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50
A Short History of Astronomy
[Ch. II.


Fig. 20.—The eclipse method of connecting the distances of the sun and moon.
it with the known angular diameters of the sun and moon, he obtained, by a simple calculation,[1] a relation between the distances of the sun and moon, which gives either when

  1. In the figure, which is taken from the De Revolutionibus of Coppernicus (chapter iv., § 85), let d, k, m represent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and let s q g, s r e denote the boundaries of the shadow-cone cast by the earth; then q r, drawn at right angles to the axis of the cone, is the breadth of the shadow at the distance of the moon. We have then at once from similar triangles

    g k—q m : a d—g k :: m k : k d.

    Hence if k d = n . m k and ∴ also a d = n. (radius of moon), n being 19 according to Aristarchus,

    g k—q m : n. (radius of moon)—g k :: I : n

    n. (radius of moon)—g k = n g kn q m

    ∴ radius of moon + radius of shadow

    = (i + i/n) (radius of earth).

    By observation the angular radius of the shadow was found to be about 40' and that of the moon to be 15', so that

    radius of shadow = 8/3 radius of moon;

    ∴ radius of moon = 3/11 (I + I/n) (radius of earth).

    But the angular radius of the moon being 15', its distance is necessarily about 220 times its radius,

    and ∴ distance of the moon

    = 60 (I + I/n) (radius of the earth),

    which is roughly Hipparchus's result, if n be any fairly large number.