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Van Heuraet's Rectification of Curves

These pages are from the brief work *On the Transformation of Curves into Straight Lines*, by Hendrick van Heuraet (1634 - 1660), published in the 1659 Latin edition of Descartes's *Geometry*, edited by van Schooten. Although van Heuraet was not the first to accomplish a rectification, a task that Descartes had said could not be done, this is the first publication of a general procedure, a procedure very close to our standard calculus procedure for finding the length of a curve.

On these two pages, van Heuraet describes his general procedure for rectification, one which tranforms the length into an integral, that is, the area under a curve. He then illustrates the procedure by calculating the length of the semi-cubical parabola, *y*^{2} = *x*^{3}/*a. * (We can take *a* = 1 for simplicity.) Note that since the procedure for finding arc length involved first finding *dy*/*dx* (or the tangent to the curve)*,* van Heuraet accomplishes this by using Descartes's normal method and Hudde's rule for finding a double root. Note also that van Heuratet uses Descartes's symbol for "equal" rather than our modern equal sign.

Van Heuraet's Rectification of Curves
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