92 Improvements in Fundamental Ideas 



and such that any two straight lines CO and BO, making 

 angle OB right to C = right, will cut MM and NN in E 

 and F, so that PE ■ QF = PA • QC = m • n. 



Sixthly. — There are scores of other particular cases, and 

 as the solution is general, the modifications for these cases 

 may be easily made, and will afford useful exercise. 



DEDUCTIONS. 



(Figures for the following to be made by the reader. ) 



In the case of this problem, in which B coincides with A, 

 we have angle PB right to M = QN right to C ; and, there- 

 fore, putting p and q for the respective points in which QC 

 and PB cut MM and NN, it is evident a circle can pass 

 through PQ pq. And we have the line BH parallel to MM, 

 the point H being in circle BOC. And L being the other 

 point in which PB cuts circle BOC, it is evident CL is paral- 

 lel to NN. 



Hence we have the following theorem : — Let PQ pq be 

 any four points in the circumference of any circle ; if on P q 

 and Qip we take any two points BC, so that PB ■ QC shall 

 be of a constant magnitude m • n, and draw BH and CL 

 parallels to Vp and Qq, to cut Qq and Yp in H and L; then 

 will the points BCHL be in the circumference of a circle ; 

 and O being any point whatever in this circumference, the 

 lines OB OC will cut Vp and Qiq in E and F, so that 

 PE • QF = PB ; QC = m • n. And the following deductions 

 are obvious. 



POBISM. 



. Given two straight lines MM NN, and two points BC 

 in position and an angular magnitude l Q right f two points 

 PQ can be found, one in each of the given lines, such that 

 BO and CO being any two lines inflected, making angle OB 

 right to C = right, and E and F the points in which these 

 lines cut MM and NN respectively ; we shall have PE.QF 

 of a constant determinable magnitude. 



For the circle BOC is determinable, and BPI and CL, and 

 therefore BL and CH, and the points P and Q, in which these 

 last cut MM and NN. And we have PE ■ QF always equal 

 PB • QC. 



PORISM. 



Given an angular magnitude & right, and two points BC 

 through two other given points P and Q, tivo lines MM and 

 NN, and but two can be drawn, such that BO and CO being 



