884
THEORY AND PRACTICE OF ENGINEERING.
Book IT.
Fig.1192.
square, so that its area may be double, that is 200 feet. By numbers this cannot be done,for if the sides are made 14 feet, these multiplied into each other give 196 feet ; if 15 feetthey give a product of 225 feet. Since, therefore, we cannot find them by the aid of numbers,in the square of 10 feet a diagonal is to be drawn from angle to angle, so that thesquare may be divided into two equal triangles of 50 feet area each : on this diagonal,another square being described, it will be found that w hereas in the first square there weretwo triangles, each containing 50 feet, so in the larger square formed on the diagonal, therewill be four triangles of equal size and number of feet to those in the larger square ; in thisway, Plato demonstrated the method of doubling the square. Rules for extracting thesquare root have since been discovered, and we no longer admit it to be impossible to an-swer this question by figures, for the square root of 200, which is 14*14211356, w*ould be thelength of the side of the square that would contain the superficial area of 200 feet.
To construct rectilineal Figures similar and double to givenFigures of the same Number of Sides. It is required to draw arectangle similar to and double the square A11CD: prolongthe line D C, and set off the base from C to E, and from E toF : find a mean proportional between E F and E D, by elevatinga perpendicular to E F, at the point E, and bisecting D F in H :from H as a centre, with the radius II D, describe the semi-circle 1) I F, cutting E G in I; the right line E I w*ill be themean proportional required: set off the mean proportional onD C prolonged from D to L, and on this length construct arectangle similar to A B C D; which is done by forming at the point I, an angle D L M,equal to D C B : draw the diagonal D B, until it cuts L M in N; through N draw a lineparallel to A B, and remark where the line NO cuts DA; prolong it in P, the figureP N L1) will be similar to and double A B C 1).
I'o double, triple , or quadruple the Circle. —To draw a circle,whose superficies shall be double that of B, draw the diameterCD; elevate the perpendicular DE at the point I); on it setoff the radius BD from D to E, in order to draw B F, whichwill serve as a radius to describe from A as a centre, a circlewhich will be double the given circle B.
If it be required to have the circle tripled : from F elevate theperpendicular F K, on which set off the radius B D from F to E,and draw BL; it will be a radius for describing a circle M N O,which will be triple the given circle B.
If it be required to have the circle quadrupled, elevate theperpendicular LB from the point L, and set off the radius BDupon it: from L to A draw B Q, and it will serve as aradius to describe a circle, which will be the quadruple of thegiven circle B.
Geodesy is synonymous with land-surveying, and compre-hends all those trigonometrical and geometrical operations whichmore particularly have for their object the determination of themagnitude and figure of the whole or any part of the earth’ssurface. The 37th prop, of the First Book of Euclid states thattriangles upon the same base, and between the same parallels,are equal to one anovher : upon this proposition, most of thefollowing rules are founded.
To divide triangular Figures into severed equal Parts whichshall all unite at one Angle . — It is required to divide the triangleA*BC into two equal parts both uniting at the angle A:divide the line C B into two equal parts at D, and draw theright line A D; this right line will divide the triangle into twoequal parts.
To divide the triangle ABC into two equal parts, unitingin the centre of the side A C : bisect A C in D ; draw D B ; itwill divide ABC into two equal parts.
To divide the Isosceles Triangle into three equal partsmeeting in the point H on the side E G •. divide tjie sideE F opposite the point H, into three equal parts in thepoints 1,2,3; draw H I and also G K parallel to it from thepoint G, cutting E F in L; from L draw LII to the givenpoint H, fron the given point H draw H 2, and from thepoint G draw G N parallel to H2, cutting E F in N, fromN draw N H : then will the triangle E F G be divided intothree equal parts by the lines H I and H N.
Fig. 1193.
v M
Fig. 1195.