731 PHILOSOPHICAL TRANSACTIONS. [aNNO 1776. 



For the square on ac, being equal to the squares on ab, bc, is equal to the 



2 



square on ab multiplied by the number ^— 1- 1, or'^—— . Therefore it ap- 



pears (from prop. 1 and cor. l to 20 e 6), that al is to lb as '•^— 1- 1 to" — ?. 



Consequently, as is equal to the part required, a. e. f. 



Thus, if the square on bc be supposed successively equal to the square on ab 

 multiplied by the terms of the series 5, 6, 7, 8, Q, 10, 11, 12, 13, 14, 15, l6, 

 17j 18, &c. the numbers of the several partsdenoted by as, will be 11, 13, 15, 

 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, &c. which series compre- 

 hends all odd numbers after 9, and might have begun from 3, had the other 

 series begun from 1 . 



Prop. 6. To cut off from a given right line a part expressed by any even 

 number. — Let m denote any even number in general. Draw any indefinite right 

 line BH, and at right angles to it another be, fig. 15, pi. 13. On be set off the 

 given right line ba, and from a, with the distance equal to a right line, the 

 square on which is equal to the square on ab multiplied by the number m — 1 

 intersect bh in some point c. From the vertex a of the triangle bac draw al 

 as was directed in prop. 1 , and draw ls parallel to ca. Then bl is such a part 

 of BC as is expressed by the number m; and bs is the same part of ab. Thus, 

 if the square on ac be successively denoted by the square on ab multiplied by 3, 

 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, &c. then bs will be successively such a 

 part of AB as is expressed by 4, 6, 8, 10, 12, 14, \6, 18, 20, 22, 24, 26, 28, &c. 



Prop. 7. If from the angles of the base of ant/ right lined tnangle, right lines 

 be drawn to the alternate angles of rhombi described on the other tivo sides, and 

 reciprocally applied to them produced; and through the intersection of these lines, 

 a right line be drawn from the vertex to the base; the rectangle contained by the 

 sines of the angles at the extremities of one of the sides, will be equal to the rect- 

 angle contained by the sines of the angles at the extremities of the other; and the 

 parallelopiped contained by the sines of the angles of one of those triangles, into 

 ivhich the original one is divided by the said line drawn from the vertex, will be 

 equal to the parallelopiped contained by the sines of the angles of the other. 



Cor. The two triangles, adjacent to the segments of the base, have to each 

 other the proportion of the two adjacent to the sides containing the vertical angle, 

 or the proportion of the two into which the original triangle is divided; and any 

 one of these pairs of triangles are as similar figures described on the sides, being 

 as the segments of the base, which have to each other the duplicate proportion 

 of the sides. 



Prop. 8. If from the angles at the hypothenuse of any right angled right lined 

 triangle, right lines be drawn to the alternate angles of squares described on the 

 sides containing the right angle; and from the point where the right line drawn 

 from the right angle, through their intersection, meets the hypothenuse, right lines 



