Muḥammad ibn Mūsā al-Khwārizmī was a Persian^[1] Muslim mathematician, astronomer, astrologer and geographer. He was born around 780 in Khwārizm^[2] (now Khiva, Uzbekistan) and died around 850. He worked most of his life as a scholar in the House of Wisdom in Baghdad.

His Algebra was the first book on the systematic solution of linear and quadratic equations. Consequently he is considered to be the father of algebra,^[3] a title he shares with Diophantus. Latin translations of his Arithmetic, on the Indian numerals, introduced the decimal positional number system to the Western world in the 12th century.^[4] He revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology.

His contributions not only made a great impact on mathematics, but on language as well. The word algebra is derived from al-jabr, one of the two operations used to solve quadratic equations, as described in his book. The words algorism and algorithm stem from algoritmi, the Latinization of his name.^[5] His name is also the origin of the Spanish word guarismo^[6] and of the Portuguese word algarismo, both meaning digit.

Biography

Few details about al-Khwārizmī's life are known; it is not even certain where he was born. His name indicates he might have come from Khwarezm (Khiva) in the Khorasan province of the Abbasid empire (now Xorazm Province of Uzbekistan).

His kunya is given as either Abū ʿAbd Allāh (Arabic: أبو عبد الله) or Abū Jaʿfar (أبو جعفر in Arabic).^[7]

The historian al-Tabari gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (Arabic: محمد بن موسى الخوارزميّ المجوسيّ القطربّليّ). The epithet al-Qutrubbulli indicates he might instead have came from Qutrubbull, a small town near Baghdad. Regarding al-Khwārizmī's religion, Toomer writes:

Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the old Zoroastrian religion. This would still have been possible at that time for a man of Iranian origin, but the pious preface to al-Khwārizmī's Algebra shows that he was an orthodox Muslim, so al-Ṭabarī's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.^[8]

In Ibn al-Nadīm's Kitāb al-Fihrist we find a short biography on al-Khwārizmī, together with a list of the books he wrote^{[citation needed]}. Al-Khwārizmī accomplished most of his work in the period between 813 and 833. After the Islamic conquest of Persia, Baghdad became the centre of scientific studies and trade, and many merchants and scientists, from as far as China and India traveled to this city--as such apparently so did Al-Khwārizmī. He worked in Baghdad as a scholar at the House of Wisdom established by Caliph al-Maʾmūn, where he studied the sciences and mathematics, which included the translation of Greek and Sanskrit scientific manuscripts.

Contributions

His major contributions to mathematics, astronomy, astrology, geography and cartography provided foundations for later and even more widespread innovation in algebra, trigonometry, and his other areas of interest. His systematic and logical approach to solving linear and quadratic equations gave shape to the discipline of algebra, a word that is derived from the name of his 830 book on the subject, al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala (Arabic الكتاب المختصر في حساب الجبر والمقابلة) or: "The Compendious Book on Calculation by Completion and Balancing". The book was first translated into Latin in the twelfth century.

His book On the Calculation with Hindu Numerals written about 825, was principally responsible for the diffusion of the Indian system of numeration in the Middle-East and then Europe. This book also translated into Latin in the twelfth century, as Algoritmi de numero Indorum. From the name of the author, rendered in Latin as algoritmi, originated the term algorithm.

Some of his contributions were based on earlier Persian and Babylonian Astronomy, Indian numbers, and Greek sources.

Al-Khwārizmī systematized and corrected Ptolemy's data in geography as regards to Africa and the Middle east. Another major book was his Kitab surat al-ard ("The Image of the Earth"; translated as Geography), which presented the coordinates of localities in the known world based, ultimately, on those in the Geography of Ptolemy but with improved values for the length of the Mediterranean Sea and the location of cities in Asia and Africa.

He also assisted in the construction of a world map for the caliph al-Ma'mun and participated in a project to determine the circumference of the Earth, supervising the work of 70 geographers to create the map of the then "known world".^[9]

When his work was copied and transferred to Europe through Latin translations, it had a profound impact on the advancement of basic mathematics in Europe. He also wrote on mechanical devices like the astrolabe and sundial.

Algebra

Main article: The Compendious Book on Calculation by Completion and Balancing

al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة “The Compendious Book on Calculation by Completion and Balancing”) is a mathematical book written approximately 830 CE.

The word algebra is derived from the name of one of the basic operations with equations (al-jabr) described in this book. The book was translated in Latin as Liber algebrae et almucabala by Robert of Chester (Segovia, 1145)^[10] hence "algebra", and also by Gerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept is Cambridge.^[11]

Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers)

• squares equal roots (ax² = bx)
• squares equal number (ax² = c)
• roots equal number (bx = c)
• squares and roots equal number (ax² + bx = c)
• squares and number equal roots (ax² + c = bx)
• roots and number equal squares (bx + c = ax²)

by dividing out the coefficient of the square and using the two operations al-ǧabr (Arabic: الجبر “restoring” or “completion”) and al-muqābala ("balancing"). Al-ǧabr is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, x² = 40x - 4x² is reduced to 5x² = 40x. Al-muqābala is the process of bringing quantities of the same type to the same side of the equation. For example, x²+14 = x+5 is reduced to x²+9 = x.

Several authors have published texts under the name of Kitāb al-ǧabr wa-l-muqābala, including Abū Ḥanīfa al-Dīnawarī, Abū Kāmil (Rasāla fi al-ǧabr wa-al-muqābala), Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, Ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr (author uncertain), and Šarafaddīn al-Ṭūsī.

Arithmetic

Al-Khwārizmī's second major work was on the subject of arithmetic, which survived in a Latin translation but was lost in the original Arabic. The translation was most likely done in the 12th century by Adelard of Bath, who had also translated the astronomical tables in 1126.

The Latin manuscripts are untitled, but are commonly referred to by the first two words with which they start: Dixit algorizmi ("So said al-Khwārizmī"), or Algoritmi de numero Indorum ("al-Khwārizmī on the Hindu Art of Reckoning"), a name given to the work by Baldassarre Boncompagni in 1857. The original Arabic title was possibly Kitāb al-Jamʿ wa-l-tafrīq bi-ḥisāb al-Hind^[12] ("The Book of Addition and Subtraction According to the Hindu Calculation")^[13]

Geography

Al-Khwārizmī's third major work is his Kitāb ṣūrat al-Arḍ (Arabic: كتاب صورة الأرض "Book on the appearance of the Earth" or "The image of the Earth" translated as Geography), which was finished in 833. It is a revised and completed version of Ptolemy's Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.^[14]

There is only one surviving copy of Kitāb ṣūrat al-Arḍ, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid. The complete title translates as Book of the appearance of the Earth, with its cities, mountains, seas, all the islands and rivers, written by Abu Ja'far Muhammad ibn Musa al-Khwārizmī, according to the geographical treatise written by Ptolemy the Claudian.^[15]

The book opens with the list of latitudes and longitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each weather zone, by order of longitude. As Paul Gallez points out, this excellent system allows us to deduce many latitudes and longitudes where the only document in our possession is in such a bad condition as to make it practically illegible.

Neither the Arabic copy nor the Latin translation include the map of the world itself, however Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduces them from the context where they were not legible. He transferred the points onto graph paper and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He then does the same for the rivers and towns.^[16]

Astronomy

Al-Khwārizmī's Zīj al-sindhind (Arabic: زيج "astronomical tables") is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values. This is one of many Arabic zijes based on the Indian astronomical methods known as the sindhind.^[17]

The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslama al-Majrīṭī (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (January 26, 1126).^[18] The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Bibliotheca Nacional (Madrid) and the Bodleian Library (Oxford).

Jewish calendar

Al-Khwārizmī wrote several other works including a treatise on the Hebrew calendar (Risāla fi istikhrāj taʾrīkh al-yahūd "Extraction of the Jewish Era"). It describes the 19-year intercalation cycle, the rules for determining on what day of the week the first day of the month Tishrī shall fall; calculates the interval between the Jewish era (creation of Adam) and the Seleucid era; and gives rules for determining the mean longitude of the sun and the moon using the Jewish calendar. Similar material is found in the works of al-Bīrūnī and Maimonides.

Other works

Several Arabic manuscripts in Berlin, Istanbul, Taschkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials, which is mentioned in the Fihirst. Other papers, such as one on the determination of the direction of Mecca, are on the spherical astronomy.

Two texts deserve special interest on the morning width (Maʿrifat saʿat al-mashriq fī kull balad) and the determination of the azimuth from a height (Maʿrifat al-samt min qibal al-irtifāʿ).

He also wrote two books on using and constructing astrolabes. Ibn al-Nadim in his Kitab al-Fihrist (an index of Arabic books) also mentions Kitāb ar-Ruḵāma(t) (the book on sundials) and Kitab al-Tarikh (the book of history) but the two have been lost.

Pascal, Blaise (1623-1662)

French mathematician, philosopher, and religious figure. He studied the region above the mercury in a barometer, Eric Weisstein's World of Physics maintaining that it was a vacuum. In his investigations of the barometer, Eric Weisstein's World of Physics he found that the height to which the mercury rose was the same regardless of shape. Based on his double vacuum experiment, he formulated Pascal's principle, Eric Weisstein's World of Physics which states that the pressure is constant throughout a static fluid. He performed an experiment in which he convinced his brother-in-law to climb the Puy-de-Dôme Mountain in France. He found the height of the mercury dropped with altitude, indicating pressure decreases with altitude.

Pascal also designed and built mechanical adding machines, and incorporated a company in 1649 to produce and market them. Unfortunately, the machines were rather highly priced and had reliability problems. Only seven of Pascal's devices survive today.

Pascal suffered from serious health problems, and spent most of his final years writing on religious philosophy.

Buffon, Georges, Comte de (1707-1788)

French naturalist who became keeper of the French botanical gardens. At first, he believed in Leibniz's Law of Continuity and disagreed with Linnaeus's system of nomenclature, maintaining that a species was an artificial category. He changed his mind, however, when he found that the offspring of some plant hybridizations were sterile. He proposed identifying species through their reproductive histories, with two animals belonging to the same species if they can produce fertile offspring. His monumental life's work, Historie Naturelle (Natural History) (1749), ran to 36 volumes and tried to show the continuity of nature. Buffon had no concept of evolution, believing that species are fixed. Buffon was a believer in Vitalism, Eric Weisstein's World of Chemistry stating that living organisms possessed "matiére vive" (living matter). He also speculated on the origin of the Earth, Eric Weisstein's World of Astronomy suggesting that it might have been created by the collision of a comet Eric Weisstein's World of Astronomy with the Sun. Eric Weisstein's World of Astronomy Based on the cooling rate of iron, he calculated that the age of the Earth Eric Weisstein's World of Astronomy was 75,000 years. This proclamation was condemned by the Catholic Church in France, who burned Buffon's books.

Buffon also proposed the Buffon's needle problem, Eric Weisstein's World of Math which asks the probability that a needle of length l will fall on a line when a piece of paper is ruled with parallel lines a distance d apart.

Einstein, Albert (1879-1955)

German-American physicist who, in 1905, published three papers, each of which had a profound effect on the development of physics. In one paper, he proposed the theory of special relativity, Eric Weisstein's World of Physics which provides a correct description for particles traveling at high speeds. The two postulates of the special theory of relativity were that the speed of light Eric Weisstein's World of Physics in a vacuum is constant and that the laws of physics are the same for all inertial reference frames. Einstein did know about the Michelson-Morley experiment Eric Weisstein's World of Physics null result, but was not familiar with Lorentz's work after 1895, so he reinvented the Lorentz transformation Eric Weisstein's World of Math for himself (Pais 1982, p. 133).

While special relativity required a modification of the laws of mechanics, the Maxwell equations Eric Weisstein's World of Physics were found to already satisfy the requirements of special relativity. Eric Weisstein's World of Physics Using special relativity, Eric Weisstein's World of Physics Einstein derived the equivalence of rest mass Eric Weisstein's World of Physics m0 and energy Eric Weisstein's World of Physics E, expressible as E2-p2c2 = m02c4, where c is the speed of light Eric Weisstein's World of Physics and p is the (relativistic) momentum Eric Weisstein's World of Physics. When relativistic mass Eric Weisstein's World of Physics is used instead , the equation reduces to the famous E = mc2.

In another 1905 paper, Einstein also explained the photoelectric effect Eric Weisstein's World of Physics by hypothesizing that light consisted of particles (called photons Eric Weisstein's World of Physics) with energy Eric Weisstein's World of Physics equal to , where h is a constant known as Planck's constant Eric Weisstein's World of Physics (named after the physicist Max Planck) and is the frequency Eric Weisstein's World of Physics of the photon. This represented an extension of Planck's quantization to light. The equation Einstein derived was verified experimentally by Millikan in 1916.

Also in 1905, Einstein provided an explanation of Brownian motion Eric Weisstein's World of Physics using kinetic theory, Eric Weisstein's World of Physics stating that it was caused by random collisions of molecules. Einstein furthermore derived an equation stating that a suspension Eric Weisstein's World of Physics of small particles should arrange itself in an exponentially decreasing manner from bottom up. Using Einstein's equations for Brownian motion Eric Weisstein's World of Physics and the distribution of particles, Perrin was able to experimentally measure the value of Boltzmann's constant. Eric Weisstein's World of Physics

Einstein subsequently developed general relativity, Eric Weisstein's World of Physics which postulated that uniform acceleration Eric Weisstein's World of Physics and a gravitational field were equivalent, a statement known as the equivalence principle of gravitation. Eric Weisstein's World of Physics It interpreted gravity Eric Weisstein's World of Physics as a warping of space-time. The general theory of relativity made extensive use of Ricci-Curbastro's tensor calculus. Eric Weisstein's World of Math Einstein investigated cosmological Eric Weisstein's World of Physics modeling, but found that the general theory of relativity would not satisfy the conditions of homogeneity, isotropy, and staticity unless an additional "cosmological constant Eric Weisstein's World of Physics" was added.

Einstein spent the latter portion of his life in an unsuccessful attempt to create a unified theory which would explain all known forces in nature as manifestions of a single fundamental force. Eric Weisstein's World of Physics Einstein's theories were highly controversial for years after he proposed them. In a recommendation for Einstein's membership in the Prussian Academy of Science, the sponsors wrote "In sum, one can say that there is hardly one among the great problems in which modern physics is so rich to which Einstein has not made a remarkable contribution. That he may sometimes have missed the targeting his speculations, as, for example, in his hypothesis of light-quanta, cannot really be held too much against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk" (Pais 1982, p. 382).

A recent study of Einstein's preserved brain (for details, see Regis 1991) has discovered that the inferior parietal region--the part thought to be related to mathematical reasoning--was 15% wider than normal (Witelson et al. 1999). In additional, the groove normally running from the front to the back did not extend all the way in Einstein's brain. However, it is unclear what the true significance of these anatomical anomalies are in connection with Einstein's scientific creativity.

Einstein was attacked by some with anti-Jewish leanings. When a pamphlet was published entitled 100 Authors Against Einstein, Einstein retorted "If I were wrong, one would be enough." Some famous Einstein quotes about God include

"Whoever undertakes to set himself up as judge in the field of Truth and Knowledge is shipwrecked by the laughter of the gods."
"I do not believe in immortality of the individual, and I consider ethics to be an exclusively human concern with no superhuman authority behind it."
"I want to know how God created this world. I am not interested in this or that phenomenon, Eric Weisstein's World of Physics in the spectrum of this or that element. I want to know His thoughts, the rest are details."
"God is subtle, but he is not malicious."
"God does not play dice with the world."
"Science without religion is lame, religion without science is blind" (Pais 1982, p. 319).

Einstein also had many insightful things to say about scientific discovery.

"Do not pride yourself on the few great men who, over the centuries, have been born on your earth through no merit of yours. Reflect, rather, on how you treated them at the time and how you have followed their teachings."
"Innovation is not the product of logical thought, even though the final product is tied to a logical structure" (Pais 1982, p. 131).
"Nature shows us only the tail of the lion. But I do not doubt that the lion belongs to it even though he cannot at once reveal himself because of his enormous size" (Pais 1982, p. 235).
"Above stands the marble smile of implacable Nature which has endowed us more with longing than with intellectual capacity" (Pais 1982, p. 343).
"One has been endowed with just enough intelligence to be able to see clearly how utterly inadequate that intelligence is when confronted with what exists. If such humility could be conveyed to everybody, the world of human activities would be more appealing."
"Politics is a pendulum whose swings between anarchy and tyranny are fueled by perennially rejuvenated illusions."
"It is a mistake often made in this country to measure things by the amount of money they cost."
"Do not worry about your difficulties in mathematics; I can assure you that mine are still greater."
"The more success the quantum theory has, the sillier it looks. How nonphysicists would scoff if they were able to follow the odd course of developments!" (Pais 1982, p. 399).
"A practical profession is a salvation for a man of my type; an academic career compels a young man to scientific production, and only strong characters can resist the temptation of superficial analysis."

Riemann, Bernhard (1826-1866)

German mathematician who studied mathematics under Gauss and physics under Wilhelm Weber. Riemann did important work in geometry, Eric Weisstein's World of Math complex analysis, Eric Weisstein's World of Math and mathematical physics. He was also a friend of Dedekind, who was later Riemann's biographer. In his thesis, Riemann urged a global view of geometry Eric Weisstein's World of Math as a study of manifolds Eric Weisstein's World of Math of any number of dimensions Eric Weisstein's World of Math in any kind of space. He defined space by a metric Eric Weisstein's World of Math

Riemann's work laid the foundations on which general relativity Eric Weisstein's World of Physics was built. He investigated the Riemann zeta function, Eric Weisstein's World of Math about which he stated the famous (and still not completely proven) Riemann hypothesis. Eric Weisstein's World of Math He also refined the definition of the integral, Eric Weisstein's World of Math invented Riemann surfaces, Eric Weisstein's World of Math and invented a function which was discontinuous at an infinite number of points in an interval, but the integral Eric Weisstein's World of Math of which still existed (Segal 1978). In Über eine Frage der Wärmeleitung, Riemann developed the theory of quadratic forms. Eric Weisstein's World of Math

Riemann maintained that the transmission of electricity Eric Weisstein's World of Physics not was instantaneous, but occurred through the luminiferous ether Eric Weisstein's World of Physics at the speed of light. Eric Weisstein's World of Physics

Riemann died of tuberculosis at the tragically early age 39.

Pythagoras of Samos (ca. 560-ca. 480 BC)

Greek philosopher and mathematician who founded the mystic Pythagorean cult. The cult he founded was devoted to the study of numbers, which the Pythagoreans saw as concrete, material objects. They studied figurate numbers, Eric Weisstein's World of Math defining them as triangular numbers, Eric Weisstein's World of Math pentagonal numbers, Eric Weisstein's World of Math hexagonal numbers, Eric Weisstein's World of Math etc., based on the patterns that numbers of regularly spaced dots formed (Boyer 1968, p. 59-61). Pythagoras's biographer Proclus ascribed two specific mathematical discoveries to Pythagoras: construction of the regular solids (known today as the Platonic solids Eric Weisstein's World of Math), and the theory of proportionals.

Pythagoras also investigated the ratios of lengths corresponding to musical harmonies, undertook studies in number theory, and developed methods of geometric proof. Among the results attributed to Pythagoras and his followers is the proof that the number (Pythagoras's constant Eric Weisstein's World of Math) is irrational, Eric Weisstein's World of Math usually attributed to Hippasus.

Pythagoras developed a modern theory of vision much simpler than that of Plato. This theory maintained that light is emitted from luminous bodies, can suffer reflections, and causes the sensation of sight when it enters the eyes. He was the first Greek to realize that the morning star and evening star were both the planet Venus. Eric Weisstein's World of Astronomy Pythagoras postulated that the Earth Eric Weisstein's World of Astronomy was spherical, and added more crystalline spheres to Anaximander's model, one for each planet, Eric Weisstein's World of Astronomy to account for the motions of the various planets.

In keeping with the assumed magical properties of the number ten, some of the Pythagoreans, led by Philolaus, added a tenth "wanderer" and proposed that there existed a counter-earth which, together with the earth and other "planets," orbited a central fire. Pythagoras believed that the planets produced sounds while tracing out their orbits, producing the "harmony of the spheres." While much of their studies were sheer mysticism, the Pythagoreans were the first to mathematicize the universe.

Fibonacci, Leonardo da Pisa (ca. 1170-ca. 1240)

Italian mathematician who was the first great Western mathematician after the decline of Greek science. The son of a merchant, Fibonacci drew the motivation to mathematical inquiry from his commercial trips to the the Orient. It was somewhere between Barbary (Maghreb) and Constantinople (now Istanbul) that he got acquainted with the Hindu-Arabic number system and discovered its enormous practical advantages compared to the Roman numerals, Eric Weisstein's World of Math which were still current in Western Europe.

Performing even the simplest arithmetical operations with a non-positional notation was a difficult endeavor: for this task the merchants were forced to resort to the abacus, a device where the numbers were represented by moving balls. Fibonacci exposed the new alternate computing method--based on written algorithms rather than on counting objects--in his Liber Abaci, first issued in 1202. The book began with a presentation of what he called the ten "Indian figures" (0, 1, 2, ..., 9). It was intended as an algebra manual for commercial use, and explained the arithmetical rules using numerical examples derived, for example, from measure and currency conversion, which were translated into proportions and solved by multiplication (rule of three). The so-called Fibonacci sequence Eric Weisstein's World of Math arose in this book from a concrete question concerning the growth of a rabbit population. Geometric progressions Eric Weisstein's World of Math also appeared in problems related to legacy and interest.

The treatise Practica Geometriae, published in 1225, is mainly inspired by Greek mathematics; it contains theorems from Euclid's Elements and also Heron's formula Eric Weisstein's World of Math for the area Eric Weisstein's World of Math of a triangle. Eric Weisstein's World of Math

Fibonacci distinguished himself in the mathematical competitions proposed at the court of Emperor Frederick II of Hohenstaufen, King of the Two Sicilies, who had his royal seat in Palermo. His striking ability in solving algebraic equations of higher degree clearly emerges from his works entitled Liber Quadratorum and Flos, both of which appeared in 1225. The first contains formulas and equations involving perfect squares, the second owes its fame to the irrational solution of a cubic equation, which Fibonacci determined with an accuracy of 10-9. Most of his solving techniques seem to be based on the algebraic works of al-Khwarizmi.

Fibonacci initiated the tradition of the maestri d'abaco, experts in practical algebra and arithmetic, who flourished in Italy during the 14th century, and can be considered as the forerunners of Cardano, Tartaglia, and Ferrari.