534 Notices lespeciing New Booh. 



from established principles, and, by a chain of reasoning deduces new prin- 

 ciples from these ; the former pioceeds from the principles tliat are to be 

 established considered as known, and from these, taken as premises, arrives, 

 by reversing the chain of reasoning, at known principles. The latter 

 method is the diilactic method used in communicating instruction; the 

 former is rather employed in the discovery of truth." — P. 205, 



It may be as well to remark, once for all, that though the descrip- 

 tion of geometrical analysis given by Pappus does certainly admit 

 (rather by implication than by direct statements) of the belief that the 

 ancients did apply the analytical method to the discovery of the de- 

 monstration of theorems, yet its principal use was to trace the re- 

 lations between the data and qusesita of problems, so as to deduce a 

 method of constructing them. We may likewise inform Mr. Bell, 

 and such as him (if indeed such there be), that the details of an 

 analytical process in the two cases had so great a dissimilarity in 

 everything but the one general idea, as to be incapable of description 

 in the same form of expression. The analysis of a theorem indeed 

 differs but little from Euclid's reductio ad absurdiim : — merely, in fact, 

 in assuming the conclusion to be true instead of false. The reason- 

 ing ended in the deduction of a known truth, instead of the contra- 

 diction of one. As, however, Mr. Pott's has very clearly explained 

 the analysis of theorems and of problems in the Appendix to his 

 Euclid (8vo ed.), we shall i-efer to that valuable tract instead of 

 quoting it at length here. 



We shall only quote two other examples of Mr. Bell's method of 

 improving geometry. Concise enough they are : but the " perspi- 

 cuity " is beyond our power of detecting. 



" Def. Pjlane Locj. — If the position of a point is determined by certain 

 conditions; also, if every point in some line, and no other point, fulfill 

 these conditions, the line is said to be a locus of the point. 



" As a simple illustration of a locus, consider that of a point which is 

 always equally distant from a given point. This is obviously a circle, whose 

 radius is equal to that distance. So the locus of a point which is always 

 equally distant from a given straight line, is a line parallel to it, and at a 

 distance equal to the given distance." — P. 207. 



" Def. PoRiSMs. — A po7-ism is a proposition of an indeterminate nature, 

 such that an indefinite number of quantities must fulfill the same conditions. 

 As a simple example of a porism, let it be required to find a point such, that 

 all straight lines drawn from it to the circumference of a given circle shall 

 be equal. This point is evidently the centre of the circle. Problems of 

 loci and porisms are in many cases convertible. The preceding problem be- 

 comes a problem of the latter [form?] kind when the centre is given, and 

 it is required to find the locus of all the points that are at a given distance 

 from it."— P. 208. 



Mr. Bell is not the only one who has blundered about the 

 "porisms;" but he is, we tlaink, the only one who has blundered 

 quite so egregiously. His description amounts to nothing, except a 

 complete proof that he does not understand what a porism is. But 

 supposing that he understood the subject ever so well, can anybody 

 but himself think that the two definitions quoted above (plane loci 

 and porisms) are adequate to give a learner the slightest conception 



