Quadrature of lunes; circles in double proportion

Taking equal parts destroys the proportion; squaring by borrowing

Removing equal parts from a proportion leaves the remainder in a different proportion, since the arc of f outside b belongs to a circle double that of b, so twice as much must be taken from f's larger arc. Leonardo works the quadrature by making b as small as needed, dividing its portion, and lending portions between the sub-double and outer circles.

Excess of the greater surface equals the lesser

If two surfaces are double one to the other and the lesser lies wholly within the greater, what remains of the greater outside the lesser equals the lesser. This is proved on the portions or sectors above, which are double one to the other. A circular ring with a sector illustrates it.

Circles in double proportion, part for part to infinity

For two circles in double proportion the half of the greater equals the whole of the lesser, the quarter equals half the lesser, the eighth the quarter, and so on infinitely, when the denomination of the greater's part is sub-double to that of the lesser's. Leonardo restates the rule and its converse condition on the denominations.

Squaring sickle-shapes a b and c d against squares e and f

Making the borrowings equal and lent to equal things, double circles yield the sickle a b and equal quarter-circles yield the sickle c d; the square e equals a b and f equals c d. Since a b is not squarable but c d is, Leonardo squares c, draws it from e, and the remainder equals the single-curved triangle b.

A pupil's running tally

Four lines in the hand of a pupil of Leonardo (the same who annotated f. 523 v) keep a tally: 'I had the first time one hundred; the second, another hundred; the third, fifty-five; the fourth, seventy; the fifth…'. The note is unrelated to the geometry filling the sheet.

Only the focal point of a reflected ray burns

In the lower-left corner, rays converge on a point a with the note that a alone burns — the centre of the percussion of the reflected ray, which is a single point. The note treats the burning focus of reflected light.

In alto a sinistra, quattro righe di mano dello stesso allievo di Leonardo che ha annotato il f. 523 v:

In alto, tre figure con crocetta, la seconda con:

Margine destro. Triangolo con cinque parallele alla base:

Corona circolare con settore e porzioni:

Verso sinistra, nella colonna centrale. Settori parzialmente sovrapposti, con:

Due figure di porzioni sovrapposte, con:

Colonna sinistra.

Due figure di falcate e due quadratini, con:

Angolo inferiore sinistro, raggi convergenti su:

Top left, four lines in the hand of the same pupil of Leonardo who annotated f. 523 v:

At the top, three figures with a little cross, the second with:

Right margin. A triangle with five lines parallel to the base:

A circular ring with a sector and portions:

Toward the left, in the central column. Partly overlapping sectors, with:

Two figures of overlapping portions, with:

Left-hand column.

Two figures of sickle-shapes and two little squares, with:

Lower left corner, rays converging on: