STEWART'S THEOREMS. 123 



We have thus gone through Dr Stewart's properties of the 

 circle, and have arrived at his results by a simple and uniform 

 method. 



It is evident that there is no limit to the number of geome- 

 trical theorems which may be deduced from the general formula : 

 almost every curve will afford interpretations, if the word may 

 be so used, of our analytical conclusions. 



Thus in the ellipse : If any n radii vectores be drawn from 

 the centre at equal angles to one another, the sum of the squares 

 of their reciprocals is equal to n times the square of the reciprocal 

 of that radius vector which is equally inclined to the major and 

 minor axes. For we have 



- 2 = p(l-e 2 cos 2 <); 



and J = cos 2 ; 



therefore &c. Q. E. D. 



It is to be regretted that we have hardly any idea by what 

 considerations Dr Stewart was led to the curious theorems which 

 bear his name. It is said, indeed, that he was engaged on geo- 

 metrical porisms when he discovered them, and we are told that 

 he would have published them under the title of porisms, but 

 for his unwillingness to interfere with a subject which the re- 

 searches of his friend, Dr Simson, seemed to have appropriated. 

 Whether they are in reality porismatic, is a question on which 

 it would not be worth while to enter. 



The fundamental formula of our analysis is perhaps not new ; 

 the geometrical applications which we have made of it appear to 

 be original. 



