In intellectual rankings, greatest mathematician ever is an epitaph given to a person, depending on ranking methodology, some rankings of which are listed below, that classify, list, or describe someone as being the greatest thinker in the field of mathematics of all time; the work-in-progress ranking of which is shown below; the IQ column shows the mathematician's real IQ and ranking position amid the top 1000 geniuses of all time:

 # Person Other Rankings Meta-analysis Ranking Publication ------------------------------------------------------------------------------ ----------------------------------------------- 1 —16 Leonhard Euler(1707-1783) (Cattell 1000:512) [RGM:101|1,500+](Murray 4000:14|CS / 1|M)(CR:103) (4,4,4,4,1) (12%) (64283) 2. —23 Carl Gauss (1777-1855) (Cattell 1000:848)[RGM:39|1,500+](Murray 4000:4|M) (CR:73) (7,3,2,5,2) (10%) (50699) 3. —2 Isaac Newton(1643-1727) (Cattell 1000:14)[RGM:3|1,500+](Murray 4000:2|CS / 1|P / 2|M) (CR:595) (1,1,1,0,10) (1%) (9199) "On the Quadrature of Curves" (1706) 4. —55 Euclid(c.340-280BC) (Cattell 1000:501)[RGM:50|1,500+] (Murray 4000:19|CS / 3|M) (CR:65) (6,6,3,0,5) (8%) Elements (c.300BC) 5. —39 Archimedes(287-212BC) (Cattell 1000:414) [RGM:9|1,330+] (Murray 4000:20|M / 5|T) (CR:28) (8,2,0,0,3) (10%) The Method of Mechanical Theorems (c.250BC) 6. —13 Rene Descartes(1596-1650) (Cattell 1000:23)[RGM:33|1,500+](Murray 4000:6|CS / 7|M / 4|WP) (CR:290) (9,17,9,0,9) (2%) "The Geometry" (1637) 7. —24 Joseph Lagrange(1736-1813) (Cattell 1000:512) [RGM:558|1,500+] (CR:114) (3,8,0,0,6) (61065) (6%) 8. —21 Gottfried Leibniz(1646-1716) (Cattell 1000:34)[RGM:32|1,500+] (Murray 4000:14|CS / 6|M / 11|WP) (CR:252) (2,10,0,0,7) (66592) (6%) "A New Method for Maxima and Minima, and Also for Tangents, Which Stops at Neither Fractions nor Irrational Quantities, and a Singular Type of Calculus for These" (1686) 9. —44 Henri Poincare(1854-1912) [RGM:471|1,500+](CR:64) (90,7,6,0,20) 10. —86 Pythagoras (c.570-490BC) (Cattell 1000:89) [RGM:15|1,500+](CR:86) (0,29,0,1,8) (3%) 11. —115 Bernhard Riemann (1826-1866) [RGM:280|1,500+] (Murray 4000:10|M) (CR:14) (38,5,7,0,21) 12. —215 David Hilbert (1862-1943) [RGM:421|1,500+](Murray 4000:11|M) (0,9,5,0,24) (20414) 13. —292 Gerolamo Cardano (1501-1576) (Murray 4000:14|M)(CR:8) (10,91,11,3,0) 14. —224 Pierre Fermat(1601-1665) (Cattell 1000:893)[RGM:628|1,500+] (Murray 4000:5|M) (CR:8) (17,12,0,0,13) 15. —394 Felix Klein(1849-1925) (9,45,0,0,11) 16. Georg Cantor(1845-1918) [RGM:478|1,500+](Murray 4000:8|M) (0,25,13,6,30) (1) 17. —455 Evariste Galois(1811-1832) [RGM:517|1,500+] (84,14,8,0,23) 18. —47 Blaise Pascal(1623-1662) [RGM:42|1,500+](Murray 4000:9|M) (19,35,10,0,41) 19. —528 Augustin Cauchy (1789-1857) [RGM:315|1,500+](Murray 4000:18|M) (CR:10) (15,21,0,0,15) 20. —307 Jacob Bernoulli (1654-1705) (Murray 4000:12|M) (CR:12) (28,53,0,0,56) 21. —276 Adrien Legendre (1752-1833) (Murray 4000:16|M) (11,0,0,0,0) 22. Diophantus(c.207-293) (Murray 4000:13|M) (26,38,0,0,0) 23. Francois Viete (1540-1603) (Murray 4000:15|M) (22,48,0,0,0) 24. Leonardo Fibonacci(1170-1250) (Murray 4000:19|M) (21,30,0,0,0) 25. John Wallis(1616-1703) (Murray 4000:17|M) (34,0,0,0,0) 26. —84 Jean d'Alembert (1717-1783) (Cattell 1000:124)(CR:28) (14,44,0,0,0) 27. William Hamilton (1805-1865) (34,39,0,0,0) 28. —33 Christiaan Huygens (1629-1695) (Eells 100:24) 39. Srinivasa Ramanujan(1887-1920) (25,0,19,0,0) 30. —30 Pierre Gassendi (1592-1655) “If we take the mathematics, and those mixed sciences to which they are applicable, it will be universally admitted that their most successful cultivators in France during the seventeenth century were Descartes, Pascal, Fermat, Gassendi, and Mersenne.”— Henry Buckle (1856), History of Civilization, Volume One (pg. 499-50)

Meta-analysis ranking
The adjacent box shows a combined meta-analysis summary ranking (combination of 5 different rankings below) of the greatest mathematicians ever (known descendants listed as well), the numbers in brackets being the rank of each mathematician of each ranking study (below), in chronological order. Meta-analysis ranking positions (a,b,c,d,e) come from the following sources:

a. W.C. Eells (1962) | mathematics professor
(100 greatest list)
b. James Allen (1998) | mathematics historian
(180 greatest list)
c. Clifford Pickover (2001) | science historian
(10 greatest list)
d. Alex Bellos (2010) | mathematician
(10 greatest list)
e. Top 10 Lists (2012) | public vote
(64 choices)

The "percentage" is the percentage of votes from the 2012 The-Top-Tens.com’s listing of the “Greatest Mathematician of All Time”, the ranking of this list shown by position five or "e" in the bracketed listing. The hyperlinked number, shown next to some mathematicians, e.g. Hilbert (20,414), shows the number of descendents, according to the data base at the mathematics genealogy project, a sixth factor taken into account in the meta analysis ranking.

Eells’ 1962 rankings
See main: Eells 100 mathematicians
In the 1962 issue of Mathematics Teacher, American mathematician W.C. Eells’ published a listing of who he considered to be 100 greatest mathematicians of all time: [1]

 1. Isaac Newton2. Gottfried Leibniz3. Joseph Lagrange (Euler genealogy)4. Leonhard Euler (Euler genealogy)5. Pierre Laplace6. Euclid7. Carl Gauss8. Archimedes9. Rene Descartes10. Gerolamo Cardano11. Adrien-Marie Legendre12. Pitagora13. Gaspard Monge14. Jean d'Alembert 15. Augustin Cauchy16. Joseph Fourier17. Pierre Fermat18. John Napier19. Blaise Pascal20. Apollonius21. Leonardo Fibonacci22. François Viete23. Ptolemy24. Christiaan Huygens (Euler genealogy)25. Regiomontanus 26. Diophantus27. Colin Maclaurin28. Jacob Bernoulli (Bernoulli family)29. Pappus30. Bonaventura Cavalieri31. Carl Jacobi32. Johann Bernoulli (Bernoulli family)33. John Wallis 34. William Hamilton35. Niccolò Tartaglia36. Heron37. Jean-Victor Poncelet38. Bernhard Riemann39. Siméon Poisson40. Niels Abel41. Michel Chasles42. Luigi Cremona43. Gilles Roberval44. Roger Boscovich45. Galileo Galilei46. Alexis Clairaut47. Johann Lambert48. Isaac Barrow49. Jacques Strum50. Simon Stevin 51. Augustus de Morgan52. Brook Taylor53. Johannes Kepler 54. Daniel Bernoulli (Bernoulli family)55. Girard Desargues56. Henri Briggs57. James Sylvester58. Lazare Carnot59. Pierre Maupertuis60. Charles Babbage61. Charles Hermite62. Thales63. Henry Smith64. Sofia Kovalevskay65. Luca Pacioli66. Hippocrates (of Chois)67. Gerbert68. Alfred Clebsch69. Julius Plucker70. Hermann Grassmann71. Peter Dirichlet72. Arthur Cayley73. Muhammed al-Khwārizmi74. Roger Cotes75. Abraham De Moivre 76. George Boole77. Karl Weierstrass78. Sophus Lie79. Nikolai Lobachevsky80. Ahmes (the scribe)81. Jean-Charles Borda82. Eugenio Beltrami83. Paolo Frisi84. Evariste Galois85. Evangelista Torricelli86. Jean-Étienne Montucla87. Otto Hesse88. Jordanus de Nemore89. Plato90. Henri Poincare91. Jakob Steiner92. Edmond Halley93. Andre Ampere94. Guillaume L'Hospital95. William Thomson96. Boethius97. Ehrenfried Tschirnhausen98. Bhaskara II99. Eratosthenes100. Zeno of Elea

The key (Eells 100:#) is shorthand for each persons W.C. Eells ranking.

Allen's 1998 ranking
In 1998, American computer programmer (turned mathematics history hobbyist) James Allen decided to practice his HTML skills by making a listing of the top thirty greatest mathematicians listing. In his own words: [2]

“When I was first learning to create html pages, for some reason I chose to build a list of great mathematicians as a practice page, even though I wasn't qualified to make such a list without a lot of advice. Since then I've invested a lot of time reading mathematical histories and biographies and revising the page. I'm proud of it now: please read it! (With 60 mini-biographies, the single page has now grown to over 100 kilobites).”

Allen's ranking, as of 2012, has become Google search top result for "greatest mathematicians" and currently lists the 100 “Greatest Mathematicians of All Time”, ranked in approximate order of greatness, born before 1930, whose work has breadth, depth, and historical importance (the numbers in brackets being the position in Eells' 1962 ranking): [3]

 1. Isaac Newton (1)2. Archimedes (8)3. Carl Gauss (7)4. Leonhard Euler (4) (Euler genealogy)5. Bernhard Riemann (38)6. Euclid (6)7. Henri Poincare (90)8. Joseph Lagrange (3)9. David Hilbert (0)10. Gottfried Leibniz (2)11. Alexandre Grothendieck (0)12. Pierre Fermat (17)13. Niels Abel (40)14. Évariste Galois (84)15. John Neumann (0)16. Karl Weierstrass (77)17. Rene Descartes (9)18. Carl Jacobi (31)19. Srinivasa Ramanujan (0)20. Brahmagupta (0)21. Augustin Cauchy (15)22. Peter Dirichlet (71)23. Hermann Weyl (0)24. Eudoxus of Cnidus (0)25. Georg Cantor (0) 26. Muhammed al-Khwārizmi (73)27. Arthur Cayley (72)28. Emma Noether (0)29. Pythagoras (0)30. Leonardo Fibonacci (21)31. Kurt Godel (0)32. Aryabhata (0)33. Charles Hermite (0)34. Apollonius of Perga (0)35. Blaise Pascal (19)36. Pierre Laplace (5)37. Richard Dedekind (0)38. Diophantus (26)39. William Hamilton (34)40. Bháscara Áchárya (0)41. Gaspard Monge (13)42. George Boole (76)43. Stefan Banach (0)44. Jean d'Alembert (14)45. Felix Klein (0)46. Ferdinand Eisenstein (0)47. Jacques Hadamard (0)48. Francois Viete (22)49. Johannes Kepler (53)50. Elie Cartan (0) 51. Jean-Victor Poncelet (37)52. Archytas of Tarentum (0)53. Jacob Bernoulli (28) (Bernoulli family)54.

The shorthand (Allen 100:#) is key for each person's Allen top 100 mathematicians ranking.

 Pickover's 2001 rankingIn American science biographer Clifford Pickover's listing in his 2001 Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, in which he devotes a chapter to "A Ranking of the 10 Most Influential Mathematicians", based on, supposedly, surveys and interviews with mathematicians (the numbers in the brackets being the position in Eells' 1962 ranking and Allen's 1998 ranking, respectively): [4] 1. Isaac Newton (1,1) 2. Carl Gauss (3,7) 3. Euclid (6,6) 4. Leonhard Euler (4,4) 5. David Hilbert (0,9) 6. Henri Poincare (90,7) 7. Bernhard Riemann (38,5) 8. Evariste Galois (84,14) 9. Rene Descartes (9,17) 10. Blaise Pascal (19,35) Runners-up Girolamo Cardano, Kurt Godel, Georg Cantor, and John Napier. Bellos 2010 top 10The following is the 2010 listing of the “10 Best Mathematicians” by Brazilian mathematician Alex Bellos (author of the recent book Alex's Adventures in Numberland), ordered via ranking of "revolutionary discoveries said to have changed the world" (the numbers in the brackets being the position in Eells' 1962 ranking, Allen's 1998 ranking, and Pickover's 2001 ranking, respectively): [5]1. Pythagoras (0,29,0)2. Hypatia (0,~140,0)3. Girolamo Cardano (10,91,11)4. Leonhard Euler (4,4,4) 5. Carl Gauss (7,3,2) 6. Georg Cantor (0,25,13)7. Paul Erdos (0,95,0)8. John Conway (0,~140,0)9. Grigori Perelman (0,~140,0)10. Terrance Tao (0,0,0)The last three of which, to note, are still reactive (living) and seem to be more of celebrity mathematicians, rather than the "greatest" as history may show.

The-Top-Tens.com’s 2012 listing
The following is the 2012 current ranking (by percentage vote) of The-Top-Tens.com’s listing of the “Greatest Mathematician of All Time”: [6]

 1. Leonhard Euler (12%) (4,4,4,4)2. Carl Gauss (10%) (7,3,2,5)3. Archimedes (10%) (8,2,0,0)4. Aryabhata (9%) (0,32,0,0)5. Euclid (8%) (6,6,3,0)6. Joseph Lagrange (6%) (3,8,0,0)7. Gottfried Leibniz (6%) (2,10,0,0)8. Pythagoras (3%) (0,29,0,1)9. Rene Descartes (2%) (9,17,9,0)10. Isaac Newton (1%) (1,1,1,0)11. Felix Klein (9,45,0,0)12. Brahmagupta (0,20,0,0)13. Pierre Fermat (17,12,0,0)14. Alexandre Grothendieck (0,11,0,0)15. Augustin Cauchy (15,21,0,0)16. Apollonius 17. Pierre Laplace (5,36,0,0)18. Niles Abel (40,13,0,0)19. John Neumann (0,15,0,0)20. Henri Poincare (90,7,6,0)21. Bernhard Riemann (38,5,7,0)22. Bhascar Acharya 23. Évariste Galois (84,14,8,0)24. David Hilbert (0,9,5,0)25. Srinivasa Ramanujan (0,19,0,0) 26. Karl Weierstrass 27. Peter Dirichlet 28. Omar Al Khayyam29. Jean Serre30. Georg Cantor (0,25,13,6)31. Muhammad Al-Khowarizmi32. Sridhar Acharya33. Johannes Kepler (53,49,0,0)34. Albert Einstein35. Hippocrates36. James Maxwell37. Galileo Galilei38. Jacob Steiner39. Aristotle40. Kurt Godel41. Blaise Pascal (19,35,10,0)42. Abu Biruni43. Christiaan Huygens44. Liu Hui45. Nicolai Lobachevsky46. Alex Clairaut47. Jean Darboux48. Panini49. Johann Lambert50. Hermann Minkowski 51. Pafnuti Chebyshev52. Charles Hermite53. Richard Dedekind54. George Boole55. Andrey Kolmogorov56. Jacob Bernoulli57. Hipparchus58. Alhazen59. Godfey Hardy60. Andrew Weil61. Jean-Victor Poncelet62. Jacques Hadamard63. Francois Viate64. Elie Cartan

The numbers in the brackets being the position in Eells' 1962 ranking, Allen's 1998 ranking, Pickover's 2001 ranking, Bellos's 2010 ranking, respectively.

 A related greatest mathematician image made by Indian mathematician Vinod Sir. [7] Euler also gets extra ranking points for blowing out both his eyes in the name of the proof; see: genius page (section: "physical and mental over-stressing").
Quotes
The following are notable and or relevant quotes:

“Taking mathematics from the beginning of the world to the time of Newton, what he has done is much the better half.”
Gottfried Leibniz (1700), told to the Queen of Prussia

“If we take the mathematics, and those mixed sciences to which they are applicable, it will be universally admitted that their most successful cultivators in France during the seventeenth century were Descartes, Pascal, Fermat, Gassendi, and Mersenne. Fermat, among these, was one of the most profound thinkers of the seventeenth century, particularly as a geometrician, in which respect he was second only to Descartes. The most important steps are those concerning the geometry of infinites, applied to the ordinates and tangents of curves; which he completed in or before 1636.”
Henry Buckle (1856), History of Civilization, Volume One (pg. 499-50)

“The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.”
Felix Klein (c.1900) (Ѻ)

“One of the properties inherent in mathematics is that any real progress is accompanied by the discovery and development of new methods and simplifications of previous procedures … The unified character of mathematics lies in its very nature; indeed, mathematics is the foundation of all exact natural sciences.”
David Hilbert (c.1910), Publication [8]

“You will see it written that Hadamard was the last of the universal mathematicians (see: last person to know everything) —the last, that is, to encompass the whole of the subject, before it became so large that this was impossible. However, you will also see this said of Hilbert, Poincare, Klein, and perhaps of one or two other mathematicians of the period. I don't know to whom the title most properly belongs, though I suspect the answer is actually Gauss.”
— John Derbyshire (2003), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics [9]

 ● Greatest chemist ever● Greatest physicist ever● Greatest thermodynamicist ever● Greatest philosopher ever● Greatest engineer ever ● Polymath● Last person to know everything● Universal genius● Last universal genius ● Genius IQs (top 1000 geniuses)● IQ: 200+ | Smartest person ever● IQ: 150+ | Smartest woman ever

References
1. Eells, W.C. (1962). “100 Greatest Mathematicians of All Time” (link), Mathematics Teacher, 7(55).
2. James Dow Allen (about) – FabPedigree.com.
3. Allen, James Dow. (1998). “The Greatest Mathematicians of All Time”, FabPedigree.com.
4. Pickover, Clifford. (2001). Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning (ch. 36: A Ranking of the 10 Most Influential Mathematicians Who Ever Lived, pg. 78-82). Oxford University Press.
5. Bellos, Alex. (2010). “The 10 Best Mathematicians”, The Guardian, Apr 10.
6. Greatest Mathematician of All Time (2012) – The-Top-Tens.com.
7. Leonard Euler (banner) – StandardTutorials.com.
8. Myint-U, TYn, and Debnath, Lokkenath. (2007). Linear Partial Differential Equations for Scientists and Engineers (pdf) (pg. vii). Springer, 2011.
9. Derbyshire, John. (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (pg. 159) (Ѻ). Joseph Henry Press.