REMAINING LAWS OF NATURE. 175 



than one necessary. It is assumed that there are 

 such things in nature as straight lines, and that any 

 two of them setting out from the same point, diverge 

 more and more without limit. This assumption, 

 (which includes and goes beyond Euclid's axiom that 

 two straight lines cannot inclose a space,) is as indis- 

 pensable in geometry, and as evident, resting upon as 

 simple, familiar, and universal observation, as any of 

 the other axioms. It is also assumed that straight 

 lines diverge from one another in different degrees ; in 

 other words, that there are such things as angles, and 

 that they are capable of being equal or unequal. It is 

 assumed that there is such a thing as a circle, and that 

 all its radii are equal ; such things as ellipses, and that 

 the sums of the focal distances are equal for every 

 point in an ellipse; such things as parallel lines, and 

 that those lines are everywhere equally distant*. 



* Geometers have usually preferred to define parallel lines by 

 the property of being in the same plane and never meeting. This, 

 however, has rendered it necessary for them to assume, as an addi- 

 tional axiom, some other property of parallel lines ; and the un- 

 satisfactory manner in which properties for that purpose have been 

 selected by Euclid and others has always been deemed the oppro- 

 brium of elementary geometry. Even as a verbal definition, equi- 

 distance is a fitter property to characterize parallels by, since it is 

 the attribute really involved in the signification of the name. If 

 to be in the same plane and never to meet were all that is meant 

 by being parallel, we should feel no incongruity in speaking of a 

 curve as parallel to its asymptote. The meaning of parallel lines is, 

 lines which pursue exactly the same direction, and which, therefore, 

 neither approach nearer nor go farther from one another ; a concep- 

 tion suggested at once by the contemplation of nature. That the 

 lines will never meet is of course implied in the more comprehensive 

 proposition that they are everywhere equally distant. And that 

 any straight lines which are in the same plane and not equi-distant 

 will certainly meet, may be demonstrated in the most rigid manner 

 from the fundamental property of straight lines assumed in the text, 

 viz., that if they set out from the same point they diverge more and 

 more without limit. 



