300 Cambridge Course of Mathematics. 



ments of the science. The author has ver) properly con- 

 fined himself to the consideration of convex lines and surfa- 

 ces, which are such, that they cannot be met by a straight 

 line in more than two points. He has completely reformed 

 the ordinary definition of similar solid polyedrons, though 

 he has followed that of similar rectilineal figures containing, 

 as it does, three superfluous conditions. 



In the latter part of the note just referred to, Legendre 

 says, "the angle formed by the meeting of two planes, and 

 the solid angle formed by the meeting of several planes in 

 the same point, are distinct kinds of magnitudes to which it 

 would be well perhaps to give particular names. With- 

 out this, it is difficult to avoid obscurity and circumlocu- 

 tions in speaking of the arrangement of planes which com- 

 pose the surface of a polyedron; and as the theory of sol- 

 ids has been little cultivated hitherto, there is less incon- 

 venience in introducing new expressions, where they are 

 required by the nature of the subject." According to the 

 suggestion here made, M. Lacroix has introduced into the 

 geometry of planes and volumes, a very convenient system 

 of new expressions, and some slight alterations in the nota- 

 tion, by which much circumlocution is avoided, and our 

 power of expression much enlarged. The translator has 

 adopted one of these changes in the notation, which con- 

 sists merely inplacing the letter designating the vertex of 

 the polyedron first, w^th a hyphen between it, and the oth- 

 er letters. This very trifling change, contributes consider- 

 ably to the facility of following the demonstrations. La- 

 croix's improvements in general, could not be conveniently 

 adopted by the translator. 



The third section of part II. relates to the sphere and 

 spherical triangles. This is an important addition to the el- 

 ements of geometry, as it is not of difficult demonstration, 

 and is of extensive utility in its applications to geography, 

 &c. as well as in the succeeding parts of mathematics. It 

 is designed in particular to be introductory to spherical 

 trigonometry. 



The fourth section of this part, is employed in investiga- 

 ting the properties and relations of the sphere, cone, and 

 cylinder. The general method of demonstration in this 

 section, is that of Maurolycus, a Sicilian geometer, who 

 flourished in the middle of the 16th century. This meth- 



