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.
