Author(s) Blondel, François
Title Resolution des quatre principaux problemes d’architecture
Imprint Paris, Imprimerie Royale, 1673
Localisation Paris, Ensba, Les 1835
Subject Mathematics, Resistance of solids, Stereotomy
Transcribed version of the text


     François Blondel is surely one of those “able men”, in Fontenelle’s words, who contributed in the seventeenth century to the rise of modern science.  His multiple talents and fields of expertise, recognized by successive ministers (Richelieu, Mazarin, Colbert) lead him to fulfill numerous missions in the service of the Crown and to meet eminent scholars throughout Europe.  The foremost among these was Galileo (1564-1642), of whom Blondel would later claim “the honor of being among his last disciples (p. 85)”.  In Paris, Blondel was close to several intellectual and scientific circles, notably the one assembled by Père Marin Mersenne at the Minim convent near the Place Royale.  According to Fontenelle, Blondel frequented Mersenne’s Academia Parisiensis in the years before 1649 along with “Messieurs Gassendi, Descartes, Hobbes, Roberval, and the two Pascals, father and son”.  Blondel may also have met at these meetings the mathematician Girard Desargues, with whom he shared an interest in the practical applications of geometry. He was in touch later with two of Desargues’s students and friends, the engraver Abraham Bosse and the astronomer Philippe de La Hire.  In addition to his numerous contemporary contacts, Blondel possessed a vast mathematical culture, moving with ease from the ancient Greek geometers (Euclid, but also Apollonius, Eutocius, and Pappus) to modern scholars such as Nicolò Tartaglia (1500-1557), Federico Commandino (1509-1575), and François Viète (1540-1603).  A curious mind, attentive observer, and able inventor, Blondel was named Lecteur du roi in mathematics at the Collège Royal in 1656 and associated with the royal Académie royale des science as a “geometer” from 1669.
Until 1673, Blondel’s scholarly and scientific work was known principally to a small circle of friends and correspondents.  It was Colbert’s patronage that effectively transformed his intellectual profile.  From that year, Blondel published a series of prestigious treatises on technical subjects allied to mathematics, each drawing on different domains of his professional activity, namely architecture, fortification, and ballistics.  His first major work was the Résolution des quatre principaux problèmes d’architecture (1673), produced under the auspices of the newly formed royal academies in a handsome, richly made in-folio volume.  Fully endorsed by the Crown, it formed part of a larger publication program sponsored by Colbert to showcase Louis XIV’s recent cultural patronage.  One of the period’s major treatises, the author saw it as a conscious response to the impact of the new science on the architectural culture of the moment.  The collection brought together studies that had occupied Blondel since the 1640s, in particular his work on the breaking resistance of beams, inspired by Galileo’s famous Discorsi e dimostrazioni matematiche intorno à due nuove scienze (1638).  Blondel followed the work ten years later with L’art de jetter les bombes (1683) and the Nouvelle manière de fortifier les places (1683).
The book’s unusual title served a twofold purpose.  In the first place, it served to distinguish the book from the existing corpus of Renaissance trattati as a work of mathematical science.  Rather than providing a personal interpretation of the five orders or any stylistic prescriptions for design, the Résolution poses a limited number of specific questions and defines them as problems to be solved, as obstacles to be overcome.  Intentionally narrow in scope and ambition, the work emphasizes incremental discovery and gradual progress within a purposefully circumscribed field of inquiry.  Descartes’s Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences (1637) provides a contemporary analogy.  The book’s appendices, La dioptrique, Les météores et La géométrie, presented a practical application of his method in three specific domains.
The title of the Résolution also served to situate the work in an august intellectual tradition, by alluding to the three historical problems of ancient geometry: the squaring of the circle, the trisection of the angle, and the duplication of the cube.  These problems were not resolvable deductively using the compass and straightedge.  Like the solutions presented in the Résolution, they required various other tools and techniques, some of which involved the construction of complex curves.  In a broad sense, Blondel’s purpose was not so much to solve particular architectural problems, but to show how the intervention of higher mathematics could transcend everyday architectural practice.  He therefore chose four domains in which this intervention seemed to him pertinent: the geometrical definition of the curved profile of columns (p. 1-15), the outline of oblique, or rampant, arches (p. 16-54), stereotomy, in relation to the previous problem (p. 55-59), and the resistance of materials (p. 60-86).

          First problem
          Describe geometrically in several ways and in a single curve the swelling and diminishing contour of           columns.

In this chapter, Blondel shows that the point-by-point construction described in Vignola’s Regola for the profile of columns produced a curve known since antiquity as the “first conchoid” of Nicomedes.  The curve had been described by Eutocius and by Pappus in relating solutions for the duplication of the cube.  This chapter describes the circumstances of Blondel’s discovery and provides the description of a compass capable of drawing the curve in a single continuous line for columns of any given dimensions.  Several contemporaries remarked on the discovery, notably Abraham Bosse in his Traité des manières de dessiner les ordres de l’architecture antique (1664) and Claude Perrault in his translation of and commentary on Vitruvius, which appeared the same year as the Résolution.  Typically abstruse, Blondel added further considerations on projectile motion, in which he also claimed to find the “first conchoid”.  In finishing, he provided several of his own compass designs for producing parabolic, elliptical, circular, and hyperbolic column profiles, “so that some could experiment” in cases where such curves were to “their taste”.  This last section reflects contemporary interest in conic sections and in the construction of mathematical instruments.

          Second Problem
          Find a conic section that touches three given lines on the same plane and two of these lines each on a           given point, or Describe geometrically rampant arches on any two piers and of any height.

In 1600 François Viète published his Apollonius Gallus, a reconstruction of the ancient Greek author’s lost work On Tangencies.  In his study, Viète dealt expertly with the problem of the construction of a circle tangent to any three given circles.  It was on this model that Blondel based his second “solution”, subtitled “The French Apollonius On Tangencies”, which examines the problem of a conic section tangent to three given lines or tangent to two lines on two given points.  The problem was of some interest to contemporaries.  In 1672, Philippe de La Hire had published his “Observations on the points of contact of three lines which touch a conic section on some of its diameters and on the center of the same section”, edited and published by Bosse (Observations sur les points d’attouchement de trois lignes droites qui touchent la section d’un cone sur quelques-uns des diametres, & sur le centre de la même section).  Although formulated in an academic way, the problem described a practical one often faced by architects or by stonemasons in needing to trace the curve of rampant arches, that is, arches that springs on either side from a different height.  In these concrete cases, two of the lines would represent the piers and the third the height and inclination of the arch.  Springing from its tangents with the two lower lines, the arch could be made to follow continuously (as an ellipse for example) from its piers.  The solution would ideally provide practitioners with precise and rational rules for all situations of this kind.
After an initial digression, Blondel solved the problem posed by using two propositions from Pappus’ Mathematical Collection (Book 8, propositions 13 and 14). The first of these propositions explains how to find the conjugate diameters of an ellipse (that is, two diameters such that each is formed by the midpoints of chords parallel to the other). The second proposition provides a method for finding the foci and the major and minor axes. Once these elements are determined, the architect can draw the ellipse using either a special compass or a string stretched between two pins at the foci.  Since conic sections share many of the same properties, the demonstration for the ellipse can be extended to the other curves of the same family. In showing how to determine conjugate diameters, Blondel presents examples of every possible case that practitioners might meet in the field.  But, as Blondel himself realized, the visual language of rigorous geometrical proof was not well-suited to practitioners: “the austerity of the demonstration has obliged me to use a great number of lines that are useless in practice and that could hinder workers who are not used to disentangling them”.  Although Blondel was careful to include a simplified, “universal” method for finding conjugate diameters, the sheer number and complexity of some of his demonstrations must have been enough to intimidate even the most resolute builder.

            Third problem
            Find geometrically the keystone divisions for any given rampant arch.

The resolution of this problem consists only of a few pages, since it is a natural extension of the preceding.  Blondel shows first that the current practice for determining the keystone joints for circular arches does not apply to rampant arches.  He then provides two alternative manners based on the steps outlined in the previous solution.  Once the curve of the rampant arch is established, both methods serve to establish a series of tangents to it.  The joints for the keystone as well as for the voussoirs are simply extended from those tangents as perpendiculars.  In contrast to contemporary methods for addressing this issue, Blondel’s, he claims, serve “as much for the security and solidity of the voussoirs as for the beauty and elegance of the lines”.

            Fourth problem
            Find the line on which beams must be cut in their length and width to render them everywhere             equally strong and resistant.

In his Discorsi e dimostrazioni of 1638, Galileo introduces a new science: the “resistance of solids”. During the second day of the dialogue, the author examines the structural effects posed by a horizontal beam embedded at one end into a vertical wall, demonstrating how and in what proportions the beam’s length, breadth, and thickness determine its resistance to fracture.  At a certain point, Galileo introduces the idea of the “solid of equal resistance”, that is, a beam shaped so that it conserves the same resistance at all points along its length, that is, only that thickness necessary to keep it from collapsing under a given force.  He finds that a cantilever of equal resistance (abstracting the weight of the beam itself) would retain a constant width, while its thickness would diminish in proportion to the square root of its length from the tip, thus forming a semi-parabola.
It is this issue that Blondel addresses in the fourth and final problem of the Résolution.  In it, Blondel shows that Galileo’s proof is valid only for cantilevered beams, not for beams supported at both ends.  He shows that the profile for such a beam should instead follow a uniform curve from end to end with the same squared ratio between thickness and length.  It could thus be either semicircular or elliptical.  The solution is presented in the form of two reprinted letters.  The first, in Latin, was originally written in 1657 to the Swedish mathematician Paul Wurz and published as a small pamphlet in 1661.  The second, addressed to the academician Jacques Buot the same year, responds to some objections and provides clarification on certain passages in the first letter.  Reprinting the letters in their original form was surely intended to recall readers to the priority of his discovery.
Although published much later, Blondel’s work on the resistance of solids places him in the milieu of Galileo’s French supporters from the first half of the century.  Descartes, for example, was cognizant of the Discorsi at an early date, perhaps even recognizing the same error.  In a letter to Mersenne, dated 11 October 1638, he wrote: “What Galileo says of timber that must be cut in a parabola, to resist everywhere equally, is almost right…”.  Mersenne’s own translation of the Discorsi, titled Les nouvelles pensées de Galilée (1639), appeared one year after the original was published.  As Blondel mentions in his letter to Buot, his work on the subject began in the 1640s, growing presumably from his contacts with this circle.  His contribution, moreover, was widely read and generally recognized as authoritative.  Huygens, Leibniz, Robert Hooke, and Vincenzo Viviani are all known to have read it, and most modern editions of the Discorsi also credit him with discovering and amending Galileo’s discrepancy.
The Résolution illustrates, far more than his other treatises, Blondel’s prominence among mathematicians of the time.  His aim was to combine the achievements of Greek geometry with the latest developments in physical-mathematical science, in order to bring contemporary practitioners in line with a modern vision of scientific progress.  For Blondel, a renewed orientation toward higher mathematics would serve to elevate architecture from the realm of rote, uninformed practice, while at the same time providing useful solutions for the kind of problems typically faced by practitioners.  Blondel is often contrasted with Claude Perrault as a partisan of the ancients, but the view advocated in this treatise is self-consciously modern.  His view was that architects must replace the rule-of-thumb procedures taught in the workshop for those based on fixed and absolute mathematical laws, integrating their working practice with the most advanced science of the time.

Anthony Gerbino (Worcester College, University of Oxford)
Jean-Pierre Manceau (Tours) – 2010


Critical bibliography

A. Birembaut, "Blondel, Nicolas-François", C. Coulston Gillispie (ed.), Dictionary of Scientific Biography, New York, Scribner, 1970-1980, 2, pp. 200-202.

M. Chaboud, Girard Desargues, bourgeois de Lyon, mathématicien, architecte, Lyon, Aléas, 1996.

J. Dhombres & J. Sakarovitch (ed.), Desargues et son temps, Paris, Blanchard, 1994.

J. V. Field & J. J. Gray, The Geometrical Work of Girard Desargues, New York, Springer, 1987.

A. Gerbino, "François Blondel and the Résolution des quatre principaux problèmes d’architecture (1673)", Journal of the Society of Architectural Historians, 64, 4, 2005, pp. 498 – 521.

A. Gerbino, François Blondel: Architecture, Erudition, and the Scientific Revolution, London/New York, Routledge, 2010.

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

R. Taton, L’œuvre mathématique de G. Desargues, Paris, Presses Universitaires de France, 1951.

R. Taton, "La première œuvre géométrique de Philippe de La Hire", Revue d’histoire des sciences, 6, 2, 1953, pp. 93-111.

R. Taton, "La Hire, Philippe de", Dictionary of Scientific Biography, 7, pp. 576-579.