The Medieval Interfaith History of Mathematics

So what image do you have of the Middle Ages? Is it that? I asked one of my granddaughters the other day what she thought when you think of the word Middle Ages. She said, “Yeah, knights and armor.”

But maybe there’s other images you might know. Here’s an image of two Muslims discussing some point in philosophy. And that was certainly something that would happen in the Middle Ages. And here is an image of a Jewish trader, even has a symbol on his cloak that indicates that he’s Jewish. In a lot of places Jews had to wear identifying symbols. Muslims and Jews generally didn’t participate in medieval battles like the previous slide. Much of what they did was intellectual, as we will see.

So what I want to talk about tonight is the way religious cultures affected, in particular, the study of mathematics. We will see that although there were religious objections to the study of science in general, and mathematics in particular, still many religious leaders claimed that God required you to learn about the world.

Why? Well, there were several reasons. There was a general idea of fairness – “How do you figure this out?” Also, to determine inheritances, especially when there are lots of kids, you certainly needed mathematics. Even more important was creating a calendar, which was very important for the Muslims and Jews early on, and later for Catholics as well. Prayer practices were also often determined by a detailed knowledge of astronomy, which in turn required mathematics.

And certainly, if you were going to debate someone on any subject at all, you needed to be able to understand, to produce, logical arguments. Mathematics was central to doing this. Finally, we will see that mathematicians working in these three religious cultures learn from each other.

All right, I’m going to begin by looking at a problem that shows up in medieval times in many places. It’s called “three men buying a horse.” So, three men want to buy a horse in common. The first says to the others that if they give him 1/3 of their money, then, with the money he already has, he can buy the horse. The second says to the first and the third that if they give him 1/4 of their money, he can buy the horse. And the third says to the first and second that if they give him 1/5 of their money, then he can buy the horse. So how much does each man has to begin with, and how much does the horse cost? Well, I guess the three men must have known each rather well, if they needed to do something to figure out how to share a horse.

In any case, I’m not going to ask you to solve this problem right now, but if you feel impelled to do it, go right ahead. But why are we interested in this problem, because it appears in mathematical textbooks coming from medieval Islam, from medieval Jewish writing, and also from medieval Catholic writing. Why were all three civilizations interested in a problem like this?

It’s hard to say specifically, but people who wrote mathematics in all three groups were interested in solving problems and teaching students how to solve problems. Somehow, this problem caught the fancy of medieval mathematicians. That it appears in texts of all three cultures, certainly get some indication that mathematicians learned from each other. The problems were, in all likelihood, not invented from scratch by the various authors.

Also, it turns out, although the problems were the same, the authors’ methods of solution were not. As we will see, this tells us something about how Muslim, Jewish and Catholic mathematicians approach mathematics.

Okay – now before we go further with the mathematics. Let me remind you of some history. Remember that Islam began in Arabia in the middle of the 7th Century, but was soon spread by Muslim armies throughout the Middle East, then East as far as India and West through North Africa into Iberia, which was conquered the second decade of the 8th Century.

Initially, the Muslim Empire was ruled from Damascus, but after many wars, the Caliphate split up. Here you can see a picture of this in the Eastern segment. The Eastern Caliphate found a new capital in Baghdad, where a new culture developed.

It was there that Harun al-Rashid established a library in Baghdad in which manuscripts, especially scientific and mathematical ones, were brought from Greece, India and elsewhere and translated into Arabic. This was the beginning of a very strong scientific movement in Islam, in which scholars not only read the old sources but also mastered them and continued to do research and make new discoveries.

Now, in Iberia,  a similar Golden Age began under the rule of the Umayyad dynasty beginning in 912. But after 1031, al-Andalus broke up into many small Islamic kingdoms, as you can see on the slide – some of which encouraged the study of science, while others did not.

Meanwhile, of course, the Catholic Reconquista was well underway. Catholics pushed down from the North and gradually pushed the Muslims out of Spain. As you probably remember, a critical date was 1085, with the reconquest of Toledo, which was right in the middle of Spain. Toledo had been one of the richest of the Islamic kingdom, but was conquered in that year by Alfonso the 6th of León and Castile. Fortunately, Alfonso was happy to leave intact the intellectual riches that had accumulated in the city. And so in the following century, Toledo became the center of the massive transfer of intellectual property undertaken by the translators of Arabic material, including previously translated written material into Latin. More on that a little bit later.

But what of the Jews? Well, there were other Jews in many parts of the Islamic world. It was mainly in Spain, in Provence, that Jews became active in mathematics starting in the 12th century. During the 11th century, with a breakup of al-Andalus and the return of Catholic rule to a part of the peninsula, Jews were often forced to make choices of where to live. Some of the small Islamic kingdoms welcomed Jews, while others were not so friendly. The Catholic monarchs at the time often welcomed us because they provided a literate and a numerate class fluent in Arabic, who could help the emerging Spanish kingdoms prosper.

So by the middle of the 12th century, most Jews in Spain lived under Catholic rule. But once the Catholic kingdoms were well established, the Jews were often persecuted. So then in the 13th century, Jews started to leave Spain, with the final blow coming in 1492. By then, many Jews had moved to Provence or to Italy, and it was there that the Jews began to fully develop their interest in science and mathematics. They also began to write in Hebrew rather than Arabic, which [had been] their intellectual language back in Muslim Spain.

Now, in this talk, we’re going to concentrate initially on Muslim and Jewish work and mathematics, since the Catholic word generally came much later. And we will break this study down to various topics of mathematics.

So we’re going to start with arithmetic. Now, as you probably know – well first of all, arithmetic was necessary for everyone. Anybody, right, to be in the marketplace had know arithmetic. But what do we mean by arithmetic? Well, calculation, to start with, and after all our base-10 place value system comes from India, but it was transmitted to Baghdad in the 8th Century. And it turned out – you know, you might think that because this system was very useful, that it would easily replace earlier systems that Islamic people had, but it wasn’t quite so simple. One problem with this medium was that calculations, at the beginning, were generally done on a dustboard, a board covered with sand on which the tradesmen wrote the figures. But one Muslim scholar wrote, “Many a man hates to show the dustboard in his hand when he needs to use this art of calculation, for fear of misunderstanding, since it since it is seen in the hands of the good-for-nothings earning their living by astrology in the streets.” So people didn’t necessarily want to use it.

Muslims actually often used finger reckoning. Now, this this slide is not Muslim finger reckoning – this comes from Europe, because we don’t know exactly how the Muslims used finger reckoning. But it was probably a system similar to this.

But finally, with the fortunate introduction of paper, Islamic domains eventually adopted the base 10 place value system, and started to write books on how to use it for calculation – how to do various algorithms. The Jews, like the Muslims early on, usually used letters of the alphabet to represent numbers, and did calculations on some kind of counting board. By the 12th century, they had adopted the Hindu system, but instead of using the numbers that we’re used to, they used Hebrew letters for the first nine digits – you can see in this circle here – and they used the circle for the zero. They also learned the appropriate algorithms to do calculation.

Although arithmetic was certainly necessary in various aspects of daily life, it was Abraham bar Hiyya, a Spanish Jewish scholar from the early 12th Century, who wrote about its religious importance, in particular for calculations regarding the Jubilee. Now, I don’t know if you remember what the Jubilee is, but here’s a quote from Leviticus 25, where it’s talked about:

“And you shall count seven weeks of years, seven times seven years; so the time of the seven weeks of year shall be to you forty-nine years. Then you should send abroad the loud trumpet on the tenth day of the seventh month, you shall hallow the fiftieth year and proclaim liberty throughout the land to all its inhabitants. It shall be a jubilee for you, when each of you shall return to his property and each of you shall return to his family.” (Leviticus 25:8-10)

What that meant was that all debts were released, and people had to give back land that they had bought, and all sorts of things like that. And so what Abraham wrote was:

[The scriptures say] “I the Lord am your God, instructing you for your own benefit, guiding you in the way you should go” … (Isaiah 48:17). From which you learn that any craft and branch of wisdom that benefit man in worldly and holy matters are worthy of being studied and practiced.

I have seen that arithmetic and geometry are such branches of wisdom, and are useful for many tasks involved in the laws and commandments of the Torah. We found many scriptures that require them, such as, “In buying from your neighbor, you shall deduct only for the number of years since the jubilee”, and “the more such years, the higher the price you pay; the fewer such years, the lower the price”, followed by: “Do not wrong one another, but fear your God” (Leviticus 25:15-17).

Further, he said, “No man can count precisely without falsification unless he learns arithmetic.” So there was a religious reason for learning arithmetic.

But what other reasons do you need to study arithmetic? Well, Abraham ibn-Ezra, who also was born in Spain and later traveled widely during his lifetime, wrote a good deal about mathematics. He talked about studying – about the study of proportions, right, here’s a problem in proportions, fairly simple “Reuben hired Simon to carry on his beast of burden 13 measures of wheat over 17 miles for a payment of 19 pashuts. But he carried seven measures over 11 miles. How much should he be paid?” You needed some methods of calculation to figure this out.

Also, Abraham – it was interesting – was an early contributor to combinatorics, ways of counting various sets. In particular, he believed that when two or more planets had a conjunction – that is, when they appear together in the sky – this was an important portent of events to happen on the Earth. So he used his skill in calculating to develop the basic method of determining how many possible conjunctions there could be of the seven wandering heavenly bodies. Remember what they were in his day – Mercury, Venus, Mars, Jupiter and Saturn, and the sun and the moon. Those were the wanderers in the sky.

Now, Muslims and Jews contributed further to this study over the next couple of centuries. Interestingly, the major question for both groups, ultimately, was not conjunctions, but how many words could be formed out of the letters of the alphabet, either the Arabic alphabet or the Hebrew alphabet. For Jews, this was a religious question, since they believe that God had created the world and everything by naming these things – in Hebrew, of course, which was God’s first language.

So, the question was “How many things could be named?” That means “How many words can you form?” By 1321, Levi ben Gerson, who we will talk about a little later in more detail, had rigorously calculated the basic combinatorial formulas, based on earlier work by Islamic mathematicians.

All right, now, growing out of arithmetic, beginning of the 9th Century, was the science of algebra. The first algebra text was written around 825 by Muhammad al-Khwarizmi in Baghdad.

Now, Muhammad al-Khwarizmi –  his name, “al-Khwarizmi,” means he came from Kwarzim, which is now in western Uzbekistan. And if you go to Urgench in Uzbekistan, you’ll see the statue of Muhammad there. Of course, there’s no picture of Mohammed, but somebody just created this statue. Doesn’t mean he looked like that.

In any case, he wrote this first algebra text around 825. As he said in the introduction, “That fondness for science, by which God is distinguished the Imam al-Ma’mun, the Commander of the Faithful, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade.”

Okay, so what he says was – that the study of algebra was to be able to solve questions of inheritance, partition, lawsuits, trade, et cetera. However, the central part of al-Khwarizmi’s algebra text was the presentation of the ways of solving quadratic equations. Now, I’m sure you all remember how to solve quadratic equations – that’s drummed into you when you started out to study algebra, back in ninth grade, or maybe earlier. And you may remember from your algebra texts that any word problem about quadratic equations was quite artificial. The textbook authors who wrote your textbooks could never figure out exactly what kind of problems – real-world problems – needed quadratic equations. And neither did al-Khwarizmi. His book has a large number of problems dealing with inheritance, because he said that’s important, but those problems don’t need quadratic equations to be solved, generally only linear equations. And all the quadratic equations he had in there were abstract. There were very few word problems that were sort of very fake.

So it’s really interesting that al-Khwarizmi was trying to justify the study of quadratic equations in those terms, but actually that study was not necessary for everyday life. Although Islamic mathematicians continued to write algebra texts over the next several centuries and made many discoveries of a rather advanced nature, none of these mathematicians could actually point to real-life problems that required quadratic equations, only to interesting theoretical questions. Because of this lack of application, the Jews generally didn’t study the subject. Here’s a quotation from a Jewish philosopher in Toledo in the middle of the 12th Century. As he says,

“Among those who spend their time on vanities, thereby depriving their soul of afterlife, is he who consumes his time with number and with strange stories like the following: A man wanted to boil fifteen quarters of new wine so that it be reduced to a third. He boiled it until a quarter thereof departed, whereupon two quarters of the remaining wine were spilled; he again boiled it until a quarter vanished in the fire, whereupon two quarters of the rest were spilled. What is the proportion between the quantity obtained and the quantity sought?”

So again, Abraham was sort of pointing out these ridiculous problems that were in algebra textbooks at the time and saying, “Why do you need to do this?”

So Jews didn’t really study algebra. In fact, many Jewish authorities at the time wrote that Jews limit their studies to the Torah and the Talmud. But the most important Jewish philosophers of the Middle Ages, namely Maimonides, who lived from 1138 to 1204, at first in Spain and later in Egypt, convinced Jews that they should in fact study mathematics. As he wrote, the study of science and philosophy was actually a religious obligation, beginning with logic.

Maimonides emphasized it was only truth that counted, and that it did not matter who discovered it. On the other hand, since it was the divine science of metaphysics that was the ultimate goal, Maimonides emphasized that science was legitimate and desirable only insofar as it contributed to the divine science. Thus, medieval Jews were to study mathematics either because it prepared the intellect to apprehend abstract truths, or because they needed it as a prerequisite for the study of mathematical astronomy. But for Maimonides, algebra was a mere technique, having no philosophical value or even practical use. Now, this picture of Maimonides, again, is traditional, although we have no idea what Maimonides actually looked like, but it turns out the signature at the bottom is his actual signature that was discovered in a Genizah in Cairo.

Okay, so let’s move on to geometry. Going back to Abraham bar Hiyya – well, first of all, there’s two pieces, two parts of geometry that were of interest. I mean, anybody who knew something about geometry knew that Euclid had written this major work, The Elements of Geometry. But it was an abstract work, and used logical proof from explicit axioms. But people were also interested in other kinds of geometry – practical geometry that was actually necessary in one’s life, including the calculation of areas and volumes. So Abraham bar Hiyya wrote, as follows:

“Geometry is useful for […] following commandments from the Torah, but is difficult to understand, so “so one has to study and interpret it for land measurement and division between heirs and partners, so much so that no one can measure and divide land rightfully and truthfully unless they depend on this wisdom.” I have seen in most contemporary scholars in Spain and Provence are not skillfully measuring land, and do not divide it cleverly. Their calculation might mete out a quarter to the owner of a third, and a third to the owner of a quarter, and there is no greater theft and falsification.”So Abraham himself – he wrote his treatise on geometry. He presented not only rules for finding areas and volumes, but also presented methods for dividing land, including what one should do when land is on a slope.

Abraham ibn Ezra also wrote a good deal about practical geometry. As for theoretical geometry, even Maimonides felt that its study was useful. After all, Euclidean geometry is itself a major exercise in logic, and was therefore important in being able to understand abstract truths – important for the study of metaphysics.

Interestingly, Maimonides explicitly cites the study of conics – that is, parabolas, ellipses, and hyperbolas – those of you who know something about mathematics know something about these – that this was a topic that is commendable only inasmuch as it “Follow[s] the aim of sharpening the intellect and getting the intellect used to demonstrations, which is the way through which man achieves the knowledge of the true existence of God.”

Now, Maimonides believed that part of the study of conics was useful in sharpening the intellect. In particular, he believed that the existence of asymptotes showed that just because you could not imagine something did not mean that it did not exist. And what he meant by that, he wrote, “It has been made clear in the second book of the conic sections” – that was this treatise by Apollonius – “That two lines between which there was a certain distance at the outset, may go forth in such a way that the farther they go, this distance diminishes, and they come nearer to one another but without it ever being possible for them to meet, even if they are drawn forth to infinity, and even though they come near to one another the farther they go.” As he said, this cannot be imagined – it can in no way enter within the net of imagination. “Of these two lines, one is straight and the other curved.”

So he was impressed by this, as he said. “Here, what the mathematical sciences have taught us and how capital are the premises we obtained from them.”

Okay. Now I mentioned Levi ben Gershon, who was the foremost Jewish mathematician of the Middle Ages. On the left here is a slide of Ralbag, this is the street sign in Tel Aviv. And those of you who have been to Tel Aviv probably know that most of the streets of Tel Aviv are named after people, and at one end of the street or the other, there’s a little blurb about this person. And here it basically says that Levi ben Gershon was a philosopher, a mathematician and an astronomer and a Biblical commentator. Those of you who’ve been to Tel Aviv probably have not run into this street – it’s a street in the south of Tel Aviv and it’s not in a very nice neighborhood, – but I went and found it a couple years ago when I was there.

Levi certainly had read Maimonides’ works – everybody read Maimonides. But he interpreted Maimonides differently from most of others. Namely, he felt there should be no restriction at all on what one could write about in science or mathematics, because all knowledge of God’s work has religious significance, the acquisition of scientific knowledge about the world is a legitimate end in itself.

So Levi explored many different aspects of science and mathematics – combinatorics, geometry, astronomy and even number theory. And his work – he actually he had the first consistent use of the technique of mathematical induction.

Now, I want to explain it to those of you who don’t know what mathematical induction is, but all the mathematicians in the audience certainly know what it is. And he was the first one that used it very consistently in proving a number of results in combinatorics.

But then he wrote this treatise on number theory that he wrote at the request of a French music theorist (so he was in contact with non-Jews, certainly). He proved the theorem that a power of 2 must differ from a power of 3 by at least 2, except for the four pairs that are written there – (1,2), (2,3) (3,4) and (8,9). And this had to do with production of tones and music. You know, these four pairs of numbers are – well, if you set these up as lengths of strings, the ratio of lengths of strings, you know that 1:2 gives you an octave, 2:3 gives you a fifth, 3:4 gives you a fourth, and 8:9 is a single tone in the normal scale.

Okay, there were other mathematicians in Spain and France who also ignored Maimonides’ strictures. The most important was, perhaps, Abner of Burgos, who lived from 1270 to 1348. He wrote a text whose aim was somehow to show that there exists a rectilinear area equal to a circular area – exactly, not just approximately. But in the only manuscript we have, the concluding chapter where this was to be accomplished is missing. Still, one of the chapters has many interesting mathematical discussions, including a study of the use of this curve. It’s called the Conchoid of Nicomedes, and again, I’m not going to discuss the details of it, but this curve had first been used in ancient Greece. But until the work of Abner was discovered in the last 30 or 40 years,  it was thought that nobody talked about nobody talked about the Conchoid of Nicomedes until the 17th century. But, Abner showed how to use the curve to solve two very famous old Greek problems – how to trisect an angle, and how to double a cube.

Now, the Muslims also studied geometry, they had fully absorbed the Greek notion of logical proof, from axioms that came from Greece, and they carried the ideas of theoretical geometry well beyond Euclid. They also did a lot of work in practical geometry of the sort that I mentioned earlier, areas and volumes and things like that. Now, usually for the Muslims, there wasn’t any religious restriction to studying geometry. But there was always, in Islam, a countercurrent among certain Islamic authorities, that although they acknowledged that the basics of the ancient sciences, such as mathematics, were important for the Islamic community, they believed that over-study of the sciences could lead to conceit and a falling away from the faith. And this conflict continued through Islam for centuries, even though Muslim scientists were often engaged in wonderful, original work. And as time went on, Islamic authorities clamped down more and more on doing things outside of the study of religion.

All right, Let’s move on to astronomy. Now, this was certainly important for Muslims and Jews – in particular, Muslims needed to understand astronomy so they could determine the direction of, and the correct times for, prayer. Jews needed astronomy to help them work out the intricacies of the calendar.

Now, the main mathematical tool for astronomy was trigonometry – the study of angles and lines and triangles. And although plain trigonometry was useful, it was spherical trigonometry that was the basic tool for studying astronomy, because the Earth was a sphere, and the heavenly bodies moved on the surface of the heavenly sphere.

And this is just a diagram of a spherical triangle, as you see we’re talking about. Now, for Muslims, one of the religious challenges was determining the direction of Mecca, because that was the direction in which one prayed. Now, by the direction of Mecca, Muslims meant the direction alone – a great circle that connected your location with Mecca.

Now, remember, a great circle, on the earth, is a curve, in the left picture, such that a plane passing through it goes through the center of the earth. In general, a “great circle” route is the shortest distance between two points on the surface of the earth. It’s the curve generally followed today by an airline or a long flight. I mean – and this is just an example – going from somewhere in the eastern United States to India, you fly, well, north and then come back south again, that’s the shortest distance.

But it was never a trivial problem to determine the correct great circle, even if you knew your geographical coordinates. And of course, you had to know the geographical coordinates of Mecca. Many Muslim mathematicians worked on this problem for years to find various methods of solution. In the process, they developed spherical and also plain trigonometry, far beyond the version they inherited from the Greeks.

Now, as it turned out, if you go look as mosques around the world, not all Muslims were willing to go through the mathematics necessary to correctly align a mosque. So many mosques were situated with a direction of prayer just approximately toward Mecca. But there were a good number of them where they did, where we have records of them doing this quite difficult calculation.

But also, astronomical knowledge was necessary for determining the right time today to pray. Muslims were supposed to pray five times a day. And this depended on when the sun rose and set, among other events.

Now, you may also know that Jews wanted to pray toward Jerusalem. Before we really go to this slide, it’s what I mentioned, that when the tradition developed, most Jews lived in European countries generally west of Jerusalem, so they oriented their synagogues toward the east, but they didn’t actually try to calculate the direction of a great circle. They didn’t do what the Muslims did. They simply decided they should pray toward the East, and therefore made clear in the synagogue which wall was the eastern wall.

Thus, praying toward the east became a tradition, even though, if you look at a map, you can see that if you were living in the Ukraine, the direction of Jerusalem is south. Nevertheless, the eastern wall was the direction that you prayed toward.

Okay. On the other hand, you didn’t need trigonometry for the astronomical knowledge to correctly determine the calendar. What did you need to determine the Jewish calendar? Well, all sorts of things. The time when Shabbat began or ended, the day when the new month began, the time when the holidays began, the times of prayer, and so on.

Abraham bar Hiyya and Maimonides himself wrote detailed treatises on the calendar, and Levi ben Gershon wrote a trigonometry text, which form the basis of some of his astronomical work, work that was used by others for the religious purposes I just mentioned.

Now, Levi was without doubt the most accomplished Jewish mathematician of the Middle Ages, even though he went beyond the standard interpretation of Maimonides in deciding that he could study and write on any topic he thought interesting. He had few followers. There was a conflict within the Jewish community regarding what subjects could legitimately be studied. And as I said earlier, there were a sizable proportion of traditionalists who disagreed even with Maimonides and insisted that only the Torah and the Talmud were worthy of study. But a further issue for the Jews was that there was no institutional infrastructure in which students could learn mathematics.You know, one could always arrange to study privately with an individual, but there were no Jewish universities, just as there were no Muslim universities, and to find a particular private teacher was not always easy.

For the Muslims, it was the same issue. They did have madrassas, these very famous schools – in this slide is this big square in Samarkand where the the two buildings on the left and the right are old madrassas. But in the madrasas, they were generally studying only the religious sciences. Occasionally, Muslims could teach scientific subjects there, but most of the time, in the later Middle Ages, Muslim religious orthodoxy more and more prevented this.

So, there were no good universities where Muslims could study. As a Muslim cleric wrote, “Study of certain aspects of the mathematical sciences require their practitioners to direct themselves to beings other than God.” And this was to be avoided.

Now, a Jewish doctor around 1400, Leon Joseph of Carcassonne, wrote about this very issue. He complained that Jews generally lack knowledge of the secular sciences, because science is as far from the fundamentals of the Torah and religious faith “as East is from West.” And furthermore, what he said those few who did study the sciences, “had no right to propound their knowledge in the squares and in the streets or discuss it or to show themselves favorable toward it, or to conduct public debates with the aim of reaching the complete truth, for knowledge of the truth can only be attained by means of the contrary.”

So Leon decided to study Latin and attend secular universities. He found that the way of studying that they practiced in his day, at the Catholic universities, was basically the same as Jews had used to practice in studying the Torah, by disputation, by argument. But, Jews weren’t allowed to study mathematics, or the sciences in general. He noted that the Christians have continued to advance in the secular sciences, while “we Jews have continued to lose ground as a consequence of distress and oppression.”

Now, I’m going to talk about the Catholics, because in fact it was the existence of universities in Catholic Europe, beginning in the 12th century, along with a concurrent flood of translations from the Arabic that provided the impetus for this study and practice of mathematics and other sciences in Europe from that time on. These translations, as I said earlier were started mainly in Spain (to begin with anyway). They included Greek material that had been preserved in Arabic, as well as much more contemporary Islamic work, and even some work originally written in Hebrew. In many cases, a particular manuscript was first translated by a Jew from Arabic into Spanish, since Jews were familiar with both languages. It was then translated by a Catholic translator from Spanish into Latin, so it would be useful throughout Europe.

It was only after this translation activity took place that Latin Christendom began to develop its own scientific and mathematical capabilities.

Now, in mathematics, we know from the existence of Latin translations that Catholics, especially, learned arithmetic and aspects of algebra from the Muslims. They learned some combinatorics from the Jews and some geometry and trigonometry from both groups. In the work of Fibonacci – officially his name is Leonardo of Pisa, but it’s usually known as Fibonacci – early in the 13th century, we can actually trace the influences of Arabic and Hebrew works in both of his major works – the Liber Abaci, which means “the book of calculation,” and the De Practica Geometrie, on practical geometry. A lot of material in there came from Muslim and Hebrew sources. And we can pretty much see where they came from, because the Muslims and the Hebrew sources are still available to us.

On the other hand, if we look at other Latin mathematical writing in the Middle Ages, we sort of can assume that Catholic mathematicians became aware of other work, whether through personal contact, translations that have not yet surfaced, or maybe just verbal transmission. There are often such great similarities between early European Latin mathematical works and trigonometry combinatorics, geometry and algebra, and work that was originally in Arabic or Hebrew, that it is difficult to believe that the Latin writers began from scratch. But there was no tradition in those days to put in footnotes and say you got this material from someone or some set of people. You just wrote.

I want to look at two particular examples of this. I mentioned that Levi ben Gershon had worked out in detail the basic combinatorial formulas in 1321 in his work Ma’asei Hoshev. Okay, these are the two formulas. I don’t want to discuss them, those of you who know the mathematics have seen these formulas many times. These are the ways a number of combinations of n things taken r at a time, and the number of permutations of n things taken r at a time. Now, this book, Ma’asei Hoshev, was relatively popular for a medieval manuscript, because we know that there are still a dozen manuscripts of this that are still in existence. So evidently some people were interested in the work. But there was nothing written about these formulas in Europe again until some works of Marin Mersenne in Paris in the 1630’s. What happened to the ideas? We don’t really know. Mersenne’s methodology bears some resemblance to Levi’s work. So did Mersenne somehow learn of this from Levi’s manuscript?

Well, we know that Levi’s work was available in Paris. There was a manuscript in Paris, we know that it was there since 1620, and that’s still in Paris – it’s in the major library in Paris. And certainly Mersenne knew Hebrew. Lots of priests, to know things, had to study Hebrew, so they were familiar with it. So, did Mersenne learn from Levi? We have no idea whether [there was] an intermediary he learned it from. We don’t know. The manuscript itself, it’s interesting, has no notes on it. So we don’t even know anybody who read it. But it’s just a question.

What a more interesting question is the following: the so-called Tusi couple. This idea occurred in Muslim and Jewish texts and later moved to Latin. The Tusi Couple is a mathematical device, invented in Maragha, Persia, by Nasir al-Din al-Tusi in the mid-13th Century, to help explain some of the motions of the heavens, and the long-accepted proposition that the Earth was the center of the universe.

Now, what this diagram shows you is that straight-line motion can come from a combination of circular motions. That’s the basic idea here – you have these two circular motions and you get a straight line. And from Greek times on, it was believed that all motion in the heavens was basically circular, and what you saw in the heavens, you had to figure out how circles would give you exactly what you saw.

Okay. Now, as you are aware, of course, Copernicus turned this Earth-centric system on its head in the 16th Century by asserting that in fact the Earth moves around the sun. But one of the important mathematical ideas that provided the basis for his theory was in fact this very device, this Tusi Couple. And there have been many speculations as to how he learned of it. Maybe it was during his studies in Italy, because he studied in Italy as a young man.

In recent years, however, historians have become aware that the Tusi couple appears in this work of Abner Burgos that I was talking about earlier. Now, Abner’s work was written in Hebrew, in Spain, early in the 14th century. The manuscript we have today was at one time owned by Mordecai Finzi who was a Jew of Mantua in Italy. He lived throughout the 15th century, and we know that he had numerous contacts with Christian scholars. We don’t know if Alfonso’s work or any piece of it was ever translated into Latin, but it’s certainly a strange coincidence that this idea that’s central to Copernicus’s work comes from Persia, appears in Spain, then in Italy, and finally shows up in 1543 in Copernicus’s Magnum Opus De Revolutionibus, written in what is now Poland.

Okay. These are just speculations, these last two slides. I can’t prove anything about these, and I don’t know if anybody will be ever be able to, but they’re interesting questions. Now, most of the newly translated material from Arabic was used in the mathematical part of the university curriculum. The centerpiece of the university curriculum, though, was the study of logic, and the primary texts for this were the logical works of Aristotle, all of which were also provided by the translators. Gradually, other works of Aristotle were added to the curriculum. Again, this is a traditional bust of Aristotle –  we have no idea what Aristotle looked like, maybe this was Plato, maybe this was Euclid, maybe this is someone else – but this is sort of the traditional Aristotle.

Often the mathematics studied in universities was just material that was started primarily as it related to Aristotle’s work in logic, or in the physical sciences. Aristotle didn’t have much actual mathematics, but he had some ideas about mathematics, in his works, and people tried to put some mathematics into it, to try to see if it made sense mathematically.

Now, Aristotle’s philosophy posed a problem for Catholic theologians. From Aristotle’s point of view, the world was eternal. It always existed and would continue to exist. But for Catholics, and for Muslims and Jews also the world have been created by God out of nothing.In fact, in 1277, the Bishop of Paris drew up a list of 219 errors which alleged that some scholars of arts in Paris were transgressing the limits of their own faculty. In particularly he wrote that “it was an error to doubt God’s omnipotence, that in fact God an absolute power to do whatever he wills, including creating the world out of nothing.” Plus, he condemned those ideas that could not be maintained in light of the revealed truth of the Catholic religion.

But this condemnation was too little, too late. In fact, a new canon law had already been developed in the 12th century saying that “anyone ought to learn secular knowledge, not just for pleasure, but for instruction, in order that what is founder therein may be turned to the use of sacred learning.” In essence, the study of the natural sciences, and the pursuit of philosophical truths, had become institutionalized in the universities, and nothing would disturb this state of affairs. The universities were not under Church control, but usually had royal charters, so mathematicians and other scientists and Catholic Europe were free to study whatever they wished. In particular, one group of mathematicians who worked at a university were the so-called “Oxford calculators” of the 14th century. They were associated with Merton College in Oxford, because they were involved in university teaching, they had to figure out how to explain difficult concepts to students, with, as I said before, the basic method of teaching being disputations with participation from both teachers and students.

First they concentrated on the logical argument based on Aristotle’s principles, and then used the argument to try to determine what Aristotle meant and his discussions of physical problems. Ultimately these mathematicians developed some interesting mathematics out of Aristotle’s speculations, the most important result being that a body  moving with a uniformly accelerating speed traverses in a given time the same distance as a body that in the same time moves with a constant speed equal to the accelerating body speed at the middle instant.

There’s a picture, not a very good one. This slide shows the graph of velocity on the vertical axis and time on the horizontal. Distance traveled on the uniform acceleration is the area under the triangle, under the line AEC, above the axis ADB, and distance traveled under constant speed at time D is the area under the line FEG. And you can see easily from the picture that those two are the same.

An obvious question here is why, since Aristotle was known both to Muslim and Jewish philosophers, these men never considered the mathematical problems connected with kinematics. But evidently, the ideas of Aristotle were never discussed in a setting in which one could debate these questions. So this lack of the ability to debate in a secular setting helped to prevent both Jews and Muslims from proceeding further in mathematics than they did, when one needs to be able to speak with a debate with colleagues to push the ideas further.

Okay, let me just review a little bit. There are clearly mathematical geniuses in all three of these medieval mathematical cultures, most of whom shared a common mathematical background of Hindu Arabic numbers system, the works of Aristotle,  and Euclid’s elements. You’re starting with the same basic information, [but] the mathematicians from the three cultures were interested in different mathematics. Algebra, of course, had been developed in eastern Islam, but it seems that the only algebra work available in Western Islam was that of al-Khwarizmi from the ninth century.

Okay, but as we have seen, there was a definite interest in theoretical and practical geometry, and Muslims also advanced the study of trigonometry, and worked in combinatorics.

As I said, the Jews were not interested in algebra at all, when on the other hand Jewish scholars were very interested in theoretical and practical geometry. There are quite a few authors who investigated advanced geometric topics and were careful to give strict Euclidean proofs, and even Leon ben Gershon, in his work on number theory, gave strictly Euclidean proofs, based on Euclid’s zone own work in number theory. As I said, there was work in combinatorics, and there was a lot of Jewish work in trigonometry.

The Catholics themselves had little interest in advanced geometry, and did little new in trigonometry. They understood algebra, but they missed out on Islamic advances beyond Al-Khwarizmi, basically. Al-Khwarizmi’s work was translated several times into Latin, in fact. But the advanced algebra that was done in Islam never made it into Catholic Europe during the Middle Ages. Catholics did not understand combinatorics at all, until what I mentioned, the work of Mersenne much later. But they did develop, as we saw, the mathematics of motion.

All right, I want to conclude. Well, I just want to mention briefly, you know, as we’ve said this already, but the Jews and Muslims learned from each other in Spain, Jews and Catholics learned from each other in Provence and Italy, and Catholics learned much from translation with the Islamic work made from Arabic, often with the assistance of the Jews.

But I want to just go back to our “men buying a horse” problem. As I said, it showed up many times. These are just a few. It shows up in the work of al-Karaji in Baghdad around 1000 as an abstract problem, not in his work as horses, but as an abstract algebra problem. And it was solved using algebra.

Muslims were fluent in algebra, as we said. The horse problem occurs, as I wrote it, in the work of ibn al-Banna in Morocco around 1300. He also solves it algebraically. It also shows up in Fibonacci’s Liber Abbaci in 1202 in Italy, but Fibonacci does not use algebra to solve it; he basically uses arithmetic. It shows up abstractly in a work of Jordanus de Nemore in Paris maybe 50 years later, and there it’s solved using algebra, although it’s a little complicated algebra that Jordanus uses. It shows up in Levi ben Gershon’s work, again as an abstract problem. And Levi’s solution is just sort of strange. He writes this not with the 1/4 and 1/5 and 1/3 that I mentioned before, but sort of “any” fractions, and he gives them a recipe for an answer that you can plug in any numbers you need, and he shows that it’s correct, but he doesn’t show how he got it.

But then it does show up in a later work in Constantinople – this is already in 1500. This is a work by Elijah Mizrahi, and it’s not a problem about men and a horse, it’s a problem about men sharing a fish.  Maybe Jews didn’t buy horses together, but they did buy fish. And again, Elijah solves it using an arithmetic method.

Okay. So as we can see in this study of mathematics in the medieval period, as in other times and places that mathematics is not, as people have often asserted, and it really can’t be, a culture-free subject. It really depends on where and when you live and what is going on around you.

I want to thank you for listening, for choosing to learn more about the history of mathematics and how it connects to everything. I want to thank my wife Phyllis for her help in designing the presentation and providing feedback, and if you have any questions, I mean I’ll take questions now obviously, but if you have any others you can send me an email and I’ll try to answer them.