158 Dr. Day on the Theory of Parallels. 



will be equal to two right angles, is open to the objection, that it 

 is not proved, or certainly known, that there are any lines which 

 wUl always exhibit this property, however cut, and therefore that 

 it contains covertly what Euclid openly assumes ; and I do not 

 think that because Euclid^s definition of a square is allowed to 

 pass unquestioned, it is therefore superfluous to investigate the 

 grounds of the parallel theory. We all know that what set- 

 tles the one settles the other ; it is not necessary to revive the 

 strife every time the question recurs. Many instances could be 

 adduced where Euclid has thought fit to be scrupulous in one in- 

 stance, and to neglect such over-nicety in others equally requiring 

 rigidity of proof; but this is only saying, that, with all his won- 

 derful discernment and care, he is not perfect. To return, how- 

 ever, to the main question. Some fourteen or fifteen years ago, 

 I published two small pamphlets on the subject of the present 

 discussion, in which I attempted to derive the doctrine of paral* 

 lels, as well as that of Proportion, from the consideration of 

 similar triangles, which I regarded as identical, or as undistin- 

 guishable by the understanding. Geometry has not to do with 

 empirical or assigned space magnitude. A figure once given is 

 the same whether the space it fills is large or small. We do not 

 by a specific triangle mean one which has one spatial dimension 

 rather than another; but just as a limited space is a part of 

 infinite space, a specific form may have regard to any spatial 

 dimension, and thus there is always an infinite number of similar 

 figures of every kind. The understanding recognizes these only 

 as identical. Form has regard only to angularity, and the rela- 

 tion of the bounding sides to one another measured by an arbi- 

 trary unit. I regarded parallel lines to be the corresponding 

 boundaries of these identical figures, and thus deduced all the 

 leading facts of geometry by attempting to show that the diffi- 

 culties owe their origin to a confusion of empirical and merely 

 intelligible conditions. 



The same thing might have been accomplished difi"erently by 

 insisting on the absolute identity of parallel lines, as regarded 

 by the understanding, without reference to their passing through 

 different points in space which are likewise undistinguishable 

 by an act of pure thought*. But further than this, the geometric 

 conception of a circle really assumes all that is supposed in the 

 rotation of a line round any point in itself; and as this may take 

 place indifferently at all points in space, for anything we cau 

 conceive to the contrary, there can be no objection to the intro- 

 duction of such a postulate. As every bounded figure can be 

 conceived to be produced by the rotation of one of its sides into 



* The full development of this line of argument i« here omitted as 

 requiring too much letter-press. 



1 



