84 GREEK GEOMETRY. 



any line that can be drawn through the given lines will be 

 cut by them in the same proportions, like the lines drawn 

 through three harmonicals in the porisin already given of the 

 harmonical curve. To this Newton had not adverted, nor to 

 the unfortunate circumstance that the case of comets is 

 actually the case in which the problem thus becomes capable 

 of an infinite number of solutions. The error was only dis- 

 covered after 1739, when it was found that the comet of that 

 year was thrown on the wrong side of the sun by the 

 Newtonian method. This enormous discrepancy of the theory 

 with observation, led to a full consideration of the subject, 

 and to a discovery of the porismatic case.* 



* The remarkable circumstance of the case of the comet's motion, for 

 which Sir I. Newton's solution was intended, proving to be the porismatic 

 case of the construction, has been .mentioned in the text. It has been 

 sometimes considered as singular, that this did not occur to himself, the 

 more especially as he evidently had observed two cases in which the 

 problem became indeterminate namely, when the lines were parallel, and 

 when they all met in one point, for he excepts those cases in express terms 

 (Prin. lib. 1. Lem. xxvii.). It may be observed, that such oversights could 

 very rarely happen to the ancient geometers, because they most carefully 

 examined each variation in the data, and so gave to their solutions such 

 a fulness as exhausted the subject. 



The commentators on the Principia (Le Seur and Jacquier) make no 

 mention of the omission. The circumstance of the porismatic c:i 

 not discovered till ten years after their publication, when F. Boscovic-h 

 found it out, in 1749. But it is very extraordinary that Montucla appears* 

 to have been unaware of the matter, although the first edition of his work 

 did not appear till 1758. Nor is the least reference made to it in the 

 second edition, which was published the year he died (1799). Then- art- 

 other omissions in both editions, and also in the continuation. He appears 

 well to have understood the ancient method, and to have read and 

 examined some of the most celebrated works upon it. He had given due 

 praise to Simson in his first edition ; and to Lord Stanhope, who sent him 

 the ' Opera Eeliqua ;' and we find in the second edition a full note upon 

 the subject, ii. 277. In the continuation iii. 11, and seq., we have 

 further indications of the attention which he had bestowed upon the 

 ancient geometry ; but it is remarkable that though Matthew Stewart's 

 Tracts, published in 1761, were known to him, he was wholly unac- 

 quainted with the ' Propositiones Geometricse,' which appeared soon after, 

 unl with the General Theorems which had been published fifteen years 



