Page:The Rhind Mathematical Papyrus, Volume I.pdf/146

130 Elementärsétze, die der allgemeinen Gestaltungslehre angehōren und von meinem verstorbenen Vater endeckt worden sind."

The father, F. G. Röber, was professor of architecture in the academy at Dresden. In studying the construction of the Temple at Edfu he conceived that he could explain it as being connected with the inscription of a regular heptagon in a circle, the first account of which is given on pages 15-16 of the above mentioned Beiträge. In this connection the following quotations may be made from a letter of William Rowan Hamilton, dated September 15, 1862, and addressed to the son-in-law of the Archbishop of Dublin.

"A wish to gratify the archbishop and yourself was the first motive for my attempting to examine to some extent the Essays of Röber which you had the goodness to leave for me a few days ago, and to form some opinion of their value, unimportant as that opinion might be. But the Memoir on the ancient temples of Egypt (Röber, Dresden, 1854) has interested me profoundly. Indeed I have scarcely been able, since I opened it, to attend to anything else; and it led me into some long calculations which I have only just completed to my satisfaction. As I have paid no special attention to Egyptian Antiquities, nor meditated much on such mystical guesses as some have made at their inner meaning, the only point which I could hope to study usefully was the geometrical discovery announced in the first memoir, namely 'the construction of the regular heptagon', which the elder Röber appears to have divined, from the study of the ancient Temple Architecture.

"I entered on the subject, perhaps with prejudice: for like most (if not all) modern geometers, I have been accustomed to hold, and indeed still do hold, that it is impossible to construct such a heptagon with the 'right line' and 'circle' alone. Yet to my great surprise. I found no error in Röber's numbers, and on repeating the calculations on another plan, with Taylor's seven-ﬁgure logarithms, I found myself quite unable to pronounce whether Röber's arc erred in excess or in defect from the exact seventh part of the circumference; for that it must err I felt assured.

"It seemed, therefore, worth while to go much more closely to work; and laying tables entirely aside, to perform the whole of the work for myself, by arithmetic alone, and especially by extractions of square roots. And to be quite sure of a high degree of accuracy in the final result, I made it a rule to work with not fewer than fifteen decimal places, besides employing all verifications that I could think of in the progress of calculation, which thus laboriously conducted, has covered many sheets of paper, and cost many hours on two or three successive days.

"At last, however, it is finished; and I should have no hesitation to commit myself publicly to the result, which is, technically expressed, that the natural cosines of the angle assigned by Röber's construction is, to thirteen decimals,