Hooke discovered that the catenary is the ideal curve for an arch of uniform density and thickness which supports only its own weight. When the centerline of an arch is made to follow the curve of an up-side-down (i.e. inverted) catenary, the arch endures almost pure compression, in which no significant bending moment occurs inside the material. If the arch is made of individual elements (e.g., stones) whose contacting surfaces are perpendicular to the curve of the arch, no significant shearforces are present at these contacting surfaces. (Shear stress is still present inside each stone, as it resists the compressive force along the shear sliding plane.) The thrust (including the weight) of the arch at its two ends is tangent to its centerline.

It was Robert Hooke who discovered that the line of an arch, for supporting any weight assigned, should be the inversion of the shape of a catenary, or hanging chain, which is bearing that weight. He apparently announced that he had made the discovery to the Royal Society of London around 1671, but he did not provide any details until 1675, and then the details were encrypted. In an appendix to his Description of Helioscopes, he stated that he had found “a true mathematical and mechanical form of all manner of Arches for Building,” and the solution was: “abcccddeeeeeefggiiiiiiii-illmmmmnnnnnooprrsssttttttuuuuuuuux.” Unlike Hooke’s law of the spring, which he announced with a similar anagram, Hooke did not provide a translation in his lifetime, but it was provided by his executor in 1705: “Ut pendet continuum flexile, sic stabit contiguum rigidum inversum–As hangs a flexible cable, so inverted, stand the touching pieces of an arch.”

We display the page showing Hooke’s announcement; the “law of the arch” anagram is in paragraph 2. Paragraph 3, which follows immediately, contains the much more succinct anagram for Hookes’ law of springs: “ceiiinossssttuu,” or “ut tensio, sic vis as the deflection, so is the force.”

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