"The "exhaustion method" (the term "exhaust" appears first in , 1647) was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... by reducing problems... to... formal logic only. ...This indirect method... the standard Greek and Renaissance mode of strict proof in area and volume computation was quite rigorous, ...It had the disadvantage that the result... must be known in advance, so that the mathematician finds it first by another less rigorous and more tentative method. ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...several mathematicians before Newton, notably Kepler, used essentially the same conceptions... our modern limit conceptions have made it possible to build this... into a theory as rigorous as... "exhaustion"... The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."