On the Fundamental Principles of Mathematics. 331 



length. If (still in the same direction) we measure from P a dis- 

 tance such as PP', greater than PO, the farther extremity of the 

 line so measured will extend beyond to P', i. e., to a distance equal 

 to the excess of PP' above PO ; or the attempt to subtract from 

 PO a quantity greater than itself } will result in a negative remain- 

 der equivalent to the excess of the greater quantity PP above 

 PO; and this remainder will extend from O itself, in the direc- 

 tion opposite to that of OP. If PO= zero, the whole of PP 7 

 will in this manner, extend from O in the negative direction. 



The negative quantities thus originating are, in all the respects 

 specified, strictly analogous to those which present themselves 

 when the attempt is made to subtract 7 from 4, or 9 from 6, or, 

 in general, the numerical quantity n from q, when n is greater 

 than q ;* the differences in the results of such subtractions being 

 no other than those which must exist in the case of quantities of 

 another species. 



The same reasoning will apply to distances measured, in like 



manner, upon either of the other axes. Hence distance outward 

 from the origin, in the direction first assumed, will in any case be 

 naturally positive ; and distance in the opposite direction, negative; 

 and will be exhibited in its isolated as well as negative character, 

 when measured in that direction, beginning at the origin. 



These conclusions being independent of any particular inclina- 

 tion of the axes among themselves, will apply to the case of three 

 axes the sum of whose three angles — that of first axis with second, 

 second with third, and third with first — differs scarce at all from 

 four right angles; and this, whether those three axes be situated 

 on the one side or the other of a given plane of reference, passing 

 through the origin O. As, therefore, the conclusions referred to 

 will be applicable, however near the state of things may approach 

 to that in which the three axes would be all in one plane, and 

 this, on either side of that state of things as a limit ; these same 

 conclusions must be regarded as true in the case of that limit 

 itself: or direction from the origin outward must be regarded as 

 positive, whichever of the three axes may be employed to indi- 

 cate it, and the contrary direction be regarded as negative, even 

 when carried beyond the origin by excess of distance extended in 

 that direction: the three axes beinsr moreover all in the same 

 plane. 



As, moreover, the conclusions, from first to last, are indepen- 

 dent of any fixed direction of one or more of the axes, they 

 w iU all be alike applicable to any other three axes, which like 

 GQ< do not coincide with any of the first three ; direction from 



>tre — 



The view here pi i will be found to coincide with that of Jr. r 



u Etoai snr la Th et VI tat d ntant > d s hn naires, Premier 



Memoire, (16.) a Pa 1845." M. Faure moreover intimates [Bum, dc<\ (19.)], 



the distance subtrae from OP may be regard I as m ured negativehj from a 



%tm arisen o+ P. rwiaolv o« OP' is measured yieaativdv from (X 





