Intersecting circles, with the rule of branching trees and waters
'Fourth of the third'; and an inverted note on branch and river cross-sections
Headed 'Fourth of the third,' the upper part of the page draws two intersecting circumferences and reasons about equal segments: 'by the definition of the circle ab and bc are equal,' and 'when two things are equal to a third, they are equal to each other.' Turned upside down at the foot, a separate note states Leonardo's branching rule: all the branches of a tree at any height, taken together, equal the thickness of the trunk, and all the ramifications of water at any length, being of equal motion, equal the thickness of their source. The sheet thus joins a circle proof with a natural-philosophy observation on trees and rivers.
On this page
Two intersecting circumferences and equal segments
Two circles are drawn so their circumferences intersect (points c, d, b, a), and 'by the definition of the circle ab and bc are equal.' A second figure adds point f and invokes the axiom that things equal to a third are equal to one another. The pair proves an equality of segments from the circles' radii.
Branches of a tree equal the thickness of its trunk
The inverted note states that all the branches of a tree at every level of its height, joined together, are equal to the thickness of the trunk. This is Leonardo's rule for the conservation of cross-section as a tree divides. It records a general law of plant growth.
Ramifications of water equal the thickness of their source
The same inverted note adds that all the ramifications of the waters at every level of their length, being of equal motion, equal the thickness of their source. Water is treated like a branching tree, its total flow conserved as it divides. The remark applies the branching rule to rivers and streams.