• 1 . Christopher Marlowe, Tamburlaine, Part ii (v.iii.123-124).

« Give me a map: then let me see how much
Is left for me to conquer all the world… »1

• 2 . Patricios, N. N., 1973.

2How individuals perceived and conceptualized urban space can be discerned in the built environment and its visual representation. Since ancient times, cityscapes offered idealized views of the world using symbolic spatial schemata that transformed natural space into inhabited place. These symbolic schemata took geometric form and displayed elements drawn from myth and metaphysics. Thus the symbol for city in ancient Mesopotamia was a simple « О », though the circle also in time signified the bounded cosmos, notions of infinity, and nothingness. Hellenistic mathematics, especially Euclidean geometry, further developed these abstractions of place referents. As a result, visual representations of place often juxtaposed the terrestrial to the cosmic, the imaginary to the real, and the sacred to the profane2. These visual traditions changed after 1400 when cartography began to construe geographic space in mathematical ratios expressed geometrically and viewed from a single perspective, an approach that also became normative in architectural and urban design. Place images – be they towns or vast territories – also served powerful military ambitions, political agendas, and commercial interests. Creating and reading these images relied on a collaborative process of reduction, choice, abstraction, and final transformation; all this required, in turn, the ordering of distance, direction, scale and shape as well as meanings inscribed by the various literary, artistic, and scientific cultures of the time. Mapping also encouraged action, such as travel and war; they also connected, in a psychic sense, the map viewer with the mapped place in terms of its social and natural settings. Maps and cityscapes thus combined mathematics, history, text, and aesthetics together to offer new ways to conceptualize and act upon space.

• 3 . MacDonald, W. L., 1986; Vasaly, A., 1993.

3Western traditions of founding towns go back to ancient Rome when building a town began through inscribing upon the land the foundational surveying lines. These first drawn schematics of towns in dirt remain in the literal meanings of the words « geography » (writing on earth) and « geometry » (measuring the earth). Other media, such as stone, parchment, paper, engraving plates, or, more recent times, aerial and satellite photography, eventually displaced raw earth. From the initial intersection of the two axial lines – the cardo and decumanus – evolved the familiar orthogonal grid pattern of the urban interior that emphasized connective regularity. It was as much a religious endeavor as a practical one, undertaken by priests – the first engineers – who performed the necessary auspices and took the vital measurements. The sacrosanct boundary along the urban edge, known as the pomerium, further demarcated the settled place from its wilder surroundings. The orientation and aspect of buildings and open places arrayed along streets converted the town into a performance space that mediated, indeed at time intentionally choreographed, the flow and interaction of the town’s inhabitants3. These basic characteristics of Western urbanism remain dominant to this day.

• 4 . Hyde, J. K., 1966.
• 5 . Lewis, S., 1987; Lavedan, P., 1954.
• 6 . Mortet, V. & Deschamps, 1913, p. 290-291.

4Mathematics continued to shape design practices and modes of representing secular and religious place in medieval France4. Much like the ancient Egyptians and Pythagoreans, the English monk Matthew of Paris, for example, envisioned towns as circular forms that connoted geometric perfection and cosmological harmony 5. Medieval treatises on geometry and surveying further betrayed their clerical origins by focusing on pictorial re-presentations of Creation, the mathematical relationships found in the Trinity, and philosophical arguments about ontology using geometry. Geometry, of course, formed one of the seven classical liberal arts, defined by Isidore of Seville as the art of measuring the world; medieval mason handbooks later echoed this topos when discussing the tools used to shape the proportions of earthly materials for use in construction. Legends circulated in these texts about its biblical origins as part of the prima theologia. St. Thomas Aquinas described God as the artifex principalis of all Creation, whose use of the divine scientia of geometry – represented visually by Thierry of Chartres, for example – could only be apperceived in the architect’s mind, not any earthly form. Yet in medieval sermons, the sin of arrogance often became embodied in the figure of the brilliant master builder Daedalus, doctor lathmorum, sapiens architectus, vir peritissimus, as related by, for example, the Dominican preacher and writer, Nicolas de Biard, in 12616. Whether master craftsmen in the building trades conceived of their buildings as reflections of the harmonic structure of the universe or as moral parables remain an open question, however.

• 7 . Zaitsev, E. A., 1999; Tannery, P., 1922; Mortet, V., 1904.
• 8 . Clagett, M., 1953; Clagett, M., 1966.
• 9 . Ullman, B. L., 1964.
• 10 . Bubnov, N. (ed.), 1899, p. 48-97, 310-313.

5The transmission of Hellenistic mathematics to medieval Europe is a vexed question. Latin translations of some of Euclid’s Elements can be found in Boethius’s Consolation of Philosophy and Pseudo-Boethius, for examples7. Fragments of Euclidean geometry could also be found in works of Roman encyclopedists and agrimensores, as well as Latin translations of Arabic works on science or directly from Greek manuscripts from Byzantium8. Some medieval monasteries apparently maintained these gromatic and Euclidean traditions9. The first Latin scholar to demonstrate any real understanding of Euclidean geometry was Gerbert of Rheims in the tenth century, though the Geometria Gerberti was likely not written by him10. These selected portions of Euclid’s Elements, excerpts from Roman surveyor manuals, and bits of ancient metaphysics in early medieval geometrical texts reflected eclecticism, not an integration of theory and praxis.

• 11 . St. Victor, H. (of), 1991.

6Improved understanding of applied geometry first became evident in Hugh of St. Victor’s Practica geometriae. Composed around 1120, this text encapsulated a late Roman and medieval computational tradition before it changed as a result of the influx of Greek and Arab science11. Primarily a teaching manual, it showed that surveying practices remained little changed since Antiquity. In the first two parts, Hugh of St. Victor demonstrated useful techniques employing astrolabes and other surveying devices. The first, altimetria, dealt with heights and depths, while the second, planimetry, treated the properties of planes. The last, cosmimetry, took up measuring the terrestrial and heavenly spheres. Subsequent treatises on practical geometry invariably retained this tripartite schema. This part of the manual was mainly a scholastic exercise rather than a direct application of surveying techniques, however. In this respect, the manual echoes Hugh of St. Victor’s broader interests in the relationship between technology and theology found in his better-known work, the Didascalicon.

• 12 . Rivolo, M.T. & Simi, A., 1998.

7After the twelfth century, the theoretical approach to geometry in the universities and the more practical one in grammar schools and craft guilds parted ways even further. Leonardo Fibonacci’s Practica Geometriae (1220), despite its title, reflects this separation as it treated primarily theoretical, not applied problems12. On the other side, craft manuals evinced little understanding about the Euclidean foundations of classical Greek geometry. While the teaching of geometry as part of the quadrivium formed a key component of medieval curriculums, masons did not gain their knowledge of it through formal schooling. For practical applications, constructive geometry only became based on Euclid later in the Renaissance, though that approach dominated in the scholastic geometry treatises. Two parallel systems, indeed cultures, of geometry thus existed until the sixteenth century.

• 13 . The portfolio consists of thirty-three parchment leaves and approximately 250 drawings, and it i (...)
• 14 . Shelby, L., 1972.
• 15 . « Car en cest livre puet o(n) trover grant consel de le grant force de maconerie (et) des engien (...)

8Craft handbooks in Latin and the vernacular reveal how medieval craftsmen used practical geometry to deal with problems in Gothic design, such as pinnacles, gables, and window mullions. Practical geometry became further distinguished from theoretical explorations of Euclid in the schools by its use of instruments manually manipulated to measure proportions and then to relate them together in a design ensemble. After 1300, a further distinction set surveyors, who simply measured, apart from craftsmen, who applied measurements to remaking materials. Such specialization likely arose due to the increasing scale of constructing cathedrals and turreted stone enceintes. Surveyors and masons learned their skills through mastering the oral traditions of the craft as apprentices; writing down such knowledge before the age of print was quite exceptional. This explains the unique importance of the Picard artist Villard de Honnecourt’s thirteenth-century sketchbook13. Villard’s familiarity with the building trades did not mean he was mason. Even so, his sketchbook provides a privileged glimpse into the cognitive foundations of the applied geometry used by medieval masons14. The few notebooks, like Villard’s, that survive reflect pedagogy more than construction techniques. In his preface, Villard emphasized that masons learn how to draw properly using geometry in order to make their work easier through these « techniques of forms » (li force des trais)15. His ideas fell in line with medieval practica geometriae that did not simply apply Euclid, but rather blended ancient methods of mensuration with craft training that substituted numerical examples based on metaphysically-informed ratios in lieu of actual demonstrations; they also emphasized convenient and manageable approximation over canonical precision using physical templates as well as compass and straightedge.

• 16 . Henry, C., 1882.
• 17 . Shelby, L., 1977.

9Later manuals approached problems in surveying and metrology more systematically than Villard. A late thirteenth-century French manual known as the Pratike de geometrie, written in the Picard dialect like Villard’s sketchbook, for example, tackled basic problems such as calculating the area of a field, the circumference of a rounded city wall, and the volume of a barrel.16 The Geometria deutsch and Büchlein by the fifteenth-century German mason, Matthias Roriczer, showed the thought-processes of medieval masons by laying out the steps to construct a heptagon and octagon using a compass and straightedge17.

• 18 . Henry, 1882.
• 19 . Shelby, 1977.
• 20 . Clagett, M., 1964.
• 21 . Curtze, M., 1896; Shelby, L., 1961 and 1965.
• 22 . Shelby, L., 1970.
• 23 . Jordanus’ treatise was finally edited and published by Nicolò Tartaglia in Venice in 1565 and se (...)

10Later manuals approached problems in surveying and metrology more systematically than Villard.  A late thirteenth-century French manual known as the Pratike de geometrie, written in the Picard dialect like Villard’s sketchbook, for example, tackled basic problems such as calculating the area of a field, the circumference of a rounded city wall, and the volume of a barrel.18  The Geometria deutsch and Büchlein by the fifteenth-century German mason, Matthias Roriczer, showed the thought-processes of medieval masons by laying out the steps to construct a heptagon and octagon using a compass and straightedge19. He also treated how to determine the circumference of a circle using the practical method found in Gerard of Cremona’s twelfth-century Latin translation of an original Arab treatise on the subject – a text that circulated widely in medieval European learned circles and contained elements of Archimedean mechanics20. None of these works, however, evidences any real understanding of actually how to use more sophisticated surveying instruments, such as the astrolabe or surveyor’s quadrant, or recently introduced Arab advances in trigonometry, all of which casts doubt on their practical utility for practitioners21. The actual knowledge of geometry necessary for effective surveying thus did not lie in these manuscript manuals but instead remained in the craft’s oral traditions of apprentice training22. The attempt by Jordanus Nemorarius to formulate in his Opusculum de ponderositate a theoretical statics to account for structural problems and solutions found in Gothic architecture was thoroughly exceptional23.

• 24 . James, J., 1982.
• 25 . It was a dynamic, evolving set of practices, not the frozen one suggested by G. Bachelard, 1964.

11These limits in craft manuals make the building they constructed the best sources of information about masons’ understanding of mathematics. Well-established indigenous traditions of using geometric principles in construction existed throughout in Europe after 1100. Gothic cathedrals, in particular, served as sites of experimental practice. The simple templates based on scaled geometric ratios that masons and carpenters used come closest to revealing their theoretical grasp of geometry’s underlying principles. The study of the simple templates based on scaled geometric ratios that masons and carpenters used to design these structures offers a place to begin. In its medieval form, the template served both an accounting function and as a design formula that integrated theory and practice. They also provided a sort of graphic functionalism that literally strove to create an image of heaven on earth, often in the form of the rose window of cathedrals24. Templates encapsulated all the essential information about design, while experience taught builders about the behavior of materials. Templates also had the advantage of portability, especially important for itinerant masons and carpenters who traveled from work site to work site25.

• 26 . Shelby, L., 1971; James, J., 1989.
• 27 . Conant, K., 1968 and 1959.

12More complex design schemas from the Romanesque to the late Gothic periods reflected the dynamic innovation of medieval craftsmen. Surviving medieval templates and, of course, the final built forms all demonstrate the use of a rational modular geometry to achieve a wide variety of forms and expressions26. Medieval constructive geometry relied on techniques for deriving ratios and proportion through the division of squares and circles with compasses, all which then became materially manifested on the template. As in planning medieval new towns known as bastides, discussed below, the use of the square also formed the basis of the ad quadratum or square schematism approach typical in Romanesque architecture. Especially popular were the use of the diagon (right rectangle) and auron (golden-section rectangle) derived from Platonic theories of forms, which carried further Christian symbolic significance27.

• 28 . Bucher, F., 1972.
• 29 . Davis, M. T.,  & Neagley, L. E., 2000.

13By the High Gothic period, these techniques had become so codified that builders increasingly departed from their norms using controlled irregularities or a forcible bending of established design procedures28. This approach skewing the canonical practice of repetitively rotating the square 45º had its advantages and drawbacks as clerical patrons sought allegory in church design while craftsmen gave priority to practical building problems. Even so, the geometry informing the designs of medieval masons reveals symbolic thinking at work in terms of ratios and proportions. In fact, most building projects indicate that clerical patrons and craftsmen collaborated to create designs at once functional and analogical. Symbolic significance could also be coded into the design and coordinated in the realization of its construction29. At once practical and symbolic, the basic rules of the ad quadratum method provided a highly flexible, economical, and effective means of constructing large edifices that eventually spread across most of Western Europe by 1400 as a new international style.

• 30 . Patetta, L., 1987, especially on the arguments in favor of the triangular approach put forward b (...)
• 31 . Ackerman, J. M., 1949.

14A new departure in medieval design schema came with the use of the equilateral triangle and its derivations ad triangulum, along side the rotating square approach. A famous debate even occurred in fourteenth-century Milan over these two approaches when it came to deciding upon the design of the new duomo’s cross-section, a debate that touched on both practical matters of ars and deeper considerations of the scientia of geometry30. The equilateral triangular schema became employed in fortification design at a variety of sites across Europe, including Poitiers, though only after 1400 did builders begin to realize the full potential of this approach31. It also later served as the basis for designing the new bastioned trace italienne of the sixteenth century, though in point of fact its origins lay in the craft techniques of late medieval builders.

• 32 . Baron, R. (ed.), 1966, p. 17.
• 33 . Frankl, P., 1960, p. 35-48 and 48-54.
• 34 . Tannery, P., (ed.), 1897.
• 35 . Turnbull, D., 1993.

15The projection method based on segments of circles as segment units, especially used for complex vault design, became the third important schematic approach in medieval architecture and the one that most fuelled the eventual development of projective geometry. The geometry of spherical solids, or stereometry, was the art of measuring and computing the cubical contents of bodies and figures, an essential skill for architects and stone masons. Scholarly treatises, such as Hugh of St. Victor’s, first raised these issues when discussing cosmimetry, but not much beyond the rather elementary problem of calculating the earth’s circumference32. The earliest craft handbook to address stereometric problems is attributed to the so-called Magister 2 in the 1230 sketchbook of Villard. It used a fairly simple squaring technique consisting of uniformly excising triangular or curved wedges in order to realize the desired shape33. Villard’s treatment clearly pointed the way to the more complex treatment of stereometry taken up briefly in Robertus Angelicus’s Tractatus quadrantis, written sometime before 1276 in Montpellier34. Medieval masons solved stereometrical problems by the practical manipulation of geometrical forms in templates using tools and instruments to work materials, usually stone, wood, and glass35. Indeed, some of the resulting geometric designs for towers, vaults, tracery, and profiles closely resemble the polygonal forms found in fifteenth-century ideal city design plans of Antonio Averlino, also known as Filarete, and celebrated bastioned enceintes of the sixteenth century. Continuities in medieval design thus persisted into the early modern period.

• 36 . These surveying techniques continued the practices established in Antiquity. See Campell, B., 20 (...)

16Craft traditions of applied geometry found in architecture carried over into medieval urban design, especially evident in the new towns in southern France known as bastides. In fact, the underlying geometry of bastides derived from their ceremonial foundation with the fixatio pali, which consisted of planting a large pole on the spot, destined to become the market place. Affixed atop these poles were banners displaying the coats-of-arms of the king or great lord who sponsored the new town as laid out in the founding charter known as the paréage. The lord’s agents then supervised the surveyors, known varyingly as lotisseurs or arpenteurs, as they traced the bastide’s initial layout from this centrally fixed point to ensure conformity with the terms of the paréage36. A more detailed process known as perticatio then ensued to delineate the building parcels (local or ayral), garden plots (ort, casal, or casau), and the surrounding arable for cereal cultivation and vineyards (arpans), separating each zone with ditches and earthen mounds. The measurements had to be done very precisely because they determined a person’s tax rates. Beginning as a geometrical trace, a bastide’s design became literally etched into the soil which then supported its growth.

• 37 . Leblond, H., 1987; Pulol, F., 1991.  
• 38 . An extensive literature exists on individual bastides; however, for synthetic, comparative studi (...)

17While planned, bastides did not subscribe to any set, ideal scheme but rather responded to pragmatic considerations of site topography, the tension between orthogonal and radial features, and the placement of the parish church37. Some bastides, such as the ones found in Quercy, became characterized by a single main avenue bisected by streets, much like rungs on a ladder. Another type found in Aquitaine usually consisted of eight residential blocks formed by longitudinal avenues bisected by streets to form an orthogonal grid. An additional type was an elongated yet still orthogonal grid laid out along a principal avenue flanked by two side streets. The use of right rectangular and golden-section rectangular schemas found in ad quadratum Romanesque church design occurred as well in bastides, such as Sainte-Foy-la-Grande and Grenade-sur-Garonne. Another type of bastide form found in Gascony possessed a central square formed by two sets of intersecting perpendicular avenues off of which went side streets to form the residential blocks38. Some bastides, as seen below in the case of Montpazier, used the ad triangulum method to realize an orthogonal layout through the application of Pythagorean triangles or triples. The sides of such right-angled triangles were whole numbers, usually units of 3:4:5. The surveyors used a simple cord with twelve knots, all equally spaced in intervals called a coudée (« elbow-length »), to lay out the triangles. Each coudée was approximately 2.5 feet in length, which made the cord some thirty feet long. Using the Pythagorean Theorem (a² + b² = c²), the surveyors could then easily extrapolate the larger overall dimensions of the planned bastide before they even walked onto the site. Besides its simplicity, this application of practical geometry to planning bastides also offered flexibility, for it could be readily adapted to fit just about any terrain.

• 39 . Salamagne, A., 2001.
• 40 . Wolfe, M., 1995. On Italy, see D. Friedman, 1988.

18The establishment of bastides continued fairly vigorously until the mid-fourteenth century, when the staggering population decline caused by the Black Death removed a powerful incentive to build new towns. With upwards of two hundred new towns founded in greater Occitania after 1200, the skills and mental outlook used to build these planned communities were quite ubiquitous, as was also the case in northern France, where urban expansion and renewal projects proliferated through the Hundred Years War39. Here, too, master masons and carpenters applied geometry to rework the existing physical fabric to accommodate huge new cathedrals and turreted enceintes40.

• 41 . Municipal artists form the focus of F. Baron, 1969.

19An excellent source to trace their techniques is municipal account books. These records tend to be spotty until the fourteenth century, where after they assumed more regular and continuous form. Construction records from Amiens, a frontier town in Picardy, for example, again reveal the continuities in indigenous building and design practice into the sixteenth century. Admittedly, these books mainly tabulated the payments for goods and services rendered to the city, so recovering the practices of the builders must be inferred. Several different kinds of accounts existed. Up to 1500, masters of works (maistres des ouvrages) kept track of such payments. Master masons mainly served in these positions, though occasionally a master carpenter filled it. Masters of works oversaw contracts and accounts; among the skilled artisans they hired were visual artists (peintres)41. They provided all this information to municipal tax receivers (receveurs) drawn from the urban elites who dominated municipal government. The receveurs actually kept the books to balance expenditures with available revenues. After 1500, a separate official, the comptroller (contrerolleur) representing the interests of the royal provincial governor, audited the accounts, a copy of which went to Paris for inspection. This new central oversight showed the rising influence of royal officials in directing local public works projects. While urban design practices remained essentially medieval, the changing composition of these records reflected the increasing bureaucratization of municipal government under the tutelage of the monarchy.

• 42 . B.M. Amiens EE 270. See also M. Wolfe, 1998.
• 43 . B.M. Amiens CC 113 (1528), fol. 12v for an example of such a « forme d’amendement » contract for (...)
• 44 . B.M. Amiens DD 568 (1573), fol. 273v: « … le grez preparez par les brizeurs estans aux champs, p (...)

20Judging from the built results, masons obviously used practical geometry to plan and execute the projects beyond cathedrals, such as fortifications. Master masons stood out as the unquestioned resident experts on urban design and construction until the mid-sixteenth century when Italian engineers begin to show up in the record. Yet the arrival of these supposed experts from the south, as discussed further below, complemented rather than supplanted these local experts. In 1565 in a ceremony akin to founding a bastide four hundred years earlier, municipal and royal officials assembled to watch the master mason, Jean Bullant, dresser l’ouvrage of a new bastion. He used stakes and rope to outline the royal state’s imminent imprint upon this portion of urban space42. Masters of works used these same techniques, sometimes with special chains made of one-inch links, to verify the proper completion of contracted work, measuring it all out to the last pouce (inch) of each toise. Excavation projects occasionally offered calculations of volume, though not always43. Traditional practical mathematics comes through elsewhere in these records. Visits to nearby forests to select timber, quarries to choose stone, and even brick works required precise calculations to procure no more than was necessary. Cutting stone still utilized templates to ensure delivery of properly dressed stone44. While new techniques of practical geometry could be found in contemporary publications, the actual methods used to take measurements had changed little since the twelfth century.

• 45 . Smith, P. H., 2004, p. 59. Smith largely concentrates on Flemish and German areas, though the cu (...)
• 46 . On the development of the royal fortification service, see the first two chapters of H. Vérin, 1 (...)

21This continuity thus confirms Pamela Smith’s contention that from 1450 to 1650 the exploration and experience of nature in Europe was dominated by « artisanal epistemology », the conviction that practice, not theory, was the primary mode of engagement with the world45. These artisans, who worked out of a longer medieval tradition, created what Smith calls a « vernacular science of matter », an immensely powerful and broad reaching engagement with nature rooted in the conviction that « knowledge is active and knowing is doing ». Rational modular geometry was part of this science and formed an essential point of continuity for design schemata – both in terms of conceptualization and representation – all the while new developments in projective mathematics and cartography got underway during the Renaissance. While construction practices still used applied mathematics in medieval ways, the novelty rested in the greater level of accountability to governmental authority and the emergence of new forms of pictorial representation. An embryonic royal fortification service certainly existed by 1500 composed of local experts, municipal officials, and an ascending hierarchy of royal officers and agents stretching from Amiens and other such towns to the Louvre46.

• 47 . See D. Turnbull, 2000, on these different visual cultures.

22Medieval constructive geometry and the new projective geometry were far from antithetical. Indeed, the eventual shift after 1500 to projective geometry – and with it, linear perspective in the visual arts – occurred because of craftsmen’s long familiarity with rudimentary Euclidean geometry. The first key change among master craftsmen was the eventual decision after 1450 to use drawings to express – and presumably solve – design problems. Indeed, novel forms of cognitive mapping joined new theories with older practices to spur new ways to build and depict cities and the wider world47.

• 48 . This is a vast subject. See D. Woodward, 1985; Smith,C. T. & Hawthorne, J. G., 1974; Chevalier, (...)

23Before 1400, visual representations of townscapes tended to draw on generic, interchangeable conventions. Chorographic views distinguished a town’s identity by emphasizing the profile (0º or head on) defined by its walls, gates, and principal buildings. Like Villard’s mapping of human faces, chorographic views sought to reveal a place’s particular intrinsic character. They also underscored a town’s enclosed nature, its walls a barrier against a hostile world. Manuscript copyists and illuminators lacked the requisite mathematical expertise (namely, trigonometry) and knowledge of surveying methods to go beyond this external view. They also likely never even set eyes on most of the places they depicted, for their emblematic views situated a town in a cosmological rather than geographic setting. This metaphysical aspect informed the techniques of constructive geometry and came through most clearly in medieval mappae mundi centered on the axes of Rome, Jerusalem, and Paradise48.

• 49 . B.M. Amiens CC 221 (1585), fol. 88v, for example.

24New ways to visualize urban space in France began not with the master masons, but rather the men referred to as peintres (painters). Since the late Middle Ages, these artists performed an eclectic mix of services for towns, from banners and elaborate stagecraft for public festivals and ceremonial entries to depictions of patron saints and coats-of-arms on walls, towers, and gates. They also created tableaux of written public prohibitions and exhortations to citizens to do their duty. They even prepared images for public display of criminals executed either in person or in effigy49. Municipal artists in Amiens, and presumably other urban areas in France, did not need to await the arrival of linear perspective techniques from Italy to create such images. To be effective, painting in public places required optimal sight lines and proper plotting. These tasks, in turn, required the use of some basic geometry and further suggest that the artist had to conceive of the actual townscape as a kind of canvas on which to leave his mark.

• 50 . Fournier, G., 1973.

25It was then only a short step to devising a design plan for discrete features of the townscape. A celebrated example is the collection of exquisite images in the Armorial of Guillaume Revel. Revel was a herald from Auvergne who served Charles vii. Its 400 drawings, most left unfinished, depicted the places with the coats-of-arms of lords in Auvergne and Forez under the king’s suzerainty50. Cadastral surveys from the early nineteenth century attest to the rough accuracy of Revel’s townscapes, situating notable public buildings, plazas and residential areas in a proper ensemble. Amateurish, but still effective, techniques of draftsmanship and color lent these portraits a sense of perspective within the confines of the walls and in relation with the hinterland. While Revel evidently did not use practical mathematics to execute his views, he clearly demonstrates a rudimentary understanding of geometrical space.

• 51 . « avoit esté chairché, veu, tézé, pourtraict et rédigé par escript en ce présent livre de la mai (...)

26Another well-known example is Jacques Le Lieur’s Livre des fontaines for Rouen. Composed in the 1520s, it also reflects the shift from chorographic profiles to visual depictions of a town’s interior. Le Lieur was city counselor who prepared a study plan to bring fresh water to Rouen. His images of the town and plans for aqueducts mainly served utilitarian purposes, but left room for aesthetic expression. He attests in the copy he presented to the city that it « had been researched, viewed, measured, portrayed and edited in the present book by the hand of me, Le Lieur »51. The accompanying narrative provided vital information about the slopes and heights of geographic features critical for this gravity-fed water system. The use of practical mathematics can thus clearly be inferred from the texts and its image as well as the technological prerequisites of the project itself.

• 52 . B.M. Amiens, CC 97 (1520), fol. 198v: « A Adrieu de Moncheaux, paintre, la somme de lxiii s., po (...)
• 53 . B.M. Amiens, CC 86 (1509), fol. 86.
• 54 . B.M. Amiens, CC 98, (1521), fol. 90 and CC 99 (1521), fol. 15v.

27Municipal account books in Amiens refer to such images of the town in a 1520 entry about a payment to Adrieu de Moncheaux, a local artist, for preparing design schematics of Amiens’ fortifications52. Yet this shift toward projective geometry to depict and refashion urban space began even earlier. Extensive records on roofing and gabling projects from the late 1400s, for example, indicate the use of at least rudimentary geometry to calculate the area and pitch of surfaces. Inspection teams not only visited the walls and towers, but actually traveled outside the fortified periphery to higher ground to gain « les veus par dehors » – a perspective that soon dominated graphic representations of towns53. Pragmatic considerations outweighed aesthetic ones until the 1540s when special presentation drawings to official patrons of the city began to be made. After 1550, builders began to use these images to guide construction projects. Until then, they still principally served to communicate basic design information to city officials54. Local artists became increasingly mentioned in the group who visited and consulted about Amiens’ fortifications.

• 55 . Firpo, L., 1974; Lang, S., 1952; Rosenau, H., 1983; Le Mollé, R., 1972.
• 56 . That said, religious symbolism continued to be important in visual representations of Renaissanc (...)
• 57 . Kemp, M., 1990, p. 9-52.
• 58 . Barkan, L., 1999, explores this new impulse to come into physical contact with both a place and (...)
• 59 . Buisseret, D., 1992.

28During the Renaissance, modes of representing urban places, together with geographical and cosmological space, underwent profound change but still retained these earlier emblematic, metaphysical features. Neoplatonic ideas informed the Vitruvian figure and images of the ideal city, capturing a vast place in a composite image with few words55. New secular meanings gradually overlaid these metaphysical ones as more scenographic representations situated towns in « real time and real space »56. Not coincidentally, new levels of mathematical expertise as well as improved surveying techniques became attained as the Renaissance notion of disegno moved from the studio into the world at large. Yet the governing geometric logic for artists remained rooted in medieval practical mathematics, even though improved techniques of measurement led to subsequent abstraction57. As a result, first in Italy and then in France around 1500, images of towns quite literally came down to earth as interest in their physicality and history became more pronounced58. In a sense, the artistic ability to integrate specific places into a broader geographic, as opposed to metaphysical, setting provided a political analog and reinforcement of the process of state formation and territorial consolidation already underway59.

• 60 . Pelletier, M., 1989.
• 61 . Edson, E., 2007.
• 62 . Besse, J.-M., 2003.
• 63 . On the connection between optics and these new forms of visual representation, see M. Kemp, 1990

29Claudius Ptolemy’s Geographia, composed in the second century, played a central role in the change. It circulated widely in manuscript form after it arrived from Byzantium in 1400 and was eventually published in the 1480s. Use of the Ptolemaic system of coordinated axes in cartography offered a new way to abstract space60. As a result, the terrestrial globe became increasingly considered as both a physical sphere and as a human oekoumène, a place to live61. These two approaches took their cue respectively from Ptolemy and then Strabo, Pomponius Mela, and Pliny, all of whose texts became published by 1500. The first, more mathematically and astronomically oriented one, mainly concerned measuring the size of the planet, determining its position in the universe and accurately locating place according to latitude and longitude. The second, both literary and descriptive in its style and historical and encyclopedic in its scope, looked at the men who live on the planet, at their environment, their traditions, beliefs, and forms of government. Yet Ptolemaic geography and Straboian chorography steadily converged in sixteenth-century cartography, as mapmakers sought to meld their mathematical and aesthetic features into accurate, yet visually pleasing forms in such works as Sebastian Münster’s influential Universal Cosmography, published in 154462. The quest to create these new maps proved difficult to realize. Ptolemy demonstrated a method to project space using a grid and basic geometry to represent territory. While mathematical precision only lay in the future, visual artists could use an array of illusionist techniques to lend perspectival verisimilitude to new-style maps63.

• 64 . Bartoli, M. T., 1978; Nuti, L., 1994; Pinto, J., 1976; Frangenberg, T., 1994.

30These innovations in cartography affected depictions of towns after 1450 first in Italy and then elsewhere. New modes of portraying urban space include ichnography, or plan; orthography, or section; and scenography, or view. Scenography was furthermore divided into two types: classical profile and military axonometric views. Each view held advantages and limits in urban design, especially used in fortifications, depending on which basic perspective it employed: profile (0º), equestrian (10º), oblique view (30º), bird’s eye view (45º), cartographic (60º), and ichnographic (90º). Reconciling the new ichnographic and older profile modes required using multiple points of view and the elaboration of new ways to symbolically represent space and the objects found in it64. Take singly, each form of visual depiction could easily rely upon on medieval traditions of applied geometry; combining different points of view, however, created challenges only solvable by more advanced mathematical theory.

• 65 . Rose, P. L., 1975.

31Changes in urban design theory and practice sprang from new directions in Renaissance mathematics and metaphysics. Renewed interest in Neoplatonic and Hermetic philosophies posited mathematics as a manifestation of God’s immanence in the world, revealing harmony and proportion at every level of creation. This flowering of Euclidean geometry gradually opened up new ways of « physical » thinking that linked harmonic consonances perceived in visual art and heard in music. New forms of notation and graphic representation broadened and transformed the means of mathematical communication which, in turn, reshaped mathematical thinking and practice, including pictorial representations based on perspective and surveying. The crowning achievements of Renaissance mathematics in these domains were the revival of Archimedean mechanics to explain motion and the development of algebraic geometry to solve problems related to complex three-dimensional forms65.

• 66 . Davis, M. D., 1977.
• 67 . These published practitioners included Daniele Barbaro, Giovanni Zamberto, Federico Commandino, (...)
• 68 . See in general J. V. Field, 1997 as well as the more specialized work of P. DubourgGlatigny, 199 (...)

32The relationship between mathematics and visual art deepened in its sophistication in the late 1400s. Piero della Francesca’s Libellus De Quinque Corporibus Regularibus, composed in the 1490s, brought a very high level of mathematical rigor to perspective techniques that became popularized, with some error, in Luca Pacioli’s De Devina Proportione, published in 150966. Subsequent work on three-dimensional geometry developed in the later writings of other Italian mathematician theorist-practitioners who strove, with varying degrees of success, to graft the complex mathematics of three-dimensional geometry onto the practical roots of the artist’s procedure67. Most of them had extensive experience in ateliers and formal, if not very extensive, training in Euclidean geometry. They also evinced a keen interest in the physiology of sight. Cartography was an area of especial interest for such applications, as were ballistics and fortification design. These works exerted extensive and enduring influence on design theory and practices in France not long after their original publication in Italy68.

• 69 . It should be mentioned that Charles de Bouelles published a mathematical treatise in Latin in 15 (...)
• 70 . Pierre Forcadel, 1564 and, 1565. Forcadel also translated several works on mechanics and hydraul (...)

33The reception of these ideas and techniques in France relied upon the already well-established interest there in Euclidean geometry. Euclid’s Elements was only surpassed by the Bible in terms of editions published during the late fifteenth and sixteenth centuries. The initial 1482 Latin pu-blication in Venice by Erhard Ratdolt was quickly followed by numerous re-printings. An edition appeared as a French imprint in Paris in 1516 when Jacques Lefèvre d’Étaples, though better known now for his biblical commentaries, edited a Latin edition. Lefèvre d’Étaples was closely connected to erudite circles affiliated with the Valois court69. A French translation of Euclid’s Elements finally appeared in 1564 by Pierre Forcadel of Béziers, who was lecteur du roy ès mathématiques70. Forcadel had spent time in Italy before returning to Paris where Pierre Ramus had him named professor of mathematics at the Collège Royal in 1560. His translation included long commentaries from the courses he taught, thus suggesting it was intended for his students. Forcadel already had an established reputation, having published in French several books on arithmetic in the mid-1550s.

• 71 . On these developments, see P. L. Rose, 1975. Rose evinces little interest in the applications of (...)

34Much more sophisticated mathematical theory became inspired by the publication of the thirteenth-century mathematician, Johannes Sacrobosco’s De Sphaera. This work elucidated the spherical geometry underlying the mathematical astronomy of Ptolemy. It first appeared in print in Ferrara in 1472 and then in a Latin edition in Venice in 1485, again edited by Erhard Ratdolt. As with Euclid, a Paris edition was published in 1507, again edited by Lefèvre d’Étaples. A French translation appeared in Paris in 1546. A woodcut of a medallion in this last edition suggested a connection between Sacrobosco’s ideas and townscapes. The works of the fifteenth-century German mathematician-astronomers, Georg von Peurbach and Johannes Müller, also known as Regiomontanus, contributed further advances to projective techniques; their works underwent much the same publication track as those of Ptolemy, Euclid, and Sacrobosco. Numerous editions of all these essential works disseminated a new projective mathematics across Europe that only became superseded a century later by the powerful new algebraic geometry of Girard Desargues and René Descartes71.

• 72 . Oronce Finé, 1532. This work appears in Italian as Della geometria sometime shortly thereafter. (...)

35This new projective mathematics first became applied in France to cartography, not townscapes, in the 1520s in the work of Oronce Finé, a mathematician and cosmographer from Briançon who held a chair in mathematics at the Collège de France. Finé fell back on geometric terms, depicting France as a diamond or a square (carré), a form rife with symbolic import, as later would be the hexagon72. Finé’s carte gallicane of 1525 reflected France’s Italian ambitions at the time in its inclusion of Piedmont and Lombardy. Finé drew inspiration from the map of Gaul published in Florence in 1481-1482 by the Florentine humanist Francesco Berlinghieri (1440-1501), who extracted his GalliaNovella from Ptolemy’s Geographia, which he translated into Italian verse. Subsequent maps of France include the Frantia, published in Ulm in 1482, and the Moderna tabula Galliæ, which appeared in Strasbourg in 1513. The latter encompassed the left bank of the Rhine in « modern Gaul ». These maps also replaced ancient place names with their contemporary equivalents and employed information from portolan charts to depict coastlines.

• 73 . Oronce Finé, 1525; Broc, N., 1983.

36Finé found inspiration in these maps for his map of Gaul, which went through five editions between 1525 and 1557. It incorporated the left bank of the Rhine and northern Italy as parts of « greater Gaul », with the Alps as little more than mole hills. His map improved upon Ptolemy’s trapezoidal projections by using longitudinal and latitudinal calculations based on his new measuring instruments, such as the « méthroscope géographique », an astrolabe modified by the addition of a compass. In his 1530 work Cosmographie, Finé provided the coordinates of 180 French towns, some taken from Ptolemy and others from his own direct observations or corrections of Ptolemy. He included both ancient and contemporary toponyms. His cordiform map of the world employed the new forms of projective mathematics discussed in his De Mundi Sphere, published in 1542, using optics and conics to transcribe geometrically in two dimensions any object in space73.

• 74 . Sébastien Münster, 1545; Drapeyron, L., 1889.

37Finé’s work soon inspired imitators and contributed to the first atlases, known as Theatrums. His map of Gaul, for example, was reprinted in Sebastian Münster’s celebrated Das gantz Frankreich, printed in Basel in 154574. These maps tended to be much less abstract than the two-dimensional maps of today, replete with symbols for water, forestland, built-up areas, etc. Instead, cartographers depicted areas from a 45° bird’s-eye-view, using a slanting projection to overlook a particular panorama. The same techniques also became used in townscapes, which appeared in these atlases. Such a view easily allowed rendering of prominent features, such as towns, buildings, and other landmarks three-dimensionally.

• 75 . Schulz, J., 1978; d’Ailly, A. E., 1934.
• 76 . Jean d’Ogerolles, 1553, p. 11. J. d’Ogerolles reused many of his woodcut blocks in A. de Pinet, (...)

38Applications of the new projective mathematics to cityscapes began with the famous early sixteenth-century image of Venice by Jacopo da Barbari (1440-1515), while the Dutch artist-cartographer Cornelis Anthonisz (c. 1500-1553) produced a bird’s-eye-view of Amsterdam, first as a painting and then in 1544 as a woodcut75. In France, cityscapes from an ichnographic perspective developed alongside maps of territories. Guillaume Gueroult’s Épitomé de la corographie d’Europe, published in Lyon in 1553, together with Antoine Du Pinet’s Plantz, pourtraitz et descriptions, printed in Lyon eleven years later, contain several dozen woodblock prints of major cities by Jean d’Ogerolles that utilize this dual perspective76.

• 77 . Derens, J., 1980; Boutier, J., 2002.
• 78 . Braun, G. & Hogenberg, F., 1572. A modern edition of this work is J. Goss (ed.), 1992. On plans (...)

39The 1550 plan of Paris by Truschet and Hoyau printed in Basel offered a richly detailed visual representation of the metropolitan area, as viewed from the west (church facades head on), depicting the allotments, buildings, and streets in the city and adjoining suburbs. Accompanying texts embedded in the maps identify specific places; other texts inscribed in cartouches celebrate the city grandeur, antiquity, and noble inhabitants.77 Georg Braun’s copper etching of Paris, published in 1572 in his Civitates orbis terrarum, assumed the same western view and closely followed the format of the Basel plan with a few differences. Most noticeable is the prominent coat-of-arms of the city that stands in lieu of the royal one. The Braun plan also makes considerably less of an effort to portray the suburbs, except the important one of Saint-Germain-des-Prés. The delineation between inside and outside the city thus becomes sharper, even though the fortified periphery again seems excessively conventional and indistinctive78.

• 79 . Harvey, P. D. A., nd. See also the vital work another sixteenth-century cartographer, Abraham Or (...)

40Although the application of trigonometric measurements became more common by 1550, thus making it technically possible to create geometrically accurate maps, the custom of rendering elevations in slanting projection continued, largely because viewers expected such images. The conventions of the bird’s-eye-view allowed artist-cartographers to tackle a variety of compositional and perspective problems by striking a balance between orthogonal exactitude and the comforting familiarity of the profile view. It also injected a sense of volume in the otherwise « flat » two-dimensionality of the orthogonal and profile views. In the seventeenth century, all three views became commonly used in representations of cities, sometimes combining all three viewpoints in one print—a technique pioneered in the illustrations commissioned by Braun and Hogenberg. The demand for this variety of topographical material grew after 1600, especially as the royal fortification and engineering service developed79. These perspective renderings of maps and cityscapes, as well as the construction of new-style fortifications à la trace italienne, all relied upon a long, complex series of surveys of the place to be depicted or built. Thus, visual depictions of urban space in France prior to 1600, mainly intended for commercial distribution and official consumption, bear ample evidence of the impact of the new projective mathematics as well as the abiding influence of long-established surveying techniques.

• 80 . Frangenberg, T., 1986; Brion-Guerry, M., 1962, gives a French translation of the 1505 Latin text (...)

42Innovations in surveying and design can be found in the many arithmetic primers and geometry manuals printed after 1500. Without denying Italian influences, French works on applied mathematics provided sources which builders could readily consult. Most of these works contained sections on solving common surveying problems. While Italians certainly blazed new trails on perspective geometry, the context in which perspective was taken up in France exhibited a less abstract nature than in Italy or Germany at the time, and was instead more closely associated with the representation of realistic architectural space. This difference can be seen in the work of an obscure canon from Toul, Jean Pélerin (1435-1524), who wrote under the name Viator (voyager). In 1505, he published the first French work exclusively devoted to Euclidean geometry, De Artificiali perspectiva. In it, he offers perspectival renderings ranging from the grandiose interiors of Notre-Dame de Paris and the Palais de Justice in Rouen to the mundane interior of his own home. These illustrations hold no relationship with the perspective geometry discussed in the text, however80. Pélerin’s method represented a much simpler approach to linear perspective than the ones already developed in Italy. He presented a formulaic but effective three-point system using a series of pyramids with the apexes along the horizon (or « pyramidal ») line. Indeed, Pélerin made no effort to maintain the illusion of a vanishing point in the rear of the picture. His form of perspective image thus did not record the objective world but instead reproduced it through a specific method of construction. Pragmatic applications mattered more for him that theoretical accuracy.

• 81 . Hart, V. & Hicks, P., 1996; 2001. Serlio relied in particular on the earlier work of Baldassare (...)

43Advances beyond Pélerin’s method came from the influence of Italian painters brought to France in the 1530s to work on Fontainebleau. They include Rosso Fiorentino, Francesco Primaticcio, and Giacomo Barrozi da Vignola, all proficient perspectivists whose work can still be seen today in the palace. Most important, however, was Sebastiano Serlio, whose books on architecture became so popular in France that he became François i’s chief architect in the early 1540s. Serlio devised another easily accessible technique to render perspective for stage scenery. He provided particularly good guidance for properly using a compass to delineate curved lines in ovoid spaces81.

• 82 . While there is no modern edition of Leçons, a modern edition of Du Cerceau’s work on architectur (...)

44Subsequent draftsmen in France, such as Philibert de l’Orme and Jacques Androuet du Cerceau, the latter of whom wrote a stand-alone work on perspective, expanded further upon Pélerin’s approach. Androuet du Cerceau’s Leçons de perspective positive, published in 1576, complemen-ted his earlier collection on architecture and the building trades, Excellents bâtiments de France. In it, he presents sixty lessons in a do-it-yourself format, complete with easy-to-follow illustrations to guide the student or builder82. Du Cerceau purposely omits mention of Euclidean geometry in his primer, as he strove for readers to master applied technique rather than bookish knowledge. By contrast, the application of more advanced perspective geometry comes through in the work – both built and written – of Philibert de l’Orme.

• 83 . J.-M. Pérouse de Montclos, 2000 surpasses A. Blount’s 1958 study.
• 84 . Rykwert, J., 1996, p. 364-376.

45A close look at the trompe that L’Orme built at the château of Anet between 1549 and 1551, which became featured in his Premier tome de l’architecture sixteen years later, reveals the ongoing innovations possible in the craft tradition.83 As the name suggests, a trompe aimed to appear to flout the laws of gravity. The techniques featured by L’Orme grew out of the stonecutting traditions of the masons; his father, in fact, was a master mason.84 In the introduction to the Premier tome, L’Orme insists on the importance for both architects and master masons to know geometry. Indeed, no built work can begin without the fundamental—and foundational—act of squaring up. In a clumsy nod to the metaphysical tradition, he supplemented his pragmatic advice with banal musings on the symbolic meaning of geometric forms.

• 85 . Cousin’s son, Jean, followed in his father’s trade as an artist-writer, publishing a short work (...)
• 86 . Jean Cousin,1560 [1974]. Auclair, V., 2003.

47The major theorist of perspective in sixteenth-century France who perfected the three-point linear technique was Jean Cousin. In his 1560 Livre de perspective in Paris, he achieved complex results using fairly simple means long used by craftsmen. Originally from Sens before taking up residence in Paris, Cousin was an artist who worked in a number of media – fabric, glass, and engravings – all of which benefited from his knowledge of perspective and basic Euclidean geometry. In fact, Cousin bridged the craft world of applied mathematics and the learned culture of court, much as Du Cerceau and L’Orme did85. What Cousin’s treatise lacked in originality was made up for in clarity of expression and practicality of application86.

• 87 . van Egmond, W., 1988. On Chuquet, see G. Flegg, C. Hay & B. Moss (eds.), 1985.
• 88 . Étienne de La Roche, 1520. Pacioli was known for Summa de arithmetica geometria proportioni e pr (...)

48Works devoted mainly to mathematical theory also contained much practical information readily applicable to problems encountered by builders and merchants. The first such book printed in French was Larismetique by Étienne de La Roche, published in Lyon in 1520. From Villefranche, La Roche studied under Nicolas Chuquet, known for his work in algebraic notation, and taught commercial mathematics for twenty-five years in Lyon. Lyons offered a well established book market which introduced mathematical work by Italians to a French audience87. Larismetique was essentially a pastiche of basic algebra drawn from Chuquet, Luca Pacioli and Philippe Frescobaldi, a banker in Lyon88. It included solutions using algebra for problems familiar to merchants and builders, such as calculating volumes.

• 89 . Charles de Bouelles, 1542, reprinted in 1547 as Géométrie practique.

49True advances in geometry only came in 1542 when Charles de Bouelles published his Géometrie practique. Bouelles was a canon at Noyons and humanist savant who had studied mathematics and theology at the University of Paris in 1493, where he became a close associate of Lefèvre d’Étaples.89 Reprinted four times in the next twenty years, he aimed to teach his readers the rudiments of Euclidean geometry, covering general principles, angular figures, angular figures with circles, the quadrature of the circle, dimensions of solids, the cubification of the sphere, the properties of sound, wind and water, the proportions of the human body and the uses of geometry in astrology. Oronce Finé provided the book’s many woodcuts, visually illustrating principles logically notated in the text. Bouelles provided a much more powerful combination of advanced projective geometry focused on a variety of practical problems.

• 90 . Jacques Peletier, 1549, with numerous editions in French and Latin. His other mathematical works (...)
• 91 . Hooykaas, R. & Minnaert, M. G. J. (eds.), 1970. Andersen, K., 1990.

50Another important contributor was Jacques Peletier, who wrote a primer on arithmetic in 1549, a short introduction to algebra in 1554, and later translated several of Finé’s manuals on practical geometry. Peletier hailed from Le Mans and had studied at the College de Navarre, where his brother Jean was a professor of mathematics. He frequented the literary circle of Marguerite de Navarre and had served as René du Bellay’s private secretary in the mid-1540s. In 1573, he composed his own work on geometry, De l’usage de géometrie, explicitly aimed at practitioners. After a brief résumé of basic Euclidean geometry, Peletier provides several dozen examples of practical applications for surveyors, such as measuring a building, a plot of land, or a mountain90. He also brought to bear a more probing acumen by pointing out flaws in some of Euclid’s original proofs. The noted Flemish fortification expert, Simon Stevin, published a French version of his own treatise on arithmetic and algebra in Leiden in 158591. His work on perspective looks at a number of innovations, such as the case of calculating the perspective for making a drawing on a canvas which is not perpendicular to the ground, and the case of inverse perspective.

• 92 . Valentin Mennher, 1565.
• 93 . Jacques Focard, 1555 & Auger Ferrier, 1559. Oronce Finé, 1551, published a short treatise on ast (...)
• 94 . Abel Foullon (1513-1563), 1555 [1561]. Foullon’s manual was translated into Italian in 1564. As (...)
• 95 . Jean des Merliers (1575). Merlier’s piece in the Bouelles’ 1608 edition is entitled L’art de me (...)
• 96 . Jean Pierre des Mesmes (1560).
• 97 . Jacques Chauvet, 1585 and 1585.
• 98 . Alexandre Guybert, Paris, 1580. It was subsequently republished in the seventeenth century in va (...)

51Treatises on advanced instrumentation further demonstrated this conjunction of mathematical theory and design practice. A Fleming named Valentin Mennher, for example, wrote a short work on practical geometry in French in 1565 detailing, among other matters, how to use geometry to prepare account books – an important task as procedures became more bureaucratic.92 Surveying formed a main topic in manuals on the use of astrolabes, mainly for astrological charts of the heavens, by Jacques Focard and Auger Ferrier93. Abel Foullon published another guide to a holometer, an instrument used to measure all sorts of angles, which appeared in the 1550s. Foullon served as director of the royal mint under Henri ii and later held the title of ingénieur du roy. Foullon even claimed he could teach the basics of measuring distances to persons lacking any mathematical training94. In 1573, Jean des Merliers, a mathematics teacher in Amiens, published a short manual describing the proper use of the square to solve practical geometry problems, a subject he further explored in another short tract in 1575. Another work by him, though only published posthumously in 1608 in a revised edition of Bouelles’s Geométrie pratique, explicitly demonstrated the Euclidean underpinnings of surveying95. This same edition also contained a short piece by Jean Pierre des Mesmes on calculating heights, distances, and volumes96. Dominique Jacquinot and Jacques Bassantin collaborated on a manual, L‘usage de l’astrolabe, published in Paris in 1573, which described how to use an astrolabe and calculate the mathematical properties of spheres. Jacques Chauvet, about whom little is known, wrote a short manual in 1585 describing how to use a new surveying gadget called the cosmomètre as well as a manual on arithmetic97. Finally, Alexandre Guybert published a tract in 1580 to help masons calculate more exactly the work involved in large excavation and construction projects98.

• 99 . Jacques Besson, 1567-1569 and 1571. Edited by Béroalde de Verville, the Théâtre (Besson, J., 157 (...)

52Advances in etching led to finely detailed illustrations of new tools, occasionally in exploded views, to accompany step-by-step guides of how to use them. The work of Jacques Besson stands out. From Grenoble, he entered French royal service under Charles ix as an ingénieur du roy. He wrote several manuals on practical geometry and mathematical instrumentation, though his masterpiece, Théâtre des instrumens mathématiques et mechaniques, appeared posthumously in Lyon in 1578.99

• 100 . Henri de Suberville, 1598.

53In 1598, Henri de Suberville published a treatise on a new surveying device that he named after the first Bourbon king100. The accompanying illustration showed its use in laying out the periphery of a bastioned trace. Again, this method did not essentially differ from the one used in founding medieval bastides.

• 101 . See Y. Pauwel, 2002, on Bullant’s role at Chenonceaux.

54One work actually brings us close to design practices at the time in Amiens. It was the Petit traicté de geometrie et d’horologiographie pratique by Jean Bullant (1515-1578), published in Paris in 1562. Perhaps born in Amiens, Bullant was likely related to Jehan Bullant, master mason of Amiens101. He served as an architect for the constable, the Anne de Montmorency, and then Catherine de Médicis; his design commissions included castles at Écouen and La Fère in the 1540s and the Château de Chenonceaux in 1566. He traveled extensively in Italy and helped to introduce the five classical Vitruvian orders – and thus classicism – to France in his Reigle generalle d’architecture, published in Paris in 1568. The practical bent of this work came through particularly in the instructions that he addressed to stonecutters working in the quarries of Écouen, a chief supply source of stone for Amiens.

• 102 . B.M. Amiens, CC 60 (1480), fol. 49v; CC 106 (1524), fol. 90v; CC 138 (1541), fol. 143v; CC 141 (...)

55There thus developed in sixteenth-century France a substantial body of publications on practical mathematics that offered merchants and builders new – and sometimes not so new – techniques to solve age-old practical problems. These printed works opened up new ways to conceptualize and express changes in design that accompanied the « chorographic shift » of the 1500s. While university-trained mathematicians predominated in printed works on practical mathematics until 1550, artisans steeped in traditional building practices increasingly came to the fore thereafter both in print and the emerging professions of architects and engineers. This mix of old and new in design practice becomes even more apparent in municipal account books on construction projects in Amiens and presumably elsewhere. These continuities in indigenous design practices only began to merge with Italian innovations in fortification science in the late 1500s. Until then, masons still dominated most phases of design formulation and execution102.

• 103 . B.M. Amiens CC 126 (1535), fol. 97. The first mention of him is for a payment for writing the na (...)
• 104 . B.M. Amiens CC 623 (1539), fol. 48.
• 105 . Cailleux, P. & Lardin, P., 2001.
• 106 . B.M. Amiens CC 141 (1543): « …pour enquérir de luy la manière comment on pourroit besongner pour (...)
• 107 . B.M. Amiens CC 151 (1548) fol. 74. On the siege of Boulogne, see P. Gressier, 1977.

56Representing design practices fell, however, to visual artists employed on public commissions. The most important such person in sixteenth-century Amiens whose career reflected these evolving techniques was the painter, Zacharie Le Sellier (or Cellier), whose career spanned from 1530 to the 1570s.103Le Sellier took part in site visitations, where he worked with the master of works to render « le tout réduict au certain »104. Although it may still have included narrative elements from the medieval tradition, this phrasing suggests a perspectival drawing based on measurements taken by the master mason, in this case Jehan Bullant, who headed up public works. Yet Le Sellier did more, much more, than simply provide a visual representation of what the masons measured. He also used his knowledge of practical mathematics to calculate volumes and building materials – and thus costs – for specific projects. His methods remained essentially medieval105. In 1543, for example, he contributed his expertise to calculate in cubic toises the amount of earth to move to widen the ditch between the Porte de Haultoie and the Porte de Beauvais. He also collaborated with the master of works to determine the most cost-effective manner to remove this earth. This collaboration included the building of a customized machine (engin) for which Le Sellier provide the technical design plan106. In 1548, Le Sellier even traveled to Boulogne to prepare site drawings of the two principal forts guarding this vital port, which the English had recently occupied107.

• 108 . B..M. Amiens CC 151 (1548), fol. 69. Marini’s title in this entry is « maistre ingénieulx du Ro (...)
• 109 . Marini became involved in designing the defenses of Saint-Dizier. See C. Paillard & G. Hérelle, (...)

57Zacharie Le Sellier thus demonstrated most of the essential knowledge sets and designs skills of an engineer prior to the actual appearance of such experts from Italy who sported this title. The first Italian engineer to appear in Amiens in these records was Girolamo Marini, who arrived in 1548 to consult about building a bastion between the Porte of Haultoye and the Porte de Beavais108. Although usually referred to as a Bolognese, Marini was actually from the vicinity of Modena. He began service for the French during François I’s 1537 campaign in the Piedmont and the 1542 siege of Perpignan. He also provided the French advice on rebuilding the city of Luxembourg after its capture in 1543, overseeing work by some 100 to 120 Italians specially sent to upgrade its defenses. Marini had most recently participated in defending places in Champagne against a series of Imperial sieges in 1544. Marini also received the celebrated commission to build a new town, Vitry-le-François, incorporating the new-style trace italienne109. Blaise de Monluc, in fact, described him as « le plus grand homme d’Italie pour assiéger une place ».

• 110 . B.M. Amiens CC 152 (1549), fol. 23.
• 111 . B.M. Amiens CC 196 (1572), fol. 83v.

58By almost any reckoning, Marini was among the most accomplished, cutting-edge engineers of his day, yet his role as consultant to Amiens essentially consisted of one single visit to meet with Zacharie Le Sellier to go over a building project that lasted for the better part of six years. Le Sellier’s standing as principal designer of this bastion can be seen when he traveled in 1549 to Paris to show his pourtraict de bolevert to Henri ii himself110. Other itinerant engineers made periodic visits to Amiens over the next several decades. These included a certain sieur Vincent in 1555 and two men, seigneur Fredau, possibly a German, and Bartholomée, all of whom visited at the behest of the provincial governor. An unwelcome inspection occurred in 1572 when municipal authorities arrested a certain maistre Henry, a Flemish engineer, on charges of espionage. Poor Henry was caught conducting measurements of the city’s bastions and also reputedly tried to steal brick making secrets, a local specialty111.

• 112 . B.M. Amiens CC 180 (1564), fol. 17: « … pour ester tousjours plus prestz à faire qu’il plaisoit… (...)
• 113 . B.M. Amiens CC 635 (1569), fol. 40v-42. Apparently William of Orange threatened to invade the r (...)

59In the eyes of local officials, Le Sellier certainly served in a subordinate status to Italian engineers sent by the king. In 1564, for example, he received a payment for assisting Bernardino Bellarmato who spent eight days examining Amiens’ defenses, especially a bastion project destined for the Porte des Rabuissons. The entry underscored Le Sellier’s humble readiness to follow Bellarmato’s commands112. Unlike earlier Italian engineers, Bellarmato visited Amiens on a fairly regular basis over the next five years; indeed, his title explicitly specified his duties and corresponding authority as royal engineer of Picardy. During this time, Jean de l’Orme, now an « architecte des bastimens du Roy », visited Amiens during a tour of Picardy, where he presented to the lieutenant general a set of specific defensive recommendations, together with five plans and accompanying drawings113. These technical experts, brought in from the outside by the budding royal fortification service, increasingly dominated the design process of urban fortifications after mid-century.

• 114 . B.M. Amiens CC 638 (1572), fol. 29 and EE 272 (1572-1577).
• 115 . B.M. Amiens CC 199 (1573), fol. 11.
• 116 . B.M. Amiens CC 201 (1574), fol. 14v. « … du dessin et pourtraict que le sieur Belarmat le jeune (...)
• 117 . B.M. Amiens CC 643 (1577), fol. 34v.  … pour donner ordre à conduire l’ouvraige et observer plu (...)

60This change continued in 1572 with the arrival of Bernardino’s son, Aineas Bellarmato, who held the same royal title. Aineas, however, seemed to spend more time in Paris than in the province. He, too, worked on the bastion at Rabuissions, now named Longueville after the new provincial governor. He provided not only technical advice, but also design drawings for casemates sent after his visit114. By the next year, officials in Amiens sent letters to the provincial governor, the marquis de Crèvecoeur, asking him to send his engineers, led by the younger Bellarmato, to the city so that they could precisely mark out the dimensions of the new curtain walls joining the bastion de Longueville to the city115. Aineas spent a sufficient amount of time in 1574 to prepare for Crèvecoeur a comprehensive design plan of Amiens’ entire circuit of fortifications. This rendering presumably superseded the one created by Le Sellier six years earlier116. Three years later, Aineas stayed three weeks in the city to ensure that the initial construction of the bastion de Longueville scrupulously followed his design. This kind of close, sustained supervision suggests that Bellarmato wished to forestall any changes by the master masons of his original vision for the work117.

• 118 . B.M. Amiens CC 215 (1582), fol. 66v and EE 273 (1583-1585).

61Finally in 1582, Bellarmato returned from Paris to conduct another full inspection which yielded a detailed report together with design drawings for municipal officials, while another copy of the report presumably went back to Paris with him. However, the next year a separate such inspection, this time conducted by the master of works and local master masons, reexamined Bellarmato’s recommendations. It appears the city received two separate streams of information about the state of its municipal fortifications, one from a representative of the provincial governor and ultimately the crown, while the other came from resident experts of long standing118. Urban design practices were clearly in transition to a system increasingly dominated by the central government and its experts. It remains open to question whether the new-style trace italienne was a foreign import or, as seems more likely in this scenario, a home-grown adaptation.

• 119 . B..M. Amiens CC 162 (1554), fol. 20v.
• 120 . B.M. Amiens CC 164 (1555), fol. 29-29v.

62Disentangling the respective areas of expertise of these outsiders from Le Sellier’s is difficult, though Le Sellier’s consistent involvement suggests he handled both the drafting and technical aspects of these sundry projects. As a result, his stature seemed to rise in the 1550s as he again went to Paris to present his plans before the king’s conseil privé, including in 1554 a high quality (but now lost) vellum portrait of the Amiens showing all its fortifications and principal buildings, together with a listing of the militia companies responsible for its defense119. Significantly, an account from 1555 detailing expenses associated with a new bastion fronting the Porte de Noyon identifies Le Sellier as the « architecteur de ladicte ville »120. He often thereafter received this appellation when he worked on building projects, but was designated as a mere peintre when took more humble commissions.

• 121 . B.M. Amiens CC 634 (1568), fol. 30. « … pour avoir toizé, mesuré, faict calcul et reduction...  (...)
• 122 . B.M. Amiens CC 180 (1564), fol. 15: « … pour y employer grand nombre des habitans oisifz et men (...)
• 123 . B.M. Amiens CC 196 (1572), fol. 87v. Further work remains to be done to determine what relation (...)

63Mastery of new perspectival techniques in cartography also became evident when Le Sellier and the master mason Jehan Bullant collaborated over two weeks in 1568 to create a double portrait of the entire city. Using a compass, theodolite and surveyor’s rods, they laid out a series of traversal lines that served as the base for triangulating various important places inside the city’s walls. These measurements then enabled Le Sellier to create a proportional plan of Amiens, containing « les distances, longueurs et largeurs des fossez, rampars, moyneaulx, bolvartz et autres forteresses, avec les retours, coulds et jaretz y estans »121. The plans described here entailed a double pourtraict, one to guide the masons on site and another to present to Gaspard de Coligny, the provincial governor of Picardy at the time122. An even more ambitious cartography commission came in the form of five maps of Picardy which Le Sellier prepared in 1572 as a presentation gift to visiting dignitaries123.

• 124 . B.M. Amiens CC 192 (1570), fol. 80v.
• 125 . B.M. Amiens CC 640 (1574), fols. 58-58v, 59v: The artisans involved were Jehan Bullan, Jehan Mar (...)
• 126 . B.M. Amiens CC 196 (1572), fol. 81v and CC 202 (1575) fol. 96. For more on this case, which drag (...)
• 127 . B.M. Amiens CC 216 (1582), fol. 18. « … pour avoir… faict deux pourtraictz en pappier… faict le (...)

64Le Sellier’s was not the only local painter-architect to demonstrate mastery of these techniques of projective geometry. A certain Jehan de l’Abbaye regularly shows up in the record during this same period. In 1570, for example, he received a commission to prepare a pourtraict of various properties at dispute in a lawsuit between the cathedral canons and local inhabitants. He presumably used, or had someone assist him in using, surveying instruments in order to realize a true delineation of the property lines124. There is even mention of a payment to him for a design plan for a new roof of the Tour de Beffroy, containing exact dimensions and a calculation of the timber necessary for the project, prepared by a master mason and master carpenter. The plan’s practicality became evident when local merchants and carpenters used it to guide their purchase of lumber in nearby forests125. Another set of local experts who demonstrated these same skills were the surveyors or mesureurs jurés. One of them, Nicholas Gargualt, for example, occasionally became hired by the city to help deal with land expropriations and excavation projects126. In 1582, the mesurier Pierre Norman submitted two copies of a design plan for iron doors intended for the bastion de Longueville, together with a precise cost estimate of all necessary materials and a model127.

• 128 . B.M. Amiens CC 643 (1577), fol. 32. These phrases came in reference to the ongoing repairs of th (...)
• 129 . B.M. Amiens CC 655 (1590), fol. 58v.

65And then there were the masons and carpenters, long fluent in medieval traditions of practical geometry. Jean de Maoeul, a master mason and frequent master of public works, and his colleague Jehan Bullant received similar mention as expert en le mesure et bastimens and catisseur juré. In fact, Bullant prepared on-site drawings of artillery platforms in Péronne and Corbie to guide the design of the bastion de Longueville in 1577, while in 1589 another master mason, Anthoine Pasturon, rendered a scale drawing of the city’s ramparts from the Tour de la Haye to the Porte de Beauvais at the behest of the provincial governor, the Duc de Nevers128. Pasturon, who appeared regularly in the municipal account books since the early 1570s, receives mention in a 1590 account not as master mason but as architecte de ceste ville, thus filling the professional role originally played by Le Sellier. The lack of any further references to Italian engineers visiting Amiens during these years may have contributed to Pasturon’s elevation in stature, as did the fact that in that instance he served as part of a team of officials sent from Amiens to Conti to oversee the demolition of a castle129.

• 130 . See M. Wolfe, 2000, p. 61-79.
• 131 . Ambroise Bachot, ?-1599, 1587. See M. T. Gnudi, 1974.

66In the 1590s, the indigenous traditions of municipal construction and practical mathematics in Amiens became increasingly subject to direct monarchical control with the emergence of a more formal royal fortification service under Henri iv. The near disastrous seizure of Amiens in 1597 by the Spanish proved a key factor in precipitating this crucial change130. The episodic visits of Italian engineers typical under the Valois now became replaced by more permanent missions to the city, as elsewhere in the kingdom, by French experts who practiced and published these new design techniques. In 1591, example, Ambroise Bachot, identified in this account simply as ingénieur en Picardi, inspected Amiens’ fortifications and prepared several design plans for their improvement. Ambroise Bachot was originally trained as an engraver; however, he had been active in fortification design and engineering since at least the early 1580s. His Le Timon, published in 1587, relates albeit somewhat confusedly, different practical applications of mathematics to solve basic problems of calculating scale and distance in perspective. It also offers what Bachot claims are several newly invented machines for moving earth and constructing fortifications. A number of the plates, however, were reminiscent of Agostino Ramelli’s justly celebrated Diverse e artificiose machine, published the next year in 1588. While Bachot extensively lauded Ramelli in his book, Ramelli bitterly accused Bachot of plagiarism and theft. It should be pointed out that Bachot, if guilty as charged, was not all that unique in the world of early modern engineering and engraving131.

• 132 . Ambroise Bachot, 1598.

67Bachot serves as an important transitional figure because he arguably became the first Frenchmen to head a post in the nascent royal fortification service when Henri iv appointed him in 1596 to serve as director of fortifications at Melun, a key town in the Seine valley. Two years later, Bachot issued, using his own press, it seems, a revised and expanded version of Le Timon under the new, more exalted title of Le gouvernail (the rudder), a nautical image usually associated with the king, who directed the ship of state.132 This work further demonstrated Bachot’s weak grasp of basic mechanical engineering and entirely derivative ideas about fortification design.

• 133 . Jean Errard, 1584; 1598; 1605; 1600.
• 134 . In general, see D. Buisseret, 2000.
• 135 . Claude Flamand, 1597 (reprinted in 1612, German edition in 1613). His other works on practical g (...)

68After recapturing Amiens in 1597, Henri iv sent not Bachot but rather Jean Errard from Bar-le-Duc to the city to oversee repairs and begin design of a royal citadel – a structure that symbolized for the city, and in time, the rest of the kingdom, the monarchy’s heightened ambitions. He served the Duke of Lorraine before entering French royal service after 1594.133 Errard is generally credited, along with Claude de Flamand, with creating the first truly French school of fortification science134. Both men also wrote on practical geometry and the art of war like earlier theorist-practitioners. Flamand’s guide to fortifications derived from his own study of Italian and Dutch treatises on the subject as well as his own war-time experience135. Errard’s publications and actual built structures, though shaped by the same influences, proved much more original.

• 136 . Vérin, H., 1998 and 2002.

69Along with the emergence of a French school of fortification design came the enhanced use of linear perspective and new cartography techniques to depict advances in fortification design136. The work of Jean de Beins and Claude Chastillon stand out in particular, providing designs for the revamped royal administration of fortifications, which underwent si-gnificant change under the leadership of the Duc de Sully. In 1599, Sully united in his hands the offices of Superintendent of Finances, Artillery, and Buildings and Fortifications. He was also royal governor of the citadels of Amiens, Calais, Ponthieu, and the Trois-Évêchés in Lorraine. One of the expressed purposes for this administrative consolidation was to improve royal military performance. In 1601, Sully proposed to the king the goal of strengthening frontier fortifications by applying recently developed theories of Errard. Three years later came a survey of the condition of all these places, along with the grand règlement of 1604, in which Sully proposed organizing military engineers, who numbered between eighteen and twenty-four along with local contrôleurs des fortifications, by defining their duties and assigning them to specific territories. He ordered them to make regular inspections of fortifications in their areas and report recommended construction or demolition. He appointed a chief engineer to direct inspections and construction projects for Picardy, Champagne, Dauphiné, and Provence. It became general practice under Henri iv for the crown to assume the initial construction costs for new urban fortifications, leaving maintenance costs thereafter as the responsibility of municipal authorities. In this manner, the crown gained much greater control over deciding where and how to modernize fortifications. The emerging fortification service became the means by which this general policy became enacted.

• 137 . See R. Carvais, 2001. This work will soon be published by Droz.
• 138 . See J.-L. Harouel, 1993.
• 139 . Carvais, R., 2005 and 2005.

70Similar developments in civil construction took place in the late sixteenth century. The Chambre des Bâtiments, for example, which since the thirteenth century had essentially been a corporate tribunal representing craft guilds under the prévôt of Paris received lettres patentes in 1574 that placed it directly under the crown, which appointed a judge to serve as Maître général des œuvres de maçonnerie137. Another major innovation occurred in 1597 when the crown reformed the statutes of the corporations of masons and carpenters in Paris. These reforms grew out of longstanding conflicts between the crown and masons, on the one side, and the Parlement of Paris and the urban notability, on the other. The new statutes affirmed through royal prescription the power of the guilds by extending them the right of appeal from its tribunal directly to the Parlement, thus bypassing the prévôt. The crown essentially confirmed the traditional hierarchy within the guilds, reaffirming the authority of the master over his apprentices and journeymen. The king also laid out rules governing salaries, work conditions, arbitration of disputes, theft from building sites, and guild elections. In time, the crown extended these new standards by decree to the rest of France. In 1607, it issued a decree that enhanced the rights and functions of the Grand-Voyer de France in an attempt to create a more centralized system of administrative control over urban planning and construction. This move sharpened tensions between masons and royal building administrators into the next century.138 Controlling the urban fabric of the walled towns required not only a growing royal monopoly over fortification design and construction, but also greater royal regulation of the building trades. Crown control over municipal fortifications and the guilds involved in construction thus proceeded in tandem139.