On Proportion: Doubling Circles and Squares
Concentric circles and inscribed squares in ratio, and uniform thickness between branch nodes
The page is headed 'On proportion' and carries a marginal diagram of two concentric circles with an inscribed and a circumscribed square. Leonardo states that two circles touching a single square at four points are double one another, as are two squares touching a single circle, and reasons that when similar parts are removed from similar wholes the ratio is preserved, so a quarter of the greater circle is double a quarter of the lesser. A red-chalk note at the foot returns to botany, arguing that between one ramification and the next a branch keeps a uniform thickness because equal nourishment yields an equal effect.
On this page
Two circles about one square are double one another
The two circles that touch a single square at four places are double one another, and likewise the two squares that touch a single circle at four places are double one another. The marginal figure shows the concentric circles with their inscribed and circumscribed squares.
Similar parts of similar wholes keep the same ratio
If similar parts are taken from two similar wholes, the ratio of part to part equals that of whole to whole. Since the two circles are double one another, a quarter of the greater is double a quarter of the lesser, and the ratio holds for remainder to remainder as well.
A branch stays uniform in thickness between ramifications
Where no side branches intervene between one ramification and the next, the stem keeps a uniform thickness. This is because the whole of the sap that feeds the branch's start keeps feeding it until the next ramification forms, and an equal cause produces an equal effect.