Proportionality law for elastic forces is one discovery at which Hooke did arrive first. He describes it in De Potentia Restitutiva (1678):
”The Power of any Spring is in the same proportion with the Tension thereof: That is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward. Now as the Theory is very short, so the way of trying it is very easie. […] it is very evident that… in every springing body… the force or power thereof to restore itself to its natural position is always proportionate to the Distance or space it is removed therefrom.”
However, Hooke does not differentiate between Boyle’s law for gases and the law for elastic bodies, his “springing body” in the quote can represent either, and his surmise is clearly inspired by the former (he worked as Boyle’s assistant in 1659). And his theoretical justification for the elastic case specifically was erroneous. He modeled the spring as a row of boxes containing vibrating microscopic particles. At the equilibrium the tendency of the particles to recede from each other is balanced by the external pressure from the surrounding subtle matter. The rate of particles' collisions with their neighbors accounts for the restoring force of springs. But according to Moyer’s Robert Hooke's Ambiguous Presentation of “Hooke's Law” his model produces an inverse square dependence, which he confuses with the observed proportionality.
“Something about Huygens” probably refers to Hooke’s dispute with him which did involve springs, it was over the priority for inventing balance-spring watches. The affair goes back to 1665 when Huygens got patent for his famous isochronous pendulum clock employing cycloidal arcs. Hooke disputed the originality of the invention and offered his own coiled spring watch as prior art:
“I my self had an other way of continuing and equalling the vibrations of a pendulum by clock work long before I heard of Monsier Zulichems way, nay through equated with a Cycloeid yet I have not either cryd eureka or publisht it and yet I think I can produce a sufficient number of Credible witnesses that can testify for it about these 12 years. Soe that the argument that he soe much Relys upon to secure to him the Invention is not of soe great force as to preswaid all the World that he was the first & sole inventor of that first particular of applying a pendulum to a clock.”
Fast forward to 1675 when Huygens published the description of his balance-spring watch in French Academy’s Journal des Sçavans. It appears that between 1665 and 1675 Royal Society’s Oldenburg and Moray shared Hooke’s balanced-spring designs with Huygens, conveniently off the record, and Hooke even accused Oldenburg of “spying” for Huygens. Nonetheless, the Royal Society ultimately sided with Huygens in 1676, formally rebuked Hooke, and commended "Oldenburg's fidelity and honesty ‘in the management of the intelligence of the Royal Society'".
Hooke was really good at making enemies, but one productive thing that came out of the affair was that it turned Hooke’s interest to the restoring force of springs, although it does not seem that he realized the relation between its proportionality and the isochrony of balance-spring watches. See Horibe’s Robert Hooke, Hooke's Law & the Watch Spring for the full story and design pictures.
