In compiling this list, I've considered contributions outside mathematics. Newton contributed little to number theory, for example, but I consider him to have great breadth because of his physics, which is also his main influence. (I do give a much lower "weight" to contributions outside pure mathematics.) With different criteria, the List would appear much different. For example, here's a different ordering for the Top Twelve that might make more sense to some:
1 Leonhard Euler  4 Archimedes  7 Bernhard Riemann  10 Isaac Newton  
2 Carl F. Gauss  5 Alexandre Grothendieck  8 David Hilbert  11 Henri Poincaré  
3 Euclid of Alexandria  6 JosephLouis Lagrange  9 Pierre de Fermat  12 Évariste Galois 
(The list has become fairly stable: The only major change in 2013 was to expand the List to a Top 105 for which the five new names ended up being Daniel Bernoulli, George Birkhoff, Felix Hausdorff, George Pólya and (after adding but then demoting Thabit ibn Qurra) Israel Gelfand. In 2014 I added Tarski and Frege, two of the greatest logicians ever. (Erdös and Lobachevsky were removed to make space.) Given these additions, it made sense to demote two other pure logicians on the list: Gödel and Boole.
During June and July of 2016 I made a large number of smallish changes. The Top 30 and Top 60 remained the same, though the ordering within these groups changed. There were changes in the Top 75, with Huygens and Poncelet demoted to make room for Vieta and Lie. In addition to the four major moves just mentioned, several others got demoted by four or five ranks.
September 2016 I suddenly made several overdue largish changes: Panini and Weil joined the Top 60 evicting Grassman and Monge. Also, Borel, Jacob Bernoulli and others were promoted, while Hadamard, Hipparchus, d'Alembert, etc. were demoted.
After expanding the List to 200 to include many modern mathematicians I was missing, I found that there was much more scope for ongoing changes. For exaqmple, I promoted Claude Shannon to the Top 200. This may offend some  the List is still missing deeper mathematicians  but Shannon's several contributions to the Information Revolution give him great historical importance.
I wanted this to be a List of Great Mathematicians of the Past, but several email correspondents insisted that I include modern mathematicians. So I now have a list of 200 that includes 24 living mathematicians and another ten who died in the 21st century. However most of these are placed at the end of the list, in slots #151#200. If someone were to allocate these "fairly" in accordance with my own rankings, one would expect to see about 15 names from the 19th century or earlier removed from the top Hundred and replaced with some of these recent mathematicians. And there are probably another two dozen or so people alive today who await further distinguishment but will belong on a future Greatest 100 Mathematicians List!
Again using my list of 200 as the reference, 35 were born before 1500 A.D., then there is, very approximately, one born every ten years during the 1500's, one every six years during the 1600's, two per decade during the 1700's, about 6 per decade during the early 1800's, and very roughly 1 per year from 1875 to 1937. After that it tapers off, with five born in the 1940's, four in the 1950's, and finally one "greatest" mathematician born in 1966 and one in 1975.
Since the List emphasizes the past, Grothendieck (who barely makes the 1930 birth "cutoff") does not appear in my Top Ten. Other Top Ten lists might omit Lagrange and Leibniz to make room for Grothendieck, Galois or Descartes, but Lagrange and Leibniz had extreme genius and historical importance.
There's overlap between top mathematicians and top practitioners of other areas, especially physics. I'm afraid that, almost unconsciously, I've allowed greatness at physics (or other areas) to blur in and affect my ratings. Newton belongs somewhere on the Top Ten, but he'd not get the #1 rank without his physics. Similarly, I've promoted Leibniz, von Neumann, Weyl, Pythagoras, Pascal, Archytas, Alhazen, Kepler, Grassmann, Huygens, and several others because of their work outside pure mathematics. (As an extreme case, Leonardo da Vinci appears on the extended list.) Does this affect the validity of my rankings? Perhaps  But I always encourage readers, who might have different criteria, to use this List as a startingpoint in constructing their own List.
Even an expert with very clear criteria would have trouble assigning exact ranks to mathematicians' "greatness," if such a thing even has meaning; and I am not an expert and do not have clear criteria. Nevertheless I am largely satisfied with the present rankings. I believe that the Top Ten are the only possible Top Ten! I think the Top Thirty (or, rather the Top ThirtyOne!), and the Top Sixty are also reasonable.
In recent years I've made minor changes to the Top Thirty, demoting Eudoxus and AlKhowarizmi and promoting Hamilton and Aryabhata. I'd also like to make room for Apollonius in the Top 30. (Despite his historical importance and fame, there were several Islamic mathematicians with more genius than AlKhowarizmi, who borrowed much from the great Hindu mathematician Aryabhata.)
The only changes to the Top 60 List since early 2013 have been to replace Boole, Fourier, Poncelet and Vieta with Borel, Grassmann, Liouville and Peano. Liouville at #54 may still be too low; he was a mathematician of outstanding breadth.
A major change to the Top 90 was to include five applied mathematicians: Galileo, Einstein, Maxwell, Aristotle, Cardano. To make room for these five, I needed small demotions of Hausdorff, Artin and Pólya; a large demotion of Wallis, and a huge demotion of Boole.
Of course there will be quibbling about the precise order within these lists. Some think Ramanujan isn't historically important enough to make the Top 30, but please judge my list by the criteria I use (breadth of work and depth of genius also considered), not the criteria you'd have used.
I'm still willing to try to improve the List. :) No List will please everybody, so please complain only if your choice for Top 75 is missing from my Top 150 (or if I have someone in the Top 75 who doesn't even belong in the Top 150). Please Email suggestions for the List (and, more importantly, corrections to the minibios) to me.
To qualify, a mathematician must have historical importance, "depth" and "breadth." Historical importance (or influence) deals with the question "Did this person change the course of mathematical history?" Many lists consider only historical influence, and will end up with a very different list from mine. Ramanujan had only minor influence, but appears high on my list because of his great genius.
When I say a mathematician has "Depth," I mean that his work was particularly creative, revolutionary, or difficult. Cantor and Gödel have great "depth" because their work was especially creative and revolutionary. Weierstrass and Dirichlet have great "depth" because they were able to prove difficult theorems that had stumped others.
With "Breadth" I promote mathematicians who did excellent work in varied fields. For example, I rank Fermat higher than many would because he was a key early developer of both analytic geometry AND number theory and also did good work in geometry, probability, and even optics. Leibniz did not prove any particularly difficult theorem, but is ranked high because of his historical importance and great breadth. Due to outstanding breadth, I also rank Von Neumann and Leonardo 'Fibonacci' higher than others might. On the other hand, Jakob Steiner was an outstanding genius whom I've "demoted" somewhat due to his focus exclusively on geometry.
Of course, this is all arbitrary and fuzzy. The exact "rankings" are just my feelings ... and I often rearrange the list on whim!
Some will object to my decision to ignore mathematicians born after 1930, but this decision seemed clearly right to me for several reasons:
Seven born between 1910 and 1930 appear in the Top 105. I'm quite unsure whether my "rankings" of these relatively recent mathematicians will meet the test of time. Adding more recent mathematicians would just increase my anxiety!
Anyway, my List isn't intended to be anything but a starting point. You're welcome to adopt it with additions and subtractions as you wish!
There are five men who are among the most important mathematical scientists ever, yet may lack the importance as pure mathematicians for a list such as this  Maxwell, Einstein, Galileo, Aristotle and Cardano. I'd definitely want to include them in any List of Greatest Mathematicans with size 100 or more, but am not sure where on the List to put them; instead I've just set them aside as an extra quintet that brings the List to 105 total.
There are several other outstanding physicists that may seem to belong in the same category, for examples: Huygens, Kepler, Alhazen. However each of these did produce work that affected the development of pure mathematics (though I've considered their importance to physics in their high rankings). The Maxwell quintet, on the other hand, lack the mathematical importance to belong on the list at all, and are mentioned solely because of their huge importance as applied mathematicians.
Other top theoretical mathematical physicists include Dirac, who is on the List of 200; and Boltzmann, Feynman, Heisenberg and Schrödinger who are not.
I've tried to do a good job of ranking the mathematicians, but realize it's a silly conceit, and that no ranking would satisfy everyone.
Some will wonder, not whether my rankings are "correct," but whether Great Mathematicians should be ranked at all, especially by someone like me, with no apparent qualifications. I ask myself this as well. But to simply list "100 Great Mathematicians" would be even less satisfying: it would seem silly to have both Archimedes and Legendre appear on the same single list. And to make, for example, two lists ("50 Greatest" and "50 Near Greats") would combine the worst of both approaches: How could one justify the distinction between the #50 slot and the #51 slot? Like it or not, the ranking is needed for the List to have any value. I've tried to justify the rankings in the minibios.
As a separate matter, some of my rankings are controversial, despite that I've tried to base them partly on expert opinions I've gleaned from books and Internet resources. One correspondent thought Pascal should rank above Fibonacci. (This was one of the correspondents who thought ranking at all was wrong!) Since this is probably a popular opinion, I'll comment here. Pascal was certainly a spectacular genius, but had very low actual influence. Wallis and Cavalieri, who are each shown far below Pascal in rank, were each more influential to the early development of calculus; moreover Pascal's brilliant geometry was inspired in large measure by Desargues' work. I wonder if I've placed Pascal too high, not too low. Leonardo `Fibonacci', on the other hand was a versatile and important teacher who was one of the very best number theorists before Fermat. It isn't widely acknowledged but Leonardo proved the n=4 case of FLT more than 400 years before Fermat did.
Many people view Newton, Gauss and Archimedes as an almost "Divine" Trinity, being the three greatest mathematicians ever, while others would add Euler and make this a Divine Quaternity. Euler was superlative in several ways and it is tempting to rank him above Gauss, but Gauss was the greatest theorem prover ever and handily solved several problems that had stumped Euler. I just keep changing my mind about the best order to rank the four greatest. If you still disapprove of my rankings within the Top Four, just pretend I've ranked them all as tied for #1 !
The criteria of "depth" (work that was particularly creative, revolutionary, or difficult) and "historical importance" are probably not controversial, but my insistence on "breadth" (excellent work in multiple fields) may result in rankings different from others. Since the fuzzy measures of "depth" and "breadth" apply unequally to the candidates, the final "rankings" become arbitrary. Leibniz may lack the "depth" for the Top Ten, but his breadth and importance are enormous. Abel, Weierstrass and Dirichlet probably have less importance and "breadth" than others in the Top Thirty but, as measured by skill at proving difficult theorems, they each had great "depth." Some think Lebesgue should rank higher because of his importance, but, lacking "breadth," he should perhaps be lower. Et cetera. In any event, I hope those commenting on my rankings will base their comments on my criteria, not the criteria they would have chosen!
There are two versions of the list; the content is identical but greatmm.htm is the start of a sevenpage set containing the List and biographies, while mathmen.htm combines the List of 100 and bios into a single, very large, page. (gmat200.htm is an even longer page with the List expanded to 200 names.)
There are many living mathematicians with extraordinary genius, who will certainly appear on a future List like this, but for several reasons it would be uncomfortable or difficult to include them on my list. Therefore I've adopted 1930 as an arbitrary cutoff for birth year. Even with this cutoff, the List includes eight people still living, or who were alive after 1986. These names are Atiyah, Chern, Gelfand, Grothendieck, Kolmogorov, Selberg, Serre, Weil. While most of those just named exhibited great breadth, many of the great geniuses born after 1930 specialized of necessity. This is one reason I include Conway and Milnor on an expanded list even though they were born after 1930  they each exhibited outstanding breadth.
The minibiographies are of different lengths. Some of the greatest (e.g. Jacobi) have biographies much shorter than less obvious candidates like Kepler. This is due in part to a desire to justify Kepler's inclusion, while Jacobi's inclusion is not in doubt.
I've tried to add a quotation to each of the minibiographies: either something that genius said, or something some other genius said about him.
I've learned a great deal while preparing this list, not only about Renaissance and Modern mathematicians, but about ancient mathematics as well. While preparing the very brief summary of ancient mathematics I stumbled upon descriptions of Babylonian Multiplication. In the note you'll see why this came as a pleasant surprise to me. I'm not really qualified to make a list like this  it started as a practice exercise while learning HTML tags!  but many Websurfers were stumbling on My List of Mathematicians, so I've devoted considerable effort to making this a list I can be proud and confident about. By now, I've devoted many hours to reading biographies, and reacting to others' opinions, and by now I'm fairly satisfied with the validity of my List of Greatest Mathematicians, but I'd be happy to make it better!
I had trouble deciding whether to include great mathematical physicists like Maxwell, Einstein, and Galileo, who would not qualify for the List if only their contributions to pure mathematics are considered. I've changed my mind backandforth about whether to include them, and finally compromised by adding these names (and Aristotle and Cardano) as a special addendum to increase the List to 105. (In the following discussion references to my "List of 100" usual refer to the complete List of 105.)
Hipparchus, Huygens, Kepler and Daniel Bernoulli are also great mathematical physicists, but I have included them in the main List of 100 because of their great historical importance in the development of mathematics.
I get many comments that Einstein doesn't belong on a List of Greatest Mathematicians, despite that I indicate my reasons in his minibio. (I admit that this is my List, and I may have occasionally allowed whims to influence some of the choices. I might not have included Omar alKhayyám if it weren't for his poetry.)
But I even receive comments that Einstein wasn't even a great physicist, despite that his 1905 papers revolutionized physics, and that his 1915 General Relativity has been called the most creative physics ever. I won't comment on this fringe iconoclastic view except to show the comments of Hilbert (not known for humility) on the matter: "Every boy in the streets of Gottingen understands more about fourdimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
And, of course, all great scientists have built on others' work: Cantor and Gödel are regarded as two of the most original thinkers ever, yet Dedekind anticipated much of Cantor's work, and von Neumann's thinking inspired Gödel. Abel's theorem of quintics was first stated and partially proved by Ruffini; and so on.
While preparing the minibios, I was struck by how many great mathematicians emphasized the beauty of their work. The quotations I've chosen by Kepler, Boole, Cayley, Hardy, Weyl, Dirac and Banach all contain the word "beauty." If words like "poetry" (or "ecstacy"  Kepler) are considered, d'Alembert, Steiner, Weierstrass, Kovalevskaya, and Weil can be added to that list.
Betrand Russell wrote "Mathematics ... possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
Buckminster Fuller once said "When I am working on a problem, I never think about beauty. I think only of how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong."
Gosta MittagLeffler wrote "The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination."
In his memorial address for Minkowski, Hilbert said "Our science, which we loved above everything, had brought us together. It appeared to us as a flowering garden. In this garden there are beaten paths where one may look around at leisure and enjoy oneself without effort, especially at the side of a congenial companion. But we also liked to seek out hidden trails and discovered many a novel view, beautiful to behold, so we thought, and when we pointed them out to one another our joy was perfect."
In the minibios of ancient mathematicians I state some traditions that are disputed. Assertions that Hindu mathematicians knew the laws of motion may be exaggerated (though recently I found that the writings of alBiruni confirm some ancient Hindu knowledge often ignored in the West.) Some historians do not believe Thales predicted a solar eclipse. Though the Pythagorean school had great influence, some historians believe Pythagoras himself was mythical. Some say that the compass/straightedge construction regime often associated with Plato was introduced only later by Apollonius. Some historians believe that alKhayyám the mathematician and alKhayyám the poet were two different people. I am certainly not qualified to arbitrate these controversies, but I will say that my readings in social sciences have taught me to be very skeptical of skeptics. And I wonder if those who think Khayyam the poet was not the astronomer and mathematician have read poems such as:
Ah, by my Computations, People say,or
Reduce the Year to better reckoning?  Nay,
'Twas only striking from the Calendar
Unborn Tomorrow and dead Yesterday.
Up from Earth's Centre through the seventh Gate
I rose, and on the Throne of Saturn sate,
And many Knots unravel'd by the Road;
But not the Knot of Human Death and Fate.
 Omar Khayyam (trans. by Edward Fitzgerald)
The simplified minibios ignore some first discoveries. Neither Huygens nor Leibniz was first to sum the reciprocals of the triangle numbers; it was Pietro Mengoli (16251686).
I don't list Lorenzo Mascheroni (17501800) who is credited with the amazing theorem that straightedge is unnecessary for Platonic constructions; that theorem was actually first discovered by the forgotten Jorgen Mohr (16401697).
Many questions of first discovery can be tied to "threads": The biggest threads in the whole of mathematical physics are the issues of optics and celestial motion. One question where these come together is: What is the speed of light? Christiaan Huygens and/or Isaac Newton seem to get credit for deducing the speed of light from Danish astronomer Ole Roemer's 1675 observation of Io's transits of Jupiter. (Galileo is somethimes shown as first, but he didn't even show it to be finite, merely at least Mach 10.)
There are several threads which flow through the stories of the greatest mathematicians. The elementary bases of science like mensuration methods, light refraction, etc. qualify. (The development and understanding of lenses by Alhazen, Galileo, Fermat, Huygens, Newton, etc. is a major thread in the early history of science . So are other issues in optics, like understanding rainbows.)
Several exercised their talents by computing π, or by trisecting the general angle, or constructing the regular heptagon. Some worked in codebreaking. Several are closely involved with the theory of celestial motions, or the theory of music.
Several of the mathematicians worked with ancient problems like Apollonius' tangent circles, or Fermat's Last Theorem. Gauss' solution of FLT3 for complex numbers led directly to great insights into algebra. Chasles' list of the six who conceived of calculus seems useful: Archimedes, Kepler, Cavalieri, Fermat, Leibniz, Newton.
A few "Fundamental Theorems" have been worked on and extended over the centuries. In this category is the work of Gauss and Bonnet whose eponymous theorem linked geometry and topology  a simple version had already been stated by Descartes, and several 20thcentury mathematicians took it far beyond Gauss's work.
One topic of applied mathematics which shows up in the minibios is map projections: representing the spherical earth on planar paper. Among those who invented new map projections are Thales, Hipparchus, Ptolemy, Alberuni, Lambert, Lagrange and Karl Gauss. (I don't mention Lagrange's map projection in his minibio. He improved one of Lambert's projections, one where all latitude and longitude lines are circular arcs.) Klein, Darboux, and other great mathematicians also worked with such projections, though their names may not appear on actual mapmaking algorithms.
The development of heliocentrism was an important thread in the development of science; I discuss it a little in the minibios of Ptolemy and Copernicus. Not only did the facts of celestial motion lead to Newton's Laws of Gravitation and Motion, but placing the Earth away from the center led Galileo and others to conclude that physical laws are constant throughout the universe, rather than differing between heaven and earth. (It may have been this notion that most offended the Church.)
Another interesting thread concerned the distinctions betwen constuctible and nonconstructible numbers. By allowing addition, then division, then square roots, then cube and higher roots, one gets integers, rationals, Euclidconstructibles, and Galoisconstructible; then come algebraic numbers; the leftover reals are called transcendentals. One of Pythagoras' disciples proved a number irrational; Omar alKhayyám speculated that many cubic polynomial roots were not Euclidconstructible; and Fibonacci demonstrated one that wasn't. Abel and Galois achieved fame by noting algebraic numbers not Galoisconstructible; and the hunt for transcendental numbers made Liouville and Hermite famous.
Of the Top 105 mathematicians, 12 fluorished during the Antique Classical Age, 8 during the Middle Ages, then (based on death year) 1, 9, 9, 28, 32 in respectively the 16th, 17th, 18th, 19th, 20th centuries, and 6 are still living (or lived into the 21st). Comparing the state of mathematics as Lagrange found it and as Cayley left it, the 19thcentury concentration should be no surprise. Most of the 20thcentury names were in the early part of that century: no matter how great the latest geniuses may be, they can't have as much importance, at least in applied math, as those that came before them.
As implied by the numbers just presented, the List includes roughly three times as many mathematicians per generation from the modern era compared with the 18th century. Based on the numbers of mathematicians and the continuing high pace of advances, this actually constitutes a bias towards the past, justified by the extreme historic influence of the earlier time.
Of the 105 Great Mathematicians, there are 23 from France; 21 from Germany; 11 from the ancient Greek world; 9 from England; 7 from Switzerland; 6 from Italy; 5 from India; 3 each from Holland, Norway, Russia, and Persian domain, 2 each from China, Hungary and Poland; and 1 each from Austria, Ireland, Lithuania, Scotland and the U.S.A. (It's not always clear which country to assign: I put Cantor in Germany, Lagrange in Italy, Einstein in Switzerland.)
I'll summarize birthyears for the List of 200. 15 were born B.C.; 5 during the first 500 years of the Common Era; 6 during the second 500 years; 11 between 1000 and 1499; 10 during the 16th century; 15 during the 17th century; 20 during the 18th century; 73 during the 19th century; and 45 born during the 20th century (10, 7, 9, 10, 5, 2, 1, 1, 0, 0 by decade). (I'm sure there are many more "greatest mathematicians" of recent birth, but the List emphasizes the past.)
Recently I learned that a team of psychologists led by Catherine M. Cox tried to estimate the childhood IQ's of certain famous people born between 1450 and 1850. This sounds "iffy" to me, but the reader may be interested to know that the team scored Leibniz and Pascal highest among mathematicians. (The top six on the Cox list had four nonmathematicians: Johann Wolfgang von Goethe, Hugo Grotius, Thomas Wolsey and Pietro Sarpi. Goethe ranked 1st, Leibniz 2nd.) Bygone mathematicians scoring 170 or higher in that survey include 10 others on my list (Newton, Laplace, Lagrange, Descartes, Kepler, Huygens, Cardano, Hamilton, d'Alembert, Galileo) and 3 not on my list (Buffon, Napier, Lazare Carnot). That famous Cox list has been revised (by whom I don't know) with five names promoted above Goethe and Leibniz: Newton, Voltaire, da Vinci, David Hume and Michelangelo. Mathematicians or physicists not mentioned above who are sometimes seen on lists of great polymaths or geniuses include two women (Hypatia and Kovalevskaya), and several men (including Alhazen, Poincaré, Ludwig Wittgenstein, Einstein, von Neumann, Gauss, Maxwell, Ettore Majorana, Rudolf Clausius and Nikola Tesla). John Stuart Mill was unusually precocious. Wittgenstein is sometimes shown with the highest IQ ever but is missing from my list since his emphasis was logic and philosophy.
Bombelli, called the Inventor of Complex Numbers (and who was also one of the first Europeas to treat negative numbers) might deserve a prime place on the list. But complex numbers were a gradual development. That some quadratics have only "fictitious" solutions was known in ancient Greece; Cardano worked with them before Bombelli; Wallis (and perhaps Vieta) began serious use of complex numbers; Leibniz, de Moivre, Johann Bernoulli and d'Alembert developed them further; yet even Euler (who popularized the symbol i) once made elementary mistakes with imaginary numbers. It was finally JeanRobert Argand (not "great" enough to appear on my list) and then Carl Gauss who integrated complex numbers into the mathematical mainstream.
The story of Euler's famous discovery may be interesting. The following facts were already well known to him ((3) was discovered by Isaac Newton):
Since 0, ±π, ± 2π, ± 3π, etc. are all zeros of the sine function, (1) suggests sin(x) = A(x) x (1  (x/π)^{2}) (1  (x/2π)^{2}) (1  (x/3π)^{2}) .... Although we've not proved that A(x) is a simple constant, if we assume that it is, (2) leads directly to A(x) = 1. Substitute this new polynomial for sin x into (3); equate the x^{3} terms on each side, and Euler's identity soon emerges!
By itself, this doesn't constitute a rigorous proof (similar sleight of hand can be used to derive 1 + 1 = 3), but it is how Euler first derived his formula, and it can be made rigorous. Euler's intuition and the simplicity of his method here may serve as inspiration!
In the bios, I refer to fields like "algebraic number theory" that I don't understand at all. I have caught a few glimpses that amazed me, however. Consider Euler's formula, that generates 40 consecutive primes for consecutive n: n^{2} + n + 41. Now consider Martin Gardner's famous "April Fool's integer": e^{√163 π} = 640320^{3} + 744. (In this "equality" the leftside is actually smaller than the rightside integer, but by less than a trillionth of a unit. Too close to be a "coincidence", right?)
163 = 4*41  1 is a Heegner number and the fact that it yields both Euler's primegeneration formula and the "April Fool's integer" (also called Ramanujan's constant) is not a coincidence (though the "April Fool's equation" might seem completely unrelated to prime numbers). If you doubt this, substitute another Heegner number, such as 67 = 4*17  1, to get 16 consecutive primes and the approximation e^{√67 π} = 5280^{3} + 744. It was Carl Gauss himself who discovered these "magical" Heegner numbers (though not the mysterious "almost equations") and conjectured (correctly) that 163 was the largest of them. (Ramanujan's formula for π is also related to the Heegner number 163.)
Names like Kepler, Einstein and da Vinci appear on the List even though they didn't specialize in mathematics. These men were important because their work inspired other mathematicians. While reading about Gaspard Monge, to judge whether he qualified for the List or not, I learned that Kepler had anticipated the invention of Descriptive Geometry when he related the dodecahedron to the "Snowflake" shown at right. 
In about 1767, Euler proved that (2^31  1), was prime. This record beat the previous highest known prime by a factor of 4096 and was not surpassed for 100 years. This recordsetting prime, whose hexadecimal value is 7FFFFFFF, is immortalized in the C Programming system as "INT_MAX"! (While Euler's prime has only 10 decimal digits, a prime discovered in 2016 has more than 22 million digits. But perhaps the prime number Gold Medal should go to Edouard Lucas whose clever 19thcentury method found a 39digit prime that held the record for 75 years. Euler's prime, Lucas' prime and the present recordholder are all Mersenne numbers but, perhaps surprisingly, neither the prime which first beat Euler's nor the prime that first beat Lucas' record was a Mersenne.)
According to tradition, Pythagoras' understanding of music began when he passed a blacksmith shop and, finding it harmonious, inspected the anvils and noticed some harmonious anvils had length ratio 2 : 1 (what we now call an "octave") and others 3 : 2 (what we now call a "perfect fifth"). Pythagoras devised a musical scale based on the same seven notes as ours, with the wholestep ratio 9 : 8, but the halfsteps BC, and EF in a (2^8 : 3^5) ratio. Five wholesteps and 2 halfsteps made an "octave"; three wholesteps and 1 halfstep a "fifth", and the interval between octave and fifth was thus a "perfect fourth" (4 : 3 ratio). In this system a wholestep exceeds two halfsteps in the ratio 3^12 : 2^19 = 1.01364; this "error" term is called the "Pythagorean comma." (For centuries, the octaves and perfect fifths of an instrument were tuned as described, and Bsharp needed to be sharper than C by a Pythagorean comma.)
(An archaelogist has deciphered ancient clay tablets from Ugarit showing a song in a diatonic scale which predates Pythagoras by eight centuries.)
Archytas considered the question of dividing the "perfect fifth" into two harmonious intervals. He had observed that a ratio is the product of arithmetic and harmonic means, suggesting that CEG should have the ratios 4 : 5 : 6 rather than the 4 : 5.0625 : 6 of Pythagoras' scheme. (5/4 and 6/5 are the arithmetic and harmonic means of the pair (1, 3/2).) Today the 5 : 4 and 6 : 5 ratios are called major and minor thirds. In the Renaissance, the notes CDEFGABC were given the ratios 24 : 27 : 30 : 32 : 36 : 40 : 45 : 48.
But neither scheme provides harmony in all keys; and the crosskey errors are much more severe in the Renaissance system than in Pythagoras' scheme. An equaltempered scale was desired; it was Simon Stevin who suggested halfsteps whose size was the twelfth root of 2. The "fifth" ratio is now 1.4983  almost "perfect," and the same in every key. Unfortunately the major and minor "thirds" were each off by almost 1%. (Marin Mersenne suggested a constructible number for the halfstep ratio, but 12 of those halfsteps would make the octave ratio 2.006 instead of 2.)
Huygen's 31tone scale gives 1.4955 as the ratio for the 18step "fifth" and 1.2506 for the 10step "third." (A 53tone equaltempered scale would be even more harmonious.) But of course, an organ with 31 keys per octave instead of just 12 is rather unwieldy!
I was slightly amazed when I learned that Babylonians used the formula xy = ((x+y)^2  (xy)^2) / 4 to perform multiplications with just additions, subtractions and table lookups. I've used the same method on computers with slow multipliers and was surprised to find, on an expert programmer messageboard, that hardly anyone else was familiar with it. (Manual multiplication today is done by memorizing products up to 9times9; the obvious Babylonian analog would have required fitting 59times59 terms onto tablets; this explains why they invented the approach requiring only the first 59 square numbers.) (See, for example, http://www.math.tamu.edu/~don.allen/history/babylon/babylon.html.)
Hardy once described himself as "for a time the 5th best mathematician in the world." What was the time, and who were the Top Four?
Hardy was doing excellent work in his 20's, but Hilbert, Poincare, Cartan, Hadamard and Cantor were established greats, and Noether, Weyl and Littlewood were hot on Hardy's heels. When Hardy was approaching his 60's, von Neumann, Weyl, Cartan, Siegel were the obvious greats with Littlewood and others also in contention. Hardy himself said he was at his peak in his early 40's, and it must be after the death of Ramanujan, that he became "fifth best in the world." At this time, the powers of Hilbert were waning and von Neumann was still very young.
The best candidates for Top Four in 1922, about when Hardy considered himself 5thbest, were Weyl, Noether, Cartan, and probably either Siegel or Littlewood.
Isaac Newton: I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
René Descartes: I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.
(I've ranked Isaac Newton a bit higher than others might, and Descartes a bit lower. This is partly because of the respective modesty they show in their selfappraisals above. By the way, Newton described his method of discovery with "I keep the subject constantly before me, and wait 'till the first dawnings open slowly, by little and little, into a full and clear light.")
Cliff Pickover has a list of the "Ten Most Influential Mathematicians." His Top Ten include seven of my top nine (all except Archimedes and Lagrange), but bypass my #10#12 to include Descartes, Galois and Pascal. His omitting Archimedes and including Descartes may make sense since his criterion emphasizes historical importance, but Pascal seems an odd choice: many mathematicians were more "influential" than Pascal.
Compared with this and other lists, I seem to overrate Fermat, and underrate Napier. Pickover names Cardano and Napier as "runnersup." I'm doubtful about John Napier, who had great influence (he discovered 'e', developed logarithms and the use of decimal point, invented "Napier's bones", and seems to have conceived of continuous functions), but proved no great theorems.
W.C. Eells produced a list of the top 100 mathematicians of all time, in order (though the list contains no mathematician born after 1843 except Poincaré). His Top Ten include two names (Laplace and Cardano) missing from my Top Twenty. His Top Forty include seven names missing from my Top Ninety; in Eells' order these are: Legendre, Napier, Ptolemy, Regiomontanus, Maclaurin, Tartaglia, Heron. Eells' List includes 27 names missing from my Top 200; only six of these (Tartaglia, Heron, Boskovic, Barrow, Sturm, De Morgan) rank in Eells' Top 55. (The other 21 on Eells' 100 missing from my 200 are Briggs, Carnot, Maupertuis, Babbage, Pacioli, Gerbert, Cotes, Ahmes, Borda, Frisi(*), Montucla(*), Hesse, Jordanus de Nemore, Plato, Halley, Ampere, L'Hospital(*), Thomson Lord Kelvin, Boethius, Tschirnhausen(*), Zeno. The four asterisked names are missing from my List of 330.)
On my List of 100 there are about 72 names born before 1843; 18 of these are missing from Eells' List of 100, including seven from the modern era: Dedekind, Eisenstein, Kummer, Liouville, Maxwell, Chebyshev, Jordan.
Andrei D. Polyanin has a List of the Top 24 Mathematicians. It contains seventeen names from my Top 21 (all but Grothendieck, Fermat, von Neumann, Dirichlet), four other names from my Top Sixty (d'Alembert, Fourier, Kolmogorov, Laplace), and three Russian mathematicians each of whom I rank near #110 (Lobachevsky, Sophie Kovalevskaya, Andrei Markov).
Ioan James has a List of the 60 Remarkable Mathematicians from Euler to Von Neumann, which restricts to mathematicians born between 1705 and 1905. (His criterion of "remarkable" isn't quite the same as "great" but let's compare the lists anyway.) My List of Hundred includes 59 mathematicians born in that period; the two lists are fairly close with 45 names in common. However there is a strong difference in the Lists' temporal biases: 25% of the names on James' List were born between 1880 and 1899.
Comparing the lists shows that mine has a bias toward the past: I'm missing only two names from James' list who were born before 1825 (Germain, Kronecker) while he is missing four of mine from the same time period (Eisenstein, Lambert, Plücker, Steiner). For the period from 1825 to 1905, James is missing eight names (Frege, Jordan, Lebesgue, Littlewood, Minkowski, Peano, Siegel, Tarski) from my List. My List of 200 is missing five names from James' List (James Alexander, Richard Courant, Gosta MittagLeffler, Robert L. Moore(*) and Teiji Takagi). another five of James' names are present on my List of 200 but missing from my Top 150: Aleksandrov, Germain, Lefschetz, E.H. Moore, Zariski. (Of these, all except Germain, MittagLeffler, E.H. Moore, Takagi were born during the the 18801899 period which James emphasizes.)
Luke Mastin lists 110 important mathematicians. These are ones he chose to summarize the story of math's development, rather than the "greatest;" and nine of his names are not even on my List of 330; nevertheless I'll compare the lists. Mastin is missing four names (Dirichlet, Weyl, Noether, Hermite) from my Top 40, and another 12 from my 41 to 60. My List of 200 is missing 18 names from Mastin's list: Hippasus (Pythagoras' student), Democritus, Sun Tzu, Yang Hui (mentioned in Qin's minibio), alFarisi, Pacioli, Ferrari, Mersenne, Goldbach, Peacock, Venn, Whitehead, Fatou, Julia, Lorenz, Julia Robinson, Cohen, Wiles.
Ian Stewart associates names with his "17 equations that changed the world." Twelve of these names appear on my List of 200: Einstein, Pythagoras, Maxwell, Napier, Gauss, Shannon, d'Alembert, Fourier, and, with two equations each, Newton and Euler.
The Royal Society's Copley Medal is among the most prestigious of science prizes (fewer are given than Nobel's). Eleven men on my List received that prize: Atiyah, Cayley, Einstein, Gauss, Klein, Hardy, Littlewood, Plücker, Poisson, Sylvester, Weierstrass. Several other great mathematicians have also won the Copley Medal, e.g. Chandrasekhar, Chasles, Clausius, Gibbs, Lamb, Le Verrier, Penrose, Rayleigh, Stokes, Sturm, Waring. (The Copley Medal is usually awarded late in a career and never posthumously; this explains the absence of certain greats who died at a relatively young age, e.g. Riemann and Maxwell.)
Prestigious prizes where preference is given to Britons include the De Morgan Prize and Sylvester Medal. From my list of 200, Atiyah, Cantor, Carleson, Cayley, Darboux, Hardy (both), Klein, LeviCivita, Littlewood (both), Poincaré, Sylvester, J.G. Thompson (both) have won one (or both) of those prizes. Among several other winners are Besicovitch (both), Coxeter, Davenport, Lamb, Mordell, Penrose, Rayleigh, Russell (both), Titchmarsh (both), Whitehead. One of the top math chairs at Oxford or Cambridge has been held by 7 on my list: Wallis, Newton, Smith, Sylvester, Hardy, Dirac, Atiyah. Other Savilian or Lucasian Professors are Briggs, Barrow, Halley, Babbage, Stokes, Titchmarsh, Hawking and 23 others.
Five mathematicians on my List won Cambridge's Smith's Prize: Cayley, Maxwell, Hardy, Littlewood and Turing. Among these, Cayley and Littlewood were Senior Wranglers the same year; Maxwell Second Wrangler, and Hardy Fourth Wrangler. Turing took the test in a year when ranks were not announced  he might have been the highest scoring Wrangler. Other famous mathematicians who won Smith's Prize and were Senior or Second Wrangler include Airy, Coxeter, Eddington, Lamb, Mordell, Stirling, Stokes, Strutt (Lord Rayleigh), Thomson (Lord Kelvin). From my List, Sylvester and Clifford were Second Wrangler.
Miss Philippa Fawcett was ineligible for Wrangler but was permitted to take the examination: she scored above the Senior Wrangler. (As an experiment, Hardy asked George Pólya to sit for the undergraduate examination when he was Visiting Professor; the results, which were not published in that year, are mysterious. Some say Pólya failed, others say that he joined Miss Fawcett in the rare feat of scoring "above the Senior Wrangler.")
The Wolf Prize, Abel Prize and Fields Medal are perhaps the most prestigious of math awards; winners from my List of 100 include Atiyah (3), Chern, Erdös, Grothendieck, Kolmogorov, Selberg (2), Serre (3), Siegel, and Weil. There are several other major prizes (e.g. the Bocher Memorial Prize) given for mathematics, or sometimes (e.g. Nobel Prize, Copley Medal) given to mathematicians. At least three dozen mathematicians not on the List of 100 have won two or more major prizes; these include 21 from my List of 200: Lars Ahlfors, Lennart Carleson, John H. Conway, Paul Cohen, Paul Dirac, Pierre Deligne (3), Simon Donaldson, Gerd Faltings, Michael Freedman, Timothy Gowers, Mikhail Gromov (3), Lars V. Hörmander, Kunihiko Kodaira, Peter Lax, Solomon Lefschetz, John W. Milnor (3), David Mumford, John Nash, Sergei Novikov, Daniel Quillen, Yakov Sinai (3), Isadore Singer, Stephen Smale, Terence Tao (3), John T. Tate, John G. Thompson (3), William Thurston, Norbert Wiener, Edward Witten (3); and some names not on my list: James W. Alexander, Enrico Bombieri, Jean Bourgain, Jesse Douglas, Charles Fefferman, Maxim Kontsevich, Gregory Margulis, Shigefumi Mori, Louis Nirenberg, Roger Penrose, Klaus Roth, Wendelin Werner, ShingTung Yau.
My own "Eight Greatest Scientists" List would be: Newton, Einstein, Galileo, Maxwell, Faraday, Pasteur, Darwin, Lavoisier. (With the exception of Galileo himself, these scientists are all postGalilean.)
To bring this up to Eleven, I'll add Aristotle, Aryabhata, and Alhazen  each a great preGalileo advancer of science, and representing the three great sources of ancient science: the Greek, Indian, and Islamic empires. (Names are forgotten from ancient Babylonian, Egyptian and Harrapan empires. East Asia and Europe (after Greece, before Galileo) contributed little compared with the Great Three.
To up the Top Eleven scientists to Thirteen, I'll add Kepler and Bohr. Eight of my Top 13 scientists (in alph. order, Alhazen, Aristotle, Aryabhata, Einstein, Galileo, Kepler, Maxwell, Newton) are also on my list of Top 75 mathematicians.
Girolamo Cardano prepared a list of the 12 men who "excelled all others in the force of genius and invention." This list includes eight from my List of 200: Archimedes, Euclid, Apollonius, alKhowârizmi, Archytas, Aristotle, Ptolemy, alKindi, and four other names: John Duns Scotus, Richard Swineshead the Calculator, Galen of Pergamum (physician), and Jabir ibn Aflah (astronomer and mathematician whose works were translated into Latin). (Sources show "Heber of Spain" as the 12th name; it is my assumption that this refers to Geber of Spain, an alternate name for Jabir ibn Aflah. There was a more famous Jabir ibn Hayyan and a "pseudoGeber;" perhaps Cardano conflated these figures. However in another work, Cardano calls specific attention to Jabir ibn Aflah and writes that Regiomontanus borrowed his ideas from that Jabir.) When Copernicus obsoleted Ptolemy's cosmology during Cardano's lifetime, Cardano removed Ptolemy and added Marcus Vitruvius Pollio (engineer) to keep the list at twelve names.
John Galbraith Simmons prepared a list of the 100 Most Influential Scientists in order. His Top Twelve are Newton, Einstein, Bohr, Darwin, Pasteur, Freud, Galileo, Lavoisier, Kepler, Copernicus, Faraday, Maxwell; with the exception of Copernicus, Freud and perhaps Bohr, these seem to me to be excellent choices. Five of these names are in my List of 100. John Balchin and John H. Tiner have each prepared their own lists of 100 influential scientists; each includes Simmons' Top Twelve. Other mathematicians who appear on all three lists are Archimedes, Euclid, Huygens. Mathematicians appearing on two of the lists are Aristotle, Descartes, Euler, Gauss, Pascal, Pythagoras, and one missing from my List: Charles Babbage.
Michael Hart's List of the 100 Most Influential People Ever seems well reasoned. His List, which considers only historical significance and not genius, includes nine mathematicians from my List (Newton, Einstein, Euclid, Descartes, Kepler, Euler, Aristotle, Galileo, Maxwell) and four other mathematical physicists (Copernicus, Heisenberg, Planck, Fermi). Hart's list also includes Pasteur, Lavoisier, Faraday, Dalton, Plato, Rutherford, Roentgen, Francis Bacon and seventynine people who did no work in pure physical science (44 political, military or religious leaders; 2 explorers; 16 inventors; 9 social philosophers; 5 artists; 3 biologists).
Among scientists, Hart omits Bohr altogether  though he included him in a First Edition of the List; he also ranks Freud and Kepler behind Aristotle, Euclid, Adam Smith, Dalton, Heisenberg, etc.; but otherwise his top scientists agree fairly well with Simmons'. (In addition to the Top 12 Scientists, seven other scientists not on my List appear on all four of the lists just described  Dalton, Fermi, Harvey, Heisenberg, Leeuwenhoek, Mendel, Rutherford.) Some names often seen near the top of other Top Scientist lists include Nikola Tesla, Marie Curie, Thomas Edison, and two names from my List: Leonardo da Vinci and Alan Turing.
Researchers at MIT's Pantheon project, using the statistics of online biographies, prepared their own list of the 100 most influential persons ever. Combining this list with Hart's yields a List of 157. This combined list includes 15 mathematicians from my List of 50; these are the nine already mentioned from Hart's List and six more: Leonardo da Vinci (ranked #6), Pythagoras (#10), Archimedes (#11), Thales (#38), Pascal (#67), Ptolemy (#80). A small sample of the nonmathematicians on Pantheon's list missing from Hart's list are Socrates (ranked #4), Mozart (#16), Dante Alighieri (#26), Cleopatra VII (#29), Herodotus (#32), Virgil (#36), Marco Polo (#40), Joan of Arc (#53), Epicurus (#56) Goethe (#62), Mahatma Gandhi (#75), Saladin (#76), Heraclitus (#84), Gilgamesh (#85), Anne Boleyn (#91), Avicenna (#97).
I hope no one takes offense that 29 of the Top Thirty and 197 of the Top 200 are males, In addition to Emmy Noether, Sophia Vasilyevna Kovalevskaya and MarieSophie Germain, other famous female mathematicians include Hypatia (the last librarian of Alexandria), and Theano (Pythagoras' wife). Part of the reason for the dearth of women among famous mathematicians may be discrimination; even the great Noether might have been almost overlooked except that colleagues like Hilbert and Weyl insisted that her greatness be recognized. (Above we mentioned Philippa Fawcett, who scored above Cambridge's Senior Wrangler.)
(In the following discussion I list some notable omissions from the List of 100. Many or most of these do appear on the List of 200.)
I started with just Ten names, and gradually grew this list up to One Hundred over several years. I know I'm finally done because there's no one left that I fervently want to include. To demonstrate this, I'll "dispose" of the most obvious remaining candidates.
Certain great geniuses are omitted because they lacked great historical influence, e.g Roberval, who didn't publish; Bolzano, who was censored by the Austrian government; or Torricelli, Clifford and Gregory, who each died young.
Regiomontanus and Ptolemy are famous mathematicians of great historic importance whom I omit. Regiomontanus' work is important primarily because he lived shortly after Gutenberg's printing press invention; his actual creative achievements in spherical trigonometry were anticipated by Nasir alDin alTusi. Similarly, Ptolemy's great fame exaggerates his genius. Careful comparison of the errors in their works reveals that Ptolemy borrowed both his data and his methods from Hipparchus. Napier, Stevin, alKashi and perhaps Chang are strong candidates based on genius and historical importance, but none of them is credited with any nontrivial theorem. I would include members of the Maxwell/Galileo group before any of these.
I may include Thales of Miletus (he produced the first known "proof" and was influential) and Panini (who anticipated both Boolean logic and modern mathematical grammars) but the list includes no other mathematicians prior to Pythagoras, though some names from India's ancient Vedic period are known, e.g. Lagadha, the first astronomer known to have used trigonometry, and Baudhayana, who taught the Pythagorean theorem, and Apastambha, perhaps the greatest named mathematician before Archytas. Another famous Vedic mathematician who left more than fragmentary work, was Pingala (a Jainist who was famous for his work in music theory and combinatorics).
I omit some famous mathematicians like Ferdinand Lindemann and Andrew Wiles (lacking in breadth, and their "revolutionary" discoveries relied heavily on earlier work). (Some other mathematicians frequently included on "Great" lists are Babbage, Barrow, Buffon, Chasles, Clebsch, Cramer, DeMoivre, Heron, and MittagLeffler.)
Two correspondents have asked me to include Oliver Heaviside among the greatest mathematicians. This selftaught man was certainly an amazing genius. It was partly to make room for him that I made an expanded list with, now, 150 names.
Finally, I've imposed an arbitrary rule: the List of 100 includes no mathematicians born after 1930. (This rule is relaxed slightly for the List of 150 and eliminated altogether for the List of 200.)
If the List of 200 still isn't long enough for you,
here is a list of about 330 of some of the most important mathematicians.
(Names in deep purple are on my Top Thirty List.
Names in boldface are candidates for the Top Two Hundred.)
(This long list is divided into three parts, based
on whether the mathematician died before 1740, died before 1950,
or was living some time after 1950.)




 




 




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