362 Remarks on the Logic of Elementarif Geometry. 



howev.^r, is the course proposed by Euclid iu his 5th def. 

 book 5, and its peculiar excellence is this, that to investigate 

 the equality of quotients, the relations of number are dismissed 

 from the mind in that comparison of two geometrical magni- 

 tudes which it is called upon to make. The employment of 

 numbet is only a preparatory step, and in drawing the con- 

 clusion our attention is entirely confined to the inquiry whether 

 there be an excess of one quantity above another. Perhaps 

 this method of investigation led Euclid to his definition of 

 ratio in terms not strictly applicable to if, when he called it 

 a relation of qiinntitxj or magnitude, whereas it is undoubtedly 

 a relation of quotity or number, taking these terms in their 

 most general sense, as including the ratio of any functions of 

 Tinmbers whether commensurable or incommensurable, as 

 »ij2 : "JJ3 &c. One of the earliest efforts of geometry un- 

 doubtedly would be to demonstrate the relations of the parts 

 of a field, a house, or a city, by means of a figure drawn on 

 a board or in the sand, and nothing else could occupy the 

 mind but the relations of determinate numbers. The pro- 

 perties and relations of quantities must have been long and 

 deeply studied, before the existence of incommensurables could 

 be sugg-ested. In fact, we do not see how it could have been 

 suggested at all, except in the investigation of the properties 

 of figures, by the discovery that there were relations among 

 parts of them, which the ratios of rational numbers could 

 not express, this I think necessarily implies the previous in- 

 vestigation of those which they can express. 



Rendering the discovery of such relations as are indicated 

 by equal ratios dependent on the occurrence of excess or differ- 

 ence, is therefore a recondite process, and would probably be 

 reached only by many endeavours, and Euclid's definition may 

 therefore be considered as a generalization from many previous 

 rules, or a reduction to unity and simplicity of the consequences 

 of many previous inquiries. Perhaps the peculiar method of 

 arithmetical notation employed by the ancient geometers aided 

 in the discovery of the rule. To reduce the investigations of 

 relations implying quotity to those of relations implying excess 

 only, is, however, the very process which is most convenient 

 for demonstrating that such relations exist between geometric 

 quantities ; and so far Euclid's process is perfect. But I do 

 not think that either Euclid or his Commentators have suc- 

 ceeded in the logical announcement of the rule. It has been 

 considered an excellence in its character, that the rule in its 

 common form is applicable both to commensurable and in- 

 commensurable quantities, but the truth is, that it is a some- 

 ■what illogical combination of two rules, one of which is appli- 

 cable only to commensurable quantities, and the other is 



