 
 24747 
 
 4.04081 
 
 2GG08 
 
 3.75828 
 
 28486 
 
 3.51053 
 
 6 
 
 55 
 
 22934 
 
 4.3G040 
 
 24778 
 
 4.03578 
 
 2GG39 
 
 3.75388 
 
 28517 
 
 3.506G6 
 
 5 
 
 50 
 
 22964 
 
 4.35459 
 
 24809 
 
 4.03076 
 
 26670 
 
 3.74950 
 
 28549 
 
 3.50279 
 
 4 
 
 57 
 
 22995 
 
 4.34879 
 
 24840 
 
 4.02574 
 
 26701 
 
 3.74512 
 
 28580 
 
 3.49894 
 
 & 
 
 58 
 
 23026 
 
 4.34300 
 
 24871 
 
 4.02074 
 
 26733 
 
 3.74075 
 
 28G12 
 
 3.49509 
 
 2 
 
 59 
 
 23056 
 
 4.33723 
 
 24902 
 
 4.01576 
 
 26764 
 
 3.73G40 
 
 28643 
 
 3.49125 
 
 1 
 
 GO 
 
 23087 
 
 4.33148 
 
 24933 
 
 4.01078 
 
 26795 
 
 3.73205- 
 
 28G75 
 
 3.48741 
 
 
 
 / 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 ' Tang 
 
 Cotang 
 
 Tang 
 
 / 
 
 
 77 
 
 76 
 
 75 
 
 74 
 
 
 238 
 
TABLE XII. TANGENTS AND COTANGENTS. 
 
 
 16 
 
 17 
 
 18 
 
 19 
 
 f 
 
 / 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 o 
 
 28675 
 
 3.48741 
 
 30573 
 
 3.27085 7 
 
 32492 
 
 3.07768 
 
 34433" 
 
 2.90421 
 
 60 
 
 1 
 
 28706 
 
 3.48359 
 
 30605 
 
 3.26745 
 
 32524 
 
 3.07464 
 
 34465 
 
 2.90147 
 
 59 
 
 L J 
 
 28738 
 
 3.47977 
 
 30637 
 
 3.26406 
 
 32556 
 
 3.07160 
 
 34498 
 
 2.89873 
 
 58 
 
 8 
 
 28769 
 
 3.47596 
 
 30669 
 
 3.26067 
 
 32588 
 
 3.06857 
 
 34530 
 
 2.89600 
 
 57 
 
 4 
 
 28800 
 
 3.47216 
 
 30700 
 
 3.25729 
 
 32621 
 
 3.06554 
 
 34563 
 
 2.89327 
 
 56 
 
 5 
 
 28832 
 
 3.46837 
 
 30732 
 
 3.25392 
 
 32653 
 
 3.06252 
 
 34596 
 
 2.89055 
 
 55 
 
 G 
 
 28864 
 
 3.46458 
 
 30764 
 
 3.25055 
 
 32685 
 
 3.05950 
 
 34628 
 
 2.88783 
 
 54 
 
 y 
 
 28895 
 
 3.46080 
 
 30796 
 
 3.24719 
 
 32717 
 
 3.05649 
 
 34661 
 
 2.88511 
 
 53 
 
 
 
 28927 
 
 3.45703 
 
 30828 
 
 3.24383 
 
 32749 
 
 3.05349 
 
 34693 
 
 2.88240 
 
 52 
 
 9 
 
 28958 
 
 3.45327 
 
 30860 
 
 3.24049 
 
 32782 
 
 3.05049 
 
 34726 
 
 2.87970 
 
 51 
 
 10 
 
 28990 
 
 3.44951 
 
 30891 
 
 3.23714 
 
 32814 
 
 3.04749 
 
 34758 
 
 2 87700 
 
 50 
 
 11 
 
 29021 
 
 3.44576 
 
 30923 
 
 3.23381 
 
 32846 
 
 3.04450 
 
 34791 
 
 2.87430 
 
 49 
 
 12 29053 
 
 3.44202 
 
 30955 
 
 3.23048 
 
 32878 
 
 3.C4152 
 
 34824 
 
 2.87161 
 
 48 
 
 13 29084 
 
 3.43829 
 
 30987 
 
 3.22715 
 
 32911 
 
 3.03854 
 
 34856 
 
 2.86892 
 
 47 
 
 14 29116 
 
 3.43456 
 
 31019 
 
 3.22384 
 
 32943 
 
 3.03556 
 
 34889 
 
 2.86624 
 
 46 
 
 15 29147 
 
 3.43084 
 
 31051 
 
 3.22053 
 
 32975 
 
 3.03260 
 
 34922 
 
 2.86356 
 
 46 
 
 16 29179 
 
 3.42713 
 
 31083 
 
 3.21722 
 
 33007 
 
 3.02963 
 
 34954 
 
 2.86089 
 
 44 
 
 17 
 
 29210 
 
 3.42343 
 
 31115 
 
 3.21392 
 
 33040 
 
 3.02667 
 
 34987 
 
 2.85822 
 
 43 
 
 IS 29242 
 
 3.41973 
 
 31147 
 
 3.21063 
 
 33072 
 
 3.02372 
 
 35020 
 
 2.85555 
 
 42 
 
 19 29274 
 
 3.41604 
 
 31178 
 
 3.20734 
 
 33104 
 
 3.02077 
 
 35052 
 
 2.85289 
 
 41 
 
 20 
 
 29305 
 
 3.41236 
 
 31210 
 
 3.20406 
 
 33136 
 
 3.01783 
 
 35085 
 
 2.85023 
 
 40 
 
 21 
 
 29337 
 
 3.40869 
 
 31242 
 
 3.20079 
 
 33169 
 
 3.01489 
 
 35118 
 
 2.84758 
 
 39 
 
 23 
 
 29368 
 
 3.40502 
 
 31274 
 
 3.19752 
 
 33201 
 
 3.01196 
 
 35150 
 
 2.84494 
 
 38 
 
 X!3 
 
 29400 
 
 3.40136 
 
 31306 
 
 3.19426 
 
 33283 
 
 3.00903 
 
 35183 
 
 2.84229 
 
 37 
 
 94 
 
 29432 
 
 3.39771 
 
 31338 
 
 3.19100 
 
 332G6 
 
 3.00611 
 
 35216 
 
 2.83965 
 
 36 
 
 96 
 
 29403 
 
 3.39406 
 
 31370 
 
 3.18775 
 
 &3298 
 
 3.00319 
 
 35248 
 
 2.83702 
 
 35 
 
 20 
 
 29495 
 
 3.39042 
 
 31402 
 
 3.18451 
 
 33330 
 
 3.00028 
 
 35281 
 
 2.83439 
 
 34 
 
 27 
 
 29526 
 
 3.38679 
 
 31434 
 
 3.18127 
 
 33363 
 
 2.99738 
 
 35314 
 
 2.&3176 
 
 33 
 
 23 
 
 29558 
 
 3.38317 
 
 31466 
 
 3.17804 
 
 33395 
 
 2.99447 
 
 35346 
 
 2.82914 
 
 32 
 
 23 
 
 29590 
 
 3.37955 
 
 31498 
 
 3.17481 
 
 33427 
 
 2.99158 
 
 35379 
 
 2.82653 
 
 31 
 
 80 
 
 29621 
 
 3.37594 
 
 31530 
 
 3.17159 
 
 33460 
 
 2.98868 
 
 35412 
 
 2.82391 
 
 30 
 
 31 
 
 29653 
 
 3.37234 
 
 31562 
 
 3.16838 
 
 33492 
 
 2.98580 
 
 35445 
 
 2.82130 
 
 29 
 
 as 
 
 29685 
 
 3.30875 
 
 31594 
 
 3.16517 
 
 33524 
 
 2.98292 
 
 35477 
 
 2.81870 
 
 ^ 
 
 83 
 
 29716 
 
 3.36516 
 
 31626 
 
 3.16197 
 
 33557 
 
 2.98004 
 
 35510 
 
 2.81610 
 
 27 
 
 ::4 
 
 29748 
 
 3.30158 
 
 31658 
 
 3.15877 
 
 33589 
 
 2.97717 
 
 35543 
 
 2.81350 
 
 26 
 
 35 
 
 29780 
 
 3.35800 
 
 31690 
 
 3.15558 
 
 33621 
 
 2.97430 
 
 35576 
 
 2.81091 
 
 25 
 
 86 
 
 29811 
 
 3.35443 
 
 31722 
 
 3.15240 
 
 33654 
 
 2.97144 
 
 35608 
 
 2.80833 
 
 24 
 
 37 
 
 29843 
 
 3.35087 
 
 31754 
 
 3.14922 
 
 33686 
 
 2.96858 
 
 35641 
 
 2.80574 
 
 23 
 
 '13 
 
 29875 
 
 3.34732 
 
 31786 
 
 3.14605 
 
 33718 
 
 2.96573 
 
 '35674 
 
 2.80316 
 
 22 
 
 :!9 
 
 29906 
 
 3.34377 
 
 31818 
 
 3.14288 
 
 33751 
 
 2.96288 
 
 35707 
 
 2.80059 
 
 21 
 
 40 
 
 29938 
 
 3.34023 
 
 31850 
 
 3.13972 
 
 33783 
 
 2.96004 
 
 35740 
 
 2.79802 
 
 20 
 
 41 
 
 29970 
 
 3.33670 
 
 31882 
 
 3.13656 
 
 33816 
 
 2.95721 
 
 35772 
 
 2.79545 
 
 19 
 
 42 
 
 30001 
 
 3.33317 
 
 31914 
 
 3.13341 
 
 33848 
 
 2.95437 
 
 35805 
 
 2.79289 
 
 18 
 
 43 
 
 30033 
 
 3.32965 
 
 31946 
 
 3.13027 
 
 33881 
 
 2.95155 
 
 35838 
 
 2.79033 
 
 17 
 
 44 
 
 80065 
 
 3.32614 
 
 31978 
 
 3.12713 
 
 33913 
 
 2.94872 
 
 35871 
 
 2.78778 
 
 16 
 
 45 
 
 30097 
 
 3.32264 
 
 32010 
 
 3.12400 
 
 33945 
 
 2.94591 
 
 35904 
 
 2.78523 
 
 15 
 
 46 
 
 30128 
 
 3.31914 
 
 32042 
 
 3.12087 
 
 33978 
 
 2.94309 
 
 35937 
 
 2.78269 
 
 14 
 
 47 
 
 30160 
 
 3.31565 
 
 32074 
 
 3.11775 
 
 34010 
 
 2.94028 
 
 35969 
 
 2.78014 
 
 13 
 
 48 30192 
 
 3.31216 
 
 32106 
 
 3.11464 
 
 34043 
 
 2.93748 
 
 36002 
 
 2.77761 
 
 12 
 
 49 i 30224 
 
 3.30SG8 
 
 32139 
 
 3.11153 
 
 34075 
 
 2.93468 
 
 36035 
 
 2.77507 
 
 11 
 
 50 30255 
 
 3.30521 
 
 32171 
 
 3.10843 
 
 34108 
 
 2.93189 
 
 36068 
 
 2.77254 
 
 10 
 
 51 
 
 30287 
 
 3.30174 
 
 32203 
 
 3.10532 
 
 34140 
 
 2.92910 
 
 36101 
 
 2.77002 
 
 9 
 
 52 
 
 30319 
 
 3.29829 
 
 32235 
 
 3.10223 
 
 34173 
 
 2.92632 
 
 36134 
 
 2.76750 
 
 8 
 
 53 
 
 30351 
 
 3.29483 
 
 32267 
 
 3.09914 
 
 34205 
 
 2.92354 
 
 36167 
 
 2.76-198 
 
 7 
 
 54 
 
 30382 
 
 3.29139 
 
 32299 
 
 3.09606 
 
 34238 
 
 2.92076 
 
 36199 
 
 2.76247 
 
 6 
 
 56 
 
 30414 
 
 3.28795 
 
 32331 
 
 3.09298 
 
 34270 
 
 2.91799 
 
 36232 
 
 2.75996 
 
 5 
 
 56 
 
 30446 
 
 3.28452 
 
 32363 
 
 3.08991 
 
 34303 
 
 2.91523 
 
 36265 
 
 2.75746 
 
 4 
 
 57 
 
 30478 
 
 3.28109 
 
 32396 
 
 3.08685 
 
 343:35 
 
 2.91246 
 
 36298 
 
 2.75496 
 
 3 
 
 58 
 
 30509 
 
 3.27767 
 
 32428 
 
 3.08379 
 
 34368 
 
 2.90971 
 
 36331 
 
 2.75246 
 
 2 
 
 59 
 
 30541 
 
 3.27426 
 
 32460 
 
 3.08073 
 
 34400 
 
 2.90696 
 
 36364 
 
 2.74997 
 
 1 
 
 GO 
 
 30573 
 
 3.27085 
 
 32492 
 
 3.07768 
 
 34433 
 
 2.90421 
 
 36397 
 
 2.74748 
 
 
 
 / 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 [Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 / 
 
 
 73 
 
 72 
 
 71 
 
 i 70 
 
 
 239 
 
TABLE XII. TANGENTS AND COTANGENTS. 
 
 20 
 
 21 
 
 22 
 
 23 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 r 
 
 
 
 36397 
 
 2.74748 
 
 ' 38386 
 
 2.60509 
 
 40403 
 
 2.47509 
 
 42447 
 
 2.35585 
 
 GO 
 
 1 
 
 3S430 
 
 2.74499 
 
 38420 
 
 2.60283 
 
 40436 
 
 2.47302 
 
 42482 
 
 2.35395 
 
 59 
 
 2 
 
 36463 
 
 2.74251 
 
 38453 
 
 2.60057 
 
 40470 
 
 2.47095 
 
 42516 
 
 2.35205 
 
 58 
 
 3 
 
 36496 
 
 2.74004 
 
 38487 
 
 2.59831 
 
 40504 
 
 2.46888 
 
 42551 
 
 2.35015 
 
 57 
 
 4 
 
 36529 
 
 2.73756 
 
 38520 
 
 2.59606 
 
 40538 
 
 2.46682 
 
 42585 
 
 2.34825 
 
 56 
 
 5 
 
 36562 
 
 2.73509 
 
 38553 
 
 2.59381 
 
 40572 
 
 2.46476 
 
 42619 
 
 2.34636 
 
 55 
 
 C 
 
 36595 
 
 2.73263 
 
 38587 
 
 2.59156 
 
 40606 
 
 2.46270 
 
 42654 
 
 2.34447 
 
 54 
 
 7 
 
 36628 
 
 2.73017 
 
 38620 
 
 2.58932 
 
 40640 
 
 2.46065 
 
 42GS8 
 
 2.34258 
 
 53 
 
 8 
 
 36661 
 
 2.72771 
 
 38654 
 
 2.58708 
 
 40674 
 
 2.45860 
 
 42722 
 
 2.34069 
 
 52 
 
 9 
 
 36694 
 
 2.72526 
 
 38687 
 
 2.58484 
 
 40707 
 
 2.45655 
 
 42757 
 
 2.33881 
 
 51 
 
 10 
 
 36727 
 
 2.72281 
 
 38721 
 
 2.58261 
 
 40741 
 
 2.45451 
 
 42791 
 
 2.33693 
 
 50 
 
 11 
 
 36760 
 
 8.72036 
 
 38754 
 
 2.58038 
 
 40775 
 
 2.45246 
 
 42826 
 
 2.33505 
 
 49 
 
 12 
 
 36793 
 
 2.71792 
 
 38787 
 
 2.57815 
 
 40809 
 
 2.45043 
 
 428GO 
 
 2.33317 
 
 48 
 
 13 
 
 36826 
 
 2.71548 
 
 38821 
 
 2.57593 
 
 40843 
 
 2.44839 
 
 42894 
 
 2.33130 
 
 47 
 
 11 
 
 36859 
 
 2.71305 
 
 38854 
 
 2.57371 
 
 40877 
 
 2.44636 
 
 42929 
 
 2.32943 
 
 46 
 
 15 
 
 36892 
 
 2.71062 
 
 38888 
 
 2.57150 
 
 40911 
 
 2.44433 
 
 42963 
 
 2.32756 
 
 45 
 
 16 
 
 36925 
 
 2.70819 
 
 38921 
 
 2.56928 
 
 40945 
 
 2.44230 
 
 42998 
 
 2.32570 
 
 44 
 
 17 
 
 36958 
 
 2.70577 
 
 38955 
 
 2.56707 
 
 40979 
 
 2.44027 
 
 43032 
 
 2.32383 
 
 43 
 
 18 
 
 36991 
 
 2.70335 
 
 38988 
 
 2.56487 
 
 41013 
 
 2.43825 
 
 43067 
 
 2.32197 
 
 42 
 
 19 
 
 37024 
 
 2.70094 
 
 39022 
 
 2.56266 
 
 41047 
 
 2.43623 
 
 43101 
 
 2.32012 
 
 41 
 
 20 
 
 37057 
 
 2.69853 
 
 39055 
 
 2.56046 
 
 41081 
 
 2.43422 
 
 43136 
 
 2.31826 
 
 40 
 
 21 
 
 37090 
 
 2.69612 
 
 39089 
 
 2.55827 
 
 41115 
 
 2.43220 
 
 43170 
 
 2.31641 
 
 39 
 
 22 
 
 37123 
 
 2.69371 
 
 39122 
 
 2.55608 
 
 41149 
 
 2.43019 
 
 43205 
 
 2.31456 
 
 33 
 
 23 
 
 37157 
 
 2.69131 
 
 39156 
 
 2.55389 
 
 41183 
 
 2.42819 
 
 43233 
 
 2.31271 
 
 37 
 
 24 
 
 37190 
 
 2.68892 
 
 39190 
 
 2.55170 
 
 41217 
 
 2.42618 
 
 43274 
 
 2.31086 
 
 30 
 
 25 
 
 37223 
 
 9.68653 
 
 39223 
 
 2.54952 
 
 41251 
 
 2.42418 
 
 43308 
 
 2.30902 
 
 35 
 
 26 
 
 37256 
 
 2.68414 
 
 39257 
 
 2.54734 
 
 41285 
 
 2.42218 
 
 43343 
 
 2.30718 
 
 34 
 
 27 
 
 37289 
 
 2.68175 
 
 39290 
 
 2.54516 
 
 41319 
 
 2.42019 
 
 43378 
 
 2.30534 
 
 33 
 
 28 
 
 37322 
 
 2.67937 
 
 39324 
 
 2.54299 
 
 41353 
 
 2.41819 
 
 43412 
 
 2.30351 
 
 32 
 
 29 
 
 37355 
 
 2.67700 
 
 39357 
 
 2.54082 
 
 41387 
 
 2.41620 
 
 43447 
 
 2.30167 
 
 31 
 
 30 
 
 37388 
 
 2.67462 
 
 39391 
 
 2.53865 
 
 41421 
 
 2.41421 
 
 43481 
 
 2.29984 
 
 30 
 
 31 
 
 37422 
 
 2.67225 
 
 39425 
 
 2.53648 
 
 41455 
 
 2.41223 
 
 43516 
 
 2.29801 
 
 29 
 
 32 
 
 37455 
 
 2.66989 
 
 39458 
 
 2.53432 
 
 41490 
 
 2.41025 
 
 43550 
 
 2.29619 
 
 28 
 
 33 
 
 37488 
 
 2.66752 
 
 39492 
 
 2.53217 
 
 41524 
 
 2.40827 
 
 43585 
 
 2.29437 
 
 27 
 
 34 
 
 37521 
 
 2.66516 
 
 39526 
 
 2.53001 
 
 41558 
 
 2.40829 
 
 43620 
 
 2.29254 
 
 26 
 
 35 
 
 37554 
 
 2.66281 
 
 39559 
 
 2.52786 
 
 41592 
 
 2.40432 
 
 43654 
 
 2.29073 
 
 25 
 
 30 
 
 37588 
 
 2.66046 
 
 39593 
 
 2.52571 
 
 41626 
 
 2.40235 
 
 43689 
 
 2.28891 
 
 21 
 
 37 
 
 37621 
 
 2.65811 
 
 39628 
 
 2.52357 
 
 41660 
 
 2.40038 
 
 43724 
 
 2.28710 
 
 23 
 
 38 
 
 37654 
 
 2.65576 
 
 39660 
 
 2.52142 
 
 41694 
 
 2.39841 
 
 43758 
 
 2.28528 
 
 2'2 
 
 39 
 
 37687 
 
 2.65342 
 
 39694 
 
 2.51959 
 
 41728 
 
 2.39645 
 
 43793 
 
 2.28348 21 
 
 40 
 
 37720 
 
 2.65109 
 
 39727 
 
 2.51715 
 
 41763 
 
 2.39449 
 
 43828 
 
 2.28167 
 
 20 
 
 41 
 
 37754 
 
 2.64875 
 
 39761 
 
 2.51502 
 
 41797 
 
 2.39253 
 
 43862 
 
 2.27987 
 
 19 
 
 42 
 
 37787 
 
 2.64642 
 
 39795 
 
 2.51289 
 
 41831 
 
 2.39058 
 
 43897 
 
 2.27806 
 
 18 
 
 43 
 
 37820 
 
 2.64410 
 
 39829 
 
 2.51076 
 
 418G5 
 
 2.38863 
 
 43932 
 
 2.27626 
 
 17 
 
 44 
 
 37853 
 
 2.64177 
 
 39862 
 
 2.50864 
 
 41899 
 
 2.38668 
 
 43966 
 
 2.27447 
 
 16 
 
 45 
 
 37887 
 
 2.63945 
 
 39S96 
 
 2.50652 
 
 41933 
 
 2.38473 
 
 44001 
 
 2.27267 
 
 15 
 
 46 
 
 37920 
 
 2.63714 
 
 39930 
 
 2.50440 
 
 41968 
 
 2.38279 
 
 44036 
 
 2.27088 
 
 14 
 
 47 
 
 37953 
 
 2.63483 
 
 39963 
 
 2.50229 
 
 42002 
 
 2.38084 
 
 44071 
 
 2.26909 
 
 13 
 
 48 
 
 37986 
 
 2.63252 
 
 39997 
 
 2.50018 
 
 42036 
 
 2.37891 
 
 44105 
 
 2.26730 
 
 12 
 
 49 
 
 38020 
 
 2.63021 
 
 40031 
 
 2.49807 
 
 42070 
 
 2.37697 
 
 44140 
 
 2.26552 
 
 11 
 
 50 
 
 38053 
 
 2.62791 
 
 40065 
 
 2.49597 
 
 42105 
 
 2.37504 
 
 44175 
 
 2.26374 
 
 10 
 
 51 
 
 38086 
 
 2.62561 
 
 40098 
 
 2.49386 
 
 42139 
 
 2.37311 
 
 44210 
 
 3.26196 
 
 9 
 
 52 
 
 38120 
 
 2.62332 
 
 40132 
 
 2.49177 
 
 42173 
 
 2.37118 
 
 44244 
 
 2.26018 
 
 8 
 
 53 
 
 38153 
 
 2.62103 
 
 40166 
 
 2.48967 
 
 42207 
 
 2.36925 
 
 44279 
 
 2.25840 
 
 7 
 
 54 
 
 38186 
 
 2.61874 
 
 40200 
 
 2.48758 
 
 42242 
 
 2.36733 
 
 44314 
 
 2.25663 
 
 G 
 
 55 
 
 38220 
 
 2.61646 
 
 40234 
 
 2.48549 
 
 42276 
 
 2.36541 
 
 44349 
 
 2.25486 
 
 5 
 
 56 
 
 38253 
 
 2.61418 
 
 40267 
 
 2 48340 
 
 42310 
 
 2.36349 
 
 44384 
 
 2.25309 
 
 4 
 
 57 
 
 38286 
 
 2.61190 
 
 40301 
 
 2.48132 
 
 42345 
 
 2.36158 
 
 44418 
 
 2.25132 
 
 3 
 
 58 
 
 38320 
 
 2.60963 
 
 40335 
 
 2.47924 
 
 42379 
 
 2.35967 
 
 44453 
 
 2.24956 
 
 2 
 
 59 
 
 38353 
 
 2.60736 
 
 40369 
 
 2.47716 
 
 42413 
 
 2.35776 
 
 44488 
 
 2.24780 
 
 1 
 
 GO 
 
 38386 
 
 2. 60509 
 
 40403 
 
 2.47509 
 
 42447 
 
 2.35585 
 
 44523 
 
 2.24604 
 
 
 
 / 
 
 Cotang 
 
 "Tang ' 
 
 Cotang' 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 / 
 
 1 69 
 
 68 
 
 67 
 
 66 
 
 
 240 
 
'TABLE XII. TANGENTS AND COTANGENTS. 
 
 
 24 
 
 25 26 
 
 27 
 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 
 
 44523 
 
 2.24604 
 
 4668T 
 
 2.1445T 
 
 48773 
 
 2.05030 
 
 50953 
 
 1.96261 
 
 GO 
 
 1 
 
 41558 
 
 2.24428 
 
 46GG6 
 
 2.14288 
 
 48809 
 
 2.04879 
 
 50989 
 
 1.96120 
 
 59 
 
 2 
 
 44593 
 
 2.24252 
 
 46702 
 
 2.14125 
 
 48845 
 
 2.04728 
 
 51026 
 
 1.95979 
 
 58 
 
 3 
 
 44627 
 
 2.24077 
 
 46737 
 
 2.139G3 
 
 48881 
 
 2.04577 
 
 51063 
 
 1.95838 
 
 57 
 
 4 
 
 446G2 
 
 2.23902 
 
 46772 
 
 2.13801 
 
 48917 
 
 2.04426 
 
 51099 
 
 1.95698 56 
 
 5 
 
 44G97 
 
 o 03707 
 
 46S08 
 
 2.13639 
 
 48953 
 
 2.04276 
 
 51136 
 
 1.95557 !55 
 
 
 
 44732 
 
 2^23553 
 
 46843 
 
 2.13477 
 
 48989 
 
 2.04125 
 
 51173 
 
 1.95417 54 
 
 7 
 
 44767 
 
 2.23378 
 
 46879 
 
 2.13316 
 
 49026 
 
 2.03975 
 
 51209 
 
 1.95277 53 
 
 8 
 
 44802 
 
 2.23204 
 
 46914 
 
 2.13154 
 
 49062 
 
 2.03825 
 
 51246 
 
 1.95137 
 
 52 
 
 9 
 
 44837 
 
 2.23030 
 
 46950 
 
 2.12993 
 
 49098 
 
 2.03675 
 
 51283 
 
 1.94997 
 
 51 
 
 10 
 
 44872 
 
 2.22857 
 
 46985 
 
 2.12833 
 
 49134 
 
 2.03526 
 
 51319 
 
 1.94858 
 
 50 
 
 11 
 
 44907 
 
 2.22683 
 
 47021 
 
 2.12671 
 
 49170 
 
 2.03376 
 
 51356 
 
 1.94718 
 
 49 
 
 18 
 
 44942 
 
 2.22510 
 
 47056 
 
 2.12511 
 
 49206 
 
 2.03227 
 
 51393 
 
 1.94579 
 
 48 
 
 13 
 
 44977 
 
 2.22337 
 
 47092 
 
 2.12350 
 
 49242 
 
 2.03078 
 
 51430 
 
 1.94440 
 
 47 
 
 14 
 
 45012 
 
 2.22164 
 
 47128 
 
 2.12190 
 
 49278 
 
 2.02929 
 
 51467 
 
 1.94301 
 
 46 
 
 15 
 
 45047 
 
 2.21992 
 
 47163 
 
 2.12030 
 
 49315 
 
 2.02780 
 
 51503 
 
 1.94162 
 
 45 
 
 1C 
 
 45082 
 
 2.21819 
 
 47199 
 
 2.11871 
 
 49351 
 
 2.02631 
 
 51540 
 
 1.94023 
 
 14 
 
 17 
 
 45117 
 
 2.21647 
 
 47234 
 
 2.11711 
 
 49387 
 
 2.02483 
 
 51577 
 
 1.93885 
 
 43 
 
 18 
 
 45152 
 
 2.21475 
 
 47270 
 
 2.11552 
 
 49423 
 
 2.02335 
 
 51614 
 
 1.93746 
 
 42 
 
 19 
 
 45187 
 
 2.21304 
 
 47305 
 
 2.11392 
 
 49459 
 
 2.02187 
 
 51G51 
 
 1.93608 
 
 41 
 
 20 
 
 45222 
 
 2.21132 
 
 47341 
 
 2.11233 
 
 49495 
 
 2.02039 
 
 51688 
 
 1.93470 
 
 40 
 
 21 
 
 45257 
 
 2.20961 
 
 47377 
 
 2.11075 
 
 49532 
 
 "2.01891 
 
 51724 
 
 1.93332 
 
 39 
 
 22 
 
 45292 
 
 2.20790 
 
 47412 
 
 2.10916 
 
 49568 
 
 2.01743 
 
 51761 
 
 1.93195 
 
 38 
 
 23 
 
 45327 
 
 2.20619 
 
 47448 
 
 2.10758 
 
 49G04 
 
 2.01596 
 
 51798 
 
 1.93057 
 
 37 
 
 24 
 
 45362 
 
 2.20449 
 
 47483 
 
 2.10600 
 
 49640 
 
 2.01449 
 
 51835 
 
 1.92920 
 
 36 
 
 25 
 
 45397 
 
 2.20278 
 
 47519 
 
 2.10442 
 
 49677 
 
 2.01302 
 
 51872 
 
 1.92782 
 
 35 
 
 26 
 
 45432 
 
 2.20108 
 
 47555 
 
 2.10284 
 
 49713 
 
 2.01155 
 
 51909 
 
 1.92645 
 
 34 
 
 27 
 
 45467 
 
 2.19938 
 
 47590 
 
 2.10126 
 
 49749 
 
 2.01008 
 
 51946 
 
 1.92508 
 
 33 
 
 X 
 
 45502 
 
 2.19769 
 
 47G26 
 
 2.09969 
 
 49786 
 
 2.008G2 
 
 51983 
 
 1.92371 
 
 32 
 
 29 
 
 45538 
 
 2.19599 
 
 47G62 
 
 2.09811 
 
 49822 
 
 2.00715 
 
 52020 
 
 1.92235 
 
 31 
 
 30 
 
 45573 
 
 2.19430 
 
 47698 
 
 2.09654 
 
 49858 
 
 2.00569 
 
 52057 
 
 1.92098 
 
 30 
 
 31 
 
 45608 
 
 2.19261 
 
 47733 
 
 2.09498 
 
 49894 
 
 2.00423 
 
 52094 
 
 1.91962 
 
 29 
 
 32 
 
 45643 
 
 2.19092 
 
 47769 
 
 2.09341 
 
 49931 
 
 2.00277 
 
 52131 
 
 1.91826 
 
 28 
 
 33 
 
 45678 
 
 2.18923 
 
 47805 
 
 2.09184 
 
 49967 
 
 2.00131 
 
 52168 
 
 1.91690 
 
 27 
 
 34 
 
 45713 
 
 2.18755 
 
 47840 
 
 2.09028 
 
 50004 
 
 1.99986 
 
 52205 
 
 1.91554 
 
 2G 
 
 35 
 
 45748 
 
 2.18587 
 
 47876 
 
 2.08872 
 
 50040 
 
 1.99841 
 
 52242 
 
 1.91418 
 
 25 
 
 36 
 
 45784 
 
 2.18419 
 
 47912 
 
 2.08716 
 
 50076 
 
 1.99695 
 
 52279 
 
 1.91282 
 
 24 
 
 37 
 
 45819 
 
 2.18251 
 
 47948 
 
 2.08560 
 
 50113 
 
 1.99550 
 
 52316 
 
 1.91147 
 
 23 
 
 38 
 
 45854 
 
 2.18084 
 
 47984 
 
 2.03405 
 
 50149 
 
 1.99406 
 
 52353 
 
 1.91012 
 
 22 
 
 39 
 
 45889 
 
 2.17916 
 
 48019 
 
 2.08250 
 
 50185 
 
 1.99261 
 
 52390 
 
 1.90876 
 
 21 
 
 40 
 
 45924 
 
 2.17749 
 
 48055 
 
 2.08094 
 
 50222 
 
 1.99116 
 
 52427 
 
 1.90741 
 
 20 
 
 41 
 
 45960 
 
 2.17.582 
 
 48091 
 
 2.07939 
 
 50258 
 
 1.98972 
 
 52464 
 
 1.90607 
 
 19 
 
 42 
 
 45995 
 
 2.17416 
 
 48127 
 
 2.07785 
 
 50295 
 
 1.98828 
 
 52501 
 
 1.90472 
 
 18 
 
 43 
 
 46030 
 
 2.17249 
 
 48163 
 
 2.07G30 
 
 50331 
 
 1.98684 
 
 52538 
 
 1.90337 
 
 17 
 
 44 
 
 46065 
 
 2.17083 
 
 48198 
 
 2.07476 
 
 50368 
 
 1.98540 
 
 52575 
 
 1.90203 
 
 16 
 
 45 
 
 46101 
 
 2.16917 
 
 48234 
 
 2.07321 
 
 50404 
 
 1.98396 
 
 52613 
 
 1.90069 
 
 15 
 
 46 
 
 46136 
 
 2.16751 
 
 48270 
 
 2.07167 
 
 50441 
 
 1.98253 
 
 52650 
 
 1.89935 
 
 14 
 
 47 
 
 46171 
 
 2.16585 
 
 48306 
 
 2.07014 
 
 50477 
 
 1.98110 
 
 52687 
 
 1.89801 
 
 13 
 
 48 
 
 46206 
 
 2.16420 
 
 48342 
 
 2.06860 
 
 50514 
 
 1.97'966 
 
 52724 
 
 1.89667 
 
 12 
 
 49 46242 
 
 2.16255 
 
 48378 
 
 2.06706 
 
 50550 
 
 1.97823 
 
 52761 
 
 1.89533 
 
 11 
 
 50 46277 
 
 2.1G090 
 
 48414 
 
 2.06553 
 
 50587 
 
 1.97681 
 
 52798 
 
 1.89400 
 
 10 
 
 51 
 
 46312 
 
 2.15925 
 
 48450 
 
 2.06400 
 
 50623 
 
 1.97538 
 
 52836 
 
 1.8926t> 
 
 9 
 
 B2 
 
 46348 
 
 2.15760 
 
 48486 
 
 2.06247 
 
 50GGO 
 
 1.97395 
 
 52873 
 
 1.89133 
 
 8 
 
 53 
 
 46383 
 
 2.15596 
 
 48521 
 
 2.06094 
 
 50696 
 
 1.97253 
 
 52910 
 
 1.89000 
 
 7 
 
 54 
 
 46418 
 
 2.15432 
 
 48557 
 
 2.05942 
 
 50733 
 
 1.97111 
 
 52947 
 
 1.88867 
 
 6 
 
 55 
 
 46454 
 
 2.15268 
 
 48593 
 
 2.05790 
 
 50769 
 
 1.96969 
 
 52985 
 
 1.88734 
 
 5 
 
 56 
 
 46489 
 
 2.15104 
 
 48629 
 
 2.05637 
 
 50806 
 
 1.96827 
 
 53022 
 
 1.88602 
 
 4 
 
 57 
 
 46525 
 
 2.14940 
 
 48G65 
 
 2.05485 
 
 50843 
 
 1.96685 
 
 53059 
 
 1.88469 
 
 3 
 
 58 
 
 46560 
 
 2.14777 
 
 4S701 
 
 2.05333 
 
 50879 
 
 1.96544 
 
 53096 
 
 1.88337 
 
 2 
 
 59 46595 
 
 2.14614 
 
 48737 
 
 2.05182 
 
 50916 
 
 1.96402 
 
 53134 
 
 1.88205 
 
 1 
 
 GO 
 
 46631 
 
 2.14451 
 
 48773 
 
 2.05030 
 
 50953 
 
 1.96261 
 
 53171 
 
 1.88073 
 
 
 
 / 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 / 
 
 
 65 
 
 64 
 
 63 
 
 62 
 
 
 241 
 
TABLE XII. TANGENTS AND COTANGENTS. 
 
 
 28 
 
 29 
 
 30 
 
 31 
 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 
 
 53171 
 
 1.88073 
 
 55431 
 
 1.80405 
 
 57735 
 
 1.73205 
 
 60086 
 
 1.66428" 
 
 60 
 
 1 
 
 53208 
 
 1.87941 
 
 55469 
 
 1.80281 
 
 57774 
 
 1.73089 
 
 60126 
 
 1.66318 
 
 59 
 
 2 
 
 53246 
 
 1.87809 
 
 55507 
 
 1.80158 
 
 57813 
 
 1.72973 
 
 60165 
 
 1.66209 
 
 58 
 
 3 
 
 53283 
 
 1.87677 
 
 55545 
 
 1.80034 
 
 57851 
 
 1.72857 
 
 60205 
 
 1.G6099 
 
 57 
 
 4 
 
 53320 
 
 1.87'546 
 
 55583 
 
 1.79911 
 
 57890 
 
 1.72741 
 
 60245 
 
 1.G5990 
 
 56 
 
 5 
 
 53358 
 
 1.87415 
 
 55621 
 
 1.79788 
 
 57929 
 
 1.72625 
 
 60284 
 
 1.65881 
 
 55 
 
 6 
 
 53395 
 
 1.87283 
 
 55659 
 
 1.79665 
 
 57968 
 
 1.72509 
 
 60324 
 
 1.65772 
 
 54 
 
 7 
 
 53432 
 
 1.87152 
 
 55697 
 
 1.79542 
 
 58007 
 
 1.72393 
 
 60364 
 
 1.65663 
 
 53 
 
 8 
 
 53470 
 
 1.87021 
 
 55736 
 
 1.79419 
 
 58046 
 
 1.72278 
 
 60403 
 
 1.65554 
 
 52 
 
 9 
 
 53507 
 
 1.86891 
 
 55774 
 
 1.79296 
 
 58085 
 
 1.72163 
 
 60443 
 
 1.G5445 
 
 51 
 
 10 
 
 53545 
 
 1.86760 
 
 55812 
 
 1.79174 
 
 58124 
 
 1.72047 
 
 60483 
 
 1.65337 
 
 50 
 
 11 
 
 53582 
 
 1.86630 
 
 55850 
 
 1.79051 
 
 58162 
 
 1.71932 
 
 60522 
 
 1.65228 
 
 49 
 
 12 
 
 53620 
 
 1.86499 
 
 55888 
 
 1.78929 
 
 58201 
 
 1.71817 
 
 60562 
 
 1.65120 
 
 48 
 
 18 
 
 53657 
 
 1.86369 
 
 55926 
 
 1.18807 
 
 58240 
 
 1.71702 
 
 60602 
 
 1.65011 
 
 47 
 
 14 
 
 53694 
 
 1.86239 
 
 55964 
 
 1.78685 
 
 58279 
 
 1.71588 
 
 60642 
 
 1.64903 
 
 46 
 
 15 
 
 53732 
 
 1.86109 
 
 56003 
 
 1.78563 
 
 58318 
 
 1.71473 
 
 60681 
 
 1.64795 
 
 45 
 
 16 
 
 53769 
 
 1.85979 
 
 56041 
 
 1.78441 
 
 58357 
 
 1.71358 
 
 607'21 
 
 1.64687 
 
 44 
 
 17 
 
 53807 
 
 1.85850 
 
 56079 
 
 1.78319 
 
 58896 
 
 1.71244 
 
 60761 
 
 1.64579 
 
 4 
 
 18 
 
 53844 
 
 1.85720 
 
 56117 
 
 1.78198 
 
 58435 
 
 1.71129 
 
 60801 
 
 1.G4471 
 
 42 
 
 19 
 
 53882 
 
 1.85591 
 
 56156 
 
 1.78077 
 
 58-174 
 
 1.71015 
 
 60841 
 
 1.G4363 
 
 41 
 
 20 
 
 53920 
 
 1.85462 
 
 56194 
 
 1.77955 
 
 58513 
 
 1.70901 
 
 60861 
 
 1.G425G 
 
 40 
 
 21 
 
 53957 
 
 1.85333 
 
 56232 
 
 1.77834 
 
 58552 
 
 1.70787 
 
 GOP21 
 
 1.64148 
 
 39 
 
 22 
 
 53995 
 
 1.85204 
 
 56270 
 
 1.77713 
 
 58591 
 
 1.70673 
 
 60960 
 
 1.64041 
 
 38 
 
 23 
 
 54032 
 
 1.85075 
 
 56309 
 
 1.77592 
 
 58631 
 
 1.70560 
 
 61000 
 
 1.63934 
 
 37 
 
 21 
 
 54070 
 
 1.84946 
 
 561347 
 
 1.77471 
 
 5867'0 
 
 1.70446 
 
 61040 
 
 1.63826 
 
 3G 
 
 25 
 
 54107 
 
 1.84818 
 
 56385 
 
 1.77351 
 
 58709 
 
 1.70332 
 
 61080 
 
 1.63719 
 
 35 
 
 26 
 
 54145 
 
 1.81089 
 
 56424 
 
 1.77230 
 
 58748 
 
 1.70219 
 
 61120 
 
 1.63612 
 
 34 
 
 27 
 
 54183 
 
 1.84561 
 
 56462 
 
 1.77110 
 
 58787 
 
 1.70106 
 
 61160 
 
 1.63505 
 
 33 
 
 28 
 
 54220 
 
 1.84433 
 
 56501 
 
 1.76990 
 
 58826 
 
 1.69992 
 
 61200 
 
 i leases 
 
 32 
 
 29 
 
 54258 
 
 1.84305 
 
 5G539 
 
 1.76869 
 
 588C5 
 
 1 .C9879 
 
 61240 
 
 1.63292 
 
 31 
 
 30 
 
 54296 
 
 1.84177 
 
 56577 
 
 1.76749 
 
 58905 
 
 1.697G6 
 
 61280 
 
 1.63185 
 
 30 
 
 31 
 
 54333 
 
 1.84049 
 
 56616 
 
 1.76629 
 
 58944 
 
 1.69G53 
 
 61320 
 
 1.G3079 
 
 29 
 
 32 
 
 54371 
 
 1.83922 
 
 56654 
 
 1.76510 
 
 58083 
 
 1.G9541 
 
 61360 
 
 1.62972 
 
 28 
 
 33 
 
 54409 
 
 1.83794 
 
 56693 
 
 1.76390 
 
 59022 
 
 1.69428 
 
 61400 
 
 1.G28G6 
 
 27 
 
 34 
 
 54446 
 
 1.83667 
 
 56731 
 
 1.76271 
 
 59061 
 
 1.69310 
 
 61440 
 
 1.62760 
 
 2G 
 
 35 
 
 54484 
 
 1.83540 
 
 56769 
 
 1.76151 
 
 59101 
 
 1.69203 
 
 61480 
 
 1.62654 
 
 25 
 
 30 
 
 54522 
 
 1.83413 
 
 56808 
 
 1.76032 
 
 59140 
 
 1.69091 
 
 61520 
 
 1.62548 
 
 24 
 
 37 
 
 54560 
 
 1.83286 
 
 56846 
 
 1.75913 
 
 59179 
 
 1.68979 
 
 61561 
 
 1.62442 
 
 23 
 
 38 
 
 54597 
 
 1.83159 
 
 58885 
 
 1.75794 
 
 59218 
 
 1.68866 
 
 61601 
 
 l! 68886 
 
 22 
 
 39 
 
 54635 
 
 1.88033 
 
 56923 
 
 1.75G75 
 
 59258 
 
 1.68754 
 
 61641 
 
 1.62230 
 
 21 
 
 40 
 
 54673 
 
 1.82906 
 
 56962 
 
 1.75556 
 
 59297 
 
 1.68643 
 
 61681 
 
 1.62125 
 
 2U 
 
 41 
 
 54711 
 
 1.82780 
 
 57000 
 
 1.75437 
 
 59336 
 
 1.68531 
 
 61721 
 
 1.62019 
 
 19 
 
 42 
 
 54748 
 
 1.82654 
 
 57039 
 
 1.75319 
 
 59376 
 
 1.68419 
 
 61761 
 
 1.61914 
 
 18 
 
 43 
 
 54786 
 
 1.82528 
 
 57078 
 
 1.75200 
 
 59415 
 
 1.68308 
 
 61801 
 
 1.61808 
 
 17 
 
 44 
 
 54824 
 
 1.82402 
 
 57116 
 
 1.75082 
 
 59454 
 
 1.68196 
 
 61842 
 
 1.61703 
 
 1G 
 
 4* 
 
 54862 
 
 1.82276 
 
 57155 
 
 1.74964 
 
 59494 
 
 1.68085 
 
 61882 
 
 1.61598 
 
 15 
 
 4G 
 
 54900 
 
 1.82150 
 
 57193 
 
 1.74846 
 
 59533 
 
 1.67974 
 
 61922 
 
 1.61493 
 
 14 
 
 47 
 
 54938 
 
 1.82025 
 
 57232 
 
 1.74728 
 
 59573 
 
 1.67863 
 
 61962 
 
 1.61388 
 
 13 
 
 48 
 
 54975 
 
 1.81899 
 
 57271 
 
 1.74G10 
 
 59612 
 
 1.G7752 
 
 62003 
 
 1.61283 
 
 12 
 
 49 
 
 S6013 
 
 1.81774 
 
 57309 
 
 1.74492 
 
 59651 
 
 1.G7641 
 
 62043 
 
 1.61179 ill 
 
 50 
 
 55051 
 
 1.81649 
 
 57348 
 
 1.74375 
 
 59691 
 
 1.07530 
 
 62083 
 
 1.61074 
 
 10 
 
 51 
 
 55089 
 
 1.81524 
 
 57386 
 
 1.74257 
 
 59730 
 
 1.67419 
 
 62124 
 
 1.60970 
 
 9 
 
 52 
 
 55127 
 
 1.81399 
 
 57425 
 
 1.74140 
 
 59770 
 
 1.67309 
 
 62164 
 
 1.60865 
 
 8 
 
 53 
 
 55165 
 
 1.81274 
 
 57464 
 
 1.74022 
 
 59809 
 
 1.67198 
 
 62204 
 
 1.60761 
 
 7 
 
 54 
 
 55203 
 
 1.81150 
 
 57503 
 
 1.73905 
 
 59849 
 
 1.67088 
 
 62245 
 
 1.60657 
 
 6 
 
 55 
 
 55241 
 
 1.81025 
 
 57541 
 
 1.73788 
 
 59888 
 
 1.66978 
 
 62285 
 
 1.60553 
 
 5 
 
 56 
 
 55279 
 
 1.80901 
 
 57580 
 
 1.73671 
 
 59928 
 
 1.66867 
 
 62325 
 
 1.60449 
 
 4 
 
 57 
 
 55317 
 
 1.80777 
 
 57619 
 
 1.73555 
 
 59967 
 
 1.66757 
 
 623G6 
 
 1.60345 
 
 3 
 
 58 
 
 55355 
 
 1.80653 
 
 57657 
 
 1.73438 
 
 60007 
 
 1.66647 
 
 62406 
 
 1.60241 
 
 2 
 
 59 
 
 55393 
 
 1.80529 
 
 57696 
 
 1.73321 
 
 60046 
 
 1.66538 
 
 62446 
 
 1.60137 
 
 1 
 
 60 
 
 55431 
 
 1.80405 
 
 57735 
 
 1.73205 
 
 .60086 
 
 1.66428 
 
 62487 
 
 1.60033 
 
 
 
 / 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 
 
 61 
 
 o 
 
 60 
 
 59' 
 
 68 
 
TABLE XII. TANGENTS AND COTANGENTS. 
 
 
 32 
 
 o 
 
 33 
 
 o . 
 
 34 
 
 35 
 
 
 
 
 Tang Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang Cotang 
 
 
 
 
 62487 
 
 1.60033' 
 
 64941 
 
 1.52986 
 
 67451 
 
 1.48256 
 
 70021 
 
 1.42815 
 
 
 
 
 62527 
 
 1.59930 
 
 64982 
 
 1.53888 
 
 67493 
 
 1.48163 
 
 70064 
 
 1.42726 
 
 9 
 
 2 
 
 62568 
 
 1.59826 
 
 65024 
 
 1.53791 
 
 67536 
 
 1.48070 
 
 70107 
 
 1.42638 
 
 8 
 
 3 
 
 62608 
 
 1.59723 
 
 65065 
 
 1.53693 
 
 67578 
 
 1.47977 
 
 70151 
 
 1.42550 
 
 7 
 
 4 
 
 62649 
 
 1.59620 
 
 65106 
 
 1.53595 
 
 67620 
 
 1.47885 
 
 70194" 
 
 1.42462 
 
 6 
 
 
 62689 
 
 1.59517 
 
 65148 
 
 1.53497 
 
 67G63 
 
 1.47792 
 
 70238 
 
 1.42374 
 
 5 
 
 5 
 
 62730 
 
 1.59414 
 
 65189 
 
 1.53400 
 
 67705 
 
 1.47699 
 
 70281 
 
 1.42286 
 
 4 
 
 7 
 
 62770 
 
 1.59311 
 
 65231 
 
 1.53302 
 
 67748 
 
 1.47607 
 
 70325 
 
 1.42198 
 
 3 
 
 8 
 
 62811 
 
 1.59208 
 
 65272 
 
 1.53205 
 
 67790 
 
 1.47514 
 
 703C8 
 
 1.42110 
 
 2 
 
 9 
 
 62852 
 
 1.59105 
 
 65314 
 
 1.53107 
 
 67832 
 
 1.47422 
 
 70412 
 
 1.42022 
 
 1 
 
 10 
 
 62892 
 
 1.59002 
 
 65355 
 
 1.53010 
 
 67875 
 
 1.47330 
 
 70455 
 
 1.41934 
 
 
 
 1 
 
 629S3 
 
 1.58900 
 
 65397 
 
 1.52913 
 
 67917 
 
 1.47238 
 
 70499 
 
 1.41847 
 
 9 
 
 
 62973 
 
 1.58797 
 
 65438 
 
 1.52816 
 
 G79GO 
 
 1.47146 
 
 70542 
 
 1.41759 
 
 8 
 
 3 
 
 63014 
 
 1.58695 
 
 65480 
 
 1.52719 
 
 68002 
 
 1.47053 
 
 70586 
 
 1.41672 
 
 7 
 
 14 
 
 63055 
 
 1.58593 
 
 65521 
 
 1.52622 
 
 68045 
 
 1.46962 
 
 70629 
 
 1.41584 
 
 6 
 
 15 
 
 63095 
 
 1.58490 
 
 65563 
 
 1.52525 
 
 68088 
 
 1.46870 
 
 70673 
 
 1.41497 
 
 45 
 
 6 
 
 63136 
 
 1.58388 
 
 65G04 
 
 1.52429 
 
 68130 
 
 1.46778 
 
 70717 
 
 1.41409 
 
 44 
 
 
 63177 
 
 1.58286 
 
 65646 
 
 1.52332 
 
 68173 
 
 1.46686 
 
 707GO 
 
 1.41322 
 
 43 
 
 8 
 
 63217 
 
 1.58184 
 
 65688 
 
 1.52235 
 
 68215 
 
 1.46595 
 
 70804 
 
 1.41235 
 
 
 
 19 
 
 63258 
 
 1.58083 
 
 65729 
 
 1.52139 
 
 68258 
 
 1.46503 
 
 70S43 
 
 1.41148 
 
 41 
 
 20 
 
 63299 
 
 1.57981 
 
 65771 
 
 1.52043 
 
 68301 
 
 1.46411 
 
 70891 
 
 1 .41061 
 
 40 
 
 21 
 
 63340 
 
 1.57879 
 
 65813 
 
 1.51946 
 
 68343 
 
 1.46320 
 
 70935 
 
 1.40974 
 
 39 
 
 22 
 
 633SO 
 
 1.57778 
 
 65854 
 
 1.51850 
 
 68386 
 
 1.46229 
 
 70979 
 
 1.40887 
 
 38 
 
 23 
 
 63421 
 
 1.57676 
 
 65896 
 
 1.51754 
 
 68429 
 
 1.46137 
 
 71023 
 
 1.40800 
 
 37 
 
 24 
 
 63462 
 
 1.57575 
 
 65938 
 
 1.51658 
 
 68471 
 
 1.46046 
 
 71066 
 
 1.40714 
 
 36 
 
 25 
 
 63503 
 
 1.57474 
 
 65980 
 
 1.51562 
 
 68514 
 
 1.45955! 
 
 71110 
 
 1.40627 
 
 35 
 
 26 
 
 63544 
 
 1.57372 
 
 66021 
 
 1.51466 
 
 68557 
 
 1. 45864 / 
 
 71154 
 
 1.40540 
 
 >4 
 
 27 
 
 63584 
 
 1.57271 
 
 66063 
 
 1.51370 
 
 68600 
 
 1.45773 
 
 71198 
 
 1.40454 
 
 53 
 
 28 
 
 63625 
 
 1.57170 
 
 66105 
 
 1.51275 
 
 68642 
 
 1.45682^ 
 
 71242 
 
 1.40367 
 
 .'1:2 
 
 23 
 
 63666 
 
 1.57069 
 
 66147 
 
 1.51179 
 
 68G85 
 
 1.45592 
 
 71285 
 
 1.40281 
 
 31 
 
 30 
 
 63707 
 
 1.56969 
 
 66189 
 
 1.51084 
 
 68728 
 
 1.45501 
 
 71329 
 
 1.40195 
 
 30 
 
 31 
 
 63748 
 
 1.56868 
 
 66230 
 
 1.50988 
 
 68771 
 
 1.45410 
 
 71373 
 
 1.40109 
 
 29 
 
 o 
 
 63789 
 
 1.56767 
 
 66272 
 
 1.50893 
 
 68814 
 
 1.45320 
 
 71417 
 
 1.40022 
 
 a 
 
 33 
 
 63830 
 
 1.56667 
 
 66314 
 
 1.50797 
 
 68857 
 
 1.45229 
 
 71461 
 
 1.89936 
 
 27 
 
 34 
 
 63871 
 
 1.56566 
 
 66356 
 
 1.50702 
 
 68900 
 
 1.45139 
 
 71505 
 
 1.39850 
 
 30 
 
 35 
 
 63912 
 
 1.56466 
 
 66398 
 
 1.50607 
 
 68942 
 
 1.45049 
 
 71549 
 
 1.39764 
 
 25 
 
 36 
 
 63953 
 
 1.56366 
 
 66440 
 
 1.50512 
 
 68985 
 
 1.44958 
 
 71593 
 
 1.39679 
 
 24 
 
 ft 
 
 63994 
 
 1.56265 
 
 66482 
 
 1.50417 
 
 69028 
 
 1.44868 
 
 71637 
 
 1.39593 
 
 03 
 
 oc 
 
 64035 
 
 1.56165 
 
 66524 
 
 1.50322 
 
 69071 
 
 1.44778 
 
 71681 
 
 1.39507 
 
 8 
 
 >9 
 
 64076 
 
 1.56065 
 
 665G6 
 
 1.50228 
 
 69114 
 
 1.44688 
 
 71725 
 
 1.39421 
 
 
 10 
 
 64117 
 
 1.55966 
 
 66608 
 
 1.50133 
 
 69157 
 
 1.44598 
 
 71769 
 
 1.39336 
 
 20 
 
 41 
 
 4158 
 
 1.55866 
 
 66650 
 
 1.50038 
 
 69200 
 
 1.44508 
 
 71813 
 
 1.39250 
 
 19 
 
 42 
 
 64199 
 
 1.55766 
 
 66692 
 
 1.49944 
 
 69243 
 
 1.44418 
 
 71857 
 
 1.39165 
 
 18 
 
 43 
 
 64240 
 
 1.55666 
 
 66734 
 
 1.49849 
 
 69286 
 
 1.44329 
 
 71901 
 
 1.39079 
 
 17 
 
 44 
 
 64281 
 
 1.55567 
 
 66776 
 
 1.49755 
 
 69329 
 
 1.44239 
 
 71946 
 
 1.38994 
 
 1G 
 
 45 
 
 64322 
 
 1.55467 
 
 66818 
 
 1.49G61 
 
 69372 
 
 1.44149 
 
 71990 
 
 1.38909 
 
 15 
 
 46 
 
 64363 
 
 1.55363 
 
 66860 
 
 1.49566 
 
 69416 
 
 1.44060 
 
 72034 
 
 1.38824 
 
 14 
 
 47 
 
 64404 
 
 1.55269 
 
 66902 
 
 1.49472 
 
 69459 
 
 1.43970 
 
 72078 
 
 1.38738 
 
 13 
 
 48 
 
 64446 
 
 1.55170 
 
 66944 
 
 1.49378 
 
 69502 
 
 1.43881 
 
 72122 
 
 1.38653 
 
 12 
 
 49 
 
 64487 
 
 1.55071 
 
 66986 
 
 1.49284 
 
 69545 
 
 1.43792 
 
 72167 
 
 1.38568 
 
 11 
 
 50 
 
 64528 
 
 1.54972 
 
 67023 
 
 1.49190 
 
 69588 
 
 1.43703 
 
 72211 
 
 1.38484 
 
 10 
 
 5 
 
 64569 
 
 1.54873 
 
 67071 
 
 1.49097 
 
 69631 
 
 1.43614 
 
 72255 
 
 1.3a399 
 
 9 
 
 5; 
 
 64610 
 
 1.54774 
 
 67113 
 
 1.49003 
 
 69G75 
 
 1.43525 
 
 72299 
 
 1.38314 
 
 8 
 
 5! 
 
 64652 
 
 1.54G75 
 
 67155 
 
 1.48909 
 
 69718 
 
 1.43436 
 
 72344 
 
 1.38229 
 
 7 
 
 54 
 
 64693 
 
 1.54576 
 
 67197 
 
 1.48816 
 
 69761 
 
 1.43347 
 
 72388 
 
 1.38145 
 
 6 
 
 5 
 
 64734 
 
 1.54478 
 
 67239 
 
 1.48722 
 
 69804 
 
 1.43258 
 
 72432 
 
 1.38060 
 
 5 
 
 5 
 
 64775 
 
 1.54379 
 
 87883 
 
 1.48G29 
 
 69847 
 
 1.43169 
 
 72477 
 
 1.37976 
 
 4 
 
 5' 
 
 64817 
 
 1.54281 
 
 67324 
 
 1.48536 
 
 69891 
 
 1.43080 
 
 72521 
 
 1.37891 
 
 3 
 
 5 
 
 64858 
 
 1.54183 
 
 67366 
 
 1.48442 
 
 69934 
 
 1.42992 
 
 72565 
 
 1.37807 
 
 2 
 
 55 
 
 64899 
 
 1.54085 
 
 67409 
 
 1.48349 
 
 69977 
 
 1.42903 
 
 72610 
 
 1.37722 
 
 1 
 
 6_ 
 
 64941 
 
 1.53986 
 
 67451 
 
 1.48256 
 
 70021 
 
 1.42815 
 
 72654 \ 
 
 1.37638 
 
 
 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 ' Tang 
 
 f 
 
 57 
 
 56 
 
 55 
 
 54 
 
 
TABLE XII. TANGENTS AND COTANGENTS. 
 
 
 36 
 
 37 
 
 88 
 
 39 
 
 
 
 Tang^ Cotang 
 
 Tang Cotang 
 
 Tang Cotang 
 
 Tang Cotang 
 
 / 
 
 
 
 72654 1.37038 
 
 ' 75355 1.3-3704 
 
 78129 1.27994 
 
 80978 1.23490 
 
 60 
 
 1 
 
 72699 1.37554 
 
 75401 1.82024 
 
 78175 1.27917 
 
 81027 1.23416 
 
 59 
 
 2 
 
 72743 1.37470 
 
 75447 1.32514 
 
 78222 1.27841 
 
 81075 1.23343 
 
 58 
 
 3 
 
 72788 1.3738G 
 
 75492 1.32464 
 
 78209 1.27764 
 
 81123 1.23270 
 
 57 
 
 4 
 
 72832 1.37302 
 
 75538 1.32384 
 
 78316 1.27688 
 
 81171 1.23190 
 
 66 
 
 5 
 
 72877 1.37218 
 
 75584 1.32304 
 
 78363 1.27611 
 
 81220 1.23123 
 
 55 
 
 C 
 
 7'2921 1.37134 
 
 75629 1.32221 
 
 78410 1.27535 
 
 81268 1.23050 
 
 54 
 
 7 
 
 72966 1.37050 
 
 75675 1.32144 
 
 78457 1.27458 
 
 81316 1.22977 
 
 53 
 
 8 
 
 73010 1.36967 
 
 75721 1.32064 
 
 78504 1.27382 
 
 81364 1.22904 
 
 52 
 
 9 
 
 73055 1.36883 
 
 75767 1.31984 
 
 78551 1.27306 
 
 81413 1.22831 
 
 51 
 
 10 
 
 73100 1.36800 
 
 75812 1.31904 
 
 78598 1.27230 
 
 81461 1.22758 
 
 50 
 
 11 
 
 73144 1.36716 
 
 75858 1.31825 
 
 78645 1.27153 
 
 81510 .22685 
 
 49 
 
 12 
 
 73189 1.36633 
 
 75904 1.31745 
 
 78092 1.27077 
 
 81558 .22012 
 
 48 
 
 13 
 
 73234 1.36549 
 
 75950 1.31666 
 
 78739 1.27001 
 
 81006 .22539 
 
 47 
 
 14 
 
 73278 1.36466 
 
 75996 1.31586 
 
 78786 1.26925 
 
 81055 .22467 
 
 46 
 
 15 
 
 73323 1.36883 
 
 76042 1.31507 
 
 78834 1.26849 
 
 81703 .22394 
 
 45 
 
 1C 
 
 73368 1.36300 
 
 76088 1.31427 
 
 78881 1.26774 
 
 81752 .22321 
 
 44 
 
 17 
 
 73413 1.36217 
 
 76134 1.31348 
 
 78928 1.26698 
 
 81800 .22249 
 
 43 
 
 18 
 
 73457 1.36134 
 
 76180 1.31269 
 
 78975 1.20622 
 
 81849 .22176 
 
 42 
 
 19 
 
 73502 1.36051 
 
 76226 1.31190 
 
 79022 1.26546 
 
 81898 .22104 
 
 41 
 
 20 
 
 73547 1.35968 
 
 76272 1.31110 
 
 79070 1.26471 
 
 81946 .22031 
 
 40 
 
 21 
 
 73592 1.35885 
 
 76318 1.31031 
 
 79117 1.26395 
 
 81995 .21959 
 
 39 
 
 22 
 
 73637 1.35802 
 
 76364 1.30952 
 
 79104 1.20319 
 
 820 It .21886 
 
 38 
 
 23 
 
 73681 1.35719 
 
 76410 1.30873 
 
 79212 1.20244 
 
 82092 .21814 
 
 37 
 
 24 
 
 73726 1.35637 
 
 76456 1.30795 
 
 79259 1.20169 
 
 82141 .21742 
 
 36 
 
 25 
 
 73771 1.35554 
 
 76502 1.30716 
 
 79306 1.26093 
 
 82190 .21670 
 
 35 
 
 20 
 
 73816 1.35472 
 
 76548 1.30637 
 
 79354 1.26018 
 
 82238 .21598 
 
 34 
 
 27 
 
 73861 1.35389 
 
 76594 1.30558 
 
 79401 1.25943 
 
 82287 .21526 
 
 33 
 
 28 
 
 73906 1.35307 
 
 76640 1.30480 
 
 79449 1.25807 
 
 82336 .21454 
 
 32 
 
 29 
 
 73951 1.35224 
 
 76686 1.30401 
 
 79496 1.25792 
 
 82385 .21382 
 
 31 
 
 30 
 
 73996 1.35142 
 
 76733 1.30323 
 
 79544 1.25717 
 
 82434 .21310 
 
 30 
 
 31 
 
 74041 1.35060 
 
 76779 1.30244 
 
 79591 1.25642 
 
 82483 .21238 
 
 29 
 
 32 
 
 74086 1.34978 
 
 76825 1.30160 
 
 79039 1.25507 
 
 82531 .21166 
 
 28 
 
 33 
 
 74131 1.34896 
 
 76871 1.30087 
 
 79036 1.25492 
 
 82580 .21094 
 
 27 
 
 34 
 
 74176 1.34814 
 
 76918 1.30009 
 
 79734 1.25417 
 
 82629 .21023 
 
 26 
 
 35 
 
 74221 1.34732 
 
 76964 1.29931 
 
 79781 1.25343 
 
 82678 .20951 
 
 25 
 
 36 
 
 74267 1.34650 
 
 77010 1.29853 
 
 79829 1.25268 
 
 82727 .20879 
 
 24 
 
 37 
 
 74312 1.34568 
 
 77057 1.29775 
 
 79877 1.25193 
 
 82776 .20808 
 
 23 
 
 38 
 
 74357 1.34487 
 
 77103 1.29696 
 
 79924 1.25118 
 
 82825 .20736 
 
 22 
 
 39 
 
 74402 1.34405 
 
 77149 1.29618 
 
 79972 1.25044 
 
 82874 .20665 
 
 21 
 
 40 
 
 74447 1.34323 
 
 77196 1.29541 
 
 80020 1.24969 
 
 82923 .20593 
 
 20 
 
 41 
 
 74492 1.34242 
 
 77242 1.29463 
 
 80067 1.24895 
 
 82972 .20522 
 
 19 
 
 42 
 
 74538 1.34160 
 
 77289 1.29385 
 
 80115 1.24820 
 
 83022 .20451 
 
 18 
 
 43 
 
 74583 1.34079 
 
 77335 1.29307 
 
 80103 1.24746 
 
 83071 .20379 
 
 17 
 
 44 
 
 74628 1.33998 
 
 T382 1.29229 
 
 80211 1.24672 
 
 83120 .20308 
 
 16 
 
 45 
 
 74674 1.33916 
 
 T423 1.29152 
 
 80258 1.24597 
 
 83169 .20237 
 
 15 
 
 46 
 
 74719 1.33835 
 
 T475 1.29074 
 
 80306 1.24523 
 
 83218 .20166 
 
 14 
 
 47 
 
 74764 1.33754 
 
 7521 1.28997 
 
 80354 1.24449 
 
 83268 .20095 
 
 13 
 
 48 
 
 74810 1.33673 
 
 T5G8 1.28919 
 
 80402 1.24375 
 
 83317 .20024 
 
 12 
 
 49 
 
 74855 1.33592 
 
 T615 1.28842 
 
 80450 1.24S01 
 
 83366 .19953 
 
 11 
 
 50 
 
 74900 1.33511 
 
 TGG1 1.28764 
 
 80498 1.24227 
 
 83415 .19882 
 
 10 
 
 51 
 
 74946 1.33430 
 
 77708 1.28687 
 
 80546 1.24153 
 
 83465 .19811 
 
 9 
 
 52 
 
 74991 1.33349 
 
 77754 1.28610 
 
 80594 1.24079 
 
 83514 .19740 
 
 8 
 
 53 
 
 75037 1.33268 
 
 77801 1.28533 
 
 80642 1.24005 
 
 83564 .19669 
 
 7 
 
 54 
 
 75082 1.33187 
 
 77848 1.28456 
 
 80690 1.23931 
 
 83613 .19599 
 
 6 
 
 55 
 
 75128 1.33107 
 
 77895 1.28379 
 
 80738 1.23858 
 
 83662 .19528 
 
 5 
 
 56 
 
 75173 1.33026 
 
 77941 1.28302 
 
 80786 1.23784 
 
 83712 .19457 
 
 4 
 
 57 
 
 75219 1.32946 
 
 77988 1.28225 
 
 808:34 1.23710 
 
 83761 .19387 
 
 3 
 
 58 
 
 75264 1.32865 
 
 78035 1.28148 
 
 80882 1.23637 
 
 83811 .19316 
 
 2 
 
 59 
 
 75310 1.32785 
 
 78082 1.28071 
 
 80930 1.23563 
 
 83860 .19246 
 
 1 
 
 GO 
 
 75355 1.32704 
 
 78129 1.27994 
 
 80978 1.23490 
 
 83910 .19175 
 
 
 
 / 
 
 Cotang Tang 
 
 Cotang Tang 
 
 Cotang Tang 
 
 Cotang Tang 
 
 / 
 
 
 53 
 
 52 
 
 51 
 
 50 
 
 
 244 
 
TABLE XII.-TANGENTS AND COTANGENTS. 
 
 
 40 
 
 41 
 
 42 
 
 43 
 
 
 
 Tang Cotang 
 
 Tang Cotang 
 
 Tang Cotang 
 
 Tang Cotang 
 
 
 o 
 
 83910 1.19175 
 
 86929 1.15037 
 
 90040 1.11061 
 
 93252 1.07237 
 
 GO 
 
 1 
 
 83960 1.19105 
 
 86980 1.14969 
 
 90093 1.10996 
 
 93306 1.07174 
 
 59 
 
 2 
 
 84009 1.19035 
 
 87031 1.14902 
 
 90146 1.10931 
 
 93360 1.07112 
 
 58 
 
 3 
 
 84059 1 . 18964 
 
 87082 1.14834 
 
 90199 1.10867 
 
 93415 1.07049 
 
 57 
 
 4 
 
 84108 1.18894 
 
 87133 1.14767 
 
 90251 1.10802 
 
 93469 1.06987 
 
 50 
 
 5 
 
 84158 1.18824 
 
 87184 1.14699 
 
 90304 1.10737 
 
 93524 1.06925 
 
 55 
 
 c 
 
 84208 1.18754 , 
 
 87236 1.14632 
 
 90357 1.10672 
 
 93578 1.06862 
 
 54 
 
 7 
 
 84258 1.18684 i 
 
 87287 1.14565 
 
 90410 1.10607 
 
 93633 1.06800 
 
 53 
 
 8 
 
 84307 1.18614 ' 
 
 87338 1.14498 
 
 90463 1.10543 
 
 93688 1.06738 
 
 52 
 
 9 
 
 84357 1.18544 
 
 87389 1.14430 
 
 90516 1.10478 
 
 93742 1.06676 
 
 51 
 
 10 
 
 84407 1.18474 
 
 87441 1.14363 
 
 90569 1.10414 
 
 93797 1.06613 
 
 50 
 
 11 
 
 84457 1.18404 
 
 87492 1.14296 
 
 90621 1.10349 
 
 93852 1.06551 
 
 49 
 
 12 
 
 84507 1.18334 
 
 87543 1.14229 
 
 90674 1.10285 
 
 93906 1.06489 
 
 48 
 
 13 
 
 84556 1.18264 
 
 8^595 1.14162 
 
 90727 1.10220 
 
 93961 1.06-127 
 
 47 
 
 14 
 
 84606 1.18194 
 
 87646 1.14095 
 
 90781 1.10156 
 
 94016 1.00365 
 
 40 
 
 15 
 
 84656 1.18125 
 
 87698 1.14028 
 
 90834 1.10091 
 
 94071 1.06303 
 
 45 
 
 1C 
 
 84706 1.18055 
 
 87749 1.13961 
 
 90887 1.10027 
 
 94125 1.06241 
 
 44 
 
 17 
 
 84756 1.17986 
 
 87801 1.13894 
 
 90940 1.09963 
 
 94180 1.06179 
 
 43 
 
 18 
 
 84806 1.17916 
 
 87852 1.13828 
 
 90993 1.09899 
 
 94235 1.06117 
 
 42 
 
 19 
 
 84856 1.17846 
 
 87904 1.13761 
 
 91046 1.09834 
 
 94290 1.06056 
 
 41 
 
 20 
 
 84906 1.17777 
 
 87955 1.13694 
 
 91099 1.09770 
 
 94345 1.05994 
 
 40 
 
 21 
 
 84956 1.17708 
 
 88007 1.13627 
 
 91153 1.09706 
 
 94400 1.05932 
 
 39 
 
 22 
 
 85006 1.17638 
 
 88059 1.13561 
 
 91206 1.09642 
 
 94455 1.05870 
 
 38 
 
 23 
 
 85057 1.17569 
 
 88110 1.13494 
 
 91259 1.09578 
 
 94510 1.05809 
 
 37 
 
 24 
 
 85107 1.17500 
 
 88162 1.13428 
 
 91313 1.09514 
 
 94565 1.05747 
 
 30 
 
 25 
 
 85157 1.17430 
 
 88214 1.13361 
 
 91366 1.09450 
 
 94620 1.05685 
 
 35 
 
 26 
 
 85207 1.17361 
 
 88265 1.13295 
 
 91419 1.09386 
 
 94676 1.05624 
 
 31 
 
 27 
 
 85257 1.17292 
 
 88317 1.13223 
 
 91473 1.09322 
 
 94731 1.05562 
 
 33 
 
 28 
 
 85308 1.17223 
 
 88369 1.13162 
 
 91526 1.09258 
 
 94786 1.05501 
 
 W 
 
 29 
 
 85358 1.17154 
 
 88421 1.13096 
 
 91580 1.09195 
 
 94841 1.05439 
 
 31 
 
 30 
 
 85408 1.17085 
 
 88473 1.13029 
 
 91633 1.09131 
 
 94896 1.05378 
 
 30 
 
 31 
 
 85458 1.17016 
 
 88524 1.12963 
 
 91687 1.09067 
 
 94952 1.C5317 
 
 29 
 
 32 
 
 85509 1.16947 
 
 88576 1.12897 
 
 91740 1.09003 
 
 95007 1.05255 
 
 28 
 
 33 
 
 85559 1.16878 
 
 88628 1.12831 
 
 91794 1.08940 
 
 95062 1.05194 
 
 27 
 
 34 
 
 85609 1.16809 
 
 88680 1.12765 
 
 91847 1.08876 
 
 95118 1.05133 
 
 20 
 
 35 
 
 85660 1.16741 
 
 88732 1.12699 
 
 91901 1.08813 
 
 95173 1.05072 
 
 
 
 38 
 
 85710 1.16672 
 
 88784 1.12633 
 
 91955 1.08749 
 
 95229 1.05010 
 
 21 
 
 37 
 
 85761 1.16603 
 
 88836 1.12567 
 
 92008 1.08686 
 
 95284 1.04949 
 
 23 
 
 38 
 
 85811 1.16535 
 
 88888 1.12501 
 
 92062 1.08622 
 
 95340 1.04888 
 
 22 
 
 39 
 
 85862 1.164JG6 
 
 88940 1.12435 
 
 92116 1.08559 
 
 95395 1.04827 
 
 21 
 
 40 
 
 85912 1.16398 
 
 88992 1.12369 
 
 92170 1.08496 
 
 95451 1.04766 
 
 20 
 
 41 
 
 85963 1.16329 
 
 89045 1.12303 
 
 92224 1.08432 
 
 95506 1.04705 
 
 19 
 
 42 
 
 86014 1.16261 
 
 89097 1.12238 
 
 92277 1.08369 
 
 95562 1.04644 
 
 18 
 
 43 
 
 86064 1.16192 
 
 89149 1.12173 
 
 92331 1.08306 
 
 95618 1.04583 
 
 17 
 
 44 
 
 86115 1.1G124 
 
 89201 1.12106 
 
 92385 1.08243 
 
 95673 1.04522 
 
 16 
 
 45 
 
 86166 1.16056 
 
 89253 1.12041 
 
 92439 1.08179 
 
 95729 1.04461 
 
 15 
 
 46 
 
 86216 1.15987 
 
 89306 1.11975 
 
 92493 1.08116 
 
 95785 1.04401 
 
 14 
 
 47 
 
 86267 1.15919 
 
 89358 1.11909 
 
 92547 1.08053 
 
 95841 1.04340 
 
 13 
 
 48 
 
 86318 1.15851 
 
 89410 1.11844 
 
 92601 1 -07990 
 
 95897 1.04279 
 
 13 
 
 49 
 
 86368 1.15783 
 
 89463 1 11778 
 
 92655 1.07927 
 
 95952 1.04218 
 
 11 
 
 50 
 
 86419 1.15715 
 
 89515 1.11713 
 
 92709 1.07864 
 
 96008 1.04158 
 
 10 
 
 51 
 
 86470 1.15047 
 
 89567 1.11648 
 
 92763 1.07801 
 
 96064 1.04097 
 
 9 
 
 52 
 
 86521 1.15579 
 
 89620 1.11582 
 
 92817 1.07738 
 
 96120 1.04036 
 
 8 
 
 53 
 
 86572 1.15511 
 
 89672 1.11517 
 
 92872 1.07676 
 
 96176 1.03976 
 
 7 
 
 54 
 
 86023 1.15443 
 
 89725 1.11452 
 
 92926 1.07613 
 
 96232 1.03915 
 
 6 
 
 55 
 
 86674 1.15375 
 
 89777 1.11387 
 
 92980 1.07550 
 
 96288 1.03855 
 
 5 
 
 56 
 
 86725 1.15308 
 
 89830 1.11321 
 
 93034 1.07487 
 
 96344 1.03794 
 
 4 
 
 57 
 
 86776 1.15240 
 
 89883 1.11256 
 
 93088 1.07425 
 
 96400 1.03734 
 
 3 
 
 58 
 
 86827 1.15172 
 
 89935 1.11191 
 
 93143 1.07362 
 
 96457 1.0:3674 
 
 2 
 
 59 
 
 86878 1.15104 
 
 89988 1.11126 
 
 93197 1.07299 
 
 96513 1.03613 
 
 1 
 
 60 
 
 86929 1.15037 
 
 90040 1.11061 
 
 93252 1.07237 
 
 96569 1.03553 
 
 
 
 / 
 
 Cotang Tang 
 
 Cotang Tang 
 
 Cotang Tang 
 
 Cotang Tang 
 
 f 
 
 
 49 
 
 48 
 
 47 
 
 46 
 
 
 245 
 
TABLE XII.-TANGENTS AND COTANGENTS. 
 
 
 44 
 
 
 
 44 
 
 
 
 44 
 
 
 
 Tang Cotang 
 
 
 
 Tang Cotang 
 
 
 
 Tang Cotang 
 
 
 
 
 9G569 .03553 
 
 60 
 
 20 
 
 97700 1.02355 
 
 40 
 
 40 
 
 98843 1.01170 
 
 90 
 
 1 
 
 96625 : .03493 
 
 59 
 
 21 
 
 97756 1.02295 
 
 39 
 
 41 
 
 98901 .01112 
 
 19 
 
 2 
 
 96681 .03483 
 
 58 
 
 22 
 
 97813 1.02236 
 
 38 
 
 42 
 
 98958 .01053 
 
 18 
 
 8 
 
 96738 .03372 
 
 57 
 
 23 
 
 97870 1.02176 
 
 37 
 
 43 
 
 99016 .00994 
 
 17 
 
 4 
 
 9(5794 : .03312 
 
 56 
 
 24 
 
 97927 1.02117 
 
 36 
 
 44 
 
 99073 .00935 
 
 Hi 
 
 5 
 
 96850 .03252 
 
 55 
 
 25 
 
 97984 1.02057 
 
 35 
 
 45 
 
 99131 .0087(5 
 
 15 
 
 6 
 
 96907 .03192 
 
 54 
 
 26 
 
 98041 1.01998 
 
 34 
 
 46 
 
 99189 .1.0818 
 
 14 
 
 7 
 
 96063 .03132 
 
 53 
 
 27 
 
 98098 1.01939 
 
 33 
 
 47 
 
 99247 : .00759 
 
 13 
 
 8 
 
 97020 .03072 
 
 5? 
 
 28 
 
 98155 1.01879 
 
 83 
 
 48 
 
 99304 .00701 
 
 12 
 
 9 
 
 97076 .03012 
 
 51 
 
 29 
 
 98213 1.01820 
 
 31 
 
 49 
 
 99362 .(XW42 
 
 11 
 
 10 
 
 9713JJ .02952 
 
 50 
 
 30 
 
 98270 1.01761 
 
 30 
 
 50 
 
 99420 .00583 
 
 10 
 
 11 
 
 97189 .02892 
 
 49 
 
 31 
 
 98327 1.01702 
 
 29 
 
 51 
 
 99478 .00525 
 
 9 
 
 12 
 
 97246 .02832 
 
 48 
 
 32 
 
 98384 1.01642 
 
 28 
 
 52 
 
 99536 .00467 
 
 8 
 
 13 
 
 97302 .02772 
 
 47 
 
 33 
 
 98441 1.01583 
 
 27 
 
 53 
 
 99594 .00103 
 
 7 
 
 14 
 
 97359 .02713 
 
 46 
 
 34 
 
 98499 1.01524 
 
 26 
 
 54 
 
 99652 .00350 
 
 6 
 
 15 
 
 97416 .02653 
 
 45 
 
 35 
 
 98556 1.01465 
 
 25 
 
 55 
 
 99710 .00291 
 
 5 
 
 1fl 
 
 97472 .02593 
 
 44 
 
 36 
 
 98613 1.01406 
 
 24 
 
 56 
 
 99768 ' .00233 
 
 4 
 
 1? 
 
 97529 .02533 
 
 43 
 
 37 
 
 98671 1.01347 
 
 23 
 
 57 
 
 99826 : .00175 
 
 3 
 
 18 
 
 97586 .02474 
 
 42 
 
 38 
 
 98728 1.01288 
 
 22 
 
 58 
 
 99884 : .00116 
 
 2 
 
 1!) 
 
 97643 .02414 
 
 41 
 
 39 
 
 98786 1.01229 
 
 21 
 
 59 
 
 99942 .00058 
 
 1 
 
 20 
 
 97700 .02355 
 
 40 
 
 40 
 
 98843 1.01170 
 
 20 
 
 60 
 
 1.00000 .00000 
 
 
 
 
 Cotang Tang 
 
 / 
 
 / 
 
 Cotang Tang 
 
 / 
 
 / 
 
 Cotang Tang 
 
 / 
 
 
 45 
 
 
 
 45 
 
 
 
 45 
 
 
 246 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 / 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 
 .00000 
 
 .00000 
 
 .00015 
 
 .00015 
 
 .00061 
 
 .00061 
 
 .00137 
 
 .00137 
 
 
 
 1 
 
 .00000 
 
 .00000 
 
 .00016 
 
 .00016 
 
 .00062 
 
 .00062 
 
 .00139 
 
 00139 
 
 1 
 
 2 
 
 .00000 
 
 .00000 
 
 .00016 
 
 .00016 
 
 .00063 
 
 .00063 
 
 .00140 
 
 .00140 
 
 2 
 
 3 
 
 .00000 
 
 .00000 
 
 .00017 
 
 .00017 
 
 .00064 
 
 .00064 
 
 .00142 
 
 .00142 
 
 3 
 
 4 
 
 .00000 
 
 .00000 
 
 .00017 
 
 .00017 
 
 .00065 
 
 .00065 , 
 
 .00143 
 
 .00143 
 
 4 
 
 5 
 
 .00000 
 
 .OJOOO 
 
 .00018 
 
 .00018 
 
 .00066 
 
 .00066 
 
 .00145 
 
 .00145 
 
 5 
 
 6 
 
 .00000 
 
 .00000 
 
 .00018 
 
 .00018 
 
 .00067 
 
 .00067 
 
 .00146 
 
 .00147 
 
 6 
 
 7 
 
 .00000 
 
 .00000 
 
 .00019 
 
 .00019 
 
 .00068 
 
 .00068 
 
 .00148 
 
 .00148 
 
 7 
 
 8 
 
 .00000 
 
 .00000 
 
 .00020 
 
 .00020 
 
 .00069 
 
 .00069 
 
 .00150 
 
 .00150 
 
 8 
 
 9 
 
 .00000 
 
 .00000 
 
 .00020 
 
 .00020 
 
 .00070 
 
 .00070 
 
 .00151 
 
 .00151 
 
 9 
 
 10 
 
 .00000 
 
 .00000 
 
 .00021 
 
 .00021 
 
 .00071 
 
 .00072 
 
 .00153 
 
 .00153 
 
 10 
 
 11 
 
 .00001 
 
 .00001 
 
 .00021 
 
 .00021 
 
 .00073 
 
 .00073 
 
 .00154 
 
 .00155 
 
 11 
 
 12 
 
 .00001 
 
 .03001 
 
 .00022 
 
 .00022 
 
 .00074 
 
 .00074 
 
 .00156 
 
 .00156 
 
 12 
 
 13 
 
 .00001 
 
 .00001 
 
 .00023 
 
 .00023 
 
 .00075 
 
 .00075 
 
 .00158 
 
 .00158 
 
 13 
 
 14 
 
 .00001 
 
 .00001 
 
 .00023 
 
 .00023 
 
 .00076 
 
 .00076 
 
 .00159 
 
 .00159 
 
 14 
 
 15 
 
 .00001 
 
 .00001 
 
 .00024 
 
 .00024 
 
 .00077 
 
 .00077 
 
 .00161 
 
 .00161 
 
 15 
 
 16 
 
 .00001 
 
 .00001 
 
 .00024 
 
 .00024 
 
 .00078 
 
 .00078 
 
 .00162 
 
 .00163 
 
 16 
 
 17 
 
 .00001 
 
 .00001 
 
 .00025 
 
 .00025 
 
 .00079 
 
 .00079 
 
 .00164 
 
 .00164 
 
 17 
 
 18 
 
 .00001 
 
 .00001 
 
 .00026 
 
 .00026 
 
 .00081 
 
 .00081 
 
 .00166 
 
 .00166 
 
 18 
 
 19 
 
 .00002 
 
 .00002 
 
 .00025 
 
 .00026 
 
 ,00082 
 
 .00082 
 
 .00168 
 
 .00168 
 
 19 
 
 20 
 
 .00002 
 
 : 00002 
 
 .00027 
 
 .00027 
 
 .00083 
 
 .00083 
 
 .00169 
 
 .00169 
 
 20 
 
 21 
 
 .00002 
 
 .00002 
 
 .00028 
 
 .00028 
 
 .00084 
 
 .00084 
 
 .00171 
 
 .00171 
 
 21 
 
 22 
 
 .0000:2 
 
 .00002 
 
 .00023 
 
 .00028 
 
 .00085 
 
 .00085 
 
 .00173 
 
 .00173 
 
 22 
 
 23 
 
 .00002 
 
 .00002 
 
 .00029 
 
 .00029 
 
 .00087 
 
 .00087 
 
 .00174 
 
 .00175 
 
 23 
 
 24 
 
 .00002 
 
 .00002 
 
 .00030 
 
 .00030 
 
 .00088 
 
 .00088 
 
 .00176 
 
 .00176 
 
 24 
 
 25 
 
 .00003 
 
 .00003 
 
 .00031 
 
 .00031 
 
 .00089 
 
 .00089 
 
 .00178 
 
 .00178 
 
 25 
 
 26 
 
 .00003 
 
 .00003 
 
 .00031 
 
 .00031 
 
 .00000 
 
 .00090 
 
 .00179 
 
 .00180 
 
 26 
 
 27 
 
 .00003 
 
 .00003 
 
 .00032 
 
 .00032 
 
 .00091 
 
 .00091 
 
 .00181 
 
 .00182 
 
 27 
 
 28 
 
 .00003 
 
 .00003 
 
 .00033 
 
 .00033 
 
 .00093 
 
 .00093 
 
 .00183 
 
 .00183 
 
 28 
 
 29 
 
 .00004 
 
 .00004 
 
 .00034 
 
 .00034 
 
 .00091 
 
 .00094 
 
 .00185 
 
 .00185 
 
 29 
 
 30 
 
 .00004 
 
 .00004 
 
 .00034 
 
 .00034 
 
 .00003 
 
 .00095 
 
 .00187 
 
 .00187 
 
 30 
 
 31 
 
 .00004 
 
 .00004 
 
 .00035 
 
 .00035 
 
 .00096 
 
 .00097 
 
 .00188 
 
 .00189 
 
 31 
 
 32 
 
 .00004 
 
 .00004 
 
 .00036 
 
 .00036 
 
 .00093 
 
 .00098 
 
 .00190 
 
 .00190 
 
 32 
 
 33 
 
 .00005 
 
 .00005 
 
 .00037 
 
 .00037 
 
 .00039 
 
 .00099 
 
 .00192 
 
 .00192 
 
 33 
 
 34 
 
 .00005 
 
 .00005 
 
 .00037 
 
 .00037 
 
 .00100 
 
 .00100 
 
 .00194 
 
 .00194 
 
 34 
 
 35 
 
 .00005 
 
 .00005 
 
 .00038 
 
 .00038 
 
 .00102 
 
 .00102 
 
 .00196 
 
 .00196 
 
 35 
 
 36 
 
 .00005 
 
 .00005 
 
 .00039 
 
 .00039 
 
 .00103 
 
 .00103 
 
 .00197 
 
 .00193 
 
 33 
 
 37 
 
 .00006 
 
 .00006 
 
 .00040 
 
 .00010 
 
 .00104 
 
 .00104 
 
 .00199 
 
 .00200 
 
 37 
 
 38 
 
 .00006 
 
 .00006 
 
 .00041 
 
 .00041 
 
 .00106 
 
 .00106 
 
 .00201 
 
 .00201 
 
 38 
 
 39 
 
 .00006 
 
 .00006 
 
 .00041 
 
 .00041 
 
 .00107 
 
 .00107 i 
 
 .00203 
 
 .00203 
 
 39 
 
 40 
 
 .00007 
 
 .00007 
 
 .00042 
 
 .00042 
 
 .00108 
 
 .00108 
 
 .00205 
 
 .00205 
 
 40 
 
 41 
 
 .00007 
 
 .00007 
 
 .00043 
 
 .00043 
 
 .0*110 
 
 .00110 
 
 .00207 
 
 .00207 
 
 41 
 
 42 
 
 .00007 
 
 .00007 
 
 .00044 
 
 .00044 
 
 .001H 
 
 .00111 
 
 .00208 
 
 .00203 
 
 42 
 
 43 
 
 .00008 
 
 .00008 
 
 .00045 
 
 .00045 
 
 .00112 
 
 .00113 
 
 .00210 
 
 .00211 
 
 43 
 
 44 
 
 .00008 
 
 .00008 
 
 .00046 
 
 .00046 
 
 .00114 
 
 .00114 
 
 .00212 
 
 .00213 
 
 44 
 
 45 
 
 .00009 
 
 .00009 
 
 .00047 
 
 .00047 
 
 .00115 
 
 .00115 
 
 .00214 
 
 .00215 
 
 45 
 
 46 
 
 .00009 
 
 .00009 
 
 .00048 
 
 .00048 
 
 .00117 
 
 .00117 
 
 .00216 
 
 .00216 
 
 46 
 
 47 
 
 .00009 
 
 .00009 
 
 .00048 
 
 .00048 
 
 .00118 
 
 .00118 
 
 .00218 
 
 .00218 
 
 47 
 
 48 
 
 .00010 
 
 .00010 
 
 .00019 
 
 .00049 
 
 .00119 
 
 .00120 
 
 .00220 
 
 .00220 
 
 48 
 
 49 
 
 .00010 
 
 .00010 
 
 .00050 
 
 .00050 
 
 .00121 
 
 .00121 
 
 .00222 
 
 .00222 
 
 49 
 
 50 
 
 .00011 
 
 .00011 
 
 .00051 
 
 .00051 
 
 .00122 
 
 .00122 
 
 .00224 
 
 .00224 
 
 50 
 
 51 
 
 .00011 
 
 .00011 
 
 .00052 
 
 .00052 
 
 .00124 
 
 .00124 
 
 .00226 
 
 .00226 
 
 51 
 
 52 
 
 .00011 
 
 .00011 
 
 .00053 
 
 .00053 
 
 .00125 
 
 .00125 
 
 .00228 
 
 .00228 
 
 52 
 
 53 
 
 .00012 
 
 .00012 
 
 .00054 
 
 .00054 
 
 .00127 
 
 .00127 
 
 .00230 
 
 .00230 
 
 53 
 
 54 
 
 .00012 
 
 .00012 
 
 .00055 
 
 .00055 
 
 .00128 
 
 .00128 
 
 .00232 
 
 .00232 
 
 54 
 
 55 
 
 .00013 
 
 .00013 
 
 .00056 
 
 .00056 
 
 .00130 
 
 .00130 
 
 .00234 
 
 .00234 
 
 55 
 
 56 
 
 .00013 
 
 .00013 
 
 .00057 
 
 .00057 
 
 .00131 
 
 .00131 
 
 .00236 
 
 .00236 
 
 56 
 
 57 
 
 .00014 
 
 .00014 
 
 .00058 
 
 .00058 
 
 .00133 
 
 .00133 
 
 .00238 
 
 .00238 
 
 57 
 
 58 
 
 .00014 
 
 .00014 
 
 .00059 
 
 .00059 
 
 .00134 
 
 .00134 
 
 .00240 
 
 .00240 
 
 58 
 
 59 
 
 .00015 
 
 .00015 
 
 .00060 
 
 .00060 
 
 .00136 
 
 .00136 
 
 .00242 
 
 .00242 
 
 59 
 
 60 
 
 .00015 
 
 .00015 
 
 .00061 
 
 .00061 
 
 .00137 
 
 .00137 
 
 .00244 
 
 .00244 
 
 60 
 
 247 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 40 
 
 5 C 
 
 6 
 
 7 
 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 
 .00244 
 
 .00244 
 
 ~003ST 
 
 .00382 
 
 .00548 
 
 .00551 
 
 .00745 
 
 .00751 
 
 
 
 1 
 
 .00246 
 
 .00246 
 
 .00383 
 
 .00385 
 
 .00551 
 
 .00554 
 
 .00749 
 
 .00755 
 
 1 
 
 2 
 
 .00248 
 
 .00248 
 
 .00386 
 
 .00387 
 
 .00554 
 
 .00557 
 
 .00752 
 
 .00758 
 
 2 
 
 3 
 
 .00250 
 
 .00250 
 
 .00388 
 
 .00390 
 
 .00557 
 
 .00560 
 
 .00756 
 
 .00762 
 
 3 
 
 4 
 
 .00252 
 
 .00252 
 
 .00391 
 
 .00392 
 
 .00560 
 
 .00563 
 
 .00760 
 
 .00765 
 
 4 
 
 5 
 
 .00254 
 
 .00254 
 
 .00393 
 
 .00395 
 
 .00563 
 
 .00566 
 
 .00763 
 
 .00769 
 
 5 
 
 6 
 
 .00256 
 
 .00257 
 
 .00396 
 
 .00397 
 
 .00566 
 
 .00569 
 
 .00767 
 
 .00773 
 
 6 
 
 7 
 
 .00258 
 
 .00259 
 
 .00398 
 
 .00400 
 
 .00569 
 
 .00573 
 
 .00770 
 
 .00776 
 
 7 
 
 8 
 
 .00260 
 
 .00261 
 
 .00401 
 
 .00403 
 
 .00572 
 
 .00576 
 
 .00774 
 
 .00780 
 
 8 
 
 * 9 
 
 .00262 
 
 .00263 
 
 .00404 
 
 .00405 
 
 .00576 
 
 .00579 
 
 .00778 
 
 .00784 
 
 9 
 
 10 
 
 .00264 
 
 .00265 
 
 .00406 
 
 .00408 
 
 .00579 
 
 .00582 
 
 .00781 
 
 .00787 
 
 10 
 
 11 
 
 .00266 
 
 .00267 
 
 .00409 
 
 .00411 
 
 .00582 
 
 .00585 
 
 .00785 
 
 .00791 
 
 11 
 
 12 
 
 .00269 
 
 .00269 
 
 .00412 
 
 .00413 
 
 .00585 
 
 .00588 
 
 .00789 
 
 .00795 
 
 12 
 
 13 
 
 .00271 
 
 .00271 
 
 .00414 
 
 .00416 
 
 .00588 
 
 .00592 
 
 .00792 
 
 .00799 
 
 13 
 
 14 
 
 .00273 
 
 .00274 
 
 .00417 
 
 .00419 
 
 .00591 
 
 .00595 
 
 .00796 
 
 .00802 
 
 14 
 
 15 
 
 .00275 
 
 .00276 
 
 .00420 
 
 .00421 
 
 .00594 
 
 . .00598 
 
 .00800 
 
 .00806 
 
 15 
 
 16 
 
 .00277 
 
 .00278 
 
 .00422 
 
 .00424 
 
 .00598 
 
 .00601 
 
 .00803 
 
 .00810 
 
 16 
 
 17 
 
 .00279 
 
 .00280 
 
 .00425 
 
 .00427 
 
 .00601 
 
 .00604 
 
 .00807 
 
 .00813 
 
 17 
 
 18 
 
 .00281 
 
 .00282 
 
 .00428 
 
 .00429 
 
 .00604 
 
 .00608 
 
 .00811 
 
 .00817 
 
 18 
 
 19 
 
 .00284 
 
 .00284 
 
 .00430 
 
 .00432 
 
 .00607 
 
 .00611 
 
 .00814 
 
 .00821 
 
 19 
 
 20 
 
 .00286 
 
 .00287 
 
 .00433 
 
 .00435 
 
 .00610 
 
 .00614 
 
 .00818 
 
 .00825 
 
 20 
 
 21 
 
 .00288 
 
 .00289 
 
 .00436 
 
 .00438 
 
 .00614 
 
 .00617 
 
 .00822 
 
 .00828 
 
 21 
 
 22 
 
 .00290 
 
 .00291 
 
 .00438 
 
 .00440 
 
 .00617 
 
 .00621 
 
 .00825 
 
 .00832 
 
 22 
 
 23 
 
 .00293 
 
 .00293 
 
 .00441 
 
 .00443 
 
 .00620 
 
 .00624 
 
 .00829 
 
 .00836 
 
 23 
 
 24 
 
 .00295 
 
 .00296 
 
 .00444 
 
 .00446 
 
 .00623 
 
 .00627 
 
 .00833 
 
 .00840 
 
 24 
 
 25 
 
 .00297 
 
 .00298 
 
 .00447 
 
 .00449 
 
 .00626 
 
 .00630 
 
 .00837 
 
 .00844 
 
 25 
 
 26 
 
 .00299 
 
 .00300 
 
 .00449 
 
 .00151 
 
 .OOG30 
 
 .00634 
 
 .00840 
 
 .00848 
 
 26 
 
 27 
 
 .00301 
 
 .00302 
 
 .00452 
 
 .00454 
 
 .00633 
 
 .00637 
 
 .00844 
 
 .00851 
 
 27 
 
 28 
 
 .00304 
 
 .00305 
 
 .00455 
 
 .00457 
 
 .00636 
 
 .00640 
 
 .00848 
 
 .00855 
 
 28 
 
 29 
 
 .00306 
 
 .00307 
 
 .00458 
 
 .00460 
 
 .00640 
 
 .00644 
 
 .00852 
 
 .00859 
 
 29 
 
 30 
 
 .00308 
 
 .00309 
 
 .00460 
 
 .00463 
 
 .00643 
 
 .00647 
 
 .00856 
 
 .00863 
 
 30 
 
 31 
 
 .00311 
 
 .00312 
 
 .00463 
 
 .00465 
 
 .00646 
 
 .00650 
 
 .00859 
 
 .00867 
 
 31 
 
 32 
 
 .00313 
 
 .00314 
 
 .00466 
 
 .00468 
 
 .00649 
 
 .00654 
 
 .00863 
 
 .00871 
 
 32 
 
 33 
 
 .00315 
 
 .00316 
 
 .00469 
 
 .00471 
 
 .00653 
 
 .00657 
 
 .00867 
 
 .00875 
 
 33 
 
 34 
 
 .00317 
 
 .00318 
 
 .00472 
 
 .00474 
 
 .00656 
 
 .00660 
 
 .00871 
 
 .00878 
 
 34 
 
 35 
 
 .00320 
 
 .00321 
 
 .00474 
 
 .00477 
 
 .00059 
 
 .00664 
 
 .00875 
 
 .00882 
 
 35 
 
 36 
 
 .00322 
 
 .00323 
 
 .00477 
 
 .00480 
 
 .00663 
 
 .00667 
 
 .00878 
 
 .00886 
 
 36 
 
 37 
 
 .00324 
 
 .00326 
 
 .00480 
 
 .00482 
 
 .00666 
 
 .00671 
 
 .00882 
 
 .00890 
 
 37 
 
 38 
 
 .00327 
 
 .00328 
 
 .00483 
 
 .00485 
 
 .00669 
 
 .00674 
 
 .00886 
 
 .00894 
 
 38 
 
 39 
 
 .00329 
 
 .00330 
 
 .00486 
 
 .00488 
 
 .00673 
 
 .00677 
 
 .00890 
 
 .00898 
 
 39 
 
 40 
 
 .00332 
 
 .00333 
 
 .00489 
 
 .00491 
 
 .00676 
 
 .00681 
 
 .00894 
 
 .00902 
 
 40 
 
 41 
 
 .00334 
 
 .00335 
 
 .00492 
 
 .00494 
 
 .00680 
 
 .00684 
 
 .00898 
 
 .00906 
 
 41 
 
 42 
 
 .00336 
 
 .00337 
 
 .00494 
 
 .00497 
 
 .00688 
 
 .00688 
 
 .00902 
 
 .00910 
 
 42 
 
 43 
 
 .00339 
 
 .00340 
 
 .00497 
 
 .00500 
 
 .OOG86 
 
 .00091 
 
 .00906 
 
 .00914 
 
 43 
 
 44 
 
 .00341 
 
 .00342 
 
 .00500 
 
 .00503 
 
 .00690 
 
 .00695 
 
 .00909 
 
 .00918 
 
 44 
 
 45 
 
 .00343 
 
 .00345 
 
 .00503 
 
 .00506 
 
 .00693 
 
 .00098 
 
 .00913 
 
 .00922 
 
 45 
 
 46 
 
 .00346 
 
 .00347 
 
 .00506 
 
 .00509 
 
 .00697 
 
 .00701 I 
 
 .00917 
 
 .00926 
 
 46 
 
 47 
 
 .00348 
 
 .00350 
 
 .00509 
 
 .00512 
 
 .00700 
 
 .00705 1 
 
 .00921 
 
 .00930 
 
 47 
 
 48 
 
 .00351 
 
 .00352 
 
 .00512 
 
 .00515 
 
 .00703 
 
 .00708 
 
 .00925 
 
 .00934 
 
 48 
 
 49 
 
 .00353 
 
 .00354 
 
 .00515 
 
 .00518 
 
 .00707 
 
 .00712 
 
 .00929 
 
 .00938 
 
 49 
 
 50 
 
 .00356 
 
 .00357 
 
 .00518 
 
 .00521 
 
 .00710 
 
 .00715 
 
 .00933 
 
 .00942 
 
 50 
 
 51 
 
 .00.358 
 
 .00359 
 
 .00521 
 
 .00524 
 
 .00714 
 
 .00719 
 
 .00937 
 
 .00946 
 
 51 
 
 52 
 
 .00361 
 
 .00362 
 
 .00524 
 
 .00527 
 
 .00717 
 
 .00722 
 
 .00941 
 
 .00950 
 
 52 
 
 53 
 
 .00363 
 
 .00364 
 
 .00527 
 
 .00530 
 
 .00721 
 
 .00726 
 
 .00945 
 
 .00954 
 
 53 
 
 54 
 
 .00365 
 
 .00367 
 
 .00530 
 
 .00533 
 
 .00724 
 
 .00730 
 
 .00949 
 
 .00958 
 
 54 
 
 55 
 
 .00368 
 
 .00369 
 
 .00533 
 
 .00536 
 
 .00728 
 
 .00733 
 
 .00953 
 
 .00982 
 
 55 
 
 56 
 
 .00370 
 
 .00372 
 
 .00536 
 
 .00539 
 
 .00731 
 
 .00737 
 
 .00957 
 
 .00066 
 
 56 
 
 57 
 
 .00373 
 
 .00374 
 
 .00539 
 
 .00542 
 
 .00735 
 
 .00740 
 
 .00961 
 
 .00970 
 
 57 
 
 58 
 
 .00375 
 
 .00377 
 
 .00542 
 
 .00545 
 
 .00738 
 
 .00744 
 
 .00965 
 
 .00975 
 
 58 
 
 59 
 
 .00378 
 
 .00379 
 
 .00545 
 
 .00548 
 
 .00742 
 
 .00747 
 
 .00969 
 
 .00979 
 
 59 
 
 60 
 
 .00381 
 
 .00382 
 
 .00548 
 
 .00551 
 
 .00745 
 
 .00751 
 
 .00973 
 
 .00983 1 60 
 
 248 
 
TABLE XIIL VERSINES AND EXSECANTS. 
 
 
 8 
 
 9 
 
 10 
 
 11 
 
 i 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 xsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 
 .00973 
 
 .00983 
 
 .01231 
 
 .01247 
 
 .01519 
 
 .01543 1 
 
 .01837 
 
 .01872 
 
 
 
 1 
 
 .00977 
 
 .00987 
 
 .01236 
 
 .01251 
 
 .01524 
 
 .01548 
 
 .01843 
 
 .01877 
 
 1 
 
 2 
 
 00981 
 
 .00991 
 
 .01240 
 
 .01256 
 
 .01529 
 
 .01553 
 
 .01848 
 
 .01883 
 
 2 
 
 3 
 
 .00985 
 
 .00995 
 
 .01245 
 
 .01201 
 
 .01534 
 
 .01558 
 
 .01854 
 
 .01889 
 
 3 
 
 4 
 
 .00989 
 
 .00999 
 
 .01249 
 
 .01265 
 
 .01540 
 
 .01564 
 
 .01860 
 
 .01895 
 
 4 
 
 5 
 
 .00994 
 
 .01004 
 
 .01254 
 
 .01270 
 
 .01545 
 
 .01509 
 
 .01865 
 
 .01901 
 
 5 
 
 6 
 
 .00998 
 
 .01008 
 
 .01259 
 
 .01275 
 
 .01550 
 
 .01574 
 
 .01871 
 
 .01906 
 
 6 
 
 7 
 
 01002 
 
 .01012 
 
 .01263 
 
 .01279 
 
 .01555 
 
 .01579 
 
 .01876 
 
 .01912 
 
 7 
 
 8 
 
 .01006 
 
 .01016 
 
 .01268 
 
 .01284 
 
 .01560 
 
 .01585 
 
 .01882 
 
 .01918 
 
 8 
 
 9 
 
 .01010 
 
 .01020 
 
 .01272 
 
 .01289 
 
 .01565 
 
 .01590 
 
 .01888 
 
 .01924 
 
 9 
 
 10 
 
 .01014 
 
 .01024 
 
 .01277 
 
 .01394 
 
 .01570 
 
 .01595 
 
 .01893 
 
 .01930 
 
 10 
 
 11 
 
 .01018 
 
 .01029 
 
 .01282 
 
 .01298 
 
 .01575 
 
 .01001 
 
 .01899 
 
 .01936 
 
 11 
 
 12 
 
 .01022 
 
 .01033 
 
 .01286 
 
 .01303 
 
 .01580 
 
 .01006 
 
 .01004 
 
 .01941 
 
 12 
 
 13 
 
 .01027 
 
 .01037 
 
 .01291 
 
 .01308 
 
 .01586 
 
 .01611 
 
 .01910 
 
 .01947 
 
 13 
 
 14 
 
 .01031 
 
 .01041 
 
 .01296 
 
 .01313 
 
 .01591 
 
 .01616 
 
 .01916 
 
 .01953 
 
 14 
 
 15 
 
 .01035 
 
 .01046 
 
 .01300 
 
 .01318 
 
 .01596 
 
 .01622 
 
 .01921 
 
 .01959 
 
 15 
 
 16 
 
 .01039 
 
 .01050 
 
 .01305 
 
 .01322 
 
 .01601 
 
 .01027 
 
 .01927 
 
 .01965 
 
 16 
 
 17 
 
 .01043 
 
 .01054 
 
 .01310 
 
 .01327 
 
 .01606 
 
 .01033 
 
 .01933 
 
 .01971 
 
 17 
 
 18 
 
 .01047 
 
 .01059 
 
 .01314 
 
 .01332 
 
 .01612 
 
 .01038 
 
 .01939 
 
 .01977 
 
 18 
 
 19 
 
 .01052 
 
 .01063 
 
 .01319 
 
 .01337 
 
 .01617 
 
 .01043 
 
 .01944 
 
 .01983 
 
 19 
 
 20 
 
 .01056 
 
 .01067 
 
 .01324 
 
 .01342 
 
 .01622 
 
 .01649 
 
 .01950 
 
 .01989 
 
 20 
 
 21 
 
 .01060 
 
 .01071 
 
 .01329 
 
 .01346 
 
 .01627 
 
 .01654 
 
 .01956 
 
 .01995 
 
 21 
 
 22 
 
 .01064 
 
 .01076 
 
 .013:33 
 
 .01351 
 
 .01032 
 
 .01059 
 
 .01961 
 
 .02001 
 
 22 
 
 23 
 
 .010G9 
 
 .01080 
 
 .01338 
 
 .01356 
 
 . .01638 
 
 .01665 
 
 .01967 
 
 .02007 
 
 23 
 
 24 
 
 .01073 
 
 .01084 
 
 .01343 
 
 .01361 
 
 .01643 
 
 .01070 
 
 .01973 
 
 .02013 
 
 24 
 
 25 
 
 .01077 
 
 .01089 
 
 .01348 
 
 .01366 
 
 .01648 
 
 .01676 
 
 .01979 
 
 .02019 
 
 25 
 
 23 
 
 .01081 
 
 .01093 
 
 .01352 
 
 .01371 
 
 .01653 
 
 .01681 
 
 .01984 
 
 .02025 
 
 26 
 
 27 
 
 .01086 
 
 .01097 
 
 .01357 
 
 .01376 
 
 .01659 
 
 .01687 
 
 .01990 
 
 .02031 
 
 27 
 
 28 
 
 .01090 
 
 .01102 
 
 .01362 
 
 .01381 
 
 .01064 
 
 .01692 
 
 .01996 
 
 .02037 
 
 28 
 
 29 
 
 .01094 
 
 .01106 
 
 .01307 
 
 .01386 
 
 .01009 
 
 .01698 
 
 .02002 
 
 .02043 
 
 29 
 
 30 
 
 .01098 
 
 .01111 
 
 ..01371 
 
 .01391 
 
 .01675 
 
 .01703 
 
 .02008 
 
 .02049 
 
 30 
 
 31 
 
 .01103 
 
 .01115 
 
 .01376 
 
 .01395 
 
 .01680 
 
 .01709 
 
 .02013 
 
 .02055 
 
 31 
 
 32 
 
 .01107 
 
 .01119 
 
 .01381 
 
 .01400 
 
 .01085 
 
 .01714 
 
 .02019 
 
 .02001 
 
 33 
 
 33 
 
 .01111 
 
 .01124 
 
 .01386 
 
 .01405 
 
 .01690 
 
 .01720 
 
 .02025 
 
 .02007 
 
 33 
 
 34 
 
 .01116 
 
 .01128 
 
 .01391 
 
 .01410 
 
 .01096 
 
 .01725 
 
 .02031 
 
 .02073 
 
 31 
 
 35 
 
 .01120 
 
 .01133 
 
 .01396 
 
 .01415 
 
 .01701 
 
 .01731 
 
 .02037 
 
 .02079 
 
 35 
 
 36 
 
 .01124 
 
 .01137 
 
 .01400 
 
 .01420 
 
 .01706 
 
 .01736 
 
 .02042 
 
 .02085 
 
 36 
 
 37 
 
 .01129 
 
 .01142 
 
 .01405 
 
 .01425 
 
 .01712 
 
 .01742 
 
 .02048 
 
 .02091 
 
 37 
 
 38 
 
 .01133 
 
 .01146 
 
 .01410 
 
 .01430 
 
 .01717 
 
 .01747 
 
 .02054 
 
 .02097 
 
 38 
 
 39 
 
 .01137 
 
 .01151 
 
 .01415 
 
 .014-35 
 
 .01723 
 
 .01753 
 
 .020GO 
 
 .02103 
 
 39 
 
 43 
 
 .01142 
 
 .01155 
 
 .01420 
 
 .01440 
 
 .01728 
 
 .01758 
 
 .02066 
 
 .02110 
 
 40 
 
 41 
 
 .01146 
 
 .01160 
 
 .01425 
 
 .01445 
 
 .01733 
 
 .01764 
 
 .02072 
 
 .02116 
 
 41 
 
 42 
 
 .01151 
 
 .01164 
 
 .01430 
 
 .01450 
 
 .01739 
 
 .01769 
 
 .02078 
 
 .02122 
 
 42 
 
 43 
 
 .01155 
 
 .01169 
 
 .01435 
 
 .01455 
 
 .01744 
 
 .01775 
 
 .02084 
 
 .02128 
 
 43 
 
 44 
 
 .01159 
 
 .01173 
 
 .01439 
 
 .01401 
 
 .01750 
 
 .01781 
 
 .02090 
 
 .02134 
 
 44 
 
 45 
 
 .01164 
 
 .01178 
 
 .01444 
 
 .01406 
 
 .01755 
 
 .01786 
 
 .02095 
 
 .02140 
 
 45 
 
 46 
 
 .01168 
 
 .01182 
 
 .01449 
 
 .01471 
 
 .01760 
 
 .01792 
 
 .02101 
 
 .02146 
 
 46 
 
 47 
 
 .01173 
 
 .01187 
 
 .01454 
 
 01476 
 
 .01766 
 
 .01798 
 
 .02107 
 
 .02153 
 
 47 
 
 48 
 
 .01177 
 
 .01191 
 
 .01459 
 
 .01481 
 
 .01771 
 
 .01803 
 
 .02113 
 
 .02159 
 
 48 
 
 49 
 
 .01182 
 
 .01196 
 
 .01464 
 
 .01486 
 
 .01777 
 
 .01809 
 
 .02119 
 
 .02165 
 
 49 
 
 50 
 
 .01186 
 
 .01200 
 
 .01469 
 
 .01491 
 
 .01782 
 
 .01815 
 
 .02125 
 
 .02171 
 
 50 
 
 51 
 
 .01191 
 
 .01205 
 
 .01474 
 
 .01496 
 
 .01788 
 
 .01820 
 
 .02131 
 
 .02178 
 
 51 
 
 52 
 
 .01195 
 
 .01209 
 
 .01479 
 
 .01501 
 
 .01793 
 
 .01826 
 
 .02137 
 
 .02184 
 
 52 
 
 53 
 
 .01200 
 
 .01214 
 
 .01484 
 
 .01506 
 
 .01799 
 
 .01832 
 
 .02143 
 
 .02190 
 
 53 
 
 54 
 
 .01204 
 
 .01219 
 
 .01489 
 
 .01512 
 
 .01804 
 
 .01837 
 
 .02149 
 
 .02196 
 
 54 
 
 55 
 
 .01209 
 
 .01223 
 
 .01494 
 
 .01517 
 
 .01810 
 
 .01843 
 
 .02155 
 
 .02203 
 
 55 
 
 56 
 
 .01213 
 
 .01228 
 
 .01499 
 
 .01522 
 
 .01815 
 
 .01849 
 
 .02161 
 
 .02209 
 
 56 
 
 57 
 
 .01218 
 
 .01233 
 
 .01504 
 
 .01527 
 
 .01821 
 
 .01854 
 
 .02167 
 
 .02215 
 
 57 
 
 58 
 
 .01222 
 
 .01237 
 
 .01509 
 
 .01532 
 
 .01826 
 
 .01860 
 
 .02173 
 
 .02221 
 
 58 
 
 59 
 
 .01227 
 
 .01242 
 
 .01514 
 
 .01537 
 
 .01832 
 
 .01866 
 
 .02179 
 
 .02228 
 
 59 
 
 60 
 
 .01231 
 
 .01247 
 
 .01519 
 
 .01543 
 
 .01837 
 
 .01872 
 
 .02185 
 
 .02234 
 
 60 
 
 249 
 
TABLE XIII.-VERSINES AND EXSECANTS. 
 
 / 
 
 12 
 
 13 
 
 14 
 
 15 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 ~0~ 
 
 .02185 
 
 .02234 
 
 .02563 
 
 .02630 
 
 .02970 
 
 .03061 
 
 .03407 
 
 .03528 
 
 
 
 1 
 
 .02191 
 
 .02240 
 
 .02570 
 
 .02637 
 
 .02977 
 
 .03069 
 
 .03415 
 
 .03536 
 
 1 
 
 2 
 
 .02197 
 
 .02247 
 
 .02576 
 
 .02644 
 
 .02985 
 
 .03076 
 
 .03422 
 
 .03544 
 
 2 
 
 3 
 
 .02203 
 
 .02253 
 
 .02583 
 
 .02651 
 
 .02992 
 
 .03084 
 
 .03430 
 
 .03552 
 
 3 
 
 4 
 
 .02210 
 
 .02259 
 
 .02589 
 
 .02658 
 
 .02999 
 
 .03091 
 
 .03438 
 
 .03560 
 
 4 
 
 5 
 
 .02216 
 
 .02266 
 
 .02596 
 
 .02665 
 
 .03006 
 
 .03099 
 
 .03445 
 
 .035G8 
 
 5 
 
 6 
 
 .02222 
 
 .02272 
 
 .02602 
 
 .02(372 
 
 .03013 
 
 .03106 
 
 .03453 
 
 .03576 
 
 6 
 
 7 
 
 .02228 
 
 .02279 
 
 .02609 
 
 .02679 
 
 .03020 
 
 .03114 
 
 .03460 
 
 .03584 
 
 7 
 
 8 
 
 .02234 
 
 .02285 
 
 .02616 
 
 .C2686 
 
 .03027 
 
 .03121 
 
 .03468 
 
 .03592 
 
 8 
 
 9 
 
 .02240 
 
 .02291 
 
 .02622 
 
 .02693 
 
 .03034 
 
 .03129 
 
 .03476 
 
 .03601 
 
 9 
 
 10 
 
 .02246 
 
 .02298 
 
 .02629 
 
 .02700 
 
 .03041 
 
 .03137 
 
 .03483 
 
 .03609 
 
 10 
 
 11 
 
 .02252 
 
 .02304 
 
 .02635 
 
 .02707 
 
 .03048 
 
 .03144 
 
 .03491 
 
 .03617 
 
 11 
 
 12 
 
 .02258 
 
 .02311 
 
 .02642 
 
 .02714 
 
 .03055 
 
 .03152 
 
 .03-498 
 
 .03625 
 
 12 
 
 13 
 
 .02265 
 
 .02317 
 
 .02649 
 
 .02721 
 
 .03063 
 
 .03159 
 
 .03506 
 
 .03633 
 
 13 
 
 14 
 
 .02271 
 
 .02323 
 
 .02655 
 
 .02728 
 
 .03070 
 
 .03167 
 
 .03514 
 
 .03642 
 
 14 
 
 15 
 
 .02277 
 
 .02330 
 
 .02662 
 
 .02735 
 
 .03077 
 
 .03175 
 
 .03521 
 
 .03650 
 
 15 
 
 16 
 
 .02233 
 
 .02336 
 
 .02669 
 
 .02742 
 
 .03084 
 
 .03182 
 
 .03529 
 
 .03658 
 
 16 
 
 17 
 
 .02289 
 
 .02343 
 
 .02675 
 
 .02749 
 
 .03091 
 
 .03190 
 
 .03537 
 
 .03666 
 
 17 
 
 18 
 
 .02295 
 
 .02349 
 
 .02682 
 
 .02756 
 
 .03098 
 
 .03198 
 
 .03544 
 
 .03674 
 
 18 
 
 19 
 
 .02302 
 
 .02356 
 
 .02689 
 
 .02763 
 
 .03106 
 
 .03205 
 
 .03552 
 
 .03683 
 
 19 
 
 20 
 
 .02308 
 
 .02362 
 
 .02696 
 
 .02770 
 
 .03113 
 
 .03213 
 
 .03560 
 
 .03691 
 
 20 
 
 21 
 
 .02314 
 
 .02369 
 
 .02702 
 
 .02777 
 
 .03120 
 
 .03221 
 
 .03567 
 
 .03699 
 
 21 
 
 22 
 
 .02320 
 
 .02375 
 
 .02709 
 
 .02784 
 
 .03127 
 
 .03228 
 
 .0357'5 
 
 .03708 
 
 22 
 
 23 
 
 .02327 
 
 .02382 
 
 .02716 
 
 .02791 
 
 .03134 
 
 .03236 
 
 .03583 
 
 .03716 
 
 23 
 
 24 
 
 .02333 
 
 .02388 
 
 .02722 
 
 .02799 
 
 .03142 
 
 .03244 
 
 .03590 
 
 .037'24 
 
 24 
 
 25 
 
 .02339 
 
 .02395 
 
 .02729 
 
 .02806 
 
 .03149 
 
 .03251 
 
 .03598 
 
 .03732 
 
 25 
 
 ' 26 
 
 .02345 
 
 .C2402 
 
 .02736 
 
 .02813 
 
 .03156 
 
 .03259 
 
 .03606 
 
 .03741 
 
 2(3 
 
 27 
 
 .02352 
 
 .02408 
 
 .02743 
 
 .02820 
 
 .03163 
 
 .03267 
 
 .03614 
 
 .03749 
 
 27 
 
 28 
 
 .02358 
 
 .02415 
 
 .02749 
 
 .02827 
 
 .03171 
 
 .03275 
 
 .03621 
 
 .03758 
 
 28 
 
 3 
 
 .02364 
 
 .02421 
 
 .02756 
 
 .02834 
 
 .03178 
 
 .03282 
 
 ,,03629 
 
 .03766 
 
 29 
 
 SO 
 
 .02370 
 
 .02428 
 
 .02763 
 
 .02843 
 
 .03185 
 
 .03290 
 
 .03637 
 
 .03774 
 
 30 
 
 31 
 
 .02377 
 
 .02435 
 
 .02770 
 
 .02849 
 
 .03193 
 
 .03298 
 
 .03645 
 
 .03783 
 
 31 
 
 S3 
 
 .02383 
 
 .02441 
 
 .02777 
 
 .02856 
 
 .03200 
 
 .03306 
 
 .03653 
 
 .03791 
 
 32 
 
 S3 
 
 .02389 
 
 .02448 
 
 .02783 
 
 .028G3 
 
 .03207 
 
 .03313 
 
 .03660 
 
 .03799 
 
 33 
 
 34 
 
 .02396 
 
 .02454 
 
 .02790 
 
 .02870 
 
 .03214 
 
 .03321 
 
 .03608 
 
 .03808 
 
 34 
 
 35 
 
 .02-402 
 
 .02461 
 
 .02797 
 
 .02878 
 
 .03222 
 
 .03329 
 
 .03676 
 
 .03816 
 
 35 
 
 36 
 
 .02408 
 
 .02468 
 
 .02804 
 
 .02885 
 
 .03229 
 
 .03337 
 
 .03684 
 
 .03825 
 
 36 
 
 37 
 
 .02415 
 
 .02474 
 
 .02811 
 
 .02892 
 
 .03236 
 
 .03345 
 
 .03692 
 
 .03833 
 
 37 
 
 38 
 
 .02421 
 
 .02481 
 
 .02818 
 
 .02899 
 
 .03244 
 
 .03353 
 
 .03699 
 
 .03842 
 
 38 
 
 ^Q 
 
 .02427 
 
 .02488 
 
 .02824 
 
 .02907 
 
 .03251 
 
 .03360 
 
 .03707 
 
 .03850 
 
 W 
 
 40 
 
 .02434 
 
 .02494 
 
 .02831 
 
 .02914 
 
 .03258 
 
 .03368 
 
 .03715 
 
 .03858 
 
 4d 
 
 41 
 
 .02440 
 
 .02501 
 
 .02838 
 
 .02921 
 
 .03266 
 
 .03376 
 
 .03723 
 
 .03867 
 
 41 
 
 42 
 
 .02447 
 
 .02508 
 
 .02845 
 
 .02928 
 
 .03273 
 
 .03384 
 
 .03731 
 
 .0387'5 
 
 42 
 
 43 
 
 .02453 
 
 .02515 
 
 .02852 
 
 .02936 
 
 .03281 
 
 .03392 
 
 .03739 
 
 .03884 
 
 43 
 
 44 
 
 .02459 
 
 .02521 
 
 .02859 
 
 .02943 
 
 .03288 
 
 .03400 
 
 .03747 
 
 .03892 
 
 44 
 
 45 
 
 .02466 
 
 .02528 
 
 .02866 
 
 .02950 
 
 .03295 
 
 .03408 
 
 .03754 
 
 .03901 
 
 45 
 
 46 
 
 .02472 
 
 .02535 
 
 .02873 
 
 .02958 
 
 .03303 
 
 .03416 
 
 .08762 
 
 .03909 
 
 46 
 
 47 
 
 .02479 
 
 .02542 
 
 .02880 
 
 .02965 
 
 .03310 
 
 .03424 
 
 .03770 
 
 .03918 
 
 47 
 
 48 
 
 .02485 
 
 .02548 
 
 .02887 
 
 .02'J7'2 
 
 .03318 
 
 .03432 
 
 .03778 
 
 .03927 
 
 48 
 
 49 
 
 .02492 
 
 .02555 
 
 .02894 
 
 .02980 
 
 .03325 
 
 .03439 
 
 .03786 
 
 .03035 
 
 49 
 
 50 
 
 .0^98 
 
 .02562 
 
 .02900 
 
 .02987 
 
 .03333 
 
 .03447 
 
 .03794 
 
 .03944 
 
 50 
 
 51 
 
 .02504 
 
 .02569 
 
 .02907 
 
 .02994 
 
 .03340 
 
 .0345E 
 
 .03802 
 
 .03952 
 
 51 
 
 52 
 
 .02511 
 
 .02576 
 
 .02914 
 
 .03002 
 
 .03347 
 
 .03463 
 
 .03810 
 
 .03961 
 
 52 
 
 53 
 
 .02517 
 
 .02582 
 
 .02921 
 
 .03009 
 
 .03355 
 
 .03471 
 
 .03818 
 
 .03969 
 
 53 
 
 54 
 
 .02524 
 
 .02589 
 
 .02928 
 
 .03017 
 
 .03362 
 
 .03479 
 
 .03826 
 
 .03978 
 
 54 
 
 55 
 
 .02530 
 
 .02596 
 
 .02935 
 
 .03024 
 
 .03370 
 
 .03487 
 
 .03834 
 
 .03987 
 
 55 
 
 56 
 
 .02537 
 
 .02603 
 
 .02942 
 
 .03032 
 
 .03377 
 
 .03495 
 
 .03842 
 
 .03995 
 
 56 
 
 57 
 
 .02543 
 
 .02610 
 
 .02949 
 
 .03039 
 
 .03385 
 
 .03503 
 
 .03850 
 
 .04004 
 
 57 
 
 58 
 
 .02550 
 
 .02617 
 
 .02956 
 
 .03046 
 
 .03392 
 
 .03512 
 
 .03858 
 
 .04013 
 
 58 
 
 59 
 
 .02556 
 
 .02624 
 
 .02963 
 
 .03054 
 
 .03-100 
 
 .03520 
 
 .03866 
 
 .04021 
 
 59 
 
 60 
 
 .02563 
 
 .02630 
 
 .02970 
 
 .03061 
 
 .03407 
 
 .03528 
 
 .03874 
 
 .04030 
 
 60 
 
 250 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 16 
 
 17 
 
 18" 
 
 19' 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .03874 
 
 .04030 
 
 .04370 
 
 .04569 
 
 .04894 
 
 .05146 
 
 .05448 
 
 .05762 
 
 
 
 1 
 
 .03882 
 
 .04039 
 
 .04378 
 
 .04578 
 
 .04903 
 
 .05156 
 
 .05458 
 
 .05773 
 
 1 
 
 2 
 
 .03890 
 
 .04047 
 
 .04387 
 
 .04588 
 
 .04912 
 
 .05166 
 
 .05467 
 
 .05783 
 
 2 
 
 3 
 
 .03898 
 
 .04056 
 
 .04395 
 
 .04597 
 
 .04921 
 
 .05176 
 
 .05477 
 
 .05794 
 
 3 
 
 4 
 
 .03906 
 
 .04065 
 
 .04404 
 
 .04606 
 
 .04930 
 
 .05186 
 
 .65486 
 
 .05805 
 
 4 
 
 5 
 
 .03914 
 
 .04073 
 
 .04412 
 
 .04616 
 
 .04939 
 
 .05196 
 
 .05496 
 
 .05815 
 
 5 
 
 6 
 
 .03922 
 
 .04082 
 
 .04421 
 
 .04625 
 
 .04948 
 
 .05206 
 
 .05505 
 
 .05826 
 
 6 
 
 7 
 
 .03930 
 
 .04091 ! 
 
 .04429 
 
 .04635 
 
 .04957 
 
 .65216 
 
 .05515 
 
 .05836 
 
 7 
 
 8 
 
 .03938 
 
 .04100 | 
 
 .04438 
 
 .04644 
 
 .04967 
 
 .05226 ! 
 
 .05524 
 
 .05847 
 
 8 
 
 g 
 
 .03946 
 
 .04108 
 
 .04446 
 
 .04653 
 
 .04976 
 
 .05236 
 
 .05534 
 
 .05858 
 
 9 
 
 10 
 
 .03954 
 
 .04117 | 
 
 .04455 
 
 .04663 
 
 .04985 
 
 .05246 
 
 .05543 
 
 .05869 
 
 10 
 
 11 
 
 .03963 
 
 .04126 
 
 .04464 
 
 .04672 
 
 .04994 
 
 .05256 
 
 .05553 
 
 .05879 
 
 11 
 
 13 
 
 .03971 
 
 .04135 1 
 
 .04472 
 
 .04683 
 
 .05003 
 
 .05266 
 
 .05562 
 
 .05890 
 
 12 
 
 13 
 
 .03979 
 
 .04144 ! 
 
 .04481 
 
 .04691 
 
 .05012 
 
 .05276 
 
 .05572 
 
 .05901 
 
 13 
 
 14 
 
 .03987 
 
 .04152 ' 
 
 .04489 
 
 .04700 
 
 .05021 
 
 .05286 
 
 .05582 
 
 .05911 
 
 14 
 
 15 
 
 .03995 
 
 .04161 i 
 
 .04498 
 
 .04710 [ 
 
 .05030 
 
 .05297 
 
 .05591 
 
 .05922 
 
 15 
 
 16 
 
 .01003 
 
 .04170 
 
 .04507 
 
 .04719 i 
 
 .05039 
 
 .05307 
 
 .05601 
 
 .05933 
 
 16 
 
 17 
 
 .04011 
 
 .04179 
 
 .04515 
 
 .04729 
 
 .05048 
 
 .05317 
 
 .05610 
 
 .05944 
 
 17 
 
 18 
 
 .01019 
 
 .04188 
 
 .04524 
 
 .04738 
 
 .05057 
 
 .05327 
 
 .05620 
 
 .05955 
 
 18 
 
 19 
 
 .04028 
 
 .04197 
 
 .04533 
 
 .04748 
 
 .05067 
 
 .05337 i 
 
 .05630 
 
 .05965 
 
 19 
 
 20 
 
 .04036 
 
 .04206 
 
 .04541 
 
 .04757 
 
 .05076 
 
 .05347 
 
 .05639 
 
 .05976 
 
 20 
 
 21 
 
 .04044 
 
 .04214 
 
 .04550 
 
 .04767 
 
 .05085 
 
 .05357 
 
 .05649 
 
 .05987 
 
 21 
 
 22 
 
 .04052 
 
 .04223 
 
 .04559 
 
 .04776 
 
 .05094 
 
 .05367 
 
 .05658 
 
 .05998 
 
 22 
 
 23 
 
 .04060 
 
 .04232 
 
 .04567 
 
 .04786 
 
 .05103 
 
 .05378 
 
 .05668 
 
 .06009 
 
 23 
 
 24 
 
 .04069 
 
 .04241 
 
 .04576 
 
 .04795 
 
 .05112 
 
 .05388 
 
 .05678 
 
 .06020 
 
 24 
 
 25 
 
 .04077 
 
 .04250 | 
 
 .04585 
 
 .04805 
 
 .05122 
 
 .05398 
 
 .05687 
 
 .06030 
 
 25 
 
 26 
 
 .04085 
 
 .04259 ! 
 
 .04593 
 
 .04815 
 
 .05131 
 
 .05408 
 
 .05697 
 
 .06041 
 
 26 
 
 27 
 
 .04093 
 
 .04268 
 
 .04602 
 
 .04824 
 
 .05140 
 
 .05418 
 
 .05707 
 
 .06052 
 
 27 
 
 28 
 
 .04102 
 
 .04277 
 
 .04611 
 
 .04834 
 
 .05149 
 
 .05429 
 
 .05716 
 
 .06063 
 
 28 
 
 29 
 
 .04110 
 
 .04286 
 
 .04020 
 
 .04843 
 
 .05158 
 
 .05439 
 
 .05726 
 
 .06074 
 
 29 
 
 30 
 
 .04118 
 
 .04295 
 
 .04628 
 
 .04853 
 
 .05168 
 
 .05449 
 
 .05736 
 
 .06085 
 
 30 
 
 31 
 
 .04126 
 
 .04304 
 
 .04637 
 
 .04863 
 
 .05177 
 
 .05460 
 
 .05746 
 
 .06096 
 
 31 
 
 32 
 
 .04ia5 
 
 .04313 
 
 .04646 
 
 .04872 
 
 .05186 
 
 .05470 
 
 .05755 
 
 .06107 
 
 82 
 
 33 
 
 .04143 
 
 .04322 
 
 .04655 
 
 .04882 
 
 .05195 
 
 .05480 
 
 .05765 
 
 .06118 
 
 33 
 
 34 
 
 .04151 
 
 .04331 
 
 .04663 
 
 .04891 
 
 .05205 
 
 .05490 
 
 .05775 
 
 .06129 
 
 34 
 
 35 
 
 .04159 
 
 .04340 
 
 .04672 
 
 .04901 
 
 .05214 
 
 .05501 
 
 .05785 
 
 .06140 
 
 35 
 
 36 
 
 .04168 
 
 .04349 
 
 .04681 
 
 .04911 
 
 .05223 
 
 .05511 
 
 .05794 
 
 .06151 
 
 36 
 
 37 
 
 .04176 
 
 .04358 
 
 .04690 
 
 .04920 
 
 .05232 
 
 .05521 
 
 .05804 
 
 .06162 
 
 37 
 
 38 
 
 .04184 
 
 .04367 
 
 .04699 
 
 .04930 
 
 .05242 
 
 .05532 
 
 .05814 
 
 .06173 
 
 38 
 
 39 
 
 .04193 
 
 .04376 
 
 .04707 
 
 .04940 
 
 .05251 
 
 .C5542 
 
 .05824 
 
 .06184 
 
 39 
 
 40 
 
 .01201 
 
 .04385 
 
 .04716 
 
 .04950 
 
 .05260 
 
 .05552 
 
 .05833 
 
 .06195 
 
 40 
 
 41 
 
 .04209 
 
 .04394 
 
 .01725 
 
 .04959 
 
 .05270 
 
 .05563 
 
 .05843 
 
 .06206 
 
 41 
 
 42 
 
 .01218 
 
 .04403 
 
 [04784 
 
 .04969 
 
 .05279 
 
 .05573 
 
 .05853 
 
 .06217 
 
 42 
 
 43 
 
 .04226 
 
 .04413 
 
 .04743 
 
 .04979 
 
 .05288 
 
 .05584 
 
 .05863 
 
 .06228 
 
 43 
 
 44 
 
 .04234 
 
 .04422 
 
 .04752 
 
 .04989 
 
 .05298 
 
 .05594 
 
 .05873 
 
 .06239 
 
 44 
 
 45 
 
 .04243 
 
 .04431 
 
 .04760 
 
 .04998 
 
 .05307 
 
 .05604 
 
 .05882 
 
 .06250 
 
 45 
 
 46 
 
 .04251 
 
 .04440 
 
 .04769 
 
 .05008 
 
 .05316 
 
 .05615 
 
 .05892 
 
 .06261 
 
 46 
 
 47 
 
 .04260 
 
 .04449 
 
 .04778 
 
 .05018 
 
 .05326 
 
 .05625 
 
 .05902 
 
 .06272 
 
 47 
 
 48 
 
 .04268 
 
 .04458 
 
 .04787 
 
 .05028 
 
 .05335 
 
 .05636 
 
 .05912 
 
 .06283 
 
 48 
 
 49 
 
 .04276 
 
 .04468 
 
 .04796 
 
 .05038 
 
 .05344 
 
 .05646 
 
 .05922 
 
 .06295 
 
 49 
 
 50 
 
 .04285 
 
 .04477 
 
 .04805 
 
 .05047 
 
 .05354 
 
 .05057 
 
 .05932 
 
 .06306 
 
 50 
 
 51 
 
 .04293 
 
 .04486 
 
 .04814 
 
 .05057 
 
 .05363 
 
 .05667 
 
 .05942 
 
 .06317 
 
 51 
 
 52 
 
 .04302 
 
 .04495 
 
 .04823 
 
 .05067 
 
 .05373 
 
 .05678 
 
 .05951 
 
 .06328 
 
 52 
 
 53 
 
 .04310 
 
 .04504 
 
 .04832 
 
 .05077 
 
 .05382 
 
 .05688 
 
 .05961 
 
 .06339 
 
 53 
 
 54 
 
 .04319 
 
 .04514 
 
 .04841 
 
 .05087 i .05391 
 
 .05699 
 
 .05971 
 
 .06350 
 
 54 
 
 55 
 
 .04327 
 
 .04523 
 
 .04850 
 
 .05097 |l .05401 
 
 .05709 
 
 .05981 
 
 .06362 
 
 55 
 
 56 
 
 .04336 
 
 .04532 
 
 .04858 
 
 .05107 
 
 .05410 
 
 .05720 
 
 .05991 
 
 .06373 
 
 56 
 
 57 
 
 .04344 
 
 .04541 
 
 .04867 
 
 .05116 
 
 .05420 
 
 .05730 
 
 .06001 
 
 .06384 
 
 57 
 
 58 
 
 .04&53 
 
 .04551 
 
 .04876 
 
 .05126 
 
 .05429 
 
 .05741 
 
 .06011 
 
 .06895 
 
 58 
 
 59 
 
 .04361 
 
 .04560 
 
 .04885 
 
 .05136 
 
 .05439 
 
 .05751 
 
 .06021 
 
 .06407 
 
 59 
 
 60 
 
 .04370 
 
 .04569 
 
 .04894 
 
 .05146 
 
 .05448 
 
 .05762 
 
 .06031 
 
 .06418 
 
 60 
 
 251 
 
TABLE XIII. VEESINES AND EXSECANTS. 
 
 / 
 
 20" 
 
 21 
 
 22 
 
 23' 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 ~0~ 
 
 .06031 
 
 .06418 
 
 .06642 
 
 .07115 
 
 .07282 
 
 .07853 
 
 .07950 
 
 .08636 
 
 ~T 
 
 1 
 
 .06041 
 
 .06429 
 
 .06652 
 
 .07126 
 
 .07293 
 
 .07866 
 
 .07961 
 
 .08649 
 
 i 
 
 2 
 
 .06051 
 
 .06440 
 
 .06663 
 
 .07138 
 
 .07303 
 
 .07879 
 
 .07972 
 
 .08663 
 
 2 
 
 3 
 
 .06061 
 
 .06452 
 
 .06673 
 
 .07150 
 
 .07314 
 
 .07892 
 
 .07984 
 
 .08676 
 
 3 
 
 4 
 
 .06071 
 
 .06463 
 
 .06684 
 
 .07162 
 
 .07325 
 
 .07904 
 
 .07995 
 
 .08690 
 
 4 
 
 5 
 
 .06081 
 
 .06474 
 
 .06694 
 
 .07174 
 
 .07336 
 
 .07917 
 
 .08006 
 
 .08703 
 
 5 
 
 6 
 
 .06091 
 
 .06486 
 
 .06705 
 
 .07186 
 
 .07347 
 
 .07930 
 
 .08018 
 
 .08717 
 
 6 
 
 7 
 
 .06101 
 
 .06497 
 
 .06715 
 
 .07199 
 
 .07358 
 
 .07943 
 
 .08029 
 
 .08730 
 
 7 
 
 8 
 
 .06111 
 
 .06508 
 
 .06726 
 
 .07211 
 
 .07369 
 
 .07955 
 
 .08041 
 
 .08744 
 
 8 
 
 9 
 
 .06121 
 
 .06520 
 
 .06736 
 
 .07223 
 
 .07380 
 
 .07968 
 
 .08052 
 
 .08757 
 
 9 
 
 10 
 
 .06131 
 
 .06531 
 
 .06747 
 
 .07235 
 
 .07391 
 
 .07981 
 
 .08064 
 
 .08771 
 
 10 
 
 11 
 
 .06141 
 
 .06542 
 
 .06757 
 
 .07247 
 
 .07402 
 
 .07994 
 
 .08075 
 
 .08784 
 
 11 
 
 12 
 
 .06151 
 
 .06554 
 
 .06768 
 
 .07259 
 
 .07413 
 
 .08006 
 
 .08086 
 
 .08798 
 
 12 
 
 13 
 
 .06161 
 
 .06565 
 
 .06778 
 
 .07271 
 
 .07424 
 
 .08019 
 
 .08098 
 
 .08811 
 
 13 
 
 14 
 
 .06171 
 
 .06577 
 
 .06789 
 
 .07283 
 
 .07435 
 
 .08032 
 
 .08109 
 
 .08825 
 
 14 
 
 15 
 
 .06181 
 
 .06588 
 
 .06799 
 
 .07295 
 
 .07446 
 
 .08045 
 
 .08121 
 
 .08839 
 
 15 
 
 16 
 
 .06191 
 
 .06600 
 
 .06810 
 
 .07307 
 
 .07457 
 
 .08058 
 
 .08132 
 
 .08852 
 
 16 
 
 17 
 
 .06201 
 
 .06611 
 
 .06820 
 
 .07320 
 
 .07468 
 
 .08071 
 
 .08144 
 
 .08866 
 
 17 
 
 18 
 
 .06211 
 
 .06622 
 
 .06831 
 
 .07332 
 
 .07479 
 
 .08084 
 
 .08155 
 
 .08880 
 
 18 
 
 19 
 
 .06221 
 
 .06634 
 
 .06841 
 
 .07344 
 
 .07490 
 
 .08097 
 
 .08167 
 
 .08893 
 
 19 
 
 20 
 
 .06231 
 
 .06645 
 
 .06852 
 
 .07356 
 
 .07501 
 
 .08109 
 
 .08178 
 
 .08907 
 
 20 
 
 21 
 
 .06241 
 
 .06657 
 
 .06863 
 
 .07368 
 
 .07512 
 
 .08122 
 
 .08190 
 
 .08921 
 
 21 
 
 22 
 
 .06252 
 
 .06668 
 
 .06873 
 
 .07380 
 
 .07523 
 
 .08135 
 
 .08201 
 
 .08934 
 
 22 
 
 23 
 
 .06262 
 
 .06680 
 
 .06884 
 
 .07393 
 
 .07534 
 
 .08148 
 
 .08213 
 
 .08948 
 
 23 
 
 24 
 
 .06272 
 
 .06691 
 
 .06894 
 
 .07405 
 
 .07545 
 
 .08161 
 
 .08225 
 
 .08962 
 
 24 
 
 25 
 
 .06282 
 
 .06703 
 
 .06905 
 
 .07417 
 
 .07556 
 
 .08174 
 
 .08236 
 
 .08975 
 
 25 
 
 26 
 
 .06292 
 
 .06715 
 
 .06916 
 
 .07429 
 
 .07568 
 
 .08187 
 
 .08248 
 
 .08989 
 
 26 
 
 27 
 
 .06302 
 
 .06726 
 
 .06926 
 
 .07442 
 
 .07579 
 
 .08200 
 
 .08259 
 
 .09003 
 
 27 
 
 28 
 
 .06312 
 
 .06738 
 
 .06937 
 
 .07454 
 
 .07590 
 
 .08213 
 
 .08271 
 
 .09017 
 
 23 
 
 29 
 
 .06323 
 
 .06749 
 
 .06948 
 
 .07466 
 
 .07601 
 
 .08226 
 
 .08282 
 
 .09030 
 
 23 
 
 30 
 
 .06333 
 
 .06761 
 
 .06958 
 
 .07479 
 
 .07612 
 
 .08239 
 
 .08294 
 
 .09044 
 
 30 
 
 31 
 
 .06343 
 
 .06773 
 
 .06969 
 
 .07491 
 
 .07623 
 
 .08252 
 
 .08306 
 
 .09058 
 
 31 
 
 32 
 
 .06353 
 
 .06784 
 
 .06980 
 
 .07503 
 
 .07634 
 
 .08265 
 
 .08317 
 
 .09072 
 
 32 
 
 33 
 
 .06363 
 
 .06796 
 
 .06990 
 
 .07516 
 
 .07645 
 
 .08278 
 
 .08329 
 
 .09086 
 
 33 
 
 34 
 
 .06374 
 
 .06807 
 
 .07001 
 
 .07528 
 
 .07657 
 
 .08291 
 
 .08340 
 
 .09099 
 
 34 
 
 35 
 
 .06384 
 
 .06819 
 
 .07012 
 
 .07540 
 
 .07668 
 
 .08305 
 
 .08352 
 
 .09113 
 
 35 
 
 36 
 
 .06394 
 
 .06831 
 
 .07022 
 
 .07553 
 
 .07679 
 
 .08318 
 
 .08364 
 
 .09127 
 
 36 
 
 37 
 
 .06404 
 
 .06843 
 
 .07033 
 
 .07565 
 
 .07690 
 
 .08331 
 
 .08375 
 
 .09141 
 
 37 
 
 38 
 
 .06415 
 
 .06854 
 
 .07044 
 
 .07578 
 
 .07701 
 
 .08344 
 
 .08387 
 
 .09155 
 
 38 
 
 39 
 
 .06425 
 
 .06866 
 
 .07055 
 
 .07590 
 
 .07713 
 
 .08357 
 
 .08399 
 
 .09169 
 
 39 
 
 40 
 
 .06435 
 
 .06878 
 
 .07065 
 
 .07602 
 
 .07784 
 
 .08370 
 
 .08410 
 
 .09183 
 
 40 
 
 41 
 
 .06445 
 
 .06889 
 
 .07076 
 
 .07615 
 
 .07735 
 
 .08383 
 
 .08422 
 
 .09197 
 
 41 
 
 42 
 
 .06456 
 
 .06901 
 
 .07087 
 
 .07627 
 
 .07746 
 
 .08397 
 
 .08434 
 
 .09211 
 
 42 
 
 43 
 
 .06466 
 
 .06913 
 
 .07098 
 
 .07640 
 
 .07757 
 
 .08410 
 
 .08445 
 
 .09224 
 
 43 
 
 44 
 
 .06476 
 
 .06925 
 
 .07108 
 
 .07652 
 
 .07769 
 
 .08423 
 
 .08457 
 
 .09238 
 
 44 
 
 45 
 
 .06486 
 
 .06936 
 
 .07119 
 
 .07665 
 
 .07780 
 
 .08436 
 
 .08469 
 
 .09252 
 
 45 
 
 46 
 
 .06497 
 
 .06948 
 
 .07130 
 
 .07677 
 
 .07791 
 
 .08449 
 
 .08481 
 
 .09266 
 
 46 
 
 47 
 
 .06507 
 
 .06960 
 
 .07141 
 
 .07690 
 
 .07802 
 
 .08463 
 
 .08492 
 
 .09280 
 
 47 
 
 48 
 
 .06517 
 
 .06972 
 
 .07151 
 
 .07702 
 
 .07814 
 
 .08476 
 
 .08504 
 
 .09294 
 
 48 
 
 49 
 
 .06528 
 
 .06984 
 
 .07162 
 
 .07715 
 
 .07825 
 
 .08489 
 
 .08516 
 
 .09308 
 
 49 
 
 50 
 
 .06538 
 
 .06995 
 
 .07173 
 
 .07727 
 
 .07836 
 
 .08503 
 
 .08528 
 
 .09323 
 
 50 
 
 51 
 
 .06548 
 
 .07007 
 
 .07184 
 
 .07740 
 
 .07848 
 
 .08516 
 
 .08539 
 
 .09337 
 
 51 
 
 52 
 
 .06559 
 
 .07019 
 
 .07195 
 
 .07752 
 
 .07859 
 
 .08529 
 
 .08551 
 
 .09351 
 
 52 
 
 53 
 
 .06569 
 
 .07031 
 
 .07206 
 
 .07765 
 
 .07870 
 
 .08542 
 
 .08563 
 
 .09365 
 
 53 
 
 54 
 
 .06580 
 
 .07043 
 
 .07216 
 
 .0777'8 
 
 .07881 
 
 .08556 
 
 .08575 
 
 .09379 
 
 54 
 
 55 
 
 .06590 
 
 .07055 
 
 .07227 
 
 .07790 
 
 .07893 
 
 .08569 
 
 .08586 
 
 .09393 
 
 55 
 
 56 
 
 .06600 
 
 .07067 
 
 .07238 
 
 .07803 
 
 .07904 
 
 .08582 
 
 .08598 
 
 .09407 
 
 56 
 
 57 
 
 .06611 
 
 .07079 
 
 .07249 
 
 .07816 
 
 .07915 
 
 .08596 
 
 .08610 
 
 .09421 
 
 57 
 
 58 
 
 .06621 
 
 .07091 
 
 .07260 
 
 .07828 
 
 .07927 
 
 .08609 
 
 .08622 
 
 .09435 
 
 58 
 
 59 
 
 .06632 
 
 .07103 
 
 .07271 
 
 .07841 
 
 .07938 
 
 .08623 
 
 .08634 
 
 .09449 
 
 59 
 
 60 
 
 .06642 
 
 .07115 
 
 .07282 
 
 .07853 
 
 .07950 
 
 .08636 
 
 .08645 
 
 .09464 
 
 60 
 
 252 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 f* 
 
 24 
 
 25 
 
 26 
 
 27 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .08645 
 
 .09464 
 
 .09309 
 
 .10338 ; 
 
 .10121 
 
 .11260 
 
 .10899 
 
 .12233 
 
 
 
 1 
 
 .08657 
 
 .09478 
 
 .09382 
 
 .10353 
 
 .10133 
 
 .11276 
 
 .10913 
 
 .12249 
 
 1 
 
 2 
 
 .08669 
 
 .09492 
 
 .09394 
 
 .10368 
 
 .10146 
 
 .11292 
 
 .10926 
 
 .12266 
 
 2 
 
 3 
 
 .08681 
 
 .09506 
 
 .09406 
 
 .10383 
 
 .10159 
 
 .11308 
 
 ! 10939 
 
 .12283 
 
 3 
 
 4 
 
 .03693 
 
 .09520 
 
 .09418 
 
 .10398 
 
 .10172 
 
 .11323 
 
 .10952 
 
 .12299 
 
 4 
 
 5 
 
 .08705 
 
 .09535 
 
 .09431 
 
 .10413 
 
 .10184 
 
 .11339 
 
 .10965 
 
 .12316 
 
 5 
 
 6 
 
 .08717 
 
 .09549 
 
 .09443 
 
 .10428 
 
 .10197 
 
 .11355 
 
 .10979 
 
 .12333 
 
 6 
 
 7 
 
 .08728 
 
 .09563 
 
 .09455 
 
 .10443 
 
 .10210 
 
 .11371 
 
 .10992 
 
 .12349 
 
 7 
 
 8 
 
 .08740 
 
 .09577 
 
 .09468 
 
 .10458 
 
 .10223 
 
 .11387 
 
 .11005 
 
 .12366 
 
 8 
 
 9 
 
 .08752 
 
 .09592 
 
 .09480 
 
 .10473 
 
 .10236 
 
 .11403 
 
 .11019 
 
 .12383 
 
 9 
 
 10 
 
 .08764 
 
 .09606 
 
 .09493 
 
 .10488 
 
 .10248 
 
 .11419 
 
 .11032 
 
 .12400 
 
 10 
 
 11 
 
 .08776 
 
 .09620 
 
 .09505 
 
 .10503 
 
 .10261 
 
 .11435 
 
 .11045 
 
 .12416 
 
 11 
 
 13 
 
 .08788 
 
 .09635 
 
 .00517 
 
 .10518 
 
 .10374 
 
 .11451 
 
 .11058 
 
 .12433 
 
 12 
 
 13 
 
 .08800 
 
 .09649 
 
 .C9530 
 
 .10533 
 
 .10287 
 
 .11467 
 
 .11072 
 
 .12450 
 
 13 
 
 14 
 
 .08812 
 
 .09663 
 
 .09542 
 
 .10549 
 
 .10300 
 
 .11483 
 
 .11085 
 
 .12467 
 
 14 
 
 15 
 
 .08824 
 
 .09678 
 
 .09554 
 
 .10564 
 
 .10313 
 
 .11499 
 
 .11098 
 
 .12484 
 
 15 
 
 13 
 
 .08836 
 
 .09692 
 
 .09567 
 
 .10579 
 
 .10326 
 
 .11515 
 
 .11112 
 
 .12501 
 
 16 
 
 17 
 
 .08848 
 
 .09707 
 
 .09579 
 
 . 10594 
 
 .10338 
 
 .11531 
 
 .11125 
 
 .12518 
 
 17 
 
 13 
 
 .08860 
 
 .09721 
 
 .09592 
 
 .10609 
 
 .10351 
 
 .11547 
 
 .11138 
 
 .12534 
 
 18 
 
 13 
 
 .08872 
 
 .09735 
 
 .09604 
 
 .10625 
 
 .10304 
 
 .11563 
 
 .11152 
 
 .12551 
 
 19 
 
 23 
 
 .08884 
 
 .09750 
 
 .09617 
 
 .10640 
 
 .10377 
 
 .11579 
 
 .11165 
 
 .12568 
 
 20 
 
 21 
 
 .08896 
 
 .09764 
 
 .09629 
 
 .10655 
 
 .10390 
 
 .11595 
 
 .11178 
 
 .12585 
 
 21 
 
 23 
 
 .08903 
 
 .09779 
 
 .09642 
 
 .10070 
 
 .10403 
 
 .11611 
 
 .11192 
 
 .12602 
 
 23 
 
 23 
 
 .03920 
 
 .09793 
 
 .09654 
 
 .10636 
 
 .10416 
 
 .11627 
 
 .11205 
 
 .12619 
 
 23 
 
 24 
 
 .08932 
 
 .09808 
 
 .09666 
 
 .10701 
 
 .10429 
 
 .11643 
 
 .11218 
 
 .12636 
 
 24 
 
 25 
 
 .08944 
 
 .09822 
 
 .09679 
 
 .10716 
 
 .10442 
 
 .11659 
 
 .11232 
 
 .12653 
 
 25 
 
 26 
 
 .08956 
 
 .09837 
 
 .09691 
 
 .10731 
 
 .10455 
 
 .11675 
 
 .11245 
 
 .12670 
 
 26 
 
 27 
 
 .08968 
 
 .09851 
 
 .097'04 
 
 .10747 
 
 .10468 
 
 .11691 
 
 .11259 
 
 .12687 
 
 27 
 
 28 
 
 .08980 
 
 .09866 
 
 .09716 
 
 .10762 
 
 .10481 
 
 .11708 
 
 .11272 
 
 .12704 
 
 28 
 
 29 
 
 .03992 
 
 .09880 
 
 .09729 
 
 .10777 
 
 . .10494 
 
 .11724 
 
 .11285 
 
 .12721 
 
 29 
 
 30 
 
 .09004 
 
 .09895 
 
 .09741 
 
 .10793 
 
 .10507 
 
 .11740 
 
 .11299 
 
 .12738 
 
 30 
 
 31 
 
 .09016 
 
 .09909 
 
 .09754 
 
 .10808 
 
 .10520 
 
 .11756 
 
 .11312 
 
 .12755 
 
 31 
 
 33 
 
 .09028 
 
 .09924 
 
 .09767 
 
 .10824 
 
 .10533 
 
 .11772 
 
 .11326 
 
 .12772 
 
 32 
 
 33 
 
 .09040 
 
 .09939 
 
 .09779 
 
 .10839 
 
 .10546 
 
 .11789 
 
 .11339 
 
 .12789 
 
 33 
 
 34 
 
 .09052 
 
 .09953 
 
 .09792 
 
 .10854 
 
 .10559 
 
 .11805 
 
 .11353 
 
 .12807 
 
 34 
 
 35 
 
 .09064 
 
 .09908 
 
 .09804 
 
 .10870 
 
 .10572 
 
 .11821 
 
 .11366 
 
 .12824 
 
 35 
 
 36 
 
 .09076 
 
 .09982 
 
 .09817 
 
 .10885 
 
 .10585 
 
 .11838 
 
 .11380 
 
 .12841 
 
 36 
 
 37 
 
 .09089 
 
 .09997 
 
 .09829 
 
 .10901 
 
 .10598 
 
 .11854 
 
 .11393 
 
 .12858 
 
 37 
 
 38 
 
 .09101 
 
 .10012 
 
 .09842 
 
 .10916 
 
 .10611 
 
 .11870 
 
 .11407 
 
 .12875 
 
 38 
 
 39 
 
 .09113 
 
 .10026 
 
 .09854 
 
 .10932 
 
 .10624 
 
 .11886 
 
 .11420 
 
 .12892 
 
 39 
 
 40 
 
 .09125 
 
 .10041 
 
 .09867 
 
 .10947 
 
 .10637 
 
 .11903 
 
 .11434 
 
 .12910 
 
 40 
 
 41 
 
 .09137 
 
 .10055 
 
 .09880 
 
 .10963 
 
 .10650 
 
 .11919 
 
 .11447 
 
 .12927 
 
 41 
 
 43 
 
 .09149 
 
 .10071 
 
 .09892 
 
 .10978 
 
 .10663 
 
 .11936 
 
 .11461 
 
 .12944 
 
 42 
 
 43 
 
 .09161 
 
 .10085 
 
 .09905 
 
 .10994 
 
 .10676 
 
 .11952 
 
 .11474 
 
 .12961 
 
 43 
 
 44 
 
 .09174 
 
 .10100 
 
 .09918 
 
 .11009 
 
 .10689 
 
 .11968 
 
 .11488 
 
 .12979 
 
 44 
 
 45 
 
 .09186 
 
 .10115 
 
 .09930 
 
 .11025 
 
 .10702 
 
 .11985 
 
 .11501 
 
 .12996 
 
 45 
 
 46 
 
 .09198 
 
 .10130 
 
 .09943 
 
 .11041 
 
 .10715 
 
 .12001 
 
 .11515 
 
 .13013 
 
 46 
 
 47 
 
 .09210 
 
 .10144 
 
 .09955 
 
 .11056 
 
 .10728 
 
 .12018 
 
 .11528 
 
 .13031 
 
 47 
 
 48 
 
 .00222 
 
 .10159 
 
 .09963 
 
 .11072 
 
 .10741 
 
 .12034 
 
 .11542 
 
 .13048 
 
 48 
 
 49 
 
 .09234 
 
 .10174 
 
 .09981 
 
 .11087 
 
 .10755 
 
 .12051 
 
 .11555 
 
 .13065 
 
 49 
 
 50 
 
 .09247 
 
 .10189 
 
 .09993 
 
 .11103 
 
 .10768 
 
 .12067 
 
 .11569 
 
 .13083 
 
 50 
 
 51 
 
 .09259 
 
 .10204 
 
 .10006 
 
 .11119 
 
 .10781 
 
 .12084 
 
 .11583 
 
 .13100 
 
 51 
 
 52 
 
 .09271 
 
 .10218 
 
 .10019 
 
 .11134 
 
 .10794 
 
 .12100 
 
 .11596 
 
 .13117 
 
 52 
 
 53 
 
 .09283 
 
 .10233 
 
 .10032 
 
 .11150 
 
 .10807 
 
 .12117 
 
 .11610 
 
 .13135 
 
 53 
 
 54 
 
 .09296 
 
 .10248 
 
 .10044 
 
 .11166 
 
 .10820 
 
 .12133 
 
 .11623 
 
 .13152 
 
 54 
 
 55 
 
 .09308 
 
 .10263 
 
 .10057 
 
 .11181 
 
 .10833 
 
 .12150 
 
 .11637 
 
 .13170 
 
 55 
 
 56 
 
 .09320 
 
 .10278 
 
 .10070 
 
 .11197 
 
 .10847 
 
 .12166 
 
 .11651 
 
 .13187 
 
 56 
 
 57 
 
 .09332 
 
 .10293 
 
 .10082 
 
 .11213 
 
 .10860 
 
 .12183 
 
 .11664 
 
 .13205 
 
 57 
 
 58 
 
 .09345 
 
 .10308 
 
 .10095 
 
 .11229 
 
 .10873 
 
 .12199 
 
 .11678 
 
 .13222 
 
 58 
 
 59 
 
 .09&57 
 
 .10323 
 
 .10108 
 
 .11244 
 
 .10886 
 
 .12216 
 
 .11692 
 
 .13240 
 
 59 
 
 60 
 
 .09369 
 
 .10338 
 
 .10121 
 
 .11260 
 
 .10899 
 
 .12233 
 
 .11705 
 
 .13257 
 
 60 
 
 253 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 2 
 
 8 
 
 2 
 
 9" 
 
 31 
 
 ) 
 
 3: 
 
 L 
 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 
 .11705 
 
 .13257 
 
 .12538 
 
 .14335 
 
 .13397 
 
 .15470 
 
 .14283 
 
 .16663 
 
 
 
 1 
 
 .11719 
 
 .13275 
 
 .12552 
 
 .14354 
 
 .13412 
 
 .15489 
 
 .14298 
 
 .16684 
 
 1 
 
 2 
 
 .11733 
 
 .13292 
 
 .12566 
 
 .14372 
 
 .13427 
 
 .15509 
 
 .14313 
 
 .16704 
 
 2 
 
 a 
 
 .11746 
 
 .13310 
 
 .12580 
 
 .14391 
 
 .13441 
 
 .15528 
 
 .14328 
 
 .16725 
 
 3 
 
 4 
 
 .11760 
 
 .13327 
 
 .12595 
 
 .14409 
 
 .13456 
 
 .15548 
 
 .14343 
 
 .16745 
 
 4 
 
 5 
 
 .11774 
 
 .13345 
 
 .12609 
 
 .14428 
 
 .13470 
 
 .15567 
 
 .14358 
 
 .16766 
 
 5 
 
 6 
 
 .11787 
 
 .13362 
 
 .12623 
 
 .14446 
 
 .13485 
 
 .15587 
 
 .14373 
 
 .16786 
 
 6 
 
 7 
 
 .11801 
 
 .13380 
 
 .12637 
 
 .14465 
 
 .13499 
 
 .15606 
 
 .14388 
 
 .16806 
 
 7 
 
 8 
 
 .11815 
 
 .13398 
 
 .12651 
 
 .14483 
 
 .13514 
 
 .15626 
 
 .14403 
 
 .16827 
 
 8 
 
 9 
 
 .11828 
 
 .13415 
 
 .12665 
 
 .14502 
 
 .13529 
 
 .15645 
 
 .14418 
 
 .16848 
 
 9 
 
 10 
 
 .11842 
 
 .13433 
 
 .12679 
 
 .14521 
 
 .13543 
 
 .15665 
 
 .14433 
 
 .16868 
 
 10 
 
 11 
 
 .11856 
 
 .13451 
 
 .12694 
 
 .14539 
 
 .13558 
 
 .15684 
 
 .14449 
 
 .16889 
 
 11 
 
 12 
 
 .11870 
 
 .13468 
 
 .12708 
 
 .14558 
 
 .13573 
 
 .157-04 
 
 .14464 
 
 .16909 
 
 12 
 
 13 
 
 .11883 
 
 .13486 
 
 .12722 
 
 .14576 
 
 .13587 
 
 .15724 
 
 .14479 
 
 .16930 
 
 13 
 
 14 
 
 .11897 
 
 .13504 
 
 .12736 
 
 .14595 
 
 .13602 
 
 .15743 
 
 .14494 
 
 .1G950 
 
 14 
 
 15 
 
 .11911 
 
 .13521 
 
 .12750 
 
 .14014 
 
 .13616 
 
 .15763 
 
 .14509 
 
 .16971 
 
 15 
 
 16 
 
 .11925 
 
 .13539 
 
 .12765 
 
 .14032 
 
 .13631 
 
 .15782 
 
 .14524 
 
 .1G992 
 
 16 
 
 17 
 
 .11938 
 
 .13557 
 
 .12779 
 
 .14351 
 
 .13646 
 
 .15802 
 
 .14539 
 
 .17012 
 
 17 
 
 18 
 
 .11952 
 
 .13575 
 
 .12793 
 
 .14070 
 
 .13660 
 
 .15822 
 
 .14554 
 
 .17033 
 
 18 
 
 19 
 
 .11966 
 
 .13593 
 
 .12807 
 
 .14GS9 
 
 .13G75 
 
 .15841 
 
 .14569 
 
 \17C5i 
 
 19 
 
 20 
 
 .11980 
 
 .13610 
 
 .12822 
 
 .14707 
 
 .13690 
 
 .15861 
 
 .14584 
 
 .17075 
 
 20 
 
 21 
 
 .11994 
 
 .13628 
 
 .12836 
 
 .14726 
 
 .13705 
 
 .15881 
 
 .14599 
 
 .17095 
 
 21 
 
 22 
 
 .12007 
 
 .13646 
 
 .12850 
 
 .14745 
 
 .13719 
 
 .15901 
 
 .14615 
 
 .17116 
 
 22 
 
 23 
 
 .12021 
 
 .13664 
 
 .12864 
 
 .14764 
 
 .13734 
 
 .15920 
 
 .14630 
 
 .17137 
 
 23 
 
 24 
 
 .12035 
 
 .13682 
 
 .12879 
 
 .14782 
 
 .13749 
 
 .15940 
 
 .14645 
 
 .17158 
 
 24 
 
 25 
 
 .12049 
 
 .13700 
 
 .12893 
 
 .14801 
 
 .13763 
 
 .15960 
 
 .146GO 
 
 .17178 
 
 25 
 
 26 
 
 .12063 
 
 .13718 
 
 .12207 
 
 .14820 
 
 .13778 
 
 .15980 
 
 .14675 
 
 .17199 
 
 26 
 
 27 
 
 .12077 
 
 .13735 
 
 .12921 
 
 .1-1839 
 
 .13793 
 
 .16000 
 
 .14690 
 
 .17220 
 
 27 
 
 28 
 
 .12091 
 
 .13753 
 
 .12DC6 
 
 .14058 
 
 .13808 
 
 .16019 
 
 .14706 
 
 .17341 
 
 23 
 
 29 
 
 .12104 
 
 .13771 
 
 .12950 
 
 .14377 
 
 .13822 
 
 .10039 
 
 .14721 
 
 .17262 
 
 29 
 
 30 
 
 .12118 
 
 .13789 
 
 .12964 
 
 .14896 
 
 .13837 
 
 .16059 
 
 .14736 
 
 .17283 
 
 30 
 
 31 
 
 .12132 
 
 .13807 
 
 .12979 
 
 .14914 
 
 .13852 
 
 .16079 
 
 .14751 
 
 .17304 
 
 31 
 
 32 
 
 .12146 
 
 .13825 
 
 .12993 
 
 .14933 
 
 .13367 
 
 .10C99 
 
 .147GO 
 
 .17325 
 
 3x5 
 
 33 
 
 .12160 
 
 .13843 
 
 .13007 
 
 .1495.'? 
 
 .13881 
 
 .16119 
 
 .14782 
 
 .17346 
 
 33 
 
 34 
 
 .12174 
 
 .13861 
 
 .13022 
 
 .14971 
 
 .13896 
 
 .16139 
 
 .14797 
 
 . 7367 
 
 34 
 
 35 
 
 .12188 
 
 .13379 
 
 .13036 
 
 .14990 
 
 .13911 
 
 ' .16159 
 
 .14812 
 
 . 7383 
 
 35 
 
 36 
 
 .12202 
 
 .13397 
 
 .13051 
 
 .15009 
 
 .13926 
 
 .16179 
 
 .14827 
 
 . 7409 
 
 36 
 
 37 
 
 .12216 
 
 .13916 
 
 .13005 
 
 .15028 
 
 .13941 
 
 .16199 
 
 .14843 
 
 . 7430 
 
 37 
 
 38 
 
 .12230 
 
 .13934 
 
 .13079 
 
 .15047 
 
 .13955 
 
 .16219 
 
 .14858 
 
 . 7451 
 
 33 
 
 39 
 
 .12244 
 
 .13952 
 
 .13094 
 
 .15086 
 
 .13970 
 
 .16239 
 
 .1487'3 
 
 . 7472 
 
 39 
 
 40 
 
 .12257 
 
 .13970 
 
 .13108 
 
 .15085 
 
 .13985 
 
 .16259 
 
 .14888 
 
 . 7493 
 
 40 
 
 41 
 
 .12271 
 
 .13988 
 
 .13122 
 
 .15105 
 
 .14000 
 
 .16279 
 
 .14904 
 
 . 7514 
 
 41 
 
 42 
 
 .12285 
 
 .14006 
 
 .13137 
 
 .15124 
 
 .14015 
 
 .16299 
 
 .14919 
 
 . 7535 
 
 42 
 
 43 
 
 .12299 
 
 .14024 
 
 .13151 
 
 .15143 
 
 .14030 
 
 .16319 
 
 .14934 
 
 . 7556 
 
 43 
 
 44 
 
 .12313 
 
 .14042 
 
 .131G6 
 
 .15162 
 
 .14044 
 
 .16339 
 
 .14949 
 
 . 7577 
 
 44 
 
 45 
 
 .12327 
 
 .14061 
 
 .13180 
 
 .15181 
 
 .14059 
 
 .16359 
 
 .149G5 
 
 .17598 
 
 45 
 
 46 
 
 .12341 
 
 .14079 
 
 .13195 
 
 .15200 
 
 .14074 
 
 .16380 
 
 .14980 
 
 .17620 
 
 46 
 
 47 
 
 .12355 
 
 .14097 
 
 .13209 
 
 .15219 
 
 .14089 
 
 .16400 
 
 .14995 
 
 .17641 
 
 47 
 
 48 
 
 .12369 
 
 .14115 
 
 .13223 
 
 .15239 
 
 .14104 
 
 .16420 
 
 .15011 
 
 .17G62 
 
 48 
 
 49 
 
 .12383 
 
 .14134 
 
 .13238 
 
 .15258 
 
 .14119 
 
 .16440 
 
 .15026 
 
 .17G83 
 
 49 
 
 50 
 
 .12397 
 
 .14152 
 
 .13252 
 
 .15277 
 
 .14134 
 
 .16460 
 
 .15041 
 
 .17704 
 
 50 
 
 51 
 
 .12411 
 
 .14170 
 
 .13267 
 
 .15296 
 
 .14149 
 
 .16481 
 
 .15057 
 
 .17726 
 
 51 
 
 52 
 
 .12425 
 
 .14188 
 
 .13281 
 
 .15315 
 
 .14164 
 
 .16501 
 
 .15072 
 
 .17747 
 
 53 
 
 53 
 
 .12439 
 
 .14207 
 
 .13296 
 
 .15335 
 
 .14179 
 
 .16521 
 
 .15087 
 
 .17768 
 
 53 
 
 54 
 
 .12454 
 
 .14225 
 
 .13310 
 
 .15354 
 
 .14194 
 
 .16541 
 
 .15103 
 
 .17790 
 
 54 
 
 55 
 
 .12468 
 
 .14243 
 
 .13325 
 
 .15373 
 
 .14208 
 
 .16562 
 
 .15118 
 
 .17811 
 
 55 
 
 56 
 
 .12482 
 
 .14262 
 
 .13339 
 
 .15393 
 
 .14223 
 
 .16582 
 
 .15134 
 
 .17832 
 
 56 
 
 57 
 
 .12496 
 
 .14280 
 
 .13354 
 
 .15412 
 
 .14238 
 
 .16602 
 
 .15149 
 
 .17854 
 
 57 
 
 58 
 
 .12510 
 
 .14299 
 
 .13368 
 
 .15431 
 
 .14253 
 
 .16623 
 
 .15164 
 
 .17875 
 
 58 
 
 59 
 
 .12524 
 
 .14317 
 
 .13383 
 
 .15451 
 
 .14268 
 
 ,16643 
 
 .15180 
 
 .17896 
 
 59 
 
 60 
 
 .12538 
 
 .14335 
 
 .13397 
 
 .15470 
 
 .14283 
 
 .16663 
 
 .15195 
 
 .17918 
 
 60 
 
 254 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 32- 
 
 33 
 
 84 
 
 35 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .15195 
 
 .17918 
 
 .16133 
 
 .19236 
 
 .17096 
 
 .20622 
 
 .18085 
 
 .22077 
 
 
 
 1 
 
 .15211 
 
 .17939 
 
 .16149 
 
 .19259 
 
 .17113 
 
 .20645 
 
 .18101 
 
 .22102 
 
 1 
 
 2 
 
 .15226 
 
 .17961 
 
 .16165 
 
 .19281 
 
 .17129 
 
 .20669 
 
 .18118 
 
 .22127 
 
 2 
 
 3 
 
 .15241 
 
 .17982 
 
 .16181 
 
 .19304 
 
 .17145 
 
 .20693 
 
 .18135 
 
 .22152 
 
 3 
 
 4 
 
 .15257 
 
 .18004 
 
 .16196 
 
 .19327 
 
 .17161 
 
 .20717 
 
 .18152 
 
 .22177 
 
 4 
 
 5 
 
 .15272 
 
 .18025 
 
 .16212 
 
 .19349 
 
 .17178 
 
 .20740 
 
 .18168 
 
 .22202 
 
 5 
 
 6 
 
 .15288 
 
 .18047 
 
 .16228 
 
 .19372 
 
 .17194 
 
 .20764 
 
 .18185 
 
 .22227 
 
 6 
 
 7 
 
 .15303 
 
 .18068 
 
 .16244 
 
 .19394 
 
 .17210 
 
 .20788 
 
 .18202 
 
 .22252 
 
 7 
 
 8 
 
 .15319 
 
 .18090 
 
 .16200 
 
 .19417 
 
 .17227 
 
 .20812 
 
 .18218 
 
 .22277 
 
 8 
 
 9 
 
 .15334 
 
 .18111 
 
 .16276 
 
 .19440 
 
 .17243 
 
 .20836 
 
 .18235 
 
 .22302 
 
 9 
 
 10 
 
 .15350 
 
 .18133 
 
 .16292 
 
 .19463 
 
 .17259 
 
 .20859 
 
 .18252 
 
 .22327 
 
 10 
 
 11 
 
 .15305 
 
 .18155 
 
 .16308 
 
 .19485 
 
 .17276 
 
 .20883 
 
 .18269 
 
 .22352 
 
 11 
 
 12 
 
 .15381 
 
 .18176 
 
 .16324 
 
 .19508 
 
 .17292 
 
 .20907 
 
 .18286 
 
 .22377 
 
 12 
 
 13 
 
 .15396 
 
 .18198 
 
 .16340 
 
 .19531 
 
 .17308 
 
 .20931 
 
 .18302 
 
 .22402 
 
 13 
 
 14 
 
 .15412 
 
 .18220 
 
 .16355 
 
 .19554 
 
 .17325 
 
 .20955 
 
 .188*9 
 
 .22428 
 
 14 
 
 15 
 
 .15427 
 
 .18241 
 
 .16371 
 
 .19576 
 
 .17341 
 
 .20979 
 
 .18336 
 
 .22453 
 
 15 
 
 16 
 
 .15443 
 
 .18263 
 
 .16387 
 
 .19599 
 
 .17357 
 
 .21003 
 
 .18353 
 
 .22478 
 
 16 
 
 17 
 
 .15453 
 
 .18285 
 
 .16403 
 
 .19622 
 
 .17374 
 
 .21027 
 
 .18369 
 
 .22503 
 
 17 
 
 18 
 
 .15474 
 
 .18307 
 
 .16419 
 
 .19645 
 
 .17390 
 
 .21051 
 
 .18386 
 
 .22528 
 
 18 
 
 19 
 
 .15489 
 
 .18328 
 
 .16435 
 
 .19668 
 
 .17407 
 
 .21075 
 
 .18403 
 
 .22554 
 
 19 
 
 20 
 
 .15505 
 
 .18350 
 
 .16451 
 
 .19691 
 
 .17423 
 
 .21099 
 
 .18420 
 
 .22579 
 
 20 
 
 21 
 
 .15520 
 
 .18372 
 
 .16467 
 
 .19713 
 
 .17439 
 
 .21123 
 
 .18437 
 
 .22604 
 
 21 
 
 22 
 
 .15536 
 
 .18394 
 
 .16483 
 
 .19736 
 
 .17456 
 
 .21147 
 
 .18454 
 
 .22629 
 
 22 
 
 23 
 
 .15552 
 
 .18416 
 
 .16409 
 
 .19759 
 
 .17472 
 
 .21171 
 
 .18470 
 
 .22655 
 
 23 
 
 24 
 
 .15567 
 
 .18437 
 
 .16515 
 
 .19788 
 
 .17489 
 
 .21195 
 
 .18-487 
 
 .22680 
 
 24 
 
 25 
 
 .15583 
 
 .18459 
 
 .16531 
 
 .19805 
 
 .17505 
 
 .21220 
 
 .18504 
 
 .22706 
 
 25 
 
 26 
 
 .15598 
 
 .18481 
 
 .16547 
 
 .19828 
 
 .17522 
 
 .21244 
 
 .18521 
 
 .22731 
 
 26 
 
 27 
 
 .15614 
 
 .18503 
 
 .16563 
 
 .19851 
 
 .17538 
 
 .21268 
 
 .18538 
 
 .22756 
 
 27 
 
 28 
 
 .15630 
 
 .18525 
 
 .16579 
 
 .19874 
 
 .17554 
 
 .21292 
 
 .18555 
 
 .227'82 
 
 28 
 
 29 
 
 .15645 
 
 .18547 
 
 .16595 
 
 .19897 
 
 .17571 
 
 .21316 
 
 .18572 
 
 .22807 
 
 29 
 
 30 
 
 .15661 
 
 .18569 
 
 .16611 
 
 .19920 
 
 .17587 
 
 .21341 
 
 .18588 
 
 .22833 
 
 30 
 
 31 
 
 .15676 
 
 .18591 
 
 .16627 
 
 .19944 
 
 .17604 
 
 .21365 
 
 .18005 
 
 .22858 
 
 31 
 
 32 
 
 .15693 
 
 .18613 
 
 .16644 
 
 .19967 
 
 .17620 
 
 .21389 
 
 .18022 
 
 .22884 
 
 32 
 
 33 
 
 .15708 
 
 .18635 
 
 .16660 
 
 .19990 
 
 .17637 
 
 .21414 
 
 .18639 
 
 .22909 
 
 33 
 
 34 
 
 .15723 
 
 .18657 
 
 .16676 
 
 .20013 
 
 .17653 
 
 .21433 
 
 .18056 
 
 .22935 
 
 34 
 
 35 
 
 .15739 
 
 .18679 
 
 .16692 
 
 .20036 
 
 .17670 
 
 .34402 
 
 .18673 
 
 .22960 
 
 35 
 
 36 
 
 .15755 
 
 .18701 
 
 .16708 
 
 .20059 
 
 .17686 
 
 .21487 
 
 .18690 
 
 .22986 
 
 36 
 
 37 
 
 .15770 
 
 .18723 
 
 .16724 
 
 .20083 
 
 .17703 
 
 .21511 
 
 .18707 
 
 .23012 
 
 37 
 
 38 
 
 .15786 
 
 .18745 
 
 .16740 
 
 .20106 
 
 .17719 
 
 .21535 
 
 .18724 
 
 .23037 
 
 38 
 
 39 
 
 .15802 
 
 .18767 
 
 .16756 
 
 .20129 
 
 .17736 
 
 .21500 
 
 .18741 
 
 .23003 
 
 39 
 
 40 
 
 .15818 
 
 .18790 
 
 .16772 
 
 .20152 
 
 .17752 
 
 .21584 
 
 .18758 
 
 .23089 
 
 40 
 
 41 
 
 .15833 
 
 .18812 
 
 .16788 
 
 .20176 
 
 .17769 
 
 .21609 
 
 .18775 
 
 .23114 
 
 41 
 
 42 
 
 .15349 
 
 .18834 
 
 .16805 
 
 .20193 
 
 .17786 
 
 .21633 
 
 .18792 
 
 .23140 
 
 42 
 
 43 
 
 .15865 
 
 .18856 
 
 .16821 
 
 .20222 
 
 .17802 
 
 .21658 
 
 .18809 
 
 .23166 
 
 43 
 
 44 
 
 .15880 
 
 .18878 
 
 .16837 
 
 .20246 
 
 .17819 
 
 .21082 
 
 .18826 
 
 .23192 
 
 44 
 
 45 
 
 .15896 
 
 .18901 
 
 .16853 
 
 .20269 
 
 .17835 
 
 .21707 
 
 .18843 
 
 .23217 
 
 45 
 
 46 
 
 .15912 
 
 .18923 
 
 .16809 
 
 .20292 
 
 .17852 
 
 .21731 
 
 .18860 
 
 .23243 
 
 46 
 
 47 
 
 .15928 
 
 .18945 
 
 .16885 
 
 .20316 
 
 .17808 
 
 .21756 
 
 .18877 
 
 .23269 
 
 47 
 
 48 
 
 .15313 
 
 .18967 
 
 .16902 
 
 .20339 
 
 .17885 
 
 .21781 
 
 .18894 
 
 .23295 
 
 43 
 
 49 
 
 .15959 
 
 .18990 
 
 .16918 
 
 .20363 
 
 .17902 
 
 .21805 
 
 .18911 
 
 .23321 
 
 49 
 
 50 
 
 .15975 
 
 .19012 
 
 .16934 
 
 .20386 
 
 .17918 
 
 .21830 
 
 48928 
 
 .23347 
 
 50 
 
 51 
 
 .15991 
 
 .19034 
 
 .16950 
 
 .20410 
 
 .17935 
 
 .21855 
 
 .18945 
 
 .23373 
 
 51 
 
 52 
 
 .16006 
 
 .19057 
 
 .16906 
 
 .20133 
 
 .17952 
 
 .21879 
 
 .18962 
 
 .23399 
 
 52 
 
 53 
 
 .16022 
 
 .19079 
 
 .16983 
 
 .20457 
 
 .17968 
 
 .21904 
 
 .18979 
 
 .23424 
 
 53 
 
 54 
 
 .16038 
 
 .19102 
 
 .16999 
 
 .20480 
 
 .17985 
 
 .21929 
 
 .18996 
 
 .23450 
 
 54 
 
 55 
 
 .16054 
 
 .19124 
 
 .17015 
 
 .20504 
 
 .18001 
 
 .21953 
 
 .19013 
 
 .23476 
 
 55 
 
 56 
 
 .16070 
 
 .19146 
 
 .17031 
 
 .20527 
 
 .18018 
 
 .21978 
 
 .19030 
 
 .23502 
 
 56 
 
 57 
 
 .16085 
 
 .19169 
 
 .17047 
 
 .20551 
 
 .18035 
 
 .22003 
 
 .19047 
 
 .23529 
 
 57 
 
 58 
 
 .16101 
 
 .19191 
 
 .17064 
 
 .20575 
 
 .18051 
 
 .22028 
 
 .19064 
 
 .23555 
 
 58 
 
 59 
 
 .16117 
 
 .19214 ; 
 
 .17080 
 
 .20598 
 
 .18068 
 
 .22053 
 
 .19081 
 
 .23581 
 
 59 
 
 60 
 
 .16133 
 
 .19236 I 
 
 .17096 
 
 .20622 
 
 .18085 
 
 .22077 1 
 
 .19098 
 
 .23607 
 
 60 
 
 255 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 t 
 
 36 
 
 37 
 
 38 
 
 39 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .19008 
 
 .23007 
 
 .201S6 
 
 .25214 
 
 .21199 
 
 .20902 
 
 .22285 
 
 .28076 
 
 ~0~ 
 
 1 
 
 .19115 
 
 .23633 
 
 .20154 
 
 .25241 
 
 .21217 
 
 .26931 
 
 .22304 
 
 .28706 
 
 1 
 
 2 
 
 .19133 
 
 .23659 
 
 .20171 
 
 .25269 
 
 .21235 
 
 .20960 
 
 .22322 
 
 .28737 
 
 2 
 
 3 
 
 .19150 
 
 .23685 
 
 .20189 
 
 .25296 
 
 .21253 
 
 .20988 
 
 .22340 
 
 .28767 
 
 3 
 
 4 
 
 .19167 
 
 .23711 
 
 .20207 
 
 .25324 
 
 .21271 
 
 .27017 
 
 .22359 
 
 .28797 
 
 4 
 
 5 
 
 .19184 
 
 .23738 
 
 .20224 
 
 .25351 
 
 .21289 
 
 .27046 
 
 .22377 
 
 .28823 
 
 5 
 
 6 
 
 .19201 
 
 .23764 
 
 .20242 
 
 .25379 
 
 .21307 
 
 .27075 
 
 .22395 
 
 .28858 
 
 6 
 
 7 
 
 .19218 
 
 .23790 
 
 .20259 
 
 .25406 
 
 .21324 
 
 .27104 
 
 .22414 
 
 .28889 
 
 7 
 
 8 
 
 .19235 
 
 .23816 
 
 .20277 
 
 .25434 
 
 .21342 
 
 .27133 
 
 .22432 
 
 .28919 
 
 8 
 
 9 
 
 .19252 
 
 .23843 
 
 .20294 
 
 .25402 
 
 .21300 
 
 .27162 
 
 .22450 
 
 .28950 
 
 9 
 
 10 
 
 .19270 
 
 .23869 
 
 .20312 
 
 .25489 
 
 .21378 
 
 .27191 
 
 .22469 
 
 .28980 
 
 10 
 
 11 
 
 .19287 
 
 .23895 
 
 .20329 
 
 .25517 
 
 .21396 
 
 .27221 
 
 .22487 
 
 .29011 
 
 11 
 
 12 
 
 .19304 
 
 .23922 
 
 .20347 
 
 .23545 
 
 .21414 
 
 .27250 
 
 .22506 
 
 .29042 
 
 12 
 
 13 
 
 .19321 
 
 .23948 
 
 .20305 
 
 .25572 
 
 .21432 
 
 .27279 
 
 .22524 
 
 .29072 
 
 13 
 
 14 
 
 .19338 
 
 .23975 
 
 .20382 
 
 .25600 
 
 .21450 
 
 .27308 
 
 .22542 
 
 .29103 
 
 14 
 
 15 
 
 .19356 
 
 .24001 
 
 .20400 
 
 .25628 
 
 .21468 
 
 .27337 
 
 .22561 
 
 .29133 1 15 
 
 16 
 
 .19873 
 
 .24028 
 
 .20417 
 
 .25656 
 
 .21486 
 
 .27366 
 
 .22579 
 
 .29164 16 
 
 17 
 
 .19390 
 
 .24054 
 
 .20435 
 
 .25683 
 
 .21504 
 
 .27396 
 
 .22598 
 
 .29195 
 
 17 
 
 18 
 
 .19407 
 
 .24081 
 
 .20453 
 
 .25711 
 
 .21522 
 
 .27425 
 
 .22616 
 
 .29226 
 
 18 
 
 19 
 
 .19424 
 
 .24107 
 
 .20470 
 
 .25739 
 
 .21540 
 
 .27454 
 
 .22634 
 
 .29256 
 
 19 
 
 20 
 
 .19442 
 
 .24134 
 
 .20488 
 
 .25767 
 
 .21558 
 
 .27483 
 
 .22653 
 
 .29287 
 
 20 
 
 21 
 
 .19459 
 
 .24160 
 
 .20506 
 
 .25795 
 
 .21576 
 
 .27513 
 
 .22671 
 
 .29318 
 
 21 
 
 22 
 
 .19476 
 
 .24187 
 
 .20523 
 
 .25823 
 
 .21595 
 
 .27542 
 
 .22090 
 
 .29349 
 
 23 
 
 23 
 
 .19493 
 
 .24213 
 
 .20541 
 
 .25851 
 
 .21613 
 
 .27572 
 
 .22703 
 
 .29380 
 
 23 
 
 24 
 
 .19511 
 
 .2-1240 
 
 .20559 
 
 .25879 
 
 .21631 
 
 .27601 
 
 .22727 
 
 .29411 
 
 24 
 
 25 
 
 .19528 
 
 .2-4267 
 
 .20576 
 
 .25907 
 
 .21649 
 
 .27630 
 
 .22745 
 
 .29442 
 
 25 
 
 26 
 
 .19545 
 
 .24293 
 
 .20594 
 
 .25935 
 
 .21667 
 
 .27660 
 
 .22704 
 
 .29473 
 
 26 
 
 27 
 
 .19562 
 
 .24320 
 
 .20612 
 
 .25963 
 
 .21685 
 
 .27689 
 
 .22782 
 
 .29504 
 
 27 
 
 28 
 
 .19580 
 
 .24347 
 
 .20029 
 
 .25991 
 
 .21703 
 
 .27719 
 
 .22801 
 
 .29535 
 
 28 
 
 29 
 
 .19597 
 
 .24373 
 
 .20647 
 
 .26019 
 
 .21721 
 
 .27748 
 
 .22819 
 
 .29566 
 
 29 
 
 30 
 
 .19614 
 
 .24400 
 
 .20665 
 
 .26047 
 
 .21739 
 
 .27775 
 
 .22838 
 
 .29597 
 
 30 
 
 31 
 
 .19632 
 
 .24427 
 
 .20682 
 
 .26075 
 
 .21757 
 
 .27807 
 
 .22856 
 
 .29628 
 
 31 
 
 32 
 
 .19049 
 
 .24454 
 
 .20700 
 
 .20104 
 
 .21775 
 
 .27837 
 
 .22875 
 
 .29659 
 
 32 
 
 33 
 
 .19666 
 
 .24481 
 
 .20718 
 
 .26132 
 
 .21794 
 
 .27867 
 
 .22893 
 
 .29090 
 
 33 
 
 34 
 
 .Iy684 
 
 .24508 
 
 .20736 
 
 .26160 
 
 .21812 
 
 .27896 
 
 .22912 
 
 .29721 
 
 34 
 
 35 
 
 .19701 
 
 .24534 
 
 .20753 
 
 .26188 
 
 .21830 
 
 .27926 
 
 .22330 
 
 .29752 
 
 35 
 
 30 
 
 .19718 
 
 .24561 
 
 .20771 
 
 .26216 
 
 .21848 
 
 .27956 
 
 .22949 
 
 .29784 
 
 SG 
 
 37 
 
 .19736 
 
 .24588 
 
 .20789 
 
 .26245 
 
 .21866 
 
 .27985 
 
 .22907 
 
 .29815 
 
 37 
 
 38 
 
 .19753 
 
 .24615 
 
 .20807 
 
 .26273 
 
 .21884 
 
 .28015 
 
 .22986 
 
 .29846 
 
 33 
 
 39 
 
 .19770 
 
 .21642 
 
 .20824 
 
 .26301 
 
 .21902 
 
 .28045 
 
 .23004 
 
 .29877 
 
 39 
 
 40 
 
 .19788 
 
 .24669 
 
 .20842 
 
 .26330 
 
 .21921 
 
 .28075 
 
 .23023 
 
 .29909 
 
 40 
 
 41 
 
 .19805 
 
 .24696 
 
 .20860 
 
 .26358 
 
 .21939 
 
 .28105 
 
 .23041 
 
 .29940 
 
 41 
 
 42 
 
 .19822 
 
 .24723 
 
 .20878 
 
 .26387 
 
 .21957 
 
 .28134 
 
 .23000 
 
 .29971 
 
 42 
 
 43 
 
 .19840 
 
 .24750 
 
 .20895 
 
 .26415 
 
 .21975 
 
 .28104 
 
 .23079 
 
 .30003 
 
 43 
 
 44 
 
 .19857 
 
 .24777 
 
 .20913 
 
 .26443 
 
 .21993 
 
 .28194 
 
 .23097 
 
 .30034 
 
 44 
 
 45 
 
 .1987'5 
 
 .24804 
 
 .20931 
 
 .26472 
 
 .22012 
 
 .28224 
 
 .23116 
 
 .30006 
 
 45 
 
 46 
 
 .19892 
 
 .24832 
 
 .20949 
 
 .26500 
 
 .22030 
 
 .28254 
 
 .23134 
 
 .30097 
 
 46 
 
 47 
 
 .19909 
 
 .24859 
 
 .20967 
 
 .26529 
 
 .22048 
 
 .28284 
 
 .23153 
 
 .30129 
 
 47 
 
 48 
 
 .19927 
 
 .24886 
 
 .20985 
 
 .26557 
 
 .22006 
 
 .28314 
 
 .28178 
 
 .30100 
 
 48 
 
 49 
 
 .19944 
 
 .24913 
 
 .21002 
 
 .26586 
 
 .22084 
 
 .28344 
 
 .23190 
 
 .30192 
 
 49 
 
 50 
 
 .19962 
 
 .24940 
 
 .21020 
 
 .26615 
 
 .22103 
 
 .28374 
 
 .23209 
 
 .30223 
 
 50 
 
 51 
 
 .19979 
 
 .24967 
 
 .21038 
 
 .26643 
 
 .22121 
 
 .28404 
 
 .23228 
 
 .30255 
 
 51 
 
 52 
 
 .19997 
 
 .24995 
 
 .21056 
 
 .26672 
 
 .22139 
 
 .28434 
 
 .23246 
 
 .30287 
 
 52 
 
 53 
 
 .20014 
 
 .25022 
 
 .21074 
 
 .20701 
 
 .22157 
 
 .28464 
 
 .23265 
 
 .30318 
 
 53 
 
 54 
 
 .20032 
 
 .25049 
 
 .21092 
 
 .20729 
 
 .22176 
 
 .28495 
 
 .23283 
 
 .30350 
 
 54 
 
 55 
 
 .20049 
 
 .25077 
 
 .21109 
 
 .26758 
 
 .22194 
 
 .28525 
 
 .23302 
 
 .30382 
 
 55 
 
 56 
 
 .20066 
 
 .25104 
 
 .21127 
 
 .26787 
 
 .22212 
 
 .28555 
 
 .23321 
 
 .30413 
 
 56 
 
 57 
 
 .20084 
 
 .25131 
 
 .21145 
 
 .26815 
 
 .22231 
 
 .28585 
 
 .23339 
 
 .30445 
 
 57 
 
 58 
 
 .20101 
 
 .25159 
 
 .21163 
 
 .26844 
 
 .22249 
 
 .28615 
 
 .23358 
 
 .30477 
 
 58 
 
 59 
 
 .20119 
 
 .25186 
 
 .21181 
 
 .26873 
 
 .22267 
 
 .28646 
 
 .23377 
 
 .30509 
 
 59 
 
 60 
 
 .20136 
 
 .25214 1 
 
 .21199 
 
 .26902 
 
 .22285 
 
 .28676 
 
 .23396 
 
 .30541 
 
 60 
 
 256 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 1 
 
 40 
 
 41 
 
 42 
 
 43 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .23396 
 
 .30541 
 
 .24529 
 
 .32501 
 
 .25686 
 
 .34563 
 
 .26865 
 
 .36733 
 
 "o" 
 
 1 
 
 .23414 
 
 .30573 
 
 .24548 
 
 .32535 
 
 .25705 
 
 .3-1599 
 
 .26884 
 
 .36770 
 
 1 
 
 2 
 
 .23433 
 
 .30605 
 
 .24567 
 
 .32568 
 
 .25724 
 
 .34634 
 
 .26904 
 
 .36807 
 
 2 
 
 3 
 
 .23452 
 
 .30636 
 
 .24586 
 
 .32602 
 
 .25744 
 
 .34669 
 
 .26924 
 
 .36844 
 
 8 
 
 4 
 
 .23470 
 
 .30668 
 
 .24605 
 
 .32636 
 
 .25763 
 
 .34704 
 
 .26944 
 
 .36881 
 
 4 
 
 5 
 
 .23489 
 
 .30700 
 
 .24625 
 
 .32669 
 
 .25783 
 
 .34740 
 
 .26964 
 
 .36919 
 
 6 
 
 6 
 
 .23508 
 
 .30732 
 
 .24644 
 
 .32703 
 
 .25802 
 
 .34775 
 
 .26984 
 
 .36956 
 
 6 
 
 7 
 
 .23527 
 
 .30764 
 
 .24663 
 
 .32737 
 
 .25822 
 
 .34811 
 
 .27004 
 
 .36993 
 
 7 
 
 8 
 
 .23545 
 
 .30796 
 
 .24682 
 
 .32770 
 
 .25841 
 
 .34846 
 
 .27024 
 
 .37030 
 
 8 
 
 9 
 
 .23564 
 
 .30829 
 
 .24701 
 
 32804 
 
 .25861 
 
 .34882 
 
 .27043 
 
 .37068 
 
 9 
 
 10 
 
 .23583 
 
 .30861 
 
 .24720 
 
 .32838 
 
 .25880 
 
 .34917 
 
 .27063 
 
 .37105 
 
 10 
 
 11 
 
 .23602 
 
 .30893 
 
 .24739 
 
 .32872 
 
 .25900 
 
 .34953 
 
 .27083 
 
 .37143 
 
 11 
 
 12 
 
 .23620 
 
 .30925 
 
 .24759 
 
 .32905 
 
 .25920 
 
 .34988 
 
 .27103 
 
 .37180 
 
 12 
 
 13 
 
 .23639 
 
 .30957 
 
 24778 
 
 .32939 
 
 .25939 
 
 .35024 
 
 .27123 
 
 .37218 
 
 13 
 
 14 
 
 .23658 
 
 .30989 
 
 24797 
 
 .32973 
 
 .25959 
 
 .35060 
 
 .27143 
 
 .37255 
 
 14 
 
 15 
 
 .23677 
 
 .31022 
 
 .24816 
 
 .33007 
 
 .25978 
 
 .35095 
 
 .27163 
 
 .37223 
 
 15 
 
 16 
 
 .23696 
 
 .31054 
 
 .24835 
 
 .33041 
 
 .25998 
 
 .35131 
 
 .27183 
 
 .37330 
 
 16 
 
 17 
 
 .23714 
 
 .31086 
 
 .24854 
 
 .33075 
 
 .26017 
 
 .35167 
 
 .27203 
 
 .37368 
 
 17 
 
 18 
 
 .23733 
 
 .31119 
 
 .24874 
 
 .33109 
 
 .26037 
 
 .35203 
 
 .27223 
 
 .37406 
 
 18 
 
 19 
 
 .23752 
 
 .31151 
 
 .24893 
 
 .33143 
 
 .26056 
 
 .35238 
 
 .27243 
 
 .37443 
 
 19 
 
 20 
 
 .23771 
 
 .31183 
 
 .24912 
 
 .33177 
 
 .26076 
 
 .35274 
 
 .27263 
 
 .37481 
 
 20. 
 
 21 
 
 .23790 
 
 .31216 
 
 .24931 
 
 .33211 
 
 .26096 
 
 .35310 
 
 .27283 
 
 .37519 
 
 21 
 
 22 
 
 .23808 
 
 .31248 
 
 .24950 
 
 .33245 
 
 .26115 
 
 .35346 
 
 .27303 
 
 .37556 
 
 22 
 
 23 
 
 .23827 
 
 .31281 
 
 .24970 
 
 .33279 
 
 .26135 
 
 .35382 
 
 .27323 
 
 .37594 
 
 23 
 
 24 
 
 .23846 
 
 .31313 
 
 .24989 
 
 .33314 
 
 .26154 
 
 .35418 
 
 .27343 
 
 .37632 
 
 24 
 
 25 
 
 .23865 
 
 .31346 
 
 .25008 
 
 .33348 
 
 .26174 
 
 .35454 
 
 .27363 
 
 .37670 
 
 25 
 
 26 
 
 .23884 
 
 .31378 
 
 .25027 
 
 .33382 
 
 .26194 
 
 .35490 
 
 .27383 
 
 .37708 
 
 26 
 
 27 
 
 .23903 
 
 .31411 
 
 .25047 
 
 .33416 
 
 .26213 
 
 .35526 
 
 .27403 
 
 .37746 
 
 27 
 
 28 
 
 .23922 
 
 .31443 
 
 .25066 
 
 .33451 
 
 .26233 
 
 .35562 
 
 .27423 
 
 .37784 
 
 28 
 
 29 
 
 .23941 
 
 .31476 
 
 .25085 
 
 .33485 
 
 .26253 
 
 .35598 
 
 .27443 
 
 .37822 
 
 29 
 
 30 
 
 ...23959 
 
 .31509 
 
 .25104 
 
 .33519 
 
 .26272 
 
 .35634 
 
 .27463 
 
 .87860 
 
 30 
 
 31 
 
 ;23978 
 
 .31541 
 
 .25124 
 
 .33554 
 
 .26292 
 
 .35670 
 
 .27483 
 
 .87898 
 
 31 
 
 32 
 
 .23997 
 
 .31574 
 
 .25143 
 
 .33588 
 
 .26312 
 
 .35707 
 
 .27503 
 
 .37936 
 
 32 
 
 33 
 
 .24016 
 
 .31607 
 
 .25162 
 
 .33622 
 
 .26331 
 
 .35743 
 
 .27523 
 
 .87974 
 
 33 
 
 34 
 
 .24035 
 
 .31610 
 
 .25182 
 
 .33657 
 
 .26351 
 
 .35779 
 
 .27543 
 
 .88012 
 
 34 
 
 35 
 
 .24054 
 
 .31672 
 
 .25201 
 
 .33691 
 
 .26371 
 
 .35815 
 
 .27563 
 
 .38051 
 
 35 
 
 36 
 
 .24073 
 
 .31705 
 
 .25220 
 
 .33726 
 
 .26390 
 
 .35852 
 
 .27583 
 
 .38089 
 
 36 
 
 37 
 
 .24092 
 
 .31738 
 
 .25240 
 
 .33760 
 
 .26410 
 
 .35888 
 
 .27603 
 
 .88127 
 
 37 
 
 38 
 
 .24111 
 
 .31771 
 
 .25259 
 
 .33795 
 
 .26430 
 
 .35924 
 
 .27623 
 
 .38165 
 
 38 
 
 39 
 
 .24130 
 
 .31804 
 
 .25278 
 
 .33830 
 
 .26449 
 
 .35961 
 
 .27643 
 
 .38204 
 
 39 
 
 40 
 
 .24149 
 
 .31837 
 
 .25297 
 
 .33864 
 
 .26469 
 
 .35997 
 
 .27663 
 
 .38242 
 
 40 
 
 41 
 
 .24168 
 
 .31870 
 
 .25317 
 
 .33899 
 
 .26489 
 
 .36034 
 
 .27683 
 
 .38280 
 
 41 
 
 42 
 
 .24187 
 
 .31903 
 
 .25336 
 
 .33934 
 
 .26509 
 
 .36070 
 
 .27703 
 
 .38319 
 
 42 
 
 43 
 
 .24206 
 
 .31936 
 
 .25356 
 
 .33968 
 
 .26528 
 
 .36107 
 
 .27723 
 
 .38357 
 
 43 
 
 44 
 
 .24225 
 
 .31969 
 
 .25375 
 
 .34003 
 
 .26548 
 
 .36143 
 
 .27743 
 
 .38396 
 
 44 
 
 45 
 
 .24244 
 
 .32002 
 
 .25394 
 
 .34038 
 
 .26568 
 
 .36180 
 
 .27764 
 
 .38431 
 
 45 
 
 46 
 
 .24262 
 
 .32035 
 
 .25414 
 
 .34073 
 
 .26588 
 
 .36217 
 
 .27784 
 
 .38473 
 
 46 
 
 47 
 
 .24281 
 
 .32068 
 
 .25433 
 
 .34108 
 
 .26607 
 
 .36253 
 
 .27804 
 
 .38512 
 
 47 
 
 48 
 
 .24300 
 
 .32101 
 
 .25452 
 
 .34142 
 
 .26627 
 
 .36290 
 
 .27824 
 
 .38550 
 
 48 
 
 49 
 
 .24320 
 
 .32134 
 
 .25472 
 
 .34177 
 
 .26647 
 
 .36327 
 
 .27844 
 
 .38589 
 
 49 
 
 50 
 
 .24339 
 
 .32168 
 
 .25491 
 
 .34212 
 
 .26667 
 
 .36363 
 
 .27864 
 
 .38628 
 
 50 
 
 51 
 
 .24358 
 
 .32201 
 
 .25511 
 
 .34247 
 
 .26686 
 
 .36400 
 
 .27884 
 
 .38666 
 
 51 
 
 52 
 
 .24377 
 
 .32234 
 
 .25530 
 
 .34282 
 
 .26706 
 
 .36437 
 
 .27905 
 
 .38705 
 
 52 
 
 53 
 
 .24396 
 
 .32267 
 
 .25549 
 
 .34317 
 
 .26726 
 
 .36474 
 
 .27925 
 
 .38744 
 
 53 
 
 54 
 
 .24415 
 
 .32301 
 
 .25569 
 
 .34352 
 
 .26746 
 
 .36511 
 
 .27945 
 
 .38763 
 
 54 
 
 55 
 
 .24434 
 
 .32334 
 
 .25588 
 
 .34387 
 
 .26766 
 
 .36548 
 
 .27965 
 
 .38822 
 
 55 
 
 56 
 
 .24453 
 
 .32368 
 
 .25608 
 
 .34423 
 
 .26785 
 
 .36585 
 
 .27985 
 
 .38860 
 
 56 
 
 57 
 
 -.24472 
 
 .32401 
 
 .25627 
 
 .34458 
 
 .26805 
 
 .36622 
 
 .28005 
 
 .38899 
 
 &: 
 
 58 
 
 .24491 
 
 .32434 
 
 .25647 
 
 .84493 
 
 .26825 
 
 .36G59 
 
 .28026 
 
 .38938 
 
 58 
 
 59 
 
 .24510 
 
 ,32468 
 
 ,25666 
 
 .34528 
 
 .26845 
 
 .36696 
 
 .28046 
 
 .38977 
 
 59 
 
 60 
 
 .24529 
 
 .32501 
 
 .25686 
 
 .34563 
 
 .26865 
 
 .36733 
 
 .28066 
 
 .39016 
 
 60 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 ' 
 
 44 
 
 45 
 
 46 
 
 47 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. , 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .28066 
 
 .39016 
 
 .29289 
 
 .41421 
 
 .305.3-4 
 
 .43956 
 
 .31800 
 
 .46628 
 
 
 
 1 
 
 .28086 
 
 .39055 
 
 .29310 
 
 .41463 
 
 .30555 
 
 .43999 
 
 .31821 
 
 .46674 
 
 1 
 
 2 
 
 .28106 
 
 .39095 
 
 .29330 
 
 .41504 
 
 .30576 
 
 .44042 
 
 .31843 
 
 .46719 
 
 2 
 
 3 
 
 .28127 
 
 .39134 
 
 .29351 
 
 .41545 
 
 .30597 
 
 .44086 
 
 .31864 
 
 .46765 
 
 3 
 
 4 
 
 .28147 
 
 .39173 
 
 .29372 
 
 .41586 
 
 .30618 
 
 .44129 
 
 .81885 
 
 .46811 
 
 4 
 
 5 
 
 .28167 
 
 .39212 
 
 .29392 
 
 .41627 
 
 .30639 
 
 .44173 
 
 .31907 
 
 .46857 
 
 5 
 
 6 
 
 .28187 
 
 .39251 
 
 .29413 
 
 .41669 
 
 .30660 
 
 .44217 
 
 .31928 
 
 .46903 
 
 6 
 
 7 
 
 .28208 
 
 .39291 
 
 .29433 
 
 .41710 
 
 .30081 
 
 .44260 
 
 .31949 
 
 .46949 
 
 7 
 
 8 
 
 .28228 
 
 .39330 
 
 .29454 
 
 .41752 
 
 .30702 
 
 .44304 
 
 .31971 
 
 .46995 
 
 8 
 
 9 
 
 .28248 
 
 .39369 
 
 .29475 
 
 .41793 
 
 .30723 
 
 .44347 
 
 .31992 
 
 .47041 
 
 9 
 
 10 
 
 .28268 
 
 .39409 
 
 .29495 
 
 .41835 
 
 .30744 
 
 .44391 
 
 .32013 
 
 .47087 
 
 10 
 
 11 
 
 .28289 
 
 .39448 
 
 .29516 
 
 .41876 
 
 .30765 
 
 .44435 
 
 .32035 
 
 .47134 
 
 11 
 
 12 
 
 .28309 
 
 .39487 
 
 .29537 
 
 .41918 
 
 .30786 
 
 .44479 
 
 .32056 
 
 .47180 
 
 12 
 
 13 
 
 .28329 
 
 .39527 
 
 .29557 
 
 .41959 
 
 .30807 
 
 .44523 
 
 .32077 
 
 .47226 
 
 13 
 
 14 
 
 .28350 
 
 .39566 
 
 .29578 
 
 .42001 
 
 .30828 
 
 .44567 
 
 .32099 
 
 .47272 
 
 14 
 
 15 
 
 .28370 
 
 .39606 
 
 .29599 
 
 .42042 
 
 .30849 
 
 .44610 
 
 .32120 
 
 .47319 
 
 15 
 
 16 
 
 .28390 
 
 .39646 
 
 .29619 
 
 .42084 
 
 .30870 
 
 .44654 
 
 .32141 
 
 .47365 
 
 16 
 
 17 
 
 .28410 
 
 .39685 
 
 .29640 
 
 .42126 
 
 .30891 
 
 .44698 
 
 .32163 
 
 .47411 
 
 17 
 
 18 
 
 .28431 
 
 .39725 
 
 .29661 
 
 .42168 
 
 .30912 
 
 .44742 
 
 .32184 
 
 .47458 
 
 18 
 
 19 
 
 .28451 
 
 .39764 
 
 .29G81 
 
 .42210 
 
 .30933 
 
 .44787 
 
 .32205 
 
 .47501 
 
 19 
 
 20 
 
 .28471 
 
 .39804 
 
 .29702 
 
 .42251 
 
 .30954 
 
 .44831 
 
 .32227 
 
 .47551 
 
 20 
 
 21 
 
 .28492 
 
 .39844 
 
 .29723 
 
 .42293 
 
 .30975 
 
 .44875 
 
 .32248 
 
 .47598 
 
 21 
 
 22 
 
 .28512 
 
 .39384 
 
 .29743 
 
 .42335 
 
 .30933 
 
 .44919 
 
 .32270 
 
 .47644 
 
 22 
 
 23 
 
 .28532 
 
 .39924 
 
 .2970 i 
 
 .42377 
 
 .31017 
 
 .44963 
 
 .32291 
 
 .47691 
 
 23 
 
 24 
 
 .28553 
 
 .39963 
 
 .29785 
 
 .42419 
 
 .31038 
 
 .45007 
 
 .32312 
 
 . 47738 
 
 24 
 
 25 
 
 .28573 
 
 .40003 
 
 .29805 
 
 .42461 
 
 .31059 
 
 .45052 
 
 .32334 
 
 .47784 
 
 25 
 
 26 
 
 .28503 
 
 .40043 
 
 .29826 
 
 .42503 
 
 .31030 
 
 .45096 
 
 .323.')5 
 
 .47831 
 
 26 
 
 27 
 
 .28614 
 
 .40083 
 
 .29847 
 
 .42545 
 
 .31101 
 
 .45141 
 
 .32377 
 
 .47878 
 
 27 
 
 28 
 
 .28634 
 
 .40123 
 
 .29868 
 
 .42587 
 
 .31122 
 
 .45185 
 
 .32398- 
 
 .47925 
 
 28 
 
 29 
 
 .28655 
 
 .40163 
 
 .29888 
 
 .42630 
 
 .31143 
 
 .45229 
 
 .32420 
 
 .47972 
 
 29 
 
 30 
 
 .28675 
 
 .40203 
 
 .29909 
 
 .42672 
 
 .31165 
 
 .45274 
 
 .32441 
 
 .48019 
 
 30 
 
 31 
 
 .28695 
 
 .40243 
 
 .29930 
 
 .42714 
 
 .31186 
 
 .45319 
 
 .32462 
 
 .48066 
 
 31 
 
 32 
 
 .28716 
 
 .40283 
 
 .29951 
 
 .42756 
 
 .31207 
 
 .45363 
 
 .32-184 
 
 .48113 
 
 Si 
 
 33 
 
 .28736 
 
 .40324 
 
 .29971 
 
 .42799 
 
 .31228 
 
 .45408 
 
 .32505 
 
 .48160 
 
 33 
 
 34 
 
 .28757 
 
 .40304 
 
 .29992 
 
 .42841 
 
 .31219 
 
 .45452 
 
 .32527 
 
 .48207 
 
 34 
 
 35 
 
 .28777 
 
 .40104 
 
 .30013 
 
 .42883 
 
 .81270 
 
 .45497 
 
 .32548 
 
 .48254 
 
 35 
 
 36 
 
 .28797 
 
 .40414 
 
 .30034 
 
 .42926 
 
 .31291 
 
 .45542 
 
 .32570 
 
 .48301 
 
 36 
 
 37 
 
 .28818 
 
 .40485 
 
 .30054 
 
 .42908 
 
 .31312 
 
 .45587 
 
 .32591 
 
 .48349 
 
 37 
 
 38 
 
 .28838 
 
 .40525 
 
 .30075 
 
 .43011 
 
 .31334 
 
 .45031 
 
 .32613 
 
 .48396 
 
 38 
 
 39 
 
 .28859 
 
 .40r,G5 
 
 .30096 
 
 .43053 
 
 .31355 
 
 .45676 
 
 .32634 
 
 .48443 
 
 39 
 
 40 
 
 .28879 
 
 .40606 
 
 .30117 
 
 .43096 
 
 .31376 
 
 .45721 
 
 .32656 
 
 .48491 
 
 40 
 
 41 
 
 .28900 
 
 .40646 
 
 .30138 
 
 .43139 
 
 .31397 
 
 .45766 
 
 .32677 
 
 .48538 
 
 41 
 
 42 
 
 .28920 
 
 .40687 
 
 .30158 
 
 .43181 
 
 .31418 
 
 .45811 
 
 .32699 
 
 .48586 
 
 42 
 
 43 
 
 .28941 
 
 .40727 
 
 .30179 
 
 43224 
 
 .31439 
 
 .45856 
 
 .32720 
 
 .48633 
 
 43 
 
 44 
 
 .28961 
 
 .40768 
 
 .30200 
 
 .43267 
 
 .31461 
 
 .45901 
 
 .32742 
 
 .48681 
 
 44 
 
 45 
 
 .28981 
 
 .40808 
 
 .30221 
 
 .43310 
 
 .31482 
 
 .45946 
 
 .32763 
 
 .48728 
 
 45 
 
 46 
 
 .29002 
 
 .40849 
 
 .30242 
 
 .43352 
 
 .31503 
 
 .45992 
 
 .32785 
 
 .48776 
 
 46 
 
 47 
 
 .29022 
 
 .40890 
 
 .30263 
 
 .433D5 
 
 .31524 
 
 .46037 
 
 .32806 
 
 .48824 
 
 47 
 
 48 
 
 .29043 
 
 .40930 
 
 .30283 
 
 .43438 
 
 .31545 
 
 .46082 
 
 .32828 
 
 .48871 
 
 48 
 
 49 
 
 .29063 
 
 .40971 
 
 .30304 
 
 .43481 
 
 .31567 
 
 .46127 
 
 .32849 
 
 .48919 
 
 49 
 
 50 
 
 .29084 
 
 .41012 
 
 .30325 
 
 .43524 
 
 .31588 
 
 .46173 
 
 .32871 
 
 .48967 
 
 50 
 
 51 
 
 .29104 
 
 .41053 
 
 .30346 
 
 .43567 
 
 .31609 
 
 .46218 
 
 .32893 
 
 .49015 
 
 51 
 
 52 
 
 .29125 
 
 .41093 
 
 .30367 
 
 .43610 
 
 .31630 
 
 .46203 
 
 .32914 
 
 .49063 
 
 52 
 
 53 
 
 .29145 
 
 .41134 
 
 .30388 
 
 .43653 
 
 .31651 
 
 .46309 
 
 .32936 
 
 .49111 
 
 53 
 
 54 
 
 .29166 
 
 .41175 
 
 .30409 
 
 .43696 
 
 .31073 
 
 .46354 
 
 .32957 
 
 .49159 
 
 54 
 
 55 
 
 .29187 
 
 .41216 
 
 .30430 
 
 .43739 
 
 .31694 
 
 .46400 
 
 .32979 
 
 .49207 
 
 55 
 
 56 
 
 .29207 
 
 .41257 
 
 .30451 
 
 .43783 
 
 .31715 
 
 .46445 
 
 .33001 
 
 .49255 
 
 56 
 
 57 
 
 .29228 
 
 .41298 
 
 .30471 
 
 .43826 
 
 .31736 
 
 .46491 
 
 .33022 
 
 .49303 
 
 57- 
 
 58 
 
 .29248 
 
 .41339 
 
 .30492 
 
 .43869 
 
 .31758 
 
 .46537 
 
 .33044 
 
 .49351 
 
 58 
 
 59 
 
 .29269 
 
 .41380 
 
 .30513 
 
 .43912 
 
 .31779 
 
 .46582 
 
 .33065 
 
 .49399 
 
 59 
 
 60 
 
 .29289 
 
 .41421 
 
 .30534 
 
 .43956 
 
 .31800 
 
 .46628 
 
 .33087 
 
 .49448 
 
 60 
 
 258 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 
 48- 
 
 49 s 
 
 50 
 
 51 
 
 r 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 o 
 
 .33087 
 
 .49448 
 
 .34394 
 
 .52425 
 
 .35721 
 
 .55572 
 
 .37068 
 
 .58902 
 
 
 
 1 
 
 .33109 
 
 .49496 
 
 .34416 
 
 .52476 
 
 .35744 
 
 .55626 
 
 .37091 
 
 .58959 
 
 1 
 
 2 
 
 .33130 
 
 .49544 
 
 .34438 
 
 .52527 
 
 .35766 
 
 .55680 
 
 .37113 
 
 .59018 
 
 2 
 
 3 
 
 .33152 
 
 .49593 
 
 .31460 
 
 .52579 
 
 .35788 
 
 .55734 
 
 .37136 
 
 .59073 
 
 3 
 
 4 
 
 33173 
 
 .49641 
 
 .34482 
 
 .52630 
 
 .35810 
 
 .55789 
 
 .37158 
 
 .59130 
 
 4 
 
 5 
 
 .33195 
 
 .49690 
 
 .34504 
 
 .52681 
 
 .35833 
 
 .55843 
 
 .37181 
 
 .59188 
 
 5 
 
 6 
 
 .33217 
 
 .49738 
 
 .34526 
 
 .52732 
 
 .35855 
 
 .55897 
 
 .37204 
 
 .59245 
 
 6 
 
 7 
 
 .33238 
 
 .49787 
 
 .34548 
 
 .52784 
 
 .35877 
 
 .55951 
 
 .37226 
 
 .59302 
 
 7 
 
 8 
 
 .33260 
 
 .49835 
 
 .34570 
 
 .52835 
 
 .35900 
 
 .56005 
 
 .37249 
 
 .59360 
 
 8 
 
 9 
 
 .33282 
 
 .49884 
 
 .34592 
 
 .52886 
 
 .35922 
 
 .56060 
 
 .37272 
 
 .59418 
 
 9 
 
 10 
 
 .33303 
 
 .49933 
 
 .34614 
 
 .52938 
 
 .35944 
 
 .56114 
 
 .37294 
 
 .59475 
 
 10 
 
 11 
 
 .33325 
 
 .49981 
 
 .34636 
 
 .52989 
 
 .35967 
 
 .56169 
 
 .37317 
 
 .59533 
 
 11 
 
 12 
 
 .33347 
 
 .50030 
 
 .34638 
 
 .53041 
 
 .35989 
 
 .56223 
 
 .37340 
 
 .59590 
 
 12 
 
 13 
 
 .33368 
 
 .50079 
 
 .34680 
 
 .53C92 
 
 .36011 
 
 .56278 
 
 .37362 
 
 .59648 
 
 13 
 
 14 
 
 .33390 
 
 .50128 
 
 .34702 
 
 .53144 
 
 .36034 
 
 .56332 
 
 .37385 
 
 .59706 
 
 14 
 
 15 
 
 .33412 
 
 .50177 
 
 .34724 
 
 .53196 
 
 .36056 
 
 .56387 
 
 .37408 
 
 .59764 
 
 15 
 
 16 
 
 .33434 
 
 .50226 
 
 .34746 
 
 .53247 
 
 .36078 
 
 .56442 
 
 .37430 
 
 .59822 
 
 16 
 
 17 
 
 .33455 
 
 .50275 
 
 .34768 
 
 .53299 
 
 .36101 
 
 .56497 
 
 .37453 
 
 .59880 
 
 17 
 
 18 
 
 .33477 
 
 .50324 
 
 .34790 
 
 .53351 
 
 .36123 
 
 .56551 
 
 .37476 
 
 .59938 
 
 13 
 
 19 
 
 .33499 
 
 .50373 
 
 .34812 
 
 .53403 
 
 .36146 
 
 .56606 
 
 .37493 
 
 .59996 
 
 19 
 
 20 
 
 .33520 
 
 .50422 
 
 .34834 
 
 .53455 
 
 .36168 
 
 .56661 
 
 .37521 
 
 .60054 
 
 20 
 
 21 
 
 .33542 
 
 .50471 
 
 .34856 
 
 .53507 
 
 .36190 
 
 .56716 
 
 .37544 
 
 .60112 
 
 21 
 
 22 
 
 .33564 
 
 .50521 
 
 .34878 
 
 .53559 
 
 .36213 
 
 .56771 
 
 .37567 
 
 .60171 
 
 23 
 
 23 
 
 .33336 
 
 .50570 
 
 .34900 
 
 .53611 
 
 .36235 
 
 .56826 
 
 .37589 
 
 .60229 
 
 23 
 
 24 
 
 .33607 
 
 .50619 
 
 .34923 
 
 .53663 
 
 .36258 
 
 .56881 
 
 .37612 
 
 .60287 
 
 24 
 
 25 
 
 .33629 
 
 .50669 
 
 .34945 
 
 .53715 
 
 .36230 
 
 .56937 
 
 .37635 
 
 .60346 
 
 25 
 
 26 
 
 .33651 
 
 .50718 
 
 .34967 
 
 .53768 
 
 .36302 
 
 .56992 
 
 .37658 
 
 .60404 
 
 26 
 
 27 
 
 .33673 
 
 .50767 
 
 .34989 
 
 .53820 
 
 .36325 
 
 .57047 
 
 .37680 
 
 .60463 
 
 27 
 
 28 
 
 .33694 
 
 .50817 
 
 .35011 
 
 .53872 
 
 .36347 
 
 .57103 
 
 .37703 
 
 .60521 
 
 28 
 
 29 
 
 .33716 
 
 .50866 
 
 .350S3 
 
 .53924 
 
 .36370 
 
 .57158 
 
 .37726 
 
 .60580 
 
 29 
 
 30 
 
 ^33738 
 
 .50916 
 
 .35055 
 
 .53977 
 
 .36392 
 
 .57213 
 
 .37749 
 
 .60639 
 
 30 
 
 31 
 
 -.183760 
 
 .50966 
 
 .35077 
 
 .54029 
 
 .36415 
 
 .57269 
 
 .37771 
 
 .60698 
 
 81 
 
 32 
 
 .33782 
 
 .51015 
 
 .35099 
 
 .54082 
 
 .36437 
 
 .57324 
 
 .37794 
 
 .607'36 
 
 32 
 
 33 
 
 .33803 
 
 .51065 
 
 .35122 
 
 .54134 
 
 .36460 
 
 .57380 
 
 .37817 
 
 .60815 
 
 2? 
 
 34 
 
 .33825 
 
 .51115 
 
 .35144 
 
 .54187 
 
 .36482 
 
 .57436 
 
 .37840 
 
 .60874 
 
 34 
 
 35 
 
 .33847 
 
 .51165 
 
 .35166 
 
 .54240 
 
 .36504 
 
 .57491 
 
 .37862 
 
 .60033 
 
 35 
 
 36 
 
 .33869 
 
 .51215 
 
 .35188 
 
 .54292 
 
 .36527 
 
 .57547 
 
 .37885 
 
 .60992 
 
 S3 
 
 37 
 
 .33891 
 
 .51265 
 
 .35210 
 
 .54345 
 
 .36549 
 
 .57603 
 
 .37908 
 
 .61051 
 
 37 
 
 38 
 
 .33912 
 
 .51314 
 
 .352S2 
 
 .54398 
 
 .36572 
 
 .57659 
 
 .37931 
 
 .61111 
 
 33 
 
 39 
 
 .33934 
 
 .51364 
 
 .35254 
 
 .54451 
 
 .36594 
 
 .57715 
 
 .37954 
 
 .61170 
 
 39 
 
 40 
 
 .33956 
 
 .51415 
 
 .35277 
 
 .54504 
 
 .36617 
 
 .57771 
 
 .37976 
 
 .61229 
 
 40 
 
 41 
 
 .a3978 
 
 .51465 
 
 .35299 
 
 .54557 
 
 .36639 
 
 .57827 
 
 .37999 
 
 .61288 
 
 41 
 
 42 
 
 .34000 
 
 .51515 
 
 .35321 
 
 .54610 
 
 .36662 
 
 .57883 
 
 .38022 
 
 .61348 
 
 42 
 
 43 
 
 .34022 
 
 .51565 
 
 .35343 
 
 .54063 
 
 .36684 
 
 .57939 
 
 .38045 
 
 .61407 
 
 43 
 
 44 
 
 .34044 
 
 .51615 
 
 .35365 
 
 .54716 
 
 .36707 
 
 .57995 
 
 .38068 
 
 .61467 
 
 44 
 
 45 
 
 .34065 
 
 .51665 
 
 .35388 
 
 .54769 
 
 .36729 
 
 .58051 
 
 .38091 
 
 .61526 
 
 45 
 
 46 
 
 .34087 
 
 .51716 
 
 .35410 
 
 .54822 
 
 .36752 
 
 .58108 
 
 .38113 
 
 .61586 
 
 46 
 
 47 
 
 .34109 
 
 .51766 
 
 .35432 
 
 .54876 
 
 .36775 
 
 .58164 
 
 .38136 
 
 .61646 
 
 47 
 
 48 
 
 .34131 
 
 .51817 
 
 .35454 
 
 .54929 
 
 .36797 
 
 .58221 
 
 .38159 
 
 .61705 
 
 48 
 
 49 
 
 .34153 
 
 .51867 
 
 .35476 
 
 .54982 
 
 .36820 
 
 .58277 
 
 .38182 
 
 .61765 
 
 49 
 
 50 
 
 .34175 
 
 .51918 
 
 .35499 
 
 .55036 
 
 .36842 
 
 .58333 
 
 .38205 
 
 .61825 
 
 50 
 
 51 
 
 .34197 
 
 .51968 
 
 .35521 
 
 .55089 
 
 .36865 
 
 .58390 
 
 .38228 
 
 .61885 
 
 51 
 
 52 
 
 .34219 
 
 .52019 
 
 .35543 
 
 .55143 
 
 .36887 
 
 .58447 
 
 .38251 
 
 .61945 
 
 52 
 
 53 
 
 .34241 
 
 .52069 
 
 .35565 
 
 .55196 
 
 .36910 
 
 .58303 
 
 .38274 
 
 .62005 
 
 53 
 
 54 
 
 .34262 
 
 .52120 
 
 .35388 
 
 .55250 
 
 .36932 
 
 .58560 
 
 .38296 
 
 .62065 
 
 54 
 
 55 
 
 .34284 
 
 .52171 
 
 .35610 
 
 .55303 
 
 .36955 
 
 .58017 
 
 .38319 
 
 .62125 
 
 55 
 
 56 
 
 .34306 
 
 .52222 
 
 .35632 
 
 .55357 
 
 .36978 
 
 .58674 
 
 .38342 
 
 .62185 
 
 56 
 
 57 
 
 A?34328 
 
 .52273 
 
 .35654 
 
 .55411 
 
 .37000 
 
 .58731 
 
 .38365 
 
 .62246 
 
 57;- 
 
 58 
 
 .'34350 
 
 .52323 
 
 .35677 
 
 .55465 
 
 .37023 
 
 .58788 
 
 .38388 
 
 .62306 
 
 58' 
 
 59 
 
 .34372 
 
 .52374 
 
 .35699 
 
 .55518 
 
 .37045 
 
 .58845 
 
 .38411 
 
 .62366 
 
 59 
 
 60 
 
 .84394 
 
 .52425 
 
 .35721 
 
 .55572 
 
 .37068 
 
 .58902 
 
 .38434 
 
 .62427 
 
 60 
 
 259 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 52 
 
 53 
 
 54 
 
 55 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 / 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .38434 
 
 .62427 
 
 .39819 
 
 .66164 
 
 .41221 
 
 .70130 
 
 .42642 
 
 .74345 
 
 
 
 1 
 
 .38457 
 
 .62487 
 
 .39842 
 
 .66228 
 
 .41245 
 
 .70198 
 
 .42666 
 
 .74417 
 
 1 
 
 2 
 
 .38480 
 
 .62548 
 
 .39865 
 
 .66292 
 
 .41269 
 
 .70267 
 
 .42690 
 
 .74490 
 
 2 
 
 3 
 
 .38503 
 
 .62609 
 
 .39888 
 
 .66357 
 
 .41292 
 
 .70335 
 
 .42714 
 
 .74562 
 
 3 
 
 4 
 
 .38526 
 
 .62669 
 
 .39911 
 
 .66421 
 
 .41316 
 
 .70403 
 
 .42738 
 
 .74635 
 
 4 
 
 5 
 
 .38549 
 
 .62730 
 
 .39935 
 
 .66486 
 
 .41339 
 
 .70472 
 
 .42762 
 
 .74708 
 
 5 
 
 6 
 
 .38571 
 
 .62791 
 
 .39958 
 
 .66550 
 
 .41363 
 
 .70540 
 
 .42785 
 
 .74781 
 
 6 
 
 7 
 
 .38594 
 
 .62852 
 
 .39981 
 
 .66615 
 
 .41386 
 
 .70609 
 
 .42809 
 
 .74854 
 
 7 
 
 8 
 
 .38617 
 
 .62913 
 
 .40005 
 
 .66679 
 
 .41410 
 
 .70677 
 
 .42833 
 
 .74927 
 
 8 
 
 9 
 
 .38640 
 
 .62974 
 
 .40028 
 
 .66744 
 
 .41433 
 
 .70746 
 
 .42857 
 
 .75000 
 
 9 
 
 10 
 
 .38663 
 
 .63035 
 
 .40051 
 
 .66809 
 
 .41457 
 
 .70815 
 
 .42881 
 
 .75073 
 
 10 
 
 11 
 
 .38686 
 
 .63096 
 
 .40074 
 
 .66873 
 
 .41481 
 
 .70884 
 
 .42905 
 
 .75146 
 
 11 
 
 12 
 
 .38709 
 
 .63157 
 
 .40098 
 
 .66938 
 
 .41504 
 
 .70953 
 
 .42929 
 
 .75219 
 
 12 
 
 13 
 
 .38732 
 
 .63218 
 
 .40121 
 
 .67003 
 
 .41528 
 
 .71022 
 
 .42953 
 
 .75293 
 
 13 
 
 14 
 
 .38755 
 
 .63279 
 
 .40144 
 
 .67068 
 
 .41551 
 
 .71091 
 
 .42976 
 
 .75366 
 
 14 
 
 15 
 
 .38778 
 
 .63341 
 
 .40168 
 
 .67133 
 
 .41575 
 
 .71160 
 
 .43000 
 
 .75440 
 
 15 
 
 16 
 
 .38801 
 
 .63402 
 
 .40191 
 
 .67199 
 
 .41599 
 
 .71229 
 
 .43024 
 
 .75513 
 
 16 
 
 17 
 
 .38824 
 
 .63464 
 
 .40214 
 
 .67264 
 
 .41622 
 
 .71298 
 
 .43048 
 
 .75587 
 
 17 
 
 18 
 
 .38847 
 
 .63525 
 
 .40237 
 
 .67329 
 
 .41646 
 
 .71368 
 
 .43072 
 
 .75661 
 
 18 
 
 19 
 
 .38870 
 
 .63587 
 
 .4026? 
 
 .67394 
 
 .41670 
 
 .71437 
 
 .43096 
 
 .75734 
 
 19 
 
 20 
 
 .38893 
 
 .63648 
 
 .40284 
 
 : 67460 
 
 .41693 
 
 .71506 
 
 .43120 
 
 .75808 
 
 20 
 
 21 
 
 .38916 
 
 .63710 
 
 .40307 
 
 .67525 
 
 .41717 
 
 .71576 
 
 .43144 
 
 .75882 
 
 21 
 
 22 
 
 .38939 
 
 .63772 
 
 .40331 
 
 .67591 
 
 .41740 
 
 .71646 
 
 .43168 
 
 .75956 
 
 22 
 
 23 
 
 .38962 
 
 .63834 
 
 .40354 
 
 .67656 
 
 .41764 
 
 .71715 
 
 .43192 
 
 .76031 
 
 23 
 
 24 
 
 .38985 
 
 .63895 
 
 .40378 
 
 .67722 
 
 .41788 
 
 .71785 
 
 .43216 
 
 .76105 
 
 24 
 
 25 
 
 .39009 
 
 .63957 
 
 .40401 
 
 .67788 
 
 .41811 
 
 .71855 
 
 .43240 
 
 .76179 
 
 25 
 
 26 
 
 .39032 
 
 .64019 
 
 .40424 
 
 .67853 
 
 .41835 
 
 .71925 
 
 .43264 
 
 .76253 
 
 26 
 
 27 
 
 .39055 
 
 .64081 
 
 .40448 
 
 .67919 
 
 .41859 
 
 .71995 
 
 .43287 
 
 .76328 
 
 27 
 
 28 
 
 .39078 
 
 .64144 
 
 .40471 
 
 .67985 
 
 41882 
 
 .72065 
 
 .43311 
 
 .76402 
 
 28 
 
 29 
 
 .39101 
 
 .64206 
 
 .40494 
 
 .68051 
 
 41906 
 
 .72135 
 
 .43335 
 
 .76477 
 
 29 
 
 30 
 
 .39124 
 
 .64268 
 
 .40518 
 
 .68117 
 
 .41930 
 
 .72205 
 
 .43359 
 
 .76552 
 
 30 
 
 31 
 
 .39147 
 
 .64330 
 
 .40541 
 
 .68183 
 
 .41953 
 
 .72275 
 
 .43383 
 
 .76626 
 
 31 
 
 32 
 
 .39170 
 
 .64393 
 
 .40565 
 
 68250 
 
 .41977 
 
 .72346 
 
 .43407 
 
 .76701 
 
 32 
 
 33 
 
 .39193 
 
 .64455 
 
 .40588 
 
 .68316 
 
 .42001 
 
 .72416 
 
 .43431 
 
 .76776 
 
 33 
 
 34 
 
 .39216 
 
 .64518 
 
 .40011 
 
 68382 
 
 .42024 
 
 .72487 
 
 .43455 
 
 .76851 
 
 34 
 
 85 
 
 .39239 
 
 .64580 
 
 .40635 
 
 ! 68449 
 
 .42048 
 
 .72557 
 
 .43479 
 
 .76926 
 
 35 
 
 36 
 
 .39262 
 
 .64643 
 
 .40658 
 
 .68515 
 
 .42072 
 
 .72628 
 
 .43503 
 
 .77001 
 
 36 
 
 37 
 
 .39286 
 
 .64705 
 
 .40682 
 
 .68582 
 
 .42096 
 
 .72698 
 
 .43527 
 
 .77077 
 
 37 
 
 38 
 
 .39309 
 
 .04768 
 
 .40705 
 
 .68648 
 
 .42119 
 
 .72769 
 
 .43551 
 
 .77152 
 
 38 
 
 39 
 
 .39332 
 
 .64831 
 
 .40728 
 
 .68715 
 
 .42143 
 
 .72840 
 
 .43575 
 
 .77227 
 
 39 
 
 40 
 
 .39355 
 
 .64894 
 
 .40752 
 
 .68782 
 
 .42167 
 
 .72911 
 
 .43599 
 
 .77303 
 
 40 
 
 41 
 
 .39378 
 
 .64957 
 
 .40775 
 
 .68848 
 
 .42191 
 
 .72982 
 
 .43623 
 
 .77378 
 
 41 
 
 42 
 
 .39401 
 
 .65020 
 
 .40799 
 
 .68915 
 
 .42214 
 
 .73053 
 
 .43647 
 
 .77454 
 
 42 
 
 43 
 
 .39424 
 
 .65083 
 
 .40822 
 
 .68982 
 
 .42238 
 
 .73124 
 
 .43671 
 
 .77530 
 
 43 
 
 44 
 
 .39447 
 
 .65146 
 
 .40846 
 
 .69049 
 
 .42262 
 
 .73195 
 
 .43695 
 
 .77606 
 
 44 
 
 45 
 
 .39471 
 
 .65209 
 
 .40869 
 
 .69116 
 
 .42285 
 
 .73267 
 
 .43720 
 
 .77681 
 
 45 
 
 46 
 
 .39494 
 
 .65272 
 
 .40893 
 
 .69183 
 
 .42309 
 
 .73338 
 
 .43744 
 
 .77757 
 
 46 
 
 47 
 
 .39517 
 
 .65336 
 
 .40916 
 
 .69250 
 
 .42333 
 
 .73409 
 
 .43768 
 
 .77833 
 
 47 
 
 48 
 
 .39540 
 
 .65399 
 
 .40939 
 
 .69318 
 
 .42357 
 
 .73481 
 
 .43792 
 
 .77910 
 
 48 
 
 49 
 
 .39563 
 
 .65462 
 
 .409G3 
 
 .69385 
 
 .42381 
 
 .73552 
 
 .43816 
 
 .77986 
 
 49 
 
 50 
 
 .39586 
 
 .65526 
 
 .40986 
 
 .69452 
 
 .42404 
 
 .73024 
 
 .43840 
 
 .78062 
 
 50 
 
 51 
 
 .39610 
 
 .65589 
 
 .41010 
 
 .69520 
 
 .42428 
 
 .73696 
 
 .43864 
 
 .78138 
 
 51 
 
 52 
 
 .39633 
 
 .65653 
 
 .41033 
 
 .69587 
 
 .42452 
 
 .73768 
 
 .43888 
 
 .78215 
 
 62 
 
 53 
 
 .39656 
 
 .65717 
 
 .41057 
 
 .69655 
 
 .42476 
 
 .73840 
 
 .43912 
 
 .78291 
 
 53 
 
 54 
 
 .39679 
 
 .65780 
 
 .41080 
 
 .69723 
 
 .42499 
 
 .73911 
 
 .43936 
 
 .7'8368 
 
 54 
 
 55 
 
 .39702 
 
 .65844 
 
 .41104 
 
 .69790 
 
 .42523 
 
 .73983 
 
 .43960 
 
 .78445 
 
 55 
 
 56 
 
 .39726 
 
 .65908 
 
 .41127 
 
 .69858 
 
 .42547 
 
 74056 
 
 .43984 
 
 .78521 
 
 56 
 
 57 
 
 .39749 
 
 .65972 
 
 .41151 
 
 .69926 
 
 .42571 
 
 .74128 
 
 .44008 
 
 .78598 
 
 57 , 
 
 58 
 
 .39772 
 
 .66036 
 
 .41174 
 
 .69994 
 
 .42595 
 
 .74200 
 
 .44032 
 
 .78675 
 
 58 
 
 59 
 
 .39795 
 
 .66100 
 
 .41198 
 
 .70062 
 
 .42619 
 
 .7427'2 
 
 .44057 
 
 .78752 
 
 59 
 
 60 
 
 .39819 
 
 .66164 
 
 .41221 
 
 .70130 
 
 .42642 
 
 .74345 
 
 .44081 
 
 .78829 
 
 60 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 56' 
 
 57 
 
 58- 
 
 59 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .44081 
 
 .78829 
 
 .45536 
 
 .83608 
 
 .47008 
 
 .88708 
 
 .48496 
 
 .94160 
 
 
 
 1 
 
 .44105 
 
 .78906 
 
 .45560 
 
 .83690 
 
 .47033 
 
 .88796 
 
 .48521 
 
 .94254 
 
 1 
 
 2 
 
 .44129 
 
 .78984 
 
 .45585 
 
 .83773 
 
 .47057 
 
 .88884 
 
 .48546 
 
 .94349 
 
 2 
 
 3 
 
 .44153 
 
 .79061 
 
 .45609 
 
 .83855 
 
 .47082 
 
 .88972 
 
 .48571 
 
 .94443 
 
 3 
 
 4 
 
 .44177 
 
 .79138 
 
 .45634 
 
 .83938 
 
 .47107 
 
 .89060 
 
 .48596 
 
 .94537 
 
 4 
 
 5 
 
 .44201 
 
 .79216 
 
 .45658 
 
 .84020 
 
 .47131 
 
 .89148 
 
 .48621 
 
 .94632 
 
 5 
 
 6 
 
 .44225 
 
 .79293 
 
 .45683 
 
 .84103 
 
 .47156 
 
 .89237 
 
 .48646 
 
 .94726 
 
 6 
 
 7 
 
 .44250 
 
 .79371 
 
 .45707 
 
 .84186 
 
 .47181 
 
 .89325 
 
 .48671 
 
 .94821 
 
 7 
 
 8 
 
 .44274 
 
 .79449 
 
 .45731 
 
 .84269 
 
 .47206 
 
 .89414 
 
 .48696 
 
 .94916 
 
 8 
 
 9 
 
 .44295 
 
 .79527 
 
 .45756 
 
 .84352 
 
 .47230 
 
 .89503 
 
 .48721 
 
 .95011 
 
 9 
 
 10 
 
 .44322 
 
 .79604 
 
 .45780 
 
 .84435 
 
 .47255 
 
 .89591 
 
 .48746 
 
 .95106 
 
 10 
 
 11 
 
 .44346 
 
 .79682 
 
 .45805 
 
 .84518 
 
 .47280 
 
 .89680 
 
 .48771 
 
 .95201 
 
 11 
 
 12 
 
 .44370 
 
 .79761 
 
 .45829 
 
 .84601 
 
 .47304 
 
 .89769 
 
 .48796 
 
 .95296 
 
 12 
 
 13 
 
 .44395 
 
 .79839 
 
 .45854 
 
 .84685 
 
 .47329 
 
 .89858 
 
 .48821 
 
 .95392 
 
 13 
 
 14 
 
 .44419 
 
 .79917 
 
 .45878 
 
 .84768 
 
 .47354 
 
 .89948 
 
 .48846 
 
 .95487 
 
 14 
 
 15 
 
 .44443 
 
 .79995 
 
 .45903 
 
 .84852 
 
 .47379 
 
 .90037 
 
 .48871 
 
 .95583 
 
 15 
 
 16 
 
 .44467 
 
 .80074 
 
 .45927 
 
 .84935 
 
 .47403 
 
 .90126 
 
 .48896 
 
 .95678 
 
 16 
 
 17 
 
 .44491 
 
 .80152 
 
 .45951 
 
 .85019 
 
 .47428 
 
 .90216 
 
 .48921 
 
 .95774 
 
 17 
 
 18 
 
 .44516 
 
 .80231 
 
 .45976 
 
 .85103 
 
 .47453 
 
 .90305 
 
 .48946 
 
 .95870 
 
 18 
 
 19 
 
 .44540 
 
 .80309 
 
 .46000 
 
 .85187 
 
 .47478 
 
 .90395 
 
 .48971 
 
 .95966 
 
 19 
 
 20 
 
 .44564 
 
 .80388 
 
 .46025 
 
 .85271 
 
 .47502 
 
 .90485 
 
 .48996 
 
 .96062 
 
 20 
 
 21 
 
 .44588 
 
 .80467 
 
 .46049 
 
 .85355 
 
 .47527 
 
 .90575 
 
 .49021 
 
 .96158 
 
 21 
 
 22 
 
 .44612 
 
 .80546 
 
 .46074 
 
 .85439 
 
 .47552 
 
 .90665 
 
 .49046 
 
 .96255 
 
 22 
 
 23 
 
 .44637 
 
 .80625 
 
 .46098 
 
 .85523 
 
 .47577 
 
 .90755 
 
 .49071 
 
 .96351 
 
 23 
 
 24 
 
 .44661 
 
 .80704 
 
 .46123 
 
 .85608 
 
 .47601 
 
 .90845 
 
 .49096 
 
 .96448 
 
 24 
 
 25 
 
 .44635 
 
 .80783 
 
 .46147 
 
 .85692 
 
 .47626 
 
 .90935 
 
 .49121 
 
 .96544 
 
 25 
 
 26 
 
 .44709 
 
 .80862 
 
 .46172 
 
 .85777 
 
 .47651 
 
 .91026 
 
 .49146 
 
 .966-11 
 
 26 
 
 27 
 
 .44734 
 
 .80942 
 
 .46196 
 
 .85861 
 
 .47676 
 
 .91116 
 
 .49171 
 
 .967'38 
 
 27 
 
 28 
 
 .44758 
 
 .81021 
 
 .46221 
 
 .85946 
 
 .47701 
 
 .91207 
 
 .49196 
 
 .96835 
 
 28 
 
 29 
 
 .44782 
 
 .81101 
 
 .46246 
 
 .b6031 
 
 .47725 
 
 .91297 
 
 .49221 
 
 .96932 
 
 29 
 
 30 
 
 .44806 
 
 .81180 
 
 .46270 
 
 .86116 
 
 .47750 
 
 .91388 
 
 .49246 
 
 .97029 
 
 30 
 
 31 
 
 .44831 
 
 .81260 
 
 .46295 
 
 .86201 
 
 .47775 
 
 .91479 
 
 .49271 
 
 .97127 
 
 31- 
 
 32 
 
 .44855 
 
 .81340 
 
 .46319 
 
 .86286 
 
 .47800 
 
 .91570 
 
 .49296 
 
 .97224 
 
 32 
 
 33 
 
 .44879 
 
 .81419 
 
 .46344 
 
 .86371 
 
 .47825 
 
 .91661 
 
 .49321 
 
 .97322 
 
 33 
 
 34 
 
 .44903 
 
 .81499 
 
 .46368 
 
 .86457 
 
 .47849 
 
 .91752 
 
 .49346 
 
 .97420 
 
 34 
 
 35 
 
 .44928 
 
 .81579 
 
 .46393 
 
 .86542 
 
 .47874 
 
 .91844 
 
 .49372 
 
 .97517 
 
 35 
 
 36 
 
 .44952 
 
 .81659 
 
 .46417 
 
 .86627 
 
 .47899 
 
 .91935 
 
 .49397 
 
 .97615 
 
 36 
 
 37 
 
 .44976 
 
 .81740 
 
 .46442 
 
 .86713 
 
 .47924 
 
 .92027 
 
 .49422 
 
 .97713 
 
 37 
 
 38 
 
 .45001 
 
 .81820 
 
 .46466 
 
 .86799 
 
 .47949 
 
 .92118 
 
 .49447 
 
 .97811 
 
 38 
 
 39 
 
 .45025 
 
 .81900 
 
 .46491 
 
 .86885 
 
 .47974 
 
 .92210 
 
 .49472 
 
 .97910 
 
 39 
 
 40 
 
 .45049 
 
 .81981 
 
 .46516 
 
 .86990 
 
 .47998 
 
 .92302 
 
 .49497 
 
 .98008 
 
 40 
 
 41 
 
 .45073 
 
 .82061 
 
 .46540 
 
 .87056 
 
 .48023 
 
 .92394 
 
 .49522 
 
 .98107 
 
 41 
 
 42 
 
 .45098 
 
 .82142 
 
 .46565 
 
 .87142 
 
 .48048 
 
 .92486 
 
 .49547 
 
 .98205 
 
 42 
 
 43 
 
 .45122 
 
 .82222 
 
 .46589 
 
 .87229 
 
 .48073 
 
 .92578 
 
 .49572 
 
 .98304 
 
 43 
 
 44 
 
 .45146 
 
 .82303 
 
 .46614 
 
 .87315 
 
 .48098 
 
 .92670 
 
 .49597 
 
 .98403 
 
 44 
 
 45 
 
 .45171 
 
 .82384 
 
 .46639 
 
 .87401 
 
 .48123 
 
 .92762 
 
 .49623 
 
 .98502 
 
 45 
 
 46 
 
 .45195 
 
 .82465 
 
 .46663 
 
 .87488 
 
 .48148 
 
 .92855 
 
 .49648 
 
 .98601 
 
 46 
 
 47 
 
 .45219 
 
 .82546 
 
 .46688 
 
 .87574 
 
 .48172 
 
 .92947 
 
 .49673 
 
 .98700 
 
 47 
 
 48 
 
 .45244 
 
 .82627 
 
 .46712 
 
 .87661 
 
 .48197 
 
 .93040 
 
 .49698 
 
 .98799 
 
 48 
 
 49 
 
 .45268 
 
 .82709 
 
 .46737 
 
 .87748 
 
 .48222 
 
 .93133 
 
 .49723 
 
 .98899 
 
 49 
 
 50 
 
 .45292 
 
 .82790 
 
 .46762 
 
 .87834 
 
 .48247 
 
 .93226 
 
 .49748 
 
 .98998 
 
 50 
 
 51 
 
 .45317 
 
 .82871 
 
 .46786 
 
 .87921 
 
 .48272 
 
 .93319 
 
 .49773 
 
 .99098 
 
 51 
 
 52 
 
 .4.5341 
 
 .82953 
 
 .46811 
 
 .88008 
 
 .48297 
 
 .93412 
 
 .49799 
 
 .99198 
 
 52 
 
 53 
 
 .45365 
 
 .83034 
 
 .46836 
 
 .88095 
 
 .4S322 
 
 .93505 
 
 .49824 
 
 .99298 
 
 53 
 
 54 
 
 .45390 
 
 .83116 
 
 .46860 
 
 .88183 
 
 .48347 
 
 .93598 
 
 .49849 
 
 .99398 
 
 54 
 
 55 
 
 .45414 
 
 .83198 
 
 .46885 
 
 .88270 
 
 .48372 
 
 .93692 
 
 .49874 
 
 .99498 
 
 55 
 
 56 
 
 .45439 
 
 .83280 
 
 .46909 
 
 .88357 
 
 .48396 
 
 .93785 
 
 .49899 
 
 .99598 
 
 56 
 
 57 
 
 -45463 
 
 .83362 
 
 .46934 
 
 .88445 
 
 .48421 
 
 .93879 
 
 .49924 
 
 .99698 
 
 57 
 
 58 
 
 .45487 
 
 .83444 
 
 .46959 
 
 .88532 
 
 .48446 
 
 .93973 
 
 .49950 
 
 .99799 
 
 58 
 
 59 
 
 .45512 
 
 .83526 
 
 .46983 
 
 .88620 
 
 .48471 
 
 .94066 
 
 .49975 
 
 .99899 
 
 59 
 
 60 
 
 .45536 
 
 .83608 
 
 .47008 
 
 .88708 1 
 
 .48496 
 
 .94160 
 
 .50000 
 
 1.00000 
 
 60 
 
 261 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 
 60- 
 
 61 
 
 62 
 
 63 
 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 
 .50000 
 
 1.00000 
 
 .51519 
 
 1.06267 
 
 .53053 
 
 1.13005 
 
 .54601 
 
 1.20269 
 
 
 
 ; 
 
 .50025 
 
 1.00101 
 
 .51544 
 
 1.06375 
 
 .53079 
 
 1.13122 
 
 .54627 
 
 1.20395 
 
 1 
 
 j 
 
 .50050 
 
 1.00202 
 
 .51570 
 
 1.06483 
 
 .53104 
 
 1.13239 
 
 .54653 
 
 1.20521 
 
 2 
 
 t 
 
 .50076 
 
 1.00303 
 
 .51595 
 
 1.06592 
 
 .53130 
 
 1.13356 
 
 .54679 
 
 1.20647 
 
 
 i 
 
 .50101 
 
 1.00404 
 
 .51621 
 
 1.06701 
 
 .53156 
 
 1.13473 
 
 .54705 
 
 1.20773 
 
 4 
 
 5 
 
 .50126 
 
 1.00505 
 
 .51646 
 
 1.06809 
 
 .53181 
 
 1.13590 
 
 .54731 
 
 1.20900 
 
 5 
 
 6 
 
 .50151 
 
 1.00607 
 
 .51672 
 
 1.06918 
 
 .53207 
 
 1.13707 
 
 .54757 
 
 1.21026 
 
 6 
 
 r< 
 
 .50176 
 
 1.00708 
 
 .51697 
 
 1.07037 
 
 .53233 
 
 1.13825 
 
 .54782 
 
 1.21153 
 
 
 8 
 
 .50202 
 
 1.00810 
 
 .51723 
 
 1.07137 
 
 .53258 
 
 1.13942 
 
 .54808 
 
 1.21280 
 
 8 
 
 1 
 
 .50227 
 
 1.0091:3 
 
 .51748 
 
 1.07246 
 
 .53284 
 
 1.14060 
 
 .54834 
 
 1.21407 
 
 9 
 
 1C 
 
 .50252 
 
 1.01014 
 
 .51774 
 
 1.07356 
 
 .53310 
 
 1.14178 
 
 .54860 
 
 1.21535 
 
 10 
 
 11 
 
 .50277 
 
 1.01116 
 
 .51799 
 
 1.07465 
 
 .53336 
 
 1.14296 
 
 .54886 
 
 1.21662 
 
 11 
 
 12 
 
 .50303 
 
 1.01218 
 
 .51825 
 
 1.07575 
 
 .53361 
 
 1.14414 
 
 .54912 
 
 1.21790 12 
 
 13 
 
 .50328 
 
 1.01320 
 
 .51850 
 
 1.07685 
 
 .53387 
 
 1.14533 
 
 .54938 
 
 1.21918 113 
 
 14 
 
 .50353 
 
 1.01422 
 
 .51876 
 
 1.07795 
 
 .53413 
 
 1.14651 
 
 .54964 
 
 1.22045 
 
 14 
 
 15 
 
 .50378 
 
 1.01525 
 
 .51901 
 
 1.07905 
 
 .53439 
 
 1.14770 
 
 .54990 
 
 1.22174 
 
 15 
 
 16 
 
 .50404 
 
 1.01628 
 
 .51927 
 
 1.08015 
 
 .53464 
 
 1.14889 
 
 .55016 
 
 1.22302 
 
 10 
 
 17 
 
 .50429 
 
 1.01730 
 
 .51952 
 
 1.08126 
 
 .53490 
 
 1.15008 
 
 .55042 
 
 1.22430 
 
 17 
 
 18 
 
 .50454 
 
 1.01833 
 
 .51978 
 
 1.08236 
 
 .53516 
 
 1.15127 
 
 .55068 
 
 1.22559 
 
 18 
 
 19 
 
 .50479 
 
 1.01936 
 
 .52003 
 
 1.08347 
 
 .53542 
 
 1.15246 
 
 .55094 
 
 1.22688 
 
 19 
 
 20 
 
 .50505 
 
 1.02039 
 
 .52029 
 
 1.08458 
 
 .53567 
 
 1.15366 
 
 .55120 
 
 1.22817 
 
 20 
 
 21 
 
 .50530 
 
 1.02143 
 
 .52054 
 
 1.08569 
 
 .53593 
 
 1.15485 
 
 .55146 
 
 1.22946 
 
 21 
 
 22 
 
 .50555 
 
 1.02246 
 
 .52080 
 
 1.08680 
 
 .53619 
 
 1.15605 
 
 .66172 
 
 1.23075 
 
 00 
 
 23 
 
 .50581 
 
 1.02349 
 
 .52105 
 
 1.08791 
 
 .53645 
 
 1.15725 
 
 .55198 
 
 1.23205 
 
 23 
 
 24 
 
 .50606 
 
 1.02453 
 
 .52131 
 
 1.08903 
 
 .53670 
 
 1.15845 
 
 .55224 
 
 1.23334 
 
 24 
 
 25 
 
 .50631 
 
 1.02557 
 
 .52156 
 
 1.09014 
 
 .53696 
 
 1.15965 
 
 .55250 
 
 1.23464 
 
 25 
 
 20 
 
 .50656 
 
 1.02661 
 
 .52182 
 
 1.09126 
 
 .53722 
 
 1.16085 
 
 .55276 
 
 1.23594 
 
 20 
 
 27 
 
 .50682 
 
 1.02765 
 
 .52207 
 
 1.09238 
 
 .53748 
 
 1.16206 
 
 .55302 
 
 1.23724 
 
 27 
 
 28 
 
 .50707 
 
 1.02869 
 
 .52233 
 
 1.09350 
 
 .53774 
 
 1.16326 
 
 .55328 
 
 1.23855 
 
 88 
 
 29 
 
 .50732 
 
 1.02973 
 
 .52259 
 
 1.09462 
 
 .53799 
 
 1.16447 
 
 .55354 
 
 1.23985 
 
 ". 
 
 30 
 
 .50758 
 
 1.03077 
 
 .52284 
 
 1.09574 
 
 .53825 
 
 1.16568 
 
 .55380 
 
 1.24116 
 
 30 
 
 31 
 
 .50783 
 
 1.03182 
 
 .52310 
 
 1.09686 
 
 .53851 
 
 1.16689 
 
 .55406 
 
 1.24247 
 
 81 
 
 32 
 
 .50808 
 
 1.03286 
 
 .52335 
 
 1.09799 
 
 .53877 
 
 1.16810 
 
 .55432 
 
 1.24378 132 
 
 33 
 
 .50834 
 
 1.03391 
 
 .52361 
 
 1.09911 
 
 .53903 
 
 1.16932 
 
 .55458 
 
 1.24509 
 
 3.3 
 
 34 
 
 .50859 
 
 1.03496 
 
 .52386 
 
 1.10024 
 
 .53928 
 
 1.17053 
 
 .55484 
 
 1.24640 
 
 84 
 
 35 
 
 .50884 
 
 1.03601 
 
 .52412 
 
 1.10137 
 
 .53954 
 
 1.17175 
 
 .55510 
 
 1.24772 
 
 35 
 
 36 
 
 .50910 
 
 1.03706 
 
 .52438 
 
 1.10250 
 
 .53980 
 
 1.17297 
 
 .55536 
 
 1.24903 
 
 83 
 
 37 
 
 .50935 
 
 1.03811 
 
 .52463 
 
 1.10363 
 
 .54006 
 
 1.17419 
 
 .55563 
 
 1.25035 
 
 87 
 
 38 
 
 .50960 
 
 1.03916 
 
 .52489 
 
 1.10477 
 
 .54032 
 
 1.17541 
 
 .55589 
 
 1.25167 
 
 3S 
 
 39 
 
 .50986 
 
 1.04022 
 
 .52514 
 
 1.10590 
 
 .54058 
 
 1.17663 
 
 .55615 
 
 1.25300 
 
 89 
 
 40 
 
 .51011 
 
 1.04128 
 
 .52540 
 
 1.10704 
 
 .54083 
 
 1.17786 
 
 .55641 
 
 1.25432 
 
 40 
 
 41 
 
 .51036 
 
 1.04233 
 
 .52566 
 
 1.10817 
 
 .54109 
 
 1.17909 
 
 .55667 
 
 1.25565 
 
 41 
 
 42 
 
 .51062 
 
 1.04339 
 
 .52591 
 
 1.10931 
 
 .54135 
 
 1.18031 
 
 .55693 
 
 1.25697 
 
 42 
 
 43 
 
 .51087 
 
 1.04445 
 
 .52617 
 
 1.11045 
 
 .54161 
 
 1.18154 
 
 .55719 
 
 1.25830 
 
 43 
 
 44 
 
 .51113 
 
 1.04551 
 
 .52642 
 
 1.11159 
 
 .54187 
 
 1.18277 
 
 .55745 
 
 1.25063 
 
 44 
 
 45 
 
 .51138 
 
 1.04658 
 
 .52668 
 
 1.11274 
 
 .54213 
 
 1.18401 
 
 .55771 
 
 1.26097 
 
 45 
 
 46 
 
 .51163 
 
 1.04764 
 
 .52694 
 
 1.11388 
 
 .54238 
 
 1.18524 
 
 .55797 
 
 1.26230 
 
 46 
 
 47 
 
 .51189 
 
 1.04870 
 
 .52719 
 
 1.11503 
 
 .54264 
 
 1.18648 
 
 .55823 
 
 1.26364 
 
 47 
 
 48 
 
 .51214 
 
 1.04977 
 
 .52745 
 
 1.11617 
 
 .5-1290 
 
 1.18772 
 
 .55849 
 
 1.26498 
 
 48 
 
 49 
 
 .51239 
 
 1.05084 
 
 .52771 
 
 1.11732 
 
 .54316 
 
 1.18895 
 
 .55876 
 
 1.26632 
 
 49 
 
 50 
 
 .51265 
 
 1.05191 
 
 .52796 
 
 1.11847 
 
 .54342 
 
 1.19019 
 
 .55902 
 
 1.26766 
 
 50 
 
 51 
 
 .51290 
 
 1.05298 
 
 .52822 
 
 1.11963 
 
 .54368 
 
 1.19144 
 
 .55928 
 
 1.26900 
 
 51 
 
 52 
 
 .51316 
 
 1.05405 
 
 .528-18 
 
 1.12078 
 
 .54394 
 
 1.10268 
 
 .55954 
 
 1.27035 
 
 52 
 
 53 
 
 .51341 
 
 1.05512 
 
 .52873 
 
 1.12193 
 
 .54120 
 
 1.19393 
 
 .55980 
 
 1.27169 
 
 63 
 
 54 
 
 .51366 
 
 1.05619 
 
 .52899 
 
 1.12309 j 
 
 .54446 
 
 1.19517 
 
 .56006 
 
 1.27304 
 
 54 
 
 55 
 
 .51392 
 
 1.05727 
 
 .52924 
 
 1.12425 
 
 .54471 
 
 1.19642 
 
 .56032 
 
 1.27439 
 
 55 
 
 56 
 
 .51417 
 
 1.05835 
 
 .52950 
 
 1.12540 
 
 .54497 
 
 1.19767 
 
 .56058 
 
 1.27574 
 
 66 
 
 57 
 
 .51443 
 
 1.05942 
 
 .52976 
 
 1.12657 
 
 .54523 
 
 1.19892 
 
 .56084 
 
 1.27710 
 
 57 
 
 58 
 
 .51468 
 
 1.06050 
 
 .53001 
 
 1.12773 
 
 .54549 
 
 1.20018 
 
 .56111 
 
 1.27845 
 
 53 
 
 59 
 
 .51494 
 
 1.06158 
 
 .53027 
 
 1.12889 
 
 .54575 
 
 1.20143 
 
 .56137 
 
 1.27981 
 
 59 
 
 60 
 
 .51519 
 
 1.06267 
 
 .53053 
 
 1.13005 
 
 .54601 
 
 1.20269 
 
 2,56163 
 
 1.28117 
 
 60 
 
 262 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 
 6 
 
 4 
 
 6 
 
 5 
 
 6 
 
 6' 
 
 6 
 
 7 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .56163 
 
 1.28117 ! 
 
 .57738 
 
 1.36620 ! 
 
 .59326 
 
 1.45859 
 
 .60927 
 
 1.55930 
 
 1 
 
 .56189 
 
 .28253 I 
 
 .57765 
 
 1.3(5768 
 
 .59353 
 
 1.46020 
 
 .60954 
 
 1.56106 
 
 2 
 
 .56215 
 
 .28390 ! 
 
 .57791 
 
 1.36916 ! 
 
 .59379 
 
 1.46181 
 
 .60980 
 
 .56282 
 
 3 
 
 .56241 
 
 .28526 i 
 
 .57817 
 
 1.37064 i 
 
 .59406 
 
 1.46342 ; 
 
 .61007 
 
 .56458 
 
 4 
 
 .56267 
 
 .28663 i 
 
 .57844 
 
 1.37212 
 
 .59433 
 
 1.46504 1 
 
 .61034 
 
 .56634 
 
 5 
 
 .56294 
 
 .28800 I 
 
 .57870 
 
 1.37361 ! 
 
 .59459 
 
 1.46665 i 
 
 .61061 
 
 .56811 
 
 6 
 
 .56320 
 
 .28937 
 
 .57896 
 
 1.37509 
 
 .59486 
 
 1.46827 
 
 .61088 
 
 .56988 
 
 7 
 
 .56346 
 
 .29074 
 
 .57923 
 
 1.37658 
 
 .59512 
 
 1.46989 
 
 .61114 
 
 .57165 
 
 8 
 
 .56372 
 
 .29211 
 
 .57949 
 
 1.37808 
 
 .59539 
 
 1.47152 
 
 .61141 
 
 .57342 
 
 9 
 
 .56398 
 
 .29349 i 
 
 .57976 
 
 1.37957 
 
 .59566 
 
 1.47314 
 
 .61168 
 
 .57520 
 
 10 
 
 .56425 
 
 .29487 
 
 .58002 
 
 1.3810? 
 
 .59592 
 
 1.47477 
 
 .61195 
 
 .57698 
 
 11 
 
 .56451 
 
 .20625 
 
 .58028 
 
 1.38256 
 
 .59619 
 
 1.47640 
 
 .61222 
 
 .57876 
 
 12 
 
 .50477 
 
 29763 
 
 .58055 
 
 1.38406 
 
 .59645 
 
 1.47804 
 
 .61248 
 
 .58054 
 
 13 
 
 .56503 
 
 .29901 
 
 .58081 
 
 1.38556 
 
 .59672 
 
 1.47967 
 
 .61275 
 
 .58233 
 
 14 
 
 .56529 
 
 .30040 
 
 .58108 
 
 1.38707 
 
 .59699 
 
 1.48131 
 
 .61302 
 
 .58412 
 
 15 
 
 .56555 
 
 .30179 
 
 .58134 
 
 1.38857 
 
 .59725 
 
 1.48295 
 
 .61329 
 
 .58591 
 
 16 
 
 .56582 
 
 .30318 ! 
 
 .58160 
 
 1.39008 
 
 .59752 
 
 1.48459 ! 
 
 .61356 
 
 .58771 
 
 17 
 
 .56608 
 
 .30457 i 
 
 .58187 
 
 1.39159 
 
 .59779 
 
 1.48624 
 
 .61383 
 
 .58950 
 
 18 
 
 .56634 
 
 .30596 i 
 
 .58213 
 
 1.39311 
 
 .59805 
 
 1.48789 
 
 .61409 
 
 .59130 
 
 19 
 
 .500(50 
 
 .30735 ! 
 
 .58240 
 
 1.39462 
 
 .59832 
 
 1.48984 
 
 .61436 
 
 .59311 
 
 20 
 
 .56687 
 
 .30875 
 
 .58266 
 
 1.39614 
 
 .59859 
 
 1.49119 
 
 .61463 
 
 59491 
 
 21 
 
 .56713 
 
 .31015 
 
 .58293 
 
 1.39766 
 
 .59885 
 
 1.49284 
 
 .61490 
 
 .59672 
 
 22 
 
 .50739 
 
 .31155 
 
 .58319 
 
 1.39918 
 
 .59912 
 
 1.49450 
 
 .61517 
 
 .59853 
 
 23 
 
 .56765 
 
 .31295 
 
 .5*345 
 
 1.40070 
 
 .59938 
 
 1.49616 
 
 .61544 
 
 .60035 
 
 21 
 
 .56791 
 
 .31436 
 
 .58372 
 
 1.40222 
 
 .59965 
 
 1.49782 
 
 .61570 
 
 .60217 
 
 25 
 
 .56818 
 
 .31576 I 
 
 .58398 
 
 1.40375 
 
 .59992 
 
 1.49948 
 
 .61597 
 
 .60399 
 
 20 
 
 .56844 
 
 .31717 
 
 .58425 
 
 1.40528 
 
 .60018 
 
 1.50115 
 
 .61624 
 
 .60581 
 
 27 
 
 .56870 
 
 .31858 
 
 .58451 
 
 1.40681 
 
 .60045 
 
 1.50282 
 
 .61651 
 
 .60763 
 
 & 
 
 .56896 
 
 .31999 
 
 .58478 
 
 1.40835 
 
 .60072 
 
 1.50449 
 
 .61678 
 
 .60946 
 
 29 
 
 .56923 
 
 .32140 
 
 .58504 
 
 1.40988 
 
 .60098 
 
 1.50617 
 
 .61705 
 
 .61129 
 
 30 
 
 .56949 
 
 .32282 
 
 .58531 
 
 1.41142 
 
 .60125 
 
 1.50784 
 
 .61732 
 
 .61313 
 
 31 
 
 .56975 
 
 .32424 
 
 .58557 
 
 1.41296 
 
 .60152 
 
 1.50952 
 
 .61759 
 
 .61496 
 
 32 
 
 .57001 
 
 .32566 
 
 .58584 
 
 1.41450 
 
 .60178 
 
 1.51120 
 
 .61785 
 
 .61680 
 
 88 
 
 .57028 
 
 .32708 
 
 .58610 
 
 1.41605 
 
 .60205 
 
 1.51289 
 
 .61812 
 
 .61864 
 
 31 
 
 .57054 
 
 .32850 
 
 .58637 
 
 1.41760 
 
 .60232 
 
 1.51457 
 
 .61839 
 
 .62049 
 
 36 
 
 .57080 
 
 .32993 
 
 .58663 
 
 1.41914 
 
 .60259 
 
 1.51626 
 
 .61866 
 
 .62234 
 
 86 
 
 .57106 
 
 .33135 
 
 .58690 
 
 1.42070 
 
 .60285 
 
 1.51795 
 
 .61893 
 
 .62419 
 
 37 
 
 .57133 
 
 .33278 
 
 .58716 
 
 1.42225 
 
 .60312 
 
 1.51965 
 
 .61920 
 
 .62604 
 
 :js 
 
 .57159 
 
 .33422 
 
 .58743 
 
 1.42380 
 
 .60339 
 
 1.52134 i 
 
 .61947 
 
 .62790 
 
 99 
 
 .57185 
 
 1.33565 
 
 .58769 
 
 1.42536 
 
 .60365 
 
 1.52304 
 
 .61974 
 
 .62976 
 
 40 
 
 .57212 
 
 1.33708 
 
 .58796 
 
 1.42692 
 
 .60392 
 
 1.52474 i 
 
 .62001 
 
 .63162 
 
 41 
 
 .57238 
 
 1.33852 
 
 .58822 
 
 1.42848 
 
 .60419 
 
 1.52645 
 
 .62027 
 
 .63348 
 
 42 
 
 .57204 
 
 1.33996 
 
 .58849 
 
 1.43005 
 
 .60445 
 
 1.52815 
 
 .62054 
 
 .63535 
 
 43 
 
 .57291 
 
 1.34140 
 
 .58875 
 
 1.43162 
 
 .604?2 
 
 1.52986 
 
 .62081 
 
 .63722 
 
 44 
 
 .57317 
 
 1.34284 
 
 .58902 
 
 1.43318 
 
 .60499 
 
 1.53157 
 
 .62108 
 
 .63909 
 
 45 
 
 .57343 
 
 1.34429 
 
 .58928 
 
 1.43476 
 
 .60526 
 
 1.53329 ! 
 
 .62135 
 
 .64G97 
 
 46 
 
 .57369 
 
 1.34573 
 
 .58955 
 
 1.43633 
 
 .60552 
 
 1.53500 
 
 .62162 
 
 .64285 
 
 47 
 
 .57396 
 
 1.34718 
 
 .58981 
 
 1.43790 
 
 .60579 
 
 1.53672 
 
 .62189 
 
 .64473 
 
 48 
 
 .57422 
 
 1.34863 
 
 .59008 
 
 1.43948 
 
 .60606 
 
 1.53845 
 
 .62216 
 
 .64662 
 
 49 
 
 .57448 
 
 1.35009 
 
 .59034 
 
 1.44106 
 
 .60633 
 
 1.54017 
 
 .62243 
 
 .64851 
 
 50 
 
 .57475 
 
 1.35154 
 
 .59061 
 
 1.44264 
 
 .60659 
 
 1.54190 
 
 .62270 
 
 .65040 
 
 51 
 
 .57501 
 
 1.35300 
 
 .59087 
 
 1.44423 
 
 .60686 
 
 1.54363 
 
 .62297 
 
 .65229 
 
 52 
 
 .57527 
 
 1.35446 
 
 .59114 
 
 1.44582 
 
 .60713 
 
 1.54536 
 
 .62324 
 
 .65419 
 
 58 
 
 .57554 
 
 1.35592 
 
 .59140 
 
 1.44741 
 
 .60740 
 
 1.54709 
 
 .62351 
 
 .65609 
 
 54 
 
 .57580 
 
 1.35738 
 
 .59167 
 
 1.44900 
 
 .60766 
 
 1.54883 
 
 .62378 
 
 .65799 
 
 55 
 
 .57606 
 
 1.35885 
 
 .59194 
 
 1.45059 
 
 .60793 
 
 1.5505? 
 
 .62405 
 
 .65989 
 
 50 
 
 .57633 
 
 1.36031 
 
 .59220 
 
 1.45219 
 
 .60820 
 
 1.55231 
 
 .62431 
 
 .66180 
 
 57 
 
 .57659 
 
 1.36178 
 
 .59247 
 
 1.45378 
 
 .60847 
 
 1.55405 
 
 .62458 
 
 .66371 
 
 58 
 
 .57685 
 
 1.36325 
 
 .59273 
 
 1.45539 
 
 .60873 
 
 1.55580 
 
 .62485 
 
 .66563 
 
 59 
 
 .57712 
 
 1.36473 
 
 .59300 
 
 1.45699 
 
 .60900 
 
 1.55755 
 
 .62512 
 
 .66755 
 
 GO 
 
 .57738 
 
 1.36620 
 
 .59326 
 
 1.45859 
 
 .60927 
 
 1.55930 
 
 .62539 
 
 .66947 
 
 263 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 68- 
 
 69 
 
 70 
 
 71 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 .62539 
 
 1.66947 
 
 .64163 
 
 1.79043 
 
 .65798 
 
 1.92380 
 
 .67443 
 
 2.07155 
 
 Q 
 
 1 .62566 
 
 1.67139 
 
 .64190 
 
 1.79254 
 
 .65825 
 
 1.92614 
 
 .67471 
 
 2.07415 
 
 
 
 2 .62593 
 
 1.67332 
 
 .64218 
 
 1.79466 
 
 .65853 
 
 1.92849 
 
 .67498 
 
 2.07675 
 
 
 3 .62620 
 
 1.67525 
 
 .64245 
 
 1.79679 
 
 .65880 
 
 1.93083 
 
 .67526 
 
 2.07936 
 
 
 4 .62647 
 
 1.67718 
 
 .64272 
 
 1.79891 
 
 .65907 
 
 1.93318 
 
 .67553 
 
 2.08197 
 
 ^ 
 
 5 .62674 
 
 1.67911 
 
 .64299 
 
 1.80104 
 
 .65935 
 
 1.93554 
 
 .677)81 
 
 2.08459 
 
 ( 
 
 6 .62701 
 
 1.68105 
 
 .64326 
 
 1.80318 
 
 .65962 
 
 1.93790 
 
 .67608 
 
 2.08721 
 
 ( 
 
 7 .62728 
 
 1.68299 
 
 .64353 
 
 1.80531 
 
 .65989 
 
 1.94026 
 
 .67636 
 
 2.08983 
 
 t 
 
 8 .62755 
 
 1.68494 
 
 .64381 
 
 1.80746 
 
 .66017 
 
 1.94263 
 
 .67663 
 
 2.09246 
 
 | 
 
 9 .62782 
 
 1.68689 
 
 .64408 
 
 1.80960 
 
 .66044 
 
 1.94500 
 
 .67691 
 
 2.09510 
 
 | 
 
 10 .62809 
 
 1.68884 
 
 .64435 
 
 1.81175 
 
 .66071 
 
 1.94737 
 
 .67718 
 
 2.09774 
 
 10 
 
 11 .62836 
 
 1.69079 
 
 .64462 
 
 1.81390 
 
 .66099 
 
 1.94975 
 
 .67746 
 
 2.10038 
 
 11 
 
 12 .62863 
 
 1.69275 
 
 .64489 
 
 1.81605 
 
 .66126 
 
 1.95213 
 
 .67773 
 
 2.10303 
 
 12 
 
 3 .62890 
 
 1.69471 
 
 .64517 
 
 1.81821 
 
 .66154 
 
 1.95452 
 
 .67801 
 
 2.10568 
 
 13 
 
 4 .62917 
 
 1.69667 
 
 .64544 
 
 1.82037 
 
 .66181 
 
 1.95691 
 
 .67829 
 
 2.10834 
 
 14 
 
 5 .62944 
 
 1.69864 
 
 .64571 
 
 1.82254 
 
 .66208 
 
 1.95931 
 
 .67856 
 
 2.11101 
 
 15 
 
 6 .62971 
 
 1.70061 
 
 .64598 
 
 1.82471 
 
 .66236 
 
 1.96171 
 
 .67884 
 
 2.11367 
 
 16 
 
 7 .62998 
 
 1.70258 
 
 .64625 
 
 1.82688 
 
 .66263 
 
 1.96411 
 
 .67911 
 
 2.11635 
 
 17 
 
 8 .63025 
 
 1.70455 
 
 .64653 
 
 1.82906 
 
 .66290 
 
 1.96652 
 
 .67939 
 
 2.11903 
 
 18 
 
 9 .63052 
 
 1.70653 
 
 .64680 
 
 1.83124 
 
 .66318 
 
 1.96893 
 
 .C7966 
 
 2.12171 
 
 19 
 
 20 .63079 
 
 1.70851 
 
 .64707 
 
 1.83342 
 
 .66345 
 
 1.97135 
 
 .67994 
 
 2.12440 
 
 20 
 
 21 .63106 
 
 1.71050 
 
 .64734 
 
 1.83561 
 
 .66373 
 
 1.97377 
 
 .68021 
 
 2.12709 
 
 21 
 
 22 .63133 
 
 1.71249 
 
 .64761 
 
 1.83780 
 
 .66400 
 
 1.97619 
 
 .68049 
 
 2.12979 
 
 00 
 
 23 .63161 
 
 1.71448 
 
 .64789 
 
 1.83999 
 
 .66427 
 
 1.97862 
 
 .68077 
 
 2.13249 
 
 23 
 
 24 .63188 
 
 1.71647 
 
 .64816 
 
 1.84219 
 
 .66455 
 
 1.98106 
 
 .68104 
 
 2.13520 
 
 24 
 
 25 .63215 
 
 1.71847 
 
 .64843 
 
 1.84439 
 
 .66482 
 
 1.98349 
 
 .68132 
 
 2.13791 
 
 25 
 
 26 .63242 
 
 1.72047 
 
 .64870 
 
 1.84659 
 
 .66510 
 
 1.9S594 
 
 .68159 
 
 2.14063 
 
 26 
 
 27 .63269 
 
 1.72247 
 
 .64898 
 
 1.84880 
 
 .66537 
 
 1.98838 
 
 .68187 
 
 2.14335 
 
 27 
 
 28 .63296 
 
 1.72448 
 
 .64925 
 
 1.85102 
 
 .66564 
 
 1.99083 
 
 .68214 
 
 2.14608 
 
 23 
 
 29 .63323 
 
 1.72649 
 
 .64952 
 
 1.85323 
 
 .66592 
 
 1.99329 
 
 .68242 
 
 2.14881 
 
 Of 
 
 30 .63350 
 
 1.72850 
 
 .64979 
 
 1.85545 
 
 .66619 
 
 1.99574 
 
 .68270 
 
 2.15155 
 
 30 
 
 31 .63377 
 
 1.73052 
 
 .65007 
 
 1.85767 
 
 .66647 
 
 1.99821 
 
 .68297 
 
 2.15429 
 
 31 
 
 32 .63404 
 
 1.73254 
 
 .65034 
 
 1.85990 
 
 .66674 
 
 2.00067 
 
 .68325 
 
 2.15704 
 
 32 
 
 33 .63431 
 
 1.73456 
 
 .65061 
 
 1.86213 
 
 .66702 
 
 2.00315 
 
 .68352 
 
 2.15979 
 
 33 
 
 34 .63458 
 
 1.73659 
 
 .65088 
 
 1.86437 
 
 .66729 
 
 2.00562 
 
 .68380 
 
 2.16255 
 
 34 
 
 35 .63485 
 
 1.73862 
 
 .65116 
 
 1.86661 
 
 .66756 
 
 2.00810 
 
 .68408 
 
 2.16531 
 
 35 
 
 36 .63512 
 
 1.74065 
 
 .65143 
 
 1.86885 
 
 .66784 
 
 2.01059 
 
 .68435 
 
 2.16808 
 
 80 
 
 37 .63539 
 
 1.74269 
 
 .65170 
 
 1.87109 
 
 .66811 
 
 2.01-308 
 
 .68463 
 
 2.17085 
 
 37 
 
 38 .63566 
 
 1.74473 
 
 .65197 
 
 1.87334 
 
 .66839 
 
 2.01557 
 
 .68490 
 
 2.17363 
 
 38 
 
 39 .63594 
 
 1.74677 
 
 .65225 
 
 1.87560 
 
 .66866 
 
 2.01807 
 
 .68518 
 
 2.17641 
 
 39 
 
 40 .63621 
 
 1.74881 
 
 .65252 
 
 1.87785 
 
 .66894 
 
 2.02057 
 
 .68546 
 
 2.17920 
 
 40 
 
 41 .63648 
 
 1.75086 
 
 .65279 
 
 1.88011 
 
 .66921 
 
 2.02308 
 
 .68573 
 
 2.18199 
 
 41 
 
 42 .63875 
 
 1.75292 
 
 .65306 
 
 1.88238 
 
 .66949 
 
 2.02559 
 
 .68601 
 
 2.18479 
 
 42 
 
 43 .63702 
 
 1.75497 
 
 .65334 
 
 1.88465 
 
 .66976 
 
 2.02810 
 
 .68628 
 
 2.18759 
 
 43 
 
 44 .63729 
 
 1.75703 
 
 .65361 
 
 1.88692 
 
 .67003 
 
 2.03062 
 
 .68656 
 
 2.19040 
 
 44 
 
 45 .63756 
 
 1.75909 
 
 .65388 
 
 1.88920 
 
 .67031 
 
 2.03315 
 
 .68684 
 
 2.19322 
 
 15 
 
 46 .63783 
 
 1.76116 
 
 .65416 
 
 1.89148 
 
 .67058 
 
 2.03568 
 
 .68711 
 
 2.19604 
 
 to 
 
 47 .63810 
 
 1.76323 
 
 .65443 
 
 1.89376 
 
 .67'086 
 
 2.03821 
 
 .68739 
 
 2.19886 
 
 [7 
 
 48 .63838 
 
 1.76530 
 
 .65470 
 
 1.89605 
 
 .67113 
 
 2.04075 
 
 .68767 
 
 2.20169 
 
 48 
 
 49 .63865 
 
 1.76737 
 
 .65497 
 
 1.89834 
 
 .67141 
 
 2.04329 
 
 .68794 
 
 2.20453 
 
 49 
 
 30 .63892 
 
 1.76945 
 
 .65525 
 
 1.90063 
 
 .67168 
 
 2.04584 
 
 .68822 
 
 2.20737 
 
 50 
 
 51 .63919 
 
 1.77154 
 
 .65552 
 
 1.90293 
 
 .67196 
 
 2.04839 
 
 .68849 
 
 2.21021 
 
 51 
 
 32 .63946 
 
 1.77362 
 
 .65579 
 
 1.90524 
 
 .67223 
 
 2.05094 
 
 .68877 
 
 2.21306 
 
 
 53 .63973 
 
 1.77571 i 
 
 .65607 
 
 1.90754 
 
 .67251 
 
 2.05350 
 
 .68905 
 
 2.21592 53 
 
 54 .64000 
 
 1 77780 
 
 .65634 
 
 1.90986 
 
 .67278 
 
 2.05607 
 
 .68932 
 
 2.21878 |54 
 
 55 .64027 
 
 1.77990 
 
 .65661 
 
 1.91217 
 
 .67306 
 
 2.05864 
 
 .68960 
 
 2.22165 
 
 55 
 
 56 .64055 
 
 1.78200 
 
 .65689 
 
 1.91449 
 
 .67333 
 
 2.06121 
 
 .68988 
 
 2.22452 
 
 56 
 
 57 .64082 
 
 1.78410 ( 
 
 .65716 
 
 1.91681 
 
 .67361 
 
 2.06379 
 
 .69015 
 
 2.22740 
 
 57 
 
 58 .64109 
 
 1.78621 
 
 .65743 
 
 1.91914 
 
 .67388 
 
 2.06637 
 
 .69043 
 
 2.23028 
 
 58 
 
 59 .641--J6 
 
 1.78832 
 
 .65771 
 
 1.92147 
 
 .67416 
 
 2.06896 
 
 .69071 
 
 2.23317 
 
 59 
 
 30 64163 
 
 1.79043 I 
 
 .65798 
 
 1.92380 
 
 .67443 
 
 2.07155 
 
 .69098 
 
 2.23607 
 
 60 
 
 264 
 
TABLE Xm. VERSINES AND EXSECANTS. 
 
 ' 
 
 72"' 
 
 73 
 
 74 
 
 75 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .69098 
 
 2.23607 
 
 .70763 
 
 2.42030 
 
 .72436 
 
 2.62796 
 
 .74118 
 
 2.86370 
 
 1 
 
 .69126 
 
 2.23897 
 
 .70791 
 
 2.42356 
 
 .72464 
 
 2.63164 
 
 .74146 
 
 2.86790 
 
 2 
 
 .69154 
 
 2.24187 
 
 .70818 
 
 2.42683 
 
 .72492 
 
 2.63533 
 
 .74174 
 
 2.87211 
 
 3 
 
 .69181 
 
 2.24478 
 
 .70846 
 
 2.43010 
 
 .72520 
 
 2.63903 
 
 .74202 
 
 2.87633 
 
 4 
 
 .69209 
 
 2.24770 
 
 .70874 
 
 2.43337 
 
 .72548 
 
 2.64274 
 
 .74231 
 
 2.88056 
 
 5 
 
 .69237 
 
 2.25062 
 
 .70902 
 
 2.43666 
 
 .72576 
 
 2.64645 
 
 .74259 
 
 2.88479 
 
 6 
 
 .69264 
 
 2.25355 
 
 .70930 
 
 2.43995 
 
 .72604 
 
 2.65018 
 
 .74287 
 
 2.88904 
 
 7 
 
 .69292 
 
 2.25648 
 
 .70958 
 
 2.44324 
 
 .72632 
 
 2.65391 
 
 .74315 
 
 2.89330 
 
 8 
 
 .69320 
 
 2.25942 
 
 .70985 
 
 2.44655 
 
 .72660 
 
 2.65765 
 
 .74343 
 
 2.89756 
 
 9 
 
 .69347 
 
 226237 
 
 .71013 
 
 2.44986 
 
 .72688 
 
 2.66140 
 
 .74371 
 
 2.90184 
 
 10 
 
 .69375 
 
 2.26531 
 
 .71041 
 
 2.45317 
 
 .72716 
 
 2.66515 
 
 .74399 
 
 2.90G13 
 
 11 
 
 .69403 
 
 2.26S27 
 
 .71069 
 
 2.45650 
 
 .72744 
 
 2.66892 
 
 .74427 
 
 2.91042 
 
 13 
 
 .69430 
 
 2.27123 
 
 .71097 
 
 2.45983 
 
 .72772 
 
 2.67269 
 
 .74455 
 
 2.91473 
 
 13 
 
 .69458 
 
 2.27420 
 
 .71125 
 
 2.46316 
 
 .72800 
 
 2.67647 
 
 .74484 
 
 2.91904 
 
 14 
 
 .69486 
 
 2.27717 
 
 .71153 
 
 2.46651 
 
 .72828 
 
 2.68025 
 
 .74512 
 
 2.92337 
 
 15 
 
 .69514 
 
 2.28015 
 
 .71180 
 
 2.46986 
 
 .72856 
 
 2.68405 
 
 .74540 
 
 2.92770 
 
 1(3 
 
 .69541 
 
 2.28313 
 
 .71208 
 
 2.47321 
 
 .72884 
 
 2.68785 
 
 .74568 
 
 2.93204 
 
 17 
 
 .69569 
 
 2.28612 
 
 .71236 
 
 2.47658 
 
 .72912 
 
 2.69167 
 
 .74596 
 
 2.93640 
 
 18 
 
 .69597 
 
 2.28912 
 
 .71264 
 
 2.47995 
 
 .72940 
 
 2.69549 
 
 .74624 
 
 2.94076 
 
 19 
 
 .69624 
 
 2.29212 
 
 .71292 
 
 2.48333 
 
 .72968 
 
 2.69931 
 
 .74652 
 
 2.94514 
 
 20 
 
 .69652 
 
 2.29512 
 
 .71320 
 
 2.48671 
 
 .72996 
 
 2.70315 
 
 .74680 
 
 2.9495JJ 
 
 21 
 
 .69680 
 
 2.29814 
 
 .71348 
 
 2.49010 
 
 .73024 
 
 2.70700 
 
 .74709 
 
 2.95392 
 
 22 
 
 .69708 
 
 2.30115 
 
 .71375 
 
 2.49350 
 
 .73052 
 
 2.71085 
 
 .74737 
 
 2.95832 
 
 23 
 
 .69735 
 
 2.30418 
 
 .71403 
 
 2.49691 
 
 .73080 
 
 2.71471 
 
 .74765 
 
 2.96274 
 
 24 
 
 .69763 
 
 2.30721 
 
 .71431 
 
 2.50032 
 
 .73108 
 
 2.71858 
 
 .74793 
 
 2.96716 
 
 25 
 
 .69791 
 
 2.31024 
 
 .71459 
 
 2.50374 
 
 .73136 
 
 2.72246 
 
 .74821 
 
 2.97160 
 
 26 
 
 .69818 
 
 2.31328 
 
 .71487 
 
 2.50716 
 
 .73164 
 
 2.72635 
 
 .74849 
 
 2.97604 
 
 27 
 
 .69846 
 
 2.31633 
 
 .71515 
 
 2.51060 
 
 .73192 
 
 2.73024 
 
 .74878 
 
 2.98050 
 
 28 
 
 .69874 
 
 2.31939 
 
 .71543 
 
 2.51404 
 
 .73220 
 
 2.73414 
 
 .74906 
 
 2.98497 
 
 29 
 
 .69902 
 
 2.32244 
 
 .71571 
 
 2.51748 
 
 .73248 
 
 2.73806 
 
 .74934 
 
 2.98944 
 
 30 
 
 .69929 
 
 2.32551 
 
 .71598 
 
 2.52094 
 
 .73276 
 
 2.74198 
 
 .74962 
 
 2.99393 
 
 31 
 
 .69957 
 
 2.32858 
 
 .71626 
 
 2.52440 
 
 .73304 
 
 2.74591 
 
 .74990 
 
 2.99843 
 
 32 
 
 .69985 
 
 2.33166 
 
 .71654 
 
 2.52787 
 
 (>>!'* 
 
 2.74984 
 
 .75018 
 
 3.00293 
 
 33 
 
 .70013 
 
 2.33474 
 
 .71682 
 
 2.53134 
 
 .73360 
 
 2.75379 
 
 .75047 
 
 3.00745 
 
 34 
 
 .70040 
 
 2.33783 
 
 .71710 
 
 2.53482 
 
 .73388 
 
 2.75775 
 
 .75075 
 
 3.01198 
 
 35 
 
 .70068 
 
 2.34092 
 
 .71738 
 
 2.53831 
 
 .73416 
 
 2.76171 
 
 .75103 
 
 3.01652 
 
 36 
 
 .70096 
 
 2.34403 
 
 .71766 
 
 2.54181 
 
 .73444 
 
 2.76568 
 
 .75131 
 
 3.02107 
 
 37 
 
 .70124 
 
 2.34713 
 
 .71794 
 
 2.54531 
 
 .73472 
 
 2.76966 
 
 .75159 
 
 3.02563 
 
 38 
 
 .70151 
 
 2.35025 
 
 .71822 
 
 2.54883 
 
 .73500 
 
 2.77365 
 
 .75187 
 
 3.03020 
 
 39 
 
 .70179 
 
 2.35336 
 
 .71850 
 
 2.55235 
 
 .73529 
 
 2.77765 
 
 .75216 
 
 3.03479 
 
 40 
 
 .70207 
 
 2.35649 
 
 .71877 
 
 2.55587 
 
 .73557 
 
 2.78166 
 
 .75244 
 
 3.03938 
 
 41 
 
 .70235 
 
 2.35962 
 
 .71905 
 
 2.55940 
 
 .73585 
 
 2.78568 
 
 .75272 
 
 3.04398 
 
 42 
 
 .70263 
 
 2.36276 
 
 .71933 
 
 2.56294 
 
 .73613 
 
 2.78970 
 
 .75300 
 
 3.04860 
 
 43 
 
 .70290 
 
 2.36590 
 
 .71961 
 
 2.56649 
 
 .73641 
 
 2.79374 
 
 .75328 
 
 3.05322 
 
 44 
 
 .70318 
 
 2.36905 
 
 .71989 
 
 2.57005 
 
 .73669 
 
 2.79778 
 
 .75356 
 
 3.05786 
 
 45 
 
 .70346 
 
 2.37221 
 
 .72017 
 
 2.57361 
 
 .73697 
 
 2.80183 
 
 .75385 
 
 3.06251 
 
 46 
 
 .70374 
 
 2.37537 
 
 .72045 
 
 2.57718 
 
 .73725 
 
 2.80589 
 
 .75413 
 
 3.06717 
 
 47 
 
 .70401 
 
 2.37854 
 
 .72073 
 
 2.58076 
 
 .73753 
 
 2.80996 
 
 .75441 
 
 3.07184 
 
 48 
 
 .70429 
 
 2.38171 
 
 .72101 
 
 2.58434 
 
 .73781 
 
 2.81404 
 
 .75469 
 
 3.07652 
 
 49 
 
 .70457 
 
 2.38489 
 
 .72129 
 
 2.58794 
 
 .73809 
 
 2.81813 
 
 .75497 
 
 3.08121 
 
 50 
 
 .70485 
 
 2.38808 
 
 .72157 
 
 2.59154 
 
 .73837 
 
 2.82223 
 
 .75526 
 
 3.08591 
 
 51 
 
 .70513 
 
 2.39128 
 
 .72185 
 
 2.59514 
 
 .73865 
 
 2.82633 
 
 .75554 
 
 3.09063 
 
 52 
 
 .705-10 
 
 2.39448 
 
 .72213 
 
 2.59876 
 
 .73893 
 
 2.83045 
 
 .75582 
 
 3.09535 
 
 53 
 
 .70568 
 
 2.39768 
 
 .72241 
 
 2.60238 
 
 .73921 
 
 2.83457 
 
 .75610 
 
 3.10009 
 
 54 
 
 .70596 
 
 2.40089 
 
 .72269 
 
 2.60601 
 
 .73950 
 
 2.83871 
 
 .75689 
 
 3.10484 
 
 55 
 
 .70624 
 
 2.40411 
 
 .72296 
 
 2.60965 
 
 .73978 
 
 2.84285 
 
 .75667 
 
 3.10960 
 
 56 
 
 .70652 
 
 2.40734 
 
 .72324 
 
 2.61330 
 
 .74006 
 
 2.84700 
 
 .75695 
 
 3.11437 
 
 57 
 
 .70679 
 
 2.41057 
 
 .72352 
 
 2.61695 
 
 .74034 
 
 2.85116 
 
 .75723 
 
 3.11915 
 
 58 
 
 .70707 
 
 2.41381 
 
 .72380 
 
 2.62061 
 
 .74062 
 
 2.85533 
 
 .75751 
 
 3.12394 
 
 59 
 
 .70735 
 
 2.41705 
 
 .72408 
 
 2.62428 
 
 .74090 
 
 2.85951 
 
 .75780 
 
 3.12875 
 
 60 
 
 .70763 
 
 2.42030 
 
 .72436 
 
 2.62796 
 
 .74118 
 
 2.86370 
 
 .75808 
 
 3.13357 
 
 265 
 
.TABLE XIII. VERSINES AND EXSECANTS. 
 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 8 
 
 7 
 
 9 
 
 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 j Vers. 
 
 Exsec. 
 
 
 
 
 .75808 
 
 3.13357 
 
 .77505 
 
 3.44541 
 
 .79209 
 
 3.80973 
 
 .80919 
 
 4.24084 
 
 
 
 1 
 
 .75836 
 
 3.13839 
 
 .77'533 
 
 3.45102 
 
 .79237 
 
 3.81633 
 
 .80948 
 
 4.24870 
 
 1 
 
 2 
 
 75864 
 
 3.14323 
 
 .77562 
 
 3.45664 
 
 .79266 
 
 3.82294 
 
 .80976 
 
 4.25658 
 
 2 
 
 3 
 
 7'5892 
 
 3.14809 
 
 .77590 
 
 3.46228 
 
 .79294 
 
 3.82956 
 
 .81005 
 
 4.26448 
 
 3 
 
 4 
 
 75921 
 
 3.15295 
 
 .77618 
 
 3.46793 
 
 .79323 
 
 3.83621 
 
 .81033 
 
 4.27241 
 
 4 
 
 5 
 
 75949 
 
 3.15782 
 
 .77647 
 
 3.47360 
 
 .79351 
 
 3.84288 
 
 .81062 
 
 4.28030 
 
 5 
 
 G 
 
 75977 
 
 3.16271 
 
 .77675 
 
 3.47928 
 
 .79380 
 
 3.84956 
 
 .81090 
 
 4.28833 
 
 6 
 
 7 
 
 76005 
 
 3.16761 
 
 .77703 
 
 8.48498 
 
 .79408 
 
 3.85627 
 
 .81119 
 
 4.29634 
 
 7 
 
 8 
 
 76034 
 
 3.17252 
 
 .77732 
 
 3.49069 
 
 .79437 
 
 3.86299 
 
 .81148 
 
 4.30436 
 
 8 
 
 9 
 
 76062 
 
 3.17744 
 
 .77760 
 
 3.49642 
 
 .79465 
 
 3.86973 
 
 .81176 
 
 4.31241 
 
 g 
 
 10 
 
 76090 
 
 3.18238 
 
 .77788 
 
 3.50216 
 
 .79493 
 
 3.87649 
 
 .81205 
 
 4.32049 
 
 10 
 
 11 
 
 .76118 
 
 3.18733 
 
 .77817 
 
 3.50791 
 
 .79522 
 
 3.88327 
 
 .81233 
 
 4.32P59 
 
 11 
 
 12 
 
 .76147 
 
 3.19228 
 
 .77845 
 
 3.51368 
 
 .79550 
 
 3.89007 
 
 .81262 
 
 4.33(171 
 
 12 
 
 13 
 
 .76175 
 
 3.19725 
 
 .77874 
 
 3.51947 
 
 .79579 
 
 3.89689 
 
 .81290 
 
 4.34480 
 
 18 
 
 14 
 
 .76203 
 
 3.20224 
 
 .77902 
 
 3.52527 
 
 .79607 
 
 3.90373 
 
 .81319 
 
 4.35304 
 
 14 
 
 15 
 
 .76231 
 
 3.20723 
 
 .77930 
 
 3.53109 
 
 .79636 
 
 3.91058 
 
 .81348 
 
 4.36124 
 
 Ifi 
 
 16 
 
 .76260 
 
 3.21224 
 
 .77959 
 
 3.53692 
 
 .79664 
 
 3.91746 
 
 .81376 
 
 4.36947 
 
 18 
 
 17 
 
 .76288 
 
 3.21726 
 
 .77987 
 
 3.54277 
 
 .79693 
 
 3.92436 
 
 1 .81405 
 
 4.37772 
 
 17 
 
 18 
 
 .76316 
 
 3.22229 
 
 .78015 
 
 3.54863 
 
 .79721 
 
 3.93128 
 
 .81433 
 
 4.38600 
 
 18 
 
 19 
 
 .76344 
 
 3.22734 
 
 .78044 
 
 3.55451 
 
 .79750 
 
 3.93821 
 
 .81403 
 
 4.3943') 
 
 19 
 
 20 
 
 .76373 
 
 3.23239 
 
 .78072 
 
 3.56041 
 
 .79778 
 
 3.94517 
 
 .81491 
 
 4.40263 
 
 20 
 
 21 
 
 .76401 
 
 3.23746 
 
 .78101 
 
 3.56632 
 
 .79807 
 
 3.95215 
 
 .81519 
 
 4.41099 
 
 21 
 
 22 
 
 .76429 
 
 3.24255 
 
 .78129 
 
 3.57224 
 
 .79835 
 
 3.95914 
 
 .81548 
 
 4.41937 
 
 O-i 
 
 23 
 
 .76453 
 
 3.24764 
 
 .78157 
 
 3.57819 
 
 .79864 
 
 3.S6616 
 
 .81576 
 
 4.42778 
 
 28 
 
 24 
 
 .76486 
 
 3.25275 
 
 .78186 
 
 3.58414 
 
 .79892 
 
 3.97320 
 
 .81605 
 
 4.43022 
 
 x!4 
 
 25 
 
 .76514 
 
 3.25787 
 
 .78214 
 
 3.59012 
 
 .79921 
 
 3.98025 
 
 .81633 
 
 4.41408 
 
 ,'.'.-> 
 
 2(5 
 
 .76542 
 
 3.26300 
 
 .78242 
 
 3.59611 
 
 .7994!) 
 
 3.98733 
 
 .81662 
 
 4.45317 
 
 26 
 
 27 
 
 .70571 
 
 3.26814 
 
 .78271 
 
 3.G0211 
 
 .79973 
 
 3.99443 
 
 1 .81691 
 
 4. 46? 09 
 
 87 
 
 28 
 
 .76599 
 
 3.27330 
 
 .78299 
 
 3.60813 
 
 .80006 
 
 4.00155 
 
 i .81719 
 
 4.47023 
 
 L'8 
 
 29 
 
 .76627 
 
 3.27847 
 
 .78328 
 
 3.61417 
 
 .80035 
 
 4.00869 
 
 1 .81748 
 
 4.47'881 
 
 28 
 
 30 
 
 .76655 
 
 3.28366 
 
 .78356 
 
 3.62023 
 
 .80063 
 
 4.01585 
 
 .81776 
 
 4.48740 
 
 *u 
 
 31 
 
 .76684 
 
 3.28885 
 
 .7838-1 
 
 3.62630 
 
 .80092 
 
 4.02303 
 
 i .81805 
 
 4.49603 
 
 81 
 
 32 
 
 .76712 
 
 3.29406 
 
 .78413 
 
 3.63238 
 
 .80120 
 
 4.03024 
 
 ! .81834 
 
 4.50408 
 
 3.^ 
 A 
 
 33 
 
 .76740 
 
 3.29929 
 
 .78441 
 
 3.63849 
 
 .80149 
 
 4.03746 
 
 .81862 
 
 4.51387 
 
 88 
 
 34 
 
 .76769 
 
 3.30452 
 
 .78470 
 
 3.61461 
 
 '.80177 
 
 4.04471 
 
 .81891 
 
 4.52208 
 
 34 
 
 35 
 
 .76797 
 
 3.30977 
 
 .78498 
 
 3.65074 
 
 .80206 
 
 4.05197 
 
 .81919 
 
 4.53081 
 
 35 
 
 36 
 
 .76825 
 
 3.31503 
 
 .78526 
 
 3.65690 
 
 .80234 
 
 4.05926 
 
 .81948 
 
 4.53958 
 
 88 
 
 37 
 
 .76854 
 
 3.32031 
 
 .78555 
 
 3.66307 
 
 .80263 
 
 4.06657 
 
 .81977 
 
 4.54837 
 
 3V 
 
 38 
 
 .76882 
 
 3.32560 
 
 .78583 
 
 3.60925 
 
 .80291 
 
 4.07390 
 
 .82005 
 
 4.55720 
 
 88 
 
 39 
 
 .76910 
 
 3.33090 
 
 .78612 
 
 3.67545 
 
 .80320 
 
 4.08125 
 
 .82034 
 
 4.56605 
 
 39 
 
 40 
 
 .76938 
 
 3.83622 
 
 .78640 
 
 3.68167 
 
 .80348 
 
 4.08863 
 
 .82063 
 
 4.57493 
 
 40 
 
 41 
 
 .76967 
 
 3.34154 
 
 .78669 
 
 3.68791 
 
 .80377 
 
 4.09602 
 
 .82091 
 
 4.58383 
 
 41 
 
 42 
 
 .76995 
 
 3.34689 
 
 .78697 
 
 3.69417 
 
 .80405 
 
 4.10344 
 
 .82120 
 
 4.59277 
 
 42 
 
 43 
 
 .77023 
 
 3.35224 
 
 .78725 
 
 3.7'0044 
 
 .80434 
 
 4.11088 
 
 .82148 
 
 4.6017'4 
 
 43 
 
 44 
 
 .77052 
 
 3.35761 
 
 .78754 
 
 3.70673 
 
 .80462 
 
 4.11835 
 
 .82177 
 
 4.61073 
 
 44 
 
 45 
 
 .77080 
 
 3.36299 
 
 .78782 
 
 3.71303 
 
 .80491 
 
 4.12583 
 
 .82206 
 
 4.61976 
 
 45 
 
 46 
 
 .77108 
 
 3.36839 
 
 .78811 
 
 3.71935 
 
 .80520 
 
 4.13334 
 
 .82234 
 
 4.62881 
 
 40 
 
 47 
 
 .77137 
 
 3.37380 
 
 .78839 
 
 3.72569 
 
 .80548 
 
 4.14087 
 
 .82263 
 
 4.63790 
 
 47 
 
 48 
 
 .77165 
 
 3.37923 
 
 .78868 
 
 3.73205 
 
 .80577 
 
 4.14842 
 
 .82292 
 
 4.64701 
 
 48 
 
 49 
 
 .77193 
 
 3.38466 
 
 .78896 
 
 8. 73843 
 
 .80605 
 
 4.15599 
 
 .82320 
 
 4.65010 
 
 49 
 
 60 
 
 .77222 
 
 3.39012 
 
 .78924 
 
 3.74482 
 
 .80634 
 
 4.16359 
 
 .82349 
 
 4.G0533 
 
 50 
 
 51 
 
 .77250 
 
 3.39558 
 
 .78953 
 
 3.75123 
 
 .80669 
 
 4.17121 
 
 .82377 
 
 4.67454 
 
 51 
 
 52 
 
 .77278 
 
 3.40106 
 
 .78981 
 
 3.75rG6 
 
 .80691 
 
 4.17886 
 
 .82406 
 
 4.08377 
 
 52 
 
 53 
 
 .77307 
 
 3.40656 
 
 .79010 
 
 3.76411 
 
 .80719 
 
 4.18652 
 
 .82435 
 
 4.69304 
 
 53 
 
 54 
 
 .77335 
 
 3.41206 
 
 .79038 
 
 3.77057 
 
 .80748 
 
 4.19421 
 
 .82463 
 
 4.70234 
 
 54 
 
 55 
 
 .77363 
 
 3.41759 
 
 .79067 
 
 3 . 77705 
 
 .80776 
 
 4.20193 
 
 .82492 
 
 4.71166 
 
 55 
 
 56 
 
 .77392 
 
 3.42312 
 
 .79095 
 
 3.78355 
 
 .80805 
 
 4.20966 
 
 .82521 
 
 4.72102 
 
 56 
 
 57 
 
 .77420 
 
 3.42867 
 
 .79123 
 
 3.79007 
 
 .80833 
 
 4.21742 
 
 82549 
 
 4.73041 
 
 57 
 
 58 
 
 .77448 
 
 3.43424 
 
 .79152 
 
 3.79661 
 
 .80862 
 
 4.22521 
 
 .82578 
 
 4.73983 
 
 58 
 
 59 
 
 .77477 
 
 3.43982 
 
 .79180 
 
 3.80316 
 
 .80891 
 
 4.23301 
 
 .82607 
 
 4.74929 
 
 59 
 
 60 
 
 .77505 
 
 3.44541 
 
 .79209 
 
 3.80973 
 
 .80919 
 
 4.24084 
 
 .82035 
 
 4.75877 
 
 60 
 
 266 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 80 
 
 81 82 83 
 
 ! 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .82635 
 
 4.75877 
 
 .84357 
 
 5.39245 
 
 .86083 
 
 6.18530 
 
 .87813 
 
 7.205*1 
 
 
 
 1 
 
 .82664 
 
 4.76829 
 
 .84385 
 
 5.40422 
 
 .86112 
 
 6.20020 
 
 .87842 
 
 7.22500 
 
 
 2 
 
 .82692 
 
 4.77784 
 
 .84414 
 
 5.41602 
 
 .86140 
 
 6.21517 
 
 .87871 
 
 7.24457 
 
 & 
 
 3 
 
 .82721 
 
 4.78742 
 
 .84443 
 
 5.42787 
 
 .86169 
 
 6.23019 
 
 .87900 
 
 7.26425 
 
 3 
 
 4 
 
 .82750 
 
 4.79703 
 
 .84471 
 
 5.43977 
 
 .86198 
 
 6.24529 
 
 .87929 
 
 7.28402 
 
 4 
 
 5 
 
 .82778 
 
 4.80667 
 
 .84500 
 
 5.45171 
 
 .86227 
 
 6.26044 
 
 .87957 
 
 7.36388 
 
 5 
 
 6 
 
 .82807 
 
 4.81635 
 
 .84529 
 
 5.46369 
 
 .86256 
 
 6.27566 
 
 .87986 
 
 7.32384 
 
 6 
 
 7 
 
 .82836 
 
 4.82606 
 
 .84558 
 
 5.47572 
 
 .86284 
 
 6.29095 
 
 .88015 
 
 7.34390 
 
 7 
 
 8 
 
 .82864 
 
 4.83581 
 
 .84586 
 
 5.48779 
 
 .86313 
 
 6.30630 
 
 .88044 
 
 7.36405 
 
 8 
 
 9 
 
 .82893 
 
 4.84558 
 
 .84615 
 
 5.49991 
 
 .86342 
 
 6.32171 
 
 .88073 
 
 7.38431 
 
 9 
 
 10 
 
 .82922 
 
 4.85539 
 
 .84644 
 
 5.51208 
 
 .86371 
 
 6.33719 
 
 .88102 
 
 7.40466 
 
 10 
 
 11 
 
 .82950 
 
 4.86524 
 
 .84673 
 
 5.52429 
 
 .86400 
 
 6.35274 
 
 .88131 
 
 7.42511 
 
 11 
 
 12 
 
 .82979 
 
 4.87511 
 
 .84701 
 
 5.53655 
 
 .86428 
 
 6.36835 
 
 .88160 
 
 7.44566 
 
 12 
 
 13 
 
 .83003 
 
 4.88502 
 
 .84730 
 
 5.54886 
 
 .86457 
 
 6.38403 
 
 .88188 
 
 7.46632 
 
 13 
 
 14 
 
 .83036 
 
 4.89497 
 
 .84759 
 
 5.56121 
 
 .86486 
 
 6.39978 
 
 .88217 
 
 7.48707 
 
 14 
 
 15 
 
 .83065 
 
 4.90495 
 
 .84788 
 
 5.57361 
 
 .86515 
 
 6.41560 
 
 .88246 
 
 7.50793 
 
 15 
 
 16 
 
 .83094 
 
 4.91496 
 
 .84816 
 
 5.58606 
 
 .86544 
 
 6.43148 
 
 .88275 
 
 7.52889 
 
 16 
 
 17 
 
 .83122 
 
 4.92501 
 
 .84845 
 
 5.59855 
 
 .86573 
 
 6.44743 
 
 .88304 
 
 7.54996 
 
 17 
 
 18 
 
 .83151 
 
 4.93509 
 
 .84874 
 
 5.61110 
 
 .86601 
 
 6.46346 
 
 .88333 
 
 7.57113 
 
 18 
 
 19 
 
 .83180 
 
 4.94521 
 
 .84903 
 
 5.62369 
 
 .86630 
 
 6.47955 
 
 .88362 
 
 7.59241 
 
 19 
 
 20 
 
 .83208 
 
 4.95536 
 
 .84931 
 
 5.63633 
 
 .86659 
 
 6.49571 
 
 .88391 
 
 7.61379 
 
 20 
 
 21 
 
 .83237 
 
 4.96555 
 
 .84960 
 
 5.64902 
 
 .86688 
 
 6.51194 
 
 .88420 
 
 7.63528 
 
 21 
 
 22 
 
 .83266 
 
 4.97577 
 
 .84989 
 
 5.66176 
 
 .86717 
 
 6.52825 
 
 .88448 
 
 7.65688 
 
 22 
 
 23 
 
 .83294 
 
 4.98603 
 
 .85018 
 
 5.67454 
 
 .86746 
 
 6.54462 
 
 .88477 
 
 7.67859 
 
 23 
 
 24 
 
 .83323 
 
 4.99633 
 
 .85046 
 
 5.68738 
 
 .86774 
 
 6.56107 
 
 .88506 
 
 7.70041 
 
 24 
 
 25 
 
 .83352 
 
 5.00666 
 
 .85075 
 
 5.70027 
 
 .86803 
 
 6.57759 
 
 .88535 
 
 7.72234 
 
 25 
 
 26 
 
 .83380 
 
 5.01703 
 
 .85104 
 
 5.71321 
 
 .86832 
 
 6.59418 
 
 .88504 
 
 7.74438 
 
 26 
 
 27 
 
 .83409 
 
 5.02743 
 
 .85133 
 
 5.72620 
 
 .86861 
 
 6.61085 
 
 .88593 
 
 7.76653 
 
 27 
 
 28 
 
 .83438 
 
 5.03787 
 
 .85162 
 
 5.73924 
 
 .86890 
 
 6.62759 
 
 .88622 
 
 7.78880 
 
 28 
 
 29 
 
 .83467 
 
 5.04834 
 
 .85190 
 
 5.75233 
 
 .86919 
 
 6.64441 
 
 .88651 
 
 7.81118 
 
 29 
 
 30 
 
 .83495 
 
 5.05886 
 
 .85219 
 
 5.76547 
 
 .86947 
 
 6.66130 
 
 .88680 
 
 7.83367 
 
 30 
 
 31 
 
 .83524 
 
 5.06941 
 
 .85248 
 
 5.77866 
 
 .86976 
 
 6.67826 
 
 .88709 
 
 7.85628 
 
 31 
 
 32 
 
 .83553 
 
 5.08000 
 
 .85277 
 
 5.79191 
 
 .87005 
 
 6.69530 
 
 .88737 
 
 7.87901 
 
 32 
 
 33 
 
 .83581 
 
 5.09062 
 
 .85305 
 
 5.80521 
 
 .87034 
 
 6.71242 
 
 .88766 
 
 7.90186 
 
 33 
 
 34 
 
 .83610 
 
 5.10129 
 
 .85334 
 
 5.81856 
 
 .87063 
 
 6.72962 
 
 .88795 
 
 7.92482 
 
 34 
 
 35 
 
 .83639 
 
 5.11199 
 
 .85363 
 
 5.83196 
 
 .87092 
 
 6.74689 
 
 .88824 
 
 7.94791 
 
 35 
 
 36 
 
 .83667 
 
 5.12273 
 
 .85392 
 
 5.84542 
 
 .87120 
 
 6.76424 
 
 .88853 
 
 7.97111 
 
 36 
 
 37 
 
 .83696 
 
 5.13350 
 
 .85420 
 
 5.85893 
 
 .87149 
 
 6.78167 
 
 .88882 
 
 7.99444 
 
 37 
 
 38 
 
 .83725 
 
 5.14432 
 
 .85449 
 
 5.87250 
 
 .87178 
 
 6.79918 
 
 .88911 
 
 8.01788 
 
 38 
 
 39 
 
 .83754 
 
 5.15517 
 
 .85478 
 
 5.88612 
 
 .87207 
 
 6.81677 
 
 .88940 
 
 8.04146 
 
 39 
 
 40 
 
 .83782 
 
 5.16607 
 
 .85507 
 
 5.89979 
 
 .87236 
 
 6.83443 
 
 .88969 
 
 8.06515 
 
 40 
 
 41 
 
 .83811 
 
 5.17700 
 
 .85536 
 
 5.91352 
 
 .87265 
 
 6.85218 
 
 .88998 
 
 8.08897 
 
 41 
 
 42 
 
 .83840 
 
 5.18797 
 
 .85564 
 
 5.92731 
 
 .87294 
 
 6.87001 
 
 .89027 
 
 8.11292 
 
 42 
 
 43 
 
 .83868 
 
 5.19896 
 
 .85593 
 
 5.94115 
 
 .87322 
 
 6.88792 
 
 .89055 
 
 8.13699 
 
 43 
 
 44 
 
 .83897 
 
 5.21004 
 
 .85622 
 
 5.95505 
 
 .87351 
 
 6.90592 
 
 .89084 
 
 8.16120 
 
 44 
 
 45 
 
 .83926 
 
 5.22113 
 
 .85651 
 
 5.96900 
 
 .87380 
 
 6.92400 
 
 .89113 
 
 8.18553 
 
 45 
 
 46 
 
 .83954 
 
 5.23226 
 
 .85680 
 
 5.98301 
 
 .87409 
 
 6.94216 
 
 .89142 
 
 8.20999 
 
 46 
 
 47 
 
 .83983 
 
 5.24343 
 
 .85708 
 
 5.99708 
 
 .87438 
 
 6.96040 
 
 .89171 
 
 8.23459 
 
 47 
 
 48 
 
 .84012 
 
 5.25464 
 
 .85737 
 
 6.01120 
 
 .87467 
 
 6.97873 
 
 .89200 
 
 8.25931 148 
 
 49 
 
 .84041 
 
 5.26590 
 
 .85766 
 
 6.02538 
 
 .87496 
 
 6.99714 
 
 .89239 
 
 8.28417 49 
 
 50 
 
 .84069 
 
 5.27719 
 
 .85795 
 
 6.03962 
 
 .87524 
 
 7.01565 
 
 .89258 
 
 8.30917 
 
 50 
 
 51 
 
 .84098 
 
 5.28853 
 
 .85823 
 
 6.05392 
 
 .87553 
 
 7.03423 
 
 .89287 
 
 8.33430 
 
 51 
 
 52 
 
 .84127 
 
 5.29991 
 
 .85852 
 
 6.06828 
 
 .87582 
 
 7.05291 
 
 .89316 
 
 8.35957 
 
 52 
 
 53 
 
 .84155 
 
 5.31133 
 
 .85881 
 
 6.08269 
 
 .87611 
 
 7.07167 
 
 .89345 
 
 8.38497 
 
 53 
 
 54 
 
 .84184 
 
 5.32279 
 
 .85910 
 
 6.09717 
 
 .87640 
 
 7.09052 
 
 .89374 
 
 8.41052 
 
 54 
 
 55 
 
 .84213 
 
 5.33429 
 
 .85939 
 
 6.11171 
 
 .87669 
 
 7.10946 
 
 .89403 
 
 8.43620 
 
 55 
 
 56 
 
 .84242 
 
 5.34584 
 
 .85967 
 
 6.12630 
 
 .87698 
 
 7.12849 
 
 .89431 
 
 8.46203 
 
 56 
 
 57 
 
 .84270 
 
 5.35743 
 
 .85996 
 
 6.14096 
 
 .87726 
 
 7.14760 
 
 .89460 
 
 8.48800 
 
 57 
 
 58 
 
 .84299 
 
 5.36906 
 
 .86025 
 
 6.15568 
 
 .87755 
 
 7.16681 
 
 .89489 
 
 8.51411 
 
 58 
 
 50 
 
 .84328 
 
 5.38073 
 
 .86054 
 
 6.17046 
 
 .87784 
 
 7.18612 
 
 .89518 
 
 8.54037 
 
 59 
 
 60 
 
 .84357 
 
 5.39245 
 
 .86083 
 
 6.18530 
 
 .87813 
 
 7,20551 
 
 .89547 
 
 8.56677 
 
 60 
 
 267 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 84 
 
 85 
 
 86 
 
 ' 
 
 Vers. 
 
 Exsec. 
 
 Yers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .89547 
 
 8.5G677 
 
 .91284 
 
 10.47371 
 
 .93024 
 
 13.33559 
 
 
 
 1 
 
 .89576 
 
 8.59332 
 
 .91313 
 
 10.51199 
 
 .93053 
 
 13.39547 
 
 1 
 
 2 
 
 .89605 
 
 8.62002 
 
 .91342 
 
 10.55052 
 
 .93082 
 
 13.45586 
 
 2 
 
 3 
 
 .89634 
 
 8.64687 
 
 .91371 
 
 10.58932 
 
 .93111 
 
 13.51676 
 
 3 
 
 4 
 
 .89663 
 
 8.67387 
 
 .91400 
 
 10.62837 
 
 .93140 
 
 13.57817 
 
 4 
 
 5 
 
 .89092 
 
 8.70103 
 
 .91429 
 
 10.66769 
 
 .93169 
 
 13.64011 
 
 5 
 
 6 
 
 . 89721 
 
 8.72833 
 
 .91458 
 
 10.70728 
 
 .93198 
 
 13.70258 
 
 6 
 
 7 
 
 .89750 
 
 8.75579 
 
 .91487 
 
 10.74714 
 
 .93227 
 
 13.76558 
 
 7 
 
 8 
 
 .89779 
 
 8.78341 
 
 .91516 
 
 10.78727 
 
 .93257 
 
 13.82913 
 
 8 
 
 9 
 
 .89808 
 
 8.81119 
 
 .91545 
 
 10.82768 
 
 .93286 
 
 13.80323 
 
 9 
 
 10 
 
 .89836 
 
 8.83912 
 
 .91574 
 
 10.86837 
 
 .93315 
 
 13.95788 
 
 10 
 
 11 
 
 .89865 
 
 8.86722 
 
 .91603 
 
 10.90934 
 
 .93344 
 
 14.02310 
 
 11 
 
 12 
 
 .89894 
 
 8.89547 
 
 .91632 
 
 10.95060 
 
 .93373 
 
 14.08890 
 
 12 
 
 13 
 
 .89923 
 
 8.92389 
 
 .91661 
 
 10.99214 
 
 .93402 
 
 14.15527 
 
 13 
 
 14 
 
 .89952 
 
 8.95248 
 
 .91690 
 
 11.03397 
 
 .93431 
 
 14.22223 
 
 14 
 
 15 
 
 .89981 
 
 8.98123 
 
 .91719 
 
 11.07610 
 
 .93460 
 
 14.28979 
 
 15 
 
 16 
 
 .90010 
 
 9.01015 
 
 .91748 
 
 11.11852 
 
 .93489 
 
 14.35795 
 
 16 
 
 17 
 
 .90039 
 
 9.03923 
 
 .91777 
 
 11.16125 
 
 .93518 
 
 14.42672 
 
 17 
 
 18 
 
 .90088 
 
 9.06849 
 
 .91806 
 
 11.20427 
 
 .93547 
 
 14.49611 
 
 18 
 
 19 
 
 .90097 
 
 9.09792 
 
 .91835 
 
 11.24761 
 
 .93576 
 
 14.56614 
 
 19 
 
 20 
 
 .90126 
 
 9.12752 
 
 .91864 
 
 11.29125 
 
 .93605 
 
 14.63679 
 
 20 
 
 21 
 
 .90155 
 
 9.15730 
 
 .91893 
 
 11.33521 
 
 .93634 
 
 14.70810 
 
 21 
 
 22 
 
 .90184 
 
 9.18725 
 
 .91922 
 
 11.37948 
 
 .93663 
 
 14.78005 
 
 22 
 
 23 
 
 .90213 
 
 9.21739 
 
 .91951 
 
 11.42408 
 
 .93692 
 
 14.85268 
 
 23 
 
 24 
 
 .90242 
 
 9.24770 
 
 .91980 
 
 11.46900 
 
 .93721 
 
 14.92597 
 
 24 
 
 25 
 
 .90271 
 
 9.27819 
 
 .92009 
 
 11.51424 
 
 .93750 
 
 14.99995 
 
 25 
 
 26 
 
 .90300 
 
 9.30887 
 
 .92038 ' 
 
 11.55982 
 
 .93779 
 
 15.07462 
 
 26 
 
 27 
 
 .90329 
 
 9.33973 
 
 .92067 
 
 11.60572 
 
 .93808 
 
 15.14999 
 
 27 
 
 28 
 
 .90358 
 
 9.87077 
 
 .92096 
 
 11.65197 
 
 .93837 
 
 15.22607 
 
 28 
 
 29 
 
 .90386 
 
 9.40201 
 
 .92125 
 
 11.69856 
 
 .93866 
 
 15.30287 
 
 29 
 
 30 
 
 .90415 
 
 9.43343 
 
 .92154 
 
 11.74550 
 
 .93895 
 
 15.38041 
 
 30 
 
 31 
 
 .90444 
 
 9.46505 
 
 .92183 
 
 11.79278 
 
 .93924 
 
 15.45869 
 
 31 
 
 32 
 
 .90473 
 
 9.49685 
 
 '.92212 
 
 11.84042 
 
 .93953 
 
 15.53772 
 
 32 
 
 33 
 
 .90502 
 
 9.52886 
 
 .92241 
 
 11.88841 
 
 .93982 
 
 15.61751 
 
 33 
 
 34 
 
 .90531 
 
 9.56106 
 
 .92270 
 
 11.93677 
 
 .94011 
 
 15.69808 
 
 34 
 
 35 
 
 .90560 
 
 9.59346 
 
 .92299 
 
 11.98549 
 
 .94040 
 
 15.77044 
 
 35 
 
 36 
 
 .90589 
 
 9.62605 
 
 .92328 
 
 12.03458 
 
 .94069 
 
 15.86159 
 
 36 
 
 37 
 
 .90618 
 
 9.65885 
 
 .92357 
 
 12.08040 
 
 .94098 
 
 15.94456 
 
 37 
 
 38 
 
 .90647 
 
 9.69186 
 
 .92386 
 
 12.13388 
 
 .94127 
 
 16.02835 
 
 38 
 
 39 
 
 .90676 
 
 9.72507 
 
 .92415 
 
 32.18411 
 
 .94156 
 
 16.11297 
 
 39 
 
 40 
 
 .90705 
 
 9.75849 
 
 .92444 
 
 12.23472 
 
 .94186 
 
 16.19843 
 
 40 
 
 41 
 
 .90734 
 
 9.79212 
 
 .92473 
 
 12.28572 
 
 .94215 
 
 16.28476 
 
 41 
 
 42 
 
 .90763 
 
 9.82596 
 
 .92502 
 
 12.33:12 
 
 .94244 
 
 16.37196 
 
 43 
 
 43 
 
 .90792 
 
 9.86001 
 
 .92531 
 
 12.38891 
 
 .94273 
 
 16.46005 
 
 43 
 
 44 
 
 .90821 
 
 9.89428 
 
 .92560 
 
 12.44112 
 
 .94302 
 
 16.54903 
 
 41 
 
 45 
 
 .90850 
 
 9.92877 
 
 .92589 
 
 12.49373 
 
 .94331 
 
 16.63893 
 
 45 
 
 46 
 
 .90879 
 
 9.96348 
 
 .92618 
 
 12.54676 
 
 .94360 
 
 16.72975 
 
 46 
 
 47 
 
 .90908 
 
 9.99841 
 
 .92647 
 
 12.60021 
 
 .94389 
 
 16.82152 
 
 47 
 
 48 
 
 .90937 
 
 10.03356 
 
 .92676 
 
 12.65408 
 
 .94418 
 
 16.91424 
 
 48 
 
 49 
 
 .90966 
 
 10.06894 
 
 .92705 
 
 12.70S38 
 
 .94147 
 
 17.C0794 
 
 49 
 
 50 
 
 .90995 ' 
 
 10.10455 
 
 .92734 
 
 12.70312 
 
 .94476 
 
 17.10262 
 
 50 
 
 51 
 
 .91024 
 
 10.14039 
 
 .92763 
 
 12.81829 
 
 .94505 
 
 17.19830 
 
 51 
 
 52 
 
 .91053 
 
 10.17646 
 
 .92?'92 
 
 12.87391 
 
 .94534 
 
 17.29501 
 
 53 
 
 53 
 
 .91082 
 
 10.21277 
 
 .92821 
 
 12.92999 
 
 .94563 
 
 17.39274 
 
 53 
 
 54 
 
 .91111 
 
 10.24932 
 
 .92850 
 
 12.98651 
 
 .94592 
 
 17.49153 
 
 54 
 
 55 
 
 .91140 
 
 10.28610 
 
 .92879 
 
 13 04350 
 
 .94621 
 
 17.59139 
 
 55 
 
 56 
 
 .91169 
 
 10.32313 
 
 .92908 
 
 13.10096 
 
 .94650 
 
 17.69233 
 
 56 
 
 57 
 
 .91197 
 
 10.36040 
 
 .92937 
 
 13.15889 
 
 .94679 
 
 17.79438 
 
 57 
 
 58 
 
 .91226 
 
 10.39792 
 
 .92966 
 
 13.21730 
 
 .94708 
 
 17.89755 
 
 58 
 
 59 
 
 .91255 
 
 10.43569 
 
 .92995 
 
 13.27620 
 
 .94737 
 
 18.00185 
 
 59 
 
 60 
 
 .91284 
 
 10.47371 
 
 .93024 
 
 13.33559 
 
 .94766 
 
 18.10732 
 
 60 
 
 268 
 
TABLE XIII. VERSINES AND EXSECANTS. 
 
 / 
 
 87 3 
 
 88 
 
 89 
 
 / 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 Vers. 
 
 Exsec. 
 
 
 
 .94766 
 
 18.10732 
 
 .96510 
 
 27.65371 
 
 .98255 
 
 56.29869 
 
 
 
 1 
 
 .94795 
 
 18.21397 
 
 .96539 
 
 27.89440 j 
 
 .98284 
 
 57.26976 
 
 1 
 
 
 .94825 
 
 18,32182 
 
 .96568 
 
 28.13917 
 
 .98313 
 
 58.27431 
 
 2 
 
 3 
 
 .94854 
 
 18.43088 
 
 .96597 
 
 28.38812 
 
 .98342 
 
 59.31411 
 
 3 
 
 4 
 
 .94833 
 
 18.54119 
 
 .96626 
 
 28.64137 
 
 .98371 
 
 60.39105 
 
 4 
 
 5 
 
 .94912 
 
 18.65275 
 
 .96655 
 
 28.89903 
 
 .98400 
 
 61.50715 
 
 5 
 
 6 
 
 .94941 
 
 18.76560 
 
 .96684 
 
 29.16120 
 
 .98429 
 
 62.66460 
 
 6 
 
 7 
 
 .94970 
 
 18.87976 
 
 .96714 
 
 29.42802 
 
 .98458 
 
 63.86572 
 
 7 
 
 8 
 
 .94999 
 
 18.99524 
 
 .96743 
 
 29.69960 
 
 .98487 
 
 65.11304 
 
 8 
 
 9 
 
 .95028 
 
 19.11208 
 
 .96772 
 
 29.97607 
 
 .98517 
 
 66.40927 
 
 9 
 
 10 
 
 .95057 
 
 19.23028 
 
 .96801 
 
 30.25758 
 
 .98546 
 
 67.75736 
 
 10 
 
 11 
 
 .95086 
 
 19.34989 
 
 .96830 
 
 30.54425 
 
 .98575 
 
 69.16047 
 
 11 
 
 12 
 
 .95115 
 
 19.47093 
 
 .96859 
 
 30.83623 
 
 .98604 
 
 70.62285 
 
 12 
 
 13 
 
 .95144 
 
 19.59341 
 
 .96888 
 
 31.13366 
 
 .98633 
 
 72.14583 
 
 13 
 
 14 
 
 .95173 
 
 19.71737 
 
 .96917 
 
 31.43671 
 
 .98662 
 
 73.73586 
 
 14 
 
 15 
 
 .95202 
 
 19.84283 
 
 .96946 
 
 31.74554 
 
 .98691 
 
 75.39655 
 
 15 
 
 16 
 
 .95231 
 
 19.96982 
 
 .96975 
 
 32.06030 
 
 .98720 
 
 77.13274 
 
 16 
 
 17 
 
 .95260 
 
 20.09838 
 
 .97004 
 
 32.38118 
 
 .98749 
 
 78.94968 
 
 17 
 
 13 
 
 .95289 
 
 20.22852 
 
 .97033 
 
 32.70835 
 
 .98778 
 
 80.85315 
 
 18 
 
 19 
 
 .95318 
 
 20.36027 
 
 .97062 
 
 33.04199 
 
 .98807 
 
 82.84947 
 
 19 
 
 20 
 
 .95347 
 
 20.49368 
 
 .97092 
 
 33.38232 
 
 .98836 
 
 84.94561 
 
 20 
 
 21 
 
 .95377 
 
 20.62876 
 
 .97121 
 
 33.72952 
 
 ,98866 
 
 87.14924 
 
 21 
 
 22 
 
 .95406 
 
 20.76555 
 
 .97150 
 
 34.08380 
 
 .98895 
 
 89.46886 
 
 22 
 
 23 
 
 .95435 
 
 20.90409 
 
 .97179 
 
 34.44539 
 
 .98924 
 
 91.91387 
 
 23 
 
 24 
 
 .95464 
 
 21.04440 
 
 .97208 
 
 34.81452 
 
 .98953 
 
 94.49471 
 
 24 
 
 25 
 
 .95493 
 
 21.18653 
 
 .97237 
 
 35.19141 
 
 .98982 
 
 97.22303 
 
 25 
 
 26 
 
 .95522 
 
 21.33050 
 
 .97266 
 
 35.57633 
 
 .99011 
 
 100.1119 
 
 26 
 
 27 
 
 .95551 
 
 21.47635 
 
 .97295 
 
 35.96953 
 
 .99040 
 
 103.1757 
 
 27 
 
 28 
 
 .95580 
 
 21.62413 
 
 .97324 
 
 36.37127 
 
 .99069 
 
 106.4311 
 
 28 
 
 29 
 
 .95609 
 
 21.77386 
 
 .97353 
 
 36.78185 
 
 .1)9098 
 
 109.8966 
 
 29 
 
 30 
 
 .95638 
 
 21.92559 
 
 .97382 
 
 37.20155 
 
 .99127 
 
 113.5930 
 
 30 
 
 31 
 
 .95667 
 
 22.07935 
 
 .97411 
 
 37.63068 
 
 .09156 
 
 117.5444 
 
 31 
 
 32 
 
 .95696 
 
 22.23520 
 
 .97440 
 
 38.06957 
 
 .99186 
 
 121.7780 
 
 32 
 
 33 
 
 .95725 
 
 22.39316 
 
 .97470 
 
 38.51855 
 
 .99215 
 
 126.3253 
 
 33 
 
 34 
 
 .95754T 
 
 22.55329 
 
 .97499 
 
 38.97797 
 
 .99244 
 
 131.2223 
 
 34 
 
 35 
 
 .95783 
 
 22.71563 
 
 .97528 
 
 39.44820 
 
 .99273 
 
 136.5111 
 
 35 
 
 36 
 
 .95812 
 
 22.88022 
 
 .97557 
 
 39.92963 
 
 .99S02 
 
 142.2406 
 
 36 
 
 37 
 
 .95842 
 
 23.04712 
 
 .97586 
 
 40.42266 
 
 .99331 
 
 148.4684 
 
 37 
 
 38 
 
 .95871 
 
 23.21637 
 
 .97615 
 
 40.92772 
 
 .99360 
 
 155.2623 
 
 38 
 
 39 
 
 .95900 
 
 23.38802 
 
 .97644 
 
 41.44525 
 
 .99889 
 
 162.7033 
 
 39 
 
 40 
 
 .95929 
 
 23.56212 
 
 .97673 
 
 41.97571 
 
 .99418 
 
 170.8883 
 
 40 
 
 41 
 
 .95958 
 
 23.73873 
 
 .97702 
 
 42.51961 
 
 .99447 
 
 179.9350 
 
 41 
 
 42 
 
 .95987 
 
 23.91790 
 
 .97731 
 
 43.07746 
 
 .99476 
 
 189.9868 
 
 42 
 
 43 
 
 .96016 
 
 24.09969 
 
 .97760 
 
 43.64980 
 
 .99505 
 
 201.2212 
 
 43 
 
 44 
 
 .96045 
 
 24.28414 
 
 .97789 
 
 44.23720 
 
 .99535 
 
 213.8600 
 
 44 
 
 45 
 
 .96074 
 
 24.47134 
 
 .97819 
 
 44.84026 
 
 .99564 
 
 228.1839 
 
 45 
 
 4G 
 
 .96103 
 
 24.66132 
 
 .97848 
 
 45.45963 
 
 .99593 
 
 244.5540 
 
 46 
 
 47 
 
 .96132 
 
 24.85417 
 
 .97877 
 
 46.09596 
 
 .99622 
 
 263.4427 
 
 47 
 
 48 
 
 .96161 
 
 25.04994 
 
 .97906 
 
 46.74997 
 
 .99651 
 
 285.4795 
 
 48 
 
 49 
 
 .96190 
 
 25.24869 
 
 .97935 
 
 47.42241 
 
 .99C80 
 
 311.5230 
 
 49 
 
 50 
 
 .96219 
 
 25.45051 
 
 .97964 
 
 48.11406 
 
 .997'09 
 
 342.7752 
 
 50 
 
 51 
 
 .96248 
 
 25.65546 
 
 .97993 
 
 48.82576 
 
 .99738 
 
 380.9723 
 
 51 
 
 52 
 
 .90277 
 
 25.86360 
 
 .98022 
 
 49.55840 
 
 .99767 
 
 428.7187 
 
 52 
 
 53 
 
 .96307 
 
 26.07503 
 
 .98051 
 
 50.31290 
 
 .OC796 
 
 490.1070 
 
 53 
 
 54 
 
 .96336 
 
 26.28981 
 
 .98080 
 
 51.09027 
 
 .99825 
 
 571.9581 
 
 54 
 
 55 
 
 .96365 
 
 26.50804 
 
 .98109 
 
 51.89156 
 
 .99855 
 
 686.5496 
 
 55 
 
 56 
 
 .96394 
 
 26.72978 
 
 .98138 
 
 52.71790 
 
 .99884 
 
 858.4369 
 
 56 
 
 57 
 
 .96423 
 
 26.95513 
 
 .98168 
 
 53.57046 
 
 .99913 
 
 1144.916 
 
 57 
 
 58 
 
 .96452 
 
 27.18417 
 
 .98197 
 
 64.45053 
 
 .99942 
 
 1717.874 
 
 58 
 
 59 
 
 .96481 
 
 27.41700 
 
 .98226 
 
 55.a5946 
 
 .99971 
 
 3436.747 
 
 59 
 
 60 
 
 .96510 
 
 27.65371 
 
 .98255 
 
 56.29869 
 
 1.00000 
 
 Infinite 
 
 60 
 
 269 
 
TABLE XIV.-CUBIC YARDS PER 100 FEET. SLOPES 
 
 Depth 
 
 Base 
 12 
 
 Base 
 14 
 
 Base 
 16 
 
 Base 
 18 
 
 Base 
 22 
 
 Base 
 24 
 
 Base 
 26 
 
 Base 
 28 
 
 1 
 
 45 
 
 53 
 
 60 
 
 68 
 
 82 
 
 90 
 
 97 
 
 105 
 
 2 
 
 93 
 
 107 
 
 122 
 
 137 
 
 167 
 
 181 
 
 196 
 
 211 
 
 3 
 
 142 
 
 163 
 
 186 
 
 208 
 
 253 
 
 275 
 
 297 
 
 319 
 
 4 
 
 193 
 
 222 
 
 252 
 
 281 
 
 341 
 
 870 
 
 400 
 
 430 
 
 5 
 
 245 
 
 282 
 
 319 
 
 356 
 
 431 
 
 468 
 
 505 
 
 542 
 
 6 
 
 300 
 
 844 
 
 389 
 
 433 
 
 522 
 
 567 
 
 611 
 
 656 
 
 7 
 
 356 
 
 408 
 
 460 
 
 512 
 
 616 
 
 668 
 
 719 
 
 771 
 
 8 
 
 415 
 
 474 
 
 533 
 
 593 
 
 711 
 
 770 
 
 830 
 
 889 
 
 9 
 
 475 
 
 542 
 
 608 
 
 675 
 
 808 
 
 875 
 
 942 
 
 1008 
 
 10 
 
 537 
 
 611 
 
 685 
 
 759 
 
 907 
 
 981 
 
 1056 
 
 1130 
 
 11 
 
 601 
 
 682 
 
 764 
 
 845 
 
 1008 
 
 1090 
 
 1171 
 
 1253 
 
 12 
 
 667 
 
 756 
 
 844 
 
 933 
 
 1111 
 
 1200 
 
 1289 
 
 1378 
 
 13 
 
 734 
 
 831 
 
 926 
 
 1023 
 
 1216 
 
 1312 
 
 1408 
 
 1505 
 
 14 
 
 804 
 
 907 
 
 1010 
 
 1115 
 
 1322 
 
 1426 
 
 1530 
 
 1633 
 
 15 
 
 875 
 
 986 
 
 1096 
 
 1208 
 
 1431 
 
 1542 
 
 1653 
 
 1764 
 
 16 
 
 948 
 
 1067 
 
 1184 
 
 1304 
 
 1541 
 
 1659 
 
 1778 
 
 1896 
 
 17 
 
 1023 
 
 1149 
 
 1274 
 
 1401 
 
 1653 
 
 1779 
 
 1905 
 
 2031 
 
 18 
 
 1100 
 
 1233 
 
 1366 
 
 1500 
 
 1767 
 
 1900 
 
 2033 
 
 2167 
 
 19 
 
 1179 
 
 1319 
 
 1460 
 
 1601 
 
 1882 
 
 2023 
 
 2164 
 
 2305 
 
 20 
 
 1259 
 
 1407 
 
 1555 
 
 1704 
 
 2000 
 
 2148 
 
 2296 
 
 2444 
 
 21 
 
 1342 
 
 1497 
 
 1653 
 
 1808 
 
 2119 
 
 2275 
 
 2431 
 
 2586 
 
 22 
 
 1426 
 
 1589 
 
 1752 
 
 1915 
 
 2241 
 
 2404 
 
 - 2567 
 
 2730 
 
 23 
 
 1512 
 
 1682 
 
 1853 
 
 2023 
 
 2364 
 
 2534 
 
 2705 
 
 2875 
 
 24 
 
 1600 
 
 1778 
 
 1955 
 
 2133 
 
 2489 
 
 2667 
 
 2844 
 
 8022 
 
 25 
 
 1690 
 
 1875 
 
 2060 
 
 2245 
 
 2616 
 
 2801 
 
 2986 
 
 3171 
 
 26 
 
 1781 
 
 1974 
 
 2166 
 
 2359 
 
 2744 
 
 2937 
 
 3130 
 
 3322 
 
 27 
 
 1875 
 
 2075 
 
 2274 
 
 2475 
 
 2875 
 
 3075 
 
 3275 
 
 3475 
 
 28 
 
 1970 
 
 2178 
 
 2384 
 
 2593 
 
 3007 
 
 3215 
 
 3422 
 
 3630 
 
 29 
 
 2068 
 
 2282 
 
 2496 
 
 2712 
 
 3142 
 
 3358 
 
 3571 
 
 3786 
 
 30 
 
 2167 
 
 2389 
 
 2610 
 
 2833 
 
 3278 
 
 8500 
 
 3722 
 
 3944 
 
 31 
 
 2268 
 
 2497 
 
 2726 
 
 2956 
 
 3416 
 
 8645 
 
 3875 
 
 4105 
 
 32 
 
 2370 
 
 2607 
 
 2844 
 
 3081 
 
 3556 
 
 8793 
 
 4030 
 
 4267 
 
 33 
 
 2475 
 
 2719 
 
 2964 
 
 3208 
 
 3697 
 
 3942 
 
 4186 
 
 4431 
 
 34 
 
 2581 
 
 2833 
 
 3085 
 
 3337 
 
 3841 
 
 4093 
 
 4344 
 
 4596 
 
 85 
 
 2690 
 
 2949 
 
 3208 
 
 3468 
 
 3986 
 
 4245 
 
 4505 
 
 4764 
 
 36 
 
 2800 
 
 3067 
 
 3333 
 
 3600 
 
 4133 
 
 4400 
 
 4667 
 
 4933 
 
 37 
 
 2912 
 
 3186 
 
 8460 
 
 8734 
 
 4282 
 
 4556 
 
 4831 
 
 5105 
 
 88 
 
 3026 
 
 3307 
 
 3589 
 
 3870 
 
 4433 
 
 4715 
 
 4996 
 
 5278 
 
 39 
 
 3142 
 
 3431 
 
 3719 
 
 4008 
 
 4586 
 
 4875 
 
 5164 
 
 5453 
 
 40 
 
 3259 
 
 3556 
 
 3852 
 
 4148 
 
 4741 
 
 5037 
 
 5333 
 
 5630 
 
 41 
 
 8379 
 
 3682 
 
 3986 
 
 4290 
 
 4897 
 
 5201 
 
 5505 
 
 5808 
 
 42 
 
 3500 
 
 3811 
 
 4122 
 
 4433 
 
 5056 
 
 5367 
 
 5678 
 
 5989 
 
 43 
 
 3623 
 
 3942 
 
 4260 
 
 4579 
 
 5216 
 
 5534 
 
 5853 
 
 6171 
 
 44 
 
 3748 
 
 4074 
 
 4400 
 
 4726 
 
 5378 
 
 5704 
 
 6030 
 
 6356 
 
 45 
 
 3875 
 
 4208 
 
 4541 
 
 4875 
 
 5542 
 
 5875 
 
 6208 
 
 6542 
 
 46 
 
 4004 
 
 4344 
 
 4684 
 
 5026 
 
 5707 
 
 6048 
 
 6389 
 
 6730 
 
 47 
 
 4134 
 
 4482 
 
 4830 
 
 5179 
 
 5875 
 
 6223 
 
 6571 
 
 6919 
 
 48 
 
 4267 
 
 4622 
 
 4978 
 
 5333 
 
 6044 
 
 6400 
 
 6756 
 
 7111 
 
 49 
 
 4401 
 
 4764 
 
 5127 
 
 5490 
 
 6216 
 
 6579 
 
 6943 
 
 7305 
 
 50 
 
 4537 
 
 4907 
 
 5278 
 
 5648 
 
 6389 
 
 6759 
 
 7130 
 
 7500 
 
 51 
 
 4675 
 
 5053 
 
 5430 
 
 5808 
 
 6564 
 
 6942 
 
 7319 
 
 7697 
 
 52 
 
 4815 
 
 5200 
 
 5584 
 
 5970 
 
 6741 
 
 7126 
 
 7511 
 
 7896 
 
 53 
 
 4956 
 
 5349 
 
 5741 
 
 6134 
 
 6919 
 
 7312 
 
 7705 
 
 8097 
 
 54 
 
 5100 
 
 5500 
 
 5900 
 
 6300 
 
 7100 
 
 7500 
 
 7900 
 
 8300 
 
 55 
 
 5245 
 
 5653 
 
 6060 
 
 6468 
 
 7282 
 
 7690 
 
 8097 
 
 8505 
 
 56 
 
 5393 
 
 5807 
 
 6222 
 
 6637 
 
 7467 
 
 7881 
 
 8296 
 
 8711 
 
 57 
 
 5542 
 
 5964 
 
 6386 
 
 6808 
 
 7653 
 
 8075 
 
 8497 
 
 8919 
 
 58 
 
 5693 
 
 6122 
 
 6552 
 
 6981 
 
 7841 
 
 8270 
 
 8700 
 
 9130 
 
 59 
 
 5845 
 
 6282 
 
 6719 
 
 7156 
 
 8031 
 
 8468 
 
 8905 
 
 9342 
 
 60 
 
 6000 
 
 6444 
 
 6889 
 
 7333 
 
 8222 
 
 8667 
 
 9111 
 
 9556 
 
 270 
 
TABLE XIV. CUBIC YARDS PER 100 FEET. SLOPES 
 
 Depth 
 
 Base 
 12 
 
 Base 
 14 
 
 Base 
 16 
 
 Base 
 18 
 
 Base 
 22 
 
 Base 
 24 
 
 Base 
 26 
 
 Base 
 28 
 
 1 
 
 46 
 
 54 
 
 61 
 
 69 
 
 83 
 
 91 
 
 98 
 
 106 
 
 2 
 
 96 
 
 111 
 
 126 
 
 141 
 
 170 
 
 185 
 
 200 
 
 215 
 
 3 
 
 150 
 
 172 
 
 194 
 
 217 
 
 201 
 
 283 
 
 306 
 
 328 
 
 4 
 
 207 
 
 237 
 
 267 
 
 296 
 
 356 
 
 385 
 
 415 
 
 444 
 
 5 
 
 269 
 
 306 
 
 343 
 
 380 
 
 454 
 
 491 
 
 528 
 
 565 
 
 6 
 
 333 
 
 378 
 
 422 
 
 467 
 
 556 
 
 600 
 
 644 
 
 689 
 
 7 
 
 402 
 
 454 
 
 506 
 
 557 
 
 661 
 
 713 
 
 765 
 
 817 
 
 8 
 
 474 
 
 533 
 
 593 
 
 652 
 
 770 
 
 830 
 
 889 
 
 948 
 
 9 
 
 550 
 
 617 
 
 683 
 
 750 
 
 883 
 
 950 
 
 1017 
 
 1083 
 
 10 
 
 630 
 
 ' 704 
 
 778 
 
 852 
 
 1000 
 
 1074 
 
 1148 
 
 1222 
 
 11 
 
 713 
 
 794 
 
 876 
 
 957 
 
 1120 
 
 1202 
 
 1283 
 
 1365 
 
 13 
 
 800 
 
 889 
 
 978 
 
 1067 
 
 1244 
 
 1333 
 
 1422 
 
 1511 
 
 13 
 
 891 
 
 987 
 
 1083 
 
 1180 
 
 1372 
 
 1469 
 
 1565 
 
 16G1 
 
 
 985 
 
 1089 
 
 1193 
 
 1296 
 
 1504 
 
 1607 
 
 1711 
 
 1815 
 
 15 
 
 1083 
 
 1194 
 
 1306 
 
 1417 
 
 1639 
 
 1750 
 
 1861 
 
 1972 
 
 16 
 
 1185 
 
 1304 
 
 1422 
 
 1541 
 
 1779 
 
 1896 
 
 2015 
 
 2133 
 
 17 
 
 1291 
 
 1417 
 
 1543 
 
 1669 
 
 1920 
 
 2046 
 
 2172 
 
 2298 
 
 13 
 
 1400 
 
 1533 
 
 1667 
 
 1800 
 
 2067 
 
 2200 
 
 2333 
 
 2467 
 
 19 
 
 1513 
 
 1G54 
 
 1794 
 
 1935 
 
 2217 
 
 2357 
 
 2498 
 
 2639 
 
 23 
 
 1630 
 
 1778 
 
 1926 
 
 2074 
 
 2370 
 
 2519 
 
 2667 
 
 2815 
 
 21 
 
 1750 
 
 1906 
 
 2061 
 
 2217 
 
 2528 
 
 2683 
 
 2839 
 
 2994 
 
 22 
 
 1874 
 
 2037 
 
 2200 
 
 2363 
 
 2689 
 
 2852 
 
 3015 
 
 3178 
 
 23 
 
 2002 
 
 2172 
 
 2343 
 
 2513 
 
 2854 
 
 3024 
 
 3194 
 
 8365 
 
 24 
 
 2133 
 
 2311 
 
 2489 
 
 2667 
 
 3022 
 
 3200 
 
 3378 
 
 3556 
 
 23 
 
 2269 
 
 2454 
 
 2639 
 
 2824 
 
 3194 
 
 3380 
 
 35G5 
 
 8750 
 
 23 
 
 2407 
 
 2GOO 
 
 2793 
 
 2985 
 
 3370 
 
 &5C3 
 
 3756 
 
 3948 
 
 27 
 
 2550 
 
 2750 
 
 2950 
 
 3150 
 
 3550 
 
 3750 
 
 3950 
 
 4151 
 
 28 
 
 2096 
 
 2904 
 
 3111 
 
 3319 
 
 8733 
 
 3941 
 
 4148 
 
 4356 
 
 29 
 
 28-16 
 
 3061 
 
 3276 
 
 3491 
 
 3920 
 
 4135 
 
 4350 
 
 45G5 
 
 30 
 
 3000 
 
 3222 
 
 3444 
 
 3667 
 
 4111 
 
 4333 
 
 4556 
 
 4778 
 
 81 
 
 8157 
 
 3387 
 
 3617 
 
 3846 
 
 4306 
 
 4535 
 
 4765 
 
 4994 
 
 32 
 
 3319 
 
 3556 
 
 3793 
 
 4030 
 
 4504 
 
 4741 
 
 4978 
 
 5215 
 
 33 
 
 3483 
 
 3728 
 
 3972 
 
 4217 
 
 4706 
 
 4950 
 
 5194 
 
 5439 
 
 84 
 
 3652 
 
 3904 
 
 4156 
 
 4407 
 
 4911 
 
 5163 
 
 5415 
 
 5667 
 
 35 
 
 8824 
 
 4083 
 
 4343 
 
 4602 
 
 5120 
 
 5380 
 
 5639 
 
 5898 
 
 36 
 
 4000 
 
 4267 
 
 4533 
 
 4800 
 
 5333 
 
 5600 
 
 5867 
 
 6133 
 
 37 
 
 4180 
 
 4454 
 
 4728 
 
 5002 
 
 5550 
 
 5824 
 
 6098 
 
 6372 
 
 38 
 
 4363 
 
 4644 
 
 4926 
 
 5207 
 
 5770 
 
 6052 
 
 6333 
 
 6615 
 
 39 
 
 4550 
 
 4839 
 
 5128 
 
 5417 
 
 5994 
 
 6283 
 
 6572 
 
 6861 
 
 40 
 
 4741 
 
 5037 
 
 5333 
 
 5630 
 
 6222 
 
 6519 
 
 6815 
 
 7111 
 
 41 
 
 4935 
 
 5239 
 
 5543 
 
 5846 
 
 6454 
 
 6757 
 
 7061 
 
 7365 
 
 42 
 
 5133 
 
 5444 
 
 5756 
 
 6067 
 
 6689 
 
 7000 
 
 7311 
 
 7623 
 
 43 
 
 5335 
 
 5654 
 
 5072 
 
 6291 
 
 6928 
 
 7246 
 
 7565 
 
 7883 
 
 44 
 
 5541 
 
 5867 
 
 6193 
 
 6519 
 
 7170 
 
 7496 
 
 7822 
 
 8148 
 
 45 
 
 5750 
 
 6083 
 
 6417 
 
 6750 
 
 7417 
 
 7750 
 
 8083 
 
 8417 
 
 46 
 
 5963 
 
 6304 
 
 6644 
 
 6985 
 
 7667 
 
 8007 
 
 8348 
 
 8689 
 
 47 
 
 6180 
 
 6528 
 
 6876 
 
 7224 
 
 7920 
 
 8269 
 
 8617 
 
 8965 
 
 41 
 
 6400 
 6624 
 
 6756 
 6987 
 
 7111 
 
 7350 
 
 .7467 
 7713 
 
 8178 
 8439 
 
 
 m 
 
 9244 
 9528 
 
 50 
 
 6852 
 
 7222 
 
 7593 
 
 7963 
 
 87C4 
 
 9074 
 
 9444 
 
 9815 
 
 51 
 
 7083 
 
 7461 
 
 7839 
 
 8217 
 
 8972 
 
 9350 
 
 9728 
 
 10106 
 
 52 
 
 7319 
 
 7704 
 
 8089 
 
 8474 
 
 9244 
 
 9G30 
 
 10015 
 
 10400 
 
 53 
 
 7557 
 
 7950 
 
 8343 
 
 8735 
 
 9520 
 
 9913 
 
 10306 
 
 10698 
 
 54 
 
 7800 
 
 8200 
 
 8600 
 
 9000 
 
 9800 
 
 10200 
 
 10600 
 
 11000 
 
 55 
 
 8046 
 
 8454 
 
 8861 
 
 9269 
 
 10083 
 
 10491 
 
 10898 
 
 11306 
 
 56 
 
 8296 
 
 8711 
 
 9126 
 
 9541 
 
 10370 
 
 10785 
 
 11200 
 
 11615 
 
 57 
 
 8550 
 
 8972 
 
 9394 
 
 9817 
 
 10661 
 
 11083 
 
 11506 
 
 11928 
 
 58 
 
 8807 
 
 9237 
 
 9667 
 
 10096 
 
 10956 
 
 11385 
 
 11815 
 
 12244 
 
 59 
 
 9069 
 
 9506 
 
 9943 
 
 10380 
 
 11254 
 
 11691 
 
 12128 
 
 12565 
 
 60 
 
 9333 
 
 9778 
 
 10222 
 
 10667 
 
 11556 
 
 12000 
 
 12444 
 
 12889 
 
 271 
 
TABLE XIV. CUBIC YARDS PER 100 FEET. SLOPES 1 1. 
 
 Depth 
 
 Base 
 12 
 
 Base 
 14 
 
 Base 
 16 
 
 Base 
 18 
 
 Base 
 20 
 
 Base 
 28 
 
 Base 
 30 
 
 Base 
 32 
 
 1 
 
 48 
 
 S6 
 
 63 
 
 70 
 
 78 
 
 107 
 
 115 
 
 122 
 
 2 
 
 104 
 
 119 
 
 133 
 
 148 
 
 163 
 
 222 
 
 237 
 
 252 
 
 3 
 
 167 
 
 189 
 
 211 
 
 233 
 
 256 
 
 344 
 
 367 
 
 389 
 
 4 
 
 237 
 
 267 
 
 296 
 
 326 
 
 356 
 
 474 
 
 504 
 
 533 
 
 5 
 
 315 
 
 352 
 
 389 
 
 426 
 
 463 
 
 611 
 
 648 
 
 685 
 
 C 
 
 400 
 
 444 
 
 489 
 
 533 
 
 578 
 
 756 
 
 800 
 
 844 
 
 7 
 
 493 
 
 544 
 
 596 
 
 648 
 
 700 
 
 907 
 
 959 
 
 1011 
 
 8 
 
 593 
 
 652 
 
 711 
 
 770 
 
 830 
 
 1067 
 
 1126 
 
 1185 
 
 9 
 
 700 
 
 767 
 
 833 
 
 900 
 
 967 
 
 1233 
 
 1300 
 
 13G7 
 
 10 
 
 815 
 
 889 
 
 963 
 
 1037 
 
 1111 
 
 1407 
 
 1481 
 
 1556 
 
 11 
 
 937 
 
 1019 
 
 1100 
 
 1181 
 
 1263 
 
 1589 
 
 1670 
 
 1752 
 
 12 
 
 10G7 
 
 1156 
 
 1244 
 
 1333 
 
 1422 
 
 1778 
 
 1867 
 
 1956 
 
 13 
 
 1204 
 
 1300 
 
 1396 
 
 1493 
 
 1589 
 
 1974 
 
 2070 
 
 2167 
 
 14 
 
 1348 
 
 1452 
 
 1556 
 
 1659 
 
 1763 
 
 2178 
 
 2281 
 
 2385 
 
 15 
 
 1500 
 
 1611 
 
 1722 
 
 1833 
 
 1944 
 
 2389 
 
 2500 
 
 2611 
 
 16 
 
 1659 
 
 1778 
 
 1896 
 
 2015 
 
 2133 
 
 2607 
 
 2726 
 
 2844 
 
 17 
 
 1826 
 
 1952 
 
 2078 
 
 2204 
 
 2330 
 
 2833 
 
 2959 
 
 3085 
 
 18 
 
 2000 
 
 2133 
 
 2267 
 
 2400 
 
 2533 
 
 3067 
 
 3200 
 
 3333 
 
 19 
 
 2181 
 
 2322 
 
 2463 
 
 2604 
 
 2744 
 
 3307 
 
 3418 
 
 35G9 
 
 20 
 
 2370 
 
 2519 
 
 2667 
 
 2815 
 
 2963 
 
 3556 
 
 3704 
 
 3852 
 
 21 
 
 2567 
 
 2722 
 
 2878 
 
 3033 
 
 3189 
 
 3811 
 
 3967 
 
 4122 
 
 
 2770 
 
 2933 
 
 3096 
 
 3259 
 
 3422 
 
 4074 
 
 4237 
 
 4144 
 
 23 
 
 2981 
 
 3152 
 
 3322 
 
 3193 
 
 3663 
 
 4344 
 
 4515 
 
 46S5 
 
 24 
 
 3200 
 
 3378 
 
 3556 
 
 3733 
 
 3911 
 
 4622 
 
 4800 
 
 4978 
 
 25 
 
 3426 
 
 3611 
 
 3796 
 
 3981 
 
 4167 
 
 4907 
 
 5093 
 
 5278 
 
 26 
 
 3659 
 
 3852 
 
 4044 
 
 4237 
 
 4430 
 
 5200 
 
 5393 
 
 5585 
 
 27 
 
 3900 
 
 4100 
 
 4300 
 
 4500 
 
 4700 
 
 5500 
 
 5700 
 
 5900 
 
 28 
 
 4148 
 
 4356 
 
 4563 
 
 4770 
 
 4978 
 
 5807 
 
 6015 
 
 6222 
 
 29 
 
 4404 
 
 4619 
 
 4833 
 
 5048 
 
 5263 
 
 G122 
 
 6337 
 
 6552 
 
 30 
 
 4667 
 
 4889 
 
 5111 
 
 5333 
 
 5556 
 
 6444 
 
 6667 
 
 6889 
 
 31 
 
 4937 
 
 5167 
 
 5396 
 
 5626 
 
 5856 
 
 6774 
 
 7004 
 
 7233 
 
 32 
 
 5215 
 
 5452 
 
 5689 
 
 5926 
 
 6163 
 
 7111 
 
 7348 
 
 7585 
 
 33 
 
 5500 
 
 5744 
 
 5989 
 
 6233 
 
 6478 
 
 7456 
 
 7700 
 
 7944 
 
 84 
 
 5793 
 
 6044 
 
 6296 
 
 6548 
 
 6800 
 
 7807 
 
 8059 
 
 8311 
 
 35 
 
 6093 
 
 6352 
 
 6611 
 
 6870 
 
 7130 
 
 8167 
 
 8426 
 
 8685 
 
 36 
 
 6400 
 
 6667 
 
 6933 
 
 7200 
 
 7467 
 
 8533 
 
 8800 
 
 90C7 
 
 37 
 
 6715 
 
 6989 
 
 7263 
 
 7537 
 
 7811 
 
 8907 
 
 9181 
 
 94C6 
 
 38 
 
 7037 
 
 7319 
 
 7600 
 
 7881 
 
 8163 
 
 9289 
 
 9570 
 
 9852 
 
 39 
 
 7367 
 
 7656 
 
 7944 
 
 8233 
 
 8522 
 
 9678 
 
 9967 
 
 10256 
 
 40 
 
 7704 
 
 8000 
 
 8296 
 
 8593 
 
 8889 
 
 10074 
 
 10370 
 
 106G7 
 
 41 
 
 8048 
 
 8352 
 
 8656 
 
 8959 
 
 9263 
 
 10478 
 
 10781 
 
 11085 
 
 42 
 
 8400 
 
 8711 
 
 9022 
 
 9333 
 
 9644 
 
 10889 
 
 11200 
 
 11511 
 
 43 
 
 8759 
 
 9078 
 
 9396 
 
 9715 
 
 10033 
 
 11307 
 
 11626 
 
 11944 
 
 44 
 
 9126 
 
 9452 
 
 9778 
 
 10104 
 
 10430 
 
 11733 
 
 12059 
 
 12385 
 
 45 
 
 9500 
 
 9833 
 
 10167 
 
 10500 
 
 10833 
 
 12167 
 
 12500 
 
 12833 
 
 46 
 
 9881 
 
 10222 
 
 10563 
 
 10904 
 
 11244 
 
 12607 
 
 12948 
 
 13289 
 
 47 
 
 10270 
 
 10619 
 
 10367 
 
 11315 
 
 11663 
 
 13056 
 
 13404 
 
 13752 
 
 48 
 
 10667 
 
 11022 
 
 11378 
 
 11733 
 
 12089 
 
 13511 
 
 13867 
 
 14222 
 
 49 
 
 11070 
 
 114:33 
 
 11796 
 
 12159 
 
 12522 
 
 13974 
 
 14337 
 
 14700 
 
 50 
 
 11481 
 
 11852 
 
 12222 
 
 12593 
 
 12963 
 
 14444 
 
 14815 
 
 15185 
 
 51 
 
 11900 
 
 12278 
 
 12656 
 
 13033 
 
 13411 
 
 14922 
 
 15300 
 
 15678 
 
 52 
 
 12326 
 
 12711 
 
 13096 
 
 13481 
 
 13867 
 
 15407 
 
 15793 
 
 16173 
 
 53 
 
 12759 
 
 13152 
 
 13544 
 
 13937 
 
 14330 
 
 15900 
 
 16293 
 
 16685 
 
 54 
 
 13200 
 
 13600 
 
 14000 
 
 14400 
 
 14800 
 
 16400 
 
 16800 
 
 17200 
 
 55 
 
 13648 
 
 14056 
 
 14463 
 
 14870 
 
 15278 
 
 16907 
 
 17315 
 
 17722 
 
 56 
 
 14104 
 
 14519 
 
 14933 
 
 15348 
 
 15763 
 
 17422 
 
 17837 
 
 18252 
 
 67 
 
 14567 
 
 14989 
 
 15411 
 
 15833 
 
 16256 
 
 17944 
 
 18367 
 
 18789 
 
 58 
 
 15037 
 
 15467 
 
 15896 
 
 16326 
 
 16756 
 
 18474 
 
 18904 
 
 19333 
 
 59 
 
 15515 
 
 15952 
 
 16389 
 
 16826 
 
 17263 
 
 19011 
 
 19448 
 
 198R5 
 
 60 
 
 16000 
 
 16444 
 
 16889 
 
 17333 
 
 17778 
 
 19556 
 
 20000 
 
 20444 
 
 272 
 
TABLE XIV, CUBIC YARDS PER 100 FEET. SLOPES l^J 1 L 
 
 Depth 
 
 Base 
 12 
 
 Base 
 14 
 
 Base 
 16 
 
 Base 
 18 
 
 Base 
 20 
 
 Base 
 28 
 
 Base 
 30 
 
 Base 
 32 
 
 1 
 
 50 
 
 57 
 
 65 
 
 72 
 
 80 
 
 109 
 
 117 
 
 124 
 
 2 
 
 111 
 
 126 
 
 141 
 
 156 
 
 170 
 
 230 
 
 244 
 
 259 
 
 3 
 
 183 
 
 206 
 
 228 
 
 250 
 
 272 
 
 361 
 
 383 
 
 406 
 
 4 
 
 267 
 
 296 
 
 326 
 
 356 
 
 385 
 
 504 
 
 533 
 
 5G3 
 
 5 
 
 361 
 
 398 
 
 435 
 
 472 
 
 509 
 
 657 
 
 694 
 
 731 
 
 6 
 
 467 
 
 511 
 
 556 
 
 000 
 
 644 
 
 822 
 
 867 
 
 911 
 
 7 
 
 583 
 
 635 
 
 687 
 
 739 
 
 791 
 
 998 
 
 1050 
 
 1102 
 
 8 
 
 711 
 
 770 
 
 830 
 
 889 
 
 948 
 
 1185 
 
 1244 
 
 1304 
 
 9 
 
 850 
 
 917 
 
 983 
 
 1050 
 
 1116 
 
 1383 
 
 1450 
 
 1517 
 
 10 
 
 1000 
 
 1074 
 
 1148 
 
 1222 
 
 1296 
 
 1593 
 
 1667 
 
 1741 
 
 11 
 
 1161 
 
 1243 
 
 1324 
 
 1406 
 
 1487 
 
 1813 
 
 1894 
 
 1976 
 
 12 
 
 1333 
 
 1422 
 
 1511 
 
 1600 
 
 1689 
 
 2044 
 
 2133 
 
 2222 
 
 13 
 
 1517 
 
 1613 
 
 1709 
 
 1806 
 
 1902 
 
 2287 
 
 2383 
 
 2480 
 
 14 
 
 1711 
 
 1815 
 
 1919 
 
 2022 
 
 2126 
 
 2541 
 
 2644 
 
 2743 
 
 15 
 
 1917 
 
 2028 
 
 2139 
 
 2250 
 
 2361 
 
 2808 
 
 2917 
 
 3028 
 
 16 
 
 2133 
 
 2252 
 
 2370 
 
 2489 
 
 2607 
 
 3081 
 
 3200 
 
 3319 
 
 17 
 
 2361 
 
 2487 
 
 2613 
 
 2739 
 
 2865 
 
 33G9 
 
 3494 
 
 90 
 
 18 
 
 2600 
 
 2733 
 
 2867 
 
 3000 
 
 3133 
 
 36G7 
 
 3800 
 
 3933 
 
 19 
 
 2850 
 
 2991 
 
 3131 
 
 3272 
 
 3413 
 
 3976 
 
 4117 
 
 4257 
 
 20 
 
 3111 
 
 3259 
 
 3407 
 
 3556 
 
 3704 
 
 4296 
 
 4444 
 
 4592 
 
 21 
 
 3383 
 
 3539 
 
 3694 
 
 3850 
 
 4005 
 
 4628 
 
 4783 
 
 4939 
 
 22 
 
 3667 
 
 3830 
 
 3993 
 
 4156 
 
 4318 
 
 4970 
 
 5133 
 
 5296 
 
 23 
 
 3961 
 
 4131 
 
 4302 
 
 4472 
 
 4642 
 
 5324 
 
 5494 
 
 56G5 
 
 24 
 
 4267 
 
 4444 
 
 4622 
 
 4800 
 
 4978 
 
 5689 
 
 58G7 
 
 6044 
 
 25 
 
 4583 
 
 4769 
 
 4954 
 
 5139 
 
 5324 
 
 60G5 
 
 6250 
 
 6435 
 
 26 
 
 4911 
 
 5104 
 
 5296 
 
 5489 
 
 5681 
 
 6452 
 
 6644 
 
 6837 
 
 27 
 
 5250 
 
 5450 
 
 5650 
 
 5850 
 
 6050 
 
 6850 
 
 7050 
 
 7250 
 
 28 
 
 5600 
 
 5807 
 
 6015 
 
 6222 
 
 6430 
 
 7259 
 
 7467 
 
 7674 
 
 29 
 
 5961 
 
 6176 
 
 6391 
 
 6606 
 
 6820 
 
 7680 
 
 7894 
 
 8109 
 
 30 
 
 6333 
 
 6556 
 
 6778 
 
 7000 
 
 7222 
 
 8111 
 
 8333 
 
 8555 
 
 31 
 
 6717 
 
 6946 
 
 7176 
 
 7406 
 
 7635 
 
 8554 
 
 8783 
 
 9013 
 
 32 
 
 7111 
 
 7348 
 
 7585 
 
 7822 
 
 8059 
 
 9007 
 
 9244 
 
 9482 
 
 33 
 
 7517 
 
 7761 
 
 8006 
 
 8250 
 
 8494 
 
 9472 
 
 9717 
 
 9962 
 
 34 
 
 7933 
 
 8185 
 
 8437 
 
 86S9 
 
 8941 
 
 9948 
 
 10200 
 
 10452 
 
 35 
 
 8361 
 
 8620 
 
 8880 
 
 9139 
 
 9398 
 
 10435 
 
 10694 
 
 10954 
 
 36 
 
 8800 
 
 9067 
 
 9333 
 
 9600 
 
 9867 
 
 10933 
 
 11200 
 
 114G7 
 
 37 
 
 9250 
 
 9524 
 
 9798 
 
 10072 
 
 10346 
 
 11443 
 
 11717 
 
 11991 
 
 38 
 
 9711 
 
 9993 
 
 10274 
 
 10556 
 
 10837 
 
 11963 
 
 12244 
 
 125?S 
 
 39 
 
 10183 
 
 10472 
 
 10761 
 
 11050 
 
 11339 
 
 12494 
 
 12783 
 
 13072 
 
 40 
 
 10667 
 
 10963 
 
 11259 
 
 11556 
 
 11852 
 
 13037 
 
 13333 
 
 13G30 
 
 41 
 
 11161 
 
 11465 
 
 11769 
 
 12072 
 
 12376 
 
 13591 
 
 13894 
 
 14198 
 
 42 
 
 11667 
 
 11978 
 
 12289 
 
 12GOO 
 
 12911 
 
 14156 
 
 14467 
 
 14778 
 
 43 
 
 12183 
 
 12502 
 
 12820 
 
 13139 
 
 13457 
 
 14731 
 
 15050 
 
 15369 
 
 44 
 
 12711 
 
 13037 
 
 13363 
 
 13689 
 
 14015 
 
 15319 
 
 15644 
 
 15970 
 
 45 
 
 13250 
 
 13583 
 
 13917 
 
 14250 
 
 14583 
 
 15917 
 
 16250 
 
 16583 
 
 46 
 
 1:3800 
 
 14141 
 
 14481 
 
 14822 
 
 15163 
 
 16526 
 
 16867 
 
 17207 
 
 47 
 
 14361 
 
 14709 
 
 15057 
 
 15406 
 
 15754 
 
 17146 
 
 17494 
 
 17843 
 
 48 
 
 14933 
 
 15289 
 
 15644 
 
 16000 
 
 16356 
 
 17778 
 
 18133 
 
 18489 
 
 49 
 
 15517 
 
 15880 
 
 16243 
 
 16606 
 
 16968 
 
 18420 
 
 18783 
 
 19146 
 
 50 
 
 16111 
 
 16481 
 
 16852 
 
 17222 
 
 17592 
 
 19074 
 
 19444 
 
 19815 
 
 51 
 
 16717 
 
 17094 
 
 17472 
 
 17850 
 
 18228 
 
 19739 
 
 20117 
 
 20494 
 
 52 
 
 17333 
 
 17719 
 
 18104 
 
 18489 
 
 18874 
 
 20415 
 
 20800 
 
 21185 
 
 53 
 
 17961 
 
 18354 
 
 18746 
 
 19139 
 
 19531 
 
 21102 
 
 21494 
 
 21887 
 
 54 
 
 18000 
 
 19000 
 
 19400 
 
 19800 
 
 20200 
 
 21800 
 
 22200 
 
 22600 
 
 55 
 
 10250 
 
 19657 
 
 20065 
 
 20472 
 
 20880 
 
 22509 
 
 22917 
 
 23324 
 
 56 
 
 19011 
 
 20326 
 
 20741 
 
 21156 
 
 21570 
 
 23230 
 
 23644 
 
 24059 
 
 57 
 
 20583 
 
 21006 
 
 21428 
 
 21850 
 
 22272 
 
 23961 
 
 24383 
 
 24805 
 
 58 
 
 21267 
 
 21696 
 
 22126 
 
 22556 
 
 22985 
 
 24704 
 
 251:33 
 
 25563 
 
 59 
 
 21961 
 
 22398 
 
 2^835 
 
 23272 
 
 23709 
 
 25457 
 
 25894 
 
 26332 
 
 60 
 
 22G07 
 
 23111 
 
 23556 
 
 24000 
 
 24444 
 
 2G222 
 
 26667 
 
 27111 
 
 273 
 
. 
 
 TABLE XTV. CUBIC YAUDS PER 100 FEET. SLOPES 2 ; 1. 
 
 Depth 
 
 Base 
 12 
 
 Base 
 14 
 
 Base 
 16 
 
 Base 
 18 
 
 Base 
 
 20 
 
 Bae 
 
 28 
 
 Base 
 30 
 
 Base 
 32 
 
 1 
 
 52 
 
 59 
 
 67 
 
 74 
 
 81 
 
 ill 
 
 119 
 
 126 
 
 2 
 
 119 
 
 133 
 
 143 
 
 1C3 
 
 178 
 
 237 
 
 252 
 
 267 
 
 3 
 
 200 
 
 222 
 
 244 
 
 267 
 
 289 
 
 378 
 
 400 
 
 422 
 
 4 
 
 296 
 
 326 
 
 356 
 
 385 
 
 415 
 
 533 
 
 563 
 
 693 
 
 5 
 
 407 
 
 444 
 
 481 
 
 519 
 
 556 
 
 704 
 
 741 
 
 778 
 
 6 
 
 533 
 
 578 
 
 622 
 
 6G7 
 
 711 
 
 889 
 
 933 
 
 978 
 
 7 
 
 674 
 
 726 
 
 778 
 
 830 
 
 881 
 
 1089 
 
 1141 
 
 1193 
 
 8 
 
 830 
 
 889 
 
 948 
 
 1007 
 
 1067 
 
 1304 
 
 1363 
 
 1422 
 
 9 
 
 1000 
 
 1067 
 
 1133 
 
 1200 
 
 1267 
 
 1533 
 
 1600 
 
 1667 
 
 10 
 
 1185 
 
 1259 
 
 1333 
 
 1407 
 
 1481 
 
 1778 
 
 1852 
 
 1926 
 
 11 
 
 1385 
 
 1467 
 
 1548 
 
 1630 
 
 1711 
 
 2037 
 
 2119 
 
 2200 
 
 12 
 
 1600 
 
 1689 
 
 1778 
 
 1867 
 
 1956 
 
 2311 
 
 2400 
 
 2489 
 
 13 
 
 1830 
 
 1926 
 
 2022 
 
 2119 
 
 2215 
 
 2600 
 
 2696 
 
 2703 
 
 14 
 
 2074 
 
 2178 
 
 2281 
 
 2385 
 
 2489 
 
 2904 
 
 3007 
 
 3111 
 
 15 
 
 2333 
 
 2444 
 
 2556 
 
 2667 
 
 2778 
 
 3222 
 
 3333 
 
 3444 
 
 16 
 
 2607 
 
 2726 
 
 2844 
 
 2963 
 
 3081 
 
 3556 
 
 3674 
 
 3793 
 
 17 
 
 2896 
 
 3022 
 
 3148 
 
 3274 
 
 3400 
 
 3904 
 
 4030 
 
 4156 
 
 18 
 
 3200 
 
 S333 
 
 3467 
 
 3600 
 
 3733 
 
 4267 
 
 4400 
 
 4533 
 
 19 
 
 3319 
 
 3659 
 
 3800 
 
 3941 
 
 4081 
 
 4644 
 
 4785 
 
 4926 
 
 20 
 
 3852 
 
 4000 
 
 4148 
 
 4296 
 
 4444 
 
 5037 
 
 5185 
 
 5333 
 
 21 
 
 4200 
 
 4356 
 
 4511 
 
 4667 
 
 4822 
 
 5444 
 
 5600 
 
 5756 
 
 22 
 
 4563 
 
 4730 
 
 4889 
 
 5052 
 
 5215 
 
 5867 
 
 6030 
 
 81D8 
 
 23 
 
 4941 
 
 5111 
 
 5281 
 
 5452 
 
 5622 
 
 6304 
 
 6474 
 
 6644 
 
 24 
 
 5333 
 
 5511 
 
 5689 
 
 5867 
 
 6044 
 
 6756 
 
 6933 
 
 7111 
 
 25 
 
 5741 
 
 5926 
 
 6111 
 
 6296 
 
 6481 
 
 7522 
 
 7407 
 
 7593 
 
 26 
 
 6163 
 
 6356 
 
 6548 
 
 6741 
 
 6933 
 
 7704 
 
 7896 
 
 KO!-'9 
 
 27 
 
 6600 
 
 6800 
 
 7000 
 
 7200 
 
 7400 
 
 8200 
 
 8400 
 
 8600 
 
 28 
 
 7052 
 
 7259 
 
 7467 
 
 7674 
 
 7881 
 
 8711 
 
 8919 
 
 9126 
 
 29 
 
 7519 
 
 7733 
 
 7948 
 
 8163 
 
 8378 
 
 9237 
 
 9452 
 
 9ii67 
 
 30 
 
 8000 
 
 8222 
 
 8444 
 
 8667 
 
 8389 
 
 9778 
 
 10000 
 
 10222 
 
 31 
 
 8496 
 
 8729 
 
 8956 
 
 9185 
 
 9415 
 
 10333 
 
 10563 
 
 10793 
 
 32 
 
 9007 
 
 9244 
 
 9481 
 
 9719 
 
 9958 
 
 10904 
 
 11141 
 
 11378 
 
 33 
 
 9533 
 
 9778 
 
 10022 
 
 10267 
 
 10511 
 
 11489 
 
 11733 
 
 11978 
 
 34 
 
 10074 
 
 10326 
 
 10578 
 
 10330 
 
 11081 
 
 12089 
 
 1-2341 
 
 12593 
 
 35 
 
 10830 
 
 10889 
 
 11148 
 
 11407 
 
 11G67 
 
 1-2704 
 
 12963 
 
 13222 
 
 36 
 
 11200 
 
 11467 
 
 11733 
 
 12000 
 
 12267 
 
 13333 
 
 13600 
 
 13867 
 
 37 
 
 11785 
 
 12059 
 
 12333 
 
 12607 
 
 12S81 
 
 131)78 
 
 14252 
 
 14526 
 
 38 
 
 12385 
 
 12667 
 
 12948 
 
 13230 
 
 13511 
 
 14ii37 
 
 14919 
 
 15200 
 
 39 
 
 13000 
 
 13289 
 
 13578 
 
 13867 
 
 14156 
 
 15311 
 
 15600 
 
 15889 
 
 40 
 
 13630 
 
 13926 
 
 14222 
 
 14519 
 
 14815 
 
 16000 
 
 16296 
 
 16593 
 
 41 
 
 14274 
 
 14578 
 
 14881 
 
 15185 
 
 15489 
 
 16704 
 
 17007 
 
 17311 
 
 42 
 
 14:83 
 
 15244 
 
 155C6 
 
 15867 
 
 16178 
 
 17.22 
 
 17733 
 
 18044 
 
 43 
 
 15607 
 
 15926 
 
 16224 
 
 16o63 
 
 16881 
 
 18156 
 
 18474 
 
 18793 
 
 44 
 
 16296 
 
 16622 
 
 16948 
 
 17274 
 
 17600 
 
 18904 
 
 19230 
 
 19556 
 
 45 
 
 irooo 
 
 17333 
 
 17C67 
 
 18000 
 
 18333 
 
 19667 
 
 20000 
 
 20333 
 
 46 
 
 17719 
 
 18059 
 
 18400 
 
 18741 
 
 19081 
 
 20444 
 
 20785 
 
 21126 
 
 47 
 
 13152 
 
 18800 
 
 19148 
 
 19496 
 
 19S44 
 
 21237 
 
 21585 
 
 21933 
 
 48 
 
 19200 
 
 19556 
 
 19911 
 
 20267 
 
 20<>22 
 
 22344 
 
 22400 
 
 22756 
 
 49 
 
 19963 
 
 20326 
 
 20689 
 
 21052 
 
 21415 
 
 22867 
 
 23230 
 
 2355)3 
 
 50 
 
 20741 
 
 20711 
 
 21481 
 
 21852 
 
 22222 
 
 23704 
 
 24074 
 
 24444 
 
 51 
 
 2U33 
 
 21911 
 
 22289 
 
 22667 
 
 23044 
 
 2-;5'6 
 
 24933 
 
 25311 
 
 52 
 
 22.341 
 
 22726 
 
 23111 
 
 23496 
 
 23881 
 
 25422 
 
 25807 
 
 26193 
 
 53 
 
 23163 
 
 23556 
 
 2-3948 
 
 24341 
 
 24733 
 
 26304 
 
 26696 
 
 27089 
 
 54 
 
 24000 
 
 24400 
 
 24800 
 
 25200 
 
 2:>600 
 
 27200 
 
 27600 
 
 28000 
 
 55 
 
 24852 
 
 25259 
 
 25667 
 
 26074 
 
 26481 
 
 28111 
 
 28519 
 
 28926 
 
 56 
 
 25719 
 
 26133 
 
 26548 
 
 26963 
 
 27378 
 
 29037 
 
 29452 
 
 29867 
 
 57 
 
 20600 
 
 27022 
 
 27444 
 
 27867 
 
 28289 
 
 2i)978 
 
 30400 
 
 30822- 
 
 58 
 
 27496 
 
 27926 
 
 28356 
 
 23785 
 
 29215 
 
 30933 
 
 31363 
 
 31793 
 
 59 
 
 28407 
 
 28844 
 
 29281 
 
 29719 
 
 30156 
 
 31904 
 
 32341 
 
 32778 
 
 CO 
 
 89333 
 
 29778 
 
 30222 
 
 30667 
 
 31111 
 
 32889 
 
 33333 
 
 33778 
 
 274 
 
TABLE XIV. CUBIC YARDS PER 100 FEET. SLOPES 3 : I. 
 
 Depth 
 
 Base 
 12 
 
 Base 
 14 
 
 Base 
 16 
 
 Base 
 18 
 
 Base 
 20 
 
 Base 
 28 
 
 Base 
 30 
 
 Base 
 32 
 
 1 
 
 56 
 
 63 
 
 70 
 
 78 
 
 85 
 
 115 
 
 122 
 
 130 
 
 2 
 
 133 
 
 148 
 
 163 
 
 178 
 
 193 
 
 252 
 
 267 
 
 281 
 
 3 
 
 233 
 
 256 
 
 278 
 
 300 
 
 322 
 
 411 
 
 433 
 
 456 
 
 4 
 
 356 
 
 385 
 
 415 
 
 444 
 
 474 
 
 593 
 
 622 
 
 652 
 
 5 
 
 500 
 
 537 
 
 574 
 
 611 
 
 648 
 
 796 
 
 833 
 
 870 
 
 6 
 
 667 
 
 711 
 
 756 
 
 800 
 
 844 
 
 1022 
 
 1067 
 
 1111 
 
 7 
 
 856 
 
 907 
 
 959 
 
 1011 
 
 1063 
 
 1270 
 
 1322 
 
 1374 
 
 8 
 
 1067 
 
 1126 
 
 1185 
 
 1244 
 
 1304 
 
 1541 
 
 1600 
 
 1659 
 
 9 
 
 1300 
 
 1367 
 
 1433 
 
 1500 
 
 1567 
 
 1833 
 
 1900 
 
 1967 
 
 10 
 
 1556 
 
 1630 
 
 1704 
 
 1778 
 
 1852 
 
 2148 
 
 2222 
 
 2296 
 
 11 
 
 1833 
 
 1915 
 
 1996 
 
 2078 
 
 2159 
 
 2485 
 
 2567 
 
 2648 
 
 13 
 
 2133 
 
 2222 
 
 2311 
 
 2400 
 
 2489 
 
 2844 
 
 2933 
 
 3022 
 
 13 
 
 2456 
 
 2552 
 
 2648 
 
 2744 
 
 2841 
 
 3226 
 
 3322 
 
 3419 
 
 14 
 
 2800 
 
 2904 
 
 3007 
 
 3111 
 
 3215 
 
 3630 
 
 3733 
 
 3837 
 
 15 
 
 3167 
 
 3278 
 
 3389 
 
 3500 
 
 3611 
 
 4056 
 
 4167 
 
 4278 
 
 16 
 
 3556 
 
 3674 
 
 3793 
 
 3911 
 
 4030 
 
 4504 
 
 4622 
 
 4741 
 
 17 
 
 3967 
 
 4093 
 
 4219 
 
 4344 
 
 4470 
 
 4974 
 
 5100 
 
 5226 
 
 18 
 
 4400 
 
 4533 
 
 4667 
 
 4800 
 
 4933 
 
 5467 
 
 5600 
 
 5733 
 
 19 
 
 4856 
 
 4996 
 
 5137 
 
 5278 
 
 5419 
 
 5981 
 
 6122 
 
 6263 
 
 20 
 
 5333 
 
 5481 
 
 5630 
 
 5778 
 
 5926 
 
 6519 
 
 6667 
 
 6815 
 
 21 
 
 5833 
 
 5989 
 
 6144 
 
 6300 
 
 6456 
 
 7078 
 
 7233 
 
 7389 
 
 22 
 
 6356 
 
 6519 
 
 6681 
 
 6844 
 
 7007 
 
 7659 
 
 7822 
 
 7985 
 
 23 
 
 6900 
 
 7070 
 
 7241 
 
 7411 
 
 7581 
 
 8263 
 
 8433 
 
 8504 
 
 24 
 
 7467 
 
 7644 
 
 7822 
 
 8000 
 
 8178 
 
 8889 
 
 9067 
 
 9141 
 
 25 
 
 8056 
 
 8241 
 
 8426 
 
 8611 
 
 8796 
 
 9537 
 
 9722 
 
 9807 
 
 26 
 
 8667 
 
 8859 
 
 9052 
 
 9244 
 
 9437 
 
 10207 
 
 10400 
 
 10593 
 
 27 
 
 9300 
 
 9500 
 
 9700 
 
 9900 
 
 10100 
 
 10900 
 
 11100 
 
 11300 
 
 28 
 
 9956 
 
 10163 
 
 10370 
 
 10578 
 
 10785 
 
 11615 
 
 11822 
 
 12030 
 
 29 
 
 10633 
 
 10848 
 
 11063 
 
 11278 
 
 11493 
 
 12352 
 
 12567 
 
 12781 
 
 30 
 
 11333 
 
 11556 
 
 11778 
 
 12000 
 
 12222 
 
 13111 
 
 13333 
 
 13556 
 
 31 
 
 12056 
 
 12285 
 
 12515 
 
 12744 
 
 12974 
 
 13893 
 
 14122 
 
 14352 
 
 32 
 
 12800 
 
 13037 
 
 13274 
 
 13511 
 
 13748 
 
 14696 
 
 14933 
 
 15170 
 
 33 
 
 13567 
 
 13811 
 
 14056 
 
 14300 
 
 14544 
 
 15522 
 
 15767 
 
 16011 
 
 34 
 
 14356 
 
 14607 
 
 14859 
 
 15111 
 
 
 16370 
 
 16622 
 
 16874 
 
 35 
 
 15167 
 
 15426 
 
 15685 
 
 15944 
 
 16204 
 
 17241 
 
 17500 
 
 17759 
 
 36 
 
 16000 
 
 16267 
 
 16533 
 
 16800 
 
 17067 
 
 18133 
 
 18400 
 
 18667 
 
 37 
 
 16856 
 
 17130 
 
 17404 
 
 17678 
 
 17952 
 
 19048 
 
 19322 
 
 19596 
 
 38 
 
 17733 
 
 18015 
 
 18296 
 
 18578 
 
 18859 
 
 19985 
 
 20267 
 
 20548 
 
 39 
 
 18633 
 
 18922 
 
 19211 
 
 19500 
 
 19789 
 
 20944 
 
 21233 
 
 21522 
 
 40 
 
 19556 
 
 19852 
 
 20148 
 
 20444 
 
 20741 
 
 21926 
 
 22222 
 
 22516 
 
 41 
 
 20500 
 
 20804 
 
 21107 
 
 21411 
 
 21715 
 
 22930 
 
 23233 
 
 23537 
 
 42 
 
 21467 
 
 21778 
 
 22089 
 
 22400 
 
 22711 
 
 23956 
 
 24267 
 
 24578 
 
 43 
 
 22456 
 
 22774 
 
 23093 
 
 23411 
 
 23730 
 
 25004 
 
 25322 
 
 25641 
 
 44 
 
 23467 
 
 23793 
 
 24119 
 
 24444 
 
 24770 
 
 26074 
 
 26400 
 
 26726 
 
 45 
 
 24500 
 
 24833 
 
 25167 
 
 25500 
 
 25833 
 
 27167 
 
 27500 
 
 27833 
 
 46 
 
 25556 
 
 25896 
 
 26237 
 
 26578 
 
 26919 
 
 28281 
 
 28622 
 
 28963 
 
 47 
 
 26633 
 
 26981 
 
 27330 
 
 27678 
 
 28026 
 
 29419 
 
 29767 
 
 80115 
 
 48 
 49 
 
 
 IS 
 
 28444 
 
 29581 
 
 m 
 
 29156 
 30307 
 
 30578 
 31759 
 
 30933 
 32122 
 
 31289 
 32485 
 
 50 
 
 30000 
 
 30370 
 
 30741 
 
 31111 
 
 31481 
 
 32963 
 
 33333 
 
 33704 
 
 51 
 
 31167 
 
 31544 
 
 31922 
 
 32300 
 
 32678 
 
 34189 
 
 34567 
 
 34944 
 
 52 
 
 32356 
 
 32741 
 
 33126 
 
 33511 
 
 33896 
 
 35437 
 
 35822 
 
 36207 
 
 53 
 
 33567 
 
 33959 
 
 34352 
 
 34744 
 
 35137 
 
 36707 
 
 37100 
 
 87493 
 
 54 
 
 34800 
 
 35200 
 
 05600 
 
 36000 
 
 36400 
 
 38000 
 
 38400 
 
 88800 
 
 55 
 
 36056 
 
 36463 
 
 36870 
 
 37278 
 
 37685 
 
 39315 
 
 39722 
 
 4C130 
 
 56 
 
 37333 
 
 37748 
 
 38163 
 
 38578 
 
 38993 
 
 40652 
 
 41067 
 
 41481 
 
 57 
 
 38633 
 
 39056 
 
 39478 
 
 39900 
 
 40322 
 
 42011 
 
 42433 
 
 42856 
 
 58 
 
 39956 
 
 40385 
 
 40815 
 
 41244 
 
 41674 
 
 43393 
 
 43822 
 
 44252 
 
 59 
 
 41300 
 
 41737 
 
 42174 
 
 42611 
 
 43048 
 
 44796 
 
 45233 
 
 45670 
 
 60 
 
 42667 
 
 43111 
 
 43556 
 
 44000 
 
 44444 
 
 46222 
 
 46667 
 
 47111 
 
 275 
 
TABLE XV. CUBIC YARDS IN 100 FEET LENGTH. 
 
 Area. 
 Sq. 
 Ft. 
 
 Cubic , 
 Yards. 
 
 Area. 
 
 I?: 
 
 Cubic 
 Yards. 
 
 Area. 
 
 Sq. 
 Ft. 
 
 Cubic 
 Yards. 
 
 Area. 
 Sq. 
 Ft. 
 
 Cubic 
 Yards. 
 
 Area. 
 
 & 
 
 Cubic 
 Yards. 
 
 1 
 
 3.7 
 
 51 
 
 188.9 
 
 101 
 
 374.1 
 
 151 
 
 559.3 
 
 201 
 
 744.4 
 
 2 
 
 7.4 
 
 52 
 
 192.6 
 
 102 
 
 377.8 
 
 152 
 
 563.0 
 
 202 
 
 748.2 
 
 3 
 
 11.1 
 
 53 
 
 196.3 
 
 103 
 
 381.5 
 
 153 
 
 566.7 
 
 203 
 
 751.9 
 
 4 
 
 14.8 
 
 54 
 
 200.0 
 
 104 
 
 385.2 
 
 154 
 
 570.4 
 
 204 
 
 755.6 
 
 5 
 
 18.5 
 
 55 
 
 203.7 
 
 105 
 
 388.9 
 
 155 
 
 574.1 
 
 205 
 
 759.3 
 
 6 
 
 22.2 
 
 56 
 
 207.4 
 
 106 
 
 392.6 
 
 156 
 
 577.8 
 
 206 
 
 763.0 
 
 7 
 
 25.9 
 
 57 
 
 211.1 
 
 107 
 
 396.3 
 
 157 
 
 581.5 
 
 207 
 
 766.7 
 
 8 
 
 29.6 
 
 58 
 
 214.8 
 
 108 
 
 400.0 
 
 158 
 
 585.2 
 
 208 
 
 770.4 
 
 9 
 
 33.3 
 
 59 
 
 218.5 
 
 109 
 
 403.7 
 
 159 
 
 588.9 
 
 209 
 
 774.1 
 
 10 
 
 37.0 
 
 60 
 
 222.2 
 
 110 
 
 407.4 
 
 160 
 
 592.6 
 
 210 
 
 777.8 
 
 11 
 
 40.7 
 
 61 
 
 225.9 
 
 111 
 
 411.1 
 
 161 
 
 596.3 
 
 211 
 
 781.5 
 
 12 
 
 44.4 
 
 62 
 
 229.6 
 
 112 
 
 414.8 
 
 162 
 
 600.0 
 
 212 
 
 785.2 
 
 13 
 
 48.1 
 
 63 
 
 233.3 
 
 113 
 
 418.5 
 
 163 
 
 603.7 
 
 213 
 
 788.9 
 
 14 
 
 51.9 
 
 64 
 
 237.0 
 
 114 
 
 422.2 
 
 164 
 
 607.4 
 
 214 
 
 792.6 
 
 15 
 
 55.6 
 
 65 
 
 240.7 
 
 115 
 
 425.9 
 
 165 
 
 611.1 
 
 215 
 
 796.3 
 
 16 
 
 59.3 
 
 66 
 
 244.4 
 
 116 
 
 429.6 
 
 166 
 
 614.8 
 
 216 
 
 800.0 
 
 17 
 
 63.0 
 
 67 
 
 248.2 
 
 117 
 
 433.3 
 
 167 
 
 618.5 
 
 217 
 
 803.7 
 
 18 
 
 66.7 
 
 68 
 
 251.9 
 
 118 
 
 437.0 
 
 168 
 
 622.2 
 
 218 
 
 807.4 
 
 19 
 
 70.4 
 
 69 
 
 255.6 
 
 119 
 
 440.7 
 
 169 
 
 625.9 
 
 219 
 
 811.1 
 
 20 
 
 74.1 
 
 70 
 
 259.3 
 
 120 
 
 444.4 
 
 170 
 
 629.6 
 
 220 
 
 814.8 
 
 21 
 
 77.8 
 
 71 
 
 263.0 
 
 121 
 
 448.2 
 
 171 
 
 633.3 
 
 221 
 
 818.5 
 
 22 
 
 81.5 
 
 72 
 
 266.7 
 
 122 
 
 451.9 
 
 172 
 
 637 
 
 222 
 
 822.2 
 
 23 
 
 85.2 
 
 73 
 
 270.4 
 
 123 
 
 455.6 
 
 173 
 
 640.7 
 
 223 
 
 825.9 
 
 24 
 
 88.9 
 
 74 
 
 274.1 
 
 124 
 
 459.3 
 
 174 
 
 644.4 
 
 204 
 
 829.6 
 
 25 
 
 92 6 
 
 75 
 
 277.8 
 
 125 
 
 403.0 
 
 175 
 
 648.2 
 
 225 
 
 833.3 
 
 28 
 
 96.3 
 
 76 
 
 281.5 
 
 126 
 
 466.7 
 
 176 
 
 651.9 
 
 226 
 
 837.0 
 
 27 
 
 100.0 
 
 77 
 
 285 2 
 
 127 
 
 470.4 
 
 177 
 
 655.6 
 
 227 
 
 840.7 
 
 28 
 
 103.7 
 
 78 
 
 288.9 
 
 128 
 
 474.1 
 
 178 
 
 659.3 
 
 228 
 
 844.4 
 
 29 
 
 107.4 
 
 79 
 
 292.6 
 
 129 
 
 477.8 
 
 179 
 
 663.0 
 
 229 
 
 848.2 
 
 30 
 
 111.1 
 
 80 
 
 296.3 
 
 130 
 
 481.5 
 
 180 
 
 666.7 
 
 230 
 
 851.9 
 
 31 
 
 114.8 
 
 81 
 
 300.0 
 
 131 
 
 485.2 
 
 181 
 
 670.4 
 
 231 
 
 855.6 
 
 32 
 
 us. 5 
 
 82 
 
 303.7 
 
 132 
 
 488.9 
 
 182 
 
 674.1 
 
 232 
 
 859.3 
 
 33 
 
 122.2 
 
 83 
 
 307.4 
 
 133 
 
 492.6 
 
 183 
 
 677.8 
 
 233 
 
 8G3.0 
 
 34 
 
 125.9 
 
 84 
 
 311.1 
 
 134 
 
 496.3 
 
 184 
 
 681.5 
 
 234 
 
 866.7 
 
 35 
 
 129 6 
 
 85 
 
 314.8 
 
 135 
 
 500.0 
 
 185 
 
 685.2 
 
 235 
 
 870.4 
 
 36 
 
 133.3 
 
 86 
 
 318.5 
 
 136 
 
 503.7 
 
 186 
 
 688.9 
 
 236 
 
 874.1 
 
 37 
 
 137.0 
 
 87 
 
 322.2 
 
 137 
 
 507.4 
 
 187 
 
 692.6 
 
 237 
 
 877.8 
 
 38 
 
 140.7 
 
 88 
 
 325.9 
 
 138 
 
 511.1 
 
 188 
 
 696.3 
 
 238 
 
 881.5 
 
 39 
 
 144 4 
 
 89 
 
 329.6 
 
 139 
 
 514.8 
 
 189 
 
 700.0 
 
 239 
 
 885.2 
 
 40 
 
 148.2 
 
 90 
 
 333.3 
 
 140 
 
 518.5 
 
 190 
 
 703.7 
 
 240 
 
 888.9 
 
 41 
 
 151.9 
 
 91 
 
 387.0 
 
 141 
 
 522.2 
 
 191 
 
 707.4 
 
 241 
 
 892.6 
 
 42 
 
 1.">.6 
 
 92 
 
 340.7 
 
 142 
 
 525.9 
 
 192 
 
 711.1 
 
 242 
 
 896.3 
 
 43 
 
 159.3 
 
 93 
 
 344.4 
 
 143 
 
 529.6 
 
 193 
 
 714.8 
 
 243 
 
 900.0 
 
 44 
 
 1G3.0 
 
 94 
 
 348.2 
 
 144 
 
 533.3 
 
 194 
 
 718.5 
 
 244 
 
 90.1.7 
 
 45 
 
 166.7 
 
 95 
 
 351.9 
 
 145 
 
 537.0 
 
 195 
 
 722.2 
 
 245 
 
 907.4 
 
 46 
 
 170.4 
 
 96 
 
 355.6 
 
 146 
 
 540.7 
 
 196 
 
 725.9 
 
 246 
 
 911.1 
 
 47 
 
 174.1 
 
 97 
 
 359.3 
 
 147 
 
 544.4 
 
 197 
 
 729.6 
 
 247 
 
 914.8 
 
 48 
 
 177.8 
 
 98 
 
 363.0 
 
 148 
 
 548.2 
 
 198 
 
 733.3 
 
 248 
 
 918.5 
 
 49 
 
 181.5 
 
 99 
 
 306.7 
 
 149 
 
 551.9 
 
 199 
 
 737.0 
 
 249 
 
 9-.2.2 
 
 50 
 
 185.2 
 
 100 
 
 370.4 
 
 150 
 
 555.6 
 
 200 
 
 740.7 
 
 250 
 
 925.9 
 
 276 
 
TABLE XV. CUBIC YARDS IN 100 FEET LENGTH. 
 
 Area. 
 
 $ 
 
 Cubic 
 Yards. 
 
 Area. 
 
 ft 
 
 Cubic 
 Yards. 
 
 Area. 
 
 ft 
 
 Cubic 
 Yards. 
 
 Area. 
 
 ft 
 
 Cubic 
 Yards. 
 
 Area. 
 
 ft 
 
 Cubic 
 Yards. 
 
 251 
 
 929.6 
 
 301 
 
 1114.8 
 
 351 
 
 1300.0 
 
 401 
 
 1485.2 
 
 451 
 
 1670.4 
 
 252 
 
 933 3 
 
 302 
 
 1118.5 
 
 352 
 
 1303.7 
 
 402 
 
 1488.9 
 
 452 
 
 1674.1 
 
 253 
 
 937.0 
 
 303 
 
 1122.2 
 
 353 
 
 1307.4 
 
 403 
 
 1492.6 
 
 453 
 
 1677.8 
 
 254 
 
 940.7 
 
 304 
 
 1125.9 
 
 354 
 
 1311.1 
 
 404 
 
 1496.3 
 
 454 
 
 1681.5 
 
 255 
 
 944.4 
 
 305 
 
 1129.6 
 
 355 
 
 1314.8 
 
 405 
 
 1500.0 
 
 455 
 
 1685.2 
 
 256 
 
 948.2 
 
 306 
 
 1133.3 
 
 356 
 
 1318.5 
 
 406 
 
 1503.7 
 
 456 
 
 1688.9 
 
 257 
 
 951 9 
 
 307 
 
 1137.0 
 
 357 
 
 1322.2 
 
 407 
 
 1507.4 
 
 457 
 
 1692.6 
 
 258 
 
 955.6 
 
 308 
 
 1140.7 
 
 358 
 
 1325.9 
 
 408 
 
 1511.1 
 
 458 
 
 1696.3 
 
 259 
 
 959.3 
 
 309 
 
 1144.4 
 
 359 
 
 1329.6 
 
 409 
 
 1514.8 
 
 459 
 
 1700.0 
 
 260 
 
 963.0 
 
 310 
 
 1148.2 
 
 360 
 
 1333.3 
 
 410 
 
 1518.5 
 
 460 
 
 1703.7 
 
 261 
 
 966.7 
 
 311 
 
 1151.9 
 
 361 
 
 1337.0 
 
 411 
 
 1522.2 
 
 461 
 
 1707.4 
 
 262 
 
 970.4 
 
 312 
 
 1155.6 
 
 362 
 
 1340.7 
 
 412 
 
 1525.9 
 
 462 
 
 1711.1 
 
 203 
 
 974 1 
 
 313 
 
 1159.3 
 
 363 
 
 1344.4 
 
 413 
 
 1529.6 
 
 463 
 
 1714.8 
 
 264 
 
 977.8 
 
 314 
 
 1163.0 
 
 364 
 
 1348.2 
 
 414 
 
 1533.3 
 
 464 
 
 1718.5 
 
 265 
 
 981.5 
 
 315 
 
 1166.7 
 
 365 
 
 1351.9 
 
 415 
 
 1537.0 
 
 465 
 
 1722.2 
 
 266 
 
 985.2 
 
 316 
 
 1170.4 
 
 366 
 
 1355.6 
 
 416 
 
 1540.7 
 
 466 
 
 1725.9 
 
 267 
 
 988.9 
 
 317 
 
 1174.1 
 
 367 
 
 1359.3 
 
 417 
 
 1544.4 
 
 467 
 
 1729.6 
 
 268 
 
 992.6 
 
 318 
 
 1177.8 
 
 368 
 
 1363.0 
 
 418 
 
 1548.2 
 
 468 
 
 1733.3 
 
 263 
 
 996.3 
 
 319 
 
 1181.5 
 
 369 
 
 1366.7 
 
 419 
 
 1551.9 
 
 469 
 
 1737.0 
 
 270 
 
 1000.0 
 
 320 
 
 1185.2 
 
 370 
 
 1370.4 
 
 420 
 
 1555.6 
 
 470 
 
 1740.7 
 
 271 
 
 1003.7 
 
 321 
 
 1188.9 
 
 371 
 
 1374.1 
 
 421 
 
 1559.3 
 
 471 
 
 1744.4 
 
 272 
 
 1007.4 
 
 322 
 
 1192.6 
 
 372 
 
 1377.8 
 
 422 
 
 1563.0 
 
 472 
 
 1748.2 
 
 273 
 
 1011. I 
 
 323 
 
 1196.3 
 
 373 
 
 1381.5 
 
 423 
 
 1566.7 
 
 473 
 
 1751.9 
 
 274 
 
 1014.8 
 
 324 
 
 1200.0 
 
 374 
 
 1385.2 
 
 424 
 
 1570.4 
 
 474 
 
 1755.6 
 
 275 
 
 1018.5 
 
 325 
 
 1203.7 
 
 375 
 
 138S.9 
 
 425 
 
 1574.1 
 
 475 
 
 1759.3 
 
 276 
 
 1022.2 
 
 326 
 
 1207.4 
 
 376 
 
 1392.6 
 
 426 
 
 1577.8 
 
 476 
 
 1763.0 
 
 277 
 
 1025.9 
 
 327 
 
 1211.1 
 
 377 
 
 1396.3 
 
 427 
 
 1581.5 
 
 477 
 
 1766.7 
 
 278 
 
 10-29.6 
 
 328 
 
 1214.8 
 
 378 
 
 1400.0 
 
 428 
 
 1585.2 
 
 478 
 
 1770.4 
 
 279 
 
 1033.3 
 
 329 
 
 1218.5 
 
 379 
 
 M03.7 
 
 429 
 
 1588.9 
 
 479 
 
 1774.1 
 
 280 
 
 1037.0 
 
 330 
 
 1222.2 
 
 3SO 
 
 1407.4 
 
 430 
 
 1592.6 
 
 480 
 
 1777.8 
 
 281 
 
 1040.7 
 
 331 
 
 1325.9 
 
 381 
 
 1411.1 
 
 431 
 
 1596.3 
 
 481 
 
 1781.5 
 
 ,'?82 
 
 1044.4 
 
 332 
 
 1229.6 
 
 382 
 
 1414.8 
 
 432 
 
 1600.0 
 
 482 
 
 1785.2 
 
 283 
 
 1048.2 
 
 333 
 
 1233.3 
 
 383 
 
 1418.5 
 
 433 
 
 1603.7 
 
 483 
 
 1788.9 
 
 284 
 
 1051.9 
 
 334 
 
 1237.0 
 
 384 
 
 1422.2 
 
 434 
 
 1607.4 
 
 484 
 
 1792.6 
 
 285 
 
 1055.6 
 
 335 
 
 1240.7 
 
 385 
 
 1425.9 
 
 435 
 
 1611.1 
 
 485 
 
 1796.3 
 
 286 
 
 1059.3 
 
 336 
 
 1244.4 
 
 386 
 
 1429.6 
 
 436 
 
 1614.8 
 
 486 
 
 1800.0 
 
 287 
 
 1003.0 
 
 337 
 
 1248.2 
 
 387 
 
 1433.3 
 
 437 
 
 1618.5 
 
 487 
 
 1803.7 
 
 288 
 
 1066.7 
 
 338 
 
 1251.9 
 
 388 
 
 1437.0 
 
 438 
 
 1622.2 
 
 488 
 
 1807.4 
 
 289 
 
 1070.4 
 
 339 
 
 1255.6 
 
 389 
 
 1440.7 
 
 439 
 
 1625.9 
 
 489 
 
 1811.1 
 
 290 
 
 1074.1 
 
 340 
 
 1259.3 
 
 390 
 
 1444.4 
 
 440 
 
 1629.6 
 
 490 
 
 1814.8 
 
 291 
 
 1077.8' 
 
 341 
 
 1263.0 
 
 391 
 
 1448.2 
 
 441 
 
 1633.3 
 
 491 
 
 1818.5 
 
 292 
 
 1081.5 
 
 342 
 
 1266.7 
 
 392 
 
 1451.9 
 
 442 
 
 1637.0 
 
 492 
 
 1822.2 
 
 293 
 
 1085.2 
 
 343 
 
 1270.4 
 
 393 
 
 1455.6 
 
 443 
 
 1640.7 
 
 493 
 
 1825.9 
 
 294 
 
 1088.9 
 
 344 
 
 1274.1 
 
 394 
 
 1459.3 
 
 444 
 
 1644.4 
 
 494 
 
 1829.6 
 
 295 
 
 1092.6 
 
 345 
 
 1277.8 
 
 395 
 
 1463.0 
 
 445 
 
 1648.2 
 
 495 
 
 1833.3 
 
 296 
 
 1096.3 
 
 346 
 
 1281.5 
 
 396 
 
 1466.7 
 
 446 
 
 1651.9 
 
 496 
 
 1837.0 
 
 297 
 
 1100.0 
 
 347 
 
 1285.2 
 
 397 
 
 1470.4 
 
 447 
 
 1655.6 
 
 497 
 
 1840.7 
 
 298 
 
 1103.7 
 
 348 
 
 1288.9 
 
 398 
 
 1474.1 
 
 448 
 
 1659.3 
 
 498 
 
 1844.4 
 
 299 
 
 1107.4 
 
 349 
 
 1292.6 
 
 399 
 
 1477.8 
 
 449 
 
 1663.0 
 
 499 
 
 1848.2 
 
 300 
 
 1111.1 
 
 350 
 
 1296.3 
 
 400 
 
 1481.5 
 
 450 
 
 1666.7 
 
 500 
 
 1851.9 
 
 277 
 
TABLE XV. CUBIC YARDS IN 100 FEET LENGTH. 
 
 Area. 
 
 ft 
 
 Cubic 
 Yards. 
 
 Area. 
 
 ft 
 
 Cubic 
 Yards. 
 
 Area. 
 
 ft 
 
 Cubic 
 
 Yards. 
 
 Area. 
 
 11: 
 
 Cubic 
 Yards. 
 
 Area. 
 
 ft 
 
 Cubic 
 
 Yards. 
 
 501 
 
 1855.6 
 
 551 
 
 2040.7 
 
 601 
 
 2225.9 
 
 651 
 
 2411.1 
 
 701 
 
 2596.3 
 
 502 
 
 1859.3 
 
 552 
 
 2044.4 
 
 602 
 
 2229.6 
 
 652 
 
 2414.8 
 
 702 
 
 2600.0 
 
 503 
 
 1863.0 
 
 553 
 
 2048.2 
 
 603 
 
 2233.3 
 
 653 
 
 2418.5 
 
 703 
 
 2603.7 
 
 504 
 
 1866.7 
 
 554 
 
 2051.9 
 
 604 
 
 2237.0 
 
 654 
 
 2422.2 
 
 704 
 
 2607.4 
 
 505 
 
 1870.4 
 
 555 
 
 2055.6 
 
 605 
 
 2240.7 
 
 655 
 
 2425.9 
 
 705 
 
 2611.1 
 
 506 
 
 1874.1 
 
 556 
 
 2059.3 
 
 606 
 
 2244.4 
 
 656 
 
 2429.6 
 
 706 
 
 2614.8 
 
 507 
 
 1877.8 
 
 557 
 
 2063.0 
 
 607 
 
 2248.2 
 
 657 
 
 24a3.3 
 
 707 
 
 2618.5 
 
 508 
 
 1881.5 
 
 558 
 
 2066.7 
 
 608 
 
 2251.9 
 
 658 
 
 2437.0 
 
 708 
 
 2622.2 
 
 509 
 
 1885.2 
 
 559 
 
 2070.4 
 
 609 
 
 2255.6 
 
 659 
 
 2440.7 
 
 709 
 
 2625.9 
 
 510 
 
 1888.9 
 
 560 
 
 2074.1 
 
 610 
 
 2259.3 
 
 660 
 
 2444.4 
 
 710 
 
 2629.6 
 
 511 
 
 1892.6 
 
 561 
 
 2077.8 
 
 611 
 
 2263.0 
 
 661 
 
 2448.2 
 
 711 
 
 2633.3 
 
 512 
 
 1896.3 
 
 562 
 
 2081.5 
 
 612 
 
 2266.7 
 
 662 
 
 2451.9 
 
 712 
 
 2637.0 
 
 513 
 
 1900.0 
 
 563 
 
 2085.2 
 
 613 
 
 2270.4 
 
 663 
 
 2455.6 
 
 713 
 
 2640.7 
 
 514 
 
 1903.7 
 
 564 
 
 2088.9 
 
 614 
 
 2274.1 
 
 664 
 
 2459.3 
 
 714 
 
 2644.4 
 
 515 
 
 1907.4 
 
 565 
 
 2092.6 
 
 615 
 
 2277.8 
 
 665 
 
 2463.0 
 
 715 
 
 2648.2 
 
 516 
 
 1911.1 
 
 566 
 
 2096.3 
 
 616 
 
 2281.5 
 
 666 
 
 2466.7 
 
 716 
 
 2651.9 
 
 517 
 
 1914.8 
 
 567 
 
 2100.0 
 
 617 
 
 2285.2 
 
 667 
 
 2470.4 
 
 717 
 
 2655.6 
 
 518 
 
 1918.5 
 
 568 
 
 2103.7 
 
 618 
 
 2288.9 
 
 668 
 
 2474.1 
 
 718 
 
 2659.3 
 
 519 
 
 1922.2 
 
 569 
 
 2107.4 
 
 619 
 
 2292.6 
 
 669 
 
 2477.8 
 
 719 
 
 2663.0 
 
 520 
 
 1925.9 
 
 570 
 
 2111.1 
 
 620 
 
 2296.3 
 
 670 
 
 2481.5 
 
 720 
 
 2666.7 
 
 521 
 
 1929.6 
 
 571 
 
 2114.8 
 
 621 
 
 2300.0 
 
 671 
 
 2485.2 
 
 721 
 
 2670.4 
 
 523 
 
 1933.3 
 
 572 
 
 2118.5 
 
 622 
 
 2303.7 
 
 672 
 
 2488.9 
 
 722 
 
 2674.1 
 
 523 
 
 1937.0 
 
 573 
 
 2122.2 
 
 623 
 
 2307.4 
 
 673 
 
 2492.6 
 
 723 
 
 2677.8 
 
 524 
 
 1940.7 
 
 574 
 
 2125.9 
 
 624 
 
 2311.1 
 
 674 
 
 2496.3 
 
 724 
 
 2681.5 
 
 525 
 
 1944.4 
 
 575 
 
 2129.6 
 
 625 
 
 2314.8 
 
 675 
 
 2500.0 
 
 725 
 
 2685.2 
 
 526 
 
 1948.2 
 
 576 
 
 2133.3 
 
 626 
 
 2318.5 
 
 676 
 
 2503.7 
 
 726 
 
 2688.9 
 
 527 
 
 1951.9 
 
 577 
 
 2137.0 
 
 627 
 
 2322.2 
 
 677 
 
 2507.4 
 
 727 
 
 2692.6 
 
 528 
 
 1955.6 
 
 578 
 
 2140.7 
 
 628 
 
 2325.9 
 
 678 
 
 2511.1 
 
 728 
 
 2696.3 
 
 529 
 
 1959.3 
 
 579 
 
 2144.4 
 
 629 
 
 2329.6 
 
 679 
 
 2514.8 
 
 729 
 
 2700.0 
 
 530 
 
 1963.0 
 
 580 
 
 2148.2 
 
 630 
 
 2333.3 
 
 680 
 
 2518.5 
 
 730 
 
 2703.7 
 
 531 
 
 1966.7 
 
 581 
 
 2151.9 
 
 631 
 
 2337.0 
 
 681 
 
 2522.2 
 
 731 
 
 2707.4 
 
 532 
 
 1970.4 
 
 582 
 
 2155.6 
 
 632 
 
 2340.7 
 
 682 
 
 2525.9 
 
 732 
 
 2711.1 
 
 533 
 
 1974.1 
 
 583 
 
 2159.3 
 
 633 
 
 2344.4 
 
 683 
 
 2529.6 
 
 733 
 
 2714.8 
 
 534 
 
 1977.8 
 
 584 
 
 2163.0 
 
 634 
 
 2348.2 
 
 684 
 
 2533.3 
 
 734 
 
 2718.5 
 
 535 
 
 1981.5 
 
 585 
 
 21G6.7 
 
 635 
 
 2351.9 
 
 685 
 
 2537.0 
 
 735 
 
 2722.2 
 
 536 
 
 1985.2 
 
 586 
 
 2170.4 
 
 636 
 
 2355.6 
 
 686 
 
 2540 7 
 
 736 
 
 2725.9 
 
 537 
 
 1988.9 
 
 587 
 
 2174.1 
 
 637 
 
 2359.3 
 
 687 
 
 2544.4 
 
 737 
 
 2729.6 
 
 538 
 
 1992.6 
 
 588 
 
 2177.8 
 
 638 
 
 2363 
 
 688 
 
 2548.2 
 
 138 
 
 2733.3 
 
 539 
 
 1996.3 
 
 589 
 
 2181.5 
 
 639 
 
 2366.7 
 
 689 
 
 2551.9 
 
 739 
 
 2737.0 
 
 540 
 
 2000.0 
 
 590 
 
 2185.2 
 
 640 
 
 2370.4 
 
 690 
 
 2555.6 
 
 740 
 
 2740.7 
 
 541 
 
 2003.7 
 
 591 
 
 2188.9 
 
 641 
 
 2374.1 
 
 691 
 
 2559.3 
 
 741 
 
 2744.4 
 
 542 
 
 2007.4 
 
 592 
 
 2192.6 
 
 642 
 
 2377.8 
 
 692 
 
 2563.0 
 
 742 
 
 2748.2 
 
 543 
 
 2011.1 
 
 593 
 
 2196.3 
 
 643 
 
 2381.5 
 
 693 
 
 2566.7 
 
 743 
 
 2751.9 
 
 544 
 
 2014.8 
 
 594 
 
 2200.0 
 
 644 
 
 2385.2 
 
 694 
 
 2570.4 
 
 744 
 
 2755.6 
 
 545 
 
 2018.5 
 
 595 
 
 2203.7 
 
 645 
 
 2388.9 
 
 695 
 
 2574.1 
 
 745 
 
 2759.3 
 
 546 
 
 2022.2 
 
 596 
 
 2207.4 
 
 646 
 
 2392.6 
 
 696 
 
 2577.8 
 
 746 
 
 2763.0 
 
 547 
 
 2025.9 
 
 597 
 
 2211.1 
 
 647 
 
 2396.3 
 
 697 
 
 2581.5 
 
 747 
 
 2766.7 
 
 548 
 
 2029.6 
 
 598 
 
 2214.8 
 
 648 
 
 2400.0 
 
 698 
 
 2585.2 
 
 748 
 
 2770.4 
 
 549 
 
 2033.3 
 
 599 
 
 2218.5 
 
 649 
 
 2403.7 
 
 699 
 
 2588.9 
 
 749 
 
 2774.1 
 
 550 
 
 2037.0 
 
 600 
 
 2222.2 
 
 650 
 
 2407.4 
 
 700 
 
 2592.6 
 
 750 
 
 2777.8 
 
 278 
 
TABLE XV. CUBIC YARDS IN 100 FEET LENGTH. 
 
 Area. 
 
 1?: 
 
 Cubic 
 Yards. 
 
 Area. 
 Sq. 
 Ft. 
 
 Cubic 
 Yards. 
 
 Area. 
 
 a 
 
 Cubic 
 
 Yards. 
 
 Area. 
 
 11: 
 
 Cubic 
 Yards. 
 
 Area. 
 
 St 
 
 Cubic 
 Yards. 
 
 751 
 
 2781.5 
 
 801 
 
 2966.7 
 
 851 
 
 3151.9 
 
 901 
 
 3337.0 
 
 951 
 
 3522.2 
 
 752 
 
 2785.2 
 
 802 
 
 2970.4 
 
 852 
 
 3155.6 
 
 902 
 
 3340.7 
 
 952 
 
 3525.9 
 
 753 
 
 27'88.9 
 
 803 
 
 2974.1 
 
 853 
 
 3159.3 
 
 903 
 
 3344.4 
 
 953 
 
 3529.6 
 
 754 
 
 2792.6 
 
 804 
 
 2977.8 
 
 854 
 
 3163.0 
 
 904 
 
 3348.2 
 
 954 
 
 3533.3 
 
 755 
 
 2796.3 
 
 805 
 
 2981. 5 
 
 855 
 
 3166.7 
 
 905 
 
 3351.9 
 
 955 
 
 3537.0 
 
 756 
 
 2800.0 
 
 806 
 
 2985.2 
 
 856 
 
 3170.4 
 
 906 
 
 3355.6 
 
 956 
 
 3540.7 
 
 757 
 
 2803.7 
 
 807 
 
 2988.9 
 
 857 
 
 3174.1 
 
 907 
 
 3359.3 
 
 957 
 
 3544.4 
 
 758 
 
 2807.4 
 
 808 
 
 2992.6 
 
 858 
 
 3177.8 
 
 908 
 
 3363.0 
 
 958 
 
 3548.2 
 
 7'59 
 
 2811.1 
 
 809 
 
 2996.3 
 
 859 
 
 3181.5 
 
 909 
 
 3366.7 
 
 959 
 
 3551.9 
 
 760 
 
 2814.8 
 
 810 
 
 3COO.O 
 
 860 
 
 3185.2 
 
 910 
 
 3370.4 
 
 960 
 
 3555.6 
 
 761 
 
 2818.5 
 
 811 
 
 3003.7 
 
 861 
 
 3188.9 
 
 911 
 
 3374.1 
 
 961 
 
 3559.3 
 
 762 
 
 2822.2 
 
 812 
 
 3007.4 
 
 862 
 
 3192.6 
 
 912 
 
 3377.8 
 
 962 
 
 3563.0 
 
 763 
 
 2825.9 
 
 813 
 
 3011.1 
 
 863 
 
 3196.3 
 
 913 
 
 3381.5 
 
 963 
 
 3566.7 
 
 764 
 
 2829.6 
 
 814 
 
 3014.8 
 
 864 
 
 3200.0 
 
 914 
 
 3385.2 
 
 964 
 
 3570.4 
 
 765 
 
 2833 3 
 
 815 
 
 3018.5 
 
 865 
 
 3203.7 
 
 915 
 
 3488.9 
 
 965 
 
 3574.1 
 
 766 
 
 2837.0 
 
 816 
 
 3022.2 
 
 866 
 
 3207.4 
 
 916 
 
 3392.6 
 
 966 
 
 3577.8 
 
 767 
 
 2840.7 
 
 817 
 
 3025.9 
 
 867 
 
 3211.1 
 
 917 
 
 8396.8 
 
 967 
 
 3581.5 
 
 768 
 
 2844.4 
 
 818 
 
 3009.6 
 
 868 
 
 3214.8 
 
 918 
 
 3400.0 
 
 968 
 
 3585.2 
 
 769 
 
 2848.2 
 
 819 
 
 3033.3 
 
 869 
 
 3v!l8.5 
 
 919 
 
 3403.7 
 
 969 
 
 3588.9 
 
 770 
 
 2851.9 
 
 820 
 
 3037.0 
 
 870 
 
 3222.2 
 
 920 
 
 3407.4 
 
 970 
 
 3592.6 
 
 771 
 
 2855.6 
 
 821 
 
 3040.7 
 
 871 
 
 3225.9 
 
 921 
 
 3411.1 
 
 971 
 
 3596.3 
 
 772 
 
 2859.3 
 
 822 
 
 3044.4 
 
 872 
 
 3229.6 
 
 922 
 
 3414.8 
 
 972 
 
 3600.0 
 
 773 
 
 2863.0 
 
 823 
 
 3048.2 
 
 873 
 
 3233.3 
 
 923 
 
 3-118.5 
 
 973 
 
 3603.7 
 
 774 
 
 2866.7 
 
 824 
 
 3051.9 
 
 874 
 
 3237.0 
 
 924 
 
 3422.2 
 
 974 
 
 3607.4 
 
 775 
 
 2870.4 
 
 825 
 
 3055.6 
 
 875 
 
 3240.7 
 
 925 
 
 3425.9 
 
 975 
 
 3611.1 
 
 776 
 
 2874.1 
 
 826 
 
 3059.3 
 
 876 
 
 3244.4 
 
 926 
 
 3429.6 
 
 976 
 
 3614.8 
 
 777 
 
 2877.8 
 
 827 
 
 3063.0 
 
 877 
 
 3248.2 
 
 927 
 
 3433.3 
 
 977 
 
 3618.5 
 
 778 
 
 2881.5 
 
 828 
 
 3066.7 
 
 87'8 
 
 3251.9 
 
 928 
 
 3437.0 
 
 978 
 
 3622.2 
 
 779 
 
 2885.2 
 
 829 
 
 3070.4 
 
 879 
 
 3255.6 
 
 929 
 
 3440.7 
 
 979 
 
 3625.9 
 
 780 
 
 2888.9 
 
 830 
 
 3074.1 
 
 880 
 
 3259.3 
 
 930 
 
 3444.4 
 
 980 
 
 3629.6 
 
 781 
 
 2892.6 
 
 831 
 
 3077.8 
 
 881 
 
 3263.0 
 
 931 
 
 3448.2 
 
 981 
 
 3633.3 
 
 782 
 
 2896.3 
 
 832 
 
 3081.5 
 
 882 
 
 3266.7 
 
 932 
 
 3451.9 
 
 982 
 
 3637.0 
 
 783 
 
 2900.0 
 
 833 
 
 3085.2 
 
 883 
 
 3270.4 
 
 933 
 
 3455.6 
 
 983 
 
 3640.7 
 
 784 
 
 2903.7 
 
 834 
 
 3088.9 
 
 884 
 
 3274.1 
 
 934 
 
 3459.3 
 
 984 
 
 3644.4 
 
 785 
 
 2907.4 
 
 835 
 
 3092.6 
 
 885 
 
 3277.8 
 
 935 
 
 3463.0 
 
 985 
 
 3648.2 
 
 786 
 
 2911.1 
 
 836 
 
 3096.3 
 
 886 
 
 3281.5 
 
 936 
 
 3466.7 
 
 986 
 
 3651.9 
 
 787 
 
 2914.8 
 
 837 
 
 3100.0 
 
 887 
 
 3285.2 
 
 937 
 
 3470.4 
 
 987 
 
 3655.6 
 
 788 
 
 2918.5 
 
 838 
 
 3103.7 
 
 888 
 
 3288.9 
 
 938 
 
 3474.1 
 
 988 
 
 3659.3 
 
 789 
 
 2922.2 
 
 839 
 
 3107.4 
 
 889 
 
 3292.6 
 
 939 
 
 3477.8 
 
 989 
 
 3663.0 
 
 790 
 
 2925.9 
 
 840 
 
 3111.1 
 
 890 
 
 3296.3 
 
 940 
 
 3481.5 
 
 990 
 
 3666.7 
 
 791 
 
 2929.6 
 
 841 
 
 3114.8 
 
 891 
 
 3300.0 
 
 941 
 
 3485.2 
 
 991 
 
 3670.4 
 
 792 
 
 2933.3 
 
 842 
 
 3118.5 
 
 892 
 
 3303.7 
 
 942 
 
 3488.9 
 
 992 
 
 3674.1 
 
 793 
 
 2937.0 
 
 843 
 
 3122.2 
 
 893 
 
 3307.4 
 
 943 
 
 3492.6 
 
 993 
 
 3677.8 
 
 794 
 
 2940.7 
 
 844 
 
 3125.9 
 
 894 
 
 3311.1 
 
 944 
 
 3496.3 
 
 994 
 
 3681.5 
 
 795 
 
 2944.4 
 
 845 
 
 3129.6 
 
 895 
 
 3314.8 
 
 945 
 
 snoo.o 
 
 995 
 
 3685.2 
 
 796 
 
 2918.2 
 
 846 
 
 3133.3 
 
 896 
 
 3318.5 
 
 946 
 
 3503.7 
 
 996 
 
 3688.9 
 
 797 
 
 2951.9 
 
 847 
 
 3137.0 
 
 897 
 
 3322.2 
 
 947 
 
 3507.4 
 
 997 
 
 3692.6 
 
 798 
 
 2955.6 
 
 848 
 
 3140.7 
 
 898 
 
 3325.9 
 
 948 
 
 3511.1 
 
 998 
 
 3696.3 
 
 799 
 
 2959.3 
 
 849 
 
 3144.4 
 
 899 
 
 3329.6 
 
 949 
 
 3514.8 
 
 999 
 
 3700.0 
 
 800 
 
 2963.0 
 
 850 
 
 3148.2 
 
 900 
 
 3333.3 
 
 950 
 
 3518.5 
 
 1000 
 
 3703.7 
 
 279 
 
TABLE XVI. 
 
 CONVERSION OF ENGLISH INCHES INTO CENTIMETRES. 
 
 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 
 
 81.28 
 
 83.82 
 
 86.36 
 
 88.90 
 
 91.44 
 
 93.98 
 
 96.52 
 
 99.06 
 
 40 
 
 101.60 
 
 104.14 
 
 106.08 
 
 109.22 
 
 111.76 
 
 114.30 
 
 116.84 
 
 119.38 
 
 121.92 
 
 124.46 
 
 50 
 
 127.00 
 
 129. 54 j 132.08 
 
 134.62 
 
 137.16 
 
 139.10 
 
 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 
 
 175.26 
 
 70 
 
 177.80 
 
 180.34 182.88 
 
 185.42 
 
 187.96 
 
 190.50 
 
 193.04 
 
 195.58 
 
 198.12 
 
 200.96 
 
 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 
 
 276.8,. 
 
 CONVERSION OF CENTIMETRES 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 
 
 4.742 5.118 
 
 5.512 
 
 5.906i 6.299 6.693 
 
 7.087 7.480 
 
 20 
 
 7.874 
 
 8.268 
 
 8.662 9.055 
 
 9.449 
 
 9.843110.236 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 15.355 
 
 40 
 50 
 
 15.748 
 19.685 
 
 16.142 
 20.079 
 
 16.530 16.929 
 20.473 20.867 
 
 17.323 
 21.260 
 
 17.717:18.111 18.504 18.898 19.292 
 21. 654122.048 22.441122.835 23.229 
 
 60 
 
 23.622 
 
 24.016 
 
 24.410 24.804 
 
 25.197 
 
 25.591 
 
 25 . 985 26 . 378 1 26 772 27 . 1 66 
 
 70 
 
 27.560 
 
 27.953 
 
 28.347 28.741 
 
 29.134 
 
 29.528 
 
 29 . 922 80. 316 80. 709 31 . 103 
 
 80 
 
 31.497 
 
 31.890! 32.284 32.678 
 
 33.071 
 
 33 . 465 33 . 8r>9 34 . 253 < 34 . 646 35 . 040 
 
 90 
 
 35.434 
 
 35. 827 i 36.221 36.615 
 
 37.009 
 
 37 . 402 ' 37 . 796 38 . 1 90 38 . 583 38 . 977 
 
 100 
 
 39.370| 39.764 40.158 40.552 
 
 40.945 
 
 41 .339J41 .733 42.126 42.520 42.914 
 
 CONVERSION OF ENGLISH FEET INTO METRES. 
 
 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 
 
 1.52391.82872.1335 
 
 2.4383 
 
 2.7431 
 
 10 
 
 3.0479 
 
 3.3527 
 
 3.6575 
 
 3.9623 
 
 4.2671 
 
 4. 5719J4. 8767:5.1815 
 
 5.4S63 
 
 5.7911 
 
 20 
 
 6.0359 
 
 6.4006 
 
 6.7055 
 
 7.0102 
 
 7.3150 
 
 7.61987.92468.2294 
 
 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 
 
 14.020 ! 14.325 
 
 14.630 
 
 14.935 
 
 50 
 
 15.239 
 
 15.544 
 
 15.849 
 
 16.154 
 
 16.459 
 
 16.763 
 
 17.068 17.373 
 
 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. 859 123.1 64 23. 469 
 
 23.774 
 
 24.079 
 
 80 
 
 24.383 
 
 24.688 
 
 24.993 
 
 25.298 
 
 25.602 
 
 25.907 
 
 26. 212 26. 517126. 822 
 
 27.126 
 
 90 
 
 27.431 
 
 27.736 
 
 28.041 
 
 28.346 
 
 28.651 
 
 28.955 
 
 29.26029.56529.870 
 
 30.174 
 
 100 
 
 30.479 
 
 30.784 
 
 31.089 
 
 31.394 
 
 31.698 
 
 32.003 32.308 32.613 32.918 
 
 33.222 
 
 CONVERSION OF METRES INTO ENGLISH FEET. 
 
 Met. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 B' 
 
 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 
 
 32.809 
 
 36.090 
 
 39.371 42.651 
 
 45.932 
 
 49. 213i52.494 
 
 55.775 
 
 59.056 
 
 62.337 
 
 20 
 
 65.618 
 
 68.899 
 
 72.179 75.461 
 
 78.741 
 
 82.02285.303 
 
 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 
 
 213.26(216.54 
 
 219.82 
 
 223.10 
 
 226.38 
 
 70 
 
 229.66 
 
 232.94 
 
 236.22 239.51 
 
 242.79 
 
 246.07249.35 
 
 252.63 
 
 255.91 
 
 259.19 
 
 8C 
 
 262.47 
 
 265.75 
 
 269.03 272.31 
 
 275.60 
 
 278.88282.16 
 
 ^85.44 
 
 288 72 
 
 293.00 
 
 90 
 
 295.28 
 
 298.56 
 
 391.84 305.12 
 
 308.40 
 
 311.69314.97 
 
 318.25 
 
 1.53 
 
 324.81 
 
 100 
 
 328.09 
 
 331.37 
 
 334.65, 337.93 
 
 341.21 
 
 344.49J 347. 78 
 
 351.06354.34 
 
 357.62 
 
 280 
 
TABLE XVH. 
 
 CONVERSION OF 
 
 ENGLISH STATUTE-MILES INTO KILOMETRES. 
 
 Miles. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 Kilo. 
 
 
 
 0.0000 
 
 1.6093 
 
 3.2186 
 
 4.82796.437* 
 
 8.0465 
 
 9.6558 
 
 11.2652 
 
 12.8745 
 
 14.4848 
 
 10 
 
 16.093 
 
 17.70219.312 
 
 20.921 22.531 
 
 24.139 
 
 25.749 
 
 27.358 
 
 28.967 
 
 30.577 
 
 20 
 
 32.186 
 
 33.795 
 
 35.405 37.014 
 
 38.62: 
 
 40.232 
 
 41.842 
 
 43.451 
 
 45.060 
 
 46.670 
 
 30 
 
 48.279 
 
 49.888 
 
 51.49853.107 
 
 54.7K 
 
 56.325 
 
 57.935 
 
 59.544 
 
 61.153 
 
 62.763 
 
 40 
 
 64.372 
 
 65.981 67.591 
 
 69.200 
 
 70.80$ 
 
 1 72.418 
 
 74.028 
 
 75.637 
 
 77.246 
 
 78.856 
 
 50 
 
 80.465 
 
 82.074 
 
 83.684 
 
 85.293 
 
 86.90; 
 
 5 88.511 
 
 90.121 
 
 91 730 
 
 93.339 
 
 94.949 
 
 60 
 
 96.558:98.16799.777 
 
 101.39 
 
 102. 9< 
 
 ) 104.60 
 
 106.21 
 
 107.82 
 
 109.43 
 
 111.04 
 
 70 
 
 112.65 
 
 114.26 
 
 115.87 
 
 117.48 
 
 119. 0* 
 
 5 120.69 
 
 122.30 
 
 123.91 
 
 125.52 
 
 127.13 
 
 80 
 
 128.74130.35131.96 
 
 133.57 
 
 135. 1' 
 
 r 136.78 
 
 138.39 
 
 140.00 
 
 141.61 
 
 143.22 
 
 90 
 
 144.85 
 
 146.44 
 
 148.05 
 
 149.66 
 
 151. 2( 
 
 5 152.87 
 
 154.48 
 
 156.09 
 
 157.70 
 
 159.31 
 
 100 
 
 100.93162.53164 14 
 
 165 75 
 
 167. & 
 
 > 168.96 
 
 170.57 
 
 172.18 
 
 173.79 
 
 175.40 
 
 CONVERSION OF 
 
 KILOMETRES INTO ENGLISH STATUTE-MILES. 
 
 Kilom. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 Miles. 
 
 Miles. 
 
 Miles. 
 
 Miles. 
 
 Miles 
 
 . Miles. 
 
 Miles. 
 
 Miles. 
 
 Miles. 
 
 Miles. 
 
 
 
 0.0000 
 
 0.6214 
 
 1.2427 
 
 1.8641 
 
 2.485 
 
 5 3.1069 
 
 3.7282 
 
 4.3497 
 
 4.9711 
 
 5.5924 
 
 10 
 
 6.2138 
 
 6.8352 
 
 7.4565 
 
 8.0780 
 
 8.699< 
 
 I 9.3208 
 
 9.9421 
 
 10.562 
 
 11.185 
 
 11.805 
 
 20 
 
 12.427 
 
 13.049 
 
 13.670 
 
 14.292 
 
 14.91, 
 
 5 15.534 
 
 16.156 
 
 16.776 
 
 17.399 
 
 18.019 
 
 30 
 
 18.641 
 
 19.263 
 
 19.884 
 
 20.506 
 
 21.12 
 
 r 21.748 
 
 23.370 
 
 22.990 
 
 23.613 
 
 24.233 
 
 40 
 
 24.855 
 
 25.477 
 
 26.098 
 
 26.720 
 
 27.34 
 
 27.962 
 
 28.584 
 
 29.204 
 
 29.827 
 
 30.447 
 
 50 
 
 31.069 
 
 31.690 
 
 32.311 
 
 32.933 
 
 33.55 
 
 I 34.175 
 
 34.797 
 
 35.417 
 
 36.040 
 
 36.660 
 
 60 
 
 37.282 
 
 37.904 
 
 38.525 
 
 39.147 
 
 39.76* 
 
 J 40.389 
 
 41.011 
 
 41.631 
 
 42.254 
 
 42.874 
 
 70 
 
 43.497 
 
 44.118 
 
 44.789 
 
 45.361 
 
 45.98- 
 
 I 46.603 
 
 47.225 
 
 47.845 
 
 48.468 
 
 49.088 
 
 80 
 
 49.711 
 
 50.332 
 
 50.953 
 
 51.575 
 
 52.19 
 
 5 52.817 
 
 53.439 
 
 54.059 
 
 54.682 
 
 55.302 
 
 90 
 
 55.924 
 
 56.545 
 
 57.166 
 
 57.788 
 
 58.40 
 
 ) 59.030 
 
 59.652 
 
 60.272 
 
 60.895 
 
 61.515 
 
 100 
 
 62.138 
 
 62.75963.380 
 
 64.002 
 
 64.62 
 
 i 65.244 
 
 65.866 
 
 66.486 
 
 67.109 
 
 67.729 
 
 
 
 TABLE XVIII. 
 
 LENGTH IN FEET OF 1' ARCS OF LATITUDE AND LONGITUDE. 
 
 Lat. 
 
 1' Lat. 
 
 V Long. 
 
 Lat. 
 
 1' Lat. 
 
 V Long. 
 
 1 
 
 
 6045 
 
 
 6085 
 
 31 
 
 6061 
 
 5222 
 
 2 
 
 
 6045 
 
 
 6083 
 
 32 
 
 6062 
 
 5166 
 
 3 
 
 
 6045 
 
 
 6078 
 
 33 N 
 
 6063 
 
 5109 
 
 40 
 
 
 6045 
 
 
 6071 
 
 34 
 
 6064 
 
 5051 
 
 5 
 
 
 6045 
 
 
 6063 
 
 35 
 
 6065 
 
 4991 
 
 6 
 
 
 6045 
 
 
 6053 
 
 36 
 
 6066 
 
 4930 
 
 7 
 
 
 6046 
 
 
 6041 
 
 37 
 
 6067 
 
 4867 
 
 8 
 
 
 6046 
 
 
 6027 
 
 38 
 
 6068 
 
 4802 
 
 9 
 
 
 6046 
 
 
 6012 
 
 39 
 
 6070 
 
 4736 
 
 10 
 
 
 6047 
 
 
 5994 
 
 40 
 
 6071 
 
 4669 
 
 11 
 
 
 6047 
 
 
 5975 
 
 41 
 
 6072 
 
 4600 
 
 12 
 
 
 6048 
 
 
 5954 
 
 42 
 
 6073 
 
 4530 
 
 13 
 
 
 6048 
 
 
 5931 
 
 43 
 
 6074 
 
 4458 
 
 14 
 
 
 6049 
 
 
 5907 
 
 44 
 
 6075 
 
 4385 
 
 15 
 
 
 6049 
 
 
 5880 
 
 45 
 
 6076 
 
 4311 
 
 16 
 
 i 
 
 6050 
 
 
 5852 
 
 46 
 
 6077 
 
 4235 
 
 17 
 
 
 6050 
 
 
 5822 
 
 47 
 
 6078 
 
 4158 
 
 18 
 
 
 6051 
 
 
 5790 
 
 48 
 
 6079 
 
 4080 
 
 19 
 
 
 6052 
 
 
 - 5757 
 
 49 
 
 6080 
 
 4001 
 
 20 
 
 
 6052 
 
 
 5721 
 
 50 
 
 6081 
 
 3920 
 
 21 
 
 
 6053 
 
 
 5684 
 
 51 
 
 6082 
 
 3838 
 
 22 
 
 
 6C54 
 
 
 5646 
 
 52 
 
 6084 
 
 3755 
 
 23 
 
 
 6054 
 
 
 5605 
 
 53 
 
 6085 
 
 3671 
 
 24 
 
 
 6055 
 
 
 5563 
 
 54 
 
 6086 
 
 3586 
 
 25 
 
 
 8056 
 
 
 519 
 
 55 
 
 6087 
 
 3499 
 
 26 
 
 
 6057 
 
 
 5474 
 
 56 
 
 6088 
 
 3413 
 
 27 
 
 
 6058 
 
 
 5427 
 
 57 
 
 6089 
 
 3323 
 
 28 
 
 
 6059 
 
 
 5378 
 
 58 
 
 6090 
 
 3233 
 
 29 
 
 
 6060 
 
 
 5327 
 
 59 
 
 6091 
 
 3142 
 
 30 
 
 
 6061 
 
 5275 I 
 
 60 6092 
 
 3051 
 
 281 
 
EXAMPLE ILLUSTRATING USE OF TABLE XIX. 
 
 Find the horizontal distance and the difference of level when 
 n= 16 30', ak= 580 feet, and the instrumental constant c= .75. 
 In column headed 16 opposite 30' in the series for " Horizontal 
 
 TABLE XIX. 
 
 SHOWING HORIZONTAL DISTANCES AND DIFFERENCES 
 LEVELS FOR STADIA MEASUREMENTS. 
 
 M. 
 
 
 
 
 1 
 
 
 2 
 
 
 3 
 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 O'.... 
 
 100.00 
 
 .00 
 
 99.97 
 
 1.74 
 
 99.88 
 
 3.49 
 
 99.73 
 
 5.23 
 
 2 
 
 
 .06 
 
 
 1.80 
 
 99.87 
 
 3.55 
 
 99.72 
 
 5.28 
 
 4 
 
 " 
 
 .1? 
 
 M 
 
 1.86 
 
 u 
 
 3.60 
 
 99.71 
 
 5.34 
 
 6 
 
 
 
 .17 
 
 99..96 
 
 1.92 
 
 u 
 
 3.66 
 
 " 
 
 5.40 
 
 P 
 
 " 
 
 .23 
 
 
 1.98 
 
 99.86 
 
 3.72 
 
 99.70 
 
 5.46 
 
 10 
 
 " 
 
 .29 
 
 *' 
 
 2.04 
 
 " 
 
 3.78 
 
 99.69 
 
 5.52 
 
 12 
 
 it 
 
 .35 
 
 M 
 
 2.09 
 
 99.85 
 
 3.84 
 
 u 
 
 5.57 
 
 14 
 
 M 
 
 .41 
 
 99.95 
 
 2.15 
 
 M 
 
 3.90 
 
 99.68 
 
 5.63 
 
 16 
 
 " 
 
 .47 
 
 
 2.21 
 
 99.84 
 
 3.95 
 
 u 
 
 5.69 
 
 18 
 
 M 
 
 .52 
 
 *' 
 
 2.27 
 
 
 
 4.01 
 
 99.67 
 
 5.75 
 
 20 
 
 II 
 
 .58 
 
 " 
 
 2.33 
 
 99.83 
 
 4.07 
 
 99.66 
 
 5.80 
 
 22... 
 
 M 
 
 .64 
 
 99 94 
 
 2.38 
 
 H 
 
 4.13 
 
 M 
 
 5.86 
 
 24 
 
 < 
 
 .70 
 
 H 
 
 2.44 
 
 99.82 
 
 4.18 
 
 99.65 
 
 5.92 
 
 26 
 
 99.^99 
 
 .76 
 
 II 
 
 2.50 
 
 M 
 
 4.24 
 
 99.64 
 
 5.98 
 
 28 
 
 
 .81 
 
 99.93 
 
 2.56 
 
 99.81 
 
 4.30 
 
 99.63 
 
 6.04 
 
 80 
 
 " 
 
 .87 
 
 u 
 
 2.62 
 
 
 4.36 
 
 H 
 
 6.09 
 
 82 
 
 
 
 .93 
 
 U 
 
 2.67 
 
 99.80 
 
 4.42 
 
 99.62 
 
 6.15 
 
 84 
 
 M 
 
 .99 
 
 U 
 
 2.73 
 
 
 4.48 
 
 u 
 
 6.21 
 
 36... . 
 
 
 
 .05 
 
 99.92 
 
 2.79 
 
 99.J9 
 
 4.53 
 
 99.61 
 
 6.27 
 
 38 
 
 M 
 
 .11 
 
 M 
 
 2.85 
 
 
 4.59 
 
 99.60 
 
 6.33 
 
 40 
 
 " 
 
 .16 
 
 " 
 
 2.91 
 
 99.78 
 
 4.65 
 
 99.59 
 
 6.38 
 
 42 
 
 M 
 
 .22 
 
 99.91 
 
 2.97 
 
 H 
 
 4.71 
 
 u 
 
 6.44 
 
 44 
 
 99.98 
 
 .28 
 
 M 
 
 3.02 
 
 99.77 
 
 4.76 
 
 99.58 
 
 6.50 
 
 46 
 
 
 .34 
 
 99.90 
 
 3.08 
 
 
 4.82 
 
 99.57 
 
 6.56 
 
 48 
 
 
 
 .40 
 
 
 3.14 
 
 99.76 
 
 4.88 
 
 99.56 
 
 6.61 
 
 60...., 
 
 
 
 .45 
 
 H 
 
 3.20 
 
 
 4.94 
 
 " 
 
 6.67 
 
 62 
 
 H 
 
 .51 
 
 99.89 
 
 3.26 
 
 99.75 
 
 4.99 
 
 99.55 
 
 6.73 
 
 54 
 
 
 
 .57 
 
 M 
 
 3.31 
 
 99.74 
 
 5.05 
 
 99.54 
 
 6.78 
 
 56 
 
 99.97 
 
 .63 
 
 u 
 
 3.37 
 
 
 5.11 
 
 99.53 
 
 6.84 
 
 58... 
 
 rt 
 
 .69 
 
 99.88 
 
 3.43 
 
 99.73 
 
 5.17 
 
 99.52 
 
 6.90 
 
 60 
 
 " 
 
 .74 
 
 M 
 
 3.49 
 
 
 5.23 
 
 99.51 
 
 6.96 
 
 c= .75 
 
 .75 
 
 .01 
 
 .75 
 
 .02 
 
 .75 
 
 .03 
 
 .75 
 
 .05 
 
 c=1.00 
 
 1.00 
 
 .01 
 
 1.00 
 
 .03 
 
 1.00 
 
 .04 
 
 1.00 
 
 .06 
 
 0=1.25 
 
 1.25 
 
 .02 
 
 1.25 
 
 .03 
 
 1.25 
 
 .05 
 
 1.25 
 
 .08 
 
 From Winslow's "Stadia Surveying." D. Van Nostrantfs Science Series. 
 282 
 
Distances," we find 91.93 as the expression for ak cos 2 when 
 ak= 100 ; therefore, when ak= 580, alrcosPn = 91.93 x 5.80 = 533.19. 
 Add to this c cos n from value of c at bottom of page, and we have 
 533.19 + .72 = 533.91, hor. dist. Similarly, 27.23 x 5.80 + .21 = 157.93, 
 diff . level. 
 
 TABLE XIX. 
 
 STADIA MEASUBEMENTS. 
 
 M. 
 
 4 
 
 5 
 
 6 
 
 7* 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 (X .... 
 
 99.51 
 
 6.96 
 
 99.24 
 
 8.68 
 
 98.91 
 
 1C. 40 
 
 98.51 
 
 12.10 
 
 2 
 
 H 
 
 7.02 
 
 99.23 
 
 8.74 
 
 9G.9C 
 
 10.45 
 
 98.50 
 
 12.15 
 
 4 
 
 99.50 
 
 7.07 
 
 99.22 
 
 8.80 
 
 98.88 
 
 10.U 
 
 98.48 
 
 12.21 
 
 6 
 
 99.49 
 
 7.13 
 
 99.21 
 
 8.85 
 
 98.87 
 
 10.57 
 
 98.47 
 
 12.26 
 
 8 
 
 99.48 
 
 7.10 
 
 99.20 
 
 8.91 
 
 98.86 
 
 10.62 
 
 98.46 
 
 12.32 
 
 10 
 
 99.47 
 
 7.25 
 
 99.19 
 
 8.97 
 
 98.85 
 
 10.68 
 
 98.44 
 
 12.38 
 
 12 
 
 99.46 
 
 7.30 
 
 99.18 
 
 9.03 
 
 98.83 
 
 10.74 
 
 98.43 
 
 12.43 
 
 14 
 
 
 7. 30 
 
 99.17 
 
 9.08 
 
 98.82 
 
 10-79 
 
 98.41 
 
 12.49 
 
 16 
 
 99.45 
 
 7.42 
 
 99.16 
 
 9.14 
 
 98.81 
 
 10.85 
 
 98.40 
 
 12.55 
 
 18 
 
 yy.44 
 
 7.48 
 
 99.15 
 
 9.20 
 
 98.80 
 
 10.91 
 
 98.39 
 
 12.60 
 
 20 
 
 99.43 
 
 7.53 
 
 99.14 
 
 9.25 
 
 98.78 
 
 10.96 
 
 98.37 
 
 12.66 
 
 22 
 
 99.42 
 
 7.59 
 
 99.13 
 
 9.31 
 
 98.77 
 
 11.02 
 
 98.36 
 
 12.72 
 
 24 
 
 99.41 
 
 7.65 
 
 99.11 
 
 9.37 
 
 98.76 
 
 11.08 
 
 98.34 
 
 12.77 
 
 26 
 
 99.40 
 
 7.71 
 
 99.10 
 
 9.43 
 
 98.74 
 
 11.13 
 
 98.33 
 
 12.83 
 
 28 
 
 99.39 
 
 7.76 
 
 99.09 
 
 9.48 
 
 98.73 
 
 11.19 
 
 98.31 
 
 12.88 
 
 80 
 
 99.38 
 
 7.82 
 
 99.08 
 
 9.54 
 
 98.72 
 
 11.25 
 
 98.29 
 
 12.94 
 
 82... 
 
 99.38 
 
 7.88 
 
 99.07 
 
 9.60 
 
 98.71 
 
 11 .30 
 
 98.28 
 
 13.00 
 
 84 
 
 99.37 
 
 7.94 
 
 99.06 
 
 9.65 
 
 98.69 
 
 11.36 
 
 98.27 
 
 13.05 
 
 86 
 
 99.36 
 
 7.99 
 
 99.05 
 
 9.71 
 
 98.68 
 
 11.42 
 
 98.25 
 
 13.11 
 
 88 
 
 99.35 
 
 8.05 
 
 09.04 
 
 9.77 
 
 98.67 
 
 11.47 
 
 98.24 
 
 13.17 
 
 40 
 
 99.34 
 
 8.11 
 
 99.03 
 
 9.83 
 
 98.65 
 
 11.53 
 
 98.22 
 
 13.22 
 
 42... 
 
 99.33 
 
 8.17 
 
 99.01 
 
 9.88 
 
 98.64 
 
 11.59 
 
 98.20 
 
 13.28 
 
 44.... 
 
 99.32 
 
 8.22 
 
 99.00 
 
 9.94 
 
 98.63 
 
 11.64 
 
 98.19 
 
 13.33 
 
 46 
 
 99.31 
 
 8.28 
 
 98.99 
 
 10.00 
 
 98.61 
 
 11.70 
 
 98.17 
 
 13.39 
 
 48 
 
 99.30 
 
 8.34 
 
 98.98 
 
 10.05 
 
 98.60 
 
 11.76 
 
 98.16 
 
 13.45 
 
 50 
 
 99.29 
 
 8.40 
 
 98.97 
 
 10.11 
 
 98.58 
 
 11.81 
 
 98.14 
 
 13.50 
 
 52 
 
 99.28 
 
 8.45 
 
 98.96 
 
 10.17 
 
 98.57 
 
 11.87 
 
 98.13 
 
 13.58 
 
 54 
 
 99.27 
 
 8.51 
 
 98.94 
 
 10.22 
 
 98.56 
 
 11.93 
 
 98.11 
 
 13.61 
 
 56 
 
 99.26 
 
 8.57 
 
 98.93 
 
 10.28 
 
 98.54 
 
 11.98 
 
 98.10 
 
 13.67 
 
 58 
 
 99.25 
 
 8.63 
 
 98.92 
 
 10.34 
 
 98.53 
 
 12.04 
 
 98.08 
 
 13.73 
 
 60 
 
 99.24 
 
 8.68 
 
 98.91 
 
 10.40 
 
 98.51 
 
 12.10 
 
 98.06 
 
 13.78 
 
 c= .75 
 
 .75 
 
 .06 
 
 .75 
 
 .07 
 
 .75 
 
 .08 
 
 .74 
 
 .10 
 
 c=1.00 
 
 1.00 
 
 .08 
 
 .99 .09 
 
 .99 .11 
 
 .99 
 
 .13 
 
 c=1.25 
 
 1.25 
 
 .10 
 
 1.24 
 
 .11 
 
 1.24 
 
 .14 
 
 1.24 
 
 .16 
 
 FromWinslow's "Stadia Surveying" 1). Van ivostrancCs Science Series. 
 
 283 
 
TABLE XIX. 
 
 STADIA MEASUREMENTS. 
 
 M. 
 
 8 
 
 9 
 
 10 
 
 11 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 0'.... 
 
 98.06 
 
 13.78 
 
 97.55 
 
 15.45 
 
 96.98 
 
 17.10 
 
 96.36 
 
 18 73 
 
 2 
 
 98.05 
 
 13.84 
 
 97.53 
 
 15.51 
 
 96.96 
 
 17.16 
 
 96.34 
 
 18.78 
 
 4 
 
 98.03 
 
 13.89 
 
 97.52 
 
 15.56 
 
 96.94 
 
 17.21 
 
 96.32 
 
 IS. 84 
 
 6.... 
 
 98.01 
 
 13.95 
 
 97.50 
 
 15.62 
 
 96.92 
 
 17.26 
 
 96.29 
 
 18 89 
 
 8 
 
 98.00 
 
 14.01 
 
 97.48 
 
 15. f7 
 
 96.90 
 
 17.32 
 
 96.27 
 
 18.95 
 
 10 
 
 97.98 
 
 14.06 
 
 97.46 
 
 15/3 
 
 96.88 
 
 17.37 
 
 96.25 
 
 19 00 
 
 12 
 
 97.97 
 
 14.12 
 
 97.44 
 
 15 .\ 3 
 
 96.86 
 
 17.43 
 
 96.23 
 
 19.05 
 
 14 
 
 97.95 
 
 14.17 
 
 97.43 
 
 15.84 
 
 96.84 
 
 17.48 
 
 96.21 
 
 19.11 
 
 16 
 
 97.93 
 
 14.23 
 
 97.41 
 
 15 89 
 
 96.82 
 
 17.54 
 
 96.18 
 
 19.16 
 
 18 
 
 97.92 
 
 14.28 
 
 97.39 
 
 15.95 
 
 96.80 
 
 17.59 
 
 96.16 
 
 19.21 
 
 20 
 
 97.90 
 
 14.34 
 
 97 37 
 
 16 00 
 
 96.78 
 
 17.65 
 
 96.14 
 
 19 27 
 
 22 
 
 97.88 
 
 14.40 
 
 97.35 
 
 16.06 
 
 96.76 
 
 17.70 
 
 96.12 
 
 19.32 
 
 24 
 
 97.87 
 
 14.45 
 
 97.33 
 
 16.11 
 
 96.74 
 
 17.76 
 
 96.09 
 
 19.38 
 
 26 
 
 97.85 
 
 14.51 
 
 97.31 
 
 16.17 
 
 96.72 
 
 17.81 
 
 96.07 
 
 19.43 
 
 28.... 
 
 97.83 
 
 14.56 
 
 97.29 
 
 16.22 
 
 96.70 
 
 17.86 
 
 96.05 
 
 19.48 
 
 80 
 
 97.82 
 
 14.62 
 
 97.88 
 
 16. 2d 
 
 96.68 
 
 17.92 
 
 96.03 
 
 19.54 
 
 32... 
 
 97.80 
 
 14.67 
 
 97.26 
 
 16.33 
 
 96.66 
 
 17.97 
 
 96.00 
 
 19.59 
 
 84 
 
 97.78 
 
 14.73 
 
 97.24 
 
 16.39 
 
 96.64 
 
 18.03 
 
 95.98 
 
 19.64 
 
 36 
 
 97.76 
 
 14.79 
 
 97.22 
 
 16.44 
 
 96.62 
 
 18.08 
 
 95.96 
 
 19.70 
 
 38 
 
 97.75 
 
 14.84 
 
 97.20 
 
 16. 50 
 
 96.60 
 
 18.14 
 
 95.93 
 
 19.75 
 
 40 
 
 97.73 
 
 14.90 
 
 97.18 
 
 16.55 
 
 96.57 
 
 18.19 
 
 95.91 
 
 19.80 
 
 42 
 
 97.71 
 
 14. 95 
 
 97.16 
 
 16.61 
 
 96.55 
 
 18.24 
 
 95.89 
 
 19.86 
 
 44. ... 
 
 97.69 
 
 1501 
 
 9T.14 
 
 16.66 
 
 96.53 
 
 18.30 
 
 95.86 
 
 19.91 
 
 46 
 
 97.68 
 
 15.06 
 
 97.12 
 
 16.72 
 
 96.51 
 
 18.35 
 
 95.84 
 
 19.96 
 
 48.... 
 
 97.66 
 
 15.12 
 
 97.10 
 
 16.77 
 
 96.49 
 
 18.41 
 
 95.82 
 
 20.02 
 
 50 
 
 97.64 
 
 15.17 
 
 97.08 
 
 16.83 
 
 96.47 
 
 18.46 
 
 95.79 
 
 20.07 
 
 52 
 
 97.62 
 
 15.23 
 
 97.06 
 
 16.88 
 
 96.45 
 
 18.51 
 
 95.77 
 
 20.12 
 
 54 
 
 97.61 
 
 15.28 
 
 97.04 
 
 16.94 
 
 96.42 
 
 18.57 
 
 95.75 
 
 20.18 
 
 56 .... 
 
 97.59 
 
 15.34 
 
 97.02 
 
 16.99 
 
 96.40 
 
 18.62 
 
 95.72 
 
 20.23 
 
 58... 
 
 97.57 
 
 15.40 
 
 97.00 
 
 17.05 
 
 96.38 
 
 18.68 
 
 95.70 
 
 20.28 
 
 60 
 
 97.55 
 
 15.45 
 
 96.98 
 
 17.10 
 
 96.36 
 
 18.73 
 
 95.68 
 
 20.34 
 
 c= .75 
 
 .74 
 
 .11 
 
 .74 
 
 .12 
 
 .74 
 
 .14 
 
 .73 
 
 .15 
 
 <r=1.00 
 
 .99 .15 
 
 .99 .16 
 
 .98 
 
 .18 
 
 .98 
 
 .20 
 
 c=1.25 
 
 1.23 
 
 .18 
 
 1.23 
 
 .21 
 
 1.23 .23 
 
 1.22 
 
 .36 
 
 FromWinslow's "Stadia Surveying." D.VanNostrantfs Science Seriet. 
 284 
 
TABLE XIX. 
 
 STADIA MEASUREMENTS. 
 
 M. 
 
 12 
 
 13 
 
 14 
 
 15 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 
 Disft. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 0' .. 
 
 95.68 
 
 20.34 
 
 94.94 
 
 21.92 
 
 94.15 
 
 23.47 
 
 93.30 
 
 25.00 
 
 2 
 
 95.65 
 
 20.39 
 
 94.91 
 
 21.97 
 
 94.12 
 
 23.52 
 
 93.27 
 
 25.05 
 
 4 
 
 95.63 
 
 20.44 
 
 94.89 
 
 22.02 
 
 94.09 
 
 23.58 
 
 93.24 
 
 25.10 
 
 6 
 
 95.61 
 
 20.50 
 
 94 86 
 
 22.08 
 
 94.07 
 
 23.63 
 
 93.21 
 
 25.15 
 
 8 
 
 95.58 
 
 20.55 
 
 94.84 
 
 22.13 
 
 94.04 
 
 23.68 
 
 93.18 
 
 25.20 
 
 10 
 
 95.56 
 
 20.60 
 
 94.81 
 
 22.18 
 
 94.01 
 
 23.73 
 
 93.16 
 
 25.25 
 
 12 
 
 95.53 
 
 20.66 
 
 94.79 
 
 22.23 
 
 93.98 
 
 23.78 
 
 93.13 
 
 25.30 
 
 14 
 
 95.51 
 
 20.71 
 
 94.76 
 
 22.28 
 
 93.95 
 
 23.83 
 
 93.10 
 
 25.35 
 
 16 
 
 95.49 
 
 20.76 
 
 94.73 
 
 22.34 
 
 93.93 
 
 23.88 
 
 93.07 
 
 25.40 
 
 18. ... 
 
 95.46 
 
 20.81 
 
 94.71 
 
 22.39 
 
 93.90 
 
 23.93 
 
 93.04 
 
 25.45 
 
 20 
 
 95.44 
 
 20.87 
 
 94.68 
 
 22.44 
 
 93.87 
 
 23.99 
 
 93.01 
 
 25.50 
 
 22 
 
 95.41 
 
 20.92 
 
 94.66 
 
 22.49 
 
 93.84 
 
 24.04 
 
 92.98 
 
 25.55 
 
 24 
 
 95.39 
 
 20.97 
 
 94.63 
 
 22.54 
 
 93.81 
 
 24.09 
 
 92.95 
 
 25.60 
 
 26 
 
 95.36 
 
 21.03 
 
 94.60 
 
 22.60 
 
 93.79 
 
 24.14 
 
 92.92 
 
 25.65 
 
 28. ... 
 
 95.34 
 
 21.08 
 
 94.58 
 
 22.65 
 
 93.76 
 
 24.19 
 
 92.89 
 
 25.70 
 
 30 
 
 95.32 
 
 21.13 
 
 94.55 
 
 22.70 
 
 93.73 
 
 24.24 
 
 92.86 
 
 25.75 
 
 32 
 
 95.29 
 
 21.18 
 
 94.52 
 
 22.75 
 
 93.70 
 
 24.29 
 
 92.83 
 
 25.80 
 
 34 
 
 95.27 
 
 21.24 
 
 94.50 
 
 22.80 
 
 93.67 
 
 24.84 
 
 92.80 
 
 25.85 
 
 86 
 
 95.24 
 
 21.29 
 
 94.47 
 
 22.85 
 
 93.65 
 
 24.39 
 
 92.77 
 
 25.90 
 
 38 
 
 95.22 
 
 21.34 
 
 94.44 
 
 22.91 
 
 93.62 
 
 24.44 
 
 92.74 
 
 25.95 
 
 40 
 
 95.19 
 
 21.39 
 
 94.42 
 
 22.96 
 
 93.59 
 
 24.49 
 
 92.71 
 
 26.00 
 
 42 
 
 95.17 
 
 21.45 
 
 94.39 
 
 23.01 
 
 93.56 
 
 24.55 
 
 92.68 
 
 26.05 
 
 44 
 
 95.14 
 
 21.50 
 
 94.36 
 
 23.06 
 
 93.53 
 
 24.60 
 
 92.65 
 
 26.10 
 
 46 
 
 95.12 
 
 21.55 
 
 94.34 
 
 23.11 
 
 93.50 
 
 24.65 
 
 92.62 
 
 26.15 
 
 48 
 
 95.09 
 
 21.60 
 
 94.31 
 
 23.16 
 
 93.47 
 
 24.70 
 
 92.59 
 
 26.20 
 
 50 
 
 95.07 
 
 21.66 
 
 94.28 
 
 23.22 
 
 93.45 
 
 24.75 
 
 92.56 
 
 26.25 
 
 52... 
 
 95.04 
 
 21.71 
 
 94.26 
 
 23.27 
 
 93.42 
 
 24.80 
 
 92.53 
 
 26.30 
 
 54 
 
 95.02 
 
 21.76 
 
 94.23 
 
 23.32 
 
 93.39 
 
 24.85 
 
 92.49 
 
 26.35 
 
 56 
 
 94.99 
 
 21.81 
 
 94.20 
 
 23.37 
 
 93.36 
 
 24.90 
 
 92.46 
 
 26.40 
 
 58 
 
 94.97 
 
 21.87 
 
 94.17 
 
 23.43 
 
 93.33 
 
 24.95 
 
 92.43 
 
 26.45 
 
 60.... 
 
 94.94 
 
 21.92 
 
 94.15 
 
 23.47 
 
 93.30 
 
 25.00 
 
 92.40 
 
 26.50 
 
 c= .75 
 
 .73 
 
 .16 
 
 .73 .17 
 
 .73 
 
 .19 
 
 .72 .20 
 
 0=100 
 
 .98 .22 
 
 .97 .23 
 
 .97 .25 
 
 .96 .27 
 
 C=1.25 
 
 1.22 
 
 .27 
 
 1.21 
 
 .20 
 
 1.21 
 
 .81 
 
 1.20 
 
 .34 
 
 "Stadia Surveying" D. ran Nostr and' s Science Series. 
 285 
 
TABLE XIX. 
 STADIA MEASUKEMENTS. 
 
 M. 
 
 16 
 
 17 
 
 18 
 
 19 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Sor. 
 
 Diff. 
 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 ist. 
 
 Elev. 
 
 (X.... 
 
 92.40 
 
 26.50 
 
 91.45 
 
 27.96 
 
 90.45 
 
 29.39 
 
 89.40 
 
 30.78 
 
 2 
 
 92.37 
 
 26.55 
 
 91.42 
 
 28.01 
 
 90.42 
 
 29.44 
 
 89.36 
 
 30.83 
 
 4 
 
 92.34 
 
 26.59 
 
 91.39 
 
 28.06 
 
 90.38 
 
 29.48 
 
 89.33 
 
 30.87 
 
 6 
 
 92.31 
 
 26.64 
 
 91.35 
 
 28.10 
 
 90.35 
 
 29.53 
 
 89.29 
 
 30.92 
 
 8 
 
 92.28 
 
 26.69 
 
 91.32 
 
 28.15 
 
 90.31 
 
 29.58 
 
 89.26 
 
 30.97 
 
 10 
 
 92.25 
 
 26.74 
 
 91.29 
 
 28.20 
 
 90.28 
 
 29.62 
 
 89.22 
 
 31.01 
 
 12 
 
 92.22 
 
 26.79 
 
 91.26 
 
 28.25 
 
 90.24 
 
 29.67 
 
 89.18 
 
 31.06 
 
 14 
 
 92.19 
 
 26.84 
 
 91.22 
 
 28.30 
 
 90.21 
 
 29.72 
 
 89.15 
 
 31.10 
 
 16 
 
 92.15 
 
 26.89 
 
 91.19 
 
 28.34 
 
 90.18 
 
 29.76 
 
 89.11 
 
 31.15 
 
 18 
 
 92.12 
 
 26.94 
 
 91.16 
 
 28.39 
 
 90.14 
 
 29.81 
 
 89.08 
 
 31.19 
 
 20 
 
 92.09 
 
 26.99 
 
 91.12 
 
 28.44 
 
 90.11 
 
 29.86 
 
 89.04 
 
 31.24 
 
 28 
 
 92.06 
 
 27.04 
 
 91.09 
 
 28.49 
 
 90.07 
 
 29.90 
 
 89.00 
 
 31.28 
 
 24 
 
 92.03 
 
 27.09 
 
 91.06 
 
 28.54 
 
 90.04 
 
 29.95 
 
 88.96 
 
 31.33 
 
 26 
 
 92.00 
 
 27.13 
 
 91.02 
 
 28.58 
 
 90.00 
 
 30.00 
 
 88.93 
 
 31.38 
 
 28... 
 
 91.97 
 
 27.18 
 
 90.99 
 
 28.63 
 
 89.97 
 
 30.04 
 
 88 89 
 
 31.42 
 
 30 
 
 91.93 
 
 27.23 
 
 90.96 
 
 28.68 
 
 89.93 
 
 30.09 
 
 88.86 
 
 31.47 
 
 32... 
 
 91.90 
 
 27.28 
 
 90.92 
 
 28.73 
 
 89.90 
 
 30.14 
 
 88.82 
 
 31.51 
 
 34 
 
 91.87 
 
 27.33 
 
 90.89 
 
 28.77 
 
 89.86 
 
 30.19 
 
 88.78 
 
 31.56 
 
 36... 
 
 91.84 
 
 27.38 
 
 90 86 
 
 28.82 
 
 89.83 
 
 30.23 
 
 88.75 
 
 31.60 
 
 38 
 
 91.81 
 
 27.43 
 
 90.82 
 
 28.87 
 
 89.79 
 
 30.28 
 
 88.71 
 
 31.65 
 
 40 
 
 91.77 
 
 27.48 
 
 9C-79 
 
 28.92 
 
 89.76 
 
 30.32 
 
 88.67 
 
 31 69 
 
 42 
 
 91.74 
 
 27.52 
 
 90.76 
 
 28.96 
 
 89.72 
 
 30.37 
 
 88.64 
 
 31.74 
 
 44.... 
 
 91.71 
 
 27.57 
 
 90.72 
 
 29.01 
 
 89.69 
 
 30.41 
 
 88.60 
 
 31.78 
 
 46 
 
 91.68 
 
 27.62 
 
 90.69 
 
 29.06 
 
 89.65 
 
 30.46 
 
 88.56 
 
 31.83 
 
 48. 
 
 91.65 
 
 27.67 
 
 90.66 
 
 29.11 
 
 89.61 
 
 30.51 
 
 88.53 
 
 31.87 
 
 50 
 
 91.61 
 
 27.72 
 
 90.62 
 
 29.15 
 
 89.58 
 
 30.55 
 
 88.49 
 
 31.92 
 
 52 
 
 91.58 
 
 27.77 
 
 90.59 
 
 29.20 
 
 89.54 
 
 30.60 
 
 88.45 
 
 31.96 
 
 54 
 
 91.55 
 
 27.81 
 
 90.55 
 
 29.25 
 
 89.51 
 
 30.65 
 
 88.41 
 
 32.01 
 
 56 
 
 91.52 
 
 27.86 
 
 90.52 
 
 29.30 
 
 89.47 
 
 30.69 
 
 88.38 
 
 32.05 
 
 58 
 
 91.48 
 
 27.91 
 
 90.48 
 
 29.34 
 
 89.44 
 
 30.74 
 
 88.34 
 
 32.09 
 
 60 
 
 91.45 
 
 27.96 
 
 90.45 
 
 29.39 
 
 89.40 
 
 30.78 
 
 88.30 
 
 32.14 
 
 c= .75 
 
 .72 
 
 .21 
 
 .72 .23 
 
 .71 .24 
 
 .71 .25 
 
 <?=1.00 
 
 .96 .28 
 
 .95 
 
 .30 
 
 .95 .32 
 
 .94 .33 
 
 c=1.25 
 
 1.20 .36 
 
 1.19 .38 
 
 1.19 
 
 .40 
 
 1.18 
 
 .42 
 
 From Winslow's "Stadia Surveying." D. Van Xostrand's Science Series. 
 286 
 
TABLE XIX. 
 
 STADIA MEASUREMENTS. 
 
 M. 
 
 20 
 
 21 
 
 22 
 
 23 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dis. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 0'.... 
 
 88.30 
 
 32.14 
 
 87.16 
 
 33.46 
 
 85.97 
 
 34.73 
 
 84.73 
 
 35.97 
 
 2..... 
 
 88.26 
 
 32.18 
 
 87.12 
 
 33.50 
 
 85.93 
 
 34.77 
 
 84.69 
 
 36.01 
 
 4 
 
 88.23 
 
 32.23 
 
 87.08 
 
 33. 64 
 
 85.89 
 
 34.82 
 
 84.65 
 
 36.05 
 
 6 
 
 88.19 
 
 32.27 
 
 87.04 
 
 33.59 
 
 85.85 
 
 34.86 
 
 84.61 
 
 36.09 
 
 8 
 
 88.15 
 
 32.32 
 
 87.00 
 
 33.63 
 
 85.80 
 
 34.90 
 
 84.57 
 
 36.13 
 
 10 
 
 83.11 
 
 32.36 
 
 86.96 
 
 33.67 
 
 85.76 
 
 34.94 
 
 84.52 
 
 36.17 
 
 12 
 
 88.08 
 
 32.41 
 
 86.92 
 
 33.72 
 
 85.72 
 
 34.98 
 
 84.48 
 
 36.21 
 
 14 
 
 88.04 
 
 32.45 
 
 86.88 
 
 33.76 
 
 85.68 
 
 35.02 
 
 84.44 
 
 36.25 
 
 16 
 
 88.00 
 
 32.49 
 
 86.84 
 
 33.80 
 
 85.64 
 
 35.07 
 
 84.40 
 
 36.29 
 
 18 
 
 87.96 
 
 32.54 
 
 86.80 
 
 33.84 
 
 85.60 
 
 35.11 
 
 84.35 
 
 36.33 
 
 20 
 
 87.93 
 
 32.58 
 
 86.77 
 
 33.89 
 
 85.56 
 
 35.15 
 
 84.31 
 
 36.37 
 
 22 
 
 87.89 
 
 32.63 
 
 86.73 
 
 33.93 
 
 85.52 
 
 35.19 
 
 84.27 
 
 36.41 
 
 24 
 
 87.85 
 
 34.67 
 
 86.69 
 
 33.97 
 
 85.48 
 
 35.23 
 
 84.23 
 
 36.45 
 
 26 
 
 87.81 
 
 32.7' 
 
 86.65 
 
 34.01 
 
 85.44 
 
 3527 
 
 84.18 
 
 36.49 
 
 28 
 
 87.77 
 
 32.7<> 
 
 86.61 
 
 34.06 
 
 85.40 
 
 35.31 
 
 84.14 
 
 36.53 
 
 80 
 
 87.74 
 
 32.80 
 
 86.57 
 
 34.10 
 
 85.36 
 
 35.36 
 
 84.10 
 
 36.57 
 
 82 
 
 87.70 
 
 32.85 
 
 86.53 
 
 34.14 
 
 85.31 
 
 35.40 
 
 84.06 
 
 36.61 
 
 34 
 
 87.66 
 
 32.89 
 
 86.49 
 
 34.18 
 
 85.27 
 
 35.44 
 
 84.01 
 
 36.65 
 
 36 
 
 87.62 
 
 32.93 
 
 86.45 
 
 34.23 
 
 85.23 
 
 35.48 
 
 83.97 
 
 36.69 
 
 88 
 
 87.58 
 
 32.93 
 
 86.41 
 
 34.27 
 
 85.19 
 
 35.52 
 
 83.93 
 
 38.73 
 
 40 
 
 87.54 
 
 33.02 
 
 86.37 
 
 34.31 
 
 85.15 
 
 35.56 
 
 83.89 
 
 36.77 
 
 42 
 
 87.51 
 
 33.07 
 
 86.33 
 
 34.35 
 
 85.11 
 
 35.60 
 
 83.84 
 
 36.80 
 
 44 
 
 87.47 
 
 33.11 
 
 86.29 
 
 34.40 
 
 85.07 
 
 35.64 
 
 83.80 
 
 36.84 
 
 46 
 
 87.43 
 
 33.15 
 
 86.25 
 
 34.44 
 
 85.02 
 
 35.68 
 
 83.76 
 
 36.88 
 
 48 
 
 87.39 
 
 33.20 
 
 86.21 
 
 34.48 
 
 84.98 
 
 35.72 
 
 88.72 
 
 36.92 
 
 50 
 
 87.35 
 
 33.24 
 
 86.17 
 
 34.52 
 
 84.94 
 
 35.76 
 
 83.67 
 
 36.96 
 
 52... 
 
 87-31 
 
 33.28 
 
 86.13 
 
 34.57 
 
 84.90 
 
 35.80 
 
 83.63 
 
 37.00 
 
 54 
 
 87.27 
 
 33.33 
 
 86.09 
 
 34.61 
 
 84.86 
 
 35.85 
 
 83.59 
 
 37.04 
 
 56 
 
 87.24 
 
 33.37 
 
 86.05 
 
 34.65 
 
 84.82 
 
 35.89 
 
 83.54 
 
 37.08 
 
 58 
 
 87.20 
 
 33.41 
 
 86.01 
 
 34.69 
 
 84.77 
 
 35.93 
 
 83.50 
 
 37.12 
 
 60 
 
 87.16 
 
 33.46 
 
 85.97 
 
 34.73 
 
 84.73 
 
 35.97 
 
 83.46 
 
 37.16 
 
 e= .75 
 
 .70 .26 
 
 .70 
 
 .27 
 
 .69 .29 
 
 .69 .30 
 
 c=1.00 
 
 .94 
 
 .35 
 
 .93 .37 
 
 .92 .38 
 
 .92 .40 
 
 c=1.25 
 
 1.17 
 
 .41 
 
 1.16 
 
 .46 
 
 1.15 
 
 .48 
 
 1.15 
 
 .50 
 
 From Winsloid's "Stadia Surveying." D. Van Nostrantfs Science Series. 
 287 
 
TABLE XIX. 
 
 STADIA MEASUREMENTS. 
 
 M. 
 
 24 
 
 25 
 
 26 
 
 27 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diflf. 
 
 Hor. 
 
 Diflf. 
 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 (X.... 
 
 83.46 
 
 37.16 
 
 82.14 
 
 38.30 
 
 80.78 
 
 39.40 
 
 79.39 
 
 40.45 
 
 2 
 
 83.41 
 
 37.20 
 
 82.09 
 
 38.34 
 
 80.74 
 
 39.44 
 
 79.34 
 
 40.49 
 
 4 
 
 83.37 
 
 37.23 
 
 82.05 
 
 38.38 
 
 80.69 
 
 39.47 
 
 79.30 
 
 40.52 
 
 6 
 
 83.33 
 
 37.27 
 
 82.01 
 
 38.41 
 
 80.65 
 
 39.51 
 
 79.25 
 
 40.55 
 
 8 
 
 83.28 
 
 37.31 
 
 81.96 
 
 38.45 
 
 80.60 
 
 39.54 
 
 79.20 
 
 40.59 
 
 10 
 
 83.24 
 
 37.35 
 
 81.92 
 
 38.49 
 
 80.55 
 
 39.58 
 
 79.15 
 
 40.62 
 
 12..... 
 
 83.20 
 
 37.39 
 
 81.87 
 
 38.53 
 
 80.51 
 
 39.61 
 
 79.11 
 
 40.66 
 
 14 
 
 83.15 
 
 37.43 
 
 81.83 
 
 38.56 
 
 80.46 
 
 39.65 
 
 79.06 
 
 40.69 
 
 16 
 
 83.11 
 
 37.47 
 
 81.78 
 
 38.60 
 
 80.41 
 
 39.69 
 
 79.01 
 
 40.72 
 
 18 
 
 83.07 
 
 37.51 
 
 81.74 
 
 38.64 
 
 80.37 
 
 39.72 
 
 78.96 
 
 40.76 
 
 20 
 
 83.02 
 
 37.54 
 
 81.69 
 
 38.67 
 
 80.32 
 
 39.76 
 
 78.92 
 
 40.79 
 
 22 
 
 82.98 
 
 37.58 
 
 81.65 
 
 38.71 
 
 80.28 
 
 39.79 
 
 78.87 
 
 40.82 
 
 24 
 
 82.93 
 
 37.62 
 
 81.60 
 
 38.75 
 
 80.23 
 
 39.83 
 
 78.82 
 
 40.86 
 
 26 
 
 82.89 
 
 37.66 
 
 81.56 
 
 38.78 
 
 80.18 
 
 39.86 
 
 78.77 
 
 40.89 
 
 28 
 
 82.85 
 
 37.70 
 
 81.51 
 
 38.82 
 
 80.14 
 
 39.90 
 
 78.73 
 
 40.92 
 
 80 
 
 82.80 
 
 37.74 
 
 81.47 
 
 38.86 
 
 80.09 
 
 39.93 
 
 78.68 
 
 40.96 
 
 82 
 
 82.76 
 
 37.77 
 
 81.42 
 
 38.89 
 
 80.04 
 
 39.97 
 
 78.63 
 
 40.99 
 
 
 82.72 
 
 37.81 
 
 81.38 
 
 38.93 
 
 80.00 
 
 40.00 
 
 78.58 
 
 41.02 
 
 se!'./! 
 
 82.67 
 
 37.85 
 
 81.33 
 
 38.97 
 
 79.95 
 
 40.04 
 
 78.54 
 
 41.06 
 
 88 
 
 8263 
 
 37.89 
 
 81.28 
 
 39.00 
 
 79.90 
 
 40.07 
 
 78.49 
 
 41.09 
 
 40 
 
 82.58 
 
 37.93 
 
 81.24 
 
 39.04 
 
 79.86 
 
 40.11 
 
 78.44 
 
 41.12 
 
 42 
 
 82.54 
 
 37.96 
 
 81.19 
 
 39.08 
 
 79.81 
 
 40.14 
 
 78.39 
 
 41.16 
 
 41 
 
 82.49 
 
 38.00 
 
 81.15 
 
 39.11 
 
 79.76 
 
 40.18 
 
 78.34 
 
 41.19 
 
 46 
 
 82.45 
 
 38.04 
 
 81.10 
 
 39.15 
 
 79.72 
 
 40.21 
 
 78.80 
 
 41.22 
 
 48 
 
 8241 
 
 38.08 
 
 81.06 
 
 39.18 
 
 79.67 
 
 40.24 
 
 78.25 
 
 41.26 
 
 BO 
 
 82.36 
 
 38.11 
 
 81.01 
 
 89.22 
 
 79.62 
 
 40.28 
 
 78.20 
 
 41.29 
 
 S2 
 
 82.32 
 
 38.15 
 
 80.97 
 
 39.26 
 
 79.58 
 
 40.31 
 
 78.15 
 
 41.32 
 
 64 
 
 82.27 
 
 38.19 
 
 80.92 
 
 39.29 
 
 79.53 
 
 40.35 
 
 78.10 
 
 41.35 
 
 56 
 
 82.23 
 
 38.23 
 
 80.87 
 
 39.33 
 
 79.48 
 
 40.38 
 
 78.06 
 
 41.39 
 
 58 
 
 82.18 
 
 38.26 
 
 80.83 
 
 39.36 
 
 79.44 
 
 40.42 
 
 78.01 
 
 41.42 
 
 60 
 
 82.14 
 
 38.30 
 
 80.78 
 
 39.40 
 
 79.39 
 
 40.45 
 
 77.96 
 
 41.45 
 
 C= .75 
 
 .68 .31 
 
 .68 .32 
 
 .67 
 
 .33 
 
 .66 .85 
 
 C=1.00 
 
 .91 
 
 .41 
 
 .90 .43 
 
 .89 .45 
 
 .89 .48 
 
 C=l 25 
 
 1.14 
 
 .52 
 
 1.13 
 
 .54 
 
 1.12 
 
 .56 
 
 1.11 .58 
 
 FromWinslow's "Stadia Surveying." D. Van Nostranffs Science Series. 
 288 
 
TABLE XIX. 
 
 STADIA MEASUREMENTS. 
 
 M. 
 
 28 
 
 29 
 
 30 
 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 Hor. 
 
 Diff. 
 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 Dist. 
 
 Elev. 
 
 (X.... 
 
 77.96 
 
 41.45 
 
 76.50 
 
 42.40 
 
 75.00 
 
 43.30 
 
 2 
 
 77.91 
 
 41.48 
 
 76.45 
 
 42.43 
 
 74.95 
 
 43.33 
 
 4 
 
 77.86 
 
 41.52 
 
 76.40 
 
 42.46 
 
 74.90 
 
 43 36 
 
 6 
 
 77.81 
 
 41.55 
 
 76.35 
 
 42.49 
 
 74.85 
 
 43.39 
 
 8 
 
 77.77 
 
 41.58 
 
 76.30 
 
 42.53 
 
 74.80 
 
 43.42 
 
 10 
 
 77.72 
 
 41.61 
 
 76.25 
 
 42.56 
 
 74.75 
 
 43.45 
 
 12 
 
 77.67 
 
 41.65 
 
 76.20 
 
 42.59 
 
 74.70 
 
 43.47 
 
 14 
 
 77.62 
 
 41.68 
 
 76.15 
 
 42.62 
 
 74.65 
 
 43.50 
 
 16 
 
 77.57 
 
 41.71 
 
 76.10 
 
 42.05 
 
 74.60 
 
 43.53 
 
 18 
 
 77.52 
 
 41.74 
 
 76.05 
 
 42.68 
 
 74.55 
 
 43.56 
 
 80 
 
 77.48 
 
 41.77 
 
 76.00 
 
 42.71 
 
 74.49 
 
 43. 59 
 
 22 
 
 77.42 
 
 41.81 
 
 75.95 
 
 42.74 
 
 74.44 
 
 43.62 
 
 4 
 
 77.38 
 
 41.84 
 
 75.90 
 
 42.77 
 
 74.39 
 
 43.65 
 
 26 
 
 77.33 
 
 41.87 
 
 75.85 
 
 42.80 
 
 74.34 
 
 43.67 
 
 28 
 
 77.28 
 
 41.90 
 
 75.80 
 
 42.83 
 
 74.29 
 
 43.70 
 
 BO 
 
 77.23 
 
 41.93 
 
 75.75 
 
 42.86 
 
 74.34 
 
 43.73 
 
 82 
 
 77.18 
 
 41.97 
 
 75.70 
 
 42.89 
 
 74.19 
 
 43 76 
 
 84 
 
 77.13 
 
 42.00 
 
 75.65 
 
 42 92 
 
 74.14 
 
 43.7? 
 
 86 
 
 77.09 
 
 42.03 
 
 75.60 
 
 42.95 
 
 74.09 
 
 43.82 
 
 88 
 
 77.04 
 
 42.06 
 
 75.55 
 
 42.98 
 
 74.04 
 
 43.84 
 
 40 
 
 76.99 
 
 42.09 
 
 75.50 
 
 43.01 
 
 73.99 
 
 43.87 
 
 42 
 
 76.94 
 
 42.12 
 
 75.45 
 
 43.04 
 
 73.93 
 
 43.90 
 
 44 
 
 76.89 
 
 42.15 
 
 T5.40 
 
 43.07 
 
 73.88 
 
 43.93 
 
 46 
 
 76.84 
 
 42.19 
 
 75.35 
 
 43.10 
 
 73.83 
 
 43.95 
 
 48.... 
 
 76.79 
 
 42.22 
 
 75.30 
 
 43.13 
 
 73.78 
 
 43.98 
 
 50 
 
 76.74 
 
 42.25 
 
 75.25 
 
 43.16 
 
 73.73 
 
 44.01 
 
 52 
 
 76.69 
 
 42.28 
 
 75.20 
 
 43.18 
 
 73.68 
 
 44.04 
 
 54.... 
 
 76.64 
 
 42.31 
 
 75.15 
 
 43.21 
 
 73.63 
 
 44.07 
 
 56 
 
 76.59 
 
 42.34 
 
 75.1,0 
 
 43.24 
 
 73.58 
 
 44.09 
 
 58 
 
 76.55 
 
 42.37 
 
 75.05 
 
 43.27 
 
 73.52 
 
 44.12 
 
 60 
 
 76.50 
 
 42.40 
 
 75.00 
 
 43.30 
 
 7347 
 
 44.15 
 
 e= .75 
 
 .66 
 
 .36 
 
 .65 
 
 .87 
 
 .65 .38 
 
 C=1.00 
 
 .88 
 
 .48 
 
 .87 
 
 .49 
 
 .86 .51 
 
 ff=1.25 
 
 1.10~ 
 
 ~^60 
 
 1.09 .62 
 
 1.08 
 
 .64 
 
 From Winsloufs "Stadia Surveying." D. Van Nostrand's Science Series. 
 
 289 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 1 
 
 Sine Cosine 
 
 Sine 
 
 Cosine 
 
 Sine Cosine 
 
 
 1 
 2 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 
 20 
 21 
 
 6.46373 
 76476 
 94085 
 
 7.06579 
 16270 
 24188 
 
 41797 
 
 .46373 
 50512 
 54291 
 57767 
 
 66784 
 69417 
 71900 
 74248 
 
 7.76475 
 78594 
 80615 
 82545 
 
 87870 
 89509 
 
 92612 
 
 7.94084 
 95508 
 96887 
 98223 
 99520 
 
 8.00779 
 02002 
 03192 
 04350 
 05478 
 
 8.06578 
 07650 
 
 09718 
 10717 
 11693 
 12647 
 13581 
 14495 
 15391 
 
 8.16968 
 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 
 
 9.99999 
 99999 
 99999 
 
 99999 
 99999 
 99999 
 
 99998 
 
 99997 
 99997 
 99997 
 
 99996 
 99996 
 
 99996 
 99996 
 
 9.99995 
 
 99995 
 99995 
 99994 
 
 99994 
 99994 
 99994 
 
 8.24186 9 
 
 25609 
 26304 
 
 27661 
 28324 
 28977 
 29621 
 
 8.30879 
 31495 
 32103 
 32702 
 
 40816 
 41307 
 
 8 41792 
 42272 
 42746 
 43216 
 
 44139 
 44594 
 45044 
 
 45930 
 
 .46366 
 46799 
 47226 
 47650 
 
 48485 
 48896 
 49304 
 49708 
 50108 
 
 .50504 
 50897 
 51287 
 51673 
 52055 
 52434 
 52810 
 53183 
 53552 
 53919 
 54282 
 
 99993 
 
 99992 
 99992 
 99992 
 99992 
 
 99990 
 99990 
 
 33875 99990 
 31450 99989 
 35018 99989 
 35578 
 36131 
 
 8.36678 
 37217 
 37750 
 38276 
 38796 99987 
 99987 
 
 99986 
 
 9998E 
 
 99984 
 99983 
 
 99981 
 
 99981 
 99980 
 999SO 
 99979 
 
 99979 
 99978 
 
 9.99978 
 99977 
 99977 
 99977 
 
 9997G 
 99975 
 99975 
 99974 
 99974 
 99974 
 
 8.54282 
 54642 
 54999 
 55354 
 55705 
 56054 
 56400 
 56743 
 57084 
 57421 
 
 58419 
 58747 
 59072 
 
 59715 
 
 8.60973 
 61282 
 61589 
 61894 
 62196 
 62497 
 62795 
 
 63385 
 
 8.63968 
 64256 
 64543 
 64827 
 65110 
 65391 
 65670 
 65947 
 66223 
 66497 
 
 8.66769 
 
 67308 
 67575 
 67841 
 68104 
 68367 
 68687 
 68886 
 69144 
 
 8.69400 
 69654 
 69907 
 70159 
 70409 
 70658 
 70905 
 71151 
 71395 
 71638 
 71880 
 
 9 99974 
 99973 
 99973 
 99972 
 99972 
 69971 
 99971 
 99970 
 
 99969 
 
 9.99991 8.57757 9.99969 
 
 99967 
 99967 
 
 99965 
 99964 
 
 9.99964 
 99963 
 99963 
 99962 
 
 99961 
 
 99959 
 9.99959 
 
 99958 
 99957 
 99956 
 99956 
 99955 
 99955 
 99954 
 99954 
 
 .99953 
 99952 
 99952 
 99951 
 99951 
 99950 
 99949 
 99949 
 99948 
 99948 
 
 .99947 
 99946 
 99946 
 99945 
 99944 
 99944 
 
 99942 
 99942 
 99941 
 
 Cosine Sine 
 89 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 88 
 
 87 C 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 40 
 41 
 42 
 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 
 50 
 51 
 5-2 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 
 3 
 
 Sine Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 8 71880 
 72120 
 72359 
 72597 
 72834 
 73069 
 73303 
 73535 
 73767 
 73997 
 
 8 74226 
 74454 
 74680 
 74906 
 75130 
 75353 
 75575 
 75795 
 76015 
 76234 
 
 8.76451 
 76667 
 
 77097 
 77310 
 77522 
 77733 
 77943 
 78152 
 78360 
 
 8.78568 
 78774 
 78979 
 79193 
 79386 
 79588 
 79789 
 79990 
 80189 
 80388 
 
 8.80585 
 80782 
 80978 
 81173 
 81367 
 81560 
 81752 
 81944 
 82134 
 82324 
 
 8.82513 
 82701 
 
 83075 
 83261 
 83446 
 83630 
 83813 
 83996 
 84177 
 84358 
 
 9.99940 
 99940 
 99939 
 99938 
 
 99935 
 99934 
 
 .99934 
 99933 
 99932 
 99932 
 
 99928 
 99927 
 
 .99926 
 99926 
 
 99921 
 99920 
 99920 
 
 9.99919 
 99918 
 99917 
 99917 
 
 99915 
 
 99913 
 99912 
 
 9.99911 
 
 99908 
 99907 
 99906 
 99905 
 99904 
 99904 
 
 99902 
 99901 
 99900 
 
 8.84358 
 84539 
 84718 
 84897 
 85075 
 85252 
 85429 
 
 85955 
 
 8.86128 
 86301 
 86474 
 86645 
 86816 
 
 87156 
 87325 
 87494 
 87661 
 
 8.87829 
 87995 
 88161 
 
 88490 
 
 8881' 
 
 89784 
 89943 
 90102 
 90260 
 90417 
 90574 
 90730 
 90885 
 
 8.91040 
 91195 
 91349 
 91502 
 91655 
 91807 
 
 93448 
 93594 
 93740 
 
 94030 
 
 9.99894 
 
 99887 
 99886 
 
 9.99885 
 
 99880 
 99879 
 99879 
 
 99877 
 
 99875 
 
 99873 
 99872 
 99871 
 99870 
 
 99857 
 9.99856 
 
 99854 
 
 99851 
 
 92110 99848 
 
 92261 99847 
 
 92411 99846 
 
 .92561 9.99845 
 
 92710 99844 
 
 92859 99843 
 
 93007 99842 
 
 93154 99841 
 
 99839 
 
 8.94030 
 94174 
 94317 
 94461 
 
 94746 
 
 95728 
 95867 
 96005 
 96143 
 96280 
 96417 
 96553 
 
 8.96825 
 96960 
 97095 
 97229 
 97363 
 97496 
 97629 
 97762 
 
 89142 99868 97894 
 89304 99867 98026 
 
 8.89464 9.99866 8.98157 
 99865 
 
 98419 
 
 98679 
 
 98808 
 
 99194 
 99322 
 
 8.99450 
 99577 
 99704 
 
 99956 
 
 9.00082 
 
 00207 
 
 00456 
 00581 
 
 9.00704 
 00828 
 00951 
 01074 
 01196 
 01318 
 01440 
 01561 
 01682 
 01803 
 01923 
 
 Cosine Sine 
 86 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Cosine 
 
 99829 
 
 99828 
 
 95029 
 95170 
 95310 
 
 8.95450 9 
 
 99822 
 
 99817 
 
 99815 
 99814 
 
 9.99812 
 99810 
 99809 
 
 99807 
 
 99802 
 99801 
 
 .99800 
 99798 
 99797 
 99796 
 99795 
 99793 
 99792 
 99791 
 99790 
 
 9.99787 
 99786 
 99785 
 99783 
 99782 
 99781 
 99780 
 99778 
 99777 
 99776 
 
 9.99775 
 99773 
 99772 
 99771 
 
 99767 
 99765 
 99764 
 99763 
 99761 
 
 Sine 
 
 85 C 
 291 
 
 84 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 / 
 
 6 
 
 7 
 
 
 8" 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 9.01923 
 
 9.99761 
 
 9.08589 
 
 9.99675 
 
 9.14356 
 
 9.99575 
 
 60 
 
 1 
 
 0*043 
 
 99760 
 
 08692 
 
 99674 
 
 14445 
 
 99574 
 
 59 
 
 2 
 
 02163 
 
 99759 
 
 08795 
 
 99672 
 
 14535 
 
 99572 
 
 58 
 
 3 
 
 02283 
 
 99757 
 
 08897 
 
 99670 
 
 14624 
 
 99570 
 
 57 
 
 4 
 
 02402 
 
 99756 
 
 08999 
 
 99669 
 
 14714 
 
 99568 
 
 56 
 
 5 
 
 02520 
 
 99755 
 
 09101 
 
 99667 
 
 14803 
 
 99566 
 
 55 
 
 6 
 
 02639 
 
 99753 
 
 09202 
 
 99666 
 
 14891 
 
 99565 
 
 54 
 
 7 
 
 02757 
 
 99752 
 
 09304 
 
 99664 
 
 14980 
 
 99563 
 
 53 
 
 8 
 
 02874 
 
 99751 
 
 09405 
 
 99663 
 
 15069 
 
 99561 
 
 52 
 
 9 
 
 02992 
 
 99749 
 
 09506 
 
 99661 
 
 15157 
 
 99559 
 
 51 
 
 10 
 
 9.03109 
 
 9.99748 
 
 9.09606 
 
 9.99659 
 
 9.15245 
 
 9.99557 
 
 50 
 
 11 
 
 03226 
 
 99747 
 
 09707 
 
 99658 
 
 15333 
 
 99556 
 
 49 
 
 12 
 
 03342 
 
 99745 
 
 09807 
 
 99656 
 
 15421 
 
 99554 
 
 48 
 
 13 
 
 03458 
 
 99744 
 
 09907 
 
 99655 
 
 15508 
 
 99552 
 
 47 
 
 14 
 
 03574 
 
 99742 
 
 10006 
 
 99653 
 
 15596 
 
 99550 
 
 46 
 
 15 
 
 03690 
 
 99741 
 
 10106 
 
 99651 
 
 15683 
 
 99548 
 
 45 
 
 16 
 
 03805 
 
 99740 
 
 10205 
 
 99650 
 
 15770 
 
 99546 
 
 44 
 
 17 
 
 03920 
 
 99738 
 
 10304 
 
 99648 
 
 15857 
 
 99545 
 
 43 
 
 18 
 
 04034 
 
 99737 
 
 10402 
 
 99647 
 
 15944 
 
 99543 
 
 42 
 
 19 
 
 04149 
 
 99736 
 
 10501 
 
 99645 
 
 16030 
 
 99541 
 
 41 
 
 20 
 
 9.04262 
 
 9.99734 
 
 9.10599 
 
 9.99643 
 
 9.16116 
 
 9.99539 
 
 40 
 
 21 
 
 04376 
 
 99733 
 
 10697 
 
 99642 
 
 16203 
 
 99537 
 
 39 
 
 22 
 
 04490 
 
 99731 
 
 10795 
 
 99640 
 
 16289 
 
 99535 
 
 38 
 
 23 
 
 04603 
 
 99730 
 
 10893 
 
 99638 
 
 16374 
 
 99533 
 
 37 
 
 24 
 
 04715 
 
 99728 
 
 10990 
 
 99637 
 
 16460 
 
 99532 
 
 36 
 
 25 
 
 04828 
 
 99727 
 
 11087 
 
 99635 
 
 16545 
 
 99530 
 
 35 
 
 26 
 
 04940 
 
 99726 
 
 11184 
 
 99633 
 
 16631 
 
 99528 
 
 34 
 
 27 
 
 05052 
 
 99724 
 
 11281 
 
 99632 
 
 16716 
 
 99526 
 
 33 
 
 28 
 
 05164 
 
 99723 
 
 11377 
 
 99630 
 
 16801 
 
 99524 
 
 32 
 
 29 
 
 05275 
 
 99721 
 
 11474 
 
 99629 
 
 16886 
 
 99522 
 
 31 
 
 30 
 
 9.05386 
 
 9.99720 
 
 9.11570 
 
 9.99627 
 
 9.16970 
 
 9.99520 
 
 30 
 
 31 
 
 05497 
 
 99718 
 
 11666 
 
 99625 
 
 17055 
 
 99518 
 
 29 
 
 32 
 
 05607 
 
 99717 
 
 11761 
 
 99624 
 
 17139 
 
 99517 
 
 28 
 
 33 
 
 05717 
 
 99716 
 
 11857 
 
 99622 
 
 17223 
 
 99515 
 
 27 
 
 34 
 
 05827 
 
 99714 
 
 11952 
 
 99620 
 
 17307 
 
 99513 
 
 26 
 
 a<5 
 
 05937 
 
 99713 
 
 12047 
 
 99618 
 
 17391 
 
 99511 
 
 25 
 
 36 
 
 06046 
 
 99711 
 
 12142 
 
 S9617 
 
 17474 
 
 99509 
 
 24 
 
 37 
 
 06155 
 
 99710 
 
 12236 
 
 99615 
 
 17558 
 
 99507 
 
 23 
 
 38 
 
 06264 
 
 99708 
 
 12331 
 
 99613 
 
 17641 
 
 99505 
 
 22 
 
 39 
 
 06372 
 
 99707 
 
 12425 
 
 99612 
 
 17724 
 
 99503 
 
 21 
 
 40 
 
 9.06481 
 
 9.99705 
 
 9.12519 
 
 9.99610 
 
 9.17807 
 
 9.99501 
 
 20 
 
 41 
 
 06589 
 
 99704 
 
 12612 
 
 99608 
 
 17890 
 
 99499 
 
 19 
 
 42 
 
 06696 
 
 99702 
 
 12706 
 
 99607 
 
 17973 
 
 99497 
 
 18 
 
 43 
 
 06804 
 
 99701 
 
 12799 
 
 99605 
 
 18055 
 
 99495 
 
 17 
 
 44 
 
 06911 
 
 99699 
 
 12892 
 
 99603 
 
 18137 
 
 99494 
 
 16 
 
 45 
 
 07018 
 
 99698 
 
 12985 
 
 99601 
 
 18220 
 
 99492 
 
 15 
 
 46 
 
 07124 
 
 99696 
 
 13078 
 
 99600 
 
 18302 
 
 99490 
 
 14 
 
 47 
 
 07231 
 
 99695 
 
 13171 
 
 99598 
 
 18383 
 
 99488 
 
 13 
 
 48 
 
 07337 
 
 99693 
 
 13263 
 
 99596 
 
 18465 
 
 99486 
 
 12 
 
 49 
 
 07442 
 
 99692 
 
 13355 
 
 99595 
 
 18547 
 
 99484 
 
 11 
 
 50 
 
 9.07548" 
 
 9.99690 
 
 9.13447 
 
 9.99593 
 
 9.18628 
 
 9.99482 
 
 10 
 
 51 
 
 07653 
 
 99689 
 
 13539 
 
 99591 
 
 18709 
 
 99480 
 
 9 
 
 52 
 
 07758 
 
 99687 
 
 13630 
 
 99589 
 
 18790 
 
 99478 
 
 8 
 
 53 
 
 07863 
 
 99686 
 
 13722 
 
 99588 
 
 18871 
 
 99476 
 
 7 
 
 54 
 
 07968 
 
 99684 
 
 13813 
 
 99586 
 
 18952 
 
 99474 
 
 6 
 
 55 
 
 08072 
 
 99683 
 
 13904 
 
 99584 
 
 19033 
 
 99472 
 
 5 
 
 56 
 
 08176 
 
 99681 
 
 13994 
 
 99582 
 
 19113 
 
 99470 
 
 4 
 
 57 
 
 08280 
 
 99680 
 
 14085 
 
 99581 
 
 19193 
 
 99468 
 
 3 
 
 58 
 
 08383 
 
 99678 
 
 14175 
 
 99579 
 
 19273 
 
 99466 
 
 2 
 
 59 
 
 08486 
 
 99677 
 
 14266 
 
 99577 
 
 19353 
 
 99464 
 
 1 
 
 60 
 
 08589 
 
 99675 
 
 14356 
 
 99575 
 
 19433 
 
 99462 
 
 
 
 / 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 t 
 
 
 83 
 
 
 82 
 
 
 81- 
 
 292 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 10 
 
 Sine Cosine 
 
 Sine 
 
 Cosine 
 
 Sine Cosine 
 
 37 
 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 
 50 
 51 
 52 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 60 
 
 9.19433 
 19513 
 
 19672 
 19751 
 
 19909 
 19988 
 20067 
 20145 
 
 9.20223 
 
 20380 
 20458 
 20535 
 20613 
 
 20768 
 20845 
 
 21076 
 21153 
 21229 
 21306 
 21382 
 21458 
 21534 
 21610 
 21685 
 
 9.21761 
 
 21836 
 21912 
 21987 
 
 22137 
 22211 
 
 22361 
 22435 
 
 9.22509 
 22583 
 22657 
 22731 
 
 22805 
 22878 
 
 23025 
 23098 
 23171 
 
 9.23244 
 
 23317 
 
 28535 
 23607 
 23679 
 23752 
 
 23895 
 23967 
 
 9.99462 
 99460 
 99458 
 99456 
 99454 
 99452 
 99450 
 99448 
 99446 
 99444 
 
 9.99442 
 99440 
 
 99434 
 99432 
 99429 
 99427 
 99425 
 
 9.99421 
 99419 
 99417 
 99415 
 99413 
 99411 
 99409 
 99407 
 99404 
 
 9.99400 
 
 00302 
 
 99390 
 99388 
 99385 
 
 9.99379 
 
 99375 
 99372 
 99370 
 
 99364 
 
 99359 
 
 9.99357 
 99355 
 99353 
 99351 
 99:348 
 99346 
 
 99337 
 99335 
 
 9.23967 
 24039 
 24110 
 24181 
 24253 
 24324 
 24395 
 24466 
 24536 
 
 9.24677 
 
 24748 
 
 24888 
 24958 
 25028 
 25098 
 25168 
 25237 
 25307. 
 
 9.25376 
 25445 
 25514 
 25583 
 25652 
 25721 
 25790 
 25858 
 
 9.26063 
 26131 
 26199 
 
 26335 
 26403 
 26470 
 
 26672 
 
 9.26739 
 26806 
 26873 
 
 27007 
 27073 
 27140 
 27206 
 27273 
 27339 
 
 .27405 
 27471 
 27537 
 27602 
 27668 
 27734 
 27799 
 27864 
 27930 
 27995 
 
 9.99335 
 99333 
 99331 
 99328 
 99326 
 
 99322 
 99319 
 99317 
 99315 
 
 99297 
 99294 
 
 99288 
 99285 
 
 99281 
 
 99276 
 99274 
 99271 
 
 9.99267 
 99264 
 
 99260 
 99257 
 99255 
 99252 
 99250 
 
 99245 
 
 99241 
 99238 
 99236 
 99233 
 99231 
 
 99221 
 
 99217 
 99214 
 99212 
 99209 
 99207 
 
 99200 
 99197 
 99195 
 
 9.28060 
 28125 
 28190 
 28254 
 28319 
 
 28448 
 28512 
 28577 
 28641 
 
 9.28705 
 28769 
 
 29087 
 29150 
 
 29277 
 .29340 
 
 29529 
 129591 
 29654 
 29716 
 29779 
 29841 
 
 30028 
 30090 
 30151 
 30213 
 30275 
 
 9.30582 
 30643 
 80704 
 30765 
 
 30887 
 30947 
 31008 
 31068 
 31129 
 
 9.31189 
 31250 
 31310 
 31370 
 31430 
 31490 
 31549 
 31609 
 
 31728 
 31788 
 
 9.99195 
 99192 
 99190 
 99187 
 
 99182 
 
 9917? 
 99175 
 99172 
 
 9.99170 
 99167 
 99165 
 99162 
 99160 
 99157 
 99155 
 99152 
 99150 
 99147 
 
 9.99145 
 99142 
 99140 
 99137 
 99135 
 99132 
 99130 
 99127 
 99124 
 99122 
 
 9.99119 
 99117 
 99114 
 99112 
 99109 
 
 99104 
 99001 
 
 99078 
 99075 
 99072 
 99070 
 
 99064 
 99062 
 99059 
 99056 
 99054 
 
 99043 
 99040 
 
 Cosine Sine 
 80 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 79 
 
 293 
 
 78 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 f 
 
 12 
 
 13 
 
 
 14 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 9.31788 
 
 9.99040 
 
 9.35209 
 
 9.98872 
 
 9.38368 
 
 9.98690 
 
 60 
 
 1 
 
 31847 
 
 99038 
 
 35263 
 
 98869 
 
 38418 
 
 98687 
 
 59 
 
 2 
 
 31907 
 
 99035 
 
 35318 
 
 98867 
 
 38469 
 
 98684 
 
 58 
 
 3 
 
 31966 
 
 99032 
 
 35373 
 
 98864 
 
 38519 
 
 98681 
 
 57 
 
 4 
 
 32025 
 
 99030 
 
 35427 
 
 98861 
 
 38570 
 
 98678 
 
 56 
 
 5 
 
 32084 
 
 99027 
 
 35481 
 
 98858 
 
 38620 
 
 98675 
 
 55 
 
 6 
 
 32143 
 
 99024 
 
 35536 
 
 98855 
 
 38670 
 
 98671 
 
 54 
 
 7 
 
 32202 
 
 99022 
 
 35590 
 
 98852 
 
 38721 
 
 986C8 
 
 53 
 
 8 
 
 32261 
 
 99019 
 
 35644 
 
 98849 
 
 38771 
 
 98665 
 
 52 
 
 9 
 
 32319 
 
 99016 
 
 35698 
 
 98846 
 
 38821 
 
 98662 
 
 51 
 
 10 
 
 9.32378 
 
 9.99013 
 
 9.35752 
 
 9.98843 
 
 9.38871 
 
 9.98659 
 
 50 
 
 11 
 
 32437 
 
 99011 
 
 35806 
 
 98840 
 
 38921 
 
 98656 
 
 49 
 
 32 
 
 32495 
 
 99008 
 
 35860 
 
 98837 
 
 38971 
 
 98652 
 
 48 
 
 13 
 
 32553 
 
 99005 
 
 35914 
 
 98834 
 
 39021 
 
 98649 
 
 47 
 
 14 
 
 32612 
 
 99002 
 
 35968 
 
 98831 
 
 39071 
 
 98646 
 
 46 
 
 15 
 
 32670 
 
 99000 
 
 36022 
 
 98828 
 
 39121 
 
 98643 
 
 45 
 
 16 
 
 32728 
 
 98997 
 
 36075 
 
 98825 
 
 39170 
 
 98640 
 
 44 
 
 17 
 
 32786 
 
 98994 
 
 36129 
 
 98822 
 
 39220 
 
 98636 
 
 43 
 
 18 
 
 32844 
 
 98991 
 
 36182 
 
 98819 
 
 39270 
 
 98633 
 
 42 
 
 19 
 
 32902 
 
 98989 
 
 36236 
 
 98816 
 
 39319 
 
 98U30 
 
 41 
 
 20 
 
 9.32960 
 
 9.98986 
 
 9.36289 
 
 9.98813 
 
 9.39369 
 
 9.98627 
 
 40 
 
 21 
 
 33018 
 
 98983 
 
 36342 
 
 98810 
 
 39418 
 
 98623 
 
 39 
 
 22 
 
 33075 
 
 98980 
 
 36395 
 
 98807 
 
 39467 
 
 98620 
 
 38 
 
 23 
 
 33133 
 
 98978 
 
 36449 
 
 98804 
 
 39517 
 
 98617 
 
 37 
 
 24 
 
 33190 
 
 98975 
 
 36502 
 
 98801 
 
 39566 
 
 98614 
 
 36 
 
 25 
 
 33248 
 
 98972 
 
 36555 
 
 98798 
 
 39615 
 
 98610 
 
 86 
 
 26 
 
 33305 
 
 98969 
 
 36608 
 
 98795 
 
 39664 
 
 98607 
 
 34 
 
 27 
 
 33362 
 
 98967 
 
 36660 
 
 98792 
 
 39713 
 
 98604 
 
 33 
 
 28 
 
 33420 
 
 98964 
 
 36713 
 
 98789 
 
 39762 
 
 98601 
 
 32 
 
 29 
 
 33477 
 
 98961 
 
 36766 
 
 98786 
 
 39811 
 
 98597 
 
 31 
 
 30 
 
 9.33534 
 
 9.98958 
 
 9.36819 
 
 9.98783 
 
 9.39860 
 
 9.98594 
 
 30 
 
 31 
 
 33591 
 
 98955 
 
 36871 
 
 98780 
 
 39909 
 
 98591 
 
 29 
 
 32 
 
 33647 
 
 98953 
 
 36924 
 
 98777 
 
 39958 
 
 98588 
 
 28 
 
 33 
 
 33704 
 
 98950 
 
 36976 
 
 98774 
 
 40006 
 
 98584 
 
 27 
 
 34 
 
 33761 
 
 98947 
 
 37028 
 
 98771 
 
 40055 
 
 98581 
 
 26 
 
 35 
 
 33818 
 
 98944 
 
 37081 
 
 98768 
 
 40103 
 
 98578 
 
 25 
 
 36 
 
 33874 
 
 98941 
 
 37133 
 
 98765 
 
 40152 
 
 98574 
 
 24 
 
 37 
 
 33931 
 
 98938 
 
 37185 
 
 98762 
 
 40200 
 
 98571 
 
 23 
 
 38 
 
 33987 
 
 98936 
 
 37237 
 
 98759 
 
 40249 
 
 98568 
 
 22 
 
 39 
 
 34043 
 
 98933 
 
 37289 
 
 98756 
 
 40297 
 
 98565 
 
 21 
 
 40 
 
 9.34100 
 
 9.98930 
 
 9.37341 
 
 9.98753 
 
 9.40346 
 
 9.98561 
 
 20 
 
 41 
 
 34156 
 
 98927 
 
 37393 
 
 98750 
 
 40394 
 
 98558 
 
 19 
 
 42 
 
 34212 
 
 98924 
 
 37445 
 
 98746 
 
 40442 
 
 98555 
 
 18 
 
 43 
 
 34268 
 
 98921 
 
 37497 
 
 98743 
 
 40490 
 
 98551 
 
 17 
 
 44 
 
 34324 
 
 98919 
 
 37549 
 
 98740 
 
 40538 
 
 98548 
 
 16 
 
 45 
 
 34380 
 
 98916 
 
 37600 
 
 98737 
 
 40586 
 
 98545 
 
 15 
 
 46 
 
 34436 
 
 98913 
 
 37652 
 
 98734 
 
 40634 
 
 98541 
 
 14 
 
 47 
 
 34491 
 
 98910 
 
 37703 
 
 98731 
 
 40682 
 
 98538 
 
 13 
 
 48 
 
 34547 
 
 98907 
 
 37755 
 
 98728 
 
 40730 
 
 98535 
 
 12 
 
 49 
 
 34602 
 
 98904 
 
 37806 
 
 98725 
 
 40778 
 
 98531 
 
 11 
 
 50 
 
 9.34658 
 
 9.98901 
 
 9.37858 
 
 9.98722 
 
 9.40825 
 
 9.98528 
 
 10 
 
 51 
 
 34713 
 
 98898 
 
 37909 
 
 98719 
 
 40873 
 
 98525 
 
 9 
 
 52 
 
 34769 
 
 98896 
 
 37960 
 
 98715 
 
 40921 
 
 98521 
 
 8 
 
 53 
 
 34824 
 
 98893 
 
 38011 
 
 98712 
 
 40968 
 
 98518 
 
 f 
 
 54 
 
 34879 
 
 98890 
 
 38062 
 
 98709 
 
 41016 
 
 98515 
 
 6 
 
 55 
 
 34934 
 
 98887 
 
 38113 
 
 98706 
 
 41063 
 
 98511 
 
 5 
 
 56 
 
 34989 
 
 98884 
 
 38164 
 
 98703 
 
 41111 
 
 98508 
 
 4 
 
 57 
 
 35044 
 
 98881 
 
 38215 
 
 98700 
 
 41158 
 
 98505 
 
 3 
 
 58 
 
 35099 
 
 98878 
 
 38266 
 
 98697 
 
 41205 
 
 98501 
 
 2 
 
 59 
 
 35154 
 
 98875 
 
 38317 
 
 98694 
 
 41252 
 
 98498 
 
 1 
 
 60 
 
 35209 
 
 98872 
 
 38368 
 
 98690 
 
 41300 
 
 98494 
 
 
 
 ] 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 t 
 
 L' 
 
 
 77 
 
 
 76 
 
 
 75 
 
 f 
 
 294 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 t 
 
 15 
 
 16 
 
 17 
 
 f 
 
 
 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 
 
 98056 
 
 59 
 
 2 
 
 41394 
 
 98488 
 
 44122 
 
 98277 
 
 46676 
 
 98052 
 
 58 
 
 3 
 
 41441 
 
 98484 
 
 44166 
 
 98273 
 
 46717 
 
 98048 
 
 57 
 
 4 
 
 41488 
 
 98481 
 
 44210 
 
 98270 
 
 46758 
 
 98044 
 
 56 
 
 5 
 
 41535 
 
 98477 
 
 44253 
 
 98266 
 
 46800 
 
 98040 
 
 55 
 
 6 
 
 41582 
 
 98474 
 
 44297 
 
 98262 
 
 46841 
 
 98036 
 
 54 
 
 7 
 
 41628 
 
 98471 
 
 44341 
 
 98259 
 
 46882 
 
 98032 
 
 53 
 
 8 
 
 41675 
 
 98467 
 
 44385 
 
 98255 
 
 46923 
 
 98029 
 
 52 
 
 9 
 
 41722 
 
 98464 
 
 44428 
 
 98251 
 
 46964 
 
 98025 
 
 51 
 
 10 
 
 9.41768 
 
 9.98460 
 
 9.44472 
 
 9.98248 
 
 9.47005 
 
 9.98021 
 
 50 
 
 11 
 
 41815 
 
 98457 
 
 44516 
 
 98244 
 
 47045 
 
 98017 
 
 49 
 
 12 
 
 41861 
 
 98453 
 
 44559 
 
 98240 
 
 47086 
 
 98013 
 
 48 
 
 13 
 
 41908 
 
 98450 
 
 44602 
 
 98237 
 
 47127 
 
 98009 
 
 47 
 
 14 
 
 41954 
 
 98447 
 
 44646 
 
 98233 
 
 47168 
 
 98005 
 
 46 
 
 15 
 
 42001 
 
 98443 
 
 44689 
 
 98229 
 
 47209 
 
 98001 
 
 45 
 
 16 
 
 42047 
 
 98440 
 
 44733 
 
 98226 
 
 47249 
 
 97997 
 
 44 
 
 17 
 
 42093 
 
 98436 
 
 44776 
 
 98222 
 
 47290 
 
 97993 
 
 43 
 
 18 
 
 42140 
 
 98433 
 
 44819 
 
 98218 
 
 47330 
 
 97989 
 
 42 
 
 19 
 
 42186 
 
 98429 
 
 44862 
 
 98215 
 
 47371 
 
 97986 
 
 41 
 
 20 
 
 9.4-2232 
 
 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 
 
 97966 
 
 36 
 
 25 
 
 42461 
 
 98409 
 
 45120 
 
 98192 
 
 47613 
 
 97962 
 
 35 
 
 26 
 
 42507 
 
 98405 
 
 45163 
 
 98189 
 
 47654 
 
 97958 
 
 34* 
 
 27 
 
 42553 
 
 98402 
 
 45206 
 
 98185 
 
 47694 
 
 97954 
 
 33 
 
 28 
 
 42599 
 
 98398 
 
 45249 
 
 98181 
 
 47734 
 
 97950 
 
 32 
 
 29 
 
 42644 
 
 98395 
 
 45292 
 
 98177 
 
 47774 
 
 97946 
 
 31 
 
 30 
 
 9.42690 
 
 9.98391 
 
 9.45334 
 
 9.98174 
 
 9.47814 
 
 9.97942 
 
 30 
 
 31 
 
 42735 
 
 98388 
 
 45377 
 
 98170 
 
 47854 
 
 97938 
 
 29 
 
 32 
 
 42781 
 
 98384 
 
 45419 
 
 98166 
 
 47894 
 
 97934 
 
 28 
 
 33 
 
 42826 
 
 98381 
 
 45462 
 
 98162 
 
 47934 
 
 97930 
 
 27 
 
 34 
 
 42872 
 
 98377 
 
 45504 
 
 98159 
 
 47974 
 
 97926 
 
 26 
 
 35 
 
 42917 
 
 98373 
 
 45547 
 
 98155 
 
 4K)14 
 
 97922 
 
 25 
 
 36 
 
 42962 
 
 98370 
 
 45589 
 
 98151 
 
 48054 
 
 97918 
 
 24 
 
 37 
 
 43008 
 
 98366 
 
 45632 
 
 98147 
 
 48094 
 
 97914 
 
 23 
 
 38 
 
 43053 
 
 98363 
 
 45674 
 
 98144 
 
 48133 
 
 97910 
 
 22 
 
 39 
 
 43098 
 
 98359 
 
 45716 
 
 98140 
 
 48173 
 
 97906 
 
 21 
 
 40 
 
 9.43143 
 
 9.98356 
 
 '9.45758 
 
 9.98136 
 
 9.48213 
 
 9.97902 
 
 20 
 
 41 
 
 43188 
 
 98352 
 
 45801 
 
 98132 
 
 48252 
 
 97898 
 
 19 
 
 42 
 
 43233 
 
 98349 
 
 45843 
 
 98129 
 
 48292 
 
 97894 
 
 18 
 
 43 
 
 43278 
 
 98345 
 
 45885 
 
 98125 
 
 48332 
 
 97890 
 
 17 
 
 44 
 
 43323 
 
 98342 
 
 45927 
 
 98121 
 
 48371 
 
 97886 
 
 16 
 
 45 
 
 43367 
 
 98338 
 
 45969 
 
 98117 
 
 48411 
 
 97882 
 
 15 
 
 46 
 
 43412 
 
 98334 
 
 46011 
 
 98113 
 
 48450 
 
 97878 
 
 14 
 
 47 
 
 43457 
 
 98331 
 
 46053 
 
 98110 
 
 48490 
 
 97874 
 
 13 
 
 48 
 
 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 
 
 i 
 
 60 
 
 44034 
 
 98284 
 
 46594 
 
 98060 
 
 48998 
 
 97821 
 
 
 
 , 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 i 
 
 
 
 74 
 
 
 73 
 
 
 72 
 
 
 295 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 / 
 
 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 
 
 49076 
 
 97812 
 
 51338 
 
 97558 
 
 53475 
 
 97289 
 
 58 
 
 3 
 
 49115 
 
 97808 
 
 51374 
 
 97554 
 
 5:3509 
 
 97285 
 
 57 
 
 4 
 
 49153 
 
 97804 
 
 51411 
 
 97550 
 
 53544 
 
 97280 
 
 56 
 
 5 
 
 49192 
 
 97800 
 
 51447 
 
 97545 
 
 53578 
 
 97276 
 
 55 
 
 6 
 
 49231 
 
 97796 
 
 51484 
 
 97541 
 
 53613 
 
 97271 
 
 54 
 
 7 
 
 49269 
 
 97792 
 
 51520 
 
 97536 
 
 53647 
 
 97266 
 
 53 
 
 8 
 
 49308 
 
 97788 
 
 51557 
 
 97532 
 
 53682 
 
 97262 
 
 52 
 
 9 
 
 49347 
 
 97784 
 
 51593 
 
 97528 
 
 53716 
 
 97257 
 
 51 
 
 10 
 
 9.49385 
 
 9.97779 
 
 9.51629 
 
 9.97523 
 
 9.53751 
 
 9.97252 
 
 50 
 
 11 
 
 49424 
 
 97775 
 
 51666 
 
 97519 
 
 53785 
 
 97248 
 
 49 
 
 12 
 
 49462 
 
 97771 
 
 51702 
 
 97515 
 
 53819 
 
 97243 
 
 48 
 
 13 
 
 49500 
 
 97767 
 
 51738 
 
 97510 
 
 53854 
 
 97238 
 
 47 
 
 14 
 
 49539 
 
 97763 
 
 51774 
 
 97506 
 
 53888 
 
 97234 
 
 46 
 
 15 
 
 49577 
 
 97759 
 
 51811 
 
 97501 
 
 53922 
 
 97229 
 
 45 
 
 16 
 
 49615 
 
 97754 
 
 51847 
 
 97497 
 
 53957 
 
 97224 
 
 44 
 
 17 
 
 49654 
 
 97750 
 
 51883 
 
 97492 
 
 53991 
 
 97220 
 
 43 
 
 18 
 
 49692 
 
 97746 
 
 51919 
 
 97488 
 
 54025 
 
 97215 
 
 42 
 
 19 
 
 49730 
 
 97742 
 
 51955 
 
 97484 
 
 54059 
 
 97210 
 
 41 
 
 20 
 
 9.49768 
 
 9.97738 
 
 9.51991 
 
 9.97479 
 
 9.54093 
 
 9.97206 
 
 40 
 
 21 
 
 49806 
 
 97734 
 
 52027 
 
 97475 
 
 54127 
 
 97201 
 
 39 
 
 22 
 
 49844 
 
 97729 
 
 52063 
 
 97470 
 
 54161 
 
 97196 
 
 38 
 
 23 
 
 49882 
 
 97725 
 
 52099 
 
 97466 
 
 54195 
 
 97192 
 
 37 
 
 24 
 
 49920 
 
 97721 
 
 52135 
 
 97461 
 
 54229 
 
 97187 
 
 36 
 
 25 
 
 49958 
 
 9T717 
 
 52171 
 
 97457 
 
 54263 
 
 97182 
 
 35 
 
 26 
 
 49996 
 
 97713 
 
 52207 
 
 97453 
 
 54297 
 
 97178 
 
 34 
 
 27 
 
 50034 
 
 97708 
 
 52242 
 
 97448 
 
 54331 
 
 97173 
 
 33 
 
 28 
 
 50072 
 
 97704 
 
 52278 
 
 97444 
 
 54365 
 
 97168 
 
 32 
 
 29 
 
 50110 
 
 97700 
 
 52314 
 
 97439 
 
 54399 
 
 97163 
 
 31 
 
 30 
 
 9.50148 
 
 9.97696 
 
 9.52350 
 
 9.97435 
 
 9.54433 
 
 9.97159 
 
 30 
 
 31 
 
 50135 
 
 97691 
 
 52385 
 
 97430 
 
 54466 
 
 97154 
 
 29 
 
 32 
 
 50223 
 
 97687 
 
 52421 
 
 97426 
 
 54500 
 
 97149 
 
 28 
 
 33 
 
 50261 
 
 97683 
 
 52456 
 
 97421 
 
 51534 
 
 97145 
 
 27 
 
 34 
 
 50298 
 
 97679 
 
 52492 
 
 97417 
 
 54567 
 
 97140 
 
 26 
 
 35 
 
 50336 
 
 97674 
 
 52527 
 
 97412 
 
 54601 
 
 97135 
 
 25 
 
 36 
 
 50374 
 
 97670 
 
 5*663 
 
 97408 
 
 54635 
 
 97130 
 
 24 
 
 37 
 
 50411 
 
 97666 
 
 52598 
 
 97403 
 
 54668 
 
 97126 
 
 23 
 
 38 
 
 50449 
 
 97662 
 
 52634 
 
 97399 
 
 54702 
 
 97121 
 
 22 
 
 39 
 
 50486 
 
 97657 
 
 52669 
 
 97394 
 
 54735 
 
 97116 
 
 21 
 
 40 
 
 9.50523 
 
 9.97653 
 
 9.52705 
 
 9.97390 
 
 9.54769 
 
 9.97111 
 
 20 
 
 41 
 
 50561 
 
 976*9 
 
 52740 
 
 97385 
 
 54802 
 
 97107 
 
 19 
 
 42 
 
 5059S 
 
 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 
 
 97358 
 
 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 
 
 50933 
 
 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 
 
 51191 
 
 97576 
 
 53336 
 
 97308 
 
 55367 
 
 97025 
 
 2 
 
 59 
 
 51227 
 
 97571 
 
 53370 
 
 97303 
 
 55400 
 
 97020 
 
 1 
 
 60 
 
 51264 
 
 97567 
 
 53405 
 
 97299 
 
 55433 
 
 97015 
 
 
 
 i 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 / 
 
 
 
 71 
 
 
 70 
 
 
 69 
 
 
 296 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 ; 
 
 21 
 
 22 
 
 23 
 
 ' 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 o 
 
 9.55433 
 
 9.97015 
 
 9.57358 
 
 9.96717 
 
 9.59188 
 
 9.96403 \ 60 
 
 i 
 
 55466 
 
 97010 
 
 57389 
 
 96711 
 
 59218 
 
 96397 
 
 59 
 
 2 
 
 55499 
 
 97005 
 
 57420 
 
 96706 
 
 59247 
 
 96392 
 
 58 
 
 3 
 
 55532 
 
 97001 
 
 57451 
 
 96701 
 
 59277 
 
 963 "7 
 
 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 
 
 55663 
 
 96981 
 
 57576 
 
 96681 
 
 59396 
 
 9(5365 
 
 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 
 
 9.96349 
 
 50 
 
 11 
 
 55793 
 
 96962 
 
 57700 
 
 96060 
 
 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 
 
 96922 
 
 57947 
 
 96619 
 
 59749 
 
 96300 
 
 41 
 
 20 
 
 9.56085 
 
 9.96917 
 
 9.57978 
 
 9.96614 
 
 9.59778 
 
 9.96294 
 
 40 
 
 21 
 
 56118 
 
 96912 
 
 58008 
 
 96608 
 
 59808 
 
 96289 
 
 39 
 
 22 
 
 56150 
 
 96907 
 
 58039 
 
 96603 
 
 59837 
 
 96284 
 
 38 
 
 23 
 
 56182 
 
 96903 
 
 58070 
 
 96598 
 
 59866 
 
 96278 
 
 37 
 
 24 
 
 56215 
 
 96898 
 
 58101 
 
 96593 
 
 59895 
 
 96273 
 
 36 
 
 25 
 
 56247 
 
 96893 
 
 58131 
 
 96588 
 
 59924 
 
 96267 
 
 35 
 
 26 
 
 56279 
 
 96888 
 
 58162 
 
 96582 
 
 59954 
 
 96262 
 
 34 
 
 27 
 
 56311 
 
 96883 
 
 58192 
 
 96577 
 
 59983 
 
 96256 
 
 33 
 
 28 
 
 56343 
 
 96878 
 
 58223 
 
 96572 
 
 60012 
 
 96251 
 
 32 
 
 29 
 
 56375 
 
 96873 
 
 58253 
 
 96567 
 
 60041 
 
 96245 
 
 31 
 
 30 
 
 9.56408 
 
 9.96868 
 
 9.58284 
 
 9.96562 
 
 9.60070 
 
 9.96240 
 
 30 
 
 31 
 
 56440 
 
 96863 
 
 58314 
 
 96556 
 
 G0099 
 
 96234 
 
 29 
 
 32 
 
 56472 
 
 96858 
 
 58345 
 
 96551 
 
 60128 
 
 96229 
 
 28 
 
 33 
 
 56504 
 
 96853 
 
 58375 
 
 96546 
 
 60157 
 
 96223 
 
 27 
 
 34 
 
 56536 
 
 96848 
 
 58406 
 
 96541 
 
 60186 
 
 Mi 18 
 
 26 
 
 35 
 
 56568 
 
 96843 
 
 58436 
 
 96535 
 
 60215 
 
 96212 
 
 25 
 
 36 
 
 37 
 
 56599 
 56631 
 
 9G838 
 96833 
 
 58467 
 58497 
 
 96530 
 96525 
 
 60244 
 60273 
 
 96207 
 96201 
 
 24 
 23 
 
 38 
 
 56663 
 
 96828 
 
 58527 
 
 96520 
 
 60302 
 
 96196 
 
 22 
 
 39 
 
 56695 
 
 96823 
 
 58557 
 
 96514 
 
 60331 
 
 96190 
 
 21 
 
 40 
 
 9.56727 
 
 9.96818 
 
 9.58588 
 
 9.96509 
 
 9.60359 
 
 9.96185 
 
 20 
 
 41 
 
 ' 56759 
 
 96813 
 
 58618 
 
 96504 
 
 60388 
 
 96179 
 
 19 
 
 42 
 
 56790 
 
 96808 
 
 58648 
 
 96498 
 
 60417 
 
 96174 
 
 18 
 
 43 
 
 56822 
 
 96803 
 
 58678 
 
 96493 
 
 60446 
 
 96168 
 
 17 
 
 44 
 
 56854 
 
 96798 
 
 58709 
 
 96488 
 
 60474 
 
 96162 
 
 16 
 
 45 
 
 56886 
 
 96793 
 
 58739 
 
 96483 
 
 60503 
 
 96157 
 
 15 
 
 46 
 
 56917 
 
 96788 
 
 58769 
 
 96477 
 
 60532 
 
 96151 
 
 14 
 
 47 
 
 56949 
 
 96783 
 
 58799 
 
 96472 
 
 60561 
 
 96146 
 
 13 
 
 48 
 
 56980 
 
 96778 
 
 58829 
 
 96467 
 
 60589 
 
 96140 
 
 12 
 
 49 
 
 57012 
 
 96772 
 
 58859 
 
 96461 
 
 60618 
 
 96135 
 
 11 
 
 50 
 
 9.57044 
 
 9.96767 
 
 9.58889 
 
 9.96456 
 
 9.60646 
 
 9.96129 
 
 10 
 
 51 
 
 57075 
 
 96762 
 
 58919 
 
 96451 
 
 60675 
 
 96123 
 
 9 
 
 52 
 
 57107 
 
 96757 
 
 58949 
 
 96445 
 
 60704 
 
 96118 
 
 g 
 
 53 
 
 57138 
 
 96752 
 
 58979 
 
 96440 
 
 60732 
 
 96112 
 
 7 
 
 54 
 
 57169 
 
 96747 
 
 59009 
 
 96435 
 
 60761 
 
 96107 
 
 6 
 
 55 
 
 57201 
 
 96742 
 
 59039 
 
 96429 
 
 60789 
 
 96101 
 
 5 
 
 56 
 
 57232 
 
 96737 
 
 59069 
 
 96424 
 
 60818 
 
 96095 
 
 4 
 
 57 
 
 57264 
 
 96732 
 
 59098 
 
 96419 
 
 60846 
 
 96090 
 
 g 
 
 58 
 
 57295 
 
 96727 
 
 59128 
 
 96413 
 
 60875 
 
 960S4 
 
 2 
 
 59 
 
 57326 
 
 96722 
 
 59158 
 
 96408 
 
 60903 
 
 96079 
 
 
 60 
 
 57358 
 
 96717 
 
 59188 
 
 96403 
 
 60931 
 
 96073 
 
 
 
 t 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 
 
 
 
 68 
 
 
 67 
 
 
 66 
 
 297 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 / 
 
 
 24 
 
 
 25 
 
 
 26 
 
 / 
 
 
 Sit.e 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 9.60931 
 
 9.96073 
 
 9.62595 
 
 9.95728 
 
 9.G4184 
 
 9.95366 
 
 60 
 
 1 
 
 60960 
 
 96067 
 
 62622 
 
 95722 
 
 64210 
 
 95360 
 
 59 
 
 2 
 
 6098S 
 
 960G2 
 
 62649 
 
 95716 
 
 64236 
 
 95354 
 
 58 
 
 3 
 
 61016 
 
 9G056 
 
 62676 
 
 95710 
 
 642G2 
 
 95348 
 
 57 
 
 4 
 
 61045 
 
 96050 
 
 62703 
 
 95704 
 
 64288 
 
 95341 
 
 56 
 
 5 
 
 61073 
 
 96045 
 
 62730 
 
 95G98 
 
 64313 
 
 95335 
 
 55 
 
 6 
 
 61101 
 
 96039 
 
 62757 
 
 95692 
 
 64339 
 
 95329 
 
 54 
 
 7 
 
 61129 
 
 96034 
 
 62784 
 
 95686 
 
 64365 
 
 95323 
 
 53 
 
 8 
 
 61158 
 
 9(3028 
 
 62811 
 
 956SO 
 
 64391 
 
 95317 
 
 52 
 
 9 
 
 61186 
 
 96022 
 
 62838 
 
 95674 
 
 64417 
 
 95310 
 
 51 
 
 10 
 
 9.61214 
 
 9.96017 
 
 9.62865 
 
 9.95668 
 
 9.64442 
 
 9.95304 
 
 50 
 
 11 
 
 61242 
 
 96011 
 
 62892 
 
 95663 
 
 64468 
 
 95298 i 49 
 
 12 
 
 61270 
 
 96005 
 
 62918 
 
 95657 
 
 64494 
 
 95292 
 
 48 
 
 13 
 
 61*98 
 
 96000 
 
 62945 
 
 95651 
 
 64519 
 
 95286 
 
 47 
 
 14 
 
 61326 
 
 95994 
 
 62972 
 
 95645 
 
 64545 
 
 95279 
 
 46 
 
 15 
 
 61354 
 
 95988 
 
 62999 
 
 95639 
 
 64571 
 
 95273 
 
 45 
 
 16 
 
 61382 
 
 95982 
 
 63026 
 
 95633 
 
 64596 
 
 95267 
 
 44 
 
 17 
 
 61411 
 
 95977 
 
 63052 
 
 95627 
 
 64622 
 
 95261 
 
 43 
 
 18 
 
 61438 
 
 95971 
 
 63079 
 
 95621 
 
 64647 
 
 95254 
 
 42 
 
 19 
 
 6 1466 
 
 95965 
 
 63106 
 
 95615 
 
 64673 
 
 95248 
 
 41 
 
 20 
 
 9.61494 
 
 9.95960 
 
 9.63133 
 
 9.95609 
 
 9.64698 
 
 9.95242 
 
 40 
 
 21 
 
 61522 
 
 95954 
 
 63159 
 
 95603 
 
 64724 
 
 952^6 
 
 39 
 
 22 
 
 61550 
 
 95948 
 
 63186 
 
 95597 
 
 64749 
 
 95229 
 
 38 
 
 23 
 
 61578 
 
 95942 
 
 63213 
 
 95591 
 
 64775 
 
 95223 
 
 37 
 
 24 
 
 61606 
 
 95937 
 
 63239 
 
 95585 
 
 64800 
 
 95217 
 
 36 
 
 25 
 
 61634 
 
 95931 
 
 63266 
 
 95579 
 
 64826 
 
 95211 
 
 35 
 
 26 
 
 61662 
 
 95925 
 
 63292 
 
 95573 
 
 64851 
 
 95204 
 
 34 
 
 27 
 
 61689 
 
 95920 
 
 63319 
 
 95567 
 
 C4877 
 
 95198 
 
 33 
 
 28 
 
 61717 
 
 95914 
 
 63345 
 
 95561 
 
 64902 
 
 95192 
 
 32 
 
 29 
 
 61745 
 
 95908 
 
 63372 
 
 95555 
 
 64927 
 
 95185 
 
 31 
 
 30 
 
 9.61773 
 
 9.95902 
 
 9.63398 
 
 9.95549 
 
 9.64953 
 
 9.95179 
 
 30 
 
 31 
 
 61800 
 
 95897 
 
 63425 
 
 95543 
 
 64978 
 
 95173 
 
 29 
 
 32 
 
 61828 
 
 95891 
 
 63451 
 
 95537 
 
 65003 
 
 95167 28 
 
 33 
 
 61856 
 
 95885 
 
 63478 
 
 95531 
 
 65029 
 
 95160 27 
 
 34 
 
 61883 
 
 95879 
 
 63504 
 
 95525 
 
 65054 
 
 95154 ! 26 
 
 35 
 
 61911 
 
 95873 
 
 63531 
 
 95519 
 
 65079 
 
 95148 25 
 
 36 
 
 61939 
 
 95S68 
 
 63557 
 
 95513 
 
 65104 
 
 95141 24 
 
 37 
 
 61966 
 
 95862 
 
 635&3 
 
 95507 
 
 65130 
 
 95135 23 
 
 38 
 
 61994 
 
 95856 
 
 63610 
 
 95500 
 
 65155 
 
 95129 
 
 22 
 
 39 
 
 62021 
 
 95850 
 
 63636 
 
 95494 
 
 65180 
 
 95122 
 
 21 
 
 40 
 
 9.6-2049 
 
 9.95844 
 
 9.63662 
 
 9.95488 
 
 9.65205 
 
 9.95116 
 
 20 
 
 41 
 
 62076 
 
 95839 
 
 63689 
 
 95482 
 
 65230 
 
 95110 
 
 19 
 
 42 
 
 62104 
 
 95833 
 
 63715 
 
 95476 
 
 65255 
 
 95103 
 
 18 
 
 43 
 
 62131 
 
 95827 
 
 63741 
 
 95470 
 
 65281 
 
 95097 
 
 17 
 
 44 
 
 62159 
 
 95821 
 
 63767 
 
 95464 
 
 65306 
 
 95090 
 
 16 
 
 45 
 
 62186 
 
 95815 
 
 63794 
 
 95458 
 
 65331 
 
 95084 
 
 15 
 
 46 
 
 62214 
 
 95810 
 
 63820 
 
 95452 
 
 65356 
 
 1)5078 
 
 14 
 
 47 
 
 62241 
 
 95804 
 
 63846 
 
 95446 
 
 65381 
 
 95071 
 
 13 
 
 48 
 
 62268 
 
 95798 
 
 63S72 
 
 95440 
 
 65406 
 
 95065 
 
 12 
 
 49 
 
 62296 
 
 95792 
 
 63898 
 
 95434 
 
 65431 
 
 95059 
 
 11 
 
 50 
 
 9.62323 
 
 9.95786 
 
 9.63924 
 
 9.95427 
 
 9.65456 
 
 9.95052 
 
 10 
 
 51 
 
 62350 
 
 95780 
 
 63950 
 
 95421 
 
 65481 
 
 95046 
 
 9 
 
 52 
 
 62377 
 
 95775 
 
 63976 
 
 95415 
 
 65506 
 
 95039 
 
 8 
 
 53 
 
 62405 
 
 95769 
 
 64002 
 
 95409 
 
 65531 
 
 95033 
 
 7 
 
 54 
 
 62432 
 
 95763 
 
 64028 
 
 95403 
 
 65565 
 
 95027 
 
 6 
 
 55 
 
 62459 
 
 95757 
 
 64054 
 
 95397 
 
 65580 
 
 95020 
 
 5 
 
 56 
 
 62486 
 
 95751 
 
 64080 
 
 95391 
 
 65605 
 
 95014 
 
 4 
 
 57 
 
 62513 
 
 95745 
 
 64106 
 
 95384 
 
 65630 
 
 95007 
 
 3 
 
 58 
 
 62541 
 
 95739 
 
 64132 
 
 95378 
 
 65655 
 
 95001 
 
 2 
 
 59 
 
 62568 
 
 95733 
 
 64158 
 
 95372 
 
 65680 
 
 94995 
 
 1 
 
 60 
 
 62595 
 
 95728 
 
 64184 
 
 95366 
 
 65705 
 
 9498 
 
 
 
 / 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 / 
 
 
 
 66 
 
 
 64 
 
 
 63 
 
 
 298 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 
 27 
 
 28 
 
 29 
 
 f 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 1 
 
 9.65705 
 
 9.94988 
 
 9.G7161 
 
 9.94593 
 
 9.68557 
 
 9.94182 
 
 60 
 
 1 
 
 65729 
 
 94982 
 
 67186 
 
 94587 
 
 68580 
 
 94175 
 
 59 
 
 2 
 
 65754 
 
 94975 
 
 67208 
 
 94580 
 
 68603 
 
 94168 
 
 58 
 
 3 
 
 65779 
 
 94969 
 
 67232 
 
 94573 
 
 68625 
 
 94161 
 
 57 
 
 4 
 
 65804 
 
 94962 
 
 67256 
 
 94567 
 
 68648 
 
 94154 
 
 56 
 
 5 
 
 65828 
 
 94956 
 
 67280 
 
 94560 
 
 68G71 
 
 94147 
 
 55 
 
 6 
 
 65853 
 
 94949 
 
 67303 
 
 94553 
 
 68694 
 
 94140 
 
 54 
 
 7 
 
 65878 
 
 94943 
 
 67327 
 
 94546 
 
 68716 
 
 94133 
 
 53 
 
 8 
 
 65902 
 
 94936 
 
 67350 
 
 94540 
 
 68739 
 
 94126 
 
 52 
 
 9 
 
 G5927 
 
 94930 
 
 67374 
 
 94533 
 
 68762 
 
 94119 
 
 51 
 
 10 
 
 9.65952 
 
 9.94923 
 
 9.67398 
 
 9.94526 
 
 9.68784 
 
 9.94112 
 
 50 
 
 11 
 
 65976 
 
 94917 
 
 67421 
 
 94519 
 
 68807 
 
 94105 
 
 W 
 
 12 
 
 66001 
 
 94911 
 
 67445 
 
 94513 
 
 68829 
 
 94098 
 
 48 
 
 13 
 
 66025 
 
 94904 
 
 67468 
 
 94506 
 
 68852 
 
 94090 
 
 47 
 
 14 
 
 66050 
 
 94898 
 
 67492 
 
 94499 
 
 68875 
 
 94083 
 
 46 
 
 15 
 
 66075 
 
 94891 
 
 67515 
 
 94492 
 
 68897 
 
 94076 
 
 45 
 
 16 
 
 66099 
 
 94885 
 
 67539 
 
 94485 
 
 68920 
 
 940G9 
 
 44 
 
 17 
 
 66124 
 
 94878 
 
 67562 
 
 94479 
 
 68942 
 
 94062 
 
 43 
 
 18 
 
 66148 
 
 94871 
 
 67586 
 
 94472 
 
 68965 
 
 94055 
 
 42 
 
 19 
 
 66173 
 
 948G5 
 
 67609 
 
 94465 
 
 68987 
 
 94048 
 
 41 
 
 20 
 
 9.66197 
 
 9.94858 
 
 9.67633 
 
 9.94458 
 
 9.69010 
 
 9.94041 
 
 40 
 
 21 
 
 66221 
 
 94852 
 
 67656 
 
 94451 
 
 69032 
 
 94034 
 
 39 
 
 22 
 
 66246 
 
 94S45 
 
 67680 
 
 94445 
 
 69055 
 
 94027 
 
 38 
 
 23 
 
 66-J70 
 
 94839 
 
 67703 
 
 94438 
 
 69077 
 
 94020 
 
 37 
 
 24 
 
 66295 
 
 94832 
 
 67726 
 
 94431 
 
 69100 
 
 94012 
 
 36 
 
 25 
 
 66319 
 
 94826 
 
 67750 
 
 94424 
 
 69122 
 
 94005 
 
 35 
 
 26 
 
 66343 
 
 94819 
 
 67773 
 
 94417 
 
 69144 
 
 93998 
 
 34 
 
 27 
 
 66368 
 
 94813 
 
 67796 
 
 94410 
 
 69167 
 
 93991 
 
 33 
 
 28 
 
 66302 
 
 94806 
 
 67820 
 
 94404 
 
 69189 
 
 93984 
 
 32 
 
 29 
 
 66416 
 
 94799 
 
 67843 
 
 94397 
 
 69212 
 
 93977 
 
 31 
 
 30 
 
 9.66441 
 
 9.94793 
 
 9.67866 
 
 9.94390 
 
 9.69234 
 
 9.93970 
 
 30 
 
 31 
 
 66465 
 
 94786 
 
 67890 
 
 94383 
 
 69256 
 
 93963 
 
 29 
 
 32 
 
 66489 
 
 94780 
 
 67913 
 
 94376 
 
 69279 
 
 93955 
 
 28 
 
 33 
 
 66513 
 
 94773 
 
 67936 
 
 94369 
 
 69301 
 
 93948 
 
 27 
 
 34 
 
 66537 
 
 947G7 
 
 67959 
 
 94362 
 
 69323 
 
 93941 
 
 26 
 
 35 
 
 665G2 
 
 94760 
 
 67982 
 
 94355 
 
 69345 
 
 93934 
 
 25 
 
 36 
 
 66K6 
 
 94753 
 
 68006 
 
 94349 
 
 69368 
 
 93927 
 
 24 
 
 37 
 
 66610 
 
 94747 
 
 C8029 
 
 94342 
 
 69390 
 
 93920 
 
 23 
 
 38 
 
 66634 
 
 947'40 
 
 68052 
 
 94335 
 
 69412 
 
 93912 
 
 22 
 
 39 
 
 66G58 
 
 94734 
 
 68075 
 
 94328 
 
 69434 
 
 93905 
 
 21 
 
 40 
 
 9.666^2 
 
 9.94727 
 
 9.68098 
 
 9.94321 
 
 9.69456 
 
 9.93898 
 
 20 
 
 41 
 
 66706 
 
 94720 
 
 68121 
 
 94314 
 
 69479 
 
 93891 
 
 19 
 
 42 
 
 66731 
 
 94714 
 
 68144 
 
 94307 
 
 69501 
 
 93884 
 
 18 
 
 43 
 
 66755 
 
 94707 
 
 68167 
 
 94300 
 
 69523 
 
 93876 
 
 17 
 
 44 
 
 66779 
 
 94700 
 
 68190 
 
 94293 
 
 69545 
 
 93869 
 
 16 
 
 45 
 
 66803 
 
 94694 
 
 68213 
 
 94286 
 
 69567 
 
 93862 
 
 15 
 
 46 
 
 66827 
 
 94687 
 
 68237 
 
 94279 
 
 69589 
 
 938f,5 
 
 14 
 
 47 
 
 66851 
 
 94680 
 
 68260 
 
 94273 
 
 69611 
 
 93847 
 
 13 
 
 48 
 
 66875 
 
 94674 
 
 68283 
 
 94266 
 
 696:^3 
 
 93840 
 
 12 
 
 49 
 
 66899 
 
 94G67 
 
 68305 
 
 94259 
 
 69655 
 
 93833 
 
 11 
 
 50 
 
 9.66922 
 
 9.94660 
 
 9.68328 
 
 9.94252 
 
 9.69677 
 
 9.93826 
 
 JO 
 
 51 
 
 66946 
 
 94654 
 
 68351 
 
 94245 
 
 69G99 
 
 93819 
 
 9 
 
 52 
 
 66970 
 
 94647 
 
 68374 
 
 94238 
 
 69721 
 
 93811 
 
 8 
 
 53 
 
 66994 
 
 94640 
 
 68397 
 
 94231 
 
 69743 
 
 93804 
 
 7 
 
 54 
 
 67018 
 
 94634 
 
 68420 
 
 94224 
 
 69765 
 
 93797 
 
 6 
 
 55 
 
 67042 
 
 94627 
 
 68443 
 
 94217 
 
 69787 
 
 93789 
 
 5 
 
 56 
 
 67066 
 
 94620 
 
 68466 
 
 94210 
 
 69809 
 
 93781 
 
 4 
 
 57 
 
 67090 
 
 94614 
 
 68489 
 
 94203 
 
 69831 
 
 93775 
 
 3 
 
 58 
 
 67113 
 
 94607 
 
 68512 
 
 94196 
 
 69853 
 
 93768 
 
 2 
 
 59 
 
 67137 
 
 94600 
 
 68534 
 
 94189 
 
 69875 
 
 93760 
 
 1 
 
 60 
 
 67161 
 
 94593 
 
 68557 
 
 94182 
 
 69897 
 
 93753 
 
 
 
 f 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 \ 
 
 
 < 
 
 82 
 
 
 61 
 
 
 60 
 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 
 30 
 
 31 
 
 
 
 82 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 9.69897 
 
 9.93753 
 
 9.71184 
 
 9.93307 
 
 9.72421 
 
 9.92842 
 
 60 
 
 1 
 
 69919 
 
 93746 
 
 71205 
 
 93299 
 
 72441 
 
 92834 
 
 59 
 
 2 
 
 69941 
 
 93738 
 
 71226 
 
 93291 
 
 72461 
 
 92826 
 
 58 
 
 3 
 
 69963 
 
 93731 
 
 71247 
 
 93284 
 
 72482 
 
 92818 
 
 57 
 
 4 
 
 69984 
 
 93724 
 
 71268 
 
 93276 
 
 72502 
 
 92810 
 
 56 
 
 5 
 
 70006 
 
 93717 
 
 71289 
 
 93269 
 
 72522 
 
 92803 
 
 55 
 
 6 
 
 70028 
 
 93709 
 
 71310 
 
 93261 
 
 72542 
 
 92795 
 
 54 
 
 7 
 
 70050 
 
 93702 
 
 71331 
 
 93253 
 
 72562 
 
 92787 
 
 53 
 
 8 
 
 70072 
 
 93095 
 
 71352 
 
 93246 
 
 72582 
 
 92779 
 
 52 
 
 9 
 
 70093 
 
 93687 
 
 71373 
 
 93238 
 
 72602 
 
 92771 
 
 51 
 
 10 
 
 9.70115 
 
 9.]680 
 
 9.71393 
 
 9.93230 
 
 9.72622 
 
 9.92763 
 
 50 
 
 11 
 
 70137 
 
 93673 
 
 71414 
 
 93&.'3 
 
 72643 
 
 92755 
 
 49 
 
 12 
 
 70159 
 
 93665 
 
 71435 
 
 93215 
 
 72663 
 
 92747 
 
 48 
 
 13 
 
 70180 
 
 93658 
 
 71456 
 
 93207 
 
 72683 
 
 92739 
 
 47 
 
 14 
 
 70202 
 
 93650 
 
 71477 
 
 93-200 
 
 72703 
 
 92731 
 
 46 
 
 15 
 
 70224 
 
 93643 
 
 71498 
 
 93192 
 
 72723 
 
 92723 
 
 45 
 
 16 
 
 70245 
 
 93636 
 
 71519 
 
 93184 
 
 72743 
 
 92715 
 
 44 
 
 17 
 
 70267 
 
 93628 
 
 71539 
 
 93177 
 
 72763 
 
 92707 
 
 43 
 
 18 
 
 70288 
 
 93621 
 
 71560 
 
 93169 
 
 72783 
 
 9-2699 
 
 42 
 
 19 
 
 70310 
 
 93614 
 
 71581 
 
 93161 
 
 72803 
 
 92691 
 
 41 
 
 20 
 
 9.70332 
 
 9.93606 
 
 9.71602 
 
 9.93154 
 
 9.72823 
 
 9.92683 
 
 40 
 
 21 
 
 70353 
 
 93599 
 
 71622 
 
 93146 
 
 72843 
 
 9267'5 
 
 39 
 
 22 
 
 70375 
 
 93591 
 
 71643 
 
 93138 
 
 72863 
 
 92667 
 
 38 
 
 23 
 
 70396 
 
 93584 
 
 71664 
 
 93131 
 
 72883 
 
 92659 
 
 37 
 
 24 
 
 70418 
 
 93577 
 
 71685 
 
 93123 
 
 72902 
 
 92651 
 
 36 
 
 25 
 
 70439 
 
 93569 
 
 71705 
 
 93115 
 
 72922 
 
 92643 
 
 35 
 
 26 
 
 70461 
 
 93562 
 
 71726 
 
 93108 
 
 72942 
 
 92635 
 
 34 
 
 27 
 
 70482 
 
 93554 
 
 71747 
 
 93100 
 
 72962 
 
 92627 
 
 33 
 
 28 
 
 70504 
 
 93547 
 
 71767 
 
 93092 
 
 72982 
 
 92619 
 
 32 
 
 29 
 
 70525 
 
 93539 
 
 71788 
 
 93084 
 
 73002 
 
 92611 
 
 31 
 
 30 
 
 9.70547 
 
 9.93532 
 
 9.71809 
 
 9.93077 
 
 9.73022 
 
 9.92603 
 
 30 
 
 31 
 
 70568 
 
 93525 
 
 71829 
 
 93069 
 
 73041 
 
 92595 
 
 29 
 
 38 
 
 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 
 
 92563 
 
 25 
 
 36 
 
 70675 
 
 93487 
 
 71932 
 
 93030 
 
 73140 
 
 92555 
 
 24 
 
 87 
 
 70697 
 
 93480 
 
 71952 
 
 93022 
 
 73160 
 
 92546 
 
 23 
 
 38 
 
 70718 
 
 93472 
 
 71973 
 
 93014 
 
 73180 
 
 9-2538 
 
 22 
 
 39 
 
 70739 
 
 93465 
 
 71994 
 
 93007 
 
 73200 
 
 92D30 
 
 21 
 
 40 
 
 9.70761 
 
 9.93457 
 
 9.72014 
 
 9.92999 
 
 9.73219 
 
 9.92522 
 
 20 
 
 41 
 
 70782 
 
 93450 
 
 72034 
 
 92991 
 
 73239 
 
 92514 
 
 19 
 
 42 
 
 70803 
 
 93442 
 
 72055 
 
 92983 
 
 73259 
 
 92506 
 
 18 
 
 43 
 
 70824 
 
 93435 
 
 72075 
 
 92976 
 
 73278 
 
 9-2498 
 
 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 
 
 78857 
 
 92465 
 
 13 
 
 48 
 
 70931 
 
 93397 
 
 72177 
 
 92936 
 
 73377 
 
 92457 I 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 
 
 53 
 
 71015 
 
 93367 
 
 72259 
 
 9-2905 
 
 73455 
 
 92425 
 
 8 
 
 53 
 
 71036 
 
 93360 
 
 72279 
 
 92897 
 
 73474 
 
 92416 
 
 7 
 
 54 
 
 71058 
 
 93352 
 
 72299 
 
 92889 
 
 73494 
 
 92408 
 
 6 
 
 55 
 
 71079 
 
 93344 
 
 72320 
 
 92881 
 
 73513 
 
 92400 
 
 5 
 
 56 
 
 71100 
 
 93337 
 
 72340 
 
 92874 
 
 73533 
 
 92392 
 
 4 
 
 57 
 
 71121 
 
 93329 
 
 72360 
 
 92866 
 
 73552 
 
 92384 
 
 3 
 
 58 
 
 71142 
 
 93322 
 
 72381 
 
 92858 
 
 73572 
 
 92376 
 
 2 
 
 59 
 
 71163 
 
 93314 
 
 72401 
 
 92850 
 
 73591 
 
 92367 
 
 1 
 
 60 
 
 71184 
 
 93307 
 
 72421 
 
 92842 
 
 73611 
 
 9-2359 
 
 
 
 / 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 / 
 
 
 
 69 
 
 J 
 
 8 
 
 
 57 
 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 
 33 
 
 84 
 
 35 
 
 
 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 9.73611 
 
 9.92359 
 
 9.74756 
 
 9.91857 
 
 9.75859 
 
 9.91336 
 
 60 
 
 1 
 
 73630 
 
 92351 
 
 74775 
 
 91849 
 
 75877 
 
 91328 
 
 59 
 
 2 
 
 73650 
 
 92343 
 
 74794 
 
 91840 
 
 75895 
 
 91319 
 
 58 
 
 3 
 
 73669 
 
 92335 
 
 74812 
 
 91832 
 
 75913 
 
 91310 
 
 57 
 
 4 
 
 73689 
 
 92326 
 
 74831 
 
 91823 
 
 75931 
 
 91301 
 
 56 
 
 5 
 
 73708 
 
 92318 
 
 74850 
 
 91815 
 
 75949 
 
 91292 
 
 55 
 
 6 
 
 73727 
 
 92310 
 
 74868 
 
 91806 
 
 75967 
 
 91283 
 
 54 
 
 7 
 
 73747 
 
 92302 
 
 74887 
 
 91798 
 
 75985 
 
 91274 
 
 53 
 
 8 
 
 73766 
 
 92293 
 
 74906 
 
 91789 
 
 76003 
 
 91266 
 
 52 
 
 9 
 
 73785 
 
 92285 
 
 74924 
 
 91781 
 
 76021 
 
 91257 
 
 51 
 
 10 
 
 9.73805 
 
 9.92277 
 
 9.74943 
 
 9.91772 
 
 9.76039 
 
 9.91248 
 
 50 
 
 11 
 
 73824 
 
 92269 
 
 74961 
 
 91763 
 
 76057 
 
 91239 
 
 49 
 
 12 
 
 73843 
 
 92260 
 
 74980 
 
 91755 
 
 76075 
 
 91230 
 
 48 
 
 13 
 
 73863 
 
 92252 
 
 74999 
 
 91746 
 
 76093 
 
 91221 
 
 47 
 
 14 
 
 73882 
 
 92244 
 
 75017 
 
 91738 
 
 76111 
 
 91212 
 
 46 
 
 15 
 
 73901 
 
 92235 
 
 75036 
 
 91729 
 
 76129 
 
 ,91203 
 
 45 
 
 16 
 
 73921 
 
 92227 
 
 75054 
 
 91720 
 
 76146 
 
 91194 
 
 44 
 
 17 
 
 73940 
 
 92219 
 
 75073 
 
 91712 
 
 76164 
 
 91185 
 
 43 
 
 18 
 
 73959 
 
 92211 
 
 75091 
 
 91703 
 
 76182 
 
 91176 
 
 42 
 
 19 
 
 73978 
 
 92202 
 
 75110 
 
 91695 
 
 76200 
 
 91167 
 
 41 
 
 20 
 
 9.73997 
 
 9.92194 
 
 9.75128 
 
 9. 91 f 86 
 
 9.76218 
 
 9.91158 
 
 40 
 
 21 
 
 74017 
 
 92186 
 
 75147 
 
 91677 
 
 76236 
 
 91149 
 
 39 
 
 22 
 
 74036 
 
 92177 
 
 75165 
 
 91669 
 
 76253 
 
 91141 
 
 38 
 
 23 
 
 74055 
 
 92169 
 
 75184 
 
 91660 
 
 76271 
 
 91132 
 
 37 
 
 24 
 
 74074 
 
 92161 
 
 75202 
 
 91651 
 
 76289 
 
 91123 
 
 36 
 
 25 
 
 74093 
 
 92152 
 
 75221 
 
 91643 
 
 76307 
 
 91114 
 
 35 
 
 26 
 
 74113 
 
 92144 
 
 75239 
 
 91684 
 
 76324 
 
 91105 
 
 34 
 
 27 
 
 74132 
 
 92136 
 
 75258 
 
 91625 
 
 76342 
 
 91096 
 
 33 
 
 28 
 
 74151 
 
 92127 
 
 75276 
 
 91617 
 
 76360 
 
 91087 
 
 32 
 
 29 
 
 74170 
 
 92119 
 
 75294 
 
 91608 
 
 76378 
 
 91078 
 
 31 
 
 30 
 
 9.74189 
 
 9.92111 
 
 9.75313 
 
 9.91599 
 
 9.76395 
 
 9.91069 
 
 30 
 
 31 
 
 74208 
 
 92102 
 
 75331 
 
 91591 
 
 76413 
 
 91060 
 
 29 
 
 32 
 
 74227 
 
 92094 
 
 75350 
 
 91582 
 
 76431 
 
 91051 
 
 28 
 
 33 
 
 74246 
 
 92086 
 
 75368 
 
 91573 
 
 76448 
 
 91042 
 
 27 
 
 34 
 
 74265 
 
 92077 
 
 75386 
 
 91565 
 
 76466 
 
 91033 
 
 26 
 
 35 
 
 74284 
 
 92069 
 
 75405 
 
 91556 
 
 76484 
 
 91023 
 
 25 
 
 36 
 
 74303 
 
 92060 
 
 75423 
 
 91547 
 
 76501 
 
 91014 
 
 24 
 
 37 
 
 74322 
 
 92052 
 
 75441 
 
 91538 
 
 76519 
 
 91005 
 
 23 
 
 38 
 
 74341 
 
 92044 
 
 75459 
 
 91530 
 
 76537 
 
 90996 
 
 22 
 
 39 
 
 74360 
 
 92035 
 
 75478 
 
 91521 
 
 76554 
 
 90987 
 
 21 
 
 40 
 
 9.74379 
 
 9.92027 
 
 9.75496 
 
 9.91512 
 
 9.76572 
 
 9.90978 
 
 20 
 
 41 
 
 74398 
 
 92018 
 
 75514 
 
 91504 
 
 76590 
 
 90969 
 
 19 
 
 42 
 
 74417 
 
 92010 
 
 75533 
 
 91495 
 
 76607 
 
 90960 
 
 18 
 
 43 
 
 74436 
 
 92002 
 
 75551 
 
 91486 
 
 76625 
 
 90951 
 
 17 
 
 44 
 
 74455 
 
 91993 
 
 75569 
 
 91477 
 
 76642 
 
 90942 
 
 16 
 
 45 
 
 74474 
 
 91985 
 
 75587 
 
 91469 
 
 76660 
 
 90933 
 
 15 
 
 46 
 
 74493 
 
 91976 
 
 75605 
 
 91460 
 
 76677 
 
 90924 
 
 14 
 
 47 
 
 74512 
 
 91968 
 
 75624 
 
 91451 
 
 76695 
 
 90915 
 
 13 
 
 48 
 
 74531 
 
 91959 
 
 75642 
 
 91442 
 
 76712 
 
 90906 
 
 12 
 
 49 
 
 74549 
 
 91951 
 
 75660 
 
 91433 
 
 76730 
 
 90896 
 
 11 
 
 50 
 
 9.74568 
 
 9.01942 
 
 9.75678 
 
 9.91425 
 
 9.76747 
 
 9.90887 
 
 10 
 
 51 
 
 74587 
 
 91934 
 
 75696 
 
 91416 
 
 76765 
 
 90878 
 
 9 
 
 52 
 
 74606 
 
 91925 
 
 75714 
 
 91407 
 
 76782 
 
 90869 
 
 8 
 
 53 
 
 74625 
 
 91917 
 
 75733 
 
 91398 
 
 76800 
 
 90860 
 
 
 54 
 
 74644 
 
 91908 
 
 75751 
 
 91389 
 
 76817 
 
 90851 
 
 6 
 
 55 
 
 74662 
 
 91900 
 
 75769 
 
 91381 
 
 76835 
 
 90842 
 
 5 
 
 56 
 
 74681 
 
 91891 
 
 75787 
 
 91372 
 
 76852 
 
 90832 
 
 4 
 
 57 
 
 74700 
 
 91883 
 
 75805 
 
 91363 
 
 76870 
 
 90823 
 
 3 
 
 58 
 
 74719 
 
 91874 
 
 75823 
 
 91354 
 
 76887 
 
 90814 
 
 2 
 
 59 
 
 74737 
 
 91866 
 
 75841 
 
 91345 
 
 76904 
 
 90805 
 
 1 
 
 60 
 
 74756 
 
 91857 
 
 75859 
 
 91336 
 
 76922 
 
 90796 
 
 
 
 f 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 J 
 
 
 
 66 
 
 
 65 
 
 
 64 
 
 
 301 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 f 
 
 
 36 
 
 
 37 
 
 
 38 
 
 i 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 9.76922 
 
 9.90796 
 
 9.77946 
 
 9.90235 
 
 9.78934 
 
 9.89653 
 
 GO 
 
 1 
 
 76939 
 
 90787 
 
 77963 
 
 90225 
 
 78950 
 
 89643 
 
 59 
 
 2 
 
 76957 
 
 90777 
 
 77980 
 
 90216 
 
 78967 
 
 89633 
 
 58 
 
 3 
 
 76974 
 
 90768 
 
 77997 
 
 90206 
 
 78983 
 
 89624 
 
 57 
 
 4 
 
 76991 
 
 90759 
 
 78013 
 
 90197 
 
 78999 
 
 89614 
 
 50 
 
 5 
 
 77009 
 
 90750 
 
 78030 
 
 90187 
 
 79015 
 
 89604 
 
 55 
 
 6 
 
 77026 
 
 90741 
 
 78047 
 
 90178 
 
 79031 
 
 89594 
 
 54 
 
 7 
 
 77043 
 
 90731 
 
 78063 
 
 90168 
 
 79047 
 
 89584 
 
 53 
 
 8 
 
 77061 
 
 90722 
 
 78080 
 
 90159 
 
 79063 
 
 89574 
 
 52 
 
 9 
 
 77078 
 
 90713 
 
 78097 
 
 90149 
 
 79079 
 
 89564 
 
 51 
 
 10 
 
 9.77095 
 
 9.90704 
 
 9.78113 
 
 9.90139 
 
 9.79095 
 
 9.89554 
 
 50 
 
 11 
 
 77112 
 
 90694 
 
 78130 
 
 90130 
 
 79111 
 
 89544 
 
 4!) 
 
 12 
 
 77130 
 
 90685 
 
 78147 
 
 90120 
 
 79128 
 
 89534 
 
 48 
 
 13 
 
 77147 
 
 90676 
 
 78163 
 
 90111 
 
 79144 
 
 89524 
 
 47 
 
 14 
 
 77164 
 
 90667 
 
 78180 
 
 90101 
 
 79160 
 
 89514 
 
 46 
 
 15 
 
 77181 
 
 90657 
 
 78197 
 
 90091 
 
 79176 
 
 89504 
 
 45 
 
 16 
 
 77199 
 
 90648 
 
 78213 
 
 90082 
 
 79192 
 
 89495 
 
 44 
 
 17 
 
 77216 
 
 90639 
 
 78230 
 
 90072 
 
 79208 
 
 89485 
 
 43 
 
 18 
 
 77233 
 
 90630 
 
 78246 
 
 90063 
 
 79224 
 
 89475 
 
 42 
 
 19 
 
 77250 
 
 90620 
 
 78263 
 
 90053 
 
 79240 
 
 89465 
 
 41 
 
 20 
 
 9.77268 
 
 9.90611 
 
 9.7S280 
 
 9.90043 
 
 9.79256 
 
 9.89455 
 
 40 
 
 21 
 
 77285 
 
 90602 
 
 78296 
 
 90034 
 
 79272 
 
 89445 
 
 39 
 
 22 
 
 77302 
 
 90592 
 
 78313 
 
 90024 
 
 79288 
 
 89435 
 
 38 
 
 23 
 
 77319 
 
 90583 
 
 78329 
 
 90014 
 
 79304 
 
 89425 
 
 37 
 
 24 
 
 77336 
 
 90574 
 
 7834G 
 
 90005 
 
 79319 
 
 89415 
 
 36 
 
 25 
 
 77353 
 
 90565 
 
 78362 
 
 89995 
 
 79335 
 
 89405 
 
 35 
 
 26 
 
 77370 
 
 90555 
 
 78379 
 
 89985 
 
 79351 
 
 89395 
 
 34 
 
 27 
 
 77387 
 
 90546 
 
 78395 
 
 89976 
 
 79367 
 
 89385 
 
 33 
 
 28 
 
 77405 
 
 90537 
 
 78412 
 
 89966 
 
 79383 
 
 89375 
 
 32 
 
 29 
 
 77422 
 
 90527 
 
 78428 
 
 89956 
 
 79399 
 
 89364 
 
 31 
 
 30 
 
 9.77439 
 
 9.90518 
 
 9.78445 
 
 9.89947 
 
 9.79415 
 
 9.89354 
 
 30 
 
 31 
 
 774E5 
 
 90509 
 
 78461 
 
 89937 
 
 79431 
 
 89344 
 
 29 
 
 32 
 
 77473 
 
 90499 
 
 78178 
 
 89927 
 
 79447 
 
 89334 
 
 28 
 
 33 
 
 77490 
 
 90490 
 
 78494 
 
 89918 
 
 79463 
 
 89324 
 
 27 
 
 34 
 
 77507 
 
 90480 
 
 78510 
 
 89908 
 
 79478 
 
 89314 
 
 26 
 
 35 
 
 77524 
 
 90471 
 
 785-27 
 
 89898 
 
 79494 
 
 89304 
 
 25 
 
 36 
 
 77541 
 
 90462 
 
 78543 
 
 89888 
 
 79510 
 
 89291 
 
 24 
 
 37 
 
 77553 
 
 90452 
 
 78560 
 
 89879 
 
 79526 
 
 89284 
 
 23 
 
 38 
 
 77575 
 
 90443 
 
 78576 
 
 89869 
 
 79542 
 
 89274 
 
 22 
 
 39 
 
 77592 
 
 90434 
 
 78592 
 
 89859 
 
 79558 
 
 89264 
 
 21 
 
 40 
 
 9.77609 
 
 9.90424 
 
 9.78609 
 
 9.89849 
 
 9.79573 
 
 9.89254 
 
 20 
 
 41 
 
 77626 
 
 90415 
 
 78625 
 
 89840 
 
 79589 
 
 89244 
 
 19 
 
 42 
 
 77643 
 
 90405 
 
 78642 
 
 89830 
 
 79605 
 
 89233 
 
 18 
 
 43 
 
 77060 
 
 90396 
 
 78658 
 
 898-20 
 
 78621 
 
 89223 
 
 17 
 
 44 
 
 77677 
 
 90386 
 
 78674 
 
 89810 
 
 79636 
 
 89213 
 
 16 
 
 45 
 
 77694 
 
 90377 
 
 78691 
 
 89801 
 
 79652 
 
 89203 
 
 15 
 
 46 
 
 77711 
 
 9036S 
 
 78707 
 
 89791 
 
 79668 
 
 89193 
 
 14 
 
 47 
 
 77728 
 
 90358 
 
 78723 
 
 89781 
 
 79684 
 
 891 S3 
 
 13 
 
 48 
 
 77744 
 
 90349 
 
 78739 
 
 89771 
 
 79699 
 
 89173 
 
 12 
 
 49 
 
 77761 
 
 90339 
 
 78756 
 
 89761 
 
 79715 
 
 89162 
 
 11 
 
 50 
 
 9.77778 
 
 9.90330 
 
 9.78772 
 
 9.89752 
 
 9.79731 
 
 9.89152 
 
 10 
 
 51 
 
 77795 
 
 90320 
 
 78788 
 
 89742 
 
 79746 
 
 89146 
 
 <) 
 
 52 
 
 77812 
 
 90311 
 
 78805 
 
 89732 
 
 79762 
 
 89132 
 
 8 
 
 53 
 
 77829 
 
 90301 
 
 78821 
 
 89722 
 
 79778 
 
 89122 
 
 7 
 
 54 
 
 77846 
 
 90292 
 
 78837 
 
 89712 
 
 79793 
 
 89112 
 
 6 
 
 55 
 
 77862 
 
 90282 
 
 78853 
 
 89702 
 
 79809 
 
 89101 
 
 5 
 
 56 
 
 77879 
 
 90273 
 
 78869 
 
 89693 
 
 79S25 
 
 89091 
 
 4 
 
 57 
 
 77896 
 
 90263 
 
 78886 
 
 89683 
 
 79840 
 
 89081 
 
 3 
 
 58 
 
 77913 
 
 90254 
 
 78902 
 
 89G73 
 
 79856 
 
 89071 
 
 2 
 
 59 
 
 77930 
 
 90244 
 
 78918 
 
 89663 
 
 79872 
 
 890(50 
 
 1 
 
 60 
 
 77946 
 
 90235 
 
 78934 
 
 89653 
 
 79887 
 
 89050 
 
 
 
 ~ 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 f 
 
 
 
 63 
 
 
 52 
 
 
 51 
 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 
 
 39 
 
 40 
 
 
 
 41 
 
 / 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 
 9.79887 
 
 9.89050 
 
 9.80807 
 
 9.88425 
 
 9.81694 
 
 9.87778 
 
 60 
 
 1 
 
 79903 
 
 89040 
 
 80822 
 
 88415 
 
 81709 
 
 87767 
 
 59 
 
 2 
 
 79918 
 
 89030 
 
 80837 
 
 88404 
 
 81723 
 
 87756 
 
 58 
 
 3 
 
 79934 
 
 89020 
 
 80852 
 
 88394 
 
 81738 
 
 87745 
 
 57 
 
 4 
 
 79950 
 
 89009 
 
 80867 
 
 88383 
 
 81752 
 
 87734 
 
 56 
 
 5 
 
 79965 
 
 88999 
 
 80882 
 
 88372 
 
 81767 
 
 87723 
 
 55 
 
 6 
 
 79981 
 
 88989 
 
 80897 
 
 88362 
 
 81781 
 
 87712 
 
 54 
 
 
 79996 
 
 88978 
 
 80912 
 
 88351 
 
 81796 
 
 87701 
 
 53- 
 
 8 
 
 80012 
 
 88968 
 
 80927 
 
 88340 
 
 81810 
 
 87690 
 
 52 
 
 9 
 
 80027 
 
 88958 
 
 80942 
 
 88330 
 
 81825 
 
 87679 
 
 51 
 
 10 
 
 9.80043 
 
 9.88948 
 
 9.80957 
 
 9.88319 
 
 9.81839 
 
 9.87668 
 
 60 
 
 11 
 
 80058 
 
 88937 
 
 80972 
 
 88308 
 
 81854 
 
 87657 
 
 49 
 
 12 
 
 80074 
 
 88927 
 
 80987 
 
 88298 
 
 81868 
 
 87646 
 
 48 
 
 13 
 
 80089 
 
 88917 
 
 81002 
 
 88287 
 
 81882 
 
 87635 
 
 47 
 
 14 
 
 80105 
 
 88906 
 
 81017 
 
 88276 
 
 81897 
 
 87624 
 
 46 
 
 15 
 
 80120 
 
 88896 
 
 81032 
 
 88266 
 
 81911 
 
 87613 
 
 45 
 
 16 
 
 80136 
 
 88886 
 
 81047 
 
 88255 
 
 81926 
 
 87601 
 
 44 
 
 17 
 
 80151 
 
 88875 
 
 81061 
 
 88244 
 
 81940 
 
 87590 
 
 43 
 
 18 
 
 80166 
 
 88865 
 
 81076 
 
 88234 
 
 81955 
 
 87579 
 
 42 
 
 19 
 
 80182 
 
 88855 
 
 81091 
 
 88223 
 
 81969 
 
 87568 
 
 41 
 
 20 
 
 9.80197 
 
 9.88844 
 
 9.81106 
 
 9.88212 
 
 9.81983 
 
 9.87557 
 
 40 
 
 21 
 
 80213 
 
 88834 
 
 81121 
 
 88201 
 
 81998 
 
 87546 
 
 39 
 
 22 
 
 80228 
 
 88824 
 
 81136 
 
 88191 
 
 82012 
 
 87535 
 
 38 
 
 23 
 
 80244 
 
 88813 
 
 81151 - 
 
 88180 
 
 8~>026 
 
 87524 
 
 37 
 
 24 
 
 80259 
 
 8S803 
 
 81166 
 
 88169 
 
 82041 
 
 87513 
 
 36 
 
 25 
 
 80274 
 
 88793 
 
 81180 
 
 88158 
 
 820.-5 
 
 87501 
 
 35 
 
 26 
 
 80290 
 
 88782 
 
 81195 
 
 88148 
 
 82069 
 
 87490 
 
 34 
 
 27 
 
 80305 
 
 88772 
 
 81210 
 
 88137 
 
 82084 
 
 87479 
 
 33 
 
 28 
 
 80320 
 
 88761 
 
 81225 
 
 
 82098 
 
 87468 
 
 32 
 
 29 
 
 80336 
 
 88751 
 
 81240 
 
 88115 
 
 82112 
 
 87457 
 
 31 
 
 30 
 
 9.80351 
 
 9.88741 
 
 9.81254 
 
 9.88105 
 
 9.82126 
 
 9.87446 
 
 30 
 
 31 
 
 80366 
 
 88730 
 
 81269 
 
 88094 
 
 82141 
 
 87434 
 
 29 
 
 32 
 
 80382 
 
 88720 
 
 81284 
 
 88083 
 
 82155 
 
 87423 
 
 28 
 
 33 
 
 80397 
 
 88709 
 
 81299 
 
 88072 
 
 82169 
 
 87412 
 
 27 
 
 34 
 
 80412 
 
 88699 
 
 81314 
 
 88061 
 
 82184 
 
 87401 
 
 26 
 
 35 
 
 80428 
 
 88688 
 
 81328 
 
 88051 
 
 82198 
 
 87390 
 
 25 
 
 36 
 
 80443 
 
 88678 
 
 81343 
 
 88040 
 
 82212 
 
 87378 
 
 24 
 
 37 
 
 80458 
 
 88668 
 
 81358 
 
 88029 
 
 82226 
 
 67367 
 
 23 
 
 38 
 
 80473 
 
 88657 
 
 81372 
 
 88018 
 
 82240 
 
 87356 
 
 22 
 
 39 
 
 80489 
 
 88647 
 
 81387 
 
 88007 
 
 82255 
 
 87345 
 
 21 
 
 40 
 
 9.80504 
 
 9.88636 
 
 9.81402 
 
 9.87996 
 
 9.82269 
 
 9.87334 
 
 20 
 
 41 
 
 80519 
 
 88626 
 
 81417 
 
 87985 
 
 82283 
 
 87322 
 
 19 
 
 42 
 
 80534 
 
 88615 
 
 81431 
 
 87975 
 
 82297 
 
 87311 
 
 18 
 
 43 
 
 80550 
 
 88605 
 
 81446 
 
 87964 
 
 82311 
 
 87300 
 
 17 
 
 44 
 
 80565 
 
 88594 
 
 81461 
 
 87953 
 
 8-J326 
 
 87288 
 
 16 
 
 45 
 
 80580 
 
 88584 
 
 81475 
 
 87942 
 
 82340 
 
 87277 
 
 15 
 
 46 
 
 80595 
 
 8^573 
 
 81490 
 
 87931 
 
 
 87266 
 
 14 
 
 47 
 
 80610 
 
 88563 
 
 81505 
 
 87920 
 
 82368 
 
 87255 
 
 13 
 
 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 
 
 9.87221 
 
 10 
 
 51 
 
 80671 
 
 88521 
 
 81563 
 
 87877 
 
 82424 
 
 87209 
 
 9 
 
 52 
 
 8(1686 
 
 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 
 
 81636 
 
 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 
 
 
 303 
 
TABLE XX. LOGARITHMIC SINES AND COSINES. 
 
 42 ' 
 
 48 
 
 440 
 
 Sine Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 
 30 
 31 
 32 
 33 
 34 
 
 37 
 
 59 
 
 82579 
 82593 
 82607 
 
 82635 
 82649 
 
 82677 
 
 9.82691 
 82705 
 82719 
 82733 
 82747 
 82r61 
 82775 
 82788 
 
 82816 
 
 9.82830 
 
 S2844 
 
 82913 
 
 9.82968 
 82982 
 
 83010 
 
 83051 
 83065 
 83078 
 
 9.83106 
 83120 
 83133 
 83147 
 83161 
 83174 
 
 83215 
 83229 
 
 83270 
 83283 
 83297 
 83310 
 83324 
 
 9.87107 
 87096 
 87085 
 87073 
 87062 
 87050 
 
 87028 
 87016 
 87005 
 
 86970 
 
 86913 
 
 86867 
 86855 
 86844 
 86832 
 
 86786 
 86775 
 
 9.86763 
 86752 
 86740 
 86728 
 86717 
 86705 
 
 86670 
 
 9.86647 
 86635 
 
 86612 
 
 86577 
 86565 
 86554 
 86542 
 
 9.86530 
 86518 
 86507 
 86495 
 86483 
 86472 
 
 83351 
 
 83365 86425 
 
 83378 86413 
 
 83405 
 83419 
 83432 
 83446 
 
 83540 
 83554 
 83567 
 
 83621 
 83634 
 
 83674 
 
 83701 
 83715 
 83728 
 83741 
 83755 
 
 9.83781 
 83795 
 
 83821 
 
 83874 
 
 83927 
 
 83967 
 
 84006 
 84020 
 84033 
 
 9.84046 
 84059 
 84072 
 84085 
 84098 
 84112 
 84125 
 84138 
 84151 
 84164 
 84177 
 
 9.86413 
 86401 
 
 86377 
 
 86354 
 86342 
 
 86318 
 
 83478 
 83486 
 83500 
 
 9.83513 9.86295 
 
 86271 
 86259 
 86247 
 86235 
 
 86211 
 
 9.86176 
 86164 
 86152 
 86140 
 86128 
 86116 
 86104 
 86092 
 
 9.86056 
 
 85984 
 85972 
 
 85948 
 
 85924 
 85912 
 85900 
 
 85876 
 
 85851 
 
 85827 
 
 9.85815 
 85803 
 85791 
 85779 
 85766 
 85754 
 85742 
 85730 
 85718 
 85706 
 
 9.84177 
 84190 
 
 84216 
 84229 
 84242 
 84255 
 
 84295 
 
 9.84308 
 84321 
 84334 
 84347 
 
 84373 
 84385 
 
 84411 
 84424 
 
 9.84437 
 84450 
 84463 
 84476 
 84489 
 84502 
 84515 
 84528 
 84540 
 84553 
 
 9.84566 
 84579 
 84592 
 84605 
 84618 
 84630 
 84643 
 84656 
 
 9.84694 
 84707 
 84720 
 84733 
 84745 
 84758 
 84771 
 84784 
 84796 
 84809 
 
 9.84822 
 84835 
 84847 
 84860 
 84873 
 84885 
 
 84911 
 84923 
 84936 
 84949 
 
 85681 
 
 85657 
 85645 
 85G32 
 
 85608 
 
 85583 
 
 1.85571 
 85559 
 85547 
 85534 
 85522 
 85510 
 85497 
 85485 
 85473 
 85460 
 
 1.85448 
 85436 
 85423 
 85411 
 85399 
 85386 
 85874 
 85361 
 85349 
 85337 
 
 85312 
 85299 
 
 85274 
 
 85250 
 85237 
 85225 
 85212 
 
 9.85200 
 85187 
 85175 
 
 85150 
 85137 
 85125 
 85112 
 85100 
 85087 
 
 9.85074 
 
 85049 
 85037 
 85024 
 85012 
 
 84974 
 84961 
 84949 
 
 Cosine Sine 
 47 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 46 a 
 
 304 
 
 45 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 
 
 
 
 
 1 
 
 
 2 
 
 f 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 
 
 oo 
 
 00 
 
 8.24192 
 
 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 
 
 42886 
 
 52 
 
 9 
 
 41797 
 
 58203 
 
 30263 
 
 69737 
 
 57452 
 
 42548 
 
 61 
 
 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 
 
 63982 
 
 36018 
 
 33886 
 
 66114 
 
 59428 
 
 40572 
 
 45 
 
 16 
 
 66785 
 
 33215 
 
 34461 
 
 65539 
 
 59749 
 
 40251 
 
 44 
 
 17 
 
 69418 
 
 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 
 
 36681 
 
 39 
 
 22 
 
 80615 
 
 19385 
 
 37762 
 
 62238 
 
 61626 
 
 38374 
 
 38 
 
 23 
 
 82546 
 
 17454 
 
 38289 
 
 61711 
 
 61931 
 
 38069 
 
 37 
 
 24 
 
 84394 
 
 15606 
 
 38809 
 
 61191 
 
 62234 
 
 37766 
 
 36 
 
 25 
 
 86167 
 
 13833 
 
 39323 
 
 60677 
 
 62535 
 
 37465 
 
 35 
 
 26 
 
 87871 
 
 12129 
 
 39832 
 
 60168 
 
 62834 
 
 37166 
 
 34 
 
 27 
 
 89510 
 
 10490 
 
 40334 
 
 59666 
 
 63131 
 
 36869 
 
 33 
 
 28 
 
 91089 
 
 08911 
 
 40830 
 
 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 
 
 85702 
 
 29 
 
 32 
 
 96889 
 
 03111 
 
 42762 
 
 57238 
 
 64585 
 
 35415 
 
 28 
 
 33 
 
 98225 
 
 01775 
 
 43232 
 
 56768 
 
 64870 
 
 35130 
 
 27 
 
 34 
 
 99522 
 
 00478 
 
 43696 
 
 56304 
 
 65154 
 
 34846 
 
 26 
 
 35 
 
 8.00781 
 
 11.99219 
 
 44156 
 
 55844 
 
 65435 
 
 34565 
 
 25 
 
 36 
 
 02004 
 
 97996 
 
 44611 
 
 55389 
 
 65715 
 
 34285 
 
 24 
 
 37 
 
 03194 
 
 96806 
 
 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 
 
 8.46385 
 
 11.53615 
 
 8.66816 
 
 11.33134 
 
 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 
 
 49325 
 
 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 
 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 i 
 
 
 
 89 
 
 
 88 
 
 
 87 
 
 
 305 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 / 
 
 
 3 
 
 
 4 
 
 
 5 
 
 / 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 
 
 8.71940 
 
 11.28060 
 
 8.84464 
 
 11.15536 
 
 8.94195 
 
 11.05805 
 
 60 
 
 1 
 
 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 
 
 73832 
 
 26163 
 
 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 
 
 745-21 
 
 25479 
 
 86417 
 
 13583 
 
 95767 
 
 04233 
 
 49 
 
 12 
 
 74748 
 
 25252 
 
 86591 
 
 13409 
 
 95908 
 
 04092 
 
 48 
 
 13 
 
 74974 
 
 25026 
 
 86763 
 
 13-237 
 
 96C47 
 
 03953 
 
 47 
 
 14 
 
 75199 
 
 24801 
 
 86935 
 
 13065 
 
 96187 
 
 03813 
 
 46 
 
 15 
 
 75423 
 
 24577 
 
 8H06 
 
 12894 
 
 96325 
 
 03675 
 
 45 
 
 16 
 
 75645 
 
 24355 
 
 87277 
 
 12723 
 
 96464 
 
 03536 
 
 44 
 
 ir 
 
 75867 
 
 24133 
 
 87447 
 
 12553 
 
 96G02 
 
 03398 
 
 43 
 
 18 
 
 76087 
 
 23913 
 
 87616 
 
 12384 
 
 96739 
 
 03261 
 
 42 
 
 19 
 
 76306 
 
 23694 
 
 877'85 
 
 12215 
 
 96877 
 
 02123 
 
 41 
 
 20 
 
 8.765-25 
 
 11.23475 
 
 8.87953 
 
 11.12047 
 
 8.97013 
 
 11.02987 
 
 40 
 
 21 
 
 76742 
 
 23258 
 
 88150 
 
 11880 
 
 97150 
 
 0'2850 
 
 39 
 
 22 
 
 76958 
 
 23042 
 
 88-287 
 
 11713 
 
 97285 
 
 02715 
 
 38 
 
 23 
 
 77173 
 
 228-27 
 
 88453 
 
 11547 
 
 97421 
 
 02579 
 
 37 
 
 24 
 
 77387 
 
 22613 
 
 88618 
 
 11382 
 
 97556 
 
 0-2444 
 
 36 
 
 25 
 
 77600 
 
 22400 
 
 88783 
 
 11217 
 
 97691 
 
 . 02309 
 
 35 
 
 26 
 
 77811 
 
 22189 
 
 88948 
 
 11052 
 
 97825 
 
 02175 
 
 34 
 
 27 
 
 78022 
 
 21978 
 
 89111 
 
 10889 
 
 97959 
 
 02041 
 
 33 
 
 28 
 
 78232 
 
 21768 
 
 89274 
 
 10726 
 
 9809-2 
 
 01908 
 
 32 
 
 29 
 
 78441 
 
 21559 
 
 89437 
 
 10563 
 
 98225 
 
 01775 
 
 31 
 
 30 
 
 8.78649 
 
 11.21351 
 
 8.89598 
 
 11.1040-2 
 
 8.98358 
 
 11.01642 
 
 80 
 
 31 
 
 78855 
 
 21145 
 
 89760 
 
 10240 
 
 98490 
 
 01510 
 
 29 
 
 32 
 
 79061 
 
 20939 
 
 89920 
 
 10080 
 
 986'22 
 
 01378 
 
 28 
 
 83 
 
 79'266 
 
 20734 
 
 90080 
 
 09920 
 
 98753 
 
 01247 
 
 27 
 
 34 
 
 79470 
 
 20530 
 
 90i>40 
 
 09760 
 
 98884 
 
 01116 
 
 26 
 
 35 
 
 79673 
 
 203'27 
 
 90399 
 
 09601 
 
 99015 
 
 00985 
 
 25 
 
 36 
 
 79875 
 
 20125 
 
 90557 
 
 09443 
 
 99145 
 
 00855 
 
 24 
 
 37 
 
 80076 
 
 19924 
 
 90715 
 
 09-285 
 
 99-275 
 
 00725 
 
 23 
 
 38 
 
 80277 
 
 19723 
 
 90872 
 
 09128 
 
 99405 
 
 00595 
 
 22 
 
 39 
 
 80476 
 
 19524 
 
 910-.'9 
 
 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 
 
 10.90954 
 
 17 
 
 44 
 
 81459 
 
 18541 
 
 91803 
 
 08197 
 
 00174 
 
 99P'26 
 
 16 
 
 45 
 
 81653 
 
 18347 
 
 91957 
 
 08043 
 
 00301 
 
 99699 
 
 15 
 
 46 
 
 81846 
 
 18154 
 
 9-2110 
 
 07890 
 
 00427 
 
 99573 
 
 14 
 
 47 
 
 82038 
 
 17962 
 
 92262 
 
 07738 
 
 00553 
 
 99447 
 
 18 
 
 48 
 
 82230 
 
 17770 
 
 9-2414 
 
 07586 
 
 00679 
 
 998-21 
 
 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 
 
 98945 
 
 9 
 
 52 
 
 8-2987 
 
 17013 
 
 93016 
 
 06984 
 
 01179 
 
 98821 
 
 8 
 
 53 
 
 83175 
 
 168-25 
 
 93165 
 
 06835 
 
 01303 
 
 98697 
 
 7 
 
 54 
 
 83361 
 
 16639 
 
 93313 
 
 06G87 
 
 01427 
 
 98573 
 
 6 
 
 55 
 
 83547 
 
 16453 
 
 93462 
 
 06538 
 
 01550 
 
 98450 
 
 5 
 
 56 
 
 83732 
 
 16268 
 
 93609 
 
 06391 
 
 01673 
 
 98327 
 
 4 
 
 57 
 
 83916 
 
 16084 
 
 93756 
 
 06244 
 
 01796 
 
 98204 
 
 3 
 
 58 
 
 84100 
 
 15900 
 
 93903 
 
 06097 
 
 01918 
 
 98082 
 
 2 
 
 59 
 
 84282 
 
 15718 
 
 94049 
 
 05951 
 
 02040 
 
 97960 
 
 1 
 
 60 
 
 84464 
 
 15536 
 
 94195 
 
 05805 
 
 02162 
 
 97838 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 / 
 
 
 
 86 
 
 
 85 
 
 
 84 
 
 
TABLE XXL LOG. TANGENTS AND COTANGENTS. 
 
 / 
 
 6 
 
 o 
 
 
 7 
 
 
 8 
 
 ; 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 o 
 
 9.02162 
 
 10.97838 
 
 9.08914 
 
 10.91086 
 
 9.14780 
 
 10.85220 
 
 60 
 
 1 
 
 02283 
 
 97717 
 
 09019 
 
 90981 
 
 14872 
 
 85128 
 
 59 
 
 2 
 
 02404 
 
 97596 
 
 09123 
 
 90877 
 
 14963 
 
 85037 
 
 58 
 
 3 
 
 025-J5 
 
 97475 
 
 09227 
 
 90773 
 
 15054 
 
 84946 
 
 57 
 
 4 
 
 02645 
 
 97355 
 
 09330 
 
 90670 
 
 15145 
 
 84855 
 
 56 
 
 5 
 
 02766 
 
 97234 
 
 09434 
 
 90566 
 
 15236 
 
 84764 
 
 55 
 
 6 
 
 02H85 
 
 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 
 
 842:23 
 
 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 
 
 04-297 
 
 95703 
 
 10756 
 
 89244 
 
 16401 
 
 83599 
 
 42 
 
 19 
 
 04413 
 
 95587 
 
 10856 
 
 89144 
 
 16489 
 
 83511 
 
 41 
 
 20 
 
 9.045-28 
 
 10.95472 
 
 9.10956 
 
 10.89044 
 
 9.16577 
 
 10.83423 
 
 40 
 
 21 
 
 04643 
 
 95357 
 
 11056 
 
 88944 
 
 16665 
 
 83335 
 
 39 
 
 22 
 
 04758 
 
 95242 
 
 11155 
 
 88845 
 
 16753 
 
 83247 
 
 38 
 
 23 
 
 04873 
 
 95127 
 
 11254 
 
 88746 
 
 16841 
 
 83159 
 
 37 
 
 24 
 
 04987 
 
 95013 
 
 11353 
 
 88647 
 
 16928 
 
 83072 
 
 86 
 
 25 
 
 05101 
 
 94899 
 
 11452 
 
 88548 
 
 17016 
 
 82984 
 
 35 
 
 26 
 
 05214 
 
 94786 
 
 11551 
 
 88449 
 
 17103 
 
 82897 
 
 34 
 
 27 
 
 053-28 
 
 94672 
 
 11649 
 
 88351 
 
 17190 
 
 82810 
 
 33 
 
 28 
 
 05441 
 
 94559 
 
 11747 
 
 88253 
 
 17277 
 
 82723 
 
 32 
 
 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 
 
 06113 
 
 93887 
 
 12332 
 
 87668 
 
 17794 
 
 82206 
 
 26 
 
 35 
 
 06224 
 
 93776 
 
 12428 
 
 87572 
 
 17880 
 
 82120 
 
 25 
 
 36 
 
 06335 
 
 93665 
 
 12525 
 
 87475 
 
 17965 
 
 82035 
 
 24 
 
 37 
 
 06445 
 
 93555 
 
 12621 
 
 87379 
 
 18051 
 
 81949 
 
 23 
 
 38 
 
 06556 
 
 93444 
 
 12717 
 
 87283 
 
 18136 
 
 81864 
 
 22 
 
 39 
 
 00666 
 
 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 
 
 07-211 
 
 92789 
 
 13289 
 
 86711 
 
 18644 
 
 81856 
 
 16 
 
 45 
 
 07320 
 
 92680 
 
 13384 
 
 86616 
 
 187'28 
 
 81272 
 
 15 
 
 46 
 
 07428 
 
 92572 
 
 13178 
 
 86522 
 
 18812 
 
 81188 
 
 14 
 
 47 
 
 07536 
 
 9:2464 
 
 13573 
 
 86427 
 
 18896 
 
 81104 
 
 13 
 
 48 
 
 07643 
 
 9-2357 
 
 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 
 
 19-229 
 
 80771 
 
 9 
 
 52 
 
 08071 
 
 91929 
 
 14041 
 
 85959 
 
 19312 
 
 80688 
 
 8 
 
 53 
 
 08177 
 
 91P'23 
 
 14134 
 
 85866 
 
 19395 
 
 80605 
 
 7 
 
 54 
 
 08288 
 
 91717 
 
 14227 
 
 85773 
 
 19478 
 
 80522 
 
 6 
 
 55 
 
 08389 
 
 91(511 
 
 14320 
 
 85680 
 
 19561 
 
 80439 
 
 5 
 
 56 
 
 08495 
 
 91505 
 
 14412 
 
 855S8 
 
 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 
 
 800-29 
 
 
 
 t 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 / 
 
 
 
 83 
 
 
 82 
 
 
 81 
 
 
 307 
 
-LOG. TANGENTS AND COTANGENTS. 
 
 
 1 
 
 
 9 
 
 
 10 
 
 
 11 
 
 1 
 
 
 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 
 
 79866 
 
 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 
 
 62 
 
 9 
 
 20701 
 
 79299 
 
 25292 
 
 74708 
 
 29468 
 
 70532 
 
 51 
 
 10 
 
 9.20782 
 
 10.79218 
 
 9.25365 
 
 10.74635 
 
 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 
 
 6967 4 
 
 38 
 
 23 
 
 21814 
 
 78186 
 
 26301 
 
 73699 
 
 30391 
 
 G9609 
 
 87 
 
 24 
 
 21893 
 
 78107 
 
 26372 
 
 73628 
 
 30457 
 
 69543 
 
 36 
 
 25 
 
 21971 
 
 78029 
 
 26443 
 
 73557 
 
 30522 
 
 69478 
 
 35 
 
 26 
 
 22049 
 
 77951 
 
 26514 
 
 73486 
 
 30587 
 
 69413 
 
 34 
 
 27 
 
 22127 
 
 77873 
 
 26585 
 
 73415 
 
 30652 
 
 69348 
 
 33 
 
 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 
 
 35 
 
 22747 
 
 77253 
 
 27148 
 
 72852 
 
 31168 
 
 68832 
 
 25 
 
 36 
 
 22824 
 
 77176 
 
 27218 
 
 72782 
 
 31233 
 
 68767 
 
 24 
 
 87 
 
 22901 
 
 77099 
 
 27288 
 
 72712 
 
 31297 
 
 68703 
 
 23 
 
 88 
 
 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 
 
 23787 
 
 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 
 
 75963 
 
 28323 
 
 71677 
 
 32248 
 
 67752 
 
 8 
 
 53 
 
 24113 
 
 75888 
 
 28391 
 
 71609 
 
 3:2311 
 
 67689 
 
 7 
 
 54 
 
 24186 
 
 75814 
 
 28459 
 
 71541 
 
 32373 
 
 67627 
 
 6 
 
 55 
 
 24261 
 
 75739 
 
 28527 
 
 71473 
 
 32436 
 
 67564 
 
 5 
 
 56 
 
 24335 
 
 75665 
 
 28595 
 
 71405 
 
 32498 
 
 67502 
 
 4 
 
 67 
 
 24410 
 
 75590 
 
 28662 
 
 71338 
 
 32561 
 
 67439 
 
 3 
 
 58 
 
 24484 
 
 75516 
 
 28730 
 
 71270 
 
 32623 
 
 67377 
 
 2 
 
 59 
 
 24558 
 
 75442 
 
 28798 
 
 71202 
 
 32685 
 
 67315 
 
 1 
 
 60 
 
 24632 
 
 75368 
 
 28865 
 
 71135 
 
 32747 
 
 67253 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 
 
 
 80* 
 
 
 79 
 
 
 78 
 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 Jj 
 
 12 
 
 13 
 
 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 
 
 60269 
 
 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 
 
 33057 
 
 66943 
 
 36624 
 
 63376 
 
 39945 
 
 60055 
 
 55 
 
 g 
 
 33119 
 
 66881 
 
 36681 
 
 63319 
 
 39999 
 
 60001 
 
 54 
 
 7 
 
 83180 
 
 66820 
 
 36738 
 
 63262 
 
 40052 
 
 59948 
 
 53 
 
 8 
 
 S3242 
 
 66758 
 
 36795 
 
 63205' 
 
 40106 
 
 59894 
 
 52 
 
 9 
 
 83303 
 
 66697 
 
 36852 
 
 63148 
 
 40159 
 
 59841 
 
 61 
 
 10 
 
 9.33365 
 
 10.66635 
 
 9.36909 
 
 10.63091 
 
 9.40212 
 
 10.59788 
 
 50 
 
 11 
 
 33426 
 
 66574 
 
 36966 
 
 63034 
 
 40266 
 
 59734 
 
 49 
 
 12 
 
 33487 
 
 66513 
 
 37023 
 
 62977 
 
 40319 
 
 59681 
 
 48 
 
 13 
 
 33548 
 
 66452 
 
 37080 
 
 62920 
 
 40372 
 
 59628 
 
 47 
 
 14 
 
 33609 
 
 66391 
 
 37137 
 
 628G3 
 
 40425 
 
 59575 
 
 46 
 
 15 
 
 33670 
 
 66330 
 
 87193 
 
 62807 
 
 40478 
 
 59522 
 
 45 
 
 16 
 
 33731 
 
 66269 
 
 37250 
 
 62750 
 
 40531 
 
 59469 
 
 44 
 
 17 
 
 33792 
 
 66208 
 
 37306 
 
 62G94 
 
 40584 
 
 59416 
 
 43 
 
 18 
 
 83853 
 
 66147 
 
 37363 
 
 62637 
 
 40636 
 
 59364 
 
 42 
 
 19 
 
 33913 
 
 66087 
 
 37419 
 
 62581 
 
 40689 
 
 59311 
 
 41 
 
 20 
 
 9.33974 
 
 10.66026 
 
 9.37476 
 
 10.62524 
 
 9.40742 
 
 10.59258 
 
 40 
 
 21 
 
 34034 
 
 65966 
 
 37532 
 
 62468 
 
 40795 
 
 69205 
 
 39 
 
 22 
 
 34095 
 
 65905 
 
 37588 
 
 62412 
 
 40847 
 
 59153 
 
 38 
 
 23 
 
 84155 
 
 65845 
 
 37644 
 
 62356 
 
 40900 
 
 59100 
 
 37 
 
 24 
 
 34215 
 
 65785 
 
 37700 
 
 62300 
 
 40952 
 
 59048 
 
 36 
 
 25 
 
 84276 
 
 65724 
 
 37756 
 
 62244 
 
 41005 
 
 58995 
 
 35 
 
 26 
 
 34336 
 
 65664 
 
 87812 
 
 62188 
 
 41057 
 
 58943 
 
 34 
 
 27 
 
 34396 
 
 65604 
 
 37868 
 
 62132 
 
 41109 
 
 58891 
 
 33 
 
 28 
 
 34456 
 
 65544 
 
 37924 
 
 62076 
 
 41161 
 
 58839 
 
 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 
 
 28 
 
 33 
 
 34755 
 
 65245 
 
 38202 
 
 61798 
 
 41422 
 
 58578 
 
 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 
 
 5&319 
 
 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 
 
 85640 
 
 64360 
 
 39027 
 
 60973 
 
 42195 
 
 57805 
 
 12 
 
 49 
 
 3569S 
 
 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 
 
 42399 
 
 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 
 
 3 
 
 58 
 
 36221 
 
 63779 
 
 39569 
 
 60431 
 
 42704 
 
 57296 
 
 2 
 
 59 
 
 36279 
 
 63721 
 
 39623 
 
 60377 
 
 42755 
 
 57245 
 
 1 
 
 60 
 
 36336 
 
 63664 
 
 39677 
 
 60323 
 
 42805 
 
 57195 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 j 
 
 
 
 77 
 
 
 76 
 
 
 75 
 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 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 
 
 54060 
 
 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 
 
 S3 
 
 8 
 
 43208 
 
 56792 
 
 46130 
 
 53870 
 
 48894 
 
 51106 
 
 53 
 
 9 
 
 43258 
 
 56742 
 
 46177 
 
 53823 
 
 48939 
 
 51061 
 
 51 
 
 10 
 
 9.43308 
 
 10.56692 
 
 9.46224 
 
 10.53776 
 
 9.48984 
 
 10.51016 
 
 50 
 
 11 
 
 43358 
 
 56642 
 
 46271 
 
 53729 
 
 49029 
 
 50971 
 
 49 
 
 12 
 
 43408 
 
 5659.2 
 
 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 
 
 53446 
 
 49296 
 
 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 
 
 H 
 
 43954 
 
 56046 
 
 46835 
 
 53165 
 
 49563 
 
 50437 
 
 37 
 
 24 
 
 44004 
 
 55996 
 
 46881 
 
 53119 
 
 49607 
 
 50393 
 
 36 
 
 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 
 
 32 
 
 29 
 
 44250 
 
 55750 
 
 47114 
 
 52886 
 
 49828 
 
 50172 
 
 31 
 
 30 
 
 9.44299 
 
 10.55701 
 
 9.47160 
 
 10.52840 
 
 9.49872 
 
 10.50128 
 
 30 
 
 31 
 
 44348 
 
 55652 
 
 47207 
 
 52793 
 
 49916 
 
 50084 
 
 29 
 
 32 
 
 44397 
 
 55603 
 
 47253 
 
 52747 
 
 49960 
 
 50040 
 
 28 
 
 33 
 
 44446 
 
 55554 
 
 47299 
 
 52701 
 
 50004 
 
 49996 
 
 
 34 
 
 44495 
 
 55505 
 
 47346 
 
 52654 
 
 50048 
 
 49952 
 
 26 
 
 35 
 
 44544 
 
 55456 
 
 47392 
 
 52608 
 
 50092 
 
 49908 
 
 25 
 
 36 
 
 44592 
 
 55408 
 
 47438 
 
 52562 
 
 50136 
 
 49864 
 
 24 
 
 37 
 
 44641 
 
 55359 
 
 '47484 
 
 52516 
 
 50180 
 
 49820 
 
 23 
 
 38 
 
 44690 
 
 55310 
 
 47530 
 
 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 
 
 49602 
 
 18 
 
 43 
 
 44933 
 
 55067 
 
 47760 
 
 52240 
 
 50442 
 
 49558 
 
 17 
 
 44 
 
 44981 
 
 55019 
 
 47806 
 
 52194 
 
 50485 
 
 49515 
 
 16 
 
 45 
 
 45029 
 
 54971 
 
 47852 
 
 52148 
 
 50529 
 
 49471 
 
 15 
 
 46 
 
 45078 
 
 54922 
 
 47897 
 
 52103 
 
 50572 
 
 49428 
 
 14 
 
 47 
 
 '45126 
 
 54874 
 
 47943 
 
 52057 
 
 50616 
 
 49384 
 
 13 
 
 48 
 
 45174 
 
 54826 
 
 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 
 
 10 
 
 51 
 
 45319 
 
 54681 
 
 48126 
 
 51874 
 
 50789 
 
 49211 
 
 9 
 
 52 
 
 45367 
 
 54633 
 
 48171 
 
 51829 
 
 50833 
 
 49167 
 
 8 
 
 53 
 
 45415 
 
 54585 
 
 48217 
 
 51783 
 
 56876 
 
 49124 
 
 7 
 
 54 
 
 45463 
 
 54537 
 
 48262 
 
 51738 
 
 50919 
 
 49081 
 
 6 
 
 K 
 
 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 
 
 457C2 
 
 54298 
 
 48489 
 
 51511 
 
 51135 
 
 48H65 
 
 1 
 
 60 
 
 45750 
 
 54250 
 
 48534 
 
 51466 
 
 51178 
 
 48822 
 
 
 
 / 
 
 Co tan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 / 
 
 
 
 74 
 
 
 73 
 
 
 72 
 
 , 
 
 310 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 / 
 
 18 
 
 19 
 
 
 20 
 
 t 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 o 
 
 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 
 
 56:224 
 
 43776 
 
 57 
 
 4 
 
 51 349 
 
 48651 
 
 53861 
 
 46139 
 
 56264 
 
 43736 
 
 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 
 
 540:25 
 
 45975 
 
 56420 
 
 43580 
 
 52 
 
 9 
 
 51563 
 
 48437 
 
 54065 
 
 45935 
 
 56459 
 
 43541 
 
 51 
 
 10 
 
 9.51606 
 
 10.48394 
 
 9.54106 
 
 10.45894 
 
 9.56498 
 
 10.43502 
 
 50 
 
 11 
 
 51648 
 
 48352 
 
 54147 
 
 45&53 
 
 56537 
 
 43463 
 
 49 
 
 12 
 
 51691 
 
 48309 
 
 54187 
 
 45813 
 
 56576 
 
 43424 
 
 48 
 
 13 
 
 51734 
 
 48266 
 
 54228 
 
 45772 
 
 56815 
 
 43385 
 
 47 
 
 14 
 
 51776 
 
 48224 
 
 54269 
 
 45731 
 
 56654 
 
 43346 
 
 46 
 
 15 
 
 51819 
 
 48181 
 
 54309 
 
 45691 
 
 56693 
 
 43307 
 
 45 
 
 16 
 
 51861 
 
 48139 
 
 54350 
 
 45650 
 
 56732 
 
 43268 
 
 44 
 
 17 
 
 51903 
 
 48097 
 
 54390 
 
 45610 
 
 56771 
 
 432-29 
 
 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 
 
 10.43113 
 
 40 
 
 21 
 
 52073 
 
 47927 
 
 54552 
 
 45448 
 
 56926 
 
 43074 
 
 39 
 
 22 
 
 52115 
 
 47885 
 
 54593 
 
 45407 
 
 56965 
 
 43035 
 
 38 
 
 23 
 
 52157 
 
 47843 
 
 54633 
 
 45367 
 
 57004 
 
 42996 
 
 37 
 
 24 
 
 52200 
 
 47800 
 
 54673 
 
 45327 
 
 57042 
 
 48958 
 
 86 
 
 25 
 
 5224-2 
 
 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 
 
 32 
 
 29 
 
 52410 
 
 47590 
 
 54875 
 
 45125 
 
 57235 
 
 42765 
 
 i 
 
 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 
 
 57351 
 
 42649 
 
 28 
 
 33 
 
 5-2578 
 
 47422 
 
 55035 
 
 44965 
 
 57389 
 
 42611 
 
 27 
 
 34 
 
 52620 
 
 47380 
 
 55075 
 
 449-25 
 
 57428 
 
 42572 
 
 26 
 
 35 
 
 52661 
 
 47339 
 
 55115 
 
 44885 
 
 57466 
 
 42534 
 
 25 
 
 36 
 
 52703 
 
 47297 
 
 55155 
 
 44845 
 
 57504 
 
 42496 
 
 24 
 
 37 
 
 52745 
 
 47255 
 
 55195 
 
 44805 
 
 57543 
 
 42457 
 
 23 
 
 38 
 
 5-2787 
 
 47213 
 
 55235 
 
 44765 
 
 57581 
 
 42419 
 
 22 
 
 39 
 
 5-2829 
 
 47171 
 
 55275 
 
 44725 
 
 57619 
 
 42381 
 
 21 
 
 40 
 
 9.5-2870 
 
 10.47130 
 
 9.55315 
 
 10.44685 
 
 9.57658 
 
 10.42342 
 
 20 
 
 41 
 
 52912 
 
 47088 
 
 55355 
 
 44645 
 
 57696 
 
 4*304 
 
 19 
 
 42 
 
 5-2953 
 
 47047 
 
 55395 
 
 44605 
 
 57734 
 
 42266 
 
 18 
 
 43 
 
 52995 
 
 47005 
 
 55434 
 
 44566 
 
 57772 
 
 42:228 
 
 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 
 
 57963 
 
 42037 
 
 12 
 
 49 
 
 53244 
 
 46756 
 
 55673 
 
 44327 
 
 5S001 
 
 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 
 
 58868 
 
 46632 
 
 55791 
 
 44209 
 
 58115 
 
 41885 
 
 8 
 
 53 
 
 53409 
 
 46591 
 
 55831 
 
 44169 
 
 58153 
 
 41847 
 
 ; 
 
 54 
 
 53450 
 
 46550 
 
 55870 
 
 44130 
 
 58191 
 
 41809 
 
 6 
 
 55 
 
 53492 
 
 46508 
 
 55910 
 
 44090 
 
 582:29 
 
 41771 
 
 5 
 
 56 
 
 53533 
 
 46467 
 
 55949 
 
 44051 
 
 58567 
 
 41733 
 
 4 
 
 57 
 
 53574 
 
 46426 
 
 55989 
 
 44011 
 
 58304 
 
 41696 
 
 3 
 
 58 
 
 53615 
 
 46385 
 
 56028 
 
 43972 
 
 5834:3 
 
 41658 
 
 2 
 
 59 
 
 53656 
 
 46344 
 
 56067 
 
 43933 
 
 58380 
 
 41620 
 
 1 
 
 60 
 
 53697 
 
 46303 
 
 56107 
 
 43893 
 
 58418 
 
 41582 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 i 
 
 
 
 71 
 
 
 70 
 
 
 69 
 
 
 311 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 t 
 
 
 21 
 
 
 22 
 
 
 23 
 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 
 
 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 
 
 3707'4 
 
 56 
 
 5 
 
 58606 
 
 41394 
 
 60823 
 
 39177 
 
 62961 
 
 37039 
 
 55 
 
 6 
 
 58644 
 
 41356 
 
 60859 
 
 39141 
 
 62996 
 
 37004 
 
 54 
 
 7 
 
 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 
 
 10.36B65 
 
 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 
 
 30690 
 
 45 
 
 16 
 
 59019 
 
 40981 
 
 61220 
 
 38780 
 
 63345 
 
 36655 
 
 44 
 
 17 
 
 59056 
 
 401)44 
 
 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 
 
 9.61864 
 
 10.38636 
 
 9.63484 
 
 10.36516 
 
 40 
 
 21 
 
 59205 
 
 40795 
 
 61400 
 
 38600 
 
 63519 
 
 36481 
 
 39 
 
 22 
 
 59243 
 
 40757 
 
 61436 
 
 38564 
 
 63553 
 
 36447 
 
 38 
 
 23 
 
 59280 
 
 40720 
 
 61472 
 
 38528 
 
 63588 
 
 86412 
 
 37 
 
 24 
 
 59317 
 
 40683 
 
 61508 
 
 3-492 
 
 63623 
 
 36377 
 
 36 
 
 25 
 
 59354 
 
 40646 
 
 61544 
 
 38456 
 
 63657 
 
 36313 
 
 35 
 
 26 
 
 59391 
 
 40609 
 
 61579 
 
 38421 
 
 63692 
 
 36808 
 
 34 
 
 27 
 
 59429 
 
 40571 
 
 61615 
 
 38385 
 
 63726 
 
 36274 
 
 33 
 
 28 
 
 59466 
 
 40534 
 
 61651 
 
 38349 
 
 63761 
 
 36239 
 
 32 
 
 29 
 
 59503 
 
 40497 
 
 61687 
 
 38313 
 
 63796 
 
 36204 
 
 31 
 
 80 
 
 9.59540 
 
 10.40460 
 
 9.61722 
 
 10.38278 
 
 9.63830 
 
 10.36170 
 
 30 
 
 31 
 
 59577 
 
 40423 
 
 61758 
 
 38242 
 
 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 
 
 38028 
 
 64072 
 
 35928 
 
 23 
 
 38 
 
 59835 
 
 40165 
 
 62008 
 
 37992 
 
 64106 
 
 35894 
 
 22 
 
 39 
 
 59872 
 
 40128 
 
 62043 
 
 37957 
 
 64140 
 
 35860 
 
 21 
 
 40 
 
 9.59909 
 
 10.40091 
 
 9.62079 
 
 10 37921 
 
 9.64175 
 
 10.35825 
 
 20 
 
 41 
 
 59946 
 
 40054 
 
 62114 
 
 87886 
 
 64209 
 
 35791 
 
 19 
 
 42 
 
 59983 
 
 40017 
 
 62150 
 
 37850 
 
 64243 
 
 35757 
 
 18 
 
 43 
 
 60019 
 
 39981 
 
 62185 
 
 37815 
 
 64278 
 
 35722 
 
 17 
 
 44 
 
 60056 
 
 39944 
 
 62221 
 
 37779 
 
 64312 
 
 35688 
 
 16 
 
 45 
 
 60093 
 
 39907 
 
 6-2256 
 
 37744 
 
 64346 
 
 35654 
 
 15 
 
 46 
 
 60130 
 
 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 
 
 30468 
 
 62680 
 
 37320 
 
 64756 
 
 35244 
 
 3 
 
 58 
 
 60568 
 
 39432 
 
 62715 
 
 37285 
 
 64790 
 
 35210 
 
 2 
 
 59 
 
 60605 
 
 39395 
 
 62750 
 
 37250 
 
 64824 
 
 35176 
 
 1 
 
 60 
 
 60641 
 
 39359 
 
 62785 
 
 37215 
 
 64858 
 
 35142 
 
 
 
 t 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 , 
 
 
 
 68 
 
 
 67 
 
 
 66 
 
 
 312 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 
 24 
 
 25 
 
 
 26 
 
 / 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 o 
 
 9.64858 
 
 10.35142 
 
 9.66867 
 
 10.33133 
 
 9.68818 
 
 10.31182 
 
 60 
 
 1 
 
 64892 
 
 35108 
 
 66900 
 
 33100 
 
 68850 
 
 31150 
 
 59 
 
 2 
 
 64926 
 
 35074 
 
 66933 
 
 33067 
 
 68882 
 
 31118 
 
 58 
 
 3 
 
 64960 
 
 35040 
 
 66966 
 
 33034 
 
 68914 
 
 31086 
 
 57 
 
 4 
 
 G4994 
 
 35006 
 
 66999 
 
 33001 
 
 68946 
 
 31054 
 
 56 
 
 5 
 
 K038 
 
 34972 
 
 67032 
 
 32968 
 
 68978 
 
 31022 
 
 85 
 
 6 
 
 65062 
 
 34938 
 
 67065 
 
 32935 
 
 69010 
 
 30990 
 
 54 
 
 7 
 
 65096 
 
 34904 
 
 67098 
 
 32902 
 
 69042 
 
 30958 
 
 53 
 
 8 
 
 65130 
 
 34870 
 
 67131 
 
 32869 
 
 69074 
 
 30926 
 
 52 
 
 9 
 
 65164 
 
 34836 
 
 67163 
 
 32837 
 
 69106 
 
 30894 
 
 61 
 
 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 
 
 82574 
 
 69361 
 
 30639 
 
 43 
 
 18 
 
 65467 
 
 34533 
 
 67458 
 
 32542 
 
 69393 
 
 30607 
 
 42 
 
 19 
 
 65501 
 
 34499 
 
 67491 
 
 32509 
 
 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 
 
 32411 
 
 69520 
 
 30480 
 
 38 
 
 23 
 
 65636 
 
 34364 
 
 67622 
 
 32378 
 
 69552 
 
 30448 
 
 37 
 
 24 
 
 656G9 
 
 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 
 
 30321 
 
 33 
 
 28 
 
 65803 
 
 34197 
 
 67785 
 
 32215 
 
 69710 
 
 30290 
 
 32 
 
 n 
 
 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 
 
 69805 
 
 30195 
 
 29 
 
 32 
 
 65937 
 
 34063 
 
 67915 
 
 32085 
 
 69837 
 
 30163 
 
 28 
 
 33 
 
 65971 
 
 34029 
 
 67947 
 
 32053 
 
 69868 
 
 30132 
 
 27 
 
 34 
 
 66004 
 
 33996 
 
 67980 
 
 32020 
 
 69900 
 
 30100 
 
 26 
 
 35 
 
 66038 
 
 33962 
 
 68012 
 
 31988 
 
 69932 
 
 30068 
 
 25 
 
 86 
 
 66071 
 
 83929 
 
 68044 
 
 31956 
 
 69963 
 
 30037 
 
 24 
 
 37 
 
 66104 
 
 33896 
 
 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 
 
 33702 
 
 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 
 
 12 
 
 49 
 
 66503 
 
 33497 
 
 68465 
 
 31585 
 
 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 
 
 33166 
 
 68786 
 
 31214 
 
 70685 
 
 29315 
 
 1 
 
 60 
 
 66867 
 
 33133 
 
 68818 
 
 31182 
 
 70717 
 
 29283 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 / 
 
 
 
 66 
 
 
 64 
 
 
 63 
 
 
 313 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS 
 
 / 
 
 27 
 
 28 
 
 
 29 
 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotau 
 
 Tan 
 
 Cotan 
 
 
 
 
 9.70717 
 
 10.29283 
 
 9.72567 
 
 10.27438 
 
 9.74375 
 
 10.25625 
 
 60 
 
 1 
 
 70748 
 
 29252 
 
 72598 
 
 27402 
 
 74405 
 
 26595 
 
 59 
 
 2 
 
 70779 
 
 29221 
 
 72628 
 
 27372 
 
 74435 
 
 255G5 . 
 
 58 
 
 3 
 
 70810 
 
 29190 
 
 72659 
 
 27341 
 
 74465 
 
 25585 
 
 57 
 
 4 
 
 70841 
 
 29159 
 
 72689 
 
 27311 
 
 74494 
 
 25306 
 
 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 
 
 70966 
 
 29034 
 
 72811 
 
 27189 
 
 74613 
 
 25387 
 
 52 
 
 9 
 
 70997 
 
 29003 
 
 72811 
 
 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 
 
 27068 
 
 74732 
 
 252G8 
 
 48 
 
 13 
 
 71121 
 
 28879 
 
 72963 
 
 27037 
 
 74762 
 
 25238 
 
 47 
 
 14 
 
 71153 
 
 28847 
 
 72993 
 
 27007 
 
 74791 
 
 252U9 
 
 46 
 
 15 
 
 71181 
 
 28816 
 
 73023 
 
 26977 
 
 74821 
 
 251 7 9 
 
 45 
 
 16 
 
 71215 
 
 28785 
 
 73054 
 
 26946 
 
 74851 
 
 25149 
 
 44 
 
 17 
 
 71246 
 
 28754 
 
 73084 
 
 26916 
 
 74880 
 
 25120 
 
 43 
 
 18 
 
 71277 
 
 28723 
 
 73114 
 
 26886 
 
 74910 
 
 25090 
 
 42 
 
 19 
 
 71303 
 
 28692 
 
 73144 
 
 26856 
 
 74939 
 
 2506 1 
 
 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 
 
 23 
 
 71431 
 
 28569 
 
 73265 
 
 26735 
 
 75058 
 
 24942 
 
 37 
 
 24 
 
 71462 
 
 28538 
 
 73295 
 
 26705 
 
 75(87 
 
 24913 
 
 36 
 
 25 
 
 71493 
 
 28507 
 
 73326 
 
 26674 
 
 75117 
 
 24883 
 
 35 
 
 26 
 
 71524 
 
 28476 
 
 73316 
 
 26644 
 
 75146 
 
 24854 
 
 34 
 
 27 
 
 71555 
 
 28445 
 
 733S6 
 
 26614 
 
 75176 
 
 24824 
 
 33 
 
 28 
 
 715S6 
 
 28414 
 
 7341 G 
 
 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 
 
 34 
 
 71771 
 
 28229 
 
 73597 
 
 26403 
 
 75382 
 
 24618 
 
 26 
 
 35 
 
 71802 
 
 28198 
 
 73627 
 
 26373 
 
 75411 
 
 24589 
 
 25 
 
 36 
 
 71833 
 
 28167 
 
 73G57 
 
 26M3 
 
 75441 
 
 24559 
 
 24 
 
 37 
 
 71863 
 
 28137 
 
 73687 
 
 26313 
 
 75470 
 
 24530 
 
 23 
 
 38 
 
 71894 
 
 2810o 
 
 73717 
 
 26283 
 
 75500 
 
 24500 
 
 
 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 
 
 27958 
 
 73867 
 
 26133 
 
 75647 
 
 24353 
 
 17 
 
 44 
 
 72078 
 
 27922 
 
 73897 
 
 26103 
 
 75676 
 
 24824 
 
 16 
 
 45 
 
 72109 
 
 27891 
 
 73927 
 
 26073 
 
 75705 
 
 24295 
 
 15 
 
 46 
 
 72140 
 
 27860 
 
 73957 
 
 26043 
 
 75735 
 
 2421 :5 
 
 14 
 
 47 
 
 72170 
 
 27830 
 
 73987 
 
 26013 
 
 75764 
 
 24236 
 
 13 
 
 48 
 
 72201 
 
 27799 
 
 74017 
 
 259S3 
 
 75793 
 
 24207 
 
 12 
 
 49 
 
 72231 
 
 27769 
 
 74047 
 
 25953 
 
 75822 
 
 24178 
 
 11 
 
 50 
 
 9.72262 
 
 10.27738 
 
 9.74077 
 
 10.25923 
 
 9.7SH52 
 
 10.24148 
 
 10 
 
 51 
 
 72293 
 
 27707 
 
 74107 
 
 25893 
 
 75881 
 
 24119 
 
 9 
 
 52 
 
 72323 
 
 27677 
 
 74137 
 
 25863 
 
 75910 
 
 24080 
 
 8 
 
 53 
 
 72354 
 
 27646 
 
 74166 
 
 25834 
 
 75939 
 
 24061 
 
 
 54 
 
 723S4 
 
 27616 
 
 7419(5 
 
 25R04 
 
 75969 
 
 24031 
 
 6 
 
 55 
 
 72415 
 
 27585 
 
 74888 
 
 2o774 
 
 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 
 
 23014 
 
 2 
 
 59 
 
 72537 
 
 27403 
 
 74345 
 
 25655 
 
 76115 
 
 2388") 
 
 1 
 
 60 
 
 72567 
 
 27433 
 
 74375 
 
 25625 
 
 76144 
 
 23856 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 / 
 
 
 
 62 
 
 
 61 
 
 
 60 
 
 
 314 
 
TABLE XXL LOG. TANGENTS AND COTANGENTS. 
 
 / 
 
 30 
 
 31 
 
 
 32 
 
 f 
 
 
 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 
 
 22065 
 
 79635 
 
 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 
 
 20 '96 
 
 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 
 
 76783 
 
 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 
 
 38 
 
 28 
 
 76957 
 
 23043 
 
 78675 
 
 21325 
 
 80363 
 
 19637 
 
 32 
 
 29 
 
 76986 
 
 23014 
 
 78704 
 
 21296 
 
 80391 
 
 19609 
 
 31 
 
 30 
 
 9.77015 
 
 10.22985 
 
 9.78732 
 
 10.212G8 
 
 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 
 
 771 88 
 
 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 
 
 77019 
 
 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 
 
 18770 
 
 1 
 
 60 
 
 77877 
 
 22123 
 
 79579 
 
 20421 
 
 81253 
 
 18748 
 
 
 
 , 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 / 
 
 
 
 69 
 
 
 58 
 
 
 67 
 
 
 315 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 
 33 
 
 34 
 
 35 
 
 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 o 
 
 9.81252 
 
 10.18748 
 
 9.82899 
 
 10.17101 
 
 9.84523 
 
 10.15477 
 
 60 
 
 1 
 
 81279 
 
 18721 
 
 82926 
 
 17074 
 
 81550 
 
 15450 
 
 59 
 
 2 
 
 81307 
 
 18693 
 
 82953 
 
 17047 
 
 8J576 
 
 15424 
 
 58 
 
 g 
 
 81335 
 
 18665 
 
 82980 
 
 17020 
 
 84603 
 
 15397 
 
 57 
 
 4 
 
 81362 
 
 18638 
 
 83008 
 
 16992 
 
 84630 
 
 15370 
 
 56 
 
 g 
 
 81390 
 
 18610 
 
 83035 
 
 16965 
 
 84657 
 
 15343 
 
 55 
 
 
 81418 
 
 18582 
 
 83062 
 
 16938 
 
 84684 
 
 15316 
 
 54 
 
 7 
 
 81445 
 
 18555 
 
 83089 
 
 16911 
 
 84711 
 
 15289 
 
 53 
 
 g 
 
 81473 
 
 18527 
 
 83117 
 
 16883 
 
 84738 
 
 152G2 
 
 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 
 
 83198 
 
 16802 
 
 84818 
 
 15182 
 
 49 
 
 12 
 
 81583 
 
 18417 
 
 83225 
 
 16775 
 
 84845 
 
 15155 
 
 48 
 
 18 
 
 81611 
 
 18389 
 
 83252 
 
 16748 
 
 84872 
 
 15128 
 
 47 
 
 14 
 
 81638 
 
 18362 
 
 83280 
 
 16720 
 
 84899 
 
 15101 
 
 46 
 
 JK 
 
 81666 
 
 18334 
 
 83307 
 
 16693 
 
 84925 
 
 15075 
 
 45 
 
 JO 
 
 81693 
 
 18307 
 
 83334 
 
 16666 
 
 84952 
 
 15048 
 
 44 
 
 17 
 
 81721 
 
 18279 
 
 83361 
 
 16639 
 
 84979 
 
 15021 
 
 43 
 
 10 
 
 81748 
 
 18252 
 
 83338 
 
 16612 
 
 85006 
 
 14994 
 
 42 
 
 JO 
 
 19 
 
 81776 
 
 18224 
 
 83415 
 
 16585 
 
 85033 
 
 14967 
 
 41 
 
 20 
 
 9.81803 
 
 10.18197 
 
 9.83142 
 
 10.16558 
 
 9.85059 
 
 10.14941 
 
 40 
 
 21 
 
 81831 
 
 18169 
 
 83470 
 
 16530 
 
 85086 
 
 14914 
 
 39 
 
 
 81858 
 
 18142 
 
 83497 
 
 . 16503 
 
 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 
 
 2H 
 
 82023 
 
 17977 
 
 83659 
 
 16341 
 
 85273 
 
 14727 
 
 32 
 
 *o 
 
 29 
 
 82051 
 
 17949 
 
 83686 
 
 16314 
 
 85300 
 
 14700 
 
 31 
 
 30 
 
 9.82078 
 
 10.17922 
 
 9.83713 
 
 10.16287 
 
 9.85327 
 
 10.14673 
 
 30 
 
 31 
 
 82106 
 
 17894 
 
 83740 
 
 16260 
 
 85354 
 
 14646 
 
 29 
 
 0.1 
 
 82133 
 
 17867 
 
 83768 
 
 16232 
 
 85380 
 
 14620 
 
 28 
 
 o-* 
 
 83 
 
 82161 
 
 17839 
 
 83795 
 
 16-205 
 
 85407 
 
 14593 
 
 27 
 
 
 82188 
 
 17812 
 
 83822 
 
 16178 
 
 85434 
 
 14566 
 
 26 
 
 OK 
 
 82215 
 
 17785 
 
 83849 
 
 16151 
 
 85460 
 
 14540 
 
 25 
 
 OD 
 
 36 
 
 82243 
 
 17757 
 
 83876 
 
 16124 
 
 85487 
 
 14513 
 
 24 
 
 07 
 
 82270 
 
 17730 
 
 83903 
 
 16097 
 
 85514 
 
 14486 
 
 23 
 
 ul 
 
 00 
 
 82298 
 
 17702 
 
 83930 
 
 16070 
 
 85540 
 
 14460 
 
 22 
 
 OO 
 
 39 
 
 82325 
 
 17675 
 
 83957 
 
 16043 
 
 85567 
 
 14433 
 
 21 
 
 40 
 
 9.82352 
 
 10.17648 
 
 9.83984 
 
 10.16016 
 
 9.85594 
 
 10.14406 
 
 20 
 
 
 
 82380 
 
 17620 
 
 84011 
 
 15989 
 
 85620 
 
 14380 
 
 19 
 
 42 
 
 82407 
 
 17593 
 
 84038 
 
 15962 
 
 85647 
 
 14353 
 
 18 
 
 
 82435 
 
 17565 
 
 84065 
 
 15935 
 
 85674 
 
 14326 
 
 17 
 
 AA 
 
 82462 
 
 17538 
 
 84092 
 
 15908 
 
 85700 
 
 14300 
 
 16 
 
 44 
 
 82489 
 
 17511 
 
 84119 
 
 15881 
 
 85727 
 
 14273 
 
 15 
 
 4fi 
 
 82517 
 
 17483 
 
 84146 
 
 15854 
 
 85754 
 
 14246 
 
 14 
 
 40 
 
 47 
 
 82544 
 
 17456 
 
 84173 
 
 15827 
 
 85780 
 
 14220 
 
 13 
 
 48 
 
 82571 
 
 17429 
 
 84200 
 
 15800 
 
 85807 
 
 14193 
 
 12 
 
 4o 
 
 49 
 
 82599 
 
 17401 
 
 84227 
 
 15773 
 
 85834 
 
 14166 
 
 11 
 
 ^ 
 
 9.82626 
 
 10.17374 
 
 9.84254 
 
 10.15746 
 
 9.85860 
 
 10.14140 
 
 10 
 
 
 82653 
 
 17347 
 
 84280 
 
 15720 
 
 85887 
 
 14113 
 
 9 
 
 
 82681 
 
 17319 
 
 84307 
 
 15693 
 
 85913 
 
 14087 
 
 8 
 
 fcO 
 
 82708 
 
 17292 
 
 84334 
 
 15666 
 
 85940 
 
 14060 
 
 7 
 
 Oo 
 fr 1 
 
 82735 
 
 17265 
 
 84361 
 
 15639 
 
 85967 
 
 14033 
 
 6 
 
 04 
 
 55 
 
 82762 
 
 17238 
 
 843S8 
 
 15612 
 
 85993 
 
 14007 
 
 5 
 
 KC 
 
 82790 
 
 17210 
 
 84415 
 
 15585 
 
 86020 
 
 13980 
 
 4 
 
 00 
 
 82817 
 
 17183 
 
 84442 
 
 15558 
 
 86046 
 
 13954 
 
 3 
 
 KQ 
 
 82844 
 
 17156 
 
 84469 
 
 15531 
 
 86073 
 
 13927 
 
 2 
 
 Oo 
 
 en 
 
 82871 
 
 17129 
 
 84496 
 
 15504 
 
 86100 
 
 13900 
 
 1 
 
 oy 
 60 
 
 82899 
 
 17101 
 
 84523 
 
 15477 
 
 86126 
 
 13874 
 
 
 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 , 
 
 
 
 50 
 
 
 55 
 
 
 54 
 
 
 316 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 
 86 
 
 
 , 
 
 J7 
 
 
 38 
 
 ' 
 
 
 Tan Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 o 
 
 9.86126 10. 
 
 13874 
 
 9.87711 
 
 10.12289 
 
 9.89281 
 
 10.10719 
 
 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 
 
 13688 
 
 87895 
 
 12105 
 
 89463 
 
 10537 
 
 53 
 
 8 
 
 86333 
 
 13662 
 
 87922 
 
 12078 
 
 89489 
 
 10511 
 
 52 
 
 9 
 
 86365 
 
 13635 
 
 87948 
 
 12052 
 
 89515 
 
 10485 
 
 51 
 
 10 
 
 9.86392 10. 
 
 13603 
 
 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 
 
 8S367 
 
 11633 
 
 89931 
 
 10069 
 
 35 
 
 26 
 
 86815 
 
 13185 
 
 8S393 
 
 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 
 
 9.80921 10. 
 
 13079 
 
 9.88498 
 
 10.11502 
 
 9.90081 
 
 10.09939 
 
 30 
 
 31 
 
 86947 
 
 130;>3 
 
 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 
 
 11307 
 
 90164 
 
 09836 
 
 26 
 
 35 
 
 87053 
 
 12947 
 
 88629 
 
 11371 
 
 90190 
 
 09810 
 
 25 
 
 36 
 
 87079 
 
 12921 
 
 88655 
 
 11345 
 
 90216 
 
 09784 
 
 24 
 
 37 
 
 87106 
 
 12894 
 
 88681 
 
 11319 
 
 90242 
 
 09758 
 
 23 
 
 38 
 
 87132 
 
 12868 
 
 88707 
 
 11293 
 
 90268 
 
 09732 
 
 22 
 
 39 
 
 87158 
 
 12842 
 
 88733 
 
 11267 
 
 90294 
 
 09706 
 
 21 
 
 40 
 
 9.87185 10. 
 
 12815 
 
 9.88759 
 
 10.11241 
 
 9.90320 
 
 10.09680 
 
 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 
 
 45 
 
 87290 
 87317 
 
 12710 
 12683 
 
 88864 
 88890 
 
 11136 
 11110 
 
 90423 
 90449 
 
 09577 
 09551 
 
 16 
 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 
 
 12525 
 
 89046 
 
 10954 
 
 90604 
 
 09396 
 
 9 
 
 52 
 
 87501 
 
 12499 
 
 89073 
 
 10927 
 
 90680 
 
 09370 
 
 8 
 
 53 
 
 87527 
 
 12473 
 
 89099 
 
 10901 
 
 90656 
 
 09344 
 
 7 
 
 54 
 
 87554 
 
 12446 
 
 89125 
 
 10875 
 
 90682 
 
 09318 
 
 6 
 
 55 
 
 87580 
 
 12420 
 
 89151 
 
 10849 
 
 90708 
 
 09292 
 
 5 
 
 56 
 
 87606 
 
 12394 
 
 89177 
 
 10823 
 
 90734 
 
 09266 
 
 4 
 
 57 
 
 87633 
 
 12367 
 
 89203 
 
 10797 
 
 90759 
 
 09241 
 
 3 
 
 58 
 
 87659 
 
 12341 
 
 89229 
 
 10771 
 
 90785 
 
 09215 
 
 2 
 
 59 
 
 87685 
 
 12315 
 
 89255 
 
 10745 
 
 90811 
 
 09189 
 
 1 
 
 60 
 
 87711 
 
 12289 
 
 89281 
 
 10719 
 
 90837 
 
 09163 
 
 
 
 , Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 t 
 
 53 
 
 
 
 62 
 
 
 51 
 
 
 317 
 
TABLE XXI. LOG. TANGENTS AND COTANGENTS- 
 
 t 
 
 39 
 
 40 
 
 
 41 
 
 / 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 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 
 
 94018 
 
 0598-2 
 
 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 
 
 91069 
 
 08931 
 
 92612 
 
 07388 
 
 94146 
 
 05854 
 
 51 
 
 10 
 
 9.91095 
 
 10.08905 
 
 9.92638 
 
 10.07362 
 
 9.94171 
 
 10.058-29 
 
 50 
 
 11 
 
 91121 
 
 08879 
 
 92663 
 
 07337 
 
 94197 
 
 05803 
 
 49 
 
 12 
 
 91147 
 
 08853 
 
 92689 
 
 07311 
 
 94222 
 
 05778 
 
 48 
 
 13 
 
 91172 
 
 088-28 
 
 92715 
 
 07285 
 
 94248 
 
 05752 
 
 47 
 
 14 
 
 91198 
 
 08802 
 
 92740 
 
 07260 
 
 94273 
 
 05727 
 
 46 
 
 15 
 
 91224 
 
 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 
 
 056-25 
 
 42 
 
 19 
 
 91327. 
 
 08673 
 
 92868 
 
 07132 
 
 94401 
 
 05599 
 
 41 
 
 20 
 
 9.91353 
 
 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 
 
 91482 
 
 08518 
 
 93022 
 
 06978 
 
 94554 
 
 05446 
 
 35 
 
 26 
 
 91507 
 
 08493 
 
 93048 
 
 06952 
 
 94579 
 
 054-21 
 
 34 
 
 27 
 
 91533 
 
 08467 
 
 93073 
 
 06927 
 
 94604 
 
 05396 
 
 33 
 
 28 
 
 91559 
 
 08441 
 
 93099 
 
 06901 
 
 94630 
 
 05370 
 
 32 
 
 29 
 
 91585 
 
 08415 
 
 93124 
 
 06876 
 
 94655 
 
 05345 
 
 31 
 
 30 
 
 9.91610 
 
 10.08390 
 
 9.93150 
 
 10.06850 
 
 9.94681 
 
 10.05319 
 
 30 
 
 31 
 
 91636 
 
 08364 
 
 93175 
 
 06825 
 
 94706 
 
 05294 
 
 29 
 
 32 
 
 91662 
 
 08338 
 
 93201 
 
 06799 
 
 94732 
 
 05268 
 
 28 
 
 33 
 
 91688 
 
 08312 
 
 93227 
 
 06773 
 
 94757 
 
 05243 
 
 27 
 
 34 
 
 91713 
 
 08-287 
 
 93252 
 
 06748 
 
 94783 
 
 05217 
 
 26 
 
 35 
 
 91739 
 
 08261 
 
 93278 
 
 06722 
 
 94808 
 
 05192 
 
 25 
 
 36 
 
 91765 
 
 08235 
 
 93303 
 
 06697 
 
 94834 
 
 05166 
 
 24 
 
 87 
 
 91791 
 
 08209 
 
 93329 
 
 06671 
 
 94859 
 
 05141 
 
 23 
 
 38 
 
 91816 
 
 08184 
 
 93354 
 
 06646 
 
 94884 
 
 05116 
 
 22 
 
 39 
 
 91842 
 
 08158 
 
 93380 
 
 06620 
 
 94910 
 
 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 
 
 08081 
 
 93457 
 
 06543 
 
 94986 
 
 05014 
 
 18 
 
 43 
 
 91945 
 
 08055 
 
 93482 
 
 06518 
 
 95012 
 
 04988 
 
 17 
 
 44 
 
 91971 
 
 08029 
 
 93508 
 
 06492 
 
 95037 
 
 04963 
 
 16 
 
 45 
 
 91996 
 
 08004 
 
 93533 
 
 06467 
 
 95062 
 
 04936 
 
 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 
 
 06361 
 
 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 
 
 53 
 
 92202 
 
 07798 
 
 93738 
 
 06262 
 
 95266 
 
 04734 
 
 7 
 
 54 
 
 92227 
 
 07773 
 
 93763 
 
 06237 
 
 95291 
 
 04709 
 
 6 
 
 55 
 
 92253 
 
 07747 
 
 93789 
 
 06211 
 
 95317 
 
 04683 
 
 5 
 
 56 
 
 92279 
 
 07721 
 
 93814 
 
 06186 
 
 95342 
 
 04658 
 
 4 
 
 57 
 
 9-2304 
 
 07696 
 
 93840 
 
 06160 
 
 95368 
 
 04032 
 
 3 
 
 58 
 
 9-2330 
 
 07670 
 
 93865 
 
 06135 
 
 95393 
 
 04607 
 
 2 
 
 59 
 
 92356 
 
 07644 
 
 93891 
 
 06109 
 
 95418 
 
 0458-2 
 
 1 
 
 60 
 
 92381 
 
 07619 
 
 93916 
 
 06084 
 
 95444 
 
 04556 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 ., 
 
 
 
 50- 
 
 
 49 
 
 
 48 
 
 
 318 
 
FABLE XXI. LOG. TANGENTS AND COTANGENTS. 
 
 
 42 
 
 43 
 
 
 440 
 
 / 
 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 
 o 
 
 9.95444 
 
 10.04556 
 
 9.96966 
 
 10.03034 
 
 9.98484 
 
 10.01516 
 
 60 
 
 1 
 
 95469 
 
 04531 
 
 96991 
 
 03009 
 
 98509 
 
 01491 
 
 &9 
 
 3 
 
 95495 
 
 04505 
 
 97016 
 
 02984 
 
 98534 
 
 01466 
 
 58 
 
 3 
 
 95520 
 
 04480 
 
 97042 
 
 02958 
 
 98560 
 
 01440 
 
 67 
 
 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 
 
 i 11 
 
 95723 
 
 04277 
 
 97244 
 
 02756 
 
 98762 
 
 01238 
 
 49 
 
 12 
 
 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 
 
 01137 
 
 45 
 
 16 
 
 95850 
 
 04150 
 
 97371 
 
 02629 
 
 98888 
 
 01112 
 
 44 
 
 17 
 
 95875 
 
 04125 
 
 97396 
 
 02604 
 
 98913 
 
 01087 
 
 43 
 
 18 
 
 95901 
 
 04099 
 
 97421 
 
 02579 
 
 98939 
 
 010G1 
 
 42 
 
 19 
 
 95926 
 
 04074 
 
 97447 
 
 02553 
 
 98964 
 
 01036 
 
 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 
 
 96068 
 
 03947 
 
 97573 
 
 0242? 
 
 99090 
 
 00910 
 
 86 
 
 25 
 
 96078 
 
 03922 
 
 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 
 
 32 
 
 29 
 
 96180 
 
 03820 
 
 97700 
 
 02300 
 
 99217 
 
 00783 
 
 31 
 
 30 
 
 9.96205 
 
 10.03795 
 
 9.97725 
 
 10.02275 
 
 9.99242 
 
 10.00758 
 
 30 
 
 31 
 
 96281 
 
 03769 
 
 97750 
 
 02250 
 
 99267 
 
 00733 
 
 29 
 
 32 
 
 96256 
 
 03744 
 
 97776 
 
 02224 
 
 99293 
 
 00707 
 
 28 
 
 33 
 
 9(5281 
 
 03719 
 
 97801 
 
 02199 
 
 99318 
 
 00682 
 
 27 
 
 34 
 
 96307 
 
 03693 
 
 97826 
 
 02174 
 
 99343 
 
 00657 
 
 26 
 
 as 
 
 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 
 
 9b003 
 
 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 
 
 01845 
 
 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 
 
 52 
 
 96763 
 
 03237 
 
 98281 
 
 01719 
 
 99798 
 
 00202 8 
 
 53 
 
 96788 
 
 03212 
 
 98307 
 
 01693 
 
 99823 
 
 00177 7 
 
 54 
 
 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 
 
 89949 
 
 00051 
 
 2 
 
 59 
 
 96940 
 
 03060 
 
 98458 
 
 01542 
 
 99975 
 
 00025 
 
 1 
 
 60 
 
 96966 
 
 03034 
 
 98484 
 
 01516 
 
 10.00000 
 
 00000 
 
 
 
 / 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 Cotan 
 
 Tan 
 
 / 
 
 I 
 
 47 
 
 
 46 
 
 
 45 
 
 
 319 
 
or THE 
 UNIVERSITY 
 
 OF 
 
 APPENDIX. 
 
 THE TRANSITION CURVE. 
 
 The true transition curve is a spiral whose radius of curva- 
 ture at the origin, A , is infinity, and at any point on the curve 
 is inversely proportional to the distance of that point from A ; 
 or, the degree of curvature is directly proportional to this dis- 
 tance. The degree of curvature will therefore have a constant 
 increase per foot. 
 
 Let n = this increase in minutes, 
 
 d = the distance of any point from A, 
 and 8 = degree of curvature at distance' d ; 
 
 then 8 = j^, (1) 
 
 and d = *. (2) 
 
 n 
 
 Now, since the curvature increases uniformly from to 8 in 
 the distance d, it is evident that the total angle turned will be 
 only half that turned by a 8 curve d feet in length. Therefore 
 the total angle turned by the spiral in the distance d will be 
 
 60 Bd , 
 
 (expressed in minutes). 
 200 
 
 If in Fig. 1 B be the point where the degree of curvature 
 becomes 8, then 
 
 AB = d, 
 
 Substituting the value of 8 from (1), we have 
 
 =^. (4) 
 
 200 
 
 321 
 
322 
 
 APPENDIX. 
 
 E 
 
 This spiral possesses the following well-known properties : 
 
 1. It is almost identical with the cubic parabola, the only 
 difference being this : in the cubic parabola the ordinates vary 
 as the cubes of the abscissas, while in this curve they vary as 
 the cubes of the corresponding lengths of the curve. 
 
 2. The spiral bisects the offset, FN, to the central curve 
 produced and is bisected by it. 
 
 3. It therefore follows that the central ordinate FB = and 
 the central curve offset FN = \ of the terminal ordinate CE. 
 
 Now, in the cubic parabola, the tangent of any deflection 
 angle, as BAF, is equal to of the tangent of the tangential 
 angle BGF. And since small angles vary nearly as their tan- 
 gents, we may assume that 
 
 (5) 
 
 and since 
 
 we have 
 
 600 
 = BGF-BAF, 
 
 = ^L_^L = ^. 
 200 600 300* 
 
 (6) 
 
 * Fig. 2 exhibits lines in their proper relation. Fig. 1 is not drawn 
 to scale but made to aid in the demonstration. 
 
APPENDIX. 323 
 
 By (5) and (6) the spiral may be laid out with the transit 
 using (5) for deflections from the tangent and (6) for turn- 
 ing tangent from any chord. If there be no obstacle in the 
 line of sight, and the terminating spiral be run . backward, 
 we need nothing more. But it may be necessary to set some 
 intermediate point, as B, and continue from that point by 
 deflections from the tangent BK ; and it is generally desirable 
 to run the terminating spiral with the transit set at its junction 
 with the central curve. It is therefore necessary to find a 
 general expression for the deflection from any point on the spiral 
 to any other point on the spiral. 
 
 From (5) we have 
 
 .00029 nd 2 
 
 smSAF - *%?" < 7 > 
 
 since .00029 = sine of 1'. 
 
 Then if d' = BC, that is any distance beyond B, we have also 
 
 *?. (8) 
 
 Hence we have for the ordinates FB and CE, by regarding the 
 curves AB and AC as equal to the chords of the same, 
 
 .00029 nd* 
 ra = 600 ' (9) 
 
 CE = - v n ; . (10) 
 
 600 
 
 TVion r'T-T" C'TT' 7772 v ' ~L_!~ / /1 1 \ 
 
 ^600~ "' (U) 
 
 and ^ - sin CBH - - 00029 n (3 c? 2 + 3 dd' + ^ /2 ) a . 
 d' ~ 600 
 
 
 
 But CBK = CBH - KBH = CBH - BGF = CBH - . 
 
 200 
 
 Therefore CBg = < 81 "' + "*> = ?g -H^. (14, 
 
 BOO 600 200 
 
324 APPENDIX. 
 
 To obtain an expression for the angle BCK, we have from (4) 
 
 and since BCK = CDS - CBH, 
 
 . 
 
 600 300 200 
 
 Eqs. (14) and (16) may be put into the following forms : 
 _ /nd . 1 . 60 d 
 
 \6Q 2 100 / 600 
 
 ~ ( ian X o X inn / ~~ ~at\n ' ( - ) 
 
 in which the first term of (17) is the deflection for a distance 
 d' of a circular curve, whose curvature is equal to the curvature 
 of the spiral at distance d (see Eq. 1) ; and the first term of 
 (18) is the deflection for a distance d' of a circular curve, whose 
 curvature is equal to the curvature of the spiral at distance 
 d -f d'. Hence the following property : 
 
 The deflection from tangent at any point on the spiral to any 
 other point on the spiral is equal to the deflection of a circular curve 
 for the same distance, whose curvature is equal to the curvature of 
 the spiral at said tangent point, PLUS or MINUS the deflection for 
 an equal distance from the initial point of the spiral, according as 
 the transit is turned TOWARD or FROM the central curve. 
 
 To find the length of semi-tangent, T, and external secant, E, 
 we have from (9), since 
 
 FN = 2 FB, 
 
 , = , 
 
 in which d = total length of spiral. Therefore if in Fig. 2 
 L = total length of spiral, V point of intersection of tangents, 
 P middle point of circular curve, and / = angle of intersec- 
 tion of tangents, then 
 
 AV= T=(R+ 0)taniI + iL, (20) 
 
 and py = E= -R. (21) 
 
 COS-J I 
 
APPENDIX. 
 
 325 
 
 FIG. 2. 
 
326 APPENDIX. 
 
 It is proposed to call n in these formulas the number of the 
 spiral. Thus if n 2, we would designate the spiral as a 
 number two spiral ; if n = 3, a number three spiral, etc. n may 
 have any value whatever, either entire or fractional. In prac- 
 tice its value will generally be between 1 and 6. 
 
 It will be seen that by this method the transition curve 
 becomes absolutely universal. By varying the value of n a 
 spiral of any length whatever can be fitted to a central curve 
 of any degree. The question of standard sub-chords, which are 
 used in a great many systems of treating the transition curve, 
 is entirely eliminated. A transit hub can be set at any point 
 on the curve and deflections may be turned from any point on 
 the curve to any other point with practically the same ease and 
 facility as in the simple curve. 
 
 HOW TO LAY OUT A SPIRAL CURVE. 
 
 1. The tables which follow will greatly facilitate the work of 
 computing and laying out the spiral. Table I gives the length 
 (Z), total angle (^4), and central curve offset (0) of a No. 1 
 spiral for central curves of different degrees. To find the cor- 
 responding elements of a spiral of different number it will be 
 observed that L and A vary inversely as the number and 
 inversely as the square of the number. The number of the 
 spiral should be so chosen that the total angle will not much 
 exceed 15, as the formula for deflections becomes less accu- 
 rate for large angles. Table II is a tabulation of Eq. 5 for a 
 No. 1 spiral. 
 
 2. Determine by inspection of Table I and the conditions 
 on the ground the number of the spiral to be used in any par- 
 ticular case, and find central curve offset, O, for this spiral. 
 
 Calculate semi-tangent by the following formula, 
 
 T - (R + O) tan I + $ L, 
 
 and locate points of the spiral A and F on each tangent. 
 
 3. Set transit at A and by use of Table II turn deflections 
 for all points from A to C. Move transit to C, backsight on 
 
APPENDIX. 327 
 
 A, and turn twice the angle that was turned at A, which will 
 bring transit on tangent at this point. Continue around the 
 central curve to D in the usual manner. The transit may then 
 be moved to F and the terminating spiral run backward in 
 the same manner as the first spiral was run forward. 
 
 FIG. 3. 
 
 4. If desired to run both spirals and central curve continu- 
 ously from beginning to end, or to turn deflections from inter- 
 mediate points on either spiral, observe the property of the. 
 spiral, stated on page 324. 
 
 5. An example will illustrate the application of this prop- 
 erty. Suppose it be required to run a 4 curve with Xo. 1 
 spirals, and the initial point, A , be established at Sta. 122 4- 80 
 as shown in Fig. 3. Table I shows that either spiral will be 
 240 feet long, and that the initial spiral will connect with the 
 central curve at Sta. 125 + 20. Suppose the other end of the 
 central curve be found at Sta. 131 + 60. The terminating 
 spiral will then unite with the tangent at Sta. 134. 
 
 6. Suppose it be desired to set transit points at B, Sta. 124, 
 and E, Sta. 133, in addition to points C and D at ends of the 
 
328 APPENDIX. 
 
 central curve, and run the entire curve continuously. Set 
 transit at A, and turn deflections as follows : 
 
 Sta. 123 : deflection for 20 feet (by Table II) 00. 7' 
 Sta. 124 : deflection for 120 feet (by table) 24.' 
 
 7. Move transit to B. Backsight on A and turn 48' for 
 tangent. Then for deflections from B to C, since the spiral has 
 attained a curvature of 2 per 100 feet at B, we have, according 
 to the property stated above, 
 
 Sta. 125 : deflection for 100 feet, 2 curve 1 00. ' 
 
 Plus deflection for 100 feet spiral (by table) 16.7' 
 
 Total ; 1 16.7' 
 
 Sta. 125 + 20 : deflection for 120 feet, 2 curve 1 12.' 
 
 Plus deflection for 120 feet spiral (by table) 24/ 
 
 Total 1 36.' 
 
 8 . Then move transit to C. Backsight on B, and to turn 
 tangent, since curvature has become 4 at C, 
 
 Deflection for 120 feet, 4 curve 2 24.' 
 
 Minus deflection for 120 feet spiral (by table) 24.' 
 
 Total 2 00.' 
 
 9. Circular curve is then run to D. Next move transit to 
 Z> and set on tangent at this point. We then have for deflec- 
 tions from D to F, since curvature is here 4, 
 
 Sta. 132 : deflection for 40 feet, 4 curve 48.' 
 
 Minus deflection for 40 feet spiral (by table) 02.7' 
 
 Total 45.3' 
 
 Sta. 133 : deflection for 140 feet, 4 curve 2 48.' 
 
 Minus deflection for 140 feet spiral (by table) 32.7' 
 
 Total 2 15.3' 
 
 10. Move transit to E. Backsight on D, and to turn tan- 
 gent, since spiral has a curvature of 1 40' at this point, we 
 have 
 
 Deflection for 140 feet, 1 40' curve 1 10.' 
 
 Plus deflection for 140 feet spiral (by table) 32.7' 
 
 Total . 142.7' 
 
APPENDIX. - 329 
 
 Then, to finish curve, we have 
 
 Deflection for 100 feet, 1 40' curve 50. ' 
 Minus deflection for 100 feet spiral (by table) 16. V 
 
 Total 033.3 / 
 
 11. By moving transit to F, and backsighting upon E, we 
 have for turning on final tangent 
 
 Deflection for 100 feet spiral (by table) 16.7' 
 
 EXAMPLES. 
 
 1. Intersection of two tangents, Sta. 127 + 65.2 ; inter- 
 section angle / = 35 24'. 
 
 Connect tangents with a 6 curve (JR = 955) terminating in 
 a No. 2 spiral. By reference to Table I we find 
 
 L = 180 feet ; A = 5 24' ; = 1.41 feet. 
 
 .-. T = (B + 0) tan i I + i L = 395.2 feet. 
 Beginning of spiral, BS = Sta. 123 + 70. 
 
 Beginning of central curve, BC = " 125 + 50. 
 End of central curve, EC = " 129 + 60. 
 
 End of spiral, ES = " 131+ 40. 
 
 Set transit at BS, Sta. 123 + 70 
 Deflection to Sta. 124 = 03. ' 
 
 Deflection to Sta. 125 = 56.4' 
 
 Deflection to Sta. 125 + 50 =1 48. ' 
 
 Move transit to Sta. 125 + 50. Backsight on Sta. 123 + 70 
 and turn 3 36' for tangent. Run central curve to Sta. 129 + 60 
 in usual manner. Move transit to this point and set on tan- 
 gent in usual way. Then 
 
 Deflection to Sta. 130 = 1 06.6' 
 
 Deflection to Sta. 131 = 3 06.6' 
 
 Deflection to Sta. 131 +40 - 3 36. ' 
 
 Move transit to Sta. 131 + 40. Backsight on Sta. 129 + 60 
 and turn on final tangent by turning 1 48'. 
 
330 - APPENDIX. 
 
 2. Illustrating the use of the spiral as a transition between 
 the branches of a compound curve. A 2 curve compounding 
 to a 6 is to be connected by a No. 1 spiral. 
 
 In this case the length of the spiral and offset between curves 
 produced is the same as for a 4 (6 2) curve. L = 240 feet, 
 O = l. 67 feet. 
 
 Deflections from Sta. 136 + 40 to Sta. 138 + 80 would be 
 deflections for a 2 curve plus tabular deflections for spiral, 
 and deflections from Sta. 138 + 80 to Sta. 136 + 40 would be 
 deflections for a 6 curve minus tabular deflections for spiral, as 
 follows : 
 
 Transit at Sta. 136 + 40 : 
 
 Deflection to Sta. 137 = 42. ' 
 
 Deflection to Sta. 138 = 2 18. 1' 
 
 Deflection to Sta. 138 + 80 = 4 OO/ 
 
 Transit at Sta. 138 + 80: 
 
 Deflection to Sta. 138 = 2 13.3' 
 
 Deflection to Sta. 137 = 4 30. ' 
 
 Deflection to Sta. 136 + 40 = 5 36. ' 
 
APPENDIX. 
 
 TABLE I. 
 
 ELEMENTS OF A No. 1 SPIKAL. 
 
 L = Length of spiral. 
 
 A = Total angle turned by spiral. 
 
 O = Central curve offset. 
 
 L and A vary inversely as the number of spiral. 
 
 O varies inversely as square of the number. 
 
 Deg. 
 
 L 
 
 A 
 
 O 
 
 Deg. 
 
 L 
 
 A 
 
 
 
 2 00' 
 
 120 
 
 1 12' 00" 
 
 .21 
 
 7 10' 
 
 430 
 
 15 24' 30" 
 
 9.61 
 
 10 
 
 130 
 
 1 24 30 
 
 .27 
 
 20 
 
 440 
 
 16 08 00 
 
 10.29 
 
 20 
 
 140 
 
 1 38 00 
 
 .33 
 
 30 
 
 450 
 
 16 52 30 
 
 11.01 
 
 30 
 
 150 
 
 1 52 30 
 
 .41 
 
 40 
 
 460 
 
 17 38 00 
 
 11.76 
 
 40 
 
 160 
 
 2 08 00 
 
 .49 
 
 50 
 
 470 
 
 18 24 30 
 
 12.55 
 
 50 
 
 170 
 
 2 24 30 
 
 .59 
 
 8 00 
 
 480 
 
 19 12 00 
 
 13.36 
 
 3 00 
 
 180 
 
 2 42 00 
 
 .70 
 
 10 
 
 490 
 
 20 00 30 
 
 14.22 
 
 10 
 
 190 
 
 3 00 30 
 
 .83 
 
 20 
 
 500 
 
 20 50 00 
 
 15.10 
 
 20 
 
 200 
 
 3 20 00 
 
 .97 
 
 30 
 
 510 
 
 21 40 30 
 
 16.03 
 
 30 
 
 210 
 
 3 40 30 
 
 1.12 
 
 40 
 
 520 
 
 22 32 00' 
 
 16.99 
 
 40 
 
 220 
 
 4 02 00 
 
 1.29 
 
 50 
 
 530 
 
 23 24 30 
 
 17.99 
 
 50 
 
 230 
 
 4 24 30 
 
 1.47 
 
 9 00 
 
 540 
 
 24 18 00 
 
 19.03 
 
 4 00 
 
 240 
 
 4 48 00 
 
 1.67 
 
 10 
 
 550 
 
 25 12 30 
 
 20.10 
 
 10 
 
 250 
 
 5 12 30 
 
 1.89 
 
 20 
 
 560 
 
 26 08 00 
 
 21.22 
 
 20 
 
 260 
 
 5 38 00 
 
 2.12 
 
 30 
 
 570 
 
 27 04 30 
 
 22.38 
 
 30 
 
 270 
 
 6 04 30 
 
 2.38 
 
 40 
 
 580 
 
 28 02 00 
 
 23.58 
 
 40 
 
 280 
 
 6 32 00 
 
 2.65 
 
 50 
 
 590 
 
 29 00 30 
 
 24.82 
 
 50 
 
 290 
 
 7 00 30 
 
 2.95 
 
 1000 
 
 600 
 
 30 00 00 
 
 26.10 
 
 5 00 
 
 300 
 
 7 30 00 
 
 3.26 
 
 30 
 
 630 
 
 33 04 30 
 
 30.21 
 
 10 
 
 310 
 
 8 00 30 
 
 3.60 
 
 11 00 
 
 660 
 
 36 18 00 
 
 34.74 
 
 20 
 
 320 
 
 8 32 00 
 
 3.96 
 
 30 
 
 690 
 
 39 40 30 
 
 39.69 
 
 30 
 
 330 
 
 9 04 30 
 
 4.34 
 
 12 00 
 
 720 
 
 43 12 00 
 
 45.10 
 
 40 
 
 340 
 
 9 38 00 
 
 5.75 
 
 30 
 
 750 
 
 46 52 30 
 
 50.98 
 
 50 
 
 350 
 
 10 12 30 
 
 5.18 
 
 13 00 
 
 780 
 
 50 42 00 
 
 57.34 
 
 600 
 
 360 
 
 10 48 00 
 
 5.64 
 
 14 00 
 
 840 
 
 58 48 00 
 
 61.62 
 
 10 
 
 370 
 
 11 24 30 
 
 6.12 
 
 15 00 
 
 900 
 
 67 30 00 
 
 8809 
 
 20 
 
 380 
 
 12 02 00 
 
 6.63 
 
 16 00 
 
 960 
 
 76 48 00 
 
 106.91 
 
 30 
 
 390 
 
 12 40 30 
 
 7.17 
 
 17 00 
 
 1020 
 
 86 42 00 
 
 128.23 
 
 40 
 
 400 
 
 13 20 00 
 
 7.73 
 
 18 00 
 
 1080 
 
 97 12 00 
 
 152.22 
 
 50 
 
 410 
 
 14 00 30 
 
 8.33 
 
 19 00 
 
 1140 
 
 108 18 00 
 
 179.02 
 
 7 00 
 
 420 
 
 14 42 00 
 
 8.95 
 
 2000 
 
 1200 
 
 120 00 00 
 
 208.80 
 
APPENDIX. 
 
 TABLE II. 
 
 DEFLECTIONS OF A No. 1 SPIRAL FOR DIFFERENT VALUES OF d. 
 (Deflections for spiral of different number in direct proportion.) 
 
 d 
 
 rf2 
 
 600 
 
 d 
 
 r/2 
 600 
 
 d 
 
 <PL 
 
 600 
 
 d 
 
 d* 
 600 
 
 
 
 0.0 
 
 50 
 
 4.1 
 
 100 
 
 16.7 
 
 150 
 
 37.5 
 
 1 
 
 0.0 
 
 51 
 
 4.3 
 
 101 
 
 17.0 
 
 151 
 
 38.0 
 
 2 
 
 0.0 
 
 52 
 
 4.5 
 
 102 
 
 17.3 
 
 152 
 
 38.5 
 
 3 
 
 0.0 
 
 53 
 
 4.7 
 
 103 
 
 17.7 
 
 153 
 
 39.0 
 
 4 
 
 0.0 
 
 54 
 
 4.9 
 
 104 
 
 18.0 
 
 154 
 
 39.5 
 
 5 
 
 0.0 
 
 55 
 
 5.0 
 
 105 
 
 18.4 
 
 155 
 
 40.0 
 
 6 
 
 0.1 
 
 56 
 
 5.2 
 
 106 
 
 18.7 
 
 156 
 
 40.6 
 
 7 
 
 0.1 
 
 57 
 
 5.4 
 
 107 
 
 19.1 
 
 157 
 
 41.1 
 
 8 
 
 0.1 
 
 58 
 
 5.6 
 
 108 
 
 19.4 
 
 158 
 
 41.6 
 
 9 
 
 0.1 
 
 59 
 
 5.8 
 
 109 
 
 19.8 
 
 159 
 
 42.1 
 
 10 
 
 0.2 
 
 60 
 
 6.0 
 
 110 
 
 20.2 
 
 160 
 
 42.7 
 
 11 
 
 0.2 
 
 61 
 
 6.2 
 
 111 
 
 20.5 
 
 161 
 
 43.2 
 
 12 
 
 0.2 
 
 62 
 
 6.4 
 
 112 
 
 20.9 
 
 162 
 
 43.7 
 
 13 
 
 0.3 
 
 63 
 
 6.6 
 
 113 
 
 21.3 
 
 163 
 
 44.3 
 
 14 
 
 0.3 
 
 64 
 
 6.8 
 
 114 
 
 21.7 
 
 164 
 
 44.8 
 
 15 
 
 0.4 
 
 65 
 
 7.0 
 
 115 
 
 22.0 
 
 165 
 
 45.4 
 
 16 
 
 0.4 
 
 66 
 
 7.3 
 
 116 
 
 22.4 
 
 166 
 
 45.9 
 
 17 
 
 0.5 
 
 67 
 
 7.5 
 
 117 
 
 22.8 
 
 167 
 
 46.5 
 
 18 
 
 0.5 
 
 68 
 
 7.7 
 
 118 
 
 23.2 
 
 168 
 
 47.0 
 
 19 
 
 0.6 
 
 69 
 
 7.9 
 
 119 
 
 23.6 
 
 169 
 
 47.6 
 
 20 
 
 0.7 
 
 70 
 
 8.2 
 
 120 
 
 24.0 
 
 170 
 
 48.2 
 
 21 
 
 0.7 
 
 71 
 
 8.4 
 
 121 
 
 24.4 
 
 171 
 
 48.7 
 
 22 
 
 0.8 
 
 72 
 
 8.6 
 
 122 
 
 24.8 
 
 172 
 
 49.3 
 
 23 
 
 0.9 
 
 73 
 
 8.9 
 
 123 
 
 25.2 
 
 173 
 
 49.9 
 
 24 
 
 1.0 
 
 74 
 
 9.1 
 
 124 
 
 25.6 
 
 174 
 
 50.5 
 
 25 
 
 1.0 
 
 75 
 
 9.4 
 
 125 
 
 26.0 
 
 175 
 
 51.0 
 
 26 
 
 1.1 
 
 76 
 
 9.6 
 
 126 
 
 26.5 
 
 176 
 
 51.6 
 
 27 
 
 1.2 
 
 77 
 
 9.9 
 
 127 
 
 26.9 
 
 177 
 
 52.2 
 
 28 
 
 1.3 
 
 78 
 
 10.1 
 
 128 
 
 27.3 
 
 178 
 
 52.8 
 
 29 
 
 1.4 
 
 79 
 
 10.4 
 
 129 
 
 27.7 
 
 179 
 
 53.4 
 
 30 
 
 1.5 
 
 80 
 
 10.7 
 
 130 
 
 28.2 
 
 180 
 
 54.0 
 
 31 
 
 1.6 
 
 81 
 
 11.0 
 
 131 
 
 28.6 
 
 181 
 
 54.6 
 
 32 
 
 1.7 
 
 82 
 
 11.2 
 
 132 
 
 29.0 
 
 182 
 
 55.2 
 
 33 
 
 1.8 
 
 83 
 
 11.5 
 
 133 
 
 29.5 
 
 183 
 
 55.8 
 
 34 
 
 1.9 
 
 84 
 
 11.8 
 
 134 
 
 29.9 
 
 184 
 
 56.4 
 
 35 
 
 2.0 
 
 85 
 
 12.0 
 
 135 
 
 30.4 
 
 185 
 
 57.0 
 
 36 
 
 2.2 
 
 86 
 
 12.3 
 
 136 
 
 30.8 
 
 186 
 
 57.7 
 
 37 
 
 2.3 
 
 87 
 
 12.6 
 
 137 
 
 31.3 
 
 187 
 
 58.3 
 
 38 
 
 2.4 
 
 88 
 
 12.9 
 
 138 
 
 31.7 
 
 188 
 
 58.9 
 
 39 
 
 2.5 
 
 89 
 
 13.2 
 
 139 
 
 32.2 
 
 189 
 
 595 
 
 40 
 
 2.7 
 
 90 
 
 13.5 
 
 140 
 
 32.7 
 
 190 
 
 60.2 
 
 41 
 
 2.8 
 
 91 
 
 13.8 
 
 141 
 
 33.1 
 
 191 
 
 60.8 
 
 42 
 
 2.9 
 
 92 
 
 14.1 
 
 142 
 
 33.6 
 
 192 
 
 61.4 
 
 43 
 
 3.1 
 
 93 
 
 14.4 
 
 143 
 
 34.1 
 
 193 
 
 62.1 
 
 44 
 
 3.2 
 
 94 
 
 14.7 
 
 144 
 
 34.6 
 
 194 
 
 62.7 
 
 45 
 
 3.4 
 
 95 
 
 15.0 
 
 145 
 
 35.0 
 
 195 
 
 63.4 
 
 46 
 
 3.5 
 
 96 
 
 15.4 
 
 146 
 
 35.5 
 
 196 
 
 64.0 
 
 47 
 
 3.7 
 
 97 
 
 15.7 
 
 147 
 
 36.0 
 
 197 
 
 64.7 
 
 48 
 
 3.8 
 
 98 
 
 16.0 
 
 148 
 
 365 
 
 198 
 
 65.3 
 
 49 
 
 4.0 
 
 99 
 
 16.3 
 
 149 
 
 37.0 
 
 199 
 
 66.0 
 
APPENDIX. 
 
 TABLE II. -Continued. 
 
 d 
 
 d 2 
 600 
 
 d 
 
 .* 
 
 600 
 
 d 
 
 </2 
 
 600 
 
 d 
 
 rf2 
 
 600 
 
 200 
 
 66.7 
 
 240 
 
 96.0 
 
 280 
 
 130.7 
 
 320 
 
 170.7 
 
 201 
 
 67.3 
 
 241 
 
 96.8 
 
 281 
 
 131.6 
 
 321 
 
 171.7 
 
 202 
 
 68.0 
 
 242 
 
 97.6 
 
 282 
 
 132.5 
 
 322 
 
 172.8 
 
 203 
 
 68.7 
 
 243 
 
 98.4 
 
 283 
 
 133.5 
 
 323 
 
 173.9 
 
 204 
 
 69.4 
 
 244 
 
 99.2 
 
 284 
 
 134.4 
 
 324 
 
 175.0 
 
 205 
 
 70.1 
 
 245 
 
 100.0 
 
 285 
 
 135.4 
 
 325 
 
 176.0 
 
 206 
 
 70.7 
 
 246 
 
 100.9 
 
 286 
 
 136.3 
 
 326 
 
 177.1 
 
 207 
 
 71.4 
 
 247 
 
 101.7 
 
 287 
 
 137.3 
 
 327 
 
 178.2 
 
 208 
 
 72.1 
 
 248 
 
 102.5 
 
 288 
 
 138.2 
 
 328 
 
 179.3 
 
 209 
 
 72.8 
 
 249 
 
 103.3 
 
 289 
 
 139.2 
 
 329 
 
 180.4 
 
 210 
 
 73.5 
 
 250 
 
 104.2 
 
 290 
 
 140.2 
 
 880 
 
 181.5 
 
 211 
 
 74.2 
 
 251 
 
 105.0 
 
 291 
 
 141.1 
 
 331 
 
 182.6 
 
 212 
 
 74.9 
 
 252 
 
 105.8 
 
 292 
 
 142.1 
 
 332 
 
 183.7 
 
 213 
 
 75.6 
 
 253 
 
 106.7 
 
 293 
 
 143.1 
 
 333 
 
 184.8 
 
 214 
 
 76.3 
 
 254 
 
 107.5 
 
 294 
 
 144.1 
 
 334 
 
 185.9 
 
 215 
 
 77.0 
 
 255 
 
 108.4 
 
 295 
 
 145.0 
 
 335 
 
 187.0 
 
 216 
 
 77.8 
 
 256 
 
 109.2 
 
 296 
 
 146.0 
 
 336 
 
 188.2 
 
 217 
 
 78.5 
 
 257 
 
 110.1 
 
 297 
 
 147.0 
 
 337 
 
 189.3 
 
 218 
 
 79.2 
 
 258 
 
 110.9 
 
 298 
 
 148.0 
 
 338 
 
 190.4 
 
 219 
 
 79.8 
 
 259 
 
 111.8 
 
 299 
 
 149.0 
 
 339 
 
 191.5 
 
 220 
 
 80.7 
 
 260 
 
 112.7 
 
 300 
 
 150.0 
 
 340 
 
 192.7 
 
 221 
 
 81.4 
 
 261 
 
 113.5 
 
 301 
 
 151.0 
 
 341 
 
 193.8 
 
 222 
 
 82.1 
 
 262 
 
 114.4 
 
 302 
 
 152.0 
 
 342 
 
 194.9 
 
 223 
 
 82.9 
 
 263 
 
 115.3 
 
 303 
 
 153.0 
 
 343 
 
 196.1 
 
 224 
 
 83.6 
 
 264 
 
 116.2 
 
 304 
 
 154.0 
 
 344 
 
 197.2 
 
 225 
 
 84.4 
 
 265 
 
 117.0 
 
 305 
 
 155.0 
 
 345 
 
 198.4 
 
 226 
 
 85.1 
 
 266 
 
 117.8 
 
 306 
 
 156.1 
 
 346 
 
 199.6 
 
 227 
 
 85.9 
 
 267 
 
 118.8 
 
 307 
 
 157.1 
 
 347 
 
 200.7 
 
 228 
 
 86.6 
 
 268 
 
 119.7 
 
 308 
 
 158.1 
 
 348 
 
 201.8 
 
 229 
 
 87.4 
 
 269 
 
 120.6 
 
 309 
 
 159.1 
 
 349 
 
 203.0 
 
 230 
 
 88.2 
 
 270 
 
 121.5 
 
 310 
 
 160.2 
 
 350 
 
 204.2 
 
 231 
 
 88.9 
 
 271 
 
 122.4 
 
 311 
 
 161.2 
 
 351 
 
 205.3 
 
 232 
 
 89.7 
 
 272 
 
 123.3 
 
 312 
 
 162.2 
 
 352 
 
 206.5 
 
 233 
 
 90.5 
 
 273 
 
 124.2 
 
 313 
 
 163.3 
 
 353 
 
 207.7 
 
 234 
 
 91.3 
 
 274 
 
 125.1 
 
 314 
 
 164.3 
 
 354 
 
 208.9 
 
 235 
 
 92.0 
 
 275 
 
 126.0 
 
 315 
 
 165.4 
 
 355 
 
 210.0 
 
 236 
 
 92.8 
 
 276 
 
 127.0 
 
 316 
 
 166.4 
 
 356 
 
 211.2 
 
 237 
 
 93.6 
 
 277 
 
 127.9 
 
 317 
 
 167.5 
 
 357 
 
 212.4 
 
 238 
 
 94.4 
 
 278 
 
 128.8 
 
 318 
 
 168.5 
 
 358 
 
 213.6 
 
 239 
 
 95.2 
 
 279 
 
 129.7 
 
 319 
 
 169.6 
 
 359 
 
 214.8 
 
(L 
 
THIS BOOK IS DUE ON THE LAST DATE 
 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. 
 
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 133 
 
YA 01415