Rectifying the Surface of a Shield-Shaped Body
Unrolling a curved solid's skin into straight parallel bands
Leonardo studies how to rectify (flatten into straight measure) the curved surface of a 'shield-shaped' body of revolution. Two teardrop-like solids at the right are hatched with parallel rings, and he argues that the plane surface a b c equals the whole skin of the curved body. Dividing the body into five parallels of equal width, he unrolls each curved ring by circular motion onto an indefinite straight line, marking the divisions before separating curve from line so that the true lengths are preserved.
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The shield-body's curved skin equal to a plane surface a b c
The plane surface a b c equals the whole skin of the shield-body a b, and the straight lines that enclose the parallels of the straight surfaces are the curved parallels which clothe the said body. The rectification thus matches each curved band to a straight one.
Unrolling five annular parallels onto straight lines
The irregular periphery a b c, stretched by motion, equals the straight line d e. The body being divided into five parallels of equal width, he divides d e likewise and turns each circle upon its indefinite straight line by circular motion, terminating it at the length reached in one full revolution. The divisions are marked on both curve and line before the two are separated.
