0.6234900759241, whereas the true cosine of the seventh part of four right angles deduced to a corresponding accuracy from a known cubic equation I find to be a little less, namely, 0.6234898018587. Admitting, though I do not believe it, that the two or three last of these decimals may be wrong after all the precautions taken, I am quite satisfied that the cosine of the Egyptian Angle—for really Röber seems to make it likely that the Egyptians did employ it—is somewhat greater, and that the cosine of the true or geometrical angle (for of course we can in geometrical conception divide the circumference into any number of equal parts) is somewhat less than 0.62349, and consequently that the supposed Egyptian rule of the heptagon is not mathematically perfect, though Röber seems to suppose it to be so. . . .
"But now let us turn to the tables and inquire how near does the supposed ancient rule come to the truth? How small in practice is the error which theory pronounces to exist? And I answer that in practice the error does not exist at all. I do not think that experiments of measurement, &c., could be so conducted by men, at least in the present age, as to prove to sight that there was any error. For practical purposes, then, the elder of the Robers, or the old Egyptian sage whose secrets he supposed himself to have divined, has done the impossible. . . Yet the practical success of the rule is to me absolutely wonderful: and it is long since any discovery in science produced in me such a sensation of surprise."
In the same month that this letter was written one finds that Hamilton and De Morgan refer to the matter in correspondence. In Hamilton's paper “On Röber's construction of the heptagon," Philosophical Magazine, series 4, vol. 27, 1864, pp. 124-132, he describes Röber's diagram as "not very complex and may even be considered elegant;" he then proceeds to indicate the derivation of the various quadratic equations leading to the result.[1]
F. Röber is also the author of Die aegyptischen Pyramiden in ihren ursprünglichen Bildungen, nebst einer Darstellung der Proportionalen Verhdltnisse im Parthenon zu Athen, Dresden, 1855, 6 + 28pp. + 1 plate. The discussion refers to the regular heptagon as fundamental.
Among other early publications in this connection perhaps the most notable are those of John Taylor: The Great Pyramid, why was it built? & Who built it? London. 1859; second edition, 1864, 327pp.; an appendix, issued also as a separate publication, was The Battle of the Standards: the ancient, of four thousand years, against the modern, of the last fifty years—the less perfect of the two, London, 1864, 79pp. Of the first, De Morgan, Budget of Paradoxes, London, 1872, p. 311, remarks that it is a "very learned" work which "may be referred to for the history of previous speculations. It professes to connect the dimensions of the Pyramid with a system of metrology which is supposed to have left strong traces in the systems of modern times; showing the Egyptians to have had good approximate knowledge of the dimensions of the earth, and of the quadrature of the circle. These are points on which coincidence is hard to distinguish from intention. Sir John Herschel noticed this work and gave several
- ↑ More details in this connection are given in my article on "Problems discussed by Huygens" in American Mathematical Monthly, vol. 28, 1921, pp. 477-479.
