this as a general theorem about magnitudes A, B. In the second part of the proof, one now
assumes that X > B. Here this case is reduced to the first, since, by (4), there will be a Y such
that Y < A and D : C = X : A = B : Y, which contradicts the first case. Hence, X Y.
So, by (1), A : B = C : D.
There is some evidence that Eudoxos also used this method for proving general theorems
in proportion theory, where the first case would be for commensurable magnitudes and the
second for incommensurable magnitudes. If so, then Elements 5, on proportion theory,
would represent a later reworking of his theory and proofs. Here, there is a general defin-
ition of “same ratio” (Elements 5. def. 5), eliminating the need for separate cases:
A:B=C:Diffn, m: (n×A)(m×B) iff(n×C)(m×D).Eudoxos was the first astronomer to attempt a general geometrical model to explain apparent
motion of planetary stars, the sun, Moon, and five visible planets. The model assumed that all
celestial bodies are spheres with the Earth as their center and whose motion is regular: circular
about an axis through the center of the Earth. Each planetary star has a system of concentric
spheres where the axis of one inner sphere is fixed to the next outer sphere. In this way,
Eudoxos could create apparent irregular motions. For example, each planetary system con-
sisted of an outer sphere whose poles would be the poles of the celestial equator and which
rotated daily east/west. Fixed to it were the poles of a sphere contained in it with the same
center. The poles of the fixed sphere would be perhaps^1 / 15 circle (the obliquity of the eclip-
tic) from the poles of the first sphere with the second sphere rotating slowly west/east, i.e.,
with the zodiacal period of the planetary star, where the net motion produced is a spherical
spiral. For the five planets, further variations in their motion would then be explained by two
Eudoxos of Knidos: Hippopede © Mendell
EUDOXOS OF KNIDOS