Question

The St. Andrews site is an invaluable resource. From that list, I picked (usually) at most one great mathematician born in each year from 1860 to 1910:

$\textbf{EDIT: By popular demand, the list now extends from 1849 to 1920.}$

1849: Felix Klein, Ferdinand Georg Frobenius
1850: Sofia Vasilyevna Kovalevskaya
1851: honorable mention: Schottky
1852: William Burnside
1853: honorable mentions: Maschke, Ricci-Curbastro, Schoenflies
1854: Henri Poincare
1856: Emile Picard (honorable mention: Stieltjes)
1857: honorable mention: Bolza
1858: Giuseppe Peano (honorable mention: Goursat)
1859: Adolf Hurwitz (honorable mention: Holder)
1860: Vito Volterra
1861: honorable mention: Hensel
1862: David Hilbert
1864: Hermann Minkowski
1865: Jacques Hadamard (honorable mention: Castelnuovo)
1868: Felix Hausdorff
1869: Elie Cartan
1871: Emile Borel (honorable mentions: Enriques, Steinitz, Zermelo)
1873: honorable mentions: Caratheodory, Levi-Civita, Young
1874: Leonard Dickson
1875: Henri Lebesgue (honorable mentions: Schur, Takagi)
1877: Godfrey Harold Hardy
1878: Max Dehn
1879: honorable mentions: Hahn, Severi
1880: Frigyes Riesz
1881: Luitzen Egbertus Jan Brouwer
1882: Emmy Amalie Noether (honorable mentions: Sierpinski, Wedderburn)
1884: George Birkhoff, Solomon Lefschetz
1885: Hermann Weyl (honorable mention: Littlewood)
1887: Erich Hecke, George Polya (honorable mentionmentions: Ramanujan, Skolem)
1888: Louis Joel Mordell (honorable mention: Alexander)
1891: Ivan Matveevich Vinogradov
1892: Stefan Banach
1894: Norbert Wiener
1895: honorable mention: Bergman
1896: Carl Ludwig Siegel (honorable mention: Kuratowski)
1897: honorable mention: Jesse Douglas
1898: Emil Artin, Helmut Hasse (honorable mentions: Kneser, Urysohn)
1899: Oscar Zariski (honorable mentions: Bochner, Krull, Ore)
1900: Antoni Zygmund
1901: Richard Brauer
1902: Alfred Tarski (honorable mention: Hopf)
1903: John von Neumann (hm's: Hodge, Kolmogorov, de Rham, Segre, Stone, van der Waerden)
1904: Henri Cartan (honorable mentions: Hurewicz, Whitehead)
1905: Abraham Adrian Albert
1906: Kurt Godel, Andre Weil (honorable mentions: Dieudonne, Feller, Leray, Zorn)
1907: Lars Ahlfors, Hassler Whitney (honorable mentions: Coxeter, Deuring)
1908: Lev Pontrjagin
1909: Claude Chevalley, Saunders Mac Lane (honorable mentions: Stiefel, Ulam)
1910: Nathan Jacobson (honorable mention: Steenrod)
1911: Shiing-shen Chern (honorable mentions: Birkhoff, Chow, Kakutani, Witt)
1912: Alan Mathison Turing (honorable mentions: Eichler, Zassenhaus)
1913: Samuel Eilenberg, Paul Erdos, Israil Moiseevich Gelfand (dis/honorable mention: Teichmuller)
1914: honorable mentions: Dantzig, Dilworth, Kac
1915: Kunihiko Kodaira (honorable mentions: Hamming, Linnik, Tukey)
1916: Claude Elwood Shannon (honorable mention: Mackey)
1917: Atle Selberg (honorable mentions: Iwasawa, Kaplansky)
1918: Abraham Robinson
1919: honorable mention: Julia Robinson
1920: Alberto Calderon