Why Forks Bulge Outward: A Three-Circle Proof
Bark splits lengthwise (save the cherry), and the joints of a fork rise at the sides but hollow toward the trunk
Leonardo notes that tree bark always cracks along the plant's length, except the cherry's, which bursts in rings. He then explains why, when a trunk divides into several boughs at one height, the joints between them rise higher on their facing sides than toward the tree's centre, leaving great concavities there. Because the outer angles between branches are narrower, the boughs thicken and meet sooner at their sides, raising the outer joint while the middle joint stays low. He proves it geometrically with three tangent circles n, m, o, which touch only along the lines n m, m o and o n and not at their centres. Pen diagrams show a forked trunk lettered a-b-c, a Y-junction, and the three-circle figure.
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Bark splits lengthwise, save the cherry's ringed bark (k)
The bark of trees always cracks along the length of the plant, except that of the cherry, which bursts in rings around it.
Fork joints rise at the sides, hollow toward the trunk (m)
When the main plant divides into several principal boughs at one height, the margins of their joints rise higher on their facing sides than toward the tree's centre, where great hollows remain. The outer angles a b and b c are narrower than a c, so those branches thicken and meet sooner at b c while the middle joint stays low.
Three tangent circles n m o prove the concave joint
He proves it by necessity with three circles n, m, o, which touch one another only at the points of the lines n m, m o and o n, and not at their middles. Unable to attach except where they touch, they join at those contacts, leaving the centre low and hollow.