Greatest Mathematicians born between 1580 and 1700 A.D.

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

Gerard   Desargues (1591-1661) France

Desargues invented projective geometry and found the relationship among conic sections which inspired Blaise Pascal. Among several ingenious and rigorously proven theorems are Desargues' Involution Theorem and his Theorem of Homologous Triangles. Desargues was also a noted architect and inventor: he produced an elaborate spiral staircase, invented an ingenious new pump, and had the idea to use cycloid-shaped teeth in the design of gears.

Desargues' projective geometry may have been too creative for his time, and was largely ignored (except by Pascal himself) until Poncelet rediscovered it almost two centuries later. (Copies of Desargues' own works surfaced about the same time.) For this reason, Desargues may not belong on our list, despite that he may have been among the greatest natural geometers ever.

René  Déscartes (1596-1650) France

Déscartes' early career was that of soldier-adventurer and he finished as tutor to royalty, but in between he achieved fame as the preeminent intellectual of his day. He is considered the inventor of analytic geometry and therefore the "Father of Modern Mathematics." Because of his famous philosophical writings ("Cogito ergo sum") he is considered, along with Aristotle, to be one of the most influential thinkers in history. He ranks #49 on Michael Hart's famous list of the Most Influential Persons in History. Déscartes developed laws of motion (including a "vortex" theory of gravitation) which were very influential, though largely incorrect. His famous mathematical theorems include the Rule of Signs (for determining the signs of polynomial roots), the elegant formula relating the radii of Soddy kissing circles, his theorem on total angular defect, and an improvement on the ancient construction method for cube-doubling. He improved mathematical notation (e.g. the use of superscripts to denote exponents). He also discovered Euler's Polyhedral Theorem (V - E + F = 2).

Déscartes has an extremely high reputation and would be ranked higher by many list makers. I've demoted Déscartes partly because he had only insulting things to say about Pascal and Fermat, each of whom was more brilliant at mathematics than Déscartes. (Some even suspect that Déscartes arranged the destruction of Pascal's lost Essay on Conics.) Déscartes' errors may have set back the cause of science, Huygens writing "in all of [Déscartes'] physics, I find almost nothing to which I can subscribe as being correct." Moreover the historical importance of the Frenchmen may be slightly exaggerated since others, e.g. Wallis and Cavalieri, were developing modern mathematics independently.

Francesco Bonaventura de  Cavalieri (1598-1647) Italy

Cavalieri did work in analysis, geometry and trigonometry; he is most famous for developing a rudimentary calculus. (Because of his calculus, Cavalieri was certainly an influential mathematician; however his work was largely anticipated by Kepler, and was soon surpassed by Fermat and Wallis.) Cavalieri also worked in theology, astronomy, mechanics and optics; he was an inventor, and published logarithm tables. He wrote several books, the first one developing the properties of mirrors shaped as conic sections. His name is especially remembered for Cavalieri's Principle of Solid Geometry. Galileo said of Cavalieri, "Few, if any, since Archimedes, have delved as far and as deep into the science of geometry."

Pierre de  Fermat (1601-1665) France

Pierre de Fermat was the most brilliant mathematician of his era and, along with Déscartes, one of the most influential. Although mathematics was just his hobby (Fermat was a government lawyer), Fermat practically founded Number Theory, and also played key roles in the discoveries of Analytic Geometry and Calculus. He was also an excellent geometer (e.g. discovering a triangle's Fermat point), and (in collaboration with Blaise Pascal) discovered probability theory. Fellow geniuses are the best judges of genius, and Blaise Pascal had this to say of Fermat: "For my part, I confess that [Fermat's researches about numbers] are far beyond me, and I am competent only to admire them." E.T. Bell wrote "it can be argued that Fermat was at least Newton's equal as a pure mathematician."

Fermat's most famous discoveries in number theory include the ubiquitously-used Fermat's Little Theorem; the n = 4 case of his conjectured Fermat's Last Theorem (he may have proved the n = 3 case as well); the fact that every natural number is the sum of three triangle numbers; and Fermat's Christmas Theorem (that any prime (4n+1) can be represented as the sum of two squares in exactly one way, also called the Fermat-Euler Prime Number Theorem). As suggested by the "Euler" in the name of this latter theorem (which Fermat records proving with difficulty using "infinite descent"), proofs for this and many other Fermat results had to be rediscovered (most of Fermat's work was never published). However it is wrong to suppose that Fermat's work comprised mostly false or unproven conjectures. (This misconception arises from his so-called "Last Theorem" which was actually just a private scribble.)

Fermat developed a system of analytic geometry which both preceded and surpassed that of Déscartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration. Solving   df(x)/dx = 0   to find extrema of f(x) is perhaps the most useful idea in applied mathematics; this technique originated with Fermat. Fermat was also the first European to find the integration formula for the general polynomial; he used his calculus to find centers of gravity, etc.

Fermat's contemporaneous rival René Déscartes is more famous than Fermat, and Déscartes' writings were more influential. Whatever one thinks of Déscartes as a philosopher, however, it seems clear that Fermat was the better mathematician. Fermat and Déscartes did work in physics and independently discovered the (trigonometric) law of refraction, but Fermat gave the correct explanation, and used it remarkably to anticipate the Principle of Least Action later enunciated by Maupertius (though Maupertius himself, like Déscartes, had an incorrect explanation of refraction). Fermat and Déscartes independently discovered analytic geometry, but it was Fermat who extended it to more than two dimensions, and followed up by developing elementary calculus.

John Brehaut  Wallis (1616-1703) England

Wallis began his life as a savant at arithmetic (it is said he once calculated the square root of a 53-digit number to help him sleep and remembered the result in the morning), a medical student (he may have contributed to the concept of blood circulation), and theologian, but went on to become perhaps the most brilliant and influential English mathematician before Newton. (James Gregory is another candidate for this honor, but Gregory's career was truncated by death at age 36.) He made major advances in analytic geometry, but also contributions to algebra, geometry and trigonometry. He is especially famous for using negative and fractional exponents (though Oresme had introduced fractional exponents 3 centuries earlier), taking the areas of curves, and treating inelastic collisions (he and Huygens were first to develop the law of momentum conservation). He was the first European to solve Pell's Equation. Like Vieta, Wallis was a code-breaker, helping the Commonwealth side (though he later petitioned against the beheading of King Charles I). He was the first great mathematician to consider complex numbers legitimate; and first to use the symbol ∞. Wallis coined several terms including "continued fraction," "induction," "interpolation," "mantissa," and "hypergeometric series."

Also like Vieta, Wallis created an infinite product formula for pi, which might be (but isn't) written today as:
    π = 2 ∏_k=1,∞ 1+(4k²-1)^-1

Blaise  Pascal (1623-1662) France

Pascal was an outstanding genius who studied geometry as a child. At the age of sixteen he stated and proved Pascal's Theorem, a fact relating any six points on any conic section. The Theorem is sometimes called the "Cat's Cradle" or the "Mystic Hexagram." Pascal followed up this result by showing that each of Apollonius' famous theorems about conic sections was a corollary of the Mystic Hexagram; along with Gerard Desargues (1591-1661), he was a key pioneer of projective geometry. He also made important early contributions to calculus. Returning to geometry late in life, Pascal advanced the theory of the cycloid. In addition to his work in geometry and calculus, he founded probability theory, and made contributions to axiomatic theory. His name is associated with the Pascal's Triangle of combinatorics and Pascal's Wager in theology.

Like most of the greatest mathematicians, Pascal was interested in physics and mechanics, studying fluids, explaining vacuum, and inventing the syringe and hydraulic press. At the age of eighteen he designed and built the world's first automatic adding machine. (Although he continued to refine this invention, it was never a commercial success.) He suffered poor health throughout his life, abandoned mathematics for religion at about age 23, and died at an early age. Many think that had he devoted more years to mathematics, Pascal would have been one of the greatest mathematicians ever.

Christiaan  Huygens (1629-1695) Holland, France

Christiaan Huygens (or Hugens, Huyghens) was second only to Newton as the greatest mechanist of his era. Although an excellent mathematician, he is much more famous for his physical theories and inventions. He developed laws of motion before Newton, including the inverse-square law of gravitation, centripetal force, and treatment of solid bodies rather than point approximations; he (and Wallis) were first to state the law of momentum conservation correctly. He advanced the wave ("undulatory") theory of light, a key concept being Huygen's Principle, that each point on a wave front acts as a new source of radiation. His optical discoveries include explanations for polarization and phenomena like haloes. (Because of Newton's high reputation and corpuscular theory of light, Huygens' superior wave theory was largely ignored until the 19th-century work of Young, Fresnel, and Maxwell. Later, Planck, Einstein and Bohr, partly anticipated by Hamilton, developed the modern notion of wave-particle duality.)

Huygens is famous for his inventions of clocks and lenses. He invented the escapement and other mechanisms, leading to the first reliable pendulum clock; he built the first balance spring watch, which he presented to his patron, King Louis XIV of France. He invented superior lens grinding techniques, the achromatic eye-piece, and the best telescope of his day. He was himself a famous astronomer: he discovered Titan and was first to properly describe Saturn's rings and the Orion Nebula. He also designed, but never built, an internal combustion engine. He promoted the use of 31-tone music: a 31-tone organ was in use in Holland as late as the 20th century. Huygens was an excellent card player, billiard player, horse rider, and wrote a book speculating about extra-terrestrial life.

As a mathematician, Huygens did brilliant work in analysis; his calculus, along with that of Wallis, is considered the best prior to Newton and Leibniz. He also did brilliant work in geometry, proving theorems about conic sections, the cycloid and the catenary. He was first to show that the cycloid solves the tautochrone problem; he used this fact to design pendulum clocks that would be more accurate than ordinary pendulum clocks. He was first to find the flaw in Saint-Vincent's then-famous circle-squaring method; Huygens himself solved some related quadrature problems. He introduced the concepts of evolute and involute. His friendships with Déscartes, Pascal, Mersenne and others helped inspire his mathematics; Huygens in turn was inspirational to the next generation. At Pascal's urging, Huygens published the first real textbook on probability theory; he also became the first practicing actuary.

Huygens had tremendous creativity, historical importance, and depth and breadth of genius, both in physics and mathematics. He also was important for serving as tutor to the otherwise self-taught Gottfried Leibniz (who'd "wasted his youth" without learning any math). Before agreeing to tutor him, Huygens tested the 25-year old Leibniz by asking him to sum the reciprocals of the triangle numbers.

Takakazu  Seki (Kowa) (ca 1637-1708) Japan

Seki Takakazu (aka Shinsuke) was a self-taught prodigy who developed a new notation for algebra, and made several discoveries before Western mathematicians did; these include determinants, the Newton-Raphson method, Newton's interpolation formula, Bernoulli numbers, discriminants, methods of calculus, and probably much that has been forgotten (Japanese schools practiced secrecy). He calculated π to ten decimal places using Aitkin's method (rediscovered in the 20th century). He also worked with magic squares. He is remembered as a brilliant genius and very influential teacher.

Seki's work was not propagated to Europe, so has minimal historic importance; otherwise Seki might rank high on our list.

Isaac  (Sir)  Newton (1642-1727) England

Newton was an industrious lad who built marvelous toys (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is most famous for his Three Laws of Motion (inertia, force, reciprocal action) and Law of Universal Gravitation. As Newton himself acknowledged, the Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle. (However, since Christiaan Huygens, the other great mechanist of the era and who had also deduced that Kepler's laws imply inverse-square gravitation, considered the action at a distance in Newton's universal gravitation to be "absurd," at least this much of Newton's mechanics must be considered revolutionary. Newton's other intellectual interests included chemistry, theology, astrology and alchemy.) Although this list is concerned only with mathematics, Newton's greatness is indicated by the wide range of his physics: even without his revolutionary Laws of Motion and his Cooling Law of thermodynamics, he'd be famous just for his work in optics, where he explained diffraction and observed that white light is a mixture of all the rainbow's colors. (Although his corpuscular theory competed with Huygen's wave theory, Newton understood that his theory was incomplete without waves, and thus anticipated wave-particle duality.) Newton also designed the first reflecting telescope, first reflecting microscope, and the sextant.

Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (which he called "fluxions"); he shares credit with Leibniz for the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus for several purposes: finding areas, tangents, the lengths of curves and the maxima and minima of functions. In addition to several other important advances in analytic geometry, his mathematical works include the Binomial Theorem, his eponymous numeric method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equation   e^x = ∑ x^k / k!   has been called the "most important series in mathematics.") He contributed to algebra and the theory of equations; he was first to state Bézout's Theorem; he generalized Déscartes' rule of signs. (The generalized rule of signs was incomplete and finally resolved two centuries later by Sturm and Sylvester.) He developed a series for the arcsin function. He developed facts about cubic equations (just as the "shadows of a cone" yield all quadratic curves, Newton found a curve whose "shadows" yield all cubic curves). He proved that same-mass spheres of any radius have equal gravitational attraction: this fact is key to celestial motions. He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. (Like some of the greatest ancient mathematicians, Newton took the time to compute an approximation to π; his was better than Vieta's, though still not as accurate as al-Kashi's.)

Newton is so famous for his calculus, optics and laws of motion, it is easy to overlook that he was also one of the greatest geometers. He solved the Delian cube-doubling problem. Even before the invention of the calculus of variations, Newton was doing difficult work in that field, e.g. his calculation of the "optimal bullet shape." Among many marvelous theorems, he proved several about quadrilaterals and their in- or circum-scribing ellipses, and constructed the parabola defined by four given points. He anticipated Poncelet's Principle of Continuity. An anecdote often cited to demonstrate his brilliance is the problem of the brachistochrone, which had baffled the best mathematicians in Europe, and came to Newton's attention late in life. He solved it in a few hours and published the answer anonymously. But on seeing the solution Jacob Bernoulli immediately exclaimed "I recognize the lion by his footprint."

In 1687 Newton published  Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, although the heliocentric system allowed Kepler to describe the orbits. In Principia Newton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. (In this work Newton also proved important theorems about inverse-cube forces, work largely unappreciated until Chandrasekhar's modern-day work.) The notion that the Earth rotated about the Sun was introduced in ancient Greece, but Newton explained why it did, and the Great Scientific Revolution began. Newton once wrote "Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things." Sir Isaac Newton was buried at Westminster Abbey in a tomb inscribed "Let mortals rejoice that so great an ornament to the human race has existed."

Newton ranks #2 on Michael Hart's famous list of the Most Influential Persons in History. (Muhammed the Prophet of Allah is #1.) Whatever the criteria, Newton would certainly rank first on any list of physicists, or scientists in general, but some listmakers would demote him slightly on a list of pure mathematicians: his emphasis was physics not mathematics, and the contribution of Leibniz (Newton's rival for the title Inventor of Calculus) lessens the historical importance of Newton's calculus. One reason I've ranked him at #1 is a comment by Gottfried Leibniz himself: "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part."

Gottfried Wilhelm von  Leibniz (1646-1716) Germany

Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals."

Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Ten who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. (And his political influence may have been huge: he was a special consultant to both the Holy Roman and Russian Emperors, and was helped arrange for the son of his patron Sophia Wittelsbach, only distantly in line for the British throne, to be crowned King George I of England.)

Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the symbols ∫, df(x)/dx; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He invented more mathematical terms than anyone, including "function," "analysis situ," "variable," "abscissa," "parameter," and "coordinate." His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation ("calculus"), because with "symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished."

Leibniz' thoughts on mathematical physics had some influence. He developed laws of motion that gave different insights from those of Newton. His cosmology was opposed to that of Newton but, anticipating theories of Mach and Einstein, is more in accord with modern physics. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves.

Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π:
        π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

Jacob   Bernoulli (1654-1705) Switzerland

Jacob Bernoulli studied the works of Wallis and Barrow; became friends with Leibniz; tutored Leibniz as well as Jacob's own brother Johann. Jacob developed important methods for integral and differential equations, coining the word "integral." He and his brother were the key pioneers in mathematics during the generations between the era of Newton-Leibniz and the rise of Leonhard Euler.

Jacob liked to pose and solve physical optimization problems. His "catenary" problem (what shape does a clothesline take?) became more famous than the "tautochrone" solved by Huygens. Perhaps the most famous of such problems was the brachistochrone, wherein Jacob recognized Newton's "lion's paw", and about which Johann Bernoulli wrote: "You will be petrified with astonishment [that] this same cycloid, the tautochrone of Huygens, is the brachistochrone we are seeking." Jacob did significant work outside calculus; in fact his most famous work was the Art of Conjecture, a textbook on probability and combinatorics which proves the Law of Large Numbers, the Power Series Equation, and introduces the Bernoulli numbers. Jacob also did outstanding work in geometry, for example constructing perpendicular lines which quadrisect a triangle.

Johann  Bernoulli (1667-1748) Switzerland

Johann Bernoulli learned from his older brother and Leibniz, and went on to become principle teacher to Leonhard Euler. He developed exponential calculus; together with his brother Jacob, he founded the calculus of variations. Johann solved the catenary before Jacob did; this led to a famous rivalry in the Bernoulli family. (No joint papers were written; instead the Bernoullis, especially Johann, began claiming each others' work.) Although his older brother may have demonstrated greater breadth, Johann had no less skill than Jacob, contributed more to calculus, discovered L'Hopital's Rule before L'Hopital did, and made important contributions in physics, e.g. about vibrations, elastic bodies, optics, tides, and ship sails.

It may not be clear which Bernoulli was the "greatest." Johann has special importance as tutor to Leonhard Euler, but Jacob has special importance as tutor to his brother Johann!

Brook  Taylor (1685-1731) England

Brook Taylor invented integration by parts, developed what is now called the calculus of finite differences, developed a new method to compute logarithms, made several other key discoveries of analysis, and did significant work in mathematical physics. His love of music and painting may have motivated some of his mathematics: He studied vibrating strings; and also wrote an important treatise on perspective and vanishing points in drawing which helped develop the fields of both projective and descriptive geometry.

Taylor was one of the few mathematicians of the Bernoulli era who was equal to them in genius, but his work was much less influential. Today he is most remembered for Taylor Series and the associated Taylor's Theorem, but he shouldn't get full credit for this. The method had been anticipated by earlier mathematicians including Gregory, Leibniz, Newton, and, even earlier, Madhava; and was not fully appreciated until the work of later mathematicians like Colin Maclaurin and Lagrange.

Daniel  Bernoulli (1700-1782) Switzerland

Johann Bernoulli had a nephew, three sons and some grandsons who were all also outstanding mathematicians. Of these, the most important was his son Daniel. Johann insisted that Daniel study biology and medicine rather than mathematics, so Daniel specialized initially in mathematical biology. He went on to win the Grand Prize of the Paris Academy no less than ten times, and was a close friend of Euler. He developed partial differential equations, anticipated Fourier series, did important work in statistics and the theory of equations, discovered and proved a key theorem about trochoids, developed a theory of economic risk (motivated by the St. Petersburg Paradox discovered by his cousin Nicholas), but is most famous for his important discoveries in mathematical physics, including the Bernoulli Principle underlying airflight. Daniel Bernoulli is sometimes called "the Founder of Mathematical Physics."