Cords and pulleys: tangency and the centre of gravity of an arc
Mechanics of a cord over pulleys, right-angle contact with the radius, and where an arc's weight acts
A mechanics sheet on cords running over pulleys, illustrated with a rod slung by cords between two pulleys, a large triangle, and several pulley diagrams. Leonardo disputes an 'adversary' over where the centre of gravity of an arched cord lies, arguing it falls along the central line L h rather than at the arc's midpoint, and that the calculating line is a h, not the chord a b. He notes that the power of a pulley's arm should be reckoned on the central line of the arched cord, and states as a rule that the cord meets the pulley's radius at a right angle. The reasoning invokes the converse of a proposition from the geometric Elements.
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Where the centre of gravity of an arched cord lies
Leonardo argues that if the arc a n b is referred to the line a b, the straight line's contact with the cord's curvature faces the semidiameter a f, so a line from one circle's centre to the other's contact does not pass through that other centre — which would falsify the converse of a proposition in the geometric Elements. He concludes that not the chord a b but the line a h serves to calculate the arc, and that the arc's centre of gravity always lies along the central line L h, not merely at the arc's midpoint as the adversary claims.
The power of the pulley's arm
Considering a thick cord wound over a pulley, Leonardo places the calculation of the power of the pulley's arm o p, with the right-angled appendage o q, on the central line f c of the arched cord. He objects to setting it on the line n S, the centre of the hanging cord, which must pass through the centre of the sustained weight, since the surface line h g would fall outside the centre of gravity S.
The cord meets the pulley's radius at a right angle
A short rule states that the line of the cord joining the circumferential line of the pulley is always at a right-angled junction with the semidiameter (radius) of that circumference.