The King of Infinite Space: Euclid and His Elements

David Berlinski’s most recent volume, The King of Infinite Space, joins a group of similar books such as The Advent of Algorithm (Harcourt, 2000) and A Tour of the Calculus (Vintage, 1997). As in the other books, Berlinski is writing for a “popular” or “general” audience. The book’s basic goals are two: to describe the contents and structure of Euclid’s most famous work, the Elements, and to show how Euclid and the Elements have shaped the entire mathematical community and the world. The two points are interleaved and juxtaposed.

The alternation between Euclid and his reception is a good idea, but I found the implementation a bit wanting. The transitions seemed generally abrupt and confusing. I could sometimes see why the author made the transitions when he did after reading a few more pages, but I fear that some readers may find the abruptness a turn-off or might not quite see where Berlinski is going.

Berlinski’s discussion of Euclid is not exhaustive, nor should it be: He presents an abstract, as it were, of the Elements and an introduction to Euclid more broadly. Readers who want to know the exact contents and structure of the whole work should, of course, turn to the Elements itself. Berlinski focuses in particular on some of the more famous theorems: Pythagoras’ Theorem as given in I.47 and the so-called “Bridge of Asses” theorem, I.5, to name a few from Book I.

Berlinski handles the reception and influence of Euclid in a very social way, using a mix of fictional dialogue with quotations from mathematicians and non-mathematicians spanning the two millennia since Euclid. I found this to be one of the book’s strengths: it does a good job of giving the reader a sense of our mathematical heritage and lore. I was less fond of the dialogue, which seemed a little too artificial and stylistically awkward.

The other aspect of Berlinski’s attempt to fulfill the second goal is a broad-strokes overview of the history of mathematics since Euclid. He spends considerable time talking about Euclid’s fifth postulate, pointing out to the reader how Euclid states it quite differently than Playfair. Kudos for doing so — I suspect many take Playfair’s version as if it were Euclid’s. He also talks about the seemingly universal discomfort with the postulate, even amongst the ancients, and the false “proofs” of it from the other axioms from ancient to modern times. This of course naturally leads to a discussion of the non-Euclidean geometries. Berlinski gives a good overview of their basic properties, including two models of hyperbolic geometry (Lobachevsky’s and Poincaré’s).

As for Euclidean geometry itself, he tends, perhaps a bit too much, towards Whig history, with the future a bit too present in the past. Berlinksi’s discussion of Pythagoras’ theorem, for example, centers on its algebraic interpretation and representation, the so-called “geometrical algebra.” Historians of mathematics will recognize this phrase well, and the associated controversies surrounding it. He writes, for example (page 96):

It is not disrespectful — is it? — to say that the geometrical algebra has in Euclid’s hands all of the elegance of bears chained and taught to dance. In his proof of the Pythagorean theorem, Euclid ignores the algebraic equation in which the facts are so easily expressed — a² + b² = c² — and occupies himself with the construction of those rather oafish squares, seeing in their area the secret to the theorem’s meaning. It is a clumsy business. The first proposition of Book II of the Elements affirms that the area of a given rectangle is equal to the sum of its subrectangles. This is in algebraic terms, the distributive law a(b + c + d) = ab + ac + ad, where a, b, c and d are numbers. The rectangles are illustrations; they get in the way.

I would say first that, yes, it is disrespectful. But more than that, it misses the point: to say that Euclid is proving that a² + b² = c² is ahistorical and a disservice to a reader untrained in historiography. He is proving that the squares on the legs together equal the square on the hypotenuse. While we can “translate” the theorem into modern algebraic notation, that translation says little about Pythagoras’ theorem itself or how Euclid understood it. Euclid is not doing algebra or “geometrical algebra” — we are interpreting it that way.

The chapter titles may be helpful to consider. Descartes is mentioned in “The Devil’s Offer”; Lobachevsky, Poincaré, Hilbert, and the lot in “The Euclidean Joint Stock Company.” In the first, Berlinski leaves to our imagination the exact offer and the Devil’s role. He is more direct with the “Stock Company”: he describes the addition of the non-Euclidean geometries as “diluting” the shares, expanding the market. The “company” is still the Euclidean one, though, perhaps appropriately: there is still something Euclidean about non-Euclidean geometries. The metaphor isn’t perfect: it makes geometry seem a bit more like some ancient multinational, a sort of mathematical equivalent of IBM.

Overall, the book is an interesting read. Mathematicians will be familiar with most of the mathematical content, but may find the story aspect and overall picture enjoyable. Active teachers will find a treasure-trove of quotations and references, ranging from stories of shipwrecked Greeks to Arabic poetry, starting with the very first sentence in the book. Students will see a tapestry of mathematics, even if Berlinski took some artistic liberties when constructing it.

Colin McKinney is an assistant professor of mathematics and computer science at Wabash College. He appreciates good music, good math, good Greek, and good Scotch… preferably all at the same time.