Rule of Weights and the Compound Motion of a Toothed Wheel
Balancing 3 against 4, a self-turning instrument, and centers of gravity of triangles
Folio 164r opens with a "rule of weight" for balancing 3 weights against 4 by spacing them along the beam, and a drawing of a mechanical instrument whose motion Leonardo sets out to define. He analyzes how a toothed wheel n m turning about a pinion c d upon a larger wheel a b acquires a compound revolving motion that prevents the uneven wear (concavity) of ordinary rolling contact. Further notes prove the triangles a m n, a r m and a n o equal in sum, relate the weights of pyramids on a common base to their axes, and locate the natural center of gravity of paired right-angled and equilateral triangles. Tapering balance-beam diagrams and a triangular figure accompany the text.
On this page
Rule of weight: 3 against 4 on the balance
Leonardo gives a rule for making three weights stand in equilibrium against four on a balance. If you want 3 weights to hold against 4, arrange 4 spaces of the beam against 3, distributing the loads by their distances from the fulcrum.
Compound revolving motion of a toothed wheel on a pinion
In revolving motion the part nearest the center is always slower than the more remote part, so wheel n m would wear its running surface concave. Leonardo notes that art has remedied this with compound motion: the toothing of the wheel turns about the pinion c d upon the larger wheel a b, giving a compound revolving motion that prevents such concavity.
Triangles a m n, a r m and a n o equal in sum
A short proposition states that the triangles a m n and a r m and a n o are equal in sum. It is keyed to the lettered triangular diagram in the lower right of the page.
Centers of gravity of pyramids and paired triangles
Pyramids of various heights raised on one and the same base have weights in the same proportion as their axes. For a right-angled and an equilateral triangle of equal length and base, their perpendicular meeting is the natural center of their gravity.