June 5, 1890] 



NATURE 



14] 



By a similar process of reasoning it can be shown that 



til — "^> 



wherefore — 



so that the lines e^e-l and d^d{ must meet in the same point on 

 line o^o{, which point of convergence is therefore the centre of 

 perspective, S. 



The perspectivity of the punctuated lines PT' and PR' may 

 be deduced from the projective relations of Fig. 3 by means of 

 the following two well-known theorems of projective geometry. 



Theorem I. — If the correlative or coharmonic points a and 

 a, of the punctuated lines u and tt^ (Fig. 4) coincide with their 

 common point of intersection, forming what may be termed a 

 coharmonic point, the lines u and Mj are in perspective. For, 

 let /'^i and cc^^ be any other two pairs of coharmonic points, and 

 meet in the centre S ; then S will be the centre of perspectivity 

 of the two lines u and u^^ ; seeing that in a harmonic or other 

 system of ratios, any three mem^rs of a compound proportion 



6^|.^^2^rr;:_.....„^.„ 



consisting of four terms suffice to determine the fourth or un- 

 known term. Then, this fourth point, being known and taken 

 in conjunction with any pair of the other three known terms, 

 will serve to determine a fifth ; and so on ad infinitum. 



Theorem II. — Similarly, if in Fig, 5 any pair of coharmonic 

 rays Sa and Sjaj of two different pencils lying in the same plane 

 are coincident or coperspective, the two pencils will be perspec- 



tive of the same line, and consequently perspective of each other. 

 For, let the coharmonic rays S^ and Sji^i meet in a point B, and 

 the rays S^ and Sj^i in C ; then the line BC will be perspective 

 of each pencil ; seeing that, if three coharmonic rays of the 

 pencils meet upon the line BC, it necessarily follows that a 

 fourth pair of such rays will meet upon the same line. 



Now it will be observed that lines PT and PR (Fig. 3) are in 



NO. 1075, VOL. 42] 



perspective, being projected from infinity on the right of the 

 figure. So also are lines PT and PT' as projected from 

 centre B, and lines PR and PR' as projected from centre C. 

 Further, the point P is a coharmonic point common to lines^ 



PT' and PR' as projected from centres B and C respectively j 

 for it will be seen that P does not change its position when pro- 

 jected through centre B from line PT to line PT', nor does it 

 change when projected, first from infinity parallel to dd', ee\ &c.,. 



