Number Theory for the Millenium

University of Illinois at Urbana-Champaign
Rare Book Room Exhibit


Felkel, Anton (1740 ----). Ta𝔣el aller Einfachen Factoren der durch 2, 3, 5 nicht theilbaren Zahlen von 1 bis 10 000 000. Vienna: 1776.

Glaisher wrote of Felkel's work: "The table is so curious and rare, and, besides, the facts connected with its calculation and publication are so remarkable that I here give an account in some detail."

According to Glaisher, Anton Felkel (b. 1740) was a schoolmaster in Vienna, who did not begin to study mathematics until his mid 30s, and was then involved in the construction of mathematical tables for the rest of his life. Felkel corresponded with the mathematician Lambert, who had promised immortality to anyone "unwearied and resolute" enough to write a factor table to 1,000,000. There is a complicated history of correspondence, announcements of future tables, and competing table-writers. At one point, Lambert wrote that Felkel always seemed to be printing circulars instead of the table itself. The last reported sighting of Felkel is in Lisbon in 1798.

The frontispiece was explained by Glaisher: "On the title page of the work is a large engraving representing Felkel turning in contempt from a disordered cabinet of books of general literature, Basedow, Cutrius, Gottsched, &c., to a neatly arranged cabinet of mathematical books, Euler, Kastner, Newton, Maclaurin, &c., the book which is open in his hand being Lambert. An apparatus, consisting of eight rods in a frame, is resting against the table; this is not doubt Felkel's machine for forming a factor table,"

Felkel gave in 1776 a table of all the prime factors (designated by letters or pairs of letters) of numbers, not divisible by 2,3 or 5, up to 408,000, requesting for entry two auxiliary tables. In manuscript, the table extended to 2 million; but as there were no purchasers of the part printed, the entire edition, except for a few copies, was used for cartridges in the Turkish war. The imperial treasury at Vienna, at the cost of which the table was printed, retained the further manuscript. (Dickson, p. 349).

In his Latin translation of Lambert's Zusatze, Felkel gave all the prime factors except the greatest of numbers not divisible by 2, 3, 5 up to 102,000, large primes being denoted by letters. In the preface, he stated that, being unable to obtain his extensive manuscript in 1785, he calculated again a factor table from 408,000 to 2,856,000. (Dickson, p. 350)

Glaisher says that he knew of two copies of the manuscript: one, with a German title page, and going up to 144,000, which belonged to the Royal Society, and another in the Graves Library of University College, with a Latin title page, and going up to 408,000. There is currently a copy in the New York Public Library.

The copy on display here resembles the former, with its German title page and upper limit of 144,000. However it also has an inscribed name "Zach". Baron Franz Xaver von Zach (1754-1832) was a prominent astronomer, and it seems plausible that he might have had a copy. In fact, the story of the table being used for cartridges in the Turkish was comes from Zach's ""Monatliche Correspondenz" of 1800. Another possible owner is Zacharias Dase (1824-181), who was a "lightning calculator" with little mathematical ability, who constructed many mathematical tables.

Dickson, Leonard Eugene, History of the Theory of Numbers, vol. 1, 1919.
Glaisher, J. W., Proceedings of the Cambridge Philosophical Society, 1878.


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