Mr. T. S. Davies on Geometry and Geometers. 383 



figure*. Other and multiplied coincidences might be pointed 

 out, all of which tend to show that the existing MSS. are only 

 copies (with sometimes unaccountable variations, it is true) 

 from some one of an earlier date, — and this, too, an imperfect 

 transcript of the original. 



The Savilian Library of Oxford contains two such MSS. 

 (Nos. 3 and 9) ; and from these Halley attempted to form a 

 text of that celebrated preface. This was prefixed to his 

 Restoration of the Section of Ratio and Section of Spsce, 

 printed in 1706; and is still the only text that is considered 

 to make any probable approach towards the original. Yet he 

 gives up in despair all attempts to elucidate what is said re- 

 specting the Porisms. 



Under these circumstances it would have been advisable 

 (at least as regards the section, Tiepl rtov TropiafxaTcov Ey/cXet- 

 8ov, pp. vi.-ix.) to give the readings of each of his MSS., and 

 his reasons for any departure from them, however slight. The 



* M. Breton, indeed, denies that this figure (or these figures) has ever 

 existed, or was at all necessnry {Comptes liendtcs, Oct. 29, 1849, p. 482) : 

 but inthishe stands alone, and offers a very unsatisfactory reason, — "puisqu'il 

 s'agissait de proprietes generales." As, however, M. Breton's paper is not 

 published, and only a slight notice given of it in the place referred to, it 

 would be obviously an inappropriate subject of comment here. Still his 

 general views are indicated with precision j and I may be (especially as my 

 thoughts are deeply occupied with the same subject) permitted to express 

 my entire dissent from his conclusions, and my conviction that his inter- 

 pretation is alike contradictory to all historical evidence, and incompatible 

 with the state of geometrical science in Euclid's time. 



I hope to be able to offer conclusive evidence that the porism of Euclid 

 must have been what Simson divined it to be, and could not possibly have 

 been anything else. I speak of it as a proposition : but I do not venture to 

 affirm that any single actual porism that has been said to be *• restored " is 

 precisely one of those which Euclid gave, whether offered as such by Sim- 

 son, Noble, or whoever else has made the attempt. They may be so : but 

 so long as the same lemma may be subservient to many porisms, how are 

 we to tell which of these many was really Euclid's individual porism ? 

 Probability would indeed rest upon the simplest, but certainty upon no one. 



The fragmentary character of the description of the several porisms in 

 the last three paragraphs of the text renders it impossible to affirm whether 

 the words refer to the predicate of the proposition, or to successive muta- 

 tions of the subject ; or, in other words, whether they express the conclu- 

 sion in which the reasoning is to terminate, or the several interchanges 

 of one single condition. The former has been uniformly assumed: but, 

 when closely examined, it seems to lead to difficulties from which the latter 

 is free. One of these difficulties is, that it is impossible to see in an enun- 

 ciation so constructed anything more than a local theorem. This assump- 

 tion, indeed, may account for the almost universal tendency (especially 

 amongst the continental geometers) to confound the porism with the local 

 theorem. There is no question, however, that the propositions given by 

 Simson, as porisms, are truly and essentially such, whether they coincide 

 with any of Euclid's or not. 



I cannot, however, enter into further details here. What I have to offer on 

 the subject will hence be reserved for another occasion and another place. 



