**Jean-Victor Poncelet**'s father was Claude Poncelet, a rich landowner who was a lawyer at the Parliament of Metz. His mother was Anne-Marie Perrein, but Jean-Victor was an illegitimate child and, although he was born in Metz, he was sent away before he was a year old to be brought up by the Olier family in Saint-Avold, a town to the east of Metz. We should add that much later Claude Poncelet married Anne-Marie Perrein making Jean-Victor legitimate from that time. He was cared for with much love and affection by the Olier family and he lived with them until 1804 when he reached the age of 15. It was a happy time for Poncelet, who showed great curiosity for all things around him, particularly a love of mechanical objects and he spent many happy hours playing with the mechanism of a clock which had been bought for him.

When he was fifteen years old, Poncelet returned to Metz where he studied at the lycée taking the special classes designed to prepare students to take the entrance examinations for the École Normale and the École Polytechnique. He entered the École Polytechnique in 1807, and there he had outstanding teachers such as the mathematicians Gaspard Monge, Lazare Carnot, Charles Brianchon, Sylvestre Lacroix, André-Marie Ampère, Louis Poinsot, and Jean Hachette. However his health was poor and he missed most of his third year of study. He graduated from the École Polytechnique in 1810 at the age of 22, older than was usual due to taking an extra year because of his health problems, and decided on a military career. He joined the Engineering Corps and went to Metz to study at the École d'Application. After two years of study he graduated, having reached the rank of Lieutenant and, in March 1812, was given as a first assignment work on the fortifications of Ramekens on the island of Walcheren in the estuary of the river Scheldt (or Escaut) [21]:-

His first engineering work here was the erection of a casemated fort in a very limited time, on a peat soil, without having at his command proper materials for a foundation.

However he was called away from that assignment in June 1812 to take part in Napoleon's Russian campaign.

Poncelet joined Napoleon's army of 600,000 men at the town of Vitepsk as it was approaching Russia. By 18 August the army was nearing Smolensk, the first genuinely Russian city, and Poncelet reconnoitred the city despite being under fire from the defending garrison. He was actively involved in the fighting later that day, then on the following day he was responsible for constructing bridges over the Dnieper River below Smolensk. Not only did he have to overcome problems of constructing bridges but he had to do so while under fire from Russian guns on the opposite bank. He devised a plan to divert the attention of the Russians to a particular crossing point while he organised building bridges at a different location. It was not until September 1812 that the Russian army fully engaged with the French and were defeated at the Battle of Borodino. Poncelet spent five frustrating weeks with the army in Moscow, then on 19 October Napoleon ordered the army to withdraw. The Russians then attacked the retreating French army and Poncelet was left for dead on the battlefield following the Battle of Krasnoi, not far from Smolensk, on 19 November [21]:-

In this battle, Poncelet charged the Russian batteries at the head of a column of sappers and miners; his horse was killed under him...

He was extremely fortunate to survive but had great hardships to come [6]:-

He was picked up by enemy soldiers only because they thought that being an officer he might be able to give useful information. As a prisoner of war, he was forced to march for nearly five months across frozen plains to his prison[Saratov]on the banks of the Volga. At first he was too exhausted, cold and hungry even to think; but when the spring came("the splendid April sun"), he resolved to utilise his time by recalling all he could of his mathematical education. Later he was to apologise that "deprived of books and comforts of all sorts, distressed above all by the misfortune of my country and my own lot, I was not able to bring these studies to a proper perfection."

He was held in the prison from March 1813 to June 1814 when he returned to France. During his imprisonment he recalled the fundamental principles of geometry but, forgetting the details of what he had learnt from Monge, Carnot and Brianchon, he went on to develop projective properties of conics. He called the notes that he made the 'Saratov notebook,' but it was only fifty years later that he incorporated much of what he had written in his treatise on analytic geometry *Applications d'analyse et de géométrie* (1862). His development of the pole and polar lines associated with conics led to the principle of duality but this, as we explain below, led to a priority dispute. He also discovered circular points at infinity. First we look at what Julian Coolidge writes about Poncelet's inspired work [15]:-

The fundamental problem which Poncelet sets himself is to study the graphical properties of figures which he defines as those which do not involve the magnitude either of distances or of angles. The distance of two points is not projectively invariant, but in looking for projectively invariant configurations he finds the harmonic one, and this he develops at length. ... In his second chapter Poncelet attacks the problem of imaginary points in pure geometry with a courage and thoroughness ahead of anything shown by his predecessors. ... he makes quite casually the historic statement that two coplanar circles should not be looked upon as completely independent figures, but as having two imaginary infinite points in common. Here we have the first announcement of one of the basic principles of metrical geometry. Later Poncelet allows, without careful definition, imaginary projections.

On 30 May 1814 the Treaty of Paris was signed making peace between France and Russia (and the other countries involved in the conflict). A few days later Poncelet was released from Saratov prison but it took him until September of that year before he reached France. From 1815 he taught at Metz. He published *Traité des propriétés projectives des figures* in 1822, which is a study of those properties which remain invariant under projection. Note that the work was subtitled "A work of utility for those studying the applications of descriptive geometry and geometric operations on land" and here one can see the influence of Monge's teaching. This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity. While writing this book he consulted with François Servois who he had known while he worked at Metz but who had moved to Paris in 1816. In *Traité des propriétés projectives des figures*, Poncelet wrote:-

If one figure is derived from another by a continuous change and the latter is as general as the former, then any property of the first figure can be asserted at once for the second figure.

He illustrated this technique by first noting the theorem from Euclidean geometry which states that the product of segments of intersecting chords in a circle is constant. Poncelet then used his principle to show that if the point of intersection is considered to be outside the circle, one obtains the theorem that the product of the secants and their external segments are constant. No proof is required, Poncelet says, for one simply uses the Euclidean theorem and invokes his principle of continuity. It is worth remarking that our term "projective geometry" comes from the title of this book, which is quite appropriate since Poncelet was one of the founders of modern projective geometry simultaneously discovered by Joseph Gergonne. Let us look briefly at Andrei Nikolaevich Kolmogorov's description [4]:-

Poncelet showed that a conic section(conic)is a projective figure and that to solve a difficult problem in conics, one should project the conic, solve the problem for the circle, and then carry out the inverse projection. Since the "points of convergence" of parallel lines on the "mapped plane" do not correspond to real points of the projective plane, Poncelet added "ideal" or "infinitely distant" points to all planes, points that project to "points of convergence." Poncelet introduced infinitely distant points using Carnot's principle of correlation, which he called "the principle of continuity." Developing an idea of Carnot on "complex correlation," Poncelet introduced imaginary points of the plane, and, in particular, imaginary infinitely distant points, such as, for example, "cyclic points" - points belonging to all circles in the plane. Two conics can intersect in four real or imaginary points, and two circles in two real and two cyclic points.

The principle of continuity caused some disputes. In particular Augustin-Louis Cauchy, writing a report on Poncelet's work on 5 June 1820, claimed that the principle of continuity was "capable of leading to manifest errors". He gave an example to show that the principle was false, but his example was not correct. This was not the only dispute that Poncelet was involved in. Articles appearing in Joseph Gergonne's *Annales des Mathématique* which used the principle of duality gave Poncelet little credit. He protested his priority to Gergonne in December 1826 and his comments were published in March 1827 accompanied by critical remarks added by Gergonne. The priority dispute about duality lasted until May 1829 and also involved Julius Plücker. It pushed Poncelet away from his work on projective geometry and towards mechanics.

From 1815 to 1825 he was a Captain of Engineers at Metz, overseeing the construction of machinery in the arsenal at Metz and teaching mechanics in the military college. During this time François Arago urged him to accept the position of Professor of Mechanics at Metz but for a while he hesitated. Finally on 1 May 1824 he agreed, taking up his duties in January 1825. He held this position for ten years. He applied mechanics to improve turbines and waterwheels more than doubling the efficiency of the waterwheel [5]:-

Poncelet was familiar with Borda's work and the necessity for an efficient water wheel of having water enter without velocity and leave without impact. The basic problem he faced in redesigning the undershot wheel was how to accomplish this while retaining the practical advantages of the traditional construction - simplicity, low construction costs, high rotational velocity. Poncelet declared: "After having reflected on this, it seemed to me that we could fulfil this double condition by replacing the straight blades on ordinary wheels with curved or cylindrical blades, presenting their concavity to the current." Thus, in1823, Poncelet took the old undershot wheel and replaced its flat, radial blades with curved blades and angled its sluice gate to bring the water as close to the lower blades as possible. These changes produced a wheel with all of the advantages of the undershot wheel plus a relatively high efficiency.

These ideas were published by Poncelet in 1826 and were awarded a prize by the French government. It is hard for us to understand how important this work was for at this time much of industry was powered by waterwheels. He also collaborated with Arthur Morin on experiments on friction beginning in May 1831. Their work confirmed and extended Coulomb's work on friction, verifying the three general laws he had proposed.

Poncelet was promoted to Chef de Bataillon in 1831, and then moved to Paris in 1834 when he was elected in March of that year to the mechanics section of the Académie des Sciences. His work on projective geometry was too controversial, particularly following the attacks made on it earlier by Cauchy, for him to enter the Academy on the strength of these contributions. In the following year he become Professor of Mechanics at the Sorbonne. He served on the Committee for Fortifications of Paris from 1835 to 1848. In 1842, Poncelet married Louise Palmyre Gaudin and his intention was to have a quieter time, but events conspired to prevent this for several years. In 1841 he became a Lieutenant-Colonel, then three years later became Colonel and, on 19 April 1848, a General of Brigade. He also became director of the École Polytechnique in April 1848, holding the post until 1850. During his time in this role there occurred in Paris the "June Days Uprising" by French workers on 23-26 June 1848. Barricades were set up and the army attacked the workers with a large loss of life. Poncelet, as director of the École Polytechnique, led his students through the barricades to the Luxembourg Palace where they protected the Provisional Government. Louis Eugène Cavaignac, who had become French head of state and led the suppression of the revolt, honoured Poncelet for his support by appointing him to take command of the National Guards of the Department of the Seine. He was later elected to the governing assembly. In 1849 Poncelet and Arthur Morin invented the dynamometer of rotation, which together with later refinements, became the basic investigative tool in the study of work.

In 1851 the Great Exhibition of the Works of Industry of all Nations was held in Hyde Park, London. Poncelet was appointed as head of the Scientific Commission for the Exhibition. On his return he wrote a report on progress in the applications of science in the first half of the nineteenth century making particular mention of the English machinery and tools he had seen exhibited. Prompted by the Great Exhibition of 1851, the French organised the first Universal Exhibition which opened in Paris in May 1855. Poncelet also played an important role in this Exhibition.

Poncelet published many articles on geometry and mechanics in addition to those we have mentioned, particularly in Gergonne's *Annales des Mathématique* and Crelle's Journal. The lectures he gave at Metz were first produced in lithographed form then, after a series of versions, were eventually published. For example the course on Mechanics Applied to Machines appeared first as lithographed notes in 1826, again as a second version in 1832, then a third definitive version with the assistance of his friend Arthur Morin in 1836. The notes were not properly published, however, until 1874. The course notes *Mécanique industrielle*

went through a similar process. He also wrote many reports and memoirs which were published in the Mémorial du Génie and the Avis du Comité des Fortifications. For example we mention *Sustaining Walls; Geometrical Constructions to Determine Their Thickness Under Various Circumstances: Geometrical Constructions to Determine Their Thickness Under Various Circumstances* (1845), and *Memoir Upon the Stability of Revetments and of Their Foundations*. Publications following his retirement in 1850 include *Applications d'analyse et de géométri*e in two volumes: 1862 and 1864. He also published a second edition of *Traité des propriétés projectives des figures* in 1865-66 which was reprinted in 1995. Here is a list of the contents of this second edition:

(1) General principles (consisting of Preliminary notions of central projection; Preliminary notions on secants and ideal chords of conic sections; and Principles related to projection of plane figures).

(2) Fundamental properties of straight lines, circles, and conic sections (consisting of Geometry of ruler and transversals; Figures inscribed in and circumscribed around conic sections. Reciprocal poles and polars; and Similarity and homothety, centre of similarity).

(3) Systems of conic sections (consisting of Homologous figures, center and axis of homology, in particular for conic sections; Complete systems of conic sections; and Double contact of conic sections).

(4) On angles and polygons (this section contains the projective definition of foci of conic sections).

(5) General theory of centers of middle harmonics.

(6) General theory of reciprocal polars.

(7) Analysis of transversals applied to geometric curves and surfaces.

(8) Properties common to systems of geometric curves and surfaces of arbitrary order.

Poncelet received many honours in addition to being elected to the Academy which we mentioned above. He was an officer of the Legion of Honour, and Chevalier of the Prussian Order. Many academies and learned societies elected him to membership including the Royal Society of London, the Berlin Academy of Science, the Imperial Academy of Sciences of St Petersburg and Academy of Sciences in Turin.

After a long and painful illness, Poncelet died in December 1867. In the following year the Prix Poncelet was endowed by his wife in carrying out Poncelet's dying wish that the sciences be advanced. The prize, augmented by a further sum of money, was awarded for work in pure mathematics or mechanics by the Academy of Sciences from 1876. His unpublished manuscripts survived until World War I when they vanished and have not been traced since. Sadly it is highly likely that they were destroyed at this time.

**Article by:** *J J O'Connor* and *E F Robertson*

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