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186 shaded and unshaded portions of the figure represent equal areas each corresponding to the motion of the planet during a month. Kepler's triumph at arriving at this result is expressed by the figure of victory in the corner of the diagram (fig. 61) which was used in establishing the last stage of his proof.

141. Thus were established for the case of Mars the two important results generally known as Kepler's first two laws:—

1. The planet describes an ellipse, the sun being in one focus.

2. The straight line joining the planet to the sun sweeps out equal areas in any two equal intervals of time.

The full history of this investigation, with the results already stated and a number of developments and results of minor importance, together with innumerable digressions and quaint comments on the progress of the inquiry, was published in 1609 in a book of considerable length, the Commentaries on the Motions of Mars.

142. Although the two laws of planetary motion just given were only fully established for the case of Mars,