i8 73 ] HEGEL AND THE CALCULUS 33 



&quot; throwing out, as Lagrange has shown, the very term 

 which for the problem in hand was wanted. Newton 

 had erred from adhering to the formal and superficial 

 principle of omission from relative smallness. This error, 

 by the way, is only in the first edition of the Principia, 

 which Hegel, one may safely affirm, had never seen. 

 The whole statement here is taken from Lagrange, and 

 applies much better to Lagrange s analytical way of 

 putting Newton s argument than to that argument in its 

 geometrical form. 



Newton, in fact, investigating the law of resistance, 

 that a body under gravity may describe a given path, 

 seeks a geometrical expression for the moment of the 

 sagitta a small quantity of the third order. It is clear, 

 therefore, that no such expression can be exact unless 

 account is taken of every small quantity of an order not 

 higher than the third in the geometrical construction 

 involved, for such quantities will not vanish in the limit, 

 or are not &quot; relatively small/ in a mathematical sense. 

 The principle of the problem, then, presents no difficulty 

 on Newton s method ; and the true account of the error is, 

 that by a mere slip in the details of a complicated process, 

 Newton failed to see that he was omitting a term (or 

 better, a line) not small relatively to the moment of the 

 sagitta. Hegel, however, conceives that so far as this 

 goes Newton was all right. The error, according to him, 

 lies in neglecting a term which, though &quot; relatively small,&quot; 

 possessed the qualitative value sought . &quot; &quot;In mechanic , 

 a particular import is attached to the terms of the series 

 in which the function of a motion is developed, so that the 

 first term, or the first function, relates to the moment of 

 velocity, the second to the accelerating force, and the 

 third to the resistance of forces.&quot; The terms are thus to 

 be regarded as &quot; qualitative moments of a whole of the 

 notion &quot; ; and, of course, in a problem about resistances 

 Newton needed the third term. Now here we have, 

 firstly, a laxness in the use of terms so gross as to make 



3 



