Transmutation of Straight-Sided into Curved Surfaces
A row of five squares with inscribed circles, biangles filled with portions, and four lunes in a square
A comparatively clean sheet headed 'On the transmutation of rectilinear surfaces into curvilinear surfaces and vice versa,' laid out as an orderly row of five squares each with an inscribed circle and the 'biangle' offcuts. The rule of the game is to fill the biangles with the circular portions, and to swap portions and biangles until a straight-sided figure becomes a curved one. Near the right margin a square holding four lunes states that the central square equals the four lunes, and an inverted diagram equates a circle with a ring.
On this page
On transmutation of rectilinear and curvilinear surfaces
The heading above the row of five squares announces the sheet's programme: converting rectilinear surfaces into curvilinear ones and the reverse. The five numbered squares below, each with an inscribed circle, are the worked stages of that conversion.
Fill the biangles with the portions
Across the five squares the instruction is simply to fill the biangles (the lens-shaped corner offcuts) with the circular portions. Doing so trades area between the square and the circle without changing the total, the essence of the transmutation.
A middle square worth four lunes
In the square holding four lunes at the right margin, the middle square is declared worth the four lunes, and the surrounding field worth the middle square plus the four lunes. A paired construction says: remove the biangles abcd and fill the four empty portions to obtain the second figure.