KEMAINING LAWS OF NATURE. 435 



say, propositions asserting the real 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 can not 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 small a number of original premises ; why 

 it is that we can set out from only one characteristic property of each kind 

 of phenomenon, and with that and two or three general truths relating to 

 equality, can travel from mark to mark ixntil we obtain a vast body of de- 

 rivative truths, to all appearance extremely unlike those elementary ones. 



The explanation of this remarkable fact seems to lie in the following cir- 

 cumstances. In the first place, all questions of position and figure may be 

 resolved into questions of magnitude. The position and figure of any ob- 

 ject are determined by determining the position of a sufticient number of 

 points in it; and the position of any point may be determined by the mag- 

 nitude of three rectangular co-ordinates, that is, of the perpendiculars drawn 

 from the point to three planes at right angles to one another, arbitrarily 

 selected. By this transformation of all questions of quality into questions 

 only of quantity, geometry is reduced to the single problem of the meas- 

 urement of magnitudes, that is, the ascertainment of the equalities which 

 exist between them. Now when we consider that by one of the general 

 axioms, any equality, when ascertained, is proof of as many other equalities 

 as there are other things equal to either of the two equals; and that by 

 another of those axioms, any ascertained equality is pi'oof of the equality 

 of as many pairs of magnitudes as can be formed by the numerous opera- 



* 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 pui-pose 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 sig- 

 nification 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 pai-allel 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 everywhere equally distant. 

 And that any straight lines which are in the same plane and not equidistant will certainly 

 meet, mny 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. 



