Three balls under a pivot beat four: wear and friction
Conical pivots seated on rolling balls, compared for contact, durability and friction
Leonardo compares low-friction pivot bearings in which a conical point rests on a set of rolling balls or pyramidal supports. He argues that three balls beneath a pivot are better than four, because three are always and equally touched by necessity, whereas with four one ball might miss contact and then grind against its neighbour. The long note ranks the pivots a, c, d and e by how their point-to-support proportions govern the number of turns, the amount of contact, and hence the wear and friction each suffers. The drawings show a funnel-like pivot dropping onto three balls (in section and in plan) and several faceted roller supports lettered a through e.
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Three balls are surer than four
The three balls under the pivot are better than four would be, because the three are with certainty and of necessity always touched, and equally moved by the pivot. With four there is the danger that one is not touched, so it is not moved and waits for its companion, and then grinds against it in friction.
Proportion of pivot point to support governs wear (a, c, d, e)
The pivot a is most praiseworthy because its three pyramidal supports are equal to one another and to the pyramid of the pivot's point, so each gives one full turn as the pivot does and lasts longer than a support of three balls, where least contact means most wear. c turns three times while its supports turn once; d and e are wretched because they are not proportioned to their supports, giving contact with greater friction.
Supports that have no pivots at all
Of the arrangements on the right Leonardo notes that some are very useful precisely because they have no pivots, naming the pair f and g. Removing the sliding pivot altogether is presented as one way to escape friction and wear.
Funnel-shaped pivot seated on rolling balls
At the top a hopper- or funnel-like pivot body descends onto a cluster of balls, shown both in section and, at the upper right, as a plan of three balls set symmetrically around a central hub. It is the physical embodiment of the three-versus-four argument in the text.