III. An Inquiry into the Geometrical Character of the Hour-Lines upon the Antique Sun-Dials

Abstract

Il resterait à résoudre une problème plus générale, à trouver la courbe hectémoreale sur la sphère même, et sur un cadran quelconque.Delambre, Conn. des Temps, an. 1820, p. 341.The nature of these lines, though a subject of repeated investigation, has never been accurately determined. The loci of the points which divide the semi-diurnal arc into n equal parts, have almost invariably been considered as great circles, and their projections upon a plane, which are the dial-lines in question, in consequence viewed as straight lines. Clavius was the first to point out, and to prove, the error of this opinion; and Montucla (or rather Lalande) not only denied that they were circles at all, but affirmed they were curves of a very fantastical kind—“tres bizarre.” This latter opinion, which is partly true and partly false, seems rather to have been inconsiderately hazarded, than derived from any satisfactory course of reasoning. They are not great circles, it is true; but, as Delambre has replied, “il y a beaucoup d'exageration à donner une forme tres bizarre à des lignes qui, dans aucun des cadrons qu'on a tracées de cette manière, n'ont jamais pas s'ecarter sensiblement de la ligne droite.”