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 .88^ .018 3449 .O2I5 .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. 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