Darkest Point of a Sphere Lit by the Sky-Hemisphere
Where an opaque sphere between the luminous hemisphere and the earth is deepest in shadow
Continuing the shadow theory, this page determines which part of an opaque sphere set between the light of the sky-hemisphere and the darkness of the earth is deepest in shadow. It argues that the point q, midway above the earth, is darker than the flanking points o and p, because o and p begin to see the horizon of the hemisphere and mingle with its light. The reasoning is supported by a tangent-and-centre construction drawn from Euclid's Elements, illustrated by two semicircular diagrams with an inscribed sphere and radiating lines labelled with letters. A new section then begins on the shadow of an opaque sphere resting upon the earth.
On this page
The midpoint q is the sphere's darkest region
For the sphere b c p o placed between the hemisphere's light and the earth's darkness, the part o q p is darker than any other. The point q, being in the middle above the earth, is of more excellent darkness than o or p, which are nearer the extremes and begin to catch the hemisphere's horizon light.
Tangent-and-radius proof from the Elements
The demonstration cites the rule that a line drawn from a circle's centre to the point of tangency is perpendicular, falling between two right angles. Hence the line from the sphere's centre x meeting s c at right angles at point o sees the whole darkness of the earth d c, and p opposite does likewise.
Semicircular diagrams of sphere, centre and tangents
Two semicircular figures illustrate the argument. The upper carries an inscribed circle with a centre marked x and radiating construction lines labelled n, E, q, s, o; the lower shows a shaded sphere m, n resting on a base line with points a, c, s.
A sphere resting on the earth casts a deeper shadow
A new heading opens the case of the shadow of an opaque sphere resting in contact with the earth, stating it will be of greater darkness than the preceding one, which sees the earth only as its object. The proof, with sphere n m s on the earth a c, continues on the following page.