Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/31

 is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out."

This principle is deduced by Euclid from the axiom that, if there are two magnitudes of the same kind, then a multiple of the smaller one can be found which will exceed the greater one. This latter axiom is given by Euclid in the form of a definition of ratio (Book v. def. 4), and is now known as the axiom of Archimedes, although, as Archimedes himself states in the introduction to his work on the quadrature of the parabola, it was known and had been already employed by earlier Geometers. The importance of this so-called axiom of Archimedes, now generally considered as a postulate, has been widely recognized in connection with the modern views as to the arithmetic continuum and the theory of continuous magnitude. The attention of Mathematicians was directed to it by O. Stolz, who shewed that it was a consequence of Dedekind's postulate relating to "sections." The possibility of dealing with systems of numbers or of magnitudes for which the principle does not hold has been considered by Veronese and other Mathematicians, who contemplate non-Archimedean systems, i.e. systems for which this postulate does not hold. The acceptance of the postulate is equivalent to the ruling out of infinite and of infinitesimal magnitudes or numbers as existent in any system of magnitudes or of numbers for which the truth of the postulate is accepted.

The example of the use of the method of exhaustions which is most familiar to us is contained in the proof given in Euclid 2, that the areas of two circles are to one another as the squares on their diameters. This theorem which is a presupposition of the reduction of the problem of squaring the circle to that of the determination of a definite ratio $$\pi$$ is said to have been proved by Hippocrates, and the proof given by Euclid is pretty certainly due to Eudoxus, whom various other applications of the method of Exhaustions are specifically attributed by Archimedes. Euclid shews that the circle can be "exhausted" by the inscription of a sequence of regular polygons each of which has twice as many sides as the preceding one. He shews that the area of the inscribed square exceeds half the area of the circle; he then passes to an octagon by bisecting the arcs bounded by the sides of the square. He shews that the excess of the area of the circle over that of the octagon is less than half what is left of the circle when the square is removed from it, and so on through the further stages of the process.