7 



186 On the Fundamental Principles of Mathematics. 



This fiction may not however be wholly harmless, when not 

 merely mathematical, but also physical relations are the subjects of 

 investigation. Such is the case in mechanics, when motion be- 

 comes the subject of investigation as a physical property, as which 

 alone, in accordance with what has already been said, is it in any 

 case (in effect) to be regarded. The motion of a body involves 

 then the motion of its atoms, and as there can be no moving 

 mathematical point, a moving point, or whatever in mechanics 

 may be spoken of as such, must be a moving atom. Its physical 

 property of motion, the laws which regulate it, and those which 

 determine an equilibrium, must be made to rest upon observation 

 experiment and induction. * 



Corollary. — There can be no " Rational Mechanics" in the 

 sense in which that phrase is often employed. 



The considerations already urged, against the doctrine of the 

 motion of a mathematical point, will apply with equal force to the 

 case of a mathematical line, surface, or solid. 



Fundamental Reason for the Existence of Incommensurability. 



(16.) From the consideration of limits, zeros, and their special 

 relations, we may pass to that which supposes the introduction of 

 new limits ; viz., the division of quantities into parts or portions; 

 fractions, properly so called, among the rest ; by the aid of which, 

 the nature of incommensurable quantities and the necessity for 

 their existence may both be made apparent. 



Tf we select as a very simple example, a finite straight line ; 

 and suppose it to be divided in the middle, into its two most sim- 

 ple fractions; viz., its two halves; each half will, of course, be 

 equal to the other. When we divide the same line into thirds, 

 three fractions will be obtained, all equal among themselves. 

 The same perfect equality of the parts will still be found when 

 we successively divide the line into fourths, fifths, &c. ; any one 

 of such fractions being an aliquot part of the whole; and any 

 fraction, such as 4 3 , f, &c. of the line, a combination, or grouping 

 together, of such aliquot parts. 



Now. however many such divisions of the whole into all the 

 several fractions of the series of halves, thirds, &c, may be made, 

 it must happen, if the process be far enough continued, that some 

 of the points of division will agree, (| of the whole being equiv- 

 alent to fo of the same, &c, &c. ;) and there no new division of 

 the line will take place. Yet some must also differ, at each new 

 division; since | cannot = *, nor i ~ f , &c. : and very many 

 other combinations, such as §, f, &e., must be different, as the 



*The French phrase," un point materiel," is descriptive of the real state of the 

 »ase j whatever may be said of the reasoning in connection with which that phrase 



case j wruuever mav 

 may sometimes occur. 







