Common measure of line a c and a doubling number series
Powers of two and a step-by-step comparison of segments a c and d e
Folio 85r carries a short geometrical study in which segments are compared by a common measure: d e is 30, and the line a c goes into it once leaving 16, which in turn measures 30 three times leaving 14, and so on down toward a unit. At the top a doubling number series runs 1, 2, 4, 8, 16, 32, and a small figure is labelled with the letters a d, b e, c. Inked circular and spiral diagrams accompany the text at the right, while faint underdrawings of an arched or domed structure show through lower on the sheet. This further untranscribed sketch is present but is not part of the transcription.
On this page
A doubling series 1, 2, 4, 8, 16, 32
At the top centre of the page a geometric progression is written out, each term twice the one before: 1, 2, 4, 8, 16, 32. It sits above the lettered figure as a reference for the proportional reasoning that follows.
Finding a common measure for line a c
The note reasons that d e is 30, and that the line a c enters it once and leaves half plus one, that is 16; 16 then measures 30 three times and leaves 14; 14 measures 16 and leaves 1; and 1 in turn measures the remainder. It is a stepwise search for the common measure of the segments.
Lettered figure a d, b e, c
A small diagram at the right is annotated with the letters a d, b e, c, keying the points of the construction to the measuring argument written beside it.