Curve equal to straight, with maxims on knowledge
Equal motions of greater and lesser curvatures; a note on learning
The verso continues the demonstration that curved figures can be made equal to straight-sided ones through equal motions: a greater curvature p q r and a lesser curvature a n m move into new positions in the same time, and equals removed from equals leave equal remainders. At the top stand two of Leonardo's maxims, that the acquisition of any knowledge is useful to the intellect and that nothing can be loved or hated without first being known, together with a principle of motion that a moving thing gains as much space as it loses. Lettered figures of arches, curved triangles and lens shapes carry the geometric argument.
On this page
On the acquisition of knowledge
In the upper margin Leonardo writes that the acquisition of any knowledge is always useful to the intellect, because it can drive out useless things and keep the good ones, and that nothing can be loved or hated unless one first has knowledge of it.
A moving thing gains as much space as it loses
A short note at the right states the principle that the thing which moves gains as much space as it loses. It underlies the argument that curve and straight, moved alike, remain equal.
Greater and lesser curvatures moved by equal motion
By the previous conclusion the greater curvature p q r moves into the position a o m, and in the same time and motion the lesser curvature a n m moves into the position S t v, while the straight line x t f likewise moves to 3 7. The figure carries these lettered points.
Equals removed from equals leave equal remainders
The greater curvature a travels to the height of b in as much time and motion as r takes to reach n. If from equal things equal things are removed the remainders are equal: remove S d v of the curve and p h o of the straight, and the remainder of the straight equals the remainder of the curve.