The lune equals the square: equal figures leave equal remainders
Pen notes on the quadrature of the lune, with circle-and-square diagrams
Pen notes on the quadrature of the lune. Leonardo argues that if several equal figures are set upon a base equal to all of them together, the surplus of the greater equals the surplus of the lesser, and concludes that 'the black square is equal to the lune.' A further note treats two tangent circles set on a larger circle, marking that points a, m, n and o are equal. Several small circle-and-square diagrams and a crescent-shaped lune run down the left margin.
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Equal figures on a common base leave equal remainders
If two, three or four equal things are placed upon a single thing equal to all of them together, what is left over of the greater equals what is left over of all the lesser ones. Leonardo points to the figure below as the worked example.
The black square equals the lune
Two squares, a greater and a lesser, are inscribed in two partly overlapping circles, and the note states simply that the black square is equal to the lune. This is the payoff of the remainder argument above.
Points a, m, n, o are equal
Two circles tangent to one another are superimposed on a larger circle, with two of their portions redrawn outside them. The note marks the segments n–a and o–m and states that a, m, n and o are equal.