Abū Rayhān al-Bīrūnī

Abū Rayḥān Muḥammad ibn Aḥmad al-Bīrūnī (September 15 973 in Kath, Khwarezm – December 13 1048 in Ghazni) was a Persian (Tājīk) Muslim polymath of the 11th century.

He was a scientist and physicist, an anthropologist and psychologist, an astronomer, a chemist, a critic of alchemy and astrology, an encyclopedist and historian, a geographer and traveller, a geodesist and geologist, a mathematician, a pharmacist and physician, an Islamic philosopher and theologian, and a scholar and teacher, and he contributed greatly to all of these fields.

He was the first Muslim scholar to study India and the Brahminical tradition, and has been described as the father of Indology, the father of geodesy, and "the first anthropologist". He was also one of the earliest leading exponents of the experimental scientific method, and was responsible for introducing the experimental method into mechanics, the first to conduct elaborate experiments related to astronomical phenomena, and a pioneer of experimental psychology.

George Sarton, the father of the history of science, described Biruni as: ""One of the very greatest scientists of Islam, and, all considered, one of the greatest of all times.""

A. I. Sabra desribed Biruni as: ""One of the great scientific minds in all history.""

The Al-Biruni crater, on the Moon, is named after Biruni. Tashkent Technical University (formerly Tashkent Polytechnic Institute) is also named after Abu Rayhan al-Biruni.

Biography
He was born in Khwarazm (formerly north-eastern part of the Persian Samanid dynasty) presently in Khiva, Uzbekistan. He studied mathematics and astronomy]] under Abu Nasr Mansur.

He was a colleague of the fellow Persian Muslim philosopher and physician Abū Alī ibn Sīnā (Avicenna), the historian, philosopher and ethicist Ibn Miskawayh, in a university and science center established by prince Abu al-Abbas Ma'mun Khawarazmshah. He also travelled to South Asia or Central Asia (Modern Day Afghanistan) with Mahmud of Ghazni (whose son and successor Masud was, however, his major patron), and accompanied him on his campaigns in India (in 1030), learning Indian languages, and studying the religion and philosophy of its people. There, he also wrote his Ta'rikh al-Hind ("Chronicles of India"). Biruni wrote his books in Arabic and his native language Persian, though he knew no less than four other languages: Greek, Sanskrit, Syriac, and possibly Berber.

He was buried in Ghazni in Afganistan.

Works


Biruni's works number 146 in total. These include 35 books on astronomy, 4 on astrolabes, 23 on astrology, 5 on chronology, 2 on time measurement, 9 on geography, 10 on geodesy and mapping theory, 15 on mathematics (8 on arithmetic, 5 on geometry, 2 on trigonometry), 2 on mechanics, 2 on medicine and pharmacology, 1 on meteorology, 2 on mineralogy and gems, 4 on history, 2 on India, 3 on religion and philosophy, 16 literary works, 2 books on magic, and 9 unclassified books. Among these works, only 22 have survived, and only 13 of these works have been published. 6 of his surviving works are on astronomy. His extant works include:


 * Critical study of what India says, whether accepted by reason or refused (Arabic تحقيق ما للهند من مقولة معقولة في العقل أم مرذولة) - a compendium of India's religion and philosophy
 * The Remaining Signs of Past Centuries (Arabic الآثار الباقية عن القرون الخالية) - a comparative study of calendars of different cultures and civilizations, interlaced with mathematical, astronomical, and historical information.
 * The Mas'udi Canon (Persian قانون مسعودي) - an extensive encyclopedia on astronomy, geography, and engineering, named after Mas'ud, son of Mahmud of Ghazni, to whom he dedicated
 * Understanding Astrology (Arabic التفهيم لصناعة التنجيم) - a question and answer style book about mathematics and astronomy, in Arabic and Persian
 * Pharmacy - about drugs and medicines
 * Gems (Arabic الجماهر في معرفة الجواهر) about geology, minerals, and gems, dedicated to Mawdud son of Mas'ud
 * Astrolabe
 * A historical summary book
 * History of Mahmud of Ghazni and his father
 * History of Khawarazm

Astronomy


Will Durant wrote the following on al-Biruni's contributions to Islamic astronomy:

""He wrote treatises on the astrolabe, the planisphere, the armillary sphere; and formulated astronomical tables for Sultan Masud. He took it for granted that the earth is round, noted “the attraction of all things towards the center of the earth,” and remarked that astronomic data can be explained as well by supposing that the earth turns daily on its axis and annually around the sun, as by the reverse hypothesis.""

Experimental observations
Biruni was the first to conduct elaborate experiments related to astronomical phenomena. He supposed the Milky Way galaxy to be a collection of numerous nebulous stars. In Khorasan, he observed and described the solar eclipse on April 8, 1019, and the lunar eclipse on September 17, 1019, in detail, and gave the exact latitudes of the stars during the lunar eclipse.

In 1031, Biruni completed his extensive astronomical encyclopaedia Kitab al-Qanun al-Mas'udi (Latinized as Canon Mas’udicus), in which he recorded his astronomical findings and formulated astronomical tables. The book introduces the mathematical technique of analysing the acceleration of the planets, and first states that the motions of the solar apogee and the precession are not identical. Biruni also discovered that the distance between the Earth and the Sun is larger than Ptolemy's estimate, on the basis that Ptolemy disregarded the annual solar eclipses.

Al-Biruni also introduced a new method of observation called the "three points observation". A later Muslim polymath astronomer, Taqi al-Din, described the three points as "two of them being in opposition in the ecliptic and the third in any desired place." Prior to al-Biruni, astronomers used the relatively inaccurate method of Hipparchus who used the intervals of seasons for calculating solar parameters. Al-Biruni's new "three points observation" was an important contribution to practical astronomy, and was still used six centuries later by Taqi al-Din, Tycho Brahe and Nicolaus Copernicus to calculate the eccentricity of the Sun's orbit and the annual motion of the apogee.

Instruments
Al-Biruni invented a number of astronomical instruments. He wrote the first treatises on the planisphere and the orthographical astrolabe, as well as a treatise on the armillary sphere, and he was able to mathematically determine the direction of the Qibla from any place in the world. He also wrote the earliest treatise on the sextant.

He also invented an early hodometer, and the first mechanical lunisolar calendar computer which employed a gear train and eight gear-wheels. These were early examples of fixed-wired knowledge processing machines.

In his Exhaustive Treatise on Shadows, he explained the calculation of Salah prayer times according to the shadow cast by the gnomon of a sundial.

The first description of an "observation tube" is found in a work of al-Biruni, in a section "dedicated to verifying the presence of the new cresent on the horizon." Though these early observation tubes did not have lens, they "enabled an observer to focus on a part of the sky by eliminating light inteference." These observation tubes were later adopted in Latin-speaking Europe, where they influenced the development of the telescope.

Theories
In 1030, Biruni discussed the Indian heliocentric theories of Aryabhata, Brahmagupta and Varahamihira in his Indica. Biruni noted that the question of heliocentricity was a philosophical rather than a mathematical problem.

Abu Said al-Sijzi, a contemporary of Biruni, suggested the possible heliocentric movement of the Earth around the Sun, which Biruni did not reject. Biruni agreed with the Earth's rotation about its own axis, and while he was initially neutral regarding the heliocentric and geocentric models, he considered heliocentrism to be a philosophical problem. He remarked that if the Earth rotates on its axis and moves around the Sun, it would remain consistent with his astronomical parameters:

""Rotation of the earth would in no way invalidate astronomical calculations, for all the astronomical data are as explicable in terms of the one theory as of the other. The problem is thus difficult of solution.""

Biruni also wrote the following on al-Sijzi's heliocentric astrolabe called the "Zuraqi":

""I have seen the astrolabe called Zuraqi invented by Abu Sa'id Sijzi. I liked it very much and praised him a great deal, as it is based on the idea entertained by some to the effect that the motion we see is due to the Earth's movement and not to that of the sky. By my life, it is a problem difficult of solution and refutation. [...] For it is the same whether you take it that the Earth is in motion or the sky. For, in both cases, it does not affect the Astronomical Science. It is just for the physicist to see if it is possible to refute it.""

Refutation of astrology
The first semantic distinction between astronomy and astrology was given by al-Biruni in the 11th century. In a later work, he wrote a refutation of astrology. His reasons for refuting astrology were both due to the methods used by astrologers being conjectural rather than empirical and also due to the views of astrologers conflicting with orthodox Islam.

Earth sciences


Biruni made a number of contributions to the Earth sciences. In particular, he is regarded as the father of geodesy, and has made significant contributions to cartography, geography, geology and mineralogy.

Cartography
By the age of 22, he had written several short works, including a study of map projections, Cartography, which included a method for projecting a hemisphere on a plane.

Geodesy and Geography
At the age of 17, Biruni calculated the latitude of Kath, Khwarazm, using the maximum altitude of the Sun. Al-Biruni also solved a complex geodesic equation in order to accurately compute the Earth's circumference, which were close to modern values of the Earth's circumference. His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.

John J. O'Connor and Edmund F. Robertson write in the MacTutor History of Mathematics archive: ""Important contributions to geodesy and geography were also made by Biruni. He introduced techniques to measure the earth and distances on it using triangulation. He found the radius of the earth to be 6339.6 km, a value not obtained in the West until the 16th century. His Masudic canon contains a table giving the coordinates of six hundred places, almost all of which he had direct knowledge.""

In mathematical geography, Biruni, around 1025, was the first to describe a polar equi-azimuthal equidistant projection of the celestial sphere. He was also regarded as the most skilled when it came to mapping cities and measuring the distances between them, which he did for many cities in the Middle East and western Indian subcontinent. He often combined astronomical readings and mathematical equations, in order to develop methods of pin-pointing locations by recording degrees of latitude and longitude. He also developed similar techniques when it came to measuring the heights of mountains, depths of valleys, and expanse of the horizon, in The Chronology of the Ancient Nations.

He also discussed human geography and the planetary habitability of the Earth. He hypothesized that roughly a quarter of the Earth's surface is habitable by humans, and also argued that the shores of Asia and Europe were "seperated by a vast sea, too dark and dense to navigate and too risky to try" in reference to the Atlantic Ocean and Pacific Ocean.

Geology
Among his writings on geology, Biruni wrote the following on the geology of India: ""But if you see the soil of India with your own eyes and meditate on its nature, if you consider the rounded stones found in earth however deeply you dig, stones that are huge near the mountains and where the rivers have a violent current: stones that are of smaller size at a greater distance from the mountains and where the streams flow more slowly: stones that appear pulverised in the shape of sand where the streams begin to stagnate near their mouths and near the sea - if you consider all this you can scarcely help thinking that India was once a sea, which by degrees has been filled up by the alluvium of the streams.""

Mineralogy
Biruni's Kitab al-Jawahir (Book of Precious Stones) described minerals such as stones and metals in depth, and was regarded as the most complete book on mineralogy in his time. He conducted hundreds of experiments to gauge the accurate measurements of items he catalogued, and he often listed them by name in a number of different languages, including Arabic, Persian, Greek, Syriac, Hindi, Latin, and other languages. In the Book of Precious Stones, he catalogued each mineral by its color, odor, hardness, density and weight. The weights for many of these minerals he measured were correct to three decimal places of accuracy, and were almost as accurate as modern measurements for these minerals.

Scientific method
In early Islamic philosophy, Biruni discussed the philosophy of science and introduced an early scientific method in nearly every field of inquiry he studied. For example, in his treatise on mineralogy, Kitab al-Jamahir (Book of Precious Stones), he is "the most exact of experimental scientists", while in the introduction to his study of India, he declares that "to execute our project, it has not been possible to follow the geometric method" and develops comparative sociology as a scientific method in the field. He was also responsible for introducing the experimental method into mechanics, the first to conduct elaborate experiments related to astronomical phenomena, and a pioneer of experimental psychology.

Unlike his contemporary Avicenna's scientific method where "general and universal questions came first and led to experimental work", Biruni developed scientific methods where "universals came out of practical, experimental work" and "theories are formulated after discoveries." In his debate with Avicenna, Biruni made the first real distinction between a scientist and a philosopher, referring to Avicenna as a philosopher and considering himself to be a mathematical scientist (see Natural philosophy below).

Biruni's scientific method was similar to the modern scientific method in many ways, particularly his emphasis on repeated experimentation. Biruni was concerned with how to conceptualize and prevent both systematic errors and random errors, such as "errors caused by the use of small instruments and errors made by human observers." He argued that if instruments produce random errors because of their imperfections or idiosyncratic qualities, then multiple observations must be taken, analyzed qualitatively, and on this basis, arrive at a "common-sense single value for the constant sought", whether an arithmetic mean or a "reliable estimate."

Natural philosophy
Biruni and Avicenna (Ibn Sina), who are regarded as two of the greatest polymaths in Persian history, were both colleagues and knew each other since the turn of the millennium. Biruni later engaged in a written debate with Avicenna, with Biruni criticizing the Peripatetic school for its adherence to Aristotelian physics and natural philosophy, while Avicenna and his student Ahmad ibn 'Ali al-Ma'sumi respond to Biruni's criticisms in writing.

This debate has been preserved in a book entitled al-As'ila wal-Ajwiba (Questions and Answers), in which al-Biruni attacks Aristotle's theories on physics and cosmology, and questions almost all of the fundamental Aristotelian physical axioms. For example, he rejects the notion that heavenly bodies have an inherent nature and asserts that their "motion could very well be compulsory"; maintains that "there is no observable evidence that rules out the possibility of vacuum"; and states that there is no inherent reason why planetary orbits must be circular and cannot be elliptical. He also argues that "the metaphysical axioms on which philosophers build their physical theories do not constitute valid evidence for the mathematical astronomer." This marks the first real distinction between the vocations of the philosopher-metaphysician (which he labelled Aristotle and Avicenna as) and that of the mathematician-scientist (which al-Biruni viewed himself as). In contrast to the philosophers, the only evidence that al-Biruni considered reliable were either mathematical or empirical evidence, and his systematic application of rigorous mathematical reasoning later led to the mathematization of Islamic astronomy and the mathematization of nature.

Biruni began the debate by asking Avicenna eighteen questions, ten of which were criticisms of Aristotle's On the Heavens, with his first question criticizing the Aristotelian theory of gravity for denying the existence of levity or gravity in the celestial spheres, and the Aristotelian notion of circular motion being an innate property of the heavenly bodies. Biruni's second question criticizes Aristotle's over-reliance on more ancient views concerning the heavens, while the third criticizes the Aristotelian view that space has only six directions. The fourth question deals with the continuity and discontinuity of physical bodies, while the fifth criticizes the Peripatetic denial of the possibility of there existing another world completely different from the world known to them. In his sixth question, Biruni rejects Aristotle's view on the celestial spheres having circular orbits rather than elliptic orbits. In his seventh question, he rejects Aristotle's notion that the motion of the heavens begins from the right side and from the east, while his eighth question concerns Aristotle's view on the fire element being spherical. The ninth question concerns the movement of heat, and the tenth question concerns the transformation of elements.

The eleventh question concerns the burning of bodies by radiation reflecting off a flask filled with water, and the twelfth concerns the natural tendency of the classical elements in their upward and downward movements. The thirteenth question deals with vision, while the fourteenth concerns habitation on different parts of Earth. His fifteenth question asks how two opposite squares in a square divided into four can be tangential, while the sixteenth question concerns vacuum. His seventeenth question asks "if things expand upon heating and contract upon cooling, why does a flask filled with water break when water freezes in it?" His eighteenth and final question concerns the observable phenomenon of ice floating on water.

After Avicenna responded to the questions, Biruni was unsatisfied with some of the answers and wrote back commenting on them, after which Avicenna's student Ahmad ibn 'Ali al-Ma'sumi wrote back on behalf of Avicenna.

Astrophysics
In astrophysics and the celestial mechanics field of physics, Biruni described the Earth's gravitation as: ""The attraction of all things towards the centre of the earth.""

He also discovered that gravity exists within the heavenly bodies and celestial spheres, and he criticized Aristotle's views of them not having any levity or gravity and of circular motion being an innate property of the heavenly bodies.

Experimental mechanics
Biruni was the first to apply experimental scientific methods to mechanics, especially the fields of statics and dynamics, particularly for determining specific weights, such as those based on the theory of balances and weighing.

In the dynamics and kinematics fields of mechanics, Biruni was the first to realize that acceleration is connected with non-uniform motion, which is part of Newton's second law of motion.

In statics, Biruni measured the specific gravities of eighteen gemstones, and discovered that there is a correlation between the specific gravity of an object and the volume of water it displaces. He also introduced the method of checking tests during experiments, measured the weights of various liquids, and recorded the differences in weight between fresh water and salt water, and between hot water and cold water.

During his experiments, he invented the conical measure, in order to find the ratio between the weight of a substance in air and the weight of water displaced, and to accurately measure the specific weights of the gemstones and their corresponding metals, which are very close to modern measurements.

Optics
In optics, Biruni was one of the first, along with Ibn al-Haytham, to discover that the speed of light is finite. Biruni was also the first to discover that the speed of light is much faster than the speed of sound.

Anthropology
In the social sciences, Biruni has been described as "the first anthropologist". He wrote detailed comparative studies on the anthropology of peoples, religions and cultures in the Middle East, Mediterranean and South Asia. Biruni's anthropology of religion was only possible for a scholar deeply immersed in the lore of other nations. Biruni has also been praised by several scholars for his Islamic anthropology.

Al-Biruni developed a sophisticated methodology for his anthropological studies. For example, he wrote the following in the opening passages of his Indica:

""No one will deny that in questions of historic authenticity hearsay does not equal eyewitness; for in the latter the eye of the observer apprehends the substance of that which is observed, both in the time when and in the place where it exists, whilst hearsay has its peculiar drawbacks.""

He was also aware that there are limitations to eye-witness accounts:

""The object of eye-witness can only be actual momentary existence, whilst hearsay comprehends alike the present, the past and the future""

Experimental psychology
In Islamic psychology, al-Biruni was a pioneer of experimental psychology, for his use of empirical observation and experimentation in his discovery of the concept of reaction time, which he described as follows:

""Not only is every sensation attended by a corresponding change localized in the sense-organ, which demands a certain time, but also, between the stimulation of the organ and consciousness of the perception an interval of time must elapse, corresponding to the transmission of stimulus for some distance along the nerves.""

History
By the age of 27, in the year 1000, he had written a book called Chronology which referred to other works he had completed (now lost) that included one book about the astrolabe, one about the decimal system, four about astrology, and two about history.

In his Kitab fi Tahqiq ma li'l-Hind (Researches on India), he was the first to distinguish between the historical method and the scientific method. He also discussed more on his idea of history in another work, The Chronology of the Ancient Nations.

Indology
Until the 10th century, history most often meant political and military history, but this was not so with Biruni (973-1048). In his Kitab fi Tahqiq ma li'l-Hind (Researches on India), he did not record political and military history in any detail, but wrote more on India's cultural, scientific, social and religious history. Biruni is now regarded as the father of Indology.

Islamic theology
In theology, Biruni was a follower of the orthodox Ash'ari school of Sunni Islamic theology. He is also thought to have associated with Maturidi theologians and was sympathetic towards Ismailis. He was critical, however, of Mutazili theologians, particularly al-Jahiz and Zurqan, and he also criticized Muhammad ibn Zakarīya Rāzi's sympathy for Manichaeanism.

Biruni also assigned to the Qur'an a separate and autonomous realm of its own and held that:

""[the Qur'an] does not interfere in the business of science nor does it infringe on the realm of science.""

He also argued that the possesion of intellect makes humans superior to animals and that God "placed humans as stewards over Earth and other terrestrial life-forms." He also considered hearing and sight to be the two most important senses, as they allow humans to "observe the signs of God's divine wisdom in his creations" and "receive the word of God and his command."

Comparative religion
In religious education, Biruni was a pioneer of comparative religion. According to Arthur Jeffery, "It is rare until modern times to find so fair and unprejudiced a statement of the views of other religions, so earnest an attempt to study them in the best sources, and such care to find a method which for this branch of study would be both rigorous and just."

In the introduction to his Indica, Biruni himself writes that his intent behind the work was to engage dialogue between Islam and the Indian religions, particularly Hinduism as well as Buddhism. He writes:

""Abu-Sahl at-­Tiflisi incited me to write down what I know about the Hindus as a help to those who want to discuss religious questions with them, and as a repertory of information to those who want to associate with them. We think now that what we have related in this book will be sufficient for anyone who wants to converse with the Hindus, and to discuss with them questions of religion, science or literature, on the very basis of their own civilisation.""

Biruni was aware that statements about a religion would be open to criticism by its adherents, and insisted that a scholar should follow the requirements of a strictly scientific method. According to William Montgomery Watt, Biruni "is admirably objective and unprejudiced in his presentation of facts" but "selects facts in such a way that he makes a strong case for holding that there is a certain unity in the religious experience of the peoples he considers, even though he does not appear to formulate this view explicitly." Biruni argued that Hinduism was a monotheistic faith like Islam, and in order to justify this assertion, he quotes Hindu texts and argues that the worship of idols is "exclusively a characteristic of the common people, with which the educated have nothing to do." He writes:

""The educated among the Hindus abhor anthropomorphisms of this kind, but the crowd and the members of the single sects use them most extensively.""

""The Hindus believe with regard to God that he is one, eternal, without beginning and end, acting by free-will, almighty, all-wise, living, giving life, ruling, preserving; one who in his sovereignty is unique, beyond all likeness and unlikeness, and that he does not resemble anything nor does anything resemble him.""

Biruni argued that the worship of idols "is due to a kind of confusion or corruption." He writes:

""The physical images are monuments in honour of certain much venerated persons, prophets, sages, angels, destined to keep alive their memory when they are absent or dead, to create for them a lasting place of grateful veneration in the hearts of men when they die.""

According to Watt, Biruni "goes on to maintain that in the course of generations the origin of the veneration of the images is forgotten, and further that the ancient legislators, seeing that the Veneration of images is advantageous, made it obligatory for the ordinary. He mentions the view of some people that, before God sent Prophets, all mankind were idol-worshippers, but he apparently does not presumably held that, apart from the messages transmitted by prophets, men could know the existence and unity of God by rational methods of philosophy." Biruni argued that "the Hindus, no less than the Greeks, have philosophers who are believers in monotheism."

Other comparatisons between Islamic theology and Indian theology include the following comparison between the Qur'an and the Indian religious scriptures in the "On the Configuration of the Heavens and the Earth According to [Indian] astrologers" chapter of the Indica:

""[The views of Indian astrologers] have developed in a way which is different from those of our [Muslim] fellows; this is because unlike the scriptures revealed before it, the Qur'an does not articulate on this subject [of astronomy], or any other [field of] necessary [knowledge] any assertion that would require erratic interpretations in order to harmonize it with that which is known by necessity.""

""[In contrast, the religious and transmitted books of the Indians do indeed speak] of the configuration of the universe in a way which contradicts the truth which is known to their own astrologers.""

Al-Biruni also had an interest in studying Hermeticism and often criticized its religious views. He showed an interest in comparative religion, comparing Islam with pre-Islamic religions, and was willing to accept certain elements of pre-Islamic wisdom which would conform with his understanding of the Islamic spirit.

Biomedical sciences
In the biomedical sciences, al-Biruni's Kitab al-Saidana fi al-Tibb was an extensive medical and pharmacological encyclopedia which synthesized Islamic medicine with Indian medicine. His medical investigations included one of the earliest descriptions on Siamese twins. The Kitab-al-Saidana was also a materia medica which was celebrated for its in-depth botanical studies of minerals and herbs. It was the earliest to describe the eating of several fungi, including truffles, which are a type of hypogeous fungi. The earliest documented description of khat also dates back to the Kitab al-Saidana, in which al-Bīrūnī wrote that khat is:

""a commodity from Turkestan. It is sour to taste and slenderly made in the manner of batan-alu. But qat is reddish with a slight blackish tinge. It is believed that batan-alu is red, coolant, relieves biliousness, and is a refrigerant for the stomach and the liver.""

Chemistry
Along with al-Kindi and Avicenna, Biruni was one of the first chemists to reject the theory of the transmuation of metals supported by some alchemists.

Law
In Islamic law and jurisprudence, Biruni understood natural law as the law of the jungle. He argued that the antagonism between human beings can only be overcome through a divine law, which he believed to have been sent through the prophets of Islam.

Linguistics
In linguistics, al-Biruni could speak, read and write in a number of different languages, including Persian, Arabic, Greek, Hebrew and Sanskrit. He was also conversant in Syriac and Turkish, and could also speak some Hindi and Latin.

Mathematics
He made significant contributions to mathematics, especially in the fields of theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, and the development of Archimedes' theorems.