Sixth

...by the diameter of the circle, it is equally exceeded. As previously stated, this occurs in three half-parts three half-parts semipartibus tribus—that is, in one part and a half. Therefore, any side of this square is equal [to the quarter-circle], and in this way, the circle and a square of this kind are made equal to one another. Our author Campano proves all of this with a single example in the following manner: If a gold four-cubit four-cubit tetracubitum A cubit is an ancient unit of length based on the distance from the elbow to the tip of the middle finger. and a silver four-cubit are both equally exceeded by an ivory five-cubit five-cubit pentacubitum—because the ivory one is longer by exactly one cubit—it follows that the gold four-cubit and the silver four-cubit are necessarily equal to each other. Indeed, through "common conceptions" This refers to the axioms or "common notions" found in Euclid's Elements, specifically that things equal to the same thing are equal to each other., any spaces or intervals that are equal to one and the same thing will also be equal to each other. Likewise, whatever things are exceeded equally by the same thing are equal among themselves. We conclude, therefore, that since any quarter of a circle and any side of the square are exceeded equally by one third thing—namely, by the diameter of the same circle—it is necessary that they be entirely equal to one another.

Minor proposition, which is the sixth conclusion.

108

¶ Every circle is a single plane figure contained by a line drawn in a circular manner original: orbiculariter, whose diameter exceeds precisely the fourth part of that same figure by three half-parts.

A large decorative woodcut initial H, featuring a central footed vase or chalice overflowing with floral and leafy designs, set within a rectangular frame.
The declaration of this point is evident in the fourth proposition. For if a circle is divided into 22 parts—according to what many mathematicians have written and in accordance with physical truth The "physical truth" refers to the Archimedean approximation of Pi as 22/7, which was the standard ratio used in practical medieval geometry.—and one part is removed (that is, the 22nd part), the remaining third, namely seven parts, is the diameter of the circle. And since a quarter of that same circle contains five and a half parts, the diameter (being seven parts) transcends the quarter of the circle (the 5.5 parts) by precisely three half-parts...