given angle; 'which proportion gives the operation as in the rule above. Q. E. D. PROB. X. To find the area of a trapezoid, viz. a figure bounded by four right-lines two of which are parallel, but unequal. RULE. Multiply the sum of the parallel sides by their perpendicular distance, and take half the product for the area. EXAMPLES. 1. Required the area of a trapezoid of which the parallel sides are respectively, 30 and 49 perches, and their perpendicular distance 61. 6. Note. On this 10th problem are founded most of the calculations of differences by latitude and departure, and those by off-sets, following in this treatise. 2. In the trapezoid ABCD the parallel sides are, AD, 20 perches, BC, 32, and their perpendicular distance, AB, 26; required the content. Answer 4A. 36P. PROB. XI. To find the content of a trapezium. RULE. Multiply the diagonal, or line joining the remotest opposite angles, by the sum of the two perpendiculars falling from the other angles to that diagonal, and half the product will be the area. Let ABCD be a field in form of a trapezium, the diagonal AC, 64.4 perches, the perpendicular Bb. 13.6 and Dd. 27.2, required the content? Proceed by the rule, and the area will be found= A. R. P 8 0 33 Note. The method of multiplying together the half sums of the opposite sides of a trapezium for the content is erroneous, and the more so the more oblique its angles are. To draw the map; set off Ab 28 perches and Ad 34.4, and there make the perpendiculars to their proper lengths, and join their extremities to those of the diagonal. PROB. XII. To find the area of a circle, or an ellipsis. RULE, Muliply the square of the circle’s diameter, of the product of the longest and shortest diameters of the ellipsis by :7854 for the area. Cr, subtract 0.10491 from the double logarithm of the circle's diameter, or from the sum of the logarithms of those elliptic diameters, and the remainder will be the logarithm of the area. Note, In any circle, the Diam. multi. 2 S produces the Circum. Circum. div. S by 3.14159, 2 quotes the diameter, EXAMPLES. 1. How many acres are in a circle of a mile diamter ? Answer, 502A. 2R. 25P. . 2. A.gentleman, knowing that the area of a circle is greater than that of any other figure of equal perimeter, walls in a circular deer-park of 100 perches diameter, in which he makes an elliptical fish pond 10 perches long by 5 wide ; required the length of his wall, content of his park, and area of his pond ? Answer, the wall 314.16 perches, inclosing 49A. 14P. of which 394 perches, or $ of an acre nearly is appropriated to the pond. 3. What is the area of an elliptical pond whose diameters are 15 and 28 perches? Answer 1. 2. 11. A. R. P. PROB. XIII. The area of a circle given to find its diameter. RULE. To the logarithm of the area add 0.10491, and half the sum will be the logarithm of the diameter. Or divide the area by .7854 and the square root of the A horse in the midst of a meadow suppose, Answer, 7.13665 per.=117F. 1In. PROB. XIV. To make the proper allowance for roads. It is customary to deduct 6 acres out of 106 for roads; the land before the deduction is made may be termed the gross, and that remaining after such deduction, the neat. RULE. The gross div. The neat mul. S quotes the neat. prod. the gross. EXAMPLES. 1. How much land must I enclose to have 850A. 2R. 20. neat. A. R. P. 2. How much neat land is there in a tract of 901 A. 2R. 26P. gross? A. R. P. Note. These two operations prove cach other. |