Combining and subtracting squares and circles
Making one circle from several, and taking a circle from a circle by the right-triangle rule
This page gathers Leonardo's rules for combining and dividing squares and their inscribed circles. He shows how to fold several circles into one by building three squares on the sides of a right triangle a b c, so that circle-to-circle keeps the same proportion as square-to-square. A longer construction takes a smaller square a f g h out of a larger one and finds, through the right triangle K L b, the side L b of the square equal to what remains. The same rule, he notes, reduces the difference of two circles to a single circle and serves for any similar figures; three diagrams of squares with inscribed circles are drawn in the right margin.
On this page
Making one circle out of several
First make the right triangle a b c; then from the square of each of its sides make three squares, and in each square draw its circle. Circle-to-circle will then hold the same proportion as square-to-square, and the like can be done for any figure enclosed in those squares.
Subtracting a square from a square, leaving a square
To take the square a f g h from a larger square and leave a square, open the compass to a b, swing the curve a K onto the side f g at K, and drop a parallel to L on a b. Then L b is the side of the square L m o b equal to the remainder, proved by the right triangle K L b whose hypotenuse K b equals a b and leg K L equals f a.
Circle-to-circle proportion equals square-to-square
To take a smaller circle from a larger and reduce the remainder to a single circle, build the three squares and inscribe the largest circle in each: because circle-to-circle keeps the proportion of square-to-square, the aim is achieved. The same rule serves for every similar surface figure.
Three construction diagrams of squares and inscribed circles
The right margin carries three figures: a square with a diagonal and the points K, L; a pair of interlocking squares with inscribed circles; and two squares each holding a circle, labelled m and o. They bear further ink labels not included in the transcription.