Capital Letters and Small Letters in Mathematics

by A. G. DRACHMA”*

It must have been thirty years ago that Dr. 0. Neugebauer and I went together from a meeting in the Philological-Historical Society in Copen- hagen, and Dr. Neugebauer asked me: “Can you tell me when they began to use capital letters and small letters in mathematics?’ and I answered: “I don’t know; but I should think that it was after the invention of the printing press.”

I do not pretend to be like the elephant that never forgets, but this only imperfectly answered question has rankled in my mind ever since, and now I have tried to find out if I was right.

It is evident that the use of two different alphabets, capitah and small letters, was not possible before the small letters came into use; but the minuscule writing turns up, both in Greek and Latin manuscripts, in the 8th century A.D. So if we find in Greek manuscripts from the 11th century A. D. a tendency to mark the figures with uncial letters, we cannot ascribe this to the author; it must be due to the scribe.

Indeed, in Heron’s Dioptra, ch. 6, where he runs out of letters, though he uses both Koppa and Sampi, he begins again with A, ,B, marking them so that they mean 1.000 and 2.000.

Turning next to the palaeotypes, the books printed before 1500, I looked through some mathematical and astronomical works and made a discovery. These early books have no title-page, but a colophon, that is a postscriptum containing the information now usuaHy found on the title-page, but eventually also an account of the sheets or signatures contained in the book. The sheets were not numbered, but marked with letters, as A-Z, AA-ZZ, or the like, but I found a book with a-h, A-I, clear evidence of the use of two alphabets, capitaIs and small letters.

* Svends All6 47, 2800 Lyngby, Denmark.

Centaurus 1969: vol. 14. no. 1 : pp. 47-48

48 A. G. Drachmann

The earliest case I have found is Augustinus: de civitate dei. Venezia, Gabriele de Pietro, 1475, with the signatures a-z, A-D. After this time we find occasionally e.g. a-z, aa-u; but most printers use two alphabets for their signatures.

There can be no doubt that the use of upper and lower case letters as two different alphabets was invented by the printers, and that the mathe- maticians, then as now in dire need of alphabets, have learned it from them.

The first clear example of the use of two alphabets by a mathematician I have found in Ottavio Fabri: Us0 del la squadro mobile, Venetia 1598. Here the areas to be measured are marked by lower case letters: a, b, c.. ., while the angles of their figures have upper case letters: A, B, C. But in another work from the same time, Ludolf van Ceulen: Van den Circkel, 1596, small letters are used in the figures, but capital letters in the text, so that at this time the distinction between two different alphabets cannot have been usual.

This is only a very cursory and imperfect investigation; but still I think that I have found the right answer to the question asked thirty years ago.