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In
the above diagram ABPD is an ellipse, S and F foci, A and P aphelion and
perihelion; the planet is at E, and angle AFE (the middle anomaly) changes
uniformly. As Seth Ward defined it, the problem is to derive angle ASE
(the true anomaly) from knowledge of angle AFE. Calculation requires plane
trigonometric functions after extending line FE to T (such that ET equals
ES). Ward came upon this Simple Elliptical Hypothesis in the process of
criticizing Boulliau's Conical Hypothesis (1645). Though easier to
employ than Kepler's Second Law, Ward's hypothesis was very weak in several
respects. In practice, Ward's Hypothesis differed from Kepler's predicted
positions for Mars by 30' or more in longitude. In 1657 Boulliau provided
a response to Ward's published criticism, and in the process, Boulliau
proposed what has come to be called the 'Modified Elliptical Hypothesis'
(1657), which had a number of subsequent variants. As with other
modified elliptical hypotheses, Ward's model in particular was weak in
predicting planetary latitudes.
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