Roots, Medial Lines and the Number 108
Squaring, roots and an "irrational" rectangle in the theory of magnitudes
The sheet works with roots and "medial lines," the Euclidean theory of irrational magnitudes. A square encloses the number 108 with the note "root of the root of 108"; a short segment marked 3 carries the instruction "make a root of it, it makes nine," and another marked "root of 12" is called a medial line. A tall rectangle labelled 12, 3 and 108 is marked "irrational" and headed "the proof of what is above," while further squares and rectangles, several labelled 108, are linked by looping pen-strokes and decomposed into a square and its complement. The page is numbered 25 at the top corner.
On this page
The fourth root of 108
A square encloses the number 108 and is annotated "root of the root of 108," that is the root taken twice over of the quantity 108.
Taking the root of the segment 3
A segment marked 3 carries the instruction "make a root of it, it makes nine," relating the marked length to a squared or rooted value.
Root of 12 as a medial line
A segment labelled "root of 12" is identified as a "medial line," the Euclidean mean between commensurable magnitudes.
The irrational rectangle and its proof
A tall rectangle carries the labels 12, 3 and area 108 and is marked "irrational," headed as "the proof of what is above." Related squares and rectangles marked 108 and "root of 12" are linked by pen-strokes and split into a square plus its complement.