A REMARKABLE FIGURE FROM THE BOOK OF LEMMAS

Abstract

The Proposition 4 from the 11-th volume (called Book of Lemmas) of „Great Books of the Western World” (printed in USA in 1952) presents the properties of a remarkable curve from Antiquity. This curve was studied by Archimedes, who called it Arbelos and it looks like a shoemaker's knife. It consists of 3 semicircles. The first part of the paper is dedicated to the description of an original mechanism, conceived to plot the above-mentioned curve. It is made of a rotating driving element and two dyads of RRR type. The curve plotted by the mechanism checks the equality of areas described by Archimedes. He proved (by using geometric reasonings) that the area of a circle whose diameter is equal to the height of the right triangle inscribed in the Arbelos curve’s big semicircle is equal to the Arbelos curve’s area. In the second part of the paper, the authors modified the mechanism’s parameters and obtained 3 particular cases of Arbelos curves. These particular curves, plotted with the modified mechanism, are also composed of 3 semicircles whilst exhibiting other equalities between areas. The authors used geometric reasonings to prove the equality between the area of each particular curve and the area of a circle whose diameter is equal to the leg of a right triangle inscribed in the big semicircle of the studied particular curve.