RESPECTING THE THEORY OF PARALLELS. 435 



SO far as they are logically correct, and make no use of the hypothesis of paral- 

 lelism. As a matter of convenience, it is best to define a straight line, with 

 Playfair, in the following way : — " If two lines are such that they cannot coin- 

 cide in any two points without coinciding altogether, each of them is called a 

 straight line ;" for Euclid's definition has no significance in a logical system. It 

 is never referred to by Euclid himself, nor indeed, could it be, seeing that it 

 expresses nothing. The real characteristics of straightness are contained in two 

 postulates, or, as Simson designates them (and it is as well to refer to Simson), 

 axioms. They constitute Simson's 10th and 11th axioms, thus breaking into 

 three parts — a definition and two axioms — what Playfair has reduced to one. 

 Euclid's definition of a square must of course be rejected. It is essentially 

 vicious, involving both a superfluity and a want of necessary consistency. It is 

 equivalent to the assumption, that the angles of a triangle are together equal to 

 two right angles, or the alternative, which is demonstrably false, that a triangle 

 may have its angles together greater than two right angles. 



It is further premised, that all the Propositions of Euclid, up to the 28th 

 inclusive, are correctly demonstrated, and are clear of any assumption relative to 

 the angles of a triangle, or to the doctrine of parallels. In the 29th Proposition, 

 Euclid has to convert the 27th and 28th, or, in other words, to show the conse- 

 quences of starting with the hypothesis that two straight lines do not meet. The 

 difficulty consists in deducing positive consequences out of negative premises. 

 And the difficulty is further increased by the fact, that straightness is only known 

 from the necessity that two lines do meet, whilst parallelism is only known from 

 the necessity that they do not meet. This difficulty renders it imperative on 

 Euclid to make some additional assumption. His assumption or postulate is 

 what Simson calls the 12th axiom : " If a straight line meets two straight lines, 

 so as to make the two interior angles on the same side of them, taken together, less 

 than two right angles, these straight lines, being continually produced, shall at 

 length meet upon that side on which are the angles which are less than two right 

 angles." The object of all that has been written on the subject of parallels has 

 been to get rid of this assumption. I shall not even allude to the history of this 

 subject. I am concerned only with that form of the argument which depends on 

 the sum of the angles of a triangle. It has been stated above, that Legendre has 

 narrowed the requirements to the discovery of some one triangle, for which the 

 angles shall be together equal to two right angles, by having proved that, if this 

 be true, the same holds good of every triangle. It may perhaps be as well to add 

 that this, once established, leads directly to Euclid's 12th axiom. 



The following appears to be the most simple and satisfactory manner of estab- 

 lishing this conclusion : — 



