We’ve had some pretty smart presidents. James Garfield devised this proof of the Pythagorean theorem in 1876, while serving in the House of Representatives:
The area of the trapezoid above is
The area of each green triangle is
And the yellow triangle is
So:
Suppose we hold an election with three candidates, X, Y, and Z. And suppose the voters fall into three groups:
Group 1 prefers, in order, X, Y, Z
Group 2 prefers, in order, Y, Z, X
Group 3 prefers, in order, Z, X, Y
Now, if Candidate X wins, his opponents can rightly object that a majority of voters would have preferred Candidate Z. And corresponding arguments can be made against the other candidates. So even though we’ve held a fair election, it’s impossible to establish majority rule.
The Marquis de Condorcet noted this oddity in the 1700s; it’s sometimes known as Condorcet’s paradox.
Epitaph in the churchyard of Llangerrig, Montgomeryshire:
— Charles Bombaugh, Facts and Fancies for the Curious From the Harvest-Fields of Literature, 1860
Phantom ships, as they have been called, have repeatedly been seen by various observers. Mr. Scoresby, in his voyage to Greenland, in 1822, saw an inverted image of a ship in the air, so well defined that he could distinguish by a telescope every sail, the peculiar rig of the ship, and its whole general character, insomuch that he confidently pronounced it to be his father’s ship, the Fame, which it afterwards proved to be.
— Charles Kingsley, The Boys’ and Girls’ Book of Science, 1881
See also The Wizard of Mauritius.
32 – 23 = 3 – 2