Triangle's inscribed and circumscribing circles, and the lune
With a note on why water and a ball run downhill, and reducing a lune to a rectilinear figure
A geometry sheet dominated by the annular band between a triangle's inscribed and circumscribing circles, whose boundaries Leonardo names the 'first' and 'second' line, arguing the triangle fills more of that ring than any other polygon. Further diagrams superimpose circular sectors and semicircles to reduce a lune to a rectilinear figure equal to a given portion. A note in the upper margin asks why both water and a ball move down a slope, answering that each is drawn by its inequality with respect to the center of the world, and a short column of figures sits at lower center.
On this page
Why water and a ball both run downhill
Leonardo asks why water and a ball alike move down a slope. He answers that the water moves because of the inequality it holds with respect to the center of the world, and that the ball descends for a like cause.
The circles inscribed in and circumscribing a triangle
The circle enclosed by the triangle and the circle enclosing it differ more than any other pair of circles, and the triangle fills more of the annular band between them than any other polygon. Of that band's two boundaries, the smaller is called the first line and the greater the second line (labels C, a, b).
Reducing the lune to a rectilinear figure
Removing a from the semicircle and b from the sector leaves p of the sector equal to c d of the semicircle. Leonardo then seeks how much portion o of the lune exceeds portion r, so as to add it to p and make the result rectilinear.
Column of figures
A short worked column of numbers accompanies the geometric notes at lower center of the sheet.