Gravity, Centre of Gravity, and Powder-to-Ball Proportion
Folio 2r: shortest-line fall, the semicircle's centre of gravity, and artillery charges
Leonardo states that gravity moves a body toward the centre by the shortest line, not by choice or attraction but because the surrounding medium cannot resist it. He then poses a proportion problem: the charge of powder needed grows with the square of a cannonball's diameter, worked out with the example one-times-one and two-times-two. A further note explains how to approximate the true centre of gravity of a semicircle by dividing it into many thin triangles, illustrated beside a balance and sector diagram, and a lettered pulley figure (b a - m c n - d e - p o) accompanies weights on inclined planes.
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Gravity moves to the centre by the shortest line
Gravity is caused by one element being situated within another, and it moves toward the centre by the shortest line. This happens not by the body's choice, nor because the centre attracts it, but because the medium in which it lies cannot resist it.
Matching the powder charge to a cannonball's size
If one ounce of powder suffices for a one-ounce ball, Leonardo asks what charge a larger ball requires. He answers that the powder must increase with the diameters of the ball multiplied by themselves, so a ball of double diameter needs four times the charge (one-times-one versus two-times-two).
Centre of gravity of a semicircle by subdivision into triangles
The centre of gravity of a triangle with two equal sides lies at one third of its length toward the base. To find the true centre of gravity of a semicircle, Leonardo divides it into so many triangles that their curved base seems almost a straight line, then applies the method figured above.
Proportion as the square of the diameters
The relation of powder to powder is set equal to the relation of square to square formed by multiplying the balls' diameters by themselves. A diameter double another gives one-times-one (1) against two-times-two (4).