[25] God--says Herodotus in the famous dialogue of Xerxes and Artabanus on the fugitive character of human happiness--God, that has given to man a sweet taste of life, is found grudging in his dole. [26] Nothing is more striking in the old churches of such towns as Lüneburg, Wismar, Rostock, than the representation of the Crucifixion on the carved wooden work used as the reredos in various chapels as well as over the high altar. The main figures are in high relief, the crowd standing in front of them in the clear, all coloured richly and in many colours as well as gilding. But for the painful subject, these triumphs of the carver are perhaps the most splendid in mediæval art. The magnificent tombs of the Dukes of Burgundy at Dijon are their worthy rivals in stone. [27] The foaming heads of the horses of the rising sun at the left corner of the east pediment of the Parthenon are a splendid exception to this calm in the angles of the pediment, and the subdued calmness of the horses of Selene, which represent her setting, accentuates this exception. [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. [Illustration: graphic] here _b_² = 2_a_² ∴ _a_ is odd and _b_ even. Now let _b_ = 2_c_ then 4_c_² = 2_a_² and 2_c_² = _a_² ∴ _a_ is even and _b_ odd, which is absurd. [34] Readers who wish to prosecute this subject further will find the best exposition of it in Prof. Smyly’s paper in the volume of Essays dedicated to Prof. Nicole of Geneva. [35] Heron, περἰ Αὐτοματοποιητικῆς, caps. iii. and iv. [36] See my _Epoch of Irish History_, last chapter. [37] His theory is laid down in several now forgotten books brimful of learning. It is the theory of four fundamental harmonies, or elements, the relations of which produce in every body health and disease. [38] πάρ γἀρ έμοἰ θάνατος,, said Agamemnon, according to the copy which Aristotle quotes. [39] ἐπἰ ῤήτοις γέρασι πατριας βασιλείας (Pol.).