Jamshid al-Kashi

Jamshid al-Kashi was a great thinker. He lived a long time ago in Iran. He loved to study numbers and the stars.

He worked with a ruler to find shapes. He found out how big the Earth is. He also studied the Moon and the Sun. He used math to find their sizes.

He made tools to help him work. One tool was like a small computer. It helped him track the planets. He was very good at solving hard puzzles. He was a remarkable scientist.

Jamshid al-Kashi was a famous Persian mathematician. He lived in the 1400s. He worked in the court of Ulugh Beg. Ulugh Beg was a ruler who loved science. He built a great university in Samarkand. Al-Kashi was a very talented scientist. Ulugh Beg said he could solve hard problems.

Al-Kashi loved to study numbers. He found a very precise way to measure a circle. He used a special number to find the circle's edge. His math was better than many people before him. He was also great at using decimals. Decimals are a way to show parts of a whole number.

He also studied the stars and planets. He made tools to help him see the sky. One tool was a mechanical computer. It helped him find where planets would be. He also wrote about the size of the Earth. He wrote about the Sun and the Moon. He even helped find ways to solve math equations.

Jamshid al-Kashi was a brilliant mathematician and astronomer. He lived during a time of great learning in Persia. He was born around 1380 in the city of Kashan. His life changed when the ruler Timur died in 1405. After that, a new leader named Shah Rokh took power. He and his wife, Goharshad, loved science and math. They encouraged people to study these subjects deeply. This created a perfect home for al-Kashi to grow.

Al-Kashi was famous for working with very precise numbers. He wanted to find the exact value of a circle constant. He calculated this number to sixteen decimal places of accuracy. This was much better than the work of earlier scholars. For example, Ptolemy in Greece only reached three decimal places. Even the great Indian mathematicians reached eleven decimal places. Al-Kashi's work was not beaten for 180 years. He even used decimal fractions with great ease. This helped him solve very hard math problems.

He also built amazing tools to study the night sky. One tool was a mechanical planetary computer. He called this invention the Plate of Zones. It could help predict where the Sun and Moon would be. It could also show the positions of the planets. He also invented a double quadrant instrument. He made a small armillary sphere with a part called an alhidade. These tools helped scientists measure the stars and planets. He wrote about these tools in a special book.

In the year 1414, al-Kashi joined the court of Ulugh Beg. Ulugh Beg was a ruler who loved to learn. He built a famous university in the city of Samarkand. Many students traveled from far away to study there. Al-Kashi worked at the university and the observatory. He wrote many books, including one called The Key to Arithmetic. He was still working on a book about sines when he died in 1429. Some say he died naturally, while others say he was murdered.

Many math ideas we use today come from his work. In France, a rule for triangles is called the theorem of al-Kashi. This is a version of the law of cosines. He also studied patterns in numbers called binomial coefficients. These are part of what some call Khayyam's triangle. His methods for solving equations were very advanced. Some of his ideas were not found in Europe for centuries. He truly was a remarkable scientist. Even Ulugh Beg said he could solve the most difficult problems.

Ghiyāth al-Dīn Jamshīd al-Kāshī was a Persian mathematician and astronomer. He lived during the Timurid era, a period of significant scientific growth. Al-Kāshī was born around 1380 in Kashan, located in central Iran. His career flourished under the patronage of the ruler Shah Rokh and his wife, Goharshad. They encouraged deep scientific study within their court. This support provided the ideal environment for al-Kāshī to become a leading figure in mathematics.

In 1414, al-Kāshī began contributing his knowledge to the court of Ulugh Beg. Ulugh Beg was the son of Shah Rokh and a great patron of the sciences. He founded a prominent university and an observatory in the capital city of Samarkand. This academy attracted students and scientists from across the Middle East. Al-Kāshī worked extensively at both the university and the observatory. He produced several important works, including the Khaqani Zij, which was based on earlier astronomical tables.

Al-Kāshī is perhaps most famous for his extreme precision in numerical computation. In his work *al-Risāla al-muhītīyya*, or *Treatise on the Circumference*, he calculated the circle constant with incredible accuracy. He computed this value to nine sexagesimal digits in 1424. This is equivalent to 16 decimal places of accuracy. This level of precision was far beyond previous scholars. For example, the Greek mathematician Ptolemy reached only three decimal places. The Chinese mathematician Zu Chongzhi reached seven places. The Indian mathematician Madhava of Kerala reached eleven places. Al-Kāshī's record was not surpassed for 180 years until Ludolph van Ceulen computed 20 places.

His mathematical contributions extended deeply into trigonometry and algebra. In his *Risālah al-watar wa’l-jaib*, or *Treatise on the Chord and Sine*, he calculated the sine of one degree with high accuracy. This was the most accurate sine value of his era. He also developed an iterative method for solving cubic equations. This technique was similar to what would later be known as Newton's method in Europe. He used this method to find the roots of specific equations. In his *Miftāḥ al-ḥisāb*, or *Key to Arithmetic*, he explained how to solve triangles using various data sets.

Al-Kāshī was also a gifted inventor of astronomical instruments. In 1416, he wrote a treatise describing many different tools. These included the triquetrum, the armillary sphere, and the sextant. He even invented a double quadrant Azimuth-altitude instrument. He also designed a small armillary sphere that used an alhidade. One of his most complex inventions was the Plate of Zones. This was a mechanical planetary computer used to solve planetary problems. It could graphically predict the positions of the Sun, Moon, and planets.

His work on the law of cosines remains a notable part of his legacy. He presented a version of this rule for solving triangles when two sides and an included angle are known. In France, this specific mathematical rule is often called the *théorème d'Al-Kashi*. He also discussed the properties of binomial coefficients. These properties are related to what is known as Pascal's triangle, or Khayyam's triangle in Persia. Al-Kāshī was also highly skilled in using both decimal and sexagesimal fractions. This allowed him to perform complex calculations with great ease.

Al-Kāshī's life ended in 1429 while he was still writing a book on sines. There are different accounts regarding his death. Some historical sources suggest he may have been murdered, possibly by order of Ulugh Beg. Other sources suggest he died of natural causes. Regardless of how he died, his impact was profound. Ulugh Beg described him as a remarkable scientist. He noted that al-Kāshī possessed the ability to solve the most difficult problems. His mathematical and astronomical discoveries helped shape the science of his time.