(504 MR THOMAS STEPHENS DAVIES ON 



STEWART did not set such an example by attempting to antiquate the forms of their enun- 

 ciation ; and he does not even call them geometrical theorems, though, as they relate to the 

 properties of geometrical figures, he might have done so without impropriety. Whether 

 he even obtained them by geometrical considerations, in the first place, is open to question, 

 when we turn to what Professor PLAYFAIR remarks (Ed. Trans., i. p. 60) concerning the 

 probable origin of his inquiries on this subject. In fact, his being in possession of SIMSON'S 

 definition of the porism (which may be inferred from his formally distinct enunciations of the 

 porismatic part of his propositions), I do not think there is a single step in the present paper, 

 which it would have been at all improbable that Dr STEWART could readily take. Any slight 

 mistake in estimating the number of final equations would create no surprise, when we 

 recollect how much he was in the habit of "thinking out" his conclusions without the aid 

 of writing. Possibly, therefore. Dr STEWART'S investigations were not much unlike those of 

 the present paper, in their essential character. But to return : 



Mr GLENIE, of the Royal Artillery, gave, in the Edinburgh Transactions (vol. vi.), demon- 

 strations of nearly all the indeterminate theorems, except those proved by Dr STEWART him- 

 self, and the 41st theorem. Of this last, he afterwards gave a proof in a small tract (1813.) 



Mr GLENIE'S course of investigation is remarkably elegant, and it discloses many curious 

 and interesting properties of the circle, of which uses, generally unsuspected, may yet be made. 

 This able geometer has, however, fallen into the prevailing error on the subject of geome- 



r" 

 trical purity : that -^r a expresses an idea recognised by the ancient geometry ; but it in no 



degree vitiates his reasonings as demonstrations, though it throws them, as all demonstrations 

 of these theorems must be thrown, into the domain of algebra. 



Mr BABBAGE, after paying a high and deserved tribute to the genius of Dr STEWART, 

 gave (Quarterly Journal of Science, vol. i.) investigations of nearly all the indeterminate theo- 

 rems, by means of trigonometry. They differ from the investigations given in this paper 

 mainly in the forms of expansion employed ; and I gladly avail myself of the opportunity 

 of acknowledging my obligations to that paper, for some useful modifications of my own 

 primary processes respecting this class of propositions. 



The late Mr THOMPSON published (in the Newcastle Magazine, 1826-27-28) investiga- 

 tions of a considerable number of these theorems. He adopted the mixed method that is, 

 the employment of the ancient geometry, algebra, and trigonometry, as best suited his imme- 

 diate purpose. These solutions bear ample testimony to the mathematical powers of their 

 author ; but they are, unfortunately, so disfigured by an awkward and irregular notation, 

 and bad style of printing, as to be not only generally unintelligible to the ordinary reader, 

 but, in many cases, to almost defy interpretation by those who look into the subject with the 

 greatest care. I was much struck, when I first met with this work (three or four months 

 ago), with the near approach made, in some of his demonstrations, to the xise of the principle 

 which forms one of the foundations of this paper the formation of conditional equations. 

 He seems, however, to use it as a matter of convenience rather than as a principle as some- 

 thing that may answer a special purpose, rather than as a general method founded in the 

 nature of the algebraic analysis, and applying to all possible cases. 



Owing to the circumstances before mentioned, had my own investigations been in a less 

 complete state than they were at that time, Mr THOMPSON'S papers could have afforded me 

 no assistance ; but I gladly embrace this opportunity of publicly recording the high estimate 



