The Visual Pyramid and How It Narrows With Distance
Perspective tables and the 'power' concentrated at a section of the pyramid
Two facing leaves headed 'Prospettiva' develop Leonardo's theory of the visual pyramid: a base of 4 braccia whose sides converge, and the rate at which the figure narrows over hundreds of braccia and over distances of 4 to 128 miles. A long table pairs each pyramid and base with a 'power' (potenzia) that quadruples as the section shrinks, and a note argues that a small cross-section far from the base carries as much 'force' as the whole base, so that a cut 1/16 of the base's diameter is 256 times more powerful. Practical advice explains how to enlarge a base eight or ten times when its true partitions are too fine to draw. The sheet is filled with columns of fractions and small edge-on diagrams of the tapering pyramid; a British Museum stamp appears at lower left.
On this page
Rate at which the visual pyramid narrows over miles
For a base of 4 braccia, Leonardo tabulates how much the pyramid shrinks per 200 braccia. A pyramid 4 miles long narrows 1/60 of its base (1/15 of a braccio); one of 8 miles narrows 1/120; one of 32 miles 1/240; one of 64 miles 1/480. He adds that every pyramid is worth as much after the intersection of its sides as before it.
The 'power' concentrated at a section of the pyramid
He asks whether a small section far behind the base, only 1/16 of the base's diameter, carries the whole base's power. Setting the base at 4 braccia and the pyramid at 8 miles, a cut made 7.5 miles from the base is 1/4 of a braccio and 256 times more powerful than the base; halving the diameter again makes it 1024, then 4096 times more powerful.
Enlarging a base whose divisions are too fine to draw
If the base of the pyramid cannot be divided into its true partitions because they are too minute, make the base eight or ten times larger and divide that well, enlarging the opposite cut of the pyramid in proportion. Then remove the borrowed part, and the remainder at top and foot will be correctly proportioned.
Converting miles into hundreds of braccia
8 miles are 24 thousands of a braccio, that is 240 hundreds, and the base of 4 braccia is divided by those 240 hundreds. At every hundred braccia the pyramid narrows by 1/240; thus in eight miles, i.e. 240 hundreds of length, the base has consumed all its hundreds entirely.