Geometry of artillery fire: shots kept within the angle
Radiating trajectories over shot-heaps and the pyramids a, b, c, d
This page works out the geometry of artillery fire, showing sheaves of trajectories that radiate over rounded emplacements studded with cannonballs. Leonardo treats the converging lines of fire as the pyramids a, b, c, d and notes that the whole arrangement exists only so that these pyramids are not struck outside their angle. A second, similar diagram below repeats the crossing radiant pattern on its own.
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Fire kept within the angle of the pyramids a, b, c, d
The scheme drawn here is arranged for one purpose only: so that the pyramids a, b, c, d are not struck outside their angle. The radiating lines mark the paths of the shots as they cross over the mounded batteries.
Radiant construction of the fire pyramids
The trajectories are laid out as straight lines fanning from a common vertex, forming crossing grids that define each pyramid of fire. A second construction below reworks the same radiating figure to test the angle.