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 signifi- 
cance— she wrote me a cheery letter only the day before yesterday; 
at all events, I shall presumably hear further details tonight or tomor- 
row 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 thor- 
ough study of your mathematical mss. !°? 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 dy = ; could never enter their 
x 

dy 

heads. And yet it is obvious that can only be the pure expres- 

sion 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 antici- 
pated 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 undoubt- 
edly 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 dy =5 is the operation 

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.* Now we 
also know why. 

When we say that in y = f(x), x and y are variables, this is an asser- 
tion 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- 

; : : A : 

lity —-come to light till then. The first derivate x shows this rela- 
x 

tion as it occurs in the course of true variation, i. e. in any given varia- 

‘ ; d : ; a 
tion; the final derivate = shows it purely and simply in its general- 

: F A ‘ 
ity and hence, from 7" we can arrive at any x we choose, while 
x x 

this itself never covers more than the particular case. But in order to 
proceed from the particular case to the general relation, the particu- 
lar 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 main- 
tained 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 for- 
mula itself requires that the two so-called irrational factors should 
originally constitute one side of the equation and only when one has 

: ee = d 
reduced the equation to this, its original form, = f(x), can one do 
xX 

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 

* G. W. F. Hegel, Wissenschaft der Logik, Book 1, Section II, Chapter 2. Note: Der Zweck 
des Differentialkalkuls aus seiner Anwendung abgeleitet.-° temporarily 

which I gave a fellow my studs to differentiate and he made off with 
the lot. 

Your 
F. E.