"The arch is one of those brilliant innovations... Spanning... with horizontal beams is a losing game. ...By converting all the stress that fractures the middle of... stone beams—technically tension—into compression on stone piers larger... spaces could be spanned. ...But shift the pressure even slightly off center, and the pillar is likely to collapse. ...In their early incarnations, the limitations of both arch and dome was the ability of craftsmen to shape the stones carefully enough to create blocks precisely in the wedge shapes needed for a particular arch. Despite their mathematical sophistication in most other respects, the architects of antiquity lacked a proper geometric solution to the ideal form of the arch. (It was not until 1675 that the English polymath Robert Hooke described mathematically the shape of an arch loaded in pure compression, that is, with no tension, by showing how it describes an upside-down version of the catenary curve of a hanging chain.) As a result, the only way they could design an arch, and its component stones, was completely by eye, and... such tolerances commanded high prices. Rome overcame this drawback with typical ingenuity, first replacing stones and mortar... and expensive stonecutters with relatively cheap bricklayers. Even more ingeniously, some anonymous Roman builder found how to combine the mortar—in Latin pulvis puteoli—with lime, sand, and gravel to make the first concrete. ...The concrete domes of Rome were not surpassed until the age of steel."