Of arithmetic.

Every odd number multiplied by an odd number remains odd.

Every odd number multiplied by an even number remains an even number.

b - a - d - a c f b - e

The lune b is worth the lune a placed below in the margin, and is worth the parallel-figure a with its horns, because their curvatures that touch are equal, and equal too are those that do not touch, and each of the two surfaces is half of the square enclosed in the circular parallel-figure. And it is proved thus: because if two things are equal to a third, they are also equal to one another, and so on.

First. — All the parts of a similar figure taken from two similar surfaces have among themselves the same proportion as that of their wholes.

Second. — And if two circles of double value are made upon one same centre, the excess of the greater circle is worth the whole of the smaller circle. And by the first, the portions a b of the greater circle are of the same value as the portions c d e f of the smaller circle, which is half of the greater. Therefore, removing the 2 greater portions from the greater circle and the 4 smaller portions from the smaller circle, the remainder of the greater will be equal to the remainder of the smaller.

Of circles double one to the other, the half of the greater is worth the whole of the smaller. And if from two equal things two equal parts are removed, the remainder will be equal. Therefore from the 2 semicircles of one same circle I shall remove c from the one and d e from the other, since c is worth as much as d e. It follows that the remainders a b remain equal.

a - c - d b e - n m o p

The square d b e m o is worth the lunes a n m o p, and the lunar portions n m o p are equal to the lune a.

Because the triangle b is equal to the whole lune a, the triangle d m is worth the half-lune n m, and the triangle e o is worth the other half-lune o n.