Squaring the circle, a pulley, and two building elevations
Extensive quadrature of lunes and circles with pulley statics and architectural studies
This crowded sheet is dominated by quadrature studies. Under the heading "Of transmutation," Leonardo argues that among portions of a circle only the semicircle can be given an equal circle, then works through dozens of small figures of circles, lunes (falcate), inscribed squares and lens-shapes (bisangoli), repeatedly invoking the axiom that if equals are removed from equals the remainders stay equal in order to reduce each figure to a squarable one. At the top he treats a pulley with cords, weight and load, reasoning that the motor's cord bears more than the load's cord, and a toothed wheel and the number 125 also appear. Two carefully drawn architectural elevations — a multi-storey arcaded building and a pedimented temple front — occupy the upper centre, and dense marginal notes surround the many diagrams.
On this page
Of the pulley, cords, weight and load (a, c, b)
A pulley lettered a, c, b heads a note titled "Of the pulley, cords, weight and moving load." Leonardo argues that the motor's cord feels more weight than the load's cord, because the motor's cord bears the weight of the load plus the excess by which the motor exceeds it.
Of transmutation: only the semicircle can be squared
Under the heading "Of transmutation," Leonardo states that to no portion of a circle is an equal circle given except the semicircle, while to every sector — greater or smaller than the semicircle — an equal circle can be given. It sets the programme for the surrounding figures.
The axiom: equals removed from equals leave equals
Above an inscribed square subdivided into small squares Leonardo writes the axiom that if from equal things one removes equals, the remainder stays equal. He reuses this rule again and again to reduce lunes and circle portions to squares.
Squaring a circle by its greatest portions (a, b)
A lettered circle (a, b) carries a full proof: the smaller circles being half the greater, the remainder of the greater circle equals the smaller circles, so that once the four greatest portions are taken away the remainder is squarable, and removing the two squares leaves a squared remainder.
The oval equals its two surrounding lunes
Beside one figure Leonardo notes that the oval is worth the two lunes (falcate) that surround it, and beside a similar figure that each lacks the value of the four greatest portions. These are among the concluding results of the squaring series.
Two elevations: an arcaded building and a temple front
The upper centre carries two finished pen elevations: a tall, multi-storey arcaded building with balustrade and a pedimented, columned temple front with an arched opening. They stand out from the surrounding geometric diagrams as architectural design studies.
