a b, c

If the whole circumference is made into 196 portions, together with the wedges answering to them, you will find that each of those portions goes 196 times into the greater portion of the greater wedge a b c. And likewise the rectilinear wedge a b c will receive within itself the 196 little wedges into which the circle was divided. Here one does not multiply the diameter by 3 and 1/7, as Archimedes prescribes, but proceeds more exactly, for I give the diameter the value 14, and then I know that the proportion between circle and circle is as the square of the one to the square of the other, each multiplied by itself. And so I make a diameter of 14 and say '14 times 14 makes 196,' and the other diameter is one, and I say 'one times one makes one'; and the other [continues at top left:]

To remove the portion a b of the small circle is as much as to remove a hundred like small portions of that small circle, because the triangle a b c is the hundredth part of its circle. It is proved that the triangle, made of two straight lines and a third curved one, is equal to the whole circle, the curvature being lacking in the triangle and as much of the circle as that portion occupies. That is, I find the whole triangle, with its portion of circle a b, to be in itself equal to the whole circle. Now I wish to take away from the triangle its portion, and to take away as much of the semicircle where that portion lies. And by the axiom which says: if two things are laid one upon the other and the one neither exceeds nor is exceeded by the other, then without doubt those quantities are equal to one another; and if from equal things equal parts are taken, the remainder stays equal. So then, removing the said portion of the triangle, and likewise as much from the circle occupied by that portion, the remainder of the triangle stays squared and is equal to the remainder of the circle. That the circle was in the first place equal to the triangle is proved thus: because the proportion between circle and circle is as that between square and square made from the multiplication of its diameter. Now, since the diameter of the small circle is one braccio, and the diameter of the circle from which the triangle is drawn is 14 braccia, which multiplied by itself makes 196 braccia for its whole surface, I divide its circumference into 196 equal parts and draw from the ends of one of those partitions two straight lines meeting at the centre of the circle. Thus I have formed a wedge holding within itself the 196th part of the whole circle, which triangle comes to be equal to the circle that was one braccio in diameter. Therefore I shall do as I said above: I shall split the circle on its diameter and shall have made a straight line, and I shall lay the whole portion of the triangle within that straight line, and so remove whatever the triangle and the semicircle touch together. Thus I shall have removed an equal part from each, and the remainder will be equal, that is, the remainder of the circle to the remainder of the triangle. And if the triangle were a millionth part of its circle, it would make the portion straight, which, being one braccio, would let you square the triangle and the circle divided into 196 triangles.