Chasles' Bad Relations

Chasles' bad relations

Christophe Ritzenthaler September 10, 2004

Michel Chasles (1793-1880)

1 Introduction

We give here a very atypical orientation to Chasles' life. In particular we will not deal with the ordinary aspects of his life and work. This can be found in many places (in particular [1]). We mention only that, as a mathematician, he synthesized and generalized the whole geometric knowledge of his time (projective geometry, conic sections, duality and homography) - even if sometimes, (as he didn't speak German), he merely rediscovered some of these results - ([3]). Also, as a historian of mathematics, he wrote a history of geometry ([5] and two controversial essays ( [2],[4]). The following list of honors should convince us of the seriousness of this character :

Membre de l'Institut, professeur de Géométrie supérieure à la faculté des Sciences de Paris, membre de la Société royale de Londres, membre honoraire de la Société royale d'Irlande, de la Société philosophique de Cambridge, de l'Académie impériale des Sciences de Saint-Pétersbourg, associé de l'Académie royale des Sciences de Brux- elles,correspondant de l'académie ponticale des Nuovi Lincei de Rome, membre de lAcadémie royale de Berlin, de l'Académie royale des Sciences de Turin, de l'Académie royale des Sciences de Naples, de l'Académie de Lincei de

Rome, de la Société royale danoise des Sciences de Copenhague, de l'Académie royale des Sciences de Stockholm, de l'Académie des Sciences de l'Institut de Bologne, de l'Institut roual lombard de Milan, de la Société italienne des Sciences de Modène, correspondant de l'Académie royale des Sciences de Madrid, de l'Institut vénitien des

Sciences, Lettres et Arts, membre honoraire de l'Académie royale des Sciences, Lettres et Arts de Modène, de l'Athénée vénitien des Sciences et Lettres, de l'Université d'Odessa, de l'Académie américaine des Sciences et

Arts de Boston, de l'Académie nationale des sciences d'Amérique.

Our purpose is not to give such a history. We will try to show how Chasles was twice a victim : rst someone stole his money and his reputation then someone `stole' his name.

1 2 The perfect victim

Firstly we will see that Chasles was the archetypical victim. He was born into an upper-middle-class Catholic family. Some anecdotes tell us that when he was young he was already a good fellow : at the Ecole Polytechnique he was accepted into the engineering corps, but he gave up the appointment in favour of a poor student. Good fellow indeed, but also a good son, for after this event he obeyed his father's wishes and entered a stock brokerage rm in Paris. A good citizen: his name when he was young, after the revolution, was Floreal and he took an active part in the defense of Paris in 1814. A good human being, at last, since his only activities outside mathematics seem to have been involvement with charitable organizations. A good husband also? Well, he was not married. But those who see in him now as a quasi-monastic character who gave his soul to Science and Fatherland, might be fooled : one source seems instead to show that his career as a trader stopped and his geometer's life started, because he was ruined by the mundane pleasures of life. Let us keep this in mind for the next story.

3 Case 1 : Chasles vs Vrain-Lucas

Chasles was a collector of autographs and manuscripts like Denis Vrain- Lucas, who was a law clerk, but is most famous as the cleverest and most prolic of historical literary forgers. From 1861 to 1869, Chasles bought him 27,000 autographed forgeries for about 140 000 francs (420 000 euros) : letters from Charles Quint to Rabelais, Cleopatra, Jeanne d'Arc, Vercinge- torix, from Alexander the Great to Aristotle and even from Mary Magdalen to Lazarus! The greatest comedy is that all these letters were written in French [7] :

Lucas also made crude approximations of Carolingian script and archaic or- thography, but his texts are all essentially in modern French.

So may we suppose that Chasles was senile at that time (remember that he was 68) [7]?

If this is so then he had not lost the ability to produce mathematics of the highest quality since his solution of the problem to determine the number of conics tangent to ve given conics which he found in 1864 was remarkable, particularly as it corrected a previous incorrect solution by the outstanding

2 mathematician Steiner.

To nd a plausible explanation one has to point out that the rst letters Chasles bought were about a correspondence between Isaac Newton, Blaise Pascal, and Robert Boyle which established that Pascal had anticipated New- ton in the discovery of the law of universal gravitation. Chasles presented these letters to the Academy in 1867 and took an active part in the furor that ensued, vigorously defending the genuineness of the letters. However in 1869 Vrain-Lucas was brought to trial and convinced. So Chasles was maybe so patriotic (we have seen that he was a loyal citizen) that his desire for the letters to be genuine overcame his common sense and he was proud to show that France beat (again) England. But we must believe that he was really excited since Pascal wrote to Newton (12 years old !) :

My young friend, I send you dierent problems...in order to exercise your genius.

However exaltation is among the features we could ascribe to our character : a lonely and fervent mathematician, a believer who liked beautiful stories. Remember he argued for a Pythagorean rather than a Hindu origin for our numerical system [2] and reconstructed the lost book of Porisms of Euclid [4].

In spite of that, some have suggested then that Chasles was indeed an ac- complice of Vrain-Lucas. One reason is that Vrain-Lucas seemed to have had little opportunity to spend the huge amount of money he made (but he claimed he did). If we suppose that Chasles didn't pay him, he was still naive to believe that the Academy would accept the unsophisticated forgeries as genuine. If not, we have to suppose that Chasles was naive and a crook, however nothing in his life would suggest that he was anything other than the most honest of men.

The question might still be open, however there is a question nobody has asked yet : where can a mathematician nd 420 000 euros ?

4 Case 2 : Chasles vs the parallelogram

Unless a French pupil goes on by studing sciences, he will not hear of a lot of famous mathematicians in his life, perhaps only of Pythagoras, Thales, Newton, Pascal and...Chasles. Yes, Chasles is quite special since he is one of the two French mathematicians that almost everybody has heard about.

3 When does this special event happen in our scholarship ? It is in the nal year before high school (`classe de Troisième'), and Chasles' name is associated to vectors through the famous relation called `relation de Chasles' : −→ −−→ −→ AB + BC = AC. A problem occurs at once : after contemplating the formula, we feel that its emptiness. Indeed it is more an axiom in the denition of ane space than a property :

Denition 1. [8] Let E be a vector space. One says that a set X is an ane space with direction E if there is a map from X × X into E denoted (x, y) 7→ xy−→ such that • For all x, y, z we have xy−→ + yz−→ = xz−→, −→ • For all x, θx : y 7→ xy is a bijection from X onto E. Of course we have to understand it in the context of primary school. There it is seen as a property of natural objects called vectors and repre- sented by couple of points in the plane.

But then we have other clues that something is amiss:

• Except French people, nobody calls this relation Chasles' relation but instead it is called the parallelogram relation.

• Even some French people take precautions when naming it : for ex- ample in the preceding denition [8] says : `relation dite de Chasles' (relation called after `Chasles'). Indeed, Chasles did not nd this relation, since he apparently did not know the vector notion ! It is Sir William R. Hamilton (1805-1865) who was one of the rst to use vectors and probably invented the word (from the Latin vehere, to carry). The German, Grassman (1809-1877) used this notion in physics problems and the American, Gibbs (1839-1903) and the Briton, Heaviside (1850-1925) put vector theory on a quasi-denitive footing. How- ever this notion took time to be introduced in France and even in the work of Hadamard who followed Chasles who wrote the standard reference, until 1947, for French geometry, leçons de géométrie élémentaires, there is no talk of vectors.

So, why Chasles ? Chasles found something close, that can be seen as the beginnings of vectors. He introduced the notion of oriented segments -

4 and oriented angles - (for the complete story, one may read his introduction in [3]) : [3, p. 1]

` Quand nous aurons à considérer sur une même droite plusieurs segments, nous indiquerons leur direction en regardant comme positifs tous ceux qui seront dirigés dans un même sens convenu, à partir de leurs origines, et comme négatifs tous ceux qui seront dirigés dans le sens contraire, c'est-à- dire que nous donnerons, dans le calcul, le signe + aux premiers etr le signe − aux autres.' and emphazised its importance [3, p. XII] :

`Le principe des signes étend même son inuence sur l'énoncé des propo- sitions ; car, lorsqu'elles ne se compliquent pas de conditions de situation, elles prennent une forme à la fois plus générale et plus concise, qui présente à l'esprit une idée plus nette et se prête mieux au raisonnement. On a donc beaucoup perdu à ne pas introduire systématiquement dans la Géométrie pure le principe des signes; le progrès de la science en ont été nécessairement re- tardés.'

This notion is in fact a clarication of the one used as a principle in Carnot's work Géométrie de position. Before this notion, the general practise was to give a proposition as many proofs and a problem as many pictures, as there are congurations. So it was indeed a huge step forward and Chasles could write with pride the `Théorème fondamental' :

Theorem 1. [3, p. 2] Let a, b, c be three points on the same line, then the sum of the three segments ab, bc, ca is always zero, which we would write nowadays as

ab + bc = ac, and which is, for most authors, the real Chasles' relation [6].

We may wonder now how the name of Chasles was eventually associated to vectors. One may suppose of course that some professor wanted to honour Chasles and slightly exaggerated the claim relating to this theorem. Usage made the rest. One might also see this, in a more polemic approach, as a consequence of French chauvinism : we wanted a French name associated to the important notion of vectors and Chasles was the perfect candidate for this job. From this point of view, Chasles was again the victim of patriotism.

5 References

[1] J. Bertrand : Eloges académiques, Paris 1902, 27-58.

[2] M. Chasles : L'histoire d'arithmétique, Paris 1843.

[3] M. Chasles : Traité de géométrie supérieure, Paris 1852.

[4] M. Chasles : Les trois livres de Porismes d'Euclide, Paris 1860.

[5] M. Chasles : Aperçu historique sur l'origine et le développement des méthodes en géométrie, particulièrement de celles qui se rapportent à la géométrie moderne, suivi d'un mémoire de géométrie sur deux principes généraux de la science, la dualité et l'homographie, Paris 1875.

[6] A.B. Ivanov : Chasles theorem, Encyclopedia of Mathematics, an up- dated and annoted translation of the Soviet Mathematical Encyclopedia, Klewer Academics Publishers, 1988.

[7] J. J. O'Connor & E. F. Robertson : The mathematician and the forger, available on http://www-gap.dcs.st-and.ac.uk/~history/ HistTopics/Forgery_1.html.

[8] C. Tisseron : géométries ane, projective et euclidienne, Actualités sci- entiques et industrielles 1408, formation des enseignants et formation continue, Hermann 1983.