It has been said that Vitellio, or Vitello, whom we shall have 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), "Ptolemeus in libro de Opticis, id est, de Aspectibus, seu in Perspectivâ sua, 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." ६ (G.) p. 110. Nicomachus says that Pythagoras found the weights to be, as I have mentioned, in the proportion of 12, 6, 8, 9; and the intervals, an. Octave, corresponding to the proportion 12 to 6, or 2 to 1; a Fifth, corresponding to the proportion 12 to 8, or 3 to 2; and a Fourth, corresponding to the proportion 12 to 9, or 4 to 3. There is no doubt that this statement of the ancient writer is inexact as to the physical fact, for the rate of vibration of a string, on which its note depends, is, other things being equal, not as the weight, but as the square root of the weight. But he is right as to the essential point, that those ratios of 2 to 1, 3 to 2, and 4 to 3, are the characteristic ratios of the Octave, Fifth and Fourth. In order to produce these intervals, the appended weights must be, not as 12, 9, 8, and 6, but as 12, 63, 5, and 3. The numerical relations of the other intervals of the musical scale, as well as of the Octave, Fifth and Fourth, were discovered by the Greeks. Thus they found that the proportion in a Major Third was 5 to 4; in a Minor Third 6 to 5; in a Major Tone 9 to 8; in a Semitone or Diesis 16 to 15. They even went so far as to determine the Comma, in which the interval of two notes is so small that they are in the proportion of 81 to 80. This is the interval between two notes each of which may be called the Seventeenth above the key-note;-the one note being obtained by ascending a Fifth four times over; the other being obtained by ascending through two Octaves and a Major Third. The want of coincidence between these two notes is an inherent arithmetical imperfection in the musical scale, of which the consequences are very extensive. The numerical properties of the musical scale were worked out to a very great extent by the Greeks, and many of their Treatises on this subject remain to us. The principal ones are the seven authors published by Meibomius*. These arithmetical elements of Music are to the present day important and fundamental portions of the Science of Harmonics. Τόδε δὲ μηδέις ποτε φοβηθῇ τῶν Ἑλλήνων, ὡς οὐ χρη περι τὰ θεῖα ποτὲ πραγματεύεσθαι θνητοὺς ὄντας· πᾶν δε τούτου διανοηθῆναι τοὐναντίον, ὡς οὔτε ἄφρον εστι ποτὲ το θεῖον, οὔτε ἀγνοεῖ που τὴν ἀνθρωπίνην φυσιν· ἀλλ ̓ οἶδεν ὅτι, διδάσκοντος αυτού, ξυνακαλουθήσει καὶ μαθήσεται τα διδάσκομενα. PLATO, Epinomis, p. 988. Nor should any Greek have any misgiving of this kind; that it is not fitting for us to inquire narrowly into the operations of superior Powers, such as those by which the motions of the heavenly bodies are produced: but, on the contrary, men should consider that the Divine Powers never act without purpose, and that they know the nature of man: they know that by their guidance and aid, man may follow and comprehend the lessons which are vouchsafed him on such subjects. INTRODUCTION. HE earliest and fundamental conceptions of THE men respecting the objects with which Astronomy is concerned, are formed by familiar processes of thought, without appearing to have in them anything technical or scientific. Days, Years, Months, the Sky, the Constellations, are notions which the most uncultured and incurious minds possess. Yet these are elements of the Science of Astronomy. The reasons why, in this case alone, of all the provinces of human knowledge, men were able, at an early and unenlightened period, to construct a science out of the obvious facts of observation, with the help of the common furniture of their minds, will be more apparent in the course of the philosophy of science; but I may here barely mention two of these reasons. They are, first, that the familiar act of thought, exercised for the common purposes of life, by which we give to an assemblage of our impressions such a unity as is implied in the above notions and terms, a Month, a Year, the Sky, and the like, is, in reality, an inductive act, and shares the nature of the processes by which all sciences are formed; and, in the next place, that the ideas appropriate to the induction in this case, are those which, even in |