The hour lines on the sundials of the ancient Greeks and Romans correspond to the division of the time between sun rise and sun-set into twelve equal parts, which was their mode of computing time. An example of these hour lines occurs in an ancient Greek sundial, forming part of the Elgin collection of marbles at the British Museum, and which there is reason to believe had been constructed during the reign of the Antonines. This dial contains the twelve hour lines drawn on two vertical planes, which are inclined to each other at an angle of 106°; the line bisecting that angle having been in the meridian. The hour lines actually traced on the dial consist of such portions only as were requisite for the purpose the dial was intended to serve: and these portions are sensibly straight lines. But the author has shown, in a paper published in the Transactions of the Royal Society of Edinburgh, that if these lines are continued through the whole zone of the rising and setting semidiurnal arcs, they will be found to be curves of double curvature on the sphere. In the present paper the author enters into an investigation of the course of these curves; first selecting as an example the lines indicating the 3rd and the 9th hours of the ancients. These lines are formed by the points of bisection of all the rising and setting semidiurnal arcs; commencing from the southern point where the meridian cuts the horizon, and proceeding till the line reaches to the first of the always apparent parallels, which, being a complete circle, it meets at the end of its first quadrant. At this point the branch of another and similar curve is continuous with it: namely, a curve which in its course bisects another set of semidiurnal arcs, belonging to a place situated on the same parallel of latitude as the first, but distant from it 180° in longitude. Continuing to trace the course of this curve, along its different branches, we find it at last returning into itself, the whole curve being characterized by four points of flexure. If the describing point be considered as the extremity of a radius, it will be found that this radius has described, in its revolution, a conical surface with two opposite undulations above, and two below the equator. The right section of this cone presents two opposite hyperbolas between asymptotes which cross one another at right angles This cone varies in its breadth in different positions of the sphere; diminishing as the latitude of the place increases. The cones to which the other ancient hour lines belong, are of the same description, having undulations alternately above and below the equator; but they differ from one another in the number of the undulations: and some of these require more than one revolution to complete their surface. The properties of the cones and lines thus generated, may be rendered evident by drawing the sections of the cones on the sphere, in perspective, either on a cylindrical or on a plane surface: several examples of which are given in the paper.
Survey of entomopathogenic nematodes and fungi in agricultural areas
