Squaring Circles and Lunes by Superposition
Geometric equivalences of circular segments and Euclid's common notion on equals
The verso continues the study of squaring circular segments, with sectors and semicircles labelled to show that one figure equals another through interchange and superposition. A central passage restates Euclid's common notion, that if equal parts are taken from equal things the remainders are equal, and extends it to the superimposition of two equal surfaces. A larger construction shows how to square a portion by drawing a line tangent to the two acute angles of a triangle, producing a figure equal to the lune. Ratios of doubled and quadrupled circles are asserted, and a lone figure 1000 stands at the lower right.
On this page
Squaring a lune by a tangent construction
To square a c, a line n m is drawn tangent to the two acute angles of triangle a, yielding the portion b, which equals the lune c; the result is proved below by drawing the line n h. The construction sits beside a large circle bearing lune and tangent lines.
From equals take equals: the remainders are equal
A column restates the axiom that if equal parts are removed from equal things the remainder is equal, and applies it to two equal surfaces superimposed part for part, so that their contacts and their remainders are equal. It is offered as a demonstration by the common notion.
Circles double and quadruple one another
Captions label sectors and semicircles as circles double one to the other, with a equal to b c and n equal to m by interchange and superposition. Of circles quadruple one another, an eighth part of the greater is said to equal the smaller.