the equation, and multiplying each term of the equation respectively by the corresponding term of the progression, we get , or . This last equation is by one degree lower than the original one. Find the G.C.D. of the two equations. This is ; hence 2 is one of the two equal roots. Had there been no common divisor, then the original equation would not have possessed equal roots. Hudde gave a demonstration for this rule.[24]
Heinrich van Heuraet must be mentioned as one of the earliest geometers who occupied themselves with success in the rectification of curves. He observed in a general way that the two problems of quadrature and of rectification are really identical, and that the one can be reduced to the other. Thus he carried the rectification of the hyperbola back to the quadrature of the hyperbola. The semi-cubical parabola was the first curve that was ever rectified absolutely. This appears to have been accomplished independently by Van Heuraet in Holland and by William Neil (1637–1670) in England. According to Wallis the priority belongs to Neil. Soon after, the cycloid was rectified by Wren and Fermat.
The prince of philosophers in Holland, and one of the greatest scientists of the seventeenth century, was Christian Huygens (1629–1695), a native of the Hague. Eminent as a physicist and astronomer, as well as mathematician, he was a worthy predecessor of Sir Isaac Newton. He studied at Leyden under the younger Van Schooten. The perusal of some of his earliest theorems led Descartes to predict his future greatness. In 1651 Huygens wrote a treatise in which he pointed out the fallacies of Gregory St. Vincent (1584–1667) on the subject of quadratures. He himself gave a remarkably close and convenient approximation to the length of a circular arc. In 1660 and 1663 he went to Paris and to London. In 1666 he was appointed by Louis XIV. member of the French
