Products as areas: squares, rectangles, gnomons, Pythagoras
The product of two numbers displayed as the area of a rectangle
The sheet is worked in two halves. The upper half sets squares and rectangles labelled with numbers (a unit square, a rectangle of 12, sides 3 and 4) to show that a rectangle's 'surface number' comes from multiplying its base and height, and that the product is far greater than the two factors that produce it; a rectangle is broken into twelve unit cells to prove 3 by 4 makes 12. The lower half turns to a parallelogram and to gnomons, with the note that the complements do not move, then separates a greater from a lesser square and sums squares by the theorem of Pythagoras.
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Linear number and surface number
Base and height 3 and 4 are called the linear numbers, while the rectangle they enclose, 12, is the surface number. Leonardo observes that the product is far greater than the two quantities that produce it, distinguishing a length from the area it generates.
Rectangle broken into twelve unit cells
A rectangle of sides 3 and 4 is subdivided into twelve smaller rectangles, giving a visual proof that three multiplied by four yields twelve. The decomposition makes the product legible as a count of equal cells.
Gnomons and their complements
Two gnomon figures accompany a note that the complements do not move. This concerns Euclid's parallelograms set about the diagonal of a figure, whose complements remain equal as the gnomon is formed.
Sum of squares by the theorem of Pythagoras
At the foot of the sheet a greater and a lesser square are separated and then combined, illustrating the summing of squares through the theorem of Pythagoras. The squares built on the sides are set against the square on the whole.