300 Cambridge Course of Mathematics. 
ments of the science. The author has very 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 oe figures containing, 
as it does, three superfluous condition 
In the latter part ‘of the note just resol 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 — 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 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, with 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 Il. 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; 
. as wellas'in the succeeding parts of mathematics. It 
is designed in particular to be aah a to spherical 
tgenomety: 
he fourth section of this part, is employed:i in sg 9 
ting the properties and relations of the sphere, cone, and 
cylinder. The general method of demonstration in = 
section, is that of Maurolycus, a Sicilian geometer, W 
flourished in the middle of the 16th century. This ips 
