KUMMER, knin'f!r, ERNST EDUARD ( 1810 93). A German mathematician, born at Sorau. in Silesia. He studied theology and mathematics in Halle (1828-31), and received the doctor's degree in 1S32. He then for ten years taught mathe matics in the gymnasium at Liegnitz, where Kroneeker (q.v.) was one of his pupils. From 1S42 to 1855 Kummer was professor of mathe matics at Breslau. and from 1853 to 1884 at Berlin. From 1874 he also taught in the military academy of Berlin. He became a member of the Academy of Sciences in Berlin in 1855. and in 1S57 was awarded the grand prize in mathematics by the Academy of Sciences in Paris, of which he became a foreign member in 1868. Kummer's chief contributions to mathematics were in the domains of the hypergeometrie (Gaussian) series (Crelle's Journal, vol. xv.), of cubic and biquad ratic remainders (in Crelle. vols. xxiii. and xxxii.), and of complex numbers. The creation of the theory of ideal numbers (see Numnr.n) is due to him, and to the theory of numbers in gen eral he was an extensive contributor. Be also devoted himself with success to the subject of pure geometry. In the Allyoneine Thew* der Strablensysteme (Crelle's Journal, vol. he
laid down the principles applicable to the so called Kummer surfaces. These are surfaces of the fourth degree with 16 knot points (Enoten punkten, corresponding to double points of a curve). and 16 singular tangent planes. The points and planes are so related that each of the 16 planes contains six of the points, and through each of the 16 points pass six of the planes. The system of these points and planes is called a Kummer configuration. The theory of these sur faces has been studied by Cayley, Beye, Lie, and others, and Borchardt and IL Weber have shown the relation of this theory to that of hyperel liptie (Abelian) functions. Besides the contri butions already mentioned. Kummer's writings include an interesting memoir entitled, Urber die Wirkung des Luftwiderstandes auf Korper ron vcrsehiedener Gestalt, inbesondrre auf die Ge sehossc (Abhandlungen der Berliner Akademie, 1875). For biographical sketch and list of works, consult the Jahresbrricht der deutsehen Nathe matiker-fereinigung. yol. iii. (Berlin. 1894).