# Girard Desargues

### Quick Info

Lyon, France

Lyon, France

**Girard Desargues**was a French mathematician who was a founder of projective geometry. His work centred on the theory of conic sections and perspective.

### Biography

**Girard Desargues**'s family (on both his mother's and his father's side) had been very rich for several generations and had supplied lawyers and judges to the Parlement in Paris as well as to that in Lyon (then the second most important city in France). Girard's father, also having the name Girard Desargues, married Jeanne Croppet in Condrieu a small town in the Rhône department in eastern France. They moved to Lyon but certainly retained the property in Condrieu since Girard (Junior) spent time there towards the end of his life. In Lyon Girard (Senior) worked as an investigator for the bailiff, then as a tax collector and royal notary, which was his occupations when his son Girard was born. Girard and Jeanne Desargues had eight children, four daughters Marie, Clemence, Francoise and Catherine, and four sons Fleury, Philippe, Antoine and Girard (the subject of this biography). It appears that their son Girard was the youngest of the eight children, which is rather surprising since he took his father's name.

Girard, the subject of this biography, was baptised in the parish church of Sainte-Croix on 2 March 1591 when he was nine days old. In fact Desargues' date of birth was unknown until the work of René Taton [28] published in 1962. Prior to Taton's research it was wrongly believed that Desargues was born in 1593 because in Adrien Baillet's 1691 biography of Descartes states that Desargues was three years older than Descartes. Taton discovered a horoscope of Desargues giving his birth at 6:30 on 21 February 1591. There is no information about Desargues' education and about his early life. He is in his middle 30s before we have any definite information about his activities.

Desargues seems to have made several extended visits to Paris in connection with a lawsuit for the recovery of a huge debt. Despite this loss, the family still owned several large houses in Lyon, a manor house (and its estate) at the nearby village of Vourles, and a small chateau surrounded by the best vineyards in the vicinity. It is thus clear that Desargues had every opportunity of acquiring a good education, could afford to buy what books he chose, and had leisure to indulge in whatever pursuits he might enjoy. In his later years, these seem to have included designing an elaborate spiral staircase, and an ingenious new form of pump, but the most important of Desargues' interests was Geometry. He invented a new, non-Greek way of doing geometry, now called 'projective' or 'modern' geometry. As a mathematician he was very good indeed: highly original and completely rigorous. He was, however, far from lucid in his mathematical style.

It is unclear when he first went to Paris but we know that he was there on 9 September 1626 for on that day he wrote to the leaders of the merchants and magistrates of the city of Paris proposing that he and a colleague François Villette build, or have built, a machine to raise the level of the water of the river Seine to distribute it to the inhabitants of Paris. It is unclear who François Villette was but there was an optician named François Villette who was born in Lyon in 1621. Clearly it could not have been that François Villette, who was only five years old at the time, but it is possible that it could have been his father. Desargues' letter, written at the Hotel de Ville of Paris on 9 September 1626, is given in full in [14]. It begins:-

François Villette and Girard Desargues, both bourgeois of Lyon, propose to the leaders of the merchants and magistrates of the city of Paris for the decoration, public convenience and embellishment of the said city, the following assertion:We note that the Samaritaine, operating from 1608, pumped water from the Seine into a reservoir above the Pont-Neuf which supplied the Louvre palace and the Tuileries gardens. The Parisian leaders would have to agree a sum in payment, with the required assurances, provide a site for Villette and Desargues' machine to be installed, and then they would build it, or have it built, and wait a month for their payment, only to be made when it is judged satisfactory. They received a reply, amazingly quickly by today's standards, on 15 September 1626 saying that the proposals [14]:-

In all the places where the Seine river can make a wheat mill grind throughout the year, to raise its water to a height of about forty feet, continuously flowing twice as much as it will rise by the pump of the Samaritaine of the Pont Neuf; and this achieved by means of a machine, which, once well established, can be maintained for less than three hundred pounds per year ...

... can be only for the good and the convenience of the public, decoration and embellishment of this city. And therefore we agree that Villette and Desargues begin to execute them at their own expense, with the charge that they will not be able in any way to plant their machines in the river nor in any place on the edges and shores of it before permission is given in our presence by the masters of works of the city and masters of the bridges of it, so that they cannot harm or prejudice the navigation path, the approach and the unloading of goods.No further correspondence survives and it is assumed that Villette and Desargues chose not to go ahead under the conditions imposed on them. We suggest that the address from which Desargues sent his letter and the fact that he offers no Paris recommendations, indicates that he had newly arrived in Paris.

We mentioned above Adrien Baillet's 1691 biography of Descartes where Desargues' incorrect year of birth is given. It is unclear, therefore, how much we should trust this work although we must not be too harsh on it because of one error. Baillet states that Desargues was an engineer involved in the siege of La Rochelle in 1628 and it was there that he first met Descartes. There is no additional evidence to substantiate this claim, although given Desargues' skills, it certainly appears plausible. Let us explain briefly about the siege of La Rochelle in 1628. This was a consequence of the Catholic-Protestant hostility at the time. The Huguenots, who were Protestant, had their stronghold at La Rochelle and were supported by the English. The Catholic side, which consisted of royal troops of Louis XIII, wanted to take La Rochelle and prevent the English landing ships in support. Fortifications were built by the Royal side, led by the King and Cardinal Richelieu, to lay siege to the city and also massive sea defences were built to prevent the English support reaching the Huguenots. That Desargues would be involved in such an undertaking would certainly seem possible. There is, however, a statement in C Adam and P Tannery (eds.),

*Oeuvres de Descartes*Ⓣ (1897) that Descartes first met Desargues in 1637.

When in Paris, Desargues became part of the mathematical circle surrounding Marin Mersenne (1588-1648). This circle included René Descartes (1597-1650), Étienne Pascal (1588-1651) and his son Blaise Pascal (1623-1662). It was probably essentially for this limited readership of friends that Desargues prepared his mathematical works, and had them printed. Some of them were later expanded into more publishable form by Abraham Bosse (1602-1676), who is now best remembered as an engraver, but was also a teacher of perspective. Bosse states that Desargues was given a royal licence to publish several of his writings in 1630. This adds a little weight to Desargues assisting Cardinal Richelieu in the siege and, probably, being involved in other work by the Royal side.

Exactly when Desargues joined Marin Mersenne's "academy" is unclear. Mersenne writes in one of his letters that Desargues met Pierre Gassendi in Paris before 1632. Mersenne states in 1635 that Desargues was a regular attender of his meetings and his comment makes it look as though he had been doing so for some time. In 1635-36 Mersenne published

*La Harmonie Universelle*Ⓣ which contains a short paper by Desargues entitled

*Une méthode aisée pour apprendre et enseigner à lire et escrire la musique*Ⓣ. Mark Schneider writes in [8]:-

Desargues' "Easy Method" is his only known writing which does not deal with geometry and its application. Here perhaps we have an indication that Desargues had been under the influence of Mersenne during the period in which his ideas on geometry were taking their definitive form.Desargues wrote on 'practical' subjects such as perspective in

*Exemple de l'une des manières universelles du S.G.D.L. touchant la pratique de la perspective sans emploier aucun tiers point, de distance ny d'autre nature, qui soit hors du champ de l'ouvrage*Ⓣ (1636), the cutting of stones for use in building in

*Brouillon project d'exemple d'une manière universelle du S.G.D.L. touchant la pratique du trait a preuves pour la coupe des pierres en l'architecture*Ⓣ (1640) and sundials in

*Manière universelle de poser le style aux rayons du soleil en quelconque endroit possible, avec la règle, le compas, l'esquerre et le plomb*Ⓣ (1640). His writings are, however, dense in content and theoretical in their approach to the subjects concerned. There is none of the wordy and elementary step-by-step explanation which one finds in texts that are truly addressed to artisans.

The title of the work on perspective translates as

*Example of one of S.G.D.L.'s general methods concerning drawing in perspective without using any third point, a distance point or any other kind, which lies outside the picture field*. One immediately wonders who or what "S.G.D.L." is, but this is simply "Desargues" from the initials of "Sieur Girard Desargues Lyonnais". This work on perspective must have led Desargues to develop a new approach to geometry. Concerning his work on stone cutting, Mark Schneider writes [8]:-

Desargues' method of stone-cutting works and is indeed a brilliant invention, but, at the same time, it must be noted that without the author's personal tutoring no mason of the time would have been likely to understand it.The description of Desargues' stone cutting method, in a form that those working on stone would understand, was produced by Desargues' disciple Abraham Bosse (1604-1676) in 1643. Bosse also describes Desargues' work on sundials and, as Desargues's original publication has not survived, this is our only information about this text.

In 1640 Blaise Pascal, who was 16 years old at the time, produced his 'mystic hexagram'. In it he referred to Desargues:-

We shall also demonstrate this property of which the original inventor is M Desargues of Lyon who is one of the great minds of this time and one of the most versed in mathematics, in particular among others in conics, whose writings on this matter, though small in number, have given ample testimony of his ability to those who have desired to become aware of it: and I will admit that I owe the little that I have found on this matter to his writings, and that I have tried to imitate as much that it is possible for me his method on this subject, ...Pascal must be referring here to Desargues' most important work, the one in which he invented his new form of geometry, which has the title

*Brouillon project d'une atteinte aux evenemens des rencontres du Cone avec un Plan*Ⓣ). A small number of copies was printed in Paris in 1639. Only one is now known to survive, and until this was rediscovered, in 1951, Desargues' work was known only through a manuscript copy made by Philippe de la Hire (1640-1718). The book is short, but very dense. It begins with pencils of lines and ranges of points on a line, considers involutions of six points (Desargues does not use or define a cross ratio), gives a rigorous treatment of cases involving 'infinite' distances, and then moves on to conics, showing that they can be discussed in terms of properties that are invariant under projection. We are given a unified theory of conics.

Desargues' famous 'perspective theorem' - that when two triangles are in perspective the meets of corresponding sides are collinear - was first published in 1648, in a work on perspective by Abraham Bosse.

You can see more about this result at THIS LINK.

It is clear that, despite his determination to explain matters in the vernacular, and without direct reference to the theorems or the vocabulary of Ancient mathematicians, Desargues is well aware of the work of ancient geometers, for instance Apollonius and Pappus. His choosing to explain himself differently may perhaps be due to his recognition that his own work was also deeply indebted to the practical tradition, specifically to the study of perspective (which is a form of conical projection). It seems highly likely that it was in fact from his work on perspective and related matters that Desargues' new ideas arose. When projective geometry was reinvented, by the pupils of Gaspard Monge (1746-1818), the reinvention was from descriptive geometry, a technique that has much in common with perspective.

Desargues' work on perspective led to a very unpleasant argument. In 1642 an anonymous work entitled

*La Perspective practique nécessaire à tous peintres, graveurs, sculpteurs, architectes, orfèvres, bordeurs, tapissiers & autres se servans du Dessein*Ⓣ was published by the publishers Melchior Tavernier and Francois l'Anglois. The work was actually written by Jean Du Breuil (1602-1670), the son of the bookseller Claude Du Breuil, who was primarily an architect. It was the first of three volumes published between 1642 and 1647. The preface to the book credited Desargues but he was very upset to see his ideas presented with many errors and his reaction was to place placards around Paris. One was headed "Incredible error" and another "Enormous faults and duplicities". One placard claimed that Du Breuil had:-

... stuffed into this book on practical perspective, a diagram [due to Desargues] which he claims is an example of his own, which he had altered and falsified with the petty claws of envy.This looks like a massive overreaction by Desargues and it prompted an equally vicious response by Du Breuil who counterattacked with a pamphlet claiming that Desargues' 1636 paper on perspective presented ideas that had been published earlier by Jean-Louis de Vauzelard in

*Perspective cilindrique et conique*Ⓣ (1630) and by Jacques Aleaume in

*Introduction a la perspective, ensemble a l'usage de compas optique et perspective*Ⓣ (1628). He also infuriated Desargues by claiming that, for all practical purposes, his work was without value.

Desargues kept up the argument by publishing

*Six erreurs des pages*87, 118, 124, 128, 132

*et*134

*. du livre intitulé 'La Perspective practique nécessarie à tous peintres ...'*Ⓣ in 1642 in which he detailed errors in Du Breuil's work. The publishers Melchior Tavernier and Francois l'Anglois then attacked Desargues by publishing a collection of articles criticising his work in

*Advis charitables sur les diverses oeuvres et feuilles volantes du Sieur Girard Desargues, Lyonnois*Ⓣ (1642). This work included a letter written by Jean Beaugrand in August 1640, shortly before his death, in which he criticised Desargues' projective study of conics.

For Mark Schneider's summary in [8] of the criticisms made against Desargues in this work, see THIS LINK.

At this point Desargues seems to have turned to Abraham Bosse to publish clarifications of his work and to defend it against these attacks. As we noted above, Bosse published two treatises in 1643 presenting in a simpler way Desargues' work on stone cutting and on sundials.

A new attack came in 1644 from Jacques Curabelle with the 81-page book

*Examen des oeuvres de Sieur Desargues, Lyonnois*Ⓣ. Curabelle attacked all of Desargues' work, including the two publications by Bosse in 1643, saying that he could find:-

... find nothing in them but mediocrity, errors, plagiarism, and information of no practical interest.Curabelle claimed that Desargues' lack of practical experience makes his work useless. He writes:-

If the said Sieur had understood and practiced the things he wanted to talk about, he probably would not fall into such errors, practice being necessary to help and strengthen our senses; it will confirm or deny what the speculation of our minds would have produced.A vicious argument between Curabelle and Desargues followed with various pamphlets attacking each other and with Desargues threatening to sue Curabelle if he did not retract. The two, during a number of bitter exchanges, set up a debate with rules and regulations, judges were to be appointed to decide on the winner who would receive a large sum from the loser. There is no evidence, however, that this ever took place.

Desargues appears to have grown tired of the continuous battles he was involved in and, from 1645, turned to architecture. In 1648 he returned to Lyon where he seems to have been more involved in architectural design and published little. He did go back to Paris in 1649-50 and again in 1657-1660 where he was responsible for the design of several mansions. One has to wonder what other brilliant mathematical work this outstanding mathematician might have done if he had not been subjected to such widespread criticism.

Let us end this biography with two quotes regarding Desargues' mathematical contributions. Florian Cajori writes in his

*History of Mathematics*(1893):-

We owe to Desargues the theory of involution and of transversals; also the beautiful conception that the two extremities of a straight line may be considered as meeting at infinity, and that parallels differ from other pairs of lines only in having their points of intersection at infinity. He re-invented the epicycloid and showed its application to the construction of gear teeth, a subject elaborated more fully later by La Hire.David Eugene Smith in Volume 2 of his

*History of Mathematics*(1958) writes:-

One of the first important steps to be taken in modern times ... was due to Desargues. In a work published in 1639 Desargues set forth the foundation of the theory of four harmonic points, not as done today but based on the fact that the product of the distances of two conjugate points from the centre is constant. He also treated the theory of poles and polars, although not using these terms

### References (show)

- R Taton, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990).

See THIS LINK. - Biography in
*Encyclopaedia Britannica.*

http://www.britannica.com/biography/Girard-Desargues - M Chaboud,
*Girard Desargues: bourgeois de Lyon, mathématicien, architecte*(IRME de Lyon, 1996). - J V Field,
*The Invention of Infinity: Mathematics and Art in the Renaissance*(Oxford 1997). - J F Field and J J Gray,
*The geometrical work of Girard Desargues*(Springer Science & Business Media, 2012). - J V Field and J J Gray,
*The geometrical work of Girard Desargues*(New York-Berlin, 1987). - O R Fullerton,
*A review of the works of Gerard Desargues with emphasis on his perspective triangle theorem and a special application of this theorem*(Ph.D. dissertation, Arkansas State Teachers College, 1963). - M Schneider,
*Girard Desargues, The architectural and perspective geometry: A study in the rationalisation of figure*(Doctoral Dissertation, Virginia Polytechnic Institute and State University, 1983). - R Taton,
*L'oeuvre mathématique de Desargue*s (Paris, 1951). - R Taton,
*La géométrie projective en France de Desargues à Poncelet*(Paris, 1951). - K Andersen, Desargues' method of perspective: its mathematical content, its connection to other perspective methods and its relation to Desargues' ideas on projective geometry,
*Centaurus***34**(1) (1991), 44-91. - M Anglade and J Y Briend, La notion d'involution dans le 'Brouillon Project' de Girard Desargues,
*Archive for History of Exact Sciences***71**(6) (2017), 543-588. - M Anglade and J Y Briend Le diamètre et la traversale: dans l'atelier de Girard Desargues,
*Archive for History of Exact Sciences***73**(4) (2019), 385-426. - A Birembaut, Quelques documents sur Desargues,
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*Revue d'histoire des sciences et de leurs applications***4**(2) (1951), 176-181. - A J Turner, Another lost work by Girard Desargues recovered,
*Arch. Internat. Hist. Sci.***34**(112) (1984), 61-67. - T Viola, Per la storia del teorema di Desargues sui triangoli omologici,
*Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat.*(8)**1**(1946), 570-575. - V Vita, Il teorema sul quadrangolo completo nel Brouillon project di Desargues,
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### Additional Resources (show)

Other pages about Girard Desargues:

### Honours (show)

Honours awarded to Girard Desargues

### Cross-references (show)

- History Topics: Mathematics and art - perspective
- Famous Curves: Cycloid
- Famous Curves: Epicycloid
- Famous Curves: Epitrochoid
- Famous Curves: Hypocycloid
- Famous Curves: Hypotrochoid
- Societies: Paris Academy of Sciences
- Other: Earliest Known Uses of Some of the Words of Mathematics (I)
- Other: Earliest Known Uses of Some of the Words of Mathematics (P)

Last Update September 2020