Quadruple proportion of circles.

Double proportion of circles.

Double.

Double; n

Here the circles must not touch, because it is impossible that the 4 portions touch at the point of contact of the two circles, where, according to Euclid, nothing touches except a straight line; so draw the circles somewhat apart from one another.

m; a; b; n -o

a equals b, because a is the 2/6 of the semicircle m, and h is 1/6 of a circle double the circle m. But since it has been said that every curvilinear surface to which a square equal to it is given is always squarable into itself, therefore remove the convex of the fronts of a, and with the double portion fill the concave, and you will have made the square n o equal to the a above.

b a e — d c – f - g

a b c d equals a b e. Take away b d, and there remains a c, a rectilinear triangle equal to the surface f g; and this is proved because c d equals e.

a b

a equals b.

b a -(a) -c

To divide the half-portion a b c into equal parts.

o -a -b n

a b equals the half of the triangle a b n o, and such a half cannot be obtained by any other way in similar triangles.

Here are treated the lunes that have the concave side shorter and of lesser curvature than the convex side. - And of each the quadrature is given.

Then will be treated the lunes that have the concave side of greater curvature and shorter length than the convex side.

Then we shall treat of the lunes with concave and convex sides similar and equal; but these will be placed in the first book. Then we shall treat of 13 ways of squaring circles.

e c a — d; b

a equals b.

The side a b c is the eighth [part] of the curvature of the circumference of the lesser circle, and the side d e c the curvature of the sixteenth part of the circumference that the greater circle has, which is double the lesser circle. - And these are infinite.

b – c n a - m

The rectilinear triangle a b c equals a b n c, a triangle of two curved sides. It is proved because I remove the lower portion m n e from that triangle and restore to it as much value with the 2 lesser portions a b and b e.

a – h – c – n – S o – b – r - h

a b c is the portion of an eighth of a circle sub-double to the circle n o b, of which circle n o b it is 1/16. It follows that the small portion c S b is the half of the greater portion n o b. It is proved by the eighth [proposition] of this book, which says: 'Such is the proportion from a similar part to a similar part of two circles as is from circle to circle.' It follows here that, n o b being the circumference of the sixteenth part of the circumference of its greater circle, the like is the circumference c S b of the lesser circle. Therefore the ...

The wind always has motion where it has its exit, and all that which finds no exit returns backward amid the wind that has no motion. The motion of the wind is not straight, but rather always revolving after its impacts; and the revolving part runs from the extremities to the centre of that whirling motion, and from the centre it issues forth through the lateral parts with a conical figure, keeping the angle at the centre of that revolving motion.