## Folios 48- 49, 163-165:

## AAL to ADM

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

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