The power of the cut: subdividing a four-braccia square
A numerical study of how ever-smaller cuts fill the base, beside a bench mechanism.
Leonardo computes the 'power of the cut' over a base of four braccia. A cut of one oncia enters the base 576 times; halving the cut multiplies the count fourfold at each step, so 1/2 oncia enters 2304 times, 1/4 gives 9216, 1/8 gives 36854, 1/16 gives 147,416, and 1/32 gives 589,684 (the cut marked with the letter O). He derives this by squaring the 48-once side into 2304 square once and dividing repeatedly by four down through thirty-two squares, working the multiplications in columns. A note above the O explains that 'a', one thirty-second of an oncia's side, enters that oncia 1024 times and stands to the four-braccia base as 4,698,545. A large bench-mounted device with a threaded rod and curved arm is drawn across the page.
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How ever-smaller cuts fill the four-braccia base
A cut of one oncia enters the base of four braccia 576 times; each halving of the cut quadruples the count, giving 2304, 9216, 36854, 147,416 and 589,684 for cuts of 1/2, 1/4, 1/8, 1/16 and 1/32 oncia. The 48-once side squared makes 2304 square once, resolved by repeated division by four into thirty-two squares.
The value of one thirty-second of an oncia's side
Above the letter O, the note explains that 'a' is one thirty-second of an oncia's side, which enters into that oncia 1024 times; and that a stands in proportion to its base of four braccia as four million, six hundred ninety-eight thousand, five hundred forty-five.
Bench-mounted mechanism with threaded rod and curved arm
A large table on splayed legs carries a mechanism whose threaded rod runs through a curved, scroll-shaped arm braced against a block. The device dominates the upper half of the sheet; the accompanying transcription records only the numerical study, not a caption for the machine.