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Jan de Witt's Elements of Curves

This image is page 263 of *Elements of Linear Curves *by Jan de Witt (1625 - 1672). De Witt was a student of Frans van Schooten, who published this work in his 1661 edition of Descartes' *Geometry*. (This copy is from the 1683 edition. And, in fact, de Witt himself probably finished the work by 1646.) The first of the two books of this treatise was devoted to developing the properties of the conic sections using synthetic methods based on the work of Apollonius. But in the second book, de Witt produced a complete algebraic treatment of the conics, beginning with equations in two variables, based on the work of Fermat and Descartes.

On this page, de Witt shows how to rotate the axes to turn a complicated second degree equation in two variables into the standard one displayed earlier. Unlike in modern treatises, de Witt does not use trigonometry, but gives the equations of the new axes in terms of the old ones. That is, he uses a transformation of coordinates based on the form of the given equation.

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