Scholars in early medieval Europe had very little access to mathematical knowledge. Greek was no longer a language most could read, and contact with Byzantium was, anyway, very limited. Few things had been translated to Latin. Though scholars said positive things about the importance and value of mathematics, their actual mathematical knowledge was quite small. Of Euclid, for example, the definitions and statements of propositions from the first five books were available in Latin, but not any of the proofs except for the first three.

Starting in the ninth century, Islamic scholars learned the ancient results from Greek and India and extended them, creating new subjects such as algebra. The *lingua franca* of the Islamic world was Arabic, into which Greek words were translated and in which the new works were written. So it was from materials in Arabic that mathematics was learned again and re-introduced in Europe. Only much later were the Greek manuscripts found and translated.

The transmission from the Islamic cultures to Europe happened in two places, almost independently. From the tenth century onward, Europeans could learn mathematics in Spain, then under Islamic control. Of course, this might require learning Arabic, or at least the help of someone who knew both Arabic and a European language, which was something of a challenge. Nevertheless, a huge translation enterprise developed in the eleventh and twelfth centuries. A little bit later, in the late twelfth century, Italian traders came into contact with the Islamic culture of North Africa and began to learn practical and theoretical mathematics from them. In the latter case, the big name is Leonardo of Pisa, later known as Fibonacci. His *Liber Abaci*, from around 1200, contained material he had learned in the Eastern Mediterranean, from the Indo-Arabic numeration system to algebra.

By contrast, it’s hard to point to a single main vector in Spain. Instead, there were many different translators who presented Latin versions of many Arabic works. Al-Khwarizmi’s book on how to calculate with Indo-Arabic numerals, for example, was translated several times, as was his algebra book. Greek philosophical works were translated from Arabic, and so were some mathematical works. But there is a “big book” from that region as well: the *Liber Mahameleth*, presented here in a magnificent edition by Jacques Sesiano.

These three volumes are the result of some forty years of work by Sesiano. The first volume contains an introduction and a critical edition of the full Latin text of the *Liber Mahameleth. *The second volume presents a translation to our current *lingua franca*, English. The third volume contains an extended mathematical commentary. Together, they represent an amazing resource for scholars that may also interest other mathematicians.

The *Liber Mahameleth* was written in Latin, probably, Sesiano argues, by John of Seville. The word “mahameleth” is a Latinized version of an Arabic word that means something like “business mathematics.” The author clearly lived in an Islamic culture and could almost certainly read Arabic. In particular, he uses and refers to Abu-Kamil’s *Algebra*, which had not yet been translated into Latin. Sesiano speculates that the book was intended for the use of the Christian community in Sevilla, and that the manuscript was eventually taken to Toledo.

Unfortunately, the book seems to have been left unfinished by the author, leading to a very messy manuscript situation that is well-described in Sesiano’s introduction. In particular, the only existing copy that seems to be in the right order is *not* our oldest copy. Sesiano has nevertheless decided to stick to that ordering of the materials, which at least produces a text that can be read sequentially. The table of contents gives some sense of what the book contains. It is divided in two parts: a shorter theoretical part and a longer applications part. As a sample, here is the first case of “addition of a fraction to a fraction”:

**(A.145)** You want to add three eighths to four fifths.

(a) You will do the following. Multiplying the denominators of the fractions produces forty, which is the principal number. Next, add three eights of the principal number, thus, fifteen, to four fifths of this same principal number, thus thirty-two; this makes forty-seven. Dividing it by the principal gives one and eighth and two fifths of an eighth, and this is result from the addition of the proposed quantities.

(b) Or otherwise. Convert the fractions of one side into the others, namely four fifths into eighths; this gives six eighths and two fifths of an eighth. Add it to three eighths; this makes nine eighths, which is one and an eighth, and two fifths of an eighth. This is the quantity you require.

Several things jump out: numbers are written out in words, as in most Arabic texts; there are more “cases”, and one wonders what they might be; there is no “least common denominator”; instead there is this curious “fraction of a fraction” construct, “two fifths of an eighth.” All of this is fertile ground for a discussion in a class for future teachers, and to my mind this is one of the ways the book may prove useful.

Similarly, part B is essentially an immense collection of “story problems,” many of which tell us interesting things about the culture in which they were born, or about the uses of mathematics, or about how mathematical problems are created and transmitted. There is a full section of problems about ladders, for example:

A ladder ten cubits long standing against an equally high wall is withdrawn from the bottom of the wall by six cubits. By how much will it come down from the top?

That could easily be assigned in school today, and probably is!

There is a lot of useful material to be mined here. Alas, buying this set is not for the faint of heart or the short of cash, so we have to hope that libraries will decide to invest.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He really likes old books.