Why doubling a figure needs more than doubled power
The auger argument: doubled size gives quadrupled surface and force
Leonardo argues that a body of doubled figure cannot be moved by merely doubled power, because a body of similar shape but doubled linear size has four times the surface of the smaller one. He illustrates the geometry with two triangles labelled a and n and with a pair of augers (trivelli): each bores out the same thickness of wood, yet the larger hole is quadruple in surface and in the power it demands. The page combines a written demonstration with a marginal triangle diagram and a sketch of the two boring tools. This is part of his running critique of the scaling of models.
On this page
Doubled figure is not moved by doubled power
A body of doubled figure cannot be moved by doubled power, because the surface of every body of like shape and doubled quantity is quadruple that of the other. The two figures a and n are offered as proof.
Simple triangle a and quadruple triangle n
A marginal diagram sets a simple triangle against a quadruple one, labelled a and n. It demonstrates that doubling the side of a similar figure yields four times the surface area.
The two augers (trivelli)
From each of two augers the same thickness of wood is removed from every hole they bore. But because the holes, or augers, are of doubled quantity relative to one another, they are of quadruple surface and of power.