# Greatest Mathematicians born between 400 and 1559 A.D.

Biographies of the greatest mathematicians are in separate files by birth year:

 Born before 400 Born betw. 400 & 1559 (this page) Born betw. 1560 & 1699 Born betw. 1700 & 1799 Born betw. 1800 & 1859 Born betw. 1860 & 1975 List of Greatest Mathematicians

Decimal system -- from India? China?? Persia???

Laplace called the decimal system "a profound and important idea [given by India] which appears so simple to us now that we ignore its true merit ... in the first rank of useful inventions [but] it escaped the genius of Archimedes and Apollonius." But even after Fibonacci introduced the system to Europe, it was another 400 years before it came into common use.

Ancient Greeks, by the way, did not use the unwieldy Roman numerals, but rather used 27 symbols, denoting 1 to 9, 10 to 90, and 100 to 900. Unlike our system, with ten digits separate from the alphabet, the 27 Greek number symbols were the same as their alphabet's letters; this might have hindered the development of "syncopated" notation. The most ancient Hindu records did not use the ten digits of Aryabhata, but rather a system similar to that of the ancient Greeks, suggesting that China, and not India, may indeed be the "ultimate" source of the modern decimal system.

The Chinese used a form of decimal abacus as early as 3000 BC; if it doesn't qualify, by itself, as a "decimal system" then pictorial depictions of its numbers would. Yet for thousands of years after its abacus, China had no zero symbol other than plain space; and apparently didn't have one until after the Hindus. Ancient Persians and Mayans did have place-value notation with zero symbols, but neither qualify as inventing a base-10 decimal system: Persia used the base-60 Babylonian system; Mayans used base-20. (Another difference is that the Hindus had nine distinct digit symbols to go with their zero, while earlier place-value systems built up from just two symbols: 1 and either 5 or 10.) The Old Kingdom Egyptians did use a base-ten system, but it was similar to that of Greece and Vedic India: 1, 10, 100 were depicted as separate symbols.

Conclusion: The decimal place-value system with zero symbol seems to be an obvious invention that in fact was very hard to invent. If you insist on a single winner then India might be it. But China, Babylonia, Persia and even the Mayans deserve Honorable Mention!

Indian mathematicians excelled for thousands of years, and eventually even developed advanced techniques like Taylor series before Europeans did, but they are denied credit because of Western ascendancy. Among the Hindu mathematicians, Aryabhata (called Arjehir by Arabs) may be most famous.

While Europe was in its early "Dark Age," Aryabhata advanced arithmetic, algebra, elementary analysis, and especially plane and spherical trigonometry, using the decimal system. Aryabhata is sometimes called the "Father of Algebra" instead of al-Khowârizmi (who himself cites the work of Aryabhata). His most famous accomplishment in mathematics was the Aryabhata Algorithm (connected to continued fractions) for solving Diophantine equations. Aryabhata made several important discoveries in astronomy, e.g. the nature of moonlight, and concept of sidereal year; his estimate of the Earth's circumference was more accurate than any achieved in ancient Greece. He was among the very few ancient scholars who realized the Earth rotated daily on an axis; claims that he also espoused heliocentric orbits are controversial, but may be confirmed by the writings of al-Biruni. Aryabhata is said to have introduced the constant e. He used π ≈ 3.1416; it is unclear whether he discovered this independently or borrowed it from Liu Hui of China. Although it was first discovered by Nicomachus three centuries earlier, Aryabhata is famous for the identity
Σ (k3) = (Σ k)2

Some of Aryabhata's achievements, e.g. an excellent approximation to the sine function, are known only from the writings of Bhaskara I, who wrote: "Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world."

No one person gets unique credit for the invention of the decimal system but Brahmagupta's textbook Brahmasphutasiddhanta was very influential, and is sometimes considered the first textbook "to treat zero as a number in its own right." It also treated negative numbers. (Others claim these were first seen 800 years earlier in Chang Tshang's Chinese text and were implicit in what survives of earlier Hindu works, but Brahmagupta's text discussed them lucidly.) Along with Diophantus, Brahmagupta was also among the first to express equations with symbols rather than words.

Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala') made great advances in arithmetic, algebra, numeric analysis, and geometry. Several theorems bear his name, including the formula for the area of a cyclic quadrilateral:
16 A2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)
Another famous Brahmagupta theorem dealing with such quadrilaterals can be phrased "In a circle, if the chords AB and CD are perpendicular and intersect at E, then the line from E which bisects AC will be perpendicular to BD." He also began the study of rational quadrilaterals which Kummer would eventually complete. Proving Brahmagupta's theorems are good challenges even today.

In addition to his famous writings on practical mathematics and his ingenious theorems of geometry, Brahmagupta solved the general quadratic equation, and worked on number theory problems. He was first to find a general solution to the simplest Diophantine form. His work on Pell's equations has been called "brilliant" and "marvelous." He proved the Brahmagupta-Fibonacci Identity (the set of sums of two squares is closed under multiplication), and generalized it further to assist in the Pell's solutions. He applied mathematics to astronomy, predicting eclipses, etc.

The astronomer Bháskara I, who takes the suffix "I" to distinguish him from the more famous Bháskara who lived five centuries later, made key advances to the positional decimal number notation, and was the first known to use the zero symbol. He preserved some of the teachings of Aryabhata which would otherwise have been lost; these include a famous formula giving an excellent approximation to the sin function, as well as, probably, the zero symbol itself.

Among other original contributions to mathematics, Bháskara I was first to state Wilson's Theorem (which should perhaps be called Bháskara's Conjecture):
(n-1)! ≡ -1 (mod n) if and only if n is prime
Bháskara's Conjecture was rediscovered by Alhazen, Fibonacci, Leibniz and Wilson. The "only if" is easy but the difficult "if" part was finally proved by Lagrange in 1771. Since Lagrange has so many other Theorems named after him, Bháskara's Conjecture is always called "Wilson's Theorem."

Al-Khowârizmi (aka Mahomet ibn Moses) was a Persian who worked as a mathematician, astronomer and geographer early in the Golden Age of Islamic science. He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography. He wrote the book Al-Jabr, which demonstrated simple algebra and geometry, and several other influential books. Unlike Diophantus' work, which dealt in specific examples, Al-Khowârizmi was the first algebra text to present general methods; he is often called the "Father of Algebra." (Diophantus did, however, use superior "syncopated" notation.) The word algorithm is borrowed from Al-Khowârizmi's name, and algebra is taken from the name of his book. He also coined the word cipher, which became English zero (although this was just a translation from the Sanskrit word for zero introduced by Aryabhata). He was an essential pioneer for Islamic science, and for the many Arab and Persian mathematicians who followed; and hence also for Europe's eventual Renaissance which was heavily dependent on Islamic teachings. Al-Khowârizmi's texts on algebra and decimal arithmetic are considered to be among the most influential writings ever.

Al-Kindi (called Alkindus or Achindus in the West) wrote on diverse philosophical subjects, physics, optics, astronomy, music, psychology, medicine, chemistry, and more. He invented pharmaceutical methods, perfumes, and distilling of alcohol. In mathematics, he popularized the use of the decimal system, developed spherical geometry, wrote on many other topics and was a pioneer of cryptography (code-breaking). (His work with code-breaking also made him a pioneer in basic concepts of probability.) (Al-Kindi, called The Arab Philosopher, can not be considered among the greatest of mathematicians, but was one of the most influential general scientists between Aristotle and da Vinci.) He appears on Cardano's List of 12 Greatest Geniuses. (Al-Khowârizmi and Jabir ibn Aflah are the other Islamic scientists on that list.)

Thabit produced important books in philosophy (including perhaps the famous mystic work De Imaginibus), medicine, mechanics, astronomy, and especially several mathematical fields: analysis, non-Euclidean geometry, trigonometry, arithmetic, number theory. As well as being an original thinker, Thabit was a key translator of ancient Greek writings; he translated Archimedes' otherwise-lost Book of Lemmas and applied one of its methods to construct a regular heptagon. He developed an important new cosmology superior to Ptolemy's (and which, though it was not heliocentric, may have inspired Copernicus). He was perhaps the first great mathematician to take the important step of emphasizing real numbers rather than either rational numbers or geometric sizes. He worked in plane and spherical trigonometry, and with cubic equations. He was an early practitioner of calculus and seems to have been first to take the integral of √x. Like Archimedes, he was able to calculate the area of an ellipse, and to calculate the volume of a paraboloid. He produced an elegant generalization of the Pythagorean Theorem:
AC 2 + BC 2 = AB (AR + BS)
(Here the triangle ABC is not a right triangle, but R and S are located on AB to give the equal angles ACB = ARC = BSC.) Thabit also worked in number theory where he is especially famous for his theorem about amicable numbers. (Thabit ibn Qurra's Theorem was rediscovered by Fermat and Descartes, and later generalized by Euler.) While many of his discoveries in geometry, plane and spherical trigonometry, and analysis (parabola quadrature, trigonometric law, principle of lever) duplicated work by Archimedes and Pappus, Thabit's list of novel achievements is impressive. Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius.

Ibn Sinan, grandson of Thabit ibn Qurra, was one of the greatest Islamic mathematicians and might have surpassed his famous grandfather had he not died at a young age. He was an early pioneer of analytic geometry, advancing the theory of integration, applying algebra to synthetic geometry, and writing on the construction of conic sections. He produced a new proof of Archimedes' famous formula for the area of a parabolic section. He worked on the theory of area-preserving transformations, with applications to map-making. He also advanced astronomical theory, and wrote a treatise on sundials.

Al-Hassan ibn al-Haytham (Alhazen) made contributions to math, optics, and astronomy which eventually influenced Roger Bacon, Regiomontanus, da Vinci, Copernicus, Kepler, Galileo, Huygens, Descartes and Wallis, thus affecting Europe's Scientific Revolution. While Aristotle thought vision arose from rays sent from the eye to the viewed object, and Ptolemy thought light rays originated from objects, Alhazen understand that an object's light is reflected sunlight. He's been called the best scientist of the Middle Ages; his Book of Optics has been called the most important physics text prior to Newton; his writings in physics anticipate the Principle of Least Action, Newton's First Law of Motion, and the notion that white light is composed of the color spectrum. (Like Newton, he favored a particle theory of light over the wave theory of Aristotle.) His other achievements in optics include improved lens design, an analysis of the camera obscura, Snell's Law, an early explanation for the rainbow, a correct deduction from refraction of atmospheric thickness, and experiments on visual perception. He studied optical illusions and was first to explain psychologically why the Moon appears to be larger when near the horizon. He also did work in human anatomy and medicine. (In a famous leap of over-confidence he claimed he could control the Nile River; when the Caliph ordered him to do so, he then had to feign madness!) Alhazen has been called the "Father of Modern Optics," the "Founder of Experimental Psychology" (mainly for his work with optical illusions), and, because he emphasized hypotheses and experiments, "The First Scientist."

In number theory, Alhazen worked with perfect numbers, Mersenne primes, and the Chinese Remainder Theorem. He stated Wilson's Theorem (which is sometimes called Al-Haytham's Theorem). Alhazen introduced the Power Series Theorem (later attributed to Jacob Bernoulli). His best mathematical work was with plane and solid geometry, especially conic sections; he calculated the areas of lunes, volumes of paraboloids, and constructed a regular heptagon using intersecting parabolas. He solved Alhazen's Billiard Problem (originally posed as a problem in mirror design), a difficult construction which continued to intrigue several great mathematicians including Huygens. To solve it, Alhazen needed to anticipate Descartes' analytic geometry, anticipate Bézout's Theorem, tackle quartic equations and develop a rudimentary integral calculus. Alhazen's attempts to prove the Parallel Postulate make him (along with Thabit ibn Qurra) one of the earliest mathematicians to investigate non-Euclidean geometry.

Al-Biruni (Alberuni) was an extremely outstanding scholar, far ahead of his time, sometimes shown with Alkindus and Alhazen as one of the greatest Islamic polymaths, and sometimes compared to Leonardo da Vinci. He is less famous in part because he lived in a remote part of the Islamic empire. He was a great linguist; studied the original works of Greeks and Hindus; is famous for debates with his contemporary Avicenna; studied history, biology, mineralogy, philosophy, sociology, medicine and more; is called the Father of Geodesy and the Father of Arabic Pharmacy; and was one of the greatest astronomers. He was an early advocate of the Scientific Method. He was also noted for his poetry. He invented (but didn't build) a geared-astrolabe clock, and worked with springs and hydrostatics. He wrote prodigiously on all scientific topics (his writings are estimated to total 13,000 folios); he was especially noted for his comprehensive encyclopedia about India, and Shadows, which starts from notions about shadows but develops much astronomy and mathematics. He anticipated future advances including Darwin's natural selection, Newton's Second Law, the immutability of elements, the nature of the Milky Way, and much modern geology. Among several novel achievements in astronomy, he used observations of lunar eclipse to deduce relative longitude, estimated Earth's radius most accurately, believed the Earth rotated on its axis and may have accepted heliocentrism as a possibility. In mathematics, he was first to apply the Law of Sines to astronomy, geodesy, and cartography; anticipated the notion of polar coordinates; invented the azimuthal equidistant map projection in common use today, as well as a polyconic method now called the Nicolosi Globular Projection; found trigonometric solutions to polynomial equations; did geometric constructions including angle trisection; and wrote on arithmetic, algebra, and combinatorics as well as plane and spherical trigonometry and geometry. (Al-Biruni's contemporary Avicenna was not particularly a mathematician but deserves mention as an advancing scientist, as does Avicenna's disciple Abu'l-Barakat al-Baghdada, who lived about a century later.)

Al-Biruni has left us what seems to be the oldest surviving mention of the Broken Chord Theorem (if M is the midpoint of circular arc ABMC, and T the midpoint of "broken chord" ABC, then MT is perpendicular to BC). Although he himself attributed the theorem to Archimedes, Al-Biruni provided several novel proofs for, and useful corollaries of, this famous geometric gem. While Al-Biruni may lack the influence and mathematical brilliance to qualify for the Top 100, he deserves recognition as one of the greatest applied mathematicians before the modern era.

Omar Khayyám (aka Ghiyas od-Din Abol-Fath Omar ibn Ebrahim Khayyam Neyshaburi) was one of the greatest Islamic mathematicians. He did clever work with geometry, developing an alternate to Euclid's Parallel Postulate and then deriving the parallel result using theorems based on the Khayyam-Saccheri quadrilateral. He derived solutions to cubic equations using the intersection of conic sections with circles. Remarkably, he stated that the cubic solution could not be achieved with straightedge and compass, a fact that wouldn't be proved until the 19th century. Khayyám did even more important work in algebra, writing an influential textbook, and developing new solutions for various higher-degree equations. He may have been first to develop Pascal's Triangle (which is still called Khayyám's Triangle in Persia), along with the essential Binomial Theorem (Al-Khayyám's Formula):       (x+y)n = n! Σ xkyn-k / k!(n-k)!

Khayyám was also an important astronomer; he measured the year far more accurately than ever before, improved the Persian calendar, built a famous star map, and believed that the Earth rotates on its axis. He was a polymath: in addition to being a philosopher of far-ranging scope, he also wrote treatises on music, mechanics and natural science. He was noted for deriving his theories from science rather than religion. Despite his great achievements in algebra, geometry, astronomy, and philosophy, today Omar al-Khayyám is most famous for his rich poetry (The Rubaiyat of Omar Khayyám).

Bháscara II (also called Bhaskaracharya) may have been the greatest of the Hindu mathematicians. He made achievements in several fields of mathematics including some Europe wouldn't learn until the time of Euler. His textbooks dealt with many matters, including solid geometry, combinations, and advanced arithmetic methods. He was also an astronomer. (It is sometimes claimed that his equations for planetary motions anticipated the Laws of Motion discovered by Kepler and Newton, but this claim is doubtful.) In algebra, he solved various equations including 2nd-order Diophantine, quartic, Brouncker's and Pell's equations. His Chakravala method, an early application of mathematical induction to solve 2nd-order equations, has been called "the finest thing achieved in the theory of numbers before Lagrange" (although a similar statement was made about one of Fibonacci's theorems). (Earlier Hindus, including Brahmagupta, contributed to this method.) In several ways he anticipated calculus: he used Rolle's Theorem; he may have been first to use the fact that dsin x = cos x · dx; and he once wrote that multiplication by 0/0 could be "useful in astronomy." In trigonometry, which he valued for its own beauty as well as practical applications, he developed spherical trig and was first to present the identity
sin a+b = sin a · cos b + sin b · cos a

Bháscara's achievements came centuries before similar discoveries in Europe. It is an open riddle of history whether any of Bháscara's teachings trickled into Europe in time to influence its Scientific Renaissance.

Leonardo (known today as Fibonacci) introduced the decimal system and other new methods of arithmetic to Europe, and relayed the mathematics of the Hindus, Persians, and Arabs. Others, especially Gherard of Cremona, had translated Islamic mathematics, e.g. the works of al-Khowârizmi, into Latin, but Leonardo was the influential teacher. (Two centuries earlier, the mathematician-Pope, Gerbert of Aurillac, had tried unsuccessfully to introduce the decimal system to Europe.) Leonardo also re-introduced older Greek ideas like Mersenne numbers and Diophantine equations. His writings cover a very broad range including new theorems of geometry, methods to construct and convert Egyptian fractions (which were still in wide use), irrational numbers, the Chinese Remainder Theorem, theorems about Pythagorean triplets, and the series 1, 1, 2, 3, 5, 8, 13, .... which is now linked with the name Fibonacci. In addition to his great historic importance and fame (he was a favorite of Emperor Frederick II), Leonardo `Fibonacci' is called "the greatest number theorist between Diophantus and Fermat" and "the most talented mathematician of the Middle Ages."

Leonardo is most famous for his book Liber Abaci, but his Liber Quadratorum provides the best demonstration of his skill. He defined congruums and proved theorems about them, including a theorem establishing the conditions for three square numbers to be in consecutive arithmetic series; this has been called the finest work in number theory prior to Fermat (although a similar statement was made about one of Bhaskara II's theorems). Although often overlooked, this work includes a proof of the n = 4 case of Fermat's Last Theorem. (Leonardo's proof of FLT4 is widely ignored or considered incomplete. I'm preparing a page to consider that question. Al-Farisi was another ancient mathematician who noted FLT4, although attempting no proof.) Another of Leonardo's noteworthy achievements was proving that the roots of a certain cubic equation could not have any of the constructible forms Euclid had outlined in Book 10 of his Elements. He also wrote on, but didn't prove, Wilson's Theorem.

Leonardo provided Europe with the decimal system, algebra and the 'lattice' method of multiplication, all far superior to the methods then in use. He introduced notation like 3/5; his clever extension of this for quantities like 5 yards, 2 feet, and 3 inches is more efficient than today's notation. It seems hard to believe but before the decimal system, mathematicians had no notation for zero. Referring to this system, Gauss was later to exclaim "To what heights would science now be raised if Archimedes had made that discovery!"

Some histories describe him as bringing Islamic mathematics to Europe, but in Fibonacci's own preface to Liber Abaci, he specifically credits the Hindus:

... as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods;
... But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, ...

Had the Scientific Renaissance begun in the Islamic Empire, someone like al-Khowârizmi would have greater historic significance than Fibonacci, but the Renaissance did happen in Europe. Liber Abaci's summary of the decimal system has been called "the most important sentence ever written." Even granting this to be an exaggeration, there is no doubt that the Scientific Revolution owes a huge debt to Leonardo `Fibonacci' Pisano.

Al-Tusi was one of the greatest Islamic polymaths, working in theology, ethics, logic, astronomy, and other fields of science. He was a famous scholar and prolific writer, describing evolution of species, stating that the Milky Way was composed of stars, and mentioning conservation of mass in his writings on chemistry. He made a wide range of contributions to astronomy, and (along with Omar Khayyám) was one of the most significant astronomers between Ptolemy and Copernicus. He improved on the Ptolemaic model of planetary orbits, and even wrote about (though rejecting) the possibility of heliocentrism. (Allegedly one of his students hypothesized elliptical orbits.)

Tusi is most famous for his mathematics. He advanced algebra, arithmetic, geometry, trigonometry, and even foundations, working with real numbers and lengths of curves. For his texts and theorems, he may be called the "Father of Trigonometry;" he was first to properly state and prove several theorems of planar and spherical trigonometry including the Law of Sines, and the (spherical) Law of Tangents. He wrote important commentaries on works of earlier Greek and Islamic mathematicians; he attempted to prove Euclid's Parallel Postulate. Tusi's writings influenced European mathematicians including Wallis; his revisions of the Ptolemaic model led him to the Tusi-couple, a special case of trochoids usually called Copernicus' Theorem, though historians have concluded Copernicus discovered this theorem by reading Tusi.

There were several important Chinese mathematicians in the 13th century, of whom Qin Jiushao (Ch'in Chiu-Shao) may have had particularly outstanding breadth and genius. Qin's textbook discusses various algebraic procedures, includes word problems requiring quartic or quintic equations, explains a version of Horner's Method for finding solutions to such equations, includes Heron's Formula for a triangle's area, and introduces the zero symbol and decimal fractions. Qin's work on the Chinese Remainder Theorem was very impressive, finding solutions in cases which later stumped Euler.

Other great Chinese mathematicians of that era are Li Zhi, Yang Hui (Pascal's Triangle is still called Yang Hui's Triangle in China), and Zhu Shiejie. Their teachings did not make their way to Europe, but were read by the Japanese mathematician Seki, and possibly by Islamic mathematicians like Al-Kashi. Although Qin was a soldier and governor noted for corruption, with mathematics just a hobby, I've chosen him to represent this group because of the key advances which appear first in his writings.

Zhu Shiejie (Chu Shih-Chieh) was more famous and influential than Qin; historian George Sarton called him "one of the greatest mathematicians ... of all time." His book Jade Mirror of the Four Unknowns studied multivariate polynomials and is considered the best mathematics in ancient China and describes methods not rediscovered for centuries; for example Zhu anticipated the Sylvester matrix method for solving simultaneous polynomial equations.

Al-Farisi was a student of al-Shirazi who in turn was a student of al-Tusi. He and al-Shirazi are especially noted for the first correct explanation of the rainbow. Al-Farisi made several other corrections in his comprehensive commentary on Alhazen's textbook on optics.

Al-Farisi made several contributions to number theory. He improved Thabit's theorem about amicable numbers, made important new observations about the binomial coefficients, and noted the N=4 case of Fermat's Last Theorem. In addition to his work with amicable numbers, he is especially noted for his improved proof of Euclid's Fundamental Theorem of Arithmetic.

Gersonides (aka Leo de Bagnols, aka RaLBaG) was a Jewish scholar of great renown, preferring science and reason over religious orthodoxy. He wrote important commentaries on Aristotle, Euclid, the Talmud, and the Bible; he is most famous for his book MilHamot Adonai ("The Wars of the Lord") which touches on many theological questions. He was likely the most talented scientist of his time: he invented the "Jacob's Staff" which became an important navigation tool; described the principles of the camera obscura; etc. In mathematics, Gersonides wrote texts on trigonometry, calculation of cube roots, rules of arithmetic, etc.; and gave rigorous derivations of rules of combinatorics. He was first to make explicit use of mathematical induction. At that time, "harmonic numbers" referred to integers with only 2 and 3 as prime factors; Gersonides solved a problem of music theory with an ingenious proof that there were no consecutive harmonic numbers larger than (8,9). (In fact, (2^3,3^2) is the only pair of consecutive powers; this is Catalan's Conjecture and was first proved in the 21st century.) Levi ben Gerson published only in Hebrew so, although some of his work was translated into Latin during his lifetime, his influence was limited; much of his work was re-invented three centuries later; and many histories of math overlook him altogether.

Gersonides was also an outstanding astronomer. He proved that the fixed stars were at a huge distance, and found other flaws in the Ptolemaic model. But he specifically rejected heliocentrism, noteworthy since it implies that heliocentrism was under consideration at the time.

Oresme was of lowly birth but excelled at school, became a young professor, and soon personal chaplain to King Charles V. (One of his early teachers was Jean Buridan (ca 1300-ca 1358), famous for positing what became Newton's First Law of Motion, contrary to Aristotelian dogma.) The King commissioned him to translate the works of Aristotle into French (with Oresme thus playing key roles in the development of both French science and French language), and rewarded him by making him a Bishop. His contributions to political science (on human rights and popular monarchy) infuenced King Charles V. He wrote several books; was a renowned philosopher and natural scientist (challenging several of Aristotle's ideas); he contributed to economics (e.g. anticipating Gresham's Law) and to optics (he was first to posit curved refraction). Although the Earth's annual orbit around the Sun was left to Copernicus, Oresme was among the pre-Copernican thinkers to claim clearly that the Earth spun daily on its axis.

In mathematics, Oresme observed that the integers were equinumerous with the odd integers; was first to use fractional (and even irrational) exponents; introduced the symbol + for addition; was first to write about general curvature; and, most famously, first to prove the divergence of the harmonic series. Oresme used a graphical diagram to demonstrate the Merton College Theorem (a discovery related to Galileo's Law of Falling Bodies made by Thomas Bradwardine, et al); it is said this was the first abstract graph. (Some believe that this effort inspired Descartes' coordinate geometry and Galileo.) Oresme was aware of Gersonides' work on harmonic numbers and was among those who attempted to link music theory to the ratios of celestial orbits, writing "the heavens are like a man who sings a melody and at the same time dances, thus making music ... in song and in action." Oresme's work was influential; with several discoveries ahead of his time, Oresme deserves to be better known.

Madhava, also known as Irinjaatappilly Madhavan Namboodiri, founded the important Kerala school of mathematics and astronomy. If everything credited to him was his own work, he was a truly great mathematician. His analytic geometry preceded and surpassed Descartes', and included differentiation and integration. Madhava also did work with continued fractions, trigonometry, and geometry. He has been called the "Founder of Mathematical Analysis." Madhava is most famous for his work with Taylor series, discovering identities like   sin q = q - q3/3! + q5/5! - ...   , formulae for π, including the one attributed to Leibniz, and the then-best known approximation π ≈ 104348 / 33215.

Despite the accomplishments of the Kerala school, Madhava probably does not deserve a place on our List. There were several other great mathematicians who contributed to Kerala's achievements, some of which were made 150 years after Madhava's death. More importantly, the work was not propagated outside Kerala, so had almost no effect on the development of mathematics.

Al-Kashi was among the greatest calculators in the ancient world; wrote important texts applying arithmetic and algebra to problems in astronomy, mensuration and accounting; and developed trig tables far more accurate than earlier tables. He worked with binomial coefficients, invented astronomical calculating machines, developed spherical trig, and is credited with various theorems of trigonometry including the Law of Cosines, which is sometimes called Al-Kashi's Theorem. He is sometimes credited with the invention of decimal fractions (though he worked mainly with sexagesimal fractions), and a method like Horner's to calculate roots. However decimal fractions had been used earlier, e.g. by Qin Jiushao; and Al-Kashi's root calculations may also have been derived from Chinese texts by Qin Jiushao or Zhu Shiejie.

Using his methods, al-Kashi calculated π correctly to 17 significant digits, breaking Madhava's record. (This record was subsequently broken by relative unknowns: a German ca. 1600, John Machin 1706. In 1949 the π calculation record was held briefly by John von Neumann and the ENIAC.)

Nicholas Kryffs (aka Nikolaus Cusanus) was an astronomer and ordained priest. His main work was in philosophy (he wrote "All we know of the truth is that the absolute truth, such as it is, is beyond our reach") and theology; but he also made proposals for scientific experimentation, e.g. the use of water-clocks to investigate the speed of falling bodies, and the use of weighing balances to quantify the transfer of mass from soil to growing plant. Nicholas was also a mathematician: he wrote on geometric problems, calendar reform, and invented the Borda Count method of balloting (although it is named after its 18th-century rediscoverer), but his work doesn't qualify him for our list. He belongs in our story because he deduced that the Earth orbited the Sun before Copernicus did, and made a deduction that Copernicus missed: that the universe is immensely huge or infinite; and the fixed stars may have their own planets with their own life forms. He was far ahead of his time, but his writings eventually influenced Galileo, Leibniz, and another mathematician-priest more famous than himself: Giordano Bruno, who wrote "If Nicholas [of Cusa] had not been hindered by his priest's vestment, he would have even been greater than Pythagoras!"

Regiomontanus was a prodigy who entered University at age eleven, studied under the influential Georg von Peuerbach, and eventually collaborated with him. He was an important astronomer; he found flaws in Ptolemy's system (thus influencing Copernicus), realized lunar observations could be used to determine longitude, and may have believed in heliocentrism. His ephemeris was used by Columbus, when shipwrecked on Jamaica, to predict a lunar eclipse, thus dazzling the natives and perhaps saving his crew. More importantly, Regiomontanus was one of the most influential mathematicians of the Middle Ages; he published trigonometry textbooks and tables, as well as the best textbook on arithmetic and algebra of his time. (Regiomontanus lived shortly after Gutenberg, and founded the first scientific press.) He was a prodigious reader of Greek and Latin translations, and most of his results were copied from Greek works (or indirectly from Arabic writers, especially Jabir ibn Aflah); however he improved or reconstructed many of the proofs, and often presented solutions in both geometric and algebraic form. His algebra was more symbolic and general than his predecessors'; he solved cubic equations (though not the general case); applied Chinese remainder methods, and worked in number theory. He posed and solved a variety of clever geometric puzzles, including his famous angle maximization problem. Regiomontanus was also an instrument maker, astrologer, and Catholic bishop. He died in Rome where he had been called to advise the Pope on the calendar; his early death may have delayed the needed reform until the time of Pope Gregory.

Leonardo da Vinci is most renowned for his paintings -- Mona Lisa and The Last Supper are among the most discussed and admired paintings ever -- but he did much other work and was probably the most talented, versatile and prolific polymath ever to live; his writings exceed 13,000 folios. He developed new techniques, and principles of perspective geometry, for drawing, painting and sculpture; he was also an expert architect and engineer; and surely the most prolific inventor of all time. Although most of his paper designs were never built, Leonardo's inventions include reflecting and refracting telescope, adding machine, parabolic compass, improved anemometer, parachute, helicopter, flying ornithopter, several war machines (multi-barreled gun, steam-driven cannon, tank, giant crossbow, finned mortar shells, portable bridge), pumps, an accurate spring-operated clock, bobbin winder, robots, scuba gear, an elaborate musical instrument he called the 'viola organista,' and more. (Some of his designs, including the viola organista, his parachute, and a large single-span bridge, were finally built five centuries later; and worked as intended.) Like another genius (Albert Einstein) da Vinci was intrigued by the science of river meanders; there is a meandering river in the background of Mona Lisa. His scientific writings are much more extensive and thorough than is generally appreciated; he made advances in anatomy, botany, and many other fields of science; he developed much mechanics including the theory of the arch; he developed an octant-based map projection; he was first to conceive of plate tectonics; and, in one cryptic reference, shows belief in heliocentrism. He was also a poet and musician.

He had little formal training in mathematics until he was in his mid-40's, when he and Luca Pacioli (the other great Italian mathematician of that era) began tutoring each other. Despite this slow start, he did make novel achievements in mathematics: he was first to note the simple classification of symmetry groups on the plane, achieved interesting bisections and mensurations, advanced the craft of descriptive geometry, and developed an approximate solution to the circle-squaring problem. He was first to discover the 60-vertex shape now called "buckyball." (Leonardo is also widely credited with the elegant two-hexagon proof of the Pythagorean Theorem, but this authorship appears to be a myth.) Along with Archimedes, Alberuni, Leibniz, and J. W. von Goethe, Leonardo da Vinci was among the greatest geniuses ever; but none of these appears on Hart's List of the Most Influential Persons in History: genius doesn't imply influence. (However, M.I.T.'s Pantheon project, using the statistics of on-line biographies, prepared a list of the Thirty-Five (or Eighty) Most Influential Persons in History; in addition to six (seven) names already on our list and Hart's -- Plato, Aristotle, Newton, Einstein, Galileo, Euclid (and Descartes) -- their list includes three (six) other mathematicians missing from Hart's list: Leonardo (#6), Pythagoras (#10), Archimedes (#11) (and Thales (#38), Pascal (#67), Ptolemy (#80)).

Leonardo was also a writer and philosopher. Among his notable adages are "Simplicity is the ultimate sophistication," and "The noblest pleasure is the joy of understanding," and "Human ingenuity ... will never discover any inventions more beautiful, more simple or more practical than those of nature."

The European Renaissance developed in 15th-century Italy, with the blossoming of great art, and as scholars read books by great Islamic scientists like Alhazen. The earliest of these great Italian polymaths were largely not noted for mathematics, and Leonardo da Vinci began serious math study only very late in life, so the best candidates for mathematical greatness in the Italian Renaissance were foreigners. Along with Regiomontanus from Bavaria, there was an even more famous man from Poland.

Nicolaus Copernicus (Mikolaj Kopernik) was a polymath: he studied law and medicine; published poetry; contemplated astronomy; worked professionally as a church scholar and diplomat; and was also a painter. He studied Islamic works on astronomy and geometry at the University of Bologna, and eventually wrote a book of great impact. Although his only famous theorem of mathematics (that certain trochoids are straight lines) may have been derived from Oresme's work, or copied from Nasir al-Tusi, it was mathematical thought that led Copernicus to the conclusion that the Earth rotates around the Sun. Despite opposition from the Roman church, this discovery led, via Galileo, Kepler and Newton, to the Scientific Revolution. For this revolution, Copernicus is ranked #19 on Hart's List of the Most Influential Persons in History; however I think there are several reasons why Copernicus' importance may be exaggerated: (1) Copernicus' system still used circles and epicycles, so it was left to Kepler to discover the facts of elliptical orbits; (2) he retained the notion of a sphere of fixed stars, thus missing the unifying insight that our sun is one of many; (3) Giordano Bruno (1548-1600), who built on Copernicus' discovery, was a better and more influential scientist, anticipating some of Galileo's concepts; and (4) the Scientific Revolution didn't really get underway until the invention of the telescope, which would have soon led to the discovery of heliocentrism in any event.

Until the Protestant Reformation, which began about the time of Copernicus' discovery, European scientists were reluctant to challenge the Catholic Church and its belief in geocentrism. Copernicus' book was published only posthumously. It remains controversial whether earlier Islamic or Hindu mathematicians (or even Archimedes with his The Sand Reckoner) believed in heliocentrism, but were also inhibited by religious orthodoxy.

Girolamo Cardano (or Jerome Cardan) was a highly respected physician and was first to describe typhoid fever. He was also an accomplished gambler and chess player and wrote an early book on probability. He was also a remarkable inventor: the combination lock, an advanced gimbal, a ciphering tool, and the Cardan shaft with universal joints are all his inventions and are in use to this day. (The U-joint is sometimes called the Cardan joint.) He also helped develop the camera obscura. Cardano made contributions to physics: he noted that projectile trajectories are parabolas, and may have been first to note the impossibility of perpetual motion machines. He did work in philosophy, geology, hydrodynamics, music; he wrote books on medicine and an encyclopedia of natural science.

But Cardano is most remembered for his achievements in mathematics. He was first to publish general solutions to cubic and quartic equations, and first to publish the use of complex numbers in calculations. (Cardano's Italian colleagues deserve much credit: Ferrari first solved the quartic, he or Tartaglia the cubic; and Bombelli first treated the complex numbers as numbers in their own right. Cardano may have been the last great mathematician unwilling to deal with negative numbers: his treatment of cubic equations had to deal with ax3 - bx + c = 0 and ax3 - bx = c as two different cases.) Cardano introduced binomial coefficients and the Binomial Theorem, and introduced and solved the geometric hypocycloid problem, as well as other geometric theorems (e.g. the theorem underlying the 2:1 spur wheel which converts circular to reciprocal rectilinear motion). Cardano is credited with Cardano's Ring Puzzle, still manufactured today and related to the Tower of Hanoi puzzle. (This puzzle may predate Cardano, and may even have been known in ancient China.) Da Vinci and Galileo may have been more influential than Cardano, but of the three great generalists in the century before Kepler, it seems clear that Cardano was the most accomplished mathematician.

Cardano's life had tragic elements. Throughout his life he was tormented that his father (a friend of Leonardo da Vinci) married his mother only after Cardano was born. (And his mother tried several times to abort him.) Cardano's reputation for gambling and aggression interfered with his career. He practiced astrology and was imprisoned for heresy when he cast a horoscope for Jesus. (This and other problems were due in part to revenge by Tartaglia for Cardano's revealing his secret algebra formulae.) His son apparently murdered his own wife. Leibniz wrote of Cardano: "Cardano was a great man with all his faults; without them, he would have been incomparable."

Bombelli was a talented engineer who wrote an algebra textbook sometimes considered one of the foremost achievements of the 16th century. Although incorporating work by Cardano, Diophantus and possibly Omar al-Khayyám, the textbook was highly original and extremely influential. Leibniz and Huygens were among many who praised his work. Although noted for his new ideas of arithmetic, Bombelli based much of his work on geometric ideas, and even pursued complex-number arithmetic to an angle-trisection method. He was the first European of his era to use continued fractions, using them for a new square-root procedure. In his textbook he also introduced new symbolic notations, allowed negative and complex numbers, and gave the rules for manipulating these new kinds of numbers. Bombelli is often called the Inventor of Complex Numbers.

François Viète (or Franciscus Vieta) was a French nobleman and lawyer who was a favorite of King Henry IV and eventually became a royal privy councillor. In one notable accomplishment he broke the Spanish diplomatic code, allowing the French government to read Spain's messages and publish a secret Spanish letter; this apparently led to the end of the Huguenot Wars of Religion.

More importantly, Vieta was certainly the best French mathematician prior to Descartes and Fermat. He laid the groundwork for modern mathematics; his works were the primary teaching for both Descartes and Fermat; Isaac Newton also studied Vieta. In his role as a young tutor Vieta used decimal numbers before they were popularized by Simon Stevin and may have guessed that planetary orbits were ellipses before Kepler. Vieta did work in geometry, reconstructing and publishing proofs for Apollonius' lost theorems, including all ten cases of the general Problem of Apollonius. Vieta also used his new algebraic techniques to construct a regular heptagon. He discovered several trigonometric identities including a generalization of Ptolemy's Formula, the latter (then called prosthaphaeresis) providing a calculation shortcut similar to logarithms in that multiplication is reduced to addition (or exponentiation reduced to multiplication). Vieta also used trigonometry to find real solutions to cubic equations for which the Italian methods had required complex-number arithmetic; he also used trigonometry to solve a particular 45th-degree equation that had been posed as a challenge. Such trigonometric formulae revolutionized calculations and may even have helped stimulate the development and use of logarithms by Napier and Kepler. He developed the first infinite-product formula for π. In addition to his geometry and trigonometry, he also found results in number theory, but Vieta is most famous for his systematic use of decimal notation and variable letters, for which he is sometimes called the "Father of Modern Algebra." (Vieta used A,E,I,O,U for unknowns and consonants for parameters; it was Descartes who first used X,Y,Z for unknowns and A,B,C for parameters.) In his works Vieta emphasized the relationships between algebraic expressions and geometric constructions. One key insight he had is that addends must be homogeneous (i.e., "apples shouldn't be added to oranges"), a seemingly trivial idea but which can aid intuition even today.

Descartes, who once wrote "I began where Vieta finished," is now extremely famous, while Vieta is much less known. (He isn't even mentioned once in Bell's famous Men of Mathematics.) Many would now agree this is due in large measure to Descartes' deliberate deprecations of competitors in his quest for personal glory. (Vieta wasn't particularly humble either, calling himself the "French Apollonius.")

```
PI := 2
Y  := 0
LOOP:
Y  := SQRT(Y + 2)
PI := PI * 2 / Y
IF (more precision needed) GOTO LOOP

```

Vieta's formula for π is clumsy to express without trigonometry, even with modern notation. Easiest may be to consider it the result of the BASIC program above. Using this formula, Vieta constructed an approximation to π that was best-yet by a European, though not as accurate as al-Kashi's two centuries earlier.

Bruno wrote on mnemonics, philosophy, cosmology and more, and further developed Nicholas of Cusa's pandeism (though it wasn't then known by that name). He embraced not only Copernicus' notion that the Earth rotates around the Sun, but thought the Sun itself moved and that, as Nicholas had conjectured, the universe was infinite and teemed with other worlds. This is a conclusion the great Kepler did not make. (Galileo's stance on this question is ambiguous: he knew the price Bruno had paid for expressing this belief.) Bruno was egotistical and obnoxious, and his teachings (most especially his belief in multiple solar systems) infuriated the Roman Church, which had him hung upside-down, spikes driven through his mouth so he could not utter further heresies, and burned alive. Before his execution he was tortured for seven years but never recanted, instead saying "Maybe you who condemn me are in greater fear than I who am condemned."

One historian claims that it was Bruno's publications in 1584 and their influence that defined the transition from medieval thought to modern scientific thought. William Gilbert (who laid the foundations of electricity and magnetism, which is called the first important scientific discovery by an Englishman), acknowledged the influence of Bruno's cosmology. The Revolution started by Nicholas and Bruno was a revolt against the Church's embracing Aristotle's teachings, as Bruno made clear when he wrote "There is no absolute up or down, as Aristotle taught; no absolute position in space.... Everywhere there is incessant relative change ... and the observer is always at the centre of things." Bruno made few mathematical discoveries, if any; but some consider him to be one of history's great geniuses, in a class with Johann von Goethe, Leonardo da Vinci, Francis Bacon, Voltaire, Nietzsche, and some of the very greatest mathematicians.

Stevin was one of the greatest practical scientists of the Late Middle Ages. He worked with Holland's dykes and windmills; as a military engineer he developed fortifications and systems of flooding; he invented a carriage with sails that traveled faster than with horses and used it to entertain his patron, the Prince of Orange. He discovered several laws of mechanics including those for energy conservation and hydrostatic pressure. He lived slightly before Galileo who is now much more famous, but Stevin discovered the equal rate of falling bodies before Galileo did; and his explanation of tides was better than Galileo's, though still incomplete. He was first to write on the concept of unstable equilibrium. He invented improved accounting methods, and (though also invented at about the same time by Chinese mathematician Zhu Zaiyu and anticipated by Galileo's father, Vincenzo Galilei) the equal-temperament music scale. He also did work in descriptive geometry, trigonometry, optics, geography, and astronomy.

In mathematics, Stevin is best known for the notion of real numbers (previously integers, rationals and irrationals were treated separately; negative numbers and even zero and one were often not considered numbers). He introduced (a clumsy form of) decimal fractions to Europe; suggested a decimal metric system, which was finally adopted 200 years later; invented other basic notation like the symbol . Stevin proved several theorems about perspective geometry, an important result in mechanics, and special cases of the Intermediate Value Theorem later attributed to Bolzano and Cauchy. Stevin's books, written in Dutch rather than Latin, were widely read and hugely influential. He was a very key figure in the development of modern European mathematics, and may belong on our List.

Napier was a Scottish Laird who was a noted theologian and thought by many to be a magician (his nickname was Marvellous Merchiston). Today, however, he is best known for his work with logarithms, a word he invented. (Several others, including Archimedes, had anticipated the use of logarithms.) He published the first large table of logarithms and also helped popularize usage of the decimal point and lattice multiplication. He invented Napier's Bones, a crude hand calculator which could be used for division and root extraction, as well as multiplication. He also had inventions outside mathematics, especially several different kinds of war machine.

Napier's noted textbooks also contain an exposition of spherical trigonometry. Although he was certainly very clever (and had novel mathematical insights not mentioned in this summary), Napier proved no deep theorem and may not belong in the Top 100. Nevertheless, his revolutionary methods of arithmetic had immense historical importance; his logarithm tables were used by Johannes Kepler himself, and led to the Scientific Revolution.

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