Fragments of the Past

The early history of mathematics is like a jigsaw puzzle missing many of its pieces. Historians and mathematicians have been painstakingly filling in the blanks, gradually constructing a richer, more complete story of how and where mathematical thought originated and spread.

One period of considerable interest is that between the decline of Greek mathematics, coinciding with the collapse of the western Roman Empire in the fifth century, and the rise of European mathematics in the fifteenth century. Mathematics professor Morris Kline of New York University's Courant Institute of Mathematical Sciences expressed a common view of that period in his 1972 book Mathematical Thought from Ancient to Modern Times. "The Arabs made no significant advance in mathematics," he wrote. "What they did was absorb Greek and Hindu mathematics, preserve it, and ultimately, ... transmit it to Europe."

In other words, Islamic scholars did little more than put Greek mathematics into cold storage until Europe was ready to accept it.

Historian George G. Joseph challenged that view in his provocative book The Crest of the Peacock: Non-European Roots of Mathematics. He asserted that mathematical knowledge originated in many parts of the world, and much of this knowledge was transmitted over the centuries to Europe, where it inspired further developments. However, Joseph had scant concrete evidence to back his claim.

Historians of mathematics now generally agree that scholars in China, India, and the Islamic world produced remarkably sophisticated mathematics during this period. However, most would probably still argue that Europeans in later centuries were unaware of this work and made advances with minimal help from the earlier efforts.

Careful detective work now hints that significant ideas in several areas of mathematics -- trigonometry, non-Euclidean geometry, number theory, and combinatorics -- were in fact transmitted from the Islamic world in time for them to play crucial roles in furthering European mathematics.

"There's no smoking gun," admits mathematician Victor J. Katz of the University of the District of Columbia, who described the evidence earlier this month in a presentation at the Joint Mathematics Meetings in San Diego. In certain areas of mathematics, however, the evidence of important interactions now looks quite convincing, he remarks.

It's clear that Arabic and Hebrew manuscripts describing mathematical advances were available to European scholars, but it's hard to tell whether anyone paid attention to them. What clues there are often take the form of remarkable similarities between passages in certain manuscripts and later European writings.

"The standards for attribution were not very high in those days," Katz notes. And there were no bibliographies.

Islamic mathematicians made important contributions to plane and spherical trigonometry, greatly improving upon and extending earlier Greek work. One of these scholars was Jabir ibn Aflah, who worked in Islamic Spain in the twelfth century. His methods for solving triangles on the surface of a sphere were translated into both Latin and Hebrew. Somehow, they also showed up in a mathematics book by Johann Muller (1436-1476), also known as Regiomontanus, probably the most influential European mathematician of the fifteenth century. The passages were so similar that physician, gambler, and mathematician Gerolamo Cardano (1501-1576) commented on the plagiarism in one of his books. The trigonometry that Nicholas Copernicus (1473-1543) outlined in the first part of his epochal work De revolutionibus was also apparently inspired by Jabir.

Islamic mathematicians were also active in trying to deduce Euclid's postulate on parallel lines from the other four postulates of Euclidean geometry. In attempting such a reconciliation, mathematicians like Nasir al-Din al-Tusi (1201-1274) invented new geometric constructions. Nasir's work was known to John Wallis (1616-1703), the most influential English predecessor of Isaac Newton, and it inspired Girolamo Saccheri (1667-1733) to try his hand at proving the parallel postulate.

The Greeks, especially the Pythagoreans, were fascinated by numbers. They defined as "perfect" numbers those equal to the sum of their parts (or proper divisors). For example, 6 (1 + 2 + 3) is the smallest perfect number. Amicable numbers are a pair in which each one is the sum of the proper divisors of the other. The smallest such pair is 220 (1 + 2 + 4 + 71 + 142) and 284 (1 + 2 + 4 + 5 +10 + 11 + 20 + 22 + 44 + 55 + 110).

In the ninth century, Thabit ibn Qurra (826-901) discovered a remarkable formula for finding amicable numbers: if p, q, and r are prime numbers, and if they are of the form p = 3[2^n] - 1, q = 3[2^(n-1)] - 1, and r = 9[2^(2n-1)] - 1, then (2^n)pq and (2^n)r are amicable numbers. A few centuries later, in a letter to Marin Mersenne (1588-1648), Pierre de Fermat (1601-1665) revealed that he had found a second pair of amicable numbers (18,416 and 17,296) using a method virtually identical to that proposed by Thabit.

What was the link? It's known that Thabit was just one in a long line of mathematicians who dabbled in number theory, and scholars have found Arab and Hebrew manuscripts of the twelfth and fourteenth centuries that contain details of these studies. "We don't know for sure whether Europeans were aware of this work," Katz says. But there's a good possibility that people like Fermat had access to a Latin edition.

The basic formulas for finding permutations and combinations can be found in the writings of Levi ben Gerson (1288-1344). There's no further mention of these rules until the sixteenth century, when Cardano takes note of them in one of his books. Then, the rules appear in a book by Mersenne on music theory, derived in a manner strikingly similar to that used by Levi ben Gerson. Curiously, Mersenne attributes this section to someone with the initials IMDMI.

It turns out that there happened to be a manuscript of Levi's work in Paris at that time, brought there by the French ambassador to Constantinople. One member of the institution where the manuscript was stored was someone named Jean Matan, who could have read the manuscript and passed the information on to Mersenne.

So, the detective work goes on, and the dusty history of mathematics begins to take on a new look.

References:

Boyer, Carl B. 1985. A History of Mathematics. Princeton, N.J.: Princeton University Press.

Joseph, George G. 1992. The Crest of the Peacock: Non-European Roots of Mathematics. New York: Viking Penguin.

Katz, Victor J. 1997. The transmission of mathematics from Islam to Europe. Abstracts of Papers Presented to the American Mathematical Society 18(No. 1):5.

Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press.

Mahoney, Michael Sean. 1994. The Mathematical Career of Pierre de Fermat, 1601-1665 (Second Ed). Princeton, N.J.: Princeton University Press.

An excellent history of mathematics archive is available at http://www-groups.dcs.st-and.ac.uk/~history/index.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.