## Folios 91-95: AAL to ADM

[beginning of letter seems to be missing]

&bnsp;

**[91r] ** (unless the limit for \(\frac{v^n-w^n}{v-w}\)

is dispensed with in the

demonstration for the Binomial

Theorem, which it is not

in your Algebra, nor am

I aware that it c__an__ be

dispensed with in any of the

e__lementary__ proofs of that

Theorem). __ __ It had not

struck me that, calling

\( (x+\theta)=v\), the form

\( \frac{(x+\theta)^n-x^n}{\theta}\) becomes \(\frac{v^n-x^n}{v-x}\) .

And by the bye, I may

here remark that the curious__transformations__ many formulae

can undergo, the unexpected

& to a beginner apparently**[91v] ** i__mpossible__ __identity__ of forms

exceedingly __dissimilar__ at first

sight, is I think one of

the chief difficulties in the

early part of mathematical

studies. I am often reminded

of certain sprites & fairies

one reads of, who are at

one's elbow in o__ne__ shape

now, & the next minute in

a form the most dissimilar,

and uncommonly deceptive,

troublesome & tantalizing are

the mathematical sprites &

fairies sometimes; like the

types I have found for them

in the world of Fiction. __ __**[92r] ** I will now go to the question

I delayed asking before :

In the development of the

Exponential Series

\( a^x=1+(\log a)x+\frac{(\log a)^2x^2}{2}+\) &c,

and the Logarithmic Series

\( \log a=(a-1)-\frac{1}{2}(a-1)^2+\) &c

deduced from it; I object

to the necessity involved of

supposing \(x\) to be __diminished____without limit__, a supposition

['obviously' inserted] quite necessary to the completion

of the Demonstration. It has

struck me that though this

supposition leaves the Demonstration

& Conclusions perfect for the

cases in which \(x\) __is__ supposed

to d__iminish without limi__t, yet**[92v] ** it makes it valueless for the

many in which \(x\) may be

anything else which does __not__

diminish. No by the bye,

I think I begin to see it now;

I am sure I do. It is as

follows : the supposition of

\( x\) diminishing without limit

is merely a __parenthetical____one__, by means of which a

limit for a certain expression

\( \frac{a^x-1}{x}\) is deduced under those

circumstances; & then the

argument proceeds, that having

already obtained in another

place, a ['different' inserted] limit for this same

expression under the same**[93r] ** circumstances, we at once

deduce the __equality of these two____limits__, from whence follows

&c, &c. Thus this supposition

of \(x\) diminishing without limit,

is not a portion of the __main____argument__, but only a totally

independent & parenthetical

hypothesis introduced in order

to prove s__omething else__ which__is\ __ a part of the __main__

a__rgument__. __ __ Yes this is

it, I am sure. I had

had the same objection to

the Demonstration in B__ourdon__,

to which I have had the

curiosity to refer. I am**[93v] ** sometimes very much interested

in seeing __how__ the s__am__e

conclusions are arrived at

in di__fferent ways__ by different

people; and I happen to

have been inclined to compare

you & Bourdon in this

case of developing Exponential

& Logarithmic Series; and

very amusing has it been to

me to see h__im__ b__egin__ exactly

where y__ou__ __end__, &c. Y__our__

demonstration is __much__ the

b__est__ for practical purposes.

His is exceedingly general, &

the vast number of s__ubstitutio__ns**[94r] ** of one thing for another make

it lengthy, & by no means very

simple to follow. __ __ But it

is very well occasionally to

make these comparisons. __ __

We are going to

Town on Monday the 25^{th}

for two or three nights, &

I will ask M^{r} De Morgan's

& your permission to spend

Monday Evening with you,

going towards 8 o'clock,

as I did before. It would

give me great pleasure, &

may perhaps be not only

agreeable to me, but of use

[94v] too, as there are one or two

points [something crossed out] relating to my future plans

which I rather think of

speaking to you upon. __ __

By the bye, Lord

Lovelace & I are both of us

much vexed, at our own

negligence in letting the Xmas

Vacation go bye [*sic*], without

proposing to you & your

lady & children to visit us

here, which you might

perhaps have been able to do

during Holiday-time. I fear

you __may__ __now__ be unable to

think of it; but pray consider

[95r] the question with her; if not

for any immediate use, at

any rate for the __next__ occasion.

The fact is, that we have

so much the habit of thinking

of you __only__ in connexion

with Town & engagements there,

that it only suddenly occurred

to us whether you __migh__t not

be able to breathe country

air like other people. __ __

You would come by Railway,

& we would send the

carriage to the Station for

you.

Yours most truly

A. A. L

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