BOOKS ON ARCHITECTURE

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French

     In the 1590s, two scholars very well-versed in geometry, François Viète (1540-1603) and Jean Errard (1554-1610), in the immediate entourage of Henri IV, published books (sometimes with the same publisher David Le Clerc) with very different emphases. Whereas the first showed decisive progress in algebra and summoned past geometricians in order to go beyond them, the second book aimed at deepening and especially popularizing their heritage. This is why La géometrie et pratique générale d'icelle published in 1594 by Jean Errard must be considered, in France, as a crowning achievement in the humanist mathematical culture of his period. He received his essential humanist training in his wealthy and influential family in Bar-le-Duc and also at the University of Heidelberg where he was enrolled as a Fench student in 1573 (perhaps he converted to Protestantism there) and later when he was in contact with Italian fortifications engineers, when he mastered Latin, German and Italian. Nevertheless out of a pedagogical concern to be understood by the greatest number, also in step with the use of the vernacular recommended by Luther and Calvin, he always wrote in French. This engineer to the king, in his training and because of his taste, a real connoisseur of Euclid and Archimedes, was a geometrician ; all his work is steeped in geometry. Particularly his most famous treatise La fortification réduicte en art et démontrée, which draws its strength and originality in part from the use of geometry. Thus his treatise on geometry, more than his editions of Euclid's Éléments, is essential to grasp the nature and breadth of his mathematical knowledge.
     In this treatise on geometry, a brief book of 96 pages, made up of three books, he proposes to convey “that which is most beautiful and most rare” in the results and strict demonstrations of the Greek geometricians. In doing so, he teaches to “lovers of the beautiful sciences” and to practitioners (fortifications engineers, surveyors, architects, etc.) the very useful art of measuring “straight lines” (book I), “the areas of plane surfaces” (book II), and “solids” (book III) linked to figures, plane, flat and spatial, simple and complex. Errard dedicated his work to his protector, dedicating books II and III to two very important persons (sometimes controversial) of the period : the duc de Bouillon, Henri de la Tour d'Auvergne, Field Marshall of France, and the governor of Paris, François d'O. A second edition, identical, came out during Errard's lifetime at the printing shop of Guillaume Auvray in 1602. After a few pages devoted to the definitions of geometric objects “for the relief of those who are untaught and less well-versed in mathematics”, the first book joins the long tradition of the treatises on surveying, such as the Liber de geometria practica by Oronce Fine (1544). In order to measure the length, the height or the depth of accessible or inaccessible objects, Jean Errard displays an instrument made up of three graduated mobile rulers allowing one to make a small triangle similar to the concrete triangle whose sides one wishes to measure. He doesn't settle for showing how to manipulate this instrument but he justifies its use with meticulous demonstrations notably based on the propositions in book VI of Euclid's Éléments. Philippe Danfrie, Engraver-General of the French coinage, perhaps benefited from it in 1597 in the opuscule on the trigometer that he added to his Déclaration de l'usage du graphomètre : this instrument “with which one can easily make measurements without being subjected to arithmetic” is no other than Errard's instrument, slightly improved.
     The second book on “measuring plane surfaces” is clearly more speculative. It is divided in 11 chapters, each devoted to a particular geometric figure (rectangle, triangle, lozenge, trapezium, polygon, circle, ellipse and spiral). Without overlooking the classical results, which he sometimes lays out with a single numeric example, Errard also chooses to examine in detail particularly attractive demonstrations where he shows his knowledge to advantage. This is how he states masterfully (p. 39) the geometric demonstration of the Heron of Alexandria formula which gives the area of a triangle depending on the lengths of its sides. This demonstration was not new ; it is found in Arab treatises and in numerous Renaissance works, particularly in the last pages of the Scholarum mathematicarum unus et triginta by Pierre de La Ramée (1569). Euclid's Éléments constitute the framework of Errard's demonstrations, but he also resorts to Archimedes' methods which he contributes to popularize particularly for those “who revel in geometric subtleties”. Thus he takes up the Archimedean method to prove that “the circumference of a circle contains three times the diameter and [a] little less than a seventh part of the diameter thereof, and more than the eighth part of the same diameter” (i.e. 3+1/8 < π < 3+1/7). This demonstration can be found in the Protomathesis by Oronce Fine in 1532 and La Pratique de la géométrie d'Oronce by Pierre Forcadel in 1570, but Jean Errard's calculations are different. Drawing from Archimedes he found the area of an oval (i.e. of an ellipse considered as the intersection of a cylinder and a plane surface) in a very elegant way. Still using the same method he proves that the area of a spiral after one revolution is equal to a third of its circumscribed circle. In France, this demonstration was a rarity, and even an Errard exclusivity ; in Venice it had already been exhibited in Federico Commandino's translation of works by Archimedes in 1558 (proposition 24). All Errard's formulae are stated simply. He uses no symbols other than the “R” of Nicolas Chuquet to denote a square root. It is probable that he used Theon's method for the rational approximations of square roots, which enabled him to use only whole numbers or fractions. Thus in this way Errard marked the upper limit of the mathematics he inherited from his predecessors.
     The third book is dedicated to the “measurement of solids” and is organized in 14 chapters. The first 12 deal with the volumes or the surfaces of the usual solids (cubes, parallelepipeds, prisms, pyramids, tetrahedrons, octrahedrons, dodecahedrons, icosahedrons, truncated pyramids, cylinders, cones, spheres, spherical sectors, ellipsoids) and less traditional solids like the cylinder and the cone with a spiral base. In the foreword “to the reader” Errard puts forth a solution to the well-known problem of the duplication of the cube “as easily and exactly as anyone could do it until now”. He introduces it (p. 77) in using “the invention of Heron & Apollonius”, admittedly apt but debatable. In terms of the surfaces and the volumes of a dodecahedron and an icosahedron inscribed in a single sphere, he formulates but does not prove proposition 4 of Euclid's book XIV which proves that the pentagons and the equilateral triangles of these two solids have circles of the same radius. In this regard, it is likely that Errard had read the work of François de Foix- Candale (1512-1594) (quite well acquainted with the court of Nérac) and in particular Foix-Candale's Élémenta by Euclid appearing in 1578 where this proposition can be found (p. 432) and many others on Platonic solids. For the surface and the volume of a sphere Errard resumed Archimedes' exhaustion method. From that he deduces the measurements of a spheroid (of the ellipsoid) and of the truncated ellipsoid (barrel-shaped). He does no more than skim the subject of spherical triangles, for, as he writes, “we have fully proved it in the treatise on the world map and the planisphere”. Consequently he is announcing a treatise which was never published. The thirteenth chapter mentions a practical method to measure the volumes of irregular bodies, referring to the legend telling that Archimedes proved “the fraud of the goldsmith who had falsified Hiero's gold crown”. The last chapter finally addresses the “manner of weighing”. The engineer refers to the “sixth theorem of the first book De æque ponderantibus” ; the terms of this reference allow one to induce that he likely had read the Graeco-Latin compilation of Archimedes' works published in 1544 by Thomas T. Gehauff (Venatorius). Next he gives two astute methods, although impractical, for distinguishing the metals in two objects having the same weight.
     The Géométrie is not the theoretical introduction to the Fortification which was to be its practical application ; the theory/practice pair is in action in the two books. In La Géométrie theory predominates, for Errard was concerned, strictly speaking, with establishing formulae according to a coherent mathematical process like Euclid's or Archimedes'. Consequently his book, expressed very clearly, was established as a reference manual. It was reissued in 1602 by Guillaume Auvray and had posthumous editions in 1619, 1620 and 1621.

Jean-Pierre Manceau (Tours) – 2019

Critical bibliography

M. Lallemend, A Boinette, Jean Errard de Bar-le-Duc “Premier ingénieur du très chrétien Roy de France et de Navarre Henry IV”…, Paris / Bar-le-Duc, Thorin & Dumoulin / Comte-Jacquet & Boinette, 1884.

A. France-Lanord (ed.), Jean Errard, Le premier livre des instruments mathématiques mechaniques, Paris, Berger-Levrault, 1979 (fac-simile edition), Introduction, pp. VII-XV.

D. Buisseret, Ingénieurs et fortifications avant Vauban. L'organisation d'un service royal aux XVI^e -XVII^e  siècles, Paris, CTHS, 2002, pp. 74-83.

A. Bernard, Jean Errard, Mnémosyne à la BnF, Spécial issue 3, Journées nationales de l'APMEP 2010, 2^nd edition, Paris, IREM de Paris Groupe MATH. Grt., BnF, 2010, pp. 40-42.

F. Métin, La fortification géométrique de Jean Errard et l'école française de fortification (1550-1650), PhD, University of Nantes, 2016.

F. Métin, “La formation de Jean Errard, de Nancy à Sedan, en passant par Heidelberg” Mobilités d'ingénieurs en Europe, XV^e  – XVIII^e  siècle, Presses Universitaires de Rennes, 2017, pp. 57-72.