A little while ago I ran across and couldn’t resist the urge to acquire a copy of the last published critical edition of Nicomachus of Gerasa‘s nearly 2000-year-old Introduction to Arithmetic (Αριθμητικη εισαγωγη), the Hoche edition published by Teubner in Leipzig in 1866.
The provenance of the volume is interesting in itself. A small imprint on the inside front cover, “Paul Koehler—Buchhändler und Antiquar—Leipzig ’05“, shows where, thirty-nine years after its publication, it went through the bookshop of the famed German antiquarian and publisher Paul Koehler; in the rear, on the blank bottom of the last page of the index, is a stamp in Hebrew, stating that it was at one point part of the library collection of the University of Jerusalem; then on the FEP, the handwritten signature of the late, renowned philologist, linguist and classical scholar, Benedict Einarson (long of U. Chicago’s classics dept.). When I found the volume, it had already made its way back to a small bookshop in the UK.
In a recent period of recovery, I kept myself occupied by reading the little volume, and while so engaged found myself entertaining myriad ideas about polygonal numbers, a dormant fascination reawakened by Nicomachus. Skipping a good bit of middle ground and jumping to the point: I generalized the idea of the factorial function, which traditionally produces only the product of the first n natural or counting numbers, to a function which returns the product of the first n polygonal numbers for k-sided polygons. For simplicity of reference, I have called these “polygorials”, merging the two words polygonal factorials. In exploring these I discovered that a few have already appeared in the literature as other sequences in the Encyclopedia of Integer Sequences, though most were novel to this research, which also yielded other surprises, such as the fact that the ratio of the nth hexagorial to the nth trigorial yields the Catalan numbers.
For those interested, a small, introductory paper on the Polygorials details the work.