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Ad Meskens
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In his masterful book, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (2014), Amir Alexander writes that the French courtier Samuel Sobrière’s (1615–70) network included “some of the greatest luminaries in France, and philosophers and scientists in Italy, the Dutch Republic and England.” For a Fleming, like myself, it is striking that Alexander omits Spain and, more importantly, the Spanish Netherlands, thus reinforcing, albeit unwittingly, the idea that the Southern Netherlands was a scientifically barren nation after the re-conquest by the Spanish. It is certainly true that the Northern Netherlands gained a great deal from the exodus from the South. Not in the least did Amsterdam flourish because of the Antwerp merchants, with their capital and commercial network, who had found refuge there. In their wake came numerous painters and scientists, Gillis Coignet (c.1542–99) and Simon Stevin (c.1548–1620) among them. But this cultural and scientific hemorrhage did not suck the lifeblood out of the South. The South was crippled, but not on its knees; on the contrary, this was the age of Peter Paul Rubens (1577–1640) and Gregorio a San Vicente (1584–1667).

On the back cover of Alexander’s book, we find:

On August 10, 1632 five Jesuits met in an austere Roman palace to pass judgment on the veracity of an apparently simple idea: that a continuous line is made up of distinct and limitless small parts. On that fateful day the judges ruled that this notion wasn’t only improbable but actually repugnant—with the stroke of a pen it was banished.

A number of mathematicians within the Jesuits’ own ranks would have vehemently disagreed with this judgment. Indeed, it was one of their own who had invented the concept of summating an infinity of terms to find the area of a plane figure or of a solid; it was one of their own who, although intuitively, defined the limit concept of a series; it was one of their own who pushed the exhaustion method to its limits. And it was some of their own, who despite the judgment, would be at the cradle of integral calculus. These Jesuits were San Vicente and his students.

The idea for this book arose when I was asked to write a contribution on mathematics for a book on the development of Antwerp’s intellectual life. Writing the chapter brought to light how little attention had been paid to these Jesuit-mathematicians, despite their remarkable and ground-breaking contributions to mathematics.

This book is about the Flemish Jesuit school of mathematics, in which San Vicente played a major role. This school would have a profound influence on the course of mathematics in the seventeenth century, an influence that has been hugely underscored up to now. Not only did San Vicente arrive at his most influential insights when working in Antwerp but he also taught students who would become renowned mathematicians in their own right.

It all began with a simple Jesuit house and college, which became a focus of a triumphant Catholicism during the Counter-Reformation. The book will show that the Flemish school of mathematics was a major force of change within mathematics and mechanics during the first part of the seventeenth century. However, although San Vicente may have been the most important Flemish Jesuit mathematician, he was by no means the only one with any influence. Joannes della Faille’s (1597–1652) theorem, for instance, is still compulsory knowledge for any mechanics student. The failure of San Vicente to bring his circle quadrature to a good end still casts a dark and undeserved shadow over his reputation and that of his students. To quote William Shakespeare (c.1564–1616): “The evil that men do lives after them; The good is oft interred with their bones” (Julius Caesar, act 3, scene 4), or in this case—“The error a mathematician makes blots his reputation, his grand theorems are oft forgotten to be his.”

This book deals with mathematics and mathematicians, but not entirely as we now know them. At the turn of the century in 1600, mathematics would have included subjects we know as accounting, astronomy, physics, engineering, wine-gauging, cryptography, and, to a lesser degree, architecture. For court mathematicians, it would, in many cases, also have included astrology and perhaps even numerology. On the other hand, some mathematical subjects, such as logic, were part of the philosophy course, while astrology was studied in conjunction with medicine. By 1700, pure mathematics was more clearly defined as an abstract subject and began to resemble our concept of mathematics. This is in no small part due to the development of infinitesimal calculus, which, in a sense, can be viewed as the fusion of algebra and geometry. Under the influence of Johannes Kepler (1571–1630) and Galileo Galilei (1564–1642), other sciences became mathematized, not only physics but also geography, stonecutting or stereotomy, and even music. The distinction between a mathematician and an engineer or a scientist remained hard to make. Some even ventured into what we would call the occult sciences, the best-known example being Isaac Newton’s (1643–1727) (al)chemical interests. Fortunately for us, our subjects conform to our image of a mathematician, their prime research interest being geometry, and with it a nascent calculus. The book will, however, also point out whether and to what extent they ventured into other sciences.

In writing this book, I sometimes had to make difficult decisions. Keeping San Vicente’s style would deter any modern reader, and I have consequently chosen to “translate” his prose into a more modern mathematical style, even though many historians of mathematics would disagree. Not doing so would result in a very hard to read and lengthy text. Moreover, San Vicente was a genius when it came to finding particular cases and proving all of them, usually without drawing a generalization. Yet every aspect of this generalization is present in his work.

For geographical names, the book uses the names that are presently in use, even though they may have been different in the seventeenth century. An exception was made for names that have become standard in English, such as Prague or Ypres, instead of Praha or Ieper.

The lack of an appropriate historical-geographical terminology for the Low Countries in English for this specific period may create some confusion. The book uses the term “Seventeen Provinces” or “the Netherlands” (de Nederlanden) for the country ruled by Charles V (1500–58, r.1515–55) and Philip II (1527–97, r.1555–97) that was created by the Pragmatic Sanction (1548), as opposed to the nation we now call the Netherlands (Nederland). The Low Countries also includes the independent prince-bishopric of Liège. In general, Flanders is used to indicate the northern Dutch-speaking part of the Southern Netherlands; however, in citations of a seventeenth-century document, Flanders refers to the then County of Flanders. The religious–civil war at the end of the sixteenth century created two nations in the Low Countries: the Southern or Spanish Netherlands on the one hand and the Northern Netherlands or the Republic of the United Provinces (the republic for short) on the other.

The people studied in this book are referred to by the name under which they were known when they were alive or by their baptismal name. An exception is made for Gregorius a Sancto Vincentio who, in this monograph, is called Gregorio a San Vicente. His students are referred to by their proper names.

Writing the book also involved navigating the difficulty of how best to arrange the material. This could have been done chronologically, in which case the theater would have had to change every few pages, going from Prague to Madrid to Stockholm, without any coherence in the mathematical subject matter. The material could have been arranged thematically, but in that case, some of the individuals’ work would had to have been torn apart, making it hard to see a line of thought in their work. It could have been arranged by person, but this would have made it difficult to see the connections between the individuals’ work. As none of the options was ideal, it was ultimately this last option that was chosen, although architecture has been kept as a separate subject. This allows the narrative to focus in particular on San Vicente.

Much of the material presented here has been published before, but it is dispersed over many articles and sometimes hard to find books. San Vicente has already been studied by Herman van Looy, Jean Dhombres, and Patricia Radelet-de Grave; Francisco de Aguilón (1567–1617) has been studied by August Ziggelaar, S.J.; and della Faille has been the subject of my own research, not to mention the earlier studies by the Jesuits Henri Bosmans (1852–1928) and Omer van de Vyver (1915–97). The same remark goes for the archival material, which is dispersed over many archives. Nevertheless, I feel I have been able to bring together the core of this material.

It is fortunate that Van Looy allowed me to use his material, as the description of San Vicente’s manuscripts is based to a significant extent on his doctoral thesis. These parts have been duly indicated.

I hope I have brought all of this material, and new archival sources, together in a coherent narrative, bringing a new view on this remarkable group of men. This book would not have been possible without the help of numerous people. At the risk of forgetting someone, we mention Fr. Raúl González, S.J. (ARSI), Van Looy, Stephen Hargreaves, Philippe Martin, and Miguel Robert (University of Barcelona).

The staff of Erfgoedbibliotheek Hendrik Conscience Antwerpen, Museum Plantin Moretus (UNESCO World Heritage), Felixarchief Antwerpen, KaDoc Leuven, the State Archives of Belgium at Beveren, and the libraries of the University of Antwerp (especially Ruusbroecgenootschap) and Artesis Plantijn University College Campus Noord were, as usual, very helpful. The cooperation with Paul Tytgat, who provided advice on the drawings, proved as invaluable as ever.

Last, but certainly not least, I have to thank my wife Nicole, without whom this book would never have materialized. Her support genuinely is a labor of love.

Ad Meskens

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