Tetragonismus idest circuli quadratura per Campanum archimedem Syracusanum atque boetium mathematicae perspicacissimos adinuenta

This text, "Tetragonism, that is, the Quadrature of the Circle, Discovered by Campanus, Archimedes the Syracusan, and Boethius," is a fascinating mathematical treatise from 1503, edited with additions by Lucas Gauricus. It tackles one of the most enduring problems in classical mathematics: the quadrature of the circle – the challenge of constructing a square with the same area as a given circle using only a compass and straightedge. Gauricus's introductory epistle immediately sets a triumphant tone, claiming that this long-sought solution, which even Aristotle deemed "knowable but not yet known," has finally been "most perfectly handed down" by Campanus and Archimedes, dismissing earlier attempts by figures like Pythagoras and Boethius as having failed to transmit the true method.

The work is presented in two main parts, each offering a distinct approach. The first, attributed to Campanus, outlines a method for circle quadrature based on dividing the circle's diameter into 7 equal parts. It then posits that a quarter of the circle, or the side of an equivalent square, corresponds to 5.5 of these parts, and thus the diameter "precisely exceeds" the quarter-circle by three "semiparts" (1.5 parts). This leads to the conclusion that a circle is equal to a square whose side relates to the circle's diameter in this specific, somewhat intuitive, proportion. The exposition is practical, even noting that full circles are not drawn to avoid obscuring the underlying square.

The second part shifts dramatically to Archimedes, focusing not on the quadrature of the circle itself, but on the quadrature of segments of *right cones* (parabolas). Archimedes’s section, introduced with a lament for his friend Konon, demonstrates that such a segment is "sesquialter" (1.5 times) the triangle sharing its base and height. This section employs more rigorous geometric demonstrations, referencing "conic elements" and even principles of mechanics like the center of gravity and the balance. Readers will gain insight into the diverse methods and conceptual frameworks employed by Renaissance mathematicians grappling with ancient problems, highlighting both precise geometric reasoning and, in Campanus's case, a compelling numerical approximation presented as exact truth, reflecting the era's engagement with and reinterpretation of classical knowledge.