# Greatest Mathematicians born between 1560 and 1699 A.D.

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

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

Galileo discovered the laws of inertia (including rudimentary forms of all three of Newton's laws of motion), falling bodies (including parabolic trajectories), and the pendulum; he also introduced the notion of relativity which later physicists found so fruitful. Galileo discovered important principles of dynamics, including the essential notion that the vector sum of forces produce an acceleration. (Aristotle seems not to have considered the notion of acceleration, though his successor Strato of Lampsacus did write on it.) Galileo is famous for dropping balls of different weights from the Tower of Pisa, but this experiment may not have taken place at all. Instead Galileo derived his famous result that weight cannot determine falling speed via one of the first thought experiments! Galileo may have been first to note that a larger body has less relative cohesive strength than a smaller body. He was a great inventor: in addition to being first to conceive of a pendulum clock and of a thermometer, he developed a new type of pump, the first compound-lens microscope, and the best telescope, 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. Galileo tried to measure the speed of light, but it was too fast for him. However 66 years after Galileo discovered Jupiter's moons and proposed using them as a clock, the astronomer Roemer inferred the speed of light from that 'clock': the clock had a discrepancy of up to seven minutes depending on the Earth-Jupiter distance.) Galileo's several other astronomical achievements also included confirming that the Milky Way was made up of stars, and discovering sunspots and lunar craters.

Perhaps Galileo's most important astronomical discovery was the phases of Venus. Ptolemy's epicycles, Copernicus' epicycles, Brahe's hybrid system, and Kepler's ellipses all gave almost the same solutions for planets' apparent positions, but Ptolemy's system gave completely wrong predictions about the phases of Venus. Galileo's observation of Venus' phases was the critical discovery which finally forced acceptance of heliocentrism. (Galileo's pursuit of Venus' phases may have been inspired in part by Galileo's student, Benedetto Castelli.) Just as modern inventors sometimes mail sealed envelopes to an arbiter to establish precedence, Galileo sent an anagram to Kepler, to later prove the date of his unpublished discovery. (The anagram was Haec immatura a me iam frustra leguntur o.y. which letters can be rearranged to Cynthiae figuras aemulatur mater amorum. These two Latin sentences translate respectively as "I am now bringing these unripe things together in vain, Oy!" and "The mother of love [Venus] copies the forms of Cynthia [the Moon].")

Galileo's contributions outside physics and astronomy were also enormous: He made discoveries with the microscope he invented, and made several important contributions to the early development of biology. Perhaps Galileo's most important contribution was the Doctrine of Uniformity, the postulate that there are universal laws of mechanics, in contrast to Aristotelian and religious notions of separate laws for heaven and earth.

Galileo is often called the "Father of Modern Science" because of his emphasis on experimentation. His use of a ramp to discover his Law of Falling Bodies was ingenious. (For his experiments he started with a water-clock to measure time, but found the beats reproduced by trained musicians to be more convenient.) He understood that results needed to be repeated and averaged (he minimized mean absolute-error for his curve-fitting criterion, two centuries before Gauss and Legendre introduced the mean squared-error criterion). For his experimental methods and discoveries, his laws of motion, and for (eventually) helping to spread Copernicus' heliocentrism, Galileo may have been the most influential scientist 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. Some of his other ideas were wrong; for example, he dismissed Kepler's elliptical orbits and notion of gravitation and published a very faulty explanation of tides.) Despite his extreme importance to mathematical physics, Galileo doesn't usually appear on lists of greatest mathematicians. However, Galileo did do work in pure mathematics; he derived certain centroids and the parabolic shape of trajectories using a rudimentary calculus, and mentored Bonaventura Cavalieri, who extended Galileo's calculus; he named (and may have been first to discover) the cycloid curve. Moreover, Galileo was one of the 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."

(In my List I try to follow a consensus of mathematical historians. Galileo made many mistakes, but top thinkers like Einstein declare that his many contributions outweigh his flaws. However one historian of mathematics argues that Galileo's flaws are huge, and his contributions exaggerated. One reason I give Galileo a high ranking is that he was apparently first to deduce correctly, 1800 years after Archimedes, how that genius measured the density of his King's gold crown -- see Archimedes' mini-bio.)

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 furthered the theory of the camera obscura, telescopes built from two convex lenses, and atmospheric refraction. The question of human vision had been considered by many great scientists including Aristotle, Euclid, Ptolemy, Galen, Alkindus, Alhazen, and Leonardo da Vinci, but it was Kepler who was first to explain the operation of the human eye correctly and to note that retinal images will be upside-down. Kepler developed a rudimentary notion of universal gravitation, and used it to produce the best explanation for tides before Newton; however he seems not to have noticed that his empirical laws implied inverse-square gravitation. Kepler noticed Olbers' Paradox before Olbers' time and used it to conclude that the Universe is finite. 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."
Kepler also once wrote "Mathematics is the archetype of the beautiful."

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 on stationary points. 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 Descartes) 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.)

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 based on the epicycloid, 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; Descartes admired Desargues but was disappointed his friend didn't apply algebra to his geometric results as Descartes did; Desargues' writing was poor; and one of his best pupils (Blaise Pascal himself) turned away from math, so Desargues' work was largely ignored (except by Philippe de La Hire, Desargues' other prize pupil) until Poncelet rediscovered it almost two centuries later. (Copies of Desargues' own works surfaced about the same time.) For this reason, Desargues may not be important enough to belong in the Top 100, despite that he may have been among the greatest natural geometers ever.

Descartes' 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 both analytic geometry and symbolic algebraic notation and is therefore called the "Father of Modern Mathematics." His use of equations to partially solve the geometric Problem of Pappus revolutionized 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. 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 (an early form of the Gauss-Bonnet result so key to much mathematics), and an improved solution to the Delian problem (cube-doubling). While studying lens refraction, he invented the Ovals of Descartes. He improved mathematical notation (e.g. the use of superscripts to denote exponents). He also discovered Euler's Polyhedral Theorem, F+V = E+2. Descartes was very influential in physics and biology as well, e.g. developing laws of motion which included a "vortex" theory of gravitation; but most of his scientific work outside mathematics was eventually found to be incorrect.

Descartes has an extremely high reputation and would be ranked even higher by many list makers, but whatever his historical importance his mathematical skill was not in the top rank. Some of his work was borrowed from others, e.g. from Thomas Harriot. He had only insulting things to say about Pascal and Fermat, each of whom was much more brilliant at mathematics than Descartes. (Some even suspect that Descartes arranged the destruction of Pascal's lost Essay on Conics.) And Descartes made numerous errors in his development of physics, perhaps even delaying science, with Huygens writing "in all of [Descartes'] physics, I find almost nothing to which I can subscribe as being correct." Even the historical importance of his mathematics may be somewhat exaggerated since others, e.g. Fermat, Wallis and Cavalieri, were making similar discoveries independently.

Cavalieri worked in analysis, geometry and trigonometry (e.g. discovering a formula for the area of a spherical triangle), but is most famous for publishing works on his "principle of indivisibles" (calculus); these were very influential and inspired further development by Huygens, Wallis and Barrow. (His calculus was partly anticipated by Galileo, Kepler and Luca Valerio, and developed independently, though left unpublished, by Fermat.) Among his theorems in this calculus was
lim (n→∞) (1m+2m+ ... +nm) / nm+1 = 1 / (m+1)
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 was the most brilliant mathematician of his era and, along with Descartes, 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. Lagrange considered Fermat, rather than Newton or Leibniz, to be the inventor of calculus. Fermat was first to study certain interesting curves, e.g. the "Witch of Agnesi". 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 (that (ap-a) is a multiple of p whenever p is prime); the n = 4 case of his conjectured Fermat's Last Theorem (he may have proved the n = 3 case as well); and Fermat's Christmas Theorem (that any prime (4n+1) can be represented as the sum of two squares in exactly one way) which may be considered the most difficult theorem of arithmetic which had been proved up to that date. Fermat proved the Christmas Theorem with difficulty using "infinite descent," but details are unrecorded, so the theorem is often named the Fermat-Euler Prime Number Theorem, with the first published proof being by Euler more than a century after Fermat's claim. Another famous conjecture by Fermat is that every natural number is the sum of three triangle numbers, or more generally the sum of k k-gonal numbers. As with his "Last Theorem" he claimed to have a proof but didn't write it up. (This theorem was eventually proved by Lagrange for k=4, the very young Gauss for k=3, and Cauchy for general k. Diophantus claimed the k=4 case but any proof has been lost.) I think Fermat's conjectures were impressive even if unproven, and that this great mathematician is often underrated. (Recall that his so-called "Last Theorem" was actually just a private scribble.)

Fermat developed a system of analytic geometry which both preceded and surpassed that of Descartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration. Although Kepler anticipated it, Fermat is credited with Fermat's Theorem on Stationary Points (df(x)/dx = 0 at function extrema), the key to many problems in applied analysis. 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 Descartes is more famous than Fermat, and Descartes' writings were more influential. Whatever one thinks of Descartes as a philosopher, however, it seems clear that Fermat was the better mathematician. Fermat and Descartes 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 Maupertuis (though Maupertuis himself, like Descartes, had an incorrect explanation of refraction). Fermat and Descartes independently discovered analytic geometry, but it was Fermat who extended it to more than two dimensions, and followed up by developing elementary calculus.

Roberval was an eccentric genius, underappreciated because most of his work was published only long after his death. He did early work in integration, following Archimedes rather than Cavalieri; he worked on analytic geometry independently of Descartes. With his analysis he was able to solve several difficult geometric problems involving curved lines and solids, including results about the cycloid which were also credited to Pascal and Torricelli. Some of these methods, published posthumously, led to him being called the Founder of Kinematic Geometry. He excelled at mechanics, worked in cartography, helped Pascal with vacuum experiments, and invented the Roberval balance, still in use in weighing scales to this day. He opposed Huygens in the early debate about gravitation, though neither fully anticipated Newton's solution.

Torricelli was a disciple of Galileo (and succeeded him as grand-ducal mathematician of Tuscany). He was first to understand that a barometer measures atmospheric weight, and used this insight to invent the mercury barometer and to create a sustained vacuum (then thought impossible). (Descartes conjectured, and Pascal later confirmed, that this same device could also be used as an altimeter.) Torricelli was a skilled craftsman who built the best telescopes and microscopes of his day. As mathematical physicist, he extended Galileo's results, was first to explain winds correctly, and discovered several key principles including Torricelli's Law (water drains through a small hole with rate proportional to the square root of water depth). In mathematics, he applied Cavalieri's methods to solve difficult mensuration problems; he also wrote on possible pitfalls in applying the new calculus. He discovered Gabriel's Horn with infinite surface area but finite volume; this "paradoxical" result provoked much discussion at the time. (At first Torricelli guessed it was a mistake, just another pitfall of calculus, though he later accepted its validity.) He also solved a problem due to Fermat by locating the isogonic center of a triangle. Torricelli was a significant influence on the early scientific revolution; had he lived longer, or published more, he would surely have become one of the greatest mathematicians of his era.

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. He made major advances in analytic geometry, but also contributions to algebra, geometry and trigonometry. Unlike his contemporary Huygens, who took inspiration from Euclid's rigorous geometry, Wallis embraced the new analytic methods of Descartes and Fermat. He is especially famous for using negative and fractional exponents (though Oresme had introduced fractional exponents three 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 a polymath; his non-mathematical work included a highly respected English grammar; he introduced the (still controversial) linguistic concept of phonesthesia.

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; he invented the symbol (and used 1/∞ to denote infinitesimal). Wallis coined several terms including momentum, 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

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 Gérard Desargues (1591-1661), he was a key pioneer of projective geometry. He also made important early contributions to calculus; indeed it was his writings that inspired Leibniz. 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, wrote the philosophical treatise Pensées ("We arrive at truth, not by reason only, but also by the heart"), and died at an early age. Pascal is ranked #67 on the Pantheon Popular/Productive List. Many think that had he devoted more years to mathematics, Pascal would have been one of the greatest mathematicians ever.

Christiaan Huygens (or Hugens, Huyghens) was second only to Newton as the greatest mechanist and theoretical physicist of his era; he inspired Newton, who praised him above the other 17th-century mathematicians. 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 was first to give the correct "equation of time" relating sundial time to absolute time. 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, was first to properly describe Saturn's rings and the Orion Nebula, and estimated the Sun-Earth distance far more accurately than any predecessor. He also designed, but never built, an internal combustion engine. He promoted the use of an equal-tempered 31-tone music scale to avoid the tuning errors in Stevin's 12-tone scale; 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 Descartes, 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.

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.

James Gregory (Gregorie) was the outstanding Scottish genius of his century. Had he not died at the age of 36, or if he had published more of his work, (or if Newton had never lived,) Gregory would surely be appreciated as one of the greatest mathematicians of the early Age of Science. Inspired by Kepler's work, he worked in mechanics and optics; invented a reflecting telescope; and is even credited with using a bird feather as the first diffraction grating. But James Gregory is most famous for his mathematics, making many of the same discoveries as Newton did: the Fundamental Theorem of Calculus, interpolation method, and binomial theorem. He developed the concept of Taylor's series and used it to solve a famous semicircle division problem posed by Kepler and to develop trigonometric identities, including
tan-1x = x   -   x3/3   +   x5/5   -   x7/7   +   ...   (for |x| < 1)
Gregory anticipated Cauchy's convergence test, Newton's identities for the powers of roots, and Riemann integration. He may have been first to suspect that quintics generally lacked algebraic solutions, as well as that π and e were transcendental. He produced a partial proof that the ancient "Squaring the Circle" problem was impossible.

Gregory declined to publish much of his work, partly in deference to Isaac Newton who was making many of the same discoveries. Because the wide range of his mathematics wasn't appreciated until long after his death, Gregory lacks the historic importance to qualify for the Top 100.

Although this list is concerned only with mathematics, Newton's greatness is indicated by the huge range of his physics: even without his Laws of Motion, Gravitation and Cooling, he'd be famous just for his revolutionary work in optics, where he explained diffraction, observed that white light is a mixture of all the rainbow's colors, noted that purple is created by combining red and blue light and, starting from that observation, was first to conceive of a color hue "wheel." (The mystery of the rainbow had been solved by earlier mathematicians like Al-Farisi and Descartes, but Newton improved on their explanations. Most people would count only six colors in the rainbow but, due to Newton's influence, seven -- a number with mystic importance -- is the accepted number. Supernumerary rainbows, by the way, were not explained until the wave theory of light superseded Newton's theory.) He noted that his dynamical laws were symmetric in time; that just as the past determines the future, so the future might, in principle, determine the past. Newton almost anticipated Einstein's mass-energy equivalence, writing "Gross Bodies and Light are convertible into one another... [Nature] seems delighted with Transmutations." Ocean tides had intrigued several of Newton's predecessors; once gravitation was known, the Moon's gravitational attraction provided the explanation -- except that there are two high tides per day, one when the Moon is farthest away. With clear thinking the second high tide is also explained by gravity but who was the first clear thinker to produce that explanation? You guessed it! Isaac Newton. (The theory of tides was later refined by Laplace.) Newton's earliest fame came when he discovered the problem of chromatic aberration in lenses, and designed the first reflecting telescope to counteract that aberration; his were the best telescopes of that era. He also designed the 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. (I've mentioned several ancient mathematicians who used Archimedes' approach or its variations to get closer and closer approximations to π. With a clever application of integral calculus, Newton found a convergent series that leap-frogged all prior approaches.) Although Descartes is renowned as the inventor of analytic geometry, he and followers like Wallis were reluctant even to use negative coordinates, so one historian declares Newton to be "the first to work boldly with algebraic equations." In addition to several other important advances in analytic geometry, Newton's mathematical works include the Binomial Theorem, his eponymous interpolation method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (The equation   ex = xk / k!   has been attributed to Newton and called the "most important series in mathematics," but, although he published some related trignometric formulae, he doesn't seem to have published the exponential series explicitly prior to Bernoulli's discoveries circa 1690.) He contributed to algebra and the theory of equations; he was first to state Bézout's Theorem; he generalized Descartes' 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, using a purely geometric argument of awesome ingenuity, that same-mass spheres (or hollowed spheres) of any radius have equal gravitational attraction: this fact is key to celestial motions. (He also proved that objects inside a hollowed sphere experience zero net attraction.) He discovered Puiseux series (and proved the associated theorem) 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 gravitation and motion, it is easy to overlook that he was also one of the very greatest geometers. He was first to fully solve the famous Problem of Pappus, and did so with pure geometry. Building on the "neusis" (non-Platonic) constructions of Archimedes and Pappus, he demonstrated cube-doubling and that angles could be k-sected for any k, if one is allowed a conchoid or certain other mechanical curves. He also built on Apollonius' famous theorem about tangent circles to develop the technique now called hyperbolic trilateration. Despite the power of Descartes' analytic geometry, Newton's achievements with synthetic geometry were surpassing. 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." His other marvelous geometric theorems included several about quadrilaterals and their in- or circum-scribing ellipses. He constructed the parabola defined by four given points, as well as various cubic curve constructions. (As with Archimedes, many of Newton's constructions used non-Platonic tools.) 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, often called 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. With the key mystery of celestial motions finally resolved, the Great Scientific Revolution began. (In his work Newton also proved important theorems about inverse-cube forces, work largely unappreciated until Chandrasekhar's modern-day work.) 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 or second 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."

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's. 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 Fifteen who was never the greatest living algorist or theorem prover. I 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. Leibniz was also one of the earliest intellectuals whose contributions to social science led to the 'Enlightenment.' Leibniz also had political influence: he consulted to both the Holy Roman and Russian Emperors; another of his patrons was Sophia Wittelsbach (Electress of Hanover), who was only distantly in line for the British throne, but was made Heir Presumptive. (Sophia died before Queen Anne, but her son was 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; but Frege, himself sometimes called the greatest logician of all, wrote that Leibniz was "in a class by himself." Mathematical innovations attributed to Leibniz include the notations ∫f(x)dx, df(x)/dx, ∛x, and even the use of a·b (instead of a X b) for multiplication; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He worked in number theory, conjecturing Wilson's Theorem. He invented more mathematical terms than anyone, including function, analysis situ, variable, abscissa, parameter and coordinate. He also coined the word transcendental, proving that sin() was not an algebraic function. 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 the 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 was one of the first to articulate the law of energy conservation and may have written on the principle of least action. He developed laws of motion that gave different insights from those of Newton; his views on cosmology anticipated theories of Mach and Einstein and are more in accord with modern physics than are Newton's views. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves. But despite his huge contributions to mathematics and physical sciences, Leibniz is most renowned for his contributions to philosophy, where his "sublime eloquence" was compared to Plato. He made too many contributions to several branches of philosophy to summarize here; perhaps most famous were his theory of monads in Monadology and his often-parodied claim that we live in "the best of all possible worlds."

Although others had 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 studied the works of Wallis and Barrow; he and Leibniz became friends and tutored each other. 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. While studying compound interest he introduced the constant e, though it was given that symbol by Euler. He is credited with the invention of polar coordinates (though Newton and Alberuni had also discovered them). Jacob also did outstanding work in geometry, for example constructing perpendicular lines which quadrisect a triangle.

Johann Bernoulli learned from his older brother and Leibniz, and went on to become principal 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'Hôpital's Rule before L'Hôpital 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. Johann's son Daniel is also a candidate for greatest Bernoulli.

De Moivre was an important pioneer of analytic geometry and, especially, probability theory. (He and Laplace may be regarded as the two most important early developers of probability theory.) In probability theory he developed actuarial science, posed interesting problems (e.g. about derangements), discovered the normal and Poisson distributions, and proposed (but didn't prove) the Central Limit Theorem. De Moivre was first to introduce the use of generating functions. He was first to discover a closed-form formula for the Fibonacci numbers; and he developed an early version of Stirling's approximation to n!. He discovered De Moivre's Theorem:       (cos x + i sin x)n = cos nx + i sin nx

He was a close friend and muse of Isaac Newton, who allegedly told people who asked about Principia: "Go to Mr. De Moivre; he knows these things better than I do."

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 in drawing which helped develop the fields of both projective and descriptive geometry. His work in projective geometry rediscovered Desargues' Theorem, introduced terms like vanishing point, and influenced Lambert.

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 crucially important Theorem. 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 Maclaurin and Lagrange.

Maclaurin received a University degree in divinity at age 14, with a treatise on gravitation. He became one of the most brilliant mathematicians of his era. He wrote extensively on Newton's method of fluxions, and the theory of equations, advancing these fields; worked in optics, and other areas of mathematical physics; but is most noted for his work in geometry. Lagrange said Maclaurin's geometry was as beautiful and ingenious as anything by Archimedes. Clairaut, seeing Maclaurin's methods, decided that he too would prove theorems with geometry rather than analysis. Maclaurin did important work on ellipsoids; for his work on tides he shared the Paris Prize with Euler and Daniel Bernoulli. As Scotland's top genius, he was called upon for practical work, including politics. Although Maclaurin's work was quite influential, his influence didn't really match his outstanding brilliance: he failed to adopt Leibnizian calculus with which great progress was being made on the Continent, and much of his best work was published posthumously. Many of his famous results duplicated work by others: Maclaurin's Series was just a form of Taylor's series; the Euler-Maclaurin Summation Formula was also discovered by Euler; and he discovered the Newton-Cotes Integration Formula after Cotes did. His brilliant results in geometry included the construction of a conic from five points, but Braikenridge made the same discovery and published before Maclaurin did. He discovered the Maclaurin-Cauchy Test for Integral Convergence before Cauchy did. He was first to discover Cramer's Paradox, as Cramer himself acknowledged. Colin Maclaurin found a simpler and more powerful proof of the fact that the cycloid solves the famous brachistochrone problem.

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