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CANTOR, GEORG:

German mathematician; born at St. Petersburg, Russia, March 3, 1845. He is distantly related to Moritz Cantor. He was only eleven years old when he went to Germany, where he received his high-school and university education. In 1862 he entered the University of Zurich, Switzerland, but at the close of the academic year moved to Berlin, where he remained until 1867, deeply interested in his studies and enthusiastically following mathematical and philosophical lectures at the university. In 1869, two years after receiving the degree of Ph.D. at the University of Berlin, he wasadmitted as privat-docent at the University of Halle, where he became assistant professor in 1872, and professor seven years later.

Without being a prolific writer, Cantor has rendered invaluable services to the progress of mathematical analysis, more especially to that of the modern theory of functions, by his epoch-making contributions to the theory of multiplicities ("Mannigfaltigkeitslehre" in German, "théorie des ensembles" in French)—a doctrine which he wholly and independently created and developed. The startling but fruitful ideas embodied in his "Grundlagen einer Allgemeinen Mannigfaltigkeitslehre," Leipsic, 1883, have become the property of the best modern text-books on mathematical analysis, despite the difficult and abstruse character of the new conceptions involved.

No mathematician could to-day dispense with the perusal of the little volume that, at a single stroke, brought universal fame to the author, and opened a new and rich field for mathematical investigation. Georg Cantor's definition of the mathematical continuum, as a particular form of a more general class of multiplicities, has been of immense benefit to the progress of mathematics, and in itself constitutes an undying monument to the name of this profound philosopher and mathematician. Much of the clearness and precision of modern mathematical methods is due to his example and instigation. He endeavored to unravel the mysteries of the infinite, and succeeded in establishing, if an indirect, nevertheless a perfectly determinate conception of the mathematical infinity—theory of transfinite numbers. His rigorous mathematical theory of irrational numbers, together with the independent investigations of Weierstrass and Dedekind, filled an important lacuna in the development of modern mathematical thought. On this subject see more especially his paper "Ueber die Ausdehnung eines Satzes aus der Theorie der Trigonometrischen Reihen," in vol. v. of the "Mathematische Annalen," 1872; and the memoir "Die Elemente der Functionenlehre," by E. Heine in Crelle's "Journal für die Reine und Angewandte Mathematik," 1871, vol. lxxii.

The articles by Georg Cantor which appeared under different titles in Crelle's "Journal," in the "Acta Mathematica," and in the "Zeitschrift für Philosophie und Phil. Kritik," are, for the greater part, either reproductions or translations of papers published in the "Mathematische Annalen," and later collected, under the title "Grundlagen einer Allgemeinen Mannigfaltigkeits lehre." His "Gesammelte Abhandlungen" were published in 1890.

S. A. S. C.
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