REMAINING LAWS OF NATURE. 



40s 



as any of the other axioms. It is 

 also assumed that straight lines di- 

 verge 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 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 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 peculi- 

 arity 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 

 i-elating to equality can travel from 



* Geometers have usually preferred to 

 define parallel Hues by the property of 

 being in tiio same plane and never meet- 

 ing. This, however, has rendered it neces- 

 sary 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 otliera has 

 always been deemed the opprobrium of 

 elementary geometry. Even as a verbal 

 definition, equidistance is a fitter pro})erty 

 to characterise parallels by, since it is the 

 attribute i-eally involved in the signifi- 

 cation of the name. If to bo 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 in 

 eluded in the more comprehensive pro- 

 position that they are everywhere equally 

 distant. And that any straight lines which 

 are in the same plane and not equidis- 

 tant will certainly meet, may be demon- 

 strated in the most rigorous manner from 

 the fundamental property of straight lines 

 assumed m the text, viz. that if they set 

 out from the same point, they diverge more 

 and more without limit. 



mark to mark until we obtain a vast 

 body of derivative truths, to all ap- 

 pearance extremely unlike those ele- 

 mentary 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 object 

 are determined by determining the 

 position of a sufficient number of 

 points in it ; and the position of any 

 point may be determined by the 

 magnitude of three rectangular co- 

 ordinates, that is, of the perpendiculars 

 drawn from the point to three planes 

 at right angles to one another, arbitra- 

 rily selected. By this transformation 

 of all questions of quality into ques- 

 tions 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 formed 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 afford a 

 more copious supply of marks of 

 marks ; and that the sciences of 

 number and extension, which are con- 

 versant w-ith little else than equality, 

 should be the most deductive of all 

 the sciences. 



There are also two or three of the 

 principal laws of space or extension 

 which are unusually fitted for rendering 

 one position or magnitude a mark of 

 anotlier, and thereby contributing to 

 render the science largely deductive. 

 First, the magnitudes of enclosed 

 spaces, whether superficial or solid, 



