Squaring the Circle: the Science of Equalization
Deriving the greatest circle from a hexagon; the night of St. Andrew memorandum
The page opens Leonardo's "science of equalization" (scientia de equiparantia): he proposes to draw the greatest possible circle from a hexagon and to square the remainder, then to equate the two squares in order to conclude the circle's quadrature with precision. A long procedure works with two equal circles divided into equal parts, superposing figures labelled a, b, c, d, n, m, p, q, S, g, h, t and v while removing matched portions, and he candidly flags a "fallacy" where he takes 4 portions from one figure but 5 from the other. Written crosswise beside the diagrams is the famous jotting that on the night of Saint Andrew he found the end of the squaring of the circle, concluded just as the candlelight, the night and the page ran out. Numerous hexagon-and-circle diagrams fill the right margin.
On this page
The science of equalization: greatest circle from a hexagon
Leonardo sets out to draw from a hexagon the greatest circle it can contain and to square what remains. He then places the lesser square upon the greater, concluding that the remainder of the greater square is exactly equal to the greatest inscribed circle.
Superposing two equal circles, and the acknowledged fallacy
Two equal circles are divided into equal parts and equal sixth-portions removed; superposing them (a, b, c, d, n, m) leaves equal remainders. Counting the portions he removes 4 from m (2 lacking, making 6) and 5 from n, and frankly notes the fallacy: he does not take as much from one as from the other.
The night of Saint Andrew
A note written crosswise records that on the night of Saint Andrew he found the end of the squaring of the circle. It was concluded, he writes, at the end of the candlelight, of the night and of the paper on which he was writing, at the end of the hour.
Equal figures give equal squares
A closing note ties the construction together: if a and b are equal, then n and m are their squares (quadratures). The result is set beneath the row of figures.