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HISTORICAL VIEW OP THE

without entering into mathematical details, to explainthe process by which the ultimate suggestion wasbrought under his consideration; and it would beequally so to convey an idea of the immense mass ofcalculation through which he toiled, in putting each ofhis successive theories to the test of agreement with theobservations. Finally, after working his way in alter-nate exultation at anticipated triumphs, and bitter dis-appointment when, one after another, they vanished inair, driving him, as he says, “ almost to insanity,”he at length had the intense gratification of finding,that an elliptic orbit described about the sun in one of thefoci, agreed accurately with the observed motions ofthe planet Mars. This is called Kepler ’s first law.His exultation knew no bounds — his very diagramwas drawn with a figure of victory in the corner. Healso found immediately that his former inference of theequable areas in the circular orbit was only an ap-proximate case of the law, which is accurately true ofmotion in an elliptic orbit about the focus; and his se-cond law is announced by saying that the radius vectorof the elliptic orbit passes over equal areas in equaltimes. A third of equal importance was discoveredafterwards.

In connection with the second law, it became a pointof importance to find some method of determining thearcs of the ellipse which correspond to equal sectorialareas. Kepler , in proposing this point to the attentionof geometers, stated his belief that no exact solution ofit could be attained. It has engaged the attention ofthe most eminent mathematicians since his time ; andhas received solutions by means of series carried to anydegree of approximation. It is distinctively known bythe name of Kepler ’s problem.

This difficulty of calculating the place of a planet,and which Kepler confessed he had not the means ofovercoming, was urged by his opponents as a seriousobjection to his system, as if he were answerable for theproceedings of nature.