Centre of gravity of a triangle and a hanging pillar
Statics of weights: a triangle's balance point, and a pillar of weight 400 borne on a cord
Leonardo studies the centre of gravity of plane figures and hanging loads. A triangle's true centre is found where its 'natural' centre of gravity along the height (line f p) meets its 'accidental' centre along the width (line a n); a pillar of weight 400 borne on a cord is reduced by geometry to 200 and then 100. At the top a masonry arch and gateway is labelled with weights 800, 400 and 100, and pulley-and-weight diagrams below illustrate the reasoning.
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Finding the true centre of gravity of a triangle
To learn how much weight the triangle gives its counterweight, take its centre of gravity at point b and measure how far it projects beyond c. The true centre is found by taking the natural centre of gravity along the height (line f p) and the accidental centre along the width (line a n); where f p and a n intersect lies the common centre. That common centre shifts to as many positions as there are motions given to the figure while it stays upright.
A pillar of weight 400 reduced on a cord to 100
If the pillar weighs 400, that weight refers wholly to the centre t. Because t stands perpendicular above the middle of line v X at point r, the weight of 400 becomes 200; and because the cord q h takes its distance from the centre equal to the pillar's distance from where it rests at v, the weight halves again, so that it becomes 100.
Arched gateway labelled with its loads
The upper drawing shows a voussoired masonry arch set over a rectangular gateway, flanked by piers, with diagonal lines drawn to the base. It is annotated with weights: 800 at the crown region, and 400 and 100 at the supports, treating the built arch as an exercise in the balancing of weights.