Archimedes pinned down π with no decimals and no computing machine, simply by trapping a circle between two regular 96-gons, one inscribed for the lower bound and one circumscribed for the upper, and concluded 3 + 10/71 < π < 3 + 1/7. The same habit of reasoning by area builds the rest: the squares on the two legs of a right triangle exactly fill the square on the hypotenuse, the three angles of any triangle detached and joined at one vertex lie flat along a straight 180-degree angle, and a disc sliced into thin wedges and interleaved settles into a rectangle of sides πr and r.
Archimedes pinned down π with no decimals and no computing machine, simply by trapping a circle between two regular 96-gons, one inscribed for the lower bound and one circumscribed for the upper, and concluded 3 + 10/71 < π < 3 + 1/7. The same habit of reasoning by area builds the rest: the squares on the two legs of a right triangle exactly fill the square on the hypotenuse, the three angles of any triangle detached and joined at one vertex lie flat along a straight 180-degree angle, and a disc sliced into thin wedges and interleaved settles into a rectangle of sides πr and r.
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