Moritz Cantor

by David Eugene Smith

The history of mathematics is a relatively late study in any university; indeed it is relatively late as a subject for serious investigation anywhere. This does not mean that the world has lacked for works which give us information about the development of the various branches of human knowledge, including mathematics; for there are numerous works that contain historical material, such as the early treatises of Pappus (probably of the third century of our era) and Proclus (who wrote about two centuries later) and Sporus of Nicaea (c. 275). There are also such later works as the Cronica de Matematici of Bernardino Baldi (1553-1617), published long after his death; the Algebra of the learned Oxford scholar, John Wallis (1616-1703), containing a wealth of historical information; and the Arab writer, Abu'l-Faradsch Mohammed ibn Ishaq, whose Kitab al-Fihrist (Book of Lists, c. 987) is a collection of biographical notes upon Greek and Mohammedan mathematicians. Not until the eighteenth century, however, was a work bearing the name of history of mathematics written, the Versuch einer Geschichte der Mathematik und Arithmetik (1739) of Johann Christoph Heilbronner (1701-c. 1747), this being followed by the same writer's more important Historia Matheseos Universae a mundo condito ad seculum post Chr. Nat. XVI (1742).

Of these works the only ones that considered the history of mathematics in any large way was the second one of Heilbronner's, so that works on the general subject are less than two hundred years old. Such studies, however, seem to go in waves, and the eighteenth century saw the publication of a number of histories relating to certain phases of the subject, such as Cossali's Origine, trasporto in Italia, primi progressi in essa dell' Algebra (1797), and one outstanding general history, the Histoire des math&ecaute;matiques (2 vols., Paris, 1758; 2d ed., 4 vols., 1799-1802) by Jean Étienne Montucla (1725-1799). This may be called the world's first general history of mathematics of distinctly high grade, of wide scope, and based largely upon original sources.

The century following the appearance of Montucla's first edition there appeared a considerable number of general histories of the subject, including those of Kästner (4 vols., 1796-1800), Franchini (Lucca, 1821, and other related works), Arneth (Stuttgart, 1852), and Zeuthen (Leipzig, 1874). There were also such special histories as Libri's Histoire des sciences mathématiques en Italie (Paris, 4 vols., 1838-1841) and the notable publications of Boncompagni (1803-1869) including his Scritti di Leonardo Pisano (1857-1862) and his Bullettino di Bibliografia e Storia delle Scienze matematiche e fisiche (Rome, 1868-1887), the greatest collection of mathematical source material of the Middle Ages and the Renaissance that has ever been brought together. Passing over the extensive literature including the works of Heath, Heiberg, and Tannery, largely on Greek mathematics, since we are discussing general rather than special histories, the above sketch will serve to show the status of the work when the subject of this article entered the field.

Moritz Benedict Cantor, to use the full name, was the author of the greatest work on the general history of mathematics that has appeared up to the present time. Like every work which covers such a vast field and contains such a large number of details, it is open to and has been subject to many criticisms; but taken as a whole, it stands out as by far the best treatise of the kind that we are likely to have for many years to come. Its errors have been listed, apparently quite exhaustively, where they can be entered in the pages of his work, and this being done a student will have a mine of information that will not easily lose its richness and value.

Cantor was born of Hebrew parents at Mannheim on August 23, 1829, and died at Heidelberg on April 10, 1920. In 1848 he entered the University of Heidelberg and later spent some time in Göttingen, where he came under the influence of Gauss in mathematics and astronomy, and of Wilhelm Eduard Weber in physics. He returned to Heidelberg for his doctor's degree (1851), the subject of his thesis being "Ueber ein wenig gebrauchten Koordinaten-system." He then spent some time in Paris, where he came in contact with Michel Chasles, whose Aperçu historique sur I'origine et développement des méthodes en géometrie (Paris, 1837, with later editions), and with Joseph Bertrand (1822-1900) whose mathematical powers were already attracting attention. He returned to Heidelberg in 1853 and became a privat-Dozent in the University.

Cantor was in his early thirties when his first historical book, the Mathematische Beiträge zum Kulturleben der Völker (1863) appeared, emphasizing the bonds formed by mathematics between nations seemingly separated in other respects. The work has long been obsolete in many respects, recent discoveries having revealed new material and changed former impressions, but in its method of approach and general treatment it still ranks as a minor classic on the subject.

In 1863 he was advanced to the rank of ausserordentliche Professor of mathematics at Heidelberg, and in 1877 to that of honorary ordentliche Professor, and here his great work as a historian was done. For this his preliminary researches were made in the fields of Greek and Roman mathematics, resulting in his Euklid und sein Jahrhundert (1867) and Die römischen Agrimensoren (1875), the former having long since been replaced by such treatises as that of Sir Thomas Heath, but the latter being still a standard authority. During this preliminary period he also began (1875) the editing of the "Historisch-literarische Abtheilung" of the Zeitschrift fur Mathematik und Physik and the Abhandlungen zur Geschichte der Mathematik (1877). The second of these publications consists largely of source material which had come to his attention in the preparation of his great work. This work was entitled Vorlesungen zur Geschichte der Mathematik, the first volume of which appeared in 1880, and the fourth in 1908. Volume I (editions in 1880, 1894, and 1907) covered the period from earliest times to 1200, and in spite of the various editions is now out of date as to the mathematics of China, India, Egypt, and Iraq, so important have been the recent discoveries. Volume II (editions in 1892 and 1899) carried the work from 1200 to 1668, the beginning of the Leibniz-Newton period. Volume III (1894-1898, with a second edition in 1901) covered the succeeding ninety years (1668- 1758). Volume IV was planned when the International Mathematical Congress met at Heidelberg. Cantor was then too far advanced in years to do more than select a body of collaborators and lay out the general plan. The work was completed in 1908 and was shown at the Congress held in Rome.

In 1899 there was published in the Abhandlungen (vol. IX) a list of his books and articles. This list fills twenty-six pages (pp. 625-650). It includes three minor works not mentioned above and having no historical significance, — a Politische Arithmetik (1898), Das Gesetz im Zufall (1877), and Grundzüge einer Elementararithmetik (1855), no one of which added to his reputation.

On the occasion of his 70th birthday a special number of the Abhandlungen was issued. It was devoted chiefly to articles on the history of mathematics, contributed by various writers, and contained a bibliography of Cantor's works compiled by M. Curtze (pp. 625-650).

I feel that I may be allowed to mention my personal impressions of Professor Cantor, gained from an acquaintance of about thirty years. It began with a visit to his home when I was planning to spend a year in his course at Heidelberg. I shall never forget his kindliness of manner when I stated that I wished to devote time to the early stages of the calculus. He asked me where I thought it best to begin and I said with Kepler or Cavalieri. The pleasant way in which he approved, with the suggestion that it might be better to start with Archimedes, impressed the young American of thirty, but now he would begin somewhat earlier still. Circumstances have a way of changing one's life rather suddenly, and the plan was never carried out in Heidelberg University, but it did not interfere with the making of several visits from time to time in his home. The last of these was made about 1910, I believe. He was then nearly blind. As I entered his study he rose from a chair near the window, held out his hands, and advanced towards me. I took them and led him back to his chair and we talked over the years since we had first met. It was a pathetic interview, the more so when he mentioned Eneström's constant publication, in the Bibliotheca Mathematica, of lists of errata in the history, or rather of notes on the text which gave the impression of errors. He remarked that it did not seem right to call attention only to changes, and never say a word in praise of his life work. To this I agreed, and I have continued to feel that Eneström's method was unfair. Each was a friend of mine through many years, but I shall always feel less kindly towards the latter when I remember my last visit to Heidelberg. When I was leaving he rose, motioned with his hand to his bookcase (not a large collection) and said, "these are my books, but I cannot see them." Then he walked with me to the door and I said the conventional "Auf Wiedersehen", knowing that it would never come.


Smith, David E.: Moritz Cantor
In: Scripta mathematica. - 1 (1932), S. 204-207
Signatur UB Heidelberg: L 29-9-10::1

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