AAL to ADM

[48r]

 Ashley

 Sun^dy 13^th Sep^r

 [1840] [added in pencil by later reader]

 Dear M^r De Morgan

I am very much
obliged by your remarks &
additions.  I believe I now
understand as much of the
points in question as I am
intended to understand at
present.  I am much inclined
to agree with the paragraph
in page 48; for though
the conclusions must be
[48v] admitted to be most perfectly
correct & indisputable, logically
speaking, yet there is a
something intangible & a
little unsatisfactory too, about
the proposition.                  

I expect to gain a good
deal of new light, & to get
a good lift, in studying
from page 52 to 58;         
though probably I shall be
a long time about this.  I
could wish I went on
quicker.  That is       I wish
a human head, or my head
at all events, could take in
[49r] a great deal more & a great
deal more rapidly than is the
case;      and if I had made
my own head, I would
have proportioned it's [sic] wishes
& ambition a little more to 
it's [sic] capacity.        When I sit
down to study, I generally
feel as if I could never
be tired;    as if I could
go on for ever.      I say
to myself constantly, ''Now today
I will get through so & so'';
and it is very disappointing
to find oneself after an
hour or two quite wearied,
& having accomplished perhaps
[49v] about one twentieth part of
one's intentions,    perhaps not 
that.  When I compare
the very little I do, with the
very much    the infinite I
may say that there is to
be done;    I can only
hope that hereafter in some
future state, we shall be
cleverer than we are now.

I am
afraid I do not understand
what you were kind enough
to write about the Curve;
and I think for this reason,
that I do not know what
[164r] the term equation to a curve
means.     Probably with some
study, I should deduce that
meaning myself; but having
plenty else to attend to of
more immediate consequence,
I do not like to give my
time to a mere digression
of this sort.    I should
much like at some future
period, (when I have got
rid of the common Algebra
& Trigonometry which at
present detain me), to
attend particularly to this
subject.     At present, you
[164v] will observe I have four
distinct things to [something crossed out]
carry on at the same
time;    the Algebra;            
Trigonometry;     Chapter 2^nd of the
Differential Calculus;    & the
mere practice in Differentiation.

This last reminds me
that my bookseller has at
last & with much difficulty
got me Peacock's Book; &
I hope it will be of
great use, for it's [sic] cost is
£2..12..6 !      It was
originally 30^s.    It is
[163r] coming here next week.

By the bye I have a
question to ask upon pages
203 & 204 of the Algebra.
In consequence of a reference
to page 203, in the 9^th line
of the 25^th page of the
Trigonometry, I was induced
to look & see what it
related to.  Reading on
afterwards to the bottom of
the page, I found
''A f u n c t i o n a l  e q u a t i o n is an
''equation which is necessarily
''true of a function or functions
''for every value of the letter
''which it contains.  Thus if,
[163v] ''\( \varphi x=ax\), we have \(\varphi(bx)=\) 
''\( abx=b\times\varphi x\), or

 ''\( \varphi(bx)=b\varphi x\) ''
''is always true when \(\varphi x\) 
''means \(ax\) .''
So far I think is clear;
but then what follows,
''Thus &c
\[\begin{array}{lll}``\text{If} & \varphi x=x^\alpha & \varphi\alpha\times\varphi y=\varphi(\alpha y)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ ``&\varphi =a^x \ldots & \varphi x\times\varphi y=\varphi(x+y)\\ ``&\varphi x=ax+b \ldots & \displaystyle{\frac{\varphi x-\varphi y}{\varphi x-\varphi z}=\frac{x-y}{x-z}}\\ ``&\varphi x=ax &\varphi x+\varphi y=\varphi(x+y))\end{array}\]

I cannot trace the
connection.  I suppose there is
something I have not
understood, in the explanation
of the Functional Equation.

 [165r] I hope before very long to
have something further to
send you upon Chapter 2^nd
of the Calculus, either of
success or of enquiry.       

Has M^rs De Morgan returned
yet, & how is M^r Frend?   

     With many thanks,
            Yours very truly
                  A. A. Lovelace