Wednesday, September 14, 2011

Edna and the Greek

Lyubov Popova, Untitled

Still working my way through Constance Reid's A Long Way from Euclid, which I heartily recommend to anybody with even the mildest interest in mathematics. Though you'd never guess from what they teach in the public schools, mathematics has a rich and fascinating history, which Reid illuminates with her lucid prose and diffusive enthusiasm for the subject.

Once or twice, Reid cites a snippet of verse from the fabulous flapper poet Edna St. Vincent Millay (1892 - 1950): "nothing, intricately drawn nowhere." I punched the line into Google, expecting to find a characteristic Millay piece about the aches and ecstasies of love, and was pleasantly surprised to discover she'd written a sonnet about Euclid himself. Well, sorta. Have a look:

"Euclid alone has looked on beauty bare"
by Edna St. Vincent Millay

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

Turning back to A Long Way from Euclid for a moment, it would seem that Millay (and most of the general populace) are misinformed in their understanding that the familiar laws of geometry were conceived by Euclid. Reid tells us that this is not exactly the case:

The Elements, from the beginning, was immediately recognized for what it was -- a masterpiece. The form of the book was not original. The logical ladder of definitions, axioms, theorems, and proofs was first erected by some earlier Greek than Euclid, perhaps a priest. The subject matter was not original. The masterly treatment of proportion which enabled the later Greeks to handle incommensurable as well as commensurable magnitudes is that of Eudoxus; and the other books are frankly based on the known work of other men. ("The picture has been handed down of a genial man of learning, modest and scrupulously fair, always ready to acknowledge the work of others," H.W. Turnbull wrote in The Great Mathematicians.) Only one proof -- that of the Pythagorean Theorem -- is traditionally ascribed to Euclid himself, although it is apparent that to fit theorems into his new arrangement he must have had to create other new proofs. Even the title, the Elements, was not original. This term did not refer, as we might think, merely to the elementary aspects of the subject but rather -- according to an early mathematical historian -- to certain leading theorems in the whole of mathematics which bear to those which follow the relation of a principle, furnishing proofs of many properties. Such theorems were called by the name of elements; and their function was somewhat like that of the letters in the alphabet in the language, letters being called by the same name in Greek. There had been many Elements before Euclid. That there were none after him is an unequivocal tribute to the sheer genius of his work.

As a mathematician, Euclid falls far behind Eudoxus, who preceded him, and Archimedes and Apollonius, who followed. The Encyclopaedia Britannica admits regretfully that he was not even a "first-rate" mathematician, but adds that there is no question but that he was a first-rate teacher. What he brought to the already great mathematics of his time was a genius for system. And system was exactly what was needed! There were many fine single works on specialized subjects. Many editors had gathered together what had seemed to them important. There were definitions, axioms, theorems, and proofs galore; and an almost equal number of organized and disorganized, overly complete and incomplete arrangements, all called the Elements. Euclid took these. He selected, substituted, added, rearranged; and what came out in his Elements was a distillation of all that had come before -- a model of systematic thought.

Note: I will reply to comments from the last two updates tomorrow, I promise.

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