De ludo geometrico: squaring lunes and curved figures
Dozens of demonstrations that curved figures equal rectilinear ones, with page-counting sums and a moral maxim.
This crowded sheet, headed 'De ludo geometrico' (On the geometric game), gathers dozens of demonstrations that curvilinear surfaces — lunes, crescents, sectors and annuli — can be shown equal to rectilinear figures and thus 'squared.' Numbered propositions (First through Tenth) build on Euclid, repeatedly using the rule that equals taken from equals leave equals, and on circles standing in double or quadruple proportion. In one corner Leonardo counts the letters, lines and pages of a manuscript in a column of sums; a compass is annotated 'make it more massive.' A left-hand column carries an unrelated maxim: superfluous money is, in the miser, a good that is never put to use.
On this page
De ludo geometrico — squaring curved surfaces
The heading 'On the geometric game' opens a program: any surface to which a square equal to it can be assigned is itself squarable by its own parts. Leonardo also restates the Pythagorean rule that in a right isosceles triangle the squares on the two sides equal the square on the third. These principles frame the many lune-and-sector demonstrations that fill the sheet.
A lune shown equal to a triangle
In a numbered set of steps Leonardo takes a portion b from the semicircle a b and the same from the sector b c, leaving the lune a equal to the triangle c. He justifies it by the rule that of circles double one another the quarter of the larger equals the half of the smaller, and that equals removed from equals leave equals.
The tenth demonstration and the doubling of circles
A central double figure of semicircles upon circles shows that the triangle and lune remaining from the circle equal a triangle and two crescents. Leonardo generalises: of circles in subduple ratio, if equal quarter-portions are laid one upon the other, the remainder of the larger equals the whole of the smaller. Each figure is quadrable within itself by exchanging its own parts.
Counting the letters and pages of a manuscript
In an inverted panel Leonardo reckons that a page is 52 lines and each line 50 letters, giving 2600 letters a page, and multiplies out across 160 pages. The result — over 400,000 letters — is set down in a column of sums with a note that 160 are the pages.
Maxim on the miser's superfluous money
A single line in the left column reads as a moral aphorism: superfluous money is, in the miser, a good that is never put to use. It stands apart from the surrounding geometry.
A compass to be made more massive
Among the geometric figures Leonardo drew compasses, one in Panel D annotated with the instruction to make it more massive. The tools serve the constructions that dominate the sheet.