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XXXYTTI. 



An Analytical Discussion of Dr MATTHEW STEWART'S General Theorems. By 

 THOMAS STEPHENS DAVIES, Esq., F.R.S.L. & Ed., F.A.S., Royal Military 

 Academy, Woolwich. 



(Read April 1. 1844.) 



PART FIRST. 

 V 



DURING the century which has nearly elapsed since Dr MATTHEW STEWART 

 published his General Theorems, many eminent geometers, both English and 

 Foreign, have attempted to discover their solutions. Those attempts have, how- 

 ever, been rewarded with but limited success, and by far the most general and 

 the most difficult of them remain still without a single published remark in the 

 way of discussion or solution. Dr STEWART did not, as far as I know, make 

 allusion to them himself in any of his subsequent writings, though he describes 

 them as " of considerable use in the higher parts of mathematics ;" and we learn 

 from the preface to Mr GLENIE'S Demonstration of the 42d Proposition (Tract, 

 1813), that in conversation, Professor DUGALD STEWART, in 1805, stated that, 

 " he had not been able to find amongst his father's posthumous papers one word 

 respecting them ; that he had, oftener than once, observed mention made of them, 

 in terms of admiration and respect, by some of the first mathematicians on the 

 continent of Europe ; but that as both they and the geometers in this country 

 had tried their strength on them without success, and they had so long remained 

 without demonstrations, he never expected to see them demonstrated." This 

 circumstance, of neither any demonstrations nor even memoranda, on the subject 

 being found amongst Dr STEWART'S papers, is readily accounted for by Professor 

 PLAYFAIR, in his biography of that distinguished geometer (Edin. Trans, vol. i. 

 p. 74), in the description which he gives of the habits of study of Dr STEWART. 

 " He rarely wrote down any of his investigations till it became necessary to do 

 so for the purpose of publication. When he discovered any proposition, he would 

 put down the enunciation with great accuracy, and on the same piece of paper 

 would construct very neatly the figure to which it referred. To these he trusted 

 for recalling to his mind, at any future period, the demonstration or the analysis, 

 however complicated it might be." 



It thus appears that no ground exists for our hoping to discover the means 

 by which Dr STEWART originally investigated these theorems ; whilst we cannot 

 but be surprised at the powers of attention and invention of that mind which 

 could carry on, without the aid of writing, such extended and complicated inqui- 

 ries as are implied in the discovery of them. Neither can we be surprised if some 

 oversights should occur in their investigations, conducted in such a manner. 

 That oversights do exist, will, however, presently appear. 



VOL. XV. PART IV. 7 Q 



