**Charles René Reyneau**'s father was a surgeon. He studied at the Oratorian College in Angers. This college was run by the Congregation of the Oratory of Jesus and Mary Immaculate, also called the Bérulliens, which was founded by Pierre de Bérulle in 1611. Its chief aim was, and still is, training candidates for the priesthood but it also aimed to provide an education to young people. De Bérulle was a friend of Descartes and when Reyneau studied at the Oratory its teaching were strongly based on Descartes' philosophy. On 17 October 1676 Reyneau entered the Maison d'Institution of the Congregation of the Oratory in Paris where he met Nicolas Malebranche and Jean Prestet (1642-1691), who had just published his

*Élements des mathématiques*Ⓣ, the first edition appearing in 1675. This book was largely an algebra text and was adopted by the Congregation of the Oratory for teaching in their Colleges. Reyneau was sent to teach philosophy at Pezenas, then, in 1679, he moved to the Collège de Toulon and, in 1681, he was ordained a priest there. At this College he was required to teach some geometry and so he studied the subject quickly falling in love with it.

In 1682 Reyneau was appointed professor of mathematics at the University of Angers, replacing Jean Prestet. However, in 1705, he had to give up teaching as be became deaf. He had been going deaf for a number of years but he managed to keep his post by having former students give his lectures for him. After giving up the struggle to continue his job in these difficult circumstances, Reyneau went to Paris and lived at the Oratorian house on rue Saint-Honorè for the rest of his life.

This was a time when there were major new mathematical ideas coming through the work of Johann Bernoulli and being brought into France through Guillaume de l'Hôpital and others. For many years Reyneau was not really abreast of these new developments, even when Johann Bernoulli visited Paris in 1692, and Reyneau did not rush to keep up to date with the important new ideas. Malebranche asked Reyneau to undertake some editorial duties in 1694 but then, in 1698, he persuaded Reyneau to write a new textbook to provide instruction in the new mathematics. Jean-Pierre Lubet writes [10]:-

Let us note that the two colleagues who helped him most were fellow Oratorians Louis Byzance and Claude Jaquemet.Until at least1698, Reyneau encountered difficulties in assimilating the new calculus. But his experience made him decide to work on elements that can replace the books of Prestet, the work to be implemented must also include an introduction to differential and integral calculus. Reyneau did not work alone. Pierre Costabel shows him belonging to a network whose documentary activity is particularly intense at times. While still living in Angers, he received from the librarian of the Oratory articles of the 'Acta eruditorum', signed notably by Leibniz and Jacques Bernoulli. The course that Johann Bernoulli gave to de l'Hôpital in1692gave rise to one or more copies, it served as a support for the personal works that Malebranche carried out to assimilate the methods of differential and integral calculus, but Reyneau also had this course in his hands. He also benefited from the help of some of his colleagues.

While he was still teaching at the University of Angers, Reyneau visited Paris in 1700 and spent from July to August there learning more mathematics from Pierre Varignon, someone who he would naturally get to know since Varignon was a leading member of the group around Malebranche. He made notes of what he learnt from Varignon and we can see from these the questions that were being discussed by mathematicians at this time. For example he writes:-

Reyneau struggled to assimilate the differential and integral calculus participating in debates provoked by Michel Rolle on these topics. In fact he kept a diary of the Varignon-Rolle dispute from 1701 to 1704 most of which was only recorded in the unpublished proceedings of the Paris Academy of Sciences. In 1705 Reyneau received Louis Byzance's papers and among them was a copy of Johann Bernoulli's "Leçons" that he had prepared for de l'Hôpital. Reyneau lent some of these papers to Pierre Rémond de Montmort who lost them. However Byzance's papers were helpful to Reyneau in working on getting his lecture notes into publishable form. He worked with other mathematicians but, mainly due to his having to learn revolutionary new ideas as he went along, the book took a long time to complete. The two volume workOn14July1700, I learnt from M Varignon that the ratios of positive and negative magnitudes of the same type are equal to the ratios of the same magnitudes all being taken as positive, the plus and the minus being only signs for calculating the magnitudes, that is to say, for adding and subtracting, and the magnitudes, supposing they are lines, are on different sides with respect to the point they start from, that is, with respect to the origin. He proves it by the proposition+2 : -4 :: -4 : +8, which, according to all mathematicians, is true, the product of the extremes being equal to that of the means. Therefore if^{+2}/_{-4}is that by which+2exceeds -4, it must be that in the equal ratio^{-4}/_{+8}, -4also exceeds +8, which cannot be the case. Thus these ratios are the same as those between positive magnitudes, thus^{+a}/_{-2}=^{-a}/_{+2}, always half of a. Thus when working, either ordinarily or by the integral and differential calculus, a negative solution is found, it only indicates that the magnitude that gives the solution is on the other side of the origin, opposite to the one that was taken to indicate positive magnitudes.

*Analyse demontrée ou la méthode de résoudre les problèmes des mathématiques et d'apprendre facilement ces sciences*Ⓣ was published in 1708 and a second enlarged edition was produced which was the text from which Jean d'Alembert learnt mathematics. In [5] there is the following description (which is almost identical to that in [2] and [9], all of them being based on [7]):-

The Preface toReyneau, not content with making himself master of every thing worth knowing, which the modern analysis, so fruitful in sublime speculations and ingenious discoveries, had already produced, undertook to reduce into one body, for the use of his scholars, the principal theories scattered here and there in Newton, Descartes, Leibniz, Bernoulli, the Leipsic Acts, the Memoirs of the Paris Academy of Sciences, and in other works; treasures which by being so widely dispersed, proved much less useful than they otherwise might have been. The fruit of this undertaking, was his "Analyse demontrée," or Analysis demonstrated, which he published in1708,2volumes. He gave it the name of "Analysis demonstrated," because he demonstrates in it several methods which had not been handled by the authors of them, with sufficient perspicuity and exactness. The book was so well approved, that it soon became a maxim, at least in France, that to follow him was the best, if not the only way, to make any extraordinary progress in the mathematics; and he was considered as the first master, as the Euclid of the sublime geometry.

*Analyse demontrée*Ⓣ begins as follows:-

Given the extract above where Reyneau is discussing what he learnt from Varignon about negative ratios, it is interesting to look at Reyneau's use of negative numbers inThe spirit of man is so limited that he cannot see distinctly from a single viewpoint many objects at once. Vivid perceptions; as all the senses and the imagination give, dazzle it, and so occupy its extent that it cannot discover the relations and the properties of the significant objects, except by considering them in parts one after the other with a painful and fatiguing application; and when he is attentive to some one, he has lost sight of the others, who are nevertheless necessary to him in order to perceive their relations. This is one of the principal causes of the little progress made by the sensible sciences. But to speak here only of mathematics, which their usefulness, beauty, evidence, and certainty have always cultivated; while people have only applied themselves to the contemplation of the figures themselves, so that they have sought the properties of figures by looking at them, or by forming them in their imagination, and thus they have not made much progress. The discoveries were very limited, and they found with great difficulty only the resolutions of particular problems. They fatigued themselves; they rebelled; and we cannot praise enough the work, the patience, and the strength of mind of the ancient geometers, for having carried Mathematics by such difficult means, to the state in which they have been left to us. Fortunately, in the last century, the lines and figures were expressed by the familiar characters of the alphabet, and these expressions were reduced to an easy calculus, which also expresses all the simple and compound relations which these lines and figures can have. A methodical Art was formed(which is what is called Analysis)to find, by known techniques the unknown magnitudes which are sought in the problems from those which are known, giving equations which express the conditions, and the nature, so to speak, of the problems; and allows one to discover the values of the unknown magnitudes in these equations; which gives the resolution of the problems. M Descartes perfected, and reduced to an extreme facility, these calculations and this new analysis. He added to it the excellent method of employing indeterminate expressions, which, however simple they were, represented an infinity of magnitudes; and to determine from all of them the particular magnitudes which they may satisfy. He gave us the method of reducing curves to equations which express their principal properties; and to derive from these equations all the things which we could desire to know about these curves, and, lastly, how to use the curves themselves to solve equations and problems.

*Analyse demontrée*Ⓣ. In the book he solves equations, always assuming that the coefficients are positive. This means that when giving the solution to a quadratic equation, he had six cases to consider:

*x*

^{2}+

*px*+

*q*= 0,

*x*

^{2}-

*px*+

*q*= 0,

*x*

^{2}+

*px*-

*q*= 0,

*x*

^{2}-

*px*-

*q*= 0,

*x*

^{2}+

*q*= 0,

*x*

^{2}-

*q*= 0.

*x*

^{3}±

*ax*

^{2}±

*bx*±

*c*= 0,

*x*

^{3}±

*bx*±

*c*= 0,

*x*

^{3}±

*ax*

^{2}±

*c*= 0,

*x*

^{3}±

*c*= 0.

*La Science du Calcul des Grandeurs en Général, ou les Éléments des Mathématique*Ⓣ, published in 1714, which [1]:-

The 7th edition of... attempted to preserve the central conceptions of the Oratorian mathematics of the end of the preceding century,[but]was less successful than the first.

*Encyclopaedia Britannica*(1842) describes the end of Reyneau's life:-

In 1716 he was elected to the Paris Academy of Sciences. On his death it was discovered that he had been working on a second volume ofTowards the close of life he was too much afflicted with sickness to give much application to study; and died in1728, at seventy-two years of age. His many virtues and extensive erudition made this event much regretted by all who had the pleasure of being acquainted with him. It was regarded as an honour and a happiness by the first men in France to number him amongst their friends, such as the chancellor of the kingdom, and Malebranche ...

*Science du Calcul des Grandeurs*Ⓣ but, although the work was incomplete, it was published in 1736. Bernard de Fontenelle wrote Reyneau's

*Eloge*[7] in which he stated:-

We note that this, like much of the... he attempted to gather, into one work for his students' use, the principal theories found in Descartes, in Leibniz, in Newton, in Bernoulli, in the Acta of Leipzig, in the Mémoires de l'Académie ...

*Eloge*, has been almost literally copied by various encyclopaedias over the years.

Although *Analyse demontrée* Ⓣ was widely used, the book contained a number of errors. In 1728, the year Reyneau died, Alexis Clairaut corrected errors in the book. A second edition of *Analyse demontrée* Ⓣ was published in 1736 but it still contained errors which were corrected by Jean Le Rond d'Alembert in 1739.

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