292 Cambridge Course of Mathematics. 
another,” which is better than any other definition of if, 
since it is of more easy and extensive application than any 
other. We think the most natural way of giving a general 
definition of a point,a line, and a surface, is, to contemplate 
a surface, as one of the limits terminating a solid, which has 
necessarily three dimensions; a dine as a limit terminating 
a surfacé; and a point, as a limit terminating a line. ‘These 
definitions flow naturally from the definition of a solid, in 
defining which there is/no difficulty. When these defini- 
tions are obtained in this way, and viewed in this light, they 
have less the nature of abstractions, than when stated m 
the quantity whether greater or less, by which they depart 
from each other as to their position, is called an angle; 
the point of meeting or intersection is the vertex of the an- 
gle; the lines (comprising the angle) are its sides.” Very 
various definitions of an angle, have been given by geome- 
ters. That of Euclid, is certainly faulty. In fact, if we 
define an angle by the inclination of its lines, the expres- 
sion is both obscure and pleonastic. If we say that an an- 
gle is the meeting of two lines, the expression directs the 
attention entirely to the vertex. On the whole, we believe 
it best to understand by the term angle, the indefinite space 
comprised between two straight lines which meet each 
other. ‘The celebrated D’Alembert proposed to limit this 
space by an arc of a circle described from the vertex as a 
centre with any convenient radius, but this is introducing 
a foreign idea into the definition. The space in question, 
is perfectly distinguished from all other space. The defi- 
nition suggested above, comprises all the properties usual- 
ly ascribed to an angle, such as addition, subtraction, coin- 
cidence by super-position, &c. But besides this, the ad- 
ditional valuable circumstance included in the idea of an 
angle, that it comprises the space included within its sides, 
prevents the awkwardness and tedious circumlocution, with 
which every one must have felt the geometry of planes and 
solids to be invested. This is a point, in which we thi 
Lacroix has the advantage over Legendre. Euclid has 
