The arithmetical and geometrical view of nature, to which such exclusive supremacy is here given by the Pythagoreans, is one of the most interesting features of Grecian philosophy. They were the earliest cultivators of mathematical science,[45] and are to be recognised as having paved the way for Euclid and Archimedes, notwithstanding the symbolical and mystical fancies with which they so largely perverted what are now regarded as the clearest and most rigorous processes of the human intellect. The important theorem which forms the forty-seventh Proposition of Euclid's first book, is affirmed to have been discovered by Pythagoras himself: but how much progress was made by him and his followers in the legitimate province of arithmetic and geometry, as well as in the applications of these sciences to harmonics,[46] which they seem to have diligently cultivated, we have not sufficient information to determine with certainty. [Footnote 45: Aristot. Metaph. A. 5, p. 985, b. 23. [Greek: oi( Puthagorei=oi tô=n mathêma/tôn a)psa/menoi _prô=toi tau=ta proê/gagon_, kai\ e)ntraphe/ntes e)n au)toi=s ta\s tou/tôn a)rcha\s tô=n o)/ntôn a)rcha\s ô)|ê/thêsan ei)=nai pa/ntôn.]] [Footnote 46: Concerning the Pythagorean doctrines on Harmonics, see Boeckh's Philolaus, p. 60-84, with his copious and learned comments.] [Side-note: Eleatic philosophy--Xenophanes.] Contemporary with Pythagoras, and like him an emigrant from Ionia to Italy, was Xenophanes of Kolophon. He settled at the Phokæan colony of Elea, on the Gulf of Poseidonia; his life was very long, but his period of eminence appears to belong (as far as we can make out amidst conflicting testimony) to the last thirty years of the sixth century B.C. (530-500 B.C.). He was thus contemporary with Anaximander and Anaximenes, as well as with Pythagoras, the last of whom he may have personally known.[47] He composed, and recited in person, poems--epic, elegiac, and iambic--of which a very few fragments remain. [Footnote 47: Karsten. Xenophanis Fragm., s. 4, p. 9, 10.] [Side-note: His censures upon the received Theogony and religious rites.]