Squaring of the Circle

A geometric diagram illustrating the rectification of a circle. It features a circle on the left labeled with points. To the right, a linear scale or ruler is bent into a semi-circular arc, divided into 22 equal segments numbered 1 through 22. A horizontal line segment AB intersects the diagram, representing the diameter or a related linear measurement.

To find the diameter of a circle and vice versa.

Although this second conclusion is clear enough on its own, it can nevertheless be explained in this way: Campanus Campanus of Novara, a 13th-century mathematician and astronomer whose commentary on Euclid was a standard textbook. says it is possible that a straight line equal to a circular line (one drawn in an orbit—that is, to the circle itself) may be given and found. According to the truth of all mathematicians and physicistsoriginal: "philosophorum"; in this period, "philosophers" included those studying the natural world and physics., a circle is divided into 22 equal portions. If, afterwards, just one part—that is, only the twenty-second part—is removed from the whole circumference original: "periferia" of the pre-established circle, then any third part of what remains—that is, the seventh part of the circle—will immediately result in the diameter, such as [line] A.B.

The author is using the Archimedean approximation of Pi (22/7). If the circumference is 22 units, removing 1 unit leaves 21. A third of 21 is 7, which is the length of the diameter.

If, however, the process is reversed and that diameter is tripled, and to that resulting product is added a seventh part of the diameter (namely, the twenty-second part [of the circumference]), and finally these tripled parts are arranged in a straight line, a straight line A.f. will immediately arise, composed of 22 parts distributed among themselves. This aforesaid straight line A.f. will be precisely equal to the line drawn circularly—that is, to the circle designated below as A.c.b.d.—and vice versa. And so we conclude that Campanus's second conclusion is true: namely, that it is possible for a line...