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
Holland that the effect of Cartesian teachings was most immediate and strongest.
The only prominent Frenchman who immediately followed in the footsteps of the great master was De Beaune (1601–1652). He was one of the first to point out that the properties of a curve can be deduced from the properties of its tangent. This mode of inquiry has been called the inverse method of tangents.
He contributed to the theory of equations by considering for
the first time the upper and lower limits of the roots of numerical equations.
In the Netherlands a large number of distinguished mathematicians were at once struck with admiration for the Cartesian geometry. Foremost among these are van Schooten, John de
Witt, van Heuraet, Sluze, and Hudde. Van Schooten (died 1660),
professor of mathematics at Leyden, brought out an edition of Descartes' geometry, together with the notes thereon by
De Beaune. His chief work is his Exercitationes Mathematicæ, in which he applies the analytical geometry to the solution of many interesting and difficult problems. The noble-hearted Johann de Witt, grand-pensioner of Holland, celebrated as a statesman and for his tragical end, was an ardent geometrician. He conceived a new and ingenious way of generating conics, which is essentially the same as that by projective pencils of rays in modern synthetic geometry. He treated the subject not synthetically, but with aid of the Cartesian analysis. René François de Sluze (1622–1685) and Johann Hudde (1633–1704) made some improvements on Descartes' and Fermat's
methods of drawing tangents, and on the theory of maxima and minima. With Hudde, we find the first use of three variables in analytical geometry. He is the author of an ingenious rule
for finding equal roots. We illustrate it by the equation . Taking an arithmetical progression 3, 2, 1, 0, of which the highest term is equal to the degree of
the equation, and multiplying each term of the equation respectively by the corresponding term of the progression, we get 3 x 3 − 2 x 2 − 8 x = 0 , or 3 x 2 − 2 x − 8 = 0 . This last equation is by one degree lower