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How did Robert Hooke’s experiments lead to Hooke’s law?

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mechanics of solids Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).

Robert Hooke discovered his law of elasticity in 1660 by observing the behavior of solid bodies under various forces, such as stretching, compressing, squeezing, bending, or twisting. Through his experiments, he determined that for relatively small deformations, the displacement or change in size of an object is directly proportional to the deforming force applied to it.12

This relationship is mathematically expressed as $F = k x$ , where $F$ is the applied force, $x$ is the displacement, and $k$ is a constant that depends on the material's properties, shape, and dimensions. Hooke’s work demonstrated that within this proportional range, an object will return to its original shape and size once the load is removed.2

These findings laid the essential foundation for the study of stress and strain in the mechanics of solids. Hooke famously applied these principles to practical engineering, most notably in his designs for the balance springs of watches.1