Squaring of the Circle

The Fifth Conclusion.

Every plane figure contained by a single circular line This is a classical way of defining a circle. whose diameter diameter diameter exceeds exactly a quarter of the circumference of that same figure by three half-parts three half-parts semipartibus tribus is equal to a square square quadrato whose side is exceeded by the diameter of that same circle by exactly three half-parts.

A large decorative woodcut initial 'H' featuring floral motifs and a central urn or vessel on a pedestal.

The truth of this is shown as follows: for whatever things are surpassed by the same equal amount are equal to each other. For example, if a golden four-cubit rod four-cubit rod tetracubitum and a silver four-cubit rod are both surpassed equally by a wooden five-cubit rod five-cubit rod pentacubito—specifically by one cubit—then the golden rod and the silver rod are necessarily equal to one another. Therefore, since any quarter of this circle and any side of this square are surpassed equally by the diameter of the circle (by the amount of three half-parts), it follows that any quarter of the circle and any side of this square are equal. Thus, the circle and a square of this kind are equal.

Addition.

This is the author’s fifth conclusion, which should be called the "major" premise of our demonstration concerning the squaring of the circle squaring of the circle circuli quadratura which we intend to show. In this, our Campanus Johannes Campanus of Novara (c. 1220–1296), a famous medieval mathematician whose commentary on Euclid was highly influential. seems to hold this view: that every plane figure contained by a single circular line—that is, which is bounded by only one line, namely the circumference—whose diameter exceeds (that is, surpasses) exactly a quarter of that same figure (namely the circle itself) by three half-parts (that is, by three half-measures, or one part plus a half), is equal to any square whose side the diameter...