## Folios 62-67: AAL to ADM

Ashley-Combe

10 Nov^{r}

Dear M^{r} De Morgan. The

last fortnight has been spent

in total idleness, m__athematically__

at least; for we have had

company & been as they say

g__addin__g about. __ __ I must

set too [*sic*] now & work up

arrears. __ __ But I have a

batch of questions & remarks

to send. __ __

First — on Peacock's Examples,

which I have only now begun**[62v] ** upon:

What d__oes__ he mean by adding

\( d\,x\) to every solution? It

appears to me a work of

supererogation. I take the

very first example in the

book as an instance, and

the same applies to all :

Let \(u=ax^3+bx^2+cx+e\) :

it's [\textit{sic}] differential, or \(du=\)

\( =3ax^2dx+2bx\,dx+cdx\)

or \((3ax^2+2bx+c)\,dx\) .__I__ should have written, & in

fact d__id__ write : it's [*sic*] differential

or \(du=3ax^2+2bx+c\) .

I s__uppos__e that this form**[63r] ** is used under the supposition

that \(x\) itself __may__ be a

function. __ ____My__ result & the b__ook's__ do

not agree in one particular

in the 9^{th }example, page 2,

& I am inclined to think

it is a misprint in the

latter : the Books says :

Let \(u=x^2(a+x)^3(b-x)^4\)

\( du=\{2ab-(6a-5b)x-9x^2\}x(a+x)^2(b-x)^3dx\)

and __I__ say :

\( du=\{2ab-(6a-5b)x-x^2\}x(a+x)^2(b-x)^3dx\)

In case it may save you

trouble, I enclose my working

out of the whole. __ __

I do not the least understand**[63v] ** the __note__ in page 2. Not

one of the three theorems it

contains is intelligible to me.

I conclude you to have

the Book by you; but if

not I can copy out the

note & send it to you.

Secondly --- to go to your

Algebra : I think there is

an evident erratum page 225,

line 8 from the bottom, where

\( 1+x+\frac{x-\frac{1}{n}}{2}+\frac{x-\frac{1}{n}}{2}\cdot\frac{x-\frac{2}{n}}{3}+\text{&c}\)

should certainly be

\( 1+x+x\,\frac{x-\frac{1}{n}}{2}+x\,\frac{x-\frac{1}{n}}{2}\cdot\frac{x-\frac{2}{n}}{3}+\text{&c}\) .

I have a little difficulty

in page 226, the l__ast__ line,**[64r] ** ''let \(\frac{1+b}{1-b}=\frac{1+x}{x}\) which gives \(b=\)

''\( =\frac{1}{2x+1}\) ''. __ __

In the first place I do not

feel satisfied that the form

\( \frac{1+b}{1-b}\) is capable of being

changed into the form

\( \frac{1+x}{x}\) . There are three

suppositions we may make

upon it, (supposing that

it __is__ capable of this second

form) . \(x\) may be __less__

than \(b\), in which case

the denominator must also

be __less__ than \(1-b\), and less

in a certain given proportion,

in order that the Fractional**[64v] ** Expression may remain the

same . \(x\) may \(=b\), in

which case the second form

can only be true on the

supposition that \(1-b=x=\)

\( =b\), or \(b=\frac{1}{2}\) .

\( x\) may be g__reater__ than \(b\),

in which case the denominator

of the second form must also

be g__reater__ than \(1-b\), in

a certain given proportion,

in order that the Fractional

expression may remain the

same. __ __

But secondly supposing \(\frac{1+b}{1-b}\)

to be under all circumstances**[65r] ** susceptible of the form \(\frac{1+x}{x}\),

I cannot deduce from this

equation \(b=\frac{1}{2x+1}\) . __ __

Your last letter, on

the Binomial Theorem, was

quite satisfactory to me, but

I have some remarks to make

on the s__econd__ proof of it,

pages 211 to 213. I think

you __well__ observe in the

note page 213, that the two

proofs supply each other's

deficiencies; for I like neither

of them t__ake__n __singly__.

The latter one is what I

should call rather cumbrous,

especially the verification of

\( \varphi n\times\varphi m=\varphi(n+m)\) by**[65v] ** actual multiplication in page

212, which is an exceedingly

awkward & inconvenient

process in my opinion. __ __

Then I am not at all

sure that I like the

assumption in the last

paragraph of page 212. __ __

It seems to me somewhat

a large one, & much

more wanting of proof than

many things which in

Mathematics are rigorously &

scrupulously demonstrated.

But these inconsistencies

have always struck me

occasionally, and are perhaps

only in reality the inconsistencies**[66r] ** in a beginner's mind, &

which long experience &

practice are requisite to do

away with. __ __

The e__nd__ of Euler's proof,

page 213, is not agreeable

to me, and for this reason,

that I cannot feel properly

satisfied as yet with the

little Chapter on Notation

of Functions, and upon the

full comprehension of this

depends the force of the

latter part of this proof. __ __

I do not know __why__ it

is exactly, but I feel I

only half understand that**[66v] ** little Chapter X, and it has

already cost me more trouble

with less effect than most

things have . I must study

it a little more I suppose.

I hope soon I

may be able to return to

your Differential Calculus. __ __

At the same time, I never

more felt the importance of

n__ot__ being in a h__urry__. __ __

I fancy great proficiency

in Mathematical Studies is

best attained by t__ime__; __ __

co__nstantly__ & c__ontinually__ doing

a li__ttle__ . If this is so,

surely then the University**[67r] ** cramming system must be

very prejudicial to a real

progress in the long run,

particularly when one considers

how v__ery__ v__ery__ little School-boys

are ['generally' inserted] prepared on first going

to the Universities, with

anything like distinct

mathematical or even

arithmetical notions of the

most elementary kind. __ __

I am now

puzzling over the C__omposition____of Ratios__, but I hope in

a day or two more I shall

get successfully over that.

It plagues me a good deal.**[67v] ** I believe I thought some

years ago, that I understood

it; but I am inclined to

think I certainly never did.

You see just at

this moment I am full of

unsatisfactory obstacles; but

I doubt not they will soon

yield . __ __

With kindest remembrances to

M^{r} De Morgan, I am

Yours very truly

A. A. L

I think there is an erratum

in your Trigonometry, page 34, line 7

from the top :

''let \(NOM=\theta\odot\), \(MOP=\varphi\odot\) &c''

should be \(\ldots\) \({\underline{N}}OP=\varphi\odot\) &c

## About this document

All Ada Lovelace manuscript images on the

Clay Mathematics Institute website are

© 2015 The Lovelace Byron Papers,

reproduced by permission of

Pollinger Limited. To re-use them in

any form, please apply to

katyloffman@pollingerltd.com.

The LaTeX transcripts of the letters

were made by Christopher Hollings

(christopher.hollings@maths.ox.ac.uk).

Their re-use in any form requires his

permission, and is subject to the

rights reserved to the owner of

The Lovelace Byron Papers.

Bodleian Library, Oxford, UK

Dep. Lovelace Byron