Quadrature of the circle by sectors and octagon
Geometric squaring studies with an arithmetical note on Venetian war spending
A dense sheet of Leonardo's attempts to square the circle, working with circle sectors, semi-portions and annular strips (the 'crowns'), and reducing an octagon that circumscribes a circle to rectilinear triangles that can be measured. Lettered diagrams of sectors, an octagon and circular strips accompany the reasoning, which repeatedly seeks the proportion between a circular strip and its equivalent rectilinear figure. A separate arithmetical note multiplies 300,000 by 120 to reach 36 million, recording that the Venetians boasted of spending that sum in gold over ten years in war against the Empire, the Church and the kings of Spain and France. Faint additional sketches, including a small round device at centre-left, are present but not covered by the transcription.
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Squaring the circle by reducing a circumscribed octagon
The portions of portions h a joined with b b below are said to equal the large figure r n, a quarter of the periphery of the greater circle. Removing them leaves two rectilinear triangles equal to surface b b, which is a quarter of the excess of an octagon whose eight sides touch an inscribed circle. Knowing the octagon's value, one squares it, removes the triangles o p, and the remainder equals the circle the octagon surrounds.
Quadrature of a circular strip (annulus) via an equivalent triangle
From the parallel-strip a b, a b another strip equal to the rectilinear triangle g (that is d f) is removed, so the strip is itself squarable. By dilating the two arcs of a b, a b one obtains the strip a b g h and its equal e d e f around the centre f. Finding the proportion of one strip to the other yields the quadrature of the circle a b g d, and one can tell whether a sector is an eighth, a tenth, or some other part of circle f.
Arc, chord and the sector of a semicircle
Of the arc and chord: make the sector at a, giving a b, then complete its circle and see what part of the circle the sector is. Square the circle, subtract the quadrature of the sector, then from that subtract the triangle b; the remainder of the sector's quadrature will equal the portion a.
Venetian war-spending: 36 million ducats over ten years
A worked multiplication of 300,000 by 120 reaching 36,000,000. The note records that the Venetians boasted they could spend 36 million in gold over ten years in the war against the Empire, the Church and the kings of Spain and France, at three hundred thousand ducats a month.