878

THEORY AND PRACTICE OF ENGINEERING.

Book IT.

Fig. 1163.

over a bill composed of upright paling, we shall discover that, although more rails might berequired to pass over the convex surface, as there are more houses shown on G than on II,there would not be more posts or upright pales.

The conversion or reducing of Plane Figures of one kind to that of another is found ex-tremely useful in practice, as it often saves considerable trouble in the calculation, par-ticularly where they are very irregular: this subject has not K

received the attention it demands in the ordinary works whichtreat upon the measurement of land.

To reduce a Triangle to a Rectangle . —If the triangle beequilateral, as A B C, and it is required to form a rectangle onthe side C B ; divide the other two sides A C and A B intotwo equal parts in the points D and E ; through them drawthe line DE, parallel to the side CB : from the point A dropa perpendicular A F to C 11; remark where it cuts D E at G ;set off DG from D to II, and EG from E to I ; draw theright lines II C and IB: the parallelogram IIIBC will beequal to the equilateral triangle A B C on the given side C B.

To reduce a Rectangle to a Triangle , as the figure A B C D to anisosceles triangle. Bisect the base D C of the square in E, andelevate a perpendicular E F, tw r ice the length of the side of thesquare; draw the upright lines FD and FC; the superficies of theisosceles triangle will be equal to that of the square.

If it be required to reduce the parallelogram GIII K to a scalenetriangle, it is only necessary to divide the base of the parallelograminto two unequal parts, as at L, and elevate Ma perpendicular LM, twice the length ofthe side of the square; then draw the rightline IvIK and MI: the superficies of thescalene will be equal to that of the rectangle.

To elevate or depress a Triangle on agiven length, without altering its superficialcontent, as to elevate the point A of thetriangle A B C, so that it may be at the dis-tance D E from the base C B. Draw a per-pendicular to the base C B, through thepoint A, and set off the given length from F to G, anddraw a line III through G at right angles to F G ; prolongA C until it cuts H I in K ; from this point K draw a rightline KB; make AE parallel to KB, and draw the right lineKL ; the triangle KLC will be equal to ABC.

Fig.1165.

To prove that the triangle KLC is equal to the triangleABC, and that it is the height of the line given : it must beremarked that the line KL has cut the side AB in Z, and thatby its construction the lines K B and AL are parallel, whichoccasions the two triangles A B L and A K L, which are on thesame base A L, and between the same parallels KB and AL,are equal to each other, agreeably to the S7th proposition of Fig. 1166.Euclid ’s First Book ; so that the triangle A Z L being cut off, the twotriangles Z B L and Z K A will remain equal to each other. If fromthe triangle A B C we cut off the triangle Z B L, to make Z K A equal,we shall have the triangle KLC equal to ABC, and the height of theline already given ; since the line II G I, which is parallel to the baseC F B, is removed from it the distance of the given line by the per-pendicular G F, which is equal to it.

To reduce Trapeziums and Trapezoids to Triangles , as the figureA B C D to a triangle, whose height shall be equal to the trapezium :fix a point in one side of the trapezium, for the summit of the triangle,as E on A B ; from this point E draw to the extremities of the baseDC the right lines ED and EC, and prolong the line DC eitherway ; from A draw a line parallel to E D, cutting the line in F, andlikewise from B draw B G, parallel to E C; draw E F and E G, tohave the triangle E F G equal to the trapezium A B C D.

To prove that the triangle E G F is equal to the trapezium A B C D,and of the same height, we have only to observe that the lines E F andF G cut the sides A D, B C at the points P and R, and that by theconstruction the lines E D and A F are parallel, which causes the twotriangles AED and FDE to be upon the same base ED, andbetween the same parallels E D and A F.

Fig. 1)67.