REMAINING LAWS OF NATURE. 371 



linos cannot inclose a space,) is as indispensable 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 ditferent degrees ; in other words, that there are 

 such things as angles, and that they are capable of being equal or un- 

 equal. 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 t;very point in an ellipse ; such things as 

 parallel lines, and that those lines are everyw^here equally distant.* 



§ S. It is a matter of something more than curiosity to consider to 

 w^hat peculiarity of the physical truths w^hich are the subject of geom- 

 etry, it is owing that they can all bo deduced from so small a number of 

 original premisses : why it is that we can set out from only one charac- 

 teristic property of each kind of phenomenon, and with that and two 

 or tln'ee general truths relating to equality, can travel from mark to 

 mark until we obtain a vast body of derivative truths, to all appear- 

 ance extremely unlike those elementary ones. 



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

 circumstances. Ih the first place, all questions of position and figure 

 may be resolved into questions of magnitude. The position and figure 

 of any object is determined, by determining the position of a sufficient 

 number of points in it ; and the position of any point may be deter- 

 mined by the magnitude of three rectangular coordinates, that is, of 

 the perpendiculars drawn from the point to three axes at right angles 

 to one another, arbitrarily selected. By this transformation of all ques- 

 tions of quality into questions only of quantity, geometry is reduced to 

 the single problem of the measurement 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 proof of the equality of as many 

 pairs of magnitudes as can be fi>rmed by the numerous operations which 

 resolve themselves into the addition of the equals to themselves or to 

 other equals: we cease to wonder that in proportion as a science is 

 conversant about equality, it should aflford a more copious supply of 

 marks ; and that the sciences of number and extension, which are con- 

 versant with little else than equality, should be the most deductive of 

 all the sciences. 



* Geometers have usually preferred to (lefinc parallel lines by the property of being in 

 the same plane and never meeting. This, however, has rendered it necess'iry for them to 

 assume, as an a'dditional axiom, some other property of parallel lines ; and the unsatisfac- 

 tory manner in which properties for that pur()Ose have been selectetl by Euclid and others 

 has always been deemed the opprol)rium of elementary geometry. Even as a verbal delini- 

 tion. 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 m 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 further 

 from one another ; a conception suggested at once by the contemplation of nature. That 

 the lines will never meet is of course implied in the more comprenensive proposition that 

 they are everywhere 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 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 lijnit. 



