ANCIENT ANALYSIS. POEISMS. 81 



given in minute particularity, he never could think that he 

 had so far elaborated and finished it as to warrant him in 

 finally resolving on its publication. 



There needs no panegyric of this most admirable perform- 

 ance. Its great merit is best estimated by the view which 

 has been taken of the extraordinary difficulties overcome by it. 

 The difficulty of some investigations the singular beauty of 

 the propositions, a beauty peculiar to the porisrn from the 

 wonderfully general relations which it discloses the sim- 

 plicity of the combinations the perfect elegance of the 

 demonstrations render this a treatise in which the lovers of 

 geometrical science must ever find the purest delight. 



Beside the general discussions in the preface, and in a long 

 and valuable scholium after the sixth proposition, and an 

 example of algebraical porisms, Dr. Simsoii has given in all 

 ninety-one propositions. Of these, four are problems, ten are 

 loci, forty-three are theorems, and the remaining thirty-four 

 are porisms, including four suggested by Matthew Stewart, 

 and the five of Fermat improved and generalized ; there are, 

 besides, four lemmas and one porism suggested by Dr. Traill, 

 when studying under the professor. There may thus be 

 said to be in all ninety-eight propositions. The four lemmas 

 are propositions ancillary to the author's own investigations ; 

 for many of his theorems are the lemmas preserved by Pappus 

 as ancillary to the porisms of Euclid. 



In all these investigations the strictness of the Greek 

 geometry is preserved almost to an excess ; and there cannot 

 well be given a more remarkable illustration of its extreme 

 rigour than the very outset of this great work presents. The 

 porissm is, that a point may be found in any given circle 

 through which all the lines drawn cutting its circumference 

 and meeting a given straight line shall have their segments 

 within and without the circle in the same ratio. This, 

 though a beautiful proposition, is one very easily demonstrated, 

 and is, indeed, a corollary to some of those in the ' Elements.' 

 But Dr. Simson prefixes a lemma : that the line drawn to the 

 right angle of a triangle from the middle point of the 



