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 equation has imaginary roots; but Descartes does not say that the equation always has but that it may have so many roots. It is true that Descartes does not consider the case of imaginaries directly, but further on in his Geometry he gives incontestable evidence of being able to handle this case also.

In mechanics, Descartes can hardly be said to have advanced beyond Galileo. The latter had overthrown the ideas of Aristotle on this subject, and Descartes simply "threw himself upon the enemy" that had already been "put to the rout." His statement of the first and second laws of motion was an improvement in form, but his third law is false in substance. The motions of bodies in their direct impact was imperfectly understood by Galileo, erroneously given by Descartes, and first correctly stated by Wren, Wallis, and Huygens.

One of the most devoted pupils of Descartes was the learned Princess Elizabeth, daughter of Frederick V. She applied the new analytical geometry to the solution of the "Apollonian problem." His second royal follower was Queen Christina, the daughter of Gustavus Adolphus. She urged upon Descartes to come to the Swedish court. After much hesitation he accepted the invitation in 1649. He died at Stockholm one year later. His life had been one long warfare against the prejudices of men.

It is most remarkable that the mathematics and philosophy of Descartes should at first have been appreciated less by his countrymen than by foreigners. The indiscreet temper of Descartes alienated the great contemporary French mathematicians, Roberval, Fermat, Pascal. They continued in investigations of their own, and on some points strongly opposed Descartes. The universities of France were under strict ecclesiastical control and did nothing to introduce his mathematics and philosophy. It was in the youthful universities of