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Okay, so obviously I would like to start by thanking Ursula not only for organizing

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this amazing meeting, but also for drawing me into this project in the first place.

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As you can see this, have I put it there, yes.

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This is joint work with Ursula and also with Adrian Rice,

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who also deserves special thanks because he was delegated to

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write the first draft of our paper on this subject.

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And this landed in our inboxes just a few weeks ago, a near perfect

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first draft which has been fantastically useful when writing this talk.

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Okay, so my title is The Mathematical Correspondence of Ada Lovelace And

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Augustus De Morgan, I'm using that as

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a way of trying to gain a proper assessment of

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how Ada was as a mathematician, just by focusing on this correspondence.

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And I'll explain how the correspondence came about, a bit later on.

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So, there have been various, different assessments of Ada, as we all know.

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We've heard about different ones over the last 24 hours.

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She's perhaps hailed as being a visionary in computer science and

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it seems to be that, that is then carried over to the mathematics.

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It's claimed that she was a brilliant mathematician also.

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And I'm not quite sure that's the case.

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And I will sort of make my argument as I go along.

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So of course, broadly speaking, the previous assessments of

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Ada's abilities both generally, both in computer science and

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in mathematics, fall very broadly into two camps.

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[LAUGH] So, we have some very occasionally extravagant claims on the one side,

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and we have the backlash against that on the other side.

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And as you go through you can perhaps see why these things have come about, so

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maybe if I have time I'll comment on why I think these views have emerged.

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But our purpose in our work has been to actually, to provide for

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the first time, because what I should say is the people who expressed these views

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haven't always gone into the mathematics in fantastic detail.

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So, this has been our goal to actually get into the archives and

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have a look and just look at what math is there.

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So, the idea was to provide a sober,

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objective assessment of how Ada was as a mathematician.

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So, we just heard a little bit about her mathematical education.

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Sorry, I'm skipping ahead.

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The truth is gonna be somewhere in the middle.

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This is gonna be the point of the talk.

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I'm going to sort of, I'm going to argue why she wasn't brilliant,

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but why she wasn't stupid, and I'm gonna end up in the middle somewhere.

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So, we've heard a bit about her early education.

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Surely, she was learning arithmetic with tutors quite early on.

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Then, a bit later she started to learn Euclid, as any mathematical education

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of any depth in the 19th century would begin with Euclid.

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And she was tutored by various people.

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Dr. King, a friend of Lady Byron.

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There was William Friend, a mathematician, who had himself tutored Lady Byron.

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There was Mary Summerville, occasionally.

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There were letters exchanged there.

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So, all of these people are tutoring Ada mostly at a distance.

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This is mostly through letters.

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Which is why we know about it, because the letters that are there in the body.

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And early on, Ada's interest in mathematics seem to have been

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as a means to an end.

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She wanted to understand more astronomy.

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She wanted to understand more optics.

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And she says this explicitly in a letter to Dr. King.

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But as it goes on, she does seem to get more absorbed by the mathematics for

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its own sake, these hints of applications drop away, to some extent anyway.

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And so, by the mid-1830's, she seems to be quite confident

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with the mathematics she'd learned with the Euclid in particular.

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So much, so that she set herself up as a tutor to a young family friend,

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Annabella Acheson, and there are some enormous letters

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proving propositions from Euclid for Annabella's benefit.

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What's not clear to me is whether Annabella actually wanted to be tutored in

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Euclid.

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[LAUGH] So then, that's as far as things went before Ada married.

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So, 1835, she married.

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Three children followed.

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And then, we get to 1839, and Ada is again

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wanting to study more mathematics to go beyond what she's already done.

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So, she's looking around for a tutor.

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She certainly asked Babbage to help her find someone.

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But it doesn't seem to have been through Babbage, that she came to

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Augustus De Morgan, the founding professor of mathematics at UCL.

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He and Babbage had been friends for many years, but the connection to Ada seems to

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have come through De Morgan's wife Sophia, who was a friend of Lady Byron.

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And so, Ada and De Morgan entered into what was essentially a correspondence

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course in mathematics.

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And we have quite a lot of that surviving in

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Box 170 of the Lovelace-Byron papers down at the end.

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There are mathematical sheets scattered throughout the various boxes.

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But Box 170 has a particular concentration of them.

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So, 357 sheets in total.

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The first 43 comprised 20 letters from De Morgan to Ada.

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The next 120 or so are the 42 letters in the other direction.

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And in the second half of the box is just assorted mathematical jottings.

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And a very large part of this doesn't seem to be relevant to this project.

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We think because of the handwriting, because of various references to

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textbooks that haven't been published by this stage, we think a lot of these papers

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are actually Ada's daughter's from when she was learning mathematics.

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But there are other things in there, in these assorted mathematical jottings,

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there are enclosures that were sent with the letters, and also,

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the Konigsberg British Sheets we've seen already, it is in there.

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Okay, so what was the nature of this correspondence?

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Course well it was a largely self-motivated one, De Morgan was

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there to help, but Ada was mostly working through De Morgan's textbooks.

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He was directing her to appropriate reading and

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I'll show you some examples of that in a moment.

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There are some gaps, but I think anybody who's ever tried to study something on

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a self-motivated basis knows that there will always be gaps because you get

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sidetracked by things and have to come back and restart and so on.

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And of course there are lots of letters going back and forth.

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Ada's asking about such-and-such a thing that she doesn't understand and

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De Morgan writes back and explains, and Ada writes back with a new question,

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and sometimes more than one letter per day.

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And occasionally, they were meeting face to face, and

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we see this littered throughout the letters, we'll see things like this

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from De Morgan to Ada, we'll be happy to see you on Monday evening and

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Lord Lovelace too if he not be afraid of the algebra.

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[LAUGH] So, this conjures the nice image I think of Ada and

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De Morgan sitting at a table doing mathematics.

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Lord Lovelay's I don't know,

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perhaps he has to amuse himself in the corner, I don't know.

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But very similar to the Kenigsberg British Sheet where we

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have the pencil going through the ink and we have a picture of Babbage and

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Ada sitting at a table again, doing mathematics.

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So, this is the sort of reading that Ada was doing,

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these are essentially sort of elementary textbooks by De Morgan.

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Or elementary as compared to the other things that Ada was doing.

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Because up to this point,

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she hadn't had a particularly systematic mathematical education.

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There were some gaps in there.

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And so, you see in the letters that De Morgan keeps having to send her back to

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these books.

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She wants to learn calculus, that's the main thrust of what she was doing.

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But it keeps turning out she doesn't have the necessary algebra,

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she doesn't have the necessary trigonometry,

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so De Morgan keeps sending her back to these books to fill these gaps.

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Other books she used, well, we have Peacock's Treatise of Algebra.

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Peacock was De Morgan's Cambridge tutor.

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So, this is a book from 1830.

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So, by the time this correspondence was going on in 1840-41,

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this book had become quite difficult to get a hold of.

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But we know that Ada managed to get it because she wrote to De Morgan

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with an element of shock that she had to pay 2 pounds, 12 shillings,

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and 6 pence for this book, which originally sold for 30 shillings.

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So, she was quite shocked by this.

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On the right hand side, we have a page from the Penny Cyclopedia,

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which is a publication that was produced by the Society for

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the Diffusion of Useful Knowledge, with which De Morgan was involved.

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And he wrote lots of the mathematical articles for this encyclopedia and

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this is a page from the article on negative and impossible quantities.

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Impossible quantities meaning complex numbers in modern terminology.

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And this is something I'll come back to in a bit.

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But the main text that Ada was using was De Morgan's calculus, because as I said,

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calculus is what she really wanted to get into.

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Okay, so that's an overview of what she was doing.

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Now, there is the question of how she was doing,

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how she was doing as a mathematical learner?

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So, what I've done is I've put together a few indications of her weaknesses and

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then, I'll follow that with an indication of her strengths.

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And hopefully we'll come to some kind of conclusion, however vague, at the end.

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Okay, oh yes, so this is what I was just commenting on.

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So this is the first of her weaknesses,

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the weakness that was rectified fairly quickly, was

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just the fact that she had insufficient foundations to study the calculus.

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So we find comments like this.

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You must make up the points in algebra and

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trigonometry that you have left behind and Ada did.

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She was very impatient to get on with things,

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but you do eventually see that she acknowledges the need to do this.

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She said, my algebra wits not having been stretched proportion,

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not having been quite stretched with some of my other wits.

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But you see things in this correspondence where

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there are things that perhaps she ought to know by this stage but she doesn't,

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because she's never been taught it.

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So, here's an example of something that she had some difficulty with.

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This is a concern with the equation of a curve and we can see it there,

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it's y equals x squared.

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And she really struggled to understand what do we mean

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by the equation of a curve?

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And this is I suppose of GCSE level type thing now, but she said,

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well what is the equation?

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Is it the sequence of values that we get from this, so

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up the scale there, or is it the spikes that we have or what?

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And she doesn't get it, and there's a sort of back and

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forth where De Morgan is really trying to explain, well, we just mean this.

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This is the curve and this is the equation and you put values in.

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And she did seem to just, I don't know, it's odd

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to read because she just seemed to label the points lightly, I don't know why.

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But this is a particularly nice example of some of the papers.

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Because at the top here we have Ada's,

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she's written out her understanding of what this means.

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For De Morgan to correct, and then, here in a slightly darker ink and

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different handwriting is De Morgan's corrections.

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His comments on her understanding,

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trying to make her see where she's misunderstanding.

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And there are similar such things certainly in the earlier letters.

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There's a very similar exchange about logarhythms she doesn't understand

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logarithms and that again, takes a few exchanges for her to pin down the idea.

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But these problems generally seem to have been sorted out in the long run.

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They perhaps in some cases take a bit longer to resolve than you would perhaps

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expect, but they do tend to be resolved.

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One thing that isn't really ever resolved, as far as I can tell, is the struggles

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that Ada had with algebra, just simply manipulating symbols on the page.

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She just frankly wasn't very good at it.

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So, here we have November 1835, she's writing to Mary Somerville for

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asking her for help, manipulating some trigonometric identities, and

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you can see it there, I think.

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Cosine A equals sine A minus B, etc.

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And we have a nice little exchange of letters there.

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But this is 1835,

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so this is still comparatively early in Ada's mathematical learning.

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She hadn't done a great deal up to this point, so

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it's perhaps understandable that she's having this difficulty at this point.

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But it persists.

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This is five years later.

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Ada confesses to De Morgan in a letter that she's completely

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baffled by this problem showing that the thing at the bottom

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satisfies that equation at the top for all values of A.

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And this is something that never really goes away.

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This struggle with algebra, and

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in fact this is a sheet from which we had a quotation from yesterday.

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These functional equations meaning the things that make the equation at the top

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5x plus y, etc., are complete will-o'-the-wisps to me.

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And like I said, this just continues throughout the correspondence.

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This sheet, incidentally, is one of the ones on display in the Bodleian,

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so if you haven't already seen it, you can go and have a look and

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see if you can understand what it is that Ada is struggling with, more specifically.

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This type of comment also,

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I should say is an example of her very whimsical turn of phrase.

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So, she's whimsical, she can also be quite long-winded and I wonder if that's perhaps

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one of the reasons why people have dismissed her in the past.

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They haven't been perhaps necessarily been prepared to work past this,

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her way of expressing herself which can be a bit exhausting at times to read it,

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but I don't know if that's the case.

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Perhaps, it's also worth mentioning here, and I didn't put this on a slide because

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it didn't look particularly interesting on a slide,

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but there are issues with dating of these letters.

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Ada was appalling at dating things,

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she was very slap dash with the dates.

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There are, for example, two consecutive lessons, I can't remember what the date on

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them is exactly, but it's something like Sunday, the 6th of June, 1841.

235
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Followed immediately by a Monday the 6th of June 1841.

236
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And then, we look at a calendar, you find that the 6th of June 1841 was a Tuesday.

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[LAUGH] So, you're never quite sure whether to believe the dates completely,

238
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if she put them on at all.

239
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But her son Ralph sorted through some of this material after her death and

240
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put some dates on them.

241
00:16:09,240 --> 00:16:12,977
But we don't think he's got it quite right.

242
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I'm not sure if it's this letter but

243
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one in this sequence has been assigned to 1842.

244
00:16:19,200 --> 00:16:21,830
So, it looks like we have Ada doing this

245
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material on function of equations in November in 1840 and

246
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then still by November of 1842 she's asking elementary questions about it.

247
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Well, the reality is we think that this is not 1842, this is 1840, so suddenly

248
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it seems a lot more reasonable that she's asking these questions about it.

249
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So, that's the sort of detail that interests me, but

250
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I don't know whether it would work so well in a slide.

251
00:16:48,100 --> 00:16:53,080
And I've already mentioned another of her weaknesses, impatience and

252
00:16:53,080 --> 00:16:58,140
occasionally over ambition so she wishes she could go on quicker.

253
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She's very disappointed to sit down and find an hour or

254
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two later she has accomplished one-twentieth part of one's intentions.

255
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I think we've all had days like that.

256
00:17:08,970 --> 00:17:13,250
So, you find De Morgan reigning her in, as well as telling her, no,

257
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you've got to go back and fill in these gaps in the algebra and trigonometry.

258
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You should never estimate progress by number of pages.

259
00:17:20,770 --> 00:17:24,650
And it's interesting to see this comment, because she was saying precisely the same

260
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thing to Annabelle Agerson in her tutorship of her just five years earlier.

261
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So, she's not heeding her own advice here, but

262
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she does sort of settle into things eventually, it does seem.

263
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Right so, so far I've been quite particular.

264
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I've shown you some examples of where I think Ada's weaknesses were,

265
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so now i want to, on a more positive note turn to her strengths.

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And this is slightly problematic because I think, as Betty Toole commented yesterday,

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the letters to De Morgan are letters about what Ada doesn't understand.

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If she understood something then, she wouldn't write about it.

269
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So, there's a certain amount of extrapolation, and

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dare I say speculation involved in saying what she was good at,

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which I think is why some authors have, in my view,

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gone slightly too far on occasion.

273
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But anyway, so the strengths that I think she had were, first of all, the thing I've

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already mentioned, the self-motivation, so she'd corresponded with Dr. King,

275
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with William Friend, with Mary Somerville, so much of her higher mathematical

276
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learning had be done at a distance and this pattern continues with De Morgan.

277
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So, she must of been determined to do this just to keep things going.

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And speaking for myself, I've not always managed to keep these sort of things going

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when I've tried to study things myself.

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She's determined to understand every last detail of things,

281
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it's something that you notice.

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As you work through the letters you get to the point where you think oh no not

283
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that integral again.

284
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But she wants to understand every detail.

285
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And she doesn't want to just apply rules, she wants to understand where the rules

286
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are coming from and so, as I've said here I need to understand why rules work,

287
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so, for example, she's presented with the naive manipulation of differentials and

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she wants to know well why does that work?

289
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Why do we write dx on the end of everything and

290
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if we have dy by dx then, why could we simply multiply by dx?

291
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Why doesn't it work like that?

292
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So, she wants to understand.

293
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And there's also some scepticism over existing rules.

294
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There was a principle which held amongst British mathematicians at this time.

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The principle of the permanence of equivalent forms which said that anything

296
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you can do in arithmetic, you can apply to other contexts.

297
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We don't hold to this rule anymore because well, we know it's not true.

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But this was presented to Ada just as a rule that she could apply and

299
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she wanted to know why.

300
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Well, why does this work?

301
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And she doesn't seem to have ever got a satisfactory explanation.

302
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At least not in the papers that survive.

303
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Perhaps because she can't give a satisfactory explanation.

304
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She also has a very critical eye.

305
00:20:19,560 --> 00:20:24,170
There's a lot of letters where she's pointing out typos in the textbooks.

306
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She's not right all the time.

307
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Sometimes it's her mistake and

308
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she hasn't seen it but she has quite a good hit rate with these things.

309
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She's really understanding what's going on and

310
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getting into the details and seeing where problems are.

311
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And this is where I get a bit more speculative, I'm afraid.

312
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I think, she had a broad view of mathematics.

313
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We've seen that a bit with the comments on her computer science,

314
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that there's a question about what she actually contributed to this paper,

315
00:20:57,456 --> 00:21:02,020
but she seems to have had this big vision of what computing might do.

316
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And so perhaps, she wasn't good at algebra.

317
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Perhaps, she wasn't good at the details but

318
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she does seem to have had a good broad view of mathematics.

319
00:21:10,780 --> 00:21:14,560
A view towards generalization possibly, I've got an example coming up.

320
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It's a feature of the mathematical minds to prove a theorem and

321
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then seek to generalize it, so she has an awareness of that process.

322
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And possibly also, there's some research awareness, and there's a letter to Mrs.

323
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De Morgan, where she comments.

324
00:21:28,390 --> 00:21:32,526
Oh, I understand that Mr. De Morgan is working on such and such.

325
00:21:32,526 --> 00:21:34,190
De Morgan, was sending at least on one occasion,

326
00:21:34,190 --> 00:21:37,620
was sending her a copy of one of his research papers.

327
00:21:37,620 --> 00:21:40,320
Now, we've no idea whether she read it, whether she understood it,

328
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what she thought of it.

329
00:21:41,130 --> 00:21:44,400
But the fact that he was sending it, makes you think, well, you know?

330
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Maybe he thought that she will get something from this.

331
00:21:48,970 --> 00:21:52,600
So, my example for generalizations, then, is this here.

332
00:21:52,600 --> 00:21:58,530
This is a much quoted passage in connection with Ada,

333
00:21:58,530 --> 00:22:01,740
and this comes after she's just been reading the article I showed

334
00:22:01,740 --> 00:22:05,060
you from the Penny Cyclopaedia, the negative and impossible quantities.

335
00:22:05,060 --> 00:22:08,900
So, she's read that, but she's made a few comments in this letter.

336
00:22:08,900 --> 00:22:12,000
And it's about complex numbers which, amongst other things,

337
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can be used to do geometry in two dimensions.

338
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And she's musing that maybe we can do something

339
00:22:17,530 --> 00:22:20,330
to enable us to do Geometry in three dimensions.

340
00:22:20,330 --> 00:22:25,450
And this is cited as her vision, because this is just two years before

341
00:22:25,450 --> 00:22:28,870
the Irish Mathematician William Rowan Hamilton did just that.

342
00:22:28,870 --> 00:22:33,270
However, if you look at

343
00:22:33,270 --> 00:22:38,960
the Penny Encyclopedia article you see that actually everything up to

344
00:22:38,960 --> 00:22:44,990
the last comma in this quotation is more or less there in that article.

345
00:22:44,990 --> 00:22:49,420
So, is she just parroting what she's read?

346
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It just seemed to be that case.

347
00:22:51,690 --> 00:22:54,420
So then, it all hinges on what we make of this last bit.

348
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This and so on, ad infinitum possibly.

349
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Is that naive speculation?

350
00:23:01,710 --> 00:23:03,400
Is it genuine insight?

351
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I don't know and I don't think we have any way of knowing.

352
00:23:06,410 --> 00:23:08,020
So, I'm just gonna leave that dangling.

353
00:23:08,020 --> 00:23:12,410
This is just sort of a matter for debate, I suppose.

354
00:23:12,410 --> 00:23:16,290
So, just to bring things to a close, then.

355
00:23:16,290 --> 00:23:20,000
I'm going to quote from another much quoted source in connection with Ada.

356
00:23:20,000 --> 00:23:23,880
This is a letter that De Morgan wrote to Lady Byron in 1844

357
00:23:23,880 --> 00:23:27,320
saying what he thought of Ada as a mathematician.

358
00:23:27,320 --> 00:23:31,550
So, he's commenting here that her abilities are so utterly out of the common

359
00:23:31,550 --> 00:23:34,320
way for any beginner which is a slightly odd compliment.

360
00:23:34,320 --> 00:23:38,020
Is he saying that she's good, or is he saying that she's good for

361
00:23:38,020 --> 00:23:39,340
a beginner, I don't know.

362
00:23:39,340 --> 00:23:41,450
At least in that paragraph anyway, but he does go on and

363
00:23:41,450 --> 00:23:43,064
it does seem to be quite complimentary.

364
00:23:43,064 --> 00:23:48,090
He comments the tract about Babbage's machine is a pretty thing enough, but

365
00:23:48,090 --> 00:23:51,750
I could I think produce a series of extracts out of Lady Lovelace's first

366
00:23:51,750 --> 00:23:56,490
queries upon a new subject which would make a mathematician see that

367
00:23:56,490 --> 00:23:59,350
it was no criterion of what might be expected of her.

368
00:23:59,350 --> 00:24:04,330
So, he thinks that she has more in her, that she doesn't really seem to have had

369
00:24:04,330 --> 00:24:07,563
a chance to pursue.

370
00:24:07,563 --> 00:24:10,280
And so, I've tried basically to do this,

371
00:24:10,280 --> 00:24:14,130
to produce a series of extracts from Ada's correspondence with De Morgan.

372
00:24:14,130 --> 00:24:19,040
And what I hope I've demonstrated is it's a bit of a stretch to say she was to say

373
00:24:19,040 --> 00:24:20,640
she was a mathematical genius.

374
00:24:20,640 --> 00:24:23,778
She was certainly competent, definitely competent in most things.

375
00:24:23,778 --> 00:24:29,780
And so, she certainly wasn't stupid.

376
00:24:29,780 --> 00:24:31,750
This is definitely not true.

377
00:24:31,750 --> 00:24:36,430
So, to go back to my diagram at the beginning,

378
00:24:36,430 --> 00:24:41,710
she certainly does lie somewhere in the middle, and

379
00:24:41,710 --> 00:24:47,235
I think that's what makes the question interesting.

380
00:24:47,235 --> 00:24:48,320
Thank you. [APPLAUSE]