BOOKS ON ARCHITECTURE

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French

     Of all Desargues’s works, the Exemple is the one receiving the largest number of commentaries. The 12-page opuscule, with only one illustrated plate, published in 1636 with a privilege dated 1630, interested firstly architects, artists and engravers, then mathematicians, historians of art, historians of science and philosophers. Although few copies were printed (presently we know that five of them exist), the learned world seized on it as soon as it was published and even before. Thus Mersenne wrote to Peiresc as early as 1634, “Perhaps you will soon see a little treatise on perspective that is more delectable than any you have ever seen, although shorter”. Between 1641 and 1643 this booklet from one “of the great minds of this period and one of the most accomplished in mathematics”, according to Pascal’s expression, provoked a turbulent scientific debate, especially with Jacques Curabelle, whom Desargues accused of “venimous malice” and “defamatory duplicity”, and with Jean Du Breuil, the author of a Perspective practique in 1642, which, according to Desargues, expounded “his Manière Universelle de Perspective [ in a way that was ] imperfect, difficult and false in this Livre de Copies”.
     Abraham Bosse’s book Manière universelle de Mr. Desargues pour pratiquer la perspective par petit-pied, comme le géométral came out in 1647-48. Desargues had instructed Bosse who not only set forth the Arguesian method in minute detail, but also taught it in the entirely new Royal Academy of Painting and Sculpture starting in 1648. This teaching was the subject of another debate on the subservience of the arts of drawing to the laws of linear and aerial perspective and concluded with the exclusion of Bosse in 1661. Desargues’ book (1636) was reprinted at the end of Bosse’s treatise and it was essentially because of this addition that Desargues’ Perspective was known. The method set forth in it is composed of two important ideas. The first is that the method is the same in locating a given point in Euclidian space as in finding the representation in perspective of that same point in a picture. He illustrates this with a cage with a square base which is slightly buried, terminating in a pointed tip. He projects the main points of this cage on a horizontal plane that Bosse calls “geometral plane” and Desargues calls assiette du sujet which produces a square. In this same plane he also indicates the projection [ ab ] of the vertical picture and that of the observor’s eye placed at a given distance from the picture. Then the points are located by their distances at two straight lines, one passing through [ ab ] and the other perpendicular to it.
     These distances are measured with a specific unit of measure called petit pied. They are transferred to the devis in accordance with a practice familiar to various guilds. For the elevations of the subject points Desargues traces segments of corresponding lengths. These points are then transposed to the drawing in perspective provided that one constructs a “scale of distances” allowing one to determine the distance of each plane parellel to the painting, and to construct directly in each of these plans a scale of measurements or as Bosse says, a corresponding scale of the pieds de front. If there is no problem in constructing the latter, on the other hand the scale of the distances is not obvious. Treatises on perspective from the beginning of the 16th century (at least since Viator) frequently used the main vanishing point on the horizon line, the projection of the eye level of the viewer on the picture and the distance points, which on either side of the main vanishing point, was at distance d from the vanishing point, equal to distance d from the eye of the viewer to the picture. According to the position of the eye these points could be outside the picture field, making the classic construction impossible.
     Desargues’ second idea is that by modifying the scale of the petits pieds used in the preliminary geometral study, it is possible to make constructions of scales of distances entirely within the picture. By diminishing in the same proportions the basic petit pied and distance d of the eye to the picture plane, he proposes a simple geometric construction to place frontal planes situated at 1, 2, 3 […] petits pieds from the plane of the picture. Furthermore, if distance d from the eye is chosen as unit of length, by using only simple numeric ratios, he very easily constructs a rather rough scale of distances which positions frontal planes situated at distances d, 2d, 3d […] from the picture plane. By playing subtly with these two constructions, he places all the main points of his cage in perspective and can therefore draw it in perspective. Desargues points out that the general rules thus expounded on this example “are demonstrated with only two evident and familiar propositions to those enclined to appreciate them”. Thales’ theorem can justify these constructions, but perhaps the author had other constructions in mind, especially the one on pages 336-337 (pl. 151) reprinted in Bosse’s treatise.
     It was especially on the constructions of the scales of distances and measures that Desargues was accused of plagiarism. Jacques Curabelle, in his Examen des œuvres du Sr. Desargues, 1643, claims that “the universal method of Sieur Desargues [… ] is in no other manner than that of most of our elders” and attributes the invention of the construction of the scale of distances to Jacques Alleaume, who described it in his 1628 treatise Traité de perspective which remained, it appears, unpublished. Curabelle also indicates that the Abregé ou racourcy de la perspective par l’imitation by Vaulezard, published in 1631, contains a perspective compass “which is nothing other than the construction of the scale of distances and without leaving the field of the work”. In other words, Desargues’ second idea was in the air ; Mersenne was to affirm later that “those who read and understood M. Desargues’ universal method […] recognize that it exceeds in an abbreviated version anything that has been given up to the present time and that he was right in 1636 to call himself inventor of the universal method…”.
     One might be astonished that Arguesian geometry is expressed in this treatise in measures of length. In fact, Desargues’ main goal was “purely practical”. As he pointed out in 1647, “having noticed that a good part of the practice of the arts is founded on Geometry as well as on an assured base…” for laborers he had “the desire and affection to relieve them” by seeking and publishing “abridged rules for each of these arts”. On the other hand, in 1636, Desargues had not yet mathematically deepened his knowledge of how more distant objects could be rendered by using “strong and weak colors”. He cautiously points out that “the demonstration is mixed in part with Geometry, in part with Physics and is not yet found anywhere in France in any public book”. But Bosse took care of it, with his support, in 1647.
Desargues completes his study by indicating the possible help of “instruments based on geometric demonstration”. In this connection he cites a treatise published in Rome, doubtless Scheiner’s, 1631, in which a pantograph appears. In 1643 he gave the construction of a compass found in plate 119 in Bosse’s treatise. The leaflet ends, for theorists, with a few general geometric propositions, here adapted for perspective. Perhaps they are premises of his subsequent mathematical work if one believes, as is sometimes disputed, that perspective is a rational anticipation of perspective geometry, (Pfeiffer 1998). Lastly, for good measure, in the last two lines he presents a geometry problem, a true challenge launched at the scientific community, in accordance with a practice fashionable in the 17th century.

Jean-Pierre Manceau (Tours) – 2017

Critical bibliography

J. Curabelle, Examen des œuvres du Sr. Desargues, Paris, Hénault, 1644.

N.-G. Poudra, Histoire de la Perspective ancienne et moderne, Paris, Corréard, 1864.

N.-G. Poudra, Œuvres de Desargues..., Paris, Leiber, 1864. 2 vols.

R. Taton, L’œuvre mathématique de G. Desargues, Paris, PUF, 1951 (reed. Paris, Vrin, 1981 & 1988).

J. Dhombres & J. Sakarovitch (eds.), Desargues en son temps, Paris, Blanchard, 1994.

R. Laurent, “La perspective et la rupture d’une tradition”, in Desargues en son temps, Paris, Blanchard, 1994, pp. 231-243.

J. Peiffer, “L’histoire de la perspective au XXe siècle : une déconstruction”, La Gazette des mathématiciens, 78, Oct. 1998, SMF, Paris 1998, pp. 63-75.

M. Le Blanc, D’acide et d’encre. Abraham Bosse (1604 ?- 1676) et son siècle en perspective, Paris, CNRS Éditions, 2004.

J.-P. Manceau, “Abraham Bosse, un cartésien dans les milieux artistiques et scientifiques du XVIIe siècle”, S. Join-Lambert & M. Préaud (eds.), Abraham Bosse savant graveur, Paris/Tours, Bibliothèque nationale de France/Musée des Beaux-Arts de Tours, 2004, pp. 53-63.