Draw any triangle, pick a point on each side, and connect these in pairs to the vertices using circles as shown.

The circles will always intersect in a single point.

Further, the angles marked in green will all be equal.

When in very good spirits he would jest in a delightful manner. This took the form of deliberately absurd or extravagant remarks uttered in a tone, and with a mien, of affected seriousness. On one walk he ‘gave’ to me each tree that we passed, with the reservation that I was not to cut it down or do anything to it, or prevent the previous owners from doing anything to it: with those reservations it was henceforth mine. Once when we were walking across Jesus Green at night, he pointed at Cassiopeia and said that it was a ‘W’ and that it meant Wittgenstein. I said that I thought it was an ‘M’ written upside down and that it meant Malcolm. He gravely assured me that I was wrong.

— Norman Malcolm, Ludwig Wittgenstein: A Memoir, 1958

Botanist George B. Hinton named the plant species Salvia leninae Epling after a saddle mule, Lenina, who had helped him to gather more than 150,000 specimens in the mountains of western Mexico.

He wrote, “What is more deserving of commemoration than the dignity of long and faithful service to science, even though it be somewhat unwitting — or even unwilling?”

See Rigged Latin.

Writing on “The Sagacity of the Bees” in fourth century, Pappus of Alexandria argued that bees had contrived the hexagonal shape of their honeycomb cells “with a certain geometrical forethought.” Irregularly shaped cells “would be displeasing to the bees,” he wrote, and only triangles, squares, or hexagons could fill the space regularly. “The bees in their wisdom chose for their work that which has the most angles, perceiving that it would hold more honey than either of the two others.”

In 1964, in a charming address titled “What the Bees Know and What They Do Not Know,” Hungarian mathematician László Tóth told the American Mathematical Society that he had found a slight improvement on the classic honeycomb design: Instead of closing the bottom of each cell with three rhombi, as bees do, it’s more efficient to use two hexagons and two rhombi.

But, he added immediately, “We must admit that all this has no practical consequence. By building such cells the bees would save per cell less than 0.35% of the area of an opening (and a much smaller percentage of the surface-area of a cell). On the other hand, the walls of the bee-cells have a non-negligible width which is, in addition, far from being uniform and even the openings of the bee-cells are far from being exactly regular. Under such conditions the above ‘saving’ is quite illusory. Besides, the building style of the bees is definitely simpler than that described above. So we would fail in shaking someone’s conviction that the bees have a deep geometrical intuition.”

(László Fejes Tóth, “What the Bees Know and What They Do Not Know,” Bulletin of the American Mathematical Society, 1964, 468-81.)

UPDATE: Wait — maybe they’re not as smart as we thought. (Thanks, Vic.)

David Hume argued that reports of miracles can never be credited, because the weight of human experience must always favor a more natural explanation. “Nothing is esteemed a miracle, if it ever happen in the common course of nature. It is no miracle that a man, seemingly in good health, should die on a sudden: because such a kind of death, though more unusual than any other, has yet been frequently observed to happen. But it is a miracle, that a dead man should come to life; because that has never been observed in any age or country. There must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appellation.”

The sun is said to have danced in the sky in 1917. Well, which is more likely, that such an extraordinary event actually occurred, or that it was really a mass hallucination, an optical illusion, or any of a hundred more familiar explanations? A miracle, a suspension of natural law, is always the least likely possibility, so as rational creatures we must always reject it.

But Alfred Russel Wallace objected, “Such a simple fact as the existence of flying fish could never be proved, if Hume’s argument is a good one; for the first man who saw and described one, would have the universal experience against him that fish do not fly, or make any approach to flying, and his evidence being rejected, the same argument would apply to the second, and to every subsequent witness.”

Hume’s argument, he said, was “radically fallacious,” because if it were sound “no perfectly new fact could ever be proved, since the first and each succeeding witness would be assumed to have universal experience against him.” Who’s right?