JOHNSON'S TABLES "REESE LIBRARY UNIVERSITY OF CALIFORNIA JOHNSON'S TABLES. STADIA AND EARTH-WORK TABLES FOUR-PLACE LOGARITHMS, LOGARITHMIC TRAVERSE TABLE, NATURAL FUNCTIONS, MAP PROJECTIONS, ETC., ETC. REPRINTED FROM THEORY AND PRACTICE OF SURVEYING. BY J. B. JOHNSON, PROFESSOR OF CIVIL ENGINEERING,' WASHINGTON UNIVERSITY, ST. LOUIS NEW YORK: JOHN WILEY & SONS, 53 EAST TENTH STREET. 1892. COPYRIGHT, 1892, BY J. B. JOHNSON. 7 NOTE BY THE AUTHOR. THE great use made by engineers of three of the following tables, viz., the Four-place Logarithmic Table, the Stadia Table, and the table giving Prismoidal Volumes, has necessi- tated the binding of these in more convenient form than that in which they first appeared in the Theory and Practice of Sur- veying. Since the cost is not materially increased by addi- tional pages, the remaining tables are also included, as well as the entire chapter on the Measurement of Volumes. The Stadia Tables were computed by Mr. Arthur Winslow, State Geologist of Missouri, and first published by the Penn- sylvania Geological Survey. The four-place logarithm tables were originally taken from Lee's Tables and Formulae, a pub- lication of the U. S. Engineer Corps. The table giving Vol- umes by the Prismoidal Formula was computed by the Author, It is the only table, he believes, giving volumes by the pris- moidal formula at one operation. It may also be used for Mean End-areas. Tables IV and VIII are also original in their ar- rangement. J. B. J. iii EXPLANATION OF TABLES. TABLES I, II, III, VI, and VII require no explanation. TABLE IV gives logarithmic sines and cosines to four places for computing latitudes and departures when the angles are read from zero to 360 degrees. It can of course be used for bearings reading from zero to 90 degrees, as is ordinarily done in compass work. In stadia work, and always in transit work where the instrument is graduated continuously to 360 degrees, this table will be found very convenient for coordinating trav- erse lines, as well as for computing latitudes and departures for closed surveys. From zero to 5 degrees, and from 85 to 90 degrees, the tables give values for each minute of arc without tabular dif- ferences. From 5 to 45 degrees values are given for each 10 minutes of arc with tabular differences for the log. sines, and from 45 to 85 degrees with tabular differences for the lo-minute increments for the log. cosines. In the other cases the tabular difference is so small as to be readily taken at sight. Table III A can of course be used in place of Table IV if preferred. TABLE V gives horizontal distance and difference of elevation for inclined sights in stadia work. The true equations of reduction are : Hor. Dist. r cos a z> + (+/) cos z/, . . . . (i) and Dif. Elev. = r cos v sin v -f- (c -)-/) sin v ; . . (2) EXPLANATION OF TABLES. where r reading of distance on stadia rod when held vertically ; v = vertical angle with the horizon ; /= focal length of objective ; c = distance from objective to centre of instrument. The tables give the values for the first term only of the second member. The values for the second term are given at the bottom of the page, the constant term (c-\-f) in the above equations being there called " c" The sum of these two dis- tances, viz., distance from centre of instrument to objective plus distance from cross-wires to objective, varies in different instruments from nine to fifteen inches. Three values of this second term are given, therefore, one corresponding to c-\-f=. 0.75 foot, one to c-\-f-=> i.oo foot, and one to c-\-f=. 1.25 foot. In ordinary work these corrections may be neglected. See chapter on Stadia Surveying in the Theory and Practice of Surveying. A Reduction Diagram, printed from an engraved plate 20 by 24 inches, has been prepared with great care, giving correc- tions to the horizontal distance read, and the differences of elevation, for inclined sights, as shown by the table, not includ- ing the (+/) term. For all angles below 6 and distances less than 1500 feet, with differences of elevation less than 50 feet, this diagram is much preferable to the table. The results are found at one operation, to the nearest tenth of a foot, with great rapidity. It can be procured from the pub- lisher of these tables, printed on heavy lithographic paper, price 50 cents, post paid. TABLE VIII gives the coordinates to be used in the poly- conic projection of maps. It is fully explained in the chapter on Projection of Maps in the Surveying. TABLES IX and X will be found very useful in sewer and hydraulic work where Kutter's formula is to be used. They Vi EXPLANATION OF TABLES. are fully explained in the chapter on Hydrographic Survey- ing. TABLE XI gives correct volumes of prismoids, by the pris- moidal formula. For the benefit of railroad engineers and others who either do not possess a copy of the Surveying, or who do not have it by them, the entire chapter on the Measurement of Volumes is here inserted. At least seven pages of this chapter is requisite to a full explanation of the table, and for the sake of completeness, and to show the superiority of this table over any table of volumes from mean end-areas, or by the use of diagonals, it has been thought best to insert the entire chap- ter. TABLE XII gives the azimuth of Polaris at any hour-angle. By its use an observation for azimuth to the nearest minute of arc can be made at any hour when the star is visible, provided the local time is known to within one or two minutes. When the observation is taken two hours from the time of elongation, the local time need not be known nearer than five minutes. A detailed explanation of its use is given in the Surveying, Art. 38 1 A. CONTENTS. EXPLANATION OF TABLES iv THE MEASUREMENT OF VOLUMES. 310. Proposition I 311. Grading over Extended Surfaces 3 312. Approximate Estimates by means of Contours 6 313. The Prismoid II 314. The Prismoidal Formula II 315. Areas of Cross-sections 13 316. The Centre and Side Heights 14 317. The Area of a Three-level Section 14 318. Cross-sectioning , 15 319. Three-level Sections, the Upper Surface consisting of two Warped Surfaces 17 320. Construction of Tables for Prismoidal Computation 19 321. Three-level Sections, the Upper Surface divided into Four Planes by Diagonals 24 322. Comparison of Volumes by Diagonals and by Warped Surfaces. . 26 323. Preliminary Estimates from the Profiles 28 324. Borrow Pits 31 325. Shrinkage of Earthwork 31 326. Excavations under Water 32 TABLES. I. TRIGONOMETRICAL FORMULAE 37 II. FOR CONVERTING METERS, FEET, AND CHAINS 41 III. LOGARITHMS OF NUMBERS TO FOUR PLACES 42 IIlA. LOGARITHMS OF TRIGONOMETRICAL FUNCTIONS TO FOUR PLACES.. 44 IV. LOGARITHMIC TRAVERSE TABLE 48 V. STADIA REDUCTIONS FOR HORIZONTAL DISTANCE AND FOR ELEVA- TION 56 VI. NATURAL SINES AND COSINES 64 VII. NATURAL TANGENTS AND COTANGENTS 73 VIII. COORDINATES FOR POLYCONIC PROJECTION 85 IX. VALUES OF COEFFICIENTS IN KUTTER'S FORMULA 86 X. DIAMETERS OF CIRCULAR CONDUITS BY KUTTER'S FORMULA 87 XI. EARTHWORK TABLE VOLUMES BY THE PRISIMOIDAL FORMULA. ... 88 XII. AZIMUTHS OF POLARIS AT ALL HOUR ANGLES 98 vii CHAPTER XIII. THE MEASUREMENT OF VOLUMES. 310. Proposition. The volume of any doubly-truncated prism or cylinder, bounded by plane ends, is equal to the area of a right section into the length of the element through the centres of gravity of the bases, or it is equal to the area of either base into the altitude of the element joining the centres of gravity of the bases, measured perpendicular to that base. Let ABCD, Fig. 107, be a cylinder, cut by the planes OC and OB, the unsymmetrical right section EF being shown in plan in E'F'. Whatever position the cutting planes may have, if they are not parallel they will intersect in a line. This line of intersection may be taken perpendicular to the paper, and the body would then appear as shown in the figure, the line of intersection of the cutting planes being projected at O. Let A = area of the right section ; A A = any very small portion of this area; x = distance of any element from O ; then ax = height of any element at a distance x from <9. An elementary volume would then be ax A A, and the total volume of the solid would be "Sax A A. Again, the total volume is equal to the mean or average height of all the elementary volumes multiplied by the area of the right section. The mean height of the elementary volumes is, therefore, OF THF UNIVERSITY SURVEYING. But A is the distance from O to the centre of gravity, G, of the right section,* and a times this dis tance is the height of the element LK through this point. Therefore, the mean height is the height through the centre of PIG. 107. gravity of the base, and this into the area of the right section is the volume of the truncated prism or cylinder. The truth of the alternative proposition can now readily be shown. Corollary. When the cylinder or prism has a symmetrical cross-section, the centre of gravity of the base is at the centre of the figure, and the length of the line joining these centres is the mean of any number of symmetrically chosen exterior elements. For instance, if the right section of the prism be a regular polygon, the height of the centre element is the mean of the length of all the edges. This also holds true for paral- lelograms, and hence for rectangles. Here the centres of gravity * This is shown in mechanics, and the student may have to take it for granted temporarily. THE MEASUREMENT OF VOLUMES. of the bases lie at the intersections of the diagonals ; and since these bisect each other, the length of the line joining the in- tersections is the mean of the lengths of the four edges. The same is true of triangular cross-sections. 311. Grading over Extended Surfaces. Lay out the area in equal rectangles of such a size that the surfaces of the several rectangles may be considered planes. For common rolling ground these rectangles should not be over fifty feet on a side. Let Fig. 108 represent such an area. Drive pegs at 4 4 4 4 4 4 4 4 4 4 4 a 2 i 123 3 1 FIG. 108. the corners, and find the elevation of the ground at each in- tersection by means of a level, reading to the nearest tenth of a foot, and referring the elevations to some datum-plane below the surface after it is graded. When the grading is completed, relocate the intersections from witness-points that were placed outside the limits of grading, and again find the elevations at these points. The several differences are the depths of excava- tion (or fill) at the corresponding corners. The contents of any partial volume is the mean of the four corner heights into the area of its cross-section. But since the rectangular areas were made equal, and since each corner height will be used as many times as there are rectangles joining at that corner, we have, in cubic yards, r= 4x27 SUX VE YING. The subscripts denote the number of adjoining rectangles the area of each of which is A. From this equation we may frame a RULE. Take each corner height as many times as there are partial areas adjoining it, add them all together, and mul- tiply by one fourth of the area of a single rectangle. Tnis gives the volume in cubic feet. To obtain it in cubic yards, divide by twenty-seven. If the ground be laid out in rectangles, 30 feet by 36 feet, A 1080 then = ~- = 10 ; and if the elevations be taken to the nearest tenth of a foot, then the sum of the multiplied corner heights, with the decimal point omitted, is at once the the amount of earthwork in cubic yards. This is a common way of doing this work. In borrow-pits, for which this method is peculiarly fitted, the elementary areas would usually be smaller. In general, on rolling ground, a plane cannot be passed through the four corner heights. We may, however, pass a plane through any three points, and so with four given points FIG. 109. on a surface either diagonal may be drawn, which with the bounding lines makes two surfaces. If the ground is quite irregular, or if the rectangles are taken pretty large, the sur- veyor may note on the ground which diagonal would most THE MEASUREMENT OF VOLUMES. 5 nearly fit the surface. Let these be sketched in as shown in Fig. 109. Each rectangular area then becomes two triangles, and when computed as triangular prisms, each corner height at the end of a diagonal is used twice, while the two other corner heights are used but once. That is, twice as much weight is given to the corner heights on the diagonals as to the others. In Fig. 109, the same area as that in Fig. 108 is shown with the diagonals drawn which best fit the surface of the ground. The numbers at the corners indicate how many times each height is to be used. It will be seen that each height is used as many times as there are triangles meeting at that corner. To derive the formula for this case, take a single rectangle, as in Fig. no, with the diagonal joining corners 2 and 4. Let A be the area of the rectangle. Then from the corollary, p. 395, we have for the volume of the rectangular prism, in cubic yards, rr _ " * '"\ I r "2 I '"* [ '"a M 4* V/' S%*V 1 /> 2X2;\ 3 3 A 6 X 27 - (k, + 2h, + ^ + 2/0 (2) For an assemblage of such rectangular prisms as shown in Fig. 109, the diagonals being drawn, we have, in cubic yards, ; . . . (3) where A is the area of one rectangle, and the subscripts denote the number of triangles meeting at a corner, SURVEYING. As a check on the numbering of the corners, Fig. 109, add them all together and divide by six. The result should be the number of rectangles in the figure. In this case, if the rectangles be taken 36 feet by 45 feet, or, better, 40 feet by 40.5. feet, then the sum of the multiplied heights with the decimal point omitted is the number of cubic yards of earthwork, the corner heights having been taken out to tenths of a foot. The method by diagonals is more accurate than that by rectangles simply, the dimensions being the same ; or, for equal degrees of exactness larger rectangles may be used with diagonals than without them, and hence the work materially reduced. In any case some degree of approximation is neces- sary. 312. Approximate Estimates by means of Contours. (A) Whenever an extended surface of irregular outline is to be graded down, or filled up to a given plane (not a warped or curved surface), a near approximation to the amount of cut or fill may be made from the contour lines. In Fig. 1 1 1 the full curved lines are contours, showing the original surface of the ground. Every fifth one is numbered, and these were the con- tours shown on the original plat. Intermediate contours one foot apart have been interpolated for the purpose of making this estimate. The figures around the outside of the bound- ing lines give the elevations of those points after it is graded down. The straight lines join points of equal elevation after grading ; and since this surface is to be a plane these lines are surface or contour lines after grading. Wherever these two sets of contour lines intersect, the difference of their elevations is the depth of cut or fill at that point. If now we join the points of equal cut or fill (in this case it is all in cut), we ob- tain a new set of curves, shown in the figure by dotted lines, which may be used for estimating the amount of earthwork. The dotted boundaries are the horizontal projections of the traces on the natural surface of planes parallel to the final THE MEASUREMENT OF VOLUMES. graded surface which are uniformly spaced one foot apart ver- tically. These projected areas are measured by the planimeter and called A l} A 2 , A s , etc. Each area is bounded by the dotted line and the bounding lines of the figure, since on these 78 FIG. in. bounding lines all the projections of all the traces unite, the slope here being vertical. For any two adjoining layers we have, by the prismoidal formula* as well as by Simpson's one- third rule, (I) where h is the common vertical distance between the pro- jected areas. * For the demonstration of the prismoidal formula see Art. 314. SURVEYING. For the next two layers we would have, similarly, r,- i =|<^.-K4A^);. .".- ..... (2) or for any even number of layers we would have, in cubic yards, V= ^ + * A * + 2A, + 4A t + 2A f + .... A.\ (3) where n is an odd number, h and A being in feet and square feet respectively. (B) Whenever the final surface is not to be a plane, but warped, undulating, or built to regular outlines like a fortifi- cation, a reservoir embankment, or terraced grounds, a differ- ent method should be employed. In the former method the areas bounded by the dotted lines were areas cut out by planes parallel to the final plane surface, passed one foot apart vertically. But since the map shows only the horizontal projections of these planes, these pro- jections, multiplied by the vertical distance between them, would give the true volumes. When the final surface is not to be a plane, proceed as fol- lows : First make a careful contour map of the ground. Then lay down on this map a system of contour lines, corresponding in elevation to the first set of contours, but in a different colored ink, which will accurately represent the final surface desired. This second set of contours would be a series of straight lines if a regular surface, composed of plane faces, was to be constructed, but would be curving lines if the ground were to be brought to a final curving or undulating surface. The closed figures bounded by the two sets of intersecting contours of the same elevation are horizontal areas of cut or fill, separated by the common vertical distance between THE MEASUREMENT OF VOLUMES. contours. The volumes here defined are oblique solids bounded by horizontal planes at top and bottom, and are a species of prismoid. The volume of one of these prismoids is found by applying the prismoidal formula to it, finding the end areas by means of a planimeter, and taking the length as the 660 Fig. Ilia. vertical distance between contours. If the contours be drawn close enough together, then each alternate contour-area may be used as a middle area, and the length of the prismoid taken at twice the vertical distance between contours ; or the volume 10 SURVEYING. may be computed by either of the formulas (12), (13), (14), or (15) of Appendix C, where the /is would here become the end areas and / the vertical distance between contours. Example : Let it be required to build a square reservoir on a hillside, which shall be partly in excavation and partly in* embankment, the ground being such as shown by the full con- tour lines in Fig. iii#.* The contours, for the sake of simplicity and brevity, are spaced five feet apart. The top of the wall, shown by the full lines making the square, is 10 feet wide and at an elevation of 660 feet. The reservoir is 20 feet deep, with side slopes, both inside and outside, of two to one, making the bottom elevation 640 feet, and 20 feet square, the top being ico feet square on the inside. The dotted lines are contours of the finished slopes, both inside and out, at elevations shown on the figure. The areas in fill all fall within the broken line marked a b c d e f g h i k, and the cut areas all fall within the broken line marked a b c def g o. These broken lines are grade lines. The horizontal sectional areas in fill and cut are readily traced by following the closed figures formed by contours of equal elevation, thus At 640 foot level sectional area in fill is/ s t. " 650 " " " " " Imn u v x I. " 650 " " " " cut is i 2 3 u x. The other areas are as easily traced. In the figure the lines have all been drawn in black. In practice they should be drawn in different colors to avoid confusion. This second method should be used in all cases where the graded area is considerable and the final relief form is not a plane. If the contours be carefully determined and be taken * This figure is taken from a paper describing the method by Prof. William G. Raymond, University of California. THE MEASUREMENT OF VOLUMES. \\ near enough together, the method will give as accurate results as may be obtained in any other way. The volume may be computed by eq. (3) of this article, where the areas are the horizontal sectional areas bounded by contours of equal ele- vation, and h is the vertical distance between contours. When these methods are used for final estimates, the con- tours should be carefully determined, and spaced not more than two feet apart on steep slopes and one foot apart on low slopes. 313. The Prismoid is a solid having parallel end areas, and may be composed of any combination of prisms, cylinders, wedges, pyramids, or cones or frustums of the same, whose bases and apices lie in the end areas. It may otherwise be defined as a volume generated by a right-line generatrix mov- ing on the bounding lines of two closed figures of any shapes which lie in parallel planes as directrices, the generatrix not necessarily moving parallel to a plane director. Such a solid would usually be bounded by a warped surface, but it can always be subdivided into one or more of the simple solids named above. Inasmuch as cylinders and cones are but special forms of prisms and pyramids, and warped surface solids may be divided into elementary forms, of them, and since frustums may also be subdivided into the elementary forms, it is sufficient to say that all prLmoids may be decomposed into prisms, wedges, and pyramids. If a formula can be found which is equally applicable to all of these forms, then it will apply to any com- bination of them. Such a formula is called 314. The Prismoidal Formula. Let A = area of the base of a prism, wedge, or pyramid ; A^ A m , A 9 = the end and middle areas of a prismoid, or of any of its elementary solids ; h altitude of the prismoid or elementary solid. 12 SURVEYING. Then we have, For Prisms, V= hA = g- (A l + 4^ TO + ^ 3 ) (i) For Wedges, 2 6^ 1 ' * ^ For Pyramids, (3) Whence for any combination of these, having all the common altitude h, we have (4) which is the prismoidal formula. It will be noted that this is a rigid formula for all prismoids. The only approximation involved in its use is in the assump- tion that the given solid may be generated by a right line moving over the boundaries of the end areas. This formula is used for computing earthwork in cuts and fills for railroads, streets, highways, canals, ditches, trenches, levees, etc. In all such cases, the shape of the figure above the natural surface in the case of a fill, or below the natural surface in the case of a cut, is previously fixed upon, and to complete the closed figure of the several cross-section areas only the outline of the natural surface of the ground at the section remains to be found. These sections should be located so near together that the intervening solid may fairly be as- THE MEASUREMENT OF VOLUMES. sumed to be a prismoid. They are usually spaced 100 feet apart, and then intermediate sections taken if the irregularities seem to require it. The area of the middle section is never the mean of the two end areas if the prismoid contains any pyramids or cones among its elementary forms. When the three sections are similar in form, the dimensions of the middle area are always the means of the corresponding end dimensions. This fact often enables the dimensions, and hence the area of the middle section, to be computed from the end areas. Where this can- not be done, the middle section must be measured on the ground, or else each alternate section, where they are equally spaced, is taken as a middle section, and the length of the prismoid taken as twice the distance between cross-sections. For a continuous line of earthwork, we would then have, in cubic yards, 0. (0 where / is the distance between sections in feet. This is the same as equation (3), p. 401. Here the assumption is made that the volume lying between alternate sections conforms sufficiently near to the prismoidal forms. 315. Areas of Cross-sections. Inmost cases, in practice at least, three sides of a cross-section are fixed by the conditions of the problem. These are the side slopes in both cuts and fills, the bottom in cuts and the top in embankments, or fills. It then remains simply to find where the side slopes will cut the natural surface, and also the form of the surface line on the given section. Inasmuch as stakes are usually set at the points where the side slopes cut the surface, whether in cut or fill, such stakes are called slope-stakes, and they are set at the time 14 SURVEYING. the cross-section is taken. The side slopes are defined as so much horizontal to one vertical. Thus a slope of i to I means that the horizontal component of a given portion of a slope- line is it^ times its vertical component, the horizontal com- ponent always being named first. The slope-ratio is the ratio of the horizontal to the vertical component, and is therefore always the same as the first number in the slope-definition. Thus for a slope of i to I the slope-ratio is I J. 316. The Centre and Side Heights. The centre heights are found from the profile of the surface along the centre line, on which has been drawn the grade line of the proposed work. These are carefully drawn on cross-section paper, when the height of grade at each station above or below the surface line can be taken off. These centre heights, together with the width of base and side slopes in cuts and in fills, are the neces- sary data for fixing the position of the slope-stakes. When these are set for any section as many points on the surface line joining them may be taken as desired. In ordinary rolling ground usually no intermediate points are taken, the centre point being already determined. In this case three points in the surface line are known, both as to their distance out from the centre line and as to their height above the grade line. Such sections are called " three-level sections," the surface lines being assumed straight from the slope-stakes to the centre stake. 317. The Area of a Three-level Section. Let d and d f be the distances out, and h and ti the heights above grade of right and left slope- stakes, respectively; D the sum ot d and d', c the centre height, r the slope-ratio, w the width of bed. THE MEASUREMENT OF VOLUMES. Then the area ABCDE is equal to the sum of the four trian- gles Aw, BCw t wCD, and wED. Or, w A = (0 This area is also equal to the sum of the triangles FCD and FED, minus the triangle AFB. Or, D it? FIG. Equation (2) can also be obtained directly from equation (1) by substituting for h and h' in (i) their values in terms of <*--* d and w, h = , and then putting D = d-\- d' . Equation (2) has but two variables, c and D, and is the most convenient one to use. 318. Cross-sectioning. It will be seen from Fig. 112 that in the case of a three-level section the only quantities to be determined in the field are the heights, h and k ', and the dis- tances out, d and d' , of the slope-stakes. These are found by trial. A levelling instrument is set up so as to read on the SURVEYING. three points C, D^ , and the rod held first atZ>. The reading here gives the height of instrument above this point. Add this algebraically to the centre height (which may be negative, and which has been obtained from the profile for each station), and the sum is the height of instrument above (or below) the grade line. If the ground were level transversely, the distance out to the slope-stakes would be w -. But this is not usually the case, and hence the distance out must be found by trial. If the ground slopes j wn 1 from the centre line in a j , > the distance out will evidently be more than that given by the above equation, and vice versa. The rodman estimates this distance, and holds his rod at a cer- tain measured distance out, d^ The observer reads the rod, and deducts the reading from the height of instrument above grade (or adds it to the depth of instrument below grade), and this gives the height of that point, h lt above or below grade. Its IV distance out, then, should bed = hf -\ . If this be more than 2 the actual distance out, d l9 the rod is set farther out ; if less, it is moved in. The whole operation is a very simple one in prac- tice, and the rodman soon becomes very expert in estimating nearly the proper position the first time. In heavy work that is, for large cuts or fills, and for irregu- lar ground it may be necessary to take the elevation and dis- tance out of other points on the section in order to better determine its area. These are taken by simply reading on the rod at the critical points in the outline, and measuring the dis- tances out from the centre. The points can then be plotted THE MEASUREMENT OF VOLUMES. on cross-section paper and joined by straight or by free-hand curved lines. In the latter case the area should be deter- mined by planimeter. 319. Three-level Sections, the Upper Surface con- sisting of two Warped Surfaces. If the three longitudinal lines joining the centre and side heights on two adjacent three- level sections be used as directrices, and two generatrices, one on each side the centre, be moved parallel to the end areas as plane directers, two warped surfaces are generated, every cross- section of which parallel to the end areas is a three-level sec- tion. These same surfaces could be generated by two longi- tudinal generatrices, moving over the surface end-area lines as directrices. The surface would therefore be a prismoid, and its exact volume would be given by the prismoidal formula. The middle area in this case is readily found, since the center and side heights are the means of the corresponding end di- mensions. The prismoidal formula, giving volumes in cubic yards, could therefore be written This equation is derived directly from eq. (i) above, and eq. (2), p. 406. The quantity is the distance from the grade-plane 1 8 SURVEYING. to the intersection of the side slopes, and is a constant for any given piece of road. It would have different values, however, in cuts and fills on the same line. For brevity, let = c - and w = - K zr 4 X 27?- ' 54 Here K is the volume of the prism of earth, 100 feet long, in- cluded between the roadbed and side slopes. It is first in- cluded in the computation and then deducted. It is also a constant for a given piece of road. Equation (2) now becomes -#; . (3) where c m and D m are the means of c^ t and DJ) V respectively. This equation involves but two kinds of variables, c and D, and is well adapted to arithmetical, tabular, or graphical com- putation. Thus if / = 100 ; w 18 ; and r = i-J- ; then c 6 ; and K = 200 ; and equation (3) becomes ', + 6) A + ('. + 6)A + 4fc. + 6) AJ - 2 oo (4) If the total centre heights (to intersection of side slopes) be represented by C lt C v and C m , then eq. (3) becomes, in general, V=lC(C l D l + CJ) % + 4CJ) m )-K t . . . (5) where K' = -J^j-, and is independent of width of bed and of slopes. For any given piece of road, the constants K, K' , and c are known, and for each prismoid the C's and D's are observed, hence for any prismoid all the quantities in eq. (5) are known. THE MEASUREMENT OF VOLUMES. 320. Construction of Tables for Prismoidal Computa- tion. If a table were prepared giving the products K'CD for various values of C and D, it could be used for evaluating equation (3), which is the same as equation (5). The argu- ments would be the total widths (Z^), and the centre heights (). Such a table would have to be entered three times for each prismoid, first with C, and D l ; second with C t and Z> a ; and finally with C m and D m . If four times the last tabular value be added to the sum of the other two, and K subtracted, the result is the true volume of the prismoid. VALUES OF c (= -) \ 2rl AND 4 X AND SLOPES. - FOR VARIOUS WIDTHS Width of Road- SCOPES. X to 1. Ntol. % to 1. 1 to 1. Ik tol. IN tol. IX tol. 2 to 1. bed. Co JC c; jr C K C K c K Co K Co K C K 1O 20 370 10 185 6.7 123 5-o 93 4.0 74 3-3 62 2.9 53 2.5 46 11 22 448 ii 224 7-3 149 5-5 112 4.4 9 3-7 75 3-i 64 2.8 56 13 24 533 12 266 8.0 I7 8 6.0 J 33 4 .8 107 4.0 89 3-4 76 3-o 67 13 26 626 13 313 8.7 209 6-5 157 5-2 125 4-3 104 3-7 89 3-2 78 14 28 725 M 363 9-3 242 7.0 z8z 5-6 *45 4-7 121 4.0 104 3-5 9* 15 3 833 15 4 T 7 IO.O 278 7-5 208 6.0 167 S-o 139 4-3 119 3-8 104 16 3 2 948 16 474 10.7 3,6 8.0 237 6.4 190 5-3 158 4.6 i35 4.0 118 17 34 1070 7 7 535 "3 357 8-5 268 6.8 214 5-7 I 7 8 4-9 i53 4-2 134 18 36 1200 18 600 12.0 400 9.0 300 7.2 240 6.0 200 S-i 171 4-5 I 5<> 19 38 '337 J 9 668 I2. 7 446 9-5 334 7.6 267 6-3 223 4-4 191 4.8 167 30 40 1481 20 740 *3-3 494 IO.O 370 8.0 296 6-7 247 5-7 212 5.0 185 31 42 1633 21 816 14.0 544 IO -5 408 8.4 327 7.0 272 6.0 233 5-2 204 33 44 1793 22 896 14.7 598 ii .0 448 8.8 359 7-3 299 6.3 2 5 6 5-5 224 33 46 J 959 23 980 iS-3 653 "5 490 9.2 392 7-7 326 6.6 280 5-8 245 34 48 2i34 2 4 1067 16.0 711 12. 534 9.6 427 8.0 356 6.9 305 6.0 267 35 5 2315 2 5 1158 16.7 772 12-5 579 IO.O 463 8-3 3 86 7-i 33 1 6.2 264 36 S* 2504 26 1252 17-3 835 13.0 626 10.4 50i 8-7 417 7-4 358 6-5 3^3 37 54 2700 27 i35o 18.0 900 13-5 675 10.8 54 9.0 450 7-7 386 6.8 338 38 56 2904 28 *452 18.7 968 14.0 726 IZ.2 581 93 484 8.0 4i5 7.0 363 39 58 3"5 2 9 1558 i9-3 1038 14.5 779 ii. 6 623 9-7 519 8 3 445 7-2 389 3O 60 3333 3 1667 20.0 IIII 15.0 833 12.0 667 IO.O 556 8.6 476 7-5 417 20 SURVEYING. Table XL* is such a table, computed for total centre heights from i to 50 feet, and for total widths from I to 100 feet. In railroad work neither of these quantities can be as small as one foot, but the table is designed for use in all cases where, the parallel end areas may be subdivided into an equal number of triangles or quadrilaterals. EXAMPLE I. Three- level Ground having two Warped Surfaces. Find the volume of two prismoids of which the following are the field-notes, the width of bed being 20 feet, and the slopes i to i. Station n. Station 12. Station 12 -f- 5^. 28. 9 f 43-0 + 12.6 + 18.6 4-22.0 27.1 40.3 4-11.4 4-14.8 4~ 20. 2 24.3 34-9 4-9.5 IO -3 4- From the table, p. 410, giving values of C and K, we find for and r = i|, C = 6. 7, and K = 247. The computation may be tabulated as follows: = 20, Sta. Width, D=d+d'. Height, C = c + c . Partial Volume. Volume of Prismoid. II 71.9 25-3 5 62 M 69.6 23-4 503 X 4 = 2012 12 67.4 21.5 447 3021 247 2774 M 63.3 19.2 374 X 4 = I49 6 12 4- 56 59-2 17.0 3ii .56(2254 - 247) 1124 * Modeled somewhat after Crandall's Tables, but adapted to give volumes by the Prismoidal Formula at once instead of by the method of mean end areas first and correcting by the aid of another table to give prismoidal volumes, as Prof. Crandall has done. f The numerators are the distances out, and the denominators are the heights above grade, 4- denoting cut and fill. THE MEASUREMENT OF VOLUMES. 21 Entering the table (No. XI.) for a width of 71 and a height of 25, we find 548, to which add 7 for the 3 tenths of height, and 7 more for the 9 tenths in width, both mentally, thus giving 562 cu. yds. for this partial volume. Simi- larly for the width 67. 4, and height 21.5, obtaining 447 cu. yds. The correspond- ing result for the middle area is 503, which is to be multiplied by 4, thus giving 2012 cu. yds. The sum of these is 3021 cu. yds., from which is to be subtracted the constant volume K, which in this case is 247 cu. yds., leaving 2774 cu. yds. as the volume of the prismoid. The next prismoid is but 56 feet long, but it is taken out just the same as though it were full, and then 56 hundredths of the resulting volume taken. The data for the I2th station is used in getting this result without writing it again on the page. EXAMPLE 2. Five-level Ground having four Warped Surfaces. Find the volume of a prismoid of which the following are the field-notes, the width of bed being 20 feet, and the slopes i? to I : II. 28.9 15-0 + 12.6 +12.0 +18.6 20.0 + 21.0 43-0 + 22. 12. 27.1 12.5 + II.4 +12.0 + 14- 18.5 40-3 + 19.6 +20.2 This is the same problem as the preceding, with intermediate heights added. To compute this from the table, it is separated into three prismoids, as shown in Fig. 113. Let ABDGCFE be the cross-section. This may be separated into the triangle ABC, and the two quadrilaterals BCGD and ACFE. The area of the triangle is %cw. That of the right quadrilateral is, from Art. 179, p. 202, 22 SURVEYING. ~ o) Similarly the area of the left quadrilateral is (* /')( tfk ) + ' may be omitted, and there will also be but three terms in each partial product. Thus, if sections n and 12 had been taken with the interior elevations, each 10 feet from the centre line, we might have had something as follows : n. 28.9 + 12.6 IO.O + 15.4 + 18.6 IO.O 19.* 43.0 + 22.0 12. 27-1 IO.O +-11.4 +12.5 +14. 10.0 4-3 + 17.4 +20.2 The computation then, by eq. (2), would have been : Sta. d' h . k>. C. k. d k . Partial Volumes. Total Volume. II 28.9 15-4 18.6 19.8 43-0 137 + 114 + 263 = 5H II 28. c I4.O 16.7 18.6 41.6 4 (121 + 102 + 239) = 1848 12 27.3 12-5 14.8 17.4 40.3 IO4 + 90 + 215 = 409 2771 By this method the computation of a five-level section is little more trouble 2 4 SURVEYING. than that of a three-level section, and yet the intermediate points taken at a dis- w tance of from the centre, are apt to increase the accuracy considerably on ordinary rolling ground. 321. Three-level Sections, the Surface divided into four Planes by Diagonals. If the surface included between two three-level sections be assumed to be made up of four planes formed by joining the centre height at one end with a side , height at the other end sec- tion on each side the centre line (Fig. 114), these lines being called diagonals, an exact computation of the volume is readily made without computing the mid-area. Two diag- onals are possible on each side the centre line but the one is drawn which is observed to most nearly fit the surface. They are noted in the field when the cross-sections are taken. FIG. The total volume of such a prismoid in cubic * yards is _-_ DC+ D'C where c lt tion and (i) and h{ are the centre and side heights at one sec- and d^ the distances out, c^ h%, h^ d^ and d^ be- * For a demonstration of this formula see Henck's Field-Book. THE MEASUREMENT OF VOLUMES. ing the corresponding values for the other end section. C and C are the centre heights, H and H' the side heights, and D and D' the distances out on the right and left diagonals. Although this formula seems long, the computations by it are very simple. Thus let the volume be found from the following field-notes for a base of 20 feet and side slopes i to i. 22 + 16 + 4 The upper figures indicate the distances out and those below the lines the heights, the plus sign being used for cuts* The computation in tabular form is as follows : Sta. d. h. c. A'. d'. <*+f f >h',2' divide 28 SURVEYING. the sides of the quadrilateral */', */, r/X' proportionally. But it is a proposition in geometry that if the four sides of a quad- rilateral, or two opposite sides and the diagonals, be divided proportionally and the corresponding points of subdivision joined, the resulting figure is a parallelogram. Therefore ef'tf g' is a parallelogram, and e'h' is one of its diagonals and hence bisects it. Whence the surface generated by this line moving along / and d^ parallel to the end areas bisects the volume formed by the four planes resulting from the use of both di- agonals on one side the centre line. Q. E. D. It is probable, therefore, that the warped surface would usually fit the ground better than either of the sets of planes formed by the diagonals. Furthermore, the errors caused by the use of the warped surface (Table XI.) are compensating errors, thus preventing any marked accumulation of errors in a series of prismoids.* There are extreme cases, however, such as that given in the example, Fig. 1 14, which are best computed by the method by diagonals. 323. Preliminary Estimate from the Profile. If the cross-sections be assumed level transversely then for given width of bed and side slopes, a table of end areas may be pre- pared in terms of the centre heights. From such a table the * The two methods here discussed are the only ones that have any claims to accuracy. The method by " mean end areas," wherein the volume is assumed to be the mean of the end areas into the length, always gives too great a volume (except when a greater centre height is found in connection with a less total width, which seldom occurs), the excess being one sixth of the volume of the pyramids involved in the elementary forms of the prismoid. This is a large error even in level sections, and very much greater on sloping ground, and yet it is the basis of most of the tables used in computing earthwork, and in some States it is legalized by statute. Thus in the example computed by Henck's method on p. 414 the volume by mean end areas is 1193 cu. yards; by the prismoidal formula it is 1168 cu. yards, while by the method by diagonals it was only 1001 cu. yards. This was an extreme case, however, and was selected to show the adaptation of the method by diagonals to such a form. UNIVERSITY THE MEASUREMENT OF VOLUMES. 29 end areas may be rapidly taken out and plotted as ordinates from the grade line. The ends of these ordinates may then be joined by a free-hand curve, and the area of this curve found by the planimeter. The ordinates may be plotted to such a scale that each unit of the area, as one square inch, shall represent a convenient number of cubic yards, as 1000. The record of the planimeter then in square inches and thou- sandths gives at once the cubic yards on the entire length of line worked over by simply omitting the decimal point. Evi- dently the scale to which the ordinates are to be drawn to give such a result is not only a function of the width of bed and side slopes, but also of the longitudinal scale to which the pro- file line is plotted. The area of a level section is A =wc + rc\ (I) where w, c, and r are the width of base, . centre height, and slope-ratio respectively. Now if h = the horizontal scale of the profile, that is the number of feet to the inch, and if one square inch of area is to represent 1000 cu. yards, the length of the ordinate must be - hA - k(M + r4 . sin J? 63. sin^4 + sinB = 2sm^M + B)cos^(^~J5) 64. sin ^4 sin B = 2 cos J^ (^1 + B) sin J (A B) 65. cos A + cos B = 2 cos ^ (^4 + B) cos H(A B 66. cosB eo8^ = 2sm^C4 + P)sin^(.4 B) 67. sin ^4 sin* B = cos a B cos 8 ^4 = sin (A + J5) sin (A B> 68. cos 8 A sin 8 J? = cos (A + J5) cos (A B) cos ^4 . cos 5 70. tan 4 - tan B = - ri ' ( /~' B) n cos ^4 . cos B TABLES. TABLE II. FOR CONVERTING METRES, FEET, AND CHAINS. METRES TO FEET. FEET TO METRES AND CHAINS. CHAINS TO FEET. Metres. Feet. Feet. Metres. Chains. Chains. Feet. I 3.28087 ! 0.304797 0.0151 0.01 0.66 2 6.56174 2 0.609595 .0303 .02 1.32 3 9.84261 3 0.914392 0455 .03 1.98 4 13.12348 4 1.219189 .0606 .04 2.64 5 16.40435 5 1.523986 .0758 05 3-30 6 19.68522 6 1.828784 .0909 .06 3.96 7 22 . 96609 7 2.I3353I .1061 .07 4.62 8 26.24695 8 2.438378 .1212 .08 5.28 9 29.52732 9 2-743I75 .1364 .09 5-94 10 32.80869 10 3-047973 .1515 . IO 6.60 20 65.61739 20 6.095946 .3030 .20 13.20 30 98.42609 30 9.143918 4545 30 19.80 40 131.2348 40 12.19189 .6o6l .40 26.40 50 164.0435 50 15.23986 .7576 50 33-00 60 196.8522 60 18.28784 .9091 .60 39-6o 70 229.6609 70 2I.3358I I. 0606 .70 46.20 80 262.4695 80 24.38378 I.2I2I .80 52.80 go 295.2782 9 27.43175 1.3636 .90 59-40 JOO 328.0869 IOO 30.47973 I.5I5I I 66.00 200 656.1739 100 60.95946 3.0303 2 132 300 984.2609 300 91.43918 4-5455 3 198 400 1312.348 400 121.9189 6.0606 4 264 500 1640.435 500 152.3986 7.5756 5 330 600 1968.522 600 182.8784 9.0909 6 396 700 2296.609 700 2I3.358I 10.606 7 462 800 2624.695 800 243.8378 12. 121 8 528 900 2952.782 900 274.3175 13.636 9 594 IOOO 3280.869 IOOO 304.7973 15.151 IO 660 2000 6561.739 2000 609 . 5946 30-303 20 1320 3000 9842 . 609 3000 914.3918 45-455 30 1980 4OOO 13123.48 4OOO 1219.189 60.606 40 2640 5000 16404.35 5000 1523.986 75-756 50 3300 6OOO 19685.22 6000 1828.784 90,909 60- 396o 7000 22966.09 7OOO 2133.581 106.06 70 4620 8000 26246.95 8000 2438.378 121. 21 80 5280 9000 29527.82 9000 2743.175 136.36. 90 5940 SURVEYING. TABLE III. LOGARITHMS OF NUMBERS. 173. Z a 1 a 3 4 5 6 ^ 8 9 Proportional Parts. 1 3 4 5 G 7 8 9 10 .0000 .0043 .0086 .0128 .0170 .0212 0253 .0294 334 0374 4 8 12 17 21 25 29 33 37 II .0414 453 .0492 .0531 .0569 .0607 .0645 .0682 .0719 0755 4 8 II 19 2 3 26 30 34 12 .0792 .0828 .0864 .0899 934 .0969 . 1004 .1038 .1072 . 1106 3 7 IO 4 21 2 4 28 3 1 I 3 .1139 "73 .1206 .1239 .1271 1303 1335 1367 .1399 .1430 3 6 10 16 *9 23 26 20 14 .1461 .1492 1523 1553 .1584 .1614 .1644 1673 I 73 ^S 2 3 6 9 2 15 18 21 24 27 IS .1761 .1790 .1818 .1847 1875 .1903 1931 1959 .1987 .2014 3 6 8 i M r 7 20 22 25 i6 ( .2041 .2068 2095 .2122 .2148 2175 .2201 .2227 . 2480 2253 .2279 3 5 8 I 13 16 18 21 2O 24 18 .2553 2577 ^355 .260! .2625 ^2648 .2430 .2672 2 455 2695 .2718 .25O4 .2742 .2529 .2765 5 7 9 12 16 21 19 .2788 .2810 2833 .2856 .2878 .2900 2923 2945 .2967 .2989 4 7 9 11 i3 16 18 20 20 .3010 .3032 354 375 3096 .3118 3139 3160 3181 3201 4 6 8 11 13 15 17 19 21 .3212 3243 3263 3284 334 3324 3345 3365 3385 344 4 6 8 10 12 H 16 si 22 .3424 3444 3464 3483 .3502 3522 3541 3560 3579 3598 4 6 8 10 12 14 15 17 23 .3617 .3636 3655 3674 -3692 37" 3729 3747 .3766 3784 4 6 7 9 II 13 15 17 2 4 .3802 .3820 .3838 3856 .3874 399 3927 3945 .3962 4 5 7 9 II 12 H 16 25 -3979 3997 .4014 .4031 .4048 4065 .4082 .4099 .4116 4133 3 5 7 S 10 12 I4 15 26 .4150 .4166 .4183 .4200 .4216 .4232 .4249 .4^65 .4281 .4298 3 5 7 8 10 II 13 15 27 -43 J 4 433 434 6 .4362 .4378 4392 .4409 4425 .4440 .4456 3 5 6 8 9 II 13 H 28 .4472 .4487 .4502 .4518 4504 4579 4594 .4609 3 s 6 8 9 II 12 ] 4 29 .4624 4 6 39 .4654 .4669 4683 .4698 47 I 3 .4728 4742 4757 3 4 6 7 9 10 12 13 3 -4771 .4786 .4800 4814 .4829 4843 4857 .4871 .4886 .4900 3 4 6 7 9 10 II 3 3 1 -49J4 .4928 .4942 4955 .4969 4983 4997 5011 .5024 .5038 3 4 6 7 8 10 II 12 32 .5051 5065 5079 .5092 5105 5"9 5132 5145 5159 S 1 ? 2 3 4 5 7 8 9 II 12 33 -5185 .5198 .5211 .5224 5237 5250 .5263 .5276 5289 53 2 3 4 5 6 8 9 10 12 34 -53*5 5328 534 5353 .5366 5378 5391 543 .5416 .5428 3 4 5 6 8 10 II 35 -5441 5453 .5465 5478 5490 5502 5514 5527 5539 555' 4 5 6 7 9 10 II 36, -5563 5575 5587 5599 .5623 5635 5 6 47 .5658 .5670 ' 4 5 6 7 S TO II 37 -5682 38 .5798 5694 .5809 5705 5821 5717 5832 .5729 5843 5740 .5855 :US 5763 5877 3S .5786 5899 ' 7 5 6 6 7 7 8 8 9 9 10 IO 39 -59" 5922 5933 5944 5955 .5966 5977 .5988 5999 .6010 3 4 5 7 8 9 10 40 .6021 .6031 .6042 .6053 .6064 .6075 .6085 .6096 .6107 .6117 3 4 5 6 8 9 IO 41 .6128 6138 .6149 .6160 .6170 .6180 .6191 .6201 .6212 .6222 3 4 5 6 7 8 9 42 .6232 43 - 6 335 .6243 6345 .6253 6 355 .6263 6365 .6274 6375 .6284 6385 .6294 6395 .6304 .6405 6 3*4 6415 6325 .6425 1-3 4 4 5 6 6 7 7 8 8 9 9 44 -6435 .6444 6454 .6464 .6474 .6484 6493 .6503 6513 .6522 3 4 5 6 7 8 9 45 - 6 53 2 .6542 655 1 6561 657' .6580 .6^90 6 599 .6609 .6618 3 4 s 6 7 8 Q 46 .6628 .6637 .6646 .6656 .6665 .6675 .6684 6693 .6702 .6712 1 3 4 5 6 f 7 8 47 -6721 .6730 6 739 .6749 .6758 .6767 .6776 .6785 .6794 .6803 . : 3 4 5 5 6 7 1 48 .6812 .6821 .6830 6839 .6848 .6857 .6866 .6875 .6884 6893 3 4 4 5 6 7 & 4g .6902 .6911 .6920 .6928 6937 .6946 6955 .6964 .6972 .6981 3 4 4 5 6 7 fa 50 -6990 .6998 .7007 .7016 .7024 733 .7042 .7050 759 .7067 3 3 4 5 6 7 8 51 -7076 7084 .7093 .7101 .7110 .7118 .7126 7135 7H3 7152 3 3 4 5 6 7 8 52 .7160 .7168 .7177 7185 7193 .7202 .7210 .7218 .7226 7235 2 3 4 5 6 7 7 53 -7243 .7251 7259 .7267 7275 7284 .7292 .7300 .7308 73 l6 2 3 4 5 6 6 7 54 -7324 7332 7340 7348 -735^ 7364 7372 .7380 .7388 7396 i 2 3 4 5 6 V TABLES. 43 TABLE III. Continued. LOGARITHMS OF NUMBERS. ! R fc 1 2 3 4 5 6 7 8 9 Proportional Parts. 1 2 3 4 5 6 7 8 9 55 -7404 .7412 .7419 .7427 7435 7443 7451 7459 .7466 7474 3 4 5 5 6 7 56 .7482 .7490 7497 7505 75*3 .7520 7528 7536 7543 7551 4 5 5 6 7 57 -7559 .7566 7574 .7582 7589 7597 .7604 .7612 .7619 .7627 i 4 5 5 6 7 5 8 -7634 .7642 .7649 7657 .7664 .7672 .7679 .7686 .7604 7701 2 4 4 5 6 7 59 -7709 .7716 7723 773^ 7738 7745 .775 2 .7760 .7767 7774 3 4 4 5 6 7 60 .7782 .7789 .7796 .7803 .7810 .7818 .7825 .7832 7839 .7846 3 4 4 5 6 6 61 .7853 .7860 .7868 7875 .7882 .7889 .7896 793 .7910 .7917 jj 4 4 5 6 6 62 .7924 6 3 -7993 7931 .8000 7938 .8007 7945 .8014 7952 .8021 3 .7966 .8055 7973 .8041 .7980 .8048 7987 8055 g 4 4 5 5 6 5 6 6 64 .8062 .8069 ,8075 .8082 .8089 .8096 .8102 .8109 .8116 .8122 ji 4 5 5 6 65 .8129 .8136 .8142 .8149 .8156 .8162 .8169 .8176 .8182 .8189 3 4 5 5 6 66 .8195 .8202 .8209 -8215 .8222 .8228 .8235 .8241 .8248 .8254 ! 4 5 5l 6 67 .8261 .8267 .8274 .8280 .8287 .8293 .8299 .8306 .8312 .8319 ; 4 5 5 6 68 .8325 8331 .8338 .8344 8351 8357 8363 .8370 .8376 .8382! J 4 5 6 69 .8388 70 .8451 8395 Q t p.* .8401 8463 .8407 8470 .8414 8476 .8420 8482 .8426 8488 .8432 8 \C\A 8439 . 8^OO 8445 8506 2 4 5 6 5 71 -8513 72 -8573 457 .8519 8579 4 u j .8525 .8585 .0470 853' .8591 47 U .8537 8597 .040^ 8543 .8603 :1SS O 494 :llll ^8561 .8621 lie I; 4 4 5 5 5 5 5 73 -8633 .8639 8645 .8651 .8657 .866-, .8669 .8675 .8681 .8686 2 4 5 5 74 .8692 .86 9 8 .8704 .8710 .8716 .8722 .8727 8733 8739 8745 2 4 5 5 75 -8751 .8756 .8762 .8768 .8774 .8779 .8785 .8791 .8797 .8802 2 3 4 5 5 76 .8808 .8814 .8820 .8825 .8831 8837 .8842 .8848 8854 .8859! 2 3 4 5 5 77 .8865 .8871 .8876 .8882 .8887 .8893 .8899 .8904 .8910 .8915! 2 3 4 4 5 78 .8921 79 .8976 .8927 .8982 .8932 .8987 .8938 .8993 .8943 .809? .8949 .900*: 8954 QOOQ .8960 QOI c: .8965 GO2O .8971 002^ j 2 3 4 4 5 80 .9031 .9036 .9042 .9047 vyiyu 9053 .9058 . y*-^y .9063 y ui j .9069 y^MK) .9074 ymg .9079 1 2 3 4 4 5 5 81 .9085 .9090 .9096 .9101 .9106 .9112 .9117 .9122 .9128 9133 ! 2 3 4 4 5 82 .9138 9143 .9149 9!54 9*59 .9165 .9170 9175 .9180 .9186 2 2 3 4 4 5 83 .9191 .9196 .920! .9206 .9212 .9217 .9222 .9227 9232 .9238 2 2 3 4 4 5 84 .9243 .9248 9253 .9258 .9263 .9269 .9274 .9279 .9284 .9289 2 3 4 4 5 85 .9294 .9299 .9304 9309 9315 .9320 9325 9330 9335 9340 2 3 4 4 c 86 -9345 9350 9355 .9360 9365 937 9375 .9380 9385 9390 2 3 4 4 5 87 '9395 .9400 9405 .9410 9415 .9420 9425 943 9435 .9440 ! 2 3 3 4 4 88 .9445 9450 9455 -9460 9465 .9469 9474 9479 .9484 .9489 O 2 3 3 4 4 89 .9494 9499 .9504 9509 9513 .9518 9523 9528 9533 9538 j 2 3 3 4 4 90 .9542 9547 9552 9557 .9562 .9566 9571 9576 .9581 .9586 O , 2 2 3 3 4 4 91 .9590 92 .9638 9595 9 6 43 .9600 .9647 .9605 .9652 . 9 609 9657 .9614 .9661 .9619 .9666 .9624 .9671 .9628 9675 9633 .9680' i ol ; 2 2 2 3 3 3 3 4 4 4 93 -9685 .9689 .9694 .9699 9703 .9708 9713 .9717 .9722 .9727 1 2 2 3 3 4 4 94 -973 1 9736 9741 9745 975 9754 9759 9763 .9768 9773 O; 2 2 3 3 4 4 95 -9777 .9782 .9786 .9791 9795 .9800 .9805 .9809 .9814 .9818 2 2 3 3 4 4 96 .9823 .9827 .9832 .9836 .9841 9845 .9850 9854 .9859 .9863 o 1 2 2 3 4 4 97 .9868 .9872 .9877 .9881 .9886 .9890 9894 .9899 9903 .9908 l 2 2 3 3 4 4 98 .99" 99 .9956 .9917 .9961 .9921 99 6 5 .9926 .9969 993 9974 9934 .9978 9939 9983 9943 .9987 .9948 .9991 9952 .9996 3 2 2 2 2 3 3 3 4 3 4 4 44 SURVEYING. TABLE IIlA. LOGARITHMS OF SINES AND TANGENTS. o" i Sin. Cos. | Tan. Cot. Sin. Cos. Tan. Cot. o' o.oooo 8.2419 9.9999 8.2419 1.7581 60' I 6.4637 .0000 6.4637 3 5363 .2490 9999 .2491 .7509 59 2 .7648 .0000 .7648 .2352 .2561 9999 2562 7438 58 3 6 9408 .0000 6.9408 3.0592 .2630 9999 2631 73 6 9 57 - 4 7-0658 .0000 7 0658 2-9342 .2699 9999 .2700 .7300 56 5 .1627 .0000 ..627 8373 .2766 9999 .2767 7233 55 6 .2419 .0000 .2419 .7581 .2832 9999 -2833 .7167 54 7 .3088 .0000 .3088 .6912 .2898 9999 .2899 .7101 53 8 .3668 .0000 .3668 6332 .2962 9999 .2963 7037 52 9 .4.80 .0000 .4180 .5820 3 02 5 9999 .3026 .6974 5 l 10 4637 .0000 4637 53 6 3 .3088 9999 .3089 .6911 50 ii SOS 1 .0000 5051 4949 3 T 5 9999 3*5 .6850 49 12 5429 .0000 5429 457 1 .3210 9999 .3211 .6789 48 13 ^777 .0000 5777 .4223 .3270 9999 3271 .6729 47 M .6099 .OOOJ .6099 .3901 3329 9999 3330 .6670 46 15 .6398 .000 ) .6398 .3602 .3388 9999 3389 .6611 45 16 .6678 .0000 ! .6678 3322 3445 9999 3446 6554 44 17 .6942 .0000 .6942 3058 .3502 9999 353 .6497 43 18 .7190 .0000 .7190 .2810 3558 9999 3559 .6 44 t 42 19 7425 .0000 7425 2575 36*3 9999 .3614 .6386 4 1 20 .7648 .0000 .7648 2352 .3668 9999 3669 6331 40 21 7859 .0000 .7860 .2140 .3722 9999 3723 .6277 39 22 .8061 .0000 .8062 .1938 3775 9999 3776 .6224 38 23 8255 .0000 .8255 '745 .3828 9999 .3829 .6171 37 24 .8439 .0000 .8439 1561 .3880 9999 3881 .6119 36 25 .8617 .0000 .8617 393 1 9999 3932 .6068 35 26 .8787 .0000 8787 .1213 .3982 9999 3983 .6017 34 2 7 895' .0000 .8951 .1049 .4032 9999 4033 5967 33 28 .9109 .0000 .9109 .0891 .4082 9999 4083 5917 32 2 9 .9261 .0000 .9261 .0739 43* 9999 4 T 3 2 .5868 3 1 3 .9408 .0000 .9409 .0591 4'79 9999 .4181 .5819 3 31 955 1 .0000 g- S i .0449 .4227 .9998 .4229 5771 29 32 .9689 .0000 .9689 .0311 4275 .9998 .4276 5724 28 33 .9822 .0000 9823 .0177 4322 9998 4323 5677 27 34 7-99S 2 .0000 7.9952 2.0048 .4368 .9998 437 .5630 26 35 8.0078 .0000 8.0078 1.9922 .4414 .9998 .4416 5584 25 36 .0200 .0000 .0200 .9800 4459 .9098 .4461 5539 24 37 .0319 .0000 .0319 .9681 4504 .9998 .4506 5494 23 38 435 .0000 0435 95 6 5 4549 .9908 4551 5449 22 39 .0548 .0000 .0548 9452 4593 .9998 4595 5405 21 40 .0658 .0000 .0658 9342 4637 .9998 .4638 SS 62 20 41 .0765 .0000 .0765 9235 .4680 .9998 .4682 5318 I 9 42 .0870 .0000 .0870 .9130 4723 .9998 4725 5275 18 43 .0972 .0000 .0972 .9028 47 6 5 .9998 .4767 5233 17 44 .1072 .0000 .1072 .8928 .4807 .9998 .4809 S'9 1 16 45 . 1169 .0000 .1170 .8830 .4848 .9998 .4851 5H9 15 46 .1265 .0000 .1265 8735 .4890 .9998 .4892 -5108 14 47 1358 .0000 1359 .8641 .4930 9998 4933 .5067 13 48 .1450 .0000 .1450 8550 .4971 .9998 4973 .5027 12 49 1539 .0000 .1540 .8460 .5011 .9998 5013 4987 II 5 .1627 .0000 . 1627 8373 5050 .9998 5053 4947 IO 51 T 7'3 .0000 I 7 I 3 .8287 .5090 .9998 .5092 .4908 9 52 '797 o.oooo 1798 .8202 .5^29 .9998 5 I 3 I .4869 8 53 .1880 9 9999 .1880 .8120 5^7 .9998 .5170 4830 7 54 . 1961 9999 .1962 .8038 .5206 9998 .5208 4792 6 55 .2041 9999 .2041 7959 .5243 .9998 .5246 4754 5 56 .2119 9999 .2120 .7880 .5281 .9998 5283 .4-717 4 57 .2196 9999 .2196 .7804 53*8 9997 5321 .4679 3 58 .2271 9999 .2272 .7728 5355 9997 5358 .4642 2 g .2346 8.2419 9999 9-9999 .2346 8.2419 7 6 54 i-758i 5392 8.5428 9997 9-9997 5394 8.5431 .4606 1-4569 I O Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 89" 88 TABLES. 45 TABLE \\\^. Continued. LOGARITHMS OF SINES AND TANGENTS. 2 3 4 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. Sin. | Cos. Tan. Cot. o' .5428 9-9997 543* 1.4569 .7188 9-9994 8.7194 .2806 8.84369.9989 8.8446 I.IS54 60' I 54 6 4 9997 54 6 7 -4533 .7212 9994 .7218 .2782 8454 .9989 .8465 1535 59 2 55o 9997 5503 4497 .7236 9994 7242 .2758 8472 .9989 .8483 1517 58 3 5535 9997 5538 .4462 .7260 9994 .7266 2734 .8490! .9989 .8501 .1499 57 4 5571 9997 5573 .4427 7283 9994 .7299 .2710 .8508! .9989 -8518 .1482 56 5 5605 9997 .5608 4392 7307 9994 7313 .2687 8525 .9989 .8536 .1464 55 6 .5640 9997 5 6 43 4357 733 9994 7337 .2663 8543 9989 8554 .1446 54 7 .5 6 74 9997 5677 4323 7354 9994 .7360 .2640 .8560 .9989 8572 .1428 53 8 .5708 9997 .5711 .4289 7377 9994 7383 .2617 .8578 .9989 8589 .1411 5 2 9 5742 9997 5745 4255 .7400 9993 .7406 2594 8595 9989 .8607 1393 5 10 577 6 9997 5779 .4221 7423 9993 .7429 2571 .8613 9989 .8624 1376 50 ii .5809 9997 .5812 .4188 7445 9993 7452 .2548 .8630 .9988 .8642 .1358 49 12 .5842 9997 .5845 4^5 .7468 9993 7475 2525 .8647 .9988 .8659 .1341 48 13 5875 9997 .5878 .4122 .7491 9993 7497 250? 8665 .9988 .8676 1324 47 *4 5907 9997 59 11 .4089 7513 9993 .7520 .2480 .8682 .9988 .8694 .1306 46 i5 5939 9997 5943 457 7535 9993 754 2 .2458 .8699 .9988 .8711 .1289 45 16 5972 9997 5975 .4025 7557 9993 .7565 2435 .8716 .9988 .8728 .1272 44 17 .6003 9997 .6007 3993 .7580 9993 7587 ' 2 4!3 8733 .9988 8745 1255 43 18 6035 .9996 .6038 .3962 .7602 9993 .7609 .2391 8749 .9988 .8762 .1238 42 *9 .6066 .9996 .6070 393 .7623 9993 763! .2369 .8766 .9988 .8778 . 1222 4 1 20 .6097 .9996 .6101 3899 7 6 45 9993 7652 .2348 .8783 .9988 8795 .1205 43 21 .6!28 .9996 .6132 .3868 .7667 9993 .7674 .2326 .8799 9987 .8812 .1188 39 22 .6159 .9996 .6163 .3837 .7688 .9992 .7696 .2304 .8816 9987 .8829 .1171 38 23 .6189 9996 .6193 .3807 .7710 .9992 .7717 .2280 -8833 .9987 -8845 1155 37 24 .6220 .9996 .6223 3777 7731 .9992 7739 .2261 .8849 9987 .8862 .1138 36 25 .6250 .9996 .6254 3746 7752 .9992 .7760 .2240 .8865 .9987 .8878 .1122 35 26 .6279 .9996 .6283 3717 7773 .9992 .778i .2219 .8882 .9987 .8895 .1105 34 27 .6309 .9996 6313 3687 -7794 .9992 .7802 .2198 .8898 .9987 .8911 .1089 33 28 6339 .9996 ^343 3657 -7815 .9992 7823 .2177 .8914 9987 .8927 1073 3 2 29 .6368 .9996 .6372 .3628 .7836 .9992 .7844 .2156 .8930 9987 8944 . 1056 3 1 30 6397 .9996 .6401 3599 7857 9992 .7865 2135 .8946 .9987 .8960 .1040 30 3 1 .6426 .9996 .6430 357 .7877 .9992 .7886 .211, .8962 .9986 .8976 . IO2j( 29 3 2 6454 .9996 6459 3541 .7898 9992 .7906 .2094 .8978 .9986 .8992 .1008 28 33 .6483 .9996 .6487 3513 .7918 9992 .7927 .2073 .8994 .9986 .9008 .0992 27 34 .6511 .9996 6s?5 -3485 7939 9992 7947 .2053 .9010 .9986 .9024 .0976 26 35 6539 .9996 6 544 .3456 7959 .9992 .7967 .2033 .9026 .9986 .9040 .0960 25 3 6 .6567 .9996 6571 34 2 y 7979 .9991 .7988 .2OI2 9342 .9986 .9056 .0944 24 37 6595 9995 .6599 .3401 .7999 .9991 .8008 .1992 957 .9986 .9071 .og2C 2 3 38 .6622 9995 .6627 3373 .8019 .9991 .8028 .1972 9073 .9986 .9087 .0913 22 39 .6650 9995 .6654 3346 .8039 .9991 .8048 1952 .9089 .9986 .9103 .0897 21 4 .6677 9995 .6682 .3318 .8059 .9991 .8067 193^ .9104 .9986 .9118 .0882 2O 4* .6704 9995 .6709 .3291 .8078 .9991 .8087 .igi. .9119 9985 9i34 .0866 19 4 2 43 673! .6758 9995 9995 .6736 .6762 3264 .3238 .8098 .8117 .9991 .9991 .8107 '8l2( .1893 .1874 9i35 .9150 9985 9985 .9150 .9165 .0850 .0835 17 44 .6784 9995 .6789 .3211 -8i37 .9991 .8146 .1854 .9166 9985 .9180 .0820 16 45 .6810 9995 .6815 .3^85 8156 .9991 8165 .1835 .9181 9985 .9196 .0804 15 46 .6837 9995 .6842 3158 8i75 .9991 8185 .181^ .9196 .9985 .9211 .0789 14 47 .6863 9995 .6868 3 T 3 2 .8194 .9991 .8204 .1796 .9211 9985 .9226 .0774 '3 48 .6889 9995 .6894 .3106 .8213 .9990 .8223 .1777 .9226 9985 .9241 0759 12 49 .6914 9995 .6920 .3080 .8232 999 .8242 !7S8 .9241 9985 .9256 0744 II 50 .6940 9995 .6945 .3055 -8251 .9990 .8261 1739 .9256 .9985 .9272 .072^ 10 5i .6965 9995 .6971 .3029 .8270 .9990 .8280 .1720 .9271 9984 .9287 .0713 9 52 .6991 9995 .6996 .3004 .8289 .9990 .8299 .170 .9286 9984 .9302 .0608 8 53 .7016 9994 .7021 2979 .8307 9990 8317 .1683 .9301 9984 .9316 .0684 7 54 .7041 9994 .7046 2954 .8326 .9990 .8336 .1664 93*5 9984 933 1 .0669 6 55 .7066 9994 .7071 .2929 8345 .9990 8355 .1645 9330 9984 9346 .06 S4 5 56 .7090 9994 .7096 .2904 .8363 .9990 8373 .1627 9345 9984 .9361 .0639 4 57 7"5 9994 .712 .2879 .8381 .9990 .8392 .1608 9359 .9984 9376 .0624 3 58 .7140 9994 7145 .2855 .8400 99QQ .8410 .1590 9374 9984 939 .0610 a 59 .7164 9994 .7170 2830 .8418 .9989 .8428 I57 2 .9388 9984 9405 595 i 60 8.7188 9-9994 8.7194 i . 2806 8 8436 9-9989 8.8446 I-I554 8 9403 9-9983 8.9420 1.0580 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 87 86 85 4 6 SUR VE YING. TABLE II lA Continued. LOGARITHMS OF SINES AND TANGENTS. Arc. Sin. jDf.| Cos. Df. Tan. IDf. Cot. Arc. Arc. Sin. Df. Con. Df. Tan. Df. Cot. Arc. 1 / / / 5 c IO 8.9403 9545 142 9.9983 .9982 8.9420 9563 143 138 1.0580 0437 85 o 5 10 4'77 47 46 9.9849 .9846 3 3 9.4281 433 1 So 50 -57i9 .5669 75 o 5 20 .9682 T4 .9981 .9701 T 35 .0299 40 20 .4223 46 9843 4 .4381 49 5619 40 30 .9816 129 .9980 9836 130 .0164 3 30 .4269 45 9839 3 443 49 5570 3 40 8-9945 125 9979 8.9966 127 1.0034 20 40 43M 45 .9836 4 4479 48 552i 20 50 9.0070 122 9977 9.0093 1230.9907 10 50 4359 44 9832 4 4S 2 7 48 5473 10 60 .0192 1 10 .9976 .0216 I2O .9784 840 16 o 4403 44 .9828 3 4575 47 5425 74 o 10 .0311 "5 9975 033 6 117 .9664 50 IO 4447 44 .9825 4 .4622 47 5378 5 20 .0426 "3 9973 0453 9547 40 20 .4491 42 .9821 4 .4669 47 5331 40 30 .0539 109 .9972 .0567 III 9431 30 3 4533 43 .9817 3 .4716 46 .5284 3 40 .0648 107 .9971 .0678 108 9322 20 40 4576 4 2 .9814 4 .4762 46 20 50 0755 104 .9969 .0786 105 .9214 IO So .4618 4 1 .9810 4 .4808 45 5192 10 7 o .0859 102 .9968 .0891 104 .9109 830 17 o .4659 4 1 .9806 4 4853 45 5 I 47 73 o 10 .0961 99 .9966 0995 101 .9005 50 10 .4700 4 1 .9802 4 .4898 45 .5102 5 20 . 1060 97 .9964 . 1096 98 .8904 40 20 4741 40 .9798 4 4943 44 .5057 40 30 57 95 9963 "94 97 .8806 30 30 .4781 4O 9794 4 4987 44 .5013 3 40 -1252 93 .9961 . 1291 94 .8709 20 40 .4821 49 .9790 4 5031 44 .4969 20 50 1345 9959 1385 93 .8615 IO 50 .4861 39 .9786 4 575 43 4925 IO 8 o 1436 89 .9958 -I478 01 .8522 82 o 18 o .4900 39 .9782 4 .5118 43 .4882 72 o IO .1525 9956 .1569 8') .8431 5 IO 4939 38 .9778 4 .5161 42 4839 5 20 . 1612 85 9954 .1658 87 .8342 40 20 4977 38 9774 4 5203 42 4797 40 30 .1697 84 9952 1745 86 8255 30 30 50' 5 37 .9770 5 5245 4* 4755 3 40 .178! 82 .9950 1831 84 8169 20 40 SOS 2 38 9765 4 .5287 4 2 4713 20 50 1863 80 9948 .1915 82 .8085 10 50 .5090 36 .9761 4 5329 41 .4671 10 9 o T 9O 79 .9946 .1997 Bi .8003 81 o 19 o .5126 37 9757 5 537 41 .4630 71 o 10 .2022 78 9944 .20; j So .7922 So 10 .5153 9752 4 ' 54" 40 4589 50 20 .2100 76 .9942 .2158 78 .7842 40 20 .5199 36 .9748 S 5451 40 4549 40 30 .2176 75 9940 2236 77 .7764 30 30 5235 3 c; 9743 4 549 1 40 4509 30 40 2251 73 9938; .2313 76 7687 20 40 .5270 36 9739 5 553 1 4 .4469 20 50 2324 73 9936 .2389 74 .7611 10 5 .5306 35 9734 4 5571 40 4429 IO IO O .2397 7 1 9934 .2463 73 7537 80 o 20 5341 34 9730 S .5611 39 .4389 70 o 10 .2468 7 9931 .2536 73 .7464 5 IO 5375 34 9725 4 .5650 39 435 5 20 2538 63 .9929 .2609 7 1 7391 40 20 5409 34 .9721 5 .5689 38 43" 40 30 .2606 68 9927 3 .2680 70 7320 30 3 5443 34 .9716 .5 5727 39 4273 3 4 .2674 66 9924 2 2750 69 .7250 2O 40 5477 33 97" 5 .5766 38 4234 20 5 .2740 66 .9922 3 .2819 68 .7181 10 50 55 10 33 .9706 4 .5804 33 .4196 IO II .2806 64 .9919 2 .2887 66 .7113 79 o 21 -5543 33 .9702 5 .5842 37 4158 69 o IO .2870 64 .9917 3 2953 67 .7047 50 10 .5576 33 .9697 5 5879 38 .4121 5 20 2934 63 .9914 2 .3020 65 .6980 40 2O .5609 32 .9692 5 5917 37 .4083 40 30 .2997 61 .9912 3 3085 64 .6915 30 30 .5641 S 2 .9687 5 5954 37 .4046 3 40 3058 61 .9909 2 3H9 63 .6851 20 40 5673 3 1 .9682 5 37 .4009 20 5 3"9 60 .9907 3 3212 63 .6788 IO 50 574 32 .9677 5 ^6028 36 3972 10 12 O 3*79 59 994 3 3275 6l 6725 78 o 22 5736 3 1 .9672 5 .6064 36 3936 68 o 10 3238 58 .Q^OIj 2 333 6 61 .6664 5 IO 5767 3 l .9667 6 .6100 36 .3900 50 20 .3296 57 .9899 3 3397 61 .6603 40 20 5798 30 .9661 5 .6136 36 3864 40 30 3353 57 .9896, 3 3458 59 .6542 3 3 .5828 3 1 9656 8 .6172 .,f .3828 3 4 .3410 56 9893' 3 35 r 7 59 6483 20 40 5859 .3 .9651 5 .6208 3S 3792 20 5 -3466 55 .9890 3 3576 58 6424 IO So .5889 30 .9646 6 .6243 36 3757 IO '3 o 3521 54 .9887 3 3634 57 .6366 77 o 23 o 59*9 29 .9640 5 .6279 35 3721 67 o IO 3575 54 988* 3 .3691 57 .6309 50 10 5948 3 .9635 6 63M 34 .3686 50 20 .3629 53 .9881; 3 3748 56 .6252 40 20 5978 9 .9629 5 .6348 35 3652 40 30 .3682 52 .9878' 3 .3804 55 .6196 30 30 .6007 29 .9624 6 .6383 34 3 6l 7 3 40 3734 52 9875! 3 3859 55 .614. 20 40 .6036 29 .9618 m .6417 35 .3583 20 50 .3786 .9872 3 39*4 54 .6086 10 50 .6065 28 96*3 6 .6452 34 3548 IO 14 o 3837 5 .9869 3 3968 53 .6032 76 o 24 o .6093 28 .9607 5 .6486 34 35*4 66 o 10 .3887 So .9866 3 .4021 53 5979 50 10 .6121 28 .9602 6 .6520 33 .3480 5 20 3937 49 .9863 4 474 53 .5926 40 20 .6149 28 9596 6 6553 34 3447 40 30 .3986 49 9859 3 .4127 52 5873 30 30 .6177 28 9590 6 .6587 33 34i3 30 40 .4035 4 8 .9856 3 .4178 -5822 20 4 .6205 27 .9584 5 .6620 34 20 50 .4083 47 9853 4 .4230 5i 577 IO 5 .6232 27 9579 6 .6654 33 3346 IO '5 o 9-4I30 47 9-9 8 49 3 9.4281 500.5719 75 o 25 o 0.6259 27 9-9573 7 9.6687 33 o-33 l 3 650 Arc. COB. Df. Sin. Df. Cot. Df. Tan. Arc. Arc. Cos. Df. Sin. Df. Cot. Df. Tan. Arc. TABLES. 47 TABLE IIU Continued. LOGARITHMS OF SINES AND TANGENTS. Arc. Sin. ->f. Cos. Df. Tan. Df. Cot. Arc. Arc. Sin. Df. Cos. Df. Tan. |Df. Cot. Arc. / / / / 25 o 9.6259 27 9-9573 6 9.6687 33 0.3313 65 o 35 o 9.7586 18 9-9*34 9 9.8452 27 0.1548 55 o 10 .6286 27 .9567 6 .6720 3 2 .3280 50 IO .7604 18 9 I2 5 9 .8479 27 .1521 5 20 6313 27 .9561 6 .6752 33 .3248 40 20 .7622 18 .9116 9 .8506 27 .1494 40 30 .6340 26 9555 6 .6785 32 3215 30 3 .7640 i7 .9107 9 8533 26 .1467 3 40 .6366 26 9549 6 .6817 33 3 lS 3 20 40 7 6 57 18 .9098 9 8559 27 .1441 20 50 .6392 26 9543 6 .6850 32 3*50 IO 50 7675 17 .9089 9 .8586 27 .1414 10 26 o .6418 26 9537 7 .6882 3 2 .3118 64 o 36 o .7692 18 .9080 IO .8613 26 1387 54 o IO .6444 26 953 6 .6914 3 2 .3086 5 IO .7710 jy .9070 9 .8639 27 .1361 50 20 .6470 25 9524 6 .6946 .3054 40 20 .7727 J 7 .9061 9 .8666 26 1334 40 3 .6495 26 .9518 6 .6977 S 2 3023 30 30 7744 17 .9052 IO .8692 26 .1308 30 40 .6521 25 .9512 7 .7009 .2991 20 40 .7761 17 .9042 9 .8718 27 .1282 20 50 .6546 24 955 6 .7040 32 .2960 IO 5 .7778 17 9033 10 8745 26 '1255 10 27 o .6570 25 9499 7 .7072 3 1 .2928 63 o 37 o 7795 16 9023 9 .8771 26 .1229 53 10 6 595 25 .9492 6 7103 .2897 5 IO .7811 17 .9014 IO 8797 27 .1203 5 20 .6620 24 .9486 7 7134 31 .2866 40 20 .7828 16 .9004 9 .8824 26 .1176 40 3 .6644 24 9479 6 7 l6 5 3 1 2835 3 3 .7844 17 8995 IO .8850 26 .1150 3 4 .6668 9473 7 .7196 30 ,2804 20 4 .7861 16 .8985 IO .8876 26 .1124 20 50 .6692 2 4 .9466 7 .7226 3' 2774 IO 5 .7877 1 6 8975 10 .8902 26 .1098 10 28 o .6716 24 94S9 6 7257 3 2743 6a o 38 o .7893 17 .8965 IO .8928 26 .1072 52 o 1.0 .6740 23 9453 7 .7287 3 2 7 I 3 5 IO .7910 16 8955 IO 8954 26 . 1046 5 20 .6763 24 .9446 7 7317 3* .2683 40 20 .7926 15 8945 10 .8980 26 .1020 40 3 .6787 23 9439 7 7348 3" .2652 3 3 .7941 16 8935 to .9006 26 .0994 3 40 .6810 23 .9432 7 7378 30 .2622 20 40 7957 16 .8925 xo .9032 af> .0968 20 50 -6833 23 9425 7 .7408 3 .2592 10 5 7973 16 .8915 10 .9058 26 .0942 10 29 o .6856 22 .9418 7 .7438 29 .2562 61 o 39 o .7989 15 .8905 10 .9084 26 .091^ 51 o 10 .6878 23 .9411 7 .7467 3 2533 50 IO .8004 16 .8895 XI .9110 25 .0890 50 20 .690! 22 .9404 7 7497 29 2503 40 20 .8020 15 .8884 IO 9*35 2fc .0865 40 3 6923 23 9397 7 .7526 3 .2474 3 3 8035 15 .8874 10 .9161 26 .0839 3 40 .6946 22 939 7 7556 29 .2444 20 4P 8050 1 6 .8864 ii .9187 25 .0813 20 50 .6968 22 9383 8 7585 29 2415 TO 5 .8066 15 -8853 10 .9212 26 .0788 10 30 o .6990 22 9375 7 .7614 3 .2386 60 o 40 o .8081 15 .8843 II .9238 26 .0762 50 IO .7012 21 .9368 7 .7644 29 2356 5 IO .8096 15 .8832 II .9264 25 .0736 5 20 7033 22 .9361 8 7673 28 .2327 40 20 .8111 M .8821 II .9289 26 .0711 40 30 7055 21 9353 7 .7701 29 .2299 3 3 .8125 15 .8810 IO 93^5 26 .0685 3 40 .7076 21 9346 8 773 29 .2270 20 40 .8140 15 .8800 11 934' 25 .0659 20 50 .7097 21 9338 7 7759 29 .2241 10 5 M .8789 I! .9366 26 .0634 10 31 o .7118 21 9331 8 .7788 28 .2212 59 o 41 o .8169 15 8778 II 9392 2S .0608 49 10 7'39 21 9323 8 .7816 29 .2184 5 10 .8184 14 .8767 11 .941? 26 .0583 50 20 .7160 21 9315 7 .7845 28 2155 40 20 .8.98 15 .8756 II 9443 2 5 557 40 30 .7181 90 .9308 8 7873 2 9 .2127 3 30 .8213 T 4 8745 12 .9468 26 0532 3 4 .7201 21 .9300 8 .7902 2H .2098 20 40 .8227 J 4 8733 II 9494 2 5 .0506 20 5 .7222 2O .9292 8 7930 23 .2070 IO 5 .8241 .8722 XX 25 .0481 IO 32 o 10 .7242 .7262 20 20 .9284 .9276 8 8 7958 .7986 28 28 .2042 .2014 58 o 5 42 o IO 8255 .8269 M M .8711 .8699 12 11 9544 957 26 2 5 .0456 .0430 480 5 20 .7282 2O .9268 8 .8014 28 .1986 4 20 .8283 M .8688 12 9595 26 .0405 40 30 .7302 2O .9260 8 .8042 28 .1958 30 3 .8297 M .8676 ii .9621 2 5 .0379 3 40 .7322 2O .9252 8 .8070 27 .1930 20 40 .8311 13 .8665 12 .9646 25 0354 20 5 7342 T 9 .9244 8 .8097 28 .1903 IO 50 8324 14 .8653! I2 .9671 26 .0329 IO 33 o .7361 19 .9236 8 .8125 28 .1875 57 o 43 o 8338 13 .8641) 12 .9697 2 5 .0303 47 o IO .7380 20 .9228 9 8153 87 .1847 50 IO 8351 14 .8629 II .9722 2 5 .0278 5 20 .7400 19 .9219 8 .8180 28 .1820 40 20 8365 .8618 12 9747 2 5 .0253 40 30 .7419 '9 .9211 8 .8208 27 .1792 3 3 .8378 13 .8606 12 .9772 26 .0228 3 40 .7438 1C .9203 9 82^5 28 1765 20 40 .8391 8594 12 .9798 25 .0202 20 50 7457 K .9194 8 .8263 27 IO 50 .8405 13 .8582 13 .9823 25 .0177 IO 34 o .7476 18 .9186 9 .8390 27 .1710 560 44 o .8418 13 .8569 12 .9848 26 .0152 46 o 10 7494 ly .9177 8 .83.7 27 .1683 5 IO -8431 I 3 8557 12 .9874 2 5 .OI26 5 20 7513 18 .9169 9 8344 27 -1656 40 20 .8444 13 .8545 13 .9899 25 .0101 40 30 7531 19 .9160 9 8371 27 .1629 30 30 8457 12 .8532 12 9924 M .0076 3 4 755 C ii 9151 9 8398 27 -I602 20 40 .8469 X 3 .8520 X 3 9949 26 .0051 20 50 .7568 18 .9142 8 .8425 27 1575 IO 5 .8482 '3 .8507 12 9-9975 25 .0025 10 35 o 9.7586 18 9-9I34 9 9.8452 27 0.1548 55 45 OJ9.8495 9-8495 o.oooo 0.0000 45 o Arc Cos. Df Sin. Df Cot. Df Tan. Arc. ArcJ Cos. Df. Sin. 'Df. Cot. Df. Tan. Arc. 48 SURVEYING. S w " I ^ s & H 2 2 g PQ < |S H o ^vO 10 Tf co (M M a ooo 5^^-tJ.o j^^^^i i m ^<3 6 I a . t^ t^ t^ t^ t^vo VO } VO VO VO VO VI W j> OWint^CvN^ir'vo QJ OOO^OOO 00 00 I I , K S K J?i > ?^- ^^ !? ? ? 5- 5- Ar and s 2: inMt^woo-^oinw O MNNPtror-in invo H ft m * mvo fxoo ON e M N fn * < MMMHMMIHMMr; MtMMMI w ooo t^.vo m . VO * M R 8 1 f: 8;%^ 5- 8. 2 ^^ 1 0>O 1 ** 0>O t^OO O HMHHHMHMM ^ I TABLES. 49 cT^ "13 " > I I M M ^ I Z~ j>ooco*~v Tf,v,o^ioe>io r ~ - M oo i- JVO^ONO CQ vo^O^O'O^OvO'OOVO $ \OvOVO\O 1 I I to i^oo o & <- ovo t^oo o ^ M N m * o C1WN w f ^^l- 1 ro o M A ooo t^vo 10 -* m N M a o>oo 1 ONOO t^vo 10 * m M M CO 05 5s i-^t-. ^-TTI/*. lovoCD vot^ r^co ooooooo O Ot-'noiNNmmro^ *-^--*-4--4-^( - Tf-vo t^ ON O M f) ^ iovO t^C rrt N M O O ONOO r^vo voxo-^-roNMMO ONOO t^-vo vo to * ro (S M O 12 vo oo O S ^-vo oo O o ^-vo oo O H^iot^o^rn ioS oo cf ?5 * Jo tQ 4th. nts rc nd . IS gar, s e ! > ^ 1 Si 3 S & ?f H i f 2 1 (A Qu K5-2- ' O O ON H-^-oo N vo O ro t~- t>00 00 ON ON ONO O O t- n ~ P4 H - -^ 2 8 ^5-5> 2 8 8>3.8, 2 8 O|H|^ N 8 a !i I MMMMM ""2,5-^8 2. 8,3- 3, 8 2 2,5-^8 8 & 5-^8 2 a 5-^,8 2 o 2,5-^8 2 I>-*O IO m N O ON t^vO 10 CO 2 "** fO(MMO ONOO t-. OVO lOlO^-^rOJpr^NMMHMOOOOONONONON ONOO OO OO OO OO OO 00 * M ^-vo oo O MVO ro f>vo -^-vo mu^wvo too w MVO o^oo to o> *- O t^ N tovo to N r^ w m co 10% w W* O o O -^-00 M '^.ONH M ro-ovo r--oo ON O w ct m * >ovo vo t-^oo ON OOOOONON t lIHsi aS-2> rt 28^5-2> H 2 8^5-S,H s 8&3-&* 88&3>$* 2 TABLES. 5 1 8 i 8 I i " oooocoo oo 2 M ooooo w oooooooooo oo ooo^ I iri i 5 1 4 i I ) i i 5 ^ H H ^ _ !^ _ _ w . o T? I H I H ooo o" o o o o o" * ro CNI M m (- ro 01 w e I O I tO ' CO I CO i/i iru-tir)Tt-Tj--^-r^, fCJ- - ^. Q ONO^ ^NO. in -r ro cV -^ o o-oo & ro oo *. " K - ^vg!g v"vo vo^cfvo vo vo Sn^ln * % & 2, ? & ONO?0?0? 0?00'0?0? 0?0?0?OJ OOWOOTOOOOOOOOoS oToO a3O30C>2S^5-2,*2g^ ' ' I i I I ! t t^O -*c N in O . o f> tooo M rovo OO w romN M M o Ov OOO t^ t^vo irivo-4-rr)rON ~ O O Ovoo t^vo vo 10^- ro ro iAinininioin- ON O ON ON ONOO OOOOOOOO vo vo vo vo vo o-oo oo M ^- ti. ON O- ON in in in M. M (N c* N r^!fOr r ^mr)--}-^-^-in i nin invo vo vo vo r^ r-^ t^ t^oo oocooo ONCNO-ONQNO O O O HM vOvOvOvOvOvOvOVOvovOvovovovovONOvovovOvovOvovov)vONOVOvovovovoo t^t^txti.lfi.tx ON ON O\ ON ON ON ON i|i|i|i|i|i|i|i:3 OOOOQQt,OOQQOQ%QOOOQQ$,O3OOGGlO OOOC^OOOOOj^OOOOOiJjOO |TH|U}|O|^|NIO|O I A-k I A-k ' A-k ' /\> A> ' A> ' AA i i i e i I I h-. t^vo vo vo ninn^ < i*- i j-romrON CN! N M ^"2 ' I ' k ,H 2 8 &S.&* 2 8 ^>3 & H 2 8 ^5-S> H 2 S 8.5-&H g o o o O H o o o^o O H o o o o oS | L i 131515 ' 5 ' 5 i 5 SURVEYING. s i a o c *. rt- M t/ > .s ?s 8- iJ <* Q \o M mO -^-oo N \O O r^vD o * rx. ? M ONO ^- ^00 S IHOOVO fOM to o*O ro ovo w O tx.f)O\ir>-t t^fi inKi'oi?; ^^ ^ ^ SS 3 > 3 ' 3 o ON oooo ON oooo ON oooo ON 1 M ' 1 TH 1 i% |H |O | i i 1 in N ot^^-woo in M oo in^-oo ^M t^roomt-t t^ moo ^ oo oo oo oo oToToo oo ocPocToToo o? oo^oo oooooooooooooooooooooooooooo mMVOMvoOmo^o- mro N gco s O W rn ^4- m'O oo O O M c* ^- tr-.vo t^oo ON M pg co Tf- u~jvc t>*oo r- m u-i 10 in iO m i/i iovo VOVOVO'O'O'OO'O N.t^.t^r>.t^.t>.t^.t^( oooooooooooooooooooooooooooooooooooooooooooooooooooo< "2 8 &$-& 2 8 ^5-a" 2 8 &S-2$ 2 8 - o ON TABLES. 53 i i jo t^-^-MVD O W rorOM t^c4 invO in N t^ O M C> inoo ON*O MOO m\O in ro*o -^vo ro f>vO i-i ON t^ in moo ^ O^ M in r-. O IN rcvo 00 S3 N To Tf .0 mO M3 m m Tj- m 01 M OM^TMHOO T|- m O -l-OO M m 10 in 10 -* 5 00 ^*- ro ro M M G ONCO t^vo vo m * ro CNI >- O O>oo f~vo m -*- ro ~ Q Ooo vo in < M O- t^^o TJ- IN o oo vo * S S MM H ^ H M M M a M O O O O O O O O C^ 0> O> 0>oo oooooooooo t^t^?.fx O O-OO 00 OOOOOOOOOO OO O N ^t-'O CO O w rO Tj"O CO O M N fO i/~JVO a I o 19 CD O O ON g 9 * 8 I I I f i 2 8 a5-S> 2 8 8,5-S, 1 I W i r. X 90 o o o o o oooo o oooo o oooo o oooo o oooo O oooo o o o o o o m^-(^)lN m-<(-roS>H m-<*-rocN)F. m- 4 I '.-',. I - O .' O in ro I? t^ "* 5 g-vg' rp i >O vb vo vo NO vo * O OCO OO hxVO ^ CO M O*\O O'O roO t^.^j-0 t^-^-M i^rt-O t^^)-o t-^o^ O'O N O> ON ' I I I I I I oooo o oooo o oooo o i i |o|o| - - " OO f> m ^ IN - O t^O ^ tN *-< O t^^O -T N w ) 00 CO CO 00 OOO OOO ro co C* W in O O OOO t^\O 10 -* w M O O txvo ^- ro H o>CO vo ^J- W O CO vo C* CO ^" ^S? t^OO CO <> O w W ro rf- IO\O VO t^OO VO O M M CN CO -^- lOVO VO t-x< o>" ***oX" *ov' ' * di * * * . * ^ 2 8 &*&* 2 8 K9. I Tl* I LO I O 2 8 ^5-S> 2 8 R>5-5> 2 8 fcfc& 2 8 2 8 &$ 2 8 ^5- 54 SURVEYING. J=2 ||| 1 i o>oo tx\o in m w M Q 1 jl ^0! e rt 05 >0 I ! Ifs s 05 rt OOOOlOOCtMOlO* 1?li! 00 00 00 OO | fa- I J W B 1 05 05 OOsOOOOO. ' ^ 2 ' o rt O H SUM w el-oil I 1 1 S ^"aS 05 X 1 " 1 1 j < M i C ^5 uji 1^1 ^rtg Sa B . m 05 *f* &S 1 ?5. ^ ^ ? 5 5. 5. o 1 1 OOO IX.N 10 re ro CO CO CO i 1 t/5 M U > o ; ? 8 ' 3 00 N W moo M rovo ^ O O S OIOMQO "^O^o M o f;* --OOoot^iowO ** O o ooo oo oo r^ t^\o CO sssn 1 > < M H rt GO 00 w _} u S i tio a j'i Q 1 00 000000000000000000 oo^ogogog^o^ CQ < H rt - 05 05 O 2 o c *c 2 GO 1 1 I a I ^i'aS CO GO 1 ^ 1 1 rt ^=2 e;! | 1 g:* !j* 5 5 5 o 1 "ll ^a CO 05 10 "* "* o i 1 N 2 o t- 3 rt ! f IEIII^MI 5gf 8 JSJsf 5 iglfl 1 ! 1 00 00 - ^^.^^^^^.,0,0 10 li 1 1 ooooooooo O iiiiiiiiii iillS 05 0) 05 ijhji 87 267 1 '*r ^ 1 M M rr> + in\O t^OO O O i i TABLES. 55 t-^O ir> -^ O q> <> <> O <> ooo oocjooooooor-ix. t-. t^vo vo NO NO in in up O OO GO O t^OO OO OO^OOO H CN ir>O M VO O f*G 3 -^- o ' o" M f ro M- U->NO t>00 ON a HI ft rn * UTO I^C o o M ' "*** kX ^ ' H M O O O ONOO 00 t^ t>. CO ^ ^S cT R. 2 2, oo S'vS ^mrONvNHl O O mnmrofOro<5 r^ lONOOOfls OOOOONO' VO 1^.00 O> OOO r\O >0 * ro tl HI otO to * ro f) I ^ooosoo ooo *" vo ir> rv ^ CO VO\OVO^O*OVO\O'O'O PQ \O'OvoiOiOinioiO*/>vjfc '& v o > o v o ^^ o ONOOoOi OOO O^OOOfls OOO S S;^! o ?? ; ^ M C* CO * IOVO t^OO O ^ - ' 1 SURVEYING. TABLE V. HORIZONTAL DISTANCES AND ELEVATIONS FROM STADIA READINGS. 204. | i 1 2 3 Minutes. Hor. DifT. Hor. Diff. Hor. Diff. Hor. DifT. Dist. Elev. Dist. Elev. Dist. Elev. Dist. Elev. o . . 100.00 o.oo 99-97 1.74 99 .88 3-49 99-73 5-23 2 . . " 0.06 " 1. 80 99.87 3-55 99.72 5.28 4 " O.I2 1.86 u 3-6o 99.71 5-34 6 . . O.I 7 99.96 1.92 3-66 u 5-40 8 . . 0.2 3 u 1.98 99.86 3-7 2 99.70 5-46 10 . . n O.29 " 2.04 u 3-/8 99.69 5.52 12 . . o-35 2.09 99-85 3-84 5-57 14 . . 0.41 99-95 2.15 it 3-90 99-68 5-63 16 . . it 0.47 " 2.21 99.84 3-95 5-69 18 . . 0.52 " 2.27 " 4.01 99.67 5-75 20 . . 0.58 2-33 99-83 4.07 99.66 5.80 22 . . 0.64 99-94 2.38 4-13 5.86 24 . . 0.70 H 2-44 99.82 4.18 99.65 5-92 26 . . 99-99 0.76 (( 2.50 4.24 99.64 5.98 28 . . (4 o.Si 99-93 2.56 99.81 4-3 99-63 6.04 3 . . 0.87 2.62 4-36 " 6.09 3 2 . . o-93 2.67 99.80 4.42 99.62 6.15 34 " 0.99 2-73 " 4.48 M 6.21 36 1.05 99.92 2-79- 99-79 4-53 99.61 6.27 38 i. ii u 2.85 4-59 99.60 6-33 40 . . " 1.16 " 2.91 99.78 4-65 99-59 6.38 42 . . 1.22 99.91 2.97 " 4.71 " 6.44 44 . 99.98 1.28 ". 3.02 99-77 4.76 99.58 6.50 46 . . (( i-34 99.90 3-o8 <( 4.82 99-57 6.56 48 . . 1.40 3- 1 * 99.76 4.88 99-56 6.61 50 . . (( i-45 " 3.20 4-94 " 6.67 52 i-5i 99.89 3.26 99-75 4-99 99-55 6-73 54 '57 " 3-3i 99-74 5-5 99-54 6.78 56 . . 99-97 1.63 <( 3-37 " 5- 11 99-53 6.84 58 . . " 1.69 99.88 3-43 99-73 5-!7 99-52 6.90 60 . . 1.74 " 3-49 " 5-23 99-51 6.96 ' = o-75 0.75 O.OI 0-75 O.O2 0.75 0.03 0-75 0.05 C= I.OO I.OO O.OI I.OO O.O3 I.OO 0.04 I.OO 0.06 r=i.2 5 125 O.O2 1-25 0.03 1.25 0.05 1.25 0.08 ^ * This table was computed by Mr. Arthur Winslow of the State Geological Survey of Pennsylvania. For description of chart for graphical reduction see p. v. TABLES. 57 TABLE V. Continued. HORIZONTAL DISTANCES AND ELEVATIONS FROM' STADIA READINGS. 40 5 6 70 Minutes. Hor. Diff. Hor. Diff. Hor. Diff. Hor. Diff. Dist. Elev. Dist. Elev. Dist. Elev. Dist. Elev. O . . 99.51 6.96 99-24 8.68 98.91 10.40 98.51 12. IO 2 . . 7.02 99- 2 3 8.74 98.90 10.45 98.50 12.15 4 99-5 7.07 99.22 8.80 98.88 10.51 98.48 12.21 6 . . 99-49 7-13 99.21 8.85 98.87 ro-57 98.47 1226 8 . . 99.48 7.19 99.20 8.91 98.86 10.62 98.46 12.32 10 . . 99-47 7-25 99.19 8.97 98.85 10.68 98.44 12.38 12 . . 99.46 7-30 99.18 9-3 98.83 10.74 98.43 12.43 14 . . " 7-36 99.17 9.08 98.82 10.79 98.41 12.49 16 . . 99-45 7-42 99.16 9.14 98.81 10.85 98.40 I2 -55 18 . . 99-44 7.48 99-15 9.20 98.80 10.91 98.39 12.60 20 . . 99-43 7-53 99.14 9.25 9 8. 7 8 10.96 98.37 12.66 22 . . 99.42 7-59 99-13 9-3 1 98.77 1 1. 02 98.36 12.72 24 . . 99.41 7-65 99.11 9-37 98.76 11.08 98.34 12.77 26 -. . 9940 7.71 99.10 9-43 98.74 11.13 98.33 12.83 28 . . 99-39 7.76 99.09 9.48 98.73 11.19 98.31 12.88 30 . . 9938 7.82 99.08 9-54 98.72 11.25 98.29 12.94 32 - 99-38 7.88 99.07 9.60 98.71 11.30 98.28 13.00 34 99-37 7-94 99.06 9.65 98.69 11.36 98.27 T 3-o5 36 99-36 7-99 99-05 9.71 98.68 11.42 98.25 13.11 38 99-35 8,05 99.04 9-77 98.67 11.47 98.24 i3-!7 40 . . 99-34 8.11 99-03 9-83 98.65 "53 98.22 13.22 42 . . 99-33 8.17 99.01 9.88 98.64 11.59 98.20 13.28 44 . . 99-3 2 8.22 9900 9-94 98-63 11.64 98.19 J 3-33 46 .. 99- 3 1 8.28 98.99 10.00 98.61 11.70 98.17 13-39 48 . . 99-30 8-34 98.98 10.05 98.60 11.76 98.16 13-45 50 . . 99.29 8.40 98.97 IO.II 98.58 11.81 98.14 !3-5o 52 . . 99.28 8-45 98.96 10.17 98.57 11.87 98.13 13-56 54 99.27 8. 5 I 98.94 10.22 98.56 11 -93 98.11 13.61 56 . . 99.26 8. 57 98.93 10.28 98.54 11.98 98.10 13-67 58 . . 99-25 8.63 98.92 10.34 98.53 12.04 98.08 13-73 60 . . 99.24 8.68 98.91 10.40 98.51 12.10 98.06 13-78 ' = 0.75 0-75 0.06 0.75 O.O7 0-75 o.oS 0.74 O.IO c i.oo I.OO 0.08 0.99 0.09 0.99 O.I I 0.99 0.13 '=1.25 1.25 O.IO 1.24 O.I I 1.24 0.14 1.24 0.16 SUR VE YING. TABLE V. Continued. HORIZONTAL DISIANCES AND ELEVATIONS FROM STADIA READINGS. 8 9 1O 11 j Minutes. Hor. Dift Hor. Diff. Hor. Diff. Hor. Diff. Dist. Elev. Dist. Elev. Dist. Elev. Dist. Elev. . . 98.06 I3-78 97-55 ws 96.98 17.10 96.36 18.73 2 . . 98.05 13.84 97-53 15-51 96.96 17.16 96.34 18.78 4 98.03 13.89 97-5 2 15-56 96.94 17.21 96.32 18.84 6 . . 98.01 13-95 97-5 15.62 96.92 17.26 96.29 l8.8 9 8 . . 98.00 I4.OI 97.48 15-67 96.90 I7.3 2 96.27 18.95 10 . . 9 7. 9 8 14.06 97.46 15-73 96.88 17-37 96.25 19.00 12 . . 97-97 14.12 97-44 15-78 96.86 17-43 96.23 19.05 I 4 . . 97-95 14.17 97-43 15.84 96.84 17.48 96.21 19.11 16 . . 97-93 14.23 97 4 1 15.89 96.82 17-54 96.18 19.16 18 .. . 97.92 14-28 97-39 15-95 96.80 T 7-59 96.16 19.21 20 . . 97.90 14-34 97-37 16.00 96.78 17-65 96.14 19.27 22 97.88 14.40 "97-35 1 6.06 96.76 17.70 96.12 19.32 24 97.87 14-45 97-33 i6.n 96.74 17.76 96.09 19.38 26 . . 97-85 I4-5 1 97-3 1 16.17 96.72 17.81 96.07 19-43 28 . . 97-83 14.56 97.29 16.22 96.70 17.86 96.05 19.48 30 . . 97.82 14.62 97.28 16.28 96.68 17.92 96.03 19.54 3 2 . . 97.80 14.67 97.26 16.33 96.66 17.97 96.00 19-59 34 97-78 M-73 97.24 16.39 96.64 18.03 95-98 19.64 36 .. 97.76 14.79 97-22 16.44 96.62 18.08 95-96 19.70 38 . . 97-75 14.84 97.20 16.50 96.60 18.14 j 95.93 J 9-75 40 . . 97-73 14.90 97.18 16.55 96.57 18.19 95-91 19.80 42 . . 97-71 14-95 97.16 1 6.6 1 96.55 18.24 95-89 19.86 44 . 97.69 15.01 97.14 1 6.66 96.53 i8'. 3 o 95.86 19.91 46 . . 97.68 15.06 97.12 16.72 96.51 18-35 95.84 19.96 48 . . 97.66 15.12 97-iQ 16.77 96.49 18.41 95.82 20.02 50 . . 97.64 I5- X 7 97.08 16.83 96.47 18.46 95-79 20.07 52 . . 97.62 15-23 97.06 1 6.88 96.45 18.51 95-77 20. 1 2 54 . 97.61 15.28 97.04 16.94 96.42 18.57 95-75 20. 1 8 56 . . 97-59 '5-34 97.02 16.99 96.40 18.62 95-72 20.23 58 . . 97-57 15.40 97.00 17.05 96.38 1 8.68 95-70 20.28 60 . . 97-55 *5-45 96.98 17.10 96.36 18.73 95.68 20-34 ' = 0.75 0.74 O.I I 0.74 0.12 0.74 0.14 0-73 0.15 c i.oo 0.99 0.15 0.99 0.16 0.98 0.18 0.98 0.20 rasIJtj 1.23 0.18 1.23 0.21 1.23 0.23 1.22 0.25 TABLES. 59 TABLE V '.Continued. HORIZONTAL DISTANCES AND ELEVATIONS FROM STADIA READINGS. 12 13 14 15 Minutes. Hor. Diff. Hor. Diff. Hor. Diff. Hor. Diff. Dist Elev. Dist. Elev. Dist. Elev. Dist. Elev. O . . 95.68 20.34 94-94 21.92 94-15 23-47 93-30 25.00 2 . . 95-65 20.39 94.91 21.97 94.12 23-52 93-27 25-05 4 95-63 20.44 94.89 22.O2 94.09 23-58 93- 2 4 25.10 6 . . 95.61 20.50 94.86 22.08 94-07 23-63 93.21 25.I5 8 . . 95-58 20-55 94.84 22.13 94.04 23.68 93.18 25.20 10 . . 95-56 20.60 94.81 22.18 94.01 23-73 93.16 25-25 12 . . 95-53 2066 94-79 2 2.2 3 93-98 23.78 93- i 3 25-30 14 . . 95-5i 20.71 94.76 22.28 93-95 23.83 93.10 25-35 16 . . 95-49 20.76 94-73 22.34 93-93 23.88 93-07 25.40 18 . . 95-46 20.8 1 9471 22.39 93-90 23-93 93-04 25-45 20 . . 95-44 20.87 94.68 22.44 93-87 23-99 93.01 25-50 22 , . 95.41 20.92 94.66 22.49 93-84 24.04 92.98 25-55 24 . . 95-39 20.97 94-63 22.54 93.81 24.09 9 2 -95 25.60 26 . . 95-36 21.03 94.60 22.60 93-79 24.14 92.92 25-65 28 . . 95-34 2 1. 08 94.58 22.65 93-76 24.19 92.89 25.70 30 . . 95-3 2 21.13 94-55 22.70 93-73 24.24 92.86 25-75 3 2 . . 95.29 2I.I8 94-52 22-75 93-70 24.29 92-83 25.80 34 95- 2 7 21.24 94-5 22.80 93- 6 7 24-34 92.80 25.85 36 95-24 21.29 94-47 22.8 5 93-65 24-39 92.77 25.90 38 . 95-22 21.34 94-44 22.91 93.62 24.44 92.74 25-95 40 . . 95- ! 9 21-39 94.42 22. 9 6 93-59 24.49 92.71 26.00 42 . . 95-^7 21-45 94-39 23.01 93-56 24-55 92.68 26.05 44 95-'4 21.50 94-36 23.06 93-53 24.60 92-65 26.10 46 . . 95.12 2i-55 94-34 23.11 93-5 24.65 92.62 26.1 5 48 . . 95-09 21.60 94-31 23.16 93-47 24.70 92.59 26.20 50 . . 95-07 21.66 94.28 23.22 93-45 24-75 92.56 26.25 52 . . 95-04 21.71 94.26 23.27 93-42 24.80 92-53 26.30 54 95.02 21.76 94.23 23-32 93-39 24-85 92-49 26.35 56 . . 94-99 21.81 94.20 2 3-37 93-36 24.90 92.46 26.40 58 . . 94-97 21.87 94.17 23.42 93-33 24-95 92-43 26.45 60 . . 94-94 21.92 94-15 23-47 93-30 25.00 92.40 26.50 ' = 0.75 0-73 0.16 0-73 0.17 0-73 0.19 0.72 O.2O c = i.oo 0.98 O.22 0.97 0-23 o-97 0.2 S 0.96 O.27 c = 1.25 1.22 0.27 1. 21 0.29 1. 21 0-31 i. 20 0-34 1 6o SURVEYING. TABLE V. Continued. HORIZONTAL DISTANCES AND ELEVATIONS FROM STADIA READINGS. mpf*__^_4._ _. 16 17 18 19 Minutes. Hor. Diff. Hor. Diff. Hor. Diff. Hor. Diff. Dist. Elev. Dist. Eiev. Dist. Elev. Dist. Elev. . . 92.40 26.50 9 r -45 27.96 9-45 29-39 89.40 30. 7 8 *> 92-37 26.55 91.42 28.01 90.42 29.44 89.36 30.83 4 - 92-34 26.59 9 T -39 28.06 90.38 29.48 89-33 30.87 6 . . 92.31 26.64 9 r 35 28.10 90.35 29-53 89.29 30.92 8 . . 92.28 26.69 91.32 28.15 90.31 29.58 89.26 30-97 10 . . 92.25 26.74 91.29 28.20 90.28 29.62 89.22 31.01 12 . . 92.22 26.79 91.26 28.25 90.24 29.67 89.18 31.06 14 . . 92.19 26.84 91.22 28.30 9O.2I 29.72 89.15 31.10 16 . . 92.15 26.89 91.19 28.34 90.IS 29.76 89.11 3I-I5 18 . . 92.12 26.94 91.16 28.39 90.14 29.81 89.08 3LI9 20 . . 92.09 26.99 91.12 28.44 9O.II 2Q.86 89.04 3L24 22 . . 92.06 27.04 91.09 28.49 90.07 29.90 89.00 31.28 24 . . 92.03 27.09 91.06 28.54 90.04 29.95 88.96 31-33 26 . . 92.00 27.13 91.02 28.58 90.00 30.OO 88.93 3I-38 23 . . 91.97 27.18 90.99 28.63 89.97 30.04 88.89 31-42 30 . . 9 T -93 27.23 . 90-96 28.68 8.9-93 30.09 88.86 3M7 32 - 91.90 27.28 90.92 28.73 89.90 30.14 88.82 3i-5i 34 91.87 27-33 90.89 28.77 89.86 30.19 88.78 3I-56 36 . . 91.84 27.38 90.86 28.82 89-83 30.23 88.75 31.60 3 8 . . 91.81 27-43 90.82 28.87 89.79 30.28 88.71 31-65 40 . . 91.77 27.48 90.79 28. 9 2 89.76 30-32 88.67 31.69 42 , . 91.74 27-52 90.76 28.96 89.72 30.37 88.64 31-74 44 91.71 27-57 90.72 29.01 89.69 30.41 88.60 3I-78 46 . . 91.68 27.62 90.69 29.06 89.65 30.46 88.56 31-83 48 . . 91.65 27.67 90.66 29.11 89.61 30- 5 i 88.53 31-87 50 . . 91.61 27.72 90.62 =9-15 89.58 30-55 88.49 31.92 52 . . 91.58 27.77 90-59 29.20 89.54 30.60 88.45 31.96 54 . - 9^-55 27.81 90-55 29.25 89.51 30-65 88.41 32.01 56 . . 91-52 27.86 90.52 29.30 89.47 30-69 88.38 3--05 58 . $ 91.48 27.91 90.48 29-34 89.44 30-74 88.34 32.09 60 . :.;' 9M5 2 7 . 9 6 90-45 29-39 89.40 30.78 88.30 32*14 r = o.75 0.72 O.2I 0.72 0.23 0.71 0.24 0.71 0.25 f = i .00 0.86 0.28 o-95 0.30 -95 0.32 0-94 Q-33 [ ' = 1.25 i. 20 0-35 1.19 0.38 1.19 0.40 1.18 0.42 J TABLES. 6l TABLE V .Continued. HORIZONTAL DISTANCES AND ELEVATIONS FROM STADIA READINGS. 2< > 2: 1 2 2 2J * Minutes. Hor. DifT. Hor. Diff. Hor. Diff. Hor. Diff. Dist. Elev. Dist. Elev. Dist. Elev. Dist. Elev. o . . 88.30 32.14 87.16 33-46 85.97 34-73 84-73 35-97 2 , . 88.26 32.18 87.12 33-50 85-93 34-77 8 4 .6 9 36.01 4 88.23 32.23 87.08 33-54 85.89 ^4.82 84.65 36-05 6 . . 88.19 32.27 87.04 33-59 85.85 34.86 84.61 36.09 8 . . 88.15 32-32 87.00 33-63 85.80 34.90 84.57 36-13 10 . . 88.11 32.36 86.96 33-6? 85-76 34-94 84.52 36-17 12 . . 88.08 32. 4 I 86.92 33-72 85.72 34.98 84.48 36.21 14 . ' 88.04 32.45 86.88 33-76 85.68 35.02 84.44 36-25 16 . . 88.00 3249 86.84 33-8o 85-64 35-07 84.40 3 6.2Q | 18 . . 87.96 32.54 86.80 33-84 85.60 35- 11 84-35 36-33 20 . . 87-93 32.58 86.77 33-89 85.56 35- i 5 84.31 36.37 22 . . 87.89 32.63 86.73 33-93 85.52 35-19 84-27 36.41 24 . . 87.85 32.67 86.69 33-97 85.48 35-23 84-23 36.45 26 . . 87.81 32.72 86.65 34.01 85-44 35- 2 7 84.18 36.49 28 . . 87.77 3 2. 7 6 86.61 34.o6 8540 35-31 84-14 36.53 3 . . 87.74 32.80 86.57 34.10 85.36 35-36 84.10 36.57 3 2 . . 87.70 32.85 86.53 34-M 85-3I 35-40 84.06 36.61 34 . 87.66 3 2.8 9 86.49 34.18 85.27 35-44 84.01 36.65 36 . 87.62 32.93 8645 34.23 85-23 3548 83-97' 36.69 38 87.58 32.98 86.41 34.27 85.19 35-S 2 83-93 36.73 40 . . 87.54 33-02 86.37 34-3 85.15 35-56 83-89 36.77 42 . . 87-51 33-07 86.33 34-35 85.11 35.60 83-84 36.80 44 87.47 33-n 86.29 34-40 85.07 35-64 83.80 36.84 46 . . 8743 33- J 5 86.25 34-44 85.02 35-68 83-76 36.88 48 . . 87.39 33-20 86.21 34-48 84-98 35=72 83.72 36.92 50 . . 87-35 33-24 86.17 34-52 84.94 35-76 83.67 36.96 52 . . 87.31 33.28 86.13 34-57 84.90 35.80 8 3 .6 3 37-oo 54 - 87.27 33-33 86.09 34.61 84.86 35-85 83.59 37-04 56 . . 87.24 33-37 86.05 34.65 84.82 35.89 83-54 37.08 58 . . 87.20 33-41 86.01 34.69 84.77 35-93 83.50 37.12 60 . . 87.16 3346 85.97 3473 84.73 35-97 83.46 37-i6 58 3 .57429 .81865 .58849 .80850 .60251 .79811 .61635 .78747 .63000 .77660 57 4 .57453 .81848 .58873 .80833 .60274 .79793 .61658 .78729 .63022 .776411 56 5 .57477 .81832 .58896 .80816 .60298 .79776 .61681- .78711 .63045 .77623; 55 6 .57501 .81815 .58920 .80799! .60321 .79758 .617'04 .78694 .63068 .77605 54 7 .57524 .81798 .58943 .80782 .60344 .79741 .61726 .78676 .63090 .77586 53 8 .57548 .81782 .58967 .80765 .60367 .79723 .61749 .78658 .63113 .77568 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640 .63135 .77550 51 10 .57596 .81748 .59014 .80730 .60414 .79688 .61795 .78622 .63158 .77531 50 11 .57619 .81731 .59037 .80713 .60437 .79671 .61818 .78604 .63180 .77513 40 12 .57643 .81714 .59061 .80696 .60460 .79653 .61841 .78586 .63203 .77494 48 13 .57667 .81698 .59084 .80679 .60483 .79635 .61864 .78568 .632251.77476 47 14 .57691 .81681 .59108 .80662 .60506 .79618 .61887 .78550 .63248 .77458! 46 15 .57715 .81664 .59131 .80644 .60529 .79600 .61909 .78532 .632711.77439 45 16 .57738 .81647 .59154 .80627i .60553 .79583 .61932 |. 78514 i . 63293 : . 77421 44 17 .57762 .81631 .59178 .80610 .60576 .79565 .61955 .78496 .63316 .77402J 43 18 .57786 .81614 .59201 .80593; .60599 .79547 .61978 .78478 .63338 .77384; 42 19 .57810 .81597 .59225 .80576 .60622 .79530 .62001 .78460 .63361 .77366! 41 20 .57833 .81580 .59248 .80558 .60645 .79512 .62024 .78442 .63383 .77347 40 21 .57857 .81563 .59272 .80541 .60668 .79494 .62046 .78424 .63406 .77329 39 22 .57881 .81546 .59295 .80524 .60691 .79477 .62069 .78405 .63428 .77310 38 23 .57904 .81530 .59318 .80507 .60714 .79459 .62092 .78387 .63451 .77292 37 24 .57928 .81513 .59342 .80489 .60738 .79441 .62115 .78369 .63473 .77273 36 25 .57952 .81496 .59365 .80472 .60761 .79424 .62138 .78351 .63496 .77255 35 23 .57976 .81479 .59389 .80455! .60784 .79406 .62160 .78333 .63518 .77236 34 27 .57999 .81462 .59412 .80438! .60807 .79388 .62183 .78315 .63540 .77218 33 28 .58023 .81445 .59436 .80420 .60830 .79371 .62206 .78297 .63563 .77199 32 29 .58047 .81428 .59459 .80403 60853 .79353 .62229 .78279 .63585 .77181 31 30 .58070 .81412 .59482 .80386 .60876 .79335 .62251 .78261 .63608 .77162 30 31 .58094 .81395 .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 29 32 .58118 .81378 .59529 .80351! .60922 .79300 .62297 .78385 .63653 .77125 28 33 : .58141 .81361 .59552 .80334 .60945 .79282 .62320 .78206 .63675 .77107,27 34 i .58165 .81344 .59576 .80316 .60968 .79264 .62342 .78188 .63698 .77088 26 35 .58189 .81327 .59599 .80299! .60991 .79247 .62365 .78170 .63720 .77070, 25 36 .58212 .81310 .59622 .80282! .61015 .79229 .62388 .78152 .63742!. 7 7051 24 37 .58236 .81293 .59646 .80264 .61038 .79211 .62411 .78134! .63765;. 77033 23 38 .58260 .81276 .59669 .80247. .61061 .79193 .62433 .78116| .63787 '.77014 22 39 .58383 .81259 .59693 .80230 .61084 .79176 '1.62456 .78098; .63810 .76996 21 40 .58307 .81242 .59716 .80212 .61107 .79158 | .62479 .78079] .63832 .76977 20 41 .58330 .81225 .59739 .80195 .61130 .79140 .62502 .78061! .63854 .76959 19 42 .58354 .81208' .59763 .80178' .61153 .79122 .62524 .78043 .63877 .76940 18 43 .58378 .811911 .59786 .80160 .61176 .79105 .62547 .78025 .63899 .76921! 17 44 .58401 .81174 .59809 .80143 .61199 .79C87 .62570 .78007! .63922 .76903 16 45 .58425 .81157 .59832 .80125 .61222 .79069 .62592 .779881 .63944 .768841 15 46 .58449 .81140 .59856 . 80108 ' .61245 .79051 .62615 .77970 .63966 .76866 14 47 .58472 .81123 .59879 .80091 ! .61268 .79033 .62638 . 77952 ' .63989 .76847 13 48 .58496 .81106 .59902 . 80073 ' .61291 .79016 .62660 .779341 .64011 .76828 12 49 .58519 .81089 .59926 .80056 .61314 .78998 .62683 .77916 .64033 .76810 11 50 .58543 .81072 .59949 . 80038 | .61337 .78980 .62706 .77897 .64056 .767S1 10 51 .58567 .81055 .59972 .80021^ .61360 .78962 .62728 .77879 .64078 .76772 9 52 .58590 .810381 .59995 . 80003 ' .61383 .78944 .62751 .77861 .64100 .76754 8 53 .58614 .81021 .60019 .79986, .61406 .78926 .62774 .77843 .64123 .76735 7 54 .58637 .81004 .60042 .79968 .61429 .78908 .62796 .77824 .64145 .76717 6 55 .58661 .80987 .60065 .79951 .61451 .78891 .62819 .77806 .64167 .76698 5 56 .58684 .80970' .60089 .79934 .61474 .78873 .62842 .77788 .641901.76679 4 57 .58708 . 80953 i .60112 .79916 .61497 .78855 .62864 .77769 .64212 .76661 3 58 .58731 .80936 .60136 .79899 .61520 .78837 .62887 .77751 .64234!. 76642 2 59 .58755 .80919 .60158 .79881 .61543 .78819 .62909 .77733 .64256|.76623 1 60 .58779 .80902 .60182 .79864; .61566 .78801 .62932 .77715 .64279|.76604 e Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine / 54 53 52 51* 50 SURVEYING. TABLE Ml. Continued. NATURAL SINES AND COSINES. 40 Sine ,61279 6J301 ,61323 64346 64368 64390 64412 64435 64457 64479 64501 64524 64546 64568 64590 64612 64657 34679 64701 64723 64746 64768 64790 64812 64834 64856 64878 64901 64945 64967 64989 65011 65033 65055 65077 65100 65122 65144 Cosin .70604 .76586 .76567 .76548 .76530 .76511 .76492 .76473 .76455 .76438 .76417 .76398 ,76361 76342 76323! 76304 76286 i 76267 76248 76210 76192 76173 76154 76135 76116 76097 76078 76059 76041 76022 76003 75984 75965| 75946 75927 75908 65188 65210 65232 65254 65342 .65430 .65452 .65474 .65496 .65518 .65540 .65562 .65584 .65606 Cosin 75870 75851 75813 75794! 75775! 75756! 75738 1 75719 75700 75630. 75661 .75642 .75604 .75585 .75566 .75547 .75509 .75490 .75471 Sine 49 41' Sine ,65606 .65628 ,65650 ,65672 65694 65716 65738 65759 65781 65803 ,65847 ,65869 ,65891 ,65913 65935 65956 65978 66000 66022 66044 66109 66131 66153 66175 66197 63218 63240 66284 66306 66327 66349 66371 66393 66414 66436 66458 60480 66523 66545 66566 66588 66610 .66875 66718 .66740 .66762 .66783 Cosin .75471 .75452 .75433 .75414 .75395 .75375 .75356 .75337 .75318 .75299 .75280 .75261 '5241 .75222 .75203 .75184 75165 .75146 .75126 .75107 75088 75069 75050 75030 75011 .66848 .66870 Cosin 74973 74934 ! 74915 i 74876 ! 74857 i 74838 74318 74799 74780! 74760 1 74741 1 74722 74703 74683 746641 74644! 74625 : 7460S 74586 1 74567 ,74548 ! ,74528! , 74509 i .74470 .74451 .74431 .74412 .74392 .74373 .74353 .74334 .74314 Sine 48 42 Sine ,66913 ,66956 ,66978 .67021 .67043 .67064 .67086 .67107 .67129 .67151 .6717 .67194 .67215 .67237 .67258 .67280 .67301 .67323 .67344 .67366 .67387 .67409 .67430 .67452 .67473 .67495 .67516 .67538 .67559 .67580 .67602 67623 67645 .67666 67709 67730 67752 ,67773 ,67795 ,67816 ,67837 .67859 .67901 .67944 .67965 .67987 .68008 .68051 .68072 .68157 .68179 Cosin Cosin .74314 .74295 .74276 .74256 .74237 .74217 .74198 .74178 .74159 .74139 .74120 .74100 ,74080 ,74061 ,74041 ,74022 ,74002 ,73983 ,73903 ,70944 73924 73904 73885 733G5 73840 7382G 7380G 73787 737C7 73747 73728 73708 73683 73649 73G29 73G10 73590 73570 73551 73531 73511 73491 73472 73452 73432 73413 73373 73353 73314 73294 .73274 .73254 .73234 .73215 .73195 .731 .73155 .73135 Sine 47 e 43 Sine ;68200 .68221 .68242 .68264 .68285 .68327 .68349 .68370 ,68391 .68413 .68434 .68455 .68476 .68497 .68518 .68539 .68561 .68582 .CSG03 .68645 .63GG6 .68688 .68730 .68751 .68772 .63793 .63814 ,68857 ,63878 ,63399 ,63920 ,63941 ,68902 ,69004 ,69025 ,69109 ,69151 ,69172 ,69193 .69214 ,69256 .69445 .t>9466 Cosin Cosin .73135 .73116 .73096 .73076 .73056 .73036 .73016 .72996 .72976 .72957 .72937 .72917 .72897 .72877 .72857 .72837 .72817 .72797 .72777 .72757 .72737 .72717 .72697 .72677 .72G57 .72637 .72017 .72597 .72577 .72557 .72537 .72517 .72497 .72477 .72457 .72437 .72417 .72397 .72377 .72357 .72337 .72317 .72297 .72277 .72257 .72236 .72216 .72196 .72176 .72156 .72136 .72116 .72095 .72075 .72055 .72035 .72015 .71995 .71974 .71954 .71934 Sine 46 Sine .69487 .69529 .69549 .69570 .69591 .69612 .69717 .69737 .60758 .69779 .69800 .69821 .69904 .69925 .69946 .69966 .69987 .70008 .70029 .70049 .70070 .70091 .70112 70132 .70153 .70174 .70195 .70215 .70257 .70277 .70298 .70319 .70339 .70360 .70381 .70401 .70422 .70443 .70463 .70484 .70505 .70525 .70546 .70567 .70587 i! | , Cosin _ .71934 60 1914 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 .71894 .71873 .71853 .71833 .71813 .71792 .71772 ,69654 .71752 ,696751.71732 .71711 .71691 .71671 .71650 .71630 .71610 .71590 .71569 .71549 .71529 .71508 .71488 .71468 .71447 .71427 .71407 .71386 .71366 .71345 .71325 .71305 .71284 .71264 .71243 .71223 .71203 .71182 .71162 .71141 .71121 .71100 .71080 24 .1059! 17 71039 16 .70628! .70649' .70670 .70711 Cosin 71019 70998 70978 70957 70937 70916 70896 70875 70855 70834 70813 70793 70772 70752 70731 70711 Sine 45 a TABLES. 73 TABLE VII. NATURAL TANGENTS AND COTANGENTS. 1 o Oo 3 / Tang Cotang Tang Cotang Tang Cotang Tang Cotang .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 60 1 .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9755 159 2 .00058 1718.87 .01804 55.4415 .03550 28.1664 .05299 18.8711 58 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7678 57 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 56 5 .00145 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 55 6 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 54 7 .00204 491.106 .01949 51.3032 .03696 27.0566 .05445 18.3655 53 8 .00233 429.718 .01978 50.5485 .03725 26.8450 .05474 18.2677 52 9 .00262 381.971 .02007 49.8157 .037'54 26.6367 .05503 18.1708 51 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 50 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 49 12 .00349 286.478 .02095 47.7395 .03842 26.0307 .05591 17.8863 48 13 .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 J46 15 .00433 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 J45 16 .00465 214.858 .02211 45.2261 .03958 25.2644 .05708 17.5205 144 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 190.984 .02269 44.0G61 .04016 24.8978 .05766 17.3432 42 19 .00553 180.932 .02298 43.5081 .04046 24.7185 .05795 17.2558 41 20 .00582 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 22 .00640 156.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 138 23 .00669 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 21 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 26 .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 88 .00815 122.774 .02560 39.0568 .04308 23.2137 .06058 16.5075 32 20 .00844 118.540 .02589 38.6177 .04337 23.0577 .06087 16.4283 31 3D .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 30 31 .00902 110.892 .02648 37.7680 .04395 22.7519 .06145 16.2722 29 :;j .00931 107.426 .02677 37.3579 .04424 22.6020 .06175 16.1952 128 33 .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 27 84 .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 K .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 :v.i .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3G78 .04658 21.4704 .06408 15.6048 20 41 .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 19 42 .01222 81.8470 .02963 33.6935 .04716 21.2049 .06467 15.4638 18 43 .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 15.3943 17 44 .01280 78.1263 .03026 33.0452 .04774 20.9460 .06525 15.3254 16 45 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.8571 15 4fi .C1338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 47 .01367 73.1390 .08114 32.1181 .04862 20.5691 .06613 15.1222 13 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0557 12 49 .01425 70.1533 .08172 31.5284 .04920 20.3253 .06671 14.9898 11 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14.7954 8 53 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 r 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 6 55 .01600 62.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 5 56 .01629 61.3829 .03376 29.6245 .05124 19.5156 .06876 14.5438 4 57 .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 3 58 .01687 59.2659 .03434 29.1220 .05182 19.2959 .06934 14.4212 2 59 .01716 58.2612 .08463 28.8771 .05212 19.1879 .06963 14.3607 1 60 .01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 Cotang Tang Cotang Tang Co tang Tang Cotang j Tang t 89 88 87* 86 74 SUR VE YING. TABLE VII. Continued. NATURAL TANGENTS AND COTANGENTS. / 4 I 5 6 70 / Tang Cotang Tang Cotang Tang Cotang i Tang Cotang .06993 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 3 .07080 14.1235 .08837 11.3163 .10599 9.43515 .12367 8.08600 57 4 .07110 14.0655 .08866 11.2789 .10628 9.40904 .12397 8.06674 56 5 .07139 14.0079 .08895 11.2417 .10657 9.38307 .15426 8.04756 55 G .07168 13.9507 .08925 11.2048 .10687 9.35724 .12456 8.02848 54 7 .07197 13.8940 .08954 11.1681 .10716 9.33155 .12485 8.00948 53 8 .07227 13.8378 .08983 11.1316 .10746 9.30599 .12515 7.99058 52 9 .07256 13.7821 .09013 11.0954 .10775 9.28058 .12544 7.97176 51 10 .07285 13.7267 .09042 11.0594 .10805 9.25530 .12574 7.95302 50 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 .12608 7.93438 49 12 .07344 13.6174 .09101 10.9882 .10863 9.20516 .12633 7.91582 48 13 .07373 13.5634 .09130 10.9529 .10893 9.18028 .12662 7.89734 47 14 .07402 13.5098 .09159 10.9178 .10922 9.15554 .12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 .10952 9.13093 .12722 7.86064 45 1G .07461 13.4039 .09218 10.8483 .10981 9.10646 .12751 7.84242 44 17 .07490 13.3515 .09247 10.8139 .11011 9.08211 .12781 7.82428 43 18 .07519 13.2996 .09277 10.7797 .11040 9.05789 .12810 7.80622 42 19 .07548 13.2480 .09306 10.7457 .11070 9.03379 .12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 .11099 9.C0983 .12869 7.77035 40 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 7.75254 39 22 .07636 13.0958 .09394 10.6450 .11158 8.96227 .12929 7.73480 38 23 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 24 .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 7.69957 36 25 .07724 12.9469 .09482 10.5462 .11246 8.89185 .13017 7.68208 35 26 .07753 12.8981 .09511 10.5136 .11276 8.86862 .13047 7.66466 34 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 .13076 7.64732 33 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13106 7.63005 32 29 .07841 12.7536 .09600 10.4172 .11364 8.79964 .13136 7.61287 |31 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 43165 7.59575 30 81 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.57872 29 32 .07929 12.6124 .09688 10.3224 .11452 8.73172 .13224 7.56176 28 33 .07958 12.5660 .09717 10.2913 .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 .13284 7.52806 26 35 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 25 36 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 7.49465 '24 37 .08075 12.3838 .09834 10.1683 .11600 8.62078 .13372 7.47806 |23 38 .08104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 i22 39 .08134 12.2946 .09893 10.1080 .11659 8.57718 .13432 7.44509 21 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 .09952 10.0483 .11718 f 53402 .13491 7.41240 19 42 .08221 12.1632 .09981 10.0187 .11747 8.51259 .13521 7.39616 18 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .11806 8.47007 .13580 7.36389 16 45 .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 7.34786 15 46 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 7.33190 14 47 .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 7.31600 13 48 .08397 11.9087 .10158 9.84482 .11924 8.38625 .13698 7.30018 12 49 .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 7.28442 11 50 .08456 11.8262 .10216 9.78817 .11983 8.34496 .13758 7.26873 10 51 .08485 11.7853 .10246 8.76009 .12013 8.32446 .13787 7.25310 9 52 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 7.23754 8 53 .08544 11.7045 .10305 9.70441 .12072 8.28376 .13846 7.22204 7 54 .08573 11.6645 .10334 9.67680 .12101 8.26355 .13876 7.20661 6 55 .08602 11.6248 .10363 9.64935 .12131 8.24345 .13906 7J9125 5 56 .08632 11.5853 .10393 9.62205 .12160 8.22344 .13935 7.17594 4 57 .08661 11.5461 .10422 9.59490 .12190 8.20352 .13965 7.16071 3 58 .08690 11.5072 .10452 9.56791 .12219 8.18370 .13995 7.14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 60 .08749 11.4301 .10510 9.51436 .12278 8.14435 .14054 7.11537 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang 1 85 84 83 82 TABLES. 75 TABLE Vll. Continued. NATURAL TANGENTS AND COTANGENTS. 8 9 10 11 t Tang Cotang Tang Cotang Tang Cotang Tang Cotang o .14054 7.11537 .15838 6.31375 .17633 5.67128 .19438 5.14455 60 1 .14084 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 59 2 .14113 7.08546 .15898 6.29007 .17693 5.65205 .19498 5.12862 58 a .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 57 4 .14173 7.05579 .15958 6.26655 .17753 5.63295 .19559 5.11279 5G 5 .14202 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 55 6 .14232 7.02637 .16017 6.24321 .17813 5.61397 .19619 5.09704 5-1 7 .14262 6.91174 .16047 6.23160 .17843 5.60452 .19649 5.08921 53 8 .14291 6.99718 .16077 6.22003 .17873 5.59511 .19680 5.08139 52 9 .14321 6.98268 .16107 6.20851 .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 5.06584 5d 11 .14381 6.95385 .16167 6.18559 .17963 5.56706 .19770 5.05809 49 12 .14410 6.93952 .16196 6.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.042G7 47 14 .14470 6.91104 .16256 6.15151 .18053 5.53927 .19861 5.03499 46 15 .14499 6.89688 .16286 6.14023 .18083 5.53007 .19891 5.02734 45 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 .14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10664 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49C56 .20012 4.99695 41 20 .14643 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 39 22 .14707 6.79936 .16495 6.06240 .18293 5.4GG48 .20103 4.97438 38 23 .14737 6.78564 .16525 6.05143 .18323 5.45751 .20133 4.96690 37 24 .14767 6.77199 .16555 6.04031 .18353 5.44857 .20164 4.95945 36 25 .14796 6.75838 .16585 6.02962 .18384 5.43966 .20194 4.95201 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14856 6.73133 .16645 6.00797 .18444 5.42192 .20254 4.93721 33 28 .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 83 29 .14915 6.70450 .16704 5.93646 .18504 5.40429 .20315 4.92249 31 30 .14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 31 .14975 6.67787 .16764 5.96510 .18564 5.38677 .20376 4.90785 29 32 .15005 6.66463 .16794 5.95448 .18594 5.37805 .20406 4.90056 28 33 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 4.89330 27 34 .15064 6.63831 .16854 5.93335 .18654 5.36070 .20466 4.88605 26 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.878S2 25 36 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 24 37 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 23 38 .15183 6.58627 .16974 5.89151 .18775 5.32631 .20588 4.85727 22 39 .15213 6.57339 .17034 5.88114 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 '4.84300 20 41 .15272 6.54777 .17063 5.86051 .18865 5.30080 .20679 4.83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4.82882 18 43 .15332 6.52234 .17123 5.84001 .18925 5.28393 .20739 4.82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4.81471 16 45 .15391 6.49710 .17183 5.81966 .18986 5.26715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 14 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 4.79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4.78673 12 49 .15511 6.44720 .17303 5.77936 .19106 5.23391 .20921 4.77978 11 50 .15540 6.43484 .17333 5.76937 .19136 5.22566 .20952 4.77386 10 51 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4.75906 8 53 .15630 6.39804 .17423 5.73960 .19227 5.20107 .21043 4.75219 7 54 .15660 6.38587 .17453 5.72974 .19257 5.19293 .21073 4.74534 6 55 .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 56 .15719 6.36165 .17513 5.71013 .19317 5.17671 .21134 4.73170 4 57 .15749 6.34961 .17543 5.70037 .19347 5.16863 .21164 4.72490 3 58 .15779 6.33761 .17573 5.69064 .19378 5.16058 .21195 4.71813 2 59 .15809 6.32566 .17603 5.68094 .19408 5.15256 .21225 4.71137 1 60 .15838 6.31375 .17633 5.67128 .19438 5.14455 .21256 4.70463 _0 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / L 81 80 79 78 SURVEYING. TABLE V\\. Continued. NATURAL TANGENTS AND COTANGENTS. 12 13 14 3 15 Tang Cotang Tang Cotang Tang Cotang Tang Cotang / .21256 4.70463 .23087 4.33148 .24933 4.01078 .26795 3.73205 60 1 .21286 4.69791 .33117 4.32573 .24964 4.00582 .26826 3.72771 59 2 .21316 4.69121 .23148 4.32001 .24995 4.00086 .26857 3.72338 58 3 .21347 4.68452 .23179 4.31430 .25026 3.99592 .26888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.99099 .26920 3.71476 56 5 .21408 4,67121 .23240 4.30291 .25087 3.98607 .26951 3.71046 55 6 .21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.70616 M .21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 g .21499 4.65138 .23332 4.28595 .25180 3.97139 .27CM4 3.69761 52 9 .8U5S9 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69335 51 10 .21560 4.63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.68485 49 l| .21621 4.62518 .23455 4.26352 .25304 3.95196 .27169 3.68061 48 13 .21651 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.67638 47 M .21682 4.61219 .23516 4.25239 .25366 3.94232 .27232 3.67217 46 15 .21712 4.60572 .23547 4.24685 .25397 3.93751 .27263 3.66796 45 16 .21743 4.59927 .2357'8 4.24132 .25428 3.93271 .27294 3.66876 4-1 17 .21773 4.59283 .23608 4.23580 .25459 3.92793 .27326 3.65957 43 18 .21804 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.65538 48 19 .21834 4.58001 .23670 4.22481 .25521 3.91839 .27388 3.65121 41 20 .21864 4.57363 .23700 4.21933 .25553 3.91364 .27419 3.64705 40 21 .21895 4.56726 .23731 4.21387 .25583 3.90890 .27451 3.64289 39 2-2 .21925 4.56091 .23762 4.20842 .25614 3.90417 .27482 3.63874 as 23 .21956 4.55458 .23793 4.20298 .25645 3.89945 .27'513 3.63461 37 4 .21986 4.54826 .23823 4.19756 .25676 3.89474 .27545 3.63048 3(3 25 .22017 4.54196 .23854 4.19215 .25707 3.89004 .27576 3.62636 35 26 .22047 4.53568 .23885 4.18675 .25738 3.88536 .27607 3.62224 34 27 .22078 4.b2941 .23916 4.18137 .25769 3.88068 .27638 3.61814 88 28 .22108 4.52316 .23946 4.17600 .25800 3.87601 .27670 3.61405 32 29 .22139 4.51693 .23977 4.17064 .25831 3.87136 .27701 3.60996 31 30 .22169 4.51071 .24008 4.16530 .25862 3.86671 .27732 360588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.60181 29 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 3.59775 2*-? 33 .22261 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 84 .22292 4.48600 .24131 4.14405 .25986 3.84824 .27858 3.58966 26 35 .22322 4.47986 .24162 4.13877 .26017 3.84364 .27889 3.58562 25 30 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.58160 24 37 .22383 4.46764 .24223 4.12825 .26079 3.83449 .27952 3.57758 23 38 .22414 4.46155 .24254 4.12301 .26110 3.82992 .27983 3.57357 23 39 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 21 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3.56557 20 41 .22505 4.44338 .24347 4.10736 .26203 3.81630 .28077 3.56159 10 42 .22536 4.43735 .24377 4.10216 .26235 3.81177 .28109 3.55761 18 13 .22567 4.43134 .24408 4.09699 .26286 3.80726 .28140 3.55364 17 44 .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3.54968 1(5 45 .22628 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.54573 15 46 .22658 4.41340 .24501 4.08152 .26359 3.79378 .28234 3.54179 14 47 .22689 4.40745 .24532 4.07639 .26390 3.78931 .28266 3.53785 13 48 .22719 4.40152 .24562 4.07127 .26421 3.78485 .28297 3.53393 12 49 .22750 4.39560 .24593 4.06616 .26452 3.78040 .28329 3.53001 11 50 .22781 4.38969 .24624 4.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 52 .22842 4.37793 .24686 4.05092 .26546 3.76709 .28423 3.51829 8 53 .22872 4.37207 .24717 4.04586 .26577 3.76268 .28454 3.51441 7 54 .22903 4.36623 .24747 4.04081 .26608 3.75828 .28486 3.51053 e 55 .22934 4.36040 .24778 4.03578 .26639 3.75388 .28517 3.50666 5 56 .22964 4.35459 .24809 4.03076 .26670 8.74950 .28549 3.50279 4 57 .22995 4.34879 .24840 4.02574 .26701 3.74512 .28580 3.49894 3 58 .23026 4.34300 .24871 4.02074 .26733 3.74075 .28612 3.49509 2 59 .23056 4.33723 .24902 4.01576 .25764 3.73640 .28643 3.49125 1 60 .23087 4.33148 .24933 4.01078 .26795 3.73205 .28675 8.48741 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 77 76 II 75 74 TABLES. 77 TABLE VII. Continued. NATURAL TANGENTS AND COTANGENTS. 31 16 .28675 .26706 .28738 .28769 .28800 .28864 .28895 .28927 .28990 .29021 .29053 .29084 .29116 .29147 .29179 .29210 .29242 .29274 .29305 .29400 .29432 .29403 .29495 .29526 .29558 .29590 .29621 .29685 .29716 .29748 .29780 .29811 .29843 .29875 .29906 .29938 .29970 .30001 .30033 .30065 .30097 .30128 .30160 .30224 .30255 .30287 .30319 .30351 .30414 .30446 .30478 .30541 30573^ Cotang Cotang 3.48741 3.48359 3.47977 3.47596 3.47216 3.46837 3.46458 3.46080 3.45703 3.45327 3.44951 3.44576 3.44202 3.43829 3.43456 3.43084 3.42713 3.42343 3.41973 3.41604 3.41236 3.40502 3.40136 3.39771 3.39406 3.39042 3.38679 3.38317 3.37955 3.37594 3.37234 3.36875 3.36516 3.36158 3.35800 3.35443 3.3508? 3.34733 3.34377 3.34023 3.33670 3.33317 3.32965 3.32614 3.32264 3.31914 3.31565 3.31216 3.30521 3.30174 3.29829 3.29483 3.29139 3.28795 3.28452 3.28109 3.27767 8.27426 3.27085 Tang 73 17 C Tang .30573 .30605 .30637 .30700 .30732 .30764 .30796 .30891 .30923 .30955 .30987 .31019 .31051 .31083 .31115 .31147 .31178 .31210 .31274 .31306 .31338 .31370 .31402 .31434 .31466 .31498 .31530 .31563 .31594 .31658 .31690 .31722 .31754 .31786 .31818 .31850 .31882 .31914 .31946 .31978 .32010 .32042 .32074 .32106 .32139 .32171 .32203 .32267 .32363 .32396 .32428 .32492 Cotang Tang 72 18 Tang .32524 .32558 .32653 .32685 .32717 .32749 .32782 .32878 .32911 .32943 .32975 .33007 .33040 .33072 .33104 .33136 .33169 .33201 .33233 .33200 .33298 .3330 .33363 .83395 .83427 .23460 .33492 .33524 .33557 .33589 .33621 .33654 .33686 .33718 .33751 .33783 .33913 .33945 .33978 .34010 .34043 .34075 .34108 .34140 .34173 .34205 .34238 .34270 .34303 .34400 .34433 Cotang Cotang 71 19 Tang .34465 .34530 .34563 .34661 .34726 .34758 .34791 .34856 .34922 .34954 .34987 .35020 .35052 .35085 .35118 .35150 .35183 .35216 .35314 .35346 .35379 .35412 .35445 .35477 .35510 .35543 .35576 .35608 .35641 .35674 .35707 .35740 .35772 .35871 .35904 .35937 .36002 .36035 .36068 .36101 .36134 .36167 .36265 .36298 .36331 Cotang 2.90421 2.90147 2.89055 2.88783 2.88511 2.88240 2.87970 2 87700 2.87430 2.871(31 2.86624 2.86356 2.85822 2.85555 2.85289 2.85023 2.84758 2.84494 2.84229 2.83702 2.83439 2.83176 2.82914 2.82653 2.82391 2.82130 2.81870 2.81610 2.81350 2.81091 2.80833 2.80574 2.79545 2. 2. 2.78778 2.78523 2.78014 2.77761 2.77507 2.77254 2.77002 2.76750 2.76498 2.76247 2.75996 2.75746 2.75496 2.75246 2.74997 2.74748 Cotang Tang 70 37 31 SURVEYING. TABLE VII. Continued. NATURAL TANGENTS AND COTANGENTS. / 20 21 22 23 Tang Cotang Tang Cotang Tang Cotang Tang Cotang / ~o .36397 2.74748 .38386 2.60509 .40403 2.47509 .42447 2.35585 60 ] .36430 2.74499 .38420 2.60283 .40436 2.47302 .42482 2.35395 59 j .36463 2.74251 .38453 2.60057 .40470 2.47095 .42516 2.35205 58 1 .36496 2.74004 .38487 2.59831 .40504 2.46888 .42551 2.35015 57 t .36529 2.73756 .38520 2.59606 .40538 2.46682 .42585 2.34825 56 5 .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.34636 55 G .36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.34447 54 \ .36628 2.73017 .38620 2.58932 .40640 2.46065 .42688 2.34258 53 8 .36661 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.34069 52 9 .36694 2.72526 .38687 2.58484 .40707 2.45655 .42757 2.33881 51 10 .36727 2.72281 .38721 2.58261 1 .40741 2.45451 .42791 2.33693 50 11 .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.33505 49 12 .36793 2.71792 .38787 2.57815 1 .40809 2.45043 .42860 2.33317 48 13 .36826 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.33130 47 14 .36859 2.71305 .38854 2.57371 .40877 2.44636 .42929 2.32943 46 15 .36892 2.71062 .38888 2.57150 .40911 2.44433 .42963 2.32756 45 1G .36925 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.32570 44 17 .36958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 43 IB .36991 2.70335 .88988 2.56487 .41013 2.43825 .43067 2.32197 42 19 .37024 2.70094 .39022 2.56266 .41047 2.43623 .43101 2.32012 41 2L .37057 2.69853 .39055 2.56046 .41081 2.43422 .43136 2.31826 40 21 .37090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641 39 22 .37123 2.69371 .39122 2.55608 .41149 2.43019 .43205 2.31456 38 23 .37157 2.69131 .39156 2.55389 .41183 2.42819 .43233 2.31271 37 24 .37190 2.68892 .39190 2.55170 .41217 2.42G18 .43274 2.31086 36 25 .37223 2.68653 .39223 2.54952 .41251 2.42418 .43308 2.30902 135 26 .37256 2.68414 .39257 2.54734 .41285 2.42218 .43343 2.30718 34 27 .37289 2.68175 .39290 2.54516 .41319 2.42019 .43378 2.30534 33 28 .37322 2.67937 .39324 2.54299 .41353 2.41819 .43412 2.30351 32 29 .37355 2.67700 .39357 2.54082 .41387 2.41620 .43447 2.30167 31 30 .37388 2.67462 .39391 2.53865 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 .39425. 2.53648 .41455 2.41223 .43516 2.29801 29 32 .37455 2.66989 .39453 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37488 2.66752 .33492 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.66516 .39526 2.53001 .41558 2.40629 .43620 2.29254 26 35 .37554 2.66281 .39559 2.52786 .41592 2.40432 .43654 2.29073 25 315 .37588 2.66046 .39593 2.52571 .41626 2.40235 .43689 2.28891 |24 37 .37621 2.65811 .39626 2.52357 .41660 2.40038 .43724 2.28710 |23 38 .37654 2.65576 .39660 2.52142 .41694 2.39841 .43758 2.28528 '22 39 .37687 2.65342 .39G91 2.51929 .41728 2.39645 .43793 2.28348 !21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 20 41 .37754 2.64875 .39761 2.51502 .41797 2.39253 .43862 2.27987 19 42 .37787 2.64642 .39795 2.51289 .41831 2.39058 .43897 2.27806 18 43 .37820 2.64410 .39829 2.51076 .41805 2.38863 .43932 2.27626 17 41 .37853 2.64177 .39862 2.50864 41893 2.38668 .43966 2.27447 |16 45 .37887 2.63945 .39896 2.50652 .41933 2.38473 .44001 2.27267 15 40 .37920 2.63714 .39930 2.50440 .41968 2.38279 .44036 2 27088 14 47 .37953 2.63483 .39963 2.50229 .42002 2.380S4 .44071 2.2G909 13 48 .37986 2.63252 .39997 2.50018 .42036 2.37891 .44105 2.26730 12 49 .38020 2.63021 .40031 2.49807 .42070 2.37G97 .44140 2.26552 11 50 .38053 2.62791 .40065 2.49597 .42105 2.37504 .44175 2.26374 10 51 .38086 2.62561 .40098 2.49386 .42139 2.37311 .44210 2.26196 9 52 .38120 2.62332 .40132 2.49177 .42173 2.37118 .44244 2.26018 8 53 .38153 2.62103 .40166 2.48967 .42207 2.36925 .44279 2.25840 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25663 6 55 .38220 2.61646 1 .40234 | 2.48549 .42276 2.36541 .44349 2.25486 5 56 .38253 2.61418 .40267 2 48340 .42310 2.36349 .44384 2.25309 4 57 .38286 2.61190 .40301 2.48132 .42345 2.36158 .44418 2.25132 3 58 .38320 2.60963 .40335 2.47924 .42379 2.35967 .44453 2.24956 2 59 .38353 2.60736 .40369 2.47716 .42413 2.35776 .44488 2.24780 1 GO .38386 2.60509 .40403 2.47509 .42447 2.35585 .44523 2.24604 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 69 1 68 67 60 TABLES. 79 TABLE VII. Continued. NATURAL TANGENTS AND COTANGENTS. 2 4 o 2 5 2 6 2 7 Tang Cotang Tang Cotang Tang Cotang Tang Cotang ~0 .44523 2.24604 .46631 2.14451 ' .48773 2.05030 .50953 1.96261 60 1 .44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 .96120 59 2 .44593 2.24252 .46702 2.14125 .48845 2.04728 .51026 .95979 58 3 .44627 2.24077 .46737 2.13963 .48881 2.04577 .51063 .95838 57 4 .44662 2.23902 .46772 2.13801 .48917 2.04426 .51099 .95698 56 5 .44697 2.23727 .46808 2.13639 .48953 2.04276 .51136 .95557 55 6 .44732 2.23553 .46843 2.1.3477 .48989 2.04125 .51173 .95417 54 7 .44767 2.23378 .46879 2.13316 .49026 2.03975 .51209 .95277 53 8 .44802 2.23204 .46914 2.13154 .49062 2.03825 .51246 .95137 52 9 .44837 2.23030 .46950 2.12993 .49098 2.03675 .51283 .94997 51 10 .44872 2.22857 .46985 2.12833 .49134 2.03526 .51319 .94858 50 11 .44907 2.22683 .47021 2.12671 .49170 2.03376 .51356 .94718 49 id .44942 2.22510 .47056 2.12511 .49206 2.03227 .51393 .94579 48 13 .44977 2.22337 .47092 2.12350 .49242 2.03078 .51430 .94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 .94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51503 .94162 45 16 .45082 2.21819 .47199 2.11871 .49351 2.02631 .51540 .94023 44 17 .45117 2.21647 .47234 2.11711 .49387 2.02483 .51577 .93885 43 18 .45152 2.21475 .47270 2.11552 .49423 2.02335 .51614 .93746 42 to .45187 2.21304 .47305 2.11392 .49459 2.02187 .51651 .93608 41 20 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 .93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 .93332 39 22 .45292 2.20790 .47412 2.10916 .49568 2.01743 .51761 .93195 38 23 .45327 2.20619 .47448 2.10758 .49604 2.01596 .51798 .93057 37 24 .45362 2.20449 .47483 2.10600 .49640 2.01449 .51835 .92920 36 25 .45397 2.20278 .47519 2.10442 .49677 2.01302 .51872 .92782 35 20 .45432 2.20108 .47555 2.10284 .49713 2.01155 .51909 .92645 34 27 .45467 2.19938 .47590 2.10126 .49749 2.01008 .51946 .92508 33 28 .45502 2.19769 .47626 2.09969 .49786 2.00862 .51983 .92371 32 29 .45538 2.19599 .47662 2.09811 .49822 2.00715 .52020 .92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 .92098 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 .91962 29 32 .45643 2.19092 .47769 2.09341 .49931 2.00277 .52131 .91826 28 33 .45678 2.18923 .47805 2.09184 .49967 2.00131 .52168 .91690 27 34 .45713 2.18755 .47840 2.09028 .50004 1.99986 .52205 .91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99841 .52242 .91418 25 36 .45784 2.18419 .47912 2.08716 .50076 .99695 .52279 .91282 24 37 .45819 2.18251 .47948 2.08560 .50113 .99550 .52316 .91147 23 38 .45854 2.18084 .47984 2.08405 .50149 .99406 .52353 .91012 22 39 .45889 2.17916 .48019 2.08250 .50185 .99261 .52390 .90876 21 40 .45924 2.17749 .48055 2.08094 .50222 .99116 .52427 .90741 20 41 .45960 2.17582 .48091 2.07939 .50258 .98972 .52464 1.90607 19 42 .45995 2.17416 .48127 2.07785 .50295 .98828 .52501 1.90472 18 43 .46030 2.17249 .48163 2.07630 .50331 .98684 .52538 1.90337 17 44 .46065 2.17083 .48198 2.07476 1 .50368 .98540 .52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 .98396 .52613 1.90069 15 46 .46136 2.16751 .48270 2.07167 .50441 .98253 .52650 1.89935 14 47 .46171 2.16585 .48306 2.07014 .50477 .98110 .52687 1.8G801 13 48 .46206 2.16420 .48342 2.06860 .50514 .97966 .52724 1.89667 12 49 .46242 2.16255 .48378 2.06706 .50550 .97823 .52761 i. 89533 11 50 .46277 2.16090 .48414 2.06553 .50587 .97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 .97538 .52836 1.89266 9 52 .46348 2.15760 .48486 2.06247 .50660 .97395 .52873 1.89133 8 53 .46383 2.15596 .48521 2.06094 .50696 .97253 .52910 1.89000 7 54 .46418 2.15432 .48557 2.05942 .50733 .97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 .50769 .96969 .52985 1.88734 5 56 .46489 2.15104 .48629 2.05637 .50806 .96827 .53022 1.88602 4 57 .46525 2.14940 .48665 2.05485 .50843 .96685 .53059 1.88469 3 58 .46560 2.14777 .48701 2.05333 .50879 .96544 .53096 1.88337 2 59 .46595 2.14614 .48737 2.05182 .50916 .96402 .53134 1.88205 1 60 .46631 2.14451 .48773 2.05030 .50953 .96261 .53171 1.88073 _0 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / - - 6 5 6 4 6 3 6 2 So SURVEYING. TABLE VII. Continued. NATURAL TANGENTS AND COTANGENTS. 2 8 2 9 3 3 1 Tang Cotang Tang Cotang Tang Cotang ; Tang Cotang .53171 1.88073 .55431 1.80405 .57735 .73205 .60086 1.66428 60 1 .53208 1.87941 .55469 1.80281 .57774 .73089 .60126 1.66318 59 2 .53246 1.87809 .5.5507 .80158 .57813 .72973 .60165 .66209 68 3 .53283 1.87677 .55545 .80034 .57851 .72857 .60205 .66099 57 4 .53320 1.87546 .55583 .79911 .57890 .72741 .60245 .65990 56 B .53358 1.87415 .55621 .79788 .57929 .72625 .60284 .65881 55 8 .53395 1.87283 .55659 .79665 i .57968 .72509 .60324 .65772 54 7 .53432 1.87152 .55G97 .79542 .58007 .72393 .60364 .65663 53 8 .53470 1.87021 .55736 .79419 .58046 .72278 .60403 .65554 R8 9 .53507 1.86891 .55774 .79296 .58085 .72163 .60443 .65445 51 10 .53545 1.86760 .55812 .79174 .58124 .72047 .60483 .65337 50 11 .53582 1.86630 .55850 .79051 .58162 .71932 .60522 .65228 49 12 .53620 1.86499 .55888 .78929 .58201 .71817 .60562 .65120 48 13 .53657 1.86369 .55926 .78807 .58240 .71702 i .60602 .65011 47 14 .53694 1.86239 .55964 .78685 .58279 .71588 1 .60642 .64903 46 15 .53732 1.86109 .56003 .78563 .58318 .71473 .60681 .64795 45 10 .53769 1.85979 .56041 .78441 .58357 .71358 .60721 .64687 44 17 .53807 1.85850 .56079 .78319 .58396 .71244 .60761 .64579 43 18 .53844 1.85720 .56117 .7-8198 .58435 .71129 .60801 .64471 42 Ifl .53882 1.85591 .56156 .78077 .58474 .71015 .60841 .64363 41 20 .53920 1.85462 .56194 .77955 .58513 .70901 .60881 .64256 40 21 .53957 1.85333 .56232 .77834 .58552 .70787 .60921 .64148 39 22 .53995 1.85204 .56270 .77713 .58501 .70673 .60960 .64041 33 23 .54032 1.85075 .56309 .77592 .58631 .70560 .61000 .63934 37 24 .54070 1.84946 .56347 : .77471 .58670 .70446 .61040 .63826 3tj 2r, .54107 1.84818 .56385 .77351 .58709 .70332 .61080 .63719 35 20 .54145 1.84689 .56424 .77230 .58748 .70219 .61120 .63612 34 27 .54183 1.84561 .56462 .77110 .58787 .70106 .61160 .63505 33 2* .54220 1.84433 .56501 .76990 .58826 .C9992 .61200 .63398 32 .54258 1.84305 .56609 .76869 .58865 .69879 .61240 .63292 31 30 .54296 1.84177 .56577 .76749 .58905 .69766 .61280 .63185 30 31 .54333 1.84049 .56616 .76629 .58944 .69653 .61320 .63079 29 82 .54371 1.83922 .56654 .76510 .58983 .69541 .61360 .62972 28 33 .54409 1.83794 .56693 .76390 .59022 .69428 .61400 .62866 on 34 .54446 1.83667 .56731 .76271 i .59061 .69316 .61440 .62760 26 35 .54484 1.83540 .56769 .76151 .59101 .69203 .61480 .62654 25 36 .54522 1.83413 .56808 .76032 .59140 .69091 .61520 .62548 24 37 .54560 1.83286 .56846 .75913 .59179 .68979 .61561 .62442 23 88 .54597 1.83159 .56885 75794 .59218 .68866 .61601 .62336 22 39 .54635 1.83033 .56923 .75675 .59258 .68754 .61641 .62230 21 40 .54673 1.83906 .56962 .75556 .59297 .68643 .61681 .62125 20 41 54711 1.82780 .57000 .75437 .59336 .68531 .61721 .62019 19 42 .54748 1.82654 .57039 .75319 .59376 .68419 .61761 .61914 18 43 .54786 1.82528 .57078 .75200 .59415 .68308 .61801 .61808 17 44 .54824 1.82402 .57116 .75082 .59454 .68196 .61842 .61703 16 45 .54862 1.82276 .57155 .74964 .59494 .68085 .61882 .61598 15 46 .54900 1.82150 .57193 .74846 i .59533 .67974 .61922 .61493 14 47 .54938 1.82025 .57232 .74728 .59573 .67863 .61962 .61388 13 48 .54975 1.81899 .57271 .74610 i .59612 .67752 .62003 .61283 12 49 .55013 1.81774 .57309 .74492 i .59651 .67641 .62043 .61179 11 50 .55051 1.81649 .57348 .74375 j .59691 .67530 .62083 .61074 10 51 .55089 1.81524 .57386 .74257 ! .59730 .67419 .62124 .60970 9 52 .55127 1.81399 .57425 .74140 .59770 .67309 .62164 .60865 8 53 .55165 1.81274 .57464 .74022 .59809 .67198 .62204 : .60761 7 54 .55203 1.81150 .57503 .73905 1 .59849 .67088 .62245 .60657 6 55 .55241 1.81025 .57541 .73788 .59888 .66978 .62285 .60553 5 56 .55279 1.80901 .57580 .73671 .59928 .66867 .62325 .60449 4 57 .55317 1.80777 .57619 .73555 .59967 .66757 .62366 .60345 3 58 .55355 1.80653 .57657 1.73438 .60007 .66647 .62406 .60241 2 59 .53893 1.80529 i .57696 1.73321 .60046 .66538 .62446 .60137 1 GO .55431 1.H0405 .57735 1.73205 .60086 .66428 .62487 60033 ; Cotang Tang Cotang Tang Cotang Tang Cotang Tang 7} 6 1 6 5 9 5 8 j TABLES. 8l TABLE Vll. Continued. NATURAL TANGENTS AND COTANGENTS. 32 33 1 34 I 35 Tang Cotang Tang Cotang Tang Cotang Tang Cotang .62487 1.60033 .64941 1.53986 .67451 1.48256 i .70021 1.42815 60 1 .62527 1.59930 .64982 1.53888 .67493 1.48163 .70064 1.42726 59 .62568 1.59826 .65024 1.53791 .67536 1.48070 .70107 1.42638 58 3 .62608 1.59723 .65065 1.53693 .67578 1.47977 .70151 1.42550 57 4 .62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 56 5 .62689 1.59517 .65148 1.53497 .67063 1.47792 .70238 1.42374 55 3 .62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 7 .62770 1.59311 .65231 1.53302 .67748 1.47607 .70325 1.42198 53 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70368 1.42110 52 9 .62852 1.59105 .65314 1.53107 .67832 1.47422 .70412 1.42022 51 10 .62892 1.59002 .65355 1.53010 .67875 1.47330 .70455 1.41934 50 11 .62933 1.58900 .65397 1.52913 .67917 1.47238 .70499 1.41847 49 2 .62973 1.58797 .65438 1.52816 .67960 1.47146 .70542 1.41759 43 3 .63014 1.58695 .65480 1.52719 .68002 1.47053 .70586 1.41672 47 4 .63055 1.58593 .65521 1.52622 ! .68045 1.4G9G2 .70629 1.41584 4G 3 .63095 1.58490 .65563 1.52525 .68088 1.43870 .70673 1.41497 *5 6 .63136 1.58388 .65604 1.52429 .68130 1.4G778 .70717 1.41409 41 .63177 1.58286 .65646 1.52332 .68173 1.40086 .70760 1.41322 43 8 .63217 1.58184 .65688 1.52235 .68215 1.4G595 .70804 1.41235 42 19 63258 1.58083 .65729 1.52139 .68258 1.40503 .70848 1.41148 41 20 63299 1.57981 .65771 1.52043 .68301 1.4G411 .70891 1.41061 40 21 63340 1.57879 .65813 1.51946 .68343 1.46320 .70985 1.40974 39 22 .63380 1.57778 .65854 1.51850 .08386 1.4G229 .70979 1.40387 23 23 63421 1.57676 .65896 1.51754 .08429 1.46137 .71023 1.40800 37 24 63462 1.57575 .65938 1.51658 .68471 1.46046 .71066 1.40714 36 25 .63503 1.57474 .G5980 1.515G2 .08514 1.45955 .71110 1.40C27 ?J5 26 .63544 1.57372 .66021 1. 51,406 .00557 1.45864 .71154 1.40540 4 27 .63584 1.57271 .66063 1.51370 .08000 1.45773 .71198 1.40454 33 28 .63625 1.57170 .66105 1.51275 .08642 1.45682 .71242 1.40367 32 29 .63666 1.57069 .66147 1.51179 1 .08085 1.45592 .71285 1.40281 31 30 .63707 1.56969 .66189 1.51084 .08728 1.45501 .71329 1.40195 20 31 .63748 1.56868 .66230 1.50988 .08771 1.45410 .71373 1 .40109 29 32 .03789 1.56767 .66273 1.G0803 .08814 1.45320 .71417 1.40022 28 33 .63830 1.56667 .66314 1.50797 .08857 1.45229 .71461 1.80936 27 34 .63871 1.56566 .663^6 1.50702 .08900 1.45139 .71505 1.39350 28 35 .63912 1.56466 .66308 1.50007 .08942 1.45049 .71549 1.39764 25 16 .63953 1.56366 .C6440 1.50512 .08985 1.44958 .71593 1.39G79 24 >7 .63994 1.56265 .66482 1.50417 .69028 1.44868 .71637 1.39593 23 38 .64035 1.56165 .66524 1.50322 .69071 1.44778 .71681 1.39507 22 39 .64076 1.56065 .665GG 1.50228 .69114 1.44688 .71725 1.39421 21 40 .64117 1.55966 .66608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55766 .66692 1.49944 .69243 1.44418 .71857 1.39165 10 43 .64240 1.55666 .66734 1.49849 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .66776 1.49755 .69329 1.44239 .71946 1.38994 16 45 .64822 1.55467 .66818 1.49661 .69372 1.44149 '. 71990 1.38909 15 46 .64363 1.55368 .66860 1.49566 .69416 1.44060 .72034 1.38824 14 47 .64404 1.55269 .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 .66944 1.49378 .69502 1.43881 .72122 1.38653 12 49 .64487 1.55071 .66986 1.49284 .69545 1.43792 .72167 1.38568 11 50 .64528 1.54972 .67028 1.49190 .69588 1.43703 .72211 1.38484 10 5 .64569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.3&399 9 5 .64610 1.54774 .67113 1.49003 .69675 1.43525 .72299 1.38314 8 5 .64652 1.54675 .67155 1.48909 .69718 1.43436 .72344 1.38229 7 54 .64699 1.54576 .67197 1.48816 .69761 1.43347 .72388 1.38145 6 5 .64734 1.54478 .67239 1.48722 .69804 1.43258 .72432 1.38060 I 5 .64775 1.54379 .67282 1.48629 .69847 1.43169 .72477 1.37976 4 5 .64817 1.54281 .67324 1.48536 .69891 1.43080 .72521 1.37891 3 & .64858 1.54183 .67366 1.48442 .69934 1.42992 .72565 1.37807 2 5 .64899 1.54085 .67409 1.48349 .69977 1.42903 .72610 1.37722 1 6 .64941 1.53986 1 .67451 1.48256 .70021 1.42815 .72654 1.37638 C Cotang | Tang Cotang Tang Cotang Tang Cotang Tang i 57 56 55 54 82 SUX VE YING. TABLE VII. Continued. NATURAL TANGENTS AND COTANGENTS. 3 6 3 7 3 8 3 9 ,! Tang Cotang Tang Cotang Tang Cotang Tang Cotang .72654 1.37638 .75355 1.32704 .78129 .27994 .80978 1.23490 60 1 .72699 1.37554 .75401 1.32624 .78175 .27917 .81027 1.23416 59 2 .72743 1.37470 .75447 1.32544 .78222 .27841 .81075 1.23343 58 3 .72788 1.37386 .75492 1.32464 .78269 .27764 .81123 1.23270 57 4 .72832 1.37302 .75538 1.32384 .78316 .27688 .81171 1.23196 56 5 .72877 1.37218 .75584 1.32304 .78363 .27611 .81220 1.23123 55 6 .72921 1.37134 .75629 1.32224 .78410 .27535 .81268 1.23050 54 7 .72966 1.37050 .75675 1.32144 .78457 .27458 .81316 1.22977 53 8 .73010 1.36967 .75721 1.32064 .78504 .27382 .81364 1.22904 52 9 .73055 1.36883 .75707 1.31984 .78551 .27306 .81413 1.22831 51 10 .73100 1.36800 .75812 1.31904 .78598 .27230 .81461 1.22758 50 11 .73144 1.36716 .75858 1.31825 .78645 .27153 .81510 1.22685 49 12 .73189 1.36633 .75904 1.31745 .78CS2 .27077 .81558 1.22612 48 13 .73234 1.36549 .75950 1.31066 .73739 .27001 .81600 1.22539 47 14 .73278 1.36466 .75996 1.31586 .78786 .26925 .81055 1.22467 46 15 .73323 1.36383 .76042 1.31507 .78834 .20849 .81703 1.22394 45 16 .73368 1.36300 .76088 1.31427 .78881 .20774 .81752 1.22321 44 17 .73413 1.33217 .76134 1.31348 .78928 .20098 .81800 1.22249 43 18 .73457 1.30134 .76180 1.31209 .7897'5 .20022 .81849 1.22176 42 19 .73502 1.3C351 .76226 1.31190 .79022 .20546 .81898 1.22104 41 20 .73547 1.35968 .76272 1.31110 .79070 .26471 .81946 1.22031 40 21 .73592 1.35885 .76318 1.31031 .79117 .26395 .81995 1.21959 39 22 .73637 1.83802 .70304 1.30952 .79104 .20319 .82044 1.21886 88 23 .73681 1.35719 .76410 1.30373 .79212 .26244 .82092 1.21814 37 24 .73726 1.35637 .76456 1.30795 .79259 .20169 .82141 1.21742 36 25 .73771 1.35554 .76502 1.30716 .79308 .20093 .82190 1.21670 35 26 .73816 1.35472 .76548 1.30637 .79354 .20018 .82238 1.21598 34 27 .73861 1.35389 .76594 1.3C558 .79401 .25943 .82287 1.21526 33 28 .73906 1.35307 .76640 1.30480 .79449 .25867 .82336 1.21454 32 29 .73951 1.35224 .70686 1.30401 .79496 .25792 .82385 1.21382 31 30 .73996 1.35142 .76733 1.30323 .79544 .25717 .82434 1.21310 30 31 .74041 1.35060 .76779 1.30244 .79591 .25642 .82483 1.21238 29 32 .74086 1.34978 .70825 1.30166 .79639 .25567 .82531 1.21166 23 33 .74131 1.34896 .70871 1.30087 .79636 .25492 .82580 1.21094 27 34 .74176 1.34814 .76918 1.30009 .79734 .25417 .82629 1.21023 26 35 .74221 1.34732 .76904 1.29931 .79781 .25343 .82G7'8 1.20951 25 36 .74267 1.34650 .77010 1.29853 .79829 .25268 .82727 1.20879 24 37 .74312 1.34568 .77057 1.20775 .79877 : .25193 .82776 1.20808 23 38 .74357 1.34487 .77103 1.29696 .79924 .25118 .82825 1.20736 22 39 .74402 1.34405 .77149 1.29618 .79972 .25044 .82874 1.20665 21 40 .74447 1.34323 .77196 1.29541 .80020 .24969 .82923 1.20593 20 41 .74492 1.34242 .77242 1.29463 .80067 .24895 .82972 1.20522 19 42 .74538 1.34160 .77239 1.29385 .80115 .24820 .83022 1.20451 18 43 .74583 1.34079 .77335 1.29307 .80163 .24746 .83071 1.20379 17 44 .74628 1.33998 .77382 1.29229 .80211 .24672 .83120 1.20308 16 45 .74674 1.33916 .77428 1.29152 .80258 .24597 .83109 1.20237 15 46 .74719 1.33835 .77475 1.29074 .80306 .24523 .83218 1.20166 14 47 .74764 1.33754 .77521 1.28997 .80354 .24449 .83268 1.20095 13 48 .74810 1.33673 .77568 1.28919 .80402 .24375 .83317 1.20024 12 49 .74855 1.33592 .77615 1.28842 .80450 .24301 .83366 1.19953 11 50 .74900 1.33511 .77661 1.28764 .80498 .24227 .83415 1.19882 10 51 .74946 1.33430 .77708 1.28687 .80546 .24153 .83465 1.19811 9 52 .74991 1.33349 .77754 1.28610 .80594 .24079 .83514 1.19740 8 53 .75037 1.33268 .77801 1.28533 .80642 .24005 .83564 1.19669 7 54 .75082 1.33187 .77848 1.28456 .80690 .23931 .83613 .19599 6 55 .75128 1.33107 .77895 1.28379 .80738 .23858 .83662 .19528 5 56 .75173 1.33026 .77941 1.28302 .80786 .23784 .83712 .19457 4 57 .75219 1.32946 .77988 1.28225 .80834 .23710 .83761 .19387 3 58 .75264 1.32865 .78035 1.28148 .80882 .23637 .83811 .19316 2 59 .75310 1.32785 .78082 1.28071 .80930 .23563 .83860 .19246 1 60 .75355 1.32704 .78129 1.27994 .80978 .23490 .83910 .19175 t Cotang Tang Cotang Tang Cotang Tang Cotang Tang / I 3 J 2 5 1 5 TABLES. TABLE \\l. Continued. NATURAL TANGENTS AND COTANGENTS. J 4 4 1 I 4 3 4 1 Tang Cotang Tang Cotang Tang Cotang Tang Cotang ~0 .83910 1.19175 .86929 1.15037 .90040 .11061 .93252 1.07237 60 1 .83960 1.19105 .86980 1.14969 .90093 .10996 .93306 1.07174 59 2 .84009 1.19035 .87031 1.14902 .90146 .10931 .93360 1.07112 58 3 .84059 1.18964 .87082 1.14834 .90199 .10867 .93415 1.07049 57 4 .84108 1.18894 .87133 1.14767 .90251 .10802 .93469 1.06987 5G B .84158 1.18824 .87184 1.14699 .90304 .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 R .84307 1.18614 .87338 1.14498 .90463 .10543 .93688 .06738 52 9 .84357 1.18544 .87389 1.14430 .90516 .10478 .93742 .06676 51 10 .84407 1.18474 .87441 1.14363 .90569 .10414 .93797 .06613 50 11 .84457 1.18404 .87492 1.14296 .00621 .10349 .93852 .06551 49 12 .84507 1.18334 .87543 1.14229 .90674 .10285 .93906 .06489 43 13 .84556 1.18264 .87595 1.14162 .90727 .10220 .93961 .06427 47 14 .84606 1.18194 .87646 1.14095 .90781 .10156 .94016 .06365 40 15 .84656 1.18125 .87698 1.14028 .90834 .10091 .94071 .06303 45 1G .84706 1.18055 .87749 1.13961 .90887 .10027 .94125 .06241 44 17 .84756 1.17986 .87801 1.13894 .90940 .09963 .94180 .06179 43 18 .84806 1.17916 .87852 1.13828 .C0993 .09899 .94235 .06117 42 19 .84856 1.17846 .87904 1.13761 .91046 .09834 .94290 .06056 41 90 .84906 1.17777 .87955 1.13694 .91099 .09770 .94345 .05994 40 21 .84956 1.17708 .88007 1.13627 .01153 .09706 .94400 .05932 3!) .85006 1.17638 .88059 1.13561 .91206 .09642 .94455 .05870 38 23 .85057 1.17569 .88110 1.13494 .91259 .09578 .94510 .05809 37 24 .85107 1.17500 .88162 1.13428 .91313 .09514 .94565 .05747 36 25 .85157 1.17430 .88214 1.13361 .91-366 .09450 .94620 .05685 35 20 .85207 1.17361 .88265 1.13295 .91419 .09386 .94676 .05624 34 27 .85257 1.17292 .88317 1.13223 .91473 .09322 .94731 .05562 33 2!i .85308 1.17223 .88369 1.13162 .91526 .09258 .94786 .05501 32 29 .85358 1.17154 .88421 1.13096 .91580 .09195 .94841 .05439 31 30 .85408 1.17085 .88473 1.13029 .91633 .09131 .94896 .05378 30 31 .85458 1.17016 .88524 1.12963 .91687 .09067 .94952 .05317 29 38 .85509 1.16947 .88576 1.12897 .91740 .09003 .95007 .05255 28 33 .85559 1.16878 .88628 1.12831 .91794 .08940 .95062 .05194 27 3-i .85609 1.16809 .88680 1.12765 .91847 .08876 .95118 .05133 26 35 .85660 1.16741 .88732 1.12699 .91901 .08813 .95173 .05072 25 36 .85710 1.16672 .88784 1.12633 .91955 .08749 .95229 .05010 24 37 .85761 1.16603 .88836 1.12567 .92008 .08686 .95284 .04949 23 88 .85811 1.16535 .88888 1.12501 .92062 .08622 .95340 .04888 22 39 .85862 1.16466 .88940 1.12435 .92116 .08559 .95395 .04827 21 40 .85912 1.16398 .88993 1.12369 .92170 .08496 .95451 .04766 20 41 .85963 1.16329 .89045 1.12303 .92224 .08432 .95506 .04705 19 42 .86014 1.16261 .89097 1.12238 .92277 .08369 .95562 .04644 18 43 .86064 1.16192 .89149 1.12172 .92331 .08306 .95618 .04583 17 44 .86115 1.16124 .89201 1.12106 .92385 .08243 .95673 .04522 16 45 .86166 1.16056 .89253 1.12041 .92439 .08179 .95729 .04461 15 40 .86216 1.15987 .89306 1.11975 .92493 .08116 .95785 .04401 14 47 .86267 1.15919 .89358 1.11909 .92547 .08053 .95841 .04340 13 48 .86318 1.15851 .89410 1.11844 .92601 07990 .95897 .04279 12 49 .86368 1.15783 .89463 1 11778 .92655 .07927 .95952 .04218 11 50 .86419 1.15715 .89515 1.11713 .92709 .07864 , .96008 .04158 10 51 .86470 1.15647 .89567 1.11643 .92763 .07801 .96064 .04097 9 52 .86521 1.15579 .89620 1.11582 .92817 .07738 .96120 .04036 8 53 .86572 1.15511 .89672 1.11517 .92872 .07676 .96176 .03976 7 54 .86623 1.15443 .89725 1.11452 .92926 .07613 .96232 .03915 6 55 .86674 1.15375 .89777 1.11387 .92980 .07550 .96288 1.03855 5 56 .86725 1.15308 .89830 1.11321 .93034 .07487 i .96344 1.03794 4 57 .86776 1.15240 .89883 1.11256 .93088 .07425 .96400 1.03734 3 58 .86827 1.15172 .89935 1.11191 .93143 .07362 .96457 1.03674 2 59 .86878 1.15104 .89988 1.11126 .93197 .07299 .96513 1.03613 1 80 .86929 1.15037 .90040 1.11061 .93252 1.07237 .96569 1.03553 J) / Cotang Tang Cotang Tang Cotang Tang Cotang Tang f i ! * 9 4 8 4 70 4 rf 8 4 SURVEYING. TABLE VII. Continued. NATURAL TANGENTS AND COTANGENTS. / 4 4 , i, 4 i 4 4 Tang Cotang Tang Cotang Tang Cotang .96569 1.03553 60 20 .97700 1.02355 40 40 .98843 1.01170 20 1 2 .966-25 .98681 1.03493 1.03433 59 58 21 22 .97756 .97813 1.02295 1.02236 89 MS (41 42 .98901 .98958 1.01112 1 01053 19 18 3 .96788 1.03372 57 23 .97870 1.02176 37 143 .99016 1.00994 17 4 .90794 1.03312 56 24 .97927 1.02117 86 44 .99073 1.00935 16 5 .9ea->o 1.03252 55 25 .97984 .02057 35 45 .99131 1.00876 15 6 .96907 1.03192 54 26 .98041 1.01998 34 !46 .99189 1.00818 14 7 .96963 1.03132 53 27 .98098 .01939 33 ^47 .99247 1.00759 13 8 .97020 1.03072 52 28 .98155 .01879 32 ! 48 .99304 1.00701 12 9 .97076 1.03012 51 29 .98213 .01820 31 149 .99362 1.00642 11 10 .97133 1.02952 50 30 .98270 .01761 30 i 50 .99420 1.00583 10 11 .97189 1.02892 49 31 .98327 .01702 20 |51 .99478 1.00525 9 12 .97246 1.02832 48 32 .98384 .01642 28 153 .99536 1.00467 8 13 .97302 1.02772 47 33 .98441 .01583 27 : 53 .99594 1.00408 7 14 .97359 1.02713 46 34 .98499 .01524 26 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 1? .97529 1.02533 43 37 .98671 .01347 23 57 .99826 1.00175 3 18 .97586 1.02474 42 38 .98728 .01288 22 58 .99884 1.00116 2 19 .97643 1.02414 41 39 .98786 .01229 21 59 .99942 1.00058 1 20 .97700 1.02355 40 40 .98843 .01170 20 60 1.00000 1.00000 Cotang Tang / / Cotang Tang / / Cotang Tang / -1 t> 4 4 5 - } TABLES. r i 4^ 4* 4^ 4* OO ON 4^- to O O O O 4^ Co Co Co Co ?>J I $ a fON CN ON vo vo vo O O O vb vb vb vb oo bo 00 ON 4>- 1-1 vo -J vj 4^ M 00 ^J Ln III Co i Ln OO t-n 4^ ^J CO M O Ln CO O OO VQ VO *" *^ CO VO 'ilH oo Co VO 00 00 OO 00 VO ON ^J ^J Ln O O O vo VO vO vO O O O Co ON OO I-H Co Ln Co O Ln O CO Ln "M 4^. VO M to O 1 g; I ^ ^J oo oo 4* ^J O M I-H Ln VO 4*. Ln to ^1 O OO CC VO VO VO VO Ln "^ O tO 4^ CN Co 00 >-i Co 4*. 4^. OO to Ln *^J OO ^-J Co to 10 Co M ON H Cn 4k 4*. 4^ 4v Ln ON OO VO "i CO IH OO ji. x (.0 Co 00 b un b > vj vb CN ^J tO O 1-1 Ln ~ 1 I i M * a a a a s a a a a i S g S o o o o o o 00 (/i C/3 C/3 C/3 C/5 ^ P ~n *b poop Lo M O 00 Ln OO to O\ o p p p p p Co Co Co . Co Co to ON Ln CO M O OO VO Co *^J O 4^- OO O O O O O O Ln Ln Ln Ln Ln Ln Ln Ln -P*- 4^ 4^ L.J CN Ln Co ^J O bo b -^ b bo 4*. 5* PI ^ to OO CO CO 00 4. ON ON Co b M M bo ^ ^J CN Ln Co to b bo 4*. *I 8 S i * N *"M m ta * i i 86 SURVEYING. . ^w*5 o-oo ^o TI*WO ^ * .'.... .... a (f> O t^ t^t^t^.0000 OOOOOOOOOO C^OOOvO g a H jwss. as ^odo' 4-u^voodov M^^-U^^ < > / CO ^oiais ff'ssy's 8sr < 8 < 8&, ?,s^^^ 3.5.5?^ r .. * h C? ,,-000* K ? W ?^t*^ --0- ^-O.r. g C -<>2S ..222 Pff.MS- ???.? H) 00 f^^O ^ H TOUMIHC* Nf^iroro^ ^ -^ m w> 10 *o i^ivo vo vo \o vo vo t^ f^. * E O g S ??>% SJSJ&R'ft ^^^K. KRi^S cSoT^S * 1 ^i&,^ 5S^RR g!?gc^& SSi^g^ S >< * C J HNW^IO er^wase ^o ^coo IHN?OO ;. B ^ HHHHOt WNCO CO CC * ^< TABLES. j C si g S 2 z g OS &* ro Q fcui J-< 3 " .H o >eo >o >00 o ^ rs.w O M u-ioo o M T^ T 1 ^ * ^* Ci M * * us O ft ft - q M --vq oo vq ON moo M*H , o> cooo N 06 ^ ft ^( Hw^io>ftOU5aooiaso 88 SURVEYING. TABLE XI. VOLUMES BY THE PRISMOID/L FORMULA. 320. 1 HEIGHTS. Corrections 1 1 2 3 4 5 6 7 8 9 10 for tenths in height. 1 1 1 1 2 2 2 2 3 3 .1 2 1 1 2 o 3 3 4 5 6 6 .2 o 3 1 2 3 4 5 6 6 7 8 9 . 3 o 4 1 2 4 5 6 7 9 10 11 12 4 1 5 2 3 5 6 -8 9 11 12 14 15 1 6 2 4 6 7 9 11 13 15 17 19 'g 1 7 2 4 6 9 11 13 15 17 19 22 .7 1 8 2 5 7 10 12 15 17 20 22 25 .8 1 9 3 6 8 11 14 17 19 22 25 28 9 1 10 3 6 9 12 15 19 22 25 28 31 11 3 n 10 14 17 20 24 27 31 34 . i o 12 4 7 11 15 19 22 26 30 33 37 .2 1 13 4 8 12 16 20 24 28 32 36 40 .3 1 14 4 9 13 17 22 26 30 35 39 43 4 o 15 5 9 14 19 23 28 32 37 42 46 .5 2 10 5 10 15 20 25 30 35 40 44 49 .6 3 17 5 10 16 21 26 31 37 42 47 52 .7 3 18 6 11 17 22 28 33 39 44 50 56 .8 4 19 6 12 18 23 29 35 41 47 53 59 9 ! 4 20 6 12 19 25 31 37 43 49 56 62 21 6 11 19 26 32 39 45 52 58 05 . T 1 22 7 It 20 27 34 41 48 54 61 68 . "2 2 23 7 14 21 28 86 43 to 57 64 71 2 24 7 15 22 30 37 44 52 59 67 74 4 3 25 -8 15 23 31 39 46 54 62 69 77 4 26 8 16 24 32 40 48 56 64 72 80 '5 5 27 8 17 25 33 42 no 58 67 75 83 .7 5 28 9 17 26 35 43 52 60 69 78 86 .8 6 29 9 18 27 3fi 45 54 63 72 81 90 9 7 80 9 19 28 37 46 56 65 74 83 93 31 10 19 29 38 48 57 67 77 86 96 ,i 1 32 10 20 30 40 49 59 69 79 89 99 .2 2 33 10 20 31 41 51 61 71 81 92 102 .3 3 34 10 21 31 42 52 63 73 84 94 105 4 4 35 11 oo -32 43 54 65 76 86 -97 -108 5 36 n 22 33 44 56 67 78 P9 100 111 g 6 37 11 23 31 46 57 69 80 91 103 114 .7 8 38 12 23 35 47 59 70 82 94 KM 117 .8 9 39 12 24 36 48 60 72 84 96 108 120 9 10 40 12 25 37 49 62 74 86 99 111 123 41 13 25 28 51 63 76 89 101 114 127 . i 1 42 13 26 39 M (55 78 91 104 117 130 .2 3 43 13 27 40 M 66 80 . 93 106 119 133 3 4 44 14 27 41 54 68 81 95 109 J vX. 136 4 6 45 14 28 42 56 69 83 -97 -111 125 139 .5 46 14 28 43 57 71 85 99 114 128 142 .6 8 47 15 29 44 58 73 87 102 116 13! 145 .7 10 48 15 30 44 59 74 89 104 119 133 148 .8 11 49 15 30 45 60 76 91 106 121 136 151 .g 13 50 15 31 46 62 77 93 108 123 139 154 1 2 3 4 5 6 7 8 9 10 .1 .2 3 4 5 .6 7 .8 9 Corrections for 1 1 1 1 1 1 tenths in width. TABLES. 89 TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. - HEIGHTS. Corrections 1 2 3 4 5 6 7 8 9 10 for tenths in height. 51 16 31 47 63 79 94 110 126 142 157 .1 2 52 16 32 48 64 80 96 112 128 144 160 .2 3 53 16 33 49 65 82 98 115 131 147 163 .3 5 54 17 33 50 67 83 100 117 133 150 167 4 7 55 17 -34 51 68 85 102 119 136 153 170 .5 8 56 17 35 52 69 86 104 121 138 156 173 .6 10 57 18 35 53 70 88 106 123 141 158 176 .7 12 58 18 36 54 72 90 107 125 143 161 179 .8 14 59 18 36 55 73 91 109 127 146 164 182 9 15 60 19 37 56 74 93 111 130 148 167 185 61 19 38 56 75 94 113 132 151 169 188 .1 2 62 19 38 57 77 96 115 134 153 172 191 .2 4 63 19 39 58 78 97 117 136 156 175 194 .3 6 64 20 40 59 79 99 119 138 158 178 197 4 8 65 20 40 60 80 -100 -120 140 160 181 -201 .5 10 66 20 41 61 81 102 122 143 163 183 204 .6 12 67 21 41 62 83 103 124 145 165 186 207 .7 14 68 21 42 63 84 105 126 147 168 189 210 .8 16 69 21 43 64 85 106 128 149 170 192 213 9 18 70 22 43 65 86 108 130 151 173 194 216 71 22 44 66 88 100 131 153 175 197 219 .1 2 72 22 44 67 89 111 133 156 178 200 222 .2 5 73 23 45 68 90 113 135 158 180 203 225 .3 7 74 23 46 69 91 114 137 160 183 206 228 4 9 75 23 46 69 93 116 139 162 -185 -208 231 .5 12 76 23 47 70 94 117 141 164 188 211 235 .6 14 77 24 48 71 95 119 143 166 190 214 238 .7 16 78 24 48 72 96 120 144 169 193 217 241 .8 19 79 24 49 73 98 122 146 171 195 219 244 9 21 80 25 49 74 99 123 148 173 198 222 247 81 25 50 75 100 125 150 175 200 225 250 .1 3 82 25 51 76 101 127 152 177 202 228 253 .2 5 83 26 51 77 102 128 154 179 205 231 256 .3 8 84 26 52 78 104 130 156 181 207 233 259 .4 10 85 26 52 79 105 131 157 -184 210 236 262 13 86 27 53 80 106 133 159 186 212 239 265 !6 16 87 27 54 81 107 134 161 188 215 242 269 .7 18 88 27 54 81 109 136 163 190 217 244 272 .8 21 89 27 55 82 110 137 165 192 220 247 275 9 24 90 28 56 83 111 139 167 194 222 250 278 91 28 56 84 112 140 169 197 225 253 281 T 3 92 28 57 85 114 142 170 199 227 256 284 .2 6 93 29 57 86 115 144 172 201 230 258 287 9 94 29 58 87 116 145 174 203 232 261 290 .4 12 95 -29 59 -88 117 -147 176 205 235 264 293 15 96 30 59 89 119 148 178 207 237 267 296 .6 18 97 30 60 90 120 150 180 210 240 269 299 .7 21 98 30 60 91 121 151 181 212 242 272 302 .8 23 99 31 61 92 122 153 183 214 244 275 306 9 26 100 31 62 93 123 154 185 216 247 278 309 1 2 3 4 5 6 7 8 9 1O .1 .2 3 4 5 .6 7 .8 9 1 1 1 1 1 1 Corrections for tenths in width. SURVEYING. TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. 1 HEIGHTS. Corrections o tor tenths i 11 12 13 14 15 16 17 18 19 20 in height. l 3 4 4 4 5 5 5 6 6 6 .1 2 7 7 8 9 9 10 10 11 12 12 .2 3 10 11 12 13 14 15 16 17 18 19 .3 4 14 15 16 17 19 20 21 22 23 25 .4 1 5 17 19 20 22 23 25 26 28 29 -31 .5 1 6 20 22 24 26 28 30 31 33 35 37 .6 1 7 24 26 28 30 32 35 37 39 41 43 .7 1 8 27 30 32 35 37 40 42 44 47 49 .8 1 9 31 33 36 39 42 44 47 50 53 56 9 1 10 34 37 40 43 46 49 52 56 59 62 11 37 41 44 48 51 54 58 61 65 68 .i 12 41 44 48 52 56 59 63 67 70 74 .2 1 13 44 48 52 56 60 64 68 72 76 80 .3 1 14 48 52 56 60 65 69 73 78 82 86 4 2 15 51 56 60 65 69 79 -83 88 93 2 16 54 59 64 69 74 ~79 84 89 94 99 g 3 17 58 63 68 73 79 84 89 94 100 105 .7 3 18 61 67 72 78 83 89 94 100 106 111 .8 4 19 65 70 76 82 88 94 100 106 111 117 9 4 20 68 74 80 86 93 99 105 111 117 123 21 71 78 84 91 97 104 110 117 123 130 .1 1 22 75 81 88 95 102 109 115 122 129 136 .2 2 23 78 85 92 99 106 114 121 128 135 142 .3 2 24 81 89 96 104 111 119 126 133 141 148 4 3 25 85 -93 100 108 116 -123 131 139 147 154 .5 4 26 88 96 104 112 120 128 136 144 152 160 .6 5 27 92 100 108 117 125 133 142 150 158 167 .7 5 28 95 104 112 121 130 138 147 156 164 173 .8 6 29 98 107 116 125 134 143 152 161 170 179 9 7 30 102 111 120 130 139 148 157 167 176 185 31 105 115 124 134 144 153 163 172 182 191 .1 1 82 109 119 128 138 148 158 168 178 188 198 .2 2 33 112 122 132 143 153 163 173 183 194 204 3 3 34 115 126 136 147 157 168 178 189 199 210 4 4 35 119 -130 140 151 162 173 184 194 -205 -216 .5 5 36 122 133 144 156 167 178 189 200 211 222 .6 6 37 126 137 148 160 171 183 194 206 217 228 .7 8 38 129 141 152 164 176 188 199 211 223 235 .8 9 39 132 144 156 169 181 193 205 217 229 241 9 10 40 136 148 160 173 185 198 210 222 235 247 41 139 152 165 177 190 202 215 228 240 253 ,i 1 42 143 156 169 181 194 207 220 233 246 259 .2 3 43 146 159 173 186 199 212 226 239 252 265 3 4 44 149 163 177 190 204 217 231 244 258 272 .4 6 45 153 -167 -181 -194 208 222 236 250 264 278 7 46 156 170 185 199 213 227 241 256 270 284 .6 8 47 160 174 189 203 218 232 247 261 276 290 ' 7 10 48 163 178 193 207 222 237 252 267 281 296 .8 11 49 166 181 197 212 227 242 257 272 287 302 9 13 60 170 185 201 216 231 247 262 278 293 309 11 12 13 14 15 16 17 18 19 20 .1 .2 -3 | -4 5 .6 7 .8 9 Corrections for 1 1 a 2 3 3 4 4 tenths in width. TABLES. gi TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. 3 HEIGHTS. Corrections 1 11 12 13 14 15 16 17 18 19 20 for tenths in height. 51 173 189 205 220 236 252 268 283 299 315 .1 2 52 177 193 209 225 241 257 273 289 305 321 .2 3 53 180 196 213 229 245 262 278 294 311 327 .3 5 54 183 200 217 233 250 267 283 300 317 333 4 7 55 -187 -204 221 238 -255 272 289 306 323 340 8 56 190 207 225 242 259 277 294 311 328 346 'g 10 67 194 211 229 246 264 281 299 317 334 352 .7 12 58 197 215 233 251 269 286 304 322 340 358 .8 14 59 200 219 237 255 273 291 310 328 346 364 9 15 60 204 222 241 259 278 296 315 333 352 370 61 207 226 245 264 282 301 320 339 358 377 .1 2 62 210 230 249 268 287 306 325 344 364 383 .2 4 63 214 233 253 272 292 311 331 350 369 389 .3 6 64 217 237 257 277 296 316 336 356 375 395 4 8 65 221 241 261 -281 301 -321 -341 -361 -381 401 .5 10 66 224 244 265 285 306 326 346 367 387 407 .6 12 67 227 248 269 290 310 331 352 372 393 414 .7 14 68 231 252 273 294 315 336 357 378 399 420 .8 16 69 234 256 277 298 319 341 362 383 405 426 9 18 70 238 259 281 302 324 346 367 389 410 432 71 241 263 285 307 329 351 373 394 416 438 .1 2 72 244 267 289 311 333 356 378 400 422 444 .2 5 73 248 270 293 315 338 360 383 406 428 451 .3 7 74 251 274 297 320 343 365 388 411 434 457 4 9 75 255 -278 -301 324 347 370 -394 417 -440 463 .5 12 76 258 281 305 328 352 375 399 422 446 469 .6 14 77 261 285 309 333 356 380 4**i 428 452 475 .7 16 78 265 289 313 337 361 385 409 433 457 481 .8 19 79 268 293 317 341 366 390 415 439 463 488 9 21 80 272 296 821 346 370 395 420 444 469 494 81 275 300 325 350 375 400 425 450 475 500 .1 3 82 278 304 329 354 380 405 430 456 481 506 .2 5 88 282 307 333 359 384 410 435 461 487 512 .3 8 84 311 337 363 389 415 441 467 493 519 .4 10 85 -289 315 341 -367 394 420 -446 -472 -498 -525 .5 13 86 292 319 345 372 398 425 451 478 504 531 .6 16 87 295 322 349 376 403 430 456 483 510 537 18 88 299 326 353 380 407 435 462 489 516 543 g 21 89 303 330 357 385 412 440 467 494 522 549 9 24 90 306 333 361 389 417 444 472 500 528 556 91 309 337 365 393 421 449 477 506 534 562 .1 3 92 312 341 369 398 426 454 483 511 540 568 .2 6 93 316 344 373 402 431 459 488 517 545 574 .3 9 94 319 348 377 406 435 464 493 522 551 580 4 12 95 323 352 -381 410 440 -469 498 528 -557 586 .5 15 96 326 356 385 415 444 474 504 533 563 593 .6 18 97 329 359 389 419 449 479 509 539 569 599 .7 21 98 333 363 393 423 454 484 514 544 575 605 .8 23 99 336 367 397 428 458 489 519 550 581 611 9 26 100 340 370 401 432 463 494 525 556 586 617 11 12 13 14 15 16 17 18 19 20 .1 .2 3 4 5 .6 7 .8 9 Corrections for 1 1 2 2 3 3 4 4 tenths in width. I 9 2 SURVEYING. TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. 1 HEIGHTS. Corrections 1 21 22 23 24 25 26 27 28 29 30 for tenths in height. 2 13 14 14 15 8 15 8 16 8 17 9 17 9 18 9 19 .1 .2 3 19 20 21 22 23 24 25 26 27 28 .3 4 26 27 28 30 31 32 33 35 36 37 -4 6 32 34 -35 -37 39 -40 42 43 45 46 .5 6 39 41 43 44 46 48 50 52 54 56 II 7 45 48 50 52 54 56 58 60 63 65 .7 8 52 54 57 59 62 64 67 69 72 74 .8 9 58 61 64 67 69 72 75 78 81 83 10 65 68 71 74 77 80 83 86 90 93 11 71 75 78 81 85 88 92 95 98 102 .1 12 78 81 85 89 93 96 100 104 107 111 .2 1 13 84 88 92 96 100 114 108 112 116 120 .3 1 14 91 95 99 104 108 112 117 121 125 130 4 2 15 -97 102 -106 111 -116 120 125 130 -134 139 .5 2 16 104 109 114 119 123 128 133 138 143 148 .6 3 17 110 115 121 126 131 136 142 147 152 157 .7 3 18 117 122 128 133 139 144 150 156 161 167 .8 4 19 123 129 135 141 147 152 158 164 170 176 9 4 20 130 136 142 148 154 160 167 173 179 185 21 136 148 149 156 162 169 175 181 188 194 . i 1 22 143 149 156 163 170 177 183 190 197 204 .2 2 23 149 156 163 170 177 185 192 199 206 213 .3 2 24 156 163 170 178 185 193 200 207 215 222 4 3 25 162 -170 177 -185 -193 -201 -208 216 -224 231 4 26 169 177 185 193 201 209 217 225 233 241 .6 5 27 175 183 192 200 208 217 225 233 242 250 .7 5 28 181 190 199 207 216 225 233 242 251 259 .8 6 29 188 197 206 215 224 233 842 251 260 269 9 7 30 194 204 213 222 231 241 250 259 269 278 31 201 210 220 230 239 249 258 268 277 287 , 1 32 207 217 227 237 247 257 267 277 286 296 .2 2 33 214 224 234 244 255 265 275 285 295 306 .3 3 34 220 231 241 252 262 273 283 294 304 315 4 4 35 227 238 248 259 270 281 292 302 313 -324 5 36 233 244 256 267 278 289 300 311 322 333 g 6 37 240 251 263 274 285 297 308 320 331 343 .7 8 38 246 258 270 281 293 305 317 328 340 352 .8 9 39 253 265 277 289 301 313 325 337 349 361 9 10 40 259 272 284 296 309 321 333 346 358 370 41 266 278 291 304 316 329 342 354 367 380 .1 j 42 272 285 298 311 324 337 350 363 376 389 .2 3 43 279 292 305 319 332 345 358 372 385 398 3 4 44 285 299 312 326 340 353 367 380 394 407 4 6 45 292 306 319 333 347 361 375 389 -403 417 .5 7 46 298 312 327 341 355 369 383 398 412 426 .6 8 47 305 319 334 348 363 377 392 406 421 435 .7 10 48 311 326 341 356 370 385 400 415 430 444 .8 11 49 318 333 348 363 378 393 408 423 439 454 9 13 50 324 340 355 370 386 401 417 432 448 463 21 22 23 24 25 26 27 28 29 30 i .2 3 4 5 .6 7 ^ .8 9 Corrections for 1 2 2 3 4 5 5 6 7 tenths in width. TABLES. 93 TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. HEIGHTS. Corrections iM 3 21 22 23 24 25 26 27 28 29 30 in height. 61 331 346 362 378 394 409 425 441 456 472 .1 2 52 337 353 369 385 401 417 433 449 465 481 .2 3 53 344 360 376 393 409 425 442 458 474 491 .3 5 54 350 367 383 400 417 433 450 467 483 500 .4 7 55 -356 -373 390 407 -424 441 -458 475 492 509 5 8 56 363 380 398 415 432 449 467 484 501 519 .6 10 57 369 387 405 422 440 457 475 493 510 528 .7 12 58 376 394 412 430 448 465 483 501 519 537 .8 14 59 382 401 419 437 455 473 492 510 528 546 9 15 60 389 407 426 444 463 481 500 519 537 556 61 395 414 433 452 471 490 508 527 546 565 .1 2 62 402 421 440 459 478 498 517 536 555 574 .2 4 63 408 428 447 467 486 506 525 544 564 583 3 6 64 415 435 454 474 494 514 533 553 573 593 4 8 65 421 441 -461 481 502 522 542 -562 582 -602 10 66 428 448 469 489 509 530 550 570 591 611 .6 12 67 431 455 476 496 517 538 558 579 600 620 .7 14 68 441 462 483 504 525 546 567 588 609 630 .8 16 69 447 469 490 511 532 , 554 575 596 618 639 9 18 70 454 475 497 519 540 562 583 605 627 648 71 460 482 504 526 548 570 592 614 635 657 .1 2 72 467 489 511 533 556 578 600 622 644 667 .2 5 73 473 496 518 541 563 586 608 631 653 676 .3 7 74 480 502 525 548 571 594 617 640 662 685 .4 9 75 486 -509 532 556 579 -601 -625 648 671 694 S 12 76 493 516 540 563 586 610 633 657 680 704 .6 14 77 499 523 547 570 594 618 642 665 689 713 .7 16 78 506 530 554 578 602 626 650 674 698 722 .8 19 79 512 536 561 585 610 634 658 683 707 731 9 21 80 519 543 568 593 617 642 667 691 716 741 81 525 550 575 600 625 650 675 700 725 750 j 3 82 531 557 582 607 633 658 683 709 734 759 .2 5 83 538 564 589 615 640 666 692 717 743 769 3 8 84 544 570 596 6-22 648 674 700 726 752 778 4 10 85 551 577 -603 630 656 682 -708 735 761 787 5 13 86 557 584 610 637 664 690 717 743 770 796 .6 16 87 564 591 618 644 671 698 725 752 779 806 7 18 88 570 598 625 652 679 706 733 760 788 815 .8 21 89 577 604 632 659 687 714 742 769 797 824 9 24 90 583 611 639 667 694 722 750 777 806 833 91 590 618 646 674 702 730 758 786 815 843 .1 3 92 596 625 653 681 710 738 767 795 823 852 .2 6 93 603 631 660 689 718 746 775 804 832 861 3 9 94 609 638 667 696 725 754 783 812 841 870 4 12 95 -616 -645 -674 704 733 762 792 821 -850 880 5 15 96 622 652 681 711 741 770 800 830 859 889 .6 18 97 629 659 689 719 748 778 808 838 868 898 .7 21 98 635 665 696 756 756 786 817 847 877 907 .8 23 99 642 672 703 733 764 794 825 856 886 917 9 26 100 G48 679 710 741 772 802 833 864 895 926 21 22 23 24 25 26 27 28 29 30 .1 .2 3 4 5 .6 7 .8 9 Corrections for 1 2 2 3 4 5 5 6 7 tenths in width. 94 SURVEYING. TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. co JS "O HEIGHTS. Corrections g 31 32 33 34 35 36 37 38 39 40 for tenths in height. 1 10 10 10 10 11 11 11 12 12 12 .1 2 19 20 20 21 22 22 23 23 24 25 .2 3 29 30 31 31 32 33 34 35 36 37 .3 o 4 38 40 41 42 43 44 46 47 48 49 , A 1 6 48 49 51 52 54 56 S7 -59 60 62 .5 1 6 57 59 61 63 65 67 68 70 72 74 .6 1 7 67 69 71 73 76 78 80 82 84 86 .7 1 8 77 79 81 84 86 89 91 94 96 97 .8 1 9 86 89 92 94 97 100 103 106 108 111 9 1 10 96 99 102 105 108 111 114 117 120 123 11 105 109 112 115 119 122 126 129 132 136 .1 12 115 119 122 126 130 133 137 141 144 148 .2 1 13 124 128 132 136 140 144 148 152 156 160 .3 1 14 134 138 143 147 151 156 160 164 169 173 4 2 15 144 148 153 157 -162 167 171 -176 181 185 .5 2 16 153 158 163 168 173 178 183 188 193 198 .6 3 17 163 168 173 178 183 189 194 199 205 210 . 7 3 18 172 178 183 189 194 200 206 211 217 222 .8 4 19 182 188 194 199 205 211 217 223 229 235 9 4 20 191 198 204 210 216 222 228 235 241 247 21 201 207 214 220 227 233 240 246 253 259 . i 1 22 210 217 224 231 238 244 251 258 265 272 .2 2 23 220 227 234 241 248 256 263 270 277 284 .3 2 24 230 237 244 252 259 267 274 281 289 296 4 3 25 239 247 255 262 -270 278 -285 293 301 309 .5 4 26 249 *57 265 273 281 289 297 305 313 321 .6 5 27 258 267 275 283 292 300 308 317 325 333 .7 5 28 90S 277 285 294 302 311 320 328 337 346 .8 6 29 277 286 295 304 313 322 331 340 349 358 9 30 287 2% 306 315 324 3.3S 343 352 361 370 31 297 306 316 825 335 344 354 364 373 383 .1 1 32 306 316 326 336 346 356 365 375 385 395 .2 2 33 316 326 336 346 356 367 377 387 397 407 .3 3 34 325 336 346 357 367 378 388 399 409 420 4 4 35 --335 346 -356 367 378 389 400 ^ilO 421 -432 .5 5 36 344 356 367 378 389 400 411 422 433 444 .6 6 37 354 365 377 388 400 411 423 434 445 457 .7 8 38 364 375 387 399 410 422 434 446 457 469 .8 9 39 373 385 S97 409 421 433 445 457 469 481 9 10 40 383 395 407 420 432 444 457 469 481 494 41 392 405 418 430 443 456 468 481 494 506 .1 1 42 402 415 428 441 454 467 480 493 506 519 .2 3 43 411 425 438 451 465 478 491 504 518 531 .3 4 44 421 435 448 462 475 489 502 516 530 543 4 6 45 -^131 444 -458 472 486 500 514= 528 -542 -556 .5 7 46 440 454 469 483 497 511 525 540 554 568 .6 8 47 450 464 479 493 508 522 537 551 566 580 .7 10 48 459 474 489 504 519 533 548 563 578 593 .8 11 49 469 484 499 514 529 544 560 575 590 605 .9 13 60 478 494 509 525 540 556 571 586 602 617 31 32 33 34 35 36 37 38 39 40 .1 .2 3 4 5 .6 7 .8 9 Corrections for 1 9 3 4 5 6 8 9 10 tenths in width. TABLES. 95 TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. j3 HEIGHTS. Corrections for tenths !2 31 32 33 34 35 36 37 38 39 40 in height. 51 488 504 519 535 551 567 582 598 614 630 T 2 62 498 514 530 546 562 578 594 610 626 642 .2 3 63 507 523 540 556 573 589 605 622 638 654 .3 5 64 517 533 550 567 583 600 617 633 650 667 4 7 66 526 543 560 577 594 611 628 645 662 679 .5 8 66 536 553 570 588 605 622 640 657 674 691 .6 10 67 545 563 581 598 616 633 651 669 686 704 .7 12 68 555 573 591 609 627 644 662 680 698 716 .8 14 69 565 583 601 619 637 656 674 692 710 728 9 15 60 574 593 611 630 648 667 685 704 722 741 61 584 602 621 640 659 678 697 715 734 753 .z 2 62 593 612 631 651 670 689 708 727 746 765 .2 4 63 603 622 642 661 681 700 719 739 758 778 .3 : 6 64 612 632 652 672 691 711 731 751 770 790 4 8 66 622 642 662 682 702 722 742 762 782 802 .5 10 66 631 652 672 693 713 733 754 774 794 815 .6 12 67 611 662 682 703 724 744 765 786 806 827 .7 14 68 651 672 693 714 735 756 777 798 819 840 .8 16 69 660 681 703 724 745 767 788 809 831 852 9 18 70 670 691 713 735 756 778 799 821 843 864 71 679 roi 723 745 767 789 811 833 855 877 , 2 72 689 711 733 756 778 800 822 844 867 889 .2 5 73 698 721 744 766 789 811 834 856 879 901 3 7 74 708 731 754 777 799 822 845 868 891 914 4 9 75 718 741 764 787 810 833 856 -80 903 926 .5 12 76 727 751 774 798 821 844 868 891 915 938 .6 14 77 737 760 784 808 832 856 879 903 927 951 .7 16 78 746 770 794 819 843 867 891 915 939 963 .8 19 79 756 780 805 829 853 878 902 927 951 975 9 21 80 765 790 815 840 864 889 914 938 963 988 81 775 800 825 850 875 900 925 950 975 1000 .1 3 82 785 810 835 860 886 911 936 962 987 1012 .2 5 83 794 820 845 871 897 922 948 973 999 1025 .3 8 84 804 830 856 881 907 933 959 985 1011 1037 4 10 85 813 840 866 892 918 944 971 997 1023 1049 13 86 823 849 876 902 929 956 982 1009 1035 1062 .6 16 87 832 859 886 913 940 967 994 1020 1047 1074 .7 18 88 842 869 896 923 951 978 1005 1032 1059 1086 .8 21 89 852 879 906 934 961 989 1016 1044 1071 1098 9 24 90 861 889 917 944 972 1000 1028 1056 1083 1111 91 871 899 927 955 983 1011 1039 1067 1095 1123 .1 3 92 880 909 937 965 994 1022 1051 1079 1107 1136 .2 6 93 890 919 947 976 1005 1033 1062 1091 1119 1148 .3 9 94 899 928 957 986 1015 1044 1073 1102 1131 1160 4 12 95 909 938 968 997 1026 1056 1085 1114 1144 1173 .5 15 96 919 948 978 1007 1037 1067 1096 1126 1156 1185 .6 18 97 928 958 988 1018 1048 1078 1108 1138 1168 1198 .7 21 98 933 968 998 1028 1059 1089 1119 1149 1180 1210 .8 23 99 947 978 1008 1039 1069 1100 1131 1161 1192 1222 9 26 100 957 988 1019 1049 1080 1111 1142 1173 1204 1235 31 32 33 34 35 36 37 38 39 40 .1 .2 3 4 5 .6 7 .8 9 Corrections for 1 2 3 4 5 6 8 9 10 tenths in width. 9 6 SURVEYING. TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. 1 *o HEIGHTS. Corrections % 41 42 43 44 45 46 47 48 49 50 for tenths in height. 1 13 13 13 14 14 14 15 15 15 15 . i o 2 25 26 27 27 28 28 29 30 30 31 .2 o 3 38 39 40 41 42 43 44 44 45 46 . -3 o 4 51 52 53 54 56 57 58 59 60 62 *J 1 6 .63 65 C6 68 69 71 73 74 76 77 "T 1 6 76 78 80 81 83 85 87 89 91 93 g 1 7 89 91 93 95 97 99 102 104 106 108 .7 1 8 101 104 106 109 111 114 116 119 121 123 .8 1 9 114 117 119 122 125 128 131 133 136 139 .Q 1 10 127 130 133 136 139 142 145 148 151 154 y 11 139 143 146 149 153 156 160 163 166 170 ,i o 12 152 156 159 163 167 170 174 178 181 185 .2 1 13 165 169 173 177 181 185 189 193 197 201 . o 1 14 177 181 186 190 194 199 203 207 212 216 " j 4 2 15 190 194 199 204 208 213 218 222 227 231 . c 2 16 203 207 212 217 222 227 232 237 242 247 .6 3 17 215 220 226 231 236 241 247 252 257 262 .7 3 18 228 23d 239 244 250 256 261 267 272 278 .8 4 19 240 246 252 258 264 270 276 281 287 293 9 4 20 253 259 265 272 278 284 290 296 302 309 21 266 272 279 285 292 298 305 311 318 324 .1 1 22 278 285 293 299 306 312 319 326 333 340 .2 2 23 291 298 305 312 319 327 334 341 348 355 .3 2 24 304 311 319 326 333 341 348 356 363 370 4 3 25 316 324 332 340 -347 355 363 370 378 386 .5 4 26 329 337 345 353 361 369 377 385 393 401 .6 5 27 342 330 358 367 375 383 392 400 408 417 .7 5 28 354 363 372 380 389 398 406 415 423 432 .8 6 29 367 376 385 394 403 412 421 430 439 448 9 7 30 380 389 398 407 417 426 435 444 454 463 31 392 402 411 421 431 440 450 459 469 478 .1 1 32 405 415 425 435 444 454 464 474 484 494 .2 2 33 418 428 438 448 458 469 479 489 499 509 .3 3 34 430 441 451 462 472 483 493 504 514 525 4 4 35 -443 -454 465 -475 -486 497 508 -519 529 -540 .5 5 36 456 467 478 489 500 511 522 533 544 556 .6 6 37 468 480 491 502 514 525 537 548 560 671 .7 ' 8 38 481 493 504 516 528 540 551 563 575 586 .8 9 39 494 506 518 530 542 554 566 578 590 602 9 10 40 506 519 531 543 556 568 580 593 605 617 41 519 531 544 557 569 582 595 607 620 633 .1 1 42 531 544 557 670 583 596 609 622 635 648 .2 3 43 544 557 571 584 597 610 624 637 650 664 .3 4 44 557 570 584 598 611 625 638 652 665 679 4 6 46 569 583 -597 611 625 639 653 -667 -681 694 .5 7 46 582 596 610 625 639 653 667 681 696 710 .6 8 47 595 609 624 638 653 667 682 696 711 725 .7 10 48 607 622 637 652 667 681 696 711 726 741 .8 11 49 620 635 650 665 681 696 710 726 741 756 9 13 60 633 648 C64 679 694 710 725 741 756 772 41 42 43" 44 45 46 47 48 49 50 .1 .2 3 4 5 .6 7 .8 9 Corrections for 1 3 4 6 7 8 10 11 13 tenths in width. TABLES. 97 TABLE XI. Continued. VOLUMES BY THE PRISMOIDAL FORMULA. i HEIGHTS. Corrections s 41 42 43 44 45 46 47 48 49 50 in height. 51 645 661 677 693 708 724 740 756 771 787 x 2 52 658 674 690 706 722 738 754 770 786 802 .2 8 53 671 687 703 720 736 752 768 785 802 818 .3 5 54 683 700 717 733 750 767 783 800 817 833 4 7 55 696 713 730 747 764 781 798 815 -832 849 8 56 709 726 743 760 778 795 812 830 847 864 '.6 10 57 721 739 756 774 792 809 827 844 862 880 .7 12 58 734 752 770 788 806 823 841 859 877 895 .8 14 59 747 765 783 801 819 833 856 874 892 910 9 15 60 759 778 706 815 833 852 870 889 907 926 61 772 791 810 828 847 866 885 991 923 941 .1 2 62 785 804 823 842 861 880 899 919 938 957 .2 4 63 797 817 836 856 875 894 914 933 953 972 .3 6 64 810 830 849 869 889 909 928 948 968 988 4 8 65 823 843 863 883 903 923 943 963 983 1003 10 66 835 856 876 896 917 937 957 978 998 1019 .6 12 67 848 869 889 910 931 951 972 993 1013 1034 .7 14 68 860 881 902 923 944 965 986 1007 1028 1049 .8 16 69 873 894 916 937 958 980 1001 1022 1044 1065 9 18 70 886 907 929 951 972 994 1015 1037 1059 1080 71 898 920 942 964 986 1008 1030 1052 1074 1096 .1 2 72 911 933 956 9^8 1000 1022 1044 1067 1089 1111 .2 3 73 924 946 969 991 1014 1036 1059 1081 1104 1127 .3 7 74 936 959 982 1005 1028 1051 1073 1096 1119 1142 .4 9 15 949 972 995 1019 1042 1065 1088 1111 1134 1157 5 12 76 962 985 1009 1032 1056 1079 1102 1126 1149 1173 .6 14 77 974 998 1022 1046 1069 1093 1117 1141 1165 1188 .7 16 78 987 1011 1035 1059 1083 1107 1131 1156 1180 1204 .8 19 79 1000 1024 1048 1073 1097 1122 1146 1170 1195 1219 9 21 80 1012 1037 1062 1086 1111 1136 1160 1185 1210 1235 81 1025 1050 1075 1100 1125 1150 1175 1200 1225 1250 .1 3 82 1038 1063 1088 1114 1139 1164 1190 1215 1240 1265 .2 5 83 1050 1076 1102 1127 1153 1178 1204 1230 1255 1281 .3 8 84 1063 1089 1115 1141 1167 1193 1219 1244 1270 1296 4 10 85 1076 1102 1128 -1154 1181 1207 1233 1259 1285 1312 .5 13 86 1088 1115 1141 1168 1194 1221 1248 1274 1301 1327 .6 16 87 1101 1128 1155 1181 1208 1235 1262 1289 1316 1343 .7 18 88 1114 1141 1168 1195 1222 1249 1277 1304 1331 1358 .8 21 89 1126 1154 1181 1209 1236 1264 1291 1319 1346 1373 9 24 90 1139 1167 1194 1222 1250 1278 1806 1333 1361 1389 91 1152 1180 1208 1236 1264 1292 1320 1348 1376 1404 .1 3 92 1164 1193 1221 1249 1278 1306 1335 1363 1391 1420 6 93 1177 1206 1234 1263 1292 1320 1349 1378 1406 1435 .3 9 94 1190 1219 1248 1277 1306 1335 1364 1393 1422 1451 .4 12 95 1202 1231 1261 1290 1319 1349 1378 1407 -1437 1466 .5 15 96 1215 1244 1274 1304 1333 1363 1393 1422 1452 1481 .6 18 97 1227 1257 1287 1317 1347 1377 1407 1437 1467 1497 .7 21 98 1240 1270 1301 1331 1361 1391 1422 1452 148fi 1512 .8 23 99 1253 1283 1314 1344 1375 1406 1436 1467 1497 1528 9 26 100 1265 1296 1327 1358 1389 1420 1451 1481 1512 1543 41 42 43 44 45 46 41 48 49 5O .1 .2 3 4 s .6 7 .8 9 Corrections for 1 3 4 6 7 8 10 11 13 tenths in width. 9 8 SURVEYING. TABLE XII. AZIMUTHS or POLARIS THE STAR AND THE AZIMUTH are W. of N. when the hour angle is less THE ARGUMENT is the star's hour angle (or 23h. 56min. To FIND THE TRUE MERIDIAN the azimuth must he laid off to the east when the jj i 0*