## Folios 100-103: AAL to ADM

Ockham Park

Friday. 19^{th} Feb^{y}

Dear M^{r} De Morgan. I have one or two

queries to make respecting the ''Calculus of

Finite Differences'' up to page 82.

Page 80, line 4 from the top, ''remembering \(.\ldots\)

''\( .\ldots\) that in \(\varphi''(x+\theta\omega)\), \(\theta\) itself __is a function__

''__of \(x\) and \(\omega\) __, &c''; Now, neither on examining

\( \theta\) as here used & introduced, nor on

referring to the first __rise & origin__ of \(\theta\) in

this capacity, (see page 69), can I discover

that it is a function of \(x\) and \(\omega\) here, or

a function of the analogous \(a\) and \(h\) in

page 69. I neither see the truth of this

assertion, nor do I perceive the importance

of it (supposing it __is__ true) to the rest of

the argument & demonstration in page 80.

There is also a point of doubt I have

relating to the conclusion in lines 15, 16 from**[100v] ** the top of page 79 :

It is very clear that the law for the Co-efficients

being proved for \(u_n\), and for \(\Delta u_n\), follows

immediately & easily for \(u_{n+1}\), or \(u_n+\Delta u_n\) .

But if we now wish to establish it

for \(u_{n+2}\), we must prove it true not

o__nly__ for \(u_{n+1}\), but also for \(\Delta u_{n+1}\) :

To retrace from the beginning : the

object in the first half of page 79 evidently

is to prove firstly, that __any__ order of \(u\),

say \(u_n\) __can__ be expressed in term ['of,' inserted] or in

a Series of all the Differences of \(u\); \(\Delta u\),

\( \Delta^2u\), \(\Delta^3u\), \(..\ldots\ldots\ldots\) \(\Delta^nu\);

Secondly, that the Co-efficients for this Series

follow the law of those in the Binomial Theorem.

Now the first part is evident from the

law of formulation of the Table of Differences;

Since all the Differences \(\Delta u\), \(\Delta^2u\), \(\Delta^3u\) &c

are made out of \(u\), \(u_1\), \(u_2\) &c, it is

obvious that by exactly retracing & reversing

the process, we can make \(u\), \(u_1\), \(u_2\) &c**[101r] ** out of \(\Delta u\), \(\Delta^2u\), \(\Delta^3u\) &c.

For the second part of the above; if we

can [something crossed out] show that the law for the Co-efficients

holds good up to a certain point, say \(u_4\);

and also that being true for any one

value, it must be true [something crossed out] for the __next__

value too; the demonstration is effected for__all__ values :

Now the __fact__ is shown that it __is__ true up

to \(u_4\) . (I must not here enquire __why__ the__fact__ is so. That is I suppose not y__our__

arranging, or any part of your affairs).

It is shown that the two parts \(u_3\), \(\Delta u_3\) of

which \(u_4\) is made up are under this law,

& __therefore__ that \(u_4\) is so. And next it is

shown that any other two parts \(u_n\), \(\Delta u_n\)

being under this law, their sum \(u_{n+1}\)

must be so. But this proves nothing

for a c__ontinued__ succession. \(u_{n+1}\) being

under this law does not prove that \(\Delta u_{n+1}\)

is under it, & therefore that \(u_{n+2}\) is under it.**[101v] ** There seems to me to be a step or condition

omitted.

I am sorry still to be obliged to trouble

you about \(f\,x\), \(f'x\), \(f''x\), I cannot yet

agree to the assertion that the r__esult\ __ would

not be affected by discontinuity or singularity

in \(f'x\), \(f''x\), &c. The r__esult__ it is true

would not be d__irectly__ affected; but it surely

would be ['__indirectly__' inserted] affected, inasmuch as the conditions

of page 69, necessary to prove that result,

could not be fulfilled unless we suppose

\( f'x\), \(f''x\) \(..\ldots\) \(f^{(n+1)}x\) continuous &

ordinary as well as \(f\,x\) . To arrive at

the equation \(\frac{\varphi(a+h)}{\psi(a+h)}=\frac{\varphi^{(n+1)}(a+\theta h)}{\psi^{(n+1)}(a+\theta h)}\)

page 69, it is a necessary condition that

\( \varphi x\), \(\varphi'x\), \(\varphi''x\) \(\ldots\ldots\ldots\) \(\varphi^{(n+1)}x\) be all

continuous & without singularity from \(x=a\) to

\( x=a+h\) . And the \(\varphi'x\), \(\varphi''x\) \(..ldots\) \(\varphi^{(n)}x\), \(\varphi^{(n+1)}x\)

of page 71, could not fulfil this condition

unless \(f'x\), \(f''x\) \(..\ldots\) \(f^{(n)}x\), \(f^{(n+1)}x\) did so**[102r] ** also. I fear I am very troublesome about

this.

I have remarks to make respecting some of

the conclusions of the Chapter on Algebraical

Development; but t__hey__ will keep, and

therefore I will delay them, as I think

I have send abundance, & I have also

some questions to put on the last 8 pages

of your ''Number & Magnitude'' on Logarithms.

On the Differential Calculus I will only

now further say that on the whole I believe

I go on pretty well with it; and that

I suppose I understand as much about it,

[something crossed out] as I am intended to do;

possibly more, for I spare no pains to do

so.

Now for the Logarithms : I had not till now

read the last pages of your Number & Magnitude,

& there are certain points I do not fully

understand. The last line of the whole, on

the n__atural__ logarithms is one. I cannot**[102v] ** i__dentify__ the c__onstituent__ quality of the n__atural__

logarithms there given, with the constituent

qualities I am already acquainted with thro'

other relations & means : I know ['for instance' inserted] that the

natural logarithms must have 2.717281828

for their Base; that is to say that the

line \(HL\), or \(A\) (\( OK\), or \(V\) being the linear

unit) should be 2.717281828 \(V\) units.

Now I do not see ['but' inserted] that the condition in the

last paragraph of the book is one that

might perfectly consist with __any__ Base whatever.

To prove that I understand

the previous part, at least to a considerable

degree, I enclose a Demonstration I wrote

out of the property to be deduced by the

Student, (see second paragraph of page 79),

& which I believe is quite correct.

Pray of what use is the Theorem

(page 75, ['& which' inserted] continues in page 76)? I do not see

that it is subservient to anything that**[103r] ** follows; and it appears to me, to say the

truth, to be rather a useless & cumbersome

addition to a subject already sufficiently

complicated & cumbersome. The passage I

mean is from line 13 (from the top) page 75, to

the middle of page 76.

Believe me

Yours very truly

A. A. Lovelace

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