Ada, the Enchantress of Numbers:Poetical Science
Page 10
Ada began her task of writing the Notes by asking pertinent questions and selecting a mathematical model that would highlight the difference between Babbage’s first calculating engine, the Difference Engine, and the Analytical Engine.
Ada’s selection of the Bernoulli numbers was a perfect example to highlight the difference. To calculate Bernoulli numbers, one must perform many operations, then take the results of those operations and use them in other operations. For example, add, then divide, then raise to a power, and on and on. No mere calculator or calculating engine like the Difference Engine could perform this feat. Only the Analytical Engine could calculate Bernoulli numbers without the intervention of a “human hand or head” because numerical information and operational instructions would be received by means of a punch card (or punched card), which Babbage had adapted from Jacquard. Jacquard’s punch card instructed the loom how to weave designs into a fabric.
Just as Ada was grounded in selecting a specific mathematical model that would show the power of the Analytical Engine, she was also developing a metaphysical understanding of the Analytical Engine. She began to see not only the technical details, but also the whole picture, the concept, of what the Analytical Engine could do. This was no easy task since Babbage had volumes of designs and many sample iterations of what today might be referred to as computer programs. Yet Ada, questioning and arguing with Babbage and distilling the information he gave her, was able to put his remarkable idea for the Analytical Engine in its proper perspective.
In the midst of this very serious undertaking, Ada wrote delightful and outlandish letters to Babbage. The letters are not only filled with discussions about Bernoulli numbers and the Medora melodrama, but visions of herself as a fairy, puzzle-pate and general. She even made references to flirtations with her Somerset neighbour, Frederick Knight. Babbage became her confidant, and she became his “interpretess.”
To Charles Babbage
Thursday Morning [1843]
Ockham
My Dear Babbage. I have read your papers over with great attention; but I want you to answer me the following question by return of post. The day I called on you, you wrote off on a scrap of paper (which I have unluckily lost), that the Difference Engine would do. . . (something or other) but that the Analytical Engine would do
. . . (something else which is absolutely general).
Be kind enough to write this out properly for me; & then I think I can make some very good Notes.
I have been considering about Prince Albert; but I much doubt the expediency of it. However there is time enough to consider of this.
I am anxious to hear how you are.
Yours ever
A.A.L.
Babbage’s goals for the Analytical Engine
Ada began to make headway with the Notes and sent some off for Babbage’s inspection. As for her Note A, Babbage replied the next day: “If you are as fastidious about the acts of your friendship as you are about those of your pen, I much fear I shall equally lose your friendship and your Notes. I am very reluctant to return your admirable & philosophic Note A. Pray do not alter it . . . All this was impossible for you to know by intuition and the more I read your notes the more surprised I am at them and regret not having earlier explored so rich a vein of the noblest metal.”
Babbage continued his compliments and wrote her that Note D was in her usual “clear style.”
In 1840 Ada had started thinking about putting games into mathematical language when she wrote Babbage about the game Solitaire. Using indices as a means of tracing each step of an algorithm was a similar conceptual skill.
The Analytical Engine would receive information about number, operations, and variables by means of the punched card. Ada wanted Babbage to be very specific about how the Analytical Engine would handle variables. In J. M. Dubbey’s The Mathematical Work of Charles Babbage, he describes how Ada had a slightly more elaborate and improved way than Menabrea of writing out the sequence of operations taking into account the variables. From Ada’s questions, and the table included and annotated in Chapter 15, I believe Ada had the skills that today are important to programmers and programming languages. The most important thing for Ada in writing this description was not fame or fortune but following the Lovelace family motto “Labor ipse voluptas” (labor is its own reward).
To Charles Babbage
Monday [10 July 1843]
Ockham
My Dear Babbage. I am working very hard for you; like the Devil in fact; (which perhaps I am).
I think you will be pleased. I have made what appear to me some very important extensions & improvements. Why I now write is to beg you will send down to the Square before tomorrow evening. Brookes’s Formulae, & also the Report of the Royal Society on your machine. I suppose you can get it easily, & I particularly want to see it, before I see you on Weddy Morg. –
It appears to me that I am working up the Notes with much success; & that even if the book be delayed in it’s [sic] publication, a week or two, in consequence, it would be worth Mr Taylor’s while to wait. I will have it well & fully done; or not at all.
I want to put in something about Bernoulli’s Numbers, in one of my Notes, as an example of how an implicit function, may be worked out by the engine, without having been worked out by human head & hands first. Give me the necessary data & formulae.
Yours ever
A.A.L.
To Charles Babbage
Saturday, 6 o’clock [undated]
My dear Babbage. I have been hard at work all day, intending to send you the Diagram & all, quite complete.
Think of my horror then at just discovering that the Table & Diagram, (over which I have been spending infinite patience & pains) are seriously wrong, in one or two points. I have done them however in a beautiful manner, much improved upon our first edition of a Table & Diagram. But unluckily I have made some errors.
I send you all of this final note H, excepting the said Table & Diagram. I also return you Note C, in which (for a wonder), I can discover nothing to alter or mend. . . .I also beg for Note A, in which I remember a wrong passage about Variable-cards. Now pray attend strictly to my requests; or you will cause me very serious annoyance.
I shall be up betimes tomorrow morning, & finish off the Table & Diagram; so as to send it [to] you by post; together with the amendments in what my servant shall bring me down tomorrow morning from you.
You will have therefore a budget on Monday Morg; and I intend to go to Town myself on Monday, for a few hours, in order to run over a few things with you finally. Be so kind as to be in the Square at two o’clock.
I fear you will think me detestably persevering.
Yours ever
A.L.
Let me know how you like my finishing up of H. Mind you scrutinise all of the n’s very carefully; I mean those of sheet 4 and 5.
To Charles Babbage
Sunday, 6 o’clock [2 July 1843]
Ockham.
I have worked incessantly, & most successfully, all day. You will admire the Table & Diagram extremely. They have been made out with extreme care, & all the indices most minutely & scrupulously attended to. Lord L – is at this moment kindly inking it all over for me. I had to do it in pencil. . .
I cannot not imagine what you mean about the Variable-Cards; since I never either supposed in my own mind, that one Variable-card could give off more than one Variable at a time; nor have (as far as I can make out) expressed such an idea in any passage whatever.
I cannot find what I fancied I had put in Note A; so I return it whole & sound, for your speedy relief. . .
Lord L – has put up, I find, in a separate cover, all that belongs to Note H. (He is quite enchanted with the beauty & symmetry of the Table & Diagram). No – I find I can put in Note D with H.
To Charles Babbage
Sunday Morning, 2 July 1843
Ockham
My Dear Babbage. Unless you really have necessary business to transact with me tomo
rrow, pray do not come to see me; for I am full of pressing & unavoidable engagements; & moreover, I shall obliged to be in Town on Friday next, & could then see you at 12 o’ clock – . . .
I am reflecting much on the work & duties for you & the engine, which are to occupy me during the next two or three years I suppose; & I have some excellent ideas on the subject.
I intend to incorporate with one department of my labours, – a complete reduction to a system, of the principles & methods of discovery, – elucidating the same with examples. I am already noting down a list of discoveries hitherto made, in order myself to examine into their history, origin, & progress. One first & main point, whenever & wherever I introduce the subject, will be first to define& to classify all that is to be legitimately included under the term discovery. Here will be a fine field for my clear, logical, & accurate, mind, to work its’ powers upon; & to develop its’ metaphysical genius, which is not least amongst its’ qualifications & –characteristics. –
Let me find a note tomorrow when I arrive, which I expect to do between twelve & one. I should like to know whether to expect you or not.
Yours
A.L.
To Charles Babbage
Tuesday Morning [4 July 1843]
Ockham
My Dear Babbage. I now write to you expressly on three points; which I have very fully & leisurely considered during the last 18 hours; & think of sufficient importance to induce me to send a servant up so that you may have this letter by half after six this evening. . .
Firstly: the few lines I enclosed you last night about the connexion of (8) with the famous Integral, I by no means intend you to insert, unless you fully approve the doing so.
It is perhaps very dubious whether there is any sufficient pertinence in noticing at all that (8) is an Integral. . .
Secondly: Lord L – suggests my signing the translation & the Notes; by which he means, simply putting at the end of the former: “translated by A. A.L;” & adding to each note the initials A.A.L.
It is not my wish to proclaim who has written it; at the same time that I rather wish to append anything that may tend hereafter to individualise, & identify it, with other productions of the said A. A. L.
My third topic, tho’ my last, is our most anxious & important: –
I have yesterday evening & this morning very amply analysed the question of the number of Variable-Cards, as mentioned in the final Note H (or G?). And I find that you & I between us have made a mess of it; (for which I can perfectly account, in a very natural manner). I enclose what I wish to insert instead of that which is now there. I think the present wrong passage is only about eight or ten lines, & is I believe on the second of the three great sheets which are to follow the Diagram. . .
It does not signify whether the operations be in cycles or not. . . . In Note H, the erroneous lines are founded on the hasty supposition that the cycle, or recurring group, of Operation-Cards (13 . . . .23), will be fed by a cycle, or recurring group, of Variable-Cards.
I enclose what I believe it ought to be.
If already gone to the printer, we must alter that passage in the proofs, unless you could call at the printers & there paste over the amendment. –
I can scarcely describe to you how very ill & harassed I felt yesterday. Pray excuse any abruptness or other unpleasantness of manner, if there were any.
I am breathing well again today, & am much better in all respects; owing to Dr. L’s remedies. He certainly does seem to understand the case, I mean the treatment of it, which is the main thing.
As for the theory of it, he says truly that time & Providence alone can develop that. It is so anomalous an affair altogether. A Singular Function, in very deed!
Think of my having to walk, (or rather run), to the Station, in half an hour last evening; while I suppose you were feasting & flirting in luxury & ease at your dinner. It must be a very pleasant merry sort of thing to have a Fairy in one’s service, mind & limbs! – I envy you! – I, poor little Fairy, can only get dull heavy mortals, to wait on me! –
Ever Yours
A.L.
Poetical Science
Ada asks Babbage questions about cycles, which in her Notes represent one of her greatest contributions. The concept of repeating a cycle saves a programmer time in writing programs. Ada helps us to understand a cycle and a repetitive pattern in Chapter 15 by using a graphic metaphor.
How do designers, architects and artists use repetitive design?
13
What a General I Would Make, An Analyst and a Metaphysician
[1843]
After all her hard work in such a short period of time, Ada made a lot of progress and wrote to William: “Wheatstone has been with me a long while today, & has taken my translation away with him, after reading it over with me.” She continued her letter saying that she hoped to receive the “proofs of it for correction, by & bye as I trust Taylor will not reject it. I am now translating a beautiful Italian scientific paper.” In the evening she went off with William’s sister Hester to see Miss Kemble in the opera Norma. She concluded she did not care about anything unless it was the mathematical and scientific.
To Charles Babbage
Tuesday [Undated]
Ockham
My Dear Babbage. I hope you will approve of what I send. I have taken much pains with it. I have explained that there would be, in this instance & in many others, a recurring group or cycle of Variable as well as of Operation Cards; . . . I think I have done it admirably & diplomatically. (Here comes in the intrigante & the politician!)
Ever yours
A.L.
To Charles Babbage
Wednesday, 5 July [1843]
Ockham Park
My Dear Babbage. I am much obliged by the contents of your letter, in all respects. Should you find it expedient to substitute the amended passage about the Variable-Cards, there is also one other short sentence which must be altered similarly. This sentence precedes the passage I sent yesterday by perhaps half a page or more. . .
“Why does my friend prefer imaginary roots for our friendship?” – Just because she happens to have some of that very imagination which you would deny her to possess; & therefore she enjoys a little play & scope for it now & then. Besides this, I deny the Fairyism to be entirely imaginary; (& it is to the fairy similes that I suppose you allude).
That brain of mine is something more than merely mortal; as time will show; (if only my breathing & some other et-ceteras do not make too rapid a progress towards instead of from mortality). –
Before ten years are over, the Devil’s in it if I have not sucked out some of the life-blood from the mysteries of this universe, in a way that no purely mortal lips or brains could do.
No one knows what almost awful energy & power lie yet undevelopped [sic]in that wiry little system of mine. I say awful, because you may imagine what it might be under certain circumstances.
Lord L, – sometimes says “what a General you would make!” Fancy me in times of social & political trouble, (had worldly power, rule, & ambition been my line, which now it never could be).
A desperate spirit truly; & with a degree of deep & fathomless prudence, which is strangely at variance with the daring & the enterprise of the character, a union that would give me unlimited sway & success, in all probability.
My kingdom however is not to be a temporal one, thank Heaven! – ...
I am doggedly attacking & sifting to the very bottom, all the ways of deducing the Bernoulli Numbers. In the manner I am grappling with this subject, & connecting it with others, I shall be some days upon it.
I shall then take in succession the other subjects that have been suggested to me during my late labours, & treat them similarly. –
Labor ipse voluptas is in very deed my motto! – And, (as I hinted just now), it is perhaps well for the world that my line & ambition is over the spiritual; & that I have not taken it into my head, or lived in times & circumstances calculated to put it int
o my head, to deal with the sword, poison, & intrigue, in the place of x, y, & z. . .
Your Fairy for ever
A.A.L...
To Charles Babbage
Thursday [6 July 1843]
Ockham
My Dear Babbage. I send you the first sheet all corrected. I have taken much pains with it, & I think it very much improved.
The printer has made one or two paragraphs where none ought to exist; & has also not put some words in Italics that ought to be so expressed. I have endeavoured to indicate all this, as I best might.
I send you four Foot-Notes, which are referred to in the proper places. I think you will like them; especially the first one about Pascal’s Machine. – I hope to send you the remainder of the translation tomorrow. – My plans are again all altered. I go on Monday for the day only. Will you come at three o’clock.
I do not suppose that the Notes will take half the corrections which the translation does. I took so much more pains with them. I hope not, for it is damnably troublesome work, & plagues me. Pray let me know if my corrections are intelligible. . .
F Knight reappeared from Holland on Tuesday, so he spent Wednesday Morg with me, & was very delightful. He is anxious about my publication, & only fears my writing anything inexperienced.
I see I am more his ladye-love than ever. He is an excellent creature, & deserves to have a nice ladye-love. –