A moving figure gains as much space as it loses
Euclidean axioms of equality applied to a triangle sliding within a square
The sheet works out a principle of the conservation of area during motion, illustrated by a square a b c d cut by the diagonal b c into two triangles, one of which is slid across so that the space it gains equals the space it leaves. Leonardo underpins this with the common notion that if equal parts are taken from equal wholes the remainders stay equal, and with the maxim that a moving thing leaves as much space as it takes. The facing leaf (219v) is nearly blank, bearing only faint construction lines showing through the parchment.
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A triangle slid across the square a b c d
The quadrangle a b c d is divided into equal parts by the diagonal b c, giving triangle a b c and triangle c d g. Moving triangle c d g as far as e f g gains the space c d e f and leaves an equal space in b c g e.
The moving body neither creates nor destroys space
The thing that moves gains as much space as it loses, and leaves as much space as it gains. Two equal things partly superimposed touch in equal parts and leave equal remainders, and a whole is restored when its removed parts are returned.
Common notion: equals taken from equals leave equals
a is equal to the quantity c. Since a b equals b c, taking b from a and b from c leaves a and c equal to one another. If from two equal things equal parts are taken, the remainder will be equal.