How Equal Arcs of the Sky Light a Sphere Evenly
The upper cap of the sphere is uniformly bright; the least-lit part is the one seen by the fewest rays
Leonardo analyses a sphere lit by the dome of the sky (the hemisphere a b c) and shadowed by the darkness of the earth (line a c). The upper region f n (or f p n) is of most intense and uniform brightness because it sees no part of the dark earth and is illuminated by three equal arcs of the hemisphere (a r e, r b s, b s c); invoking the Euclidean common notion that things equal to a third are equal to one another, he concludes the points p, f, n are equal in clarity. A new proof, [750], then addresses which part of the spherical opaque body is least illuminated, namely the one seen by the fewest luminous rays, and cites 'the 7th of the 9th' book: parts of bodies lit at equal distance by equal and similar lights are always equally bright. A semicircular diagram of the sky-hemisphere over the sphere accompanies the argument.
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Uniform brightness of the sphere's upper cap
The region f n is of most intense clarity because it sees no part of the earth's darkness a c, and is uniformly bright because lit by equal arcs of the hemisphere: arc a r e equals arc r b s and arc b s c.
Equal arcs and the common notion of equality
By the conception that things equal to a third thing are equal to one another, the three equal arcs give equal illumination, so the points p, f and n are equal in clarity. The proof appeals to 'the 7th of the 9th,' that parts lit at equal distance by equal and similar lights are always of equal brightness.
The least-illuminated part of the sphere
Chapter [750] asks which part of the spherical opaque body is of darkest shadow: it is the one seen by the least sum of luminous rays. Though close to the earlier proof, Leonardo restates it, letting the shadowed body be f n o under the hemisphere a b c.