Squaring circular segments by adding and removing triangles
Semicircle, quadrant and five figures proving equal areas by lending and subtracting portions.
This closely written sheet works through the equivalence of circular segments and lunes. In a semicircle labelled b d a c, a quadrant of a circle, and a rectangle set on a triangle, the notes track how one figure 'loses' or 'gains' the value of certain portions, resting on the relation a b equals c. A sequence of five figures labelled with n, m, S, o, a, b, f, t, c, d and L demonstrates a method of 'lending' pairs of triangles and then removing equal portions to prove two areas equal, closing with an 'et cetera'.
On this page
Semicircle divided into portions b d a c
A semicircle in the first column is divided into three portions labelled b, d, a, c. The note reasons that d 'loses' the value of two portions equal to the portion c, because a b equals c.
Quadrant and rectangle on a triangle
A quadrant of a circle labelled n, g is said to lose only one portion, n; and in a rectangle set on a triangle the square n is stated to equal the g above — one portion less than the d above — which will be removed below so that it remains equal to d.
Five figures: equating areas by lending triangles
Five figures, labelled first through fifth with n, m, S, b, o, a, t, f, c, d and L, set out a method of 'lending' pairs of triangles (a b, then f t) to a deficient figure, then subtracting equal portions so the remainders are equal. The demonstration ends 'because I lent things equal to equal things, et cetera'.