84 LECTURES AND ESSAYS [1869- 



able to do, and will endeavour to give fresh proof of my 

 assertions in a more popular form. 1 



I begin with the conceptions of variation and continuity, 

 conceptions which in the calculus occur associated, in the 

 notion of continuous variation. A variable quantity is one 

 which can take various values. A quantity continuously 

 variable, which is the only kind of quantity that admits of 

 differentiation, is one which between certain limits can 

 have any value you please, which, in fact, runs on between 

 these limits, not by leaps, but by a continuous flow. 

 Time, for example, is such a quantity ; for since the birth 

 of Christ it has run through every possible value between 

 o and 1872 years. But the time which a planet takes to 

 go round the sun is not continuously variable ; for you 

 cannot find a planet to fit any periodic time you may please 



1 One or two remarks may be allowed by way of note on certain 

 criticisms directly affecting the use of symbols. At the foot of p. 113, 

 Dr. Stirling seems to take an objection in limine against the translation 

 of &quot; perfectly general &quot; Hegelian language into symbols which Hegel 

 had not before him. He might as well object to translation from 

 German into English. Again, to my statement, that a differential 

 coefficient, or, what is the same thing, the ratio of the fluxion of y to the 

 fluxion of x, is a fraction with definite numerator and denominator, 

 it is replied that Newton himself cannot have regarded mere symbols as 

 possessing &quot;definite values&quot; (p. 114). Here perhaps Dr. Stirling 

 and I are at cross purposes from attaching different senses to the word 

 definite, which I do not use as equivalent to constant. But questions of 

 language apart, the matter is clear. What I assert, and what of course 

 is a mathematical truism, is that the differential coefficient of y with 

 respect to x is no more indeterminate, no less fully quantitative than 

 x and y themselves. This Hegel denies, when he says (iii. 292) that on 

 proceeding to the limit the ratio vanishes &quot; insofern es ein Quantum 

 ware.&quot; This is what I mean when I speak of Hegel as setting up an 

 indeterminate j. Dr. Stirling, not observing that the &quot; nerve &quot; of this 

 expression lies in the word indeterminate, allows himself (p. 125) to be 

 sarcastic on me for &quot; seeing Hegel set up |.&quot; Again, this false 

 conception of ^ vanishing, quatenus Quantum, leads Hegel to find 

 certain insoluble logical difficulties in the mathematical use of a binomial 

 expansion to determine the actual value of ^- This I have called 

 &quot; knocking down the indeterminate J which he has just set up.&quot; Dr. 

 Stirling calls these words &quot;incredibly inapposite&quot; (p. 130). But if a 

 philosopher insists on holding a quantity to vanish when it does not do 

 so, and then, a few pages later, expatiates on the contradictions arising 

 from his own conceptions, he surely is knocking down what he set up. 



