83 The Three Sections, Tangencies, 



line LL through G and H is known. The determinable ratio 

 can be found by drawing CD and BIT parrallel to LL^ to cut 

 NN and MM in D and U : it is that of PU to QD. 



PORISM. 



Given the point P in a given straight line MM., and a point 

 Ci in a given straight line NN^ given also a point B, and a 

 straight line LL ; a point C can be determined such that lines 

 drawn from any pioint I in the third given line LL, through 

 the given point B and the determined p)oint C to cut MM and 

 NN, respectively in E and F, loe shall have PE to QF in a 

 constant determinable ratio. 



For BPH is known, and BG parallel to MM is known, and 

 therefore GC through G parallel to jN'N is known in position; 

 and hence the point C where GC cuts QH. And I being any- 

 assumed point in LL, and E, F, the pohits in which BI and 

 CI cut MM and NN, the required ratio is evidently the same 

 of that of PE to QF. Or, drawing CD and BU parallel 

 to LL to cut NN and MM in D and U, the required ratio is 

 evidently the same as that of PU to Q,D {the lines CD, BU 

 intersect at infinity on LL) . 



PORISM. 



Two poiiits B, C, and two straight lines MM, LL, are 

 given in position, and a point P in the first of these lines MM ; 

 a straight line NN and a point Q, in it can be found, such that 

 if lines IB and IC be drawn from any point I m the second 

 LL of the given lines through B and C, to cut MM and NN, 

 in E and F respectively, then ivill PE have to QF the given 

 ratio of m to n. 



For we can draw BG parallel to MM to cut LL in G, and 

 therefore we have CG to which NN must be parallel. Again 

 we have BP and point H where it cuts LL ; and therefore 

 HCQ through H and C is known in position. And we have 

 PU, U being the point in which BU parallel to LL cuts MM. 

 And if D be the point in which CD parallel to LL cuts NN,- 

 then since QD : PU : : m : n, and that QD is parallel to CG, 

 it follows QD is known, and therefore NN with which it is 

 coincident. 



If the ratio ~ be not restricted in sign in respect to direc- 

 tions on CG and MM, it is evident there are two positions 



