DR MATTHEW STEWART'S GENERAL THEOREMS. 603 



Expand, therefore, in the ordinary form, and bearing in mind that 2 n is 

 even, and the angular functions vanish, we have for each value of k from 1 to m 

 inclusive, the expression 



1.3.5.... (2-l) s 



1.2.3 w 



and as these are m lines, we have at once the expression in question. 



This is Prop. 45: also, when rc=2, it becomes Prop. 34; and when n=l it 

 becomes Prop. 14. 



NOTES UPON THE PRECEDING DISCUSSION. 



NOTE A, page 574. 



The first case of the solution of any one of Dr STEWART'S General Theorems being pub- 

 lished, that I remember to have met with, is that of the 41st Proposition, by Professor 

 PLAYFAIR, in his paper on the Arithmetic of Imaginary Quantities, Phil. Trans. 1777 ; and 

 I am not aware, impressed as he was with the great beauty of these propositions, that he 

 published anything more on the subject, and even this one is taken up incidentally. 



Dr SMALL, in the Edinb. Trans., vol. ii., gave solutions of Propositions 9, 10, 11, 12, 13, 

 14, 15, 16, 17, 19, 30, 31. 



Professor Lo WRY gave solutions of 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 

 in vols. i. ii., O.S., of Leybourn's Repository ; and annexed to them several interesting, but 

 tolerably obvious deductions. 



Mr SWALE of Liverpool also gave, in the same work, solutions of 15, 16, 17, 18, 19, 

 20, 21. 



The three last-mentioned authors treat the subject by methods purely geometrical ; but 

 Dr SMALL has been inconsiderately censured for a " lack of geometrical purity," merely 

 because he mentions the " centre of gravity," notwithstanding he builds nothing upon it 

 deduced from its physical character. (Repos. i. p. 131.) Geometrical purity, however, is 

 not vitiated by the use of an injudicious term, but by the employment of methods which are 

 unrecognised by geometry, or inconsistent with those which are recognised. The standard 

 itself was originally arbitrary ; but being once recognised and generally admitted, it is the 

 proper criterion by which to judge of the purity of geometrical processes. Taking, however, 

 the strictest view of the subject, I confess I know of no flaw in Dr SMALL'S argument ; and 

 there is no doubt that to Dr SMALL'S paper we are in reality indebted for all that has been 

 effected concerning the porismatic part of Dr STEWART'S Theorems. It may be further 

 remarked, too, that the ordinary sense of the word geometrical is altogether inapplicable to 

 the greater part of theorems themselves, since they relate to magnitudes which the ancient 

 geometry does not recognise (viz., to fourth and higher powers of lines) ; and hence, as far 

 as purity is concerned, it seems to savour of the " gnat and camel" character, to affect a 

 rigid adherence to even the forms of the ancient geometry in attempting their solution. Dr 



