Legendre's Geometry. 301 



od is indicated in Euclid B. XII. prop. 16th. It consists 

 in assuming two concentric circumferences, and circum- 

 scribing about the smaller a regular polygon which does 

 not touch the greater; or inscribing a regular polygon in 

 the greater, which does not touch the smaller. On these 

 polygons, as bases, in case circumstances require it, regular 

 polyedrons are supposed to be constructed, and it is de- 

 monstrated, that the assigned measure of the solid or sur- 

 face in question,cannot be that of one greater or less, with- 

 out falling into the absurdity of concluding that a figure con- 

 tained by another is th.-s greater of the two. These de- 

 monstrations are long, and on account of the frequent repe- 

 tition of the same constructions, become somewhat tedious. 

 But they have the merit of being plain, and satisfactory, 

 although indirect ; and we think, on the whole, that the 

 method pursued by Legendre is the best. Lacroix's man- 

 ner as it respects these demonstrations, is much more con- 

 cise, but it is too abstract, and difficult to he seized ; and 

 Cavalleri's method of indivisibles is not sufficiently rigorous 

 to be used in elementary geometry. 



After what has been said, it is scarcely necessary to ob- 

 serve, that American mathematical science, is under great 

 obligations to the translator, for giving Legendre's elements 

 in so handsome an English dress. The only fault we have 

 to charge him with, is, that he did not furnish us with the 

 entire work, as it came from the hands of the author- Con- 

 siderable of that part which in the original is printed in fine 

 type, and almost all the notes, are omitted in the transla- 

 tion. These omissions we very much regret. By pre- 

 serving the original difference in type, the work would 

 have been equally convenient for academical instruction, 

 and the additional expense of printing the parts omitted, 

 would have been quite trifling. As it now is, we are pur- 

 suaded, that all lovers of mathematical learning, after hav- 

 ing perused the translation, will feel induced to go to the 

 expense of sending out for the original, for the sake of those 

 parts, which the translation does not contain. The notes 

 are a great curiosity, and would be likely to inspire a taste 

 for the higher mathematics. 



The translation is executed faithfully, and it is accurate- 

 ly printed. A list of errata, however, to some extent 

 might be made out, though none is given. In the demon- 



