Circular sectors, semicircles, and crescents made equal
Comparing a quarter-circle's segments with a semicircle to square a crescent.
Two circles are set in quadruple ratio, so that the whole of the smaller equals a quarter of the larger, and half the smaller equals a quarter of the larger (1). Working from a figure lettered a g b e f g h d c, Leonardo shows that the sector a b c d e f equals the semicircle f g h, and by removing and then restoring equal parts he makes the crescent b e d and a rectilinear 'pyramid' equal and square (3). A second study asks the difference between the segments p and o, testing whether the rectilinear triangle n o equals m just as the crescent p n did (4, 5).
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Two circles in quadruple ratio
Leonardo starts from two circles standing in quadruple ratio. The whole of the smaller equals a quarter of the larger, and, it follows, half the smaller equals a quarter of the larger.
Sector equals semicircle; crescent b e d squared
The sector a b c d e f equals the semicircle f g h; removing f from both leaves equal remainders. After exchanging h in the semicircle for c in the eighth-circle, the crescent b e d is shown equal in length and substance, and adding a rectilinear pyramid b to a produces a square figure equal to c, h and d.
Difference between the segments p and o
Leonardo seeks the difference between portion p and portion o, certain that p and n together equal the rectilinear m. He straightens p into o and tests whether the triangle n o equals m, so that the excess of m over n o measures how much p exceeds o.
