Euclid - A Man Who Shaped Our World

Euclid's geometrical treatise "The Elements" has been a touchstone for more than 2,000 years.

One of the more curious historical revelations of Steven Spielberg's "Lincoln" is that America's 16th president was obsessed by a Greek mathematician from the fourth century B.C. While traveling from town to town as a young lawyer riding the Eighth Circuit in Illinois, Abraham Lincoln kept a copy of Euclid's geometrical treatise, "The Elements," in his saddlebag; his law partner Billy Herndon related that at night Lincoln would lie on the floor reading it by lamplight. Lincoln said he was moved to study Euclid by his desire to understand what a "demonstration" was, and how it differed from any other kind of argument.

Euclid's notion of a demonstration, or proof, and related topics are explored in David Berlinski's "The King of Infinite Space." He wants to goad the reader into appreciating a textbook that has endured for over 2,000 years. But he is also concerned to convince the reader that the "Euclidean style" endures as a "way of life" and that this style consists in Euclid's fusion of beauty and logic in his proofs. Along the way we learn about the axioms (or "postulates"), definitions and theorems of Euclid's system.

Reading this brief, lively work is like sitting with the author in a French café with too many carafes of red wine and the smoke of hundreds of Gauloises swirling inside your head. "Mathematicians are fussy as cats. And almost as conservative," your host says. "Counting would seem to come first, no? . . . Long live the numbers then. But then there is seeing. Shapes are metaphysically as compelling as numbers. . . . Long live the shapes too." The book's style will put off some readers, especially during the first half, in which we are subjected to numerous stream-of-consciousness musings.

Readers, however, should persist, because in Chapters VI and VII we get to the heart and soul of this book, a discussion of Euclid's proofs. Here Mr. Berlinski depicts Euclid as a mountain climber whose theorems or conclusions are the peaks, and the proofs the record, of his treks upward. In yet more colorful writing, Mr. Berlinski depicts the different kinds of demonstrations we find in the "Elements": "In some, Euclid gets to the top quickly; in others, he is obliged to grunt and slog, and in these he is like some grizzled old climber recalling how he was threatened by congelation of the anus."

In these chapters Mr. Berlinski makes two provocative claims about Euclid's notion of a proof. One concerns the illustrations or diagrams that accompany his arguments. What are Euclid's diagrams for? Are they essential parts of the proofs, illustrative guides, helpful if occasionally misleading visualizations, or something else? Mr. Berlinski proposes that the diagrams and the logical structure of the proofs come together to form a unified whole; they "undergo fusion." This combination of illustration and argument, indeed, "lies at the heart of Euclid's method."

But how can a visual image play a role in a logical argument? Mr. Berlinski defines for us what a proof should do: It "compels belief." But Mr. Berlinski seems to have a controversial understanding of this compulsion. For him it is more psychological than logical, more rhetorical than mathematical, more feeling than fact. Take the author's discussion of Euclid's first proof, a demonstration of the proposition that on a given straight line segment it is always possible to construct an equilateral triangle. After agreeing with modern mathematicians that there are numerous logical lapses in Euclid's argument, Mr. Berlinski nevertheless concludes that his proof succeeds—because, again, its combination of logic and illustration "compels belief."

This view contrasts markedly with the view of Aristotle (who, Mr. Berlinski postulates, might conceivably have known Euclid). For Aristotle, and philosophers ever since, a proof or demonstration is an argument that compels rational belief in the sense of showing that the conclusions must be drawn from the premises, that is, that the conclusion is a necessary consequence of the premises. If there are logical holes in Euclid's first proof, it cannot compel rational belief, even if it persuades some people to believe it.

Even when Euclid's proofs make no logical leaps, they are based on assumptions about points, lines and shapes in space as he conceived it. This "Euclidean space" was taken for nearly 2,000 years to describe the "invisible scaffolding of spheres, cones and cylinders in nature," as the painter Paul Cézanne put it. Non-Euclidean geometries, first developed in the 19th century, showed that there are other ways to imagine space, in which the Euclidean postulates are not only not necessary but false.

For ages, doubts about the necessity of Euclidean geometry focused on the fifth of Euclid's axioms—the one claiming that parallel lines do not converge at a point. This had always been recognized, even by Euclid, as being "shady" (in Mr. Berlinski's phrasing) compared with the other postulates. By the end of the 19th century, many mathematicians could no longer share the robust confidence of Immanuel Kant, who had once proclaimed that Euclidean geometry was not only true but necessarily true, "an aspect of the human mind, the expression of the way the mind confronted the sensuous world of shapes."

"The King of Infinite Space" comes alive in the penultimate chapter, on the rise of non-Euclidean geometries. Mr. Berlinski introduces us to the Hungarian mathematician János Bolyai (1802-60), whose father was obsessed with the parallel postulate and tried to warn his son: "I beg you in the name of God. Leave the parallels in peace." His son would not desist. Neither would the Russian mathematician Nicolai Lobachevsky (1792-1856) or the Italian Eugenio Beltrami (1835-1900), who took things further by devising a physical model of non-Euclidean space, the Beltrami pseudosphere. Later, the French philosopher and mathematician Henri Poincaré (1854-1912) added another pictorial model to the non-Euclidean gallery, the Poincaré disk. In the end, the combination of illustration and logical proof, so important to Euclid, is what led to his undoing.

This sea change in geometry had far-reaching consequences, especially in the 19th century. In an age of political upheaval, this mathematical revolution provoked a great deal of anxiety. If the axioms of Euclidean geometry were not necessarily true but only true of Euclidean space—which was only one type of possible space among many—what could be necessarily true? As the philosopher Bertrand Russell succinctly put it: "If mathematics was doubtful, how much more doubtful ethics must be." It may be that what Lincoln found in Euclid was not only the notion of a logical demonstration but also the comfort of certainty, necessity and stability in a transformative and tumultuous time.