2l0 Review of the Cambridge Course of Mathematics. 



and from which spring trees whose numerous branches are 

 loaded vvitli fruit ; the others, which have scarcely pierced 

 the ground, soon disappear, after having offered a sterile 

 aliment to vanity." 



To speak more particularly, M. Lacroix appears to have 

 been governed in preparing his mathematical works, by the 

 following principles : — 



1 . To give a demonstration as rigorous a? the nature of 

 the case would admit, of every rule and principle of which 

 any use is made. This is very different from the course 

 pursued in most American and English books upon math- 

 ematics. In our treatises upon Arithmetic and Algebra, 

 with a very few honourable exceptions, the rules are given 

 in a very concise and purely didactic form, and whatever 

 attempt there is at an investigation of them, is thrown into 

 notes which are seldom much consulted. Nor is the stu- 

 dent generally in blame for not consulting them, as they are 

 usually so ill adapted to the state of his knowledge, that he 

 finds it impossible to understand them. In Dr. Mutton's 

 " course of mathematics," which will be admitted to be a 

 pretty fair example, the rules of Arithmetic are demonstra- 

 ted by algebraic methods, and the rules of algebra are usu- 

 ally without demonstrations. Now, by Lacroix and Euler, 

 on the contrary, every thing is demonstrated in as rigorou? 

 a manner, as the state of the student's ip.formation will war- 

 rant. He sees at each step, the ground on which he is pro- 

 ceeding, and forms from the beginning, a habit of demon- 

 stration. The greatest part of the mathematical learning 

 among us, we believe, is not much unlike that of the mere 

 practical navigator, who knows what !ii« book says, and how 

 to apply what it says, but who is in Egyj)tian darkness with 

 respect to the reason upon which any thing is said. A 

 young man who accustoms himself from the first, to de- 

 monstrating every thing which he receives as truth, and to 

 developing fully the conditions expressed or implied, of ev- 

 ery problem which he solves, will soon form a habit of re- 

 searches of this kind. Such a habit constitutes true knowl- 

 edge in mathematics, and will furuish resources for unex- 

 pected cases where no rules are provided, or where a new 

 combination of them is required ; and where the mere 

 mechanical calculator would find himself totally unable to 

 proceed. But what is above all price, such a course most 



