Enough concerning my plans for tomorrow – I want to tell you everything we saw and did today. We began quite early by leaving the hotel to have our breakfast outside; café crème and croissants, namely, at a lovely sunny terrasse de café. If only Annabel would speak for us, all this would have been easy enough, but she would not, and obliged us all most severely to order for ourselves in French, saying that all our studies, and the many lessons she has given me, ought not to be wasted. I tried my best to overcome my shyness and remember much that I had carefully learnt, and was quite pleased at the unexpectedly reasonable result. As for Arthur and Charles, they are unashamedly British. In fact, it makes very little difference whether the words they speak are French or English really, as they sound exactly the same in both languages. The garçon de café looked down his nose upon hearing them, and pretended not to understand, while simultaneously leering in Annabel’s and my direction.
‘Bother the “confident and over-lusty French”,’ observed Arthur coldly, unconsciously touching his pocket.
‘Hm,’ I said as I made out the oblong flat shape within it, ‘I really don’t think you should be using Henry the Fifth as your guidebook to France. Is that what you’ve been doing? I ought to confiscate it! You can’t possibly learn to love the country with that as your inspiration!’
After this breakfast, we walked about the Louvre palace and the Tuileries gardens, and then purchased and consulted a map which we used to find our way to our prearranged meeting place with Mr Korneck.
‘I tried to persuade him to meet us somewhere else,’ said Charles with a guilty smile, ‘but he would insist on meeting us in front of the Academy of Sciences. It’s the epicentre of Paris as far as he’s concerned, and he clearly can’t think of anything more exciting than having a look around it. Girls, I promise you we won’t talk shop – will we, Arthur?’
‘No, no, we shall resist at all costs,’ he laughed. ‘I must admit that I’m longing to see the place myself, though, even just for a moment. It will make me feel less of a complete stranger here, to glimpse the place where so much of the history of mathematics has been made, and where we ourselves will spend most of our time for the next two weeks.’
After consulting our map, we crossed a delicate bridge arching over the Seine – a glorious great mass of water running between banks of royal stone, plied by large flat boats going seriously about their business; a true city river, so different from our secretive, lovely Cam, dotted with little pleasure crafts as it winds amongst the green fields and hedgerows. The famous Academy was not far; it lies on the southern bank of the Seine, which they call the left bank, on the Quai de Conti. We had not yet arrived at the main entrance before we perceived our portly friend, puffing and looking about him with great impatience.
‘Ah, what a pleasure, what a pleasure,’ he said, shaking hands with an irrepressible air of pride and proprietorship. ‘It is such a lovely day, we shall walk about Paris and see many sights. I promise these young ladies that we shall not dally too long within these illustrious walls, but let us walk inside briefly and see the main hall.’
We entered. The interior was cool and dim. Few people were present, a lone figure here and there crossed the hallway, loaded with books. Mr Korneck led us to an imposing round room with a podium and seats all around it.
‘Here is where the meetings and announcements of the Members take place,’ he pronounced respectfully. ‘The history of this room, with all the events of importance that occurred here, is an astonishing one.’
‘Has Fermat’s mysterious theorem ever been discussed here?’ I asked politely, seeing that he was longing to recount what he knew, but hesitating at the idea of boring us to tears with his pet topic.
‘Indeed it has!’ he replied with an air of delight. ‘The terrible intellectual duel between Gabriel Lamé and Augustin Cauchy took place, week after week, in this very room!’ He took a deep breath, assumed a special, dramatic expression, lowered his voice, and began to speak, stopping now and then to search for sufficiently impressive words.
‘It was more than forty years ago, and at that time, my dear ladies, you should be aware that mathematics was a subject that was discussed and debated in the most elegant salons of Paris, and books were written expressly for young ladies to learn about it, so as not to appear ignorant when the theme arose naturally in conversation. The work of Sophie Germain had appeared so promising to the Academy that they established a glorious prize, to be awarded to anyone who could finally solve the mysterious theorem, and many of the most renowned mathematicians of the day were struggling for it, and dropping hints about their progress. And one fine day in the year 1847, during the regular Academy meeting, Gabriel Lamé arose and announced that he had solved the problem! At that time Lamé was one of the most illustrious mathematicians in all of France, and yet, it was an extraordinary thing for him to be interested in Fermat’s theorem, for he was really more of a physicist than a mathematician, and deeply involved in designing and building the railroads that criss-cross France today. But his researches in physics had led him, a few years before, to study Fermat’s beautiful equation xn + yn = zn when n is equal to the number 7, and he had succeeded in discovering a brilliant proof of the expected result that no solution to this equation can exist. This result had led him to new ideas, and he believed that he had solved the entire problem once and for all, and would soon become the winner of the newly established prize! He had not yet written his proof completely, but he was in the process of doing it, and expected to be finished within a few short weeks.
‘No sooner had he finished making his announcement to the assembled company, who were stunned by the magnitude of the news, than another mathematician arose and pushed his way forward to the podium. It was Augustin-Louis Cauchy, devoted Catholic, ardent Royalist, unpleasant personage (if I may say so), but one of the most prolific mathematicians this country has ever known. Cauchy was a dangerous character; his understanding of mathematics was so gigantically vast, his ideas were so astoundingly varied and prolific, and his speed and eagerness so immense that he easily crushed any smaller or less influential person that crossed his path, and very possibly he never even noticed it. It had happened before and would happen again, and on this day, he could not endure Lamé’s declaration, and the astonished and admiring faces of the audience, and their murmurs of approval. No sooner had Lamé returned to his seat, than Cauchy announced that he, too, had solved the problem, and that he, too, would have written down a full proof within the next few weeks. Look!’
Taking down one of the fat volumes of Proceedings of the Academy which lined one of the walls, he turned the pages and showed us a passage that I must admit I found more amusing and revealing than I would have thought possible for a dry scientific tome.
‘You see – it was the first of March, 1847,’ he said, showing us the date, ‘and here is the report of Lamé in which he sketched out his proof. And Cauchy could not bear to stand by and hear that! He could never endure someone else’s making a discovery on anything to which he had already bent his fertile mind.
A la suite de la lecture faite par M. Lamé, M. CAUCHY prend aussi la parole et rappelle un Mémoire qu’il a présenté à l’Académie dans une précédente séance (19 octobre 1846), et qui a été paraphé, à cette époque. Dans ce Mémoire, M. Cauchy exposait une méthode et des formules qui étaient, en partie, relatives à la théorie des nombres, et qui lui avaient semblé pouvoir conduire à la démonstration du dernier théorème de Fermat. Détourné par d’autres travaux, M. Cauchy n’a pas eu le temps de s’assurer si cette conjecture était fondée. D’ailleurs, la méthode dont il s’agit était très-différente de celle que M. Lamé paraît avoir suivie, et pourra devenir l’objet d’un nouvel article.
‘Monsieur Cauchy says that he had deposited a memoir – when was it? half a year earlier – in which he gave a method that he thought could also lead to a proof of Fermat,’ translated Arthur. ‘He had turned to other work and hadn’t had time to check it
, but his method was very different to Monsieur Lamé’s, and would be the subject of a new article.’
‘Doesn’t he sound jealous!’ exclaimed Charles. ‘Just in the way it’s written, you can almost imagine the stenographer thinking so.’
‘He was jealous,’ agreed Mr Korneck. ‘After this, a race began between the two men. Lamé deposited a sealed envelope at the Academy, in case there should be some dispute over priority later on, and on the very same day, Cauchy did the same! Lamé published some small portions of his proof in these Comptes-Rendus, and Cauchy did the same. Each of the two claimed to be smoothing out the final details of their proofs. They continued to publish at regular intervals for two months – and then came the catastrophe, in the form of a letter from Germany! Ernst Kummer of Berlin had been reading the rival articles by Cauchy and Lamé as they appeared, and he soon recognised that both mathematicians had fallen into a trap that was perfectly familiar to him, for he had fallen into it himself not long before, but had subsequently realised the error of the method. When he read the announcements of the proofs, which I just showed you, he had his suspicions, and when he saw their subsequent articles, he became certain. He wrote a letter to Liouville, to be read out in front of the whole of the Academy, in which he detailed what he believed was their error. Kummer showed that there were two kinds of prime numbers, those called regular and those called irregular. The proofs Cauchy and Lamé were developing could only be applied to the regular primes – but Kummer himself had already understood how to deal with these many months earlier! As for the obstacle of the irregular primes, he felt that the methods proposed in the Academy publications could not succeed in vanquishing it.
‘Cauchy could not endure to read Kummer’s work – he reacted exactly as he had reacted to Lamé’s announcement. Look here, what he wrote in May 1847,’ and he turned several dozen pages in the same volume.
Dans la dernière séance, M. Liouville a parlé de travaux de M. Kummer, relatifs aux polynômes complexes. Le peu qu’il en a dit me persuade que les conclusions auxquelles M. Kummer est arrivé sont, au moins en partie, celles auxquelles je me trouve conduit moi-même par les considérations précédentes. Si M. Kummer a fait faire à la question quelques pas de plus, si même il était parvenu à lever tous les obstacles, j’applaudirais le premier au succès de ses efforts; car ce que nous devons surtout désirer, c’est que les travaux de tous les amis de la sciences concourent à faire connaître et à propager la vérité.
‘What a hypocrite!’ said Annabel. ‘Just look – he says that Kummer appears to have found results which he himself had already discovered – but that if Kummer had done more than he had, then he would be the first to applaud! Why, he can’t have been a good mathematician, can he? He sounds like he needed to attribute everybody else’s work to himself!’
We all smiled.
‘Yet he was very good, more than good; one of the great geniuses,’ said Mr Korneck. ‘It proves something, does it not? A man’s satisfaction does not depend upon his abilities or accomplishments.’
‘So how did the story end?’ I wondered.
‘After Kummer’s letter was read, Lamé understood instantly that he had made an error. Cauchy continued stubbornly for several more months to insist on his success, publishing further morsels, but finally, under the guise of new interests, and without ever making a public retraction, he turned to other topics.
‘This story represents the final flare of the history of Fermat’s last theorem. Since Kummer’s devastating result, no one has dared make another attempt at solving it.’
Mr Korneck fell silent and looked at us expectantly.
‘It reminds me of the three-body problem,’ I said doubtfully. ‘These mathematical competitions really seem to do more harm than good. Did anybody ever win the prize?’
‘Ten years later, Cauchy recommended that it be attributed to Kummer himself, for his remarkable discovery,’ he responded. ‘It was the closest he ever came to admitting he had made a mistake.’
‘So Kummer won the prize for proving that it couldn’t be done – more like the three-body problem than ever!’
‘No, no, do not say that,’ he answered, his heavy-featured face animated with passion. ‘Kummer did not show that it could not be done, on the contrary; he showed that it could be done for regular primes, and developed a new and extraordinary notion in mathematics, even while explaining that the road adopted by Cauchy and Lamé was a bad one for the irregular primes. But it can be done in general, I am convinced of it – it can be done, and Fermat surely did it! My dream is to rediscover what he did. And I hope it is more than a dream.’
I glanced at Charles and Arthur, but they wore studied looks of expressionlessness. We followed the hallway to the main door and emerged once again into the glare of sunlight beating down on the stone buildings.
‘Enough of mathematics for today,’ said Mr Korneck kindly, but with a trace of wistfulness, ‘let us now follow the traces of kings and queens. I shall take you to visit the Louvre.’ And we spent the remainder of the afternoon and evening exploring and admiring many astonishing sights, with oases during which he guided us to his favourite restaurants, in which we tasted various dishes interestingly smothered in sauce. In the evening he took his leave of us (to return, no doubt, to a place of superior comfort and beauty), and we wandered back together along the quays, the river twinkling and purling below us, reflecting back the many lights which shone upon its wrinkled surface. Arthur and I soon fell behind the others, to offer ourselves the dreamy pleasure of walking hand in hand in the fresh evening air, and for a brief moment, I thought I was in heaven.
It has been a magical day. But tomorrow I must return to my business, and force myself to recall things which would be, perhaps, better forgotten …
Your loving sister,
Vanessa
Paris, Wednesday, July 6th, 1892
My dearest Dora,
I have been very active, and I shall take advantage of writing to you to give a complete description of all that I have seen and done. To begin with, the day before yesterday, I left a small note with the concierge of Mrs Clemming’s luxurious immeuble, whose address I was able to locate in the post office (there was no possibility of error, as there are no other Clemmings in Paris). You must imagine a solid and imposing, but most refined building made of heavy stone blocks, pierre de taille as they are called, with a giant wrought-iron and glass door leading into an imposing tiled hallway, and a wide crimson-carpeted staircase leading upwards. The only room opening into this entrance belongs to the concierge, who peers out a little window to examine all comers and goers. She accepted my note together with the gift of a coin and grumbled something in French which I did not understand, but optimistically took to mean that the note would be duly delivered to ‘Madame Clemming, la dame anglaise’ as I assiduously explained.
It must have succeeded, for the very next day I received a reply, in the form of a card printed with her name and the legend At Home Wednesdays 4:30 in both French and English; underneath, she had written by hand ‘Please do come and visit’.
Although I had mentioned my travelling comrades in my note to her, the card was addressed to me alone, so that I had not the reassurance of bringing them with me. However, I imagined that this British lady would have many English speakers amongst her guests, and this, together with the idea that it was extremely likely that they should have encountered Sylvia during her winter visit, and could perhaps even share a great deal of information on her score, sufficed to put me into a state of great excitement. By three o’clock, I was already hovering over the clothes laid out upon my bed, hesitating because nothing seemed quite elegant enough for the occasion. Annabel watched me a little wistfully.
‘I do wish you were coming with me!’ I exclaimed warmly.
‘Oh, it doesn’t matter,’ she said, smiling. ‘I won’t miss it. I am not so used to social occasions. After all, I am merely a governess.’
I glanced up in surprise. I h
ad never thought that Annabel felt resentment about her place in society. On the contrary, since I had met her four years ago, I had frequently heard her express the sincerest gratitude towards Mrs Burke-Jones for having offered to a penniless orphan whose only accomplishment was an excellent education in a French convent, a position which in spite of being nominally inferior had all the charms of an authentic family life. As far as I could observe, Annabel had been treated with kindness and respect; on occasion, she was even invited to join Mrs Burke-Jones’ guests for supper, to even out the numbers. It is true, I suppose, that there is no real equality in all of this, but I had never thought that Annabel felt it as a sting.
‘You were a governess,’ I observed, ‘now you are a schoolteacher, exactly as I am. I really see no difference, apart from the fact that you cannot claim the acquaintance of Mrs Bryce-Fortescue. But that means nothing.’
‘Oh,’ she answered, jumping up vigorously. ‘I really didn’t mean to hint that I wasn’t invited because of my station; I just meant that I was used to missing out on social things because of it. But I don’t care about them – not at all!’
‘Well,’ I said stoutly, ‘as far as I am concerned, I’m proud of working for my living, and I love the work, and I’m proud of all the ideas I’ve had for it over these last years, and how well it is turning out. And so should you be! And to the D with high society!’
‘Oh, I know you’re right,’ she sighed.
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