Range of a Culverin and How to Measure It
Projectile arcs, the Rule of Three, air density, and doubling the charge
This densely worked page attacks the range of firearms from several sides. A central diagram plots trajectories of a ball fired from a culverin or fusil planted slightly off vertical, the arcs rising to heights S, h, K and l and landing at n, m, o and p, with mouth, ground and butt of the piece at b, c and d. The columns argue that the 'pyramids' of trajectory keep base, side and height in proportion, to be solved by the Rule of Three, and note an exception whereby a ball rising into ever thinner air keeps gaining velocity. In the lower half Leonardo sets out an experiment: fix the piece along bc, fire measured charges, and read off how the landing distances and pyramid heights double together.
On this page
Trajectories of a planted culverin or fusil
A firearm is set into the ground slightly off vertical, its mouth, ground level and butt marked b, c and d. Four trajectories carry the ball up to heights S, h, K and l and out to ground distances n, m, o and p, laying out the arcs of successive shots for comparison.
Solving the range by the Rule of Three
Base stands to base as side to side and height to height, so the problem can be worked by the Rule of Three. Leonardo phrases it: if a base of one braccio comes from a pyramid of ten braccia, from what pyramid will a base of sixty-five braccia come?
Thinner air and acquired velocity
An exception is noted: a ball that rises only a hundred braccia passes through denser air than one that rises three thousand. Because the higher shot works through ever thinner air at each hundred-braccio interval, it must go on acquiring ever greater velocity.
Doubling the powder to gauge the range
To compare two pieces, Leonardo plants his instrument firmly along line bc, gives it so little powder that the ball travels two braccia to Sb and falls at n, then doubles the charge and sees it fall at m. If base mc is twice base nc, the height of pyramid hc is twice that of Sc.