## Folios 96-98: AAL to ADM

Ockham

Sat^{dy} 6^{th} Feb^{y}

['1841' added by later reader]

Dear M^{r} De Morgan. Had I

waited a day or two longer,

I need not have troubled

you with my letter of Wed^{dy},

& I can only reproach

myself now with having been

a little too __hasty\,__ in my

examination of the Theorem in

pages 68, 69, and having

sent you an enquiry which

certainly indicates some

negligence. I fear this letter**[96v] ** may not be in time to

stop one from you. [something crossed out]

However I will try to

send it by an opportunity this

afternoon.

But, to show you that I

now understand the matter

completely :

In the first place the question

of the Denominator, or the

Numerator, being all of the__same__ sign, in such [something crossed out] collection of

expressions as

\( \frac{a-b}{m-n}\), \(\frac{c-a}{p-m}\), \(\frac{d-c}{r-p}\), \(\frac{e-d}{q-r}\) &c

has __nothing whatever to do__

with the letters __effacing each____other__ when the above are**[97r] ** put into the form,

\( \frac{(a-b)+(c-a)+(d-c)+(e-d)}{(m-n)+(p-m)+(r-p)+(q-r)}\) &c;

whether \((a-b)\), &c be positive

or negative, or some one &

some the other, still

\( \frac{a-b+c-a+d-c+e-d}{m-n+p-m+r-p+q-r}\) &c

must \(=\frac{e-b}{q-n}\)

In the second place, the

Denominator __mus__t be all of

the s__am__e sign, in order

to fulfil the conditions of

the Lemma in page 48;

& t__his__ is the reason why

the condition is made respectively

\( \psi\ x\) __always__ increasing or**[97v] ** always decreasing &c.

For \(\varphi\ x\), it matters not

whether it alternately increases

& decreases (provided always

that it be __continuous__).

I believe I now

have the whole quite clear;

& I shall be more careful

in future.

I enclose a paper upon

pages 70, 71, 72, 73.

It is merely the general

argument, put into my __own__

order & from; & I send

it in order to know if

you think I understand as

much about the matter as**[98r] ** I am intended to do.

You know I always have

so many metaphysical

enquiries & speculations which

intrude themselves, that I

never am really satisfied

that I understand a__nything__;

because, understand it as

well as I may, my

comprehension __can__ only be

an infinitesimal fraction of

all I want to understand

about the many connexions

& relations which occur to

me, __how__ the matter in

question was first thought of**[98v] ** or arrived at, &c, &c.

I am particularly curious

about this wonderful Theorem.

However I try to keep

my metaphysical head in

order, & to remember Locke's

two axioms.

You should receive this about

6 o'clock this evening, if not

before. I fear you will

have written to me today

however. Believe me

Yours most truly

A. A. Lovelace

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