80 Review ofT. P. Thomp.ion's Geometry without A^rioms. 



rately. No good reason appears for treating on ratios in a geometrical 

 treatise at all; but if any where, it should be in the introduction; because 

 at present he is forced to delay to his sixth book what should have been 

 partly in the first and partly in the third. It is objected that the fifth 

 book is too hard for a beginner. But that is the autlior's own fault, for 

 taking so extraordinary a method. Oh ! but, say the advocates of this 

 extolled book, it is so geometrical ! and so philosophical ! — We apprehend 

 that if it be really "geometrical," this is a fault; for a treatise on ratios 

 ought no more to be geometrical, than astronomical or optical. But if it 

 be meant, that it is adapted to those who understand nothing of Arith- 

 metic, then we reply that such will remain incurably ignorant of ratios, 

 thougli they read Euclid fifty times. And this brings us to the boasted 

 philosophy of the method, which we venture to describe thus. Having 

 given a useless definition of ratio, (which R. Simson rejects,) he then gives 

 his serviceable definitions of the phrases, same ratio, greater ratio, less 

 ratio, compound ratio ; in such wise, that the words greater, less, &c. have 

 new senses,* or rather no sense at all as isolated words, any more than 

 cir and cle of the word circle. Accordingly it becomes necessary to prove 

 that " Ratios which are the same with the same ratio, are the same with 

 one another." Thus the upshot of this treatise on ratios, is, that we do not 

 and may not know what " ratio" means. For if we do know, then since 

 greater and less and same (or equal) are well known terms, it is un- 

 lawful to lay down new definitions of same ratio, greater ratio, &c., and 

 every proposition in the book that concerns proportion is vitiated. t 



Again, Euclid has chosen to limit his proofs to figures which he can 

 construct with rule and compass, for which two instruments he makes 

 humble request in three postulates. The compass we willingly give him ; 

 concerning the ruler we demur, until he has shown how the straight line is 

 generated. But all are aware that he might as reasonably ask for elliptic 

 trammels or any other instrument, as for the ruler; in short, in construct- 

 ing his figures, it is enough to show that his description of the figure con- 

 tains nothing self-contradictory. His squeamishness here has entailed on 

 him much disorder, besides the frequent introduction of petty problems, 



* (Note by R. Simson on Euclid, V. 10.) — "It was necessary to give another 

 demonstration of this proposition, because that which is in the Greek and Latin and 

 other editions, is not legitimate ; for the words greater, the same, or equal, lesser, have 

 a quite different meaning when appliC'l to magnitudes and ratios, as is plain from 

 tlie fifth and seventh definitions of Book V." As usual, this editor makes sure that 

 the error could not have been Euclid's own ! Theon is generally the name on which 

 he heaps the odium. 



f R. Simson's additions at the end of this book are quite a literary curiosity. For 

 ourselves, we cannot read the statement of Prop. K. with gravity. It contains the 

 word ratio twenty-three times, as in school-boy days we remember to have counted. 

 He says " they are frequently made use of by both ancient and modern geometers :" 

 which we believe on his word ; but what he called modern is not now modern. 



