## Folios 54-57: AAL to ADM

S^{t}James' Square

Friday Morning

Dear M^{r} De Morgan. I

send you a large packet

of papers :

1 : Some Remarks & Queries on

the subjects of a portion of

pages 75 & 76 (Differential Calculus)

2 : an __Abstract__ of the demonstration

of the Method of finding the

\(n^{\text{th}}\) Differential Co-efficient by

means of the Formula

Limit of \(\frac{\Delta^nu}{(\Delta x)^n}=u^{(n)}\) **[54v] ** 3 : Some objections & enquiries

on the subjects of pages 83,

84, 85.

4 : Two enquiries on two

Formulae in page['s' crossed out] 35 of the

''__Elementary Illustrations__''.

In addition to all

this, I have a word to

say on two points in your

last letter.

Firstly : that \(\theta\) is a function

of \(a\) & \(h\), (or in all cases

a function of __one__ at any

rate of these quantities), is

very clearly shown by you

in reply to my question.

But I still do not see exactly**[55r] ** the __use__ & __aim__ of this fact

being so particularly pointed

out in the parenthesis at the

top of page 80. It does

not appear to me that

the subsequent argument is

at all affected by it.

Secondly : I still am not

satisfied about the Logarithms,

I mean about the peculiarity

which constitutes a __Naperian__

Logarithm in what I call

the Geometrical Method,

the method in your

Number & Magnitude. I

am ['now' inserted] satisfied of the following :

that there is nothing in

the Geometrical Method to**[55v] ** lead to the precise __determination__

of \(\varepsilon\); that \(\varepsilon\) is arrived

at by __other__ means, Algebraical

means; & t__hen__ identified

with the \(k\) on \(HL\) of the

Geometrical Method. What

constitutes a __Naperian__ Logarithm

in the __Geometrical view__, is

''taking \(k\) so that \(x\) shall

''expound \(1+x\), __or rather that__

''__the smaller \(x\) is, the more__

''__nearly shall \(x\) expound \(1+x\) __.

But in this definition there

are two points that are still

misty to me : I do not

see in w__ha__t, (beyond the

mere fact itself), t__hese__

Logarithms differ from those**[56r] ** in which \(x\) does __not__

expound \(1+x\) . I cannot

perceive __how__ this one

peculiarity in them, involves

any others, or imparts to

them any particular use,

or simplicity, not belonging

to other logarithms.

Also, I do not comprehend

the d__oubt__ implied as to

the absolute theoretical

strictly-mathematical existence

of a construction in which

\( x\) shall expound \(1+x\) .

It appears to me that,

whether practically with a

pair of good compasses, or

theoretically with a pair of**[56v] ** m__enta__l compasses, I can as

easily as may be take any

[diagram in original]

line I please \(MQ\) greater than

\( OK\) or \(V\), measure their difference

\( PQ\) which call \(x\), then on

\( OH\) (\( =OK\) ) lay down a portion

\( OM\) equal to this difference \(x\)

(not that I pretend this is

correctly done in my figure,

which is only roughly inked

down at the moment), &

['finally' inserted]} stick up \(MQ\) on the point

\( M\) . Then \(x\) expounds \(MQ\) or**[57r] ** \(V+x\), or \(1+x\) . I can see

no difficulty in accomplishing

this, or any reason why these

can be only an a__pproximation__

to it. __Neither do I very____clearly perceive that the____Base \(k\) would be____necessarily influenced by this____proceeding__.

In short I take the real

truth to be that this view

of Exponents being wholly n__ew__

to me, there is some little

l__ink__ which has escaped me,

or to which at any rate I

have not given it's [*sic*] due

importance. But I think

I have now fully explained**[57v] ** __what__ it is that I do __not__

understand.

Believe me

Yours very truly

A. A. Lovelace

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