jRemarhs on the Logic of Elementary Geometry. 357 



our attention is withdrawn to the investigation of a property of 

 Plane Triangles ; this ought to have been settled in its natural 

 place, (i.e ) with the other truths relating to plane figures. 

 Euclid's investigations are on the whole free from this defect; 

 but the reverse is the case, as might be expected, with regard 

 to those who have added Supplements to his Books, because 

 Euclid's Elements do not always afford sufficient lemmas for 

 their arguments. 



Truths are presented to us either in axioms or definitions or 

 theorems or corollaries, and it may be well to distinguish what 

 ought to be our idea of these things. 



A xioms are said to be self evident truths ; which perhaps means, 

 those of which the certainty is intimated in the logical definition 

 of the terms employed. 1 should rather say, that axioms are 

 truths relating to quantity generally, or are the relations of 

 magnitude common to all its modes and kinds, and are therefore 

 either theorems or definitions. Belonging to the latter class, are 

 such as "The whole is equal to all its parts, or is greater than 

 any of its parts," &c. &c. and among the axiomic theorems ought 

 to be included, all the relations expressible by numbers or func- 

 tions of numbers which it is intended to make use of, such as the 

 one before mentioned, that half the sum added to half their 

 difference makes the greater of two quantities, which is, it is 

 obvious, as much an axiom with regard to geometrical relations 

 as "that things equal to the same are equal to each other." The 

 same is the|factin regard to such truths as this — that "proportional 

 quantities taken alternatelyjare proportional," and all of the same 

 kind — every thing Algebraic is axiomic in regard to Geometry, 

 and nothing strictly geometrical can be an axiom. 



A definition is an announcement of a single property of an 

 object which distinguishes it from all other objects. A definition 

 cannot include two independent properties, for this would present 

 two definitions, since it requires demonstration that both charac- 

 terise or belong to the same thing. Since the aim of our demon- 

 strations is to establish the properties of objects or combinations 

 of objects as distinguished from others, they must all be de- 

 duced from those which characterise or distinguish the indi- 

 viduals. Hence the definition is the single original or elemen- 

 tary hypothesis in all demonstration. This rule is violated in 

 the common method of deducing the properties of parallel lines ; 

 part of them are derived from the definition that, — " Parallel 

 lines never meet," and part from another definition, viz: That 

 parallel lines are such that only one line so related to another 

 can pass through any point. This latter is called an axiom, in 

 the form given by Playfair, viz : "That two straight lines which 

 meet cannot be parallel to the same." But giving it this name 

 does not make it an axioni; or make it cease to be a sufficient 



