Squaring the Lunules; a Night Water-Clock
Right triangles inscribed in semicircles, the rule of Zenophon, and a graduated night lamp
This leaf continues Leonardo's attack on squaring the lune: semicircles each containing a right triangle are compared, and he argues from the equality of the greater right triangle to two lunules that a lesser and a middle right triangle can be made proportional to their lunules, invoking the 'rule of Zenophon' to reach the true quadrature. Between the geometrical passages he draws a lit lamp graduated 1 to 9 and labels it 'night clock'. A mutilated note in the lower-left corner concerns water being drawn or pushed.
On this page
Right triangles in semicircles equal to two lunules
If the semicircle c f g is worth the two others (as proved through the right triangle c a g), then the right triangle c a g must be worth the two right triangles a b c and a q g; and if the greater right triangle is worth the two lunules, the two lesser right triangles equal to it are also equal to those lunules. Yet this does not prove that a lesser right triangle is the quadrature of its lesser lunule.
The rule of Zenophon and the true squaring of the lunules
Leonardo proposes to make the lesser and middle portions by the rule of Zenophon, detaching another lunule inside the lesser and adding one to the middle, then squaring it and giving it to the greater right triangle. Dividing the greater triangle in the proportion of the middle semicircles and adding the lesser quadrature within the division, he expects always to reach the true squaring of the lunules.
Comparing the portions around the right triangle
Here the portions are equal around the right triangle, and one must consider by how much the portion c d e has grown beyond the portions h i K and h g f, more than its due. It is manifest that it has taken in half the right triangle and about three quarters of the opposite portion.
A night clock: a lit lamp graduated 1 to 9
Set among the geometry is a drawing of a burning lamp with a scale of the numbers 1 through 9, labelled 'night clock'. The graduation lets falling oil or light mark off the hours of darkness.
Fragmentary note on water drawn or pushed
In the mutilated lower-left corner a broken note speaks of water that comes and goes, or is drawn or pushed. Too little survives to reconstruct the argument.
