Hooke's cubico–parabolical conoid

In 1675 Robert Hooke published, as one of his ‘Inventions’, a Latin anagram concerning the ‘true...form of all manner of arches for building’. His discovery was that the shape of a light flexible cord subjected to specified loads would, when inverted, give the required shape of the perfect (masonry) arch to carry those same loads. Hooke knew that the catenary curve was not given by the parabola y = ax2, but he was unable to solve the problem mathematically, and the decipherment of the anagram was not published until after his death. Four years earlier Hooke had stated to the Royal Society that the solution to the corresponding three–dimensional problem, that of the shape of the perfect dome, was the cubico–parabolical conoid; that is, the dome was formed by rotating the cubic parabola y = ax3 about the y–axis. It is shown that the correct form of dome may be evaluated in terms of the integrals erf(t) and erg(t). Moreover, an alternative solution as a power series is rapidly convergent, and has a leading term in x3 followed by a much smaller term in x7. Wren's design for the dome of St Paul's Cathedral made use of the idea of Hooke's ‘hanging chain’.