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of the lines P' P" give the altitudes of the several points P : this 

 is also a method in ordinary use. 



But it is to be observed that the points P', P" are both of them 



projections, and that the general theory is as follows : we repre- 

 sent the position of the point P by means of its projections P', P", 

 from two fixed points £1', X2" respectively ; the line joining these 

 points passes, it is clear, through a fixed point fl which is the 

 intersection of the plane of projection by the line which joins 

 the two points XI', 12". 



Hence we say that a point P in space is represented in piano 

 by any two points P', P" which are such that the line joining 

 them passes through a fixed point O. And we have thus a 

 system of constructive geometry which is the more simple on ac- 

 count of the generality of its basis, and which is at once appli- 

 cable to any of the special projections above referred to. I esta- 

 blish the fundamental notions of such a geometry, and by way 

 of illustration apply it to the solution of the well-known problem 

 of finding the lines which meet four given lines in space. 



A point P (as already mentioned) is given by its projections 

 P', P", which are points such that the line joining them passes 

 through the fixed point O. 



A line L is given by its projections 1/, I/', which are any two 

 lines in the plane. We speak of the point (P', P"), meaning 

 the poiut P whose projections are P', P ,; ; and similarly of the 

 line (L/, I/'), meaning the line whose projections are L', I/'. 



If P', P" coincide, then the point P is in the plane of projec- 

 tion ; and so if L/, U ! coincide, then the line L is in the plane 

 of projection. 



If through 12 we draw a line meeting L/, L" in the points 

 P', P" respectively, these are the projections'of a point P on the 

 line L. In particular the intersection of L/, L" (considered as 



