Divifion of Right Lines , Surfaces , Solids. £5 
number m- 1, interfe6l bh in fome point c. From the 
vertex a of the triangle bac draw al as was directed in 
prop. i. and draw ls parallel to ca. I fay bl is fuch a 
part of bc as is expreffed by the number m\ and that bs 
is the fame part of ab. For it appears (from prop, i.) 
that bl is to cl as i to m— i. Wherefore bl is the 
part of bc. And bs is the fame part of ab that bl is of 
bc (4. E. 6.) 
Thus if the fquare on ac be fucceffively denoted by 
the fquare on ab multiplied by 3, 5, 7, 9, 11, 13, 13, 
17, 19, 21, 23, 25, See. bs will be fucceffively fuch a 
part of ab as is expreffed by 4, 6, 8, 10, 12, 14, 16, 1 8, 
20., 22, 24, 26, 28, &:c. 
PROPOSITION VII. THEORE M II. 
If from the angles at the bafe of any right lifted triangle , 
right lines be drawn to the alternate angles of rhombi 
defer ibed on the other two Jides , and reciprocally applied to 
them produced, and through the interfedlion of thefe lines, 
a right line be drawn from the vertex to the bafe ; the 
redi angle contained by the fines of the angles at the ex- 
tremities of one of the fides, will be equal to the redi angle 
contained by the fines of the angles at the extremities of 
the other and the parallelepiped contained by the fines of 
the 
(b) The author means, that fin. acl x fin. CALirfin. bcl xfin. cbl (fee 
fig. prop, 1.). For fin. acl : fin. l = al : ac, and fin. l : fin. bcl— bc :bl. Take 
Mj a third in proportion to al, ac. Then, becaufe ac* ; bc 1 =:al : lf, n will 
o like wife 
