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, adopting small letters of the alphabet in place of the capitals used by Vieta. The symbols of inequality $$\scriptstyle{>}$$ and $$\scriptstyle{<}$$ were introduced by him. Harriot's work, Artis Analyticæ praxis, was published in 1631, ten years after his death. William Oughtred (1574–1660) contributed vastly to the propagation of mathematical knowledge in England by his treatises, which were long used in the universities. He introduced $$\scriptstyle{\times}$$ as symbol of multiplication, and $$\scriptstyle{::}$$ as that of proportion. By him ratio was expressed by only one dot. In the eighteenth century Christian Wolf secured the general adoption of the dot as a symbol of multiplication, and the sign for ratio was thereupon changed to two dots. Oughtred's ministerial duties left him but little time for the pursuit of mathematics during daytime, and evenings his economical wife denied him the use of a light.

Algebra was now in a state of sufficient perfection to enable Descartes to take that important step which forms one of the grand epochs in the history of mathematics,—the application of algebraic analysis to define the nature and investigate the properties of algebraic curves.

In geometry, the determination of the areas of curvilinear figures was diligently studied at this period. Paul Guldin (1577–1643), a Swiss mathematician of considerable note, rediscovered the following theorem, published in his Centrobaryca, which has been named after him, though first found in the Mathematical Collections of Pappus: The volume of a solid of revolution is equal to the area of the generating figure, multiplied by the circumference described by the centre of gravity. We shall see that this method excels that of Kepler and Cavalieri in following a more exact and natural course; but it has the disadvantage of necessitating the determination of the centre of gravity, which in itself may be a more difficult problem than the original one of finding the