## Folios 74-76: AAL to ADM

## Folios 70-73: AAL to ADM

Ockham

Tuesday. 22^{nd} Dec^{r}

Dear M^{r} De Morgan I now see

exactly my mistake. I had

overlooked that the Series in

question is not one in s__uccessive__

Powers of \(x\) ['like that in page 185' inserted], but only in

s__uccessive__ __even__ powers of \(x\) .

## Folios 68-69: AAL to ADM

## 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. __ __

## Folios 149-151: AAL to ADM

Ashley-Combe

Nov^{r }27^{th}

## Folios 146–148: AAL to ADM

Ashley-Combe

Wed^{dy}. 16^{th} Nov^{r}

['1842' or '1847' added by later reader]

[CH: this might date from earlier than 1842, as it contains material that seems to fit better with the earlier letters]

## Folios 48- 49, 163-165:

## AAL to ADM

Ashley

Sun^{dy} 13^{th} Sep^{r}

[1840] [added in pencil by later reader]

Dear M^{r} De Morgan

## Correspondence with De Morgan

## AAL to ADM

**Folios 48, 49, 164, 163, 165, 13 Sept 1840**

Dissatisfied with progress; seeks clarification of meaning of 'equation of a curve' and 'functional equation'**Folios 146–148, 16 Nov 1840**

Differential coefficients; manipulation of functions; a little trigonometry**Folios 149–151, 27 Nov 1840**

Functional equations**Folios 62–67, 10 Nov 1840**

Difficulties over the meaning of differentials; fractional expressions; proofs of the binomial theorem; comments on the necessarily slow and painstaking nature of learning mathematics, with a remark on how the 'University' method must be prejudicial to understanding in the long run; is puzzling over composition of ratios**Folios 68–69, 17 Dec 1840**

Remarks on how the poor weather is keeping her indoors doing mathematics; persists on a point of convergence that was apparently discussed earlier in the week (Monday evening)**Folios 70–73, 22 Dec 1840**

Now sees her mistake regarding a series previously discussed; learns from, and no longer regrets, such mistakes; seeks clarification over incommensurables and ratio**Folios 74–76, 4 Jan 1841**

More-or-less direct continuation of the previous: series and ratio**Folios 77–83, 10 Jan 1841**

Series; differential coefficients**Folios 91–95, Jan 1841**

Details of proof of binomial theorem; more series, inc. exponential and logarithmic; Christmas plans**Folios 84–87, 17 Jan 1841**

Limits of series of fractions; remaining difficulties over differential coefficients, which she feels she ought to have got to grips with by now**Folio 156, Jan 1841**

Rearrangement of meeting**Folio 88, 22 Jan 1841**

Nothing mathematical: arrangement to call round on Monday evening, without Lord Lovelace**Folios 54-57, Feb 1841**

Seeks clarification of various issues: functions of two variables, logarithms (Napierian in particular)**Folios 89–90, 3 Feb 1841**

More on limits of fractions**Folios 96–98, 6 Feb 1841**

Was too hasty in sending letter of Weds, for now understands the matter at hand (on fractions); resolves to rein in her tendency towards metaphysical musing**Folios 100–103, 19 Feb 1841**

Differential coefficients; brief question about logarithms**Folios 104–105, 22 Feb 1841**

Problems over the coefficients of a particular series**Folios 58–60, Mar/Apr 1841**

No mathematical content; hopes to see the De Morgans before

going to Paris; has a bundle of papers for ADM; recounts recent

social life (inc. trips to the opera)**Folios 50–52, May/Jun 1841**

No mathematical content: bemoans the fact that she hasn't had

much time for mathematics, asks to see ADM**Folios 159–160, Jun 1841**

Annoyance that ADM was turned away by footman; arrangements

for meeting**Folios 161–162, Jun 1841**

Follows on from previous; arrangements for meeting**Folios 106–107, 4 Jul 1841**

Follows on from previous; arrangements for meeting**Folios 108–109, 6 Jul 1841**

Many very specific queries about the calculation of various integrals in the textbook, including the pointing out of possible misprints**Folios 110–111, 6 Jul 1841**

Continuation of 108–109, with brief mention also of the dynamics in 106–107**Folios 152–153, Jul 1841**

Derivatives: primitive and derived functions; definite vs. indefinite integrals**Folios 112–114, 11 Jul 1841**

Continuation of the last two letters on integration; mention also of the mechanics problem; query over differentials – that dV/dx does not represent a division**Folios 115–118, 15 Aug 1841**

Queries over the solution of particular integrals; logarithms of negative quantities; differential coefficients; accelerating force as an application of the calculus**Folio 119, 16 Aug 1841**

Sends proof of result which she has carried out in a opposite manner to that in the book; more on accelerating force**Folio 166, 20 Aug 1841**

Developments of integrals; accelerating force**Folios 121–122, 21 Aug 1841**

Same integral as in preceding letters; manipulation of differentials; more on logarithms of negative quantities; another integral from book**Folios 144–145, 28 Aug 1841**

Various problems involving integrals, apparently in a dynamical context**Folios 46–47, 4 Sept 1841**

Misapprehensions over meaning of differentials**Folios 123–125, 9 Sept 1841**

Asks whether she should be able to prove a certain formula, or whether she is overreaching herself; questions the point of complicated method in the book; determination of maxima and minima; nature of logarithms and differential coefficients; accelerated force**Folios 127–129, 19 Sept 1841**

Cubic equations; complex numbers; asks whether there can be a

three-dimensional analogue; asks about Monge's descriptive geometry**Folios 130–131, 27 Oct 1841**

Will be coming to town for a few days from next Tuesday (without

her mother's knowledge); hopes to see Mr. and Mrs. De Morgan at some point**Folios 132–133, 4 Nov 1841**

Manipulation of differentials; calculus involving trigonometric expressions;

differential equations?**Folios 134–135, 8 Nov 1841**

Continuation of the last letter; logarithms; queries notation for

(and notion of) the inverse of a function**Folio 136, 10 Nov 1841**

Continuation of the last letter; logarithms; queries notation for

(and notion of) the inverse of a function**Folios 138–139, 11 Nov 1841**

Journey to town has been unexpectedly delayed**Folios 140–141, Nov 1841**

Revised plan for being in town; mention of an integral that has a

ppeared in previous letters**Folios 142–143, 21 Nov 1841**

Arrangements for meeting; assorted problems involving

expressions for/operations on functions**Folios 154–155, 14 Jan 1842**

Arrangements for meeting; has been amusing self with

book named in next note**Folios 157-158, 1848**

Promises to send ADM an unspecified paper which is

about to be published

## ADM to AAL

**Folios 1-2, 1840**

Advises on readings on acoustics; clarifies the nature of zero; notes foundation of society for publication of scientific manuscripts and suggests that AAL should join**Folios 3–4, 1 Aug 1840**

Comments that AAL has understood the problem of the stone (?); explains the notion of a variable and its passing to a limit; defines coordinates; again promotes manuscript publishing society**Folios 5–6, Aug 1840**

Going to see Frend as his health is not good, so may not be able to see AAL; reminds AAL that she is not supposed to know about differentiation for the material at hand (CH: she has been skipping ahead?); defines, and suggests reading in connection with, logarithms**Folios 7–8, 17 Aug 1840**

Clarifies theorem involving inequalities and fractions; discusses right-angled triangles and defines tangent; discusses nature of differential and integral calculus, with historical comments; recommends further reading in differential calculus, but warns that it will also require extra reading in algebra and trigonometry (which he recommends) – "mechanical expertness in the differential calculus is of the utmost consequence"; W. Frend is better; Lord Lovelace's name added to list of members (presumably of historical manuscript society)**Folios 9–10, Aug/Sept 1840**

Responds to enquiry about book (?): directs AAL to 'Algebra' or 'Trigonometry', rather than 'Study of Mathematics'; advises that continuity must be studied well**Folios 12-13, Aug/Sept 1840**

Discussion of equations and curves; gives example of finding the equation of a curve passing through given points; thanks AAL for partridge**Folios 14–15, 15 Sept 1840**

Comments on Nicolas Occam and similarity to AAL's son's title – asks whether Lord Lovelace has any Occam manuscripts; general advice on making progress in mathematical study; distinction between quantity and form; explains the equation of a curve; W. Frend neither better nor worse**Folios 16-17, 27 Sept 1840**

Recommends continuing with incommensurables, and notes that progress in differential calculus may have to be paused from time to time to fil in gaps in algebra and trigonometry; notes on limits and meaning of differentials; AAL should try some examples of differentiation from Peacock's book; should also read Proportion in Penny Cyclopaedia when it comes out**Folios 18–19, 15 Oct 1840**

Muses on differing national notions of of guilt and innocence; explanations on convergence of functions; algebraic truth vs. arithmetical truth of expressions**Folios 20–22, 14 Nov 1840**

Points out sign error; handling differentials; combinations; validity of integer formulae for fractions also; value vs. form**Folios 24–26, Nov 1840**

Comments on writing of d; formulae true for whole numbers and for fractions**Folios 27-8, Nov/Dec 1840**

Clarifies problem concerning solution of equation; believes that AAL has got everything out of the chapter on functions that she ought to have done**Folios 29-30, Nov/Dec 1840**

On the meaning of the ratio θ/sin θ**Folios 31–32, Dec 1840**

Correct form of terms in series; convergence of series**Folios 33–34, Jan 1841**

Returns papers on series; responds to some comments on limits and continuity – AAL must wait until she has studied discontinuous functions; notes AAL's circular reasoning in a derivation of a binomial coefficient (assumes binomial theorem); thanks for pheasants and hare**Folios 35–36, Jan 1841**

Points out that if it is necessary for x to diminish without limit in order to prove a conclusion, it is not necessarily the case that that conclusion holds only for small x or x=0; illustrates this with an example; politely declines invitation to Ockham, as he is too busy with lectures**Folio 37, 6 Feb 1841**

Comments on bounds that may be placed on a particular fraction using those on ithe various terms making it up; comments on proof of Taylor's Theorem; it is not necessary for a beginner to know what use differential coefficients are**Folios 38–39, 11 Feb 1841**

Just because a theorem is true for continuous functions, this does not mean that it is false for discontinuous ones; discontinuous functions are excluded for the time being because they have no language to describe them – AAL will have enough of them when she comes to study the theory of heat; acknowledges a letter from AAL to Mrs DM**Folios 42–43, 22 Feb 1841**

Discussion of logarithms; more functions – justification of the fact that a particular function depends on certain variables; (finite?) differences; advises AAL to leave discontinuous functions alone for the time being**Folios 40–41, 24 Feb 1841**

Notes misprint in book; asks AAL to call one night other than Sunday, as Mrs DM will be at her mother's then (the day after Frend's funeral)

## Notes and Exercises

**Folio 168-170**

Explanations relating to differential equations

168r, 168v, 169r, 169v, 170r, 170v, Transcription**Folios 171-172**

Scrap featuring trigonometric calculations involving surds; handwriting uncertain; 172v: manipulation of trig identities (possibly connected with trig identities in letters to/from Mary Somerville: Box 174 fols. 29–34)**Folio 174, Sept 1841**

Chain rule**Folio 175**

Definition of complete derivative and instructions for reading

175, Transcription**Folios 176-177**

176r: scribbling; 176v–177r: Königsberg bridges; 177v: more trig, much like 172v (both hands)

176, 177, Transcription**Folio 178**

Calculus- and exponential function-related jottings

178, Transcription**Folio 179**

Appears to be applying L'Hôpital's rule to the generating function for the Bernoulli numbers

179, Transcription

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## Folios 42-43: ADM to AAL

**[42r] ** My dear Lady Lovelace

Mr. Frend's death (which took place on Sunday Morning)

has made me answer your letter later than I should otherwise have

done. The family are all well, and have looked forward to this termination

for some time. My wife will answer your letter on a part of this.