358 JRemarks oji the Logic of Elementary Geometry/. 



definition of parallels, and no one has shown that these two 

 definitions describe the same thing. In all methods which 1 

 have seen, the object is either imperfectly accomplished or else 

 this error is committed, that there are two definitions assumed, 

 or the definition includes two properties. 



Since a definition is to consist of a single property of any 

 object, if there be several properties susceptible of independent 

 enunciation, it is a matter of indifference as to the precision of 

 our reasoning which of them is assumed as the definition. Thus 

 assuming as the definition of a triangle, that it is a figure with 

 three sides, we may deduce from that fact, another, viz. that its 

 angles are equal to two right angles : but it would be equally 

 logical to reverse the process, taking the latter fact as the defini- 

 tion, and making It a matter of demonstration that the figure 

 most have three sides. It is not enough, then, to object to a 

 definition that it may be demonstrated, or that it requires demon- 

 stration : every property may be demonstrated and requires 

 demonstration, when some other is assunied as the hypothesis or 

 made the definition. The only circumstance, then, which ought 

 to guide us in our choice, is the ease or difficulty of enunciating 

 the property, and of making it the ground of reasoning The 

 first of these circumstances is of comparatively little importance, 

 except that a useful definition ought to be short and plain ; but 

 the second circumstance is deserving of every attention*. There 

 are, for instance, two properties of straight lines, either of which 

 it is perfectly logical to choose as distinguishing it, they are — 

 1. Universal coincidence when two points coincidcw 2. Being 

 shorter than any other line connecting two points. Assuming 

 either as our definition, it can be demonstrated that the other 

 is the property of the object so defined : but as a matter of con- 

 venience, we ought to adopt the latter one as the definition, on 

 this account that the other is instantly deduced from it ; if we 

 reverse the process, wo do not get at our conclusion with respect 

 to its minimum length till we consider lines as boundaries of 

 figures, and till we have demonstrated se\eral properties of 

 fi^gures. The only question then which requires consideration 

 in our selection of a definition is simply this: AVhat property 

 affords the most convenient hypothesis ? If such a coui^e do- 

 not dispel all difficulty from our proceeding in any instance, it 

 does at least clearly set forth wherein the difficulties lies. We 

 will see, for instance, that the difficulty in respect to itarallel 

 lines may be obviated by a rightly chosen definition of them, to 

 constitute an original hypothesis ; which is its use. The difficulty 

 is perfectly independent of any definition of the term by which 

 the relation is indicated, it consists really in the nature of the 

 hypothesis on which we have to reason. It is easily demonstrated 

 that two straight lines equally iuclijicd to another towards its 



