I am interested in the early history of curvature. Who defined it first and when, who came up with the name, how was it calculated before mathematicians used calculus to define $k=|α''(s)|$? Are there books or articles containing such early calculations?
Apollonius (c. 262–190 BC) "calculated" curvature of conic sections implicitly when solving the problem of drawing normals to them in book V of Conica, but he did not think of it as a property of a curve, and his "calculations" are constructions of segments. The first person to "see" curvature was Oresme (c. 1320-1382), Descartes's precursor in introducing coordinates. He described it as a local measure of curve's bending, and christened it with the Latin "curvitas". Later he proposed that for circles it can be quantified by the reciprocal of the radius, our modern convention. Kepler vaguely suggested how to define curvature for general curves by considering the "closest" circle at a point, named osculating ("kissing") circle by Leibniz in 1680s. But it was Huygens, who first found a way to calculate curvature for general curves, and Newton who gave the concept its modern form.
The calculus was yet to be invented so Huygens relied on Descartes's ideas about multiplicity of intersections and Fermat's ideas of infinitely small. Descartes in Geometry (1637) gives a method of drawing tangents and normals to a curve by finding circles with centers on the axis, whose two intersection points with the curve "coalesce", and so intersection gives a double root. This was a foundation of Descartes's algebraic calculus, that preceded Newton's and Leibniz's. Huygens realized in 1653-4 that for any point on a curve one can consider two "coalescing" normals, and gave a semi-geometric way to calculate their point of intersection. That of course was the center of curvature, and its distance to the curve was the radius of curvature, although Huygens did not use these terms. Van Schooten included the method in appendices to the second edition of Descartes' Geometry (1659), which is where Newton learned it from. Later in a remarkable tour de force of Horologium Oscillatorum (1673) Huygens named the locus of the centers of curvature to a curve its evolute, and showed how to construct a perfect pendulum, whose period does not depend on its amplitude. The construction was based on the fact that evolute to a cycloid is congruent to it.
It was Newton who developed Huygens's ideas into a general method for computing curvature (which he initially called "crookedness") in 1664. He used that two nearby normals "coalescing" is equivalent to three nearby intersection points with the circle "coalescing" on the curve, and used Descartes's method, with Hudde's 1659 improvements to find its center by looking for a triple root. Newton also realized that at inflexion points, where the radius of curvature "blows up", one should assign to curvature value zero. Later in the Method of Fluxions and Infinite Series; with its Applications to the Geometry of Curve-Lines (1671) he transformed the algebraic approach to curvature into a more recognizable one of calculus. But it is interesting that Newton was well aware, and facile with, the "lost" algebraic calculus when developing his own. Some modern textbooks, e.g. Struik's Lectures on Classical Differential Geometry, still use the language of coalescing points and normals when defining curvature.
Lodder's Curvature in Calculus Curriculum gives a step by step guide through Huygens's calculation of curvature and evolute of the cycloid, he also describes Euler's 1760 calculation of the principal curvatures of a surface. An overview of early history (with some inaccuracies) is given in Curvature of Surfaces in 3-Space, the source is Margalit's History of Curvature. Nauenberg's Huygens and Newton on Curvature and its Applications to Dynamics gives a detailed account based on original sources, it briefly describes Newton's alternative method for calculating curvature a la Descartes.
From Curvature of Surfaces in 3-Space by Michael Garman & Jessica Bonnie published in Verge 6:
The notion of curvature first began with the discovery and refinement of the principles of geometry by the ancient Greecks circa 800-600 BCE. Curvature was originally defined as a property of the two classical Greek curves, the line and the circle.
It was noted that lines do not curve amd that every point on a circle curves the same amount. The actual study of curvature began when Aristotle expanded upon these two points and declared that there are three kinds of loci: straight, circular and mixed. It was from this premise that the true study of curvature began.
Appollonius of Perga devised methods for calculating the radius of curvature in the 3rd BCE. These methods were similar to Newton and Huygens (discovered some 2,000 years ago later) ...
The next momentous advancement in the study of curvature came from Nicole Orseme in the 14C. Oresme was the first person to hint at an actual definition of curvature. He also assumed that there was a specific measure of twist which he called 'curvitas'. By observing multiple curves at once, Oresme eventually proposed that the curvature of a circle was proportional to the multiplicative inverse of its radius. This would eventually provide the driving force behind the quest of finding the curvature of a general curve ...
Johannes Kepler made the next contribution to the notion of curvature. While working on the problem of al Hazin, (finding the image of a brilliant point when reflected off a circle), Kepler arrived at the notion of using a circle to measure the general curvature at a point of reflection. This approximating circle would come to be known as a curvea 'circle of curvature' at a point ...
After this Huygens, Newton & Liebniz reified Keplers discovery through calculus into the modern notion of a curvature of a curve. A similar definition can be given for the curvature of a surface. But in higher dimensions a scalar measure of curvature isn't sufficient and tensors are required. This was Riemann's achievement after Gauss discovered his eponymous curvature. It's worth noting that calculus in Newton's hands actually began, as Garman & Bonnie note, with his study of curvature rather than that of motion as is commonly assumed.