96 Improvements in Fundamental Ideas, &;c. 



to draw a'G b'G, making angle Gb' right to a' of a given mag- 

 nitude, and through c' and d' draw c'H and d'H making angle 

 Hd' right to c' of a given magnitude, so that a, b, c, d being the 

 points in ivhich a'G, b'G, c'H, d'H cut the straight lines 1, 2, 

 3, 4, iv e shall have Aa to Dd in a given ratio, and the rectangle 

 under Bb and Cc of a given magnitude. 



Draw a' A b'B. Suppose we draw CE making angle CE 

 right to c = to angle Bb rigbt to V , and CE-B6' = CcBb; 

 then E is given, and we have also the angle cC right to E 

 = b'B right to b or G. 



Again, drawing DF, making angle DF right to d = 

 angle Aa right to a, and DF to Aa' as Dd is to A«; then F 

 is a given point, and the angle FD right to d is = angle a' A 

 right to a or G. 



Now EF are given points, and FD, Cc given lines in posi- 

 tion ; and a'b' are given points, and a' A b'B lines given in 

 position ; hence as G is in the circumference of a given circle 

 b'a'G, it follows (from improved theorems) that the intersec- 

 tion I of c-E and d¥ is in che circumference of a determinable 

 circle through E and F. 



But (by porism No. 1 of second problem) we can find 

 points P and Q in lines 3 and 4, corresponding to circle EFI, 

 such that Vc.Qd shall be of a constant determinable magni- 

 tude ; similarly we can find points M and N in the same 

 lines corresponding to the given circle c'Hd', such that the 

 rectangle Mc.Nrf shall be of a constant determinable magni- 

 tude : therefore by problem 3rd Ave can find the points c 

 and d. 



If instead of the ratio Aa to J)d we were given the rect- 

 angle under them, or that instead of the rectangle under Bb 

 and Cc we were given their ratio, the method of solution is 

 obvious from the above method of proceeding. 



