344 NECESSARY TRUTH 



no matter how far they be produced. Given these real things 

 (surfaces and lines), those qualities (plane and straight) and that 

 relation (perpendicular), then, that other relation (parallelism) 

 necessarily follows and from that also the necessary truth that 

 parallel straight lines will never meet for the notion of straight- 

 ness, combined with that of parallelism, implies that lines having 

 that quality and that relation will always preserve the same distance, 

 and, therefore, will never approach, and, therefore, never meet. 1 It 

 matters not that I have never seen surfaces that are absolutely 

 plane nor lines that are quite straight, perpendicular and parellel, 

 but only surfaces and lines that nearly approach perfection in 

 these particulars. 2 ' Absolutely ' and ' nearly ' are relative terms 

 indicating, not different kinds of qualities, but only different 

 degrees in qualities. Therefore, when I have experience of things 

 that are nearly straight, perpendicular and parallel, I can think 

 in terms of things that are absolutely so. 



580. Again, I can through direct experience, through actual 

 measurement, reach the notion that the three angles of a certain 

 triangle, which is bounded by straight lines on a plane surface, are 

 equal to two right angles. Moreover, by repeating these observa- 

 tions on other triangles I can, through simple enumeration, reach 

 the notion that what is true of one such triangle is probably true 

 of all similar triangles. But a notion so reached will appear to me 

 merely an expectation, not a necessary truth. It will only seem to be 

 the latter, if I deduce it, as Euclid does, from the previously known 

 and admitted properties and relations of lines, angles and triangles. 



581. In mathematics we are able to discover many necessary 

 truths, for here the conditions are, speaking comparatively, so little 

 complicated that we are able to perceive and conceive things and 

 their properties and relations with greater completeness and 

 certitude than elsewhere, and, therefore, can think more exactly 

 and confidently. Nevertheless, in everyday life we are continually 

 endeavouring to deduce such truths. That is, we are continually 

 endeavouring to deduce from the previously known and admitted 

 properties and relations of things, effects that, given those pro- 



1 I am informed by Professor H. H. Turner that much or all of this is con- 

 troverted by the " New Euclidians." But the example even if faulty will serve to 

 illustrate my meaning. It may be noted also that in mathematics we do not 

 deal with sequences of events. We suppose, however, that, given certain facts, 

 certain other facts must necessarily be true also, and we proceed to discover the 

 latter. The thinking, therefore, is of the same nature as when we deal with 

 sequences of events with cause and effect. 



2 See Mill, Logic, II. v. 147. 



