“The surface of the sun has 6×10^5 times the brightness of the full moon (incidentally the sun is 5×10^6 times as bright as the *half* moon).”

]]>André probably wanted to say “zenith” which indeed sounds a lot like “azimuth” Both words entered our language via Arabic: سمت الرأس (path above the head) and السمت (direction). Source: Wikipedia.

]]>“[…] inaccurate reading of the Arabic expression سمت الرأس (samt ar-ra’s) meaning “direction of the head”/”path above the head”, by Medieval Latin scribes in the Middle Ages, probably through Old Spanish. It was incorrectly reduced to ‘samt’ (“direction”) and imprecisely written as ‘senit’/’cenit’ by those scribes. Through Old French ‘cenith’, Middle English ‘senith’ and finally ‘zenith’ first appears in the 17th century.

]]>Once you’ve shown that a particular pyramid has volume (1/3)(height)(base), you can use affine shifts, dissections, etc. to convince yourself that the formula holds for ALL pyramids and cones.

]]>John, your generalization of “shadow area” to take account of multiplicity corresponds exactly to something standard in integral geometry.

Good — it seemed pretty natural, so it would be depressing if people were so silly as not to have noticed it before.

By the way, I now think the answer to my “Challenge” is very easy if one lets oneself use a bit of machinery. Taking a suitable limit, Descartes’ theorem gives a special case of the Gauss-Bonnet theorem, which says that the integral of the scalar curvature over a surface is 2π times its Euler characteristic. Namely, it gives the special case where our surface is a sphere embedded in **R**^{}^{3}: it says that the integral of this sphere’s scalar curvature must be 2π × 2 = 4π. But a round sphere of radius 1 has scalar curvature 1. So, the area of this sphere must be 4π.

This could be seen as a pathetically obscurantist way of computing the area of a sphere, but my point is rather to illustrate that the 4π in Descartes’ theorem is directly connected to the 4π in the formula for the area of the sphere.

And probably all this generalizes enormously, and probably that’s already been done.

]]>http://www.theonion.com/articles/historians-admit-to-inventing-ancient-greeks%2C18209/

“WASHINGTON—A group of leading historians held a press conference Monday at the National Geographic Society to announce they had “entirely fabricated” ancient Greece, a culture long thought to be the intellectual basis of Western civilization. … Emily Nguyen-Whiteman, one of the young academics who “pulled a month’s worth of all-nighters” working on the project, explained that the whole of ancient Greek architecture was based on buildings in Washington, D.C., including a bank across the street from the coffee shop where they met to “bat around ideas about mythology or whatever.”

“We picked Greece because we figured nobody would ever go there to check it out,” Nguyen-Whiteman said. “Have you ever seen the place? It’s a dump. It’s like an abandoned gravel pit infested with cats.”

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