Rectangles of equal area, Euclid, and surd roots
As a rectangle lengthens it narrows; square and surd roots to 100
A torn sheet of mathematical reasoning, illustrated at the right by a column of rectangles of equal area and a circle over a lune. Leonardo lays down the rule that a rectangle lengthens exactly as much as it is narrowed, holding for every rational division, and cites Euclid's Elements that the diagonal of a parallelogram with equal opposite angles divides it into two equal parts. A lower passage lists the square roots from 1 to 100 — ten rational roots and ninety "surd" roots that were never heard named — arguing that arithmetic cannot handle continuous quantities with surd roots and that one must therefore proceed geometrically. A brief memorandum in the top corner notes blue sheets and nutmegs.
On this page
Rectangles of equal area: length grows as width shrinks
A rectangle lengthens exactly as much as it is narrowed: narrow it by half and it lengthens by half; narrow it by a third of its width and it lengthens by a third of its length, and so on to infinity for every named division. Because the divisions without a numerical denomination (those with surd roots) are infinite, one must proceed geometrically.
Euclid: the diagonal bisects a parallelogram
Citing Euclid's Elements, Leonardo notes that in every quadrilateral whose opposite angles are equal, the diagonal divides it into two equal parts. The figure is keyed to the single letter a.
Square roots and surd roots from 1 to 100
Between 1 and 100 there are only ten square (rational) roots and ninety surd roots "which were never heard named." Leonardo concludes that arithmetic, a mental science of exact denomination, has no power over continuous quantities with surd roots, which divide magnitudes without any numerical denomination.
Memorandum: blue sheets, nutmegs
A short jotting in the upper right corner, unrelated to the geometry, notes items to remember: blue sheets and nutmegs. It reads as a domestic memorandum.