Jamshīd al-Kāshī

Jamshīd al-Kāshī was born around 1380 in Kāshān, a town on the eastern edge of the Central Iranian Range. He died on 22 June 1429 in Samarkand. The first dated event of his adult life is his observation of a lunar eclipse from Kāshān on 2 June 1406. From there the record picks up quickly: astronomical tables dedicated to a local vizier in 1407, a compendium on astronomy dated 1410 to 11 for the Timurid prince Iskandar of Isfahan, then the Khāqānī Zīj, a set of astronomical tables dedicated to Ulugh Beg and based on the earlier tables of Naṣīr al-Dīn al-Ṭūsī (see the previous entry). Around 1420 he moved to Samarkand, where Ulugh Beg had founded a madrasa and would soon begin, in 1424, the construction of the observatory that would carry his name.

The Samarkand school

Ulugh Beg, a Timurid ruler and himself a working astronomer, assembled about sixty scientists at Samarkand, including Qāḍī Zāda al-Rūmī and al-Kāshī. MacTutor puts it plainly: “There is little doubt that al-Kāshī was the leading astronomer and mathematician at Samarkand,” and two later fifteenth century historians called him “the second Ptolemy.” Two undated letters he sent home to his father describe the observatory's construction, its instruments, and daily scholarly life, letters that also record his closeness to Ulugh Beg despite what MacTutor calls his “ignorance of the correct court behaviour and lack of polished manners.”

π to sixteen decimal places

In July 1424 al-Kāshī finished the Treatise on the Circumference. He works entirely in sexagesimal (base 60). He inscribes a regular polygon of \(3 \cdot 2^{28}\) sides in a circle, iterates the chord doubling formula through twenty eight halvings, and obtains \(2\pi\) accurate to nine sexagesimal digits. He then translates the result into base 10 for the reader, at which point it lands correct to sixteen decimal places.

MacTutor is direct about the historical scale of this: “far beyond anything which had been obtained before, either by the ancient Greeks or by the Chinese (who achieved six decimal places in the 5th century).” The record would stand for nearly two centuries, until Ludolph van Ceulen surpassed it with twenty decimal places around the turn of the seventeenth century.

The Key to Arithmetic

Miftāḥ al-Ḥisāb (“Key to Arithmetic”), completed on 2 March 1427, is his most complete book. It was written as a teaching text for the Samarkand madrasa, aimed at students training for astronomy, surveying, architecture, accounting, and trade. That's a real cross section of the mathematics a working person of the period would actually need. Two chapters in particular still get quoted for their applied geometry: one gives methods for the surface area of a muqarnas, the stalactite shaped architectural vaulting used in mosques and palaces, modelled here as a collection of three dimensional polygons with plane and curved faces, and another gives approximation methods for the surface area and volume of a qubba, the dome of a funerary monument. He uses decimal fractions throughout.

Decimal fractions, credit where it is due

For a long time the standard European story credited Simon Stevin (1585) with the introduction of decimal fractions. In 1948 Paul Luckey showed that al-Kāshī's Key to Arithmetic already gives “as clear a description of decimal fractions as Stevin does.” So the older attribution to Stevin requires correction. Roshdi Rashed has argued that the essential idea was already present in the work of al-Karajī's school, particularly with al-Samaw'al. MacTutor quotes Rashed directly:

Al-Kāshī can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samaw'al and represents an important dimension in the history of decimal fractions.

Rashed identifies two specific advances that are al-Kāshī's own: the analogy between the sexagesimal and decimal systems as parallel ways of writing fractions. He also used decimal fractions for real numbers such as \(\pi\), extending their role beyond approximations of algebraic reals. Both are substantial developments within an established tradition.

The sine of one degree

Al-Kāshī's last work is the Treatise on the Chord and Sine. It may have been unfinished at his death and completed by Qāḍī Zāda al-Rūmī. In it he computes \(\sin 1^{\circ}\) to ten sexagesimal places. This precision is what Ulugh Beg's later astronomical tables needed, and it is what distinguishes those tables from earlier ones.

The technique is worth naming. From the triple angle identity applied to a \(3^\circ\) angle, computing \(\sin 1^\circ\) reduces to solving a specific cubic equation of the form \(\sin 3^\circ = 3\sin\theta - 4\sin^{3}\theta\) for \(\theta = 1^\circ\). Al-Kāshī solves it iteratively, refining the solution to whatever precision he wants; MacTutor calls the method “one of the best achievements in medieval algebra.” This is the same trisection related cubic that al-Bīrūnī had worked on earlier (see the archive entry for al-Bīrūnī); al-Kāshī's advance lies in the iterative refinement of the cubic solution.

Death and its shadow

Al-Kāshī died on 22 June 1429 outside the Samarkand observatory. Britannica reports that he was “murdered” and that the killing was “probably on the command of Ulugh Beg.” MacTutor uses more limited wording, and the surviving sources leave the circumstances contested. Ulugh Beg's own testimony after the death describes him in warm terms: “a remarkable scientist, one of the most famous in the world, who had a perfect command of the science of the ancients, who contributed to its development, and who could solve the most difficult problems.”

Related curriculum links in the curriculum

Al-Kāshī sits at the end of the arc this archive traces, and he touches most of MOTA's curriculum strands at once. Trigonometry: the triple angle identity is still the cleanest example of an algebraic relation between circular functions that reduces a trigonometric problem to a polynomial one. Algebra: iterative refinement of a root of a specific cubic is exactly what a modern numerical solver does, and it is what olympiad students meet the first time they use Newton's method or a fixed point argument. Geometry: the muqarnas and qubba calculations show applied solid geometry inside a working teaching book.

Sources

1. MacTutor History of Mathematics, University of St Andrews, al-Kāshī biography (birth c. 1380 Kāshān; death 22 June 1429 Samarkand; 2 June 1406 lunar eclipse observation; the Khāqānī Zīj dedicated to Ulugh Beg and based on Naṣīr al-Dīn al-Ṭūsī's tables; the sixty scientist Samarkand school; the July 1424 Treatise on the Circumference giving \(2\pi\) to 9 sexagesimal / 16 decimal places; the comparison with the Chinese fifth century six decimal result and van Ceulen's later twenty decimal record; the 2 March 1427 Key to Arithmetic; the muqarnas and qubba calculations; the Treatise on the Chord and Sine possibly completed by Qāḍī Zāda; the iterative sin 1° cubic method extending earlier work by al-Bīrūnī; the Rashed quotation on decimal fractions and al-Samaw'al's priority). https://mathshistory.st-andrews.ac.uk/Biographies/Al-Kashi/
2. Encyclopædia Britannica, “al-Kāshī” (author Yvonne Dold-Samplonius), secondary verification of birth and death dates; the polygon of \(3 \cdot 2^{28}\) sides used in the 1424 π computation; the description of the Key to Arithmetic as an encyclopedic teaching work solving problems from astronomy, surveying, and finance; and the report that he was “murdered outside the Samarkand observatory on June 22, 1429, probably on the command of Ulugh Beg.” This final wording belongs specifically to Britannica. https://www.britannica.com/biography/al-Kashi
3. Roshdi Rashed, The Development of Arabic Mathematics: Between Arithmetic and Algebra, Boston Studies in the Philosophy of Science 156, Dordrecht: Kluwer, 1994, the source for the priority argument on decimal fractions (al-Karajī's school and al-Samaw'al before al-Kāshī) and for the two specific advances Rashed credits to al-Kāshī: the sexagesimal to decimal analogy, and the extension of decimal fractions to real numbers such as \(\pi\).
4. Paul Luckey, Der Lehrbrief über den Kreisumfang von Ğamšīd b. Masʻūd al-Kāšī, Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin, 1953 (posthumous), and Luckey's 1948 studies of the Key to Arithmetic, the mid twentieth century work that overturned the older attribution of decimal fractions to Stevin, and to which MacTutor and Rashed both refer.