Before getting into the textbook, let me consider some ways that one might approximate the arc length of a curve.
Firstly, I can always use linear approximation: lay out points along the curve, draw straight lines between adjacent points, and add up the distances. But perhaps I can do better than this.
The lenght of the circular arc through three consecutive points on the curve responds to both first and second derivatives, and should be a much better approximation. Let me try it.

NOTES:
Assuming the curve is real and continuous through the 2nd derivative... Let P, Q, R, be thee points on the curve, in order of increasing u. In the limit as R→Q→P, the radius of the circle through them converges to a unique value (the radius of curvature), and the circle (the osculating circle) can be drawn.
The osculating circle at M is shown in orange.
The chapter on Interpolation was concerned primarily with polynomials, whose values, integral, and derivative, can be calculated directly (and exactly, when applicable) from the tabular points. In this worksheet, I must calculate the length of a circular arc from straight-line measures (sine, cosine, tangent). I have used the arctan() function, Here is a hint about how to obtain a convergent series for the arc tangent function.