Remarks on the Logic of Elementary Geometrij. 869 



same extremity will never meet, but the converse of this has 

 proved universally a stumbling block, viz.— if there be two 

 straight lines which never meet they are equally inclined to a 

 third line meeting them. The hypothesis in this case is singularly 

 unfit for being the foundation of an inference, inasmuch as the 

 position "that they never meet" caimot be presented in a distinct 

 form to the mind, nor is it in any way the object of representation : 

 it can neither be thought nor figured. The conclusion cannot 

 be deduced but by reasoning of this nature : the only reason of 

 the inequality of the angles is, that the lines meet at a finite 

 distance, — but they do not meet at a finite distance, therefore 

 any reason that would show the one to be the greater would also 

 show the other to be the gi eater. 



The property of parallels, however, which is here assumed as 

 the definition, is, in comparison with all others, of the least 

 practical value, and we do'not see how it should have been intruded 

 as the test of parallelism, except for a reason purely theoretic, 

 that is to say, to conceal without overcoming, a difficulty occur- 

 ing in the process of reasoning. After all, it is very remarkable, 

 that there should be so much difficulty in giving a reason for a 

 position so simple as this, viz, that straight lines equally inclined 

 to one straight line are also equally inclined to any other. 

 There seems something wanting in regard to the position of lines 

 which needs previous elucidation. That elucidation ought to be 

 presented in definitions — but they, in regard to this matter, are 

 very deficient. The thing understood by the term angle is not 

 defined at all, by any words representing it as a species of 

 magnitude or mode of extension. We have no commonly 

 recognised general term, to which wc can refer it, as we have ia 

 the case of the word line, which is referable to the general term 

 length ; nor does it seem to me that angle can be defined at all, 

 so as to admit of reasoning on the definition of it, without intro- 

 ducing the idea of revolution, which thus becomes a general 

 term to angle as length docs to line. It is absurd to object to 

 this, as introducing an idea not logically connected with the 

 subject, insomuch as revolution implies motion. We must reason 

 on our rational conceptions, and it will be found that the concep- 

 tion of motion has a necessary connection with ideas of difference 

 of magnitude in extension ; and therefore, though avoided iu 

 words, cannot be excluded from among the elementary hypotheses 

 of geometrical reasoning. Even Euclid talks of the production 

 of a line, and others speak of the opening between two lines. 

 The idea of motion, however, should be only introduced in 

 reference to a changed state or position, and not in reference to 

 a changing one. 



Perhaps the difficulty above-mentioned may bo obviated in 

 the following method ; 



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