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The French Revolution is usually said to have begun in 1789, when the National Constituent Assembly, a parliament-like body of aristocrats, churchmen and commoners, alarmed by the violent behaviour of the peasantry — a Parisian rabble had already destroyed the symbolic but by then almost empty prison of the Bastille on 14 July 1789 — hoped to appease them by abolishing the feudal regime (4 August 1789) and passing the Declaration of the Rights of Man and of the Citizen (26 August).

Following the example of a celebrated contemporary novelist, whose statue stands at the entrance to the room where our academicians hold their private meetings to glorify, without doubt, a political and religious system from the day before yesterday and which is still fashionable today, I could have entitled this work, which is purely mathematical, . It is, in fact, the fruit of the meditations of a young lieutenant of the engineers, left for dead on the fatal battlefield of Krasnoy, not far from Smolensk, and for a long time strewn with the bodies of the French army. There, in that terrible retreat from Moscow, seven thousand Frenchmen, exhausted by hunger, cold and fatigue, under the orders of the unfortunate Marshal Ney, came, deprived of all artillery, on the 18th of November 1812, the anniversary of the Russian Saint Michael, to fight a furious, bloody and final combat with twenty-five thousand soldiers, fresh and equipped with forty cannons of Field-Marshal Prince Miliradowitch, who himself would soon become the victim of a military conspiracy hatched in the bosom of the modern capital of the Muscovite Tsars. But the adoption of such an ostentatious title, however justifiable it might seem, would seem with good reason to be a ridiculous plagiarism, an overweening imitation with perhaps a permitted licence, of the avowed leader of the romantic novel in our France, at a time of moral perturbation as much political as literary. A similar title, besides, would suggest of this modest book neither the serious and reserved habits of the author, still less the character, the aptitudes, the tastes which presume a sincere love of the truths of geometry, whose profound culture calls for a spirit disengaged from all foreign passion and, one might say, of any earthly interest.

The flavour of this chapter will be very different from the previous two. It is chiefly devoted to giving an account of some theorems which establish that there is a subject worthy of investigation, and which Poncelet was rediscovering. I shall state what they say, and indicate how they might be proved. Then I shall indicate a way of proving them by the tactic of establishing them in a special case (when the argument is easy) and then showing that the general case reduces to this special one. I shall prove them in the special case, and indicate how the reduction from general to special can be carried out. This method of reduction is the key idea in projective geometry, and in that way we shall begin our study of the subject. Towards the end of the section we shall work our way back to Poncelet and see what he required of projective geometry.

The singular novelty of Poncelet’s geometry is that he was seldom interested in the metrical properties of figures. Everything is studied at the level of what properties a figure has in common with its shadows (its projections). This was not the first time such an idea had been presented, but this time the message caught on. Poncelet was quite explicit, and reasonably clear, that there was a class of geometric properties that deserved to be singled out, and these were the projective ones. For example, and this needs to be shown, the property of being pole and polar is a projective property.

We have seen that Poncelet used a way (in a sense, discovered by Brianchon) of pairing each point of the plane with a line, called the polar of the point, and each line in the plane with a point, called its pole. All that was needed for this was a conic. Any conic will do, and if a different conic is chosen the details of which point is paired with which line is changed, but nothing else. We also saw that the process of starting from a point (a pole) and producing a line (its polar) is the inverse of the process of starting from a line (a polar) and producing a point (its pole). We also saw that if three points lie on a line then their polars meet in a point and, conversely, if three lines meet in a point then their poles lie on a line. (See Figure 5.1.)

To hear Poncelet tell it, what he invented was a uniform method of tackling problems in geometry, brought about by the use of ideal and imaginary elements, that reduced calculation to a minimum (and thereby made geometry easier to understand). To hear Cauchy tell it, Poncelet’s methods were at best heuristic, and liable to mislead, and insofar as they led to correct conclusions it would be safer to use algebra (in particular, complex numbers) at least to validate the use of ideal methods if not to replace them entirely.

Euclid’s is a set of books of great antiquity, written around 300 bc. Our knowledge of them derives from a few copies of copies from shortly before 1000 ad — nearer to us in the present day than to Euclid himself. We rely, and have relied for four hundred years, on a succession of editions and commentaries by many people. This transmission with commentaries was reasonably well established by 1600. The best modern edition was published just before 1900. It is the work of a tireless Danish scholar, J. L. Heiberg, and is the basis for the standard English edition, that of Sir T. L. Heath.

Carl Friedrich Gauss (1777–1855) is one of the few truly brilliant mathematicians to deserve the label “genius”. In later life his mother and he used to like to say that he taught himself to read with little instruction beyond learning the individual letters and that he had more or less taught himself arithmetic. These stories have some plausibility, because as an adult Gauss learned several languages, including English and Russian, and was a phenomenal calculator. His prodigious gifts as a child brought him to the attention of Martin Bartels, himself a good mathematician, and through him to the Duke of Brunswick, who was happy to sponsor the child’s education at the impressive local Collegium Carolinum. He amply repaid them for their support, and all his life was ever the loyal subject.

János Bolyai, who was born in Klausenburg, Transylvania, Hungary on 15 December 1802, was educated at home by his father Wolfgang (Farkas). Wolfgang had become the professor of mathematics at the Evangelical Reformed College in Maros-Vàsérhely (now Târgu-Mures, Romania) in April 1804. The college dated back to 1557, the town was a pleasant one in the wine district, and Wolfgang showed himself to be a widely educated man. He had a picture of Gauss on the wall, alongside one of Shakespeare, and one of Schiller. He also wrote plays himself and translated several English and German works into Hungarian.

The new mathematical world of a geometry other than Euclid’s, was the independent discovery of Nicolai Ivanovich Lobachevskii, born in Kasan in Russia in 1792 and his Hungarian contemporary, János Bolyai (1802–1860). Although little understood in their lifetimes, their work eventually helped to overturn almost every belief about the mathematics of space, and to open the way to the modern geometries of Hilbert and Einstein. After his death, Lobachevskii became known as the Copernicus of geometry. On the occasion of his centennial he was taken up by Malevich and the Russian futurists, and if his bicentennial was a quieter affair it was nonetheless worth noting.