Eselsbrücke

The German word of my title is usually translated as “the bridge of donkeys” into English; but no sooner read, we are told its Latin derivation, pons asinorum, and referred to Euclid’s fifth proposition, the one that deals with isosceles triangles.

I learned that phrase in German; in German usage the word is simply a mnemonic—as it also does in Dutch and Czech—with an implied confession, when we use it, of our close relation to donkeys. The discovery that it has quite other meanings in other languages, meanings more closely related to Euclid, had to await my reading of Frederick W.H. Myer’s book, Human Personality and Its Survival of Bodily Death, a late-nineteenth century work, where the following sentence startled me into awakeness:

He [see below] may thus be ranked as the only man who has ever done valuable service to Mathematics without being able to cross the Ass’s Bridge.

     [Myers, p. 68]

While the phrase instantly brought the word eselsbrücke to my mind, the context was alien—and set me to investigating the phrase. I’d never encountered it in a book before. Well…

It turns out that Euclid’s Fifth Proposition—which simply asserts that triangles with at least two equal sides will have angles under the base equal to one another—was the last proposition medieval students had to study. If they managed to understand Euclid’s proof, they had then crossed the bridge from donkey to student status.

I am showing the diagram Euclid used in his proof of this assertion. The “base” here is BC—and the angle under the base would seem to be greater than the angle above. Therefore some serious proving is involved. How Euclid does it may be found here, on page 11.

I won’t go into the proof itself except to say that it is more complex than that for the first four propositions. And I’d point out that this graphic, if you look at it poetically, may suggest a bridge. Its top is BC and it is supported by F and G. Thus the graphic may have suggested the name.

The man to whom Myers refers was Dase (no first name given), a math wizard from childhood until his death. He could not even understand the simplest arithmetic as taught in first grade—yet the Academy of Sciences at Hamburg employed him to produce tables of factors and prime numbers out to nearly 8 million. (Factors are whole numbers that divide exactly into another whole number leaving no remainder; primes only ever have two factors: 1 and themselves.) Now as for that bridge, Dase never even came close to crossing it. Ah, the mystery of mind…