| Author(s) | Friedrich Engels |
|---|---|
| Written | 18 August 1881 |
ENGELS TO MARX
IN LONDON
Bridlington Quay,
18 August 1881
1 Sea View
Dear Moor,
Not until last night did I get your Argenteuil letter explaining your sudden arrival. I trust Tussy's indisposition is of no real significance—she wrote me a cheery letter only the day before yesterday; at all events, I shall presumably hear further details tonight or tomorrow morning, and also whether your wife accompanied you as far as Boulogne or Calais or whether she stopped off before that.
Yesterday, then, I at last plucked up the courage to make a thorough study of your mathematical mss.[1] without any reference to manuals and was glad to find I had no need of them. I offer you my congratulations. The thing is so crystal clear that one can only marvel at the obstinacy with which mathematicians insist on shrouding it in mystery. But that is what comes of those gentry's one-sided mentality.
To write firmly and categorically —- = - could never enter their
dy heads. And yet it is obvious that — can only be the pure expression of a process undergone by x and y when the last trace of the terms x and y has disappeared and all that remains is the expression, free from all quantity, of the process of variation they are undergoing. There is no need to fear that some mathematician may have anticipated you in this. The above method of differentiating is, after all, much simpler than any other — so much so that I myself have just used it to deduce a formula that had momentarily slipped my mind, afterwards verifying it in the usual way. The process would undoubtedly create a great stir, especially since it clearly demonstrates that the usual method, ignoring dx dy, etc., is positively wrong. And the particular beauty of it is that only when — = -
is the operation
dx
0 absolutely correct mathematically.
So old Hegel was quite right in supposing that the basic premiss for differentiation was that both variables must be of varying powers and at least one of them must be to the power of at least 2 or '/2-[2] Now we also know why.
When we say that in y = f(x), x and y are variables, this is an assertion which, so long as we continue to maintain it, has no implications whatsoever and x and y still remain, pro tempore factual constants. Only when they really change, i. e. within the function, do they become variables in fact, nor does the relationship implicit in the original equation — not of the two quantities as such, but of their variabi-
Ay lity — come to light till then. The first derivate -— shows this rela-; 5
Ax tion as it occurs in the course of true variation, i. e. in any given varia-
dy tion; the final derivate = —
shows it purely and simply in its general-
ly
.
A
y ity and hence, from —
we can arrive at any — we choose, while
dx
Ax this itself never covers more than the particular case. But in order to proceed from the particular case to the general relation, the particular case as such has to be eliminated. Hence, after the function has gone through the process from x to x' with all this implies, one can simply let x' revert to x; it is no longer the old x, a variable only in name; it has undergone real variation, and the result of that variation remains, even if we again eliminate that variation itself.
Here at last we are able to see clearly what has long been maintained by many mathematicians who were unable to produce rational grounds for it, namely that the differential quotient is the prototype, while the differentials dx and dy are derived: the derivation of the formula itself requires that the two so-called irrational factors should originally constitute one side of the equation and only when one has dy reduced the equation to this, its original form, — = f(x), can one do
dx anything with it, is one rid of the irrational factors, replacing them with their rational expression.
The thing has got such a hold over me that it not only keeps going round in my head all day, but last night I actually had a dream in
which I gave a fellow my studs to differentiate and he made off with the lot.
Your
F. E.