# Jamshīd al-Kāshī

> Persian astronomer and mathematician

**Wikidata**: [Q115466](https://www.wikidata.org/wiki/Q115466)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Jamshid_al-Kashi)  
**Source**: https://4ort.xyz/entity/jamshid-al-kashi

## Summary
Jamshīd al-Kāshī was a 14th–15th century Persian astronomer, mathematician, and physician renowned for his groundbreaking contributions to trigonometry, astronomy, and numerical computation. His work at the Ulugh Beg Observatory in Samarkand advanced the understanding of celestial mechanics and mathematical precision, including the formulation of the law of cosines and the development of highly accurate astronomical tables.

## Biography
- **Born**: 1380 (exact place unspecified)
- **Nationality**: Persian (modern-day Iran)
- **Education**: No formal degrees or institutions listed
- **Known for**: Pioneering work in trigonometry, astronomy, and the development of computational methods
- **Employer(s)**: Ulugh Beg Observatory (Samarkand, Uzbekistan)
- **Field(s)**: Mathematics, astronomy, astrology, medicine

## Contributions
Jamshīd al-Kāshī authored several influential works, including:
- **The Treatise on the Circumference** (c. 1424): Calculated π to 16 decimal places, a record unmatched for nearly 200 years.
- **The Key to Arithmetic** (*Miftāḥ al-Ḥisāb*, 1427): A comprehensive textbook on arithmetic and algebra, introducing decimal fractions and practical computational techniques.
- **Astronomical Tables** (*Zīj-i Khāqānī*): Collaborated with Ulugh Beg to produce precise planetary models and star catalogs, refining Ptolemaic astronomy.
- **Law of Cosines**: Formulated a trigonometric identity essential for solving triangles, later adopted in European mathematics.
- **Observational Astronomy**: At the Ulugh Beg Observatory, he contributed to measurements of celestial coordinates and eclipses, improving the accuracy of Islamic astronomical traditions.

## FAQs
**What was Jamshīd al-Kāshī’s most famous mathematical achievement?**
Al-Kāshī’s calculation of π to 16 decimal places (3.1415926535897932) in *The Treatise on the Circumference* was unparalleled in precision until the 16th century, demonstrating his mastery of numerical approximation.

**Where did Jamshīd al-Kāshī work?**
He was affiliated with the Ulugh Beg Observatory in Samarkand (modern Uzbekistan), a leading center of Islamic astronomy during the Timurid era, where he collaborated with Ulugh Beg on astronomical research.

**Did Jamshīd al-Kāshī contribute to fields beyond mathematics?**
Yes, he was also a physician and astrologer, reflecting the interdisciplinary nature of medieval Islamic scholarship. His works blended theoretical mathematics with practical applications in astronomy and medicine.

**What is *Miftāḥ al-Ḥisāb*?**
*Miftāḥ al-Ḥisāb* (*The Key to Arithmetic*) is a seminal textbook that systematized arithmetic operations, introduced decimal fractions, and provided methods for solving equations, influencing later mathematical pedagogy in the Islamic world and beyond.

**How did al-Kāshī’s work influence later scientists?**
His trigonometric methods, including the law of cosines, were adopted by European mathematicians like François Viète. His astronomical tables and computational techniques laid groundwork for the Copernican Revolution and early modern astronomy.

## Why They Matter
Jamshīd al-Kāshī bridged medieval Islamic science and the European Renaissance, advancing mathematical rigor and observational astronomy. His calculation of π set a benchmark for numerical precision, while his trigonometric innovations became foundational in both Eastern and Western mathematics. The Ulugh Beg Observatory’s astronomical tables, co-developed by al-Kāshī, were among the most accurate pre-telescopic star catalogs, directly influencing Tycho Brahe and Johannes Kepler. Without his contributions, the transition from Ptolemaic to heliocentric models might have been delayed, and computational mathematics would have lacked key methodological tools.

## Notable For
- First to compute π to 16 decimal places (1424).
- Author of *Miftāḥ al-Ḥisāb*, a cornerstone of Islamic arithmetic.
- Formulated the law of cosines for plane triangles.
- Key collaborator at the Ulugh Beg Observatory, producing the *Zīj-i Khāqānī* astronomical tables.
- Pioneered the use of decimal fractions in systematic calculations.
- Recognized as a polymath in mathematics, astronomy, medicine, and astrology.

## Body
### Early Life and Background
Jamshīd al-Kāshī was born in 1380 in Persia (present-day Iran), though specific details about his early education remain undocumented. His nisba, *al-Kāshī*, suggests origins in Kashan, a city known for its scholarly traditions. He adopted the honorific *Ghiyāth al-Dīn* ("Helper of the Faith"), reflecting his stature in Islamic intellectual circles.

### Mathematical Innovations
Al-Kāshī’s mathematical legacy is defined by two major works:
1. **The Treatise on the Circumference** (c. 1424): Using a polygon with 805,306,368 sides, he calculated π to 16 decimal places—a feat that surpassed all prior attempts and remained unmatched until the 16th century. His method combined Archimedean exhaustion with original computational techniques.
2. **The Key to Arithmetic** (*Miftāḥ al-Ḥisāb*, 1427): This encyclopedic text covered arithmetic, algebra, and measurement, introducing decimal fractions (e.g., expressing 1/7 as 0.142857) and practical algorithms for root extraction. It became a standard reference in madrasas across the Islamic world.

His trigonometric work included the law of cosines, expressed as:
\[ c^2 = a^2 + b^2 - 2ab \cos C \]
This formula, later formalized in Europe, was critical for solving spherical triangles in astronomy.

### Astronomical Work
At the **Ulugh Beg Observatory** (founded 1420s), al-Kāshī collaborated with Ulugh Beg to compile the *Zīj-i Khāqānī*, a set of astronomical tables that corrected Ptolemy’s *Almagest*. Key contributions included:
- **Star Catalogs**: Measured the positions of 992 stars with unprecedented accuracy.
- **Planetary Models**: Refined parameters for planetary orbits, including the Moon’s inclination and solar apogee.
- **Eclipse Predictions**: Developed methods to forecast lunar and solar eclipses, integrating empirical data with theoretical models.

The observatory’s 40-meter sextant, designed under al-Kāshī’s guidance, enabled measurements precise to within seconds of arc.

### Interdisciplinary Roles
Beyond mathematics and astronomy, al-Kāshī practiced **medicine** and **astrology**, typical of medieval Islamic scholars. His astrological writings, though less emphasized today, were contemporary with his scientific work, reflecting the era’s holistic view of knowledge.

### Death and Legacy
Al-Kāshī died on **June 22, 1429**, in Samarkand. His influence persisted through:
- **Transmission to Europe**: Latin translations of his works (e.g., via the *Zīj-i Khāqānī*) reached Renaissance mathematicians like Regiomontanus.
- **Impact on Computation**: His decimal fraction system foreshadowed Simon Stevin’s 16th-century formalization.
- **Astronomical Continuity**: The Ulugh Beg Observatory’s data was later used by Copernicus in *De Revolutionibus*.

### Cultural and Historical Context
Al-Kāshī’s career flourished during the **Timurid Renaissance**, a period of scientific patronage under rulers like Ulugh Beg. His work exemplifies the synthesis of Persian, Arabic, and Greek knowledge, which shaped the scientific foundations of both the Islamic world and early modern Europe.

### Notable Works (with Identifiers)
- *The Treatise on the Circumference* (1424)
- *Miftāḥ al-Ḥisāb* (*The Key to Arithmetic*, 1427)
- *Zīj-i Khāqānī* (astronomical tables, co-authored)
- *Risāla al-Watar wa’l-Jayb* (*Treatise on the Chord and Sine*), advancing trigonometry.

### Aliases and Naming Conventions
- Full name: **Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī**
- Alternate spellings: **al-Kāshī**, **al-Kashi**, **Kashi**
- Arabic: **غياث الدين جمشيد بن مسعود بن محمد الكاشي**

### Institutional Affiliations
- **Ulugh Beg Observatory** (Samarkand, Uzbekistan): Primary workplace from the 1420s until his death.

### Recognition and Identifiers
- **Wikidata**: Q315217
- **VIAF**: 169807882
- **Library of Congress**: n88168314
- **MathSciNet**: 118975498
- **Google Scholar**: /m/055vmf

Al-Kāshī’s life and work embody the zenith of Islamic scientific achievement, demonstrating how empirical rigor and theoretical innovation could transcend cultural and temporal boundaries.

## References

1. Integrated Authority File
2. BnF authorities
3. International Standard Name Identifier
4. Virtual International Authority File
5. CiNii Research
6. MacTutor History of Mathematics archive
7. Complete Dictionary of Scientific Biography
8. Freebase Data Dumps. 2013
9. CERL Thesaurus
10. [Source](https://islamansiklopedisi.org.tr/kasi)
11. HMML Authority File