Sharaf al-Dīn al-Ṭūsī

Sharaf al-Dīn al-Ṭūsī was born around 1135 in the region of Ṭūs in Khorāsān and died in 1213 somewhere in Iran. His exact place of death remains unknown. Most surviving biographical information comes through remarks by later scholars who studied under him or with him. What we do know is where he taught: Damascus around 1165, then Aleppo for at least three years, then Mosul, and finally Baghdad late in life. His most consequential student, Kamāl al-Dīn ibn Yūnus, would in turn teach Naṣīr al-Dīn al-Ṭūsī , the astronomer who runs the next page of this archive. Two Ṭūsīs. One teaching lineage.

The Treatise on Equations

Written around 1209 in Baghdad, the Treatise on Equations is his central mathematical work. An anonymous medieval commentary preserves quotations and worked passages from the original. In 1986 Roshdi Rashed published a critical edition and French translation as Sharaf al-Dīn al-Ṭūsī: Œuvres mathématiques. Algèbre et géométrie au XII^e siècle (Paris: Les Belles Lettres, 2 vols.). Rashed's edition is the source almost every modern account draws on.

The treatise classifies equations of degree at most three into twenty five canonical forms. The first twelve are of degree at most two. The next eight are cubics that always admit a positive root. The last five may admit zero, one, or two positive roots, and this final group carries the work's most original mathematics.

Existence conditions for the last five cubics

For each of the five contested cases, al-Ṭūsī rewrites the equation as \(f(x) = c\), where \(f\) is a specific cubic polynomial and \(c\) a positive constant. He then locates the point \(x_0\) where \(f\) attains its maximum on the positive reals, proves that \(f(x) < f(x_0)\) for every positive \(x \neq x_0\), and reads off the number of positive roots from a comparison between \(c\) and \(f(x_0)\):

• if \(c > f(x_0)\), zero positive roots;
• if \(c = f(x_0)\), one positive root (a double root at \(x_0\));
• if \(c < f(x_0)\), two positive roots, one on either side of \(x_0\).

The signature example is \(x^3 + a = bx\), with \(a, b > 0\). Al-Ṭūsī considers \(y = bx - x^3\), locates its maximum at \(x_0 = \sqrt{b/3}\) with peak value \(2\sqrt{(b/3)^3}\), and expresses the existence condition as \(a \leq 2\sqrt{(b/3)^3}\). Squaring and rearranging gives \(\tfrac{1}{27}b^3 - \tfrac{1}{4}a^2 \geq 0\), the same nonnegativity condition on the quantity we now call the discriminant of the cubic.

Was it a derivative?, a scholarly dispute

Rashed argues, in the introduction to his 1986 edition and in later papers, that al-Ṭūsī's way of locating the maximum is best understood as a derivative computation, and that this makes him an early user of what would become differential calculus. This reading is influential and often quoted, it's the reading behind the older claim that al-Ṭūsī “discovered the derivative.”

Jan P. Hogendijk offers a competing interpretation in Historia Mathematica 16 (1989), pp. 69 to 85, argues that al-Ṭūsī almost certainly reached his maxima through geometric manipulation of squares and rectangles in the style of Book II of Euclid's Elements. The surviving text presents the maximum point and proves its status as a maximum. A general rule of differentiation lies outside the surviving description.

Both readings are defensible, the two historians are working from the same primary text and reaching different conclusions about which modern concept comes closest to al-Ṭūsī's procedure. Both readings agree on the mathematics itself: the existence conditions are correct, the maximum values are correct, and locating a polynomial's maximum in order to count its roots is a genuinely original move in the twelfth century, whichever conceptual vocabulary we use to describe how he found it.

Numerical solution, a Ruffini Horner method

Once existence is settled, al-Ṭūsī gives an iterative numerical method for approximating the root of a general cubic. The method is, to a reader today, the Ruffini Horner scheme in almost every detail, six centuries before Ruffini (1804) and Horner (1819). Earlier Arabic mathematicians had used similar iteration to approximate \(n\)th roots of an integer; al-Ṭūsī is the first for whom we have a clear record of extending the method from root extraction to a general cubic equation. MacTutor's biography and Rashed's introduction both make this point explicitly.

Algebraic geometry, avant la lettre

Rashed reads the treatise as a break from the algebra of al-Karajī's school and as the beginning of a different programme, one that studies curves through their equations, which is why he calls it, in a phrase now widely quoted, the inauguration of “algebraic geometry” in the Arabic tradition. This strong claim comes from a specialist historian working with a full critical edition. We present it as Rashed's reading within an open discussion because the term algebraic geometry means several different things in different periods.

The linear astrolabe (staff of al-Ṭūsī)

Separately from his algebra, al-Ṭūsī invented a linear astrolabe, a graduated wooden rod with a plumb line and a perforated pointer, which reduces the traditional planispheric astrolabe to a single one dimensional instrument. It is easier to construct than the disc shaped version and circulated for a time in al-Andalus. The planispheric form remained dominant.

What survives to the classroom

Two threads. First: the idea that you can decide whether an equation has a positive root by inspecting where a polynomial takes its maximum. That habit, test the extremum before you try to construct the root, is exactly what olympiad students do when they use AM GM or convexity to bound a symmetric expression before attempting an algebraic identity. Second: the Ruffini Horner iteration itself, which sits inside every polynomial evaluation routine in modern numerical software. Both are part of the algebra strand in MOTA's curriculum.

Sources

1. MacTutor History of Mathematics, University of St Andrews, Sharaf al-Dīn al-Ṭūsī biography (dates c. 1135 to 1213, the teaching path Damascus → Aleppo → Mosul → Baghdad, the twenty five form classification of cubic and subcubic equations, the treatise dated to 1209, the Ruffini Horner style iteration applied to general cubics, the invention of the linear astrolabe). https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf/
2. Wikipedia, Sharaf al-Dīn al-Ṭūsī, consolidated dates and student lineage (Kamāl al-Dīn ibn Yūnus taught Naṣīr al-Dīn al-Ṭūsī), the reformulation of the last five cubics as \(f(x) = c\) with maximum \(x_0\), and the identification of the quantity \(D = f(x_0) - c\) with the modern discriminant. https://en.wikipedia.org/wiki/Sharaf_al-Din_al-Tusi
3. Roshdi Rashed, Sharaf al-Dīn al-Ṭūsī: Œuvres mathématiques. Algèbre et géométrie au XII^e siècle, 2 vols., Paris: Les Belles Lettres, 1986, the critical Arabic edition, French translation, and commentary on which the modern “derivative” reading of the maximum computation rests, and the source of Rashed's reading of the treatise as inaugurating algebraic geometry in the Arabic tradition.
4. Jan P. Hogendijk, “Sharaf al-Dīn al-Ṭūsī on the number of positive roots of cubic equations,” Historia Mathematica 16 (1989), pp. 69 to 85. This alternative reading interprets al-Ṭūsī's procedure through the manipulation of squares and rectangles in the style of Book II of Euclid's Elements. https://research-portal.uu.nl/en/publications/sharaf-al-din-al-tusi-on-the-number-of-positive-roots-of-cubic-eq/
5. Encyclopedia.com (drawn from the Dictionary of Scientific Biography updated entry), secondary verification of the death date c. 1213 to 1214, the three extant treatises published in Rashed's 1986 Œuvres (the Equations, On the Construction of a Geometric Problem, and On Two Lines that Approach Each Other but Do Not Meet), and the linear astrolabe. https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/al-tusi-sharaf-al-din-al-muzaffar-ibn-muhammad-ibn-al-muzaffar