156 INDUCTION. 



existence of the various objects therein designated, together 

 with some one property of each. In some cases more than 

 one property is commonly assumed, but in no case is more 

 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 indispensable in geometry, and as evident, resting on 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.* 



8. It is a matter of more than curiosity to consider, to 

 what peculiarity of the physical truths which are the subject 

 of geometry, it is owing that they can all be deduced from so 



* 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 additional axiom, some other property of 

 parallel lines ; and the unsatisfactory manner in which properties for that 

 purpose have been selected by Euclid and others has always been deemed the 

 opprobrium of elementary geometry. Even as a verbal definition, equidistance 

 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 draw nearer nor go farther from one another ; a conception suggested 

 at once by the contemplation of nature. That the lines will never meet is of 

 course included in the more comprehensive proposition that they are every- 

 where equally distant. And that any straight lines which are in the same 

 plane and not equidistant will certainly meet, may be demonstrated in the most 

 rigorous 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. 



