Equal-volume pyramids and a construction for doubling the cube
Pyramids of equal height on equal bases are equal; a graphic method for roots of doubled cubes
The leaf opens with two theorems on pyramids: those of equal altitude standing between the same bases are equal to one another, and pyramids on an equal base are in the same proportion of volume as their heights. A longer note, which Leonardo heads an 'ortogonio giudiziale', gives a graphic method for finding the roots of a series of cubes each double the previous one, by cutting a pyramid a b c at f g, swinging semicircles such as d m b from the base line, and drawing chords like h i behind each. Three diagrams accompany the text: twin pyramids between parallels at top, a single pyramid, and a long triangle inscribed with successive semicircles.
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Pyramids of equal height on equal bases are equal
All pyramids of equal altitude, constituted between the same bases, are equal to one another. This first proposition is illustrated by two pyramids sharing a base and set between parallel lines.
Pyramids on an equal base scale as their heights
Among pyramids placed on an equal base, there is the same proportion of volume as there is between their heights. Thus one of double height on the same base has double the quantity.
Graphic construction for roots of successively doubled cubes
To find the roots of infinitely many cubes each double the next, cut pyramid a b c at f g, dividing it into two equal parts, then set the compass foot on line a b to draw semicircle d m b touching the cut f g. If width a d remains outside the semicircle, extend that excess to the pyramid's angle a d c; then make further semicircles and draw a straight line such as h i behind each, continuing as far as the pyramid lasts.