Abu Mahmud Hamid ibn Khidr Khojandi (known as Abu Mahmood Khojandi, Alkhujandi or al-Khujandi, was a Central Asian astronomer and mathematician with Mongol orijin who lived in the late 10th century and helped build an observatory, near the city of Ray (near today’s Tehran), in Iran. He was born in Khujand; a bronze bust of the astronomer is present in a park in modern-day Khujand, now part of Tajikistan. The few facts about Khujandi’s life that are known come from his surviving writings as well as from comments made by Nassereddin Tusi. From Tusi’s comments it is fairly certain that Khujandi was one of the rulers of the Mongol tribe in the Khudzhand region, and thus must have come from the nobility
In Islamic astronomy, Khujandi worked under the patronage of the Buwayhid Amirs at the observatory near Ray, Iran, where he is known to have constructed the first huge mural sextant in 994 AD, intended to determine the Earth’s axial tilt (“obliquity of the ecliptic”) to high precision. He determined the axial tilt to be 23°32’19” for the year 994 AD. He noted that measurements by earlier astronomers had found higher values (Indians: 24°; Ptolemy 23° 51′) and thus discovered that the axial tilt is not constant but is in fact (currently) decreasing. His measurement of the axial tilt was however about 2 minutes too small, probably due to his heavy instrument settling over the course of the observations.
In Islamic mathematics, he stated a special case of Fermat’s last theorem for n = 3, but his attempted proof of the theorem was incorrect. The spherical law of sines may have also been discovered by Khujandi, but it is uncertain whether he discovered it first, or whether Abu Nasr Mansur, Abul Wafa or Nasir al-Din al-Tusi discovered it first..
Sayyed Mahmoud Hessaby (February 23, 1903, Tehran – September 3, 1992, Geneva) was an Iranian scientist, researcher and professor of University of Tehran. During the congress on “60 years of physics in Iran” the services rendered by him were deeply appreciated and he was called “the father of modern physics in Iran”
Hessaby was born in Tehran to Abbas and Goharshad Hessaby. When he was seven, the family moved from Iran to Beirut in Lebanon where he attended school.
At seventeen he obtained his Bachelor’s in Arts and Sciences from the American University of Beirut. Later he obtained his B.A. in civil engineering while working as a draftsman. He continued his studies and graduated from Engineering school of Beirut.
Hessaby was admitted to the École Superieure d’Electricité and in 1925 graduated while he was employed by the SNCF (French National Railway). He started working in the electric locomotive maintenance department. Hew was a scientific mind and continued his research in Physics at the Sorbonne University and obtained his Ph.D. in Physics from that University at the age of twenty-five.
Dr Hessaby was a Polymath, having held five Bachelor’s degrees in literature, civil engineering, mathematics, electric engineering and mining engineering. He continued lecturing at University of Tehran for three working generations, teaching seven generations of students and professors.
In 1947, he published his classic paper on “Continuous particles”. Following this, in 1957 he proposed his model of “Infinitely extended particles”.
As Hessaby wished, he was buried in his hometown, Tafresh.
Prof. Mahmoud Hessaby was fluent in five living languages: Persian, French, English, German and Arabic. He was also familiar with Sanskrit, Latin, Greek, Pahlavi, Avestan, Turkish and Italian, which he used for etymological studies.
The Dr. Hessaby Museum
The Museum Of Dr. Hessaby is a collection of some of personal belongings and communications with various scientific cultural figures.
The museum has been established by his family, colleagues and students in order to value his 60 years of scientific, educational and cultural activities, and to set an example for young generation of Iran, students in particular, of a hard-working contemporary scientist, who despite his difficult childhood led a successful life and contributed greatly towards his country’s progress by establishing many scientific, industrial, cultural, and research centers in Iran, including the University of Tehran, the first modern university in the country.
Every item of the museum is a reminder of a corner of his life and bears a valuable lesson of life.
The museum is situated in his personal home, north of Tehran, and visited daily by many visitors from different scientific, cultural, and educational institutes and organisations, free of charge.
Dr. Hessaby Foundation
The Dr. Hessaby Foundation was established by his son to continue all the various aspects of his work, highlighting his belief that giving priority to research and researchers is the basis of the scientific and industrial progress of a country.
He had a son and a daughter. His son graduated in political Science from Melli University and is currently in charge of the Dr. Hessaby Institute.
According to the Dr Hessaby Institute, essentially an institution run by his son, the following were some of his accomplishments:
Founding the Highway Engineering school and teaching there from 1928
Survey and drawing of the first coastal road-map between Persian Gulf ports
Founding the “teachers college” and teaching there from 1928
Construction of the first radio-set in Iran (1928)
Construction of the first weather-station in 1931
Installation and operation of the first radiology center in Iran in 1931
Calculation and setting of Iranian time (1932)
Founding the first private hospital in Iran (Goharshad Hospital) in 1933
Writing the University carechair and founding Tehran University (1934)
Founding the Engineering school in 1934 and acting as the dean of that school until 1936 and teaching there from then on
Founding the faculty of science and acting as its dean from 1942 to 1948
Minister of Education in the cabinet of Dr Mossadegh from 1951 to 1952
Opposing the contract with the consortium while in the Senate of Iran in 1954
Opposing the membership of Iran in CENTO
Founding the Telecommunication Center of Assad-Abad in Hamedan (1959)
Writing the standards charter for the standards Institute of Iran (1954)
Founding the Geophysical Institute of Tehran University (1961)
Title of distinguished professor of Tehran University from 1971
Founding the atomic research center and atomic reactor at Tehran University
Founding the atomic Energy center of Iran, member of the UN scientific sub-committee of peaceful use of nuclear power, member of the international space committee (1981)
Establishment of Iran’s space research committee and member of the international space committee (1981)
Establishment of the Iranian music society and founding the Persian language Academy
Awards and honours
Father of Iranian Physics, By Iran’s Physical Society
Hessaby M, Model of an Infinite Particle, Journal de Physique et le Radium 18 (5): 323-326 1957. Times Cited: 0 University of Tehran.
Hessaby M, Theoretical Evidence for the Existence of a Light-Charged Particle of Mass Greater than That of the Electron, Physical Review, Vol. 73, Issue 9, p. 1128 (1948). Times Cited: 1 While at Institute for Nuclear Studies, University of Chicago, Chicago, Illinois.
Hessaby M, Continuous Particles, Proceedings of the National Academy of Sciences of the United States of America, Vol. 33, No. 6, pp. 189–194 (1947). Times Cited: 0 University of Tehran and Princeton University
Hessaby M, Continuous Particles, Proceedings of the American Physical Society, Minutes of the Meeting at Montreal, June 19–21, 1947,
Research and writing
Abu ‘Ali Ahmad ibn Muhammad ibn Ya’qub Ibn Miskawayh also known as Ibn Miskawayh (932–1030) or Ebn Meskavayh, was a Persian chancery official of the Buwayhid era, and philosopher and historian from Rey, Iran. As a neo-platonist, his influence on Islamic philosophy is primarily in the area of ethics. He was the author of the first major Islamic work on philosophical ethics, entitled Tahdhib al-akhlaq (تهذيب الأخلاق: Refinement of Morals), focusing on practical ethics, conduct, and refinement of character. He separated personal ethics from the public realm, and contrasted the liberating nature of reason with the deception and temptation of nature.
Ebn Meskavayh was a prominent figure in the intellectual and cultural life of his time. Miskawayh may have been a Mazdaean convert to Islam, but it seems more likely that it was one of his ancestors who converted He was fluent enough in Middle Persian to have translated some pre-Islamic texts in that language into Arabic. He worked as a secretary and librarian for a sequence of viziers, including Adud al-Dawla. Some contemporary sources associated him with the Brethren of Purity, claiming that some of his writings were used in the compilation of the Encyclopedia of the Brethren of Purity.
Ibn Miskawayh was one of the first to clearly describe a version of the idea of evolution. Muhammad Hamidullah describes the evolutionary ideas found in Ibn Miskawayh’s al-Fawz al-Asghar as follows:
“[These books] state that God first created matter and invested it with energy for development. Matter, therefore, adopted the form of vapour which assumed the shape of water in due time. The next stage of development was mineral life. Different kinds of stones developed in course of time. Their highest form being mirjan (coral). It is a stone which has in it branches like those of a tree. After mineral life evolves vegetation. The evolution of vegetation culminates with a tree which bears the qualities of an animal. This is the date-palm. It has male and female genders. It does not wither if all its branches are chopped but it dies when the head is cut off. The date-palm is therefore considered the highest among the trees and resembles the lowest among animals. Then is born the lowest of animals. It evolves into an ape. This is not the statement of Darwin. This is what Ibn Maskawayh states and this is precisely what is written in the Epistles of Ikhwan al-Safa. The Muslim thinkers state that ape then evolved into a lower kind of a barbarian man. He then became a superior human being. Man becomes a saint, a prophet. He evolves into a higher stage and becomes an angel. The one higher to angels is indeed none but God. Everything begins from Him and everything returns to Him.”
Arabic manuscripts of the al-Fawz al-Asghar were available in European universities by the 19th century. This work is believed to have been studied by Charles Darwin, who was a student of Arabic, and it is thought to have had an influence on his inception of Darwinism.In his Tajarib al-umam (Experiences of Nations) he was one of the first major Muslim historians to write a chronicle of contemporary events as an eyewitness. As a Buwayhid bureaucrat, he worked under the vizier al-Muhallabi and had access to the internal happenings of the court. The chronicle is a universal history from the beginning of Islam, but it cuts off near the end of the reign of Adud al-Dawla.
His major work in the field of philosophy is his Tahḏib al-aḵlāq wa-taṭhir al-aʿrāq. The book is meant to provide students of philosophy and ethics an exposition of the main elements of philosophy.
Ketāb al-ḥekma al-ḵāleda (Book of Eternal Wisdom) is an Arabic translation of a Persian work called Jāvidān ḵerad (“Eternal Wisdom”). One manuscript of which bears the title Ketāb ādāb al-ʿArab wa’l-Fors (lit. “Book of Literatures of the Arabs and Persians”).
Abu Mūsā Jābir ibn Hayyān , fl.c.721–c.815) was a prominent Persian or Arab polymath: a chemist and alchemist, astronomer and astrologer, engineer, geographer, philosopher, physicist, and pharmacist and physician. Born and educated in Tus, he later traveled to Kufa. Jābir is held to have been the first practical alchemist.As early as the 10th century, the identity and exact corpus of works of Jābir was in dispute in Islamic circles. His name was Latinized as “Geber” in the Christian West and in 13th-century Europe an anonymous writer, usually referred to as Pseudo-Geber, produced alchemical and metallurgical writings under the pen-name Geber
In 988 Ibn al-Nadim compiled the Kitab al-Fihrist which mentions Jabir as a spiritual follower and as a companion to Jafar as-Sadiq . In another reference al-Nadim reports that a group of philosophers claimed Jabir was one of their own members. Another group, reported by al-Nadim, says only The Large Book of Mercy is genuine and that the rest are pseudographical. Their assertions are rejected by al-Nadim. Joining al-Nadim in asserting a real Jabir; Ibn-Wahshiyya (“Jaber ibn Hayyn al-Sufi …book on poison is a great work..”) Rejecting a real Jabir; (the philosopher c.970) Abu Sulayman al-Mantiqi claims the real author is one al-Hasan ibn al-Nakad al-Mawili. 14th century critic of Arabic literature, Jamal al-Din ibn Nubata al-Misri declares all the writings attributed to Jabir doubtful.
Life and background
Jabir was a Natural Philosopher who lived mostly in the 8th century; he was born in Tus, Khorasan, in Iran (Persia), then ruled by the Umayyad Caliphate. Jabir in the classical sources has been entitled differently as al-Azdi al-Barigi or al-Kufi or al-Tusi or al-Sufi. There is a difference of opinion as to whether he was a Persian from Khorasan who later went to Kufa or whether he was, as some have suggested, of Syrian origin and later lived in Persia and Iraq. His ethnic background is not clear, but most sources reference him as a Persian. In some sources, he is reported to have been the son of Hayyan al-Azdi, a pharmacist of the Arabian Azd tribe who emigrated from Yemen to Kufa (in present-day Iraq) during the Umayyad Caliphate. while Henry Corbin believes Geber seems to have been a client of the ‘Azd tribe.
Jābir became an alchemist at the court of Caliph Harun al-Rashid, for whom he wrote the Kitab al-Zuhra (“The Book of Venus”, on “the noble art of alchemy”). Hayyan had supported the Abbasid revolt against the Umayyads, and was sent by them to the province of Khorasan (present day Afghanistan and Iran) to gather support for their cause. He was eventually caught by the Umayyads and executed. His family fled to Yemen, where Jābir grew up and studied the Quran, mathematics and other subjects. Jābir’s father’s profession may have contributed greatly to his interest in alchemy. After the Abbasids took power, Jābir went back to Kufa. He began his career practicing medicine, under the patronage of a Vizir (from the noble Persian family Barmakids) of Caliph Harun al-Rashid. His connections to the Barmakid cost him dearly in the end. When that family fell from grace in 803, Jābir was placed under house arrest in Kufa, where he remained until his death. It has been asserted that Jābir was a student of the sixth Imam Ja’far al-Sadiq and Harbi al-Himyari, however other scholars have questioned this theory.
The Jabirian corpus
In total, nearly 3,000 treatises and articles are credited to Jabir ibn Hayyan. Following the pioneering work of Paul Kraus, who demonstrated that a corpus of some several hundred works ascribed to Jābir were probably a medley from different hands, mostly dating to the late 9th and early 10th centuries, many scholars believe that many of these works consist of commentaries and additions by his followers, particularly of an Ismaili persuasion.The scope of the corpus is vast: cosmology, music, medicine, magic, biology, chemical technology, geometry, grammar, metaphysics, logic, artificial generation of living beings, along with astrological predictions, and symbolic Imâmî myths.
The 112 Books dedicated to the Barmakids, viziers of Caliph Harun al-Rashid. This group includes the Arabic version of the Emerald Tablet, an ancient work that proved a recurring foundation of and source for alchemical operations. In the Middle Ages it was translated into Latin (Tabula Smaragdina) and widely diffused among European alchemists.
The Seventy Books, most of which were translated into Latin during the Middle Ages. This group includes the Kitab al-Zuhra (“Book of Venus”) and the Kitab Al-Ahjar (“Book of Stones”).
The Ten Books on Rectification, containing descriptions of alchemists such as Pythagoras, Socrates, Plato and Aristotle.
The Books on Balance; this group includes his most famous ‘Theory of the balance in Nature’.
Jābir states in his Book of Stones (4:12) that “The purpose is to baffle and lead into error everyone except those whom God loves and provides for”. His works seem to have been deliberately written in highly esoteric code (see steganography), so that only those who had been initiated into his alchemical school could understand them. It is therefore difficult at best for the modern reader to discern which aspects of Jābir’s work are to be read as symbols (and what those symbols mean), and what is to be taken literally. Because his works rarely made overt sense, the term gibberish is believed to have originally referred to his writings (Hauck, p. 19).
Jābir’s interest in alchemy was inspired by his teacher Ja’far as-Sadiq. When he used to talk about alchemy, he would say “my master Ja’far as-Sadiq taught me about calcium, evaporation, distillation and crystallization and everything I learned in alchemy was from my master Ja’far as-Sadiq.” Imam Jafar was famed for his depth and breadth of knowledge. In addition to his knowledge of Islamic sciences, Imam Jafar was well educated in natural sciences, mathematics, philosophy, astronomy, anatomy, chemistry (alchemy), and other subjects. The foremost Islamic alchemist Jabir bin Hayyan was his most prominent student. Other famous students of his were Imam Abu Hanifa and Imam Malik Ibn Anas, the founders of two Sunni schools of jurisprudence, and Wasil ibn Ata, the founder of the Mutazilite school of Islamic thought. Imam Jafar was known for his liberal views on learning, and was keen to debate with scholars of different faiths and of different beliefs. Imam Abu Hanifa is quoted by many souces as having said “My knowledge extends to only two years. The two I spent with Imam Jafar Sadiq”, some Islamic scholars have gone so far as to call Imam Jafar Saddiq as the root of most of Islamic jurisprudence, having a massive influence on Hanafi, Maliki and Shia schools of thought extending well into mainstream Hanbali and Shafi’i thought. Imam Jafar also attained a surpassing knowledge in astronomy and in the science of medicine. it is said that he wrote more than five hundred books on health care which were compiled and annotated by another great scholar and scientist of Islam, Jabir bin Hayyan Jābir professes to draw his inspiration from earlier writers, legendary and historic, on the subject.
In his writings, Jābir pays tribute to Egyptian and Greek alchemists Zosimos, Democritus, Hermes Trismegistus, Agathodaimon, but also Plato, Aristotle, Galen, Pythagoras, and Socrates as well as the commentators Alexander of Aphrodisias Simplicius, Porphyry and others. A huge pseudo-epigraphic literature of alchemical books was composed in Arabic, among which the names of Persian authors also appear like Jāmāsb, Ostanes, Mani, testifying that alchemy-like operations on metals and other substances were also practiced in Persia. The great number of Persian technical names (zaybaq = mercury, nošāder = sal-ammoniac) also corroborates the idea of an important Iranian root of medieval alchemy. Ibn al-Nadim reports a dialogue between Aristotle and Ostanes, the Persian alchemist of Achaemenid era, which is in Jabirian corpus under the title of Kitab Musahhaha Aristutalis. Ruska had suggested that the Sasanian medical schools played an important role in the spread of interest in alchemy. He emphasizes the long history of alchemy, “whose origin is Arius … the first man who applied the first experiment on the [philosopher’s] stone… and he declares that man possesses the ability to imitate the workings of Nature” (Nasr, Seyyed Hussein, Science and Civilization of Islam).
Jābir’s alchemical investigations ostensibly revolved around the ultimate goal of takwin — the artificial creation of life. The Book of Stones includes several recipes for creating creatures such as scorpions, snakes, and even humans in a laboratory environment, which are subject to the control of their creator. What Jābir meant by these recipes is unknown. Jābir’s alchemical investigations were theoretically grounded in an elaborate numerology related to Pythagorean and Neoplatonic systems. The nature and properties of elements was defined through numeric values assigned the Arabic consonants present in their name, ultimately culminating in the number 17.
By Jabirs’ time Aristotelian physics, had become Neoplatonic. Each Aristotelian element was composed of these qualities: fire was both hot and dry, earth, cold and dry, water cold and moist, and air, hot and moist. This came from the elementary qualities which are theoretical in nature plus substance. In metals two of these qualities were interior and two were exterior. For example, lead was cold and dry and gold was hot and moist. Thus, Jābir theorized, by rearranging the qualities of one metal, a different metal would result. Like Zosimos, Jabir believed this would require a catalyst, an al-iksir, the elusive elixir that would make this transformation possible — which in European alchemy became known as the philosopher’s stone.
According to Jabir’s mercury-sulfur theory, metals differ from each in so far as they contain different proportions of the sulfur and mercury. These are not the elements that we know by those names, but certain principles to which those elements are the closest approximation in nature. Based on Aristotle’s “exhalation” theory the dry and moist exhalations become sulfur and mercury (sometimes called “sophic” or “philosophic” mercury and sulfur). The sulfur-mercury theory is first recorded in a 7th-century work Secret of Creation credited (falsely) to Balinus (Apollonius of Tyana). This view becomes widespread. In the Book of Explanation Jabir says
the metals are all, in essence, composed of mercury combined and coagulated with sulphur [that has risen to it in earthy, smoke-like vapors]. They differ from one another only because of the difference of their accidental qualities, and this difference is due to the difference of their sulphur, which again is caused by a variation in the soils and in their positions with respect tothe heat of the sun
Holmyard says that Jabir proves by experiment that these are not ordinary sulfur and mercury.
The seeds of the modern classification of elements into metals and non-metals could be seen in his chemical nomenclature. He proposed three categories:
The origins of the idea of chemical equivalents might be traced back to Jabir, in whose time it was recognized that “a certain quantity of acid is necessary in order to neutralize a given amount of base.”[verification needed] Jābir also made important contributions to medicine, astronomy/astrology, and other sciences. Only a few of his books have been edited and published, and fewer still are available in translation..
Ahmad ibn Muhammad al-Nahavandi was a Persian astronomer of the 8th and 9th centuries. His name indicates that he was from Nahavand, a city in Iran. He lived and worked at the Academy of Gundishapur, in Khuzestan, Iran, at the time of Yahya ibn Khalid ibn Barmak, who died in 803AD, where he is reported to have been making astronomical observations around the year 800AD. He and Mashallah ibn Athari were among the earliest Islamic era astronomers who flourished during the reign of al-Mansur, the second Abbasid Caliph. He also compiled tables called the comprehensive (Mushtamil).
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) (Persian: غیاثالدین جمشید کاشانی Ghiyās-ud-dīn Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer and mathematician.
Al-Kashi was one of the best mathematicians in the Islamic world. He was born in 1380, in Kashan, in central Iran. This region was controlled by Tamurlane, better known as Timur. Al-Kashi lived in poverty during his childhood and the beginning years of his adulthood. The situation changed for the better when Timur died in 1405, and his son, Shah Rokh, ascended into power. Shah Rokh and his wife, Goharshad, a Persian princess, were very interested in the sciences, and they encouraged their court to study the various fields in great depth. Consequently, the period of their power became one of many scholarly accomplishments. This was the perfect environment for al-Kashi to begin his career as one of the world’s greatest mathematicians.
Eight years after he came into power in 1409, their son, Ulugh Beg, founded an institute in Samarkand which soon became a prominent university. Students from all over the Middle East, and beyond, flocked to this academy in the capital city of Ulugh Beg’s empire. Consequently, Ulugh Beg harvested many great mathematicians and scientists of the Muslim world. In 1414, al-Kashi took this opportunity to contribute vast amounts of knowledge to his people. His best work was done in the court of Ulugh Beg, and it is said that he was the king’s favourite student. Al-Kashi was still working on his book, called “Risala al-watar wa’l-jaib” meaning “The Treatise on the Chord and Sine”, when he died in 1429. Some scholars believe that Ulugh Beg may have ordered his murder, while others say he died a natural death. The details are unclear.
Al-Kashi produced a Zij entitled the Khaqani Zij, which was based on Nasir al-Din al-Tusi‘s earlier Zij-i Ilkhani. In his Khaqani Zij, al-Kashi thanks the Timurid sultan and mathematician-astronomer Ulugh Beg, who invited al-Kashi to work at his observatory (see Islamic astronomy) and his university (see Madrasah) which taught Islamic theology as well as Islamic science. Al-Kashi produced sine tables to four sexagesimal digits (equivalent to eight decimal places) of accuracy for each degree and includes differences for each minute. He also produced tables dealing with transformations between coordinate systems on the celestial sphere, such as the transformation from the ecliptic coordinate system to the equatorial coordinate system.
Astronomical Treatise on the size and distance of heavenly bodies
He wrote the book Sullam al-Sama on the resolution of difficulties met by predecessors in the determination of distances and sizes of heavenly bodies such as the Earth, the Moon, the Sun and the Stars.
Treatise on Astronomical Observational Instruments
In 1416, al-Kashi wrote the Treatise on Astronomical Observational Instruments, which described a variety of different instruments, including the triquetrum and armillary sphere, the equinoctial armillary and solsticial armillary of Mo’ayyeduddin Urdi, the sine and versine instrument of Urdi, the sextant of al-Khujandi, the Fakhri sextant at the Samarqand observatory, a double quadrant Azimuth–altitude instrument he invented, and a small armillary sphere incorporating an alhidade which he invented.
Plate of Conjunctions
Al-Kashi invented the Plate of Conjunctions, an analog computing instrument used to determine the time of day at which planetary conjunctions will occur, and for performing linear interpolation.
Al-Kashi also invented a mechanical planetary computer which he called the Plate of Zones, which could graphically solve a number of planetary problems, including the prediction of the true positions in longitude of the Sun and Moon, and the planets in terms of elliptical orbits; the latitudes of the Sun, Moon, and planets; and the ecliptic of the Sun. The instrument also incorporated an alhidade and ruler.
Law of cosines
In French, the law of cosines is named Théorème d’Al-Kashi (Theorem of Al-Kashi), as al-Kashi was the first to provide an explicit statement of the law of cosines in a form suitable for triangulation.
The Treatise on the Chord and Sine
In The Treatise on the Chord and Sine, al-Kashi computed sin 1° to nearly as much accuracy as his value for π, which was the most accurate approximation of sin 1° in his time and was not surpassed until Taqi al-Din in the 16th century. In algebra and numerical analysis, he developed an iterative method for solving cubic equations, which was not discovered in Europe until centuries later.A method algebraically equivalent to Newton’s method was known to his predecessor Sharaf al-Dīn al-Tūsī. Al-Kāshī improved on this by using a form of Newton’s method to solve to find roots of N. In western Europe, a similar method was later described by Henry Biggs in his Trigonometria Britannica, published in 1633.In order to determine sin 1°, al-Kashi discovered the following formula often attributed to François Viète in the 16th century:
The Key to Arithmetic
Computation of 2π
In his numerical approximation, he correctly computed 2π (or ) to 9 sexagesimal digits in 1424, and he converted this approximation of 2π to 17 decimal places of accuracy. This was far more accurate than the estimates earlier given in Greek mathematics (3 decimal places by Archimedes), Chinese mathematics (7 decimal places by Zu Chongzhi) or Indian mathematics (11 decimal places by Madhava of Sangamagrama). The accuracy of al-Kashi’s estimate was not surpassed until Ludolph van Ceulen computed 20 decimal places of π nearly 200 years later. It should be noted that al-Kashi’s goal was not to compute the circle constant with as many digits as possible but to compute it so precisely that the circumference of the largest possible circle (ecliptica) could be computed with highest desirable precision (the diameter of a hair).
“The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548-1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).“
In considering Pascal’s triangle, known in Persia as “Khayyam’s triangle” (named after Omar Khayyám), Struik notes that (p. 21):
“The Pascal triangle appears for the first time (so far as we know at present) in a book of 1261 written by Yang Hui, one of the mathematicians of the Sung dynasty in China. The properties of binomial coefficients were discussed by the Persian mathematician Jamshid Al-Kāshī in his Key to arithmetic of c. 1425. Both in China and Persia the knowledge of these properties may be much older. This knowledge was shared by some of the Renaissance mathematicians, and we see Pascal‘s triangle on the title page of Peter Apian‘s German arithmetic of 1527. After this we find the triangle and the properties of binomial coefficients in several other authors.“
In 2009 IRIB produced and broadcast (through Channel 1 of IRIB) a biographical-historical film series on the life and times of Jamshid Al-Kāshi, with the title The Ladder of the Sky  (Nardebām-e Āsmān ). The series, which consists of 15 parts of each 45 minutes duration, is directed by Mohammad-Hossein Latifi and produced by Mohsen Ali-Akbari. In this production, the role of the adult Jamshid Al-Kāshi is played by Vahid Jalilvand.