74 Mr. glenie’s Proportions . 
thefe lines, a right line 
col be drawn from the 
vertex c to meet the 
bale ab ; the fegments 
al, lb, of the bafemade 
TP 
thereby, will have to r 
each other the dupli- 
cate proportion of the 
lides ac, cb. For 
through the vertex c 
let there be a right line 
drawn parallel to ab, to meet bf, ag produced, if ne- 
ceffary. Then, fince the triangles cqh, cpi, are re- 
fpedtively equiangular to the triangles aqb, apb (15. 
and 29. E. 1.); the proportions of ch to ab and of ab 
to ic are refpedtively equal to the proportions of cq^ 
to qb and of ap to pc (4. E. 6.). But the proportion 
of ch to ic is compounded of the proportion of ch to 
ab, and of ab to ic ; and confequently is equal to the 
proportion compounded of the proportions of CQ^to qb, 
and of ap to pc. And, fince the triangles acq^, apf, 
are refpedfively equiangular to the triangles bqg, bpc 
(15. and 29. E. 1.); the proportions of CQ^to qb and of 
ap to pc are each equal to the proportion of ac to cb 
(4. E. 6.); and when compounded are equal to the du- 
plicate proportion of ac to cb. Wherefore the propor- 
tion of ch to ic, which hath been fhewn to be equal to 
the proportion compounded of the proportions of cqJo 
qb and of a p to pc, is alfo equal to the duplicate propor- 
4 tion 
E 
