1 10 PHILOSOPHICAL TRANSACTIONS. [aNNO iS/S. 



Again, because CL is to LG as HF to FG, it will be, by permutation and 

 division, as CL — HF to HF so LF to FG, that is n — y\y wc — x : 



'±IlIL. which taken from AF = x, leaves GA = !i^Ilf^. And EA = 

 — , because FA, AH, AE are proportionals: Therefore EA — GA, that is 

 EG = -^ - ai-liiy-, and KA - EA or KE = i^^±^ - ^. But we said 



X n — y b + y X 



that KE is to EG as KA to AG, 



,, , . ay + bx dd dd nx — cy ay + bx nx — cy 



that is -f-f : : : -^^— : ^; 



+ y X X fi — y + y n — x 



hence is r lanxxy + 2 hnx^ — ddhnx — ddnxy ■> naddy -|- nbddx 



found 1 — 2acxyy — ibcxxy + ddhcy + ddcyy j — addyy — hddxy. 



. , , be ., . 2bbcx^ bbddcx Qbbcyyx , , , 



And because n = — , it is ^-^ , because xx =z ad — ijy, 



a ' a a a ' J J 



_, ., . Ibbcx^ 2bbcddx 2bbcyyx •, , , 



But it IS = ^^^, because xx = dd — ?/?/. 



a a a ^ ^^ 



Therefore — ^^^ __c£j/ __ ^acxyy + ddcyy = — addyy — hddxy. 



Then dividing all by y and multiplying by a, it 



is — ihhcxy — ddhcx — laacxy + ddcay = — aaddy — hddax, 



or ahddx — chddx + acddy + aaddy = laacxy -j- ibbcxy, 



. , abddx — cbddx + acddy + aaddy ,. , .i i i i 



^^^^ 2aac + 2bbc — ^^ ^^ equation to the hyperbola. 



_ , , abddx — anddx + acddy 4- aaddy 



Or, because be = na, , ,,,/ = xy. 



^ ' 2aac + 2bbc -J 



Pnf _f^l_ - /,. then P^^'^-P^'^'+P'^y + P^y _ ^y 



Hence also is easily found the following construction : Join BA, AC (fig. 7), 

 and applying to each separately the square of the radius AD, they give AP, 

 AQ ; then joining PQ, bisect it in R, through which draw RD parallel to AD, 

 (which bisects the angle BAC) and RN perpendicular to it. Then will RD, 

 RN be the asymptotes of the opposite hyperbola, the one of which ought to 

 pass through the centre A, and they will cut the circumference in the points 

 H sought. The hyperbolas will also pass through the points P, Q. 



The reason of the construction is manifest, by drawing P-y and Q(^ perpen- 

 dicular to AM : for it makes Ay = ^J ^^ or p; and AJ = — . Also Py = 



^, and QC = ^. Therefore AG = ■^^^, and OR = ^^-^. Hence 

 c ' ^^ c 2c ' 2c 



the rest is easy. 



To the above M. Slusius replies as below. 



You will not wonder, learned Sir, that in Alhazen's problem, the same con^ 

 struction may be deduced from different equations, since all those that we have 



