NOTES TO BOOK If. 117 



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 Perspec- 

 tiva sua, qui prius quam Alhazen dedit hanc sententiam, 

 quam a Ptolemaeo acceptam Alhazen exposuit." This re- 

 fers 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." 



(G.) p. 110. Nicomachus says that Pythagoras found 

 the weights to be, as I have mentioned, in the pro- 

 portion of J 2, 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 de- 

 pends, 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, 



