He entered Heidelberg University in 1848 where he was taught by Franz Ferdinand Schweins (1780-1856) and Arthur Arneth (1802-1858). Given Cantor's fame as an historian of mathematics, we should note at this point that Arneth was an excellent historian of mathematics who wrote the important work History of Pure Mathematics in its relation to the History of the Human Mind (1852). He was one of the first to try to base the history of mathematics on general philosophical principles, rather than seeing it as merely compiling facts and recording past events. There is, however, no sign that Cantor was influenced towards the history of mathematics at this stage in his career. After a year at Heidelberg, Cantor went to the University of Göttingen where he spent the years from 1849 to 1851. At Göttingen he was taught mathematics and astronomy by Carl Gauss, physics by Wilhelm Weber and mathematics by Moritz Stern, who was particularly interested in number theory. He returned to Heidelberg where he presented his doctoral thesis in 1851. His advisor at the Ruprecht-Karls University of Göttingen was Ferdinand Schweins and he was awarded a doctorate on 6 May 1851 for his thesis entitled Über ein weniger gebräuchliches Koordinatensystem Ⓣ. He did not take the state examination to qualify as a gymnasium teacher, which most students took at this stage, but he went to Berlin where he spent the summer semester of 1852 attending courses by Lejeune Dirichlet and Jakob Steiner. On 30 April 1853 he was appointed as a docent at the University of Heidelberg having submitted his habilitation thesis Grundzüge einer Elementararithmetik Ⓣ. He was to remain at Heidelberg for the rest of his life.
Cantor's early work was not on the history of mathematics but he did write a short paper on Ramus, Stifel and Cardan which he presented to a scientific meeting in Bonn. It looked at the introduction of Hindu-Arabic numerals into Europe, and was published in 1857. Later Cantor told Cajori, see , that his paper was:-
... so well received that he felt encouraged to continue his historical work.However, he was influenced at Heidelberg by Arneth, whom we mentioned above, and by the cultural philosopher Eduard Maximilian Röth. Although Röth was the professor of philosophy and Sanskrit at Heidelberg, he had studied mathematics, physics and chemistry in Paris with teachers such as François Arago, Jean-Baptiste Biot, Pierre-Louis Dulong (1785-1838) and Jean-Baptiste-André Dumas (1800-1884). He may well have encouraged Cantor to visit Paris which he did in the late 1850s. During this visit, which Cantor made shortly after his encouraging Bonn meeting, he became friendly with Michel Chasles and Joseph Bertrand. Chasles was an acknowledged leading expert on the history of geometry and encouraged Cantor to publish further historical material in Comptes Rendus.
From 1860 Cantor lectured on the history of mathematics and became one of the leading German historians of mathematics at the end of the 19th Century. His first significant work was Mathematische Beiträge zum Kulturleben der Völker Ⓣ (1863) (Mathematical Contributions to the Cultural Life of the People) which, like his earlier work, concentrated on the introduction of Hindu-Arabic numerals into Europe. Not only did 1863 mark the first important work by Cantor, but it also was the year in which he was promoted to extraordinary professor at Heidelberg. On his birthday, 23 August 1868, he married Telly Gerothwohl (1847-1873); they had one son and one daughter.
Cantor's second book was Euclid und sein Jahrhundert (Euclid and his century) in which he summarised the work of Euclid, Archimedes and Apollonius. It was published in 1867. His next major contribution was Die römischen Agrimensoren und ihre Stellung in der Geschichte der Feldmesskunst Ⓣ . This may seem a strange topic at first sight since it is recognised that the Romans added little to the development of mathematics. However, Cantor saw that the Roman surveyors had played an important role in transmitting Egyptian and ancient Greek practical geometric methods to Europe in the Middle Ages.
Cantor is best remembered for the four volume work Vorlesungen über Geschichte der Mathematik which traces the history of mathematics up to 1799. The first volume was published in 1880 and the last volume appeared in 1908. The first volume traces the general history of mathematics up to 1200. The second volume traces the history up to 1668. The year 1668 was chosen by Cantor because in this year Newton and Leibniz were just about to embark on their mathematical researches. The third volume continues the overview of the history up to 1758, again chosen because of the significance of Lagrange's work which began shortly after this date. The review of these three volumes in  is full of praise:-
It would be impossible to do justice to this monumental work within the brief limits of a book review, even if the task were not rendered supererogatory by the high standing of the work and the acknowledged authority of its author. Cantor's 'Lectures on the History of Mathematics' are the work of a man who has unswervingly devoted a life-time to this single task, who thirty-three years ago was well known for his important contributions to this subject, and who can now in the second edition of the first volume of his great work point with pride to the impulse and awakened interest which his endeavours have aroused in the historical studies of his science. He has had many predecessors, each of whom has distinguished himself in certain branches and by certain excellences ... Nevertheless, it may safely be said that profundity, accuracy, and extensiveness of treatment have never before in any history of mathematics been so thoroughly and intimately united as in the three volumes constituting these Lectures of Moritz Cantor.George Gibson, in the review , writes:-
It hardly requires to be stated that this history is certain to remain for many years the standard work on the subject with which it deals; in completeness, in accuracy, in clearness of arrangement, it stands unrivalled, and for the period which it covers is bound to be a permanent work of reference.After completing the third volume Cantor realised that, at the age of 69, he was not up to the task of completing another volume, so at the Congress of 1904 in Heidelberg he organised a team with nine further contributors to collaborate on the fourth volume. As editor-in-chief Cantor set high standards and insisted that the style and impartiality of the first three volumes be retained. This fourth volume again stopped just before a highly significant development since 1799 is the year of Gauss's doctoral thesis. The review  regrets that Cantor could not have written this himself:-
It is fifteen years since the second edition of the first volume of Cantor's 'Vorlesungen' made its appearance, and last year a third edition was called for. After the completion of Vol. III, which brought this monumental work up to 1758, the author felt that the time had come when the claims of advancing years were strong enough to force him to leave the completion of his magnum opus to others. He must have felt that he had not lived in vain and that an historical school worthy of German traditions had grown up around him, when, inspired by his example and enthusiasm, men were found able and eager to continue his great work under his direction. We are unfortunate in that the health of the aged savant has prevented him from giving that full personal supervision which is so essential in securing a general unity of treatment when the mathematical labours of half a century are divided among so many hands, however individually competent they may be. And when we say that no less than nine monographs by nine men constitute this fourth volume, it is not surprising to find, that although there is less diversity of treatment than might be expected, yet the volume is not what it would have been had Cantor been able to continue his colossal task. As it is, he had intended to append to this portion of the work a general treatment of the progress of mathematics during the latter part of the eighteenth century. We are, however, disappointed at the absence of what would of course have been a masterly survey of the development of ideas in that period, and greatly regret that the plan had to be abandoned. The greatest mathematical historiographer found himself compelled to be content with a chronological index. Even so, it is one of the most useful features in the book. His 'Überblick' gives a mass of information in a couple of dozen pages, fortified with ample cross references to the sections of this volume.Cantor was an excellent linguist and Cajori describes in  some amusing incidents of English speaking mathematicians speaking foreign languages at Congresses told to him by Cantor. Let us note at this point that Cantor was invited to give a plenary lecture at the International Congress of Mathematicians in Paris in 1900. He gave the lecture L'historiographie des mathématiques Ⓣ which he delivered in French. His command of the language was so good that some believed him to be French.
In  the historian of mathematics David Smith relates some touching personal memories of Cantor:-
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. ... 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.Cajori describes in  meeting Cantor, then 86, in 1915. Cantor described himself as:-
... a hewer of timber who with a big axe and with powerful strokes roughly cut the timber to proper form and dimension, but left it for those who follow him to dress, polish and finish.Cajori relates details of that visit in . We give an extract:-
Almost complete blindness had compelled him to stop research. He said he could see the general outline of a person's body, but could not make out the features. He could move about in his house without assistance. His hearing was still good and his mind fairly clear. Neither of us touched upon war issues, except that he once referred to me as coming from a nominally neutral country. Later in the day I met his son and daughter, who were free in expressing to me the hope that the United States would soon come to observe real neutrality. Cantor made inquiries about certain American and English mathematicians and spoke of special historical researches then in progress in Germany. ... After a dinner which was served in the garden in the open, the writer, having been told that it was Cantor's habit to lie down after the mid-day meal, departed.Moritz Cantor was no relation to Georg Cantor who referred to him as his Namensvetter (cousin by name only). Moritz Cantor was honoured with election to the German Academy of Scientists Leopoldina in 1877. He is buried in the Heidelberg Bergfriedhof Cemetery.
Article by: J J O'Connor and E F Robertson