First conclusion

The squaring of the circleoriginal: "tetragonismus idest circuli quadratura"; the geometric challenge of constructing a square with the exact same area as a given circle. by the most illustrious Campanus, published in Rome with additions by Gaurico.

To demonstrate the squaring of the circle, our Campanus first sets forth four conclusions—and indeed very easy ones—and secondly, from these is derived a fifth which, together with the sixth, most clearly completes the entire demonstration regarding the squaring of the circle.

First conclusion.

A line drawn in a circular fashion is divided into four equal parts by two diameters.

A diameter is a straight line drawn from one edge to the other through the center, dividing figures into equal parts. If, therefore, there are two diameters intersecting each other at the center at right angles, they would divide the figure into four equal parts. And it should be noted that "diameter" is said [to come] from dia which is "two" and metros which is "measure"—as if it were the measure of two halves. The author uses a medieval folk etymology; "diameter" actually comes from the Greek "dia" (through) and "metron" (measure). Thus says Campanus.

For a greater understanding of this first theorem, it must be noted that a "figure" (to use the words of EuclidA Greek mathematician (c. 300 BCE) often called the "Father of Geometry." His work "Elements" was the primary textbook for geometry for centuries.) is that which is enclosed by a boundary or boundaries. A circle, moreover, is a plane figure contained by a single line which is called the circumferenceoriginal: "circunferentia", in the middle of which is a point from which all straight lines extending to the circumference are equal to one another. And this point indeed is called the center of the circle.