Squaring of the Circle

Conclusion

And because a semicircle, by its very definition original: "diffinitionē", is a plane figure contained by the diameter of the circle and half of the circumference, it must necessarily consist of those same parts. There are 14 parts, since that is half of the line which we previously established as being 28 parts. Furthermore, since a straight line—as also stated in its definition above—is the shortest extension from one point to another, the diameter f. g. will be smaller than the semicircle f. h. g. Because half the circumference is 14 parts, as has been said, the diameter cannot reach that many parts; thus, it will be a number less than 14.

Again, since the line K. l. is equal to the side of the square, and the side of the square is seven parts (as is clear and has been demonstrated), it is therefore necessary that the segment K. l. be 7 parts. But the diameter f. g. is greater than K. l.; for it is neither smaller nor equal, since it would then follow that the whole was less than or equal to its part, which is manifestly impossible original: "metrū [manifestum] impossibile"; a fundamental rule of geometry is that the whole must be greater than its parts. And since the side of the square claims for itself a fourth part of the circle's perimeter, while the diameter original: "Diameter vero dimidiā" bisects the circle in half, the diameter will therefore be greater than the side of the square. We conclude, then, that the diameter of the circle is less than 14 parts and greater than 7. This was to be demonstrated: that the diameter tripled, along with a ninth part The author calculates the circumference as (3 × 9) + (9/9) = 28, completes the circular line. Thus, the tripled diameter comes very close to 28, because it reaches 27; but with the addition of one unit, it is completed.

A geometric diagram showing a circle with an inscribed square. A vertical diameter is labeled with points F at the top and G at the bottom. Within the square, a vertical segment is labeled K at the top and L at the bottom, marked with several horizontal tick marks along its length to represent units of measurement. The left side of the circular arc is labeled H and the right side M.