Saturday, June 30, 2012

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, y2 = x3/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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