Subdivide the curve and connect points in the subdivision by line segments. For a typical line segment, draw a right triangle with hypotenuse \(\Delta s_i\) and legs \(\Delta x_i\) and \(\Delta y_i\), and apply the Pythagorean theorem. Arc length is approximately \[\begin{array}{rcl} L \approx \sum_{i=1}^n \Delta s_i & = & \sum_{i=1}^n \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2} \\ & = & \sum_{i=1}^n \sqrt{1 + \left(\frac{\Delta y_i}{\Delta x_i}\right)^2} \Delta x_i \\ & = & \sum_{i=1}^n \sqrt{\left(\frac{\Delta x_i}{\Delta y_i}\right)^2+1} \Delta y_i. \end{array}\] Note: the subdivisions of the curve simultaneously define \(\Delta x\) and \(\Delta y\), so we can use either.