-&: -' LIBRARY ' OF'THB I *& & " UNIVERSITY OF CALIFORNIA. |. Ei -* . ' '%? da?* ' WORKS OF PROF. W. H. SEARLES PUBLISHED BY JOHN WILEY & SONS. Field Engineering. A Handbook of the Theory and Practice of Rail- way Surveying, Location, and Construction, de- signed for Class-room, Field, and Office Use, and containing a large number of Useful Tables, Original and Selected. 12010, morocco, $3.00. The Railroad Spiral. The Theory of the Compound Transition Curve reduced to Practical Formulae and Rules for Appli- cation in Field Work, with complete Tables of Deflections, Ordinates, Tangents and Long Chords for five hundred Spirals. 12010, morocco, $1.50. FIELD ENGINEERING. A HAND.BOOK OF THE THEORY AND PRACTICE OF RAILWAY SURVEYING, LOCATION, AND CONSTRUCTION, DESIGNED FOR THE CLASS-ROOM, FIELD, AND OFFICE, AND CONTAINING I LARGE NUMBER OF USEFUL TABLES, ORIGINAL AND SELECTED. BT WILLIAM H. SEAELES, C.E. MEMBER AMERICAN SOCIETY OF CIVIL ENGINEERS. SIXTEENTH EDITION, FORTY-THIRD THOUSAND OF THE \ UNIVERS/TV NEW YORK: ^HN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1910. Copyright, 1880, BY JOHN WILEY & Sons. Copyright renewed, 1908, BY JOHN WILEY & SONS. PKESS OF BRAUNWORTH ft CO. BOOKBINDERS AND PRINTERS BROOKLYN, N. Y PEEFACE. ALTHOUGH the modern railway system is but about fifty years old, yet its growth has been so rapid, and the progress in the science of railway construction so great, as to render the earlier technical books on this subject inadequate to the needs of the engineer of to-day. In the course of his practical experience as a railway engi- neer, the author was strongly impressed with the want of a more complete hand-book for field use, and finally concluded, at the solicitation of his friends, to undertake the preparation of the present volume. The aim in this work has been : First To present the general subject of railway field work in a progressive and logical order, for the benefit of beginners. Second To classify the various problems presented, so that they may be readily referred to. Third To embrace discussions of all the more important practical questions while avoiding matters non-essential. Fourth To employ throughout the work a uniform and systematic notation, easily understood and remembered, so tliat after one perusal the formulae may be intelligible at a glance wherever referred to. Fifth To express the resulting formula of everv problem in the shape best adapted to convenient numerical compu- tation. Sixth To furnish a large variety of useful tables, more com- plete and extended than any heretofore published, especially adapted to the wants of the field engineer. An elementary knowledge of algebra, geometry and trigono- metry on the part of the reader has been taken for granted, as a command of these instrumentalities is deemed essential to the education of the civil engineer. The few references to mechanics, analytical geometry, optics and the calculus may be assumed correct by those not conversant with these branches. 204830 IV PREFACE. Many of the problems in curves are new, yet there is hardly one that has not presented itself to the author in the course of his practice. The investigation of the valvoid curve is original, and though the mathematical discussion is somewhat difficult, yet the resulting formulae, taken in connection with Table X, are exceedingly simple and convenient for the solution of a certain class of problems. The treatment of compound curves is novel and exhaustive. A few general equations are established, which, by slight modifications, solve all the problems that can occur. No discussion of reversed curves is given, because these are inconsistent with good practice, except in turnouts, under which head they are noticed. The chapter on levelling includes a discussion of stadia measurements, with practical formulae. The chapter on earth- work contains a review of several methods for calculating quantities, and states the conditions under which these suc- ceed or fail in giving correct results. Among the tables, numbers 3, 5, 6, 10, 18, 19, 26 and 29 are original. The adoption of versed sines and external secants throughout the work, wherever these would simplify the formulae, rendered necessary the preparation of tables of these functions. The table of logarithmic versed sines and external secants has been computed from ten-place logarithmic tables of sines and tangents, so that the last decimal is to lie relied on, and no pains have been spared to make the table thoroughly accurate. Tables numbers 4, 7, 8, 9, 11, 12, 13, 14 and 30 have been recalculated, enlarged, and some of them carried to more deci- mal, places than similar tables heretofore published. The intention has been to give one more decimal than usual, so that in any combination of figures the result of calculation might be reliable to the last figure usually required. The tables which have been compiled and rearranged are numbers 1, 2, 15, 16, 17, 24, 25 and 31. The tables of log. sines and tangents here given are the only six-place tables which give the differences correctly for seconds. The table of logarithms of numbers is accompanied by a complete table of proportional parts, which greatly facilitates interpolation for the fifth and sixth figures. In all the tables, whether new or old, scrupulous care has PKEFACE. ( 7 been taken to make the last figure correct, and the greatest diligence has been exercised by various checks and compari- sons to eliminate every error. It is, therefore, hoped and believed that a very high degree of accuracy has been ob- tained, and that these tables will be found to stand second to none in this respect. The preparation of this work has extended over several years, as time could be spared to it from other engagements. It is, therefore, the expression of deliberate thought, based on experience, and as such is submitted to the judgment of brother engineers. If it shall prove to have even partially met the aim herein announced, and so shall serve to smooth the way of the ambitious student, or to assist the expert in his responsible duties, the labors of the author will not have been in vain. WM. H. SEARLES, C.E. NEW YORK, March 1st, 188 turn a right angle with the transit, and measure accurately one offset, putting a transit point at its extremity, where another right angle will give a parallel line. If the offset prove too short for an accurate backsight, a temporary point at a sufficient distance may be established for that purpose on the offset line produced before the instrument is removed from the main line. If any other angle than 90 is used it should be selected, when circumstances permit, so that the distances on the inter- cepted part of the main line may be in some simple ratio to the distances measured on the auxiliary line. Thus a deflection of 60 gives a distance on the main line equal to half the length of the auxiliary course, that is, 60 gives a ratio of *, = 0.5 53 08' " " " 0.6 nearly 45 34' " " " 0.7 " 36 52" " " " 0.8 " 25 60V " " " 0.9 " the angles being taken to the nearest half minute. 49. If it be desired that the stakes on the auxiliary line should stand on perpendiculars through the true stations on the main line, a certain correction must be added to each chain length depending on the angle which the auxiliary makes with the main line. If there is a fraction of the chain at either end of the course, a proportional addition must be made for this. Thus, by referring to the table of external secants, we find that we must add a correction as follows : 233V. .0.1 ft. per chain. 6 45V- . 0.7 ft. per chain. 3 37' .. .0.2" 7 13V-. .0.8 " 4 26' .. .0.3 " 7 39V.. 0.9 " 5 07' . . .0.4 " 8 04' . . .1.0" 5 43' :'. .0.5 " 9 52 .. .1.5 " 6 15*'.. 0.6" 11 22' . . .2.0" These methods of suiting the angle to an even measure are much superior to assuming an even number of degrees deflec- tion, and then calculating the distance by trigonometry. The last table, which may be extended indefinitely by reference to the table of Ex. secants, is perfectly adapted to chaining: by surface measure on regular slopes when the slope angle is 22 W,D ENGINEERING. known, t'ae chain being lengthened by the correction corre- sponding to the slope angle. 50. If the chain is lengthened as per above table on auxil- iary lines, the numbering of the stakes goes on us usual, but they should have an additional mark as X to show that they are off the main line; and they may stand facing the true stations which they represent, and the length of offset, if known, may also be recorded on them. The leveller will then understand that he is to read the rod not only at the stakes as they stand, but also at the true stations, as nearly as may be. The assistant engineer will always make a diagram in his field book, showing exactly the method pursued in reference to auxiliary lines. Having passed the obstacle, it is advisable to return to the main line by a course equal in length to the first auxiliary, and making an equal angle with the main line. If this cannot be done from the end of the first course, a parallel to the main line may be run any convenient distance, and the return line then put in, forming a trapezoid. 51. When there is no obstruction to sight on the main line, but only to measurement, a transit point should be set in line beyond the obstacle before the transit leaves the main line, as a check on the other operations, and the main line should be afterward produced from this point by back- sight on the main line, rather than by deflection from an auxiliary line. 52. The main line should always be resumed as soon as practicable, making the auxiliary lines the mere exception. When a number of courses at a large angle are likely to be required before the main line can again be reached, it may be better to consider these as regular courses of the survey, and to note them as such. The simplest method is always the best, because least likely to involve mistakes. 53. When the natural obstacles are so numerous and of such magnitude as to render any continuous line of sur vey or location extremely difficult, if not impossible, as in the case of a bold rocky shore, all the data necessary to a location should be gathered with precision on the preliminary survey, the measurements and angles being taken with the greatest care, and as many checks as possible should be introduced to verify the work. In meandering such a shore it is probable that a large number of short courses will be used, which may be measured PRELIMINARY SURVEY. 23 correctly, but there is liability to error in the angles. To verify the latter the more conspicuous transit stations on prominent points of the shore are selected, and these being named by the letters of the alphabet, the deflections between them are taken by careful observations repeated a number of times, as for a triangulation. These points, joined by tie- lines, then form a survey of themselves, much simpler than the full traverse. To obtain the length of these tie-lines, the angles between them and the courses meeting at the same station are measured. Then since each tie-line forms the closing side of a field, in which all the bearings are known, and all the distances, save one, that one may be calculated by latitude and departures. But the angles should first be tested for error in each complete field, and if the error be large the angles must all be remeasured until the error is found and cor- rected, but if very small it may be distributed among the angles, or among those most probably inaccurate. Before cal- culating the traverse of any of these fields, it will be advanta- geous to assume, for an artificial meridian, a line parallel to the average direction of the shore for several miles, and to refer all courses to this meridian for their bearing. This meridian is called the ax-i* of the survey, and all bearings referred to it are called axial bearings, as distinguished from magnetic bearings. The magnetic bearing of the axis should be some exact number of degrees, so as to facilitate the reduc- tion from one system to the other. 54. In plotting the map, the axis is first laid down, and then the lettered stations in their respective positions, after which the meandering surveys can be filled in. The map being drawn on a scale of one hundred feet to an inch, and the con- tours constructed from the notes of the level and cross-level parties, the engineer may project the location upon it with great certainty and economy of result. But he should calcu- late the traverse of the location as projected, and compare it with the traverse of the preliminary, to eliminate all errors in drafting, before taking his notes to the field to reproduce the location on the ground. Any point where the location crosses the preliminary should have the same latitude and longitude by the traverse of either line. This system, though laborious, is the only one that will ensure a successful location under the circumstances supposed. Advantage may sometimes be taken 24 FIELD ENGINEERING. of cold weather to cross bays and inlets on the ice, but there is great liability to error in angles taken upon the ice, due both to its motion and to the sinking of the feet of the tripod into the ice as soon as exposed to the rays of the sun. CHAPTER III. THEORY OF MAXIMUM ECONOMY IN GRADES AND CURVES. 55. Before commencing the field work of location it de- volves upon the engineer to decide as to which of the surveyed routes shall be adopted as being most advantageous in all respects, and also to establish the maximum grade in each direction and the minimum radius of curve on that route. The general considerations which guide the engineer in the selection of one of several routes for location are such as were hinted at in the chapter on reconnoissance, but upon the com- pletion of the preliminary surveys he has at hand a large amount of information which enables him to consider this important question much more in detail. Unless his instruc- tions are explicitly to the contrary, he may assume it to be his duty to find the best line, or that one which, for a series of years following the completion of the road, will require the least annual expense, including interest on first cost. The finances of the company may be so limited as not to permit the construction of the best line at once, and it may then be the duty of the engineer to select the cJieapest line, or that of least first cost, as a temporary expedient, with the expectation of building the road at its best when the improved credit of the company will permit. But generally he will be able to build the cheaper portions of the best line at once, only making deviations and introducing heavier grades at the expensive points to avoid a cost beyond the present means at his com- mand. The selection of the best line may be a question as between different routes or as between different grades and curves on the same route. We will consider the latter case first. 56. To solve the problem of true economy we must determine the actual expense both of building and operating MAXIMUM ECONOMY IN GRADES, ETC. 25 f the line at a given maximum grade, and also what changes will be made in these expenses by a change in that maximum. We have then, on one hand, the annual interest upon the original cost, and, on the other, the annual expense of operating the road. \ The best grade is that which will render the sum of these two a \ minimum. Both forms or expense consist of two parts: one that is affected by a change in grade, and the other that is not. Clearly the former is the only one we have to consider in either, since when the sum of the variable portions is a minimum, the sum total will be a minimum also. The varying portions then are functions of the grade, though independent of each other. If, therefore, we let z' represent the maximum grade in feet per mile, and let x represent the corresponding value of that portion of the annual expense which varies with the grade, and establish the relation existing between the two, we shall have x =f(z'). Similarly if we let y represent the interest on so much of the first cost as is affected by grade, we shall have y=f (z'). The problem then is to find that value of z' whick shall render x -\- y a minimum. Let us now seek the complete expression represented by =/- The elements that enter into this expression are numerous, and will be considered in succession. 57. The traction of an engine is the force with which it pulls a train, and is limited by the reaction of the drivers against the rails. It depends on the weight upon each driver, the number of drivers, and the coefficient of friction. The weight on one driver should not exceed 12,000 Ibs., and is usually less than this. If the exact proportions of engine that will be used on the road are not known, the weight per driver may be assumed at 10,000 Ibs., with 4 drivers for ordinary grades and traffic, or at 11,000 Ibs. with 6 drivers, if the grades are steep and the tonnage large. For extraordinary grades special engines are required, having 8 or 10 drivers. The coefficient of friction, called also the adhesion, varies from .09 to .37, these being the extremes on record. The lowest is due to extremely unfavorable circumstances, as sleet and frost ; the highest doubtless to the use of sand, though not so stated in the record. The more common range of values is from .15 to FIELD ENGINEERING. .25. For our present purpose it will be assumed at .20, so that if a 4-driver engine has 10,000 Ibs. on each driver, its traction is 40,000 X .20 = 8000 Ibs. when hauling its maximum train. 58. The expense of running an engine one mile, hauling a train, on the proposed road, can only be estimated from the experience on other roads similarly situated. The expense is composed of the items of fuel, water, oil and waste, repairs (including renewals), wages of conductor, engineer, and fire- man, engine-house expenses, and interest on first cost of engine and engine-stall. The range and approximate average of these items is here given: ITEMS. 4-DRivER ENGINE. 4-DRIVER 6-DnrvER 8-DRIVER Lowest, j Highest. Average. Average.' Average. Fuel.. . $0.050 .001 .004 .050 050 025 .0^5 $0.210 .010 .030 .150 .100 .060 .038 $0.100 .004 .006 .080 .075 .035 .030 $0.165 .006 .008 .104 .075 .050 038 $0.213 .008 .010 .133 075 .060 .047 Water Oil and waste Repairs and renewals Wages Engine-house Interest Totals .205 .598 .330 .446 546 In a given case the probable value of each item should be estimated separately, and the sum taken afterwards. In the above averages each engine is supposed to haul its maximum train. The relative expense of the several classes of engines has not been established conclusively. 59. The resistance offered to the motion of a railway train is occasioned by a variety of causes, concerning which a great deal of uncertainty exists as to their relative effect. An investigation which should seek to determine the exact amount of each partial resistance, and then by a summation derive the total, would be tedious, and, in the present state of our knowledge, unsatisfactory. We shall therefore simply group the resistances under three general heads, namely: Resistance due to uniform motion on a straight, level track; " Resistance due to grade ; Resistance due to curvature. MAXIMUM ECONOMY IN GRADES, ETC. 27 6O. The first of these, considered as an aggregate of the various items of friction in engine and train, of oscillations and impacts, and of resistance of the atmosphere, is found to vary nearly or quite as the square of the velocity. The fric- tion of an engine is greater in proportion to iis weight than that of a car, owing to its many moving parts, so that the resistance of a short train is greater in proportion to its total vveight than that of a long train. The resistance of the atmos- phere is greater also in proportion to the weight of a short train than of a long one. An empty train will offer more resistance in proportion to its weight than a loaded one. A formula which shall express the resistance of a train to uni- form motion must include at least the velocity and the weight of the train and engine. The following empirical formula is based upon a careful investigation of all such records of experiments on the subject, several hundred in number, as have come to the author's notice, and is believed to give results agreeing closely with the average experience and practice of the present day. It is designed to give the resistance per ton for all trams, whether freight or passenger, and at any velocity, under ordinary circumstances. Accidental circumstances, such as the state of the weather, and the condition of the road-bed, rails, and rolling stock, may largely modify the resistance, but these, of course, are not taken account of in the formula, Let V = velocity of train in miles per hour, " E = weight of engine and tender in tons, " W = weight of cars in tons, " T = weight of freight in tons, " q = resistance to uniform motion in Ibs. per to^. We then have the formula = 5.4 +(.006 + ^ (6 61. The second resistance considered is that due to gravity in grades. It varies in the exact ratio of the rise to the length of the grade. Let G 8 = rise of grade in feet per station. " Gr m rise of grade in feet per mile. " q' resistance in pounds per ton due to grade. 28 FIELD ENGINEERING. Then, ?' = 2240 ^ = 23.40. ( (2) 62. The third resistance considered is that due to curvature of the track. This resistance is due to the friction of the wheels upon the top of the rail, and of their flanges upon the side of the rail. The top friction is lateral, due to the oblique position of the wheel on the rail, and longitudinal, due to the greater length of the outer rail, since both wheels are rigidly attached to the axle. The flange friction is due to the reaction of the top friction, which, combined with the parallel- ism of the axles, throws the truck into an oblique position on the track. A forward flange presses the outer rail, while a rear flange is usually in contact with the inner rail. The centri- fugal force of the car will increase the pressure on the outer rail, unless the ties are inclined at an angle sufficient to coun- terbalance this force. But if the ties are inclined too muCh, or the velocity is less, the pressure on the inner rail will be increased. An uneven track will cause" the truck to pursue a zigzag course, increasing the resistance considerably. Experiments for determining the amount of curve resistance have been neither numerous nor very satisfactory, but the generally accepted conclusion is that the resistance is a little less than half a pound per ton on a one-degree curve, and that it varies as the degree of curve. On European roads, how- ever, it is estimated at about one pound per ton per degree of curve, owing largely to the form of rolling stock used. 63. Let q" = curve resistance in pounds per ton on any curve, * and I) = degree of curve. Then, assuming the resistance per ton on a one-degree curve at 0. 560, we have for any other curve q" - 0.56Z* (3) To ascertain what grade upon a straight line will offer the same resistance as a given curve; substitute the value of q" for q' in eq. (2) and solve for G ; whence (4) f MAXIMUM ECONOMY IX GRADES, ETC. 29 For definition of degree of curve, see Art. 84. 64. It is evident that grades and curves, by their resistances, fix a limit to the weight of a train which a given engine can haul over them. A locomotive is usually built with such a surplus of boiler and cylinder capacity that its power, at ordinary velocities, is limited by the adhesion of the drivers, so that the adhesion is the proper measure of the tractive force. To Jind an expression for the maximum train which a given engine can haul over a given grade and curve: Let P = tractive force of engine in pounds, " T' = weight of paying load in tons per maximum train, " W- = weight in tons of cars carrying the load T'. Then for uniform motion, at a given velocity, (E -{- W + T') (q -f- q' + q") = P (5) Let t = average load of one car in tons " w = average weight of one car and load in tons. Then TV" -j- T' = -rT' t substituting which in eq. (5) we derive t + (6) In this equation q = the resistance per ton due to uniform motion, q' = the resistance per ton due to the maximum grade opposed to the direction of the train, and q" = the resistance per ton due to the sharpest curve on that grade. For accelerated motion the reaction of inertia of the train must be added to the above resistances. This is estimated at l) for station/, 3(D) for station g, etc., as before. When the end of the curve is reached, a transit-point is set at the Point of Tangent, after which it only remains to find the direction of the tangent, by the above rule. Thus if g is to be 54 FIELD the point of tangent, we obtain the direction of the tangent by deflecting from the chord gd an angle equal to xdg, or to | dOg. If this tangent VB was already established, the line gx thus obtained should coincide with it; and if it does so, the correctness of our work is proved. 104. The centre line is measured, and the stations num- bered regularly and continuously through tangents and curves from the starting point to the end of the work. It therefore frequently happens that a curve will neither begin nor end at an even station, but at some intermediate point, or plus distance. If the Point of Curve occurs a certain number of feet beyond a station, the first chord on the curve is composed of the remaining number of feet required to make 100. Any chord less than 100 feet is called a subchord. If a curve ends with a subchord, the remainder of the 100 feet must be laid off on the tangent from the Point of Tangent to give the position of the next station, so that the stations may everywhere be 100 feet apart. 105. The deflection to be tnadefor a subchord is equal to one Jialf the arc it subtends. Let c length of any subchord in feet. * " d = angle at centre subtended by subchord. Then, from eq. (22), by analogy But by eq. (16) 2R = (29) 100 sin^Z) = 100^ (30) (81) When D does not exceed 8 or 10, we may assume without serious error that the angles are to each other as their sines, and the last two equations become (approx.) c = 100 - (32) SIMPLE CUEVES. 55 and i^-fliJ) (33) In curves sharper than 10 per station, the error involved in this assumption becomes apparent and must be corrected. 106. If curves were measured on the actual arc, then eqs. (32) and (33) would be true in all cases ; but since a curve is measured by 100-ft. chords, it is evident that if a 100-ft. chord between any two stations were replaced by two or more subchords, these taken together would be longer than 100 feet, since they are not in the same straight line. Let us conceive the actual arc of one station to be divided into 100 equal parts; since the arc is longer than the chord, each part will be slightly longer than one foot. Now if we take an arc contain- ing any number of these parts (less than 100), the nominal length of the corresponding subchord in feet will equal the number of parts, and the deflection for the subchord will be proportional to the number of parts which the arc contains. The deflection therefore will be exactly given by eq. (33) if in that equation we let c equal the number of parts in the arc, or the nominal length of the subchord in feet. Having thus obtained the correct value of (| d\ we may introduce it into eq. (29) or (30), and obtain the true value of the subchord, which will always be a little greater than its nominal value. Suppose, for instance, that the arc of one station is to be divided into four equal portions ; then each subchord will be nominally 25 feet long; and by eq. (33) OK f*=^(W=i(40) (34) which is the correct value of the deflection, whatever be the degree of curve. Substituting this value in eq. (29) or (30) we obtain the true value of the subchord, c, a little greater than 25 ; the excess is called the correction of the nominal length. 107. This correction for any given subchord bears an almost constant ratio to the excess of arc per station, what- ever be the degree of curve. These ratios are shown in the following table for a series of subchords, and Table VII. gives the length of actual arc per station for various degrees of curve. Subtracting 100 we have the excess of arc per station, and multiplying this excess by the ratio corresponding to the 56 FIELD nominal length of subchord we obtain as a product the proper correction for the subchord. TABLE OF THE RATIOS OF CORRECTIONS OF SUBCHORDS TO THE EXCESS OF ARC PER STATION. Nominal Length of Subchord. Ratio. Nominal Length of Subchord. Ratio. Nominal Length of Subchord. Ratio. 5 10 15 30 25 90 .000 .050 .099 .147 .192 .234 .373 35 40 45 50 55 60 65 .307 .336 .358 .374 .383 .383 .374 70 75 80 85 90 95 100 .356 .327 .287 .235 .169 .092 .000 We observe that the largest correction is required by a sub- chord between 55 and 60 feet in length. Example. It is proposed to run a 14 curve with a 50-ft. chain. "What correction must be added to the chain? K.f\ By eq. (30) = 100 Ans. Correction = .093 Or, by Table VII., length of arc = 100 . 249 excess of arc = . 249 and by above table, ratio for 50 feet = . 374 Ans. Correction = product = .093 Example. The, P.O. of an 18 curve is fixed at -f- 55 feet beyond a station. What are the nominal and true values of the first subchord, and what the proper deflection? Nominal value = 100 55 45 feet AK Deflection = $d = ^ X 9 = 4. 05 = 4 03' 100 and by eq. (30) True value = e = 100 sin . 4 ?- = 45.148 sin 9 SIMPLE CURVES. 57 Or, by Table VII., excess of arc = .412 by above table, ratio for 45 feet = . 358 Correction product = .147 Ans. True value of subcliord = 45.147 Example. The last deflection at the end of a 40 curve is found to l)e 6 30'. What are the nominal and true values of the last subcliord? Here %d = 6 30', and by eq. (32) Nominal value, c = 100 ^ = 33.5 f ee t By eq. (30) True value, c = 100 ^-^^- = 33.098 feet sin 20 Or by Table VII., excess of arc 40 = 2.060 by above table, ratio for 32 . 5 feet = . 290 Correction = product . 597 Nominal value of subchord = 32 . 5 True value -33.097 1O8, For convenience in making deflections, the zeros of the instrument should always be together when the line of collimation coincides with a tangent to the curve. Thus, in beginning a curve, the transit being set at the P. C. zeros together, and line of collimation on the tangent, the rea,d- ing of the limb for any station on the curve has simply to be made equal to the proper deflection from the tangent for that station. After the transit is moved forward from the P. C. and set at another point of the curve, the vernier is set to a reading equal to the reading used to establish that point, but on the opposite nide of the zero of tlie limb, and the line of collimation is set on the P. C. just left. Then by simply turn- ing the zeros together again, the line of collimation will be made to coincide with a tangent to the curve through the new point, and the deflections for the succeeding stations can be read off directly, as before. Thus any number of transit points may be used in locating a curve by finding the direc- tion of the tangent through each by a deflection from the pre- ceding point, until finally the P. T. is reached, where another deflection gives the direction of the located tangent. 58 FIELD ENGINEERING. 109. The assistant engineer keeps neat and systematic field-notes of all his operations with the transit in running curves. The numbers of the stations are written in regular order up the first column of the left-hand page of the field- book, using every line, or every other line, as may be pre- ferred. The second column contains the initials of each transit point on the same line as the number of its station, or between lines, if the point occurs between two stations. In the third column, and opposite the initials in the second, is recorded the station and plus distance, if any, of each transit point. The fourth. column contains, opposite the "P.C7.," the degree of curve used, and an R or L, showing whether the curve deflects to the right or left ; the fifth column contains the readings or deflections made from a tangent to set each station or point, written on the same line as the number of that station or point; and the sixth column contains the cen- tral angle of the whole curve, A, written opposite the " P.T." The plus distances recorded in the third column are always the V nominal lengths of subchords, but if the true lengths have been calcu^ lated and laid off on the ground, these should also be recorded in parenthesis. On the right-hand page are recorded the calculated bearings of the tangents and their magnetic bearings; and on the A centre line of the page, opposite FlG 7 the record of each transit point, a dot is made with a small circle around it, to show the relative position of the several points on the ground. Some slight topographical sketches may be made, indicating the more prominent objects, but the full sketches should be taken by the topographer in a separate book. 110. Since the deflections start from zero at each new transit point, the sum of the deflections by which the transit points are located will be equal to one half the central angle of the curve. 111. The stations on a curve may be located by deflec- tions only, without linear measurements. For this purpose two transits are set at two transit points on the curve, as A SIMPLE CURVES. 59 and B, Fig. 7, and the proper deflections for any station are made with both instruments, the station being located by find- ing the intersection of the two lines of collimation. This method requires that the two transit points shall have been previously established, that their distance from each other shall be known, that they shall be visible from each other, and that they shall both command a view of the stations to be located. It is not therefore generally useful, but may be resorted to to set stations which fall where chaining cannot be accurately done, as in water or swamps. The chord join- ing the two transit points becomes, in fact, a base-line, and the deflections form a series of triangulations. C. Location of Curves by Offsets. 112. A curve may be located by linear measurement only, without angular deflections. There are four general methods, viz. : By offsets from the chords produced, By micCdle-ordinates, By offsets from the tangents, and By ordmates from a long chord. To locate a curve by offsets from the chords produced. When the curve begins and ends at a station. 1 13. Let A, Fig. 8, be the P. C. of a curve taken at a station, to locate the other stations, a, b, c, etc. The chords Aa, ab, be, etc., each equal 100 feet, and since the angle A Oa = D, the angle VAa iD. (Tab. I. 20.) Taking an off- set ax = t, perpendicular to the tangent, we have in the right- angled triangle Axa. ax = AaX sin ^D or t = 100 sin \D (34) The offset t is called the tangent offset, and its value is givenfor all degrees of curve in Tab. IV. col. 4. F. a If the curve were produced backward from A, 100 feet to station z, the offset zy would 60 FIELD EHGIKEEKIKG. equal I', and if the chord zA were produced 100 feet from A to a', the offset a'x would also equal t. Therefore the distance aa' = 2t, and the angle aAa' = D. 80 if we produce the chord Aa 100 feet to V, the distance bb' = 21. To lay out the curve, stretch the chain from A, keeping the forward end at a perpendicular distance, t, from the line of the tangent to locate station a. Then find the point b' by stretch- ing the chain from a in line with a and A, and then stretching the chain again from a, fix its forward end at a distance from b' equal to 2t. This gives station b. In the same way find other stations. When the last station, as d, of the curve is reached, produce the curve one station farther to e". Then the tangent through d is parallel to the chord ce\ and laying off t from c and e" per- pendicular to this chord, the tangent c"e is found. If the work has been correctly done the tangent c"e will coincide with the given tangent VB. When the curve begins or ends with a subchord. 114. Let ^1, Fig. 9, be the PC. and Aa the first sub^ chord = c, and the angle VAa = \d t and let the offset ax = ti, Then ti = c sin %d (35) Producing the curve backward to the nearest station z, we have another subchord Az = (100 c), and the angle yA& = ^ (D d), and putting the offset yz ~ t u Laying off the two subcliords on the ground, and making the proper offsets, t t and t lit at the same time, we fix the position of the two stations a and z on the curve ; after which we may pro- duce the chord sa 100 feet to b', and proceed as before until the curve is finished. If the curve ends with a sub- chord, as dB, produce the curve to the first station beyond B, as e", then calculate the two offsets for the two subchords Bd and Be", fM g t and lay them off from d and e' SIMPLE CURVES. 61 perpendicular to the supposed direction of the tangent. If the line d"e so obtained coincides with the given tangent, VB, the work is correct. 115. We may find the values of t, and t u otherwise than by the formulae above, for in Fig. 8 we have shown that the angle aAa' = aOA, and since these triangles are isosceles, they are similar; therefore Fig. 8, OA : Aa:: Aa : aa' or Ri 100:: 100 :2t t = -^f- (37) and similarly, Fig. 9, <- = T5 > Hence (89) Thus t t may be found by multiplying the square of the sub- chord by the value of t given in Tab. IV. , and dividing the product by 10000. As c is always less than 100, so t t is always less than t. 1 16* In eqs. (35), (38), and (39) it is customary to use the nominal values of c, and this can produce no error in t or t, exceeding -005, when the degree of curve does not exceed ten degrees. In the case of a very sharp curve, the formulae eqs. (40) and (41) are preferable. To locate a curve by middle-ordinates. When the curve begins and ends at a station. 117. In Fig. 10, let A be the P. G. at a station, and let a and g be the next stations on the curve either way from A. Then, since zy = ax = t, the chord za is parallel to the tangent A V, and Ag = t. Hence, having any two consecutive stations on the curve, as z and A, we may lay off the tangent offset t from A to g on the radius, and find the next station, a, 100 feet from A on the line eg produced. Then laying off ah = t on the radius aO, a point on the line Ah produced and 100 feet from a will be the next station b. 62 FIELD On reaching the end of the curve, the tangent is found precisely as described in the method by chords produced, 113. In Fig. 10, we observe that if the radius OA were unity, gA would be the versed sine of the angle aOA = D. But gA = t, . t = R vers D (40) When the curve begins or ends with a subchord. 118. Let A, Fig. 11, be the P.O., and a and z the neare&a *r>^. o z' y Fia. 10. FIG. It stations. Then Aa = c, the first subchord, and aOA = d, and by analogy, we have from the last equation, if ax = t t and t, = Rversd t, = or eq. (39) may be used if preferred. Having found the two stations, a and z, on the curve, lay off from the forward station a, ah = t on the radius, and so continue the curve as described above. When the end of the curve is reached, produce the curve to the next station beyond, and find the tangent by offsets as described in the previous method, 114. To locate a curve by offsets from the tangents. Wfien the curve begins at a station. 1 19. Let A, Fig. 12, be the P. C. at a station. Then the next station a is located by the tangent offset t, taken from SIMPLE CURVES. 63 Tab. IV., or calculated by eq. (40). To calculate the distance* and offsets for the following stations, b, c, etc. , in the diagram draw lines through the points b, c, etc., parallel to the tangent AV, intersecting the radius AO in g', g", etc., and draw the lines bx', ex", etc., perpendicular to the tangent. Then Ax' = g'b = Ob sin bOA or and Also, or and Ax' = R sin %m Ax" = R sin 3D i etc. etc. j M = g'A = Ob vers. bOA f = R vers 22) t" = R vers 3D etc. etc. (43) (43) But these calculations maybe avoided, for as twice ag equals the chord of two stations, so twice bg' equals the chord of four stations, and twice eg" the chord of six stations, etc. So also as Ag is the middle-ordinate of two sta- tion, Ag' is the middle-ordinate of four, and Ag" the middle-ordinate of six stations, etc. Hence the rule: The distance on the tangent from the tangent point to the perpendicu- lar offset for the extremity of any arc is equal to one half the long chord for twice that arc,' and the offset from the tangent to the ex- tremity of any arc is equal to the middle-ordinate of twice that arc. The long chords and middle-ordinates may be taken from Tables VII. and VIII. for 2, 4, 6, 8, etc., stations, when the P.O. is at a station, or for 1, 3, 5, 7, etc., stations, when the P.O. is at -f- 50, or half a station. If the offsets from the first tangent A V prove inconveniently long, the second half of the curve may be located from the other tangent BY, beginning at the point of tangent B, and closing on a station located from the first tangent. Fia. 12. 64 FIELD ESTGIKEERIKG. When the curve begins with a subchord. 12O. If d=the angle at centre, subtended by the first subchord, we have for the distances on the tangent (Fig. 13) Ax = R sin d 1 Ax' R sin (d -f D) Ax" = R sin (d -j- 2D etc. etc. and for the offsets (Fig. 11) t, = R vers d t = R vers (d + D) , f = R vers (d + 2D) | etc. etc. 1 (44) (45) If the first subchord equals 50 feet (nominal), then d =^D, and the Tables VII. and VIII. may be used as explained FIG. 13. FIG. 14. above. These tables may be used in any case, by adopting a temporary tangent through any station, and laying off the dis- tances on this, and making the offsets from it. When a curve is located by offsets the chain should be car- ried around the curve, if possible, to prove that the stations are 100 feet apart. To locate a curve by ordinates from a long chord. Wlien tJie curve begins and ends at a station. 121. In Fig. 14 draw the long chord AB, joining the tan- gent points, and from this draw ordinates to all the stations oa SIMPLE CURVES. 65 the curve. We then require to know the several distances on the long chord Aa', a'b', be', etc., and the length of ordinate at each point. Let C = the long chord AB, then eq. (22) C=2$ sin | A If a is the second station and i next to the last on the curve., join ai, and let the chord ai = C'. Then since the arc Aa t'k D t the angle at the centre subtended by C' is ( A 2Z>) Again, if we join b and h the next stations and let bh = C" C"'=2,Rsini(A - 4D) and so on for other chords. Since Aa' = ki, C = C' -f- 2Aa' C- C' .'.^= Similarly, , ,, C' - C" ** = 2 Thus we continue to find the distances up to the middle of the curve, after which they repeat themselves in inverse order. 122. When the long chord C, subtends an even number of stations (as 10 in Fig. 14), the middle ordinate of the chord is the ordinate of the middle station, as e. Since the chords AB and ai are parallel, the ordinate a' a or i'i is evidently equal to the difference of the middle ordinates of these chords. Let M, M', M", etc., be the middle-ordinates of the chords G, C', C", etc. Then eq. (23) M = R vers \ A M 1 = #vers|(A - 2D) M" = R vers \ (A - 4D) etc., etc. And a' a i'i = M M' b'b =h'h = M-M" etc. etc. etc. The values of the chords and middle-ordinates may be taken at once from Tables VII, and VUL 66 FIELD ENGINEERING. Example. It is required to locate a 4 degree curve of ten stations by offsets from the long chord. By Table VII.: Diff. ^Diff. 10 sta. G =980.014 190.211 95 105 """"" A.QJ rm Jd' 8 " C l =789.803 194,059 97.030 = a'b' = i'h 6 " 0" = 595.744 196.962 98.481 = b'e' = h'g' 4 " C m = 398.782 198.904 99.452 = c') = i (A - 2d - D) cb c " = (A - 2d-D) - i (2D) = i( A - 2d - 3Z>) cdd" = | ( A - 2d -3D) - * (2#) = 1 ( A - 2d - 5D) etc. etc. etc. 68 FIELD ENGINEERING. Solving the several right-angled triangles we have, Fig. 15. Aa' = c. cos i ( A d) ~] ab" = 100 cos $ ( A 2d D) \ be" = 100 cos \ (A - 2d - 3D) Y (46) dd" = 100 cos i(A - 2d - 5D) etc., etc., j b"b = 100 sin i ( A - 2d - D) c"c - 100 sin i (A - 2d - 3D) |* (47) d"c = 100 sin 1 (A 2d 5D) etc., etc., And also When the middle point of the curve is passed the minus quantities in the parentheses become greater than A, making the parentheses negative, and, therefore, the sines negative, and indicating that such values as are determined by them must be laid off toward the long chord AB. By a proper summation of the quantities determined by eqs. (46) and (47) we obtain the distances Aa', Ab', Ac', etc., and the ordinates a' a, b'b, c'c, etc., and the curve may be located accordingly. It is well to make all the necessary calculations before beginning to lay down the lines on the ground, thus avoiding confusion and mistakes. Example. The P.O. of a 3 20' curve is fixed at -f 25 feet beyond a station, and the central angle is 16 24' A. It is required to locate the curve by ordinates from the long chord, We have c = 100 - 25 = 75 and d = 2 30' and D = 3 20', Hence, eqs. (46) Aa'= 75 cos 6 57' =74.449 ab" = 100 cos 4'02' =99.752 be" = 100 cos 042' =99.993 d'd = 100 cos (- 2 38') = 99.894 e"e = 100 cos (- 5 58') = 99.458 e'B = 17 cos (- 7 55') = 16.838 By eqs. (47) a'a = 75 sin 6 57' = b'b = 100 sin 4 02' = c"c = 100 sin 42' = cd" = 100 sin (- 2 38') = de" = 100 sin (- 5 58') = ee' = 17 sin (-7 55') = 74.449 - Aa' 174.201 = AV 274.194 = Ac' 374.088 = Ad! 473.546 = Ae' 490.384 = AB 9.075 9.075 = a'a 7.034 16.109 = b'b 1.222 17.331 = c'c 4.594 12.737 = d'd 10.395 2.342 = e'e 2.341 0.000 . . . SIMPLE CUEVES. 69 The same formulae can be used when the curve begins at a Station by making c = 100 and d D. 126. The methods of locating curves by linear measure- ments do not require the use of a transit, although one may be used to advantage for giving true lines, turning right angles, etc. When a transit is not used the alignments should be made across plumb-lines suspended over the exact points previously marked on top of the stakes. A rig'ht angle may easily be obtained, without an instrument, by laying off on the ground the three sides of either of the right-angled triangles represented in the following table (or any multiples of them), always making the base coincide with the given line. TABLE OP RIGHT-ANGLED TRIANGLES. Base. Hypothenuse. Perpendicular. 453 12 13 5 20 29 21 24 25 7 40 41 9 60 61 11 84 85 13 D. Obstacles to the Location of Curves. 127. To locate a curve joining two tangents when the in- tersection F is inaccessible. Fig. 16. From any transit point p on one tangent run a line pq to intersect the other tangent; measure pq and the angles it makes with the tangents. Then the sum of the de- flections at p and q equals the central angle A. Solve the triangle pqV and find Vp. Having decided on the radius R of the curve, calculate the tangent distance VA by eq. (21), and lay off from p the distance pA = VA - Vp to locate the point FlG - 16 - of curve. The point p being as- sumed at random, Vp may exceed VA, in which case the differ- ence pA is to be laid off toward V. In case obstacles prevent the direct alignment, of any line pq, a line of several courses may be substituted for it (as 70 FIELD ENGINEERING. explained in 46, 47, 48,) from which the length of pq will be deduced. The algebraic sum of the several deflections will equal A. 128. To locate a curve when the point of curve is inaccessible. Fig. 17. Assume any distance Ap on the curve which will reach to an accessible point p. Then by eq. (19) the angle 100 Ap' = R sin pOA p'p = R \ers pOA Vp' = VA- Ap' Measure Vp' and p'p to locate a transit point at p; and meas ure an equal offset from some transit point on the tangent, as qq'. This gives a line pq' , parallel to the tangent, from which deflect at p an angle equal to pOA for the direction of a tangent through the point p. Instead of measuring the second offset qq' we may deflect from pq an angle found by tan qpq' = ~ and so obtain the line pq' parallel to the FIG. 17. tangent. Or we may deflect from p V the angle found by tan p Vp' =-y~, to obtain the line q'p pro- duced, from which the tangent to the curve at p is found as above. Again, we may lay off from V, the external distance Vh found by eq. (24) or Tab. VI on a line bisecting the angle A VB. This gives us h, the middle point of the curve, and a line at right angles to h V is tangent to the curve at 7i, from which the curve may be located in either direction. 129. To locate a curve when both the Vertex and Point of curve are inaccessible. Fig. 18. From any point p on the tangent run a line pq' to the other SIMPLE CURVES. 71 tangent, and so determine pA as in 127. Suppose the curve produced backward to p' on the perpendicular offset pp'. Then t)A sin p' OA t-- and pp' = R vers p' OA Having located the point p', a parallel chord p'q may be laid off, giving a point q on the curve, since p'q = 2 X pA. At q deflect from qp' an angle equal to p' OA for a tangent to the curve at q. If any obstacle prevents using fhe chord p'q, any other Fia. 18. Fio. 19. chord as p's may be used, by deflecting from p'q the angle qp's = (qOs) and laying off its length, p's = 2R sin (p'OA -f qp's). At s a deflection from the chord sp' of (p'OA + ^'s)will give the tangent at s. If obstacles prevent the use of any chord, the methods de- scribed in 131 may be resorted to. 13O. To pass from a curve to the forward tangent when the Point of Tangent is inaccessible. Fig. 19. From any transit point p on the curve, near the end of the curve, run a chord parallel to the tangent. The middle point g of the chord will be on the radius through the point of tan- gent B. At any convenient point beyond this an offset equal to pp' = R vers pOB may be made to the langent, and at some other point an equal offset will fix the direction of the tangent. FIELD Otherwise, if an unobstructed line pq can be found inter- secting the tangent at a reasonable distance from B, measure the angle q'pq = pqp', and lay off the distance , = _&_ sin q'pq to fix the point q. Then Bq p'q p'B pp' cot q'pq R sin pOB. Otherwise ; assume an arc pf any number of stations from p to q" on the curve produced, and take the length of chord from Tab. VII. Lay off pq", and from q" lay off q"q = E vers q"OB, perpendicular to the tangent, to locate q. The angle pq"q = 90 q'pq", and the distance qB H sin q" OB. 131, To pass an obstacle on a curve. Fig. 20. From any transit point A on the curve take the direction of a long chord which will miss the obstacle, as A'B'. The length of this chord is 272 sin V'A'B',V'A' being tangent to the curve at A (see eq. 22), and by measuring this distance, the point B' on the curve is obtained. If the angle V'A 'B' is made equal to the deflection for an exact number of stations, the chord may be taken from Tab. VII. If the chord which will clear tl^e obstacles would be too long for con- venience, as A'q' , we may measure a part of it as A'p', and then, by an ordinate to some station, regain the curve at p. The distance on the curve from A' to p being assumed, the distances A'p and p'p are calculated by the methods given in 121 to 125. Jf p'p can be made a middle ordinate the work will be much simplified. If more convenient the middle ordinate may first be laid off from A ' to p", and the half chord afterwards measured from p" to locate p. Again, we may calculate the auxiliary tangent A ' V for any assumed length of curve A 'B', and lay off the distance A'V and V'B', deflecting at V an angle eaual to twice o Fio. 20. SIMPLE CUB YES. 73 V'A'B'. But if the point V should prove inaccessible, we may conceive the auxiliary tangents to be revolved about the chord A B' as an axis, so that V will fall at V", and the lines A'V" and V"B' may be laid out accordingly. If these in turn meet obstructions, we may run a curve from A ' to B' of same radius as the given curve, but tangent to A'V" and V"B'. Again, the entire curve or any portion of it may be laid out by offsets from the tangents, or by ordinates from a long chord, as already explained, 119 to 126. In case any distance on a curve must be measured by a tri- angulation, as in crossing a stream, a long chord inay be chosen, either end of which is accessible, and the triangula- tion is then performed with respect to this chord or a part of it, as upon any other straight line. SPECIAL PROBLEMS IN SIMPLE CURVES. 132* Given: a curve joining two tangents, to find the change required in the radius K, and external distance E, for an assumed change in the value of the tangent distance T. Fig. 21. O G FIG. 21. T = AV= VB and T' = A' V- R = AO " R' = A'0! Then T- T' = A A' = the given change. Byeq. (25) It = T cot^A R' = T cot |A = R-R' = T- (48) 74 FIELD ENGINEERING. By eq. (26), similarly, HE' =E-.E' = (T- T')taniA (49) Eqs. (48) (49) give the changes in R and E for any change in T. When T is increased R and E will be increased also, and vice versa. Example. A 4 curve joins two tangents, making an angle of 38 = A , and it is necessary to shorten the last tangent dis- tance 80 feet. What will be the change in the radius and in the external distance? Eq. (48) T-T' = 80 log 1.903090 iA 19 log cot 0.463028 Ans. R -R' 232.34 log. 2.366118 R 1432.69 R 1 = 1200.35 or about 4 46' =D'. If the tangent distance had been increased 80 feet we should add the above to JK. R' = 1665.03 or about 3 26' = I) ' Eq. (49) T- T' = 80 log 1.903090 i-A 9 30' log tan 9.223607 Ans. E-E' 13.387 log 1.126697 133. Given: a curve joining two tangents, to find the change required in the radius K, and tangent distance T, for any assumed change in the value of the external distance E. Fig. 21. We suppose EH' given to find OG and AA '. By eq. (24) E =R ex sec % A E' = R' ex sec-J-A (50) ex sec A By eq. (49) AA = T- T' = (E-E')cotA (51) SIMPLE CURVES. 75 Example. A 4 curve joins two tangents, making an angle of 38 = A , and it is necessary to bring the middle point of the curve 25 feet nearer the vertex F. What changes are re- quired in the radius and point of curve? Eq. (50) E-E'= 25 log 1.397940 | A 19 log ex sec 8.760578 Ans. R-R' 433.87 log 2.637362 R 1432.69 R ' 998.82 or about 5 44' = D' Eq. (51) E-E' 25 log 1.397940 i-A 9 30 log cot 0.776393 T 'T 149.39 2.174333 or the P. C. will be moved toward the vertex 149.39 feet. But if the point H, Fig. 21, were to be moved 25 feet further from the vertex V, then R' 1866.56 or about 3 04' = D ' and the P.C. will be moved 149.39 feet further from the vertex. It is preferable to assume some radius from Table IV. near the value of R ' found as above, and from this calculate the value of T' by eq. (21). 134. Given: a curve joining two tangents, to find the change made in the tangent distance T, and external distance E, by any assumed change in the value of the radius R. Fig. 21. By eq. (48) AA' = T- T' = (#-.') tan A (52) By eq. (50) EH' = E-E' =(R-R ') ex sec A (53) The changes calculated by eqs. (52) (53) will be added to or subtracted from T and E respectively, according as the radius i? increased or diminished. 135. Since for a constant value of the central angle A, FIELD ENGINEERING. the homologous parts of any two curves are proportional to each other, we may write at once T' T R 1 = R (54) rri^ J T rp C E C M etc. etc. etc. 136. Given: a curve joining two tangents, to change the position of the Point of curve BO that the curve may end in a parallel tftiigeiit. Fig. 22. Let AB be the given curve, AV, VB the tangents, and V'B ' the parallel tangent. Then VV is the distance from one vertex to the other; and since there is no change in the form or dimensions of the curve, we may conceive it to be moved bodily, parallel to the line AV, until it touches the line VB', when every point of the curve will have moved a distance equal to VV. Hence AA ' = 00 ' = BB ' = VV. There- fore, run a line from B parallel to A V, intersecting the new tangent in B\ measure BB', and lay off the dis- tance from A to find A. _In the figure the new tangent is taken outside the curve, and so A ' falls beyond A, but if the new tangent were taken inside the curve at V"B U , the new P. C. would fall back of A at some point A". If the parallel tangent is defined by a perpendicular offset from B, as Bp; since the angle BB'p = A (55) FIG. 82. sin A 137. Given: a curve joining two tangents, to find the radius of a curve that, from the same Point of curve, will end in a parallel tangeut. Fig. 23. Let AB be the given curve, AV, VB the tangents, and V'B' the parallel tangent; and let AO = R and AO' R'. SIMPLE CURVES. 77 Since the central angle A remains unchanged, the angle | A between the tangent and long chord remains unchanged ; therefore V 'A B ' = VAB, and the new point of tangent is on the long chord AB produced. Find on the ground the inter- section of V'B' with AB produced and measure BB'. In the diagram draw Be parallel to AO, then BeB -- A , and by eq. (22) BB' 2Besm A Be= 00' =R' - R but The + sign is used when B ' is be- FlG - 23 - yond B, as in the figure; but if the parallel tangent is within the given curve it will cut the chord in some point B ", and then the sign must be used, since R ' will evidently be less than R. If the parallel tangent is defined by a perpendicular offset, ^Bp B '/; since BeB' = A Bp Be vers A = (R ' R) vers A vers A (57) Add or subtract as explained above. If the long chord C = AB is known, then the new long chord C' = AB' or AB" = C BB', and by eq. (54) (58) 138. Given: a curve joining two tangents, to change the radius, and also the Point of curve, so that the new curve may end in a parallel tangent directly opposite the given Point of tangent. Fig. 24. Let AB be the given curve, A V, VB the tangents, VB' the parallel tangent, and B ' the given tangent point on the radius OB produced. 78 FIELD ENGINEERING. In the diagram, produce the tangent J.Fand the radius OB to intersect at K. Then BK = R exsec A B ' K = R ' exsec A Subtracting we have SB' =(R- R') exsec A (59) exsec A from which R' is easily determined, as in 132 and 133. FIG. 24. FIG. 25. To find the change AA of the P.O., in the diagram draw O'G parallel to A A; then or 0'G= OQ tan A AA = (R-R')i&n A (60) By substituting the value of (R R') from eq. (59) and ob serving Table II. 42 we have AA=BB' X cot $ A (61) Observe that eqs. (59), (60), and (61) may be derived directly from eqs. (50), (52), and (51) respectively by writing A for | A. 139. Given: a curve join-ing two tangents; to find the new tangent points after each tangent has been moved parallel to itself any distance in either direction, Fig. 25 SIMPLE CURVES. 79 Let A and B be the given tangent points, and A and B ' the new tangent points required. Let the known perpendicu- lar distances Aq = a, and Bp = b. We then require the unknown parallel distances qA = x and pB ' = y. Since the form and dimensions of the curve remain un- changed we may conceive the curve to be moved bodily into its new position on lines parallel and equal to the line VV joining the vertices. Then AA = 00' = BB' . = FF. In the diagram draw VK parallel and equal to Bp = b and VH parallel and equal to Aq = a. Then VH= qA x, and V'K= B'p = y. Since VG-V = A, we have and since and GH= sin A tan A VH= VG-GH=x Similarly a sin A tan A b a (62) tan A sin A When the new tangents are outside of the given curve, the offsets a and b are considered positive ; if either new tangent were inside of the given curve its offset would be considered negative. In solving eqs. (62) if x and y are found to be positive they are to be laid off forwards from q and p, as in Fig. 25; if either is found to be negative it is to be' laid off in the opposite direction. Example. A certain curve has a central angle of 50 A , and it is proposed -to move the first tangent in 20 feet and the second tangent out 12 feet. Required, the distances on the tangents from the old tangent points to the new. Fig. 26. FIG. 26. 80 FIELD ENGIKEERIHG. Here a = - 20 and b - -f 12 12 1.079181 50* log sin 9.884254 15.665 1.194927 -a 20 1.301030 A 50 log taa 0.076186 16.782 1.224844 x = 15.665 ~ (- 16.782) = -f 32.450 12 1.079181 50 log tan 0.076186 10.069 1.002995 -a 20 1.301030 A 50 log sin 9.884254 - 26.108 1.416776 y = 10.069 - (- 26.108) = + 36.177 f = - 32.450 For -f a and - b y=- 36.177 For + a and -f b ^ = _ 1 5 '.939 = 1.120 If we have a and x given to find b and y: Solving eqs. (62) for b and y we obtain b = x sin A -4-- & cos A > ,0,,-v r (63) y = x cos A sm A ) In which the algebraic signs of the quantities must be ob- served as above. 14O. Given: a curve joining two tangents, to find a new Radius and new position of the Point of curve, such that the curve may end at the same point as before, but with a given change in the direction of the forward tangent. Fig. 27. Let AB be the given curve, AV, VB the given tangents, VB the new tangent, and VBY' the given change in direc lion. Let A' = A + VBV . SIMPLE CURVES. 81 In the diagram draw## perpendicular to AV produced; then BG = R vers A = R' vers A' Hence vers A R' = and vers A (64) AA = AG- A'G = Rsin A - R' sin A' (65) ( In the figure the change in direction of tangent makes A' greater than A ; therefore V falls beyond F, and A beyond A' Fio. 27. FIG. 28. A; but if the change made A' less than A, then V and A would fall behind V and A respectively, and R ' would be greater than R. The same formulae apply to the converse problem in which B is taken as the point of curve, and A and A as points of tangent. 1.41. Owen a curve joining two tangents, to find the change in the Point of curve when tht forward tangent takes a new direction from the vertex V. Fig. 28. By eq. (21) AA = R (tan |A - tan |A') (66) 142. Given: a curve joining two tangents, to find the now FIELD EHGINEEKIKG. radius, R', when the forward tangent takes a new direc- tion from the vertex, V. Fig. 29. By eqs. (21) (25) (67) ' = H tan | A cot ; A' 143. Given: a curve joining two tangents, and a given change in the direction of the forward tangent from, the vertex, to find the radius and point of curve of a curve that shall pass at the same distance, VH, from th^ vertex. Fig. 30. Let AB be the given curve, BVB' the given change in O 0' FIG. 29. Fi. 30. direction of tangent, and VH' = VH. Let A' = A -f BVB'. then eq. (24) VH= E ex sec A = VH' = R ' ex sec i A' _ exsec jA exsec -J- A ' By eq. (28) VA = VHcot i A, F-4' = FH"' cot i A' ^1^1' = VH (cot i A - cot (68) (69) But in case A' = A BVB', AA becomes negative and must be laid off backward from A. SIMPLE CURVES. 83 Example. Given a 2 curve, A = 80 and BVB 1 10 ;. A' = 70 R log 3.457114 | A 40 log exsec 9.484879 VH 874.97 2.941993 | A' 35 log exsec 9.343949 R' 1 27' nearly 3.598044 iA 20 cot 2.74748 A' 17 30' cot 3.17159 - 0.42411 AA = 874.97 X (- .42411) = - 371.08 and must be laid off backward from A. 144. Given: two indefinite tangents, a point situated be- tween them, and the angle A , to find the radius R, and tan- gent distance T of a curve joining the tangents which shall pass through the given point. Fig. 31. If the given point is on the bisecting line VO, as H, meas- ure VH=E, and find R and Tas in 97, 98. When the given point, as P is not on the bisecting line VO; if a line GK is passed through P per- pendicular to VO, it will be parallel to any long chord, as AB, and the angle VGK \ A . The curve pass- ing through P will intersect GK in some other point P'; the line GK is bisected by the line VO at /, and PI= P'L If the given point P is located by a perpendicular offset from the tangent, Fl as Pi; in the triangle PLG, LG = PL cot |A. Lay off LG, and at G deflect VGK= A, and measure GP and PK. Since by Geom. (Tab. I. 24) GA* = GP' X GP, and GP' = PK ; GA = V GP X PK (70) 84 FIELD ENGINEERING. Lay off GA; and A is the Point of curve, AV T, and If the given point were located by an offset from BV, find B first, and make VA = BV. If the given point P is located by a perpendicular offset IP from the bisecting line VO; produce IP to intersect the tangent at G and measure PG. Since P'G =; GP + 2PI GA= VGP (GP -f 2PJ) (71) whence we have the point of curve A, as before. 145. Given: a curve, AP, and the radial offset PP' to find a curve which shall pass through the point P ', start ing from the same point of curve A. Fig. 32. V' 0' _ O 0" Q' FIG. 38. Let b = PP', and in the diagram draw P'G' parallel to the common tangent AX, and join AP'. Then P'G' = (H &)sin A G'A = R-(R d)cos A cot A (72) sin A sin A' sin A' (73) When the offset is outward use R -(- b, when it is inward use J? - b. Example. Given: a 3 curve of 16 stations and a radial offset of 205 feet inward from the P.T. to fincf the radius of the curve passing through the extremity of the offset. SIMPLE CURVES. 85 Here A = 3 X 16 - 48; and b = 205. .R 3= 1910.08 R-b 1705.08 log 3.231745 A 48 log sin 9.871073 P'G' 3.102818 R3 log 3.281051 1.50742 0.178233 A 48 cot .90040 I A' tan .60702 = 31 15V 2 A' 62 31' log sin 9.947995 P'G' log 3.102818 R ' (about 4 01'). Am. 3. 154823 If the same offset were made outride of the curve we should find R' log 3.438350, or about a 2 05' curve. This solution is inconveniently long for ordinary field prac- tice. When the offset is small compared with the length of curve, we may use the following Approximate Kule : Divide twice the offset b by the length of curve, look for the quotient in the table of nat. sines, and take out the corresponding angle, which multiply by 100, and divide by the length of curve. The quotient is the correction for the given degree of curve ; to be subtracted when the offset is made outward, and added when the offset is made inward. This rule is expressed by the formula " ~ZT S ~L Taking the same example, we have ^- - sin 14 51' 100 and correction = 14 51' X =- \r = T 56' loUU Hence D' = 3 56' or 1)' = 2 04' 8g FIELD THE VALVOID. 146. Owen: any number of circular curves of equal length L, all starting from a common point of curve A, in a common tangent AX, to find the equation of the curve joining their extremities. Fig. 33. Let AP be any one of the given curves, " -R- its radius AO, " D = its degree of curve, '' A = its central angle AOP, " C = its Ion? chord AP. P' FIG. 33. By substituting the value of R from eq. (16) in eq. (22) we have tf=100^_i (75) Substituting in this the value of D from eq. (20) and letting heta) 6 = | A, (rho) p = ~ and J 1UU polar equation of the required curve (theta) 6 = | A, (rho) p = ~ and JV = ?7S r, we have for the 1UU iOO sm F in which p is the radius- vector AP, Q the variable angle XAP. the unit of measure is one side of the inscribed polygon by which the circular curve AP is measured, and N the num ber of these sides in the length of the curve AP. By the SIMPLE CURVES. 87 conditions of the problem N is constant, but may have any value whatever. If we let 6 vary from to -f- 180 and from to 180 the point X will describe the curve XP'PA shown in the figure, which is called the Valvoid from its re- semblance to the shell of a bivalve. All circular curves tan- gent to AX at A and having a length L = AX will terminate in the valvoid, and the line PP' joining the extremities of any two of them is a chord of the valvoid. 147. To find a tangent to the valvoid atanypoini P. Fig. 34. See Appendix. Differentiating eq. (76) (77) which is essentially negative, since p is a decreasing function of 6. Let (phi) q> APG, the angle between the radius vector and the normal PG. tan

for the point P, through an angle p'op". Now p'op = Xop> - Xop" = (-+ " we may assume u' = w" = ^ ; hence cp' y Table IV. PP' OPP' = *' - ~ ' id nearly. PP' Eq. (86). A' = A + ^=-. Table X. 3. if Instead of taking I, from Table X. 3 for the exact value of A it is well to take it for the estimated value of 5~ . 100 Eq. (20). D' = ?. A' j When P' is outside of the given curve : = UA, r = vL, PP' 180 - OPP' = f'+ g IS nearly. A' PP 7)' 10 A' A = A--^-, D = T -A Example. Given, a 4 curve of 800 feet, or A = 32 to find SIMPLE CURVES. 93 a curve from the same P. 0. which shall shift the last station, in, about 55 feet. (Fig. 36.) i= 32 X .3355 =10. 736 r = 800 X .7450 = 596, .'. S = 9 36' = 9.6 OPP' = 10.736 - f^ X 4-8 = 8 06' * 2)' = = 5. Ans. o For a 5 curve, the true distance PP ' = 55.53 " 459' " " " " PP' = 546 which proves this method practically correct. 152. Given: a tangent and curve, and a straight line intersecting: them, making a given angle with the tangent at a given point, to determine the distance on the line from the tangent to the curve. Fig. 37. We have OA, AG, and the angle AGP to find OP. R tan AGO = AG PGO = AGO - AGP sin PGO PO=R sm(OPI-PGO) sin PGO 94 FIELD ENGINEERING. When AGP = AGO, eq. (24), GP = R exsec (90 - AGO) When AGP= 90, (92), (119), GP=R vers POA, sin POA When AGP' > AGO, we have P'GO = AGP' -AGO but the other formulas remain unchanged. Example. Let R 955.37, AG = 350, R ~ JK 955.37 350. AGP PGO 69 52' 47" OPI POG R PG 69 52' 47" 40 29 52' 47" 32 02' 36" 2 09' 49" = 40 log 2.980170 log 2.544068 log tan 0.436102 log sin 9.697387 log sin 9.972653 log sin 9. 724734 log sin 8.576953 8.879566 log 2. 980170 72.40 log 1.859736 This problem may be used in passing from a tangent to a curve when the tangent point is obstructed. The distance AP on the curve is defined by the angle AOP, which is readily found. If AGP' > 2 AGO the line will not cut the curve. 153. Given : a curve and a distant point to find a tangent that sJiall pass through the point. Fig. 38. We have the curve adg and the point P visible, but distance unknown, to find the point of tangent B. SIMPLE CURVES. 95 Any chord, as bf, parallel to the required tangent, if pro- duced will pass the point P at a perpendicular distance equal to the middle ordiuate of that chord. Ranging across every two consecutive stakes on the curve we at first find the range falling outside of the required tangent, as bcG, cdH, etc. ; but finally the range falls inside, as deK. We then know that the required point is between c and e. i If the range ce falls inside the point P, a perpendicular distance equal to the middle ordinate of ce, the tangent point is at d. If the perpendicular distance is greater than this, the point B is between c and d. If less, or if the range ce falls outside of P, the point B is between d and e. The middle ordinate for ce (200 feet) equals the tangent offset for 100 feet, given in Tab. IV. , and it is generally so small that it can be estimated at P without going to lay it off. To find the exact point B, when it falls between d and e, find by trial a point x on the arc cd in range with e and a point inside of P a perpendicular distance equal to the middle ordinate of ex. The point B is at the middle point of the arc ex. If the point B is between c and d, stand at c and find a point x on the arc de in the same way. B is at one half the arc ex. The middle ordinate of any chord ex is Fi -. 38. less than M for 200 feet, and greater than m for 100 feet, necessary, its exact value m' can be found by m = m x J loooo (87) and this equation is nearly true when ex is as great at 300 or 400 feet. That is, middle ordinates on the same curve are to each other as the squares of their chords very nearly. By this method the point B is found without the use of the transit, so that the plug can be driven at B before the transit FIELD is brought up from the rear. It is therefore preferable to the following solution. Fig. 89. From any two points a and c of the curve measure the angles to the point P, so that with the chord ac as a base, and the measured angles, we may find cPby the formula sin caP cP ac -. sm c-Pa Knowing the angle c that cP makes with a tangent at c, we find the length of the chord cd by cd = 2J2 sin c. By Geom. Tab. I. 24, PB = Pe VcP X dP whence we know ce. Opposite e, or on tile arc eB described with the radius Pe, we find B. '11 FIG. 40. 154. Given: two curves exterior to each other, to find tlie tangent points of a line tangent to both and 6? length between tangent points. Fig. 40. Let B and A be the required tangent points. Let OB E, aud#'^-.R'. On the curve of greater radius R select a point H supposed to be near the unknown tangent point B, and knowing the SIMPLE CURVES. 97 direction of the radius OH, find on the other curve a point K having a radius 'K parallel to OH, and measure HK. In the diagram draw Ob and O'a perpendicular to HK. Then the angle KO'a = 90 HKO = KO'A nearly, which is the angle required. "We have therefore to find the correction aO'A x, and apply it to KO'a. Aa R ' vers KO 'a; Bb = R vers KO 'a nearly. Ka = R' sin KO 'a Hb = R sin KO 'a Bb Aa = (RR'} vers KO'a ab = HK -f (R R') sin KO'a (R R') vers KO'a (88; KO'A = (KO'a - a-) = HOB Observe that KO 'a = the angle between the tangent at K or H and the line HK ; and KO 'A the angle between the tangent at K or H and the required tangent BA. If, instead of H and K, the points H' and K' had been selected, then (R-R 1 ) vers H'Ob . (QQ ., (88) and H'OB = K'O'A = H'Ob -f x. The length of BA should be obtained by measurement, but it may be calculated by AB - ab - (R - R') sin a? (89) When R = R', x = 0, and HKis parallel to BA. In case the curves are reverse to each other, as in Fig. 41, (R4-R') vers KO'a sm * = nearl >'- (90) KO'A = HOB = KO'a - x If the points H' and K' are selected, Fig. 41, (R 4- R') vers H'Ob c\r\ <~p < ' _^ ...._, H'K' ~(R-\-R') sin H'Ob H'OB = K'O'A = H'Ob + x. nearly. (91) 98 FIELD The lines HK, AS, and 00' all intersect in a common point J, Fig. 41. (93) (94) IB = YHI(HI+2RsmHOV) These last three equations furnish another method of solving the same problem. They may be applied to Fig. 40 by changing the sign of M'. In Fig. 41, itft = R' t then HI= |//JTand AB = 2IS. H K FIG. 41. Fia. 42. 155. Given: two curves, O and O', reverse to each other, joined by a tangent BA', and terminating in another tangent, B'F ; to change the position of the Point of Tangent B of the first curve, so that the second curve may terminate in a given parallel tangent, B"F'. Fig. 42. Let X be the required new position of B. " 0" be the corresponding position of 0'. " A' = AO'B' and A" = A"0"B\ Since the radii and the connecting tangent are unchanged in length, and all rotate together about as a centre, " will be on a circle passing through 0', described with a radius 00', and the require^ angle BOX= O'QO". SIMPLE CURVES. 99 In the diagram, produce O'A' and draw the perpendicular OG, -and let a the angle 00' G. Also, draw OK parallel and 0"K and 'H perpendicular to B'O'. In the triangle 00' G- we have (95) (96) cos a The angle KOO' 00' B 1 = a -f- A'. The angle KOO" = 00 "B" = a -f A". KO = 00". cos (a -f A"), HO = 00'. cos (a -f A'X /. HK 00' [cos(a + A") cos (a -f A')] = B'F' B'F' cos (a -f A") = cos (a -}- A') -f- -QQ? (97) BOX = O'OO" = ( 4- A') - (a -f A") (98) If we conceive a line to be drawn through bisecting the arc O'O", the angle it makes with B"0" is a mean between B'0'0 and B"0"0 ; hence the chord O'O", perpendicular to this line, makes an angle with O'P perpendicular to B' 0' of and since 0'P = PO" cotPO'O" F'B" = B 'F' cot | [ (a + A') -f (a + A")] (99) which gives the distance, measured on the parallel tangent, between the old tangent point and the new. This problem occurs in practice when both the connecting tangent and the radius of the last curve are at their minimum limit, and the parallel tangent is inside of the old one, as in the figure. Should the new tangent be outside, the same for- mulae apply, only changing the sign of B'F' in eq. (97). But in this last case it is usually preferable to employ problem 136 or 137. Example. A 1 40' curve is followed by a tangent of 200 ft., and that by a 4 curve of 10 stations ending: in a tangent ; 100 FIELD ENGINEERING. and the offset to the given parallel tangent is 80 ft. on the inside. Required, the position of the new tangent points X and B". Here It = 3437.87, R' - 1432.69, BA' = 200, B'F' = 80. Eq. (95) R + R' 4870.56 log 3.687579 BA' 200. log 2. 301030 .-.a 2 21' log cot 1.386549 Eq. (96) a 2 21' log cos 9. 999635 00' 3.687944 Eq. (97) B'F' 80 1.903090 .01641 8.315146 + A '42 21' cos. 73904 a + A" 40 56' cos .75545 Eq. (98) BOX 1 25' . . BX = 85 ft. Ans. Eq. (99) PO '0 * 41 38' 30" cot 1.12468 X 80 = 89.97 = J?'B " 156. When the tangents of a proposed road are to be in general much longer than the curves, it is desirable to estab- lish the tangents first in making the location, and afterwards determine suitable curves. On the other hand, if the curves necessarily predominate, they should be first selected and adjusted to the ground with reference to grade and easy alignment, and afterwards joined by tangents. In the latter case the field work cannot be successfully accomplished unless the location has been previously worked out upon a correct map constructed from the preliminary surveys. The map sliould show contours of the surface, and also the grade contour, or intersection of the surface and plane of the grade. In side-hill work the grade contour indicates approximately the degree and position of the necessary curves. In the work of selecting proper curves upon the map, templets or pattern curves are almost indispensable. The templets are cut to form a series of curves, the radii being taken from Table IV. to a scale corresponding to the scale of the map, which ranges from 400 to 100 feet per inch, according to the difficulty of the location. The templets should represent convenient curves, or those in which the number of minutes SIMPLE CURVES. 101 per station bear a simple ratio to 100. Curves of 50' and multiples of 50' are most convenient; 40' curves and multi- ples standing next in order, and 30' curves and multiples next. TABLE OP CONVENIENT CURVES. D. Ratio of Min. to Feet. D. Ratio of Min. to Feet. D. Ratio of Min. to Feet. 50' 1 :2 40' 2 5 30' 3: 10 1 40' 1:1 1 20' 4 5 1 00' 3:5 2 30' 3:2 2 00' 6 5 1 30' 9: 10 3 20' 2: 1 2 40' 8 5 2 00' 6:5 4 10' 5:2 3 20' 2 1 2 30' 3:2 5 00' 3:1 4 00' 12 5 3 00' 9:5 5 50' 7:2 40 4Q/ 14 5 3 30' 21 : 10 6 40' 4:1 5 20' 16 5 4 00' 12:5 7 30' 9:2 6 00' 18 5 4 30' 27:10 8 20' 5: 1 6 40' 4 1 5 00' 3: 1 9 10' 11:2 7 20' 22 5 5 30' 33:10 10 00' 6:1 8 00' 24 5 6 00' 18:5 After drawing the curves and tangents upon the map, the tangent points and central angles are carefully determined, the latter being compared with the lengths of the curves ob- tained by a pair of stepping dividers set precisely by scale to the length of one station. Field notes are then prepared from the map, and if the work has been well done these notes may be followed in the field with scarcely any alterations. No ordinary protractor will measure the angles closely enough for this purpose ; it is better to use a radius as large as convenient, of 50 parts. The chord of any arc drawn with this radius equals 100 times the sine of one half the angle subtended. The importance of having absolutely straight-edged rulers in such work is obvious. In case a very long line is to be projected upon the map, it is well to use a piece of fine sewing silk for the purpose. See 53, 54. 102 FIELD CHAPTER VI. COMPOUND CURVES. A. Theory. 157. A compound curve consists of two or more consecu- tive circular arcs of different radii, having their centres on the same side of the curve ; but any two consecutive arcs must have a common tangent at their meeting point, or their radii at this point must coincide in direction. The meeting point is called the point of compound curve, or P.C.C. Compound curves are employed to bring the line of the road upon more favorable ground than could be done by the use of any simple curve. When a compound curve of two arcs connects two tangent lines, the tangent points are at unequal distances from the intersection or vertex, the shorter distance being on the line which is tangent to the arc of shorter radius. 158. Let VA, VB (Fig. 43) be any two right lines inter- secting at V, and let A be the deflection angle between them. Let A and B be the tangent points of a compound curve ( VA less than VB), and let AP, PB be the two arcs of the curve. The centre Oj of the arc AP will be found on AS, drawn per- pendicular to VA ; the centre 2 of the arc PB will be found on BS produced perpendicular to VB ; and the angle ASB will evidently equal A . Join VS, and on VS as a diameter describe a circle ; it will pass through the points A and B, since the angles VA8, VB8 are right angles in a semicircle. Draw the chord VQ, bisecting the angle A VB, and join A Q, BQ. Then AQ, BQ are equal, since they are chords subtend- ing the equal angles AVQ, BVQ. From Q as a centre, and with radius QA, describe a circle ; it will cut the tangent lines at A and B, and also at two other points G and Y, such that VG = VA,\ and VY= VB. Hence BG = AY, and the parallel chords AG, BY are perpendicular to VQ. Join AB; then AQB = 'ASB = A , since both angles are subtended by the same chord AB. In the triangle VAB, the sum of the angles at A and B is equal to the exterior angle A between the tangents ; while their difference (A B) is equal to the angle at the centre Q COMPOUND CURVES. 103 subtended by the chord BG, which is the difterence of the sides (VB - VA). For the angle VAB = VAG + GAB, and the angle VBA = VBT - ABY. But VAG = VBY and GAB - ABY, and by subtraction VAB - VBA = 2 GAB = GQB, since A is on the circumference and Q at the centre. 150. THEOREM. The circle YAGB, whose centre is Q, is the locus of the point of compound curve P, whatever be the relative lengths of the arcs AP, PB composing the curve. FIG. 43. On the circle YAGB, and between A and G, take any point P, and on AS find a centre 0,, from which a circular arc may be drawn cutting the circle at A and P ; also on B8 produced find a centre 2 , from which a circular arc may be drawn cutting the circle at B and P. Join PQ, PO, and P0 2 . Since when two circles intersect, the angles are equal be- tween radii drawn to the points of intersection, QPOi = QAO l 104 FIELD ENGINEERING. and QPOv = QBO^. Draw the chord QS and it subtends the equal angles QAO, = QBO^. Hence QPO, = QPO* and the radius P0\ coincides in direction with the radius P0 2 , which is the condition essential to a compound curve. Now, if we imagine another point P ' to be taken on QP or on QP produced, and the arcs AP' HP', drawn from centres found on AS and B/S, it is evident that the equality of angles found in respect to P could not exist in respect to P'. Hence the arcs would intersect in P' at some angle OiPO^ and would not form a compound curve. Therefore, Q. E. I). 16O. THEOREM. In any compound curve the radial lines passing through the three tangent points A, P, and B are all tangent to a circle having the point Q for its centre, and for its diameter the difference of the sides VB and VA. Draw the three lines QN, QL, QM perpendicular to the radial lines BO*, P0 2 , and A8 respectively. Then the three right-angled triangles BQN, PQL, and AQM are equal, since BQ = PQ = AQ = radius of the circle AGB, and the angles at B, P, and A are equal by the last theorem. Hence QM = QL = QN, and if a circle be described with this radius about Q, the three lines B0 2 , P0 2 , and AOi produced will be tan- gent to it. Draw Q2 perpendicular to VB; it will bisect the chord GB in /; and QN = BI = %BG. Hence the diameter 2Q1T= BG = VB VA; which was to be proved. Corottary 1. The compound curve intersects the circle AGB in the point P, at an angle equal to half the difference of the angles VAB, VBA. For QPL = QBtf= BQJ = $BQG. The arc AP is exterior, and the arc PB interior to the circle AGB. Cor. 2. Since both centres are on the line PL, the position <)f the point P fixes the lengths of the radii of a compound curve. As P is moved toward G both radii are increased, until when P reaches G, AOi becomes AK, a maximum, while 7?0 2 becomes infinite. As P moves toward A both radii are diminished, but the least value of the arc AP depends upon the least radius allowed on the road. If in the diagram we make AOi equal to the least radius allowed, a right line drawn through the point O l tangent to the circle LMN fixes the corresponding minimum value of the arc AP, and also of the radius B0 2 for given values of VA, VB, and A. Be- COMPOUND CURVES. 105 tween these limits any desired values of the radii may be em- ployed. Cor. 3. In the triangle SOi 2 , the sum of the two central angles AOiP&nd PO-^B is equal to the exterior angle A8B = A ; consequently, as the central angle of one arc is increased by any change in the position of the point P, the central angle of the other will be diminished an equal amount. Cor. 4. Only one value of the angle AOiP is consistent with a given value of the radius AOi, since both depend on the variable position of the line PL; and for the same reason only one value of the angle BO^P is consistent with a given value of the radius BOi. Hence only one radius or one central angle can be assumed at pleasure, the remaining parts being deducible therefrom in terms of the sides VA, VB, and the angle A. B. General Equations. 161. Let 8, = the side VA, # 2 = the side VB Let Ei = the radius AOi JR 2 = the radius BOi " y = diff. VAB - VBA, A = the sum VAB+ VBA " Ai = central angle AOiP, A 2 = central angle BO^P. In the triangle BQI, cot BQI = ^-. But IQ = VI X JDJ. cot IQV= i( a -\- &) cot I A, and BI = $(8* - &). cot \y = |^| cot | A (100) By Cor. 3, Ai + A 2 = A (101) In the triangle AQM, AOi = AM MOi. But AM = Mq cot \y, and MO^ = MQ cot | A i. ^i = -H& - ^) (cot ir - cot 1 AO ) (102) Similarly, ^ a = *(& ~ ^) (cot \y + cot i A 9 ) ) Subtracting, ^ 2 - & = |(^ 2 - ^) (cot i A a -f COt i A (108) - ft) 106 FIELD cot I A i = cot \y From (102), i _ & In the triangle ABG, D ~ AB sin BAG sinAGV or ~ sin 4 (104) by which we find ^(/S' 2 Si), when, instead of the sides and A , we have given AB, and the angles VAB and VBA. From (103), M _ft) = __ (106) D cot \y = ^ ^_ -f cot ^ AI From (102), j- (107) From(100X W. + ft) = (108) (Sa and & are found by adding and subtracting the values found by eqs. (106), (108). From (105), ^ = ^ "^ * A - d> which may be used instead of (108) when the sides are not re- quired. VAB = i( A + y) and VBA - \( A - y\ 162. Given : the sides VA = & ^140') 3.536290 (IS) .'. A, =42 54', Li = 715; A 2 = 31 06', 2 = 1866. 163. Given: the line AB, and the angles VAB, VBA; assuming the longer radius R^, to find A 2 , Aj, and Si. e. Let AB = 2437.82, TAB = 48 31', VBA = 25 29', and assume R* = 3437.87. (105) \AB J 218.91 \Y .-. i(8.2 - &) 404.38 (104) .B 2 lr (101) i A 2 11 31' 37 11 31' 15 33' 37 log sin " < 8.50166 " cot 4.90785 3.085972 9.300276 2.386248 9.779463 2.606785 3.536289 0.929504 cot 3.59381 cot 2.54516 '" 4.90785 21 27' 2.36269 log 0.373407 2.606785 2.980192 108 FIELD 164. Usually a compound curve is fitted by trial to the shape of the ground, after which it may be desirable to calculate the sides VA, VB, or the line AB, and the angles VAB, VBA. Example. From the point of curve A, a 6 curve is run 715 feet to the P.O.C.; thence a 1 40' curve is run 1866 feet to the P.T. Required, the sides VA, VB, and the line AB, and angles VAB, VBA. Here R, = 955.37, A! =42 54', J2 3 = 3437.87, A 2 = 3106'. (106) 2 - R, 2482.50 iAi *A a 21 27' 15 33' cot 2. 54516 " 3.59370 (107) .-. \r (108) X -&) 404.39 - A) 21 27' 6.13886 2.36248 cot 2.54516 ll31'0r.7 "4.90764 log 3.394889 0.788088 37 cot " 1495.51 1899.90 1091.12 (109) K& I A 48 31' 25 29' 37 11 31' 01 ".7 2.606801 2.980170 0.373369 0.690873 2.606801 3.297674 0.122886 3.174788 " 2.606801 sin 9.779463 AB 1218.91 2437.82 2.386264 9.300294 3.085970 165. Given: the radii R lt R*, the angle A, and one side, VA, or VB, to find the other side and the central angles AI, A 3 . Fig. 43. COMPOUND CURVES. 109 In the triangle AMQ, A0 l = AM - M0 l = IQ - M Q cot MOiQ\ or Si = i(& + Si) COt | A - $(8* -S^ COt | A! whence i(& -f- &) = |(S 2 8i)~ m cot * A ; tan A -[- .R, tan iA By eq. (106) px sinjA 2 sinjAi Substituting this above, subtracting and reducing = .B 2 - A sin ^A 2 Ri tan But i(A AI) = -J A a and 2 sin 2 1 A 2 = vers A 2 , whence ^ = (J? 2 - JgQ vers A , + E l vers A (n()) . Transposing, /S'j sin A Hi vers A Similarly, from the triangle J2 a rr ^(5a + ^) COt i A + K& - ^) COt | A from which and eq. (106) we derive vers A ~ (^^ JgQ vers A i 110 sin A and 2? 2 vers A 8 9 sin A vers A! = 110 FIELD Example. Given : VA = S t = 1091.12, A = 74, and thb radii R, = 955.37, J? 2 = 3437.87, to find AI, A 2 , and 2 . (Ill) & 1091.12 log 3.037873 A 74 sin " 9.982842 1048.85 " 3.020715 " 2.980170 74 vers " 9.859956 692.03 " 2.840126 356.82 " 2.552449 --Ri 2482.50 " 3.394889 A 2 31 06' vers " 9.157560 A! 42 54' " " 9.427254 Rt-Ri " 3.394889 663.96 " 2.822143 R* " 3.536289 A vers " 9.859956 2490.26 " 3.396245 1826.30 " 3.261572 sin " 9.982842 1899.90 " 3.278730 166. Given : one side, and the radius and central angle of tJie adjacent arc, to find the other radius and side. From eqs. (Ill), (113) we have Si sin A Ri vers A vers A a RZ vers A Si sin A vers AI (114, by one of which the required radius may be found ; the required side is then found by eq. (110) or (112), as in the last problem. Example. Given : VA =81 = 1091.12 A = 74, S l =955.37 and A: = 4254'; to find R, A 2 = 74 - 43 54' = 81 06'. Ui4) & COMPOUND CURVES. 1091.12 74 1048.85 955.37 A 2 ."! -B! ~" 74 3106 Ill log 3.037873 " sin 9.982843 692.03 356.82 2482.52 3437.89 3.020715 2.980170 ' : vers 9.859956 2.840126 2.55244C- vers 9. 157556 3.394893 Fio. 44. Otherwise : Fig. 44. If convenient in the field, a tan-, gent PV-t may be run from the point P to intersect the farther tangent. The distance PFj multiplied by cot -JA 2 will equal the radius -R 2 by eq. (25). 167. Remarks. If the first arc AP be produced to G, Fig. 44, so that AOiG A, then G is the tangent point of a tangent parallel to VB, and by 137, the tangent point B must be on the line PG produced. Conversely, if the point B is assumed, and the arc AG given, the point P must be on the line BG produced. The radius Ri may be found by 112 FIELD B^GIKEERI^G. jR a = - : , BP being measured on the ground ; or by 2 sin -J A a similar triangles R* : fa :: BP : GP. The distance VD, Fig. 43, from the vertex to the circle AOB is expressed by the formula I (tan | - tan ^-~} (115) VD = & cos If the point P falls at D, then VD is also the distance of the curve from the vertex measured on the line VQ. But when P falls at D, the radius P0 2 is perpendicular to the line AB, and AI = VAB, and A 2 = VBA. When AI is greater than VAB, the arc AP, being exterior to the circle, cuts the line VD; but when AI is less than VAB, the arc PB cuts the line DQ. If the line 0,,P produced passes through V, we have sin QVL = 7-f. sin i A (116) giving AI i A + QVL and A 2 |A QVL. When A! is greater than this, we have for the external distance of the vertex EI = R! ex sec AOi V in which the angle AO^ Fis found by the formula cot AOi V= r> , and EI is measured on a line VOi, making the angle AVO, =90 - A0 1 V. When A i is less than (| A -f- Q VL), we have similar expres- sions with respect to the arc BP and centre 2 . 168. To locate a compound curve when the point of com- pound curve is inaccessible. Fig. 45. Each arc being in itself a simple curve is located as such. When the P. C. G. is accessible, the transit is placed over it, and the direction of the common, tangent found, from which the second arc is then located. When the P,G.C. is not accessible, the common tangent V\ V-i may be found by locating the points V\ and F 2 , which may be easily done, since V\A V\P Hi tan AI, and COMPOUND CURVES. 113 V^B FaP Ri tan A 2 , from which each arc may then be located by offsets or otherwise, as in the case of simple curves. Should the points Fi F 2 be obstructed, the common tangent may be found by an offset HG = LP from any convenient point H, for knowing the angle HOiP, we have HG = Ri vers HOiP, and GP = & sin HO,P. If the entire tangent Fi F 2 is too much obstructed for use, the parallel line HK may be employed, observing that the LP angle PO t K is found by vers PO?K -^-, and the distance xt a LK by LK R* sin PO?K, by which a point K on the second arc is found having a tangent offset Kl HG. FIG. 45. FIG. 46. Should the line HK be also obstructed, we may run the in- verted curve HP' = HP and P'K PKto find the point K from which so much of the second arc as is accessible may be located. (7. Special Problems in Compound Curves. 169. Given: a compound curve ending in a tangent; to change the P.C.C. so that the curve may end in a given parallel tangent. Fig. 46. Let APB be the given curve ending in VB, " VB' be the given parallel tangent, " p = perpendicular distance between tangents. It is required to change the point P, and with it the values of A i and A 2 , so that with the same radii E\ and R* the new curve APB' may end in the parallel tangent 114 FIELD ENGINEERING. a. When the tangent V'B ' is inside of VB : Let A i = ^.01 P, Ai'=AO l P', A 2 = P0 2 #, Aa' - P'Oi'B t and in the diagram draw 0^ perpendicular to 50.,; then GOt = 0i 2 cos A 2, -KYV 0i 2 ' cos A 2 '. Subtracting, since X 2 = 0!0 2 ' = (R. - RJ, and JT0 2 ' - GO* = GB - KB'=p, p = (R 2 RJ (cos A a' cos A 2 ) whence cos A' = R ^_ R - + cos A 2 (117) POiP' = (Aa A a') and the point Pis advanced. b. When the tangent V'B' is outside of VB: p = (Ri RJ (cos A 2 cos A a ') whence COS A a' = COS A 2 - ^~~- (118) POiP' = (A a ' As) and the point P is mwed: Jc^ and the arc AP diminished. FIG. 47. In case the curve terminates with the arc of shorter radius, or R t follows R z . Fig. 47. c. When V'B' is inside of VB: p = (R.t R$ (cos A i cos A/) whence cos A i' = cos A i - ^ (119; -it2 ^Vl ' (A/ ^0 and the point P is moved COMPOUND CURVES. 115 d. When V'B' is outside of VB : p = (fit Mi) (cos AI' cos Ai) whence cos A ,' = cos A i + TT-^T- (120) /C 2 -til P0 2 P' (Ai A/) and the point P is advanced. Example. Let It = 2292.01, R l = 1432.69, A 2 = 28, and p - 20.07 inside of VB ; case a. p 20.07 log 1.302547 (117) R* - Ri 859.32 " 2.934155 .023356 " 8.368392 A 2 28 cos .88295 .-. A'Q 2JT " T906306 .-. POiP' 3 17O. Given: a compound curve terminating in a tangent^ to change the P.C.C. and also the last radius, so that the curve shall end in a parallel tangent at a point on the same radial line as before. Fig. 48. Let APE be the given curve ending in the tangent VB; let V'B' be the given parallel tangent; and let p = BB' = HI = tne perpendicular distance between tangents. It is required to change the point P to P', and also the value of -R 2 to R% ', so that the new curve may end in V'B' at B' inside of VB on the same radial line 502. In the diagram produce the arc AP to G to meet O^G drawn parallel to 0?B; then POiG = A 2 . Draw the chord PB, and it will pass through G. Lay off the distance p from 116 FIELD ENGINEERING. B on BO* to find B'; draw B'G and produce it to intersect the arc APG in P'. Then P' is the P. C. C. required. Join P ' Oi and produce it to meet J90 2 produced in T>t COt i A ' ~ COt -jAa ,TT? (131) Or/L 120 FIELD ENGINEERING. GR being obtained from eq. (127) and E, r from eq. (128). In eq. (131) use the + sign when B' is beyond B as in the Fig. 50. II. When the given curve ends with the smaller radius -Ri. Fig. 51. G FIG. 51. We have by a similar reasoning GK = (R* - .ft,) vers Ai (132) = A- , (133) vers AI vers AI' /-K\A\ (134) BB = GK(cot i A, - cot | A /) (135) T>T>' COt* A/ = COtiAi -FT* (136) using the sign when B is beyond B. Example. Fig. 51. Let Ki - 2292.01, B l = 1432.69, A! = 46, and let the P. C. C. be moved back 200 feet from P to P ; hence P0 2 'P' = 5 and A/ = 51; to find the new radius Ri and the distance BB. Eq. (132) It* - Ai .-. OK eq. (133) A/ # 2 - Si' COMPOUND CURVES. ?i 859.32 46 51 9 707.85 2292.01 121 log 2.934155 " vers 9.484786 log 2.418941 " vers 9. 568999 1584.16 and 7) =3 87' eq. (135) OK cot !AJ' 2.35585 2.09654 23 25 30' 0.25931 .-. BB' 68.04 log log 2.849942 2.418941 9.413819 1.832760 172. Given: a compound curve ending in a tangent, the last radius being the greater, to change the last radius and also the position oftheJ*C.C. so that the curve may end at the same tangent point, but with a given difference in the direction of the tangent. Fig. 52. Fio. 52. Let APB be the given compound curve, POi = Mi and PO. = R, > EL Let V'B be the new tangent, and the angle VBV = , the given difference in direction : to find J?0 2 ' = It*', BO^'P' = A,' and the angle PO^P'. 122 FIELD We have 2 - 0!0 2 = R* - O2 2 - #0 = R l BO* - 0,0* = R* - (R,'~ R,) = R, From which we see that whatever may be the value of the new radius, the difference of the distances from B and t to the new centre is constant, and equal to R>. We therefore conclude that the centres 0. 2 and a ' are on an hyperbola of which B and A are the foci, and R\ the major axis. This suggests an easy graphical method of solving the problem. Through B draw a line perpendicular to the new tangent VB which will give the direction of the required centre 2 '. On this line lay off BK equal to Ri, and since (R* R>) = 0i 2 ' = KO*', if we join KO,, the triangle K0*0i is isosceles; therefore bisect KO, and erect a perpendicular from the mid- dle point to intersect the line BK produced in 2 '. Draw 2 '0i and produce it to intersect the arc AP (produced if necessary) in P'. Then P is the new P.G.C. required, and BO* P'O* Ri', the new radius. The analytical solution is as follows : Adopting the usual notation of the hyperbola Let 2a = Ri the major axis, " 2c = BOi = the distance between foci. Produce the arc AP and through B draw the tangent BH. and join HO, = R,. Then m the right-angled triangle BHO, BH* = BO? - RS = 4c 8 -^4a 2 Now by Anal. G-eom., c 2 a 9 = J 2 . Therefore 2b = BH = the minor axis. Draw the -chord PB and produce the arc AP to cut it in Q. Then by Geom. (Table I. 34) BH* = PB X GB = 2R> sin i A 3 X 2(R, - Ri} sin * A a . '. BH = 2 sm i A a VR, (R, -~RT) (137) COMPOUKB CURVES. Let a = the angle HOiB, then toa^andBftzz-sA- (138 ) In the triangle BO^O* let OiBO* = ft ; then sin ft = ^* 1 sin A 3 (139) The polar equation of the hyperbola for the branch 10M> taKing the pole at B and estimating the variable angle from tne line BOi, is c . cos a When v = ft i, r = RJ, and substituting the values of a, b, and c found above, we have , _ _ ~ ! cos (/? t) - ^j) using (/? + i) when F' falls between F and A, as in the figure, and (ft i) when V falls beyond F. In the triangle B0i0 2 ', the angle BO^'Oi = A 2 ' and sin A,' = Finally PO.P' = A a ~(A 3 ' )' (142) Remark, When F' falls between F and A, as in Fig. 52, if the angle i be greater than the angle VBH, the curve ceases to be a compound, and becomes reversed. Therefore VBH = a ft is the maximum value of / possible in this case. When F' falls beyond F, the point P will fall between Pand A\ and the largest possible value of I will then be that which renders POiP' A i, and makes the point P' coincide with A. 124 FIELD ENGINEERING. Example. Fig. 52. Let 7?, = 1432.69 (137) 12, - 12, 12 a = 6 859.32 2292.01 (138) (139) 12 3 - (iO) 12z 1432.69 a a BO, A 2 BH- 12 a ' (141).'. (142) .-. 1? 2 = 2292.01 28 42 36' 23". 7 42' 36' 23". 7 56 21 28' 06".3 27 28' 06".3 Ai = 31 A a = 56 log 2.934155 3.360217 2 ) 6.294372 3T47186 log sin 9.671609 0.301030 3.119825 3.156151 log tan 9.963674 log cos 9.866889 3.289262 2.934155 9.644893 log sin 9.918574 log sin 9.563467 log cos 9.948053 3.289262 1727.09 1432.69 294.40 X 2 = 588.80 2949.05 A 3 ' = 3618'26" POjP' = 13 41' 34" = 342.3 feet. 3.237315 2.769968 6.239650 3T469682 Remark This problem may also be solved by first finding the new sides V'A, V'B, from which and the new central angle (A t), and the radius 12j, may be found A/, A 2 ', and l/ 2 ', as in 162. The new sides are readily found from the old ones by solving the triangle VBV '. If the original sides are not given, they must be calculated as in 164. 173. Given: a compound curve ending in a tangent, the last radius being the less, to change the last radius and the position of the P.C.C. so that the curve may end at the same tangent point, but direction of tangent. with a given Fig. 53. difference in the COMPOUKD CURVES. 125 Let APE be the given curve, and P0 3 = .#3, and JPOi = R l < R 2 . Let F'J5be the new tangent, and VBV = i, the given angle; to find BOi - Ri, BO^P' = A,', and PO a P'. We have BO, + 0,0, = B, + (R* - Ri) = R* BOi + 0/02 = R,' + CR a - J2i') = -R 2 from which we infer that the locus of the centre 0/ is an ellipse, of which B and 2 are the foci, and jf? the major axis, FIG. 53. since the sum of the distances .B6V and O^Oi is always equal to # 2 . This suggests an easy graphical solution of the prob- blem, as follows : Perpendicular to V'B draw the indefinite line BK, which will contain the required centre 0\ , and layoff BK = -R 2 . Join KOi, bisect it, and from the middle point erect a perpen- dicular to intersect BK in Oi. Join 2 0,', and produce the line to intersect the arc AP (produced if necessary) in P', which is the new P.C.C. required. P'Oi = BOi Ri ', the required radius, and P'O^B = Ai'. The analytical solution is as follows : Adopting the usual notation of the ellipse, let 2a = Rz = the major axis, " 2c = BOi = the distance between foci. At B erect BH perpendicular to BOi to intersect the arc AP 126 FIELD ENGINEERING. (produced if necessary) in H, and join HO* = 72 a . Then BE* = RJ - BOJ = 4a 2 - 4c 2 jSut by Anal. Geom., a 2 c 2 = 6 2 . Hence 26 = BE = the minor axis. In the triangle BO^O* we know BO, #,, and 0,0^ = J? 2 Jf?i, and the included angle J90i0 2 = 180 Ai; hence by Trig. (Tab. II. 25) O T> D tan |(0 X 2 - O^Oa) = -- L ^~- tan i A , (143) The angles at j5 and 2 are then found by (Tab. II. 26). Let/? = the angle 0,BO^\ then m = (^-*,)i^ (t44) The value of BE* above may be written BE* = (R., + BO*) (R* - BO*) (145) The polar equation of the ellipse, taking the pole at B, anO estimating the variable angle v from the axis BO*, is a c . cos When v = ft T *, then r = .&', and substituting the values of a, b, and c, given above, we have 7? ' - -- using (ft i} when F' falls between Fand ^4, as in Fig. 53, and (/?+ ) when F' falls beyond F In the triangle BOi'O^ the angle Oi'j50 2 = (/?T ') and tne exterior angle BOi'P' = AI'; hence 2 p> sin (/? T ) (147) Finally P0 2 P' = (Ai T *) - A,' (148) When V is on AV, then P0 2 P' is negative, showing that it must be laid off from P toward A; but when V is beyond COMPOUND CURVES. 127 V, then P0 2 P' is positive, and P' will be on AP produced. The only limits imposed on the angle i are that the resulting value of PP' shall not exceed PA, and tnat Ri shall not be less than a practical minimum. Example. Fig. 53. Let D., = 3 20' .R a = 1719.12 A 2 = 23 20' D, = 6 R, = 955.37 A, = 48 The resulting values are as follows: 21 09' 32".6 BO, BH* 5' Ax' '0 a P f PP 1 1572.42 1273.65 440.5 * = 7 45' 3.196567 5.683829 3.105052 54 56' 14 41' (See also remark at end of 172.) 174. Given a simple curve joining two tangents, to re- place it by a three-centred compound curve between the same tangent points. Fig. 54. Fia. 54. Let R = AO radius of simple curve. R, = P0 l = P'0 l < R A! = POiP 1 Ri = AOv = B0 3 > R A a = AO*P = SO S P' A = A073 Since A0 R; whence In selecting values for R^ and 7? 2 , the degree of curve D v should be but little greater than D of the simple curve, say from 30 to 60 minutes, while 7> 2 may be taken at \D to D. Example. Given: R = 1719.12 D 3 20' A = 40 Let R, = 1432.69 D, = 4 " Rz = 5729.65 />, = 1 20 - R l 4296.96 A 2 AP - P'B 138.4 ft. 18 36' 57" 37 13' 54" 123'03" log 3.603202 " 3.633161 " 9.970041 log sin 9.534052 " " 9.504093 Again we may assume A 2 and R\, whence Ai = A 2A S and = sin | sin Example. Given: R = 1719.12 A = 40 Let R, = 1432.69 A 2 = 1 .-. AI = 38 Am. R* = 7387.24 . -. Z> 2 = 46f AP = 129. Finally we may assume A a and R?, and deduce AI and R\ from eqs. (149) (150); but this is the least desirable because COMPOUND CURVES. the value of 7?i so found will not usually give a convenient value to the degree of curve J)\. 175. To determine the distance HH' between the middle points of a simple curve and a three-centred compound curve joining the same tangent points AB. Fig. 54. In the triangle OO^O*, we have sin A a HH' = 00, + 0>H' - OH .-. HE' = (*, - *.) - (B - *0 (153) In the first example given above HH' = 14.55, and in the second HH' = 17.05 ft. In many instances the distance HH' is so great as to render this problem practically useless, unless the distance HHi is discounted beforehand by putting the simple curve AHB a sufficient distance inside of the proper location through the point H'. But the problem given below is usually preferable. 170. Given, a simple curve joining two tangents to re- place it by a three-centred compound curve which shall pass through the same middle point H. I. The curve flattened at the tangents. Fig. 55. Let R = AO, the radius, and A = the central angle of the simple curve AHB, and let H be the middle point. Let R! = P0! = HO, A i = P0.P' " A' and B' be the new tangent points required. We have at once, as in the last problem, 2A a -f Ai = A. (154) 130 FIELD Since the curve is to be symmetrical about VO, HP = HP 1 . PA = P'B, and A A' = SB'. FIG. 55. In the diagram produce the arc HF to G, and draw Oi parallel to OA, and produce it to K. Then a tangent line al G will be parallel to VA\ and by 137 the point G will be on the long chord HA, and on the long chord PA. GK is the perpendicular distance between parallel tangents, and the problem is similar to that given in 171 ; whence by eq. (57) we have, in this case, GK = (B t - Bi) vers A 2 = (R - JB,) vers i A. (155) for the general equation in which R and A are given. Analagous to eq. (130) we have AA = KA - KA = GK cot OA'K - GK cot GAK. .'. AA = GK(coi |A 2 -cotiA) (156) in which GK is obtained from (155). We may now assume values for Ri and Ri, making R l < R and Ri > R, and deduce the values of A 2 , Ai, and AA. Solving eq. (155) (R R t ) vers A GK .-1 & j^ - = g Eq. (154) gives AI, and eq. (156) gives AA'. COMPOUND CURVES. 131 Example. Fig. 55. Given: R = 764.489 D = 7 30' A = 40 Let.fr = 716.779 A = 8 " Rs = 3437.870 D 2 = 1 40' (155) R - Ri 47.71 log 1.678609 ^A 20 log vers 8. 780370 GK log 0.458979 R - R, 2721.091 " 3.434743 A 2 (say) 2 38' log vers 7.024236 AP 158.00 Ai = 3444' (156) iA 2 43.5081 = cot 1 19' iA 5.6713 cot 10 37.8368 log 1.577914 GK " 0.458979 AA 108.87 " 2.036893 Again, we may assume A 2 and Ri < R; whence Ai = A 2A 3 and eq. (155) GK = (R 120 vers A and < 158 > Eq. (156) gives AA'. Again, we may assume A 3 and the distance AA' ; whence, from eq. (156) GK=- AA ' ^ (159) eq. (155) M l = R - GK vers iA eq. (158) gives R*. Again, we may assume R t < R and AA ; then, eq. (155) GK= (R- 120 vers i A and eq. (156) cot | A a = cot iA + -f^ (160) and eq. (158) gives R 2 . 132 FIELD ENGINEERING. Example. Fig 55. Given: 12 = 764.489 Let R, = 716.779 ' A A = 110. Hence by last example, D = 7 30' Z>i = 8 A =40 OK eq. (160) AA (158) COt^A A 2 GK AP' 110. 38.2309 5.6713 43.9022 2759.5 3476.3 157, log 0.458979 2.041393 10' (say) 2 37' D* = 1 39' Ai = 34 46' 1.582414 18" log 1.642486 log vers 7.018147 0.458979 3.440832 II. The curve sharpened at the tangents. Fig. 56. This case will only occur when, with a given external dis- tance VH, a simple curve would absorb too much of the tan- gents. OF THE UNIVERSITY OF FIG. 66. Let AHB be the simple curve, and " A'PHP'B'thQ required compound curve. " R* = P0 2 = HO*; A 2 = P0 2 P' " JBi = POi = AO, = B'0 3 ; AI = A'0,P= P'0,B'. We have from the figure, 2Ai + A, = A. (161) COMPOUKD CURVES. 133 In the diagram draw O^G parallel to OA cutting the tan- gent at K, and produce the arc HP to G, Draw the chords GH and GP, passing through A and A respectively. We have then a discussion similar to the preceding case, and to the problem 171, Fig. 51, whence we derive the general formulas : GK = (R, - M^ vers A ! = (M 9 - R) vers \ A (162) and A A = GK (cot i A i - cot i A ) (163) 1. Assuming Ri < R and R* > R f Tf 7? 7? vers Ai = ST^A = vers * A (164) 2. Assuming Ai < |A and Mi < R vers | A Ri vers A i vers i A -vers A a 3. Assuming A i < \ A and A A A A' (166) (167) (169) 4. Assuming Ri > R and AA GK = (Ri R) vers A The third assumption will usually secure most readily the ! desired curve. AA' should be assumed as small as the nature i of the case will allow, and A i should not be much smaller jthaniA. It is evidently not necessary that the new curve should be i symmetrical ; for having laid out the curve APH, the simple curve HB may then be used, or, if desirable, some compound Curve HP'S' determined by an assumed value of BB' not equal to A A'. 134 FIELD These formulae (154) to (169) are readily adapted to the case of substituting a compound for a simple curve when it is necessary to keep one tangent point fixed, but to move the other a certain distance in either direction on the tangent. For if in Figs. 55, 56, we draw a tangent at //, and make 11 the fixed point of tangent, it is evident that the central angle of the curve will then be AOH. The only change necessary, therefore, to adopt the formulae to this case is to write A in place of A, and to observe, instead of eqs. (154) (161), that Example. Fig. 55. Let R = 1910.08 A Assume AA = 260. A Eq. (166) A A' = 260. cotiAi 2.90421 cotA 2.60509 A 2 = A. = 84 38 19" 21 A 2 8 log 2.414973 -.29912 log GK Eq. (167) i A R Eq. (168) GK 3384.07 1910.08 5294.15 42 38 9.475846 2.939127 vers 9.409688 3.529439 log 2.939127 " vers 9.326314 4100.27 R l 1193.88 A'P 791.67 3.612813 Pff=369.23 177. Given, two curves joined by a common tangent to replace the tangent by a curve compounded with the given curves. Fig. 57. Let HI = BOi the radius of one curve, " R 3 = A0 3 the radius of the other curve, > Ri, " I = BA the common tangent, " J? 2 = POi = P O y the radius of connecting curve. " A 2 = P0 2 P' the central angle of " " " a= A0 3 P and ft = BO.P. " i = AOiOt. COMPOUND CURVES. 135 In the diagram join Oi0 3 and draw OiG parallel to BA. Then in the right-angled triangle OiG0 3 we have, cot * = 0i0, = O0 3 GO, cos i I I sin (170) (171) which gives the distance between the centres of the given curves. FIG. 57. We shall now assume the following geometrical truths, which may be easily demonstrated. If two circles intersect in one point, they intersect in two points; and the line joining the two points is the common chord. The common chord is perpendicular to the line joining the centres, and when produced it bisects the common tangents. If a third circle is drawn touching the two circles, a tangent to the third circle, parallel to the common tangent, will have its tangent point on the common chord produced. Conversely, therefore, if the tangent BA be bisected at K, and a line, KI, drawn perpendicular to Oi 3 , KI will coincide with the common chord produced, and the angle IKA A0 3 0i = i. If on KI we assume a point / through which it is desirable that the connecting curve should pass, then / is the tangent point of a tangent parallel to BA; consequently a line through / perpendicular to BA contains the required centre 0,. 136 FIELD ENGINEERING. I. Let p HI = the perpendicular distance between the tangents. If in the diagram we join IA and IB, and produce the chords to intersect the given curves in Pand P', then Pand P' are the points of compound curvature; and the lines POi and P 3 produced will intersect /0 2 in the same point 2 ; and the angles P'OJ = a and P0 a / = /3. In the triangle AIB the line KI bisects the base AB, and we have by Geom. Tab. I. 25. AP- + P/ 2 = 2AK* + ZKI* By eq. (56) AI = 2(P 2 - P 3 ) sin la BI = 2(P a - Pi) sin 1/5 AK = U and KI = P sin a .-. 4(P a - P 3 ) 2 sin 2 la -f 4(P, - Pi) 2 sin 2 1 ft = I? + ^-. Dividing by 2 and putting vers a = 2 sin 2 la and vers ft = 2 sin 2 1/3 (Tab. II. 46) (P 2 - P 3 ) 2 vers a -\- (P 8 - Pi) 2 vers ft - JP + ^--. But by eq (57) (P 3 - P 3 ) vers a - (P 2 - P t ) vers fi>= p (172) . . p (2P 9 - (P 3 + &))= V + 5^7 (173) From (172) verg a = p p p ; vers /? = P p (174) Pa PS JVa "! and from the figure A, = a + /3 (175) These formulae solve the problem when p is assumed. If desirable we may find a and /5 independently of P a , for in COMPOUND CURVES. 137 the triangle AIB, IAB = ia and ISA = \fa and since HK = p cot i. rot iff - ^L - **-t-^ _ ' _i_ cot i (177) " HI ' p %P II. In case a or ft is assumed, we have from the last equa- tion 7 I 2(cot i<* + cot i) 2(cot i/J - cot i) (178) III. Ift case the radius R* is assumed, then in the triangle i02#3 we know all three sides; for 0i0a = (R* &) 3 = (5. - ,), and 0,0 3 = ' By Trig. (Table II. 31.) - 0i0 2 )(s - a 8 ) vers A 2 = X in which s = $ sum of the three sides. Substituting values, and reducing, observing that, and that CR 3 JRi) tan i = I, we have li vers A, = ^Tp -- W\Tn - W\ 2(2f a -ft) (^ta J^s) In the same triangle. sin 0, 0j 02 = sin A , %% = sin (i - /?) (179)' C/iC/3 for from the figure 3 0i0 2 i ft, and taking the value of 0, 0s from eq. (171). 138 FIELD ENGINEERING. (180) We then find a from eq. (175) and p from (172). The angles a and (3 may be found otherwise, for by Trig (Tab. II. 27) we have in the triangle OiO,O s sin KOiOaOa - 0,0,0,) = --~-~-cos iA 2 or /o / ' i a ~~ AV (-^3 Mi ) cos i cos | A 2 sin 190 (4 -j s )l = p~n~7? . . cos I* -j g-^- 1 = cos * . cos \ A 2 (181) which is a convenient formula when i and A 2 are not too small. Having obtained 5-^-, we have (182) For a constant value of I the less the difference of Jf? 3 ^ t the greater will be the value of the angle i. When M 3 = MI, cot i = and i = 90 and the tangent point I will be on a per- pendicular to BA drawn through the middle point K; and a = ft. On the contrary, as (JR 3 Mi) increases, i becomes less, and the foot, H, of the perpendicular HI moves toward B, the tangent point of the curve of smaller radius Mi. The distance HK = p cot i. The connecting curve is farthest from the tangent BA at /. To find the ordiiiate from BA to the curve at any other point, subtract from p the tangent offset for the length of curve from I to the ordinate in ques- tion. 115, eq. (39) may be used on flat curves with tolera- ble accuracy, even when the distance equals several hundred feet. IV. It is evident that in this problem E* must be greater than either Ri or R*. As the centre 2 is taken nearer the COMPOUND CURVES. 139 line Oi 3 , 7? 2 grows less, and is a minimum when 2 falls on the line 0i0 3 . In this case we have A 2 = 180, and fit = K^ -f R, -1-0! S ); a minimum. (183) This limit must be regarded in assuming the value of _K 2 . Since 0!0 2 - 2 3 = (fa - #,) - (# 2 - E 3 ) = (#3 - -Ri) a constant value, independent of R z , we infer that the centre 0-ji is always on a hyperbola of which Oi and 3 are the foci; (R 3 Hi) equals the diameter on the axis joining the foci; and I equals the diameter at right angles to it, for in the tri- angle OiGOs, I* = 0707 - (R 3 -Erf (184) Example. Fig. 57. Given : Ri = 1432.69 1 ? 3 = 1910.08 and I = 400. Assume P 11.4 to find R 2 , a and P' Eq. (170) 8 - R \ 477.39 400. log 2.678873 " 2.602060 .'. i 39 57' 34" log cot 0.076813 Eq. (173) i i 39 57' 34" 39 57' 34" " sin 9.807701 " sin 2 9.615402 P 11.4 log 1.056905 * 27.64 " 1.441503 ^l 2 " 4.602060 P " 1.056905 * 3508.77 " 3.545155 Us -f R l 3342.77 2) 6879.18 R* 3439.59 (say) 3437.87 Eq. (174) p 11.4 " 1.056905 R* - R 3 1527.79 " 3.184064 a 7 00' log vers 7^872841 p 11.4 log 1.056905 JS a - R l 2005.18 ' 3.302153 ft (nearly) 6 07' log vers 7. 754752 A 2 13 07' 140 FIELD ENGINEERING. Example. Fig 57. Given: 7? t 1432.69, R 3 = 1910.08, and I = 400. Assume R* = 3437.87, to find A a , ft, a and;?. Eq. (179) 2. log 0.301030 Ri - R, 2005.18 " 3.302153 R* - R* 1527.79 " 8.184064 A a Eq.(170)5,-5 Eq. (180) i t A 2 - R 9 Z -/? ytf Eq. (175) a Eq. (172) R* -.ft, 477.39 400. 1527.79 400. 13 07' 22" 39 57' 34" 39 57' 34" 13 07' 22" 33 50' 39" 6 06' 55" 7 00' 27" 7 00' 27" " 6.787247 5.204120 3og vers 8.416873 log 2.678873 " 2.602060 11.41 log cot 0.076813 log sin 9.807701 " " 9.356099 log 3.184064 log sin 2.347864 log 2. 602060 log sin 9.745804 log 3. 184064 log vers 7.873309 1.057373 178. Given: a three-centred compound curve to replace the middle arc by an arc of differed radius. I. When the radius of the middle arc is the greatest. Fig. 57. First find the length and direction of the common tangent AB. Let A a = central angle of the middle arc, J? 2 = its radius, and RI and R 3 the radii of the other arcs. From eq. (179). I = V2Cff a - .BO (# ~ ^3) vers A, (1 85) Then find t by eq. (170), a and ft by eqs. (179)' (175) and p by eq. (172). For the new arc we may now assume a new value for p, 'or for R A 3 = 2*, a minimum, and the long chord PP' is perpendicular to 0,0 3 . When jR 2 is greater than this, a is greater than ft, and vice versa. What- ever be the value of R 2> the long chord PP' always cuts the line Oi0 3 produced in the same point 8, at a distance from Zof Z8 R, vers *; or from 0, of OiS = Ri cos i. This item will be found useful in solving the problem graphically. Example. Let R, = 781.84 D, = 7 20' " Ri - 1375.40 D 3 = 4 10' A a = 48 " R 3 = 1910.08 D 3 = 3 00' Let pp' = 11.30 Eq. (194) 2 log 0.301030 R 3 - Ri 584.68 " 2.728094 P 2 - R, 593.56 " 2.773465 A 2 48 log vers 9.519657 2) 5T2~2246 I 458.27 log 2. 661123 (192) R 3 - R, 1128.24 " 3.052402 23 57' 55" log sin 9. 608721 (193) i 23 57' 55" log cos 9. 960847 R 3 - Ri log 3.052402 0,0, 1030.98 log* 3. 013249 (195) R z - R, log 2.773465 A a 48 log sin 9.871073 log * 2^644538 a 25 19' 52" log sin 9T631289 (196) ft 22 40' 08" (203) 0,0, log 3.013249 (200) q 97.26 1.987934 ^- 1.025315 p-p' 11.30 log 1.053078 ' - R, 119.78 " 2.078393 R*' 149.118 (say) 1494 95 for 3 50' curve. 146 FIELD , (201 ) R 3 - RS - ihas been prop- erly taken. The distance DK through which the point moves is called the throw of the switch. It varies on different roads from 4| to 6 inches, but is usually made about 5 inches, or 0.42 feet. A turnout should be a simple curve from the heel of the switch to the point of the frog. 18O. Given: a main track, straight, and a frog angle F, to determine the distance BF, on the main track from the heel of switch to point of frog, the radius, r, of Hie centre line of the turn- out, the length of chord af, and the proper length of switch AD. Fig. 61. FIG. 61. Let C be the centre of the turnout. " F - the frog angle, HFJ = FOB. " g the gauge of track AB. ' r = radius, aO = fC. the throw of switch. Then the radius of the gauge side of the outer rail is (r -}- and we have TURNOUTS. 149 = FO. versFCB or, 9 = (r + ty} vers F whence The angle and #F = ^.5 cot AFB = g . cot %F (208) Again, in the triangle FOB BF = FG . sin FOB = (r -f &) sin F (209) The chord of is evidently af = 2r sin $F (210) Similar to eq. (207), we have DK DK But since the inside rail has the same throw, while its radius is (r $g), we may, if convenient, drop the \g, and hence the length of switch is AD -T . sin ACD (211) The degree of curve corresponding to r is found from Table IV., or by eq. (17), and the centre line of the turnout may be located by transit deflections from the tangent point a, using chords of 20 or 25 feet + the correction found in 106, 107; or the deflection for a 20-foot chord may be calculated at once by 181. Simple as these formulae are, they may be rendered still more convenient by introducing the number of the frog, n. By eq. (206) we have cot \F = 2n, which substi- tuted in eq. (208) gives BF=2gn (213) Drawing the chord AF to the outer rail, AF = VAB* + JSF 2 = ff V 1 + 4n (214) 150 FIELD ENGINEERING. Make BA' = AS and join FA' ; then by similar triangles, AA'FmdAFO, AA : AF :: AF : FG whence or (r + ft) = \g (1 -f 4n*) (215) whence r = 2gn* = BF . n (216) The chord of to the arc of the centre line is to AFas r is to ( r _j_ %g)- t hence of = t ' , and substituting values from sqs. (214) (215) we have af= 2r (217) 4/1 -f n* Issuming that, for small angles, the tangent offsets vary as .he squares of their distances from the tangent point, which will lead to no material error in this case; AB : DK :: BF* : AD* Whence AD=I (218) or AD = y4n* g . DK = It is not necessary to determine the degree of curve in order to locate the turnout, for having fixed the position of BF, the position of af is found by laying off Ba, and F/, each equal to \g. Whatever be the length of the chord af, found by eq. (217) or (210), its middle ordinate is always ^g, and the orclin- nates at the quarter points, f . %g = T \#. 'Thus for the stan- dard gauge of 4.708 the middle ordinate is 1.177, and the side ordinates 0.883. By the preceding formulae Table XI. has been calculated, Which gives the required parts of a turnout for various frogs j When the gauge is 4 feet 8 inches and the throw 5 inches; j also for a gauge of 3 feet and throw of 4 inches. For any other throw, onfy AD must be calculated. For a different gauge the engineer will do well to construct a similar table, I adapted to the frogs used on the road. TURNOUTS. 151 In the table the frog angle is given to seconds, in order that the results may agree, whether found by equations in 180 or 181; but in practice the nearest minute is sufficiently exact. The frogs most used for single turnouts are those from .No. 7 to No. 9, inclusive. 182. In case of a double turn out from the same switch three frogs are required, as at F,F' and F\ Fig. 62., and the switch is called a three-throw switch, because its point takes three positions. The frogs F and F' are usually alike, and placed exactly opposite each other in the main track. The other frog F" is placed on the centre line of the main track. Its angle F" and its distance from a are now to be determined in terms of F. A.a In the figure we have vers F"Ca = -^7771 or If vers The distance also 2(r+ ig) aF" = (r -f- iff) sin aF" = r . tan IF" (219) (220) (221) All the parts of the turnout required to locate the frogs F and F" are calculated by the formulae in the preceding sec- tions, or are taken from Table XI. If we let n" = the number of the frog F", then by eq.(206) tan $F" = -^-77-, which substituted in eq. (221) gives aF" = (222) 152 FIELD ENGINEERING. Also, in the triangle aF"C, aF" = V(r -f #) - r> = Equating these and replacing r by 2#/i 2 , we obtain n" = If we neglect the , we have (approx.) n" = = . 707172. (223) (224) (225) Example. It F = F' = 6 44', or n = n' = 8.5, then n" =* 6.0 + or F" = 9 32'. 183. In case no frog is at hand of the angle or number given by eq. (219) or (225), we may select one as nearly like it as pos- sible, and locate the turnout as a compound curve, pro- vided that F" is less than 2F. Fig. 63. Let r" = C"a, and r = r' = Cf Of Then analogous to the equations of 180, ' vers iF" .'. r" = aF" = (r" exsect./' sin IF" = r" (227) (228) TURNOUTS. 153 The length of the switch, by eq. (218), is AD= The curvature of the rail between the frogs F" and F is F"CF = (F- $F"). Draw the chord ^".Fand the perpendicular F"L; then the angle LFF" = F- %(F - F") = i(F -f $F"); and since LF" = .-. F"F= -- =ig. cot i(F + $F") (230) (331) Example. Let F = 6 44' J7"' = 10 24' Eq. (226) & 2.354 log 0.371806 %F" 5 12' log exs 7.616224 r" 569.616 2/755582 Eq. (228) iF" 5 12' log tan 8.959075 aF" 51.839 1.714657 Eq. (229) \g 2.354 log 0.371806 ") 5 58' log sin 9.016824 F"F 22.645 1.354982 Eq. (231) $(F- iF") 46' log sin 8.126471 %T + iff) 1692.432 3^228511 r 843.862 When n" > . 707^, r will be less than r". Should F ' not equal F, (F" being given), then r' and Z'^'must be calculated also, by substituting F' for Fin eqs. (230) and (231). 184. Fi*om the same switch in a straight track it is required to lay two turnouts 011 the same side. Fig. 64. If we assume F' = F, and that these two frogs shall be opposite each other, we calculate all the distances of the first turnout for the angle F (or number n) by 180, 181, whence we have the radius r = Ca, 154 FIELD ENGINEERING. Let r' = C'a, the radius of the centre line of the second turnout. The angle ACF = F, and since F' = F, the angle OF'C' = F, and the triangle CF'C' is isosceles, and G'F' = O'O. But G'F' = tf'J. = or (232) (233) c' FIG. 64. To calculate the remaining frog at F", we have from eq. (207) or from eq. (216) n" - V Jl ^) gin ^ = of" = 2r' sin and since AC'F' = 2F, af - 2r' sin F (235) (236) (337) (238) The length of switch may be calculated by either r or r', since for r', which is about \r, the throw of switch is double, thus giving practically identical results. If we compare the values of F" as obtained by eqs. (234) and (219), we shall find them almost identical for given values TUll^OUTS. 155 of F and g ; and since this may also be proved analytically by assuming that vers ^F" = vers F" ', which is very nearly true for ordinary values of F" , we conclude that a set of frogs (F = F' , and F"} which is adapted to a double turnout in opposite directions from a straight line (as in Fig. 62) is also adapted to a double turnout on one side (as in Fig. 64), the curves being simple curves in every case. But this being true, the set is also adapted to a double turnout in opposite directions from any curved track the radius of which is not less than r as given for F, since any such case is intermediate between the two cases named. When, therefore, a certain frog, F, is adopted for general use on any road, another frog should also be adopted, whose angle, F ", is determined by eq. (219), or whose number n is determined by eq. (225). Thus, if F= 6 44', or n = 8, then F" should be 9 32', or n" = 6. 185. In case no frog is at hand of the angle or number given by eqs. (234) (235), we may select one as near the same angle as possible, and, calling this F", calculate the distance BF" and the radius G"F" (Fig. 65) as for a single turnout; 180. C"K FIG. 65. Then assuming any other frog F', whether equal to .For not, it is required to find the chord F"F', and the radius C'F' of the arc F"F' . The point F' may fall either s : de of the radius CF, according to the values given to F" and .F'. a. In case F' falls beyond the radius CF, we will assume first, that the entire rail from B to F' is laid with the same- radius BC, and centre C. (This investigation also applies to the case when F' falls between B and the line CF.} In the diagram (Fig. 65) draw CF". We then have 156 FIELD ENGI^EEKLSTG. = f: = ^ (238 ) and GF" = (r - $g) exsec BCF" (240) In the triangle F"GF'. F"C- F'C :F"G + F'V:: tan i(F"F'C- F'F"C) :cotF"CF' Now, since C'F'C = F', and BC"F" = F", .'. F"F'C= F"F'G" + F' and F'F"C = F'F"C" - C"F"C= F"F'C' (F" BGF"\i Letting U = C"F"C = (F" - BCF") and subtracting, we have F"F'C - F'F"G = F' + U Hence the above proportion may be written GF" : 2BC + GF" :: tan $(F' + U) : cot iF"CF' whence cot tF"CF' = * SC + F ? F " tan W +U) (241) (Since BCF" + F"GF' = BCF', and we know the radius BC, the chord or arc 'BF' is easily obtained, which fixes the position of the frog F'; and the problem may end here, frequently, in practice.) Now in the same triangle F"CF', the half sum of F" F'C and F'F"C is 90 %F"CF'; while, as we have just seen, the half difference is $(F' -j- U); and by subtracting we have the less, or F'F'O = 90 - #F' + V+ F"CF') (243) *r Now _ BC.BmF"CF' F * -' TUKtfOUTS. 157 To find the angle F"C'F'; produce the lino F'C' in the dia- gram to intersect the line BC at K. Then the two triangles KC"C' and KCF' have the angle K common, and the sum of the other angles will be equal; that is, KO"G' + C"C'K = KCF' -f CF'K or F" -f- F"C'F' = BCF 1 -J- F' and since BCF' = BCF" + F"CF' . p" C ' F ' = F " C F' + F' - U (244) If we denote the radius F'C' by r' + iff < 245) Example. Given: the three frogs F = 6 43' 59", F' = 6 01' 32", and F' = 8 47' 51" to lay a double turnout on one side of a straight track. Fig. 65. By Tab. XI. BF = 80.036 r = 680.306 AD = 23.82 BF" = 61.204 r" = 397.826 Eq. (239) BF" 61.204 log 1.786779 (r - %g) 677.952 " 2.831199 BCF" 5 09' 38" log tan 8.955580 Eq. (240) BCF" 5 09' 38" log exsec 7.609587 (r - M) 677.952 . log 2.831199 GF" 2.760 " 0.440786 Eq. (241) (2J3(7-f OF") 1358.664 " 3.133112 (CT=338'13") 2.692326 1(F" + U) 4 49' 52".5 log tan 8.! Eq. (243) Eq. (245) l(F "C\ p) 1 22' 35" " cot 1 .619294 F"CF' 2 45' 10" " sin 8 .681481 r iff 677, 952 2 .831199 1 J512680 -F it sy ?') 6 12' 27". 5 " cos 9 .997446 F" F' 32.752 1 .515234 if "C 1 F' 2 34' 14".5 " sin 8.651781 Kr' + lg) 730.219 2.863453 r' 362.755 158 FIELD ENGINEERING. b. We assume, secondly, that the middle track is straight beyond F, and tangent to the curve at F. Fig. 66. Then whenever the value of F" is less than that given by eq. (234), the arc AF", produced with the same radius AC', will intersect the straight rail HF' at some point F 1 , and the frog angles F and F' will be equal. FIG. 66. For the straight rail HF' produced backwards, passes through the point A, making an angle F with the main track, since the triangles CBF and CHA are equal, and AH >= BF. Now any circle, tangent to the main rail at A, will intersect the line AH in some point F', and since AF' is the chord of the arc, the angle at F' equals the angle at A, which is F. Hence F = F'- and the angle AC"F' = 2F, The length of the chord AF' is AF' = ZAC" sin F (246) The chord F"F' = 2F"C" sin XF'O'F 1 ) = 2 A G" sin 1(2 F - F") Hence, F'F 1 = 2(r" -f tg) sinC^- ^F") (247) Example. Let F' = F = 6 43' 59" and F" = 8 47' 51" By Table XL r" = 397.826 Eq.(247)2(r" + ^) = 800.360 2 20' 03". 5 F"F' 32.60 log 2.903285 log sin 8.609915 1.513200 If the frog F' is required to be different from F, then the inside curve must be compounded at F", giving other values to the length and radius of the arc F"F'. TURNOUTS. 159 c. We assume, thirdly, that the curve of tne middle track is reversed at F. Fig. 67. In the diagram, let Q be the centre of the reversed portion, and F' the proper position of the frog F', and C' the centre of the required arc F 'F'. Then Q is on the radial line CF, produced, and C' is on the radial line F"C" produced. Join FQ and F'Q, and produce C"F" to intersect these lines in L and M respectively. Also join F"Q, and denote the angle LF 'Q by tfand the angle F'QF" by Q. C /C" B A FIG. 67. In the triangle FF"Q we know F"F BF BF", and the side FQ is given; and the included angle F"FQ = 90 -j- F. Hence we may calculate (Tab. II. 25) the angle F"QF&nd the side F"Q. The triangle CC"L gives the angle at L = F" F; and the triangle F"LQ gives LF"Q = L - F"QF .-. U= F" - F - F"QF (248) In the triangle F'QF" we have tan "Q - F"F'Q) But F'F"Q = F'F"L -f CTand F"F'Q = F ''F'N - F', and since F"F'N = F'F"L, we have by subtraction, F'F"Q - F"F'Q = U+F' Xlence cot = f^^*W+-X'i (249) 160 FIELD ENGINEERING. (Now the angle FQF = Q F"QF, and is subtended by the chord HF', which is therefore easily found, and serves to locate the frog F', and frequently this is all that will be required.) In the triangle F"QF', the half sum of QF"F' and QF'F". is 90 iQ, while, as we have just seen, the half difference is F'); hence by adding, we have the greater, or QF"F' = 90+K^+^' ~ Q) (350) The triangle G'F'M gives F"C'F' = P' - M, while the triangle F"MQ gives M = U + Q; lience F"C'F' = F' - (U--\- Q); and denoting the radius G 'F' by r' -f- $g (251) . Let F = F' = 6 43' 59", F" = 8 47' 51", and FQ = 953.012. Then by Tab. XL, BF = 80.036 and BF " = 61.204 ; hence F"F = 18,882; and the included angle is 96 43' 59". Solving the triangle FF"Q we find F"QF.= 1 07' 18", FF"Q = 82 08' 43", and F"Q = 955.402. Now F'Q = ^4-^ = 957.720. (249) F"Q + F"Q 1913.122 log 3.281743 F'Q-F"Q 2.318 " 0.365113 F 1 ) 3 50' 16".5 log tan 8.826231 1 02' 08".4 " cot 1.742861 $50) Q 2 04' 16".8 " sin 8.558033 F 1 - ) 2 48' 08". 1 " cos 9. 999480 F'Q 957.720 log 2.981239 F"F' 34.633 1.539892 (251) K*" - U- Q) 1 51' 34".l log sin 8.511191 2(/ + &0 1068.82 r' 531.81 TURNOUTS. 161 186. Given: a main track, curved, and a frog-angle F, to locate a turnout on the inside of the curve. Fig. 68. Let R = Oa = radius of main track. " r = Ca radius of turnout. " F = GFO = the frog angle. In the diagram draw the chord AF and produce it to inter- sect the outer rail at G; and draw FO and GO. Since the 'chords AF and AG coincide, and the radii AC and AO coincide, the chords subtend equal angles at C and respec- tively, and GO is parallel to FC. (See 137.) Hence, FOG = CFO = F. Let = the angle FOA. In the triangle FOA, = GFO - FAO = GFO - FGO; and in the triangle GFO, GO+FO : GO FO :: tan $(GFO + FGO) : tan \(GFO - FGO}, or 2R : g :: cot \F : tan |6 .'. tan |9 = cot \W = In the triangle CFO, In the triangle EOF, BF =r 2(fi - \g) sin |fl (254) In the triangle aCf, af=2r sin ^JF-j- 6) (255) 162 FIELD The length of switch AD, for a given throw DK, may be found thus : from Table IV. take the tangent offsets, t and f, corresponding to E and r respectively, and assuming that the offsets may vary as the squares of their distances from the tangent point, we have t - t' : DK:: (100) 2 : AD* AD = (256) t - t' This result is practically the same as that found for length of switch in a turnout from a straight line with the same frog, when R is large. Example. Let R - 1432.69 and F = 6 43' 59". Eq. (252) ig 2.354 log 0.371806 $F 3 21' 59". 5 log cot 1.230440 R (Tab. IV.) log 1.602246 " 3.156151 1 35' 59". 8 log tan 8.446095 Eq. (254) 6 F+e 3 11' 59". 6 9 55' 58".6 sin 8. 746786 " 9.236778 (254) (255) R-ig 1430.336 r + lg 462.856 r 460.502 2 R \g) 1430.336 P BF 79.872 921.004 1 35' 59".8 4 57' 59". 3 of 79.734 9.510008 3.155438 2.665446 log 0.301030 " 3.155438 log sin 8.445924 log 002392 " 2.964262 log sin 8.937381 log 1.901643 The values of BF and of sue found to be so nearly identical in this case with those determined in case of a turnout from a straight line, that the values given in Table XI may be used at once for ordinary values of JR; and the degree of curve of the turnout in this problem is approximately the sum of the degree of curve of the main track and the degree of curve given in Table XI. opposite F. Thus, in the example 4 + 8 26' =? 13 26' .-. r = 401. 7 nearly. TURNOUTS. 163 187. Given: a main track, curved, and a frog-angle F, to locate a turnout on the outside of the curve. Fig. 69. In the diagram draw the chord AF, and produce it to meet the inner rail at G; and draw FO and GO. The triangles CAP and OAG are both isosceles, and have the angles at A equal; hence they are similar, and FCA = AOG. Hence FOG = HFO = F. Let M = Oa, r = Ca, and = FIG. 69. In the triangle FOA, Q b= OAG - AFO = FGO - GFO; and in the triangle FOG; FO + GO : FO - GO :: tan k(FGO + GFO) : tan i(FGO - GFO), or 2R : g :: cot $F : tan |9 which is identical with (252). In the triangle CFO sm (ft V) In the triangle EOF, BF = 2(E + ig) sin 46 In the triangle aCf, of 2r . sin 1(F 6) For a given throw, the length of switch will be AD _ , /10000 DK (257) (258) (259) (260) (261) 164 FIELD ENGINEERING. in which t and t' are the tangent offsets (Tab. IV.) corre- sponding to R and r. In this problem, as in the preceding, we may, for ordinary values of R, assume the values for BF and of given in Tab. XI. The degree of curve of this turnout is, approximately, d D, taking d from Tab. XL and D from Tab. IV. corresponding to R. Should D d, this turnout becomes a straight line; Fio. 70. and when D > d, or when R is less than r given in Tab. XI. , the centre falls on the same side as 0. Fig. 70. In this case, using the same notation, is given by eq. (257). Eq. (259) BF = 2(R + ig) sin |0 af=2rsmi(Q-F) (263) 188. A tongue-switch is a short, stiff switch which, when moved, revolves at the heel as on a pivot. When it is thrown over to the turnout track, it makes an abrupt angle with the main track, called the switch angle; but in this posi- tion it should be tangent to the turnout curve. The use of this switch is generally confined to yards and warehouses, where but little space can be afforded, and where the motion of the cars is always slow. 189. Gixen: a straight track, a frog-angle F, and tJie length and throw of a tongue-switch, to locate the turnout. Fig. 71. TURNOUTS. 165 Let AD be the length, and DK the throw of switch, and let S denote the switch-angle DAK. Then sin 8 = or 8 = 57.3 -- (264) (Compare 86.) Let C be the centre of the required turnout, and in the dia- gram draw CK and CF; also draw DG perpendicular to the straight track. Then DGF = F; and in the triangle KGCi KCF = KGF GKC, and since CKA is a right-angle, GKC - 8 .-. KGF= F- 8. Draw the chord KF, and since the triangle KCF is isosceles, the angle CFK = 90 - $(F - 8). Now, CFI = 90 - F; hence by subtraction, KFI - i(F+ 8). FIG. 71. If g denote the gauge, we know KI = g DK; and in the right-angled triangle KIF, we have IF=KI. cot i(F + 8) (265) KI KF = (266) IKF (267) These equations are analogous to eqs. (229) (230) (231). 19O. Given .' a double turnout with tongue- switch, from a straight track; to find the angle, F", of the middle frog. Assuming F' = F calculate (? -f ig) by the last equations. Since the rails of the turnouts intersect on the centre line of 166 FIELD ENGINEERING. the straight track, as in Fig 63; if we substitute the value of F" F 1 , eq. (229) in eq. (231), we have sn - and by Trig. Table II. cos F" -cos!? whence cos^F" = + iff) (268) If the angle of the middle frog to be used does not agree with F" found by the last equation, the turnout will be com- pounded at F". 191. Given : a straight track, the frog-angles F, F' and F", and the switch angle S, to locate a double turnout. Fig. 72. FIG. 72, Assuming that F" shall be placed on the centre line of the straight track, let h be a point on the centre line at the point of switch. Then liK = \g DK, and since the angle F" is bisected by the centre lino the necessary formulse in this case are obtained from 189 by simply replacing Fby \F" and KI by hK\ and in the first members /Fby hF" and r by r". This is obvious by the similarity of the figures. TURNOUTS. Hence KF" = = hK . cot hK ' + 8} sin -f- 8) 167 (269) (270) an Hi*"- 8) (271) The location of the remaining frogs is a problem already discussed, 183, eq. (229), etc. 192. Given: a straight track, the frog angles F, F', F", and the switch angle S, to locate a double turnout on one side. Fig. 73. FIG. 73. The frog ^is located by 189; but for the frog F" we have evidently a double throw; hence eqs. (265) (266) (267) become IF" = (g - 2DK) cot i(F" + 28) (272) (273) , sin KJ--W) P"> To locate the remaining frog F' : when F 1 falls beyond the line CF, there are three cases. a. The middle track reversed beyond F. We find the distance F"Fby subtracting IF", eq. (272) fron? IF, eq. (265) : after which the solution is identical with that given 185, c., Fig. 67. 168 FIELD ENGItfEEKIKG. b. The middle track compounded at F. Let Q be the centre of the curve beyond F, and also let Q the angle F'QF"; and let U ' = the angle C"F"Q. Then by a course of reasoning analogous to that of case a, we derive U= F" F+ F"QF (275) cot *Q = F-QFIQ tan Now since the radius F'Q is given, and the angle FQF' Q FQF", we readily determine the distance HF', and so locate the frog F'. In the triangle F"QF', the half sum of QF"F' and QF'F" is 90 IQ, while the half difference is K#"+ #"); hence by subtraction we have the less, or F'F"Q = 90 - $(U+ F' -f Hence ** = *" cos F , + Q) (277) Join C"$, and the quadrilateral C'QF'F" gives J" + Q= U+F"C'F' hence F H G'F' = F' U-\-.Q; and denoting the radius C/'- by ^' + 1^, we have Cor. Since the centre Q is assumed at pleasure, it may be made to coincide with the centre C, and then the compound curve becomes a simple curve. Then also, the above formulae will apply when F' is such that the frog will come on the arc IK But as FQF" will be greater than Q, the difference FQF' will be negative, indicating that the distance HF' is to be laid off backwards from H. c. The middle track straight Ibeyoiid F, and tan- gent to the curve at F. Fig. 74. Let F' be the required position of the frog F' . A tangent to the curve at F' makes an angle (F 1 -\- F) with the main track, and a tangent at F" makes an angle of F" with the same ; hence the angle they make with each other is TUBKOUTS. 169 F"), and this is the curvature of the arc F"F\ and equals the angle F"C'F'. Produce the straight line F'H backwards to G, and draw F"G- perpendicular to it. Then F"G = FH F"F. sin F, or = - F"F.sinF (279) FIG. 74. In the right-angled triangle F'GF", the angle F"F'G = - F") = i(F' -f F" - F). (280) F'F' = v sin i(F' + F" - F) and GF' = ^ 7ff ^ 7 ' . cos \(F' + F" - F) (281) Observe that GF' cannot be less than GH F"F.cos, F. 193. Given : a turnout with a frog angle F, and the perpen. dieular distance p between the centre lines of the main and side tracks ; to find the radius r of the curve connecting the turnout with the side track. Fig. 75 170 FIELD Let the reversing point be taken at F, and let Q on CF pro- duced be the centre of the required curve, and draw QM per- pendicular to the main track. Then QM QF= r -%g; the point M is the point of tangent, and the angle FQM = F. Now -ZV being the intersection of the rail .RZ^with the radius QM, we have MN= QFveis F, but MN = p g; hence The distance FN\& evidently FN=(r-ig)smF (283) and the chord to the centre line is fm = 2r sin $F (284) Should the distance FN consume too much of the track, it may be lessened by introducing a short tangent at F, denoted by k; then by eq. (48) the radius will be shortened by an amount equal to k . cot %F, and the distance FN will be shortened by k. Since the tangent k reduces the length of the tangent offset of the entire curve by k . sin F, we have for the new radius r' When r' is fixed by a limit, we obtain k by resolving eq. (285) a (r f 4r/7) vers F _ AJ ; -- ;pi sin F In case the main track is but slightly curved, we may at first assume it to be straight, and find r as above, eq. (282), and the degree of curve corresponding to r; but this degree of curve must then be increased or diminished by the degree of curve of the main track, according as the track is concave or convex toward Q. 194. Given : the perpendicular distance p between the centre lines of a curved main track and a parallel side track, and the frog angle Fofa turnout; to find the radius r of the connecting curve, and the length FN, or fm, of tlie curve. Fig. 76. TURNOUTS. 171 Let FN be the rail of the main track, and GM the rail of the siding, adjacent to each other; let be the centre of the main track, and Q the centre of the connecting curve. Then the connecting curve will terminate at in, on the line OQ pro- duced. In the diagram draw MF, and produce it to intersect the rail MG at G, and join GO, FO, and FQ. Let H = radius of centre line of the main track; r = radius of centre line of the connecting curve; and B = the angle FOM. Case a. The siding outside the main track. Fig. 76. FIG. 76. By similarity of the triangles GO M and FQM, GO is paral- lel to FQ, and the angle GOF F; and by a process similar to that of 186, we have tan O = cot sinQ " sin (F+ 6) FN =2(R + i#) sin \B fm = 2r . sin (287) (288) (289) (290) Case 1>. The siding inside the main track. Fig. 77. By a process entirely similar to 187, we have (291) 172 FIELD ENGINEERING. W = 2(5-40 sin fm = 2r sin %(F 6) (292) (293) (294) ' When 6 = Fin the last equations, sin (F - 6) = 0, and r - \g is infinite, and the curve FM becomes a straight line. FIG. 77. When > F, sin (F 6) is negative, and the centre Q falls on the same side of the track as 0, and we have /m = 2r . sin |(0 F) Equations (291) and (293) remain unchanged. (296) 195. To locate a crossing between parallel tracks. Fig. 78. When a turnout from one track enters a parallel track by means of another frog and switch, the whole is called a cross- ing. The frogs are alike, and the calculation for one end of the crossing answers for the other. 180, 181. We have only to tind the length of track between the two frogs. In the diagram let AF be one turnout, and A'F' the other, connected by the straight track F'@. It is required to deter- mine the length F'G, or the distance FN measured on the main track from F to a perpendicular through F'. Produc- ing the line F'G to intersect the rail NF&t H, we have two TURNOUTS. 173 right-angled triangles GFH and F'NH, having the common angle at H = F. Let p = the perpendicular distance between centre lines of main tracks, and g = gauge. Then GF = g, = (p-g.) F'G = F'H - GH= 7 smF - GFcot F (297) So FN - NH - FH = (pg) cot F - (298) Wlien the main tracks are curved the distance F'G may be calculated by the same formula (297) which gives a value only a fraction too small, but in laying the track the rail F'G must be curved to a radius which is to jR of the main track as F'G : NF. 1 OO. When p is large, or the tracks are very wide apart, it will effect some saving of room to lay the crossing in the form of a reversed curve; and the frogs being alike, the two arcs will be equal, and the point of reversed curve P will be midway between F and F'. Fig. 79. In the diagram we have a Pa' the centre line of the cross- ing, and PL the centre line between tracks; aL = \p, and aC = a'C' r. The radius r having been found by 180 or 181, we have and versa OP = ~ r PL r sin a f!P (299) (300) 174 FIELD EtfGIXEERLN'G. The distance between frogs, FN, measured on the main track is evidently FN=2(PL-BF) (301) in which .RFis determined by eqs. (209), (213), or by Tab. XL 197. To lay a crossing in the form of a reversed curve, when the parallel tracks are oil a cui've. Fig. 80. FIG. 80. Let be the centre of the main curve, G and G' the centres of the reversed curve. Then in the triangle GOG' we know all three sides; for CO = R + r- CG' = r + r', and G'O = R -f- p - r' ; and the half sum of the three sides is * = R + r + $p. Denoting the angle COG by cp, we have (Trig. Tab. II. 31) " " (303) The angle q> determines the length of the arc BN described with the radius (R -\- %y} and so fixes the position of the point A' from A. By a formula similar to the above, TUBHOUTS. 175 The angle G'CO determines the length of the arc aP described with the radnis r; the angle (

(Q -f- 6'), which determines the distance between the frogs, measured on the main track. 198. To find the middle ordinate m, for 1 sta- tion, or 100 feet, on any curve, in terms of the degree of curve D. Referring to Fig. 4 we have in the right triangle AGH OH = GA . tan GAH But GA = \AE = \C, and (Tab. I. 18) GAH = AOB = A ; hence Jf = iC7. taniA (304) a general expression for the middle ordinate of any chord. If in this equation we make C = 100, A becomes D; and denoting the corresponding value of M by ra, we have m = ilOO tan D (305) whence the rule, Multiply the nat. tangent of the degree of curve by 100 and divide by 2. Thus the values of m in the 5th column of Tab. IV. have been calculated 199. To find the middle ordinate for any chord in terms of the chord and radius Referring to Fig. 4 we have GH = OE - OG = OE - VAO* - GA* M=R- j j$- - (306) 176 FIELD ENGINEERING. When C = 100 we have for the middle ordinate of one station m = R - VW - 2500 (307) For any subchord c, less than 100, we have for the middle ordinate, or __ 1- (308) By adding TJDF to the quantity under the radical in eq. (308) it becomes a perfect square, giving Wl = M near ty' < 309 ) which is a very useful formula, although approximate. The error in m\ does not exceed .002 for any subchord c when the radius is greater than 800. On a 20 curve the error will be .002 for a chord of 50 feet; and on a 40 curve the error in mi will be only .003 fora chord of 33 feet. Equation (309) io therefore practically correct in all cases for finding the middle ordinates of rails. Table XII. is calculated by eq. (308). 2OO. Curving Rails. Before any rail is spiked to its place in a curve, it must be evenly bent from end to end, so that it will assume the proper curvature when lying free. The bending may be done by using sledges, but is best accom- plished, especially for turnouts and other sharp curves, by using a bending machine made especially for this purpose. The proper curvature of a rail is tested by measuring its middle ordinate from a small cord stretched from end to end and touching the side of the rail-head. The cord should also be stretched from the middle point of the rail to either end, and the middle ordinate of each half length measured, to test the uniformity of curvature. From the last equation it appears that, with a given radius, the middle ordinate varies nearly as the square of the chord. TURNOUTS. 17? We may therefore find the middle ordinate ot a rail whose length is c by the proportion (100) 2 : c 2 :: in : m^ or, m l = nearly, (310) in which ra is obtained from Tab. IV., col. 5, for the given radius or degree of curve. Example. What is the middle ordinate of a 30 ft. rail tvhen curved for a 20 curve? When a long rail is bent for a sharp curve, observe that c is the length of the chord of the rail not of the rail itself. For the chord of half a rail the middle ordinate is one-fourth the middle ordinate of the whole rail. Thus, in the above ex- ample it would be .099 or 1 T 3 ^ inches. Instead of using the chord of the whole rail, it may be more convenient to assume a chord shorter than the rail, especially when the chord is not an exact number of feet, knotting the string to the length assumed, and applying it to different por- tions of the rail successively. 20 1. Elevation of the outer rail on curves. When a car passes around a curve, a centrifugal force is developed which presses the flanges of the wheels against the outer rail. This force acts horizontally, and varies as the square of the velocity, and inversely as the radius of the curve. Denoting the centrifugal force by/, we have from the theory of mechanics /= vr^rr o~~rJ m which w = weight of o. lob H loaded car in pounds, = velocity in feet per second, and R = radius of curve in feet. In Fig. 81, let ah represent a level line at right angles to the track, let a and c be the tops of rails on a curve, let be = e = elevation of outer rail c, and let the point d be the centre of gravity of the car. The force / acts in the direction ab, and if/' = the component of /in the direction ac, then /' :/ :: ab : ac. 178 FIELD ENGINEERING, The weight w, resting on the inclined plane at, developes a component in the direction ca, and denoting this by w', we have by similar triangles, w' : w :: be : etc. Since equilibrium requires that w' shall equal/', we have after dividing one proportion by the other = ^-, or /= ~j~> Equating this value of/ with that given above we find, ab.tf ~32.166.fl But db 4/^2 _ c 2 , and ac = distance between rail centres = gauge -f one rail head = g -f 0. 188. Also = - V, if V de- oOUU note the velocity in miles per hour. Making these substitu- tions and reducing, we have F 2 .06688 . 188) S (311) / / F 2 \ '* |/l+f.06688-gj By this formula Table XIII. is calculated for the standard gauge g = 4' 8T, =4.708. An approximate formula may be obtained by assuming that ah = g for practicable values of e. Substituting this in the first value of e given above, and replacing v by -ognn J 72 we have (approx.) e = .06688^- (312) which is the formula generally employed. TUKKOUTS. 179 In laying a new track, the transverse inclination is first given to the ballast by grade pegs driven either side of the centre line at a distance of (g + .188) each side of the centre; the outside peg being set higher, and the inside peg lower than the grade of ballast on the centre line, by the proper elevation selected from Table XIII. But in re-surfacing an old track, the inner rail is taken as grade and the outer rail is raised the necessary amount. 2O2. The proper elevation may be found mechan- ically by the following method: To find, on a curved track, the length of a chord whose middle ordinate shall equal the proper elevation of the outer rail for any velocity V ifi miles per hour. By the conditions of the problem, we have m l in eq. (309) equal to e in eq. (312), or . 06688 c = .73144 V^g (313) When# = 4.708, c = 1.587 V (314) Lay off the chord, c, upon the rail of the track, stretch a piece of twine between the points so found, and measure the middle ordinate; it will equal the proper elevation. 2O.3. The velocity assumed in the preceding formulae should be that of the fastest regular trains which will pass over the curve in question, since the flanges would be forced against the outer rail were there no centrifugal force devel- oped, by reason of the wheels being rigidly attached to the axles, and the axles being parallel. The rails on tangents should be level transversely, except near curves, where for 50 or 100 feet from the curve one rail is gradually raised, so that at the P. C. or P. T. it may have the full elevation due to the curve. At a P. C. G. the elevation should be an average of the elevations due to the two arcs. Owing to the difficulty of properly adjusting the elevation of rail, it is objectionable to have arcs of very dissimilar radii join each other; and the objection is much greater in the case of reversed curves unless separated by a short tangent. See 180 IELD On the other hand, a short tangent between arcs which curve in the same direction should be avoided, since it makes a "flat place" both in line and levels, at once unsightly and injurious to the rolling stock. In the case of turnouts, however, no elevation of rail is pos sible (except when both tracks curve in the same direction); hence reversed curves are allowable, the speed of trains being usually quite low also. 2O4. The coning of the wheels, by which the wheel on the outer rail gains a diameter enough larger than the other to compensate for the superior length of the outer rail, although a theoretically perfect device, is gradually going into disuse. To be effective for the sharpest curves, the coning must be so great as to produce an unsteady motion on tan- gents, very objectionable at high speeds. Moreover, it is un- desirable to seek for an equilibrium of lateral forces in a car on a curve, since the flanges are then sure to strike the inner and outer rails alternately with damaging' force, as that equi librium is momentarily disturbed. It is far better that the flange should press steadily against the outer rail, while that pressure is modified and reduced somewhat by the elevation of the rail. For these and other reasons, car- wheels are now made nearly cylindrical. LEVELLING. 181 CHAPTER VIII. LEVELLING. 205. The field operations with the Engineer's Level are of a more simple character than those performed with the transit, yet require equal skill and nicety of manipulation in order to produce trustworthy results. The transit is used to ascertain the relative horizontal position of points, the level to obtain their relative vertical position. 206. In order to express the elevation of points, they must be referred to some level surface of known (or assumed) eleva- tion; and in order that the elevations may all be positive up- ward, this surface of reference should be selected below all the points to be considered. The level surface of reference is called the datum. The elevation of the datum is always zero. The elevation of any point is its vertical height above the datum. Near the coast the sea level is usually adopted as the datum ; inland, the low water mark of a river or lake, etc. ; but it is not necessary that the datum should coincide with a water surface. If any points whose elevations are to be ascertained are below the water surface, the latter may be assumed to have an eleva- tion of 100 or 1000 feet instead of zero; that is, we remove the datum, in imagination, to 100 or 1000 feet below the level of the water surface. 207. In case of a survey commencing at a point quite re- mote from any important water surface, any 'permanent point may be selected as the original point of reference, and its ele- vation may be assumed at 100 or any other number of feet; that is, we fix the datum at the same number of feet below that point. The point of reference is called a bench, or bench- mark, and is designated by the initials B. M. Other benches are established at intervals during a survey, and their eleva- tions determined instrumentally. They are then convenient )82 FIELD ENGINEERING. points of known elevation for future reference. We cannot assume the elevation of more than one bench on the same sur- vey, else, we should have more than one datum, and all the results would be thrown into confusion. 208. Having established the first bench and recorded its elevation, the next^step is to set up the instrument firmly at a moderate distance from the bench, so that the telescope shall be somewhat higher than the bench, and in full view of a rod held vertically upon it. The instrument having been tested for its several adjustments, and found to be correct, the line of sight through the intersection of the cross-hairs is known to be hori- zontal when the bubble stands at the middle of its tube. Turn- ing the line of sight upon the rod, the point of the rod covered by the horizontal cross-hair is known to be on a level with the cross-hair ; and the latter is therefore higher than the bench by the distance intercepted on the rod from its lower end. Add- ing this distance to the elevation of the bench, we obtain the elevation of the cross-hair, known technically as the " Height of Instrument," and designated by the initials J7./. 209. The distance intercepted on a rod from its lower end by the line of sight, when the rod is held vertically on any given point, is called the reading- of the rod at that point. 2 1C. Having obtained the height of instrument, the eleva- tion of any point somewhat lower than the cross-hair is easily ascertained by taking a reading of the rod upon it. The read- ing subtracted from the height of instrument gives the eleva- tion of the point above the datum. The elevation of any num- ber of other points may be similarly obtained. But the eleva- tion of points on the ground higher than the cross-hair, or farther below it than the length of the rod, cannot be deter- mined, because in either case the line of sight will not cut the rod, and hence there can be no reading. In order to observe such points, the instrument must be removed to a new posi- tion, higher or lower than before, as the case may require. 211. Before the instrument is removed to a new position, a temporary bench, called a Turning Point (and designated by T.P.or "Peg"} must be established, and its elevation ascer- LEVELLING. 183 tained as for any other point, but with more care. A turning point must be a firm and definite point whose position cannot readily be altered in the least, nor lost sight of. A small stake firmly driven, or a point of rock projecting upward, is fre- quently used. The reading having been taken on the turning point, the instrument is carried forward to a new position, levelled up properly, and the new Height of Instrument ob- tained by a new reading on the same turning point. Since the cross-hair is higher than the point (otherwise there could be no reading) the reading, added to the elevation of the point, gives the Height of Instrument. 212. In general, the intersection of the cross-hairs being higher than any point on which a reading is taken: Tojitid the Height of Instrument, acid the reading on a point to the elevation of the point; and To find the Elevation of a point, subtract tfte reading on it from the Height of Instrument. A reading taken for the purpose of finding the Height of Instrument is called a Backsight (B.S). A reading taken for the purpose of finding the elevation of a turning-point (or of a bench used as such) is called a Foresight (F.S). Hence Backsights are always plus, and Foresights always minus. 213. The form of field-book used for the survey of a railroad, or other continuous line, is shown below. The first column contains the numbers of the stations on the line and of plus distances to other points on the line where readings are taken also the initials of benches and turning points, in order, as they occur. The second column contains the back- sights, taken on points of known elevation only. The third column contains the height of instrument, recorded on the same line as the elevation of the turning point (or bench) from which it is calculated. The fourth column contains the fore- sights, taken on new turning points, and benches used as such, only. The fifth column contains the readings taken on all other points noted in the first column. The sixth column con- tains the elevations of all points observed. The right-hand page is reserved for remarks, descriptive of the benches and their locationof objects crossed by the line, as roads, streams, swamps, ditches, etc. ; the depths of streams, etc. 184 FIELD LEVEL BOOK. Sta. B.S. H.I. F.S. Rod. Elev. Remarks. B.M. 4.683 204.683 200.000 White oak, 115 R. 2.1 202.6 1 3.4 201.3 + 50 5.2 199.5 Peg 1.791 197.260 9.214 3.7 195.469 193.6 + 25 7.0 190.3 Brook 5 wide ; 1 deep + 50 3.1 194.2 3 0.5 196.8 Peg 11.750 308.574 0.436 196.824 Peg 11.933 219.528 0.979 207.595 + 90 3.5 216.0 4 2.6 216.9 B.M. 2.075 217.453 Maple, 78 L. 5 1,7 217.8 6 0.9 218.6 Peg 9.005 227.801 0.732 218.796 7 6.2 221.6 1 39.162 | 11.361 i When a bench is not used as a turning point, the reading on it is recorded in the fifth column. The numbers in the second, fourth, and fifth columns come directly from the rod, those in the third are obtained by addition, those in the sixth by subtraction, according to the rule given above. The additions and subtractions made on each page should be proved before proceeding to the calcula- tions of the next. When correct, the difference of the sums of the backsights and foresights on the page equals the differ- ence of the first and last elevations on the page. Thus, in the form given (39.162 - 11.361) = (227.801 - 200.000) = 27.801 In this proof we ignore all elevations except those of turn- ing points, and benches used as such, and the height of instru- ment. At the end of the survey, as well as at the end of each day's work, a bench is established from which the survey may be resumed at any future time See 28, 29, and 80. 2 14. The object of making such a survey with level and rod is to furnish a profile or vertical section of the entire line, showing in detail the rise aad fall of the feurface over LEVELLING. 185 which it passes. The profile is plotted on profile-paper pub- lished for the purpose, the horizontal scale being usually 400 feet to an inch, and the vertical scale 30 feet to an inch. This distortion of scale magnifies the vertical measures so that slight changes in the elevation of the surface may be seen distinctly. 215. When only the difference of level of two extreme points is required, the survey is more simple. No readings are taken except on turning-points, the backsights and fore- sights being recorded in separate columns. No calculation is required until the survey is finished, when the first reading having been taken on one of the given points, and the last on the other the difference of the sums of the backsights and foresights is the difference in elevation of the two points, ac cording to the method of proof mentioned in 213. Thus the difference in level of any two benches established on a previ- ous survey may be tested, and, if found correct, all the inter- mediate elevations on the line may be assumed to be correct also. The discrepancy should not exceed one tenth of a foot in any case, and is usually much less. 216. Any lack of adjustment in the instrument gives the line of sight a slight angle of elevation or depression, causing a slight error in every reading, proportional to the distance of the rod from the instrument. But the errors being equal for equal distances, and the backsights and foresights having opposite signs in our calculations, the errors cancel when the distances are equal. Hence, to avoid errors in ele- vation, each new turning-point should be as nearly as possible at the same distance from the instrument as the point on which the last backsight was taken. For precise reading, the rod should not be more than 400 feet from the instrument. 217. Another cause of error in readings Is want of verti- cality in the rod. This may be avoided by the use of a disk- level, or in the absence of wind, by balancing the rod. The rod may be plumbed one way by the vertical cross-hair of the level, and to ensure a vertical reading in the plane of the line of sight, the rod may be gently waved each side of the vertical toward and from the instrument, the shortest reading being 186 FIELD the correct one; or in case of a target rod, the target should rise to, but not above the horizontal cross-hair, as the rod is waved. 218. When very long sights are required to be taken with the level, another source of error must be considered, namely, the curvature of the earth. A level line is parallel to a great circle of the earth, and is therefore an arc of a circle, or may be so considered. A horizontal line is a straight line parallel to the plane of the horizon. Therefore the line of sight, being a horizontal line, is tangent to the circle of a level line passing through the in strument. To find the correction in elevation due to curvature of the earth for any distant station. Fig. 82. I E C\ FIG. 82. Let A be the station of the instrument /, and B the distant station observed. Let R CT= the radius of curvature of the earth, or of the parallel arc ID. Let L = ID = the level distance between A and B. Then IE, perpendicular to CI, is the line of sight, BE is the reading of \he rod, and DE = E a = the correction due to curvature. By Tab. I., 24, IE* = DE (DE + 2# ); but since DE is very small compared with 2R , it may be omitted from the parenthesis, and since IE ID L very nearly, because the angle ACB is very small, we have L? = 2R E . (315) is to be added to the apparent elevation of station B. LEVELLING. 18? 219. Refraction. In observing distant stations the line of sight passing through the atmosphere is refracted from the straight line IE, Fig. 82, and takes the form of a curve, which, for practical purposes, may be considered as the arc of a circle, concave downwards. Its radius, depending on the conditions of the atmosphere, varies from 5 to 7 times the radius of curvature of the earth. 1R is considered a good average value. Refraction causes the observed object to appear too high, while the curvature of the earth causes it to appear too low ; the effects being contrary, the correction for curvature is re- duced by the correction for refraction. If we let H = the total correction for both curvature and refraction, to be added to the apparent elevation of the observed object, then H.=JS,= (316) Table XVII. is calculated by this formula, assuming a mean value of R = 20,913,650 feet. 22O. The form of the earth is approximately an el- lipsoid of revolution. Its meridian section at the mean level of the sea is an ellipse, the semi-axes of which are, according to Clarke, at the equator A = 6378206 metres [6.8046985] at the poles B = 6356584 " [6. 8032238] According to the same authority 1 metre = 3.280869 feet [0.5159889] Therefore the semi-axes expressed in feet are A = 20 926 058 feet [7.3206874] 5 = 20855119 " [7.3192127] Then the radius of curvature of the meridia. at the equator, - = R 20 784 422 ft. [7.3177379] at the poles, - = R = 20 997 240 " [7.3221622] 188 FIELD ENGINEERING. In latitude 40 the radius of curvature of the meridian is 20 871 900, and of a section at right angles to the meridian, 20 955 400; the mean value, or R - 20 913 650 [7.320430] , be- ing adopted for general use. The error in the correction H eq. (316) due to this assumption will usually be much less than that due to the assumed value of the radius of refraction. 221. Levelling by Transit or Theodolite. When a transit has a level-tube attached to the telescope, it may be used as a Theodolite for levelling, and for taking vertical angles. If the instrument be in perfect adjustment, the line of sight will be horizontal when the bubble stands at the middle point of the tube, and the reading of the vertical circle will be zero. Should there be a small reading when the line of sight is horizontal it is called the index error. "When the line of sight is not horizontal, the angle which it makes with the plane of the horizon is called an angle of elevation, or of de- pression, according as the object upon which the line of sight is directed is above or below the telescope. This angle is measured on the vertical circle, being the difference of the reading and the index error, when both are on the same side of the zero mark, and their sum, when they are on opposite sides. When the distance to an observed object is known, and its angle of elevation or depression is measured, we may calculate its vertical height above or below the telescope. , ( elevation Let a = angle of ] . t depression " L the horizontal distance " L' = the distance parallel to line of sight " h = difference in elevation of object and instrument. Then for short distances, h = L tan a = L sin a (317) FIG. 83. For long distances the curvature of the earth and refraction must be considered. Fig. 83. Let I be the place of the instrument, and F the object observed. LEVELLING. 189 Let L = the distance, measured on the chord of the level arc ID, passing through the instrument; and let if) = the number of seconds in the arc ID; hence, since for ordinary distances the chord and arc are sensibly equal, w, = A 206264".8 [5.314425] xv or giving to E its mean value, 220, il> - L X .0098627 [7.993995] or a fraction less than 1" per 100 feet. Let IF be the arc of the refracted ray, and assuming that its radius is 1R , the arc will contain jth the number of seconds of the arc IF IF' , tangent to IF, is the direction of the telescope; IF is the chord of the arc IF, and IE is the horizontal. Let a = EIF' = observed angle of elevation. Then EIF true angle of elevation = EIF' - F'IF= a - \ . $if> = a - .071^. The angle EID = W .'. DIF = $if> + a - .071^; and IDF = 90 + $if> . .'. IFD = 90 -(# + <*- .071^). We now solve the triangle IFD for the side DF = h, and find For an observed angle of depression make a negative in the formula. The coefficient .071 is called the coefficient of refraction, this being a fair average value, while its extreme range is from .067 to .100 under varying conditions of the atmosphere, and values of the angle a. When the difference in elevation of two or more distant objects is required, we obtain the elevation of each separately, and subtract one elevation from another. The elevation of the observed object is given by (H. I.) h. 222. To find the Height of Instrument of a transit or theodolite by an observation of the horizon. Fig. 84. 190 FIELD ENGINEERING. Let / be the place of the instrument, and let a = observed angle of depression of the horizon. Let F be the point where the refracted ray meets the level surface, and draw the chords IF and. AF. Let ip = the angle ACF, let li = AI, and let k = the coeffi- cient of refraction. In the triangle IAF, IAF = 90 + tip, AFI = $ip - kip, AIF = 90 - (ip - kip) Hence FIE = ip - kip. But FIE = a + kip ip=Y^T2k ( 319) Let F" be the tangent point of a right line drawn through /; FIG. 84 then AI = OF' exsec AOF', but OF' = R , and, since ip is 1 k always very small, ACF' = K$ -f a) very nearly = 1 _^ a a (320) Giving to J? its mean value, 220, and assuming & = ^ \og /i = 7.320430 + log exsec 1.0801 a (321) LEVELLING. 191 Otherwise, we may solve the triangle AIF since AF = 2M sin W = 2R sin When k - & h = 2R sin Aa . r| TI cos If a Example. The observed dip of the sea horizon is 24' = a- What is the height of the instrument above the sea? By eq. (321) 1.0801 X a X 60 = 1555".34 6.383650 Table XXVI. (q - 2 1) 9.070130 R 7.320430 h =594.58 2.774210 Methods of determining heights by distant observations can- not be relied on for more than approximate results, since they necessarily involve the uncertain element of refraction, and usually a lack of precision in the vertical angle, the arc reading only to minutes in ordinary instruments. These methods, how- ever, are useful where no great accuracy is required, as for a temporary purpose until levels can be taken in the regular way, or for interpolating between points of established elevation. 223. Stadia Measurements. It is sometimes convenient to determine distances by instru- mental observation For this purpose two additional cross liairs may be placed in the telescope parallel to each other and equidistant from the central cross-hair. These are called stadia hairs, and distances determined by them are called stadia measurements. The stadia hairs are adjusted so as to inter- cept a certain space on a rod held at a certain distance from the instrument and perpendicular to the line of sight. For any 192 FIELD ENGINEERING. other place of the rod, the distances and intercepted spaces are nearly proportional. The exact relation is given below. Fig. 85. Let I = AB, the distance of the rod from the vertical axis of the instrument ; c = the distance from the axis to the ob- ject glass of the telescope; a = the distance from the object- glass to the rod ; i = the space between the stadia hairs ; = CD the space intercepted by them on the rod; and/= the focal distance of the object-glass. We then have by optics, g (I f f '-. = , whence a f = -s-, and since a I c .'. I (/+ c) = -.s. Now in any given instrument the focal distance /, and the space between the stadia hairs i are constant, while * and c vary with I. For any other distance I', we then have I' (/-f- c') = -.s', and combining the two equations s ' is usually assumed at 1 foot and I' (f-\-c r ) at 100 feet, and the stadia hairs are then adjusted accordingly. The focal distance /may be found by removing the object glass and ex- posing it to the rays of the sun and noting at what distance from the surface of the lens the rays form a perfect and min ute image of the sun on a smooth surface ; the distance c' is measured on the telescope when the rod is clearly in focus, at the assumed distance. To measure any other distance, the rod is again observed at the desired point, and the space s noted, which, placed in eq. (324), gives I (/-f- c) at once. We then measure c on the telescope, and adding (/-f- c}, obtain I, the distance re- quired. LEVELLING. 193 But inasmuch as c has but a small range of values, it will usually be sufficient to assume for it a mean value, as a con- stant. In this case we may find the value of (/ -f- c) = IF for the instrument used. Making c' = c in eq. (324), and solv- ing for (/-|- c), we have and by laying off on level ground any two distances from the instrument for I' and I, as 100 and 500, and observing the corresponding spaces ' and s intercepted on a rod, we insert them in eq. (325) and find (/-f c). Having found (/-f c), lay off (100 -f-/-f- c) from the instru- ment and adjust the stadia hairs to inclose just one foot on the rod at that distance. Any other distance is then found by the formula, (326) Example. At I' - 100 we finds' = 1.00, and at I = 500 we find s = 5.061. Kf\a 1 _ KfkA Hence, eq. (325) /+ c = =***! = 1.502 and eq. (326) I = 100 * + 1.5; provided the stadia hairs be ad- justed so as to intercept 1 foot at 101.5 feet distance from the centre of the instrument. 224. The foregoing formulae are all that are necessary for horizontal sights, but since the line of collimation is generally inclined more or less to the horizon, it follows that the stadia hairs will intercept a larger space on the vertical rod than that due to the true horizontal distance. We therefore require a formula for reducing inclined measurements to the horizontal. Fig. 86. Let a = EFG = the angle of Inclination of the line of colli- mation IG\ ** = CFD the visual angle defined by the stadia hairs, " s CD ~ space intercepted on a vertical rod. Then (Fig. 85), 104 FIELD ENGINEERING. Ill Fig. 86 s = CE - DE - EF [tan (a -f |0) tau (a $ while the true value (for the same distance) would be Dividing one by the other we derive C'D' 2tani0 s tan (a -f- |0) tan (a |0) By giving to s' and I' (/+ c) in eq. (327) their customary FIG. 86. values, viz., 1 and 100, we have tan |0 = .005 . . 6 = 34' 22". 63 and by Trig. Table II. 70, tan - tan (a - W = (g Since 6 is small, we have sensibly sin 6 2 tan -JO, and cos (a + 6) cos (a $0) = cos 2 a and the last equation reduces sensibly to which is the coefficient of reduction required by which to multiply the observed space s in case of inclined sights. Hence the formula for distance (eq. 326) becomes in this case without sensible error I = 100 s cos 1 a + (/+ c) (329) Tables XVIII. and XIX. have been calculated by the exact formula for the coefficient. LEVELLING. 195 Kmmplc. Given : a = 8 20' and s = 9.221: what is the horizontal distance to the rod? Eq. (329) s a 100 9.221 902.7 1.5 log. 2. " 0.964778 8 20' Tab. XIX. ' " 9.990780 2.955558 .-. Ans. 904.2ft. The rodman should have a disk level to insure keeping the rod vertical. 225. Another method of procedure is that in which the rod is always held perpendicular to the line of collimation, however much inclined the latter may be. To secure this posi- tion of the rod, a small brass bar is attached, having sights upon it through which the rodman watches the instrument during an observation, the line of sight being at right angles to the rod. The distance thus obtained is of course parallel to the line of collimation, and requires to be reduced to the hori- zontal. For this purpose, we have (Fig. 87). or rr jfi 11 Mi FIG. 87. IE = ia cos a -f EG sin a IE = (100 * +/+ c) cos a-\-r sin a (330) in which r is the reading of the rod by the line of collimation. For the elevation of the point 7? above /, tEB = IIG- GB cos a EB - (100 s +/+ c) sin a - r cos a (331) 19(3 FIELD ENGINEERING. When the distances are sufficiently great, correction must be made for curvature of the earth and refraction, as already ex- plained. This method is employed by the topographical parties of the U. S. Coast Survey in connection with the plane table. Their instruments, however, are so constructed as to give distances in metres, and heights in feet, requiring a modification of the above formulae. CHAPTER IX. CONSTRUCTION. 226. The engineer department of a railway com- pany is usually reorganized for the construction of the road, as follows : Chief engineer, Division engineers, Resident engineers, Assistant engineers. On some roads the division engineers are styled "Principal Assistants;" the resident engineers, "Assistants;" and the assistant engineers are de- signated according to their duties, as "leveller," " rodman," etc. A resident engineer has charge of a few miles of line, limited to so much as he can personally superintend and direct. He has one or more assistants and an axman in his party. All instrumental work is done and all measurements taken by the resident engineer and his assistants. A division engineer has charge of several residencies, and inspects the progress of the work on his division once or twice a week. In his office, which should be centrally located, all maps, profiles, plans, and most of the working drawings required on his division are prepared. To him the resident engineers make detailed reports once a month, or ofteuer if necessary, which he passes upon as to their cor- rectness, and from which he makes up a monthly report, or estimate, of the amount and value of the work done and ma- terials provided by each contractor on his division. The esti- mates are forwarded about the first of each month to the chief engineer, who examines and approves them, returning for modification any that seem to require it. CONSTRUCTION. 19? The chief engineer has charge of the entire work, and directs the general business of the engineer department. He occasionally inspects the work along the line. 227. Clearing and Grubbing. The first step in the work of construction is to clear off all growth of timber within the limits of the right of way. The resident engineer with his party passes over the line, making offsets to the right and left, and blazing the trees which stand on, or just within, the limits of the company's property. The blazed spot is marked with a letter C, as a guide to the contractor. After felling, the valuable timber should be piled near the boun- dary lines, to be saved as the property of the company. The brushwood is burned. Where a deep cut is to be made, the stumps are left to be removed as the earth is excavated. In very shallow cuts and fills the contractor will generally prefer to tear up the trees by their roots at once, rather than to grub out the stumps after clearing. Where the embankments will be over three feet high, grubbing is not necessary; but the trees require to be low-chopped, leaving no stump above the roots. The engi- neer should indicate to the contractor the localities where each process is suitable. 228. While the clearing is in progress, the engineer should run a line of test levels touching on all the benches to verify their elevations ; he may also rerun the centre line, replacing any stakes that may have disappeared, and setting guard plugs to any important transit points which may not have been previously guarded. If any changes in the alignment have been ordered, these may be made at the same time. 229. Cross Sections. The resident engineer is fur- nished with a profile of the portion of the line in his charge, upon which is plainly indicated by line and figures the estab- lished grade. From this he calculates the elevation of grade at each station, and by subtracting this from the elevation of the surface, he derives the depth of cut or fill (-|- or ) to be made at each point. The grade given on the profile is that which is subsequently called the subgrade, being the surface of the road-bed. The final or true grade is the upper surface of the ties after the track is laid. 198 FIELD Tlie base of a cross section is identical with the width of the road-bed. It is made wider in cuts than in fills to allow for the side ditches. Six feet should be allowed in earth, and four feet in rock cuts. The ratio of the side slopes depends upon the material. The usual slope ratio for earth is 1$ horizontal to 1 vertical for both excavation and embank- ment, Damp clay and solid gravel beds will stand for a time in cuts at 1 to 1, or an angle of 45, but this cannot be perma- nently depended on. On the other hand, fine sand and very wet clay may require slopes of If to 1 or 2 to 1. Exceptional cases require slopes of 3 or 4 to 1. In rock work the slopes are usually made at to 1 for solid, to 1 for loose, and 1 to 1 for very loose rock, liable to disintegrate. Rock embankments stand at 1 to 1. 23O. All cross sections are taken in vertical planes at right angles to the direction of the centre line. Figs. 88, 89. Formulae, Let 6 = AB, the base of section, or road-bed. ~DJT -* s = -jijf = the slope ratio JDH. " d CG the cut (or fill) at the centre stake. " h = DII or EN = the cut (or fill) at the side stake. " x = CD the "distance out ' from centre to side stake. " y = h-d = KD. We have at once from the figures the general formula x= $b + 8h (332) When the ground is level transversely; h = d, and x ib -f- id. For embankment use the same formula, considering d or h as positive in this case also, the figure being simply inverted. When the ground is inclined transversely; h = CG + DK = d -f- y on the upper side in cuts; <333) and h = EN=d y on the lower side in cuts sy (334) CONSTRUCTION. 199 For embankments use the same formulas, but apply eq. (333) to the lower side and eq. (334) to the upper side, the figure being inverted. The points D and E on the ground are usually found by trial, such that the corresponding values of x and y will verify the formulas. When the natural slope FD or LE is uniform its ratio s' may be found by measuring along the section the horizontal dis- tance necessary to change the reading of the rod 1 foot (or halt the distance necessary to change it 2 feet, etc.). Then,, having found the depths of cut (or fill) at /''and L, distant $b from the centre 0, we have BE = sh = s'(h - BF) and AN = sh = s'(AL - h) From these we have, for the upper side in cuts, and lower side in fills. h = -^ BF . -. x = ib + -j^ BF (335) s s ' s s also, for the lower side in cuts, and upper side in fills, h = -r^ AL .'. x = & + -^ AL (336) S -{- S ' 8 -\-. S We also have h - BF = Y^ BF and } (337) AL-h = -rl AL s -f- s whence the points D and E may be found by the level. But points D and E thus calculated should have their posi- tions verified by the general formula, eq. (332), lest the slope ' may not have been perfectly uniform. When the natural surface intersects tJie base between the points A and B, the section is said to be in side hill work, Fig. 90. Both portions of the section are then determined by eq. (333), or where the slope s' is regular, by eq. (335) measuring In every case from the centre stake C; but observing that when the centre is in cut and one side in fill, or vice versa, that d must be considered negative for that side, whence eq. (333) becomes for this case x $b sd -f sy {333V soo FIELD ENGINEERING. 231. Staking out Earthwork. Beginning at a point on the centre line where the grade cuts the natural sur- face, the engineer drives a grade stake (marked 0.0) and notes the point in the cross-section book. If the line of intersection of the road-bed and surface would make an acute angle with the centre line, he also finds the points where the edges of the proposed road-bed will intersect the surface, drives grade stakes, and also stakes out a cross section through each of those points, if necessary. Then advancing to the next point on the centre line where a section is required, he finds its elevation with the level (veri- fying or correcting the elevation taken on the location), calcu- lates the depth of cut or fill CG, which is then marked upon the back of a stake there driven; a cut being designated by G and a fill by F. If the ground is level transversely (Fig. 88), he calculates x by eq. (332) and lays off this distance at right angles to the centre line, driving slope stakes at the points D and E, marked with the depth of cut or fill. The marked side of slope stakes should face the centre line. If the ground is inclined transversely (Fig. 89), he first measures Fia. 89. the distance, $b, to F, and finds the depth BFfor record. He then proceeds to find the point D. If the natural slope be uni- form, 2) may be found by eq. (335) or (337), verifying the result by eq. (332). The point E oi the other slope maybe found similarly, using eq. (336) or eq. (337) ; verifying by eq. (332). CONSTRUCTION. 201 232. If the (/round be irregular, the depth of cut or fill is found not only at the centre and edges of the road-bed, but also at every other point along the cross section where tJie sur- fact slope changes, all of which depths are recorded, together with their respective distances from the centre. To Jind the point D : assume a point supposed to be near D, and there take a reading of the rod. The difference of the readings at that point and at G equals y' for that point, which inserted in eq. (333) gives a value x. If x' agrees with the horizontal dis- tance of the assumed point from (J, the true position of D has been found. If x' be greater than this, by subtracting the eq. x' = ib -j- sd -{- *y' from eq. (333) we derive x = x' + s(y-y') (338) the last term of which shows the correction to be added to x'. Now in advancing from the assumed point to the extremity of x', the rise of the surface is approximately (y y'}, and if, in going the additional distance, $(y y'), a further rise is en- countered, this last, multiplied by s, must also be added to x', and so on until the additional advance makes no change in the value of y. The point thus found, verified by eq. (332), is the point D required. But if x' be less than the distance of the assumed point from C, we have x = x'-s(y'-y) (338)' the corrections being subtractive. The point E on the other slope is found in a similar manner, using eq. (334) for the value of x ; if x' be greater than the as- sumed distance, we have x = x - s(y - y'} (339) the corrections being subtractive ; but if x' be less than the as sumed distance, x=x' + s(y' -y) (339)' the corrections being additive. 233. In side-hill work (Fig. 90) proceed in the same manner, using eqs. (333) or (333)' and (338) in all cases of un- even ground. When the surface slope s' is uniform, cq. (335) may be used, if preferred, on either side. In addition to the 202 FIELD ENGINEERING. centre and side stakes, a grade stake is driven at the point 0, where the surface intersects the grade, the stake facing down hill. To find a grade point, set the target to a reading equal to the height of instrument less the elevation of grade, and stand the rod at various points along the given line until the target coin- cides with the line of collimation. JL FIG. 90. 234. When two materials are found in the same section, as rock overlaid with earth, each material requires its own slope, and a compound section is the result. To stake out work of this description, the depth of earth to the rock must be known, and may be nearly ascertained by reference to an adjacent section already excavated. Fig. 91. N,-~Nr^ Let a l be the depth of earth at C " , " " " Por Q " Si be the ratio of rock slope " * 2 " " earth slope Then (340) in which y^ = difference of rod readings on the rock at 6 T and Di, or C and E\ ; and y y difference of rod readings on the surface at P and D 2 , or at Q and J 2 . The upper sign applies to the upper side, the lo.ver sign to the lower, CONSTRUCTION. 203 It is better, however, to make an indefinite cross profile at first, driving two reference stakes quite beyond the section limits; and when the contractor has removed the earth from between D l and E l} indicate to him those exact points by marks on the rock, and also set the slope stakes at D^ and E^. 235. The frequency with which cross sections should be taken depends entirely upon the form of the surface ; where this is regular, a section at each station is sufficient. A cross section should be taken, not only at every point on the centre line where there is an angle in the profile, but also wherever an angle would be found in the profile of a line joining a series of slope stakes on either side, even though the profile of the centre line maybe quite regular at the corresponding point: the object being, not only to indicate the proper outlines of the earthwork, but to furnish the data necessary to calculate correctly the quantities of material removed. Rockwork will generally require more frequent sections than earthwork. 236. Vertical Curves. The grades as given on the profile are right lines, which intersect each other with angles more or less abrupt. These angles require to be replaced by vertical curves, slightly changing the grade at and near the point of intersection. A vertical curve rarely need extend more than 200 feet each way from that point. Fig. 92. Let AB, BO, be two grades in profile, intersecting at station fi, and let A and C be the adjacent stations. It is required to join the grades by a vertical curve extending from A to C. Suppose a chord drawn from A to C; the elevation of the middle point of the chord will be a mean of the elevations of r:-ade at A and C; and one half of the difference between this 204 FIELD ENGINEERING. and the elevation of grade at B will be the middle ordinate of the curve. Hence we have X = * (S**AS!**. _ grade B) (341) in which M = the correction in grade for the point B. The correction for any other point is proportional to the square of its distance from A or G, Thus the correction at A -j- 25 is &M ; at A -f 50 it is $M; at A -\- 75 it is &M; and the same ior corresponding points on the other side of B. The correc- tions in the case shown are subtractive, since M is negative. They are additive when M is positive, and the curve concave upward. These corrections are made at the time the cross sections are taken, and the corrected grades are entered in the field- book opposite the numbers of the respective stations. 237. Form of Field-book. A complete record of all cross-section work is kept in the cross-section book. On the left-hand page is recorded, in the first column, the numbers of the stations and other points where sections are taken ; in the second, the elevations of those points, copied in part from the location level-book, but verified or corrected at the time the section is taken ; in the third, the elevation of the grade for the same points; in the fourth, the width of base b; in the fifth, the slope ratios, s; and in the sixth, the surface ratio s' when uniform. The right-hand page has a central column, in which, and opposite the number of the station, is recorded the centre^depth of the section, marked -f- or , to indicate cut or fill, as the case may require. To the right of this are recorded the notes of that portion of the section which lies on the right of the centre 'line, as the line was run, and to the left, the notes of the left side. The distance from the centre to each point noted is recorded as the numerator of a fraction, and the cut or fill at the point as the denominator, prefixed by a -{- or as the case may require. The denominator for a grade point is zero. The numbers of the stations should increase up the page, as in a transit book, so that there may be no confusion as to the right and left side of the line. The several points being noted in order as they occur from the centre outwards, the notes far CONSTRUCTION. 205 thest from the centre of the page usually appertain to the slope stakes; but in case the cross profile is extended beyond the slope stake, the note of the latter should be surrounded by a circle to distinguish it. The following form is a specimen, of a right-hand page, with the first column only of the left- hand page: Sta. Cross Sections. 83 + 60 82 + 38 + 27 + 19 81 80 22.9 16.5 10 5 + 21^ 10 20 32 55.6 + 8.G +14 +17.7+21 17.5 10 4 +20.8 +25.6 +28.3 10 24 42.6 +30.4 + 5.0 + 10 +13.2 14.2 10 + 14 +14.7 +20.1 +21.7 6 10 31.6 + 2.8 + 5.4 10 21.7 7 + 9.4 + 8.5 +11.6 +14.4 10 19.3 + 2.8 ^177 + 3.8 + 6.2 10 ~T~ 7 15 - 9.8- 5.6 25.9 7 -12.6-11.2 33.4 7 - 12 -10.6 - 5.3 7 18 25.6 -17.6-16.4 -17.6 -19.6 -19.1 -12.4 238. In case there is a liability to land-slips, the profiles of cross sections should be carried beyond the slope stakes, on the upper side of the cut, to any distance thought neces- sary to reach firm ground, and stakes driven for future refer- ence. When a number of consecutive cross profiles are to be considerably extended, it is well to first run, instrumentally, a line parallel to the centre line, and set stakes opposite the stations, taking their elevations. The intermediate surface of the sections may then be taken with cross-section rods if more convenient. See 37. 239. In case of inaccessible ground, preventing a regular staking out, an indefinite profile of the section may generally be obtained, referred to the datum for elevation and to the centre line for position, which being plotted on cross- section paper, and the grade line and side slopes added, shows to scale where the slope stakes should be. 206 FIELD ENGINEERING. 240. Any isolated mass of rock or earth which oc- curs within the limits of the slope stakes, but not included in the regular notes, is separately measured and noted, so that its contents may be computed and added to the sum of the same material found in the cross sections. 241. Borrow-pits. When the excavations will not suffice to complete the embankments, material may be taken from other localities, termed borrow-pits. These should be staked out by the engineer and their contents calculated, unless the contractor is to be paid for work by embankment measurements. A number of cross profiles are taken of the original surface, and (on the same lines) of the bottom of the pit after it is excavated, which furnish the depth of cutting at each required point. Borrow-pits should be regularly ex- cavated, so that they may not present an unsightly appear- ance when abandoned. Borrow-pits may be avoided by widening the cut uniformly at the time it is staked out, so that it may furnish sufficient material; provided the material is suitable, the embankment accessible, and the distance not too great. When the excavation is in excess, the surplus ma- terial should be uniformly distributed by widening the adja- cent embankments, if possible; otherwise it is deposited at convenient places indicated by the engineer and is said to be wasted. 242. Shrinkage. In estimating the relative amounts of excavation and embankment required, allowance must be made for difference in the spaces occupied by the material before ex- cavation and after it is settled in embankment. The various earths will be more compact in embankment, rock less so. The difference in volume is called shrinkage in the one case, and increase in the other. Shrinkage in 1000 cu. yds. Material. Of excavation. Of settled emhkt. Sand and gravel 80 C. Yds. 87 C. Yds. Clay 100 " 111 " Loam 120 " 136 " Wet soil 150 " 200-"* Increase in 1000 cu. yds. Rock, large fragments. 600 C. Yds. 375 C. Yds. " medium fragments 700 " 413 " " small 800 " 444 " CONSTRUCTION. 207 Thus, an excavation of sand and gravel measuring 1000 cubic yards will form only about 920 cubic yards of embankment; or an embankment of 1000 cubic yards will require 1087 cubic yards of sand or gravel measured in excavation to fill it; but will require only 58? cubic yards of rock excavation, the rock being broken into medium-sized fragments; while 1000 cubic yards of the latter, measured in excavation, will form 1700 cubic yards of embankment. The lineal settlement of an earth embankment will be about in the ratio given above, therefore the contractor should be instructed in setting his poles to guide him as to the height of grade on an earth embankment, to add 10 per cent (average) to the fill marked on the stakes. In rock embankments this is not necessary. The engineer should see that all embank- ments are made full width at first, out to the slope stakes, and by measure at or above grade, so that the whole may settle in a compact mass. Additions to the width made subsequently are likely to slide off. 243. The cross-section notes should be traced in ink at the first opportunity to secure their permanence. An office copy should also be made to serve in case of loss or damage to the original. 244. Alteration of Line. Inasmuch as the centre line at grade is the base of reference for all measurements and cal- culations in earthwork, any change made in it after the work of grading has begun should be most carefully recorded and explained. The centre stakes of the old line should be left standing until after the new line is established, so that the per- pendicular offset from the old line to the new, at each station, may be measured, as also the distance that the new station may be in advance of, or behind the old one. The date of the change should be recorded. The original cross sections are extended any amount requisite, the distance out being stil 1 reckoned from the old centre, while a marginal note states the amount by which the centre has been shifted. The difference in length of the lines will make a long or short station at the point of closing. The exact length of such a station should be recorded, so that it may be observed in re- tracing the line at any time, and in calculating the quantity of 208 FIELD earthwork. The original transit notes of the altered line should be preserved, but marked as "abandoned," with a reference to the notes of the new line on another page. 245. Drains and Culverts. The engineer should ex amine the nature and extent of each depression in the profile with reference to the kind of opening required for the passage of water. For small springs, and for a limited surface of rain- fall, cement pipes, in sizes varying from 12 to 24 inches diame- ter, serve an excellent purpose as drains. These are easily laid down, and if properly bedded, with the earth tamped about them, are very permanent; but their upper surface should be at least 2^ feet below grade. The embankment is protected at the upper end of the drain by a bit of vertical wall, enclosing the end of the pipe. If necessary, a paved gutter may lead to it. Where stone abounds, the bed of a dry ravine may be partly filled with loose stone, extending beyond the slopes a few feet, which will prevent the accumulation of water. When the flow of water is estimated to be too great for two lines of the largest cement pipe, or when the embankment is too shallow to admit them safely, a culvert is required. A pavement is laid one foot thick, protected by a curb of stone or wood 3 feet deep at each end, and wide enough to allow the walls to be built upon it. It should have a uniform slope, usu- ally between the limits of 50 to 1 and 100 to 1 to ensure the ready flow of water. In firm soils the foundation pit is exca- vated one foot below the bed of the stream, but if mud is found this must be removed and the space filled with riprap, the up- per course of which is arranged to form the pavement at the proper level. In a V-shaped ravine, requiring too much ex- cavation at the sides, and where the fall is considerable, riprap may be used to advantage, the bed of the stream above the culvert being graded up by the same material to meet the pave- ment. In some cases a curtain, or cross wall, is necessary on the lower end to retain the riprap. Culverts should be laid out at right angles to the centre line whenever practicable, the bed of the stream being altered if necessary. The length of an open culvert is the entire distance between slope stakes, the walls being parallel throughout, or the length may be taken somewhat less than this, and the walls CONSTRUCTION. 209 turned at right angles on the upper end, forming a facing to the foot of the slope. The walls are carried up to grade for the width of the road-bed, and are stepped down to suit the slopes. A course is afterwards added to retain the ballast. 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. The walls terminate in short head-walls built parallel to the centre line, the top course being a continuation of the cover. The length of a head-wall, measured on the outer face, is equal to the height of the culvert in the clear multiplied by the slope ratio of the embankment. The perpendicular distance from the centre line to the face of a head-wall is equal to one half the road-bed, plus the depth of the top of the wall below grade multiplied by the slope ratio, or $ -(- sk. A coping is sometimes added. 246. Arch culverts are used 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. 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 line 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 foundations are laid broader and deeper than in box cul- verts, each abutment having its own pit, carried to any depth found necessary. The half length of the culvert is \b -}- sk, in which k is the depth of the crown of the arch below grade. The abutments are carried up half way from the spring to the level of the crown of the arch, and thence sloped off toward the crown. The face walls arc carried up to the crown, and coped. The wing walls stand at an angle of 30 with the axis of the culvert, they receive a batter on the face, and are stepped (or sloped) down to suit the embankment. Then- thickness, at the base, is the same as that of the abutment; at the outer end 3 feet. They stop about 3 feet short of the foot of the slope. They need not be curved in plan. Any stone structure of dimensions greater than those given above, scarcely comes under the head of culverts, and should be made the subject of a special design by the engineer. 310 FIELD ENGINEERING. 247. Staking out Foundation Pits. For bor culverts. The engineer having decided upon the size of cul- vert required, makes a diagram of it in plan, on a page of his masonry book, recording all the dimensions, stating the sta- tion and plus at which its centre is taken, the span and height of the opening, etc. He then sets the transit at the centre A, Fig. 93, measures the angle between the centre line and axis, Fio. 93. (making it 90 if practicable) ; on the a'xis he lays off the dis- tances to the ends of the culvert and drives stakes at B and G. Perpendicular to BChe lays off the half widths of the pit, set- ting stakes at D and E, and laying off Z^and EH AB; and D G and El = AC. On IG produced he lays off CJ = OK, and perpendicular to this JM and KL, and finds the intersections and N. A stake is driven at each angle, and upon it is marked the cut required to reach the assumed level for the foundation. These cuts are recorded on the corresponding angles of the diagram. The pit is thus no larger than the plan of the proposed masonry, and the sides are vertical, which answers the purpose for shallow pits. For arch culverts. The pit for each abutment when shallow may be of the same dimensions as the lower founda tion course ; if more than five feet deep, it should be enlarged by an extra space of one foot all around. In Fig. 94 the inside CONSTRUCTION. 211 lines show the plan of the abutments at the neat-lines ; the outside lines represent the pits. Having prepared a plan of the structure suited to the locality, and made a diagram of the same in the masonry book, set the transit at A, and drive stakes at D, E, N and on the centre line. Then turning to the axis BC, lay off AC, and set stakes at G and /. With G as a centre, and a radius equal to 2DE, describe on the ground D FIG. 94. an arc cutting El in X or (IX DE . cot 30) may be calcu- lated; and on XG produced lay off GK, and perpendicular to this, KL. From N lay off NP, parallel to AC, and measure PL as a check. Drive a stake at each angle, marked with the proper cutting, and record the same on the diagram. The locality may require the wings to be of different lengths and angles, of which the engineer will judge. Guard-plugs should be driven in line with the intended face of one or both abut- ments, so that the neat-lines can be readily given when re quired. In case the material is not likely to stand vertically, the pit must be staked out with sloping sides, as described below. For bridge abutments. A design for every impor- tant structure is usually prepared in the office after a survey of the site. The foundation pit is then laid out from dimen- sions furnished on a tracing, but a diagram of the pit should be made in the masonry book as usual. When the bridge is on a tan- gent, Fig. 95, set the transit at A on the centre line at its inter- section with the axis 5Cof the abutment ttttlie level of the seat. 212 FIELD ENGINEERING. Deflect from the tangent the angle giving the direction of BC, and lay off AC, AB, setting plugs at B and C, and reference plugs (two on each side) on BC produced. After staking out the sides of the pit parallel to BC, set the transit at C, and deflect the angle for the wing, laying off CD, and driving stakes at the corners E and F. Two reference points are then set on the line CD produced. The other wing being Fio. 95. staked out in the same manner, the cut is found at each stake and marked and recorded. Cross sections are then taken near each corner, perpendicular to each side, and slope stakes (marked "slope") are driven where the slope runs out. Inter- mediate sections are taken when the unevenness of the ground makes it necessary, and the lines joining the slope stakes are produced to intersect, and other stakes are driven at the inter- sections. The position of each stake is shown on the diagram, and the cut recorded. A slope of 1 to 1 is usually sufficient for pits. If the material will not stand at 1^ to 1 , or if space cannot be spared for the slope, the sides may be carried down vertically, supported by sheet piling braced from within. The reference points should be so chosen that the points A, Band C may be found by intersection, on any course of the masonry, during the progress of construction. When the bridge is 011 a curve, the bridge-chord should be found and the abutments laid out from this. Fig. 96. The bridge-chord is a line AB, midway between the chord of the curve CD, joining the centres of the abutments, and a tan- gent to the curve at the middle point of the span. Hence COKSTEUCTIO^. GA = DB = %MN, which may be laid off, and A and B are the true centres of the abutments, from which the foundations are staked out as before. The distance CE = D F to the points where the bridge-chord cuts the curve is 0.147 CD. Should an abutment site on a curve be inaccessible, as when Fia. 96. under water, from any transit point P on the curve lay off. PX perpendicular to the tangent at M, observing that PX = M Q - A = R (vers PM- $ vers CM) and AX = PQ- %AB = R(sin PM - The point A may then be found by intersection, or by direct measurement .with a steel tape or wire, driving a long stout stake to show the point above the water. Other points may then be approximately found, sufficient to begin operations. In case of a bridge of several spans, the piers are laid out in the same manner, from a centre point and axis. If on a curve, each span has its own bridge-chord, but for convenience, the centre of a pier may be taken on the centre line during its con- struction, and the bridge-chord only found for the purpose of placing the bridge ; the piers being long enough to allow of the shift. 214 FIELD To locate the centres of piers, a base line is re quired on one or both shores, and two transits are used to give the intersections by calculated angles. When practicable the spans should also be measured with a steel tape or wire. The bed of a pit for any sort of structure should receive the closest scrutiny of the engineer, it being his duty to judge whether the material will resist the load to be im- posed upon it. A pit may require to be excavated to a greater depth than first ordered, while sometimes a less depth will Answer, as when solid rock is found. When a good material is reached, if any doubt exist as to its thickness, or as to the character of the underlying stratum, borings should be made or sounding rods driven down. Piles may be driven to gain the requisite firmness, and a layer of riprap, of beton, or of timber may be used to afford a uniform bearing. When satis- fied of the stability of the bed, the engineer finds the original centres, and gives points for the courses of masonry. A com- plete record is kept of the amount and kind of excavation, the materials used in foundation under the masonry, and of the size and thickness of each foundation course of masonry ; the notes should be taken at the time the work is done, it being generally impossible to take measurements thereafter. 248. Cattle-guards are shallow pits placed at right, angles across the road at the fence lines to prevent the passage of cattle. They are either entirely open, in which case they should be at least 4 feet deep, or they are covered in part with Wooden rails laid a few inches apart. The open guard is preferred. It is built like an open culvert except that no pavement is required. The stringers carrying the rails over any opening should be no longer than the span plus the thick- ness of the walls. 249. Trestle Work. No wooden culverts should ever be used. If stone cannot be had at first, two trestle bents may be erected, leaving between them a space sufficient to contain the stone structure to be built when the material for it can be brought by rail. The bents may be backed by plank to retain the embankment, and the stringers are then notched down an inch on the caps to receive the pressure of the earth, and render the bents mutually sustaining. The sills are prevented from yielding to the pressure of the earth by being sunk iii CONSTRUCTION. a trench, or by sheet piling. Should the span be too long, a central bent may be used, so as not to interfere with building the wall. Sometimes pile-bents may be used with greater ad- vantage, the piles being driven in rows of four each, and cap- ped to receive the stringers. In districts where suitable stone is entirely wanting, pile or trestle abutments and piers are used for the support of bridges, the piles or posts being arranged in groups and capped to receive the direct weight of the trusses. They should not sustain the embankment, but should be connected with it by a short trestle work. Trestle work is frequently used as a substitute for embank- ment, either to lessen the first cost, or to hasten the completion of the line, or for lack of suitable material with which to form an embankment. The cost of trestle work, however, is not less than that of an earth embankment formed from borrow pits, unless its height exceeds about 15 feet, depending on the relative prices of materials and labor. "When not exceeding 30 feet in height, the bents, for single track, are usually composed of two posts, a cap and sill, each 12 X 12, and two batter posts, 10 X 12, inclined at th to 1, all framed together. Two lengths of 3-inch plank are spiked on diagonally on opposite sides of the bent as braces. The length of the caps should equal the width of the embankment; the posts should be 5 feet from centre to centre, aud the batter posts 2 feet from the posts at the cap. The sill should extend about two feet beyond the foot of the batter post. A masonry foundation for the bent is preferable, though pile foundations are not uncommon, and some temporary structures are placed directly on a firm soil, supported only by mudsills laid crosswise under the sill. The spans, or distance between bents, may vary from 12 to 16 feet. The stringers should consist of 4 pieces, 2 under each rail,, bolted together, with packing blocks to separate them 2 or 3-, inches. Over each bent and at the centre of each span a piece of thick plank about 4 feet long should be placed on edge between the two pair of beams to preserve the proper distance between them, while rods pass through the beams and strain them up to the ends of the plank, to increase the stability of the beams and prevent their buckling under a load. The string- ers should be able to carry safely the heaviest load without bracing against the posts. The bents, however, if high, must be braced against each other. The stringers should be con- 216 FIELD tinuous, tne two pieces breaking joints with each other at the bents, to which they are firmly bolted. They may rest directly on the caps, or corbels may intervene. The spans on a curve should be shorter than on a tangent. The ties should be notched down to fit the stringers closely, and guard rails, either 'wood or iron, secured to them firmly. Unless the spans are very short, horizontal bracing should be employed consisting of 3-inch plank, extending from the centre of each span to the ends of the caps, which are notched down to receive the plank. For trestles much higher than 30 feet the cluster bent is preferable, so termed because each vertical post is composed of a cluster of four pieces, 8x8, standing a little apart to allow the horizontal members to pass between them. The verticals are continuous, breaking joints, two and two, while the hori- zontals pass the posts and are bolted to them at the joints; the framing is accomplished entirely by packing blocks and bolts. The batter posts consist each of two pieces 8X8; the horizon- tals may be 4 X 10, and extend not only across the bent, but from one bent to another. Proper bracing is also used in every direction. When very high, a secondary pair of batter posts may be introduced in the lower part of the structure. The batter need not exceed |th to 1. In some instances two adjoin- ing bents are strongly braced together, forming a tower or pier, and the piers placed from 50 to 100 feet apart, the roadway being carried on trussed bridges. The cluster bent admits of any piece being removed and a new one inserted when neces- sary. Iron trestles are now adopted where a permanent struc- ture is desired. Owing to the expansion of the metal by heat, the bents cannot be continuously connected with each other as in a wooden trestle ; hence the pier form is resorted to, having Q)ans varying from 30 to 150 feet, covered by trussed bridges, and the whole structure is more properly styled a viaduct. 2>O. Tunnels. Tunnels are adopted in certain cases to avoid excessive excavations, steep grades, high summits, and circuitous routes. Their disadvantages are the increased time and cost of their construction compared with an open line, and their lack of light and fresh air when in use. It is desirable that they should be on a tangent throughout, both for the ad- mission of light and for convenience of alignment. Many CONSTRUCTION. 217 tunnels, however, have been built with a curve at one or both ends.* The location of a tunnel, other things being equal, should be such as to make not only the tunnel proper, but also its im- mediate approaches by open cut as short as possible ; and the 'latter should be selected so as not to be subject to overflow, nor liable to land slides. The material to be encountered may frequently be determined with tolerable accuracy by a study of the geological formation in the vicinity, or by actual borings. The most favorable material for tunnelling is a homogeneous self-supporting rock, devoid of springs, which does not disin- tegrate on exposure to the atmosphere. The worst materials are saturated earth and quicksands. The presence of water in any material increases the cost considerably. The alignment of a tunnel is made the subject of special survey, after the general location is decided, and this is more or less elaborate according to the length of tunnel. A perma- nent station is established at the highest point crossed by the tunnel tangent, from which, if possible, monuments are set in each direction at points beyond the ends of the tunnel. If there are two principal summits, stations on these will define the tangent, which may then be produced. The monuments established beyond the tunnel should be sufficiently distant to afford a perfect backsight from the ends of the tunnel, where other monuments are also established. The first quality of in- struments only should be used, and these perfectly adjusted, and the observations should be repeated many times until it is certain that all perceptible errors are eliminated. Since the line of collimation will be frequently inclined to the horizon at a considerable angle, it is important that it should revolve in a vertical plane ; and to secure this, a sensitive bubble tube should be attached to the horizontal axis, at right angles to the telescope of the transit. The distance may be obtained by tri- augulation, though direct measurement is to be preferred. A steel tape is convenient and accurate, providing that allowance be made for variations due to temperature, from an assumed standard. The rods described in 43 may be used instead of *The Mont Cenis tunnel, requiring a curve at each end, was first opened on the tangent produced, giving a straight line through, and the curves were excavated subsequently. 218 FIELD plumb lines, the tape being held at right angles to them, and therefore horizontal. A plug should be driven for each rod to stand on, and a centre set to indicate the line and measure- ment. As the excavation of the tunnel proceeds, the centre line ia given at short intervals by points either on the floor or roof. Overhead points are generally preferred, from which short plumb lines may be hung, constantly indicating the line, with little danger of being disturbed. When a new transit point is required in the tunnel, it should be established directly under an overhead point, which serves as a check upon its perma- nence, and as a backsight when needed. Shafts are sometimes opened to give access to several points of the tunnel at the same time, thus facilitating the work, though at an increased cost. They also serve for ventilation during the progress of the work, though they are worse than useless foi this purpose afterward, except possibly in the case of a single shaft near the centre of the tunnel. Some of the longest tun- nels have been formed without shafts, while many shorter ones have had several, which have generally been closed after the tunnel was completed. Shafts are either vertical, inclined, or nearly horizontal ; in the latter case they are called adits. In- clined shafts should make an angle of at least 60 with the ver- tical. Vertical shafts may be either rectangular, round, or oval. Their dimensions vary, depending on their depth and the material encountered, between 8 and 25 feet. They are usually sunk on the centre line of the tunnel, though some- times at one side. When over the tunnel the alignment below is obtained directly from two plumb lines of fine wire suspended on opposite sides of the shaft from points very carefully deter- mined at the surface. The plummets are suspended in water .to lessen their vibrations, and as soon as the transit can be set up at a sufficient distance to bring the lines into focus, it is shifted by trial into exact line with the mean of their oscilla- tions, the latter being very limited. Permanent points may then be set, but should be repeatedly verified. As soon as the workings from a shaft communicate with those from either end, or from another shaft, the alignment thus found is tested, and revised if necessary. These operations require the greatest nicety of observation and delicacy of manipulation to obtain satisfactory results. CONSTRUCTION. 219 From plumb lines in the central shaft of the Hoosac tunnel, die line was produced three tenths of a mile, and met the line produced 2.1 miles from the west end with an error in offset of five sixteenths of an inch. In the Mont Cenis tunnel the lines met from opposite ends with " no appreciable" error in alignment, while the error in measurement was about 45 feet in a total length of 7.6 miles. When a curve occurs in a tunnel it is usually near one end. The tunnel tangent is produced and established as before described, and a second tangent from some point on the curve outside the tunnel is produced to intersect it, the inter- section being precisely determined and the angle measured with many repetitions. The tangent distances are then calcu- lated, and the position of the tangent points corrected by precise measurements, and permanent monuments are estab- lished. As the tunnel advances, points may be set at short intervals on the curve in the usual manner; but at intervals of 100 feet the regular stations should be defined with finely centred monuments, using a 100-foot steel tape carefully sup- ported in a horizontal position. When it is necessary to use a subchord, its exact length should be calculated as shown in 107. When the curve has advanced so far as to render a new transit point necessary, this should be established at a full station. The subtangents from the two transit points should then be produced to intersect, and measured for equality with each other and with their calculated length. The distance from their intersection to the middle of the long chord should also be measured as a check on the deflections. When no perceptible errors remain, the curve may be produced as before until the P. T. is reached. It is evident that correct measure is indispensable to correct alignment on curves. Should obstacles on the surface necessitate triaugulation, more than ordinary care must be exercised, and as many checks introduced as possible. The triangles should be so arranged that all of the angles and most of the sides m ly be measured. Test levels are carried over the surface with great care, each turning point being made a permanent bench, and its elevation determined with a probable error not exceeding 0.005 foot. Levels may be carried down a shaft on a series of bolts or spikes about 12 feet apart in the same vertical line, the distances being measured by the same level rod as that 220 FIELD ENGINEERING. with which the benches are determined. The measures should be taken between two graduations of the rod, not using the end of the rod, which may be slightly worn. Fine horizontal lines on the heads of the bolts may be used to mark the exact distances. After the shaft reaches the level of the tunnel, the depth may be measured more directly with a steel tape, the entire length of which has been corrected at the given tem- perature, by comparison with the same rod. If the grade of a tunnel is to be continuous, it should be assumed at something less than the maximum of the road, but not less than 0.10 per station, which is required for drainage. If a summit is to be made in the tunnel, the grade from the upper end should not exceed 0.10 per station. Grades are given in the tunnel from day to day, or as often as required by the progress of the work, the marks being made on the sides at some arbitrary distance above grade. Turning points should be taken on permanent benches. The least width of a tunnel in the clear should be, for single track about 15 feet, and for double track 26 feet. The least height in the clear above the tie should be 18.5 feet for single track, and 16.5 feet at the outside rails for double track, allowing for tie and ballast; the roof at the centre of the section should be at least 20 feet above subgrade, and with a full, centred arch 22 or 23 feet for double track. The form of section depends somewhat on the material traversed. In perfectly solid rock a nearly rectangular section may be used, the roof being slightly rounded. In dry clay, and stratified rock, a flat arch may be used, and in olher cases a full-centred arch. The latter form is rather to be preferred on account of the better ventilation afforded. The sides are made vertical, battered or curved, as necessity or taste may dictate. In wet and infirm soil an invert floor may be required, otherwise it ,is made level transversely. When a lining is required the original section must of course be made large enough to allow for the masonry, and the temporary timber supports behind it. Hard burned brick is usually adopted for arching, being durable and easily handled. In loose rock the arching may be from 13 to 26 inches thick, in wet and yielding soil a thickness of from 26 to 39 inches may be necessary. The walls may be from 3 to 6 feet thick. In forming a tunnel, a heading or gallery of smaller cross section is first driven and afterwards enlarged to the full size required. In firm clay or loose rock which will tem- porarily support itself until the masonry can be put in, it is better to drive the heading along the floor (at subgrade) of the tunnel, the remaining material being then easily thrown down in sections as the arching is advanced. In solid rock, or wet earth, a top-heading (along the roof) is generally preferred. The dimensions of a heading driven by hand are usually 8 feet high by 8 or 10 feet wide, but in solid rock where drilling machinery is introduced, it is advantageous to make the head- ing as wide as the tunnel at once. By drilling holes into the face at points about five feet each side of the centre, and con- verging on the centre line at a depth of about ten feet, a tri- angular mass of rock may be blown out, and the space thus gained facilitates the blasting of the adjacent rock on either side. An advance of about 10 feet in each day of 24 working hours may thus be made, using nitroglycerine in some form as the explosive agent. Owing, however, to unavoidable delays from various causes, this rate of progress cannot always be maintained. At the Hoosac tunnel the greatest advance in one week was 50 feet; in one month 184 feet at one heading. At the Musconetcong tunnel a heading 8 X 22 feet in syenitic gneiss was advanced at the average rate of 137 feet per month for 6 months, the maximum being 144 feet the enlargement of the tunnel to full size going on at the same time, a few hundred feet behind. At the St. Gothard tunnel the north heading 2. 5 X 3 metres was advanced in mica gneiss, during the year 1875 at the average daily rate of 3.71 metres, with a maximum of about 4 metres, but the en- largement was not made. The south heading advanced at the rate of 2 metres a day, timbering being at times necessary. In ordinary clay a heading may be driven at from 75 to 180 ft. per month, according to circumstances, where timbering is put in. The enlargement, including timbering and masonry, may be advanced at from 20 to 60 ft. per month. Small tun- nels for water conduits are driven through dry clay at the rate of 10 ft. per day, the masonry following at once without tim- bering. The compressed air used to drive the drilling machinery serves to supply ventilation also. When this is wanting or proves insufficient, exhaust fans are used. At Mont Cenis a 222 FIELD E horizontal brattice or partition was built in the tunnel, dividing it so as to secure a circulation of air. When foul gases are en- countered, ventilation becomes a serious question, and in one instance an important work was abandoned for this cause. Cross sections of the heading, and also of the tunnel en largement, should be measured at intervals of about 20 feet, as soon as opened, to see that the sides, roof, and floor are taken out to the prescribed lines, at the same time that the latter are exceeded as little as possible. In solid rock, since some ma- terial outside of the true section will necessarily be thrown down, leaving an irregular outline, it is well to take two cross sections at the same point, one following the projections and the other the recesses of the rock, from which an average sec- tion may be estimated. A daily, or at least a weekly, record of operations should be kept in tabular form, and the progress indicated by a profile and cross sections drawn on a sufficiently large scale to show details. The drainage of a tunnel is best secured by a line of stoneware or cement pipe laid in a trench along each side, and covered with ballast or other loose material. The entire floor is thus made available for the use of the trackmen. When an invert is used, the drain is placed in the centre between tracks. If the amount of water is large, drain pipe may be laid behind the walls, and the back of the arch may be covered with as- phaltum, or coal tar, to prevent a constant dripping on the track. 251. Retracing the "Line. As the grading pro- gresses, in either excavation or embankment, the principal transit points are established on the road-bed from the points of reference, and the centre line is retraced, setting stakes at every 50 feet. Transit points on grade should be fixed upon stout, durable posts firmly set in the ground, and standing high enough to be easily reached after the ballast is laid. To recover the old line, any discrepancies in measurement must be left between the transit points where they occur, and not carried forward. In retracing a curve, if the transit is placed at the forward point, allowing the chain to ad- vance toward it, slight differences in measurement will not affect the position of the curve. If any short or long sta- CONSTRUCTION. 223 tions have been introduced on the location, their position on the line must not be changed in retracing. The chain may be adjusted so that its measures will agree with the recorded distances between transit points. Offsets are made right and left from the new stakes to see that the road-bed is of the full width at all points. The levels are also carried over the grade, and any remaining cut or fill found necessary is marked on the back of the stakes, due allowance being made for the probable settlement of embankments. 252. As the work approaches completion the contractor goes over the line dressing it to grade and opening the side ditches if this has not been previously done. Drain-tile should be laid at the bottom of these ditches and lightly covered with earth, particularly if the cut be wet. These not only prevent the water from reaching the ballast, but by keeping the foot of the slope comparatively dry pre- vent the earth from sliding down and filling up the cut. There is also a marked economy in their use, as the cost is trifling, and all further excavation of mud and water from the cut is generally obviated. Should any springs appear in the slope a branch line of smaller tile may be laid to meet it. If the slope is liable to be overflowed from the surface above, an open ditch should be dug a few feet beyond the slope stakes, leading the surface water to discharge elsewhere. 253. The road-bed being prepared, ballast stakes are driven at every half station, giving the width of the ballast at its base, while the tops of the stakes indicate the proper level of its upper surface, which is the under side of the tie. These stakes should be set so as to give the proper elevation to the outer rails on curves when the ballast is graded to them. The ballast should be about one foot deep before the ties are laid. Broken stone or a mixture of coarse and fine gravel is the best material, affording elasticity and good drainage. The side slopes of the ballast are made 1 to 1 ; its width at the under side of the tie should be one foot greater than the length of the tie. 254. Track-laying. After the ballast has been laid and graded, the centre line is retraced upon it ; short stakes 234 FIELD ENGINEERING. are used, each of which is centred. On long tangents, one stake in every 200 feet is sufficient, on ordinary curves one in every 50 feet, and on very sharp curves one In every 25 feet. The ties are then spaced evenly according to the number prescribed per mile, or per rail length ; but a tie should not be allowed to cover a transit point. Ties for the standard gauge are 8 or 9 feet long; they should be sawed off square at the ends and in uniform lengths for appearance sake when laid. Specifications usually call for ties having a thickness of 6 inches and a width of from 7 to 10 inches. The ends of the ties arc aligned on one side of the road, though if cut into uniform lengths both ends will be equally well aligned. The rails are then laid on, and spiked to gauge. The first spikes are driven in the ties near a centre stake, the centre mark of the gauge bar being kept over the centre on the stake. Upon curves the rails must be sprung to the proper arc before they are laid ( 199). All the ties required in a given distance should be laid before the rails are brought upon them. The practice of laying only joint and middle ties at first subjects the rails to the danger of bending from passing loads. Owing to the expansion of the rails by heat, a space must be left at the rail- joints. The highest temperature of a rail in the summer sun is about 130 Fah. The expansion of iron or steel per 100 is .0007 per foot; or for a 30-foot rail .021 foot or .252 inch. Therefore when 30-foot rails are laid at a temperature near the freezing point, or 100 below the maximum, the space allowed must be at least a quarter of an inch. At 80 Fah. or 50 below the maximum, it need be only half as much. The space required is also proportional to the length of rail used. The exact space should be given, as less would result in the rails being forced up by expansion, while more than necessary space gives a rough road, and hastens the destruction of the rail. Wherever sidings are required, the necessary frogs and long switch- ties should be provided in advance, so that they may be put in place at the time of laying the main track. For every road crossing at grade, heavy oak plank should be pro- vided, and laid upon the ties as soon as the jails are spiked, so that the highway travel may not be impeded. CALCULATION OF EARTHWORK, 225 CHAPTER X. CALCULATION OF EARTHWORK. I 254. The first step toward finding the cubical content of an excavation is to divide it into a number of prismoids by several cross sections. A prismoicl is a solid having plane parallel bases or ends, and bounded on the sides either by planes, or by such surfaces as may be generated by a right line moving continuously along the edges of the bases as directrices. The positions of the cross sections must be so selected that the solid included between any two consecutive sections may.be a prismoid as nearly as possible. Upon a tangent the road-bed and side slopes are planes, so that the prismoidal character of a given solid depends upon the shape of the natu- ral surface. When the natural surface is a plane, the sections are taken only at the regular stations, 100 feet apart; when it is curved, warped, irregular, or broken, the sections must be more numerous, so that the surface limited by any two shall be composed substantially of right-lined elements extending from one section to the other. If two end sections of a prismoid are somewhat similar, we infer that the corresponding points are connected by right- lined elements, forming in each case the axis of a ridge or of a hollow. If one section has less breaks than the next, some of these ridges or hollows must vanish; and in order that the solid may be a prismoid, they must vanish in the section of least breaks ; therefore a cross section must be taken on the ground through the point where each ridge or hollow vanishes, and the distance of that point from the centre line noted, so that it may be coupled with the proper point in the next section for exact calculation of content. When ridges or hollows run diagonally across the line of road, cross sections must be taken where they are intersected not only by the centre line but also by the side slopes ; that is, sections must be taken so that a side stake may stand on top of 226 FIELD ENGINEERING. each ridge and at bottom of each hollow. In case the centre line intersects at right angles a retaining wall or other vertical surface, two cross sections are required at the same point, one at top and the other at base of wall, in order to furnish the data necessary to calculate the content each way from the ver- tical surface. (See Art. 235.) Every thorough cut terminates in either side-hill cutting, a pyramid, or a wedge ; the latter happens only when the con- tour of the natural surface is at right angles to the line of road. Sections should always be taken through the points where the edges of the road-bed meet the surface, as these are the points of separation between thorough and side hill-work. Such sec- tions also serve to define terminal pyramids when they occur as is illustrated by Fig. 97. In side-hill work the foregoing FIG. 97. rules apply as well, but sections will generally be more numer- ous than in thorough cuts. The same rules apply also to em- bankment, but as grading is preferably paid for in excavation, the same precision in determining the quantities in embank- ment js not usually necessary, CALCULATION OF EAKTHWOiiJv. 255. Formulae for Sectional Areas* Let b = base of section or width of road-bed, horizontal " s = slope ratio = -- : vertical ' ' d = depth at centre stake. " h, Jc = depths at side stakes. " m, n= horizontal distances from centre to side stakes. For ground level transversely, the section is a parallelogram, and the area is evidently (342) or directly from the field notes, A = i(b + m-\-n)d (343) For ground of uniform transverse slope between slope stakes, "? I N A G B~~ H FIG. 98. Fig. 98, the section consists of the parallelogram ABOE and the triangle EOD. Hence A = &AB+ EO)EN+ \EO(DR-EN) A = %(AB . EN H- EO . DH) or also ( (344) From which also and (345) A = \bh 4- mk ) These formulae are independent of the centre depth. They are convenient for calculating the area of a plotted section 228 MELD EXGLBf BERING. having an irregular surface after the surface line has been averaged by stretching a silk thread over it. The points where the thread intersects the slope lines determine the values of h, k, m, and n respectively. When, the ground has uniform slopes transversely from thd centre to the side stakes: Fig. 99: If in the diagram we draw D N. G /B U P Fio. 99. EG and DG, the section will be divided into four triangles, two having the common base CG = d and respective heights GN-* m and GR n, and two having the equal bases AG GB = %b and the respective heights EN = h and DH = k. Hence we have for the area of section (346) Otherwise, if the slope lines are produced to meet below grade at P, then GP = - = ^r-. The area of CEPD is S M (m -f- 7i). The area of ABP is AG X OP = -r- Hence we have for the area of the section 4:8 ~ (347) Both these formulae are convenient, and as the values of the several letters can be substituted directly from the field notes, it is unnecessary to plot such sections. When the surface of the (/round is irregular, verticals are con- ceived to be drawn to the grade line through the slope stakes, CALCULATION OF EARTHWORK. 229 and through each break in the surface line, giving a number of trapezoids, the areas of which are severally calculated, and from their sum is subtracted the area of the two triangles ENA and DHB. The remainder is the area of section required. This calculation may be made directly from the data furnished by the field notes without plotting ; but if the ground has a number of small breaks, it is generally better to plot the sec- tions and stretch an averaging line over them, finding the areas by eq. (345). Or two averaging lines may be employed extend- ing from the centre stake, each way, when the area may be cal- 5 culated by eq. (B4C) or (347). 25G. Prismoidal Formulae for Solid Contents. The content of a prismoid may be exactly calculated by means of the Prismoidal Formula, which is (348) 8 ==. cubic yards, I = length in feet, A, A = the areas at the two parallel ends, and M the area of a section midway be- tween the ends. This area is not a mean of the other two, but the linear dimensions of the mid-section are means of the cor- responding dimensions severally of the end sections; from which therefore the area of the mid-section may be computed. The labor of calculating the middle area may be avoided in many instances by substituting in the prismoidal formula, eq. (348), for A, A, and M, their values as given in eq. (342) for ground level transversely : ,.. A bd-\- sd? A = bd -}- sd 3 M = . ' . 8= 6 27 & s in which S is expressed in terms of the end dimensions. 25-7. Tables of cubic yards may be constructed upon this formula which are very convenient in practice. The constant values in any one table are I which is taken at 100, and b and s which are given values corresponding to the road-bed and slope ratio. The variables are d and d'. The columns in the table 230 FIELD ENGINEERING. will be headed by the successive values of d', while each hori- zontal line will be headed by a value of d. For any one column therefore d' is constant, and the only variable is d. Assuming any value for d', the values of 8 in that column may be computed, letting d take a series of values differing by unity from zero upwards, and the corresponding values of 8 will be placed in the column d' opposite the several values of d. But instead of solving the eq. (349) for each value of 8 re- quired, the process of filling the table may be much abbre- viated by observing that since the equation is of the second degree with respect to the variable d, the second difference of the values of 8 will be a constant and equal to twice the co- efficient of d~*, or d" = - - Also the first term in the series o X 7 of first differences of 8 in the column d' (i.e. between d and d = 1) is expressed by the sum of the coefficients of dJ 1 and d; or The first value of 8 in any column d ' is found by solving eq. (349) after making d = 0; or, Starting with these values we may fill any column d ' simply by successive additions. The values of d' for the several columns should also differ by unity. The final value of 8 in each column should be calculated by formula as a check ; or ^ince all the final quantities in the same line d of the table form a series of which the second difference is d", if on taking their differences this result is obtained, the quantities are proved to be correct. Example. Given a base of 18 feet and slopes H to 1, to fill the column of d' = 6 in a table of cubic yards for level cross sections. Here I = 100, ft = 18, * = f , d' = 6. Hence 8" = 3.7037-f, then d l = d-\-(c l - c} A table may be prepared giving (Ci c) for various inclina- tions of surface with given base and side slopes. It is then only necessary to add this correction to the centre depth to obtain the equivalent depth. Such tables of correction usually accompany the published tables of cubic yards. This method of obtaining quantities is particularly applicable to preliminary estimates, where the ground has not been cross-sectioned, and only the centre depth and transverse inclination is known. 260. The use of the earthwork tables described gives correct results ; 1st. When the surface of the prismoid is a plane, however much inclined; provided it does not intersect the road-bed within the limits of the prismoid. 3d. When with regular, or ihree-levelend sections, generally similar to each other, the surface is regularly warped from one end to the other; provided that the side lines and centre line of the surface are straight, and that no two of them are in- clined to grade in opposite directions. 3d. When the ridges or hollows of an undulating surface are parallel to the line of the road. 4th. When a surface of numerous irregularities may be averaged by planes or warped surfaces so as to comply with one of the preceding conditions. But the method fails on undulating ground when the ridges or hollows run obliquely to the line of road, even though the sections may appear quite regular. In 'general, the method of equivalent depths holds good! when the mid-section of the equivalent level end sections equals in area the actual mid-section or the prismoid; other- wise it fails. 261. The content of a prismoid may be approximately ob- tained by the method of mean areas, the formula for which is i Although approximate, this method is much employed on 234 FIELD ENGINEERING. account of its convenience. It is approved by statute to be used upon the public works of the State of New York. If the values of A and A' derived from eq. (342) be substi- tuted in eq. (352), and then eq. (349) be subtracted from it there remains which is the correction by which 8 obtained by eq. (352) must be diminished to make it equal to 8 obtained by eq. (349) when the ground is level transversely. Again, for three-level sections, if the values of A, A, and M derived from eq. (347) be substituted in both eq. (348) and (352), and one subtracted from the other, there remains [(d -d')(m + n -(m + ri) } ] which is the correction by which 8 obtained by eq . (352) must be diminished to make it equal to 8 obtained by eq. (349). Hence we may write at once, for three level-sections, the cor- rect formula: 8= This formula gives results identical with eq. (349), is applica- ble to the same cases, and gives correct results or fails to do so according to the conditions stated in the previous section. 262. When the conditions of the surface are such that eq. (349) or eq. (353) will not give correct results, the area of the mid-section may be derived from its calculated linear dimensions as stated in 256. The contents of the prisinoid are then given by eq. (348). Example \. (Fig. 101.) Base 20. Slopes li : 1. 22 , , 47.5 .'. A'=. 443 sq. ft. : .'. A = 200 sq.ft. ,, , 28 11 8 31.75 . k M + T2 + IT + IT + -T + -14V CALCULATION OF EARTHWORK. 235 If I = 100, eq. (348) 100 S = jf=- - (443 -V 967 -f 200) = 1001 c. yds. o X &7 Had this been calculated by eq. (349) or eq. (353) or by the FIG. 101. tables, the result would be 1167 c. yds., showing an error of 166 cubic yards in excess. Example 2. (Fig. 102.) Base 20. Slopes 1-J : 1. A' J- . J J-JL 284 sq f 88 sq. i .-|_-1?JL- 164.5sq. 1. 13 ^ +T If Z = 100, eq. (348) 100 8 = 6X2^ + 658 -f 234] = 605 c. yds. Had this been calculated by eq. (349) or eq. (353) or by the 236 FIELD tables, the result would be 584 c. yds., showing an. error of 21 cubic yards in deficit. 263. At the termination of a cut or fill we have usually either a wedge or a pyramid. To a wedge the pre- ceding formulae ami tables based on them apply by making FIG. 102. one end depth equal zero. In the case of a pyramid, the content is equal to the area of the section forming the base multiplied by one third the length of the solid, and divided by 27; or IA < 354) 264. Side-hill Work. When the natural surface has a regular transverse slope and intersects the road-bed, the cross section is reduced to a triangle. If w = the intercepted portion of the road-bed, and k = the side height, then A = \wk. Similarly A' = Iw'k 1 and 4M = $(w -f w') (k -f k'\ which substituted in'eq. (348) give 8 = 12 X (355) which is convenient for direct calculation from the field notes. It is not adapted to the construction of tables, since it contains four independent variables. CALCULATION OF EARTHWORK. 237 If the slope of the natural surface is given, let s be the sur- face slope ratio at one section, and s" that at the other, and s the ratio of the side slope. Then w = k (s' s) and w' = &'(/ _ g) t which substituted in eq. (355) give If the surface is a plane, then s" = s', and we have for thisl ise 7 /' A\ + 7-7,' | 7*'2~1 /QP*^ KK "~f fC I ^OOUJ which is a formula of quite limited application ; yet it is the one on which tables and diagrams are usually constructed. Consequently the latter will not give correct results, except when the surface is a plane. 265. When the natural surface is broken the sections may be plotted, and the values of w and k taken from the points where an averaging line intersects the grade and side slope respectively. Finding values for w' and k' in the same way, the content may then be obtained by eq. (355) as before. The averaging line should not only cut off the same area as the original section, but should also have in each case a slope agreeing as nearly as possible with the general slope of the natural surface. The slope is determined simply by inspection of the diagram, but the area may be had pre- cisely, for, taking w from the averaging line, and knowing A, 2 A we may calculate k by the formula k = ; or k may be taken from the plot ami w calculated. Otherwise, the actual mid-section may be calculated and the cubic contents determined by the method illustrated in 262. 266. To express side-hill areas and cubic yards in terms of the centre depth, d, and transverse slope-ratio s'. Fig. 103. 238 FIELD ENGINEERING. For any depth d, add to this area s'd and there results, ' o> 2(8 - S) (357) 2 X 27(*' -s) J Observe that d may be phis or minus, and that its limits are Tables of cubic yards may be constructed on this formula, making d and *' the variables, which would be extremely con Fia. 108. venient for making up estimates upon preliminary lines on which the profile of centre line and angle of transverse slope lonly are known. Since s' is the cotangent of the slope angle ; the columns of the table may be headed by the angles in a series of degrees, while the corresponding values of s' are used in the formula. The values of d should vary by tenths of a foot. The results obtained by eq. (356) and eq. (357) will be identical for the same sections. 267. Several different systems of diagrams have been devised and published for determining quantities in earthwork by a sort of graphical method. These diagrams, which are substitutes for tables are preferred by some engineers. They CALCULATION OF EARTHWORK. 239 are based on the same principles, and are constructed on modi- fications of the same formulae. 268. Correction of Earthwork for Curvature. The preceding calculations are based on the assumption that the centre line is straight, with cross sections at right angles to it. When an excavation is on a curve, the cross sections, be- ing in radial planes, are inclined to each other, so that the con- dition of a prismoid is not exactly fulfilled. But by the proper- ty of Guldinus,. if any plane area is made to revolve about an axis in the same plane, the volume of a solid generated by the area is equal to that of a prism having a base equal to the given area, and a height equal to the length of path described by the centre of gravity of the area. The path, being the arc of a cir- cle, is proportional to the radius drawn to the centre of gravi- ty. If therefore a cross section is symmetrical with respect to the centre line, the path of the centre of gravity is equal to the measured length of the centre line, and no correction for cur- vature is required. But when the ground is inclined transversely, the centre of gravity is one side of the centre line, and its path, if we con- ceive it to sweep around the curve, from one end of a prismoid to the other, is longer or shorter than the distance measured on the centre line, according as the centre of gravity is outside or inside of the centre line curve. Let C = correction in cubic yards due to curvature. " S = cubic yards as obtained by prism oidal formula. " R radius of centre line. " e = eccentricity of centre of gravity of section. = horizontal distance from centre line to centre ol gravity. We then have the proportion, 8C : 8:: Ee : R -I As the sections of a solid are seldom similar and equal, we 11 usually have a different value of e for every section, from 240 FIELD ENGINEEKIKG. which, however, a mean average value may be deduced, and used in the above formula. But it will be more convenient to correct the areas themselves for eccentricity before finding 8, which will then require no correction. For the same result will ensue whether we multiply S by -~-, or multiply one of the component factors of 8 by the same ratio. If then c = correction of area in square feet due to eccentri- city, we have at once Ae and the corrected area equals A c according as the cut is deeper on the outside or inside of the curve. Each area used in determining the solid contents should, on a curve, be first corrected in this manner. To find the value of e for any three-level section. Fig. 104. D FIG. 104. Find the areas either side of the centre line separately, call ing them Hand K, and take their sum and difference. Using the same notation as in 255, H = \md -f- $bh, K = \nd -j- /iT = A. K -H = & (k - h) In the figure draw OE' equal to OE, and the triangle CE'D will represent the area (K H). Bisect the side E'D, and draw a line from C to the middle point. Then the centre of CALCULATION" OF EARTHWORK. 241 gravity of the triangle will be on this line at two thirds its length from 0, and the horizontal distance of the centre of gravity from C is f X \(m -f n) - %(m -f ). The centre of gravity of the remainder of the section is on the centre line CG, so that the value of e is found from the proportion :: K - H : A n-\-m - Hence (358) Sections which are more irregular may be plotted and reduced by averaging lines to three-level sections, in order that the formula may be applied. If the ground is so irregular as to require the computation of the middle section, the correc- tion c should be found and applied to this area (M ) also I'before introducing it into the prismoidal formula. As the j correction for curvature is always relatively small, it is usually ; ignored in practice for thorough cuts, except where deep cut- - tings with steep transverse slope occur on sharp curves. The correction is of more importance relatively in side-hill work as the centre of gravity of the section is more remote from the centre line. Let the section be reduced FIG. 105. a triangle by an averaging line (Fig. 105), and w be the base of the triangle formed by the averaging line. The centre of gravity is at one third the horizontal distance from the middle point of ic to the srde stake D, while the distance of this middle point from the centre stake C is evidently |5 $w. %% FIELD Hence 6 = & - fr + i[ - (& - or e = H& + n - ^4d 6 + ra w and 6 = - = - -- X The correction c will be plus or mVm as before explained. This formula applies to all side-hill triangular sections, whether there be cut or fill at the centre stake Example 1. Thorough cut; base 20; slopes 1| : 1. Notes. I = 100; 8 curve, left; R = 716.78 16 . 12 . 58 Then i-=ix68Xl3 + iX20X82 = 508 H=$ X 16 X 12 .+ i X 20 X 4 = 116 .-. A = 634 K-H= 392 E q .(358) C = (^1 + c) 637.49 ^' = | X 40 X 8 -j- i X 20 X 20 = 260 fi-' = | X 13 X 8 + i X 20 X 2 = 62 . '. ^' = 322 JT-JI= 198 _ _13 + 40 4 gv ~ 3 X 716.78 (^ + c') = 326. 8? From which we obtain /fif = 1758 cub. yds. Ans. Without correction we have 1726 " Showing a difference of 32 " " Had the curve been to the right with same notes, c would have been minus, and 8 would = 1694. CALCULATION OF EARTHWORK. 243 Example 2. Side-hill cut; base 20; slopes U : I I = 60; 10 curve, right; R - 573.69 6 , 40 Notes _0 _2 jtf 0.8""~0.0 + 18 A = | X 16 X20 - 160 > -Siro 5 ' _^ (A - c) - 156.42 A' 1 X 8 X 18 = 72 20 + 37-8 _ O r> . . ~^. nr. t& *.UO (A' - c)= 69.95 : Hence 8 = 248 cub. yds. Without correction 8 would = 255 " " Difference 7 " " 269. Haul. The cost of removing excavated material, when the distance does not exceed a certain specified limit, is included in the price per cubic yard of the material as meas- ured in the cutting. But when the material must be carried beyond this limit, the extra distance is paid for at a stipulated price per cubic yard, per 100 feet. The extra distance is known by the name of haul, and is to be computed by the engineer with respect to so much of the material as is affected by it. The contractor is entitled to the benefit of all short hauls (less than the specified limit), and material so moved should not 1 be averaged against that which is carried beyond the limit. Therefore, in all cuts, the material of which is all deposited within the limiting distance, no calculation of haul is to be nade. On the other hand, the company is entitled, in cases of long laul, to free transportation for that portion of the cutting, no me yard of which is carried beyond the specified limit. There- ore, this portion is first to be determined in respect to its ex- ent; and the number of cubic yards contained in it is to be de- 244 FIELD ducted from the total content of the cutting, before estimating the haul upon the remainder. Find on the profile of the line two points, one in excavation, and the other in embankment, such, that while the distance between them equals the specified limit, the included quantities of excavation and embankment shall just balance. These points are easily found by trial, with the aid of the cross sections and calculated quantities, and be- come the starting points from which the haul of the remainder of flie material is to be estimated. Fig. 106 represents a cut and fill in profile. The distance AB is the limit of free haul. The materials taken from A just make the fill OB and without charge for haul ; but the haul of every cubic yard taken from AG, and carried to the fill BD, is subject to charge for the distance it is carried, less AB. It would be impossible to find the distance that each separate yard is carried, but we know from mechanics that the average dis- tance for the entire number of yards is the distance between the centres of gravity of the cut AC, and of the fill BD which is made from it. If, therefore, JTand T represent the centres of gravity, the actual average haul is the sum of the distances (AX-\-BT), and this (expressed in stations) multiplied by the number of cubic yards in the cut AC, gives the product to which the price for haul applies. But the product of AX by the number of cubic yards in AG is equal to the sum of the products obtained by multiplying thei contents of each prismoid in AC by the distance of its own centre of gravity from A. The distance of the centre of gravity of a prismoid from its mid-section is expressed by the formula^ If we replace 8 by its approximate value, "0 which will produce no important error in this case, we have 1 A A CALCULATION OF EAKTHWOBK. in which A should always represent the more remote end area from the starting point A, fig. 106. Hence, x may be -f- or , and it must be applied, with its proper sign, to the distance of the mid-section from the starting point A, before multiplying by the contents 8. Each partial product is thus obtained. By a similar process with respect to the prismoids composing the mass BD, and using the point B as the starting point, we obtain finally a sum of the products representing this portion of the haul. If a cut is divided, and parts are carried in opposite direc- tions, the calculation of each part terminates at the dividing line. If a portion of the material in A C is wasted, it must be deducted, and the haul calculated only on the remainder. The specified limit is sometimes made as low as 100 feet, sometimes as high as 1000 feet. A limit of about 300 feet, how- ever is usually most convenient, as it includes the wheelbarrow work, and a large part of the carting, while it protects the con- tractor on such long hauls as may occur. 27O. The Final Estimate is a complete statement in detail, of the amount of work done and materials provided, in the construction of the road, and is the basis of final settlement between the company and contractor. Its preparation should be begun as soon as possible after the work is in progress, and should be continued, as fast as the necessary data are accumu- lated, while the circumstances are still fresh in mind, and when any omissions in the field notes may be readily supplied. The content of each prismoid, the classification of its material, and the length of haul to which it is subject, should be matters of j special record in a book provided for that purpose. These re- j suits having been carefully computed by exact methods form I a standard of comparison for those approximate results which I must be had from time to time during the progress of the work, and furnish a limit to the amounts of the monthly estimates. i The same remark applies to all other items of labor and mate- rial. The notes and record of the final estimate should be par- ticularly full and exact in respect to all such items as will be inaccessible to measurement at the completion of the work, such as foundation pits, foundation courses of masonry, cul- verts, and works under water. 24G FIELD E^GI^EERIKG. 271. Monthly Estimates. On or before the last day of every month during the progress of construction, measure- ments are taken to determine the total amount of work done and material provided up to that date. The estimates based on these measurements are called Monthly Estimates. It is fre- quently necessary to take measurements for both monthly and final estimates at other times than the end of the month, as in the case of foundations which are not long accessible. With respect to each piece of work satisfactorily completed, the monthly estimate should be exact, and identical in amount with the final estimate. With respect, however, to items of work in progress at the time of measurement, the monthly estimate is only approximate, yet should be as precise as the nature of the case will allow; and the quantities stated should not be in excess of fair proportion of the total quantities given on the final estimate for the same piece of work. A special field l>ook is devoted to monthly estimate notes. Each page should be dated with the day on which the notes upon it were taken. The notes consist of measurements of all sorts, principally of cross sections partially excavated. These sections should be at the same points on the line as the original sections, so that comparisons may be made. Where- ever the excavation is finished to grade, it is only necessary to write "completed" opposite such stations, and the quantities may be taken from the final estimate or computed from the original notes. It is frequently necessary to retrace portions of the centre line in taking estimate notes, so that all the field instruments are required, but a party of three or four men is usually sufficient. If the contractor has provided materials, such as stone, lum- ber, etc., which are not as yet put into any structure when the estimate is taken, these should be measured and entered under the head of temporary allowance, an arbitrary price be- ing used somewhat below the actual value of the material as delivered. Such allowances should never be copied from one month's estimate to the next, but made anew on such material as may be found that seems to require it. But all completed items of contract work, and of extra work when ordered by the engineer, are necessarily copied from one monthly esti- mate to the next during the continuance of the contract. A blank form is used by the resident engineer in report TOPOGRAPHICAL SKETCHING. 247 ing monthly estimates, on which a column is provided for each class of material and work required by the contract, while the several lines, headed by the numbers of the proper stations, are devoted to the different cuttings, structures, etc., in consecu- tive order as they occur on the line of road. The estimates are made out and reported separately for the several sections into which the line of road is divided for letting. These reports are reviewed by the division engineer, and the footings copied upon another blank, which is the monthly estimate proper ; the prices are attached to the items, and the amounts extended and summed up. This sum indicates ap- proximately the total amount earned by the contractor up to date, from which is deducted a certain percentage (usually 15 per cent.), which is retained by the company until the comple- tion of the contract. From the remainder is deducted the amount of previous payments, which leaves the amount due the contractor on the present estimate. A blank form of re- ceipt is appended, to be signed by the contractor. CHAPTER XI. TOPOGRAPHICAL SKETCHING. 272. Topographical sketches taken on preliminary surveys are usually of great value in projecting a line for location; they should be made therefore as accurate and complete as possible. In too many instances sketches are presented having a picturesque appearance, but conveying little information, if not tending to mislead the map-maker. The aim of the topog- rapher should be to record the topographical features either side of the line with as much precision as those directly upon iihe line, without taking actual measurements, except in rare instances. The eye and the judgment must be usually depended oii for distances and dimensions. The sketch of a tract ex- tending to 400 feet each side of the line ought to be accurate enough to warrant its being copied literally upon the map. If a much wider range is required it may be advisable to use the plane-table; but an approximation to nta.D^able methods may be employed in ordinary sketching. 248 FI^LD ENGINEERING. 273. As artificial features are the most readily de- fined and located these should first receive attention in making a sketch. When recorded they form a skeleton upon which the natural features can be drawn with more precision than if the order were reversed. The point where each fence crosses the line and the angle between the two may be sketched exact- ly. The distance along the fence to any object may be esti- mated, and checked (in case of an oblique angle) by observing where a line from the object perpendicular to the centre line would intersect the latter. Tlie book may be rested on any support, the centre-line of the page coinciding with the line of survey, and the direction of objects defined by a small ruler laid on the page. This operation being repeated from another point gives intersections which locate the several objects on the sketch. If the bearings are taken they may be plotted on the page as well as recorded, giving the same results. The eye may be trained to estimate distances correctly by observ- ing the appearance of objects along the measured line, the dis- tances to which are therefore known. 274. After the artificial objects the more distinct natural features are to be sketched, as streams, shores, margins of swamps, forests, etc., ravines, ridges, and bluffs, taking care that all these outlines intersect the features of the sketch already delineated at the proper points. The correct repre- sentation of contours is the most difficult part of sketching, since these lines are quite imaginary, yet for railroad maps they are usually as important as any others. It is desirable to know not only the locality of a hill or slope, but also its shape, steepness, and height. This information is best given by con- tour lines. A contour is the intersection of the surface of the ground by an imaginary level surface. When the surface is real, like that of a lake, the intersection is called a shore. If the water should rise a certain height a new shore would be defined, and rising double that height still another shore would result, each of which, on the subsidence of the water, would be a contour. A practiced eye is able to follow on the ground the course of a contour with all its windings; but m sketching them due allowance must be made for the fore- shortening effect of distance. All contours on the same sketch should have the same vertical interval, so that by counting TOPOGRAPHICAL SKETCHING. 249 fliem the height of the hill may be known. The spaces on the sketch between contours vary as the cotangent of the slope angle, so that the width of the spaces indicates the degree of steepness. The contours nearest the topographer should gene- rally be sketched first, although if there be a shore that is apt to be the best guide to the shape of the slopes. If the height of the hill is known and the upper contour located, the other contours can be spaced between with less difficulty, the proper number being ascertained by dividing the height by the assumed verti- cal interval. A special line of levels up an inclined ravine or sloping ridge to fix the contour points is often of the greatest service in obtaining correct results. Other random lines are sometimes run to locate the contours more definitely. These should be made to cross several contours rather than to trace a single one. Old preliminary lines which have proved useless in themselves often furnish by their profiles valuable informa- tion in respect to contours. The use of hatchings should be avoided in the sketch-book, except to represent precipitous banks, or slight terraces, which would not be sufficiently defined by the contour system. Hatchings freely used consume too much time, and fail to give an accurate idea of either slope or height, while they obscure the page for the representation of other objects. 275. Tlie centre line on the page is straight, and for sketching purposes the surveyed line on the ground is assumed to be so also. Slight deflections in the course of a preliminary line may be ignored in the sketch; but if a large angle occurs it is better to terminate the sketch with the course, and begin again, leaving a few blank lines between the two sketches. jOn a located line with curves, the sketch is continuous. The I curved line in the field is represented by the straight line on I the page, and the radial lines through the stations are repre- Isccted by the parallel lines ruled across the page. All objects [lire sketched at the proper offset distance by scale from the jpentre line; but longuudinally the sketch is necessarily dimin- lushed outside of the curve, and magnified inside of the curve. I; Consequently topographical lines which are straight in fact ap- j pear curved in the sketch, concave to the centre line if inside hhe curve, and convex if outside of it. Such features are cor- [irectiy sketched by means of offsets estimated or measured 250 FIELD from each station of the curve on the radial lines. This kind of distortion creates no confusion if properly done, for in mak- ing the map, after drawing the curve and the radial lines, the same offsets will give the correct positions of the objects delin- eated. This method is preferable to drawing a curved line on the page to represent the centre line, as it is difficult to draw it correctly; it will cross the ruled lines obliquely, rendering them of no service for offsets or scale, and moreover is likely to run off the page altogether. CHAPTER XII. ADJUSTMENT OF INSTRUMENTS. Every adjustment consists of two processes: first the test, and second the correction. Inasmuch as the amount of correction is made by estimation, the test must always be repeated until no further lack of adjustment is observable. 276. THE TRANSIT. The level tubes should be parallel to the vernier plate. Test : Place the tubes in position over the levelling screws, and turn the latter till the bubbles are centred; revolve the plate 180. The bubbles should remain centred; if they have retreated Correction : Bring them half way back to the centre by turning the adjusting screws which attach the tubes to the plate. The line of collimation should be perpendi- cular to the horizontal axis. Test : Clamp the limb, and by the tangent screws bring the intersection of the cross-hairs to cover a well-defined point about on a level with the telescope ; plunge the telescope to look in the opposite direction, and note any point about on a level with the telescope and about equidistant with the first point, which the intersection of the cross-hairs now happens to cover. Now unclamp the limb and turn through 180, and repeat the above operation, using the same first point as before. ADJUSTMENT OF INSTRUMENTS. 251 ^ The third point obtained should be identical with the second; if not- Correction : Move the vertical cross-hair over one fourth of the apparent distance from the third to the second point, by turning the adjusting screws at the side of the telescope. The horizontal axis should be parallel to the vernier plate. Test : After completing the above adjustments level the limb, clamp it, and bring the intersection of the cross-hairs to cover some high point so that the telescope may be elevated to a large angle; depress the telescope and note sonic point on the ground now covered by the intersection of the cross-hairs. Now unclamp the limb, turn it through 180, and repeat the above operation, using the same high point as before. The third point found should be identical with the second; if not Correction : Raise the end of the axis opposite the second point (or lower the other end) by a small amount, by turning the adjusting screws in the standard. The amount of motion required is only determined by repeated trials until the test is satisfied. The intersection of the cross-hairs should appear in the centre of the field of view. Test : Bring the cross-hairs into focus and direct the tele- scope toward the sky, or hold a sheet of blank paper in front of it. If the intersection appear eccentric Correction : Turn the screws (by pairs) which support the end of the eyepiece until the desired result is obtained. If there be a level on the telescope it should be parallel to the line of collimatioii. Drive two stakes equidistant from the instrument in exactly opposite directions, and having perfected the previous adjust- ments, tevel the plate carefully, clamp the telescope in about a horizontal position, and observe a rod placed on each stake. Have the stakes driven by trial until the rod reads alike on both. The heads of the stakes are then on a level. Re- move the instrument beyond one stake, and set it up in line with the two, level the plate, and elevate or depress the telescope to a position which will again give equal readings on the stakes. The line of collimation is now level 252 FIELD Test : While in this position the bubble of the attached level should stand centred; if not Correction : Bring the bubble to the centre by turning the nuts at one end of the tube, while the cross-hair continues to give equal readings. 277. THE Y LEVEL. The line of collimation should coincide with the axis of the telescope. Test : Clamp the spindle, and bring the intersection of the cross-hairs to cover a well-defined point by the tangent and levelling screws ; revolve the telescope half over in the Ys, so that the level tube is on top. The intersection of the cross- hairs should still cover the point. If either hair has departed Correction : bring it half way back by means of the pair of adjusting screws at the extremities of the other hair. The attached level should be parallel to the axis of the telescope. Test : Bring the telescope over one pair of levelling screws, clamp the spindle, open the clips, and bring the bubble to the centre. Then gently remove the telescope from the Ys, and replace it end for end. If the Ys have not been disturbed, the bubble should return to the centre. If it does not Correction : bring the bubble half way back by turning the nuts at one end of the tube. But as now the level tube and telescope may only lie in parallel planes, and yet not be parallel to each other Test : bring the bubble to the centre as before, and turn the telescope on its axis so as to bring the level tube out to one side. The bubble should remain centred. If it has departed Correction : bring it back to the centre by the adjusting ^screws at one end. The axis of the telescope should be at .right angles to the spindle. Test: Having completed the above adjustments (and not before), fasten down the clips, unclamp the spindle, and bring the bubble to the centre over each pair of levelling screws in succession, then swing the telescope end for end on the spin- dle. The bubble should settle at the centre. If it do not Correction : bring it half way back by the large nuts at one end of the bar. EXPLAJSAT1OX OF TABLES. 253 278. THE THEODOLITE. This instrument being a combination of Transit and Level, its several adjustments are to be made according to the rules already given for those instruments. CHAPTER XIII. EXPLANATION OF TABLES. TABLE I. Contains concise statements of such geometrical truths as are applicable to the various discussions in this volume. References are given to Davies' Geometry, in which the demon- strations of the propositions may be found. TABLE II. Contains all the formulae necessary to the solu- tion of any plane triangle; also, a select list of miscellaneous formulae. A few formula? with respect to the versed sine and external secant are new. TABLE III. Contains a complete list of formulae expressing the relations between the radius, tangent, chord, versed sine, external secant, and central angle of a railway curve; also, the relations between the radius, degree of curve, length of curve, and central angle. The notation is identical with that used elsewhere in the book. TABLE IV. Contains the radius, and its logarithm, tor every degree of curve to single minutes up to 10 degrees, and thence by larger intervals up to 50 degrees. With the radius is given also the perpendicular off-set, t, from the tangent to a point on the curve at the end of the first 100-foot chord from the tan- gent-point, and the middle ordinate, m, of a 100-foot chord. See eqs* (16, 34, 37, 40, and 305). TABLE V. Contains tl3 corrections to be added to the tan- gents and externals of any railroad curve, as obtained by refe- rence to Table VI., according to the degree of the given curve (found at head of columns), and its central angle, (found in the 254 FIELD ENGIKEEK1KG. first column.) If the given degree of curve, or central angle. does not appear in the table, the exact value of the correction may be easily obtained by interpolation. TABLE VI. Contains the exact values of the tangents, T, and externals, E, to a 1 degree curve, for every 10 minutes of central angle, from 1 to 120 50'. Approximate values of the tangent and external to any other degree of curve may be had by simply dividing the tabular values opposite the given cen- tral angle by the given degree of curve, expressed in degrees. These approximations may be made exact by adding the proper corrections taken from Table V. See eqs. (21) and (24). TABLE VII. Contains the value of Long Chords of from 2 to 12 stations, for every 10 minutes of degree of curve from to 15, and of a less number of stations for degrees of curve be- tween 15 and 30. As the chord of one station is always 100 feet, the column of the first station gives instead the length of arc subtended by the chord of 100 ffeet. See 121, 122, 123, 124, 125. TABLE VIII. Contains the values of Middle Ordinates to long chords of from 2 to 12 stations, for every 10 minutes of degree of curve from to 10, and of from 2 to 6 stations for every curve from 10 to 20, at 10-minute intervals. The table may be used, not only to fix the middle point of an arc, but also, in conjunction with the table of long chords, to locate in- termediate stations. See 121, 122, 123, 124, 125. TABLE IX. Contains the chords of a series of angles vary- ing by half degrees up to 30 for radii varying by 100 feet up to 1000 feet. It shows, therefore, the linear opening between the extremities of two equal lines at any given number of hun- dred feet from their intersection, when the angle does not ex- ceed 30. For any distance exceeding 1000 we have only to add to the value found in that column, the value found in the column headed by the excess of distance over 1000 feet. Con- versely, the table gives the angular deflection required between two equal lines, in order that at a given distance from the point of intersection they may be separated a given amount. EXPLANATION OF TABLES. TABLE X. 1. Contains values of the ratio u = , accord- A ing to the notation of 147 for finding the angle i (Fig. 34) between the radius PO of the curve at any point P, and the tangent PK to the valvoid arc PX by the simple formula eq. (80) i = 11 A . The table embraces lengths of curve from 300 to 2000 feet, and central angles from 10 to 120. When -- = 60 u = , and for hasty approximation this 1000 value of u may be assumed in any case without consulting the table. v 2. Contains values of the ratio v = - for finding the radius of the valvoid arc at the point P (Fig. 35) in terms of the length of curve L AP by the simple formula, eq. (82), r vL. 3. Contains values of the length l t of a valvoid arc limited by two curves of equal length laid out from the same tangent and same P. C. , but whose central angles differ by 1. The length L of each curve is given in the first column, and the half sum of their central angles I ~ I is given at the head of the other columns. When the central angles of two curves of equal length differ by x degrees the length I of the valvoid arc joining their extremities is expressed by the simple formula, Fig. 36, eq. (86) I = P'P" = (A' - A"K in which I, is taken from the column headed by - -J- and opposite the given value of Z;or l t is found by inter polation if necessary. See 150 and example. TABLE XL Contains the measurements necessary to lay down a turnout with frogs of given numbers or angles for both a standard and a three-foot gauge. The distance BF is measured on the rail of the given track from the heel of the switch to the point of the frog, while af is the chord of the centre line of the turnout between the same points. The radius r applies to the centre line of the turnout. The dis- tance aF" is measured on the centre line of the straight track 25G FIELD ENGINEERING. from tJie heel of the switch to the point of the middle frog. The length of switch AD should conform to the tabular values unless the throw is to be different from that assumed in the table. See 180, 181, 182. TABLE XII. Contains the middle ordinates of chords vary- ing in length from 10 to 83 feet, and for degrees of curve vary- ing from 1 to 50. The use of the table is obvious. See 199. TABLE XIII. Gives the proper difference in elevation of rails on curves of various degrees from 1 to 50 for veloci- ties varying from 10 to 60 miles per hour. See 201. TABLE XIV. Gives the rise of grades in feet per mile and their angle of inclination corresponding to a rise per station (100 feet) varying from 0.01 foot to 10 feet. TABLE XV. Contains values of the formula (log h 1) 60384.3 in which h reading of the barometer in inches. The inches and tenths of the readings are in the left-hand column, while the hundredths are found at the top of the other columns. The difference of any two values corresponding to two read- ings taken simultaneously at any two stations is the differ- ence in elevation in feet of those stations. But the differ- ence in height so found is subject to a correction for tempera- ture given in the next table. See 10. TABLE XVI. Contains coefficients of correction for atmos- pheric temperature, by which the approximate heights ob tained by Table XV. are to be multiplied for a correction of these heights, which correction is to be added or subtracted according as the coefficient given in the table is marked for-. See 11. TABLE XVII. Contains corrections in feet, required by the curvature of the earth and the refraction of the atmosphere, to be applied to the elevation of a distant object as obtained by a level or theodolite observation for distances ranging from 300 feet to 10 miles. See 219. TABLE XVIII. Contains the coefficients for reducing the space on a vertical rod intercepted by the stadia hairs when EXPLANATION" OF TABLES. 25? the line of collimation is inclined to the horizon, to the space that would be intercepted were the line of collimation horizon- tal; provided, that the visual angle defined by the stadia hairs is such that tan 40 = .005 or = 34' 22". 63, which is its customary value in surveying instruments. The angle of in- clination a is taken at every 10 minutes through half a quad- rant. TABLE XIX. Contains the logarithms of the eoefticien' : given in Table XVIII. TABLE XX. Gives the lengths of circular arcs to a radius = 1. To find the length of any arc expressed in degrees, minutes, and seconds, take from the table the lengths of the given num- ber of degrees, minutes, and seconds respectively, and multi- ply their sum by the length of the radius. The product is the length of arc required. TABLE XXI. Contains the values of minutes and seconds expressed in decimals of a degree, for every 10 seconds of arc, and also for quarter minutes up to one degree. TABLE XXII. Contains the values of inches and fractions expressed in decimals of a foot for every 32d of an inch up to one foot. TABLE XXIII. Contains the squares, cubes, square roots, cube roots, and reciprocals of numbers from 1 to 1054. Its use may be greatly extended by observing that if any number is multiplied by n its square is multiplied by n*, its cube by 7i s , and its reciprocal by . TABLE XXIV. The logarithm of a number consists of two parts, a whole number called the characteristic, and a deci- mal called the mantissa. All numbers which consist of the same figures standing in the same order have the same man- tissa, regardless of the position of the decimal point in the number, or of the number of ciphers which precede or follow the significant figures of the number. The value of the char- acteristic depends entirely on the position of the decimal point in the number. It is always one less than the number of JJ5b FIELD ENGINE BRING. figures in the number to the left of the decimal point. The value is therefore diminished by one every time the decimal point of the number is removed one place to the Jefl, and wte versa. Thus Number. Logarithm, 13840. 4.141136 1384.0 3.141136 138.40 2.141136 13.84 1.141136 1.384 0.141136 .1384 1.141136 .01384 2.141136 .001384 3.141136 etc. etc. The mantissa is always positive even when the characteristic is negative. We may avoid the use of a negative characteristic by arbitrarily adding 10, which may be neglected at the closf of the calculation. By this rule we have Number. Logarithm. 1.384 0.141136 .1384 9.141136 .01384 8.141136 .001384 7.141136 etc. etc. No confusion need arise from this method in finding a number from its logarithm; for although the logarithm 6.141136 repre- sents either the number 1,384,000, or the decimal .0001384, yet these are so diverse in their values that we can never be uncer- tain in a given problem which to adopt. The table XXIV. contains the mantissas of logarithms, car- ried to six places of decimals, for numbers between 1 and 9999, inclusive. The first three figures of a number are given in the first column, the fourth at the top of the other columns.- The first two figures of the mantissa are given only in the second column, but these are understood to apply to the remaining four figures in either column following, which are comprised between the same horizontal lines with the two. If a number (after cutting off the ciphers at either end) con- sists of not more than four figures, the mantissa may be taken direct from the table ; but by interpolation the logarithm of a number having six figures may be obtained. The last column contains the average difference of consecutive logarithms on EXPLANATION OF TABLES. 259 the same line, but for a given case the difference needs to be verified by actual subtraction, at least so far as the last figure is concerned. The lower part of the page contains a complete list of differences, with their multiples divided by 10. To find the logarithm of a number having- six figures : Take out the mantissa for the four superior places directly from the table, and find the difference between this mantissa and the next greater in the table. Add to the man tissa taken out the quantity found in the table of proportional parts, opposite the difference, and in the column headed by the fifth figure of the number; also add y^ the quantity in the col- umn headed by the sixth figure. The sum is the mantissa required, to which must be prefixed a decimal point and the proper characteristic. Example. Find the log of 23.4275. For 2342 mantissa is 369587 " diff. 185 col. 7 129.5 " " " " 5 9.2 Ans. For 23.4275 log is 1.369726 The decimals of the corrections are added together to deter- mine the nearest value of the sixth figure of the mantissa. To find the number corresponding to a given logarithm. If the given mantissa is not in the table find the one next less, and take out the four figures corresponding to it; divide the difference between the two mantissas by the tabu- lar difference in that part of the table, and annex the figures of the quotient to the four figures already taken out. Finally, place the decimal point according to the rule for characteristics, prefixing or annexing ciphers if necessary. The division re- quired is facilitated by the table of proportional parts, which furnishes by inspection the figures of the quotient. Example. Find the number of which the logarithm is 8.263927 8.263927 First 4 figures 1836 from 263873 Diff. 540 Tabular diff. =236 .'. 5th fig. =2 47.2 6.80 6th fig. = 3 _7.0_8 Ans. No. .0183628 or 183,623.000. 260 FIELD ENGINEERING. The number derived from a six-place logarithm is not reliable beyond the sixth figure. At the end of table XXIV. is a small table of logarithms of numbers from 1 to 100, with the characteristic prefixed, for easy reference when the given number does not exceed two ? digits. But the same mantissas may be found in the larger table. TABLE XXV. The logarithmic sine, tangent, etc. of an arc is the logaritum of the natural sine, tangent, etc. of the same arc, but with 10 added to the characteristic to avoid negatives. This table gives log sines, tangents, cosines, and cotangents for every minute of the quadrant. With the number of degrees at the left side of the page are to be read the minutes m the left-hand column ; with the degrees on the right -hand side are to be read the minutes in the right-hand column. When the degrees appear at the top of the page the top headings must be observed, when at the bottom those at the bottom. Since the values found for arcs in the first quad- rant are duplicated in the second, the degrees are given from to 180. The differences in the logarithms due to a change of one second in the arc are given in adjoining columns. To find the log. sin, cos, tan, or cot of a given arc. : Take out from the proper column of the table the log- arithm corresponding to the given number of degrees and minutes. If there be any seconds multiply them by the ad- joining tabular difference, and apply their product as a cor- rection to the logarithm already taken out. The correction is to be added if the logarithms of the table are increasing with the angle, or subtracted if they are decreasing as the angle in- creases. In the first quadrant the log sines and tangents in- crease, and the log. cosines and cotangents decrease as tbt angle increases. Exompk Find the log sin of 9 88' 20". Log sin of 9 28' is 9.216097 Add correction 20 X 12.62 252 Ana. 9.216349 Example. Find the log cot of 9 28' 20". Log cotan of 9 28' is 10.777948 Subtract correction 20 X 12.97 259 Ana. 10777689 EXPLANATION- OF TABLES. 261 To find the angle or arc corresponding to a given logarithmic sine, tangent, cosine, or co- tangent. If the given logarithm is found in the proper column take out the degrees and minutes directly; if not, find the two consecutive logarithms between which the given logarithm would fall, and adopt that one which corresponds to the least number of minutes; which minutes take out with the degrees, and divide the difference between this logarithm and, the given one by the adjoining tabular difference for a quo- tient, which will be the required number of seconds. With logarithms to six places of decimals the quotient is not reliable beyond the tenth of a second. Example. 9.383731 is the log tan of what angle? Next leas 9.383682 gives 13 36' Diff. 49.00 -h 9.20 = 05 ".3 Ans. 13 36' 05".3 Example. 9.249348 is the log cos of what angle? Next greater 583 gives 79' 46' Diff. 235 -* 11.07 = 20 M Ans. 79 40' 20M The above rules do not apply to the first two pages of this table (except for the column headed cosine at top) because here the differences vary so rapidly that interpolation made by them in the usual way will not give exact results. On tbe first two pages, the first column contains the number of seconds for every minute from I'to2; the minutes are given in the second, the log. sin. in the third, and in the fourth are the last three figures of a logarithm which is the difference between the log sin and the logarithm of the number of sec onds in the first column. The first three figures and the clur acteristic of this logarithm are placed, once for all, at the iicac of the column. To find the log sin of an arc less than 2 given to seconds. Reduce the given arc to seconds, and take the logarithm of the number of seconds from the table of loga- rithms, and add to this the logarithm from the fourth column opposite the same number of seconds. The sum is the log sin required. The logarithm in the fourth column may need a slight inter- 262 FIELD ENGINEERING. polation of the last figure, to make it correspond closely to the, given number of seconds. Example. Find the log sm of 1 39' 14". 4. 1 39' 14".4 = 5954".4 . log 3.774838 add (q-t) 4.685515 Ans. log sin 8.460353 Log tangents of small arcs are found in the same way, only taking. the last four figures of (q I) from thejifth column. Example. Find the log tan of 52' 35". 52' 35" = (3120" + 35") = 3155" log 5.498999 add (q - I) 4.685609 Ans. log tan 8. 184608 To find the log cotangent of an angle less than 2 given to seconds. Take from the column headed ( q-\- 1) the logarithm corresponding to the given angle, interpolating for the last figure if necessary, and from this subtract the loga- rithm of the number of seconds in the given angle. Example. Find the log cotan of 1 44' 22". 5. q + I 15.314292 6240" + 22". 5 = 6262.5 log 3.796748 Ans. 11.517544 These two pages may be used in the same way when the given angle lies between 88 and 92, or between 178 and 180; but if the number of degrees be found at the bottom of the page, the title of each column will be found there also; and if the number of degrees be found on the right hand side of the page, the number of minutes must be found in the right hand col- umn, and since here the minutes increase upward, the number of seconds on the same line in the first column must be dimin- ished by the odd seconds in the given angle to obtain the nurn ber whose logarithm is to be used with (q I) taken from the table. Example. Find the log cos of 88 41' 12". 5 (q -I) 4.685537 4740" - 12".5 = 4727.5 log 3.674631 Ans. 8.360168 EXPLANATION OF TABLES. 263 Example. F'mtl the log tan of 90 30' 50'. q-\-l 15.314413 1800" -f 50" = 1850' log_3._267172 Ans. T2J047241 To find the arc corresponding- to a given log sin, cos, tan, or cotaii which falls within the limits of the first two pages of Table XXV. Find in the proper column two consecutive logarithms be- tween which the given logarithm falls. If the title of the given function is found at the top of that column read the degrees from the top of the page; if at the bottom read from the bottom. Find the value of (q I) or (q -J- 1), as the case may require, corresponding to the given log (interpolating for the last figure if necessary). Then if q = given log and I = log of number ot seconds, n, in the required arc, we have at once I = q (q 1} or I (q-\-l) q, whence n is easily found. Find in the first column two consecutive quantities between which the number n falls, and if the degrees are read from the left hand side of the page, adopt the less, take out the minutes from the second column, and take for the seconds the difference between the quantity adopted and the number n. But if the degrees are read from the right-hand side of the page, adopt the greater quantity, take out the minutes on the same line from the right-hand column, and for the seconds take the difference between the number adopted and the num- ber n. Example. 11.734268 is the' log cot of what arc? q -f I 15.314376 q . 11.734268 .'. n- 3802". 8 3.580108 For 1 adopt 3780. giving 03' Difference 22". 8 Ans. 1 03' 22".8 or 178 56' 37".2. Example. 8.201795 is the log cos of what arc? q - I 4.685556 q 8.201795 .'. n = 3282".8 3.516239 For 89 adopt 3300. giving 05' Difference 17".2 Ans. 89 05' 17".2 or 90 54' 42".8. 264 FIELD TABLE XXVI. Contains logarithmic versed sines and ex- ternal secants for every minute of the quadrant, with the differences of the same corresponding to a change of 1 second in the arc or angle. Interpolation for seconds is made in the .same manner as with log sines of the preceding table, except on the lir.it two pages. For angles less than 4 the differences vary so rapidly that interpolation by direct proportion will not give exact values. On the first two pages the column headed q 21 contains the difference between the log versed sine (or log ex secant) of an arc and twice the logarithm of the number of seconds in the same arc. The characteristic, and first three decimals (9.070) are common to all the logarithms in these columns up to 3 19' for log vers sines, where it changes to (9.069), as shown at the foot of the column; and up to A for log ex secants, where it changes to (9.071). At the point of change a cipher is replaced by the mark 4- to call attention. To find the log- vers sin, or log ex sec of an angle given to seconds. Reduce the angle to seconds, take the logarithm of this number, multiply it by 2, and add the product to the logarithm in the column (q 2') found op posite the given angle. The log (q 21) should be corrected by interpolation for the fractional part of a minute in the give.h angle. Example. What is the log ex secant of 2 14' 43". 7? 2 14' 43".7 = 8040" + 43.7 = 8083".7 log 3.907610 2 21 7.815220 (q-2T) 9.070398 Ans. .', 6.885618 'To find the arc corresponding to a given vers, or log 1 ex sec. Find in the column of log vers, 01- log ex sec the two values between which the given log falls, and take out from the column (q 21) the logarithm corres- ponding to the given log (interpolating for the value of the last figure if necessary). Subtract this from the given logarithm and divide by 2. The quotient is the logarithm of the num- ber of seconds in the required arc, EXPLANATION OF TABLES. 265 Example. 7.344728 is the log vers of what arc? q 7.344728 348'-f i ft -2Q 9.069960 2)8.274768 13720". 9 .'. I 4.137384 13G80. Ana. 3 48' 40".9 To find the log ex sec of an arc greater than 88 given to seconds. Take from the column (q-{-l) the logarithm corresponding to the given arc, interpolating for the fraction of a minute. From this subtract the logarithm of the number of seconds in the complement of the given arc. Example. What is the log ex sec of 88 24' 20". 5? For 88 24' q + 1 15.302183 Correction 129 X -~= 44 q -f I 15302227 Comp. 88 24' 20".5 = 5739".5 log 3.758874 Ana. log ex sec 11.543353 To find the angle corresponding to a given log ex sec when the angle is greater than 88. Find in the table the two consecutive log ex secants between which the given one falls, and then find by interpolation the value of the log (q-\-l) corresponding to the given log ex sec and subtract the latter from it. The difference will be the logarithm of the number of seconds in the complement of the required angle, which is then easily found. Example. 11.924368 is the log ex sec of what arc? Given log ex sec 11.924368 Next less 11.918290 q l 15.309225 'Diff. 6078 15.309296 Given log ex sec 11.924368 40' 26".2 = 2426".2 .-. log 3.384928 Ans. 89 19' 33".8. 266 FIELD TABLE XXVII. Contains natural sines and cosines, to five places of decimals for every minute of the quadrant. Correc- tions for fractions of a minute are made directly proportional to the difference of consecutive values in the table ; positive for sines, negative for cosines. TABLE XXVIII. Contains natural tangents and cotangents to five places of decimals for every minute of the quadrant. Corrections for fractions of a minute are made directly propor- tional to the difference of consecutive values in the table ; positive for tangents, negative for cotangents. TABLE XXIX. ^-Contains natural versed sines and external secants to five places of decimals for every minute of the quadrant. Corrections for fractions of a minute are made directly proportional to the difference of consecutive values. They are positive in every case. TABLE XXX. Contains the number of cubic yards con- tained in prismoids of various side slopes, bases, and depths, as indicated by the title and the numbers in the first column. Each prismoid is supposed to have a uniform level cross sec- tion throughout. These tables are chiefly useful in making up preliminary estimates from the profile, or in other cases where only approximate results are required. For monthly and final estimates more elaborate tables are required, such as are des- cribed in 257. To make an approximate estimate of quanti- ties from a profile by use of Table XXX. Select the proper column for base and slopes, and if the outline of a cut on the profile is roughly a four-sided figure, stretch a fine silk thread over the surface line to average it, note the depth from thread to grade line midway of the cutting, and multiply the tabular number opposite this depth by the average length of the cutting in stations of 100 feet. (By average length is meant the half sum of the length of the grade line in the cutting and of so much of the surface line as is covered by the thread.) If the area of a cutting as seen on the profile is approximately triangular, stretch an averaging line over each incline, and note the depth from the intersection of these lines to grade, and multiply the tabular number opposite this depth by one- EXPLANATION OF TABLES. lialf the length of the cut measured on the grade line in sta- tions. The resulting quantities will be slightly in excess if the ground is level transversely, but may be too small if the trans- verse slope is steep, and cutting on the centre line is small. In general they furnish a good approximation. Quantities in embankments may, of course, be found similarly. A cut or fill may be divided on the profile into several portions, and the contents of each portion found separately if preferred. The content of a prismoid, level transversely, but having different end depths, may be found correctly by this table thus: add together the quantities opposite each end-depth and 4 times the quantity opposite the half sum of the depths; multiply the sum by the length in feet, and divide by 600. TABLE XXXI. Contains a variety of useful numbers and formulae. The logarithms are here gi ven to seven places of decimals. TABLES. TABLE I. GEOMETRICAL PROPOSITIONS. The References are to Davies' Legendre, Revised Edition. No. REFERENCE. HYPOTHESES. CONSEQUENCES. 1 IV., XI If a triangle is right angled, The square on the hypothe- nuse is equal to the sum of the squares on the other two sides. 2 I., XI., Cor. 1.... If a triangle is equilateral, It is equiangular. 3 L, XI If a triangle is isosceles, The angles at the base are equal. 4 L,XL, Cor. 2.... If a straight line from the vertex of an isosceles triangle bisects the base, It bisects the vertical angle. And is perpendicular, to the .base. 5 I., XXV., Cor. 6.. If one side of a tri- angle is pro- duced, The exterior angle is equal to the sum of the two interior and opposite angles. 6 7 IV., XX I , XXVII If two triangles are mutually equian- gular, If the sides of a polygon are pro- duced in the same order, They are similar. And their corresponding sides are proportional. The sum of the exterior angles equals four right angles. 8 I., XXVI., Cor. 1. If a figure is a quadrilateral, The sum of the interior angles equals four right angles. 9 10 L, XXVIII I., XXXI. III., VII If a figure is a parallelogram, If three points are not in the same straight line, The opposite sides are equal. The opposite angles are equal. It is bisected by its diagonal. And its diagonals bisect each other. A circle may be passed through them. 11 in., xvn If two arcs are in- tercepted on the same circle, They are proportional to the corresponding angles at the centre. 12 13 14 15 V., XIII. , Cor. 2.. V., XIII If two arcs are similar, If two areas are circles, If a radius is per- pendicular to a chord, If a straight line is tangent to a circle, They are proportional to their radii. They are proportional to the squares on their radii. It bisects the chord. And it bisects the arc subtended by the chord. It meets it in only one point. And it is perpendicular to the radius drawn to that point. Ill VI HI , IX 16 in., xrv., cor. . . If from a point without a circle tangents are drawn to touch the circle, There are but two. They are equal. And they make equal angles with the chord joining the tangent points. [270] TABLE I. GEOMETRICAL PROPOSITIONS. The References are to Davies* Legendre, Revised Edition. No. 17 REFERENCE. HYPOTHESES. CONSEQUENCES. III., X If two lines are parallel chords or a tangent and parallel chord, They intercept equal arcs of a circle. 18 IIL, xvni If an angle at the circumference of a circle is sub- tended by the same arc as an angle at the cen- tre, The angle at the circumfer- ence is equal to half the angle at the centre. 19 HI.,XVIII.,Cor.2 If an angle is in- scribed in a semi- circle, It is a right angle. 20 III., XXI If an angle is formed by a tan- gent and chord, It is measured by one half of the intercepted arc. 21 IV.,XXVin.,Cor. If two chords in- tersect each oth- er in a circle, The rectangle of the seg- ments of the one, equals the rectangle of the segments of the other. 22 IV.,XXIII.,Cor.2 And if one chord is a diameter, and the other per- pendicular to it. The rectangle of the seg- ments of the diameter is equal to the square on half the other chord, ^nd the half chord is a mean pro- portional between the seg- ments of the diameter. 23 IV., XXIX., Cor.. If two secants meet without the circle, The rectangles of each secant and its external segment are equal. 24 IV., XXX If a secant and tangent meet, The rectangle of the secant and its external segment is equal to the square on the tangent. And the tangent is a mean proportional be- tween the secant and its external segment. 25 IV., XIV If a straight line from the vertex of a triangle bi- sects its base, T l ie sum of the squares on the two sides is equal to twice the square of half the base increased by twice the square of the bisecting line. 26 rv., xii If a perpendicular be drawn from the vertex of a triangle to the base, The square of a side opposite an acute angle is equal to the sum of the squares of the other side and the base, diminished by twice the rectangle of the base and the distance from the ver- tex of the acute angle to the foot of the perpendicu- lar. TABLE II. -TRIGONOMETRIC FORMULAE. H TRIGONOMETRIC FUNCTIONS. Let A (Fig. 107) = angle BAG = arc BF, and let the radius AF ~ AB --- We then have sin A - BG cos A = AG tan A = DF cot A = HO sec A = AD cosec A = AG versin A = CF = BE covers A = BK = HL exsec A = BD coexsec A = BG chord A = BF chord 2 .4 = BI = 2BC FIG. 107. In the right-angled triangle ABC (Fig. 107) Let AB = c,AC = b, and BC = a. We then have : 1. sin A 2. cos A 3. tan A 4. cot A 5. sec A = cos B - = sin B ft cosec ^4 = = a 7. vers A = = covers B c b 8. exsec A = - = coexsec 5 covers ^4 = = versin B c 11. a = c sin A b tan A 12. & = c cos.4 = a cot ^4 13. c = -" 7 = T-^ r sin ^4 cos ^4 14. a = c cos B = b cot jB 15. 6 = c sin B = a tan 5 16. C = --- fi = . - cos ^ sin 17. a 18. & = y"(^" 19. c 20. C = 90 = 4 + B 21. area = - [272] TABLE II. TRIGONOMETRIC FORMULA SOLUTION OF OBLIQUE TRUN&LES. FIG. 108. 22 24 25 26 27 28 29 30 31 32 83 A, B,a A, , b C, a, b a, 6, c B,C, c X (A - B) area A FORMULjE. C=180-04-|-J5), b = stoTA' SinB = ^-7^(^ + 5) sin p sin ^. = 90 - ^ (7 . sin C. tan 14 (.4 - B) = 1| tan %(A + B) = (a ~ 6) -.- . ... COS 7-j c = (a + &)-= JKT = ^ a 6 sin C. Let s = U sin J. vers ^1 6c : = V s (s - a) (s - b) (s - c) a 2 sin J? . sin (7 Tf y sin ^1 [273] TABLE II. TRIGONOMETRIC FORMULA. GENERAL FORMULA. 84 35 36 37 38 39 40 41 42 43 44 45 46 47 48 *49 50 51 52 sin A sin A = sin A = V 1 cos 2 A tan A cos A cosecA ~ 2 sin y% A cos X A = vers A cot J^ A V % vers 2A = t% (1 cos 2 .4) cos A cos A = cos A = tan A = j VI sin 2 .4 cot A sin A SPC -4. cos 3 l&A sin 2 X j. =' V X -f X cos 2 .4 sin .4 cot^ " cos^ /I V 1 cos 2 .4 sin 2 A y cos 2 -4 cos A 1 + cos 2 ^i ^^ vers cot ^4 A = = = 1 1 7 vers A sin A cosec A cot A 57. 2 cot .4 ^3 vers J. 1 cos A 1 -|_ 4/1 _ ^ vers ^. 2 -J- V2 (1 + cos A) 58. vers 2 A 2 sin 9 .4 - 2 sin A cos A tan <4 59. --" ' 1-cos^ 60. exsec 2A= 2 tan .1 1 - tana ^ 61. sin (A B) = sin A . cos jff sin B . cos ^ 62. cos (A B) = cos A . cos B =F sin ^ . sin B 63. sin ^ + sin B = 2 sin ^ (.4 + B) cos }&(A B) 64. sin .4 sin B = 2 cos J^ (A + B) sin ^(A B) 65. cos ^ + cos B = 2 cos ^ {A + B) ccs }(A B) 66. cos B cos A = 2 sin ^ (A -f B) sin 14 (4 3) 67. sin 9 ^1 sin' B = cos 2 B cos 2 A - sin (^ + B) sin (.4 B) 68. cos 2 A sin 2 B = cos (^ + B) cos (4 B) [275] TABLE III. -CURVE FORMULAE. GIVEN. SOUGHT. FORMULA. 1 D R B = sin~^D 2 R D 8lnHZ>=-*L 3 A,D L = 100 -A. 4 D, L A A= TwT 5 A, L D D = 100^- 6 R, A T T = R tan ^ A 7 " C C = 2 /2 sin / A 8 " M Jf = # vers }g A 9 E = R exsec ^ A 10 T, A R #=Tcot^A 11 " E E- Ttan/4 A 12 " C C = 2 T cos ^ A 13 14 E, A M R M = T cot ^ A . vers J^ A ~ exsec J4 A 15 " T T= E cot % A 16 C C = 2 -slc^T 17 " M M = E cos }^j A 18 C, A R ~ ^^in J^ A 19 ", M M = \& C tan J4 A 20 II T C 3 2 cos ^ A 21 It E E _ ^ c ,_exse_cJ^A 22 M, A R ~ vers ^ A 23 M C C = 2 M cot M A 24 25 L K T E _ tan ^j A vers J^ A ~ cos /^ A [276] TABLE III. -CURVE KOUMULuE. GIVEN. SOUGHT. FORMULA. 26 R, T A tan^A=^- 28 R, C A 29 30 R, M M A u M vers T nn A A COS ^ A = ^ 2^:4 3/2 M COS J^j A = - - 43 If MR 44 " E T _ CR 45 R, C T /(*+!)(*-?) 46 M .R 2 47 [277] TABLE III. -CURVE FORMULAE. GIVEN. SOUGHT. FORMULAE. 48 R M T R VM(2R-^M) R-M 49 KA C J7I C = 2VM(2R -M) RM -R-M 51 R, E T T = \'E(2R-\-E) 52 u n 2RyE(2R-{-E) 53 M CT 54 T, C R ~ VQT+cy&TCJ 55 u Hf /23 1 C 56 57 58 59 60 T, E ri -\r E R C M R '* \ 2T+C \/ ~2~T+C~ (T+E)(T-E) 2'E C - *T_( T .l. ~ E ^ E(T*- E*) Tu -I- E* 61 T R 2M 62 Tjl 2( cl+*M? 63 lit E R EM E- M 64 77 F / E-\- M \ E- M 65 66 67 68 fiQ T, M C R E C T> E s _ ^ 2 ^+11 +Br -MMT* = 2 M E 3 -}- E* M ET Z + MT* C 8 + 2 TC 2 + 4 M 2 C - 8 M 2 T = A 7^2 r<2 n^ r>2 Tf< pair>2*- G '~^ p^ o^ - 70 71 " T M 1 SR 1 4. o .c/ 4 o 2T 3 T"* C2TE* CE* = Q 4 4 [278] TABLE IV.-RAD1I, LOGARITHMS, OFFSETS, ETC. Deg. D. Radius. R. Loga- rithm. log. K. Tan. Off. t. Mid. Ord. m. Deg. D. Radius. K. Loga- rithm. log. K. Tang. Off t. Mid. Ord. in. 0' Infinite Infinite .000 .000 1 0' 5729.65 3.758128 .873 ! .218 1 343775. 5 536274 .015 .004 5635.72 .750950 .887 .222 2 171887. 5235244 .029 .007 2 5544.83 .743888 .902 .225 3 114592. 5.059153 i .044 .011 3 5456.82 .736939 .916 .229 4 85943.7 4.934214 .058 .015 4 5371.56 .730100 .931 .233 5 68754.9 .837304 j -073 ;< 5 5288.92 .72a367 .945 .236 6 57295.8 .758123 ' .087 .022 6 5208.79 .716737 .960 .240 7 49110.7 .691176 .102 .025 7 5131.05 .710206 .974 .244 8 42971.8 .633184 .116 .029 8 5055.59 .703772 .989 .247 9 38197.2 .582031 .131 .033 9 4982.. 33 .697432 1.004 .251 10 34377.5 4.536274 .145 .036 10 4911.15 3.691183 1.018 .255 11 31252.3 4.494881 .100 .040 11 4841.98 3.685023 1.0.33 .258 12 28647.8 .457093 .175 .044 12 4774.74 .678949 1.047 .262 ' 13 26444.2 .422331 .189 .047 13 4709. as .672959 1.062 .265 14 24555.4 .390146 .204 .051 14 4645.69 .667051 1.076 .269 15 22918.3 .360183 .218 .055 15 4583.75 .661221 1.091 .273 16 21485.9 .332154 .233 .658 16 4523.44 .655469 1.105 .276 17 20222.1 .305825 .247 .062 17 4464.70 .649792 1.120 .280 18 19098.6 .281002 .065 18 4407.46 .644189 1.184 .284 19 18093.4 .257521 .276 .069 19 4351.67 .638656 1.149 .287 20 17188.8 4.235244 .291 .073 20 4297.28 3.633194 1.164 .291 21 16370.2 4.214055 .305 .076 21 4244.23 3.627799 1.178 .295 22 15626.1 .193852 .320 .080 22 4192.47 .622470 1.193 .298 23 14946.7 .174547 .335 .084 23 4141.96 .617206 1.207 .302 24 14323.6 .156064 .349 .087 24 4092.66 .612005 1.222 .305 25 13751.0 .138335 .364 .091 25 4044.51 .606866 1.236 .309 26 18922.1 .121302 .378 .095 26 3997.49 .601787 1.251 .313 27 12732.4 .104911 .393 .098 27 3951.54 .596766 1.265 .316 28 12277.7 .089117 .407 .102 28 3906.54 .591803 1.280 .320 29 11854.3 .073877 .422 .105 29 3862.74 .586*66 1.294 .324 3 11459.2 4.059154 .436 .109 30 3819.83 3.582044 1.309 .327 31 11089.6 4.044914 .451 .113 31 3777.85 3.577245 1.324 .331 32 10743.0 .031125 .465 .116 32 3736.79 .572499 1.338 .886 .33 10417.5 .017762 .480 .120 33 3696.61 .567804 I.a53 .m 34 10111.1 4.004797 .495 .124 34 31)57.2!) .563160 1.367 .342 35 9322.18 .3.993208 .509 .127 35 31)18. SO .55851)4 1.382 .345 36 9549.34 .979973 .524 .131 36 3581.10 .5.54017 1.396 .349 37 9291.29 .968074 .538 .135 37 3544.19 .545)517 1.411 .353 38 9046.75 .956493 .553 .138 ! 38 3508.02 .545063 1.496 .356 39 8814.78 .945212 .567 .142 39 3472.59 .540654 1.440 .360 40 8594.42 3.934216 .582 .145 40 3437.87 3.536289 1.454 .364 41 8384.80 3.923493 .596 .149 41 3403.83 3.531968 1.469 .367 42 8185.16 .913027 .611 .153 42 3370.46 .527690 1.483 .371 43 7994.81 .902808 .625 .156 43 aw. 74 .523453 1.498 .375 44 7813.11 .892824 .640 .160 44 3305.65 .519257 1.513 .378 45 7639.49 .883065 .654 .164 45 &J74.17 .515101 1.527 .382 46 7473.42 .873519 .669 .167 46 3243.29 .510985 1.542 .sa5 47 7314.41 .864179 .684 .171 47 3212.98 .506908 1.556 .389 48 7162.03 .855036 .698 .174 48 3183.23 .502868 1.571 .393 49 7015.87 .846082 .713 .178 49 3154.03 .498866 1.585 .396 50 6875.55 3.837308 .727 .182 50 3125.36 3.494900 1.600 .400 51 6740.74 3.828708 .742 .185 51 3097.20 3.490970 1.614 .404 52 6611.12 .820275 .756 .189 52 3069.55 .487075 1.629 .407 53 6486! 88 .812002 .771 .193 53 3042.39 .483215 1.643 .411 54 6366. ?*> .803885 .785 .196 54 3015.71 .479389 1.658 .414 55 6250.51 .795916 .800 .200 55 2989.48 .475596 1.673 .418 56 6138. 9>) .788091 .814 .204 56 2963.71 .471836 1.687 .4.22 57 6031.20 .780404 .829 .207 57 2938.39 .468109 1.702 .425 58 5927.22 .772851 .844 .211 58 2913.49 .464413 1.716 .429 59 5826.76 .765427 .858 .215 59 2889.01 .460749 1.731 .433 60 5729.65 3.758128 .873 .218 60 2*!i4 03 3.457115 1 . 745 .436 [279] TABLE IV.-RADH, LOGARITHMS, OFFSETS, ETC. Deer. I). Radius. B. Loga- rithm. log. B. Tang. Off t. Mid. Ord. ni. Beg. . Radius. B. Loga- rithm. log. B. T t. Mid. Ord. ^ go / 2864.93 3 457115 1.745 .436 3 0' 1910.08 3.281051 2.618 .654 1 2841.26 .453511 1.760 .440 1 1899.53 .278646 2.632 658 2 2817.97 .449937 1.774 .444 2 1889.09 .276253 2.647 .662 3 2795.06 .446392 1.789 .447 3 1878.77 .273874 2.661 665 4 2772.53 .442876 1.803 .451 4 1868.56 .271508 2.676 .669 5 2750.35 .439^388 1.818 .454 5 1858.47 .269155 2.690 .673 6 2728.52 .435928 .832 .458 6 1848.48 .266814 8.705 .676 7 2707.04 .482495 .847 .462 7 1838.59 .264486 2.719 .680 8 2685.89 .429089 .862 .465 8 1828.82 .262170 2.734 .684 9 2665.08 .425710 .876 .469 9 1819.14' .259867 2.749 .687 10 2644.58 3 422356 .891 .473 10 1809.57 3.257576 2.763 .691 11 2624.39 8.419029 .905 .476 11 1800.10 3.255296 2.778 .694 12 2604.51 .415727 .920 .480 12 1790.73 .253029 2.792 .698 13 2584.93 .412449 .934 .484 13 1781.45 .250774 2.807 ,702 14 2565.65 .409197 .949 .487 14 1772.27 .248530 2 821 .705 15 2546.64 .405968 .963 .491 15 1763.18 .246297 2.836 .709 16 2527.92 .402763 .978 .494 16 1754.19 .244077 2.850 .713 17 2509.47 .399582 .992 .498 17 1745.26 .241867 2.865 .716 18 2491.29 .390424 2.007 .502 18 1736.48 .239669 2.879 .720 19 2473.37 .393289 2.022 .505 19 1727.75 .237481 2.894 .7-C:3 20 8455.70 3 390176 2.036 .509 20 1719.12 3.235305 2.908 .727 21 2438.29 3.387085 2.051 .513 21 1710.56 3.238140 2.923 .731 22 2421.12 .384016 2.065 .516 22 1702.10 .230985 2.938 .734 23 2404.19 .380969 2.080 .520 23 1693.72 .228841 2.952 .738 24 2387.50 .877943 2.094 .524 24 1685.42 .226707 2.967 .742 25 2371.04 .374938 2.109 .527 25 1677.20 .224584 2.981 .745 26 2854. 8J .371954 2.123 .531 26 1669.06 222472 2.996 .749 27 2338.78 .368990 2.138 .534 27 1661.00 .220369 3.010 .753 28 2322.98 .366046 2.152 .538 28 1653.01 .218277 3.025 .756 29 3307.39 .363122 2.167 .542 29 1645.11 .216195 3.039 .760 30 2292.01 3.360217 2.181 .545 30 1637.28 3.214122 3.054 .763 31 2276.84 3.357332 2.196 .549 31 1629.52 3.212060 3.068 ,767 32 2261.86 .854466 2.211 .553 32 1621.84 .210007 3.083 .771 33 2247.08 .351618 2.225 .556 33 1614.22 .207964 3.097 .774 34 2232.49 .348789 2.240 .560 34 1606.68 .205930 3.112 .778 35 2218.09 .345979 2.254 .564 35 1599.21 .203906 3.127 .782 36 2203.87 .343187 2.269 .567 36 1591.81 .201892 3. 141 .785 37 2189.84 .340412 2.283 .571 37 1584.48 .199886 3.156 .789 38 2175.98 .337655 2.298 .574 38 1577.21 .197890 3.170 .793 39 2162.30 .334916 2.312 .578 89 1570.01 .195903 3.185 .796 40 2148.79 3.332193 2.327 .582 40 1562.88 3.193925 3.199 .800 41 2135.44 3.329488 2.341 .585 41 1555.81 3.191956 3.214 .803 42 2122.26 [886799 2.356 .589 42 1548.80 .189996 3.228 .807 43 2109.24 .324127 2 371 .593 43 1541.86 .188045 3.243 .811 44 2096.39 .321471 2.385 .596 44 1534.98 .186103 3. 57 .814 45 2083.68. .318832 2.400 .600 45 1528.16 .184169 3.272 818 46 2071.13 .316208 2.414 .604 46 1521.40 .182244 3.286 .822 47 2058.73 .313600 2.429 .607 47 1514.70 .180327 3,301 .825 48 2046.48 .311008 2.443 .611 48 1508.06 .178419 3.316 .829 49 2034.37 .308431 2.458 .614 49 1501.48 .176519 3.330 832 50 2022.41 3.305869 2.472 .618 50 1494.95 3.174627 3.345 .'836 51 2010.59 3.30.3323 2 487 .622 51 1488.48 3.172744 3.359 .840 52 1998.90 .300791 2.501 625 52 1482.07 .170868 3.374 .843 53 1987.35 .298274 2.516 .629 53 1475.71 .169001 3.388 .847 54 1975.93 .295771 2.530 .633 54 1469.41 .167142 3.403 .851 55 1964.64 .293283 2.545 .636 55 1463.16 .165291 3.417 .854 56 1953.48 .290809 2.560 .640 56 1456.96 .163447 3 .432 .858 57 1942.44 .288349 2.574 .644 57 1450.81 .161612 3.446 .862 58 1931.53 .265902 2.589 .647 58 1444.72 .159784 3.461 .865 59 1920.75 .'283470 2.603 .651 59 1438.68 .157963 3.475 .869 60 1910.08 3.281051 2.618 .654 60 1432.69 3.156151 3.490 .872 [280] TABLE IV.-RADII, LOGARITHMS, OFFSETS, ETC. Deg. D. Radius. K. Loga- rithrn. log. B. -ff t. Mid. Ord. in. Deg. D. Radius. B. Loga- rithm. log. B. Tang. Off t. Mid. Ord. in. 4 0' 1432.69 3.156151 3.490 .872 5 0' 1146.28 3.059290 4.302 091 l 1426.74 .154346 3.505 .870 i 1142.47 .057840 4.376 094 2 1420.85 .152548 3.519 .880 2 1138.09 . 056407 4.391 098 3 1415.01 .150758 3.534 .883 3 1134.94 .054972 4.405 102 4 1409.21 .148975 3.548 .887 4 1131.21 .053542 4.420 105 5 1403.40 .147200 3.503 .891 5 11S7-.50 .052116 4.435 10! U 1397.70 .14.5431 3.577 .894 1123.82 .050696 4.449 11 7 1392.10 .1-13670 3.592 .898 1120.10 .049280 4.464 11 8 1386.49 .141916 3.606 .902 8 1110.52 .047868 4.478 12f 9 1380.92 .140170 3.621 .905 9 1112.91 .046462 4.493 12;; 10 1375.40 3.138430 3.635 .909 10 1109.33 3.045059 4.507 127 11 1369.92 3.136697 3.650 .912 11 1105.76 3.043662 4.522 .131 12 1364.49 . K34971 3 064 .916 12 1102.22 .0422(58 4.536 1:34 13 1359.10 .133251 3.679 .920 13 1098.70 .040880 1.551 .138 14 1353.75 .131539 3.693 .923 14 1095.20 039495 4.565 .142 15 1848.45 .129833 3.708 .927 15 1091.73 .038115 4.580 .146 16 1:343.15 .128184 3.723 .931 16 1088.28 .030740 4 . .V.M .149 17 1337.05 .126442 3.736 .934 17 1084.85 .035308 4.609 .153 18 1332.77 .124756 3.752 .938 18 1081.44 .0:34002 4.623 157 19 1327. 03 .123077 3.766 .942 19 tore. OB .032039 4.038 .160 20 1322.53 3 .121404 3.781 .945 20 nin.r.s 3.031281 4.653 .164 21 1317.46 3.119738 3.795 .949 21 1071.34 3.029927 4.067 .108 22 1312.43 .118078 3.810 .952 22 1008.01 .028577 4.682 .171 23 1307.45 .116424 3.824 .956 23 iixn.n .027231 4.098 .175 24 1302.50 .114777 3.839 .960 24 11 Hi 1.43 .025890 4.711 .179 25 1297. 58 .113136 3.853 .903 25 1058.1(5 .024652 4 . 725 .182 20 12&2.7J .111501 3.868 .967 26 1054.92 .023219 4.740 .180 "7 1287.87 .109872 3.882 .971 27 1051.70 .021890 4.754 .190 28 1283.07 . 108249 3.897 .974 28 1048.48 .0205(55 4.769 .193 29 1278.30 .106632 3.911 .978 .)() 1045.31 .019244 4.783 .197 30 1273.57 3.105022 3.926 .982 30 1042.14 3.017927 4.798 .200 31 1268.87 3.103417 3.941 .985 31 1039.00 3.016614 4.812 .204 32 1264.21 .101818 3.955 .989 32 10&5.87 .015305 4.827 .208 88 1259.58 .100225 3.970 .993 33 1032.70 .013999 4.841 .211 34 125-1.98 .098638 3.984 .996 34 1029.07 .012098 4.856 .215 35 1-.T.H. 12 .097057 3.W9 1.000 5 1026.60 .011401 4.870 .218 96 1845.89 .09.5481 4.013 1.003 30 1023.55 .010107 4.885 .222 37 1241.40 .093912 4.028 1.007 37 1020.51 .008818 4.900 .226 3H 1230.94 .092:347 4.042 1.011 38 1017.49 .007532 4.914 229 39 1232.51 .090789 4.057 1.014 39 1014.50 .000250 4.929 ^233 40 122S.11 3.089236 4.071 1.018 40 1011.51 3.004972 4.943 .237 11 1223.74 3.087689 4.086 1.022 41 1008.55 3.003698 4.958 .240 12 1219.40 .086147 4.100 1.025 42 1005.60 .002427 4.972 .244 43 1215.30 .084010 4.115 1.029 43 1002.07 3.001160 4.987 .247 U 1210. S2 .083079 4.129 1.032 44 !'.. 7W 2.999897 5.001 .251 1 i:> i->ot;.:>7 .081553 4.144 1.036 i:. 998.867 .998687 5.016 .255 40 1202.30 .080033 4.159 1.040 10 993. 98S .997381 5.030 .258 47 1198.17 .078518 4.173 1.043 47 991.126 .996129 5.045 .202 48 1194.01 .077008 4.188 1.047 48 ! 188. 280 .994880 5.059 .2. Radius. R. Loga- rithm. log.B. Tang. Off t. Mid. Ord. in. Deg. D. Radius. K. Loga- rithm. log. B. T ss- t. Mid. Ord. in. 6 0' 955.366 2.980170 5.234 1.309 7 o ' 819.020 2.913295 6.105 1.528 1 952.722 .978966 5.248 1.313 1 817.077 .912263 6.119 1.531 2 950.093 .977766 5.263 1.317 2 815.144 .911234 6.134 1.535 3 947.478 .976569 5.277 1.320 3 813.238 .910208 6.148 1.539 4 944.877 .975375 5.292 1.324 4 811.303 .909183 6.163 1.543 5 942.291 .974185 5.306 1.327 5 809.397 .908162 6.177 1.546 6 939.719 .972998 5.321 1.331 6 807.499 .907142 6.192 1.550 7 937.161 .971814 5.335 1.335 7' 805.611 .906125 6.206 1.553 8 934.616 .970633 5.350 1.338 8 803.731 .905111 6.221 1.557 9 932.086 .969456 5.364 1.342 9 801.860 .904098 6.236 1.561 10 929.569 j 2. 968282 5.379 1.346 10 799.997 2.903089 6.250 1.564 11 927.066 2.967111 5.393 1.349 11 798.144 2.902081 6.265 1.568 12 924.576 .965943 5.408 1.353 12 796.299 .901076 6.279 1.572 13 922.100 .964778 5.422 1.356 13 794.462 .900073 6.294 1.575 14 919.637 .963616 5.437 1.360 14 792.634 .899073 6.308 1.579 15 917.187 .962458 5.451 1.364 15 790.814 .898074 6.323 1.582 16 914.750 .961303 5.466 1.368 16 789.003 .t>97078 6.337 1.586 17 912.326 .960150 5.480 1.371 17 787.210 .896085 6.352 1.590 18 909.915 .959001 5.495 1.375 18 785.405 .895094 6.366 1.593 19 .907.517 .957855 5.510 1.378 19 783.618 .894105 6.381 1.597 20 905.131 2.956711 5.524 1.382 20 781.840 2.893118 6.395 1.600 21 902.758 2.955571 5.539 1.386 21 780.089 2.892133 6.410 1.604 22 900.397 .954434 5.553 1.389 22 778.307 .891151 6.424 1.608 23 898.048 .95&300 5.568 1.393 23 776.552 .890171 6.439 1.611 24 895.712 .952168 5.582 1.397 24 774.806 .889193 6.453 1.615 25 893.388 .951040 5.597 1.400 25 773.067 .888217 6.468 1.619 26 891.076 .949915 5.611 1.404 26 771.336 .887244 6.482 1.623 27 888.776 .948792 5.626 1.407 27 769.613 .886272 6.497 1.626 28 886.488 .947673 5.640 1.411 28 767.897 .885303 6.511 1.630 29 884.211 .946556 5.655 1.415 29 766.190 .884336 6.526 1.633 30 881.946 2.945442 5.669 1.418 30 764.489 2.883371 6.540 1.637 31 879.693 2.944331 5.684' 1.422 31 762.797 2.882409 6.555 1.641 32 877.451 .943223 5.698 1.426 32 761.112 .881448 6.569 1.644 33 875.221 .942118 5.713 1.429 33 759.434 .880490 6.584 1.648 34 873.002 .941015 5.727 1.433 34 757.764 .879534 6.598 1.651 35 870.795 .939916 5.742 1.437 35 756.101 .878580 6.613 1.655 36 868.598 .938819 5.756 1.440 36 754.445 .877627 6.627 1.659 37 866.412 .93JT725 5.771 1.444 37 752.796 .876678 6.642 1.662 38 864.238 936633 5.785 1.447 38 751.155 .875730 6.656 1.666 39 862.075 !935545 5.800 1.451 39 749.521 .874784 6.671 1.670 40 859.922 2.934459 5.814 1.455 40 747.894 2.873840 6.685 1.673 41 857.780 '2.933376 5.829 1.458 41 746.274 2.872898 6.700 1.677 42 855.648 .932295 5.844 1.462 42 744.661 .871959 6.714 1.680 43 853.527 .931218 5.858 1.466 43 743.055 .871021 6.729 1.684 44 851.417 .930142 5.873 1.469 44 741.456 .870086 6.743 1.688 45 849.317 .929070 5.887 1.473 45 739.864 .869152 6.758 1.691 46 847.228 .928000 5.902 1.476 46 738.279 .868221 6.773 1.695 47 845.148 .926933 5.916 1.480 47 736.701 .867291 6.787 1.699 48 843.080 .925869 5.931 1.484 48 735.129 .866363 6.802 1.702 49 841.021 .924807 5.945 1.487 49 733.564 .865438 6.816 1.706 50 838.972 2.923747 5.960 1.491 50 732.005 2.864514 6.831 1.710 51 836.933 2.922691 5.974 1.495 51 730.454 2.863593 6.845 1.713 52 834.904 .921637 5.989 1.498 52 728.909 .862673 6.860 1.717 53 832.885 .920585 6.003 1.502 53 727.370 .861755 6.874 1.720 54 830.876 .919536 6.018 1.505 54 725.838 .-860840 6.889 1.724 55 828.876 .918489 ! 6.032 1.510 55 724.312 .859926 6.903 1.728 58 826.886 .917446 6.047 1.513 56 722.793 .859014 6.918 1.731 57 824.905 .916404 6.061 1.517 57 721.280 .858104 6.932 1.735 58 822.934 .915365 6.076 1.520 ^8 719.774 .857196 6.947 1.739 59 820.973 .914329 6.090 1.524 59 718.273 .856290 6.961 1.742 60 819.020 2.913295 6.105 1.528- 60 716.779 2.855385 6.976 1.746 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. I>. Radius. K. Loga- rithm. log. B. Tang. Off. | t. Mid. Ord. m. Deg. D. Radius. R. Loga- rithm. log. R. Tang. Oft! t. Mid. Ord. m. 8 0' 716.779 2.855385 6.976 1.746 9 0' 637.275 2.804327 .J846 1.965 1 715.291 .854483 6.990 1.749 1 636.099 .80:3525 .'860 .968 2 ! 713.810 .853583 7.005 1.753 2 634.928 .802724 .875 .972 3 712. .3-35 .852684 7.019 1.756 i 3 633.761 .801926 .889 .975 4 710.865 .851787 7.034 1.761 i 4 632.599 .801128 .904 .979 5 709.402 .850892 7.048 1.764 1 5 631.440 .800*32 .918 .9&3 6 707.945 .849999 7.063 1.768 i 6 630.286 .799538 .933 .987 7 706.493 .849108 7.077 1.771 7 629.136 .798745 .947 .990 8 705.048 .848219 7.092 1.775 i 8 627.991 .797953 ^.962 .994 9 703.609 .847331 7.106 1.778 9 626.849 .797163 7.976 1.998 10 702.175 2.846445 7.121 1.782 10 625.712 2.796374 .991 2.001 11 700.748 2.845562 7.135 1.786 11 624.579 2.7'95587 8.005 2.005 12 699.326 .844679 7.150 1.790 12 623.450 .794801 8.020 2.008 13 697.910 .843799 7.164 1.793 13 622.325 .794017 8.034 2.012 14 696.499 .842921 7.179 1.797 14 621.208 .793234 8.049 2.016 15 695.095 .842044 7.193 1.801 15 620.087 .792453 8.063 2.019 16 693.696 .841169 7.208 1.804 16 618.974 .791673 8.078 2.023 17 692.302 .840296 7.222 1.807 17 617.865 .790894 8.092 2.026 18 690.914 .839424 7.237 1.811 18 616.760 .790117 8.107 2.030 19 689.532 .838555 7.251 1.815 19 615.660 .789341 8.121 2.034 20 688.156 2.837687 7.266 1.819 20 614.563 2.788566 8.136 2.037 21 686.785 2.836821 7.280 1.822 21 613.470 2.787793 8.150 2.041 22 685.419 .835956 7.295 1.826 22 612.380 .787021 8.165 2.045 23 684.059 .835093 7.309 1.829 23 611.295 .786251 8.179 2.048 24 082.704 .834232 7.324 1.833 24 610.214 .785482 8.194 2.052 25 681.354 .833373 7.338 1.837 25 609.136 .784714 8.208 2.056 26 680.010 .832515 7.353 1.840 26 608.062 .783948 8.223 2.060 27 678.671 .HSNJIiO 7.367 1.844 27 606.992 .783183 8.237 2.063 28 677.338 .830805 .382 1.848 28 605.926 .782420 8.252 2.066 29 676.008 .829953 .396 1.851 29 604.864 .781657 8.266 2.070 30 674.686 2.829102 .411 1.855 30 603.805 2.780897 8.281 2.074 31 673.369 2.828253 .425 1.858 31 602.750 2.780137 8.295 2.077 32 672.056 .827405 .440 1.862 32 601.698 .779379 8.310 2.081 33 670.748 .826560 .454 1.866 33 600.651 .778022 8.324 2.084 34 669.446 .825715 .469 1.869 34 599.607 .777867 8.339 2.088 35 668.148 .824873 .483 1.873 35 598.567 .777112 8.353 ,2.092 36 666.856 .824032 .598 1.877 36 5! 7.530 .776360 8.368 2.096 37 665.568 .823193 .512 1.880 37 596.497 .775608 8.382 2.099 38 664.286 .822355 .527 1.884 38 595.467 .774858 8.397 2.103 39 663.008 .821519 .541 1.887 39 594.441 .77410!) 8.411 2.106 40 661.736 2.820685 .556 1.892 40 593.419 2.773361 8.426 2.110 41 660.468 2.819852 .570 1.895 41 592.400 2.772615 8.440 2.113 42 659.205 .819021 .585 1.899 42 591.384 .771870 8.455 2.117 43 657.947 .818191 .599 1.903 43 590.372 .771126 8.469 2.12i 44 656.694 .817363 .614 1.906 44 589.364 .770383 8.484 2.125 45 655.446 .816537 .628 1.910 45 588.359 .769642 8.498 2.128 46 654.202 .815712 .643 1.914 46 587.357 .768902 8.513 2.132 47 652.963 .814889 .657 1.918 47 586.359 .768164 8.527 2.135 48 651.729 .814067 ".672 1.921 48 585.3(54 .767426 8.542 2.139 49 &50.499 .813247 "".686 1.924 49 584.373 .766690 8.556 2.142 50 649.274 2.812428 7.701 1.928 50 583.385 2.765955 8.571 2.147 51 648.054 2.811611 7.715 1.932 51 582.400 2.765221 8.585 2.150 52 646.838 .810796 7.730 1.935 52 581.419 .764489 8.600 2.154 53 645.627 .809982 7.744 1.939 53 580.441 .763758 8.614 2.1.58 54 644.420 .809169 7.759 1.943 54 579.466 .763028 8.629 2.161 55 643.218 .808358 7.773 1.946 55 578.494 .762299 8.643 2.165 56 642.021 .807549 7.788 1.950 56 577.526 .761572 8.658 2.168 B7 640.828 .806741 7.802 1.953 57 576.561 .760845 8.672 2.172 58 (539.639 .8059-35 7.817 1.957 58 575.599 .760120 8.687 2.175 59 (5518.455 .805130 7.831 1.961 59 574.641 .759397 8.701 2.179 60 637.275 2.804327 7.846 1.965 60 573. (586 2.758674 8.716 2.183 [283] TABLE IV.RADII, LOGARITHMS, OFFSETS, ETC. Deg. Radius. Loga- rithm. Off.' Mid. Ord. Deg. Radius. Loga- rithm. Tang. Oft' Mid. Ord. I>. K. log. B. t. in. 13. K. log. B. t. in. 10 0' 573.686 2.75867'4 8.716 2.ias 12 0' 478. 339 2.679735 10.453 2 620 e 571.784 .757232 8.745 2.190 2 477.018 .6785-35 10.482 2.628 4 569.896 .755796 8.774 2.198 4 475.705 .677338 10.511 2.635 P 568.020 .754364 8.803 2.205 6 474.400 .676145 10.540 2.642 8 566.156 .752937 8.831 2.212 8 473.102 .674954 10.569 2.650 10 564.305 .751514 8.860 2.219 10 471.810 .673767 10.597 2.657 12 562.466 .750096 8.889 2.227 12 470.526 .67'2584 10.626 2.664 14 560.638 .748683 8.918 2.234 14 469.249 .671403 10.655 2 671 16 558.823 .747274 8.947 2.241 16 467.978 .670286 10.684 2.679 18 557.019 2.745870 8.976 2.234 18 466.715 2.669052 10.713 2.686 20 555.227 2.744471 9.005 2.256 20 465.459 2.667881 10.742 2.693 22 553.447 .743076 9.034 2.263 22 464.209 .666713 10.771 2.701 24 551.678 .741686 9.063 2.270 24 462.966 .665549 10.800 2.708 26 549.920 .740300 9.092 2.278 26 461.729 .664387 10.829 2 715 28 548.174 .738918 0.121 2.285 28 460.500 .663229 10.858 2.722 30 546.438 .737541 9.150 2.293 30 459.276 .662074 10.887 2.730 32 544.714 .736169 9.179 2.300 32 458.060 .660922 110.916 2.737 34 543.001 .734800 9.208 2.307 34 456.850 .659773 10.945 2.744 36 541.298 .733436 9.237 2.314 36 455.646 .658628 10.973 2 752 38 539.606 2.732077 9.266 2.321 38 454.449 2.657485 11.002 2.759 40 537.924 2.730721 9.295 2.329 40 453.259 2.656345 11.031 2.766 12 536.253 .729370 9.324 2.336 42 452.073 .655208 11.060 2 774 44 534.593 .728023 9.353 2.343 44 450.894 .654075 11.089 2.781 46 532.943 .726681 9.382 2.351 46 449 722 .652944 11.118 2.788 48 531.303 .725342 9.411 2.358 48 448.556 .651816 11.147 2.795 50 529.673 .724008 9.440 2.365 50 447.395 .650691 11.176 2.803 52 528.053 .722677 9.469 2.372 52 446.241 .649570 11.205 2.810 54 526.443 .721351 9.498 2.380 54 445.093 .648451 11.234 2.817 56 524.843 .720029 9.527 2.387 56 443.951 .647335 11.263 2.825 58 523.252 2.718711 9.556 2.394 58 442.814 2.646221 11.291 2.832 11 o' 521.671 2.717397 9.585 2.402 13 0' 441.684 2.645111 11.320 2.839 2 520.100 .716087 9.614 2.409 2 440.559 .644004 11.349 2.846 4 518.539 .714781 9.642 2.416 4 439.440 .642899 11.378 2.854 6 516.986 .713479 9.671 2 423 6 438.326 .641798 11.407 2.861 8 515.443 .712181 9.700 2.431 8 437.219 .640699 11.436 2.868 10 513.909 .710887 9.729 2.438 10 436.117 .639603 11.465 2.876 12 512.385 .709596 9.758 2.445 12 435.020 .638510 11.494 2.883 14 510.869 .708310 9.?'87 2.453 14 493.999 .637419 11.523 2.890 16 509.363 .707027 9.816 2.460 16 432.844 .636331 11.552 2.898 18 507.865 2.705748 9.845 2.467 18 431.764 2.635246 11.580 2.905 20 06.376 2.704473 9.874 2.475 20 430 690 2.634164 11.609 2.912 22 504.896 .703202 9.903 2.482 22 429.620 .633085 11.638 2.919 24 503.425 .701934 9.932 2.489 24 428.557 .632008 11.667 2.927 26 501.962 .700671 9.961 2.496 26 427.498 .630934 11.696 2.934 28 500.507 .699410 9.990 2.504 28 426.445 .629863 11 725 2.941 30 499.061 .698154 10.019 2.511 30 425.396 .628794 11.754 2.949 32 497.624 .696901 10.048 2.518 32 424.354 .627728 11.783 2.956 34 496.195 .695652 10.077 2.526 34 423.316 .626665 11.812 2.963 36 494.774 .694407 10.106 2.533 36 422.283 .625604 11.840 2.971 38 493.361 3.693165 10.135 2.540 38 421.256 2.624546 11.869 2.978 40 491.956 2.691926 10.164 2.547 40 420.233 2.623490 11.898 2.985 42 490.559 .690692 10.192 2.555 42 419 215 .622437 11.927 2 992 44 489.171 .689460 10.221 2.562 44 418.203- .621387 11.956 3.000 46 487.790 .688233 10.250 2.569 46 417.195 .620339 11.985 3.007 48 486.417 .687008 110.279 2.577 48 416.192 .619294 12.014 3.014 50 485051 ,685788 '10.308 2.584 50 415.194 .618251 ; 12. 043 3.022 52 483.694 .684570 10.337 2.591 52 414.201 .617211 112.071 3.029 54 482.344 .683357 10.366 2.598 54 413.212 .616173 12.100 3.036 56 481.001 .682146 10.395 2.606 56 412.229 .615138 12.129 3.044 58 479.666 .680939 10.424 2.613 58 411.250 .614106 12.158 3.051 80 478.339 2.679735 10.453 ! 2.620 60 410.275 .613075 12.187 3.058 [284] TABLE IV. BADII, LOGARITHMS, OFFSETS, ETC. Deg. D. Radius. B. Loga- rithm. log. B. Tan. Off. t. Mid. Ord. m. Deg. D. Radius. B. Loga- rithm. log. B. Tan. Off. t. Mid. Ord. m. 14 0' 410.275 2.613075 12.187 3.058 16 0' 850.368 2.555415 13.917 3.496 409. 306 .612048 12.216 3.0(55 2 358.528 .554517 13.946 3.504 4 408.341 .611023 12.245 3.073 4 857.784 .553621 13.975 3.511 6 407.380 .610000 12.274 3.080 6 857.041 .552727 14.004 3.518 8 406.424 .608980 12.302 3.087 8 356.315 .551834 14.0a3 3.526 10 405.473 .607962 12.331 3.095 10 a55.585 .550944 14.061 8.588 12 404.526 .606946 12. 3(30 13.102 12 a54.859 .550055 14.090 3.540 14 403.583 .605933 12.389 3.109 14 354.135 .549169 14.119 3.547 402.645 .604923 12.418 3.117 16 a53.414 .548284 14.148 3.555 18 401.712 .603914 12.447 3.124 18 352.696 .547401 14.177 3.562 20 400.782 2.602908 12.476 3.131 20 351.981 2.546519 14.205 3.569 22 399.857 .601905 12.504 3.138 22 351.269 .545640 14.234 3.577 24 398.937 .600904 12.533 3.14(5 24 350.560 .544762 14.263 3.584 26 398.020 .599905 12.562 3.153 26 349.854 .54:3887 14.292. 3.591 28 397.108 .598908 12.591 3.160 28 349.150 .54:3013 14.320 3.599 30 396.200 .597914 12.620 3.168 30 348,450 .542140 14.349 3.606 32 395.296 .596922 12. (549 3.175 32 847. 75 .541270 14.378 3.613 34 394.396 .595933 12.678 3.182 34 1 347.057 1 .540401 14.407 8.621 36 393.501 .594945 12.706 3.190 36 (346.3(35 .539535 14.436 3.628 38 392.609 .593960 12.735 3.197 38 345.676 .5:38670 14.464 3.635 40 391.722 2.592978 12'. 764 3.204 40 344.990 2.537806 14.493 3.643 42 390.838 .591997 12.793 3.211 42 344.306 .53(5945 14.522 3.650 44 389.959 .591019 12.822 3.219 44 343.625 .536085 14.551 3.657 46 389.084 .590043 12.851 3.226 46 342.947 .535227 14.580 3.664 48 388.212 .589069 ! 12.880 3.2:33 48 342.271 .534370 14.608 8.672 50 387.345 .588097 12.908 3.241 50 -341.598 .533516 14.637 8.679 52 386.481 .587128 12.937 3.248 52 | 340.928 .532663 14.666 3.686 54 385.621 .586161 12.966 3.255 54 340.280 .531811 14. (595 3.694 56 384.765 .585196 12.995 3.263 56 339.595 .530962 14.723 3.701 58 383.913 .584233 13.024 3.270 58 338.933 .530114 14.752 3.708 15' 383.0(55 2.583272 13.053 3.277 17 o 338.273 2.529268 14.781 3.716 2 382.220 .582314 13.081 3.284 2 337.616 .528424 14.810 3.723 4 381.380 .5813.58 13.110 3.292 4 3:3(5. 9(52 .527581 14.838 3.730 6 380.543 .580403 13.139 3.299 6 3:36.310 .52(5740 14.867 3.7:38 $ 379.709 .579451 13.168 3.306 8 1335.660 .525900 14.89(5 3.745 10 378.880 .578501 13.197 3.314 10 j a35.013 .525062 14.925 3.752 12 378.054 .577553 13.226 3.321 12 834.360 .524226 14.954 3.760 14 377.231 .576608 13.254 3.328 14 333.727 .523392 14.982 3.767 16 376.412 .575664 13.283 3.a36 16 338.088 .522559 15.011 3.774 18 375.597 .5747'22 13.312 3.343 18 332.451 .521728 15.040 3.781 20 374.786 8.573783 13.341 3.350 20 331.816 2.520898 15.069 3.789 , 22 373.977 .572845 13.370 3.358 22 331.184 .520070 15.097 3.796 24 373.173 .571910 13.3SK) 3.365 24 330.555 .519244 15.126 3.803 26 372.372 .570977 13.427 3.372 26 32.!>2rt .518419 15.155 3.811 28 371.574 ..-V70IH5 13.456 3.379 28 B0.80B .517596 15.184 3.818 30 370.780 .569116 13.4S5 3.387 30 328 689 .516774 15.212 3.825 32 869.089 .568189 13.514 3.394 32 328. (Mil .515954 15.241 3.833 34 369.202 .5(5721)4 13.543 3.401 34 327.443 . 515136 15.270 3.840 36 368.418 .566340 13.572 3.409 36 32(5.828 .514319 15.299 3.847 38 367.637 .565419 13.600 3.416 38 326.215 .513504 15.827 3.855 40 366.859 2.564500 13.629 3.423 40 325.604 2.512690 16.856 3.862 42 366.085 .563582 13.658 3.431 42 324.996 .511878! 15.385 3.869 44 365.315 .562667 13.687 3.438 44 324.390 .511067 15.414 3.877 46 364.547 .561754 13.716 3.445 46 323.786 .510258 15.442 3.884 48 363.783 .5(50843 13.744 3.452 48 323.184 .5094511 15.471 3.891 50 363.022 .559933 13.773 3.460 50 322.585 .5086451 15.500 ?.899 52 362.2(54 559028 13.802 3.467 52 321.989 .507840' 15. 52W 3.906 54 361.510 '. 5581 20 13.831 3.474 54 321.394 .507037) 15.557 3.913 56 360.758 .557216 13.860 3.482 5(5 320.801 .506236 15.586 3.920 58 360.010 .556315 13. 889 13. 489 5S 320.211 .5054361 15.615 3.928 60 359.265 2.555415 13.J.17 3.496 60 319 . 623 2 . 5046381 15. 643 8 935 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg ! Radius. B. Loga- rithm. log. K. t. Mid. Ord. m. Deg. Radius. K. Loga- rithm. log.R. Tang. Off t. Mid. Ord. in. 18 0' 319.623 2.504638 15.643 3.935 20 0' 287.939 2.459300 17.365 4.374 2 319.037 .503841 15.672 3.942 10 285. 583 .455733 17.508 4.411 4 318.453 .503045 15.701 3.950 20 283.267 .452195 17.651 4.448 6 317.871 .502251 15.730 3.957 30 280.988 .448688 17.794 4.484 8 317.292 .501459 15.758 3.964 40 278.746 .445209 17.937 4.521 10 316.715 .500668 15.787 3.972 50 276.541 .441759 18.081 4.558 12 316.139 .499879 15.816 3.979 21 0' 274.370 2.438337 18.224 4.594 14 315.566 .499091 15.845 3.986 10 272.234 .434943 18.367 4.631 16 314.993 .498304 15.873 3.994 20 270.132 .431576 18.509 4.668 18 314.426 .497519 15.902 4.001 30 268.062 .428235 18.652 4.704 20 22 313.860 12.496736 313.295 .495953 15.931 15.959 4.008 4.016 40 50 266.024 264.018 .424921 .421633 18.795 18.938 4.741 4.778 24 312.732 .495173 15.988 4.023 22 0' 262.042 2.418371 19.081 4.814 26 312.172 .494393 16.017 4.030 10 260.098 .415134 19.224 4.851 28 311.613 .493616 16.046 4.038 20 258.180 .411922 19.366 4.888 30 311.056 .492839 16.074 4.045 30 256.292 .408734 19.509 4.925 32 310.502 34 309.949 .492064 .491291 16.103 16.132 4.052 4.060 40 50 254.431 252.599 .405571 .402431 19.652 19.794 4.961 4.998 36 38 309.399 308.850 .490518 .489748 16.160 16.189 4.067 4.074 23 0' 10 250.793 249.013 2.399315 .396222 19.937 20.079 5.035 5.071 40 42 44 46 308.303 307.759 307.216 306.675 2.488978 .488210 .487444 .486679 16.218 16.246 16.275 16.304 4.081 4.089 4.096 4.103 20 30 40 50 247.258 245. 529 243.825 242.144 .393151 .390103 .387077 .384074 20.222 20.364 20.507 20.649 5.108 5.145 5.182 5.218 48 50 52 54 56 58 306.136 305.599 305.064 304.531 304.000 303.470 .485915 .485152 .484391 .483632 .482873 .482116 16.333 16.361 16.390 16.419 16.447 16.476 4.111 4.118 4.125 4.133 4.140 4.147 24 0' 10 20 30 - 40 50 240.487 238.853 237.241 235.652 234.084 232.537 2.381091 20.791 .378130 20.933 .375190 21.076 .372270 21.218 . 369371 ' 21. 360 .366492 21.502 5.255 5.292 5.329 5.366 5.402 5.439 19 0' 302.943 2.481361 16.505 4.155 25 0' 231.011 2.363633 21.644 5.476 2 302.417 .480607 16.533 4.162 10 229.506 .360794 21.786 5.513 4 301.893 .479a54 16.562 4.169 20 228.020 .357974 21.928 5.549 6 301.371 .479102 16.591 4.177 30 226.555 .355173 22.070' 5.586 8 300.851 .478352 16.620 4.184 40 225.108 .352391 22". 212 5.623 10 300.333 .477603 16.648 4.191 50 223.680 .349627 .22.353 5.660 12 14 16 18 299.816 299.302 298.789 298.278 .476855 .476109 .475364 .474621 16.677 16.706 16. 734 16.763 4.199 4.206 4.213 4.221 26 0' 10 20 30 222.271 220.879 219.506 218.150 2.346882 .344155 .341446 .338755 22.495 22.637 ; 22. 778 22.. 920 5.697 5.734 5.770 5.807 20 297.768 2.473878 16.792 4.228 40 216.811 .336081 23.062 5.844 22 297.260 .473137 16.820 4.235 50 215.489 .333424 23.203 5.881 24 296.755 .472398 16.849 4.243 27 0' 214.183 2.330785 23.345 5.918 26 296.250 .471659 16.878 4.250 10 212.893 .328162 23.486 5.955 28 295.748 .470922 16.906 4^257 20 211.620 .325556 23.627 5.992 30 295.247 .470186 16.935 4.265 30 210.362 .322967 23.760 6.029 32 294.748 .469452 16.964 4.272 40 209.119 .320393 23.910 6.065 34 294.251 .468718 16.992 4.279 50 207.891 .317836 24.051 6.102 36 38 293.756 293.262 .467986 .467256 17.021 17.050 4.287 4.294 28 0' 10 206.678 205.480 2.315295 24.192 .312769 24.333 6.139 6.176 40 292.770 2.466526 17.078 4.301 20 204.296 .310259 24.474 6.213 42 292.279 .465798 17.107 4.308 30 203.125 .307764 24.615 6.250 44 291.790 .465071 17.136 4.316 40 201.969 .305285 24.756 6.287 46 291.303 .464345 17.164 4.323 50 200.826 .302820 24.897 6.324 48 290.818 .463621 17.193 4.330 29 0' 199.696 2.300370 25.038 6.360 50 290.334 .462897 17.222 4.338 10 198.580 .297935 25.179 6.398 52 289.851 .462175 17.250 4.345i 20 197.476 .295515 25.320 6.435 54 289.371 .461455 17.279 4.352 ! 30 196.385 .293108 25.460 6.472 56 288.892 .460735 17.308 4.360: : 40 195.306 .290716 25.601 6:509 58 288.414 .460017 17.336 4.367i 50 194.240 '.288338 25.741 6.545 60 287.939 2.459300 17.365 4.374 30 0' 193.185 2.285974 25.882 6.583 TABLE IV. -RADII, LOGARITHMS, OFFSETS, ETC. Deg. Radius. Loga- rithm. Tang. Off Mid. Ord. | Deg. Radius. Loga- 1 rithm. ssr- Mid. Ord. D. B. log.B. t. in. D. B. log.B. t. in. 30 20' 191.111 2.281286 26.163 6.657 3830' 151.657 2.180863 3 2.969 8.479 40 189.083 .276652 26.443 6.731 ,39 0'| 149.787 .175475 33.381 8.592 31 / 18 ".099 .2721 )71 26.724 6.805 30 1 47.965 .170 160 3 3.792 8.704 2 IS SJ58 .267f >41 27.004 6.879 40 > 0' 1 46.190 .164 918 3 4.202 8.816 4 181 3.^58 .263( K52 27.284 6.958 30 1 44.460 .159 747 3 4.612 8.929 32 0' 181.398 .258632 27.564 7.027 41 0' 142.773 .154645 35.021 9.041 17 ).577 .254: !50 27.843 7.101 30 1 41.127 .149 610 3 5.429 9.154 40 177.794 .249916 28.123 7.175 42 3 ' 1 39.521 .144641 35.837 9.267 33 0' 17 3.047 .245f >28 28.402 7.250 30 1 37.955 .139 736 3 6.244 9.380 20 174.336 .241386 28.680 7.324 43 o 0' 136.425 .134895 36.650 9.493 4 IT, 2.659 .2371 88 28.959 7.398 30 11 U.9S2 .130 114 3 7.056 9.606 34 O'j 171.015 2.233035 29.237 7.473 44 0' 133.473 2.125395 37.461 9.719 2 16 ).404 .228' >24 29.515 7.547 30 1 1 32. 04!) .120 734 3 7.865 9.832 40 167.825 .224855 I 29.793 7.621 45 o ' 130.656 .116130 38.268 9.946 35 J' 16 3.275 .22$ as 30.071 7.696 30 1 29.296 .111 584 3 8.671 10.059 20 164.756 .216842 30.348 7.770 46 0' 127.965 .107092 39.073 10.173 4 16 J.266 .212* $5 30.625 7.845 30 1 2(5.664 .102 (;:,:, : 9.474 10.286 36 0' 161.803 .208988 30.902 7.919 47 125.392 .098270 39.875 10.400 2 16 ).368 .205 L19 31.178 7.994 30 1 24.148 .093 9:38 4 0.275 10.516 40 158.960 .201288 31.454 8 068 48 0' 122.930 .089657 40.674 10.628 37 0' 15 i*.577 .197- 194 31.7:30 8.143 30 1 21.7:38 .085 425 4 1.072 10.742 20 156.220 .193736 32.006 8.218 49 0' 120.571 .081243 41,469 10.856 4 I) Ifr 1.887 .1901 )14 32.282 8.292 30 1 19.429 .077 109 4 1.866 10.970 38 0' 153.578 2.186, 328 32.557 8.367 50 0' 118.310 2.073022 4 2.262 11.085 TABLE V.~ CORRECTIONS FOR TANGENTS AND EXTERNALS. FOR TANGENTS , ADD FOR EXTERNALS, ADD Ang 5 10 15 20 25 30 Ang 5 10 15 20 25 30 A Cur. Cur. Cur. Cur . Cur. Cur. A GUI'. Cur. Cur. Cur Cur. Cur. 10 .03 .06 .09 .13 .16 | .19 10 001 .003 .004 .006 .007 .008 20 . i3 .13 .19 .26 .32 .39 20 .01 K3 .011 .017 .028 .034 30 10 .19 .29 .39 .49 .59 30 .013 .025 .038 '.051 .065' .078 40 13 .26 .40 .& 67 .80 40 .0$ S3 .046 .070 .093 .117 .141 50 17 .34 .51 -.85' 1 02 50 .037 .075 .116 .151 .189 .227 60 g 21 .42 .63 '.84 1.05 1.27 60 .O'i >6 .112 .168 .225 .283 .340 70 25 .51 .76 1.0$ 1.28 1.54 70 .0? <0 .159 .240 .321 .403 .485 80 30 .61 .91 1.2$ 1.53 1.84 80 .110 .220 .332 .44 .558 .671 90 36 .72 1.09 1.4E 1.83:2.20 90 .1' 19 .299 .450 .60? .756 .910 100 43 .86 1.30 1.74 t 2 18 i 2. 62 100 .2( K> .401 .604 80S 1.015 1.221 110 51 1.03 1.56 2 Oi * 2.61 ;3.14 110 .21 58 .536 .806 1.08$ 1.355 1.633 120 62 1.25 1.93 2.5$ 5 3 16 3.81 120 .31 50 .721 1.086 1.45f 1.825 2.197 1 [287] TABLE VI TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. A T. E. A T. E. " 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 10' 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 83.34 .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 i 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 40 686.60 40.992 40 1200.5 124.41 50 191.74 3.207 50 695.06 42.004 50 1209.2 126.20 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 1244.0 133.50 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 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 1428.6 175.41 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 1464.0 184.08 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 24.911 40 1044.7 94.462 40 1571.0 211.48 50 543.29 25.700 50 1053.3 96.013 50 1580.0 213.86 [288] TABLE VI.-TANGENTS AND EXTERNALS TO A 1 CURVE, Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. A T. E. A T. E. A T. 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 ,>l> 1606.9" 221.08 ID 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 1634.9 225.96 40 2180.3 400.82 40 2773.9 636.17 50 1633.9 228.42 50 21 89. 9 404.22 50 2784.2 640.60 32 1048.0 230.90 42 2199.4 407.04 52 2794.5 045.17 10 1652.0 233.39 i 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 1670.0 238.43 | 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 50 2247.3 424.98 50 2846.3 668.03 33 1697.2 246.08 43 2257.0 428.50 53 2856.7 672.66 10 1706.3 248.66 i 10 2260.6 432.04 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 i 30 2285.9 439.10 30 2888.0 686.68 40 1733.5 256.50 40 2295.6 442.75 40 2898.4 691.40 50 1742.6 259.14 50 2305.2 440.35 50 2908.9 696.13 34 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 2353.8 404.04 40 2961.5 720.11 .50 1797.4 275.31 50 2303.5 468.35 j 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 2993.3 734.76 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 1843.3 289.20 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 4<)X.i;7 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 37 1917.1 312.22 47 2491.3 518.90 57 3110.9 790.08 10 1926.4 315.17 10 2501.2 522.10 10 3121.7 795.24 20 19.35.7 318.13 20 2511.2 520.13 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 2581.1 534.15 40 I 3154.2 i 810.85 50 1963.6 327.12 50 2541.0 538.18 50 1 3165.1 816.10 38 1972.9 a30.15 48 2551.0 542.23 58 3176.0 821.37 10 1982.2 333.19 10 2561.0 540.30 10 3186.9 826.66 20 1!)!>1.:> 336.25 20 2571.0 550.39 20 3197.8 881.98 30 2000.9 339.32 30 2581.0 554.50 30 3208.8 837.31 40 2010.2 342.41 40 2591.1 558.03 40 3219.7 842.67 50 2019.6 345.52 50 2601.1 502.77 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 2621.2 571.12 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 2041.4 579.54 30 3274.8 869.82 40 2066.6 361.29 40 2651.5 583. 7'8 40 3285.8 875.32 50 2076.0 364.50 50 2001.0 688.04 50 3296-. 9 880.84 40 2085.4 367.72 50 2671.8 592.32 60 3308.0 886.38 10 2094.9 370.95 i 10 2681.9 596.62 10 3319.1 891.95 20 2104.3 374.20 20 2692.1 600.93 20 3330.3 897.54 30 2113.8 377.47 30 2702.3 605.27 30 3341.4 903.15 40 2123.3 380.76 40 2712.5 609.62 40 3352.6 908.79 50 2182.7 384.06 50 2722.7 614.00 50 3363.8 914.45 [289] TABLE VI. TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. A Tan- gent. T. Exter- nal. E. 1 Angle. A Tan- gent. T. Exter- nal. E. Angle. A 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 90 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 5099.6 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 5143.9 1970.3 64 3580.3 1026.6 74 4317.6 1444.6 84 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 j 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 5281.0 2062,5 30 3685.4 1082.9 30 4436.4 1516.7 30 5296.4 2073.0 40 3697.2 1089.3 40 4449.7 1524.9 40 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 3878.8 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 69 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 20 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 [290 TABLE VI. TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. Tan- gent. Ex- ternal. Angle. Tan- gent. Ex- ternal. Angle. Tan- gent. Ex- ternal A T. E. A T. E. A T. E. 91 5830.5 2444.9 101 , 6950.6 3278.1 1 111 8336.7 4386.1 10' 1 5847.5 2457.1 10' 6971.3 3294.1 10' 83(52.7 4407.6 20 5864.6 2469.3 20 6992.0 3310.1 20 8388.9 442D.2 30 5881.7 2481.5 30 7012.7 3326.1 30 8415.1 4450.9 40 5898.8 2493.8 40 70:33.6 3342.3 40 8441.5 4472.7 50 5916.0 2506.1 50 7054.5 3358.5 50 8468.0 4494.6 92 5933.2 i 2518.5 102 7075.5 ! 3374.9 112 8494.6 4516.6 10 5950.5 2531.0 10 7096.6 3391. > 10 8521.3 4538.8 20 5967.9 2543.5 20 7117.8 3407.7 1 20 8548.1 4561.1 30 5985.3 2556.0 30 7139.0 3424.3 30 8575.0 4583.4 40 6002.7 2568.6 40 7160.3 3440.9 40- 8602.1 4606.0 50 6020.2 2581.3 50 7181.7 3457.6 50 8629.3 4628.6 93 6037.8 2594.0 103 7203.2 3474.4 113 8656.6 4651.3 10 6055.4 2606.8 10 7224.7 3491.3 10 8684.0 4674.2 20 6073.1 2619.7 20 7246.3 3508.2 20 8711.5 4697.2 30 6090.8 2632.6 30 7268.0 3325.2 30 8739.2 4720.3 40 6108.6 2645.5 40 7289.8 3542.4 40 8767.0 4743.6 50 6126.4 2658.5 50 7311.7 3559.6 50 8794.9 47(56.9 94 6144.3 2671.6 104 7*33.6 3576.8 114 8822.9 4790.4 10 6162.2 2684.7 10 7355.6 3594.2 10 8851.0 4814.1 20 6180.2 2697.9 20 7377.8 3611.7 20 8879.3 4837.8 30 6198.3 2711.2 30 7399.9 3629.2 30 8907.7 4861.7 40 6216.4 2724.5 40 7422.2 3646.8 40 8936.3 4885.7 50 6234.6 i 2737.9 50 7444.6 3664.5 50 8965 4909.9 95 6252.8 2751.3 105 7467.0 3682.3 115 8993.8 4931 1 10 6271.1 27(54.8 10 7489.6 3700.2 10 1)022.7 1858.6 20 6289.4 2778.3 20 7512.2 3718.2 20 9051.7 4983.1 30 6307.9 2792.0 30 7534.9 3736.2 30 9080.9 5007.8 40 6326.3 2805.6 40 7557.7 3754.4 40 9110.3 5032.6 50 6344.8 2819.4 50 7580.5 3772.6 50 9139.8 5057.6 96 6363.4 2833.2 106 7603.5 3791.0 116 9169.4 5082.7 10 6382.1 j 2847.0 10 7626.6 3809.4 10 9199.1 5107.9 20 6400.8 ; 2861.0 20 7649.7 3827.9 20 9229.0 5133.3 30 6419.5 2875.0 30 7672.9 3846.5 30 9259.0 5158.8 40 6438.4 2889.0 40 7696.3 3865.2 40 9289! 2 5184.5 50 6457.3 2903.1 50 7719.7 3884.0 50 9319.5 5210.3 97 6476.2 2917.3 107 7743.2 3902.9 117 9349.9 5236.2 10 6495.2 2931.6 10 7766.8 3921.9 10 9380.5 5262.3 20 6514.3 2945.9 20 7790.5 3940.9 20 9411.3 5288.6 30 6533.4 2960.3 30 7814.3 3960.1 30 9442.2 5315.0 40 6552.6 2974.7 40 7838.1 3979.4 40 9473.2 5341.5 50 6571.9 2989.2 50 7862.1 3998.7 50 9504.4 5368.2 98 6591.2 3003.8 108 7886.2 4018.2 118 9535.7 5395.1 10 6610.6 3018.4 10 7910.4 4037 8 10 9567.2 5422.1 20 6630.1 3033.1 20 7934.6 4057.4 20 9598.9 5449.3 30 6649.6 3047.9 30 7959.0 4077.2 30 9630.7 5476.5 40 6669.2 3062.8 40 7983.5 4097.1 40 9662.6 5504.0 50 6688.8 3077.7 50 8008.0 4117.0 50 9694.7 5531.7 99 6708.6 3092.7 109 8032.7 4137.1 119 9727.0 5559.4 10 6728.4 3107.7 10 8057.4 4157.3 10 9759.4 5587.4 20 6748.2 3122.9 20 8082.3 4177.5 20 9792.0 5615.5 30 6768.1 3138.1 30 8107.3 4197.9 30 9824.8 5643.8 40 6788.1 3153.3 40 8132.3 4218.4 40 9857.7 5672.3 50 (5808.2 3168.7 50 8157.5 4239.0 50 9890.8 57C0.9 100 6828.3 3184.1 110 8182.8 4259.7 120 9924.0 5729.7 10 6848.5 3199.6 10 8208.2 4280.5 10 9957.5 5758! 6 20 6868.8 3215.1 20 8233.7 4301 . 1 20 9991.0 5787.7 30 6889.2 3230.8 30 8259.3 4322.4 30 10025.0 5817.0 40 1 6909.6 3246.5 40 8285.0' i:m.ii 40 10059.0 5846.5 50 6930.1 3262.3 50 8310.8 4364.8 50 10093.0 5876.1 TABLE VII. LONG CHORDS. Degree of Curve. Actual Arc, One Station. LONG CHORDS. 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. 010' 100.000 200.000 299.999 899 998 499.996 599.993 30 .000 199.999 299.997 399.992 499.983 599.97'0 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 .002 199.990 299.959 399.896 499.793 599.687 30 .002 199.986 299.946 399.865 499.729 599.526 30 .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. 75G 90 .oor 199.959 299.884 399.586 499.171 598.550 30 .008 199.952 299.810 399.524 499.049 598.336 40 .009 199.946 299.7'83 399.459 498.918 598.106 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 90 .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 30 .024 199.857 299.428 398.571 497.145 595.007 30 .026 199.846 299.383 398.459 496.921 594.617 40 .028 199.834 399.337 398.343 496.689 594.212 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 30 .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 .048 199.710 398.843 397.110 494.227 689.913 20 .051 199.695 298.779 396.952 493.912 589.364 30 .054 199.678 298.714 396.790 493.588 688.800 40 .056 199.662 298.648 396.623 493.257 C88.221 60 .059 199.644 298.57'9 396.453 492.917 587.628 7 100.062 199.627 298.509 396.278 492.568 687.021 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 .071 199.572 298.289 395.729 491.474 685.115 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 30 .088 199.471 297.888 394.731 489.486 581.654 30 .092 199.450 397.803 394.518 489.064 580.920 40 .095 199.428 297.716 394.302 488.634 580.172 50 ' .099 199.406 297.628 394.082 488.196 579.409 9 100.103 199.383 397.538 393.857 487.749 578.633 10 .107 199.360 397.446 893.629 487.294 577.848 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 T2021 iOf THE UNIVERSIT OF ,V B LE VII. -LONG CHORDS. LONG CHORDS. Degree i of CurvG. 7 8 9 10 11 13 Stations. Stations. Stations. Stations. Stations. Stations. 0io 699.988 7'!)!). 1)82 899.974 999.965 ; 1099.95 1199.94 20 699. 953 71M). 1)29 899.899 999.860 | 1099.81 1199.76 30 699.898 799.840 81(9.772 999.686 1090.58 1199.46 40 699.810 799.716 899.594 999.442 1099.25 i 1199.03 50 699.704 799.556 899.365 999.128 1098.84 1196.49 1 699.574 799.360 899.086 998.744 1098.33 1197.82 10 699.420 799.130 898.757 998.290 1097.72 1197.04 20 699.242 798.863 898.376 997.768 1097.02 1190.13 30 699.041 798.562 897.945 997.175 1096.23 1195.11 40 698.816 798.224 897.464 996.513 1095.35 1193.96 50 698.567 797.852 8%. 931 995.782 1094.38 1192.69 2 698.295 797.444 896.349 904.981 1093.31 1191.31 10 698.000 797 000 8'.),-). 716 994.112 1092.15 1189.80 20 697. 680 796.522 895.0*3 993.173 1090.90 1188.18 30 697.388 796.008 81)4.299 992.165 1089.56 1186.43 40 698.971 795.459 81)3.515 991.088 1088.12 1184.57 50 696.581 794.874 81)2.681 989.943 1086.60 1182.59 3 696.168 794.255 811.788 988.799 3084.98 1180.49 10 695.731 71)3.600 890.801 987.447 1083.28 1178.28 20 695.271 792.911 889.880 986.096 1081.48 1175.94 80 604.787 792.186 888.846 984.677 1079.59 1173.49 40 694.280 791.427 887.763 983.190 1077.61 1170.93 50 693.750 790.632 886.630 981.636 1075.54 1168.25 4 693.196 789.803 885.448 980.014 1073.38 1165.45 10 692.619 788.939 884.217 978.325 1071.14 1162.54 20 693.018 788.040 882.936 976.569 1068.81 1159.51 30 691.395 787.108 881.606 974.746 1066. 38 1156.37 40 690.748 786.140 880.228 972.856 1063.87 1153.12 50 690.079 785.138 87-8.800 970.900 1061.27 1149.76 5 089.386 784.101 877.324 968.877 1058.59 1140.28 10 688.670 783.030 870.800 966.788 1055.81 1142.69 20 687.930 781.925 8T4.2B7 964.0:34 1052.95 1188.99 30 687.169 780.786 872. ISO.") 962.415 1050.01 1135.18 40 686.384 779.612 871). 936 960.130 1046.97 1131.26 50 685.576 778.406 869.219 957.780 1043.86 1127.24 6 684.745 777.165 867.454 955.366 1040.66 1123.10 10 683.892 775.890 865.642 952.888 1037.37 1118.86 20 683.016 774.582 863. 7S2 950.345 1034.01 1114.51 30 682.117 773.240 861.875 947.739 1030.55 1110.05 40 681.195 771.864 859.922 945.069 1027.02 1105.49 50 680.251 770.455 867.881 942.337 1023.40 1100.83 7 679.285 769.014 855.874 939.542 1019.70 1096.06 10 678.296 767.539 853.780 936.684 1015.93 1091.19 20 677.284 766.030 851. 640 933.; T64 1012.07' 1086.22 30 676.250 764.490 849. 4.V> 930>83 1008.13 1081.15 40 675.194 762.916 817.224 1)27.741 1004.11 1075.98 50 674.116 761.809 844.947 924. 638 1000.01 1070.71 8 673.015 759.670 842.625 121.171 995.834 1065.34 10 671.81)2 757.999 840.258 918.250 991.580 1059.88 20 670.748 756.295 837.845 914.966 987.250 1054.32 30 669.581 754.560 835.389 911.623 982.844 1048.66 40 668.393 752.792 832.888 908.221 978.362 1042.91 50 667.182 750.998 830.342 904.761 973.806 1037.06 9 065.' 50 749.161 827.754 901.242 969.175 1031.18 10 664.697 747.299 825.121 81)7 667' 964.471 1025.11 20 663.421 7'45.404 822.445 894^038 959.694 1018.99 30 662.124 743.479 819.726 8H0.343 954.844 1012.79 40 660.806 7H..VJ2 816.905 886.597 949.924 ! 1006.49 50 659.466 739. 58B 814.160 882.71)5 944.933 1000.12 10 658.105 737.516 811.314 878.9J38 939.871 993.653 [293] TABLE VII. -LONG CHORDS. Degree of Curve. Actual Arc, One Station. LONG CHORDS. 2 Stations. 3 4 Stations. 1 Stations. 5 Stations. 6 Stations. 10 10' 100.131 199.213 296.860 392.171 484.397 572.813 20 .136 199.187 296.756 391.914 483.886 571.926 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 i 389.998 480.086 565.343 40 .173 198.964 295.868 389.708 479.511 564.349 50 .178 198.935 i 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 1 295.508 388.814 477.740 561 287 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 10 255 198.474 293.918 384.875 469.958 647.867 20 -261 198.437 293.77'4 384.521 469.260 546.666 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 -306 198.173 292.726 881.931 464.160 537.908 40 .312 198.134 292.570 381.546 463.401 536.608 50 .319 198.094 292.412 381.156 462.635 535.296 16 100.326 198.054 292.252 380.763 461.862 533.972 10 .333 198.013 292.091 380.365 4S1.081 532. 635 20 .339 497.972 291.928 379.964 460.293 531.287 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 17 100.368 197.803 291.261 378.320 457.069 525.776 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 .390 197.672 290.743 377.04? 454.574 521.519 40 .397 197.628 290.568 376.615 453.728 520.078 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 .436 197.399 289.665 374.397 449.392 512.699 40 .444 197.352 289.479 373.942 448.504 51t!l90 50 .452 197.305 289.292 373.483 447.608 509. (170 19 100.460 197.256 289.104 373.021 446.706 508.139 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 n 50 .501 197.012 288.137 370.652 442.091 500.320 20 100.510 196.962 287.939 370.167 441.147 498.724 TABLE VII. LONG CHORDS. Degree of Curve. .LiONG UHORDS. 7 Stations. 8 Stations. 9 Stations. 1O Stations. 11 Stations. 12 Stations. 10 10 656.723 735.467 808.426 875.025 934.741 987.105 20 655.320 733.387 805.495 871.058 929.542 980.473 30 653.895 731.277 802.524 867.038 924.276 973.760 40 652.450 729.137 799.512 862.963 918.943 966.967 50 650.983 1 726.967 796.458 858.836 913.544 960.093 11 649.496 724.767 793.364 854.656 908.080 953.141 10 647.989 722.537 790.230 850.425 902.550 946.112 20 646.460 720.278 787.056 846.140 896.957 939.007 30 644.911 717.990 783.843 841.808 891.303 931.828 40 643.342 715.672 780.590 837.424 "885.586 924.575 50 641.752 713.325 777.298 832.990 879.807 917.250 12 640.142 710.950 773.968 828.507 873.968 909.854 10 638.512 708.546 770.600 823.974 868.070 902.389 20 636.862 706.113 767.193 819.394 862.113 894. a55 30 635.191 703.653 763.749 814.766 856.099 887.254 40 6:33.501 701.164 760.268 810.092 850.028 879.588 50 631.792 698.647 756.749 805.370 843.900 871.857 13 630.062 696.103 753.194 800.602 837.718 864.063 10 628.313 693.531 749.603 795.790 831.482 856.208 20 626.544 690.932 745.976 790.932 825.192 848.293 30 624.756 688.306 742.313 786.0:30 818.850 840.318 40 622.949 685.653 738.616 781.085 812.457 832.286 50 621.123 682.974 734.883 776.096 806.013 824.198 14 619.278 680.268 731.116 771.066 799.520 816.056 10 617.413 677.535 727.315 765.993 792.979 807.860 20 615.530 674.777 723.480 760.879 786.389 799.612 30 613.628 671.993 719.612 755.725 779.753 791.313 40 611.708 669.183 715.711 750.531 773.072 782.966 50 609.769 666.348 711.777 745.297 766.345 774.571 15 607.812 663.488 707.811 740.024 759.575 766.130 10 605.836 660.603 703.814 7:34.714 752.763 20 603.842 657.693 699.785 729.366 745.908 30 601.831 654.758 695.725 723.982 739.014 40 599.801 651.799 691.634 718.561 732.078 50 597.753 648.817 687.513 713.105 725.104 16 595.688 645.810 683.362, 707.614 718.092 10 593.605 642.780 679.182 702.088 711.043 20 591.505 639.727 674.973 696.529 703.959 30 589.388 636.6.50 670.7:35 690.9:38 40 587.253 683.550 666.469 685.314 50 585.101 630.438 662.175 679.659 17 582.933 627.283 657. 854 673.972 10 580.747 624.117 653.506 668.256 20 578.545 620.928 649.131 6C 3.510 30 576.326 617.717 644.7:30 656.735 40 574.091 614. 45 640.304 650.933 50 571.839 611.232 635.852 645.103 18 569.571 607.958 631.375 639.245 10 567.287 604.664 626.874 20 564.988 601.349 622.349 30 562.673 598.013 617.801 40 560.342 . 594.658 613.229 50 557.996 591.283 608.635 19 555.634 587.888 604 ..018 10 553.257 584.475 599.379 20 550.864 581.042 594.720 30 548.457 577.591 590.039 40 546.035 574.121 585.339 50 543.599 570.634 580.618 20 541.147 567.128 575.877 .1*95} TABLE VII.-LONG CHORDS. LONG CHORDS. Degree of Actual Arc, Curve. One Station. 9 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. 21 100.562 196.651 286.716 367.179 435.345 488.931 22 100.617 196.325 285.437 364.060 429.305 478.775 23 100.675 195.985 284.101 360.810 423.033 468.270 24 100.735 195.630 282.709 357.433 416.535 457.433 25 100.798 195.259 281.262 353.930 409.819 446.280 26 100.863 194.874 279.759 350.303 402.891 434.827 27 100.931 i 194.474 278.201 346.555 395.758 423.092 28 101.002 194.059 276.589 342.688 388.428 411.092 29 101.075 193.630 274.924 .338.704 380.908 398.846 30 101.152 193.185 273.205 334.607 373.205 386.370 31 101.230 192.726 271.433 330.397 365.328 373.685 32 101.312 192.252 269.610 328.078 357.284 360.808 33 101.396 191.704 267.734 321.654 349.081 347.759 34 101.482 191.261 265.808 317.125 340.729 334.556 35 101.572 190.743 2(53.830 312.496 332.234 321.220 36 101.664 190.211 261.803 307.768 323.607 307.768 37 101.759 189.665 259.727 302.946 314.855 294.222 38 101.857 189.104 257.602 298.032 305.987 280.600 39 101.957 188.528 255.429 293.028 297.012 266.923 40 102.060 187.939 JS3.209 287.938 287.939 253.209 41 102.166 187.334 860.942 282.766 278.777 239.478 42 102.275 186.716 248.629 277.514 269.535 225.750 43 102.386 186.084 846.271 272.186 260.222 212.045 44 45 102.500 102.617 185.437 184. 77 6 243.868 241.421 2H6.784 261.313 250.848 241.421 198.380 184.776 46 102.737 184.101 338.932 255.775 231.952 171 .251 47 102.860 183.412 236.400 250.173 222.448 157.824 48 102.985 182.709 233.826 244.512 212.920 144.512 49 103.114 181.992 231.212 238.795 203.377 131.335 50 103.245 181.262 228.558 33.025 193.828 118.310 CORRECTIONS FOR SUBCHORD LENGTHS. 106. D 50 85 20 10 D 60 25 20 10 go .004 .008 .002 .001 27 .349 .218 .179 .092 4 .008 .005 .004 .002 28 .376 .235 .192 .099 5 .012 .008 .006 .003 29 .403 .252 .206 .106 6 .017 .011 .009 .005 30 .431 .270 .221 .114 7 .023 .015 .012 .006 31 .461 .288 .236 .122 8 .030 .019 .016 .008 32 .491 .307 .252 .130 9 .039 .024 .020 .010 33 .523 .327 .268 .138 10 .048 .030 .024 .013 84 .555 .347 .285 .147 11 .058 .036 .030 .015 85 .589 .368 .302 .156 12 .069 .043 .035 .018 86 .623 .390 .320 .165 13 .081 .050 .041 .021 37 .659 .412 .338 .174 14 .093 .058 .048 .025 38 .695 .435 .356 .184 15 .107 .067 .055 .028 39 .733 .458 .376 .194 16 .122 .076 .063 .032 40 .771 .483 .898 .204 17 .138 .086 .071 .036 41 .811 .507 .416 .214 18 .155 .097 .079 .041 42 .851 .533 .437 .225 19 .172 .108 .088 .045 43 .893 .559 .458 .236 20 .191 .119 .098 .050 44 .936 .586 .480 .248 21 .211 .132 .108 .056 45 .980 .613 .502 .259 22 .231 .144 .118 .061 46 .094 .641 .525 .271 23 .253 .158 .130 .067 47 .070 .670 .549 .283 24 .275 .172 .141 .073 48 .117 .699 .573 .996 25 .299 .187 .153 .079 49 .165 .729 .598 .808 26 .323 .202 .166 .085 50 .214 .760 .623 .321 TABLE VIII. MIDDLE ORDINATES. Degree 1 a 3 4 5 6 of Curve. Station. Stations. Stations. Stations. Stations. Stations. 21 4.594 18.224 40.431 70.478 107.344 149.809 22 4.814 19.081 42.275 73.545 111.741 155.460 23 5.035 19.937 44.108 W.577 116.042 160.917 24 5.255 20.791 45.929 79.570 120.248 166.172 25 5.476 21.644 47.738 82.520 124.842 171.221 26 5.697 22.495 49.534 85.427 128.335 176.058 27 5.918 23.345 51.317 88.289 132.219 180.677 28 6.139 24.192 53.086 91.105 1.35.990 185.075 29 6.360 25.038 54.842 93.873 139.647 189.245 30 6.583 25.882 56.583 96.593 143.185 193.185 31 6.805 26.724 58.309 99.261 146.603 32 7.027 27.564 .60.019 101.878 149.898 33 7.250 28.402 61.714 104.442 153.068 34 7.473 29.237 63.392 106.95^ 156.110 35 7.69(5 30.071 65.053 109.406 159.023 36 7.919 30.902 B6.698 111.808 161.803 37 8.143 31.730 68.325 114.143 38 8.367 32.557 (59.933 116.424 39 8.592 33.381 71.524 118.645 40 8.816 34.202 73.095 120.805 41 9.041 35.021 74.647 122.902 42 9.267 35.837 76.180 124.937 43 9.493 36.650 77.693 126.909 44 9.719 37.461 79.185 128.815 45 9.946 38.268 80.056 130.656 46 10.173 39.073 82.107 47 10.400 39.875 83.535 48 10.628 40.674 84.942 49 10.856 41.469 86.327 50 11.085 42.262 87.689 TABLE VIII.-MIDDLE OEDINATES. Degree of Curve. 1 Station. g Stations. 8 Stations. 4 Stations. 5 Stations. 8 Stations. O^lO 7 .036 .145 .327 .582 .909 1.309 20 -.073 .291 .654 1.164 1.818 2.618 30 .109 .436 .982 1.745 2.727 3.926 40 .145^ .582 1.309 2.327 3.636 5.235 60 .182 .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 .291 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 n .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 1 .836 3.345 7.522 13.363 20.863 30.012 4 .872 3.490 7.848 13.943 21.766 31.308 10 .909 3.635 8.175 14.522 22.668 32.603 20 .945 3.781 8.501 15.101 23.570 33.896 30 .982 3.926 8.828 16.680 24.471 35.189 40 1 1.018 4.071 9.154 16.258 25.372 36.480 50 1.054 4.217 9.480 16.837 26.272 37.770 5 1.091 4.362 9.807 17.415 27.171 39.059 10 1.127 4.507 10.133 17.992 28.070 40.346 20 1.164 4.653 10.459 18.570 28.968 41.631 30 j 1.200 4.798 10.785 19.147 29.866 i 42.916 40 ! 1.237 4.943 11.111 19.724 30.762 ! 44.198 50 | 1.273 5.088 11.436 20.301 31.658 i 45.479 6 1.309 5.234 11.762 20.877 32.553 46.759 10 1.346 5.379 12.088 21.453 33.448 48.037 20 1 1.382 5.524 12.413 22.029 34.341 49.313 30 i 1.418 5.669 12.739 22.604 35.234 ! 50.587 40 , 1.455 5.814 13.064 23.179 36.126 51.860 50 1.491 5.960 13.389 23.754 37.017 53.130 7 1.528 6.105 13.715 24.328 - 37.907 ! 54.399 10 1.564 6.250 14.040 ' 24.902 38.796 55.666 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 6.685 15.014 26.622 41.458 59.454 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 2d 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 FaoSl TABLE TOILMIDDLE ORDINATES. Degree of Curve. 7 Stations. 8 Stations. 9 Stations. 1O Stations. 11 Stations. 1* Stations. 10' 1.782 2.327 2.945 3.636 '4.400 5.236 20 3.563 4.654 5.890 7 272 8.799 10.471 30 5.345 6.981 8.8535 10 ! 907 13.197 15.704 40 7.120 9.307 11.778 14.540 If. 593 20.936 50 8.907 11.632 14.721 18.173 21.987 26.164 1 10.687 18.957 17.663 21.803 26.378 31.388 to 12.4(57 16.281 20.603 25.431 30.766 36.607 20 14.246 18.604 23.541 29.057 35.150 41.821 30 16.024 20.925 26.477 32.679 39.530 47.028 40 17.802 23.246 29.411 36.298 43.905 52.229 50 19.579 - 25.564 32.343 39.914 48.274 57.422 2 21.355 27.881 35.272 43.525 52.637 62.606 10 23.130 30.197 38.198 47.131 56.993 67.780 20 24.903 32.510 41.121 50.733 61.343 72.945 30 26.676 34.821 44.040 54.330 65.684 78.098 40 28.447 37.130 46.956 57.921 70.018 as. 240 50 30.216 39.436 49.868 61.506 74.342 88.370 3 31.984 41.740 52.77'6 65.084 7'8.657 93.486 10 33.751 44.041 55.679 68.656 82.963 98.588 20 35.516 .46.339 58.577 72.221 87.258 103.675 80 37.279 48.634 61.471 75.778 91.542 108.747 40 39.040 50.926 64.3(50 79.328 95.814 113.803 50 40.800 53.215 67.243 82.869 100.07'5 118.841 1 42.557 55.500 70.121 86.402 104.323 123.862 10 44.312 57.781 7'2.992 89.925 108.558 128.864 20 46.065 60.059 75.858 93.440 112.779 133.847 30 47.816 62.333 78.717 96.945 116.986 138.810 40 49.564 64.602 81.570 100.439 121.178 143.753 50 51.310 66.868 84.416 103.924 125.356 148.674 5 53.053 69.129 87.255 107.397 129.517 153.572 10 54.794 71.386 90.087 110.860 133.663 158.448 20 56.532 73.638 92.911 114.311 137.791 163.300 30 58.267 75.885 95.728 117.751 141.903 168.128 40 59.999 78.127 98.536 121.178 145.997 172.931 50 61.729 80.364 101.337 124.593 150.072 177.708 6 63.455 82.596- 104.129 127.995 154.129 182.459 10 65.178 84.832 106.912 131.384 158.166 187.182 20 66.898 87.043 109.686 134.759 162.184 191.878 30 68.615 89.258 112.452 138.120 166.182 196.545 40 70.328 91.468 115.208 141.468 170.159 301.183 50 72.037 93.671 117.954 144.800 174.114 205.792 7 73.744 95.868 120.691 148.118 178.048 210.370 10 Y5.446 98.059 123.417 151.421 181.960 214.916 20 77.145 100.244 126.134 154. W6 185.850 219.431 30 78.840 102.422 128.840 157.979 189.716 223.914 40 80.531 104.594 131.535 161.234 193.559 228.363 50 82.218 106.758 131.219 164.473 197.377 232.779 8 83.901 108.916 136.893 167.695 201.171 237.160 10 85.580 111.067 139.555 170.899 204.941 241.507 20 87.254 113.210 142.205 174.086 208.685 245.818 30 88.924 115.346 144.844 177.255 212.403 250.093 40 90.590 117.475 147.470 180.407 216.095 254.331 50 92.252 119.596 150.085 183.539 219.760 258.531 9 93.909 121.709 152.687 186.653 223.398 262.694 10 95.561 123.814 155.277 189.748 227.008 266.818 20 97.208 125.911 157.854 192.824 230.591 270.904 30 98.851 128.000 160.417 195.880 234.145 274.949 40 100.489 130.081 162.968 198.916 237.670 278.955 50 102.122 132.153 165.505 201.932 241.167 282.919 10 103.750 134.217 168.029 204.928 244.6*3 286.843 [290] TABLE VIII. MIDDLE ORDINATES. 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.&59 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 30 2.511 10.019 22.448 39.673 61.521 87.772 40 2.547 10.164 22.769 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 60 2.803 11.176 25.014 44.145 68.328 97.260 13 2.839 11.320 25.334 44.701 69.171 98.481 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 72.529 103.079 50 3.022 12.043 26.931 47.472 73.364 104.232 14 3.058 12.187 27.250 48.024 74.197 105.381 10 3.095 12.331 27.569 48.575 75.029 106.527 do 3.131 12.476 27.887 49.126 75.859 107.669 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 79.979 118.319 20 3.350 13.341 29.794 52.414 80.798 114.438 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 PU8.870 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 87.274 123.235 50 3.679 14.637 32.638 57.294 88.074 124.315 17 3.716 14.781 32.953 57.832 88.872 125.391 10 3.752 14.925 33.267 58.369 89.668 126.463 20 3.789 15.069 33.582 58.906 90.462 127.530 30 3.825 15.212 33.896 59.441 91.254 128.593 40 3.862 15.356 34.210 59.976 92.043 129.651 50 3.899 15.500 34.523 60.510 92.830 130.704 18 3.985 15.643 34.837 61 .042 93.616 131.753 10 3.972 15.787 35.150 61.574 94.398 182.797 20 .008 15.931 35.463 62.106 95.179 133.837 30 .045 16.074 35.775 62.636 95.957 134.872 40 .081 16 218 36.088 63.165 96.733 135.902 50 .118 16.361 36.400 63.693 97.506 136.928 19 .155 16.505 36.712 64.221 98.278 137.948 10 .191 16.648 37.023 64.747 99.047 138.964 20 .228 16.792 37.334 65.273 99.813 139.975 30 .265 16 9a5 37.645 65.797 100.577 140.981 40 .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 [300] TABLE IX. LINEAR DEFLECTION TABLE. Deflec- tion. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. SO' 0.87 1.75 2.62 3.49 4.36 5.24 6.11 6.98 7.85 8.73 1 1.75 3.49 5.24 6.98 8.73 10.47 12.22 13.96 15.71 1745 30 2.62 5.24 7.85 10.47 13.09 15.71 is. as 20.94 23.56 26.18 2 3.49 6.98 10.47 13.96 17.45 20.94 24.43 27.92 31.41 34.90 30 4.36 8.72 13.09 17.45 21.81 26.18 30.54 34.90 39.27 43.63 3 5.24 10.47 15.71 20.94 26.18 31.41 36.65 41.88 47! 12 52.35 30 6.11 12.22 18.32 24.43 30.54 36.65 42.75 48.86 54.97 61.08 4 6.98! 13.96 20.94 27.92 34.90 41.88 48.86 55.84 66.82 p. 80 30 7.85 15.70 23.56 31.41 39.26 47.11 54.96 62.82 70.67 78.52 5 8.73 17.45 26.17 ai.89 43.62 52.34 61.07 69.79 78.51 87.24 30 9.60 19.19 28.79 38.38 47.98 57.57 67.17 76.76 86.36 95.96 6 10.47 20.93 31.40 41.87 52.34 62.80 73.27 83.74 94.20 104.67 30 11.34 22.68 34.02 45.35 56.67 68.03 79.37 90.71 102.05 113.39 7 12.21! 24.42 36.63 48.84 61.05 73.26 85.47 97.68 109.89 122.10 30 13.08 26.16 30.24 52.32 65.40 78.48 91.56 104.64 117.73 130.81 8 13.95 27.90 41.85 55.80 69.76 83.71 97.66 111.61 125.56 139.51 30 14.82 29.64 44.47 59.29 74.11 88.93 103.75 118.57 133.40 148.22 9 15.69 31.38 47.08 62.77 78.46 94.15 109.84 125.53 141.23 156.92 30 16.56 as. 12 49.68 66.25 82.81 99.37 115.93 132.49 149.05 165.62 10 17.43 34.86 52.29 69.72 87.16 104.59 122.02 139.45 156.88 174.31 30 18.30 36.60 54.90 73.20 91.50 109.80 128.10 146.40 164.70 183.00 11 19.17 38.34 57.51 76.68 95. S5 115.01 134.18 153.35 172.52 191.68 30 20.04 40.08 60.11 80.15 100.19 120.23 140.26 160.30 180.34 200. 38 12 20.91 41.81 62.72 83.62 104.53 125.43 146.34 167.25 188.15 209.06 30 21.77 43.55 65.32 87.09 108.87 130.64 152.41 174.19 195.96 217.73 13 22.64 45.28 67.92 90.56 113.20 135.84 158.48 181.13 203.77 226.41 30 23.51 47.01 70.52 94.03 117.54 141.04 164.55 188.06 211.57 235.07 14 24.37 48.75 73.12 97.50 121.87 146.24 170.62 194.99 219.36 243.74 30 25.24 50.48 75.72 100.96 126.20 151.44 17'6.68 201.92 227.16 252.40 15 26.11 52.21 78.32 104.42 130.53 156.63 182. 74 208.84 234.95 261.05 30 26.97 53.94 80.91 107.88 134.85 161.82 188.79 215.76 242.73 269.70 16 27.83 55.67 83.50 111.34 139.17 167.01 194.84 222.68 250.51 278.35 30 28.70 57.40 86.10 114.79 143.49 172.19 200.89 225). 59 258.29 286.99 17 29.56 59.12 88.69 118.25 147.81 177.37 206.93 236.50 266.08 295.62 30 30.42 60. 85 91.27 121.70 152.12 182.55 212.97 243.40 273.82 304.25 18 31.29 62.57 93.86 125.15 156.43 187. 7'2 219.01 250.30 281.58 S12.87 30 32.15 64.30 96.45 28. 59 160.74 192.89 225.04 257.19 289. 34 321.49 19 33.01 66.02 99.03 32.04 165.05 198.06 231.07 264.08 297.08 330.09 30 33.87 67. 7< 101.61 35.48 169. 35 203.22 237.09 270.96 304.83 338.70 20 34.73 69.46 104.19 138.92 173.65 208.38 243.11 277.84 312.57 347.30 30 35.59 71.18 106.77 142.35 177.94 213.53 249.12 284.71 320.30la55.89 21 36.45 72.89 109.34 145.79 82.24 218.68 255.13 291.58 328.02 364.47 30 37.30 74.61 111.91 149.22 88.52 223.83 261.13 298.44 3a5.74 373.05 22 38.16 76.32 114.49 152.65 90.81 228.97 "'57.13 305.29 343.46 381.62 30 39.02 78.04 117.05 156.07 95.09 234.11 273.13 312.14 151,16 390.18 23 39.87 79 . 75 119.62 159.49 99.37 239.24 279.12 318.99 398.74 30 40.73 81.46 122.19 162.91 203.64 244.37 2K5.10 325.83 366^56 407.28 24 41.58 83.16 124.75 166.33 207.91 249.49 291.08 a32.66 374.24 415.82 30 42.44 84.87 127.31 169.74 212.18 254.61 297.05 339.48 381.92 424.36 25 43.29 86.58 129.86 173.15 216.44 259.73 303.02 364.30 389.59 432.88 30 44.14 88.28 132.42 176.56 820.70 264.84 308.98 353.12 397.26 441.39 26 44.99 89.98 134. 97 j 179. 96 824.95 266.94 314.93 359.92 404.91 449.90 30 45.84 91.68 137.52 183.36 229.20 275.04 320.88 366.72 412.56 458.40 27 46.69 93.38 140.07 186 7(5 233.45 280.14 326.82 373.51 420.20 466.89 30 47.54 95.07 142.61)190.15 ,237.69 285.22 332.76 380.30 427. 83 475.37 28 48.38 96. 77; 145. 15 193.54 841.93 290.31 338.69 387.08 435.46 483. 84 30 49.23 98.46 147.69 196.92 246.15 295.38 844.62 393.85 443. 08 1492. 31 29 50.08 100.15 150.23 200.30 250.38 300.46 350.63 400.16 450.68 500.76 30 50.92 101.84 152.7(5 203.68 254.60 305.52 a56.44 407.36 458.28 509.20 30 51.76 103.53 155.29 20V. 06 258.82 310.59 362.35 414.11 465.87 517.64 TABLE X. -COEFFICIENTS FOR VALVOID ARCS I. RATIO OF u = A L 10 20 30 40 50 60 70 80 90 100 110 I 220 300 .3518 .3516 .3514 .3510 .3506 .3500 .3498 .3485 .3476 .3466 .3455 .3444 400 .3437 .3436 .3433 .3430 .3426 .34211.3415 .3408 .3399 .3390 .3380 .3368 500 .3400 .3398 .a396 .3393 .3389 .3383 .3379 .3372 .3364 .3356 .3345 3335 600 .3379 .3378 .3370 .3373 .8369 .3365 a'i")'i .3353 ..3345 .3337 .3327 .8817 700 .3367 .a306 .3364 .3361 .8858 :.3347 .3841 .3334 .3326 .3316 .asoe 800 .3359 .8858 .3356 .3353 '.3349 .3:^45 .3340 .333:-S .3320 .3318 .3309 .3299 900 .3353 .3352 .3350 .3348 .3344 .3340 .w-m as28 .3331 .3313 .3304 .3294 1000 .3350 .3348 .3346 .3344 .3340 .3330 .3331 !3324 .3317 .3310 .3301 .3291 1200 .3345 .8848 .3341 .8339 .3336 .3331 .3326 .3320 .8313 .3:305 .3296 .3286 1500 .3340 .3339 .3387 .3335 .8331 .$327 .3322 .3316 ..3309 .3301 .3292 .3283 2000 .3337 .3*36 .3333 .3381 .3328 .3324 .3319 .3313 .3:306 .3298 .3289 .3280 II. RATIO OP v = LI L 10 20 30 40 50 60 70 80 90 100 110 120 300 .7706 .7683 .7643 .7588 .7518 .7432 .7a32 .7218 .7090 .6949 .6795 -6630 400 .7611 .7588 .7549 .7495 .7425 .73411.7243 .7130 .7004 .6865 .6714 6551 500 .7506 .7'545 .7506 .7452 .7384 . 7300 1. 7202 .7091 .0966 .6828 .6678 -6516 600 .7545 .7522 .7483 .7430 .7361 .7278 .7181 .7070 .6946 .6808 .6659 .6498 700 .7531;. 7508 .7469 .7416 .7348 .7265 .71681.7057 .0933 .6797 .6648 6487 800 .7522;. 7499 .7461 .7407 .7339 .7257 .7100 .7049 .6926 .6789 .6640 .6480 900 . 7516 \. 7492 1.7454 .7401 .7338 .7251 .7154 .7044 .6920 .6784 .6635 ,6475 1000 .7512 .7489 .7450 .7397 .7329 .7247 .7150 .7040 .6917 .6780 .6632 .6472 1200 .7505 .7483 .7444 .7391 .7324 .7241 .7145 .7035 .6912 .6775 .6627 .6468 1 1500 .7501 .7478 .7440 .7387 .7319 .7237 .7141 .7031 .69081.6772 .6624 .6464 2000 .7497 .7474 .7436 .7383 .7316 .7234 .7137 .7028 .6904 .6769 .6621 .6461 jjj RATIO OF Z ' ' LENCTII OF VALVOID ARC CORRESPONDING A x - A" TO A CHANGE OF ONE DEGREE IN THE ANGLE A . L 10 20 30 40 50 60 70 80 90 100 110 120 300 2.62 2.61 2.60 2.59 2.57 2.55 2.52 2.49 2.46 2 42 2.38 2.34 400 3.49 3.48 3.46 3.44 3.42 3.38 3.35 3.30 3.25 3^20 3.14J 3.08 500 4.36 4.35 4.33 4.30 4.26 4.22 4.17 4.11 4.05 3.98 3.90 3.81 600 5.23 5.22 5.19 5.16 5.11 5.06 4.99 4.92 4.84 4.75 4.65 4.55 700 6.10 6.09 6.06 6.02 5.96 5.90 5.83 5.74 5.65 5.54 5.43 5.31 800 6.97 6.95 6.92 6.87 6.82 6.74 6.66 6.56 6.45 6.33 6.20 6.00 900 7.85 7.82 7.79 7.73 7.67 7.59 7.49 7.38 7.26 7.13 6.98 6.82 1000 8.72 8.69 8.65 8.59 8.52 8.43 8.32 8.20 8.07 7.92 7.75 7.58 1100 9.59! 9.56 9.52 9.45 9.37 9.27 9.16 9.02 8.87 8.71 8.53| 8.34 1200 10.46 10.43 10.38 10.31 10.22 10.11 9.99 9.84 9.68 9.50 9.31 9.09 1300 11.38 11. 30|11 .25111. 17,11.07 10.96 10.82 10.66jlO.49 10.29 10.08 9.85 1400 12.21 12.17,12.11112.03 11.93 11.80 11.65 ll.48ill.29 11.08 10.86 10.61 1500 13.08 13.04 12.98 12.89 12.78 12.64 12.48 12.30 12.10 11.88 11.63: 11.37 1600 13.95 13.91 13.84 13.75 13.63 13.49 13.32 13.12 12.91 12.67 12.41 12.13 1700 14.82 14.78 14.71 14.61 14.48 14.33 14.15 13.94 13.71 13.46 13.18 12.88 1800 15.69 15.65S15.57 15.47 15.33 15.17 14.98 14.76 14.52 14.25 13.96 13.64 1900 16.57 16.52 16.44 16.33 16.19 16.01 15.81 15.58 15.33 15.04 14.73 14.40 2000 17.44 17.39 17.30 17.19 17.04 16.86 16.65 16.40 16.13 15.83 15.51 15.16 TABLE XI. TURNOUTS AND SWITCHES FROM A STRAIGHT TRACK. 180, 181, 182. GAUGE, 4 FEET 8J4 INCHES = 4 .708. THROW, 5 INCHES = 0.417. No. n. Angle Dist. BF. Chord a/. Switch AD. Radius r. Log'thm. log. r. D of egree Curve. 4 14 15' 00" 87 664 37.373 11.209 150.656 2.177986 38 45' 57" 12 4 49 42 372 4 C , !.113 1 2.610 190. < 574 2.281 1292 30 24 09 5 11 25 16 47.080 46.846 14.012 235.400 2.371806 24 31 36 10 2 3 20 51 788 51 .575 1 j 413 284. j 2.418 602. 524 2.78 J046 9 31 07 6 43 59 80 036 79.898 23.820 680.306 2.832704 8 25 47 9 6 2 1 35 84 744 84 .613 2. 5.221 762. M\ 2.88 2:15-2 7 31 04 6 01 32 89.452 89 328 26.622 849.794 2.929314 6 44 46 10 a 5 4 3 29 94 160 p. .043 2J 3.023 941. 500 2.97 1866 (5 05 16 5 27 09 98.868 3.756 29.424 1038.114 3.016245 5 31 17 11 " 5 1 2 18 103 576 10? .469 81 3.825 1139. m 3.05 5(552 5 01 50 4 5 8 45 108 284 ioe .182 3 2.227 1245., _'(,<; 3.09 5262 4 36 08 12 4 46 19 112 992 112.894 a i.628 1355.904 3.132229 4 13 36 GAUGE, 3 FEET. THROW, 4 INCHES = O.a33. No. Anerle Dist. Chord Switch Radius Log'thm. D egrfee n. F. BF. a/. AD. T. log. r. of Curve. 4 14 15' 00" 24 2.' 5.815 8 96.0 1.982271 62 46' 34" 12 40 49 27 21 5.835 9 121. > 2.084 576 48 36 04 5 11 25 16 30 21 >.851 10 150. ( ) 2.176 091 88 56 a3 10 23 20 a3 32.865 11 181.5 2.258877 31 '58 55 6 2 9 31 39 36 a i.876 12 216. 3 2.334 454 26 46 07 8 47 51 39 38.885 13 253.5 2.403978 22 45 04 7 8 10 16 42 4 L.893 14 294. 3 2.468 347 19 S5 01 7 37 41 45 44.900 15 837.5 2.528274 17 02 21 8 2 7 09 10 48 4 r.906 16 384. 3 2.584 331 14 57 48 6 43 59 51 51 3.912 17 433. 5 2.636 989 13 14 47 9 3 6 21 35 54 53.917 18 486.0 2.686636 11 48 37 9Vo 6 01 32 57 5 3.921 19 541. 5 2.7=33 598 10 35 46 10 ~ 5 43 29 60 5 3.925 20 600. ) 2.778151 9 33 38 5 27 09 63 6 2.'. 1-2!) 21 661. 5 2.820 530 8 40 12 11 ^ 5 12 18 66 65.932 22 726.0 2.860937 7 53 54 4 58 45 69 6 3.935 23 793. 5 2.899 547 7 13 82 12 3 4 46 19 72 1.938 24 864.0 2.936514 ( 38 06 ANGLE AND DISTANCE OF MIDDLE FROG, F" Gauge Gauge Gauge Gauge No. No. 4,8^. 3. No. No. An^lo 4 8^j> 3. 71 * 7l". " Dist. Disr. n. n" . F" Dist.' Dist. aF". aF". aF". aF". 4 I 2.817 20 07 86' 26.736 17.037 8 5.651 10 06' 44" 53.317 a3.974 4V 3.172 17 54 52 30.054 19.151 8Vo 6.005 9 31 08 56.643 36.094 5 r 1 3.527 1(5 OS 19 a3.374 21.266 9 6.359 8 59 30 59.969 38.213 5UI 3.881 14 40 58 36.695 23.383 9^ 6.713 8 31 10 63.296 40.3.33 6 4.235 13 27 57 40.018 25.500 10 7.067 8 05 40 66.623 42.453 " 6^ 4.589 12 26 07 43.342 27.618 10J4 7.420 7 42 35 69.950 44.573 7 19 13 11 33 04 46.666 29.736 11 7.774 7 21 36 73.277 46.693 7J^>' 5 '297 10 47 02 49.991 31 855 11U 8.128 7 02 26 76.605 48.813 8 ~ 5. (551 10 06 44 53.317 33.974 12 S.4H2 6 44 51 79.932 50.98* TABLE XII. -MIDDLE ORDINATES FOR CURVING RAILS. 8199- \ LENGTH OF RAIL-CHORD. D D 32 30 28 j J6 24 2 3 so 18 16 1 14 12 10 * .022 .020 .017 .015 .013 .011 .009 ~.oo7 .006~ .004 .003 ,002 l c Q .045 .039 .034 030 ,1 n .017 .014 .011 .009 .006 .004 2 g .067 .059 .051 'm .( (38 .1 62 .026 .021 .017 .013 .009 '.007 3 4 .089 .079 .068 ),-,() ,| )50 .( 42 .035 .028 i.022 .017 .013 .009 4 g .112 .098 .086 J74 .( MS:-J .( m .044 035 .028 .021 .016 ,011 5 Q .134 .118 .103 IHS .075 i$ .052 '.042 .034 .026 .019 6 7 .156 .137 .120 103 .( XSS .( 74 .061 .049 .039 .030 .032 !W5 7 8 .179 .157 .137 118 .100 .( 84 .070 .057 .045 ,034 .025 .017 8 9 .201 .177 .154 133 . 18 .0 5 .078 .064 .050 .038 .028 .020 9 10 .223 .196 .171 147 .126 .105 .087 .071 .056 .043 .031 .022 10 11 .245 .216 .188 162 . 88 .1 16 .096 .078 .061 .047 .035 .024 11 12 .268 .235 .205 177 .151 .127 .105 .085 .067 .051 .038 .026 12 14 .312 .274 .238 J06 L7r, .1 R .122 .099 .078 .rao .044 .030 14 16 .356 .313 .273 J35 i wo 1 5H .139 .113 .089 .068 .050 ,035 16 18 .400 .352 .307 jlj.j. ;>:> 8< .156 .127 .100 .077 .056 .039 18 20 .445 .391 .340 -"):> '.250 jti .174 .141 111 .085 .063 .043 20 24 .531 .467 .407 i-.l t ><><) .2 :>! .207 .168 .133 .102 .075 .052 24 28 .618 .543 .473 408 it7 .j \ti .241 .195 .154 .118 .087 .060 S8 32 .705 .619 .539 ] 465 i m 4 ; .275 .823 .176 .135 .099 .069 32 36 .791 .696 .606 -,00 j 145 !s 73 .309 .250 .197 .151 .111 .077 36 40 .878 .772 .672 -,7<) ]t m .414 .342 .277 .219 .168 .123 .086 40 45 .983 .863 .752 6 IS VW A 63 .383 .305 .245 .188 .137 .096 45 50 1.087 .955 .831 716 !eio .512 .423 .343 .271 .207 .152 .106 50 TABLE XIII.-DIFFERENCE IN ELEVATION OF RAILS ION CURVES. 901. VELOCITY XH MILES PER HOUR. D D 10 15 20 25 30 35 40 45 50 60 ^~ .006 .018 .023 .036 .051 .070 .091 .116 .143 .2oa ~T 2 .011 .026 .046 .071 .1 s .140 .183 .231 .2* So .410 2 3 .017 .039 .069 .107 .154 .210 .274 .346 .427 .612 3 4 .023 .051 .091 .143 .21 Mi .280 .365 .461 .5 58 .811 4 5 .029 .064 .114 .179 .2, 37 .349 .455 .574 .7( )7 1.006 5 6 .034 .077 .137 .214 .8 18 .418 .545 .687 .844 1.196 6 7 .040 .090 .160 .250 .9 3!) .487 .634 .798 .979 8 .046 .103 .183 .285 A 10 .556 .723 .908 1.1 [2 9 .051 .116 .206 .320 A 10 .624 .811 1.017 10 .057 .129 .228 .356 .511 .692 .898 1.124 11 .063 .142 .251 .391 .561 .760 .984 12 .069 .154 .274 .427 .6 11 .826 1.069 14 .080 .180 .319 .497 .7 11 .959 16 .091 .206 .365 .567 19 1.088 18 .102 .231 .410 .637 .91 X) 20 .114 .256 .455 .707 1.002 25 .141 .318 .563 .775 30 .168 .880 ,672 .844 35 .195 .441 .778 40 .222 .501 .881 50 .276 .618 TABLE XIV.-GRADES AND GRADE ANGLES. Feet Feet Feet per Sta- Feet per Mile. Inclina- tion. Sta- Feet per Mile. Inclina- tion. per Sta- Feet per Mile. Inclin- ation. tion. tion. tion. / / / .01 .528 21 .51 26.928 17 32 .01 53.328 34 43 .02 1.056 41 .52 27.456 17 53 .02 53.856 35 Q4 .03 1.584 1 02 | .58 27.984 18 13 .03 54.384 a5 24 .04 2.112 1 23 .54 28.512 18 34 .04 54.912 35 45 .05 2.640 1 43 .55 29.040 18 54 .05 55.440 36 05 .06 3.168 2 04 .56 29.568 19 15 .06 55.968 36 26 .07 3.696 2 24 .57 30.096 19 36 .07 56.496 36 47 .08 4.224 2 4:, .58 30.624 19 56 .08 57.024 37 08 .09 4.752 3 06 | .68 31.152 20 17 .09 57.552 37 28 = 10 5.280 3 26 .60 31.680 20 38 .10 58.080 3749 .11 5.808 3 47 .61 32.208 20 58 .11 58.608 38 09 .12 6.336 4 08 .62 32.7'36 21 19 .12 59.136 38 30 .13 6.864 4 28 .63 as. 264 21 39 .13 59.664 38 51 .14 7.392 4 49 .64 as. 792 22 00 .14 60.192 39 11 .15 7.920 5 09 ,65 34.320 22 21 .15 60.720 39 32 .16 8.448 5 30 .66 34.848 2241 .16 61.248 39 53 .17 8.976 5 51 .87 35.376 23 02 .17 61.776 40 13 .18 9.504 6 11 .68 35.904 23 23 .18 62.304 40 34 .19 10.032 6 32 .69 36.4S2 23 43 .19 62.832 40 54 .20 10.560 6 53 .70 36.960 24 04 .30 63.360 41 15 .21 11.088 7 13 .71 37.488 24 24 .21 63.888 41 35 .22 11.616 7 34 .72 38,016 24 45 .22 64.416 41 56 .23 12.144 7 54 .73 38.544 25 06 .23 64.944 42 17 .24 12.672 8 15 .74 39.072 25 26 .24 65.47'2 42 38 .25 13.200 8 30 .75 39.600 25 47 .25 66.000 42 58 .26 13.728 8 56 .76 40.128 26 08 .26 66.528 43 19 .27 14.256 9 17 .77 40.656 26 28' .27 67.056 43 39 .28 14.784 9 38 .78 41.184 26 49 .28 67.584 44 00 .29 15.312 9 58 .79 41.712 27 09 .29 68.112 44 21 .30 15.840 10 19 .80 42.240 27 30 .30 68.640 44 41 .31 16.368 10 39 .81 42.768 27' 51 .31 69.168 4502 .32 16.896 11 00 .82 43.296 28 11 .32 69.696 45 23 .33 17.424 11 21 .83 43.824 28 32 .33 70.224 45 43 .34 17.952 11 41 .84 44.352 28 53 .34 70.752 46 04 .85 18.480 12 02 .85 44.880 29 13 .85 71.280 46 24 .36 19.008 12 23 .86 45.408 29 :tt .36 71.808 46 45 .37 19.536 1243 .87 45.936 29 54 .37 72.aS6 47 06 .38 20.064 13 04 .88 46.464 30 15 .38 72.864 47 26 .39 20.592 13 24 .89 46.992 30 36 .39 73.392 47 47 .40 21.120 13 4G .90 47.520 30 57 .40 73.920 48 08 .41 21.648 14 06 .91 4S.048 31 17 .41 74.448 4828 .42 22.176 14 26 .98 48.576 31 as .42 74.976 48 49 .43 22.704 14 47 .93 4!). 104 31 58 .43 75.504 49 09 44 23.232 15 08 .94 49.032 32 19 44 76.032 49 30 .45 23.76'J 15 28 .95 50.160 32 3!) .45 76.560 49 51 .46 24.288 15 49 .96 50.688 33 00 .46 77.088 50 11 .47 24.816 1609 .97 51.216 as 21 .47 77.616 50 32 .48 25.344 . 16 30 .98 51.744 33 41 .48 78.144 50 52 .49 25.872 16 51 99 52.272 34 02 1.49 78.672 51 13 .50 26.400 17 11 1.00 52.800 3423 1.50 79.200 51 34 [305] TABLE XIV. GRADES AUD GRADE ANGrLES. Feet! per Sta- tion. Feet per Mile. Inclina- tion. Feet pei- Sta- tion. Feet per Mile. 11 [nclina- tion. Feet P er Sta- tion. Feet per Mile. , Inclina- tion. o / o / O / " 1.51 79.728 51 54 2.05 108.240 1 10 28 5.10 269.280 1 2 55 10 1.52 80.256 52 15 2.10 110.880 1 12 11 5.20 274.560 2 58 36 1.53 80.784 52 36 2.15 113.520 i 1 13 54 5.30 279.840 3 02 09 1.54 81.312 52 56 2.20 116.160 1 15 37 5.40 285.120 3 05 27 1.55 81.840 53 17 2.25 118.800 1 17 20 5.50 290.400 3 08 53 1.56 82.368 5337 2.30 121.440 1 19 03 5.60 295.680 3 12 19 1.57 82.896 53 58 2.35 124.080 1 20 46 5.70 300.960 i 3 15 44 1.58 83.424 54 19 2.40 126.720 1 22 29 5.80 06.240 3 19 10 1.59 83.952 54.39 2.45 189.360 1 24 12 5.90 311.520 3 22 36 1.60 84.480 5500 2.50 132.000 1 25 56 6.00 316.800 i 3 26 01 1.61 85.008 5521 2.55 134.640 1 27 39 6.10 322.080 ! 3 29 27 1.62 85-536 55 41 2.60 137.280 1 29 22 I 6.20 327.360 3 32 52 1.63 86-064 56 02 2.65 139.920 1 31 05 j| 6.30 332.640 3 36 18 1.64 '86.592 56 22 2.70 142.560 1 32 48 1 6.40 337.920 3 39 43 1.65 87.120 56 43 2.75 145.200 1 34 31 6.50 343.200 3 43 08 1.66 87.648 57 04 2.80 147.840 1 36 14 6.60 348.480 346 34 1.67 88.176 5724 2.85 150.480 1 37 57 6.70 353.760 3 49 59 1.68 i 88.704 57 45 2.90' 153.120 1 39-40 6.80 359.040 3 53 24 1.69 89.232 58 06. 2.95 155.760 1 41 '3 6.90 364.320 3 56 50 1.70 89.760 58 26 3.00 158.400 1 43 06 7.00 369.600 4 00 15 1.71 90.288 58 47 3.05 161.040 1 44 49 7.10 374.880 4 0340 1 72 90.816 59 07 3.10 163.680 1 46 32 7.20 380.160 4 07 06 1.73 9i.344 59 28 3.15 166.320 1 48 15 7.30 385.440 4 10 31 1.74 91.872 59 49 3.20 168.960 1 49 58 7.40 390.720 4 13 56 1.75 92.400 1 00 09 3.25 171.600 1 51 41 7.50 396.000 4 1721 1.76 92.928 1 00 30 3.30 174.240 1 53 24 7.60 401.280 4 20 46 1.77 : 93.456 1 00 51 3.35 176.880 1 55 07 7.70 406.560 4 24 11 1.78 93.984 1 01 11 3.40 179.520 1 56 50 7.80 411.840 4 27 36 1.79 94.512 1 01 32 3.45 182.160 1 8 33 7.90 417.120 4 31 01 1.80 95.040 1 01 52 3.50 184.800 2 00 16 i 8. CO 422.400 4 34 26 1.81 95.568 1 02 13 3.55 187.440 2 01 59 8.10 427.680 4 37 51 1.82 96.096 1 02 34 3.60 190.080 2 03 42 8.20 432.960 4 41 16 l.8 96.624 1 02 54 3.65 192.720 2 05 25 8.30 438.240 4 44 41 1.84 97.152 1 03 15 3.70 195.360 2 07 08 8.40 443.520 4 4806 1.85 | 97.680 1 03 35 3.75 198.000 2 08 51 8.50 448.800 4 51 30 1.86 98.208 1 03 56 3.80 200.640 2 10 34 II 8.60 454.080 4 54 55 1.87 98.736 1 04 17 3.85 203.280 2 12 17 1! 8.70 459.360 4 58 SO 1.88 99.264 1 04 37 3.90 205.920 2 14 00 8.80 464.640 5 01 44 1.89 99.792 1 04 58 3.95 208.560 2 15 43 8.90 469.920 5 05 10 1.90 100.320 1 05 19 4.00 211.200 2 17 26 9.00 475.200 5 08 34 1.91 100.848 1 05 39 4.10 216.480 2 20 52 9.10 480. 4SC 5 11 59 1.92 101.376 1 06 00 4.20 221.760 2 24 18 9.20 485.760 5 15 23 1.93 101.904 1 06 20 4.30 227.040 2 27 44 9 30 491.040 5 18 48 1.94 102.432 1 06 41 4.40 232.320 2 31 10 9.40 496.320 5 22 12 1.95 102.960 1 07 02 4.50 237.600 2 34 36 9 50 501.600 5 25 37 1.96 103.488 1 0722 4.60 242.880 2 38 01 9.60 506.880 5 29 01 1.97 104.016 1 07 43 4.70 248.160 2 41 27 9.70 512.160 5 32 25 1 98 104.544 1 08 04 4.80 253.440 2 44 53 9.80 517.440 5 35 50 1.99 105.072 1 0824 4.90 258.720 2 48 19 9.90 522.720 5 39 .14 2.00 105.600 1 08 45 5.00 264.000 2 51 45 10.00 528.000 5 42 38 [306] TABLE XV.-FOR OBTAINING BAROMETRIC HEIGHTS IN FEET. Barom- eter. Inches = h. 0.00 0.02 0.04 06 0.08 Diff. per .002 in. 19.0 .1 16832 16970 16860 16997 16888 17025 16915 17052 16943 17080 2.8 2.8 .2 17107 17134 17162 17189 17216 2.7 *8 17243 17270 17298 17325 17352 2.7 .4 17379 17406 17'433 17460 17487 2.7 5 17514 17540 17567 17594 17621 2.7 .6 17648 17674 17701 177'28 17755 2.7 .7 17781 17808 17834 17861 17887 2.7 .8 17914 17940 17967 17993 18020 2.7 .9 18046 18072 18099 18125 18151 2.6 20. o 18178 18204 18230 18256 18282 2.6 .1 18:308 18334 18360 18386 18413 2.6 .2 18438 18464 18490 18516 18542 2.6 ,8 18568 18594 18620 18645 18671 2.6 .4 18697 18723 18748 18774 18799 2.6 .5 ' 18825 18851 18876 18902 18927 2.6 .6 18953 18978 19004 19029 19054 2.5 .7 19080 19105 19130 19156 19181 2.5 .8 19206 19231 1925 19282 19307 2.5 .9 19332 19357 19382 19407 19432 2.5 21. 19457 19482 19507 19532 19557 2.5 .1 19582 19606 19631 19656 19681 2.5 .2 19706 19730 19755 19780 19804 2.5 .3 19829 19854 19878 19903 19927 2.5 .4 19952 19976 20001 20025 20050 2.5 .5 20074 20098 20123 20147 20172 2.5 .6 20196 20220 20244 20269 20293 2.4 20317 20341 20365 20389 20413 2.4 ."s 20438 20462 20486 20510 20534 2,4 .9 20558 20581 20605 20629 20653 2.4 22.0 20677 20701 20725 20748 20772 2.4 .1 20796 20820 20843 20867 20891 2.4 .2 20914 20938 20962 20985 21009 2.4 .3 21032 21056 21079 21103 21126 2.4 .4 21150 21173 21196 21220 21243 2.3 .5 21266 21290 21313 21.336 21859 2.3 .6 21383 21406 21429 21452 21475 2.3 .7 21498 21522 21545 21568 21591 2.3 .8 21614 21637 21660 21683 21706 2.3 .9 21728 21751 21774 21797 21820 2.3 23. o 21843 21866 21888 21911 219^34 2.3 .1 21957 21979 22002 22025 22C47 2.3 .2 22070 220!)2 22115 22138 22160 2.3 .3 22183 22205 22228 22250 22272 2.2 .4 22295 82817 22340 .22362 22384 2.2 .5 22407 22429 22451 22474 22496 2.2 .6 22518 22540 22562 22585 22607 2.2 7 22629 22651 22673 22695 22717 2.2 .8 22739 22761 227m 22805 22827 2.2 .9 22849 22871 22893 22915 22937 2.2 24 ,0 22959 22981 23003 23024 23046 2.2 .1 23068 23090 23111 23133 23155 2.2 .2 23176 23198 23220 23241 23263 2.2 .3 23285 23306 23328 23349 23371 2.2 .4 23392 23414 23435 23457 23478 2.2 .5 23500 23521 23542 23564 23585 2.1 .6 23606 23628 23649 23670 23692 2.1 .7 23713 23734 23755 23776 23798 2.1 .8 23819 23840 23861 23882 23903 2.1 .9 23924 23945 23966 , 23987 24008 2 1 [307] TABLE XV. FOR OBTAINING BAROMETRIC HEIGHTS IN FEET. Barom- eter. Inches = h. 0.00 0.02 0.04 0.06 0.08 Diff. per .002 in. 25. .1 24029 24134 24050 24155 24071 24176 24092 24197 24113 24217 2.1 2.1 .2 24238 24259 24280 24301 24321 2.1 .3 24342 24363 24384 24404 24425 2.1 .4 24446 24466 24487 24508 24528 2.1 .5 24549 24569 24590 24610 24631 2.1 .6 24651 24672 24692 24713 24733 2.0 .7 24754 24774 24794 24815 24835 2.0 .8 24855 24876 24896 24916 24937 2.0 .9 24957 24977 24997 25018 25038 2.0 26. o 25058 25078 25098 25118 25138 2.0 .1 25159 25179 25199 25219 25239 2.0 .2 25259 25279 25299 25319 25339 2.0 .3 25359 25379 25399 25419 25438 2.0 .4 25458 25478 25498 25518 25538 2.0 .5 25557 25577 25597 25617 25637 2.0 .6 25656 25676 25696 25715 25735 2.0 .7 25755 25774 25794 25813 25833 2.0 .8 25S53 25872 25892 25911 25931 2.0 .9 25950 25970 25989 26009 26028 2.0 27. 26048 26067 26086 26106 26125 1.9 .1 26145 26164 26183 26203 26222 1.9 .2 26241 26260 26280 26299 26318 1.9 .3 26337 26357 26376 26395 26414 1.9 .4 26433 26452 26472 26491 26510 1.9 .5 26529 26548 26567 26586 26605 1.9 .6 26624 26643 26662 26681 26700 1.9 26719 26738 26757 26776 26795 1.9 .'8 26813 26832 26851 26870 26889 1.9 .9 26908 26926 26945 26964 26963 1.9 28.0 7001 27020 27039 27058 27076 1.9 .1 27095 27114 27132 27151 27169 1.9 .2 27188 27207 27225 27344 27262 1.9 .3 27281 27299 27318 27336 27355 1.8 .4 27373 27392 27410 27429 27447 1.8 .5 27466 27484 27502 27521 27539 1.8 .6 27557 27576 27594 27612 27631 1.8 .7 27649 27667 27685 27704 27722 1.8 .8 27740 27758 27777 27795 27813 1.8 .9 27831 27849 27867 27885 27904 1.8 29. 27922 27940 27958 27976 27994 1.8 .1 28012 28030 28048 28066 28084 1.8 .2 28102 28120 28138 28156 28174 1.8 .3 28192 28209 28227 28245 28263 1.8 .4 28281 28299 28317 28334 28352 1.8 .5 28370 28388 28405 28423 28441 1.8 6 28459 28476 28494 28512 28529 1.8 .7 28547 28565 28582 28600 28618 1.8 .8 28635 28653 28670 28688 28706 1.8 .9 28723 28741 28758 28776 28793 1.8 30. 28811 28828 28846 28863 28881 1.8 .1 28898 28915 28933 28950 28968 1.8 .2 28985 29002 29020 29087 29054 1.7 .3 29072 29089 29106 29124 29141 1.7 .4 29158 29175 29192 29210 29227 1.7 .5 29244 29261 29278 29296 29313 1.7 .6 29330 29347 29364 29381 29398 1.7 .7 29416 29433 29450 29467 29484 1.7 .8 29501 29518 29535 29552 29569 1.7 .9 29586 29603 9620 29637 29654 1.7 [308] TABLE XVI. COEFFICIENT OF CORRECTION FOR TEMPERATURE. t+4^ 1 i %&T* , I 900 ||* + t + t' - 64 900 4- V t + t' - 64 900 20 .0489 65 _j_ .0011 1 110 _j_ .0511 155 .1011 21 _ .0478 66 .0022 111 .0522 156 .1022 22 .0467 67 .0033 112 .0533 157 .1033 23 .0456 68 .0044 113 .0544 158 .1044 24 .0444 69 .0056 114 .0556 159 .1056 25 .0433 70 .0067 115 .0567 160 .1067 26 .0422 71 .0078 J1 r '.0578 161 .1078 27 .0411 72 .0089 .0581) 162 .1089 28 .0400 73 .0100 llh .0000 163 .1100 29 .0:389 74 .0111 119 .0611 164 .1111 30 .0378 75 .0122 121 4- .0622 165 .1122 31 _ .0367 76 4- .0133 121 .0633 166 4- .1133 , 32 .0350 77 .0144 122 .0644 167 .1144 33 .0:344 .78 .0156 123 .0656 168 .1156 34 .0333 79 .0167 124 .0667 169 .1167 35 .0322 80 .0178 125 .0678 170 .1178 36 .0311 81 .0180 126 .0689 171 .1189 37 .0300 82 .0200 12? .0700 172 .1200 38 .0289 .0211 128 .0711 173 .1211 39 .0278 84 .0222 12S .0722 I 174 .1222 40 .0267 85 .0233 130 4- .0733 175 .1233 41 _4 .0256 86 _t_ .0244 131 .0744 176 4- .1244 42 .0244 87 .0256 132 .0756 177 .1256 43 .0233 88 .0267 133 .0767 178 .1267 44 .0222 89 .0278 134 .0778 17'9 .1278 45 .0211 90 .0289 135 .0789 180 .1289 46 .0200 91 .0300 136 .0800 181 .1300 47 .0189 92 .0311 137 .0811 182 .1311 48 .0178 93 .0322 138 .0822 183 .1322 49 .0167 94 .0333 139 .0833 184 .1333 50 _ .0156 95 .0344 140 + .0844 185 .1344 51 .0144 96 4- .0356 141 .0856 186 4- .1356 52 .0133 97 .0367 142 .0867 187 .1367 53 .0122 98 .0378 143 .0878 188 .1378 54 .0111 99 .0:389 144 .0889 189 .1389 55 .0100 100 .0400 145 .0900 190 .1400 56 .0089 101 .0411 146 .0911 191 .1411 57 .0078 102 .0422 147 .0922 192 .1422 5S .0067 103 .0433 148 .0933 193 .1433 59 .0056 104 .0444 149 .0944 194 .1444 60 .0044 105 .0456 150 4- .0956 195 .1456 61 - .0033 106 jf. .0467 151 .0967 198 4- .1467 62 .0022 107 .0478 152 .0978 197 .1478 63 .0011 108 .0489 153 .0989 198 .1489 64 .0000 109 .0500 154 .1000 199 .1590 TABLE XVII. CORRECTION FOR EARTH'S CURVATURE AND REFRACTION. 119. L H u L IP L H L H L H Miles H 300 .002 1:300 .035 2300 .108 8300 .223 4300 | .379 1 .571 400 500 .003 . 005 1400 15IK) .010 .046 2400 2500 118 .128 3400 3500 .237 .251 ;4400| .397 4500 .415 2 2.285 3 5.142 600 .007 1000 .052 2600 .139 3000 .266 4600 .434 4 9.141 700 .010 1700 .059 2700 .149 3700 .281 |4700 .453 5 14.282 800 .013 : 1800 .066 i 2800 .101 3800 .296 '4800 .472 6 20.567 900 .017 1900 .074 2900 .172 oSKX) .312 1900 .492 7 27.994 1000 020 2000 082 1 8000 .184 4000 .328 5000 .512 8 36.563 1100 .025 2100 .090 3100 .197 4100 345 5100 .533 9 46.275 1200 .030 2200 .099 | 8200 .210 4200 .302 5200 .554 10 57.130 [309] TABLE XVIII COEFFICIENT FOR REDUCING INCLINED STADIA MEASUREMENTS TO THE HORIZONTAL. 224. a 0' 10' 20' 30' 40' 50' 1.000000 .999992 .999967 .999924 .999865 .999789 l 999696 .999586 .999459 .999315 .999154 .998977 2 .998782 .998571 .998343 .998098 .997836 .997557 3 997261 .996949 .996619 .996273 .995910 .995531 4 .995134 .99 4721 .994291 .993844 .99:3381 .992901 5 .992404 .991891 .991360 .990814 .990250 .989670 6 .989074 .988461 .987831 .987185 .986522 .985843 7 .985148 .984436 .983708 .982963 .982202 .981424 8 980631 .979821 .978995 .978152 .97T;y4 .976419 9 .975528 .974621 .973698 .972759 .9"i804 .970833 10 .969846 .968843 .967824 .966790 .' J5739 .964673 11 .963591 .962494 .961380 .960252 959107 .957948 12 956772 .955581 ! .954375 .9.53153 .951916 .950664 13 1949396 .948113 .946815 .945502 .944174 .942831 14 .941473 .940100 .938711 .937:309 .935891 .934459 15 .933011 .931550 .930073 .928582 .927077 .925557 16 .924022 .922474 .920911 .919334 .917742 .916137 17 .914517 .912883 .911236 .909574 .907899 .906209 18 .904507 .902790 .901060 .899316 .897558 .895787 19 .894003 .892206 .890395 .888571 .886733 .884883 20 .883020 .881143 .879254 .877352 .875437 .873510 21 .871569 .869617 | .867652 .865674 .863684 .861681 22 .859667 .857640 .855601 .853550 .851487 .849412 23 .847326 .845227 .843117 .840996 .838862 .836718 24 .834561 .832394 .830215 .828025 .825825 .823613 25 .821390 .819156 .816911 .814656 .812390 .810113 26 .807826 .805529 .803221 .800903 .798575 .796236 27 .793888 .791529 .789161 .786783 .784396 .781998 28 .779591 .777175 .774749 .772314 .769870 .767416 29 .764954 .762483 .760002 .757513 .755015 .752509 30 .749994 .747471 .744939 .742399 .739850 .737294 31 .734729 .732157 .729577 .726989 .724393 721790 32 .719179 .716561 .713935 .711302 .708662 .706015 33 .703361 .700700 .698033 ' .695358 .692677 .689990 34 .687296 .684595 .681889 .679176 .676457 .6737'33 35 .671002 .668266 .665524 .662776 .660023 .657264 36 .654500 .651731 .648957 .646177 .643393 .640604 37 .637810 .635011 .632208 .629401 .626588 .623772 38 .620952 .618127 .615299 .612466 .609630 .606790 39 .603946 .601099 .598248 .595395 .592537 .589677 40 .586814 .583948 .581079 .578207 .575332 .572455 41 .569576 .566694 .563810 .560924 .558036 .555145 42 .552253 .549359 .546464 .543567 .540668 .537768 43 .534867 .531964 .529061 .526156 .523251 .520:345 44 .517438 .514530 .511622 .508714 .505805 .502897 45 .499988 .497079 .494170 .491261 .488353 .485445 TABLE XIX. LOGARITHM OF COEFFICIENT FOR REDUCING IN- CLINED STADIA MEASUREMENTS TO THE HORIZONTAL. 224. a 0' 10' 20' 30' 40' 50' 0.000000 9.999996 9.999985 9.999967 9.999941 9.999908 i 9.999868 .999820 .999765 .999702 .999633 .999555 2 .999471 .999379 .999280 .999173 .999059 .998938 3 .998809 .998673 .998529 .998379 .998220 .998055 4 .997882 .997701 .997514 .997318 .997116 .996906 5 .996689 .996464 .996232 .995992 .995745 .995491 6 9.995229 9.994959 9.994683 9.994399 9.994107 9.993808 T .993501 .993187 .992866 .992537 .992201 .991857 8 .991506 .991147 .990780 .990406 .990025 .989636 9 .989240 .988886 .988424 .988005 .987579 .987144 10 .986703 .986253 .985797 .985332 .984860 .984380 11 9.983893 9.983398 9.982895 9.982885 9.981867 9.981342 12 .980808 .980268 .979719 .979163 .978599 .978027 13 .977447 .976860 .976265 .975663 .975052 .974434 14 .973808 .973174 .972532 .971883 .971225 .970560 15 .969887 .969206 .968517 .967820 .967116 .966403 16 9. 965683 9.964954 9.964218 9.963473 9.962721 9.961960 17 .961192 .960415 .959631 .958838 .958037 .957229 18 .956412 .955587 .954753 .953912 .953063 .952205 19 .951339 .9.504(55 .949583 948692 .947793 .946886 20 .945970 .945047 .944114 .943174 .942225 .941268 21 9.940302 j 9.939328 9.9:38345 9.937354 9.936355 9.935347 22 .934330 .933:305 .932271 .931229 .930178 .929119 23 .928050 .926974 .925888 .924794 .923691 .922579 24 .921458 .920329 .919191 .918044 .916888 .915723 25 .914549 .913366 .912175 .910974 .909764 .908546 26 9.907318 9.906081 9.904835 9.903580 9.902316 9.901042 27 .899759 .898467 .897166 .895855 ..894535 .893206 28 .891867 .890519 .889161 .887794 .886417 .885031 29 .883635 .882230 .880815 .879390 .877956 .876512 30 .875058 .873594 .872121 .870637 .869144 .867641 31 9.866127 9.864604 9.863071 9.861528 9.859974 9.a58411 32 .856837 .855253 .853659 .859054 .850439 .848814 33 ' .847178 .845532 .843876 .842209 .840531 .838843 34 .837144 .835484 .833714 .881988 .830240 .828488 35 .826724 .824949 .823163 .821367 .819559 .817740 36 9.815910 9.814068 9.812216 9.810a52 9.808476 9.806589 37 .804691 .802781 .800860 .798927 .796982 .795026 38 .793058 .791078 .789086 .787082 .785066 .783038 39 .780998 .778946 .776882 .774805 .772716 .770614 40 .768500 .766374 .7642*5 .762083 j .759919 .757742 41 9.755552 9.753349 9.751133 9.748904 9.746662 9.744407 42 .742138 .789857 .737561 .735258 .732931 .730595 43 .728246 .725883 .72.3506 .721115 .718710 .716291 44 .713858 .711411 .708950 . "06474 .703983 .701479 45 9.698959 9.696425 9.693876 9.691313 9.688734 9.686140 TABLE XX.-LENGTHFS OF CIRCULAR ARCS; RADIUS = 1. Sec. Length, i Min. Length. Deg. Length. Beg. Length. 1 2 .0000048 .0000097 1 2 .0002909 - .0005818 1 2 .0174533 .0349066 61 62 1.0646508 1.0821041 3 0000145 3 .0008727 3 .0523599 63 1.0995574 A .0000194 4 .0011636 4 .0698132 64 1.1170107 5 .0000242 5 .0014544 5 .0872665 . 65 1.1344640 6 .0000291 6 .0017453 6 .1047198 66 1.1519173 7 .0000339 7 .0020362 7 .1221730 67 1.1693706 8 .0000388 8 .0023271 8 . 1396263 68 1.1868239 9 .0000436 9 .0026180 9 .1570796 69 1.2042772 10 .0000485 10 .0029089 10 .1745329 70 1.2217305 11 .0000533 11 .0031998 11 .1919862 71 1.2391838 jj> .0000582 12 .0034907 12 .2094395 72 1.2566371 13 .0000630 13 .0037815 13 .2268928 73 1.2740904 14 .0000679 14 .0040724 14 .2443461 74 1.2915436 15 .0000727 15 .0043633 15 .2617994 75 1. 3089969 16 .0000776 16 .00-46542 16 .2792527 76 1.3264502 17 .0000834 17 .0049451 17 .2967060 77 1.3439035 18 .0000873 18 .0052360 18 .3141593 78 1.3613508 19 .0000921 19 .0055269 19 .3316126 79 1.3788101 20 .0000970 20 .0058178 20 .3490659 80 1.3962634 21 .0001018 21 .0061087 21 .3665191 81 1.4137167 22 .0001067 22 .0063995 22 .3839724 82 1.4311700 23 .0001115 23 .0066904 23 .4014257 83 1.4486233 24 .0001164 24 .0069813 24 .4188790 84 1.4660766 25 , .0001212 25 .0072722 25 .4363323 85 1.4835299 26 .0001261 26 .0075631 26 .4537856 86 1.5009832 27 .0001309 27 .0078540 27 .4712389 87 1.5184364 28 .0001357 28 .0081449 28 .4886922 88 1.5358897 29 .0001406 29 .0084358 29 .5061455 89 1.5533430 30 .0001454 30 .0087266 30 .5235988 90 1.5707963 31 .0001503 31 .0090175 31 .5410521 91 1.5882496 32 .0001551 32 .0093084 32 .5585054 92 4.6057029 33 .0001600 33 .0095993 33 .5759587 93 1.6231562 34 .0001648 34 .0098902 34 .5934119 94 1.6406095 35 .0001697 35 .0101811 35 .6108652 95 1.6580628 36 .0001745 36 .0104720 36 .6283185 96 1.6755161 37 .0001794 37 .0107629 37 .6457718 97 1.6929694 38 .0001842 38 .0110538 38 .6632251 98 1.7104227 39 .0001891 39 .0113446 39 .6806784 99 1.7278760 40 .0001939 40 .0116355 40 .6981317 100 1.7458293 41 .0001988 41 .0119264 41 .7156&50 101 1.7627825 42 .0002036 42 .0122173 42 .7330383 102 1.7802358 43 .0002085 43 .0125082 43 .7504916 103 1.7976891 44 .0002133 44 .0127991 44 .7679449 104 1.8151424 45 .0002182 45 .0130900 45 .7853982 105 1.8325957 46 .0002230 46 .0133809 46 .8028515 106 1.8500490 47 .0002279 47 .0136717 47 .8208047 107 1.8675023 48 .0002327 48 .0139626 48 .8377580 108 1.8849556 49 .0002376 49 .0142535 49 .8552113 109 1.9024089 50 .0002424 50 .0145444 50 .8726646 110 1.9198622 51 .0002473 51 0148353 51 .8901179 111 1.9373155 52 .0002521 52 .0151262 52 .9075712 112 1.9547688 53 .0002570 53 .0154171 53 .9250245 113 1.9722221 54 .0002618 54 .0157080 54- .9424778 114 1.9896753 55 .0002666 55 .0159989 55 .9599311 115 2.0071286 56 .0002715 56 .0162897 56 .9773844 116 2.0245819 57 .0002763 57 .0165806 57 .9948377 117 2.0420352 58 .0002812 58 .0168715 58 1.0122910 ! 118 2.0594885 59 .0002860 59 .0171624 59 1.0297443 119 2.07(5!)41S 60 .0002909 60 .0174533 60 1.0471976 120 2.0943951 [312] TABLE XXi. -MINUTES IN DECIMALS OF A DEGREE. ' 0" 10" 15" 20" j | 30" 40" 45" 50" ' .00000 00278 .00417 .00556 .00833 .01111 .01250 .01389 1 .01667 .01944 .02083 .02222 .02500 02778 .02917 .03055 I 2 .03333 .03611 .03750 .03889 .04167 .04444 j .04583 .04722 2 3 .05000 .0527-8 .05417 .05556 .05838 .06111 .06250 i .06389 3 4 06867 .00944 .07083 .07222 .07500 .07778 .07-917 ' .08056 4 .08333 .08611 .08750 .08889 .09167 .09444 .09583 i .09722 5 6 10000 . 10278 .10417 .1055(1 . 10833 .11111 .11250 .11389 6 .11667 11944 .12083 i .12222 .12500 .12778 .12917 .13056 7 8 . 13333 .13611 .13750 1 .13889 .14167 .14444 . 14583 .14722 8 9 15000 . 1527H .15417 .15556 .15833 .16111 . 16250 . 16389 9 10 .16667 . 16944 .17083 .17222 .175 JO' .17778 1 .17917 .18056 10 11 . 18333 .18611 . 18750 .18889 .19167 j 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 237 50 .23889 .24167 .24414 .24583 , .24722 14 15 .25000 .2527'8 ! 2541 7 .25556 .25833 26111 .26250 i .26389 15 16 .206(H .26944 .2;-os;i .27222 .27500 \27778 .27917 ! .28056 16 17 .28333 .28611 .28750 .28889 .29167 .29444 .29583 .29722 17 18 .30000 30278 .30417 .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 85889 .36111 .36250 .36389 21 22 .36667 36944 .37083 .37222 .37500 .37778 .37917 .38056 22 23 .38333 .38611 .38750 .38889 ! 89167 .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 .43750 ,4:3889 .14167 11111 .44583 .44722 26 27 45000 .45278 .45417 .-15551) .45833 46111 .46250 .46389 27 28 .46667 46944 .47083 .47-222 .47500 ! 47778 .47917 .48056 28 29 .48333 .48611 .48750 .48889 .49167 .49444 .49583 .49722 29 30 .50000 .50278 .50417 .50556 .50833 .51111 .51250 .51389 30 31 .51667 .51944 .52083 .52222 .5250,1 .52778 .2917 .53056 31 32 .53333 53611 .53750 .53889 .54167 .54444 .54583 .54722 32 33 .55000 .55278 .55417 .55556 .55833 .56111 .56250 .56389 33 34 .56667 .56944 .57083 .57222 .57500 .57778 .57917 .58056 34 35 .58333 .58611 .58750 .58889 .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 .63750 .03889 .64167 .64444 .64583 .64722 38 39 .65000 .65278 .1)5417 .65556 .658:33 .66111 .66250 .66389 39 40 .66667 .66944 .67'083 .67222 .67500 .67778 .67917 .68056 40 41 .68333 .68611 .68750 .68889 .69167 .69444 .69583 .69722 41 42 7'0000 .70278 .70417 .70556 .70833 .71111 .71250 .71389 42 43 .716li7 .71944 .7-20H3 .72222 .72500 .72778 .72917 .73056 43 44 .73333 .73611 .73750 .73889 .74167 .74444 .74583 7'47'22 44 45 .77)000 .75278 .75417 .75556 .758:33 .76111 .76250 ! 76889 45 46 .76667 .715944 .770S3 .77222 .77-5(30 .77778 .77917 .78056 46 47 .78333 ! 78611 ] 78750 .78889 .79167 .79444 .79583 .79722 47 48 .80000 .80278 .80417 .80556 .80888 .81111 .81250 .81389 48 49 .81667 .81944 .82083 .82222 .82500 .82778 .82917 .83056 49 - 50 .83333 .83611 188750 .8:3889 .84167 .84444 .84583 .84722 50 51 .85000 .8527B .85417 .85556 .85833 86111 .86250 .86389 51 52 .86C.07 .86944 .87088 .cS7222 .87500 .87778 .87917 .88056 52 53 ! 88333 .88611 .88750 .88889 .89167 .89444 .89583 .89722 53 54 ,!K)000 .90278 .90417 .90556 .90833 .91111 .91250 .91389 54 55 .91667 .91944 .92083 .92222 .92500 .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 ! 96667 I 96944 .9708S .97222 .97-500 .97778 .97917 .98056 58 59 .98333 .98611 .96750 .98889 .99167 .99444 .99583 .99722 59 '1 0" 10" 15" 20" 30" 40" 45" 50" ' [313] TABLE XXII. --INCHES IN DECIMALS OF A FOOT. I ; ~1 la. 1 2 3 4 ! 5 6 7 8 9 10 11 In. ! 1 Foot .0833 .1667 .2500 .3333 .4167 .5000 .5833 .6667 .7500 .8333 .9167 1-32 .0026 .0859 .1693 .2526 .3359 .4193 .5026 .5859 .6693 .7526 .8359 .9193 1-32 1-16 .0052 .0885 .1719 .2552 .3385 .4219 .5052 .5885 .6719 .7552 .8385 .9219 1-16 3-32 .0078 .09111.1745 .2578 .3411 .4245 .5078 .5911 .6745 .7578 .8411 .9245 3-32 1-8 .0104 .0938i. 1771 .2604 .3438 .4271 .5104 .5938 .6771 .7604 .8438 .9271 1-8 5-32 .0130 .0964 .1797 .2630 .3464 .4297 .5130 .5964 .6797 .7630 .8464 .9297 5-32 3-16 .0156 .0990 .1823 .2656 .3490 .4323 .5156 .5990 .6823 .7656 .8490 .9323 3-16 7-32 .0182 .1016 .1849 .2682 .3516 .4349 .5182 .6016 .6849 .7682 .8516 .9349 7-32 1-4 .0208 .1042 .1875 .2708 .3542 .43751.5208 .6042 .6875 .7708 .8542 .9375 1-4 9-32 .0234 .1068 .1901 .2734 .3568 .4401 .5234 .6068 .6901 .7734 .8568 .9401 9-32 5-16 .0260 .1094 .1927 .2760 .3594 .4427 .5260 .6094 .6927 .7760 .8594 .9427 5-16 11-32 .0286 .1120 .1953 .2786 .3620 .4453.5286 .6120 .6953 .7786 .8620 .9453 11-32 3-8 .0313 .1146 .1979 .2813 .3646 .4479L5313 .6146 .6979 .7813 .8646 .9479 3-8 13-32 .0339 .11721.2005 .2839 .3672 .4505 .5339 .6172 .7005 .7839 .8672 .9505 13-32 7-16 .0365 .1198 .2031 .2865 .3698 .4531 .5365 .6198 .7031 .7865 .8698 .9531 7-16 15-32 .0391 .1224 .2057 .2891 .3724 .4557 .5391 .6224 .7057 .7891 .8724 .9557 15-32 1-2 17-32 .0417 .0443 .1250 .1276 !2083 .2109 .2917 .2943 .3750 .3776 .4583 .4609 .5417 .5443 ,6250 .6276 .7083 .7109 .7917 .7943 .8750 .8776 .9583 .9609 1-2 17-32 9-16 .0469 .1302 .2135 .2969 .3802 .4635 .5469 .6302 .7135 .7969 .8802 .9635 9-16 ' 19-32 .0495 .1328 !2161 .2995 .38281.4661 .5495 .6328 .7161 .7995 .8828 .9661 19-32 5-8 .0521 .1354 .2188 .3021 .3854! .4688 .5521 .6354 .71881.8021 .8854 .968S 5-8 21-32 .0547 .1380 .2214 .3047 .3880 .4714 .5547 .6880 .7214 '.8047 .8880 .9714 21-32 11-16 .0573 .1406 .2240 .3073 .3906 .4740 .5573 .6406 .7240 .8073 .8906 .9740 11-16 23-32 .0599 .1432 .2266 .3099 .3932 .4766 .5599 .6432 .7266 .8099 .8932 .9766 23-32 3-4 .0625 .1458 .2292 .3125 .3958 .4792 .56251.6458 .7292 .81861.8968 .9792 3-4 25-32 .06511.1484 .2318 .3151 .3984 .4818 . 5651 '. 6484 .7318 .8151 .8984 .9H1H 25-32 13-16 .0677 .1510 .2344 .3177 .40101.4844 .5677 .6510 .7344 .8177 .9010 .9844 13-16 27-32 .0703 .1536 .2370 .3203 . 4036 1. 4870 .5703 .6536 .7370 .8203 .9036 .9870J 27-32 7-8 .0729 .1563 2396 3229 40631 489fi 5729 fiftfiS .7396 *>><) 9063 7-8 29-32 .0755!. 1589 .2422 .3255 .4089 .4922 .5755 .6589 .7422 .8255 .9089 !9922 29-32 15-16 .0781 .1615 .2448 .3281 .4115 .4948 .5781 .6615 . 7448 l . 8281 .9115 .9948 15-16 31-32 .0807 .1641 .2474 .3307;. 4141 .4974 . 5807 T 6641 j. 7474 .8307 .9141 .9974 31-32 1 2 3 4 5 6 7 8 9 10 11 4 e < ' [314] TABLE xxin. SQUARES, CUBES, SQUARE ROOTS No. Squares. Cubes. Square Roots. Cube Roots. 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 1:331 3.3166248 2.2239801 .090909091 12 144 1728 3.4641016 2.2894286 .0833:33333 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 .0526:31579 20 400 8000 4.4721360 2.7144177 .050000000 21 441 9261 4.5825757 2.7589243 .047619048 22 484 10648 4. -6904158 2.8020393 .04545151.-) 23 529 12167 4.7958315 2.8438670 .043478261 24 576 113824 4.89897U5 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. 65085 UJ 3.1748021 .031250000 33 1089 35937 5.7445626 3.2075:343 .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 (,.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 79502 (i. 5574385 3.5033981 .023255814 44 1986 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 6.8656546 3.6088261 .021276600 48 2304 110592 6.9282032 3.6342411 .020833333 49 2401 1176*9 7.0000000 3.6593057 .020408163 50 2500 125000 7.0710678 3.684C314 .020000000 51 2601 132651 7.1414284 3.7084298 .019607843 52 2704 140608 7.2111026 3.7325111 .019230769 5:3 280ft 148877 7.2801099 3.7562858 .0188(57925 54 2916 ' 157464 7.3484692 3.7797631 .018518519 55 3025 166375 7.4161985 3.8029525 .018181818 56 8186 175616 7.4833148 3.8258624 .017857143 57 3249 185193 7.5498344 3.8485011 .017543860 58 3864 195112 ".6157731 3.8708766 .017241379 59 3481 205379 7.6811457 3.8929965 .016949153 60 3600 216000 ".7459667 3.9148676 . 0166(56(567 61 3731 226981 ".8102497 3.9364972 .01 6303 n:', (),' -! 4 238:328 ".8740079 3.9578915 .016129032 [315] CUBE ROOTS, AND RECIPROCALS. 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 274625 8.0622577 4.0207256 .015384615 66 4:356 287496 8.1240384 4.0412401 .015151515 67 4489 300763 8.1853528 4.0(315480 .014925373 68 4624 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 4.160167'6 .013888889 73 5329 389017 8.5440037 4.1793390 .013698630 74 5476 405224 8.6023253 4.1983364 .013513514 75 5625 421875 8.6602540 4.2171633 .01&333383 76 5776 438976 8.7177979 4.2358236 .013157805 77 5929 456533 8.7749644 4.2543210 .0125*87013 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 6561 531441 9.0000000 4.3267487 .012345679 82 6724 551368 9.0553851 4.3444815 .012195122 83 ($889 571787 9.1104336 4.3620707 .012048193 84 7056 592704 9.1651514 4.3795191 .0119047'62 85 7225 614125 9.2195445 4.3968296 .011764706 86 7396 636056 9.2736185 4.4140049 .011627907 87 7569 658503 9.3273791 4.4310476 .011494253 88 7744 681472 9. 3808315 4.4479602 .011363636 89 7921 704969 9.4339811 4.4647451 .011235955 90 8100 729000 9.4868330 4.4814047 .011111111 91 8281 753571 9.5393920 4.4979414 .010989011 92 8464 778688 9.5916630 4.5143574 .010869565 93 8649 804357 9.6436508 4.5306549 .010752688 94 8836 830584 9.695a597 4.5468359 .010638298 95 9025 85737'5 9.7467943 4.5629026 .010526316 96 9216 884736 9.7979590 4.5788570 .010416667 97 9409 912673 9.8488578 4.5947009 .010309278 98 9604 941192 9.S994&49 4.6104363 .010204082 99 9801 970299 9.9498744 4.6260650 .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 11664 1259712 10.3923048 4.7622032 . 00.1259259 109 11881 1295029 10.4403065 4.7768562 .009174312 110 12100 1331000 10.4880885 4.7914199 .009090909 111 12321 1367631 10.5356538 4.8058955 .OOOOOiWOfl 112 12544 1404928 10.5830052 4.8202845 .0089285;! 113 12769 1442897 10.630145R 4.8345881 .008849558 114 12996 1481544 10. (,770783 4.8488076 .008771930 115 13225 1520875 10.7238053 4.8629442 .00865)51552 116 13456 1560896 10.7703296 4.87GW90 .008620690 117 13689 1601613 10.81665:38 4.8909732 .008547009 118 13924 1643032 10.8627805 4.9048681 .008474576 119 14161 1685159 10.9087121 4.9186847 .008403361 120 14400 1728000 10.9544512 4.9324242 .008333333 121 14641 1771561 ll.OOJOOOO 4.9460874 .0082(34463 1<22 14884 1815848 11.0453610 4.9596757 .008196721 123 15129 1860867 11.0905365 4.9731898 .008130081 124 15376 190WW4 11.1355287 4.9866310 .008064516 i 61 TABLE XXIII.-SQUARES, CUBES, SQUARE ROOTS. No. Squares. Cubes. Square Roots. 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 1:34 17956 2406104 11.5758369 5.1172299 .007462687 135 18225 2460375 11.6189500 5.1299278 .007407407 130 18496 2515456 11.66190:38 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 14t 19881 2803221 11.8743421 5.2048279 .007092199 142 20164 2863288 11.9163753 5.2171034 .007042254 143 20449 2924207 11.9582607 5.2293215 .006993007 144 20736 2985984 12.0000000 5.2414828 .006944444 145 21025 3048625 12.0415946 5.2535879 .006896552 14(5 21316 3112136 12.0830460 5.2656374 .006849315 147 21609 3176523 12.1243557 5.2776321 .006802721 148 21904 3241792 12.1655251 5.2895725 .00675(5757 149 22201 3307949 12.2065556 5.3014592 .006711409 150 22500 3375000 12.^474487 5.3132928 .006666667 151 22801 3442951 12.2882057 5.3250740 .006622517 152 23104 3511808 12.3288280 5.a368033 .006578947 153 23409 8581577 12.3693169 5.3484812 .006535948 154 23716 3652264 12.4096736 5.3601084 .006493506 155 24025 3723875 12.4498996 5.3716854 .006451613 168 24336 3796416 12.4899960 5.3832126 .006410256 157 24649 3869893 12.5299641 5 3946907 .006369427 158 24964 3944312 12.5698051 5.4061202 .006329114 159 25281 4019(57!) 12.6095202 5.4175015 .006289308 160 25600 4096000 12.6491106 5.4288352 .006250000 161 25921 4173281 12.6885775 5.4401218 .006211180 162 26244 4251528 12.7279221 5.4513618 .006172840 163 26569 4330747 12.7671453 5.4625556 .006134969 11)4 26896 44105)44 12.8062485 5.4737037 .006097561 166 27225 4492125 12.8452326 5.4848066 .00(5060606 166 27556 4574296 12.8840987 5.4958647 .006024096 167 27889 4657463 12.9228480 5.5068784 .005988024 168 28224 4741632 12.9614814 5.5178484 .005952381 169 28561 4826809 13.0000000 5.5287748 .005917160 170 28900 4913000 13.0384048 5.5396583 .005882353 171 29241 5000211 13.0766968 5.5504991 .005847!)5:: 172 29584 5068448 13.1148770 5.5612978 .005813953 173 29929 5177717 13.1529464 5.5720548 .005780347 174 30276 5268024 13.1909060 5.5827702 .005747126 175 30625 5359375 13.2287566 5.5934447 . 00571 4286 176 80976 5451776 13.2664992 5.6040787 .005681818 177 3132!) 55 15233 13.3041347 5.6146724 .005649718 178 31684 5(539752 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 5.6670511 .005494505 183 33489 6128487 13.5277493 5 6774114 .005464481 184 33856 6229504 13.5646600 5.6877340 .005434783 185 34225 6331625 13.6014705 5.6980192 .005405405 186 34596 64:34856 13.6381817 5.7082675 .005:376344 [317] CUBE ROOTS, AND RECIPROCALS. No. , Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 1 187 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. 13527001 5.9059406 .004854369 207 42849 8869743 14.3874946 ! 5.9154817 .004830918 208 43264 8998912 14.4222051 5.9249921 .004807692 209 43681 9129329 14.4568323 5.9344721 .004784689 210 44100 9261000 14.4913767 5.9439220 .004761905 211 44521 9393931 14.5258390 5.9533418 .004739336 212 44944 9528128 14.5602198 5.9627320 .004716981 213 45369 9663597 14.5945195 5.9720926 .00469-1836 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 6.0822020 .004444444 226 51076 11543176 15.0332964 6.0911994 .004424779 227 51529 11697083 15.0665192 6.1001702 .004405286 228 51984 11852352 15.0996689 6.1091147 .004385965 229 52441 12008989 15.1327460 6.1180332 .004366812 230 52900 12167000 15.1657509 6.1269257 .004347826 231 53361 12326391 15.1986842 6.1357924 .004329004 232 53824 12487168 15.2315462 6.1446337 ,004310345 233 54289 12649337 15.2643375 6.1534495 .004291845 234 54756 12812904 15.2970585 6.1622401 .004273E04 235 55225 12977875 15. 3297'097 6.1710058 .004255319 236 55696 13144256 15.3622915 6.1797466 .004237288 237 56169 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 15.4919334 6.2144650 .004166667 241 58081 13997521 15 5241747 6.2230843 .004149378 242 58564 14172488 15.5563492 6.2316797 .004132231 243 59049 14348907 15.5884573 6.2402515 .004115226 244 59536 14526764 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 [318] TABLE XXIII.-SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. 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 .003952569 254 64516 16387064 15.9373775 6.3330256 .003937008 255 65025 16581375 15.9687194 6.3413257 .003921569 256 65536 10777216 16.0000000 6.3496042 .003906250 257 66049 16974593 10.0.312195 6.3578611 .003891051 258 66564 17173512 16.0623784 6.3660968 .00:3875969 259 67081 17373979 16.0934769 6.3743111 .003861004 260 67600 17576000 16.1245155 6.3825043 .003846154 261 68121 17779581 16.1554944 6.3906765 .003831 41 S 262 68644 17984728 16.1864141 6.3988279 .008816794 263 69169 18191447 16.2172747 6.4069585 .003802281 264 69696 18399744 16.2480768 6.4150687 .003787879 265 70225 18609625 16.2788206 6.4231583 .003773585 266 70756 18821096 16.3095064 6.4312276 .003759398 267 71289 19034163 16.3401346 6.4392767 .003745318 268 71824 19248832 16.3707055 6.4473057 .003731343 269 72361 19465109 16.4012195 6.4553148 .003717472 270 72900 19683000 16.4316767 6.4633041 .003703704 271 73441 19902511 16.4620776 6.4712736 .003690037 272 73984 20123648 16.4924225 6.4792236 .003676471 273 74529 20346417 16.5227116 6.4871541 .003663004 274 75076 20570824 16.5529454 6.4950653 .003649635 275 75625 20796875 16.5831240 6.5029572 .003636364 276 76176 21024576 16.6132477 6.5108300 .003623188 277 767-29 212539:33 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 .008571429 281 78961 22188041 16.7630546 6.5499116 .003558719 282 79524 22425768 16.7928556 6.5576722 .004540099 283 80089 2266.5187 16.8226038 6.5654144 .003533569 284 80656 22906304 16.8522995 6.5731385 .003521127 285 81225 23149125 16.8819430 6.5808443 .003508772 286 81796 23393656 16.9115345 6.5885323 .00349650:3 287 82369 23639903 16.9410743 6.5962023 .003484321 288 82944 28887872 16.9705627 6.6038545 .003472222 289 83521 21 1- '57569 1 .0000000 6.6114890 .003460208 290 84100 24389000 1 .0293864 6.6191060 .003448276 291 84681 24642171 1 .0587221 6.6267054 .003436420 292 85264 24897088 1 .0880075 6.6:342874 .003424658 293 85849T 85158757 1 .1172428 6.6418522 .003412%'.) 294 86436 25412184 1 .14642*2 6.6493998 .003401301 295 87025 25672375 1 .1755640 6.6569302 .0033898:31 296 87616 25934:336 1 .2046505 6.6644437 .003378378 297 88209 26198073 1 .2330s;;) 6.6719403 .003367003 298 88804 26403593 1 .2i267<>5 6.6794200 .003355705 299 89401 26730899 1 .2916165 6.6868831 .003344482 300 90000 27000000 1 .3205081 6.6943295 .004333333 301 90601 27270901 1 .3493516 6.70175'.)3 .003:322259 302 91204 27543008 1 .3781472 6.7091729 .003311.258 303 91809 27818127 1 .4068952 6.7165700 .003300330 304 92416 280944IJ4 1 .4:355958 6.7239508 .003289474 305 93025 28372625 1 .4642492 6.7313155 .003278689 306 93636 28652616 * 1 .4928557 6.7386641 .003267974 307 94249 28934443 1 .5214155 6.7459967 .003257329 308 94864 29218112 1 .5499288 6.7533134 .003246753 309 95481 29503629 17. 571*35 15 6.7606143 .003236246 310 96100 29791000 17.6068169 6.7678995 .003225806 CUBE HOOTS, AND RECIPROCALS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 311 96721 30080231 17.6351921 6.7751690 .003215434 312 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 103041 33076161 17.9164729 6.8470213 .003115265 322 103684 33386248 17.9443584 6.8541240 .003105590 323 104329 33698267 17.9722008 6.8612120 .003095975 325 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 35611289 18.1383571 6.9034359 .003039514 ' 330 108900 35937000 18.1659021 6.9104232 .003030303 331 109561 36264691 18.1934054 6.9173964 .003021148 332 110224 36594368 18.2208672 6.9243556 .003012048 333 110889 36926037 18.248287'6 6.9313008 .003003003 334 111556 37259704 18.2756609 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 .002967*59 338 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 40353607 18.5202592 7.0000000 .002915452 344 118386 40707584 18.5472370 7.0067962 .002906977' 345 119025 41063625 18.5741756 7.0135791 .002898551 346 119716 41421736 18.6010752 7.0203490 .002890173 347 120409 41781923 18.627'9360 7.0271058 .002881844 348 121104 42144192 18.6547581 7.0338497 .002873563 349 121801 42508549 18.6815417 7.0405806 .002865330 350 122500 42875000 18.7'082869 7.0472987 .002857143 351 123201 43243551 18.7349940 7.0540041 .002849003 352 123904 43614208 18.76166:30 7.0606967 .002840909 353 124609 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 128164 45882712 18.920887'9 7.1005885 .002798296 359 128881 46268279 18.9472953 7.1071937 .002785515 360 129(500 46656000 18.9736660 7.1137866 .002777778 361 130321 47045881 19 0000000 7.1203674 .002770083 362 131044 47437'928 19.026297'6 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 .0027J0027 370 136900 50653000 19.2353841 7.1790544 ( .002702703 371 " 137641 51064811 19.2613603 7.1&55162 .002695418 372 138384 51478848 19.2873015 7.1919663 1 .002688172 TABLE XXIII. -SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 373 139129 51895117 19.3132079 i 7.1984050 .002680965 874 139876 52313624 19.3390796 ; 7.2048322 .002673797 375 140625 52734375 19.3649167 7.2112479 .002666667 376 141376 53157376 19.3907194 7.2176522 .002659574 377 142129 5:35826:33 19.4164878 7.2240450 .002652520 378 142884 54010152 19.4422221 7.2:304268 .002645503 379 143641 54439939 19.4679223 7.2367972 .002638522 380 144400 54872000 19.4935887 7.2431505 .002631579 381 145161 55306:341 19.5192213 ! 7.2495045 .002624672 382 145924 55742968 19.5448203 7.2558415 .002617801 383 146689 56181887 19.5703858 .2621675 .002610966 384 147456 56623104 19.5959179 .2684824 .002604167 385 148225 57066625 19.6214169 .2747864 .002597403 386 148996 57512456 19.6468827 .2810794 .002590674 387 149769 57960603 19.6723156 .2873617 .002583979 388 150544 58411072 19.6977156 .2936330 .002577320 389 151321 58863869 19.7230829 .2998936 .002570894 390 152100 59319000 19.7484177 .3061436 .002564103 391 152881 59776471 19.7737199 .3123828 .002557'545 392 153664 60236288 19.7989899 .3186114 .002551020 393 154449 60698457 19.8242276 .3248295 .002544529 394 155236 61162984 19.8494332 .3310369 .002538071 395 156025 61629875 19.8746069 .3372339 .002531646 396 156816 62099136 19.8997487 7.3434205 .00^525253 397 157609 62570773 19.9248588 7.3495966 .002518892 398 158404 63044792 19.9499373 7.3557624 .002512563 m 159201 63521199 19.9749844 7.3619178 .002506266 400 160000 6400000G 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 7.3864373 .002481390 404 163216 65939264 20.0997512 7.3925418 .002475248 405 164025 66430125 20.1246118 7.3986363 .002469136 406 164836 66923416 20.1494417 7.4047206 .C02463054 407 165649 67419143 20.1742410 7.4107950 .00245'; 002 408 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 7.4:349938 .002433090 412 169744 09! 134528 20.2977831 7.4410189 .002427184 413 170569 70444997 20.3224014 7.4470342 .002421308 414 171396 70957944 20.3469899 7.4530399 .C02415459 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 20.4450488 7.4769664 .002392344 419 175561 73500059 20.4694895 -.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 20.56696:38 7.5066607 .002364066 434 179776 $225024 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 002386449 429 184041 78953589 20.7123152 7.5419867 .002331002 430 184900 79507000 20.7364414 7.5478423 .002325581 431 185761 80062991 20.7605395 7.5536888 .002320186 432 186624 80621568 20.7846097 7.5595263 .002314815 433 187489 81182737 20.8086520 7.5653548 .002309469 434 188356 81746504 20.8820007 7.5711743 .002304147 CUBE ROOTS, A.ND RECIPROCALS. No. Squares. Cubes. Square Roots. Cube Roots. Recipror *ls. 435 189225 82312875 20.8566536 7.5769849 .002298851 436 190096 82881856 20.8806130 7.5827865 .002293578 437 190969 83453453 20.9045450 7.5885793 .002288330 438 ' 191844 84027672 20.9284495 7.5943633 .002283105 439 192721 84604519 20.9523268 7.6001385 .002277904 440 193600 85184000 20.9761770 7.6059049 .002272727 441 194481 85766121 21.0000000 7.6116626 .002267574 442 195364 86:350888 21.0237960 -.6174116 .002262443 443 196249 86938307 21.0475652 "".6231519 .002257336 444 197136 87528384 21.0713075 .6288837 .002252252 445 198025 88121125 21.0950231 .6346067 .002247191 446 198916 88716536 21.1187121 .6403213 .002242152 447 199809 89314623 21.1423745 .6460272 .002237136 448 200704 89915392 21.1660105 7.6517247 .002232143 449 201601 90518849 21.1896201 7.6574133 .002227171 450 202500 91125000 21.2132034 7.6630943 .002222222 451 ' 203401 91733851 21.2367606 7.6687665 .002217295 452 204304 92:345408 21.2602918 7.6744303 .002212389 453 205209 92959677 21.2837967 7.0800857 .002207506 454 206116 93576664 21.3072758 7.6857328 .002202643 455 207025 94196375 21.3:307290 7.6913717 .002197802 456 207936 94818816 21.3541565 7.6970023 .002192982 457 208849 95443993 21.3775583 7.7026246 .002188184 458 209764 96071912 21.4009346 7.7082388 .002183406 459 210681 96702579 21.4242853 7.7138448 .002178649 460 211600 97336000 21.4476106 7.7194426 .002173913 461 212521 97972181 21.4709106 7.7250325 .002169197 462 213444 98611128 21.4941853 7.7306141 .002164502 4m 214369 99252847 21.5174348 7.7361877 .002159827 464 215296 99897344 21.5406592 7.7417532 .002155172 465 216225 100544625 21.5638587 7.7473109 .002150538 466 217156 101194696 21.5870331 7.7528606 .002145923 467 218089 101847563 21.6101828 7.7584023 .002141328 468 219024 102503232 21.63-33077 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 002123142 472 2227S4 105154048 21.7'255610 7.7&5992S .002118644 473 223729 105823817 21.7485632 7.7914875 .002114165 474 224676 106496424 21.7715411 7.7969745 .002109705 475 225625 107171875 21.7944947 7.8024538 .002105263 476 226576 107850176 21.8174242 .8079254 .002100840 477 227529 108531333 21.8403297 .8133892 .002096436 478 228184 109215352 21 8632111 .8188456 .002092050 479 229441 109902239 21.8860686 .8242942 .002087683 480 230400 110592000 21.9089023 .8297353 .002088333 481 231361 111284641 21.9317122 .8351688 .002079002 482 232324 111980168 21.9544984 .8405949 .002074689 483 233289 112678587 21.9772610 .8460134 .002070393 484 234256 113379904 22.0000000 7.8514244 .002066116 485 235225 114084125 22.0227155 7.8568281 .002061856 486 236196 114791256 22.0454077 7.8622242 .002057613 487 237169 115501303 22.0680765 7.8676130 .002053388 488 238144 116214272 22.0907220 7.8729944 .002049180 489 239121 116930169 22.11:33444 7 8783684 .002044990 490 240100 117649000 22.1359436 7.8837352 .002040816 491 241081 11&370771 22.1585198 7.8890946 .002036660 492 242064 119095488 22.1810730 7.8944468 .002032520 493 243049 119823157 22.2036033 7.8997917 .002028398 494 244036 120553784 22.2261108 7.9051294 .002024291 495 245025 121287375 22.2485955 7.9104599 .002020202 496 246016 122023936 22.2710575 7.9157832 .002010129 [322] TABLE XX11L SQUARES, CUBES, SQUARE ROOTS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 497 247009 122768473 22.2934968 7.9210994 .002012072 498 248004 1*3505992 22.3159136 7.9264085 .002008032 499 249001 124251499 32.3383079 7.9317104 .002004008 500 250000 125000000 22.3606798 7.9370053 .002000000 501 251001 125751501 22.3830293 7.9422931 .001996008 603 252004 126506008 22.4053565 7.9475739 .001992032 503 253009 127263527 22.4276615 7.9528477 .001988072 504 254016 1280240(54 22.4499443 7.9581144 .001984127 505 255025 128787625 22.4722051 7. 9633743 .001980198 506 256036 129554216 22.4944438 7.9686271 .001976285 507 257049 13032:3843 22.5166605 7.9738731 .001972387 508 258064 131096512 22.5388553 7.9791122 .001968504 509 259081 131872229 22.5610283 7.9843444 .001964637 510 260100 132651000 22.5831796 7.9895697 .001960784 511 261121 133432831 22.6053091 7.9947883 .00195(5947 512 262144 134217728 22.6274170 8.0000000 .001953125 518 263169 135005697 22.6495033 8.0052049 .0019403 is 514 264196 135796744 22.6715681 8.0104032 .001945525 515 2(55225 136590875 22.6936114 8.0155946 .001941748 516 266256 137388096 22.7156334 8.0207794 .001937984 517 267289 138188413 22.7376340 8.0259574 .001934236 518 268324 138991832 22.7596134 8.0311287 .001930502 519 269361 139798359 22.7815715 8.0362935 .001926782 520 8T0400 140608000 22.8035085 8.0414515 .001923077 521 271441 1414207'61 22.8254244 8.0466030 .001919386 522 272484 142236648 22.8473193 8.0517479 .001915709 523 273529 143055667 22.8691933 8.0568862 .001912046 524 274576 14:3877824 22.8910463 8.0620180 .001908397 525 275625 144703125 22.9128785 8.0671432 .001904762 526 276676 145531576 22.9346899 8.0722620 .001901141 527 277729 146363183 22.9564806 8.0773743 .001897533 528 278784 147197952 22.9782506 S.OS24800 .001893939 529 279841 148035889 23.0000000 8.0875794 .001890359 530 280900 148877000 23.0217289 8.0926723 .001886792 531 281961 149721291 23.0434372 8.0977589 .00188:3239 532 283024 150568768 23.0651252 8.1028390 .001879699 533 284089 151419437 23.0867928 8.1079128 .001876173 584 285156 152273304 23.1084400 8.1129803 .001872659 535 286225 153130375 23.1300670 8.1180414 .00186915!) 536 287'296 153990656 23.1516738 8.1230962 .001865672 537 288369 15IS54153 23.1732605 8.1281447 .001862197 538 289444 155720872 23.1948270 8.1331870 .001858736 539 290521 156590819 23.2163735 8.1382230 .001855288 540 291600 1574(54000 23.2379001 8.1432529 .001851852 541 292681 158840421 23.2594067 8.1482765 .001848429 542 293764 159220088 23.2808935 8 1532939 .001845018 543 294849 16C103007 23.3023604 8.15H3051 .001841621 544 295936 160989184 23.3238076 8.1633102 .001838235 545 297025 161878625 23.3452351 8.1683092 .0018:34862 546 298116 1(52771336 23.3666429 8.1733020 .001831502 547 299209 163607323 23.3880311 8.1782888 .001828154 548 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 16819(5(508 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 .001801802 556 309136 171879616 23.5796522 8.2228985 .001798561 557 310249 172808693 23.6008474 8.2278254 .001795332 558 311364 173741112 23.6220236 8.2327463 .001792115 CUBE HOOTS, AND RECIPROCALS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 559 312481 174676879 23.&431808 8.2376614 ..001788909 560 313600 175616000 23.6643191 8.2425706 .001785714 561 314721 176558481 23.6854386 8.2474740 .001782531 562 315844 177504:328 23.7065392 8.2523715 .001779359 563 316969 17845:3547 23.7276210 8.2572633 001776199 564 318096 179406144 23.7486842 8.2621492 .001773050 565 319225 180362125 23.7697286 8.2670294 .001769912 566 320356 181321496 23.7907545 8.2719039 .001766784 567 321489 182284263 23.8117618 8.2767726 .001763668 568 322624 183250432 23.8327506 8.2816355 .001760563 569 323761 184220000 !>. 8537209 8.264928 .001757469 570 324900 185193000 23.8746728 8.2913444 .001754386 571 326041 186169411 23.8956063 8.2961903 .001751313 572 327184 187149248 23.9165215 8.3010304 .001748252 573 328329 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 8.3203353 .001736111 577 332929 192100033 24.0208243 8.3251475 .001733102 578 334084 193100552 24. 041 630(5 8.3299542 .001730104 579 335241 194104539 24.0624188 8.3347553 .001727116 580 a36400 195112000 24.0831891 8.3395509 .001724138 581 337561 196122941 24.1039416 8.3443410 .001721170 582 338724 197137368 24.1246762 8.3491256 .001718213 583 339889 198155287 24.1453929 8.3539047 .001715266 584 341056 199176704 24.1660919 8.3586784 .001712329 585 342225 200201625 24.1867732 8.3634466 .001709402 586 34*396 201230056 24.2074369 8.3682095 .001706485 587 344569 202262003 24.2280829 8.37'29668 .001703578 588 345744 203297472 24.2487113 8.3777188 .0017'00680 589 346921 204336469 24.2693222 8.3824653 .001697793 590 348100 205379000 24.2899156 8.3872065 .001694915 591 349281 206425071 24.3104916 8.3919423 .001692047 592 350464 207474688 24.3310501 8.3966729 .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 8.4343267 .001666667 601 361201 217081801 24.5153013 8.4390098 .001663894 602 362404 218167208 24.5356883 8.4436877 .001661130 603 363609 219256227 24.5560583 8.4483605 .001658375 604 364816 220348864 24.5764115 8.4530281 .001C55629 605 366025 221445125 24.5967478 8.4576906 .001652893 606 367236 222545016 24.6170673 8.4623479 .001650165 607 368449 223648543 24.6373/00 8.4670001 .001(547446 608 369664 224755712 24.6576560 8.4716471 .001644737 609 370881 225866529 24.6779254 8.4762892 .001642036 610 372100 226981000 24.6981781 8.4809261 .001639344 611 373321 228099131 24.7184142 8.4855579 .001636661 612 374544 229220928 24.7386338 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.&596058 8.5178403 .001618123 619 383161 237176659 24.8797106 8.5224321 .001615509 620 384400 238:328000 24.8997992 8.5270189 .001612903 [324] TABLE XXIII.- SQUARES, CUBES, SQUARE ROOTS, No. t Squares Cubes. Square Roots. Cube Roots. Reciprocals. 621 385641 239483061 24.9198716 8.5316009 .001610306 622 386884 240641848 24.9399278 8.5361780 .001607717 623 388129 241804367 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 .001597441 627 393129 246491883 25.0399681 8.5589899 .001594896 628 394384 247073152 25.0599282 8.5635377 .001592357 629 395641 248858189 25,0708724 8.5680807 .001589825 630 396900 250047000 25.0998008 8.5726189 .001587302 631 398161 251239591 25.1197134 8.5771523 .001584786 632 399424 252435968 25.1396102 8.5816809 .001582278 633 400689 253636137 25.1594913 8.5862047 .001579779 634 401956 254840104 85.1798566 8.5907238 ,001577287 635 403225 256047875 25.1992063 8.5952:380 .001574803 636 404496 257259456 25.2190404 *. 5997476 .00157'2327 637 405769 258474853 25.2388589 8.6042525 .001569859 638 407044 259694072 25.2586619 8.6087526 .001567398 639 408321 260917119 25.2784493 8.6132480 .001564945 640 409600 262144000 25.2982213 8.6177388 .001562500 641 410881 263374721 25.3179778 8.6222248 .001560062 643 412164 264609288 25.3377189 8.6267063 .001557632 (543 413449 265847707 85.8574447 8.6311830 .001555210 <)44 414736 267089984 85.8771551 8.6356551 .001552795 645 416025 268336125 25.3968502 8.6401226 .00155038S 646 417316 269586136 85.4165801 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 428801 275894451 25.5147016 H.6IH58810 .001536098 652 425104 277167808 25.5342907 8.6712665 .0015:33742 653 426409 2:8445077 25.5538647 8.6756974 .001531394 654 427716 2797262(54 85.5784837 8.6801237 .001529052 655 429025 281011375 25.5929678 8.6845456 .001526718 656 430,336 282:300416 25.6124969 8.6889630 .001524390 657 431649 283593393 25.6320112 8. 698375! 1 .001522070 658 432964 284890312 85.6515107 8.6977843 .001519757 659 434281 28(5191179 25.6709953 8.7021882 .001517451 660 435600 287496000 25.6904652 8.7065877 .001515152 661 436921 288804781 25^ 7099203 S. 7409827 .001512859 662 438244 290117528 25.7293607 8.7153734 .001510574 663 439569 21)1434247 25.7487864 8.7197596 .001508296 064 440896 292754944 85.7681975 8.7241414 .001506024 0(35 442225 294079625 25.7875939 8.72~85187 .001503759 606 443556 295408296 25.8069758 8 7328918 .001501502 66T 444889 29(5740963 25.8263431 8.7372604 .001499250 6f>8 441)224 298077032 25.8456960 8.7416246 .001497006 869 447561 299U8309 25.8650343 8.7459S40 .001494768 670 448900 300763000 85.8848582 8.7508401 .001492537 671 450241 302111711 25.903C.077 8. 754691 3 .001490313 678 451584 :>,03K;4448 25.922902X 8.7590388 .001488095 673 452929 304821217 25.9422435 8.7688809 .001485881 674 454276 .'506182024 25. '.Mil 51 00 8.7677192 .001483680 975 455625 307546875 25.9807621 8.7720532 .001481481 676 456976 308915776 26.0000000 8.7763830 .001479290 677 458329 310288733 26.0192237 8.7807084 .001477105 678 459684 311665752 26.0384331 8.7850296 .001474926 679 461041 313046839 20.0576284 8.7893466 .001472754 680 462400 314432000 20.0768090 8.7936593 .001470588 681 463761 315821241 26.0959767 8.7979679 .001468429 682 465184 317214568 86.1151297 8.8022721 .001466276 CUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 6&3 46&4S9 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 473344 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 387153536 26.3818119 8.8620952 .001436782 697 485809 338608873 26.4007576 8.866*375 .001434720 698 487204 340068392 26.4196896 8.8705757 0014:32665 699 488601 341532099 26.4386081 8.8748099 .0014:30615 700 490000 343000000 26.4575131 8.8790400 .001428571 S'Ol 491401 344472101 26.4764046 8.8832661 .001426534 70S 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 8.925307'8 001406-470 712 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 00139275H 719 516961 371694959 26.8141754 8.9586581 .001390821 720 518400 373248000 26.8328157 8.9628095 .001388889 721 519841 374805361 26.8514432 8.9669570 .(,01386963 722 521284 376367048 26.870057F 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 .001362398 735 540225 397065375 27.1108834 9.0246239 .001360544 736 541696 398688256 27.1293199 9.0287149 .001358696 737 543169 400315553 27.1477439 9.0328021 .001356852 738 544644 401947272 27.1661554 9.0368857 .001355014 739 546121 403583419 27.1845544 9.0409655 .001353180 740 547600 405224000 27.2029410 9.0450419 .00ia51351 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 411830784 27.2763034 9.0613098 | .001344086 [.326] TABLE XXIII. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 745 555025 413493625 27.2946881 9.0653677 .001342282 746 556516 415100930 27.3130006 9.0094220 .001340483 747 558009 416832723 27.3313007 9.0734726 .0013:38688 748 559504 418508992 27.3495887 9.0775197 .001J336898 749 561 001 420189749 27.3678644 9.0815631 .001335113 750 562500 421875000 27.3861279 9.08560:30 .001333333 751 564001 42*564751 27.404.3792 9.0890392 .001331558 752 565504 425259008 27.4220184 9.0936719 .001329787 753 567009 426957777 27.4408455 9.0977010 .001328021 754 568516 428661064 27.4590004 9.1017268 .001326260 755 570025 430368875 27.47720:33 9.1057485 .001324503 756 571536 432081216 27.4954542 9.10976US) .001322751 757 578049 4133798093 27.5136330 9.1137818 .001321004 758 574564 435519512 27.5317998 ! l.l 177 931 .001319261 759 576081 437245479 27.5499546 9.1218010 .001317528 760 577600 438976000 27.5680975 9.1258053 .001315789 761 579121 440711081 27.5862284 9.1298061 .1:01 314000 762 580644 442450728 27.0043475 9.13380:34 .001312336 763 582169 444194947 27.62.4546 9.1377971 .001310616 764 583696 44594:574 1 27.6405499 9.1417874 .001308901 766 585225 447097125 27.6586334 9.1457742 .001307190 766 586756 449455096 27.0707050 9.1497576 .001305483 767 588289 451217603 27.0947'648 9.1537375 .001303781 708 589824 ' 452984832 27.7128129 9.1577139 .001302083 769 591361 454756609 27.7308492 9.1610809 .001300390 770 592900 456533000 27.7488739 9.1656505 .001298701 771 594441 458:314011 27.70088(58 9.1696225 .001297017 772 595984 460099648 27.7848880 9.1735852 .001295837 773 597529 461889917 27.8028775 9.177.5445 .00129J3661 774 599076 463684824 27.8208555 9.1815003 . 001291 990 775 600625 465484375 27.8388218 9.1854527 .001290323 778 602176 467288576 27.8567766 9.1894018 .001288060 777 603729 469097433 27.8747197 9.1933474 .001287001 778 605284 470910952 27.8920514 9.1972897 .001285347 779 606841 472729139 27.9105715 9.2012280 .001283097 780 608400 474552000 27.9284801 9.2051641 .001282051 781 609961 470379541 27.9403772 9.2090962 .001280410 782 611524 47'8211708 27.9642629 9.2130250 .001278772 783 61308!) 480048087 23 .HS21372 9.2169505 .001277139 784 614656 481890:304 28.0000000 9.2208726 .001275510 785 616225 483730625 28.0178515 9.2247914 .001273885 786 617796 185587050 28.0:356915 9.2287068 .001272265 787 619369 48744:3403 28.0585208 9.2326189 .001270648 788 620944 489808872 28.0713377 9.2365277 .001269036 789 622521 491169069 28.0891438 9.2404333 .001267427 790 624100 493039000 28.1069386 9.2448355 .001265823 791 625681 494913671 28.1247222 9.2482344 .001264223 792 627264 496793088 28.1424946 9.2521300 .001262620 793 628849 498677257 28.1602557 9.2560224 .001261034 794 630436 500506184 28.1780056 9.2599114 .001259446 795 632025 502459875 28.1957444 9.2637973 .001257862 796 633616 504358336 28.2134720 9.2676798 .001256281 797 6:35209 506201573 28.2311884 9.2715592 .001254705 798 636804 508169592 28.2488938 9.2754352 .001253133 799 638401 510082399 28.2665881 9.2793081 .001251564 800 640000 512000000 28.2842712 9.2831777 .001250000 801 641601 513922401 28.3019434 9.2870440 .001248439 802 643204 515849608 28.3196045 9.2909072 .001240883 803 644809 517781627 28.8372546 9.2947071 .001245330 804 646416 519718464 28.35489:38 9.2986239 .00124b781 805 648025 521660125 ! 28.3725219 9.3024775 .0012J>3<; 806 649636 523606616 | 28.3901391 9.3063278 .001240695 [327] CUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes." Square Roots. I Cube Roots. Reciprocals. 807 651249 525557943 28.4077454 9.3101750 .001239157 808 652864 527514112 28.4253408 9.3140190 .001237624 809 654481 529475129 28.4429253 9.3178599 .0012:36094 810 656100 531441000 28.4004989 9.3216975 .001234568 811 05V721 588411731 28.4780617 9.3255320 .00123:3046 812 659; J44 585887838 28.4956137 9.3293034 .001231527 813 660969 537367797 28.5131549 9.8881916 .0012:30012 814 062596 539353144 28.5306852 9.3370167 .001228501 815 664225 541343375 28.5482048 9.3408380 .001226994 816 66585(5 548888498 28.5657137 9.3446575 .001225490 817 667489 545338513 I 28.5832119 9.3484731 .001223990 818 669124 547343432 j 28.6006993 9.3522857 .001222494 819 670761 549.353259 28.6181760 9.3."60y52 .001221001 820 672400 551368000 28.6356421 9.3599016 .001219512 821 674041 558387661 28.6530976 9.3637049 .001218027 822 675684 555412348 28.67'05424 9.3675051 .001216545 823 677329 557441767 28 68797 00 9.3713022 .001215067 824 678976 559476221 28.7054002 9.3750963 .001213592 825 680625 561515625 28. 7228132 - 9.3788873 .001212121 826 682276 56355997(5 28.7402157 9.3820752 .001210654 827 683929 565(509283 ! 28.7576077 9.3864600 .001209190 828 685584 567663552 ! 28.7749891 9.3902419 .001207729 829 687241' 569722789 j 28.7923601 9.3940206 .001206273 830 688900 571787000 28.8097206 9.3977964 .001204819 831 690561 573856191 28.8270706 9.4015691 .001203369 832 692224 575930368 28.8444102 9.4053387 .001201923 833 693889 578009537 28.8617394 9.4091054 .001200480 &34 695556 580093704 28.8790582 9.4128690 .001199041 835 697225 58218287'5 28.8963660 9.4166297 .001197605 836 698896 584277056 28.9136646 9.4203873 .001196172 837 700569 586376253 28.9309523 9.42-11420 .001194743 838 702244 588480472 28.9482297 9.4278936 .001193:317 839 703921 590589719 28.9654967 9.4316423 .001191895 840 705600 592704000 28.9827535 9.4353880 .001190476 841 707281 594823321 29.0000000 9.4391307 .0011890(51 842 708964 596947688 29.0172363 9.442H704 .001187648 843 710649 599077107 29.0344623 9.4466072 .001186240 844 712336 601211584 29.0516781 9.4503410 .001184834 845 714025 603351125 29.0688837 9.4540719 .001183432 846 715716 605495736 29.0860791 9.4577999 .001182033 847 717409 607645423 29.1032644 9.4615249 .001180638 848 719104 609800192 29.120439(5 9.4652470 .001170215 849 720801 611960049 29.1376046 9.4689661 .001177850 850 722500 614125000 29.1547595 9.4726824 .001176471 851 724201 616295051 29.1719043 9.4763957 .001175088 852 725904 618470208 29.1890390 9.4801061 .00117370;) &53 727609 620650477 29.2061637 9.4838136 .001172333 89* 729316 622835864 29.2232784 9.4875182 .001170960 855 731025 625026375 29.2403830 9.4912200 .001169591 856 732736 627222016 29.2574777 9.4949188 .001108224 a57 734449 629422793 29.2745623 9.4986147 .0011IS6W51 858 736164 631628712 29.2916370 9.5023078 .001105501 859 737881 633839779 29.3087018 9.5059980 .001164144 860 739603 636056000 29.3257566 9.5096854 .001162791 861 741321 638277381 29.3428015 9.5133009 .0011614-10 862 743044 640503928 29.3598365 9.5170515 .001160093 863 744769 642735647 29.3768616 9.5207303 .001158749 864 746496 644972544 29.3938769 9.5244063 .001157407 865 748225 647214625 89.4108888 9.5280794 .001 1560(59 866 749956 649461896 29.4278779 ! 9.5317497 .0011547*1 867 751689 651714363 29.4448637 9.5354172 .001153403 868 L 753424 6539720:32 29.4618397 9.5390818 ,001152074 TABLE XXIII.- SQUARES, CUBES, SQUARE ROOTS. No. Squares. Cubes. Hoots 6 Cube Roots> Reci l jrocals - 869 755161 656234909 ! 29.4788059 9.5427437 .001150748 870 7'56900 658503000 : 29.4957624 9.5464027 .001149425 871 758641 660776311 ! 29.5127091 9.55005S!) .001148106 872 760384 663054848 29.5296461 9.5537123 .001146789 873 762129 665338617 29.5465734 9.5573630 .001145475 874 763876 667627024 2 ( .).5(i:;4U10 9.5610108 .001144165 875 765625 669921875 29.5803989 9.5646559 .001142857 876 767376 672221376 i 29.5972972 9.5682982 .001141553 877 769129 674526133 29.6141858 9.5719377 .001140251 878 770884 (7(iS:}6152 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 .0011:35074 882 777924 686128968 29.6984848 9.5900939 .001183787 883 779689 6884(55387 29.7453159 9.5937169 .001132503 sst 781456 690807104 i 29.7321375 9.5973373 .001131222 886 783225 693154125 ! 29.7489496 9.6009548 .001129941 886 784996 695506456 29.7(557521 9.6045696 .001128668 887 786769 697864103 ! 29.7825452 9.6081817 .001127396 sss 788544 700227072 i 29.7993289 9.6117911 .001126126 889 790321 702595369 } 29.8161030 9.6153977 .001124859 890 792100 704969000 ' 29.8328678 9.6190017 .001123590 891 793881 707347971 ' 29.8496231 9.6226030 ,.001122834 892 795664 7097'32288 29.8663690 9.6262016 .001121076 893 797449 712121957 29.8831056 9.6297975 .001119821 894 799336 714516984 29.8998328 J)! 6383907 .001im568 895 801025 716917375 29.9165506 9.6369812 .001117318 896 802816 719323136 29.9332591 9.6405690 .001116071 89? 804609 721734273 29.9499583 9.6441542 .001114827 888 806404 7'24150792 29.9666481 9.6477367 .001113586 899 808201 720572699 29.9833287 9.6513166 .001112347 900 810000 729000000 30.0000000 9.6548938 .001111111 901 811801 731432IU1 30.0106620 9.6584684 .001109878 902 813604 73387080$ 30.0333148 9.6620403 .001108647 903 815409 730.514327 30.0499584 9.6656096 .001107420 904 817216 73 81 8700 OIK; uvw uuir 0301 0775 A-ta 5 021189 1603 2016 2428 2841 3252 3064 4075 4486 4896 41o 412 6 5:306 5715 ! 6125 6533 0942 7350 7757 8164 j 8571 | 8978 408 Q.KM 9789 ' ' if'So^t 0195 0600 1004 1 108 1812 2216 2019 3021 A(\A 8 ( 3424 3826 4227 4028 5029 54:30 5830 6230 6029 7028 4U4 400 g 7426 7825 8223 8020 9017 9414 9811 ( )4 0207 ] 0602 0998 397 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 434 43.4 86.8 130.2 173.6 217.0 260.4 303.8 347.2 390.6 433 43.3 86.6 129.9 173.2 216.5 259.8 303.1 346.4 389.7 432 43.2 86.4 129.6 172.8 216.0 259.2 302.4 345.6 aS8.8 431 43.1 86.2 129.3 172.4 215.5 258.0 301.7 344.8 387.9 43J 43.0 86.0 129.0 172.0 ! 215.0 258.0 1 301.0 344.0 387.0 429 42.9 85.8 128.7 171.6 i 214.5 257.4 S 300.3 343.2 380.1 428 42.8 85.6 128.4 1 171.2 i 214.0 256.8 299.6 342.4 385. a 427 42.7 85.4 128.1 170.8 i 213.5 250.2 298.9 341.0 384.3 420 42.6 85.2 127.8 .1 170.4 213.0 255 6 298.2 340.8 383.4 425 42.5 85.0 127.5 170.0 212.5 255.0 297.5 340.0 382.5 424 42.4 84.8 127.2 169.6 212.0 254.4 296.8 339.2 381.6 423 42.3 84.6 126 '9 169.2 211.5 253.8 296.1 338.4 380.7 422 42.2 84.4 126.6 168.8 211.0 253.2 295. 4 337.0 379.8 421 42.1 84.2 120.3 168.4 210.5 252.0 394.7 380. S 378.1) 420 42.0' 84.0 126.0 168.0 210.0 252.0 294.0 330.0 378.0 419 41.9 83.8 125.7 167.6 209.5 251.4 393.3 335. ;j 377.1 418 41.8 83.6 125.4 167.2 209.0 250.8 292.0 884^4 376.2 417 41.7 83.4 125.1 166.8 208.5 250.2 291.9 333.0 375.3 410 41.6 83.2 124.8 166.4 208.0 249.0 291.2 332.8 :J7'4 . I 415 41.5 88.0 124.5 106.0 207.5 219.0 290.5 332.0 I 373.5 414 I 41.4 82.8 124.2 10.-). 207.0 248.4 289.8 331.2 ! 372.6 413 41.8 82.6 123.9 165.2 206.5 247.8 28!). 1 330.4 371.7 412 41.2 82.4 123.6 164.8 206.0 247.2 288.4 329.0 i 370.8 411 41.1 82.2 123.3 164.4 205.5 246.6 287.7 328.8 369.9 410 41.0 * 82.0 123.0 104.0 205.0 240.0 287.0 328.0 369.0 409 ! 40.9 81.8 122.7 103.6 204.5 245.4 280.3 327.2 : 308.1 408 40.8 81.6 122.4 103.2 204.0 244.8 285.0 320.4 307.2 407 40.7 81.4 122.1 162.8 203.5 244.2 284.9 325.0 806.3 400 40.6 81.2 121.8 162.4 203.0 243 284.2 324.8 3(15.4 405 40.5 81.0 121.5 162.0 202.5 243.0 283.5 324.0 364.5 404 40.4 80.8 121.2 101.6 202.0 242.4 282.8 323.2 363.6 403 40.3 80.6 120.9 161.2 201.5 241.8 282.1 322.4 302.7 402 40.2 80.4 120.6 160.8 201.0 241 2 2S1.4 321.6 361.8 401 40.1 80.2 120.3 160.4 200.* 240.6 280.7 320.8 360.9 400 40.0 80-0 120.0 160.0 200.0 240.0 280.0 320.0 360.0 399 39.9 79.8 119.7 159.6 199.5. 239.4 279.3 319.2 359.1 398 39.8 79.6 119.4 159.2 199.0 238.8 278.6 318.4 358.2 397 39.7 79.4 119.1 158.8 198.5 238.2 277.9 317.0 357.3 390 39.6 79.2 118.8 158.4 198.0 237.0 277.2 310.8 350.4 395 39.5 79.0 118.5 158.0 K)7 5 237.0 276.5 316 355.5 [332] TABLE XXTV. LOGARITHMS OF NUMBERS. No. 110 L. 041.] [No. 119 L. 078. 7 1 T\s~ce . UlLL. 110 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 393 1 5323 5714 6105 6495 6885 7275 7C.C4 8053 8442 8830 390 2 9218 9606 9993 0380 0766 1153 1538 1924 2809 2694 386 3 053078 3463 3846 4230 4013 4990 5378 5760 0142 6524 383 4 6905 7286 7660 8040 8420 8805 9185 9563 9942 5 01 1075 3709 4083 376 50698 1452 1829 2206 2582 2958 3333 6 4458 4832 5200 5580 5953 Ii320 6699 7071 7443 7815 373 7 8186 8557 8928 9298 9068 0038 ' 0407 07'76 1145 1514 370 8 071882 2250 2017 2985 3352 3718 4085 4451 4816 5182 36(5 9 5547 5912 (5270 6640 7004 7368 7731 8094 8457 8819 363 PROPORTIOI*.L PARTS. Diff. 1 2 3 4 5 6 7 8 9 395 804 39.5 39.4 79.0 78.8 118.5 118.2 158.0 157.6 197.5 197.0 937 236 .0 .4 276.5 275.8 316.0 315.2 355.5 354.6 ] 393 39.3 78.6 117.9 157.2 190.5 235 .8 275.1 314.4 353.7 392 39.2 78.4 117.6 156.8 . 196.0 235 .2 2 74.4 313.6 &52.8 391 39.1 78.2 117.3 156.4 A05.5 234 .6 2 73.7 312.8 351.9 390 39.0 78.0 117.0 156.0 195.0 234.0 273.0 312.0* 351.0 38( 38.9 77.8 116.7 155.6 194.5 233 .4 2 72.3 311.2 350.1 38H 38.8 77.6 116.4 155.2 194.0 232.8 271.6 310.4 349.2 387 38.7 77.4 116.1 154.8 193.5 232 .2 2 70.9 309.6 348.3 38< 38. 77.2 115.8 154.4 193.0 231 .0 2 70.2 308.8 347.4 385 38.5 77.0 115.5 154.0 192.5 231 .0 269.5 308.0 346.5 384 38.4 76.8 115.2 153.6 192.0 230.4 2 68.8 307.2 345.6 388 38.3 70.0 114.9 153.2 191.5 229.8 268.1 306.4 344.7 38* 38. a 76.4 114.6 152.8 191.0 220 .2 2 67.4 305.6 343.8 381 384 76.2 114.3 152.4 190.5 22*- .0 60.7 304.8 342.9 58C 38.0 76.0 114.0 152.0 190.0 228 .0 2 66.0 304.0 342.0 ' 379 37.9 75.8 113.7 151.0 189.5 227.4 265.3 303.2 341.1 378 37.8 75.6 113.4 151.2 189.0 226.8 264.6 302.4 340.2 37" 37.7 75.4 113.1 150.8 188.5 221 .2 2 63.9 301.6 339.3 87f > 37.6 75.2 112.8 150.4 188.0 22E .0 2 63.2 300.8 338.4 375 37.5 75.0 112.5 150.0 187.5 225.0 262.5 300.0 337.5 374 37.4 74.8 112.2 149.6 187.0 224.4 261.8 299.2 336.0 373 37.3 74.6 111.9 149.2 186.5 223.8 261.1 298.4 335.7 37: > 37.2 74.4 111.6 148.8 186.0 22' 2 2 60.4 297.6 334.8 371 37.1 74.2 111.3 148.4 185.5 222.6 259.7 296.8 333.9 371 ) 37.0 74.0 111.0 148.0 185.0 222 .0 2 59.0 296.0 333.0 361 > 30.9 73.8 110.7 147.6 184.5 221 .4 2 58.3 295.2 332.1 308 36.8 73.6 110.4 147.2 184.0 220.8 257.6 294.4 331.2 36' 30.7 73.4 110.1 146.8 183.5 22C .2 2 56.9 293.6 830.3 860 30.0 73.2 109.8 146.4 183.0 219.6 256.2 292.8 329.4 805 36.5 73.0 109.5 146.0 182.5 219.0 255.7 292.0 328.5 364 36.4 72.8 109.2 145.6 ' 182.0 218.4 254.1 291.2 327.6 30- i 36.3 72.6 108.9 145.2 181.5 217 .8 2 54.1 290.4 326.7 305 J 30.2 72.4 108.6 144.8 181.0 217 .2 2 53.4 289.6 325.8 301 30.1 72.2 108.3 144.4 180.5 216.6 252.7 288.8 324.9 36( ) 36.0 72.0 108.0 144.0 180.0 21( .0 2 52.0 288.0 324.0 359 71.8 107.7 143.6 179..-) 215.4 251.3 287.2 323 1 35i J 35^8 71.6 107.4 143.2 179.0 214 .8 x 50.0 280.4 328.2 357' .35.7 71.4 107.1 142.8 178.6 214.2 249.9 285.0 321.3 356 35.6 71.2 106.8 112.4 178.0 213.6 249.2 284.8 320.4 TABLE XXTV. LOGARITHMS OF NUMBERS. No. 120 L. 079.] [No. 134 L. 130. N. 1 2 3 45 6 7 8 9 Diff. 1 f 1 120 u/ioi i yo*3 9904 0266~ 0626 0987 1347 1707 2067 2426 360 1 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 357 6360 6716 : 7071 7426 9900 7781 8136 8490 8845 9f98 9552' 355 0258 0611 0963 1315 1667 2018 2370 2721 3071 853 4 093422 3772 4122 4471 4820 1 5169 5518 5866 , 6215 6562 349 5 6910 7257 7604 7951 8298 8644 8990 9335 9681 6 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 343 y 3804 4146 4487 , 4828 5169 5510 5851 6191 6531 6871 341 8 7210 7549 7888 8227 8565 8903 9241 9579 9916 OOQ 9 110590 0926 : 1263 | 1599, 1934 2270 2605 2940 3275 3609 OOo 335 130 3943 4277 4611 494*4* 5278 ! 5611 5943 6276 6608 6940 333 j 7271 7603 ?QSJ. Rojy; 8595 8926 9256 9586 9915 noi QQA 2 120574 0903 1231 1560 1888 * 2216 2544 2871 3198 USraD 3525 cfifU 328 3 3852 4178 4504 4830 5156 5481 5806 6181 6456 6781 325 4 7105 7429 7753 8076 8399 8722 9045 JHIW 9690 13 0012 ! 323 PROPORTIONAL PARTS. Diff. i 2 3 4 < 5 r 6 7 8 9 .355 35.5 71.0 106.5 142.0 177.5 213.0 248.5 284.0 319.5 354 35.4 70.8 106.2 1 41.6 177.0 212.4 24 7 . K '^83 2 318.6 853 35.3 70.6 105.9 1 41.2 176.5 211.8 24 7. i 282 .'4 317.7 352 35.2 70.4 105.6 140.8 176.0 211.2 246.4 281.6 316.8 35 j 35.1 70.2 105.3 1 40.4 175.5 210.6 24 5.7 280.8 31 5.0 350 35.0 70.0 105.0 140.0 175.0 | 210.0 | 245.0 280.0 315.0 349 34.9 69.8 104.7 1 39.6 174.5 209.4 24 i, -a 279.2 314.1- 348 34.8 69.6 104.4 1 39.2 174.0 208.8 24 ].!' 27'8.4 313.2 347 34.7 69.4 104.1 138.8 173.5 208.2 242.9 277.6 312.3 346 34.6 69.2 103.8 138.4 173.0 207.6 242.2 276.8 311.4 345 34.5 69.0 103.5 138.0 172.5 207.0 241.5 276.0 310.5 344 34.4 68.8 103.2 137.6 172.0 206.4 240.8 275.2 309.6 343 34.3 68.6 102.9 1 37.2 171.5 205.8 24 1.1 274.4 308.7 34-2 34.2 68.4 102.6 136.8 171.0 205.2 239.4 27'3.6 307.8 341 34.1 68.2 102.3 1 36.4 170.5 ! 204.6 231 3.7 272.8 306.9 340 34.0 68.0 102.0 1 36.0 170.0 204.0 23, 3.0 272.0 306.0 339 as. 9 67.8 101.7 135.6 169.5 203.4 23 271.2 305.1 338 33.8 67.6 101.4 1 35.2 169.0 202.8 23 L8 270.4 804.2 337 33.7 67.4 101.1 134.8 168.5 202.2 235.9 269.6 303.3 336 33.6 67.2 100.8 134.4 168.0 201.6 235.2 268.8 302.4 885 33.5 67.0 100.5 134.0 167.5 201.0 23- 1.5 268.0 301.5 334 .33.4 66.8 100.2 1 33.6 167.0 200.4 23 lifl 267.2 300.6 333 33.3 66.6 99.9 133.2 166.5 199.8 233.1 266; 4 299.7 332 .33.2 66.4 99.6 1 32.8 166.0 199.2 SK 2.4 265.6 298.8 331 33.1 66.2 99.3 132.4 165.5 198.6 23- .7 264; 8 297.9 330 33.0 66.0 99.0 1 32.0 165.0 198.0 231 .0 264.0 297.0 329 32.9 6?. 8 98.7 131.6 164.5 197.4 230.3 263.2 296.1 328 32.8 65.6 98.4 1 31,2 164.0 196.8 22' . 262.4 295.2 327 32.7 65.4 98.1 130!8 163.5 196.2 228.9 261.6 294.3 326 32.6 65.2 97.8 130.4 163.0 195.6 22* {.a 260.8 293.4 325 32.5 65.0 97.5 130.0 162.5 195.0 227.5 260.0 292.5 324 32.4 64.8 97.2 129.6 162.0 194.4 226.8 259.2 1 291.6 323 32.3 64.6 96.9 1 29.2 161.5 193.8 22( .1 ! J58.4 290.7 322 32.2 64.4 96.6 128.8 161.0 193.2 225.4 257.6 289.8 TABLE XXTV. LOGARITHMS OF NUMBERS. No. 135 L. 130.] [No. 14f> L. 175. N. 1 2 8 4 5 6 7 8 9 Diff. 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 321 fi 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 ' 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 316 g 9879 0194 0508 0822 1136 1450 1763 2076 2389 2702 314 9 143015 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 140 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 309 919 95<>7 9835 0142 0449 0756 10(53 1370 1 076 1982 307 2 152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 305 8 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 303 8362 8664 8965 9567 9868 . 04fiQ 0769 1068 nf>1 5 161368 1667 1967 2266 2564 2863 3161 3460 3758 4055 299 6 4353 4650 4947 5244 5541 5838 6131 6430 6726 7083 297 7 731? 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 8 170262 0555 0848 1141 1434 1726 2019 2311 2(503 2895 293 9 3186 8478 3769 4060 1351 4641 4932 5222 5512 5802 291 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 821 32.1 64.2 9(5 .3 128.4 160.5 192 6 224.7 256.8 288.9 320 32.0 64.0 96 .0 128.0 160.0 192 2$ 4.0 256.0 288.0 319 31.!) 63.8 95 .7 27.6 159.5 191 4 2x 3.8 255.2 287.1 318 31.8 63.6 95.4 127.2 150.0 190 8 2x 2.6 251.4 286.2 317 31.7 63.4 95 .1 26.8 158.5 190 2 2$ 51.9 253.6 285.3 316 31.6 (Ki.2 94 .8 26.4 ir,8.o 18!) 6 2$ !1 2 252.8 284.4 315 I 31.5 (W.O 94 .5 126.0 157.5 189 220.5 252.0 283.5 314 | 31.4 62.8 94 .2 25.6 157.0 188 4 2 9 8 251.2 282.6 313 i 31.3 62.6 93 .9 125.2 156.5 187.8 219.1 250.4 281.7 312 31.2 62.4 93 .6 124.8 156.0 187.2 218.4 249.6 280.8 311 31.1 62.2 93 .3 124.4 155.5 186 6 217.7 248.8 279.9 310 31.0 62.0 93 24.0 155.0 186 21 7 248.0 279.0 309 30.9 61.8 92 .7 123.6 154.5 185.4 216.3 ! 247.2 278.1 308 30.8 r.i 6 92 .4 -, 2 154.0 184 8 21 5 r> 246.4 , 277.2 307 30.7 61.4 92 .1 122.8 153.5 184.2 214.9 I 245.6 1 276.3 306 30.6 61.2 91 .8 - 22.4 issio 183 6 2 4 2 244.8 1 275.4 305 30.5 61.0 9k 5 122.0 152.5 183.0 213.5 214.0 2745 304 30.4 60.8 91 .2 21.6 152.0 182 4 21 2 8 243.2 273.6 803 - 30.3 60.6 90 .9 121.2 151.5 181.8 i 212.1 242.4 272.7 302 30.2 60.4 90 .6 120.8 151.0 181 .) 211.4 241.6 271.8 301 .'50.1 60.2 90 .3 1.20.4 150.5 180.6 210.7 240.8 270.9 300 30.0 60.0 9(! .0 20.0 1 150.0 180 2 240.0 27'0.0 299 29.9 59!8 89 .7 119.6 ! 149.5 179 4 209.3 239.2 269.1 298 29.8 59.6 8S .4 .19.2 149.0 178 8 2( )8 6 i 238.4 268.2 297 29.7 59.4 8? 1 .18.8 148.5 178 2 2( 17 9 237.6 267.3 296 29.0 59.2 88 .8 118.4 148.0 177 207.2 236.8 266.4 296 29.5 59. ( 88 .5 1B.O 147.5 177 2( )6 5 236.0 265.5 294 29.4 58.8 88 .2 117.6 147.0 176 4 205.8 235.2 264.6 293 29.3 58.6 87 .9 117.2 146.5 175 8 2( )5 1 234.4 263.7 21)2 29.2 58.4 87-. 6 116.8 146.0 175.2 204.4 233.6 262.8 291 29.1 5S.2 87.3 116.4 145.5 174 6 203.7 232.8 261.9 290 29.0 58 87 .0 116.0 145.0 174.0 203.0 i Stt.O 261.0 289 28.9 57.8 86 .7 115.6 144.5 173 4 2( }> 3 231.2 200.1 288 28.8 57.6 86 .4 115.2 111.0 172.8 201.6 230.4 259.2 287 28.7 57.4 86 .1 114.8 143.5 172 2 2( 10.9 229.6 258.3 2S6 28.6 57.2 85.8 114.4 143.0 III 6 200.2 228.8 ! 257.4 [335] TABLE XXTV. LOGARITHMS OK No. 150 L. 176.1 [No. 169 L. 230. N. 1 2 8 9 Diff. | 150 1 2 3 4 5 6 7 8 9 160 1 2 3 4 5 6 8 9 176091 8977 6381 9264 6670 9552 6959 9839 7248 0126 2985 5825 , 8647 7536 7825 8113 0986 3839 6674 9490 8401 8689 1558' 4407 7239 289 287 285 283 281 279 878 276 274 272 271 269 267 266 264 262 261 259 258 256 0413 3270 6108 8928 0699 3555 6c91 9209 1272 4123 6066 9771 181844 4691 7521 2129 4975 7803 2415 5259 8084 2700 5542 836G 0051 2846 5623 8382 190332 3125 5900 8657 201397 4120 6806 9515 0612 3403 6176 d932 1670 4391 7096 9783 0892 3681 6453 9206 1171 3959 6729 9481 1451 4237 7005 9755 2488 5204 7904 1730 4514 7281 2010 4792 7556 2289 5069 7832 0577 3305 6016 8710 2567 5346 8107 0029 2761 5475 8173 0303 3033 5746 8441 0850 3577 6286 89?'9 1124 3848 6556 9247 1921 4579^ 7221 9846 1943 4663 7365 0051 2720 5373 8010 2216 4934 7634 0319 2986 5638 8273 0892 3496 6084 8657 0586 3252 5902 8586 1153 3755 6342 8913 0853 3518 6 ]0(> i 8798 1121 3783 6430 9060 1388 4049 6694 9323 1654 4314 6957 9585 212188 4844 7484 2454 5109 7747 220108 2716 5309 7887 23 a370 2!7(i 5568 8144 0(581 32536 5H36 8400 1414 175 4015 | 4274 6600 6858 9170 9426 i 1936 4533 7115 9682 2196 4792 7372 99:38 2456 5051 7630 ~0193 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 285 I 28.5 57.0 85.5 114.0 142.5 171.0 199.5 228.0 256.5 284 28.4 56.8 a5.2 118.6 142.0 i 170.4 198.8 227.2 255.6 283 28.3 56.6 84.9 113.2 141.5 169.8 198,1 226.4 254.7 282 28.2 i 56.4 84.6 112.8 141.0 169.2 197.4 225.6 i 253.8 281 28.1 56.2 84.3 112 4 140.5 168.6 196.7 224.8 252.9 280 28.0 56.0 84.0 112.0 I 140.0 168.0 196.0 224.0 252.0 279 27.9 55.8 83.7 111.6 139.5 167.4 195.3 223.2 251.1 278 27.8 55.6 83.4 111.2 139.0 166.8 194.6 222.4 250.2 277 27.7 55.4 83.1 110.8' 138.5 1 166.2 193.9 221.6 249.3 276 2 ;'.('. 55.2 82.8 110.4 138.0- 165.6 j 193.2 220.8 248.4 275 27.5 1 55.0 82.5 110.0 137.5 165.8 192.5 220.0 247.5 274 27.4 ! 54.8 82.2 109.6 137.0 164.4 191.8 219.2 i 246.6 273 27.3 54.6 81.9 109.2 136.5 163.8 191.1 218.4 245.7 272 27.2 54.4 81.6 108.8 136.0 163.2 1 190.4 217.6 244.8 271 27.1 54.2 81.3 108.4 135.5 162.6 189.7 216.8 243.9 270 27.0 54.0 81.0 108.0 135.0 162.0 189.0 216.0 243.0 269 26.9 53.8 80.7 107.6 134.5 161.4 188.3 215.2 242.1 268 26.8 53.6 80.4 107.2 134.0 160.8 i 187.6 214.4 241.2 267 26.7 '53.4 : 80.1 106.8 133.5 160.2 186.9 213.6- 240.3 2ti6 26.6 53.2 79.8 106.4 133.0 159.6 186.2 212.8 239.4 265 26.5 53.0 79.5 106.0 132.5 159.0 185.5 212.0 238.5 2(54 26.4 52.8 79.2 105.6 132.0 158.4 184.8 211.2 237.6 263 26.3 52.6 78.9 105.2 131.5 15V. 8 184.1 210.4 236.7 262 26.2 52.4 78.6 104.8 131.0 157.2 183.4 209.6 235.8 261 26.1 52.2 78.3 104.4 130.5 156.6 182.7 208.8 234.9 260 26.0 52.0 78.0 104.0 130.0 156.0 182.0 208.0 234.0 259 25.9 51.8 77.7 103.6 129.5 155.4 181.3 207.2 233.1 258 25.8 51.6 77.4 103.2 129.0 154.8 180.6 206.4 232.2 257 25.7 51.4 77.1 102.8 128.5 154.2 179.9 205.6 231.3 256 25.6 51.2 76.8 108.4 128.0 153.6 179.2 204.8 230.4 255 25.5 51.0 76.5 io:l o 1S7.5 153.0 i 178.5 204.0 229.5 [33*1 TABLE XXIV. LOGAKITHMS OF NUMBLliS. No. 170 L. 230.] [No. 189 L. 278. 80 T)ifF N. 170 230449 V JJlIt, 255 0704 0980 1215 1470 1724 1979 2234 2488 2742 1 2996 3250 3504 .",;:,; 4011 4264 4517 4770 5023 5276 253 2 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 262 8046 8297 8548 8799 9049 9299 ' 9550 9800 0050 0300 250 4 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 5 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 (5 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 7 7973 8219 8464 8703 8954 9198 9443 9687 9932 0176 245 8 250420 0664 0908 1151 1395 |j 1638 188U 2125 2388 2610 243 9 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 180 5273 5514 5755 5906 6287 6477 6718 6958 7198 7439 241 1 7679 71)18 8158 NV.KS 8637 8877 9116 9355 9594 98*3 239 2 260071 0810 0548 0787 1025 1263 1501 1739 1976 2214 238 3 2451 26SS 29-J5 3162 3399 8636 3873 4109 4346 4582 237 4 4818 5054 5290 5525 5761 5996 6232 (.467 6708 6937 235 g 7172 9513 7406 9746 7641 '.I'.iSO 7875 8110 8344 8578 8812 9046 927'9 234 0213 Ailfi 0679 0912 1 1 ,1.1 1*377 1fiOQ OQQ 7 271842 2074 2808 2538 U-Jr*U 2770 3001 3233 J I'M: 3464 JLO/I 3696 IvUp 3927' ,-wOo 232 8 4158 438!) 4620 4850 5081 5311 55 12 5773 6002 6232 230 g 6462 6t)92 6921 7151 7880 7609 7838 8067 8296 8525 229 I'lKipoirnoNA], TARTS. Diff. 1 2 3 4 5 6 7 8 9 255 25.5 25 1 25.4 51.0 50.8 76.5 76.2 102.0 101.6 i 127.5 127.0 153.0 152.4 '178.5 177.8 204 203.2 'q.5 228.6 253 25. 3 50.15 r.->. ( . 101.2 12(5.5 151.8 177.1 202.4 227.7 252 .25.2 50. I 75.8 100. 8 12ii.O 151.2 176.4 20 1. 6 226.8 851 25.1 | 50.2 ;:, -; 100.4 125.5 150.6 175.7 200.8 225.9 250 25 50.0 75.0 100.0 125.0 150.0 17'5.0 200.0 225.0 249 24.9 49.8 74.7 99. (i 124.5 149.4 174.3 199.2 224.1 &8 24.8 49.6 74.4 99.2 ; 124.0 148.8 173.6 198.1 338.2 247 21.7 49.4 74.1 9S.8 123.5 us 2 172.9 197 6 222.3 246 24.6 19.2 73. H 98. -1 123.0 147.0 I7'2.2 196.8 221.4 2 !.- 24.5 49.0 78! 5 98.0 122.5 147.0 171.5 196.0 220.5 ,44 24.4 48.8 7'3.2 97.6 122.0 146.4 170.8 195.2 219.6 2 13 24.3 48.6 72.9 97,3 121.5 145.8 170.1 194.4 218.7 2 1-2 24.2 48.4 72.6 96.8 121.0 14;.. 2 169.4 193.6 217.8 241 24.1 48.2 72.3 96.4 120.5 144.6 168.7 192.8 216.9 240 24.0 48.0 72.0 98.0 120.0 .1(1.0 I6S.O 192.0 216.0 289 23.!) 47.8 71.7 95.6 119.5 143.4 167.3 191.2 215.1 238 23.8 47.6 71.4 95.2 119.0 142.8 166. 6 190.4 214.2 237 23.7 47.4 71.1 01. S 118.5 142.2 165.9 189.6 213.3 236 23.6 47.2 ro! 8 94.4 118.0 141.6 165.2 188.8 212.4 235 23.5 47.0 70.5 94.0 117.5 141.0 164.5 188.0 211.5 234 23.4 46.8 70.2 93.6 117.0 140.4 163.8 187.2 210.6 233 23.3 46.6 89.9 93.2 116.5 139.8 163.1 186.4 209.7 232 23.2 40.4 69.6 92.8 116.0 139.2 162.4 185.6 208.8 231 23.1 46.2 69^3 92.4 115.5 138.6 161.7 184.8 207.9 230 23.0 46.0 69.0 92.0 115.0 138.0 161.0 184.0 207.0 229 22.9 45.8 68.7 91.6 114.5 137.4 160.3 183.2 206.1 228 22.8 45.6 68.4 91.2 114.0 136.8 159.6 182.4 205.2 227 22.7 45.4 6S.I 90.8 113.5 136.2 158.9 181.6 204.3 226 22.6 45.2 87.8 90.4 113.0 135.6 158.2 180.8 203.4 TABLE XXTV. LOGARITHMS OF NUMBERS. No. 190 L. 278.] - [No. 214 L. 832. 1C 1 2 8 4 6 6 7 8 Diff. -, 1 '. 1 1 190 278754 8982 9211 9439 9667 9895 AQAfi OOQ 1 2810-33 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 2 3301 a527 3753 3979 4205 4431 4656 4882 5107 5332 226 3 557 5782 6007 6232 6456 6681 6905 7 130 7354 7578 225 4 7802 i 8026 | 8249 8473' 8696 8920 9143 9366 9589 9812 223 5 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 222 6 2256 2478 2699 2920 3141 3363 ,3584 3804 4025 4246 221 7 4466 4687 4907 5127 5347 5567 5787 6 M)7 6226 6446 220 8 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 g 8853 9071 9289 9507 9725 9943 01fi1 Ou QC-C 0595 0813 218 200 301030 1247 1464 1681 1898 ! 2114 2331 2547 2764 2980 217 1 3196 3412 3628 3844 4059 4275 4491 4 roil 4921 5136 216 2 5351 5566 5781 5996 6211 t>425 6639 6854 7068 7282 215 3 7496 7710 7924 8137 8351 8564 8778 8 991 9204 9417 213 4 9630 9843 - 0056 068 AICI fUJQO 5 311754 1966 2177 2389 2600 2812 3023 3234 8445 3656 ' 211 6 3867 4078 4289 4499 4710 4920 5130 5 340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 436 7646 7854 209 8 8063 8272 8481 8689 8898 9106 9314 9522 97:30 9938 208 9 320146 0354 0562 0769 0977 1184 4391 1598 1805 2012 * 207 210 2219 2426 2633 28S9 3046 3252 3458 3665 3871 4077 206 1 4282 4488 4694 4899 5105 5310 5516 5 721 5926 6131 205 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 8 8380 8583 8787 8991 9194 9398 i 9601 () 805 4 330414 0617 0819 1022 1225 ! 1427 ! 1630 1832 | 2034 | 2236 j 202 PROPORTIONAL PARTS. Diff. i | * * 4 5 6 7 ' 8 j 225 22.5 45.0 67.5 90.0 112.5 135.0 157.5 180.0 202.5 224 22.4 44,8 67.2 89.6 112.0 134.4 156.8 179.2 201.6 223 22.3 44.6 66.9 89.2 111.5 133.8 156.1 178.4 200.7 222 22.2 44.4 66.6 88.8- 111.0 133.2 155.4 I 177.6 199.8 221 22.1 44.2 66.3 88.4 110.5 132.6 154.7 176.8 198.9 220 22.0 44.0 66.0 88.0 110.0 132.0 154.0 176.0 198.0 219 21.9 43.8 65.7 87.6 109.5 131.4 153.3 175.2 197.1 218 21.8 43.6 65.4 87.2 ' 109.0 i:30.8 152.6 174.4 196.2 217 21.7 43.4 65.1 86.8 108.5 130.2 151.9 173.0 195.3 216 21.6 43.2 64.8 86.4 108.0 129.6 151.2 172.8 194 4 215 21.5 43.0 64.5 86.0 107.5 129.0 150.5 172.0 193.5 214 21.4 42.8 64.2 85.6 107.0 128.4 149.8 171.2 192.6 213 21.3 42.6 63.9 85. -2 106.5 127.8 149.1 170.4 191.7 212 21.2 42.4 63.6 84.8 106.0 127.2 148.4 169.6 190.8 211 21.1 42.2 63.3 84.4 105.5 126.6 147.7 168.8 189.9 210 21.0 42.0 63.0 84.0 105.0 126.0 147.0 168.0 189.0 209 ! 20.9 41.8 62.7 83.6 104.5 125.4 146.3 167.2 188.1 208 20.8 41.6 62.4 83.2 104.0 124.8 145.6 166 4 187.2 207 20.7 41.4 62.1 82.8 103.5, 124.2 144.9 165.6 186.3 206 20.6 41.2 61.8 82.4 103.0 123.6 144 . 164.8 185.4 205 20.5 4i.O C1.5 82.0 102.5 123.0 143. S 164.0 184.5 204 20.4 40.8 61.2 81.6 102.0 122.4 142.8 163.2 183.6 203 20.3 40.6 60.9 81.2 101.5 121.8 142.1 162.4 182.7 202 20.2 40.4 60.6 '0.8 101.0 121.2 141.4 161.6 181.8 TABLE XXTV. LOGARITHMS OF NUMBERS. No. 215 L. 332.] [No. 239 L. 380. N. 1 2 3 4 5 6 7 8 9 Diff. 215 6 8 9 220 1 2 3 4 5 6 7 8 9 230 1 2 3 4 5 6 8 9 332438 4454 (MOO 8456 2640 4655 6660 8656 2842 4856 6860 8855 3044 50;:7 7060 9054 3246 5257 7200 9253 344? 5458 7459 9451 3649 5658 7659 9650 3850 5859 7858 9849 4051 6059 8058 0047 coss 3999 5962 7915 9860 4253 6260 8257 0246 2225 4196 6157 8110 202 ,'01 200 199 198 197 196 195 194 193 193 192 191 190 189 188 188 187 186 185 184 184 183 182 181 340444 2423 4392 ti J.VJ 8305 0642 2620 4589 6549 8500 0841 2817 4785 6744 8694 1039 3014 4981 6939 8889 1237 3212 5178 7135 9083 1435 3409 5374 7330 9278 1632 3606 5570 7'525 947'2 1410 3339 saw 7172 9076 1830 3802 5766 7720 9666 1603 3532 5452 7363 9266 0054 1989 3916 5834 7744 9646 1539 3424 5301 7169 9030 350248 2183 4108 6026 7986 9836 361728 3612 5488 7356 9216 0442 2375 4301 6217 8125 0025 1917 3800 5675 7542 9401 0636 2568 4493 (5408 8310 0215 2105 o988 6862 7729 9587 0829 2761 4685 6599 8506 0404 2294 4176 6049 7915 9772 1023 2954 4876 '! orih) 8696 1216 3147 50(58 6981 8886 1796 3724 51543 7554 9456 1350 3236 5113 6983 8845 0698 2544 4382 6212 8034 9849 0593 2482 4363 6236 8101 9958 0783 2671 4551 6423 8287 097'2 2859 4739 6610 8473 1161 3048 4926 67% 8659 0513 2360 4198 6029 7852 9608 0143 1991 3831 56(54 7488 9306 0328 2175 4015 5846 7670 9487 0883 j;-js 4565 6394 8216 0030 371008 2912 47'48 0577 8398 38 i-r>3 3096 4932 5750 8580 1437 3280 .5115 r.;)42 8761 1622 3464 5298 7124 8043 1806 3647 5481 7806 9124 PROPORTIONAL PARTS. , Diff. 1 2 3 4 5 6 ' 7 8 9 202 20.2 40.4 60.6 80.8 101.0 121.3 141.4 161.6 181.8 201 S0.1 40.2 60.3 80.4 100.5 120.6 140.7 160.8 180.9 200 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 199 19.9 39.8 59.7 79.6 99.5 119.4 1*1. 3 159 179.1 198 19.8 39.6 59.4 79.2 99.0 118.8 138.6 158.4 178.2 197 19.7 I 39.4 59.1 78.8 us.:, 118.2 137.9 157.6 177.3 196 19.6 39.2 58.8 7S.4 OS.O 1176 137.2 ir.ti.s 176.4 195 19.5 39.0 58.5 78.0 97.6 J17.0 136.5 156.0 175.5 194 19.4 38.8 58.2 77.6 97.0 116.4 135.8 155.2 174.6 193 19.3 38.6 57.9 77.2 96.5 115.8 185:1 154.4 173 7 192 19.2 38.4 57.6 76.8 96.0 115.2 134.4 153.6 172. S 191 19.1 38.2 57.3 76.4 95.5 114.6 133.7 152.8 171.9 190 19.0 38.0 57.0 76.0 95.0 114.0 133.0 152.0 171.0 189 18.9 37.8 56.7 75.6 94.5 113.4 132.3 151.2 17'0.1 188 18.8 37.6 56.4 75.2 94.0 112.8 131.6 150.4 169.2 187 18.7 37 4 56.1 74.8 93.5 112.2 130.9 149.6 168.3 186 18.6 37.2 55.8 74.4 93.0 111.6 130.2 J48.8 1(57.4 185 18.5 37.0 55.5 74.0 92.5 111.0 109.5 148.0 166.5 184 18.4 36.8 55.2 73.6 92.0 110.4 128.8 147.2 165.6 183 18.3 36.6 54.9 73.2 91.5 109.8 128.1 146.4 164.7 182 18.2 36.4 54.6 72.8 S-1.0 109.2 127.4 i 145.6 163.8 181 18.1 36.2 54.3 72.4 90.5 108.6 126.7 144.8 162.9 180 18.0 36.0 54.0 72.0 90.0 108.0 126.0 144.0 162.0 179 17.9 &5.8 53.7 71.6 89.5 107.4 125.3 143.2 161.1 TABLE XXIV. LOGARITHMS OF NUMBEBS. r No. 240 L. 380.] [No. 269 L. 431. N. 1 . 2 3 4 5 7 8 9 Diff. 240 380211 0392 0573 0754 0934 1115 1296 1476 1656 1837 181 1 2017 2197 2377 25f r 2737 2917 3097 3 >;7 3456 3636 180 2 3815 3995 4174 4a53 4533 4712 4891 5070 5249 5428 179 3 5606 5785 5964 614 2 6321 6499 6677 6 m 7034 7212 178 4 7390 7568 7746 79X t 8101 8279 8456 s 134 8811 8989 178 O1 (\l\ 9520 (JO; 8 9875 yiou 0051 0228 0405 0582 0759 177 6 390935 1112 1288 1464 I 1641 1817 1993 2169 2345 2521 176 7 2697 2873 3048 32$ !4 3400 3575 3751 3 1-20 4101 4277 176 8 4452 4627 4802 49' 7 5152 5326 5501 5 >76 5850 6025 175 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 7940 8114 8287 8461 8634 8808 8981 9154 9328 9501 173' 9674 9847 0020 0192 0365 0538 oni 388 1056 1228 173 2 401401 1573 1745 191 I? 2089 2S1 ; 2433 2 (05 2777 2949 172 3 3121 3292 3464 36f 55 .3807 3978 4140 . 4 o 4492 4663 171 4 4834 5005 5176 5346 5517 5688 5858 6029 6199 6370 171 5 6540 6710 6881 70f >l 7221 7391 7561 7 m 7901 8070 170 6 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 0102 0271 0440 0609 ; 0777 0946 1114 1m 1451 16Q 8 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 9 3300 3467 3635 m ft 3970 4137 4305 4472 4639 4806 167 260 4973 5140 5307 5474 5641 5808' 5974 6141 6308 6474 167 1 '6641 6807 6973 m 59 7306 7472 7638 7 m 7970 8135 166 2 8301 8467 8633 8798 8964 ! 9129 9295 9460 9625 9791 165 g 9956 s 0121 0286 0451 0616 0781 0945 1110 1275 1439 165 4 421604 1768 1933 ! m ir 2261 2426 2590 2 754 2918 3082 164 5 3246 3410 3574 I arsr 3901 4065 4228 4392 4555 4718 164 6 4882 5045 5208 53' i 55:34 ! 5697 5860 6 >.': 6186 6349 163 7 6511 6674 6836 t!9 Ml 7161 1 7324 7486 7648 7811 7973 162 8 8135 1 8297 O^^O OQ1 A 8459 8621 8783 8944 9106 9268 9429 9591 162 43 t/7AT 0075 0236 0398 0559 0720 0881 1042 1203 161 PROPORTIONAL PARTS. Diff 178 1 2 3 4 5 6 7 8 9 '.8 a5.6 53.4 71.2 89.0 106.8 124.6 142.4 ! 1150.2 177 ".7 35.4 53.1 70.8 88.5 106.2 123.9 141.6 j 15!). 3 176 .6 a5.2 52.8 70.4 88.0 105.6 123.2 140.8 158.4 175 .5 a'j.O 52.5 70.0 87.8 105.0 132.5 140.0 157.5 174 .4 34.8 52.2 89.6 87.0 104.4 121.8 139.2 156.6 173 1 .3 34.6 51.9 69.2 86.5 103.8 121.1 138.4 155.7 172 1 .2 34.4 51.6 68.8 ' 86.0 103.2 120.4 137.6 154.8 171 1 .1 34.2 51.3 68.4 85.5 102.6 119.7 136.8 153.9 170 1 .0 34.0 51.0 68.0 85.0 102.0 119.0 136.0 153.0 169 16.9 38.8 50.7 67.6 84.5 101.4 118.3 135.2 152.1 168 16.8 33.6 50.4 67.2 84.0 100.8 117.6 134.4 151.2 167 16.7. 33.4 50.1 66.8 83.5 100.2 116.9 133.6 150.3 166 16,6 33.2 49.8 66.4 83.0 99.6 116.2 132.8 149.4 165 16.5 a3.0 49.5 66.0 82.5 99.0 115.5 132 148.5 164 16.4 32.8 49.2 65.6 82.0 98.4 114,8 131.2 147.6 163 16.3 32.6 48.9 65. 2 81.5 97.8 114.1 130.4 146.7 16.2 32.4 48.5 64.8 81.0 97.2 113.4 129.6 145.8 1CJI 1J.1 32.2 48.8 64.4 .80.5 96.6 112.7 128.8 144 J HO] TABLE XXIV. SWGARITIIMS OF NUMBERS. No. 270 L. 431.] [No. 299 L. 4TO. N. * 1 2 3 4 5 6 7 8 9 Diff. 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 161 1 2969 3130 3290 3450 3610 3770 3930 4090 4249 4409 160 2 4569 4729 4888 5048 5207 5367- 5526 - 5685 5844 6004 159 3 6163 6322 6481 6640 6799 6957 7116 7275 7433 7592 159 4 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 5 9333 9491 9648 1 9806 9964 0122 0279 0437 0594 0752 1KU 6 440909 ~1066 1224~ 1381 1538 1695 1852 2009 2166 2323 1OO 157 7 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 8 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 156 9 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 $80 7158 7313 7468 7623 7778 7933 8088 8242 8397 8552 155 1 8706 8861 9015 9170 1)324 9478 9633 9787 9941 2 450249* 0403 0557 0711 0*05 1018 1172 1326 1479 1633 154 3 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 4 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 5 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 152 6 6366 6518 6670 6831 6973 7125 727'6 7428 7579 7731 152 7 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 151 g 9392 9543 9694 9845 9995 0146 0296 0447 OV)7 074ft . 9 460898 1048 1198 1348 1499 1649 1799 1948 2098 \Jt 'rlO 2248 150 290 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 150 1 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 2 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 3 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 4 | 8347 8495 8643 8790 8938 908o 9233 9380 9527 9675 148 9822 9969 ' 0116 0263 0410 0557 0704 0851 0998 1145 147 6 47T292 1438~ 1585 1732 1878 2025 2171 2318 2464 2610 146 7 27518 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 8 4216 4362 4508 4(553 4799 4944 5090 5235 5381 5526 146 5671 5816 6962 6)07 6252 6397 6542 6687 6832 6976 145 PROPORTIONAL, PARTS. Diff. 1 2 3 4 5 6 7 8 9 161 16.1 32.2 48.3 64.4 80.5 96.6 112.7 128.8 144.9 160 16.0 32.0 48.0 64.0 80.0 96.0 112.0 128.0 144.0 15!) 15.9 31.8 47.7 63.6 79.5 95.4 111.3 127.2 143.1 158 15.8 81.6 47.4 63.2 79.0 94.8 110.6 126.4 142.2 157 ' 15.7 31.4 47.1 62.8 78.5 94.3 109.9 125.6 141.8 15(1 15.6 31.2 46.8 62.4 78.0 109.2 124.8 140.4 155 15.5 31.0 46.5 62.0 77.5 v&'.o 108.5 124.0 139.5 154 15.4 30.8 46.2 61.6 ' 77.0 92.4 107.8 123.2 138.0 153 15.3 soie 45.9 C1.2 76.5 1)1.8 107.1 122.4 137.7 152 15.2 30. ! 45.6 60.8 76.0 91.2 106.4 121.6 136.8 151 15.1 30.2 45.3 60.4 75.5 90.6 105.7 120.8 135.9 150 15.0 30.0 45.0 60.0 75.0 90.0. 105.0 120.0 135.0 11!) 14.9 29.8 ' 44.7 59.6 74.5 89.4 104.3 119.2 134.1 148 14.8 29.6 44.4 59.2.. 74.0 88.8 103.6 118.4 133.2 147 14.7 29.4 44.1 58.8 73.5 88.2 102.9 117.6 132.3 146 14.6 29.2 43.8 58.4 73.0 87.6 102.2 116.8 131.4 145 14.5 29.0 43.5 58.0 72.5" 87.0 101.5 116.0 180.5 144 14.4 28.8 -43.2 57.6 72.0 86.4 100.8 115.2 129.6 143 14.3 28.6 42.9 57.2 71.5 85.8 100.1 114.4 128.7 112 14 2 28.4. 42.6 56.8 71.0 S5.2 99.4 113.6 127.8 141 14.1 28.2 42.3 56.4 70.5 XI (5 98.7 112.8 126.9 140 14.0 28.0 42.0 56.0 70.0 84.0 98.0 112.0 I 126.0 [341] TABLE XXIV. LOOATUTJTMS OF NUMBERS. No. 300 L. 477.1 [No. 339 L. 531. N. 300 1. 2 3 4 5 6 8 9 310 1 2 3 4 5 6 8 9 320 1 2 3 4 5 6 8 9 330 1 o 3 4 5 ' 6 7 8 9 1 2 9 8 9 *m. 477121 8566 7266 8711 7411 8855 7555 8999 7700 9143 7844 9287 7989 9431 8133 9575 8278 9719 8422 9863 145 144 144 143 143 142 142 141 141 140 140 139 139 139 138. 138 137 137 136 ^136* 186 135 135 134 134 133 133 133 132 132 131 131 131 130 130 129 129 129 128 128 480007 1443 2874 4300 5721 7138 8551 9958 0151 1586 3016 4442 5863 7280 8692 0294 17'29 3159 4585 6005 7421 8833 0438 1872 3302 4727 6147 7563 8974 0582 2016 3445 4869 6289 7704 9114 0725 2159 3587 5011 6430 7845 9255 0661 2062 3458 4850 6238 7621 8999 0869 2302 3730 5153 6572 7986 9396 0801 2201 3597 4989 6376 7759 9137 1012 2445 3872 5295 6714 8127 9537 1156 2588 4015 5437 6855 8269 9677 1081 2481 3876 5267 6653 8035 9412 1299 ! 2731 4157 557'9 6997 8410 9818 0099 1502 2900 4294 5683 7068 8448 9824 0239 1642 3040 4433 5822 7206 8586 9962 0380 1782 3179 4572 5960 7'344 8724 0520 1922 3319 4711 6099 8862 0941 2341 3737 5128 6515 7897 9275 1222 2621 4015 5406 6791 8173 9550 491362 2760 4155 5544 6930 8311 9687 501059 2427 3791 5150 6505 9203 510545 1883 3218 4548 5874 7196 8514 9828 0099 1470 2837 4199 5557' 6911 8260 9606 0236 1607 2973 4335 5693 7046 8395 9740 i 0374 i 1744 1 3109 4471 5828 7381 8530 9874 0511 1880 3246 4607 5964 7316 8664 0648 2017 3382 4743 6099 7451 8799 07R5 2154 4878 6234 7586 8934 0922 2291 3655 5014 6370 7721 9068 1196 2564 3927 5286 6640 7991 9337 2700 4063 5421 6776 8126 9471 0009 1349 268-1 4016 5344. 0668 7987 9303 0615 1922 3226 4526 5822 7114 8402 9687 0143 1482 2818 4149 5476 6800 8119 9434 0745 2053 3356 4656 5951 7243 8531 9815 0277 1616 2951 4282 5609 6932 8251 9566 '0876 2183 3486 4785 6081 7372 8660 9943 0411 1750 3084 4415 5741 7064 8382 9697 1007 2314 3616 4915 6210 7501 8788 0679 2017 3351 4681 6006 7328 8646 9959 0813 2151 3484 4813 6139 7460 8777 0947 2284 3617 4946 6271 7592 8909 1081 2418 3750 5079 6403 7724 9040 1215 2551 3883 5211 6535 7855 9171 0484 1792 3096 4396 5693 6985 i 8274 i 9559 0090 1400 2705 4006 5304 6598 7888 9174 0221 1530 2835 4136 54:34 6727 8016 9302 0584 0353 1661 2966 4266 5563 6856 8145 9430 521138 2444 3746 5045 6339 7630 8917 1269 2575 3876 5174 6469 7759 9045 0072 1351 530200 ! 0328 1 0456 0712 || 0840 0968 | 1096 1223 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 139 13.9 138 13.8 137 13.7 136 13.6 135 13.5 134 13.4 133 13.3 132 13.2 131 13.1 130 13.0 129 12.9 128 12.8 127 12 7 27.8 27.6 27.4 27.2 27.0 26.8 26.6 26.4 26.2 26.0 25.8 25.6 25.4 41.7 55.6 41.4 55.2 41.1 54.8 40.8 54.4 40.5 54.0 40.2 58.6 39.9 53.2 39.6 52.8 39.3 52.4 39.0 52.0 38.7 51.6 38.4 ! 51.2 38.1 50.8 69.5 69.0 68.5 . (58.0 67.5 67.0 66.5 66.0 65.5 65.0 64.5 64.0 63.5 83.4 82.8 82.2 81.6 81.0 80.4 79.8 79.2 78.6 78.0 77.4 76.8 76.2 97.3 111.2 96.6 110.4 95.9 109.6 95.2 108.8 94.5 108.0 93.8 107.2 93.1 106.4 92.4 105.6 91.7 104.8 91.0 104.0 90.3 103.2 89.6 1 102.4 88.9 1 101.6 125.1 124.2 123.3 122.4 121.5 120.6 119.7 118.8 117.9 117.0 116.1 115.2 114.3 TABLE XXIY. LOGARITHMS OF NUMBERS. No. 340 L. 531.] [No. 379 L. 579. N. 340 1 2 3 4 5 6 7.. 8 9 350 1 2 3 4 5 6 7 8 9 360 1 2 3 4 5 6 7 8 9 sro 1 2 3 4 5 6 8 9 1 2 3 4 5 6 7 8 9 Diff. 128 127 127 126 126 126 125 125 125 124 12-4 124 123 123 123 122 122 121 121 121 120 120 120 119 119 119 119 118 118 118 117 117 117 116 116 116 115 115 115 114 531479 2754 4026 5294 6558 7819 9076 1607 2882 4153 5421 6685 7945 9202 1734 3009 4280 5547 6811 8071 9327 1862 3136 4407 5674 6937 8197 9452 1990 3264 4534 5800 7063 : 8322 9578 2117 3391 4661 5927 7189 8448 9703 2245 a5is i 4787 6053 7315 8574 9829 2372 3645 4914 6180 7441 8699 9954 2500 3772 5041 6306 7567 8825 0079 1330 2576 3820 5060 6296 7529 8758 9984 2627 3899 5167 6432 7693 8951 0204 1454 2701 3944 51&3 6419 7652 1 8881 540329 1579 2825 4068 5307 6543 7775 9003 0455 1704 2950 4192 5431 6666 7898 9126 0580 1829 3074 4316 5555 6789 8021 9249 0473 1694 2911 4126 5336 6544 7748 8948 0705 1953 3199 4440 5078 6913 8144 9371 0830 2078 3323 4564 5802 7036 8267 9494 0955 2203 3447 4688 5925 7159 8389 9616 1080 2327 3571 4812 6049 7282 8512 9739 1205 2452 3096 4936 6172 7405 8635 9861 0106 1328 2547 3762 4973 6182 7387 8589 9787 550228 1450 2668 3883 5094 6303 7507 8709 9907 561101 2293 3481 4666 5848 7026 8202 9374 0351 1572 2790 4004 5215 6423 7(527 8829 0595 1816 3033 4247 5457 6664 7868 9068 0717 19:38 3155 4368 5578 6785 7988 9188 0840 2060 3276 4489 5699 6905 8108 9308 0962 2181 3398 4610 5820 7026 8228 9428 1084 2303 3519 4731 5940 7146 8349 9548 1206 2425 3640 4852 6061 7267 8469 9667 0026 1221 2412 3000 4784 5966 7144 8319 9491 0146 1340 2531 3718 4903 6084 7262 8436 9608 0265 1459 2650 3837 5021 6202 7379 8554 97'25 0385 1578 2769 3955 5139 6320 7497 8671 9842 0504 1698 ,-2887 4074 5257 0437 j 7614 8788 9959 0624 1817 3006 4192 5376 6555 7732 8905 0743 1936 3125 4311 5494 6673 7849 9023 0863 2055 3244 4429 5612 6791 7967 9140 0982 2174 3362 4548 5730 6909 8084 9257 0076 1243 2407 3568 4?'26 58HO 7032 8181 9326 0193 1359 2523 3684 4841 5996 7147 8295 9441 0309 1476 2639 3800 4957 6111 7262 8410 9555 0426 1592 2755 3915 5072 6226 7377 8525 9669 570543 1709 2872 4031 5188 6341 7492 8339 0660 1823 2988 4147 5303 6457 7607 8754 0776 1942 3104 4263 5419 6572 7722 8868 0893 2058 3220 4379 5534 6687 7836 8983 1010 2174 3336 4494 5650 6802 7951 9097 1126 2291 3452 4610 ! 5765 6917 800G 9212 PROPORTIONAL PARTS. DiflP. 1 2 3 4 5 6 7 8 9 128 12.8 127 12.7 126 12.6 125 12.5 124 12.4 123 12.3 122 12.2 121 12.1 120 12.0 119 11.9 25.6 25.4 25.2 25.0 24.8 24.6 24.4 24.2 24.0 23.8 38.4 38.1 37.8 37.5 37.2 36.9 36.6 36.3 36.0 85.7 51.2 50.8 50.4 50.0 49.6 49.2 48.8 48.4 48.0 47. 6 64.0 63.5 63.0 62.5 62.0 01.5 61.0 60.5 60.0 69.5 76.8 76.2 75.6 75.0 74.4 73.8 73.2 72.6 72.0 71.4 89.6 88.9 88.2 87.5 ' 86.8 86.1 85.4 84.7 84.0 83.3 102.4 101.6 100.8 100.0 9!). 2 98.4 97.6 96.8 96.0 95.3 115.2 114.3 113.4 112.5 111.6 110.7 109.8 108.9 108.0 107. J [343] TABLE XXIV. LOGARITHMS OF NUMBERS. No. 880. L. 579.] [No. 414 L. 617. N. 1 2 3 4 5 6 7 8 a Diff. 380 579784 9898 0012~ 0126 ~0469~ ~7l7 0241 0355 0583 | 0697 0811 1 580925 1039 1153 1267 1381 1495 1608 1722 1836 1950 2 2063 2177 2291 24X M 2518 2631 2745 2 858 2972 3085 3 3199 3312 3426 3539 3652 3765 3879 3 .m 4105 4218 4 4331 4444 4557 4670 4783 -4896 5009 5122 5235 5348 113 5 5461 5574 5686 57 ( (!) 5912 6024 6137 6 250 6362 647'5 6 6587 6700 6812 695 ;r> 7037 7149 7262 7 374 7486 75U9 7 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 112 8 8832 8944 9056 91( 8 9279 9391 9503 9 615 9726 9838 g 9950 0061 0173 Q9 u 0396 0507 0619 0730 0842 0953 390 591065 1176 1287 18 i 1510 1621 1732 1843 1955 2066 1 2177 2288 2399 25 2621 2732 2843 2 954 3064 3175 111 2 3286 3397 3508 3618 3729 3840 3950 4061 4171 4282 3 4393 4503 4614 475 >i 4834 4945 5055 5 165 5276 5386 4 5496 5606 5717 58; 5937 6047 6157 6 267 6377' 6487 5 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 110 6 7695 7805 7914 805 H 8134 8243 8353 8 402 8572 8681 7 8791 8900 9009 9119 9228 9337 9446 9556 9665 9774 8 9883 9992 0101 0210 0319 0428 0537 0646 0755 0864 109 9 "600973 1082 1191 1299 1408 1517 1625 1734 1843 1951 400 2060 2169 2277 2& * 2494 2603 2711 2 819 2928 3036 1 144 3253 3361 34( i!) 3577 3686 3794 3 902 4010 108 2 4226 4334 4442 4550 4658 4766 4874 4982 5089 5197 3 5305 5413 5521 565 >8 5736 5844 5951 6 >.-,<) 6166 G274 4 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 5 7455 75ti2 7669 7884 7991 8098 8 m 8312 8419 107 6 8526 8633 8740 8847 8954 9061 9167 9274 9381 9488 7 9594 9701 9808 99 j /\ 3A-t t\A A*? 8 6106GO 0767 0873 0979 1086 1192 1298 Uoj. 1405 U44< 1511 1617 9 1723 1829 1936 2042 2148 2254 2360 2466 2572 2G78 106 410 27B4 2890 2996 3102 3207 3313 3419 U 525 363G 3736 1 3842 3947 4053 41? ,'.) 4264 4370 4475 4 581 4686. 4792 2 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 3 5950 6055 6160 62( )5 6370 ! 6476 6581 6 isi; 6790 6895 105 4 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 PROPORTIONAL PARTS. Diff. I' 2 3 4 5 6 7 8 9 118 11.8 23.6 35.4 47.2 59.0 70.8 82.6 94.4 106.8 117 11.7 23.4 35.1 46.8 58.5 70.2 81.9 93.6 105.3 116 11.6 23.2 24.8 46.4 58.0 69.6 81.2 92.8 104.4 115 11.5 23.0 34.5 46.0 57.5 69.0 80.5 92.0 103.5 114 11.4 22.8 34.2 45.6 57.0'' 68.4 79.8 91.2 102.6 113 11.3 22.6 33.9 45.2 56.5 67.8 79.1 90.4 101 7 112 11.2 22.4 33.6 44.8 56.0 67.2 78.4 89.6 100.8 111 11.1 22.2 a3. 3 44.4 55.5 66.6 77.7 88.8 99.9 110 11.0 22.0 33.0 44.0 55.0 6G.O 77.0 88.0 99.0 109 10.9 21.8 32.7 43.6 54.5 65.4 76.3 87.2 98,1 108 10.8 21.6 32.4 43.2 54.0 64.8 75.6 86.4 97.2 107 10.7 21.4 32.1 42.8 53.5 64.2 74.9 85!6 96.3 106 10.6 21.2 31.8 42.4 53.0 63.6 74.2 84.8 95.4 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5* 84.0 94.5 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 94.5 104 10.4 20.8 31.2 41.6 52.0 62.4 72.8 83.2 93.6 [344] TABLE XXIV. LOGARITHMS OF NUMBERS. No. 415 L. 618.] [No. 459 L. 662 N' 7 9 Diff . , A/HI. 415 618048 8153 8257 8302 8466 8571 8676 8780 8884 8989 105 6 9093 9198 9302 9406 9511 9615 9719 !)! 24 9928 i flAQO n> 620136 0240 0344 0448 0552 0656 0760 0864 0968 1072 104 8 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 9 2214 2318 2421 2535 2628 2732 2835 2i 39 3042 3146 420 3249 3353 H 456 3559 3663 3766 3869 3973 4076 4179 1 4282 4385 4488 4591 4095 i 4798 4901 5( )04 5107 5210 103 2 5312 5415 5518 5621 5724 5827 5929 ' 6( )32 ! 6135 6238 3 6340 6443 0546 1 6648 6751 .6853 6956 71 )58 7161 7263 4 7300 740S 7571 1073 7775 7878 7980 8 )82 8185 H2S7 5 8389 8491 8593 QA1 ^ 8695 9*15 8797 J81< 8900 OO1 9002 9104 9206 9308 102 6 9410 9512 yoio 0021 0123 H9O4 O3r> 630428 0530 0631 0733 0835 0936 1038 1139 1241 1842 8 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 9 2457 2559 2660 2701 2862 2963 3064 3165 3206 3367 430 3468 3569 3670 3771 3872 3973 4074 1-4175 4276 4376 101 4477 4578 4679 4779 '1880 4981 5081 5 82 5283 5383 2 5484 5584 5685 5785 5SS6 5986 6087 6187 6287 6388 3 OISS 6588 6688 6789 6SS9 6989 7089 71 89 7290 7390 4 7490 7590 7690 7790 7890 i 7990 8090 8190 8290 8389 100 5 8489 8589 8689 8789 SSS8 s:83 0382 7 640481 0581 i 0680 0779 ' 0879 0978 1077 1177 i 1276 1375 8 1474 1573 1672 1771 1811 1910 2069 2 08 8867 2366 9 2405 2563 2602 2761 2860 2959 3058 3156 3255 3354 99 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 1 4439 4537 4030 4734 4882 4931 5029 5 27 5226 5324 5422 5521 5619 5717 5M5 5913 6011 6110 6208 0300 3 0401 6502 6600 6698 8796 6894 0992 7( )89 7187 7285 98 1 7383 7481 7579 7676 7774 i 7872 1969 8( )67 8165 8262 5 8360 8458 8555 8053 S7'50 8848 8945 9043 9140 9237 6 9335 9432 9530 9027 9724 || 9821 9919 ____' ! I OC m; 0113 ! 0210 7 650308 0405 0502 0599 0696 0793 0890 0987 1084 1181 8 1278 1375 1472 1569 1006 1762 1859 1< )5J 2053 2150 97 9 2240 2343 2440 2536 2633 2730 2826 2923 3019 3116 450 3213 3309 3405 3502 3598 3695 3791 3i 8 3984 4080 1 4177 4273 4369 4405 4562 4658 4754 4850 4946 5042 2 5138 5235 5331 5427 5523 6619 5715 51 310 5906 6002 96 3 6098 6194 6290 03SO 6482 6577 6673 6 r69 6864 6960 4 7056 7152 7247 7343 7438 7584 7629 1 '25 7820 7916 5 8011 8107 8202 8298 8393 8488 8584 8679 8774 8870 8965 9060 9155 9250 9346 9441 953 ' 9681 9726 9821 W 9916 ' 0011 0106 0201 0296 0391 ! 0486 0581 0676 0771 95 8 600865 O'.MH) 1055 1150 1245 1339 1434 ] .-2'.) 1623 1718 9 1813 1907 2002 18096 2191 22S6 2380 2475 2569 2663 PROPORTIONAL PARTS. Diff. 1 234 5 6 7 8 9 , 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 94.5 104 10.4 20.8 31.2 41.6 52.0 62.4 72.8 83.2 93.6 103 10.3 20.6 30.9 41.2 51.5 61.8 72.1 82.4 92.7 102 10.2 20.4 30.6 40.8 51.0 6J.2 71.4 81.6 91.8 101 10.1 20.2 30.3 40.4 50.5 60.6 70 7 80.8 90.9 100 10.0 20.0 30.0 40.0 50.0 60.0 70 80.0 90.0 99 9.9 19 8 29.7 39.6 49.5 59.4 69.3 79.2 89.1 [345] TABLE XXIV. LOGARITHMS OF -NUMBERS. No. 460 L. 662.1 [No. 499 L. 698. i NA * DifF . V i 2 3418 LJllL. 460 662758 2852 2947 3041 3135 3230 3324 3512 3607 1 3701 3795 '3889 398 18 4078 4172 4266 4360 4454 4548 2 i 4642 4736 4830 492 4 5018 5112 5206 5299 5393 5487 94 3 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 4 6518 i 6612 6705 675 6892 6986 7079 7173 7266 7360 5 7453 i 7546 7640 7733 7826 7920 8013 8106 8199 8293 6 8386 ; 8479 8572 9o03 8665 q^o 8759 968J 8852 9i82 8945 98} 6145 6236 6328 6419 ! 6511 6602 5 6694 6785 6876 i 69t ,s 7059 7151 7242 7333 7424 7516 6 7607 7698 7789 ! 7881 7972 i 8063 8154 8245 8336 8427 8518 8609 8700 : 87< )1 8882 8973 9064 9155 9246 9337 91 Q ui-i-.: fKKI Oft 1O : QTl X) 9791 9882 9973 O nj | wi/xw | irv/j-v/ | ai\ 0063 0154 no/i R 9 1 680336 0426 0517 0607 0698. 0789 0879 0970 1060 1151 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 1 2145 2235 2326 ! 2416 2506 2596 2686 2777 2867 2957 2 3047 3137 3227 33 7 3407 3497 3587 3677 3767 3857 90 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 4 4845 4935 5025 51 4 5204 5294 5383 5473 5563 5652 5 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 6 6636 6726 6815 69( 14 6994 7083 7172 7261 7351 7440 7 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 89 8 8420 8509 8598 86! J7 8776 8865 8953 9042 9131 9220 y 9309 Q.QQ 9486 !).") 5 9664 9753 9841 0090 | 0019 i Mtw 490 (190196 0285 0373 0462 0550 0639 0728 0816 0905 0993 1 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 2 1905 2053 2142 2 5(1 2318 2406 2494 2583 2671 2759 3 2847 i 2935 3023 3111 3199 3287 3375 3463 3551 3639 88 4 3727 3815 3903 391 )1 4078 4166 4254 4342 4430 4517 5 4605 4693 4731 4Sf J8 4956 5044 5131 5219 5307 5394 6 5482 55*9 5657 57' 4 5832 5919 6007 6094 6182 6269 6356 6444 6531 66] 8 6706 1 6793 6880 6968 | 7055 7142 8 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 . 9 8100 8188 8275 8362 8449 1 8535 8622 8709 8796 8883 87 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 678 9 98 9.8 19.6 29.4 39.2 49.0 58.8 68.6 78.4 1 88 2 97 9.7 19.4 29-1 38.8 48.5 58.2 67.9 77.6 87.3 96 9.6 19.2 28.8 38.4 48.0 57.6 67.2 76.8 86.4 95 9.5 1 19.0 28.5 38.0 47.5 57.0 66.5 76.0 85.5 94 9.4 18.8 28. 2 ' 37.6 47.0 56.4 65.8 75.2 84.6 93 9.3 18.6 27.9 37.2 46.5 55.8 65.1 74.4 83.7 92 9.2 18.4 27.6 36.8.' 46.0 55.2 64.4 73.6 82.8 91 9.1 18.2 27.3 36.4 45.5 54.6 63.7 72.8 81.9 90 9.0 ; 18.0 27.0 36.0 45.0 54.0 63.0 72.0 81.0 89 8.9 17.8 I 26.7 35 6 44.5 53.4 62.3 71.2 80.1 88 1 8.8 17.6 i 26.4 35^2 44.0 52.8 61.6 70.4 79.2 87 8.7 17.4 i 26.1 34.8 ! 43.5 52.2 60.9 ! 69.6 78. 3 86 ! 8.6 17.2 25.8 34.4 I 43.0 51.6 60.2 1 68.8 77.4 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 500 L. 698.] [No. 544 L. 736. N. 6 7 8 9 Diff. ! 500 1 9 3 4 5 6 7 8 9 510 1 2 3 4 5 6 8 520 I I I 7 5 1 3 i 6 7 8 9 540 1 2 3 4 698970 9057 9838 9924 9144 9281 9317 9404 9491 9578 9664 9751 0011 0877 1741 2603 3403 4322 5179 ooa5 6888 7740 8591 9440 0098 0963 1827 2689 3549 4408 5265 6120 0974 7826 8076 9524 0184 1050 1913 2775 3685 4494 5350 62G6 7059 7911 87'01 9609 0271 1136 1999 2801 3721 4579 5436 6291 7144 7996 8846 9694 0358 1222 2086 2947 3807 4665 5522 6376 7229 8081 8931 9779 0444 1309 2172 3033 3893 4751 5607 6462 7315 8166 9015 9863 0531 1395 2258 3119 3979 4837 5693 6547 7400 8251 9100 9948 0617 1482 2344 3205 4065 4922 5778 6632 7485 8336 9185 86 85 84 83 82 81 80 700704 1568 2431 3291 4151 5008 5864 6718 7570 8421 9270 0790 1654 j 2517 3377 4236 5094 5949 6803 7655 8506 9355 0033 0879 1723 2566 3407 4246 5084 5920 6754 7587 8419 9248 710117 0963 1807 2650 3491 4330 5167 6003 . 6838 7671 8502 9881 0202 1048 1892 2734 3575 4414 5251 6087 6921 7T> 1 8585 9414 0287 1132 1976 2818 3659 4497 5335 6170 7004 7837 8068 9497 0371 1217 2060 2902 3742 4581 5418 6254 7088 7990 8751 9580 0450 1301 2144 2986 3826 4665 5502 6337 7171 I 8003 8834 i 9663 0540 1885 2229 3070 3910 4749 5586 6421 7254 8086 8917 9745 0625 1470 2313 3154 3994 4833 5009 6504 7338 8169 9000 9828 0710 1554 2397 3238 4078 4916 5753 6588 7421 8253 9083 9911 0794 1639 2481 3323 4162 5000 5836 6671 7504 8336 9165 999-1 0077 0903 1728 2552 3374 4194 5013 5830 6646 7460 8273 9084 9893 720159 0986 1811 2634 3456 4276 5095 5912 6727 7541 8354 9165 9974 0242 1068 1893 2716 35:38 4358 5176 5993 6809 7623 8435 9240 0325 1151 1875 2798 3620 4440 5258 6075 ess*? 7704 8516 9327 0407 12:33 2058 2881 3702 4522 5840 oir.o 6972 7785 8597 1)108 0490 1316 2140 2963 8784 4604 ! 5422 623S 71)53 ' 7'800 8678 9489 0573 1398 2222 3045 3866 4685 5503 6320 71:34 7 >( .M8 8759 9570 0655 1481 2305 8127 3948 4767 5585 0401 7216 SI >>'. 8841 9651 0738 1563 2:387 3209 4030 4849 0007' 6483 7297 8110 8922 9732 0821 1640 2409 3291 4112 49ol 5748 0504 7379 8191 9003 9813 0055 ' 0863 1669 2474 3278 4079 4880 5679 0136 0944 1750 2555 8858 4160 4960 5759 0217 1024 1830 2635 3438 4240 5040 5838 0298 1105 1911 2715 3518 i 4320 5120 5918 0378 1186 1991 2796 3598 4400 5200 5998 0459 1266 2072 2876 30.9 4480 5279 6078 0540 i:347 2152 2956 3759 4560 5359 6157 0621 1428 2233 3037 3839 4640 5439 6237 0702 1508 2313 3117 3919 4720 5519 6317 730782 1589 2394 3197 3999 4800 5599 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 87- ' 8.7 86 8.6 a5 8.5 84 8.4 17.4 17.2 17.0 16.8 86.1 25.8 25.5 25.2 34.8 |:!.n 34.4 Ui.ii 34.0 i n.5 33.6 i 42.0 52.2 60.9 51.6 60.2 51.0 59.5 50.4 58.8 69.6 78.3 68.8 77 4 68.0 7'O.r, 67.2 I 75.6 [347] TABLE XXIV. LOGARITHMS OF NUMBERS. No. 545 L. 736.] LNo. 584 L. 767. ' N. 1 2 8 4 6 C 7 8 9 I Diff. 545 736397 6476 6556 6635 6715 6795 6874 6954 7034 7113 6 7193 i 7272 7352 7431 7511 7590 7670 '749 7829 ! 7908 7 7987 8067 ; 8146 82 1 26 8305 ! 8384 8463 ! 5543 862 2 8701 8 8781 8860 8939 90 IS 9097 9177 9256 J 335 941 4 9493 9 9572 9651 9731 98 10 9889 9968 0047 ( >10 fion K r/f\ 550 740363 0442 0521 0600 0678 0757 0836 \jiAi\j 0915 U/I&Utj 0994 1073 (y 1 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 2 1939 2018 2096 21 5 2254 2332 2411 s !489 256 8 2647 3 2725 2804 2882 29 11 3039 3118 3196 275 335 i 3431 4 3510 3588 3667 3745 3823 3902 3980 4058 4136 4215 5 4293 4371 4449 45 JS 4606 4684 4762 4 840 491 9 4997 6 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 7 5855 5933 6011 60) 39 6167 6245 6323 401 647 9 6556 8 6634 6712 6790 ' 681 38 6945 7023 7101 7179 7256 1 7334 9 7412 7489 7567 7645 7722 7800 j 7878 7955 | 8033 8110 560 8188 8266 8343 8421 8498 8576 ! 8653 8731 ! 8808 8885 1 8963 9040 9118 91 5 9272 ! 9350 9427 504 958 3 2 9736 9814 9891 !!) 38 nruf; ~~^ 7^ 3 750508 0586 0663 0740 0817 i 0894 0971 1048 ucw* u-toi 1125 : 1202 4 1279 1356 1433 1510 1587 16U4 1741 1818 1895 1972 : , 5 2048 2125 2202 22 9596 ')( )('>*! 9741 9813 9SS5 9957' v 0029 0101 01 7^ O'M^ 3 rsosi 1 ; Q389 0461 0533 0605 0677 074!) UlvJJl 0821 Ul 10 0893 U.-34O 0965 72 4 1087 1109 1181 1253 1324 1390 1468 1540 1612 1084 5 1755 1887 1S99 1971 8042 2114 2186 2258 2329 2401 6 2473 25 14 2010 26S8 2759 2831 290-2 2974 3046 3117 3189 32(50 3332 3403 :)475 3546 3618 3689 3761 3832 8 3901 3975 4046 4118 41S9 42(51 4332 4403 4475 4546 9 4017 4689 4700 4831 4902 4974 5045 5116 5187 5259 610 5330 5401 5472 5543 5615 5686 5757 5828 5899 5970 1 0041 0112 61 S3 0251 6325 6396 6407 6538 6609 6UHO 71 2 6751 6822 6893 (59(51 7035 7106 7177 7248 7319 7390 8 7400 7531 700-2 7073 7744 7S15 7885 7956 8087 8098 1 8108 8239 8310 8381 8151 8522 8593 8663 8734 8804 5 ! 8875 S940 9016 9087 9157 9228 9299 9309 9440 9510 (5 9581 9051 9722 9792 9S03 9933 0004 nnr ' (MAA 7 790285 0356 0426 0496 ! 0507 0637 0707 V7VI * 0778 0848 0918 S 0988 1059 11-29 1199 1209 1340 1410 1480 1550 1620 9 1691 1701 1831 1901 1971 2041 2111 2181 2252 2822 620 2392 2102 2532 2602 2(572 27'42 2812 2882 2952 3022 70 1 3092 3162 3231 3301 3371 3441 3511 3581 3651 3721 2 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 a 1488 1558 4627 4097 4707 1S36 1900 4976 5045 5115 4 5185 5254 5324 5:i!t:5 5 103 5532 5002 5072 5741 5S11 5 ! 5880 5949 6019 j 6088 6158 6227 6297 (5306 6436 6505 j 6 ! (5574' 0044 6713 i 6782 6S52 0921 6990 7060 7129 7198 7 7268 7337 740(5 , 7475 7545 7014 7683 7752 7821 7890 8 7960 8029 8098 8167 8236 s:; s.r t 8443 8513 8582 9 ! 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 1! i PROPORTIONAL PARTS. tt 1 2 3 4 5 6 7 8 9 75 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 7.4 14.8 22.2 29.6, 37.0 44.4 51.8 59.2 66.6 7.3 14.6 21.9 29.2 36.5 48.8 51 .1 58.4 65.7 72 7 2 14.4 21.6 28.8 36.0 43.2 5( 1.4 57.6 64.8 71 7.1 14.2 21.3. 28.4 x .35.5 42.6 49.7 56.8 63.9 70 7.0 14.0 21.0 28.0 35.0 42.0 4< .0 56.0 63.0 6.9 13.8 20.7 27.6 34.5 41.4 48.3 55.2 62.1 [349] TABLE XXIV. LOGARITHMS OF NUMBERS. 1 I No. 630 L. 799.] . [No. 674 L. 829. N. 1 2 3 4 5 6 7 8 9 ! Diff. 1 630 799341 9409 9478 9547 9616 9085 9754 9823 9892 9961 1 800029 0098 0167 0236 0305 ! 0373 0442 0511 0580 0648 2 0717 0786 0854 0923 0992 1 1061 1129 1198 1266 1335 i 3 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 ! 4 2089 2158 2220 2295 2363 2432 2500 2568 2637 2705 i 5 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 6 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 1 7 4139 4208 4270 4344 4412 4480 4548 4616 4685 4753 8 4821 4889 4957 5025 5093 5161 5229 5297 5365 54.33 68 9 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 640 806180 6248 6316 6384 6451 6519 6587 6655 6723 6790 1 6858 6926 6994 7061 7129 7197 7264 7.332 7400 7467 2 75:i5 7603 7670 7738 7806 I 7873 7941 8008 8076 8143 3 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 5 9560 9627 9694 9762 9829 ! 9896 9964 Ofiftl OOQ8 6 810233 0300 0367 0434 0501 0569 0636 UUOl 0703 uuyo 0770 088? 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 8 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 9 2245 2312 2379 2445 2512 2579 2046 2713 2780 2847 650 2913 2980 3047 3114 3181 3247 3314 3381 3448 3514 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 2 4248 4314 4381 1 4447 4514 4581 ! 4647 4714 4780 4847 3 4913 4980 5046 ! 5113 5179 5246 5312 5378 5445 5511 ! 4 5578 5644 ! 5711 5777 5843 5910 5976 6042 1 6109 6175 i 5 6241 6308 ! 6374 ' 6440 6506 i 6573 6639 0705 6771 6838 6 6904 6970 703(5 7102 7169 7235 7301 7367 7433 7499 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 8 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 ! Rr 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 ; 660 9544 9610 9676 9741 9807 9873 9939 0004 0070 01 ^fi 1 820201 0267 0333 0399 0464 0530 0595 UlAH; 0061 uu*u 0727 UloO 0792 o 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 3 3614 1579 1645 1710 1775 1841 1906 1972 2037 2103 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 7 4126 4191 4256 4321 4386 4451 4516 4581 4040 4711 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 60 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 2 7369 7434 7499 7563 7628 7692* 7757 7821 7886 7951 3 8015 8080 8144 8209 8273 8338 1 8402 8467 8531 ; 8595 4 8660 8724 8789 8853 8918 8982 1 9046 9111 9175 i 9239 1 PROPORTIONAL PARTS. Diff. 1 234 5 678 9 68 6.8 13.6 20.4 27.2 34.0 40.8 47.6 54.4 61.2 67 6.7 13.4 20.1 26.8 33.5 40.2 46.9 53.6 00.3 66 6.6 13.2 19.8 26.4 33.0 39.6 46.2 52.8 59.4 65 I,. 5 13.0 19.5 26.0 32.5 39.0 45.5 52.0 58.5 64 (5.4 IS. 8 19.2 25.6 32.0 38.4 44.8 51.2 57.6 [350] TABLE XXIV. LOGARITHMS OF NUMBERS. No. 675 L. 829.] [No. 719 L. 857. N. 1 1 * 1 4 6 7 8 * Diff- 675 829304 9368 9432 9497 9561 9625 9690 9754 9818 9882 Q 9947 0011 0075 0139 0204 0268 naap> 0396 0460 0525 7 830589 0663 0717 0781 0845 : 0909 0973 1037 1102 1166 8 1230 1294 1358 1422 1486 1550 1614 1678 1742 1806 64 9 1870 1934 1998 2062 2126 2189 2253 2317 2381 2145 380 2509 2573 2637 2700 2764 2828 2892 2956 3020 8088 1 3147 3211 3275 333 8 3402 3466 3530 3593 3C.5 3721 1 2 3784 3848 3912 397 5 4039 4103 4166 4230 429^ 1 4357 i 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 ; 4 5056 5120 5183 524 7 5310 5373 5437 5500 556 i 5627 i 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 j 6 6324 6387 6451 651 4 6577 6641 6704 6767 683 ) 6894 i 6957 7020 7083 714 6 7210 7273 7336 7399 746 > 7525 j 8 7588 7652 7715 7778 7841 7904 7967 ! 8030 8093 8156 9 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 8849 891S 8975 fU?A4 8088 9101 9164 9227 9289 9352 OOC1 9415 1 94/8 9541 juv* 966 1 9<29 9<92 18 ; oav 0043 2 840106 0169 , 0232 0294 0357 0420 0482 0545 0608 0671 3 0733 0796 0859 0921 0984 1046 1109 1172 12:34 1297 4 1359 1422 1485 154 7 1610 1672 1735 1797 186 ) 1922 5 1985 2047 2110 2172 2235 2297 2360 2422 26^V 6 2609 2672 2734 S79 6 Ogg 2821 2983 3046 310! 3 3170 7 3233 3295 8357 3420 5482 3544 3606 3669 3731 3793 8 3855 3S18 S980 4042 4104 4166 4229 4291 4353 4415 9 4477 4539 4801 4664 4726 4788 4850 4912 4974 5036 700 5098 5160 5222 5284 5346 5408 5470 5533 o594 562G 62 1 5718 5780 5842 590 4 5966 6028 6090 6151 621, 3 fc275 6399 6461 3523 6585 6646 6708 6770 683, I 6894 3 6955 7017 r cG?9 1 714 1 7202 7264 7326 7388 744 ) 7511 4 7573 7634 7o96 7?58 7819 7881 7943 8004 ! 8066 8128 5 8189 8251 8312 837 4 8435 8497 8559 8620 ! 868 2 8743 6 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 7 9419 9481 9542 9604 9665 5726 9788 9849 9911 9972 8 ! ssooas 0095 0155 ' 0217 0279 0340 0401 0462 0524 0585 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 710 1258 132.0 1381 1442 1503 1564 1625 1686 1747 1809 1 1870 1931 1992 205 3 2114 2175 2236 2297 235! i 2419 2 2480 2541 2603 2663 2724 2785 2846 2907 296! 3 3029 61 3 3090 3150 3211 327 3333 3394 3455 3516 357' r 3037 4 3698 3759 3820 388 1 3941 4002 4063 4124 418. 5 4245 5 ! 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 6 4913 4974 5034 509 ~) 5156 5216 5277 5337 539* J 5459 5519 5580 5640 570 1 5761 5822 .: 's-j 5943 600. J 6064 8 6124 61&5 6245 6:306 ! 6366 6427 6487 6548 6608 (J668 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 PROPOBTIONAL PABTS. Difi. 1 2 3 4 5 6 7 8 9 65 6.5 13.0 19.5 26.0 32.5 39.0 45.5 52.0 58.5 64 6.4 12.8 19.2 25.6 32.0 38.4 44.8 51.2 57.6 63 6.3 12.6 18.9 25.2 31.5 37.8 44.1 50.4 56.7 62 6.2 12.4 18.6 24.8 31.0 37.2 43.4 49.6 55.8 61 6.1 12.2 18.3 24.4 30.5 86.6 42.7 48.8 54.9 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 [351] fABLE XXIV. LOGAKITHMS OF NUMBERS. No. 720 L. 857.] [No. 764 L. 883. T\lff . Din. 720 857332 7393 7453 7513 7574 7634 7694 7755 7815 7875 1 7935 7995 8056 8116 8176 8236 8297 8887 8417 8477 2 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 3 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 A 9739 9799 ' 9859 9918 9978 ! ' ( yqs 0098 0158 rtoie noco 5 860338 0398 0458 0518 0578 0637 0697 0757 0817 \J6iO 0877 6 0937 0996 1056 1116 1176 1.236 1295 1355 1415 1475 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2608 9 2728 2787 2847 2906 2966 3025 3085 3114 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3858 1 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 3 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 5 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 6 6878 6937 6996 7055 ! 7114 7173 7232 7291 7350 7409 59 7 7467 7526 7585 7644 j 7703 7762 7821 7880 7939 7998 8 8056 8115 8174 8233 8292 8350 8-109 8468 8527 8586 9 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 1 9818 9877 9935 9994 0053 01 1 1 0170 022ft Oi>87 Otaat 2 870404 0462 0521 0579 0638 U1II 0696 Ul t (J 0755 \j>4Xj 0813 LKo< 0872 Uo4O 0930 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 4 1573 1631 1690 1748 1806 1865' 1923 1981 2040 2068 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 6 2739 2797 2855 2913 2972 3030 3088 8146 3204 3262 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 8 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 9 4482 4540 4598 4656 4714 ! 4772 4830 4888 4945 5003 750 5061 5119 5177 5235 5293 5351 5409 5466 5524 5582 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 3 6795 6853 6910 6968 7026 | 7083 7141 7199 7256 7314 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 5 7947 8004 8062 '8119 8177 8234 8292 8349 8407 8464 6 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 7 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 g 9669 9726 9784 9841 9898 ; 9956 001^ 0070 0127 fi1 ti 9 880242 0299 0356 0413 0471 , 0528 UUlO 0585 \J\Ji U 0642 Ul/ci 0699 UloO 0756 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 1 1385 144:2 1499 1556 1613 1670 1727 1784 1841 1898 2 1955 2012 2069 2126 2183 i 2240 2297 2354 2411 2468 57 3 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 PROPORTIONAL PARTS. Diff 1 2 3 4 5 678 9 59 5.9 11.8 17.7 23.6 29.5 35.4 41.3 47.2 53.1 58 5.8 11.6 17.4 23.2 29.0 34.8 40.6 46.4 52 2 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 56 5.6 11.9 16.8 22.4 28.0 33.6 39.2 44.8 50.4 [352! TABLE XXTV. LOGARITHMS OF NUMBERS. No. 765 L. 883.] [No. 809 L. 908. H.| 1 2 3 4 5 6 7 8 9 Diff. 765 883661 3718 3775 3832 3888 3945 4002 4059 4115 4172 6 42-29 4285 4342 4399 4455 4512 4569 4625 4682 4739 7 4?95 4852 4909 4965 5022 5078 5135 5192 5248 5305 B 53(il 5418 5474 5531 5587 5644 5700 5757 5813 5870 9 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 770 6491 6547' 6604 6600 6716 6773 6829 6885 6942 6998 1 7054 7111 7167 7223 7280 7336 7392 7449 7505 7561 .> 7617 7(174 7730 7786 7842 7898 7955 8011 8067 8123 3 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 5 9302 9:358 9414 9470 9526 9582 9638 9694 9750 9806 56 m 9862 9918 9974 0030 0086 0141 0197 0253 0309 0365 7 890421 0477 0533 0589 0645 0700 0756 0812 0868 0924 8 09HO 1035 1091 1147 1203 1259 1314 1370 1426 1482 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2595 1 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 2 3207 3262 3318 3373 3439 3484 3540 3595 3651 3706 3 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 4 4316 4371 4427 4482 4538 . 4593 4648 4704 4759 4814 5 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 6 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 7 5975 6030 (IDS.-, 6140 6195 6251 6306 6361 6416 6471 8 052(i 6581 6636 8692 6747 6802 0857 6912 6967 7022 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 7627 7682 7737 7792 7847 7902 7957 8012 8067 8122 1 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 2 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 3 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 4 9821 9875 9930 9985 0039 0094 0149 003 0258 0312 5 900367 0422 0476 053} 0586 0640 0695 0749 0804 0859 0918 0968 1022 1077 1131 1186 1240 1295 1349 1404 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 8 2003 2057 2112 8106 2221 2275 2329 2384 2438 2492 9 2547 2601 2655 2710 2764 2818 2873 2987 2981 3036 800 3090 3144 3199 S253 3307 asei 3416 3470 3524 3578 1 3633 8687 3741 3795 3849 3904 3958 4012 4066 4120 2 4174 I229 4283 4337 4:i!l 4445 4499 4553 4607 4661 3 4716 4770 4824 4878 4932 4986 5040 5094 5148 ! 5202 KA 4 5256 5310 5364 5418 5472 5526 5580 5(534 5688 5742 O4 5 57% 588 1 5904 5968 6013 6066 6119 6173 6227 6281 6 6885 (>38 0443 6497 6561 6604 6658 6712 6766 6820 7 6874 6987 6981 7035 70S!) 7143 719J 7250 7304 7358 8 7411 7465 7619 7573 7686 7680 7734 7787 7841 7895 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 PROPORTIONAL PARTS. Diff. 1 234 5 678 9 57 :> 7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 56 :>.(; 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 .", .-)..-> ll.i) Ni.5 ->-'.<) 27.5 33.0 38.5 44.0 49.5 54 5.4 10.8 , Hi. 2 21.6 27.0 32.4 37.8 43.2 48.6 [35.0 TABLE XXTV. T,OGAT!TTirMS OP NTBfBEHS. No. 810 L. 908.] [No. 854 L. 931. N. 1 2 3 * " * 6 7 8 9 Diff. 810 908485 8539 8592 864 i 8699 , 8753 8807 88 50 8914 8967 1 9021 9074 9128 918 9235 9289 9342 <>:; >t; 9449 9503 2 9556 9610 9663 971 j 9770 I 9823 9877 9S -50 9981 0037 3 910091 0144 0197 0251 0304 0358 0411 0464 0518 i 0571 4 0624 0678 0731 0784 0838 0891 0944 0998 1051 [ 1104 5 1158 1211 1264 131 r 1371 i 1424 1477 15 30 1584 ] 1637 6 1690 1743 1797 1851 D 1903 | 1956 2009 ; 20 63 2116 2169 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 8 2753 2806 2859 291 ? 2966 3019 3072 i 31 25 3178 3231 9 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 820 3814 3867 3920 3973 4026 4079 4132 4184 4237 4290 1 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 2 4872 4925 4977 503 i 5083 i 5136 5189 K 41 5294 5347 . 3 5400 5453 5505 555 8 5611 |; 5664 5716 57 BO 5822 5875 4 5927 5980 6033 6085 6138 ' 6191 6243 6296 6349- 6401 5 6454 6507 6559 661 2 6664 i 6717 6770 ie 22 6875 6927 6 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 7 7506 7558 7611 766 3 7716 7768 7820 7 33 7925 7978 8 8030 8083 8135 818 8 8240 ; 8293 8345 97 8450 8502 9 8555 8607 8659 8712 8764 ! 8816 8869 8921 8973 9026 1 &30 9078 9130 9183 9235 9287 ' 9340 9392 9444 9496 9549 -j 9601 9653 9706 975 g 9810 9862 9914 (W 0019 0071 2 920123 0176 0228 0280 0332 0384 0436 1 0489 0541 0593 3 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 4 1166 1218 1270 132 2 1374 1426 1478 1. ; ;)0 ! 1582 1634 "* 5 1686 1738 1790 184 2 1894 1946 1998 2( i:,0 2102 2154 6 2206 2258 2310 2362 2414 2466 2518 2570 ; 2622 2674 7 2725 2777 2829 28 1 2933 1 2985 3037 3( IS9 3140 3192 8 3244 3296 3348 399 3451 || 3503 3555 3607 3658 3710 9 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 1 4796 4848 4899 49f 1 5003 5054 5106 5 57 5209 5261 o 5312 5364 t415 5467 5518 5570 5621 5673 5725 5776 3 5828 5879 5931 59 2 6034 6085 6137 6 18S ! 6240 6291 4 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 5 6857 6908 6959 701 1 7062 7114 7165 ft jit; 7268 7319 6 7370 7422 7473 75$ 4 7576 7627 7678 7 30 7781 7832 7 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 8 8396 8447 8498 854 B 8601 8652 8703 8 T>4 8805 8857 9 8908 8959 9010 9061 9112 9163 9215 9266 ; 9317 9368 850 9419 9470 9521 9572 9623 9674 97'25 9776 9827 9879 51 J 0032 0083 0134 0185 0236- 0287 0338 0389 o 930440 0491 0542 0592 0(543 0694 0745 0796 0847 i 0898 3 0949 1000 1051 111 B 1153 1204 1254 1 *>: 1356 1407 4 1458 1509 1560 1610 1661 .1712 1763 1814 1865 1915 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 53 5.3 10.6 15.9 21.2 26.5 31.8 37.1 42.4 47.7 52 5.2 10.4 15.6 20.8 26.0 31.2 3 3.4 41.6 46.8 51 5.1 10.2 15.3 20.4 25.5 30.6 35.7 40.8 45.9 50 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 [354] TABLE XXIV. - LOGAKITHMS OF NUMBERS. N T o. 855 L. 931.1 *, [No. 899 1,. 954. 1 . Ng\ | 9 r; 6075 6121 6167 6212 625 8 6304 7 6350 6396 1 6442 6488 6533 6579 6625 6671 6717 6763 8 6808 6854 , 6900 69 46 6992 7037 7083 7129 717 5 7220 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 950 7724 1 7769 7815 7861 7906 7952 7998 8043 8089 8135 1 8181 8226 ; 8272 8317 8363 8409 8454 8500 8546 8591 2 8637 8683 i 8728 87 74 8819 8865 8911 8956 900 2 9047 3 9093 9138 ; 9184 9230 9275 9321 9366 9412 9457 9503 4 9548 9594 9639 9685 9730 ! 9770 9821 9867 9912 9958 5 980003 : 0049 0094 0140 0185" 0231 0276 0322 0367 0412 6 0458 ! 0503 0549 0594 0640 oea5 0730 0776 0821 0867 7 0912 i 0957 1003 10 18 1093 1139 < 1184 1229 127 s 1320 8 1366 1411 1456 15! 31 1547 1592 1637 1683 172 8 1773 9 1819 1864 : 1909 1951 2000 2045 2090 2135 2181 2226 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 1 2723 2769 2814 28 59 2904 2949 2994 3040 308 5 3130 2 3175 3220 3265 33 10 3356 3401 3446 3491 353 6 3581 3 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 4 4077 4122 4167 42 12 4257 4302 4347 4392 443 7 4482 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 6 4977 5022 5067 51 12 5157 5202 5247 5292 533 7 5382 7 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 8 5875 5920 5965 60 10 6055 6100 6144 6189 623 4 6279 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6951 6996 7040 7085 7130 7175 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 2 7666 7711 7756 78( X) 7845 7890 7934 7979 802 4 8068 3 8113 8157 8202 82- 17 8291 8336 8381 8425 8470 8514 4 8559 8604 8648 86 )3 8737 8782 8826 8871 891 8 8960 5 9005 9049 9094 91 38 9183 9227 927'2 316 9361 9405 6 7 9450 QKQX 9494 9539 9939 9983 9583 9628 9672 9717 9761 9806 9850 005^ > 0072 0117 0161 0206 0250 0294 8 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 980 1226 1270 1315 1359 1403 1448 1492 1536 1580 1625 1 1669 1713 1758 18( 12 1846 1890 1935 1979 202 i 2067 2 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 3 2554 2598 2642 26* M 2730 2774 2819 2863 290 7 2951 4 2995 3039 3083 3127 3172 3216 3200 3304 3348 3392 5 3436 3480 3524 35( (8 3613 3657 3701 3745 378 ) 3833 6 3877 3921 3965 40( )9 4053 i 4097 4141 4185 422 > 4273 4317 4361 4405 4449. 4493 4537 45P 1 4625 4669 4713 44 8 4757 4801 4845 48* ' 4977 5021 5065 510 S 5152 9 5196 5240 5284 5328 5372 5416 5460 5504 554 5591 PROPORTIONAL PARTS. Diflf i 2 3 13.8 4 18.4 5 23.0 6 27.6 7 8 9 46 4.6 9.2 32.2 36.8 41.4 45 4.5 9.0 13.5 18.0 22.5 27.0 31. 5 36.0 40.5 44 4.4 8.8 13.2 17.6 22!0 26.4 30.8 35.2 39.6 43 4.3 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 TABLE XXTV. LOGARITHMS OP NUMBERS. No. 990 L. 995.] [No. 999 L. 999. N. 1 2 3 4 5 6 7 8 9 Diff. 990 995635 6074 5679 6117 5723 5767 6205 5811 6249 5854 6293 5898 6337 5942 6380 5986 6424 6030 6468 44 2 6512 6555 6599 6643 6687 6' '31 6774 6818 6862 6906 3 6949 6993 7037 7080 7124 T L6S 7212 725, 5 7299 7343 4 7386 7430 7474 7517 7561 7 5or> 7648 769 1 > 7736 7779 5 7823 7867 7910 7964 7998 8041 8085 8129 8172 8216 6 8259 8303 8347 8390 8434 & 177 8521 856' 1 8608 8652 7 8fi95 8739 8782 8826 8869 S >i:j 8956 9001 ) 9043 9087 8 9131 9174 9218 9261 9305 9 54H 9392 9435 9479 9522 9 9565 9609 9652 9696 9739 9783 9826 9870 991S 9957 43 LOGARITHMS OF NUMBERS FROM 1 TO 100. N. Log. N. Log. N. Log. N. Log. N. Log. 1 0.000000 21 1.322219 41 1.612784 61 1.785330 81 1. 908485 2 0.3010:30 22 1.34 2423 42 1.6; 2324 1 62 1.7 ',) 3!W 82 1. 913814 3 0.477121 23 1.8f 1728 43 1.633468 63 1.799341 83 1. 919078 4 0.602060 24 l. S0211 44 l.fr i:u:> 5 64 1.8 in 180 84 1. 924879 5 0.698970 25 1.397940 : 45 1.653213 65 1.812913 85 1. 929419 6 0.778151 26 1.414973 ! 46 1.662758 66 1.819544 86 1.934438 0.845098 27 1.431364 47 1.672098 67 1.826075 87 1.939519 8 0.903090 28 1.44 7158 48 1.6 >l-,'t 1 68 l.f :',-, '509 88 1. 944483 0.954243 29 1.462398 i 49 1.690196 69 l. ft 5849 89 1. 9-19390 10 1.000000 30 1.4" 7121 50 1.6 sir; ) 70 1.845098 90 1.954243 11 1.041393 31 1.491362 51 1.707570 71 1.851258 91 1. 959041 12 1 .079181 32 LJ5C >5150 52 1.7 800 J > 72 1> .V 332 92 1. 963788 '13 .113943 33 1.51&514 53 .724276 73 1.86.3323 93 1. 968483 14 .146128 34 1.5f 11479 54 ! 7 W3-.I 4 74 t:t ti! 232 94 1. 97'3128 15 .176091 35 1.544068 55 '. .7 Wtt<5 75 1.875061 95 1. 977724 16 .204120 36 1.556303 56 .748188 76 1.880814 96 1. 982271 17 .230449 37 1.568202 ; 57 .755875 77 1.886491 97 1. 986772 18 .255273 38 1.5' '9784 58 .7 Wtt 3 ! 78 i.e 11- !0!5 98 1. 991226 19 .278754 39 1.591065 59 .770852 79 1.897627 99 1. 995635 20 .301030 40 1.602060 60 .778151 80 1.908090 ,00 2.000000 Value at 0. Sign in 1st Quad. Valu< at 90 3 in2d Quad. Value at 180. Sign in 3d Quad. Value at 270 Sign in 4th Quad. Value at 360. Sin o R -f o R O Tan __ 00 _!_ 00 _ O Sec __ 00 R _ 00 -. R Versin O __ R - - 2 R _j_ R _L Cos .. . R o I I _ o t R Cot 00 __ a 3 _l_ _ 00 Cosec 00 -- R n - 00 R 00 R signifies equal to rad; oo signifies infinite ; O signifies evanescent. [358] TABI.K XXV. LOGARITHMIC STNE8. 179 ' Sine. 9f-l Tang. Cotang. q + l'Vr Cosine. / 4.685 15.314 1 Inf. neg. 575 |575 Inf. neg. } Inf. pos. 425 ten 1 60 60 1 6.463726 575 575 6.463726 13.536274 425 ten | 59 120 2 .764756 575 575 .764756 .235244 425 ten 58 180 3 6.940847 575 575 6.940847 13.059153 425 ten 57 240 4 7.0(55786 575 575 7.065786 12.934214 425 ten 56 300 5 .162696 575 575 .162696 " .837304 425 no ten 55 360 6 .241877 575 575 .241878 .758122 425 .02 9.999999 54 420 .308824 575 575 .308825 I .691175 425 .00 .999999 58 480 8 .306816 574 576 .366817 .633188 424 .00 .999999 52 540 9 .417968 574 570 .417970 .582060 424 .00 .91)9999 51 600 10 .463726 574 576 .468737 .536273 424 .02 .999998 50 660 11 7.505118 574 J576 7.505120 12.494880 424 .00 9.999998 4K TOO 12 .542906 574 577 .542909 .457091 423 .02 .999997 48 780 13 .577668 574 577 .577672 .422^328 423 .00 .999997 47 840 14 .609853 574 577 .609857 .390143 423 .02 .999996 4(5 900 15 .639816 573 578 .639820 .360180 422 .00 .999996 45 960 16 .667845 573 578 .667849 .332151 422 .02 .999995 44 1020 17 .694173 573 578 .694179 .305821 422 .00 .999995 43 1080 18 .718997 573 579 .719003 .280997 421 .02 .999994 42 1140 19 .742478 573 579 .742484 .257516 421 .02 .999993 41 1200 20 .764754 572 ,580 .764761 .235239 420 - w .999993 40 1260 21 7.785943 572 580 7.785951 12.214049 420 -g 9.999992 39 1320 22 .806146 572 '581 .806155 .193845 419 -p* .999991 38 1380 23 .825451 572 581 .825466 .174.5-40 419 -* .999990 37 1440 24 .843934 571 582 .843944 1 .156056 418 ,\xs .999989 36 1500 25 .861662 571 583 .861674 .138326 417 \ .00 .999989 35 1560 26 .878695 571 583 .878708 ! t .121292 417 .02 .999988 34 1620 27 .895085 570 584 .895099 .104901 416 .02 .999987 33 1680 28 .910879 570 584 .91081)1 .089106 416 .02 .999986 32 1740 29 .926119 570 585 .926134 .073866 415 .02 .999985 31 1800 30 .940842 569 586 .940858 .059142 414 .03 .999983 30 1860 31 7.955082 569 687 7.955100 12.044900 413 .02 9.999982 29 1920 32 .968870 561* 5S7 .968889 .031111 413 .02 .999981 28 1980 33 .982233 568 588 .982253 .017747 412 .02 .999980 27 2040 34 7.995198 568 |j 589 7.995219 12.004781 411 .02 .999979 26 2100 a5 8.007787 567 \\ 590 8.007'809 11.992191 410 .03 .999977 25 2160 36 .020021 567 5!ll .020044 .979956 409 .02 .999976 24 2220 87 .031919 566 592 .031945 .968055 408 i .02 .999975 23 2280 88 .043501 566 593 .043527 .956478 407 .03 .999973 22 2340 39 .054781 566 ' 593 1 .054809 .945191 407 .02 .999972 21 2400 40 .065776 565 j 594 .065806 .93419-4 406 .999971 20 2460 41 8.076500 565 595 8.076531 11.923469 405 1.03 9.999969 19 2520 42 .086965 56 5% .086997 .913003 404 .02 .999968 18 2580 43 .097183 5(i4 5!)S .097217 .902783 402 -S .999966 17 2640 44 .107167 56: i .Y.IK .107203 .892797 401 i ,V4 .999964 16 2700 45 .116926 562 600 .11696o .888087 400 .02 no .999963 15 2760 46 .126471 562 601 .126510 .s;;',(!)0 399 ,08 .99996] 14 2820 47 .135810 56! 6U2 .135851 | .864141 398 .03 .999959 18 2880 48 .144953 501 603 .144996 .855004 397 .02 .999958 1& 2940 49 .153907 560 604 .153952 .846048 396 .03 .999958 11 3000 50 .162681 560 605 .162727 .K37'273 395 .03 .999954 10 3060 51 8.171280 559 607 j 8.171 328 1 1 . 828672 393 .03 9.999952 9 3120 52 . 179713 558 608! .179763 .S20237 392 .03 .999950 8 3180 58 .187985 55S (iOK .1SWKJI5 .811964 391 .03 .999948 7 3240 54 .196102 557 611 . .196156 .803844 389 .03 .999946 6 3300 55 .204070 556 151 -J .204126 .79587'4 388 .03 999944 5 3360 56 .211895 556 613 .211953 .788047 387 .03 .999942 4 3420 57 .219581 555 615 .219641 .780359 385 .03 .999940 3 3480 .58 .227134 554 616 .227195 .772805 384 .03 .999938 2 8.>40 59 .234557 554 ; 618 .234621 .765379 382 .03 .999936 1 k 600 60 8.2-11855 553 619 8.241921 11.758079 381 i* 08 9.999934 4.685 15.314 ! // / Cosine. q I Cotang. Tang. q + l Dl" Sine. / TABLE XXV. LOGARITHMIC SINES, 178' r I It / Sine, a-l Tang. Cotang. q + l Dl' Cosine. / 4.685 15.314 3600 8.241855 553' 619 1 8.241921 11.758079 381 | 9.999934 60 3660 1 .249033 552 620 .249102 .750898 380 .06 .999932 59 3720 2 .256094 551 622 .256165 .743835 378 25 .999929 58 3780 3 263042 551 I 623 .263115 .736885 377 -JS .999927 57 3840 4 .269881 550 625 .269956 .730044 375 -JS .999925 56 i 3900 5 .276614 549 627 .276691 .723309 373 { .999922 55 3960 6 .283243 548 028 .283323 .716677 372 -Jg .999920 54 4020 7 .289773- ! 547 630 .289856 .710144 qr.rv i : .Od O P.8 9.999257 19 8.767417 Ofi AO 11.232583 39 22 23 24 25 26 27 28 29 30 .768828 .770970 .773101 .775223 .777333 .779434 .781524 .783605 .785675 OO. Oo a5.70 35.52 35.37 35.17 35.02 34.83 34.68 34.50 34.35. .999250 .999242 .999235 .999227 .999220 .999212 .999205 .999197 .999189 . I - .13 .12 .13 .12 .13 .12 13 .13 .13 .769578 .771727 .773866 .775995 .778114 .780222 .782320 .784408 .786486 OO .\T 85.83 35.65 86.48 35.32 35.13 34.97 34.80 34.63 34.47 .230422 .228273 .226134 .224005 .21886 .219778 .217680 .215592 .213514 38 37 36 35 34 33 32 31 30 31 32 33 8.787736 .789787 .791828 34.18 34.02 9.999181 .999174 .999166 .12 .13 10 8.788554 .790613 .792662 34.32 34.15 QO f\Q 11.211446 .209387 .207aS8 29 28 27 34 35 36 .793859 .795881 .797894 as. 85 as. 70 as. 55 oo QC .999158 .999150 .999142 .lo . .13 .13 1 Q .794701 .796731 .798752 OO.HO 33.83 33.68 OQ KO .205299 .203269 .201248 26 25 24 37 38 39 40 .799897 .801892 .803876 .805852 OO . OO as. 25 as. 07 32.93 32.78 .999134 .999126 .999118 .999110 . lo .13 .13 .13 .13 .800763 .802765 .804758 .806742 OO. O/i 33.37 33.22 33.07 32.92 .199237 .197235 .195242 .193258 23 22 21 20 41 8.807819 QO f*O. 9.999102 1Q 8.808717 oo 77 11.191283 19 42 43 44 .809777 .811726 .813667 o^.oo 32.48 32.35 .999094 .999086 .999077 . ItJ .13 .15 1 o. .810683 .812641 .814589 32 '.63 32.47 QO QQ .189317 .187359 .185411 18 17 16 45 46 47 48 49 .815599 .817522 .819436 .821343 .823240 83! 05 31.90 31.78 31.62 .999069 .999061 .999053 .999044 .999036 .lo .13 .13 .15 .13 .816529 .818461 .820384 .822298 .824205 O/w . OO 32.20 32.05 31.90 31.78 .183471 .181539 .179616 .177702 .175795 15 14 13 12 11 50 .825130 31 .50 31.35 .999027 .15 .13 .826103 31 .'48 .173897 10 51 52 53 54 8.827011 .828884 .830749 .832607 31.22 31.08 30.97 on QO 9.999019 .999010 .999002 .998993 .15 .13 .15 1 Pi 8.827992 .829874 .831748 .833613 31.37 31.23 31.08 QA Q'*' 11.172008 .170126 .168252 .166387 9 8 7 6 55 56 57 58 59 60 ..834456 .836297 .838130 .839956 .841774 8.843585 oU. o^ 30.68 '30.55 30.43 30.30 30.18 .998984 .998976 .998967 .998958 .998950 9.998941 . lo .13 .15 .15 .13 .15 .835471 .837321 .839163 .840998 .842825 8.844644 ou. y i 30. as 30.70 30.58 30.4'. 30.32 .164529 .162679 .160837 .159002 .157175 11.155356 5 4 3 2 1 ~^~ Cosin^. ~D~rTi Sine, i D. 1". Cotang. D. 1". Tang. ~ COSINES, TANGENTS, AND COTANGENTS. i' 175 Sine. D.r. Cosine. D.I". Tang. D. 1". Cotang. ' 1 8.843585 .845387 30.03 9.998941 .998932 .15 8.844644 .846455 30.18 11.155356 .153545 60 59 2 .847183 OQ n .998923 .15 1 PC .848260 i|8 .151740 58 5 6 8 9 10 .848971 .850751 .852525 .854291 .856049 .857801 .859546 .861283 2V 67 29.57 29.43 29.30 29.20 29.08 28.95 28.85 .998914 .998905 .998896 .998887 .998878 .998869 .998860 .998851 . 10 .15 .15 .15 .15 .15 .15 .15 .17 .850057 .851846 .853628 .855403 .857171 .858932 .860686 .862433 29^82 29.70 29.58 29.47 29,35 29.23 29.12 29.00 .149943 .148154 . 140372 .144597 .142829 .141068 .139314 .137567 57 - 56 55 54 53 52 51 50 11 12 13 8.863014 .864738 .866455 28.73 28.62 9.998841 .998832 .998823 .15 .15 8.864173 .865906 .867632 28.88 28.77 OQ CK 11.135827 .134094 132368 49 48 47 14 15 .868165 .869868 28^38 .998813 .998804 .17 .15 .869351 .871064 5J8.DO 28.55 .130649 .128936 46 45 16 17 .871565 .873255 28!l7 .998795 .998785 .15 .17 .872770 .874469 28.43 28.32 .127230 .125531 44 43 18 19 20 .874938 .876615 ,878285 27 !95 27.83 27.73 .998776 .998766 .998757 .15 .17 .15 .17 .876162 .877849 .879529 28 'l2 28.00 27.88 .123838 .122151 .120471 42 41 40 21 22 23 24 25 8.879949 .881607 .883258 .884903 .886542 27.63 27.52 27.42 27.32 or* on 9.998747 .998738 .998728 .998718 .998708 .15 .17 .17 .17 8.881202 .882869 .884530 .886185 .887833 27.78 27.68 27.58 27.47 Or* OQ 11.118798 .117131 .115470 .113815 .112167 39 38 37 36 35 26 27 28 29 .888174 .889801 .891421 .893035 Zl .zO 27.12 27.00 26.90 OA QH .998699 .998689 .998679 .998669 .15 -,.17 Aft .889476 .891112 .892742 .894366 4i . oo 27.27 27.17 27.07 2fi q^ .110524 34 .108888 33 .107258 32 .105634 31 30 .894643 /sb.oU 26.72 .998659 .it .17 .895984 26 '.87 .104016 30 31 8.896246 9.998649 Of 8.897596 11.102404 29 32 33 34 35 36 37 38 .897842 .899432 .901017 .902596 .904169 .905736 .907297 26^50 26.42 26.32 26.22 26.12 26.02 .998639 .998629 .998619 .998609 .998599 .998589 .998578 .'l7 .17 .17 .17 .17 .18 iff .899203 .900803 .902398 .903987 .905570 .907147 .908719 26^67 26.58 26.48 26.38 26.28 26.20 .100797 .099197 .097602 .096013 .094430 .092853 .091281 28 27 26 25 24 23 22 39 40 .908853 .910404 25 . 93 25.85 25.75 .998568 .998558 .17 .17 .17 .910285 .911846 26 '.02 25.92 .089715 .088154 21 20 41 8.911949 OK AX 9.998548 1 Q 8.913401 0" 11.086599 19 42 43 44 45 .913488 .915022 .910550 .918073 iK) . DO 25.57 25.47 25.38 OK QA i .998537 I .998527 .998516 .998506 .lo .17 .18 .17 1 Q .914951 .916495 .918034 .919568 25>3 25.63 25.57 .085049 .083505 .081966 .080432 18 17 16 15 46 47 48 49 50 .919591 .921103 .922610 .924112 .925609 4b.Q{) 25.20 25.12 25.03 24.95 24.85 .998495 .998485 .998474 .998464 ; .998453 . lo .17 .18 .17 .18 .18 .921090 .922619 .924136 .925649 .927156 25^38 25.28 25.22 25.12 25.03 .078904 .077381 .075864 .074351 .072844 14 13 12 11 10 51 52 53 54 55 56 57 58 59 60 8.927100 .928587 .930068 .931544 .933015 .934481 .935942 [937398 .938850 8.940296 24.78 24.68 24.60 24.52 24.43 24.35 24.27 24.20 24.10 9.998442 .998431 .998421 .998410 .998399 .998388 .998377 .998366 .998355 9.998344 .18 .17 .18 .18 .18 .18 .18 .18 .18 8.928658 .930155 .931647 .933134 .934616 .936093 .937565 .939032 .940494 8.941952 24.95 24.87 24 . 78 24.70 24.62 24.53 24.45 24.37 24.30 11.07134.2 .069845 .068353 .066866 .065384 .063907 .062435 .060968 .059506 11.058048 9 8 7 6 5 3 2 1 ' Cosine. D.r. i Sine. D.r. Cotang. D.r. Tang. ' 94 85 C TABLE XXV. LOGARITHMIC SINES, 174 / Sine. D. r. Cosine. D. r. Tang. D. 1". Cotang. / 1 2 3 8.940296 .941738 .943174 .944606 , m 9.998344 x4.(Jo U O o (jo ! .WOOOO ~X'Q .998322 ~H' ! .998311 .18 .18 .18 1Q 8.941952 .943404 .944852 .946295 24.20 24.13 24.05 OQ OQ 11.058048 .056596 .055148 .053705 60 59 58 57 4 5 .946034 .947456 2o. oO 23.70 .998300 .998289 .lo .18 on .947734 .949168 !&5.9o 23.90 OQ UO .052266 .050832 56 55 6 7 8 9 10 .948874 .950287 .951696 .953100 .954499 23.63 23.55 ' 23.48 i 23.40 23.32 23.25 .998277 .998266 .998255 .998243 .998232 .&) .18 .18 ! .20 .18 .20 .950597 .952021 .953441 .954856 .956267 xSo.oa 23.73 23.67 23.58 23.52 23.45 .049403 .047979 .046559 .045144 .043733 54 4 53 52 51 50 11 8.955894 9.998220 1ft 8.957674 90 OK 11.042326 49 12 .957284 23.17 OQ 1ft .998209 . lo on .959075 o . oo OQ QA .040925 48 13 14 .958670 .960052 /So . 1U 23.03 .998197 .998186 ..vU .18 on .960473 .961866 &j . oU 23.22 OQ 1 ~ .039527 .038134 47 46 15 16 17 18 19 20 .961429 .962801 .964170 .965534 .966893 .968249 22.95 22.87 22.82 22.73 22.65 22.60 22.52 .998174 .998163 .998151 .998139 .998128 .998116 ,^U J8 .20 .20 .18 .20 .20 .963255 .964639 .966019 .967394 .968766 .970133 /o.lo 23.07 23.00 22.92 22.87 22.78 22.72 .036745 .035361 .033981 .032606 .031234 .029867 45 44 43 42 41 40 21 22 23 24 25 26 8.969600 .970947 .972289 .973628 .974962 .976293 22.45 22.37 22.32 22.23 22.18 OO 1 A 9.998104 .998092 .998080 .998068 .998056 .998044 .20 .20 .20 .20 .20 on 8.971496 .972855 .974209 .975560 .976906 .978248 22.65 22.57 22.52 22.43 22.37 OO ~>i\ 11.028504 .027145 .025791 .024440 .023094 .021752 39 38 37 36 35 34 27 28 .977619 .978941 sSs.lU 22.03 .998032 .998020 ,vJ .20 on .979586 .980921 . - oU 22.25 oo 1*7 .020414 .019079 33 32 29 30 .980259 .981573 21 .97 21.90 21.83 .998008 .997996 ,<&} .20 .20 .982251 .983577 . . . 1 1 22.10 22.03 .017749 .016423 31 30 81 32 33 8.982883 .984189 .985491 21.77 21.72 9.997984 .997972 .997959 .20 .22 on 8.984899 .986217 .987532 21.97 21.92 O1 <> 11.015101 .013783 .012468 29 28 27- 34 35 36 37 38 39 40 .986789 .988083 .989374 .990660 .991943 .993222 .994497 21.63 21.57 21.52 21.43 21.38 21.32 21.25 21.18 .997947 .997935 .997922 .997910 .997897 .997885 .997872 ,&) .20 .22 .20 .22 .20 .22 .20 .988842 .990149 .991451 .992750 .994045 .995337 .996624 \ .00 21.78 21.70 21.65 21.58 21.53 21.45 21.40 .011158 .009851 .008549 .007250 .005955 .004663 .003376 26 25 24 23 22 21 20 41 42 43 44 8.995768 .997030 .998299 8.999560 21.13 21.05 21.02 9.997860 .997847 .997835 .997822 .22 .20 .22 OO 8.997908 8.999188 9.000465 .001738 "21.33 21.28 21.22 01 i r; 11.002092 11.000812 10.999535 .998262 19 18 17 16 45 46 9.000816 .002069 20 . 93 20.88 .997809 .997797 JB0 .20 OO .003007 .004272 il . JLO 21.08 01 no .996993 .995728 15 14 47 48 .003318 .004563 20.82 20.75 .997784 .997771 ,jfo .22 O.) .005534 .006792 41 .06 20.97 on no .994466 .993208 13 12 49 50 .005805 .007044 20.70 20.65 20.57 .997758 .997745 ."< .22 .22 .008047 .009298 xju.yia 20.85 20.80 .991953 .990702 11 10 51 52 9.008278 .009510 20.53 on AX. 9.997732 .997719 .22 OO 9.010546 .011790 20.73 on Co 10.989454 .988210 9 8 53 54 .010737 .011962 ^u.4o 20.42 .997706 .997693 JBB .22 .013031 .014268 &j . Do 20.62 OA K'7 .986969 .985732 7 6 55 56 57 58 .013182 .014400 .015613 .016824 20.33 20.30 20.22 20.18 .997680 .997667 .997654 1 .997641 .22 .22 .22 .22 OO .015502 .016732 .017959 .019183 &J.OI 20.50 20.45 20.40 on QQ .984498 .983268 .982041 .980817 5 4 3 59 .018031 20.12 OA rvy 1 .997628 .IBS no .020403 /w.OO on oft .979597 1 60 9.019235 ii(}.(j( 9.997614 ,Q 9.021620 - ' ' . - 10.978380 / Cosine. D. r, i Sine. D. r. Cotang. D.I". Tang. I ' 95 [364] 84 COSINES. TANGENTS, AND COTANGENTS. 173 C ' Sine. D.I". Cosine. D.I". Tang. D.I". Cotang. ' 1 2 3 4 5 6 8 9 10 9.019235 .020435 .021632 .022825 .024016 .025203 .026386 .02756? :OS8744 (W.KM8 .031089 20.00 19.95 19.88 19.85 19.78 19.72 | 19.68 19.62 19.57 19 52 19.47 9.997614 .997601 .997588 .997574 .997561 .997547 .997534 .997520 .997507 .997493 .997480 .22 .22 .23 .22 .23 .22 .23 .22 .23 .22 .23 9.021620 .022834 .024044 .025251 .026455 .027655 .028852 ! .030046 .031237 .032425 .033609 20.23 20.17 20.12 20 07 20.00 19.95 19.90 19.85 19.80 19.73 19.70 10.978380 .977166 .975956 .974749 .973545 .972345 .5171 148 .969954 .968763 .967575 .966391 60 59 58 57 56 55 54 53 52 51 50 11 12 9.032257 .033421 19.40 ; tO OR 9.997466 .997452 .23 22 9.034791 .035969 19.63 19 58 10.965209 .9&4031 49 48 13 14 15 16 17 .034582 .035741 .086896 .038048 .039197 iy . oo 19.32 19.25 19.20 19.15 .997439 .997425 .997411 .997397 .997383 !23 .23 .23 .23 Mi .037144 .038316 .039485 .040651 .041813 19^53 19.48 19.43 19.37 1 O Q*-* .962856 .961684 .960515 .959349 .958187 47 46 45 44 43 18 19 20 040342 .041485 .042625 19.08 19.05 19.00 18.95 .997369 .997355 .997341 .^"5 .23 .23 .23 .042973 .044130 .045284 iy .00 19.28 19.23 19.17 .957027 .955870 .954716 42 41 40 21 22 23 24 9.043762 .044895 .046026 .047154 18.88 18.85 : 18.80 i 1 < C'Pi 9.997327 .997313 .997299 .997285 .2% .23 .23 too 9.046434 .047582 .048727 .049869 19.13 19.08 19.03 18 98 10.953566 .952418 .951273 .950131 39 38 37 36 25 27 .048279 .049400 .050519 J.O. id 18.68 18.65 1 tt AA .997271 .997257 .997242 ,&J .23 OQ .051008 .052144 .053277 18^93 18.88 10 OQ .948992 .947856 .946723 35 34 33 28 29 30 .051635 .052749 .053859 iD.fJU 18.57 18.50 18.45 .997228 .997214 .997199 JvO .23 .25 .23 .054407 055535 .056659 lo oo 18.80 18.73 18.70 .945593 .944465 32 31 30 31 32 33 34 35 36 37 38 39 9.054966 .056071 .057172 .058271 .059367 .060460 .061551 .062639 .063724 18.42 18.35 18.32 18.27 18.22 18.18 18.13 18.08 1 fi fV-t 9.997185 .997170 .997156 .997141 .997127 .997112 .997098 .997083 .H97068 .25 .23 .25 .23 .25 .23 .25 .25 Me 9.057781 .058900 .060016 .061130 .062240 .063348 .064453 .065556 .066655 18.65 18.60 18.57 18.50 18.47 18.42 18.38 18.32 -JO OQ '.941100 .939984 .938870 .937760 .936652 .985547 .934444 .933345 29 28 27 26 25 24 23 22 21 40 .064806 lo.Uo 17.98 .997053 . ^-> .23 .067752 18^25 .932248 20 41 42 43 44 45 46 47 48 9.065885 .066962 .068030 .069107 .070176 .071242 .072301) .073366 17.95 17.90 17.S5 17.82 17.77 17.73 17.67 9 997039 .997024 .997009 .996994 .996979 .996964 .996949 .996934 25 .85 25 .25 .25 .25 .25 OK 9.068846 .069938 .071027 .072113 .073197 .074878 ! 076432 18.20 18.15 18.10 18.07 18.02 17.97 17,93 1V* OQ 10.931154 .930063 .988978 .927887 .926803 .925722 .924644 .923568 19 18 17 16 15 14 13 12 49 50 .074424 .075480 17 ! 60 17.55 .996919 .996904 ,) .25 .25 .077505 .078576 1 .OO 17.85 17.80 .922495 .921424 11 10 51 52 53 54 9.076533 .077583 .078631 .079676 17.50 17.47 17.42 -l ^ OU 9.996889 .996874 .99685S .996843 .25 .27 .25 9.079644 .080710 .081773 .082833 17.77 17.72 17.67 1r* iQ 10.920356 .919290 .918227 .917167 9 8 7 6 55 .080719 1< .00 1 ^ *3Q .996828 .27 sw .083891 < .DO Irf f*f\ .916109 5 56 57 58 59 60 .081759 .082797 .083833 .084864 9.085894 1 * .00 17.30 17.25 17.20 17.17 .996812 .996797 .996782 .996766 9.996751 .Hi .85 .25 .27 .85 .081-.H7 .086000 .us; <>:,(> . .088098 9.089144 i . t>U 17.55 17.50 17.47 17.43 .915053 .914000 ,912950 .911902 10.910856 4 3 2 1 ' Cosine. D.I". Sine. D. 1". i Cotang. D. 1". Tang. i 83' TABLE XXV. LOGARITHMIC SINES, Sine. D. 1'. Cosine. D. 1". || Tang. D. 1". Cotang. ' 1 2 3 9.085894 .086922 .087947 .088970 nwooon 17.13 17.08 17.05 17.00 9.996751 .996735 .996720 .996704 QQftAQQ .27 .25 .27 .27 9.089144 .090187 .091228 .092266 AQQOAO 17.38 17.35 17.30 17.27 10.910856 .909813 .908772 .907734 CO 59 58 57 5 6 . uoyyyu .091008 .092024 16.97 16.93 1fi K& .yyDooo .996673 .996657 .25 .27 fcV~ .uyoou/i .094336 .095367 17.23 17.18 .906698 .905664 .904633 56 55 54 7 3 9 .093037 .094047 .095056 ID .00 16.83 16.82 1 A r ' ( 7 .996641 .996625 .996610 .6t .27 .25 .9949:35 .994916 .994896 [32 .33 i OO .186439 .187280 .188120 14 . 03 14.02 14.00 .813561 .812720 .811880 16 15 14 47 .183834 13. DO 1 Q fiO .994877 .06 oo .188958 13.97 .811042 13 48 .184651 1 o . O*Q .994857 . oo .189794 13.93 .810206 12 49 50 .185466 .186280 13.58 13.57 13.53 .994838 .994818 [33 .33 .190629 1 .191462 13.92 13.88 13.87 .809371 .808538 11 10 51 52 9.187092 .187903 13.52 9.994798 .994779 .32 oo 9.192294 .193124 13. as 10.8077'06 .806876 9 8 53 54 55 .188712 .189519 .190325 13.48 13.45 ! 13.43 1 Q yiO .994759 .994739 .994720 .00 .33 .32 00 .1 93953 .194780 .195606 i 13.82 13.78 13.77 1i 7. .806047 ; .805220 .804394 7 6 5 56 57 .191130 .191933 lo.4/& 13.38 1 Q Q?; .994700 .994680 .00 .33 33 .196430 .197253 1 . i O 13.72 10 f>0 .80&570 .802747 4 8 58 .192734 1 o . oo 1 Q QQ .994660 qq .19H074 lo. Do iq pn .801926 2 59 60 .193534 9.1943*2 1 . OO 13.30 .994640 9.994620 .00 .33 .198894 9.199713 1 o . O t 13.65 .801106 10.800287 1 ' Cosine. D. r. 1 Sine. D.r.l Cotang. j ^7rT Tang. ' [067] 81 TABLE XXV.-LOGARITHMIC SINES, 170 ' Sine. D. 1". Cosine. D.I". Tang. D. 1". Cotang. ' 9.194332 9.994620 QQ 9.199713 1fi An ! 10.800287 60 1 2 3 4 5 6 7 8 9 10 .195129 .195925 .196719 .197511 .198302 .199091 .199879 .200666 .201451 .202234 IO.4O 13.27 13.23 13.20 13.18 13.15 13.13 13.12 13.08 13.05 13.05 1 .994600 I .994580 i .994560 ! .994540 .994519 ! .994499 .994479 .994459 .9944:38 .994418 .66 .33 .33 .33 .35 .33 .33 .33 .35 .33 .33 .200529 .201345 .202159 .202971 .203782 .204592 .205400 .206207 .207013 .207817 1 o . bU 13.60 13.57 13.53 13.52 13.50 13.47 13.45 13.43 13.40 13.37 .799471 .798655 .797841 .797029 .796218 .795408 .794600 .793793 .792987 .792183 59 58 57 56 55 54 53 52 51 50 11 12 9.203017 .203797 13.00 10 AA 9.994398 .994377 .35 QO 9.208619 .209420 13.35 1 Q QO 10.791381 .790580 49 48 18 14 15 16 17 18 19 20 .204577 .205354 .206131 .206906 .207679 .208452 .209222 .209992 10. uu 12.95 12.95 12.92 12.88 12.88 12.83 12.83 12.80 .994357 1 .994336 .994316 .994295 .994274 .994254 .994233 .994212 .OO .35 .33 .35 .35 .33 .35 .35 .35 .210220 .211018 .211815 .212611 .213405 .214198 .214989 .215780 lO.OO 13.30 13.28 13.27 13.23 13.22 13.18 13.18 13.13 .789780 .788982 .788185 .787389 .786595 .785802 .785011 .784220 47 46 45 44 43 42 41 40 21 22 23 24 25 26 27 28 9.210760 .211526 .212291 .213055 .213818 .214579 .215338 .216097 12.77 12.75 12.73 12.72 12.68 12.65 12.65 1O AO 9.994191 .994171 .994150 .994129 .994108 .994087 .994066 .994045 .as .35 .35 .35 .35 .35 .35 QC 9.216568 .217a56 .218142 i .218926 .219710 .220492 .221272 .222052 13.13 13.10 13.07 13.07 13.03 13.00 13.00 10.7&3432 .782644 .781858 .781074 .780290 .779508 .778728 .777948 39 38 37 36 a5 34 33 32 29 30 .216854 .217609 1*. D-* 12.58 12.57 .994024 .994003 .BO .35 .35 .222830 .223607 12.97 12.95 12.92 .777170 .776393 31 30 31 32 33 34 35 36 37 38 9.2ia363 .219116 .219868 .220618 .221367 .222115 .222861 .223606 12.55 12.53 12.50 12.48 12.47 12.43 12.42 i ,) OQ 9.993982 .993960 .993939 .998918 . 99:3897 i .993875 .993854 .993832 .37 .35 .35 .35 .37 .35 .37 or 9.224382 .225156 .225929 .226700 .227471 .228239 .229007 .229773 12.90 12.88 12.85 12.85 12.80 12.80 12.77 10.775618 .774844 .77*71 .773300 .772929 .7719ft] .770993 .770227 29 28 27 26 25 24 23 22 39 40 .224349 .225092 l^.oo 12.38 12.35 .993811 .993789 .00 .37 .a5 .230539 .231302 12.77 12.72 12.72 .769461 .768698 21 20 41 42 9.225833 .226573 12.33 i o QH 9.993768 . S)i74l> .37 QK 9.232065 .232826 12.68 10 r/ 10.767935 .767174 19 18 43 44 45 46 47 48 49 50 .227311 ,228048 .228784 .229518 .230252 .230984 .231715 .232444 Itf.oU 12.28 12.27 12.23 12.23 12.20 12.18 12.15 12.13 .993725 .993703 .993681 .993660 .993638 .993616 .993594 .993572 .00 .37 .37 .35 .37 .37 .37 .37 .37 .2aS586 .234345 .235103 .235859 .236614 .237368 .238120 .238872 iti.vt 12.65 12.63 12.60 12.58 12.57 12.53 12.53 12.50 .766414 .766665 .764897 .764141 .763386 .762632 .761880 .761128 17 16 15 14 13 12 11 10 51 52 53 54 55 56 57 58 59 9.233172 .233899 .234625 .235349 .236073 .236795 .237515 .238235 .238953 12.12 12.10 12.07 12.07 12.03 12.00 12.00 11.97 9.99a550 .993528 .993506 .993484 .993462 .993440 .993418 .993396 .99a374 .37 .37 .37 .37 .37 .37 .37 .37 9.239622 .240371 .241118 .241865 .242610 .243354 .244097 .244839 .245579 12.48 12.45 12.45 12.42 12.40 12.38 12.37 12. as 10.760378 .769629 .758882 .758135 .757390 .756646 .755903 .755161 .754421 9 8 6 5 4 3 2 1 60 9.239670 11 .95 9! 998851 .38 9.246319 12. as 10.753681 ' Cosine. 1 D. 1". J Sine. D. 1". Cotang. D. 1". Tang. ' , -.681 80 COSINES, TANGENTS, AND COTANGENTS. 169" ' Sine. D. 1". Cosine. D.I". Tang. D. 1". Cotang. ' 1 9.239670 .240386 11.93 9.993351 .993329 .37 9.246319 .247057 12.30 10.753681 .752943 60 59 3 .241101 11.92 nor> .993307 %, i .247794 12.28 .752206 58 3 .241814 .00 .993284 ! .248530 12.27 .751470 57 4 6 6 I 9 10 .242526 .24323? I 343047 .244656 .2453(13 .2460(59 .246775 11.87 11.85 11.83 11.82 11.78 11.77 11.77 11.72 .993262 .993240 .993217 .993195 .993172 .993149 .993127 .01 .37 .38 .37 ,38 .38 .37 .38 .249264 .249998 .250730 .251461 .252191 .252920 .253648 12.23 12.23 12.20 12.18 12.17 12.15 12.13 12.10 .750736 .750002 .749270 .748539 .747809 .747080 .746352 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 20 9.247478 .248181 .248883 .249583 .250282 .250980 .251677 .252373 .253067 .253761 11.72 11.70 11.67 11.65 11.63 11.62 11.60 11.57 11.57 j 11.53 9.993104 .993081 .993059 .993036 .993013 .992990 .992967 .992944 .992921 .992898 .38 .37 .38 ! .38 i .38 .38 .38 .38 .38 .38 j 9.254374 .255100 .255824 .256547 .257269 .257990 .258710 .259429 .260146 .260863 12.10 12.07 12.05 12.03 12.02 12.00 11.98 11.95 11.95 11.92 10.745626 .744900 .744176 .743453 .742731 .742010 .741290 .740571 .739854 .739137 49 48 47 46 45 44 43 42 41 40 21 22 to 24 9.254453 .255144 .255834 .256523 11.52 11.50 11.48 9.992875 .992852 .992829 .992806 .38 .38 .38 9.261578 .262292 .263005 .263717 11.90 11.88 11.87 10.738422 .737708 .736995 .736283 39 38 37 36 25 26 .257211 .257898 11.47 11.45 .992783 .992759 .38 .40 .2(54428 .265138 11 .85 11.83 noo .735572 .734862 35 34 27 .258583 11.42 .992736 .38 OQ .265847 .0% .734153 33 28 .2592(58 11 .42 .992713 .OO .26jK>55 11 .80 .733445 32 29 30 .259951 .260633 11 .38 11.37 11.35 .992690 .992666 .38 .40 .38 ! 267967 11 .77 11.77 11.73 .732739 .732033 31 30 31 32 9.261314 .261994 11.33 9.992643 .992619 .40 9.268671 .269375 11.73 10.731329 .730625 29 28 33 34 35 36 37 .262673 .263a51 .264027 .2647'03 .265377 11.32 11.30 11.27 11.27 11.23 .992596 .992572 .992549 .992525 .992501 .38 .40 .38 .40 .40 00 .270077 .270779 .271479 .272178 .272876 11 .70 11.70 11.67 11.65 11.63 HAO .729923 .729221 .728521 .727822 .727124 27 26 25 24 23 38 39 .266051 .266723 11 .23 11.20 .992478 .992454 .00 .40 .273573 .274269 .0x5 11.60 .726427 .725731 22 21 40 .267395 11.20 11.17 .992430 .40 .40 .274964 11.58 11.57 .725036 20 41 42 43 44 45 46 47 48 49 50 9.268065 .2687:34 .269402 .2700(59 .270735 .27141X1 .272064 .272726 .273388 .274049 11.15 11.13 11.12 11.10 11.08 11.07 11.03 11.03 11.02 10.98 9.992406 .992382 .992359 .992335 .992311 .992287 .992263 .992239 .992214 .992190 .40 .38 .40 .40 .40 .40 ' .40 j .42 .40 ! .40 9.275658 .276351 .277043 .277734 .278424 .279113 .279801 .280488 .281174 .281858 11.55 11.53 11.52 11.50 11.48 11.47 11.45 11.43 11.40 11.40 10.724342 .723649 .722957 .722266 .721576 .720887 .720199 .719512 .718826 .718142 19 18 17 16 15 14 13 12 11 10 51 52 53 54 55 9.274708 .275367 .276025 .276681 .277337 10.98 10.97 10.93 10.93 9.992166 .992142 .992118 .992093 .992069 .40 .40 .42 .40 9.282542 .283225 .283907 .284588 .2852(58 11.38 11.37 11.35 11.33 noo 10.717458 .716775 .716093 .715412 .714732 9 8 7 6 5 56 58 59 .277991 .278645 .279297 .279948 10.90 10.90 10.87 10.85 .992044 .992020 .991996 .991971 .'40 .40 .42 .285947 .286624 .287301 .2*7977 .64 11.28 11.28 11.27 714053 .713376 .712699 .712023 4 3 2 1 60 9.280599 10.85 9.991947 .40 9.288652 11.25 10.711348 ' Cosine. D. 1". i Sine. D. 1". 1 Cotang. v.r. Tang. ' 79 11' TABLE XXV. LOGARITHMIC SINES, 168- | Sine. D. 1". Cosine. D. 1". Tang. D. r. Cotang. ' 9.280599 9.991947 9.288652 10.711348 60 1 2 3 4 5 6 7 8 9 10 .281248 .281897 .282544 .283190 .283836 .284480 .285124 .285766 .286408 .287048 10.82 10.78 10.77 10.77 10.73 10.73 10.70 10.70 10.67 10.67 .991922 .991897 .991873 .991848 .991823 .991799 .991774 .991749 .991724 .991699 .42 .40 .42 .42 .40 .42 .42 .42 .42 .42 .289326 .289999 .290671 .291342 .292013 .292682 .293350 .294017 .294684 .295349 11.22 11.20 11.18 11.18 11.15 11.13 11.12 11.12 11.08 11.07 .710674 .710001 .709329 .708658 .707987 .707318 .706650 .705983 .705316 .704651 59 58 57 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 20 9.287688 .288326 .288964 .289600 .290236 .290870 .291504 .292137 .292768 .293399 10.63 10.63 10.60 10.60 10.57 10.57 10.55 10.52 10.52 10.50 9.991674 .991649 .991624 .991599 .991574 .991549 .991524 .991498 .991473 .991448 .42 .42 .42 .42 .42 .42 .43 .42 .42 .43 9.296013 .296677 .297339 .298001 .298662 .299322 .299980 .300638 .301295 .301951 11.07 11.03 11.03 11.02 11.00 10.97 10.97 10.95 10.93 10.93 10.703987 .703323 .702661 .701999 .701338 .700678 .700020 .699362 .698705 .698049 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 26 27 28 29 30 9.294029 .294658 .295286 .295913 .290539 .297164 .297788 .298412 .299034 .299655 10.48 10.47 10.45 10.43 10.42 10.40 10.40 10.37 10.35 10.35 ! 9.991422 .991397 .991372 .991346 .991321 .991295 .991270 .991244 .991218 .991193 .42 .42 .43 .42 .43 .42 .43 .43 .42 .43 9.302607 .303261 [304567 .305218 .305869 .306519 .307168 .307816 .308463 10.90 10.88 10.88 10.85 10.85 10.83 10.82 10.80 10.78 10.77 10.697393 .65)6739 .696086 .695433 .694782 .694131 .693481 .692832 .692184 .691537 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 9.300276 .300895 .301514 .302132 .302748 .303364 10.32 10.32 10.30 10.27 10.27 9.991167 .991141 .991115 .991090 .991064 .991038 .43 .43 .42 .43 .43 9.309109 .309754 .310399 .311042 .311685 .312327 10.75 10.75 10.72 10.72 10.70 10.690891 .690246 .689601 .688958 .688315 .687673 29 28 27 26 25 24 37 38 39 40 .303979 .304593 .305207 .305819 10.23 10.23 10.20 10.18 .991012 .990986 .990960 .990934 .43 .48 .43 .43 .312968 .313608 .314247 .314885 10.67 10.65 10.63 10.63 .687032 .686392 .685753 .685115 23 22 21 20 41 42 43 44 45 46 47 48 49 50 9.306430 .307041 .307650 .308259 .308867 .309474 .310080 .310685 .311289 .311893 10.18 10.15 10.15 10.13 10.12 10.10 10.08 10.07 10.07 10.03 9.990908 .990882 .990855 .990829 .990803 .990777 .990750 .990724 .990697 .990671 .43 .45 .43 .43 .43 .45 .43 .45 .43 .43 9.315523 .316159 .316795 .317430 .318064 .318697 319330 .319961 .320592 .321222 10.60 10.60 10.58 10.57 10.55 10.55 10.52 10.52 10.50 10.48 10.684477 .683841 .683205 .682570 .681936 .6813a3 .680670 .680039 .679408 .678778 19 18 17 16 15 14 13 12 11 10 51 52 53 54 55 56 57 58 59 60 9.312495 .313097 .313698 .314297 .314897 .315495 .316092 .316689 .317284 9.317879 10.03 10.02 9.98 10.00 9.97 9.95 9.95 9.92 9.92 9.990645 .990618 .990591 .990565 .990538 .990511 .990485 .990458 .990431 9.990404 .45 .45 .43 .45 .45 .43 .45 .45 .45 9.321851 .322479 .323106 .323733 .324358 .324983 .325607 .326231 .326853 9.327475 10.47 10.45 10.45 10.42 10.42 10.40 10.40 10.37 10.37 10.678149 .677521 - .676894 .676267 .675642 .675017 .674393 .673769 .673147 10.672525 9 8 7 6 5 4 3 2 1 ' Cosine. D. r. Sine. D. r. Cotang. D. 1'. 1 Tang. ' 101 s [370] COSINES, TANGENTS, AND COTANGENTS. 167' 34 37 Sine. D. 1". Cosine. I D. 1". Tang. D. 1". Cotang. 9.317879 .318473 .319066 .319658 .320840 .321430 .322019 .322607 .323194 .328780 9.324366 .324950 .325534 .326117 .326700 .327281 .328442 .329021 9.330176 .330753 .881903 .334195 .334767 .336475 .337043 .337610 .3404:34 .341558 .342119 .342679 .343797 .344&55 .345469 .346579 .3471:34 .347687 .348240 .350443 .350992 .351540 Cosine. 9.87 9.85 9.85 9.80 9.78 9.77 9.77 9.73 9.73 9.72 9.72 9.68 9.68 9.67 9.65 9.63 9.60 9.57 9.58 9.55 9.55 9.52 9.53 9.50 9.48 9.48 9.47 9.45 9.43 9.43 9.42 9.40 9.38 9.37 9.37 9.33 9.33' 9.30 9.25 9.25 9.25 9.18 9.17 9.17 9.15 9.13 9.13 .990404 .<<)< >37S .990351 .990297 .990270 .990161 .990134 .990107 .990079 .990052 ! 990025 .98<)7'49 .989721 .989441 .989413 9.989271 .983243 .989214 .989157 .989128 .989071 .989042 .989014 .988811 .988753 9.988724 .43 .45 .45 .45 .45 .45 .47 .45 .45 .45 I .45 | .47 j .45 I .45 .47 .45 -47 j .45 .47 .45 .47 .47 .45 .47 .47 .47 .47 .47 .45 .47 .48 .47 .47 .47 .47 .47 .47 .48 .47 .47 .48 .47 .48 .47 .48 .48 .47 .48 .48 .48 .48 .48 9.327475 .328715 .329334 .329953 .330570 .331187 .331803 .332418 9.334259 .334871 .335482 .336702 a37311 .337919 .3:38527 .339739 9.340344 .340948 .341552 .342155 . 342757 .343358 .343958 .344558 .345157 .345755 9.346:353 .346949 .347545 .348141 .348735 .349922 .350514 .351106 .351697 9.a52287 .353465 .354053 .354640 .355227 .355813 .'357566 9.358149 .358731 .,359313 .860474 .3(51053 .362210 .362787 t. 303364 10.33 10.33 10.32 10.32 10.28 10.28 10.27 10.25 10.25 10.22 10.22 10.20 10.18 10.18 10.15 10.15 10.13 10.13 10.10 10.10 10.08 10.07 10.07 10.05 10.03 10.02 10.00 10.00 9.98 9.97 9.97 9.93 9.93 9.93 9.90 9.90 9.88 9.87 9.87 9.85 9.83 9.82 9.82 9.80 9.78 9.78 9.77 9.75 9.73 9.73 9.72 9.70 9.70 9.67 9.68 9.65 9.65 9.63 9.62 10.672525 .671905 .671285 .670666 .670047 .669430 .668813 .668197 .666354 10.665741 .665129 .664518 .663907 .663298 .662689 .662081 .661473 .660867 .660261 10.659656 .659052 .658448 .657845 .657243 .656642 .656042 .655442 .654843 .654245 10.653647 .653051 .652455 .651859 .651265 .650671 .650078 .649486 10.647713 .647124 .646535 .645947 .645360 .644773 .644187 .643602 .643018 .642434 10.641851 .641269 .640687 .640107 . .639526 .637790 .637213 10.636636 D. 1". II Sine, I D.I", il Cotang. | D.I", j Tang. 102 13 'TABLE XXV. LOGARITHMIC SINES, Sine. .353726 .354271 .354815 .355358 .355901 .356443 .357524 .358603 .359141 .359678 .360215 .360752 .361287 .36235(5 9,363422 .364485 .365016 .365546 .367131 .369761 .370285 .370808 .371&30 .371852 .372373 .372894 .373414 9.373933 .374452 .374970 .375487 .376003 .376519 .377035 .377549 .ay; .378577 .379601 .380113 .381134 .381643 .382152 Cosine. D.I". 9.12 9.10 9.08 9.08 9.07 9.05 9.05 9.03 9.02 9.00 9.00 8.98 8.97 8.95 8.95 8.95 8.92 8.92 8.90 8.88 8.88 8.87 8.85 8.85 8.83 8.82 8.82 8.78 8.80 8.77 8.77 8.75 8.75 8.72 8.72 8.70 8.70 8.68 8.68 8.67 8.65 8.65 8.63 8.62 8.60 8.57 8.57 8.57 8.53 8.53 8.53 8.52 8.50 8.48 8.48 8.48 8.45 8.45 Cosine. 9.988724 .988695 D. 1" .988607 .988578 .988519 .988312 .988073 .988043 .988013 .987953 .987862 9.987801 .987771 .987740 .987710 .987679 .987649 .987618 .987588 .987557 .987526 9.987496 .987465 .987434 .987403 .987372 .987341 .987310 .987279 .987248 .987217 9.987186 .987155 .987124 .987061 .987030 Sine. D.I'. .48 .48 .50 .48 .48 .50 .48 .50 .48 .50 .48 .50 .48 .50 .50 .50 .48 .50 .50 .50 .50 .50 .50 .50 .50 .50 .52 .50 .50 .50 .52 .50 .52 .50 .52 .50 .52 .50 .52 .52 .50 .52 .52 .52 .52 .52 .52 .52 .52 .52 .52 .52 .52 .53 .52 .52. .53 .52 .52 .53 Tang. .364515 .365090 .365664 .366810 .367382 .367953 .368524 y. .370799 .371367 .371933 .372499 .373064 .373629 .374193 .374756 9.375319 .375881 .376442 .377003 .377563 .378122 .378681 .379239 .379797 ). 380910 .381466 .383129 .383682 .384234 .384786 .387536 .390270 9.391903 .392447 .392989 .393531 .394614 .395154 .395694 .396233 D. 1". ! i Cotang. D. 1'. 9.58 9.58 9.57 9.55 9 55 9.53 9.52 9.52 9.50 9.48 9.48 9.45 9.47 9.43 9.43 9.42 9.42 9.40 9.37 9.35 9.35 9.33 9.32 9.32 9.30 9.80 9.28 9.27 9.27 9.23 9.25 9.23 9.22 9.20 9.20 9.18 9.18 9.17 9.15 9.15 9.13 9.12 9.12 9.10 9.10 9.08 9.08 9.05 9.07 9.03 9.03 9.03 9.02 9.00 9.00 8.98 8.97 Cotang. 10.636636 .636060 .635485 .634910 .634336 .633763 .633190 .632618 .632047 .631476 .628067 .627501 .626371 .624119 .623558 .622997 .622437 .621878 .621319 .620761 .620203 .619646 10.619090 .618534 .617980 .617425 .616871 .616318 .615766 .615214 22 .614663 | 21 .614112 | 20 10.613562 .613013 .612464 .611916 i 16 .611369 I 15 .610822 14 .610276 13 .609730 i 12 .609185 ! 11 .608640 ! 10 10.608097 .607553 .607011 .604846 3 .604306 ! 2 .603767 1 10.603229 : D. 1". | Tang. 103 [372] 76' 14o COSIXSS, TANGENTS, AND COTANGENTS. t Sine. D. 1'. Cosine. D. 1\ Tang. D. 1'. Cotang, ' 9.383675 8.45 9.986904 .52 9.396771 Q(Y?Qf\n 8.97 10.603229 60 1 2 ! 384687 8.42 84,-v ! 986841 .53 KO . oy t ouy .397846 8.95 8 OK 1602154 59 58 3 4 .385192 .385697 .42 8.42 .986809 .986778 .OO .52 .398383 .398919 ,9o 8.93 8MB .601617 .601081 57 56 5 .386201 8.40 800 .986746 KO .399455 .yo 8QO .600545 55 6 7 8 9 10 .386704 .387207 .387709 .388210 .388711 .00 8.38 8.37 8.35 8.35 8.33 .986714 .986683 .986651 .986619 .986587 .UO .52 .53 .53 .53 .53 .399990 .400524 .401058 .401591 .402124 .we 8.90 8.90 8.88 8.88 8.87 .600010 .599476 .598942 .598409 .597876 54 53 52 51 50 11 9.389211 8QO 9.986555 to 9.402656 8 OK 10.597344 49 12 13 14 15 .389711 .390210 .390708 .391206 .OO 8.32 8.30 8.30 8Osi .986523 .986491 .986459 .986427 .Do .53 .53 .53 to .403187 .403718 .404249 .404778 .OO 8.85 8.85 8.82 800 .596813 .596282 .595751 .595222 48 47 46 45 16 17 18 19 20 .391703 .392199 .392695 .393191 .393685 SO 8.27 8.27 8.27 8.23 8.23 .986395 .986363 .986331 .986299 .986266 ,OO .53 .53 .53 .55 .53 .405308 .405836 .406364 .406892 .407419 .OO 8.80 8.80 8.80 8.78 8.77 .594692 .594164 .593636 .593108 .592581 44 43 42 41 40 21 9.394179 Q .)-) 9.986234 to 9.407945 877 10.592055 39 22 23 24 25 26 27 .394673 .395166 .395658 .396150 .396641 .397132 .397621 .398111 o.4o 3.22 8.20 8.20 8.18 8.18 8.15 8.17 81 " .986202 .986169 .986137 .986104 .986072 .986039 .986007 .985974 .UO .55 .53 .55 .53 .55 .53 .55 K.O .408471 .408996 .409521 .410045 .410569 .411092 .411615 .412137 . I * 8.75 8.75 8.73 8.73 8.72 8.72 8.70 .591529 .591004 .590479 .589955 .589431 .588908 .588385 .587863 38 37 36 35 34 33 32 31 30 .398600 .15 8.13 .985942 .DO .55 .412658 8^68 .58?342 30 31 32 33 9.399088 .399575 .400062 8.12 8.12 ft 19 9.985909 .985876 ..985843 .55 .55 CO 9.413179 .413699 .414219 8.67 8.67 8KK 10.586821 .586801 .585781 29 28 27 34 .400549 8-1 A .985811 .00 frK .414738 .OO 80K .585262 35 .401035 111 8AQ .985778 .OO .415257 . DO 8M .584743 25 36 37 38 39 40 .401520 .402005 .402489 .402972 .403455 .Uo 8.08 8.07 8.05 8.05 8.05 .985745 .985712 .985679 .985646 .985613 .55 .55 .55 .55 .55 .55 .415775 .416293 .416810 .417326 .417842 .DO 8.63 8.62 8.60 8.60 8.60 .584225 .583707 .583190 .582674 .582158 24 23 22 21 20 41 42 43 9.403938 .404420 .404901 8.03 8.02 8A41 9.985580 .985547 .985514 .55 .55 9.418358 .418873 .419387 8.58 8.57 8f* 10.581642 .581127 .580613 19 18 17 44 45 4(J .405382 .405862 .406341 .\JK 8.00 7.98 .985480 .985447 .985414 .57 .55 .55 .419901 .420415 .420927 .Of 8.57 8.55 8tK .580099 .579585 .579073 16 15 14 47 48 .406820 .407299 7.98 7.98 .985381 .985347 .55 .57 .421440 .42195* .55 8.53 8 Kit .578560 .578048 13 12 49 50 .407777 .408254 7.97 7.95 7.95 .985314 .985280 .55 .57 .55 .422463 .422974 .Oil 8.52 8.50 .577537 .577026 11 10 51 52 53 9.408731 .409207 .409682 7.98 7.93 9.985247 .985213 .985180 .57 .55 9.423484 .423993 .424503 8.48 8.50 8 A r* 10.576516 .576007 .575497 9 , 8 54 .410157 7.92 7 no .985146 .57 .425011 .4< 8/in- .574989 6 55 . .410632 . y/ .985113 .55 .425519 .4* 8AV .574481 5 56 .411106 7.90 f~ QQ .985079 ' .426027 .4i 8AK .573973 4 57 .411579 1 .OO ** QQ .985045 *5 .426534 .40 Q Jfr .573466 3 58 59 .412052 .412524 i .88 7.87 r- 017 .985011 .984978 '.55 .427041 .427547 o.4D 8.43 .572959 .572453 2 1 60 9.412996 i . o7 9.984944 ' 9.428052 ' 10.571948 / P y Cosine. D. r. Sine. D. r. Cotang. i D. 1'. Tang. ' [373] .TABLE XXV. LOGARITHMIC SINES, 164" ' Sine. D.I". Cosine. D.l". Tang. D. r. Cotang. . / 1 2 9.412996 .413467 .413938 7.85 7.85 r* oo 9.984944 .984910 .984876 .57 .57 9.428052 .428558 .429062 8.43 8.40 10.571948 60 .571442 59 .570938 58 3 4 5 6 7 8 9 10 .414408 .414878 .415347 .415815 .416283 .416751 .417217 .417684 1 .OO 7.83 7.82 7.80 7.80 7.80 7.77 7.78 7.77 .984842 .984808 .984774 .984740 .984706 .984672 .984638 .984603 .57 .57 .57 .57 .57 .57 .57 .58 .57 .429566 .430070 .430573 .431075 .431577 .432079 .432580 .433080 8.40 8.40 8.38 8.37 8.37 8.37 8.35 8.33 8.33 .570434 .569930 .569427 .568925 .568423 .567921 .567420 .566920 57 56 55 54 53 52 51 50 11 9.418150 9.984569 9.433580 800 10.566420 49 12 13 14 15 .418615 .419079 .419544 .420007 7.75 7.73 7.75 7.72 r* rv> .984535 .984500 J .984466 .984432 !58 .57 .57 to .434080 .434579 .435078 .435576 ,OO 8.32 8.32 8.30 .565920 .565421 .564922 .564424 48 47 46 45 16 17 18 19 20 .420470 .420933 .421395 .421857 .422318 7^72 7.70 7.70 7.68 7.67 .984397 .984363 .984328 .984294 .984259 .00 .57 .58 .57 .58 .58 .436073 .436570 .437067 .437563 .438059 8^28 8.28 8.27 8.27 8.25 .563927 .563430 .5629*3 .562437 .561941 44 43 42 41 40 21 9.422778 9.984224 Kff 9.438554 8 no 10.561446 39 22 23 .423238 .423697 7.67 7.65 .984190 .984155 .at .58 to .439048 .439543 .&} 8.25 800 .560952 .560457 38 37 24 25 .424156 .424615 7.65 7.65 7/n .984120 .984085 .UO .58 KQ .440036 .440529 ,xS 8.22 800 .559964 .559471 36 35 26 27 .425073 .425530 .DO 7.62 f. 984050 .984015 .OO .58 ecv .441022 .441514 JH 8.20 .558978 .558486 34 33 28 425987 7 Art .983981 .Of KQ .442006 81R .557994 32 29 .426443 .OU .983946 . Oo .442497 . lo .557503 31 30 .426899 7.60 7.58 .983911 .58 .60 .442988 8.18 8.18 557012 30 31 32 33 34 35 36 37 38 39 40 9.427354 .427809 .428263 .428717 .429170 .429623 .430075 .430527 .430978 .431429 7.58 7.57 7.57 7.55 7.55 7.53 7.53 7.52 7.52 7.50 9.983875 .983840 .983805 .983770 .983735 .983700 .983664 .983629 .983594 .983558 .58 .58 .58 .58 .58 .60 .58 .58 .60 .58 9.443479 .443968 .444458 .444947 .445435 1 .445923 .446411 .446898 .447384 .447870 8.15 8.17 8.15 8.13 8.13 8.13 8.12 8.10 8.10 8.10 10.556521 .556032 .555542 .555053 .554565 .554077 .553589 .553102 .562616 .552130 29 28 27 26 25 24 23 22 21 20 41 42 43 9.431879 .432329 .432778 7.50 7.48 9.983523 .983487 .983452 .60 .58 ff\ 9.448356 .448841 .449326 8.08 8.08 10.551644 .551159 .550674 19 18 17 44 45 .433226 .433675 7.47 7.48 .983416 .983381 .OU .58 .449810 .450294 8.07 8.07 .550190 .549706 16 15 46 47 48 .-434122 .434569 .435016 7.45 7.45 7.45 .983345 .983309 .983273 .60 .60 .60 to .450777 .451260 .451748 8.05 8.05 8.05 8AQ .549223 .548740 .548257 14 13 12 49 .435462 i .43 .983238 .00 .452225 .Oo .547775 11 50 .435908 7.43 7.42 .983202 .60 .60 .452706 8.02 8.02 .547294 10 51 9.436353 9.983166 9.453187 10.546813 9 52 .436798 7.42 .9a3130 .60 .453668 8.02 .546332 8 53 54 .437242 .437686 7.40 7.40 7 no .983094 .983058 .60 .60 Gf\ .454148 .454628 8.00 8.00 r* no .545852 .545372 i 55 56 .438129 .438572 .OO 7.38 7 an .983022 .982986 .00 .60 Art i .455107 .455586 < . yo 7.98 ri AT* .544893 .544414 5 4 57 .439014 .67 .982950 . OU .456064 7.y< .543936 3 58 .439456 7.37 .982914 .60 .456542 7.97 .543458 2 59 60 .439897 9.440338 7.35 7.35 .982878 9.982842 .60 .60 .457019 9.457496 7.95 7.95 .542981 10.542504 1 ' Cosine. D.I". Sine. D. 1". Cotang. | D. 1'. Tang. ' 105 C 74 16 , COSINES, TANGENTS, AND COTANGENTS. 163' Sine. D. 1*. Cosine. D. 1". Tang. D. 1". Cotang 9.440338 .440778 .441218 .441658 .442535 .442973 .443410 .443847 .444284 .444720 9.445155 .445590 .446025 .446459 .447326 .447759 .448191 .449054 9.449485 .449915 .450345 .450775 .451204 .451632 .452060 .452915 .453342 31 9.453768 .454194 .454619 .455044 .456316 .456739 .457162 .457584 9.458006 .458427 .458848 .460108 .460946 .461364 .461782 .463448 51 9.462199 52 53 54 55 .56 57 58 59 60 .464279 .465108 .465522 9.465935 7.33 7.33 7.33 7.30 7.32 7.30 7.28 7.27 7.25 r.25 r!23 7.20 7.18 7.18 7.17 7.17 7.17 7.15 7.13 7.12 7.12 7.10 7.10 7.08 7.08 7.08 7\05 7.05 7.05 7.03 7.03 7.02 7!oo 7.00 7.00 6.98 6.97 6.95 6.95 6.90 6.90 Cosine. 1 D. 1". 9.982842 .982733 .982587 .982551 .982514 .982477 9.982441 .982404 .982294 .982257 .982146 .982109 ). 982072 .981961 .981924 .981849 .981812 .981774 .981737 9.981700 .981662 .981625 .981587 .981549 .981512 .981474 .981436 .981399 .981361 9.981323 .981285 .981247 .981171 .981133 .981057 .981019 9.980942 .980904 .980827 .980789 .980750 .980712 .980673 980635 9.980596 Sine. .62 .62 .62 .62 .63 D. r. 9.457496 .457973 .458449 .458925 .459400 .459875 .460349 .460823 .461297 .461770 .462242 9.462715 .463186 .463658 .464128 .464599 .465069 .466008 .466477 9.467413 .467880 .468814 .469746 .470211 .470676 .471141 .471605 9.472069 .472532 .472995 .473457 .474381 .474842 .475303 .475763 .476223 9.476683 .477142 .477601 .478059 .478517 .47<432 .479889 .480345 .480801 9.481257 .481712 .482167 .482621 .483075 .484435 .484887 9.485339 Cotang. .93 .90 7.87 7.88 .87 .78 .75 .75 .75 .73 .73 .70 .68 .68 .67 .67 7.67 7.65 7 65 7.63 7'63 7^62 7.60 7.58 7.58 7.57 7.57 7.55 7.53 7.53 D. r. 10.542504 ! 60 .542027 59 .541551 ! 58 .541075 57 .540600 56 .540125 55 .539651 54 .539177 53 .538703 52 .538230 51 .537758 50 10.5372&5 49 .536814 ' 48 .536:342 47 .535872 46 .535401 45 .534931 44 .534461 43 .533992 42 .533523 ! 41 .533055 I 40 10.532587 39 .532120 \ 38 .531653 j 37 .531186 36 .530720 35 .530254 ! 34 .529789 33 .529324 | 32 .528859 31 .528395 30 10.527931 .527468 .527'005 .526543 i 26 .526081 ! 25 .525619 24 .525158 ' 23 .524697 22 .524237 21 .523777 20 10.523317 19 .522858 ! 18 .522399 17 .521941 16 .521483 15 .521025 14 .5205(58 13 .520111 12 .519655 11 .519199 10 10.518743 ! 9 .518288 .517833 .517379 .516925 .516471 .516018 .515565 .515113 10.514661 Tang. 106 73 -TABLE XXV. -LOGARITHMIC SINES, 107' 9.465935 .466348 .466761 .467173 .467585 .468407 .469637 .470046 9.470455 .470863 .471271 .471679 .472492 .472898 .47*304 .473710 .474115 9.474519 .474923 .475327 .475730 .476133 .47653(5 .477340 .477741 .478142 9.478542 .479342 .479741 .480140 .481334 .481731 .482128 9.482525 .482921 .483712 .484107 .484501 .486075 9.486467 .487251 .487643 .488034 .488424 D. 1". 6.88 6.88 6.87 6.87 6.85 6.85 6.83 6.82 6.82 6.80 6.80 6.80 6.78 6.77 6.77 6.77 6.77 6.75 6.73 6.73 6.73 6.72 6.72 6.72 6.70 6.70 6.67 6.67 6.67 6.65 6.65 6.65 (5.63 6.62 6.60 6.58 6.60 6.58 6.57 6.57 6.57 6.55 6.55 6.53 6.55 6.52 6.53 6.52 6.50 6.50 6.50 6.48 6.48 Cosine. Ccsine. D 1". .980558 .980519 .980480 D.I'. .980247 .980208 .9801130 .980052 .980012 .979973 .979K5o .979816 .979776 .979737 .979658 .979618 .979579 .979499 .979459 .979420 .979380 .979340 .979300 .979220 .979180 .979140 .979100 .979059 .979019 9.978979 .978858 .978817 .978777 .978737 .978655 .978615 9.978574 .978493 .978452 .978411 .978370 .978247 9.978206 .65 .65 .63 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .67 .65 .65 .65 .67 .65 .67 .65 .67 .65 .67 .65 .67 .67 .67 .65 .67 .67 .67 .67 .67 .67 .67 .67 .68 .67 .67 .67 .OY .67 .67 .485791 .486242 .487143 .487593 .488043 .488941 .4;K)733 .491180 .491627 .492.519 .493410 .494299 9.494743 .495186 .496073 .496515 Tang. D.I". .27 ,87 .27 .95 .25 .25 43 .23 .28 .22 M .22 .20 .18 .20 7.18 7.18 7.17 7.17 .497:399 .497'841 .498722 9.499163 .499603 .500042 .500481 .501359 .501797 .502235 .502672 .503109 9.503546 .504418 .504854 .505289 .505724 .506159 .506593 .507027 .507460 .508759 .509191 .510054 .510485 .510916 .511346 9.511776 Cotang. .512857 .512407 10.514661 j 60 .514209 j 59 .513758 68 .5ia307 56 55 .511957 | 54 .511508 .511059 .510610 .510162 10.509714 .509267 .508820 .508373 .507927 I 45 .507481 44 .507035 I 43 .506590 ! 42 .506146 j 41 .505701 j 40 10.505257 39 .504814 i 38 .504370 ; 37 .503927 36 .503485 .35 .508048 34 .502601 I 33 .502159 ' 32 .501718 31 .501278 30 10.500837 .500397 .499958 .499519 .499080 .498641 .498203 .497765 .497328 10.496454 .496018 .495582 .495146 .494711 .494276 .493841 .493407 .492973 .492540 10.492107 .491674 .491241 .490809 .490378 .489946 .489515 .489084 .488654 10.488224 Sine. | D.I". || Cotang. | D.I'. | Tang. 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 ? I 72 18 COSINES, TANGENTS, AND COTANGENTS. UN- Sine. 9.489982 .490371 .490759 .491147 .491535 .491922 9.494236 .495005 .495388 .495772 .496154 .497301 .497682 9.498064 .498444 .499204 .499584 .500342 .500721 .501099 .501476 9.501854 .502231 .502607 .503735 .504110 .504485 .504860 .505234 9.505608 .505981 .506354 .506727 .507099 .507471 .507843 .508214 .508585 .508956 9.509326 .510065 .510434 .510803 .511172 .511540 .511907 .512275 9.512642 Cosine. D. r. 6.48 6.47 6.47 6.47 6.45 6.43 6.45 6.43 6l42 6.42 6.42 6.40 6.37 6.38 6.87 6.37 6.35 6.35 6.33 6.35 6.32 6.33 6.32 . 6.30 6.28 6.30 6.27 6.25 6.25 6.23 6.23 6.22 6.22 6.22 6^20 6.20 6.18 6.18 6.18 6.17 6.17 6.15 6.15 6.15 6.15 6.13 6.12 6.13 6.12 Cosine. D. r. 9.978206 .978165 .978124 .978042 .978001 .977959 .977918 .977877 .977835 .977794 9.977752 .977711 .977628 .977586 .977544 .977503 .977461 .977419 .977377 9.977335 .977293 .977251 .977209 .977167 .977125 .977083 .977041 .976999 .976957 9.976914 .976872 .976787 .976745 .976660 .976617 .976574 .976532 .976446 .976104 .976361 .976:318 .976275 .976146 .976103 9.976060 .976017 .975974 .975930 .975887 .975844 .975800 .975757 .975714 9.975670 D. r. .70 .68 .70 .68 .70 .73 .72 !73 .72 .72 .73 Sine. D. 1". Tang. 9.511776 .512206 .512635 .513064 .513493 .513921 .514349 .514777 .515204 .515631 .516057 9.516484 .516910 .517335 .517761 .518186 .518610 .519034 .519458 .519882 .520305 9.520728 .521151 .521573 .521995 .522417 .523259 .524100 .524520 9.524940 .525359 .525778 .526197 .526615 .527033 .527451 .527868 .528702 9.529119 .529535 .530781 .531196 .531611 .532025 .532439 .532853 9.533266 .533679 .534092 .534504 .534916 .5:35739 .536150 .586561 9.536972 D. Cotang. 6.97 6.95 6.95 6.95 6.95 6.93 6.92 6.92 6.90 6.90 6.90 6.88 6.88 6.88 6.87 6.87 6.87 6.85 6.85 6.85 6,85 .487794 .487365 .484796 10.483516 .483090 .481814 .481390 .480542 .480118 .479695 10.479272 .478849 .478427 .478005 .477583 .477162 .476741 .476320 .475900 .475480 10.475060 '. 474222 .473385 .472967 .472549 .472132 .471715 .471298 10.470881 .47046;-, .470049 .468804 .467975 .467561 .467147 10.466734 Cotang. D. I', .465084 .464672 .464261 .46:3850 Tang. 19* TABLE XXV.-LOGARITHMIC SINES, 160* ' Sine. D. 1". Cosine. D. 1". Tang. D.r. Cotang. ' 9.512642 610 9.975670 9.536972 OO 10.463028 6C 1 2 .513009 .513375 .136 6.10 .975627 .975583 !TS ro .537382 .537792 D.OO 6.83 600 .462618 .462208 59 58 3 .513741 6-IA .975539 . 10 .538202 .00 .461798 57 4 .514107 .1U .975496 .72 .538611 6.82 .461389 56 5 .514472 6.08 6 no .975452 .73 .539020 *%* .460980 55 o .514837 .IR5 6 no .975408 .73 .539429 !* .460571 54 7 8 9 .515202 .515566 .515930 .(JO 6.07 6.07 .975365 .975321 .975277 .72 .73 .73 .539837 .540245 .540653 o.ou 6.80 6.80 .460163 .459755 .459347 53 52 51 10 .516294 6.07 6.05 .975233 .73 .73 .541061 6.80 6.78 .458939 50 11 12 9.516657 .517020 6.05 f* no 9.975189 .975145 .73 9.541468 .541875 6.78 10.458532 .458125 49 48 13 .517382 O.UO * f\K .975101 .73 1 *VQ .542281 Q 'll .457719 47 14 .517745 O.UO a AQ .975057 .10 ' r'n I .542688 . Xr .457312 46 15 16 .518107 .518468 O.Ua 6.02 .975013 .974969 .To 1 .73 .543094 .543499 6 '.75 .456906 .456501 45 44 17 .518829 6.02 / ru> .974925 .73 .543905 K- .456095 43 18 .519190 o.U# .974880 .75 .544310 ! M .455690 42 19 20 .519551 .519911 6.02 6.00 6.00 .974836 .974792 .73 .73 .73 .544715 .545119 O.YO 6.73 6.75 .455285 .454881 41 40 21 22 9.520271 .520631 6.00 5(\Q 9.974748 .974703 .75 9.545524 .545928 6.73 10.454476 .454072 39 38 23 24 .520990 .521349 .9o 5.98 .974659 .974614 .73 .75 .546X31 .546735 6.72 6.73 .453669 .453265 37 36 25 26 27 28 29 .521707 .522066 .522424 .522781 .523138 5.97 5.98 5.97 5.95 5.95 .974570 .974525 .974481 .974436 .974391 .73 .75 .73 .75 .75 .547138 .547540 .547943 .548345 .548747 6.72 6.70 6.72 6.70 6.70 .452862 .452460 .452057 .451655 .451253 35 34 33 32 31 30 .523495 5.95 5.95 .974347 .73 .75 .549149 6.70 6.68 .450851 30 31 32 33 34 35 36 9.523852 .524208 .524564 .524920 .525275 .525630 5.93 5.93 5.93 5.92 5.92 9.974302 .974257 .974212 .974167 .974122 .974077 .75 .75 .75 .75 .75 9.549550 .549951 .550352 .550752 .551153 ,551552 6.68 6.68 6.67 6.68 6.65 10.450450 .450049 .449648 .449248 .448847 .448448 29 28 27 26 25 24 37 .525984 5.90 .974032 .75 .551952 6.67 .448048 23 38 39 40 .526339 .526693 .527046 5^90 5.88 5.90 .973987 .973942 .973897 .75 .75 .75 .75 .552X51 .552750 .553149 6.65 6 65 6.65 6.65 .447649 .447250 .446851 22 21 20 41 9.527400 500 9.973852 9.553548 10.446452 19 42 .527753 .00 .973807 .75 .553946 6.63 .446054 18 43 .528105 5.87 .973761 .77 .554344 6.63 .445656 17 44 45 .528458 .528810 5.88 5.87 5ot .973716 ,. 973671 .75 .75 .554741 .555139 6.62 6.63 6/>o .445259 .444861 16 15 46 .529161 .oO .973625 .77 .555536 .v<4 .444464 14 47 48 49 .529513 .529864 .530215 5.87 5.85 5.85 5 on .97X580 .9735X5 .973489 .75 .75 .77 .555933 .556329 .556725 6.62 6.60 6.60 .444067 .443671 .443275 13 12 11 50 .530565 .00 5.83 .973444 .75 .77 .557121 6.60 6.60 .442879 10 51 9.530915 K 00 9.973398 77 9.557517 6AA 10.442483 9 52 53 .531265 .531614 D.oo 5.82 .973352 .973307 . < .75 .557913 .558308 bu 6.58 .442087 .441692 8 54 55 .531963 .532312 5 '.82 .973261 .973215 .77 .77 fiff .558703 .559097 6.58 6.57 .441297 .440903 6 5 56 57 58 .532661 .533009 .533357 siso 5.80 .973169 .973124 .973078 . it .75 .77 .559491 .559885 .560279 6.57 6.57 6.57 .440509 .440115 .439721 4 3 2 59 60 .533704 9.534052 5^80 .973032 9.972986 .77 .77 .560673 9.561066 6.57 6.55 .439327 10.438934 1 1 Cosine. D.r. Sine, i D. 1". |l Cotang. D. r. Tang. 1 ' 109* [378] 70 20 COSINES, TK, AND COTA.XOENTS. 159 i , Sine. D. 1'. Cosine. D. 1". 1 1 Tang. D.I*. Cotang. , 1 ! 1 9.534052 .534399 5.78 i 9.972986 .972940 .77 9.561066 .561459 6.55 / to 10.438934 .438541 60 59 2 3 .534745 .535092 5.77 5.78 .972894 .972848 .77 .77 .561851 .562244 O.5o 6.55 fi *vi .438149 .437756 58 57 4 .535438 "'ii .97'2802 .77 70 .562636 6fcQ .437364 56 5 .535783 2- 2 .972755 . fO .563028 .Ou ? (* KO .436972 55 6 .536129 , -ii .972709 " fyrr .563419 ; 2'2S .436581 54 7 .536474 2'Xo 1 .972663 r*rf .563811 u.uo 6 to .436180 53 8 .536818 5'^ .972617 . i i *VQ .564202 ,k)Sa 6XO .4357'98 52 9 .537163 ?-JLiJ .972570 . to f*ry .564593 .06 .435407 51 10 .537507 . J'j| .972524 . ii .77 .564983 6.50 6.50 .435017 50 11 9.5J37851 9.972478 ^0 9.565373 6 en 10.434627 49 12 .538194 ?-, .972431 . lo .565763 . OU 6tn .434237 48 13 14 15 16 17 18 .538538 .5:38880 .539223 .539565 .539907 .540249 5.70 5.72 5.70 5.70 5.68 fr fo .972385 .972338 .972291 .972245 .972198 .972151 .77 .78 .78 .77 .78 .78 ! .566153 ! .566542 I .566932 .567320 .567709 ! .568098 .oU 6.48 6.50 6.47 6.48 6.48 6A**r .433847 .433458 .433068 .432680 .432291 .431902 47 46 45 44 43 42 19 20 .540590 .540931 5. Do 5.68 5.68 .972105 .972058 178 .568486 .568873 .4i 6.45 6.47 .431514 .431127 41 40 21 22 23 9.541272 .541613 .541953 5.68 5.67 9.972011 .971964 .971917 .78 .78 ! 9.569261 ! .569648 1 .570035 6.45 6.45 10.430739 .430352 .429965 39 38 37 24 25 26 27 .542293 .542632 .542971 .543310 5.67 5.65 5.65 5.65 .971870 .971823 .971776 .971729 '.78 .78 .78 no .570422 .570809 .571195 .571581 6.45 6.45 6.43 6.43 6.JO .429578 .429191 .428805 .428419 36 35 34 33 28 .543649 5.65 500 .971682 -7o r>o .571967 .4o .428033 32 29 .543987 .DO " 00 .971635 . 1 170 .572352 6 .42 6AO .427648 31 30 .544325 j jj-|g .971588 . i .80 .572738 .4o 6.42 .427'262 30 31 32 9.544663 : .545000 5.62 9.971540 .971493 .78 r*o 9.573123 .573507 6.40 6 /tO 10.426877 .426493 29 28 33 .545338 .971446 . i o QA .573892 .4* 64f\ .426108 27 34 .545674 ! .971398 .OU tyo .574276 .4U fi 40 .425724 26 35 36 .546011 .546347 5^60 5fiA .971351 .971303 i o .80 78 .574660 .575044 O.4U 6.40 600 .425340 .424956 25 24 37 .546683 . OU .971256 . i u QA .575427 .00 6OQ .424573 23 38 .547019 5. 60 .971208 .oU .575810 .OO 6QO .424190 i 22 39 40 .547354 .547689 5.58 5.58 5.58 .971161 .971113 .78 .80 .78 .576193 .576576 .OO 6.38 6.38 .423807 .423424 21 20 41 42 9.548024 .548359 5.58 9.971066 .971018 .80 9.576959 .577341 6.37 10.423041 19 .422659 18 43 44 45 46 .548693 .549027 .549360 .549693 5.57 5.57 5.55 5.55 Sire .970970 .970922 .970874 .970827 .80 .80 .80 .78 OA .577723 .578104 .578486 .57886? 6.37 6.35 6.37 6.35 6 OK .422277 17 .421896 i 16 .421514 15 .4211:33 14 47 48 .550026 .550359 .55 5.55 5tt .970779 .970731 .oU .80 Qf\ .57924* .579629 .DO 6.35 6OQ .420752 .420371 13 12 49 50 .550692 .551024 .OO 5.53 5.53 .970683 .970635 .oU .80 .82 .580009 .580389 . 3O 6.33 6.33 .419991 .419611 11 10 51 9.551356 (r trt 9.970586 QA 9.580769 6QA 10.419231 9 52 .551687 5.5. .970538 .oU .581149 .OO 6.32 .418851 8 53 54 55 .552018 .552349 .552680 5. '52 5.52 .970490 .970442 .970394 !&o .80 .581528 .581907 .582286 6.32 6.32 .418472 .418093 .417714 7 6 5 56 57 .553010 .553341 5^52 5AQ .970345 .970297 !80 on .582665 .583044 6^32 6o/\ .417335 .416956 4 3 58 59 60 .553(570 .554000 9.554329 .4o 5.50 5.48 .970249 .97'0200 9.970152 .oO .82 .80 .58:3422 .583800 9.584177 .60 6.30 6.28 .41G578 .410200 10.415823 2 1 ' Cosine. D. r. Sine. D. r. 1 Cotang. 1 D. 1". Tang. ' 110 69" si- TAfeUC XXV. -LOGARITHMIC SINES, 158 t Sine. D. 1'. Cosine. D.I'. Tang. D. r. Cotang. ' 9.554329 9.970152 9.584177 10.415823 60 1 2 3 4 5 .554658 .554987 .555315 .555643 .555971 5.48 5.48 5.47 5.47 5.47 54*7 .970103 .970055 .970006 .969957 .969909 .82 .80 .82 i .82 ! .80 .584555 .584932 .585309 .585686 .586062 6.30 6.28 6.28 6.28 6.27 6QQ .415445 , 59 .415068 58 .414691 57 .414314 i 56 .413938 55 6 .556299 .4* .969860 . o<^ i .586439 .160 .413561 54 8 .556626 .556953 5.45 5.45 .969811 .969762 .82 .82 .586815 .587190 6.27 6.25 .413185 .412810 53 52 g .557280 5.45 5 JO .969714 .80 .587566 6.27 .412434 51 10 .557606 .4o 5.43 .969665 .82 .82 .587941 6.25 6.25 .412059 50 ll 12 13 14 9.557932 .558258 .558583 .558909 5.43 5.42 5.43 9.969616 .969567 .969518 .969469 .82 .82 .82 oo 9.588316 .588691 .589066 .589440 6.25 6.25 6.23 10.411684 .411309 .410934 .410560 49 48 47 46 15 .559884 5.42 .969420 .o2 .589814 6.23 .410186 45 16 17 .559558 .559883 5.40 5.42 5A(\ .969370 .969321 .83 .82 .590188 .590562 6.23 6.23 .409812 .409438 44 43 18 .560207 .40 .969272 .82 .5909:35 6.22 .409065 42 19 .560531 5.40 5AC\ .969223 .82 .591308 6.22 .408692 41 20 .560855 .40 5.38 .969173 .83 .82 .591681 6.22 6.22 .408319 40 21 9.561178 5OQ 9.969124 9.592054 10.407946 39 22 .561501 .do e QQ .969075 .82 .592426 6.20 600 .407574 38 23 24 25 .561824 .562146 .562468 5. GO 5.37 5.37 .969025 .968976 .968926 .83 .82 .83 .592799 .593171 .593542 .& 6.20 6.18 .407201 37 .406829 36 .406458 35 26 27 28 29 .562790 .563112 .563433 .563755 5.37 5.37 5.35 5.37 500 .968877 .968827 .968777 .968728 .82 .83 .83 .82 .593914 .594285 .594656 .595027 6.20 6.18 6.18 6.18 61Q .406086 .405715 .405344 .404973 34 33 32 31 30 .564075 .00 5.35 .968678 .83 .83 .595398 .lo 6.17 .404602 30 31 9.564396 500 9.968628 9.595768 10.404232 29 32 .564716 .00 t QQ .968578 .83 .596138 6.17 6*n .403862 28 33 .565036 5.OO .968528 .83 .596508 .17 .403492 j 27 34 35 36 37 38 39 .565356 .565676 .565995 .566314 .566632 .566951 5.33 5.33 5.32 5.32 5.30 5.32 OA .968479 .968429 .968379 .968329 .968278 .968228 .82 .83 .83 .83 .85 .83 .596878 .5S7247 .597616 .597985 .598354 .598722 6.17 6.15 6.15 6.15 6.15 6.13 6-tK .403122 .402753 .402384 .402015 .401646 .401278 26 25 24 23 22 21 40 .567269 O.OU 5.30 .968178 .83 .83 .599C91 .15 6.13 .400909 20 41 9.567587 5OQ 9.968128 9.599459 10.400541 19 42 .567904 .!eo 5 on .968078 .83 OK .599827 6.13 610 .400173 18 43 .568222 ,oU .968027 .OO .600194 .1* 61 Q .399806 17 44 45 .568539 .568856 5^28 .967977 .967927 .83 .83 ! .600562 .600929 .lo 6.12 610 .399438 .399071 16 15 46 47 .569172 .569488 5^27 50*^ .967876 .967826 .85 .83 .601296 .601663 .12 6.12 61 A .398704 .398337 14 13 48 49 .569804 .570120 .XI 5.27 .967775 .967725 .85 .83 QK .602029 .602395 .10 6.10 .397971 .397605 12 11 50 .570435 5.25 5.27 .967674 .OO .83 .602761 e!io .397239 10 51 52 53 54 9.570751 .571066 .571380 .571695 5.25 5.23 5.25 9.967624 .967573 .967522 .967471 .85 .85 .85 9.603127 .603493 .603858 .604233 6.10 6.08 6.08 10.396873 .396507 .396142 . 395777 9 8 7 6 55 .572009 5.23 .967421 .83 QK .604588 6.08 6 no .395412 5 56 57 .572323 .572636 5^22 5OQ .967370 .967319 .OO .85 QK .604953 .605317 .Uo 6.07 / AC .395047 .394683 4 3 58 .572950 .figf .967268 .OO QK .605682 u.Uo 607 .394318 2 59 60 .573263 9.573575 5. '20 .967217 9.967166 .OO .85 .606046 9.606410 .Ul 6.07 .393954 10.393590 1 ' ' Cosine. D.I'. Sine. D. 1". '. \ Cotang. D. 1". Tang. ' 111 68' COSINES, TANGENTS, AND COTANGENTS. 157' Sine. D. 1'. Cosine. D.I'. Tang. D.I'. Cotang. 1 2 3 4 9.573575 .573888 .574200 .574512 .574824 5.22 5.20 5.20 5.20 9.967166 .967115 .967064 .967013 .966961 .85 .85 .85 .87 Ofr 9.606410 .606773 .607137 .607500 .607863 6.05 6.07 6.05 6.05 A AQ 10.393590 .393227 .392863 .392500 .392137 60 59 58 57 56 5 6 .575136 .575447 5.20 5.18 54.Q .966910 .966859 .OO .85 ox .608225 .608588 O.Uo 6.05 A AQ .391775 .391412 55 54 7' .575758 .la 51ft .966808 .00 8ty .608950 O.Uo 6 no .391050 53 8 .576069 . lo .966756 t .609312 .Uo .390688 52 9 10 .576379 .576689 5.17 5.17 5.17 .966705 .966653 .85 .87 .85 .609674 .610036 6.03 6.03 6.02 .390326 .389964 51 50 11 12 9.576999 .577309 5.17 51 X 9.966602 .966550 .87 QX 9.610397 .610759 6.03 10.389603 .389241 49 48 13 14 15 .577618 .577927 .578236 .15 5.15 5.15 .966499 .966447 .966395 .OO .87 .87 .611120 .611480 .611841 e!oo 6.02 .388880 .388520 .388159 47 46 45 16 17 18 19 20 .578545 .578853 .579162 .579470 .579777 5.15 5.13 5.15 5.13 5.12 5.13 .966344 .966292 .966240 .966188 .966136 .85 .87 .87 .87 .87 .85 .612201 .612561 .612921 .613281 .613641 Q.OO 6.00 6.(0 6.00 5.98 .387799 44 .387439 43 .387079 42 .386719 41 .386359 40 21 22 9.580085 .580392 5.12 9.966085 .966033 .87 9.614000 .614359 5.98 5f\Q 10.386000 39 .385641 38 23 24 .580699 .581005 5.12 5.10 .965981 .965929 .87 .87 QO .614718 .615077 .98 5.98 .385282 37 .384923 36 25 26 27 28 29 .581312 .581618 .581924 .582229 .582535 5.12 5.10 5.10 5.08 5.10 5 AQ .965876 .965824 .965772 .965720 .965668 .OO .87 .87 .87 .87 .615435 .615793 .616151 .616509 .616867 5^97 5.97 5.97 5.97 5O.X .384565 35 .384207 34 .383849 33 .383491 32 .383133 31 30 .582840 .(Jo 5.08 .965615 .88 .87 .617224 .95 5.97 .382776 30 31 32 9.583145 .588449 5.07 5 no 9.965563 .965511 .87 QQ 9.617582 .617939 5.95 10.382418 28 .382061 28 33 .583754 .Uo .965458 .OO .618295 5 ox .381705 27 34 .584058 5.07 5(\K .965406 .87 QQ .618652 .95 .381348 26 35 .584361 .Uo .965353 .OO .619008 .380992 25 36 37 38 39 40 .584665 .584968 .585272 .585574 .585877 5^05 5.07 5.03 5.05 5.03 .965301 .965248 .965195 .965143 .965090 .87 .88 .88 .87 .88 .88 .619364 .619720 .620076 .620432 .620787 5^93 5.93 5.93 5.92 5.92 .380636 24 .380280 23 .379924 22 .379568 21 .379213 20 41 9.586179 K. f\ t " 9.965037 QQ 9.621142 10.378858 19 42 .586482 O.U5 .964984 .OO .621497 5.92 5 no .378503 18 43 44 .586783 .587085 5.02 5.03 .964931 .964879 .88 .87 .621852 .622207 .92 5.92 .378148 .377793 17 16 45 .587386 5.02 r f\f) .964826 '0% .622561 5 on .377439 15 46 I587888 O .Uo K AO .964773 -8J .622915 .yu Son .377085 14 47 48 49 .5B7989 .588289 .588590 5.U2 5.00 5.02 .964720 -X .964666 uo .964613 'Co .623^69 .623623 .623976 .yu 5.90 5.88 5 on .376731 .376377 .376024 13 12 11 50 .588890 5.00 5.00 .964660 .00 .88 .624330 .yu 5.88 .375670 10 51 9.589190 4 no 9.964507 QQ 9.624683 500 10.375317 9 52 53 .589489 .589789 .yo 5.00 .964454 .964400 .OO .90 QO .625036 .625388 .00 5.87 5 CO .374964 .374612 8 54 .590088 4.98 .964347 .OO OO .625741 .00 5 or* .374259 6 55 56 .590387 .590686 4.98 4 - 9 S .964294 .964240 .00 .90 QQ .626093 .626445 .O< 5.87 5 or* .373907 5 4 57 58 59 60 .590984 .591282 .591580 9.591878 4^97 4.97 4.97 .964187 .964133 .964080 9.964026 .OO .90 ' .88 .90 .626797 .627149 .627501 9.627852 .O< 5.87 5.87 5.85 ! 373803 .372851 .372499 10.372148 3 2 1 ' Cosine. I D. 1". Sine. D. 1'. || Cotang. D. 1'. Tang. ' 112 67 23 TABLE XXV.-LOGARITHMIC SINES, 156' ; Sine. D. r. Cosine. D.I". I Tang. D.I'. Cotang. / 1 9.591878 .592176 4.97 4 OK 9.964026 .963972 .90 00 9.627852 .628203 5.85 5- QK 10.372148 .371797 60 59 2 .592473 .yo .963919 .00 OA .628554 .00 .371446 58 3 4 5 .592770 .593067 .593362 4.95 4.95 4.93 4QO .963865 .963811 .963757 .yo .90 .90 QQ .628905 .629255 .629606 5.85 5.83 5.85 5 CO .371095 .370745 .370394 57 56 55 6 7 .593659 .593955 .yo 4.93 4f\0 .963704 .963650 .00 .90 AA .629956 .630306 .OO 5.83 5QO .370044 .369694 54 53 8 .594251 .yo 4 no .963596 .y(j OA .630656 .OO rr QO .369344 52 9 10 .594547 .594842 .yo 4.92 ! 4.92 .963542 .963488 .yu .90 .90 .631005 .631355 O.O/* 5.83 5.82 .368995 .368645 51 50 11 12 13 14 15 16 17 18 19 20 9.595137 .595432 .595727 .596021 .596315 .5966J9 .596903 .597196 .597490 .597783 4.92 4.92 4.90 4.90 4.90 4.90 4.88 4.90 4.88 4.87 9.963434 .963379 .963325 .963271 .963217 .963163 .963108 .963054 .962999 .962945 .92 .90 .90 .90 .90 .92 .90 .92 .90 .92 9.631704* .632053 .632402 .632750 .633099 .633447 - .6*3795 .634143 .634490 .634838 5.82 5.82 5.80 5.82 5.80 5.80 5.80 5.78 5.80 5.78 10.368296 .367947 .367598 .367250 .366901 .366553 .366205 .365857 .365510 .365163 49 48 47 46 45 44 43 42 41 40 21 22 23 9.598075 .598368 .598660 4.88 4.87 9.962890 .962836 .962781 .90 .92 9.635185 .635532 .635879 5.78 5.78 5r*Q 10.364815 .364468 .364121 39 38 37 24 .598952 4.87 4 Off .962727 *oo .636226 . <0 5 fay .363774 36 25 .599244 .cfi 4017 .962672 .y# QO .636572 . i t 578 .363428 35 26 27 28 29 30 .599536 .599827 .600118 .600409 .600700 .m 4.85 4.85 4.&5 4'.83 .962617 .962562 .962508 .962453 .962398 ,lMi .92 .90 .92 .92 .92 .636919 .637265 .637611 .637956 .638302 . 9.957276 OQ 9.668673 10.331327 60 1 2 .626219 .626490 .O/s 4.52 4 fen .957217 .957158 .yo .98 QO .669002 .669332 5.48 5.50 " AQ .330998 .330668 59 58 3 4 .626760 .627030 .OU 4.50 4K/1 .957099 .957040 . 7O .98 Oft .669661 .669991 o.4o 5.50 5JQ .330339 .330009 57 56 5 6 .627300 .627570 .OU 4.50 4K/\ .956981 .956921 .yo 1.00 no .670320 .670649 .4o 5.48 .329680 .329351 55 54 7 .627840 .OU A JO .956862 .yo QO .670977 5.47 5AQ .329023 53 8 9 10 .628109 .628378 .628647 4 .4O 4.48 4.48 4.48 .956803 .956744 .956684 ,yo .98 1.00 .98 .671306 .671635 .671963 .4o 5.48 5.47 5.47 .328694 .388365 .328037 52 51 50 11 12 13 14 15 16 17 18 19 20 9.628916 .629185 .629453 .629721 .629989 .630257 .630524 .630792 .631059 .631326 4.48 4.47 4.47 4.47 4.47 4.45 4.47 4.45 4.45 4.45 9.956625 .956566 .956506 .956447 .956387 .956327 .956268 .956208 .956148 .956089 .98 1.00 .98 1.00 1.00 .98 1.00 1.00 .98 1.00 9.672291 .672619 .672947 .673274 .673602 .673929 .674257 .674584 .674911 .675237 5.47 5.47 5.45 5.47 5.45 5.47 5.45 5.45 5.43 5.45 10.327709 .327381 .327053 .326726 .326398 .326071 .325743 .325416 .325089 .324763 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 26 27 28 9.631593 .631859 .632125 .632392 .632658 .632923 .633189 .633454 4.43 4.43 4.45 4.43 .42 .43 .42 Af> 9.956029 .955969 .955909 .955849 .955789 .955729 .955669 .955609 1.00 1.00 1.00 1.00 1.00 1.00 1.00 flQ 9.675564 .675890 .676217 .676543 .676869 .677194 .677520 .677846 5.43 5.45 5.43 5.43 5.42 5.43 5.43 10.324436 .324110 .323783 .323457 .323131 .322806 .322480 .322154 39 38 3T 36 35 34 33 32 29 .633719 ' .4/5 .955518 .9o .678171 5.42 .321829 31 30 .633984 .42 .42 .955488 1 .00 1.00 .678496 5.42 5.42 .321504 30 31 32 9.634249 .634514 .42 j(\ 9.955428 .955368 1.00 1/V) 9.678821 .679146 5.42 10.321179 .320854 29 28 33 34 .634778 .635042 ' .W .40 A(\ .955307 .955247 .(JK 1.00 ]AO .679471 .679795 5.42 5.40 5 An .320529 .320205 27 26 35 36 .635306 .635570 ' .4U - .40 .955186 .955126 .Ux5 1.00 .680120 .680444 ,C8 5.40 .319880 .319556 25 24 37 38 .635834 .636097 < .40 .38 .955065 .955005 1.02 1.00 .680768 .681092 5.40 5.40 .319232 .318908 23 22 39 .636360 ' 38 t QQ .954944 1.02 1 02 .681416 5.40 5 An .318584 21 40 .636623 *-. .GO .38 .954883 1 .U 1.00 .681740 .4U 5.38 .818260 20 41 42 43 44 45 46 47 48 49 50 9.636886 .637148 .637411 .637673 .637935 .638197 ! 638720 .638981 .639242 .37 .38 .37 .37 .37 .35 .37 .35 4.35 4.35 9.954823 .954762 .954701 .954640 .954579 .954518 .954457 .954396 .954335 .954274 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 9.682063 .682387 .682710 .683033 .683356 .683679 .684001 .684324 .684646 .684968 5.40 5.38 5.38 5.38 5.38 5.37 5.38 5.37 5.37 5.37 10.317937 .317613 .317290 .316967 .316644 .316321 .315999 .315676 .315354 .315032 19 18 17 16 15 14 13 12 11 10 51 52 53 54 55 56 9.639503 .639764 .640024 .640284 .640544 .640804 4.35 4.33 4.33 4.33 4.33 9.954213 .954152 .954090 .954029 .953968 .953906 1.02 1.03 1.02 1.02 1.03 9. 685290 .685612 .685934 .686255 .686577 .686898 5.37 5.37 5.35 5.37 5.35 10.314710 .314388 .314066 .313745 .313423 .313102 9 8 7 6 5 4 57 58 59 60 .641064 .641324 .641583 9.641842 4.33 4.33 4.32 4.32 .953845 .953783 .953722 9.953660 1.02 1.03 1.02 1.03 .687219 .687540 .687861 9.688182 5.35 5.35 5.35 5.35 .312781 .312460 .312139 10.311818 3 2 1 ' Cosine. D. r. Sine. D. 1. 1 Cotang. D. r. Tang. 1 115= 6i* 26 COSINES, TANGENTS, AND COTANGENTS. 153* ' Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. 9.641842 4QO 9.953660 9.688182 500 10.311818 fifl 1 2 3 4 5 .642101 .642:360 .642618 .642877 .643135 .96 4.32 4.30 4.32 4.30 .953599 .953537 .953475 .953413 .953352 li03 i.as 1.03 1.02 i OH .688502 .688823 .689143 .689463 .689783 .00 5.32 5.33 5.33 5.33 500 .311498 .311177 .310857 .310537 .310217 59 58 57 56 55 6 .643:393 \0 .953290 1 . UO .690103 .00 500 .309897 54 .643050 4.28 .953228 1 .03 1 OH .690423 .00 5QO .309577 53 8 .643908 4OQ .953166 1 . Uo .690742 .0% 500 .309258 f\9. 9 .644165 ./O 4 on .953104 ' Ao .691062 .00 .308938 *T 10 .644423 .oU 4.28 .953042 I'M .691381 5.32 5.32 .308619 50 11 12 13 9.644680 .644936 .645193 4.27 4*00 9.952980 .952918 952855 1.03 1.05 1 03 9.691700 .692019 .6923_8 5.32 5.32 5 on lo.soasoo .307981 .307662 49 48 47 14 15 .645450 .645706 4 .*o 4.27 437 .952793 .952731 im '. 692975 BU 5.32 5BVI .807:344 .807025 46 45 16 17 18 19 20 .645962 .646218 .646474 .646729 .646984 . ffj 4.27 4.27 4.25 4.25 4.27 .952669 .952606 .952544 .952481 .952419 l'05 1.03 1.05 1.03 1.05 .693293 .693612 .693930 .694248 .694566 . oil 5.32 5.30 5.30 5 30 5.28 .808707 .306388 .30(i070 .305752 .305434 44 43 42 41 40 21 9.647240 A 7Q 9.952356 OH 9.694883 10.305117 39 22 23 24 25 .647494 .647749 .648004 .648258 4 .&o 4.25 4.25 4.23 400 .952294 .952231 .952168 .952106 ros 1.05 1.03 1 nx .695201 .695518 .695836 .696153 5^28 5.30 5.28 .304799 .304482 .304164 .303847 38 37 36 35 26 27 28 29 30 .648512 .648766 .649020 .649274 .649527 .o 4.23 4.23 4.23 4.22 4.23 .952043 ! 931917 .951854 .951791 1 .Uo 1.05 1.05 1 05 1.05 1.05 .696470 .696787 .697103 .697420 .697736 5.28 5.28 5.27 5.28 5.27 5.28 .303530 .303213 .302897 .302580 .302264 34 33 32 31 30 31 32 33 34 35 36 9.649781 .650034 .650287 .650539 .650792 .651044 4.22 4.22 4.20 4.22 4.20 4 on 9.951728 .951665 .951602 .951539 .951476 .951412 1.05 1.05 1.05 1.05 1.07 9.698053 .698369 1 .698685 .699001 i .699316 .699632 5.27 5.27 5.27 5.25 5.27 10.301947 .301681 .301315 .300999 .300684 .300368 29 28 27 26 25 24 37 38 39 40 .651297 .651549 .651800 .652052 . v 4.20 4.18 4.20 4.20 .951849 ! 951286 .951322 .951159 i;s 1.0? 1.05 1.05 .699947 .700263 .700578 .700893 5.25 5.27 5.25 5.25 5.25 .300053 .299787 .299422 .299107 23 22 21 20 41 42 9.652304 .652,555 4.18 9.951096 .951032 1.07 1 07 9.701208 .701523 5.25 10.298792 .298477 19 18 43 44 .652806 .653057 4 . 18 4.18 4 -JO .950968 .950905 1 ."' 1.05 1 07 .701837 .702152 5.23 5.25 .298163 .297848 17 16 45 46 47 48 49 50 653308 .653558 .653808 .654059 .654309 .654558 . lo 4.17 4.17 4.18 4.17 4.15 4.17 .950841 .950778 .950714 .950650 .950586 .950522 1 ."' 1.05 1.07 1.07 1.07 1.07 1.07 .70246(5 ! 70309.3 .703409 .703722 .704036 5 . 23 5.25 5.23 5.23 5.22 5.23 5.23 .297534 .297219 ! .296905 .296591 .296278 .295964 15 14 13 12 11 10 51 52 53 54 55 9.654808 .655058 .655307 .655556 .655805 4.17 4.15 4.15 4.15 4-tK 9.950458 .950394 .950330 .950266 .950202 1.07 1.07 1.07 1.07 9.704350 .71)4663 .704976 .705290 .705603 5.22 5.22 5.23 5.22 10.295650 .295337 ; .295024 .294710 .294397 9 8 7 6 5 56 57 .656054 656302 .15 4.13 .950138 .950074 1.'07 .705916 .706338 5.' 20 .294084 .293772 4 3 58 59 60 .656551 .656799 9.657047 4. 15 4.13 4.13 .950010 .949945 9.949881 1 .07 1.08 1.07 .706541 .706854 V. 7071 66 5.23 5.22 5.20 .29:3459 .293146 10.292834 2 1 ' Cosine. D. r. Sine. D. 1'. I Cotang. D. 1", 1 Tang, ' 116* 63' 27 TABLE XXV. -LOGARITHMIC SINES, 152 1 Sine. D. 1". Cosine. D. r. Tang. D.I". Cotang. ' 1 2 9.657047 .657295 .657542 4.13 4.12 A 19 9.949881 .949816 .949752 1.08 1.07 1" (Y? 9.707166 .707478 .707790 5.20 5.20 10.292834 60 .292522 59 .292210 58 3 4 5 6 7 8 9 10 .657790 .658037 658284 .658531 .658778 .659025 .659271 .659517 i. lu 4.12 4.12 4.12 4.12 4.12 4.10 4.10 4.10 .949688 .949623 .949558 .949494 .949429 .949364 .949300 .949235 .UY 1.08 1.08 1.07 1.08 1.08 1.07 1.08 1.08 .708102 .708414 .708726 .709037 .709349 .709660 .709971 .710282 5.20 5.20 5.20 5.18 5.20 5.18 5.18 5.18 5.18 .291898 .291586 .291274 .290963 .280651 .290340 .290029 .289718 57 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 20 9.659763 .660009 .660255 .660501 .660746 .660991 .661236 .661481 .661726 .661970 4.10 4.10 4.10 4.08 4.08 4.08 4.08 4.08 4.07 4.07 9.949170 .949105 .949040 .948975 .948910 .948845 .948780 .948715 .948650 .948584 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.10 1.08 9.710593 .710904 .711215 .711525 .711836 .7"12146 .712456 .712766 .713076 .713386 5.18 5.18 5.17 5.18 5.17 5.17 5.17 5.17 5.17 5.17 10.289407 .289096 .288785 .288475 .288164 .287854 .287544 .287234 .286924 .286614 49 48 47 46 45 44 43 42 41 40 21 9.662214 A no 9.948519 Ino 9.713696 10.286304 39 22 23 .662459 .662703 5.UU 5.00 4.98 .261029 .260729 .260430 16 15 14 47 .682595 3.83 .942726 1.15 .739870 5.00 .260130 13 48 49 50 .682825 .683055 .683284 3.83 3.83 3.82 3.83 .942656 .942587 .942517 1.17 1.15 1.17 1.15 .740169 .740468 .740767 4 .98 4.98 4^98 .259831 .259532 .259233 12 11 10 51 52 9.6&3514 .683743 3.82 9.942448 .942378 1.17 ! 9.741066 .741365 10.258934 .258635 9 8 53 54 .683972 .684201 3.82 3.82 .942308 .942239 1.17 1.15 .741664 .741962 4.98 4.97 .258336 .258038 7 6 55 .684430 3.82 .942169 1 .17 .742261 4.98 4 on .257739 5 56 57 58 .684658 .684887 .685115 3.80 3.82 3.80 3Qf\ .942099 .942029 .941959 1.17 1.17 .742559 .742858 .743156 .97 4.98 4.97 A 07 .257441 .257142 .256844 4 3 2 59 60 .685343 9.685571 .oU 3.80 .941889 9.941819 1 .17 1.17 .743454 9.743752 4. tfi 4.97 .256546 10.256248 1 ' Cosine. D.I". Sine. D. r. Cotang. D.I". Tang. ' 61' TABLE XXV. LOGARITHMIC SINES, 37 '9 Sine. 9.6a5571 .686027 .680709 .687163 .687616 .687843 .688747 .688972 .690548 .690772 .691220 .691444 .691892 .692115 .692785 .693453 .694342 .694564 9.694786 .695229 .695450 .696113 .696554 .696775 .697215 .697435 .697654 .69801)4 .698751 Cosine. D.I'. 3.80 3.78 3.80 3.78 3.78 3.78 3.77 3.78 3.78 3.77 3.77 3.77 3.77 3.75 3.77 3.75 3.75 3.75 3.75 3.75 3.75 3.73 3.73 3.73 3.73 3.73 3.73 3.72 3.73 3.72 3.72 3.72 3.72 3.70 3.72 3.70 3.70 3.70 3.70 3.70 3.68 3.70 3.68 3.67 3.68 3.67 3.67 3.67 3.65 3.67 3.67 3.65 3.65 3.65 3.65 D. 1". Cosine. 9.941819 .941749 .941679 .941609 .941539 .941469 .941398 .941328 .941258 .941187 .941117 9.941046 .940975 .940763 .940693 .940622 .940551 .940480 .940409 .940267 .940196 .940125 .940054 9.939625 .939554 .939410 .938475 9.938185 .938113 .937967 D. 1". .937749 .937676 .937604 9.937531 1.17 1.17 1.17 1.17 1.17 1.18 1.17 1.17 1.18 1.17 1.18 1.18 1.17 1.18 1.18 1.17 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.20 1.18 1.18 1.20 1.18 1.20 1.18 1.20 1 20 1.18 1.20 1.20 1.20 1.18 1.20 1.20 1.20 1.22 1.20 1.20 1.20 1.20 1.22 1.20 1.20 1.22 1.20 1.22 1.22 1.20 1.22 1.22 1.22 1.20 1.22 Tang. 9.743752 .744050 .744348 .744645 .744943 .745240 .745538 .745835 .745132 .746726 ). 747023 .747319 .747616 .747913 .748209 .748505 .748801 .749097 .749689 ). 749985 .750281 .750576 .750872 .751167 .751462 .751757 .752052 .752347 .752642 1.752937 .753231 .753526 .753820 .754115 .754409 .754703 .754997 .755291 .755585 1.755878 .756172 .756465 .756759 .757052 .757345 .757638 .757931 .758224 .758517 .758810 .759102 D.I' .759687 .759979 .760273 .760856 .761148 I 9.761439 Sine. I D. 1". il Cotang. 4.97 4.97 4.95 4.97 4.95 4.97 4.95 4.95 4.95 4.95 4.95 4.93 4.95 4.95 4.93 4.93 4.93 4.93 4.93 4.93 4.93 4.93 4.92 4.93 4.92 4.92 4.92 4.92 4.92 4.92 4.92 4.90 4.92 4.90 4.92 4.90 4.90 4.90 4.90 4.90 4.88 4.90 4.88 4.90 4.88 4.88 4.88 4.88 4.88 4.88 4.88 4.87 4.88 4.87 4.87 4.88 4.87 4.87 4.87 4.85 Cotang. 10.256248 .255950 .255652 .255355 .255057 .254760 .254462 .254165 .253868 .253571 .253274 10.252977 .252681 .252384 .252087 .251791 .251495 .251199 .250903 .250607 .250311 10.250015 .249719 .249424 .249128 .248243 33 .247948 32 .24765,3 j 31 .247358 I 30 10.247063 i 29 .246769 i 28 .246474 27 .246180 .245885 .245591 .245297 .245003 .244709 .244415 10.244122 .243828 .243535 16 .242948 I 15 .242655 | 14 .242362 I 13 .242069 i 12 .241776 ! 11 .241483 10 10.241190 .240898 .240605 .240313 .239436 .239144 .238852 10.238561 Tang. 60 COSINES. TAW GENTS, AND COTANGENTS. ' Sine. * Cosine. D.I'. Tang. D. r. Cotang. ' 1 C. 698970 .699189 3.65 3 CO 9.937531 .937458 .22 9.761439 .761731 4.87 4o>~ 10.238561 .238269 60 59 2 .699407 .DO 3f*K .937385 .22 .762023 .87 4 OK .237977 58 3 4 .699626 .699844 .DO 3.63 31*0 .937318 .937238 !33 .762314 .762606 .00 4.87 .237686 .237394 57 56 5 .700062 .DO 3/o .937165 ! .22 .762897 4.85 4 OK. .237103 55 6 7 .700280 .700498 .DO 3.63 .937092 .937019 '22 .763188 .763479 .OO 4.85 4 OK .236812 .236521 54 53 8 .700716 Q PO .936946 ' .763770 .o> 4 OK .236230 52 9 .700933 ass .936872 09 .764061 .OO 4 OK .235939 51 10 .701151 3^02 .936799 '.23 .764352 .oO 4.85 .235648 50 11 9.701368 SAO 9.936725 98 9.764643 4QO 10.235357 49 12 13 14 .701585 .701802 .702019 . U-v 3.62 3.62 3 62 .936652 .936578 .936505 . - .23 .22 .764933 .765224 .765514 .00 4.85 4.83 4QK .235067 .234776 .234486 48 47 46 15 .702236 .936431 o'i .765805 .oO 400 .234195 45 16 .702452 Q-f*O .936357 OO .766095 .00 400 .233905 44 17 .702609 3fiO .936284 el .766385 . OO 4QO .233615 43 18 .702885 . OU 3AA .936210 'ol .766675 .OO 400 .233325 42 19 20 .703101 .703317 . OU 3.60 3.60 .936136 .936062 '.<& .23 .700905 .767255 .00 4.83 4.83 .233035 .232745 41 40 21 9.703533 3fiO 9.9a5988 9.767545 10.232455 39 22 .703749 . OU 3 pro .9:35914 OQ .767834 4QQ .232166 38 23 .703964 .OO 3 to .935840 *oo .768124 .OO .231876 37 24 .704179 .00 .935766 .46 .768414 4.83 .231586 36 25 .704395 3 to .935692 i Q ' .768703 4.82 .231297 35 26 .704610 .00 3 to .985618 0" .7'689!)2 4.82 4 82 .231008 34 27 .704825 . OO 3 to .935543 t>o ' .769281 400 .230719 33 28 29 30 .705040 .705254 .705409 .OO 3.57 3.58 3.57 .9:35469 .935395 .935320 .M .23 .25 .23 i .769571 .769860 .770148 .83 4.82 4.80 4.82 .230429 .230140 .229852 32 31 30 31 9.705683 3KQ 9.9-35246 OK 9.770437 10.229563 29 32 33 .705898 .706112 . Oo 3.57 .9*5171 .935097 . vO .23 .770726 .771015 4.82 .229274 .228985 28 27 34 .706326 3.57 3Kt .9a5022 OQ ! .77130:3 4.80 4QO .228697 26 35 .706539 . OU .934948 ./*O .771592 .06 4 on .228408 25 36 37 38 39 40 .706753 .706967 .707180 .707393 .707606 3^57 3.55 3'55 3.55 3.55 .934873 .9:34798 .934723 .9:34049 .934574 1^25 1.25 1.23 1.25 1.25 .771880 .772168 .772457 .772745 .7730:3:3 .8U 4.80 4.82 4.80 4.80 4.80 .228120 .227832 .227543 .227255 .226967 24 23 22 21 20 41 42 43 44 45 46 47 48 49 50 9.707819 .708032 .708245 .708458 .708670 .708882 .709094 .709306 .709518 .709730 3.55 3.55 3.55 3.53 3.53 3.53 3.53 3.53 3.53 3.52 9.934499 .93-1424 .9:34349 .934274 .934199 .934123 .9:340-48 .933973 .9a3898 .933822 1.25 1.25 1.25 1.25 1.27 1.25 1.25 1.25 1.27 1.25 9.773331 .778608 .773890 .774184 .774471 .774 ->'.( .775040 .775833 .775631 .775908 4.78 4.80 4.80 4.78 4.80 4.78 4.78 4.80 4.78 4.78 10.226679 .226392 .226104 .225816 .225529 .225241 .224954 .224667 .224379 .224092 19 18 17 16 15 14 13 12 11 10 51 52 53 54 55 56 57 58 59 60 9.709941 .710153 .710364 .710575 .710786 .710997 .711208 .711419 .711629 9.711839 3.53 3.52 3.52 3.52 3.52 3.52 3.52 3.50 3.50 9.9a3747 .933071 .933596 .93:3520 933445 .9a3369 .933293 .9:33217 .9.33141 9.9*3066 1.27 1.25 1.27 1.25 1.27 1.27 1.27 1.27 1.25 9.776195 .776482 .770708 .777055 .777342 .777628 .777915 .778201 .778488 9.778774 4.78 4.77 4.78 4.78 4.77 4.78 4.77 4.78 4.77 10.223805 .223518 .223232 .222945 .222658 .222372 .222085 .221799 .221512 10.221226 9 8 7 6 5 4 3 2 1 ' Cosine. D. 1". Sine. D. r. I i Cotang. D.I". Tang. 120 31 TABLE XXV. -LOGARITHMIC SINES, 148 ' Sine. D. 1". Cosine. D.I". Tang. D.I". Cotang. / 1 2 3 4 5 6 7 8 9 10 9.711839 .712050 .712260 .712469 .712679 .712889 .713098 .713308 .713517 .713726 .713935 3.52 3.50 3.48 3.50 3.50 3.48 3.50 3.48 3.48 3.48 3.48 9.933066 .932990 .932914 .932838 .932762 .932685 .932609 .932533 .932457 .932380 .932304 1.27 .27 .27 .27 .28 .27 .27 .27 .28 .27 .27 9.778774 .779060 .779346 .779632 .779918 .780203 .780489 .780775 .781060 .781346 .781631 4.77 4.77 4.77 4.77 4.75 4.77 4.77 4.75 4.77 4.75 4.75 10.221226 .220940 .220654 .220368 .220062 .219797 .219511 .219225 .218940 .218654 .218369 ft 60 59 58 57 56 55 54 53 52 51 50 11 12 13 14 9.714144 .714352 .714561 .714769 3.47 3.48 3.47 3AO 9.932228 .932151 .932075 .931998 .28 : .27 OQ 9.781916 .782201 .782486 .782771 4.75 4.75 4.75 10.218084 .217799 .217514 .217229 49 48 47 46 15 16 .714978 .715186 ,4o 3.47 3Aiy .931921 .931845 ,&j .27 ->w .783056 .783341 4.75 4.75 4(~N. .216944 .216659 45 44 17 18 19 .715394 .715602 .715809 ,4i 3.47 3.45 Q A*y .931768 .931691 .931614 . &$ .28 .28 >W .783626 .783910 .784195 . id 4.73 4.75 .216374 .216060 .215805 43 42 41 20 .716017 O.4i 3.45 .931537 .&$ .28 .784479 4.73 4.75 .215521 40 21 22 9.716224 .716432 3.47 3AK 9.931460 .931383 .28 QQ 9.784764 .785048 4.73 4r*o 10.215236 .214952 39 38 23 24 .716639 .716846 .40 3.45 .931306 .931229 ,*O .28 oo .785332 .785616 :. 10 4.73 .214668 .214384 37 36 25 26 .717053 .717259 3.45 3.43 AK .931152 .931075 .o .28 *s .785900 .786184 4.73 4.73 4r*o .214100 .213816 35 34 27 28 29 30 .717466 .717673 .717879 .718085 o.4O 3.45 3.43 3.43 3.43 .930998 .930921 .930843 .930766 ,/o .28 30 .28 .30 .786468 .786752 .787036 .787319 . (O 4.73 4.73 4.72 4.73 .213532 .213248 .212964 .212681 33 32 31 30 31 32 33 34 35 9.718291 .718497 .718703 .718909 .719114 3.43 3.43 3.43 3.42 3/flQ 9.930688 .930611 .930533 .930456 .930378 .28 .30 .28 30 on 9.787603 .787886 .788170 .788453 .788736 4.72 4.73 4.72 4.72 4 TO 10.212397 .212114 .211830 .211547 .211264 29 28 27 26 25 36 38 39 40 .719320 .719525 .719730 .719935 .720140 .4o 3.42 3.42 3.42 3.42 3.42 .930300 .930223 .930145 .930067 .929989 .ou .28 .30 30 .30 .30 .789019 .789302 .789585 .789868 .790151 .TX 4.72 4.72 4.72 4.72 4.72 .210981 .210698 .210415 .210132 .209849 24 23 22 21 20 41 42 43 44 45 46 9.720345 .720549 .720754 .720958 .721162 .721366 3.40 3.42 3.40 3.40 3.40 3Af\ 9.929911 .929833 .929755 .929677 .929599 .929521 .30 30 '.30 30 30 OQ 9.790434 .790716 .790999 .791281 .791563 .791846 4.70 4.72 4.70 4.70 4.72 4r*{\ 10.209566 .209284 .209001 .208719 .208437 .208154 19 18 17 16 15 14 47 .721570 .4U 3ACI .929442 t O& 30 .792128 . A) A rv\ .207872 13 48 49 .781774 .721978 .4U 3.40 .929364 .929286 !w QO .792410 .792692 4. t(j 4.70 .207590 .207306 12 11 50 .722181 3.38 3.40 .929207 Qi 1.30 .792974 4.70 4.70 .207026 10 51 9.722385 Q QQ 9.929129 100 9.793256 4r?f\ 10.206744 9 52 .722588 O.OO QQ .929050 .06 1 /> .793538 . i\J 4ao .206462 8 53 54 55 56 57 58 59 .722791 .722994 .723197 .723400 .723603 .723805 .724007 O.OO 3.38 3.38 3.38 3.38 3.37 3.37 QQ .928972 .928893 .928815 .928736 .928657 .928578 .928499 1 ."V 1.32 1.30 1.32 1.32 1.32 1.32 1 QO .793819 .794101 .794383 .794664 .794946 .795227 .795508 .DO 4.70 4.70 4.68 4.70 4.68 4.68 4AQ .206181 .205899 .205617 .205336 .205054 .204773 .204492 7 6 5 4 3 2 1 60 9.724210 O.OO 9.928420 1 . .926671 .OO 1 ^ .801955 7 Arc .198045 38 23 .728825 , - KO .857537 ' .40 .142463 14 47 .766949 ??* .909146 .0* .857803 4.43 .142197 13 48 .767124 o'of .909055 .52 .858069 4.43 4AK .141931 12 49 .767300 o oo .908964 j** .858336 .4o 4 MM .141664 11 50 .767475 2^90 ..908873 .53 .858602 .4o 4.43 .141398 10 51 52 53 54 9.767649 .767824 .767999 .768173 2.92 2.92 2.90 9.908781 .908690 .908599 .908507 1.52 1.52 1.53 9.858868 .859134 .859400 .859666 4.43 4.43 4.43 10.141132 .140866 .140600 .140334 9 8 7 6 55 56 57 58 59 .768348 .768522 .768697 .768871 .769045 2^90 2.92 ! 2.90 2.90 .908416 .908324 .908233 .908141 .908049 1 .52 1.53 1.32 1.53 1.53 .859932 .860198 .860464 .860730 .860995 4.43 4.43 4.43 4.43 4.42 .1400G8 .139802 .139536 .139270 .139005 5 4 3 2 1 60 9.769219 2.90 9.907958 1 .52 9.861261 4.43 10.138739 ' Cosine. D. 1". ! Sine. D. 1". Cotang. D. 1". Tang. ' 54 36 . TANGENTS. AND COTANGENTS. 143* ' Sine. D. 1". Cosine. D. 1'. Tang. D. 1". Cotang. ' 1 9.769219 .769393 2.90 9.907958 .907866 .53 to 9.861261 .861527 4.43 10.138739 .138473 60 59 2 3 4 .769566 .769740 .769913 2.88 2.90 2.88 .907774 .907682 .907590 .Do .53 .53 Kf> .861792 .862058 .862323 4^43 4.42 .138208 .137942 .137677 58 57 56 5 6 7 8 9 10 .770087 .770260 .770433 .770606 .770779 .770952 2.90 2.88 2.88 2.88 2.88 2.88 2.88 .907498 .907406 .907314 .907222 .907129 .907037 .53 .53 .53 .53 .55 .53 1.53 .862589 .862854 .863119 .863385 .863650 .863915 4^42 4.42 4.43 4.42 4.42 4.42 .137411 .137146 .136881 .136615 .136350 .136085 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 ,20 9.771125 .771298 .771470 .771643 .771815 .771987 .772159 .772331 .772503 .772675 2.88 2.87 2.88 2.87 2.87 2.87 2.87 2.87 2.87 2.87 9.906945 .906852 .906760 .906667 .906575 .906482 .906389 .906296 .906204 .906111 1.55 1.53 1.55 1.53 1.55 1.55 1.55 1.53 1.55 1.55 9.864180 .864445 .864710 .864975 .865240 .865505 .865770 .866035 .866300 .866564 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4.40 4.42 10.135820 .135555 .135290 .135025 .134760 . 134495 .134230 .1:33965 .133700 .133436 49 48 47 46 45 44 43 42 41 40 21 0.772847 2QK. 9. 06018 IKK 9.866829 10.133171 39 22 23 24 25 26 27 28 29 30 .773018 .773190 .773361 .773533 .773704 .773875 .774046 .774217 .774388 .85 2.87 2.85 2.87 2.85 2.85 2.85 2.85 2.&5 2.83 .905925 .905832 .905739 .905645 .905552 .905459 .905366 .905272 .905179 .55 1.55 1.55 1.57 1.55 1.55 1.55 1.57 1.55 1.57 .867094 .867358 .867623 .867887 .868152 .868416 .868680 .868945 .869209 4^40 4.42 4.40 4.42 4.40 4.40 4.42 4.40 4.40 . 132906 .132642 .132377 .132113 .131848 .131584 . 131320 .131055 .130791 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 40 9.774558 .774729 .774899 .775070 .775240 .775410 .775580 .775750 .775920 .776090 2.85 2. as 2.85 2.83 2.83 2.83 2.83 2.83 2.83 2.82 9.905085 .904992 .904898 .904804 .904711 .904617 .904523 .904429 .904335 .904241 1.55 1.57 1.57 1.55 1.57 1.57 1.57 1.57 1.57 1.57 9.869473 .869737 .870001 .870265 .870529 .870793 .871057 .871321 .871585 .871849 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.40 4.38 10.130527 .130263 .129999 .129735 .129471 . 129207 . 128943 . 128679 .128415 .128151 29 28 27 26 25 24 23 22 21 20 41 42 43 44 9.776259 .776429 .776598 .776768 2.83 2.82 2.83 9.904147 .904053 .903959 .903864 1.57 1.57 1.58 9.872112 .872376 .872640 .872903 4.40 4.40 4.38 4 Aft 10.127888 .127624 .127360 . 127097 19 18 17 16 45 46 47 48 49 50 .776937 .777106 .777275 .777444 .777613 .777781 2'82 2.82 2.82 2.82 2.80 2.82 .903770 .903676 i .903581 .903487 .903392 .903298 l!57 1.58 1.57 1.58 1.57 1.58 .873167 .873430 .8736M .873957 ; .874220 .874484 .4U 4.38 4.40 4.38 4.38 4.40 4.38 .126833 .126570 .126306 .126043 .125780 .125516 15 14 13 12 11 10 51 9.777950 9.903203 1 5ft 9.874747 A QQ 10.125253 9 52 .778119 2' on 1 .903108 IKTf .875010 4.oo 4OQ .1249UO 8 53 .778287 .oO 2 on .903014 .o< ItQ .875273 .00 .124727 7 54 .778455 .oU 2QO .902919 .Do IfcQ .875537 4OQ .124463 6 55 56 .778624 .778792 .ox 2.80 2QA .902824 .902729 .Oo 1.58 1 58 .875800 .876063 . Oo 4.38 4 38 .124200 .123937 5 4 57 778960 . oU .902634 .876326 .123674 3 58 59 60 .779128 .779295 9.779463 2.80 2.78 2.80 .902539 .902444 9.902349 1.58 1.58 1.58 .876589 .876852 9.877114 4.oo 4.38 4.37 .123411 .123148 10.122886 2 1 ' Cosine. D/r. Sine. D. 1". Cotang. D. 1". Tang. 1 126' [395] 53 TABLE SINF.S 142' Sine. 9.779463 .779631 .779798 .779966 .780133 .780300 .780467 .780634 .780801 .780968 .781134 9.781301 .781468 .781634 .781800 .781966 .782132 .782298 .782464 .782796 9.782961 .783127 .783458 .783953 .784118 .784282 .784447 9.784612 .784776 .784941 .785105 .785597 .785761 .785925 9.786252 .786416 .786579 .786742 .786906 .787069 .787232 .787395 .787557 .787720 .788045 .788370 .788532 .789018 .789180 ! Cosine. D. 1" 2.78 2.78 2.78 2.78 2.78 2.78 2.77 2.78 2.78 2.77 2.77 2.77 2.77 2.77 2.77 2.77 2.77 2.75 2.77 2.75 2.77 2.75 2.75 2.75 2.75 2.73 2.75 2.75 2.73 2.75 2.73 2.73 2.73 2.73 2.73 2.73 2.73 2.72 2.73 2.72 2.72 2.73 2.72 2.72 2.72 2.70 2.72 2.72 2.70 2.72 2.70 2.70 2.70 2.70 2.70 2.70 2.70 Cosine. 9.902349 .902158 .901967 .901872 .901776 .901681 .901585 .901490 .901394 .901202 .901106 .901010 .900914 .900818 .900626 .900529 .900240 .900144 .900047 .899757 .899660 .899564 .898787 .898202 .898104 .897810 .897712 .897614 .897516 9.897418 .897222 .897123 .896729 .896631 9.896532 Sine. D. 1". 1.60 1.58 1.58 1.60 1.58 1.60 1.58 1.60 1.58 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.62 1.60 1.60 1.62 1.60 1.62 1.60 1.62 1.62 1.62 1.60 1.62 1.62 1.62 1.62 1.63 1.62 1.62 1.62 1.63 1.62 1.63 1.62 1.63 1.62 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.65 1.63 1.65 1.63 1.65 1.63 1.65 Tang. 9. 8771 14 .877377 .877640 .877903 .878165 ^78428 .878691 .878953 .87'921(5 .879478 .879741 .880790 .881052 .881314 .881577 .881839 .883148 .883410 .883672 .884196 .884457 .884719 .884980 .885504 .885765 .886549 .887072 .887333 9.887855 .889421 D.I". .890204 9.890465 .890725 .891507 .892549 9.892810 4.38 4.38 4.38 4.37 4.38 4.38 4.37 4.38 4.37 4.38 4.37 4.37 4.38 4.37 4.37 4.37 4.38 4.37 4.37 4.37 4.37 4.37 4.a r > 4.37 4.37 4.37 4.37 4.35 4.37 4.35 4.37 4.37 4.35 4.35 4.37 4.35 4.37 4.35 4.35 4.35 4.35 4.35 4.37 4.35 4.35 4.35 4.33 4.35 4.35 4.35 4.35 4.33 4.35 4.35 4. as 4.35 4.33 4.35 4.33 4.35 Cotang. 10.122886 .122623 .122360 .122097 .121835 .121572 .121309 .121047 .120784 .120522 .120259 10.119997 .119735 .119472 .119210 .118948 .118686 .118423 .118161 .117899 .117637 10.117375 .117113 .116852 .116590 .116328 .116066 .115804 .115543 .115281 .115020 10.114758 29 .114496 .114235 .113974 .113712 .113451 .113189 .112928 ] 22 .112667 21 .112406 20 10.112145 .111884 .111622 .111361 .111100 .110839. | 14 .110579 13 .110318 12 .110057 11 .109796 10 10.109535 .109275 .109014 .108753 .108493 .108232 .107972 .107711 .10?'451 10.107190 Cotang. I D. 1". Tang. 127 [396] 52' 38 COSINES, TANGENTS, AND COTANGENTS. 141. / Sine. D. 11. Cosine. D. 1\ Tang. D.I". Cotang. / 1 2 3 4 5 6 7 8 9 10 9.789342 .789504 .789665 .789827 .789988 .790149 .790310 .790471 .790632 .790793 .790954 2.70 2.68 2.70 2.68 2.68 2.68 2.68 2.68 2.68 2.68 2.68 9.896532 .896433 .896335 .896236 .896137 .896038 .895939 .895840 .895741 .895641 .895542 1.65 1.63 1.65 1.65 1.65 1.65 1.65 1.65 1.67 1.65 1.65 9.892810 .893070 .893331 .893591 .893851 .894111 .894372 .894632 .894892 .895152 .895412 4.33 4.35 4.33 4.33 4.33 4.35 4.33 4.33 4.33 4.33 4.33 10.107190 .106930 .106669 .106409 .106149 .105889 .105628 .105368 .105108 .104848 .104588 60 59 58 57 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 20 9.791115 .791275 .791436 .791596 .791757 .791917 .792077 .792237 .792397 .792557 2.67 2.68 2.67 2.68 2.67 2.67 2.67 2.67 2.67 2.65 9.895443 .895343 .895244 .895145 .895045 .894945 .894846 .894746 .894646 .894546 1.67 1.65 1.65 1.67 1.67 1.65 1.67 1.67 1.67 1.67 9.895672 .895932 .896192 .896452 .896712 .896971 .897231 .897491 .897751 .898010 4.33 4.33 4.33 4.33 4.32 4.33 4.33 4.33 4.32 4.33 10.104328 .104068 .103808 .103548 .103288 .103029 .102769 .102509 .102249 .101990 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 26 27 28 29 30 9.792716 .792876 .793035 .793195 .793354 .793514 .793673 .793832 .793991 .794150 2.67 2.65 2.67 2.65 2.67 2.65 2.65 2.65 2.65 2.63 9.894446 .894346 .894246 .894146 .894046 .893946 .893846 .893745 .893645 .893544 1.67 1.67 1.67 1.67 1.67 1.67 1.68 1.67 1.68 1.67 9.898270 .898530 .898789 .899049 .899308 .899568 .899827 .900087 .900346 .900605 4.33 4.32 4.33 4.32 4.33 4.32 4.33 4.32 4.32 4.32 10.101730 .101470 .101211 .100951 .100692 .100432 .100173 .099913 .099654 .099395 39 38 37 36 35 34 33 32 31 30 31 9.794308 32 i .794467 33 .794626 34 .794784 35 .794942 36 .795101 37- i .795259 38 ! .795417 39 .795575 40 .795733 2.65 2.65 2.63 2.63 2.65 2.63 2.63 2.63 2.63 2.63 9.893444 .893343 .893243 .893142 .893041 .892940 .892839 .892739 .8926:38 .892536 1.68 1.67 1.68 1.68 1.68 1.68 1.67 1.68 1.70 1.68 9.900864 .901124 .901383 .901642 .901901 .902160 .902420 .902679 .902938 .903197 4.33 4.82 4.32 4.32 4.32 4.33 4.32 4.32 4.32 4.32 10.099136 .098876 .098617 .098358 .098099 .097840 .097580 .097321 .097062 .096803 29 28 27 26 25 24 23 22 21 20 41 42 43 44 45 9.795891 .796049 .796206 .796364 .796521 2.63 2.62 2.63 2.62 9.892435 .892334 .892233 .892132 .892030 1.68 1.68 1.68 1.70 9.903456 .903714 .903973 .904232 .904491 4.30 4.32 4.32 4.32 10.096544 .096286 .096027 .095768 .095*09 19 18 17 16 15 46 .796679 2K .891929 . 904750 .095250 14 48 49 50 .796836 .796998 .797150 .797307 2.62 2.62 2.62 2.62 .891827 .891726 .891624 .891523 1.68 1.70 1.68 1.70 .9050 '3 .905267 .905526 .905785 4.32 4.32 4.32 4.30 .094992 .094733 .004474 .094215 13 12 11 10 51 52 53 9.797464 .797621 . 797777 2.62 2.60 9.891421 .891319 .891217 1.70 1.70 9.906043 .906302 .906560 4.32 4.30 10.093957 .093698 .093440 9 8 54 55 .797934 .798091 2.62 .891115 .891013 1.70 .906819 .907077 4.30 .093181 .092923 6 5 56 57 58 59 60 .798247 .798403 .798560 .798716 9.798872 2.60 2.62 2.60 2.60 .890911 .890809 .890707 .890605 9.890503 1.70 1.70 1.70 1.70 | .907&36 .907594 .907853 .908111 9.908369 4.30 4.32 4.30 4.30 .092664 .092406 .092147 .091889 10.091631 4 3 2 1 ' Cosine. D. 1'. Sine. 1 D.I". II Cotang. D. 1". Tang. / 128* [397] 39 TABLE XXV. LOGARITHMIC SINES, 140' / Sine. D. 1". Cosine. D.I'. Tang. D. r. Cotang. / 9.798872 9.890503 1 "9 9.908369 , Q0 10.091631 60 1 2 3 4 5 6 8 9 10 .799028 .799184 .799339 .799495 .799651 .799806 .799962 .800117 .800272 .800427 2.60 2.58 2.60 2.60 2.58 2.60 2.58 2.58 2.58 2.58 .890400 .890298 .890195 .890093 .889990 .889888 .889785 .889682 .889579 .889477 1.70 1.72 1.70 1.72 1.70 1.72 1.72 1.72 1.70 1.72 .908628 .908386 .909144 .909402 .909660 .909918 .910177 .910435 .910693 .910951 4.30 4.30 4.30 4.30 4.30 4.32 4.30 4.30 4.30 4.30 .091372 .091114 .090856 .090598 .090340 .090082 .089823 .089565 .089307 .089049 59 58 57 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 20 9.800582 .800737 .800892 .801047 .801201 .801356 .801511 .801665 .801819 .801973 2.58 2.58 2.58 2.57 2.58 2.58 2.57 2.57 2.57 2.58 9.889374 .889271 .889168 .889064 .888961 .888858 .888755 .888651 .888548 .888444 1.72 1.72 1.73 1.72 1.72 1.72 1.73 1.72- 1.73 1.72 9.911209 .911467 .911725 .911982 .912240 .912498 .912756 .913014 .913271 .913529 4.30 4.30 4.28 4.30 4.30 4.30 4.30 4.28 4.30 4.30 10.088791 .088533 .088275 .088018 .087760 .087502 .087244 .086986 .086729 .086471 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 26 27 28 29 30 9.802128 .802282 .802436 .802589 .802743 .802897 .803050 .803204 .803357 .803511 2.57 2.57 2.55 2.57 2.57 2.55 2.57 2.55 2.f-7 2.55 9.888341 .888237 .888134 .888030 .887926 .887822 .887718 .887614 '. 887406 1.73 1.72 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 9.913787 .914044 .914302 .914560 .914817 .915075 .915332 .915590 .915847 .916104 4.28 4.30 4.30 4.28 4.30 4.28 4.30 4.28 4.28 - 4.30 10.086213 .085956 .085698 .085440 .085183 .084925 .084668 .084410 .084153 .083896 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 40 9.803664 .803817 .803970 .804123 .804276 .804428 .804581 .804734 .804886 .805039 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.53 2.55 2.53 9.887302 .887198 .887093 .886989 .886885 .886780 .886676 .886571 .886466 .886362 1.73 1.75 1.73 1.73 1.75 1.73 1.75 1.75 1.73 1.75 9.916362 .916619 .916877 .917134 .917391 .917648 .917906 .918163 .918420 .918677 4.28 4.30 4.28 4.28 4.28 4.30 4.28 4.28 4.28 4.28 10.083638 .083381 .083123 .082866 .082609 .082352 .082094 .081837 .081580 .081323 29 28 27 26 2o 24 23 22 21 20 41 42 43 44 45 46 47 48 49 50 9.805191 .805343 .805495 .805647 .805799 .805951 .806103 .806254 .806406 .806557 2.53 2.53 2.53 2.53 2.53 2.53 2.52 2.53 2.52 2.53 9.886257 .886152 .886047 .885942 .885837 .885732 .885627 .885522 .885416 .885311 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1 .77 1.75 1.77 9.918934 .919191 .919448 .919705 .919962 .920219 .920476 .920733 .920990 .921247 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.27 10.081066 .080809 .080552 .080295 .080038 .079781 .079524 .079267 .079010 .078753 19 18 17 16 15 14 13 12 11 10 51 52 53 54 55 56 57 58 59 60 9.806709 .806860 .807011 .807163 .807314 .807465 .807615 .807766 .807917 9.808067 2.52 2.52 2.53 2.52 2.52 2.50 2.52 2.52 2.50 9.885205 .885100 .884994 .884889 .884783 .884677 .884572 .884466 .884360 9.884254 1.75 1.77 1.75 1.77 1.77 1.75 1.77 1.77 1.77 9.921503 .921760 .922017 .922274 .922530 .922787 .923044 .923300 .923557 9.923814 4.28 4.28 4.28 4.27 4.28 4.28 4.27 4.28 4.28 10.078497 .078240 .077983 .077726 .077470 .077213 .076956 .076700 .076443 10.076186 9 8 6 5 4 3 2 1 ' Cosine. D. 1'. Sine. D. 1". i Cotang. D. 1'. Tang. ' 129 [308! 50' 40 COSINES, TANGENTS, AND COTANGENTS. 139'* ' Sine. D. 1". Cosine. D. r. Tang. D. r. Cotang. / 1 2 3 4 5 6 7 8 9 10 9.808067 .808218 .808368 .808519 .808669 .808819 .808969 .809119 .809269 .809419 .809569 2.52 2.50 2.52 2.50 2.50 2.50 2.50 2.50 2.50 2.50 2.48 9.884254 .884148 .884042 .883936 .883829 .883723 .883617 .883510 .883404 .883297 .883191 1.77 1.77 1.77 1.78 1.77 1.77 1.78 ! 1.77 ! 1.78 ; 1.77 1.78 9.923814 .924070 .924327 .924583 .924840 .925096 .925352 .925609 .925865 .926122 .926378 4.27 4.28 4.27 4.28 4.27 4.27 4.28 4.27 4.28 4.27 4.27 10.076186 .075930 .075673 .075417 .075160 .074904 .074648 .074891 .074135 .073878 .073622 60 59 58 57 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 20 9.809718 .809868 .810017 .810167 .810:316 .810465 .810614 .810763 .810912 .811061 2.50 2.48 2.50 2.48 2.48 2.48 2.48 2.48 2.48 2.48 9.883084 .882977 .882871 .882764 .882657 .882550 .882443 .882336 .882229 .882121 1.78 1.77 1.78 1.78 i 1.78 1.78 1.78 1.78 1.80 1.78 i 9.926634 .926890 .927147 .927403 .927659 .927915- .928171 .928427 .928684 .928940 4.27 4.28 4.27 4.27 4.27 4.27 4.27 4.28 4.27 4.27 10.073366 .073110 .072853 .072597 .072341 .072085 .071829 .071573 .071316 .071060 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 26 9.811210 .811358 .811507 .811655 .811804 .811952 2.47 2.48 2.47 2.48 2.47 9.882014 .881907 .881799 .881692 .881584 .881477 1.78 1.80 1.78 1.80 1.78 ! 9.929196 .929452 .929708 .929964 .930220 .930475 4.27 4.27 4.27 4.27 4.25 10.070804 .070548 .070292 .070036 .069780 .069525 39 38 37 36 35 34 27 28 .812100 .812248 2.47 .881369 .881261 1.80 .930731 .930987 4.27 .069269 .069013 33 32 29 30 .812396 .812544 2.47 2.47 .881153 .881046 1.80 1.78 1.80 .931243 .931499 4.27 4.27 4.27 .068757 .068501 31 30 31 32 33 34 35 36 38 39 40 9.812692 .812840 .812988 .8131:35 .813283 .813430 .813578 .813725 .813872 .814019 2.47 2.47 2.45 2.47 2.45 2.47 2.45 2.45 2.45 2.45 9.880938 .880830 .880722 .880613 .880505 .880397 .880289 .880180 .880072 .879963 1.80 i 1.80 ' 1.82 1.80 1.80 i 1.80 1.82 i 1.80 1.82 1.80 9.931755 .932010 .932266 .932522 .932778 .933033 .933289 .933545 .933800 .934056 4.25 4.27 4.27 4.27 4.25 4.27 4.27 4.25 4.27 4.25 10.068245 .067990 .0677:34 .067478 .067222 .066967 .066711 .066455 .066200 .065944 29 28 27 26 25 24 23 22 21 20 41 42 43 44 45 46 47 48 49 50 9.814166 .814313 .814460 .814607 .814753 .814900 .815046 .815193 .815339 .815485 2.45 2.45 2.45 2.43 2.45 2.43 2.45 2.43 2.43 2.45 9.879855 i .879746 : .879637 .879529 .879420 .879311 .879202 .879093 .878984 .878875 .82 .82 .80 .82 .82 .82 .82 .82 .82 .82 9.934311 .934567 .934822 .985078 .935333 .9355PQ .935844 .936100 .936355 .936611 4.27 4.25 4.27 4.25 4.27 4.25 4.27 4.25 4.27 4.25 10.065689 .065433 .065178 .064922 .064667 .064411 .064156 .063900 .063645 .063389 19 18 17 16 15 14 13 12 11 10 51 52 9.815632 .815778 2.43 9.878766 .878656 .83 9.936866 .937121 4.25 10.063134 .062879 9 8 53 54 55 56 57 .815924 .816069 .816215 .816361 .816507 2.42 2.43 2.43 2.43 .878547 .878438 .878328 .878219 .878109 .82 .&3 .82 .83 .937377 .937632 .937887 .938142 .938398 4.25 4.25 4.25 4.27 .062623 .062368 .062113 .061858 .061602 7 6 5 4 3 58 .816652 .877999 .938653 .061347 2 59 60 .816798 9.816943 2.42 .877890 9.877780 .83 .938908 9.939163 4.25 .061092 10.060837 ' Cosine. D. 1". Sine. D. r. Cotang. D. 1'. Tang. ' 130 49- TABLE XXV. -LOGARITHMIC SINES, , Sine. D. r. Cosine. D.I'. Tang. D. 1*. Cotang. ' 1 9.816943 .817088 2.42 9.877780 .877670 1.83 100 9.939163 .939418 4.25 4 OK 10.060837 I .060582 60 59 2 .817233 i'lfl .877560 .OO 1QQ .939673 :.!O 4 OK .060327 58 3 4 .817379 .817524 2^42 .877450 .877340 .OO .83 QQ .939928 .940183 .300 4.25 4 Of .060072 .059817 57 56 5 6 .817668 .817813 2^42 .877230 .877120 .OO .83 QQ .940439 .940694 .wf 4.25 .059561 .059306 ' 55 54 7 .17958 O l-> .877010 .OO Qe .940949 4.25 4 OK .05901 53 j 8 .818103 Sr.wl 21 1 1 .876899 . ot> QQ .941204 .99 .058796 52 ' 9 .818247 .4U .876789 .OO 1 ftR .941459 4OQ .058541 51 10 .818392 2^40 .876678 1 -oO 1.83 .941713 .309 4.25 .058287 50 11 13 9.818536 .818681 2.42 2AI\ 9.876568 .876457 1.85 IGO 9.941968 .942223 4.25 4 OK 10.058032 .057777 49 48 13 14 15 .818825 .818989 .819113 .40 2.40 2.40 .876347 .876236 .876125 .00 1.85 .85 QK i .942478 .942733 .942988 ,aa 4.25 4.25 .057522 .057267 .057012 47 46 45 16 17 18 19 20 .819257 .819401 .819545 .819689 .819832 2.40 2.40 2.40 2.40 2.38 2.40 .876014 .875904 .875793 .875682 .875571 -OO .83 .85 .85 .85 .87 .943243 .943498 .943752 .944007 .944262 4.25 4.25 4.23 4.25 4.25 4.25 .056757 .056502 .056248 .055993 .055738 44 43 42 41 40 21 9.819976 9.875459 OK 9.944517 400 10.0554a3 39 22 23 .820120 .820263 2.40 2.38 2QQ .875348 .875237 .OO .85 DK .944771 .945026 .46 4.25 4 OK .05522!) .054974 38 37 24 .820406 .OO 2A(\ .875126 .00 art ; .945281 .Sso 4OQ .054719 36 25 26 27 .820550 .820693 .820836 .4U 2.38 2.38 290 .875014 .874903 .874791 .of .85 .87 oe ! .945535 .945790 .946045 .&J 4.25 4.25 4OQ .054465 .054210 .053955 35 34 33 28 .820979 .OO .874680 .OO 07 .946299 .M .053701 32 29 30 .821122 .821265 2^38 2.37 .874568 .874456 . Ol .87 .87 .946554 .946808 4^23 4.25 .053446 .053192 31 30 31 9.821407 2QQ 9.874344 Li- 9.947063 4 OK 10.052937 29 32 .821550 .OO 2OQ .874232 .o7 QE .947318 .&) 4 00 .052682 28 33 34 .821693 .821835 .00 2.37 .874121 .874009 .o& .87 .947572 .947827 4 . o 4.25 4OO .052428 .052173 27 26 35 .821977 2.37 .873896 .88 or* .948081 .<> 4QQ .051919 25 36 37 38 .822120 .822262 .822404 2~37 2.37 2 or/ .873784 .873672 .873560 .or .87 .87 .948335 .948590 .948844 . . 4.22 4.23 4.22 400 .037440 .037187 .036933 .036680 28 27 26 25 36 or* .830509 oon/ \ & .tvj 2.28 .866935 QAAQ1 O !93 .963574 Gl-tQQOtt .&o 4.23 .036426 nOftl ^O 24 0o O 38 39 ,ooUO4O .830784 .830921 2.30 2.28 .ouooiy .866703 .866586 .93 .95 . yooo^o .964081 .964335 4.22 4.23 . UoOl 1 4i .035919 .035665 aBa 22 21 40 .831058 2.28 2.28 .866470 .93 .95 .964588 4.22 4.23 .035412 20 41 42 43 9.831195 .831332 .831469 2.28 2.28 2QQ 9.866353 .866237 .866120 .93 .95 no 9.964842 .965095 .965:349 4.22 4.23 4OO 10.035158 .034905 .034651 19 18 17 44 45 46 47 .831606 .8531742 .831879 .832015 .&$ 2.27 2.28 2.27 2OQ .866004 .865887 .865770 .865653 .Ho .95 .95 .95 .965602 .965855 .966109 .966362 . ~-J 4.22 4.23 4.22 400 .034398 .034145 .033891 .033638 16 15 14 13 48 49 50 .832152 .832288 .832425 .&*> 2.27 2.28 2.27 .865536 .865419 .865302 .95 .95 .95 .95 .966616 .966869 .967123 .o 4.22 .23 .22 .033384 .033131 .032877 12 11 10 51 52 53 54 55 56 57 58 59 60 9-832561 .832697 .832833 . 832969 .833105 .833241 .833377 .833512 .833648 9.833783 2.27 2.27 2.27 2.27 2.27 2.27 2.25 2.27 2.25 9.865185 .865068 .864953 .864838 .864716 .864598 .864481 .864363 .864245 9.864127 .95 .97 .95 .95 .97 .95 .97 .97 .97 9.967376 .967629 .967883 .968136 .968389 .968643 .968896 .969140 .969403 9.969656 .22 .23 .22 .22 .23 4.22 4.22 4.23 4.22 10.032624 .032371 .032117 .031864 .031611 .031357 .031101 .030851 .030597 10.030344 9 8 7 6 5 4 3 8 -I ' Cosine. D. 1'. Sine. D. r. \ Cotang. I). 1'. Tang. , i ! 132 C [401] 47' TABLE xxv. -L.OUAKITHMIU SIINES, / Sine. D. 1". Cosine. D.I'. Tang. D. 1". Cotang. ' 1 2 3 4 5 6 7 8 9 10 9.833783 .833919 .834054 .834189 .834325 .8344.60 .834595 .834730 .834865 .834999 .835134 2.27 2.25 2.25 2.27 2.25 2.25 2.25 2.25 2.23 2.25 2.25 9.864127 .864010 .863892 .863774 .863656 .863538 .863419 .863301 .863183 .863064 .862946 1.95 1.97 1.97 1.97 i 1.97 ' 1.98 1.97 i 1.97 1.98 1.97 1.98 9.969656 QfSQQQQ '. 9701 62 .970416 .970669 .970922 .971175 .971429 .971682 .971935 .972188 4.22 4.22 4.23 4.22 4.22 4.22 4.23 4.22 4.22 4.22 4.22 10.030344 .030091 .OS9838 .029584 .029*31 .029078 .028825 .028571 .028318 .028065 .027812 60 59 58 57 56 55 54 53 52 51 50 11 12 13 14 15 16 17 18 19 9.835269 .835403 .835538 .835672 .835807 .835941 .836075 .836209 .836343 2.23 2.25 2.23 2.25 2.23 2.23 2.23 2.23 9.862827 .862709 .862590 .862471 .862353 .862234 .862115 .861996 .861877 1.97 1.98 1.98 1.97 1.98 1.98 1.98 1.98 9.972441 .972695 .972948 .973201 .973454 .973707 .973960 .974213 .974466 4.23 4.22 4.22 4.22 4.22 4.22 4.22 4.22 10.027559 .027305 .027052 .026799 .026546 .026293 .026040 .025787 .025534 49 48 47 46 45 44 43 42 41 20 .836477 2.23 2.23 .861758 1.98 2.00 .974720 4.23 4.22 .025280 40 21 22 23 24 25 26 27 28 29 30 9.836611 .836745 .836878 .837012 .837146 .837279 .837412 .837546 .837679 .837812 2.23 2.22 2.23 2.23 2.22 2.22 2.23 2.22 2.22 2.22 9.861638 .861519 .861400 .861280 .861161 .861041 .860922 .860802 .860682 .860562 1.98 1.98 2.00 1.98 2.00 1.98 2.00 2.00 2.00 2.00 9.974973 .975226 .975479 .975732 .975985 .976238 .976491 .976744 .976997 .977250 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4 22 10.025027 .024774 .024521 .024268 .024015 .023762 .023509 .023256 .023003 .022750 39 38 37 36 35 34 33 32 31 30 31 32 00 9.837945 .838078 QOQO1 1 2.22 2.22 9.860442 .860322 ftAAOHO 2.00 2.00 9.977503 Q77? 4 - 22 . yt 1 1 oo x 99 Qr~o/YKl 4 . 64 10.022497 .022244 HO1 QQ1 29 28 or* do 34 35 36 37 38 39 40 . 000/411 .838344 .838477 .838610 .838742 .838875 .839007 .839140 2.22 2.22 2.22 2.20 2.22 2.20 2.22 2.20 .ODi^U^ .860082 .859962 .859842 .859721 .859601 .859480 .859360 2.00 2.00 2.00 2.02 2.00 2.02 2.00 2.02 .y 10.019967 19 42 43 44 45 46 47 48 .839404 .839536 .839668 .839800 .839932 .840064 .840196 .&) 2.20 2.20 2.20 2.20 2.20 2.20 .859119 .858998 .858877 .858756 .858635 .858514 .858393 z.uu 2.02 2.02 2.02 2.02 2.02 2.02 .980286 .980538 .980791 .981044 .981297 .981550 .981803 .2 4.20 4.22 4.22 4.22 4.22 4.22 .019714 .019462 .019209 .018956 .018703 .018450 .018197 18 17 16 15 14 13 12 49 50 .840328 .840459 2.20 2.18 2.20 .858272 .858151 2.02 2.02 2.03 .982056 .982309 4.22 4.22 4.22 .017944 ,017691 11 10 51 52 9.840591 .840722 2.18 2OA 9.858029 .857908 2.02 9.982562 .982814 4.20 400 10.017438 .017186 9 8 53 54 55 56 57 58 59 60 .840854 .840985 .841116 .841247 .841378 .841509 .841640 9.841771 .SO 2.18 2.18 2.18 2.18 2.18 2.18 2.18 .857786 .857665 .857543 .857422 .857300 .857178 .857056 9.856934 2.03 2.02 2.03 2.02 2.03 2.03 2.03 2.03 .983067 .983320 .983573 .983826 .984079 .984332 .984584 9.984837 .vti 4.22 4.22 4.22 4.22 4.22 4.20 4.22 .016933 .016680 .016427 .016174 .015921 .015668 .015416 10.015163 7 6 5 I 2 1 / Cosine. 1 D. 1'. Sine. D. 1". Cotang. D. 1". Tang. ' 133 [402] 14= COSINES, TANGENTS, AND COTANGENTS. 135* ,' Sine. D. r. Cosine. D. 1". Tang. D. r. Cotang. > 1 2 9.841771 .841902 .842033 2.18 2.18 o 1 r* 9.856934 .856812 .856690 2.03 2.03 2AO 9.984837 .985090 .985343 4.22 4.22 4OO 10.015163 .014910 .014657 60 59 58 3 4 5 6 .842163 S'Jo .842294 J'g .8424.24 *'}! .842555 ~' J .856568 .856446 .856323 .856201 . Uo 2.03 2.05 2.03 .985596 .985848 .986101 986354 .SSe 4.20 4.22 4.22 .014404 .014152 .013899 .013646 57 56 55 54 .843685 i o 17 .856078 ~ ":> .986607 4.22 .01&393 53 8 .842815 S'ia .855956 S'JS .986860 4.22 .013140 52 9 .842946 |-}2 .855833 j'jfi .987112 4.20 .012888 51 10 .843076 ~-{X OKK--I i 2.06 .855311 2.05 .987365 4.22 4.22 .012635 50 11 12 13 14 15 16 17 9.843206 .843336 .843466 .84595 .843725 .843855 .843984 2.17 2.17 2.15 2.17 2.17 2.15 O 1*7 9.855588 .855465 .855342 .855219 .855096 .854973 .854850 2.05 2.05 2.05 2.05 2.05 2.05 9.987618 .987871 .988123 .988376 .988629 .988882 .989134 4.22 4.20 4.22 4.22 4.22 4.20 10.012382 .012129 .011877 .011624 .011371 .011118 .010866 49 48 47 46 45 44 43 18 .844114 ^ .if 2-tK .854727 2.05 .989387 4.22 .010613 42 19 20 .844243 .844372 . 10 2.15 2.17 .854603 .854480 2.07 2.05 2.07 .989640 .989893 4.22 4.22 4.20 .010360 .010107 41 40 21 22 23 24 9.844502 .844631 .844760 .844889 2.15 \ 2.15 t 2.15 21 f^ 9.854356 .854233 .854109 .853986 2.05 2.07 2.05 9.990145 .990398 .990651 .990903 4.22 4.22 4.20 10.009855 .009602 .009349 .009097 39 38 37 36 25 26 27 28 29 30 .845018 .845147 .845276 .845405 .845533 .845662 . 10 2.15 2.15 2.15 2.13 2:15 2.13 .853862 .853738 .853614 .853490 .858866 .853242 2.07 2.07 2.07 2.07 2.07 2.07 2.07 .991156 .991409 .991662 .991914 .992167 .992420 4.22 4.22 4.22 4.20 4.22 4.22 4.20 .008844 .008591 .008338 .008086 .007833 .007580 35 34 33 32 31 30 31 32 33 34 35 36 .37 38 39 40 9.845790 .845919 .846047 .846175 .846304 .846432 .846560 .846688 .846816 .846944 2.15 2.13 2.13 2.15 2.13 2.13 2.13 2.13 2.13 2.12 9.853118 .852994 .852869 .852745 .852620 .852496 .852371 ! 85224? .852122 .851997 2.07 2.08 2.07 2.08 2.07 2.08 2.07 2.08 2.08 2.08 9.992672 .992925 .993178 .993431 .993683 .993936 .994189 .994441 .994694 .994947 4.22 4.22 4.22 4.20 4.22 4.22 4.20 4.22 4.22 4.20 10.007328 .007075 .006822 .006569 .006317 .006064 .005811 .005559 .005306 .005053 29 28 27 26 25 24 23 22 21 20 41 42 43 44 45 46 47 9.847071 .847199 .847327 .847454 .847582 .847709 .847836 2.13 2.13 2.12 2.13 2.12 2.12 9.851872 .851747 .851623 .851497 .851372 .851246 .851121 2.08 2.08 2.08 2.08 2.10 2.08 9.995199 .995452 .995705 .995957 .996210 .996463 .996715 4.22 4.22 4.20 4.22 4.22 4.20 10.004801 .004548 .004295 .004043 .003790 .003537 ; 003285 19 18 17 16 15 14 13 48 .847964 2. 13 (t -tn .850996 2.08 .996968 4.22 .003032 12 49 50 .848091 g'iS .848218 };g .a50870 .850745 2.10 2.08 2.10 .997221 .997473 4.22 4.20 4.22 .002779 .002527 11 10 51 52 9.848345 19 .848472 S to 9.850619 .850493 2.10 9.997726 .997979 4.22 10.002274 .002021 9 8 53 54 55 56 57 .848599 i S'iS .848726 g-g .848852 i |'}J .848979 *}* .849106 i | g ! 850368 .850242 .850116 .849990 .849864 2 08 2.10 i 2.10 - 2.10 2.10 2 in .998231 .998484 .'.MI8T37 .998989 .999242 4.20 4.22 4.22 4.20 4.22 .0017(59 .001516 .001263 .001011 .000758 7 6 5 4 3 58 .849383 5-J2 .849738 .10 .999495 4.22 .000505 2 59 60 .849359 9.849485 a. MX 2.10 .849611 9 849485 <,' in .999747 g ' 10 10.000000 4.20 4,22 .000253 10.000000 1 ' Cosine. D. 1". Sine. 1 D. 1". \\ Cotang. D.I". Tang. ' TABLE XXVI -LOGARITHMIC VERSED SINES. TABLE XXVI.- LOGARITHMIC VERSED SINES 0- 1 // ' \ Vers. q-2l Ex. sec. // ' Vers. q-2l Ex. sec. 1 9.070 9.070 o Inf. neg. 1201 120 Inf. neg. 3600 6.182714 109 175 6.182780 60 1 2.626422 120 120 2.626422 3660 1 .197071 108 : 177 .197139 120 2 3.228482 120 120 3.228482 3720 2 .211194 108 179 .211264 180 3 1 .580665 120 120 i .580665 3780 3 .225091 108. 181 ,.225164 240 4 ! 3. 830542 120 i 130 8.880542 3840 4 .238770 107 182 .238845 300 5 4.0-JI.-MW 120 120 4.0243(53 3900 5 .252236 107 i | 184 .252314 360 6 . 182725 120 120 .182725 3960 6 .265497 106 ! 186 .265577 420 7 .316618 120 120 .316619 4020 7 .278558 106 ! 1 188 .278641 480 8 .432002 120 121 .432603 4080 8 .291426 106 i 191 .291511 540 9 .534907 119 121 .534908 4140 9 .304106 105 i 193 .304193 600 10 .626422 119 121 .626424 4200 10 .316603 105 j 195 .316693 660 11 4.709207 119 122 4.709209 4260 11 6.328923 104 197 6.329016 720 12 .784784 119 122 .784787 4320 12 .341071 104 1 199 .341167 780 13 .854308 119 122 .854312 4380 13 .353052 103 201 .353150 840 14 .918678 119 123 .918681 4440 14 .364869 103 I i 204 .364970 900 15 4.978604 119 123 4.978608 4500 15 .376528 103 ; 206 .376631 960 16 5.034661 119 124 5.034666 4560 16 .388032 102 208 .388138 1020 17 .087319 119 124 .087325 4620 17 .399386 102 211 .399494 1080 18 .136966 119 125 .136972 4680 18 .410593 101 i 213 .410705 1140 19 .183928 119 125 .1839a5 4740 19 .421657 101 j i 215 .421772 1200 20 .228481 119 126 .228488 4800 20 .432583 100 218 .432700 1260 21 5.270859 118 126 5.270868 4860 21 6.443372 100!!220 6.443493 1320 22 .311266 118 127 .311275 4920 22 .454029 099 1 223 .454153 1380 23 .349876 118 128 .349886 4980 23 .464557 099 i 225 .464684 1440 24 .386843 118 129 .386854 5040 24 .474959 098 i228 .475089 1500 25 .422:300 118 129 .422312 5100 25 .485238 098 230 .485371 1560 26 .456367 118 130 .456379 5160 26 .495396 097 233 .495532 1620 27 .489148 118 131 .489161 5220 27 .505438 097 236 .505577 1680 28 .520736 117 132 .520750 5280 28 .515364 096 238 .515506 1740 29 .551216 117 133 .551231 5340 29 .525178 095 :241 .525324 1800 30 .580662 117 134 .580679 5400 30 .534882 095 244 .535031 1860 31 5.609143 117 134 5.609160 5460 31 6.544480 094 247 6.544632 1920 32 .636719 117 135 .636738 5520 32 .553972 094 249 .554128 1980 33 .663447 116 136 .663467 5580 33 .563362 093 252 -.563521 2040 34 .689377 116 137 .689398 5640 34 .572651 C93 255 .572813 2100 35 .714555 116 138 .714577 5700 35 .581842 092 258 .582008 2160 36 .739023 116 140 .739047 5760 36 .590936 092 261 .591106 2220 37 .762821 116 141 .762847 5820 37 .599937 091 i264 .600110 2280 38 .785985 115 142 .786012 5880 38 .608845 090 |267 .609021 2340 39 .808547 115 143 .808575 5940 39 .617663 090 270 .617843 2400 40 .830537 115 144 .830567 6000 40 .626392 089 273 .626575 2460 41 5.851985 115 145 5.852016 6060 41 6.635034 089 276 6.635221 2520 42 .872915 114 147 .872948 6120 42 .64:3591 088 279 1 .643782 2580 43 .893353 114 148 .89&S87 6180 43 .652064 087 282! .652259 2640 44 .913322 114 149 .913357 6240 44 .660456 087 285 .660655 2700 45 .932841 114 151 .932878 6300 45 .668767 086 289 .668970 2760 46 .951931 113 152 .951970 1 6360 46 .677000 085 292 .677206 2820 47 .970611 113 154 .970652 6420 47 .685155 085 295 .685365 2880 48 5.988898 113 155 5.988940 6480 48 .693234 084 298 .693448 2940 49 6.006807 112 157 6.006851 6540 49 .701239 083 302 .701457 3000 50 .024355 112 158 .024401 6600 50 .709171 083 305 .709393 3060 51 6.041555 112 160 6.041602 6660 51 6.717030 082 ! 308 6.717257 3120 52 .058421 111 161 .058470 6720 52 .724820 081 312 .725050 3180 53 .074965 111 163 .075017 6280 53 .732540 081 315 .732775 3240 ! 54 .091201 111 164 .091254 6MO 54 .740192 080 319 .740431 3300 55 .107138 110 166 .107194 6900 55 .747777 079 322 .748020 3360 56 .122789 110 168 .122846 6960 56 .755297 079 326 .755544 3420 57 .138162 110 169 .138222 7020 57 .762752 078 329 .763004 3430 58 .153268 109 171 .153330 7080 58 .770144 077 888 .770400 3540 59 .168116 109 173 .168180 7140 59 .777473 076 1337 .777733 3600 60 6.182714 109 175 6.182780 7200 60 6.784741 076 340 6.785005 n 1 1 " ' 1 [406] - AND EXTERNAL SECANTS. 3 "^ " r Vers. q-2l Ex. sec. " Vers. q-2l Ex. sec. r 9.070 , 9.070* 7200 6. 784741 076 340 6.785005 10800 7.136868 021 616 7.137464 7260 1 .791948 075 344 .792217 10860 1 .141679 019 622 .142281 7320 2 .799096 074 348 .799370 10920 2 .146464 018 627 .147072 7380 3 .806186 073 351 .806464 10980 3 .151222 017 633 .151837 7440 4 .813219 073 355 .813501 11040 4 .155954 016 638 .156577 7500 5 .820194 072 .820482 11100 5 .160661 015 644 .161290 7560 6 .827115 071 363 .827406 11160 6 .165342 014 650 .165978 7620 7 .833980 070 367 .834277 11220 7 .169998 013 655 .170641 7680 8 .840792 070 371 .841093 11280 8 .174630 Oil 661 .175279 7740 9 .847551 069 375 .847857 11340 9 . 179236 010 667 .179893 7800,10 .854257 068 379 .854568 11400 10 .183819 009 673 .184483 7860 11 6.860912 067 3836.861228 11460 11 7.188377 008 679 7.189048 7920 12 .867517 066 387 .867837 11520 12 .192912 007 685 .193589 7980 13 .874071 066 391 .874396 11580 13 .197423 006 690 .198108 8040 14 .880577 065 395 .880907 11640 14 .201910 004 696 .202602 8100:15 .887034 064 399 .887369 11700 16 .206375 003 702 .207074 8160 16 .893443 063 403 .893783 11760 16 .210817 002 708 .211523 8220 17 .899806 062 407 .900151 11820 17 .215236 001 714 .215949 8280 18 .906122 061 411 .906472 11880 18 219633 4-00 720 .220353 8840 19 .912393 061 416 .912748 11940 19 .224007 998 727 .224735 8400 20 .918618 060 420 .918979 12000 20 .228360 997 733 .229095 8460 21 6.924800 059 424 6.925165 12060 21 7.232691 996 739 7.233433 &520 22 .930937 058 429 .931308 12120 22 .237000 995 745 .237750 8580 23 .937032 057 433 .937408 12180 23 .241288 994 751 .242046 8640:24 .943084 056 437 .943465 12240 24 .245555 992 757 .246320 8700125 .949094 055 442 .949480 12300 25 .249801 991 764 .250574 8760 86 .955063 054 446 .955455 12360 26 .254027 990 770 .254807 8820127 .960991 054 451 .961388 12420 27 .258232 989 776 .259019 8880 28 .966879 053 455 .967281 12480 js .262416 987 783 .263212 8940 29 .972727 052 460 .9731-35 12540 29 .266581 986 789 .267384 9000,30 .978536 051 464 .978949 12600 30 .270726 985 795 .271537 9060 '31 6.984306 050 469 6.984725 12660 31 7.274851 983 802 7.275669 9120 32 .990039 049 474 .990463 12720 82 .278956 982 808 .279783 9180 33 6.9957:33 048 478 6.996164 13780 33 .283043 981 815 .283877 9240 34 7.001391 047 483 7.001827 12840 34 .287110 980 821 .287952 9300 35 .007012 046 488 .007454 12900 :-J5 .291158 978 828 .292007 9360,36 .012597 045 493 .013044 12960 3(5 .295187 977 835 .296045 9420,37 .018146 044 497 .018599 13020 37 .299197 976 841 .300063 9480 j 38 .023660 043 502 .024119 13080 38 .303190 974 848 .304063 9540 39 .029139 042 507 .029604 13140 39 .307164 973 855 .308045 9600 40 .034584 041 512 .035054 13200 40 .311119 972 861 .312009 9660 Ul 7.039995 040 517 7.040471 13260 41 7.315057 970 868 7.315955 9720 42 .045372 039 522 .045854 13320 42 .318977 969 875 .319883 9780 43 .050716 038 527 .051204 18380 43 .322880 967 882 .323794 9840 44 .056028 037 532 .056522 13440 44 .32670" 966 889 .327687 9900 i 45 .061307 036 537 .061807 13500 45 .330632 965 896 .331563 9960 46 .066554 035 542 .067061 13560 46 .334483 963 902 .335422 10020 i 47 .071770 034 547 .072282 13620 47 .&38316 962 908 .339263 10080:48 .076954 033 552 .077473 13680 48 .342133 961 916 .343089 10140 49 .082108 032 557 .082633 13740 49 .345933 959 923 .346897 10200 50 .087232 031 562 .087763 13800 50 .349716 958 930 .350689 10260 51 7.092325 030 568 7.092862 13860 51 7.353483 956 938 '7.354464 10320 52 .097389 029 573 .097932 13920 .357233 955 945 .358223 10380,53 .102423 028 578 .102973 13980 W .3(i()9()S 953 952 .361966 1 10440 54 .107428 027 J 584 .1079a5 14040 54 .364686 952 959 .365693 10MXV55 .112405 026 589 .112968 14100 55 .36838$) 951 966 .869404 10560 56 .117.353 025 594 .117922 14160 5(5 .:-572<>7<> 949 973 .373100 10620 57 .122273 024 600 .122849 14220 57 .375747 948 981 .376780 10680 58 .127165 023 605 .127748 14280 58 .379403 946 988 .380444 10710 59 .132030 022 611 .132619 14340 59 .383043 945 995 .384094 10800 60 7.136868 021 616 7.137464 14400 i;i) 7.:N<;r,f;s 943 + 03 7.387728 " 1 ' '97089* 1 OTI* TABLE XXVI.^-LOGARITHMIO VERSED SINES 4 o 5 / Vers. D. 1". Ex. sec. D.I". / Vers. D.I". Ex. sec. D. 1". 7.386668 60.17 7.387728 60.32 7.580389 48.15 7.582045 48.33 1 ,390278 59.93 .391347 60.07 i .583278 47.98 .584945 48.17 2 .393874 59.67 .394951 59.82 2 .586157 47.82 .587835 48.00 3 .397454 59.43 .398540 59.57 3 .589026 47.67 .590715 47.87 4 .401020 59.18 .402114 59. as 4 .591886 47.52 .593587 47.70 5 .404571 58.93 .405674 59.10 5 .594737 47.35 .596449 47.53 6 .406107 58.70 .409220 68.85 6 .597578 47.20 .599301 47.38 7 .411629 58.47 .412751 58.62 7 .600410 47.05 .602144 47.25 8 .415137 58.23 .416268 58.38 8 .603233 46.90 .604979 47.08 9 .418631 58.00 .419771 58.15 9 .606047 47.73 .607804 46.92 10 .422111 57.77 .423260 57.92 10 .608851 46.60 .610619 46.78 11 7.425577 57.53 7.426735 57.70 11 7.611647 46.43 7.613426 46.63 12 .429029 57.30 .430197 57.45 12 .614433 46.30 .616224 46.48 13 .432467 57.08 .433644 57.25 13 .617211 46.15 .619013 46. a5 14 .435892 56.85 .437079 57.00 14 .619980 45.98 .621794 46.18 15 .439303 56.63 .440499 56.80 15 .622739 45.87 .624565 46.05 16 .442701 56.42 .443907 56.57 16 .625491 45.70 .627328 45.90 17 .446086 56.20 .447301 56.35 17 .628233 45.57 .630082 45.75 18 .449458 55.97 .450682 56.13 18 .630967 45.42 .632827 45.62 19 .452816 55.77 .454050 55.92 19 .633692 45.28 .635564 45.48 20 .456162 55.55 .457405 55.72 20 .636409 45.13 .638293 45. 33 21 7.459495 55. as 7.460748 55.48 21 7.639117 44.98 7.641013 45.18 22 .462815 55.12 .464077 55.28 22 .641816 44.87 .643724 45.07 23 .466122 54.92 .467394 55.08 23 .644508 44.72 .646428 44.90 24 .469417 54.70 .470699 54.87 24 .647191 44.57 .649122 44.78 25 .472699 54.50 .473991 54.65 25 .649865 44.45 .651809 44.65 26 .475969 54.28 .477270 54.47 26 .652532 44.30 .654488 44.50 27 .479226 54.10 .480538 54.25 1 27 .655190 44.17 .657158 44.37 28 .482472 53.88 .483793 54.05 28 .657840 44.05 .659820 44.23 29 .485705 53.70 .487036 53.85 i 29 .660483 43.90 .662474 44.12 30 .488927 53.48 .490267 53.67 30 .663117 43.77 .665121 4S.97 31 7.492136 53.28 7.493487 53.45 31 7.665743 43.63 7.667759 43.83 32 .495333 53.10 .496694 53.27 32 .668361 43.50 .670389 43.73 33 .498519 52.90 .499890 53.07 as .670971 43.38 .673012 48.57 34 .501693 52.72 .503074 52.88 34 .673574 43.23 .675626 43.45 35 .604856 52.52 .506247 52.68 35 .676168 43.12 .678233 43. as 36 .508007 52.33 .509408 52.50 36 .678755 42.98 .680833 43.18 37 .611147 52.13 .512558 52.32 37 .681334 42.87 ... 683424 43.07 38 .514275 51.95 .515697 52.12 38 .683906 42.73 .686008 42.95 39 .517392 51.77 .518824 51.93 ! 39 .686470 42.60 .68&585 42.82 40 .520498 51.58 .521940 51.77 40 .689026 42.48 .691154 42.68 41 7.523593 51.40 7.525046 51.57 41 7.691575 42.35 7.693715 42.57 42 526677 51.22 .528140 51.38 42 .694116 42. x3 .696269 42.43 43 529750 51.03 .531223 51.22 ! 43 .696650 42.12 .698815 42.33 44 ,532812 50.85 .534296 51.02 44 .699177 41.98 .701355 42.20 45 .535863 50.68 .537357 50.85 45 .701696 41.87 .703887 42.07 46 .538904 50.50 .540408 50.68 46 .704208 41.73 .706411 41.97 47 .541934 50.32 .543449 50.50 47 .706712 41.63 .708929 41.83 48 .544953 50.15 .546479 50.33 48 .709210 41.50 .711439 41.72 49 .547962 49.98 .549499 50.15 49 .711700 41.38 .713942 41.60 50 .550961 49.80 .552508 49.98 50 .714183 41.27 .716438 41.48 51 *. 553949 49.63 7.555507 49.80 51 7.716659 41.15 7.718927 41.37 52 .556927 49.43 .558495 49.65 52 .719128 41.03 .721409 41.25 53 .559895 49.28 .561474 49.47 53 .721590 40.92 ! 738884 41.13 54 .562852 49.13 .564442 49.32 54 .724045 40.80 .788353 41.02 55 .565800 48.95 .567401 49.13 55 .726493 40.68 .728813 40.90 56 .568737 48.80 .570349 48.98 56 .728934 40.57 .731267 40.78 57 .571665 48.63 .573288 48.82 57 .731368 40.47 .7aS714 40.68 58 .574583 48.47 .576217 48.65 58 .7aS796 40.33 .736155 40.57 59 .577491 48.30 .579136 48.48 59 .736216 40.23 .738589 40.45 60 7.580389 48.15 7.582045 48.33 60 7.738630 40.13 7.741016 40.33 [408] AND EXTERNAL SECANTS. 6 7 / Vers. D. r. Ex. sec. D.I". / Vers. D. r. Ex. sec. D. 1'. 7.738630 40.13 7.741016 40.33 7.872381 34.38 7.875630 34.63 1 .741038 40.00 .743436 40.23 1 .874444 34.30 .877708 34.57 2 .743438 39.90 .745850 40.13 2 .876502 34.22 .879782 34.48 3 745832 39.78 .748258 40.00 3 .878555 34.13 .881851 34.40 4 748219 39.68 .750658 39.90 4 .880603 34.07 .883915 34.32 5 750600 39.57 .753052 39.80 5 .882647 33.98 .885974 34.25 6 752974 39.47 .755440 39.68 6 .884686 33.90 .888029 34.15 755342 39. a> .757821 39.58 7 .886720 33.82 .890078 34.08 8 757703 39.25 .760196 39.48 8 .888749 as. 73 .892123 34.02 9 760058 39.13 .762565 39.37 9 .890773 33.67 .894164 33.92 10 .762406 39.05 .764927 39.25 10 .892793 a3. 58 .896199 33.85 11 7.764749 38.92 7.767282 39.17 11 7.894808 33.50 7.898230 33.77 12 .767084 38.83 .769632 39.05 12 .896818 33.43 .900256 33.70 13 .769414 38.72 .771975 38.95 13 .898824 as. 35 .902278 33.62 14 .771737 38.62 .774312 38.85 14 .900825 33.27 .904295 33.53 15 .774054 38.52 .776643 38.75 15 .902821 33.20 .906307 33.47 16 .776365 38.42 .778968 38.63 16 .904813 33.12 .908315 33.40 17 .778670 38.30 .781286 38.55 17 .906800 83.05 .910319 33.30 18 .780968 38.22 .783599 38.43 18 .908783 32. 9f .912317 33.25 19 .783261 38.10 .785905 38.35 19 .910761 32.90 .914312 33.17 20 .785547 38.02 .788206 38.23 20 .912735 32.82 .916302 33.08 21 7.787828 37.90 7.790500 38.15 21 7.914704 32.73 7.918287 33.02 22 .790102 37.82 .792789 38.03 22 .916668 32.68 .920268 32.95 23 .792371 37.70 .795071 37.95 23 .918629 32.58 .922245 '32.87 24 .794633 37.62 .797348 37.85 24 .920584 32.53 .924217 32.78 25 .796890 37.52 .799619 37.75 25 .922536 32.45 .926184 32.73 26 .799141 37.40 .801884 37.65 26 .924483 32.37 .928148 32.65 27 .801385 37.33 .804143 37.57 27 .926425 32.32 .930107 32.58 28 .803625 37.22 .806397 37.45 28 .928364 32.22 .932062 32.50 29 .805858 37.13 .808644 37.37 29 .930297 32.17 .934012 32.43 30 .808086 37.03 .810886 37.28 30 .932227 32.08 .935958 32.37 31 7.810308 36.93 7.813123 37.17 31 7.934152 32.02 7.937900 32.30 32 .812524 36.83 .815353 37.08 32 .936073 31.95 .939838 32.23 33 .814734 36.75 .817578 37.00 33 .937990 31.88 .941772 32.15 34 .816939 36.67 .819798 36.90 34 .939903 31.80 .943701 32.08 35 .819139 36.55 .822012 36.80 35 .941811 31.73 .945626 32.02 36 .821332 36.48 .824220 36.72 36 .943715 31.67 .947547 31.95 37 .823521 36.37 .826423 36.62 37 .945615 31.60 .949464 31.87 38 .825703 36.28 .828620 36.53 38 .947511 31.52 .951376 31.82 39 .827880 36.20 .830812 36.45 39 .949402 31.47 .953285 31.73 40 .830052 36.10 .832999 36.35 40 .951290 31.38 .955189 31.68 41 7.832218 36.02 7.835180 36.27 41 7.953173 81.32 7.957090 31.60 42 .834379 35.93 .837356 36.17 42 .955052 31.27 .958986 31.53 43 .836535 35.83 .839526 36.08 43 .956928 31.18 .960878 31.48 44 .838685 35.75 .841691 36.00 44 .958799 31.12 .962767 31.40 45 .840830 35.65 .843851 35.90 45 .960666 31.05 .964651 31.85 46 .842969 35.58 .846005 35.83 46 .962529 30.98 .966531 31.28 47 .845104 a5. 48 .848155 35 . 73 47 .964388 30.92 .968408 31.20 48 .847233 85.40 .850299 35.68 48 .966243 30.85 .970280 31.13 49 .849357 85.30 .852437 35.57 49 .968094 30.78 .972148 31.08 50 .85147:5 35.23 .854571 35.48 50 .969941 30.73 .974013 31.02 51 7.853589 as. is 7.856700 a5.38 51 7.971785 30.65 7.975874 30.93 52 .855697 35.05 .858823 35.32 52 .973624 30.58 .977730 30.88 53 .857800 34.97 .860942 35.22 53 .975459 30.53 .979583 30.82 54 .859898 34.88 .863055 35.13 54 .977291 30.45 .981432 30.75 55 .861991 34.80 .865163 35.05 55 .979118 30.40 .983277 30.70 56 .864079 34.72 .867266 34.98 56 .980942 30.33 .985119 30.62 57 .866162 34.63 .869365 34.88 57 .982762 30.27 .986956 30.57 58 .868240 34.55 .871458 34.80 58 .984578 30.22 .988790 30.50 59 .870313 34.47 .87a546 34.73 59 .9WWJU 30.13 .990620 30.43 60 7.872381 34.38 7.875630 34.63 60 7.988190 30.08 7.992446 30.38 [409] TABLE XXVI. LOGARITHMIC VERSED SINES 8 9 / Vers. D.I". Ex. sec. D. 1". ' Vers. D. 1". Ex. sec. D. r. 7.988199 30.08 7.992446 30.38 8.090317 26.72 8.095697 27.05 1 .990004 30.02 .994269 30.32 1 .091920 26.68 .097320 27.02 2 .991805 29.95 .996088 30.25 2 .093521 26.63 .098941 26.97 3 .993602 29.88 .997903 30.18 3 .095119 26.58 .100559 26.92 4 .995395 29.83 7.999714 30.13 4 .096714 26.52 .102174 26.87 5 .997185 29.77 8.001522 30.07 5 .098305 26.48 1 .103786 26.82 6 7.998971 29.72 .003326 30.00 6 .099894 26.43 . 105395 26.77 7 8.000754 29.63 .005126 29.95 7 .101480 26.40 .107001 26.73 8 .002532 29.60 .006923 29.88 8 .103064 26.33 .108605 26.67 9 .004308 29.52 .008716 29.83 9 .104644 26.28 .110205 26.63 10 .006079 29.47 .010506 29.73 10 .106221 26.25 .111803 26.58 11 8.007847 29.40 8.012292 29.70 11 8.107796 26.18 8.113398 26.53 12 .009611 29.35 .014074 29.65 12 .109367 26.15 .114990 26.48 13 .011372 29.28 .015853 29.58 13 .110936 26.10 .116579 26.45 14 .013129 29.22 .017628 39.53 14 .112502 26.05 .113166 26.38 15 .014882 29.17 .019400 29.47 15 .114065 26.00 .119749 26.35 16 .016632 29.10 .021168 29.42 16 .115625 25.95 .121330 26.30 17 .018378 29.05 .0*2933 29.35 17 .117182 25.92 .122908 26.25 18 .020121 29.00 .024694 29.30 18 .118737 25.87 .124483 26.22 19 .021861 28.93 .026452 29.23 19 .120289 25.82 .126056 26.17 20 .023597 28.87 .028206 29.18 20 .121838 25.77 .127626 26.12 21 8.025329 28.82 8.029957 29.13 21 8.123384 25.72 8.129193 26.07 22 .027058 28.75 .031705 29.07 22 .124927 25.68 .130757 26.02 23 .028783 28.70 .033449 29.00 23 .126468 25.65 .132318 25.98 24 .030505 28.65 .035189 28.97 24 .128006 ! 25.58 .133877 25.93 25 .032224 28.58 .036927 28.90 25 . 129541 25.55 .135433 25.90 26 .033939 28.53 .038661 28.83 26 .131074 25.50 .136987 25.85 27 .035651 28.47 .040391 28.78 27 .132604 25.45 .138538 25.80 28 .037359 28.42 .042118 28.73 28 .134131 25.40 .140086 25.75 29 .039064 28.37 .043842 28.68 29 .135655 ! 25.37 .141631 25.72 30 .040766 28.30 .045563 28.62 30 .137177 25.32 .143174 25.67 31 8.042464 28.25 8.047280 28.57 31 8.138696 25.27 8.144714 25.63 32 .044159 28.20 .048994 28.50 32 .140212 25.23 .146252 25.58 33 .045851 28.13 .050704 28.47 33 .141726 25.18 .147787 25.53 34 .047539 28.08 .052412 28.40 34 .143237 25.13 .149319 25.50 35 .049224 28.03 .054116 28.35 35 .144745 25.10 .150849 25.45 36 .050906 27.98 .055817 28.28 36 .146251 25.05 .152376 25.40 37 .052585 27.92 .057514 28.25 37 .147754 ! 25.02 .153900 25.37 38 .054260 27.87 .059209 28.18 38 .149255 I 24.95 .155422 25.33 39 .055932 27.82 .060900 28.13 39 .150752 24.93 .156942 25.27 40 .057601 27.75 .062588 28.08 40 .152248 24.88 .158458 25.25 41 8.059266 27.72 8.064273 28.03 41 8.153741 24.83 8.159973 25.18 42 .060929 27.65 .065955 27.97 42 .155231 24.78 .161484 25.17 43 .062588 27.60 .067633 27.93 43 .156718 24.75 .162994 25.10 44 .064244 27.55 .069309 27.87 44 .158203 24.72 .164500 25.07 45 .065897 27.48 .070981 27.82 45 .159686 24.67 .166004 25.03 46 .067546 27.45 .072650 27.77 46 .161166 24.62 .167506 24.98 47 .069193 27.38 .074316 27.72 47 .162643 24.58 .169005 24.95 48 .070836 27.33 .075979 27.67 48 .164118 24.53 .170502 24.90 49 .072476 27.30 .077639 27.60 49 .165590 24.50 .171996 24.87 50 .074114 27.23 .079295 27.57 50 .167060 24.45 .173488 24.82 51 8.07:',748 27.18 8.080949 27.52 51 8.168527 24.42 8.174977 84.78 52 .077'379 27.13 .082600 27.45 52 .169992 24.37 .176464 24.73 53 .079007 27.07 .084247 27.42 53 .171454 24.33 .177948 24.70 54 .080631 27.03 .085892 27.37 54 .172914 24.30 .179430 24.65 55 .082253 26.98 .087534 27.30 55 .174372 24.25 .180909 24.62 56 .083872 26.93 .089172 27.27 56 .175827 24.20 .182386 24.58 57 .085488 26.87 .090808 27.20 57 .177279 24.17 .183861 24.53 58 .087100 26. as .092440 27.17 58 .178729 24.13 .185388 24.50 59 .088710 26.78 .094070 27.12 59 .180177 24.08 .186F03 24.47 60 8.090317 26.72 8.095697 27.05 60 8.181622 24.05 18.188271 2-1.42 AND EXTERNAL SECANTS. 10- 11 ' Vers. D.I'. Ex. sec. D.I". t Vers. D. 1". Ex. sec. D. 1'. 8.181622 24.05 8.188271 24.42 8.264176 21.85 8.272229 22.27 1 .183065 24.00 .189736 24.37 1 .265487 21.82 .273565 22 22 2 .184505 23.97 .191198 24.35 2 .266796 21.78 .274898 22^20 3 .185943 23.93 .192659 24.30 3 .268103 21.75 .276230 22.17 4 .187379 23.88 .194117 24.25 4 .269408 21.72 .277560 22.13 5 .188812 23.85 .195572 24.22 5 .270711 21.68 .278888 22.08 6 .190243 23.80 .197025 24.18 6 .272012 21.65 .280213 22.07 7 .191671 ! 23.77 .198476 24.15 7 .273311 21.62 .281537 22.03 8 .193097 23.73 .199925 24.10 8 .274608 21.58 .282859 22.00 Jt .194521 I 23.68 .201371 24.07 9 .275903 21.57 .284179 21.98 10 .195942 23.65 .202815 24.03 10 .277197 21.53 .285498 21.93 11 8.197361 23.62 8.204257 23.98 11 8.278488 21.48 8.286814 21.90 12 .198778 23.57 .205696 23.95 12 .279777 21.47 .288128 21.88 13 .200192 23.53 .207133 23.92 13 .281065 21.42 .289441 21.83 14 .201604 23.50 .208(568 23.88 14 .282350 21.40 .290751 21.82 15 .203014 23.45 .210001 23.83 15 .283634 21.37 .292060 21.78 16 .204421 23.42 .211431 23.80 16 .284916 21.33 293367 21.75 17 .205826 23.38 .212859 23.77 17 .286196 21.28 .294672 21.72 18 .207229 23.35 .214285 23.72 18 .287473 21.27 .295975 21.08 19 .208630 23.30 .215708 23.70 19 .288749 21.25 .297276 21.67 20 .210028 23.27 .217130 23.65 20 .290024 21.20 .298576 21.62 21 8.211424 23.23 8.218549 23.62 21 8.291296 21.17 8.299873 21.60 22 .212818 23.18 .219966 93.57 22 .292566 21.15 .301169 21.57 23 .214209 23.17 .221380 23.55 23 .293886 21.10 .302463 21.53 24 .215599 23.12 .222793 23.50 24 .295101 21.08 .303755 21.50 25 .216986 23.08 .224203 23.47 25 .296366 21.05 .305045 21.48 26 .218371 23.03 .225611 23.43 26 .297629 21.02 .306334 21.43 27 .219753 23.00 .227017 23.40 27 .298890 20.98 .307620 21.42 28 .221133 22.98 .228421 23.35 28 .300149 20.95 .308905 21.38 29 .222512 22.93 .221)832 23.32 29 .301406 20 93 .310188 21.35 30 .223888 22.88 .231221 23.30 30 .302662 20.90 .311109 21.33 31 8.225261 22.87 8.232019 23.25 31 8.303916 20.85 8.312749 21.28 32 .226633 23.82 .234014 23.22 32 .305167 20.85 .314026 21.27 33 .228002 22.78 .235407 23.17 88 .806418 20.80 .315302 21.23 34 .229369 28.77 .236797 23.15 34 .307666 20.77 .318576 21.22 35 .230735 22.70 .238186 23.10 35 .308912 20:75 .317849 21 17 36 .232097 22.68 .239572 23.08 36 .310157 20.72 .319119 21.15 37 .233458 2:3.65 .240957 23.03 37 .311400 20.68 .320388 21.12 38 .234317 22.60 .242339 23.00 38 .312641 20.155 .321655 21.08 39 .231)173 22.57 .243719 22.97 39 .313880 20.62 .322920 21.05 40 .237527 22.55 .245097 22.93 40 .315117 20.60 .324183 21.03 41 8.238880 22.50 8.246473 22.90 41 8.316353 20.57 '8. 325445 21.00 42 .240230 22.47 .217847 22 87 42 .317587 20.53 .320705 20.98 43 .241578 22.43 .249219 22.83 43 .31S819 20.50 .327964 20.93 44 .242924 22.38 .250589 2*. 80 44 .320049 1 20.48 .329220 .20.92 45 .244267 22.37 .251957 22.75 45 .321278 20.45 .330475 20.88 46 .245609 22.32 .253322 22.73 46 .322505 20.42 .331728 20.87 47 ,240948 22.30 .254086 22.68 47 .323730 20.38 .332980 20.82 48 .248286 22.25 .256047 22.67 48 .824958 20.37 .334229 20.80 49 .219021 22.23 .257407 22.62 49 .326175 20.33 .335477 20.78 50 .250955 22.18 .258764 22.60 50 .327395 20.30 .336724 20.73 5l 8.252286 22.15 8.260120 22.55 51 8.328613 20.27 S. 337968 20.72 52 ,253015 22.12 .261473 22.53 52 .329829 20.25 .339211 20.70 53 , 254942 22.10 .262825 22.48 5.3 .331044 20.22 .340453 20.65 54 256208 22.05 .264174 22.47 54 .332257 20.18 .341692 20.63 55 .257591 22 02 .265522 22.42 55 .333108 20.17 .342930 20.00 56 .258912 21.98 .266867 22.40 56 .334678 20.13 .aniofi 20.58 57 .200-331 21.95 .268211 22.35 57 .335886 20.10 .34M01 20.55 58 .261548 21.92 .269552 22 .33 ! 58 .337092 20.07 .346034 20.52 59 .262863 21.88 .270892 22 28 ' 59 ^338296 20.05 ,347805 20.50 60 8.264176 21.85 8.272229 22 27 . 60 } 8.339499 20.02 8.349095 20.47 L4HJ TABLE XXVI. LOGARITHMIC VERSED SINES 12 13 Vers. D.I". Ex. sec. D. 1'. ' Vers. D. 1". Ex. sec. D. 1". 8.339499 20.02 8.349095 20.47 8.408748 18.47 8.420024 18.95 .340700 20.00 .350323 20.43 1 .409856 18.43 .421161 18.98 .341900 19.95 .351549 20.42 2 .410962 18.42 .422297 18.90 .343097 19.95 .352774 20.38 3 .412067 18.40 .423431 18,88 .344294 19.90 .353997 20.35 4 .413171 18.38 .424564 18.87 .345488 19.88 .355218 20.33 5 .414274 18.35 .425696 is. as .346681 19.85 .356438 20.30 6 .415375 18.32 .426826 18.82 .347872 19.82 .357656 20.28 7 .416474 18.30 .427955 18.80 .349061 19.80 .358873 20.25 8 .417572 18.28 .429083 18.77 .350249 19.77 .360088 20.22 9 .418669 18.25 .430209 18.75 .351435 19.75 .361301 20.20 10 .419764 18.23 .431334 18.73 8.352620 19.72 8.362513 20.18 11 8.430858 18.22 8.432458 18.70 .353803 19.68 .363724 20.13 12 .421951 18.18 .433580 18.67 .354984 19.67 .364932 20.12 13 .423042 18.17 .434700 18.67 .356164 19.63 .366139 20.10 14 .424132 18.13 .435820 18.63 .357342 19.60 .367345 20.07 15 .425220 18.12 .436938 18.62 .358518 19.58 .368549 20.03 16 .426307 18.10 .438055 18.58 .359693 19.55 .369751 20.02 17 .427393 18.07 .439170 18.57 .360866 19.53 .370952 19.98 18 .428477 18.05 .440284 18.55 .362038 19.50 .372151 19.95 19 .4295(50 18.02 .441397 18.53 .363208 19.48 .373348 19.95 20 .430641 18.02 .442509 18.50 8.364377 19.43 8.374545 19.90 21 8.431722 7.97 8.443619 18.47 .365543 19.43 .375739 19.88 22 .432800 7.97 .444727 18.47 .366709 19.38 .376932 19.85 23 .433878 7.93 .445835 18.43 .367872 19.37 .378123 19.83 24 .434954 7.92 .446941 18.42 .369034 19.35 .379313 19.82 ! 25 .436029 7.88 .448046 18.38 .370195 19.32 .380502 19.78 i 26 .437102 7.87 .449149 18.38 .371354 19.28 .381689 19.75 27 .438174 7.85 .450252 18.35 .372511 19.27 .382874 19.73 28 .439245 7.82 .451353 18.32 .373667 19.25 .384058 19.70 29 .440314 7.80 .452452 18.32 .374822 19.20 .385240 19.68 30 .441382 7.78 .453551 18.28 8.375974 19.18 8.386421 19.65 31 8.442449 7.75 8.454648 18.25 .377125 19.17 .387600 19.63 32 .443514 7.73 .455743 18.25 .378275 19.13 .388778 19.60 33 .444578 7.72 .456838 18.22 .379423 19.12 .389954 19.58 34 .445641 7.68 .457931 18.20 .380570 19.08 .391129 19.55 85 .446702 7.68 .459023 18.18 .381715 19.05 .392:302 19.53 86 .447763 7.63 .460114 18.15 .382858 19.03 .393474 19.50 37 .448821 7.63 .461203 18.13 .384000 19.02 .394644 19.48 38 .449879 7.62 .462291 18.12 .385141 18.98 .395813 19.45 39 .450935 7.58 .463378 18.10 .386280 18.95 .396980 19.43 40 .451990 7.55 .464464 18.07 8.387417 18.93 8.398146 19.42 41 8.453043 7.55 8.465548 18.05 .388553 18.92 .399311 19.38 42 .454096 7.52 .466631 18.03 .389688 18.88 .400474 19.35 43 .455147 7.48 .467713 18.00 .390821 18.85 .401635 19.33 44 .456196 7.48 .468793 18.00 .391952 18.83 .402795 19.32 45 .457245 7.45 .469873 1 .97 .393082 18.82 .403954 19.28 46 .458292 7.43 .470951 1 .95, .394211 18.78 .405111 19.27 47 .459338 7.40 .472028 .m .395338 18.75 .406267 19.23 48 .460382 7.40 .473103 .90 .396463 18.73 .407421 19.22 49 .461426 7.37 .474177 .90 .397587 18.72 .408574 19.18 50 .462468 17.35 .475251 .85 8.398710 18.68 8.409725 19.17 51 8.463509 17.32 8.476322 .85 .399831 18.67 .410875 19.13 52 .464548 17.30 .477393 .83 .400951 18.63 .412023 19.13 53 .465586 17.28 .478463 .80 .402069 18.62 .413171 19.08 54 .466623 17.27 .479531 .78 .403186 18.58 .414316 19.08 55 .467659 17.23 .480598 .77 .404301 18.57 .415461 19.03 56 .468693 17.23 .481664 .73 .405415 18.53 .416603 19.03 57 .469727 17.20 .482728 .78 .406527 18.52 .417745 19.00 58 .470759 17.17 .483792 .70 .407638 18.50 .418885 18.98 59 .471789 17.17 .484854 .68 8.408748 18.47 8.420024 18.95 60 8.472819 17.13 8.485915 .67 AND EXTERNAL SECANTS. 14 ) 15 i Vers. D. r. Ex. sec. D. r. / Vers. D. r. Ex. sec. D. 1". 8.472819 17.13 8.485915 17.67 8.532425 15.98 8.547482 16.53 1 .473847 17.12 .486975 17.63 1 .533384 15.97 .548474 16.53 2 .474874 17.10 .488033 17.63 2 .534342 15. 95 .549466 16.52 3 .475900 17.08 .489091 17.60 3 .535299 15.93 .550457 16.50 4 .476925 17.05 .490147 17.58 4 .536255 15.92 .551447 16.48 5 .477948 17.03 .491202 17.57 5 .537210 15.88 .552436 16.47 6 .478970 17.02 .492256 17.53 6 .538163 15.88 .553424 16.43 7 .479991 17.00 .493308 17.53 .539116 15.87 .554410 16.43 8 .484011 16.97 .494360 17.50 8 .540068 15.83 .555396 16.42 9 .482029 16.95 .495410 17.48 9 .541018 15. as .556:381 16.38 10 .483046 16.93 .496459 17.47 10 .541968 15.80 .557364 16.38 11 8.484062 16.92 8.497507 17.45 11 8.542916 15.78 8.558347 16.37 12 .485077 16.90 .498554 17.43 12 .543863 15.78 .559329 16.33 13 .486091 16.87 .499600 17.40 13 .544810 15.75 .560309 16.33 14 .487103 16.87 .500644 17.38 14 .545755 15.73 .561289 16.30 15 .488115 16.83 .501687 17.38 15 .546699 15.72 .562267 16.30 16 .489125 16.82 .502730 17.35 16 .547642 15.70 .563245 16.28 ir .490134 16.78 .503771 17.32 17 .548584 15.68 .564222 16.25 18 .491141 16.78 .504810 17.32 18 .549525 15.67 .565197 It;. 25 19 .492148 16.75 .505849 17.30 19 .550465 15.65 .566172 16.22 20 .493153 16.73 .506887 17.27 20 .551404 15.63 .567145 16.22 21 8.494157 16.72 8.507923 17.25 21 8.552342 15.62 8.568118 16.20 22 ,495160 16.70 .508958 17.25 22 .553279 15.60 .569090 16.17 23 .496162 16.67 .509993 17.22 23 .554215 15.58 .570060 16.17 24 .497162 16.67 .511026 17.18 24 .555150 15.57 .571030 16.15 25 .498162 16.63 .512057 17.18 25 .556084 15.55 .571999 16.12 26 .499160 16.62 .513088 17.17 26 .557017 15.53 .572966 16.12 27 .500157 16.60 .514118 17.13 27 .557949 15.50 .573933 16.10 28 .501153 16.58 .515146 17.13 28 .558879 15.50 .574899 16.08 29 .502148 16.57 .516174 17.10 29 .559809 15.48 .575864 16.05 30 .503142 16.53 .517200 17.08 30 .560738 15.47 ,576827 16.05 31 8.504134 16.52 8.518225 17.07 31 8.561666 15.43 8.577790 16.03 32 .505125 16.52 .519849 17.05 32 .562592 15.43 .578752 16.02 33 .506116 16.48 .520272 17.03 33 .563518 15 42 .579713 16.00 34 .507105 16.47 .521294 17.02 34 .564443 15.40 .580673 15.98 35 .508092 16.47 .522315 16.98 35 .565367 15.37 .581632 15.97 36 .509079 16.43 .523334 16.98 36 .566289 15.37 .582590 15.95 37 .510065 16.40 .524353 16.95 37 .567211 15.35 .583547 15.93 38 .511049 16.40 .525370 16.95 38 .568132 15.33 .584503 15.92 39 .512033 16.37 .526387 16.92 39 .569052 15.30 .585458 15.90 40 .513015 16.35 .527402 16.90 40 .569970 15.30 .586412 15.88 41 8.513996 16 33 8.528416 16.88 41 8.570888 15.28 8.587365 15.88 42 ,514976 16.32 .529429 16.87 43 .571805 15.27 .588318 15.85 43 ,515955 16.28 .530441 16.85 43 .5727'21 15.25 .589269 15.83 44 .516932 16.28 .531452 16.83 44 .57TvJ6 15.22 .590219 15.83 45 .517909 16.25 .532462 16.82 45 .574549 15.22 .591169 15.80 46 .518884 16.25 .533471 16.78 46 .575462 15.20 .592117 15.80 47 .519859 16.22 .534478 16.78 47 .576374 15.18 .593065 15.78 4S .590888 16.20 .535-185 16.75 48 .577285 15.17 .594012 15.75 49 ! 531804 16.18 .536490 16.75 49 .578195 15.15 .594957 15 75 50 .522775 16.17 .537495 16.72 50 .579104 15.13 .595902 15.73 51 8.523745 16.15 8.538498 16.72 51 8.58001S 15.12 8.596846 15.72 52 .524714 16.13 .539501 16.68 52 .580919 15.10 .597789 15.70 53 .525682 16.10 .540502 16.67 &a .581825 15.08 .598731 15.68 54 .526648 16.10 .541502 16.65 54 .582730 15.07 .599672 15.67 55 .527614 16.07 .542501 16.63 55 .583634 15.05 .600612 15.65 56 .528578 16.07 .543499 16.63 56 .584537 15.05 .601551 15.65 57 .529542 16.03 .544497 16.60 57 .585440 15.02 .602490 15.62 58 .530504 16.02 .545493 16.58 58 .586341 15.00 .603427 15.60 59 .531465 16.00 .546488 16.57 59 .587241 15.00 .604:363 15.60 60 8.532425 1 15.98 8.547483 16.53 60 8.588141 14.97 8.605299 ! 15.58 1 Ui3] TABLE XXVI. -LOGARITHMIC VERSED SINES 16 17 / Vers. D. 1". Ex. sec. D.I". / Vers. D. i". Ex. sec. D. r. 8.588141 14.97 8.605299 15.58 ~~0~ 8.640434 14.08 8.659838 14.72 i .589039 14.95 .606234 15.55 1 .641279 14.07 .660721 14.72 2 .589936 14.95 .607167 15.55 2 .642123 14.05 .661604 14.70 3 .590833 14.93 .608100 15.53 3 .642966 14.05 .662486 14.68 4 .591729 14.90 .609032 15.52 4 .643809 14.02 .663367 14.68 5 .592623 14.90 .609963 15.50 5 .644650 14.02 .664248 14.65 6 .593517 14.88 .610893 15.50 6 .645491 14.00 .665127 14.65 7 .594410 14.87 .611823 15.47 7 .646331 13.98 .666006 14.63 8 .595302 14.83 .612751 15.45 8 .647170 13.97 .666884 14.62 9 .596192 14.83 .613678 15.45 9 .648008 13.95 .667761 14.60 10 .597082 14.82 .614605 15.43 10 .648845 13.95 .668637 14.60 11 8.597971 14.82 8.615531 15.42 11 8.649682 13.93 8.669513 14.58 12 .598860 14.78 .616456 15.38 12 .650518 13.92 .670:388 14.57 13 .599747 14.77 .617379 15.38 13 .651353 13.90 .671262 14.55 14 .600633 14.75 .618302 15.38 14 .652187 13.88 .672135 14.55 15 .601518 14.75 .619235 15.35 15 .653020 13.87 x 673008 14.52 16 .602403 14.72 .620146 15.33 16 .653852 13.87 .673879 14.52 '17 .603286 14.72 .621066 15.33 17 .654684 13.85 .674750 14.50 18 .604169 14.70 .621986 15.30 18 .655515 13.83 .675620 14.50 19 .605051 14.67 .622904 15.30 19 .656345 13.82 .676490 14.47 20 .605931 14.67 .623822 15.28 20 .657174 13.82 .677358 14.47 21 8.608811 14.65 8.624739 15.27 21 8.658003 13.78 8.678226 14.45 22 .607690 14.63 .625655 15.25 22 .658830 13.78 .679093 14.45 23 .608568 14,62 .626570 15. 23 23 .659657 13.77 .679960 14.42 24 .609445 14.60 .627484 15.23 24 .660483 13.75 .680825 14.42 25 .610321 14.60 .628398 15.20 25 .661308 13.73 .681690 14.40 26 .611197 14.57 .629310 15.20 26 .662132 13.73 .682554 14.38 27 .612071 14.57 .630222 15.18 27 .662956 13.72 .683417 14.38 28 .612945 14.53 .631133 15.17 28 .663779 13.70 .684280 14.35 29 .613817 14.53 .632043 15.15 29 .664601 13.68 .685141 14. a5 30 .614689 14.52 .632952 15.13 30 .665422 13.67 .686002 14.35 31 8.615560 14.50 8.633860 15.13 31 8.666242 13.67 8.686863 14.32 32 .616430 14.48 .634768 15.10 32 .667002 13.65 .687722 14.32 33 .617299 14.47 .635674 15.10 33 .667881 13.63 .688581 14.30 31 .618167 14.45 .636530 15.08 34 .668699 13.62 .689439 14.28 35 .619034 14.45 .637485 15.07 35 .669516 13.60 .690296 14.28 36 .619901 14.42 .63a389 15.05 36 .670332 13.60 .691153 14.25 37 .620766 14.42 .639292 15.05 37 .671148 13.58 .692008 14.25 38 .621631 14.40 .640195 15.02 38 .671963 13.57 .692863 14.25 39 .622495 14.38 .641096 15.02 39 .672777 13.55 .693718 14 22 40 .623358 14.37 .641997 15.00 40 .673590 13.55 .694571 14! 22 41 8.624220 14.35 8.642897 14.98 41 8.674403 13.53 8.695424 14.20 42 .6.25081 14.33 .643796 j 14.97 42 .675215 13.52 .696276 14.18 43 .625941 14.33 .644694 14.95 43 .676026 13.50 .697127 14.18 44 .626801 14.30 .645591 14.95 44 .676836 13.48 .697978 14.17 45 .627659 14.30 .646488 14.93 45 .677645 13.48 .698828 14.15 46 .628517 14.28 .647384 14.92 46 .678454 13.47 .699677 14.13 47 .629374 14.27 .648279 14.90 47 .679262 13.45 .700525 14.13 48 .630230 14.25 .649173 14.88 48 .680069 13.43 .701373 14.12 49 .631085 j 14.23 .650066 14.87 49 .680875 13.43 .702220 14.10 50 .631939 14.22 .650958 14.87 50 .681681 13.42 .703066 14.10 51 8.632792 14.22 8.651&50 14.85 51 8.682486 13.40 8.703912 14.07 52 .633645 14.18 .652741 14.83 52 .683290 13.38 .704756 14.07 53 .634496 14.18 .653631 14.82 53 .684093 13.38 .705600 14.07 54 .635347 14.17 .654520 14.80 54 .684896 13.35 .706444 14.03 55 .636197 14.15 .655408 14.80 55 .685697 13.35 .707286 14.03 56 .637046 14.13 .656296 14.77 56 .686498 13.35 .708128 14.02 57 .637894 14.13 .657182 14.77 57 .687299 13.32 .708969 14.02 58 .638742 14.10 .658068 14.77 58 ! .688098 13.32 .709810 14.00 59 .639588 14.10 .658954 14.73 59 .688897 13.30 i .710650 i 13.98 60 8.640434 14.08 ' 8.659838 14.72 60 ' 8.689695 i 13.28 8.711489 13.97 [414] AND EXTERNAL SECANTS. 18 19 ' Vers. D. r. Ex. sec. D. 1". / Vers. D. 1". Ex. sec. D. 1". ft. 689695 13.28 8.711489 13.97 8.736248 12.58 8.760578 13.30 1 690492 13.28 .712327 13.95 1 .737003 12.57 .761376 13.30 2 .691289 13.25 .713164 13.95 2 .737757 12.55 .762174 13.28 3 .692084 13.25 .714001 13.95 3 .738510 12.55 .762971 13.27 4 .692879 13.25 .714838 13.92 4 .739263 12.53 .763767 13.27 5 .693674 13.22 .715673 13.92 5 .740015 12.52 .764563 13 25 6 .694-167 13.22 .716508 13.90 6 .740766 12.50 .765.T> 13.23 7 .695260 13.20 .717342 13.88 7 .741516 12.50 .766152 1:5.23 8 .696052 13.18 .718175 13.88 8 .742266 12.50 .760946 1.-J.22 9 .696843 13.18 .719008 13.87 9 .743016 12.47 .767739 13.20 10 .697634 13.17 .719840 13.85 10 .743764 12.47 .768531 13.20 11 8.698424 13.15 8.720671 18.85 11 8.744512 12.45 8.769323 13.18 12 .699213 13.13 .721502 13.83 12 .745259 12.43 .770114 13.18 13 .700001 13.13 .722332 13.82 13 .746006 12.45 .770905 13.17 14 .700789 13.1? .723161 13.80 14 .746752 12.42 .771695 13.15 15 701576 13.10 .723989 13.80 15 .747497 12.42 .772484 13.15 16 . .702362 13.08 .724817 13.78 16 .748242 12.40 .773273 13.13 17 .703147 13.08 .725644 13.78 17 .748986 12.38 .774061 13.13 18 .703932 13.07 .726471 13.77 18 .749729 12.38 .774849 13.12 19 .704716 13.05 .727297 13.75 19 .750472 12.37 .775636 13.10 20 .705499 13.05 .728122 13.73 20 .751214 12.35 .776422 13.08 21 8.706282 13.02 8.728946 13.73 21 8.751955 12.85 8.777207 13.10 22 .707063 13.i2 .729770 13.72 22 .752696 12.33 .777993 13.07 23 .707844 13.02 .730593 13.70 23 .753436 12.32 .778777 13.07 24 .708625 12.98 .731415 13.70 24 .754175 12.32 .779561 13.05 25 .709404 12.98 .732237 13.68 25 .754914 12.30 .780344 13.05 26 .710183 12.98 .733058 13.67 26 .755652 12.28 .781127 13.03 27 .710961 12.95 .733878 13.67 27 .756389 12.28 .781909 13.02 28 .711739 12.95 .734698 13.65 ! 28 .757126 12 27 .782690 13.02 29 .712516 12.93 .735517 13.63 ; 29 .757862 12^27 .783471 13.00 30 .713292 12.92 .736*35 13.63 30 .758598 12.25 .784251 13.00 31 8.714067 12.92 8.737153 13.62 31 8.759333 12.23 8.785031 12.98 32 .714842 12.90 .737970 13.60 32 .760067 12.23 .785810 12.97 33 .715616 12.88 .738786 13.60 33 .760801 12.22 .786588 12.97 34 .716389 12.87 .739602 13.58 34 .761534 12.20 .787366 12.97 35 .717161 12.87 .740417 13.57 35 .762266 12.20 .788144 12.93 36 .717933 12.85 .741231 13.57 36 .762998 12.18 .788920 12.93 37 .7187'04 12.85 .742045 13.55 37 .763729 12.17 .789696 12.93 38 .719475 12.82 .742858 13.53 38 .764459 12.17 .?.W472 12.92 39 .720244 12.82 .743670 13.53 39 .765189 12.15 .791247 12.90 40 .721013 12.82 .744482 13.52 40 .765918 12.15 .792021 12.90 41 8.721782 12.78 8.745293 13.50 41 8.766647 12.12 8.792795 12.88 42 .722549 12.78 .746103 13.50 42 .767374 12.13 .793568 12.87 43 .723316 12.78 .746913 13.48 43 .768102 12.10 .794340 12.87 44 .724083 12.75 .747722 13.47 44 .7688-8 12.10 .795112 12.87 45 .724848 12.75 . 748530 13.47 45 .769554 12.10 .795884 12.83 46 .725613 12.73 .749338 13.45 46 .770280 12.08 .7'96654 12.85 47 .726377 12.72 .750145 13.43 47 .771005 12.07 .797425 12.82 48 .727140 12.72 .750951 13.43 48 .771729 12.05 .798194 12.82 49 .727903 12.70 .751757 13.42 4!) .772452 12.05 .798963 12.82 50 .728665 12.70 .752562 13.42 50 .773175 12.05 .799732 12.80 51 8.729427 12.67 8.753367 13.40 51 8.773898 12.02 8.800500 12.78 52 .780187 12.67 .754171 13.38 52 .774619 12.02 .801267 12.78 53 .730947 12.67 .754974 13.37 53 .775340 12.02 .802034 12.77 54 .731707 12.63 .755776 13.37 54 .776061 12.00 .802800 12.75 55 .732465 12.63 .756578 13.37 55 .776781 11.98 .803565 12.75 56 .733223 12.63 .757380 13.33 56 .777500 11.97 .804330 12.75 57 .733981 12.60 .758180 13.33 | 57 .778218 11.97 .805095 12.73 58 .734737 12.60 .758980 13.33 58 .778936 11.97 .805859 12.72 59 .735493 12.58 .759780 13.30 59 .779654 11.98 .806622 12.72 60 8.736248 12.58 8.760578 13.30 60 8.780370 11.95 8.807385 12.70 TABLE XXVI. LOGARITHMIC VERSED SINES 20 21 / Vers. D. r. Ex. sec. D.I". ' Vers. D. r. Ex. sec. I). 1*. 8.780370 11.95 8.807385 12.70 8.822296 11.35 8.852144 12.17 1 .781087 11.92 .808147 12.68 1 .822977 11.85 .852874 12.17 2 .781802 11.92 .808908 12.68 2 .823658 11.33 .853604 12.13 3 .782517 11.90 .809669 12.68 3 .824338 11.33 .854332 12.15 4 .783231 11.90 .810430 12.67 4 .825018 11.32 .855061 12.13 5 .783945 11.88 .811190 12.65 5 .825697 11.32 .855789 12.12 .784658 11.88 .811949 12.65 6 .82637(5 11.30 i .856516 12.12 7 .785371 11.87 .812708 12.63 .827051 11.28 .857243 12.10 8 .786083 11.85 .813466 12.63 8 .82 r <731 11.28 .857969 12.10 9 .786794 11.85 .814224 12.62 9 .828408 11.28 .858695 12.08 10 .787505 11.83 .814981 12.60 10 .829086 11.27 .859420 12.08 11 8.788215 11.82 8.815737 12.60 11 8.829761 11.25 8.860145 12.07 12 .788924 11.82 .816493 12.60 12 .830436 11.25 .860869 12.07 13 .789633 11.82 .817249 12.58 13 .831111 11.23 .861593 12.05 14 .790342 11.78 .818004 12.57 14 .831785 11.23 .862316 12.05 15 .791049 11.78 .818758 12.57 15 .832459 11.22 .863039 12.03 16 .791756 11.78 .819512 12.55 16 .833132 11.20 .863761 12.03 17 .792463 11.77 .820265 12.55 17 .833804 11.20 .864483 12.02 18 .793169 11.75 .821018 12.53 18 .834476 11.20 .865204 12.02 19 .793874 11.75 .821770 12.52 19 .835148 11.18 .865925 12.02 20 .794579 11.73 .822521 12.52 20 .835819 11.17 .866646 11.98 21 8.795283 11.73 8.823272 12.52 21 8.836489 11.17 8.867365 12.00 22 .795987 11.72 .824023 12.50 22 .837159 11.17 .868085 11.98 23 .796690 11.70 .824773 12.48 23 .837829 11.15 .868804 11.97 24 .797392 11.70 .825522 12.48 24 .838498 11.13 .869522 11.97 25 .798094 11.68 .826271 12.47 25 .839166 11.13 .870240 11.95 26 .798795 11.68 .827019 12.47 26 .839834 11.12 .870957 11.95 27 .799496 11.67 .827767 12.45 27 .840501 11.12 .871674 11.93 28 .800196 11.67 .828514 12.45 28 .841168 11.10 .872390 11.93 29 .800896 11.63 .820261 12.43 29 .841834 11.10 .873106 11.93 30 .801594 11.65 .830007 12.43 30 .842500 11.08 .873822 11.92 31 8.802293 11.63 8.830752 12.42 31 8.843165 11.07 8.874537 11.90 32 .802991 11.62 .831497 12.42 32 .843829 11.07 .875251 11.90 33 .803688 11.60 .832242 12.40 33 .844493 11.07 .875965 11.88 34 .804384 11.60 .832986 12.38 34 .845157 11.05 .876678 11.88 35 .805080 11.60 .833729 12.38 35 .845820 11.05 .877391 11.88 36. .805776 11.58 .834472 12.38 36 .846483 11.03 .878104 11.87 37 .806471 11.57 .835215 12.37 37 .847145 11.02 .878816 11.87 38 .807165 11.57 .835957 12.35 38 .847806 11.02 .879528 11. 85 39 .807859 11.55 .836698 12.35 39 .848467 11.00 .880239 11.83 40 .808552 11.53 .837439 12.33 40 .S49127 11.00 .880949 11.83 41 8.809244 11.53 8.838179 12.33 41 8.849787 11.00 8.881659 11.83 42 .809936 11.53 .838919 12.32 42 .850447 10.98 .882369 11.82 43 .810628 11.52 .839658 12.30 43 .851106 10.97 .883078 11.82 44 .811319 11.50 .840396 12.32 44 .851764 10.97 .883787 11.80 45 .812009 11.50 .841135 12.28 45 .852422 10.95 .884495 11.80 46 .812699 11.48 .841872 12.28 46 .853079 10.95 .885203 11.78 47 .813388 11.48 .842609 12.28 47 .853736 I 10.93 .885910 11.78 48 .814077 11.47 .843:346 12.27 48 .854392 10.93 .886617 11.77 49 .814765 11.45 .844082 12.25 49 .855048 10.92 .887323 11.77 50 .815452 11.45 .844817 12.25 50 .855703 10.92 .888029 11.75 51 8.816139 11.43 8.845552 12.25 51 8.856358 10.90 8.888734 11.75 52 .816825 11.43 .846287 12.23 52 .857012 10.90 .889439 11.75 53 .817511 11.42 .847021 12.22 53 .857666 10.88 .890144 11.73 54 .818196 11.42 .847754 12.22 I 54 .858319 10.88 .890848 11.72 55 .818881 11.40 .848487 12.22 !! 55 .858972 10.87 .891551 11.72 56 .819565 11.40 .849220 12.20 i! 56 .859624 10.87 .81)2254 11.72 57 .820249 11.38 .849952 12.18 57 .860276 10.85 .892957 11.70 58 .820932 11.37 .850683 12.18 58 .860927 10.85 .893659 11.70 59 .821614 11.37 .851414 12.17 59 .861578 10.83 .894361 11.68 60 8.822296 11.35 8.852144 12.17 60 8.862228 10.82 8.895062 11.08 AND EXTERNAL SECANTS. 22 23 "~1 | ' Vers. D. r. Ex. sec. D. 1". / Vers. D.I". Ex. sec.: D. 1'. 8.862228 10.82 8.895062 11.68 c 8.900341 10.33 8.936315 11.23 1 .862877 10.83 .895763 11.67 1 .900961 10.35 .936989 11.23 .863527 10.80 .896463 11.67 2 .901582 10.32 .937663 11.22 3 .864175 10.80 .897163 11.65 3 .902201 10.33 .938336 11.22 4 .864823 10.80 .897862 11.65 4 .902821 10.32 .939009 11.22 5 .865471 10.78 .898561 11.63 5 .903440 10.30 .939682 11. SO .866118 10.78 .899259 11.63 6 .904058 10.30 .940354 11.20 7 .86676-. 10.77 .899957 11.63 7 .904676 10.28 .941020 11.20 8 .867411 10.77 .900655 11.62 8 .905293 10.28 .941698 11.18 9 .868057 10.75 .901352 11.60 9 .905910 10.28 .942369 11.17 10 .868702 10.73 .902048 11.62 10 .906527 10.27 .943039 11.18 11 8.869346 10.75 8.902745 11.58 11 8.907143 10.27 8.943710 11.15 12 .869991 10.72 .903440 11.60 12 .907759 10.25 .944379 11.17 13 .870634 10.72 .904136 11.57 13 .908374 10.25 .945049 11.15 14 .871277 10.72 .904830 11.58 14 .908989 10.23 .945718 11.13 15 .871920 10.70 .905525 11.57 15 .909603 10.23 .946386 11.13 16 .872562 10.70 .906219 11.55 16 .910217 10.22 .947054 11.13 17 .873204 10.68 .906912 11.55 17 .910830 10.22 .947722 11.12 18 .873845 10.68 .907605 11.53 18 .911443 10.22 .948389 11.12 19 .874486 10.67 .908298 11.53 19 .912056 10.20 .949056 11.12 20 .875126 10.67 .908990 11.52 20 .912668 10.18 .949723 11.10 21 8.875766 10.65 8.909681 11.52 81 8.913279 10.20 8.950389 11.10 22 .876405 10.65 .910372 11.52 22 .913891 10.17 .951055 11.08 23 .877044 10.63 .911063 11.52 23 .914501 10.17 .951720 11.08 24 .877682 10.63 .911754 11.48 24 .915111 10 17 .952385 11.07 25 .878320 10.62 .912443 11.50 25 .915721 10.17 .953049 11.07 26 .878957 10.62 .9131*3 11.48 26 .916331 10.15 .953713 11.07 27 .879594 10.60 .913822 11.47 27 .916940 10.13 .954377 11.05 28 .880230 10.60 .914610 11.47 28 .917548 10.13 i .95.5040 11.05 29 .880866 10.58 .915198 11.47 29 .918156 10.13 .955703 11.05 30 .881501 10.58 .915886 11.45 30 .918764 10.12 .956366 11.03 31 8.882136 10.58 8.916573 11.45 31 8.919371 10.10 '8.957028 11.03 32 .882771 10.57 .917260 11.43 32 .919977 10.12 .957690 11.02 33 .883405 10.55 .917946 11.43 33 .920584 10.10 .958351 11.02 34 .884038 10.55 .918632 11.43 34 .921190 10.08 .959012 11.00 35 .884671 10.53 .919318 11.42 35 .921795 10.08 .959672 11.00 36 .885303 10.53 .920003 11.40 36 .922400 10.07 .960332 11.00 37 .885935 10.53 .920687 11.42 37 .923004 10.07 .960992 10.98 38 .886567 10.52 .921372 11.38 38 .923608 10.07 .961651 10.98 39 .887198 10.52 .923055 11.40 89 .924212 10.05 .962310 10.98 to .887829 10.50 .922739 11.37 40 .924815 10.05 .962969 10.97 41 8.888459 10.48 8.923421 11.38 41 8.925418 10.03 8.963627 10.97 42 .889088 10.48 .924104 11.37 42 .926020 10.03 .964285 10.95 43 .889717 10.48 .924786 11.35 43 .926622 10.03 .964942 10.95 44 .S'.HKUli 10.47 .925467 11.37 44 .927224 10.02 .965599 10.95 45 .890974 10.47 .926149 11.33 45 .927825 10.00 .966256 10.93 46 .891602 10.45 .926829 11.35 46 .928425 10.00 .966912 10.93 47 .892229 10.45 .927510 11.32 47 .929025 10.00 .967568 10.92 48 .892856 10.43 .928189 11.33 48 .929625 9.98 .968223 10.92 49 .893482 10.43 .928869 11.32 49 .930224 9.98 .968878 10.92 50 .894108 10.42 .929548 11.30 5D .930823 9.97 .969533 10.90 51 8.894733 10.42 8.930226 11.32 51 8.931421 9.97 8.970187 10.90 52 .895358 10.42 .930905 11.28 52 .932019 9.97 .970841 10.88 53 895983 10.40 .931582 11.30 53 .933617 9.95 .971494 10.88 54 .896607 10.38 .932260 11.27 54 .9&3214 9.95 .972147 10.88 55 .897230 10.38 .932936 11.28 55 .9.33811 9.93 .972800 10.87 56 .897853 10.38 .933613 11.27 56 .984407 9.93 .973452 10.87 57 .898476 10.37 .934289 11.27 57 .935003 9.92 .974104 10.87 58 899098 10.35 .934965 11.25 58 .935598 >.!I2 .974756 10.85 59 .899719 10.37 .935640 11.25 59 ! 936193 0.93 .97'5407 10.85 60 8.900341 10.33 8.936315 11. S3 GO s.; .066859 8.45 .120723 9.57 '">-; O'iiv! IT 8.78 .086338 .067366 8.47 .121297 9 5; .VI (136874 60 9.037401 8.78 8.77 .086929 9.087520 9.ffi 5ii .o;;x'4 !).S3 (30 9.068380 8.43 8.45 .121871 9.122145 9.57 9! 57 TABLE XXVI. LOGARITHMIC VERSEb SINES J 28 29 / Vers. D. 1'. Ex. sec. D. 1'. / Vers. D.r Ex. sec. D. 1". 9.068380 8.45 9.122445 9.57 9.098229 8.15 9.156410 9 30 1 .068887 8.43 .123019 9.57 1 .098718 8.13 .156968 9.32 2 .069393 8.43 .123593 9.55 2 .099206 8.12 .157527 9.28 3 .069899 .8.43 .124166 9.55 3 .099693 8.13 .158084 9 30 4 .070405 8.42 . 124739 9.53 4 .100181 8.12 .158642 9.30 5 .070910 8.42 .125311 9.55 5 .100668 8.12 .159200 9 28 6 .071415 8.40 .125884 9.53 6 .101155 8.12 .159757 9.28 7 .071919 8.42 .126456 9.53 .101642 8 10 . 160314 9.27 8 .072424 8.40 .127028 9.52 8 .102128 8.10 .160870 9^28 9 .072928 8.40 . 127599 9.53 9 .102614 8.10 .161427 9.27 10 .073432 8.38 .128171 9.52 10 .103100 8.08 .161983 9.27 11 9.073935 8.38 9.128742 9.52 11 9.103585 8.08 9.162539 9.27 12 .074438 8.38 .129313 9.50 12 .104070 8.08 .163095 9 25 13 .074941 8.37 .129883 9.50 13 .104555 8.08 . 163650 9.2c 14 .075443 8.38 .130453 9.50 14 .105040 8.07 .164205 9^25 15 .075946 8.35 .131023 9.50 15 . 105524 8.07 .164760 9.25 16 .076447 8.37 .131593 9.50 16 .106008 8.05 .165315 9 25 17 .076949 8.35 .132163 9.48 17 .106491 8.07 . 165870 9.23 18 .077450 8.35 .132732 9.48 18 . 106975 8.05 .166424 9.23 19 .077951 8.35 .13.3301 9.48 19 .107458 8.05 .166978 9 23 30 .078452 8.33 .133870 9.47 20 .107941 8.03 .167532 9.22 21 9.078952 8.33 9.134438 8.47 21 9.108423 8.05 9.168085 9.23 22 .079452 8.33 .135006 9.47 22 .108906 8.03 .168639 9.22 23 .079952 8.32 .135574 9.47 23 .109388 8.02 .169192 9.22 24 .080451 8.32 .136142 9.45 24 . 109869 8.03 .169745 9.20 25 .080950 8.32 .136709 9.47 25 .110351 8.02 .170297 9 22 26 .081449 8.32 .137277 9.45 26 .110832 8.02 170850 9 20 27 .081948 8.30 .137844 9.43 27 .111313 8.00 .171402 9^20 28 .082446 8.30 .138410 9.45 28 .111793 8. CO . 171954 9.18 29 .082944 8.28 . 138977 9.43 29 .112273 8.00 .172505 9 20 30 .083441 8.30 .139543 9.43 30 .112753 8.00 .173057 9.18 31 9.083939 8.28 9.140109 9.42 31 9.113233 8.00 9.173608 9 18 32 .084436 8.27 .140674 9.43 32 .113713 7.98 .174159 9.18 33 .084932 8.28 .141240 9.42 33 .114192 7.98 . 174710 9 17 34 .085429 8.27 .141805 9.42 34 .114671 7.97 .175260 9.17 35 .085925 8.25 .142370 9.40 35 .115149 7.97 .175810 9.17 36 .086420 8.27 .14.2934 9.42 36 .115627 7.97 .176360 9.17 37 .086916 8.25 .143499 9.40 37 .116105 7.97 .176910 9.17 38 .087411 8.25 .144063 9.40 38 .116583 7.97 .177460 9.15 39 .087906 8.23 .144627 9.38 39 .117061 7.95 .178009 9.15 40 .088400 8.25 . 145190 9.40 40 .117538 7.95 .178558 9.15 41 9.088895 8.23 9.145754 9.38 41 9.118015 7.93 9.179107 9.15 42 .089389 8.22 .146317 9.38 42 .118491 7.95 .179656 9 13 43 .089882 8.23 .146880 9.37 43 .118968 7.93 .180204 9.13 44 .090376 8.22 . 147442 9.38 44 .119444 7.92 .180752 9.13 45 .090869 8.22 .148005 9.37 45 .119919 7.90 .181300 9.13 46 .091362 8.20 .148567 9.37 46 . 120395 7.92 .181848 9.12 47 .091854 8.20 .149129 9.35 47 . 120870 7.92 .182395 9.13 48 .092346 8.20 .149690 9.35 48 .121345 7.92 . 182943 9.12 t 49 .092838 8.20 .150251 9.37 49 .121820 7.90 .183490 9.10 50 .093330 8.18 . 150813 9.33 50 .122294 7.90 .184036 9.12 51 9.093821 8.18 9.151373 9.35 51 9.122768 7.90 9.184583 9.10 58 .094312 8.18 .151934 9.33 52 .123242 7.88 .185129 9.10 53 094803 8.17 152494 9.35 53 .123715 7.90 .185675 9.10 54 .095293 8.17 .153055 9.32 54 .124189 7.88 .186221 9.10 55 .095783 8.17 .153614 9.33 55 .124662 7.87 .186767 9.08 56 .096273 8.17 .154174 9.32 56 .125134 7.88 .187312 9.10 57 .096763 8.15 .154733 9.33 1 57 .125607 7.87 .187858 9.08 58 .097252 8.15 .155293 9.30 i 58 .126079 7.87 .188403 9.07 59 .097741 8.13 .155851 9.32 II 59 .126551 7.85 .188947 9.08 60 9.098229 8.15 i 9.156410 i 9.30 || 60 9.127022 7.87 9.189492 9.07 [420] AND EXTERNAL SECANTS. 30 31 ' Vers. D.I". Ex. sec. D. 1*. / Vers. D.I". Ex. sec. D. 1". 9.127022 .87 9.189492 9.07 9.154828 7.58 9.221762 8.85 1 .127494 .85 .190036 9.07 1 .155283 7.58 .222293 8.87 2 .127965 .85 .190580 9.07 2 .155738 7.58 .222825 8.83 3 .128436 .83 .191124 9.07 3 .156193 7.58 .223355 8.85 4 .128906 .83 .191668 9.05 4 .156648 7.57 .223886 8.85 5 .129376 .83 .192211 9.05 5 .157102 7.57 .224417 8.83 6 .129846 .83 .192754 9.05 6 .157556 7.57 .224947 8.83 .130316 .82 .193297 9.05 .158010 7.57 .225477 8.&3 8 .130785 .83 .193840 9.03 8 .158464 7.55 .226007 8.83 9 .131255 .82 .194382 9.05 Q .158917 7.55 .226537 8.82 10 .131724 7.80 .194925 9.03 10 .159370 7.55 .227066 8.82 11 9.132192 7.80 9.195467 9.03 11 9.159823 7.55 9.227595 8.83 12 .132660 7.82 .196009 9.02 12 .160276 7.53 .228125 8.80 13 .133129 7.78 .196550 9.03 13 .160728 7.53 .228653 8.82 14 .133596 7.80 .197092 9.02 14 .161180 7.53 .229182 8.82 15 .134064 7.78 .197633 9.02 15 .161632 7.52 .229711 8.80 16 .134531 7.78 .198174 9.02 16 .162083 7.53 .230239 8.80 it .134998 7.78 .198715 9.00 17 .162535 7.52 .230767 8.80 18 . 135465 7.77 .199255 9.00 18 .162986 7.52 .231295 8.78- 19 .135931 7.77 .199795 9.00 19 .163437 7.50 .231822 8.80 20 .136397 7.77 .200335 9.00 20 .163887 7.52 .232350 8.78 21 9.136863 7.77 9.200875 9.00 21 9.164338 7.50 9.232877 8.78 22 .137329 7.75 .201415 8.98 22 .164788 7.48 .233404 8.78 23 .137794 7.77 .201954 9.00 23 .165237 7.50 .233931 8.78 24 .138260 7.73 .202494 8.97 24 .165687 7.48 .234458 8.77 25 .138724 7.75 .203032 8.98 25 .166136 7.48 .234984 8.77 26 .139189 7.73 .203571 8.98 26 .166585 7.48 .235510 8.77 27 .139653 7.73 .204110 8.97 27 .167034 7.48 .236036 8.77 28 .140117 7.73 .204648 8.97 28 .167483 7.47 .236562 8.77 29 .140581 7.73 .205186 8.97 29 .167931 7.47 .237088 8.75 30 .141045 7.72 .205724 8.97 30 .168379 7.47 .237613 8.77 31 9.141508 7.72 9.206262 8.95 31 9.168827 7.47 9.238139 8.75 32 .141971 7.72 .206799 8.97 32 .169275 7.45 .238664 8.75 33 .142434 7.70 .207337 8.95 as .169722 7.45 .239189 8.73 34 .142896 7.70 .207874 8.93 34 .170169 7.45 .239713 8.75 35 .14*358 7.70 .208410 8.95 ,35 .170616 7.43 .240238 8.73 36 .143820 7.70 .208947 8.93 36 .171062 7.45 .240762 8.73 37 .144282 7.68 .209483 8.95 37 .171509 7.43 .241286 8.73 38 .144743 7.68 .210020 8.93 38 .171955 7.42 .241810 8.72 39 .145204 7.68 .210556 8.92 39 .172400 7.43 .242333 8.73 40 .145665 7.68 .211091 8.93 40 .172846 7.42 .242857 8.72 41 9.146126 7.67 9.211627 8.92 41 9.173291 7.42 9.243380 8.72 42 .146586 7.67 .212162 8.92 42 .173736 7.42 .243903 8.72 43 .147046 7.67 .212697 8.92 43 .17481 7.42 .244426 8.72 44 .147506 7.67 .213232 8.92 44 .174626 7.40 .244949 8.70 45 .147966 7.65 .213767 8.90 45 .175070 7.40 .245471 8.72 46 .148425 7.65 .214301 8.92 46 .175514 7.40 .245994 8.70 47 .148884 7.65 .214836 8.90 47 .175958 7.40 .246516 8.70 48 .149343 7.153 .215370 8,90 48 .J76408 7.38 .247038 8.68 49 .149801 7.63 .215904 8.88 49 .176845 7.38 .247559 8.70 50 .150259 7.63 .216437 8.90 50 .177288 7.38 .248081 8.68 51 9.150717 7.63 9.216971 8.88 51 9.177731 7.38 9.248602 8.68 52 .151175 7.63 .217504 8.88 52 .178174 7.37 .249123 8.68 53 .i:.1(i:53 7.62 .218037 8.88 53 .178616 7.37 .249644 8.68 54 .152090 7.62 .218570 8.87 54 .179058 7.37 .250165 8.68 55 .152547 7.60 .219102 8.88 55 .179500 7.37 .250686 8.67 56 .153003 7.62 .2196.35 8.87 56 .179942 7.35 .251206 8.67 57 .153460 7.60 .220167 8.87 57 . I80SH3 7.37 .251726 8.67 58 153916 7.60 .220699 8.87 58 .180825 7.33 .252246 8.67 59 .154372 7.60 .221231 8.85 50 .181265 7.35 .252766 8.67 60 9.154828 7.58 9.221762 8.85 60 9.181706 7.35 9.253286 8.65 [421] TABLE XXVI.-LOGARITHMIG VERSED SINES 32 33 / Vers. D.I". Ex. sec. D.r. / Vers. D.I". Ex. sec. D.l". 9.181706 .35 9.253286 8.65 9.207714 7.10 9.284122 8.48 1 .182147 .33 .253805 8.65 1 .208140 7.10 .284631 8.47 2 .182587 .33 .254324 8.65 2 .208566 7.10 .285139 8.47 3 . 183027 .32 .254843 8.65 3 .208992 7.10 .285647 8.47 4 .183466 .33 .255362 8.65 4* .209418 7.08 .286155 8.47 5 .183906 .32 .255881 8.63 1 5 .209843 7.08 .286663 8.45 6 .184345 .32 .256399 8.65 6 .210268 .08 .287170 8.47 7 .184784 .32 .256918 8.63 7 .210693 .08 .287678 8.45 8 .185223 .32 .257436 8.63 8 .211118 .08 .288185 8.45 9 .185662 7.30 .257954 8.62 9 .211543 .07 .288692 8.45 10 .186100 7.30 .258471 8.63 ! 10 .211967 .07 .289199 843 11 9.186538 7.30 9.258989 8.62 11 9.212391 .07 9.289705 8.45 12 .186978 7.28 .259506 8 62 12 .212815 .07 .290212 8.43 13 .187413 7.30 .260023 8^62 13 .213239 .05 .290718 8.43 14 .187851 7.28 .260540 8.62 14 .213662 .05 .291224 8.43 15 .188288 7.27 .261057 8.62 15 .214085 .05 .291730 8.43 16 .188724 7.28 .261574 8.60 16 .214508 .05 .292236 8.43 17 .189161 7.27 .262090 8.60 17 .214931 .05 .292742 8.42 18 .189597 7.28 .262606 8.60 18 .215354 .03 .293247 8.43 19 .190034 7.25 .263122 8.60 19 .215776 .03 .293753 8.42 20 .190469 7.27 .263638 8.60 20 .216198 .03 .294258 8.42 21 9.190905 7.27 9.264154 8.58 21 9.216620 .03 9.294763 8.42 22 .191341 7.25 .264669 8.58 22 .217042 .02 .295268 8.40 23 .191776 7.25 .265184 8.60 23 .217463 .02 .295772 8.42 24 .192211 7.23 .265700 8.57 24 .217884 7.02 .296277 8.40 25 .192645 7.25 .266214 8.58 25 .218305 7.02 .296781 8.40 26 .193080 7.23 .266729 8.58 26 .218726 7.00 .297285 8.40 27 .193514 7.23 .267244 8.57 27 .219146 7.02 .297789 8.40 28 .193948 7.23 .267758 8.57 28 .219567 7.00 .298293 8.40 29 .194382 7.22 .268272 8.57 29 .219987 7.00 .298797 8.38 30 .194815 7.23 .268786 8.57 30 .220407 6.98 .299300 8.38 31 9.195249 7.22 9.269300 8.57 31 9.220826 7.00 9.299803 8.40 32 .195682 7.22 .269814 8.55 32 .221246 6.98 .300307 8.37 33 .196115 7.20 .270327 8.55 33 .221665 6.98 .300809 8.38 34 .196547 7.22 .270840 8.57 34 .222084 6.98 .301312 8.38 35 .196980 7.20 .271354 8.53 35 .222503 6.97 .301815 8.37 36 .197412 7.20 .271866 8.55 36 .222921 6.98 .302317 8.38 37 .197844 7.18 .272379 8.55 37 .223340 6.97 .302820 8.37 38 .198275 7.20 .272892 8.53 38 .223758 6.97 .303322 8.37 39 .198707 7.18 .273404 8.53 39 .224176 6.95 .303824 8.35 40 .199138 7.18 .273916 8.53 40 .224593 6.97 .304325 8.37 41 9.199569 7.18 9.274428 8.53 41 9.225011 6.95 9.304827 8.35 42 .200000 7.17 .274940 8.53 42 .225428 6.95 .305328 8.37 43 .200430 7.18 .275452 8.52 43 .225845 6.95 .305830 8.35 44 .200861 7.17 .275963 8.52 44 .226262 6.93 .306331 8.35 45 .201291 7.15 .276474 8.53 45 .226678 6.95 .306832 8.35 46 .201720 7.17 .276986 8.50 46 .227095 6.93 .307333 8.33 47 .202150 7.15 .277496 8.52 47 .227511 6.93 .307833 8.35 48 .202579 7.15 .278007 8.52 48 .227927 6.92 .308334 8.33 49 .203008 7.15 .278518 8.50 49 .228342 6.93 .308834 8.33 50 .203437 7.15 .279028 8.50 50 .228758 6.92 .309334 8.33 51 9.203866 7.13 9.279538 8.50 51 9.229173 6.92 9.309834 8.33 52 .204294 7.15 .280048 8.50 52 .229588 6.92 .310334 8.33 53 .204723 7.13 .280558 8.50 53 .230003 6.92 .310834 8.32 54 .205151 7.12 .281068. 8.48 54 .230418 6.90 .311333 8.32 55 .205578 7.13 .281577 8.50 55 .230832 6.90 .311832 8.32 56 .206006 7.13 .282087 8.48 56 .231246 6.90 .312331 8.32 57 .206433 7.12 .282596 8.48 57 .231660 6.90 .312830 8.32 58 .206860 7.12 .283105 8.48 58 .232074 6.88 .313329 8.32 59 .207287 7.12 .283614 8.47 59 .232487 6.90 .313828 8.30 60 9.207714 7.10 9.284122 8.48 60 9.232901 6.88 9.314326 8.32 L22l AND EXTERNAL SECANTS. 34- 35 ' j Vers. I D. 1". Ex. sec. D. 1". ' Vers. D.I". Ex. sec. D. 1". 9.232901 6.88 9.314326 8.32 9.257314 6.67 9.343949 8.15 1 .233314 6.88 .314825 8.30 1 .257714 6.68 .344438 8.15 2 .233727 6.87 .315323 8.30 2 .258115 6.67 .344927 8.15 3 .234139 6.88 .315821 8.30 3 .258515 6.67 .345416 8.13 4 .2:34552 6.87 .316319 8.30 4 .258915 6.65 .345904 8.15 5 .234964 6.87 .316817 8.28 5 .259314 6.67 .346393 8.13 6 .235376 6.87 .317314 8.28 6 .259714 6.65 .346881 8.13 7 .235788 6.85 .317811 8.30 7 .260113 6.65 .347369 8.13 8 .236199 6.87 .318309 8.28 8 .260512 6.65 .347857 8.13 9 .236611 6.85 .318806 8:28 9 .260911 6.65 .348345 8.13 10 .237022 6.85 .319:303 8.27 | 10 .261310 6.65 .348833 8.13 11 9.237433 6.85 9.319799 8.28 11 9.261709 6.63 9.349321 8.12 12 .237844 6.83 .320296 8.27 12 .262107 6.63 .349808 8.12 13 .238254 6.85 .320792 8.28 13 .262505 6.63 .350295 8.12 14 .238665 6.83 .321289 8.27 14 .262903 6.63 .350782 8.12 15 .239075 6.83 .321785 8.27 15 .263301 6.62 .351269 8.12 16 .239485 6.82 .322281 8.25 16 .263698 6.63 .351756 8.12 17 239894 6.83 .322776 8.27 17 .264096 6.62 .352243 8.12 18 .240304 6.82 .323272 8.27 18 .264493 6.62 .3527:30 8.10 19 .240713 6.82 .323768 8.25 19 .264890 6 62 .353216 8.10 20 .241122 6.82 .324263 8.25 20 .265287 6.60 .8*3702 8.10 21 9.241531 6.82 9.324758 8.25 21 9.265683 1 6.62 9.354188 8.10 22 .241940 6.82 .325253 8.25 22 .266080 6.60 .354674 8.10 23 .242348 6.80 .325748 8.25 23 .266476 6.60 .355160 8.10 24 .242756 6.80 .326243 8.23 24 .266872 6.58 .355646 8.08 25 .243164 6.80 .326737 8.25 ; 25 .267267 6.60 .356131 8.10 26 .243572 6.80 .327232 8.23 26 .267663 6.58 .a56617 8.08 27 .243980 6.78 .327726 8.23 27 .268058 6.58 .357102 8.08 28 .244387 6.78 .328220 8.23 28 .268453 6.58 .357587 8.08 29 .244794 6.78 .328714 8.22 29 .268848 6.58 .358072 8.08 30 .245201 6.78 .329207 8.23 30 .269243 6.58 .358557 8.08 31 9.245608 6.7? 9.329701 8.23 31 9.269638 6.57 9.359042 8.07 32 .246014 6.78 .330195 8.22 32 .270032 6.57 .359526 8.08 33 .246421 6.77 .330688 8.22 33 .270426 6.57 .360011 8.07 34 .246827 6.77 .331181 8.22 34 .270820 6.57 .360495 8.07 35 .247233 6.77 .331674 8.22 35 .271214 6.57 .360979 8.07 36 .247639 6.75 .332167 8.20 36 .271608 6.55 .361463 8.07 37 .248044 6.75 .332659 8.22 37 .272001 6.55 .361947 8.07 38 .248449 6.75 .333152 8.20 38 .272394 6.55 .362431 8.05 39 .248854 6.75 .333644 8.22 39 .272787 6.55 .362914 8.07 40 .249259 6.75 .334137 8.20 40 .273180 6.53 .36:3398 8.05 41 9.249664 6.73 9.334629 8.20 41 9.273572 6.55 9.363881 8.05 42 .250068 6.75 .335121 8.18 42 .273965 6.53 .364364 8.05 43 .250473 6.73 .335612 8.20 43 .ST4857 6. S3 .364847 8.05 44 .250877 6.73 .336104 8.18 44 .274749 6.53 .305330 8.05 45 .251281 6.72 .336595 8.20 45 .275141 0.52 .365813 -8.0:3 46 .251684 6.73 .337087 8.18 46 .275532 6.53 .366295 8.05 47 .252088 6.72 .337578 8.18 47 .275924 6.52 .366778 8.03 48 .252491 6.72 .S38069 8.18 48 .270315 6.52 .367260 8.0:3 49 .252894 6.72 .assseo 8.17 49 .276706 6.52 .367742 8.03 50 .253297 6.70 .339050 8.18 50 277097 6.52 .368224 8.03 51 9.253699 6.72 9.339541 8.17 51 9.277488 6.50 9.368706 8.03 52 .254102 6.70 .340031 8.18' 52 .277878 6.50 .369188 8.03 53 .254504 6.70 .340522 8.17 53 .278268 6.50 .369670 8.02 54 .254906 6.70 .341012 8.17 54 .278658 6.50 .370151 8.02 55 .255308 6.68 .341502 8.15 55 .279048 6.50 .370632 8 03 56 .255709 6.70 .341991 8.17 56 .279438 6.48 .371114 8.02 57 .256111 6.68 .342481 8.17 57 .279827 6.50 .371595 ' 8.02 58 .256512 6.68 .342971 8.15 58 .280217 6.48 .372076 8.00 59 .256913 6.68 .343460 8.15 59 .280606 6.48 .372556 8.02 60 9.257314 6.67 9.343949 8.15 60 9.280995 6.47 9.373037 8.02 [423] TABLE XXVI.-LOGARITHMIC VERSED SINES 36 37 e 31 Yers. 9.280995 .281383 .281772 .282160 .282548 .283712 .284873 9.285260 .287191 .287577 .289118 .290271 .290655 .291039 .291423 .291807 .293722 .294104 .294486 .295250 .295632 .296014 9.296776 .297157 .297538 .297918 .300198 300577 .300957 .301335 .301714 .302471 D.I 1 .303227 .303605 6.47 6.48 6.47 6.47 6.47 6.47 6.47 6.45 6.45 6.45 6.45 6.45 6.43 6.43 6.43 6.43 6.43 6.42 6.40 6.42 6.42 6.40 6.42 6.40 6.40 6.40 6.40 6.40 6.38 6.38 6.38 6.38 6.38 6.37 6.37 6.37 6.37 6.37 6.37 6.35 6.35 6.35 6.33 6.32 6.32 6.33 6.30 6.30 6.30 6.30 6.30 Ex. sec. .373037 .373518 .373998 .374478 .374958 .375438 .375918 .376877 .377357 .377836 .378315 .378794 .379273 .379752 .380231 .381188 .381666 .382144 9.383100 .383577 .384055 .384532 .385010 .385487 .385964 .386441 .387394 9.387871 .390727 .392154 9.392629 .393104 .393579 .394054 .395478 ). 397374 .399742 .400215 .400688 .401161 9.401634 D. 1". 8.02 8.00 8.00 8.00 8.00 8.00 8.00 7.98 8.00 7.98 7.98 7.98 7.98 7.98 7.98 7.97 7.98 7.97 7.97 7.97 7.97 7.95 7.97 7.95 7.97 7.95 7.95 7.95 7.95 7.93 7.95 7.93 7.95 7.93 7.93 7.93 7.92 7.93 7.92 7.93 7.92 7.92 7.92 7.92 7.92 7.90 7.92 7.90 7.90 7.90 7.90 7.90 7.90 7.88 Vers. D. 1" .304738 .305115 .305492 .305868 .306245 .306621 .307374 .307749 9.308125 .308501 .308876 6.28 6.27 6.28 6.27 6.28 6.27 6.25 6.27 6.27 6.25 6.25 6.25 6.25 6.23 6.25 6.23 6.23 6.23 6.22 6.23 6.22 .313365 ! 6.22 .310001 .310375 .310750 .311124 .311498 9.311872 .312245 .312619 .313738 .314111 .314484 .314856 .315228 .315600 .315972 .316344 .316716 .317087 .317458 .317829 .318200 .318571 .318941 .319311 .319682 .320051 .320421 .320791 .321160 .321530 6.18 6.15 6.17 6.17 6.15 6.17 6.15 6.13 6.15 .322636 | 6.15 .323005 .32-3373 .323741 .324109 .324477 .324845 .325580 .325947 6.13 6.13 6.13 6.13 6.13 6.12 6.13 6.12 6.12 6.12 6.22 6.22 ; 6.20 6.20 6.20 I 6.20 | 6.20 6.20 6.18 6.18 6.18 6.18 6.18 6.17 6.17 .426053 .426520 427452 .427918 .428850 .429316 Ex. sec. D. I* .401634 .402107 .402580 .403052 .403524 .403997 .404469 .404941 .405412 .405884 .407298 .407770 .408241 .408712 .409183 .409653 .410124 .410594 .411065 .411535 .412005 .412475 .412945 .413415 .413884 .414354 .414823 .415293 .415762 .416231 .416700 .417168 .417637 .418106 .418574 .419042 .419511 .419979 .420447 .420915 .421382 .421850 .422317 .422785 .423252 .423719 .424186 .424653 .425120 AND EXTERNAL SECANTS. 38* 39 ' Vers. D. 1". Ex. sec. D. r / Vers. D.I". 12x. sec. D. 1". 9.326314 6.12 9.429782 7.75 9.348021 5.93 9.457518 7.65 1 .326681 6.10 .430247 7.77 1 .348377 5.95 .457977 7.65 2 .327047 6.12 .430713 7.75 2 .348734 5.93 .458436 7.65 3 .327414 6.10 .431178 7.75 3 .349090 5.93 .458895 7.63 4 .327780 6.10 .431643 7.75 4 .349446 5.93 .459353 7.65 5 .328146 6.10 .432108 7.75 5 .349802 5.93 .459812 7.63 6 .328512 6.10 .432573 7.75 6 .850158 5.93 .460270 7.65 7 .328878 6.08 .433038 7.75 7 .350514 5.92 .460729 7.63 8 .329243 6.10 .433503 7.73 8 .350869 5.93 .461187 7.63 9 .3.29609 6.08 .433967 7.75 9 .351225 5.92 .461645 7.63 10 .329974 6.08 .434432 7.73 10 .351580 5.92 .462108 7.63 11 9.330339 6.08 9.434896 7.75 11 9.351935 5.92 9.462561 7.63 12 .330704 6.08 .435361 7.73 12 .352290 5.90 .463089 7.63 13 .331069 6.07 .435825 7.73 13 .352644 5.92 .463477 7 62 14 .331433 6.08 .436289 7.73 14 .352999 5.90 .463934 7.63 15 .331798 6.07 .436753 7.73 15 .353353 5.90 .464392 7.62 16 .332162 6.07 .437217 7.72 16 .353707 5.92 .464849 7.63 17 .332526 6.07 .437680 7.73 17 .354062 5.88 .465307 7.62 18 .332890 6.07 .438144 7.73 18 .354415 5.90 .465764 7.62 19 .333254 6.05 .438608 7.72 19 .354769 5.90 .466221 7.62 20 .333617 6.07 .439071 7.72 20 .355123 5.88 .466678 7.62 21 9.333981 6.05 9.439534 7.72 21 9.1 355476 5.88 9.467135 .62 22 .334344 6.05 .439997 7.72 ' 22 .355829 5.88 .467592 .62 23 .334707 6.05 .440460 7.72 23 .356182 5.88 .468049 .62 24 .335070 6.03 .440923 7.72 24 .356535 5.88 .468506 .60 25 .335432 6.05 .441386 7.72 25 .356888 5.88 .468962 .60 26 .335795 6.03 .441849 7.72 26 .357241 5.87 .469418 .62 27 .336157 6.03 .442312 7.70 27 .357593 5.87 .469875 .60 28 .336519 6.03 .442774 7.72 28 .357945 5.87 .470331 7.60 29 .336881 6.03 .443237 7.70 29 .358297 5.87 .470787 7.60 30 .337243 6.03 .443699 7.70 30 .$58649 5.87 .471243 7.60 31 9.337605 6.02 9.444161 7.70 31 9.359001 5.87 9.471699 7.60 32 .337966 6.03 .444623 7.70 32 .359353 5.85 .472155 7.60 as .338328 6.02 .445085 7.70 33 .$59704 5.87 .472611 7.60 34 .338689 6.02 .445547 7.70 I 34 .360056 5.85 .473067 7.58 35 .339050 6.02 .446009 7.68 35 .360407 5.85 .473522 7.60 36 .339411 6.00 .446470 7.70 36 .360758 5.83 .473978 7.58 37 .339771 6.02 .446932 7.68 37 .361108 5.85 .474433 7.58 38 .340132 6.00 .447393 7.70 i 38 .361459 5.85 .474888 7.58 39 .340492 6.00 ,147855 7.68 39 .361810 5.83 .475343 7.58 40 340852 6.00 .448316 7.68 40 .362160 5.83 .475798 7.58 41 9.341212 6.00 9.448777 7.68 41 9.362510 5.83 9.476253 7.58 42 .341572 6.00 .449238 .68 42 .362860 5.83 .476708 7.58 43 .341932 5.98 .449699 .68 43 .363210 5.83 .477163 7.58 44 .342291 6.00 .450160 .67 44 .363560 5.82 .477618 7.57 45 .342651 5.98 .450620 68 45 .363909 5.83 .478072 7.58 46 .343010 5.98 .451081 .67 46 .364259 5.82 .478527 7.57 47 .1343369 5.98 .451541 .68 47 .364608 5.82 .473981 7.57 48 .343728 5.97 .452002 .67 48 .364957 5.82 .4794a5 7.58 49 .344086 5.98 .452462 .67 49 .365306 5.82 .479890 7.57 50 .344445 5.97 .452922 .67 50 .365655 5.80 .480344 7.57 51 9.344803 5.97 9.453382 .67 51 9.36(5003 5.82 9.480798 7.57 52 .345161 5.97 .453842 .67 52 .366352 5.80 .481252 7.55 53 .345519 5.97 .454302 .67 53 .366700 5.80 .481705 7.57 54 .345877 5.97 .454762 .65 54 .367048 5.80 .482159 7.57 55 .346235 5.95 .455221 .67 55 .367396 5.80 .482613 7.55 56 .346592 5.97 .455681 .65 56 .367744 5.78 .483066 7.57 57 .346950 5.95 .456140 .67 57 .368091 5.80 .483520 7.55 58 .347307 5.95 .456600 .65 58 .368439 5.78 .483973 7.55 59 .347664 5.95 .457059 .65 59 .368786 5.78 .484426 7.55 60 9.348021 5.93 9.457518 .65 60 9.369133 5.78 9.484879 7.55 [425] TABLE XXVL-LOGARITHMIC VERSED SINES 40 41 / Vers. D. 1". Ex. sec. D.I'. / Vers. D.I". Ex. sec. D. r. 9.369133 5.78 9.484879 7.55 ~0~ 9.389681 5.62 9.511901 .45 1 .369480 5.78 .485332 7.55 1 .390018 5.63 .512348 .47 2 .369827 5.78 .485785 7.55 2 .390356 5.63 .512796 .45 3 .370174 5.77 .486238 7.55 3 .390694 5.62 .513243 .47 4 .370520 5.78 .486691 7.55 4 .391031 5.62 .513691 .45 5 .370867 5.77 .487144 7.53 6 .391368 5.62 .514138 .45 6 .371213 5.77 .487596 7.55 6 .391705 5.62 .514585 .47 7 .371559 5.77 .488049 7.53 7 .392042 5.62 .515033 .45 8 .371905 5.77 .488501 7.53 8 .392379 5.62 .515480 .45 9 .372251 5.75 .488953 7.55 9 .392716 5.60 .515927 .45 10 .372596 5.77 .489406 7.53 10 .393052 5.60 .516374 .43 11 9.372942 5.75 9.489858 7.53 11 9.393388 5.60 9.516820 .45 12 .373287 5.75 .490310 7.53 12 .393724 5.62 .517267 .45 13 .373632 5.75 .490762 7.53 13 .394061 5.58 .517714 .43 14 .373977 5.75 .491214 7.52 14 .394396 5.60 .5181(10 .45 15 .374322 5.75 .491665 7.53 15 .394732 5.60 .518607 .43 16 .374667 5.73 .492117 7.53 16 .395068 5.58 .519053 .45 17 .375011 5.75 .492569 7.52 17 .395403 5.58 .519500 .43 18 .375356 5.73 .493020 7.52 18 .395738 5.60 .519946 .43 19 .375700 5.73 .493471 7.53 19 .396074 5.58 .520392 .43 20 .376044 5.73 .493923 7.52 20 .396409 5.57 .520838 .43 21 9.376388 5.73 9.494374 7.52 21 9.396743 5.58 9.521284 .43 22 .376732 5.72 .494825 7.52 21 .397078 5.58 .521730 .43 23 .377075 5.73 .495276 7.52 23 .397413 5.57 .522176 .42 24 .377419 5.72 .495727 7.52 24 .397747 5.57 .522621 .43 25 .377762 5.72 .496178 7.50 25 .398081 5.57 .523067 .43 26 .378105 5.72 .496628 7.52 26 .398415 5.57 .523513 .42 27 .378448 5.72 .497079 7.52 27 .398749 5.57 .523958 .43 28 .378791 5.70 .497530 7.50 28 .399083 5.57 .524404 .42 29 .379133 5.72 .497980 7.52 29 .399417 5.55 .524849 .42 30 .379476 5.70 .498430 7.50 30 .399750 5.57 .525294 .42 31 9.379818 5.72 9.498881 7.48 31 9.400084 5.55 9.525739 .42 32 .380161 5.70 .499331 7.52 32 .400417 5.55 .526184 .42 33 .380503 5.70 .499781 7.50 33 .400750 5.55 .526629 .42 34 .380845 5.68 .500231 7.50 34 .401083 5.55 .527074 .42 35 .381186 5.70 .500681 7.50 35 .401416 5.53 .527519 .42 36 .381528 5.68 .501131 7.50 36 .401748 5.55 .527964 .42 37 .381869 5.70 .501581 7.48 37 .402081 5.53 .528409 .40 38 ,382211 5.68 .502030 7.50 88 .402413 5.53 .528853 .42 39 .382552 5.68 .502480 7.48 39 .402745 5.53 .529298 .40 40 .382893 5.68 .502929 7.50 40 .403077 5.53 .529742 .42 41 9.383234 5. 67 9.503379 7.48 41 9.403409 5.53 9.530187 .40 42 .383574 5.68 .503828 7.48 42 .403741 5.53 .530631 .40 43 .383915 5.67 .504277 7.48 43 .404073 5.52 .531075 .40 44 .384255 5.67 .504726 7.48 44 .404404 5.53 .531519 .40 45 .384595 5.67 .505175 7.48 45 .404736 5.52 .531963 .40 46 .384935 5.67 .505624 7.48 46 .405067 5.52 .532407 .40 47 .385275 5.67 .506073 7.48 47 .405398 5.52 .532851 .40 48 .385615 5.67 .506522 7.48 48 .405729 5.50 .533295 .40 49 .385955 5.65 .506971 7.47 49 .400059 5.52 .533739 .38 50 .386294 5.67 .507419 7.48 50 .406390 5.52 .534182 .40 51 9.386634 5.65 9.507868 7.47 51 9.406721 5.50 9.534626 .40 52 .386973 5.65 .508316 7.48 52 .407'051 5.50 .535070 .88 53 .387312 5.65 .508765 7.47 53 .407381 5.50 .535513 .38 54 .387651 5.63 .509213 7.47 54 .407711 5.50 .535956 .40 55 .387989 5.65 .509661 7.47 55 .408041 5.50 .536400 .38 56 .388328 5.63 .510109 7.47 56 .408371 5.48 .536843 .38 57 .388666 5.65 .510557 7.47 57 .408700 5.50 .537286 .38 58 .389005 5.63 .511005 7.47 58 .409030 5.48 .537729 .38 59 .389343 5.63 .511453 7.47 59 .409359 5.48 .538172 .33 60 9.389681 5.62 9.511901 7.45 60 9.409688 5.48 9.538G15 .as AND EXTERNAL SECANTS. 42 43 / Vers. D.r. Ex. sec. D.r. / Vers. D.r. Ex. sec. D. 1". 9.409688 5.48 '9.538615 7.38 1 9.429181 5.35 9.565053 7.32 1 .410017 5.48 .539058 7.37 1 .429502 5.33 .565492 7.30 2 .410346 5.48 .539500 7.38 2 .429822 5., 33 .565930 .32 3 .410675 5.48 .539943 .38 3 .430142 5.35 .566369 .30 4 .411004 5.47 .540386 .37 4 .430463 5.33 .566807 .30 5 .411332 5.47 .540828 .38 5 .430783 5.33 .567245 .30 6 .411660 5.48 .541271 .37 6 .431103 5.32 .567683 .30 7 .411989 5.47 .541713 .37 7 .431422 5.33 .568121 .30 8 .412317 5.45 .542155 .37 8 .431742 5.33 .568559 .30 9 .412644 5.47 .543597 .38 9 .432062 5.32 .568997 .30 10 .412972 5.47 .54:3040 .37 10 .432381 5.32 .569435 .30 11 9.413300 5.45 9.543482 .37 11 9.432700 5.33 9.569873 .30 12 .413627 5.47 .543924 7.37 12 .4&3020 5.32 .570311 .28 13 .413955 5.45 .544:366 7.35 13 .433339 5.30 .570748 .30 14 .414282 5.45 .544807 7.37 14 .433657 5.32 .571186 .30 15 .414609 5.45 .545249 7.37 15 .433976 5 32 .571624 .28 16 .414936 5.45 .545691 7.35 16 .434295 5.30 .572061 .28 17 .415263 5.43 .546132 7.37 17 .434613 5.32 .572498 .30 18 .415589 5.45 .546574 7. .35 | 18 .434932 5.30 .572936 .28 19 .415916 5.43 .547015 7.37 19 .435250 5.30 .573373 .28 20 .416242 5.43 .547457 7.35 20 .435568 5.30 .573810 .28 21 9.416568 5.43 9.547898 7.35 21 9.435886 5.30 9.574247 .30 22 .416894 5.43 .548339 7 37 | 22 .436:204 5.28 .574685 .28 23 .417220 5.43 .548781 7.35 23 .436521 5.30 .575122 .27 24 .417546 5.42 .549222 7.35 24 .436839 5.28 .575558 .30 25 .417871 5.43 .549663 7.35 25 .437156 5.28 .575995 .28 26 .418197 5.42 .550104 7.33 26 .437473 5.30 .576432 .28 27 .418522 5.43 .550544 7.35 27 .437791 5.27 .576869 .28 28 .418848 5.42 .550935 7.35 28 .438107 5.28 .577306 .27 29 .419173 5.42 .551426 7-85 29 .438424 5.28 .577742 .28 30 .419498 5.40 .551867 7.33 30 .438741 5.28 .578179 .27 31 9.419822 5.42 9.552307 7.35 31 9.439058 5.27 9.578615 .28 32 .420147 5.40 .552748 7.33 32 .439374 5.27 .579052 .27 33 .420471 5.42 .553188 7.35 33 .439690 5.28 .579488 .27 34 .420796 5.40 .553029 7.33 34 .440007 5.27 .579924 .28 35 .421120 5.40 .554069 7.33 35 .440323 5.27 .580361 .27 36 .421444 5.40 .554509 7.33 36 .440639 5.25 .580797 27 37 .421768 5.40 .554949 7.33 37 .440954 5.27 .581233 .27 38 .422092 5.40 .555389 7.33 38 .441270 5.25 .581669 .27 39 .422416 5.38 .555829 7.33 39 .441585 5.27 .582105 .27 40 .422739 5.40 .556269 7.33 40 .441901 5.25 .582541 .27 41 9.423063 5.38 9.556709 7.33 41 9.442216 5.25 9.582977 .27 42 .423386 5.38 .557149 7.33 42 .442531 5.25 .583413 .2 43 .423709 5.38 .557589 7.32 43 .442846 5.25 .583848 ,2V 41 .424032 5.38 .558028 7.33 44 .443161 5.25 .584284 27 45 .424355 5.37 .558468 7.32 45 .443476 5.23 .584720 '.25 46 .424677 5.38 .558907 7.33 46 .443790 5.25 .585155 27 47 .425000 5.37 .559347 7.32 47 .444105 5.23 .585591 !25 48 .425322 5.38 .559786 7.33 48 .444419 5.23 .586026 .ft 49 .425645 5.37 .560226 7.32 49 .444733 5.23 .586462 .25 50 .425967 5.37 .560665 7.32 50 .445047 5.23 .586897 .25 51 9.426289 5.37 9.561104 7.32 51 9.445361 5.23 9.587332 .2? 52- .426611 5.37 .561543 7.32 1 52 .445675 5.23 .587767 .>,' 53 .426933 5.35 .561982 7.32 53 .445989 5.22 .sassjrf 7.25 54 .427254 5.37 .562421 7.32 1 54 .446302 5.23 .388638 7.25 55 .427576 5.35 .562860 7.32 55 .446616 5.22 .589073 7.25 56 .427897 5.35 .563299 7.32 56 .446929 5.22 .589508 7.23 57 .428218 5.35 .563738 7.30 57' .447242 5.22 .589942 7.25 58 .428539 5.35 .564176 7.32 58 .447555 5.22 .590377 7.25 59 .428860 5.35 .564615 7.30 59 .447868 5.22 .590812 7.25 60 9.429181 5.33 9.565053 7.32 i 60 9.448 1 . Q 1 5.20 9.591247 7.23 [427] TABLE XXVI. -LOGARITHMIC VERSED SINES 440 45 / Vers. D. r. Ex. sec. D. 1'. / Vers. D. r. Ex. sec. D. r. 9.448181 5.20 9.591247 7.23 9.466709 5.08 9.617224 ".20 1 .448493 5.22 .591681 7.25 1 .467014 5.08 .617656 ".18 2 .448806 5.20 .592116 7.25 2 .467319 5.08 .618087 7.18 3 .449118 5.22 .592551 7.23 3 .467624 5.07 .618518 ".18 4 .449431 5.20 .592985 7.23 4 .467928 5.08 .618949 ".18 5 .449743 5.20 .593419 7.25 5 .468233 5.07 .619380 7.18 6 .450055 5.18 .593854 7.23 6 .468537 5.07 .619811 7.18 7 .450366 5.20 .594288 7.23 *7 .468841 5.07 .620242 MS 8 .450678 5.20 .594722 7.23 8 .469145 5.07 .620673 ".18 9 .450990 5.18 .595156 7.25 9 .469449 5.07 .621104 M8 10 .451301 5.18 .595591 7.23 10 .469753 5.07 .621535 ".18 11 9.451612 5.20 9.596025 7.23 11 9.470057 5.05. 9.621966 7.17 12 .451924 5.16 .596459 7.23 12 .470360 5.07 .622396 7.18 13 .452235 5.18 .596893 7.22 13 .470664 5.05 .622827 7.18 14 .452546 5.17 .597326 7.23 14 .470967 5.05 .623258 7.17 15 .452856 5.17 .597760 7.23 i 15 .471270 5.05 .623688 7.18 16 .453167 5.18 .598194 7.23 16 .471573 5.05 .624119 7.17 17 .453478 5.17 .598628 7.22 17 .471876 5.05 .624549 7.18 18 .453788 5.17 .599061 7.23' 18 .472179 5.05 .624980 7.17 19 .454098 5.17 .599495 7.22 19 .472482 5.03 .625410 7.18 20 .454408 5.17 .599928 7.23 20 .472784 5.05 .625841 7.17 21 9.454718 5.17 9.600362 7.22 21 9.473087 5.03 9.626271 7.17 22 .455028 5.17 .600795 7.23 22 .473389 5.03 .626701 7.17 23 .455338 5.17 .601229 7.22 23 .473691 5.03 .627131 7.17 24 .455648 5.15 .601662 7.22 24 .473993 5.03 .627561 7.1 25 .455957 5.17 .602095 7.22 25 .474295 5.03 .627991 7.1 26 .456267 5.15 .602528 7.23 26 .474597 5.03 .628421 7.1 27 .456576 5.15 .602962 7.22 27 .474899 5.02 .628851 7.1 28 .456885 5.15 .603395 7.22 28 .475200 5.03 .629281 7.1 29 .457194 5.15 .603828 7.22 29 .475502 5.02 .629711 7.1 30 .457503 5.13 .604261 7.22 30 .475803 5.02 .630141 7.1 31 9.457811 5.15 9.604694 7.20 31 9.476104 5.02 9.630571 7.1 32 .458120 5.15 .605126 7.22 ! 32 .476405 5.02 .631001 7.1 33 .458429 5.13 .605559 7.22 33 .476706 5.02 .631430 7.1 34 .458737 5.13 .605992 7.22 34 .477007 5.02 .631860 ".1 35 .459045 5.13 .606425 7.20 35 .477308 5.00 .632290 7!l 36 .459353 5.13 .606857 7.22 36 .477608 5.02 .632719 7.1 37 .459661 5.13 .607290 7.20 37 .477909 5.00 .633149 ".1 38 .459969 5.13 .607722 7.22 38 .478209 5.00 .633578 .1 39 .460277 5.12 .608155 7.20 39 .478509 5.00 .634008 .1 40 .460584 5.13 .608587 7.22 40 .478809 5.00 .634437 .1 41 9.460892 5.12 9.609020 7.20 41 9.479109 5.00 9.634866 .1 42 .461199 5.12 .609452 7.20 42 .479409 5.00 .635296 .1 43 .461506 5.12 .609884 7.20 43 .479709 5.00 .6>725 .15 44 .461813 5.12 .610316 7.22 44 .480009 4.98 .636154 .15 45 .462120 5.12 .610749 7.20 45 .480308 5.00 .636583 .15 46 .462427 5.12 .611181 7.20 46 .480608 4.98 .637012 7.15 47 .462734 5.10 .611613 7.20 47 .480907 4.98 .637441 7.15 48 .463040 5.12 .612045 7.20 48 .481206 4.98 .637870 7.15 49 .463347 5.10 .612477 7.18 49 .481505 4.98 .638299 7.15 50 .463653 5.10 .612908 7.20 50 .481804 4.98 .638728 7.15 51 9.463959 5.10 9.61&340 7.20 51 9.482103 4.97 9.639157 7.15 52 .464265 5.10 .613772 7.20 52 .482401 4.98 .639586 7.15 53 .464571 5.10 .614204 7.18 53 .482700 4.97 .640015 7.13 54 .464877 5.10 .614635 7.20 54 .482998 4.97 .640443 7.15 55 .465183 5.08 .615067 7.20 55 .483296 4.98 .640872 7.15 56 .465488 5.10 .615499 7.18 56 .483595 4.97 .641301 7.13 57 .465794 5.08 .615930 7.20 57 .483893 4.97 .641729 7.15 58 .466099 5.08 .616362 7.18 58 .484191 4.95 .642158 7.13 59 .466404 5.08 .616793 7.18 59 .484488 4.97 .642586 7.15 60 9.466709 5.08 9.617224 7.20 60 9.484786 4.97 9.643015 7.13 [428] AND EXTERNAL SECANTS. 46 47 ' Vers. D. 1'. Ex. sec. D.I". / Vers. D.l". Ex. sec. D. 1'. 9.484786 4.97 9.643015 .13 9.502429 4.85 9.668646 7.10 1 .485084 4.95 .643443 .15 1 .502720 4.83 .669072 7.10 2 .485381 4.95 .643872 .13 2 .503010 4.83 .669498 7.10 3 .485678 4.97 .644300 .13 3 .503300 4.85 .669924 7.10 4 .485976 4.95 .644728 .13 4 .503591 4.83 .670350 7.10 5 .486273 4.95 .645156 .15 5 .503881 4.83 .670776 7.08 6 .486570 4.93 .645585 .13 6 .504171 4.82 .671201 7.10 .486866 4.95 .646013 .13 7 .504460 4.83 .671627 7.10 8 .4871(53 4.95 .646441 .13 8 .504750 4.83 .672053 7.10 9 .4ST4IK) 4.93 .646869 .13 9 .505040 4.82 .672479 7.08 10 .487756 4.95 .647297 .13 10 .505329 4.82 .672904 7.10 11 9.488053 4.93 9.647725 .13 11 9.505618 4.83 9.673330 7.10 12 .488349 4.93 .648153 .13 12 .505908 4.82 .673756 7.08 13 .488645 4.93 .648581 .13 13 .506197 4.82 .674181 7.10 14 .488941 4.93 .649009 .12 14 .506486 4.82 .674607 7.08 15 .489237 4.93 .649436 .13 15 .506775 4.80 .675032 7.10 16 .489533 4.92 .649864 .13 16 .507063 4.82 .675458 7.08 17 .489828 4.93 .650292 .13 17 .507352 4.80 .675883 7.10 18 .490124 4.92 .(550720 .12 18 .507640 4.82 .676309 7.08 19 .490419 4.92 .651147 .13 19 .507929 4.80 .6767:34 7.08 20 .490714 4.93 .651575 .12 20 .508217 4.80 .677159 7.08 21 9.491010 4.92 9.652002 .13 21 9.508505 4.80 9.677584 7.10 22 .491:505 4.92 .IJ52130 .12 82 .508793 4.80 .678010 7.08 23 .491600 4.90 .652857 .13 23 .509081 4.80 .678435 7.08 24 .491894 4.92 .653285 .12 24 .509369 4.80 .678860 7.08 25 .492189 4.92 .653712 .13 25 .509657 4.80 .679285 7.08 26 .492484 4.90 .654140 .12 26 .509945 4.78 .679710 7.10 27 .492778 4.90 .654567 .12 27 .510232 4.80 .680136 7.08 28 .493072 4.92 .654994 .12 28 .510520 4.78 .680561 7.08 29 .493367 4.90 .655421 .13 29 .510807 4.78 .680986 7.08 30 .493661 4.90 .655849 .12 30 .511094 4.78 .681411 7.08 31 9.493955 4.90 9.656276 .12 31 9.511381 4.78 9.681836 7.07 32 .494249 4.88 .656703 .12 32 .511668 4.78 .682260 7.08 33 .494542 4.90 .657130 .12 33 .511955 4.77 .682685 7.08 34 .494836 4.90 .657557 .12 34 .512241 4.78 .683110 7.08 35 .495130 4.88 .657984 .12 35 .512528 4.78 .683535 7.08 36 .495423 4.88 .658411 .12 36 .512815 4.77 .683960 7.08 37- .495716 4.88 .658838 .12 37 .513101 4.77 .684385 7.07 38 .496009 4.88 .659265 .10 38 .513387 4.77 .684809 7.08 39 .496302 4.88 .659691 .12 39 .513673 4.77 .685234 7.08 40 .496595 4.88 .660118 .12 40 .513959 4.77 .685659 7.07 41 9.496888 4.88 9.660545 .12 41 9.514245 4.77 9.686083 7.08 42 .497181 4.87 .660972 .10 42 .514531 4.77 .686508 7.08 43 .497473 4.88 .661398 .12 43 .514817 4.75 .686933 7.07 44 .497766 4.87 .661825 .12 44 .51 r il02 4 .77 .687357 7.08 45 .498058 4.87 .668852 .10 45 .515388 4,75 .687782 7.07 46 .498:350 4.88 .662678 .12 46 .515673 4.77 .688206 7.08 47 .498643 4.87 .663105 .10 47 .515959 4.75 .688631 7.07 48 .498935 4.85 .663531 .12 48 .516344 4.75 .689055 7.07 49 .499226 4.87 .663958 iii) 49 .516529 4.75 .689479 7.08 50 .499518 4.87 .664384 .10 50 .516814 4.73 .689904 7.07 51 9.499810 4.85 9.664810 7.12 51 9.517098 4.75 9.690328 7.07 52 .500101 4.87 .665237 7.10 52 .517383 4.75 .690752 7*! 08 53 .500393 4.85 .665663 7.10 68 .517668 4.73 .691177 7.07 54 .500684 .85 .666089 7.10 54 .517952 4.73 .691601 7.07 55 .500975 .85 .666515 7.12 55 .518236 4.75 .692025 7.07 56 .501266 .85 .666942 7.10 56 .518521 4.73 .692449 7.07 57 .501557 .85 .Gl.TSOH 7.10 57 .518805 4.73 .692873 7.08 58 .501848 .85 .667794 7.10 58 .519089 4.73 .693298 7.07 59 .502139 .83 .668220 7.10 59 .519373 4.73 .693722 7.07 60 9.502429 4.85 9.r>(i8(>4<: 7.10 60 9.519657 4.72 9.694146 7.07 [429] TABLE XXVI. LOGARITHMIC VERSED SINES 48 1 j 49 ' Vers. D.I". Ex. sec. D.I'. I , Vers. D. r. Ex. sec. D. 1'. 9.519657 4.72 9.694146 .07 1 9.536484 4.62 9.719541 7.05 1 .519940 4.73 .694570 .07 1 .536761 4.62 .719904 7.03 2 .520224 4 72 .694994 .07 2 .537038 4.62 .720386 7.05 3 .520507 4^73 .695418 .07 3 .537315 4.62 .720809 7. as 4 .520791 4.72 .695842 .07 4 .537592 4.62 .721231 7.03 5 .521074 4.72 .696266 .05 5 .537869 4.60 .721653 7.05 6 .521357 4.72 .696689 .07 6 .538145 4.62 .722076 7.03 7 .521640 4.72 .697113 .07 ( ij" .538422 4.60 .722498 7.05 8 .521923 4.72 .697537 .07 8 .538698 4.60 .722921 7.03 9 .522206 4.70 .697961 .07 9 .538974 4.62 .723343 7.03 10 .522488 4.72 .698:385 .07 10 .539251 4.60 .723765 7.05 11 9.522771 4.72 9.698809 .05 11 9.539527 4.60 9.724188 7.03 12 .523054 4.70 .699232 .07 12 .539803 4.60 .724610 7.03 13 .523336 4.70 .699656 .07 ! 13 .540079 4.58 .725032 7.03 14 .523618 4.70 .700080 .05 14 .540354 4.60 .725454 7.05 15 .523900 4.70 .700503 .07 15 .540630 4.60 .725877 7.03 16 .524182 4.70 .700927 .05 16 .540906 4.58 .726299 7.03 17 .524464 4.70 .701350 .07 17 .541181 4.58 .726721 7.03 18 .524746 4.70 .701774 ^.07 18 .541456 4.60 .727143 7.03 19 '.525028 4.68 .702198 .05 19 .541732 4.58 .727565 7.05 20 .525309 4.70 .702621 .07 80 .542007 4.58 .727988 7.03 21 9.525591 4.68 9.703045 .05 ! 21 9.542282 4.58 9.728410 7.03 22 .525872 4.68 .703468 .05 22 .542557 4.58 .728832 7.03 23 .526153 4.70 .703891 .07 23 .542832 4.57 .729254 7.03 24 .526435 4.68 .704315 .05 24 .543106 4.58 .729676 7.03 25 .526716 4.68 .704738 ".07 25 .543:381 4.57 .730098 7.03 26 .526997 4.67 .705162 ".05 26 .543655 4.58 .730520 7.03 27 .527277 4.68 .705585 .05 27 .543930 4.57 .730942 7.03 28 .527558 4.68 .706008 ".05 ! 28 .544204 4.57 .731364 7.03 29 .527839 4.67 .706431 .07 29 .544478 4.57 .731786 7.03 30 .528119 4.68 .706855 7.05 30 .544752 4.57 .732208 7.03 31 9.528400 4.67 9.707278 7.05 31 9.545026 4.57 9.732630 7.03 32 .528680 4.67 .707701 7.05 32 .545300 4.57 .733052 7.03 33 .528960 4.67 .708124 7.05 33 .545574 4.57 .733474 7.03 34 .529240 4.67 .708547 7.07 34 .545848 4.55 .733896 7.02 35 .529520 4.67 .708971 7.05 : 35 .546121 4.57 .734317 7.03 36 .529800 4.67 .709394 7.05 i 36 .546395 4.55 .734739 7.03 37 .530080 4.65 .709817 7.05 37 .546668 4.55 .735161 7.03 38 .530359 4.67 .710240 7.05 ! 38 .546941 4.55 .735583 7.03 39 .530639 4.65 .710663 7.05 39 .547214 4.55 .736005 7.03 40 .530918 4.67 .711086 7.05 40 .547487 4.55 .736427 7.02 41 9.531198 4.65 9.711509 7.05 41 9.547760 4.55 9.736848 7.03 42 .531477 4.65 .711932 7.05 42 .548033 4.55 .737270 7.03 43 .531756 4.65 .712355 7.05 43 .548306 4.55 .737692 7.03 44 .532035 4.65 .712778 7.03 44 .548579 4.53 .738114 7.02 45 .532314 4.63 .713200 7.05 45 .548851 4.55 .738535 7.03 46 .532592 4.65 .713623 7.05 ! 46 .549124 4.53 .738957 7.03 47 .532871 4.65 .714046 7.05 ! 47 .549396 4.53 .739379 7.02 48 .533150 4.63 .714469 7.05 48 .5-19668 4.53 .739800 7.03 49 .533428 4.63 .714892 7.05 49 .549940 4.53 .740222 7.03 50 .533706 4.65 .715315 7.03 50 .550212 4.53 .740644 7.02 51 9.533985 4.63 9.715737 7.05 51 9.550484 4.53 9.741065 7.03 52 .534263 4.63 .716160 7.05 52 .550756 4.53 .741487 '7.02 53 .534541 4.63 .716583 7.03 53 .551028 4.52 .741908 7.03 54 .534819 4.63 .717005 7.05 54 .551299 4.53 .742330 7.02 55 .535097 4.62 .717428 7.05 i 55 .551571 4.52 .742751 7.03 56 .535374 4.63 .717851 7.03 56 .551842 4.52 .743173 7.03 57 .535652 4.62 .718273 7.05 57 .552113 4.52 .743595 7.02 58 .535929 4.63 .718696 7.03 58 .552384 4.53 .744016 7.03 59 .536207 4.62 .719118 7.05 59 .552656 4.52 .744438 7.02 60 9.536484 4.62 9.719541 7.05 60 9.552927 4.50 9.744859 7.02 [430] -LOGARITHMIC VERSED SINES 50 51 Vers. D. 1". Ex. sec. D. 1" / Vers. D. r. Ex. sec. D. r. 9.552927 4.50 9.744859 7.02 9.568999 4.42 9.770127 7.02 1 .553197 4.52 .745280 7.03 1 .569264 4.40 .770548 7.02 2 .553468 4.52 .745702 7.02 2 .569528 4.42 .770969 7.00 3 .553739 4.50 .746123 7.03 3 .569793 4.40 .771389 7.02 4 .554009 4.52 .746545 7.02 4 .570057 4.42 .771810 7.02 5 .554280 4.50 .746966 7.03 5 .570322 4.40 .77-2231 7.02 6 .554550 4.50 .747388 7.02 6 .570586 4.40 .772652 7.G2 .554820 4.52 .747809 7.02 .570850 4.40 .773073 7.02 8 .555091 4.50 .748230 7.03 8 .571114 4.40 .773494 7.00 9 .555361 4.50 .748652 7.02 9 .571378 4.40 .773914 7.02 10 .555631 4.48 .749073 7.02 10 .571642 4.40 .774335 7.02 11 9.555900 4.50 9.749494 7.03 11 9.571906 4.40 9.774756 7.02 12 .556170 4.50 .749916 7.02 12 .572170 4.40 .775177 7.02 13 .556440 4.48 .750337 7.02 13 .572434 4.38 .775598 7.00 14 .556709 4.50 .750758 7.03 14 .572697 4.38 .776018 7.02 15 .556979 4.48 .751180 7.02 15 .572960 4.40 .776439 7.02 10 .557248 4.48 .751601 7.02 16 .573224 4.38 .776860 7.02 17 .557517 4.48 .752022 7.02 17 .57M87 4.38 .777281 7.02 18 .557786 4.48 .7'52443 7.03 18 .573750 4.38 .777702 7.00 19 .558055 4.48 .752865 7.02 19 .574013 4.38 .778122 7.02 20 .558324 4.48 .753286 7.02 20 .574276 4.38 .778543 7.02 21 9.558593 .48 9.753707 7.02 21 9.574539 4.38 9.778964 7.02 L> .558862 .48 .754128 7.02 22 .57'4802 4.37 .779385 7. CO 23 .559131 .47 .754549 7.03 23 .575064 4.38 .779805 7.02 24 .559399 .47 .754971 7.C2 24 .575327 4.37 .780226 7.02 23 .559667 .48 .755392 7.02 25 .575589 4.38 .780647 7.02 2(5 .559936 4.47 .755813 7.02 26 .575852 4.37 .781068 7.00 27 .560204 4.47 .756234 7.02 27 .576114 4.37 .781488 7.02 28 .560472 4.47 .756655 7.02 28 .576376 4.37 .781909 7.02 29 .5(50740 4.47 .757076 7.03 29 .576638 4.37 .782330 7.02 30 .561008 4.47 .757498 7.02 30 .576900 4.37 .782751 7.00 31 9.561276 4.47 9.757919 7.02 31 9.577162 4.37 9.783171 7.02 32 .561544 4.45 .758340 7.02 32 .577424 4.35 .783592 7.02 33 .561811 4.47 758761 7.02 33 .577685 4.37 .784013 7.00 34 .562079 4.45 759182 7.02 34 .577947 4.35 .784433 7.02 85 .562346 4.45 759603 7.02 35 .578208 4.37 .784854 7.02 36 .562613 4.47 7'60024 7.02 36 .578470 4.35 .785275 7.02 37 .562881 4.45 760445 7.02 37 .578731 4.35 .785696 7.00 38 .563148 4.45 760866 7.02 38 .578992 4.35 .786116 7.02 39 .563415 4.45 .761287 7.02 39 .579253 4.35 .786537 7.02 40 .563682 4.43 .761708 7.02 40 .579514 4.35 .786958 7.00 41 9.563948 4.45 9.762129 7.02 41 9.579775 4.35 9.787378 7.02 42 .564215 4.45 .762550 7.02 42 .580036 4.35 .787799 7.02 43 .564482 4.43 .762971 7.02 43 .580297 4.33 .788220 7.02 44 .564748 4.45 .763392 7.02 44 .580557 4.35 .788641 7. CO 45 .565015 4.43 .763813 7.02 45 .580818 4.38 .789061 7.02 46 .565281 4.43 .764234 7.02 46 .581078 4.35 .789482 7.02 47 .565547 4.43 .764655 7.02 47 .581339 4.33 .789903 7.00 48 .565813 4.43 .765076 7.02 48 .581599 4.33 .790323 7.C2 49 .56(5079 4.43 .765497 7.02 49 .581859 4.33 .7907'44 7.02 50 .566345 4.43 .765918 7.02 50 .582119 4.33 .791165 7.02 51 9.566611 4.43 9.766339 7.02 51 9.582379 4.33 9.791586 7.00 52 .566877 4.42 .766760 7.02 52 .582639 4.32 .792006 7.02 53 .567142 4.43 .767181 7.02 53 .582898 4.33 .792427 7.02 54 .567408 4.42 .767602 7.00 54 .583158 4.33 .792848 7.00 55 .567673 4.42 .768022 7.02 55 .583418 4.32 .793268 7.02 56 .567938 4.43 .768443 7.02 56 .583(577 4.32 .793689 7.02 57 .568204 4.42 .768864 7.02 57 .583936 4.33 .794110 7.02 58 .568469 4.42 .769285 7.02 58 .584196 4.32 .794531 7.00 59 .568734 4.42 .769706 7.02 59 .584 4.T) 4.32 .794951 7.02 GO 9.568999 4.42 9.770127 7.02 60 9.584714 4.32 9.795372 7.02 431] TABLE XXVL-LOGfARITmnC VERSED SINES 52 63 / Vers, D. r. Ex. sec. D.r. t Vers. D.r. Ex. sec. D.I". 9.584714 4.32 9.795372 7.02 9.600085 4.22 9.820622 7.02 1 .584973 4.32 .795793 7.00 1 .600338 4.22 .821043 7.02 2 .58>J232 4.32 .796213 7.02 2 .600591 4.23 .821464 7.02 3 .585491 4.30 .796634 7.02 3 .600845 4.22 .821885 7.02 4 .585749 4.32 .797055 7.02 4 .601098 4.22 .822306 7.02 5 .586008 4.30 .797476 7.00 ! 5 .601351 4.20 .822727 7.02 6 .586266 4.32 .797896 7.02 6 .601603 4.22 .823148 7.02 7 .586525 4.30 .798317 7.02 7 .601856 4.22 .823569 7.02 8 .586783 4.30 .798738 7.00 8 .602109 4.22 .823990 7.02 9 .587041 4.30 .799158 7.02 9 .602362 4.20 .824411 7.03 10 .587299 4.30 .799579 7.02 10 .602614 4.20 .824833 7.02 11 9.587557 4.30 9.800000 7.02 i Jl 9.602866 4.22 9.825254 7.02 12 .587815 4.30 .800421 7.00 12 .603119 4.20 .825675 7.02 13 .588073 4.30 .800841 7.02 13 .603371 4.20 .826096 7.02 14 .588331 4.38 .801262 7.02 14 .603623 4.20 .826517 7.02 15 .588588 4.28 .801683 7.02 15 .603875 4.20 7.03 16 .588846 4.28 .802104 7.00 16 .604127 4.20 ! 827360 7.02 17 .589103 4.30 .802524 7.02 17 .604379 4.20 .827781 7.02 18 .589361 4.28 .802945 7.02 18 .604631 4.20 .828202 7.02 19 .589618 4.28 .80:3366 7.02 19 .604883 4.18 .828623 7.02 20 .589875 4.28 .803787 7.00 20 .605134 4.20 .829044 7.03 21 9.590132 4.28 9.804207 7.02 21 9.605386 4.18 9.829466 7.02 22 .590389 4.28 .804628 7.02 22 .605637 4.18 .829887 7.02 23 .590646 4.28 .805049 7.02 23 .605888 4.20 .830308 7.02 24 .590903 4.28 .805470 7.02 24 .606140 4.18 .830729 7.03 25 .591160 4.27 .805891 7.00 25 .606391 4.18 .831151 7.02 26 .591416 4.28 .806311 7.02 26 .606642 4.18 .831572 7.02 27 .591673 4.27 .806732 7.02 27 .606893 4.18 .831993 7.03 28 .591929 4.27 .807153 7.02 28 .607144 4.17 .832415 7.02 29 .592185 4.28 .807574 7.02 29 .607394 4.18 .832836 7.02 30 .592442 4.27 .807995 7.00 30 .607645 4.18 .833257 7.03 31 9.592698 4.27 9.808415 7.02 31 9.607896 4.17 9.833679 7.02 32 .592954 4.27 .808836 7.02 32 .608146 4.18 .834100 7.03 33 .593210 4.27 .809257 7.02 33 .608397 4.17 .834522 7.02 34 .593466 4.25 .809678 7.02 34 .608647 4.17 .834943 7.02 35 .593721 4.27 .810099 7.02 35 .608897 4.17 .835364 7.03 36 .593977 4.27 .810520 7.00 I 36 .609147 4.17 .835786 7.02 37 .594233 4.25 .810940 7.02 37 .609397 4.17 .836207 7.03 38 .594488 4.25 .811361 7.02 38 .609647 4.17 .836629 7.02 39 .594743 4.27 .811782 7.02 39 .609897 4.17 .837050 7.03 40 .594999 4.25 .812203 7.02 40 .610147 4.17 .837472 7.02 41 9.595254 4.25 9.812624 7.02 41 9.610397 4.15 9.837893 7.03 42 .595509 4.25 .813045 7.02 42 .610646 4.17 .838315 7.02 43 .595764 4.25 .813466 7.00 43 .610896 4.15 .838736 7.03 44 .596019 4.25 .813886 7.02 44 .611145 4.15 .839158 7.02 45 .596274 4.23 .814307 7.02 45 .611394 4.17 .839579 7.03 46 .596528 4.25 .814728 7.02 46 .611644 4.15 .840001 7. as 47 .596783 4.25 .815149 7.02 47 .611893 4.15 .840423 7.02 48 .597038 4.23 .815570 7.02 48 .612142 4.15 .840844 7.03 49 .597292 4.23 .815991 7.02 49 .612391 4.15 .841266 7.02 50 .597546 4.25 .816412 7.02 50 .612640 4.13 .841687 7.03 51 9.597801 4.23 9.816833 7.02 51 9.612688 4.15 9.842109 "".OS 52 .598055 4.23 .817254 7.02 52 .613137 4.15 .842531 ?. 53 .598309 4.23 .817675 7.02 53 .613386 4.13 .842953 7.02 54 .598563 4.23 .818096 7.02 54 .613634 4.15 .843374 .03 55 .598817 4.23 .818517 7.02 55 .613883 4.13 .843796 .03 56 .599071 4.22 .818938 7.02 56 .614131 4.13 .844218 .02 57 .599324 4.23 .819359 7.02 57 .614379 4.13 .844639 .03 58 .599578 4.22 .819780 7.02 58 .614627 4.15 .845061 .03 59 .599831 4.23 .820201 7.02 59 .614876 4.13 .845483 .03 60 9.600085 4.22 9.820622 7.02 60 9.615124 4.12 9.845905 .03) .[432! AND EXTERNAL SECANTS. 54 55 ' Vers. D. 1". Ex. sec. D. 1". / Vers. D.I'. Ex. sec. D. 1". 9.615124 4.12 9.845905 7.03 9.629841 4.05 9.871250 7.05 i .615371 4.13 ! 846327 7.03 1 .630084 4.03 .871673 7.05 2 .615619 4.13 .8-40749 7.02 2 630326 4.05 .872096 7.05 3 .615867 4.13 .847170 7.03 3 .630569 4.03 .872519 7.05 4 .616115 4.12 .847592 7.03 4 .6130811 4.05 .872942 7.07 5 .616362 4.13 .848014 7.03 5 .631054 4.03 .873366 7.05 6 .616610 4.12 .8484% 7.03 6 .631296 4.03 .873789 7.05 fy .616857 4.12 .848858 7.03 7 .631538 4.03 .874212 7.07 8 .617104 4.12 .849280 7.03 8 .631780 4.03 .874636 7.05 9 .617351 4.13 .849702 7.03 9 .632022 4.03 .875059 7.05 10 .617599 4.10 .850124 7.03 10 .632264 4.02 .875482 7.07 11 9.617845 4.12 9.850546 7.03 11 9.632505 4.03 9.875906 7.05 12 .618092 4.12 .850968 7.03 12 .632747 4.03 .876329 7.05 13 .618339 4.12 .851390 7.03 13 .632989 4.02 .876752 7.07 14 .618586 4.12 .851812 7.03 14 .633530 4.03 .877176 7.05 15 .618833 4.10 .852234 7.03 15 .633472 4.02 .877599 7.07 16 .619079 4.12 .852656 7.03 16 .633713 4.02 .878023 7.05 IT .619326 4.10 .853078 7.03 17 .633954 4.03 .878446 7.07 18 .61&572 4.10 .853500 7.05 18 .634196 4.02 .878870 7.07 19 .619818 4.12 .853923 7.03 19 .634437 4.02 .879294 7.05 20 .620065 4.10 .854345 7.03 20 .634678 4.02 .879717 7.07 21 9.620311 4.10 9.a54767 7.03 21 9.634919 4.00 9.880141 7.07 22 .620.557 4.10 .855189 7.05 22 .635159 4.02 .880565 7.05 23 .620803 4.08 .855612 7.03 23 .635400 4.02 .880988 7.07 24 .(521048 4 10 .856034 7.03 24 .635641 4.00 .881412 7.07 25 .621294 4.10 .856456 7.03 25 .635881 4.02 .881836 7.07 26 .621540 4.10 .856878 7.05 26 .636122 4.00 .882260 7.05 27 .621786 4.08 .857301 7.03 27 .63(5362 4.02 .882683 7.07 28 .622031 4.08 .857723 7 03 28 .636603 4.00 .883107 7.07 29 .622276 4.10 .858145 7.05 29 .636843 4.00 .883531 7.07 30 .622522 4.08 .858568 7.03 30 .637083 4.00 .883955 7.07 31 9.622767 4.08 9.858990 7.05 31 9.637323 4.00 9.884379 7.07 32 .623012 4.08 .859413 7.03 32 .637563 4.00 .88-1803 7.07 33 .623257 4.08 .859835 7.05 33 .637803 4.00 .885227 7.07 34 .623502 4.08 .860258 7.03 34 .638043 4.00 .885651 7.07 35 .623747 4.08 .860680 7.05 35 .638283 3.98 .886075 7.07 36 .623992 4.08 .861103 7.03 36 .638522 4.00 .886499 7.07 37 .624237 4.07 .861525 7.05 37 .638762 3.98 .886923 7.07 38 .624481 4.08 .861948 7.03 38 .639001 4.00 .887347 7.08 39 .624726 4.07 .862370 7.05 39 .639241 3.98 .887772 7.07 40 .624970 4.08 .862793 7.03 40 .639480 3.98 .888196 7.07 41 9.625215 4 07 9.863215 7.05 41 9.639719 3.98 9.888620 7.07 42 .625459 4.07 .863638 7.05 42 .639958 3.98 .889044 7.08 43 .625703 4.07 .864061 7.03 43 .64019i 3.98 .889469 7.07 44 .625947 4.07 .864483 7.05 44 .640436 3.98 .889893 7.07 45 .626191 4.07 .864906 7.05 45 .640675 3.98 .890317 7.08 46 .626435 4.07 .865329 7.05 46 .640914 3.98 .890742 7.07 47 .620079 4.07 .865752 7.03 47 .641153 3.97 .891166 7.08 48 .62W23 4.05 .866174 7.05 48 .641391 3.98 .891591 7.07 4!) .627166 4.07 .866597 7.05 49 .641630 3.97 .892015 7.08 50 .627410 4.07 .867020 7.05 50 .641868 3.98 .892440 7.07 51 9.627054 4.05 9.867443 7.05 51 9.642107 3.97 9.892864 7.08 52 .627897 4.05 .867866 7.05 52 .642:345 3.97 .893289 7.08 53 .628140 4.07 .868289 7.05 53 .642583 3.98 .893714 7.07 54 688884 4.05 .868712 7.05 54 .642822 3.97 .894138 7.08 55 .628627 4.05 .869135 7.05 55 .643060 3.97 .894563 7.08 56 .628870 4.05 .869558 7.05 56 .643298 3.95 .894988 7.07 57 629113 4.05 .869981 7.05 57 .643535 3.97 .895412 7.08 58 .629356 4.03 .870404 7.05 58 .643773 3.97 .895837 7.08 59 .629598 4.05 .870827 T.05 59 .644011 3.97 .896262 7.08 60 9.629841 4.05 9.871250 7.05 60 9.644249 3.95 9.896687 7.08 TABLE XXVI LOGARITHMIC VERSED SINES 56 57 ' Vers. D. r. Ex. sec. D r. / Vers. D.I'. Ex. sec. D. 1'. 9.644249 3.95 9.896687 .08 9.658356 3.87 9.922247 7.12 1 .644486 3.97 .897112 .08 1 .658588 3.88 .922674 7.13 2 .644724 3.95 .897537 .08 2 .658821 3.87 .923102 7.12 3 .644961 3.95 .897962 .08 3 .659053 3.88 .923529 7.12 4 .645198 3.95 .898387 .08 4 .659286 3.87 .923956 7.13 5 .645435 3.97 .898812 7.08 5 .659518 3.87 .924384 7.12 6 .645673 3.95 .899237 7.08 6 .659750 3.88 .924811 7.13 7 .645910 3.95 .899662 7.08 7 .659983 3.87 .925239 7.12 8 .646147 3.95 .900087 7.08 8 .660215 3.87 .925666 7.13 9 .646384 3.93 .900512 7.10 9 .660447 3.87 .926094 7.12 19 .646620 3.95 .900938 7.08 10 .660679 3.85 .926521 7.13 11 9.646857 3.95 9.901363 7.08 11 9.660910 3.87 9.926949 7.13 12 .647094 3.93 .901788 7.08 12 .661142 3.87 .927377 7 12 13 .647330 3.95 .902213 7.10 13 .661374 3.85 .927804 7J3 14 .647567 3.93 .902639 7.08 14 .661605 3.87 .928232 7.13 15 .647803 3.93 .903064 7.10 15 .661837 3.85 .928660 7.13 16 .648039 3.95 .903490 7.08 16 .662068 3.87 '.929088 7.13 17 .648276 3.93 .903915 7.10 17 .662300 3.85 .929516 7.13 18 .648512 3.93 .904341 7.08 18 .662531 3.85 .929944 7.13 19 .648748 3.93 .904766 7.10 19 .662762 3.85 .930372 7.13 20 .648984 3.93 .905192 7.08 20 .662993 3.85 .930800 7.13 21 9.649220 3.93 9.905617 7.10 21 9.663224 3.85 9.931228 7.13 22 .649456 3.92 .906043 7.10 22 .663455 3.85 .931656 7.15 23 .649691 3.93 .906469 7.08 23 .663686 3.85 .932085 7.13 24 .649927 3.93 .906894 7.10 24 .663917 3.85 .932513 7.13 25 .650163 3.92 .907320 7.10 25 .664148 3.83 .932941 7.13 26 .650398 3.92 .907746 7.10 26 .664378 3.85 .933369 7.15 27 .650633 3.93 .908172 7.10 27 .664609 3.83 .933798 7.13 28 .650869 3.92 .908598 7.10 28 .664839 3.85 .934226 7.15 29 .651104 3.92 .909024 7.10 29 .665070 3.83 .934655 7.13 30 .651339 3.92 .909450 7.10 30 .665:300 3.83 ,935083 7.15 31 9.651574 3.92 9.909876 7.10 81 9.665530 3.83 9.935512 7.15 32 .651809 3.92 .910302 7.10 32 .6657'60 3.83 .935941 7.13 33 .652044 3.92 .910728 7.10 33 .665990 3.83 .936369 7.15 34 .652279 3.92 .911154 7.10 34 .666220 3.83 .936798 7.15 35 .652514 3.90 .911580 7.10 35 .666450 3. as .937227 7.15 36 .652748 3.92 .912006 7.10 36 .666680 3.83 .937656 7.15 37 .652983 3.90 .912432 7.12 37 .666910 3.82 .938085 7.13 38 .653217 3.92 .912859 7.10 38 .667139 3.83 .938513 7.15 39 .653452 3.90 .913285 7.10 39 .667369 3. as .938942 7.15 40 .653686 3.90 .913711 7.12 40 .667599 3.82 .939371 7.17 41 9.653920 3.92 9.914138 7.10 41 9.667828 3.82 9.939801 7.15 42 .654155 3.90 .914504 7.12 42 .668057 3.83 .940230 7.15 43 .654389 3.90 .914991 7.10 43 .668287 3.82 .940659 7.15 44 .654623 3.90 .915417 7.12 44 .668516 3.82 .941088 7.15 45 .654857 3.88 .915844 7.10 45 .668745 3.82 .941517 7.17 46 .655090 3.90 .916270 7.12 46 .668974 3.82 .941947 7.15 47 .655324 3.90 .916697 7.12 47 .669203 3.82 .942376 7.15 48 .655558 3.90 .917124 7.10 48 .669432 3.82 .942806. 7.15 49 .655792 3.88 .917550 7.12 49 .669661 3.80 .943235 7.17 50 .656025 3.88 .917977 7.12 50 .669889 3.82 .943665 7.15 51 9.656258 3.90 9.918404 7.12 51 9.670118 3.82 9.944094 7.17 52 .656492 3.88 ,918831 7.12 52 .670347 3.80 .944524 7.15 53 .656725 3.88 .919258 7.12 53 .670575 3.82 .944953 7.17 54 .656958 3.88 .919685 7.12 54 .670804 3.80 .945383 7.17 55 .657191 3.88 .920112 7.12 55 .671032 3.80 .945813 7.17 56 .657424 3.88 .920539 7.12 56 .671260 3.80 .946243 7.17 57 .657657 3.88 .920966 7.12 57 .671488 3.80 .946673 7.17 58 .657890 3.88 .921393 7.12 58 .671716 3.82 .947103 7.17 59 .658123 3.88 .921820 7.12 59 .671945 3.78 .947533 7.17 60 9.658356 3.87 9.922247 7.12 60 9.672172 3.80 9.947963 7.17 AND EXTERNAL SECANTS. 58 59 ' Vers. D. 1*. Ex. sec. D.I". / Vers. D.I". Ex. sec. D. 1". 9.672172 3.80 9.947963 7.17 9.685708 3.72 9.973868 7.23 1 .672400 3.80 .948:393 7.17 1 .685931 3.72 .974302 7.23 2 .672628 3.80 .948823 7.17 2 .686154 3.72 .974736 7.22 3 .672856 3.78 .949253 7.17 3 .686377 3.72 .975169 7.23 4 .673083 3.80 .949683 7.18 4 .686600 3.72 .975603 7.23 5 .673311 3.78 .950114 7.17 5 .686823 3.72 .976037 7.23 6 .673538 3.80 .950544 7.18 6 .687046 3.72 .976471 7.23 7 .673766 3.78 .950975 7.17 7 .687269 3.72 .976905 7.23 8 .673993 3.78 .951405 7.18 8 .687492 3.70 .977339 7.23 9 .674220 3.80 .951836 7.17 9 .687714 3.72 .977773 7.23 10 .674448 3.78 .9522613 7.18 i 10 .687937 3.70 .978207 7.23 11 9.674675 3.78 9.952697 7.18 ' 11 9.688159 3.72 9.978641 7.23 12 .674902 3.78 .953128 7.17 12 .088383 3.70 .979075 7.25 13 .675129 3.78 .953558 7.18 13 .688604 3.70 .979510 7.23 14 .675356 3.77 .953989 7.18 14 .683826 3.70 .979944 7.25 15 .675582 3.78 .954420 7.18 15 .689048 3.72 .980379 7.23 16 .675809 3.78 .954851 7.18 16 .689271 3.70 .980813 7.25 17 .676036 3.77 .955282 7.18 17 .689493 3.70 .981248 7.23 18 .676262 3.78 .955713 7.18 18 .689715 3.70 .981682 7.25 19 .676489 3.77 .956144 7.18 19 .689937 3.68 .982117 7.25 20 .676715 3.77 .956575 7.18 20 .690158 3.70 .982552 725 21 9.676941 3.78 9.957006 7.20 21 9.690380 3.70 9.982987 7.25 22 .677168 3.77 .957438 7.18 83 .690602 3.68 .983422 7.25 23 .677394 3.77 .957869 7.18 23 .690823 3.70 .983857 7.25 24 .677620 3.77 .958300 7.20 24 .691045 3.68 .984292 7.25 25 .677846 3.77 .958732 7.18 25 .691266 3.70 .984727 7.25 26 .678072 3.77 .959163 7.20 26 .691488 3.68 .985162 7.25 27 .678298 3.75 .959595 7.18 27 .691709 3.68 .985597 7.27 28 .678523 3.77 .960026 7.20 28 .691930 3.68 .986033 .25 29 .678749 3.77 .960458 .20 29 .692151 3.68 .986468 .27 30 .678975 3.75 .960890 .18 30 .692372 3.68 .986904 .25 31 9.679200 3.77 9.961321 .20 31 9.692593 3.68 9.987339 .27 32 .679426 3.75 .961753 .20 32 .692814 3.68 .987775 .25 33 .679651 3.75 .962185 .20 33 .693035 8.68 .988210 .27 34 .679876 3.77 .962617 .20 34 .693256 3. US .988646 .27 35 .680102 3.75 .963049 .20 35 .693177 3.67 .989082 .27 36 .680327 8.75 .96'USl .20 36 .693697 3.68 .989518 .27 37 .680552 3.75 .963',) 13 .20 37 .693918 3.67 .989954 .27 38 .680777 3.75 .96*345 .22 38 .694138 3.68 .990390 .27 39 .681002 3.75 .964778 .20 39 .691859 3.67 .990826 .27 40 .681227 3.73 .965310 7.20 40 .694579 3.67 .'(912U2 .27 41 9.681451 3.75 9. 965642 7.22 41 9.694799 3.67 9.991698 .27 42 .681676 3.75 .966075 7.20 42 .695019 3.68 .992134 .28 43 .681901 3.73 .966507 7.22 43 .695240 3.67 .992571 .27 44 .682125 3.75 .966940 7.20 41 .695460 3.67 .993007 .28 45 .682350 3.73 .967372 7.22 45 .695680 3.65 .993444 .27 46 .682574 3.73 .967805 7.22 46 .695899 3.67 .993880 .28 17 .682798 3.75 .968238 7.20 47 .696119 3.67 .994317 .28 48 .683023 3.73 .968670 7.22 48 .696339 3.67 .994754 .28 49 .1583247 3.73 .969103 7.22 49 .696559 3.65 . 1)95191 27 50 .083471 3.73 .969536 7.22 50 '696778 3.67 .995627 !28 51 9.683695 3.73 9.969969 7.22 51 9.696998 3.65 9.996064 7.28 52 .683919 3.73 .970402 7.22 52 .697217 3.67 .996501 7.28 53 .'SS4143 3.73 .970835 7.22 53 .697437 3.65 .996938 7.30 54 .684367 3.72 .971268 7.22 54 .697656 3.65 .997376 7.88 55 .684590 3.73 .971701 7.23 55 .69787'5 3.65 .997813 7.28 56 . 684814 3.72 .972135 7.22 56 .698094 3.65 .998250 7.28 07 .685037 3.73 .972568 7.22 57 .698313 3.65 .998687 7.30 58 .685261 3.?2 .973001 7.23 58 .698532 3.65 .999125 7.28 59 .685484 3.73 .973435 7.22 59 .698751 3.65 9.999562 7.30 60 9. 685708 3.72 9.973868 7.23 60 9.698970 3.63 10.000000 7.30 [43. "51 TABLE XXVI. -LOGARITHMIC VERSED SINES 60 fc 61 ' Vers. D. 1". Ex. sec. D. 1", ' Vers. D. 1". Ex. sec. D.I'. 9.698970 3.65 10.000000 7.30 9.711968 3.57 10.026397 7.37 1 .699189 3.63 '.000438 7.28 1 .712182 3.58 .026839 7.37 2 .699407 3.65 .000875 7.30 2 .712397 3.57 .027281 7.38 3 .699626 3.65 .001313 7.30 3 .712611 3.57 .027724 7.38 4 .699845 3.63 .001751 7.30 4 .712825 3.57 .028167 7.37 5 .700063 3.65 .002189 7.30 5 .71:3039 3.57 .028609 V.38 6 .700282 3.63 .002627 7.30 6 .713253 3.57 .029052 7.38 7 .700500 3.63 .003065 7.30 7 .713467 3.57 .029495 7.38 ? .700718 3.63 .003503 7.32 8 .713681 3.57 .029938 7.38 9 .700936 3.63- .003942 7.30 9 .713895 3.57 .030381 7.40 10 .701154 3.63 .004380 7.30 10 .714109 3.57 .030825 7.38 11 9.701372 3.63 10.004818 7.32 11 9.714323 3.55 10.031268 7 38 U .701590 3.63 .005257 7.30 J2 .714536 3.57 .031711 7^40 13 .701808 3.03 .005695 7.32 13 .714750 3.55 .032155 7.38 14 .702026 3.63 .006134 7.32 14 .714963 3.57 .032598 7.40 15 .702244 3.63 .006573 7.32 15 .715177 3.55 .033042 7.40 16 .702462 3.G2 .007012 7.30 16 .715390 3.55 .033486 7.38 17 .702679 3.03 .007450 7.32 17 .715603 3.57 .033929 7.40 18 .702897 3.62 .007889 7.32 18 .715817 3.55 .034373 7.40 19 .703114 3.63 .008328 7.32 19 .716030 3.55 .034817 7.40 20 .703332 3.62 .008767 7.33 20 .716243 3.55 .035261 7.40 21 9.703549 3.62 10.009207 7.32 21 9.716456 3.55 10.035705 7.42 22 .703766 3.62 .009646 7.32 22 .716669 3.55 .036150 7.40 23 .703983 3.62 .010085 7.33 23 .716882 3.55 .036594 7.40 24 .704200 3.62 .010525 7.32 24 .717095 3.53 .037038 7.42 25 .704417 3.62 .010904 7.33 25 .717307 3.55 .037483 7.42 26 .704634 3.62 .011404 7.32 26 .717520 3.53 .037928 7.40 27 .704851 3.62 .011843 7.33 27 .717732 3.55 .038372 7.42 28 .705068 3.62 .012283 7.33 28 .717945 3.53 .038817 7.42 29 .705285 3.60 .012723 7.33 29 .718157 3.55 .039262 7.42 30 .705501 3.62 .013163 7.33 30 .718370 3.53 .039707 7.42 31 9.705718 3.62 10.013603 7.33 31 9.718582 3.53 10.040152 7.42 32 .705935 3.60 .014043 7.33 32 .718794 3.55 .040597 7.42 33 .706151 3.60 .014483 7.33 33 .719007 8.53 .041042 7.43 34 .7063G7 3.62 .014923 7.33 34 .719219 3.53 .041488 7.42 35 .706584 3.60 .015363 7.35 35 .719431 3.53 .041933 ^.43 36 .706800 3.60 .015804 7.33 36 .719643 3.53 .042379 7.42 37 .707016 3.60 .016244 7.33 37 .719855 3.52 .042824 7.43 38 .707232 3.60 .016684 7.33 38 .720066 3.53 .043270 7.43 39 .707448 3.60 .017125 7.35 39 .720278 3.53 .043716 7.43 40 .707664 3.60 .017566 7.35 40 .720490 3.52 .044162 7.43 41 9.707880 3.60 10.018007 7.23 41 9.720701 3.53 10.044608 7.43 44 .708096 3.58 .018447 7.35 42 .720913 3.52 .045054 7.43 43 .708311 3.60 .018888 7.35 43 .721124 8.53 .045500 7.43 44 .708527 3.60 .019329 7.35 44 .721336 3.52 .045946 7.45 43 .708743 3.58 .019770 7.37 45 .721547 ! 3.52 .046393 7.43 46 .708958 3.60 .020212 7.35 46 .721758 3.53 046839 7.45 47 .709174 3.58 .020653 7.35 47 .721970 3.52 .047286 7.43 48 .709389 3.58 .021094 7.a5 48 .722181 3.52 .047732 7.45 49 .709604 3.58 .021535 7.37 49 .722392 3.52 .048179 7.45 50 .709819 3.58 .021977 7.37 50 .722603 3.52 .048626 7.45 51 9.710035 3.58 10.022419 7.35 51 9.722814 3.50 1 0.049073 7.45 52 .710250 3.58 .022860 7.37 52 .723024 3.52 .049520 7.45 53 .710465 3.58 .023302 7.37 53 .723235 3.52 .049967 7.45 54 .710680 3.58 .023744 7.37 54 .723446 3.52 .050414 7.45 55 .710895 3.57 .024186 7.37 55 .723657 3.50 .050861 7.47 56 .711109 3.58 .024628 7.37 56 .723867 3.52 .051309 7.45 57 .711324 3.58 .025070 7.37 57 .724078 3.50 .051756 7.47 58 .711539 3.57 .025512 7.37 58 .724288 3.50 .052204 7.47 59 .711753 3.58 .025954 7.38 59 .724498 3.52 .052652 7.45 60 I 9.711968 3.57 10.026397 7.37 60 9.724709 3.50 10.053099 7.47 [436] AND EXTERNAL SECANTS. 62 63 / Vers. D. r, Ex. sec. D.I". / Vers. D.r. Ex. sec. D.r. 9.724709 3.50 10.053099 7.47 9.737200 3.43 10.060153 7.58 1 .724919 3.50 .053547 7.47 1 .737406 3.43 .080608 .57 2 .725129 3.50 .053995 7.47 2 .737612 3.43 .081062 .57 3 .725339 3.50 .054443 7.48 3 .737818 3.43 .081516 .58 4 .725549 3.50 .054892 7.47 4 .7:38024 3.43 .081971 .57 5 .725759 3.50 .055340 7.47 5 .738230 3.43 .082425 .58 6 .725969 3.50 .055788 7.48 6 .7:38436 3.43 .082880 .58 if .726179 3.48 .056237 7.47 7 . 7:38642 3.42 .083335 .58 8 .726388 3.50 .056685 7.48 ! 8 .738847 3.43 .083790 .58 9 .726598 3.50 .0571:34 7.48 9 .739053 3.42 .084245 .58 10 .726808 3.48 .057583 7.48 10 .739258 3.48 .084700 .58 11 9.727017 3.50 10.058032 .48 11 9.739464 3.42 10.085155 .60 13 .727227 3.48 .058481 .48 12 .739669 3.43 .085611 .58 13 .727436 3.48 .0589:30 .48 13 .739875 3.42 .086C66 .60 14 .727645 3.50 .059379 .48 14 .740080 3.42 .086522 .58 15 .727855 3.48 .059828 .50 15 .740285 3.42 .086977 .60 16 .728064 3.48 .060278 .48 i 16 .740490 3.42 .0874*3 .60 ir .728273 3.48 .060727 .50 17 .740695 3.42 .087889 .60 18 .728482 3.48 .061177 .48 18 .740900 3.42 .088345 .60 19 .728691 3.48 .061626 .50 19 .741105 3.42 .088801 .62 20 .728900 3.48 .062076 .50 20 .741310 3.42 .089258 .60 21 9.729109 3.47 10.062526 .50 21 9.741515 3.40 10.089714 .62 22 .729317 3.48 .062976 .50 22 .741719 3.42 .090171 .60 23 .729526 3.48 .063426 .50 j 23 .741924 3.42 .090627 .62 24 .729735 3.47 .063876 .52 24 .742129 3.40 .091084 .62 25 .729943 3.48 .064327 .50 j 25 .742333 3.42 .091541 .62 26 .730152 8...':7 .064777 .50 26 .742538 3.40 .091998 .62 27 .730360 3.48 .065227 .52 27 .742742 3.40 .092455 .62 28 .730569 3.47 .065678 .52 28 .742946 3.40 .092912 .63 29 .730777 3.47 .066129 .52 29 .743150 3.42 .093370 .62 30 .730985 3.47 .066580 .50 j 30 .743355 3.40 .093827 .63 31 9.731193 3.47 10.067030 .53 31 9.74,35:>9 3.40 10.094285 .63 32 .731401 3.47 .067482 .52 32 .743763 3.40 .094743 .62 33 .731609 3.47 .067933 >>2 33 .743967 3.40 .095200 .63 34 .731817 3.47 .068384 .52 31 .744171 3.40 .095658 .63 35 .732025 3.47 .068835 .53 35 .744375 3.38 .096116 .65 36 .732233 3.47 .069287 .52 3C .744578 3.40 .096575 .63 37 .732441 3.45 .069738 .53 37 .744782 3.40 .097038 .63 38 .732648 3.47 .07'0190 .53 38 .744986 3.38 .097491 .65 39 .732856 3.47 .070642 .52 39 .745189 3.40 .097950 .63 40 .7*3064 3.45 .071093 .53 40 .745393 3.38 .098408 .65 41 9.7*3271 3.45 10.071545 .55 41 9.745596 3.40 10.098867 .65 42 .7*3478 8.47 .071998 .68 42 .745800 3.38 .099326 .65 43 .733686 8.45 .072450 .53 43 .74G003 3.38 .099785 .65 44 .733893 3.45 .072902 .53 44 .74(>20tJ 3.38 .100244 .67 45 .7'34100 3.45 .07-3354 . 55 45 .746409 3.40 .100704 .65 46 .734307 3.47 .073807 .55 46 .740613 3.38 .101163 .67 47 .734515 3.43 .074260 ^53 47 .746816 3.38 .101623 .65 48 .734721 3.45 .074712 .55 48 .747019 3.38 .102082 .67 49 .734928 3.45 .075165 .55 49 ! 747222 3.37 .102542 .67 50 .735135 3.45 .075618 .55 50 .747424 3.38 . 103002 .67 51 9.735342 3.45 10.076071 .55 51 9.747687 3.38 10.103462 .67 52 .735545) 3.43 .076524 .55 52 .747830 3.38 .103922 .67 53 .735755 3.45 .076977 .57 53 .748033 3.37 .104382 .68 54 .735962 3.45 .077431 7.55 54 .748235 3.38 .104843 .67 55 .736169 3.43 .077884 7.57 55 .748438 3.37 .105303 .68 56 .736375 3.43 .078338 7.57 56 .748640 3.38 .105764 .67 57 .736581 3.45 .078792 7.55 57 .748843 3.37 .106224 .68 58 .736788 3.43 .079245 7.57 58 .749045 3.37 .106685 .68 59 .736994 3.43 .079699 7.57 59 .749247 3.37 .107146 .68 60 9.737200 3.43 10.080153 7.58 60 9.749419 3.38 10.107607 7.70 TABLE XXVI.-LOGARITIIMIC VERSED SINES 64 65 ' Vers. D. r. Ex. sec. D 1". ' Vers. D. 1". Ex. sec. D 9.749449 3.38 10.107607 .70 9.761463 3.30 10.135515 .82 1 .749652 3.37 .108069 .68 1 .761661 3.32 .135984 .83 2 .749854 3.37 .108530 .70 2 .761860 3.30 .136454 .82 3 .750056 3.37 .108992 .68 3 .762058 3.30 .136923 .83 4 .750258 3.35 . 109453 .70 4 .762256 3.30 .137393 .83 5 .750459 3.37 .109915 .70 5 .762454 8.80 .137863 .83 .750661 3.37 .110377 .70 6 .762652 3.30 .188333 .83 .750863 3.37 .110839 .70 7 .762850 3.28 .1:38803 .83 8 .751065 3.35 .111301 .70 8 .763047 3.30 .139273 .85 9 .751266 3.37 .111763 .72 9 .763245 3,'iO .139744 .83 10 .751468 3.35 .112226 .70 10 .763443 3.30 .140214 .85 11 9.751669 3.37 10.112688 .72 11 9.763641 3.28 10.140685 .85 12 .751871 3.35 .113151 .72 12 .763838 3.30 .141156 .85 13 .752072 3.35 .113614 .72 13 .764036 3.28 .141627 .85 14 .752273 3.37 .114077 .72 14 .764233 3.28 .142098 .85' 15 .752475 3.35 .114540 72 15 .764430 3.30 .142569 .87 16 .752676 3.35 .115003 '.72 16 .764628 3.28 .143041 .85 17 .752877 3.35 .115166 .72 17 .764825 3.28 .143512 .87 18 .753078 3.35 .115929 .73 18 .765022 3.28 .143984 .87 19 .753279 3.35 .116393 .73 19 .765219 3.28 .144456 .87 90 .753480 3.35 .116857 .73 20 .765416 3.28 .144928 .87 21 9.753681 3.33 10.117321 .73 21 9.765613 3.28 10.145400 .87 22 .753881 3.35 .117785 .73 22 .765810 3.28 .145872 .88 23 .754082 3.35 .118249 .73 23 .766007 3.28 . 146345 .88 24 .754283 3.33 .118713 .73 24 .766204 3.28 .146818 .87 25 .754483 3.35 .119177 .75 25 .766401 3.27 .147290 .88 26 .754684 3.33 .119642 .73 26 .766597 3.28 .147763 .88 27 .754884 3.35 .120106 .75 27 .766794 3.28 .148236 .90 28 .755085 3.33 .120571 .75 28 .766991 3.27 .148710 .88 29 .755285 3.33 .121036 .75 29 .767187 3.28 .149183 .90 30 .755485 3.33 .121501 ".75 30 .767384 3.27 .149657 .88 31 9.755685 3.35 10.121966 .75 31 9.767580 3.27 10.150130 .90 32 .755886 3 33 .1^2431 1 t r>i" 32 .767776 3.27 .150604 .90 33 .756086 3.33 .122897 "'.75 33 .767972 3.28 .151078 .90 34 .756286 3.33 .123362 34 .768169 3.27 .151552 .92 35 .756486 3.32 .123828 1 ' .77 3T, .768365 3.27 .152027 .90 36 .756685 3.33 .124294 ' '.77 36 .768561 3.27 .152501 .92 37 .756885 3.33 .124760 .77 37 .768757 3.27 .152976 .90 38 .757085 3.33 .125226 38 .768953 3.27 .153450 .92 39 .757285 3.32 1^5692 !77 39 .769149 3.25 .153923 .92 40 .757484 3.33 ! 126158 .78 40 .769344 3.27 .154400 .93 41 9.757684 3.32 10.126625 .78 41 9.769540 3.27 10.154876 .92 42 .757883 3.33 .127092 .77 42 .769736 3.25 .155351 .92 43 .758083 3.32 .127558 .78 43 .769931 3.27 .155826 .93 44 .758282 3.32 .128025 .78 44 .770127 3.27 .156302 .93 45 .758481 3.33 .128492 .80 45 .770323 3.25 .156778 .93 46 .758681 3.32 .128960 .78 46 .770518 3., '25 .157254 .93 47 .758880 3.32 .129427 .78 47 .770713 3.27 .157730 .93 48 .759079 3.32 .129894 .80 48 .770909 3.25 .158206 .95 49 .759278 3.32 .130362 .80 49 .771104 3.25 . 158683 .93 50 . 759477 3.32 .130830 .80 50 .771299 3.25 .159159 .95 51 9.759676 3.32 10.131298 .80 51 9.771494 3.25 10.159636 .95 52 .759875 3.30 .131766 .80 58 .771689 3.25 .160113 .95 53 .760073 3.32 132234 .80 53 .771884 3.25 .160590 .95 54 .760272 3.32 .132702 .80 54 .772079 3.25 .161067 .97 55 .760471 3.30 .133170 .82 55 .772274 3.25 .161545 .95 56 .760669 3.32 .133639 .82 56 .772469 3.25 .162022 .97 57 .760868 3.30 .134108 .82 57- .772664 3.23 .162500 .97. 58 .761066 3.32 .134577 .82 58 .772858 3.25 .162978 7.97 59 .761265 3.30 . 135046 .82 59 . 773053 3.25 .163456 7.97 60 9.761463 3.30 10.135515 7.82 1 1 60 9. 773248 3.23 10.163934 7.98 AND EXTERNAL SECANTS. 66 67 / Vers. D. r. Ex. sec. D. 1". ' Vers. D. r. Ex. sec. D.r. 9.773248 3.23 10.163934 ".98 9.784809 3.18 10.192931 8.15 1 .773442 3.23 .164413 7.97 1 .785000 3.18 .193420 8.13 2 .773636 3.25 .164891 ".98 2 .785191 3.17 .193908 8.15 3 .773831 3.23 .165370 ".98 3 .785381 3.18 .194397 8.15 4 .'"74025 3.23 .165849 7.98 4 .785572 3.18 .194886 8.17 5 ."74219 3.25 .166328 7.98 5 .785763 3.17 .195376 8.15 6 .774414 3.23 .166807 7.98 6 .785953 3.18 .195865 8.17 7 ."74608 3.23 .167286 8.00 .786144 3.17 .196355 8.17 8 ."74802 3.23 .167766 7.98 8 .786334 3.17 .196845 8.17 9 ."74996 3.23 .168245 8.00 9 .786524 3.18 .197335 8.17 10 ."75190 3.23 .168725 8.00 10 .786715 3.17 .197825 8.17 11 9.775384 3.22 10.169205 8.00 11 9.786905 S.17 10.198315 8.18 12 . 775577 3.23 .169685 8.00 12 .787095 3.17 .198806 8.18 13 .775771 3.23 .170165 8.02 13 .787285 3.17 .199297 8.18 14 .775965 3.23 ! 170646 8.02 14 .787475 3.17 .199788 8,18 15 .776159 3.22 .171127 8.00 15 .787665 3.17 .200279 8.18 16 .776352 3.23 .171607 8.02 | 16 .787855 3.17 .200770 8.20 17 .776546 3.22 .172088 8.02 17 .788045 3.17 .201262 8.18 18 .776739 3.23 .172569 8.03 ! 18 .788235 3.17 .201753 8.20 19 .776933 3.22 .173051 8.02 19 .788425 3.15 .202245 8.20 20 .777126 3.22 .173532 8.03 20 .788614 3.17 .202737 8.20 21 9.777319 3.22 10.174014 8.03 21 9.788804 3.15 10.203229 8.22 22 .777512 3.22 .174496 8.03 22 .788993 3.17 .203722 8.22 23 .777705 3.23 .174978 8.03 23 .789183 3.15 .204215 8.20 24 .777899 3.22 .175460 8.03 24 .789372 3.17 .204707 8.22 25 .778092 3.22 .175942 8.05 1 25 .789562 3.15 .205200 8.23 26 .778285 3.20 .176425 8.03 26 .789751 3.15 .205694 8.22 27 .778477 3.22 .176907 8.05 27 .789940 3.17 .20(5187 8.23 28 .778(570 3.22 .177390 8.05 28 .790130 3.15 .206681 8.22 29 .778863 3.22 .177873 8.05 29 .790319 3.15 .207174 8.23 30 .779056 3.20 .178356 8.05 30 .790508 3.15 .207668 8.23 31 9.779248 3.22. 10.178839 8.07 31 9.790697 3.15 10.208162 8.25 32 .779441 3.22 .179323 8.07 32 .790886 3.15 .208657 8.23 33 .779634 3.20 .179807 8.05 33 .791075 -3.15 .209151 8.25 34 .779826 3.20 .180290 8.07 34 .791264 3.15 .209646 8.25 35 .780018 3.22 .180774 8.08 35 .7914.% 3.13 .210141 8.25 36 .780211 3.20 .181259 8.07 36 .791641 3.15 .210636 8.26 37 .780403 3.20 .181743 8.07 37 .791830 3.15 .211131 8.27 38 .780595 3.20 .182227 8.08 38 .798019 3.13 .211627 8.27 39 .780787 3.22 .182712 8.08 39 .792207 3.15 .212123 8.25 40 .780980 3.20 .183197 8.08 40 .792396 3.13 .212618 8.28 41 9.781172 3.20 10.183682 8 08 41 9.792584 3.13 10.213115 8.27 42 .781364 3.20 .184167 8.10 I-J .792772 3.15 .213611 8.27 43 .781556 3.18 .184653 8.08 43 .792961 8.18 .214107 8.28 44 .781747 3.20 .185138 8.10 n .79314: 3.13 .214604 8.28 45 .781939 3.20 .185624 8.10 45 .793337 3.13 .215101 8.28 46 .782131 3.20 .186110 8.10 46 .793525 3.15 .215598 8-28 47 .782323 3.18 .186596 8.10 47 .793714 3.13 .216095 8.30 48 .782514 3.20 .187082 8.10 48 .793902 3.13 .216593 8.28 49 .782706 3.18 .187568 8.12 49 .794090 3.12 .217090 8.30 50 .782897 3.20 .188055 8.12 50 .794277 3.13 .217588 8.30 51 9.7&3089 3.18 10.188542 8.12 51 9.794465 3.13 10.218086 8.32 52 .783280 3.18 .189029 8.12 52 .794653 3.13 .218585 8.30 53 .783471 3.20 .189516 8.12 58 .794841 3.12 .219083 8.32 54 .783663 3.18 .190003 8.13 54 .795028 3.13 .219582 8.32 55 .783854 3.18 .190491 8.12 .-,:, .795216 3.13 220081 8.32 56 .784045 3.18 .190978 8.13 66 .795404 3.12 .220580 8.32 57 .784236 3.18 .191466 8.13 57 .796591 3.13 .221079 8.32 58 .784427 3.18 .191954 8.15 58 .795779 3.12 .221578 8.33 59 784618 3.18 .192443 8.1'} 59 .795966 8.13 .229078 8.33 (50 9.784809 3.18 10.192931 (U5 r,o n. 79(51 ::{ 3.13 10.222578 8.533 [430] TABLE XXVI. -LOGARITHMIC VERSED SINES 68 69 / Vers. D. r. Ex. sec. D. r. / Vers. D. r. Ex. sec. D.r. o 9.796153 3.13 10.222578" 8.33 9.807286 3.07 10.252957 8.55 i .796341 3.12 .223078 8.33 1 .807470 3.07 .253470 8.55 2 .796528 3.12 .223578 8.35 2 .807654 3.05 .253983 8.57 3 .796715 3.12 .224079 8.33 3 .807837 3.07 .254497 8.55 4 .796902 3.12 .224579 8.35 4 .808021 3.05 .255010 8.57 5 .797089 3.12 .225080 8.35 5 .808204 3.07 .255524 8.58 6 .797276 3.12 .225581 8.37 6 .808388 3.05 .256039 8.57 7 .797463 3.12 .226083 8.35 7 .808571 3.07 .256553 8.58 8 .797650 3.12 .226584 8.37 8 .808755 3.05 .257068 ! 8.57 9 .797837 3.10 .227086 8.37 9 .808938 3.05 .257582 8.60 10 .798023 3.12 .227588 8.37 10 .809121 3.07 .258098 8.58 11 9.798210 3.12 10.228090 8.37 11 9.809305 3.05 10.258613 8.60 12 .798397 3.10 .228592 8.38 12 .809488 ! 3.05 .259129 8.58 13 .798583 3.12 .229095 8.38 13 .809671 3.05 .259644 8.60 14 .798770 3.10 .229598 8.38 14 .809854 3.05 .260160 8.62 15 .798956 3.10 .230101 8.38 15 .810037 3.05 .260677 8.60 16 .799142 3.12 .230604 8.38 16 .810220 3.05 .261193 8.62 17 .799329 3.10 .231107 8.40 17 .810403 3.03 .261710 8.62 18 .799515 3.10 .231611 8.40 18 .810585 3.05 .262227 8.62 19 .799701 3.10 .232115 8.40 19 .810768 3.05 .262744 8.63 20 .799887 3.12 .232619 8.40 20 .810951 3.05 .263262 8.62 21 9.800074 8.10 10.233123 8 40 21 9.811134 3.03 10.263779 8.63 22 .800260 3.10 .233627 8.42 22 .811316 3.05 .264297 8.63 23 .800446 3.08 .234132 8.42 23 .811499 3.03 .264815 8.65 24 .800631 3.10 .234637 8.42 24 .811681 3.05 .265334 8.65 25 .800817 3.10 . 235142 8.42 25 .811864 3.03 .265853 1 8.63 26 .801003 3.10 .235647 8.43 26 .812046 3.03 .266371 i 8.67 27 .801189 3.10 .236153 8.42 27 .812228 3.03 .266891 8.65 28 .801375 3.08 .236658 8.43 28 .812410 3.05 .267410 8.67 29 .801560 3.10 .237164 8.43 29 .812593 3.03 .267930 8.65 30 .801746 3.08 .237670 8.45 30 .812775 j 3.03 .268449 8.68 31 9.801931 8.10 10.238177 8.43 31 9.812957 3.03 10.268970 8.67 32 .802117 3.08 .238683 8.45 32 .813139 3.03 .269490 ; 8.68 33 .802302 3.08 .239190 8.45 33 .813321 3.03 .270011 3.67 34 .802487 3.10 .239697 8.45 34 .813503 i 3.03 .270531 8.68 35 .802673 3.08 .240204 8.47 35 .813685 ! 3.02 .271052 8.70 36 .802858 3.08 .240712 8.45 36 .813866 3.03 i .271574 8.68 37 .803043 3.08 .241219 8.47 37 .814048 3.03 1 .272095 8.70 38 .803228 3.08 .241727 8.47 i! 38 .814230 3.02 j .272617 ' 8.70 39 .803413 3.06 .242235 8.48 39 .814411 3.03 .273139 8.72 40 .803598 3.08 .242744 8.47 40 .814593 3.03 .273662 i 8.70 41 9.803783 3.08 10.243252 8.48 41 9.814775 3.02 10.274184 8.72 42 .803968 3.08 .243761 8.48 42 .814956 3.02 .274707 8.72 43 .804153 3.08 .244270 8.48 43 .815137 3.03 .275230 8.72 44 .804338 3.07 .244779 8.50 44 .815319 3.02 j .275753 i 8.73 45 .804522 3.08 .245289 8.48 45 .815500 3.02 .276277 8.73 46 .804707 3.08 .245788 8.50 46 .815681 3.02 .276801 8.73 47 .804892 3 07 .246308 8.50 47 .815862 3.03 .277325 8.73 48 .805076 3.08 .246818 8.52 48 .816044 3.02 .277849 8.75 49 .805261 3.07 .247329 8.50 49 .816225 3.02 .278374 8.75 i50 .80*445 3.07 .247839 8.52 50 .816406 3.02 .278899 8.75 51 9.805629 3.08 10.248350 8.52 51 9.816587 3.00 10.279424 8.75 '52 .805814 3.07 .248861 i 8.52 52 .816767 3.02 .279949 8.77 53 .805998 3.07 .249372 8.52 53 .816948 3.02 .280475 8.75 54 .806182 3.07 .249883 8.53 54 .817129 3.02 .281000 , 8.78 55 .806366 3.07 .250395 8.53 5*5 .817310 3.00 .281527 i 8.77 56 . 806550 3.07 .250907 8.53 56 .817490 3.02 .282053 8.78 57 .806734 3.07 .251419 8.55 57 .817671 3.02 .282580 8.77 58 .806918 3.07 .251932 8.58 58 .817852 3.00 .283106 8.80 59 .807102 3.07 .252444 8.55 59 .818032 3.02 .283634 8.78 60 9.807286 8.07 10.252957 ! 8.55 00 9.818213 3.00 10.284161 8.80 ^_^ [440] AND EXTERNAL SECANTS 70 71 / Vers. D. r. Ex. sec. D. 1" > Vers. D. r. Ex. sec. D.I". 9.818213 3.00 10.284161 8.80 9.828938 2.95 10.316296 9.07 i .818393 3.00 .284689 8.78 j i .829115 2.95 .316840 9.08 2 .818573 3.02 ...'S.V'16 8.82 2 .829292 2.95 .317385 9.07 3 .818754 3.00 .285745 8.80 3 .829469 2.95 .317929 9.10 4 .818934 3.00 .286273 8.82 4 .829646 2.95 .318475 9.08 5 .819114 ! 3.00 .286802 8.82 5 .829823 ! 2.95 .319020 9.08 6 I .819294 I 3.00 .287331 8.82 6 .830000 ! 2.95 .319565 9.10 7 .819474 3.00 .287860 8.82 r* .830177 2.93 .320111 9.12 8 .819654 j 3.00 .288389 8.83 8 .830858 2.95 .320658 9.10 9 .819834 3.00 .288919 8.83 9 .830r>30 2.93 .321204 9.12 10 .820014 3.00 .289449 8.83 10 .830706 2.95 .321751 9.12 11 9.820194 3 00 10.289979 8.85 11 9.830883 2.93 10.322298 9.12 ia .820374 2.98 .290510 8.85 12 .831059 2.95 .322845 9.13 13 .820553 3.00 .291041 8.85 13 .831230 2.93 .323393 9.13 14 .820733 3.00 .291572 8.85 14 .831412 2.95 .323941 9.13 15 .820913 2.98 .292103 8.87 15 .831589 2.93 .324489 9.15 10 .821092 3.00 .292635 8.85 16 .831765 2.93 .325038 9.15 17 .821272 2.98 .293166 8.87 17 .831941 2.93 .325587 9.15 18 .821451 3.00 .293698 8.88 18 .832117 2.93 .326136 9.17 19 .821631 2.98 .294231 8.88 19 .832298 2.93 .326686 9.15 20 .821810 2.98 .294764 8.87 20 .832469 2.93 .327235 9.18 21 9.821989 2.98 10.295296 8.90 21 9.832645 2.93 10.327786 9.17 22 .822168 3.00 .295830 8.88 22 .832821 2.93 .328336 9.18 23 .822348 2.98 .296363 8.90 23 .832997 2.93 .328887 9.18 24 .822527 2.98 .296897 8.90 24 .833173 2.93 .329438 9.18 1 25 .822706 2.98 .297431 8.90 25 .&33S49 2.93 .329989 9.20 20 .822885 2.98 .297965 8.92 26 .833525 2.92 .330541 9.20 27 .823064 2.98 .298500 8.90 27 .833700 2.93 .1331093 9.20 28 .823243 2.97 .299034 8.93 28 .833876 2.92 .331645 9.22 29 .823421 2.98 .299570 8.92 29 &34051 2.93 .332198 9.20 30 .823600 2.98 .300105 8.93 30 .834227 2.92 .332750 31 9.823779 2.98 10.300641 8.92 31 9.834402 2.93 10.333304 9.22 32 .823958 2.97 .301176 8.95 32 .834578 2.92 .333857 9.23 33 .824136 2.98 .301713 8.93 33 .834753 2.92 .334411 9.23 34 .824315 2.97 .302249 8.95 34 .834928 2.93 .334965 9.25 35 .824493 2.98 .302786 8.95 35 .835104 2.92 .335520 9.23 36 .824672 2.97 .303323 8.95 36 .835279 2.92 .336074 9.25 37 .824850 2.97 .303860 8.97 37 .835454 2.92 .336629 9.27 38 .825028 2.98 .304398 8 97 38 .835629 2.92 .337185 9.27 39 .825207 2.97 .304936 8.97 39 .835804 2.92 .337741 9.27 40 .825385 2.97 .305474 8.97 .;>><> 107 2.90 10.349485 9.38 TABLE XXVI.- LOGARITHMIC VERSED SINES 72 73 > Vers. D.I'. Ex. sec. D. r. / Vers. D. 1". Ex. sec. D. r. 9.839467 2.90 10.349485 9.38 9.849805 2.85 10.383870 9.73 1 .839641 2.90 .350048 9.38 1 .849976 2.85 .384454 9.73 2 .839815 2.90 .350611 9.40 2 .850147 2.83 .385038 9.75 3 .839989 i 2.88 .351175 9.38 3 .850317 2.85 .385623 9.77 4 .840162 2.90 .351738 9.42 4 .850488 2.83 .386209 9.75 5 .840&36 2.90 .&52303 9.40 5 .850658 2.85 .386794 9.77 6 .840510 2.88 .352867 9.42 6 .850829 2.83 .387380 9.78 7 .840683 2.90 .353432 9.42 >j .850999 2.83 .387967 9.78 8 .840857 2.88 .353997 9.43 8 .851169 2.85 .388554 9.78 9 .841030 2.90 .354563 9.43 9' .851340 2.83 .389141 9.78 10 .841204 2.88 .355129 9.43 10 .851510 2.83 .389728 9.80 11 9.841377 2.88 10.355695 9.43 11 9.851680 2.83 10.390316 9.82 152 .841550 2.88 .356261 9.45 12 .851850 2.83 .390905 9.80 13 .841723 2.88 .356828 9.45 13 .852020 2.83 .39)493 9.82 14 .841896 2.90 .357395 9.47 14 .852190 2.83 .392082 9. as 15 .842070 2.88 .357963 9.47 15 .852360 2.83 .392672 9.83 16 .842243 2.88 .358531 9.47 16 .852530 2.83 .393262 9.83 17 .842416 2.88 .359099 9.48 17 .852700 2.83 .393852 9.85 18 .842589 2.88 .359668 9.48 18 .852870 2.83 .394443 9.85 19 .842762 2.87 .360237 9.48 19 .853040 2.82 .395034 9.85 20 .842934 2.88 .360806 9.50 20 .853209 2.83 .395625 9.87 21 9.843107 2.88 10.361376 9.50 21 9.853379 2.83 10.396217 9.87 22 .843280 2.88 .361946 9.50 22 .853549 2.82 .396809 9.88 23 .8-13453 2.87 .362510 9.52 23 .853718 2.83 .397402 9.88 24 .843625 2.88 .363087 9.52 24 .853888 2.82 .397995 9.90 25 13798 2.87 .363658 9.52 25 .854057 2.83 .398589 9.88 26 .*J43970 2.88 .364229 9.53 26 .854227 2 82 .399182 9.92 27 .844143 2.87 .364801 9.53 27 .854396 2^82 .399777 9.90 28 -.844315 2.88 .365373 9.53 I 28 .854565 2.83 .400371 9.92 29 .844488 2.87 .365945 9.55 29 .854735 2.82 .4001166 i 9.93 30 .844660 2.87 .366518 9.55 30 .854904 2.82 .401562 9.93 31 9.844832 2.87 10.367091 9.57 31 9.855073 2.82 10.402158 9.93 32 .845004 2.88 .367665 9.57 32 .855242 2.82 .4027M 9.95 33 .845177 2.87 .368239 9.57 ! 33 .855411 2.82 .403351 9.95 34 .845349 2.87 .368813 9.57 1 34 .855580 2 82 .408948 9.95 35 .845521 2.87 .369387 9.58 ! 35 ..S55749 2! 82 .404545 9.97 36 .845693 2.87 .369962 9.60 36 .855918 2.82 .405143 9.98 37 .845865 2.87 .370538 9.58 37 .856087 2.80 .405742 9.97 38 .846037 2.85 .371113 9.60 38 : > 50255 2.82 .406340 9.98 39 .846208 2.87 .371689 9.62 39 .856424 2.82 .406939 10.00 40 .846380 2.87 .372266 9.60 40 .856593 2.82 .407539 10.00 41 9.846552 2.87 10.372842 9.62 41 9.856762 2.80 10.408139 10.00 42 .846724 2.85 .373419 9.63 42 .856930 2.82 .-408739 10.02 43 .846895 2.87 .373997 9.63 43 i .857-099 2.80 .409340 10.02 44 .847067 2 85 .374575 9.63 ' 44 .857267 2.82 .409941 1 10.03 45 .847238 2 87 .375153 9.03 45 ! .857436 2.80 .410543 10.03 46 .847410 2.85 .375731 9.65 46 .857604 2.80 .411145 10.03 47 .847581 2.87 .376310 9.67 i 47 .857772 2.82 .411747 10.05 48 .847753 2.85 .376890 9.65 1 48 I .857941 2.80 .412350 10.07 49 .847924 2.85 .377469 9.67 49 .858109 2.80 .412954 10.05 50 .848095 2.87 .378049 9.68 50 .858277 2.80 .413557 10.07 51 9.848267 2.85 10.378630 9.67 51 9.858445 2.80 10.414161 10.08 52 .848438 2.85 .379210 9.70 52 .858613 2.80 .414766 j 10.08 53 .848609 2.85 .379792 9.68 53 | .858781 2.80 .415371 10.08 54 .848780 2.85 .380373 9.70 54 ! .858949 2.80 .415976 10.10 55 .848951 2.85 .380955 9.70 55 .859117 2.80 .416582 10.12 56 .849122 2.85 .381537 9.72 56 .859285 2.80 .417189 10.10 57 .849293 2.85 .382120 9.72 57 .859453 2.80 .417795 10.12 58 .849464 2,83 .382703 9.72 58 ! .859621 2.78 .418402 10.13 59 .849634 2.85 .383286 9.73 59 .859788 2.80 .419010 10.13 60 9.849805 2.85 10.383870 9.73 68 9.859956 2.80 10.419618 10.13 AND EXTERNAL SECANTS. 74 75 f Vers. D. 1". Ex. sec. D. r. / Vers. D. 1". Ex. sec. D. 1". 9.859956 2.80 10.419618 10.13 i 9.869924 2.75 10.456928 10.60 1 .860124 2.78 .420226 10.15 \ 1 .870089 2.73 .457564 10.62 2 .860291 2.80 .420835 10.17 2 .870253 2.75 .458201 10.63 3 .860459 2.78 .421445 10.15 3 .870418 2.73 .458839 10.62 4 i .860626 2.80 .422054 10.17 4 .870582 2.75 .459476 10.65 5 .860794 2.78 .422664 10.18 5 .870747 2.73 .460115 10.65 6 .860961 2.78 .423275 10.18 6 .870911 2.75 .460754 10.65 7 .861128 2.80 .423886 10.20 i 7 .871076 2.73 .461S93 10.67 8 .861296 2.78 .424498 10.20 ! 8 .871240 2.73 .462033 10.67 9 .861463 2.78 .425110 10.20 9 .871404 2.73 .462673 10.68 10 .861630 2.78 .425722 10.22 10 .871568 2.73 .463314 10 70 11 9.861797 2.78 10.426335 10.22 11 9.871732 2.73 10.463956 10.70 12 .861964 2.78 .426948 10.23 12 .871896 2.73 .464598 10.70 13 : 862131 2.78 .427562 10.23 13 .872060 2.73 .465240 10.72 14 .862298 2.78 .428176 10.23 11 .872224 2.73 465883 10.73 15 .862465 2.78 .428790 10.27 15 .872388 2.73 .466527 10.73 16 .862632 2.78 .429406 10.25 1 16 .872552 2.73 .467171 10.73 17 .862799 2.77 .430021 10.27 17 .872716 2.73 .467815 10.75 18 .862965 2.78 .430637 10.27 18 .872880 2.72 .468460 10.77 19 .863132 2.78 .431253 10.28 19 .873043 2.73 .469106 10.77 20 .863299 2.77 .431870 10.30 20 .873207 2.73 .469752 10.77 21 9.863465 2.78 10.432488 10.28 21 9.87aS71 2.72 10.470398 10.78 22 .863632 ! 2.78 ! 433105 10.32 22 .873534 2.73 .471045 10.80 23 .863799 2.77 .433734 io.;io 23 .873698 2 72 .471693 10.80 24 .863965 2 77 .434342 10.32 24 .873861 2>3 .472341 10.82 25 .864131 2! 78 .434961 10.33 25 .874025 2.72 .472990 10.82 26 .864298 2 77 .435581 10.88 26 .874188 2.72 .473639 10.83 27 .864464 2! 77 .436201 10.33 27 .874351 2.73 .474289 10.83 28 .864630 2.78 .436821 10.35 28 .874515. 2.72 .474939 10.85 29 .864797 2.77 .437442 10.37 29 .874678 2.72 .475590 10.87 30 .864963 2.77 .438064 10.37 30 .874841 2.72 .476242 10.85 31 9.865129 2.77 10.438686 10.37 31 9.875004 2.72 10.476893 10.88 32 .865295 2.77 .439308 10.38 32 .875167 2.72 .477546 10.88 33 .865461 2.77 .439931 10.38 33 .875830 2.72 .478199 10.88 34 .865627 2.77 .440554 10.40 34 .875493 2.72 .478852 10.90 35 .865793 2.77 .441178 10.40 35 .875656 2.72 .479506 10.92 36 .865959 2.75 .441802 10.42 36 .875819 2.72 .480161 10.92 37 .866124 2.77 .442427 10.42 37 .875982 2.72 .480816 10.93 38 .866290 2.77 .413052 10.43 38 .876145 2.72 .481472 10.93 39 .866456 2.77 . 1 W(i7S 10.43 39 .876308 2.70 .482128 10.95 40 .866622 2.75 .444304 10.45 40 .876470 2.72 .482785 10.95 41 9.866787 2.77 10.444931 10.45 41 9.876633 2.72 10.483442 10.97 42 .866953 2.75 .445558 10.45 12 .876796 2.70 .484100 10.98 43 .867118 2.77 .446185 10.47 43 .876958 2.72 .484759 10.98 44 .867284 2.75 .446813 10.48 44 .817121 2.70 .485418 10.98 45 .867449 2.75 .447442 10.48 15 .H7.2K3 ' 2.70 .486077 10.98 46 .867614 2.77 .448071 10.48 46 .877445 a. 72 .486738 11.00 47 .867780 2.75 .448700 10.50 47 .877608 i 2.70 .487398 11 02 48 .867945 2.75 .441)330 10.52 48 .877770 ! 2.70 .488059 11.03 49 .868110 2.75 .449961 10.52 49 .877932 2.72 .488721 11.05 50 .868275 2.77 .450592 10.52 50 .8780!).-) 2.70 .489384 11.05 51 9.868441 2.75 10.451223 10.53 51 9.878257 2.70 10.490047 11.05 52 .868606 2.75 .451855 10.53 52 .878419 2.70 .490710 11.07 53 .868771 2.75 .452487 1 10.55 53 .878581 2.70 .491374 11.08 54 .868936 2.73 .453120 10.57 54 .878743 2.70 .492039 11.08 55 .869100 2.75 .453754 10.57 55 .878905 2.70 .492704 11.10 56 .869265 2.75 .454388 10.57 56 .S7'.)or,7 2.70 .493870 11.10 ; 57 .8(59430 2.75 .455032 10.58 57 879229 2.68 .49-1036 11.12 58 .SC.9.V.I5 2.75 .455657 10.58 58 .879390 2.70 .494703 j 11.13 59 .869780 2.73 .456292 10.60 59 .879552 2.70 .495371 11.13 60 9.869924 2.75 10.456928 10.60 60 9.879714 2.70 i 10. 496039 1 11.13 [443] TABLE XXVI. LOGARITHMIC VERSED SINES 76 77 / Vers. D. 1'. Ex. sec. D. r. ' Vers. D.I". Ex. sec. D. 1". o 9.879714 2.70 10.496039 11.13 9:889329 2.65 10.537241 11.77 i .879876 2.68 .496707 11.17 1 .889488 2.65 .537947 11.78 2 .880037 2.70 .497377 11.17 2 .889647 2.63 .538654 11.80 3 .880199 2.68 .498047 11.17 3 .889805 2.65 .539362 11.82 4 .880360 2.70 .498717 11.18 4 .889964 2.65 .540071 11.82 5 .880522 2.68 .499388 11.20 5 .890123 2.63 .540780 11.83 6 .880683 2.70 .500060 11.20 6 .890281 2.65 .541490 11.83 7 .880845 2.68 .500732 11.22 7 .890440 .2.63 .542200 11.85 8 .881006 2.68 .501405 11.22 8 .890598 2.65 .542911 11.87 9 .881167 | 2.70 .502078 11 23 9 .890757 2.63 .543623 11.88 10 .881329 2.68 .502752 11.23 10 .890915 2.63 .544336 11.88 11 9.881490 2.68 10.503426 11.27 11 9.891073 2.65 10.545049 11.90 12 .881651 | 2.68 .504102 11.25 12 .891232 2.63 .545763 n.90 13 .881812 ! 2.68 .504777 11.28 13 .891390 2.63 .546477 11.93 14 .881973 2.68 .505454 11.28 14 .891548 2.63 .547193 11.93 15 .882134 2.68 .506131 11.28 15 .891706 2.63 .547909 11.95 16 .882295 2.68 .506808 11.30 16 .891864 2.63 .548626 11.95 17 .882456 2.68 .507486 11.32 17 .892022 2.63 .549343 11.97 18 .882617 2.67 .508165 11.32 18 .892180 2.63 .5.50061 11.98 19 .882777 2.68 .508844 11. as 19 .892338 2.63 .550780 12.00 20 .8829:38 2.68 .509524 11.35 20 .892496 2.63 .551500 12.00 21 9.883099 2.68 10.510205 11.35 21 9.892654 2.63 10.552220 12.02 22 .883260 2.67 .510886 11.37 22 .892812 2.62 .552941 12. as 23 .883420 2.68 .511568 11.37 23 .892969 2.63 .553663 12.03 24 .883581 2.67 .512250 11.38 24 .893127 2.63 .554385 12.07 25 .883741 2.68 .512933 11.40 25 .893285 2.62 .555109 12.07 28 .883902 2.67 .513617 11.40 26 .893442 2.63 .555833 12.07 27 .884062 2.68 .514301 11.42 27 .893600 2.63 .556557 12.10 23 .884223 2.67 .514986 11.43 28 .893758 2.62 .557283 12.10 29 .884383 2.67 .515672 11.48 29 .893915 2 62 .558009 12.12 30 .884543 2.67 .516358 11.45 30 .894072 2^63 .558736 12.12 31 9.884703 2.68 10.517045 11.45 31 9.894230 2.62 10.559463 12.15 32 .884864 2.67 .517732 11.47 32 .894387 2.62 .560192 12.15 33 .885024 2.67 .518420 11.48 as .894544 2.63 .580921 12.17 34 .885184 2.67 .519109 11.48 34 .894702 2.62 .561651 12.17 35 .885344 2.67 .519798 11.50 35 .894859 2.62 .562381 12.20 36 .885504 2.67 .520488 11.52 36 .895016 2.62 .563113 12.20 37 .885664 2.67 .521179 11.50 37 .895173 2.62 .563845 12.20 33 .885824 2.65 .521870 11.53 38 .895:330 2.62 .564577 12.23 39 .885983 2.67 .522562 11.53 39 .895487 2. 62 .565311 12.23 40 .886143 2.67 .523254 11.55 40 .895644 2.62 .566045 12.27 41 9.886303 2.67 10.523947 11.57 41 9.895801 2.62 10.566781 12.25 42 .886463 2.65 .524641 11.57 42 .895958 2.62 .567516 12.28 43 .886622 2.67 .525335 11.58 43 .896115 2.62 .568253 12.28 44 .886782 2.65 .526030 11.60 44 .896272 2.60 .568990 12.32 45 .886941 2.67 .526726 11.62 45 .896428 2.62 .569729 12.32 46 .887101 2.65 .527423 11.62 46 .8965a5 2.62 .570468 12.32 47 .887260 2.67 .528120 11.62 47 .896742 2.60 .571207 12.35 48 .887420 2.65 .528817 11.65 48 .896898 2.62 .571948 12.35 49 .887579 2.67 .529516 11.65 49 .897055 2.60 .572689 12.37 50 .887739 2.65 .530215 11.65 50 .897211 2.62 .573431 12.38 51 9.887898 2.65 10.530914 11.67 51 9.897368 2.60 10.574174 12.38 52 .888057 2.65 .531614 11.68 52 .897524 2.60 .574917 12.42 53 .888216 2.65 .532315 11.70 53 .897680 2.62 .575662 12.42 54 .888375 2.65 .533017 11.70 54 .897837 2.60 .576407 12.43 55 .888534 2.65 .533719 11.72 55 .897993 2.60 .577153 12.45 56 .888693 2.65 .534422 11.73 56 .898149 2.60 .5771)00 12.45 57 .888852 2.65 .535126 11.73 57 .898305 2.60 .578647 12.48 58 .889011 2.65 .535830 11.75 58 .898461 2.62 .579396 12.48 59 .889170 2.65 .536535 11.77 59 .898618 2.60 .580145 12.50 60 9.889329 2.65 10.537241 11.77' 60 9.898774 2.60 10.580895 12.50 [444] AND EXTERNAL SECANTS. 78 79 / Vers. D. r. Ex. sec. D. r. ' Vers. D. r. Ex. sec. D. 1". i 9.898774 2.60 10.580895 12.50 9.908051 2.55 10.627452 13.40 1 .898930 2.60 .581645 12.53 i .908204 2.55 .628256 13.40 2 .899086 2.58 .582397 12.53 2 .908357 2.57 .629060 13.43 3 .899241 2.60 .583149 12.57 3 .908511 2.55 .629866 13.45 4 .899397 2. GO .583903 12.57 4 .908664 2.55 .630673 13.45 5 .899553 2.60 .584657 12.57 5 .908817 2.55 .631480 13.48 6 .899709 2.60 .585411 12.60 6 .908970 2.55 .632289 13.48 7 .899865 2.58 .586167 12.60 7 .909123 2.55 .633098 13.52 8 .900020 2.60 .586923 12.03 8 .909276 2.53 .633909 13.52 .900176 2.58 .587681 12.63 g .909428 2.55 .634720 13.55 1C .900331 2.60 .588439 12.65 10 .909581 2.55 .635533 13.55 11 9.900487 2.58 10.589198 12.65 11 9.909734 2.55 10.636346 13.58 , 12 .900642 2.60 .589957 12.08 12 .909887 2.53 .637161 13.58 13 .900798 2.58 .590718 12.08 13 .910039 2.55 .637976 13.60 14 .900953 2.58 .591479 12.72 14 .910192 2.55 .638792 13.03 15 .91)1108 2.60 .592242 12.72 15 .910345 2.53 .639610 13.63 16 .901264 2.58 .593005 12.73 It! .910497 2.55 .640428 13.67 17 .901419 2.58 .593769 12.73 17 .910650 2.53 .641248 13.67 18 .901574 2.58 ,5945aS 12.77 18 .910802 2.55 .642068 13.70 19 .901729 2.58 .595299 12.78 19 .910955 2.53 .642890 13.72 20 .901884 2.60 .596066 12.78 20 .911107 2.53 .643713 13.72 21 9.902040 2.58 10.596833 12.80 21 9.911259 2.55 10.644536 13.75 22 .902195 2.58 .597601 12.82 00 .911412 2.53 .645361 13.75 23 .902350 2 57 .698370 12.83 23 .911504 2.53 .646180 13.78 21 .902504 2.58 .599140 12.85 24 .911716 2.53 .647013 13.80 25 .902659 2.58 .599911 12.85 25 .911868 2.53 .647841 13.82 26 .902814 2.58 .600682 12.88 26 .912020 2.53 .648670 13.82 27 .902969 2.58 .601455 12.88 27 .91217'2 2.53 .649499 13.85 28 .903124 2.57 .002228 12.92 28 .912324 2.53 .650330 13.87 29 .903278 2.58 .603003 12.92 29 .912476 2.53 .651162 13.88 30 .-903433 2.58 .603778 12.93 30 .912628 2.53 .651995 13.90 31 9.903588 2.57 10.604554 12.95 31 9.912780 2.53 10.652829 13.92 32 .903742 2.58 .605331 12.95 32 .912932 2.53 .653664 13.95 33 .903897 2.57 .606108 12.98 33 .913084 2.52 .654501 13.95 34 .904051 2.58 .606887 13.00 34 .913235 2.53 .655338 13.97 35 .904206 2.57 .607667 13.00 35 .91:3387 2.53 .656176 14.00 36 .904360 2.57 .608447 13.02 36 .913539 2.52 .657016 14.00 37 .904514 2.57 .609228 13.03 37 .913690 2.53 .657856 14.03 38 .904008 2.58 .610010 13.07 38 .913842 2.52 .658698 14.03 39 .904823 2.57 .610794 1307 39 .913993 2.53 .659540 14.07 40 .904977 2.57 .611578 13.08 40 .914145 2.52 .660384 14.08 41 9.905131 2.57 10.612363 13.08 41 9.914296 2.53 10.661229 14.10 42 .905285 2.57 .613148 13.13 42 .914418 2.52 .662075 14.12 43 .905439 2.57 .613935 13.13 43 .914599 2.52 .662922 14.13 44 .905593 2.57 .614723 13.13 44 .91475. 2.53 .663770 14.15 45 .905747 2.57 .615511 13.17 45 .914902 2.52 .004619 14.18 46 .905901 2.57 .610301 13.17 46 .915053 2.52 .665470 14.18 47 .906055 2.57 .617091 13.20 47 .915204 2.52 .666321 14.22 48 .900209 2.57 .617883 13.20 IS .915355 2.52 .667174 14.23 49 .906363 2.55 .618675 13.22 49 .91 5506 2.52 .668028 14.25 50 .906516 2.57 .619468 13.23 50 .915657 2.52 .668883 14.27 51 9.906670 2.57 10.620262 13.25 51 9.915808 2.52 10.669739 14.28 52 .906824 2.55 .621057 13.27 58 .915959 2.52 .670596 14.30 53 .906977 2.57 .621853 13.28 53 .916110 2.52 .671454 14.33 54 .907131 2.55 .622050 13.30 54 .916261 2.52 .672314 14.33 55 .907284 2.57 .623448 13.32 55 .916412 2.50 .673174 14.37 56 .W74S8 2.55 .634247 13.33 56 .910562 2.52 .674036 14.38 57 .907591 2.55 .025047 13.35 57 .930713 2.52 .674899 14.40 58 .907744 2.57 .635848 13.37 58 .916864 2.50 .675763 14.42 59 .907898 2.55 .626650 13.37 59 .917014 2.52 .670028 14.45 60 9.908051 2.55 10.027452 13.40 00 9. 91 7 105 2.52 10.677495 14.45 .[445] TABLE XXVI AND EXTERNAL SECANTS. 80 81 ' Vers. D.I". Ex. sec. D. r. / Vers. D. r. Ex. sec. D. 1". 9.917165 2.52 10.677495 14.45 9.926119 2.47 10.731786 15.78 1 .917316 2.50 .678362 14.48 1 .926267 2.47 .732733 15.78 2 .917466 2.50 .679231 14.50 2 .926415 2.45 .733680 15.83 3 .917616 2.52 .680101 14.52 3 .926562 2.47 .734630 15.83 4 .917767 2.50 -680972 14.55 4 .926710 2.47 .735580 15.87 5 .917917 2.52 .681845 14.55 5 .926858 2.47 .736532 15.90 6 .918068 2.50 .682718 14.58 6 .927006 2.45 .737486 15.92 7 .918218 2.50 .683593 14.60 .927153 2.47 .7138441 15.95 8 .918368 2.50 .684469 14.62 8 .927301 2.45 .739398 15.97 9 .918518 2.50 .685346 14.63 9 .927448 2.47 .740356 16.00 10 .918608 2.50 .686224 14.67 10 .927596 2.45 .741316 16.02 11 9.918818 2.50 10.687104 14.68 11 9.927743 2.47 10.742277 16.03 12 .918968 2.50 .687985 14.70 12 .927891 2.45 .743239 16.08 13 .919118 2.50 .688867 14.72 13 .928038 2.45 .744204 16.08 14 .919268 2.50 .689750 14.73 14 .928185 2.47 .745169 16.13 15 .919418 o 50 .690634 14.77 15 .928333 2.45 .746137 16.13 16 .919568 2^50 .691520 14.78 16 .928480 2.45 .747105 16.18 17 .919718 2.50 .692407 14.80 17 . 1)28627 2.45 .748076 16.20 18 .919868 2.50 .693295 14. as 18 .928774 2.45 .749048 16.22 19 .920018 2.48 .694185 14.83 19 .928921 2.45 .750021 16.25 20 .920167 2.50 .695075 14.87 20 .929068 2.45 .750996 16.28 21 9.920317 2.48 10.695967 14.90 21 9.929215 2.45 10.751973 16.30 22 .920466 2.50 .696861 14.90 22 .929362 2.45 .752951 16.33 23 .920616 2.. r .697755 14.93 23 .929509 2.45 .753931 16. 35 24 .920766 2.48 .698651 14.95 24 .929656 2.45 .754912 16.38 25 .920915 2.48 .699548 14.97 25 .929803 2.45 .755895 16.42 26 .921064 2.50 .700446 15.00 26 .929950 2.45 .756880 16.43 27 .921214 2.48 .701346 15.02 27 .930097 2.43 .757866 16.47 28 .921363 2.48 .702247 15.03 28 .930243 2.45 .758854 16.50 29 .921512 2.50 .703149 15.05 29 .930390 2.45 .759844 16.52 30 .921662 2.48 .704052 15.08 30 .930537 2.43 . 760835 _ 16.53 31 9.921811 2.48 10.704957 15.10 31 9.930683 2.45 10.761827 16.58 32 .921960 2.48 .705863 15.13 32 .930830 2.43 .762822 16.60 33 .922109 2.48 .706771 15.15 33 .930976 2.45 .763818 16.62 34 .922258 2.48 .707680 15.17 34 .931123 2.43 .764815 16.67 35 .922407 2.48 .708590 15.18 35 .931269 2.45 .765815 16.68 36 .922556 2.48 .709501 15.22 36 .931416 2.43 .766816 16.72 37 .922705 2.48 .710414 15.23 37 .931562 2.43 .767819 16.73 38 .922854 2.48 .711328 15.25 38 .931708 2.45 .768823 16.77 39 .923003 2.48 .712243 15.28 39 .931855 2.43 .769829 16.80 40 .923152 2.48 .713160 15.30 40 .932001 2.43 .770837 16.82 41 9.923301 2.47 10.714078 15.33 41 9.932147 2.43 10.771846 16.87 42 .923449 2.48 .714998 15.35 42 .932293 2.43 .772858 16.87 43 .923593 2.48 .715919 15.37 43 .932439 2.43 .773870 16.92 44 .923747 2.47 .716841 15 38 44 .932585 2.43 .774885 16.95 45 .923895 2.48 .7177'64 15.42 45 .932731 2.43 .775902 16.97 46 .924044 2.47 .718689 35.45 46 .932877 2.43 .776920 I'.CO 47 .924192 2.48 .719616 15.45 47 .933023 2. "43 .777940 1 .02 48 .924341 2.47 .720543 15.48 48 .933169 2.43 . .778961 .07 49 .924489 2.47 .721472 15.52 49 .933:315 2.42 .779985 .08 50 .924637 2.48 .722403 15.53 50 .933460 2.43 .781010 7.18 51 9.924786 2.47 10.723335 15.55 51 9.9a3606 2.43 10.782037 .13 52 .924934 2 47 .724268 15.58 52 .933752 2.42 .783065 .18 53 .925082 2.48 .725203 15.60 53 .933897 2.43 .784096 .20 54 .925231 2.47 .726139 15.63 54 .934043 2.43 .785128 .23 55 .925379 2.47 .727077 15.65 55 .934189 2.42 .786162 .27 56 .925527 2.47 .728016 15.67 56 .934334 2 43 .787198 .30 57 .925675 2.47 .728956 15.70 57 .934480 2.42 .788236 .33 58 .925823 2.47 .729898 15.73 58 .934625 2.42 .789276 .35 59 .925971 2.47 .730842 15.73 59 .934770 2.43 .790317 1 .40 60 9.926119 2.47 10.731786 15.78 60 9.934916 2.42 10.791361 1 .42 [446.1 AND EXTERNAL SECANTS. 82 83' ' Yers. D. 1'. Ex. sec. D. 1'. ' Vers. D. r. Ex. sec. D. 1'. 9.934916 2.42 10.791361 17.42 9.943559 2.38 10.857665 i 19.55 1 .935061 2.42 .792406 17.45 1 .943702 2.38 .858838 ! 19.58 2 .935206 2.43 .793453 17.48 2 .943845 2.37 .860013 19.63 3 .935352 2.42, ,794502 17.50 3 .943987 2.38 .861191 19.67 4 .935497 2.42 .795552 17.55 4 .944130 2.38 .862371 19.72 5 .935642 2.42 .796605 17.58 5 .',144273 2.37 .863554 19.75 6 .935787 2.42 .797660 17.60 6 .944415 2.38 .864739 19.80 7 .935932 2.42 .798716 17.63 .1(44558 2.37 .865927 19.83 8 .936077 2.42 .799774 17.68 8 .944700 2.38 .867117 19.88 9 .936222 2.42 .800835 17.70 9 .944843 2.37 868310 19.92 10 .936367 2.42 .801897 17.73 10 .944985 2.37 1869505 19.97 11 9.936512 2.42 10.802961 17.77 11 9.945127 2.38 10.870703 20.00 12 .936657 2.40 .804027 17.80 12 .945270 2.37 .871903 20.05 13 .936801 2.42 .805095 17.83 13 .945412 2.37 .873106 20.10 14 .936946 2.42 .806165 17.87 14 .945554 2.37 .874312 20.13 15 .937091 2.42 .807237 17.90 15 .945696 2.37 .875520 20.18 16 .937236 2.40 .808311 17.93 16 .945838 ! 2.38 .876731 20.23 17 .937380 2.42 .809387 17.97 17 .945981 2.37 .877945 20.27 18 .937525 2.40 .810465 18.00 18 .946123 2.37 .879161 1 20.30 19 .937669 2.42 .811545 18.03 19 .946265 2.37 .880379 20.37 20 .937814 2.40 .812627 18.07 20 .946407 2.37 .881601 20.40 21 9.937958 2.42 10.813711 18.10 21 9.946549 2.35 10.882825 20.45 22 .938103 2.40 .814797 18.13 22 .946690 2.37 .884052 20.48 23 .938247 2.40 .815885 18.17 23 .946&S2 2.37 .885281 20.55 24 .938391 2.42 .816975 18.20 24 .946974 2.37 .886514 20.58 25 .938536 2.40 .818067 18.23 25 .947116 2.37 .887749 20.62 26 .938680 ' 2.40 .819161 18.27 26 .947258 2.35 .888986 20.68 27 .938824 2.40 .820257 18.32 27 .947399 2.37 .890227 20.72 28 .938968 2.40 .821356 18.33 28 .947541 2.37 .891470 20.77 29 .939112 2.38 .822456 18.38 29 .947683 2.35 .892716 20.82 30 .939257 2.40 .823559 18.42 30 .947824 2.37 .893965 20.87 31 9.939401 2.40 10.824664 18.43 31 9.947966 2.35 10.895217 20.92 32 .939545 2.38 .825770 18.48 32 .948107 2.37 .896472 20.95 33 .939688 2.40 .826879 18.52 33 .948249 2.35 .897729 21.00 34 .939832 2.40 .827990 18.57 34 .948390 2.35 .898989 21.07 35 -939976 2.40 .829104 18.58 35 .948531 2.37 .900253 21 . 10 36 .940120 2.40 .&30219 i 18.63 36 .948673 2.35 .901519 21.15 37 .'.141)264 2.40 .831337 18.65 37 .948814 2.35 .902788 21.20 38 .940408 2.38 .832456 18.70 38 .948955 2.35 .904060 21.25 39 .940551 2.40 .833578 18.75 39 .949096 2.35 .905335 21.30 40 ! .940695 2.40 .834703 18.77 40 .949237 2.37 .906613 21.33 41 9.940839 2.38 10.835829 18.80 41 9.949379 2.35 10.907893 21.40 42 .940982 2.40 .836957 18.85 42 .949520 2.35 .909177 21.45 43 i .941126 2.38 .838088 18.88 43 .949661 2.35 .910464 21.50 44 .941269 2.40 .839221 18.93 44 .94'Jb 2 2.35 .911754 21.55 45 .941413 2.38 .840357 18.95 45 .949943 2.33 .913047 21.60 46 .941556 2.38 .841494 19.00 46 .950083 2.35 .914343 21.65 47 .941699 2.40 .842634 19.03 47 .950224 2.35 .915642 21.70 48 .941843 2.38 .843776 19.08 48 .950365 2.35 .916944 21.75 49 .941986 2.38 .844921 19.12 49 .950506 2.35 .918249 21.82 50 .942129 2.38 .846068 19.15 50 .950647 2.33 .919558 21.85 51 9.942272 2.38 10.847217 19.18 51 9.950787 2.35 10.920869 21.92 52 .942415 2.40 .848368 19.23 52 1950928 2.35 .922184 21.97 53 .942559 2.38 .849522 19.27 53 .951069 2.33 .923502 22.02 .54 .942702 2.38 .850678 19.30 54 .951209 2.35 .924823 22.07 55 .942845 2.38 .851836 19.35 55 .951350 2.33 .926147 22.13 56 .942988 2.38 .852997 19.40 56 .951490 2.35 .927475 22.17 57 .943131 2.37 .854161 19.42 57 .951631 2.33 .928805 22.23 58 .943273 : 2.38 .855326 19.47 58 .951771 2.33 930139 22.30 59 .943416 2.38 .856494 19.52 59 .951911 2.35 .'931477 22.33 60 I 9.943559 2.38 10.8. r .7665 19.55 il 60 9.952052 2.33 10.932817 22.40 .[447] T^BLE XXVI. LOGARITHMIC VERSED SINES 84 85 / Vers. D. 1". Ex. sec. D. 1". / Vers. D. 1". Ex. sec. D.I'. 9.952052 2.33 10.932817 22.40 9.960397 2.30 11.020101 26.40 i .952192 2.33 .934161 22.45 1 .960535 2.28 .021685 26.48 2 .952332 2.35 .935508 22.52 2 .960672 2.30 .023274 26.57 3 .952473 2.33 .936859 22.57 3 .960810 2.30 .024868 26 65 4 .952613 2.33 .938213 22.62 4 .960948 2.30 .026467 26.73 5 .952753 2.33 .939570 22.68 5 .961086 2.28 .028071 26.80 6 .952893 2.33 .940931 22.75 6 .961223 2.30 .029679 26.90 7 .953033 2.33 .942296 22.78 7 .961361 2.28 .031293 26.98 8 .953173 2.33 .943663 22.85 8 .961498 2.30 .032912 27.07 9 .953313 2.33 .945034 22.92 9 .961636 2.28 .034536 27.13 10 .953453 2.33 .946409 22.97 10 .961773 2.30 .036164 27.23 11 9.953593 2.32 10.947787 23.03 11 9.961911 2.28 11.037798 27.33 12 .953732 2.33 .949169 23.08 12 .962048 2.30 .039438 27.40 13 .953872 2.33 .950554 23.15 13 .962186 2.28 .041082 1 27.50 14 .954012 2.33 .951943 23.22 14 .962323 2.28 .042732 27.58 15 .954152 2.32 .953336 23.27 15 .962460 2.28 .044387 27.67 16 .954291 2.33 .954732 23.33 ; 16 .962597 2.30 .046047 27.77 17 .954431 2.33 .956132 23.38 17 .962735 2.28 .047713 27.85 18 .954571 2.32 .957535 23.45 18 .962872 2.28 .049384 27.93 19 .954710 2.33 .958942 23.52 19 .963009 2.28 .051060 28.03 20 .954850 2.32 .960353 23.57 20 .963146 2.28 .052742 28.13 21 9.954989 2.33 10.961767 23.65 21 9.963283 2.28 11.054430 28.22 22 .955129 2.32 .963186 23.70 22 .963420 2.28 .056123 28.30 23 .955268 2.32 .964608 23.77 23 .963557 2.28 .057821 28.40 24 .955407 2.33 .966034 23.82 24 .963694 2.28 .059525 28.50 25 .955547 2.32 .967463 23.90 25 .963831 2.28 .061235 28.60 26 .955686 2.32 .968897 23.95 26 .963968 2.27 .062951 28.68 27 .955825 2.32 .970334 24.02 27 .964104 2.28 .064672 28.78 28 .955964 2.32 .971775 24.10 28 .964241 2.28 .066399 28.88 29 .956103 2.33 .973221 24.15 29 .964378 2.28 .068132 28.98 30 .956243 2.32 .974670 24.22 30 .964515 2.27 .069871 29.08 31 9.956382 2.32 10.976123 24.28 31 9.964651 2.28 11.071616 29.18 32 .956521 2.32 .977580 24.35 32 .964788 2.27 .073367 29.28 33 .956660 2.32 .979041 24.42 33 .964924 2.28 .075124 29.38 34 .956799 2.30 .980506 24.48 34 .965061 2.27 .076887 29.48 35 .956937 2.32 .981975 24.55 35 .965197 2.28 .078656 29.58 36 .957076 2.32 .983448 24.63 36 .965334 2.27 .080431 29.68 37 .957215 2.32 .984926 24.68 37 .965460 2.28 .082212 29.80 38 .957354 2.32 J986407 24.77 38 .965607 2 27 .084000 29.90 39 .957493 2.30 .987893 24.83 39 .965743 2^27 .085794 30.00 40 .957631 2.32 .989383 24.90 40 .965879 2.28 .087594 30.12 41 9.957770 2.32 10.990877 24.97 41 9.966016 2.27 11.089401 30.22 42 .957909 2.30 .992375 25.03 42 .966152 2.27 .091214 30.32 43 .958047 2.32 .993877 25.12 43 .966288 2.27 .093033 30.43 44 .958186 2.30 .995384 25.18 44 .966424 2.27 .094859 30.55 45 .958324 2.32 .996895 25.27 45 .996560 2.27 .096692 30.67 46 .958463 2.30 .998411 25.33 46 .966696 2.27 .098532 30.77 47 .958601 2.30 .999931 25.40 47 .966832 2.27 .100378 30.87 48 .958739 2.32 11.001455 25.48 48 .966968 2.27 .102230 31.00 49 .958878 2.30 .002984 25.52 49 .967104 2.27 .104090 31.12 50 .959016 2.30 .004517 36.63 60 .967240 2.27 .105957 31.22 51 9.959154 2.30 11.006055 25.70 51 9.967376 2.27 11.107830 31.35 52 .959292 2.32 .007597 25.78 52 .967512 2.25 .109711 31.45 53 .959431 2.30 .009144 25.85 53 .967647 2.27 .111598 31.58 54 .959569 2.30 .010695 25.93 54 .967783 2.27 .113493 31.68 55 .959707 2.30 .012251 26.00 55 .967919 2.25 .115394 31.82 56 .959845 2.30 .013811 26.10 56 .968054 2.27 .117303 31.93 57 .959983 2.30 .015377 26.17 57 .968190 2.27 .119219 32.07 58 .960121 2.30 .016947 26.23 58 .968326 2.25 .121143 32.18 59 .960259 2.30 .018521 26.33 59 .968461 2.27 .123074 32.30 60 9.960397 2.30 11.020101 26.40 II 60 9.968597 2.25 11.125012 32.43 I.44J AND EXTERNAL SECANTS. 86 87 ' Vers. D. 1". Ex. sec. D IV ' Vers. D. r. Ex. sec. D. r. 9.968697 2.25 11.125012 32.43 9.976654 2.23 11.257854 42.52 1 .968732 2.27 .126958 32.55 1 .976788 2.22 .260405 42.73 .968868 2.25 .128911 32.70 2 .976921 2.22 .262969 42.95 3 .969003 2.25 .130873 32.80 3 .977054 2.22 .265546 43.20 4 .909138 1 2.27 .188841 32.95 4 .977187 2.22 .2681:38 43.42 5 .969274 2.25 .134818 33.07 5 .977320 2.20 .270743 43.67 ti .969409 2.25 .136802 33.22 6 .977452 2 22 .27:3363 43.88 .969544 S.9B .138795 83.88 .977585 2^22 .275996 44.15 8 .969679 2.25 . 140795 33.47 8 .977718 2.22 .278645 44.38 9 .9(59814 2.25 .142803 83.62 i 9 .977851 2.22 .281308 44.03 10 .969949 2.25 .144820 33.73 10 .977984 2.20 .283986 44.88 11 9.970084 2.27 11.146844 33.88 11 9.978116 2 22 11.286679 45.13 i 12 .970220 2.23 .148877 34.02 12 .978249 2^22 .289387 45.38 13 .970354 2.25 .150918 34.17 13 .978382 2.20 .292110 45.65 14 .970489 2.25 .152968 34.30 14 .978514 2.22 .294849 45.92 15 .970624 2.25 .155026 34.43' 15 .978617 2.20 .297604 46.17 16 .970759 2.25 . 157092 34.60 16 .978779 2.22 .300374 46.45 17 .970894 2.25 .159168 34.73 17 .978912 2.20 .303161 46.72 18 .971029 2.25 .161252 34.87 IS .979044 2 22 .305964 i 47.00 19 .971164 2 23 .168844 35.03 ID .979177 2^20 .3087'84 47.27 20 .971298 2^25 . 165446 35.17 20 .979309 2.22 .311620 47.55 21 9.971433 2.25 11.167556 35.33 21 9.979442 2.20 11 .314473 47.83 22 .971568 2.23 .169676 35.48 22 .979574 ! 2.20 .317343 48.13 23 .971702 2.25 .171805 85.68 23 .979706 2.20 .320231 48.43 24 .971837 2.23 .173943 35.78 24 .979838 2.20 .323137 48.72 25 .971971 2.25 .176090 35.93 25 .979970 2.22 .326060 49.02 26 .972106. 2.23 .178246 36.10 26 .980103 2.20 .329001 49.33 27 .972240 2.23 .180412 36.27 27 .980235 2.20 .331961 49.63 28 .972:374 2.25 .182588 36.42 28 .980367 i 2.20 .334939 49.93 29 .972509 2.23 .184773 36.58 29 .SIS04JKI 2.20 .337935 50.27 30 .972643 2.23 .186968 36.75 1 30 .980631 2.20 .340951 50.58 31 9.972777 2.25 11.189173 36.90 ' 31 9.980763 2.20 11.343986 50.92 32 .972912 2.23 .191387 37.08 32 .980895 2.18 .347041 51.23 33 .973046 2.23 .193612 37.25 33 .881026 2.20 .350115 51.58 34 .973180 2.23 .195847 37.42 34 .981158 2.20 .353210 51.92 35 .973314 2.23 .198092 37.58 35 .981290 2.20 .356325 52.25 36 .97'3448 2.23 .200347 37.77 36 .981422 2.20 .359460 52.62 37 .973582 2.23 .202613 37.93 37 .981554 2.18 .362617 52.95 88 .973716 2.23 .204889 38.12 38 .981685 2.20 .365794 53.32 39 .973850 2.23 .207176 38.28 39 .981817 2.20 .368993 53.68 40 .973984 2.23 .209473 38.47 40 .981949 2.18 .372214 54.07 41 9.974118 2.23 11.211781 38.67 41 9.982080 2.20 11.375458 54.42 42 .974252 2 23 .214101 38.83 42 .982212 2.18 .378723 54.80 43 .974386 2.22 .216431 39.03 43 .982343 2.20 .382011 55.20 44 .974519 2.33 .218773 39.20 44 .982, 5 2.18 .385323 55.58 45 .974653 2.23 .221125 39.42 45 .982606 2.18 .388658 55.97 46 .974787 2 22 .223490 39.58 46 .982737 2.20 .392016 50.38 47 .974920 2 '.23 .225865 39.80 47 .982869 2.18 .395399 56.80 48 .975054 2.23 .228253 39.98 48 .983(X)0 2.18 .398807 57.20 49 .975188 2.22 .230652 40.18 49 ! .983131 2.18 .402239 57.62 50 .975321 2.23 .233063 40.38 50 .983262 2.20 .405696 58.07 51 9.975455 2.22 11.235486 40.58 51 9.983394 2.18 11.409180 58.48 52 .975588 2.23 .237921 40.78 52 .!s:i.V.'5 2.18 .412689 58.93 53 .975722 2.22 .240368 41.00 53 .983656 2.18 .416225 59.38 54 .975855 2.22 .242828 41.20 54 .983787 2.18 .419788 59.83 55 .975988 2.23 .245300 41.42 55 .983918 2.18 .423378 60.28 56 .976122 2.22 .247785 41.63 mi .984049 2.18 .426995 60.77 57 .976255 2.22 .250283 41.83 57' .984180 2.18 .430641 61.25 58 .976:388 2.22 .252793 42.07 58 .984311 2.18 .434316 61.73 59 .976521 2.22 .255317 42.28 59 .984442 2.18 .438020 62.22 60 9.976654 1 2.23 11.257854 42.52 00 9.984573 2.17 11.- 44 1753 62.73 [449] TABLE XXVI. LOGARITHMIC VEliSED SIGNS AND SECANTS. 88 89 / Vers. D.I". Ex. sec. q+l JL Vers. D.r. Ex. sec. q+l ' 15 29* 15. 30* o 9.984573 2.17 11.441753 9086 9.992354 2.13 11.750498 6801 i 984703 2.18 .445517 9215 1 .992482 2.15 .757925 6929 2 .984834 2.18 i .449311 9345 2 .992611 2.13 .765477 7056 3 .984965 2.18 .453137 9474 3 .992739 2.15 .773158 7184 4 .985096 2.17 .456994 9603 4 .992868 2.13 .780973 7312 5 .985226 2.18 .460883 9732 5 .992996 2.13 .788926 7440 6 .985357 2.17 .464805 9862 6 .993124 2.15 .797022 7567 7 .985487 2.18 .468761 9991 .993253 2.13 .805268 7695 8 .985618 2.17 .472751 4-120 8 .993381 2.13 .813668 7823 9 .985748 2.18 .476775 0249 n .9935(39 2.13 .822229 7950 10 .985879 2.17 .480834 0378 10 .993637 2.13 .830956 8078 11 9.986009 2.18 11.484929 0507 11 9.993765 2.15 11.839858 8205 12 .986140 2.17 .489061 0636 ]2 .993894 2.13 .848940 8333 13 .986270 2.17 .493230 0765 13 .994022 2.13 .858811 8460 14 .986400 2.18 .497437 0894 14 .9941(50 2.13 .867679 8588 15 .986531 2.17 .501683 1023 15 .994278 2.13 .877351 8715 16 .986661 2.17 .505968 1152 16 .994406 2.13 .887239 8843 17 .986791 2.17 .510293 1281 .994534 2.13 .897350 8970 18 .986921 2.17 .514659 1410 18 .994662 2.12 .907697 9097 19 .987051 2.17 .519066 1539 19 .994789 2.13 .918290 9225 20 .987181 2.17 .523516 1668 20 .994917 2.13 .929141 9352 21 9.987311 2.17 11.528010 1797 21 9.995045 2.13 11.940264 9479 22 .987441 2.17 .532548 1925 22 .995173 2.13 .951072 9607 23 .987571 2.17 .537131 2054 23 .995301 2.12 .963381 9734 24 .987701 2.17 .541760 2183 24 .995428 2.13 .975408 9862 25 .987831 2.17 .546437 2312 25 .995556- 2.12 11.987769 9988 26 .987961 2.17 .551161 2440 26 .995683 2.13 12.000485 -me 27 .988091 2.17 .555935 2569 27 .995811 2.13 .013578 0243 28 .988221 2.15 .560759 2698 28 .995939 2.12 .027069 0370 29 .988350 2.17 .565634 2826 29 .996066 2.12 .040984 0497 30 .988480 2.17 j .570561 2955 30 .996193 2.13 .055352 0624 31 9.988610 2.15 11.575542 3083 31 9.996321 2.12 12.070202 0751 32 .988739 2.17 ! .580578 3212 32 .996448 2.13 .085569 0878 33 .988869 2.15 .585670 3340 33 .996576 2.12 .101490 i 1005 34 .988998 2.17 .590819 3469 34 .996703 2.12 .118008 1132 35 .989128 2.15 .596027 3597 35 .996830 2.12 .135168 1259 36 .989257 2.17 / .601295 37'26 1 36 .996957 2.13 .153024 1386 37 .989387 2.15 .606625 3854 I 37 .997085 2.12 .171634 1513 38 .989516 2.17 .612018 3983 38 .997212 2.12 .191066 1640 39 .989646 2.15 .617475 4111 39 .997339 2.12 .211396 1767 40 .9897i'5 2.15 .622998 4239 40 .997466 2.12 .232712 1894 41 9.989904 2.17 11.628589 4368 41 9.997593 2.12 12.255116 2020 42 .990034 2.15 .634250 4496 42 .997720 2.12 .278723 2147 43 .990163 2.16 i .639982 4624 43 .997847 2.12 .303674 2274 44 .990292 2.15 ! .645788 4752. 44 .997974 2.12 .330129 2401 45 .990421 2.15 .651668 4881 45 .998101 2.12 .358285 2527 46 .990550 2.15 .657626 5009 46 .998228 2.12 .388375 2654 4? .990679 2.15 .663663 5137 47 .998355 2.10 .420686 2781 48 .990808 I 2.15 .669781 5265 48 .998481 2.12 .455575 2907 49 .990937 2.15 .675984 5393 49 .998608 2.12 .493490 i 3034 50 .991066 2.15 .682272 5521 50 .998735 2.12 .535009 ! 3161 51 9.991195 2.15 11.688649 5649 51 9.998862 2.10 12.580893 3287 52 .991324 2.15 .695117 5777 52 .998988 2.12 .632172 3414 53 .991453 2.15 .701679 5905 53 . 999115 2.10 .690291 3540 54 .991582 2.13 .708388 6033 54 .999241 2.12 .757364 ' 3667 55 .991710 2.15 .715097 6161 55 .999368 2.10 .a36672 ' 3793 56 .991839 2.15 .721958 6289 56 999494 2.12 12.933708 3920 57 .991968 2.13 1 .728925 6417 57 .999621 2.10 13.058774 4046 58 .992096 2.15 j .736002 6545 58 .999747 2.12 .234991 4172 59 .992225 2.15 1 .743192 6673 59 .999874 2.10 .536148 t 4299 60 9.992354 2.13 11.750498 6801 60 10.000000 2.10 Inf. pos. i 4426 15.30*!! 15.81* TABLE XX VII. -NAT URAL SINES AND COSINES. o i 1 2 3 4 Sine 'Cosin Sine Cosin Sine Cosin Sine Cosin Sine 1 Cosin ' "0 .00000! One. .01745 .99985 .03490 .99939 .05234 .99863 .06976!.997'56 ; 60 1 .00029 j One. .01774 .99984 .03519 .99938 .05203 .99861 .07005 .99754! 59 .00058; One. .01803 .99984 .03548 .99937 .05292 .99860 .07034 .99752 58 8 .00087! One. .01&32 .99983 .03577 .99930 .05321 .99858 .07063 .99750 57 4 .00116'! One. .01862 99983 .03006 . 99985 .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 .00204: One. .01949 .99981 .03(593 .1)9982 .05437 .99852 .07179 .99742 53 8 .00233 One. .01978 .1)9980 .03723 .99931 .05466 . 99851 ! .07208 .99740 52 9 . 00262 . One. .02007 .99980 .03752 .99930 .05495 . 99849 : .07237 .99738 51 10 . 00291 j One. .02036 .99979 .03781 .99929 .05524 .99847 .07266 .99736 50 11 .00320 '.99999 .02065 .99979! .03810 .99927 .05553 .99846 .07295 .99734 49 12 .00:349 .99999 .02094 .99978 .03839 .99920 .05582 .99844 .07324 .99731 48 13 .00378 .99999 .02123 .99977 .0:3868 .99925 .05611 .99842 .07353 .99729! 47 14 .00407 .9999!) .02152 .99977 .0:3897 .99924 .05640 .99841 .07382 .99727 46 15 .00436 .99999 .02181 .99970 .03926 .99923 .05669 .99839 .07411 .99725 45 16 .00465 .99999 .02211;. 9997 (5 .03955 .99902 .05698 .99838 .07440 .99723' 44 17 .00495 .99999 .02240 .99975 .08981 .99901 .05727 .99836 .07469 .99721 43 18 .00524 .99999 .02269 .99974 .04013 .9991!) .05750 .99834 .07498 .99719 42 19 .00553 .99998 .02298 .99974 .01012 .99918 .05785 .99833 .07527 .99716 41 20 .00582 .99998 .02327 .1)9973 .04071 .99917 .05814 .99831 .07556 .99714 40 21 .00611 .99998 .02356 .99972 i. 04 100 99916 .06844 .99829 .07585 .99712 39 22 .00640 .99998 .02385 [.99972 .04129 '.99915 .05873 .07614 .99710 38 23 .00669 .99998 .02414 .99971 .04159 .99913 .051)02 ! 99826 .(.7643 .99708 37 24 .00698 .99998 .02443 .99970 .04188 .99912; .05931 .99821 .07672 .99705 36 25 .007271.99997 .02472 .99969 i .04217 .99911 .05900 .99802 .07701 .99703 35 26 27 .00756 |.99997; .00785 .99997: .02501 [.99969 1.04246 .02530 .99968 .04275 .99910 .05989 .99909 .06018 .99821 .99819 .07730 .07759 .99701 .99699 34 33 28 .00814:. 99997' .02560 .99907! .04304 .99907 .00017 .99817 .07788 .99696! 32 29 .00844 .99990 .02589 .99900 .04333 .99900 .1)0070 .99815 .07817 .99( !94 31 30 .00873 j.99996 .02618 .999661 .04362 .99905 .06105 .99813 .07846 .99692 30 31 .00902 .99996 .02647 .99965 .04391 .99904' .06134 .99812 .07875 .99689 29 32 .009311.99996 .02676 .99904!!.04420 .99902' .06163 .99810 .07904 .99687 28 33 .00960 .99995 .02705 .999Go ; .04449 .99901 .06192 .99808 .07933 .99085 27 34 .00989 .99995 .02734 .99963 .04478 .99900 .00221 .99800 .07962 .99083 26 35 .01018 .99995 .02763 .99962 .04507 . 91)81)8 .0(5250 .99804 .07991 .99080 25 36 .01 047 '.99995 .02792 .99961 .04536 .99897 .00079 .99803 .08020 .99678 24 37 .01076 .99994 .02821 .99960 .04565 .99890 .06308 .99801 .08049 .99676 23 38 .01105 .99994 .02850 99959 .04594 .99894 .00887 .99799 .08078 .99673 22 39 .01134 .99994 .02879 ! 99959! .04623 .99898 .00306 .99797 .08107 .996711 21 40 .01164 .99993 .02908 .99958 .04653 .99892 .00395 .99795 .08136 .1)9008 20 41 .01193 .99993 .00988 .99957 .04682 .99890 .06424 .99793 .08165 .99666 19 42 .01222 .99993 .02%; .99956 .04711 .9988!) .06453 .99792 .08194 .99004 18 43 .01251 .99992 .0299(5 .99955 .04740 .99888 .00180 .99790 .08223 .99661 17 44 .01280 .999921 .03025 .99954 047T,'.) .99880 .06511 .99788 .08252 .99659 16 45 .01309 .99991 .03054 .99953 .0171)8 .99885 .06540 .99786 .08281 .99657 15 46 .01338 .99991 .08083 .99952 .01807 .99883 .06569 .99784 .08310 .99654 14 47 .013(57 .99991 .03112 .99952 .04856 1)9880 .06598 .99782 .08339 .99652! 13 48 .01396 .99990 .03141 .99951 .04885 .99881 .('007 .99780 .08368 .99049 12 49 .01425 .99990 .03170 .99950 .04914 .99879 .06656 .99778 .08397 .99047 11 50 .01454 .99989 .03199 .9991!) .04913 .99878 .06685 .99776 .08426 .99044 10 51 .01483 .99989 .03228 .99948 .0(972 .99870 ".06714 .99774 .08455 .99642 9 52 .01513 .99989 .03-257 .99947' .05001 .99875 .06743 .99772 .08484 .99039 8 53 .01542 .99988 .03286 .99910 .05080 .99873 .00773 .99770 .08513 .99637 7 54 I 01571 .99988 .03316 .99915 .05059 .99872 .06802 .99768 .08542 .991185 6 5S! .01600 .99987 .03345 .99944 .05088 .99870 .06831 .99766 .08571 .99(532 5 50 .0102!) .99987 .03374 .99913 .05117 .99869 .06860 .99764 .08600 .99630 4 57 j. 01658 .1)998(1 .08403 .99942 .05140 .99867 .06889L99702 .08629 .99627 3 58 .01687 .99980 .03432 .99911 .05175 ,!)98(i(; .069181.99760 .08658 .99005 2 59 .01716 .99985 .03461 .999(0 .05205 .99S04 .00917 .99758 .08687 .99000 1 H . .01745 .9998.-, .03490 .99989 .05-84 .!I9SO:5 .0097'0 .1)9750 .08716 .99619 Cosin Sine Cosin Sine ('<><; u Sine Cosin Sine Cosin "Sine 7 89 88 87 86* 85 TABLE XXVII. NATURAL SINES AND COSINES. | 5 6 T II e 9 i Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin f .08716 .99619 1 .087451.99617 .10453 .10482 . 99452 ! .99449' .12187 .12216 .99255 .1KW51 713917 . 13946 .99027 .99023 1 . 15643 j. 98769 .156721.98764 60 59 2 .08774 .99614 .10511 .99446 .12245 .99248 .18975 .99019 .15701!. 98760 58 3 .08803 .99612 .10540 .99443 .12274!. 99244 .14004 .99015 .15730 .98755 57 4 .08831 .99609 .10569 .99440 .12302 .99240 .14033 .99011 I .15758 .98751 56 5 .08860 .99607 .10597 .99437 .12331 .99237 .14061 .99006 ! .15787:. 98746 55 6 .08889 .99604 .10626 .99434 .12360 .99233 .14090!. 99002 .15816 .98741 54 7 .08918 99602 .10655 .99431; .12:389 .99230, .14119 .98998 .15845 .98737 53 8 .08947 .99599 .10684 .99428 .12418 .99226 .14148 .98994 .15873 .98732 52 9 .08976 '.99596 .10713 .99424 .12447 .99222' .14177 .98990 .15902 .98728 51 10: .09005 .99594 .10742 .99421 .12476 .99219 .14205 .98986 1 .15931 .98723 50 11 .09034 .99591 .10771 .99418 . 12504 ! . 99215! .14234 .98982 .15969 .98718 49 12 .09063 .99588 .10800 .99415 .12533!. 99211! .14263 .98978 L15988 .98714 48 13 ! .09092 .99586 .10829 .99412 .12562 .99208 .14292!. 98973 .16017 .98709 47 14 .09121 .99583 .10858 .99409 .12591 .99204 .14320 .98969 .16046 .98704 46 15 .09150 .99580 .10887 .99406 .12620 .99200 .14349 .98965 .160741.98700 45 16 I .09179 .99578 .10916 .99402 .12649 .99197 .14378 .98961 .16103 .98695 44 17 .09208 .99575 .10945 .99399 .12678 .99193 .14407 .98957 .16132 .98690 43 18 .09237 .99572 .10973!. 99396 .12706 .99189 .14436 .98953 .16160 .98686 42 19 .09266 .99570 .110021.99393 .127a5 .99186 .14464 . 98948 1.16189 .98681 41 20 .09295 .99567 .11031 .99390 .12764 .99182 .14493 .98944 .16218 .98676 40 21 .09324 .99564 .11060 . 99386 i .12793 .99178 .14522 .98940 ! .16246 .98671 39 22 .09353 .99562 .11089 .90383 .12822 .99175 .14551 .98936, .16275 .98667 38 23 ' .09382 .99559 .11118 .998801 .12851 .99171 .14580 .989311 .16304 .98662 37 24 | .09411 .99556 .11147 .99377 .12880 .99167 .14608 .98927 h. 16333 .98657' 36 25 j. 09440 .99553 .111761.99374 .12908 .99163 .14637 .98928 .16361 .98652 35 26 .094695.99551 .112051.99370 .12937 .99160 .14666 . 98919 j .16390 .98648 34 27 .09498 .99548 .1 1234 j. 99367 .12966 .99156 .14695 .98914 .16419 .98643 33 28 .09527 .99545 .11263 .99364 .12995 .99152 .147231.98910 ' .16447 .98638 32 29 '.09556 .99542 .11291 .99360 .13024 .99148 .14752 .98906 .16476 .98633 31 30 .09585 .99540 .11320 .99357 .13053 .99144 .14781 .98902 .16505 .98629 30 31 .09614 .99537 .11349 .99354 .13081 .99141 .14810 .98897 .16533 .98624 29 32 .09642 .99534 .11378 .99351 ' .13110 .99137 .14838 .98893! .16562 .98619 28 33 .09671 .99531 .11407 .99347 .13139 .99133 .14867L98889 .16591 .98614 27 34 .09700 .99528 .11436 .99344 .13168 .99129 .14896 .98884 i .16620 .98609 26 35 .09729 .99526 .11465 .99341 .13197 .99125 .14925 .98880 .16648 .98604 25 36 .09758 .99523 .11494 .99337 .13226 .99122 .14954 .98876 . 16677 .98600 24 37 .09787 .99520 .1 1523 j. 99334 .13254 .99118 .14982 .98871 :. 16706 .98595 23 38 .09816 99517 .11552 .99331 .13283 .99114 .15011 .98867 .16734 .98590 22 89 .09845 .99514 .11580 .99327 .13312 .99110 .15040 .98863 .16763 .98585 21 40 .09874 .99511 .11609 .99324! .13341 .99106 .15069 .98858 .16792 .98580 20 41 .09903 .99508 .11638 .99320 .13370 .99102 .15097 .98854 .16820 .98575 19 42 .09932 .99506 .11667 .99317: .13899 .99098 .15126 .98849 .16849 .98570 18 43 .09961 .99503 .11696 .99314 .13427 .99094 .15155 .98845 .16878 ,!)Nr,ti5 17 44 .09990 .99500 .11725 .99310 .13456 .99091 .15184 .98S41 .16906 .98561 16 45 .10019 .99497 .11754 99307 .13485 .99087 .15212 .98836 .16935 .98556 15 46 .10048 .99194 .11783 .99303 . 13514 .99083 .15241 .98832 .16964 .98551 14 47 .10077 .99491 .11812 .99300 .13543 .99079 .15270 .98827 .16992 .98546 13 48 .10106 .99488 .11840 .992971 .1:3572 .99075 .15299 .98823 .17021 .98541 12 49 .10135 .99485 .11869 .99293 .136001.99071 .15327 .98818 .17050 .9S.K5H 11 50 . 10164 .99482 .11898 .99290 .13629 .99067 .15356 .98814 j. 17078 j. 98531 10 51 .10192 .99479 .11927 .99286 .13658 .99063 .15.385 .98809 .17107 .98526 9 52 53 .10221 .10250 .99176 .99473 .11956 .99283 .11985 .99279 .13687 .13716 .99059 .1)9055 .1541 4!. 98805 .15442 .98800 .17136 .9S.V.M .171 64 i. 98516 8 7 54 .10279 .99470 .12014 .99276 .13744!. 99051 .15471 .'.lS79(i .17193 .98511 6 55 .10308 .99467 .12043 .99272 .13773 .99047 .15500 .98791 .17222 .9850(5 5 56 . 10337' .994(54 1 . 12071 .99269 . 13802 [ . 99043 .15529 .98787 .17250 .98501 4 57 10366 .99461 I .12100 .99265 .13831!. 99039 .15557 .98782 .17279 .98496 3 58 .10395 .99458 I .12129 .99262 .13860 1.99035 .15.586 .98778 j. 17308 .98491 2 59 .10424:. 99455 .12158 .WT>8 .18889 .99031 .1.-615 .98773 .17336 .98486 1 60 .10453 .'.)<) :>2 . 12187 ! . *Mxi-)-> .13917 .99027 .1 .-itf-13 .>S7r><) .17365 .1)8481 t Cosin Sine Cosin Sine Cosin Sine Cosin Sine i Cosin Sine f 84 83- 82 81 80 TABLE XXVII. NATURAL SINES AND COSINES. 1 10 || 11 12 ! 13 || 14 ' Sine Cosin Sine Cosin Sine Cosin Sine Cosin 1 Sine Cosin .17365 .98481 .19081 .98163 .207911.97815 .22495 . 97437 i .24192 .970:30 60 1 .17393 .98476 .19109 .98157 .20820 .97809 .22523 .97430 .24220 .97023 59 2 .17422 .98471 .19138 .98152 .20848 .97803 .22552 .97424 1 .24249 .97015 58 3 .17451 .98466 .19167 .98146 .20877 .97797 .22580 .97447 1 .24277 .97008 57 4 .17479 .98461 .19195 .98140 .20905 . 97791 : .22608 .97411 .24305 .97001 56 5 .17508 .98455 .19224 .98135 .20933 .97784 .22637 .97404 .24333 .96994 55 6 .17537 .98450 .19.252 .98129 .20962 .97778 .22665 .07398 .24362 .96987 54 .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 .24440 .96966 51 10 .17651 .98430 .193*56 .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 .22920 .973:38 .24615 .96923 45 16 .17823!. 983!):) .19538 .98073 .212415 .97717 . 22948 .97331 .24644 .96916 44 17 .17852|. 98391 .19566 .98087 .21275 .97711 .22977 .97325 .24672 .96909143 18 .17880 ! 98389 II. 19595 .1)8001 .21303 .97705 .23005 .97318 .24700 .96902! 42 19 .17909 .98383 .19623 .98050 .21831 .97098 .23033 .97311 .24728 .96894 41 20 .17937 [98378 .l')05--2 .98050, .21860 .97692 .23062 .97304 .2475(5 . 96887 40 21 .17966 .98373 .11)0*0 .98044 .21388 . 97686 ' .23090 .97298 .24784 .96880 39 22 .17995 ! 98368 .19709 .98039 .21417 .97'680 .23118 .97291 .24813 .!)687'3 38 23 . 18023 .98362 . 19737 .1)8033 .21445 .97673 .23146 .97284 .24841 .96866 37 24! .18052 ! 98357 .19700 98027 .21474 .97667 .23175 .97278 .24869 .11(5858 30 25 .18081 .98352 .19794 .98081 .2150:2 .97001 .23203 .97271 .24897 .9(5854 35 26 .18109 .1)8847 '.19823 .98016 .21530 .97655 .23231 .97264 .24925 .96844 34 27 .18138 .1)8341 .19851 .1)8:110 .21559 .97648 : -28-200 .97257 .241)54 .1 1(5837 33 28 .18166 .98336 .19880 .9801)4 .21587 .97042 .23288 .97251 .24982 9(5829 32 29 . 18195 .98331 .19908 979D8 .21616 .97636 .23316 . 97'24 4 .25010 .1)0822 31 30 .18224 .98825 .19937 .!)7!)'.)2 .21644 .97630 .23345 .97237 .25038 .96815 30 31 . 18252 .98320 .19965 .97987 .21072 . 97628 .23373 .97280 .25068 .96807 29 32 .18281 .1)8815 . 19!K)4 .1)71)81 .21701 .97017 .23401 .97223 .25094 .96800 28 33 . 18309 .1)8310 .20022 ()7'<)7"i > t **>!) .97(511 .23429 .97217 .25122 .96793 27 34 .18338 .98304 .20054 ! 97969 .21758 .1)7(501 .23458 .97210 .25151 .96786 26 35 .18307 .1)82!)!) .20,)7<) .979(33 .21786 .97598 .23186 .1)7203 .25179 .96778 25 36 ! 18895 .98294 .20108 .!)7!)58 .21814 ^ 97592 .23514 .!)71!)0 .25207 .96771 37 38 : . 18424 ; . 1)8288 . 20136 . 971)52 .21843 . 975S5 .18452 .1)8283 .201(55 1)71)40 .21871 .97'57!) .23542 .23571 .9718!) .25235 .1)718:2 .25263 .967'64 .96756 22 39 ; .181811.98277; .20193 .97940 .218iH) .1)7573 .23599 .97170 .25291 .96749 21 40 .97934 .21928 .97566 .23627 .97169 .25820 .!)07'12 20 41 .18538 .98267 .20250 .97938 .211)50 .1)7500 .23656 .97102 .25348 '.96734 19 42 ! 18567 1 98261 .20279 .!)7i22 .21985 .97553 .23684 .97155 .25376 .90727 18 43 . 185U5 .1)8-250 .20:5')7 .Ii7'.)10 .22013 .!);:,17 .23712 .97148 .'25104 .90719! 17 44 ! 18624 .982.50 .20330 l)7HI) .22041 .97511 .23740 .97141 .25432 .90712 16 45 . 18652 .1)8215 .20304 .1)71)05 .22070 ! 97534 .23709 .97134 .254(50 .D0705 15 46 .18681 .1)8240 .20:593 .!)7S1K) .2201)8 - .23797 .97127 .25488 .9(5(51)7 14 47 .18710 98231 .201-21 97X)3 .22126 . 97521 i .-.>88-,>5 97120 .25516 .9001)0 13 IS .18738 .1)8-2-2!) .20150 .1)7887 .22155 .Q?515 .23853 .97113 .25515 .96682 12 49 .18707 .98223 .20178 .35 .97869 .22240 .9741)0 .23938 97093 .25629 .96660 9 52 .188:,- .DS-207 .205(53 .97803 .22208 .9748!) .231)00 ! 1)708(5 .25(557 .96653 8 58 .48881 .'I8001 .-031)2 .D7857 .2221)7 .!)743 .-281)1)5 .!)7'071) .25085 .90045 7 54 .18910 .98190 .200.2*.) .97851 .22325 .1)747(5 .240-28 ,97'07-2 .23713 .90038 56 .181)88 .1)811)0 .2004!) .97815 .22353 .9717'0 .21051 .97005 .25741 .9(5030 5 56 .18967 .11818.-, .-J0077 .97881) .2-238-2 .1)74(53 21079 .97058 .25709 .90(523 4 57 . 18995 , . 98179 1 . 20706 . 97833 .22410 .97457 .24108 .97051 .25798 .96615 3 58 .19024 .98174 .207:34 .97827 .22438 .1)7150 .24136 .970-44 .25826 .96608 2 59 .19052 .1)8108 .307(53 .!)7821 2-2107 1)7141 .211(54 .!)7'037 .25854 .96600 1 .11)081 .!)SKi8 .207'!)! .1)7815 .2-211)5 .97437 .21192 .1)7030 .25882 .96593 Cosin Sine : Cosin Sine Cosin Sine Cosin Sine Cosin Sine ~ , 79 ii 78 77 76 75^ [453] TABLE XXVII. NATURAL SINES AND COSINES. 15" 16 17 |i 18 19 1 Sine j Cosin jj Sine Cosin Sine Cosin Sine Cosin Sine Cosin i .25882!. 96593 .27564 .96126 .29237 .95630 .30902 .95106' .32557 .94552 60 1 .25910 '.96585 .27592 .96118 .29265 .95622 .30929 .95097 .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 .96563 .27676 .96094 ; .29:348 .95596 .31012 .95070 .32667 .94514 56 5 .26022 .96555 .27704 .96086; .29376 .95588 .31040 .95061 .32694 .94504 55 6 .26050 .96547 .27731 .96078! 1.29404 .95579 .31068 .95052 .32722 .94495 54 7 .26079 .96540 .27759 96070 li .29432 .95571 .31095 .95043 .32749 .94485 53 8 .26107i.96532 .27787 .96062 .29460 .95562 .31123 .95033 .32777 .94476 52 9 .26135 .96524 .27815 .96054 i .29487 .95554 .31151 .95024 .32804 .94466 51 10 .26163 .96517 .27843 .96046 j .29515 .95545 .31178 .95015 .32832 .94457 50 11 1. 26191 '. 96509 ! .27871 . 96037 ! . 29543 . 95536 ' . 31206 .95006 .32859 .94447 49 12 1. 26219 .96502 .27899 .960291 .29571 .H.VW8 .31233 .94997 .32887 .94438! 48 13 .26247 .96494 .27927 .96021 .29599 .95519 .31261 .94988 .32914 .94428 ! 47 141.26275 .96486 .27955 .96013 .29626 . 95511 i ! . 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 .263591.96463 .28039 .95989 .297101.95485 i .31372 .94952 .33024 .94390 43 18 .26387 .96456 ! .28067 .95981 .29737 .95476 i .31399 .94943 .33051 .94380: 42 19 .26415;. 96448 .28095 .95972 .29765 .95467!! .31427 .94933 i .33079 .94370 41 20 . 26443 j. 96440 .28123 .95964 .29793 .95459 .31454 .94924 .33106 .94361 40 21 .26471 .96433 .28150 .95956 i .29821 .95450 .31482 .94915 " .33134 .94351 39 22 .26500 .96425 .28178 . 95948 li. 29849 .95441 .31510 .94906 .38161 .94:342 38 23 . 26528 ! . 96417 .28200 .95940 i. 29876 .95433 .31537 .94897 :. 33189 .94332 37 24 .26556 .96410 .28234 .95931 1 .29904 .95424 .31565 .94888 .33216 .94322 36 25 .26584 .96402 .;.'S-.'(-J .95923 .29932 .95415 .31593 .94878 i .33244 .94313 35 26 .266121.96394 .28290 .95915 .29960 .95407 1' .31620 .94869 .33271 .94303 34 27 .26640 .96386 .28318 .95907 .299871.95398 .31648 .94860 .33298!. 94293 33 28 .26668 .96379 .28846 .95898 i.30015 .95389; .31675 .94851 .3*320 94284 32 29 .J<;r,% .96371 .28374 .95890 .30043 .95380 .31703 .94842 .3335:3 .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 . 963-17 .28457 .95865 ! .30126 .95354 .317861.94814 : . 33436 .94245! 28 33 34 .26808 :. 9(5340 .26836 .96332 .28485 .28513 .95857 .95849 .30154 .30182 .95345 .31813 .95337 .31841 .94805 .94795 -33463;. 94235 27 :. 33490 i. 942251 26 35 .26864 .96834 .28541 .95841 .30209 453281 .31868 .94786 I. 33518 .94215 25 36 .26892 .96316 .28569 .95832 .30237 .95319: .31896 .94777 (.33545 .94206 24 37 .26920 .96308 . 28597 i. 95824 .30265 .95310 .31923 .94768 .33573 .94196 23 38 1 .26948 .96301 .28625 .95816 .30292 .95301 .31951 .94758 .33600 .94186 22 39 .20976 .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 .27060j.96269; .28736 .95782 .30403 .95266i .32061 .94721 .33710 .94147 18 43 .270881.96261! .28764 .95774 .'30431 .9525711.32089 .94712 .38737 .94137 17 44 .27116 .96253 .28792 .95766 .30459 .95248 ; .32116 .94702 .33764 .94127 16 45 .27144 .96240 .28820 .95757 .30486 .95240 i .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 .338461.94098 13 48 .27228 .96222 .28903 .95732 .30570 .95213 .32227 .94665 !. 33874 '.94088 12 49 .27256 .96214 .28931 .95724 1.30597 '.95204 .32254 .94656 i 33901 .94078 11 50 .27284 .96206^ '.28959 .95715 1 .30625 .95195 .32282 .94646 -.33929 .94068 10 51 .27312 .96198 .28987 .95707 1 .30653 .95186 .32309 .94637 L 33956 L 94058 9 52 .27340 .96190 .29015 .95698 : .30680 .951 77 i .32337 .94627 .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 i .32419 .94599 .34065 .94019 5 56 .27452 .96158 .29126 .95664 .30791 .95142! .32447 . 94590 .34093 .94009 4 ! 57 .27480 .96150 .29154 .9565(5 .30819 .95133 .32474 .94580 .34120 .93999 3 58 .27508 .96142 .29182 .95647 .308461. 95124: .32502 .94571 .34147 .93989 2 59 .27536 .96134 .29209 .95639 .30874 .951 15 ; .32529 .94561 .341751.93979 1 60 .27564 .96126 .29237 .95630 .30902 .95106 .32357 94552 .34202 .93969 \ Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine t 74 73 72 71 70 [454] TABLE XXVII.-NATURAL SINES AND COSINES.. 20 21 22 ! 23 24 | j Sine Cosin i Sine Cosin Sine Cosin Sine Cosin Sine Cosin .34202 .93969 735837 .93358 .37461 .1)2718 .39073 .92050 740674 .91355 60 1 .34229 .93959; .35864 .93348 .37488 .1)2707 .39100 .92039 .40700 .91343 59 2 .34257 .93949 .35891 .93337 .37515 .92697 .39127' .92028 .40727 .91331 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 .39234 .91982' .40833 .91283 54 7 .34393 .93899 .36027 .93285 .37649 .92642 .39260 . 91971 j .40860 .91272 53 8 .34421 .93889 .36054 .9327'4 .37676 .92681 .39287 .91959 .40886 .91260 52 9 .34448 .93879 .36081 .93264 .37703 .92620 .39314 .91948 .40913 .91248 51 10 ;. 34475 .93869 .36108 .93253 .37730 .92609 .39341 .91936 .40939 .91236 50 11 .34503 .93859 ! .36135 .93243 .37757 .92598 .39367 .91925 .40966 .91224 49 12 .34530 .93849 .36162 . 93232 .37784 .92587 .39394 .91914 .40992 .91212 48 i 13 .34557 .93839 .36190 .93222 .37811 .92576 .39421 .91902 .41019 .91200 47 14 .34584 .36217 .93211 .37838 .92565 .39448 .91891 .410451.91188 46 15 .341)12 ; 93819 .36244 .93201 .37865 .92554 .39474 .91879 .41072 .91176 45 16 .34639 .93809 ; .36271 .93190 .37892 .92543 .39501 .91868 .41098 .91164 44 ' 17 ! .34666 .93799 .36298 .93180 .37919 .92532 .39528 .91856 .41125 .91152 43 18 .34694 .93789 .36325 .93169 .37946 .92521 .39555 .91845 .41151 .91140 42 19 .34721 .93779 .36352 .93159 .37973 .92510 .39581 .91833 .41178 .91128 41 20 .31718 .93769 .36379 .93148 .37999 .92499 .39608 .91822 .41204 .91116 40 21 .34775 .93759 .36406 .93137 .38026 .92488 .39635 .91810 .41231 .91104 39 22 .34803 .93748; .36434 .93127 .38053 .92477 .39(561 .1)1791) .41257 .91092 38 23 .34830 .93738 .36461 .98116 .38080 .92 166 .39688 .91787 1 .41284 .91080 37 24 .34857 .93728 .36488 .93106 .38107 .92155 .39715 . 91775 1| .41810 .1)1068 36 25 .34884 .93718 .36515 .93095 .38134 .92444 .39741 .91764 .41337 .91056 35 26 .34912 .93708 .36542 .D30S4 .38161 .112132 .39768 .91752 .41363 .91044 34 27 .34939 .93698 .36569 .93074 .38188 .92421 .31)795 .91741 .41390 .91032 33 28 .34960 .93688 .36596 .93063 .38215 .92410 .39822 .91729 .41416 .1)1020 32 29 '! i!)i)3 .93677 .36623 .93052 .38241 .92399 .39H4H .91718 .41443 .91008 31 30 .35021 .93667 j .36650 .93042 ; .38268 .92388 .39875 .91706 i. 41469 ^90996 80 31 .35048 .9365? .36677 .93031 .38295 .92377 .39902 .91694 .41496 .90984 29 32 .35075 .93647 .36704 .930201' .38322 .92300 .31)1)28 .91 OS- 3 .41522 .90972 28 33 .35102 .93637 .36731 .93010 38341 ( .92355 .35)955 .91671 .41549 .90960 27 34 .35130 . 93626 .36758 .92999 138376 .92343 .39982 .91660 .41575 .90948 26 38 .35157 .93616 .36785 .92! IKS .38403 92332 .40008 .91648 .41602 .90936 25 36 .35184 .93606 .36812 .92978 .38430 .92321 .40035 .91636 .41628 .90924 2 37 .35211 93596 .36839 .1)21)67 .38456 .92310 .40062 .91625 .41655 .90911 28 38 .3523!) .93585 .36867 .92956 .38483 192299 .40088 .91613 .41681 .90899 22 89 .35266 .93575 .36894 .92945 .38510 .92287 .40115 .91601 .41707 .90887 21 40 .35293 .93565 .36921 .92935 .38537 .92276 .40141 .91590 .41734 .90875 20 ! 41 .35320 .93555 1 .36948 ooc^oj 38564 .92265 .40168 .91578 .41760 .90863 19 42 .35347 .93544 .36975 ! 92913 ! 88591 .92254 .40195 .91566 .41787 .90851 18 43 .3537'5 .93534 .37002 .92902 .38617 .1)2213 .40221 .91555 .41813 .90839 17 44 .35402 .93524 .37029 .92892 .38641 .92231 .40248 .91543 .41840 .90826 16 45 .35429 .93514 ! .37056 .92881 .38671 92220 .40275 .91531 .41866 .90814 15 46 .35456 .93503 ! 87083 .92870 .38698 11 12209 .40301 .91519 .41892 .90802 14 47 .35484 .93493 .37110 .92859 .38725 .92198 .40328 .91508 .41919 .90790 13 48 .35511 .93483 .37137 ! 92849 .38752 .92186 , .4U355 .91496 .41945 .90778 12 49 .35538 .93472 .37164 .92838 .3S7TS .92175 .40381 .1)1484 .41972 .90766 11 50 .35565 .93462 .37191 192827 .38805 .92164 .40408 .91472 .41998 .90753 10 51 .35592 .93452 .37218 .92816 .38832 .92152 i .40434 .91461 .42024 .90741 9 52 .35619 .93441 .37245 .92S05 .:-5S8->9 .92141 .40461 .1)1449 : .42051 .90729 8 53 .35647 .93431 .37272 .92794 .38886 .92130'' .40488 .1)1437 .42077 .90717 54 .35674 .93420 .37299 .92784 .38912 .92119 1 .40514 .91425 : .42104 .90704 6 55 .35701 .93410 .37326 .92773 .38! 139 .1)2107 .40541 .91414 i .42130 .90692 5 56 .35728 .93400 .37353 927ti2 381166 .92096 .40507 .91402 .42156 .90680 4 57 .357'55 .93389 .37:380 192751 138993 .92085 .405! It .91390 ! .42183 .90668 3 58 35782 . 93379 .37407 .92740 .3! 1020 .92073 .40*321 .91378 .42209 .90655 2 59 .35810 .93368 .37434 .92729 .311016 .1)2062 .40647 .91366 .42235 .90643 1 (50 .35837 .93358 .37461 .92718 .31)073 .92050 .40674 .1)1355 .42262 .90631 ~ ! Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine ; 69 68 67 66 65 J4SS] TABLE XXVII. NATURAL SINES AND COSINES. 25 I 26 27 28 29 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin .42262 "90631 .43837 789879 745399 .89101 .46947 788295 748481 787462 60 1 .42288 .90618 .43863 .89867 .45425 .89087 .4697'3 .88281 .48506 .87448 59 2 .42315 .90606 .43889 .89854 .45451 .89074 .46999 .88267 .48532 .87434 58 3 .42341 .90594 .43916 .89841 .45477 .89061 .47024 .88254 . 48557 j. 87420 4 . 42367 i. 90582 .43942 .89828 .45503 .890481 .47050 .88240 .48583 .87406 56 5 ' .42394 .'(O.VJ'J .43968 .89816 .45529 .890:35 .47076 .8-8226 .48608 .87391 55 6 1 .42420 .90557 i .43994 .89803 .45654 .89021 .47101 .88213 .48634 .87377 54 7 i .42446 .90545 .44020 .89790 .45580 .89008 .47127 .88199 .48659 .87363 53 8 ! .42473 .'XI532 < .44046 .89777 .45606 .88995 .47153 .88185 .48684;. 87349 52 9 : .42499 .90520 1 .44072 .89764 .45632 .88981 .4717'8l.88172 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 :. 47204 .88158 .48735 .87321 50 11 .42552 .90495 1.44124 '.89789 .45684 .88955 .47229 .88144 .48761 .87306 49 12 .42578!. 90483 .44151 .89726 .45710 .88942 .47255 .88130 .48786 ! 87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 .47281 .88117 .48811 .87278 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .872(54 46 15 .42657 .90446 .44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 .47:383 .88062 .48913 .87221 43 18 .42736 .90408 !. 44307 .89649 .45865 .88862 .47409 i. 88048 .48938 .87.207 42 19 .42762 .90396 .44333 .89636: .45891 .88848 . 47434 ! . 88034 .48964 .87193: 41 20 .42788 .90383 .44359 .89623 .45917 .88835 .47460j.88030 .48989 .87178 40 21 .42815 .90371 .44385 .89610 .45942 .88822 .47486 .88006 .49014 .87164 39 22 1 .42841 .90358 .44411 .89597! .45968 . 88808 . 4751 1 . 87993 i . 49040 i . 8*150 38 23 i .42867 .90346 '.44437 .89584! .45994 .88795 .47537' .87979 .49065 .87136 37 24 ! .42894 .903=34 .44464 .89571' .46020 .88782 .47562 .87965 .49090 .87121 36 25 .42920 .90321 .44490 .89558 .46046 .88768 1.475881.87951 .49116 .87107 35 26 .42946 .90309 .44516 .89545 .46072 .88755 .47614:. 87937 .49141 ,87093 34 27 .42972 .90296 .44542 .89532 .46097 .88741 .47 639 !. 87923 .49166 .87079 33 28 .42999 .90284 .44568 .89519 .46123 .88728 .47665 ! .87909 .49192 .87064 32 29 .43025 .90271 .44594 .89506 .46149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 ' .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .90233 .44672 .89467 .46226 .88674 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44698 ,.89464 .46252 .88661 .47793 .87840 .49318 ,86993 27 34 .43156 .90203 .44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 26 35 .43182 .901SK5 .447501.89428 .46304 .8863-4 .47844 .8781 2 : .49369 .86964 25 36 .43209 .90183 .44776 .89415 .46330 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .448021.89402 .46355 .88607 .47895 .87784 .49419 .86935 23 38 .43261!. 90 15S . 44828 j. 89389 '.46381 .88593 .47920 .87770 .49445 .86921 22 90 .43287 .90146 .44854 ,.89376 .46407 .88580 .47946 .87756 .49470 .86906 21 40 .43313 .90133 .44880J.89363 .46433 .88566 .47971 .87743 .49495 .86892 20 41 .43340 .90120 .44906 .89350 .46458 .88553 .47997 .87729 .49521 .86878 19 42 .43366 .90108 .44932 .89337 .46484 .88539 .480221.87715 .49546 .86863 18 43 .43392^90095 .44958 .89324 .46510 .88526 .480481.87701 .49571 .86849 17 44 .434181.90082 .44984 .89311 .46536 .885121 .48073 .87687 .49596 .86834 16 45 .43445 1.90070 .45010 .89298 .46561 .88499 .48099 .87673 .49622 .86820 15 46 .43471 .90057 .450361.89285 .465871.88485(1 .48124 .87659 .49647 .86805 14 .43497 .90045 .45062 .89272 .46613 .88472 .48150 .87645 .49672 .86791 13 l 48 .43523 .90032 .45088 .89259 .46639 L 88458 .48175 '.871m .49697 . 86777 12 49 .43549 .90019 .45114 .89245 . 46664 . 88445 . 48201 1 . 87617 .49723 .86762 11 50 .43575 .90007 .45140 .89232 .46690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 9 52 .43628 .899811 .45192 . 8920(i : .46742 .Sj(!t .48277 .87575 .49798 .86719 8 53 .43654 .89968 .45218 .89193 .46767 .88390 .48303 .87561 .49824 .86704 7 54 .43680 .89956 .45243 .89180 .46793 .88377 .48328 .87546 .49849 .86690'' 6 55 .43706 .89943 .45269 .46819 .HN363 .48.354 .87532 .49874 .86675 5 56 .43733 .89930 .45295 ! 89153 .46844 .88345) .48379 .87518 .49899 .86661 4 57 .43759 .89918i .45321 . 89140 .46870 ,88336.i .48405 .87504 .49924 .86646 3 58 .43785 .89905 .45347 .89127 .46896 .88322 i .48430 .87490 .49950 .86632 2 59 . 4381 li. 89892 .45373 .89114 .46921 .HS.-JIIK : .48456 .87476 .49975 .86617 1 60 .43837 .89879 .45399 .89101 .46947 .881295 .48481 .87462 .50000 .86603 f Cosin i Sine Cosin Sine Cosin Sine Cosin | Sine Cosin Sine" ' 64 63 62 61 60 [456] TABLE XXVII. NATURxVL SINES AND COSINEb. 30 |l 31 32 | 33 1 34 Sine Cosin Sine Cosin Sine Cosin Sine i Cosin Sine Cosiu o .50000 '.86603 .51504 .85717 .52992 .84805 .54464 83867 .55919 .S-2'.IOI 60 l .50025 .86588 .51529 .85702 .53017 .84789 .54488 83851 .55943 .S2S87 59 2 .50050 .86573 .51554 .85687 .53041 .84774 .54513 838351 .559(58 .82871 58 3 .50076 .86559 .51579 .85672 .530(56 .84759 .54537 83819 .55992 .S-2S55 57 4 .50101 .86544 .51604 .85657; :. 5:5091 .84743 .54561 .83804 .560161.82839 56 5 .50126 .86530 .51628 .85642 .Ml 15 .84728 .54586 .83788 .56040 .S2S-22 55 6 .50151 .86515 .51653 .85627 .53140 .84712 .54610 .83772 .56064 .82806 54 7 .50176 .86501 .51678 .85612 .53164 .84697 .54635 S375(5 .56088 .82790 53 8 .50201 .86486 .51703 .85597 .53189 .84681 .54659 .83740 .56112 .82773 52 9 .50227 .86471 .51728 .85582 I 53214 .84666 .54683 .83724 .56136 .82757 51 10 .50252 .86457 .51753 .85567 .532=38 .84650 .54708 .837u8 .56160 .82741 50 11 .50277 .86442 .51778 .85551 .53263 .84635" . 54732 ! .83692' .56184 .82724 49 12 .50302 .86427 .51803 . S.V>:](i .53288 .84619 .54756 .S3676 .5(5-21 IS .S270S 48 13 .50327 .86413 .51828 .85521 .53312 .84604 .54781! .83(560 .56232 .S2692 47 14 .50352 .86398 .51852 .85506 . 53337 i. 84588 .54805, .S3615 .5(525(5 .82675 46 15 .50377 .86:384 .51877 .85191 .53361 .84573 .54829| .83629 .5(5'2SO .S-2059 45 16 .50403 .86369 .51902 .8547(5 .53:386 .84557 .54854' .83(513 .56305 .82643 44 17 .50428 .86.354 .519-27 .85461 .53411 .84542 .54878 .83597 .5(5329 .S-2(5-2(5 43 | 18 .50453 .8(5340 .51952 .85446 .53435 .84526 .5 190-2 .83581 .5(5353 .S-2010 42 19 .50478 S(i:-J25 .511)77 .85431 .534(50 .84511 .54927 .835(55 .5(5377 .S-259.", 41 20 .50503 .86310 .52002 .85416 .53484 .84495 .54951 .83549 .56401 .82577 40 21 .50528 .86395 .52026 .85401 .53509 .84480 .54975 .83533 .56425 .82561 39 22 .50553 .86281 .52051 .85385 .53534 .84464 .54999 183517 .56449 .S25J4 38 23 .5057S .8(52(5(5 .52076 1 85370 .53558 .84448 .55021 .83501 .56173 S'-NV'S 37 21 .50603 .86-251 .52101 .85355 1535S3 .84433 .55048 1S34S5 .56497 1 82511 36 >.-, "(M528 .86237 .52126 .85340 .53(507 .84417 .55072 .S3 4(59 .5(55-21 .82 1'.5 35 26 ! 50654 .86222 .52151 .85325 .53632 .84402 .55097! .83453 .5(55-45 .8-2 17S 34 27 .50679 .86207 .52175 .85310 .53656 .84386 .551-21 .83437 .56569 .S-2K52 33 28 .50704 .86192 .52200 .85294 .53681 .84370 .55145 83421 .56593 .8.211(5 32 29 .50729 .86178 .52225 .85279 .53705 .84355 .551(59 .8.-5405 .56617 .S-21-20 31 30 .50754 .86163 .52250 .85264 .537:30 .81339 .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 183356 28 33 .50829 .86119 .52324 .85218 .53804 .8129-2 .55266 .83340 156713 Is2:5(53 27 34 .50854 .86104 .52349 185203 .53828 .84277 .55291 .83321 .50730 .8:231 7 26 35 .50879 .86089 .52374 .85188 .53853 .84261 .55315 .83308 .56760.82380 25 36 .50904 .86074 .52399 .85173 .538771.84245 .55339 .S3292 .567S4 .S2314 24 37 .50929 .86059 .52423 .85157 .53902 .84230 .553(53 .S3276 .56808 L82297 23 38 .50954 .8(5015 .52448 .85142 .53926 .81-214 .553SS 832(50 5(5832 1.82281 22 39 .5097!) .86030 .52473 '.85127 .53951 .84198 .5541-2 .s:;-jH .5C,S5(5 .S2.264 21 40 .51004 .86015 .52498 .S5112 .53975 .84182 .55436 183238 .56880 .82248 20 41 .51029 .86000 .52522 .85090 .54000 '.84167 .554(50 .83212 .5(5901 .S223I 19 42 . 51054 I.S59S5 .52547 .S50S1 .54024 .84151 155484 .S3 195 .5(5928 .S2214 IS 43 .51079 .85970 .52572 .850(5(5 .54049 .84135 .55509 S3 179 .56952 S21!l* 17 44 .51104 .S595U .52597 .85051 .54073 .84120 .83163 .569761.82181 16 45 .51129 .85941 .52621 .85035 .54097 .84104 155657 S3 147 .57000 .82165 15 46 .51154 .S59-2I) .52646 .85020 .54122 .84088 .55581 .83131 .57024 .S21 IS 14 47 .51179 .85911 .53671 .85005 .54146 .84072 .55(505 .83115 .57047 .8213-2 13 48 .51204 .S5S9U .52696 .84989 .54171 .84057 .55630 .83098 .57071 .82115 12 49 .51229 .S.-.SM .5-27-20 .84974 .54195 .81041 .55(554 .S30S2 .57095 8-2098 11 50 .51254 .85866 .52745 .84959 .54220 1.84025 .55678 .83066 .57119 .82082 10 51 .51279 .85851 .52770 .84943 .54244 . 84009 : .55702 .83050 57143 .82065 9 52 .51304 .S5S36 .527941.84928 .54209 .s::<)<)4 .55726 .83034 .5 71 (57 .82048 , 8 53 .51329 .85821 .52819 .S 101 M .54293 .83978 .55750 .83017 .57191 .S-203-2 7 54 .51354 .85806 .5-2*11 .SJS97 .54317 .8:59(52 .55775 .83001 .57215 .82015 6 55 .51379 .85792 .52869 .S4SS2 .54342 .83916 .55799 .S29S5 .57238 .81999 5 56 .51404 .85777 .52893 .848(5(5 .54366 .83930 : .55823 .829(59 .5720-2 .81982 4 57 .51429 .85762 .52918 .84851 .54391 .83915 .55847 .S2'.).V, .572845 .81965 3 58 .51454 .85747 .52943 .54415 S3S'.I9 155871 .S2930 .57310 .81949 2 5!) .51479 .85732 60 i . 51504 ! . 85717 .52967 .84820 .52992 .84805 .:,i in .54464 .83883 .83867 .55895 .55919 .82920 .57334 .*29iU .57358 .81932 1 .81915 | Cosin Sine CosTn Sine' Cosin Sine Cosin Sine Cosin | Sine I ,- 59 58 57 56 55 I '[457] TABLE XXVir. -NATURAL SIXES AND COSINES. 35 36 37 38 M / Sine iCcsin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ~0 117358 j 781915 .58779 .80902 .60182 .79864 761566 778801 .62932 .77715 60 1 I .57381 .81899 .58802 .80885: .60205!. 79846 .61589 .78783 .62955 .77696 591 2 : .57405 .81882 .58826 .80867 .60228 .79829 .61612 .78765J .62977 .77678 58 3 ; .57429 .81865 .58849 .80850 ! 60251 .79811 .61635 .78747' .63000 .77660 57 4 1.57453 .81848; .58873 .80888, .60274 .79793 .61658 .78729! .63022 .77641 56 6 | .57477 .81832: .58896 . 80816 i .60298 .79776 .61681 .78711 .63045 .77623 55 6 .57501 .81815! .58920 . 80799 : .60321 .79758 .61704 .786941 .63068 .77605 54 7 .57524 . 81798 ' .58943 .80782; .60344 .797411 .61726 . 78676 ! .63090 .77586 53 8 i .57548 .81782 .58967 .80765 .60367 .79723 .61749 .78658 .63113 .77568 52 9 .57572 .81765' .58990 .80748 i .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 49 12 .57643 .81714 .59061 .80696 .604601.79653 .61841 .78586; .63203 .77494 48 , 13 .57667 .81698 .59084 . 80679 : .60483 .79635 .01864 .78568 .63225 .77476 47; 14 .57691 .81681 .59108 .80662 .60506 .7961 8 ! .61887 .78550 .63248 .77458 46 15 .57715 .81664 .59131 .80(544 .60529 .79600 .61909 .78532 .63271 .77439 45 ! 16 .57738 .81647 .59154 .80627 .60553 .79583 .61932 .78514 .63293 .77421 44 17 .57762 .81631 .59178 .80610 .60576 .79565 .61955 .78496 .6:3316 .77402 43 18 .57786 .81614 .59201 .80593 .60599 .79547' .61978 .78478 .63338 .77384 42 19 .57810 .81597 .59225 .80576 .60622 .79680 .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 ' .55)295 .80524 .60(591 .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 .7&S69 .63473 .77273 36 25 .57952 .81496 .59365 .80472 .60761 .79424 ! .62138 .78351! .63496 .77255 35 26 .57976 .81479! .59:589 .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 .782971 .63563 .77199 32 29 . 58047 j. 81428 ''' .59459 .80403 60853 .7985;* .62229 .78279 .63585 .77181 31 30 .58070 .81412! .59482 .80386 .608761.79335 .62251 .78261 .63608 .77162 30 31 .58094 .81 395 ! .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 29 32 .581181.81378; .59529 .808511 .60922 .79300 .62297 .78225 .63653 .77125 28 33 .58141 .S13(il! .59552 .80334 .609451.79282 .62320 .78206 .63675 .77107 27 34 '.58165 1.81344' .59576 .80316 .60968 .78864 .62342 .78188 .63698 .77088 26 35 .58189i.81327 .59599 .80299 .60991 ; 792 17 .62365 .78170 .63720 .77070 25 36 .58212 .81310 .59622 .H02H2 j .61015 .79229! .62388 .78152 .63742 .77051 24 37 .58236 .81293 .59646 .80264 i .61038,. 79211 .62411 .78184' .63765 .77033 23 38 .58260 .8127(5 .59669 .80247 ! .61 (Mil .7911)3 .68488.78116 .(53787 .77014 22 39 '.58288 .81259 .59693 80230 .01084 .791 7(5 .6245(5 .78098 .63810 .76996 21 40 .58307 .81242 .59710 .80212 j . 61107!. 79158 ;; .62479 .78079 .63832 .76977 20 41 ! .58330 .81225^ .59739 . 80195 ! . 61 130 ! . 79140 ' . 62502 . 78061 .6:3854 .76959 19 42 ; .58354 .81208 ! .59763 .80178 .61153 .79122 .62524 .78043 ! .6:3877 .76940 18 43 : .58878 .81191 .59786 .80160 .61176 .791 05 .62547 .78025 .6:3899 .76921 1 17 ! 44 ; . 58401 1. 81 174 j .59809 .80143 .61199 .79087 .62,570 .78007 .63922 .76903 I 16 45 .58425 .81157, .59832 .80125 .61222 .79069 .62592 .77988 .63944 .76884 1 15 46 . 58449 j. 811401 .59856 .80108 .61245 j.79051 .62615 .77970 1 .63966 .76866 14 ; 47 !. 58472 1.81 123 .59879 .80091 .61268:. 79033J .62638 .77952 .63989 .76847 13 1 48 .58496 .81106 .59902 .80073' . 61291 .79016 . 62(560 . 77934 .64011 .76828 12 49 i .58519 .81089 .59926 .80056 . 61314 '.78!)<>S .02(583" .77916 .64033 .76810 11 50 .58543 .81072; .59949 .80038 .61337 '.78980 .62706 .77897 .64056 .76791 10 i 1 j i 51 .58567 .81055 .59972 .80021 .61360 .78962 . 62728 !. 77879 .64078 .76772 9i 52 .58590 .81038 .59995 80003 .61383 .78944 .62751 .77861 .64100 .76754 8 53 ! .58614 .810211 .60019 .79986 .61406 .78926 .62774 .77843 .64123 .76735 7'! 54 .58637 .81004 .60042 .79968 .61429 .78908!^ .62796 .77824 .64145 .76717 6 55 .58661 . 80987 ; .60065 .79951 .61451 .78891 .62819!. 77806 .64167 .76698 ! 5 56 .58684 .80970 .60089 .799:34 .61474 .78873 .62842 .77788 .64190 .76679 4 57 .58708 .80953 .60112 .79916 .61497 .?'8855 .62864 .77769 .64212 .76661 3 58 .58731 .80936 i .60135 .79899 | .615201.78837 .62887 .77751i .64234 .76642 2 59 .58755 .80919 .60158 .79881 i. 61543 . 7881 9 .62909 .77733!! .64256 .7(5623 1 60 .58779 .80902 .60182 .79S64 .61566 .78801 | .62932 .77715 .64279 .76604 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine \ 54 53 52 51' 50 [458] TABLE XXVII. NATURAL SIXES AND COSINES. 40 II 41 42 43 44 ' Sine ('osiii Sine Cosin Si)i< Cosin I Sine Cosin Sine Cosin / ~0 .84279 776604 70 53 8 .(54157 .76456 .65781 .75318 .67088 .74159 .68370 .72976 .69633 ! 71772 52 9 .64479 .76436 .65803 .75299 .67107 .74139 .68391 .72957 .69654 .71752 51 10 .64501 .76417 .65825 .75280 .67129 .74120 .68412 .72937 .69675 .71732 50 11 .64524 .7639811.65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 49 12 .61546 .76380; .65839 .75241 .67172 .74080 .68455 .72897 .69717 .716911 48 13 .64588 .76361! .65891 ! 75222 .67194 .74061 .68476 .72877 .69737 .71671 47 14 .64590 7(5342 .65913 .75203 .67215 .74041 .68497 .7'2857 .69758 .71650 46 15 .64612 .76323 .65935 .75184! .67237 .74022 .68518 .72837 .69779 .71630 45 16 .646:35 .76331 .65956 . 75165 ! .67258 .74002 .68539 .72817 .69800 .71610 44 17 .64657 .76386 .65!) 78 .75146 .67280 .73983 .68561 .72797 .69821 .71590 43 18 .64679 .76267 .66000 .75128 .67301 .73963 .68582 .72777 .69842 .71569 42 19 .64701 .76218 j .66022 .75107 ! .67323 .73944 .68603 .72757 .69862 .71549 41 20 .64723 .76229 .66044 .75088 .67344 .73924 .68624 .72737 .69883 .71529 40 21 .64746 .76210 .66066 .75059 .67366 .73904 .68645 .72717 \69904 .71508 39- 22 .64768 .76192 .68088 .75050 j .67387 .73885 .68666 .72697 .69925 .71488 38 23 .64790 .76173 .66109 .75030 i .67409 .73805 .68688 .72677 .69946 .71468 37 24 .64812 .76154 .66131 .75011 .67430 .73846 .68709 .72657 .69966 .71447 36 35 .64834 .76135 .66153 .74992 .67452 .73826 .68730 .72637 .69987 .71427 35 26 .64856 .76116 .66175 .74373 .67473 .738(36 .68751 .72617 .70008 .71407 34 27 .64878 .76097 j .66197 .74953 .67495 .73787 .68772 .72597 .70029 .71386 33 28 .64901 .76078 .66218 .74034 .67516 .73767 .68793 .72577 .70049 .71366 32 29 .64923 .76059 .66240 .74915 .67538 .73747 .68814 .72557 .70070 .71345 31 30 .64945 .76041: .66262 .74896 .67559 .73728 .68835 .72537 .70091 .71325 30 31 .64967 .76022' .66284 .74876 .67580 .73708 .68857 . 72517 : .70112 .71305 29 32 .64980 .76003 ...68 306 .74857 .67602 .73688 .68878 .72497 70132 .71284 28 33 .65011 .75984 .66327 .74838 .67623 .73669 .68899 .72477 .70153 .71264 27 34 .65033 .75965 .68349 .74818 .67645 .73649 .68920 .70457 .70174 .71243 26 35 .65055 .75946 .66371 .74799 .67666 .73629 .68941 .72437 .70195 71223 25 36 .6.5077 .75927 .68393 .74780 .67688 .73610 .68962 .72417 .7C215 .71203 24 37 .65100 .75908 .0(5111 .74760 .67709 .73590 .68983 .72397 7023(5 .71182 23 38 .65122 .75889 .(515136 .74741 .07730 73570 .69004 . 72377 , .70257 .71162 22 39 .65144 .75870 .66458 .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 .690(57 .72317 .70819 .71100 19 42 .652101.75813 .66523 .74664 .67810 .73491 .ll'IOSS .72297 .70339 .71080 18 43 .65232 .75794 .66545 .74(511 .(57837 .73172 .69109 .72277 .70360 .71059 17 44 .65254 .75775 .(iOil. .74(525 .07S59 .73452 .69130. .7'2v!57 .70381 .71039 16 45 i 65276 .75756 .66588 .746JI5 078SO .73432 .69151 .72236 .70401 .71019 15 4G .65298 .75738 .66610 .74586 .(57!HH .73413 .69172 .72216 .70422 .70998 14 47 .65320 .75719 .66632 .74567 .(57U23 .73393 .69193 .7219(5 .70443 . 70978 13 48 ! 05342 .75700 .66653 .74548 .(571)44 .73373 .692,4 .72176 .70463 70957 12 49 .65864 .75680 .66675 .74528 .67965 .78353 .0! 12:55 .72156 .70484 .70937 11 50 .65386 .75661 .66697:. 7 4509 .67187 .73333 .69256 .72136 .70505 .70916 10 51 .65408 .75642 .66718 '.74489 .68008 .73314 .69277 .72116 .70525 .70890 9 ! 52 .65430 .75623 .00740 .74470 .OSH-2'.t .73291 .69298 .7-.>o .705 it; 70875 8 53 .65452 .75604 .66762 .74451 .(5S051 73274 .69319 .7'2075 .70507 .70855 7 54 .65474 .755S5 .(507S3 .74431 .68072 ! 78254 .6(1340 ! 72055 ! 70587 .70S:! 4 6 55 .65496 .75566 .66805 .74412 .68093 .73231 .6JI301 .72085 .70608 .70813 5 56 .65518 .75547 .6(5827 .743<>2 .68115 . 73215 ll .69382 .72015 .70628 .70793 4 57 .65540 .75528 .66848 .74373 .(5813(5 .731 95 .69403 .71995 .706-49 .70772 3 58 .65562 .75509 .66870 .74353 .68157 .73175 .('.) 4:M .711174 .70670 .70752 2 59 .65584 .75490 .6(5891 .74334 .68179 .73155 .6'.M45 .71954! .70690 .70731 1 60 .65606 .66913 .74314 .68200 .73135 .69466 .71934 .70711 .70711 ; C.'Sill Sine ! Cosin Sine ; Cosin Sine ! Cosin Sine Cosin Sine / t 49 48 47 46 45 [459] TABLE XXVIII. NATURAL TANGENTS AND COTANGENTS. 1 II 2 ! 3 Tang Cotang Tang Cotang Tang Cotang Tang Cotang .00000 Infinite. .01746 57.2900 .03492 28.6363 ~~T0524r 19.0811 i60 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 1 58 g .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 6 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 54 y .00204 491.106 .01949 51.3032 .03696 27.0566 .05445 18.3655 '53 8 .00233 429.718 .01978 50.5485 .03725 26.8450 .05474 18.2677 i52 9 .00.262 381.971 .02007 49.8157 .03754 26.6367 .05503 18.1708 51 10 .00291 343.774 .020:36 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 in .00378 264.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05649 17.7015 46 15 .00436 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 45 16 .00465 214.858 .02211 45.2261 .03958 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 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 3Q .00582 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 8J .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 22 .00640 156.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 38 28 .00669 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 40.4:358 .04220 23.6945 .05970 16.7496 35 30 .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 96 .00815 122.774 .02560 39.0568 .04308 23.2137 .06058 16.5075 |32 29 .00844 118.540 .02589 38.6177 .04337 23.0577 .06087 16.4283 31 80 .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 30 81 .00902 110.892 .02648 37.7686 .04395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .04424 22.6020 .06175 16.1952 28 38 .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 27 34 .00989 101.107 .02735 36.5627 .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 38 .01105 90.4633 .02851 35.0695 .04599 21.7426 .06350 15.7483 22 39 .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 |20 n .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 19 12 .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 15.4638 18 48 .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 15.3943 17 44 .01280 78.1263 .0^)26 33.0452 .04774 20 9460 .06525 15.3254 16 15 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.2571 .15 4(5 .C1338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1g9'3 14 17 .01307 73.1390 .03114 32.118-1 .04862 20.5691 .06613 15.1222 :13 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0557 ,12 49 .01425 70.1533 .03172 31.5284 .04920 20.3253 .06671 14.9898 ill 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.92-14 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 ' 9 52 .01513 66.1055 .03259 30. (583:3 .05007 19.9702 .06759 14.7954 8 53 .01542 64? 8580 .03288 30. 41 1C) .05037 19.8546 .06788 14.7317 7 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 6 55 .01600 62.4992 .0*346 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 o 5!) .01716 68.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 1 UO .01746 57.2900 .03492 28.6363 .05241 19.0811 .Oli'.WS 14.3007 / Cotangl Tang Cotang Tang Cotang Tang Cotang | Tang / 89 88 87 i 86 [460! TABLE XXVIII. NATURAL TANGENTS AND COTANGENTS. < P ! ( i e i 5 o Tang Cotang Tang Cotang Tang Cotang Tang Cotang .0695)3 14.3007 .08749 11.4301 .10510 9.51436 .12278 8.14435 60 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 8.10536 58 B .07080 14.1235 .08837 11.3163 . 10599 9.43515 .1^367 8.08600 57 4 .07110 14.0655 .08866 11.2789 ' .10628 9.40904 .12397 8.06674 5(5 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 .07197 13.8940 .08954 11.1(581 .10716 9.33155 .12485 8.00948 53 6 .07227 13.8378 .OH9S3 11.1316 .10746 9.30599 .12515 .99058 52 9 .07256 13.7821 .09013 11.0954 . 10775 9.28058 .12544 .97176 51 10 .07285 13.7267 .09042 11.0594 .10805 9.25530 .12574 .95302 50 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 .12603 .93438 49 12 .07344 13.6174 .09101 10.9882 1 10863 9.2051(5 .12(533 .91582 4S 13 .07373 13.5634 .09130 10.9529 .10893 9.18028 .12662 .89734 47 14 .07402 13. 5098 .09159 10.9178 .10922 9.15554 .12692 .87895 46 15 .07431 i;;. 506 .09189 10.8829 .10952 9.13093 .12722 .86064 45 Hi .074(51 13.4039 .09218 10.8483 .10981 9.10646 .12751 .84242 44 ir .07490 13. 351 5 .09247 10.8139 .11011 9.08211 .12781 .82428 43 18 .07519 13. 29911 .09277 10.7797 .11040 9.05789 .12810 .80622 42 19 .07548 13.2480 .09306 10.7457 .11070 9.03379 .12840 .78825 41 20 .07578 13.1969 .09335 10.7119 .11099 9.00983 .12869 .77035 40 31 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 .75254 39 22 .07636 13.0958 .09394 10.6450 .11158 8196227 .12929 .7:3480 38 23 .07665 13.0458 1 .09423 10.6118 . .11187 8.93867 .12958 .71715 37 24 .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 .69957 36 35 .07724 12.9469 i .09482 10.5462 .11246 8.89185 .13017 .68208 85 20 .07753 12.8981 .09511 10.5136 ! .11276 8.86862 .13047 .66466 34 27 .07782 12.8496 .09541 10.4813 .mo5 8.84551 .13076 .64732 33 28 .07812 12.8014 .09570 10.4491 1 .11335 8.82252 .13106 .631X15 32 29 .07841 12.7530 .09600 10.4172 .11364 8.79964 .13136 .61287 31 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 .59575 30 31 .07'899 12.6591 .09658 10.3538 .11423 8.75425 .13195 .57872 29 32 .07929 12.6124 .09688 10.3224 .11452 8.73172 .13224 .56176 28 83 .07958 12.5660 .09717 10.2913 .11482 8.70931 .13254 .54487 27 34 1 07987 12.5199 .09746 10.2602 | .11511 8.68701 .13284 .52806 2(5 35 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 .51132 25 :',(; .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 .49465 24 37 .08075 12! 3838 .09834 10.1688 i .11600 8.62078 .13372 .47806 88 3C .08104 12.3390 .098(54 10.13H1 : .11629 8.59898 .13402 .46154 22 Hi) .08134 1212946 .09893 10.1080 .IK ' 8.57718 .13432 .44509 21 40 .08103 12.2505 .09923 10.0780 .11688 8.55555 .13461 .42871 20 41 .08192 12.2067 .09952 10.0483 .11718 ! 53402 .13491 .41240 19 42 .08221 12.HJ32 .00981 10.01*7 .11747 8.51259 .13521 .39616 18 43 .082.-)! 12.1201 .10011 9.98931 .11777 8.49128 .13550 .37999 17 44 .08280 12.0772 .10040 9.96007 .11806 S. 47007 .13580 .3(5389 16 45 .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 .34780 15 40 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13(139 .33190 14 47 .08368 11.9504 .10128 9.87338 .11895 8 40705 .18668 .31600 13 48 .08397 11 19087 .10158 9184482 .11924 8.38625 .13698 .30018 12 4!) .08427 11.8673 .10187 9. SIC, 11 .11954 S. 36555 .1372H .28442 11 50 .08456 11.8368 .10216 9.78817 .11983 8.34496 . 137'58 .26873 10 51 .08485 11.7853 I .10246 9.76009 .12013 8.32446 .13787 .25310 7 4 .72832 1.37302 .75538 1.32884 .78316 1.27688 .81171 1.23196 56 5 .72877 1.37218 .75684 1.32304 .78363 1.27611 .81220 1.23123 55 6 .72921 1.37134 .75629 1.32224 .78410 1.27535 .81268 1.23050 54 .729(56 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 1.22685 49 12 .73189 1.36633 .75904 1.31745 .78692 1.27077 .81558 1.22612 48 13 .73234 1.36549 .75950 1.31666 1 .78739 1.27001 .81606 1.22539 47 14 .73278 1.36466 .75996 1.31586 .78786 1.26925 .81655 1.22467 46 15 .73323 1.36383 .76042 1.31507 .78834 1.26849 .81703 1.22394 45 6 .73368 1.36300 .76088 1.31427 .78881 1.26774 .81752 1.22321 44 .73413 1.36217 .76134 1.31348 .78928 1.26698 .81800 1.22249 43 8 .73457 1.36134 ! .76180 1.31269 .78975 1.26622 .81849 1.22176 42 19 .73502 1.36051 .76226 1.31190 .79022 1.26546 .81898 1.22104 41 20 .73547 1.35968 I .76272 1.31110 .79070 1.26471 .81946 1.22031 40 21 .73592 1.35885 .76318 i.siasi .79117 1.26395 .81995 1.21959 39 22 .73637 1.35802 .76364 1.30952 .71)104 1.2(5319 .82044 1.21886 38 23 .73(381 1.35719 .76410 1.30873 .79212 1.26244 .82092 1.21814 37 24 .73726 1.35637 .76456 1.30795 .79259 1.26169 .82141 1.21742 36 25 .73771 1.35554 .76502 1.30716 .79306 1.26093 .82190 1.21670 35 2(5 .73816 1.35472 .76548 1.30637 .79354 1.26018 .82238 1.21598 34 .73861 1.35389 .76594 1.30558 .79401 1.25943 .82287 1.21526 33 28 .73906 1.35307 .76640 1.30480 .79449 1.25867 .82336 1.21454 32 29 .73951 1.35224 .76686 1.30401 .79496 1.25792 .88885 1.21382 31 30 .73996 1.35142 .76733 1.30323 .79544 1.25717 .82434 1.21310 30 31 .74041 1.35060 .76779 1.30244 .79591 1.25642 .82483 1.21238 29 32 .74086 1.34978 .76825 1.30166 .79639 1.25567 .82531 1.21166 28 33 .74131 1.34896 .76871 1.30087 .79686 1.25492 .82580 1.21094 27 34 .74176 1.34814 .76918 1.30009 .79734 1.25417 .82629 1.21023 26 35 .74221 1.34732 .76964 1.29931 .79781 1.25343 .82678 1.20951 25 36 .74267 1.34650 .77010 1.29853 .79829 1.25268 .82727 1.20879 24 37 .74312 1.34568 .77057 1.29775 .79877 1.25193 .82776 1.20808 23 38 .74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 .74402 1.34405 .77149 1.29618 .79972 1.25044 .82874 1.20665 21 40 .74447 1.34323 .77196 1.29541 .80020 1.24969 .82923 1.20593 20 41 ,74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1.20522 19 42 .74538 1.34160 .77289 1.29385 .80115 1.24820 .8:3022 1.20451 18 43 .74583 1.34079 .77335 1.29307 180163 1.24746 .83071 1.20379 17 44 .74628 1.33998 .77382 1.29229 .80211 1.24672 .83120 1.20308 16 45 .74674 1.33916 .77428 1.29152 .80258 1.24597 .83169 1.20237 15 46 .74719 1.33835 . 77475 1.29074 .80306 1.24523 .83218 1.20166 14 47 .74764 1.33754 .77521 1.28997 .80&54 1.24449 .83268 1.20095 13 48 .74810 1.33673 .77568 1.28919 .80402 1.24375 .83317 1.20024 12 49 .74855 1.33592 .77615 1.28842 .80450 1.24301 .83366 1 . 19953 11 50 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 1.19882 10 51 .74946 1.33430 .77708 1.28687 .80546 1.24153 .83465 1.19811 9 52 .74991 1.33349 .77754 1.28610 .80594 1.24079 .83514 1.19740 8 68 .75037 1.33268 .77801 1.28533 .80642 1.24005 .83564 1.19669 7 54 .75082 1.33187 .77848 1.28456 .80690 1.23931 .83613 1.19599 6 55 .75128 1.33107 .77895 1.28379 .80738 1.23858 .83662 1.19528 5 56 .75173 1.33026 .77941 1.28302 .80786 1.23784 .83712 1.19457 4 57 .75219 1.32946 .77988 1.28225 .80834 1.23710 .83761 1.19387 3 58 .75264 1.32865 .78035 1.28148 .80882 1.23637 .83811 1.19316 2 59 .75310 1.32785 .78082 1.28071 .80930 1.23563 .83860 1.19246 1 6C .75:555 1.32704 .78129 1.27994 .80978 1.23490 .83910 1.19175 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang f 53 52 51 50 [469] TABLE XXVIII.-NATURAL TANGENTS AND COTANGENTS. 40 41 !| ' 42 43 Tang I Cotang Tang i Cotang Tang Cotang Tang I Cotang .83910 1.19175 .86929 1.15037 .90040 1.11061 ~ 93252 1.07237" 60 1 .83960 1.19105 .86980 1. -14969 .90093 .10996 .93306 1.07174 59 o .84009 1.19035 .87031 1.14902 .90146 .10931 .93360 1.07112 58 8 .84059 1.18964 .87082 1.14834 .90199 .10867 .93415 1.07'049 57 4 .84108 1.18894 .87133 1.14767 .90251 .10802 .93469 1.06987 56 5 .84158 1.18824 .87184 1.14699 .90:304 .10737 .93524 1.06925 55 6 .84208 1.18754 .87236 1.14632 .90357 .10672 .93578 1.06862 54 7 .84258 1.18684 .87287 1.14565 .90410 .10607 .93633 1.06800 53 8 .84307 1.18614 .87338 1.14498 .90463 .10543 .93688 1.06738 52 9 .84357 1.18544 .87389 1.144:30 .90516 .10478 .93742 1.06676 51 10 .84407 1.18474 .87441 1.14363 .90569 .10414 .93797 1.06613 50 11 .84457 1.18404 .87492 1.14296 .90621 .10349 .93852 1.06551 41) 12 .84507 1.18334 .87543 1.14229 .90674 .10285 i .93906 1.06489 48 13 .84556 1.18264 .87595 1.14162 .90727 .10220 .93961 1.06427 47 14 .84606 1.18194 .87646 1.14095 .90781 ! .10156 .94016 1.06365 46 15 .84656 1.18125 .87698 1.14028 .90834 ! .10091 .94071 1.06303 45 16 .84706 1.18055 .87749 1.13961 .90887 ! .10027 i .94125 1.06241 44 17 .84756 1.17986 .87801 1.13894 .90940 ! .09963 ! .94180 1.06179 43 18 .84806 1.17916 ,87852 1.13828 .90993 i .09899 1 .94235 1.06117 13 19 .84856 1.17846 .87904 1.13761 .91046 .09834 .94290 1.06056 41 20 .84906 1.17777 .87955 1.13694 .91099 .09770 .94345 i 1.05994 to 21 .84956 1.17708 .88007 1.13627 .91153 .09706 .94400 1.05932 J39 22 .85006 1.17638 .88059 1.13561 .91206 .09642 .94455 1.05870 38 23 .85057 1.17569 .88110 1.13494 .91259 .09578 .94510 1.05809 37 24 .85107 1 . 17500 .88162 1.13428 .91313 .09514 .94565 1.05747 86 25 .85157 1.17430 .88214 1.13361 .91366 .09450 .94620 1.05685 85 26 .85207 1.17361 .88265 1.13295 .91419 .09386 .94676 1.05624 .31 2T .85257 1.17292 .88317 1.13228 .91473 ; .09322 .94731 1.05562 33 28 .85308 1.17223 .88369 1.13162 .91526 : .09258 .94786 1.05501 32 29 .85358 1.17154 .88421 1.13096 .91580 .09195 .94841 1.05439 31 30 .85408 1.17085 .88473 1.13029 .91633 .09131 .94896 1.05378 80 31 .85458 1.17016 .88524 1.12963 .91687 .09067 .94952 1.05317 89 32 .85509 1.16947 .88576 1.12897 .91740 .09003 .95007 1.05255 28 33 .85559 1.16878 .88628 1.12831 .91794 .08940 .95062 1.05194 27 34 .85609 1.16809 .88680 1.12765 .91847 .08876 .95118 1.05133 20 35 .85660 1.16741 .88732 1.12699 .91901 .08813 .95173 1.05072 25 36 .85710 1.16672 .88784 1.12633 .91955 .08749 .95229 1.05010 24 37 .85761 1.16603 .88836 1.12567 .92008 .08686 .95284 1.04949 2:3 38 .85811 1 . 16535 .88888 1.12501 .92062 ! .08622 .95340 1.04888 22 39 .85862 1.16466 .88940 1.12435 .92116 .08559 .95395 1.04827 21 40 .85912 1 . 16398 .88992 1.12369 .92170 .08496 .95451 1.04766 90 41 .85963 1.16329 .89045 1.12303 .92224 .08432 .95506 1.04705 1!) 42 .86014 1.16261 .89097 1 . 12238 .92277 .08369 .95562 1.04644 18 43 .86064 1.16192 .89149 1.12172 .92331 .oasoe .95618 1.04583 17 44 .86115 1.16124 .89201 1.12106 .92385 .08243 .95673 1.04522 16 45 .86166 1.16056 .89253 1.12041 .92439 .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 18 48 .86318 1.15851 .89410 1.11844 .92601 1 -07990 .95897 1.04279 12 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 1.15647 .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 .86623 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 .9:3088 1.07425 .96400 1.03734 3 58 .86827 1.15172 .89935 1.11191 .93143 1.07362 .96457 1.03674 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 j Cotang Tang Cotang Tang | j Cotang Tang \ 49 |l 48 47 46 [470] TABLE XXVIII.-NATURAL TANGENTS AND COTANGENTS. 4 40 4 40 4 40 Tang Cotang Tang Cotang Tang Cotang .96569 1.03553 60 20 .97700 1.02355 40 J40 .98843 1.01170 20 1 .96625 1.03493 59 21 .97756 .02295 m 41 .98901 1.01112 19 2 .96681 1.03433 58 22 .97813 : .02236 38 42 .98958 1.01053 18 3 .96738 1.03372 57 23 .97870 .02176 37 43 .99016 1.00994 17 4 .967*1 .03312 56 24 .97927 .02117 36 44 .99073 1.00935 16 5 .96850 .03252 55 25 .97984 .02057 35 45 .99131 1.00876 15 6 .96907 .03192 54 26 .98041 .01998 34 46 .99189 1.00818 14 7 .96963 .03132 53 27 .98098 .01939 47 .99247 1.00759 13 8 .97020 .03072 52 28 .98155 .01879 *i 48 .99304 1.00701 12 9 .97076 .03012 51 29 .98213 : .01820 31 49 .99362 1.00642 11 10 .97133 .02952 50 30 .98270 .01761 30 50 .99420 1.00583 10 H .97189 .02892 49 31 .98327 .01702 29 51 .99478 1.00525 9 12 .97246 .02832 48 32 .98384 .01642 28 52 .99536 1.00467 8 13 .97302 .02772 47 -33 .98441 .01583 27 53 .99594 1.00408 7 14 .97359 .02713 46 34 .98499 .01524 2(1 54 .99652 1.00350 6 15 .97416 1.02653 45 ,35 .98556 .01465 25 55 .99710 1.00291 5 16 .97472 1.02593 44 36 .98613 .01406 24 56 .99768 1.00233 4 17 .97529 1.02533 43 37 .98671 1.01347 23 57 ,99826 1.00175 3 18 .97586 1.02474 42 38 .98728 1.01288 22 58 .99884 1.00116 2 19 .97643 1.02414 41 39 .98786 1.01229 21 59 .99942 1.00058 1 20 .97700 1.02355 40 40 .98843 1.01170 20 '60 1.00000 1.00000 Cotang Tang Cotang Tang Cotang Tang 4 5 4 5 4 5 X - *L' 'Y < ^ t 1 - i ^ a (471] TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANT? ) 1 1 2 3 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .00000 .00000 I .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 .00000 .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 8 .00000 .00000 .00020 .00020 .00069 .00069 .00150 .00150 8 9 .00000 .00000 .00020 .00020 .00070 .00070 .OQJ51 .00151 9 10 .00000 .00000 .00021 .00021 .00071 .00072 .00153 .00153 10 11 .00001 .00001 .00021 .00021 .00073 .00073 .00154 .00155 11 12 .00001 .00001 .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 .00026 .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 .00002 .00002 .00028 .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 .00090 .00090 .00179 .00180 26 27 .00003 .00003 .00032 .00032 .00091 .00091 .00181 .00182 27 28 .00003 .00003 .00033 .00033 .00098 .00093 .00183 .00183 28 29 .00004 .00004 .00034 .00034 .0009< k . 00094 .00185 .00185 29 30 .00004 .00004 .00034 .00034 .00095 .00095 | .00187 .00187 30 31 .00004 .00004 .00035 .00035 .00096 .00097 .00188 .00189 31 32 .00004 .00004 .00036 .00036 .00098 .00098 .00190 .00190 32 33 .00005 .00005 .00037 .00037 .00099 .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 .00198 36 37 .00006 .00006 .00040 .00040 .00104 .00104 .00199 .00200 37 38 .00006 .00006 .00041 .00041 .00106 .00106 .00201 .00201 38 39 .00006 .00006 .000-41 .00041 .00107 .00107 i .00203 .00203 39 40 .00007 .00007 .00042 .00042 .00108 .00108 .00205 .00205 40 41 .00007 .00007 .00043 .00043 .00110 .00110 .00207 .00207 41 42 .00007 .00007 .00044 .00044 .00111 .00111 .00208 .00209 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 .00049 .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 .00(.161 .00137 .00137 1 .00244 .00244 60 | [472! ABLE xxix. -NATURAL VERSED SINES AND EXTERNAL SECANTS / 4 1 1 e ; 1 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .00244 .00244 .00381 .00382 .00548 .00551 .00745 .00751 1 .00246 .00246 .00383 .00385 .00551 .00554 .00749 .00755 1 .00248 .00248 .00386 .00387 .00554 .00557 .00752 .00758 2 3 .00250 .00250 .003K8 .00390 .00557 .00560 .00756 .00762 3 4 .00352 .00252 .00391 .00392 .005(50 .00563 .00760 .00765 4 5 .00254 .00254 .00393 .00395 .00563 .00566 .00763 .00769 5 6 .00256 .00257 .00386 .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 .007a5 .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 .00.282 .00428 .00429 .00604 .00608 .00811 .00817 18 19 .00284 .00284 .00430 .00432 .00607 .00611 .00814 .00821 19 30 .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 .00451 .00630 .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 .00:306 .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 3-,' .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 .00659 .00664 .00875 .00882 35 30 .00322 .00323 .00477 .00480 .00063 .00667 .00878 .00886 36 87 .00324 .00326 .00480 .00482 .00666 .00671 .00882 .00890 37 38 .00327 .00328 .00483 .00485 .00609 .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 .00683 .00688 .00902 .00910 42 43 .00339 .00340 .00497 .00.500 .00686 .OC'91 .00906 .00914 43 44 .00341 .00342 .00500 .00503 .00690 .00695 .00909 .00918 44 45 .00343 .00345 .00503 .00506 .00693 .00698 .00913 .00922 45 46 .00346 .00347 .00506 .00509 .00697 .00701 .00917 .00926 46 47 .00348 .00350 .00509 .00512 .00700 .00705 .00921 .00930 47 48 .00351 .00352 .00512 .00515 .00703 .00708 .00925 .00934 48 48 .00353 .00354 .00515 .00518 .00707 .00712 .00929 .00938 49 ;>i I .00350 .00357 .00518 .00521 .00710 .00715 .00933 .00942 50 51 .00.358 .00359 .00521 .00524 .00714 .00719 .00937 .00946 51 52 .00361 !00862 .00524 .00527 .00717 .00722 .00941 .00950 52 59 .00363 .00864 .00537 .00530 .00721 .00726 .00945 .00954 53 54 .00365 .00867 .00530 .00533 .(0724 .00730 .00949 .00958 54 55 .00368 .00369 .005*3 .00536 .00728 .00733 .00953 .00962 55 56 .00370 .00372 .00536 .00539 .00731 .00737 .00957 .00966 56 57 .00373 .00374 .00539 .00542 .007:35 .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 .00881 .00382 .00548 .00551 .00745 .00751 .00973 .00983 60 TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. / 8 9 10 11 ' Vers. Ex. sec. Vers. x. sec. Vers. Ex. sec. Vers. Ex. sec. .00973 .00983 .01231 .01247 .01519 .01543 .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 .01261 .01534 .01558 .01854 .01889 3 4 .00989 .00999 .01249 .01265 .01540 .01564 .01860 .01895 4 5 .00994 .01004 .01254 .01270 .01545 .01569 .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 .01294 .01570 .01595 .01893 .01930 10 11 .01018 .01029 i .01282 .01298 .01575 .01601 .01899 .01936 11 12 .01022 .01033 .01286 .01303 .01580 .01606 .01904 .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 .01627 .01927 .01965 16 17 .01043 .01054 .01310 .01327 .01606 .01633 .01933 .01971 17 18 .01047 .01059 .01314 .01332 .01612 .01638 .01939 .01977 18 19 .01052 .01063 .01319 .01337 .01617 .01643 .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 .01333 .01351 .01632 .01659 .01961 .02001 22 23 .01069 .01080 .01338 .01356 .01638 .01665 .01967 .02007 23 24 .01073 .01084 .01343 .01361 .01643 .01670 .01973 .02013 24 25 .01077 .01089 .01348 .01366 .01648 .01676 .01979 .02019 25 26 .01081 .01093 .01352 .01371 .01653 .01681 .01984 .02025 26 27 .01086 .01097 .01357 .01376 .01659 .01687 .01990 .02031 27 28 .01090 .01102 .01362 .01381 .01664 .01692 .01996 .02037 28 29 .01094 .01106 .01367 .0-1386 .01669 .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 .01=381 .01400 .01685 .01714 .02019 .02061 32 33 .01111 .01124 .01386 .01405 .01690 .01720 .02025 .02067 33 34 .01116 .01128 i .01391 .01410 .01696 .01725 .02031 .02073 31 35 .01120 .01133 i .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 .01435 .01723 .01753 .02000 .02103 39 40 .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 .01461 .01750 .01781 .02090 .02134 44 45 .01164 .01178 .01444 .01466 .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 .0123* .01504 .01527 .01821 .01854 i .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 1 .01519 .01543 .01837 .01872 .02185 .02234 60 ABLE XX1X.-NATURAL VERSED SINES AND EXTERNAL SECANTS, 1 / 12 13 14 15 / 1 Vers. Ex. see. 1 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .02185 .02234 ! .02563 .026:30 .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 .03568 5 6 .02222 .02272 .02602 .02672 .03013 .03106 .03453 .03576 6 7 .02228 .02279 : .02609 .02679 .03020 .03114 .03460 .03584 7 8 .02234 .02285 .02616 .02686 .03027 .03121 .03468 .0-3592 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 .03498 .03625 12 13 .02265 .02317 .02649 .02721 .03063 .03159 .03506 .03658 13 14 .02271 .02323 .02655 .02728 .03070 .03167 .03514 .03642 14 15 .02277 .02330 .02662 .02735 .03077 .03175 .03521 .03650 15 16 .02283 .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 .0&567 .03699 21 22 .02320 .02375 .02709 .02784 .03127 .03228 .03575 .03708 22 23 .02327 .02:382 .02716 .02791 .03134 .03236 .03583 .03716 23 24 .02333 .02388 .02722 .02799 .03142 .03244 .03590 .03724 24 25 .02339 .02395 .02729 .02806 .03149 .03251 .03598 .03732 25 26 .02345 .02402 .02736 .02813 .03156 .03259 .03606 .03741 26 27 .02352 .02408 .02743 .02820 .03163 .03267 .03614 .03749 27 28 .08358 .02415 .02749 .02827 .03171 .03275 .03621 .03758 28 29 .02364 .02421 .02756 .02834 .03178 .03282 .03629 .03766 29 30 .02370 .02428 .02763 .02842 .03185 .03290 .03637 .03774 30 31 .02377 .02435 .02770 .02849 .03193 .03298 .03645 .03783 31 32 .02383 .02441 .02777 .02856 .03200 .03306 .03653 .03791 32 33 .02389 .02448 .02783 .02863 .03207 .03313 .03660 .03799 33 34 .02396 .02454 .02790 .02870 .03214 .03321 .03668 .03808 34 35 .02402 .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 39 .02427 .02488 .02824 .02907 .03251 .03360 .03707 .03850 39 40 .02434 .02494 .02831 .02914 .03258 .03368 .03715 .03858 40 41 .02440 .02501 .02838 .02921 .03266 .03376 .03723 .03867 41 42 .02447 .02508 .02845 .02928 .03273 .03384 .03731 .03875 42 415 .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 .0247'2 .02535 1 .02873 .02958 .03303 .03416 .03762 .03909 46 47 .02479 .02542 ! .02880 .02965 .03310 .03424 .03770 .03918 47 48 .02485 .02548 .02887 .02972 .03318 .03432 .03778 .03927 48 49 .02492 .02555 .02894 .02980 .03325 .03439 .03786 .03935 49 50 .02498 .02562 .02900 .02987 .03333 .0:3447 .03794 .03944 50 51 .02504 .02569 .02907 .02994 .0-3340 .03455 .03802 .03952 51 52 .02511 .02576 .02914 .03002 .03347 .03463 .0:3810 .03961 52 53 .02517 .02582 .02921 .03009 .03355 .03471 .03818 .03969 53 54 .02524 .02589 .02928 .03017 .03362 .03479 .03826 .0:3978 54 . 55 .02530 .02596 .02935 .03024 .03370 .03487 .03884 .03987 55 56 .02537 .02603 .02942 .03032 .03377 .03495 .03842 .03995 56 57 .02543 .02610 .02949 .03039 .0: !:}K5 .03508 .03850 .04004 57 58 .02550 .02617 .02956 .IBM) 1C, .WM .03512 .03858 .04013 58 59 .02556 .02624 .02963 .03054 .03400 .03520 .03866 .04021 59 60 .02563 .02630 .02970 .03061 . .0:)1()7 .03528 .03874 .04030 60 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 1( 5 r r 1! 1 1! ) Ve/s. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .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 .05486 .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 .05216 .05515 .05836 8 .03938 .04100 .04438 .04644 .04967 .05226 .05524 .05847 8 9 .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 12 .03971 .04135 .04472 .04682 .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 .04498 .04710 .05030 .05297 .05591 .05922 15 16 .04003 .04170 .04507 .04719 .05039 .05307 .05601 .05933 16 17 .04011 .04179 .04515 .04729 .05048 .05317 .05610 .05944 17 18 .04019 .04188 .04524 .04738 .05057 .05327 .05620 .05955 18 19 .04028 .04197 .04533 .04748 .05067 .05337 .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 .04620 .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 .04135 .04313 .04646 .04872 .05186 .05470 .05755 .06107 32 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 I .04699 .04930 .05242 .05532 .05814 .06173 38 39 .04193 .04376 i .04707 .04940 .05251 .05542 .05824 .06184 39 40 .04201 .04385 .04716 .04950 .05260 .05552 .05833 .06195 40 41 .04209 .04394 .04725 .04959 .05270 .05563 .05843 .06206 41 42 .04218 .04403 .04734 .04969 .05279 .05573 .05853 .06217 42 43 .04226 .04413 .04748 .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 .OC283 48 49 .04276 .04468 .04796 .050:38 .05344 .05646 .05922 .06295 49 50 .04285 .04477 .04805 .05047 .05364 .05657 .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 i .05382 .05688 .05961 .06339 53 54 .04319 .04514 .04841 .05087 .05391 .05699 .05971 .06&50 54 55 .04327 .04523 .04850 .05097 .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 .04353 .04551 .04876 .05126 i .05429 .05741 .06011 .06395 58 59 .04361 .04560 .04885 .05136 .05439 .05751 .06021 .06407 59 60 .04370 .04569 .04894 .05146 .05448 .05762 .06031 .06418 60 TABLE XXIX. .NATURAL VERSED SINES AND EXTERNAL SECANTS. 20 21 22 23 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .08031 .06418 .06642 .07115 .07282 .07853 .07950 .08636 1 .06041 .06489 .06652 .07126 .07293 .07866 .07961 .08649 1 2 .06051 .06440 .00003 .07138 .07303 .07879 .07972 .08663 2 3 .06061 .06453 .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 .00715 .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 .0806-4 .08771 10 11 .06141 .00542 .06757 .07247 .07402 .07994 .08075 .08784 11 13 .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 .00022 .06831 .07'332 .07479 .08084 .08155 .08880 18 1!) .06221 .00034 .06841 .07344 .07490 .08097 .08167 .08893 19 20 .013231 .06645 .06852 .07356 .07501 .08109 .08178 .08907 20 21 .06241 .06657 .06863 .07368 .07512 .08122 .08190 .08921 21 22 .06852 .06668 ! 08878 .07380 .07523 .08135 .08201 .08934 22 23 .06262 .08680 ! 08884 .07393 .07534 .08148 .08213 .08948 23 24 .06272 .06691 .08894 .07405 .07545 .08161 .08225 .08962 24 25 .06282 .06703 .06905 .07417 .07556 .08174 .08236 .08975 25 26 .06292 .06715 .08916 .07429 .07568 .08187 .08248 .08989 26 27 .00302 .06726 !06926 .07142 .07579 .08200 .08259 .09003 27 28 .06312 .06738 .06987 .07454 .07590 .08213 .08271 .09017 28 29 .06323 .06749 .0(5948 .07466 .07601 .08226 .08282 .09030 29 30 .06333 .06761 .00958 .07479 .07612 .08239 .08294 .09044 30 31 .06343 .06773 .06969 .07491 .07623 .08252 .08306 .09058 31 32 .06353 .06784 .06980 .07503 .07634 108265 .08317 .09072 32 33 ' .06363 .06796 .06990 .07516 .07645 .08278 .08329 .09080 33 34 .06374 .06807 .07001 .07528 ! 07657 .08291 .08340 .09099 34 35 .06384 .06819 .07012 .07540 .07668 .08305 .08352 .09113 35 36 i .06394 .06831 .07022 .07553 .07679 .08318 .08364 .09127 36 37 .06404 .06843 ' .070*1 .07565 .07690 .08331 .08375 .09141 37 38 .06415 .06854 .07044 .07578 .07701 .08344 .08387 .09155 38 39 i .06425 .06866 .07055 .07o90 .07713 .08357 .08399 .09169 39 40 .06435 .06878 .07065 .07002 .07724 .08370 .08410 .09183 40 41- .06445 .06889 .07076 .07615 .07735 .08383 .08422 .09197 41 42 .06456 .06901 .07087 .07627 .07746 .08897 .08434 .09211 42 43 .06466 .06913 .07098 .07640 .07757 .(F410 .08445 .09224 43 44 .06476 .06925 .07108 .07652 .077'69 .08423 .08457 .09238 44 15 .(HUSH .06986 .07119 .07665 .07780 .08436 .08469 .09252 45 46 ! .0(H!)7 .00(148 .07130 .07077 .0771)1 .08449 .08481 .09266 46 47 .<>(;:>:>; .06960 .07141 .07690 .07802 .08403 .08492 .09280 47 48 .IK 551 7 .06972 .07151 .07702 .07814 .08476 .08504 .09294 48 1'.) .06528 .06984 i .07162 .07745 .0782.-) .08489 .08510 .09308 49 50 .00538 .06905 .071 7'3 .07727 .07886 .08503 .08528 .09323 50 :>i .06548 .07'007 .07184 .07740 .07848 .08516 .08539 .09337 51 52 .06559 .07019 .07195 .07752 ! 07859 .08529 .08551 .09351 52 58 .06569 .07031 .07206 .07765 .07'87'0 i .08542 .08563 .09365 53 51 .06:580 .07043 .07216 .07778 .07881 .08556 .08575 .09379 54 .V> .06590 .07055 .07227 .07790 .07893 1 .08569 ; .08586 .09393 55 56 .06600 .07067 .07238 .07803 .07904 .08582 ! .08598 .09407 56 57 .06611 .07079 .07249 .07'816 .07915 .OH.V.IO i .08610 .09421 57 58 .06621 .07091 ! .07260 i07828 .07927 .08(50!) ! .08622 .09435 58 59 .06632 .07103 ,07271- .07841 .07938 .08623 .08634 .09449 ! 59 60 .06642 .07115 : .07282 .07853 .0795) .08636 ! .08645 .09464 ! 60 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS, 2 1 2 5 2 3 2 7 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .08645 .09464 .09369 .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 .08693 .09520 .09418 .10398 .10172 .11323 .10952 .12299 4 5 .08705 .09535 .09431 .10413 .10184 .1133b .10965 .12316 5 6 .08717 .09549 .09443 .10428 .10197 .11355 .10979 .12333 6 .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 12 .08788 .09635 .09517 .10518 .10274 .11451 .11058 .12433 12 13 .08800 .09649 .09530 .10533 .10287 .11467 .11072 .12450 13 14 .08812 .09663 .09542 .10549 .10300 .11483 .11085 .12467 14 15 .08824 .09678 i .09554 .10564 .10313- .11499 .11098 .12484 15 16 .08836 .09692 ! .09567 .10579 .10326 .11515 .11112 .12501 16 17 .08848 .09707 .09579 .10594 .10338 .11531 .11125 .12518 17 18 .08860 .09721 .09592 .10609 .10351 .11547 .11138 .12534 18 19 .08872 .09735 .09604 .10625 .10364 .11563 .11152 .12551 19 20 .08884 .09750 .09617 .10640 .10377 .11579 .11165 .12568 20 21 .08896 .09764 .09629 .10655 .10390 .11595 .11178 .12385 21 22 .08908 .09779 .09642 .10670 1 .10403 .11611 .11192 .12602 22 23 .08920 .09793 .09654 .10686 .10116 .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 .09704 .10747 .10468 .11691 .11259 .12687 27 28 .08980 .09866 .09716 .10762 .10481 .11708 .11272 .12704 28 29 .08992 .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 32 .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 .09968 .09804 .10870 .1057'2 .11821 .113(56 .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 i .10650 .11919 .11447 .12927 41 42 .09149 . 10071 .09892 .10978 1 .10663 .11936 .11461 .12944 42 43 .09161 .10085 .09905 . 10994 .10676 .11952 ; .11474 .12961 43 44 .09174 .10100 .09918 .11009 .10689 .11968 . 1 1488 .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 .1:3031 47 48 .09222 .10159 .09968 .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 .00259 .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 .09JM5 ,10308 .10095 .11229 .10873 .12199 .11678 .13222 58 59 .09357 .10323 .10108 .11244 .10886 .12216 .11692 .13240 59 60 .09369 .10338 .10121 .11260 .10899 .12233 .11705 .13257 60J [478] TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 28 29 30 31 ' Yers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .11705 .13257 .12538 .14335 .1.3397 .15470 .14283 .16663 .11719 .13275 .12552 .14354 .13412 .15489 .14298 .16684 1 2 .11788 . 13292 .12566 .14372 .1:3427 .15509 .14313 .16704 2 a .11746 .13310 .12580 .14391 .1:3441 .15528 .14328 .16725 3 4 .11760 .13327 .12595 .1-1409 .1:3456 .15548 .14343 .16745 4 5 .11774 .13345 .12609 .14428 .18470 . 155(57 .14358 .16766 5 6 .11787 .13362 .12(523 .14446 .13485 .15587 .14873 .16786 6 .11801 .13380 .12637 .14465 .134!*!) .15606 .14388 .16806 7 8 .11815 .13398 .12651 .14483 .1%14 .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 38558 .15684 .14449 .16889 11 12 .11870 .13468 .12708 .14558 .18578 .15704 .14464 .16909 12 13 .11883 .13486 .12722 .14576 .13587 .15724 .14479 .16930 13 14 .11897 .13504 .12736 .14595 .13602 .15743 .14494 .16950 14 15 .11911 .13521 .12750 .14614 .13616 .15763 .14509 .16971 15 16 .11925 .13539 .12765 .14632 .13631 .15782 .14524 .16992 16 17 .11938 .13557 .12779 .14651 .13646 . 15802 .14539 .17012 17 18 .11952 .13575 .12793 .14670 .13660 .15822 .14554 .17033 18 19 .11966 .13593 .12807 .14689 .13675 .15841 .14569 .17C54 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 .14660 .17178 25 26 .121X33 .13718 .12907 .14820 .13778 .15980 .14675 .17199 26 27 .12077 .13735 .13921 .14839 .13793 .16000 .14690 .17220 27 28 .12091 .13753 .12936 .14858 .1:3808 .16019 .14706 .17241 28 29 .12104 .13771 .12950 .14877 .13822 .16039 .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 ! 18825 .12993 .149:33 .13867 .16099 .14766 .17325 32 33 .12160 .K3843 .13007 .14952 .13881 .16119 .14782 .17346 33 34 .12174 .13861 .13022 .14971 .13896 .16139 .14797 .17367 34 35 .12188 .13870 .13036 .14990 .13911 .16159 .14812 .17388 35 36 .12202 . 13897 .13051 .15001) .13926 .16179 .14827 .17409 36 37 .12216 .13916 . 13065 .15028 .13941 .16199 .14843 .17430 37 38 .12230 .13934 .13079 .15047 .13955 .16219 .14858 .17451 38 39 .12244 . 13952 .13094 .150(56 .13970 .16239 .14873 .17472 39 40 .12257 .13970 .13108 .15085 .13985 .16259 .14888 .17493 40 41 .12271 .13988 .13122 .15105 .14000 .16279 .14904 .17514 41 42 .12285 .14006 .13137 .15124 .14015 .16299 .14919 .17535 42 43 .12299 .14024 .18151 .15143 .14030 .16319 .14934 .17556 43 44 .12313 .14042 .131(56 . 15102 .14044 . i 6339 .14949 .17577 44 45 .12327 " .14061 .13180 .15181 .14059 .16359 .14965 .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 .17662 48 49 .12383 .14134 .13238 .15258 .14119 .16440 .15026 .17683 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 52 53 .12439 .14207 .13296 .15335 .14179 .16521 .15087 .17768 5:3 54 .12454 .14225 .13310 . 15354 .14191 .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 .1.3354 .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 .13897 .15470 .11283 .16663 .15195 .17918 60 [479] TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 32 33 34 35 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .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 .19301 .17145 .20693 .18135 .22152 3 4 .15257 .18004 .16196 .19327 .17161 .20717 .18152 .22177 4 5 .15272 .18025 .16212 .19:349 .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 .16260 .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 It .15365 .18155 .16308 .19485 .17276 .20883 .18269 .22352 11 12 .15381 .18176 .16324 .19508 .17'292 .20907 .18286 .22377 12 13 .15396 .18198 .16340 .19531 .17308 .20931 .18302 .22402 13 14 .15412 .18220 .16355 .19554 .17325 ; .20955 .183^9 .22428 14 15 .15427 .18241 .16371 .19576 .17341 .20979 .18336 .22453 15 16 .15443 .18263 .16387 .19599 .17357 .21003 .18353 .22478 16 ir .15458 .18285 .16403 .19622 .17374 ! .21027 .18369 .22503 17 18 .15474 .1830? .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 .18872 .16467 .19713 | .17439 .21123 .18437 .22604 21 22 .15536 .18394 .16483 .19736 .17456 .21147 .18454 .22629 22 23 .15552 .18416 .16499 .19759 .17472 .21171 .18470 .22655 23 24 .15567 .18437 .16515 .19782 .17489 .21195 .18487 .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 i .17538 .21268 .18538 .22756 27 28 .15630 .18525 .16579 .19874 .17554 .21292 .18555 ' .22782 28 29 .15645 .18547 .16595 .19897 .17571 .21316 .18572 j .22807 29 30 .15661 .18569 .16611 .19920 .17587 .21341 .18588 , .22833 30 31 .15676 .18591 .16627 .19944 .17604 .21365 .18605 .22858 31 32 .15692 .18613 .16644 .19967 .17620 .21389 .18622 .22884 32 33 .15708 .18635 .16660 .19990 .17637 .21414 .18639 .22909 33 34 .15723 .18657 .16676 .20013 ! .17653 .21438 .18656 .22935 34 35 .15739 .18679 .16692 i .20036 ! .17670 .21462 .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 .21560 .18741 .23063 39 40 .15818 .18790 .16772 .20152 .17752 .21584 .18758 .23089 40 41 .15833 .18812 .16788 1 .20176 .17769 .21609 .18775 .23114 41 42 .15849 . 18834 .16805 .20199 .17786 .216:33 .18792 .23140 42 43 .15865 .18856 .16821 .20222 .17802 .21658 .18809 .23166 43 44 .15880 .18878 .16837 .20246 .17819 .21682 .18826 .23192 44 45 .15896 .18901 .16853 .20269 .17835 .21707 .18843 .23217 45 i 46 .15912 .18923 .16869 .20292 .17852 .21731 .18860 .23243 46 47 .15923 .18945 .16885 .20316 .17868 .21756 .18877 .23269 47 48 .15943 .18967 | .16902 .20339 .17885 .21781 .18894 .23295 48 49 .15959 .18990 || .16918 .20363 .17902 .21805 .18911 .23321 49 50 .15975 .19012 .16934 .20386 .17918 .21830 .18928 .23347 50 51 .15991 .19034 .16950 .20410 .17935 .21855 .18945 .2&S73 51 52 .16006 .19057 .16966 .20433 .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 .ro3i .20527 .18018 .21978 .19030 .2350% 56 57 .16085 .19169 .1704? .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 .17096 ,20622 .18085 .22077 1 .19098 .23607 60 [480] , TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 36 37 38 39 i Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .19098 .23007 .301S6 .25214 .21199 .26902 .22285 .28676 1 .19115 .23(533 .20154 .25241 .21217 .26931 .22304 .287'06 1 o .19183 .23659 .30171 .2520!* .21235 .38960 .22322 .28737 2 3 .19150 .23(585 .20189 .25296 .21253 .26988 .22340 .28767 3 4 .19167 .23711 .20207 .25324 .21271 .37017 .22359 .28797 4 5 . 19184 .23738 .20224 .25351 .2128!) .27046 .22377 .28828 5 6 .19201 .237(54 .20242 .25379 .21307 .27075 .22395 .28858 6 .19218 .23790 .20259 .25406 .21324 .27104 .22414 '28889 7 8 .19235 .23810 .20277 .25434 .21342 .27133 .22432 .28919 8 9 .19252 123843 .20294 .25462 .21360 .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 .25545 .21414 .27250 .22506 .29042 12 13 .19321 .23948 .20365 .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 15 Ifi .19373 .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 .257:!'.) .21540 .27454 .22634 .29256 19 20 .19442 .24134 .20488 .25767 .21558 .27483 .22653 .29287 20 21 .19459 .24160 .20506 .25795 .21576 .27513 .22071 .29313 21 22 .19476 .24187 .20523 .25823 | .21595 .27542 .22090 .29349 22 23 .19493 .24213 .20541 .25851 .21013 .27572 .227'OS .29380 23 24 .10511 .24240 .20559 .25879 ! .21631 .27001 .22727 .29411 24 25 .19528 .24267 .20576 .25907 .21649 .27630 .22745 .29442 25 2G .19545 .24293 .20594 .25935 .21667 .27660 .22764 .29473 26 27 .19562 .24320 .20612 .25963 .21685 .27689 .22782 .29504 27 28 .19580 .24347 .20629 .25991 .21703 .27719 .22801 .29535 28 29 .19597 .24373 .20647 .20019 .91731 .27748 .22819 ! 26666 29 30 . 19614 .24400 .20665 .26047 .21739 .27778 .22838 .29597 30 31 .19632 .24427 .20682 .26075 .21757 .27807 .22856 .29628 31 32 ] 19649 .24454 .20700 .20104 .21775 .27837 .22875 .29659 32 33 .19666 .24481 .20718 .261:32 .21794 .27807 .22893 .2%<)0 33 34 .Iy684 .24508 .20736 .26160 .21812 .27896 .22912 .29721 34 35 .197W .24534 .20753 .26188 .21830 .27926 .22930 .39752 35 30 .19718 .24561 .20771 .20216 .21848 .27950 .22949 .29784 36 37 .19736 .24588 .20789 .26245 .21866 .279R5 .22967 .29815 37 38 .19753 .24615 .20807 .20273 .21884 .28015 .22980 .29846 38 39 .19770 .24642 .20824 .26301 .21902 .28045 .23004 .29877 39 40 .19788 .24669 .20842 .26330 .21921 ,28075 .2:J023 .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 .20415 .21975 .88164 ! 23079 .80003 43 44 .19857 .24777 .20913 .26143 .21993 .Mrfl94 .23097 .30034 44 45 .19875 .24804 .20931 .26472 .22012 .28224 .23116 .30006 45 46 .19892 .24S32 .20949 .26500 .22030 .28254 .23134 .30097 46 47 .19909 .24859 .20967 .26529 .22048 .28284 .23153 .30129 47 48 .19927 I 24886 .20985 .26557 .22066 .28314 .23172 .30160 48 49 .19944 .24913 .21002 .26586 .22084 .28344 .23190 .30192 49 50 .19962 .21940 .21020 .26615 .22103 .28374 .23209 .30223 50 51 .19979 .24967 .21038 .26643 221 ^1 .28404 .23228 .30255 51 52 .19997 .24!)95 .21056 .26072 ! 22139 .284:34 .23246 .30287 52 53 .20014 .25002 .21074 .86701 .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 .2(5787 .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 .20S73 .22367 .28(546 .23377 .30509 59 GO .20136 .25214 .21199 .20902 .222S5 .28076 .23390 .30541 60 [481] TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 40 41 42 43- r Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .23396 .30541 .24529 .32501 .25686 .34563 .26865 .36783 1 .23414 .30573 .24548 .32535 .25705 .34599 126884 .36770 1 2 .23433 .30605 .24567 .32568 .25724. .34634 .26904 .36807 2 3 .23452 .30636 .24586 .32602 .25744 .34669 .26924 .36844 3 4 .23470 .30668 .24605 .32636 .25763 .34704 .26944 .36881 4 5 .23489 .30700 .24625 .32669 .25783 .347'40 .26964 .36919 5 6 .23508 .30732 .24644 .32703 .25802 .34775 .26984 .36956 6 7 .23527 .30764 .24663 .32737 .25822 .34811 .27004 .36993 19 8 .2:3545 .30796 .24682 .32770 .25841 .34846 i .27024 .37030 8 9 .23564 .30829 .24701 .32804 .25861 .34882 .27043 .37068 9 10 .23583 .30861 .24720 .32838 .25880 .34917 .27063 .37105 10 II .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 .37293 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 30 .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 .37860 30 31 .23978 .31541 .25124 .33554 .26292 .35670 .27483 .37898 31 32 .23997 .31574 .25143 .33588 .26312 .357'07 .27503 .37936 32 33 .24016 .31607 .25162 .33622 .2easi .35743 .27523 .37974 33 34 .24035 .31640 .25182 .33657 .26351 .35779 .27543 .38012 34 35 .24054 .31672 .25201 .33691 .26371 .35815 .27503 .38051 35 36 .24073 .31705 .25220 .33726 .26390 .35852 .27583 .38089 36 37 .24092 .31738 .25240 .33760 .26410 .35888 .27003 .38127 37 38 .24111 .31771 .25259 .33795 .26480 .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 .3&319 42 43 .24206 .31936 .25356 .33968 ..26528 .36107 .27723 .38357 43 44 .24225 .31969 .25375 .34003 .26548 .36143 .27743 .38396 44 45 .24244 .32008 .25394 .34038 .26568 .36180 .27764 .38434 45 46 .24262 .32035 .25414 .34073 .26588 .36217 .27784 .38473 46 47 .24281 .32068 .25433 .34108 .26607 .36253 .27'804 .38512 47 48 .24300 .32101 .25452 .34142 .26627 .36290 .27824 .38550 48 49 .24820 .32134 .25472 .34177 .26647 .36327 .27'844 .38589 49 50 .24339 .32168 .25491 .34212 .26667 .36363 .27864 .38628 50 51 .24358 .32201 .25511 .34247 .26686 .36400 .27884 .38668 51 52 .24377 .32234 .25530 .34282 .26706 .36437 .27905 .387'05 52 53 .24396 .32267 .25549 .34317 .267-26 .36474 .27925 .38744 53 54 .24415 .32301 .25569 .34352 .26746 .36511 .27945 .38783 54 55 .24434 .32334 .25588 .34387 ! .26766 .36548 .27'965 .38822 55 56 .24453 .32368 .25608 .34423 | .26785 .36585 .27985 .38860 56 57 .24472 .32401 .25627 .34458 i .26805 .36622 .28005 .38899 57 58 .24491 .32434 .25647 .34493 I .26825 .36659 .28026 .38938 58 59 .24510 .32468 .25666 .34528 ! .26845 .36696 .28046 .38977 59 60 .24529 .32501 .25686 .34563 .26865 .36733 .28066 .39016 60 [482] TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 440 . ! 46> 47 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .28066 .39016 .29289 .41421 .30534 .43956 .31800 .46628 1 .28086 .39055 .29310 .41463 .30555 .43999 .31821 .46674 1 2 .28106 .39095 .29330 .41504 .30576 .44042 i .31843 .46719 2 3 .28127 .39134 .29351 ' .41545 .30597 .44086 ; .31864 .46765 3 4 .28147 .39173 .29372 .41586 .30618 .44129 , .31885 .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 .30681 .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 i 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 .474 58 18 19 .28451 .39764 .29681 .42210 .30933 .44787 .32205 .47504 19 20 .28471 .39804 .29702 ,42251 .30954 .44831 .32227 .47551 20 21 .28492 .39844 .29723 .42293 .30975 .44875 .32248 .47598 21 22 .28512 .39884 .29743 .42335 .30996 .44919 .3227'0 .47644 22 23 .28532 .39924 .29764 .42377 .31017 .44963 .32291 .47691 23 24 .28553 .39963 .29785 .42419 .31038 .45007 .32312 .47738 24 25 .28573 .40003 .29805 .42401 .31059 .45052 .32334 .47784 25 26 .28593 .40043 .29826 .42503 .31080 .45096 .32355 .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 .26695 .40243 .29930 .42714 .31186 .45319 .32462 .48066 31 32 .28716 .40283 .29951 .42756 .31207 .45363 .32484 .48113 32 33 .28736 .40324 .29971 .42799 .31228 .45408 .32505 .48160 33 34 .28757 .40364 .29992 .42841 .31249 .45452 .32527 .48207 34 35 .28777 .40404 .30013 .42883 .31270 .45497 .32548 .48254 35 36 .28797 .40444 .30034 .42926 .31291 .45542 .32570 .48301 36 37 .28818 .40485 .80054 .42968 .31312 .45587 .32591 .48349 37 38 .28838 .40525 .30075 .43011 .31334 .45631 .32613 .48396 38 39 .28859 .40565 .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 .40087 .30158 .43181 .31418 .45811 .32699 .48586 42 43 .28941 .40727 .30179 .43224 j .31439 .45856 .32720 .48633 43 44 .28961 .40768 .30200 .432(57 i .31461 . 45901 .32742 .48681 44 45 .28981 .40808 .30221 .43310 .31482 .45946 .32763 .48728 45 46 .29002 .40849 .30242 .43352 I .31503 .45992 .32785 .48776 46 47 .29022 .40890 .30263 .43305 .31524 .46037 .32806 .48824 47 48 .29043 .40930 .30283 .43438 .31545 .46082 .32828 .48871 48 49 .29063 .40971 .30304 .43181 .31567 .46127 .32819 .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 .31030 .46263 .32914 .49063 52 53 .29145 .41134 .30388 .43653 .31651 .46309 .32936 .49111 53 54 .29166 .41175 .30409 .43696 .31673 .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 .31730 .46491 .33022 .49303 57 58 .29248 .41339 .30492 .43KIS!) .31758 .46537 .33044 .49351 5S 59 .29269 .41380 .30513 .43912 .31779 .46582 .33065 .49399 59 60 .29289 .41421 .30534 .43956 .31800 .46628 .33087 .494-48 60 [483] TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. / 48 49 5(h 51 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .33087 .49448 if .34394 .52425 .35721 .55572 .37068 .58902 1 .33109 .49496 1 -34416 .52476 .35744 .55626 .37091 .58959 1 2 .33130 .49544 .34438 .52527 .35766 .55680 .37113 .59016 2 3 .33152 .49593 .34460 .52579 .35788 .557:34 .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 .871804 .59245 6 7 .33238 .49787 .34548 .52784 .35877 .55951 .37226 .59302 7 8 .33260 .49835 .34570 .52835 .35900 .56005 .37249 .59360 8 9 .a3282 .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 .34658 .53041 .35989 .56223 .37340 .59590 12 18 .33368 .50079 .34680 .53092 .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 18 19 .33499 .50373 .34812 .53403 .36146 .56606 .37498 .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 22 23 .33586 .50570 .34900 .53611 .36235 .56826 .37589 .60229 23 24 . 33607 .50619 .34923 .53663 .36258 .56881 .37612 .60287 24 25 .'33629 .50669 .34945 .53715 .36280 .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 .35033 .53924 .36370 .57158 .37726 .60580 29 30 .33738 .50916 .35055 .53977 .36392 .57213 .37749 .60639 30 31 .33760 .50966 .35077 .54029 .36415 .57269 .37771 .60698 31 32 .33782 .51015 .35099 .54082 .36437 .57324 .37794 .60756 32 33 .33803 .51065 .35122 .54134 .36460 .57380 .37817 .60815 33 34 .33825 .51115 .35144 .54187 .36482 .57436 .37840 .60874 34 35 .33847 .51165 .35166 .54240 .36504 .57491 .37862 .60933 35 30 .33869 .51215 .35188 .54292 .36527 .57547 .37885 .60992 36 37 .33891 .51265 .35210 .54345 .36549 .57603 .37908 .61051 37 38 .33912 .51314 .35232 .54398 .36572 .57659 .37931 .61111 38 39 .33934 .51364 .35254 .54451 .36594 .57715 .37954 .61170 39 40 .33956 .51415 .35277 .54504 .366l'7 .57771 .37976 .61229 40 41 .33978 .51465 .35299 .54557 .36639 .57827 .37999 .61288 41 42 .34000 .51515 .35321 .54610 .36662 .57883 .38022 .61348 42 43 '.34022 .51565 .35343 .54663 .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 .3417'5 .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 .58503 .38274 .62005 53 54 .34262 .52120 .35588 .55250 .36932 .58560 .38296 .62065 54 55 .34284 .52171 .35610 .55303 .36955 .58617 .38319 .62125 55 56 .34306 .52222 .35632 .55357 . 36978 .58674 .38342 .62185 56 57 .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 .34394 .52425 .35721 .55572 .37068 .58902 1 .38434 .62427 60 [484] TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS / 52 S3 54 55 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .38434 .62427 .39819 .66164 1 .41221 .70130 .42642 .74345 1 .38457 .62487 .39842 .66228 .41245 .70198 .42666 .74417 1 o .38480 .62548 .39865 .66292 .41269 .70267 .42690 .74490 2 3 .38503 .62609 .39888 .66357 .41292 .70:335 i .42714 .74562 3 4 .38526 .62669 .39911 .66421 .41316 .70403 .427:38 .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 i .42809 .74854 8 .38617 .62913 .40005 .66679 .41410 .70677 .42833 .74927 j 8 9 .38640 .62974 .40028 .66744 .41433 .70746 1 .42857 .75000 ! 9 10 .38663 .63035 .40051 .66809 I .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 j .43048 ! 75587 17 18 .38847 .63525 .40237 .67829 .41646 .71368 .43072 .75661 18 19 .38870 .63587 .40261 .67394 .41670 .71437 .43096 .75734 19 20 .38893 .636-48 .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 .40:354 .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 .433&S .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 i .40611 .68382 .42024 .72487 .43455 .76851 34 36 .39239 .64580 j .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 .64768 i .40705 .68648 .42119 .72769 .43551 .77152 38 39 .39332 .64831 .40728 .687J5 .42143 .72840 : .43575 .77227 39 40 .39355 .64894 .40752 .68782 .42167 .72911 .43599 .77303 40 41 .39378 .64957 .40775 .68848 .42191 .72982 .43623 ..77878 41 42 .39401 .65020 .40799 .68915 .42214 .73053 .43647 .77454 42 43 .39424 .65083 .40822 .68982 .42238 .73^24 .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 .{5533IJ .40916 .69250 .42.333 .73409 .43768 [77883 47 48 .39540 .053!)!) .40939 .69318 .42357 .73481 .43792 .77916 48 49 .39563 .65462 .40963 .69385 .42381 .73552 .43816 .77986 49 50 .39586 .65526 .40986 .69452 .42404 .73624 .43840 .78062 50 51 .39610 .65589 .41010 .69520 .42428 .73696 .43864 .78138 51 52 .39633 .65653 .41033 .69587 .42452 .73768 ! .43888 .78215 52 53 .390:jfi .W717 .41057 .69655 .42476 .73840 .43912 .78291 53 54 .39679 .65780 .41080 .69723 .42499 .73911 .43936 .78368 54 55 .39702 .85844 .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 . WHO .74272 .44057 .78752 59 60 .39819 .66164 .41221 .70130 .42642 .74345 .44081 .78829 60 [485] TABLE XXIX.-NATURAL VERSED SINES AND EXTERNAL SECANTS 5 5 5 r 51 jo 5( > Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .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 i .48546 .94349 t> 3 .44153 .79061 .45609 .83855 i .47082 .88972 .48571 .94443 3 4 .44177 .79138 .45634 .83938 .47107 .89060 .48596 .94537 4 5 .44201 .79216 .45658 .84020 j .47131 .89148 .48621 .94632 5 6 .44225 .79293 .45683 .84103 ! .47156 .89237 .48646 .94726 6 7 .44250 .79371 .45707 .84186 1 .47181 .89325 .48671 .94821 7 -8 .44274 .79449 .45731 .84269 i .47206 .89414 .48696 .94916 8 9 .44298 .79527 .45756 .84352 .47230 .89503 i .48721 .95011 9 10 .44322 .79604 .45780 .84435 .47255 .89591 1 .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 1 .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 1 .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 i .47527 .90575 .49021 .96158 21 22 .44612 .80546 .46074 .85439 i .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 .44685 .80783 .46147 .85692 .47626 .90935 .49121 .96544 25 26 .44709 .80862 .46172 .85777 .47651 .91026 .49146 .96641 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 .86031 .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 1 .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 .86970 .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 i .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 .45341 .82953 .46811 .88008 .48297 .93412 .49799 .99198 52 53 .45365 .83034 .46836 .88095 .48322 .93505 .49824 .99298 53 54 .45390 .83116 .46860 .88183 ! .48347 .93598 .49849 .99398 54 55 .45414 .&3198 .46885 .88270 i .48372 .93692 .49874 .99498 55 56 .45439 .83280 .46909 .88357 ! .48396 .93785 .49899 .99598 56 57 .45463 .83362 .4<;t)34 .88445 i .48421 .93879 .49924 .99698 57 58 .45487 .83444 .46959 .88532 i .48446 .93973 .49950 .99799 58 59 .45512 .83526 .46983 .88620 ; .48471 .94066 .49975 .99899 59 60 .45536 .83608 .47008 .88708 .48496 .94160 .50000 1.00000 60 [486] TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. l 6 6 1 6 2 3 Vers. Ex .se Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .50000 1.00000 ; .51519 .06267 .53053 .13005 .54601 1.20269 1 .50025 .00101 : .51544 .06375 .53079 .13122 .54627 .20395 1 2 .50050 .00202 i .51570 .06483 .53104 .13239 .54653 .20521 2 8 .50076 .00303 i .51595 .06592 .53130 .13356 .54679 .20647 3 4 .50101 .00404 ! .51621 .06701 .53156 .13473 .5471)5 .20773 4 5 .50126 .00505 .51646 .06809 .53181 .13590 .51731 .20900 5 6 .50151 .00607 .51672 .06918 .53207 .13707 .54757 .21026 6 1 .50176 .00708 .51697 .07027 .53233 .13825 .54782 .21153 7 8 .50202 .00810 .51723 .07137 .53258 .13942 .54808 .21280 8 9 .50227 .00912 .51748 .07246 .53284 .14060 .54834 .21407 9 10 .50252 .01014 .51774 .07356 .53310 .14178 .54860 .21535 10 11 .50277 .01116 .51799 .07465 .53336 .14296 .54886 .21662 11 IS .50303 .01218 ! .51825 .07575 : ! .58861 .14414 .54912 .21790 12 18 .50328 .01320 .51850 .07685 .5.3387 .14533 .54938 .21918 13 14 .50353 .01422 | .51876 .07795 .53413 .14651 .54964 .22045 14 15 .50378 .01525 ! .51901 .07905 .5343!) .14770 .54990 .22174 15 16 .50404 .01628 .51927 .08015 . 534154 .14889 .55016 .23302 16 17 .50429 .01730 .51952 .08126 .53490 .15008 .55042 1.22430 17 18 .50454 .01833 .51978 .08236 .53516 .15127 : .55068 1.22559 18 19 .50479 .01936 .52003 .08347 .53542 .15346 ! .55094 1.22688 19 20 .50505 .02039 .52029 .08458 .5*567 .15366 .55120 i. 2281 7 20 21 .50530 .02143 .52054 .08569 .53593 .15485 1 .55146 1.22946 21 22 .50555 .02246 .52080 .08680 .53619 .15605 i .55172 1.23075 22 28 .50581 .02349 .52105 .08791 .53645 .15725 ; .55198 1.23205 23 24 .50606 .02453 .52131 .08903 .53670 .15845 i .55224 1.23334 24 25 .50631 .02557 .52156 .09014 .53696 .15965 .55250 1.23464 25 20 .50656 .02661 .52182 .09126 .53722 .16085 .55276 1.23594 2(5 27 .50682 .02765 .52207 .09238 .53748 .16206 .55302 1.23724 27 28 .50707 .02869 .52233 .09350 [63774 .16326 .55328 1.23855 28 29 .50732 .02973 .52259 .09462 .53799 .16447 .55354 1.23985 29 30 .50758 .03077 .52284 .09574 .53825 .16568 .55380 1.24116 30 81 .50783 .03182 .5.2310 .09686 .53851 .16689 .55406 1.24247 31 83 .50808 .03286 .52335 .09799 .53877 .16810 .55432 1.24378 32 33 .50834 .03391 .52361 .09911 .53903 .16932 .55458 1.24509 33 34 .50859 .03496 .52386 .10024 .53928 .17053 .55484 1.24 WO 34 88 .50884 .03601 .52412 .10137 .53954 .17175 .55510 1.24772 35 36 .50910 .03706 .52438 .10250 .53980 .17297 .55536 1.24903 36 87 .50935 .03811 .52463 .10363 .54006 .17419 .55563 1.25035 37 38 .50960 1.03916 .52489 .10477 .54032 .17541 .55589 1 25167 38 39 .50986 1.04022 .52514 .10590 .54058 .17663 .55615 1.25300 39 40 .51011 1.04128 .52540 .10704 .54083 .17786 .55641 1.25432 40 41 .51036 1.04383 .52566 .10817 .54109 .17909 .55667 1.25565 41 42 .51062 1.04339 .52591 .10031 .54185 .18031 .55693 1.25(597 42 43 .51087 1.04445 .52617 .11045 .54101 18154 .55719 1.25880 43 44 .51113 1.04551 .52642 .11159 .54187 .18277 .55745 1.25963 44 45 .51138 1.04658 .52668 .11274 .54213 .18401 .55771 1.2(5097 45 46 .51163 1.04764 .52694 .11388 ..54238 .18524 .55797 1.26230 40 47 .51189 1.04870 .52719 .11503 .54264 .18648 .55823 1.26364 47 48 .51214 1.04977 .52745 .11617 54290 .18772 .55849 1.26498 48 49 .51239 1.05084 .52771 1.11782 .54316 .18895 .55876 1.26632 49 50 .51265 1.05191 .52796 1.11847 .54342 .19019 .55902 1.26766 50 51 .51290 1.05298 .52822 1.11963 .54368 .19144 .55928 1.26900 51 52 .51316 1.05405 .52848 1.12078 .54394 . 19268 .55954 1.27035 52 53 .51341 1.05512 .52873 1.12193 .54420 .19393 .55980 1.27169 53 54 .51366 1.05619 .52899 1.12309 .54446 .19517 .56006 1.27304 54 55 .51392 1.05727 .52924 1.12425 i .54471 .19642 .56032 1.27439 55 56 .51417 1.05835 .52950 1.12540 .54497 .19767 .56058 1.27574 56 57 .51443 1.05942 .52976 1.12657 .54523 .19892 .56084 1.27710 57 58 .51468 1.06050 .53001 1.12773 .54549 .20018 .56111 1.27845 58 59 .51494 1.06158 .53027 1. 12889 .54575 .20143 .56137 1.27981 59 60 .51519 1.06267 .53053 1.13005 .54601 1.20269 .56163 1.28117 60 [487! TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. 6 4 i 6 5 6 6" 6 7 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .56163 1.28117 .57738 .36620 ! .59326 .45859 .60927 1.55930 1 .56189 1.28253 .57765 .36768 .59353 .46020 .60954 1.56106 1 8 .56215 1.28390 .57791 .36916 .59379 .46181 .60980 1.56282 2 8 .56241 1.28526 .57817 .37064 .59406 .46342 .61007 1.56458 3 4 .56267 1.28663 .57844 .37212 .59433 .46504 .61034 1.56634 4 '6 .56294 1.28800 .57870 .37361 .59459 .46665 .61061 1.56811 5 6 .56320 1.28937 ! .57896 .37509 .59486 .46827 .61088 1.56988 6 7 .56346 1.29074 .57923 .37658 i .59512 .46989 .61114 1.57165 7 8 .56372 1.29211 1 .57949 .37808 .59539 .47152 .61141 1.57342 8 9 .56398 1.29349 i .57976 .37957 ! .59566 .47314 .61168 1 57520 9 10 .56425 1.29487 .58002 .3810? | .59592 .47477 .61195 1.57698 10 11 .56451 1.29625 .58028 .38256 .59619 .47640 .61222 1.57876 11 }g .50477 1 29763 .58055 .38406 .59645 .47804 .61248 1.58054 IS & .56503 1.29901 ' .58081 .38556 .59672 .47967 .61275 1.58233 18 14 .56529 1.30040 .58108 .38707 .59699 .48131 ; .61302 1.58412 H 15 .56555 1.30179 .58134 .88857 .59725 .48295 ; .61329 1.58591 18 10 .56582 1.30318 .58160 .39008 i .59752 .48459 .61356 1.58771 Hi 17 .56608 1.30457 ! .58187 .39159 .59779 .48624 .61383 1.58950 if 18 .56634 1.30596 .58213 .39311 .59805 .48789 .61409 1.59130 18 1!) .56660 1.30735 ! .58240 .39462 .59832 .48954 .61436 1.59311 1!) 20 ,56687 1.30875 .58266 .39614 j .59859 .49119 .61463 1.59491 20 a .56713 1.31015 .58293 .39766 .59885 .49284 .61490 1.59672 2\ 22 .56739 1.31155 .58319 .895)1 S .59912 .49450 .61517 1.59853 > 2:5 .56765 1.31295 .58345 .40070 .59938 .49616 i .61544 1.60035 !>8 24 .56791 1.31436 .58372 .40222 i .59965 .49782 .61570 1.60217 4 25 .56818 1.31576 .58398 .40375 .59992 .49948 .61597 1.60399 26 20 .56844 1.31717 .58425 .40528 .60018 .50115 .61624 1.60581 26 t>7 .56870 1.31858 .58451 .40681 .60045 .50282 .61651 1.60763 27 88 .56896 1.31999 .58478 .40835 .60072 .50449 .61678 1.60946 1$ 2<) .56923 1.32140 .5&504 .40988 .60098 .50617 i .61705 1.61129 x><) Qp .56949 1.32282 .58531 .41142 .60125 .50784 .61732 1.61313 30 8] .56975 1.32424 .58557 .41296 .60152 .50952 .61759 1.61496 81 32 .57001 1.32566 .58584 .41450 .60178 .51120 .61785 1.61680 3-,' 38 .57028 1.32708 .58610 .41605 .60205 .51289 .61812 1.61864 38 34 .57054 1.32850 .58637 .41760 .60232 .51457 .61839 1.62049 34 8S .57080 1.32993 .58663 .41914 .60259 .51626 .61866 1.62234 85 8(5 .57106 1.33135 .58690 .42070 .60285 .51795 .61893 1.62419 86 87 .57133 1.33278 .58716 .42225 .60312 .51965 .61920 1.62604 37 88 .57159 1.33422 .58743 .42380 .60339 .52134 .61947 1.62790 38 8'J .57185 1.33565 ! .58769 .42536 .60365 .52304 .61974 1.62976 81) 40 .57212 1.33708 ! .58796 .42692 .60392 .52474 .62001 1.63162 40 41 .57238 1.33852 .58822 .42848 .60419 .52645 .62027 1.63348 41 42 .57264 1.38996 .58849 .43005 .60445 .52815 .62054 1.63535 48 .57291 1.34140 .58875 .43162 .60472 .52986 .62081 1.63722 48 44 .57317 1.34284 .58902 .43318 .60499 .53157 .62108 1.63909 44 [45 .57343 1.34429 .58928 .43476 .60526 .53329 .62135 1.64097 45 40 .57369 1.34573 .58955 .43633 .60552 .53500 .62162 1.64285 46 47 .57396 1.34718 .58981 .43790 j .60579 .53672 .62189 1.64473 47 4H .57422 1.34863 ' .59008 .43948 ! .60606 .53845 .62216 1.64662 48 4!) .57448 1.35009 .59034 .44106 i .60633 .54017 .62243 1.64851 4'J 00 .57475 1.35154 .59061 .44264 .60659 .54190 .62270 1.65040 50 51 .57501 1.85300 .59087 .44423 .60686 .54363 ' .62297 1.65229 51 52 .57527 1.35446 .59114 .44582 .60713 .54536 .62324 1.65419 6$ 58 .57554 1.35592 .59140 .44741 .60740 .54709 .62351 1.65609 58 54 .57580 1.35738 .59167 .44900 .60766 .54883 .62378 1.65799 54 55 .57606 1.35885 .59194 .45059 \ .60793 .55057 .62405 1.65989 55 50 .57633 1.36031 .59220 .45219 i .60820 .55231 .62431 1.66180 50 57 .57659 1.36178 .59247 .45378 .60847 .55405 .62458 1.66371 57 58 .57685 1.36325 .59273 .45539 .60873 .55580 .62485 1.66563 58 51) .57712 1.36473 .59300 1.45699 .60900 1.55755 .62512 1.66755 5!) 00 .57738 1.36620 .59326 1.45859 .60927 1.55930 .02539 1.06947 00 [4381 TABLE xxix. NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 68 69 i II TO- [ 71" I Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec ( .62539 1.60947 .64163 1.79043 .65798 1.92380 .67443 2.07155 1 .62560 1.67139 .64190 1.79354 .65825 1.92614 .67471 2.07415 t 2 .62593 1.67332 .64218 1.79466 .65853 1.92849 .67498 2.07'675 2 3 .62620 1.67525 i .64245 1.79679 .65880 1.93083 .67526 3.07986 3 4 .62647 1.67718 .64272 i 1.79891 .65907 1.93318 .67553 2.08197 4 5 .62674 1.67911 .64299 1.80104 .65935 1.93554 .67581 2.08459 5 6 .62701 1.68105 I .64326 1.80318 .65962 1.93790 .67608 2.08721 6 7 .62723 1.68299 .64353 1.80531 .65989 1.94026 .67636 2.08983 7 8 .62755 ; 1.68494 .64381 1.80746 i .66017 1.94263 .67663 2.09246 8 9 .62782 ! 1.68689 .64408 1.80960 1 .66044 1.94500 .67691 2.09510 9 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 i .66126 1.95213 .67773 2.10303 12 13 .62890 1.69471 .64517 1.81821 i .66154 1.95452 .67801 2.10568 13 14 .62917 1.69667 .64544 1.82037 .66181 1.95691 .67829 2.10834 14 15 .62944 1.69864 .64571 1.82254 .66208 1.95931 .67856 2.11101 15 16 .62971 1.70061 .64598 1.82471 .66236 1.95171 .67884 2.11367 i!6 17 .62998 1.70258 .64625 1.82688 .66263 1.96411 .67911 2.11635 (17 18 .63025 1.70455 .64653 1.82906 ! .66290 1.96652 .67939 2.11903 18 19 .63052 1.70653 .64680 1.83124 1 .66318 1.96893 .07968 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 22 23 .63161 1.71448 .64789 1.83999 I .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 1.98594 .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 28 29 .63323 1.72649 .64952 1.85323 .66592 1,99329 .68242 2.14881 29 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 .66074 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 1.80437 .66729 2.00562 .68380 2.16255 34 35 .63485 1.73862 .65110 1.86661 .66756 2.00810 .68408 2.16531 35 36 .63512 1.74065 .65143 1.86885 .66784 2.01059 .68435 2.16808 36 37 .63539 1.74269 .65170 1.87109 .66811 2.01308 .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 1 .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 .63675 1.75292 .65306 1.88238 .66949 2.02559 .68601 2.18479 42 43 .63702 1.75497 .65334 1.88465 .66976 1 2.02810 .68628 ; 2.18759 43 44 .63729 1.75703 .65361 1.88692 .67003 >. -W>-> .68656 1 2.19040 44 45 .63756 1.75909 .65388 1.88920 .67031 2.03315 .68084 2.19322 45 .63783 1.76116 .65416 1.89148 .67058 2.03568 .68711 2.19604 46 'i .63810 1.76333 .65443 1.89376 .67086 2.03821 .68739 2.19886 47 48 .63838 1.76530 .05470 1.89605 .67113 2.04075 .68767 2.20169 48 49 .63865 1.76737 .65497 1.89834 .67141 2.04329 .68794 2.30463 49 50 .63892 1.76945 .655&J 1.90063 .67168 2.04584 .68822 2.20737 50 51 .63919 1.77154 .65552 1.90293 .67196 2.04839 .68849 2.21021 51 52 .63946 1.77302 .65579 1.90524 .67223 2.05094 .68877 2.21306 52 53 .63973 1.77571 .65607 1.90754 .67'251 2.053(50 .68905 2.21592 53 54 .64000 1 77780 .65634 1.90986 .67278 2.05607 .68932 2.21878 54 x3 .64027 1.77990 .65661 1.91217 .07306 2.05864 .68960 2.22165 55 56 .64055 1.78200 .65689 1.91449 .67333 2.06121 .68988 : 2.22452 i 56 >7 .64082 1.78410 .65716 1.91681 I .67361 S.0637'9 .69015 2.22740 57 58 .64109 1.78621 .05743 1.91914 !| .67388 2.06637 .69043 2.23028 58 59! .64136 1.78832 .65771 1.92147 , .07410 2.06896 .69071 2.23317 59 60 i .64163 1.79043 .65798 1.92380 .07443 2.07155 .69098 2.23607 60 [489] TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 72 73 74 75 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .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 j 2 .69154 2.24187 .70818 2.42683 .72492 2 63533 .74174 2.87211 2 3 .69181 2.24478 .70846 2.43010 .72520 2.63903 .74202 2.87633 3 4 .69209 2.24770 .70874 2.43337 .72548 2.64274 .74231 2.88056 4 5 .69237 2.25062 .70902 2.43666 .72576 2.64645 .74259 2.88479 5 6 69264 2.25355 .70930 2.43995 .72604 2.65018 .74287 2.88904 6 7 .69292 2.25648 .70958 2.44324 .72632 2.65391 .74315 2.89330 7 8 .69320 2.25942 .70985 2.44655 .72660 2.65765 .74343 2.89756 8 9 .69347 2 26237 .71013 2.44986 .72688 2.66140 .74371 2.90184 9 10 .69375 2.26531 .71041 2.45317 .72716 2.66515 I .74399 2.90613 10 11 .69403 2.26827 .71069 2.45650 .72744 2.66892 .74427 2.91042 11 12 .69430 2.27123 .71097 2.45983 .72772 2.67269 .74455 2.91473 12 i* .69458 2.27420 .71125 2.46316 .72800 2.67647 .74484 2.91004 13 14 .69486 2.27717 .71153 2.46651 .72828 2.68025 .74512 2.92337 14 18 .69514 2.28015 .71180 2.46986 .72856 2.68405 .74540 2.92770 15 16 .69541 2.28313 .71208 2.47321 .72884 2.68785 .74568 2.93204 re 17 .69569 2.28612 .71236 2.47658 .72912 2.69167 .74596 2.93640 17 18 .69597 2.28912 .71264 2.47995 .72940 2.69549 .74624 2.94076 is 10 .69624 2.29212 .71292 2.48333 .72968 2.69931 .74652 2.94514 19 80 .69652 2.29512 .71320 2.48671 .72996 .?.. 70315 .74680 2.94952 20 21 .69680 2.29814 .71348 2.49010 .73024 2.70700 .74709 2.95392 21 88 .69708 2.30115 .71375 2.49350 .7:3052 2.71085 .74737 2.95832 22 88 .69735 2.30418 .71403 2.49691 .73080 2.71471 .74765 2.96274 23 24 .69763 2.307'21 .71431 2.50032 .73108 2.71858 .74793 2.96716 21 25 .69791 2.31024 .71459 2.50374 .73136 2.72246 .74821 2.97160 25 96 .69818 2.31328 .71487 2.50716 .73164 2.72635 1 .74849 2.97604 86 27 .69846 2.31633 .71515 2.51060 .73192 2.73024 , .74878 2.98050 K 28 .69874 2.31939 .71543 2.51404 .73220 2.73414 ! .74906 2.98497 88 2!) .69902 2 32244 .71571 2.51748 .73248 2.73806 : .74934 2.98944 89 80 .69929 2.32551 .71598 2.52094 .73276 2.74198 .74962 2.99393 80 31 .69957 2.32858 .71626 2.52440 .73304 2.74591 .74990 2.99843 31 82 .69985 2.33166 .71654 2.52787 .73332 2.7'4984 .75018 3.00293 32 33 .70013 2.33474 .71682 2.53134 .73360 2.75379 i .75047 3.00745 33 34 .70040 2.33783 .71710 2.53482 .73388 2.75775 .75075 3.01198 31 35 .70068 2.34092 .71738 2.53831 .73416 2.76171 .75103 3.01652 35 86 .70096 2.34403 .71766 2.54181 .73444 2.76568 .75131 3.02107 30 87 .70124 2.34713 .71794 2.54531 .73472 2.76966 .75159 3.02563 37 88 .70151 2.33085 .71822 2.54883 .73500 2.77365 .75187 3.03020 38 89 .70179 2.35336 .71850 2.55235 .73529 2.77765 .75216 3.03479 89 40 .70207 2.35649 .71877 2.55587 .73557 2.78166 .75244 3.03938 10 41 .70235 2.35962 .71905 2.55940 .73585 2.78568 .75272 3.04398 41 42 .70263 2.36276 .71933 2.56294 .73613 2.78970 .75300 3.04860 42 4.3 .70290 2.36590 .71961 2.56649 .73641 2.79374 .75328 3.05322 43 44 .70318 2.36905 .71989 2.57005 .73669 2.79778 .75356 3.05786 44 45 .70346 2.37221 .72017 2.57361 .73697 2.80183 .75385 3.06251 45 46 .70374 2.37537 .72045 2.57718 .73725 2.80589 .75413 3.06717 46 47 .70401 2.37854 .72073 2.58076 .73753 2.80996 .75441 3.07184 47 4 .70429 2.38171 .72101 2.58434 .73781 2.81404 .75469 3.07652 48 49 .70457 2.38489 .72129 2.58794 .73809 2.81813 .75497 3.08121 49 50 .70485 2.38808 .72157 2.59154 .73837 2.82223 .75526 3.08591 50 51 .70513 2.39128 .72185 2.59514 .73865 2.82633 .75554 3.09063 51 52 .70540 2.39448 .72213 2.59876 .73893 2.83045 .75582 3.09535 52 53 .70568 2.39768 .72241 2.60238 .73921 2.&S457 .75610 3.10009 53 54 .70596 2.40089 .72269 2.60601 .73950 2.83871 .75639 3.10484 54 55 .70624 2.40411 .72296 2.60965 .73978 2.84285 .75667 3.10960 55 5(5 .70652 2.40734 .72324 2.61,330 .74006 2.84700 .75695 3.11437 56 57 .70679 2.41057 .72352 2.61695 .74034 2.85116 .75723 3.11915 57 58 .70707 2.41381 .72380 | 2.62061 .74062 2.85533 .75751 3.12394 58 59 .707&5 2.41705 .72408 2.62428 .74090 2.85951 .75780 3.12875 59! 60 .70763 2.42030 .72436 2.6279G .74118 2.86370 .75808 3.13357 60 J l493 TABLE XXIX.-NATURAL VERSED SINES AND EXTERNAL SECANTS, ! '1 1! 76 77 78 79 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ! Vers. Ex. sec. .75808 3.13357 .77505 3.44541 .79209 3.80973 .80919 4.24084 1 .75836 3.13839 .77533 3.45102 .79237 3.81633 .80948 4.24870 1 2 .75864 3.14323 .77562 3.45664 .79266 3.82294 .80976 4.25658 2 3 .75892 3.14809 .77590 i 3.46228 .79294 3.82956 .81005 4.26448 3 4 .75921 3.15295 .77618 3.46793 .7932? 3.83621 .81033 4.27241 4 5 .75949 3.15782 .77647 3.47360 .79351 3.84288 : .81062 4.28.036 5 6 .75977 3.16271 .77675 3'. 47928 .79380 3.84956 .81090 4.28833 6 7 .76005 3.16761 .77703 3.48498 .79408 3.85627 1 .81119 4.29634 r* 8 1 .76034 3.17252 .77732 i 3.49069 .79437 3.86299 .81148 4.30436 8 9 .76062 3.17744 .77760 3.49642 .79465 3.86973 i .81176 4.31241 9 10 .76090 3.18238 .77788 3.50216 .79493 3.87649 I .81205 4.32049 10 11 .76118 8.18788 .77817 3.50791 .79522 3.88327 .81233 4.32859 11 12 .76147 3.19228 .77845 3.51368 .79550 3.89007 : .81262 4.33671 |12 13 76175 3.19725 .77874 3.51947 .79579 3.89689 .81290 4.34486 13 14 .76203 3.20224 .77902 3.52527 || .79607 3.90373 .81319 4.35304 14 15 .76231 3.20723 ! .77930 3.53109 ! .79036 3.91058 .81348 4.36124 15 16 .76260 3.21224 1 .77959 3.53692 .79664 3.91746 .81376 4.36947 16 17 .76288 3.21726 .77987 3.54277 .79693 3.92436 .81405 4.37772 17 18 .76316 3.22229 i .78015 3.54863 .79721 3.93128 .81433 4.38600 18 19 ' .76344 3.22734 ! .78044 3.55451 !79750 3.93821 .81462 4.39430 19 20 | .76373 3.23239 .78072 3.56041 .78778 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 .79885 3.95914 .81548 4.41937 22 23 .76458 3.24764 .78157 3.57819 .79864 3.96616 .81576 4.42778 23 24 .76486 3.25275 .78186 3.58414 .79892 3.97320 .81605 4.43622 24 25 .76514 3.25787 .78214 3.59012 .79921 3.98025 .81633 4.44468 25 26! .76542 3.26300 .78242 3.59611 .79949 3.98733 .81662 4.45317 '26 27 .76571 3.26814 .78271 3.60211 .7997'8 3.99443 .81691 4.46169 27 28 .76599 3.27330 .78299 3.60813 .80006 4.00155 .81719 4.47023 128 2'.) .76627 3.27847 .78328 3.61417 .80035 4.00869 .81748 4.47881 39 30 .76655 3.28366 .78356 3.62023 .80063 4.01585 ! .81776 4.48740 30 31 .76684 3.28885 .78384 3.62630 .80092 4.02303 .81805 4.49603 31 32 .76712 3.29406 .78413 3.63238 .80120 4.03024 i .81834 4.50468 32 33 .76740 3.29929 .78441 3.63849 .80149 4.03746 i .81862 4.51337 m 34 .76769 3.30452 .78470 3.64461 .80177 4.04471 .81891 4.52208 34 351 .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 36 37 .76854 3.32031 '78565 3.66307 .80263 4.06657 .81977 4.54837 37 38 .76882 3.32560 .78583 3.66925 .80291 4.07390 .82005 4.557'20 138 39 .76910 3.33090 .78612 3.67545 .80320 4.08125 .82034 4.56605 39 40 .76938 3.33622 .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 i .82120 4.59277 42 43 .77023 3.35224 .78725 3.70044 ! .80434 4.11088 .82148 4.60174 43 44l .77052 3.35761 .78754 3.7-0673 || .80462 1.1. ttr> .82177 4.61073 44 45 .77080 3.36299 .78782 3.71303 .8041)1 4.12583 .82206 4.61970 45 46 .77108 3.36839 .78811 3.71935 j| .80520 4.13334 .82234 4.02881 40 47 .77137 3.37380 .78839 3.72569 ! .80548 4.14087 .82263 4.63790 47 48 .77165 3.37923 .78868 3.73205 i .80577 4.14842 .82292 4.64701 48 49 .77193 3.38466 .78896 3.73843 il .80005 4.15599 .82320 4.65010 4!) 501 .77222 3.39012 .78924 3.74482 i .80634 4.16359 .82349 4.00533 50 511 .77250 3.39558 .78953 3.75123 .80662 4.17121 .82377 4.67454 51 52 .77278 3.40106 .78981 3.75766 .80691 4.17886 .82406 4.68377 152 53 .77307 3.40656 .79010 3.70411 .80719 i 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 4.711(50 ro 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 1! .80891 4.23301 .82607 4.74929 59 60 .77505 3.44541 .79209 3.80973 .80919 4.24084 .82635 4.75877 60 [4Qt] TABLE XXTX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 80 81 ! 82 83 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .82635 4.75877 .84357 5.39245 .86083 6.18530 .87813 7,20551 1 .82664 4.76829 .84385 5.40422 .86112 6.20020 .87842 7.22500 1 8 .82692 4.77784 .84414 5.41602 .86140 6.21517 .87871 7.24457 2 8 .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 B .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 Y .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 it .82950 4.86524 .84673 5.52429 .86400 6.35274 .88131 7.42511 11 19 .82979 4.87511 .84701 5.53655 .86428 6.36835 .88160 7.44566 12 ia .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 l(j .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 IS .83151 4.93509 .84874 5.61110 .86601 6.46346 .88333 7.57113 18 1!) .83180 4.94521 .'84903 5.6?369 .86630 6.47955 .88362 7.59241 19 BQ .83208 4.95536 .84931 5.63633 .86659 6.49571 .88391 7.61379 20 u .83237 4.96555 .84960 5.64902 .86688 6.51194 .88420 7.63528 21 3B .83266 4.97577 .84989 5.66176 .86717 6.52825 .88448 7.65688 22 88 .88894 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 as .83352 5.00666 .85075 5.70027 .86803 6.57759 .88535 7.72234 25 aa .83380 5.01703 .85104 5.71321 .86832 6.59418 .88564 7.74438 26 -.27 .83409 5.02743 .85133 5.72620 .86861 6.61085 .88593 7.76653 27 88 .83438 5.03787 .85162 5.73924 .86890 6.62759 .88622 7.78880 28 2!) .83467 5.04834 .85190 5.75233 .86919 6.64441 .88651 7.81118 29 30 .83495 5.05886 .85219 j. 76547 .86947 6.66130 .88680 7.83367 30 31 .83524 5.06941 .85348 5.77866 .86976 6.67826 .88709 7.85628 31 88 .83553 5.08000 .85277 5.79191 .87005 6.69530 .88737 7.87901 32 33 .83581 5.0J062 .85305 5.80521 .87034 6.71242 .88766 7.90186 33 34 .83010 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 3 3 .89834 8.64687 .91371 10.58932 .93111 13.51676 3 4 .89868 S. 07387 .91400 10.6.2837 .93140 13.57817 4 5 .89692 8.70103 .91429 10.66769 .93169 13.64011 6 6 .89721 8.72838 .91458 10.70728 ! 93198 13.70258 6 7 .89750 8.75579 .91487 10.74714 .93227 13.76558 8 .89779 8.78341 .91516 10.78727 .93257 13.82913 8 9 .89808 8.81119 .31545 10.82768 .98286 13.89323 9 10 .89836 8.83912 .91574 1C. 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 .93102 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 . ill 748 11.11852 .93489 14. 35795 16 17 .90039 9.03923 .91777 11.16125 .93518 14.42672 17 18 .90068 9.06849 .91806 11.20427 .93547 14.49611 18 19 .90097 9.09792 .91835 11.24761 .'.13576 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 .91928 11.87948 .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 .83750 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 9.37077 .92096 11.65197 .93837 15.22607 28 29 .90386 9.40201 .92125 11.69356 .93866 15.30287 29 30 .90415 9.43343 .92154 11.74550 .93895 15.38041 30 31 .90444 9.46505 .92183 11.78878 .93924 15.45869 31 32 .90473 9.49685 .92212 11.84042 .93953 15.53772 32 33 .90502 9.52886 .922-11 11.88843 .93982 15.61751 33 34 .90531 9.56106 .92270 11.93(577 .94011 15.69808 34 35 .90560 9.59346 .92299 11. 98:> 111 .94040 15.77944 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 <. 69186 .92386 12.13388 .1)4127 16.02835 38 39 .90676 9.78507 .92415 12.18411 .94156 16.11297 39 40 .90705 9.75849 .92414 12.23472 .94186 16.19843 40 41 .90784 9.79212 .92473 12.28572 .94215 16.28476 41 42 .90703 9.82506 .92502 12.33712 .94244 16.37196 42 43 .90792 9.8600J .92531 12.38SD1 .94273 16.46005 43 44 .90821 9.89428 .92560 12.44112 .94302 16.54903 44 45 .90850 9 92877 .92589 12.49373 .94331 16.63893 45 40 .90879 9.96348 .92618 12.54676 .94360 16.72975 46 47 .90908 9.99841 ! 92647 12.60021 .94:389 16.82152 47 48 .90937 10.0.^f; .92676 12.65408 .94418 16.91424 48 49 .90966 10.06894 ! 93705 12] 70838 .94447 17.00794 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 [92792 12.87391 .94534 17.29501 52 53 .91082 10.21277 .92821 12.92999 .94563 17.39274 53 54 .91111 10.24932 .92850 12.98651 .94592 17.49153 54 55 01140 I0.2S01:) .92879 13 04: VK) .94621 17.59139 55 56 .91169 10.32313 .92908 18.10096 .94650 17.69233 56 57 .91197 10.36040 .92937 13.15889 .94679 17.79438 57 58 .91226 10.39702 .92966 13.21730 .94708 17.89755 58 59 .91355- 10.43569 .92995 13.27620 .94737 18.00185 59 60 .91284 10.47371 .93024 13.33559 .947'66 18.10732 60 [493J TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. 1 87 88 89 e Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .94766 18.10732 .96510 27.65371 .98255 56.29869 1 .94795 18.21397 .96539 27.89440 .98284 57.26976 1 2 .94825 18,32182 .96568 28.13917 .98313 58.27431 2 3 .94854 18.43088 .96597 28.38812 .98342 59.31411 3 4 .94883 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 e 7 .94970 18.87976 .96714 29.42802 .98458 63.86572 7 8 .94999 18.99524 .96743 29.69960 .98487 65.11304 9 .95028 19.11208 .96772 29.97607 .98517 66.40927 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 18 .95289 20.22852 .9703} 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 .95486 20.90409 .97179 34.44539 .98924 91.91387 23 21 .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 &5. 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 .99098 109.8966 29 30 .95638 21.92559 .97382 37.20155 .99127 113.5930 30 31 .95667 22.07935 .97411 37.63068 .99156 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 .95754 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 .99302 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 .99389 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 46 .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 .99680 311.5230 49 50 .96219 25.45051 .97964 48.11406 .99709 342.7752 50 51 .96248 25.65546 .97993 48.82576 .99738 380.97'23 51 52 .96277 25.86360 .98022 49.55840 .99767 428.7187 52 53 .96307 26.07503 .98051 50.31290 .99796 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 54.45053 .99942 1717.874 58 59 .96481 27.41700 .98226 55.35946 .99971 3436.747 59 60 .96510 27.65371 .98255 56.29869 i 1.00000 Infinite 60 TABLE XXX. CUBIC YARDS PER 100 FEET. SLOPES r~ Depth d 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 4 193 222 252 281 341 370 400 430 5 245 282 319 356 431 468 505 542 6 300 344 389 433 522 567 611 656 a56 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 10 537 611 085 759 907 981 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 1&53 2023 2364 2534 2705 2875 24 1600 1778 1955 2133 2489 2667 2844 3022 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 3:556 3571 3786 30 2167 2389 2610 2833 3278 3500 3722 3944 31 2208 2497 2726 2956 3416 3645 3875 4105 32 23 ro 2607 2844 3081 3556 3793 4030 4267 33 2475 2719 2964 3208 3697 3942 4186 4431 34 2581 2833 3085 3337 3841 4093 4344 4596 35 2690 2949 3208 3468 3986 4245 4505 4764 36 2800 3067 3333 3600 4133 4400 4667 4933 37 2912 3186 3460 3734 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 3379 3662 3986 4290 4897 5201 5505 5808 42 ,3500 3811 4122 4433 5050 5307 567'8 5989 43 3023 :!',) l-> 4260 4579 5216 5534 5853 6171 44 9748 4074 4400 4726 5378 5704 6030 6356 45 3875 4208 4541 4875 5542 5875 6208 6542 46 4004 4:^44 4684 5020 5707 6048 6389 6730 47 4134 4482 4830 5179 587: 6223 6571 6919 48 4267 4622 4978 i 5333 6044 6400 6756 7111 49 4401 4764 5127 5490 6216 6579 6942 7305 50 4537 4907 5278 5648 6389 6759 7130 7500 51 4675 5053 5430 5808 6504 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 6000 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 [495! TABLE XXX. CUBIC YAHDS TEH 100 FEET. SLUFES Depth Base Base Base Base Base Base Base Base d 12 14 16 18 22 24 26 28 1 46 54 61 f G9 83 91* 98 100 2 96 111 126 f 141 170 185 200 215 3 150 172 194 217 261 283 306 328 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 12 800 889 978 1067 1244 1.333 1422 1511 13 891 987 1083 1180 1372 1469 1565 1661 14 985 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 18 1400 1533 1667 1800 2067 2200 2333 2467 19 1518 1654 1794 1935 2217 2357 2498 2639 20 1630 1778 1928 2074 2370 2519 2667 2815 21 1750 1906 2061 2217 2528 26&3 2839 2994 22 1874 2037 2200 2363 2689 2852 3015 3178 23 2002 2172 2343 2513 2854 3024 3194 3365 24 2188 2311 2489 26(57 3022 3200 3378 3556 25 2269 2454 2639 2824 3194 3380 3565 3750 26 2407 2600 2793 2985 3370 a5C3 3756 3948 27 2550 2750 2950 3150 a550 3750 3950 4151 28 2696 2904 3111 3319 37as 3941 4148 4356 29 2846 3061 3276 3491 3920 4135 4350 4565 30 3000 3222 3444 3667 4111 4333 4556 4778 31 3157 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 34 3652 3904 4156 4407 4911 5163 5415 5667 ' 35 3824 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 6611 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 7622 43 5335 5654 5972 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 C644 6985 7667 8007 8348 8689 47 6180 6528 6876 7224 7920 8269 8617 8965 48 49 6400 6624 6756 6987 m m 8178 8439 88 in which substitute for - d ^ . and for ^ , and lefc cote- cot - =-a _ 1 _ - C ot cotQ.cot -; C0t (1 + a 2 - cot (1 + a 2 ~ - a cot 6 BOOKS FOR CIVIL ENGINEERS PUBLISHED BY JOHN WILEY SONS. 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