heat, and this use of it was greatly developed by Rankine
in his work on the steam engine.The use of diagrams in thermodynamics has been very completely illustrated by Prof. J. Willard Gibbs (Connecticut Acad. Sci., vol. iii.), but though his methods throw much light on the general theory of diagrams as a method of study, they belong rather to thermodynamics than to the present subject.
DIALLING
DIALLING, sometimes called gnomonics, is a branch of applied mathematics which treats of the construction of sun-dials, that is, of those instruments, either fixed or portable, which determine the divisions of the day by the motion of the shadow of some object on which the sun s rays fall.
It must have been one of the earliest applications of a knowledge of the apparent motion of the sun ; though for a long time men would probably be satisfied with the division into morning and afternoon as marked by sun-rise, sun-set, and the greatest elevation.
History.—The earliest mention of a sun-dial is found in Isaiah xxxviii. 8 : " Behold, I will bring again the shadow of the degrees which is gone down in the sun-dial of Ahaz ten degrees backward." The date of this would be about 700 years before the Christian era, but we know nothing of the character or construction of the instrument.
The earliest of all sun-dials of which we have any certain knowledge was the hemicycle, or hemisphere, of the Chaldean astronomer Berosus, who probably lived about 340 B.C. It consisted of a hollow hemisphere placed with its rim perfectly horizontal, and having a bead, or globule, fixed in any way at the centre. So long as the sun remained above the horizon the shadow of the bead would fall on the inside of the hemisphere, and the path of the shadow during the day would be approximately a circular arc. This arc, divided Into twelve equal parts, deter mined twelve equal intervals of time for that day. Now, supposing this were done at the time of the solstices and equinoxes, and on as many intermediate days as might be considered sufficient, and then curve lines drawn through the corresponding points of division of the different arcs, the shadow of the bead falling on one of these curve lines would mark a division of time for that day, and thus we should have a sun-dial which would divide each period of daylight into twelve equal parts.
These equal parts were called temporary hours; and, since the duration of daylight varies from day to day, the temporary hours of one day would differ from those of another ; but this inequality would probably be disregarded at that time, and especially in countries where the variation between the longest summer day and the shortest winter day is much less than in our climates.
The dial of Berosus remained in use for centuries. The Arabians, as appears from the work of Albategnius, still followed the same construction about the year 900 A.D. Four of these dials have in modern times been found in Italy. One, discovered at Tivoli in 1746, is supposed to have belonged to Cicero, who, in one of his letters, says that he had sent a dial of this kind to his villa near Tusculum. The second and third were found in 1751—one at Castel-Nuovo, and the other at Rignano; and a fourth was found in 1762 at Pompeii. G. H. Martini, the author of a dissertation in German on the dials of the ancients, says that this dial was made for the latitude of Memphis ; it may therefore be the work of Egyptians, perhaps constructed in the school of Alexandria.
It is curious that no sun-dial has been found among the antiquities of Egypt, and their sculptures give no indication of any having existed. It has, however, been supposed that the numerous obelisks found everywhere were erected in honour of the sun and employed as gnomons.
Herodotus has recorded that the Greeks derived from the Babylonians the use of the gnomon, but the great pro gress made by the Greeks in geometry enabled them in later times to construct dials of great complexity, some of which remain to us, and are proofs, not only of extensive knowledge, but also of great ingenuity.
Ptolemy's Syntaxis treats of the construction of dials by means of his analemma, an instrument which solved a variety of astronomical problems. The constructions given by him were sufficient for regular dials, that is, horizontal dials, or vertical dials facing east, west, north, or south, and these are the only ones he treats of. It is certain, however, that the ancients were able to construct declining dials, as is shown by that most interesting monument of ancient gnomonics the Tower of the Winds which is still in existence at Athens. This is a regular octagon, on the faces of which the eight principal winds are represented, and over them eight different dials four facing the cardinal points and the other four facing the intermediate directions. The date of the dials is long subsequent to that of the tower ; for Vitruvius, who describes the tower in the sixth chapter of his first book, says nothing about the dials, and as he has described all the dials known in his time, we must believe that the dials of the tower did not then exist. The tower and its dials are described by Stuart in his Antiquities of Athens. The hours are still the temporary hours, or, as the Greeks called them, hedemoria.
As already stated, the learning and ingenuity of the Greeks enabled them to construct dials of various forms—among others, dials of suspension intended for travellers; but these are only spoken of and not explained; they may have been like our ring-dials.
The Romans were neither geometers nor astronomers, and the science of gnomonics did not flourish among them. The first sun-dial erected at Rome was in the year 290 B.C., and this Papirius Cursor had taken from the Samnites. A dial which Valerius Messala had brought from Catania, the latitude of which is five degrees less than that of Rome, was placed in the forum in the year 261 B.C. The first dial actually constructed at Rome was in the year 164 B.C., by order of Q. Marcius Philippus, but, as no other Roman has written on gnomonics, this was perhaps the work of a foreign artist. If, too, we remember that the dial found at Pompeii was made for the latitude of Memphis, and consequently less adapted to its position than that of Catania to Rome, we may infer that mathematical knowledge was not cultivated in Italy.
The Arabians were much more successful. They attached great importance to gnomonics, the principles of which they had learned from the Greeks, but they greatly simplified and diversified the Greek constructions. One of their writers, Abul-Hassan, who lived about the beginning of the 13th century, taught them how to trace dials on cylindrical, conical, and other surfaces. He even intro duced equal or equinoctial hours, but the idea was not supported, and the temporary hours alone continued in use.
conceived by Abul-Hassan, and perhaps by others, of reckoning by equal hours was generally adopted cannot now be determined. The history of gnomonics from the 13th to the beginning of the 16th century is almost a
blank, and during that time the change took place. We
