## Folios 123-125: AAL to ADM

Ashley Combe

Thurs^{dy} Mor^{g}

9^{th} Sep^{r} ['1841' added by later reader]

Dear M^{r} De Morgan. I have rather a large batch

now for you altogether :

1^{ly} : I am in the middle of the article on Negative &

Impossible Quantities; & I have a question to put on

page 134, (Second Column, lines 1, 2, 3, 4, 5 from the bottom)

\( (a+bk)^{m+nk}=\varepsilon^A\cos B+k\,\varepsilon^A.sin B\) &c

I have tried a little to demonstrate this Formula;

but before I proceed further in spending more time

upon it, I think I may as well ask if it is __intended__

to be __demonstrable by the Student__. For you know I

sometimes try to do more than anyone means me to

attempt. I have as yet only got thus far [something crossed] with

the above formula : If in \((a+bk)^{m+nk}\),

\( r\) is given \(=\sqrt{a^2+b^2}\), ['and' inserted] \(\tan\theta=\frac{b}{a}\); then \(\sin\theta=b\)

\(\cos\theta=a\)

and \((a+bk)^{m+nk}=(\cos\theta+k.sin\theta)^{m+nk}=\)

\(=(\varepsilon^{k\theta})^{m+nk}=\varepsilon^{k(m\theta)}\times\left\{\varepsilon^{k(n\theta)}\right\}^k\)

or \(=(\cos.m\theta+k.sin.m\theta)\times (\cos.n\theta+k.sin.m\theta)^k\), and**[123v] ** I dare say that from some of these transformations,

the __Second Side__ of the given equation, with the

determination of \(A\) and \(B\), may be deduced. But

it appears to me ['it must be' inserted] a very c__omplicated process__; &

therefore I should like to know before I undertook it,

that I was not w__asting__ t__ime__ ['in' deleted] doing so.

2^{dly} : I am plagued over page 135 of the Calculus.

It is not that there is __any one thing__ in it which

I do not clearly see. But it is the d__epth of the____whole argument__ which I cannot manage to discover.

I should say that whole argument from ''We now know &c''

page 134, to ''We can therefore take a function,

''which, for a particular value of \(x\), &c, &c'' page 135.

It seems to me all to be __much ado about nothing__;

and I do not see __what is arrived at__ by means

of it [something crossed out]. A very complicated process appears to be

used in the 1^{st} Paragraph of page 135, to prove

that when __\( h\) is small__ then the Increment in \(\varphi x\)

is v__ery nearly__ represented by \(\varphi'a+h\), which was

already shown in page 134. And then suddenly in

the __Second Paragraph__ the Formula \(\varphi a+\varphi'a(x-a)+\)

\( +\varphi''a\frac{(x-a)^2}{2}\) is introduced, & I do not understand__\'{a} quoi bon__ the closing conclusion drawn from it.

3^{dly} : I am not sure that I agree to what you

say in preference (for ascertaining Maxima & Minima)

of the __direct ascertainment of__ __\ the value of \(\varphi'\) __\) x\), over**[124r] ** the ordinary method. Because it seems to me in

many cases impossible __after__ you have determined \(0\) or

\( \alpha\) values of \(\varphi'x\), to __determine__ further that the sign

d__oes__ change at them & __how__ it changes, __unless by means____of the ordinary rule__. I have written out and

enclose an example from Peacock, in which unless

I had used the ordinary rule, a__fter__ I had

determined \(0\) values for \(\varphi'x\), I should have been

at my wits' end how to bring out the conclusion.

4^{thly} : I send you a little Maxima & Minima

Theorem of my own, which occurred to me by accident;

It is for \(\varphi x=x^2-mx\) . After proving it by the

Differential Calculus, I have given a d__irect__ __proof__

of another sort. I merely wrote this ['__direct proof__' inserted], because it

[something crossed out] occurred to me; but it gave me __a great deal of____trouble__, & I think was rather a __work of supererogation__;

but __I believe__ it is quite correct. You will find

enclosed in the same sheet the demonstration of

''What is the number whose excess above it's [*sic*] Square

''Root is the __least possible__?'' (see page 133 of the Calculus);

and on the reverse side of this latter [something crossed out]

is the ''verification round the 4 Right Angles'' for the__continual increase together__ of \(x\) and it's [*sic*] tangent (See

page 132). But here I have something further to

add. In this Chapter VIII, we hear of Differential

Co-efficient which become \(=0\), or \(=\alpha\) . In this very**[124v] ** instance, \(1+\tan^2x\) is alternately \(=1\), and \(=\alpha\) .

Now according to my p__revious ideas__, the terms__Differential Co-efficient__ was only applied to some

f__init__e quantity; and referring to pages 47, 48,

where one acquired one's f__irst ideas__ of a Differential

Co-efficient, I think it is there clearly explained

that the term is only used with reference to a__finite limit__. But in this Chapter VIII, there

seems to be a considerable __extension__ of __meaning__ on

the subject.^{thly} . In page 132, it is very clearly deduced that

the Ratio of a [something crossed out] Logarithm to it's [*sic*] number is increasing

as long as \(x\) is \(<\varepsilon\), and afterwards decreases.

The proof is most obvious. But, unluckily, the

conclusion seems to me to be c__ontrary to the fact__; at

least the f__irst half__ of the conclusion, not the l__atter__ __half__.

On this principle : from the very nature of a

Logarithm, it is obvious that (\) x\) being \(>\varepsilon\) ), for__equal increments__ to the \(\log,x\), there will be__larger & larger__ __Increments__ to \(x\) . The one being in__arithmetical__, the other in g__eometrical__ progression.

Therefore clearly the Ratio of the Logarithm to the

number, is a d__iminishing__ one. But then the

same thing seems to me to apply [something crossed out] when

\) x<\varepsilon\) . Surely there is then just the same**[125r] ** __arithmetical & geometrical__ progression for equal

Increments of the Logarithms. I suppose there is

some link that I have over-looked.

I send you two Integrations worked out. They

are from Peacock. I in vain __spent hours__ over the

one marked 2, of which I could make nothing by

any method that I devised; until in despair, I

looked thro' your Chapter XIII to see if I could there

find any hints; & accordingly at page 277, I

found a general formula which included this

case. But I do not believe I should ever have

hit upon it by myself. The Integral marked 1,

might of course be proved also in the s__ame way__;

tho' ['my' crossed out] the method ['I have used' inserted] is sufficient in this instance.

I have written out no more papers on

Forces. In fact there __is__ only one more that is

left for me, viz: \(f=v\frac{dv}{ds}\) . And for this I see

no occasion; for I am sure that I __must__ __thoro'ly__

understand it, after all I h__av__e written.

I quite see ['the truth' inserted] your remarks on my having treated__Acceleration of Velocity__ as being i__dentical__ with__Force__; whereas, (as I now understand it), it is

simply the m__easure__ __of Force__, & our only way of

g__etting at__ __expressions for this latter__. On the subject**[125v] ** of \(v^2=2\int fds+C\); I have considered it __a great____deal__; and any d__irect demonstration__ of it, after the

manner of my other papers, seems to me to be__quite impracticable__. Neither \(\int v.dv\), nor \(\int f.ds\)

[something crossed out] now appear to me to have any __actual proto-type__s in the__real motion__. Here then suggests itself to me the

question : ''then are there certain truths & conclusions

''which can be arrived at by __pure analysis__, & in

''__no other way__?'' And also, how far __abstract____analytical expressions__ __must__ express & mean__something real__, or not. In short, it has

suggested to me a good deal of enquiry, which

I am desirous of being put in the way of

satisfying.

By the bye, I fear that one little paper of mine__dropped out of the last packet__. It was a little

pencil memorandum on ['the meaning of' inserted] \(\int f.ds\); & there were

remarks upon it, (if you remember) in my accompanying

letter. It bears upon the above question.

I could write it out again, if it has been lost.

Is not t__his__ a __budget indeed__?

Yours most truly

A. A. Lovelace

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