292 Cambridge Course of Mathematics. 



another," which is better than any other definition of it, 

 since it is of more easy and extensive apphcafion 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 line as a limit terminating 

 a surface; 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 in 

 the common way ; since the real existence of limits of these 

 different kinds, can no more be called in question, than that 

 of the solid from which they are all ultimately derived. 



His definition of an angle, is more happily expressed 

 than usual. " When," says he, " two straight lines meet, 

 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 think 

 Lacroix has the advantage over Legendre. Euclid has 



