74 Length and Space. 



description; Length, breadth, thickness; a body, a surface, 

 a line, a point ; a straight line, a curve, an angle. The 

 meaning of these terms must be settled by description, 

 and by exhibiting examples to the eye. When this is done, 

 all the other terms used in geometry may be logically 

 defined, so that their precise application can never be 

 mistaken, nor admit of any ambiguity. But, besides 

 determining by convention and example, the meaning of 

 the abovementioned terms, it is further necessary, before 

 proceeding to teach the science of geometry, to assume 

 some property of a straight line. For this purpose, differ- 

 ent properties of the straight line, have been assumed by 

 different geometers. One of the properties most commonly 

 assumed for this purpose is, that '• if two straight lines 

 coincide in two points, they will coincide throughout." — 

 This property, though not assumed by Euclid, is implied 

 in the fourth proposition of the first book ; for, if it be not 

 presupposed, the bases of the two triangles, though coin- 

 ciding in the two angular points, may not coincide in other 

 points, and consequently may not be equal. Another 

 property sometimes employed for this purpose is, that a 

 straight line is the shortest distance between two points. — 

 But each of these is a theorem, and not a definition. They 

 are indeed both theorems which cannot be demonstrated or 

 proved, and therefore 1 have said, they must be assumed. 

 What I have said of the impossibility of defining the terms 

 straight line and angle, refers only to the present state of 

 the science, and docs not preclude the possibility of discov- 

 ering definitions of one or both of them. A logical definition 

 of an angle, would add a new beauty, and anew degree of 

 simplicity to the subject; but a correctly logical definition 

 of a straight line would greatly elucidate the elen^ents of 



