When I reviewed Victor Katz’s sourcebook of The Mathematics of Egypt, Mesopotamia, China, India, and Islam some years ago, I pointed out that it covered all but one of the important non-Greek pre-modern mathematical traditions. The missing one was early medieval mathematics in Latin, i.e., the mathematics of the period often referred to as the “Dark Ages.” Many of the important works in this tradition have received proper critical editions, but very few of them have been translated into a modern language. As a result (and despite the heroic efforts of Menso Folkerts), the mathematics of this period remains under-studied or treated as not really worthy of study. If medieval mathematics is discussed at all, the discussion usually begins with the first translations of mathematical works from Arabic, leaving the whole period between the fall of the Western Roman Empire to the 1100s untreated except for a mention of one person: Gerbert of Aurillac, who became Pope Sylvester II.
We learn very little about Gerbert from the standard sources: he was born around 945, he studied in Spain, he taught in Reims, he knew the Hindu-Arabic numerals, he became pope, he died in 1003. It would be easy to conclude from this that there was very little information about Gerbert’s life and works, leaving us with little else to say about him. This is far from true. Gerbert’s mathematical works were edited in 1899 by Nicolaus Bubkov; it’s a well-made volume of 600+ pages, almost all in Latin. More importantly, Gerbert seems to have saved copies of his letters beginning around 976, so that we have quite a bit of information about the last thirty years of his life. (The letters have been translated into both English and French.) Finally, since Gerbert was pope at the end of the first millennium, he received some attention from both academic and popular historians at the end of the second millennium.
Nancy Marie Brown has accessed all of this material to put together The Abacus and the Cross, a biography of Gerbert aimed at the “educated public.” The result is a lovely book, interesting and a pleasure to read.
Brown has written a work of serious popular history: popular in that it is aimed at the general public rather than professional historians or mathematicians, serious in that it has been carefully researched. Brown includes a lot of background on life in medieval Europe and on the state of scientific knowledge at the time, which should make her book very useful to teachers and students who are interested in the history of mathematics but are not medievalists. One of the book’s goals, of course, is to tell the story of Gerbert’s life and to give us some insight into the man and his ideas. Brown has another point to make, however: she wants to convince us that our notion of what those “dark ages” were like is mostly mistaken. For example, she explains that every educated person knew that the Earth is a sphere, and takes the time to investigate the source of the myth that people believed the Earth was flat. (The answer seems to be a book about Columbus written by Washington Irving in the 1800s.) She explains the trivium and quadrivium, discusses what a monk’s life was like, tells us what it was like to be pope in the 900s, and so on. The result is a rich picture of life in early medieval Europe, presented as the background to a very interesting life.
The Gerbert that emerges from The Abacus and the Cross is fundamentally a teacher. While he wrote well, he seems to have been most comfortable in the classroom. Brown cites, for example, a text on the abacus in which Gerbert seems exasperated at how hard it is to explain in writing what he could easily have demonstrated in class. She also notes Gerbert’s use of visual aids: a celestial sphere, for example, and a large abacus intended for teaching. Gerbert’s mathematical work, however, never really comes into focus. Gerbert clearly loved the subject. He introduced several new ideas to European mathematics, most notably the Hindu-Arabic numerals, and he wrote a work on geometry that seems to have had some influence. But exactly how those symbols were used in Gerbert’s abacus and what is in Gerbert’s geometrical work remains unclear in Brown’s book. Perhaps we are still waiting for historians of mathematics to sort that out.
Becoming pope turns out to have been anything but a blessing. The teacher ended up embroiled in politics partly because of the influence of his mentor, Bishop Adelbero of Reims, and partly due to his own dreams of a rational and well-ordered empire. Gerbert, as it turns out, was not always a successful politician or administrator. He would have been happier if he had remained a teacher in Reims.
Brown has read widely, but her treatment of the scholarship on Gerbert and his time is shaped at least in part by the kind of book she is writing. Whenever scholars are uncertain about something, Brown tends to go for the option that will make the best story. That makes sense, but it does mean that she takes as known things that scholars would probably want to leave undecided. Her account of the middle ages at times (for example, when she discusses the theft of relics) veers just a bit into a kind of exoticism that seems to ask us to marvel at how strange people of the time were. The effect is like a form of “orientalism” applied to the early middle ages.
Brown clearly likes Gerbert, and as a result his political enemies come off rather badly; don’t look here for a balanced treatment of Abbo of Fleury. To emphasize the importance of Gerbert’s work, she sometimes overstates his prominence. Her account of his death, in particular makes it seem as if he was the last bastion of rationality and common sense. After him, the flood: a time of oppression and intolerance culminating in the Crusades and the Inquisition.
There are occasional flubs. At one point, for example, Brown discusses a letter of Gerbert that attempts to explain why the idea of using the triangular numbers to compute the area of an equilateral triangle does not work. She explains Gerbert’s analysis (p. 110), referring in particular to an article by G. A. Miller in the Monthly. What she does not say is that Miller’s article explains why Gerbert’s entire argument is bogus; my guess is that Brown did not quite understand his point. (I must admit that the article is quite badly written.)
Most of the errors are about tangential matters. An example is a remark on page 111: in the middle of an irrelevant but irresistible discussion of the Archimedes Palimpsest Brown says that mathematicians had not seen Archimedes’ Method until 1999, which is kind of surprising given that it appears in English translation in Heath’s 1912 book. Brown often says that the Greek originals of certain texts “were lost,” giving the impression that even the modern texts derive from the Arabic, which in most cases is not true. My favorite is a theological mistake: she describes on page 154 a mosaic of Christ’s baptism as “a heretical version that showed him naked and fully human.” I do not know if the mosaic is heretical, but if so it is certainly not for that reason! So, as I keep telling my history of mathematics students, “trust, but verify.”
The Abacus and the Cross gives us a well-written account of a fascinating man. May it encourage more work on Gerbert and his mathematics.
Gerberti postea Silvestri II papae Opera Mathematica, ed. By Nicolaus Bubnov. Friendländer & Sohn, Berolini, 1899.
The Letters of Gerbert, with his Papal Privileges as Sylvester II, translated with an introduction by Harriet Pratt Lattin. Columbia University Press, 1961.
“The Formula ½a(a+1) for the Area of an Equilateral Triangle,” by G. A. Miller. American Mathematical Monthly, June-July 1921, 256–258.
Correspondance de Gerbert d’Aurillac, ed. by P. Riché and J.-P. Callu. Les Belles Lettres, 1993. (Not seen; citation as given in The Abacus and the Cross.)
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College; he wishes his Latin were better.