He is finding the area inside a parabola by the method of exhaustion, and he gets a series of triangles; each stage of the construction adds an area times the area of the previous stage.
22.
This represents the most sophisticated use of the method of exhaustion in ancient mathematics, and remained unsurpassed until the development of integral calculus in the 17th century, being succeeded by Cavalieri's quadrature formula.
23.
The case of the parabola was proven in antiquity by the ancient Greek mathematician Archimedes in his " The Quadrature of the Parabola " ( 3rd century BC ), via the method of exhaustion.
24.
This formula cannot be proven without using such infinitesimal arguments unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the method of exhaustion.
25.
Greek geometers, even before Euclid's time, informally calculated areas of two dimensional figures by regarding those figures as being composed of an infinite number of lines; but, after determining the area of a figure they proved their answer to be correct rigorously using Eudoxus'method of exhaustion, without the use of infinitesmals.
26.
Out of this necessity, Eudoxus developed his method of exhaustion, a kind of reductio ad absurdum that & established the deductive organization on the basis of explicit axioms & as well as & reinforced the earlier decision to rely on deductive reasoning for proof . This method of exhaustion is the first step in the creation of calculus.
27.
Out of this necessity, Eudoxus developed his method of exhaustion, a kind of reductio ad absurdum that & established the deductive organization on the basis of explicit axioms & as well as & reinforced the earlier decision to rely on deductive reasoning for proof . This method of exhaustion is the first step in the creation of calculus.
