504 MR W. ROBERTSON SMITH ON HEGEL 



know to become not relatively but absolutely zero in proceeding to the limit. 

 The motive for using such expressions as " minuatur quantitas o in infinitum, 1 ' 

 instead of simply saying, let o = zero, is merely to show that o becomes zero not 

 by a discontinuous process, as subtraction, but by a continuous flow. Nay, cries 

 Hegel, for in the 3d Problem of Book ii. of the " Principia," Newton fell into an 

 error, by " 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 " relatively small," in a mathe- 

 matical 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 "relatively small," 

 " possessed the qualitative value sought." " 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." The 

 terms are thus to be regarded as " qualitative moments of a whole of the 

 notion," 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 it hardly possible to criticise our author fairly. Luckily, we can see 

 that Hegel is leaning entirely on Lagrange, and that " the series in which the 

 function of a motion is developed," must therefore mean the series which expresses 

 space in ascending powers of time. And this enables us to ask, secondly, What 

 reason Hegel has for supposing that it is in this series that we are to find the 

 basis for a truly philosophical view of kinetics ? It was Hegel's misfortune to live 

 at a time when, among other fruits of the " Aufklarung," Lagrange's " formal 

 and superficial" method of treating physics was in great repute ; and surely it 

 was a cruel fate that the great enemy of the Aufklarung should, through a 

 defective mathematical education, be made a willing captive to a mathematical 



