The Greek writers also cultivated the subject of Perspective speculatively, in mathematical treatises, as well as practically, in pictures. The whole of this theory is a consequence of the principle that vision takes place in straight lines drawn from the object to the eye.
“The ancients were in some measure acquainted with the Refraction as well as the Reflection of Light,” as I have shown in Book ix. Chap. 2 [2d Ed.] of the Philosophy. The current knowledge on this subject must have been very slight and confused; for it does not appear to have enabled them to account for one of the simplest results of Refraction, the magnifying effect of convex transparent bodies. I have noticed in the passage just referred to, Seneca’s crude notions on this subject; and in like manner Ptolemy in his Optics asserts that an object placed in water must always appear larger then when taken out. Aristotle uses the term ἀνακλάσις (Meteorol. iii. 2), but apparently in a very vague manner. It is not evident that he distinguished Refraction from Reflection. His Commentators however do distinguish these as διακλάσις and ἀνακλάσις. See Olympiodorus in Schneider’s Eclogæ Physicæ, vol. i. p. 397. And Refraction had been the subject of special attention among the Greek Mathematicians. Archimedes had noticed (as we learn from the same writer) that in certain cases, a ring which cannot be seen over the edge of the empty vessel in which it is placed, becomes visible when the vessel is filled with water. The same fact is stated in the Optics of Euclid. We do not find this fact explained in that work as we now have it; but in Ptolemy’s Optics the fact is explained by a flexure of the visual ray: it is [103] noticed that this flexure is different at different angles from the perpendicular, and there is an elaborate collection of measures of the flexure at different angles, made by means of an instrument devised for the purpose. There is also a collection of similar measures of the refraction when the ray passes from air to glass, and when it passes from glass to water. This part of Ptolemy’s work is, I think, the oldest extant example of a collection of experimental measures in any other subject than astronomy; and in astronomy our measures are the result of observation rather than of experiment. As Delambre says (Astron. Anc. vol. ii. p. 427), “On y voit des expériences de physique bien faites, ce qui est sans exemple chez les anciens.”
Ptolemy’s Optical work was known only by Roger Bacon’s references to it (Opus Majus, p. 286, &c.) till 1816; but copies of Latin translations of it were known to exist in the Royal Library at Paris, and in the Bodleian at Oxford. Delambre has given an account of the contents of the Paris copy in his Astron. Anc. ii. 414, and in the Connoissance des Temps for 1816; and Prof. Rigaud’s account of the Oxford copy is given in the article Optics, in the Encyclopædia Britannica. Ptolemy shows great sagacity in applying the notion of Refraction to the explanation of the displacement of astronomical objects which is produced by the atmosphere,—Astronomical Refraction, as it is commonly called. He represents the visual ray as refracted in passing from the ether, which is above the air, into the air; the air being bounded by a spherical surface which has for its centre “the centre of all the elements, the centre of the earth;” and the refraction being a flexure towards the line drawn perpendicular to this surface. He thus constructs, says Delambre, the same figure on which Cassini afterwards founded the whole of his theory; and gives a theory more complete than that of any astronomer previous to him. Tycho, for instance, believed that astronomical refraction was caused only by the vapors of the atmosphere, and did not exist above the altitude of 45°.
Cleomedes, about the time of Augustus, had guessed at Refraction, as an explanation of an eclipse in which the sun and moon are both seen at the same time. “Is it not possible,” he says, “that the ray which proceeds from the eye and traverses moist and cloudy air may bend downwards to the sun, even when he is below the horizon?” And Sextus Empiricus, a century later, says, “The air being dense, by the refraction of the visual ray, a constellation may be seen above the horizon when it is yet below the horizon.” But from what follows, it [104] appears doubtful whether he clearly distinguished Refraction and Reflection.
In order that we may not attach too much value to the vague expressions of Cleomedes and Sextus Empiricus, we may remark that Cleomedes conceives such an eclipse as he describes not to be possible, though he offers an explanation of it if it be: (the fact must really occur whenever the moon is seen in the horizon in the middle of an eclipse:) and that Sextus Empiricus gives his suggestion of the effect of refraction as an argument why the Chaldean astrology cannot be true, since the constellation which appears to be rising at the moment of a birth is not the one which is truly rising. The Chaldeans might have answered, says Delambre, that the star begins to shed its influence, not when it is really in the horizon, but when its light is seen. (Ast. Anc. vol. i. p. 231, and vol. ii. p. 548.)
It has been said that Vitellio, or Vitello, whom we shall [hereafter] have to speak of in the history of Optics, took his Tables of Refractions from Ptolemy. This is contrary to what Delambre states. He says that Vitello may be accused of plagiarism from Alhazen, and that Alhazen did not borrow his Tables from Ptolemy. Roger Bacon had said (Opus Majus, p. 288), “Ptolemæus in libro de Opticis, id est, de Aspectibus, seu in Perspectivâ suâ, qui prius quam Alhazen dedit hanc sententiam, quam a Ptolemæo acceptam Alhazen exposuit.” This refers only to the opinion that visual rays proceed from the eye. But this also is erroneous; for Alhazen maintains the contrary: “Visio fit radiis a visibili extrinsecus ad visum manantibus.” (Opt. Lib. i. cap. 5.) Vitello says of his Table of Refractions, “Acceptis instrumentaliter, prout potuimus propinquius, angulis omnium refractionum . . . invenimus quod semper iidem sunt anguli refractionum: . . . secundum hoc fecimus has tabulas.” “Having measured, by means of instruments, as exactly as we could, the whole range of the angles of refraction, we found that the refraction is always the same for the same angle; and hence we have constructed these Tables.” ~Additional material in the [3rd edition].~ [105]
CHAPTER III.
Earliest Stages of Harmonics.
AMONG the ancients, the science of Music was an application of Arithmetic, as Optics and Mechanics were of Geometry. The story which is told concerning the origin of their arithmetical music, is the following, as it stands in the Arithmetical Treatise of Nicomachus.
Pythagoras, walking one day, meditating on the means of measuring musical notes, happened to pass near a blacksmith’s shop, and had his attention arrested by hearing the hammers, as they struck the anvil, produce the sounds which had a musical relation to each other. On listening further, he found that the intervals were a Fourth, a Fifth, and an Octave; and on weighing the hammers, it appeared that the one which gave the Octave was one-half the heaviest, the one which gave the Fifth was two-thirds, and the one which gave the Fourth was three-quarters. He returned home, reflected upon this phenomenon, made trials, and finally discovered, that if he stretched musical strings of equal lengths, by weights which have the proportion of one-half, two-thirds, and three-fourths, they produced intervals which were an Octave, a Fifth, and a Fourth. This observation gave an arithmetical measure of the principal Musical Intervals, and made Music an arithmetical subject of speculation.
This story, if not entirely a philosophical fable, is undoubtedly inaccurate; for the musical intervals thus spoken of would not be produced by striking with hammers of the weights there stated. But it is true that the notes of strings have a definite relation to the forces which stretch them; and this truth is still the groundwork of the theory of musical concords and discords.