Abstract
Between 1508 and 1515, mathematical book production in Paris was particularly high and largely addressed to the Parisian calculatores: a group of masters and students mostly related to Parisian colleges of the Iberian tradition, such as SainteBarbe, Coqueret, and Montaigu. The mathematical publications of Pedro Sánchez Ciruelo, who taught in Paris during the last years of the fifteenth century, were a main inspiration for these Parisian calculatores; they also published several other books by themselves, mostly on arithmetic, proportions, and astronomy. This induced the development of a Parisian market for the publishing of the calculatores’ mathematical books, mainly supported by printers and booksellers who did not published scholarly mathematical texts in other periods. This paper will question who the actors of this market were, why they published mathematical books, and how they contributed to the development of mathematical teaching in Paris.
Keywords
 Johannes de Sacrobosco
 Mathematical books
 Pedro Sánchez Ciruelo
 Paris
 Quadrivial education
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1 Introduction
Johannes de Sacrobosco’s (d. 1256) Sphaera took many forms during the Parisian Renaissance, from the first editions of this text in the late 1480s to the standard model printed by the middle of the sixteenth century (Table 1). These forms are associated with different ways of teaching mathematics, and they are particularly numerous by the 1510s, when the teaching of mathematics began to be more regular in Parisian colleges. This can be explained by the fact that there were two mathematical currents taught in Paris in this period: a traditional current, inspired in the work of ancient philosophers (Oosterhoff 2018), and the calculatores current, inspired in the work of fourteenthcentury mathematicians (Biard and Rommevaux 2008; Calderon 1990; Wallace 1969). Both were taught in Paris between the 1500s and the middle of the 1510s, after the development of two mathematical programs, respectively, prepared by Jacques Lefèvre d’Étaples (ca. 1450–1536) and Pedro Sánchez Ciruelo (1470–ca. 1560), at the end of the fifteenth century.
This paper will focus on mathematical book production in the calculatores current, between 1508 and 1515. We will mainly be interested on publishers (i.e., printers and booksellers) and their relationship with authors and the public. Although the calculatores were also interested in physics and philosophy, we will be mainly concentrated here on their mathematical works, namely works related to the quadrivium as well as proportions and movement. Through the study of mathematical books as material objects, we will be primarily attentive to how publishers influenced the development of mathematical teaching in Paris.
2 Pedro Sánchez Ciruelo and His Mathematical Program (1492–1500)
The development of mathematical teaching in Paris begins in the 1490s after the arrival of two scholars: Jacques Lefèvre d’Étaples (Chap. 2) and Pedro Sánchez Ciruelo (Chap. 7). At this time, there was no specific regulation on the teaching of mathematics at the University of Paris, even if Sacrobosco’s Sphaera seems to have been regularly studied (Beaujouan 1997). Between 1489 and 1494, we know of at least four editions of this text (Sacrobosco 1489, 1493a, b, 1494) all printed in a small quarto format without commentaries, following Erhard Ratdolt’s (fl. 1476–1528) publication (Sacrobosco 1485). However, unlike their model, the first Parisian editions of Sacrobosco’s Sphaera are poorly illustrated and do not contain Peuerbach’s complementary texts on the motion of the planets (Sacrobosco 1482, 1485).
Between 1495 and 1500, Lefèvre and Ciruelo improved mathematical teaching in Paris with two advanced mathematical programs. On the one hand, Lefèvre gave Parisian students seven mathematical texts organized in two books: the first on arithmetic and music, and the second on geometry and astronomy (Jordanus Nemorarius 1496; Sacrobosco 1500). Meanwhile, Ciruelo offered five mathematical texts, organized in four books, on theorical arithmetic, practical arithmetic, geometry, and astronomy (Ciruelo 1495; Bradwardine 1495, 1496; Sacrobosco 1498).^{Footnote 1} In both programs, astronomy is represented by an edition of Sacrobosco’s Sphaera, with their respective commentaries. Most of these editions are folios and are widely illustrated. They were still regularly published in Paris in the middle of the 1510s.
2.1 Ciruelo’s Arrival in Paris and His Mathematical Books
We do not seem to have administrative information about Ciruelo’s stay in Paris, unlike his friend Pedro de Lerma (ca. 1461–1541) and other international students there, and we do not know much about the conditions of his Parisian studies. From his time in Spain, his Parisian publications, and some letters, we know that he stayed in Paris at least ten years, from 1492 to 1502, where he pursued his doctorate in theology and taught mathematics (Lorente y Péres 1921). Nevertheless, Ciruelo is quite unclear in his texts when he talks about his own activity. In a letter of 1526, published twentyfour years after his departure from Paris, he says that he went “through the most prestigious Parisian colleges of theology,” but he does not specify the names of these colleges or his precise situation (Sacrobosco 1526). It is well known that the main Parisian colleges of theology were Sorbonne and Navarre, but students of these establishment generally appear in their administrative sources (Farge 1980).
Information about Ciruelo’s teaching activity is also unclear. In this letter of 1526, Ciruelo says that “the profession of mathematician was necessary to buy food and clothes,” but he does not describe the nature of this profession. The colophon of Thomas Bradwardine’s (ca. 1290–1349) Arithmetica speculativa designates him as a “lecturer of mathematics” (mathematica legente), but so far as we know, there was no position in Paris for the teaching of mathematics before the 1530s (Bradwardine 1495). In the other hand, in his letter of 1526, Ciruelo says that his edition of Sacrobosco’s Sphaera was published “to be read in public” (publice legeram), which might imply that his teaching was recognized by the university. It is also common to find in secondary bibliography that Ciruelo taught at the college of Beauvais, but this information seems to come from a misunderstanding related to the interpretation of some colophons containing the mention in Bellovisu (Ciruelo 1505a); this actually means that the book was printed at Jean Marchant’s (fl. 1504–ca. 1515) workshop, the Beauregard, named in Latin the Bellovisu (Renouard 1965, 294). The Latin translation of Beauvais would be Belvacensis, but it would be equally possible to designate the college by the adjoining street, In clauso Brunelli.
Despite this lack of administrative information, we have at least four mathematical books by Ciruelo published during his Parisian stay. Two of these books, the theorical arithmetic and the geometry, are editions of Bradwardine’s texts, an Oxfordian calculator of the fourteenth century. The practical arithmetic, on the other hand, is an original publication. Finally, the astronomical book, as we said, is an edition of Sacrobosco’s Sphaera, with Ciruelo’s commentaries.
Nevertheless, unlike Lefèvre’s publications—almost always justified and wellstructured following a traditional organization of mathematical knowledge—Ciruelo’s mathematical textbooks do not seem to be conceived as a planned program. For most of his books, Ciruelo does not explain the reasoning behind their publication. His edition of Sacrobosco’s Sphaera is the only one to contain personal considerations, mostly on the importance of mathematical studies, but without references to his previous mathematical works. In addition, his books are printed in different formats, quarto and folio, which means that they most probably did not circulate together, as books of different formats were rarely bound together. This difference between Lefèvre’s and Ciruelo’s projects could be perhaps explained by the fact that Ciruelo was much younger than Lefèvre. In his following works, published after his departure to Spain, Ciruelo would be deeply marked by Lefèvre’s Parisian program before taking some distance, probably for theological reasons (Ciruelo 1516; Sacrobosco 1526).
2.2 Guy Marchant, Publisher of Ciruelo’s Mathematical Books
During his ten years at the University of Paris, Ciruelo always published his books at the same place: Guy Marchant’s (fl. 1483–1505) workshop. Marchant was one of Paris’ first printers: he began his activity in 1483, in a workshop named the Champ Gaillard, at Clopin Street, next to the college of Navarre (Renouard 1965, 293). He is mainly known for his role in the introduction of illustrated books in Paris, such the Danse macabre and the Calendrier des bergers, as well as the Danse macabre des femmes and the Calendrier des bergères (Hindman 1991). He seems to have obtained his master’s degree by 1497, when he starts to sign colophons using the formula a Magistro Guidone Mercatore, instead of per Guidonem Mercatoris. His academic background as well as Marchant’s interest in illustrated books could perhaps explain why he agreed to publish Ciruelo’s mathematical texts. In fact, the publication of these books involved at least two technical problems: the acquisition of a specialized printing material and the reproduction of mathematical figures (Chap. 2). Printing mathematical texts involves mathematical figures, such geometrical and astronomical woodcuts, but also typographical characters representing Arabic numbers. In the beginning of the 1490s, these characters were still rare in Paris and often replaced by blank spaces, abbreviations, or Roman numbers (Levy 2020, 201–203).
The first three mathematical works prepared by Ciruelo, namely the two arithmetical texts and the geometry, were printed by Guy Marchant between 1495 and 1496. So far, we do not know if the arithmetical texts were printed before or after the geometry, as they are dated February 1495, which could also be, according to the Easter calendar, February 1496. The geometry, published in a folio format, contains multiple woodcuts representing mathematical diagrams. It was probably a costly book to print. The arithmetical texts do not contain so many mathematical figures, but the theorical arithmetic presents diagrams reproduced with typographical characters. They also include some historiated woodcuts, next to the title page and the colophon, coming from other books published in the same workshop. These illustrations do not have an explicit relationship with the content; they mainly represent persons talking or listening to others, which could evoke the scholarly dimension of these texts. In addition, both have a common illustration representing the nativity of the Christ, perhaps linking the two texts as separate volumes.
2.3 Jean Petit and Guy Marchant
From 1497, Guy Marchant began to regularly publish his works with another Parisian bookseller, Jean Petit I (ca. 1495–1540), who was active from 1495 at SaintJacques Street—the main street of the Parisian booksellers—and who quickly became one of the most influent publishers of the Parisian university neighborhood. However, unlike most of the publishers of his time, Petit was not a printer, which means that all the books that he published and/or sold were printed in workshops independent of his own bookshop (Renouard 1896). In the association between Jean Petit and Guy Marchant, Petit does not seem always to have had an editorial responsibility in the production of the books. In most cases, his name is mentioned on the title page but only Guy Marchant is credited in the colophon. This is the case of the last of Ciruelo’s mathematical books published by Guy Marchant, Sacrobosco’s Sphaera. The publication is dated February 1498, but according to Denise Hillard, Petit’s device is from the beginning of 1499 (Hillard 1989).^{Footnote 2} The colophon only designates Marchant in the colophon, as the financier and the printer of the book. The name of Petit only appears in the copies sold by him, which were probably bought in advance to be printed with his personal device.
On technical matters, Sacrobosco’s Sphaera presents more difficulties than the other mathematical books printed by Marchant, as it has not only text and images, but also extensive numerical tables, composed with metal strips. Pages of tables took longer to be produced than pages of text, as they required a lot of precision and a careful proofreading. Bibliographical analyses in Lefèvre’s editions of Sacrobosco’s Sphaera show that Wolfgang Hopyl (fl. 1489–1522) organized the typographical composition of the edition based on the difficulty of the sheets to compose (Levy 2020, 243–244). Therefore, compositors in charge of pages with a lot of tables and mathematical data worked on a smaller number of sheets than compositors in charge of pages mostly constituted of text. From this perspective, pages of tables seem to have required twice the composition time of a page of text. In addition, the printing process of these tables was a demanding operation: copies of Marchant’s edition show how these tables could sometimes split out.^{Footnote 3}
Finally, this edition also contains a few decorative illustrations, again mainly from other publications of Guy Marchant. Two of them are used because of their astronomical or astrological subject: an astronomer holding an armillary sphere and a young man offering flowers to a lady under the astrological symbols of the month of April. The other illustration has a more hypothetical relation to the work, as it comes from a poem of the Calendrier des bergers entitled “The scream of death” and represents a young man blowing in a horn. In the original illustration, the letters “TO TO TO,” cut as a part of the image, are coming out of the horn to express the sound of the instrument. In the Sphaera edition, the printer covers the two last “TOs” leaving only the first one. “TO” was the common representation of the Earth in medieval astronomy and a symbol generally used in early modern astronomical books. Therefore, it does not seem that this image was selected randomly: more likely, it was used because the detail echoed the astronomical nature of the work.
3 The Parisian Current of the Calculatores and Its Publishers (1508–1515)
In fact, Pedro Ciruelo did not publish mathematical books in Paris related to calculatores studies, such as texts on proportions or movement, but he was interested in the calculatores’ works, notably in Thomas Bradwardine’s texts. Mostly, he offered Parisian students an alternative to Jacques Lefèvre d’Étaples’ mathematical program, still regularly published in the university neighborhood after his departure to Spain.
3.1 Republishing Ciruelo’s Arithmetical Texts Before the Calculatores
Even before Ciruelo’s departure, his arithmetical texts seem to have been reprinted, but these imprints are so fragile and ephemeral that we are not sure we know them all.
To begin with, we know of an edition of Ciruelo’s practical arithmetic also signed by Guy Marchant and presenting the same date as the first edition, February 1495, but containing a different composition of the text and the address of the Beauregard.^{Footnote 4} The Catalogue of books printed in the fifteenth Century now in the British Museum [British Library] (from now on referred to as BMC) points out that this workshop seems to have been acquired by Guy Marchant at the end of the 1490s and reports the date of this edition as 1498 (BMC 1963, 69). Philippe Renouard argues on the other hand that some books of Guy Marchant were already signed from the Beauregard before he officially acquired the place, just in front his original workshop, the Champ Gaillard (Renouard 1965, 293). In any case, the book was necessarily printed by 1505, before the end of Guy Marchant’s activity (Renouard 1965, 293).
The subsequent editions of Ciruelo’s arithmetical texts were published between 1502 and 1505 by Denis Roce (fl. 1490–1517) and Jean Lambert (fl. 1493–1514). Denis Roce began his publishing activity in 1490, but he did not print himself (Renouard 1965, 375–376). In March 1502, he published an edition of Bradwardine’s arithmetical text following the 1495 publication (Bradwardine 1502): according to the colophon, the book was still printed by Guy Marchant, but all the title pages were to be distributed under Roce’s device. In April 1505, Roce’s also published an edition of Ciruelo’s practical arithmetic, now printed by Jean Marchant, nephew and successor of Guy Marchant (Ciruelo 1505a). However, this edition presents a difference compared to previous publications: the addition of a mathematical problem at the end of the book. In this problem, students are invited to help Parisian scholars to administrate their finances, when, according to Ciruelo’s own words, Parisian scholars were not particularly good in mathematics (Ciruelo 1521). Yet Ciruelo was no longer in Paris in 1505, which could mean that another edition of this book was published in the meantime.
Jean Lambert was active from 1503 as a printer and a bookseller. From at least the beginning of his career, he produced books in his own workshop (Renouard 1965, 133–234). Until the beginning of the 1510s, he was a neighbor of Denis Roce at SaintJacques Street, in front of the SaintBenoît cloister (Renouard 1965, 133–234). In November 1505, he published two new editions of Ciruelo’s and Bradwardine’s arithmetical texts, also printing the mathematical problem published in Roce’s 1505 edition in the practical arithmetic (Ciruelo 1505b; Bradwardine 1505). Finally, between Roce’s and Lambert’s publications, we know of at least three editions of Ciruelo’s and Bradwardine’s arithmetical texts, showing that these books were probably largely studied in Paris between the departure of Ciruelo and the main period of the calculatores. Nevertheless, Ciruelo’s other mathematical books were mainly reprinted during the calculatores period.
3.2 The Parisian Calculatores and the Colleges of Iberian Tradition
The Parisian calculatores current, as we said, followed the works of medieval theologians and mathematicians primarily active at Oxford and Paris in the fourteenth century. Unlike ancient philosophers, the calculatores argued that physics and mathematics were complementary disciplines, and they were particularly interested in studies related to proportions and movement. These studies were quickly diffused in Europe, and by the fifteenth century, works on proportions also led to the improvement of algebraic principles (Veltman 2000, 401–404).
The calculatores works seem to have interested scholars from universities where these disciplines were taught at an advanced level at the end of the fifteenth century. In Paris, mathematical works from the calculatores began to be published by scholars who were primarily in Spain, such the Sicilian Renaud Montoro (15th cent–16th cent.), who did most of his theological studies at the University of Salamanca, but also at the University of Paris (Beltrán de Hereda 2001). By the beginning of the 1480s, during his Parisian stay, Montoro published an edition of Albertus de Saxonia’s (ca. 1316–1390) Proportiones; Albertus Saxonia was a Parisian calculator of the fourteenth century (Saxonia ca. 1485). So too, as we said, Pedro Ciruelo also published mathematical works from Thomas Bradwardine, an Oxfordian calculator of the same period.
Even if the University of Paris was constantly attractive for international students, the end of the fifteenth century was an important moment for the constitution of an Iberian scholarly community. In fact, by the end of the 1490s, their arrival started to be recorded by three Parisian colleges, namely Coqueret, SainteBarbe, and Montaigu, neighbors at the SaintHilaire Mount, and progressively known for the attendance of Iberian scholars. These scholars were students of the liberal arts, as well as masters who were teaching these disciplines while finishing their doctorate, mostly on theology, but also on law or medicine (Quicherat 1860, 77).
The Iberian scholarly community at Paris developed primarily for diplomatic reasons. In 1498, John Standonck (1453–1504), master of the college of Montaigu, was reforming the establishment with the financial support of the admiral Louis Mallet de Granville (1438–1516), a former student of the college (De Matos 1950). Nevertheless, part of this financial support was taken from booty confiscated from a French privateer who attacked a Portuguese merchant ship; eventually, Emmanuel I, King of Portugal (1469–1521), took the money back (De Matos 1950). In consequence, John Standonck proposed to Emmanuel I to direct the money to the reformation of the college and promised in exchange to provide two grants for Portuguese students and to make the king himself a benefactor of the establishment (De Matos 1950). The king accepted, and soon Montaigu became the first destination for Portuguese students in Paris (De Matos 1950). By the beginning of the sixteenth century, the college also attracted Spanish students, and they were so numerous that the college of SainteBarbe, next to Montaigu, and the college of Coqueret, next to SainteBarbe, became two other main destinations for Iberian scholars.
The Parisian calculatores of the sixteenth century were not all Iberian scholars, but most of them were attached to colleges of the Iberian tradition. Thus, the calculatores current was first defined by a geographical and cultural space. Of the authors who published mathematical books, four of them were Iberian scholars—Juan Martinez Siliceo (ca. 1486–1557), Gaspar Lax (1487–1560), Juan Luis Vives (1492–1540), and Alvaro Thomás—but two of them were from other regions, namely Juan Dullaert (ca. 1480–1513), from The Netherlands and Jérôme de Hangest (d. 1538), from France. Extant scholarlship is not always clear about these authors’ places of study and teaching, but according to Jules Quicherat, Lax and Vives studied together at SainteBarbe (Quicherat 1960, 88). In addition, it seems that Lax and Dullaert taught at the college of Montaigu, and Álvaro Tomás taught at the college of Coqueret (Farge 1980; Leitão 2000). Jérôme de Hangest, who was the first author to publish a mathematical book related to the calculatores current, was on the other hand not attached to a college of Iberian tradition but rather taught at the college of Reims, which was immediately next to SainteBarbe and Coqueret (Farge 1980).
3.3 Publishers of Mathematical Books During the Calculatores Current
Parisian mathematical production related to the calculatores current was mainly concentrated between 1508 and 1515. The calculatores themselves were principally interested in publications related to proportions, arithmetic, and astronomy, and their books were first published for their students. Because of this important Parisian demand for mathematical books, several Parisian printers and booksellers published mathematical books during this period. For the only time between the end of the fifteenth century and the middle of the sixteenth century, Parisian mathematical book production exceeded two percent of Parisian printing production in general.^{Footnote 5}
3.3.1 Jean Petit (1508)
The first mathematical book related to the Parisian calculatores current was Jérôme de Hangest’s Liber proportionum, published by Jean Petit in June 1508 (Hangest 1508). Petit had been the main publisher of Hangest’s works since the beginning of the sixteenth century. Two months later, however, Petit also published a second edition of Sacrobosco’s Sphaera by himself, including Ciruelo’s commentaries (Sacrobosco 1508), not reprinted in Paris since the first edition of 1498. Unlike the 1498 edition, shared with Guy Marchant, Petit alone seems to have been responsible for the 1508 publication: he is the only one explicitly mentioned in the colophon, and all the copies were distributed under his device. Jean Marchant, the printer, only appears trough the address of his workshop.
The 1508 edition of Sacrobosco’s Sphaera presents two main differences compared to the 1498 publication. First, redrawing the historical woodcuts was not related to the subjects of the book: From the beginning of the 1500s, scholarly publication began to be formally different from books addressed to a larger audience. Then, the choice of a twocolumn layout, instead of a single line presentation of the text. This can perhaps be explained by way of financial reasons: as we said, Sacrobosco’s Sphaera with Ciruelo’s commentaries was particularly bulky, and therefore expensive. However, a twocolumn layout allowed a more readable text for smaller types. By choosing smaller characters, the publisher reduced the printing area of the text and saved fourteen sheets per copy (eighty folios instead of one hundred and eight folios), which is enough to print, for example, an equivalent run of fiftyfolio quarto books.
3.3.2 The Marnef Brothers (1509)
In September 1509, the brothers Marnef (fl. 1485–1533) also started publishing mathematical books for the calculatores. They are known for their large network of bookshops around the country and their diversified production of printed books. The first mathematical text published by them was an edition of Ciruelo’s Algorismus printed by Jean Marchant (Ciruelo 1509). So far, we do not know of any other edition of this text published between 1505 and 1513, and the Marnef brothers’ edition is only known by one copy. Perhaps, other editions of Ciruelo’s arithmetical books existed at the same time.
3.3.3 Poncet Le Preux and Guillaume Anabat (1509–1510)
By 1510, a neighbor of the Marnef brothers, Poncet Le Preux (fl. 1498–1559), published Álvaro Tomás’ Liber de triplici motu, another book related to the calculatores current (Tomás ca. 1510). It is also one of the most advanced mathematical books published in Paris in the first half of the sixteenth century (Leitão 2000). The text is addressed to the students of the college of Coqueret, as is noted by a colleague of the author, Georges Bruneau de Vendôme, in a letter published at the end of the same edition. It presents all the formal characteristics of the calculatores books: gothic characters, twocolumn layout, red and black composition, and a heavy frame on the title page. The title page also explicitly refers to the original calculatores current as it mentions the fourteenthcentury Oxfordian mathematician Richard Swineshead (d. 1354) and his calculationes.
The Liber de triplici motu does not contain a date of publication, but the explicit says that the author finished his work in February 1509, which could also be February 1510 according to the Easter calendar. The state of Le Preux’s device is anterior to October 1510 (it does not contain a damage in the right side of the frame), so the book is probably published before this date (Valla 1510). The printing process is assured by Guillaume Anabat (fl. 1505–1510), who is mainly known for the publishing of religious texts. However, in the two last years of his career, from 1509 to 1510, he mostly printed scholarly books, especially for Iberian authors (Renouard 1964, 29–46). This could perhaps be explained by the fact that the printing of religious books required skills in red and black composition, a delicate process based on two press runs, also demanded for the publishing of Iberian scholarly books.
Individual copies of the Liber de triplici motu are particularly interesting because they present multiple states: some copies contain sheets entirely recomposed (i.e. four folio pages) but with no modification of the content. However, recomposed sheets differ from one copy to another and present red and black elements, while regular sheets are only printed in black. In addition, copies containing recomposed sheets, less numerous, were distributed under Guillaume Anabat’s device, while regular copies were distributed under Poncet Le Preux’s. The probable explanation here is that once all the sheets were printed, Le Preux allowed Anabat to keep the incomplete copies and to reprint the missing sheets so he could sell them under his own device. But Anabat went further, printing these sheets in red and black, and thereby highlighting his own run.
3.3.4 Thomas Kees (1510–1512)
For some printers, the publishing of mathematical books was indeed a way to stand out. Many examples can be found among the most important printers of the time, such as Erhard Ratdolt, Wolfgang Hopyl, or Henri Estienne I (fl. 1502–1520). But littleknown printers also published mathematical books to highlight their technical skills and the quality of their work. Such is the case of Thomas Kees (fl. 1507–1515), a mysterious printer who had an important role in the publishing of mathematical books during the calculatores current. We do not know much about Kees, but according to Philippe Renouard, he was a modest printer, active from 1507 to 1515 (Renouard 1965, 223). He mainly printed for other publishers and never had his own device. By 1510, he had begun printing mathematical books. For the printing of these books, he acquired a large set of astronomical and astrological woodcuts often requested for the mathematical texts of the calculatores current.
The first part of this set was utilized in the publishing of an edition of Paulus Venetus’ (ca. 1372–1429) De compositione mundi (Paulus Venetus, ca. 1510), an introduction to astronomy extracted from a Philosophia naturalis of the same author: an extensive commentary on Aristotle’s (384–322 BCE) works. The Parisian edition was prepared by Juan Dullaert based on a Venetian edition published by Ottaviano Scotto (fl. 1490–1501) in 1498 (Paulus Venetus 1498). It was printed and distributed by Thomas Kees, who replaced the publisher’s device with a historiated woodcut. The edition is not dated, but thanks to the second part of the set of illustrations, we know that it was not published after 1512.
The second part of the set was acquired for the publishing of an edition of Hyginus’ (ca. 1st cent.) Poeticon astronomicon (Hyginus 1512), dated from May 1512. It was still printed by Thomas Kees, but shared with Olivier Senant (fl. 1505–1526), neighbor of Denis Roce and Jean Lambert. There is no secondary author mentioned in this edition, but a letter from Juan Luis Vives published in 1514 says that this book was also prepared by Juan Dullaert. For the printing of this text, Kees used woodcuts from his edition of the De compositione mundi and acquired new woodcuts based on a Venetian edition of the Poeticon astronomicon published by Giovanni Battista Sessa (fl. 1489–1505) in 1502 (Hyginus 1502). In other words, while the edition of the De compositione mundi was entirely realized on the basis of Scotto’s edition, the Poeticon astronomicon was based on Scotto’s De compositione mundi, as well as on Sessa’s Poeticon astronomicon. The Poeticon astronomicon was therefore probably not published before the De compositione mundi.
3.3.5 Jean Petit (1511–1512)
Jean Petit sold and published mathematical books from the beginning of his career and especially during the calculatores current: after the two astronomical books published in 1508, he also published a philosophical and geometrical work with Henri Estienne in 1511, as well as a second edition of Bradwardine’s Geometria speculativa. In fact, geometry was a discipline rarely taught in Paris compared to arithmetic and astronomy: Bradwardine’s Geometria, first published in 1495 by Guy Marchant, was the only mathematical book from Ciruelo’s bibliography not to be reprinted before the 1510s. So too, the beginning of Euclid’s (323–285 BCE) geometry, regularly printed at the end of Lefèvre d’Étaples’ editions of Sacrobosco’s Sphaera, was entirely contained in three folios (Sacrobosco 1500). Finally, Charles de Bovelles’ (1479–1567) introduction on geometry, published in a collective mathematical textbook of 1503 (Lefèvre d’Étaples 1503), when Bovelles was probably teaching in Paris, was not reprinted with the first part on the textbook in 1511 (Lefèvre d’Étaples 1511), after the departure of the author. On Bovelles’ life, see (KlingerDollé 2016).
Because of the lack of teaching in this discipline, geometry was the first mathematical subject in Paris to be published by scholars for audiences other than students. In 1511, two geometrical texts were published addressed to other readers. Both were printed by Henri Estienne, who was the main publisher of Bovelles’ works. The first was a collection of letters, mostly on philosophy, known by the title Liber de intellectu and addressed to an advanced scholarly audience (Bovelles 1511a). The four last letters, all on geometry, are regrouped under the name Mathematicum opus quadripartitum. The book is a bulky volume constituted by some two hundred folios richly illustrated with woodcuts especially realized for the publication. The second text, published in September of the same year, was the Geometrie française, a compendium of mathematical principles written and published in vernacular (Bovelles 1511b).
The Liber de intellectu was not only published by Henri Estienne. Even if all the title pages only mention his name, the colophon says that the book is printed by Henri Estienne and financed by Henri Estienne and Jean Petit I “associated in the art of copper.” Estienne and Petit did not print many books together but their association with this publication might potentially be explained by the high costs of the printing of the book because of the amount of paper and the illustrations. On the other hand, the book was also too specialized to be sold by Jean Petit, who indeed does not seem to have sold any copies, at least under his own device. Instead, his name appears in the title pages of another book financed by Henri Estienne but sold by Estienne and Petit: Aristotle’s (ca. 385–323 BCE) Moralia, published some weeks before Bovelles’ Liber de intellectu, at the end of 1510 (Aristotle 1510).
Probably because of the publication of various geometrical books in Paris at the beginning of the 1510s, Jean Petit also financed the second edition of Bradwardine’s Geometria speculativa (Bradwardine 1511) between 1511 and 1512. However, this edition does not appear to have been prepared by a professor of the university, as it contains many mistakes in Latin and mathematics, mainly related to abbreviations and the technical diagrams. In addition, this edition presents a more modern layout, with roman characters and long line disposition of the text (instead of two columns), far from the calculatores books of the beginning of the 1510s. Therefore, it does not seem that Petit was publishing this book for the calculatores and their students, but rather for Estienne’s and Bovelles’ audience: perhaps this seemed like a good time to publish a geometrical text.
3.3.6 The Marnef Brothers (1512–1515)
The Marnef brothers also saw an opportunity to publish mathematical books in the calculatores period. As early as 1509, an edition of Ciruelo’s arithmetical text had already been financed by them. In the first half of the 1510s, two other mathematical books were published under their device: a compilation of three medieval texts on proportions, respectively from Albertus Saxonia (1316–1390), Thomas Bradwardine, and Nicole Oresme (ca. 1322–1382), and an edition of Sacrobosco’s Sphaera following the first Parisian model (Saxonia et al. [1512–1515]; Sacrobosco [1512–1515]). These two books seem to have been published on the initiative of the booksellers, as they do not present the name of a scholar or any paratexts that could indicate that they were prepared for a specific course. The Sphaera book is very faulty in terms of typographical composition.
These two books published by the Marnef brothers are not dated, but they present the same publisher device (Renouard 1926, no. 718). The only dated occurrences of this device that we know of are from 1512 (Barletta 1512; Lemaire de Belges 1512). The two mathematical books published by the Marnef brothers were probably published by this date according to the state of their device. In addition, the text on proportions was probably published before 1515—before the end of the main period of the calculatores current—according to the subject of the book and the presentation of the text. The printer is not mentioned, but both editions include decorated letters used by Pierre Vidoue (fl. 1510–1543) in later publications (Erasmus 1520; Lefèvre d’Étaples 1533). We do not know of any book signed and dated by Vidoue before 1516, but according to Renouard and Jean de La Caille, he was active from 1510 (Renouard 1965, 428–429).
3.3.7 Thomas Kees, Jean Petit, and Jean Lambert (1513)
In 1513, Thomas Kees, Jean Petit, and Jean Lambert, all publishers of mathematical books, started a partnership together. They publish several books related to different subjects, among them three mathematical books: another edition of Ciruelo’s Algorismus (Ciruelo 1513a), Juan Martinez Siliceo’s Liber arithmetica practice (Siliceo 1513), and another edition of Paulus Venetus’ De compositione mundi (Paulus Venetus 1513).
Unlike Ciruelo’s Algorismus, published several times in Paris, Siliceo’s Liber arithmetice practice was a previously unpublished text. Mostly known for his career in religion after his return to Spain (Quero 2014), Siliceo also published arithmetical texts reprinted several times in Paris until the middle of the sixteenth century. The first version of his arithmetic, on practical issues, was published in June 1513 by the three associated publishers. The text is explicitly addressed to the calculatores, as is mentioned in the complete title of the work and in the text itself. It is also the first time that special mathematical typographical characters, representing crossed numbers, were used in a Parisian printed book. The text is signed “Juan Martinez Blasius,” a pseudonym of Juan Martinez Siliceo, who was named, in fact, Juan Martinez Guijarro (however, it does not seem that Juan Martinez Siliceo was the same person as Juan Martinez Población, who also published mathematical books in Paris in the 1510s (Levy 2020, 95–98).
Finally, Paulus Venetus’ De compositione mundi was not only distributed by the three associated publishers, but by at least seven publishers, including Poncet Le Preux, Gilles de Gourmont (fl. 1499–1533), Claude Chevallon (fl. 1506–1537), and François Regnault (fl. 1501–1540). It was published three months after the premature death of John Dullaert (Elie 1951, 222), who prepared and taught the first Parisian edition. The title page announces the entire Summa philosophia naturalis of Paulus Venetus, specifying however that “the book starts with the De compositione mundi,” which is the only text actually printed. In fact, the Summa philosophia naturalis, from which the De compositione mundi is extracted, was published in 1514 by most of the same associated printers and booksellers. The main party responsible for these publications seems to have been Thomas Kees, who is mentioned in the colophon as the printer and proofreader of the text. Jean Petit, on the other hand, was not among the publishers of this second part of the work.
3.3.8 Hémon Le Fèvre (1514–1515)
From 1514, Thomas Kees, Jean Petit, and Jean Lambert were no longer working as an association. However, another printer based in front of the SaintBenoît cloister, next to Roce and Senant, started printing mathematical books: Hémon Le Fèvre (fl. 1509–1525). It is with Le Fèvre and Kees that Juan Martinez Siliceo published the second version of his arithmetical text, revised and extended, the Arithmetica theoricen et praxim (Siliceo 1514). This book presents a common second part to the Liber de arithmetice practice, but with many differences. First, it was printed in a smaller quarto format with roman characters, arringed in long lines, and without red. It also includes both theorical and practical issues and an additional introduction from the author, in which he states that Jacques Lefèvre d’Étaples and Josse Clichtove (ca. 1472–1543)—the greatest representants of the classical mathematical current—are among the most important mathematicians of the time.
Nevertheless, Kees and Le Fèvre also published many other books for the calculatores, before and after Siliceo’s Arithmetica theoricen et praxim, much closer to the calculatores expectations. Therefore, the modern presentation of this book seems to have been realized on the initiative of the author himself, who was openly inspired by the authors and publishers of the classical current. In addition, Siliceo appears in this book as the main party responsible for its publication, as all the copies were distributed under his author’s device and signed with his own initials “JMS” (Renouard 1926, no. 601). The address is however from Le Fèvre’s workshop, and the colophon says that the book is published by Le Fèvre.
After the publication of Siliceo’s text, Hémon Le Fèvre became a wellknown Parisian publisher of mathematical books; he also published at least to other texts on these subjects: Tommaso Tedeschi’s (1488–1527) Sideralis abyssus prepared by Nicolas Bérault (ca. 1473–ca. 1550) in 1514, and two books on arithmetic and proportions written in Paris by the Spanish calculator Gaspar Lax in 1515. The Sideralis abyssus is a text on the relation between astronomy and theology first published in Pavia in 1511 (Tavuzzi 1994). It is the first modern mathematical text published in another country before being reprinted in Paris (Radini Tedeschi 1514). It was also one of the first books published by Nicolas Bérault, who was preparing his doctorate on law and teaching liberal arts in Parisian colleges (Delaruelle 1902). The text was printed again by Thomas Kees with his mathematical illustrations. The content of the book is not precisely related to the calculatores intellectual current, but the book was printed following the conventions of the mathematical texts published for Iberian scholars: in the middle of the 1510s, Parisian scholarly mathematical book production was still divided in two main editorial currents.
Finally, between October and December 1515, Hémon Le Fèvre published Gaspar Lax’s Arithmetica speculativa and Proportiones (Lax 1515a, b). Thomas Kees was not active anymore and was replaced by his neighbor, Nicolas de La Barre (fl. 1496–1528). The texts were published as two separate editions with their own title page and colophon but were probably meant to circulate together as they appear in Parisian copies of this book (Paris, Bibliothèque nationale de France, RESR141; Paris, Bibliothèque Mazarine, 2° 4621; Paris, Bibliothèque Mazarine, 2° 4621 bis). There is no Parisian copy of the Proportiones bound without the Arithmetica speculativa. The introduction of the Proportiones says that this text was published as a complement to the Arithmetica speculativa, but according to colophons, the Proportiones was printed two months before the arithmetical text: the formal distinction between these books was then premeditated and reinforced by a different arrangement of the text for each book. In addition, the red and black title page for the Arithmetica speculativa appears as an invitation to place this text before the Proportiones. Thereby, even if Lax’s publications were close to the standard expectations of the calculatores, the book as an object is not less important, but considered part of the intellectual content.
3.3.9 Jean Lambert (1514)
Meantime, Jean Lambert kept publishing mathematical books, sometimes with other colleagues. By 1514, two astronomical texts, first prepared by John Dullaert, had been republished by Lambert: Hyginus’ Poeticon astronomicon and Paulus Venetus’ De compositione mundi (Hyginus 1514; Paulus Venetus 1514). Those were then revised by Juan Luis Vives, who was a student of John Dullaert and who writes about his project in a letter to his friend Johannes Fortius, published at the end of the Poeticon astronomicon (González 2015). The books are not explicitly dated but Lambert’s vignette on the title page suggests a date before November 1513 and Vives’ letter is signed from March 1514 (Paulus Venetus 1513). In addition, we do not know of any book published by Jean Lambert after 1514.
The form of these two books seems to have been influenced by Siliceo’s Arithmetica in theoricen et praxim, as they were both printed in a small quarto with roman typographical characters. In his letter, Vives says that he himself asked the printer to publish the book in more readable types. Moreover, the two editions present Siliceo’s author device, redesigned twice for their publication: in the Poeticon astronomicon, Siliceo’s initials are replaced by a rooster and three faces, and in the De compositione mundi, the rooster and the three faces are replaced by the words “Spes mea Deus” printed in typographical characters. The woodcut was considerably damaged between these modifications, so we can situate the publications in this order chronologically. Lastly, both publications were printed with Thomas Kees material, but his name is not mentioned. We also do not know if he was still active at this moment, or if these two texts were printed by a successor. The De compositione mundi presents a messy application of the signature system (A^{4} [A2 sig. A1] A^{4} BD^{4} E^{6} [E5 sig. F2] GI^{4}); as this was an important element in the printing process of a book, it was perhaps produced by someone less experienced.
3.3.10 Following Editions of Ciruelo’s Arithmetical Texts (1514–1515)
In the middle of the 1510s, Ciruelo’s arithmetical texts were still regularly printed in Paris by several publishers, such as Denis Roce, Jean Lambert, and Olivier Senant, but also Jean de Gourmont (fl. 1506–1522) and Michel Lesclancher (fl. 1512–1520). From the end of the fifteenth century, Lambert and Roce offered many editions of Ciruelo’s arithmetical texts, and between 1513 and 1514, they both published editions of Ciruelo’s Algorismus (Ciruelo 1514a, b). Roce’s edition, dated May 1514, does not present the name of a printer, but the material seems to come from Guillaume Des Plains (fl. 1512–1521), who only signed a few publications (Benoît 1521). Lambert’s edition, on the other hand, presents the name of printer Antoine Aussourd (14th–15th cent.), but is dated from the beginning of the year, March 1513, which could also be March 1514 according to the Easter calendar.^{Footnote 6}
Meantime, Olivier Senant (probably influenced by the mathematical production of his neighbors) published an edition of Bradwardine’s Arithmetica speculativa, following Ciruelo’s editions of this text but in a folio format (Bradwardine 1514). This is an interesting choice, as all the other known Parisian editions of Ciruelo’s and Bradwardine’s arithmetical texts were published in a quarto format. For the most part, the format of an edition of a scholarly book was chosen according to the number of sheets used in the production of each copy. However, by choosing a folio format, Senant proposed something else: an edition of Bradwardine’s Arithmetica speculativa that could be bound with other mathematical texts printed in the same format. The book is not explicitly dated, but the state of Senant’s device is posterior to May 1514 (Jean de Jandun 1514). In addition, we do not know of any occurrence of this device after 1514, so the book was probably published that year.
Finally, the two editions published by Jean Gourmont and Michel Lesclancher, respectively on practical and theorical arithmetic, are different from the previous editions, as they are printed “with corrections and additions” (Ciruelo ca. 1515; Bradwardine ca. 1515). Neither are explicitly dated: Gourmont’s edition was definitely published after 1513, as we find the same device in a much better state printed in a book containing a privilege from this year (Sabellico [1513]). In addition, the corrections to Gourmont’s edition are not reported on Roce’s editions of 1514, while they are present in a later edition (Ciruelo 1524) published in 1524 by Prigent Calvarin (fl. 1518–1566). Lesclancher’s edition of Bradwardine’s Arithmetica speculativa also contains modifications that are not reported in Senant’s edition, which could perhaps indicate that his publication was printed after 1514. Between 1515 and the beginning of 1516, he also printed two mathematical books for other publishers, Jean Petit and Regnault Chaudière (fl. 1509–1554).
3.3.11 Jean Petit and Michel Lesclancher (1515–1516)
During his career, from 1512 to 1520, Michel Lesclancher mainly printed for other publishers. Therefore, he only signed a few books and he did not seem to have his own device. In 1515, he replaced Jean Marchant in printing Sacrobosco’s Sphaera with Ciruelo’s commentaries (Sacrobosco 1515), again under the responsibility of Jean Petit. However, in this new edition, Lesclancher’s name and address are completely omitted, and his presence is only recognizable by the material employed for the printing of the book (Peuerbach 1515). In addition, for this publication, Jean Petit obtained the main historiated diagram used on Estienne’s editions of Sacrobosco’s Sphaera, representing Urania, Astronomia, and Ptolemy under an armillary sphere. This suggests that these two editions were not published concurrently, as they are indeed addressed to different audiences.
Moreover, in January 1515 or 1516 (probably the latter), Petit and Michel Lesclancher also published the first Parisian edition of Georg Peuerbach’s (1423–1461) Theoricae novae planetarum (Peuerbach 1515), revised by Oronce Fine (1494–1555) and following the same disposition of text and paratexts used on the 1515 edition of Sacrobosco’s Sphaera. Peuerbach’s Theoricae novae was not related to the calculatores current, but it was probably designed to be gathered with the 1515 edition of Sacrobosco’s Sphaera, as both were published by the same printer for the same bookseller, with the same general layout of the text. The colophon mentions Regnault Chaudière as copublisher of this text, but the title pages seem to be distributed in the name of Jean Petit. In fact, it is possible that Chaudière’s participation was mostly related to the presence of the technical diagrams, realized by Oronce Fine for his own edition of Sacrobosco’s Sphaera (Sacrobosco 1516). Following editions were exclusively published by him.
4 The End of the Calculatores Current
Oronce Fine, Regnault Chaudière, and Simon de Colines (fl. 1520–1546) were among the main representatives of a new generation of actors in Parisian mathematical book production, which divided the first fifteen years of the sixteenth century between a classical teaching of the quadrivium and the calculatores current. The calculatores current, as we said, developed in a particular geographical and cultural space—the Parisian colleges of Iberian tradition—but it is also the product of a generation of scholars who taught liberal arts in these colleges. It spawned a local market for the publishing of mathematical scholarly books, notably around some personalities of the Parisian book trade, such as Jean Petit or Thomas Kees, and in some specific parts of the university neighborhood, such as in front the SaintBenoît cloister.
The teaching of liberal arts was generally a provisory activity for graduated masters while preparing their doctorate (Quicherat 1860, 77). Between the middle of the 1510s and the beginning of the 1520s, most of these masters who were publishing mathematical books were no longer teaching liberal arts in these colleges: Vives moved to Flanders between 1512 and 1514 (González 2015, 45), Siliceo went back to Spain in 1517 (Villoslada 1938, 191), Hangest left Paris for Le Mans in 1519 (Bietenholz and Deutscher 2003, 409), Tomás had obtained his medical doctorate by 1520—after which we have no other information about him (Leitão 2000)—and Lax was in Spain in 1524 (Villoslada 1938, 406). From 1515, there are no more mathematical books published for the Parisian calculatores current, except a last reprint of Hyginus’ Poeticon astronomicon in 1517 (Hyginus [1517]). By the middle of the 1520s, there were no more calculatores from the previous generation teaching mathematics in Paris.
In general, most of the books published for the calculatores current were not republished after the departure of the authors. This includes Ciruelo’s books, even if the practical arithmetic was still republished in the 1520s by Prigent Calvarin (fl. 1523–1566). Bradwardine’s Geometria speculativa was also republished in 1530 by Regnault Chaudière following the previous editions, but the name of Ciruelo is completely absent (Bradwardine 1530). The other books related to the calculatores were not reprinted after the end of the current, except Siliceo’s Arithmetica theoricen et praxim, republished by Oronce Fine in 1519 and Thomas Rhaetus in 1526, both teachers of liberal arts in Paris (Siliceo 1519, 1526). In 1540, the Parisian publisher Jean Loys, personally interested in mathematics, gave a summary of this text, reprinted by Prigent Calvarin in 1542 (Siliceo 1540, 1542).
Most of the publishers who produced mathematical books for the calculatores current stopped publishing scholarly mathematical texts after this period. Some of them were not active anymore by the second half of the 1510s, such as Denis Roce, Jean Lambert, and Thomas Kees, but others were still publishing in the 1520s, for example, Olivier Senant, Hémon Le Fèvre, and Jean Petit. In the following years, mathematical texts were sometimes present in their publications addressed to a larger audience, like Anianus’ (14th cent.) Computus manualis, the Coeur de philosophie, or the Calendrier des bergers, but from the second half of the 1510s mathematical scholarly books were mostly published by Henri Estienne’s family network, including Regnault Chaudière from 1516, and Simon de Colines from 1520.
Finally, in the long term, mathematical publications related to the Parisian calculatores do not stand out as a reference in the teaching of these disciplines. However, they presented an alternative to the traditional conception of the quadrivium and had a great influence on the development of a properly modern conception of mathematics. Moreover, the calculatores current also led to an important increase in the production of mathematical textbooks in Paris, in the number of publishers interested in these publications, and in the presence of these disciplines in the Parisian market. In consequence, during the next fifteen years, Parisian professors and publishers would not look so much for the publication of new mathematical textbooks, but would invest their time and skills in the publication of more advanced mathematical books, primarily addressed to an international audience: from the 1520s, Paris would be a main pole of the European mathematical book trade.
Notes
 1.
Dates given in bibliographical references are those which appear on the printed book. However, in France, until the middle of the 16th century, the year could begin on January 1 (new style) or on Easter (ancient style). There was no established dating system for printed books even at the level of a particular workshop (VeyrinForrer 2017).
 2.
The publisher’s device is an image, generally printed in the title page, giving information about the publisher.
 3.
Bibliothèque nationale de France, RESV203, f. H8.
 4.
This information is given by the ISTC, ic00699600.
 5.
Numbers related to Parisian printed production are based on the Incunabula Short Title Catalogue (ISTC) and the Bibliographie des éditions parisiennes du 16^{e} siècle (BP16).
 6.
Abbreviations
 BP16:

Bibliographie des éditions parisiennes du 16^{e} siècle. Bibliothèque nationale de France. https://bp16.bnf.fr/. Accessed 07 June 2021
 ISTC:

Incunabula Short Title Catalogue. Consortium of European Research Libraries. https://data.cerl.org/istc/. Accessed 07 June 2021
 Sphaera CorpusTracer:

Max Planck Institute for the History of Science. https://db.sphaera.mpiwgberlin.mpg.de/resource/Start. Accessed 07 June 2021
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Levy, A. (2022). Publishing Mathematical Books of Parisian Calculatores (1508–1515). In: Valleriani, M., Ottone, A. (eds) Publishing Sacrobosco’s De sphaera in Early Modern Europe. Springer, Cham. https://doi.org/10.1007/9783030866006_13
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