Squaring of the Circle

Conclusion

...is called. A straight line, however, is the shortest extension from one point to another, that is receiving each of them into its extremities. Moreover, the diameter of a circle, which we call the "measurer through" measurer throughoriginal: "dimentientem"; a literal Latin translation of the Greek "diameter.", is a certain straight line which, passing through its center and attaching its extremities to the circumference, divides the circle into two halves. From which (as I judge) it is called in Greek diameter original: "$Διαμετρ oσ$", that is, "diameter," from the preposition dia original: "$Δια$", namely "through," and metros original: "$μετρ oσ$", that is "measure"—as if it were dia meson metros original: "$Δια$ $μεσ oν$ $μετρ oσ$", namely, the equal division and measure of two halves.

If, therefore, says Campanus, there were two diameters—as in the following figure: A-B and C-D—intersecting each other at the circle's center F at right angles, the line drawn circularly (that is, the circle A-B-C-D) will be cut into four equal portions. This is because the arcs A-C and B-D, which are two portions of the circle, are equal to one another. And since they are positioned opposite each other, it is necessary that the angles themselves also be equal and right angles.

Since (as Euclid says), when two straight lines contain an angle, it is named a "rectilinear angle." And when a straight line stands upon another straight line, and the two angles on both sides are equal, each of them will be a right angle. And the line standing upon the other line is called perpendicularoriginal: "perpendicularis"; a line meeting another at a 90-degree angle. to that which it stands upon. Let the same be said concerning the arcs of the same circle, B-C and D-A, and their angles. And thus it happens that the circular line is cut into four equal portions by a pair of diameters, just as is clear in the following figure.