Legend7-e's Geometry. 289 



Enjrlish writers on geometry to the present time, while the 

 French mathematicians have for a long period investigated 

 the principles of ratios and proportion by arithmetical and 

 algebraic methods. Even the Edinburgh Encyclopoedists, 

 who have made Legendre's Elements the basis of their ar- 

 ticle Geometry, have in this part entirely deserted him, and 

 have introduced the theory of ratiosand proportion essential- 

 ly after the manner of Euclid. Atthe same time, the author 

 of the article referred to, confesses, that "it might with pro- 

 priety be inserted, rather as a preliminary theory, than as 

 forming a part of geometry.* It wais necessity, and notchoice, 

 that led Euclid to connect the theory of ratios and propor- 

 tion with geometry. In his time algebra, to which we have 

 before said that this theory in all its extent belongs, was 

 unknown. When, therefore, Euclid wished to apply pro- 

 portion to geometrical figures, it was necessary for him 

 to iiivestigate its principles by geometry, as the only means 

 with which he was furnished. Euclid is not in fault for the 

 course which be pursued. He did all that could be done, 

 in the circumstances in which he was placed. But for us, 

 who are in possession of algebraic methods, at once easy 

 and elegant, to pursue the same course, is entirely a different 

 thing. To do so, is not less absurd than it would be to set 

 about determining the obliquity of the ecliptic to the equator 

 by means of the gnomon, when we have the theodolite and 

 repeating circle; or to pursue Aristotle's method of j)hiloso- 

 phising, when we have so long followed that of the illustri- 

 ous Bacon, with such splendid success. The theory of pro- 

 portion as given by Euclid, is extremely tedious, circuitous 

 a.nd difficult to be understood by beginners. The reason of 

 Vbiis is, that geometry in its nature is of very little generality, 

 ^nd in its construction is not sufficiently y?exi6/e to admit of 

 easy application to the subject. But by making use of al- 

 gebra, which at the same time accommodates itself to the 

 suject with great facility, and is a language vastly more gen- 

 eral than geometry, the whole theory of ratios and propor- 

 tion flows in the most natural and easy manner, from the 

 simplest properties of equations. 



Again, the Elements of Euclid contain too many proposi- 

 tions merely subsidiary, and propositions which are of almost 

 no practical utility, and have no connection with the sue- 



* EJinb. Encyc. Vol. IX. pp. 658, 669. 



Vol. VI.— No. 2. 37 



