I 



On the Fundamental Principles of Mathematics. 185 



terior space begins. Now as the pyramid, during its motion, con- 

 tinually forsakes the place it may happen at any instant to occupy, 

 the point at the vertex being just icithout the pyramid, or ac the 

 limit of the space thus occupied, will be at once left behind; the 

 motion by hypothesis being such that the vertex should not be 

 stationary; i. e., the particular cases of a rotation about the ver- 

 tex, without a progression from its position in space, being exclu- 

 ded. That the mathematical point at the vertex will be thus 

 left behind or forsaken, will moreover appear, incontrovertibly, 

 from the fact that its position as determined by fixed standards 

 of reference will be found to be invariable. 



It is nevertheless true, that the line in which, or precisely at 

 which, the vertex, durin 



istmctly marked out : it being the limit up to which that space 

 extends, which was either occupied or passed through by the 

 pyramid. 



The like must be true of the centre of gravity of a sphere in 

 motion through space, or which has so moved. A new point in 

 space will be found to be the position of the centre of gravity, as 

 the sphere advances. Still more obviously must the like be true 

 of the centre of gravity of two or more bodies, when they so move 

 as to change its position, that centre moreover being throughout 

 supposed to be without the bodies themselves. When the masses, 

 tec, of the bodies are known, the successive positions of the centre 

 of gravity of two or more may be computed, or even prescribed ; 

 yet such a mere position, at any instant in space, is not pushed 

 forward or drawn backward by or with those bodies ; and all this, 

 while, moreover, the entire curve in which all the successive (but 

 certainly different) positions of the centre of gravity are situated, 

 may throughout admit of being accurately defined, and its limits 

 therefore precisely settled. Indeed, lastly, should we suppose the 

 contrary to all this to be true, we could not escape from the seem- 

 J ng contradiction, that a point which (8.) is the absolute zero 

 of space should become somewhat in space, that is, should be 

 drawn out into a line which has length, by the introduction of 

 we foreign element of motion. 



We must, in view of all that has been advanced, regard the 

 motion of a mathematical point, as a pleasant fiction ; the result 

 ^regards position, magnitude, and of the quantities concerned, 

 being the same as it would be if such motion were possible : 

 while the actual description of a mat hematica I line in space would 

 require the motion of a pointed atom, if such a thing may be* 



This does not militate against the mathematical existence (I.) of such curves as 

 the cycloid, & c .; since it is only necessary to suppose the generating circle, or other 

 curve, & c ^ to De drawn on a material substance; that it take successively the sev- 

 er al positions required ; and that the point at the edge, or elsewhere, be assumed 

 successively where the describing point ought to be. 



Second Series, Vol. VII, No. 20— March, 1849. 24 



