ff~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ot e8 aca l~~~~~~C 0 X.- off on it a number of equal parts, each representing a chain to the desired scale. Sub-divide the left hand division into ten equal parts, each of which will therefore represent ten links to this scale. Through each point of division on the central line, draw (with the T square) perpendiculars extending to the edges, and the scale is made. It explains itself. The above figure is a scale of 2 chains to 1 inch. On it the distance 220 links would extend 3 34 FUNDAMENTAL OPERATIONS. [PART I. between the arrow-heads above the line in the figure; 560 links extends between the lower arrow-heads, &c. A paper scale has the great advantage of varying less from a plat which has been made by it, in consequence of changes in the weather, than any other. The mean of many trials showed the difference between such a scale and drawing paper, when exposed alternately to the damp open atmosphere, and to the air of a warm dry room, to be equal to.005, while that between box-wood scales and the paper was.012, or nearly 21 times as much. The difference with ivory would have been even greater. Some of the more usual platting scales are here given in their actual dimensions. In these five figures, different methods of drawing the scales have been given, but each method may be applied to any scale. The first and second, being the most simple, are generally the best. In the third the subdivisions are made by a diagonal line: the distances between the various pairs of arrow heads, beginning with the uppermost, are, respectively, 310, 54Q, and 270 links. Fig. 18. Scale of 1 chain to 1 inch. 0 1 0oo 50 I j Fig. 19. Scale of 2 chains to 1 inch. l ml[lLlj -,,- I2.. 3. 4, g",; - Fig. 20. Scale of 3 chains to 1 inch. o 1 2 3 4 5 6 7 8 9 In the fourth figure the distances between the arrow heads are respectively 310, 270, and 540 links. Fig. 21. Scale of 4 chains to I inch. 0 1 2 3 4 I 6 7 8 9 10 11 12 18 E- 0,-1 - X,_ -1_ _= __- _- - CHAP. II.] Drawing the Map. 36 In the fifth figure the scale of 5 chains to 1 inch is subdivided diagonally to only every quarter chain, or 25 links. The distance between the upper pair of arrow-heads on it is 123 chains, or 12.25; between the next pair of arrow-heads, it is 6.50; and between the lower pair, 14.75. Fig, 22. Scale of 5 chains to 1 iwnch. a a i o 5. 10 A diagonal scale for dividing an inch, or a half'inch, into 100 equal parts, is found on the "c Plain scale" in every case of instruments. (5t) Vernier Scale. This is an ingenious substitute for the diagonal scale. The one given in the following figure divides an inch into 100 equal parts, and if each inch be supposed to represent a chain, it gives single links. Fig. 23. 00 5'0 0 1 2()().3 22.86 4. 22 Make a scale of an inch divided into tenths, as in the upper scale of the above figure. Take in the dividers eleven of these divisions, and set off this distance from the 0 of the scale to the left of it. Divide the distance thus set off into 10 equal parts. Each of them will be one tenth of eleven tenths of one inch; i. e. eleven hundredths, or a tenth and a hundredth, and the first division on the short, or vernier scale, will overlap, or be longer than the first division on the long scale, by just one hundredth of an inch; the second division will overlap two hundredths, and so on. The principle will be more fully developed in treating of" Verniers," Part IV, Chapter II. Now suppose we wish to take off from this scale 275 hundredths of an inch. To get the last figure, we must take five divisions on the lower scale, which will be 55 hundredths, for the reason just given. 220 will remain which are to be taken from the upper 36 FUNDAMENTAL OPERATIONS. [PART I scale, and the entire number will be obtained at once by extending the dividers between the arrow-heads in the figure from 220 on the upper scale (measuring along its lower side) to 55 on the lower scale, 254 would extend from 210 on the upper scale to 44 on the lower. 318 would extend from 230 on the upper scale to 88 on the lower. Always begin then with subtracting 11 times the last figure from the given number; find the remainders on the upper scale, and the number subtracted on the lower scale. (51) A plat is sometimes made by a nominally reduced scale in the following manner. Suppose that the scale of the plat is to be ten chains to one inch, and that a diagonal scale of inches, divided into tenths and hundredths, is the only one at hand. By dividing all the distances by ten, this scale can then be used without any further reduction. But if the content is measured from the plat to the same scale, in the manner explained in the next chapter, the result must be multiplied by 10 times 10. This is called by old Surveyors " Raising the scale," or " Restoring true measure." (52) Sectoral Scales. The Sector, (called by the French " Compass of Proportion"), is an instrument sometimes convenient for obtaining a scale of equal parts. It is in two portions, turning on a hinge, like a carpenter's pocket rule. It contains a great number of scales, but the one intended for this use is lettered at its ends L in English instruments, and consists of two lines running from the centre to the ends of the scale, and each divided into ten equal parts, each of which is again subdivided into 10, so that each leg of the scale contains 100 Fig. 24. equal parts. To illustrate its use, suppose that a scale of 7 chains to 1 inch is re- ~ quired. Take 1 inch in the dividers, and open the sec- tor till this distance will just reach from the 7 on one leg to the 7 on the other. The sector is then " set" for this CHAP. III.] Drawing the Map. 87 scale, and the angle of its opening must not be again changed. Now let a distance of 580 links be required. Open the dividers till they reach from 58 to 58 on the two legs, as in the dotted line in the figure, and it is the required distance. Again, suppose that a scale of 21 chains to one inch is desired. Open the sector so that 1 inch shall extend from 25 to 25. Any other scale may be obtained in the same manner. Conversely, the length of any known line to any desired scale can thus be readily determined. (53) Whatever scale may be adopted for platting the survey, it should be drawn on the map, both for convenience of reference, and in order that the contraction and expansion, caused by changes in the quantity of moisture in the atmosphere, may affect the scale and the map alike. When the drawing paper has been wet and glued to a board, and cut off when the map is completed, its contractions have been found by many observations to average from one-fourth to one-half per cent. on a scale of 3 chains to an inch, (1:2376), which would therefore require an allowance of from one-half perch to one perch per acre. A scale made as directed in Art. (49), if used to make a plat on unstretched paper, and then kept With the plat, will answer nearly the same purpose. Such a scale may be attached to a map, by slipping it through two or three cuts in the lower part of the sheet, and will be a very convenient substitute for a pair of dividers in measuring any distance upon it. (54) Scale omitted. It may be required to find the unknown scale to which a given map has been drawn, its superficial content being known. Assume any convenient scale, measure the lines of the map by it, and find the content by the methods to be given in the next chapter, proceeding as if the assumed scale were the true one. Then make this proportion, founded on the geometrical principle that the areas of similar figures are as the squares of their corresponding sides: As the content found Is to the given content So is the square of the assumed scale To the square of the true scale. 38 FUNDAMENTAL OPERATIONS. [PART I. CHAPTER IV. CALCULATING THE CONTENT. (55) The CONTENT of a piece of ground is its superficial area, or the number of square feet, yards, acres, or miles which it contains. (56) Horizontal Measurement. All ground, however inclined or uneven its surface may be, should be measured horizontally, or as if brought down to a horizontal plane, so that the surface of a hill, thus measured, would give the same content as the level base on which it may be supposed to stand, or as the figure which would be formed on a level surface beneath it by dropping plumb lines from every point of it. This method of procedure is required for both Geometrical and Social reasons. aeometrically, it is plain that this horizontal measurement is absolutely necessary for the purpose of obtaining a correct plat. In Fig. 25, let ABCD, and BCEF, Fig. 25. be two square lots of ground, platted A I horizontally. Suppose the ground to slope in all directions from the point \E. C, which is the summit of a hill. Then the lines BC, DC, measured on D the slope, are longer than if measur- Fig. 2. ed on a level, and the field ABCD, of Fig. 25, platted with these long lines, would take the shape ABGD in Fig. 26; and the field BCEF, ", ----- of Fig. 25, would become BHEF of Fig. 26. The two adjoining fields would thus overlap each other; and the same difficulty would occur in every case of platting any two adjoining fields by the measurements made on the slope. CHAP. IV.] Calculating the Content. 39 Let us suppose another case, Fig. 27. Fig. 28. more simple than would ever oc- 1 cur in practice, that of a threesided field, of equal sides and composed of three portions each sloping down uniformly, (at the rate of one to one) from one point in the centre, as in Fig. 27. Each slope being accurately platted, the three could not come together, but would be separated as in Fig. 28. We have here taken the most simple cases, those of uniform slopes. But with the common irregularities of uneven ground, to measure its actual surface would not only be improper, but impossible. In the Social aspect of this'question, the horizontal measurement is justified by the fact that ~no more houses can be built on a hill than could be built on its flat base; and that no more trees, corn, or other plants, which shoot up vertically, can grow on it; as is represented by the vertical lines in the Fig. 29. Figure.* Even if a side hill should produce more of certain creeping plants, the: i | | increased difficulty in their cultivation;might perhaps balance this. For this reason the surface of the soil thus measured is sometimes called the productive base of the ground. Again, a piece of land containing a hill and a hollow, if measured on the surface would give a larger content than it would after the hollow had been filled up by the hill, while it would yet really be of greater value than before. Horizontal measurement is called the " Method of Cultellation," and Superficial measurement, the "Method of Developement."t An act of the State of New-York prescribes that " The acre, for land measure, shall be measured horizontally." * This question is more than two thousand years old, for Polybius writes, "Some even of those who are employed in the administration of states, or placed at the head of armies, imagine that unequal and hilly ground will contain more houses than a surface which is flat and level. This, however, is not the truth, For the houses being raised in a vertical lihue, form right angles, not with the declivity of the ground, but with the flat surface which lies below, and upon which the hills themselves also stand." t The former from Cultellum, a knife, as if the hills were sliced off; the latter so named because it strips off or unfolds, as it were, the surface. 40 FUNDAMENTAL OPERATIONS. [PART I (57) Unit of Content. The Acre is the unit of land-measure. ment. It contains 4 Roods. A Rood contains 40 Perches. A Perch is a square Rod; otherwise called a Perch, or Pole. A Rod is 5~ yards, or 164 feet. Hence, 1 acre =4 Roods = 160 Perches = 4,840 square yards = 43,560 square feet. One square mile = 5280 x 5280 feet = 640 acres. Since a chain is 66 feet long, a square chain contains 4356 square feet; and consequently ten square chains make one acre.* In different parts of England, the acre varies greatly. The statute acre, as in the United States, contains 160 square perches of 161 feet, or 43,560 square feet. The acre of Devonshire and Somersetshire, contains 160 perches of 15 feet, or 36,000 square feet. The acre of Cornwall is 160 perches of 18 feet, or 51,840 square feet. The acre of Lancashire is 160 perches of 21 feet, or 70,560 square feet. The acre of Cheshire and Staffordshire, is 160 perches of 24 feet, or 92,160 square feet. The acre of Wiltshire is 120 perches of 161 feet, or 32,670 square feet. The acre in Scotland consists of 10 square chains, each of 74 feet, and therefore contains 54,760 square feet. The acre in Ireland is the same as the Lancashire. The chain is 84 feet long. The French units of land-measure are the Are= 100 square Metres, = 0.0247 acre, = one fortieth of an acre, nearly; and the Hectare 100 Ares = 2.47 acres, or nearly two and a half. Their old land-measures were the "Arpent of Paris," containing 36,800 square feet; and the "Arpent of Waters and Woods," containing 55,000 square feet. (58) When the content of a piece of land (obtained by any of the methods to be explained presently) is given in square links, as is customary, cut off four figures on the right, (i. e. divide by 10,000), to get it into square chains and decimal parts of a chain; cut off the right hand figure of the square chains, and the remaining figures will be Acres. Multiply the remainder by 4, and the figure, if any, outside of the new decimal point will be Roods. * Let the young student beware of confounding 10 square chains with 10 chains square. The former make one acre; the latter space contains ten acres. CHAP. IV.] Calculating the Content. 41 Multiply the remainder by 40, and the outside figures will be Perches. The nearest round number is usually taken for the Perches; fractions less than a half perch being disregarded.* Thus, 86.22 square chains =8 Acres 2 Roods 20 Perches. Also, 64.1818 do. = 6 A. 1 R. 27 P. " 43.7564 do. =4 A. 1 R. 20 P. " 71.1055 do. -7 A. 0 R. 18 P. 82.50 do. 8 A. 1 R. 0 P. " 8.250 do. = A. 3 R. 12 P. " 0.8250 do. 0 A. 0 R. 13 P. (59) The following Table gives by mere inspection the Roods and Perches corresponding to the Decimal parts of an Acre. It explains itself. ROODS. ROODS. 0 | 1 | 1 | 3 Peees. 0 1 | | 3 Perches..000.250.500.750 + 0.131.381.631.881 +21.006.256.506.756 + 1.137.387.637.887 +22.012.262 512.762 + 2.144.394.644.894 +23.019.269.519.769 + 3.150.400.650.900 +24.025.275.525.775 + 4.156.406].656.906 +25.031.281.531.781 + 5.162.412.662.912 +26.037.287.537.787 + 6.169.419.669.919 +27 c.044.294.44.794 + 7.175.425.675.925 +28.050.300.550.800 + 8.181.431.681.931 +29 0.056.306.556.806 + 9.187.437.687.937 +30 1.0.31256 2.812 +10.194.444.694.944 +31 ^.069.319.569.819 +11 -.200.450.700.950 +32 ~.075.325.575.825 +12 E.2061.456.706.956 +33;.081.331.581.831 +13.2121.4621.712.962 +34 a.087.337.587.837 +14 1.219.469.719.969 +35.094.344.594.844 I +15.2251.475.725.975 +36.100.350.600.850 +16.2311.481.731.981 +37.106.356.606.856 +17.237.487.737.987 +38.112.362.612.862 +18.244.494.744.994 +39.119.369.619.869 +19.250.500.7501.000 +40 _1251.375.625.875 +.20 _ 1 _____ (60) Chain Correction. When a survey has been made, and the plat has been drawn, and the content calculated; and after* To reduce square yards to acres, instead of dividing by 4840, it is easier, and very nearly correct, to multiply by 2, cut off four figures, and add to this product one-third of one-tenth of itself. 42 FUNDAMENTAL OPERATIONS. [PART I. wards the chain is found to have been incorrect, too short or too long, the true content of the land, may be found by this proportion: As the square of the length of the standard given by the incorrect chain Is to the square of the true length of the standard So is the calculated content To the true content. Thus, suppose that the chain used had been so stretched that the standard distance measured by it appears to be only 99 links long; and that a square field had been measured by it, each side containing 10 of these long chains, and that it had been so platted. This plat, and therefore the content calculated from it, will be smaller than it should be, and the correct content will be found by the proportion 992: 1002:: 100 sq. chains: 102.03 square chains. If the chain had been stretched so as to be 101 true links long, as found by comparing it with a correct chain, the content would be given by this proportion: 1002: 1012:: 100 square chains: 102.01 square chains. In the former case, the elongation of the chain was 14- true links; and 1002: (10141)2: 100 square chains: 102.03 square chains. (61) Boundary Lines. The lines which are to be considered as bounding the land to be surveyed, are often very uncertain, unless specified by the title deeds. If the boundary be a brook, the middle of it is usually the boundary line. On tide-waters, the land is usually considered to extend to low water mark. Where hedges and ditches are the boundaries of fields, as is almost universally the case in England, the dividing line is generally the top edge of the ditch farthest from the hedge, both hedge and ditch belonging to the field on the hedge side. This varies, however, with the customs of the locality. From three to six feet from the roots of the quickwood of the hedges are allowed for the ditches. CHAP. IV.] Calculating the Content, 43 METHODS OF CALCULATION. (62) The various methods employed in calculating the content of a piece of ground, may be reduced to four, which may be called Arithmetical, Geometrical, Instrumental, and Trigonometrical. (63) FIRST METHOD.-ARITHMETICALLY. From direct measurements of the necessary lines on the ground. The figures to be calculated by this method may be either the shapes of the fields which are measured, or those into which the fields can be divided by measuring various lines across them. The familiar rules of mensuration for the principal figures which occur in practice, will be now briefly enunciated. (64) Rectangles, If the piece of ground be rectangular in shape, its content is found by multiplying its length by its breadth. (65) Triangles. When the given quantities are one side of a triangle and the perpendicular distance to it from the opposite angle; the content of the triangle is equal to half the product of the side and the perpendicular. When the given quantities are the three sides of the triangle; add together the three sides and divide the sum by 2; from this half sum subtract each of the three sides in turn; multiply together the half sum and the three remainders; take the square root of the product; it is the content required. If the sides of the triangle be designated by a, b, c, and their sum by s, this rule will give its area- = s (a -a) (a s b) (s - c)]* " When two sides of a triangle, and the included Fig. 30. angle are given, its content equals half the product B of its sides into the sine of the included angle. Designating the angles of the triangle by the capital C \ letters A,B,C, and the sides opposite them by the corresponding small letters a,b,c, the area = A be sin. A., \A. When one side of a triangle and the adjacent an- - ID gles are given, its content equals the square of the given side multiplied by the sines of each of the given angles, and divided by twice the sine of the sum of sin. B. sin. C these angles. Using the same symbols as before, the area =a2 sin. (B + C) When the three angles of a triangle and its altitude are given, its area, referring sin. B to the above figure, = ~ BD2. i A sin. 0 —C 44 FUNDAMENTAL OPERATIONS. [PART I. (66) Parallelograms; or four-sided figures whose opposite sides are parallel. The content of a Parallelogram equals the product of one of its sides by the perpendicular distance between it and the side parallel to it. (67) Trapezoids; or four-sided figures, two opposite sides of which are parallel. The content of a Trapezoid equals half the product of the sum of the parallel sides by the perpendicular distance between them. If the given quantities are the four sides a, h, c, d, of which b and d are parallel; then, making q = 9 (a + b + c - d), the area of the trapezoid will = b + d ) (q c) (q-b + d).]* b - d (68) Quadrilaterals, or Trapeziums; four-sided figures, none of whose sides are parallel. A very gross error, often committed as to this figure, is to take the average, or half sum of its opposite sides, and multiply them together for the area: thus, assuming the trapezium to be equivalent to a rectangle with these averages for sides. In practical surveying, it is usual to measure a line across it from corner to corner, thus dividing it into two triangles, whose sides are known, and which can therefore be calculated by Art.(65).t * When two parallel sides, b and d, and a third side, a, are given, and also the angle, C, which this third side makes with one of the parallel sides, then the content of the trapezoid — d a. sin. C. t When two opposite sides, and all the angles are given, take one side and its adjacent angles, (or their supplements, when their sum exceeds 180~), consider them as belonging to a triangle, and find its area by the second formula in the note on page 43. Do the same with the other side and its adjacent angles. The difference of the two areas will be the area of the quadrilateral. When three sides and their two included angles are given, multiply together the sine of one given angle and its adjacent sides. Do the same with the sine of the other given angle and its adjacent sides. Multiply together the two opposite sides and the sine of the supplement of the sum of the given angles. Add together the first two products, and add also the last product, if the sum of the given angles is more than 180~, or subtract it if this sum be less, and take half the result. Calling the given sides, p, q, r; and the angle between p and q = A; and the angle between q and r = B; the area of the quadrilateral ^ [p.qsin. A + q. r. sin. B 4 p. sin. (180 - A - B)]. Wihen the four sides and the sum of any two opposite angles are given, proceed thus: Take half the sum of the four given sides, and from it subtract each side in turn Multiply together the four remainders, and reserve the product. Multiply together the four sides. Take half their product, and multiply it by the cosine of tae given sum of the angles increased by unity. Regard the sign of CHAP, IV.] Calulating the Contents 46 (69) Surfaces bounded by irregularly curved lines. The rules for these will be more appropriately given in connection with the surveys which measure the necessary lines; as explained in Part II, Chap. III. (70) SECOND METHOD.-GEOMETRICALLY. From measurements of the necessary lines upon the plat. (71) Division into Triangles. The plat of a piece of ground having been drawn from the measurements made by any of the methods which will be hereafter explained, lines may be drawn upon the plat so as to divide it into a number of triangles. Four Fig. 31. Fig. 32. Fig. 33. Fig. 34. -...... ways of doing this are shown in the figures: viz. by drawing lines from one corner to the other corners; from a point in one of the sides to the corners; from a point inside of the figure to the corners; and from various corners to other corners. The last method is usually the best. The lines ought to be drawn so as to make the triangles as nearly equilateral as possible, for the reasons given in Part V. One side of each of these triangles, and the length of the perpendicular let fall upon it, being then measured, as directed in Art. (43,) the content of these triangles can be at once obtained by multiplying their base by their altitude, and dividing by two. The easiest method of getting the length of the perpendicular, without actually drawing it, is, to set one point of the dividers at the angle from which a perpendicular is to be let fall, and to the cosine. Multiply this product by the reserved product, and take the square root of the resulting product. It will be the area of the quadrilateral. When the four sides, and the angle of intersection of the diagonals of the quadrilateral are given; square each side; add together the squares of the opposite sides; take the difference of the two sums; multiply it by the tangent of the angle of intersection, and divide by four. The quotient will be the area. When the diagonals of the quadrilateral, and their included angle are given, mul. tiply together the two diagonals and the sine of their included angle, and divide by two. The quotient will be the area. 46 FUNDAMlENTAL OPERATIONS. [PART I open and shut their legs till an arc described by the other point will just touch the opposite side. Otherwise; a platting scale, (described in Art. (49) may be placed so that the zero point of its edge coincides with the angle, and one of its cross lines coincides with the side to which a perpendicular is to be drawn. The length of the perpendicular can then at once be read off. The method of dividing the plat into triangles is the one most commonly employed by surveyors for obtaining the content of a survey, because of the simplicity of the calculations required. Its correctness, however, is dependant on the accuracy of the plat, and on its scale, which should be as large as possible. Three chains to an inch is the smallest scale allowed by the English Tithe Commissioners for plats from which the content is to be determined. In calculating in this way the content of a farm, and also of its separate fields, the sum of the latter ought to equal the former. A difference of one three-hundredth (300) is considered allowable. Some surveyors measure the perpendiculars of the triangles by a scale half of that to which the plat is made. Thus, if the scale of the plat be 2 chains to the inch, the perpendiculars are mea, sured with a scale of one chain to the inch. The product of the base by the perpendicular thus measured, gives the area of the triangle at once, without its requiring to be divided by two. Another way of attaining the same end, with less danger of mistakes, is, to construct a new scale of equal parts, longer than those by which the plat was made in the ratio /V2:1; or 1.414:1* When the base and perpendicular of a triangle are measured by this new scale and then multiplied together, the product will be the content of the triangle, without any division by two. In this method there is the additional advantage of the greater size and consequent greater distinctness of the scale. When the measurement of a plat is made some time after it has been drawn, the paper will very probably have contracted or expanded so that the scale used will not exactly apply. In that case a correction is necessary. Measure very precisely the present length of some line on the plat, of known length originally. Then CHAP. IV.] Calculating the Contents 47 make this proportion: As the square of the present length of this line Is to the square of its original length, So is the content obtained by the present measurement To the true content. (72) Graphical Multiplication. Prepare a strip of drawing paper, of a width exactly equal to two chains on the scale of the plat; i. e. one inch wide, as in the figure, for a scale of two chains to 1 inch; two-thirds of an inch wide for a scale of 3 chains; half an inch for 4 chains; and so on. Draw perpendicular lines across the paper at distances representing one-tenth of a chain on the scale of the triangle to be measured, thus making a platting scale. Apply it to the triangle so that one edge of the scale shall pass through one corner, A, of the triangle, and the other edge through another Fig. 35. K' C as / I """ 11 sI I II 111 II r TB corner, B; and note very precisely what divisions of the scale are at these points. Then slide the scale in such a way that the points of the scale which had coincided with A and B, shall always remain on the line BA produced, till the edge arrives at the point C. Then will A'C, that is, the distance, or number of divisions on the scale, from the point to which the division A on the scale has arrived, to the third corner of the triangle, express the area of the triangle ABC in square chains.* *For, from C draw a parallel to AB, meeting the edge of the scale in C', and draw C'B. Then the given triangle ABC - ABC'. But the area of this last triangle = AC' multiplied by half the width of the scale, i. e. = AC' X 1 = AC'. But, because of the parallels, A'C = AC'. Therefore the area of the given triangle ABC = A'C i. e. it is equal in square chains to the number of linear chains read off from the scale. This ingenious operation is due to M. Cousinery. 48 FUNDAMENTAL OPERATIONS. [PART I. (73) Division into Trapezoids. A line may be drawn across the field, as in Fig. 36, and perpen- Fig. 36. diculars drawn to it. The field will thus be divided into trapezoids, (ex- / \,. cepting a triangle at each end), \ and their content can be calculated 7 by Art. (67)..Otherwise; a line may be drawn outside of the figure, and per- Fig. 37. pendiculars to it be drawn from each angle. In that case the difference between the trapezoids \ formed by lines drawn to the /i outer angles of the figure, and,' those drawn to the inner angles,' a' a I will be the content.'' -. This method is very advantageously applied to surveys by the compass; as will be explained in Part III, Chap. VI. (74) Division Into Squares, Two sets of parallel lines, at right angles to each other, Fig. 38. one chain apart (to the scale " of the plat) may be drawn - - - - over the plat, so as to divide -- it into squares, as in the --- - figure. The number of i _ squares which fall within the plat represent so many square chains; and the triangles and \ trapezoids which fall outside | of these, may then be calcu- i - lated and added to the entire square chains which have been counted. Instead of drawing the parallel lines on the plat, they may better be drawn on a piece of transparent " tracing paper," which is simply laid upon the plat, and the squares counted as before. The CoHP. iv.] Calculating the Content. 49 same paper will answer for any number of plats drawn to the same scale. This method is a valuable and easy check on the results of other calculations. To calculate the fractional parts, prepare a piece of tracing paper, or horn, by drawing on it one square of the same size as a square of the plat, and subdividing it, by two sets of ten parallels at right angles to each other, into hundredths. This will measure the fractions remaining from the former measurement, as nearly as can be desired. (75) Division into Parallelograms. Draw a series of pairal lel lines across the plat at equal distances depending on the scale. Thus, for a plat made to a scale of 2 chains to 1 inch, the distance between the parallels should be 21 inches; for a scale of 3 chains to 1 inch, 19 inch; for a scale of 4 chains to 1 inch, ~ inch; for a scale of 5 chains to 1 inch, 4A inch; and for any scale, make the distance between the parallels that fraction of an inch which would be expressed by 10 divided by the square of the number of chains to the inch. Then apply a common inch scale, divided on the edge into tenths, to these parallels; and every inch in length of the spaces included between each pair of them will be an acre, and every tenth of an inch will be a square chain.* To measure the triangles at the ends of the strips between the parallels, prepare a piece of transparent horn, or stout tracing paper, of a width equal to the width between the parallels, and draw a line through its middle longitudinally. Apply it to the oblique line at the end of the space between Fig. 39. two parallels, and it will bisect the line, and thus reduce the triangle to an equivalent A/ rectangle, as at A in the figure. When an angle occurs between two parallels, as at B B in the figure, the fractional part may be measured by any of the preceding methods. * For, calling the number of chains to the inch, = n, and making the width be tween the parallels inch, this width will represent 2 X n chains; and as the inch length represents n chains, their product, 1- X n = 10 square chains -=1 acre. 4 5D) FUNDAMENTAL OPERATIONS. [PART I. A somewhat similar method is much used by some surveyors, particularly in Ireland: the plat being made on a scale of 5 chains to 1 inch, parallel lines being drawn on it, half an inch apart, and the distances along the parallels being measured by a scale, each large division of which is TO inch in length. Each division of this scale indicates an acre; for it represents 4 chains, and the distance between the parallels is 21 chains. This scale is called the "Scale of Acres." (76) Addition of Widths. When the lines of the plat are very irregularly curved, as in the Fig. 40. figure, draw across it a num- ber of equi-distant lines as'near together as the case may seem to require. Take a straightedged piece of paper, and apply one edge of it to the middle of the first space, and mark its length from one end; apply the same edge to the middle of the next space, bringing the mark just made to one end, and making another mark at the end of the additional length; so go on, adding the length of each space to the previous ones. When all have been thus measured, the total length, multiplied by the uniform width, will give the content. (77) THIRD METHOD,-INSTRIMENTALLY. By performing certain instrumental operations on the plat. (78) Reduction of a many sided figure to a single equivalent triangle. Any plane figure bounded by straight lines may be reduced to a single triangle, which shall have the same content. This can be done by any instrument for drawing parallel lines, such as those described in Art. Fig. 41. ($9). Let the trapezium, or four sided figure, shown in Fig. 41, be required to be reduced to a single equivalent triangle. Produce one side of the figure, as 4.-1. Draw a line from the first to tae third angle of 1 CHAP. iv.] Calculatig the Content..61 the figure. From the second angle draw a parallel to the line just drawn, cutting the produced side in a point 1'. From the point 1' draw a line to the third angle. A triangle (1'- 3-4 in the figure) will thus be formed, which will be equivalent to the original trapezium.* The content of this final triangle can then be found by measuring its perpendicular, and taking half the product of this perpendicular by the base, as in the first paragraph of Art. (65). (79) Let the given figure have five sides, as in Fig. 42. For brevity, the angles Eig. 42. of the figure will be inamd as numbered / / ^ in the engraving. Produce 5-1. \ Join 1- 3. From 2 draw a parallel to. 1-3, cutting the s' produced base in 1'. Join 1' -4. From 3 draw a parallel to it, cutting the base in 2'. Join 2'- 4. Then will the triangle 2' —4 —5 be equivalent to the five sided figure 1 — 2- 3-4-5, for similar reasons to those of the preceding case. (80) Let the given figure be 1-2 —3-4-5 —6 —7 —8, as shown in Fig. 43, given at the top of the following page. All the operations are shown by dotted lines, and the finally resulting triangle 6'-7-8, is equivalent to the original figure of eight sides. It is best, in choosing the side to be produced, to take one which has a long side adjoining it on the end not produced; so that this long side may form one side of the final triangle, the base of which will therefore be shorter, and will not be cut so acutely by the final line drawn, as to make the point of intersection too indefinite. For, the triangle 1-2-3 taken away from the original figure is equivalent to the triangle 1'-1-3 added to it; because both these triangles have the same base and also the same altitude, since the vertices of both lie in the same line parallel to the base, 62 FUNDAMENTAL OPERATIONS. [a.. 6 Fig. 43. 8 3s 8 i\' fs i (81) General Rule. When the given figure has many sides, with angles sometimes salient and sometimes re-entering, the operations of reduction are very liable to errors, if the draftsman attempts to reason out each step. All difficulties, however, will be removed by the following General Rule: 1. Produce one side of the figure, and call it a base. Call one of the angles at the base the first angle, and number the rest in regular succession around the figure. 2. Draw a line from the 1st angle to the 3d angle. Draw a parallel to it from the 2d angle. Call the intersections of this parallel with the base the 1st mark. 3. Draw a line from the 1st mark to the 4th angle. Draw a parallel to it from the 3d angle. Its intersection with the base is the 2d mark. 4. Draw a line from the 2d mark to the 5th angle. Draw a parallel to it from the 4th angle. Its intersection with the base is the 3d mark. 5. In general terms, which apply to every step after the first, draw a line from the last mark obtained to the angle whose number is greater by three than the number of the mark. Draw a parallel to it through the angle whose number is greater by two than that of the mark. Its intersection with the base will be a mark whose number is greater by one than that of the preceding mark.* In the concise language of Algebra, draw a line firom the nth mark to the n+ 3 angle. Draw a parallel to it through the n+2 angle, and the intersection with the base will be the n-+1 mark. CHP. v.] falculating the Conte;. 5a 6.: Repeat this process for each angle, till you get a mark whose number is such that the angle having a number greater by three is the last angle of the figure, i. e. the angle at the other end of the base. Then join the last mark to the angle which precedes the last angle in the figure, and the triangle thus formed will be the equivalent triangle required. In practice it is unnecessary to actually draw the lines joining the successive angles and marks, but the parallel ruler is merely laid on so as to pass through them, and the points where the parallels cut the base are alone marked. (82) It is generally more convenient, for the reasons given at the end of Art. (80), to reduce Fig. 44. half of the figure on one side and half on the other, as is shown in / Fig. 44, which represents the same / field as Fig. 42. The equivalent / / triangle is here 11 —8-2'. When the figure has many angles, they should not be numbered con- 3 l -5 a, secutively all the way around, but, after the numbers have gone around as far as the angle where it is intended to have the vertex 104IL y/\^ Fig. 45. of the final triangle, the numbers should be continued from the 54 FUNDAMENTAL OPERATIONS. [PART I. other angle of the base, as is shown in Fig. 45. In it only the intersections are marked.* (83) It is sometimes more convenient, not to produce one of the sides of the figure, but to draw at one end of it, as at the point 1 in Fig. 46, an indefinite line, usually a perpendicular to a line Fig. 46. ing two disant angles of the figure, and make this line the base joining two distant angles of the figure, and make this line the base of the equivalent triangle desired. The operation is shown by the dotted lines in the figure. The same General Rule applies to it, as to the previous figures. (84) Special Instruments. A variety of instruments have been invented for the purpose of determining areas rapidly and correctly. One of the simplest is the "Computing Scale," which is on the same principles as the Method of Art. (75). It is represented in Fig. 47, given on the following page. It consists of a scale divided for its whole length from the zero point into divisions, each representing 21 chains to the scale of the plat. The scale carries a slider, which moves along it, and has a wire drawn across its centre at right angles to the edges of the scale. On each side of this wire, a portion of the. slider equal in length to one of the primary, or 24 chain, divisions of the scale, is laid off and divided into 40 equal parts. This instrument is used in connection with a sheet of transparent paper, ruled with parallel lines at distances apart each equal, to one chain on the scale of the plat. It is plain, that when the A figure with curved boundaries may be reduced to a triangle in a sinilar manner. Straight lines must be drawn about the figure, so as to be partly in it and partly out, giving and taking about equal quantities, so that the figure which these lines form, shall be about equivalent to the curved figure. This having been done, as will be further developed in Art. (124), the equivalent straight lined figure is reduced by the above method. CHAP. IV.].Caculating the Content. 55 instrument is laid on this paper,with its edgeononeofthe v;g 47. parallel lines, and the slider is moved over one of the divi:- -- sions of 21 chains, that one rood, or a quarter of an acre, has been measured between two, of the parallel lines onthe paper (since 10 square chains make one acre),; and.thatone of the smaller divisions measures one perch between. the same parallels. Four of the larger divisions give one acre. The scale is generally made long enough to measure at once five acres. To apply this to the, plat of a field,, or farm, lay the transparent paper over it in such a position that two of the ruled lines shall touch two of the exterior points of the boundaries, as Fig. 48. at A andB. Lay A the scale, with the slide set to zero,, - on the paper, in a direction parallel f to the ruled lines, and so that the wire of the slide cuts the left hand oblique line so as. to make the spaces c and d about equal, Hold the scale firm, and move the slider till the wire cuts the right hand oblique line in such a way as, to equalize the spaces e and f. Without changing the slide, move the scale down the width of a space, and to the left hand end of the next space; begin there again, and proeee'as before. So go on, till the whole length of the scale is run out, (five acres having been measured), and then begin. at the right hand side and work backwards to the left,'reading the lower divisions, which run up to 10 acres. By coniinuing this process, the content of plats of any size can be obtained. A still simpler substitute for this is a scale similarly divided, but without an attached slide. In place of it there is used a piece of 56 FUNDAMENTAL OPERATIONS. [PART I. horn having a line drawn across it and rivetted to the end of a short scale of box-wood, divided like the former slide. It is used like the former, except that at starting, the zero of the short scale and not the line on the horn is made to coincide with the zero of the long scale. The slide is to be held fast to the instrument when this is moved. The Pediometer is another less simple instrument used for the same object. It measures any quadrilateral directly. (85) Some very complicated instruments for the same object have been devised. One of them, Sang's Planometer, determines the area of any figure, byrmerely moving a point around the outline of the surface. This causes motion in a train of wheel work, which registers the algebraic sum of the product of ordinates to every point in that perimeter, by the increment of their abscissas, and therefore measures the included space. Instruments of this kind have been invented in Germany by Ernst, Hansen, and Wetli. (86) A purely mechanical means of determining the area of any surface by means of its weight, may be placed here. The plat is cut out of paper and weighed by a delicate balance. The weight of a rectangular piece of the same paper containing just one acre is also found; and the "Rule of Three" gives the content. A modification of this is to paste a tracing of the plat on thin sheet lead, cut out the lead to the proper lines and weigh it. (87) FOURTH METHOD.-TRIGONOMETRICALLY. By cal culating, from the observed angles of the boundaries of the piece of ground, the lengths of the lines needed for calculating the content. This method is employed for surveys made with angular instruments, as the compass, &c., in order to obtain the content of -the land surveyed, without the necessity of previously making a plat, thus avoiding both that trouble and the inaccuracy of any calculations founded upon it. It is therefore the most accurate method; but will be more appropriately explained in Part HI, Chapter VI, under the head of "Compass Surveying." PART II. CHAIN-SURVEYING; By the tFirst and Second:MHodw: OR DIAGONAL AND PERPENDICULAR SURVEYING. (88) The chain alone is abundantly sufficient, without the aid of any other instrument, for making an accurate survey of any surface, whatever its shape or size, particularly in a district tolerably level and clear. Moreover, since a chain, or some substitute for it, formed of a rope, of leather driving reins, &c., can be obtained by any one in the most secluded place, this method of Surveying deserves more attention than has usually been given to it in this country. It will, therefore, be fully developed in the following chapters. CHAPTER I. SURVEYING BY DIAGONALS: OR By the First Method. (89) Surveying by Diagonals is an application of the First Method of determining the position of a point, given in Art. (5,) to which the student should again refer. Each corner of the field or farm which is to be surveyed is "determined" by measuring its distances from two other points. The field is then "platted" by repeating this process on paper, for each corner, in a contrary order, and the "content" is obtained by some of the methods explained in Chapter IV of Part I. 58 CHAIN SURVEYING. [PAT II. The lines which are measured in order to determine the corners of the field are usually diagonals of the irregular polygon which is to be surveyed. They therefore divide up this polygon into triangles; whence this mode of surveying is sometimes called " Chain Triangulation." A few examples will make the principle and practice perfectly clear. Each will be seen to require the three operations of measuring, platting, and calculating. (90) A three-sided field; as Fig. 49. Fig. 49. Field-work. Measure the three sides, AB, BC, and CA. Measure also, as a proof line, the distance from one of the cor- A --- Dners, as C, to some point in the opposite side, as D, at which a mark should have been left, when measuring from A to B, at a known distance from A. A stick or twig, with a slit in its top, to receive a piece of paper with the distance from A marked on it, is the most convenient mark. Platting. Choose a suitable scale as directed in Art. (44). Then, by Arts. (42) and (49), draw a line equal in length, on the chosen scale, to one of the sides; AB for example. Take in the compasses the length of another side as AC, to the same scale, and with one leg in A as a centre, describe an arc of a circle. Take the length of the third side BC, and with B as a centre, describe another arc, intersecting the first arc in a point which will be the third corner C. Draw the lines AC and BC; and ABC will be the plat, or miniature copy -as explained in Art. (35)of the field surveyed. Instead of describing two arcs to get the point C, two pairs of compasses may be conveniently used. Open them to the lengths, respectively, of the last two sides. Put one foot of each at the ends of the first side, and bring their other feet together, and their point of meeting will mark the desired third point of the triangle. To " prove " the accuracy of the work, fix the point D, by setting off from A the proper distance, and measure the length of the line CHAP. I.] Surveying by Diagonals, 59 DC, by Art. (43). If its length on the plat corresponds to its measurement on the ground, the work is correct.* Calculation. The content of the field may now be found as directed in Art. (65), either from the three sides, or more easily though not so accurately, by measuring on the plat, by Art. (43), the length of the perpendicular CE, let fall from any angle to the opposite side, and taking half the product of these two lines. Example 1. Figure 49, is the plat, on a scale of two chains to one inch, of a field, of which the side AB is 200 links, BC is 100 links, and AC is 150 links. Its content by the rule of Art. (65), is 0.726 of a square chain, or OA. OR. 12P. If the perpendicular AD be accurately measured, it will be found to be 72" links. Half the product of this perpendicular by the base will be found to give the same content. Ex. 2. The three sides of a triangular field are respectively 89.39, 54.08, and 45.98. Required its content. Ans. 100A. OR. 10P. (91) A four-sided field; Fig. 50. as Fig. 50. nD ec Field-work. Measure the four sides. Measure also a diagonal, as AC, thus di- viding the four-sided field __ into two triangles. Mea- A B sure also the other diagonal, or BD, for a "Proof line." Platting. Draw a line, as AC, equal in length to the diagonal, to any scale, by Arts. (42) and ( i). On each side of it, construct a triangle with the sides of the field, as directed in the preceding article. To prove the accuracy of the work, measure on the plat the length of the "proof line," BD, by Art. (43), and if it agrees with the length of the same line measured on the ground, the field work and platting are both proved to be correct. * It is a universal principle in all surveying operations, that the work must le tested by some means independent of the original process, and that the same result must be arrived at by two different methods. The necessary length of this proof line can also easily be calculated by the principles of Trigonometry. 60 OHIN SURVETYING [PART 1 Calculation. Find the content of each triangle separately, as in the preceding case, and add them together; or, more briefly, multiply either diagonal (the longer one is preferable) by the sum of the two perpendiculars, and divide the product by two. Otherwise: reduce the four-sided figure to one triangle as in Art. (78); or, use any of the methods of the preceding chapter. Example 3. In the field drawn in Fig. 50, on a scale of 3 chains to the inch, AB= 588 links, BC= 210, CD = 430, DA = 274, the diagonal AC = 626, and the proof diagonal BD =500. The total content will be 1A. OR. 17P. Ex. 4. The sides of a four-sided field are AB = 12.41, BC =5.86, CD = 8.25, DA -4.24; the diagonal BD =11.55, and the proof line AC = 11.04. Required the content. Ans. 4A. 2R. 38P. -Ex. 5. The sides of a four-sided field are as follows: AB 8.95, BC = 5.33, CD = 10.10, DA = 6.54; the diagonal from A to C is 11.52; the proof diagonal from B to D is 10.92. Required the content. Ans. Ex. 6. In a four-sided field, AB = 7.68, BC = 4.09, CD = 10.64, DA = 7.24, AC= 10.32, BD = 10.74. Required the content. Ans. (92) A many-sided field, as Fig. 51. Fig. 51. B.. CD\^ J /"c F?-< CHAP. I] Surveyng by Diagoals. 6 Field-Work. Measure all the sides of the field. Measure also diagonals enough to divide the field into triangles; of which there will always be two less than the number of sides. Choose such diagonals as will divide the field into triangles as nearly equilateral as possible. Measure also one or more diagonals for "Proof lines." It is well for the surveyor himself to place stakes in advance at all the corners of the field, as he can then select the best mode of division. Platting. Begin with any diagonal and plat one triangle, as in Art. (90). Plat a second triangle adjoining the first one, as in Art. (91). Plat another adjacent triangle, and so proceed, till all have been laid down in their proper places. Measure the proof lines as in the last article. Calculation. Proceed to calculate the content of the figure, precisely as directed for the four-sided field, measuring the perpendiculars and calculating the content of each triangle in turn; or taking in pairs those on opposite sides of the same diagonal; or using some of the other methods which have been explained. Example 7. The six-sided field, shown in Fig. 51, has the lengths of its lines, in chains and links, written upon them, and is divided into four triangles, by three diagonals. The diagonal BE is a " proof-line." The Figure is drawn to a scale of 4 chains to the inch. The content of the field is 5A. 3R. 22P. Ex. 8. In a five-sided field, the length of the sides are as follows: AB = 2.69, BC = 1.22, CD = 2.32, DE = 3.55, EA = 3.23. The diagonals are AD = 4.81, BD = 3.33. Required its content. Ans. (93) A field may be divided up into triangles, not only by measuring diagonals as in the last figure, but by any of the methods shown in the four figures of Art. (71). The one which we have been employing, corresponds to the last of those figures. Still another mode may be used when the angles cannot be seen from one another, or from any one point within. Take three or more convenient points within the field, and measure from them to the corners, and thus form different sets of triangles. 62 CHAIN SURVEYING. [P.ART It. KEEPING THE FIELD NOTES. (94) By Sketch. The most simple method is to make a sketch of the field, as nearly correct as the unassisted hand and eye can produce, and note down on it the lengths of all the lines, as in Fig. 51. But when many other points require to be noted, such as where fences, or roads, or streams are crossed in the measurement, or any other additional particulars, the sketch would become confused, and be likely to lead to mistakes in the subsequent platting from it. The following is therefore the usual method of keeping the Field-notes. A long narrow book is most convenient for it. (95) In Columns. Draw two parallel lines about an inch apart from the bottom to the top of the page of the field-book, as in the margin. This column, or pair of lines, may be conceived to represent the measured line, split in two, its two halves being then separated. an inch apart, merely for convenience, so that the distances measured along the line, may be written between these halves. Hold the book in the direction of the measurement. At the bottom of the page write down the name, or number, or letter, which represents the station at which the survey is to begin. A " station" is marked with a triangle or circle, as in the margin. The latter is more easily made. In the complicated cases, which will be hereafter explained, and in which one long base line is measured, and also many other subordinate lines, it will be well, as a help to the memory, to mark the stations on the Base line with a triangle, and the stations on the other lines with the ordinary circle. The station from which the measure- ( to ments are made is usually put on the left 562 of the column; and the station which is measured to, is put on theFrom A measured to, is put on the right. CHAP. I.] Surveying by Diagonals. 68 But it is more compact, and avoids interfering with the notes of "offsets" (to be explained here- B after) to write the name or number of the station A in the column, as in the margin. B The measurements to different points of a line are 400 written above one another. The numbers all refer 250 to the beginning of the line, and are counted from it. 100 A The end of a measured line is marked by a line drawn across the page above the numbers which - indicate the measurements which have been made. If the chaining does not continue along the adjoining line, but the chain-men go to some _ other part of the field to begin another measurement, two lines are drawn across the page. When a line has been measured, the marks -r r or 1 are made to show whether the follow- lowing line turns to the right or to the left. -1 A line is named, either by the names of the stations between which it is measured, as the line AB; or by its length, a line 562 links long, being called the line 562; or it is recorded as Line No. 1, Line No. 2, &c; or as Line on page 1, 2, &c., of the Field-book. When a mark is left at any point of a line, 562 as at D, in Fig. 49, with the intention of com- 200. S. ing back to it again, in order to measure to 0 some other point, the place marked is called a 562 False Station, and is marked in the Field-book 2__ F. S.; or has a line drawn around it, to distinguish it; or has a station mark A placed outside _ of the column, to the'right or left, according to 562 the direction in which the measurement from it is 200 to be made. Examples of these three modes are given in the margin. 64 CHAIN SURVEYING. [PART I, A False Station is named by its position on the line where it belongs; as thus. —" 200 on 562." When a gate occurs in a measured line, the distance from the beginning of the line to the side of. the gate first reached, is the one noted. When the measured line crosses a fence, brook, road, &c., they are drawn on the field-notes in their true direction, as nearly as possible, but not in a continuous line across the column, as in the first figure in the margin, but as in the second figure, so that the two parts would form a continuous straight line, if the halves of the " split line" were brought together. It is convenient to name the lines, in the margin, as being Sides, Diagonals, Proof lines, &c. (96) The Field-notes of the triangular field platted in Fig. 49, are given below, according to both the methods mentioned in the preceding Article, pages 62 and 63. In the Field-notes in the colunin on the right hand, it is not absolutely necessary to repeat the B and C. * ~ rJ z z C.q |89 to C 8 o 0 oFrom D F; S. OFrom ( on 200 0._ _ __ 0. - a 150 to A A a Q 150 ~.From C ~ 0 C C d _ Li 100 to C Li C From B ) B -B 200 to B 200 Fo n A|a _S 80 S _ s From A 0 A CHAP. I.] Surveying by Diagonals. 65 (97) The Field-notes of the survey platted in Fig. 51, are given below. They begin at the bottom of the left hand column. E - ------- Iu.6 770 0 F' ~ B 300 rate. |__E___ o0 1142 \ E E Li \.662 m. 5 _____ a P 400 Brook.. 0 1 0 775 / — _ D r / 480 Road. 3*0 I /~ t,, \420 300 / a2 50 Goad. -. - = o210 E on0~ ^-" j ~ J 737 __ iz 280 or I 2 ____ 210 Road. c 210 Li a A a 703 ____ 70 150 Gate. A B r 270 b " "| B a 130 Road. a 662 80 - 562 F A; -: 210~~ CHAPTER II. SURVEYING BY TIE-LINES. (98) Surveying by Tie-lines is a modification of the method explained in the last chapter. It frequently happens that it is impossible to measure the diagonals of a field of many sides, in tonse. quence of obstacles:to measurements, such as woods, water, houses, &c. In such cases, "Tielines," (so called because they tie the sides together), are employed as substitutes for diagonals. Thus, in the four-sided field shown in the Figure, the diagonals cannot be measured because of woods inter- Fig. 52. vening. As a substitute, measure off from D any convenient corner of the field, as B, any /J;. \ distances, BE, BF, along the sides of the 4$T':' field. Measure also the "tie-line" EF. Measure all the sides of the field as usual. A To plat this field, construct the triangle BEF, as in Art. (90). Produce the sides BE and BF, till they become respectively equal to BA and BC, as measured on the ground. Then with A and C as centres, and with radii respectively equal to AD and CD, describe arcs, whose intersection will be D, the remaining corner of the field. (99) It thus appears that one tie-line is sufficient to determine a four-sided field; two, a five-sided field, and so on. But, as a check on errors, it is better to measure a tie-line for each angle, and the agreement, in the plat, of all the measurements will prove the accuracy of the whole work. Since any inaccuracy in the length of a tie-line is increased in proportion to the greater length of the sides which it fixes, the tielines should be measured as far from the point of meeting of these sides as possible, that is, they should be as long as possible. The radical defect of the system is that it is " working from less to greater," (which is the exact converse of the true principle), thus magnifying inaccuracies at every step. ChAP. I.] Surveying by Tie-lnes. 67 A tie-line may also be employed as a " proof line," in the place of a diagonal, and tested in the same manner. Fig. 53. If any angle of the field is re-entering, as at: B in the figure, measure a tie-line across the \ 1 salient angle ABC. A (100) Chain Angles. It is convenient, though not necessary, to measure equal distances along the sides; BE, BF, in Fig. 52, and BA, BC, in Fig. 53. "Chain Angles" are thus formed.* (101) Inaccessible Areas. The method of tie-lines can be applied to measuring fields which cannot be entered. Thus, in the Figure, ABCD is an inac- Fig. 54. cessible wooded field, of four sides. To E C survey it, measure all the sides, and at, \D: any corner, as D, measure any distance DE, in the line of AD produced. Mea-r sure also another distance DF in the line'.. A. of CD produced. Measure the tie-line EF, and the figure can be platted as in the case of the field of Fig. 52, the sides of the triangle being produced in the contrary direction. The same end would be attained by prolonging only one side, as shown at the angle A of the same figure, and measuring AG, AH, and GH. It is better in both cases to tie all the angles in a similar manner. This method may be applied to a figure of any number of sides by prolonging as many of them as are necessary; a11 of them, if possible, * Chain angles may be reduced to angles measured in degrees, by observing that the tie-line is the chord of the angle to a radius equal to one of the equal distances measured on the sides. Therefore, divide the length of the tie-line by the length of this distance. The quotient will be the chord of the angle to a radius of one. In the TABLE OF CHORDS, at the end of this volume, find this quotient, and the number of degrees and minutes corresponding to it gives the angle required. Otherwise; since the chord of any angle equals twice the sine of half the angle, we have this rule: Divide half the tie-line by the measured distance; findl in a table of natural sines the angle corresponding to the quotient, and ultiply this angle by two, to get the angle desired. e6- @BCHIIN SURVEYING., [~ART n (102) If the sides CD and AD were prolonged by their full length, the content of the figure could be calculated without any plat; for the new triangle DEF would equal the triangle DAC; and the sides of the triangle ACB would then be known. Fig. 55 This principle may be extended still farther. ae For a five-sided field, as in Fig. 55, produce \'. two pairs of sides, a distance equal to their g \ length, forming two new triangles, as shown by e r the dotted lines, and measure the sides B'D', t', and A'D". The three sides of each of these A -- triangles will thus be known, and also the three sides of the triangle BAD, since AD = A'D" ", and BD -=B'D'. Fig. 56. The method of this article may be employed -----— = for a figure of six sides as shown in Fig. 56, ^ \ / (in which the dotted lines within the wooded f'!,, l\/t field have their lengths determined by the tri- -, //f angles formed outside of it,) but not for figures, \ of a greater number of sides " CHAPTER III SURVEYING BY PERPENDICULARS:' OR By the Second Method. (103) THE method of fukrveying by Perpendiculars is founded on the Second Method of determining the position of a point, explained in Art. (6). It is applied in two ways, either to making a complete Survey by " Diagonals and Perpendiculars," or to measuring a crooked boundary by "Off-sets." Each will-be considered in turn. CHAP. I,.] Surveylng by Perpendiculars. 6, The best methods of getting perpendiculars on the ground must, however, be first explained. TO SET OUT PERPENDICULARS. (104) Surveyor's Cross. The simplest instrument Fig. 5. for this purpose is the Surveyor's Cross, or Cross-Staff, shown in the figure. It consists of a block of wood, of any shape, having in it two saw-cuts, made very precisely at right angles to each other, about half an inch deep, sad with centre-bit holes made at the bottom of the cuts to assist in finding the objects. This block is fixed on a ri pointed staff, on which it can turn freely, and which 6 should be precisely 8 links (631 inches) long, for the convenience of short measurements. To use the Cross-staff to erect a perpendicular, set it 4 at the point of the line at which a perpendicular is want- 3 ed. Turn its head till, on looking through one saw-cut, 2 you see the ends of the line. Then will the other sawcut point out the direction of the perpendicular, and thus guide the measurement desired. To find where a perpendicular to the line, from some object, as a corner of a field, a tree, &c., would meet the line, set up the cross-staff at a point of the line which seems to the eye to be about the spot. Note about how far from the object the perpendicular at this point strikes, and move the cross-staff that distance; an4 repeat the operation till the correct spot is found. (105) To test the accuracy of the in- Fig. 58. BC BE strument, sight through one slit to some / point A, and place a stake B in the line of sight of the other slit. Then turn its A -- ------ i head a quarter of the way around, so that the second slit looked through, points to A. Then see if the other slit covers B again, as it will if correct. If it does not do so, but sights to some other point, as B', the apparent error is double the real one, for it now points as far to the right of the true point, C as it did before to its left. TO ICHAIN SUIRVEYING. LPART II. This is the first example we have had of the invaluable principle of Reversion, which is used in almost every test of the accuracy of Surveying and Astronomical instruments, its peculiar merit being that it doubles the real error, and thus makes it twice as easy to perceive and correct it. (106) The instrument, in its most finished form, is made of a hollow brass cylinder, which has two pairs of slits exactly opposite to each other, one of each pair being narrow and the other wide, with a horse-hair stretched from the top to the bottom of the latter, It is also, sometimes, made with eight faces, and two more pairs of slits added, so as to set off half a right angle. Fig. Another form is a hollow brass sphere, as in the figure. This enables the surveyor to set off perpenpendiculars on very steep slopes. Another form of the surveyor's cross consists of two pairs Fig. 60 of plain " Sights," each shaped as in the figure, placed at the ends of two bars at right angles to each other. The slit, and the opening with a hair stretched from its top to its bottom, are respectively at the top of one sight and at the bottom of the opposite sight.* This is used in the same manner as the preceding form, but is less portable and more liable to get out of order. A temporary substitute for these instruments may be Fig. 61. made by sticking four pins into the corners of a square piece of board; and sighting across them, in the direction of the line and at right angles to it. (107) Optical Square. The most convenient and accurate instrument is, however, the Optical Square. The figures give a perspective view of it, and also a plan with the lid removed. It is a small circular box, containing a strip of looking-glass, from the upper half of which the silvering is removed. This glass is placed * The French call the narrow opening aeilleton, and the wide one croisee. CHAP. III.] To set out Perpendiculars 71 so as to make precisely half a right Fi. 62. angle with the line of sight, which passes through a slit on one side -.'.. of the box, and a vertical hair stretched across the opening on the other side, or a mark on the glass. S The box is held in the hand over the spot where the perpendicular is desired, (a plumb line in the hand will give perfect accuracy) and the observer applies his eye to the slit A, looking through the upper or unsilvered part of the glass, and turns the box till he sees the other D end of the line B, through the opening C. The assistant, with a rod, E moves along in the direction where the perpendicular is desired, being seen in the silvered parts of the glass, by reflection through the opening D, till his rod, at E, is seen to coincide with, or to be exactly under, the object B. Then is the line DE at right angles to the line AB, by the optical principle of the equality of the angles of incidence and reflection. To find where a perpendicular from a distant object would strike the line, walk along the line, with the instrument to the eye, till the image of the object is seen, in the silvered part of the glass, to coincide with the direction of the line seen through the unsilvered part. The instrument may be tested by sighting along the perpendicular, and fixing a point in the original line; on the principle of " Reversion." The surveyor can make it for himself, fastening the glass in the box by four angular pieces of cork, and adjusting it by cutting away the cork on one side, and introducing wedges on the other side. The box should be blackened inside. Another form of the optical square contains two glasses, fixed at an angle of 45~, and giving a right angle on the principle of the Sextant. 72 CHAIN SURVEYING. [PART I. (108) Chain Perpendiculars. Perpendiculars may be set out with the chain alone, by a variety of methods. These methods generally consist in performing on the ground, the operations executed on paper in practical geometry, the chain being used, in the place of the compasses, to describe the necessary arcs. As these operations, however, are less often used for the method of surveying now to be explained, than for overcoming obstacles to measurement, it will be more convenient to consider them in that connection, in Chapter V. DIAGONALS AND PERPENDICULARS. (109) In Chapter I, of this Part, we have seen that plats of surveys made with the chain alone, have their contents most easily determined by measuring, on the plat, the perpendiculars of each of the triangles, into which the diagonals measured on the ground have divided the field. In the Method of Surveying by Diagonals and Perpendiculars, now to be explained, the perpendiculars are measured on the ground. The content of the field can, therefore, be found at once, (by adding together the half products of each perpendicular by the diagonal on which it is let fall,) without the necessity of previously making a plat, or of measuring the sides of the field. This is, therefore, the most rapid and easy method of surveying when the content alone is required, and is particularly applicable to the measurement of the ground occupied by crops, for the purpose of determining the number of bushels grown to the acre, the amount to be paid for mowing by the acre, &c. (110) A three-sided field. Measure the Fig. 63. longest side, as AB, and the perpendicular, A~ CD, let fall on it from the opposite angle C. \ Then the content is equal to half the product of the side by the perpendicular. If obsta- E cles prevent this, find the point, where a perpendicular let fall from an angle, as A, to the opposite side produced, as BC, would meet it, as at E in the figure. Then half the product of AE by CB is the content of the triangle. CHAP. III.] Diagonals and Perpendiculars. 73 (111) A four-sided field. Fig 64. Measure the diagonal AC. Leave marks at the points on this diagonal at which perpendiculars from B A -- and from D would meet it; finding these points by trial, as previ- ously directed in Arts. (101) and (107). The best marks at these D "False Stations," have been described in Art. (90). Return to these false stations and measure the perpendiculars. When these perpendiculars are measured before finishing the measurement of the diagonal, great care is necessary to avoid making mistakes in the length of the diagonal, when the chainmen return to continue its measurement. One check is to leave at the mark as many pins as have been taken up by the hind-chainman in coming to that point from the beginning of the line. Example 9. Required the content of the field of Fig. 64. Ans. OA. 2R. 29P. The field may be platted from these measurements, if desired, but with more liability to inaccuracy than in the first method, in which the sides are measured. The plat of the figure is 3 chains to 1 inch. The field-notes may be taken by writing the measurements on a sketch, as in the figure; or in more complicated cases, by the column method, as below. A new symbol may be employed, this mark, —, or -A, to show the False Station, from which a perpendicular is to be measured. EExample 10. Calculation. 110 toB sq. lks. From 200 on 480 F. S. - ABC= x 480 x 110 = 26400 175 to D X 280 48/ 75 - to D ADC = x 480 x 175 —42000 _ From 280 on 480 F. S. }- sq. chains 6.8400 40to ^ CO Acres 0.684 280 F- It is still easier to take the two 8' -1 200 triangles together; multiplying S From A (0 E the diagonal by the sum of the perpendiculars and dividing by two. 74 CHAIN SURVEYING. [PART It (112) A many-sided field. Fig. 65, and the accompanying field-notes represent the field which was surveyed by the First Method and platted in Fig. 51. Fig. 65. B A 5.5 /. \. I _/'\ E Example 11. Calculation. 1.54 to 1s; The content of the triangles may From 5.07 on 7.37 F. S. be expressed thus:'2.53 to D sq. lks. From 160 on 7.75 F. S. ABC = X 1142 X 267=152457 - 93- to AEC = -X 1142x493=281503 From 5.45 on 11.42? S. CDE = IX 775x253= 98037.:' to, B AEF = x 737 x154= 56749 From 4.95 on 11.42, S sq. chains 58.8746 7.37 to A Acres 5.88746 H 5.07 or, 5A. 3R. 22P. From E - -175 to E The first two triangles might I.75 Ito 1 - -, i 1.60 o have been taken together, as in From C i) r the previous field. 11.42 to C Content calculated from the 5.45 [I5.45 F perpendiculars will generally va-!1 4.95 From A O ry slightly from that obtained by measuring on the plat. CHAP. III.] Offsets. 75 (113) A small field which has many sides, may sometimes be conveniently surveyed by taking one diagonal and measuring the perpendiculars let fall on it from each angle of the field, and thus dividing the whole area into triangles and trapezoids; as in Fig. 36, page 48. The line on which the perpendiculars are to be let fall, may also be outside of the field, as in Fig. 37, page 48. Such a survey can be platted very readily, but the length of the perpendiculars renders the plat less accurate. This procedure supplies a transition to the method of " Offsets," which is explained in the next article. OFFSETS. (114) Offsets are short perpendiculars, measured from a straight line, to the angles of a crooked or zigzag line, near which the straight line runs. Thus, in the figure, Fi. 66. let ABCD be a crooked fence, D bounding one side of a field. Chain A --.- B along the straight line AB, which runs from one end of the fence to the other, and, when opposite each corner, note the distance from the beginning, or the point A, and also measure and note the perpendicular distance of each corner C and D from the line. These corners will then be "determined" by the Second Method, Art. (6). The Field-notes, corresponding to Fig. 66, are as in the margin. The measure- __ ments along the line are written in the 0 3001to B. column, as before, counting from the be- ginning of the line, and the offsets are DI 25 250 written beside it, on the right or left, oppo- CI 30 100 site the distance at which they are taken. A sketch of the crooked line is also usually From A 0 ( made in the Field-notes, though not abso- lutely necessary in so simple a case as this. The letters C and D would not be used in practice, but are here inserted to show the connection between the Field-notes and the plat. 76 CHAIN SURVEYING. [PART II In taking the Field-Notes, the widths of the offsets should not be drawn proportionally to the distances between them, but the breadths should be greatly exaggerated in proportion to the lengths. (115) A more extended example, with a little different notation, is given below. In the figure, which is on a scale of 8 chains to one inch for the distances along the line, the breadths of the offsets are exaggerated to four times their true proportional dimensions. Fig. 67. B 1500 0 1250 20 0 1000 0 30 750 50 500 40 250 0 A \ (116) The plat and Field-notes,f the position of two nouses, determined by offsets, are given below, on a scale of 2 chains to 1 inch. |l__ _[ l Fig. 68. 250 to B 30 185 30 201 150 90 101 50 10[-] 30 From A. 0 A (117) Double offsets are sometimes convenient; and sometimes triple and quadruple ones.'Below are given tile notes and the plat, 1 chain to 1 inch, of a road of varying width, both sides of which are determined by double offsets. It will be seen that the line AB crosses one side of the road at 160 links from A, and the other side of it at 220. CHAP. III.] Ofsietse 77 Two methods of keeping the Field-notes are given. In the first form, the offsets to each side of the road are given separately and connected by the sign +. In the second form, the total distance of the second offset is given, and the two measurements connected by the word "to." This is easier both for measuring and platting. Fig. 69 * \*: B B 260 30+60 260 30 to 90 240 10+70 240 10 to 80 0 220 50 0 220 50 20 200 30 20 200 30 40 180 10 40 180 10 45 160 0 45 160 0 60+ 0 140 50 to 0 140 55+ 5 120 60 to 5 120 50 +20 100 70 to 20 100 45+15 80 60 to 15 80 50+10 60 60 to 10 60 50 +20 40 70 to 20 40 55 +20 20 75 to 20 20 60+ 01 A 60 to 0 A (118) These offsets may generally be taken with sufficient accuracy by measuring them as nearly at right angles to the base line as the eye can estimate. The surveyor should stand by the chain, facing the fence, at the place which he thinks opposite to the corner to which he wishes to take an offset, and measure " square" to it by the eye, which a little practice will enable him to do with much correctness. 78 CHAIN SIRtVEYING. [PARR It. The offsets may be measured, if short, with an Offset-staff, a light stick, 10 or 15 links in length, and divided accordingly; or if they are long, with a tape. They are generally but a few links in length. A chain's length should be the extreme limit, as laid down by'the English " Tithe Commissioners," and that should be employed only in exceptional cases. When the " Cross-staff" is in use, its divided length of 8 links, renders the offset-staff needless. When offsets are to be taken, the method of chaining to the end of a line, described in Art. (23), page 21, is somewhat modified. After the leader arrives at the end of the line, he should draw on the chain till the follower, with the back end of the chain, reaches the last pin set. This facilitates the counting of the links to the places at which the offsets are taken. The offsets are to be taken to every angle of the fence or other crooked line; that is, to every point where it changes its direction. These angles or prominent bends can be best found by one of the party walking along the crooked fence and directing another at the chain what points to measure opposite to. If the line which is to be thus determined is curved, the offsets should be taken to points so near each other, that the portions of the curved line lying between them may, without much error, be regarded as straight. It will be most convenient, for the subsequent calculations, to take the offsets at equal distances apart along the straight line from which they are measured. In the case of a crooked brook, such as is shown in the figure given below, offsets should be taken to the most prominent angles, such as are marked a a a in the figure. and the intermediate bends may be merely sketched by eye. Fig. 70. When offsets from lines measured around a field are taken inside of these bounding lines, they are sometimes distinguished as Insets. BHAP. I.] Offsets. 79 (119) Platting. The most rapid method of platting the offsets, is by the use of a Platting Scale (described in Art. 49) and an Offset Scale, which is a short scale divided on its edges like a platting scale, but having its zero in the middle, as in the figure. Fig. 71 CIA ccti cp te e ~h eo' The platting scale is placed parallel to the line, with its zero point opposite to the beginnihg of the line. The offset scale is slid along the platting scale, till its edge comes to a distance on the latter at which an offset had been taken, the length of which is marked off with a needle point from the offset scale. This is then slid on to the next distance, and the operation is repeated. If one person reads off the field-notes, and another plats, the operation will be greatly facilitated. The points thus obtained are joined by straight lines, and a miniature copy of the curved line is thus obtained; all the operations of the platting being merely repetitions of the measurements made on the ground. If no offset scale is at hand, make one of a strip of thick drawing paper, or pasteboard; or use the platting scale itself, turned crossways, having previously marked off from it the points from which the offsets had been taken. In plats made on a small scale, the shorter offsets are best estimated.by eye. On the Ordnance Survey of Ireland, the platting of offsets is facilitated by the use of a combination of the offset scale and the platting scale, the former being made to slide in a groove in the latter, at right angles to it. (120) Calculating Contentb When the crooked line determined by offsets is the boundary of a field, the content, enclosed 80 CHAIN SURVEYING. [PART I. between it and the straight line surveyed, must be determined, that it may be added to, or subtracted from, the content of the field bounded by the straight lines. There are various methods of effecting this. The area enclosed between the straight and the crooked lines is divided up by the offsets into triangles and trapezoids, the content of which may be calculated separately by Arts. (65) and (67), and then added together. The content of the plat on page 75, will, therefore, be 1500 + 4125 + 625 = 6250 square links = 0.625 square chain. The content of the plat on page 76, will in like manner be found to be, on the left of the straight line 30,000 square links, and on its right 5,000 square links. (121) When the offsets have been taken at equal distances, the content may be more easily obtained by adding together half of the first and of the last offset, and all the intermediate ones, and multiplying the sum by one of the equal distances between the offsets. This rule is merely an abbreviation of the preceding one. Thus, in the plat of page 76, the distances being equal, the content of the offsets on the left of the straight line will be 120 x 250 = 30,000 square links, and on the right 20 x 250 = 5,000 square links; the same results as before. When the line determined by the offsets is a curved line, " Simpson's rule" gives the content more accurately. To employ it, an even number of equal distances must have been measured in the part to be calculated. Then add together the first and last offset, four times the sum of the even offsets, (i. e. the 2d, 4th, 6th, &c.,) and twice the sum of the odd offsets, (i. e. the 3d, 5th, 7th, &c.,) not including the first and the last. Multiply the sum by one of the equal distances between the offsets, and divide by 3. The quotient will be the area. Example 12. The offsets from a straight line to a curved fence, were 8, 9, 11, 15, 16, 14, 9, links, at equal distances of 5 links. What was the content included between the curved fence and the straight line? Ans. 371.666 OHAP. II.] OffSets. 81 (122) Many erroneous rules have been given on this part of the subject. One rule directs the surveyor to divide the sum of all the offsets by one less than their number, and multiply the quotient by the whole length of the straight line; or, what is the same thing, to multiply the sum of all the offsets by the common distance between them. This will be correct only when the offsets at each end of the line are nothing, i. e. when the curved line starts from the straight line and returns to it at the beginning and end of one of the equal distances. In all other cases it will give too much. A second rule directs the surveyor to divide the sum of all the offsets by their number, and then to multiply the quotient by the whole straight line. This may give too much, or too little, according to circumstances. Suppose offsets of 10, 30, 20, 80, 50, 30, links, to have been taken at equal distances of a chain. The correct content of the enclosed space is 200 x 100 = 2 square chains. The first of the above rules would give 2.2 square chains, and the second would give 1.8333 chains. (123) Reducing to one triangle the many-sided figure which is formed by the offsets, is the method of calculation sometimes adopted. This has been fully explained in Part I, Art. (78), &c. The method of Art. (83) is best adapted for this purpose. (124) Equalizing, or giving and taking, is an approximate mode of calculation much used by practical surveyors. A crooked line, determined by offsets, having been platted, a straight line is drawn on the plat, across the crooked line, leaving as much space outside of the straight line as inside of it, as nearly as can be estimated by the eye, "Equalizing" it, or "Giving and taking" equal Fig. 72.........., I - --- —..- ~ portions. The straight line is best determined by laying across the irregular outline the straight edge of a piece of transparent born, or tracing paper, or glass, or a fine thread or horse-hair 6 82 CHAIN SURVEYING. [PARr I. stretched straight by a light bow of whalebone. In practical hands, this method is sufficiently accurate in most cases. The student will do well to try it on figures, the content of which he has previously ascertained by perfectly accurate methods. Sometimes this method may be advantageously combined with the preceding; short lengths of the croooked boundary being "Equalized," and the fewer resulting zigzags reduced to one line by the method of Art. (78), &c. CHAPTER IV. SURVEYING BY THE PRECEDING METHODS COMBINED. 125) All the methods which have been explained in the three preceding chapters-Surveying by Diagonals, by Tie-lines, and by Perpendiculars, particularly in the form of offsets-are frequently required in the same survey. The method by Diagonals should be the leading one; in some parts of the survey, obstacles to the measurement of diagonals may require the use of Tie-lines; and if the fences are crooked, straight lines are to be measured near them, and their crooks determined by Offsets. (126) Offsets are necessary additions to almost every other method of surveying. In the smallest field, surveyed by diagonals, unless all the fences are perfectly straight lines, their bends must be determined by offsets. The plat (scale of 1 chain to 1 inch), and field-notes, of such a case are given below. A sufficient num CHAP. iv.] Diagonals, Tie-lines and Offsets, 83 ber of the sides, diagoals, and proof-lines, to prove the work, should be platted before platting the offsets. Fig. 73. B 3. 60..___ ~,~-"' T "'-'?'" —'C-_ -, —.- ~. —-.- —:- - ~ ~... ~2\~ -,A~~..~- — J 0 360 Z 40 6 315 ~ z Di 10 275 — __-. / - c 5 215 -J C 0 150 0 310 115 10 a A |r 80 5 A 65 8 80 248 B Or L 11 180 ---- -- 0 105 0 B 65 5 0 125 D Or 23 62 135 12 22 15 110 0 A 5._ A, 13 90 - --.50 0 Example 13. Required the con- 30 9 tent of the above field. Ans. C 0 r (127) Field-books. The difficulty and the importance of keeping the Field-notes clearly and distinctly, increase with each new combination of methods. For this reason, three different methods of keeping the Field-notes of the same survey will now be given, (from Bourns' Surveying), and a careful comparison by the student of the corresponding portions of each will be very profitable to him. 84 CHAIN SURVEYING. [PART I. FIELD-BOOK No. 1. /t ~ so X 00 /' I. X',, \, Q OT ~,,0x -.,x /:.0 ) \, ( ) ed in A0 62rt. 1260 (94)30. o, | 0'a', | "f \ v\ v50:3O CD Field-Book 1No. 1 (Fig. 74) shews the Sketch method, explained in Art. (91). CHAP. xv.] Diagonals, Tie-lines and Offsets. 85 FIELD-BOOK No. 2.,'4570" A - 4-080 - (1 4000o ~ 1300 o 12 50 (3650. e 1 3- 480 / 1020 200 3060 / 140 680 190 (13020' a / 240 From0 c to A ( 2450, o 1390 1 2360 2 1 5 %.. 230, - 160 120 760 x- ace —' T_00 1'30\ 7 OO ~ -- 00 - 620 i J20 626 260 2 / 580 fromA o tooX X a A Field-Book No. 2 (Fig. 75) shews the Column method, explain ed in Art. (95). 86 CHAIN SURVEYING. [CPAR I. FIELD-BOOK No. 3. + I 12 12 /\ IJ - 12110 700 13 (' 00) 64-0 _ / 1'20 625 140 OT8 00A 20 580 o S I /. O Ca O./.,,.... 4, ~9t X \ H ) N b \g t -V \, / to,/ ~.. 1100 -~ I'/il \ \ \/,3 /, I 1220 \50 y / /icV / ~ 10 ~ H Field Book No. 3 (Fig. 76) is a convenient combination of the two preceding methods. The bottom of the Book is at the side of this figure, at A. CHAP. iv.] Diagonals, Tie-laes and Offsets. 87 (128) It will easily appear from the sketch of Field-book No. 1. how much time and labor may be saved, or lost, by the manner of doing the work. Thus, beginning at A, and measuring 750 links, a pole should be left there, and the line to the right measured tc 17 chains, or C, leaving a pole at 12.30 as a new starting point by and by. Then from C measure 19 chains to A again; then measure from A to B, and from B back to the pole left at 7.50 on the main line. (129) The example which will now be given shows part of the Field-notes, the plat, (on a scale of 6 inches to 1 mile [1: 10,560]), and a partial calculation of the "Filling up" of a large triangle, the angular points of which are supposed to have been determined by the methods of Geodesic Surveying. They should be well studied.* Fig. 77. D XXA 0 IL C \ Capt. FROME, in his "Trigonometrical Survey," from which this example has been condensed, remarks, "It may, perhaps, be thought that too much stress is laid on forms; but method is a most essential part of an undertaking of magnitude: and without excellent preliminary arrangements to ensure uniformity in all the most trifling details, the work never could go on creditably." 88 CHAIN SURVEYING. [PART II ______' C l 2564 80 D F 2452 o0 100 4050 Q 1700 0 62 3890 *N 2324 1530 84 42 3730 1420 40 0 3540 -0 0 1340 0 3420 30 H 1264 0 FromA A to D A A72 2484 S 1240 52 A -- 1140 86 40 2332 950 100 60 2206 772 60 t 0 2056 0 0 604 0 1805 40 34 502 M 3296 0 1550 50 50 450 -- 70 342 3275 54 X 1442 0 82 220 3120 62 FromC!A toA 2940 85 FromD A to C. I C 2572 60 D In the above specimen of a field-book, (which resembles that on page 85), all offsets, except those having relation to the boundary lines, are purposely omitted, to prevent confusion, the example being given solely to illustrate the method of calculating these larger divisions. Rough diagrams are drawn in the field-book not to any scale, but merely bearing some sort of resemblance to the lines measured on the ground, for the purpose of showing, at any period of the work, their directions and how they are to be connected; and also of eventually assisting in laying down the diagram and content plat. On these rough diagrams are written the distinctive letters by which each line is marked in the field-book, and also its length, and the distances between points marked upon it, from which other measurements branch off to connect the interior portions of the district surveyed. (130) Calculations. The calculation of one of the figures,, is given below in detail. It is composed of the triangle DPQ, with offsets along the sides PQ; and of the triangle DWX, with offsets CHAP. IV.] Diagonals, Tie-lines and Offsets. 89 along the sides PW and WX. From the content thus obtained must be subtracted the offsets on PQ, belonging to the figure A, and those on WX belonging to the figure S. When the offsets are triangles, (right angled, of course), the base and perpendicular are put down as two sides; when they are trapezoids, the two parallel sides and the distance between them occupy the columns of" sides." TRIANGLE 1ST 2 D CONTENT DIVISION. OR IN ON TRAPEZOID SIDE. SIDE. SIDE. IN TRAPEZOID. CHAINS. DPQ 11680 1698 1078 86.2650 ( - 52 250.6500) PQ 52 30 80.3280 30- 216.3240 1.8020 Additives. DWX 13701442 770 51.8339 PW 30 310.4650 ~( -- 56 114.3192) WX 56 36 104.4784 36 90.1620.9596 Total Additives, 140.8255 50 1741.4350 PQ ( 50 30 292 1.1680. 1.7020 M2~ ( -- 52 142.3692) Subtractives. WX 52 64 232 1.3456 ( 64- 88.2816) 1.9964 Total Subtractives, 3,6984 Total Additives, 140.8255 Difference, 137.1271 _~~~~~~.. j, 90 CHAIN SURVEIYING. [PART II. The other figures, comprised within the large triangle, are recorded and calculated in a similar manner. An abridged register of the results is given below. DIFFERENCE IN DIVISION. ADDITIVES. SUBTRACTIVE. SQUARE and off sets. _SQUARE CHAINS. ( DNS DWX ) and offsets. NUV 140.4893 and offsets. DNO DPQ 1001882 IL { and offsets. and offsets. _}100__ _ a ANO anRM 103.9778 and offsets. and offsets. HTN NUV H NRM Offsets. 81.6307 and offsets. CNS HTN 09.504 and offsets. and offsets. 109.5064 DPQ DWX Offsets. 137.1271 and offsets. Total, - - - 672.9195 The accuracy of the preceding calculations of the separate figures must now be tested by comparing the sum of their areas with that of the large triangle ACD, which comprises them all. Their area must previously be increased by the offsets on the lines CS and CH, which had been deducted from I, and which amount respectively to 3.5270 and 2.8690. The total areas will then equal 679.3155 square chains. That of the triangle ACD is 679.5032; a difference of less than a fifth of a square chain, or a fiftieth of an acre; or about oneAfortieth of one per cent. on the total area. (131) The six lines, In most cases, great or small, six fundamental lines will need to be measured; viz. four approximate boundary lines, forming a quadrilateral, and its two diagonals. Small triangles, to determine prominent points, can be formed within and without these main lines by the FIRST METHOD, Art. (5), and the lesser irregularities can be determined by offsets, AP. vv.] M Diagonailst Militesv a ft Offsets. 81 Fig. 78. A, —~ 4,' V, II % Di Thus, in the above figure, two straight lines AB and CD are measured through the entire length and breadth of the farm, or township, which is to be surveyed. The connecting lines AC, OB, BD and DA are also measured, uniting the extremities of the first two lines. The last four lines thus form a quadrilateral, which is divided into two triangles by one of the first measured lines, while the second serves as a proof-line. The distance from the intersection of the two diagonals to the extremities of each, being measured Qn the ground and on the plat, affords an additional test. Other points of the district surveyed (as E, G, K., &c., in the figure,) are determined by measuring the distances from them to known points (as M, N, P, R, &c., in the figure) situated on some of the six fundamental lines, thus forming the triangles T, T. The intersection; 0 of the main diagonals, and also the intersections'of the various minor lines with. the mai, lines and with each other, should all be carefully noted, as additional checks when the work comes to be platted. 92 CHAIN SURVEYING. [PART it The larger figures are determined first, and the smaller ones based upon them, in accordance with this important principle in all surveying operations, always to work from the whole to the parts, and from greater to less. The unavoidable inaccuracies are thus subdivided and diminished. The opposite course would accumulate and magnify them. These additional lines, which form secondary triangles, should be so chosen and ranged as to pass through and near as many objects as possible, in order to require as few and as short offsets as the position of the lines will permit; the smaller irregularities being determined by offsets as usual. It is better to measure too many lines than too few, and to establish unnecessary "false stations," rather than not to have enough. (132) Exceptional cases. The preceding arrangement of lines, though in most cases the best, may sometimes be varied with advantage. Unless the farm surveyed be of a shape nearly as broad as long, the two diagonals will cross each other obliquely, instead of nearly at right angles, as is desirable. Fig. 79. When the farm is much B longer than it is wide, two..., (A systems, of six lines each, H e;...- ^ Y - \\. may be used with much (i.... advantage, as in Fig. 79.'"'''f Several such may beco- /..I. bined when necessary. klg. 80. In a case-like that in Fig. 80, five lines will be better \,than six, and will tie one an-. other together, their points of. l intersection being carefully noted. CHAP. IV.] Diagonals, Tie-lines and Offsets. 9S Fig. 81. In the farm represented in Fig. 81, the system of lines / there shown is the best, and. ^_ / i' they will also tie one another. = >: B (133) Much difficulty will often be found in ranging and measuring the long lines required by this method in extensive surveys. Various contrivances for overcoming the obstacles which may be met with, will be explained in the following chapter. It will often be convenient to measure the minor lines along roads, lanes, paths, &c., although they may not lie in the most desirable directions. Steeples, chimneys, remarkable trees, and other objects of that character, may often be sighted to, and the line measured towards them, with much saving of time and labor. The point where the measured lines cross one another should always be noted, and they will thus form a very complete series of tie-lines.* A view of the district to be surveyed, taken from some elevated position, will be of much assistance in planning the general direction of the lines to be measured. (134) Inaccessible Areas. Fig. 82. A combination of offsets and,,\ —. --... tie-lines supplies an easy me-. —.. thod of surveying an inacces- sible area, such as a pond, l swamp, forest, block of houses, / ".J| ):i &c., as appelrs from the figure; in which external bound- ing lines are taken at will and * To find the exact point of intersection of these lines, which are only visual lines," (explained in Art. (19),) three persons are necessary: one stands at some point of one of the lines and sights to some other point on it; a second does the same on the second line; by signs they direct, to right or left, the movements of a third person, who holds a rod, till he is placed in both of the lines and thus at their intersection, on the principle of Art. (11). j4' GEIJIN 8U SURVEVING. [PARET I.measured, and tied by " tie-lines"' neasured between these lines, prolonged when necessary, as in Art. (101), while offsets from them determine the irregularities of the actual boundaries of the pond, &c. These offsets are insets, and their content is, of course, to be subtracted from the content of the principal figure. Even a circular field might thus be approximately measured from the outside. If the shape of the field admits of Fig. 83. it, it will be preferable to measure four lines about the field in such,' / o directions as to enclose it in a rectr.: g angle, and to measure offsets from the:. sides of this to the angles of the: field. (135) When one of the lines with which Fig. 81 an inaccessible field is surrounded, as in the last two figures, cuts a corner of the --- field, as in Fig. 84, the triangle ABC is c to be deducted from the content of the enclosing figure, and the triangle CDE added to it. The triangle DEF is also D. —i — to be added, and the triangle FGH de- ducted. To do this directly, it would be necessary to find the points of intersection' C and F. But this may be difficult, and can be dispensed with by obtaining the difference of each pair of triangles. The ---- difference of ABC and CDE will be obtained at once by multiplying the differ-..-i...~.... ence of the offsets AB and DE by half of BE; and the difference of DEF and FGH by multiplying the difference of DE and GH by half of EG.* For, making the triangle Dmn = ABC, then mnEC - En X j (mn + CE) = (DE - AB) X j EB; and so with the other pair of triangles. CHAP. iv.] Diagonals, Tie-lines and Perpendiculars. 95 (136) Roads. A winding Road may also be surveyed thus, as is shown in Fig. 85; straight lines being measured in the road, Fig. 85. *,,/ their changes in direction determined by tie-lines, tying one line to the preceding one prolonged, as explained in Chapter II, of this Part, and points in the road-fences, on each side of these straight lines, being determined by offsets. A River may also be supposed to be represented by the above winding lines; and the lower set of lines, tied to one another as before, and with offsets from them to the water's edge, will be sufficient for making an accurate survey of one side of the river. (137) Towns. A town could be surveyed and mapped in the same manner, by measuring straight lines through all the streets, determining their angles by tie-lines, and taking offsets from them to the blocks of houses. 96 CHAIN SURVEYING. [PART II. CHAPTER V. OBSTACLES TO MEASUREMENT IN CHAIN SURVEYING. (138) In the practice of the various methods of surveying which have been explained, the hills and valleys which are to be crossed, the sheets of water which are to be passed over, the woods and houses which are to be gone through-all these form obstacles to the measurement of the necessary lines which are to join certain points, or to be prolonged in the same direction. Many special precautions and contrivances are, therefore, rendered necessary; and the best methods to be employed, when the chain alone is to be used, will be given in the present chapter. (139) The methods now to be given for overcoming the various obstacles met with in practice, constitute a LAND-GEOMETRY. Its problems are performed on the ground instead of on paper: its compasses are a chain fixed at one end and free to swing around with the other; its scale is the chain itself; and its ruler is the same chain stretched tight. Its advantages are that its single instrument, (or a substitute for it, such as a tape, a rope, &c.) can be found anywhere; and its only auxiliaries are equally easy to obtain, being a few straight and slender rods, and a plumb-line, for which a pebble suspended by a thread is a sufficient substitute. Many of these problems require the employment of perpendicular and parallel lines. For this reason we will commence with this class of Problems. The Demonstrations of these problems will be placed in an Ap pendix to this volume, which will be the most convenient arrangement for the two great classes of students of surveying; those who wish merely the practice without the principles, and those who wish to secure both. The elegant " Theory of Transversals" will be an important element in some of these demonstrations. All of them will constitute excellent exercises for students. CHAP. V.] Obstacles to Measurement. 97 PROBLEMS ON PERPENDICULARS.* Problem 1. To erect a perpendicular at any point of a line. (140) First Method. Let A be the Fig. 86. point at which a perpendicular to the line is to be set out. Measure off equal distances AB, AC, on each side of the point. Take _B \C a portion of the chain not quite 1~ times as long as AB or AC, fix one end of this at B, and describe an arc with the other end. Do the same from C. The intersection of these arcs will fix a point D. AD will be the perpendicular required. Repeat the operation on the other side of the line. If that is impossible, repeat it on the side with a different length of chain. (141) Second Method. Measure off as be- Fig. 87 fore, equal distances AB, AC, but each about only one-third of the chain. Fasten the ends of the chain with two pins at B and C. Stretch B it out on one side of the line and put a pin at the middle of it, D. Do the same on the other side of the line, and set a pin at E. Then is DE a perpendicular to BC. If it is impossible to perform the operation on both sides of the line, repeat it on the same side with a different length of chain, as shown by the lines BF and CF in the figure, so as to get a second point. (142) Other Methods. All the methods to be given for the next problem may be applied to this. * Many of these methods would seldom be required in practice, but cases sometimes occur, as every surveyor of much experience in Field-work has found to his serious inconvenience, in which some peculiarity of the local circumstances forbids any of the usual methods being applied. In such cases the collection here given will be found of great value. In all the figures, the given and measured lines are drawn with fine full lines, the visual lines, or lines of sight, with broken lines, and the lines of the result with heavy full lines. The points which are centres around which the chain is swung, are enclosed in circles. The alphabetical order of the letters attached to the points shows in what order they are taken. 7 98 CHAIN SURVEYING. [PART II. Problem 2. To erect a perpendicular to a line at a given point, when the point is at or near the end of the line. (143) First Method. Measure Fig. 88. 50 40 links along the line. Let one assistant hold one end of the chain at that point; let a second hold the 20 link mark which is nearest the other end, at the given point A, and let a 4o A third take the 50 link mark, and ~ tighten the chain, drawing equally on both portions of it. Then will the 50 link mark be in the perpendicular desired. Repeat the operation on the other side of the line so as to test the work. The above numbers are the most easily remembered, but the longer the lines measured the better; and nearly the whole chain may be used, thus: Fix down the 36th link from one end at A, and the 4th link from the same end on the line at B. Fix the other end of the chain also at B. Take the 40th link mark from this last end, and draw the chain tight, and this mark will be in the perpendicular desired. The sides of the triangle formed by the chain will be 24, 32 and 40. (144) Otherwise: using a 50 feet Fig. 89. tape, hold the 16 feet mark at A; hold the 48 feet mark and the ringend of the tape together on the line; take the 28 feet mark of the tape, and draw it tight; then will the 28 feet 1X8 16 A mark be in the perpendicular desired. 0~ 1 (145) Second Method. Hold one end Fig. 90. of the chain at A and fix the other end at a D point B, taken at will. Swing the chain around B as a centre, till it again meets the line at C. Then carry the same end around B (the other end remaining at B) till it comes in the line of CB at D. AD is the perpendicular required. A CHAP. v.] Obstacles to Measurement. 99 (146) Third Method. Let A be the given Fig. 91. point. Choose any point B. Measure BA. D Set off, on the given line, AC = AB. On CB 2 AC2 B produced set off from C, a distance = CB. This will fix the point D, and AD will be the perpendicular required. c — / (147) Fourth Method. From the Fig. 92. given point A set off on the given line p p any distance AB. From B, in any convenient direction, set off BC = AB. D Then on the given line, set off AD = AC. On CB prolonged, set off CE = C AD. Join DE; and on DE, from D, set off DF = 2 AB. Then will the line AF be perpendicular to the line AD at the point A. Problem 3. To erect a perpendicular to an inaccessible line, at a given point of it. (148) First Method. Get points in the direction of the inaccessible line prolonged, and from them set out a parallel to the line, by methods which are given in Art. (165), &c. Find by trial the point in which a perpendicular to this second line (and therefore to the first line) will pass through the required point. (149) Second Method. If the line is not only inaccessible, but cannot have its direction prolonged, the desired perpendicular can be obtained only by a complicated trigonometrical operation. Problem 4. To let fall a perpendicular from a given point to a given line. (150) First Method. Let P be Fig.93. the given point, and AB the given line. Measure some distance, a chain or less, from C to P, and then fix one end of the chain at P, and swing it A- -, around till the same distance meets "-. 100 CHAIN SURVEYING. [PART rt. the line at some point D. The middle point E of the distance CD will be the required point, at which the perpendicular from P would meet the line. (151) Second Method. Stretch a chain, or a portion of it, from the given point P, to some point, as A, of the Fig. 94. given line. Hold the end of the distance at A, and swing round the other end of the chain from P, so as to set off the same distance along the given line from A to some point B. Mea- ~ sure BP. Then will the distance BC from B to the ft of the BP2 desired perpendicular = 2-AB (152) Other Methods. All the methods given in the next problem can be applied to this one. Problem 5. To let fall a perpendicular to a line, from a point nearly opposite to the end of the line. (153) First Method. Stretch a chain from the given point P, to some point, as A, of the given line. Fix to Fig. 95. the ground the middle point B of the chain AP, and swing around the end which was at P, or at A, till it meets the given line in a B point C, which will be the foot of the re- quired perpendicular..A,, /. (154) Second Method. Takeanypoint, Fig. 96 as A, on the given line. Measure a dis- tance AB. Let the end of this distance /" - on the chain be held at B, and swing around the end of the chain, till it comes in the A B D line of AP at some point C, thus making BC = AB. Measure AC and AP. Then the distance AD, from A to the foot of the AP x A-'C A perpendicular required= o A CHAP. v.] Obstacles to Measurement. 101 (155) Third Method. At any convenient Fig. 97. p point, as A, of the given line, erect a perpen- B dicular, of any convenient length, as AB, and mark a point C on the given line, in the line of P and B. Measure CA, CB and CP. A. -- D Then the distance from C to the foot of the CAxCP perpendicular, i. e. CD CB Problem 6. To let fall a perpendicular to a line, from an inaccessible point. (156) First Method. Let P be the given Fig. 98. point. At any point A, on the given line, set out a perpendicular AB of any convenient length. Prolong it on the other side of the line the same distance. Marjk on the given line a. / point D in the line of PB; and a point E in D A, A. /~. the line of PC. Mark the point F at the in-.X \ tersection of DC and BE prolonged. The line c FP is the line required, being perpendicular to the given line at the point G. (157) Second Method. Let A and B Fig. 99 be two points of the given line. From A i let fall a perpendicular, AC, to the visual line BP; and from B let fall a perpendicular, BD, to the visual line AP. Find the point at which these perpendiculars / intersect, as at E (seeArt.(133)), and the A B line PE, prolonged to F, will give the perpendicular required. Problem 7. To letfall a perpendicular from a given point to an inaccessible line. 102 CHAIN SURVEYING. [PART I. (158) First Method. Let P be Fig. 100. the given point and AB the given A- _ line. By the preceding problem, let fall perpendiculars from A to BP, at' C C; and from B to AP, at D; the line PE, passing from the given point P to the intersection of these perpendiculars, is the desired perpendicular to the inaccessible line AB. This method will apply when only two points of the line are visible. (159) Second Method. Through the given point, set out, by the methods of Art. (165), &c., a line parallel to the inaccessible line. At the given point erect a perpendicular to the parallel line, and it will be the required perpendicular to the inaccessible line. PROBLEMS ON PARALLELS. Problem 1. To run a line, from a gzven point, parallel to a given line. (160) First Method. Let fall a perpendicular from the point to the line. At another point of the line, as far off as possible, erect a perpendicular, equal in length to the one just let fall. The line joining the end of this line to the given point will be the parallel required. (161) Second Method. LetABbe Fig.101. the given line, and P the given point. A --- D Take any point, as C, on the given line, and from it set off equal distances, as long as possible, CD on the given line, and CE, on the line CP. Measure DE. From P set off PF = CE; and from F, with a distance = DE, and from P, with a distance = CD, describe arcs intersecting in G. PG will be the parallel required. If it is more convenient, PC may be prolonged, and -the equal triangle, CDE, be formed on the other side of the line AB. cMor. v.] Obstacles to Measurement. 103 (162) Third Method. Measure from Fig' 102. P to any point, as C, of the given line, and A-_- put a mark at the middle point, D, of that line. From any point, as E, of the given line, measure a line to the point D, and continue it till DF = DE. Then will the line P PF be parallel to AB. (163) Fourth Method. Measure from Flg. 103. P to any point C, of the given line, and D continue the measurement till CD = CP. From D measure to any point E of the given line, and continue the measurement A till EF = ED. Then will the line PF be parallel to AB. If more convenient, P F CD may be made one-half, or any other fraction, of CP,, and EF be then made twice, &c., DE. (164) Fifth Method. From any Fig. 14. point, as C, of the line, set off equal A —, B distances along the line, to D and E.' Take a point F, in the line of PD. Stake out the lines FC and FE, and also the line EP, crossing the line CF E in the point G. Lastly, prolong the line DG, till it meets the line EF in the point H. PH is the parallel required. Problem 2. To run a line from a given point parallel to an inaccessible line. (165) First Method. Let AB Fig. l05. be the given line, and P the given A point. Set a stake at C, in the line of PA, and another at any conven-''' \ ient point, D. Through P, set out, \ - D by the preceding problem, a parallel to DA, and set a stake at the point, as E, where this parallel intersects DC prolonged. Through E 104 CHAIN SURVEYING. [PART I. set out a parallel to BD, and set a stake at the point F, where this parallel intersects BC prolonged. PF is the parallel required (166) Second Method. Set a stake Fig. 106. at any point, C, in the line of AP, and A>:. / another at any convenient place, as at D. - Through P set out a parallel to AD, P- ^' intersecting CD in E. Through E set \. -\ -D out a parallel to DB, intersecting CB in'-" E F. The line PF will be the parallel re- C quired. (167) Alinement and Measurement. We are now prepared, having secured a variety of methods for setting out Perpendiculars and Parallels in every probable case, to take up the general subject of overcoming Obstacles to Measurement. Before a line can be measured, its direction must be determined. This operation is called Ranging the line; or Alining it; or Boning it.* The word Alinementt will be found very convenient for expressing the direction of a line on the'ground, whether between two points, or in their direction prolonged. This branch of our subject naturally divides itself into two parts, the first of which is preliminary to the second; viz: I. Of Obstacles to Alinement; or how to establish the direction of a line in any situation. II. Of Obstacles to Measurement; or how to find the length of a line whicA, cannot be actually measured. 1. OBSTACLES TO ALINEMENT. (168) All the cases which can occur under this head, may be reduced to two; viz: A. To find points in a line beyond the given points, i. e. to prolong the line. B. To find points in a line between two given points of it, i. e. to interpolate points in the line. * This word, like many others used in Engineering, is derived from a French word, Borner, to mark out, or limit; indicating that the Normans introduced the art of Surveying into England. t Slightly modified from the French Alignement. CHAP. v.1 Obstacles to Measurement. 106 A. TO PROLONG A LINE (169) By ranging with rods. When two points in a line are given, and it is desired to Fig. 107. prolong the line by ranging.I-'2Kg ~' it out with rods, three per- A sons are required, each furnished with a straight slender rod, and with a plumb-line, or other means of keeping their rods vertical. One holds his rod at one of the given points, A, in the figure, and another at B. A third, C, goes forward as far as he can without losing sight of the first two rods, and then, looking back, puts himself " in line" with A and B, i. e. so that when his eye is placed at C, the rod at B hides or covers the rod at A. This he can do most accurately by holding a plumb-line before his eye, so that it shall cover the first two rods. The lower end of the plumb-bob will then indicate the point where the third rod should be placed; and so with the rest. The first man, at A, is then signalled, and comes forward, passes both the others, and puts himself at D, " in line" with C and B. The man at B, then goes on to E, and " lines" himself with D and C: and so they proceed, in this "hand over hand" operation, as far as is desired. Stakes are driven at each point in the line, as soon as it is determined. (170) The rods should be perfectly straight, either cylindrical or polygonal, and as slender as they can be without bending. They should be painted in alternate bands of red and white, each a foot or link, in length. Their lower ends should be pointed with iron, and a projecting bolt of iron will enable them to be pressed down by the foot into the earth, so that they can stand alone. When this is done, one man can range out a line. A rod can be set perfectly vertical, by holding a plumb-line before the eye at some distance from the rod, and adjusting the rod so that the plumb-line covers it from top to bottom; and then repeating the operation in a direction at right angles to the former. A stone dropped from top to bottom of the rods will approximately attain the same end. When the lines to be ranged are long, and great accuracy is required, the rods may have attached to them plates of tin with open 106 CHAIN SURVEYING. [PART I. ings cut out of them, and black horse-hairs stretched from Fig. 108. top to bottom of the openings. A small telescope must then be used for ranging these hairs in line. In a hasty survey, straight twigs, with their tops split to receive a paper folded as in tle figure, may be used. (171) By perpendiculars. Fi 109. The straight line, AB in the - LL —-- -T. —figure, is supposed to be stop- C D E F ped by a tree, a house, or other obstacle, and it is desired to prolong the line beyond this obstacle. From any two points, as A and B, of the line, set off (by some of the methods which have been given) equal perpendiculars, AC and BD, long enough to pass the obstacle. Prolong this line beyond the obstacle, and from any two points in it, as E and F, measure the perpendiculars EG and FH, equal to the first two, but in a contrary direction. Then will G and H be two points in the line AB prolonged, which can be continued by the method of the last article. The points A and B should be taken as far apart as possible, as should also the points E and F. Three or more perpendiculars, on each side of the obstacle, may be set off, in order to increase the accuracy of the operation. The same thing may also be done on the other side of the line, as another confirmation, or test, of the accuracy of the prolonged line. (172) By equilateral triangles. Fig. 110 The obstacles, noticed in the last arti- A B K cle, may also be overcome by means of three equilateral triangles, formed by the chain. Fix one end of the chain, C and also the end of the first link from / its other end, at B; fix the end of the 33d link at A; take hold of the 66th D link, and draw the chain tight, pulling equally on each part, and put a pin at the point thus found, C, in the figure. An equilateral triangle will thus be formed, each side being 33 links. Prolong the line AC, past the obstacle, to some point, as D. Make another CHAP. V.] Obstacles to Measurement. 107 equilateral triangle, DEF, as before, and thus fix the point F. Prolong DF, to a length equal to that of AD, and thus fix a point G. At G form a third equilateral triangle GHK, and thus fix a point K. Then will KG give the direction of AB prolonged. (173) By symmetrical triangles. Let AB be the line to be prolonged. Take any conve- Fig. 111. nient point, as C. Range A B - 2?' P out the line AC, to a point., \. A', such that CA'= CA..\ " Range out CB, so that CB' = CB. Range backwards A'B', to some point D, such that DC prolonged will pass B the obstacle. Find, by ranging, the intersection, at E, of DB and AC. From C, measure, on CA', the distance CE'= CE. Then range out DC and B'E' to their intersection in P, which will be a required point in the direction of AB prolonged. The symmetrical points are marked by corresponding letters. Several other points should be obtained in the same manner. In this, as in all similar operations, very acute intersections should be avoided as far as possible. (174) By transversals. Let AB be Fig. 112. the given line. Take any two points C P and D, such that the line CD will pass the obstacle. Take another point, E, in the intersection of CA and DB. E Measure AE, AC, CD, BD and BE. Then the distance from D to P, a point in the required prolongation, will be DP CDxBDxAE A BE xAC-BD x AE Other points in the prolongation may be obtained in the same manner, by merely moving the single point C, in the C line of EA; in which case the new distances CA and CD will alone require to be measured. 108 CHAIN SURVEYING. [PART I. CD xBD If AE be made equal to AC, then is DP = BE-BD CDx AE If BE be made equal to BD, then is DP =C Ac-AE The minus sign in the denominators must be understood as only meaning that the difference of the two terms is to be taken, without regard to which is the greater. (175) By harmonic conjugates. lig. 113. Let AB be the given line. Set a. _^ P stake at any point C. Set stakes at'::-.s;- I,,,r points, D, on the line CA, and at \. ^*^^.'!''/ E, on the line CB; these points, \ /,>tE / D and E, being so chosen that the. \ -.' i line DE will pass beyond the obsta- cle. Set a fourth stake, F, at the, intersection of the lines AE and \ DB. Set a fifth stake, G, any- c where in the line CF; a sixth stake, H, at the intersection of CB and DG prolonged; and a seventh, K, at the intersection of CA and EG prolonged. Finally, range out the lines DE and KH, and their intersection at P, will be in the line AB prolonged. (176) By the complete quadrilateral. Let AB be the given line. Take any conven- Fig. 114 ient point C; measure F C. from it to B, and onward, /' in the same line prolong- \ ed, an equal distance to D. Take any other convenient I) point, E, such that CE and H DE produced will clear the obstacle. Measure from E to A, and onward, an equal distance, to F. Range out the lines FC and DE to their intersection in G. Range out FD and CE to intersect in H. Measure GH. Its middle point, P, is the required point in the line of AB prolonged. The unavoidable acute intersections in this construction are objectionable. CHAP. v.] Obstacles to Measurement. 109 B. TO INTERPOLATE POINTS IN A LINE. (177) The most distant given point of the line must be made as conspicuous as possible, by any efficient means, such as placing there a staff, bearing a flag; red and white, if seen against woods, or other dark back-ground; and red and green, if seen against the sky. A convenient and portable signal is shown in the figure. Fig. 115. Front View. Side View. Back View. C.3C \ sXI ) S I The figure represents a disc of tin, about six inches in diameter, painted white and hinged in the middle, to make it more portable. It is kept open by the bar, B, being turned into the catch, C. A screw, S, holds the disc in a slit in the top of the pole. Another contrivance is a strip of tin, which has its ends bent horizontally in contrary directions. As the wind will take strongest hold of the side which is concave towards it, the bent strip will continually revolve, and thus be very conspicuous. Its upper half should be painted red and its lower half white. A bright tin cone set on the staff, can be seen at a great distance when the sun is shining. 178) Ranging to a point, thus made conspicuous, is very simple when the ground is level. The surveyor places his eye at the nearest end of the line, or stands a little behind a rod placed on it, and by signs moves an assistant, holding a rod at some point as nearly in the desired line as he can guess, to the right or left, till his rod appears to cover the distant point. 110 CHAIN SURVEYING. [PART II (179) Across a valley. When a valley, or low spot, intervenes between the two ends Fig. 116. of the line, A and Z in the figure, a rod held in the low place, as at B, would seldom be high enough to _ be seen, from A, to cover — = — - -: the distant rod at Z. In such a case, the surveyor at A should hold up a plumb-line over the point, at arm's length, and place his eye so that the plumb-line covers the rod at Z. He should then direct the rod held at B to be moved till it too is covered by the plumb-line. The point B is then said to be "in line" between A and Z. In geometrical language, B has now been placed in the vertical plane determined by the vertical plumb-line and the point Z. Any number of intermediate points can thus be " interpolated," or placed in line between A and Z. (180) Over a hill. When a hill rises between two points and prevents one being seen from the other, as in the figure, (the upper Fig. 117. A S Z of which shows the hill in "Elevation," and the lower part in "Plan"), two observers, B and C, each holding a rod, may place themselves on the ridge, in the line between the two points, as nearly as they can guess, and so that each can at once see the other and the point beyond him. B looks to Z, and by signals puts C CHAP. v.] Obstacles to Measurement. 111 "in line." C then looks to A, and puts B in line at B'. B repeats his operation from B', putting C at C', and is then himself moved to B", and so they alternately " line" each other, continually approximating to the straight line between A and Z, till they at last find themselves both exactly in it, at B"' and C"'. (181) A single person may put himself in line between two points, on the same principle, by laying a straight stick on some support, going to each end of it in turn, and making it point successively to each end of the line. The " Surveyor's Cross," Art. (104), is convenient for this purpose, when set up between the two given points, and moved again and again, until, by repeated trials, one of its slits sights to the given points when looked through in either direction. (182) On water. A simple instru- Fig. 118. ment for the same object, is represented X in the figure. AB and CD are two tubes, about 1 inches in diameter, con- M A nected by a smaller tube EF. A piece ~ of looking-glass, GiH, is placed in the lower part of the tube AB, and another, - - KL, in the tube CD. The planes of < the two mirrors are at right angles to I each other. The eye is placed at A, and the tube AB is directed to any distant i object, as X, and any other object be- hind the observer, as Z, will be seen, apparently under the first object in the mirror GH, by reflection from the mirror KL, when the observer has succeeded in getting in line between the two objects. M, N, are screws by which the mirror KL may be adjusted. The distance between the two tubes will cause a small parallax, which will, however, be insensible except when the two objects are near together. 112 CHAIN SURVEYING. [PART II. (183) Through a wood. When a wood intervenes between any two given Fig. 119. points, pre- Z venting one.B'-, Cr;' from being. —-- l S- V ~ -,.~. seen from the Z' other, as in the figure, in which A and Z are the given points, proceed thus. Hold a rod at some point B' as nearly in the desired line from A as can be guessed at, and as far from A as possible. To approximate to the proper direction, an assistant may be sent to the other end of the line, and his shouts will indicate the direction; or a gun may be fired there; or, if very distant, a rocket may be sent up after dark. Then range out the " random line" AB', by the method given in Art. (169), noting also the distance from A to each point found, till you arrive at a point Z', opposite to the point Z, i. e. at that point of the line from which a perpendicular there erected would strike the point Z. Measure Z'Z. Then move each of the stakes, perpendicularly from the line AZ', a distance proportional to their distances from A. Thus, if AZ' be 1000 links, and Z'Z be 10 links, then a stake B', 200 links from A, should be moved 2 links to a point B, which will be in the desired straight line AZ; if C' be 400 links from A, it should be moved 4 links to C, and so with the rest. The line should then be cleared, and the accuracy of the position, of these stakes tested by ranging from A to Z. (184) To an invisible intersection. Let AB and CD be two lines, which, if prolong Fig. 120. Fig. lB,0. ed, would meet in a 4'',-,^,;point Z, invisible from -' either of them; and let \, E,. P be a point, from which,/ f - a line is required to be / // set out, tending to this' i %: invisible intersection. C Set stakes at the five given points, A, B, C, D, P. Set a sixth stake at E, in the alinements of AD and CP; and a seventh stake CHAP. v.] Obstacles to Measurement. 11s at F, in the alinements of BC and AP. Then set an eighth stake at G, in the alinements of BE and DF. PG will be the required line. Otherwise; Through P range out a parallel to the line BD. Note the points where this parallel meets AB and CD, and call these points Q and R. Then the distance from B, on the line BD, to a point which shall be in the required line running from P to the *BDxQP invisible point, will be = xQP QR II. OBSTACLES TO MEASUREMENT. (185) The cases, in which the direct measurement of a line is prevented by various obstacles, may be reduced to three. A. When both ends of the line are accessible. B. When one end of it is inaccessible. C. When both ends of it are inaccessible. A. WHEN BOTH ENDS OF THE LINE ARE ACCESSIBLE, (186) By perpendiculars. On Fi. 21. reaching the obstacle, as at A in A D the figure, set off a perpendicular, AB; turn a second right angle at B, C and measure past the obstacle; turn a third right angle at C; and measure to the original line at D. Then will the measured distance, BC, be equal to the desired distance, AD. If the direction of the line is also unknown, it will be most easily obtained by the additional perpendiculars shown in Fig. 109, of Art. (171). Fig. 121'. (187) By equilateral triangles. A, K The method given in Art. (172), for determining the direction of a line through an obstacle, will also give its c length; for in Fig. 121' (Fig. 110 re- peated) the desired distance AGis equal to the measured distances AD, or DG. 8 114 CHAIN SURVEYING. [PART II (188) By symmetrical triangles. Fig. 122. Let AB be the distance required. Measure from A obliquely to some B point C, past the obstacle. Measure onward, in the same line, till CD is as long as AC. Place stakes at C and D. From B measure to C, and from C measure onward, in E D the same line, till CE is equal to CB. Measure ED, and it will be equal to AB, the distance required. If more convenient, make CD and CE equal, respectively, to half of AC and CB; then will AB be equal to twice DE. (189) By transversals. Let Fig. 123. AB be the required distance. Set C A a stake, C, in the line prolonged; set another stake, D, so that C and B can be seen from it; and a third stake, E, in the line of BD prolonged, and at a distance from D equal to the distance from D to B. E Set a foirth stake, F, at the intersection of EA and CD. Measure AC AC, AF and FE. Then is AB = A (FE — AF). Fig. 124. (190) In a Town. Cases may occur, \ in the streets of a compactly built town, in which it is impossible to measure along any other lines than those of the streets. The figure represents such a case, in which is required the distance, AB, be tween points situated on two streets which meet at the point C, and between which runs a cross-street, DE. In this case measure AC, CE, CD, DE and CB. Then is the required distance CHAP. v.] Obstacles to Measurement. 115 AB = (AC -BC)2 + [DE2-(E - CD)2] ACBC CDxCE As this expression is somewhat complicated, an example will be given: Let AC = 100, CE 40, CD =30, DE = 21, and CB 80; then will AB = 51.7. B. WHEN ONE END OF THE LINE IS INACCESSIBLE. (191) By perpendiculars. This principle Fig. 125. may be applied in a variety of ways. In Fig. B 125, let AB be the required distance. At the point A, set off AC, perpendicular to AB, and of any convenient length. At C, set off a perpen- C dicular to CB, and continue it to a point, D, in the line of A and B. Measure DA. Then is AC2 D AB-A (192) Otherwise: At the point A, in Fig. Fig. 126. 126, set off a perpendicular, AC. At C set off another perpendicular, CD. Find a point, E, in the line of AC, and BD. Measure AE and EC. Then is AB -- AE CD CE C u —— = If EC be made equal to AE, and D be set in the line of BE, and also in the perpendicular D from 0, then will CD be equal to AB. If EC - a AE, then CD= a AB. Fig. 127. (193) Otherwise: At A, in Fig. 127, mea- B sure a perpendicular, AC, to the line AB; and at any point, as D, in this line, set off a perpendicular to DB, and continue it to a point E, in the line of CB. Measure DE and also DA. A Te i A AC x AD Then is AB = x AD DE -AC' 116 CHAIN SURVEYING. [PART Ir Fig. 128. (194) By parallels. From A measure B AC, in any convenient direction. From a /\ point D, in the line of BC, measure a line parallel to CA, to a point E, in the line of AB. Measure also AE.__ C A AC x AE Then is AB = ADC AE_ (195) By a parallelogram. Set a stake, C, Fig. 129 in the line of A and B, and set another stake, D, wherever convenient. With a distance equal to CD, describe from A, an arc on the ground; and, with a distance equal to AC, describe another arc from D, intersecting the first arc in E. Or,'F A take AC and CD, so that together they make one chain; fix the ends of the chain at A and D; take hold of the chain at such a link, that one part of it equals AC, and the other CD, and draw it tight to fix the point E. Set a stake at F, in the intersection of AE and DB. Measure AF and AC x AF AC x CD EF. Then is AB - --; or, CB EF EF (196) By symmetrical triangles. Fig.130. Let AB be the required distance. From / A measure a line, in any convenient di- 2 rection, as AC, and measure onward, in the same direction, till CD = AC. Take' A any point E in the line of A and B. D-4-C i Measure from E to C, and onward in theD, E same line, till CF- CE. Then find by trial a point G, which shall be at the same time in the line of C and B, and in G the line of D and F. Measure the distance from G to D, and it will be equal to the required distance from A to B. If more convenient, make CD = AC, and CF - - CE, as shown by the finely dotted lines in the figure. Then will DG = ~ AB. CHAP. v.] Obstacles to Measurement. 117 (197) Otherwise: Prolong BA to Fig. 131. some point C. Range out any convenient line CA', and measure CA' =_ CA. The triangle CA'B, is now to be reproduced in a symmetrical triangle,,/ -'/ situated on the accessible ground. i' / X — D For this object, take, on AC, some point ^: D, and measure CD' CD. Find the C point E, at the intersection of AD' and A'D. Find the point F, at the intersection of A'B and CE. Lastly, find the point B', at the intersection of AF and CA'. Then will A'B' = AB. The symmetrical points have corresponding letters affixed to them. (198) By transversals. Set a stake, C, Fig. 132. in the alinement of BA; a second, D, at any convenient point; a third, E, in the line CD; _ _ _ and a fourth, F, at the intersection of the alinements of DA and EB. Measure AC, { 1'\ 7 CE, ED, DF and FA. Then is \ AC x AF x DE D __ / AB-CE x DF -AF x DE' E C If the point E be taken in the middle of CD, (as it is in the figure) then AB = A x AF DF — AF AC x DE If the point F be taken in the middle of AD, then AB = AC DE CE —DE' The minus signs must be interpreted as in Art. (174). y (199) By harmonic division. Set Fig.133. stakes, C and D, on each side of A, and ~,s so that the three are in the same straight line. Set a third stake at any point, E, of the line AB. Set a fourth, F, at the F / — intersection of CB and DE; and a fifth, \ G, at the intersection of DB and CE. /'- Set a sixth stake, H, at the intersection / —'D AE x All of AB and FG. Measure AE and EH. Then is AB AE x A AE - EH' 118 CHAIN SURVEYING. [PART II. (200) To an inaccessible line. The Fig. 134. shortest distance, CD, from a given point, _ T C, to an inaccessible straight line AB, is required. From C let fall a perpendicular' to AB, by the method of Art. (158).\' / Then set a stake at any point, E, on the I line AC; set a second, F, at the inter- \ section of EB and CD; a third, G, at c the intersection of AF and CB; and a fourth, H, at the intersection of EG and CD. Measure CH and HF. Then is CHxCF CH+HF _ CHx CF CD H-HF; or, CD — CH CH For CD C CHl HF CHH-] r C 2-CH-CF Otherwise; When the inaccessible line is determined by the method of Art. (205) or (206), the distance from any point to it, can be at once measured to its symmetrical representative. (201) To an inaccessible intersection. When two lines (as AB, CD, in the figure) meet in a Fig. 135. river, a building, or any other inaccessible point, the distance ____ G from any point of either to their \A- intersection, DE, for example, d - O may be found thus. From any - -\ point B, on one line, measure C_/ _ BD, and continue it, till DF = DB. From any other point, G, of the former line, measure GD, and continue the line till DH = GD. Continue HF to meet DC in some point K. Measure KD. KD will be equal to the desired distance DE. BE can be found by measuring FK, which is equal to it. If DF and DH, be made respectively equal to one-half, or onethird, &c., of DB and DG, then will KD and KF be respectively equal to one-half or one-third, &c., of DE and BE. CHAP. v.] Obstacles to Measurement. 119 C. WHEN BOTH ENDS OF THE LINE ARE INACCESSIBLE, (202) By similar triangles. Let AB Fi,. 136. be the inaccessible distance. Set a stake at any convenient point C, and find the distan- V.. ces CA and CB, by any of the methods just / given. Set a second stake at any point, D, on the line CA. Measure a distance, equal to CB x, from C, on the line CB, to some point E. Measure CA DE. Then is AB A x DE CD Fig. 137. If more convenient, measure CD in the A -,B contrary direction from the river, as in Fig. 137, instead of towards it, and in other respects proceed as before. E' —— D (203) By parallels. Let AB be the in- Fig. 138. accessible distance. From any point, as C, A.-B range out a parallel to AB, as in Art. (165), &c. Find the distance CA, by Art. (191), / &c. Set a stake at the point E, the inter- / section of CA and DB, and measure CE. D/ C Then is AB CD x (AC -CE) Then is AB CE CE (204) By a parallelogram. Set Fig. 139. a stake at any convenient point C. A _... Set stakes D and E, anywhere in the alinements CA and CB. With \ / D as a centre, and a length of the chain equal to CE, describe an arc; and with E as a centre, and a length of the chain equal to CD, describe another arc, intersecting the former one at F. A parallelogram, CDEF, will thus be formed. Set stakes at G and H, where the alinements DB and EA intersect the sides of this parallelogram. Measure CD, DF, GF, FH, 120 CHAIN SURVEYING. [PART II. and HG. The inaccessible distance AB CD x DF x GH FG x FHI If CD = CE, then AB = CD2 x GH FG x FH (205) By symmetrical triangles. Take any convenient point, as C. Set stakes at two other Fig. 140. points, D and D', in the same A. line, and at equal distances _ from C. Take a point E, in _- - the line of AD; measure from it to C, and onward till CE'. -CE. Take a point F in, — the line of BD; measure from\ ", it to C, and onward till CF' =,,' /i. CF. Range out the lines AC /, and E'D', and set a stake at,,/ N their intersection, A'. Range " ----' out the lines BC and F'D', and set a stake at their intersection, B'. Measure A'B'. It will be equal to the desired distance AB. (206) Otherwise: Take Fig, 141. any convenient point, as C, A = and set off equal distances Z g - _ — on each side of it, in the A-N,:!E line of CA, to Dand D'. Set \\, off the same distances from \ / C, in the line of CB, to E and, E'. Through C, set out a parallel to DE, or D'E', and /. —-D \ set stakes at the points F / \ and F' where this parallel/ \\ intersects AE' and BD'. B',A Range out the lines AD' and EF', and set a stake at their intersection A'. Range out the lines BE' and DF, and set a stake at their intersection B'. Measure A'B', and it will be equal to the desired distance AB. CHAP. v.] Obstacles to Measurement. 121 The easiest method of setting out the parallel in the above case, is to fix the middle of the chain at the point C, and its ends on the lines CD, CE'; then carry the middle of the chain from C towards F, and mark the point to which it reaches;. and repeat this on the other side of C, as shown by the finely dotted lines in the figure. INACCESSIBLE AREAS (207) Triangles. In the case of a triangular field, in which one side cannot be measured, or determined by any of the methods just given, the two accessible sides may be prolonged to their full length, and an equal symmetrical triangle formed, all of whose sides can be measured. Thus in Fig. 102, page 103, if CDE be the original triangle, of which the side EC is inaccessible, DFP will be equal to it. But if this also be impossible, por- Fig.142. tions of the sides may be measured, as AD, AE, B \-" -- in the figure in the margin, and also DE, and the area of this triangle found by any of the E methods which have been given. Then is the desired area of the triangle ABC = area of AB xAC A AD x AE (208) Quadrilaterals. In the case Fig. 143. of a four-sided field, whose sides cannot be measured, or prolonged, but whose __ diagonals can be measured, the area may be obtained thus. Measure the - kc diagonals AC and BD; and also the portions AE, EC, into which one of L them is divided by the other. Calcu- I late the area of the triangle BCE,by the preceding method, or any of those heretofore given. Then the area of the quadrilateral ABCD - area ofBCE x AC x BD BE x CE (209) Polygons. Methods for obtaining the areas of inaccessible fields of more than four sides, have been given in Arts. (101,) &c. PART III. COMPASS SURVEYING; OR By the Third Method. CHAPTER I. ANGULAR SURVEYING IN GENERAL. (210) Angular Surveying determines the relative positions of points, and therefore of lines, on the THIRD PRINCIPLE, as explained in Art. (7), which should now be referred to. (211) When the two lines which form an angle lie in the same horizontal or level plane, the angle is called a horizontal angle.* When these lines lie in a plane perpendicular to the former, the angle is called a vertical angle. When one of the lines is horizontal and the other line from the eye of the observer passes above the former, and in the same vertical plane, the angle is called an angle of elevation. When the latter line passes below the horizontal line, and in the same vertical plane, the angle is called an angle of depression. When the two lines which form an angle, lie in other planes which make oblique angles with each of the former planes, the angle is called an oblique angle. Horizontal angles are the only angles employed in common land surveying. * A plane is said to be horizontal, or level, when it is parallel to the surface of standing water, or perpendicular to a plumb-line. A line is horizontal when it lies in a horizontal plane. [CHAP. I. Angular Surveying in general. 123 (212) The angles between the directions of two lines, which it is necessary to measure, may be obtained by a great variety of instruments. All of them are in substance mere modifications of the very simple one which will now be described, and which any one can make for himself. (213) Provide a circular piece of Fi. 144. wood, and divide its circumference 0 (by any of the methods of Geometri- \ cal Drafting) into three hundred and sixty equal parts, or " Degrees," and number them as in the figure. The divisions will be like those of a watch face, but six times as many. These divisions are termed graduations. The figure shows only every fifteenth one. In the centre of the circle, fix a needle, or sharp-pointed wire, and upon this fix a straight stick, or thin ruler placed edge-wise, (called an alidade), so that it may turn freely on this point and nearly touch the graduations of the circle. Fasten the circle on a staff, pointed at the other end, and long enough to bring the alidade to the height of the eyes. The instrument is now complete. It may be called a Goniometer, or Angle-measurer. (214) Now let it be required to measure Fig. 145. the angle between the lines AB and AC. Fix, the staff in the grand, so that its centre shall be exactly over the intersection of the two d lines. Turn the alidade, so that it points, (as 0. —-;. determined by sighting along it) to a rod, or C other mark at B, a point on one of the lines, and note what degree it covers, i. e. " The Reading." Then, without disturbing the circle, turn the alidade till it points to C, a point on the other line. Note the new reading. The difference of these readings, (in the figure, 45 degrees), is the difference in the directions of the two lines, or is the angle which one makes with the other. If the dis 124 COMPASS SURVEYING. [PART Itl. tance from A to C be now measured, the point C is " determined," with respect to the points A and B, on the Third Principle. Any number of points may be thus determined. (215) Instead of the very simple and rude alidade, which has been supposed to be used, needles may be fixed on each end of the alidade; or sights may be added, such as those described in Art. (106).; or a small straight tube may be used, one end being covered with a piece of pasteboard in which a very small eye hole is pierced, and threads, called " cross-hairs," being stretch- Fig. 146. ed across the other end of it, as in the figure; so that () their intersection may give a more precise line for determining the direction of any point. (216) When a telescope is substituted for this tube, and supported in such a way that it can turn over, so as to look both backwards and forwards, the instrument (with various other additions, which however do not affect the principle), is called the Engineer's Transit. With the addition of a level, and a vertical circle, for measuring vertical angles, the instrument becomes a Theodolite; in which, however, the telescope does not usually admit of being turned over. (217) The Compass differs from the instruments which have been described, in the following respect. They all measure the angle which one line makes with another. The compass measures the angle which each of these lines makes with a third line, viz: that shown by the magnetic needle, which always points (approximately) in the same direction, i. e. North and South, Fig. 147. in the Magnetic Meridian. Thus, in the figure, the N line AB makes an angle of 30 degrees with the line?o0/ AN, and the line AC makes an angle of 75 de- fs o grees with AN. The difference of these angles, 4 I " — or 45 degrees, is the angle which AC makes with AB, agreeing with the result obtained in Art. (214). S [CHAP. I. Angular Surveying in general. 125 (218) Surveying with the compass is, therefore, a less direct operation than surveying with the Transit or Theodolite. But as the use of the compass is much more rapid and easy (only one sight and reading at each station being necessary, instead of two, as in the former case), for this reason, as well as for its smaller cost, it is the instrument most commonly employed in land surveying in this country, in spite of its imperfections and inaccuracies. As many may wish to learn " Surveying with the Compass," without being obliged to previously learn " Surveying with the Transit," (which properly, being more simple in principle, though less so in practice, should precede it, but which will be considered in Part IV), we will first take up COMPASS SURVEYING. (219) Angular Surveying in general, and therefore Compass Surveying, may employ either of the 3d, 4th and 5th methods of determining the position of a point, given in Part I; that is, any instrument which measures angles may be employed for Polar, Triangular, or Trilinear Surveying. The first of these, Polar Surveying, is the one most commonly adopted for the compass, and is therefore the one which will be specially explained in this part. The same method, as employed with the Transit and Theodolite, will be explained in the following part. The 4th and 5th methods will be explained in the next two parts. (220) The method of Polar Surveying embraces two minor methods. The most usual one consists in going around the field with the instrument, setting it at each corner and measuring there the angle which each side makes with its neighbor, as well as the length of each side. This method is called by the French the method of Cheminement. It has no special name in English, but may be called (from the American verb, To progress), the Method of Progression. The other system, the Method of Radiation, consists in setting the instrument at one point, and thence measuring the direction and distance of each corner of the field, or other object. The corresponding name of what we have called Triangular Surveying is the Method of Intersections; since it determines points by the intersections of straight lines. 126 COMPASS SURVEYING. [PART II. ni f 0 0 0l-1(0 CHAPTER II. TIlE COMPASS. (221) The Needle. The most essential part of the compass is the magnetic needle. It is a slender bar of steel, usually five or six inches long, strongly magnetized, and balanced on a pivot, so that it may turn freely, and thus be enabled to continue pointing in the same direction (that of the " Magnetic Meridian," approximately North and South) however much the " Compass Box," to which the pivot is attached, may be turned around. As it is important that the needle should move with the least possible friction, the pivot should be of the hardest steel ground to a very sharp point; and in the centre of the needle, which is to rest on the pivot, should be inserted a cap of agate, or other hard material. Iridium for the pivot, and ruby for the cap, are still better. If the needle be balanced on its pivot before being magnetized, one end will sink, or " Dip," after the needle is magnetized. To bring it to a level, several coils of wire are wound around the needle SQ that they can be slid along it, to adjust the weight of its two ends and balance it more perfectly. The North end of the needle is usually cut into a more ornamental form than the South end, for the sake of distinction. The principal requisites of a compass needle are, intensity of directive force and susceptibility. "Shear steel" was found by Capt. Kater to be the kind capable of receiving the greatest magnetic force. The best form is that of a rhomboid, Fig. 149. or lozenge, cut out in the middle, so as to dimi-.... nish the extent of surface in proportion to the mass, as it is the latter on which the directive force depends. Beyond a certain limit, say five inches, no additional power is gained by increasing the length of the needle. On the contrary, longer ones are apt to have their strength diminished by several consecutive poles being formed. Short needles, made very hard, are therefore to be preferred. 128 COMPASS SURVEYING. [PART III The needle should not come to rest very quickly. If it does, it indicates either that it is weakly magnetized, or that the friction on the pivot is great. Its sensitiveness is indicated by the number of vibrations which it makes in a small space before coming to rest. A screw, with a milled head, on the under side of the plate which supports the pivot, is used to raise the needle off this pivot, when the instrument is carried about, to prevent the point being dulled by unnecessary friction. (222) The Sights. Next after the needle, which gives the direction of the fixed line, whose angles with the lines to be surveyed are to be measured, should be noticed the Sights, which show the directions of these last lines. At each end of a line passing through the pivot is placed a " Sight," consisting of an upright bar of brass, with openings in it of various forms; usually either slits, with a circular aperture at their top and bottom*; or of the form described in Art. (106); all these arrangements being intended to enable the line of sight to be directed to any desired object, with precision. (223) A Telescope which can move up and down in a vertical plane, i. e. a plunging telescope, or one which can turn completely over, is sometimes substituted for the sights. It has the great advantage of giving more distinct vision at long distances, and of admitting of sights up and down very steep slopes. Its accuracy of vision is however rendered nugatory by the want of precision in the readings of the needle. If a telescope is applied to the compass, a graduated scale and index should also be added, thus converting the compass into a " Transit." The Telescope will be found minutely described in Part IV, " Transit Surveying." (224) The divided circle. We now have the means of indicating the directions of the two lines whose angle is to be measured. The number of degrees contained in it is to be read from a circle, divided into degrees, in the centre of which is fixed the An inside and an outside view, or " Elevation," of such sights, are given on each side of the figure of the Compass, on page 126. It is itself drawn in " Military Perspective." OHAP. ii.] The Compass. 129 pivot bearing the needle. The graduations are usually made to half a degree, and a quarter of a degree or less can then be " estimated." The pivot and needle are sunk in a circular box, so that its top may be on a level with the needle. The graduations are usually made on the top of the surrounding rim of the box, but should also be continued down its inside circumference so that it may be easier to see with what division the ends of the needle coincide. The degrees are not numbered consecutively from 0~ around to 3600; but run from 0~ to 90~, both ways from the two diametrically opposite points at which a line, passing through the slits in the middle of the sights, would meet the divided circle. The lettering of' the Surveyor's Compass has one important difference from that of the Mariner's Compass. When we stand facing the North, the East is on our right hand, and the West on our left. The graduated card of the Mariner's Compass which is fastened to the needle, and turns with it, is marked accordingly. But, in the Surveyor's compass, one of the 0 points being marked N, or North, (or indicated by a fleur-delis,) and the opposite one S, or South, the 90-degrees-point on the right of this line, as you stand at the S end and look towards the N, is marked W, or West; and the left hand 90-degrees-point is marked E, or East. The reason of this will be seen when the method of using the compass comes to be explained in the following chapter. (225) The Points. In or- Fig. 150. dinary land surveying, only four points of the compass have-' names, viz: North, South, East and West; the direction of a /i, ^v~t line being described by the an- [2?i "E \ gle which it makes with a North.w _ and South line, to its East or to / sca, sV its West. But for nautical pur- \0. lA poses, the circle of the compass. is divided into 32 points, the names of which are shown; ih L S___ 9 180 COMPASS SURVEYING. [PART III. the figure. Two rules embrace all the cases. 1O When the letters indicating two points are joined together, the point half way between the two is meant; thus, N. E. is half way between North and East; and N. N. E. is half way between North and North East. 2~ When the letters of two points are joined together with the intermediate word by, it indicates the point which comes next after the first, in going towards the second; thus, N. by E, is the point which follows North in going towards the East; S. E. by S. is the next point from South East, going towards the South. (226) Eccentricity. The centre-pin, or pivot of the needle, ought to be exactly in the centre of the graduated circle; the needle ought to be straight; and the line of the sights ought to pass exactly through this centre and through the 0 points of the circle. If this is not the case, there will be an error in every observation. This is called the error of eccentricity. When the maker of a compass is about to fix the pivot in place, he is in doubt of two things; whether the needle is perfectly straight, and whether the pivot is exactly in the cen- Fig. 151. tre. In figures 151 and 152, both of these are represented as being excessively in error. Firstly, to examine if the needle be ____ straight. Fix the pivot temporarily, so that the ends of the needle may cut opposite degrees, i. e. degrees differing by 1800. The condition of things at this Fig 152. stage of progress, will be represented by Fig. 151. Then turn the compass-box half way around. The error will now be doubled, as is shown by Fig. 152, in which the former position of the needle is indi- cated by a dotted line.* Now bend the needle, as in Fig. 153, till it cuts divisions midway between those cut by it in * This is another example of the fruitful vrinciole of Reverswn, first noticed in Art. (105). CHAP. n.] The Compass. 131 its present and in its former position Fig 153. This makes it certain that the needle is straight, or that its two ends and its centre lie in the same straight line. Secondly, to put the pivot in the cen-____ tre. Move it till the straightened needle cuts opposite divisions. It is then certain that the direction of the needle passes through the centre. Turn the compass box one-quarter around, and if the needle does not then cut opposite divisions, move the pivot till it does. Repeat the operation in various positions of the box. It will be a sufficient test if it cuts the opposite divisions of 00, 45~ and 90~. To fix the sights precisely in line, draw a hair through their slits and move them till the hair passes over the 0 points on the circle. The surveyor can also examine for himself, by the principle of Reversion, whether tje line of the sights passes through the centre or not. Sight to any very near object. Read off the number of degrees indicated by one end of the needle. Then turn the compass half around, and sight to the same object. If the two readings do not agree, there is an error of eccentricity, and the arithmetical mean, or half sum of the two readings is the correct one. Fig. 154. Fig. 155. In Fig. 154, the line of sight AB is represented as passing to one side of the centre, and the needle as pointing to 46~. In Fig. 155, the compass is supposed to have been turned half around and the other end of the sights to be directed to the same object. Suppose that the needle would have pointed to 450, if the line of 132 COMPASS SURVEYING. [PART nII sight had passed through the centre. The needle will now point to 440, the error being doubled by the reversion, and the true reading being the mean. This does not, however, make it certain that the line of the sights passes through the 0 points, which can only be tested by the hair, as mentioned above. (227) Levels. On the compass plate are two small spirit levels. They consist of glass tubes, slightly curved upwards, and nearly filled with alcohol, leaving a bubble of air within them. They are so adjusted that when the bubbles are in the centres of the tubes, the plate of the compass shall be level. One of them lies in the direction of the sights, and the other at right angles to this direction. (228) Tangent Scale. This is a convenient, though not essential, addition to the compass, for the purpose of measuring the slopes of ground, so that the proper allowance in chaining may be made. In the figure of the compass, page 126, may be seen, on the edge of the left hand sight, a small projection of brass with a hole through it. On the edge of the other sight are engraved lines numbered from 0~ to 20~, the 0~ being of the same height above the compass plate that the eye-hole is. To use this, set the compass at the bottom of a slope, and at the top set a signal of exactly the height of the eye-hole from the ground. Level the compass very carefully, particularly by the level which lies lengthwise, and, with the eye at the eye-hole, look to the signal and note the number of the division on the farther sight which is cut by the visual ray. That will be the angle of the slope; the distances of the engraved lines from the 0~ line being tangents (for the radius equal to the distance between the sights) of the angles corresponding to the numbers of the lines. (229) Vernier. The compass box is connected with the plate, which carries it and the sights, so that it can turn around on this plate. This motion is given to it by a screw, (called a slow-motion, or Tangent screw), the head of which is the nearest one in CHAP II.] The Compass. 133 the figure on page 126. If two marks be made opposite to each other, one on the projecting part of the compass box, and the other on the plate to which the sights are fastened, these marks will separate when the slow-motion screw is turned. Their distance apart (in angular measurement, i. e. fractions of a circle), in any position, is measured by a contrivance called a Vernier, which is the minutely divided arc of a circle seen between the left hand sight and the compass box. It will be better to defer explaining the mode of reading the vernier for the present, since it is rarely used with the compass, and an entire chapter will be given to it in Part IV. Its principle is similar to that of the Vernier Scale, described in Art. (50). Its applications in "Field-work" will be noticed under that head. (230) Tripod. The compass, like most surveying instruments, is usually supported on a Tripod, consisting of three legs, shod with iron, and so connected at top as to be movable in any direction. There are many forms of these. Lightness Fig 156. Fig 157. and stiffness are the qualities desired. The most usual form is < shewn in the figures 1 liM of the Transit and the Theodolite, at the beginning of Part IV. Of the two represented in Figs. 156 and 157, the first has the advantage of being very easily and cheaply made; and the second that of being light and yet capable of very firmly resisting horizontal torsion. The joints, by which the instrument is connected with the tripod,. are also various. Fig. 158 is the " Ball-and-socket joint," most usual in this country. It takes its name from the ball, in which I34 COMPASS SURVEYING. [PART II. Fig. 158. Fig. 159. Fig. 160 terminates the covered spindle which enters a corresponding cavity under the compass plate, and the socket in which this ball turns. It admits of motion in any direction, and can be tightened or loosened by turning the upper half of the hollow piece enclosing it, which is screwed on the lower half. Fig. 159 is called the " Shelljoint." In it the two shell-shaped pieces enclosing the ball are tightened by a thumb-screw. Fig. 160, is " Cugnot's joint." It consists of two cylinders, placed at right angles to each other, and through the axes of which pass bolts, which turn freely in the cylinder and can be tightened or loosened by thumb-screws at their ends. The combination of the two motions which this joint permits, enables the instrument which it carries, to be placed in any desired position. This joint is much the most stable of the three. (231) Jacob's Staff. A single leg, called a " Jacob's Staff," has some advantages, as it is lighter to carry in the field, and can be made of any wood on the spot where it is to be used, thus sav ing the expense of a tripod and the trouble of its transportation Its upper end is fitted into the lower end of a brass head which has a ball and socket joint, and axis above. Its lower end should be shod with iron, and a spike running through it is useful for pressing it into the ground with the foot. Of course it cannot be conveniently used on frozen ground, or 6n pavements. It may, however, be set before or behind the spot at which the angle is to be mea CHAP. II.] The Compass. 135 sured, provided that it is placed very precisely in the line of direction from that station to the one to which a sight is to be taken. (232) The Prismatic Compass. The peculiarity of this instrument (often called Schmalcalder's) is that a glass triangular prism is substituted for one of the sights. Such a prism has this peculiar property that at the same time, it can be seen through, so that a sight can be taken through it, and that its upper surface reflects like a mirror, so that the numbers of the degrees immediately under it, can be read off at the same time that a sight to any object is taken. Another peculiarity, necessary for profiting by the last one, is, that the divided circle is not fixed, but is a card fastened to the needle and moving around with it, as in the Mariner's Compass. The minute description, which follows, is condensed from Simms. In the figure, A repre- Fig. 161. sents the compass box, and B the card, which, being attached to the magnetic needle, moves as it moves, around the agate centre, a, on which it is suspend- c ed. The circumference of the card is usually divided to; or ~ of a de- D gree. C is a prism, which -b the observer looks through. The perpendicular thread A of the sight-vane, E, and the divisions on the card, appear together on looking through the prism, and the division with which the thread coincides, when the needle is at rest, is the " Bearing" of whatever object the thread may bisect, i. e. is the angle which the line of sight makes with the direction of the needle. The prism is mounted with a hinge joint, D. The sight-vane has a fine thread stretched along its opening, in the direction of its length, which is brought to bisect any object, by turning the box around horizontally. F is a mirror, made to 136 COMPASS SURVEYING. [PART In. slide on or off the sight-vane, E; and it may be reversed at pleasure, that is, turned face downwards; it can also be inclined at any angle, by means of its joint, d; and it will remain stationary on any part of the vane, by the friction of its slides. Its use is to reflect the image of an object to the eye of an observer when the object is much above or below the horizontal plane. The colored glasses represented at G, are intended for observing the sun. At e, is shown a spring, which being pressed by the finger at the time of observation, and then released, checks the vibrations of the card, and brings it more speedily to rest. A stop is likewise fixed to the other side of the box, by which the needle may be thrown off its centre. The method of using this instrument is very simple. First raise the prism in its socket, b, until you obtain a distinct view of the divisions on the card. Then, standing over the point where the angles are to be taken, hold the instrument to the eye, and, looking through the slit, C, turn around till the thread in the sight-vane bisects one of the objects whose bearing is required; then by touching the spring, e, bring the needle to rest, and the division on the card which coincides with the thread on the vane, will be the bearing of the object from the north or south points of the magnetic meridian. Then turn to any other object, and repeat the operation; the difference between the bearing of this object and that of the former, will be the angular distance of the objects in question. Thus, suppose the former bearing to be 40~ 30', and the latter 10~ 15', both east, or both west, Fi. 162. from the north or south, the angle - o[ will be 30~ 15'. The divisions are generally numbered 5~, 10~, 15~, / \ &c. around the circle to 360~, The figures on the compass card _are reversed, or written upside down, as in the figure (in which only every fifteenth degree is mark- ed), because they are again re- versed by the prism. CHAP. II.] The Compass. 187 (233) The prismatic compass is generally held in the hand, the bearing being caught, as it were, in passing; but more accurate readings would of course be obtained if it rested on a support, such as a stake cut flat on its top. In the former mode, the needle never comes completely to rest, particularly in the wind. In such cases, observe the extreme divisions between which the needle vibrates, and take their arithmetical mean. (234) Defects of compass. The compass is deficient in both precision and correctness.* The former defect arises from the indefiniteness of its mode of indicating the part of the circle to which it points. The point of the needle has considerable thickness; it cannot quite touch the divided circle; and these divisions are made only to whole or half degrees, though a fraction of a division may be estimated, or guessed at. The Vernier does not much better this, as we shall see when explaining its use. Now an inaccuracy of one quarter of a degree in an angle, i. e. in the difference of the directions of two lines, causes them to separate from, each other 51 inches at the end of 100 feet; at the end of 1000 feet nearly 41 feet; and at the end of a mile, 23 feet. A difference of only one-tenth of a degree, or six minutes, would produce a difference of 13 feet at the end of 1000 feet; and 91 feet at the distance of a mile. Such are the differences which may result from the want of precision in the indications of the compass. But a more serious defect is the want of correctness in the compass. Its not pointing exactly to the true north does not indeed affect the correctness of the angles measured by it. But it does not point in the same or in a parallel direction, during even the same day, but changes its direction between sunrise and noon nearly a quarter of a degree, as will be fully explained in Chapter VIII. The effect of such a difference we have just seen. This direction * The student must not confound these two qualities. To say that the sun ap. pears to rise in the eastern quarter of the heavens and to set in the western, is correct, but not precise. A watch with a second hand indicates the time of day precisely, but not always correctly. The statement that two and two make five, is precise, but is not usually regarded as correct. 138 COMPASS SURVEYING. [PART Im. may also be greatly altered in a moment, without the knowledge of the surveyor, by a piece of iron being brought near to the compass, or by some other local attraction, as will be noticed hereafter. This is the weak point in the compass. Notwithstanding these defects, the compass is a very valuable instrument, from its simplicity, rapidity and convenience in use; and though never precise, and seldom correct, it is generally not very wrong. CHAPTER III. THE FIELD WORK. (235) Taking Bearings. The "Bearing" of a line is the angle which it makes with the direction of the needle. Thus, in Fig. 147,'page 124, the angle NAB is the Bearing of the line AB, and NAG is the Bearing of AC. The Bearing and length of a line are named collectively the Course. To take the Bearing of any line, set the compass exactly over any point of it by a plumb-line suspended from beneath the centre of the compass, or, approximately, by dropping a stone. Level the compass by bringing the air bubbles to the middle of the level tubes. Direct the sights to a rod held truly vertical, or " plumb," at another point of the line, the more distant the better. The two ends are usually taken. Sight to the lowest visible point of the rod. When the needle comes to rest, note what division on the circle it points to; taking the one indicated by the North end of the needle, if the North point on the circle is farthest from you, and vice versa. In reading the division to which one end of the needle points, the eye should be placed over the other end, to avoid the error which might result from the " parallax," or apparent change of place, of the end read from, when looked at obliquely. CHAP. III.] The Field Work. 138 The bearing is read and recorded by noting between what letters the end of the needle comes, and to what number; naming, or writing down, firstly, that letter, N or S, which is at the 0~ point nearest to that end of the needle from which you are reading; secondly, the number of degrees to which it points, and thirdly, the letter, E or W, of the 90~ point which is nearest to the same end of the needle. Thus, in the figure, if when the sights were directed along a line, (the North Fig. 163. point of the compass being most distant from the observer), the A4 North end of the needle was at the point A, the bearing of the line sighted on, would be North 45~ B ] East; if the end of the needle was at B, the bearing would be East; if at C, S. 30~ E; if at D, South; if at E, S. 60~ W; if at F, West; if C at G, N. 60~ W; if at H, North. D (236) We can now understand why W is on the right hand of the compass-box, and E on the left. Let the direction from the centre of the compass to the point Fig. 164. B in the figure, be required, and / suppose the sights in the first place " to be pointing in the direction of the needle, S N, and the North sight t to be ahead. When the sights (and E 7^- the circle to which they are fasten- ed) have been turned so as to point in the direction of B, the point of the circle marked E, will have come round to the North end of the needle, (since the needle remains immovable,) and the reading will therefore be " East," as it should be. The effect on the reading is the same as if the needle had moved to the left the same quantity which the sights have moved to the right, and the left side is therefore properly marked " East," and vice versa. So, too, if the bearing of the line to C be desired, half-way between North and 140 COMPASS SURIVEYING. [PART II East, i. e. N. 450 E.; when the sights and the circle have turned 45 degrees to the right, the needle, really standing still, has apparently arrived at the point half-way between N. and E., i. e. N. 450 E. Some surveyors' compasses are marked the reverse of this, the E on the right and the W on the left. These letters must then be reversed in the mind before the bearing is noted down. (237) Reading with Vernier. When the needle does not point precisely to one of the division marks on the circle, the fractional part of the smallest space is usually estimated by the eye, as has been explained. But this fractional part may be measured by the Vernier, described in Art. (229), as follows. Suppose the needle to point between N. 31~ E. and N. 31I~ E. Turn the tangent screw, which moves the compass-box, till the smaller division (in this case 31~) has come round to the needle. The Vernier will then indicate through what space the compass-box has moved, and therefore how much must be added to the reading of the needle. Suppose it indicates 10 minutes of a degree. Then the bearing is N. 31~ 10' E. It is, however, so difficult to move the Vernier without disturbing the whole instrument, that this is seldom resorted to in practice. The chief use of the Vernier is to set the instrument for running lines and making an allowance for the variation of the needle, as will be explained in the proper place. A VernierA Vernier arc is sometimes attached to one end of the needle and carried around by it. (238) Practical Hints. Mark every station, or spot, at which the compass is set, by driving a stake, or digging up a sod, or piling up stones, or otherwise, so that it can be found if any error, or other cause, makes it necessary to repeat the survey. Very often when the line of which the bearing is required, is a fence, &c., the compass cannot be set upon it. In such cases, set the compass so that its centre is a foot or two from the line, and set the flag-staff at precisely the same distance from the line at the other end of it. The bearing of the flag-staff from the compass will be the same as that of the fence, the two lines being parallel. ConP. II.] The Field Work. 141 The distances should be measured on the real line. If more convenient the compass may be set at some point on the line prolonged, or at some intermediate point of the line, " in line" between its extremities. In setting the compass level, it is more important to have it level crossways of the sights than in their direction; since if it be not so, on looking up or down hill through the upper part of one sight and the lower part of the other, the line of sight will not be parallel to the N and S, or zero line, on the compass, and an incorrect bearing will therefore be obtained. The compass should not be levelled by the needle, as some books recommend, i. e. so levelled that the ends of the needle shall be at equal distances below the glass. The needle should be brought so originally by the maker, but if so adjusted in the morning, it will not be so at noon, owing to the daily variation in the dip. If then the compass be levelled by it, the lines of sight will generally be more or less oblique, and therefore erroneous. If the needle touches the glass, when the compass is levelled, balance it by sliding the coil of wire along it. The same end of the compass should always go ahead. The North end is preferable. The South end will then be nearest to the observer. Attention to this and to the caution in the next paragraph, will prevent any confusion in the bearings. Always take the readings from the same end of the needle; from the North end, if the North end of the compass goes ahead; and vice versa. This is necessary, because the two ends will not always cut opposite degrees. With this precaution, however, the angle of two meeting lines can be obtained correctly from either end, provided the same one is used in taking the bearings of both the lines. Guard against a very frequent source Fig. 165. of error with beginners, in reading from the wrong number of the two between which the needle points, such as reading 34~ for 26~, in a case like that in the figure. 142 COMPASS SURVEYING. [PART II, Check the vibrations of the needle by gently raising it off the pivot so as to touch the glass, and letting it down again, by the screw on the under side of the box. The compass should be smartly tapped after the needle has settled, to destroy the effect of any adhesion to the pivot, or friction of dust upon it. All iron, such as the chain, &c., must be kept at a distance from the compass, or it will attract the needle, and cause it to deviate from its proper direction. The surveyor is sometimes troubled by the needle refusing to traverse and adhering to the glass of the compass, after he has briskly wiped this off with a silk handkerchief, or it has been carried so as to rub against his clothes. The cause is the electricity excited by the friction. It is at once discharged by applying a wet finger to the glass. A compass should be carried with its face resting against the side of the surveyor, and one of the sights hooked over his arm. In distant surveys an extra centre pin should be carried, (as it is very liable to injury, and its perfection is most essential), and, also, an extra needle. When two such are carried, they should be placed so that the north pole of one rests against the south pole of the other. (239) When the magnetism of the needle is lessened or destroyed by time, it may be renewed as follows. Obtain two bar magnets. Provide a board with a hole to admit of the axis, so that its collar may fit fairly, and that the needle may rest flat on it, without bearing at the centre. Place the board before you, with the north end of the needle to your right. Take a magnet in each hand, the left holding the North end of the bar, or that which has the mark across, downwards; and the right holding the same mark upwards. Bring the bars over the axis, about a foot above it, without approaching each other within two inches:-bring them down vertically on the needle, (the marks as directed) about an inch on each side of its axis; slide them outwards to its ends with slight pressure; raise them up; bring them to their former position, and repeat this a number of times. CHAP. II.] The Field WWok.- 143 (240) Back Sights. To test the accuracy of the bearing of a line, taken at one end of it, set up the compass at the other end, or point sighted to, and look back to a rod held at the first station, or point where the compass had been placed originally. The reading of the needle should now be the same as before. If the position of the sights had been reversed, the reading would be the Reverse Bearing; a former bearing of N. 30~ E. would then be S. 30~ W., and so on. (241) Local attraction. If the Back-sight does not agree with the first or forward sight, this latter must be taken over again. If the same difference is again found, this shows that there is local attraction at one of the stations; i. e. some influence, such as a mass of iron ore, ferruginous rocks, &c., under the surface, which attracts the needle, and makes it deviate from its usual direction. Any high object, such as a house, a tree, &c., has recently been found to produce a similar effect. To discover at which station the attraction exists, set the compass at several intermediate points in the line which joins the two stations, and at points in the line prolonged, and take the bearing of the line at each of these points. The agreement of several of these bearings, taken at distant points, will prove their correctness. Otherwise, set the compass at a third station; sight to each of the two doubtful ones, and then from them back to this third station. This will show which is correct. When the difference occurs in a series of lines, such as around a field, or along a road, proceed Fig. 166. thus. Let C be the station at C D which the back-sight to B dif- A fers from the foresight from B to C. Since the back-sight from B to A is supposed to have agreed with the foresight from A to B, the local attraction must be at C, and the forward bearing must be corrected by the difference just found between the fore and back sights, adding or subtracting it, according to circumstances. An easy method is to draw a 144 COMPASS SURVEYING. [PART II. figure for the case, as in Fig. In g. 167. it, suppose the true bearing of BC, as N given by a fore-sight from B to C, to be,\ / D N. 40~ E., but that there is local at- traction at C, so that the needle is drawn aside 10~, and points in the direction S'N', instead of SN. The back-sight from C to B will then give a bearing / t of N. 50~ E.; a difference, or correc- A~ tion for the next fore-sight, of 10~. If the next fore-sight, from C to D, be N. 70~ E, this 10~ must be subtracted from it, making the true fore-sight N. 600 E. A general rule may also be given. When the back-sight is greater than the fore-sight, as in this case, subtract the difference from the next fore-sight, if that course and the preceding one have both their letters the same (as in this case, both being N. and E.), or both their letters different; or add the difference if either the first or last letters of the two courses are different. When the back-sight is less than the fore-sight, add the difference in the case in which it has just been directed to subtract it, and subtract it where it was before directed to add it. (242) Angles of deflection. When the compass indicates much local attraction, the difference between the directions of two meeting lines, (or the " angle of deflection" of one from the other), can still be correctly measured, by taking the difference of the bearings of the two lines, as observed at the same point. For, the error caused by the local attraction, whatever it may be, affects both bearings equally, inasmuch as a "Bearing" is the angle which a line makes with the direction of the needle, and that here remains fixed in some one direction, no matter what, during the taking of the two bearings. Thus, in Fig. 167, let the true bearing of BC, i. e. the angle which it makes with the line SN, be, as before, N. 40~ E., and that of CD N. 60~ E. The true " angle of deflection" of these lines, or the angle B'CD,is therefore 20~. Now, if local attraction at C causes the needle to point in the direcS'N', 10~ to the left of its proper direction, BC will bear N. 50~ CHAP. III.] The Field Work. 145 E., and CD N. 70~ E., and the difference of these bearings, i. e. the angle of deflection, will be the same as before. (243) Angles between Courses. To determine the angle of deflection of two courses meeting at any point, the following simple rules, the reasons of which will appear from the accompanying figures, are sufficient. Fig. 168. Case 1. When the first letters of the bearing are alike, (i. e. both N. or both / S.), and the last letters also alike, (i. e. / both E. or both W.), take the difference of the bearings. Example. IfAB bears we ------- N. 30~ E. and BC bears N. 10~ E., the angle of deflection CBB' is 20~.'/0o 0' S JFig. 169. C) Case 2. When the first letters are \2~ alike and the last letters different; take \ / the sum of the bearings. Ex. If AB \, bears N. 40~ E. and BC bears N. 20~ w —. W.; the angle CBB' is 60~. /?v S Fig. 170. N Case 3. When the first letters are / different and the last letters alike, sub- / tract the sum of the bearings from 180~. i Ex. If AB bears N. 30~ E. and BC w bears S. 40~ E.; the angle CBB'is 110/ 10 146 COMPASS SURVEYING. [PART III Fig. 170. Case 4. When both the first and last letters are different, subtract the / i\ C difference of the bearings from 180~. /I Ex. If AB bears S. 30o W. and BC w —- ---- E bears N. 70~ E.; the angle CBB' is /i 140~. / a,/ SI If the angles included between the courses are desired, they will be at once found by reversing one bearing, and then applying the above rules; or by subtracting the results obtained as above from 180~; or an analogous set of rules could be formed for them. (244) To change Bearings. It is convenient in certain calculations to suppose one of the lines of a survey to change its direction so as to become due North and South; that is, to become a new Meridian line. It is then necessary to determine what the bearings of the other lines will be, supposing them to change with it. The subject may be made plain by supposing the survey to be platted in the usual way, with the North uppermost, and the plat to be then turned around, till the line to be changed is in the desired direction. The effect of this on the other lines will be readily seen. A (eneral Rule can also be formed. Take the difference between the original bearing of the side which becomes a Meridian and each of those bearings which have both their letters the same as it, or both different from it. The changed bearings of these lines retain the same letters as before, if they were originally greater than the original bearing of the new Meridian line; but, if they were less, they are thrown on the other side of flie N. and S. line, and their last letters are changed; E. being put for W. and W for E. Takelthe sumof the original bearing of the new Meridian line, and each of' hose bearings which have one letter the same as one letter of the former bearing, and one different. If this sum exceeds CHAP. Inr.] The Field Work. 147 90~, this shews that the line is thrown on the other side of the East or West point, and the difference between this sum and 180~ will be the new bearing and the first letter will be changed, N. being put for S. and S. for N. Example. Let the Bearings of the sides of a field be as follows: N. 820 E.; N. 80~ E.; S. 480 E.; S. 183 W.; N. 73go W.; North. Suppose the first side to become due North; the changed bearings will then be as follows: North; N. 480 E.; S. 800 E.; S. 140 E.; S. 74 ~ W.; N. 32~ W. To apply the rule to the " North" course, as above, it must be called N. 0~ W.; and then by the Rule, 323 must be added to it. The true bearings can of course be obtained from the changed bearings, by reversing the operation, taking the sum instead of the difference, and vice versa. (245) Line Surveying. This name may be given to surveys of lines, such as the windings of a brook, the curves of a road, &c., by way of distinction from Farm Surveying, in which the lines surveyed enclose a space. To survey a brook, or any similar line, set the compass at, or near, one end of it, and take the bearing of an imaginary or visual line, running in the general average direction of the brook, Fig. 17o2. such as AB in the figure. Measure this line, taking offsets to the various bends of the brook, as to the fence explained in Art.(115). Then set the compass at B, and take a back-sight to A, and if they agree, take a fore-sight to C, and proceed as before, noting particularly the points where the line crosses the brook. To survey a road, take the bearings and lengths of the lines Fig. 173. y^^^^^^SQJ 148 COMPASS SURVEYING. [PART III. which can be most conveniently measured in the road, and measure offsets on each side, to the outside of the road. When the line of a new road is surveyed, the bearings and lengths of the various portions of its intended centre line should be measured, and the distance which it runs through each man's land should be noted. Stones should be set in the ground at recorded distances from each angle of the line, or in each line prolonged a known distance, so as not to be disturbed in making the road. In surveying a wide river, one bank may be surveyed by the method just given, and points on the opposite banks, as trees, &c., may be fixed by the method of intersections, founded on the Fourth Method of determining the position of a point; and fully explained in Part IV. (246) Checks by intersecting bearings. At each station at which the compass is set, take bearings to some remarkable object, such as a church steeple, a distant house, a high tree, &c. At least three bearings should be taken to each object to make it of any use: since two are necessary to determine it, (by our Fourth Method), and, till thus determined, it can be no check. When the line is platted, by the methods to be explained in the next chapter, plat also the lines given by these bearings. If those taken to the same object from three different stations, intersect in the same point, this proves that there has been no mistake in the survey or platting of those stations. If any bearing does not intersect a point fixed by previous bearings, it shows that there has been an error, either between the last station and one of those which fixed the point, or in the last bearing to the point. To discover which it was, plat the following line of the survey, and, at its extremity, set off the bearing from it to the point; and if the line thus platted passes through the point, it proves that there was no error in the line, but only in the bearing to the point. If otherwise, the error was somewhere in the line between the stations from which the bearings to that point were taken. CHAP. III.] The Field Work. 149 (247) Keeping the Field-notes. The simplest and easiest method for a beginner is to make a rough sketch of the survey by eye, and write down on the lines their bearings and lengths. An improvement on this is to actually lay down the precise bearings and lengths of the lines in the field-book in the manner to be explained in the chapter on Platting, Art. (269). (248) A second method is to draw a straight line up the page of the field-book, and to write on it the bearings and lengths of the lines. The only advantage of this method is that the line will not run off the side of the page, as it is apt to do in the preceding method. (249) A third method is to represent the line surveyed, by a double column, as in Part II, Chapter I, Art. (95), which should be now referred to. The bearings are written obliquely up the columns. At the end of each course, its length is written in the column, and a line drawn across it. Dotted lines are drawn across the column at any intermediate measurement. Offsets are noted as explained in Art. (114). The intersection-bearings, described in Art. (246), should be entered in the field-book before the bearings of the line, in order to avoid mistakes of platting, in setting off the measured distances on the wrong line. (250) A fourth method is to write the Stations, Bearings, and Distances in three columns. This is compact, and has the advantage, when applied to farm surveying, of presenting a form suitable for the subsequent calculations of Content, but does not give facili. ties for noting offsets. Examples of these four methods are given in Art. (254); which contains the field-notes of the lines bounding a field. (251) New-York Canal Maps. The following is a description of the original maps of the survey of the line of the New-York Erie Canal, as published by the Canal Commissioners. The figure represents a portion of such a map; but, necessarily, with all its lines black; red and blue lines being used on the real map. 150 COMPASS SURVEYING. rPART III. Fig. 174. _ _ t~ Cs r C -S^; _ _ lair/S 2P-E S 3~ E S 33IE ~ 41 1~ E | S 5 o o o o "The RED LINE described along the inner edge of the towing path is the base line, upon which all the measurements in the direction of the length of the canal were made. The bearings refer to the magnetic meridian at the time of the survey. The lengths of the several portions are inserted at the end of each, in chains and links. The offsets at each station are represented by red lines drawn across the canal in such a direction as to bisect the angles formed by the two contiguous portions of the red or base line, upon the towing path. The intermediate offsets are set off at right angles to the base line; and the distances on both are given from it in links. The intermediate offsets are represented by red dotted lines, and the distances to them upon the base line are reckoned, in each case, from the last preceding station. The same is likewise done with the other distances upon the base line; those to the Bridges being taken to the lines joining the nearest angles, or corner posts of their abutments; those to the Locks extending to the lines passing through the centres of the two nearest quoin posts; and those to the Aqueducts, to the faces of their abutments. The space enclosed by the BLUE LINES represents the portion embraced within the limits of the survey as belonging to the state; and the names of the adjoining proprietors are given as they stood at the time of executing the survey. The distances are projected upon a scale of two chains to the inch." (252) Farm Surveying. A farm, or field, or other space included within known lines, is usually surveyed by the compass thus. Begin by walking around the boundary lines, and setting stakes at all the corers, which the flag-man should specially note, CHAP. II.] The Field Worl, 151 so that he may readily find them again. Then set the compass at any corner, and send the flag-man to the next corner. Take the bearing of the bounding line running from corner to corner, which is usually a fence. Measure its length, taking offsets if necessary. Note where any other fence, or road, or other line, crosses or meets it, and take their bearings. Take the compass to the end of this first bounding line; sight back, and if the back-sight agrees, take the bearing and distance of the next bounding line; and so proceed till you have got back to the point of starting. (253) Where speed is more important than accuracy in a survey, whether of a line or a farm, the compass need be set only at every other station, taking a forward sight, from the 1st station to the 2d; then setting the compass at the 3d station, taking a backsight to the 2d station (but with the north point of the compass always ahead), anda fore-sight to the 4th; then going to the 5th, and so on. This is, however, not to be recommended. (254) Field-notes. The Field-notes of a Farm survey may be kept by any of the methods which have been described with reference to a Line survey. Below are given the Field-notes of the same field recorded by each of the methods. JFirst Method. Fig. 175. 2N 83T~ JE 1.29 1 6 o 44 *a< 4/ 7~ 152 COMPASS SURVEYING. [PART III Second Third Fourth Method. Method. Method.* o (1) -(1)- --. J ( )3.23 STATIONS. BEARINGS. DISTANCES. 2kqm 1 N. 35o E. 2.70 c: g 2 N. 8310 E. 1.29 z; 3| S. 57~ E. 2.22 0 (5) i 4 S.34~W. 3.55 5 I N. 561c W. 3.23 o X1 3.54 1.40 X, Fig. 176. GQ 16~ 6 2.77 0 -(4)r g.. 2.22 th cI m a i -0.25 —6 o (3) L i — l-~- V0Cf (2) ls 1.34 F ~ 2.-0.70 —8 0 C> 2.70 — 1-1-1 0 (255) The Field-notes of a field, in which offsets occur, may be most easily recorded by the Third Method; as in Fig. 176. When the Field-notes are recorded by the Fourth Method, the offsets may be kept in a separate Table; in which the 1st column will contain the stations from which the measurements are made, the 2d column the distances at which they occur, the 3d * In the " Third Method," the bearings should be written obliquely upward, as directed in Art, (249), but are not so printed here, from typographical diffi. culties, CHAP. II.] The Field Work. 153 column the lengths of the offsets, and the 4th column the side of the line, "Right," or " Left," on which they lie. For calculation, four more columns may be added to the table, containing the intervals between the offsets; the sums of the adjoining pairs; and the products of the numbers in the two preceding columns, separated into Right and Left, one being additive to the field, and the other subtractive. (256) Tests of accuracy. 1st. The check of intersections described in Art. (246), may be employed to great advantage, when some conspicuous object near the centre of the farm can be seen from most of its corners. 2nd. When the survey is platted, if the last course meets the starting point, it proves the work, and the survey is then said to "close." 3d. Diagonal lines, running from corner to corner of the farm, like the " Proof-lines" in Chain Surveying, may be measured and their bearings taken. When these are laid down on the plat, their meeting the points to which they had been measured, proves the work. 4th. The only certain and precise test is, however, that by "Latitudes and Departures." This is fully explained in Chapter V, of this Part. (257) A very fallacious test is recommended by several writers on this subject. It is a well-known proposition of Geometry, that in any figure bounded by straight lines, the sum of all the interior angles is equal to twice as many right angles, as the figure has sides less two; since the figure can be divided into that number of triangles. Hence this common rule. " Calculate [by the last paragraph of Art. (243)] the interior angles of the field or farm surveyed; add them together, and if their sum equals twice as many right angles as the figure has sides less two, the angles have been correctly measured." This rule is not applicable to a compass survey; for, in Fig. 167, page 144, the interior angle BCD will contain the same number of degrees (in that case 160~) whether the bearings of the sides have been noted correctly, as being the 154 COMPASS SURVEYING. [PART III. angles which they make with NS —or incorrectly, as being the angles which they make with N'S'. This rule would therefore prove the work in either case. (258) Method of Radiation. A field may be surveyed from one station, either within it or without it, by taking the bearings and the distances from that point to each of the corners of the field. These comers are then " determined," by the 3d method, Art. (7). This modification of that method, we named, in Art. (220), the Method of Radiation. All our preceding surveys with the compass have been by the Method of Progression. The compass may be set at one corner of the field, or at a point in one of its sides, and the same method of Radiation employed. This method is seldom used however, since, unlike the method of Progression, its operations are not checks upon each other. (259) Method of Intersection. A field may also be surveyed by measuring a base line, either within it or without it, setting the compass at each end of the base line, and taking, from each end, the bearings of each corner of the field; which will then be fixed and determined, by the 4th method, Art. (8). This mode of surveying is the Method of Intersections, noticed in Art. (220). It will be fully treated of in Part V, under the title of Triangular Surveying. (260) Running out old lines. The original surveys of lands in the older States of the American Union, were exceedingly deficient in precision. This arose from two principal causes; the small value of land at the period of these surveys, and the want of skill in the surveyors. The effect at the present day is frequent dissatisfaction and litigation. Lots sometimes contain more acres than they were sold for, and sometimes less. Lines which are straight in the deed, and on the map, are found to be crooked on the ground. The recorded surveys of two adjoining farms often make one overlap the other, or leave a gore between them. The most difficult and delicate duty of the land-surveyor, is to run out these old boundary lines. In such cases, his first business is to find CHAP. III.] The Field Work. 155 monuments, stones, marked trees, stumps, or any other old'" corners," or landmarks. These are his starting points. The owners whose lands join at these corners should agree on them. Old fences must generally be accepted by right of possession; though such questions belong rather to the lawyer than to the surveyor.* His business is to mark out on the ground the lines given in the deed. When the bounds are given by compass-bearings, the surveyor must be reminded that these bearings are very far from being the same now as originally, having been changing every year. The method of determining this important change, and of making the proper allowance, will be found in Chapter VIII, of this Part. (261) Town Surveying. Begin at the meeting of two or more of the principal streets, through which you can have the longest prospects. Having fixed the instrument at that point, and taken the bearings of all the streets issuing from it, measure all these lines with the chain, taking offsets to all the corners of streets, lanes, bendings, or windings; and to all remarkable objects, as churches, markets, public buildings, &c. Then remove the instrument to the next street, take its bearings, and measure along the street as before, taking offsets as you go along, with the offset-staff. Proceed in this manner from street to street, measuring the distances and offsets as you proceed. Fig. 177. X LL_ XII / * "In the description of land conveyed, the rule is, that known and fixed monuments control courses and distances. So, the certainty of metes and bounds will include and pass all the lands within them, though they vary from the given quantity expressed in the deed. In New-York, to remove, deface or alter landmarks maliciously, is al indictable offence."-Kent's Commentaries, IV, 515. 156 COMPASS SURVEYING. [PART IIJ Thus, in the figure, fix the instrument at A, and measure lines in the direction of all the streets meeting there, noting their bearings; then measure AB, noting the streets at X, X. At the second station, B, take the bearings of all the streets which meet there; and measure from B to C, noting the places and the bearings of all the cross-streets as you pass them. Proceed in like manner from C to D, and from D to A, " closing" there, as in a farm survey. Having thus surveyed all the principal streets in a particular neighborhood, proceed then to survey the smaller intermediate streets, and last of all, the lanes, alleys, courts, yards, and every other place which it may be thought proper to represent in the plan. The several cross-streets answer as good check lines, to prove the accuracy of the work. In this manner you continue till you take in all the town or city. (262) Obstacles in Compass Surveying. The various obstacles which may be met with in Compass Surveying, such as woods, water, houses, &c., can be overcome much more easily than in Chain Surveying. But as some of the best methods for effecting this involve principles which have not yet been fully developed, it will be better to postpone giving any of them, till they can be all treated of together; which will be done in Part VII. CHAPTER IV. PLATTING THE SURVEY. (263) The platting of a survey made with the compass, consists in drawing on paper the lines and the angles which have been measured on the ground. The lines are drawn " to scale," as has been fully explained in Part I, Chapter III. The manner of platting angles was referred to in Art. (41), but its explanation has been reserved for this place. (264) With a Protractor. A Protractor is an instrument made for this object, and is usually a semicircle of brass, as in the figure, with its semi-circumference divided into 180 equal parts, or Fig. 178. degrees, and numbered in both directions. It is, in fact, a miniature of the instrument, (or of half of it), with which the angles have been measured. To lay off any angle at any point of a straight line, place the Protractor so that its straight side, the diameter of the semi-circle, is on the given line, and the middle of this diameter, which is marked by a notch, is at the given point. With a needle, or sharp pencil, make a mark on the paper at the required number of degrees, and draw a line from the mark to the given point. 158 COMPASS SURVEYING. [PART III Sometimes the protractor has an arm turning on its centre, and extending beyond its circumference, so that a line can be at once drawn by it when it is set to the desired angle. A Vernier scale is sometimes added to it to increase its precision. A Rectangular Protractor is sometimes used, the divisions of degrees being engraved along three edges of a plane scale. The semi-circular one is preferable. The objection to the rectangular protractor is that the division corresponding to a degree is very Fig. 179.!WO _1~O 1*60 7/0 0 _ _ _ __ __0 0QlO l O J0 6 nIO r o f te s b u to or t_ unequal on different parts of the scale, being usually two or three times as great at its ends as at its middle. A Protractor embracing an entire circle, with arms carrying verniers, is also sometimes employed, for the sake of greater accuracy. (265) Platting Bearings. Since "Bearings" taken with the Compass are the angles which the various lines make with the Magnetic Meridian, or the direction of the compass-needle, which, as we have seen, remains always (approximately) parallel to itself, it is necessary to draw these meridians through each station, before laying off the angles of the bearings. The T square, shown in Fig. 14, is the most convenient instrument for this purpose. The paper on which the plat is to be made is fastened on the board so that the intended direction of the North and South line may be parallel to one of the sides of the board. The inner side of the stock of the T square being pressed against one of the other sides of the board and slid along, the edge of the long blade of the square will always be parallel to itself and to the first named side of the board, and will thus represent the meridian passing through any station. CHAP. Iv.] Platting the Surey. 159 If a straight-edged drawing Fig. 18. board or table cannot be procured, nail down on a table of any shape a straight-edged ru- / ler, and slide along against it the outside of the stock of a T square, one side of the stock being flush with the blade. A parallel ruler may also be used, one part of it being _ screwed down to the board in the proper position. If none of these means are at hand, approximately parallel meridians may be drawn by the edges of a common ruler, at distances apart equal to its width, and the diameter of the protractor made parallel to them by measuring equal distances between it and them. (266) To plat a survey with these instruments, mark, with a fine point enclosed in a circle, a convenient spot in the paper to represent the first station, 1 in the figure. Its place must be so chosen Fig. 181. 2 8!|z310 2 jr^ y} ^ /~~~~~~~~~~~D 160 COMPASS SURVEYING. [PART III that the plat may not " run off" the paper. With the T square draw a meridian through it. The top of the paper is usually, though not necessarily, called North. With the protractor lay off the angle of the first bearing, as directed in Art. (264). Set off the length of the first line, to the desired scale, by Art. (42), from 1 to 2. The line 1 —— 2 represents the first course. Through 2, draw another meridian, lay off the angle of the second course, and set off the length of this course, from 2 to 3. Proceed in like manner for each course. When the last course is platted, it should end precisely at the starting point, as the survey did, if it were a closed survey, as of a field. If the plat does not " close," or " come together," it shows some error or inaccuracy either in the original survey, if that have not been " tested" by Latitudes and Departures, or in the work of platting. A method of correction is explained in Art. (268). The plat here given is the same as that of Fig. 175, page 151. This manner of laying down the directions of lines, by the angles which they make with a meridian line, has a great advantage, in both accuracy and rapidity, over the method of platting lines by the angles which each makes with the line which comes before it. In the latter method, any error in the direction of one line makes all that follow it also wrong in their directions. In the former, the direction of each line is independent of the preceding line, though its position would be changed by a previous error. Instead of drawing a meridian through each station, sometimes only one is drawn, near the middle of the sheet, and all the bearings of the survey are laid off from some one point of it, as shown in the figure, and numbered to correspond with the stations from which these bearings were taken. The circular protractor is convenient for this. They are then transferred to the places where they are wanted, by a triangle or other parallel ruler, as explained on page 27. The figure at the top of the next page represents the same field platted by this method. A semi-circular protractor is sometimes attached to the stock end of the T square, so that its blade may be set at any desired angle with the meridian, and any bearing be thus protracted without drawing a meridian. It has some inconveniences. CHAP Iv.1 Platting the Survey. 161 Fig. 182. N 3 AY S 5 (267) The Compass itself may be used to plat bearings. For this purpose it must be attached to a square board so that the N and S line of the compass box may be parallel to two opposite edges of the board. This is placed on the paper, and the box is turned till the needle points as it did when the first bearing was taken. Then a line drawn by one edge of the board will be in a proper direction. Mark off its length, and plat the next and the succeeding bearings in the same manner. (268) When the plat of a survey does not " close," it may be corrected as follows. Let Fig. 183. ABCDE be the boundary B' lines platted according to L'- C the given bearings and distances, and suppose that the last course comes to E, Ac^, / / instead of ending at A, as _ / / it should. Suppose also' / that there is no reason to E -- suspect any single great D error, and that no one of the lines was measured over very rough 11 162 COMPASS SURVEYING. [PART 11I. ground, or was specially uncertain in its direction when observed. The inaccuracy must then be distributed among all the lines in proportion to their length. Each point in the figure, B, C, D, E, must be moved in a direction parallel to EA, by a certain distance which is obtained thus. Multiply the distance EA by the distance AB, and divide by the sum of all the courses. The quotient will be the distance BB'. To get CC', multiply EA by AB + BC, and divide the product by the same sum of all the courses. To get DD', multiply EA by AB + BC + CD, and divide as before. So for any course, multiply by the sum of the lengths of that course and of all those preceding it, and divide as before. Join the points thus obtained, and the closed polygon AB'C'D'A will thus be formed, and will be the most probable plat of the given survey.* The method of Latitudes and Departures, to be explained hereafter, is, however, the best for effecting this object. (269) Field Platting. It is sometimes desirable to plat the courses of a survey in the field, as soon as they are taken, as was mentioned in Art. (247), under the head of "Keeping the fieldnotes." One method of doing this is to have the paper of the Field-book ruled with parallel lines, at unequal distances apart, and to use a rectangular pro- Fig. 184. tractor (which may be made of Bristol-board, or other stout -/ drawing paper,) with lines ruled across it at equal distances / 9 of some fraction of an inch. A - bearing having been taken and O |._ noted, the protractor is laid on X the paper and its centre placed at the station where the bearing is to be laid off. It is then turned till one of its cross-lines coincides with some one of the lines on the paper, which represent East and West lines. The long side of, the protractor will then be on a meridian and the proper angle (40~ in the figure) can be at once marked off. The length of the course can also be set off by the equal spaces between the cross-lines, letting each space represent any convenient number of links. * This was demonstrated by Dr. BOWDITCH, in No. 4, of " The Analyst' CHAP. IV.] Platting the Survey. 163 (270) A common rectangular protractor without any cross-lines, or a semi-circular one, can also Fig. 185. be used for the same purpose. The parallel lines on the paper (which, in this method, may be equi-distant, as in common ruled writing paper) will now S represent meridians. Place the centre of the protractor on the meridian nearest to the station at which the angle is to belaid off, and turn it till the - given number of degrees is cut by the meridian. Slide the protractor up or down the meridian (which must continue to pass through the centre and the proper degree) till its edge passes through the station, and then draw by this edge a line, which will have the bearing required. (271) Paper ruled into squares, (as are sometimes the righthand pages of surveyors' field-books), may be used for platting bearings in the field. The lines running up the page may be called North and South lines, and those running across the page will then be East and West lines. Any course of the survey will be the hypothenuse of a right-angled triangle, and the ratio of its other two sides will determine the Fig.'16. angle. Thus, if the ratio of i C 1 _ " B the two sides of the right-an- _ I_ A -_ gled triangle, of which the line AB in the figure is the hypoth-i enuse, is 1, that line makes an angle of 45~ with the meridian. / - If the ratio of the long to the /L 7 short side of the right-angled - - triangle of which the line AC -// - - is the hypothenuse, is 4 to 1, _ the line AC makes an angle A of 14~ with the meridian. The line AD, the hypothenuse of an 164 COMPASS SURVEYING. [PART mI. equal triangle, which has its long side lying East and West, makes likewise an angle of 14~ with that side, and therefore makes an angle of 760 with the meridian.* To facilitate the use of this method, the following table has been prepared. TABLE FOR PLATTING BY SQUARES. -) cc) Ratio of g.k cc Ratio of M Ratio of Ratio of a of lngsideto o O" long side to lnsideto too shot side. short side.,A A short side. A I1. 57 3 to 1 893 16W 3.49 to 1 74' 31~ 1.664 to 1 590 2~ 28.6 to 1 880 17~ 3.27 to 1 730 32~ 1.600 to 1 580 30 19.1 to 1 870 18~ 3.08 to 1 72~ 333 1.540 to 1 570 4~ 14.3 to 1 86~ 19~ 2.90 to 1 71~ 340 1.483 to 1 56~ 50 11.4 to 1 85~ 20~ 2.75 to 1 700 350 1.428 to 1 550 6~ 9.5 to 1 84~ 210 2.61 to 1 69~ 36c 1.376 to 1 540 70 8.1 to 1 83~ 22~ 2.48 to 1 68~ 370 1.327 to 1 530 80 7.1 to 1 820 23~ 2.36 to 1 670 38~ 1.280 to 1 520 9~ 6.3 to 1 810 24~ 2.25 to 1 660 390 1.235 to 1 510 10~ 5.7 to 1 80~ 25~ 2.14 to 1 650 400 1.192 to 1 500 11~ 5.1 to 1 790 260 2.05 to 1 640 41~ 1.150 to 1 490 12~ 4.9 to 1 78~ 270 1.96 to 1 63~ 42~ 1.111 to 1 48~ 130 4.3 to 1 77~ 280 1.88 to 1 620 430 1.072 to 1 470 14~ 4.0 to 1 76~ 29 1.80 to 1 610 440 1.036 to 1 460 153 3.7 to 1 750 300o1.73 to 1 60~ 45~ 1.000 to 1 450 To use this table, find in it the ratio corresponding to the angle which you wish to plat. Then count, on the ruled paper, any number of squares to the right or to the left of the point which represents the station, according as your bearing was East or West; and count upward or downward according as your bearing was North or South, the number of squares given by multiplying the first number by the ratio of the Table. Thus; if the given bearing from A in the figure, was N. 200 E. and two squares were counted to the right, then 2 x 2.75 = 51 squares, should be counted upward, to E, and AE would;be the required course. (272) With a paper protractor. Engraved paper protractors may be obtained from the instrument-makers, and are very conve* Ti}fs ind all the following ratios may be obtained directly from Trigonometrical Tables; for the ratio of the long side to the short side, the latter being tak' n as unity, is the natural cotangent of the angle. CHAP. IV.] Platting the Survey. 165 nient. A circle of large size, divided into degrees and quarters, is engraved on copper, and impressions from it are taken on drawing paper. The divisions are not numbered. Draw a straight line to represent a meridian, through the centre of the circle, in any convenient direction. Number the degrees from 0 to 90~, each way from the ends of this meridian, as on the compass-plate. The protractor is now ready for Fig. 187. use. Choose a convenient point for the first station. Suppose the first bearing to ~ / beN.30~E. The line pass- ing through the centre of the 2w W E circle and through the oppo- site points N. 30~ E. and S. 3 300 W. has the.bearing re-/ o, quired. But it does not pass S through the station 1. Transfer it thither by drawing through station 1 a line parallel to it, which will be the course required, its proper length being set off on it from 1 to 2. Now suppose the bearing from 2 to be S. 60~ E. Draw through 2 a line parallel to the line passing through the centre of the circle and through the opposite points S. 60~ E., and N. 60~ W., and it will be the line desired, On it set off the proper length from 2 to 3, and so proceed. When the plat is completed, the engraved sheet is laid on a clean one, and the stations " pricked through," and the points thus obtained on the clean sheet are connected by straight lines. The pencilled plat is then rubbed off from the engraved sheet, which can be used for a great number of plats. If the central circle be cut out, the plat, if not too large, can be made directly on the paper where it is to remain. The surveyor can make such a paper protractor for himself, with great ease, by means of the Table of Chords at the end of this volume, the use of which is explained in Art. (275). The engraved ones may have shrunk after being printed. Such a circle is sometimes drawn on the map itself. This will be particularly convenient if the bearings of any lines on the map, 166 COMPASS SURVEYING. [PART I[I. not taken on the ground, are likely to be required. If the map be very long, more than one may be needed. (273) Drawing-Board Protractor. Such a divided circle. as has just been described, or a circular protractor, may be placed on a drawing board near its centre, and so that its 0~ and 90~ lines are parallel to the sides of the drawing board. Lines are then to be drawn, through the centre and opposite divisions, by a ruler long enough to reach the edges of the drawing board, on which they are to be cut in, and numbered. The drawing board thus becomes, in fact, a double rectangular protractor. A strip of white paper may have previously been pasted on the edges, or a narrow strip of white wood inlaid. When this is to be used for platting, a sheet of paper is put on the board as usual, and lines are drawn by a ruler laid across the 0~ points and the 90~ points, and the centre of the circle is at once found, and should be marked 0. The bearings are then platted as in the last method. (274) With a scale of chords. On the plane scale contained in cases of mathematical drawing instruments will be found a series of divisions numbered from 0 to 90, and marked CH, or C. This is a scale of chords, and gives the lengths of the chords of any arc for a radius equal in length to the chord of 60~ on the scale. To lay off an angle with this scale, as for Fig. 188. example, to draw a line making at A an angle B of 40~ with AB, take, in the dividers, the dis- C tances from 0 to 60 on the scale of chords; with D this for radius and A for centre, describe an in- g o definite arc CD. Take the distance from 0 to 40 on the same scale, and set it off on the arc as a chord, from C to some point D. Join AD, and A prolong it. BAE is the angle required. The Sector, represented on page 36, supplies a modification of this method, sometimes more convenient. On each of its legs is a scale marked C, or CH. Open it at pleasure; extend the compass from 60 to 60, one on each leg, and with this radius describe an arc. Then extend the compasses from 40 to 40, and the dis CHAP. iv.] Platting the Survey. 167 tance will be the chord of 40~ to that radius. It can be set off as above. The smallness of the scale renders the method with a scale of chords practically deficient in exactness; but it serves to illustrate the next and best method. (275) With a Table of chords. At the end of this volume will be found a Table of the lengths of the chords of arcs for every degree and minute of the quadrant, calculated for a radius equal to 1. To use it, take in the compasses one inch, one foot, or any other convenient distance (the longer the better) divided into tenths and hundredths, by a diagonal scale, or otherwise. ~ With this as radius describe an arc as in the last case. Find in the table of chords the length of the chord of the desired angle. Take it from the scale just used, to the nearest decimal part which the scale will give. Set it off as a chord, as in the last figure, and join the point thus obtained to the starting point. This gives the angle desired. The superiority of this method to that which employs a protractor, is due to the greater precision with which a straight line can be divided than can a circle. A slight modification of this method is to take in the compasses 10 equal parts of any convenient length, inches, half inches, quarter inches, or any other at hand, and with this radius describe an arc as before, and set off a chord 10 times as great as the one found in the Table, i. e. imagine the decimal point moved one place to the right. If the radius be 100 or 1000 equal parts, imagine the decimal point moved two, or three, places to the right. Whatever radius may be taken or given, the product of that radius into a chord of the Table, will give the chord for that radius. This gives an easy and exact method of getting a right angle; by describing an arc with a radius of 1, and setting off a chord equal to 1.4142. If the angle to be constructed is more than 90~, construct on the other side of the given point, upon the given line prolonged, an angle equal to what the given angle wants of 180~; i. e. its Supplement, in the language of Trigonometry. 168 COMPASS SURVEYING. [PART III This same Table gives the means of measuring any angle, With the angular point for a centre, and 1, or 10, for a radius, describe an arc. Measure the length of the chord of the arc between the legs of the angle, find this length in the Table, and the angle corresponding to it is the one desired.* (276) With a Table of natural sines. In the absence of a Table of chords, heretofore rare, a table of natural sines, which can be found anywhere, may be used as a less convenient substitute. Since the chord of any angle equals twice the sine of half the angle, divide the given angle by two; find in the table the natural sine of this half angle; double it, and the product is the chord of the whole angle. This can then be used precisely as was the chord in the preceding article. An ingenious modification of this method has been much used. Describe an arc from the given point as centre, as in the last two articles, but with a radius of 5 equal parts. Take, from a Table, the length of the natural sine of half the given angle to a radius of 10. Set off this length as a chord on the arc just described, and join the point thus obtained to the given point.' (277) By Latitudes and Departures. When the Latitudes and Departures of a survey have been obtained and corrected, (as explained in Chapter V), either to test its accuracy, or to obtain its content, they afford the easiest and best means of platting it. The description of this method will be given in Art. (285). * This Table will also serve to find the natural sine, or cosine, of any angle. Multiply the given angle by two; find, in the Table, the chord of this double angle; and half of this chord will be the natural sine required. For, the chord of any angle is equal to twice the sine of half the angle. To find the cosine, proceed as above, with the angle which added to the given angle would make 90~. Another use of this Table is to inscribe regular polygons in a circle by setting off the chords of the arcs which their sides subtend. Still another use is to divide an arc or angle into any number of equal parts, by setting off the fractional arc or angle. Fig. 189. t The reason of this is apparent from the figure. DE is the sine of half the angle 0 BAC, to a radius of 10 equal parts, and'O=~ / BC is the chord directed to be set off, to a radius of 5 *qual parts. BC is equal to DE; / for BC = 2.BF, by Trigonometry, and DE' = 2.BF, by similar triangles; hence BC = DE. CHAPTER V. LATITUDES AND DEPARTURES. (278) Definitions. The LATITUDE of a point is its distance North or South of some " Parallel of Latitude," or line running East or West. The LONGITUDE of a point is its distance East or West of some'"Meridian," or line running North and South. In Compass-Surveying, the Magnetic Meridian, i. e. the direction in which the Magnetic Needle points, is the line from which the Longitudes of points are measured, or reckoned. The distance which one end of a line is due North or South of the other end, is called the Difference of Latitude of the two ends of the line; or its Northing or Southing; or simply its Latitude. The distance which one end of the line is due East or West of the other, is here called the Diference of Longitude of the two ends of the line; or its Easting or Westing; or its Departure. Latitudes and Departures are the most usual terms, and will be generally used hereafter, for the sake of brevity. This subject may be illustrated geographically, by noticing that a traveller in going from New-York to Buffalo in a straight line, would go about 150 miles due north, and 250 miles due west. These distances would be the differences of Latitude and of Longitude between the two places, or his Northing and Westing. Returning from Buffalo to New-York, the same distances would be his Southing and Easting.* In mathematical language, the operation of finding the Latitude and Longitude of a line from its Bearing and Length, would be called the transformation of Polar Co-ordinates into Rectangular Co-ordinates. It consists in determining, by our Second Principle, the position of a point which had originally been determined by the Third Principle. Thus, in the figure, (which is the same as * It should be remembered that the following discussions of the Latitudes and Longitudes of the points of a survey will not always be fully applicable to those of distant places, such as the cities just named, in consequence of the surface of the earth not being a plane. 170 COMPASS SURVEYING. [PART III. that of Art.(9)), the point S is determin- Fig. 190. ed by the angle SAC and by the distance AS. It is also determined by the distances AC and CS, measured at right' \ angles to each other; and then, supposing A CS to run due North and South, CS will be the Latitude, and AC the Departure of the line AS. (279) Calculation of Latitudes and Departures. Let AB be a given line, of which the length Fig. 191. AB, and the bearing (or angle, BAC, N which it makes with the Magnetic Meridian), are known. It is required to find the differences of Latitude and of Longitude between its two extremities A and B: that is, to find AC and CB; or, what is the same thing, BD. v —-— _E — and DA. It will be at once seen that AB is the hypothenuse of a right-angled tri- angle, in which the " Latitude" and the " Departure" are the sides about the right angle. We therefore know, from the principles of trigonometry, that AC = AB. cos. BAC, BC AB. sin. BAC. Hence, to find the Latitude of any course, multiply the natural cosine of the bearing by the length of the course; and to find the Departure of any course, multiply the natural sine of the bearing by the length of the course. If the course be Northerly, the Latitude will be North, and will be marked with the algebraic sign +, plus, or additive; if it be Southerly, the Latitude will be South, and will be marked with the algebraic sign -, minus, or subtractive. If the course be Easterly, the Departure will be East, and marked +, or additive; if the course be Westerly, the Departure will be West, and marked -, or subtractive. CHAP. V.] Latitudes and Departures. 171 (280) Formulas. The rules of the preceding article may be expressed thus; Latitude = Distance x cos. Bearing, Departure = Distance x sin. Bearing.* From these formulas may be obtained others, by which, when any two of the above four things are given, the remaining two can be found. When the Bearing and Latitude are given; Distance = Latitnle = Latitude x sec. Bearing, cos. Bea.ig ~ Departure = Latitude x tang. Bearing. When the Bearing and Departure are given; Distance = Depart'le = Departure x cosec. Bearing, sin. Bearig Latitude = Departure x cotang. Bearing. When the Distance and Latitude are given; Latitudl Cos. Bearing = atitudce Departure = Latitude x tang. Bearing. When the Distance and Departure are given; Sin. Bearin Departure Sin. Bearing = Distance' Latitude = Departure x cotang. Bearing. When the Latitude and Departure are given; Depalrtullre Tang. of Bearing = patiture Distance = Latitude x sec. Bearing. Still more simply, any two of these three-Distance, Latitude and Departure-being given, we have Distance = V(Latitude2 + Departure2) Latitude_ = / (Distance2 -Departure2) Departure = V/(Distance2 -Latitude2) (281) Traverse Tables. The Latitude and Departure of any distance, for any bearing, could be found by the method given in Art. (279), with the aid of a table of Natural Sines. But to * Whenever sines, cosines, tangents, &c., are here named, they mean the natu ral sines, &c., of an arc described with a radius equal to one, or to the unit by which the sines. &c., are measured. 172 COMPASS SURVEYING. [PART I1 facilitate these calculations, which are of so frequent occurrence and of so great use, Traverse Tables have been prepared, origin. ally for navigators, (whence the name Traverse), and subsequently for surveyors.* The Traverse Table at the end of this volume gives the Latitude and Departure for any bearing, to each quarter of a degree, and for distances from 1 to 9. To use it, find in it the number of degrees in the bearing, on the left hand side of the page, if it be less than 45~, or on the right hand side if it be more. The numbers on:he same line running across the page,f are the Latitudes and Departures for that bearing, and for the respective distances-1, 2, 3, 4, 5, 6, 7, 8, 9,which are at the top and bottom of the page, and which may represent chains, links, rods, feet, or any other unit. Thus, if the bearing be 15~, and the distance 1, the Latitude would be 0.966 and the Departure 0.259. For the same bearing, but a distance of 8, the Latitude would be 7.727, and the Departure 2.071. Any distance, however great, can have its Latitude and Departure readily obtained from this table; since, for the same bearing, they are directly proportional to the distance, because of the similar triangles which they form. Therefore, to find the Latitude or Departure for 60, multiply that for 6 by 10, which merely moves the decimal point one place to the right; for 500, multiply the numbers found in the Table for 5, by 100, i. e. move the decimal point two places to the right, and so on. Merely moving the decimal point to the right, one, two, or more places, will therefore enable this Table to give the Latitude and Departure for any decimal multiple of the numbers in the Table. For compound numbers, such as 873, it is'only necessary to find separately the Latitudes and Departures of 800, of 70, and of 3, and add them together. But this may be done, with scarcely any risk of error, by the following simple rule. * The first Traverse Table for Surveyors seems to have been published in 1791, by John Gale. The most extensive table is that of Capt. Boileau, of the British army, being calculated for every minute of bearing, and to five decimal places, for distances from 1 to 10, The Table in this volume was calculated for it, and then compared with the one just mentioned. t In using this or any similar Table, lay a ruler across the page, just above or below the line to be followed out. This is a very valuable mechanica. assistance. CHAP. v.] Latitudes and Departures. 173 Write down the Latitude and Departure for the first figure of the given number, as found in the Table, neglecting the decimal point; write under them the Latitude and Departure of the second figure, setting them one place farther to the right; under them write the Latitude and Departure of the third figure, setting them one place farther to the right, and so proceed with all the figures of the given number. Add up these Latitudes and Departures, and cut off the three right hand figures. The remaining figures will be the Latitude and Departure of the given number in links, or chains, or feet, or whatever unit it was given in. For example; let the Latitude and Departure of a course having a distance of 873 links, and a bearing of 20~, be required. In the Table find 20~, and then take out the Latitude and Departure for 8, 7 and 3, in turn, placing them as above directed, thus: Distances. Latitudes. Departures. 800 7518 2736 70 6578 2394 3 2819 1026 873 820.399 298.566 Taking the nearest whole numbers and rejecting the decimals, we find the desired Latitude and Departure to be 820 and 299.* When a 0 occurs in the given number, the next figure must be set two places to the right, the reason of which will appear from the following example, in which the 0 is treated like any other number. Given, a bearing of 35~, and a distance of 3048 links. Distances. Latitudes. Departures. 3000 2457 1721 000 0000 0000 40 3277 2294 8 6553 4589 3048 2496.323 1748.529 Here the Latitudes and Departures are 2496 and 1749 links. * It is frequently doubtful, in many calculations, when the final decimal is 5, whether to increase the preceding figure by one or not. Thus, 43.5 may be called 43 or 44 with equal correctness. It is better in such cases not to increase the whole number, so as to escape the trouble of changing the original figure, and the increased chance of error. If, however, more than one such a case occurs in the same column to be added up, the larger and smaller number should be taken alternately. 174 COMPASS SURVEYING. [PART III. When the bearing is over 45~, the names of the columns must be read from the bottom of the page, the Latitude of any bearing, as 50~, being the Departure of the complement of this bearing, or 40~, and the Departure of 40~ being the Latitude of 50~, &c. The reason of this will be at once seen on inspecting the last figure, (page 170), and imagining the East and West line to become a Meridian. For, if AC be the magnetic meridian, as before, and therefore BAC be the bearing of the course AB, then is AC the Latitude, and CB the Departure of that course. But if AE be the meridian and BAD (the complement of BAC) be the bearing, then is AD (which is equal to CB) the Latitude, and DB, (which is equal to AC), the Departure. As an example of this, let the bearing be 638~, and the distance 3469 links. Proceeding as before, we have Distances. Latitudes. Departures. 3000 1350 2679 400 1800 3572 60 2701 5358 9 4051 8037 3469. 1561.061 3097.817 The required Latitude and Departure are 1561 and 3098 links. In the few cases occurring in Compass-Surveying, in which the bearing is recorded as somewhere between the fractions of a degree given in the Table, its Latitude and Departure may be found by interpolation. Thus, if the bearing be 10 ~, take the half sum of the Latitudes and Departures for 10'~ and 10~. If it be 10~ 20', add one-third of the difference between the Lats. and Deps. for 101 and for 10"~, to those opposite to 10~~; and so in any similar case. The uses of this table are very varied. The principal applications of it, which will now be explained, are to Testing the accuracy of surveys; to Supplying omissions in them; to Platting them, and to Calculating their content.* * The Traverse Table admits of many other minor uses. Thus, it may be used for solving, approximately, any right-angled triangle by mere inspection, the tearing being taken for one of the acute angles; the Latitude being the side adjacent, the Departure the side opposite, and the Distance the hypothenuse. Any two of these being given, the others are given by the Table. The Table will therefore serve to show the allowance to be made in chaining on slopes (see Art. CHAP. V.] Latitudes and Departures. 175 (282) Application to Testing a Survey. It is self-evident, that when the surveyor has gone completely arouni a field or farm, taking the bearings and distances of each boundary line, till he has got back to the starting point, that he has gone precisely as far South as North, and as far West as East. But the sum of the North Latitudes tells how far North he has gone, and the sum of the South Latitudes how far South he has gone. Hence these two sums will be equal to each other, if the. survey has been correctly made. In like manner, the sums of the East and of the West Departures must also be equal to each other. We will apply this principle to testing the accuracy of the survey of which Fig. 175, page 151, is a plat. Prepare seven columns, and head them as below. Find the Latitude and Departure of each course to the nearest link, and write them in their appropriate columns. Add up these columns. Then will the difference between the sums 9f the North and South Latitudes, and between the sums of the East and West Departures, indicate the degree of accuracy of the survey. LATITUDE.. DEPARTURE. STATION. BEARING. DISTANCE. _______ N. S. E. W. 1 N. 35~ E. 2.70 2.21 1.55 2 N. 831 E. 1.29.15 1.28 3 S. 57 E. 2.22 1.21 1.86 4 S. 34 0W. 3.55 2.93 2.00 5 N.56a0W. 3.23 1.78 2.69 4.14 4.14 4.69 4.69 The entire work of the above example is given below. 350 1638 1147 34~O 2480 1688 57340 40150 4133 2814 --—' -~ — 4133 2814 270. 221.140 154.850 355. 293.463 199.754 (26)); for, look in the column of bearings for the slope of the ground, i. e. the angle it makes with'the horizon, find the given distance, and the Latitude corresponding will be the desired horizontal measurement, and the difference between it and the Distance will be the allowance to be made. 176 COMrPASS SURVETING. [PART inr 88~ 113 994 5610 1656 2502 226 1987 1104 1668 1019 8942 1656 2502 129. 14.579 128.212 323. 178.296 269.382 570 1089 1677 5~ 1089 16777 The nearest link is taken 1089 1677 to be inserted in the Table, ____ and the remaining Decimals 222. 120.879 186.147 are neglected.: In the preceding example the respective sums were found to be exactly equal. This, however, will rarely occur in an extensive survey. If fie difference be great, it indicates some mistake, and the survey must be repeated with greater care; but if the difference be small it indicates, not absolute errors, but only inaccuracies, unavoidable in surveys with the compass, and the survey may be accepted. How great a difference in the sums of the columns may be allowed, as not necessitating a new survey, is a dubious point. Some surveyors would admit a difference of 1 link for every 3 chains in the sum of the courses: others only 1 link for every 10 chains. One writer puts the limit at 5 links for each station; another at 25 links in a survey of 100 acres. But every practical surveyor soon learns how near to an equality his instrument and his skill will enable him to come in ordinary cases, and can therefore establish a standard for himself, by which he can judge whether the difference, in any survey of his own, is probably the result of an error, or only of his customary degree of inaccuracy, two things to be very carefully distinguished.* (283) Application to supplying omissions. Any two omissions in the Field-notes can be supplied by a proper use of the method of Latitudes and Departures; as will be explained in Part VII, which treats of " Obstacles to Measurement," under which head this subject most appropriately belongs. But a knowledge of the fact that any two omissions can be supplied, should not lead * A French writer fixes the allowable difference in chaining at 1-400 of level lines; 1-200 of lines on moderate slopes; 1-100 of lines on steep slopes. CHAP V.] Latitudes and Departures. 177 the young surveyor to be negligent in making every possible measurement, since an omission renders it necessary to assume all the notes taken to be correct, the means of testing them no longer existing. (284) Balancing a Survey. The subsequent applications of this method require the survey to be previously Balanced. This operation consists in correcting the Latitudes and Departures of the courses, so that their sums shall be equal, and thus " balance." This is usually done by distributing the differences of the sums among the courses in proportion to their length; saying, As the sum of the lengths of all the courses Is to the whole difference of the Latitudes, So is the length of each course To the correction of its Latitude. A similar proportion corrects the Departures.* It is not often necessary to make the exact proportion, as the correction can usually be made, with sufficient accuracy, by noting how much per chain it should be, and correcting accordingly. In the example given below, the differences have purposely been made considerable. The corrected Latitudes and Departures have been here inserted in four additional columns, but in practice they should be written in red ink over the original Latitudes and Departures, and the latter crossed out with red ink. LATITUDES.lDE'TURES. CORRECTED CORRECTED STA. BEARING. DIST. STA. BE.RINO. DIST. LATITUDES. D EP.LTZUDES. DEPARTURES. N.+ S.- E.+ W.- N.+ S. E.+ W.1 N. 52~ E. 10.63 6.54 8.38 6.58 8.34 2 S. 29~0 E. 4.10 3.56 2.03 3.65 2.01 3 S. 31~0W. 7.69 6.54 4.05 6.51 4.08 4 N. 61~ W.'7.13 3.46 6.24 3.48 6.27 29.55 1 -10 10.10 10.41 9 10.06 110.06 10.35 10.35 The corrections are made by the following proportions; the nearest whole numbers being taken: For the Latitudes. For the )Departures. 29.55: 10.63:: 10: 4 29.55: 10.63:: 12: 4 29.55: 4.10: 10: 1 29.55: 4.10: 12: 2 29.55: 7.69::10: 3 29.55: 7.69:: 12: 3 29.55: 7.13: 10: 2 29.55 7.13: 12: 3 10 12 *A demonstration of this principle was given by Dr. Bowditch, in No. 4 of "The Analyst." 12 178 COMPASS SURVEYING. [PART II1 This rule is not always to be strictly followed. If one line of a survey has been measured over very uneven and rough ground, or if its bearing has been taken with an indistinct sight, while the other lines have been measured over level and clear ground, it is probable that most of the error has occurred on that line, and the correction should be chiefly made on its Latitude and Departure. If a slight change of the bearing of a long course will favor the Balancing, it should be so changed, since the compass is much more subject to error than the chain. So, too, if shortening any doubtful line will favor the Balancing, it should be done, since distances are generally measured too long. (285) Application to Platting. Rule three columns; one for Stations; the next for total Latitudes; and the third for total Departures. Fill the last two columns by beginning at any convenient station (the extreme East or West is best) and adding up (algebraically) the Latitudes of the following stations, noticing that the South Latitudes are subtractive. Do the same for the Departures, observing thatthe Westerly ones are also subtractive. Taking the example given on page 175, Art. (282), and beginning with Station 1, the following will be the results: TOTAL LATITUDES TOTAL DEPARTURES AFROM STATION 1. FROM STATION 1. 1 0.00 0.00!T o.o o oo 2 +2.21 N. +1.55 E. 3 +2.36 N. +2.83 E. 4 +1.15 N. +4.69 E. 5 -1.78 S. +2.69 E. 1 0.00 0.00 It will be seen that the work proves itself, by the total Latitudes and Departures for Station 1, again coming out equal to zero. To use this table, draw a meridian through the point taken for Station 1, as in the figure on the following page. Set off, upward from this, along the meridian, the Latitude, 221 links, to A, and from A, to the right perpendicularly, set off the Departure, 155 links.* This gives the point 2. Join 1....2. From 1 again, set " This is most easily done with the aid of a right-angled triangle, sliding one of the sides adjacent to the right angle along the blade of the square, to which the other side will then be perpendicular. CHAP. V.] Latitudes and Departures. 179 off, upward, 236 Fig. 192. links, to B, and from B, to the right, per- __ As pendicularly, set off A 283 links, which will fix the point 3. Join 2....3; and so pro- c — - --------- ------- 4 ceed, setting off North Latitudes along the Meridian / upwards, and South Latitudes along it downwards; East Departures perpendicularly to the right, D' and West Depar- tures perpendicularly to the left. The advantages of this method are its rapidity, ease and accuracy; the impossibility of any error in platting any one course affecting the following points; and the certainty of the plat "coming together," if the Latitudes and Departures have been "Bal CHAPTER VI. CALCULATING THE CONTENT. (286) Methods. WHEN a field has been platted, by whatever method it may have been surveyed, its content can be obtained from its plat by dividing it up into triangles, and measuring on the plat their bases and perpendiculars; or by any of the other means explained in Part I, Chapter IV. But these are only approximate methods; their degree of accuracy depending on the largeness of scale of the plat, and the skill of the draftsman. The invaluable method of Latitudes and Departures gives another means, perfectly accurate, and not requiring the previous preparation of a plat. It is sometimes called the Rectangular, or the Pennsylvania, or Rittenhouse's, method of calculation.* (287) Definitions. Imagine a Meridian line to pass through the extreme East or West corner of a field. According to the definitions established in Chapter V, Art. (278), (and here recapitulated for convenience of reference), the perpendicular distance of each Station from that Meridian, is the Longitude of that Station; additive, or plus, if East; subtractive, or minus, if West. The distance of the middle of any line, such as a side of the field, from the Meridian, is called the Longitude of that side.t The difference of the Longitudes of the two ends of a line is called the Departure of that line. The difference of the Latitudes of the two ends of a line is called the Latitude of the line. * It is, however, substantially the same as Mr. Thomas Burgh's "Method to determine the areas of right lined figures universally," published nearly a century ago. t The phrase " Meridian Distance," is generally used for what is here called "Longitude"; but the analogy of " Differences of Longitude" with " Differences of Latitude," usually but anomalously united with the word " Departure," borrowed from Navigation, seems to put beyond all question the propriety of the innovation here introduced. CHAP. VI.] Calculating the Content. 181 (288) Longitudes. To give more definiteness to the develop ment of this subject, the figure in the margin will be referred to, and may be considered to represent any space enclosed by straight lines. Let NS be the Meridian passing through the extreme Westerly Station of the field ABCDE. From Fij. 193. the middle and ends of each side N C draw perpendiculars to the Meridi- / an. These perpendiculars will be J —.. \ the Longitudes and Departures of/ the respective sides. The Longitude, FG, of the first course, AB, is evidently equal to half its Depar- - ture HB. The Longitude, JK, of the second course, BC, is equal to A JL + LM + MK, or equal to the, -__ _y_. - Longitude of the preceding course, - z. / plus half its Departure, plus half I the Departure of the course itself. iT- -- --- The Longitude, YZ, of some other |E course, as EA, taken anywhere, is $ equal to WX - VX - UV, or equal to the Longitude of the preceding course, minus half its Departure, minus half the Departure of the course itself; i. e. equal to the Algebraic sum of these three parts, remembering that Westerly Departures are negative, and therefore to be subtracted when the directions are to make an Algebraic addition. To avoid fractions, it will be better to double each of the preceding expressions. We shall then have a GENERAL RULE FOR FINDING DOUBLE LONGITUDES. The Double Longitude of the FIRST COURSE. is equal to its Departure. The Double Longitude of the SECOND COURSE is equal to the Double Longitude of the first course, plus the Departure of that course, plus the Departure of the second course. The Double Longitude of the THIRD COURSE is equal to the Double Longitude of the second course, plus the Departure of that course, plus the Departure of the course itself. 182 COMPASS SURVEYING. [PART III. The Double Longitude of ANY course is equal to the Double Longitude of the preceding course, plus the Departure of that course, plus the Departure of the course itself.* The Double Longitude of the last course (as well as of the first) is equal to its Departure. Its " coming out" so, when obtained by the above rule, proves the accuracy of the calculation of all the preceding Double Longitudes. (289) Areas. We will now proceed to find the Area, or Content of a field, by means of the " Double Longitudes" of its sides, which can be readily obtained by the preceding rule, whatever their number. (290) Beginning with a three-sided field, ABC in the figure, draw a Meridian through A, and draw perpendi- Fig. 194. culars to it as in the last figure. It is N plain that its content is equal to the differ- X ence of the areas of the Trapezoid DBCE, D B and of the Triangles ABD and ACE. The area of the Triangle ABD is equal A to the product of AD by half of DB, or to H - the product of AD by FG; i. e. equal to x — the product of the Latitude of the 1st course by its Longitude. -- The area of the Trapezoid DBCE is equal C to the product of DE by half the sum of DB and CE, or by HJ; i. e. to the product of S the Latitude of the 2d course by its Longitude. The area of the Triangle ACE is equal to the product of AE by half EC, or by KL; i. e. to the product of the Latitude of the 3d course by its Longitude. Calling the products in which the Latitude was North, North Products, and the products in which the Latitude was South, South Products, we shall find the area of the Trapezoid to be a South Product, and the areas of the Triangles to be North Pro* The last course is a " pr:eceding course" to the first course, as will appear on remembering that these two courses join each other on the ground. CHAP. VI.] Calculating the Content. 183 duets. The Difference of the North Products and the South Products is therefore the desired area of the three-sided field ABC. Using the Double Longitudes, (in order to avoid fractions), in each of the preceding products, their difference will be the double area of the Triangle ABC. (291) Taking now a four-sidedfield, ABCD in the figure, and drawing a Meridian and Longitudes as be- Fig. 195. fore, it is seen, on inspection, that its area N would be obtained by taking the two Trian-, B gles, ABE, ADG, from the figure EBCDGE, or from the sum of the two Trapezoids EBCF and FCDG. ----- The area of the Triangle AEB will be found, as in the last article, to be equal to A the product of the Latitude of the 1st course - by its Longitude. The Product will be North. The area of the Trapezoid EBCF will be -. —- D found to equal the Latitude of the 2d course by its Longitude. The product will be South. The area of the Trapezoid FCDG will be found to equal the product of the Latitude of the 3d course by its Longitude. The product will be South. The area of the Triangle ADG will be found to equal the product of the Latitude of the 4th course by its Longitude. The product will be North. The difference of the North and South products will there. fore be the desired area of the four-sided field ABCD. Using the Double Longitude as before, in each of the preceding products, their difference will be double the area of the field. (292) Whatever the number or directions of the sides of a field, or of any space enclosed by straight lines, its area will always be equal to half of the difference of the North and South Products 184 COMPASS SURVEYINIG. [PART III. arising from multiplying together the Latitude and Double Longitude of each course or side. We have therefore the following GENERAL RULE FOR FINDING AREAS. 1. Prepare ten columns, headed as in the example below, and in the first three write the Stations, Bearings and Distances. 2. Find the Latitudes and Departures of each course, by the Traverse Table, as directed in Art. (281), placing them in the four following columns. 3. Balance them, as in Art. (284), correcting them in red ink. 4. Find the Double Longitudes, as in Art. (288), with reference to a Meridian passing through the extreme East or West Station, and place them in the eighth column. 5. Multiply the Double Longitude of each course by the corrected Latitude of that course, placing the North Products in the ninth column, and the South Products in the tenth column. 6. Add up the last two columns, subtract the smaller sum from the larger, and divide the difference by two. The quotient will be the content desired. (293) To find the most Easterly or Westerly Station of a sur^vey, without a plat, it is best to make a rough hand-sketch of the survey, drawing the lines in an approximation to their true directions, by drawing a North and South, and East and West lines, and considering the Bearings as fractional parts of a right angle, or 90~; a course N. 45~ E. for example, being drawn about half way between a North and an East direction; a course N. 28~ W. being not quite one-third of the way around from North to West; and so on, drawing them of approximately true proportional lengths. (294) Example 1, given below, refers to the five-sided field, of which a plat is given in Fig. 175, page 151, and the Latitudes and Departures of which were calculated in Art. (282), page 175. Station 1 is the most Westerly Station, and the Meridian will be supposed to pass through it. The Double Longitudes are. best CHAP.,VI.] Calculating the Content 185 found by a continual addition and subtraction, STA. as in the margin, where they are marked D. L. - + 1.55 D. L ~ 1.55 The Double Longitude of the last course comes + 1.28 out equal to its Departure, thus proving the 2 + 4.38 D. L. + 1.28 work. + 1.86 + 1.86 The Double Longitudes being thus obtained, 3 + -.7.52 D.L are multiplied by the corresponding Latitudes, - 2.00 and the content of the field obtained as directed 4. 7.38 D0 L0 in the General Rule. - 2.69 This example may serve as a pattern for the 5 + 2.69 D. L. most compact manner of arranging the work. DIS- LATITUDES. DEP'TURES DOUBLE DOUBLE AREAS. STATIN. BEARNGS. TANCES. BN. + S. —E.+ W.- LONGITUDES. N.+ S. - 1 N. 35~ E. 2.70 2.21 1.55 + 1.55 3.4255 2 N. 83^o E. 1.29.15 1.28 + 4.38 0.6570 3 S. 57~ E. 2.22 1.21 1.86 + 7.52 9.0992 4 S. 34i~ W. 3.55 2.93 2.00 + 7.38 21.6234 5 N. 56J~ W. 3.23 1.78 2.69 +- 2.69 4.7882 4.14 14.141 4.69 1 4.69] 118.8707 130.7226 8.8707 (Content- 1A. OR. 15P. 2)21.8519 Square Chains, 10.9259 (295) The Meridian might equally well have STA. been supposed to pass through the most Easterly -4 -.00 D. L. station, 4 in the figure. The Double Longitudes - 2.69 could then have been calculated as in the mar- 5 6.:69 D. L. gin. They will of course be all West, or minus. + 1 55 The products being then calculated, the sum of 1 - 783 D. L. 5+ t.55 the North products will be found to be 29.9625, + 1.28 and of the South products 8.1106, and their + -.5.00 D. L. difference to be 21.8519, the same result as be- + 1.86 fore. 3 -1.86 (296) A number of examples, with and without answers, will now be given as exercises for the student, who should plat them by some of the methods given in the preceding chapter, using each of them at least once. He should then calculate their content by the method just given, and check it, by also calculating the area of the plat by some of the Geometrical or Instrumental methods given in Part I, Chapter IV; for no single calculation is ever reliable. 186 COMPASS SURVEYING. [PART IIl. All the examples (except the last) are from the author's actual surveys. 3 Fig. 196. Example 2, given below, is also fully worked out, as anoth- / er pattern for the student, who 4 need have no difficulty with any possible case if he strictly follows the directions which have been given. The plat is on a scale of 2 chains to 1 inch, (=:1584). DIS- LATITUDES. DEP'TURES. DOUBLE DOUBLE AREAS. sTATIsON. BEARINGS. STATION. TANCES. N. + S.= E.+ W.- LONGITUDES. N. + S. - 1 N. 12i~ E. 2.81 2.75.60 - 6.56 18.0400 2 N. 76~ W. 3.20.77 3.11 + 4.05 3.1185 3 S. 244~ W. 1.14 1.04.47 +.47.4888 4 S. 48~ E. 1.53 1.02 1.14 + 1.14 1.1628 5 S. 120~ E. 1.12 1.09.24 + 2.52 2.7468 6 S. 77~ E. 1.64.37 1.60 +'4 4.36 1.6132 3.52 1 3.52 3.58 13.58 1121.158516.0116 6.0116 C'ontent OA. 3R. IP. 2)15.1469 Square Chains, 7.5734 _ Example 3. -Example 4. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 N. 520 E. 10.64 1 S. 21~ W. 12.41 2 S. 290 E. 4.09 2 N. 83~ E. 5.86 3 S. 31|~ W. 7.68 3 N. 120 E. 8.25 4 N. 61~ W. 7.24 4 N. 470 W. 4.24 Ans. 4A. 3R. 28P. Ans. 4A. 2R. 37P. _ Example 5. Example 6. STA. BEARING. DISTANCE. STA. BEARING. ]DISTANCE. 1 N. 34~0E. 2.73 1 N. 350 E. 6.49 2 N. 850 E. 1.28 2 S. 56~0 E. 14.15 3 S. 56~ E. 2.20 3 S. 34~ W. 5.10 4 S34 W. 3.53 4 N. 56 W. 5.84 5 N. 56~0 W. 3.20 5 S. 291~ W. 2.52 Ans. 1A. OR. 14P. 6 N. 48~ W. 8. 3 CHAP. vI.] Calculating the Content. 187 Example 7. Example 8. STA. BEARING. DISTANCE. STA. BEARING. IDISTANCE. 1 S. 21a W. 17.62 1 S. 65~0 E. 4.98 2 S.-340 W. 10.00 2 S.. 58 E. 8.56 3 N. 56~ W. 14.15 3 S. 1410 W. 20.69 4 N. 340 E. 9.76 4 S. 47 W. 0.60 5 N. 67 E. 2.30 5 S. 571~ W. 8.98 6 N. 230 E. 7.036 N. 56~ W. 12.90 7 N. 18~0 E. 4.43 7 N. 34 E. 10.00 8 S. 761~ E. 12.41 8 N. 21~ E. 17.62 Example 9. Example 10. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 S. 57 E. 5.77 1 N. 63~ 51' W. 6.91 2 S. 361~ W. 2.25 2 N. 63~ 44' W. 7.26 3 S. 39~ W. 1.00 3 N. 69~ 35' W. 3.34 4 S. 70~ W. 1.04 4 N. 77~ 50' W. 6.54 5 N. 683~ W. 1.23 5 N. 31~ 24' E. 14.38 6 N. 56~ W. 2.19 6 N. 31~ 18' E. 16.81 7 N. 338~ E. 1.05 7 S. 68~ 55' E. 13.64 8 N. 561~ W. 1.54 8 S. 68~ 42' E. 11.54 9 N. 33~ E. 3.18 9 S. 33~ 45' W. 31.55 Ans. 2A. OR. 32P. Ans. 74 Acres. Example 11. Example 12. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 N. 18~ E. 1.93 1 N. 72~0 E. 0.88 2 N. 90 W. 1.29 2 S. 201~ E. 0.22 3 N. 140 W. 2.71 3 S. 630 E. 0.75 4 N. 740 E. 0.95 4 N. 51~ E. 2.35 5 S4. 48~.59 5 N. 44~ E. 1.10 6 S. 140 E. 1.14 6 N. 25o0 W. 1.96 7 S. 19~o E. 2.15 7 N. 8~ W. 1.05 8 S. 23~0 W. 1.22 8 S. 29 W. 1.63 9 S 5.5 W. 1.40 9 N. 71I~ W. 0.81 10 S. 30~ W. 1.02 10 N. 137~ W. 1.17 11 S. 81~0 W. 0.69 11 N. 63 W. 1.28 12 N. 32~1 W. 1.98 12 West. 1.68...-..t.. _. 13 N. 49~ W. 0.80 14 S. 19 E. 6.20 188 COMPASS SIRVEYING, [PART'ti. Example 13. A farm is described in an old Deed, as bounded thus. Beginning at a pile of stones, and running thence twentyseven chains and seventy links South-Easterly sixty-six and a half degrees to a white-oak stump; thence eleven chains and sixteen links North-Easterly twen- Fig. 197. ty and a half degrees to a e hickory tree; thence two chains and thirty-five links North-Easterly thirty-six degrees to the South-East- 5 erly corner of the homestead; thence nineteen chains and thirty-two links North-Easterly twenty-six degrees to a stone set in 1< the ground; thence twentyeight chains and eightylinks North-Westerly sixty-six degrees to a pine stump; thence thirty-three chains and nineteen links South-Westerly twenty-two degrees to the place of beginning, containing ninety-two acres, be the same more or less. Required the exact content. (297) Mascheroni's Theorem. The surface of any polygon is equal to half the sum of the products of its sides (omitting any one side) taken two and two, into the sines of the angles which those sides make with each other. Fig. 198. Thus, take any polygon, such as the fivesided one in the figure. Express the angle which the directions of any two sides, as AB, CD, make / with each other, thus (ABACD). Then will A< the content of that polygon be, as below; E = [AB. BC. sin (AB A BC) + AB. CD. sin (AB A CD) + AB. DE. sin (AB A DE) + BC. CD. sin (BC A CD) 4- BC. DE. sin (BO A DE) + CD. DE. sin (CD A DE)] CHAP. vii.] Variation of the Magnetic Needle. 189 The demonstration consists merely in dividing the polygon into triangles by lines drawn from any angle, (as A); then expressing the area of each triangle by half the product of its base and the perpendicular let fall upon it from the above named angle; and finally separating the perpendicular into parts which can each be expressed by the product of some one side into the sine of the angle made by it with another side. The sum of these triangles equals the polygon. The expressions are simplified by dividing the proposed polygon into two parts by a diagonal, and computing the area of each part separately, making the diagonal the side omitted.* CHAPTER VII. THE VARIATION OF THE MAGNETIC NEEDLE. (298) Definitions. The Magnetic Meridian is the Fig. 199 direction indicated by the Magnetic Needle. The True V N Meridian is a true North and South line, which, if produced, would pass through the poles of the earth. The Variation, or Declination, of the needle is the angle which one of these lines makes with the other.t In the figure, if NS represent the direction of the True Meridian, and N'S' the direction of the Magnetic Meridian at any place, then is the angle NAN' the Variation S of the Needle at that place. (299) Direction of Needle. The directions of these two meridians do not generally coincide, but the needle in most places points to the East or to the West of the true North, more or less The original Theorem is usually accredited to Lhuillier, of Geneva, who published it in 1789. But Mascheroni, the ingenious author of the " Geometry of the Compasses," had published it at Pavia, two years previously. The method is well developed in Prof. Whitlock's " Elements of Geometry." t " Declination" is the more correct term, and " Variation" should be reserved for the change in the Declination which will be considered in the next chapter; but custom has established the use of Variation in the sense of Declination. 190 COMPASS SURVEYING. [PART II according to the locality. Observations of the amount and the direction of this variation have been made in nearly all parts of the world. In the United States the Variation in the Eastern States is Westerly, and in the Western States is Easterly, as will be given in detail, after the methods for determining the True Meridian, and consequently the Variation, at any place, have been explained. TO DETERMINE THE TRUE MERIDIAN. (300) By equal shadows of the Sun. On the South side of any level surface, erect an up- Fig. 200. right staff, shown, in horizon- tal projection, at S. Two or three hours before noon,mark the extremity, A, of its shadow. Describe an arc of a circle with S, the foot of the staff, for centre, and SA, the distance to the extremity of the shadow, for radius. About as many hours after noon as it had been before noon when the first mark was made, watch for the moment when the end of the shadow touches the arc at another point, B. Bisect the arc AB at N. Draw SN, and it will be the true meridian, or North and South line required. For greater accuracy, describe several arcs before hand, mark the points in which each of them is touched by the shadow, bisect each, and adopt the average of all. The shadow will be better defined, if a piece of tin with a hole through it be placed at the top of the staff, as a bright spot will thus be substituted for the less definite shadow. Nor need the staff be vertical, if from its summit a plumb-line be dropped to the ground, and the point which this strikes be adopted as the centre of the arcs. This method is a very good approximation, though perfectly correct only at the time of the solstices; about June 21st and December 22d. It was employed by the Romans in laying out cities. To get the Variation, set the compass at one end of the True Meridian line thus obtained, sight to the other end of it, and take CHAP. vII.] Variation of the Magnetic Needle. 191 the Bearing as of any ordinary line. The number of degrees in the reading will be the desired variation of the needle. (301) By the North Star, when in the Meridian. The North Star, or Pole Star, (called by astronomers Alpha Ursce Minoris, or Polaris), is not situated precisely at the North Pole of the heavens. If it were, the Meridian could be at once determined by sighting to it, or placing the eye at some distance behind a plumbline so that this line should hide the star. But the North Star is about 110 from the Pole. Twice in 24 hours, however, (more precisely 23h. 56m.), it is in the Meri- Fig. 201. dian, being then exactly above or below A the Pole, as at A and C in the figure. To,/' know when it is so, is rendered easy by the / aid of another star, easily identified, which D D at these times is almost exactly above or below the North Star, i. e. situated in the same vertical plane. If then we watch for'....... the moment at which a suspended plumb- C line will cover both these stars, they will then be in the Meridian. The other star is in the well known constellation of the Great Bear, called also the Plough, or the Dipper, or Charles's Wain. Fig. 202.'Fig. 203. -OX K PO S% ~*7 Two of its five bright stars (the right-hand ones in Fig. 202) are known as the " Pointers," from their pointing near to the North 192 COMPASS SURVEYING. [PART III. Star, thus assisting in finding it. The star in the tail or handle, nearest to the four which form a quadrilateral, is the star which comes to the Meridian at the same time with the North Star, twice in 24 hours, as in Fig. 202 or 203. It is known as Alioth, or Epsilon Ursce Majoris.* To determine the Meridian by this method, suspend a long plumb-line from some elevated point, such as a stick projecting from the highest window of a house suitably situated. The plumbbob may pass into a pail of water to lessen its vibrations. South of this set up the compass, at such a distance from the plumb-line that neither of the stars will be seen above its highest point, i. e. in Latitudes of 40~ or 50~notquite asfarfrom the plumb-line as it is long. Or, instead of a compass, place a board on two stakes, so as to form a sort of bench, running East and West, and on it place one of the compass-sights, or anything having a small hole in it to look through. As the time approaches for the North Star to be on the Meridian (as taken from the table given below) place the compass, or the sight, so that, looking through it, the plumb-line shall seem to cover or hide the North Star. As the star moves one way, move the eye and sight the other way, so as to constantly keep the star behind the plumb-line. At last Alioth, too, will be covered by the plumb-line. At that moment the eye and the plumb-line are (approximately) in the Meridian. Fasten down the sight on the board till morning, or with the compass take the bearing at once, and the reading is the variation.t Instead of one plumb-line and a sight, two plumb-lines may be suspended at the end of a horizontal rod, turning on the top of a pole. The line thus obtained points to the East of the true line when the North Star is above Alioth, and vice versa. The North Star is exactly in the Meridian about 17 minutes after it has been in the same vertical plane with Alioth, and may be sighted to after that interval of time, with perfect accuracy. tf * The North Pole is very nearly at the intersection of the line from Polaris to Alioth, and a perpendicular to this line from the small star seen to the left of it in Fig. 202. t If a Transit or Theodolite be used, the cross-hairs must be illuminated by throwing the light of a lamp into the telescope by its reflection from white paper. CHAP. vii.] Variation of the Magnetic Needle. 198 Another bright star, which is on the opposite side of the Pole, and is known to astronomers as Gramma Cassiopeice, also comes on the Meridian nearly at the same time as the North Star, and will thus assist in determining its direction. (302) The time at which the North Star passes the Meridian above the Pole, for every 10th day in the year, is given in the following Table, in common clock time.* The upper transit is the most convenient, since at the other transit Alioth is too high to be conveniently observed. MONTH. 1st DAY. 11th DAY. 21st DAY. II. M. H. M. H. M. | January, 6 21 P. M. 5 41 P. M. 5 02 P. M. t February, 4 18. M. 3 39 P. M. 3 00 P.M. e March, 2 28 P. M. 1 49 P. M. 1 09 P. M..I April, 0 26 P. M. 11 47 A. M. l1 08 A. M. A May, 10 28 A. M. 9 49 A. M. 9 10 A. M. June, 8 27 A. M. 7 48 A. M. 7 08 A. M. July, 6 29 AM. 50A. 5 50 A.M. 5 11A.M. h August, 4 28 A. 49 A.M. 49 M.3 09 A. M. - September, 2 26 A. M. 1 47 A. M. 1 07 A. M. X October, 0 28 A. M. 11 45 P. M.11 06 P. M.. November, 10 22 P. M. 9 43 P. M. 9 04 p. M. December, 8 24 P. M. 7 45 P. M. 7 06 P. M. * To calculate the time of the North Star passing the Meridian at its upper culmination: Find in the "American Almanac," (Boston), or the " Astronomical Ephemeris," (Washington), or the " Nautical Almanac," (London), or by interpolation from the data at the end of this note, the right ascension of the star, and from it (increased by twenty-four hours if necessary to render the sub'actionpossible) subtract.the Right ascension of the Sun at mean noon, or the sidereal time at mean noon, for the given day, as found in the" Ephemeris of the Sun," in the same Almanacs. From the remainder subtract the acceleration of sidereal on mean time corresponding to this remainder, (3m. 56s. for 24 hours), and the new.remainder is the required mean solar time of the upper passage of the star across the Meridian, in "Astronomical" reckoning, the astronomical day beginning at noon of the common civil day of the same date. The right ascension of the North Star for Jan. 1, 1850, is lh. 05m. 01.4s.; for 1860, lh. 08m. 02.8s.; for 1870, lh. llm. 16.9s.; for 1880, lh. 14m. 45.1s.; for 1890, lh. 18m. 29.2s.; for 1900, lh. 22m. 31s. 13 194 COMPASS SURVEYING. [PART II. To find the time of the star's passage of the Meridian for other days than those given in the Table, take from it the time for the day most nearly preceding that desired, and subtract from this time 4 minutes for each day from the date of the day in the Table to that of the desired day; or, more accurately, interpolate, by saying: As the number of days between those given in the Table is to the number of days from the next preceding day in the Table to the desired day, so is the difference between the times given in the Table for the days next preceding and following the desired day to the time to be subtracted from that of the next preceding day. The first term of the preceding proportion is always ten, except at the end of months having more or less than 30 days. For example, let the time of the North Star's passing the Meridian on July 26th be required. From July 21st to August 1st being 11 days, we have this proportion: 11 days: 5 days:: 43 minutes: 19-T minutes. Taking this from 5h. 11m. A. M., we get 4h. 511m. A. M. for the time of passage required. The North Star passes the Meridian later every year. In 1860, it will pass the Meridian about two minutes later than in 1854; in 1870, five minutes, in 1880, eight minutes, in 1890, twelve minutes, and in 1900, sixteen minutes, later than in 1854: the year for which the preceding table has been calculated. The times at which the North Star passes the Meridian below the Pole, in its lower Transit, can be found by adding 11h. 58m. to the time of the upper Transit, or by subtracting that interval from it.* (303) By the North Star at its extreme elongation. When the North Star is at its greatest apparent angular distance East or West of the Pole, as at B or D in Fig. 201, it is said to be at its extreme Eastern, or extreme Western, Elongation. If it be observed at either of these times, the direction of the Meridian can be easily * The North Star, which is now about lo 28' from the Pole, was 12~ distant from it when its place was first recorded. Its distance is now diminishing at the rate of about a third of a minute in a year, and will continue to do so till it approaches to within half a degree, when it will again recede. The brightest star in the Northern hemisphere, Alpha Lyrae, will be the Pole Star in about 12,000 years, being then within about 50 of the Pole, though now more than 51~ distant from it CHAP. vII.] Variation of the Magnetic Needle. 195 obtained from the observation. The great advantage of this method over the preceding is that then the star's motion apparently ceases for a short time. (304) The following Table gives the TIMES OF EXTREME ELONGATIONS OF THE NORTH STAR.* MONTH.! 1ST DAY. 11TH DAY. 21ST DAY. EASTERN. WESTERN. EASTERN. WESTERN. EASTERN. WESTERN. H. M. H. M. H. M. H. M. H. M. H. M. Jan'y, 0 27 P.M. 0 19 A.M. 11 47 A.M. 11 35 P.M. 11 08 A.M. 10 56 P.M. Feb'y, 10 24AM. 1013. M.1 P.M. 945A.M. 933 P.M. 906A.M. 8 54 P.M. March, 8 34A.M. 8 22 P.M. 7 55 A.M. 7 43 P.M. 715 A.M. 7 04 P.M. April, 632A.M. 6 20,P.M. 553A.M. 5 41 P.M. 514A.M. 5 02 P.M. May, 4 34A.M. 4 22 P.M. 3 55 A.M. 3 43 P.M. 316 A.. 3 04 P.M. June, 2 33 A.M. 2 21 P.M. 1 53 A.M. 142 P.M. 1 14A.M. 102 P.M. uly, 0 35 A.M. 0 23 P.M.:1 52 P.M. 1144A.M. 11 13 P.M. 11 05 A.M. August, 10 30 P.M. 10 22A.M. 9 51 P.M. 9 43 A.M. 9 11 P.M. 903 A.M. Sept'r, 8 28 P.M. 8 20 A.M. 7 49 P.M. 7 41 A.M. 7 09 P.M. 7 01 A.M. Oct'r, 6 30 P.M. 622A.M. 5 51 P.M. 543A.M. 512 P.M. 504A.M. Nov'r, 4 28 P.M. 4 21A.M. 3 49 P.M. 3 41A.M. 310 P.M. 3 02A.M. Dec'r, 2 30 P.M. 2 22A.M. 151 P.M. 143A.M. 112 P.M. 1 04A.M, The Eastern Elongations from October to March, and the Western Elongations from April to September, occurring in the day time, they will generally not be visible except with the aid of a powerful telescope. * To calculate the times of the greatest elongation of the North Star: Find in one of the Almanacs before referred to, or from the data below, its Polar distance at the given time. Add the logarithm of its tangent to the logarithm of the tangent of the Latitude of the place, and the sum will be the logarithm of the cosine of the Hour angle before or after the culmination. Reduce the space to time; correct for sidereal acceleration (3m. 56s. for 24 hours) and subtract the result fiom the time of the star's passing the meridian on that day, to get the time of the Eastern elongation, or add it to get the Western. The Polar distance of the North Star, for Jan. 1, 1850, is 1~ 29' 25"; for 1860, 1~ 26' 12".7; for 1870, 1~ 23~ 01"; for 1880, 10 19' 50".4; for 1890, 10 16' 40".7; for 1900, l1 13' 32".2. 196 COMPASS S UREYING. [PART II. The preceding Table was calculated for Latitude 40~. The Time at which the Elongations occur vary slightly for other Latitudes. In Latitude 50~, the Eastern Elongations occur about 2 minutes later and the Western Elongations about 2 minutes earlier than the times in the Table. In Latitude 26~, precisely the reverse takes place. The Times of Elongation are continually, though slowly, becoming later. The preceding Table was calculated for July 1st, 1854. In 1860, the times will be nearly 2 minutes later; and in 1900, the Eastern Elongations will be about 15 minutes, and the Western Elongations 17 minutes later than in 1854. (305) Observations. Knowing from the preceding Table the hour and minute of the extreme Elongation on any day, a little before that time suspend a plumb-line, precisely as in Art. (301), and place yourself south of it as there directed. As the North Star moves one way, move your eye the other, so that the plumbline shall continually seem to cover the star. At last the star will appear to stop moving for a time, and,.then begin to move backwards. Fix the sight on the board (or the compass, &c.) in the position in which it was when the star ceased moving; for the star was then at its extreme apparent Elongation, East or West, as the case may be. (306) Azimuths. The angle which the line from the eye to the plumb-line, makes with the True Meridian (i. e. the angle between the meridian plane and the vertical plane passing through the eye and the star) is called the Azimuth of the Star. It is given in the following Table for different Latitudes, and for a number of years to come. For the intermediate Latitudes, it can be obtained by a simple proportion, similar to that explained in detail in Art. (302).* To calculate this Azimuth: From the logarithm of the sine of the Polar dis tance of the star, subtract the logarithm of the cosine of the Latitude of the place; the remainder will be the logarithm of the sine of the angle required. The Po. lar distance can be obtained as directed in the last note. CHAP. vii.] Variation of the Magnetic Needle. 197 AZIMUTHS OF THE NORTH STAR. Latitudes. 18,54 1855 1856 1857 1858 ) 1859 1860 j 1870 500 2o 16'20 1 6120 16' 2 20 15' 20 15/ 2314-120.14'2 O09I 490 20 14' 20 13' 20 13'1202 12' 12 1 20 1 2 06g' 480 20111'2011' -2o 10/2 20 10k' 2o 09' 20091'0 20 09/ 2~ 04' 470 20 09' 2~ 08' 12 08' 2~ 073 2~ 0717 2~ 063' 20 061' 2~ 01-' 460 2o 063' 20 06' 2o 05'/ 20 051' 12 05 20 041 A20 041~ 10 59'I 450 2~ 04' 20 04' 23031' 20 031' 0 023/ 2^021'20 02' 10 571' 440 2 02' 2~ 02' 2~ 01 23 o.01' o o1 2~ 00/ 200/ l0 55 430o 20 00' 20 00' lo 59' fo1 59t 10 581' 1 58'10 58 10 531/ 420 10 581'-10 58' 1' 1- 57O 3'10 51? 65'l lo SGo — 51o W 410o L0 56'1 56' 110 o 55'0 510 5 1 54' 10 541k o0 50' 400 10 55 10o 541' 10 54/ 10 53l' 10 53' 10 53' 1o 523'10 48k' 390 10 53 10O 52' 1 521' 19 52' 10 513'0 511/ 10 51- 46/' 380~ 1O 51X1( Sl'10 515'.10 50lo 1~ 50.1~ 491'10 49'1~ 45o' 37o 1o 50o'10 o49/ o0. 491 1o 49/ 10 -48W' 1:.48-' 1o 48' 11 454' 36~0 1O 480' 1 48,' 10.,48' 1~ 47 6'1j 1~ 47' 1T 46 Po 42/' 35~ u o 47,' l~ 47/ lo 4656',1~ 465:,146o 1~,}~:-5rJ 1~ ~-4x- 41~, 340 1o 46k'10 45,1o~ ~ 4591 1o 45'' 10 44,' 1o 441' 1. ~ 44' 10404' 330 45 1 44 445 Io 43' 10 43/ 1043 1o 42' 1 39/ 32 10o 44'i 1~ 43/ 1043' 104' 1~ 42 110 42/ o 1l 4 1 38' 31~ 10 424'110 42k' 10 42' 1~ 41' 1~ 41'/ 1 404' 1, 404'12 37/ tion of the North Star at its extreme elongation have been Fig. 204. obtained, as in Art. (305)4 the True Meridian can be * found thus. Let A and B be the two points. Multiply the natural tangent of the Azimuth given in the Table, by the distance AB. The product will be the length of a line which is to be set off from B, perpendicular to AB, to / some point C. A and C will then be points in the True | Meridian. This operation may be postponed till morning. ch'B If the directions of both the extreme Eastern and extreme Western elongations be set out, the line lying midway between them will be the True Meridian., A 198 COMPASS SURVEYING. [PART III. (308) Determining the Variation. The variation would of course be given by taking the Bearing of the Meridian thus obtained, but it can also be determined by taking the Bearing of the star at the time of the extreme elongation, and applying the following rules. When the Azimuth of the star and its magnetic bearing are one East and the other West, the sum of the two is the Magnetic Varition, which is of the same name as the Azimuth; i. e. East, if that be East, and West, if it be West. When the Azimuth of the star and its Magnetic Bearing are both East, or both West, their difference is the Variation, which will be of the same name as the Azimuth and Bearing, if the Azimuth be the greater of the two, or of the contrary name if the Azimuth be the smaller. Fig. 205. All these cases are presented together in the C is much exaggerated. Set /' - the instrument at any conve- / - -.. nient point, as O; level it, G -- L/4 i and fix a needle (having a / head of sealing-wax) in the board to represent the sta- F D tion. Direct the alidade to any corner of -the field, as A, the fiducial edge of the ruler touching the needle, and draw an indefinite line by it. Measure OA, and set off the distance, to any desired scale, from the needle point, along the line just drawn, to a. The line OA is thus platted on the paper of the table as soon as determined in the field. Determine and plat in the same way, OB, OC, &c., to b, c, &c. Join ab, be, &c., and a complete plat of the field is obtained. Trees, houses, hills, bends of rivers, &c., may be determnied in the same manner. The corresponding method with the Compass or Transit, was described in Articles (258) and (391). The table may be set at one of the angles of the field, if more convenient. If the alidade has a telescope, the method of measuring distances with a stadia, described in Art. (375), may be here applied with great advantage, 20 306 PLANE TABLE SURVEYING. [PART VIII. (453) Method of Progression. Let ABCD, &c., be the line to be surveyed. Fig. 308. Fix a needle at a convenient point z. E of the Plane-table, A.( near a corner so as to leave room for the plat, and D set up the table at' 6 B, the second an- B gle of the line., so that the needle, whose point repre- C sents B, and which should be named b, shall be exactly over that station. Sight to A, pressing the fiducial edge of the ruler against the needle, and draw a line by it. Measure BA, and set off its length, to the desired scale, on the line just drawn, from b to a point a, representing A. Then sight to C, draw an indefinite line by the ruler, and on it set off the length of BC from b to c. Fix the needle at c. Set upat C, the point c being over this station, and make the line cb of the plat coincide in direction with CB on the ground, by placing the edge of the ruler on eb, and turning the table till the sights point to B. The compass, if the table have one, will facilitate this. Then sight forward from C to D, and fix CD, cd on the plat, as be was fixed. Set up at D, make de coincide with DC, and proceed as before. The figure shews the lines drawn at each successive station. The Table drawn at A shews how the survey might be commenced there. In going around a field, the work would be proved by the last line " closing" at the starting point;and, during the progress of the survey, by any direction, as from C to A on the ground, coinciding with the corresponding line, ca, on the plat. This method is substantially the same as the method of surveying a line with the Transit, explained in Art. (372). It requires all the points to be accessible. It is especially suited to the sur vey of a road, a brook, a winding path through woods, &c. The offsets required may often be sketched in by eye with sufficient precision. PART VIII.] PLANE TABLE SURVEYING. 307 When the paper is filled, put on a new sheet, and begin by fixing on it two points, such as C and D, which were on the former sheet, and from them proceed as before. The sheets can then be afterwards united, so that all the points on both shall be in their true relative positions. (454) Method of Intersection. This is the most usual and the most rapid method of using the Plane-table. The principle was referred to in Articles (259) and (392). Set up the instrument at any convenient point, as X in the figure, and sight to all Fig. 309. B c X -Y the desired points A, B. C, &c., which are visible, and draw indefinite lines in their directions. Measure any line X~, Y being one of the points sighted to, and set off this line on the paper to any scale. Set up at Y, and turn the table till the line XY on the paper lies in the direction of XY, on the ground, as at C in the last method. Sight to all the former points and draw lines in their directions, and the intersections of the two lines of sight to each point will determine them, by the Fourth Method, Art. (S). Points on the other side of the line XY could be determined at the same time. In surveying a field, one side of it may be taken for the base XY. Very acute or obtuse intersections should be avoided. 80~ and 150~ should be the extreme limits. The impossibility of always doing this, renders this method often deficient in precision. When the paper is filled, put on a new sheet, by fixing on it two known points, as in the preceding method. 308 PLANE TABLE SURVEYING. [PART VIII. (455) Method of Resection. This method (called by the French Recoupement) is a modification of the preceding method of InterFig. 310. section. It requires the measurement of only one distance, but all the points must be accessible. Let AB be the measured distance. Lay it off on the paper as ab. Set the table up at B, and turn it till the line ba on the paper coincides with BA on the ground, as in the Method of Progression. Then sight to C, and draw an indefinite line by the ruler. Set up at C, and turn the line last drawn so as to point to B. Fix a needle at a on the table, place the alidade against the needle and turn it till it sights to A. Then the point in which the edge of the ruler cuts the line drawn from B will be the point c on the table. Next sight to D, and draw an indefinite line. Set up at D, and make the line last drawn point to C. Then fix the needle at a or b, and by the alidade, as at the last station, get a new line back from either of them, to cut the last drawn line at a point which will be d. So proceed as far as desired. (456) To orient the tabled The operation of orientation consists in placing the table at any point so that its lines shall have the same directions as when it was at previous stations in the same survey. * The French phrase, To orient one's self, meaning to determine one's position, usually with respect to the four quarters of the heavens, of which the Orient is the leading one, well deserves naturalization in our language. PART VIII.] PLANE TABLE SURVEYING. 309 With a compass, this is very easily effected by turning the table till the needle of the attached compass, or that of the Declinator, placed in a fixed position, points to the same degree as when at the previous station. Without a compass the table is oriented, when set at one end of a line previously determined, by sighting back on this line, as at C in the Method of Progression, Art. (453). To orient the table, when at a station unconnected with others, is more difficult. It may be Fig. 311. effected thus. Let ab on the ta-:t:> ble represent a line AB on the iground. Set up at A, make ab coincide with AB, and draw a --- line from a directed towards a s:Isteeple, or other conspicuous ob- ject, as S. Do the same at B. Draw a line ed, parallel to ab, and intercepted between aS, and bS. Divide ab and ed into the same number of equal parts. The table is then prepared. Now let there be a station, P, p on the table, at which the table is to be oriented. Set the table, so that p is over P, apply the edge of the ruler to p, and turn it till this edge cuts ed in the division corresponding to that in which it cuts ab. Then turn the table till the sights point to S, and the table will be oriented. (457) To find one's place on the grould. This problem may be otherwise expressed as Interpolating a point in a plat. It is most easily performed by reversing the Method of Intersection. Set up the table over the station, Fig. 312. 0 in the figure, whose place on the plat already on the table is desired, and orient it, by one of A / the means described in the last \ / article. Make the edge of the \ruler pass through some point, a on the table, and turn it till the sights point to the corresponding station, A on the ground. Draw a line by the ruler. The desired 310 PLANE TABLE SURVEYING. [PART VIII. point is somewhere in this line. Make the ruler pass through another point, b on the table, and make the sights point to B on the ground. Draw a second line, and its intersection with the first will be the point desired. Using C in the same way would give a third line to prove the work. This operation may be used as a new method of surveying with the plane-table, since any number of points can have their places fixed in the same manner. This problem may also be executed on the principle of Trilinear Surveying. Three points being given on the table, lay on it a piece of transparent paper, fix a needle any where on this, and with the alidade sight and draw lines towards each of these three points on the ground. Then use this paper to find the desired point, precisely as directed in the last sentence of Art. (398), page 277. (4I8) Inaccessible distances. Many of the problems in Part VII. can be at once solved on the ground by the plane-table, since it is at the same time a Goniometer and a Protractor. Thus, the Problem of Art. (435) may be solved as follows, on the principle of the construction in the last paragraph of that article. Set the table at C. Mark on it a point, c', to represent C, placing c' vertically over C. Sight to A, B and D, and draw corresponding lines from c'. Set up at D, mark any point on the line drawn from c' towards D, and call it d'. Let d' be exactly over D, and direct d'c' toward C. Then sight to A and B, and draw corresponding lines, and their intersections with the lines before drawn towards A and B will fix points a' and b'. Then on the line joining a and b, given on the paper to represent A and B, ab being equal to AB on any scale, construct a figure, abed, similar to a'b''d', and the line od thus determined will represent CD on the same scale as AB. PART IX. SURVEYING WITHOUT INSTRUMENTS. (459) THE Principles which were established in Part I, and subsequently applied to surveying with various instruments, may also be employed, with tolerable correctness, for determining and representing the relative positions of larger or smaller portions of the earth's surface without any Instruments but such as can be extemporized. The prominent objects on the ground, such as houses, trees, the summits of hills, the bends of rivers, the crossings of roads, &c., are regarded as "points" to be "determined." Distances and angles are consequently required. Approximate methods of obtaining these will therefore be first given. (460) Distances Dy pacing. Quite an accurate measurement of a line of ground may be made by walking over it at a uniform pace, and counting the steps taken. But the art of walking in a straight line must first be acquired. To do this, fix the eye on two objects in the desired line, such as two trees, or bushes, or stones, or tufts of grass. Walk forward, keeping the nearest of these objects steadily covering the other. Before getting up to the nearest object, choose a new one in line farther ahead, and then proceed as before, and so on. It is better not to attempt to make each of the paces three feet, but to take steps of the natural length, and to ascertain the value of each by walking over a known distance, and dividing it by the number of paces required to traverse it. Every person should thus determine the usual len'gth of his own steps, repeating the experiment sufficiently often. The French "Geographical Engineers" accustom themselves to take regular 312 StUVEYING WITHOUT INSTRUMENTS. [PART IX. steps of eight-tenths of a metre, equal to two feet seven and a half inches. The English military pace is two feet and six inches. This is regarded as a usual average. 108 such paces per minute give 3.07 English miles per hour. Quick pacing of 120 such paces per minute gives 3.41 miles per hour. Slow paces, of three feet each and 60 per minute, give 2.04 miles per hour.* An instrument, called a Pedometer, has been contrived, which counts the steps taken by one wearing it, without any attention on his part. It is attached to the body, and a cord, passing from it to the foot, at each step moves a toothed wheel one division, and some intermediate wheelwork records the whole number upon a dial. (461) Distances by visual angles. Prepare a scale, by marking off on a pencil what length of it, when it is held off at arm's length, a man's height appears to cover at different distances (previously measured with accuracy) of 100, 500, 1000 feet, &c. To apply this, when a man is seen at any unknown distance, hold up the pencil at-arm's length, making the top of it come in the line from the eye to his head, and placing the thumb nail in the line from Fig. 313. the eye to his feet, as in Fig. 313. The pencil having been previously graduated by the method above explained, the portion of it now intercepted between these two lines will indicate the corresponding distance. If no previous scale have been prepared, and the distance of a man be required, take a foot-rule, or any measure minutely divided, hold it off at arm's length as before, and see how much a man's height covers. Then knowing the distance from the eye to the rule, a statement by the Rule of Three (on the principle of similar triangles) will give the distance required. Suppose a man's height, of 70 inches, covers 1 inch of the rule. He is then 70 times as far * A horse, on a walk, averages 330 feet per minute, on a trot 650, and on a common gallop 1040. For longer' times, the difference in horses is more apparent. PART ix.] J URVEYING WITHOUT INSTRUMENTS. 313 from the eye as the rule; and if its distance be 2 feet, that of the man is 140 feet. Instead of a man's height, that of an ordinary house, of an apple-tree, the length of a fence-rail, &c., may be be taken as the standard of comparison. To keep the arm immovable, tie a string of known length to the pencil, and hold between the teeth a knot tied at the other end of the string. (462) Distances by visibility. The degree of visibility of various well-known objects will indicate approximately how far distant they are. Thus, by ordinary eyes, the windows of a large house can be counted at a distance of about 13000 feet, or 2~ miles; men and horses will be perceived as points at about half that distance, or 1I miles; a horse can be clearly distinguished at about 4000 feet; the movements of men at 2600 feet, or half a mile; and the head of a man, occasionally, at 2300 feet, and very plainly at 1300 feet, or a quarter of a mile. The Arabs of Algeria define a mile as " the distance at which you can no longer distinguish a man from a woman." These distances of visibility will of course vary somewhat with the state of the atmosphere, and still more with individual acuteness of sight, but each person should make a corresponding scale for himself. (463) Distances by sound. Sound passes through the air with a moderate and known velocity; light passes almost instantaneously. If, then, two distant points be visible from each other, and a gun be fired at night from one of them, an observer at the other, noting by a stop-watch the time at which the flash is seen, and then that at which the report is heard, can tell by the intervening number of seconds how far apart the points are, knowing how far sound travels in a second. Sound moves about 1090 feet per second in dry air, with the temperature at the freezing point, 32~ Fahrenheit. For higher or lower temperatures add or subtract -1- foot for each degree of Fahrenheit. If a wind blows with or against the movement of the sound, its velocity must be added or subtracted. If it blows obliquely, the correction will evidently equal its velocity multiplied by the cosine of the angle which the direction of the wind makes 314 SURVEYING WITHOUT INSTRUMENTS. [PART IX. with the direction of the sound.* If the gun be fired at each end of the base in turn, and the means of the times taken, the effect of the wind will be eliminated. If a watch is not at hand, suspend a pebble to a string (such as a thread drawn from a handkerchief) and count its vibrations. If it be 391 inches long, it will vibrate in one second; if 93 inches long, in half a second, &c. If its length is unknown at the time, still count its vibrations; measure it subsequently; and then will the time of its vibration, in seconds, = /( strin (464) Angles. Right angles are those most frequently required in this kind of survey, and they can be estimated by the eye with much accuracy. If other angles are desired, they will be determined by measuring equal distances along the lines which make the angle, and then the line, or chord, joining the ends of these distances, thus forming chain angles, explained in Art. (100). (465) Methods of operation. The "First Method" of determining the position of a point, Art. (5), is the one most generally applicable. Some line, as AB in Fig. 1, is paced, or otherwise measured, and then the lines AS and BS; the point S is thus determined. The " Second Method," Art. (6), is also much employed, the right angles being obtained by eye, or by the easy methods given in Part II, Chapter, Arts. (140), &c. It is used for offsets, as in Part II, Chapter III, Arts. (114), &c. The " Third Method," Art. (7), may also be used, the angles being determined as in Art. (46-). The "Fourth Method," Art. (8), may also be employed, the angles being similarly determined. The " Fifth Method," Art. (10), would seldom be used, unless by making an extempore plane-table, and proceeding as directed in the last paragraph of Art. (457). * A gentle, pleasant wind has a velocity of 10 feet per second; a brisk gale 20 feet per second; a very brisk gale 30 feet; a high wind 50 feet; a very high wind 70 feet; a storm or tempest 80 feet; a great storm 100 feet; a hurricane 120 feet; and a violent hurricane, that tears up trees, &c., 150 feet per second PART Ix.] SURVEYING WITHOUT INSTRUMENTS. 315 The method referred to in Art. (11) may also be employed. When a sketch has made some progress, new points may be fixed on it by their being in line with others already determined. All these methods of operation are shown in the following figure. AB is a line paced, or otherwise measured approximately. Fig. 314. M The hill C is determined by the first method. The river on the other side of AB is determined by offsets according to the Second Method. The house D is determined by the Third Method, EBF being a chain angle. The house G is determined by the Fourth Method, chain angles being measured at B and H, a point in AB prolonged. The pond K is determined, as in Art. (I ), by the intersection of the alinements CD and GH prolonged. The bend of the river at L is determined by its distance from H in the line of AH prolonged. A new base line, HM, is fixed by a chain angle at II, and employed like the former one so as to fix the hill at N, &c. All these methods may thus be used collectively and successively. The necessary lines may always be ranged with rods, as directed in Art. (169), and very many of the instrumental methods already explained, may be practiced with extempore contrivances. The use of the Plane-table is an admirable preparation for this style of surveying or sketching, which is most frequently employed by Military Engineers, though they generally use a prismatic Compass, or pocket Sextant, and a sketching case, which may serve as a Plane-table. PART X. MAPPING. CHAPTER 1. COPYING PLATS. (466) THE Plat of a survey necessarily has many lines of construction drawn upon it, which are not needed in the finished map. These lines, and the marks of instruments, so disfigure the paper that a fair copy of the plat is usually made before the map is finished. The various methods of copying plats, &c., whether on the same scale, or reduced or enlarged, will therefore now be described. (467) Stretching the paper. If, the map is to be colored, the paper must first be wetted and stretched, or the application of the wet colors will cause its surface to swell or blister and become uneven. Therefore, with a soft sponge and clean water wet the back of the paper, working from the centre outward in all directions. The 4" water-mark" reads correctly only when looked at from the front side, which it thus distinguishes. When the paper is thoroughly wet and thus greatly expanded, glue its edges to the drawing board, for half an inch in width, turning them up against a ruler, passing the glue along them, and then turning them down and pressing them with the ruler. Some prefer gluing down opposite edges in succession, and others adjoining edges. The paper must be moderately stretched smooth during the process. Hot glue is best. Paste or gum may be used, if the paper be kept wet by a damp cloth, so that the edges may dry first. " Mouth-glue " may be used CHAP. i.] Copying Plats. 317 by rubbing it (moistened in the mouth or in boiling water) along the turned up edges, and then rubbing them dry by an ivory folder, a piece of dry paper being interposed. As this is a slower process, the middle of each side should first be fastened down, then the four angles, and lastly the intermediate portions. When the paper becomes dry, the creases and puckerings will have disappeared, and it will be as smooth and tight as a drum-head. (468) Copying by tracing. Fix a large pane of clear glass in a frame, so that it can be supported at any angle before a window, or, at night, in front of a lamp. Place the plat to be copied on this glass, and the clean paper upon it. Connect them by pins, &c. Trace all the desired lines of the original with a sharp pencil, as lightly as they can be easily seen. Take care that the paper does not slip. If the plat is larger than the glass, copy its parts successively, being very careful to fix each part in its true relative position. Ink the lines with India ink, making them very fine and pale, if the map is to be afterwards colored. (469) Copying on tracing paper. A thin transparent paper is prepared expressly for the purpose of making copies of maps and drawings, but it is too delicate for much handling. It may be prepared by soaking tissue paper in a mixture of turpentine and Canada balsam or balsam of fir (two parts of the former to one of the latter), and drying very slowly. Cold drawn linseed oil will answer tolerably, the sheets being hung up for some weeks to dry. Linen is also similarly prepared,- and sold under the name of " Vellum tracing paper." It is less transparent than the tracing paper, but is very strong and durable. Both of these are used rather for preserving duplicates than for finished maps. (470) Copying by transfer paper. This is thin paper, one side of which is rubbed with blacklead, &c., smoothly spread by cotton. It is laid on the clean paper, the blackened side downward, and the plat is placed upon it. All the lines of the plat are then gone over with moderate pressure by a blunt point, such as the eye-end of a small needle. A faint tracing of these lines will then be found 318 MAPPING. [PART X. on the clean paper, and can be inked at leisure. If the original cannot be thus treated, it may first be copied on tracing paper, and this copy be thus transferred. If the transfer paper be prepared by rubbing it with lampblack ground up with hard soap, its lines will be ineffaceable. It is then called " Camp-paper." (471) Copying by punctures. Fix the clean paper on a drawing board and the plat over it. Prepare a fine needle with a sealing-wax head. Hold it very truly perpendicular to the board, and prick through every angle of the plat, and every corner and intersection of its other lines, such as houses, fences, &c., or at least the two ends of every line. For circles, the centre and one point of the circumference are sufficient. For irregular curves, such as rivers, &c., enough points must be pricked to indicate all their sinuosities. Work with system, finishing up one strip at a time, so as not to omit any necessary points nor to prick through any twice, though the latter is safer. When completed, remove the plat. The copy will present a wilderness of fine points. Select those which determine the leading lines, and then the rest will be easily recognized. A beginner should first pencil the lines lightly, and then ink them. An experienced draftsman will omit the pencilling. Two or three copies may be thus pricked through at once. The holes in the original plat may be made nearly invisible by rubbing them on the back of the sheet with a paper-folder, or the thumb nail. (472) Copying by intersections. Draw a line on the clean paper equal in length to some important line of the original. Two starting points are thus obtained. Take in the dividers the distance from one end of the line on the original to a third point. From the corresponding end on the copy, describe an arc with this distance for radius and about where the point will come. Take the distance on the original from the other end of the line to the point, and describe a corresponding arc on the copy to intersect the former arc in a point which will be that desired. The principle of the operation is that of our "First Method," Art. (5). Two pairs of dividers may be used as explained in Art. (90). " Tri CHAP. I.] Copying Plats. 319 angular compasses," having three legs, are used by fixing two of their legs on the two given points of the original, and the third leg on the point to be copied, and then transferring them to the copy. All the points of the original can thus be accurately reproduced. The operation is however very slow. Only the chief points of a plat may be thus transferred, and the details filled in by the following method. (473) Copying by squares. On the original plat draw a series of parallel and equidistant lines. The T square does this most readily. Draw a similar series at right angles to these. The plat will then be covered with squares, as in Fig. 38, page 48. On the clean paper draw a similar series of squares. The important points may now be fixed as in the last article, and the rest copied by eye, all the points in each square of the original being properly placed in the corresponding square of the copy, noticing whether they are near the top or bottom of each square, on its right or left side, &c. This method is rapid, and in skilful hands quite accurate. Instead of drawing lines on the original, a sheet of transparent paper containing them may be placed over it; or an open frame with threads stretched across it at equal distances and at right angles. This method supplies a transition to the Reduction and _Enlargement of plats in any desired ratio; under which head Copyingby the Pantagraph and Camera Lucida will be noticed. (474) Reducing by squares. Begin, as in the preceding article, by drawing squares on the original, or placing them over it. Then on the clean paper draw a similar set of squares, but with their sides one-half, one third, &c., (according to the desired reduction), of those of the original plat. Then proceed as before to copy into each small square all the points and lines found in the large square of the plat in their true positions relative to the sides and corners of the square, observing to reduce each distance, by eye or as directed in the following article, in the given ratio. 320 MAPPING. [PART X. (475) Reducing by proportional scales. Many graphical methods of finding the proportionate length on the copy, of any line of the original, may be used. The "Angle of reduction" is con structed thus. Draw any line AB. With it for radius and A Fig. 315.. for centre, describe an indefinite - x \\ arc. With B for centre and a \\\\\\ radius equal to one-half, one-third, \ \\ \\ \\\\\ &c., of AB according to the de- A, \,. \ \,, \ \ \\\ sired reduction describe another A D B arc intersecting the former arc in C. Join AC. From A as centre describe a series of arcs. Now to reduce any distance, take it in the dividers, and set it off from A on AB, as to D. Then the distance from D to E, the other end of the arc passing through D, will be the proportionate length to be set off on the copy, in the manner directed in Art. (472). The Sector, or " Compass of proportion," described in Art. (52), presents such an "Angle of reduction," always ready to be used in this manner. The " Angle of reduction" may be simplified Fig. 316. thus. Draw a line, AB, parallel to one side B1 C of the drawing board, and another, BC, at right angles to it, and one-half, &c., of it, as desired. Join AC. Then let AD be the distance re- quired to be reduced. Apply a T square so as to pass through D. It will meet AC in some point E, and DE will be the reduced length required. J Another arrangement for the same object is shown in Fig. 317. Draw two lines, AB, AC, at any angle, and de- Fig. 317. scribe a series of arcs from their intersection, A, C as in the figure. Suppose the reduced scale is to,... be half the original scale. Divide the outermost _ arc into three equal parts, and draw a line from A to one of the points of division, as D. Then each arc will be divided into parts, one of which is twice the other. Take any distance on the ori- ginal scale, and find by trial which of the arcs on A CHAP. i.] Copying Plats. 321 the right hand side of the figure it corresponds to. The other part of that arc will be half of it, as desired. " Proportional compasses," being properly set, reduce lines in any desired ratio. A simple form of them, known as " Wholes and halves," is often useful. It consists of two slender bars, pointed at each end, and united by a pivot which is twice as far from one pair of the points as from the other pair. The long ends being set to any distance, the short ends will give precisely half that distance. (476) Reducing by a pantagraph. This instrument consists of two long and two short rulers, connected so as to form a parallelogram, and capable of being so adjusted that when a tracing point attached to it is moved over the lines of a map, &c., a pencil attached to another part of it will mark on paper a precise copy, reduced on any scale desired. It is made in various forms. It is troublesome to use, though rapid in its work. (477) Reducing by a camera lucida. This is used in the Coast Survey Office. It cannot reduce smaller than one-fourth, without losing distinctness, and is very trying to the eyes. Squares drawn on the original are brought to apparently coincide with squares on the reduction, and the details are then filled in with the pencil, as seen through the prism of the instrument. (478) Enlarging plats. Plats may be enlarged by the principal methods which have been given for reducing them, but this should be done as seldom as possible, since every inaccuracy in the original becomes magnified in the copy. It is better to make a new plat from the original data. 21 322 IMAPPING. [PART X CHAPTER II. CONVENTIONAL SIGNS. (479) Various conventional signs or marks have been adopted, more or less generally, to represent on maps the inequalities of the surface of the ground, its different kinds of culture or natural products, and the objects upon it, so as not to encumber and disfigure it with much writing or many descriptive legends. This is the purpose of what is called Topographieal Mapping. (480) The relief of ground. The inequalities of the surface of the earth, its elevations and depressions, its hills and hollows, constitute its " Relief." The representation of this is sometimes called " Hill drawing." Its difficulty arises from our being accustomed to see hills sideways, or "in elevation," while they must be represented as they would be seen from above, or " in plan." Various modes of thus drawing them are used; their positions being laid down in pencil as previously sketched by eye or measured. If light be supposed to fall vertically, the slopes of the ground will receive less light in proportion to their steepness. The relief of ground will be indicated on this principle by making the steep slopes very dark, the gentler inclinations less so, and leaving the level surfaces white. The shades may be produced by tints of India ink applied with a brush, their edges, at the top and bottom of a hill or ridge, being softened off with a clean brush. If light be supposed to fall obliquely, the slopes facing it will be light, and those turned from it dark. This mode is effective, but not precise. In it the light is usually supposed to come from the upper left hand corner of the map. Horizontal contour lines are however the best convention for this purpose. Imagine a hill to be sliced off by a number of equidistant horizontal planes, and their intersections with it to be drawn as they would be seen from above, or horizontally projected on the CHAP 11.] Conventional Signs. 323 map. These are'" Contour lines." They are the same lines as would be formed by water surrounding the hill, and rising one foot at a time (or any other height) till it reached the top of the hill. The edge of the water, or its shore, at each successive rise, would be one of these horizontal contour lines. It is plain that their nearness or distance on the map would indicate the steepness or gentleness of the slopes. A right cone would thus be repreFig. 318. Fig. 319. Fig. 320. sented by a series of concentric circles, as in Fig. 318; an oblique cone by circles not concentric, but nearer to each other on the steep side than on the other, as in Fig. 319; and a half-egg, somewhat as in Fig. 320. Vertical sections, perpendicular to these contour lines, are usually combined with them. They are the " Lines of greatest slope," and may be supposed to represent water running down the sides of the hill. They are also made thicker and nearer together on the steeper slopes, to produce the effect required by the convention of vertical light Fig. 321. already referred to. i\\ The marginal figure %?,.x.. \ shews an elongated —'. \\ half-egg, or oval hill, — L- thus represented. i The spaces between:/ the rows of vertical "Hatchings" indicate the contour lines, which are not actually drawn. The beauty of the graphical execution of this work depends on the uniformity of the strokes representing uniform slopes, on their perfectly regular gradation in thickness and nearness for varying slopes, and on their being made precisely at right angles to the contour lines between which they are situated. 324 MAPPING. [PART X The methods of determining the contour lines are applications of Levelling, and will therefore be postponed, together with the farther details of " Hill-drawing," to the volume treating of that subject, which is announced in the Preface. (481) Signs for natural surface. Sand is represented by fine dots made with the point of the pen; gravel by coarser dots. Rocks are drawn in their proper places in irregular angular forms, imitating their true appearance as seen from above. The nature of the rocks, or the aeology of the country, may be shown by applying the proper colors, as agreed on by geologists, to the back of the map, so that they may be seen by holding it up against the light, while they will thus not confuse the usual details. (482) Signs for vegetation. Woods are represented by scolloped circles, irregularly disposed, Fig. 322. imitating trees seen in plan," and I A; l, closer or farther apart according tot ), the thickness of the forest. It is 0^/ / / usual to shade their lower and right S hand sides and to represent their AS'S' Wshadows, as in the figure, though, in strictness, this is inconsistent with the hypothesis of vertical light, adopted for " hill-drawing." For pine and similar forests, the signs may have a star-like form, as on the right hand side of the figure. Trees are sometimes drawn " in elevation," or sideways, as usually seen. This makes them more easily recognized, but is in utter violation of the principles of mapping in horizontal projection, though it may be defended as a pure convention. Orchards are represented by trees arranged in rows. Bushes may be drawn like trees, but smaller. Grass-land is drawn with irregularly Fig. 323. scattered groups of short lines, as in the' t.L..."'./ figure, the lines being arranged in odd 4, -^,j.''.. vl numbers, and so that the top of each group is. v Vt convex and its bottom horizontal or parallel 9t-l to the base of the drawing. Meadows are At " ""ll a v sometimes represented by pairs of diverging lines, (as on the right CHAP. ii.] Conventional Signs. 325 of the figure), which may be regarded as tall blades of grass. Uncultivated land is indicated by appropriately intermingling the signs for grass land, bushes, sand and rocks. Cultivated land is shown by parallel rows of broken and dotted Fig. 324. lines, as in the figure, representing furrows.,I,,,' i, Crops are so temporary that signs for them are unnecessary, though often used. They are usu- 1Il.1: 1 l ally imitative, as for cotton, sugar, tobacco, rice,.. — vines, hops, &c. Gardens are drawn with cir- cular and other beds and walks. (483) Signs for water, The Sea-coast is represented by drawing a line parallel to the shore, following all its windings and indentations, and as close to it as possible, then another parallel line a little more distant, then a third still more distant, and so on. Examples are seen in figures 287, &c. If these lines are drawn from the low tide mark, a similar set may be drawn between that and the high tide mark, and dots, for sand, be made over the included space. Rivers have each shore treated like the sea shore, as in the figures of Part VII.* Brooks would be shown by only two lines, or one, according to their magnitude. Ponds may be drawn like sea shores, or represented by Fig. 325. parallel horizontal lines ruled across them. ____ Marshes and Swzamps are represented by an __1_, _ irregular intermingling of the preceding -_ _......__ sign with that for grass and bushes, as in the ~ Z''_ _:. figure. - (484) Colored Topography. The conventional signs which have been described, as made with the pen, require much time and labor. Colors are generally used by the French as substitutes for them, and combine the advantages of great rapidity and effectiveness. Only three colors (besides India ink) are required; viz. Gamboge (yellow), Indigo (blue), and Lake (pink). Sepia, Burnt Sienna, Yellow ochre, Red lead, and Vermillion, are also sometimes used. The last three are difficult to work with. To * Those in Part II, Chapter V, have the lines too close together in the middle. 326 MAPPING. [PART X use these paints, moisten the end of a cake and rub it up with a drop of water, afterwards diluting this to the proper tint, which should always be light and delicate. To cover any surface with a uniform flat tint, use a large camel's hair or sable brush, keep it always moderately full, incline the board towards you, previously moisten the paper with clean water if the outline is very irregular, begin at the top of the surface, apply a tint across the upper part, and continue it downwards, never letting the edge dry. This last is the secret of a smooth tint. It requires rapidity in returning to the beginning of a tint to continue it, and dexterity in following the outline. Marbling, or variegation, is produced by having a brush at each end of a stick, one for each color, and applying first one, and then the other beside it before it dries, so that they may blend but not mix, and produce an irregularly clouded appearance. Scratched parts of the paper may be painted over by first applying strong alum water to the place. The conventions for colored Topography, adopted by the French Military Engineers, are as follows. WooDS, yellow; using gamboge and a very little indigo. GRASS-LAND, green; made of gamboge and indigo. CULTIVATED LAND, brown; lake, gamboge, and a little India ink. "Burnt Sienna" will answer. Adjoining fields should be slightly varied in tint. Sometimes furrows are indicated by strips of various colors. GARDENS are represented by small rectangular patches of brighter green and brown. UNCULTIVATED LAND, marbled green and light brown. BRUSH, BRAMBLES, &c., marbled green and yellow. HEATH, FURZE, &C., marbled green and pink. VINEYARDS, purple; lake and indigo. SANDS, a light brown; gamboge and lake. " Yellow ochre" will do. LAKES and RIVERS, light blue, with a darker tint on their upper and left hand sides. SEAS, dark blue, with a little yellow added. MARSHES, the blue of water, with spots of grass green, the touches all lying horizontally. ROADS, brown; between the tints for sand and cultivated ground, with more India ink. HILLS, greenish brown; gamboge, indigo, lake and India ink, instead of the pure India ink, directed in Art. (480). WOODS may be finished up by drawing the trees as in Art. (482) and coloring them green, with touches of gamboge towards the light (the upper and left hand side) and of indigo on the opposite side. CHAP. II.] Conventional Signs. 327 (485) Signs for detached objects. Too great a number of these will cause confusion. A few leading ones will be given, the meanings of which are apparent. Figs. Figs. Court house, t 326. Wind mill, e 334. Post office, f 327. Steam mill, A 335. Tavern, E328. Furnace, H 336. Blacksmith's shop, 3 329. Woollen factory, a 337. Guide board, t 330. Cotton factory, - 338. Quarry, R 331. Glass works, A 339 Grist mill, - 332. Church, d 340. Saw mill, i 333. Graveyard, L341. An ordinary house is drawn in its true position and size, and the ridge of its roof shown if the scale of the map is large enough. On a very small scale, a small shaded rectangle represents it. If colors are used, buildings of masonry are tinted a deep crimson, (with lake), and those of wood with India ink. Their lower and right hand sides are drawn with heavier lines. Fences of stone or wood, and hedges, may be drawn in imitation of the realities; and, if desired, colored appropriately. Mines may be represented by the signs of the planets which were anciently associated with the various metals. The signs here given represent respectively, Gold, Silver, Iron, Copper, Tin, Lead, Quicksilver. ~ D d.? 2 T? D A large black circle, 0, may be used for Coal. Boundary lines, of private properties, of townships, of counties, and of states, may be indicated by lines formed of various combinations of short lines, dots and crosses, as below.*........................ ++ + -+- + +++++ -- - -- -+- - -- ++++ * Very minute directions for the execution of the details described in this chapter, are given in Lieut. R. S. Smith's " Topographical Drawing." Wiley, N. Y. 328 MAPPING. [PART X. CHAPTER III. FINISHING THE MAP. (486) Orientation. The map is usually so drawn that the top of the paper may represent the North. A Meridian line should also be drawn, both True and Magnetic, as in Fig. 199, page 189. The number of degrees and minutes in the Variation, if known, should also be placed between the two North points. Sometimes a compass-star is drawn and made very ornamental. (487) Lettering. The style in which this is done very much affects the general appearance of the map. The young surveyor should give it much attention and careful practice. It must all be in imitation of the best printed models. No writing, however beautiful, is admissible. The usual letters are the ordinary ROMAN CAPITALS, Small Roman, ITALIC CAPITALS, Small talic, and GOTHIC OR EGYPTI AN. Thislast, when well done, is very effective. For the Titles of maps, various fancy letters may be used. For very large letters, those formed only of the shades of the letters regarded as blocks (the body being rubbed out after being pencilled as a guide to the placing of the shades) are most easily made to look well. The simplest lettering is generally the best. The sizes of the names of places, &c., should be proportional to their importance. Elaborate tables for various scales have been published. It is better to make the letters too small than too large. They should not be crowded. Pencil lines should always be ruled as guides. The lettering should be in lines parallel to the bottom of the map, except the names of rivers, roads, &c., whose general course should be followed. (488) Borders. The Border may be a single heavy line, enclosing the map in a rectangle, or such a line may be relieved by a finer line drawn parallel and near to it. Time should not be wasted in ornamenting the border. The simplest is the best. CHAP. III.] Finishing the Map. 329 (489) Joining paper. If the map is larger than the sheets of paper at hand, they should be joined with a feather-edge, by proceeding thus. Cut, with a knife guided by a ruler, about onethird through the thickness of the paper, and tear off on the under side, a strip of the remaining thickness, so as to leave a thin sharp edge. Treat the other sheet in the same way on the other side of it. When these two feather edges are then put together, (with paste, glue or varnish), they will make a neat and strong joint. The sheet which rests upon the other must be on the right hand side, if the sheets are joined lengthways, or below if they are joined in that direction, so that the thickness of the edge may not cast a shadow, when properly placed as to the light. The sheets must be joined before lines are drawn across them, or the lines will become distorted. Drawing paper is now made in rolls of great length, so as to render this operation unnecessary. (490) Mounting maps. A map is sometimes required to be mounted, i. e. backed with canvas or muslin. To do this, wet the muslin and stretch it strongly on a board by tacks driven very near together. Cover it with strong paste, beating this in with a brush to fill up the pores of the muslin. Then spread paste over the back of the paper, and when it has soaked into it, apply it to the muslin, inclining the board, and pasting first a strip, about two inches wide, along the upper side of the paper, pressing it down with clean linen in order to drive out all air bubbles. Press down another strip in like manner, and so proceed till all is pasted. Let it dry very gradually and thoroughly before cutting the muslin from the board. Maps may be varnished with picture varnish; or by applying four or five coats of isinglass size, letting each dry well before applying the next, and giving a full flowing coat of Canada balsam diluted with the best oil of turpentine. PART XI. LAYING OUT, PARTING OFF, AND DIVIDING UP LAND.* CHAPTER I. LAYING OUT LAND. (491) Its nature. This operation is precisely the reverse of those of Surveying properly so called. The latter measures certain lines as they are; the former marks them out in the ground where they are required to be, in order to satisfy certain conditions. The same instruments, however, are used as in Surveying. Perpendiculars and parallels are the lines most often employed. The Perpendiculars may be set out either with the chain alone, Arts. (140) to (159); still more easily with the Cross-staff, Art. (104), or the Optical-square, Art. (107); and most precisely with a Transit or Theodolite, Arts. (402) to (406). Parallels may also be set out with the chain alone, Arts. (160) to (166); or with Transit, &c., Arts. (407) and (408). The ranging out of lines by rods is described in Arts. (169) and (178), and with an Angular instrument, in Arts. (376), (409) and (415). (492) To lay out squares. Reduce the desired content to square chains, and extract its square root. This will be the length of the required side, which is to be set out by one of the methods indicated in the preceding article. An Acre, laid out in the form of a square, is frequently desired by farmers. Its side must be made 316k links of a Gunter's X The Demonstrations of the Problems in this part, when required, will be found in Appendix B. CHAP. I.] Laying out Land. 331 chain; or 208-Ol- feet; or 69]57 yards. It is often taken at 70 paces. The number of plants, hills of corn, loads of manure, &c., which an acre will contain at any uniform distance apart, can be at once found by dividing 209 by this distance in feet, and multiplying the quotient by itself; or by dividing 43560 by the square of the distance in feet. Thus, at 3 feet apart, an acre would contain 4840 plants, &c.; at 10 feet apart, 436; at a rod apart, 160; and so on. If the distances apart be unequal, divide 43560 by the product of these distances in feet; thus, if the plants were in rows 6 feet apart, and the plants in the rows were 3 feet apart, 2420 of them would grow on one acre. (493) To lay out rectangles, The content and length being given, both as measured by the same unit, divide the former by the latter, and the quotient will be the required breadth. Thus, 1 acre or 10 square chains, if 5 chains long, must be 2 chains wide. The content being given and the length to be a certain number of times the breadth. Divide the content in square chains, &c., by the ratio of the length to the breadth, and the square root of the quotient will be the shorter side desired, whence the longer side is also known. Thus, let it be required to lay out 30 acres in the form of a rectangle 3 times as long as broad. 30 acres = 300 square chains. The desired rectangle will contain 3 squares, each of 100 sq. chs., having sides of 10 chs. The rectangle will therefore be 10 chs. wide and 30 long. An Acre laid out in a rectangle twice as long as broad, will be 224 links by 448 links, nearly; or 1472 feet by 295 feet; or 49yards by 98- yards. 50 paces by 100 is often used as an approximation, easy to be remembered. The content being given, and the difference between the length and breadth. Let c represent this content, and d this difference. Then the longer side = d + V (d2 + 4 c). Example. Let the content be 6.4 acres, and the difference 12 chains. Then the sides of the rectangle will be respectively 16 chains and 4 chains. 332 LAYING OUT AND DIVIDING UP LAND. [PART XI. The content being given, and the sum of the length and breadth. Let c represent this content, and s this sum. Then the longer side= s + 1 V(s2 -4 c). Example. Let the content be 6.4 acres, and the sum 20 chains. The above formula gives the sides of the rectangle 16 chains and 4 chains as before. (494) To lay out triangleso The content and the base being given, divide the former by half the latter to get the height. At any point of the base erect a perpendicular of the length thus obtained, and it will be the vertex of the required triangle. The content being given and the base having to be m times the height, the height will equal the square root of the quotient obtained by dividing twice the given area by m. The content being given and the triangle to be equilateral, take the square root of the content and multiply it by 1.520. The product will be the length of the side required. This rule makes the sides of an equilateral triangle containing one acre to be 480h links. A quarter of an acre laid out in the same form would have each side 240 links long. An equilateral triangle is very easily set out on the ground, as directed in Art. (90), under " Platting," using a rope or chain for compasses. (495) The content and base being given, and one side having to make a given angle, as B, with the base Fig. 342. 2x ABC O AB,the length of the side BO =2AB. sin. B AB.. s1i. Bi Example. Eighty acres are to be laid out in the form of a triangle, on a base, AB, of sixty chains, bearing N. 80~ W.| the bearing of the side BC being N. 70~ E. Here the angle B is found from the Bearings (by Art. (243), reversing one of them) to be 30~. Hence BC = 53.33. The figure is on a scale of 50 chains to 1 inch -- 1: 39600. Any right-line figure may be laid out by analogous methods. (496) To lay out circles. Multiply the given content by 7, divide the product by 22, and take the square root of the quotient. CHAP. I.] Laying out Land. 33b This will give the radius, with which the circle can be described on the ground with a rope or chain. A circle containing one acre has a radius of 1781 links. A circle containing a quarter of an acre will have a radius of 89 links. (497) Town lots. House lots in cities are usually laid off as rectangles of 25 feet front and 100 feet depth, variously combined in blocks. Part of New-York is laid out in blocks 200 feet by 800, each containing 64 lots, and separated by streets, 60 feet wide, running along their long sides, and avenues, 100 feet wide, on their short sides. The eight lots on each short side of the block, front on the avenues, and the remaining forty-eight lots front on the streets. Such a block covers almost precisely 3| acres, and 171 such lots about make an acre. But, allowing for the streets, land laid out into lots, 25 by 100, arranged as above, would contain only 11.9, or not quite 12 lots per acre. Lots in small towns and villages are laid out of greater size and less uniformity. 50 feet by 100 is a frequent size for new villages, the blocks being 200 feet by 500, each therefore containing 20 lots. (498) Land sold for taxes. A case occurring in the State of New-York will serve as an application of the modes of laying out squares and rectangles. Land Fig. 343. on which taxes are unpaid is B ~ G sold at auction to the lowest bidder; i. e. to him who will accept the smallest portion of it in return for paying the taxes on the whole. The lot in question was originally the east half of the square lot ABCD, containing 500 acres. At a sale for taxes in 1830, 70 acres were bid off, and this area was L ---- set off to the purchaser in a square lot, from the north-east corner. Required the side of the square in links. Again, in 1834, 29 acres more were thus sold, to be set off in a strip of equal width 384 LAYING OUT AND DIVIDING UP LAND. [PART XI around the square previously sold. Required the width of this strip. Once more, in 1839, 42 acres more were sold, to be set off around the preceding piece. Required the dimensions of this third portion. The answer can be proved by calculating if the dimensions of the remaining rectangle will give the content which it should have, viz. 250 - (70 + 29 + 42) = 109 Acres. The figure is on a scale of 40 chains to 1 inch =1:31680. (499) New countries. The operations of laying out land for the purposes of settlers, are required on a large scale in new countries, in combination with their survey. There is great difficulty in uniting the necessary precision, rapidity and cheapness. "Triangular Surveying" will ensure the first of these qualities, but is deficient in the last two, and leaves the laying out of lots to be subsequently executed. " Compass Surveying" possesses the last two qualities, but not the first. The United States system for surveying and laying out the Public Lands admirably combines an accurate determination of standard lines (Meridians and Parallels) with a cheap and rapid subdivision by compass. The subject is so important and extensive that it will be explained by itself in Part XII. CHAPTER II. PARTING OFF LAND. (500) It is often required to part off from a field, or from an indefinite space, a certain number of acres by a fence or other boundary line, which is also required to run in a particular direction, to start from a certain point, or to fulfil some other condition. The various cases most likely to occur will be here arranged according to these conditions. Both graphical and numerical methods will generally be given.* * The given lines will be represented by fine full lines; the lines of consttaction by broken lines, and the lines of the result by heavy full lines. CHAP. II.] Parting off Land, 335 The given content is always supposed to be reduced to square chains and decimal parts, and the lines to be in chains and decimals. A. BY A LINE PARALLEL TO A SIDE. (501) To part off a rectangle. If the sides of the field adjacent to the given side make right angles with it, the figure parted off by a parallel to the given side will be a rectangle, and its breadth will equal the required content divided by that side, as in Art. (493). If the field be bounded by a curved or zigzag line outside of the given side, find the content between these irregular lines and the given straight side, by the method of offsets, subtract it from the content required to be parted off, and proceed with the remainder as above. The same directions apply to the subsequent problems. (502) To part off a parallelogram. If the sides adjacent to the given side be parallel, the Fig. 344. figure parted off will be a parallel- D ogram, and its perpendicular width, CE, will be obtained as above. The length of one of the parallel A -. B CE ABDC sides, as AC - AB sin. A AB. sin. A (503) To part off a trapezoid. When the sides of the field adjacent to the given side are not parallel, the figure parted off will be a trapezoid. When the field or figure is given on the ground, or on a plat, begin as if the sides were parallel, Fig. 345. dividing the given content by the / base AB. The quotient will be c ------- an approximate breadth, CE, or DF; too small if the sides con- verge, as in the figure, and vice' \ A/ versa. Measure CD. Calculate E the content of ABDC. Divide the difference of it and the required 336 LALYING OUT AND DIVIDING UP LAND. [PART xI. content by CD. Set off the quotient perpendicular to CD, (in this figure, outside of it,) and it will give a new line, GlH, a still nearer approximation to that desired. The operation may be repeated, if found necessary. (504) When the field is given by Bearings, de- Fig. 346. duce from them, as in Art. (243), the angles at A B and B. The required sides will then be given by these formulas: CD = /(AB2 2 x ABD. sin. (A + B)) V ^ sin. A sin. B sin. B AD = (AB- CD)si. (A + B BC = (AB - CD) sitA ( )' sin. (A + B) A- -D When the sides AD and BC diverge, instead of converging, as in the figure, the negative term, in the expression for CD, becomes positive; and in the expressions for both AD and BC, the first factor becomes (CD - AB). The perpendicular breadth of the trapezoid =AD. sin. A; or — BC. sin. B. Example. Let AB run North, six chains; AD, N. 80~ E.; BC, S. 60~ E. Let it be required to part off one acre by a fence parallel to AB. Here AB= 6.00, ABCD =10 square chains, A = 80~, B = 60. Ans. CD = 4.57, AD 1.92, BC = 2.18, and the breadth= 1.89. The figure is on a scale of 4 chains to 1 inch = 1: 3168. B. BY A LINE PERPENDICULAR TO A SIDE. (505) To part off a triangle. Let FG be the required line. When the field is given on the Fig. 347. ground, or on a plat, at any point, as E rC D, of the given side AB, set out a " guess line," DE, perpendicular to AB, and calculate the content of B D r A DEB. Then the required distance BF, from the angular point to the foot of the desired perpendicular, = BD. (BFG). CHAP. II.] Parting off Land. 337 Example. Let BD 30 chains; ED=12 chains; and the desired area - 24.8 acres. Then BF -- 35.22 chains. The scale of the figure is 30 chains to 1 inch =1: 23760. (506) When the field is given by Bearings, Fig. 348. find the angle B from the Bearings; then is G/c B= (2 x BFG tang. B B Example. Let BA bear S. 75~ E., and BC N. 60~ E., and let five acres be required to be parted off from the field by a perpendicular to BA. Here the angle B = 45~, and BF = 10.00 chains. The scale of the figure is 20 chains to 1 inch = 1:15840. (507) To part off a quadrilateral. Produce the converging sides to meet at B. Calculate the Fig. 369. content of the triangle HKB, whe- c ther on the ground or plat, or from K Bearings. Add it to the content -.... _ of the quadrilateral required to be B —- A parted off, and it will give that of the triangle FGB, and the method of the preceding case can then be applied. (508) To part off any figure, If the field be very irregularly shaped, find by trial any line which will part off a little less than the required area. This trial line will represent HK in the preceding figure, and the problem is reduced to parting off, according to the required condition, a quadrilateral, comprised between the trial line, two sides of the field, and the required line, and containing the difference between the required content and that parted off by the trial-line. C. BY A LINE RUNNING IN ANY GIVEN DIRECTION. (509) To part off a triangle. By construction, on the ground or the plat, proceed nearly as in Art. (505), setting out a line in the required direction, calculating the triangle thus formed, and obtaining BF by the same formula as in that Article. 22 338 LAYING OUT AND DIVIDING UP LAND. [PART XI. (510) If the field be given by Bearings, find Fig. 350. from them the angles CBA and GFB; then is G BF= j2 x BFG sin. (B + F) B sin. B. sin. F \ Example. Let BA bear S. 30~ E.; BC, \ N. 80~ E.; and a fence be required to run,from \ some point in BA, a due North course, and to part off one acre. Required the distance from \ B to the point F, whence it must start. Ans. The angle B = 70~, and F = 30~. Then BF - 6.47. The scale of Fig. 350 is 6 chains to 1 inch = 1:4752. (511) To part off a quadrilateral. Let it be required to part off, by a line running in a Fig. 351. given direction, a quadrilar- ---- teral from a field in which are given the side AB, and. the directions of the two B 13 / other sides running from A \ and from B. On the ground or platD produce the two converging sides to meet at some point E. Calculate the content A of the triangle ABE. Measure the side AE. From ABE subtract the area to be cut off, and the remainder will be the content of the triangle CDE. From A set out a line AF parallel to the given direction. Find the content of ABF. Take it from ABE, and thus obtain AFE. Then this formula, ED =AE /DE will fix V EAE the point D, since AD = AE - ED. (512) When the field and the dividing line are given by Bearings, produce the sides as in the last article. Find all the angles from the Bearings. Calculate the content of the triangle ABE, by the formula for one side and its including angles. Take the CHAP. II.] Parting off Land 339 desired content from this to obtain CDE. Calculate the side A_ sin. B 2 X CDE.sin.DCE) AE =AB. B. ThenisAID=AE-, -I. x -:- — ^7-11 sin. E ThenisAD E — sin. E. sin. DCE Example. Let DA bear S. 20}~ W.; AB, N. 510~ W., 8.19; BC, N. 738~ E.; and let it be required to part off two acres by a fence, DC, running N. 45~ W. Ans. ABE = 32.50 sq. chains; whence CDE- 12.50 sq. chs. Also, AE== 8.37; and finally AD = 8.37 - 5.49 = 2.88 chains. The scale of Fig. 351 is 5 chains to 1 inch = 1:3960. If the sum of the angles at A and B was more than two right angles, the point E would lie on the other side of AB. The necessary modifications are apparent. (513) To part off any figure. Proceed in a similar manner to that described in Art. (508), by getting a suitable trial-line, producing the sides it intersects, and then applying the method just given. D. BY A LINE STARTING FROM A GIVEN POINT IN A SIDE. (514) To part off a triangle. Let it be required to cut off from a corner of a field a triangu- Fig. 352. lar space of given content, by a D line starting from a given point A.... on one of the sides, A in the figure, the base, AB, of the desired tri- B / \ \ angle being thus given. If the field be given on the ground or on A a plat, divide the given content by half the base, and the quotient will be the height of the triangle. Set off this distance from any point of AB, perpendicular to it, as from A to C; from C set out a parallel to AB, and its intersection with the second side, as at D, will be the vertex of the required triangle. Otherwise, divide the required content by half of the perpendicular distance from A to BD, and the quotient will be BD. * This original formula is very convenient for logarithmic computation. 340 LAYING OUT AND DIVIDING UP LAND. [PART XI (515) If the field be given by the Bearings of two sides and the length of one of them, deduce the angle B (Fig. 352) from the 2 x ABD Bearings, as in Art. (213). Then is BD =. B. AB. sin. B' If it is more convenient to fix the point D, by the Second Method, Art. (6), that of rectangular co-ordinates, we shall have BE = BD. cos. B; and ED =BD. sin. B. The Bearing of AD is obtained from the angle BAD; which is ED ED known, since tang. BAD. Example. Eighty acres are to be set off from a corner of a field, the course AB being N. 80~ W., sixty chains; and the Bearing of BD being N. 700 E. Ans. BD= —53.33; BE 46.19; ED = 26.67; and the Bearing of AD, N. 60 48' W. The scale of Fig. 352 is 40 chains to 1 inch =1: 31680. 2 ABD If the field were right angled at B, of course BD -2 ABD (516) To part off a quadrilateral. Imagine the two converging sides of the field produced to meet, as in Art. (511). Calculate the content of the triangle thus formed, and the question will then be reduced to the one explained in the last two articles. (517) To part off any figure. Proceed as directed in Art.(513). Otherwise, proceed as follows. The field being given on the ground or on a plat, find on which side of it the required line will end, by drawing or running " guess lines" from the given point to various angles, and roughly measuring the content thus parted off. Fig. 353. If, as in the figure, A being the D given point, the guess line AD c/ parts off less than the required con- / / tent, and AE parts off more, then the desired division line AZ will B /.' end in the side DE. Subtract the -- area parted off by AD from the / required content, and the difference will be the content of the tri CHAP. II.] Parting off Land. 341 angle ADZ. Divide this by half the perpendicular let fall from the given point A to the side DE, and the quotient will be the base, or distance from D to Z. Or, find the content of ADE and make this proportion; ADE: ADZ:: DE. DZ. (518) The field being given by Bearings and distances, find as before, by approximate trials on the plat, or otherwise, which side the desired line of division will terminate in, as DE in the last figure. Draw AD. Find the Latitude and Departure of this line, and thence its length and Bearing, as in Art. (440). Then calculate the area of the space this line parts off, ABCD in the figure, by the usual method, explained in Part III, Chapter VI. Subtract this area from that required to be cut off, and the remainder will be the area of the triangle ADZ. Then, as in Art. (515), D2 ADZ AD. sin. ADZ' This problem may be executed without any other Table than that of Latitudes and Departures, thus. Find the Latitude and Departure of DA, as before, the area of the space ABCD, and thence the content of ADZ. Then find the Latitude and Departure of EA, and the content of ADE. Lastly, make this proportion: ADE: ADZ:: DE: DZ.' Example. In the field ABCDE, &c., part of which is shown in Fig. 353, (on a scale of 4 chains to 1 inch=== 1: 3168), one acre is to be parted off on the west side, by a line starting from the angle A. Required the distance from D to Z, the other end of this dividing line.t The only courses needed are these. AB, N. 53~ W., 1.55; BC, N. 200 E., 2.00; CD, N. 531~ E., 1.32; DE, S. 57~ E., 5.79. A rough measurement will at once shew that ABCD is less than an acre, and that ABCDE is more; hence the desired line will fall * The problem may also be performed by making the side on which the division line is to fall, a Meridian, and changing the Bearings as in Art. (244). The difference of the new Departures will be the Departure of the Division line. Its position. can then be easily determined, by calculations resembling those in Part VII, Chapter IV, Arts. (443), &c. t If the whole field has been surveyed and balanced, the balanced Latitudes and Departures should be used. We will here suppose the survey to have proved perfectly correct. 342 LAYING OUT AND DIVIDING UP LAND. [PART XI. on DE. The Latitudes and Departures of AB, BC and CD are then found. From them the course AD is found to be N. 80 E., 3.63. The content of ABCD will be 3.19 square chains. Subtracting this from one acre, the remainder, 6.81 sq. chs., is the content of ADZ. AP= 3.63 x sin. 65~ = 3.29. Dividing ADZ by half of this, we obtain DZ = 4.14 chains. By the Second Method, the Latitude and Departure of DA, the area of ABCD, and of ADZ, being found as before, we next find the Latitude and Departure of EA, from those of AD and DE, and thence the area of ADE = 9.53. Lastly, we have the proportion 9.53: 6.81:: 5.79: DZ 4.14, as before. E. BY A LINE PASSING THROUGH A GIVEN POINT WITHIN THE FIELD. (519) To part off a triangle. Let P be a point within a field through which it is required to Ml B run a line so as to part off from Fig. 354. the field, a given area in the/ \ form of a triangle. When the field is given on the//' ground or on a plat, the division / \ can be made by construction, // thus. From P draw PE, paral- / / p/ lel to the side BC. Divide the //''\ given area by half of the perpen-,/, B," dicular distance from P to AC, \ and set off the quotient from C /! / I \ to G. Bisect GC in H. On <- i G A L r~., c HE describe a semi-circle. On C it set off EK = EC. Join KH. Set off HL -= HK. The line LM, drawn from L through P, will be the division line required.' If HK be set off in the contrary direction, it will fix another line L'PM', meeting CB produced, and thus parting off another triangle of the required content. Example. Let it be required to part off 31.175 acres by a fence passing through a point P, the distance PD of P from the * As some lines in the figure are not used in the construction, though needed for the Demonstration, the student should draw it himself to a large scale. CHAP. II.] Parting off Land. 343 side BC, measured parallel to AC, being 6 chains, and DC 18 chains. The angle at C is fixed by a "tie-line" AB= 48.00 BC being 42.00, and CA being 30.00. Ans. CL= 27.31 chains, or CL' - 7.69 chains. The figure is on a scale of 20 chains to 1 inch = 1: 15840. (520) If the angle of the field and the position of the point P are Fig. 355. given by Bearings or angles, proceed p thus. Find the perpendicular dis- tances, PQ and PR, from the given point to the sides, by the formulas / i \ PQ=PC. sin. PCQ; and PR= PC. sin. PCR. Let PQ= q, PR,_/ _____ =p, and the required content= c. R aL' c T V sin. LCM _Example. Let the angle LCM 82~. Let it be required to part off the same area as in the preceding example. Let PC = 19.75, PCQ =170 30'1, PCR=64~ 291'. Required CL. Ans. PQ = 5.94, PR = 17.82, and therefore, by the formula, CL-= 27.31, or CL'=- 7.69; corresponding to the graphical solution. The figure is on the same scale. If the given point were without the field, the division line could be determined in a similar manner. (521) To part off a quadrilateral. Conceive the two sides of the field which the division line will intersect, Fig. 356. DA and CB, produced till they meet at a C point G, not shown in the figure. Calculate the triangle thus formed outside of the field. \ Its area increased by the required area, will be that of the triangle EFG. Then the problem is identical with that in the last article. The following example is that given in Gummere's Surveying. The figure E represents it on a scale of 20 chains to 1 inch =: 15840. 344 LAYING OUT AND DIVIDING UP LAND. [PART XI. Example. A field is bounded thus: N. 14~ W., 15.20; N. 70-1 E., 20.43; S. 6~ E., 22.79; N. 86~1 W., 18.00. A spring within it bears from the second corner S. 75~ E., 7.90. It is required to cut off 10 acres from the West side of the field by a straight fence through the spring. How far will it be from the first corner to the point at which the division fence meets the fourth side? Ans. 4.6357 chains. (522) To part off any figure. Let it be required to part off from a field a certain area by Fig 357. a line passing through a given point P within the field. Run a guess-line AB through P. Calculate the area which it c' parts off. Call the difference between it and the required area - d. Let CD be the desired line of division, and let P represent the angle, APC or BPD, which it makes with the given line. Obtain the angles PAC = A, and PBD = B, either by measurement, or by deduction from Bearings. Measure PA and PB. Then the desired angle P will be given by the following formula. Cot. P= — (cot. A + cot. B - AP. BP) I [AP2. cot. B -BP2. cot. A 2d _ —cot. A. cot. B +: (cot. A + cot. B- AP2-BP2)2] If the guess line be run so as to be perpendicular to one of the sides of the field, at A, for example, the preceding expression reduces to the following simpler form. Cot. P= - (cot. B - AP2-BP2) 2 otL —-B — VL.2 d X 2 d CEp. ii.] Parting off Land. 345 Example. It was required to cut off from a field twelve acres by a line passing through a spring, P. A guess-line, AB, was run making an angle with one side of the field, at A, of 55~, and with the opposite side, at B, of 81~. The area thus cut off was found to be 13.10 acres. From the spring to A was 9.30 chains, and to B 3.30 chains. Required the angle which the required line, CD, must make with the guess line, AB, at P. Ans. 20~ 45'; or - 86~ 25'. The heavy broken line, C'D', shows the latter. The scale of the figure is 10 chains to 1 inch = 1: 7920. If the given point were outside of the field, the calculations would be similar. F. BY THE SHORTEST POSSIBLE LINE. (523) To part off a triangle. Let it be required to part off a triangular space, BDE, of given content, from the Fig. 358. corner of a field, ABC, by the shortest possible 1 line, DE. / From B set off BD and BE each equal to / pVe/ nBDE The line DE thus obtained ill be perpendicular to the line, BF, which bisects the anr (2. DBE. sin. B) gle B. The length of DE = c(2 s. B. sn. /) Example. Let it be required to part off 1.3 acre from the corner of a field, the angle, B, being 30~. Ans. BD = BE = 7.21; and DE = 3.73. The scale of the figure is 10 chains to 1 inch =1: 7920. G. LAND OF VARIABLE VALUE. (521) Let the figure represent a field in which Fig. 359. the land is of two qualities and values, divided by B C the " quality line" EF. It is required to part off from it a quantity of land worth a certain sum, by E a straight fence parallel to AB. Multiply the value per acre of each part by its length (in chains) on the line AB, add the pro- ducts, multiply the value to be set off by 10, divide A D 346 LAYING OUT AND DIVIDING UP LAND. [PART XI. by the above sum, and the quotient will be the desired breadth, BC or AD, in chains. -Example. Let the land on one side of EF be worth $200 per acre, and on the other sisle $100. Let the length of the former, BE, be 10 chains, and EA be 30 chains. It is required to part off a quantity of land worth $7500. Ans. The width of the desired strip will be 15 chains. The scale of the figure is 40 chains to 1 inch = 1: 31680. If the " quality line" be not perpendicular to AB, it may be made so by " giving and taking," as in Art. (124), or as in the article following this one. The same method may be applied to land of any number of different qualities; and a combination of this method with the preceding problems will solve any case which may occur. H. STRAIGHTENING CROOKED FENCES. (525) It is often required to substitute a straight fence for a crooked one, so that the former shall part off precisely the same quantity of land as did the latter. This can be done on a plat by the method given in Art. (83), by which the irregular figure Fig. 360. 2...... ---- — I — 1...2..3...4...5 is reduced to the equivalent triangle 1.. 5...3', and the straight line 5...3' therefore parts off the same quantity of land on either side as did the crooked one. The distance from 1 to 3/, as found on the plat, can then be set out on the ground and the straight fence be then ranged from 3' to 5 The work may be done on the ground more accurately by running a guess line, AC, Fig. 361, across the bends of the fence which crooks from A to B, measuring offsets to the bends on each side of the guess line, and calculating their content. If the sums of these areas on each side of AC chanced to be equal, that would be the line desired; but if, as in the figure, it passes too far on one CHAP. III.] Dividing up Land. 347 Fig. 361.,~ ~T —— T,_ side, divide the difference of the areas by half of AC, and set it off at right angles to AC, from A to D. DC will then be a line parting off the same quantity of land as did the crooked fence. If the fence at A was not perpendicular to AC, but oblique, as AE, then from D run a parallel to AC, meeting the fence at E, and EC will be the required line. CHAPTER III. DIVIDING UP LAND. (526) MOST of the problems for " Dividing up" land may be brought under the cases in the preceding chapter, by regarding one of the portions into which the figure is to be divided, as an area to be "Parted off" from it. Many of them, however, can be most neatly executed by considering them as independent problems, and this will be here done. They will be arranged, firstly, according to the simplicity of the figure to be divided up, and then sub-arranged, as in the leading arrangement of Chapter II, according to the manner of the division. DIVISION OF TRIANGLES. (527) By lines parallel to a side. Sup- Fig.362. pose that the triangle ABC is to be divided into two equivalent parts by a line parallel to AC. The desired point, D, from which this line is to A start, will be obtained by measuring BD = AB V/. So, too, E is fixed by BE = BC V/. 348 LAYING OUT AND DIVIDING UP LAND. [PART X. Generally, to divide the triangle into two parts, BDE and ACED, which shall have to each other a ratio = m: n, we have BD = AB /nz This may be constructed thus. Describe a Fig. 363. semicircle on AB as a diameter. From B'set - -/\ off BF m=.BA. At F erect a perpendi- \ j / \ m+n cular meeting the semicircle at G. Set off BG / / from B to D. D is the starting point of the divi- A C sion line required. In the figure, the two parts are as 2 to 8, and BF is therefore = 2 BA. To divide the triangle ABC into five Fig. 364. equivalent parts, we should have, similarly, BD - AB V1; BD'=AB V/; BD" AB,/; BD"'/-AB V4. D E The same method will divide the trian- gle into any desired number of parts hav- " ing any ratios to each other, A c (528) By lines perpendicular to a side, Suppose that ABC is to be divided into two parts having Fig. 365. B a ratio = m: n, by a line perpendicular to AC. Let EF be the dividing line whose position is required. Let BD be a perpendicular let fall from B to E. AC. ThenisAE= (AC x AD x ). In this figure, m / nAFE: EFBC::: n:: 1:2. If the triangle had to be divided into two equivalent parts, the above expression would become AE = V (/ AC x AD). (529) By lines running in any given direction. Let a triangle, ABC, be given to be divided into two parts, having a ratio = m: n, by a line making a given angle with a side. Part off, as in Art. (509) or (510), Fig. 350, an area BFG - = ABC. m -+ CHAP. III.] Dividing up Lando 349 (530) By lines starting from an angle. Divide the side opposite to the given angle into the required num- Fig. 366. her of parts, and draw lines from the angle to the points of division. In the figure the triangle is represented as being thus divided into two equivalent parts. A /..c If the triangle were required to be divided into two parts, having m to each other a ratio = m: n, we should have AD = AC - and DC = AC nm +- n If: the triangle had to be divided into three Fig. 367. parts which should be to each other: m n: p, we should have AD AC m+ +, DE = AC,and EC=AC P A E m -+ n - p m + n + p Suppose that a triangular field ABC, had to be divided among five men, two of them to have a quarter each, and three of them each a sixth. Divide AC into two equal parts, one of these again into two equal parts, and the other one into three equal parts. Run the lines from the four points thus obtained to the angle B. (531) By lines starting from a point in a side, Suppose that the triangle ABC is to be divided into two Fig. 368 equivalent parts by a line starting from a point D in the side AC. Take a point E in the middle of AC. Join BD, and from E draw a parallel to it, meeting AB in F. DF will be A -E the dividing line required. The point F will be most easily obtained on the ground by the proportion AD: AB:: AE = A AC: AF. The altitude of AFD of course equals ~ ABC -- AD. If the triangle is to be divided into two parts having any other ratio to each other, divide AC in that ratio, and then proceed as AB x AC m before. Let this ratio = m: n, then AF = AD m+ n 350 LAYING OUT AND DIVIDING UP LAND. [PART xI. (532) Next suppose that the trian- Fig. 369. gle ABC is to be divided into three equivalent parts, meeting at D. The F' altitudes, EF and GH, of the parts\ / ADE and DCG, will be obtained by c In dividing I ABC, by half of the respective bases AD and DC. If one of these quotients gives an altitude greater than that of the triangle ABC, it will shew that the two lines DE and DG would both cut the same side, as in Fig. 370, in Fig. 370. which EF is obtained as above, and GH = GB ABC + I AD.,\ In practice it is more convenient to de- termine the points F and G, by these proportions; F HK BK: AK:: EF: AF; and BK: AK:: GH: AH. The division of a triangle into a greater number of parts, having any ratios, may be effected in a similar manner. (533) This problem admits of a more elegant solution, analogous to that given for the division into two Fig. 371. parts, graphically. Divide AC into G three equal parts at L and M. Join BID, and from L and M draw paral- lels to it, meeting AB and BC in E A - DM C and G. Draw ED and GD, which will be the desired lines of division. The figure is the same triangle as Fig. 369. The points E and G can be obtained on the ground by measuring AD and AB, and making the proportion AD: AB:: - AC: AE. The point G is similarly obtained. The same method will divide a triangle into a greater number of parts. (534) To divide a triangle into four equivalent triangles by lines terminating in the sides, is very Fig. 372. easy. From D, the middle point of AB, draw DE parallel to AC, and from F, the middle of AC, draw FD and FE. The problem is now solved. A CHAP. III.] Dividing up Land. 351 (535) By lines passing through a point within the triangle. Let D be a given point (such as a well, Fig. 373. &c.) within a triangular field ABC, from which fences are to run so as to divide'D the triangle into two equivalent parts. Join AD. Take E in the middle of BC, Ai' SC and from it draw a parallel to DA, meeting AC in F. EDF is the fence required. (536) If it be required to di- d B vide a triangle into two equiva- Fig. 374. lent parts by a straight line pass- / ing through a point within it, pro-/ / \ ceed thus. Let P be the given /' point. From P draw PD paral- 1 lel to AC, and PE parallel to BC. / / Bisect AC at F. Join FB. From // / B draw BG parallel to DF. Then // < bisect GC in H. On HE de- /,,/ \ \ scribe a semicircle. On it set off, / / i / i \ EI EC. Join KH. Set off,/,, / I,/' HL = HK. The line LM drawn / —-/ G AL 3^ L71 C from L, through P, will be the ". / division line required. This figure is the same as that of Art. (519). The triangle ABC contains 62.35 acres, and the distance CL = 27.31 chains, as in the example in that article. (537) Next suppose that the trian- Fig. 375. gle ABC is to be divided into three- equivalent parts by lines starting from a point D, within the triangle, given by. the rectangular co-ordinates AE and and ED. Let ED be one of the lines K E of division, and F and G the other points required. The point F will be determined if AH is known; AH and HF being its rectangular co-ordinates. From B let fall the perpendicular BK on AC. 352 LAYING OUT AND DIVIDING UP LAND. [PART XI. AK (4 ABC - AE x ED) Then isA- - A-AE x BC - A x D) The position of the AE x BK-ED x AK other point, G, is determined in a similar manner. (538) Let DB, instead of DE, Fig. 376. be one of the required lines of division.. Divide ABC by half of the perpendicular DH, let fall from D to AB, and the quotient will be the distance BF. To find \ G, if, as in this figure, the trian- A. - -^- gle BDC (= BC x 1 DK) is less than ~ ABC, divide the excess of the latter (which will be CDG) by ~ DE, and the quotient will be CG. Example. Let AB = 30.00; BC = 45.00; CA = 50.00. Let the perpendiculars from D to the sides be these; DE = 10.00; DR = 20.00; DK =5.17~. The content of the triangle ABC will be 666.6 square chains. Each of the small triangles must therefore contain 222.2 sq. chs., BD being one division line. We shall therefore have BF = 222.2' 1 DH -- 22.2 chains. BDC -=45 x ~ x 5.17~ = 116.4 sq. chs., not enough for a second portion, but leaving 105.8 sq. chs. for CDG; whence CG = 21.16 chs. To prove the work, calculate the content of the remaining portion, GDFA. We shall find DGA = 144.2 sq. chs., and ADF = 78.0 sq. chs., making together 222.2 sq. chs., as required. The scale of Fig. 376 is 30 chains to 1 inch = 1: 23760. (539) The preceding case may Fig. 377. be also solved graphically, thus. Take CL = AC. Join DL, and from B draw BG parallel to DL. Join DG. It will be a second line of division. Then take a point, M, in the middle of BG, and from A- - Gi — it draw a line, MF, parallel to DA. DF will be the third line of division. This method is neater on paper than the preceding; but less convenient on the ground. CHAP. III.] Dividing up Land. 353 (540) Let it be required to divide Fig. 378. the triangle ABC into three equivalent triangles, by lines drawn from the three angular points to some un- known point within the triangle. This F point is now to be found. On any Ac side, as AB. take AD = l AB. From D draw DE parallel to AC. The middle, F, of DE, is the point required. If the three small triangles are not to be equivalent, but are to have to each other the ratios:: m: n: p, Fig. 379. divide a side, AB, into parts having these ratios, and through each point of division, D, E, draw a parallel to the side nearest to it. The intersec- tion of these parallels, in F, is the Ac —---- point required. In the figure the parts ACF, ABF, BCF, are as 2: 3: 4. (541) Let it be required to find Fig. 380. the position of a point, D, situated within a given triangle, ABC, and equally distant from the points A, B, C; and to determine the ratios to each other of the three triangles into Ac which the given triangle is divided. By construction, find the centre of the circle passing through A, B, C. This will be the required point. AB x BC x CA By calculation, the distance DA = DB = DC = x BCa x A 4 x area ABC The three small triangles will be to each other as the sines of their angles at D; i. e. ADB: ADC BDC:: sin. ADB: sin. ADC: sin. BDC. These angles are readily found, since the sine of half of each of them equals the opposite side divided by twice one of the equal distances. 23 354 LAYING OUT AND DIVIDING UP LAND. [PART XI (542) By tihe shortest possible line. Let it be Fig. 381. required to divide the triangle ABC by the short / est possible line, DE, into two parts, which shall be to each other:: m: n; or DBE: ABC:: m: m + n. 1 From the smallest angle, B, of the triangle, / measure along the sides, BA and BC, a distance BD=BE /(m-X ABxBC). DEis the A/ C line required. It is perpendicular to the line BF which bisects the angle ABC; and it is =-. B I( x AB x cos. ~ B m+ n X AB x BC) DIVISION OF RECTANGLES. (543) By lines parallel to a side. Divide two opposite sides into the required number of parts, either equal or in any given ratio to each other, and the lines joining the points of division will be the lines desired. The same method is applicable to any parallelogram. Example. A rectangular field Fig. 382. ABCD, measuring 15.00 chains B -— E_ F' C by 8.00, is bought by three men, who pay respectively $300, $400 and $500. It is to be divided among them in that proportion. Ans. The portion of the first, A E F D AEE'B, is obtained by making the proportion 300 + 400 + 500: 300:: 15.00: AE = 3.75. EF is in like manner found to be 5.00; and FD = 6.25. BE' is made equal to AE; E'F' to EF; and F'C to FD. Fences from E to E', and from F to F', will divide the land as required. The scale of the figure is 10 chains to 1 inch = 1: 7920. The other modes of dividing up rectangles will be given under the head of " Quadrilaterals," Art. (548), &c. CHAP. II.] Dividing up Land. 355 DIVISION OF TRAPEZOIDS (544) By lines parallel to the bases. Given the bases and a third side of the trapezoid, ABCD, to be Fig. 383. divided into two parts, such that BCFE: RB _ EFDA:: m: n. I / The length of the desired dividing line, i (Qm x AD2 + n x BC2\ EF flL+ ^ E;n; +' T, _ AB (EF - BC) The distance BE= A — -( B- ) B AD — BC'.Example. Let AD = 30 chains; BC = 20 chs.; and AB = 541 chs.; and the parts to be as 1 to 2; required EF and BE. Ans. EF= 23.80; and BE =20.65. I The figure is on a scale of 30 chains to 1 A H D inch = 1: 23760. (545) Given the bases of a trapezoid, and the perpendicular distance, BH, between them; it is required to divide it as before, and to find EF, and the altitude, BG, of one of the parts. Let BCFE: EFDA::: m: n. Then BG = - BCxBI + AD-BC mirm 2 x ABCI)X BH (BC x BHI 2] AD-BC EF=BC +BGx B x Ecxample. Let AD = 30.00; BC 20.00; BH = 54.00; and the two parts to be to each other:: 46: 89. The above data give the content of ABCD = 1350 square chains. Substituting these numbers in the above formula, we obtain BG = 20.96, and EF = 23.88. (546) By lines starting from points in a side. To divide a trapezoid into parts equivalent, or having any ratios, divide its parallel sides in the same ratios, and join the corresponding points. 356 LAYING OUT AND DIVIDING UP LAND. [PART XT. If it be also required that the division lines shall start from given points on a side, proceed Fig. 384. thus. Let it be required to B/ \ X divide the trapezoid ABCD / \ into three equivalent parts by fences starting from P and Q Divide the trapezoid, as above directed, into three equivalent A. r P o —, trapezoids by the lines EF and GH. These three trapezoids mu now be transformed, thus. Join EP, and from F draw FR paral lel to it. Join PR, and it will be one of the division lines required. The other division line, QS, is obtained similarly. (547) Other cases. For other cases of dividing trapezoids, apply those for quadrilaterals in general, given in the following articles.* DIVISION OF QUADRILATERALS. (548) By lines parallel to a side. Let ABCD be a quadrila teral which it is required to Fig. 385. divide, by a line EF, paral- G lel to AD, into two parts, BEFC and EFDA, which shall be to each other as m: n. Prolong AB and CD, to intersect in G. Let a be the area of the triangle B" \ ADG-, obtained by any me- K thod, graphical or trigono- / metrical, and a'= the area A B H O of the triangle BCG, obtained by subtracting the area of the given quadrilateral from that of the triangle ADG. Then GK = GH /( (ma -+ n ). Having measured this length of GK from G on (m + n) af' GH, set off at K a perpendicular to GK, and it will be the required line of division. * If a line be drawn joining the middle points of the parallel bases of a trapezoid, any line drawn through the middle of the first line, and meeting the parallel bases, will divide the trapezoid into two equivalent parts. CHAP. III.] Dividing up Land. 357 Otherwise, take GE = GA ( ma na; and from E run V (m + n) a! a fo E a parallel to AD. If the two parts of the quadrilateral were to be equivalent, m =n, and we have GK = GH 2(a +- a - and consequently GE to 2i a r' GA in the same ratio. Example, Let a quadrilateral, ABCD, be required to be thus divided, and let its angles, B and C, be given by rectangular co-ordinates, viz: AB' = 6.00; B'B = 9.00; DC' = 8.00;.C'C = 13.00; B'C' = 24.00. Here GH is readily found to be 29.64; ADG = 563.16 square chains; and BGC = 220.16 square chains. Hence, by the formula, GK = 24.72; whence KH = GH - GK = 4.92; and the abscissas for the points E and F can be obtained by a simple proportion. The scale of the figure is 20 chains to 1 inch = 1: 15840. If the quadrilateral be given by Bearings, part off the desired area = --. ABCD, by the formulas of Art. (504). Suppose now that a quad- Fig. 386. rilateral, ABCD, is to be di- vided into p equivalent parts,' \ by lines parallel to AD. Measure, or calculate by Trigonometry, AG. Let Qbe F- the quadrilateral ABCD, and, N -- as before, a' = BOG. Then A D,( Q ),(, 2Q) GE = AG p GL=AG a; a' + Q a' + Q, 3Q GN=A G /o + p; &c. (=J a' + Q a+Q If the quadrilateral be given by Bearings, part off, by Art. (504), 1 2 -. ABCD, then part off -. ABCD; &c.; so in any similar case. P P 358 LAYING OUT AND DIVIDING'UP LAND, [PART X1 (549) By lines perpendicular to a side. Let ABCD be a quadrilateral which is to be divided, by Fig. 387. a line perpendicular to AD, into two: parts having a ratio = mn: n. By hypo- / thesis, ABEF m. ABCD. Taking away the triangle ABG, the A a E X remainder, GBEF, will be to the rest of the figure in a known ratio, and the position of EF, parallel to BG, will be found as in the last article. (559) By lines running in any given direction. To divide a quadrilateral ABCD into two parts:: mn: n, part off from it an area =. ABCD, by the methods of Arts. (509) or (510), if the area parted off is to be a triangle, or Arts. (511) or (512), if the area parted off is to be a quadrilateral. (551) By lines starting from an angle. ABCD is to be divided, by the line CE, into two Fig. 388. parts having the ratio m: n. Since the area of the triangle CDE.ABCD,DEwill m+ n / be obtained by dividing this area A- -- by half of the altitude CF. (552) By lines starting from points in a side. Let it be required to divide ABCD into two Fig. 389. parts::: n, by a line starting from the point E. The area ABFE is'known, (being. ABCD) as / also ABE; AB, BE, and EA be- A E ing given on the ground. BEF will then be known = ABFE - BEF ABE. Then GF = BE, and the point F is obtained by running a parallel to BE, at a perpendicular distance from it = GF. CHAP. III.] Dividing up Land. 359 To divide a quadrilateral, ABCD, Fig. 390. graphically, into two equivalent partsE by a line from a point, E, on a B \ side, proceed thus. Draw the diago-\ nal CA, and from B draw a parallel / / \ to it, meeting DA prolonged in F. / \' \ Mark the middle point, G, of FD. A; H Join GE. From C draw a parallel to EG, meeting DA in i. EPI is the required line. The quadrilateral could also be divided in any ratio - m: n by dividing FD in that ratio. If the quadrilateral be given by Bearings, proceed to part off the desired area, as in Art. (515) or (516). (553) Let it be required to divide a quadrilateral, ABCD, into three equivalent parts. Fig. 391. From any angle, as C, B draw CE, parallel to DA. / H Divide AD and EC, each E/ —-' —-- —'- C into three equal parts, at F, F', and G, G'. Draw / BF, BF'. From G draw GH, parallel to FB, and_ from G' draw G'H', pa- F F rallel to F'B. FH and F'II' are the required lines of division. Let it be required to make Fig. 392. the above division by lines C starting from two given L points, P and Q. Reduce D the quadrilateral to an equi- \ valent triangle CBE, as in \ Art. (87). Divide EB into \ three equal parts at F-and' I ------— _. G. Join CQ, and, from G, E A B draw GK parallel to it. Join CP, and from F draw FL parallel to it. Join PL and QK, and they will be the division lines required. (554) By lines passing through a point within the figure, Proceed to part off the desired area as in Arts. (519), (520), or (521), according to the circumstances of the case. 360 LAYING OUT AND DIVIDING UP LAND. [rAnT xI. DIVISION OF POLYGONS. (555) By lines running in any direction. Let ABCDEFG be a given polygon, and BH the di- Fig. 393. rection parallel to which is to be \R drawn a line PQ, dividing the, polygon into two parts in any de- sired ratio = m: n. The area \ PCDEQ -= m. ABCDEFG. P - \ m +n \ Taking it from the area BCDEH, B —- - \ the remainder will be the area \ BPQH. The quadrilateral A s F BCEH, CE being supposed to be drawn, can then be divided by the method of Art. (548), into two parts, BPQH and PQEC, having to each other a known relation. If DK were the given direction, at right angles to the former, the position of a dividing line RS could be similarly obtained. (556) By lines starting from an angle, Produce one side, AB, Fig. 394. z X A' V A P la equivalent triangle, XYZ, by the method of Art. (82). Then I \ \ \% divide the base, XY, in the required ratio, as at W, and draw ZW, which will be the division line desired. In this figure the polygon is divided into two equivalent parts. /: —'~~":~-~T~~"-""'I of~~~~~~~~ thI'II Iyon ohMasadrdceteplgn oasn equiale I \rage XYb h Io fAt.(~. Te divid / th Nae P nterqie rto sa,add IW Ihc wilb h iiinln eie. I hsfgr h CHAP. III.] Dividing up Land. 361 If the division line should pass outside of the polygon, as does ZP, through P draw a parallel to BZ, meeting the adjacent side of the polygon in Q, and ZQ will be the division line desired. (557) By lines starting from a point on a side. See Articles (517) and (518) in the preceding chapter. (558) By lines passing through a point within the figure. Part off, as in Arts. (519) or (522) in the preceding chapter, if a straight line be required; or by guess lines and the addition of triangles, as in Art. (538) of this chapter, if the lines have merely to start from the point, such as a spring or well. (559) Other problems. The following is from Gummere's Surveying. Question. A tract of land is Fig. 395. bounded thus: N. 35~ E., 23.00; N. A r F 750~ E., 30.50; S. 310 E., 46.49; N.. 661~ W., 49.64. It is to be divided into four equivalent parts by two straight lines, G -_ one of which is to run parallel to the third A side; required the distance of the parallel division line from the first corner, measured on the fourth side; also the Bearing of the other division line, and its distance from the same corner measured on the first side. Ans. Distance of the parallel division line from the first corner, 32.50; the Bearing of the other, S. 88~ 22' E.; and'its distance from the same corner 5.99. The scale of the figure is 40 chains to 1 inch = 1: 31680. An indefinite number of problems on this subject might be proposed, but they would be matters of curiosity rather than of utility, and exercises in Geometry and Trigonometry rather than in Surveying; and the youngest student will find his life too short for even the hastiest survey of merely the most fruitful parts of the boundless field of Mathematics. 362 F. S. PUBLIC LANDS. [PART XII. Fig. 396. + I IT//T e /I 1'^' ==~~~~~~~~~~~~~~~~~~~~~~~~~^^t Sc S V<,S IPART XII. THE PUBLIC LANDS OF THE UNITED STATES.* (560) General system. The Public Lands of the United States of America are generally divided and laid out into squares, the sides of which run truly North and South, or East and West. This is effected by means of Meridian lines and Parallels of Latitude, established six miles apart. The principal meridians and base lines are established astronomically, and the intermediate ones are run with chain and eompass. The squares thus formed are called TOWNSHIPS. They contain 36 square miles, or 23040 acres, " as nearly as may be." The map on the opposite page represents a portion of the Territory of Oregon thus laid out. The scale is 10 miles to 1 inch = 1: 633600. On it will be seen the "Willamette Meridian," running truly North and South, and a " Base line," which is a " Parallel of Latitude," running truly East and West. Parallel to these, and six miles from them, are other lines, forming Townships. All the Townships, situated North or South of each other, form a RANGE. The Ranges are named by their number East, or West of the principal Meridian. In the figure are seen three Ranges East and West of the Willamette Meridian. They are noted as R. I. E., R. I. W., &c. The Townships in each Range are named by their number North or South of the Base line. In * The substance of this Part is mainly taken from " Instructions to the Surveyor General of Oregon, being a Manual for Field Operations," prepared, in March, 1851, by John M. Moore, "Principal Clerk of Surveys," by direction of Hon. J. Butterfield, " Commissioner of the General Land Office," and communicated to the author by Hon. John Wilson, the present Commissioner. The aim of the " Instructions" is stated to be " simplicity, uniformity and permanency." They seem admirably adapted for these objects, and the lasting importance of the subject in this country has led the author to reproduce about half of them in this place. They were subsequently directed to be adopted for the Surveying service in Minnesota and California. 364 U. S. PUBLIC LANDS. [PART XII. the figure along the principal Meridian are seen four North and five South of the Base line. They are noted as T. 1 N., T. 2 N., T. 1 S., &c.* Each Township is divided into 36 SEC- N TIONS, each 1 mile square, and therefore 6 51 3 2 1 containing, " as nearly as may be," 640 7 8 9 10 11 12 acres. The sections in each Township are 18 17 16 15 14 13 numbered, as in the margin, from 1 to 36, W19 2 21 2223 24 beginning at the North-east angle of the 30 29 28 2726 25 Township, and going West from 1 to 6, 3132 33134 3136 then East from 7 to 12, and so on alter- nately to Section 36, which will be in the South-east angle of the Township. The Sections are sub-divided into Quarter-sections, half-a-mile square, and containing 160 acres, and sometimes into halfquarter-sections of 80 acres, and quarter-quarter-sections of 40 acres. By this beautiful system, the smallest subdivision of land can be at once designated; such as the North-east quarter of Section 31, in Township two South, in range two East of Willamette Meridian. (561) Difficulty. " The law requires that the lines of the public surveys shall be governed by the true meridian, and that the townships shall be six miles square,-two things involving in connection a mathematical impossibility-for, strictly to conform to the meridian, necessarily throws the township out of square, by reason of the convergency of meridians; hence, adhering to the true meridian renders it necessary to depart from the strict requirements of law as respects the precise area of townships, and the subdivisional parts thereof, the township assuming something of a trapezoidal form, which inequality developes itself, more and more as such, the higher the latitude of the surveys. In view of these circumstances, the law provides that the sections of a mile. square shall contain the quantity of 640 acres, as nearly as may be; and, moreover, provides that'In all cases where the exterior lines of the townships, thus to be subdivided into sections or half-sections, shall exceed, or shall not extend, six miles, the excess or deficiency * The marks 0, + and A, merely refer to the dates of the surveys. They are sometimes used'to point out lands offered for sale, or reserved, &c. PART XIi.] Difficulty. 365 shall be specially noted, and added to or deducted from the western or northern ranges of sections or half-sections in such township, according as the error may be in running the lines from east to west, or from south to north.'";" In order to throw the excesses or deficiencies, as the case may be, on the north and on the west sides of a township, according to law, it is necessary to survey the section lines from south to north on a true meridian, leaving the result in the northern line of the township to be governed by the convexity of the earth and the convergency of meridians." Thus, suppose the land to be surveyed, lies between 46~ and 47~ of North Latitude. The length of a degree of Longitude in Lat. 46~ N. is taken as 48.0705 statute miles, and in Lat. 47~ N. as 47.1944. The difference, or convergency per square degree = 0.8761 = 70.08 chains. The convergency per Range (8 per degree of Longitude) equals one-eighth of this, or 8.76 chains; and per Township (111 per degree of Latitude) equals the above divided by 11g, i. e. 0.76 chain. We therefore know that the width of the Townships along their Northern line is 76 links less than on their Southern line..The townships North of the base line therefore become narrower and narrower than the six mile width with which they start, by that amount; and those South of it as much wider than six miles. " STANDARD PARALLELS (usually called correction lines), are established at stated intervals (24 or 30 miles) to provide for or counteract the error that otherwise would result from the convergency of meridians; and, because the public surveys have to be governed by the true meridian, such lines serve also to arrest error arising from inaccuracies in measurements. Such lines, when lying north of the principal base, themselves constitute a base to the surveys on the north of them; and where lying south of the principal base, they constitute the base for the surveys south of them." The convergency or divergency above noticed is taken up on these Correction lines, from which the townships start again with their proper widths. On these therefore there are found Double Corners, both for Townships and Sections, one set being the Closing Corners of the surveys ending there, and the other set being the Standard Corners for the surveys starting there. 366 U. S. PUBLIC LANDS. [PART XII. (506) Running Township lines. "The principal meridian, the base line, and the standard parallels having been first astronomically run, measured, and marked, according to instructions, on true meridians, and true parallels of latitude, the process of running, measuring, and marking the exterior lines of townships will be as follows. Townships situated NORTH of the 6ase line, and WEST of the principal meridian.* Commence at No. 1, being the southwest corner of T. 1 N. —R. 1 W., as established on the base line; thence run north, on_a true meridian line, four hundred and eighty chains, establishing the mile and half-mile corners thereon, as per instructions, to No. 2, (the northwest corner of the same township), whereat establish the corner of Tps. 1 and 2 N.-Rs. 1 and 2 W.; thence east, on a random or trial line, setting temporary mile and half-mile stakes to No. 3, (the northeast corner of the same township), where measure and note the distance at which the line intersects the eastern boundary, north or south of the true or established corner. Run and measure westward, on the true line, (taking care to note all the land and water crossings, &c., as per instructions), to No. 4, which is identical with No. 2, establishing the mile and half-mile PERMANENT CORNERS on said line, the last half-mile of which will fall short of being forty chains, by about the amount of the calculated convergency per township, 76 links in the case above supposed. Should it ever happen, however, that such random line materially falls short, or overruns in length, or intersects the eastern boundary of the township at any considerable distance from the true corner thereon, (either of which would indicate an important error in the surveying), the lines must be retraced, even if found necessary to remeasure the meridional boundaries of the township (especially the western boundary), so as to discover and correct the error; in doing which, the true corners must be established and marked, and the false ones destroyed and obliterated, to prevent confusion in future; and all the facts must be distinctly set forth in the notes. Thence proceed in a similar manner north, from No. 4 to No. 5, (the N. W. corner of T. 2 N. -R. 1 W.), east from No. 5 to No. 6, (the N. E. corner of the same township), west from No. 6 to No. 7, (the same as No. 5), north from No. 7 to No. 8, (the N. W. corner of T. 3 N., R. 1 W.), east from No. 8 to No. 9, (the N. E. corner of same township), and thence west to No. 10, (the same as No. 8), or the southwest corner T. 4 N.-R. 1 W. Thence north, still on a true meridian line, establishing the mile and half-mile corners, until reaching the STANDARD PARALLEL or correction line, (which is here four town* The Surveyor should prepare a diagram of the townships, with the numbers here referred to. in their proper places, as here indicated PART XII.] Running Township Lines..3'7 ships north of the base line); throwing the excess over, or deficiency under, four hundred and eighty chains, on the last halfrmile, according to law, and at the intersection establishing the " CLosING CORNER," the distance of which fromn the standard corner must be measured and noted as required by the instructions. But should it ever so happen that some impassable barrier will have prevented or delayed the extension of the standard parallel along and above the field of present survey, then the surveyor will plant, in place, the corner for the township, subject to correction thereafter, should such parallel be extended. Townships situated NORTH of the base line, and EAST of the principal meridian. Commence at No. 1, being the southeast corner of T. 1 N.-R. 1 E., and proceed as with townships situated " north and west," except that the random or trial lines will be run and measured west, and the true lines, east, throwing the excess over or deficiency under four hundred and eighty chains on the west end of the line, as required by law; wherefore, the surveyor will commence his measurement with the length of the deficient or excessive half-section boundary on the west of the township, and thus the remaining measurements will all be even miles and half-miles. Townships situated SOUTH of the base line, and WEST of the principal meridian. Commence at No. 1, the northwest corner of township 1 S., range 1 W', and proceed due south in running and measuring line, establishing and marking the mile, half-mile, and township corners thereon, precisely in the method prescribed for running NORTH and WEST, with the exception that, in order to throw the excess or deficiency (over or under four hundred and eighty chains) of the western boundaries of such of those townships as close on the standard parallel on the south, upon the most nor liern half-mile of the townships, according to law, the proceeding will be as follows. The western (meridional) boundary line of every township, closing on the standard parallel, (being every fifth one in this case), will be carefully run south, on a true meridian, until it intersects the standard, planting stakes and making distinctive marks on line trees, in sufficient number to serve as guides in afterwards retracing the line north with ease and certainty. At the point of the line's intersection of the standard, the surveyor will establish the' closing" (southwest) corner of the township, noting in his field-book its distance and direction from the " standard corner." Then starting from such " closing corner," he will proceed north on the line identified by the guide stakes and marks, measuring such line, and establishing thereon the mile and half-mile stations, and noting, as he goes, all the land and water crossings, &c. 368 U. S. PUBLIC LANDS. [PART II. Townships situated SOUTH of the base line, and EAST of the principal meridian. Commence at No. 1, at the northeast corner of township 1 S., range 1 E., and proceed precisely as with the townships situated " south and west," except that the random lines will be run and measured west, and the true lines east; the deficiency or excess of the measurements being, as in all other cases, thrown upon the most western half-mile of line." (563) Running Section lines. The interior or sectional lines of all townships, however situated in reference to the BASE and MERIDIAN lines, are laid off and surveyed as below. 31 32 33 34 35 36 97 71 53 35 17 6 5 4 3 2 1 6 99 98 96 72 70 54 52 36 34 18 16 100 94 9568 6950 5132 3314 15 12 7 8 9 10 11 12 7 92 93 91 67 49 31 13 89 90 65 66 47 4829 3011 12 13 18 17 16 15 14 13 18 87 86 64 46 28 10 88 84 85 62 63 44 4526 278 9 24 19 20 21 22 23 24 19 82 81 61 43 25 7 83 79 8059 6041 4223 24 5 6 25 30 29 28 27 26 25 30 77 76 58 40 22 4 7874 75 56 57 38 39 20 22 3 36 31 32 33 34 35 M6 31 73 55 37 19 1 6 5 1 3 2 1 In the above Diagram, the squares and large figures represent sections, and the small figures at their corners are those referred to in the following directions. " Commence at No. 1, (see small figures on diagram), the corner established on the township boundary for sections 1, 2, 35, and PART xII.] Running Section Lines. 369 36; thence run north on a true meridian; at 40 chains setting the half-mile or quarter-section post, and at 80 chains (No. 2) establishing and marking the corner of sections 25, 26, 35, and 36. Thence east, on a random line, to No. 3, setting the temporary quarter-section post at 40 chains, noting the measurement to No. 3, and the measured distance of the random's intersection north or south of the true or- established corner of sections 25, 36, 30, and 31, on the township boundary. Thence correct, -west, "on the true line to No. 4, setting the quarter-section post on this line exactly at the equidistant point, now known, between the section corners indicated by the small figures Nos. 3 and 4. Proceed, in like manner, from No. 4 to No. 5, 5 to 6, 6 to 7, and so on to No. 16, the corner to sections 1, 2, 11, and 12. Thence north, on a random line, to No. 17, setting a temporary quarter-section post at 40 chains, noting the length of the whole line, and the measured distance of the random's intersection east or west of the true corner of sections 1, 2, 35, and 36, established on the township boundary; thence southwardly from the latter, on a true..line, noting the course and distance to No, 18, the established corner to sections 1, 2, 11, and 12, taking care to establish the quarter-section corner on the true line, at the distance of 40 chains from said section corner; so as to throw the excess or deficiency on the northern half-mile, according to law. Proceed in like manner through all the intervening tiers of sections to No. 73, the corner to sections 31, 32, 5, and 6; thence north, oon a true meridian line, to.No. 74, establishing the quarter-section corner at 40 chains, and at 80 chains the corner to sections 29, 30, 31, and 32; thence east, on a random line to No. 75, setting a temporary quarter-section post at 40 chains, noting the measurement to No. 75, and the distance of the random's intersection north or south of the established corner of sections 28, 29, 32, and 33; thence wvest from said corner, on the true line, setting the quarter-section post at the equidistant point, to No. 76, which is identical with 74; thence west, on a random line, to No. 77, setting a temporary quarter-section post at 40 chains, noting the measurement to No. 77, and the distance of the random's intersection with the western boundary, north or south of the established corner of sections 25, 36, 30, and 31; and from No. 77, correct, eastward, on the true line, giving its course, but establishing the quarter-section post, on this line, so as to retain the distance of 40 chains from the corner of sections 29, 30, 31, and 32; thereby throwing the excess or deficiency of measurement on the most western half-mile. Proceed north, in a sinilar manner, from No. 78 to 79, 79 to 80, 80 to 81, and so on to 96, the south-east corner of section 6, where having established the corner for sections, 5, 6, 7, and 8, run thence, successively, on 24 370 U. S. PUBLIC LANDSo [PART XII. random line east to 95, north to 97, and west to 99; and by reverse courses correct on true lines back to said south-east corner of section 6, establishing the quarter-section corners, and noting the courses, distances, &c., as before described. In townships contiguous to standard parallels, the above method will be varied as follows. In every township SOUTH of the principal base line, which closes on a standard parallel, the surveyor will begin at the south-east cbrner of the township, and measure west on the standard, establishing thereon the mile and half-mile corners, and noting their distances from the pre-established corners. He then will proceed to subdivide, as directed under the above head. In the townships NORTH of the principal base line, which close on the standard parallel, the sectional lines must be closed on the standard by true meridians, instead of by course lines, as directed under the above head for townships otherwise situated; and the connexions of the closing corners with the pre-established standard corners are to be ascertained and noted. Such procedure does away with any necessity for running the randoms. But in case he is unable to close the lines on account of the standard not having been run, from some inevitable necessity, as heretofore mentiond, he will plant a temporary stake, or mound, at the end of the sixth mile, thus leaving the lines and their connexions to be finished, and the permanent corners to be planted, at such time as the standard shall be extended." (564) Exceptional methods. Departures from the general system of subdividing public lands have been authorized by law in certain cases, particularly on water-fronts. Thus, an act of Congress, March 3, 1811, authorized the surveyors of Lousiana, "' in surveying and dividing such of the public lands in the said territory, which are or may be authorized to be surveyed and divided, as are adjacent to any river, lake, creek, bayou, or water course, to lay out the same into tracts, as far as practicable, of fifty-eight poles in front, and four hundred and sixty-five poles in depth, of such shape, and bounded by such lines, as the nature of the country will render practicable and most convenient." Another act, of May 24, 1824, authorizes lands similarly situated " to be surveyed in tracts of two acres in width, fronting on any river, bayou, lake, or water course, and running back the depth of forty acres; which tracts of land, so surveyed, shall be offered for sale entire, instead of in half-quarter-sections." The " Instructions" from which we have quoted say, " In those localities where it would best subserve the interests of the people to have fronts on the navigable streams, and to run back into the PART xiI.] Exceptional Methods. 371 uplands for quantity and timber, the principles of the act of May 24th, 1824, may be adopted, and you are authorized to enlarge the quantity, so as to embrace four acres front by forty in depth, forming tracts of one hundred and sixty acres. But in so doing it is designed only to survey the lines between every four lots, (or 640 acres), but to establish the boundary posts, or mounds, in front and in rear, at the distances requisite to secure the quantity of 160 acres to each lot, either rectangularly, when practicable, or at oblique angles, when otherwise. The angle is not important, so that the principle be maintained, as far as practicable, of making the work to square in the rear with the regular sectioning. The numbering of all anomalous lots will commence with No. 37, to avoid the possibility of conflict with the numbering of the regular sections." The act of Sept. 27, 1850, authorized the Department, should it deem expedient, to cause the Oregon surveys to be executed according to the principles of what is called the "Geodetic Method." The complete adoption of this has not been thought to be expedient; but " it was deemed useful to institute on the principal base and meridian lines of the public surveys in Oregon, ordered to be established by the act referred to, a system of triangulations from the recognized legal stations, to all prominent objects within the range of the theodolite; by means of which the relative distances of such objects, in respect to those main lines, and also to each other, might be observed, calculated, and protracted, with the view of contributing to the knowledge of the topography of the country in advance of the progressing linear surveys, and to obtain the elements for estimating areas of valleys intervening between the spurs of the mountains." " Meandering" is a name given to the usual mode of surveying with the coinpass, particularly as applied to navigable streams. The " Instructions" for this are, in part, as follows. " Both banks of navigable rivers are to be meandered by taking the courses and distances of their sinuosities, and the same are to be entered in the'Meander field-book.' At those points where either the township or section lines intersect the banks of a navigable stream, POSTS, or, where necessary, MOUNDS of earth or stone, (as noted in Art. (566,)) are to be established at the time of running these lines. These are called " meander corners;" and in meandering you are to commence at one of those corners on the township line, coursing the banks, and measuring the distance of each course from your commencing corner to the next' meander 372 U. S. PUBLIC LANDSe [PART XII. corner, upon the same or another boundary of the same township; carefully noting your intersection with all intermediate meander corners. By the same method you are to meander the opposite bank of the same river. The crossing distance between the MEANDER CORNERS, on same line, is to be ascertained by triangulation, in order that the river may be protracted with entire accuracy. The particulars to be given in the field-notes. The courses and distances on meandered navigable streams, govern the calculations wherefrom are ascertained the true areas of the tracts of land (sections, quarter sections, &c.) known to the law as fractional, and bounding on such streams." You are also to meander, in manner aforesaid, all lakes and deep ponds of the area of twenty-five acres and upwards; also navigable bayous. The precise relative position of islands, in a township made fractional by the river in which the same are situated, is to be determined trigonometrically. Sighting to a flag or other fixed object on the island, from a special and carefully measured base line, connected with the surveyed lines, on or near the river bank, you are to form connexion between the meander corners on the river to points corresponding thereto, in direct line, on the bank of the island, and there establish the proper meander corners, and calculate the distance across." (565) Marking Lines. "All lines on which are to be established the legal corner boundaries, are to be marked after this method, viz: Those trees which may intercept your line, must have two chops or notches cut on each side of them without any other marks whatever. These are called' sight trees,' or' line trees.' A sufficient number of other trees standing nearest to your line, on either side of it, are to be blazed on two sides, diagonally or quartering towards the line, in order to render the line conspicuous, and readily to be traced, the blazes to be opposite each other, coinciding in direction with the line where the trees stand very near it, and to approach nearer each other, the further the line passes from the blazed trees. Due care must ever be taken to have the lines so well marked as to be readily followed." (566) Marking Corners, "After a true coursing, and most exact measurements, the corner boundary is the consummation of the work, for which all the previous pains and expenditure have been incurred. A boundary corner, in a timbered country, is to be a tree, if one be found at the precise spot; and if not, a post is to be planted thereat; and the position of the corner post is to be PART XII.] Marking Corners. 373 indicated by trees adjacent, (called Bearing trees) the angular bearings and distances of which from the corner are facts to be ascertained and registered in your field book. In a region where stone abounds, the corner boundary will be a small monument of stones along side of a single marked stone, for a township corner-and a single stone for all other corners. In a region where timber is not near, nor stone, the corner will be a mound of earth, of prescribed size, varying to suit the case. Corners are to be fixed, for township boundaries at intervals of every six miles; for section boundaries at intervals of every mile, or 80 chains; and, for quarter section boundaries at intervals of every half mile, or 40 chains. MEANDER CORNER POSTS are to be planted at all those points where the township or section lines intersect the banks of such rivers, lakes, or islands, as are by law directed to be meandered," as explained in Art. (5614) When posts are used, their length and size must be proportioned to the importance of the corner, whether township, section, or quarter-section, the first being at least 24 inches above ground, and 3 inches square. Where a township post is a corner common to four townships, N it is to be set in the earth diagonally, thus: W E, and the cardis nal points of the compass are to be indicated thereon by a cross line, or wedge, (one-eighth of an inch deep at least), cut or sawed out of its top, as in the figure. On each surface of the post is to be marked the number of the particular township, and its range, which itfaces. Thus, if the post be a common boundary to four townships, say one and two, south of the base line, of range one, west of the meridian; also to townships one and two, south of the base line, of range two, west of the meridian, it is to be marked thus:!R. 1 W.) The position of the post which From N. to E. T. 1 S. is here taken as an example, is S. 31 shewn in the following diagram..( 2RW. from N. to W.. 1 S. R. 2 W. 1 W. ( 36 T. S. T. 1 S. 1 W. 36 31 6 from E. to S. i 2 S. 2 W.) 1 6 from W. to S. 2 S. R2. I. W. ( ) T. 2S. T. 2S. 374 U. S, PUBLIC LANDS. [PART X1I. These marks are to be distinctly and neatly chiselled into the wood, at least the eighth of an inch deep; and to be also marked with red chalk. The number of the sections which they respectively face, will also be marked on the township post. Section or mile posts, being corners of sections, when they are common to four sections, are to be set diagonally in the earth, (in the manner provided for township corner posts), and with a similar cross cut in the top, to indicate the cardinal points of the compass; and on each side of the squared surfaces is to be marked the appropriate number of the particular one of the four sections, respectively, which such side faces; also on one side thereof are to be marked the numbers of its township and range; and to make such marks yet more conspicuous, (in manner aforesaid), a streak of red chalk is to be applied. In the case of an isolated township, subdivided into thirty-six sections, there are twenty-five interior sections, the south-west corner boundary of each of which will be common to four sections. On all the extreme sides of an isolated township, the outer tiers of sections have corners common only to two sections then surveyed. The posts, however, must be planted precisely like the former, but presenting two vacant surfaces to receive the appropriate marks when the adjacent survey may be made. A quarter-section or half-mile post is to have no other mark on it than X S., to indicate what it stands for. Township corner posts are to be NOTCHED with six notches on each of the four angles of the squared part set to the cardinal points. All mile posts on township lines must have as many notches on them, on two opposite angles thereof, as they are miles distant from the township corners, respectively. Each of the posts at the corners of sections in the interior of a township must indicate, by a number of notches on each of its four corners directed to the cardinal points, the corresponding number of miles that it stands from the outlines of the township. The four sides of the post will indicate the number of the section they respectivelyface. Should a tree be found at the place of any corner, it will be marked and notched, as aforesaid, and answer for the corner in lieu of a post; the kind of tree and its diameter being given in the field-notes. The position of all corner posts, or corner trees of whatever description, which may be established, is to be perpetuated in the following manner, viz: From such post or tree the courses shall be taken, and the distances measured, to two or more adjacent trees, in opposite directions, as nearly as may be, which are called'Bearing trees,' and are to be blazed near the ground, with a large blaze facing the post, and having one notch in it, neatly and plainly PART XII.] Marking Corners, 375 made with an axe, square across, and a little below the middle ot the blaze. The kind of tree and the diameter of each are facts to be distinctly set forth in the field-book. On each bearing tree the letters B. T., must be distinctly cut into the wood, in the blaze, a little above the notch, or on the bark, with the number of the range, township, and section. At all township corners, and at all section corners, on range or township lines, four bearing trees are to be marked in this manner, one in each of the adjoining sections. At interior section corners four trees, one to stand within each of the four sections to which such corner is common, are to be marked in manner aforesaid, if such be found. From quarter section and meander corners two bearing trees are to be marked, one within each of the adjoining sections. Stones at township corners (a small monument of stones being alongside thereof) must have six notches cut with a pick or chisel on each edge or side towards the cardinal points; and where used as section corners on the range and township lines, or as section corners in the interior of a township, they will also be notched by a pick or chisel, to correspond with the directions given for notching posts similarly situated. Stones, when used as quari'-....;ion corners, will have X cut on them; on the west side on north and south lines, and on the north side on east and west lines. Whenever bearing trees are not found, MOUNDS of earth, or stone, are to be raised around posts on which the corners are to be marked in the manner aforesaid. Wherever a mound of earth is adopted, the same will present a conical shape; but at its base, on the earth's surface, a quadranrgular trench will be dug; a spade deep of earth being thrown up from the four sides of the line, outside the trench, so as to form a continuous elevation along its outer edge. In mounds of earth, common to four townships or to four sections, they will present the angles of the quadrangular trench (diagonally) towards the cardinal points. In mounds common only to two townships or two sections, the sides of the quadrangular trench willface the cardinal points. Prior to piling up the earth to construct a mound, in a cavity formed at the corner boundary point is to be deposited a stone, or a portion of charcoal, or a charred stake is to be driven twelve inches down into such centre point, to be a witness for the future. The surveyor is farther specially enjoined to plant midway between each pit and the trench, seeds of some tree, those of fruit trees adapted to the climate being always to be preferred. DOUBLE CORNERS are to be found nowhere except on the Standard Parallels or Correction lines, whereon are to appear both the cor 376 U. S. PUBLIC LANDS. [PART XII. ners which mark the intersections of the lines which close thereon, and those from which the surveys start in the opposite direction. The corners which are established on the standard parallel, at the time of running it, are to be known as'Standard Corners,' and, in addition to all the ordinary marks, (as herein prescribed), they will be marked with the letters S. C. The' closing corners' will be marked C. C." (567) Field Books, There should be several distinct and separate field-books; viz.: "1. Field-notes of the MERIDIAN and BASE lines, showing the establishment of the township, section or mile, and quarter-section or half-mile, boundary corners thereon; with the crossings of streams, ravines, hills, and mountains; character of soil, timber, minerals, &c. These notes will be arranged, in series, by mile stations, from number one to number 2. Field-notes of the'STANDARD PARALLELS, or correction lines,' showing the establishment of the township, section, and quarter-section corners, besides exhibiting the topography of the country on line, as required on the base and meridian lines. 3. Field-notes of the EXTERIOR lines of TOWNSHIPS, showing the establishment of the corners on line, and the topography, as aforesaid. 4. Field notes of the SUBDIVISIONS of TOWNSHIPS into sections and quarter-sections; at the close whereof will follow the notes of the MEANDERS of navigable streams. These notes will also show, by ocular observation, the estimated rise and fall of the land on the line. A description of the timber, undergrowth, surface, soil, and minerals, upon each section line, is to follow the notes thereof, and not to be mixed up with them." 5. The " Geodetic Field-book," comprising all triangulations, angles of elevation and depression, levelling, &c. The examples on the next two pages, taken from the " Instructions" which we have followed throughout, will shew what is required. The ascents and descents are recorded in the right-hand columns. PA.IT XII.] Field-Notes. 377 FIELD NOTES OF THE EXTERIOR LINES OF AN ISOLATED TOWNSHIP. Field notes of the Survey of township 25 north, of range 2 west, of the WVillamelte mnerzdian, in the Territory of OREGON, by Robert Acres, deputy surveyor, under his contract No. 1, bearing date the 2d day of January, 1851.; Chs. lks. Feet. C TOWNSHIP LINES commenced January 20, 1851. a Southern boundary variation 180 41. E. East. On a random line on the south boundaries of sections 31, 32, 33, 34, 35, and 36. Set temporary mile and half-mile posts, and intersected the eastern boundary 2 chains 20 links north of the true corner 5 miles 74 chains 53 links. Therefore the correction will be 5 chains 47 links W. 37.1 links S. per mile. C Tm.UE SOUTHERN BOUNDARY variation 18~ 41' E. West On the southern boundary of sec. 36, Jan. 24, 1851. 40.00 Set qr. sec. post from which a 10 a beech 24 in. dia. bears N. 11 E. 38 Iks. dist. a a do 9 do do S. 9 E. 17 do 62.50 a brook 8 1. wide, course NW............................. d 10 o 80.00 Set post cor. of sees. 35 & 36, 1 & 2, fiom which............ a aIS a beech 9 in. dia. bears S. 46 E. 8 1. dist. a a do 8 do do S. 62 W. 7 do aW. oak 10 do do N. 19 W. 14 do H a B. oak 14 do do N. 29 E. 16 do Land level, part wet and swampy; timber beech, oak, ash, hickory, &c. M West. On the S. boundary of sec. 3540.00 Set qr. sec. post, with trench, from which a 10 a beech 6 in. dia. bears N. 80 E. 8 1. dist. planted SW. a yellow locust seed. m 65.00 To beginning of hill............................... a 5 80.00 Set post, with trench, cor. of sees. 34 & 35, 2 & 3, from whicL a 20 a beech 10 in. dia. bears S. 51 E. 13 1. dist. E do 10 do do N. 56 W. 9 do planted SW. a white oak acorn, NE. a beech nut. 3' Land level, rich, and good for farming; timber same. P West. On the S. bolundary of sec. 3440.00 Set qr. sec. post, with trench, from which a 5 a B. oak i0 in. dia. bears N. 2 E. 635 1. dist. Planted SW. a beech nut. S 80.00 To corner of sections 33, 34, 3 and 4, drove charred stake. a 10 raised mound with trench as per instructions, and Planted NE. a W. oak ac'n; NW. a yel. locust seed. oX1 ~ ~ SE. a butternut; SW. a beech nut.,3 lLand level, rich and good for farming, some scattering oal and walnut. El &c., &c., &c. 378 U. S. PUBLIC LANDS. PART XII.] FIELD NOTES OF THE SUBDIVISIONAL OR SECTIONAL LINES, AND MEANDERS. Towunship 25 N., Range 2 W., Willamette Mer. Chs. Iks. Feet. SUBDIVISIONS. Commenced February 1, 1851. North. Between sees. 35 and 369.19 A beech 30 in. dia.....d.............1................... d 10 29.97 A beech 30 ill. dia -...................... —.. d 5 40.00 Set qr. sec. post, from which d 5 - a beech 15 in. dia. bears S. 48 E. 12 1. dist.' — a do 8 do do N. 23 W. 45 do = 51.90 A beech 18 in. diaa.................. d 5 _ 76.73 A sugar 30 in. dia...................................... d 8 80.00 Set a post cor. of sees. 25, 26, 35, 36, from which d 2 a beech 24 in. dia. bears N. 62 W. 17 1. dist. a poplar 36 do do S. 66 E. 34 do. a do 20 do do S. 70 W. 50 do. a beech 28 do do N. 60 E. 45 (do. Land level, second rate; timber beech, poplar, sugar, and nnd'gr. spice, &c. East. On random line between sees. 25 and 369.00 A brook 30 1. wide, course N................................. d 10 15.00 To foot of hill..-..-......-............................. d 10 _ 40.00 Set temporary qr. sec. post —...... —........... ------- a 60 s 55.00 To opposite foot of hill.-.......... d 40 P 72.00 A brook i5 1. wide, course N.. —............ d 20 80.00 Intersect E. boundary at post-........ a............. a 10 Land level, second rate; timber, beech, on(k, nsh, &c. &c., &c., &c. MEANDERS OF CHICKEELES RIVER. Beginning at a meander post in the northern township boundary, and thonce on the left bank down stream. Comrnnenced February 11, 1851. Dist. Courses. s ls. REMARKIS. Ohs. Ike-. S. 76 W. 18.46 In section 4 bearing to corner sec. 4 on right bank N. 70~ W. S. 61 W. 10.00 Bearing to cor. sec. 4 and 5, right bank N. 52~ W. S. 61 W. 8.18 To post in line between sections 4 and 5, breadth of river by triangulation 9 chains 51 links. S. 54 W. 10.69 In section 5. S. 40 W. 5.59 S. 50 W. 8.46 S. 37 W. 16.50 To upper corner of John Smith's claim, course E. S. 44 W. 21.96 S. 36 W. 27.53 To post in line between sections 5 and 8, breadth of river by triangnllation 8 chains 78 links. &c., &c. &c. APPENDIX. APPENDIX A. SYNOPSIS OF PLANE TRIGONOMETRY.* (1) Definition. Plane Trigonometry is that branch of Mathematical Science which treats of the relations between the sides and angles of plane triangles. It teaches how to find any three of these six parts, when the other three are given and one of them, at least, is a side. (2) Angles and Arcs. The angles of a triangle are measured by the arcs described, with any radius, from the angular points as centres, and intercepted between the legs of the angles. These arcs are measured by comparing them with an entire circumference, described with the same radius. Every circumference is regarded as being divided into 360 equal parts, called degrees. Each degree is divided into 60 equal parts, called minutes, and each minute into 60 seconds. These divisions are indicated by the marks ~' ". Thus 28 degrees, 17 minutes, and 49 seconds, are written 28~ 17' 49". Fractions of a second are best expressed decimally. An arc, including a quarter of a circumference and measuring a right angle, is therefore 90~. A semicircumference comprises 180~. It is often represented by r, which equals 3.14159, &c., or 38 approximately, the radius being unity. The length of 1~ in parts of radius = 0.01745329; that of 1'= 0.00029089; and that of 1"= 0.00000485. The length of the radius of a circle in degrees, or 360ths of the circumference - 570.29578 = 57~ 17' 24".8 = 3437'.747 = 206264".8.f An arc may be regarded as generated by a point, M, moving from an origin, A, around a circle, in the direction of the arrow. The point may thus describe arcs of any lengths, such as AM; AB = 90~ = r; ABC = 1800~ =; M ABCD = 270~0 - 3 r; ABCDA= 360~ = 2 r. The point may still continue its motion, and generate C A arcs greater than a circumference, or than two circumferences, or than three; or even infinite in length. While the point, M, describes these arcs, the radius, OM, indefinitely produced, generates corresponding angles. * For merely solving triangles, only Articles (1), (2), (3), (5), (6), (10), (11), and (12), are needed. t The number of seconds in any arc which is given in parts of radius, radius being unity, equals the length of the arc so given divided by the length of the arc of one second; or multiplied by the number of seconds in radius. 380 TRIGONOMETRY. [APP. A. If the point, M, should move from the origin, A, in the contrary direction to its former movement, the arcs generated by it are regarded as negative, or minus; and so too, of necessity, the angles measured by the arcs. Arcs and angles may therefore vary in length from 0 to +- m in one direction, and from 0 to - oo in the contrary direction. The Complement of an arc is the arc which would remain after subtracting the arc from a quarter of the circumference, or from 90~. If the arc be more than 90~, its complement is necessarily negative. The Supplement of an arc is what would remain after subtracting it from half the circumference, or from 180~. If the arc be more than 180~, its supplement is necessarily negative. (3) Trigonometrical Lines. The relations of the sides of a triangle to its angles are what is required; but it is more convenient to replace the angles by arcs; and, once more, to replace the arcs by certain straight lines depending upon them, and increasing and decreasing with them, or conversely, in such a way that the length of the lines can be found from that of the arcs, and vice versa. It is with these lines that the sides of a triangle are compared.* These lines are called Trigonometrical Lines; or Circular Functions, because their length is a function of that of the circular arcs. The principal Trigonometrical lines are Sines, Tangents, and Secants. Chords and versed sines are also used. The SINE of an arc, AM, is the perpendicular, MP, let fall, from one extremity of the arc, upon Fig. 898. the diameter which passes through the other ex- B _ tremity. The TANGENT of an arc, AM, is the distance, AT, intercepted, on the tangent drawn at one q______ - extremity of the arc, between that extremity and the prolongation of the radius which passes through the other extremity. The SECANT of an are, AM, is the part, OT, of the prolonged radius, comprised between the centre and the tangent. The sine, tangent, and secant of the complement of an arc are called the CoSINE, CO-TANGENT, and CO-SECANT of that arc. Thus, MQ is the cosine of AM, BS its cotangent, and OS its cosecant. The cosine MQ is equal to OP, the part of the radius comprised between the centre and the foot of the sine. The chord of an arc is equal to twice the sine of half that arc. The versed-sine of an arc, AM, is the distance, AP, comprised between the origin of the arc and the foot of the sine. It is consequently equal to the difference between the radius and the sine. The Trigonometrical lines are usually written in an abbreviated form. Calling the are AM = a, we write, MP = sin. a. AT = tan. a. OT = sec. a. MQ = cos. a. BS = cot.'a. OS = cosec. a. The period after sin., tan., &c., indicating abbreviation, is frequently omitted. The arcs whose sines, tangents, &c., are equal to a line =a, are written, sin.-1 a, or arc (sin. = a); tan.-l a, or arc (tan.= a); &c. * For the great value of this indirect mode of comparing the sides and angles of triangles, see Comte's " Philosophy of Mathematics," (Harpers', 1851,) page 225. APP. A.] TRIGONOMETRY, 381 (4) The lines as ratios. The ratios Fig. 899. between the trigonometrical lines and the radius B are the same for the same angles, or number of degrees in an arc, whatever the length of the ra- \ \ dius or arc. Consequently, radius being unity, \ these lines may be expressed as simple ratios. Thus, in the right-angled triangle A BC, we / would have A C. BC opposite side c AC adjacent side sin. A= -- cos. A= AB hypothenuse' AB hypothenuse'. BC opposite side A AC adjacent side t. A- adjacent side' BC opposite side' AB hypothenuse AB hypothenuse sec. A - cosec. A = - - AC adjacent side' Bc opposite side When the radius of the arcs which measure the angles is unity, these ratios may be used for the lines. If the radius be any other length, the results which have been obtained by the above supposition, must be modified by dividing each of the trigonometrical lines in the result by radius, and thus rendering the equations of the results "homogeneous." The same effect'would be produced by multiplying each term in the expression by such a power of radius as would make it contain a number of linear factors equal to the greatest number in any term. The radius is usually represented by r, or R. (5) Their variations in length. As the point M moves around the circle, and the arc thus increases, the sines, tangents, and secants, starting from zero, also increase; till, when the point M has arrived at B, and the arc has Fig. 400. become 90~, the sine has become equal to S' B S radius, or unity, and the tangent and secant have become infinite. The comple- r/ { / mentary lines have decreased; the cosine being equal to radius or unity at starting and becoming zero, and the cotangent and cosecant passing from infin- C /- A ity to zero. When the point M has passed the first quadrant at B and is / / proceeding towards C, the sines, tan- N'Q gents, and secants begin to decrease, till, when the point has reached C, they have D the same values as at A. They then begin to increase again, and so on. The Table on page 382 indicates these variations. The sines and tangents of very small arcs may be regarded as sensibly proportional to the arcs themselves; so that for sin. a", we may write a. sin. 1"; and similarly, though less accurately, for sin. a', we may write a. sin. 1'. The sines and tangents of very small arcs may similarly be regarded as sensibly of the same length as the arcs themselves.* * Consequently, the note on page 379 may read thus: The number of seconds in any very small arc given in parts of radius, radius being unity, is equal to the length of the arc so given divided by sin. 1". 382 TRIGONOMETRY. [App. A a being the length of any arc expressed in parts of radius, the lengths of its sine and cosine may be obtained by the following series: as3 a5 a sin. a =a- - + _ +, etc. 2.3 2.3.4.5 2.3....7 a2 a4 a6 cos. a - - - -- - - etc. 2 2.3.4 2....6 Let it be required to find cos. 30~, by the above series. 30~ — Tr -- X 3.1416 =.5236. 180 Substituting this number for a, the series becomes, taking only three terms of it, 1 (.5286)2 (.5286)4 1 (.523) + (54, etc. = 1 - 0.13708 + 0.003130 -=.866052; which is the correct value of cos. 30~ for the first four places of decimals. The lengths of the other lines can be obtained from the mutual relations given in Art. (7.) Some particular values are given below. sin. 30~ =-. sin. 450 = 12. sin. 600 = ~v/3. tan. 30~ = 41/3. tan. 45~ = 1 tan. 60~ = /3. sec. 30~ = 2/3. sec. 45~ = V2. sec. 60= 2. (6) Their changes of sign. Lines measured in contrary directions from a common origin, usually receive contrary algebraic signs. If then all the lines in the first quadrant are called positive, their signs will change in some of the other quadrants. Thus the sines in the first quadrant being all measured upward, when they are measured downward, as they are in the third and fourth quadrants, they will be negative. The cosines in the first quadrant are measured fiom left to right, and when they are measured from right to left, as in the second and third quadrants, they will be negative. The tangents and secants follow similar rules. The variations in length and the changes of sign are all indicated in the following table, radius being unity. The terms "increasing" and "decreasing" apply to the lengths of the lines without any reference to their signs. Lengths and Signs of the Trigonometrical Linesfor Arcs from 0~ to 360~. Arcs. 0o Between c0 and 90~. 900 Between 90~ and 1800. 1800 Sine... +, and increasing, +1 -, and decreasing, 0 Tangent.. 0 +, and increasing, 4z -, and decreasing, 0 Secant.. + 1 +, and increasing, ~r o -, and decreasing, -1 Cosine.. +1 -, and decreasing, 0 -, and increasing, -1 Cotangent.:t o -, and decreasing, 0 -, and increasing,:Foo Cosecant.. ~ oo -, and decreasing, -+1 +, and increasing, ~t o Arcs. 1800 Between 1800 and 2700. 2700 Between 2700 and 8600. 8600 Sine... 0 -, and increasing, -1 -, and decreasing, 0 Tangent. 0 -, and increasing, 4:- -, and decreasing, 0 Secant.. -1 -, and increasing, F +, and decreasing, +1 Cosine... -1 -, and decreasing, 0 +, and increasing, +-1 Cotangent.=F +, and decreasing, 0 and increasing, F o0 Cosecant.. ot -, and decreasing, -1 -, and increasing, F0 APP. A.] TRIGONOMETRY. 385s From this table, and Fig. 400, we see that an arc and its sulpplement have the same sine; and that their tangents, secants, cosines, and cotangents are of equal length but of contrary signs; while the cosecants are the same in both length and sign. We also deduce from the figure the following consequences: sin. (a —+ 180) =-sin. a~. cos. (a~+ 180) — cos. a~ tan. (a~- 180~) = tan. a~. cot. (0~+ 180~) = cot. a0. sec. (a~+ 180~) =-sec. a~. cosec. (a~+ 180~) =-cosec. a~. sin. (-a~) =-sin. a~. cos. (-a~)= cos. a~. tanl. (-a~) =-tan. a~. cot. (-a~) =-cot. a0. sec. (-a) = sec. a0. cosec. (-a0) =-cosec. a~. An infinite number of arcs have the same trigonometrical lines; for, an arc a, the same are plus a circumference, the same are plus two circumferences, and so on, would have the same sine, &c. "To bring back to the first quadrant" the trigonometrical lines of any large arc, proceed thus: Let 1029~ be an arc the sine of which is desired. Take from it as many times 860~ as possible. The remainder will be 309~. Then we shall have sin.3090=sin.(1800- 3090)=sin.-129=- sin.1290=-sin.(180 -1290)=-sin.510. (7) Their munta al relatioaas. Radius being unity, sin. a cos. a~ tan. a = — cot. a = COS. a sin. a~ 1 1 sec. a =. 0 cosec. a = cos. a sin. a~ tan. a X cot. a0 = 1. (sin. a0)2 + (cos. a0)2 = 1. 1 + (tan. a0) = (sec. a0)2. 1 + (cot. a0)2 = (cosec. a0)2. Hence, any one of the trigonometrical lines being given, the rest can be found from some of these equations. (~) Two ares. Let a and b represent any two arcs, a being the greater Then the following formulas apply: sin. (a - b) = sin. a. cos. b + cos. a. sin. b. sin. (a -- b) = sin. a. cos. b - cos. a. sin. b. cos. (a + b) = cos. a. cos. b - sin. a. sin. b. cos. (a - b) = cos. a. cos. b + sin. a. sin. b. tan. a + tan. b tan. (a b) = --— tan..tan. - tan. atan. b tan. a - tan. 5 tan. (a- b) = 1 an.- n. 1 +- tan. a. tan. b cot. a. cot. b- 1 cot. (a + b) cot. cot. 6 cot. b + cot. a cot. a. cot. b + 1 cot. (a - b)= cot. b - cot. a * The square, &c., of the sine, &c., of an arc, is often expressed by placing the exponent between the abbreviation of the name of the trigonometrical line and the number of the degrees in the arc; thus, sin.a ao, tan.? aO, &c. But the: notation given above, places the index as used by Gauss, Delambre, Arbogast, &c., though the first two omit the parentheses. 384 TRIGONOMETRY. [APP. A. sin. a. sin. b =. cos. (a - b) - cos. (a + b). cos. a. cos. b = A. cos. (a - b) +- i cos. (a - b). sin. a. cos. b =. sin. (a + b) +- sin. (a - b). cos. a. sin. b = -. sin. (a + b) - sin. (a - b). sin. a + sin. b = 2 sin. - (a + b) cos. ~ (a - b). cos. a + cos. b = 2 cos. i (a + b) cos. (a - b). sin. a - sin. b = 2 sin. ~ (a -b) cos. (a +- b). cos. -cos. a =2 sin. - (a -b) sin. i (a + b). sin. (a + b) tan. a + tan. 6. cos. a. cos. b tan. a - tan. b sin. (a - b) cos. a. cos. b sin. (a +- b) cot. b +- cot. a in. (a + b sin. a. sin. b sin. (a - b) cot. b - cot. a -- --- sin. a. sin. b (9) Double and half arcs. Letting a represent any arc, as before, we have the following formulas: sin. 2 a = 2 sin. a.cos. a. cos. 2 a = (cos. a)2 - (sin. a)2 = 2 (cos. a)2 - 1 = 1 - 2 (sin. a)2. 2 tan. a 2 cot. a 2 tan. 2 a -- 1 - (tan. a) (cot. a)2 - 1 cot, a - tan. a (cot. a)2 - 1 cot. 2 a 2 a (cot. a - tan. a). 2 cot. a sin. a= /[. (1- cos a)]. cos. a= /[(1+cos. a)]. sin. a 1 - cos. a /1 - cos. a\ tan. =. a=t-. 1 + cos. a sin. a 1 + cos. a 1 + cos. a sin. a _ / + cos. a\ cot. a -- sin. a 1- cos. a -cos. a (10) Trigonometrical Tables. In the usual tables of the natural Trigonometrical lines, the degrees from 0~ to 45~ are found at the top of the table, and those from 45~ to 90~ at the bottom; the latter being complements of the former. Consequently, the columns which have Sine and Tangent at top have Cosine and Cotangent at bottom, since the cosine or cotangent of any arc is the same thing as the sine or tangent of its complement. The minutes to be added to the degrees are found in the left-hand column, when the number of degrees at the top of the page are used, and in the right-hand column for the degrees when at the bottom of the page. The lines for arcs intermediate between those in the tables are found by proportion. The lines are calculated for a radius equal unity. Hence, the values of the sines and cosines are decimal fractions, though the point is usually omitted. So too are the tangents from 0~ to 45~, and the cotangents from 90~ to 450. Beyond those points they are integers and decimals. The calculations, like all others involving large numbers, are shortened by the use of logarithms, which substitute addition and subtraction for multiplication and division; but the young student should avoid the frequent error of regarding loga rithms as a necessary part of trigonometry. APP. A.] TRIGONOMETRYo 885 SOLUTION OF TRIANGLES. Fig. 401. (11) Bight-angled Triangles. Let B ABC be any right-angled triangle. Denote the sides opposite the angles by the corresponding small letters. Then any one side and one acute angle, or any two sides being given, the other parts can be obtained by one of the following equations: AC Given. Required. Formulas. a a a, 6 c, A, B c =/(a2 + ); tan. A = b cot. B = a a a, c b, A, B b =/(e2 -a2); sin. A -; cos. B -. c c a a, A b, c, B b==a.cot. A; c= i; B =90~-A. sin. A b, A a, c, B a =b.tan.A; c=.; B =90 -A. c, A a, b, B a =c.sin. A; b=c cos. A; B =90 - A. (12) Oblique-angled Trian- Fig. 402. gles. Let ABC be any oblique-angled B triangle, the angles and sides being noted as in the figure. Then any three of its six parts being given, and one of them being a side, the other parts can be obtained by one /\ of the following methods, which are found- ed on these three theorems. THEOREM I.-In every plane triangle, the sines of the angles are to each other as the opposite sides. THEOREM II.-In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference. THEOREM III.-In every plane triangle, the cosine of any angle is equal to a fraction whose numerator is the sum of the squares of the sides adjacent to the angle, minus the square of the side opposite to the angle, and whose denominator is twice the product of the sides adjacent to the angle. All the cases for solution which can occur, may be reduced to four. CASE 1.-Given a side and two angles. The third angle is obtained by subtracting the sum of the two given angles from 180~. Then either unknown side can be obtained by Theorem I. sin. B sin. C Calling the given side a, we have b = a. si. and c = a s-. sin. A' s. A 25 386 TRIGONOMETRY. [APP. A. CASE 2.-Given two sides and an angle opposite one of them. The angle opposite the other given side is found by Theorem I. The third angle is obtained by subtracting the sum of the other two from 180~. The remaining side is then obtained by Theorem I. Calling the given sides a and b, and the given angle A, we have sin. B = sin. A. -. a Since an angle and its supplement have the same sine, the result is ambiguous; for the angle B may have either of the two supplementary values indicated by the sine, if b > a, and A is an acute angle. C= 180~ -(A + B). c= sin. Csin. A' CASE 3.-Given two sides and their included angle. Applying Theorem II. (obtaining the sum of the angles opposite the given sides by subtracting the given included angle from 180~), we obtain the difference of the unknown angles. Adding this to their sum we obtain the greater angle, and subtracting it from their sum we get the less. Then Theorem I. will give the remaining side. Calling the given sides a and b, and the included angle C, we have A+-B=180~ -C. Then a -b tan. I (A - B) = tan. g (A + B). - a -- b' sin. C ~ (A+B)+ - (A —B)=A. (A+B) - (A-B)=B. c = a sin. A sin. A In the first equation cot. ~ C may be used in the place of tan. - (A + B). CASE 4.-Given the three sides. Let s represent half the sum of the three sides =j (a + b + c). Then any angle, as A, may be obtained from either of the following formulas, founded on Theorem III.: in. AT=/ [(s — b) (s- c) sin. A sin. = (Sa) (s-be b) (s - c) b2+c2 -a2 cos. A -= - The first formula should be used when A < 90~, and the second when A > 90~. The third should not be used when A is nearly 180~; nor the fourth when-A is nearly 900; nor the fifth when A is very small. The third is the most convenient when all the angles are required. APPENDIX B. DEMONSTRATIONS OF PROBLEMS, ETC. MANY of the problems, &c., contained in the preceding pages, require Demonstrations. These will be given here, and will be designated by the same numbers as those of the Articles to which they refer. As many of these Demonstrations involve the beautiful Theory of Transversals, &c., which has not yet found its way into our Geometries, a condensed summary of its principal Theorems will first be given. TRANSVERSALS. THEOREM I.-If a straight line be drawn so as to cut any two sides of a triangle, and the third side prolonged, thus dividing them into six parts (the prolonged side and its prolongation being two of the parts), then will the product of any three of those parts, whose extremities are not contiguous, equal the product of the other three parts. That is, in Fig. 403, ABC being the triangle, and Fig. 408. DF the Transversal, BEX AD X CF=EA XDC XBF. A To prove this, from B draw BG, parallel to CA. From the similar triangles BEG and AED, we have BG: BE:: AD: AE. From the similar triangles BFG and CFD, we have CD: F:: BG: BF. Multiplying these proportions together, we have F BGXCD:BEXCF::ADXBG: AEXBF. Multi- B c plying extremes and means, and suppressing the common factor BG, we have BEXADXCF=EAXDCXBF. These six parts are sometimes said to be in involution. If the Transversal passes entirely out- Fig. 404. side of the triangle, and cuts the prolonga- A tions of all three sides, as in Fig. 404, the theorem still holds good. The same demonstration applies without any change.* THEOREM II.-Conversely: If three points \C F be taken on two sides of a triangle, and on the third side prolonged, or on the prolon- gations of the three sides, dividing them into six parts, such that the product of three non-consecutive parts equals the prodnct of the other three parts; then will these three points lie in the same straight line. This Theorem is proved by a Reductio ad absurdum. * This Theorem may be extended to polygons. 388 TRANSVERSALS. [APP. B. THEOREM III.-Iffrom the summits of a trzangle, lines Fig. 405. be drawn, to a point situated either within or without the triangle, and prolonged to meet the sides of the triangle, or their prolongations, thus dividing them into six parts; then will the product of any three non-consecutive parts be E/ P equal to the product of the other three parts. B FO That is, in Fig. 405, or Fig. 406, AE X BF X CD=EB X FC X DA Fig.406. A For, the triangle ABF being cut by the transversal EC, gives the relation (Theorem I.), AE X BC X FP=EB X FC X PA. The triangle ACF, being cut by the B/ " C transversal DB, gives /. A DC X FB X PA =AD X CB X FP.,/ -- Multiplying these equations together, E and suppressing the common factors PA, CB, and FP, we have AE X BF X CD =EB X FC X DA. THEOREM IV.-Conversely: If three points are situated on the three sides of a triangle, or on their prolongations (either one, or three, of these points being on the sides), so that they divide these lines in such a way that the product of any three non-consecutive parts equals the product of the other three parts, then will lines drawn from these points to the opposite angles meet in the same point. This Theorem can be demonstrated by a Reductio ad absurdum. COROLLARIES OF THE PRECEDING THEOREMS. COR. 1.-The MEDIANS of a triangle (i. e., the lines drawn from its summits to the middles of the opposite sides) meet in the same point. For, supposing, in Fig. 405, the points D, E, and F to be the middles of the sides, the products of the non-consecutive parts will be equal, i. e., AE X BF X CD = DA XEB XFC; since AE = EB, BF =FC, CD = DA. Then Theorem IV. applies. CoR. 2.-The BISSECTRICES of a triangle (i. e., the lines bisecting its angles) meet in the same point. For,, in Fig, 405, supposing the lines AF, BD, CE to be Bissectrices, we have (Legendre IV. 17): BF: FC:: AB: AC, BF X AC=F X AB, CD:DA:: BC: BA, whence CD X BA=DA X.Bi, AE: EB:: CA: CB, ( AE X B'=EB X CA. Multiplying these equations together, and omitting the common factors, we have BF X CD X AE = FC X DA X EB. Then Theorem IV. applies. APP. B.] TRANSVERSALS. 389 COR. 3.-The ALTITUDES of a triangle (i. e., the lines drawn from its summits perpendicular to the opposite sides) meet in the same point. For, in Fig. 405, supposing the lines AF, BD, and CE, to be Altitudes, we have three pairs of similar triangles, BCD and FOA, CAE and DAB, ABF and EBC, by comparing which we obtain relations from which it is easy to deduce BF X CDXAE =EB XFC X DA; and then Theorem IV. again applies. CoR. 4.-If, in Fig. 405, or Fig. 406, the point F be taken in the middle of BC, then will the line ED be parallel to BC. For, since BF = FC, the equation of Theorem III. reduces to AE X CD=EB XDA; whence AE: EB:: AD: DC; consequently ED is parallel to BC. CoR. 5.-Conversely: If ED be parallel to BC, then is BF = FC. For, since AE: EB:: AD: DC, we have AE X DC = EB X AD; whence, in the equation of Theorem III., we must have BF = FC. Con. 6.-From the preceding Corollary, we derive the following: If two sides of a triangle are divided proportionally, Fig. 407. starting from the same summit, as A, and lines are drawn A from the extremities of the third side to the points of division, the intersections of the corresponding lines will all lie in the same straight line joining the summit A, and the / middle of the base. /' \ CoR. 7.-A particular case of the preceding corollary /~ ]. is this: B F C In any trapezoid, the straight line which joins the intersection of the diagonals and the point of meeting of the non-parallel sides produced, passes through the middle of the two parallel bases. Con. 8.-If the three lines drawn through the corresponding summits of two triangles cut each other in the same point, then the three points in which the corresponding sides, produced if necessary, will meet, are situated in the same straight line. This corollary may be otherwise enunciated, thus: If two triangles have their summits situated, two and two, on three lines which meet in the same point, then, &c. This is proved by obtaining by Theorem I. three equations, which, being multiplied together, and the six common factors cancelled, give an equation to which Theorem II. applies. Triangles thus situated are called homologic; the common point of meeting of the lines passing through their summits is called the centre of homology; and the line, on which the sides meet, the axis of homology. 390 HARMONIC DIVISION. [APP. B. HARMONIC DIVISION. DEFINITIONS.-A straight line, AB, is said to Fig. 408. be harmonically divided at the points C and D, 1 I ---- when these points determine two additive seg- A C D ments, AC, BC, and two subtractive segments, AD, BD, proportional to one another; so that AC: BC:: AD: BD. It will be seen that AC must be more than BC, since AD is more than BD.* This relation may be otherwise expressed, thus: the product of the whole line by the middle part equals the product of the extreme parts. Reciprocally, the line DC is harmonically divided at the points B and A; since the preceding proportion may be written DB: CB: DA: CA. The four points, A, B, C, D, are called harmonics. The points C and D are called harmonic conjugates. So are the points A and B. When a straight line, as AB, is divided harmonically, its half is a mean proportional between the distance from the middle of the line to the two points, C and D, which divide it harmonically. If, from any point, 0, lines be drawn so as to Fig. 409. divide a line harmonically, these lines are called 0 an harmonic pencil. The four lines which compose it, OA, OC,'OB, OD, in the figure, are called its radii, and the pairs which pass through the conjugate points are called conjugate radii. / / A C B D THEOREM V.-In any harmonic pencil, a line drawn parallel to any one of the radii, is divided by the three other radii into two equal parts. Let EF be the line, drawn parallel to Fig. 410. OA. Through B draw GH, also parallel ~ to OA. We have, GB: OA:: BD: AD; and BH: OA:: BC: AC. But, by hypothesis, AC: B:: AD: BD. E- / Hence, the first two proportions reduce to _/ " - \ GB = BH; and consequently, EK = KF. / The Reciprocal is also true; i. e., If four lines radiating from a point are such that a line drawn parallel to one of' them is divided into two equal parts by the other three, the four lines form an harmonic pencil. * Three numbers, m, n, p, arranged in decreasing order of size, form an harmonic proportion, when the difference of the first and the second is to the difference of the second and the third, as the first is to the third. Such are the numbers 6, 4, and 8; or 6, 8, and 2; or 15, 12, and 10; &c. So, in Fig. 408, are the lines AD, AB, and AC, which thus give BD: CB:: AD: AC; or AC: CB:: AD: BD. The series of fractions, 1, 1,,-1, -, &c., is called an harmonicprogression, because any consecutive three of its terms form an harmonic proportion. APP. B.] THE COMPLETE QUADRILATERAL. 391 THEOREM VI.-If any transversal to an harmonic pencil be drawn, it will be divided harmonically. Let LM be the transversal. Through K, where LM intersects OB, draw EF parallel to OA. It is bisected at K by the preceding theorem; and the similar triangles, FMK and LMO, EKN and LNO, give the proportions LM: KM:: OL: FK, and LN: NK::: OL: EK; whence, since FK = EK, we have LN: NK:: LM: KM. COROLLARY. — he two sides of any angle, together with the bissectrices of the angle and of its supplement, form an harmonic pencil. THEOREM VII.-If, from the summits of any Fig. 411. triangle, ABC, through any point, P, there be C drawn the transversals AD, BE, OF, and the trans- E versal ED be drawn to meet AB prolonged, in F', / i din the points F and F' will divide the base AB harmonically. ", B. A F B P' This may be otherwise expressed, thus: The line, CP, which joins the intersection of the diagonals of any quadrilateral, ABDE, with the point of meeting, 0, of two opposite sides prolonged, cuts the side AB in a point F, which is the harmonic conjugate of the point of meeting, F', of the other two sides, ED and AB, prolonged. For, by Theorem I., AF' X BD X CE = F'B X DO X EA; and by Theorem III., AF X BD X CE = FB X DC X EA; whence AF: FB:: AF': F'B. TIE COMPLETE QUADRILATERAL. A Complete Quadrilateral is formed by Fig. 412. drawing any four straight lines, so that each of them shall cut each of the other three, so as to give six different points of intersection. L It is so called because in the figure thus formed are found three quadrilaterals; viz., N \ in Fig. 412, ABCD, a common convex quadri- i lateral; EAFC, a uni-concave quadrilateral; P \C and EBAFD, a bi-concave quadrilateral, com-' \ i posed of two opposite triangles. \ \ / The complete quadrilateral, AEBCDF, has / three diagonals; viz., two interior, AC, BD; and one exterior, EF. E THEOREM VIII.-In every COMPLETE QUADRILATERAL the middle points of its three diagonals lie in the same straight line. AEBCDF is the quadrilateral, and LMN the middle points of its three diagonals. From A and D draw parallels to BC, and from B and C draw parallels to 392 THE COMPLETE QUADRILATERAL. [APP. B. AD. The triangle EDO being cut by the transversal BF, we have (Theorem I.), DF X CB X EA = CF X EB X DA. From the equality of parallels between parallels, we have CB =E'B', EA = CA', EB = DB', DA = EA'. Hence, the above equation becomes DF X E'B' X CA' = OF X DB' X E'A'; therefore, by Theorem II., the points, F', B', A', lie in the same straight line. Now, since the diagonals of the parallelogram ECA'A bisect each other at N, and those of the parallelogram EBB'D at N, we have EN: NA':: EM: MB'. Then MN is parallel to FA'; and we have EN: NA':: EL: LF, or EL = LF, so that L is the middle of EF, and the same straight line passes through L, M, and N. THEOREM IX.-In every complete quadrilateral each of the three diagonals is divided harmonically by the two others. OEBADF is the complete quadrilateral. Fig. 413. The diagonal EF is divided harmonically at A G and H by DB and AC produced; since AH, DE, and FB are three transversals drawn from the summits of the triangle AEF through the same point C; and there- B fore, by Theorem VII., DBG and ACH di- / vide EF harmonically. --— E G E I HF So too, in the triangle ABD, CB, CA, OD, are the three transversals passing through 0; and G and K therefore divide the diagonal BD harmonically. So too, in the triangle, ABO, DA, DB, DO are the transversals, and H and K the points which divide the diagonal AC harmonically. THEOREM X.-If from a point, A, any num- Fig. 414. her of lines be drawn, cutting the sides of an A angle POQ, the intersections of the diagonals of the quadrilaterals thus formed will all lie in the same straight line passing through the summit of the angle. By the preceding Theorem, the diagonal BC' of the complete quadrilateral, BAB''CO, O~ 0 C C" Q is divided harmonically at D and E. Hence, OA, OP, OD, and OQ, form an harmonic pencil. So do OA,, O OD', and OQ. Therefore, the lines OD, OD' coincide. So for the other intersections. If the point A moves on OA, the line OD is not displaced. If, on the contrary, OA is displaced, OD turns around the point O. Hence, the point A is said to be a pole with respect to the line OD, which is itself called the polar of the point A. Similarly, D is a pole of OA, which is the polar of D. OD is likewise the polar of any other point on the line OA; and this property is necessarily reciprocal for the two conjugate radii OA, OD, with respect to the lines OP, OQ, which are also conjugate radii. Hence; In every harmonic pencil, each of the radii is a polar with respect to each point of its conjugate; and each point of this latter line is a pole with respect to the former. DE X ON S TR ATION S* PART II.; CHAPTER V. (140), (14i) The equality of the triangles formed in these methods proves their correctness. (143), (144) These methods depend on the principle of the square of the hypothenuse. (145) CAD is an angle inscribed in a semicircle. (146) Let fall a perpendicular from B to AC, meeting it at a point E, not marked in Fig. 91. Then (Legendre, IV. 12), AO2 -4- BC - AB2 AB2 = AC2 + BC2 - 2 AC.. E; whence CE = + A —2 AG BC2 When AC =AB, this becomes CE = 2. The similar triangles, BCE and DOA, giveEC CB::AC: CD; whence D CB X A BC 2 AC. CD== =CB xAC — -- CE 2ACBC (14L7) Mark a point, G, in the middle of DF, and join GA. The triangle AGD will then be isosceles, since it is equal to the isosceles triangle ABC, having two sides and the included angle equal. Then AG = GD = AB = GF. The triangle AGF is then also isosceles. Now the angle FAG = ~ AGD; and GAD- = FGA. Therefore FAG + GAD = FAD = i (AGD + FGA) = ~ (180~) = 90~. (149) See Part VII., Art. (403). (150) The proof follows from the equal triangles formed. (151) The proof is found in the first half of the proof of Art. (146). (153) ACP is an angle inscribed in a semicircle. (154) Draw from C a perpendicular to the given line, meeting it at a point E. AC2 As in the proof of Art. (146), changing the letters suitably, we have AE =2 ABThe similar triangles AEC and ADP give AP AP AC2 AP XAC AC: AE:: AP: AD =A X AE A X 2AB-=AP- X (155) Similar triangles prove this. (156) The equal triangles which are formed give BP = CF. Hence FP is parallel to BC, and consequently perpendicular to the given line DG. (157) The proof of this is found in the " Theory of Transversals," corollary 3. (158) The proof of this is the same as the last. (161) The lines are parallel because of the equal angles formed. * Additional lines to the figures in the text will sometimes be employed. The student should draw them on the figures, as directed. 394 DEMONSTRATIONS [APP. B. (162) The equal triangles give equal angles, and therefore parallels. (163) AB is parallel to PF, since it cuts the sides of the triangle proportionally. (164) The proof is found in corollary 4 of " Transversals." (165) From the similar triangles, CAD and CEP, we have E: CD:: CP: CA. From the similar triangles, CEF and CBD, we have CE: CD:: CF: CB. These two proportions give tle following; CP: CA:: CF: CB. Therefore PF is parallel to AB. (166) Draw PE. The similar triangles PCE and ACD give PE: CE:: AD: CD. The similar triangles CEF and CDB give EF: CE:: DB: CD. These proportions produce PE: EF:: AD: DB. Hence PEF is similar to ADB, and PF is parallel to AB. (173) The equality of the symmetrical triangles which are formed, proves this method. (174) ABP is a transversal to the triangle CDE. Then, by Theorem I. of "' Transversals," CA X EB X DP = AE X BD X CP; whence we have CP: DP:: CA X EB: AE X BD. By "division," CP - DP: DP:: CA X EB- AE X BD: AE X BD. DC X AE X BD Hence, since CP -DP = CD, we obtain DP= E B CA X EB-AE x BD' The other formulas are simplified by the common factors obtained by making AE =AC, or BE = BD. (175) By Theorem VII. "Harmonic Division," in the quadrilateral ABED, the line CF cuts DE in a point, L, which is the harmonic conjugate of the point at which AB and DE, produced, would meet. So too, in the quadrilateral DEHK, this same line, CG, produced, cuts DE in a point, L, which is the harmonic conjugate of the point at which DE and KH, produced, would meet. Consequently, AB, DE, and KH must meet in the same point. Otherwise; this problem may be regarded as the converse of Theorem X. of "Transversals," BCA being the angle, and P the point from which the radiating lines are drawn. (176) EGCFDH is the "Complete Quadrilateral." Its three diagonals are FE, DC, and HG; and their middle points A, B, and P lie in the same straight line, by our Theorem VIII. (182) This instrument depends on the optical principle of the equality of the angles of incidence and reflection. (184) The first method given, Fig. 120, is another application of the Theory of Transversals. The second method in the article is proved by supposing the figure to be constructed, in which case we should have a triangle QZR, whose base, QR, and a parallel to it, BD, would be cut proportionally by the required line PSZ; BD x QP so that QR: BD:: QP: BS=BD = QP QR (189) By "Transversals," Theorem I., we obtain, regarding CD as the transversal of the triangle ABE, CB X AF X ED = AC XFE XDB; and since ED = DB, this becomes CB X AF = AC X FE; whence the proportion CB: AC:: FE: AF. By "division," we have CB-AC: AC:: FE —AF: AF. Observing that AC (FE AF). CB - AC =AB, we obtain AB.(FE -F AF). AF APP. B.] For Part II., Chapter V. 395 (190) Take CH = CB; and from B let fall a perpen- Fig. 124, bis. dicular, BK, to AC. Then, in the triangle CBH, we have A B (Legendre IV. 12),.. CH2K - +B2-BC2 BI2 RK= 20 [1] 2 CH 2 BC' since CH = BC. In the triangle ABH, we have (Leg. IV. 13) -, AB2 = AH2 +- BH12 +- 2 AH. HK. Substituting for HK, its value from [1], we get AB = AH + BH (1 + -AH C But AH=AC - CH=AC —BC.. B AB 2+ B=2 A + BC ) =A + Bi1H2.. [2] In the above expression for AB, BH is unknown. To find it, proceed thus. Take OF = CD. Then DF is parallel to BH; and we have CD: B:: DF: BH; whence CB2 BH2 = DF2. = D. [8] In this equation DFa is unknown; but by proceeding as at the beginning of this CE investigation, we get an equation analogous to [2], giving ED2 =EF2 + DF2. whence DF2 = (DE'- EF2). C. Substituting this value of DFg in [3], we have CB2 BH2 = (DE' - EF2)CD XCE Substituting this value of BH2 in [2], we have AC X BC ACXBC AB2=AH2+(DE2 -EF2). CX C (AC-BC)2+[DE2- (CE-CD)2]X CDXCE. (191) Since BCD is a right angle, AC is a mean proportional between AB and AD. (192) The proof follows from the similar triangles constructed. (193) The similar triangles give DE: AC:: DB: AB; whence, by "division," DE- AC: AC: DB - AB: AB; whence, since DB - AB = AD, we have ACXAD AB — DE-AC' (194) From the similar triangles, we have DE: CA:: EB: AB; whence DE-CA: CA:: EB-AB:AB; whence, since EB-AB=AE, we get ABAC X AE DE — AC' (195) The triangles DEF and BAF, similar because of the parallelogram which ED X AF AC XAF is constructed, give FE: ED:: AF: AB = F- E- FE AC XDC The triangles DEF and BCD give similarly FE: ED:: DOC: OB = FE FE 396 DEMONSTRATIONS [APP. B. (196) The equality of the triangles formed proves this problem. (197) The proof of this problem also depends on the equality of the triangles constructed. The details of the proof require attention. (198) EB is the transversal of the triangle ACD. Consequently, CB X AFxDE = AB XFD X CE; or, since CB =AB+AC, (AB+AC)XAFXDE=AB XFDXCE; AC XAFXDE whence AB = — AF FDXCE AFXDE' Taking E, in the middle of CD, CE = DE, and those lines are cancelled. Taking F in the middle of AD, AF = FD, and those lines are cancelled. (199) The line BE is harmonically divided at the points H and A, from Theorem IX., ECFBGD being a " Complete QuadriIateral." Consequently, AE: EH:: AB: HB. Hence, by "division," AE- EH: AE:: AB -HB AB. We therefore have, since AB - HB = AH, AB = AE x AH AE - EIf (200) For the same reasons as in the last article, CF is harmonically divided at H and D; and we have CH: HF:: CD: DF; whence CH-HF: CH:: CD -DF: CD. CH X CF Hence, since CD - DF = CF, CD = C- -HF CH - HFJ The other two expressions come from writing CF as CH - HF, and HF as CF - H. (201) The equality of the triangles formed proves the equality of the corresponding sides KD and DE, &c. (202) The similar triangles (made so by the measurement of CE) give ACXDE CD: DE:: CA: AB = D CD (203) The similar triangles (made so by the parallel) give CE: EA:: CD: AB CD x EA CD X(AC-CE) CE CE DF X CD (204) The similar triangles DFH and BCD give HF: FD:: DC: BC = FXH'GHI The similar triangles FGH and ABC give FG: GH:: BC: AB =BC -G. DF x CD x GH Substituting for BC, its above value, we have AB = F X FG FH XIFG' When CD = CE, DF = CD, whence the second formula. (205) The equality of the symmetrical triangles which are formed, proves the equality of A'B' to AB. (206) The proof of this is similar to the preceding. (207) Because the two triangles ABC and ADE have a common angle at A, we have ADE: ABC:: AD X AE: AB X AC; whence the expression for ABC. (208) From B let fall a perpendicular to AC, meeting it at a point B'. Call this perpendicular BB' = p. From D let fall a perpendicular to AC, meeting it at a point D'. Call this perpendicular DD' q. [APP. B.] For Part V. 397 The quadrilateral ABCD = AC X i (p + q). 2. BCE The triangle BCE =CE X - p; whence p- =.E The similar triangles EDD' and BEB' give p: q:: BE: DE, whence DE 2.BCE X DE PBE CE X BE The ( + BCE B+CE XDE BE+DE BD Then % (p i- +) - 4OXI BCE X BCE X CE CEXBE CE X BE CEXBE' BD AOXBD Lastly, ABOD =AC X BCE X BCE X CEXBE CEXBE DEMONSTRATIONS FOR PART V. (3~2) Let B = the measured inclined length, b =this length reduced to a horizontal plane, and A =the angle which the measured base makes with the horizon. Then b =B. cos. A; and the excess of B over b, i. e., B - b = B (1 - cos. A). Since 1 - cos. A = 2 (sin. ~ A)2 [Trigonometry, Art. (9)], we have B -b =2 B (sin. i A)2. Substituting for sin. ~.A, its approximate equivalent, I A X sin. 1' [Trigonometry, Art. (5)], we obtain B-b = 2 B (i A X sin. l')2 = (sin. 1')2. A. B, =0.00000004231 A2 B. By logarithms, log. (B - b) - 2.626422 + 2 log. A + log. B. The greater precision of this calculation than that of b = B. cos. A, arises from the slowness with which the cosines of very small angles increase or decrease in length. (3S6) The exterior angle LER = LCR + CLD. Also, LER = LDR + CRD...LOR+CLD =LDR+CRD, and LCR -=LDR+CRD — LD. CD From the triangle CRD we get sin. CRD = sin. CDR X C. CD From the triangle CLD we get sin. CLD = sin. LDC X C. CL' As the angles CRD and CLD are very small, these values of the sines may be called the values of the arcs which measure the angles, and we shall have CD CD LCR = LDR + sin CDR X R —sin. LDC X -. CR CL The last two terms are expressed in parts of radius, and to have them in seconds, they must be divided by sin. 1" [Trigonometry, Art. (5), Note], which gives the formula in the text. Otherwise, the correction being in parts of radius, may be brought into seconds by multiplying it by the length of the radius in seconds; i. e., 1800 X 60 X 60 180X = 206264".80625 [Trigonometry, Art. (2)]. 3.14159, &c. (391) The triangles AOB, BOO, COD, &c., give the following proportions [Trigonometry, Art. (12), Theorem I.]; AO: OB: sin. (2): sin. (1); OB: OC:: sin. (4): sin. (3); OC: OD:: sin. (6): sin. (5); and so on around the polygon. Multiplying together the corresponding terms of all the proportions, the sides will all be cancelled, and there will result 1: 1:: sin. (2) X sin. (4) X sin. (6) X sin. (8) X sin. (10) X sin. (12) X sin. (14): sin. (1) X sin. (3) X sin. (5) X sin. (7) X sin. (9) X sin. (11) X sin. (13). Hence the equality of the last two terms of the proportion. 398 DEMONSTRATIONS [APP. B. DEMONSTRATION FOR PART VI. (399) In the triangle ABS, we have AB. sin. BAS c. sin. U sin. ASB: sin. BAS:: AB:SB = sin- = -. [1] sin. ASB sin. S [ In the triangle CBS, we have BC. sin. BCS a.sin. V sin. BSC: sin. BCS: BC: SB = B sinBCS a. [2] sin. BSC sin. S' - 2] c. sin. U a. sin. F Hence, sin.S in.; whence, c. sin. S'. sin. U-a. sin. S. sin. V=0. [3] In the quadrilateral ABCS, we have BCS=360~-ASB-BSC- ABC- BAS; or V 360~ — S-S'-B-U. Let T = 3600 - S -S'- B, and we have V = T — U. [4] Substituting this value of V, in equation [3], we get [Trig., Art. (8)], c. sin S' sin. U- a. sin. S (sin. T. cos. U- cos. T. sin. U) =0. Dividing by sin. U, we get cos. U c. sin. S'- a. sin. S sin. T. - n. U- cos. T) = 0. Whence we have cos. U c. sin. S' + a. sin. S. cos. T -- = cot. U = — sin. U a. sin. S. sin. T Separating this expression into two parts, and cancelling, we get cc. sin. S' cos. T a. sin. S. sin. T sin. T Separating the second member into factors, we get =cos. T c. sin. S' sin. T a. sin. S. cos. T 1; cot. U = cot. T (+s. in.). a. sin. S. cos. THaving found U, equation [4] gives V; and either [1] or [2] gives SB; and SA and SC are then given by the familiar " Sine proportion" [Trig., Art. (12)]. APP. B.] For Part VII. 399 DEMONSTRATIONS FOR PART VII. CP (403) If APC be a right angle, - = cos. CAB [Trigonometry, Art. (4)]. CA (405) AC = PC. tan. APC; and CB = PC. tan. BPC [Trigonometry, Art. (4)]. Hence AC CB:: tan. APC: tan. BPC; and AC: AC + CB:: tan. APC: tan. APC + tan. BPC. Consequently, since AC + CB = AB, AC = AB. tan. APC tan. APC +- tan. BPC (414) The equal and supplementary angles formed prove the operation. (421) In Fig. 285, CA:EG::AB:GB. By "division," CA-EG:EG:: AB -GB: GB. Hence, observing that AB —GB =AG, we shall have G GB (CA- EG) A EG (423) Art. (12), Theorem III., [Trigonometry, Appendix A,] gives; cos. C -e + 6 -; or c2 = a2 + b2 - 2 ab. cos. C. This becomes [Trig., Art. 2 ab (6)], K being the supplement of C, c2 = a2 + b2 + 2 ab. cos. K. The series [Trig. Art. (5)] for the length of a cosine, gives, taking only its first two terms, since K is very small, cos. K = 1 - K2. Hence, cs=a2+ b2 2 ab-ab (a+ b)-abK2=(a b)2 (1 -( + b; whence, c=(a+ 6) /(1- (a+^ )2) Developing the quantity under the radical sign by the binomial theorem, and neglecting the terms after the second, it becomes ab 1K2 1 — ~ ~ i +, &c. (a + b) Substituting for K minutes, K. sin. 1' [Trig., Art. (5)], and performing the multiplication by a + b, we obtain c = a + b - b (. ) Now ( 2 =0.0000000423079; 2(a+b) 2 ab Ka whence the formula in the text, c = a +b - 0.000000042308 X —. a -- b (430) In the triangle ABC, designate the angles as A, B, C; and the sides opposite to them as a, b, c. Let CD = d. The triangle BCD gives [Trig., Art. (12), sin. BDC sin. ADC Theorem I], a = dsin. BD The triangle ACD similarly gives b = d. sin. ADC sin. CBD' sin. CAD In the triangle ABC, we have [Trig., Art. (12), Theorem II.], tan. C (A - B): cot. i C:: a - b: a + b; a-b whence tan. ~ (A -B) = a- - cot. i C. [1] a Ka b b Let K be an auxiliary angle, such that b = a. tan. K; whence tan. K = a 400 DEMONSTRATIONS [APP. B, Dividing the second member of equation [1], above and below, by a, and substituting tan. K for-, we get tan. ~ (A - B) cot. i C. a 1 tan. K. Since tan. 45~ = 1, we may substitute it for 1 in the preceding eqdation, and tan. 450 - tan. IK we get tan. i (A- B) = tan. 45~ + tan.' cot. I C. From the expression for the tangent of the difference ol two arcs [Trig., Art. (8)], the preceding fraction reduces to tan. (45~ - K); and the equation becomes tan. A (A - B) = tan. (45~ - K). cot. i C. [2] In the equation tan. K =, substitute the values of b and a from the formulas a at the beginning of this investigation. This gives sin. ADC sin. BDC sin. ADO. sin. CBD tan. K - d. + d - * sin. CAD sin. CBD sin. CAD. sin. BDC' (A - B) is then obtained by equation [2]; (A + B) is the supplement of C; therefore the angle A is known. a. sin. C d. sin. BDC. sin. ACB Then c = AB =- -- -- - sin. A sin. CBD. sin. CAB The use of the auxiliary angle K, avoids the calculation of the sides a and b. (434) In the figure on page 292, produce AD to some point F. The exterior angles, EBC=A+P; ECD=A+Q; EDF=A+R. The triangle ABE BE sin. A. CE sin. A. gives - = i. The triangle ACE gives Dividing member a sin. P a x sin. mber BE a. sin. Q by member, we get BE (a sin. P CE (a + x) sin. P' BE sin. (A +- R) In the same way the triangles BED and CED give b- =si. ( — P b + x sin. (R - P) CE sin. (A + R) Whenc BE (b + x) sin. (R - Q) and - -- / Whence as before, - b. sin. -— Pb sin.(R -Q) CE b. sin. (R — P) Equating these two values of the same ratio, we get a. sin. Q (b + x) sin. (R - Q) and then (a + x) sin. b sin. (R - P) ab. sin. Q. sin. (R- P) sin. P.sin.(R - Q) ( )( ) To solve this equation of the 2d degree, with reference to x, make 2 4 ab sin. Q (sin. R - P) -(a — bf sin. P (sin. R —Q)' Then the first member of the preceding equation = - * (a - b)2 X tan.2 K; and we get x2 + (a + b) x = (a -b)2. tan.2 K - ab, and xz= - - (a + b) ~ / [4 (a -b). tan.2 RK- ab + - (a+ b)2] =- a (a + b) / [4 (a - b)2 tan.2 K + i (a - b)2] --- - ^ a b) (a- b) (tan K + 1). 1 a +-b a -b Or, since / (tan.2 K + 1) = secant KI= -— K w, e have x. —-+b a cs. ICWe2 2. cos. K' APP. B.] For Part XI. 401 DEMONSTRATIONS FOR PART XI. (493) The content being given, and the length to be n times the breadth; Breadth X i times breadth = content; whence, Breadth =/ Given the content = c, and the difference of the length and breadth = d; to find the length 1, and the breadth b. We have I X b = c; and I - b = d. From these two equations we get I = ~ d +- I v/ (d2 ~- 4 c). Given the content =c, and the sum of the length and breadth = s; to find I and b. We have I X b = c; and 1 + b = s; whence we get I = s - (s2- 4 c). (494) The first rule is a consequence of the area of a triangle being the product of its height by half its base. To get the second rule, call the height h; then the base =- h; and the area =- h X mh; whence h = (2 Xrea. For the equilateral triangle, calling its side e, the formula for the area of a triangle V [ (s) s — a) ( s- - )(4 s — c)] reduces to en /3. Hence e = 2/(^ - -1.5197 / area. (495) By Art. (65), Note, J. AB X BC X sin. B = content of ABC; whence, 2 X ABC AB. sin. B' (496) The area of a circle = radius2 X 2; whence radius = /X area( ) 7'a22 (49'7) The blocks, including half of the streets and avenues around them, are 900 X 260 = 234000 square feet. This area gives 64 lots; then an acre, or 43560 feet, would give not quite 12 lots. (502) The parallelogram ABDC being double the triangle ABC, the proof for Art. (495), slightly modified, applies here. (504) Produce BC and AD to meet in E. Fig. 346, bis. By similar triangles, ABE: DCE:: AB2: DC2. - ABE-DCE:ABE: AB2 -DC: AB2. F -- Now ABE - DCE = ABCD; also, by — Art. (65), Note,. sin. A. sin. B ABE AB2 *2.sin. (A+B)' A The above proportion therefore becomes sin. A. sin. B ABCD: AB2.:: AB2 -CD2: A32. 2. sin. (A + B) Multiplying extremes and means, cancelling, transposing, and extracting the square root, we get CD = [AB2 _ 2. ABCD. sin. ( + B)] f V L sin. A. sin. B J 402 DEMONSTRATIONS [APP. B. When A +- B > 180~, sin. (A + B) is negative, and therefore the fraction in which it occurs becomes positive. CF being drawn parallel to DA, we have sin. B sin. B sin. B AD - C - FI. CB. (AB -- CD) sin. BCF sin. (1800 A B) F( sin. (A - B) sin. A BC = (AB - CD) ) sin. (A + B) (505) Since similar triangles are as the squares of their homologous sides, BDE: BFG:: BD2: BF2; whence BF = BD /BG(506) BFG =. BF X FG =. BF X BF.tan. B; V 2. BFG whence, BF = /. B F \tan.;' (510) By Art. (65), Note, BFG - BF2 in. B. sin. F 2. sin. (B +- F); whence, BF = (2. sin. (B F). BFG \ asin. B. sin. F (511) The final formula results from the proportion FAE: CDE:: AE2: ED2. (512) Since triangles which have an angle in each equal, are as the products of the sides about the equal angles, we have ABE: CDE:: AE X BE: CE X DE. sin. A. sin. B sin. B ABE -. AB2. sin.A s B AE AB. sin.(A -- B) sin. E' sin A sin. ODE BE = AB.-lA. CE = DE. sin. E' sin. DCE' Substituting these values in the preceding proportion, cancelling the common factors, observing that sin. (A + B) = sin. E, multiplying extremes and means, and //2. CDE. sin. DCE\ dividing, we get DE =,/ sin. E. sin. CDE DE V\ sin. E. sin. CDE /' (515) The first formula is a consequence of the expression for the area of a triangle, given in the first paragraph of the Note to Art. (65). (517) The reasons for the operations in this article (which are of very frequent occurrence), are self-evident. (51~) The expression for DZ follows from Art. (65), Note. The proportion in the next paragraph exists because triangles having the same altitude are as their bases. (519) By construction, GPO = the required content. Now, GPC = GDC, since they have the same base and equal altitudes. We have now to prove that LMC = GDC. These two triangles have a common angle at C. Hence, they are'to each other as the rectangles of the adjacent sides; i. e., GDC: LMC:: GC X CD:: LC X CM. Here CM is unknown, and must be eliminated. We obtain an expression for it by means of the similar triangles LCM and LEP, which give LE: LC:: EP = CD: CM. APP. B.] For Part XI. 403 CD XLC Hence, CM LE -. Substituting this value of CM in the first proportion, and cancelling CD in the last two terms, we get LC2 GDC: LMC:: GC: L-; or GDC: LMC:: GC X LE: LC2. LE' LC2 = (LH + HC)2 = LH2 + 2 LH X HO + HC2. But, by construction, LH2 = HK2 = HE - EK2 = HE2 - EC2 = (HE+EC) (HE -EC) = HC (HE -EC) Also, GC = 2 HC; and LE = LH + HE. Substituting these values in the last proportion, it becomes GDC: LMC:: 2.HC (LH + HE): HC (HE- EC) + 2 LH X HC + HC2.::2 HI + 2 HE: HE- EC + 2 LH + HC.: HE - EC + 2 LH + HE + EC.:2 HE- 2 LH. The last two terms of this proportion are thus proved to be equal. Therefore, the first two terms are also equal; i. e., LMO = GDO = the required content. Since HK = / (HE2 - EK2), it will have a negative as well as a positive value. It may therefore be set off in the contrary direction from L, i. e., to L'. The line drawn from L' through P, and meeting CB produced beyond B, will part off another triangle of the required content. (520) Suppose the line LM drawn. Then, by Art. (65), Note, the required content, c = I. CL X CM. sin. LCM. This content will also equal the sum of the two triangles LCP and lCP; i. e., c= ~, CL X p + ~ ~ CM X q. The first of 2c these equations gives CM = -C. si. Substituting this in the second equation, we have cq c = ~. CL X +^ -CL. sin. LCM' Whence, i p. CL2. sin. LCM t- cq = c. CL. sin. LCM. Transposing and dividing by the coefficient of CL2, we get 2 c 2 cq CL' -. CL = 2 - p p. sin. CLM CL / ( p. sin. LCM' If the given point is outside of the lines CL and CM, conceive the desired line to be drawn from it, and another line to join the given point to the corner of the field. Then, as above, get expressions for the two triangles thus formed, and put their sum equal to the expression for the triangle which comprehends them both, and thence deduce the desired distance, nearly as above. (522) The difference d, between the areas parted off by the guess line AB, and the required line CD, is equal to the difference between the triangles APO and BPD sin. A. sin. P By Art. (65), Note, the triangle APC-.AP2.' A sn sin. (A + P)J sin. B sin. P Similarly, the triangle BPD = BP2 sin. (B. -P)~' d=J- AP`2 sin. A. sin. P sin. B. sin. P sin. (A + P) sin. (B +- P)' 404 DEMONSTRATIONS [APP. B By the expression for sin. (a - b) [Trigonometry, Art. (8)], we have sin. A. sin. P sin. B. sin. P d= AP2 - — BP2. sin. A. cos. P + sin. P. cos. A sin. B. cos. P + sin. P. cos. B COs. a Dividing each fraction by its numerator, and remembering that -- = cot. a, we sin. a have i AP2 BP" cot. P + cot. A cot. P + cot. B' For convenience, let p = cot. P; a = cot. A; and b = cot. B. The above equation will then read, multiplying both sides by 2, AP2 BP2 2d - p + a p + b Clearing of fractions, we have 2 dp2 + 2 dap + 2 dbp + 2 dab =p. AP2 + b. AP' -p.BP2- a. BP. Transposing, dividing through by 2 d, and separating into factors, we get p2+ (a+ b - P= 2 d -ab. /,, AP2 BP2 tb.AP2 a.BPa. p —a b AP2BP\ L b AP2- a.BPa AP+- + BP-2..p=,a+b- 2 )~d _ —-- --- -+2+d a 2cl ) If A = 90~, cot. A = a= 0; and the expression reduces to the simpler form given in the article. (523) Conceive a perpendicular, BF, to be let fall from B to the required line DE. Let B represent the angle DBE, and 3 the unknown angle DBF. The angle BDF = 90~ --; and the angle BEF = 90~ - (B - 3) = 90~ - B + 3. By Art. sin. BDE. sin. BED (65), Note, the area of the triangle DBE= i DE2. (BDE BED) sin. (BDE + BED). DE2. sin. (90~ -1 ) sin. (90 - B +- ) sin. B Hence, DE2- 2 X DBE X sin. B 2 X DBE X sin. B Hence, DE2 = -= - sin. (90~ - 13). sin. (90~ - B + 13) cos. /3. cos. (B - 3)' Now in order that DE may be the least possible, the denominator of the last fraction must be the greatest possible. It may be transformed, by the formula, cos. a. cos. b = cos. (a + b) + i. cos. (a - ) [Trigonometry, Art. (8) ], into ~ cos. B + -. cos. (B -2 3). Since B is constant, the value of this expression depends on its second term, and that will be the greatest possible when B - 2 = 0, in which case3 i = i B. It hence appears that the required line DE is perpendicular to the line, BF, which bisects the given angle B. This gives the direction in which DE is to be run. Its starting point, D or E, is found thus. The area of the triangle DBE =. BD. BE. sin. B. Since the triangle is isosceles, this becomes DBE = B BD2. sin. B; whence BD= ( sB). DE is obtained from the expression for DE2, which becomes, making S = i B, 2 X DBE X sin. B /(2. DBE. sin. B) DE s = - -— B. s --; whence, DE ------- cD =Cs. I B.cos. cos. B APP. B.] For Part XI. 405 (524) Let a=value per acre of one portion of the land, and b that of the other portion. Let x=-the width required, BC or AD. Then the value of BCFE = a X XB, and the value of ADFE =b X 1-.A 10 10 Putting the sum of these equal to the value required to be parted off, we obtain value required X 10 a X BE+b X AE (525) All the constructions of this article depend on the equivalency of triangles which have equal bases, and lie between parallels. The length of AD is derived from the area of a triangle being equal to its base by half its altitude. (52') Since similar triangles are to each other as the squares of their homolo. gous sides, ABC: DBE:: AB'1: BD2; whence BD = AB ADB = AB ABC Vrn + a The construction of Fig. 363 is founded on the proportion BF: BG:: BG: BA; when BD = BG -= (BA X BF) = BA m+ -[- n (528) By hypothesis, AEF: EFBC m: n; whence AEF: ABC:: m: r+n; az AC X DB m and AEF =ABC - — A. Also, AEF =. AE X EF. m + n 2 m J- n DB XAE The similar triangles AEF and ABD give AD: DB:: AE: EF = AID. The AD DB X AE second expression for AEF then becomes AEF = AE AID. Equating this with the other value of AEF, we have AC X DB m whence AE2 X DB ADX mm 2 m - nt 2. AD m -n) (530) In Fig. 366, the triangles ABD, DBC, having the same altitude, are to each other as their bases. In the next paragraph, we have ABD:DB C: AD DC:: m: n; whence AD: AC:: m: m - n; and AC:D D:: m + n: t; whence the expressions for AD and DC. In Fig. 367, the expression for AD is given by the proportion AD: AC:: m: - +- n. Similarly for DE, and EC. (531) In Fig. 368, conceive the line EB to be drawn. The triangle AEB = I ABC, having the same altitude and half the base; and AFD = AEB, because of the equivalency of the triangles EFD and EFB, which, with AEF, make up AFD and AEB. The point F is fixed by the similar triangles ADB and AEF The expression for AF, in the last paragraph, is given by the proportion, ABC: ADF:: AB X AC: AD X AF; AB X AC ADF AB X AC whence, AF AD ABC AD AD ABC AD m - n (532) The areas of triangles being equal to the product of their altitudes by half their bases, the constructions in Fig. 369 and Fig. 370 follow therefrom. 406 DEMONSTRATIONS [APP. B. (533) In Fig. 371, conceive the line BL to be drawn. The triangle ABL will be a third of ABC, having the same altitude and one-third the base; and AED is equivalent to ABL, because ELB =- ELD, and AEL is common to both. A similar proof applies to DOG. (534) In Fig. 372, the four smaller triangles are mutually equivalent, because of their equal bases and altitudes, two pairs of them lying between parallels. (535) In Fig. 373, conceive AE to be drawn. The triangle AEC =. ABC, having the same altitude and half the base; and EDFC = AEC, because of the common part FEC and the equivalency of FED and FEA. (536) In Fig. 374, in addition to the lines used in the problem, draw BF and DG. The triangle BFC = - ABC, having the same altitude and half the base. Also, the triangle DFG = DFB, because of the parallels DF and BG. Adding DFC to each of these triangles, we have DCG = BFC =- ABC. We have then to prove LMC = DCG. This is done precisely as in the demonstration of Art. (519), page 402. (537) Let AE = x, ED = y, AH =x', HF =y', AK=a, KB =b. The quadrilateral AFDE, equivalent to ~ ABC, but which we will represent, generally, by zs2, is made up of the triangle AFH and the trapezoid FHED. AFH = 4. x'y'. FHED = i (x - x') (y + y'). AFDE = 2:= 2. x'y' + (x - X') (y + y') =- i x (y - y) -'y. The similar triangles, AHF and AKB, give bx' a: b:: x' y -=-. Substituting this value of y' in the expression for m2, we have m2 = x (y+ b- x'y; a (2 m2 xy) AK ( ABC - AE X ED) whence, x- bx- ay KB XAE-AK XED' The formula is general, whatever may be the ratio of the area m2 to that of the triangle ABC. (538) In Fig. 376, FD is a line of division, because BF = the triangle BDF divided by half its altitude, which gives its base. So for the other triangles. (539) In Fig. 378, DG is a second line of division, because, drawing BL, the triangle BLC = I ABC; and BDGC is equivalent to BLC, because of the common part BOLD, and the equivalency of the triangles DLG and DLB. To prove that DF is a third line of division, join MD and MA. Then BMA=- BGA. From BMA take MFA and add its equivalent MFD, and we have MDFB = i BGA = (ABDG- BDG) =- (2 ABC - BDG) = ABC- i BDG. To MDFB add MDB, and add its equivalent, ~ BDG, to the other side of the equation, and we have MDFB - MDB = 1 ABC - i BDG + i BDG; or, BDF = ~ ABC. (540) In Fig. 378, the triangle AFC = l ABC, having the same base and onethird the altitude. The triangles AFB and BFC are equivalent to each other, each being composed of two triangles of equal bases and altitudes; and each is therefore one-third of ABC. APP. B.] For Part XI. 407 In Fig. 379, AFC: ABC:: AD: AB; since these two triangles have the common base AC, and their altitudes are in the above ratio. So too, BFC: ABC: BE: BA. Hence, the remaining triangle AFB: ABC:: DE: AB. (541) By. Art. (65), Note, ABC = i AC X CB X sin. ACB. But the anglt ACB= ACD+DCB =- (180~-ADC)+0 (180~-CDB)= 180~ — (ADC+ CDB). Hence, ABC a= AC X CB X sin. n (ADC + CDB) =1 AC X CB X sin. j ADB. Let r = DA = DB = DC. Since AB is the chord of ADB to the radius r, and therefore equal to twice the sine of half that angle, we have AB AB AB X BC X CA sin. ]. ADB = -; whence, ABC - AC X CB X; and r AB Also, since the area of each of the three small triangles equals half the product of one of the equal sides (=r), by the sine of the included angle at D, these triangles will be to each other as the sines of those angles. These angles are found thus: AB BC AC sin. 1 ADB =-; sin. BDC =; sin.n ADC =. 2 r 2r 2r (542) The formulas in this article are obtained by substituting, in those of Art. (523), for the triangle DBE, its equivalent X ~ AB X BC X sin. B. m ABXBCXsin. B' n _?f ABBC BD thus becomes= ( ABXB B /( n XABXBC); V\m -+- n sin. B - X \ A+ n B; and ( DE -B ( iXABXBCXsin.2 B) sin. B and DE = IW +_,. m -XABXBO). cos. ~ B cos. B n n (543) The rule and example prove themselves. (544) In Fig. 383, conceive the sides AB and DC, produced, to meet in some point P. Then, by reason of the similar triangles, ADP: BCP:: AD2: BC2; whence, by "division," ADP - BCP = ABCD: BCP AD2 - BC2: BC2. In like manner, compaling EFP and BCP, we get EBCF: BCP:: EF2-BC2: BC2Combining these two proportions, we have ABOD: EBCF:: AD2 - BC2: EF2 - BC2; or, m + n: m:: AD2 - BC2: EF2 - BC2. Whence, (m + n) EF2 - m. BC2 - n BC2 =- m. AD2 - m. BC2;.. EF-= /(m X AD2 +n X BC2) Also, from the similar triangles formed by drawing BL parallel to CD, we have AL:EK BA BA X EK AB (EF -BC) AL: EK": BA: BE AL AD BC (545) Let BEFC-=. ABCD = a; let BO = b; BH = h; and rn 4- n AD - BC = c. Also let BG = x; and EF = y. Draw BL parallel to CD. By sim. ilar triangles, AL: EK:: BA: BE:: BH: BG; or, AD-B: EF-BC:: BH -BG; h (y- b) i.e.,: y - b::h: x; whence x —. Also, the area BEFC = a =. BG (EF + BC) = y x (y + b); whence y = —b. X 408 DEMONSTRATIONS. [APP. B Substituting this value of y in the expression for x, and reducing, we obtain 2 bh 2 ah b/i 2 a + b2h2 X22 + x- =-; whence we have x = — i / - 2 - The second proportion above gives y- b =- T; whence y = b +- x. h h Replacing the symbols by their lines, we get the formulas in the text. (546) ABEF - ~ ABCD. But ABRP = ABEF, because of the common part ABRF, and the triangles FRP and FRE, which make up the two figures, and which are equivalent because of the parallels FR and PE. So for the other parts. (547) The truth of the foot-note is evident, since the first line bisects the trapezoid, and any other line drawn through its middle, and meeting the parallel sides, adds one triangle to each half, and takes away an equal triangle; and thus does not disturb the equivalency. (54S) In Fig. 385, since EF is parallel to AD, we have ADG: EGF:: GH2: GK2. EGF is made up of the triangle BOG = a', and the quadrilateral BEFC= m.. ABCD -= (a - a'). Hence the above proportion becomes m — n m + — n a a' + m (a -a'):: GH12: GK2; or, a'+m n (m n+ ) a: ma + a':: GHi: GK2; whence GK - GH (ma + ( a). y \(m n) a/ GE is given by the proportion GH: GK:: GA: GE = GA * GGIH In Fig. 386, the division into p parts is founded on the same principle. The triangle EFG = GBC + EFCB = a' + Q-. Now ADG: EFG:: AG2: EG2; P or, a' +Q: a'+ Q::AG2: EG2; whence GE= AG ( p\ a'+ Q/ 2Q GL is obtained by taking the triangle LMG = a' + -; and so for the rest. (552) In Fig. 390, join FO and GC. Because of the parallels CA and BF, the triangle FCD will be equivalent to the quadrilateral ABCD, of which GOD will therefore be one half; and because of the parallels GE and CH, EHDC will be equivalent to GCD. (553) In Fig. 391, by drawing certain lines, the quadrilateral can be divided into three equivalent parts, each composed of an equivalent trapezoid and an equivalent triangle. These three equivalent parts can then be transformed, by means of the parallels, into the three equivalent quadrilaterals shown in the figure. The full development of the proof is left as an exercise for the student. In Fig. 892, draw CG. Then CBG = I ABCD. But CKQ = CGQ. Therefore OKQB = ~ ABCD. So for the other division line. (556) The division of the base of the equivalent triangle, divides the polygon similarly. The point Q results from the equivalency of the triangles ZBP and ZBQ, PQ being parallel to BZ. APPENDIX C. INTRODUCTION TO LEVELLING. (1) The Principles. LEVELLING is the art of finding how much one point is higher or lower than another; i. e., how much one of the points is above or below a level line or surface which passes through the other point. A level or horizontal line is one which is perpendicular to the direction of gravity, as indicated by a plumb-line or similar means. It is therefore parallel to the surface of standing water. A level or horizontal surface is defined in the same way. It will be determined by two level lines which intersect each other.* Levelling may be named VERTICAL SURVEYING, or Up-and-down Surveying; the subject of the preceding pages being Horizontal Surveying, or Right-and-left and Fore-and-aft Surveying. All the methods of Horizontal Surveying may be used in Vertical Surveying. The one which will be briefly sketched here corresponds precisely to the method of "Surveying by offsets," founded on the Second Method, Art. (6), "Rectangular Co-ordinates," and fully explained in Arts. (114), &c. The operations of levelling by this method consist, firstly, in obtaining a level line or plane; and, secondly, in measuring how far below it or above it (usually the former) are the two points whose relative heights are required. (2) The Instruments. A level Fig.415. line may be obtained by the following -- ru simple instrument, called a " Plumb-line level." Fasten together two pieces of wood at right angles to each other, so as to make a T, and draw a line on the upright one so as to be exactly perpendicular to the top edge of the other. Suspend a plumb-line as in the figure. Fix the T against a staff stuck in the ground, by a screw through the middle of the crosspiece. Turn the T till the plumb-line exactly covers the line which was drawn. Then will the upper edge of the cross-piece be a level line, and the eye can sight across it, and note how far above or below any other point this level line, prolonged, would strike. It will be easier to look across sights fixed on each end of the cross-piece, making them of horsehair stretched across a piece of wire, bent into three sides of a square, and stuck into each end of the cross-piece; taking care that the hairs are at exactly equal heights above the upper edge of the cross-piece. * Certain small corrections, to be hereafter explained, will be ignored for the present, and we will consider level lines as straight lines, and level surfaces as planes. 410 LEVELLING. [APP. C. A modification of this is to fasten a common Fig.416. carpenter's square in a slit in the top of a staff, -- - by means of a screw, and then tie a plumb-line at the angle so that it may hang beside one arm. When it has been brought to do so, by turning the square, then the other arm will be level. Another simple instrument depends upon the principle that "water always finds its level," corresponding to the second part of our definition of a level line. If a tube be bent up at each end, and nearly filled with water, the surface of the water in one end will always be at the same height as that in the other, however the position of the tube may vary. On this truth depends the " Water-level." It may be easily constructed with a tube of tin, lead, copper, &c., by bending up, at right angles, an inch or two of each end, and supporting the tube, if too Fig. 417. flexible, on a wooden bar. In these - ------ ends cement (with putty, twine dipped in white-lead, &c.), thin phi- als, with their bottoms broken off, so as to leave a free communication between them. Fill the tube and the phials, nearly to their top, with colored water. Blue vitriol, or cochineal, may be used for coloring it. Cork their mouths, and fit the instrument, by a steady but flexible joint, to a tripod. Figures of joints are given on page 134, and of tripods on page 133. To use it, set it in the desired spot, place the tube by eye nearly level, remove the corks, and the surfaces of the water in the two phials will come to the same level. Stand about a yard behind the nearest phial, and let one eye, the other being closed, glance along the right-hand side of one phial and the left-hand side of the other. Raise or lower the head till the two surfaces seem to coincide, and this line of sight, prolonged, will give the level line desired. Sights of equal height, floating on the water, and rising above the tops of the phials, would give a better-defined line. The " Spirit-level" consists essentially Fig. 41. of a curved glass tube nearly filled with alcohol, but with a bubble of air left i within, which always seeks the highest spot in the tube, and will therefore by --- -- its movements indicate any change in the position of the tube. Whenever the bubble, by raising or lowering one end, has been brought to stand between two marks on the tube, or, in case of expansion or contraction, to extend an equal distance on either side of them, the bottom of the block (if the tube be. in one), or sights at each end of the tube, previously properly adjusted, will be on the same level line. It may be placed on a board fixed to the top of a staff or tripod. When, instead of the sights, a telescope is made parallel to the level, and various contrivances to increase its delicacy and accuracy are added, the instrument becomes the Engineer's spirit-level. APP. C.] The Practice. 411 (3) The Practice. By whichever of these various means a level line has been obtained, the subsequent operations in making use of it are identical. Since the "water-level" is easily made and tolerably accurate, we will suppose it to be employed. Let A and B, Fig. 419, represent the two points, the Fig. 419. difference of the heights of which is required. Set the instrument on any spot from which both the points can be seen, and at such a height that the level line will pass above the highest one. At A let an assist- 2 H —----- ---- -- ant hold a rod graduated into feet, tenths, &c. Turn the instrument to- X wards the staff, sight along the level line, and note what division on the B staff it strikes. Then send the staff to B, direct the instrument to it, and note the height observed at that point. If the level line, prolonged by the eye, passes 2 feet above A and 6 feet above B, the difference of their heights is 4 feet. The absolute height of the level line itself is a matter of indifference. The rod may carry a target or plate of iron, clasped to it so as to slide up and down, and be fixed, at will. This target may be variously painted, most simply with its upper half red and its lower half white. The horizontal line dividing the colors is the line sighted to, the target being moved up or down till the line of sight strikes it. A hole in the middle of the target shows what division on the rod coincides with the horizontal line, when it has been brought to the right height. If the height of another point, C, Fig. 420, not visible from the first station, be required, set the instrument so as to see B and C, and proceed exactly as with A Fig. 420. A4t 3 C and B. If C be 1 foot below B, as in the figure, it will be 5 feet below A. If it were found to be 7 feet above B, it would be 3 feet above A. The comparative height of a series of any number of points, can thus be found in reference to any one of them. The beginner in the practice of levelling may advantageously make in his notebook a sketch of the heights noted, and of the distances, putting down each as it is observed, and imitating, as nearly as his accuracy of eye will permit, their pro 412 LEVELLING. [APP. C. portional dimensions.* But when the observations are numerous, they should be kept in a tabular form, such as that which is given below. The names of the points, or " Stations," whose heights are demanded, are placed in the first column; and their heights, as finally ascertained, in reference to the first point, in the last column. The heights above the starting point are marked +, and those below it are marked -. The back-sight to any station is placed on the line below the point to which it refers. When a back-sight exceeds a fore-sight, their difference is placed in the column of "Rise;" when it is less, their difference is a "Fall." The following table represents the same observations as the last figure, and their careful comparison will explain any obscurities in either. Stations. Distances. Back-sights. Fore-sights. Rise. Fall. Total Heights. A I 0.00 B 100 2.00 6.00.0 - 4.00 - 4.00 0 60 3.00 4.00 - 1.00 - 5.00 D 40 2.00 1.00 + 1.00 - 4.00 E 70 6.00 1.00 + 5.00 + 1.00 F 50 2.00 6.00 - 4.00 - 3.00 15.00 18.00 - 3.00 The above table shows that B is 4 feet below A; that C is 5 feet below A; that E is 1 foot above A; and so on. To test the calculations, add up the back-sights and fore-sights. The difference of the sums should equal the last " total height." Another form of the levelling field-book is presented below. It refers to the same stations and levels, noted in the previous form, and shown in Fig. 420. Stations. Distances. Back-sights. Ht. Inst. above Datum. Fore-sights. TotalHeights. A 0.00 B 100 2.00 + 2.00 6.00 - 4.00 C 60 3.00 - 1.00 4.00 - 5.00 D 40 2.00 - 3.00 1.00 - 4.00 E 70 6.00 + 2.00 1.00 + 1.00 F 50 2.00 + 3.00 6.00 - 3.00 15.00 18.00 - 3.00 In the above form it will be seen that a new column is introduced, containing the Height of the Instrument (i. e., of its line of sight), not above the ground where it stands, but above the D)atum, or starting-point, of the levels. The former columns of "Rise" and "Fall" are omitted. The above notes are taken thus: The height of the starting-point or " Datum," at A, is 0.00. The instrument being set up and levelled, the rod is held at A. The back-sight upon it is 2.00; therefore the height of the instrument is also 2.00. The rod is next held at B. The fore-sight to it is 6.00. That point is therefore 6.00 below the instrument, or 2.00 -6.00 =-4.00 below the datum. The instrument is now moved, and again set up, and the back-sight to B, being 3.00, the Ht. Inst. is -4.00 + 3.00 =-1.00' * In the figure, the limits of the page have made it necessary to contract the horizontal distances to one-tenth of their proper proportional size. APP. a.] The Practice. 413 and so on: the Ht. Inst. being always obtained by adding the back-sight to the height of the peg on which the rod is held, and the height of the next peg being obtained by subtracting the fore-sight to the rod held on that peg, from the Ht. Inst. The level lines given by these instruments are all lines of apparent level, and not of true level, which should curve with the surface of the earth. These level lines strike too high; but the difference is very small in sights of ordinary length, being only one-eighth of an inch for a sight of one-eighth of a mile, and diminishing as the square of the distance; and it may be completely compensated by setting the instrument midway between the points whose difference of level is desired; a precaution which should always be taken, when possible. It may be required to show on paper the ups and downs of the line which has been levelled; and to represent, to any desired scale, the heights and distances of the various points of a line, its ascents and descents, as seen in a side-view. This is called a "Profile." It is made thus. Any point on the paper being assumed for the first station, a horizontal line is drawn through it; the distance to the next station is measured along it, to the required scale; at the termination of this distance a vertical line is drawn; and the given height of the second station above or below the first is set off on this vertical line. The point thus fixed determines the second station, and a line joining it to the first station represents the slope of the ground between the two. The process is repeated for the next station, &c. But the rises and falls of a line are always very small in proportion to the distances passed over; even mountains being merely as the roughnesses of the rind of an orange. If the distances and the heights were represented on a profile to the same scale, the latter would be hardly visible. To make them more apparent it is usual to "exaggerate the vertical scale" ten-fold, or more; i. e., to make the representation of a foot of height ten times as great as that of a foot of length, as in Fig. 420, in which one inch represents one hundred feet for the distances, and ten feet for the heights. The preceding Introduction to Levelling has been made as brief as possible; but by any of the simple instruments described in it, and either of its tabular forms, any person can determine with sufficient precision whether a distant spring is higher or lower than his house, and how much; as well as how deep it would be necessary to cut into any intervening hill to bring the water. He may in like manner ascertain whether a swamp can be drained into a neighboring brook; and can cut the necessary ditches at any given slope of so many inches to the rod, &c., having thus found a level line; or he can obtain any other desired information which depends on the relative heights of two points. To explain the peculiarities of the more elaborate levelling instruments, the precautions necessary in their use, the prevention and correction of errors, the overcoming of difficulties, and the various complicated details of their applications, would require a great number of pages. This will therefore be reserved for another volume, as announced in the Preface. ANALYTICAL TABLE OF CONTENTS. PART I. GENERAL PRINCIPLES AND FUNDAMENTAL METHODS. CHAPTER I. Definitions and Methods. ETWIOLE PAGE ARTICLE PAGE 1) Surveying defined........... 9 Division of the subject. 2) When a point is determined.. 9 (12) By the methods employed.. 14 3) Determining lines and surfaces 10 (13) By the instruments........ 14 To determine points. (14) By the objects............. 14 5) First Method............... 10 (15) By the extent............. 15 6) Second do................ 11 (16) Arrangement of this book... 15 [7) Third do................ 11 (17) The three operations common'~) Fourth do................ 12 to all surveying........ 15 10) Fifth do................ 131 CHAPTER II. Making the Measurements. Measuring straight lines. (25) Chaining on slopes......... 21 (19) Actual and Visual lines.... 16 (28) Tape..................... 23'20) Gunter's Chain............ 16 (29) Rope, &c.................. 24 21) Pins..................... 19 (30) Rods................... 24 22) Staves................... 19 (32) Measuring-wheel.......... 24 23) How to chain............. 19 (33) Measuring Angles 25 [24) Tallies................ 21 (34) Noting the Measurements... 25 CHAPTER III. Drawing the Map. [35) A Map defined............ 25 (45) Scales for farm surveys... 29 (36) Platting................... 25 (46) Scales for state surveys..... 31 (37) Straight lines............. 26 (47) Scales for railroad surveys.32 (38) Arcs..................... 26 (49) How to make scales........ 33 (39) Parallels.................. 26 (50) The Vernier scale.......... 35 (40) Perpendiculars............ 27 (51) A reduced scale............ 36 (41) Angles................... 28 (52) Sectoral scales............. 36 (42) Drawing to scale........... 28 (53) Drawing scale on map...... 37 (44) Scales.................... 29 (54) Scale omitted.............. 37 CHAPTER IV. Calculating the Content. (55) Content defined........... 38 (6S) Quadrilaterals............ 44 (56) Horizontal measurement.... 38 (69) Curved boundaries...... 45 (57) Unit of content............ 40 (70) Second Method, Geometrically 45 (58) Reductions................ 40 (71) Division into triangles..... 45 (59) Table of Decimals of an acre. 41 (72) Graphical multiplication... 47 (60) Chain correction.......... 41 (73) Division into trapezoids.... 48 (61) Boundary lines............ 42 (74) Do. into squares...... 48 (75) Do. into parallelograms 49 METHODS OF CALCULATION. (76) Addition of widths........ 50 (63) First Method, Arithmetically. 43 (77) Third Method, Instrumentally 50 (64) Rectangles.............. 43 (78) Reduction to one triangle.. 50 (65) Triangles................ 43 (S4) Special instruments....... 54 (66) Parallelograms........... 44 (87) Fourth Method, Trigonometri(67) Trapezoids............ 44 cally.................... 56 416 CONTENTS. PART II. CHAIN SURVEYING. CHAPTER I. Surveying by Diagonals. ARTICLE PAGE ARTICLE PAGE (90) A three-sided field......... 58 Keeping thefield-notes...... 62 (91) A four-sided field.......... 59 (94) By sketch........... 62 (92) A many-sided field......... 60 (95) In columns............. 62 (93) How to divide a field...... 61 (96-97) Field-books...........64 CHAPTER II. Surveying by Tie-lines. (98) Surveying by tie-lines.... 66 (101) Inaccessible areas.... 67 (100) Chain angles........... 67 (102) Without platting...... 67 CHAPTER III. Surveying by Perpendiculars. To set out Perpendiculars. Offsets. (104) By Surveyor's Cross..... 69 (114) Taking offsets....... 75 (107) By Optical Square...... 70 (117) Double offsets..... 76 (108) BytheChain........... 72 (118) Field work............. 77 Diagonals and Perpendiculars. (119) Platting............. 79 (110) A three-sided field...... 72 (120) Calculating content.... 80 (111) A four-sided field....... 73 (121) When equidistant..... 80 (112) A many-sided field...... 74 (122) Erroneous rules....... 81 (113) By one diagonal........ 75 (123) Reducing to one triangle 81 (124) Equalizing........ 81 CHAPTER IV. Surveying by the methods combined, (125) Combination of the three (132) Exceptional cases........ 92 preceding methods...... 82 (134) Inaccessible areas......... 93 (127) Field-books............. 83 (136) Roads............ 95 (130) Calculations............. 88 (137) Towns................. 95 (131) The six-line system....... 90 CHAPTER V. Obstacles to Measurement in Chain Surveying. (138) The obstacles to Alinement and Measurement..................... 96 (139) LAND GEOMETRY............................................... 96 Problems on Perpendiculars. (140) PROBLEM 1. To erect a perpendicular at any point of a line......... 97 (143) 2. " when the point is at or near the end of the line............ 98 (148) 3. s" " when the line is inaccessible... 99 (150) 4. To let fall a perpendicular from a given point to a given line 99 (153) 5. " " when the point is nearly opposite to the end of the line... 100 (156) 6. " " when the point is inaccessible.. 101 (158) 7. " " when the line is inaccessible... 101 Problems on Parallels. (160) PROBLEM 1. To run a line from a given point parallel to a given line. 102 (165) 2. Do. when the line is inaccessible............. 103 CONTENTS. 417 OBSTACLES TO ALINEMENT. ARTICLE PAGE A. To prolong a line.................................. 105 (169) By ranging with rods..... 105 (174) By transversals.......... 107 (171) By perpendiculars....... 106 (175) By harmonic conjugates... 108 (172) By equilateral triangles. 106 (1T6) By the complete quadri(173) By symmetrical triangles, 107 lateral............ 108 B. To interpolate points in a line.......................... 109 (177) Signals................. 109 (1lS1) With a single person..... 111 (17S) Ranging................ 109 (182) On water............... 111 (179) Across a valley.......... 110 (183) Through a wood......... 112 (180) Over a hill.............. 110 (184) To an invisible intersection. 112 OBSTACLES TO MEASUREMENT. A. When both ends of the line are accessible..................... 113 (186) By perpendiculars........ 113 (189) By transversals.......... 114 (187) By equilateral triangles... 113 (190) In a town............... 114 (188) By symmetrical triangles.. 114 B. When one end of the line is inaccessible....................... 115 (191) By perpendiculars........ 115 (198) By transversals.......... 117 (194) By parallels............ 116 (199) By harmonic division... 117 (195) By a parallelogram....... 116 (200) To an inaccessible line.118 (196) By symmetrical triangles.. 116 (201) To an inacc. intersection.. 118 C. When both ends of the line are inaccessible.................... 119 (202) By similar triangles....... 119 (204) By a parallelogram....... 119 (203) By parallels............. 119 (205) By symmetrical triangles.. 120 INACCESSIBLE AREAS.......................... 121 (207) Triangles............... 121 (29) Polygons......... 121 (208) Quadrilaterals........... 121 PART III. COMPASS SURVEYING. CHAPTER I. Angular Surveying in general. (210) Principle................ 122 (217) The Compass............ 124 (211) Definitions.............. 122 (219) Methods of Angular Sur(213) Goniometer............. 123 veying................ 125 (2114) How to use it........... 123 (220) Subdivisions of Polar Sur(215) Improvements........... 124 veying................ 125 CHAPTER 1I. The Compass. (221) The Needle.............. 127 (228) Tangent Scale........... 132 (222) The Sights.............. 128 (229) The Vernier............. 132 (223) The Telescope........... 128 (230) Tripods................ 133 (224) The divided Circle........ 128 (231) Jacob's Staff............. 134 (225) The Points.............. 129 (232) The Prismatic Compass... 135 (226) Eccentricity............. 130 (234) The defects of the Compass. 137 (227) Levels................. 132 27 418 CONTENTS. CHAPTER III. The Field-work. ARTICLE PAGE ARTICLE PAGE (235) Taking Bearings......... 138 (242) Angles of deflection... 144 (236) Why E. and W. are re- (243) Angles between courses..145 versed............ 139 (244) To change Bearings...... 146 (237) Reading with Vernier.. 140 (245) Line Surveying.......... 147 (238) Practical Hints....... 140 (246) Checks by intersecting bearings.......... 148 Mark stations. Set beside (247) Keeping the Field-notes 149 fence. Level crossways. Do (a) New York CanalMaps. 149 not level by needle. Keep ( ) New York Canal Maps 149 same end ahead. Read from (252) Farm Surveying.......... 150 same end. Caution in read- (254) Field-notes.......... 151 ing. Cheek vibrations. ap (256) Tests of accuacy. 153 compass. Keep iron away. aca Electricity. To carry com- (258) Method of Radiation.. 154 pass. Extra pin and needle. (259) Method of Intersection. 154 (260) Running out old lines.. 154 (239) To magnetize a Needle. 142 (261) Town Surveying......... 155 (240) Back-sights........... 143 (262) Obstacles in Compass Sur(241) Local Attraction...... 143 veying................ 156 CHAPTER IV. Platting the Survey. (263) Platting in general........ 157 (273) Drawing-board protractor. 166 (264) With a protractor........ 157 (274) With a scale of chords.... 166 (265) Platting bearings......... 158 (275) With a table of chords.... 167 (268) To make the plat close... 161 (276) With a table of natural sines 168 (269) Field platting........... 162 (277) By Latitudes and Depart(272) With a paper protractor.164 ures................... 168 CHAPTER V. Latitudes and Departures. (278) Definitions.............. 169 Applications. (279) Calculation of Latitudes (282) Testing survey.......... 175 and Departures......... 170 (283) Supplying omissions...... 176 (280) Formulas.................171 (284) Balancing............... 177 (281) Traverse Table......... 171 (285) Platting................. 178 CHAPTER VI. Calculating the Content, (286) Methods................ 180 (292) General rule............ 184 (28~7) Definitions............. 180 (293) To find east or west station 184 (288) Longitudes............. 181 (294) Example 1............. 184 (289) Areas.................. 182 (296) Examples 2 to 13........ 186 (290) A three-sided field...... 182 (297) Mascheroni's Theorem.... 188 (291) A four-sided field........ 183 CHAPTER VII. The Variation of the Magnetic Needle, (298) Definitions.............. 189 (306) Table of Azimuths....... 196 (299) Direction of the needle.... 189 (307) Setting out the meridian.. 197 To determine the true meridian. To determine the variation. (300) By equal shadows of the (308) By the bearing of the star. 198 sun................. 190 (309) Other methods.......... 199 (301) By the North Star when in (310) Magnetic variation in the the meridian.......... 191 United States........ 199 (302) Times of crossing the me- Line of novariation.... 199 ridian:......... 193 Lines of equal variation.. 200 (303) By the North Star when at Magnetic Pole.......... 200 its extreme elongation.. 194 (311) To correct magn. bearings. 200 (304) Table of times....... 195 (312) To survey a line with true (305) Observations.......... 196 bearings.............. 202 CONTENTS. 419 CHAPTER VIII. Changes in the Variation. ARTIOLE PAGE ARTICLE PAGE (314) Irregular changes........ 203 (31 ) By interpolation...... 205 (315) The Diurnal change...... 203 (319) By old lines.......... 206 (316) The Annual Change...... 204 (320) Effects of this change.... 207 (317) The Secular change...... 204 (321) To run out old lines.... 208 Tables for United States. 205 (322) Example............. 208 To determine the secular (323) Remedy for the evils of change.............. 205 the secular change..... 210 PART IV. TRANSIT AND THEODOLITE SURVEYING. (BY THE 3d METHOD.) CHAPTER I. The Instruments. (324) General description of the (333) Supports..... 221 Transit and Theodolite.. 211 (334) The Indexes. Eccentricity. 221 The Transit.......... 212 (335) The graduated circle..... 223 The Theodolite....... 213 (336) Movements. Clamp and (325) Distinction between them. 214 Tangent screw......... 223 (326) Sources of their accuracy.. 214 (337) Levels................. 224 (327) Explanation of the figures. 215 (338) Parallel plates........... 225 (32~) Sectional view........... 216 (339) Watch Telescope......... 226 (329) Telescopes.............. 217 (340) The Compass............ 226 (330) Cross hairs.............. 218 (341) Theodolites............. 226 (331) Instrumental parallax..... 220 (342) Gdniasmometre.......... 227 (332) Eye-glass and object-glass.. 221 CHAPTER II. Verniers. (343) Definition............... 228 (351) Circle divided to 20'.... 235 (344) Illustration.............. 228 (352) Circle divided to 15'....... 236 (345) General rules............ 229 (353) Circle divided to 10'....... 237 (346) Retrograde Verniers...... 230 (354) Reading backwards....... 237 (347) Illustration..2.......... 231 (355) Arc of excess............ 238 (34S) Mountain Barometer...... 231 (356) Double Verniers......... 238 (349) Circle divided into degrees. 232 (357) Compass Verniers........ 239 (350) Circle divided to 30'....... 233 CHAPTER III. Adjustments. (358) Their object and necessity. 240 Rectification......... 243 (359) The three requirements in (362) In the Theodolite...... 245 the Transit............. 240 (363) Third Adjustment. To cause (360) First Adjustment. To cause the line of collimation to the circle to be horizontal revolve in a vertical plane 246 in every position........ 241 Verification(plumb-line; Verification.......... 241 star; steeple and stake) 246 Rectification......... 241 Rectification......... 246 (361) Second Adjustment. To (364) Centring eye-piece........ 247 cause the line of collima- (365) Centring object-glass...... 247 tion to revolve in a plane 242 Adjusting line of colliVerification......... 242 mation............. 248 420 CONTESTS. CtIAPTER IV. The Field-work. ARTICLE PAGE ARTICLE PAGE (366) To measure a horizontal (372) Line-surveying........... 254 anlgle.................. 250 (373) Traversing, or surveying (367) Reduction of high and by the back angle... 254 low objects......... 251 (374) Use of the Compass.... 255 (36S) Notation of angles..... 252 (375) Measuring distances with (369) Probable error........ 252 a telescope and rod.. 256 (370) To repeat an angle.... 252 (376) Ranging out lines......... 257 (371) Angles of deflection.... 253 (377) Farm-surveying.......... 258 (378) Platting................. 259 PART V. TRIANGULAR SURVEYING. (BY THE 4th METHOD.) (379) Principle................ 260 (385) Observations of the angles.. 267 (380) Outline of operations...... 260 (386) Reduction to the centre... 268 (381) Measuring a base......... 261 (387) Correction of the angles... 270 Materials.............. 261 (388) Calculation and platting... 270 Supports............... 262 (389) Base of Verification....... 271 Alinement............. 262 (390) Interior filling up........ 271 Levelling.............. 262 (391) Radiating Triangulation... 272 Contacts............... 262 (392) Farm Triangulation..... 272 (382) Corrections of Base....... 263 (393) Inaccessible Areas........ 273 (383) Choice of stations......... 263 (394) Inversion of the Fourth U. S. Coast Survey Ex- method................ 273 ample............... 265 (395) Defects of the Method of In(3~4) Signals............ 266 tersections............. 274 PART VI. TRILINEAR SURVEYING. (BY THE 5th METHOD.) (396) The Problem of the three (398) Instrumental Solution.... 277 points................ 275 (399) Analytical "..... 277 t397) Geometrical Solution..... 275 (400) Maritime Surveying...... 278 PART VII. OBSTACLES IN ANGULAR SURVEYING. CHAPTER I. Perpendiculars and Parallels. (402) To erect a perpendicular to'a line at a given point................. 279 (403) To erect a perpendicular to an inaccessible line, at a given point of it 280 (404) To let fall a perpendicular to a line, from a given point............. 280 (405) To let fall a perpendicular to a line, from an inaccessible point.... 280 (406) To let fall a perpendicular to an inaccessible line from a given point.. 281 (407) To trace a line through a given point parallel to a given line........ 281 (408) To trace a line through a given point parallel to an inaccessible line.. 281 CONTENTS. 421 CHAPTER II. Obstacles to Alinement. ARTICLE PAGE A. To prolong a line..................... 282 (409) General method......... 282 (413) When the line to be pro(410) By perpendiculars....... 282 longed is inaccessible... 283 (411) By an equilateral triangle. 282 (414) To prolong a line with only (412) By triangulation......... 288 an angular instrument... 283 B. To interpolate points in a line........ I.... 284 (415) General method.......... 284 ('18) By Latitudes and Depart(416) By a random line........ 284 ures, with transit...... 285 (417) By Latitudes and Depart- (419) By similar triangles...... 286 ures, with compass..... 285 (420) By triangulation......... 286 CHAPTER III. Obstacles to Measurement. A. When both ends of the line are accessible........... 287 (421) Previous means.......... 287 (422) By triangulation......... 287 (424) By angles to known points. 288 B. When one end of the line is inaccessible............ 288 (425) By perpendiculars........ 288 (429) To find the distance from a (426) By equal angles......... 288 given point to an inacces(427) By triangulation.......... 289 sible line......... 289 (428) When one point cannot be seen from the other.... 289 C. When both ends of the line are inaccessible.......... 290 (430) General method.......... 290 (433) When no point can be found (431) To measure an inaccessible from which both ends can distance, when a point in be seen............... 292 itsline can be obtained.. 291 (434) To interpolate a base..... 292 (432) When only one point can be (435) From angles to two points.. 293 found from which both (436) From angles to three points 293 ends of the line can be (437) From angles to four points. 294 seen................. 291 (438) Problem of the eight points 296 CHAPTER IV. To Supply Omissions. (439) General statement....................................... 297 (440) CASE 1. When the length and bearing of any one side are wanting.... 298 CASE 2. When the length of one side and the bearing of another are wanting..... 298 wanting............................................ 298 (441) When the deficient sides adjoin each other............... 298 (442) When the deficient sides are separated from each other..... 299 (443) Otherwise: by changing the meridian............. 299 CASE 3. When the lengths of two sides are wanting........ 300 (444) When the deficient sides adjoin each other................ 300 (445) When the deficient sides are separated from each other... 301 (446) Otherwise: by changing the meridian............... 301 CASE 4. When the bearings of two sides are wanting................ 302 (447) When the deficient sides adjoin each other....... 802 (448) When the deficient sides are separated from each other..... 02 422 CONTENTS. PART VIII. PLANE TABLE SURVEYING. ARTIOLE PAGE (449) General description....... 303 (455) Method of Resection...... 308 (450) The Table............... 303 (456) To Orient the Table...... 308 (451) The Alidade............. 304 (457) To find one's place on the (452) Method of Radiation...... 305 ground.............. 309 (453) Method of Progression.... 306 (458) Inaccessible distances..... 310 (454) Method of Intersection.... 307 PART IX. SURVEYING WITHOUT INSTRUMENTS. (459) General principles........ 311 (463) Distances by sound...... 313 (460) Distances by pacing...... 311 (464) Angles.................. 314 (461) Distances by visual angles. 312 (465) Methods of operation...... 314 (462) Distances by visibility.... 313 PART X. MAPPING. CHAPTER I. Copying Plats. (466) Necessity.............. 316 (474) Reducing by squares...... 319 (467) Stretching the paper..... 316 (475) " by proportional (468) Copying by tracing..... 31 scales....... 320 (469) " on tracing-paper..317 (476) " by a pantagraph 321 (470) " by transfer-paper.. 31 (477) " by a camera luci(471) " by punctures.... 318 da........... 321 (472) " by intersections.. 318 (478) Enlarging plats.......... 321 (473) " by squares...... 319 CHAPTER II. Conventional Signs. (479) Object........ 322 (483) Signs forater........ 325 (480) The relief of ground..... 322 (484) Colored topography......325 (481) Signs for natural surface... 324 (485) Signs for detached objects. 327 (482) Signs for vegetation...... 324 CHAPTER III. Finishing the Map. (486) Orientation............. 328 (49) Joining paper............ 329 (487) Lettering............... 328 (490) Mounting maps.......... 329 (488) Borders............. 328 CONTENTS. 423 PART XI. LAYING OUT, PARTING OFF, AND DIVIDING UP LAND. CHAPTER I. Laying out Land. ARTICLE PAGE ARTICLE PAGE (491) Its object......330 (496) To lay out circles....... 332 (492) To lay out squares.... 330 (497) Town lots.............. 333 (493) To lay out rectangles..... 331 (498) Land sold for taxes....... 333 (494) To lay out triangles...... 332 (499) New countries...3... 334 CHAPTER II. Parting off Land. (500) Its object.................................................. 334 A. By a line parallel to a side. (501) To part off a rectangle...................................... 335 (502) " a parallelogram..................................... 335 (503) ( " a trapezoid......................................... 335 B. By a line perpendicular to a side. (505) To part off a triangle....................................... 336 (507). ( " a quadrilateral.................................... 337 (50 ) " " any figure........................................ 33 C. By a line running in any given direction. (509) To part off a triangle.............................337 (511) " " a quadrilateral.................... 338 (51.3) " " any figure........................................ 339 D. By a line starting from a given point in a side. (514) To part off a triangle.......................................... 339 (516) "( ( a quadrilateral.................................... 340 (517) " " any figure........................................ 340 E. By a line passing through a given point within the field. (519) To part off a triangle..........................................342 (520) "a quadrilateral......................... 43 (522) ( " any figure........................................ 344 F. By the shortest possible line. (523) To part off a triangle........................................ 345 (524) G. Land of variable value.......................... 345 (525) H. Straightening crooked fences.................... 346 CHAPTER III. Dividing up Land. (526) Arrangement........................................... 34 Division of Triangles. (527) By lines parallel to a side.................................... 34 (528) By lines perpendicular to a side................. 348 (529) By lines running in any given direction................. 348 (530) By lines starting from an angle............................... 349 (531) By lines starting from a point in a side.349 (535) By lines passing through a point within the triangle............. 351 (540) Do. the point being to be found....................... 53 (541) Do. the point to-be equidistant from the angles.............. 353 (542) By the shortest possible line................................... 354 Division of Rectangles. (543) By lines parallel to a side..................................... 354 424 CONTENTS. ABTICLB PAGE Division of Trapezoids. (544) By lines parallel to the bases.................. 355 (546) By lines starting from points in a side......3................... 355 (5417) Other cases................................. 356 Division of Quadrilaterals. (54S) By lines parallel to a side................................ 856 (549) By lines perpendicular to a side................................. 358 (550) By lines running in any given direction......................... 358 (551) By lines starting from an angle............................... 358 (552) By lines starting from points in a side............................. 358 (554) By lines passing through a point within the figure................. 859 Division of Polygons. (555) By lines running in any given direction........................... 360 (556) By lines starting from an angle................. 360 (5517) By lines starting from a point on a side....................... 361 (55S) By lines passing through a point within the figure................. 361 (559) Other Problems.3.............................. 6... 361 PART XII. UNITED STATES' PUBLIC LANDS. (560) General system.......... 36 Meandering............ 371 (561) Difficulty............... 64 (565) Marking lines............ 372 (562) Running township lines.... 366 (566) Marking corners.......... 372 (563) Running section lines 368 (567) Field-books.............. 76 (564) Exceptional methods...... 370 Township lines... 377 Water fronts.......... 370 Section lines........... 378 Geodetic method........ 3'71 Meandering............. 78 APPEN DIX. APPENDIX A. Synopsis of Plane Trigonometry. (1) Definition.................. 379 (1) Their mutual relations...... 383 (2) Angles and Arcs............ 379 (S) Two arcs......3...... 383 (3) Trigonometrical lines........ 380 (9) Double and half arcs....... 384 (4) The lines as ratios......3..... 81 (10) The Tables............... 384 (5) Their variations in length.... 381 (11) Right-angled triangles...... 85 (6) Their changes of sign....... 82 (12) Oblique-angled triangles.... 85 APPENDIX B. Demonstrations of Problems, &c. Theory of Transversals........... 387 Proofs of Problems in Part V,. 397 Harmonic division.............. 390 " " in Part VI..... 398 The Complete Quadrilateral...... 391 " in Part VII..... 399 Proofs of Problems in Part II., " in Part XI.... 401 Chapter V.................... 393 APPENDIX C. Introduction to Levelling. (1) The Principles............. 409 (3) The Practice............... 411 2) The Instruments............ 409 TRAVERSE TABLES: OR, LATITUDES AND DEPARTURES OF COURSES, CALCULATED TO THREE DECIMAL PLACES: FOR EACH QUARTER DEGREE OF BEARING. LATITUDES AND DEPARTURES. Lat. Dep. Lat. Dep. Lt. Dep. Lat. Dep. Lat. OC I 000 0O000 2.000 0*o000 3.ooo o.ooo 40ooo'o ooo 5.0ooo 90 i I. 0oo o.o4 2.000 0.009 3.ooo o.oi3 4.000 0.017 5.ooo 891 01 1o 0 -00 9 Ooo 2.000 o.OI7 3.000 0.026 4.ooo o.o35 5.ooo 891 o 1.000 o0.o3 2.000 0.026 3.000 o.o39 4.ooo 0.052 5.ooo 894 I Io000 0.017 2.000 o.o35 3.ooo o.o52 3.999 0.070 4.999 89~ i I.ooo.022 2.000 o.o44 2.999 o.o65 3.999 00o87 4-999 88&.oo000o o.o26 1.999 0.052 2.999 0.079 3.999 o.io5 4.998 884 I.oo000 o.o3I i.999 o.o6I 2.999 0.o92 3.998 0.122 4.998 884 20 o.999 o.o35 1-999 0.070 2.998 o.io5 3.998 o.I4o 4-997 8~~ 2 o0.999 o0o39 1.998 0.079 2.998 o.iI8 3.997 o.1.57 4.996 874 24 0.999 o.o44 1.998 o.o87 22.997 o.i3i 3.996 0.174 4.995 87i 2J 0'999 o.o48 1.998 0.096 2.997 o.i44 3.995 0.192 4'994 874 30 ~0999 o0.52 I 997 o.io5 2.996 o.157 3.995 0.209 4.993 8 0 34 o0998 o..057 1.997 o. I 3 2.995 o.I70 3.994 0.227 4-992 864 34 0o998 o.o6I 1.996 0.122 2.994 o.i83 3.993 0-244 4-99I 864 3i'0o998 o.o65 1.996 o.i3I 2.994 o0I96 3.99I 0.262 4.989 864 40 0.998 0.070 I-995 o.4o 2.993 o.209 3.990 0.279 4 988 86~ 4i 0.997 0.074 I1995 o.I48 2.992 0.222 3.989 o0.296 4.986 854 44 0997 oo78 I994 o-.57 2 991 0.235 3.988 o-3i4 4.985 854 4i 0.997 o.o83 I.993 o. 66 2.990 0.248 3.986 o.33I 4.983 854 5\ 0.996 o0.87 1.992.174 2.989 0.261 3.985 0.349 4.981 85~ 5 J 0996 0.092 1992 o.I83 2.987 0.275 3.983 0.366 4-979 844 5j 0.995 0.096 -199 o0. 92 2.986 0.288 3-982 0.383 4-977 844 54 0.995 o.-oo3 1.990 0.200 2.985 o.3oi 3-980 o.4oi 4-975 84i 6~ 0.995 o.0b 1.989 0.209 2.984 o-3i4 3-978 o-4i8 4.973 ~40 6 o0.994 0.109 I.988 0.218 2.982 0.327 3-976 0o435 4 970 836J 0o994 o. I3 1.987 o.226 2.981 o.34o 3.974 0.453 4-968 834 64 0.993 o.-I8 1.986 0.235 2.979 6.353 3-972 0-470 4-965 834 70 o.993 0.I22 1.985 0.244 2.978 0o366 3-970 0.487 4-963 83~ 74 0.992 0.126 1.984 0.2.52 2.976 0.379 3-968 o.505 4.960 824 74 0.991 o.I3I 1.983 o.261 2.974 0o392 3-966 0.522 41957 824 7i o.991 o.i35 1.982 0.270 2.973 o.4o5 3-963 0.539 4-954 824 o o.99g o0.39 1.981 0.278 2.97.1 o.4i8 3-961 0.557 4.951 82~ 8J o0990 o.i43 1.979 0.287 2-969 o-430 3-959 o0574 4.948 8i4 8j 0.989 o.i48 1.978 0.296 2-967 0-443 3-956 o0591 4-945 81i 8J 0.988 o.I52 I.977 0.3o4 2.965 0o456 3-953 o.608 4-942 814 9 0.988 o.I56 i 975 o.3i3 2.963 0.469 3-95i 0.626 4-938 81~ 9i 0.987 o.i6I 1.974 o.321 2,96I 0-482 3.948 o.643 4-933 8o0 94 o0986 o.i65 1.973 o.33o 2.959 0.495 3-945 o.66o 4-93I 80o'94 0.986.1I69 1'971 0.339 2.957 o.508 3.942 o-677 4-928.8o4 100 09-85 o.174'-970 0.347 2.954 0.52i 3.939 0-695 4-924 8~0 ioJ 0.984 0.178 I.968 o.356 2.952 o.534 3.936 0712 4-920 794 io- 0.*983 0. 182 1.967 o.364 2.950 0.547 3.933 0.729 4-9I6 794 io 0.o982 o.I87 1965 0.373 2.947 o.56o 3-930 0-746 4-912 79i 11~ o.982 0.191 1.963 0.382 2.945 0.572 3.927 0.763 4.908 790 I1i o098I o0.95 1.962 o.39o 2.942 0.585 3-923 0-780 4-904 784 i o0.-980.1o99g 1960 0.399 2.940 o.598 3.920 0.797 4.90o 784 -11 0.979 o.204 I.958 0.407 2.937 o.6Ii 3-9I6 o.8i5 4-895 784 12~ 0.978 0.208 1.956 o.4i6 2.934 0.624 3.913 0-832 4-891 78g I224 0977 0.212 I.954 0.424 2.932 0.637 3-909 0o849 4.886 771 I2 0.976 0o.26 I.953 o.433 2.929 o0649 3-905 o-866 4.88I 774 I2i 0.975 0.221 1.951 o.44I 2.926 o0662. 3-901 0o883 4-877 774 130 0.974 0.225 1.949 o.450 2.923 0.675 3-897 0.900 4-872 770 I3i 0.973 0.229 I1947 0.458 2.920 0.688 3.894 0.917 4-867 764 i341 0.972 0.233 1.945 0.467 2.9I7 0.700 3-889 0.934 4.862 761 I31 0.97I 0.238 1.943 0.475 2.914 0.713 3.885 o.95I 4.857 764 14~0 0970 0.242 1.94I 0.484 2.9T1 0.726 3-88i 0.968 4 85I 760 i4i 0o969 0.246 1.938 0.492 2.908 0.738 3.877 0-985 4.846 754 i4 o 09681 o0250.936 o.5oi 2.904 0o75i 3.873 1I002 4-84I 754 i4i o.967 0.255 1.934 o.509 2.901 0.764 3-868 i.oi8 4-835 754 150 0.966 0.259. i.932 o.5i8 2.898 0.776 3.864 i-o35 4-83o0 75' Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. i - 1 i -'. _,_.__ o,- "F LATITUDES AND DEPARTURES. _5. 7 8 __ Dep. Lat. Dep. Lat. Dep. Lat. Pep. Lat. Dep. o o ooo 6o000 oooo 7.00 o.oo 8.000 o.ooo 9.000 o.ooo 90~ 0o 0022 6oo00 0.026 7.000 o.o3i 8.000 o.o35 9.000 0.039 894 o o.o44 6.ooo 0.052 7.000 0.061 8.ooo 0.070 9.000 0.079 894 o4 o.o65 5.999 0.079 6.999 0.092 7.999 o.io5 8.999 o.ii8 89~ 1 0.087 5.999 o. o5 6.999 0.122 7.999 o.i4o 8.999 o.i57 89~ 1 o0.o09 5.999 o.i3i 6.998 o.i53 7-998 o.175 8.998 cI96 884 I 1 o.i3I 5.998 o.I57 6.998 o.i83 7*997 0.209 8-997 o0236 884 o.i53 5.997 o.83 6997 7 0.2i4 7996 0.244 8.996 0*275 88+'o I0174 5.996 0.209 6.996 0.244 7'995 0.279 8.995 o.3i4 88~ 2i 0.196 5.995 0.236 6.995 0.275. 7.994 o.3i4 8.993 0o353 874 2 0o.2I8 5.994 0.262 6.993 o.305 7*992 0.349 8.99I 0.393 871 24 0.240 5.993 0.288 6.992 0.336 7.991 0o384 8.990 0.432 874 3o 0.262 5-992 0o3I4 6.990 o.366 7.989 0.4I9 8.988 0o471 85~ 34 0.283 5.990 0o340 6.989 0.397 7.987 0.454 8.986 o.5io 864 34 o.305 5.989 0o366 6.987 0.427 7-985 0~488 8.983 0.549 864 34 0.327 5.987 0.392 6.985 0.458 7.983 o0523 8.98I 0.589 864 40 0.349 5.985 o0419 6.983 o.488 7.'98I o0558 8.978 0.628 96i 44 o-371 5.984 0.445 6'98I 0.519 7-978 0*593 8.975 0o667 854 41 o.392 5.982 0-47 16.978 0.549 7'975 0o628 8.972 0.706 854 4 0o4414 5.979 0'497 6.976 o.58o 7'973 0.662 8-969 0.745 854 50 1.436 5.977 0o523 6.973 o.610 7-970 0.697 8.966 0.784 8,5 51 0o458 5.975 0.549 6.971 o.64i 7.966 0.732 8.962 0.824 844 51 0-479 5.972 0.575 6.968 0.671 7.963 0.767 8.959 o0863 844 54 o.-5o 5.970 o.6oi 6.965 0.701 7.960 0.802 8.955 0.902 844 0 o.523 5.967 0.627 6.962 0.732 7.956 0.836 8.95I 0o941 840 64 o-544 5.964 o.653 6.958 0.762 7.952 0*87I 8.947 0.980 834 64 o0566 5.961 o0679 6.955 0.792 7'949 0~906 8'942 Io019 834 6 o-588 5.958 o.705 6.951 0.823 7.945 0.940 8.938 i.o58 834 y0 o060o 5-955 o073I 6.948 o.853 7'940 0.975 8.933.og97 853 7i o-63i 5.952 0.757 6.944 o.883 7.936 I*OIO 8.928 I.iI36 824 7' o0.653 5-949 0.783 6.940 0.914 7.932 I'o44 8. 923 I.175 824,7 0 o674 5.945 o0809 6.936 0.944 7'927 Io079 8'918 1.214 8,24 S~ o0696 5-942 o.835 6.932 0.974 7.922 1'113 8-912' 1.253 82~ 84 o07I7 5.938 o.86i 6-928 i.oo4 7-917 I*I48 8-907 1.29I 81i 84 0.739 5.934 o0887 6.923 i.o35 791I2 II.82 8*90I I 33o 814 81 0.761 5.930 o.913 6.919 I*.o65 7.907 I*2I7 8-895 1.369 8i~ 90 0.782 5.926 o.939 6.914 1-o95 7.902 I 25I 8-889 i-4o8 81~ 9i o.8o4 5-922 0.964 6.909 I. 125 7.896 I.286 8-883 I.447 804 91 0.825 5.918 o0990 6.904 i-i55 7.890 I*320 8-877 I.485 80o 91 o0847 5.9I3 i.oi6 6.899 I.i85 7.884 I.355 8-870 1.524 80o 10~0 o.868 5.909 1.o42 6.894 1.216 7.878 I*389 8.863 1.563 80~ io o 0.890 5.904 i.o68 6.888 1.I246 7.872.424 8.856 i.6oi 794 io o0.9r1 5-900 1.093 |6.883 1.276 7.866 1.458 8.849 i.640 79o io ~0.933 5.895 I.ii9 6.877 1i3o6 7.860 1.492 8.842 1.679 794 11 o0.954 5.890 I.i45 6.871,-336 7.853 1.526 8.835 1.7I7 79~ I1i 0.975 5.885 I.I71 6.866 1.366 7.846 i.56i 8.827 1.756 784 il4 0.997 5.880 I1I96 6.859 1.396 7*839 I1595 8-819 1 794 784 Iji i.o8 5.874 1-222 6.853 1.425 7.832 1.629 8.8xI I.833 784 12~ |i.o4o 5.869 1-247 6.847 I-455 7.825 I.663 8.8o3 I.871 708 I24 i-o6i 5.863 1.273 6.84i I.485 7.8I8 1.697 8.795 1.910 774 12 i1.082 5.858 1.299 6.834 i.5I5 7.81o 1.732 8.787 1.948 774 I24 I-io3 5.852 1.324 6 6.827 I-545 7-8o3 1*766 8.778 1.986 77i 130 11 II25 5.846 i.350 6.82I 1.575 7-795 800oo 8.769 20.25 770 i34 IxI46 5.840 1.375 6.8i4 i.6o4 7'787 1.834 8-760 2.063 764 i3 | 1I167 5.834 -.40o 6.807 1-634 7'779 I.868 8.75I 2.I01 761 i3|I.188 5.828 I.426 6.799 I.664 7-77I 1.902 8.742 2.I39 764 14~i I2IO 5.822 1452 6.792 I. 693 7'762.935 8-733 2.177 76 I44 I.23| 5.8i5 I'477 6-785 1723 7-754 1.969 8-723 2.2I5 754 i4 1 252 5. 809 I.502 6.777 1-753 7-745 2o003 8-713 2.253 754 i44 1273 5.802 1.528 6.769 1.782 7-736 2.037 8'703 2.29I 754 150 I'.294 5.:796 1.553 6.76I I.8I2 7-727 2.071 8-693 2.329 75~ p Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. I c5i (3i.70 0 8M 9 3 _; _. _i: LATITUDES AND DEPARTURES., I I, -g. I[4 |[Q ^ es - --- -. 2 _ 8 __ _ Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. a. 15 0.966 o.259 1.932 o.5i8 2.898 08.776 3.864.o35 4.83o 75o i5i 0.965 0.263 1.930 0.526 2.894 0.789 3.859 I-o52 4.824 744 i5 0.964 o.267 I 927 o 534 2.891 o.802 3.855.0o69 4-8T8 744 15i o.962 0.27I I 925 o.543 2.887 o.8i4 3.85o I.o86 4.8I2 744 160 o.961 0.276 1.923 o.55I 2.884 0.827 3.845 I.io3 4.806- 740 i64 0.960 0.280.920 z o.560 2.880 0.839 3.840 IrI.9 4.800 734 I6j o0.959 o0284 I.918 0.568 2.876 0o852 3.835 i.i36 4-794 73 i6i 0.958 o.288 I.915 0.576 2.873 0.865 3-830 I.i53 4.788 734 170 0.956 0.292 I.913 0.585 2.869 0.877 3-825 1.169 4.782 730 17 o0.955 0.297 1-90o 0.593 2.865 0.890 3.820 I.I86 4-775 724 i7 o0.954 o.3oI 1907 o.6oI 2.861 o0902 3-8I5 I,.2o3 4.769 72 I74 o.952 o-3o5 Igo95 o.6io 2.857 og915 3.8Io 1.220 4.762 724 18~ o.95I 0o309 I.902 o.6i8 2.853 0.927 3.804 1-236 4.755 720 i84 0.950 o-3i3 1.899 0.626 2.849 0.939 3.799 I*253 4.748 714 i8 qo.948 o03I7 1.897 o0635 2.845 0-952 3.793 1-269 4-742 71I i84 0.947 0o32I 1.894 o0643 2.84I o 964 3.788 1*286 4.735 71i 190 o.946 0.326 1.89I o.65I 2.837 0.977 3.782 I.302 4-728 710 19 o0.944 o0330 i.888 0o659 2.832 0o989 3.776 i'3I9 4.720 7o0 194 0o943 o-334 i1885 o.668 2.828 I ooI 3.77I 1.335 4-7I3 704 I94 o.941 0o338 1.882 0.676 2.824 I-OI4 3.765 1.352 4.706 704 200 0~940 0.342 1-8,79 0.684 2.8I9 1 026 3.759 I-368 4.698 700 20o 0.938 o.346 1.876 0.692 2.8i5 I.o38 3.753 I.384 4.691 691 206 0.937 o.35o 1.873 0.700 2.8io i0o5i 3.747 I-4o0 4.683 694 204 0-935 o.354 I.870 0.709 2.805 i.o63 3.741 I.4I7 4.676 694 210 0.934 o0358 I.867 0.717 2.80I 1.075 3.734 I.433 4.668 69~ 21i 0.932 o0.362 z.864 0.725 2.796 I.o87 3.728 i-45o 4.660 684 2I1 0.930o 0.367 i.86i 0.733 2.79I I.IOO 3.722 I.466 4.652 684 21i 0.929 0.37 1.858 0.74I 2.786.iI1a 3.7I5 1-482 4.644 684 220 0-927 0.375 z.854 0.749 2-782 1. 24 3.709 1-498 4.636 68~ 224 0.926 0.37-9 i.85i 0757 2.777 i.i36 3.702 i.5i5 4.628 674 22 0 0-924 0.383 i.848 0.765 2 772 i.i48 3.696 I 53 4.6I9 674 224 0-922 o0387 I.844 0.773 2.767 i.i6o 3.689 I-547 4-'6I 674 23 o.9g21 o.391 I.84i 0.781 2.762 I.I72 3.682 i 563 4.603 670 23I 0.919 0.395 I.838 0.789 2.756 i.i84 3.675 1I579 4.594 664 23 0 o91I7 0.399 x.834 0.797 2.75I I.96 3.668 I.595 4-585 66j 234 o0.95 o.403 i.83 o0.805 2.746 z20o8 3.66i I.6rI 4-577 66t 40~ o0.94 0.407 1.827 o.8i3 2.741 1.-220 3 654 1.627 4 568 660 24 0o9I2 o0.4Ir 1-824 0.821 2-735 I1232 3.647 I'643 4.559 654 244 og910 o.45 1.820 0.829 2-730 1244 3.64o 1-659 4.550 65J 244 0.908 o.4i9.-8i6 0.837 2.724 I-256 3.633 I1675 4-54I 654 25 o0.906 0.423 I.8i3 0.845 2.7I9 1-268 3.625 1.690 4.532 650 25- o0.94 0.427 1.809 o.853 2.7i3 I1280 3.6i8 1.706 4.522 64i 254 o.9o3 o.43I i.8o5 o.86i 2.708 1.292 3.6io I.722 4-5i3 64J 25 o.o90 0o.434 i.8oi 0.869 2.702 i.3o3 3.6o3 1.738 4.5o3 64i 260.0899 0.438 1.798 o.877 2.696 I.3I5 3.595 I.753 4.494 640 261 0.897 0.442 1.794 o.885 2.691 1327 3-587 1.769 4-484 634 26 o0.895 o.446 1.790 0.892 2.685 1.339 3.58o 1.785 4-475 634 264 o.893 o.45o 1.786 0.900o 2.67,9 135e 3.572 i.8oo 4.465 634 27 o0.891 o.454 1.782.9go8 2.673 -.362 3.564 i-8i6 4.455 630 27i. 0.889 o.458 1.778 0o.96 2.667 I 374 3.556 i.83I 4.445 624 274 o 0887 0.462 1.774 0.923 2.66I I.385 3.548 1.847 4 435 624 27 o0.885 0.466 1.770 o.93I 2.655 1.397 3.54o 1.862 4.425 624 280 o.883 0-469 1.766 0.939 2.649 I-408 3.532 1.878 4-4i5 620 284 o.88i 0.473 1.762 0.947 2.643 i.420 3.524 1-893 4-404 6i| 28 o0.879 0.477 I1-758 o0954 2.636 I.43I 35 5 1.909 4.394 6i, 28 o 0877 o 48I I.7753 0.962 2.630 I.443 3.507 1.924 4 384 6i4 290 0.875 o.485 1.749 0.970 2.624 I-454 3.498 I 939 4.373 610 294 0.872 0.489 I 745 0.977 2.617 I.466 3.490 I 954 4.362 6o0 29 o0.870 0 492 I-74L o985 2.6II I 477 3'-48i I.970 4.352 6o0 29. o0868 0.496 I.736' b.992 2.605 1.489 3.473 i-985 4.34I 60o 300 o.866 o.5oo0 -.732 I 0ooo 2.598 i.5oo 3 464 22.000 4-33 600 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. b:' 1 _! _a_. 46_ _' 3m0 --...... I''.- -...... LATITUDES AND DEPARTURES. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep.. 150 I 294 5-796 I.553 6.76I I.8I2 7.727 2.07I 8.693 2.329 750 i54 I.3 15 5.789 1.578 6.754 r.84I 7.718 2.io4 8.683 2.367 744 i54 i.336 5.782 i.6o3 6.745 1.871 7.709 2.I38 8.673 2.405 744 154 I1357 5.775 1.629 6.737 1.900 7.700 2.172 8.662 2.443 744 16~ 1.r378 5.768 I.654 6.729 1.929 7.690 2.205 8.65i 2.48I 740 I64 1.399 5.760 I.679 6.720 1.959 7.680 2.239 8 64o 2.5i8 734 i6. 1.420 5.753 I.704 6.712 1.988 7.67I 2.272 8.629 2.556 73A i64 I.44r 5.745 1.729 6.703 2.017 7.661 2.3o6 8.6i8 2.594 734 170 I.462 5.738 I.754 6.694 2.047 7.650 2.339 8.607 2.63I 73~ 174 I.483 5.730 I'779 6.685 2.076 7.64o 2.372 8.595 2.669 724 174 I.5o4 5.722 I.8o4 6.676 2.io5 7.630 2.406 8.583 2.706 724 17i I.524 5.7i4.I829 6.667 2.I34 7.6i9 2.439 8.572 2.744 724 18~ 1.545 5-706 r.854 6.657 2.i63 7.608 2.472 8.560 2.78I 172 i84 i.566 5.698 1.879 6.648 2.192 7.598 2.505 8.547 2.8i8 71i i8 1 I.587 5-690 1.9o4 6.638 2.221 7.587 2.538 8.535 2.856 71A i84 1.607 5.682 I.929 6.629 2.250 7.575 2.572 8.522 2.893 71i 19~ 1.628 5.673 I-953 6.6I9 2.279 7.564 2.605 8-5Io 2.930 7-1~ 9i4 i.648 5.665 I.978 6.609 2.308 7.553 2.638 8.497 2.'967 704 194 1.669 5.656 2-003 6.598 2.337 7.54i 2.670 8.484 3.00oo4 70 I94 1.690 5.647 2.028 6.588 2.365 7.529 2.703 8.471 3.o41 704 200 I-7Io 5.638 2.052 6.578 2.394 7-.58 2.736 8.457 3.078 70~ 20o 1.73i 5.629 2.077 6.567 2.423 7.506 2.769 8.444 3.II5 694 209 I.75I 5.620o 2.10I 6.557 2.451 7.493 2.802 8.43o 3.I52 694 204o 77I 5-6i 2.126 6.546 2.480 7.481 2.834 8.4I6 3.189 694 210 1-792 5.6oi 2.150 6.535 2.509 7.469 2,867 8.402 3.225 69~ 214 i.8i2 5.592 2.I75 6.524 2.537 7456 2.9oo 8388 3.262 684 2I1 I.833 5.582 2.199 6.5i3 2.566 7.443 2.932 8.374 3.299 684 21r i 853 5.573 2.223 6.5o2 2.594 7.430 2.964 8.359 3.335 684 220 I.873 5.563 2.248 6.490 2.622 7.4I7 2.997 8.345 3.37I 68~ 224 1.893 5.553 2.272 6.479 2.65I 7.404 3o.29 8.33o 3.4o8'674 22 I1.913 5.543 2.296 6.467 2.679 7.391 3.o6i 8.3i5 3.444 674 229 I.934 5.533 2.320 6.455 2.707 7.378 3.094 8.300 3.480 674 2,3o 1.954 5.523 2.344 6.444 2.735 7.364 3I.26 8.285 3.5I7 67~ 234 I.974 5.5I3 2.368 6.432 2.763 7.350 3.i58 8.269 3.553 664 23'1-994 5.502 2.392 6.4I9 2.791 7.336 3.190 8.254 3.589 664 234 2.014 5.492 2.4I6 6.407 2.8I9 7.322 3.222 8.238 3.625 664 240 2.o34 5.48i 2.440 6.395 2.847 7.308 3.254 8.222 3.66i 66~ 24 2.054 5.471 2.464 6.382 2.875 7.294 3-286 8.206 3.696 654 244 2.073 5.460 2.488 6.370 2.903 7.280 3-3i8 8.190 3-732 654 244 2.0o93 5.449 2.5i2 6.357 2.931 7.265 3-349 8.173 3.768 654 250 2.II3 5.438 2.536 6.344 2.958 7.250 3.38i 8I.57 3.804 650 251 2.133 5.427 2.559 6.331 29986 7.236 3.4I3 8.i4o 3.839 644 254 2.I53 5.4i6 2.583 6.3I8 3.oi4 7*22i 3.444 8.I23 3.875 644 254 2I.72 5.404 2.607 6.305 30o4I 7.206 3.476 8.io6 3.910 644 260 2.192 5.393 2.630 6.292 3.069 7I.90 3.507 8.089 3.945 640 264 2.2II 5.38i 2.654 6-278 3.096 7.175 3.538 8.072 3.981 634 264 2.23i 5.370 2.677 6.265 3i.23 7I.60 3.570 8.o54 4.oi6 63j 26 2.250 5 5.358 2.701 6.25I 3.i5i 7-144 3-60o 8.037 4o05i 634 2'~ 2 2.270 5.346 2.724 6.237 3.178 7.I28 3.632 8.019 4o.86 63~ 2741 2.289 5.334 2.747 6.223 3.205 7II12 3.663 8.ooI 4121z 624 271 2.309 5.322 2.770 6.209 3.232 7.096 3.694 7.983 4.-56 624 27 2.328 5-3Io 2.794 6-I95 3.259 7.o80 3.725 7.965 4.90o 624 280 2.347 5-298 2.817 6-i8i 3.286 7-o64 3.756 7'947 4-225 620 284 2.367 5.285 2.840 6.I66 3.3i3 7'047 3-787 7.928 4.260 6i' 284 2.386 5-273 2.863 6.I52 3.34o 7.o3I 3.817| 7.909 4-294 6i4 284 2.405 5-260 2.886 6.i37 3.367 7o014 3.848 7.891 4.329 64I 29~ 2.424 5.248 2.909 6.I22 3.394 6.997 3.878 7-872 4.363 6 1 294 2.443 5.235 2.932 6.107 3.420 6.980 3.909 7.852 4.398 604 294 2.462 5 5.222 2.955 6.093 3.447 6.963 3.939 7.833 4-432 60o 29- 2.48I 5.209 2.977 6.077 3.474 6.946 3.970 78 14 4.466 60o 30~0 2.50o 5I.96 3oo000 6o62 3.5oo 6.928 4-o00 7-794 4-500 60~ Mial Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. = _LATITUDES AND DEPARTURES. 11 1. 2 |. - _ _ Lat p. Lat. Dep. Lat. Dep. Lat. Dep. Lat.. 300 o.866 o5 oo 1.732 I 000 2.598 I.5oo 3.464 2oo00 4.330 60 30o o.864 o.504 1.728 i.oo8 2.592 I.5II1 3.4555 2.oI5 4.319 594 3o0 o.862 o.508 1.723 I.oi5 2.585 1.523 3.447 2.o30 4.-308 594 3o4 0.859 o.51 1.719 1-023 2.578 I.534 3.438 2;o45 4.297 594 31~ 0.857 o.5i5 1.714.o3o 2.572 1.545 3.429 2.o60 4.286 59~ 3I 0o.855 0.5I9g 1.710 I.o38 2.565 1.556 3.420 2.075 4.275 584 3i 0 o.853 0.522 1.705 I.o45 2.558 1.567 3.41I 2.090 4.263 584 3ii o085o 0.526 1.701 i.o52 2.55i 1-579 3.4oi 2o105 4.252 584 320 o.848 o.530 1.696 i.o6o 2.544 1.-590 3.392 2.120 4.240 5S8 324 0.846 0.534 1.69I I1067 2.537 T-6oi 3.383 2.I34 4-229 574 32 o0.843 0.537 1.687 1075 2.530 1.6I2 3.374 2.149 4-217 571 324 o084i 0o541 I-682 1.082 2.523 1.623 3.364 2.I64 4.205 574 330 0.839 0.545 1.677 1-089 2.5I6 I.634 3.355 2.179 4-193 570 334 o.836 o0548 1.673 1.o97 2.509 ir645 3.345 2.193 4-i8i 561 334 0o.834 0.552 1.668 i.-o4 2.502 I 656 3.336 2.208 4-I69 56 33 o0.83I 0o556 1.663 II.I1 2.494 1.667 3.326 2.222 4.-57 564 34 o0.829 0.559 I.658, i.i I8 2.487 1.678 3.3i6 2.237 4-.45 560 341 0.827 0.563 1.653 1.126 2.480 1-688 3.306 2.251 4-I33 554 344 0.824 0-566 1.648 I I33 2.472 1699 3.297 2*266 4.121 554 34' 0.o822 0.570 1.643 i.I4o 2.465 1.7Io 3.287 2.280 4.Io8 554 350 o.819 0.574 1-638 1.I47 2.457 1-721 3.277 2.294 4-096 5,5 351 o.817 0.577 I1633 i.i54 2.450 1.73i 3.267 2.309 4-o83 544 354 o.8i4 o.58I 1.628 1.i6i 2.442 1I742 3.257 2.323 4-071 544 354 o.8I2 0.584 1.623 i.i68 2.435 1.753 3.246 2.337 4-o58 54i 360 o.809 0.588 I.6I8 1.176 2.427 1.763 3.236 2.351 4-o45 540 364 o0806 0.59i I.6i3 i. 83 2.419 1.774 3.226 2.365 4.032 534 364 o.804 0.595 1.608 1.I90 2.412 1.784 3.2i5 2.379 4.019 534 36 o-8oi 0.598 i.6o3 II97 2.404 1.795 3.205 2.393 4-oo6 534 370 0-799 0.602 1.597 I.204 2.396 I 805 3I.95 2.407 3.993 53 37i 0-796 o.605 1.592 I.211 2.388 i1816 3.i84 2.421 3.980 524 37 o0 793 o0609 1.587 1.218 2.380 1.826 3.I73 2.435 3.967 52437i 0;791 o.6I2 i.58i 1.224 2.372 I-837 3.i63 2.449 3.953 524 38~ 0-788 o-6i6 1.576 1.231 2.364 I-847 3.152 2.463 3.940 52~ 384 0-785 0.6Ig 1.57I 1.238 2.356 I 857 3.i/I 2.476 3.927 5i1 38 0-783 0.623.565 1.245 2.348 I 868 3-i3o 2.490 3.9I3 5Ij 384 0780 0.626 1.56o 1.252 2.340 1878 3.120 2504 3.899 51i 390 -0777 0.629 I.554 1.259 2.33i I-888 3.109 2.5I7 3.886 51~ 39 o 0774 o 633 I.549 1-265 2.323 1I898 3.098 2.53I 3.872 5o0 394 0-772 0.636 I 543 1.272 2.3I5 1908 3.o86 2.544 3.858 50o 394 0-769 0.639 1.538 1.279 2.307 I 918 3.075 2.558 3-844 5o0 40~ 0.766 0.643 1.532 1.286 2.298 1.928 3.o64 2.571 3.83o 500 40 o 0.763 0.646 1.526 1.292 2.290 1.938 3.o53 2.584 3.8I6 494 4o0 0.760 0.649 1.521 1.299 2.281 1.948 3.o42 2.598 3.802 494 40o 0 758 0.653 I.5i5 i-306 2.273 1.958 3.o30 2.611 3.788 494 410 0.755 0.656 1.509 -.312 2.264 1.968 3.0I9 2.624 3.774 490 41 0.752 0.659 i.504 1.319'2 256 1 978 3.007 2.637 3.759 481 -4 o 0749 o.663 1.498 1.325 2.247 1.988 2.996 2.650 3.745 484 4Ii o0746 0.666 1[492 I-332 2.238 I.998 2.984 2.664 3-730 /48 42~ 0743 0.669 [.486 I-338 2.229 2.oo7 2.973 2.677 3.716 480 424 0o740 0o672 i.48o I.345 2-221 2.017 2.961 2.689 3.70I 471 429 0-737 0.676 1I475 i.35i 221I2 2.027 2.949 2.702 3.686 474 424 0-734 0.679 1.469 I-358 2.203 2.036 2.937 2.715 3.672 47~ 43~ 0o73I 0.682 1.463 Ir364 2.194 2.046 2.925 2.728 3.657 470 43 o0.728 0.685 1.457 1.370 2. 85 2.o56 2g913 2.741 3.642 464 434 0.725 o0688 1.45I 1'377 2.176 2.o65 2.9 0 2.753 3.627 464 4304 0722 0.692 1-445 [r383 2.167 2.075 2.889 2. 766 3.612 464 440 o 719 0.695 1.439 I1.389 2.158 2.084 2.877 2.779 3.597 460 444 o07I6 0.698 I.433 1.396 2149 2.093 2.865 2.79 3582 45 444 0o7I3 0-701 1.427 I.402 2 I40 2.I03 2.853 2.8o4 3.566 451 444 0.710 0-704. 1420 I.4o8 2.I3I 2.II2 2.841 2.816 3.55I 454 450 0.707 0.707 I.414 1.414 2.12I 2.I2I 2.828 2.828 3-536 450 1 Dep. Lat epL at. Dep. Lat. Dep. Lat. Dep. t 1 2 3 4 LATITUDES AND DEPARTURES. c (Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. p 30~ 2.500 5.196 3.ooo 6.062 3.5oo 6928 4.o 794 4o o 600 3o' 2..519 5.183 3-023 6.047 3.526 6.911 4.030 7.775 4.534 594 3o 2.2538 5.170 3.o45 6.o3i 3.553 6.893 4.0o6 7.755 4.568 594 3oi 2.556 5-i56 3.o68 6.oi6 3.579 6.875 4o090 7.735 4.602 594 310 2.575 5.i43 3.090 6.000 3.6o5 6.857 4-120 7.715 4.635 590 3ii 2.594 5.129 3.ii3 5.984 3.63I 6.839 4-i5o 7.694 4.669 584 31i 2.612 5.ii6 3.I35 5./968 3.657 6.821 4-i8o 7.674 4-702 584 3i4 2.63i 5.102 3'i57 5.952 3.683 6.8o3 4.210 7.653 4.736 584 320 2.65o 5-o88 3.i8o 5-936 3.709 6.784 4-239 7.632 4.769 580 324 2.668 5.074 3.202 5.920 3.735 6.766 4.269 7.612 4.802 574 324 2.686 5,o6o 3.224 5.90o4 3761 6.747 4-298 7.591 4.836 571 32 2.705 5.o46 3.246 5.887 3.787 6.728 4.328 7.569 4.869 574 330.2.723 5o032 3.268 5.87I 3.8I2 6.709 4.357 7.548 4902o.570 334 2.741 5-oi8 3.290 5.854 3.838 6.690 4-386 7.527 4-935 561 334 2.760 5-oo3 3.3i2.5.837 3.864 6.671 4-4i6 7.505 4-967 564 334 2.778 4.989 3.333 5.820 3.889 6.652 4-445 7.483 5'ooo 56~ 34' 2.796 4-974 3-355 5-8o3 3-914 6.632 4-474 7-461 5-o33 560 344 2.814 4'960 3-377 5-786 3-940 6.6i3 4.502 7.439, 5.o65 551 344 2.832 4-945 3-398 5-769 3.965 6.593 4.53r 7-417- 5-098 554 344 2.85o0 4930 3-420 5.752 3.990 6.573 4-560 7-395 5.3o0 55i 350 2.868 4-915 3.44I 5.734 4.oi5-! 6.553 4.589 7.372 5.162- 550 35i 2.886 4-900 3.463 5.716 4-o4o 6.533 4.617 7-35o 5.194 541 35 2.90o4 4.885 3.484 5.699 4.o65 6.5i3 4-646 7.327 5.226 541 35 2. 92I 4.869 3.5o5 5.68i 4.090o 6.493 4.674 7-304 5.258 544 360 2.939 4.854 3.527 5.663.4-ii5 6.472 4-702 7-281 5.290 540 364 2.957 4-839 3.548 5.645 4.-39 6.452 4.73o 7.258 5.322 534 364 2.974 4.823 3.569 5.627 4.i64 6.43i 4-759 7-235 5.353 534 361 2.992 4-8o8 3.590 5.609 4-i88 6.4io 4-787 7-2ii 5.385 534 370 3.009 4-792 3-6ii 5.590 4.-23 6.389 4.8x5 7.188 5.4i6,530 37i 3.026 4-776 3.632 5.572 4.237 6.368 4.842 7-i64 5.448 524 374 3.o44 4-760 3.653 5'554 4-261 6.347 4.870 7.40o 5.479 524 374 3.o6i 4,744 3.673 5.535 4-286 6.326 4,898 7.-16 5.5io 524 38 3.078 4-728 3.694 5-5i6 4.31o 6.3o4 4-925 7-092 5.54i 52~ 384 3.095 471r2 3.7i5 5-497 4.334' 6.283 4.953 7-o68 5.572 5i4 381 3.ii3 4-696 3.735 5.478 4.358 6.261 4.980 7'o43 5.6o3 5i4 38 3.i3o 4679 3.756 5.459 4.38i 6.239 5.007 7.o019 5.633 5i4 390 3.147 4.-663 3.776 5.440 44o5 6.217 5.o35 6.994 5.664 510 394 3.i64 4.646 3.796 5.421 4-429 6.x95 5.062 6.970 5.694 5o| 394 3.i8o 4.63o 3-8i6 5-4or 4.453 6.173 5.089 6.945 5.725 50o 394 3.197 4.6i3 3.837 5.382 4.476 6.i5i 5.i6 6.920 5.755 5ot 400 3.214 4.596 3.857 5.362 4.5oo 6.128 5.I42 6.894 5.785 500 4o4 3.23i 4.579 3.877 5.343 4.523 6.io6 5.169 6.869 5.8i5'491 4o4 3.247 4.562 3.897 5.323 4.546 6.o83 5.196 6.844 5.845 494 40o 3.264 4.545 3.917 5.3o3 4.569 6.o6i 5.222 6.8i8 5.875 494 410 3.28o 4.528 3.936 5.283 4.592 6.o38 5.248 6.792 5.9o5 490 414 3.297 4.5TI 3.956 5.263 4.6i5 6.oi5 5.275 6.767 5.934 481 41i 3.3i3 4-494 3.976 5.243 4.638 5.992 5.3oi 6.74i 5.964 484 4ii 3.329 4-476 3.995 5.222 4.66i 5.968 5.327 6.7i5 5.993 484 421 3.346i 4.459 4.oi5 5.202 4.684 5.945 5.353 6.688 6.022 480 424 3..~362 4.44i 4.o34 5.182 4.707 5.922 5.379 6.662 6.o5i 474 424 3.378 4-424 4-o54 5.i6i 4-729 5.898 5.4o5 6.635 6.o8o 474 424 3.394 4.4o6 4.073 5.i4o 40752 5.875 5.43o 6.609 6.109 474 430 3.4io 4.388 4.092 5.119 4.774 5.85i 5.456 6.582 6.x38 470 434 3.426 4.370 4.I11 5.099 4.796 5.827 5.48i 6.555 6.167 464 434 3.442 4.352 4-i3o 5.078 4.8i8 5.8o3 5.507 6.528 6.195 464 434 3.458 4.334 4.149 5.057 4.84i 5.779 5.532 6.5oi 6.22.4 464 440 3.473 4-3i6 4-i68 5.o35 4-863 5.755 5.557 6.474 6.252 460 44i 3.489 4.298 4.-87 5-oi4 4.885 5.73o 5.582 6.447 6.280 454 444 3.5o5 4-28o 4.2o6 4.993 4.906 5.706 5.607 6.419 6.3o8 454 444 3.520 4.261 4.224 4'971 4.928 5.68i 5.632 6.392 6.336 454 450 3.536 4-243 4-243 4.950 4.950 5.657 5.657 6..364 6.364 450 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. M' SQ w?0 Q. Jl~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~cm 93~~~~ TABLE OF CHORDS: [RADIUS=.1.0000]. M. 0 10 20 30 40~ 5 6~ O 70 s 90 10~ M. o'. 0oooo.I75.o349 *o524.0698.0872 ~io47 1I22I.I395.1569 ~i743 o' r.ooo3.0177.o352.o526.0701.o875 Io5o I224 1398.1572 1746 I 2.ooo6.oi80.o355.0529.0704.0878 Io53.1227.I4oI.I575'I749 2 3 ooo9.oI83.o358.532.0707.088I I055.I230.i4o4.1578.1752 3 4.ooI2.o086.o36i.o535.0710.0884.io58.I233.1407.i58i.1755 4 5.ooi5.0189.o364.o538.0713.0887.o06I.1235.I4io.i584 *.758 5 6.0017 0192.o366 o54i.0715.0890go 64.I238.i4i3,I587 1761 6 7 oo0020.195.0369.o544.0718.0893 1067.I241.I4i5.1589 I.763 7 8.0023.198.0372 *o547.072I.o896.1070.1244.i4i8 I.592 I766 8 9 o0026 o0201.o375.o550.0724.0899.I073 1247.I42I I595 *I769 9 IO.0029.0204.0378.o553.0727.090I 1076.1250.1424.1598 1.772 IO 11.oo32.0207.o38i.o556.0730 0904.1079.1253.I427.i6oi.I775 II 12.oo35.0209.o384.o558.0733.0907.Io82.1256.i43o.i6o4.1778 I2 13.oo38.0212.0387.o56i.0736.9gIo.io84.1259.433.1607.1781 13 I4.oo4i.0o25.o3go.o564.o739.0913.0o87.1262.436.i6io.1784 I4 i5.oo44.0218.0393 o0567.0742 o0916.109o.I265 I.439.i6i3.1787 I5 6.00oo47.022I 0396.o570.0745 o.0g9.I093.1267.I442.I616.1789 i6 7.00oo49.0224.0398.o573.0747.0922.1o96.1270.I444.16I8 -I792 17 I8.oo52.0227.o4oi.0576.0750 *0925.1099.1273.I447.162I.I795 I8 19.oo55.0230.o404.0579.0753.0928.II02.I276.i45o.1624 I798 19 20.oo58.0233 0407.o582.0756 o.93I.io5.1279.i453.1627.i8oI 20 21.oo6.0236.o4io.o585.0759.0933.rio8.1282.i456 i63o.i8o4 21 22.oo64:o239.o4i3 o588.0762.0936.IIII.1285.1459.i633.1807 22 23.0067.024.o4i6.0590.0765.0939.II4.1288.1462.i636.181O 23 24 oo0070.0244.0419.0593.0768 o.942. Ii6 I.29I.i465. 639..i83 24 25.0073.0247.o422.0596.0771.0945.1119.1294.i468.1642.i8i6 25 26.0076.o250.0425.o599.0774.0948.1122.1296.I47I.I645.i8i8 26 27.0079.0253.0428.o602.0776.0951.i25 *T299.1473 I.647.1821 27 28.oo8i.o256.o43o o6o5.0779 o954.II28 3o02.1476.i65o I.824 28 29.oo84.o259.o433.o6o8.0782.0957.II3i.i3o5.479 *i653.1827 29 30.0087.0262.o436.o6ii.0785 o0960.i34.i3o8.1482.656 *i83o 30 31.0090.0265 o.439.o6i4.0788.0962.137.13I.i485.1659.i833 3i 32.0093.0268.o442.0617 -0791.0965.ii4o.i3i4.i488.1662 i836 32 33.o096.0271.o445.06I9.0794.0968.II43.i3I7.1491.i665.1839 33 34.0099.0273.o448.0622.0797 *097I II45.1320 I.494 *i668.1842 34 35.0102.0276.o45i.o625.o800.0974.*48.323.1497.167I.x845 35 36.oo05.0279 -o454.0628.o803.0977 *15i.I325.i5oo.1674.1847 36 37.oio8.0282.o457.o63i.o806.og80.154.I328.1502.1676.i85o 37 38.oI0I.0285.o46o.o634..o88.o983.1157.133i.i5o5.I679 *I853. 38 39.oii3.o288 o.462.o637 *o8ii.0986.Ix60o.334.i5o8.1682.i856 39 40.o16.029I.o465.o640.084.0989.ii63.1337.i51.i685.1859 4o 41.oI09.0294.o468.o643.0817 o 992.ii66.i34o.I5I4.I688.1862 4 42 OI22.0297 *o471.o646.0820.0994.1169.i343.i157.1691.i865 42 43.25.o3oo00.0474.o649.0823.0997.1172.i346.1520 I.694.I868 43 44.0128.o3o3 0477 *o65i.o826.iooo.II75.I349.I523.1697 1871 44 45.oi3i.o3o5.o48o.,o654 o.829.1oo 3.1177.i352.526.1700 I.873 45 46.oI34.o3o8.o483.0657.o832 1.oo6.180.i355.1529 I.703.1876 46 47.oI37 *o3I.o486.o660.o835.o009. I83.i357 i53i.1705 I1879 47 48.oi4o0 *.34.0489.o663.o838.112.ii86.i36o i534.1708.1882 48 49.oi43.o3I7.0492.o666.o84o.ioi5.1189.i363.i537.1711.i885 49 50.oI45 0o320.0494.0669.o843.1io8 I.192.i366.i54o.17I4.I888 50 51.oI48 *o323.0497.0672.o846.102I.i95.1369.i543.1717.I89g 5i 52 oi5i.0o326.o5oo.o675.o849.Io23.1198 I.372.i546.1720 I.894 52 53.oI54.o329.o503.0678.o852.1026.120I.i375.549.1723.1897 53 54.oi57.o332.o5o6 068i.o855.I029.2o4.1378.1552.1726 I.900 54 55.oi6o.o335 o5o9 *o683.o858.io32.1206.i38i.i555.1729.1902 55 56.oi63.0337.o5I2.o686.o86i.io35.209.i384.i558.1732.19o5 56 57.oi66.o34o.o5i5 o.689.o864.io38.2 386.i56o.1734.1908 57 58.o169.o343 o5i8 o0692.0867.io04i.I25.1389.i563.1737 g911t 58 59 *OI72 o346.o52I.o695 o0869.io44.1218.1392.i566.1740.1914 59 60o *o75 *o349.o524 0698.0872.1o47.I22.i1395.I569.1743 *I9I7 60 8 TABLE OF CHORDS: [RADIUS=1.0000]. M. 1~0 120 13~ 14~0 15~ I16 ~ 17~ 18~ 9~0 20~ 21~ M. o' 1917.2091.2264.2437.261I.2783.2956.3I29.33oi.3473.3645 o 1.1920.2093.2267.2440.2613.2786.2959.3132.33o4.3476.3648 I 2.1923.2096.2270.2443.2616.2789.2962.3i34.3307.3479.3650 2 3 I.926.2099.2273.2446.2619.2792.2965.3I37.33Io.3482.3653 3 4 I.928.2102.2276.2449.2622.2795.2968.3i4o.33I2.3484.3656 4 5 I.93I.2I05.2279.2452.2625.2798.297I 3143.33i5.3487.3659 5 6 I.934.2I08.228I.2455.2628.2801.2973.3i46.33i8.3490 3662 6 7 1937.21III 2284.2458.263i.2804 2976.3149 *332I.3493.3665 7 8.1940.2II4.2287.2460.2634,2807.2979.3152.3324.3496.3668 8 9 I1943.2117.2290.2463.2636.2809.2982.3I55.3327.3499.3670 9 I0.1946 *219 9 -2293.2466.639.92812.2985.3i57.3330.3502.3673 I0 1I.1949 *2122 *2296.2469.2642.28I5.2988.3i6o.3333.3504.3676 II I2.1952.2I25.2299 -2472.2645.2818.2991.3i63.3335.3507.3679 I2 13.1955.2I28.2302.2475.2648.282I.2994.3i66.3338.35io.3682 13 i4.1957.2I3I.23o5.2478.2651.2824.2996.31 69 334i.35i3.3685 i4 i5 I.960.2I34.2307.2481.2654.2827.2999.3172.3344.35i6.3688 15 16.1963.2I37.23o1.2484.657. 2830.3002.3175.3347.35i9.3690 i6 I7.1966.2I40.2313.2486.2660.2832.3oo5.3178.335o.3522.3693 I7 8.1969.2143.23r6.2489.2662.2835.3oo8.3r8o.3353.3525.3696 i8 19.I972.2I46 -23i9.2492.2665.2838.3oii 3r83.3355.3527.3699 19 20.1975.2148 -2322.2495.2668.284I.3oi4.3i86.3358.353o.3702 20 21 -I978.2i5I.2325.2498.2671.2844.30I7.3189.336i.3533.3705 21 22 *198I.2I54.2328.25oi.2674.2847.3019.3192.3364.3536.3708 22 23.1983.2I57.233I.2504.2677.285o.3022.3195.3367.3539.3710 23 24.1986.2I60o 2333.2507 2680.2853.3025.3198.3370.3542.37i3 24'25 1I989.2I63.2336 -25io.2683.2855.3028.3200.3373.3545.37I6 25 26.1992.2I66.2339.25I2.2685.2858.3o3i.3203.3376.3547.3719 26 27.1995.2I69.2342.25i5.2688.2861.3034.3206.3378.355o.3722 27 28.1998 -2172.2345.25I8.2691.2864.3037.3209.338T.3553.3725 28 29.2001.2I74.2348 *2521.2694.2867 3040.3212.3384.3556.3728 29 30.2004 21I77.235i.2524.2697.2870.3042.32I5.3387.3559.3730 30 31.2007 -298o.2354.2527.2700.2873.3045.32i8.3390.3562.3733 31 32.2010.2183.2357.2530.2703.2876.3048.3221.3393.3565.3736 32 33.2012.2i86.2359.2533.2706.2878.3o5i.3223.3396.3567.3739 33 34.20I5.2189.2362.2536.2709.2881.3o54.3226.3398.3570.3742 34 35.20I8.2192.2365.2538.2711.2884.3057.3229.34oi.3573.3745 35 36.202.2I195.2368.254I *2714.2887.3060.232.3404.3576.3748 36 37.2024.2198.2371.2544.2717.2890.3063.3235 3407.3579.375o 37 38.2027.2200.2374 *2547 2720.2893.3065.3238.34io0 3582.3753 38 39.2030o 2203.2377.2550.2723.2896.3068.3241.34i3.3585.3756 39 40.2033.2206.2380.2553.2726.2899.307i.3244.34i6.3587.3759 40 41.2036.2209.2383.2556.2729.2902.3074.3246.3419.3590.3762 4i 42.2038.2212.2385.2559.2732.2904.3077.3249.3421.3593.3765 42' 43-.204I.22I5.2388.256i.2734 2907 3080o.3252.3424.3596.3768 43 44.2044.2218.2391.2564.2737.29.10 3o83.3255.3427.3599.3770 44 45.2047.222.92394.2567.2740.2913.3086.3258.3430.3602.3773 45 46.2050.2224.2397.2570.2743.2916.3088.326I.3433.3605.3776 46 47.5..3 2226.2400.2573.2746.2919.3091.3264.3436.36o8.3779 47 48.2056.2229.24o3.2576 2749.2922.3094.3267.3439.36io.3782 48 49 20o59.2232.2406.2579 2752.2925.3097.3269.344i *36I3.3785 49 50.2062.2235.2409.2582.2755.2927.3Ioo.3272.3444.36i6.3788 5o 51.2065.2238.24II.2585.2758.2930.3io3.3275.3447 *36I9.3790 5I 52.2067.224i.2414.2587 2760.2933.3Io6.3278.345o.3622.3793 52 53.2070.2244.2417.2590.2763.2936.3o09.3281.3453.3625.3796 53 54.2073.2247.2420.2593.2766.2939.3 II.3284.3456.3628.3799 54 55.2(-6.2250.2423.2596.2769.2942.3II4.3287.3459.3630.3802 5 56.2o279.2253.2426.2599.2772.2945.3117.3289.3462.3633.38o5 56 57.2082.2255.2429.2602.2775.:2948.3I20.3292.3464.3636.38o8 57 58.2085.2258.2432.2605.2778.2950.3123.3295.3467.3639.38io 58 59.2088.2261.2434.2608.278I.2953.3126.3298.3470.3642.38i3 59 601.2091.2264.2437.2611.2783.2956.3I29.33oi.3473.3645.38i6 60 9 TABLE OF CHORDS: [RADIUS = 1.0000]. M. 220~ 23 ~ 24 25~ 260 27~ 28~ 29 300~ 31 320 M. o'.38i6.3987.4i58.4329.4499.4669.4838.5oo8.5 76.5345.-55i3 0o I.38I9.3990.4i6I.4332.4502.4672.484i.5oio.5179.5348.55i6 2.3822.3993.4i64.4334.45o5.4675.4844.5oi3.5i82.5350.55i8 2 3.3825.3996.4167.4337.45o8.4677.4847.5oi6.5i 85.5353.5521 3 4.3828.3999.4170.434o.45io.468o.485o.50I9.5i88.5356.5524 4 5.383o.4002.4172.4343.45I3.4683.4853.5022.5190.5359.5527 5 6.3833.4oo4 *4175.4346.45i6.4686.4855.5024.5I93.5362.553o 6 7.3836.4007.4I78.4349.45i9.4689.4858.5027.5196.5364.5532 7 8.3839.4oio.4i8i.4352.4522.4692.486i.5o3o.5199.5367.5535 8 9.3842 -4oi3 -4i84.4354.4525.4694 *4864.5o33.5202.5370.5538 9 io.3845.4oi6.4187.4357.4527.4697 -4867 -5o36.5204.5373.554I I0 11.3848.40o9 *4190 *436o.453o.4700.4869.5039.5207.5376.5543 II I2.3850.4022.492.4363.4533.4703.4872 504o4.52I0.5378.5546 12 13.3853.4024.4195.4366.4536.4706.4875.5044.52I3.538I.5549 13 I4.3856.4027.4198.4369.4539.4708.4878.5047.52I6.5384.5552 i4 I5.3859.4o3o.420I.4371.4542 -4711.488.5o50o.5219-.5387.5555 i5 I6.3862 -4o33.4204 -4374.4544.4714.4884.5o53.5221.5390.5557 i6 17.3865.4036.4207.4377.4547.4717.4886.5055.5224.5392.5560 17 i8.3868.4039.4209.4380.4550.4720.4889.5058.5227.5395.5563 i8 19.3870.4042.4212.4383.4553.4723.4892.506i.5230.5398.5566 19 20.3873.4044 *42f5.4386.4556.4725.4895 5o64.5233.54o0.5569 20 21.3876.4047.4218.4388.4559.4728.4898.5067.5235.5404.557I 21 22.3879.4o50o 4221.4391.456i.473i.4901.5070.5238.5406.5574 22 23.3882.4053.4224.4394.4564.4734.4903.5072.524i.5409.5577 23 24.3885.4056.4226.4397.4567.4737.4906.5075.5244 *54I2.5580 24 25.3888.4059 4229.44oo00 4570.4740.4909.5078.5247.54I5.5583 25 26.3890.4o06i 4232.44o3.4573.4742.492 508o8i.5249.54i8.5585 26 27.3893 -4064.4235.4405.4576.4745.49I5.5o84.5252.5420.5588 27 28.3896.4067.4238.44o8.4578.4748.49I7 *5o86.5255.5423.5591 28 29.3899.4070.424I *44II.458I.475i.4920.5089.5258.5426.5594 29 30.3902.4073.4244.4414.4584.4754.4923.5092.5261.5429.5597 30 3I.3905.4076.4246.44I7.4587.4757.4926.5095.5263.5432.5599 3i 32.3908.4079.4249.4420.4590.4759.4929.5098.5266.5434.5602 32 33.39I0.4o8i.4252.4422.4593.4762.4932.5Ioo.5269.5437.56o5 33 34.39I3.4084.4255.4425.4595.4765.4934.5io3.5272.5440.5608 34 35.3916.4087.42,58.4428.4598.4768.4937.5io6.5275.5443.56ii 35 36.3919.4090.4261.443i.46o0.477.4940.5I109.5277.5446.56i3 36 37.3922.4093.*4263.4434.4604.4773.4943.5ii2.5280.5448.56i6 37 38.3925.4096.4266.4437.4607.4776.4946.5lI5.5283.545i.56I9 38 39.3927.4098.4269.4439.4609.4779.4948.5II7.5286.5454.5622 39 40.3930.41Io,.4272.4442.4612.4782.495i.5I20.5289.5457.5625 4o 4i.3933.4io4.4275.4445.46i5.4785.4954.5123.5291.5460.5627 41 42.3936.4i07.4278.4448.46I8.4788.4957.5126.5294.5462.5630 42 43.3939.4I10.4280.445i.4621.4790.4960.5I29.5297.5465.5633 43 44.3942.4 4 23.4283.4454.4624.4793.4963.5I3i.53oo.5468.5636 44 45.3945.4i.i6.4286.4456.4626.4796.4965.5i34.5303.5471.5638 45 46.3947.4i18.4289.4459.4629.4799.4968.5i37.53o6.5474 564i 46 47.3950 412I.*4292.4462.4632.4802 *4971.540o.5308.5476.5644 47 48.3953.4.24 -.4295.4465.4635.48o5.4974.5i43.53ii.5479.5647 48 49.3956.4127.4298.4468.4638.4807.4977.5i45.53i4.5482.5650 49 50 3959 -.43o.43oo.4471 -464i.48Io.4979.5i48.5317.5485 5652 50 5i. 3962.4i33.43o3.4474.4643.48i3.4982.5 5..5320.5488.5655 5i 52.3965 -4i35.43o6.4476.4646.48i6.4985.554.5322.5490.5658 52 53.3967.4i38.4309.4479.4649.48I9.4988.5i57.5325 5493.566i 53 54.3970 441i4.4312.4482.4652.4822.499I.5i6o.05328.5496.5664 54 55 3973.4144.43i 5.4485.4655.4824.4994 * 562.533i.5499.5666 55 56.3976.4I47.43I7.4488.4658.4827.4996.5i65.5334.5502.5669 56 57.3979.450o.4320 *4491.466o.483o.4999.568.5336.5504.5672 57 58.3982.4153.4323.4493.4663.4833.5002.5171.5339.5507.5675 58 59.3985 -4i55.4326.4496.4666.4836.5005.5174.5342.55io.5678 59 60.3987.4i58.4329.4499.4669.4838.5oo8.5176.5345.55i3.5680 60 10 TABLE OF CHORDS: [RADIUS =1.0000]. X. 330 340 350 36~ 370 380 390 400 41~ 420 430 M o'.568o.5847.6oi4.6180.6346.65II.6676.6840.7004.7167 *7330 O I.5683.5850.6017.6i83.6349.65i4.6679.6843.7007.7170 -7333 2.5686.5853.6020.6i86.6352.6517.6682.6846 7010.7173.7335 2 3.5689.5856.6022.6189.6354.6520.6684.6849.7012.7176.7338 3 4.569I.5859.6025.619I.6357.6522.6687.685I.70o5.7178 734I 4 5.5694.586i.6028.6194.6360.6525.6690.6854 -70I8.7181'7344 5 6.5697.5864.6o3i.6I97.6363.6528.6693.6857.7020.7184 7346 6 7.5700.5867.6034.6200.6365.653i.6695.6860.7023.7186 7349 7 8.5703.5870 6o6036.6202.6368.6533.6698.6862.7026.7189 7352 8 9.5705.5872.6039.6205.637I.6536.6701.6865.7029.7192 7354 9 I0.5708.5875.6042.6208.6374.6539.6704.6868 703i.7195.7357 10 II.57II.5878.6o45.62II.6376.6542.6706.6870 7034 7197 7360 II 12.5714.588I.6047.6214.6379.6544.6709.6873.7037.7200 -7362 I2 I3.57I7.5884.6050.62I6.6382.6547.6712.6876.7040. 7203.7365 13 I4.5719.5886.6o53.6219.6385.6550.67I5.6879.7042.7205.7368 I4 15.5722.5889 -6o56.6222.6387.6553.6717.688i.7045.7208.737I 5 I6.5725.5892.6058.6225.6390.6555.6720.6884.7048 72II 7373 I6 I7.5728.5895.6o6I.6227.6393.6558.6723.6887.7050.7214 7376 17 18.5730.5897 *60o64 16230.6396.656I.6725.6890.7053.7216.7379 I8 19.5733.5900.6067.6233.6398.6564.6728.6892.7056.7219.738I 19 20.5736.5903.6070 6236.640I.6566.6731.6895.7059.7222.7384 20 21.5739.5906.6072.6238.6404.6569.6734.6898.706I.7224.7387 2I 22.5742.5909.6075.6241.6407.6572.6736.690 I.7064.7227.7390 22 23.5744.5911.6078.6244.64Io.6575.6739.6903.7067.7230.7392 23 24.5747.5914.608I.6247.64I2.6577.6742.6906.7069.7232.7395 24 25.5750.5917.6083.6249.64i5.658o.6745.6909.7072.7235 -7398 25 26.5753.5920.6o86.6252.64I8.6583.6747.6911 7075.7238.7400 26 27.5756.5922.6089.6255.6421.6586.6750.6914.7078.7241.7403 27 28.5758.5925.6092.6258.6423.6588.6753.69I7.7080.7243.7406 28 29.576I.5928.6095.6260.6426.659I.6756.6920.7083.7246 7408 29 30.5764.593I.6097.6263.6429.6594.6758'6922.7086.7249.7411 30 31.5767.5934.6ioo.6266.6432.6597.6761.6925.7089.7251.7414 3I 32.5769.5936.6io3.6269.6434.6599.6764.6928.709I.7254.7417 32 33.5772.5939.6Io6.6272.6437.6602.6767.6931.7094.7257 *74i9 33 34.5775.5942.6io8.6274.644o.66o5.6769.6933.7097.7260.7422 34 35.5778.5945.6iii.6277 6443.66o8.6772.6936.7099.7262.7425 35 36.578I.5947.6r14.6280.6445.66io.6775.6939.7102.7265.7427 36 37.5783.5950.6117.6283.6448.6613.6777.6941 -710o 5 7268 7430 37 38.5786.5953.61I9.6285.645I.66r6.6780.6944.7108.7270.7433 38 39.5789.5956. 6I22.6288.6454.66I9.6783.6947 -7I10.7273.7435 39 4o.5792.5959.6125.6291.6456.6621.6786.6950.7ii3.7276.7438 40 41.5795.596I.6128.6294.6459.6624.6788.6952.7116.7279.7441 41 42.5797.5964 *6i3o..6296.6462.6627.67.9: 6955.7118 728I'.7443 42 43.58oo.5967.6i33.6299.6465.663o.6794.6958.7I21.7284.7446 43 44.58o3.5970 *6i36.6302.6467.6632.6797.696I.7124.7287 7449 44 45.5806.5972.6139.63o5.6470.6635.6799.6963.71727..7289.7452 45 46.5808' 5975 -6142.6307.6473.6638.6802.6966.7129.7292.7454 46 47.58I-.5978.6I44 *.63Io.6476.664o.68o5.6969.7I32.7295.7457 47 48.58i4.5981.6147.63i3.6478.6643.68o8.6971.7I35. 7298.7460 48 49.5817.5984.6i5o.63i6.648I.6646.68io -6974 *7i37.7300.7462 49 50.58o2.5986 6153.6318.6484.6649.68i3.6977 -714o.7303.7465 50 51.5822.5989.6i55.6321.6487.665i.6816.6980.7143.7306.7468 5I 52.5825.5992.6i58.6324.6489.6654.68I9.6982.7146.7308.7471 52 53.5828.5995.6i6I.6327.6492.6657.6821.6985.7148.73II.7473 53 54.583i.5997.6i64.633o.6495.666o.6824.6988.71 5.73 4. 7476 54 55.5834.6o00oo.6i66.6332.6498.6662.6827.6991.7154.73i6 7479 55 56.5836.6o03.6169.6335.65oo.6665.6829.6993.7r56.739.7481, 56 57.5839.6oo6.6I72.6338..65o3.6668.6832.6996.7159.7322.7484 57.58.5842.6009.6175.634i.65o6.6671.6835.6999 7162.7325.7487 58 59.5845.6oir.6i78.6343.650.6673.6838.7001.7165.7327.7489 59 60,5847.6o.6i618o.6346 *65 I.6676.6840 *7004.7167.7330.7492 6o 11 TABLE OF CHORDS: [RADIUS =1.0000]. M. 44~' 45 46~ 4y70 4S~ 49~ 50~0 51~ 52~ 53~ 54 M. o'.7492 -7654.78i5.7975.8135.8294.8452.86io.8767.8924.9080 0o I.7495 -7656.7817 -7978.8I37.8297.8455.86I3.8770.8927.9082 2.7498.7659.7820.7980.8i4o.8299.8458.86i5.8773 8929.9085 2 3.7500.7662.7823.7983 *8I43.8302.8460.86i8.8775.8932.9088 3 4.7503.7664.7825.7986.*845.8304.8463'.8621.8778.8934.9090 4 5.7506.7667.7828.7988.8i48.8307.8466.8623.8780.8937.9093 5 6.7508.7670 -783I.7991.8I51.83io.8468.8626.8783.8940.9095 6 7.75r.7672.7833'7994.8i53.83I2.847I.8629.8786.8942.9098 7 8.7514.7675.7836.7996.8i56.83i5.8473.863i.8788.8945.910o 8 9.75I6.7678.7839.7999.8I59.83i8.8476.8634.8791.8947.9Io3 9 10.75I9.7681.784I.8002.8i6i.8320.8479.8635.8794.8950.9Io6 io 11.7522 -7683.7844.8oo4.8I64.8323.848i.8639.8796.8953.9-108 I I2.7524.7686.7847.8007.8167.8326.8484.8642.8799.8955.911I 12 I3.7527.7689.7849.8oio.8169.8328.8487.8644.88oi.8958.9113 3 14.7530.769I.7852.8012.8172.833i.8489.8647.88o4.8960.9116 i4 i5.7533.7694.7855.8oi5.8I75.8334.8492.8650.8807.8963.9119 i5 i6.7535.7697.7857.8oi8.8I77.8336.8495.8652.8809.8966.9121 16'7.7538.7699.7860 8o020.8i8o.8339.8497.8655.88I2.8968.9124 17,8.7541'7702.7863.8023.8i83.834I.85oo.8657.88i4.897I.9126 8 I9.7543.7705 *7865.8026.8i85.8344.8502.8660.8817.8973.9129 19 20.7546.7707.7868.8028.8i88.8347.85o5.8663.8820.8976.9132 20 21.7549.7710 o787I.8o3i.8190.8349.85o8 8665.8822.8979.9I34 21 22.755I.7713.7873.8o34.8I93.8352.85io.8668.8825.898I.9137 22 23.7554.7715.7876.8o36.8I96.8355.85i3.867I.8828.8984 *9I39 23 24.7557.7718.7879.8039.8198.8357.85i6.8673.8830.8986.9142 24 25.756o.772I.7882.8o42.8201.836o.85i8.8676.8833.8989 -9I45 25 26.7562'7723 7884.8044.8204.8363.8521.8678.8835.8992.9147 26 27.7565.7726.7887.8047.8206.8365.8523.868i.8838.8994 -.950 27 28.7568.7729.7890.8050.8209.8368 -.8526 *8684.884i.8997 9152 28 29.7570.773I.7892.8052.8212.8371.8529.8686.8843.8999.9155 29 30.7573.7734.7895.8055.82i4.8373.853i.8689.8846.9002 9157 3o 31.7576.7737.7898.8o58.8217.8376.8534.8692.8848.9005.9160 3i 32.7578.7740.7900.8060.8220.8378.8537.8694.885.9007.9163 32 33 -758i.7742.7903.8o63.8222.838i.8539 8697.8854.9010.965 33 34.7584.7745.7906 *8066.8225.8384.8542.8699.8856.9012.9I68 34 35.7586.7748.7908.8068.8228.8386.8545.8702.8859.9015'.970 35 36 -7589.7750.791I.8071.8230.8389.8547.8705.886I.9018.9173 36 37.7592.7753.7914.8074.8233.8392.8550.8707.8864 -9020.9I76 37 38.7595.7756.7916.8076.8236.8394.8552.8710.8867.9023'9178 38 39 -7597 -7758.7919.8079.8238.8397.8555.8712.8869.9025.9I8I 39 40 -7600.7761.7922.8082.8241.8400.8558 8715.8872.9028.9183 40 41.7603'7764 -7924.8084.8244.8402.8560.8718.8874.903I.9186 4i 42 -7605.7766.7927.8087.8246.8405.8563.8720.8877 9033.9188 42 43.7608 -7769.7930.8090.8249.84o8.8566.8723.8880.9036.9191 43 44.7611.7772.7932 "8092.825I.84io.8568.8726.8882.9038 -9r94 44 45.76I3.7774.7935.8095.8254.84i3.857I.8728.8885.904I.9196 45 46.7616.7777.7938.8098.8257.84I5.8573'873I.8887.9044 19I99 46 47.7619 -7780.7940.8ioo.8259.84i8.8576 8734.8890 9046.9201 47 48 -7621.7782.7943.8io3.8262.8421.8579.8736.8893 -9049.9204 48 49.7624.7785.7946.8io5.8265.8423.858i 8739.8895.905i -9207 49 50 -7627. 7788. 7948 *8io8.8267.8426.8584.874I.8898.9054 -9209 50 5.'7629 *7791.795I.8iii.8270.8429.8587.8744.8900.9056.9212 5i 52.7632.7793.7954.8ii3.8273.843i.8589.8747.8903.9059.924 52 53.7635.7796.7956.8ii6.8275.8434.8592.8749.8906.9062.92I7 53 54.7638.7799.7959.8119.8278.8437.8594.8752.8908.9064.9219 54 55.7640.7801.7962.812I.828I.8439.8597.8754.891.9067.9222 55 56.7643.7804 -7964.8124.8283.8442.86oo.8757.8914.9069.9225 56 57.7646.7807.7967.8127.8286.8444.8602.8760.89I6.9072.9227 57 58 -7648.7809.797.8129.8289.8447.86o5.8762.8919.9075.9230 58 59.7651.7812.7972.8i32.8291.845o.8608.8765.892I.9077.9232 59 60.7654.7815.7975.8i35.8294.8452.86io.8767 8924.9080.9235 60 12 TABLE OF CHORDS: [RADIUS =1.0000]. M. 550 560 570 5S 590 600 61-~ 620 630 64~ M. o'.9235.9389.9543 *9696.9848 I-0oooo i.oi5i i.o3oi i.o45o 1.0598 o' I 9238.9392.9546.9699.985i I.ooo3 i.oi53 i.o3o3 I.o452 I.o60o I 2.9240.9395.9548.9701.9854 I.ooo5 i.oi56 i.o3o6 i.o455.0o603 2 3 9243.9397.955I.9704.9856 1.0oo8.0o58.o308 I.457 I.0606 3 4.9245.9400.9553.9706.9859 i.ooio 1.0161 i. o3I I.o46o 1.0608 4 5 9248.9402.9556.9709.9861 i.ooi3 i.o063 i.o3i3 1.0462 I.o6rI 5 6.9250.9405.9559 *971r.9864I.ooi5 I.o66 i.o3i6.o0465 i.o6i3 6 7.9253.9407 956I *9714.9866.ooi8 I 0168 I.o38 I 0467 i.o6i6 7 8.9256.9410.9564 *97I7.9869 1.0020 I.0171 1.321 1.0470 I.o068 8 9.9258.94i3.9566.9719;9871. oo23.173 I.323 I 472 1.062 I 9 10.9261.94I5.9569.9722.9874 10025.o0176 i.o326 I.0475 1.0623 io II. 9263.9418.9571.9724.9876 0oo028 1.0178 i.o328 10o477 10626 II 12.9266 9420.9574.972.7'9879.oo3o I oi8i 1.o33i i o480 1.0628 2 13.9268.9423.9576.9729.9881 i.oo33 i.oi83 i.o333. o482 i.o63o 13 I4.9271.9425.9579 -9732.9884 i-oo35 i.oI86 i.o336 i.o485 io633 I4 I5.9274.9428.9581.9734.9886 i.oo38 I.oI88 i.o338 1. 487 i.o635 i5 I6.9276.9430.9584 -9737.9889.oo4o 1.o019I i.o341 1.490 i.o638 16 17.9279.9433.9587.9739.989I i.oo43 1.0193 i.o343 1.o492 i.o64o 17 i8.9281.9436.9589.9742.9894.oo45 1.o0196 i.o346 1.0495 i.o643 18 I9.9284.9438.9592.9744'9897 i.oo48 Io0198 i.o348.o0497 i.o645 I9 20.9287.944I -95944 9747.9899 i-oo5o 1.0201 i.o35i i.o5oo i.o648 20,2I.9289.9443.9597.9750.9902 1.oo53.0o203 i.o353.0o502 i.o65o 21 22.9292.9446.9599.9752.9904 i.oo55.0o206 i.o356 i.oo5o4.o653 22 23.9294.9448 *9602.9755.9907 i.oo58 1.0208 i.o358 i.o507 i.o655 23 24.9297.9451.9604 9757.9909 i.oo6o 1.0211 I.o36i t.o5o9 I.o658 24 25 92 9454 607 9760 99 94912 2i.oo63 I.o023 i.o363 I.0o52 i.o660 25 26 *9302.9456.96o1.9762.9914 i-oo65 1.0216 i.o366 i.o5i4 1.o662 26 27.9305.9459.96I2.9765 *9917 i.oo68 1.0218 i.o368 i.o5I7 i.o665 27 28.9307.946I ~96I5 -9767.9919 1.0070 1.022I 1.0370 1.o5I9.0o667 28 29.93io.9464'9617.9770.9922 1.0073 I*0223 i.o373 i.0522 I.o670 29 30 *93I2.9466.9620.9772.9924 1-0075 1-0226 I.o375 i.o524 *.o672 30 31.9315.9469.9622.9775 -9927 1.0078 1.0228 1.0378 1.0527 1.0675 31 32.9317.9472.9625.9778.9929 00.oo8o I023i i.o38o 1.0529 I.0677 32 33.9320 *9474.9627.9780 -.9932 r.oo83 J.0233 i.o383 1.o532 I.o680 33 34.9323'9477.9630.9783.9934 i.oo86 1.0236 i.o385 i.o534 i-o682 34 35.9325'9479.9633 -9785.9937 -.oo88 1.0238 i.o388 I.o537 i-o685 35 36.9328.9482.9635.9788.9939 i'oo91 1o024I 1 o390 I.o539 1.0o687 36 37.9330 -9484.9638 9790'9942 1.0093 I10243 1.0393I io542 i.o690 37 38.9333.9487.9640 *9793.9945.0oo96 I.o246 1.0395 i.o544 I.o692 38 39.9335.9489 *9643.9795' 9947 1-oo 98 10o248 1.o398 I.o547 1.0694 39 40o 9338 9492.9645.9798 *9950 ti*oioi 10251 i-o4oo 1.0549 1-0697 40 41.9341 -9495.9648.9800.9952 i.oio3 1.0253 i.o4o3 i.055i 1 o699 4I 421 9343 *9497.9650.9803.9955 i.oio6 1 0256 i.o4o5 i.o554 I.0702 42 43.9346.95oo.9653.9805.9957 1.o[o8.o0258 i.o4o8 i.o556 1.o704 43 44.9348.9502.9655.9808.9960 I-oii I-0261 [.o4io I.0559 1-0707 44 45.935I.95o5.9658.o8io.9962 i.oii3 10o263 1.04i3 i.o56i I-0709 45 46.9353.9507.9661.9813.9965 i.oii6 1.0266 i.o4i5 i.o564 I-0712 46 47.9356.951o.9663.9816.9967'.oii8 1.0268 i.o4i8 i.o566 I-0714 47 48.9359.9512.9666.9818.9970.1OI2I 1.027I 1.0420 I.o569 1I07I7 48 49.9361.95i5.9668.9821.9972 I1OI23 1.0273 1.o423 i.o571 I 0719 49 50 *9364.95i8.96Q7I 9823.9975 1.0126 1-0276 1Io425 i.o574 1I0721 5o 5I.9366.9520.9673.9826.9977 1.0128 I.0278 1.0428 I.0576 1.0724 5I 52.9369.9523.9676 9828 988 9980 ioi3i 1.281 i.o43o I.o579 I.0726 52 53.9371.9525.9678.983 I.9982 i.oi33 1.0283 i.o433 i.o58i 1.0729 53 54.9374.9528.9681.9833.9985 I.o36 1.0286 i.o435 i.o584 I.073I 54 55.9377.9530.9683.9836.9987 i.oi38 1.0288 i.o438 i.o586 10o734 55 56.9379.9533.9686.9838.999gg.oi4I 1.0291 i.o44o I.o589 1.0736 56 57.9382.9536.9689.9841 -9992 I.oI43 1.0293 i.o443 I 059I 1.0739 57 58.9384.9538.9691.9843.9995 i.oi46 1.0296 i.o445 -o593 1.074I 58 59.9387.954I.9694.9846.9998 I.oI48.o0298 I.o447 I-o596 I.0744 59 60.9389.9543.9696.9848 0000 i.ooi5I i.o3oi I.o450o.o598 I o746 6o 13 TABLE OF CHORDS: [RADIUS =1.0000]. M. 65~ 66~ 67~ 68~ 690 700 710 720 730 M 0o I.0746 I. 893 I-.Io39 i. i.84 1.1328 1.1 472 I 4 I64 I.756 I.I896 o' I I.0748 10895 I.io4i i. 186 I.33i.1474.i6i6I.1758 I.899 I 2 I 075 I.0898 i io44 1.1189 i.i333 1.1476 1.169 1.1760 I.I901 2 3. 1.0753 I.oo900 i.o46 I.1191 I.I335 I.i479 1.I62I I.763 I I903 3 4 I 0756. 0903 i io48 I.I 94 I.i338I i.48 I. 624I I1765 11906 4 5 1.0758 1.0905 I.i5i 1.1196 I.i340 i.483 1.1626 I 11767 I 1908 5 6 1.076I [.0907 i.io53 1.II98 I.342 I.486 I1.628 I11770 1.1910 6 7 I.0763 1.09g0 i.io56 I.I20I i.i345 I.i488 i.i63i 1I772 11g913 7 8 1.0766 1.0912 i.io58 1.12o3 1. 347 1.1491 i.i633 I 1775 I IgI5 8 9 I 0768 IO.915 I.o61 i I206 i.I35o I.1493 i.i635 I 1777 III917 9 10 I.0771 1.0917 I io63 1.12o8 i.1352 1.1495 i.i638 1.1779 1.1920 IO II r.0773 1.0920 iio65 i.12o 0 z.I354 1.1498 1.16401.1I782 11922 II 12 10775 1.0922 io68 i.i2i3 1.1357 I 5oo 1.1642 I 1784 I.924 I2 13 1-0778 1.0924 11070 I.215 I.359I 1502 I.i645 1.1786 1.1927 13 14 I o78o 1.0927 1.1073 1. 1218 1.1362 I i5o5 I.1647 1.1789 1.1929 i4 i5 1.0783 1.0929 1.1075 I.I220 i.i364 1.1507 i.i65o I.1791 1.1931 15 16 1.0785.09o32 1.1078.I222.i1366 I i5io I.i652 1.1793 1.1934 i6 77 o0788 1.o934 1ii080 1.1225 1.1369 i 1512 i.i654 I1I796 1.1936 17 i8 1.0790 1.0937 1.I082 1.I227 1.1371 I.1514I 1.I657 I.1798 i.i938 18 19 I.0793 1.939 i io85 1.1230 i 1374 i-1517 1.1659 i.i8oo 1-I941 19 20 I.0795 1.0942 I I087 I.1232 I 1376 I.15I9 I 66I I I8o3 I1943 20 2I 1-0797 1.o944 1.1o090.I234 1 I378 1. 522 i.i664 I. 805.1946 21 22 I.o8oo 1.o946 1.1092 1.1237 i.i38i i.i524 1.1666 1.1807 1.1948 22 23 I.o802 1.9g49 1.1094 I1.239 i.i383 1.I526 I.i668 1.1810 11950 23 24 I o805 Io095I I.1097 1 1242 i.i386 *1529. 1671 1.1812 I. I952 24 25.0807 1.954 II099 1. I244 i.i388 I.53 I. 1673 i.i8i4 1.I955 25 26 i-o8io I.0956I 1 I02 1.1246 1.139o0 I.1533 1.1676 1.1817 1.1957 26 27 1.o812 I.o 959 i.iio4 II.249 I*1393 1.1536 1.I678 1.1819g 1.959 27 28 I.o8i5 1.0961 I II07 I I251 1.1395 I.538 I.i68o I 182I 1.1962 28 29 I 0817 1.0963 I I09 1. 254 1.1398 I.541.i683 1.1824 1.1964 29 30 1.0820 1.0966 1i-i I I.I256 i I4oo I.i543 i.i685 1.1826 1.1966 30 3I I.0822 1.0968 1.114 I-I258. I402 I.1545 1.1687 1.1829 1.1969 31 32 I.0824 1.097I I.II6 1.126I i.i4o5 i.i548 1.1690 i.i83i 1. 971 32 33 827 973. I63.47.55o 62 833.i.o27 I973 91263 1407 o 692 i83 973 33 34 I.0829 1.0976 I.II2I 1.1266 11.1409 1.552 I1.694 i.i836 11976 34 35 I.o832 10978 I II23 1.1268 1.412 i.i555 I1.697 i.i838 I1978 35 36 i.o834 1.o980 I.II26.1.271 i.4I44 I.I557 i.I699 i.i840 1.1980 36 37 I o837 1.0983 I.128 I1I273 I.147 i.i56o 1.1702 i.i843 1.1983 37 38 1 o839 1.0985 113 1I1275 1.1419 i.562 1.1704 i.i845 1.1985 38 39.o84I.0988.1I 33 1.1278 I.i42I i.i564 I.1706 I.1847 1.1987 39 40o.o844 1.o990 i.ii36 I.i280.I1424 I.567 1.1709 i.i85o 1.1990 4o 41 i.o846 1.0o993 I.I38 1.1283 1.1426 1.1569 1I17II i.i852 11992 4i 42 I.0849 1.0995 i. I40 11285 *I429 1.1571. 1713 i.i854 I*I994 42 43 I.o85i 1.0997 i. i43 [.1287 i 143i 1.1574. 1.716 1.1857'1997 43 44 i.o854 1.000 I.I45 11290 i.I433 1.i576 II718 1.1859 II999 44 45 i.o856 1.1002 1.I148 1.1292 I.I436 1.1579 1.1720 I.i861 I.2001 45 46 1.0859 i.ioo5 i.II50o 11295 i.i438 i.58i 1.1723 i.i864 1.2004 46 47 I.o86i 1.1007 i.1152 1.1297 I.1441 i.i583 1.1725 i.186 206 12 47 48 I.o863 I.IIo i.ii55 1.1299 I.I443 i.i586 I.I727 i*.868 1.2008 48 49 I.o866 1. 012 1.Ii57 1.1302 ii445 i.i588 I.1730 I.187I I2011 49'5o i.o868 I.IoI4 1I1160 i.i3o4 i.4148 1.159 1.1732 1.I873 I.20 3 50 5 1.0871 1. I07 1.1162.307 I I45o I.1593 I.i735 1.1875 I.2oI5 5i 52 I 0873 1.11O9. ii65 i.i3o9I i.452 i.1595 I.1737 1.1878 I.2018 52 53 I o876 1.1022.1167 i.I3ii i.i455 1.1598 I.1739 i-i88o 1.2020 53 54 i.0878 1.1024 1 1169 i.i3I4 I11457 i.i6oo 1.1742 1.1882 1.2022 54 55 I.o881 I.ro27 I 1172 i.i3i6.i46o 1.1602 I.1744 i.i885 1.2025 55 56.o883 1 1029 I.1174 I I3i9 I 1462 i.i6o5 I.1746 1.887 1. 2027 56 57 I.o885 I.o03II 1177 I.32 I.i464 1.1607 11749 I'1889 1.2029 57 58 I.o888. I034 III79 I. 323 i.i467 I.i6o09 1 75 Ii.1892 1.2032 58 59 1.o890 I.io36 I.I8I I iI326 1I.469 i.1t62 I.1753 1.1894.2o34- 59 60 I.o8931. o39 i.ii84 I. 328 i.I472 I.I614 1.1756 1.1896 1.2036 60 14 TABLE OF CHORDS: [RADIUS- 1.0000]. 4L~ 14 5~ 760 770 7y0 79~ 80 81~ 820 M.'0 I 2036 I2I75 I.23I3 I.2450 I.2586 I.2722 -.2856.2989 1.3121 o I I 2039 1 2178.2316 1.2453 1.2589 1.2724 I.2858 I.299I I*3I23 I 2 1.204I 1.2180 I.23I8 1.2455.259I 1.2726 1.2860 1.2993 1.3126 2 3 I.2043 I.2182 I.2320 1.2457 1.2593 I.2728 1.2862 I.2996 1.3128 3 4 I.2o46 1.2I84 I.2322 I. 2459I59 595 I.273I I.2865 I.2998 1.330 4 5 1.2048 1.2187 1.2325 1.2462 1.2598 1.2733 I.2867 I.30ooo.3332 5 6 I.2050 1.2I89 I.2327 I.2464 1.2600 1.2735 1.2869 I.3002 I.3i34 6 7 I.2053 I.2191 I.2329 1.2466 1.2602 I.2737 I.2871 I.3004 1*3137 7 8 i.2055 1.2I94 I-2332 1.2468 1.2604 1.2740 1.2874 -13007 I.3i39 8 9 I.2057 I.2I96 1.2334 1.2471 1.2607 1.2742 1.2876 i.3009 i.3i4I 9 I0 I.2060 I.2198 I.2336 I.2473 1 2609 12744 1.2878 I.30I i.3i43 Io II 1.2062 1.220I. 2338 1.2475 1.2611 1.2746 1.2880 i.3oi3 1.3145 II I2 1. 2064 1.-2203 I.234I 1.2478 I.26r4 1.2748 1.2882 I 3oi5 1.3I47 I2 13 1.2066 1.2205 I.2343 1.2480 1.2616 1.2751 1.2885 i.3oi8 1.3i50 i3 14.2069 1. 2208 I.2345 1 2482 1.2618 I.2753 1.2887 1.3020 i.3152 I4 15 1.207I 1.2210 1.2348 1.2484 1.2620 1.2755 1.2889 I.3022 1.3I54 15 i6 1.2073 1.2212 I.2350oI.2487 1.2623 1I2757 1.2891 1.3024 I.3156 I6 I7.2076 1.22i4.I2352 I.2489 1.2625 1.2760 1.2894 I.3027 i.3i58 17 18.2078 1.2217 1.2354 I.249I [.2627 1.*2762 1.2896 I.3029 i.3i6i 18 19 I.2o80 1.2219 I.2357 1.2493 1.2629 I.2764 1.2898 I*3o3i i.3i63 I9 20 I.2083 1.222 I.2359 1-2496 I.2632 I.2766 1.2900 i.3o33 i.3i65 20 21 I.2085 1.2224 I.236I 1.2498 1.2634 I.2769 i.2903 i.3o35 1.3167 21 22 1.2087 I.2226.2364 I.2500 1.2636 1.2771 1.2905 I.3o38 1.3169 22 23 I.2090 I.2228 I.2366 I.2503 I.2638 I 2773 1 *2907 i. 3o4o 1.3172 23 24 I.2092 1.2231 I.2368 1.2505 1.2641 1.2775 1.2909 I.3042 1.3174 24 25 1.2094 I.2233 1.2370 1.2507 I,2643 1.2778 1.2911 I.3o44 1.3176 25 26 1.2097 I.2235 I.2373 1.2509 I.2645.2780 1.2914 I.3o46 I.3I78 26 27 I 2099 1.2237 1.2375 I 25I2 1.2648 1.2782 1.2916 I.3049 i.3i8o 27 28 I.2101 I. 2240 I.2377 1.25I4 I.265o 1 2784 1.2918 i.3o5 I i.383 28 29 1*2o04 I-2242 i.238o I.25I6 1.2652 1.2787 1.2920 I.3o53 i.3i85 29 30 1.2106 1.2244 1.2382 I.25i8 1.2654 12789 1.2922 I.3o55 1.*387 30 31 1.2108 1.2247 I.2384 1.2521 1.2656 1'2791 1.2925' i.3057 1.3189 31 32 I.2III 1.2249 1.2386 1.2523 1.2659 1.2793 1.2927 I.3o6o 1.319 32 33 I.2II3 I.2.25I 1.2389 I.2525 1.266I 1.2795 1.2929 1.3062 1.3I93 33 34 I.2115 1.2254 i.2391 1.2528 1.2663 1*2798 I.293I I.3o64 1.3196 34 35 I. 217 1.2256 I.2393 1.2530 I.2665 I.2800 1.2934 I.3066 1.3198 35 36 1.2I20 1.2258 I.2396 1.2532 I.2668 1*2802 I.2936 I.3o68 1.3200 36 37 1.2122 1.2260 I.2398 I.2534 I.2670 I.2804 I.2938 1.3071 I.3202 37 38 1.*2124 1.2263 I.24o0 1.2537 *1.2672 I.2807 I-2940 I*3073 1.3204 38 39 1.*2127 1.2265 I.2402 1.2539 1.2674 1*2809 1*2942 1.3075 1.3207 39 40 1.2129 I.2267 I.24o5 I*254i 1.2677I 12811 1.2945 1.3077 1.3209 4o 4I I.2i3i 1.2270 1.2407 1*2543 I*2679 I*28I3 1I2947 I.3079 1.3211 41 42 I.2I34I.2272 1.2409 1*2546 1.268I I.28I6 I.2949 I[3082 1.32i3 42 43.2I36 1.2274 1.2412 1.2548 1.2683.2818 1.2951 I.3o84.3215 43 44: I.2I38 I.2277 I.24I4 I.255o 1.2686 1.2820 1.2954 I.3o86 1.328 44 45.2a41 I.2279 I.24I.6 I.2552 1.2688 1.2822 1.2956 I.3o88 1.3220 45 46 I1.2143 I.228i I.24I8 1.2555 1.2690 1.2825 1.2958 1.3090I.3222 46 47 1.2I45 1.2283 1.2421 I.2557 1.2692 1.2827 1.2960 I13093 I.3224 47 48 1.2I48 I.2286 I1.2423 I*2559 1.2695 1.2829 1.2962 I.3095 1.3226 48 49 I.2I50.I2288 I.2425 I.2562 1.2697 I.283I I.2965 1.3097 1.3228 49 50 I*2152 1.2290 1.2428 1.2564 I.2699 1.2833 1.2967 1.3099 1.323i 50 5 1.2I54 1.2293 I.2430 I.2566 1.2701 I.2o836 1.2969 1.3Ioi I.3233 5i 52 121 57 I.2295 1.2432 I.2568 I.2704 I.2838 1.297I I.3io4 I.3235 52 53 1.2159 I*2297 I*2434 1.2571 I 2706 I * 2840 i 2973 1310o6 1.3237 53 54 i 2I6r 1.2299 I12437 1-2573 1.2708 1.2842 I.2976 I13io8 1.3239 54 55 I.2i64 1.2302 I.2439 1*2575 1.2710 1.2845 1.2978 i.3rIo I.3242 55 56 I 266 1.23o4.244I 1-2577 1*2713 1 2847 12980 I.3I12 1.3244 56 57 I*2i68 1.23o6 I.2443 1.258o 1.2715 I.2849 1*2982 I*.315 1.3246 57 58 1-2171 I2309 1.2446 1.2582 I.2717 I-285i 1.2985 I.31I7 i.3248 58 59 I*2173 i*23I I.2448 I*2584 I[2719 1*2854 1.2987 I*.319 i*3250 59 6b ['2175 1-2313 I.245o 1.2586 1.2722 I.2856 1.2989 *3I2I 1.3252 60 5I 15 TABLE OF CHORDS: [RADInS =1.0000]. _. 83~0 ~~ 50 S6~0 S0 s 8 9~ M. o' 1.3252 1.3383 1.35i2 i.364o I1.3767 1.3893 I.4oi8 o' I 1.3255 1.3385 I.35I4 1.3642 1.3769 I 3895 1.402 I 2 1.3257 1.3387 I.35i6 I.3644 1.3771 1.3897 1.4022 2 3 1.3259 1.3389 i'.35I8 pi.3646 1.3773 1.3899 1.4024 3 4 I.326I 1.3391 i.3520 i.3648 1.3776 1.3902 1.4026 4 5.3263 1.3393.3523 i365i 1 3778 I 3904 1.4029 5 6 1.3265 1.3396 I.3525 1.3653 I 3780 1.3906 i.4o3i 6 7 I.3268 1.3398 1.3527 i.3655 1.3782 1.3908 i.4o33 7 8 1.3270 i.34oo 1.3529 1.3657 1.3784 1.3916 1.4035 8 9 1.3272 1.3402 I.353I 1.3659 1.3786 1.3912 1.4037 9 10 I.3274.34o4 I.3533.366i 1.3788 1.3914 i.4039 Io 11 1.3276.34o6 I.3535 I.3663 1.3790 I.3916 i.4o41 I1 I2 1.3279 1.3409 1.3538 I.3665. I.3792 1.398 1.4o43 12 13 1.3281 I.341I 1.354o I.3668 1*3794 1.3920 i.4o45 13 14 1.3283 i.:3413 I 3542 1.3670 1.3797 1.3922 I.4047 I4 15 1.3285.34i5 1.3544 1.3672 13799 1-3925 1.4049 15 16 1.3287 I.34I7 I.3546 I.3674 I.38oi I.3927 I.4051 16 17 1.3289 1.349. 1.3548 1.3676 i.38o3 1.3929 I.4o53 17 i8 I.3292 I 3421 I 355o0.3678.38o5 1.3931.4o55 18 19 1.3294 I 3424 I.3552 I.3680 1.3807 I.3933 I.4o58 19 20.3296 13426 I.3555 1.3682 1.3809 I.3935 i.4o6o 20 21 1.3298 1.3428 1.3557 I.3685 I 38i I 1.3937 I 4062 21 22.3300oo.343o' I 3559 I.3687 i.38i3 1.3939 1.4064 22 23 1' 3302 1.3432 I 356 i1.3689 i.38i6 I 3941 i.4o66 23 24 I.3305 I.3434 I.3563 1.369I I.38i8 1.3943 I.4068 24 25 i.3307 I.3437 I.3565 1.3693 I.3820 1.3945 1.4070 25 26 I 3309 1.3439 I13567 1.3695 I.3822 1.3947 1.4072 26 27 I 33 I I 344r 1.3570 1.3697 1.3824 I.3950 14074 27 28 I.3313 r.3443 1.3572 1.3699 I.3826 1.3952 1.4076 28 29 I.3315 I.3445 1.3574 1.3702 1.3828 1.3954 1.4078 29 30o.33I8 1.3447 1.3576 I.370o4.383o 1.3956.4o8o0 30 31.3320 1. 3449.3578 1.3706 1.3832.1.3958 1.4082 31 32 1.3322 1.3452 I.358o 1.3708 I.3834 1.3960 i.4o84 32 33 1.3324 I.3454 1.3582 1.3710 1.3837 1.3962 I.4o86 33 34 1.3326 i.3456 1.3585 1.3712 1.3839 1.3964 1.4089 34 35 1.3328 I.3458 r.3587 1.37I4 i.384I I 3966 I 4091 35 36 1.333I I.346o0.3589 1.3716 I3843 I 3968 I 4093 36 37 I.3333 1.3462 1.3591 I.3718 I.3845 I 3970 I 4095 37 38 1.3335 I.3465 1.3593 1.3721 1.3847 1.3972 1.4097 38 39 i 3337 1.3467 I.3595 1.3723 I 3849 I 3975 1.4099 39 4o.3339 13469 1.3597 1.3725.3851 1.3977 I -41io 4o 4I I.334I I 3471 1.3599 I13727 1.3853 1.3979 i.4io3 41 42 I.3344 1.3473 I.3602 1.3729 1.3855 1.3981 i.4io5 42 43 I.3346 1.3475 i.36o4 1 373I.3858 1.3983 1.4107 43 44 1.3348 1.3477 I.36o6 1.3733 i.3860 1.3985 1.409 44 45 i.335o i.348o i.36o8 1.3735 I13862 1.3987 I 411I 45 46 1.3352 1.3482 i.36io 1.3738 i.3864 1.3989 1.41[3 46 47 1.3354.3484 I.36I2 1.I3740 I.386 i1.3991 1.41[5 47 48 1.3357 1.3486 I.3614 1.3742 1.3868 1.3993 1.4i17 48 49 I.3359 i.3488, 1.3617 1.3744 1.3870 1.3995 1'4I[9 49 50o I.336I 1.3490 1.3619 I.3746 1.3872 1.3997 I14122 50 5I 1.3363 1.3492 I1.3621 I.3748 1.3874 1.3999 I-4124 5i 52 I 3365 1.3495 1.3623 1.3750 1.3876 1.4002 1.4126 52 53 1.3367 1.3497 1.3625 1.3752 1.3879 i.4oo4 1.4128 53 54 1.3370 I13499 1.3627 1.3754 I.3881 i.4oo6 i.4i3o 54 55 1.3372 1.350I 1.3629 1.3757 I.3883 i.4oo8 I.4I32 55 56 i 1. 3374 I.35o3 i.363I 1.3759 i.3885 i.4oIo I.4I34 56 57 1.'3376 i.35o5 1.3634 1.3761 1.3887 I.4o02 i.4i36 57 58! I13378 I.3508 I1.3366 I3763 1.3889 i.4oi4 i.4i38 58 59 I-3380 I.35io I.3638 I.3765 1.3391 I.4oi6 i.4140 59 6o0 I3383 I.35i 2.3640 I 3767 I 3893 I 4oi8 I.4i42 60 16 TABLE OF NATURAL SINES AND TANGENTS; TO EVERY DEGREE AND MINUTE OF THE QUADRANT. IF the given angle is less than 45~, look for the degrees and the title of the column, at the top of the page; and for the minutes on the left. But if the'angle is between 450 and go~, look for. the degrees and the title of the column, at the bottom; and for the minutes on the rig7t. The Secants and Cosecants, which are not inserted in this table, may.be easily supplied. If I be divided by the cosine of an arc, the quotient will be the secant of that arc. And if I be divided by the sine, the quotient will be the cosecant. The values of the Sines and Cosines are less than a unit, and are given in decimals, although the decimal point is not printed. So also, the tangents of arcs less than 450, and cotangents of arcs greater than 45~, are less than a unit and are expressed in decimals with the decimal point omitted. NATURAL SINES AND COSINES. 0~ 1'~ 2~ 30 40 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o ooooo Unit. o1745 99985 03490 99939 o5234 99863 o6976 99756 6o I 00029 Unit. 01o74 99984 o35i9 99938 o5263 9986i 07005 99754 5 2 oo00058 Unit. oi8o3 99984 o3548 99937 05292 99860 07034 99752 58 3 oo00087 Unit. 832 99983 03577 99936 0532I 99858 o7063 99750 57 4 ooi 6 Unit. 01862 99983 o366 99935 o5350 99857 07092 99748 56 5 ooI45 Unit; oi89g 99982 o3635 99934 05379 99855 07121 99746 55 6 ooI75 Unit. o0920 99982 o3664 99933 o5408 99854 0715.0 99744 54 7 00204 Unit. 0194 99981 03693 99932 05437 99852 07I79 99742 53 8 oo233 Unit. 0978 99980 o3723 99931 o5466 9985I o7208 99740 52 9 00262 Unit. 02007 99980 03752 99930 05495 99849 07237 99738 5i 10 00291 Unit. 02036 99979 0378I 99929 05524 99847 07266 99736 5o 1 Ioo32o0 02065 99965 999727 o5553 99846 07295 99734 49 12 o0349 9999902094 99978 o383g 99926 o5582 99844 07324 9973 48 I3 00378 99999 0223 99977 o3 9999992.5 o56 99842 07353 99729 47 14 00407 99999 02152 99977 o3897 99924 o5640 9984, 07382 99727 46 15 oo436 99999 0218i 99976 03926 99923 05669 99839 07411 99725 45 i6 oo465 99999 0221I 99976 03955 99922 05698 99838 07440 99723 44 00495 99999 o02240 99975 o3984 99921 05727 99836 o7469 9972I 43 1 00524 99999 02269 99974 o40i3 99919 05756 99834 07498 99719 42 I9 00553 99998 02298 99974 04042 99918 05785 99833 07527 99716 4i 20 00582 99998 02327 99973 0407I 99917 o58i4 9983 o07556 99714 40 21 0oo6i 99998 02356 99972 04I00 99916 05844 99829 07585 99712 39 22 oo64o 99998 02385 99972 04I29 99915 05873 99827 07614 99710 38 23 00669 99998 02414 9997I 04I59 99913 o5902 99826 07643 99708 37 24 00698 99998 02443 99970 o0488 999I2 05931 99824 07672 99705 36 25 00727 99997 02472 99969 04217 99911 05960 99822 07701 99703 35 26 o00756 99997 0250I 99969 04246 999I0 05989 99821 07730o 9970I 34 27 o785 99997 02530 99968 04275 99909 o6oI8 99819 07759 99699 33 2 oo8I4 99997 02560 99967 o4304 99907 o6047 99817 07788 99696 32 29 oo844 9999 0258 99966 o4333 99906 06076 99815 07817 99694 330 00873 99996 02618 99966 04362 99905 o6io5 9983 o7846 99692 3o 31 00902 99996 02647 99965 o4391 99904 o6i34 9982 07875 99689 29 32 oo931 99996 02676 99964 04420 99902 o6i63 9981 07904 99687 28 33 00960 99995 02705 99963 4449 99901 06192 99808 07933 99685 27 34 oo989 99995 02734 99963 0447 99900 06221 99806 07962 99683 26 35 oioi8 99995 02763 99962 o4507 99898 o625o 99804 o7 99 9680 25 36 I047 99995 02792 99961 o4536 99897 06279 99803 08020 99678 24 37 01076 99994 02821 99960 o4565 99896 o63o8 99801 08049 99676 23 3 o o5 99994 o2850 99959 04594 99894 o6337 99799 08078 99673 22 39 oii34 99994 02879 9995 04623 99893 o6366 9979 0817 99671 21 40 o0164 99993 02908 99958 o4653 99892 o6395 99795 o836 99668 20 41I 093 99993 02938 99957 04682 99890 06424 99793 08i65 99666 I 42 01222 99993 02967 99956 047II 9988 o6453 99792 0894 99664 I 43 0o25I 99992 02996 99955 04740 99888 o6482 99790 8223 99661 17 44 O,280 997?092 02 44 01280 99992 o3025 99954 04769 99886 o65II 9978 o8252 99659 i6 45 oi309 9999,1 3o054 99953 04798. 99885 o654o 99786 o828i 99657 I5 46 oi338 99991 o3o83 99952 04827 99883 o6569 99784 o83o 99654 14 47 0367 9999' o31I2 99952 o4856 99882 o6598 99782 o8339 99652 I3 8 oi396 99990 o3I4t 99951 o4885 99881 06627 99780 o8368 99649 12 49 oI425 99990 o3I70 99950 o4914 99879 o6656 99778 08397 99647 II 50o 0454 99989 03199 99949 04943 99878 o6685 99776 08426 99644 10 51 01483. 99989 3228 99948 04972 99876 06714 99774 o8455 99642 52 01513 99989 o3257 99947 0500o 99875 06743 99772 08484 99639 8 53 0o542 99988 03286 99946 o5030 99873 06773 99770 o85I3 99637 7 54 o1571 99988 o336 99945 05056 99872 06802 99768 o8542 99635 6 55 oi6oo 99987 o3345 99944 05o88 99870 o683I 99766 08571 99632 5 56 o0629 99987 3374 99943 o5II7 99869 o686o 99764 o86oo 99630 4 57 o658 99986 o34o3 99942 o5i46 99867 06889 99762 08629 99627 3 58: o!I687 99986 o3432 99941 o0575 99866 06918 99760 o8658 99625 2 59 0I716 99985 0346I 99940 o52o5 99864 o6947 99758 08687 99622 I 60 oi745 9998 o03490 99939 05234 99863 o6976 99756 08716 99619 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine, Sine. Cosine. Sine. 890 88~ 81~ 86~ 85~ 18. NATURAL SINES AND COSINES. 50 6~ 70 80 90 Sine Cosine. Sine. osine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 0 08716 99619 io453 99452 12187 99255 I39I7 99027 15643 98769 6o o08745 996i7 10482 99449 12216 99251 I3946 99023 I5672.98764 59 2 08774 996I4 0o511 99446 12245 99248 i3975 990I9 15701 98760 58 3 o88o3 99612 io54o 99443 12274 99244 14004 99010 15730 98755 57 4 0883i 99609 10569 99440 I2302 99240 14033 99011 15758 98751 56 5 08860 99607 10597 99437 I233I 99237 I4o6I 99006 I5787 98746 55 6 08889 99604 10626 99434 I2360 99233 14090 99002 I58i6 98741 54 7 08918 99602 io655 9943I I2389 99230 141 98998 I5845 98737 53 8 08947 99599 io684 99428 12418 99226 I4148 98994 i5873 98732 52 9 08976 99596 10713 99424 12447 99222 I4177 98990 15902 98728 5I 10 09005 99594 10742 99421 12476 992I9 I42o5 98986 593 98723 5o II 09034 99591 1077I 99418 I2504 99215 I4234 98982 I5959 98718 49 12 09063 99588 1o8oo 99415 12533 99211 14263 98978 15988 98714'48 13 09092 99586 I0829 99412 I2562 99208 14292 98973 I6017 98709 47 14 09121 99583 io858 99409 12591 99204 14320 98969 i6046 98704 46 I5 09150 99580 10887 99406 12620 99200 14349 98965 16074 98700 45 I6 09179 99578 I0916 99402 12649 99197 14378 98961 i6io3 98695 44 I7 09208 99575 10945 99399 12678 99193 I4407 98957 I6132 98690 43 I8 09237 99572 10973 99396 12706 99189 14436 98953 i66o 98686 42 19 09266 99570 1I002 99393 12735 99186 14464 98948 I6189 9868I 4I 20 09295 99567 11o3i 99390 12764 99182 14493 98944 I6218 98676 40 21 09324 99564 IIo6o 99386 12793 99178 I4522 98940 16246 98671 39 22 09353 99562 I1089 99383 12822 99175 I455I 98936 16275 98667 38 23 09382 99559 III18 99380 I285I 9917I i458o 9893I i63o4 98662 37 24 09411 99556 I1147 99377 12880 99167 14608 98927 i6333 98657 36 25 09440 99553 11176 99374 12908 99163 I4637 98923 i636i 98652 35 26 09469 99551 11205 99370 I2937 99160 I4666 98919 16390 98648 34 27 09498 99548 1234 99367 12966 99156 I4695 98914 I6419 98643 33 28 09527 99545 11263 99364 12995 99152 14723 98910 I6447 98638 32 29 og556 99542 II29I1 99360 I3024 99148 14752 98906 I6476 98633 3 30 o9585 99540 11320 99357 i3053 99144 14781 98902 i65o5 98629 3o 31 096I4 99537 II349 99354 I308i 9914I 148o1 98897 i6533 98624 2o 32 09642 99534 11378 99351 I3IIo 99137 14838 98893 16562 98619 28 33 09671 99531 11407 99347 I3139 99133 14867 98889 16591' 98614 27 34 09700 99528 11436 99344 I3i68 99129 4896 98884 16620 98609 26 35 09729 99526 II465 99341 I3197 99125 14925 98880 16648 98604 25 3609758 99523 11494 99337 13226 99122 14954 98876 16677 98600 24 37 0987 99520 11523 99334 13254 99118 I4982 98871 16706 98595 23 38 09816 99517 11552 99331 13283 99114 I5o0I 98867 16734 98590 22 39 09845 99514 ii580 99327 13312 99110 I5040 98863 16763 98585 21 40 09874 99511 11609 69324 i334I 99106 I5069 98858 16792 98580 20 41 09903 99508 I 638 99320 I3370 99102 I5097 98854 I6820 98575 19 42 09932 99506 11667 99317 13399 99098 15126 98849,16849 98570 i8 43 09961 99503 11696 99314 13427 99094, 15I55 98845 16878 98565 17 44 09990 99500 I1725 99310 i3456 99091 51i84 9884I 16906 98561 i6 45 10019 99497 11754 99307 13485 99087 I52I2 98836 16935 98556 15 46 10048 99494 11783 99303 I35i4 99083 1524I 98832 16964 9855I I4 47 10077 99491 11812 99300 i3543 99079 15270.98827 16992 98546 3 48 ioio6 99488 1I840 99297 I3572 99075 15299 98823 17021 98541 12 49 ioi35 99485 11869 99293 i36oo 99071 I5327 988I8 17050 98536 I 50 10164 99482 I1898 99290 i3629 9907 i5356 98814 17078 9853i 1o 5I 1I092 9947911927 99286 i3658 99063 i5385 98809 17107 98526 9 52 I022I |99476 I956 99283 I3687 99059 I54I4 98805 17I36 9852 8 53 10250 99473 11985 99279 13716 99055 I5442 98800 17164 985i6 7 54 10279 99470 120I4 99276 13744 9905I 15471 98796 17193 98511 6 55 o0308 99467 12043 99272 13773 99047 I55oo 9879I 17222 98506 5 56 o0337 99464 12071 99269 3802 99043 15529 98787 17250 9850o 4 57 io366 9946I 12100 99265 I383I 99039 I5557 98782 I7279 98496 3 58 10395 99458 2129 99262 i386o 99035 i5586 98778 17308 98491 2 59 10424 99455 12158 99258 13889 99031 I565 98773 17336 98486 I 60 I0453 99452 12187 99255 13917 99027 I5643 98769 i7365 98481' Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine Sine. 84. 88 820 810 80,84~ d 830 820 810 800-19 NATURAL SINES AND COSINES. Q100 110 120 130~ 14~ Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o 17365 98481 19081 98163 2079I 978I5 22495 97437 24192 97030 60 I 17393 98476 I9I09 98157 20820 97809 22523 9743o 24220 97023 59 2 17422 98471 19138 98152 20848 97803 22552 97424 24249 97015 58 3 i7451 98466 19167 98146 2Q877 97797 22580 97417 24277 97008 57 4 17479 98461 19195 98140 20905 97791 22608 97411 24305 97001 56 5 7175 98455 96994 5 1758 98455 19224 98135 20933 97784 22637 97404 24333 96994 55 6 17537 98450 19252 98129 20962 97778 22665 97398 24362 96987 54 7 I7565 98445 1928I 98124 20990 97772 22693 9739I 24390 96980 53 8 17594 98440 19309 98118 21019 97766 22722 97384 24418 96973 52 9 17623 98435 i9338 981I2 21047 97760 22750 97378 24446 96966 5I 10 17651 98430 19366 98107 21076 97754 22778 97371 24474 96959 5o I1 17680 98425 19395 98101 21104 97748 22807 97365 245o3 96952 49 I2. 17708 98420 19423 98096 21132 97742 22835 97358 24531 96945 48 I3 I7737 98414 19452 98090 2Ii6i 97735 22863 9735i 24559 96937 47 14 17766 98409 19481 98084 21189 97729 22892 97345 24587 96930 46 I5 17794 98404 I9509 98079 21218 97723 22920 97338 24615 96923 45 I6.17823 98399 19538 98073 21246 97717 22948 9733I 24644 96916 44 17 17852 98394 19566 98067 21275 97711 22977 97325 24672 96909 43 18 7880 98389 9595 98061 21303 97705 23005 97318 24700 96902 42 9 17909 98383 19623 98056 2133i 97698 23033 97311 24728 96894 4 20 17937 98378 19652 98056 2136o 97692 23062 97304 24756 96887 40 21 17966 98373 19680 98044 21388 97686 23090 97298 24784 96880 39 22 17995 98368 19709 98039 21417 97680 23 I8 97291 24813 96873 38 23 I8023 98362 I9737 98033 2I445 97673 23I46 97284 24841 96866 37 24 18052 98357 19766 98027 21474 97667 23175 97278 24869 96858 36 25 i8o8i 98352 19794 98021 21502 97661 232o3 97271 24897 96851 35 26 18109 98347 19823 98016 2I530 97655 2323I 97264 24925 96844 34 27 I8138 98341 i985i 98010 21559 97648 23260 97257 24954 96837 33 28 i8i66 98336 19880 98004 21587 97642 23288 97251 24982 96829 32 29 18195 98331 19908 97998 216I6 97636 233I6 97244 250IO 96822 3I 30 18224 98325 19937 97992 21644 97630 23345 97237 2503k 96815 30 31 18252 98320 19965 97987 2I672 97623 23373 97230 25066 96807 29 32 18281 983 5 19994 97981 2I701 97.617 23401 97223 25094 96800 28 33 i8309 98310 20022 97975 21729 97611 23429 97217 25122 96793 27 34 i8338 98304 200o5 97969 21758 97604 23458 97210 2515i 96786 26 35 18367 98299 20079 97963 21786 97598 23486 97203 25179 96778 25 36 18395 98294 201i8 97958 2I8I4 97592 235i4 97196 25207 96771 24 37 18424 98288 2oI36 97952 2I843 97585 23542 97I89 25235 96764 23 38 I8452 98283 2o065 97946 21871 97570 2357I 97182 25263 96756 22 39 I8481 98277 20193 97940 21899 97573 23599 97176 25291 96749 2I 40 I8509 98272 20222 97934 21928 97566 23627 97169 25320 96742 20 4I I8538 98267 20250 97928 21956 97560 23656 97162 25348 96734 19 42 I8567 98261 20279 97922 21985 97553 23684 97155 25376 96727 I8 43 18595 98256 20307 97916 22013 97547 23712 97148 25404 96719 I7 44 18624 98250 20336 97910 22041 97'54I 23740 97141 25432 96712 6 45 18652 98245 20364 97905 22070 97534 23769 97134 25460 96705 I5 46 i868i 98240 20393 97899 22098 97528 23797 97I27 25488 96697 I4 47 18710 98234 20421 97893 22126 97521 23825 97120 255i6 96690 I3 48 8738 98229 20450 97887 22155 97515 23853 97I13 25545 96682 I 49 8767 98223 20478 97881 22183 97508 23882 97106 25573 96675 It 50 18795 98218 20507 97875 22212 97502 23910 97100 25601 96667 1I 51 18824 98212 20535 97869 22240 97496 23938 97093 25629 96660 52 18852 98207 20563 97863 22268 97489 23966 97086 25657 96653 53 i888i 98201 20592 97857 22297 97483 23995 97079 25685 96645 7 54 18910 98196 20620 97851 22325 97476 24023 97072 25713 96638 6 55 18938 98100 20649 97845 22353 97470 2405i 97065 25741 96630 5 56 18967 98185 20677 97839 22382 97463 24079 97058 25769 96623 4 57 18995 98I79 20706 97833 22410 97457 24108 9705 25798 665 3 58 19024 98174 20734 97827 22438 97450 24I36 97044 25826 96608 2 59 19052 98168 20763 97821 22467 97444 24164 97037 25854 96600 I 60 19081 98163 20791 978I5 22495 97437 24192 97030 25882 96593 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. | 1 i i ii i 90 1 78'70 760 76~5 20 NATURAL SINES AND COSINES. 15~ 16~ 170 180 190 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. O 25882 96593 27564 96126 29237 95630 30902 95I06 32557 94552 60 I 25910 96585 27592 96118 29265 95622 30929 95097 32584 94542 59 2 25938 96578 27620 9610 29293 19561 30957 95088 32612 94533 58 3 25966 96570 27648 96102 29321 95605 30985 95079 32639 94523 57 425994 96562 27676 96094 29348 95596 310I2 95070 32667 94514 56 5 26022 96555 27704 9686 29376 95588 3104o 9506I 32694 94504 55 6 26050 96547 27731 96078 29404 95579 3io68 95052 32722 94495 54 726079 96540 27759 96070 29432 95571 31095 95043 32749 94485 53 26107 96532 27787 96062 29460 95562 3iI23 95033 32777 94476 52 9 2635 96524 27815 96054 29487 95554. 3i5i 95024 32804 94466 5I 10 26i63.96517 27843 96046 29515 95545 31178 95015 32832 94457 50 11 2619I 96509 27871 96037 29543 95536 3I206 95006 32859 94447 49 12 262I9 96502 27899 96029 29571 95528 31233 94997 32887 94438 48 I3 26247 96494 27927 96021 29599 95519 3I26i 94988 32914 94428 47 14 26275 96486 27955 96013 29626 95511 31289 94979 32942 94418 46 15 26303 96479 27983 96005 29654 95502 3i3i6 94970 32969 94409 45 I6 2633i 96471 28011 95997 29682 95493 31344 9496I 32997 94399 44 I7 26359 96463 28039 95989 29710 95485 3372 94952 33024 94390 43 8 26387 96456 28067 95981 29737 95476 31399 94943 33o5i 94380 42 19 26415 96448 28095 95972 29765 95467 31427 94933 33079 94370 4I 20 26443 96440 28123 95964 29793 95459 31i454 94924 33io6 94361 4o 21 26471 96433 28150 95956 29821 95450 31482 94915 33134 9435I 39 22 26500 96425 28178 95948 29849 9544I 3I5io 94906 33I6i 94342 38 23 26528 964I7 28206 95940 29876 95433 3537 94897 3389 9433237 24 26556 96410 28234 95931 29904 95424 3i565 94888 33216 94322 36 25 26584 96462 28262 95923 29932 95415 3i593 94878 33244 94313 35 26 26612 96394 28290 95915 29960 95407 3620 94869 33271 94303 34 27 26640 96386 283I8 95907 29987 95398 3I648 9486o 33298 94293 33 28 26668 96379 28346 95898 3oo5 95389 31675 94851 33326 94284 32 29 26696 96371 28374 95890 3oo43 95380 31703 94842 33353 94274 31 30 26724 96363 28402 95882 30071 95372 3I730 94832 3338x 94264 3o 31 26752 96355 28429 95874 30098 95363 31758 94823 33408 94254 29 32 26780.96347 28457 95865 30126 95354 31786 94814 33436 94245 28 33 26808 96340 28485- 95857 3oi54 95345 3i8i3 94805 33463 94235 27 34 26836 96332 285I3 95849 30I82 95337 3i841 94795 33490 94225 26 35 26864 96324'2854I 9584I 30209 95328 3i868 94786 335i8 94215 2 36 26892 96316 28569 95832 30237 953I9 31896 94777 33545 94206 -24 37 26920 96308 28597 95824 30265 9 53 31923 94768 33573 94196 23 38 26948 96301 28625 95816 30292 9530o 31951 94758 336oo 94186 22 39 26916 96293 28652 95807 30320 95293 31979 94749 33627 94176 21 40 27004 96285 28680 95799 30348 95284 32006 94740 33655 94167 20 41 27032 96277 28708 957 30376 95275 32034 94730 33682 9417 I9 42 27060 96269 28736 95782 3403 95266 3206 -9472i 33710 94147 18 43 27088 96261 28764 95774 3o43I 95257 32089 94712 33737 94137 17 44 27116 96253 28792 95766 30459 95248 3216 94702 33764 94127 I6 45 27144 96246 28820 95757 3o486 95240 32144 94693 33792 94118 i5 46 27172 96238 28847 95749 3o5i4 9523I 32171 94684 338I9 94I08 14 47 27200 96230 28875 95740 3o042 95222 32199 94674 33846 94098 13 48 27228 96222 28903 95732 30570 95213 32227 94665 33874 94088 12 49 27256 96214 28931 95724 30597 95204 32254 94656 33901 94078 ii 50 27284 96206 28959 95715 3625 95195 32282 94646 33929 94068 io 1 27312 96198 28987 95707 30653 95186 32309 94637 33956 94058 52 27340 96190 29015' 95698 30680 95177 32337 94,627 33983 94049 53 27368 96I82 29042 95690 30708 95I68 32364 94618 340i 94039 7 54 27396 96174 29070 95681 30736 95159 32392 94609 34o38 94029 6 55 27424 96166 29098 95673 30763 95.50o 32419 94599 34065 94019 5 56 27452 96i58 29126 95664 30791 95142 32447 94590 34093 94009 4 57 27480 96I50 29154 95656 30819 95I33 32474 94580'34I20 93999 3 58 27508 96142 29182 95647 30846 95124 32502 9457I 34147 93989' 59 27536 96134 29209 95639 30874 95115 32529 94561 34175 93979 I 60 27564 96126 29237 9563o 30902 95I06 32557 94552 34202 93969 o.... - _ - _ Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 740 i730 1 2~ 110 70 21 NATURAL SINES AND COSINES. 200 210 1 220 230 240 Sine. Cosine. Sine. Cosine. Sine sine. Sine. Cosine. Sine. Cosine. o 34202 93969 35837 93358 37461 927I8 39073 92050 40674 9I355 60 I 34229 93959 35864 93348 3748 92707 39100 9203 40700 9I343 59 2 34257 93949 35891 93337 375I5 92697 3927 92028 40727 9I33 58 3 34284 93939 35918 93327 37542 92686 39i53 92016 40753 9I3I9 57 4 343II 93929 35945 933i6 37569 92675 39180 92005 40780 91307 56 5 34339 9399 35973 93306 37595 92664 39207 91994. 4o8o6 912.95 55 6 34366 93909 36ooo 93295 37622 92653 39234 91982 40833 91283 54 34393 9399 36027 93285 37649 92642 39260 9I97I 40860 91272 53 834421 93889 36054 93274 37676 92631 39287 91959 40886 9I260 52 9 34448 93879 36o8i 93264 37703 92620 393i4 9I 948 40913 9248 5i o1 34475 93869 36i08 93253 37730 92609 3934i 91936 40939 91236 50 II 34503 93859 36i35 93243 37757 9259 39367 91925 40966 9I224 49 12 34530 93849 36162 93232 37784 92587 39394 919I4 40992 91212 48 3' 34557 93839 36190 93222 37811 92576 39421 91902 41019 91200 47 14 34584 93829 36217 93211 37838 92565 39448 9I89I 4o45 91188 46 I5 34612 93819 36244 93201 37865 92554 39474 9I879 41072 91176 45 I6 34639 93809 36271 93I90 37892 92543 3950o 91868 4o198 91 64 44 I7 34666 3799 36298 93I8o 37919 92532 39528 91856 4 I25 9II52 43 34694 93789 36325 9369 37946 92521 39555 91845 41151 91140 42 19 34721 93779 36352 93I59 37973 925I0 3958i 9I833 41I78 9II28 4I 20 34748 93769 36379 93148 37999 9249 39608 91822 41204 91116 40 21 34775 93759 364o6 93137 3826 92488 39635 91810 4i.23I 9I104 39 22 348o3 93748 36434 93127 38o53 92477 39661 91799 41257 91092 38 23 -34830 93738 3646i 93116 3.8080 92466 39688 91787 41284 91080 37 24 34857 93728 36488 93106 3.8.07 92455 39715 91775 4i3io 91068 36 25 34884 93718 365i5 93095 38i34 92444 39741 91764 4I337 91056 35 26 349I2 93708 36542 93084 38i6i 92432 39768 91752 41363 91044 34 34939 93698 36569 93074 38i88 92421 39795 91741 41390 9o032 33 2 34966 9368 36596 93o63 382I5 92410 39822 91729 4i4I6 91020 32 29 34993 93677 36623 93052 38241 92399 39848 9171I 41443 91008 31 30 3502I 93667 3665o 93042 38268- 92388 39875 91706 41469 90996 30 3I 35048 93657 36677 9303I 38295 92377 39902 9694 41496 90984 2 32 35075 93647 36704 93020 38322 92366 39928 91683 41522 90972 28 33 35102 93637 36731 93010 38349 92355 39955 91671 4I549 90960 27 34 3530o 93626 36758 92999 38376 92343 39982 9i660 41575 90948 26 35 3557 93616 36785 92988 38403 92332 40008 91648 41602 90936 25 36 35L84 93606 36812 9297.8 3843o 92321 40035 91636 4I628 90924 24 3 3521 93596 36839 92967 38456 92310 40062 91625 4I655 90o91 23 3 35239 93585 6867 92956 38483 9229 40oo88 91613 4i68I 90899 22 39 35266 93575 36894 92945 385io 922 7 4oII5 91601 41707 90887 21 40 35293 93565 36921.92935 38537 92276 40II 9 590 41734 90875 20 41 35320 93555 36948 92924 38564 92265 40o68 91578 41760 90863 19 *42 35347 93544 36975 92913 3859I 92254 40195 9I566 41787 9085 18 43 35375 93534 37002 92902 38617 92243 40221 91555 4i8I3 90839 I 44 35402 93524 37029 92802 38644 92231 40248 91543 41840 90826 I 45 35429 935I4 37056 9288 38671 92220 40275 91531 4I866 90814 15 46 35456 93503 37083 92870. 38698' 9220 40o30I 915I9 4I892 90802 14 47 35484 93493 37110 92859 38725 92I1 4o328 9I508 41919 90790 13 48 355I 93483 37I37 92849 38752 9216 40355 91496 41945 90778 12 49 35538 93472 37164 92838 38778 92175 4o38i 91484 41972 90766 II 50 35565 93462 3791 92827 388o5 92I64 40408 91472 41998 90753 1O 51 35592 93452 37218 92816 38832 92152 4o434 91461 42024 90741 9 52 356i9 9344I 37245 92805 38859 92141 4o46i 91449 42051 90729 8 53 35647 9343I 37272 92794 38886 92130 40488 91437 42077 9077 1 7 54 35674 93420 37299 92784 38912 92119 44o54 91425 42104 90704:55 35701 93410 37326 92773 38939 9207 4054i 91414 42I30 90602 5 56 35728 93400 37353 92762 38966 92096 40567 91402 42I56 90680 57 35755 93389 37380 9275I 38993 92085 40594 9390 42I83 90668 3 58 35782 93379 37407 92740 39020 92073 40621 91378 42209 o0655 2 59 358io 93368 37434 92729 39046 92062 40647 9366 42235 90643 60 35837 93358 37461 9271 39073 92050 40674 9i355 42262' 90631 o Cosine. Sine. I ineCosine. Sine. osine. Sine. Cosine. I Sine. Cosine. Sine. 690 68~ 6~ 66~ 65~ 22 NATURAL SINES AND COSINES. 25~. 26~ 270 28~ - 29 Sine. Cosiie. oe Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 0 42262 90631 43837 89879 45399 89101, 46947 88295 48481 87462 6o 1 42288 90618 43863 89867 45425 89087 46973 88281 485o6 87448 59 2 42315 90606 43889 89854 4545i 89074 46999 88267 48532 87434 58 3 42341 90594 43916 89841. 45477 89061 47024 88254 48557 87420 57 4 42367 90582 43942 89828 455o3 89048 47050 88240 48583 87406 56 5 42394 90569 43968 89816 45529 89035 47076 88226 48608 8739I 55 6 42420 90557 43994 89803 45554 89021 47101 88213 48634 87377 54 7 42446 90545 44020 89790 4558o 89008 47127 8899 48659 87363 53 8 42473 90532 44o46 89777 456o6 88995 47153 88I85 48684 87349 52 9 42499 90520 — 44072 89764 45632 88981 47178 88172 48710 87335 5i 10 42525 90507 44098 89752 45658 88968 47204 88158 48735 87321 50 iI 42552 90495 44124 89739 45684 88955 47229 88144 4876I 87306 49 12 42578 90483 44i5i 89726 45710 88942 47255. 88i3o 48786 87292 48 13 42604 90470 44177 89713 45736 88928 47281 88117 48811 87278 47 14 42631 90458 44203 89700 45762 88915 47306 88io3 48837 87264 46 15 42657 90446 44229 89687 45787 88902 47332 88089 48862 87250 45 16 42683 90433 44255 89674 458I3 88888 47358' 88075 48888 87235 44 I7 42709 90421 44281 89662 45839 88875 47383 88062 48913 87221 43 8 42736 90408 44307 89649 45865 88862 47409 88o48 48938 87207 42 19 42762 90396 44333 89636 4589I 88848 47434 88034 48964 87193 4I 20 42788 90383 44359 89623 45917 88835 47460 88020 48989 87178 40 21 42815 90371 44385 89610 45942 88822 47486 88006 49014 87164 39 22 42841 90358 44411 89597 45968 888o8 47511 87993 49040 87150 38 23 42867 90346 44437 89584 45994 88795 47537 8979 49065 -87136 37 24 42894 90334 44464 89571 46020 88782 47562 87965 49090 87121 36 25 42920 90321 44490 89558 46046 88768 47588 87951 49116 87107 35 26 42946 90309 445i6 89545 46072 88755 47614 87937 4914I 87093 34 27 42972 90296 44542 89532 46097 88741 47639 87923 49166 87079 33 2 42999 90284 44568 895I9 46I23 88728 47663 87909 49I92 87064 32 29 43025 90271 44594 895o06 46149 887I5 47690 87896 49217 87050 3i, 30 43o5i 90259 44620 89493 46I75 88701 47716 87882 49242 87036.30 3I 43077 90246 44646 89480 46201 88688 4774I 87868 49268 8702I 29 32 43io4 90233 44672 89467 46226 88674 47767 87854 49293 87007 28.33 43i30 90221 44698 89454 46252 88661 47793 87840 493i8 86993 27 34 43I56 90208 44724 8944I 46278 88647 47858 87826 49344 86978 26 35 43i82 90196 44750 89428 46304 88634 47844 87812 49369 86964 25 36 43209 90183 44776 89415 4633o 8860 47869 87798 49394 86949 24 37 43235 9017I 44802 89402 46355 88607 47895 87784 494I9 86935 23 38 4326i 90I58 44828 89389 46381 88593 47920 87770 49445 86921 22 39 43287 90I46 44854 89376 46407 8858o 47946 87756 49470 86906 21 40 43313 90133 4488o 89363 46433 88566 4797I 87743 49495 86892 20 41 43340 90120 449o0 89350 46458 88553 47997 87729 4952I 86878 19 42 43366 90108 44932 89337 46484 88539 48022 87715 49546 86863 i8 43 43392 90095 44958 89324 465Io 88526 48048 87701 4957I 86849 -7 44 434i8 90082 44984 89311 46536 88512 48073 87687 49596 86834 i6 45 43445 90070 450io 89298 46561 88499 48099 87673 49622 86820 i5 46 43471 90057 45o36 89285 46587 88485 48124 87659 49647 868o5 14. 47 43497 90045 45o62 89272 466I3 88472 48i5o 87645 49672 86791 i3 4 43523 90032 45o88 89259 46639 88458 48175 87631 49697 86777 2 49 43549 900o9 1 5I4 89245 46664 88445 48201 87617 49723 86762 II 50 43575 90007 4514o 89232 46690 88431 48226 87603 49748 86748 io 5I 43602 89944 45i66 89219 46716 88417 48252 87589 49773 86733. 52 43628 89.9g1 45I92 89206 46742 88404 48277 87575 49798 86719 53 43654 89968 452i8 89193 46767 88390 483o3 8756i 49824 86704 7 54 4368o 89956 45243 89180 46793 88377 48328 87546 49849 86690 6 55 43706 89943 45269 89167 468i9 88363 48354 87532 49874 86675 5 56 43733 89930 45295 89153 46844 88349 48379 875I8 49899 86661 4 57 43759 89918 4532I 89140 46870 88336 484o5 87504 49924 86646 3 58 43785 8905 45347 89I27 46896 88322 4843o 87490 4995o 86632 2 59 438iI 89892 45373 89114 46921 883o8 48456. 87476 49975 86617 I 60 43837 89879 45399 89o10 46947 88295 4848I 87462 5oooo 866o3 o Cosine. I Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 640 63~ 620 61~ 60~ 23 NATURAL SINES AND COSINES. 300 31~ 32~ 330 340 Sine. Cosine. Sine. Cosine. Sine. Cosine. Cosine. Sine. Cosine. o 5oooo 866o3 5I504 857I7 52992 84805 54464 83867 5599 82904 60 I 50025 86588 5i529 85702 53oi7 84789 54488 8385- 55943 8287 59 2 5oo5o 86573 5I554 85687 53o4i 84774 545I3 83835 55968 82871I 5 3 50076 86559 51579 85672 53066 84759 54537 8389 55992 82855 57 4 50o10 86544 5I604 85657 53091 84743 5456i 83804 56i6 82839 56 5 5oI26 8653o 5I628 85642 53I51 84728 54586 83788 56040 82822 55 6 50oi5 865x5 5i653 85627 5340o 84712 546io 83772 56o64 82806 54 7 50176 865oi 51678 85612 53i64 84697 54635 83756 56o88 82790 53 8 5020I 86486 5I703 85597 53I89 8468I 54659 83740 56II2 82773 52 9 50227 86471 51728 85582 53214 84666 54683 83724 56i36 82757 5I 1O 50252 86457 5I753 85567 53238 84650 54708 83708 56i6o 82741 5o 11 50277 86442 51778 8555i 53263 84635 54732 83692 56I84 8272449 12 50302 86427 5i803 85536 53.288 8469 54756 *83676 56208 82708 4 I3 50327 864I3 51828 85521 533I2 84604 5478I 8366o 56232 82692 47 I4 5o352 86398 5i852 855o6 53337 84588 548o5 83645 56256 82675 46 15 5o377 86384 51877 85491' 5336i 84573 54829 83629 56280 82659 45 I6 50403 86369 51902 85476 53386 84557 54854 836I3 56305 82643 44 I7 50428 86354 51927 8546I 534II 84542 54878 83597 56329 82626 43 I 50453 86340 5I952 85446 53435 84526 54902 8358I 56353 82610 42 19 50478 8,6325 5i977 8543I 5346o 8451 54927 83565 56377 82593 41 20 50503 863o 52002 854i6 53484 84495 54951 83549 564oi 82577 40 21 50528 86295 52026 854oi 53509 8448o 54975 83533 56425 8256I 39 22 5o553 86281 5205I 85385 53534 84464 54999 83517 56449 82544 38 23 50578 86266 52076 85370 53558 84448 55024 835oI 56473 82528 3 24 5o6o3 8625I 52IOI 85355 53583 84433 55048 83485 56497 82511 36 25 50628 86237 52126 85340 53607 84417 55072 83469 56521 82495 35 26 5o654 86222 5215i 85325 53632 84402 55097 83453 56545 82478 34 2 50679 86207 52175 853io 53656 84386 5512I 83437 5656 82462 33 28 50704 86192 52200 85294 5368i 84370 55i45 8342 56593 82446 32 29 50729 86178 52225 85279 53705 84355 55I69 834o5 566I7 82429 3i 30 50754 86i63 52250 85264 53730 84339 55I94 83389 5664i 824I3 3o 31 50779 86i48 52275 85249 53754 84324 55218 83373 56665 82396 29 32 5o8o04 86i33 52299 85234 53779 843o8.55242 83356 56689 82380 28 33 50829 86II9 52324 852I8 53804 84292 55266 83340 5673 82363 27 34 50854 8604 52349 85203 53828 84277 5529I 83324 56736 82347 26 3.5 50879 86089 52374 85i88 53853 84261 553i5 833o8 56760 82330 25 36 5090o4 86074 52399 85I73 53877 84245 55339 83292 56784 82314 24 37 50929 86o59 52423 85I57 53902 84230 55363 83276 568o8 82297 23 38 50954 86o45 52448 85I42 53926 84214 55388 83260 56832 8228 1 22 39 50979 86630 52473 85127 5395i 84I98 55412 83244 56856 82264 2i 40 5ioo4 86oi5 52498 8512 53975. 84 82 55436 83228 56880 82248 20 41 51029 86ooo 52522 85096 540oo 84I67 5546o 83212 56904 8223I I 42 5io54 85985 52547 85o8 54024 84I5i 55484 83I95 56928 822I4 I 43 51079 85970 52572 85o66 54o49 84I35 55509 83179 56952 82198 17 44 5IIo4 85956 52597 85o5i 54073 84120 55533 83i63 56976 82181 I6 45 5129 8594I 52621 85o35 54097 8404 55557 8347 57000 82i65 i5 46 5I54 -85926 52646 85020 54I22 84088 5558 83i3i 57024 82148 14 47 51179 8591 52671 85oo5.54I46 84072 556o5 83ii5 57047 82132 3 48 51204 8586 52696 84989 54171 84057 5563o 83098 57071 82115 2 49 51229 8588 52720 84974 54195 8404 55654 83o82 57095 82098 i 50 51254 85866 52745 84959 54220 84025 55678 83o66 57II1 82082 I 5I 51279 8585I 52770 84943 54244 84009 5702 83o5o 57143 82o65 52 53o04 85836 52794 84928 54269 83994 55726 83o34 57I67 82048 8 53 5i329 8582I 5289 8491 54293 83978 55750 83017 5719 820321 7 54 5i354 858o6 52844 84897 54317 83962 55775 83ooi 5721 82015 6 55 5i379 85792 52869 84882 54342 83946 55799 82985 57238 81999 5 56 5I4o4 85777 52893 84866 54366 8393o 55823 82969 57262 81982 4 5 51429 85762 52918 8485I 5439 83915 55847 82953 57286 81965 3 5 5454 85747 52943 84836 544,5 83899 55871 82936 573o 81949 2 59 51479 85732 52967 84820 54440 83883 55895 82920 57334 81932 I 60 5I504 857I.7 52992 84805 54464 83867 55919 82904 57358 81915 o 0 Cosine. Sine. Cosine. Sine. Cosine.'Sine. Cosine. Sine. Cosine. Sine. 24 NATURAL SINES AND COSINES. 85~ 36~ 37~ 388~ 1 39~ Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. SineCosine. o 57358 81Y5 58779 80902 6082 79864 6i566 7880 62932 77715 6o 1 57381 81899 588o2 8o885 60205 79846 61589 78783 62955 77696 59 2 57405 81882 58826 80867 60228 79829.61612 78765 62977 77678 58 5742 8865 58849 80850 6025I 79811 6i635 78747 630oo00 77660 57 4 57453 81848 58873 8o833 60274 79793 6i658 78729 63022 77641 i 56 5 57477 81832 58896 80816 60298 79776 6i68i 78711 63045 77623 55 6 575oi 8i8i5 58920 80799 60321 79758 6I704 78694 63068 77605 54 7 57524 81798 58943 80782 60344 7974I 6I726 78676 63090 77586 53 8 57548 81782 58967 80765 60367 79723 61749 78658 63113 77568 02 9 57572 81765 58990 80748 60390 79706 61772 78640 6335 77550 5i 10 57596 81748 59014 80730 6o414 79688 61795 78622 63i58 77531 50 I 5769 8173I 50o37 80713 60437 79671 6i8i8 78604 63i8o 77513 I49 2 57643 81714 5906 80696 60460 79653 61841 78586 63203 774941 48 13 57667 8I698 59084 80679 60483 79635 6i864 78568 63225 77476 47 14 5769I 8168i 59io8 80662 60506 79618 61887 78550 63248 77458 46 I5 57715 81664 59i3I 80644 6o529 79600 61909 78532 63271 774391 45 I6 57738 81647 59154 80627 60553 79583 61932 78514 63293 77421 44 17 57762 8i63i 59178 806io 60576 79565 61955 78496 63316 77402 43 18 57786 8I6I4 59201 80593 60599 79547 61978 78478 63338 77384! 42 19 57810 81597 59225 80576 60622, 79530 62001 78460 6336i 77366 41 20 57833 8i58o 59248 80558 60645 79512 62024 78442- 63383 77347 40 2I 57857 81563 59272 8054i 60668 79494 62046 78424 63406 77329 39 22 57881 8546 59295 80524 60691 179477 62069 784o5 63428 77310! 3 23 57904 8i530 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 62L38 7835I 63496 77255 35 26 57976 81479 59389 80455 60784 79406 62160 78333 635i8 77236 34 2 57999 81462 59412 80438 60807 79388 6213 83i5 63540 77218 33 2 5802 8445 59436 -80420 60830 7937I 62206 78297 63563 77199 32 29 58047 81428 59459 8o4o3 6o853 79353 62229 78279 63585 77181 3i 30 58070 81412 59482 8o386 60876 79335 6225I 78261 636o8 77162 30 31 58094 81395 59506 8o368 60899 79318 62274 78243 6363o 77144 29 32 58Ii8 81378 59529 8035i 60922 79300 62297 78225 63653 77125 28 33 58i4i 8i36i 59552 80334 60945' 79282 62320 78206 63675 77107 27 34 58i65 81344 59576 8o3i6 60968 79264 62342 78188 63698 77088 26 35 58I89 8i327 59599 80299 60991 79247 62365 78170 63720 77070! 25 36 58212 8i3io 59622 80282 6ioi5 79229 62388 78I52 63742 7705i 24 37 58236'81293 59646 80264 6o1038 79211 62411 78134 63765 77033 23 38 5826o 81276 59669 80247 60o6I 79193 62433 78116 63787 77014 22 39 58283 81259 59693 80230 6I084 79176 62456 78098 638io 76996 21'40 58307 81242 59716 80212 61107 79158 62479 78079 63832 76977 20 41 5833o 81225 59739 80195 6130o 79140 62502 78061 63854 76959 19 42 58354 81208 59763 80178 61153 79122 62524 78043 63877 76940 i8' 43 58378 81191 59786 8oi60 61176 79105 62547 78025 63899 76921 17 44 584o0 81174 59o09 8o043 61199 79087 62570 78007 63922 76903 I6 45 58425 8i57 59832 8o025 61222 79069 62592 77988 63944 76884 I5 46 58449 8II40 59856 80oo8 61245 7905I 62615 77970 63966 76866 I4 47 58472 81123 59879 80091 61268 79033 62638 77952 63989 76847 I3 48 58496 8I06 59902 80073 6129I 79016 62660 77934 64oII 76828. 12 49 5859 81089 59926 80056 6i314 78998 62683 77916 64033 76810 II 50 58543 81072 59949 80038 61337 78980 62706 77897 64056 7679I io 5I 58567 8io55 59972 80021 6i360 78962 62728 77879 64078 76772. 9 52 58590 8io38 59995 80003 61383 78944 62751 7786I 64i00 76754 8 53 58614 81021 60019.79986 61406 78926 62774 77843 64123 76735 7 54 58637 81oo004 60042 79968 61429 78908 62796 77824 64i45 76717 6 55 5866i 80987 60o65 79951 61451 78891 62819 77806 64I67 76698 5 56 58684 80970 60089 79934 61474 78873 62842 77788 64190 76679 4 57 58708 80953 60II2 799I6 61497 78855 62864 77769 64212 76661 3 58 58731 80936 60o35 79899 6i520 78837 62887 7775i 64234 76642 2 59 58755 80919 6oi58 79881 61543'78819 62909 77733 64256 76623 I 60 58779 80902 60182 79864 6i566 78801 62932 77715 64279 76604 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. / --- -.... ----- -, 54~.530 52~ 51~ 50 25 NATURAL SINES AND COSINES. 400 41~ 4420 430 44~ Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o 64279 76604 65606 7547 66913 74314 68200 73135 69466 71934 6o 1 643o0 76586 65628 75452 66935 74295 68221 73116 69487 71914 59 2 64323 76567 65650 75433 66956 74276 68242 73096 69508 71894 58 3 64346 76548 65672 75414 66978 74256 68264 73076 69529 71873 57 4 64368 76530 65694 75395 66999 74237 68285 73056 69549 71853 56 5 64390 76511 65716 75375 67021 74217 6830o 736 69570 71833 55 6 64412 76492 65738 75356 67043 74198 68327 73016 69591 71813 54 7 64435 76473 65759 75337 67064 74178 68349 72996 69612 71792 53 8 64457 76455 65781 753I8 67086 74159 68370 72976 69633 71772 52 9 64479 76436 658o3 75299 67107 74139 6839I 72957 69654 71752 5i 1o. 645o0 76417 65825 75280 67129 74120 68412 72937 69675 71732 50 ii 64524 76398 65847 75261 67151 74100 68434 72917 69696 71711 I 4 12 64546 76380 65869 7524I 67172 74080 68455 72897 69717 7169 48 13 64568 76361 65891 75222 67194 74061 68476 72877 69737 71671 47 14 64590 76342 65913 75203 67215 7404I 68497 72857 69758 7I650 46 I5 64612 76323 65935 75184 67237 74022 685I8 72837 69779 71630 45 16 64635 76304 65956 75i65 67258 74002 68539 72817 69800 71610 44 I7 64657 76286 65978 75146 67280 73983 6856i 72797 69821 71590 43 I8 64679 76267 66000 75126 67301 73963 68582 72777 69842 71569 42 19 64701 76248 66022. 75107 67323 73944 68603 72757 69862 71549 4i 20 64723 76229 66044 75088 67344 73924 68624 72737 69883 71529 40 21 64746 76210 66066 75069 67366 73904 68645 72717 69904 7I508 39 22 64768 76192 66088 75050 67387 73885 68666 72697 69925 71488 38 23 64790 76173 66109 75030 67409 73865 68688 72677 69946 7 468 37 24 64812 76154 66I3I 750II 67430 73846 68709 72657 69966 71447 36 25 64834 763:5 66i53 74992 67452 73826 68730 72637 69987 71427 35 26 64856 76116 66175 74973 67473 73806 68751 72617 70008 71407 34 27 64878 76097 66I97 74953 67495 73787 68772 72597 70029 71386 33 28 6490 76078 66218 74934 67516 73767 68793 72577 70049 71366 32 29 64923 76059 66240 74915 67538 7747 688i4 72557 70070 71345 3i 30 64945 7604I 66262 74896 67559 73728 68835 72537 70091 71325 3o 3I 64967 76022 66284 74876 67580 73708 68857 72517 70112 71305 2 32 64989 76003 663o6 74857 67602 73688 68878 72497 70132 71284 28 33 650oI 75984 66327 74838 67623 73669 68899 72477 70I53 7I264 27 34 65033 75965 66349 74818 67645 73649 68920 72457 70I74 71243 26 35 65055 75946 66371 74799 67666 73629 68941 72437 70195 7I223 25i 36 65077 75927 66393 74780 67688 73610 68962 72417 702I5 71203 24 37 65ioo 75908 664I4 74760 67709 73590 68983 72397 70236 71182 23 38 6522 75889 66436 7474I 67730 73570 69004 72377 70257 71162 22 39 65I44 75870 66458 74722 67752 7355I 69025 72357 7027 71141 21 40 65i66 7585i 66480 74703 67773 73531 69046 72337 70298 71121 20 41 65i88 75832 665oi 74683 67795 735 1 69067 72317 70319 7I100 I 42 65210 75813 66523 74664 67816 73491 69088 72297 70339 71080 8 43 65232 75794 66545 74644 67837 73472 69109 72277 70360 71059 17 44 65254 75775 66566 74625 67859 73452 69130 72257 70381 71039 i6 45 65276 75756 66588 74606 67880 73432 69151 72236 70401 71019 15 46 65298 75738 66610 74586 67901 734I3 69172 72216 70422 70998 I4 47 65320 75719 66632 74567 67923 73393 69193 72I96 70443' 70978 I3 48 65342 75700 66653 74548 67944 73373 69214 72176 70463 70957 12 49 65364 75680 66675 74528 67965 73353 69235 72156 70484 70937 I7 50 65386 7566i 66697 74509 67987 73333 69256 72136 70505 70916 io 5I 65408 75642 66718 74489 68008 73314 69277 72116'0525 70896 52 65430 75623 66740 74470 68029 73294 69298 72095 70546 70875. 53 65452 75604 66762 7445I 68o5i 73274 69319 72075 70567 70855 7 54 65474 75585 66783 74431 68072 73254 69340 72055 70587 70834 55 65496 75566 66805 74412 68093 73234 69361 72035 70608 70813 5 56 655i8 75547 66827 74392 68iI5 73215 69382 72015 70628 70793 4 57 65540 75528 66848 74373 68i36 73195 69403 71995 70649 70772 3. 58 65562.75509 66870 74353 68157 73175 69424 71974 70670 70752 2 59- 65584 75490 66891 74334 68179 73155 69445 71954 70690 70731 I 60 65606 7547I 66913 743I4 68200 73i35 69466 71934 70711 70711 0 Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 49 ~,.........; I" _ 490 ~ 480 1~ 40~ 46~' 460 26 NATURAL TANGENTS AND COTANGENTS. 00 10 20 30! Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. an. g. 0 0000ooooo Infinite. 01746 57.2900 o3492 28-6363 0524I I9.08ii 60 I 00029 3437.75 o0775 56.35o6 o352i 28-3994 o5270 I8755 59 2 ooo58 7i8.87 Oi804 55.44i5 o3550 28.1664 o5299 18- 71 58 3 00087 1145.92 oi833 54.5613 03579 27-9372 o5328 18.7678 57 4 ooI16 859*436 01862 53.7086 o36o 27.7117 o5357 i8.6656 56 5 ooI45 687-549 01891 52.882I o3638 27-4899 05387 I8.5645 55 6 00175 572.957 01920 52.0807 03667 27-2715 054I6 I8.4645 54 7 00204 49I.I06 I0949 51.3032 03696 27o0566 05445 I8-3655 53 8 oo233 429.718 01978 50.5485 03725 26-8450 05474 18.2677 52 9 00262 38i.971 02007 49-8157 03754 26-6367 o5503 18.1708 5i 10 0029I 343-774 02036 49I039 03283 26 436 o5533 18.0750 5o II 00320 3I252I 02066 48-4I2I 038I2 26-2296 05562 17-9802 49 12 00349 286.,478 02095 47-7395 o3842 260307 05591 I7-8863 48 i3 00378 264-441 02124 47-0853 03871 25-8348 o5620 I7.7934 47 14 00407 245.552 02153 46-4489 03900 25-64I8 o5649 17-7015 46 15 oo436 229.182 02182 45-8294 03929 25.4517 05678 17.6io6 45 I6 00465 2I4-858 02211 45.226I 03958 25-2644 05708 17.5205 44 17 00495 202-2I9 02240 44-6386 03987 25-0798 05737 I7.43I4 43 18 00524 I90.984 02269 44.066I 040o6 24.8978 05766 17.3432 42 19 oo553 80.932 02298 43.508i o4046 24.7185 05795 17.2558 4I 20 00582 I7I.885 02328 42.9641 04075 24-54I8 o5824 I7-1693 4o 21 oo6II- 63.700 02357 42-4335 -o4o4 243675 o5854 17-0837 39 22 oo640 I56-259 02386 4I.9I58 o4I33 24-I957 o5883 16.9990 38 23 oo669 149-465 0245 4I.4io6 04I62 24-0263 05912 16. i50 37 24 o0698- 143.237 02444 40.9174 04191 23.8593 o5941 I6.83I9 36 25 00727 I37.507 02473 40.4358 04220 23.6945 o5970 I6.7496 35 26 oo756 I32.2I9 02502 39.9655 04250 23.532I o5999 i6.668i 34 27 00785 I2.7.321 0253I 395o5 o04279 23.3718 o602g I6.5874 33 28 oo814 122-774 02560 39.o568 o4308 23.2137 o6o58 6.5075 32 29 oo844 II8.54o o2589 38.6r77 o4337 23.577 06087 I6.4283 31 30 o00873 I4.589 02619 38.885 o4366 22.9038 06116 I6.3499 30 31 00902 110892 o02648 37.7686 04395 22.7519 06I45 I6.2722 29 32 00931 107.426 02677 37-3579 04424 22-6020 06175 I6.1952 28 33 oo960 104.171 02706 36.956o o4454 22-454I 06204 16.1190 27 34 o00989 101.07 02735 36.5 627 04483 22.308I o6233 i6.o435 26 035 1ooI8 98.2179 02764 36.1776 04512 22.1640 o06262 5.9687 25 36 o0o47 95.4895 o2793 35.8oo6 o454i 22.02 o06291 15.8945 24 37 o1076 92.985 02822 35.433 o4570 21.883 o6321 I5.82i1 23 38 oiio5 90.4633 0285i 35.o695 04599 21.7426 o635o i5.7483 22 39 oii35 88.I436 02881 34,715i o4628 2I.6o56 o6379 15.6762 2z 40 o0164 85.9398 02910 34.3678 o4658 21.4704 o6408 I5.6o48 20 41 OII93 83.8435 02939 34-0273 o4687 2I-3369 06437 i5.534o0 1 42 01222 8.8470 02968 33.6935 04716 21.2049 o6467 i5.4638 18 43 oI25I 79.9434 02997 33.3662 04745 21.0747 06496 i5.3943 17 44 01280 78.1263 o3026 33.o452 04774 20.9460 o6525 5.3254 i6 45 oI3o9 76.3900 o3o55 32.73o3 o48o3 20.8188 o6554 I5.257I i5 46 oi338 74.7292 o3o84 32.4213 o4832 20.6932 o6584 15.1893 14 47 o0367 73.390o o3II4 32.1181 04862 20-569I o66i3 I5 222 13 48 o1396 71.6i51 0o343 3i.82o5 04891 20.4465 06642 I5.o557 12 49 01425 70o.533 03172 3I.5284 04920 20.3253 06671 14-9898 ii 50 oi455 68-750I 0320I 3I.24I6 04949 20.2056 o6700 14-9244 o1 5I 01484 674019 03230 30.9599 04978 20.0872 06730 148596 9 52 oI5i3 66.o55 o3259 3o.6833 o5007 19.9702 o6759 147954 8 53 0I542 648580 o3288 3o.4I6 o5o37 I9.8546 o6788 I4-737 7 54 oi57I 63.6567 o33i7 30o.446 o5o66 19-7403 06817 I4-6685 6 55 oi6oo 6a24992 o3346 29.8823 o5o95 19g6273 o6847 14.6059 5 56 oI629 61-3829 3376 29-6245 o5I24 i9-5I56 06876 14-5438 4 57 oi658 6o-305 o34o5 29-3711 05I53 i9-4o05 o6905 14.4823 3 58 016$7 59-2659 o3434 29.1220 o5I82 9 2959 06934 I4-42I2 2 59 01716 58.2612 o3463 28.8771 0o5212 I9.1879 o6963 I4-3607 60 oI746 57-2900 o3492 28.6363 0524I 19-081 o06993 I4.30o7 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. -90S C 88~0 s~ 86~ 2i7 NATURAL TANGENTS AND COTANGENTS. j 40 50 6~0' Tangent. Cotang. Tangent Ctang. angent. Cotang. Tangent. Cotang. o o6993 I4- 3o7 0 8749 I I 43oI io5io 9.5I4,36 12278 8. 4435 6o I 07022 I4-24II o8778 II 399 Io54o 9-4878I 12308 8. I2481 59 2 07051 I4.I821 08807 II.3540 10569 9.46i4I 12338 8.io536 58 3 07080 I4-I235 08837 I I.3 63 Io590 9.43515 12367 8 o86oo 57 4 07I10 I4.0655 o8866 II 2789 10628 9.40904 12397 8.06674 56 5 o0739 14.o0079 o8895 II-2417 0o657 9.38307 12426 804756 55 6 0o768 I3.9507 08925 11.2048 I0687 9.35724 12456 8.02848 54 7 0797 138940 08954 ii.iI68 o1071 6 9.33I54 12485 8.00948 53 07227 I3-8378 o8983 ii.I3i6 10746 9-3.0599 12515 7-99058 52 9 o72561 3.7821 o90i3 I0og954 10775 9.28058 12544 7-97176 5i 10 07285 13-7267 o09042 11-o594 io8o5 9.2553.o 12574 7.95302 50 II 07314 I3.6719 09071 I-o0237 io834 9.23oi6 12603 7.93438 49 12 07344 13-6I74 09101 I109882 io863 9.2o5i6 12633 7-91582 48 13 07373 I3.5634 09130 I0o9529 10893 9-I8028 12662 7.89734 47 I4 07402 13.5098 09159 o10-178 10922 9-15554 12692 7.87895 46 15 07431 I3-4566 09189 IO.8829 10952 9-13093 12722 7.86064 45 I6 0746i I34o3 4 09218 Io08483 109g81 910646 1275I 7-84242 44 17 07490 I3.355 o09247 0o-8I39 I-IOII 9-08211 12781 7.,82428 43 18 07519 13.2996 09277 10-7797 1io4o 99.o5-789 12810 7.80622 42 19 07548 3-24o0 09306 107457 11070 9o3379 12840 7.78825 4I 20 07578 I3-1969 09335 -711 9 II099 -9oo983 12869 7.77035 40 21 07607 I3.I46i o09365 Io06783 11128 898598 I2899 7-75254 39 22 07636 J39o58 09394 io.6450 iii58 8.962.27 12929 7.73480 38 23 07665 I3.o458 09423 io.6 I8 11187 8.93867 12958 7-71715 37 24 07695 12.9962 09453 105789 II2I7 8.9I520 12988 7.69957 36 25 07724 12.9469 09482 10-5462 11246 8.89185 i30I7 768208 35 26 07753 I 128981 09511 Io5i36 11276 8.86862 13047 7.66466 34 27 07782 12.8496 09541 1io48I3 i305 8.8455I 13076 7.64732 33 28 07812 12.8i4 09570 10-4491 ii335 8.82252 i3io6 7.63005 32 29 0784I 12-7536 09600 Io0.4172 II364 8-79964 i3i36 7.61287 3i 3o 07870 12.7062 09629 Io.3854 II394 8.77689 i3i65 7.59575 30 31 07899 12.6591 09658 io.3538 11423 8.75425 13195 7.57872 29 32 07929 2.6i24 09688 10.3224 I1452 8.73172 13224 7-56I76 28 33 07958 I2.566o 09717 102913 11482 8.7093I I3254 7.54487 27 34 07987 I2.5I99 o9746 10-2602 I 5.ii 8.68701 i3284 7-52806 26 35 o80o7 12.4742 09776 10-2294 ii54i 8.66482 i33i3 7.5I132 25 36 08046 12-4288 o9805 10.1988 II570 8.64275' I3343 7-49465 24 37 08075 12.3838 09834 io-i683 ii6oo 8-62078 i3372 7.47806 23 38 08o14 I2.3390 09864 io.i38i 11629 8.59893 13402 7.46154 22 39 o8I34 12-2946 o09893. ioio8o0 i659 8.577 I8 3432 7.44509 21 40 o8i63 12.2D05 09923 O10o.780 ii688 8 *5555555 346I 7.42871 20 41 08192 12.2067 09952 Io.o483 11718 853402 13491 7.41240 19 42 08221 I2-i632 09981 O1-0187 11747 8-5i259 i3521 7-39616 18 43 08251 12.I201 IOOII 9-98930 11777 8.49128 1355o 7.37999 17 44 o8280 2.0772 00oo4o 9.96007 118o6 847007 i358o 7.36389 16 45 o8309' 2.o346 10069 9.93101 I836 8.44896 136o9 7.34786 15 46 o8330 II.9923 1009o 9.0211 i865 842795 I3639 7.33190 14 4 o8368 li,95o4 o1028 9.87338 11895 8.40705 i3669 7.3i6oo I3 4 08397 11.9087 ioi58 9-84482 11924 8.38625 I3698 7.30o08 12 49 08427 11.8673 11o87 9'8I641 11954 8.36555 I3728 7.28442 II 50o 8456 I1.8262 10216 9.78817 11983 8 3,4496 i3758 7.26873 io 5i o8485 II.7853 10246 9.76009 I20o3 8.32446 I3787 7.253Io 52 085I4 11.7448 10275 9.732I7 12042 8.30406 I3817 7.23754 8 53 08544 11.7045 Io3o5 9.70441 I2072 8.28376 13846 7 22204 7 54 08573 I.6645 io334 9.67680 12ior 8.26355 13876 7.20661 6 55 o8602 1 1.6248 io363 9.64935 12I3i 8-24345 3906 7.19125 5 56 o8632 II.5853 I0393 9.62205 12i60 8.22344 I3935 7.-7594 4 57 866I 11.546 I10422 9-59490 29 0 2 8. 20352 I3965 7.-16071 3 58 08690 I.5072 10452 9.5679i 12219 8-18370 13995 7-I4553 2 59 08720 11 4685 I048I 9.54o06 I2249 88i6398 14024 7'13042 I I I I4 0 60 08749 -430o io5io 9-5i436 12278 8[I4435 i4o54 7.11537 o Cotang. Tangent. Cotang. Tangent. Cotang. -Tangent. Cotang. Tangent. 850 840 880 82~ 28 NATURAL TANGENTS AND COTANGENTS. 8o 90 10~ 11~ Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 14054 7.11537 i5838 6.3i375 17633 5.67I28 19438 5.I4455 6o I 14084 7.0oo38 I5868 6.30189 I7663 5.66i65 19468 5.i3658 5 2 i413 7.08546 15898 6.29007 17693 5.65205 19498 5.I2862 58 3 4143 7.07059 15928 6.27829 17723 5.64248 19529 51I2069 57 4' 1473 7-05579 I5958 6.26655 17753 5.63295 19559 5.11279 56 5 14202 7.04105 15988 6.25486 17783 5.62344 19589 5.10490 55 6 14232 7-02637 16017 6.2432I 17813 56I397' 19619 5.09704 54 7 4262 7'OII74 I6047 6.23I60 I7843 5.60452 I9649 5.08921 53 14291 6.997I8 16077 6.22003 17873 5.595II 19680 5o08i39 52 9 I432I 6.98268 16107 6.2085I 17903 5.58573 197IO 5.07360 5I o1 I435i 6.96823 16137 6.19703" 17933 5.57638 19740 5.06584 5o I 438i 6.95385 16167 - 6.8559 17963 5.56706 19770 5.o5809 49 12 I4410 6.93952 16196 6.17419 17993 5.55777 19801 5.o5037 48 13 I4440 6.92525 16226 6.16283 18023 5.5485i 1983 5.04267 47 14 14470 6.9II04 16256 6.i5i5i i8053 5.53927 19861 5.03499 46 I5 14499 6.89688 16286 6.-4023 i8o83 5.53007 19891 5.02734 45 16 14529 6.88278 163I6 6.12899 18i13 5.52090 19921 5.01971 44 17 14559 6.86874.6346 6.II779 18143 5.5II76 19952 5.01210 43 8 14588 6.85475 16376 6.1o664 18173 5.50264 19982 5.oo45I 42 19 14618 6.84082 16405 6.o9552 18203 5.49356 20012 4-99695 41 20 14648 6.82694 i6435 6.08444 i8233 5.4845I 20042 4-98940 4o 21 14678 6.8I312 16465 6.07340 18263 5.47548 20073 4-98188 39 22 14707 6.79936 16495 606240 18293 5.46648 200o3 4-97438 38 23 14737 6.78564 16525 6.05i43 18323 5.45751 20133 4-96690 37 24 14767 6.77I99 6555 6.o4o5i 18353 5.44857 20164 4-95945 36 25 14796 6.75838 16585 6.02962 18383 5.43966 20194 4-9520I 35 26 I4826 6.74483 I66I5 6.0878 i84i4 5-43077 20224 4-94460 34 27 14856 6.73I33 I6645 6.00797 I8444 5.42192 20254 4.93721 33 28 14886 6.71789 16674 5.99720 18474 5.41309 20285 4-92984 32 29 14915 6.70450 16704 5.98646 18504 5.40429 20315 4.92249 31 30 14945 6.69116 16734 5.97576 18534 5.39552 20345 4-9I516 30 31 14975 6.67787 16764 5.965o1 18564 5.38677 20376 4-90785 29 32 5oo5 6.66463 16794 5;95448 18594 5.37805 20406 4.90056 33 i5o34 6.65I44 I6824 5.94390 18624 5.36936 20436 4.89330 27 34 15o64 6-6383I 16854 5.93335 18654 5.36070 20466 4.88605 26 35 15094 6.62523 16884 5.92283 18684 5.35206 20497 4-87882 25 36 15124 6.61219 16914 5.91235 18714 5.34345 20527 4.87162 24 37 15153 6.59921 I6944 5.9019I 18745 5.33487 20557 4.86444 23 38 i583 6.58627 16974 5.89151 18775 5.3263I 20588 4-85727 22 39 152i3 6.57339 17004 5.88 I4 i8805 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 5.8605I i8865 5.30080 20679 4.83590 1 42 I5302 6.53503 17093 585024 8895 529235 9 42882 43 15332 6.52234 17123 5.84001 18925 5.28393 20739 4.82I75 17 44 15362 6.50970 17153 5.82982 18955 5.27553 20770 4.81471 i6 45 15391 6.49710 17183 5.81966 18986 526715 20800 4.80769 5 46 1542i 6.48456 17213 5.80953 19016 5.25880 2o830 4.80068 I4 47 1545i 6.47206 17243 5 79944 19046 5.25048 20861 4-79370 13 48 i548i 6.45961 17273 5.78938 19076 5.242i8 20891 4-78673 12 49 i55iI 6.44720 1i3o3 5.77936 19106 5.2339I 20921 4.77978 1 50 15540 6.43484 17333 5 76937 19136 5.22566 20952 477286 10 51i 5570 6.42253 17363 575941 19166 5.2744 20982 4.76595 5 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 i5660 6.38587 17453 5.72974 I9257 519293 21073 4-74534 6 55 15689 6.37374 17483 5.71992 19287 5-I8480 2104 4-7385i 5 5 15749 6-3496 17543 5.70037 I9347 5.16863 21164 4-72490 3 155779 6.3376 17573 569064 19378 5.i6o58 21195 4.7I8I3 2 59 580 6.32566 17603 5-68094 19408 5 5256 21225 4.7137 I 60o 5838 6.3I375 17633 5.67128 19438 5.I4455 21256 4.70463 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 810 800 9~ 78~0 29 NATURAL-TANGENTS AND COTANGENTS. 120 130 140 150 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. 0 21256 4170463 23087 4-33i48 24933 4.01078 26795 3. 732o5 6o I 21286 4.69791 23117 4 32573 24964 40oo582 26826 3.7277I 59 2 21316 4.6912I 23148 4-3200I 24995 40ooo86 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 4-3029I 25087 3-98607 26951 3-7I046 55 6 21438 4,66458 2327I 4-29724 25I18 3-98II7 26982 3.70616 54 7 21469 4.65797 233o0 4229159 25I49 3-97627 270I3 3-70I88 53 8 21499 4-65i38 23332 4 28595 25180 3-97139 27044 3.6976I 52 9 2I529 4.64480 23363 4.28032 25211 3-96651 27076 3-69335 5i 10 21560 4.63825 23393 4-27471 25242 3-96'65 27107 3-68909 50 II 21590 4.6317I 23424 4-26911 25273 3-95680 27138 3'68485 49 12 2162I 4.62518 23455 4.26352 25304 3-95196 27169 3.6806i 48 13 2I65I 4-6I868 23485 4-25795 25335 3-94713 27201 3.67638 47 14 21682 4-6I2I9 235i6 4,2523 25366'3 94232 27232 3'67217 46 15 21712 460572 23547 4-24685 25397 393751 27263 3-66796 45 i6 21743 4-59927 23578 4-24I32 25428 3.9327I 27294 3.66376 44 17 21773 4.59283 23608 4,23580 25459 3.92793 27326 3.65957 43 i8 21804 4.5864I 23639 4*23030 25490 3.92316 27357 3-6538 42 19 2I834 4-58ooi 23670 4-2248I 2552I 3.91839 27388 3.65I2I 41 20 21864 4-57363 23700 4,2io33 25552 3.9i364 274I9 3.64705 4o 21 2189.5 4.56726 23731 4-21387 25583 3.90890 27451 3-64289 39 22 21925 4.56091 23762 4,20842 256i4 3-90417 27482 3.63874 38 23 2i956 4 55458 23793 4 20298 25645 3.89945 275I3 3.6346 37 24 21986 4-54826 23823, 4'19756 25676 389474 27545 3.63048 36 25 22017. 4.54196 23854 4. 9215 25707 389004 27576 362636 35 26 22047 453568 23885 4.18675 25738 388536 27607 362224 34 27 22078 4*52941 23916 41i8i37 25769 3.88o68 27638 3.6i8I4 33 2 22108 4-52316 23946 4-17600 25800 3.8760I 27670 3.6I4o50 32 29 22139 4'51693 23977 41-7064 2583i 3.87136 27701 3.60996 3i 30 22169 4.51071 24008 4-i6530 25862 3,8667I 27732 3-60588 30 31 22200 4.5045i 24039 4.15997 25893 3,86208 27764 3-6oi8i 2 32 22231 4-49832 24069 4i5465 25924 3-85745 27795 3.59775 28 33 22261 4-492I5 241oo 4 I4934 25955 3 85284 27826 3 59370 27 34 2292 4-48600 2413i 4- 44o5 25986 3.84824 27858 3.58966 26 35 22322 4.47986 2462 413877 26017 384364 27889 3.58562 25 36 22353 4-47374 24193 4-i335o 26048 3-83906 27920 3 58i6o 24,3 22383 4-46764 24223 4-12825 26079 3.83449 27952 357758 23 38 224I4 4-46i55 24254 4-I230I 26110 3.82992 27983 3.57357 22 39 22444 4.45548 24285 4-11778 26141 3.82537 28oi5 3.56957 21 40 22475 4-44942 243i6 411256 26172 3.82083 28046 3.56557 20 41 22505 4.44338 24347 4o10736 26203 3-8i630 28077 356159 19 42 22536 4-43735 24377 4-02I6 26235 3-8177 28109 3-55761 i8 43 22567 4-43i34 24408 4.o96g9 26266 3.80726 28140 3-55364 17 44 22597 4-42534 24439 40o912 26297 3.80276 28I72 3.54968 16 45 22628 4-4I936 24470 40o8666 26328 3.79827 28203 3-54573 i5 46 22658 4-4I34o 245o0 4-08152 26359 3.79378 28234 3'54179 14 47 22689 4-40745 24532 4-07639 26390 3.78931 28266- 3-53785 I3 48 22719 4-40o52 24562 4-07127 26421 3.78485 28297 3.53393 12 49 22750 4-39560 24593 4-o66i6 26452 3.78040 28329 3-53ooi II 50 22781 4-38969 24624 4-06107 26483 3-77595 2836o 3-52609 Io 51 228 1 4-3838i 24655 4o05599 265I5 3-77152 28391 3-522I9 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 1'3-544I 7 54 22903 4.36623 24747 4-o4o88 26608 3.75828 28486 3-5Io53 6 55 22934 4-36040 24778 4-03578 26639 3.75388 28517 3-50666 5 56 22964 4.35459 24809 4.03075 26670 3.74950 28549 3-50279 4 57 22995 4.34879 24840 4-02574 26701 3.74512 28580 3-49894 3 23026 4.343oo 24871 4-02074 26733 3.74075 28612 3-49509 2 59 23056 4.33723 24902 4-o0576 26764 3.73640 28643 3-49125 I 60 23087 4-33148 24933 4-0I078 26795 3-73205 28675 3.48741 0 Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 7~ 760I.750 14 30 NATURAL TANGENTS AND COTANGENTS. 160 1~o i 180 19~ Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. 0 28675 3 48741 30573 3.27085 32492 3.07768 34433 2 90421 6o I 28706 3.48359 3o6o5 3.26745 32524 3.07464 34465 2.90147 5 2 28738 3.47977 30637 3.26406 32556 3.0o760 34498 2 89873 58 3 28769 3.47596 30669 3.26067 32588 3.06857 3453o 2.89600 57 4 28800 3472 6 30700 3.25729 32621 3.o6554 34563 2.89327 56 5 28832 3.46837 30732 3.25392 32653 3.o6252 34596 2.89055 55 6 28864 3.46458 30764 3. 25055 32685 3.-o5950 34628 2.88783 54 7 28895 3.46080 30796 3.24719 32717 3.05649 3466 2. 885 I 53 28927 345703 30828 3.24383 32749 3o05349 34693 2.88240. 52 9 28958 3.45327 3o86o 3 24049 3282 3 o5049 34726 2.87970 5i 10 28990 3. 4495I 30891 3.23714 328 4 3.04749 34758 2.87700 50. 2902 3 44576 30923 3.2338i 32846 3.o445o 3479I 2.87430 49 12 29053 3.44202 30955 3.23048 32878 3.o4I52 34824 2.87I6I 48 13 29084 3.43829 30987 3.227I5 32911 3 o3854 34856 2.86892 47 i4 29116 3.43456 31019 3.22384 32943 3.o3556 3 34889 2.86624 46 I5 2947 3,43o84 3io5i'3.22o53 32975 3:o3260 o 34922 2.86356 45 i6 29179 3.42713 3o83 3. 21722 33007 3.02963 34954 2.86089 44 I7 2920 3.42343 3III5 3.21392 33o4o 3.02667 34987 2.85822 43 18 29242 3.41973 3II47 3'2Io63 33072 3.02372 35019 2.85555 42 19 29274 3.4I604 31178l 3.20734 33I04 3.02077 35052 2.85289 41 20 29305 3.4I236 31210 3.20406 3336 3.01783 35o85 2.85023 40 21 29337 3.40869 3I242 3.20079 33i69 3o01489 35II7 2.84758 39 22 29368 3.40502 3I274 3.19752 33201 3.0II96 3550o 2.84494 38 23 29400 3.4oi36 3I3o6 3.19426,33233 3.00903 -35i83 2-84229 37 24 29432 3.39771 3338 3.190o 33266. 3.oo6I 35216. 2.83965 36 25 29463 3.39406 31370 3.18775 33298 3.oo3iq 35248 2.83702 35 26 29495 3.39042 31402 3. 8451 33330 3.o00028 35281 2.83439 34 27 29526 3.38679 434 344.18127 33363 2a99738 35314 2.83176 33 2 29558 3.383i7 31466 3.17804 33395 2 99447 35346 2.82914 32 29 29590 3 37955 3498 3.17481 33427 2. 9958 35379 2.82653 3i 30 29621 3.37594 3i53o 3.17159 33460 2.98868 35412 2.82391 3o 31 29653 3.37234 3I562 3.r6838 33492 2-98580 35445 2.823o0 29 32 29685 3.36875 31594 3.16517 33524 298292 35477 2.8I870 28 33 29716 3.365i6 31626 3.I6I97 33557 2'98004 355io 2.816o1 27 34 29748 3.36i58 3i658 3.15877 33589 2.977I7 35543 2.8350 26 35 29780 3.358oo 31690 3.;5558 33621 2.97430 35576' 2.8Io09g 25 36 29811 3.35443 3 31722 3.5240 33654 2.97144 356o8 2.80833'24 37 29843 3.35087 31754 3.14922 33686 2'96858 3564I 2.80574 23 38 29875 3.34732 31786 3. 46o5 33718 2.96573 35674 2.8o3i6 22 39 29906 3.34377 3i8i8 3.14288 33751 2 96288 35707 2.80059 21 40 29938 3.34023 385o 3.3972 33783 2.96004 35740 2.79802 20 41 29970 3.33670 3I882 3 I3656 338i6 2.95721 35772 2-79545 I9 42 3oooi 3-33317 31914 3.1334I 33848 2.95437 358o5 2.79289 I8 43 30033 3.32965 31946 3.I3027 3388i 2.95I55 35838 2.79o33 17 44 3oo65 3.32614 31978 3.12713 33913 2 94872 35871 2.787788 6 45 30097 3.32264 320io 3.i24oo 33945 2.94590 35904 2-78523 i5 46 30I28 3.3I9I4 32042 3.12087 33978/ 2-94309 35937 2-78269 14 47 3oi60 3. 3i565 32074 3.11775 34oo 2'94028 35969 2.78014 i3 48 39192 3-.3I126 32o16 3.11464 34o43 293748 36oo2 2-77761 12 49 30224 3.3o868 32139 3.iii53 34075 2.93468 36o35 2.77507 I 50 30255 3.3052I 32aI7 3.10842 34o18 2.93189 36o68 2.77254 10 5I 30287 3.3o074 32203 3.io532 344o0 2.92910 36ioI 2.77002 9 52 30319 3.29829 32235 3.10223 34I73 2.92632 36134 276750 8 53 3o35i 3.29483 32267 3.09914 34205 2.92354 36I67 2.76498 7 54 30382 3.29139 32299 3.0o9606 34238 2.92076 36199 2.76247 6 55 304i4 3.28795 32'33I 3.09298 34270 2.91799 36232 2.75996 5 56 3o446 3.28452 32363 3.08991 343o3 2.9I523 36265 2.75746 4 57 30478 3.28I09 32396 3.o8685 34335 2-91246 36298 2.75496 3 5 30509 3.27767 32428 30o8379 34368 2.9097I 3633I 2.75246 2 59 3054i 3-27426 32460 3.08073 34400 290696 36364 2.74997 I 6o 30573 3.27085 32492 3.07768 34433 2.90421 36397 2 74748 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent.. 730~ 7'2~~ 710 700 31 NATURAL TANGENTS AND COTANGENTS. 200 21~ 220 230 Tangent a. g. angent g. angent. Ctang. g. gangent. Cotang. o 36397 2.747.48 38386 2.60509 40403 2.47509 42447 2.35585 6o 36430 2.74499 38420 2.60283 40436 2.47302 42482 2-35395 59 2 36463. 2.7425I 38453 2.60057 4047.0 2 47095 42516 2.35205 58 3 36496 2.74004 38487 2.5983I 40504 2.46888 4255I 2.35o05 57 4 36529 2.73756 38520 2'.59606 40538 2.46682 42585 2.34825 56 5 36562 2.73509 38553 2.59381 40572 2.46476 42619 2.34636 55 6 36595 2.73263 38587 2.59156 40606 2.46270 42654 2.34447 54 7 36628 2.73oi7 38620 2.58932 40640 2.46065 42688 2.34258 53 8 3666i 2 72771 38654 2.58708 40674 2.45860 42722 2.34069 52 9 36694 2. 72526 38687 2.58484 40707 2.45655 42757 2.3388I 5i 10 36727 2.7228I 38721 2.5826I 40741 2 4545I 4279I 2.33693 50 I 36760 2.72036 38754 2.58038 40775 2.45246 42826 2.33505 49 12 36793 2.71792 38787 2.57815 40809 2.45043 42860 2.33317 48 I3 36826 2.71548 38821 2.57593 4o843 2.44839 42894 2.3313o 47 14 36859 2.7I305 38854 2.5737I 40877 2.44636 42929 2.32943 46 15 36892 2.7I062 38888 2.57I50 40911 2.44433 42963 2.32756 45 i6 36925 2.70819 38921 2.56928 40945 2.44230 42998 2.32570 44 I7 36958 2.70577 38955. 2.56707 4979 2 44027 43032 232383 43 i8 36991 2.70335 38988 2-56487 4IoiJ 2.43825 43067 2.32I97 42 19 37024 2.70094 39022 2.56266 41047 2.43623 43ioi 2.320I2 41 20 37057 2.69853 39055. 2.56046 40o8i 2.43422 43I36 2.3I826 4o 21 37090 2.69612 39089 2.55827 4I I5 2.43220 43170 2.3164I 39 22 37124 2.6937I 39122 2.55608 41149 2.43019 43205 2.31456 38 23 37157 2.69131 39I56 2.55389 4I183 2-428I1 43239 2.3I27I 37 24 37190 2.68892 39190 2.55I70 412I7 2.42618 43274 2.3I086 36 25 37223 2.68653 39223 2.54952 41251 2.42418 433o8 2.30902 35 26 37256 2.68414 39257 2.54734 4I285 2.42218 43343 2.307I8 34 2 37289 2.68I75 39290 2.545I6 413I9 2.42019 43378 2.30534 33 28 37322 2.67937 39324 2.54299 4i353 2.41I8I9 434I2 2.3035i 32 29 37355 2.67700 39357 2.54082 4I387 2.41620 43447 2.30167 31 30 37388 2.67462 39391 2.53865 41421 2.41421 4348i 2.29984 3o 31 37422 2.67225 39425 2.53648 41455 2.41223 435i6 2.2980I 29 32 37455 2.66989 39458 2.53432 41490 2.40o25 43550 2.296I9 28 33 37488 2.66752 39492 2 53217 4i524 2.40827 43585 2.29437 27 34 37521 2.665i6.39526 2.53001 4I558 2.40629 43620 2.29254 26 35 37554 2.66281 39559 2.52786 41592 2.40432 43654 2.29073 25 36 37588 2.66046 39593 2.5257I 41626 2.40235 43689 2.2889I 24 37 37621 2.658ii 39626 2.52357 41660 240oo38 43724 2.28710 23 38 37654 2.65576 39660 2.52142 41694 2.39841 43758 2.28528 22 39 37687 2.65342 39694 2.51929 41728 2.39645 43793 2.28348 2I 40 37720 2.65io9 39727 2.5I71I 4I763 2.39449 43828 2.28167 20 4I 37754 2.64875 3976I 2.5I502 4I797 2.39253 43862 2.27987 I9 42 37787 2.64642 39795 2.51289 4I83I 2.39058 43897 2.278o6 i8 43 37820 2.644IO 39829 2.5176 41865 2.38862 43932 2.27626 17 44 37853 2.64177 39862 2.50864 41899 2.38668 43966 2.27447 i6 45 37887 2.63945 39896 2.50652 41933 2.38473 44001 2.27267 15 46 37920 2.63714 39930 2.50440 41968 2.38279 44o36 2.27088 14 47 37953 2.63483 39963 2.50229 42002 2.38084 44.071 2*.26909 i3 48 37986 2.63252 39997 2.500I8 42036 2.37891 44o05 2-26730 12 49' 38020 2.6302I 4oo3i 2.49807 42070 2.37697 44I4o 2.26552 II 50 38o53 2.62791 4oo65 2.49597 42105 2.37504 44175 2.26374 Io1 5I 38o86 2.62561 40098 2.49386 42139 2.3731 44210 2.26I96 9 52 38i20 2.62332 4o032 2.49177 42I73 2.37118 44244,J 2.26018 8 53 38i53 2.62io3 4oi66 2.48967 42207 2.36925 44279 2.25840 7 54 38i86 2.61874 40200 2.48758 42242 2.36733 4434 2.25663 6 55 38220 2-6I646 40234 2.48549 42276 2.3654I 44349 2.25486 5 656 38253 2.6I4I8 40267 2.48340 42310 2.36349 44384 2.25309 4 -57 38286 2.61190 4o3oi 2.48I32 42345 2.36158 44418 2.25i32 3 58 38320 2. 60963 4o335 2 47924 42379 2.35967 44453 2.24956 2 59 38353 2.60736 4o369 2 47716 4243 2.35776 44488 2.24780 1 60 38386 2.6o509 40403 2.47509 42447 2.35585 44523 2.24604 Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 690 68 670 66~ 32 NATURAL TANGENTS AND COTANGENTS. /t 24~ 25~ 260 _ _ _ Tangen Cota Tangent. Cotang. Tangent. ta T t Cotang. TangentCong. o 44523 2.24604 4663 2I14451 48773 2-o5030 50953 i.96261 6o I44558 2-24428 46666 214288 48809 2.04879 -5098 -96120 5 2 44593 2-24252 46702 2.I4I25 48845 2-04728 51026 1.95979 58 3 44627 2.24077 46737 2.I363 48881 2-04577 5io63 1.95838 57 4 44662 2-23902 46772 2.I3801 48917 2.04426 510o9 I195698 56 5 44697 2-23727 46808 2.13639 48953 2.04276 5II36 1.95507 55 6 44732 2.23553 46843 2..3477 48989 2-04125 51173 I.95417 54 7 44767 2.23378 46879 2-.33I6 49026 2.o3975 51209 1.95277 53 8 44802 2.23204 46914 2i3154 49062 2o3825 51246 1.95137 52 9 44837 2.23030 46950 2-1 2993 49098 2.03675 51283 I.9997 5i 10 44872 2.22857 46985 2I12832 49134 2.o3526 513I9 1.94838 5o II 44907 2.22683 47021 2.12671 49I70 2.03376 5I356 1.947I8 4 12 44942 2.22510 47056 2.I25II 49206 2.03227 5I393 194579 48 I3I 44977 2.22337 47092 2.i2350 49242 2.03078 51430 1.94440 47 14 456I2 2.22164 47128 (2.12190 49278 2 02929 51467 Ii94301 46 15 45047 2.21992 47i63 2.I2030 493i5 2.02780 5I503 1.94162 45 i6 45682 2.218g9 47199 2.ii871 49351 2.0263i 5i540 1.94023 44 17 45ii7 2.-2647 47234 2.11711 49387 2.02483 5i577 I.93885 43 I8 45152 2.-2475 47270 2.I552 49423 2.02335 5i6i4 1.93746 42 19 45I87 2.2I304 447305 2.1392 49459 2'02187 5i65I 1.936o8 41 20 45222 2.21132 4734I 2.1233 494952 2.02039 5688 -.93470 40 21 45257 2.20961 47377 2.075 49532 2.01891 51724 1.93332 39 22 45292 2.20790 47412 2;0916 49568 2I.1743 517611 -i93195 38 23 45327 2.206I9 47448 2.10758 49604 2.01596 51798 1.93057 37 24 45362 2.2044q 47483 2.io600 49640 2.01449 51835 1.9920o 36 25 45397 2.20278 47519 2I0442 496772 2.O302 51872 1.92782 35 26 45432 2.20I08 47555 2.10284 49713 2;oII55 51909 1.92645 34 27 45467 2I.9938 47590 2*IOI26 49749 2.oIoo8 51946 1i925o8 33 28 455o2 2g19769 47626 2.09969 49786 2.00862 51983 1.9237I 32 29 45537 2.19599 47662 2.098II 49822 2.00715 52020 1.92235 3i 30 45573 2.19430 47698 2o09654 49858 2oo569 52057 1.92098 30 31 456o8 2.1926I 47733 2.09498 49894 2 423 594 9962 29 32 45643 2.19092 47769 209341 4993 1 2o00277 52 13i.9I826 28 33 45678 2I8g23 478o52 2.09O84 49967 2;ooi3I 52I68.91690 27 34 457I3 2.18755 47840 2.09028 5ooo4 I 99986 52205 1.9554 26 35 45748 2.18587 47876 2.08872 50040 1.99841 522421 I.9418 25 36 45784 2.18419 47912 2.08716 50076 1.99695 52279. I;9I282 24 37 45819 2.-8251 47948 2o.8560 o 5o03 1.99550 523I6 1.91147 23 38 45854 2.18084 47984 2-o8405'50o49.99406 52353 1.91012 22 39 45889 2;I7916 4809 2-o8250 50185 Ig19926I 52390 1.90876 2i 40 45924 2.17749 48055 208094 50222 1.99116 52427 1.90741 20 4i 45960 2.17582 48091 2o7939 50258 I 98972 52464 190607 19 42 45995 2.17416 48127 207785 50295 1.98828 52501 1.90472 18 43 46o3o 2.I7249 48I63 2.07630 5o331 I.98684 52538 -90337 17 44 46065' 2.17083 48198 2.07476 5o368 I 98540 52575 1-90203 i6 45 46ioI 2.16917 48234 2.07321 50404 1.98396 52613 1.90069 i5 46 46i36 2.16751 48270 2.07167 5044I I198253 52650 189935 1,4 47.46171 2.6585 48306 2.07014 50477 I.98IIO 52687 1.89801 i3 48 46206 2.16420 48342 2.06860 5o514 I.97966 52724 1.89667 12 49 246242 2.I6255 48378 2.o6706 5055o 1.97823 52761 189533 II 50 46277 2I.6090 48414 2.o6553 50587 1.97680 52798 1.89400o o 5I 463I2 2,15925 4845o 2.06400 50623 1.97538 52836 89266 Q 52 46348 2;.5760 48486 2.o6247 5o66o 1.97395 52873.89133 8 53 46383 2.15596 48521 2.06094 5066 1.97253 52910 1.89000o 54 464i8 2.15432 48557 2.05942 50733 1.97111 52947 I.88867 6 55 46454 2.I5268 48593 2.05790 50769 1.96969 52984 1.88734 5 56 46489 2.I50o4 48629 2o05637 5o8o6 1;96827 53o22 1.88602 4 57 46520 2.14940 48665' 2o5485 50843 196685 53o59 1.88469 3 5 46560 2.4777 48701 2o5333 50879 I.96544 53o06 I.88337 2 5 46595 24 4 | 487373 205i82 50916I 96402 53I34 8825 6o 4663i 2-i445I 48773 2o5030 56953 I.9626i 53171 |88073 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 650 _ 64~ 1 630 620 06o- 3 0~ o 0~33 NATURAL TANGENTS AND COTANGENTS. 280 29~ 30~ 31~ Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 53171 1.88073 5543i i.80405 57735 I.73205 60o86 1.66428 6o 1 53208 1.87941 55469 1.8028I 57774 1.73089 60126 i.663i8 59 2 53246 1.87809 55507 i.8oi58 57813 1I72973 6oi65 166209 58 3 53283 1.87677 55545 i-8oo34 5785I 1.72857 60205 1.66099 5 4 53320 1.87546 55583 1.79911 57890 1.72741 60245 1.65990 56 5 53358 i-87415 55621 I.79788 57929 1-72625 60284 i-6588I 55 6 53395 1.87283 55659 1.79665 57968 1.72509 60324 1.65772 54 7 53432 1.87152 55697 1.79542 58007 1.72393 60364 i.65663 53 8 53470 1.8702I 55736 1.79419 58046 1.72278 60403 I.65554 52 9 535o7 186891 55774 1.79296 58085 1.72163 60443 I.65445 51 Io 53545 -186760 55812.79174 58i24 1.72047 60483 I.65337 50 i 53582 I.86630 55850 1.7o951 58162 I.7I132 60522 I.65228 4 12 53620 1.86499 55888 1.78929 58201 1.718I7 60562 -165120 48 3, 53657 1.86369 55926 1.78807 58240 1.71702 60602 I.650 I 47 14 53694 1.86239 55964 1-78685 58279 I171588 60642 i-649o3 46 15 53732 1.86109 56003 I 78563 583 8 I.71473 6068i I.64795 45 I6 53769 1.85979 5604i I17844, 58357 1i7I358 60721 1.64687 44 17 53807 i.85850 56079 1-78319 58396 1.71244 6076I 164579 43 18 53844 1.85720 56117 1.78198 58435 1.7129 608oi 1.64471 42 19 53882 1.8559I 56i56 1.78077 58474 i.7115 60841 1.64363 4I 20 53920 1.85462 56194 1I77955 585i3 1.7090I 6088I I.64256 4o 21 53957 1.85333 56232 1.77834 58552 1.70787 60921.64148 3 22 53995 1.85204 56270 1.77713 58591 I70673 60960 I.64041 38 23 54032 I.85075 56309 1.77592 5863i I.70560 6o1000 63934 37 24 54070 I'84946 56347 1-77471 58670.70446 61040o i63826 36 25 54107 I.84818 56385 1.7735I 58709 1.70332 6io80 i.6379 35 26 54I45. 84689 56424 I 77230 58748 1-70219 61120 I 63612 34 27 54 83 I.8456I 56462 1.77110 58S87 70I06 6160 i.63505 33 2 54220 1.84433 565o00.7699o 58826.69992 61200 1.63398 32 29 54258 I.84305 56539 1-76869 58865 1-69879 61240 I 63292 3i 30 54296 1.84177 56577 1*76749 58904 I.6976 61280.63 85 3o 3I 54333 1.84049 566i6 -176630 58944 I 69653 6132o 163079 29 32 54371 1.83922. 56654 1I76510 58983 -169541 6i360 1-62972 28 33 54409 1.83794 56693 1.76390 59022 169428 6,400 1I62866 27 34 54446 1-83667 56731 1I76271 5906I 169316 6i440 1I62760 26 35 54484 1.83540 56769 i.7651 59101 1-69203 61480 I.62654 25 36 54522 I.834i3 568o8 1.76032 59140 -1.6091 61520 1.62548 24 37 5456o 1.83286 56846 1759I3 I79 -68979 6I56i 1-62442 23 3 54597 I 83i59 56885 1.75794 59218 I.6886 6i60o I62336 22 39 54635 I83033 56923 1.75675 59258 I168754 6i64i I.62230 21 40 54673 1.82906 56962 I.75556 59297 I-68643 6i68i.62I25 20 41 54711 1I82780 57000 1.75437 59336 i.6853i 61721 1.62019 19 42 54748 1.82654 57039 I.753i9 59376 1.68419 61761 I-6I914 8 43 54786 I.82528 57078 I.75200 59415 i.683o8 6i8oi i.68O8 I 44 54824 1.82402 57116 I.75082 59454 1.68i96 6I842.161703 Ib 45 54862 1.82276 57155 1 74964 59494 i.68o85 61882 I.6 598 i5 46 54900 1.82150 57193 1-74846 59533 1.67974 6I922 I61 493 14 47 54938.82025 57232 1.74728 59573 1.67863 61962 i.6i388 13 48 54975 181i899 57271 I-746I0 59612 1.67752 62003 I.6I283 12 49 550o3 1I81774 57309 1.74492 5965i 1.67641 62043 1.61I79 II 50 5505I I 8I649 57348 174375 59691 1.67530 62083 1.61074 io 51 55089 I-8I524 57386 174257 59730 1.674I9 62124 160970 9 52 55i27.8I399 57425 I 74140 59770 1-67309 62164.60865 8 53 55i65 I8I 2j4 57464 1.74022 5989.678 62I204 60761 7 54 55203 i.8ii50 57503 1.73905 59849 1-67088 62245 I.60657 6 55 5524I i-8i025 57541 1.73788 59888 1.66978 62285 I.6o553 5 56 55279 1 80901 57580 1.73671 59928 I.66867 62325 i.60449 4 57 553I 7 I8077 57619 I*73555 59967 1I66757 6366 I.6o345 3 58 55355 I28o653 57657 1.73438 6ooo0007 166647 62406 I.60241 2 59 55393 180529 57696 I.73321 60046.66538 62446 i-60o37 I 60 5543I 1.80405 57735.73205 6oo86 1.66428 62487 i.6oo33 o Cotang Tangent. Cotang. Tangent. Cotang. Tangent. Cotang, Tangent. t____....., —--- _. - 61~ 60~ 69~6~ 61 34 600 590 5 84 NATU RAL TANGENTS AND COTANGENTS. I | 320 3^0 340 3650 Tangent. Cotang. Tangent, Cotang. Tangent. Cotang. Tangent. Cotang. o 62487 i.6oo.33 64941 1.53986 67451 I-48256 70021 I.42815 6o 1 62527 I.59930 6498~2 I.53888 67493 I.48I63 70064 1.42726 59 2 62568 1.59826 65023 1.53791 67536 48070 70107 42638 58 3 62608.59723 65065 153693 67578 1I47977 70151 I.42550 57 4 62649 1.59620 65io6 1.53595 67620 I 47885 70I94 I142462 56 5 62689.59517 65I48 I 53497 67663 1.47792 70238 1.42374 55 6 62730 1.59414 65189 I.534oo 67705 I1.47699 7,o28 I.42286 54 7 6270.593 I I 6523;I I.53302 67748 1.4760 7o3-25 142198 53 8 6281I 1.59.208 65272 i 532o5 67790 1.47514 70368 1.42110 52 9 62852.59105 653I4 1.53107 67832 1.47422 70412.42022 51 0 62892 1.59002 65355, I.53oio 67875.47330 70455 I.4I934 50 I 62933. 58900 65397 152913 67917 147238 7049-9 I 4I847 49 I2 62973 1.58797 65438.528I6 67960 1.47146 7054.41759 48 13 63oi4 158695 65480.52719 68002.47053 70586 1.41672 47 14 63055 i.58593 65521 1. 52622 68o45 1.46962 70629 I.41584 46 15 63095 1.58490 65563 I.52525 68o88 1.46870 70673 I 4I497 45 i6 63i36 I.58388 65604 1.52429 68I3o0 146778 70717 1.41409 44 17 63177 1.58286 65646 15.2332 68173 I.46686 70760 1i.4i32.2 43 i8 63217 i.58i84 65688 1.52235 68.215 1.46595 70804 1.41:235 42 19 63258 i.58o83 65729 1.5-2139 68258 i.465o3 70848 1.41148 41 20 63299 1.5798 6577 I152043 6830o 1-464 1 70891 141o61 40 21 63340 1.57879 658i3.51946 68343.4632o 70935 1.40974 39 22 6338o 1.5777 65854 I 585o 68386 1.46229 70979 140887 38 23 6342I I 57676 65896 1I51754 68429 1.46i37 7I023. 140800 37 24 63462 1.57575 65938 1.5658 68471 1.46046 70o66 1.40714 36 25 63503 157474 65980 I.5I562 685I4 1.45955 71110 1.40627 35 26 63544 1.57372 66021 i 5i466 98557 I.45864 71154 I.4o540 34 27 63584 1-57271 66063 I.51370 68600 1.45773 71198 I.4o454 33 28 63625 1.57170 66xo5 I.5I275 686.42 1.45682 71242 I.40367 32 29 63666 1.57069 66I47 1.51179 68685 1.45592 71285 I.4028I 3i 30 63707 1.56969 66189 I.5o84 68728 I.4550I 71329 1.40195 30 31 63748 1.56868 66230.50988.68771 i.454o 7 71373 1.40109 29 32 63789 1.56767 66272 I.5o893 68814 I.45320 71417 I140022 28 33 6383o I 56667 663I4 1.50797 68857 1.45229 7146I 139936 27 34 63871 J.56566 66356 150702 68900 145 5539 785o 26 35 63912 I.56466 66398 I 50607 68942 I 45o49 7I549 139764 25 36 63953 I.56366 66440 I 50512 68985 1.44958 71593.39679 24 37 639 4 I.56265 66482 I.504I7 69028 1.44868 71637 1.39593 23 38 64o35.5665 66524 I.50322.69071 I 44778 716811. 39507 22 39 64076 I.156065 66566 1.50228 69114 1.44688 71725 1.39421 21 39 64076 i.56o65 J 40.641I7 1.55966 666o8 i.5oi33 69157 1.44598 71769.1339336 20 41 64158 i.55866 6665o0.5oo38.69200o 445o8 7i.8I3.39250 19 42,64199 1.55766 66692 I1.49944 69243 1.44418 71857 1.39165 18 43 64240 i.55666 66734 1.4949j 69286.44329 71.90. I.39079 17 44 64281 I.55567 66776 1.49755 69329 1.44239 71946 1.38994 i6 45 64322 1I55467 668i8 1.4966I 69372 I 4449g 71990 1.38909 i5 46 64363 I.55368 66860 I.49566 69416 1.44o6o 72034 1.38824 14 47 64404 1.55269 66902 149472 69459 1.43970 72078 1.38738 I3 48 64446 I *55i7o 66944 1.49378 69502 i.4388 72122 1-38653 12 49 64487 I 5507i 66986 1.49284 69545 1.43792 72166 I38568 ii 50 64528 1.54972 67028 1.49190 69588 1.43703 72211 1.38484 Io 51 64569.54873 67071 1.49097 6963I I.436I4 72255 I38399 91 52 646io I.54774 67113.49o003 69675 I 43525 72299 i.383I4 8 53 64652 I.54675 67I55 1.48qo9 6971I8 I.43436 72344 I 38 29.7 54 64693 1.54576 67!97 I.488I6 69761 1.43347 72388..38145 6 55 64734 1.54478 67239 I.48722 69804 I.43258 72432.38o6o 5 56 64775 1 54379 67282 148629 69847 143169 72477,37.976 4 57 64817 1.5428 67324 I.48536 69891 i 43o8o 72521 1.37891 3 58 64858 i.54i83 67366 I.48442 69934 1.42992 72565 I.37807 2 59 64899 1.54o85 67409 1.48349 69977 1.42903 72610 137722 60 64941 1-53986 67451 1.48256 70021 1I4285 72654 1.37638 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 57~0 56~0 66 54~ p w!! ~~~~~~r *-,w.. NATURAL TANGENTS AND COTANGENTS. 360 __3___ 380 390, TTangent. Cotang. Tangent. Cotang. Tangent. angent.Cotang. o 72654 1.37638 75355 i 1.32704 78I29 1-27994 80978 1-23490. 60 I 72699 1.37554 75401 1.32624 78I73 I 27I97 81027 1I234I6 2 72743 1 3747~ 75447 132544 78222 12784I 81075.123343 5 3 72788 1 37386 75492 1 32464 78269 I 27764 8ii23 I. 23270 57 4 1 72832 I-37302 75538 I.32384 783i6 1.27688 81171 I.23196 56 5 72877 137218 75584 1.32304 78363 1-27611 8I220 I 23I23 55 6 72921 1 37134 75629 1.32224 78410 I.27535 81268 I 23o5o 54 7 72966 i-37050 75675.32144 78457 1.27458 8i3i6 122977 53 8 730I0 1.36967 7572I i 32064 78504 1 27382 81364 I 22904 52 9 73055 1.36883 75767 13984 78551 1.27306 8i4i3 1.22831 5i.0 73100 i-368oo 75812 i.3i94 78598 1-27230 81461 I-22758 5o II 73144 1.367I6 75858 I.31825 78645 1.27153 85io0 1.22685 49 12 73189 I.36633 75904 i.31745 78692 1.27077 8i558 1.22612 48 I3 73234 i1.36549 75950.i3I666 78739 12700 86o6 I.22539 47 14 73278 1.36466 75996 I.3586 78786 1.26925 8I655 1.22467 46 5 73323 I.36383 76042 i.3I507 78834 1.26849 81703 1.22394 45 i6 73368 i.363oo 76088 I.31427 7888I 1.26774 81752 1.2232I 44 17 734I3 1.36217 76134 i.3i348 78928 1.26698 8i8oo 1.22249 43 i8 73457 i.36i33 76180 1.31269 78975 I.26622 81849 1221761 42 19 73502 i-36o5I 76226.31190 79022 1 26546 81898 1.22I04 4I 20 73547 I.35968 76272 i.3IIo 79070 1.2647I 81946 1.2203I 40 21 73592 I 35885 76318 i.3io3I 79117 1.26395 81995 I.21959 39 22 73637 1.35802 76364 1.30952 79164 1.26319 82044 1.2886 38 23 7368i 1.357I9 76410 I.30873 792I2 I126244 82092 I.21814 3 24 73726 1.35637 76456 1.30795 79259 1.26169 82141 1-2I742 36 25 73771 I.35554 76502.30716 79306.26093 82190 I.2670 35 26 738i6 1.35472 76548 I.30637 79354 I.260I8 82238 1.21598 34 27 7386. I 35389 76594 i.3o558 79401 1-25943 82287 I.2I526 33 28 73906.35307 76640 I.3o480 79449 1.25867 82336 I.2454 32 29 73951 I.35224 76686 I.30401 79496 1.25792 82385 1.2I382 3i 30 73996 I135I42 76733 I130323 79544 125717 82434 I-2i3io 30 31 74041 I 35o6o 76779 1.30244 79591 1.25642 82483 1I2I238 29 32 74086 1.-34978 76825 i3oi66 79639 1.25567 82531 1.21166 28 33 7413i.34896 7687i I.30087 79686 1.25492 82580 I.21094 27 34 74176.34814 76918.3o0009 79734 125417 82629 121023 26 35 74221 I.34732 76964 12993 1 79781 1.25343 82678 12095I 25 36 74267 I.34650 77010 1.29853 79829 1.25268 82727. I20879 24 37 74312 I.34568 77057 1.29775 79877 I125I93 82776 120808 23 38 74357 1.34487 77103 1.29696 79924 I.25II8 82825 1.20736 22 39 74402 I1.34405 77149 1296I8 79972 1.25044 82874 1.20665 21 40 74447 |I-34323 77196 1.29541 80020 1.2496 82923 -20593 20 41 74492.34242 77242 1-29463 80067 24895 82972.20522 19 42 74538 I.34I60 77289 1.29385 8o05 1.24820 83022 1.2045I 18 43 74583 1.34079 77335 1.29307 8oi63 1.24746 83071 1.20379 17 44 74628 1.33998 77382 1.29229 80211 1.24672 83120 I.20308 16 45 74674 1.33916 77428 1.29152 80258 1.24597 83169 1.20237 15 46 74719 i.33835 77475 1.29074 8o3o6 1.24523 83218. 1.2066 14 47 74764 1 33754 77521 128997 8o354 1I24449 83268 1.20095 i3 48 74810.33673 77568 1.28919 80402 1.24375 83317 1-20024 12 49 74855.33592 77615 1.28842 8o45o 1.2430I 83366 I1.I953 II 50o 74900 i.335II 77661 1.28764 80498 I.24227 834i5 1.19882 I0 51 74946 i.3343o 77708 1.28687 8o546 I.24153 83465 1I981I'52 7499I I-33349 77754 1.28610 80594 1.24079 835i4 I.9740 8 53 75037 I.33268 77801 I.28533 80642 i.2400oo 83564 1.19669 7 54 75082 1.33187 77848 I.28456 80690 1.2393I 836I3 1.19599 6 55 75128 I.33107 77895 I-28379 80738 1.23858 83662 1t.9528 5 56 75173 -133026 77941 I.28302 80786 1.23784 83712 1.19457 4 57 5219.32946 77988 1-28225 8o834 1.2370o 83761 i 9387 3 58 75264 132865 78035 1.28I48 80882 1.23637 8381 II931i6 2 59 753o1 1.32785 78082 1.28071 80930 I.23563 8386o II19246 i 60 75355 1.32704 78129 1.27994 80978 123490 83910 I.19175 0 otang. Tangent Cotang. TTngent. Cotang. Tangent. Cot. 530 520 51~ 500 3R NATURAL TANGENTS AND COTANGENTS. 400 410 420 430 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 83910o.19175 86929 1.15037 90040 I.IIo6i 93252 I.07237 60 I 83960 I. 9io5 86980 1.I4969 90093 I I0996 93306 107174 59 2 84009 I -I9035 87031 I14902 90146 I 109I 93360 I07II2 5 3 84059 1.18964 87082 I1I4834 90199 I. 0867 93415 I.7049 57 4 84168 II8894 87I33 I.14767 90251 1.10802 93469 I.06987 56 5 84158 1.18824 87184 I114699 90304 1.10737 93524 1.06925 55 6 84208 I.I8754 87236 1.4632 90357 1.10672 93578 I.06862 54 7 84258 I I8684 87287 I*4565 90410 110607 93633 i.o68oo 53 8 84307 i.186I4 87338 1.I4498 90463 i.io543 93688 1.66738 52 9 84357 I*I8544 87389 I.I443o 90516 1.10478 93742 I.o6676 5I 10 84407 1.18474 87441 I.I4363 90569 1.10414 93797 I.66i3 50 11 84457 i.-84o4 87492 I1I4296 9062I 1.10349 93852 i.o655I 49 12 84507 I.i8334 87543 1.14229 90674 1.10285 93906 I.o6489 48 13 84556 I.18264 87595 I1.4162 90727 1.10220 9396I 1.06427 47 14 84606 1I18194 87646 I 14095 90781 I.ioi56 94016 I.o6365 46 I5 84656 I.I8I25 87698 I1I4028 90834 I.10091 94071 I.o6303 45 16 84706 ii8055 87749 I.13961 90887 I.I0027 94125 1.06241 44 17 84756 I.17986 87801 1.13894 90940 1.09963 94180 1o06179 43 i8 84806 I.17916 87852 I.13828 90993 I1098g9 94235 1.06117 42 19 84856 1.17846 87904 I.1376I 91046 I.09834 94290 I.o6056 41 20 84906 II7777 87955 I.13694 91099 I0o9770 94345 1.05994 4o 21 84956 I.17708 88007 I.3627 9II53.09706 94400 I.o5932 39 22 85006 I.17638 88059 I 356I 91206 1.09642 94455 I.o5870 38 23 85057 1,7569 88I I 13494 91259 I 09578 94510 Io5809 37 24 85107 1-17500 88162 1 13428 91313 1.09514 94565 1.05747 36 25 85i57 II.7430 88214 I I336I 91366 109450 94620 I.o5685 35 26 85207 1'I736I 88265 I. 3295 914I9 109386 94676 I*o5624 34 27 85257 1-17292 883-7 I.I3228 91473 1.og322 9473I I.o5562 33 28 85307 I.17223 88369 I.I3-62 91526 1 09258 94786 i.o55oi 32 29 85358 I'.7154 88421 I.13096 91580 1-09135 94841 I 5439 31 30 85408 1.17085 88473 I I3029 91633 1-09I I 94896 I0o5378 30 31 85458 I.17016 88524 I12q63 91687 I1.9067 94952 I.o5317 29 32 85509 I.I6947 88576 I I287 91740 I.oo 95007 Io5255 28 33 85559 I.16878 88628 I 128 I 91794 1.08940 95062 I05I94 27 34 85609 I.6869 8868o0.12765 91847 I o8876 95 i8 Io5i33 26 35 8566o I.16741 88732 II2699 91901 I.o88I3 9517.3.o5072 25 36 85710 I.16672 88784 I.12633 91955 1-o8749 95229 I.050oi 24 37 8576I i.i6603 88836 1.12567 92008 i-o868 95284 1.0494 23 3 858I Ii I.6535 88888 1.12501 92062 1.08622 95340 1.04888 22 39 85862 I.i6466 88940 I*I2435 92I16 i.o8559 95395 1.04827 21 40 85912 I*16398 88992 1-12369 92170 I.o8496 95451 I-04766 20 41 85963 I.16329 89045 I.2303 92223 I.o8432 95506 I 04705 1 42 86014 I-1626I 89097 1 12238 92277 1.08369 95562 1.04644 8 43 86064 I.6192 89149 1.12172 92331 i.o83o6 95618 I.o4583 I7 44 8615 I.16124 89201 I.12I06 92385 1.o8243 95673 1.04522 i6 45 86I66 I.I6o56 89253 I1I204I 92439.o08179 95729 I.04461 15 46 862I6 1.15987 89306 1.11975 92493 I.o8ii6 95785 I.o440I I4 47 86267 59 9 893 58 I.909 92547 i.o8053 95841 i.o4340 i3 48 863i8 I.I585I 894o10 I 44 92601 1.07990 95897 1.04279 12 49 86368 1.15783 89463 1.II778 92655 1.07927 95952 I.4218 II 50 86419 I.157I5 89515 I.171I3 92709 I0o7864 96008 1.04158 10 51 86470 I.I5647 89567 1.11648 92763 1.o7801 96064 1.o4097 52 86521 I.15579 89620 I.11582 92817 1.07738 96120 I.o4036 53 86572 I.i55II 89672 I.II517 92872 I1.07676 96176.03976 7 54 86623 I.i5443 89725 1.XI452 92926 1.07613 96232.039I5 6 55 86674 i.15375 89777 I.11387 92980 I.07550 96288 I.03855 5 56 86725 I.i5308 89830 -1I132I 93034 I1o7487 96344 1.03794 4 57 86776.I15240 89883 III.256 93o88 107425 96400.o3734 3 58 86827 I I5172 89935 I.1119 93143 1.07362 96457 I.03674 2 59 86878 1I5i1o4 89988 I.1ii26 93197 1.07299 965I3 I.o36I3 I 60 86929 i.i5o37 90040 ix Io6x 93252 1.07237 96569 I.o3553 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 490 480 41o 460 3 7 NATURAL TANGENTS AND COTANGENTS. 440 440 - - ----- / / --------- I Tangent. Cotang Tangent. Cotang. o 96569 I.o3553 6o 3i 98327.01702 29 I 96625 I03493 59 32 98384 I.01642 28 2 96681.o343 3 58 33 98441 i.oi583 27 3 96738.03372 57 34 98499 I.o524 26 4 96794 I.o33i2 56 35 98556 I.oI465 25 5 96850.03252 55 36 98613 I.oi4o6 24 6 96907 I.03192 54 37 98671 I oI347 23 7 96963 I o3I32 53 38 98728 I.01288 22 8 97020 0Io3072 52 39 98786.o01229 2I 9 97076 1030o2 51 4o 98843 I.01170 20 10 97133 1.0252 5o 4I 98901 011I2 I9 11 97189 I.02892 49 42 98958 I oo53 I8 12 97246 1.02832 48 43 99016 1.00994 17 13 97302 I.02772 47 44 99073 I.oo35 16 14 97359 1 o27.13 46 45 993I I.00876 15 I5 974I6 102653 45 46 99189 I oo8i8 I4 16 97472 1.o2593 44 47 99247 I.00759 3 17 97529 1.02533 43 48 99304 I.00701 12 18 97586 I.02474 42 49 99362 oo00642 II 19 97643.02414 41 50 99420 I.oo583 i 20 97700 1.02355 40 5I 99478.0oo525 21 97756 102295 39 52 99536 1.00467 22 97813 1-.02236 38 -53 99594 I-oo408 7' 23 97870.I02I76 37 54 99652 I-oo35 6 24 97927 1 02117 36 55 99710.I 0021 5 25 97984 I o2057 35 56 99768.00oo23 4 26 98041 I01998 34 57 99826 I 00175 3 27 98098 o01 39 33 58 99884.ooI6 2 28 98155 IOI. 79 32 59 99942 i 0ooo58 I 29 98213 1 01820 3I 60 Unit. Unit. 30 98270 10176I 3o0 Cotang. Tangent. Cotang. Tangent. / ___.__________ _____________ / 450 450 TABLE OF CONSTANTS. Base of Napier's system of logarithms =................. = 2.718281828459 Mod. of common syst. of logarithms =.... com. log. ~ = M = 0.4342944819o 3 Ratio of circumference to diameter of a circle =........... 7r= 3. 41592653590 log. r = 0.497149872694 Xr2-9.869604401089............ V = 1.772453850906 Arc of same length as radius =.......... 800 - r = 10800' - 7 = 648000" - r 8o0 - 7r= 570.295779530............................log. = 1.758122632409 io8oo' - -= 3437'.7467707849..........................log. = 3.536273882793 648000" - r = 206264".8062470964,.....................log. = 5.3I4425133i76 Tropical year = 365d. 5h. 48m. 47s. 588 = 365d..242217456, log. = 2.5625810 Sidereal year= 365d. 6h.. o9m. lo.742 = 365d..256374332, log. = 2.5625978 24h. sol. t.=24h. 3m. 56s. 555335 sid. t.=24h.X I o00273791, log. I.002=0.o0011874 24h. sid. t.=24h. -(3m.55s. 90944) sol. t.=24h. X o 997 2696, log. o 997=9 9988126 British imperial gallon = 27.274 cubic inches,...............log. = 2.4429091 Length of sec. pend., in inches, at London, 39.13929; Paris, 39.1285; New York, 39.1285. French metre = 3. 2808992 Englishfeet = 39.3707904 incwes. I cubic inch of water (bar. 30 inches, Fahr. therm. 620) = 252.458 Troy grains..388 BY THE SAME AUTHOR. THE PHILOSOPHY OF MATHEMATICS: TRANSLATED FROM THE "COURS DE PHILOSOPHIE POSITIVE" OF AUGUSTE COMTE. 1 vol. 8vo., pp. 260. 1851. "We rejoice that an American scholar has undertaken and so ably executed a translation of a work which has been so favorably received and so highly commended by the first scientific men of Europe. For comprehensiveness of scope, for clearness of statement and exposition, for breadth of inquiry and depth of thought, we esteem it superior to any work in this department of science with which we are acquainted."Evangelical Review. "We think the book is a desideratum, the want of which, has long been felt. We feel confident that it will be received by the mathematical public with much pleasure and satisfaction...... It is an admirable translation. The French idioms are well managed, the spirit and argument well sustained, and fidelity to the original seems to have been the constant aim."-h-ChrcAh Review. "We admit M. Comte to be second only to Bacon and Aristotle among the mighty intellects of all time."-Methodist Quarterly Review. "In giving to the public Comte's Philosophy of the Mathematics, Professor Gillespie is entitled to high commendation, and that in two important respects: the first is, the excellence of his translation, which is remarkable for its clearness as well as its fidelity; the other and chief merit consists in his wise appreciation of the work itself, or his scientific discernment in bringing it out from that great mass of writings by the same author, which, in consequence of their bulk, the forbidding nature of some of their subjects, and certain difficulties of style, have as yet remained comparatively unknown.......Beyond all doubt, Comte is one of our profoundest thinkers on purely scientific subjects; and of this the work before us, had he never written any thing else, would have furnished most convincing evidence. It does not, however, require that one should be a professed or skilful mathematician to be deeply interested in it. Others may read understandingly, and even with delight, a work which thus sets forth the philosophy of the mathematics...... In a scientific point of view, it is a work of the profoundest interest; and its extensive perusal will aid in giving a much higher idea of mathematical science than could have been entertained, as long as it is regarded as mainly subservient to what is called business or immediate practical utility. In this respect, it is worthy of introduction into all our colleges."-T. L.-Literary World. "It speaks well for the devotion to science in this country, that the present admirable analysis of the primary principles of mathematics should be issued from the American press under such worthy auspices. Professor Gillespie has thus rendered a service to the cause of intellectual culture, which we are bound to acknowledge with grateful emphasis. The work, although not intended nor adapted for popular reading,-addressing the highest faculty of abstraction which, the intellect can exercise,-assuming a conversancy with the most recondite details of mathematical calculation,-and clothed in the austere language of logical deduction,-cannot fail to win the attention of thinking men, and to be classed among the few standard treatises of science which are read for the pure gratification which they afford to the reflective powers...... The work aims to solve a similar problem in regard to mathematics, with that proposed in Humboldt's Cosmos in relation to the material universe; that is, the detection of the principle of unity amid the variety of phenomena...... We recognize the comprehensive wisdom, the masterly power of generalization, and the exquisite critical keenness with which Comte has performed the task of contributing towards an ultimate solution of the problem."-G. R.-N. Y Tribune. "A work of profounder insight and keener analysis than the original has not often come from the Press. It is a masterpiece of scientific thinking." —. I. Courier. 89 BY THE SAME AUTHOR. ROADS AND RAILROADS: A MANVUAI Or ROAD-M3AXING: COMPRISING THE PRINOIPLES AND PRACTICE OF THE LOCATION, CONSTRUCTION, AND IMPROVEMENT OF R 0 A D S (Common, 2facadam, Paved, Plank, &c.), and R AI L R 0 A D S. 1 vol. 8vo., pp. 872. 8th edition, 1854. "I have very carefully looked over Professor Gillespie's Manual of Road-Making. It is, in all respects, the best work on this subject with which I am acquainted; being, from its arrangement, comprehensiveness, and clearness, equally adapted to the wants of students of civil engineering and the purposes of persons in any way engaged in the construction or supervision of roads. The appearance of such a work, twenty years earlier, would have been a truly national benefit; and it is to be hoped that its introduction into our seminaries may be so general as to make a knowledge of the principles and practice of this branch of engineering as popular as it is important to all classes of the community." D. H. MAHAN, Professor of Civil Engineering in the Military Academy of the United States. "If the well-established principles of Road-Making, which are so plainly set forth in Professor Gillespie's valuable work, and so well illustrated, could be once put into general use in this country, every traveller would bear testimony to the fact that the author is a great public benefactor."-Silliman's American Joeurnal of Science. "This small volume contains much valuable matter, derived from the best authorities, and set forth in a clear and simple style. For the want of information which is contained in this Manual, serious mistakes are frequently made, and roads are badly located and badly constructed by persons Ignorant of the true principles which ought to govern in such cases. By the extensive circulation of such books as that now before us, and the imparting of sound views on the subject to the students of our collegiate institutions, we may hope for a change for the better in this respect." —Journ. of the Franklin Institute. " oad-making, in any country, is one of the active means and first-fruits of civilization. The public, therefore, is much indebted to Mr. Gillespie for the work now published. It is very remarkable, that in a matter where so much has been done, so little has been known. The present full and complete work has in some measure reconciled us to this delay. Perhaps no work so perfect could have been prepared at an earlier day. The author having visited Europe a few years since for the purpose of collecting information, has since been employed both as a practical engineer and as a professor of the science of engineering; and the subject of road-making, which has been overlooked by others, has engaged his special attention."-L. Y. Courier and Enquirer. "This elaborate and admirable work combines in a systematic and symmetrical form the results of an engineering experience in all parts of the Union, and of an examination of the great roads of Europe, with a careful digestion of all accessible authorities. The six chapters into which it is divided comprehend a methodical treatise upon every part of the whole subject; showing what roads ought to be in the vital points of direction, slopes, shape, surface, and cost, and giving methods of performing all the necessary measurements of distances, directions, and heights, without the use of any instruments but such as any mechanic can make and any farmer use."-Neowark Daily Advertiser. "It would astonish many'path-masters' to see how much they don't know with regard to the very business they have considered themselves such adepts in. Yet all is so simple, so lucid, so straight-forward, so manifestly true, that the most ordinary and least-instructed mind cannot fail to profit by it. We trust this useful and excellent volume may find its way into every village library, if not into every school library, as well as into the hands of every man interested in road-making. Its illustrations are very plain and valuable, and we cannot doubt that the work will be a welcome visitor in many a neighborhood, and that bad roads will vanish before it."-New York Tribune. 40