Greatest Mathematicians born between 400 and 1580 A.D.

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

Decimal system -- from China?

It's still hard to believe that the "obvious" and so-convenient decimal system didn't catch on in Europe until almost the Renaissance. 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 Aryabhatta, 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.

Aryabhatta (476-550) Ashmaka & Kusumapura (India)

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, Aryabhatta (called Arjehir by Arabs) may be most famous.

While Europe was in its early "Dark Age," Aryabhatta advanced arithmetic, algebra, elementary analysis, and especially trigonometry, using the decimal system. Aryabhatta is sometimes called the "Father of Algebra" instead of al-Khowârizmi (who himself cites the work of Aryabhatta). His most famous accomplishment in mathematics was the Aryabhatta Algorithm (connected to continued fractions) for solving Diophantine equations. Aryabhatta made several important discoveries in astronomy; for example, his estimate of the Earth's circumference was more accurate than any achieved in ancient Greece. He was among the 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. Aryabhatta 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. Among theorems first discovered by Aryabhatta is the famous identity
     Σ (k³) = (Σ k)²

Brahmagupta `Bhillamalacarya' (589-668) Rajasthan (India)

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." 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 Diophantine and Pell's equations. 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). He applied mathematics to astronomy, predicting eclipses, etc.

Muhammed `Abu Jafar' ibn Musâ  al-Khowârizmi (ca 780-850) Persia, Iraq

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 presented general methods. (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. There were several Muslim mathematicians who contributed to the development of Islamic science, and indirectly to Europe's later Renaissance, but Al-Khowârizmi was one of the earliest and most influential.

Ya'qub `Abu Yusuf' ibn Ishaq  al-Kindi (803-873) Iraq

Al-Kindi (called Alkindus 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). (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.)

Al-Sabi Thabit  ibn Qurra al-Harrani (836-901) Harran, Iraq

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 one of the most important translaters of ancient Greek writings.) 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 earlier practitioner of calculus and seems to have been first to take the integral of √x. He also may have been first to calculate the area of an ellipse, and first 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. While some 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 quite impressive. Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius.

Mohammed ibn al-Hasn (Alhazen) `Abu Ali'   ibn al-Haytham al-Basra (965-1039) Iraq, Egypt

Al-Hassan ibn al-Haytham (Alhazen) made contributions to math, optics, and astronomy which eventually influenced Roger Bacon, Regiomontanus, da Vinci, Copernicus, Kepler and Wallis, among others, thus affecting Europe's Scientific Revolution. 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 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; the Chinese Remainder Theorem, and stated Wilson's Conjecture (sometimes called Al-Haytham's Theorem though it was first proven by Lagrange). He introduced the Power Series Theorem (later attributed to Bernoulli). 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 Déscartes' 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.

Abu al-Rayhan Mohammed ibn Ahmad  al-Biruni (973-1048) Khwarizm (Uzbekistan)

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; was called the Father of Arabic Pharmacy; and was one of the greatest astronomers. He was also noted for his poetry. He invented (but didn't build) a mechanical 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 applied scientific methods; and 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 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; 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 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 our List, he deserves recognition as one of the greatest applied mathematicians before the modern era.

Omar  al-Khayyám (1048-1123) Persia

Omar Khayyám (aka Ghiyas od-Din Abol-Fath Omar ibn Ebrahim Khayyam Neyshaburi) is sometimes called the greatest Islamic mathematician. 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, along with the essential Binomial Theorem which is sometimes called Al-Khayyám's Formula:       (x+y)ⁿ = n! Σ x^ky^n-k / k!(n-k)!

Khayyám was also an important astronomer; he measured the year far more accurately than ever before, improved the Persian calendar, and built a famous star map. He emphasized science over religion and proved that the Earth rotates around the Sun. His symbol ('shay') for an unknown in an algebraic equation might have been transliterated to become our 'x'. He also wrote treatises on philosophy, music, mechanics and natural science. Despite his great achievements in algebra, geometry, and astronomy, today Omar al-Khayyám is most famous for his rich poetry (The Rubaiyat of Omar Khayyám).

Bháscara Áchárya (1114-1185) India

Bháscara (also called Bhaskara II or 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. (Another mathematician, Bháscara I who lived five centuries before Bháscara II, was also outstanding. He was famous for advancing the positional decimal number notation, for a formula giving an excellent approximation to the sin function, and for being first to state Wilson's Conjecture.)

Leonardo `Bigollo' Pisano (Fibonacci) (ca 1170-1245) Italy

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 had translated Islamic mathematics, e.g. the works of al-Khowârizmi, into Latin, but Leonardo was the influential teacher. He also re-introduced older Greek ideas like Mersenne numbers and Diophantine equations. Leonardo's 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'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.) 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.

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.

Abu Jafar Muhammad  Nasir al-Din al-Tusi (1201-1274) Persia

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 important astronomers between Ptolemy and Copernicus. He improved on the Ptolemaic model of planetary orbits, and even wrote about (though rejecting) the possibility of heliocentrism.

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.

Qin Jiushao (1202-1261) China

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. 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, and Zhu Shiejie (who may be the most famous and influential of these). 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.

Nicole  Oresme (ca 1322-1382) France

Oresme was of lowly birth but excelled at school (where he was taught by the famous Jean Buridan), became a young professor, and soon personal chaplain to King Charles V. 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. He wrote several books; and was a renowned philosopher, natural scientist (challenging several of Aristotle's ideas), and important economist (anticipating Gresham's Law). 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. 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 Déscartes' coordinate geometry.) Oresme was also first to use fractional exponents; first to write of general curvature; and, most famously, first to prove the divergence of the harmonic series. His work was largely ignored, so may have had little historic importance, but with several discoveries ahead of his time, Oresme deserves recognition.

Madhava of Sangamagramma (1340-1425) India

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 Déscartes', 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.

Ghiyath al-Din Jamshid Mas'ud  Al-Kashi (ca 1380-1429) Iran, Transoxania (Uzbekistan)

Al-Kashi was among the greatest calculaters 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 his decimal fractions built on prior work, and some historians think his method for calculating roots derived from reading 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.)

Nicolaus  Copernicus (1472-1543) Poland

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. Setting aside Leonardo da Vinci, who began serious math study only very late in life, perhaps the best "Italian" candidate for greatness who lived between Fibonacci and Cardano in time would be a foreigner. This certain Pole studied Islamic works on astronomy and geometry at the University of Bologna, and eventually wrote a book of huge impact.

Nicolaus Copernicus (Mikolaj Kopernik) was a polymath; he started by studying law and medicine; later published poetry and contemplated astronomy, while working professionally as a church scholar/diplomat. Although his only famous theorem of mathematics (that certain trochoids are straight lines) seems to be copied from Nasir al-Tusi, it was mathematical thought that led Copernicus to the conclusion that the Earth rotates around the Sun. (It was left to Kepler to discover the true facts of planetary orbits, as Copernicus' system still used circles and epicycles, rather than ellipses.) 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. It remains controversial whether earlier Islamic or even Hindu mathematicians believed in heliocentrism, but were also inhibited by religious orthodoxy.

Girolamo  Cardano (1501-1576) Italy

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 unaware of 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 hypocyloid 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."

François  Viète (1540-1603) France

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 Déscartes and Fermat. He laid the groundwork for modern mathematics; his works were the primary teaching for both Déscartes 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. 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 Déscartes 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.

Déscartes, 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 Déscartes' 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.

Simon  Stevin (1549-1620) Flanders, Holland

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 travelled 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 correctly explained the influence of the moon on tides (which Galileo later got wrong). He was first to write on the concept of unstable equilibrium. He invented improved accounting methods, and 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 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 the Intermediate Value Theorem attributed to Cauchy. Stevin's books, written in Dutch, 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.

John  Napier  8th of Merchistoun (1550-1617) Scotland

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, including war machines.

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 on a list of great mathematicians. Nevertheless, his revolutionary methods of arithmetic had immense historical importance; his tables were used by Johannes Kepler himself, and led to the Scientific Revolution.

Galileo  Galilei (1564-1642) Italy

Galileo discovered the laws of inertia, falling bodies (including parabolic trajectories), and the pendulum; he also introduced the notion of relativity which Einstein later found so fruitful. He was a great inventor: in addition to being first to conceive of the pendulum clock, he developed a new type of pump, and the best telescope, thermometer, hydrostatic balance, and cannon sector of his day. As a famous astronomer, Galileo pointed out that Jupiter's Moons, which he discovered, provide a natural clock and allow a universal time to be determined by telescope anywhere on Earth. (This was of little use in ocean navigation since a ship's rocking prevents the required delicate observations.) His discovery that Venus, like the Moon, had phases was the critical fact which forced acceptance of Copernican heliocentrism. Galileo is also renowned for early discoveries with microscope.

Galileo is often called the "Father of Modern Science" because of his emphasis on experimentation. He understood that results needed to be repeated and averaged (he used mean absolute difference as his curve-fitting criterion, two centuries before Gauss and Legendre introduced the mean squared-difference criterion). For his experimental methods and discoveries, his laws of motion, and for (eventually) helping to spread Copernicus' heliocentrism, Galileo is considered to be one of the most influential scientists ever; he ranks #12 on Hart's list of the Most Influential Persons in History. (Despite these comments, it does appear that Galileo ignored experimental results that conflicted with his theories. For example, the Law of the Pendulum, based on Galileo's incorrect belief that the tautochrone was the circle, conflicted with his own observations.) Despite his extreme importance to mathematical physics, Galileo doesn't usually appear on lists of greatest mathematicians. However Cavalieri and Torricelli were his students; to encourage them, he probably avoided competing with them. Galileo derived certain centroids using a rudimentary calculus before Cavalieri did, named (and may have been first to discover) the cycloid curve, and did other mathematical work. Moreover, Galileo may have been first to write about infinite equinumerosity (the "Hilbert's Hotel Paradox"). Galileo once wrote "Mathematics is the language in which God has written the universe."

Johannes  Kepler (1571-1630) Germany

Kepler was interested in astronomy from an early age, studied to become a Lutheran minister, became a professor of mathematics instead, then Tycho Brahe's understudy, and, on Brahe's death, was appointed Imperial Mathematician at the age of twenty-nine. His observations of the planets with Brahe, along with his study of Apollonius' 1800-year old work, led to Kepler's three Laws of Planetary Motion, which in turn led directly to Newton's Laws of Motion. Beyond his discovery of these Laws (one of the most important achievements in all of science), Kepler is also sometimes called the Founder of Modern Optics. He was first to study the operation of the human eye, telescopes built from two convex lenses, the theory of the camera obscura, and atmospheric refraction. Kepler was first to explain tides correctly. (Galileo dismissed this as well as Kepler's elliptical orbits, and later published his own incorrect explanation of tides.) Kepler ranks #75 on Michael Hart's famous list of the Most Influential Persons in History. This rank, much lower than that of Copernicus, Galileo or Newton, seems to me to underestimate Kepler's importance, since it was Kepler's Laws, rather than just heliocentrism, which were essential to the early development of mathematical physics.

According to Kepler's Laws, the planets move at variable speed along ellipses. (Even Copernicus thought the orbits could be described with only circles.) The Earth-bound observer is himself describing such an orbit and in almost the same plane as the planets; thus discovering the Laws would be a difficult challenge even for someone armed with computers and modern mathematics. (The very famous Kepler Equation relating a planet's eccentric and anomaly is just one tool Kepler needed to develop.) Kepler understood the importance of his remarkable discovery, even if contemporaries like Galileo did not, writing:

"I give myself up to divine ecstasy ... My book is written. It will be read either by my contemporaries or by posterity — I care not which. It may well wait a hundred years for a reader, as God has waited 6,000 years for someone to understand His work."

Besides the trigonometric results needed to discover his Laws, Kepler made other contributions to mathematics. He generalized Alhazen's Billiard Problem, developing the notion of curvature. He was first to notice that the set of Platonic regular solids was incomplete if concave solids are admitted, and first to prove that there were only 13 "Archimedean solids." He proved theorems of solid geometry later discovered on the famous palimpsest of Archimedes. He rediscovered the Fibonacci series, applied it to botany, and noted that the ratio of Fibonacci numbers converges to the Golden Mean. He was a key early pioneer in calculus, and embraced the concept of continuity (which others avoided due to Zeno's paradoxes); his work was a direct inspiration for Cavalieri and others. He developed the theory of logarithms and improved on Napier's tables. He developed mensuration methods and anticipated Fermat's theorem (df(x)/dx = 0 at function extrema). Kepler once had an opportunity to buy wine, which merchants measured using a shortcut; with the famous Kepler's Wine Barrel Problem, he used his rudimentary calculus to deduce which barrel shape would be the best bargain.

Kepler reasoned that the structure of snowflakes was evidence for the then-novel atomic theory of matter. He noted that the obvious packing of cannonballs gave maximum density (this became known as "Kepler's Conjecture"; optimality was proved among regular packings by Gauss, but it wasn't until 1998 that the possibility of denser irregular packings was disproven). In addition to his physics and mathematics, Kepler wrote a science fiction novel, and was an astrologer and mystic. He had ideas similar to Pythagoras about numbers ruling the cosmos (writing that the purpose of studying the world "should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics"). Kepler's mystic beliefs even led to his own mother being imprisoned for witchcraft.

Johannes Kepler (along with Galileo, Fermat, Huygens, Wallis, Vieta and Déscartes) is among the giants on whose shoulders Newton was proud to stand. Some historians place him ahead of Galileo and Copernicus as the single most important contributor to the early Scientific Revolution. Chasles includes Kepler on a list of the six responsible for conceiving and perfecting infinitesimal calculus (the other five are Archimedes, Cavalieri, Fermat, Leibniz and Newton). (www.keplersdiscovery.com is a wonderful website devoted to Johannes Kepler's discoveries.)