Geometry: squaring curved-sided figures
With a note on gauging a window by its shadow, and on proportion theory
The sheet carries a series of curvilinear geometric figures in which Leonardo equates a curved-sided quadrilateral to a curved triangle, subtracting known equal parts until a remainder equal on both sides is left. Margin notes add a method for finding a window's distance, width, height and proportion from the shadow it casts inside a room, and, on the inverted sheet, a rule on how two unequal quantities relate in proportion to a third. Letter-labelled figures (a b o; p q m o S r t; a d d b; a b s; a b) accompany the demonstrations. Some further faint text is present beyond what is transcribed.
On this page
Equating a curved quadrilateral to a curved triangle
Leonardo sets the quadrilateral r m equal to the triangle of equal curved sides m o p q. Removing the surface m from each leaves o p q equal to S r t; then taking p from the triangle and t from the quadrilateral (known equal, since t is the squaring of p) leaves o q equal to r S, and further subtraction leaves q equal to S.
Figure a b o equal to d f
A top-right figure lettered a – b (c) o – d – e – f carries the statement that a b o is equal to d f, followed by a broken-off note ('I shall draw ...').
Gauging a window from the shadow in a room
A left-margin note proposes using the shadow cast within a house to determine the distance, width and height of a window and its proportion.
Two unequal quantities in proportion to a third
On the inverted sheet Leonardo states that if two unequal quantities are set in proportion to one quantity, the greater obtains the greater proportion and the lesser the lesser — then qualifies that to the lesser the proportion will be greater and to the greater lesser.