Double the Height, Eight Times the Solid
Octuple growth of similar solids; halving a triangle from its base
Folio 45r carries rows of pyramids and tetrahedra in red chalk with two short geometric rules. The first states that a body of the same figure but double the height becomes eight times as large. The second treats a triangle divided from the middle of its hypotenuse to a side of its base, showing it is split into two equal parts because the base d c b is halved by the line a b, giving two triangles of equal height.
On this page
Doubling the height makes a similar solid eightfold
Every body of the same figure and of double height will be octuple one to the other. The rule captures how volume grows as the cube of the linear scale for similar solids.
A triangle halved from the midpoint of its hypotenuse
A triangle divided from the middle of the hypotenuse to one of the sides of its base is split into 2 equal parts. This is because the base d c b is divided into two equal parts by the line a b, which serve as bases of two triangles of equal height, a b e c and a b d c.