574 MR THOMAS STEPHENS DAVIES ON 



It is curious enough that, among the geometers who have written concern- 

 ing these theorems, very few have seen their true character. In fact, Professor 

 PLAYFAIR is the only one who has distinctly stated that " they are, for the most 

 part, porisms," but he nowhere enters into any discussion of them, either under this 

 or any other aspect. In one place, however, (Ed. Rev. vol. xvii. p. 129,) he recom- 

 mends to the attention of geometers these propositions, as fitting subjects for the 

 employment of the trigonometrical analysis, and speaks of " the difficulties which 

 they will present even to those who come armed with that powerful instrument." 

 Amongst the other authors who have spoken of these propositions as to logical 

 character, it may be sufficient to quote two merely ; but the scientific rank and 

 high acquirements of these two will prove that very precise views are not enter- 

 tained by mathematicians, even in this country, respecting these propositions. 

 Mr BABBAGE says, that " many of them are capable of forming, with a slight alte- 

 ration in their enunciations, the most beautiful porisms," (Quarterly Journal of 

 Science, vol. i. ; and Mr ELLIS affirms that, " whether they are in reality porismatic, 

 is a question on which it would not be worth while to enter." (Cambr. Journal, 

 May 1841.) Adopting SIMSON'S definition, however, of the porism, it will be quite 

 clear that a considerable number of them that is, all which are really porismatic 

 have the strictly porismatic form of enunciation. Of the remaining ones, a very 

 small number are local theorems ; and the rest are given in the ordinary form of 

 indeterminate theorems. 



In all the attempts at solution of the porismatic part of these propositions that 

 I have met with, they have invariably been treated as indeterminate theorems, 

 the porismatic constructions being first supplied; and in supplying these, the 

 authors, having no mode of analysis adapted to their object (except from con- 

 jecture), had to encounter difficulties which would inevitably render their success 

 impossible. In fact, the skill and address manifested by Dr SMALL (Ed. Trans. 

 vol. ii.), and Messrs LOWRY and SWALE (Leyb. Repos., O.S., vols. i. ii.), manifest 

 the most profound geometrical sagacity, and will reflect a lasting honour on their 

 names : but, at the same time, it must always be regretted that their degree of suc- 

 cess was not proportioned to the labour and ability employed in their researches. 

 (Note A.) 



At a very early period of my own studies, the porisms engaged much of my 

 attention, and excited a deep interest in the inquiry. This interest, in the outset, 

 was created by the paper of Professor PLAYFAIR, in the Edinburgh Transactions, 

 one of the most luminous and philosophical discussions of a mathematical sub- 

 ject it has ever been my good fortune to read. His suggestion of an algebraical 

 analysis of the porism, which unfortunately he never published, led me to 

 attempt such an application myself; and it could not long escape notice, under 

 these circumstances, that the method of treatment must be identical with that 

 employed in the " method of indeterminate co-efficients ;" in fact, that this latter 

 method always occurs in the shape of a porism, and all the propositions in 



