The series of problems in al-Khwārizmī’s Algebra

There are two collections of problems in al-Khwārizmī’s Algebra. The first is part of what I call the “algebra proper”, and consists of the thirty-nine worked out arithmetic problems in the first half of the book. This set of problems is the object of my study. It is not to be confused with the collection of inheritance problems that make up the second half of the book, after the chapters on business arithmetic and mensuration.

I have written much more than I expected, so I will summarize my arguments here and give links to my longer study and to my translation of the whole corpus. First, the links:

“The series of problems in al-Khwārizmī’s Algebra”. 13 pages, plus bibliography. here

My translation of the 39 problems. 12 pages: here

The study is divided into four sections. In (1) I discuss oral and written features of medieval Islamic scholarship. I summarize the modes of knowledge production and transmission from the studies of Jonathan Berkey and Michael Chamberlain, and I relate them to al-Khwārizmī’s corpus to gain some understanding of the ways his problems might have been utilized.

With this in mind, I continue with a brief look at algebraic notation in medieval Islam, even if there is none in al-Khwārizmī’s book. The purpose here is to uncover how problems might have been solved in his time. Briefly, problems would have been worked out on some ephemeral surface like a dust board, using Hindi numerals. To communicate the solution of a problem, whether orally or in a written book, one would compose a rhetorical version of the calculations in which even the numbers are written out in words. It is these rhetorical versions that have come down to us.

In (2) I address the overall structure of al-Khwārizmī’s Algebra. I give not only a table of contents of his book, but also the contents of the algebra books of Abū Kāmil, al-Karajī, cAlī al-Sulamī, and bn al-Bannā’, in order to show variations. From there I give a short comentary on the contents of al-Khwārizmī’s book, including an explanation of the structure of a solution by algebra.

In (3) I look at two aspects of the corpus of problems that shed light on their origin and on al-Khwārizmī’s contribution. First, many problems are not solved by algebra, but by a method I call “arithmetical reasoning”. Here one merely reasons through the stated operations to find the unknown. No unknown is named, and no equation is set up. The presence of these non-algebrac solutions suggest that al-Khwārizmī did not write all the problems himself. He seems to have taken over a corpus from an oral tradition.

Some later algebraists also include problems solved by “arithmetical reasoning”. I list them on page 9.

The second aspect has to do with the first six problems. Like in some later books, a al-Khwārizmī starts with six problems whose equations simplify to the six canonical types. He singles out these first six with special introductory and concluding phrases. For example, he begins the second problem “And the second problem”, and ends it with “This problem led you to one of the six types, which is māls equal number.” The 33 problems that follow are not distinguished like this.

Now, algebraists after al-Khwārizmī routinely give explicit assignment statements at the beginning of their solutions by algebra, such as “So make one of the parts a thing”. By contrast al-Khwārizmī gives standard assignments in only five of his first six problems. Where an assignment is made in the later problems it is worded in an odd way. This suggests that al-Khwārizmī either wrote or adjusted the first six problems, and that the remaining problems are closer to what practitioners had spoken before al-Khwārizmī wrote his book.

In part (3) I discuss al-Khwārizmī’s proofs. Following mainly the work of Høyrup, I note the “subscientific” character of the proofs, especially the first proof for the rule to solve the type 4 equation. But I cannot make a reasonable conjecture as to the extent of al-Khwārizmī’s contribution.

I review scientific developments in algebra in part (4). I note how many authors retained the basic structure of al-Khwārizmī’s book, adding scientific elements where they saw fit, while other authors broke off to produce purely scientific treatises. The texts of those who retained the traditional structure are interesting because we can identify many innovations that would have been of little use to surveyors or merchants, but they also retain many practical features that would not appeal to theorists.

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