Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrationals [or 'proportions'] and the construction of the cosmic figures.

According to the account of Proclus (Book II. c. 4 ), Pythagoras was the first who gave to Geometry the form of a deductive science, by shewing the connexion of the geometrical truths then known, and their dependence on certain first principles. ...The traditionary account, that Pythagoras was the founder of scientific mathematics, is in some degree, supported by the statement of Diogenes Laertius, that he was chiefly occupied with the consideration of the properties of number, weight, and extension, besides music and astronomy. The passage of Cicero (De Nat. Deor. III. 36) may be referred to as evidence that later writers were unable to give any precise account of the mathematical discoveries of Pythagoras. To Pythagoras, however, is attributed the discovery of some of the most important elementary properties contained in the first book of Euclid's Elements. The very important truth contained in Prop. 47, Book I. is also ascribed to Pythagoras. ...Proclus attributes to him the discovery of that right-angled triangle, the three sides of which are respectively 3, 4, and 5 units. To Pythagoras also belongs the discovery, that there are only three kinds of regular polygons which can be placed so as to fill up the space round a point; namely, six equilateral triangles, four squares, and three regular hexagons. Proclus attributes to him the doctrine of incommensurables, and the discovery of the five regular solids, which, if not due to Pythagoras, originated in his school. In Astronomy he is reputed to have held, that the Sun is the centre of the system, and that the planets revolve round it. This has been called, from his name, the Pythagorean System, which was revived by Copernicus, A.D.1541, and proved by Newton.

Such things taught he, though advising above all things to speak the truth, for this alone deifies men. For as he had learned from the Magi, who call God Oremasdes, God's body is light, and his soul is truth. He taught much else, which he claimed to have learned from Aristoclea at Delphi.

He ordained that his disciples should speak well and think reverently of the Gods, muses and heroes, and likewise of parents and benefactors; that they should obey the laws; that they should not relegate the worship of the Gods to a secondary position, performing it eagerly, even at home; that to the celestial divinities they should sacrifice uncommon offerings; and ordinary ones to the inferior deities. (The world he Divided into) opposite powers; the "one" was a better monad, light, right, equal, stable and straight; while the "other" was an inferior duad, darkness, left, unequal, unstable and movable.

The following became universally known: first, that he maintains that the soul is immortal; second, that it changes into other kinds of living things; third, that events recur in certain cycles and that nothing is ever absolutely new; and fourth, that all living things should be regarded as akin. Pythagoras seems to have been the first to bring these beliefs into Greece.

It was through philosophy, he said, that he had come to be surprised at nothing.

The votaries of Pythagoras of Samos have this story to tell of him, that he was not an Ionian at all, but that, once on a time in Troy, he had been Euphorbus, and that he had come to life after death, but had died as the songs of Homer relate. And they say that he declined to wear apparel made from dead animal products and, to guard his purity, abstained from all flesh diet, and from the offering of animals in sacrifice. For that he would not stain the altars with blood; nay, rather the honey-cake and frankincense and the hymn of praise, these they say were the offerings made to the Gods by this man, who realized that they welcome such tribute more than they do the hecatombs note and the knife laid upon the sacrificial basket. For they say that he had of a certainty social intercourse with the gods, and learnt from them the conditions under which they take pleasure in men or are disgusted, and on this intercourse he based his account of nature.

What appeared here, at the center of the Pythagorean tradition in philosophy, is another view of psyche that seems to owe little or nothing to the pan-vitalism or pan-deism (see theion) that is the legacy of the Milesians.

Ah, Pythagoras' metempsychosis, were that true,This soul should fly from me, and I be changedUnto some brutish beast!All beasts are happy, for when they die,Their souls are soon dissolved in elements;But mine must live still to be plagued in hell.

It may perhaps help us to realize the human side of our Masters if we remember that many of Them in comparatively recent times have been known as historical characters. The Master K.H. for example, appeared in Europe as the philosopher Pythagoras. Before that He was the Egyptian priest Sarthon, and on yet another occasion chief-priest of a temple at Agade, in Asia Minor, where He was killed in a general massacre of the inhabitants by a host of invading barbarians who swooped down upon them from the hills