Angles of the Balance and the Diagonal of a Cube
Plumb pendants converge to the earth's centre; the cube's axis is the root of 300
The upper note treats the balance: the plumb-line pendants never make true right angles with the beam because their lines converge toward the centre of the world, and the angles are equal or unequal in the same proportion as the weights they carry. A drawing of a balance beam with hanging weights sits at the upper right. Below, cube diagrams labelled with roots show that joining the roots of 100 and 200 at a right angle yields the cube's interior diagonal a d as the root of 300.
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Angles the pendants make with the balance beam
The plumb-line pendants never form true right angles with the beam, because their lines converge toward the centre of the world. The angles are equal when the suspended weights are equal, and unequal in the same proportion as unequal weights; yet the potential arms of the balance always join the pendants at right angles.
The cube's interior diagonal from three roots
Three roots a b, a c and a d are given: a b is the root of 100 and a c the root of 200. Joining to them at a right angle the root of 100 (d c) gives a d as the root of 300, the cube's interior diagonal or axis, since a d squared equals d c squared plus c a squared.
Roots of 100, 200 and 300
A cube figure is labelled with the roots of 300, 200 and 100 alongside the value 10, tabulating the quantities used in the diagonal construction. These labels quantify the sides and diagonals of the cube discussed in the surrounding text.