30 The Three Sections, TangendeSy 



THE SECTION OF SPACE. 



Given the points Q, and U in the given straight lines MM 

 and ISIN; through a given i^oint A to draw a straight line 

 CAD culting MM and NN in C and D, so that QC.UD shall 

 be equal to m.n (where m.ii is a given magnitude of knoivn sign 

 in respect to the directions on MM and NN^. 



ANALYSIS. 



Suppose that in the given lines we take the segments Q,P 

 and UR such that QP.UH = QC.UD = m.n, and that we 

 draw PO and QO making the angles PO right to M, and QO 

 right to M; respectively equal to All right to U, and UR right 

 to A. Then the similar triangles POQ, ARU, give QO.UA 

 = QP.UR, and therefore QO.UA = QC.UD; and .-. as the 

 angle QC right to O is equal the angle UA right to D, the 

 triangles COQ and ADU are similar^ and the angle CO right 

 to Q is equal AD right to U ; hut PO right to Q is equal 

 AR right to U ; therefore it follows that angle AR- right to 

 D or C is equal OP right to C. Hence, if H be the point in 

 Avhich RA and PO intersect, a circle can pass through AOH 

 and C : hut the points AOH are knov/n ; therefore tlie point 

 C in which the circle AOH cuts T\IM is known, and hence 

 CAD. 



COMPOSITION. 



In the given lines MM and NN take any two segments QP 

 and UR, such that QP.UR = m.n-, draw PO and QO, mak- 

 ing the angles PO right to Q, and QO right to P equal re- 

 Epectirely to AE right to U/and UR right to A; through A; 



