Proving two surfaces equal without motion
Equal areas by subtracting a doubly-counted overlap
This note proves the quantity c d e f equal to the quantity a b c d without invoking motion, relying only on the axiom that equal quantities removed from equal things leave equal remainders. Taking two equal and similar surfaces, a c e and b d f, Leonardo removes from each the shared part b d e, which is counted double where the two overlap, so that the remainder e f c d is left equal to a b c d. A lettered figure of a rectangle crossed by curved lines stands at the upper right; further faint sketches and untranscribed writing show through the lower part of the sheet.
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Equal surfaces proved by removing a doubled overlap
Two equal and similar surfaces a c e and b d f are proposed. Removing from each the shared part b d e, imagined double because there the surfaces are superposed, leaves the remainder e f c d equal to the remainder a b c d, without any appeal to motion.