Thursday, March 7, 2013

Book V, Book V

I have spent the day working through the propositions in Book V of Euclid's The Elements. Augustus De Morgan says of the book's opening propositions that they are "simple propositions of concrete arithmetic, covered in language which makes them unintelligible to modern ears. The first, for instance, states no more than that ten acres and ten roods make ten times as much as one acre and one rood." To give you an idea of what De Morgan means by the book's unintelligible language, here is Heath's translation of the enunciation of Proposition V.1 If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all. Once you sit down with the diagram and the text of the proof, these propositions are easy to work through. They are, after all, just as simple as De Morgan says. But the enunciations of the book's twenty-five propositions—the opening bits of text that tell you what the proposition is setting out to prove—are just as opaque as that of the first.

Among historic editions of Euclid, the illustrated printings are most famous but there were many beautiful editions printed in the Renaissance that contained only the enunciations—no diagrams, no proofs or conclusions. Antonio Blado printed at least two such editions, one in Greek, one in Latin. (Blado had a penchant for printing lists; the lists of banned books that he printed for the Vatican are models of typographic ingenuity.) Blado's Euclids are exquisite little pocket books, indispensable calling cards for the cosmopolitan humanist. One can only imagine the excruciating difficulty by which these books were attended. Imagine sitting down at your desk and trying to parse a proof for the proposition I quoted above, using only the enunciation. It makes me wonder how many owners of Blado's books pitched themselves head first out of their library windows in frustration.

The enunciations are not impossible to parse, of course, and once you immerse yourself in the language of Euclid his obscure geo-babble shines with an eerie legibility; but they are meant to be illustrated—by their readers if not by their printers. The diagrams that accompany each proposition are not illustrations, they are text. To properly understand Euclid you have to draw them. This singular quality of The Elements, that it is a text equally reliant upon image and language, sets it apart as a model for the contemporary artist book.