In his book “A History of Mathematics”, Carl Boyer mentions that both AlKhwarizmi and Abd ElHamid Ibn Turk wrote their books on Algebra (“Aljabr w Almuqabla” and “Logical Necessities” respectively) at nearly the same time. He also states that they use the same methodology and even an example used for demonstration is exactly the same in both books. He then concludes that this indicates that these books weren't the first on the subject as it's widely believed and that Algebra was already mature enough at that time for them to share the same methodology.

Quoting from the book:

The Algebra of al-Khwarizmi is usually regarded as the first work on the subject, but a publication in Turkey raises some question about this. A manuscript of a work by ‘Abd-al-Hamid ibn-Turk, titled “Logical Necessities in Mixed Equations,” was part of a book on Al-jabr wa’l muqabalah, which was evidently very much the same as that by al-Khwarizmi and was published at about the same time— possibly even earlier. The surviving chapters on “Logical Necessities” give precisely the same type of geometric demonstration as al- Khwarizmi’s Algebra and in one case the same illustrative example, $x^2+21=10x$. In one respect, ‘Abd al-Hamid’s exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the work of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. Successors of al-Khwarizmi were able to say, once a problem had been reduced to the form of an equation, “Operate according to the rules of algebra and almucabala.” In any case, the survival of al-Khwarizmi’s Algebra can be taken to indicate that it was one of the better textbooks typical of Arabic algebra of the time. It was to algebra what Euclid’s Elements was to geometry—the best elementary exposition available until modern times—

Did any further investigation go into this and try to trace the steps that led to this claimed maturity?

  • $\begingroup$ The final sentence of the quoted text highlights the problem of identifying previous formulations of al-Khwarizmi and Abd-al-Hamid ibn-Turk's work. The superior nature of Euclid rendered all previous texts obsolete, just as appears to be the case for al-Khwarizmi. Unless we are lucky enough to find older texts at some point in the future, then we will never know if al-Khwarizmi drew on previous authors. Of course Boyer is assuming that the symbolic methods and algorithmic techniques of al-Khwarizmi constitute what we call algebra. The Babylonians studies some "spoken" forms of algebra.... $\endgroup$
    – nwr
    Dec 14 '18 at 18:52
  • 1
    $\begingroup$ ... Diophantus studies certain types of symbolic algebraic equations. As did others such as Brahmagupta. $\endgroup$
    – nwr
    Dec 14 '18 at 18:53
  • $\begingroup$ @Nick R Can you turn that into an answer so that I can accept it? $\endgroup$ Dec 15 '18 at 7:20
  • $\begingroup$ If Carl Boyer is claiming that AlKhwarizmi "books weren't the first on the subject as it's widely believed and that Algebra was already mature enough at that time for them to share the same methodology" then he should support his statements. Remember that paper never refused an ink. Always consult plenty to references to obtain multiple viewpoints. Does he give a reference to "publication in Turkey?" $\endgroup$
    – M. Farooq
    Dec 16 '18 at 16:07
  • $\begingroup$ I haven't checked the references yet but I think he is referring to "logical necessities". Anyway, he is making a clever educated guess and is stating it as being so. $\endgroup$ Dec 16 '18 at 17:22

Nothing new was discovered after Boyer published his textbook. In 2014, Victor J.Katz and Karen Parshall published their book Taming the Unknown: A history of algebra from antiquity to the early twentieth century, in which they write (chapter 7):

While influencial, al-Khwārizmī's was evidently not the ony algebra text of the time. A fragment of a contemporaneous work remains, namely, the section Logical Necessities in Mixed Equations from a longer work Kitāb al-jabr wa'l muqabalah […] by ʿAbd al-Hamīd ibn Wase ibn Turk Jili.

So, no other algebra textbook from that time was discovered meanwhile.


Perhaps the following papers may help clarifying the issue: Hoyrup, Jens : i) “The Formation of Islamic Mathematics”: Sources and Conditions. Preprints og Reprints 1987 Nr 1. (available online) ii) “Algebraic Traditions behind Ibn Turk and al-Khwārizmī'. In Acts of the International Symposium on Ibn Turk, Khwarezmī, Farabī, Beyronī and Ibn Sīna (Ankara 9-12 september 1985). Available online.

I am working on a paper related to the subject and it will published on Slideshare within a few days


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