730 PHILOSOPHICAL TRANSACTIONS. [aNNO 1776. 



the duplicate proportion of the sides ac, cb. This Mr. G. demonstrates geo- 

 metrically, and then adds the following corollaries. 



Cor. 1 . If the triangle be isosceles, the right line drawn from the vertex to the 

 base is perpendicular to it, and the segments of the base are equal to each other. 



Cor. 2. When the triangle is right-angled, the line drawn from the vertex to 

 the base is always perpendicular to it (as appears from e. 8, 6, and its cor.) ; and 

 the rhombi become squares on the sides comjirehending the right angle. 



Cor. 3. The segments of the sides adjacent to the base, are respectively 3d 

 proportionals to the sum of the sides, and the sides themselves. 



Cor. A. The segments of the sides adjacent to the vertex are equal to each 

 other, and each of them is a 4th proportional to the sum of the sides and the 

 sides themselves.* 



Cor. 5. The segments of the base are proportional to the segments of the 

 sides, which are adjacent to them. 



Prop. 1. Let there be any two right lines given : there is an angle which may 

 made by these lines, such, that if from their extremities which do not meet, right 

 lines be drawn to the alternate angles of rhombi described on them, and recipro- 

 cally applied to them when produced ; and/rom the said angle through the inter- 

 section of these lines, a right line be drawn to meet the right line joining the said 

 extremities ; the segments of this line thus made, shall be respectively equal to the 

 adjacent segments of the given lines. — Let ac, cb, be any two given right lines, 

 fig. 12, pi. 13 ; and let cd, in ac produced, be equal to cb. On ad describe 

 a semicircle ; draw on at right angles to ad, and equal to cd ; join a, n, and 

 apply a right line am in the semicircle equal to an. From the point m draw the 

 right line ms at right angles to ad. Make a triangle acb, having its sides equal 

 to AC, AS, and cb ; and acb is the angle required to be found ; and the seg- 

 ments AL, lb, of the right line ab joining the extremities a and b, of the given 

 lines, are respectively equal to the segments ap, bs, of the given lines, which 

 are adjacent to them. This Mr. G. demonstrates as before. 



Prop. 3. To multiply the square of a given finite right line by any number. — 

 On an indefinite right line ap set off the given right line ab, fig. 13, pi. 13 ; 

 draw bc at right angles to ap and equal to ab ; and from a through c draw an 

 indefinite right line Aa. Take ad equal to ac, and draw de parallel to bc ; af 

 equal to ae , and draw pg parallel to bc, and so on. Then it appears, from 



* And it may be added, a mean in proportion between the two segments adjacent to the base. 

 For if a right line ab be any how divided in c, and from the two segments ca, cb, 3d proportionals 



j- j — -j to the whole line and each segment respectively, cd, ce, betaken 



AD c E B away, the remainders ad, eb, are equal, and each is a mean 



in proportion between the two cd, ce.— Orig. S. Horslet. 



