Interstices & Intersections: The Dreaded Book X & Super Deluxe Paper

The story has it that the first of the Pythagoreans to publicize irrational numbers perished in a shipwreck. The scholium on Euclid's Book X in which this story appears admits that the tale may have been an allegory, "hinting that everything irrational and formless is properly concealed, and, if any soul should rashly invade this region of life and lay it open, it would be carried away into the seas of becoming and be overwhelmed by its unresting currents." I am sure that I am not the first student of Euclid to feel that this is true, to reach Proposition 9 of Book X and try literally and figuratively to close the book and pretend that nothing has happened. Maybe I'll print the Bible or pass handgun legislation in the USA. Something easy like that.

The study of irrational numbers is thought to have begun with the application of the Pythagorean theorem to the diagonal of a square whose side is 1, resulting in a diagonal whose length is √2. By following the implications of this result to their logical conclusions, the side of the square is shown to be both odd and even, a proposition which would lead, if not to shipwreck, then surely to migraine in any rational Pythagorean. The discovery of irrationals, or what Euclid calls incommensurables, lead to a re-casting of geometric thought, which in turn produced Euclid's gargantuan Book X. The book contains 116 of the 450 Euclidean propositions and is veiled in a similar opacity as I described in my post on Book V.

The Euclidean Books of Lines, as I call them,—Books, V, VII, VIII, IX, and the beginning of X—use straight lines to represent number and magnitude. It is a simple enough system from which our contemporary use of x, y, etc. was developed, designed to steer clear of assigning any specific values to the formulas. For those of us who love the simple things in life, circles, triangles, rhombi, etc., the system of lines can feel more like an army of tiny little sabres slowly bleeding one to death. Take, for instance, Proposition x.10: To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line. The traditional diagram for this proposition is pictured below, five straight lines of ambiguous length, standing in for the measures and magnitudes. I get intellectual brain freeze when I stare at these diagrams. I understand them, even crave them, but they make me hurt for the pleasure. Below the traditional diagram is an image of my sketched proof which I think is an accurate portrait of how my mind deals with these problems. I assign value and build the square, both of which go against the Euclidean grain.

* * * * * * * * * * * * * * *

On other fronts, Travis Becker from Twinrocker Handmade Paper sent me a sample making of paper for the deluxe edition of Interstices & Intersections. He was trying to make a paper using cotton rag and abaca fibers that would approximate a linen and cotton paper I made with Mina Takahashi last year. The results were beautiful. Yesterday I proofed a variety of plates to test line quality and paper stretch and the sheets performed perfectly. In a few weeks Travis will begin work on the 800 sheets required for the deluxe copies.

The traditional diagram for Proposition x.10, using straight lines to represent number and magnitude.

My parsing of the proof using assigned values and forms.

Alberti & Wittkower (& Euclid)

Through an idea that lead to a web search that lead to another web search that lead to an idea, I stumbled upon Rudolph Wittkower's wonderful book, Architectural Principles in the Age of Humanism. I wish I had found it sooner. Wittkower's ability to articulate the humanists' take on classical geometry is unparalleled and his extended discussion of the architectural symbolism of the circle is something I could read every day, aloud, like a chant. In the book's second part, Alberti's Approach to Antiquity in Architecture, Wittkower says of Alberti's first ecclesiastical architectural work, San Francesco in Rimini (aka Il Tempio Malatestiano):

To bury people under the arches of the exterior of a church was actually a mediaeval custom; examples are numerous and were well known to Alberti. The tombs planned for the façade and the side fronts of S. Francesco derive from such mediaeval models. But by placing sarcophagi with classically styled inscriptions under serene Roman arches Alberti created an impressive pantheon for heroes rather than a burial-ground with its traditional funereal associations.

If we parse Wittkower's paragraph, what he is actually saying is that the decisive difference between Alberti's first church and medieval ones is the style of lettering on the inscriptions, which effectively transform a graveyard into a pantheon. Medieval Italian churches abound with sarcophagi, or noted graves at least, under "serene Roman arches;" their choice of surface treatment differed from Alberti's but the over all architectural style is the same. The lettering on S. Francesco provides the transfigurative graphic content of the work, elevating an earthy medieval model to the reserved example of a new style.

From the standpoint of lettering history, San Francesco figures prominently in the (endless, tiring) debate over who in the Renaissance first made letters that approximated classical ones. Built as a vanity project for Sigismondo Malatesta in the 1450s and 60s, only the exterior of S. Francesco can be attributed to Alberti. Which is fine because the exterior is where all the faux classical lettering appears. The building itself, as Wittkower implies, only hints at Alberti's mature architectural vision, but the inscriptions are a clarion call for the coming generation. They place Alberti firmly in the company of Andrea Mantegna and Felice Feliciano, two other potential Adams in the creation myth of humanist lettering.

You may have guessed that I am not terribly interested in who first made classically inspired letters. History just doesn't happen that way. There is no Adam or, if there is, there is only one and he is long dead. Everything else is swept up in the zeitgeist of generational change. To suggest that the greatest architect of the Quattrocento borrowed ideas (from Vitruvius) and style (from the middle ages) but that he (or Mantegna or Feliciano for that matter) somehow produced ex nihilo the lettering of the modern age is absurd. Further, to place such emphasis on the Patient X of a revival of a millennium-old lettering style is to discount the millennium of lettering that interposed the two exemplars. To disassociate Alberti's inscriptions on San Fracnesco from medieval examples such as those on the Duomo of Salerno (1081), Santi Giovanni e Paolo al Celio (1150s), or San Giorgio in Velabro (first half of the 13th century) is to miss out on the true grist of creation: the friction and dialogue between generations, the revival and rejection that defines and energizes new styles.

Somehow, this relates to Euclid.

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.

Interstices & Intersections in Progress

It has been a busy week of working on Interstices & Intersections. On Monday, 5,000 sheets of paper for the standard edition (all 1,320lbs worth) arrived from Germany, filling every available shelf and the entire surface of one of my two work tables. This morning Travis Becker at Twinrocker Handmade Paper made the first trial batch of paper for the deluxe edition. Travis is trying to create a paper that has similar qualities to one I made with Mina Takahashi on her farm last Spring. Between these two paper events I have been steadily working my way through the 115 proofs of the first four books of Euclid—drawing each proof, writing the Euclidean enunciation beneath it, and painting a title page for each volume of my Euclid notebooks as I go. I have settled on which proposition I will annotate from each of the first four books. Only 325 more proofs to go before all thirteen propositions are chosen.

The title pages for the first four volumes of my Euclid notebook, spread out on 1,000 of Zerkall paper.