PROBLEM 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.) by 1.06, $ quotes the neat. EXAMPLES 1. How much land must I inclose to have 850 A. 2R. 20P. neat? 40120. 41 2.5 Acres. A. R. P. 850.625 X 1.06= 901.6625=901. 2. 26. the ans. 2. How much neat land is there in a tract of 901A. 2R. 26P. gross ? 40126. 41 2.65 Acres. A. R. P. 1.06)901.6625(850.625=850. 2. 20. the answ. 848 &c. Note. These two operations prove each other. PROBLEM XV. To find the area of a piece of ground be it ever so irregular, by dividing it into triangles and trapezia. Plate VII. fig. 4. We here admit the survey to be taken and protracted ; by having therefore the map, and knowing the scale by which it was laid down, the content may be thus obtained. Dispose the given map into triangles, by fine pencilled lines such as are here represented by pop'd lines in the scheme, and number the triangle with 1, 2, 3, 4, 8c. Your map being thus prepared, rule a table with four columns ; the first of which is for the number of the triangle, the second for the base of it, the third for the perpendicular, and the fourth for the content in perches. Then proceed to measure the base of number 1, from the scale of perches the map was laid down, and place that in the second column of the table ; under the word base; and from the angle opposite to the base, open your compasses so, as when one foot is in the angular point, the other being moved backwards, and forwards, may just touch the base line, and neither go the least above or beneath it; that distance in the compasses measured from the same scale, is the length of that perpendicular, which place in the third column, under the word perpendicular. If the perpendicular of two triangles fall on one and the same base, it is unnecessary to put down the base twice, but insert the second perpendicular opposite to the number of the triangles in the table, and join it with the other perpendicular by a brace as No. 1 & 2, 4 & 5, 6 & 7, 9 & 10, &c. have mea Proceed after this manner, till you sured all the triangles ; and then by prob. 6. find the content in perches of each respective triangle, which severally place in the table opposite to the number of the triangle, in the fourth column, under the word content. But where two perpendiculars are joined together in the table, by a brace having both one and the same base ; find the content of each (being a trapezium) in perches, by prob. 11, which place opposite the middle of those perpendiculars, in the fourth column, under the word content. Having thus obtained the content of each respective triangle and trapezium which the map contains, add them all together, and their sum will be the content of the map in perches ; which being divided by 160, gives the content in acres. Thus, for EXAMPLES. No. Base. Content. } Perpend. 412.92 712.42 1086.8 15.05 600. 481.5 259.2 12:3}|481.5 10:0} Con. in perch. 4142.57 This being divided by 160, will give 25A. 3R. 22P. the content of the map. Let your map be laid down by the largest scale your paper will admit, for then the bases and perpendiculars can be measured with great accuracy than when laid down by a smaller scale, and if possible measure from scales divided diagonally. If the bases and perpendiculars were measured by four-pole chains, the content of every triangle and trapezium, may be had as before, in prob lem 6 and 11, and consequently the whole content of the map. If any part of your map has short or crooked bounds, as those represented in plate VII. fig. 5, then by the straight edge of a transparent horn, · draw a fine pencilled line as AB to balance the parts taken in and left out, as also another, BC: these parts when small, may be balanced very nearly by the eye, or they may be more accurately balanced by method the third. Join the points A and C by a line, so will the content of the triangle ABC, be equal to that contained between the line AC, and the crooked boundary from A to A, and to C: by this method the number of triangles will be greatly lessened, and the content become more certain ; for the fewer operations you haye, the less subject will you be to err: and if an error be committed, the sooner it may be discovered. The lines of the map should be drawn small," * and neat, as well as the bases; the compasses neat !, ly pointed, and scale accurately divided; without all which you may err greatly. The multiplications should be run over twice at least, as also the addition of the column content. From what has been said, it will be easy to survy a field, by reducing it into triangles, and neasuring the bases and perpendiculars by the chain. To ascertain the content only, it is not material to know at what part of the bases the perpendicular was taken: since it has been shewn (in cor. to theo. 13, sect. 1.) that triangles on the same base, and between the same parallels are equal : but if you would draw a map from the bases and perpendiculars, it is evident that you must know at |