Clay Mathematics Institute

Dedicated to increasing and disseminating mathematical knowledge

Correspondence with De Morgan

Sat, 2016-03-12 20:21 -- Nick Woodhouse
Banner image: 
Left text: 

 

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

 

Right text: 

 

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
    168r168v169r169v170r170vTranscription
  • 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)
    176177Transcription
  • 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

About this document

Copyright. 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. All notes and
transcriptions of material from The Lovelace
Byron Papers are © 2015 Christopher Hollings.
 

Return to Ada Lovelace's Mathematical papers

 

Folios 42-43: ADM to AAL

Sat, 2016-03-12 17:44 -- Nick Woodhouse

[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.

Folios 38-39: ADM to AAL

Sat, 2016-03-12 17:22 -- Nick Woodhouse

[38r] My dear Lady Lovelace

I have added a note or two
to your papers.

As to the subject of continuity, it
must be as much as possible your
object now to remember while proving
the things which are true of continuity
to remember that they are not false
of ['conti' crossed out?] dis continuous [sic] functions, be-
cause true of continuous ones.  Thus,
you will afterwards see that

Folios 35-36: ADM to AAL

Sat, 2016-03-12 17:17 -- Nick Woodhouse

[35r] My dear Lady Lovelace

We shall be happy to 
see you on Monday Evening, and
Lord Lovelace too if he be not
afraid of the algebra

 Your points in your letter are
are [should be 'I'] think, clear enough in
your own head.  A little addition
however may be made as follows.

Folios 31-32: ADM to AAL

Sat, 2016-03-12 17:09 -- Nick Woodhouse

[31r] My dear Lady Lovelace

You are right about
the writing down of the
terms:

 \(\frac{z}{(2n-2)(2n-3)}\) 

 is the \(n\) th term divided
by the \((n-1)\) th and the
\( \overline{n+1}\) th divided by the 
\( n\) th is \(\frac{z}{2n(2n-1)}\) as you
make it.

Folios 29-30: ADM to AAL

Sat, 2016-03-12 17:08 -- Nick Woodhouse

[29r] My dear Lady Lovelace

I have made some additional
notes on your papers.

[diagram in original] The meaning of \(\frac{\theta}{\sin\theta}\) is as follows

 \(\theta:1\) and \(1:\sin\theta\) compounded
give it in arithmetic

In fact \(\frac{a}{b}\) in arithmetic is another way of writing
\( a:b\) .

In geometry \(AB:AO\) is \(\theta[:]1\) 

 and \(AO\) or \(OB:BM\) is \(\sin\theta\) 

Pages

Subscribe to Clay Mathematics Institute RSS