222 
THE SQUARING OF THE CIRCLE 
in a circle, ‘ suppose, if it so happen, that the inscribed polygon 
is a square ’.) On each side of the inscribed triangle or square 
as base describe an isosceles triangle with its vertex on the 
arc of the smaller segment of the circle subtended by the side. 
This gives a regular inscribed polygon with double the number 
of sides. Repeat the construction with the new polygon, and 
we have an inscribed polygon with four times as many sides as 
the original polygon had. Continuing the process, 
‘Antiphon thought that in this way the area (of the circle) 
would be used up, and we should some time have a polygon 
inscribed in the circle the sides of which would, owing to their 
smallness, coincide with the circumference of the circle. And, 
as we can make a square equal to any polygon ... we shall 
be in a position to make a square equal to a circle.’ 
Simplicius tells us that, while according to Alexander the 
geometrical principle hereby infringed is the truth that a circle 
touches a straight line in one point (only), Eudemus more 
correctly said it was the principle that magnitudes are divisible 
without limit; for, if the area of the circle is divisible without 
limit, the process described by Antiphon will never result in 
using up the whole area, or in making the sides of the polygon 
take the position of the actual circumference of the circle. 
But the objection to Antiphon’s statement is really no more than 
verbal ; Euclid uses exactly the same construction in XII. 2, 
only he expresses the conclusion in a different way, saying 
that, if the process be continued far enough, the small seg 
ments left over will be together less than any assigned area. 
Antiphon in effect said the same thing, which again we express 
by saying that the circle is the limit of such an inscribed 
polygon when the number of its sides is indefinitely increased. 
Antiphon therefore deserves an honourable place in the history 
of geometry as having originated the idea of exhausting an 
area by means of inscribed regular polygons with an ever 
increasing number of sides, an idea upon which, as we said, 
Eudoxus founded his epoch-making method of exhaustion. 
The practical value of Antiphon’s construction is illustrated 
by Archimedes’s treatise on the Measurement of a Circle, 
where, by constructing inscribed and circumscribed regular 
polygons with 96 sides, Archimedes proves that 3y >n > 34x> 
the lower limit, tt > 3^5-, being obtained by calculating the