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How did Isaac Newton and Gottfried Leibniz calculate the curvature of plane curves?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In the 17th century, Isaac Newton and Gottfried Wilhelm Leibniz independently developed the foundational methods of calculus that allowed for the precise calculation of the curvature of plane curves.12

Their approach utilized the concept of the derivative to measure the rate of change of a curve's tangent as one moves along the curve. Specifically, Leibniz introduced the concept of the osculating circle in 1686, which is the circle that best approximates a curve at a given point. He defined the curvature of the curve at that point as the reciprocal of the radius of this approximating circle, expressed as $1 / r$ .2

As a curve becomes straighter, the radius of the osculating circle increases, causing the curvature to decrease; for a straight line, the radius is considered infinite, resulting in a curvature of zero. This method provided a systematic way to quantify how "curvy" a plane curve is at any specific location, moving beyond the qualitative descriptions used by mathematicians in antiquity.2