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  • Writer's pictureDale DeBakcsy

The Algebraist of Baghdad: Sutayta Al’ Mahamali’s Medieval Mathematics.

It is a thousand years ago. Europe is a stumbling, superstition-addled giant, depleting its energies on visions of holy violence and shuddering at half memories of former greatness. It has lost its past, and despairs at its present, and if you are a person seeking to know the ways of nature, it is not the continent for you.

For you, a curious soul of the mid-tenth century, there is but one place you must go: Baghdad, the Islamic capital built by the legendary Abu Jafar al-Mansur, the House of Wisdom.

In this city, science rules. The Caliph sends his scholars to scour the globe for manuscripts bearing lost Greek wisdom. He invites delegations from India to explain their curious ten-digit number system, and dispatches his mathematicians on missions to measure the curvature of the Earth. Paper technology imported from Asia is pressed into service to fuel a bustling book industry that itself supports a small army of translators, commentators, poets, and researchers, all rushing to feed a population and government drunk on the glories of the written word.

If you happen to find your way to this glimmering, learning-mad city, and if you have a mind for mathematics, then there is one particular call you must make: to the home of judge Abu Abdallah al-Hussein. Here you will find his daughter, Sutayta al-Mahamali (d. 987), as renowned for her legal mind as for her mathematical mastery, a woman of genius widely celebrated as such by her culture, praised for her abilities by three of the era’s greatest historians, and today sadly reduced to the status of a historical footnote.

We know more about her father, son, and grandson, who were all well regarded judges and scholars, than we do about her, meaning we must piece her life and work together from a few scraps of information and a wealth of intellectual context if we are to know her at all. The ‘hamal’ in her name comes from the Arabic word ‘to carry’, pointing to her family’s origins as carriers of goods and people. By her own time, however, her family was well-established in scholarly circles, and Sutayta had the benefit of a string of learned teachers.

She studied Arabic literature, jurisprudence, the interpretation of sacred texts, and mathematics a full two hundred years before Europe produced women of comparably broad education and fame in the form of Heloise of Argenteuil and Trota of Salerno. We know that she was widely consulted for her legal and mathematical insight, and that she solved problems of inheritance that imply an advanced knowledge of that era’s hot new field – ALGEBRA.

Questions of inheritance, of how to correctly distribute the proceeds of an estate between people of varying relation to the deceased, were mathematical hornet’s nests until they fell under the keen mind of Muhammad ibn Musa al-Khwarizmi (d. 850), the man who combined the Greek algebraic problem solving methods and proof formalism of Diophantus with the base-ten numerical system of Indian scholars to create what we recognise as the forerunner of modern algebra. (The name algebra, in fact, comes from the term al-jabr that al-Khwarizmi coined for the operation of rewriting an expression to eliminate negative terms.) In his works, he applied his techniques to the complicated systems of equations that result when you try and mathematise a web of competing claims on an estate.

Sutayta, we are told, made original contributions to this field, and to the theory of arithmetic as well which was truly coming into its own thanks to the adoption of a numerical system that wasn’t aggressively hostile to computation. (To get an idea of how fortunate we are to live in the world of Indian/Arabic numerals employing base 10, just look up how awful it was trying to multiply with Roman numerals, or ponder for a while what it would be like to use the Babylonian base-60, which would require children to learn 1711 times table products instead of the 36 required by our base-10. Something to use the next time your kiddo complains about having to learn their nines.) Unfortunately, the exact solution types she contributed to mathematics have been lost in the intervening centuries. We know that she memorised the Koran, that she was praised for her virtue and modesty, but the historical record doesn’t mention a single specific equation of her invention, in spite of the fact that we know that later mathematicians referenced her work.

To know her research at all, then, we must make a comparison to mathematicians of her time to get a sense of her areas of study. She lived a hundred years after al-Khwarizmi and just after the time of Abu Kamil (850–930), the two titans of classical Arabic mathematics. Both of them were interested in categorising solutions to entire systems of equations, like al-Khwarizmi’s breaking of all quadratics (equations that have a squared variable) into six solvable categories (or so he thought), or Abu Kamil’s solutions to problems involving irrational quantities.

It is mentioned by her biographers that she was known not only for solving individual problems, but for creating general solutions to types of problems, which would be a logical extension of the work of al-Khwarizmi and Abu Kamil. And, since her work both followed theirs and earned enough esteem to be mentioned by later sources, it is at least probable that she achieved a remarkable level of algebraic sophistication that opened to Arabic mathematicians groups of equations extending beyond those solved by the great Abu Kamil, perhaps even including the solution of the cubic type equations that gripped the imaginations of her near successors Ibn al-Haytham (965–1040) and Omar Khayyam (1048–1130).

All this, however, is speculation based on interpolating from the work of those algebraists just before and just after her time, and from a few tantalising clues left by historians from the next century. With so little to go on, you might fairly wonder why we bother to include her at all in a history of women’s contributions to mathematics- why talk about somebody we only know vaguely from reputation when there are so many current women of science whose work we are well familiar with?

Simply put, there is historical value in the question, How was Sutayta even possible? The relation between Islam and women’s education is a deeply intricate tangle of general principles losing themselves in the messiness of local and ancient tradition. What scholars of the era are in general agreement upon is that Islam was the victim of its own early successes, and women more so than men. The phenomenal rate of expansion of the early Islamic Empire meant that, before they were entirely ready, the original Islamic faithful, including a number of powerful women, found themselves outnumbered in their own suddenly massive state.

The majority of the Empire’s population now consisted of far-flung tribes and peoples that clung to a more repressive and traditional pre-Islamic conception of a woman’s place, and used their influence to retroactively write their beliefs into the core principles of the Islamic state. Once those beliefs were well integrated with current practice, Islamic historians, who considered pre-Islamic Arabic civilisation to be little more than a collection of barbarisms, then rewrote the lineage of those beliefs to make them Islamic inventions rather than pre-Islamic holdovers.

Meanwhile that edited version of the roots of Islamic practice was subjected to the mangle of Islam’s roving capital, as Spain, Egypt, and Mesopotamia all tried their hand at melding its tenets with their dominant cultural practices. The result was a complicated spectrum of approaches to women that belies the monolithic Western narrative of veil and darkness.

That spectrum, at least for a time, had enough breadth to include a mathematician like Sutayta, a female poet of erotica like Wallada, and a wealthy political powerhouse like Khayzuran. The constriction of that breadth, and the decline of the House of Wisdom, would come in time, but the knowledge that, once upon an Empire, there was a learned city wherein walked a woman who wove law and algebra with equal ease and to general acclaim can give us hope that what once was, might be again, and perhaps already is.


Tesneem AlKiek has made a brief YouTube video about Sutayta that is a good starting point for the curious. For more on the history of Arabic mathematics and the early history of algebra, the best book is Jacques Sesiano’s An Introduction to the History of Algebra which includes original Greek and Arabic texts in a super-neat appendix. Jonathan Lyons’s The House of Wisdom contains a lovely chapter on the intellectual atmosphere of late tenth century Baghdad, and Ehsan Masood’s Science and Islam: A History has as good a history of Arabic mathematics as you’ll get outside of Sesiano’s harder to find book.

For the history of women in the early Islamic Empire, Wiebke Walther’s Women in Islam from Medieval to Modern Times has a good account of the complex evolution of women’s position throughout the Empire, and some engaging portraits of women active in politics and writing, though not so much on science. Finally, Women in Iran: From the Rise of Islam to 1800, edited by Guity Nashat and Lois Beck, has a chapter by Richard Bulliet on how and why different types of women were covered by the great Islamic biographers which sheds light on the difficulty of knowing more about Sutayta than we do.

Finally, Sutayta is one of the over a hundred mathematicians contained in my History of Women in Mathematics, now available from Pen and Sword Books, each purchase of which helps keep the Archive afloat, so why not buy several?


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