c. 1300, from Old French transmigracion "exile, diaspora" (13c.) and directly from Late Latin transmigrationem (nominative transmigratio) "change of country," noun of action from past-participle stem of Latin transmigrare "to wander, move, to migrate," from trans "across, beyond; over" (see trans-) + migrare "to migrate" (see migration). Originally literal, in reference to the removal of the Jews into the Babylonian captivity; general sense of "passage from one place to another" is attested from late 14c.; sense of "passage of the soul after death into another body" first recorded 1590s.
1580s, "passing of the soul at death into another body, human or animal," from Late Latin metempsychosis, from Greek metempsychosis, from meta, here indicating "change" (see meta-) + empsykhoun "to put a soul into," from en "in" (see in- (2)) + psychē "soul" (see psyche). A Pythagorean word for transmigration of souls at death. Related: Metempsychose (v.) "transfer from one body to another" (1590s).
1540s, from Latin Pythagoreus "of or pertaining to Pythagoras" of Samos, Greek philosopher (6c. B.C.E.) said to have travelled to Egypt and Babylon, whose teachings included transmigration of the soul and vegetarianism (these are some of the commonest early allusions in English).
Also in reference to a school he supposedly founded in Crotona in Italy. As a noun, "a follower of Pythagoras," by 1540s. The Pythagorean theorem is the 47th of the first book of Euclid: The area of the square whose side is the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the other two sides.