Legendre’s Geometry. 289 
English 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 Encyclopeedists, 
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- 
priely be inserted, rather asa preliminary theory, than as 
orming a part ofgeometry,* It was necessity, and not choice, 
thai ied 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 investigate its principles by geometry, as the only means 
_ with whi e 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 philoso- 
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 
and difficult to be understood by beginners. The reason of 
this is, that geometry in its nature is of very little generality, 
and in its construction is not sufficiently flexible to admit o 
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 suc- 
* Edinb. Encye. Vol. IX. pp. 658,669. 
Vor. VI.—No. 2. 37 
