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Let us consider a surface in space, given by an equation of the form F (x, y, z) = 0. We shall assume we have a map from a region of the plane, with coordinates (u, v) onto part of the surface, and that we can differentiate this map as often as we like. We assume that at each point of the surface the directions u-increasing and v-increasing are distinct. [ABSTRACT FROM AUTHOR]

It is oddly hard to think of true statements once you allow doubt to play a role. What could another person say to you that you would absolutely have to believe? Statements people make about themselves are surely not acceptable (any follower of detective fiction will know that that is hopeless). The weather? The date? Are you sure you haven't been hoodwinked, kidnapped, drugged? If we decide to cut these speculations short of paranoia and allow a reasonable degree of interaction with the world in good faith, we enter an old and vexed distinction between certain and merely probable knowledge. Merely probable knowledge was once known as scientia, the word from which science derives, and it had an aspect of unreliability about it. Today, statements of science are among the most highly regarded (and may well figure in any answer to the opening question). [ABSTRACT FROM AUTHOR]

The course at Warwick ended with a longer piece of work that students were to do over the Christmas holidays, in which they were asked to look back over the course and tease out the significance of various parts. [ABSTRACT FROM AUTHOR]

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.) [ABSTRACT FROM AUTHOR]

Although we shall not discuss the point in any detail, you might ask if non-Euclidean geometry passes or fails this test: can there be a mechanics in such a space? Can one set up such things as an inverse square law of gravity, or conservation of energy and momentum? Could one recover, doubtless in an altered form, everything that physicists had discovered about Euclidean space if it turned out that space obeyed non-Euclidean geometry? The answer is, as you might suspect, yes, and this was shown by Lipschitz, a follower of Riemann, in the 1870s. [ABSTRACT FROM AUTHOR]

We have seen a number of types of argument used in geometry: algebraic, differential geometric, projective and, especially, axiomatic arguments. An interesting paper published by the American historian and philosopher of mathematics Ernest Nagel, in 1939, made the provocative suggestion that the principle of duality in plane projective geometry was an important stimulus to thinking of geometry in a purely logical way, independent of any appeal to intuition [171]. [ABSTRACT FROM AUTHOR]

From 1870 to 1914, public interest grew in the idea that space might not be Euclidean. This interest was inextricably linked with the idea that space might be four-dimensional, which was also mixed up with the idea that time could be considered as a dimension. All these ideas are separate, as matters of mathematical fact, but they attracted a tremendous amount of interest, not least because of the prospect that one could travel in the fourth dimension, or in time, in ways denied to us hitherto. [ABSTRACT FROM AUTHOR]

Jules Henri Poincaré was born at Nancy, a town in Lorraine in the East of France, on 29 April 1854. His father was professor of medicine at the university there, his mother, a very active and intelligent women, consistently encouraged him intellectually, and his childhood seems to have been very happy, at least until the war intervened. The town was surrendered to the Germans as part of the settlement of the Franco-Prussian war 1870—1871, and Poincaré remembered seeing German troops occupying it. This may have been one reason for his choosing the military school, the École Polytechnique, over the increasingly popular civilian École Normale, when the time came. As a child he did not at first show an exceptional aptitude for mathematics, but towards the end of his school career his brilliance became apparent, and he entered the École Polytechnique at the top of his class. Even then he displayed what were to be life-long characteristics: a capacity to immerse himself completely in abstract thought, seldom bothering to resort to pen and paper, a great clarity of ideas, a dislike for taking notes so that he gave the impression of taking ideas in directly, and a perfect memory for details of all kinds. When asked to solved a problem he could reply, it was said, with the swiftness of an arrow. He had a slight stoop, he could not draw at all, which was a problem more for his examiners than for him, and he was totally incompetent in physical exercises. [ABSTRACT FROM AUTHOR]

David Hilbert, who came to dominate German mathematics between 1890 and the 1920s, and who many would say was the leading mathematician in the world of his generation, was born in Königsberg on 23 January 1862.1 Königsberg was a small town in East Prussia, best known for being the home town of the philosopher Immanuel Kant, and Hilbert's schooling there was unremarkable. He later said that "I did not particularly concern myself with mathematics at school because I knew that I would turn to it later". He then went to the university in Königsberg in 1880. Although small, it had a strong tradition in mathematics and physics, beginning with Carl Jacobi and the physicist Franz Neumann, who pioneered the teaching of experimental physics. A steady stream of mathematicians and scientists came to or graduated from Königsberg: the physicist Gustav Kirchhoff, the geometer Otto Hesse and Heinrich Weber, who held the chair in mathematics at Königsberg from 1875 to 1883 and was succeeded by Ferdinand Lindemann, who had just become famous for his proof that π is transcendental. [ABSTRACT FROM AUTHOR]

When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert's Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert's work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. Recent historians of mathematics have shown that, in Italy at least, Fano's point of view on the nature of geometrical entities had been a generally accepted theory for at least a decade, but it was not in fact axiomatic in Hilbert's manner, and that other Italian mathematicians were, however, ahead of Hilbert in this manner. How did this come about, what did they do, and why did they lose out? [ABSTRACT FROM AUTHOR]

By the 1870s, geometry was beginning to be a well-understood subject, with clear research programmes. The study of algebraic curves could be undertaken in several ways, projective geometry was acquiring its status as a fundamental branch of geometry, devoted to the most basic properties of figures (those deeper than the metrical ones), non-Euclidean geometry was acceptable to, and accepted by, most mathematicians (but probably not philosophers). This is indeed the perception of most mathematicians at the time, and most historians of mathematics since, but, as we shall see later, there is another aspect to the story too. But first let us document that projective geometry had become central to many mathematicians perception of geometry. [ABSTRACT FROM AUTHOR]

What Beltrami did in his Saggio, as the extracts given in Chapter 19 will have suggested, was to take the usual metric for a map of the sphere on the plane and modify it so that it was defined only within the unit disc, but the space mapped onto the interior of that disc had the following properties. [ABSTRACT FROM AUTHOR]

Work in the history of mathematics can, and often should, involve understanding some quite hard mathematics, as it does in questions like these. [ABSTRACT FROM AUTHOR]

Riemann was the archetype of the shy mathematician, not much drawn to topics other than mathematics, physics and philosophy, devout in his religion, conventional in his tastes, close to his family and awkward outside them.1 As a child, he was taught by his father, a pastor, and then for some years at school before going to Göttingen University. There he had initially intended to study theology, in accordance with his father's wishes — Göttingen was the only university in Riemann's native Hanover with strong links to the Hanover church — but his remarkable ability at mathematics led him to switch subjects. He was always inclined to the conceptual side of things, rather than the computational or algorithmic. His written German is scholarly and old-fashioned — Victorian, one might say — and his Latin (required for academic purposes) is no easier. [ABSTRACT FROM AUTHOR]

In all this welter of original work on the geometry of curves, one matter has remained stubbornly unclear, although to a modern mathematician the need for it is painfully apparent. Is the subject the algebraic geometry of real curves in the real projective plane, or has everything migrated to the complex projective plane? In a sense, the question is anachronistic. It does not arise for Poncelet because, as we saw, he had his own way of talking about imaginary points and resisted very strongly Cauchy's suggestion of making everything algebraic. But Plücker and Hesse were avowedly algebraic, yet they do not seem to have openly confronted the issue. At one stage, Plücker even outlined a way in which complex coordinates could be written out of the theory in favour of certain symmetry consideration (a topic there is not room to explore here). Generally speaking, he and Hesse seem to have pursued a policy of quiet acceptance: intersection points of one curve with another, tangents to a curve, and all manner of objects may be complex if the algebra forces them to be so — but one will not explore too closely. This is a curious position. A cubic curve, let us say, was thought of as a real object in the real plane with nine inflection points, six of which at least were necessarily imaginary. A cubic and a quartic curve meet in 12 points, of which in any given case quite a number might be complex. But a curve was not made up of complex points — points with complex coordinates — or if it was one did not enquire too closely how this could be so. [ABSTRACT FROM AUTHOR]

We are interested in the local behaviour of a curve, that is to say what small pieces of them look like. Either they will have one branch passing through a point, or more than one. If only one, then the curve will have a tangent there. To study the nature of a curve near a particular point we may always suppose that we have moved the point to the origin. [ABSTRACT FROM AUTHOR]

This chapter provides a summary of the basic facts about what are called singular points on a curve, and an account of how they resolve the duality paradox. The next chapter gives the mathematical theory behind these facts. [ABSTRACT FROM AUTHOR]

An algebraic plane curve is, by definition, the locus in the plane corresponding to a polynomial equation of some degree, n: f (x,y) = 0. For example x3y + y3 + x = 0 represents a quartic. The equation may be written in homogeneous coordinates [x,y,z] by setting x = x′/z′, y = y′/z′ and multiplying through by the lowest power of z′ that produces a polynomial equation (if you wish you may then remove the primes) when the curve is considered to lie in the projective plane. The example above becomes x3y + y3z + z3x = 0 in homogeneous form. [ABSTRACT FROM AUTHOR]

In 1827 Möbius published his Der barycentrische Calcul [162] or The Barycentric Calculus.1 The word "barycentre" means centre of gravity, but the book is not about mechanics but geometry. It is most concerned with, and is best remembered for, introducing a new system of coordinates, the barycentric coordinates. [ABSTRACT FROM AUTHOR]

This is an unusual chapter, one of three, that offers advice on how to tackle a piece of assessed work.1 An example of suitable assessment questions will be found at the end of this chapter, along with yet more advice. [ABSTRACT FROM AUTHOR]

An interesting surface was discussed by H. F. Minding in the 1830s in papers he published in Crelle's Journal [159] although these papers were then largely forgotten for 30 years; this was a surface of constant negative curvature with the property that geodesics between points are not necessarily unique. Minding's surface is formed by rotating a tractrix about its vertical axis. A tractrix — the name is due to Huyghens — is the curve of the obstinate dog. A point Q on a line ℓ is attached to a point P by a curve of fixed length. P, the dog, is dragged behind Q, the owner, who walks along ℓ, and the path of P is called the tractrix. It is conventional for the walk to start with PQ perpendicular to ℓ. Minding's surface, which is sometimes called the pseudosphere, is formed by rotating the tractrix about ℓ; the point P then generates a circle of singular points (see Figure 11.1). [ABSTRACT FROM AUTHOR]

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.1 [ABSTRACT FROM AUTHOR]

János Bolyai, who was born in Klausenburg, Transylvania, Hungary1 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.2 [ABSTRACT FROM AUTHOR]

Carl Friedrich Gauss (1777-1855) is one of the few truly brilliant mathematicians to deserve the label "genius".1 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. [ABSTRACT FROM AUTHOR]

Euclid's Elements 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.1 [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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, Memoirs from beyond the tomb. 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Such have been the transformations in geometry between the 1840s, when we last took stock, and 1900 that it seems advisable to pass it briefly under review before moving to some final considerations. In Chapter 13, we looked at Möbius's algebraic version of projective geometry and his introduction of barycentric coordinates. All of projective geometry was thereby turned into algebra, including the line at infinity in the projective plane, and projective transformations. [ABSTRACT FROM AUTHOR]