Legendre^s Geometry, 235 



ests of science and humanity, had fallen a victim of the 

 most sanguinary tyranny, before their e^'es. The subse- 

 quent organization of a system of public instruction, gave 

 the French mathematicians an opportunity of rendering 

 eminent services to the government. The utility of the 

 exact sciences to the views of the French nation, as con- 

 stituting the basis of the science of war, called into full ex- 

 ercise all. the mathematical talents in the kingdom. But 

 above all, the establishment of the National Institute con- 

 centrated the talents of the nation, and the pensions and 

 high honors which were liberally bestowed, especially up- 

 on those who successfully cultivated the exact sciences, 

 gave an astonishing impulse to mathematical learning. To 

 these circumstances we owe the geometry of Legendre, 

 the numerous elementary treatises of Lacroix, Laplace's 

 System of the World, Lagrange's Theory of Analytical 

 Functions, Poisson's Mechanics, and an immense number 

 of other works of the highest merit, which cannot now be 

 mentioned. The exact sciences are vastly indebted to the 

 French revolution and its long train of consequences, what- 

 ever may be its ultimate effect upon the progress of knowl- 

 edge in general. The science of calculation is now invest- 

 ed with such resources, that almost nothing is too compli- 

 cated, or too stubborn to yield to its power. 



Before proceeding to a particular examination of the 

 work before us, we feel called upon to say a few words up- 

 on the enquiries, — what ought an elementary treatise of 

 geometry to contain, in the present state of the pure and 

 applied mathematics? — and why we should adopt M. Le- 

 gendre's treatises, or that of any other modern writer, in 

 preference to "Euclid's Elements,'' which have been 

 used, for the most part, as a text-book in the American 

 colleges. 



With respect to the first enquiry, it is plain, that an ele- 

 mentary treatise cannot contain all the truths within the 

 compass of elementary geometrical investigation. The 

 properties which belong to the figures of elementary ge- 

 ometry, and the relations which these properties sustain to 

 each other, are innumerable. Some of these properties 

 and relations have never been applied to any practical ob- 

 ject, others form links more or less important in a long 

 chain of connected truths, others are truths important in 



