...induction.

There is in Pappus' The Collection a treatment of a variant to the Shoemaker's knife problem that has been argued to contain the first proof containing the two elements of induction.

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...Beg

Ulugh Beg was the grandson of the Asian (Mongol) conquerer Timur. His father captured the city of Samarkand and gave it to Ulugh Beg. He was primarily an astronomer and he built an observatory, the construction beginning in 1428. In his observations he discovered a number of errors in the computations of Ptolemy. He compiled tables of sins and tans at 1 intervals. These displayed a high degree of accuracy, being correct to at least 8 decimal places. Ulugh Beg's politics were not up to his science and, on his father's death, he was unable to achieve power despite being an only son. He was eventually being put to death at the instigation of his own son.

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