Square Roots by a Semicircle, and the Fortress of Piombino
A demonstration that the line f K is the root of 12, a dated plan of the Piombino citadel, and a note on light from two apertures
Geometry fills the upper leaf: using a semicircle lettered f, g, K with squares, Leonardo shows that the line f K is the root of 12 (g K being the root of 16 less f g, the root of 4) and that the square on f K equals the rectangle f a i h. A large red-chalk study in the middle records the citadel (cassero) of Piombino, dated 20 November 1504, with its covered way, the ditch that he is straightening, and a gate. Lower notes and lettered figures return to optics, arguing that two apertures in a room double the light on a wall, just as two eyes strengthen sight. Much of the surrounding text runs in dense mirror-script.
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The line f K is the root of 12
g K equals f K together with f g, and is the root of 16 (as are g a and g m). Take away f g, the root of 4, and there remains f K, the root of 12, matching the elongated figured square f i a h.
The square on f K equals the rectangle f a i h
The three semi-diameters of the semicircle are equal, and m f into f a (with the help of f g) equals g K by itself. Removing f g from each side, m f into f a equals f K by itself; and since m f into f K makes the square f a i h, the square of the line f K equals f a i h.
The citadel of Piombino: covered way, ditch and gate (20 Nov 1504)
A red-chalk plan is inscribed cassero di Piombino, 20 November 1504, with a covered way (strada coperta), the ditch that I am straightening, and a gate (porta). It records Leonardo's fortification work at the Tyrrhenian stronghold.
Two apertures double the light on a wall, like two eyes
Take the square c d e f as a dwelling closed on every side. Open the aperture a and it lights the whole wall c d by one degree; then open a second aperture b and it doubles the light on that wall, adding a second degree upon the first, the same doubling that two eyes give to sight.