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Updated: June 11, 2025
The closing years of the last century were those of rapid progress in the art of navigation, and Captain Vancouver gives this striking summary of testimony: "By the introduction of nautical astronomy into marine education, we are taught to sail on the hypothenuse, instead of traversing two sides of a triangle, which was the usage in earlier times; by this means the circuitous course of all voyages from place to place is considerably shortened; and it is now become evident that sea officers of the most common rate abilities who will take the trouble of making themselves acquainted with the principles of this science, will, on all suitable occasions, with proper and correct instruments, be enabled to acquire a knowledge of their situation in the Atlantic, Indian or Pacific Oceans, with a degree of accuracy sufficient to steer on a meridianal or diagonal line, to any known spot, provided it be sufficiently conspicuous to be visible at any distance from five to ten leagues.
"And thought I was lost," continued the man of science too much bent on his own ideas, to understand her interruption. "Little danger of that! I made my own base, knew the length of the perpendicular by calculation, and to draw the hypothenuse had nothing to do but to work my angle.
Then comes the triangular city whose hypothenuse, fronting east, is on the sea; its chief fault is having been laid out on too small a scale. By its side runs a tramway for transporting the huge blocks of concrete intended to prolong the pier. The inner town also shows a new palace, a new hospital, and a host of improvements.
Beneath the swirl of a gown that lay in an iridescent avalanche of sequins about her feet, her foot, tilted to an unbelievable hypothenuse off a cloth-of-silver heel, beat a small and twinkling tattoo, her fingers tattooing, too, along the chair-sides. How insidiously do the years nibble in! how pussy-footed and how cocksure the crow's-feet!
We might just as well turn the argument the other way, and infer from the variety of religious opinions that there must necessarily be hostile mathematical sects, some affirming, and some denying, that the square of the hypothenuse is equal to the squares of the sides. But we do not think either the one analogy or the other of the smallest value.
To the tired student, the idea that he must give it up seemed sheer senility. As long as he could whisper, he would go on as he had begun, bluntly refusing to meet his creator with the admission that the creation had taught him nothing except that the square of the hypothenuse of a right-angled triangle might for convenience be taken as equal to something else.
"Well," said Sam smiling, "I know the length of that line because I know that the square described on the hypothenuse of a right angled triangle is equal to the sum of the squares described on the other two sides." "Whew! it fairly takes the breath out of a fellow to hear you rattle that off," replied Sid. "Come," resumed Sam, "we aren't getting on with what we undertook. Now look and listen.
You and I, reader, were we called on to superintend the education of girls of sixteen, might not select, as favorite points either the hypothenuse or the ancient methods of populating young colonies.
Take twice the sum of the squares of the minutes on the two legs and get the square root and then you have the hypothenuse of the two sides of the triangle; that is, you have the number of minutes' steaming you make good on your real course." The old skipper knew nothing of square roots or hypothenuses or anything that looked like 'em, and he had always laid his course out by compass points.
And it is reported that Pythagoras, upon the discovery of this problem, offered a sacrifice to the gods; for this is a much more exquisite theorem than that which lays down, that the square of the hypothenuse in a right-angled triangle is equal to the squares of the two sides. Right, said Diogenianus, but what is this to the present question?
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