456 ^^^^- On the Theory of Parallel Lines. 



instance solely from inadequate definition) by an indirect course. 

 All straight lines drawn in the same plane are lines which will 

 (1) either meet (2) or not meet ; and, instead of stating some 

 positive property of the second class, Euclid gives a property of 

 the first class. His definition, for such we may call it, of lines 

 that will meet, may have involved, as its differentia, other pro- 

 perties instead of that which he has chosen. There can how- 

 ever be little doubt that for the immediate purpose in view — a 

 definition of lines that will meet — no property could be better 

 chosen than the one which appears as the differentia of the so- 

 called 12th axiom. * '^"*'" '^ '''^'^ Juiuur// unh ; -".Mtii k> lu 



With these remarks we tliay|yrotiefed'to'66nsid^t'th6inith6diate 

 difficulty with which geometers have to deal. It appears to be this : 

 Euclid's 35th definition is acknowledged to be defective, the 12th 

 axiom has also been condemned, and the question arises, with 

 what single definition can we replace both of these ? Upon the 

 answer to this question must depend our method of treating the 

 theory of parallels. Enough has been stated in this, and in 

 former communications, to indicate the course 1 would take the 

 liberty of suggesting. It is, that a full and adequate definition 

 should be laid down ; such a definition, for instance, as that given 

 by Vaiignon, in which a case of the 29th proposition is proposed 

 as the differentia. In substituting a definition for the 12th 

 axiom, the example of Euclid may, however, to a certain extent, 

 be followed with advantage. I venture to think that we should 

 deviate from the course adopted by him, to the extent of stating 

 a positive property of parallel lines instead of stating a property 

 of lines that are not parallel; but, in other respects, the defini- 

 tions may be, as nearly as possible, similar. If so, the defini- 

 tions, compared together, would stand thus : — 

 Lines which wee/,— make, with an intersecting line, two interior 



angles together less than two right angles. 



For which we are to substitute. 

 Parallel lines, — make, with an intersecting line, two interior 



angles together equal to two right angles. 



This definition of parallels has no practical advantage over 

 Varignon's ; but, as it is probably the one which Euclid himself 

 would have chosen had he to remove the 12th axiom, we ought 

 perhaps to retain it, on the ground that any alteration of his 

 ' Elements ' should be as slight as possible. 



This definition is not arbitrary ; it is not merely an assumed 

 form of words. It is necessarily involved in a fundamental con- 

 dition under which the mind contemplates the first truths of 

 geometry, a condition prior to experience, and independent of it, 

 — the idea of space. It may therefore, perhaps, be said that it 

 implies an axiom as well as a definition. Such an objection, I 



