184 On the Fundamental Principles of Mathematics. 



carefully considered, lead to the conclusion, that zero is, in no 

 case, to be regarded as the absence of all quantity — for then 

 there could scarce be any occasion to consider it at all — but only* 

 as, in any case, the absence of the quantity in question. 



[If then we should conclude, from considerations abundantly 

 adequate, that before all that we here call somethings, there must 

 have existed that which is not something j what we thus arrive at 

 cannot be represented as a zero, except in the very respect that 

 is not something, as the others are ; but, even in so far as these 

 considerations, thus exclusive, can determine, that which was 

 before all these somethings may have been, and still may be infi- 

 nite in its own way.] 



Has Motion any place in Pure Mathematics ? 



(15.) What has already been said of a point, or the absolute 

 zero of space, and rest, or the absolute zero of motion, may be 

 found to have prepared the way for the consideration of the ques- 

 tion : — how far, if at all, motion may be predicated of a mathe- 

 matical point ; or indeed, how far motion may have place, when 

 what is to be moved, is a point, a line, a surface, &c, or any 

 other quantity of those specially recognized by geometers. 



A body changes its 



ft 



Mot 



which it occupied in space. It is transferred during the motion 

 from place to place : and when the motion has ceased, the body 

 is at rest ; i. e., no farther change of place occurs, but the body 

 continues to occupy the place to which, at the end of the pro- 

 gressive chancre, it was transferred. The body itself was thus 

 transferred, and not the place occupied by it : and all the bounda- 

 ries, limits, or points situated in or about that space would be 

 found to retain their positions, upon a reference to fixed standards. 

 Neither space, then, nor the limits of it, are found to be the sub- 

 jects of motion ; that being, in so far as we can investigate it, a 

 physical property of body, or, at most, of that which is in any 

 sense connected with a body ; as in the example of our own 

 selves. 



Yet a point, under certain circumstances, is, as it were, trans- 

 ferred along a line. 



Thus when a pyramid so moves as to change the position of 

 its vertex, the mathematical point at that vertex, is successively 

 to be found at different places in the line which marks the limit 

 of the whole space, either occupied, or passed through by the 



solid. 



be 



mathema 



point, as already described, (8.) is not somewhat in any sense, 



but only somewhere ; and the place of the point, in this instance 

 is precisely where the pyramid ceases to be found at all, and ex 



• 



^H 



