[28] This I tell you from the first-hand evidence of my son who was for years in the Solomon Islands and saw it done a hundred times by natives, but never even after desperate efforts by any European.
[29] A curious exception is now to be made for the panels with portraits of the dead found on mummy cases of the first and second centuries in Roman Egypt.
[30] And yet there were several ignorant and random attempts made to reproduce it with modern harmonies.
[31] This is the view advocated in the Tract on Music by Philodemus, large fragments of which were recovered on a charred papyrus from Herculaneum. There is another fragment of a similar character recently discovered and printed by Messrs. Grenfell and Hunt.
[32] Viz.: 1 + 3 = 2²²; 1 + 3 + 5 = 3²; 1 + 3 + 5 + 7 = 4², and so on.
[33] There is even an arithmetical proof mentioned by Aristotle that the ratio of side to diagonal of a square cannot be one of whole numbers.
If it were, the ratio will be that of two numbers in its lowest terms, and hence one must be even and the other odd, else both were still divisible by 2.
here b² = 2a² ∴ a is odd and b even.
Now let b = 2c then 4c² = 2a² and 2c² = a² ∴ a is even and b odd, which is absurd.