But it is amazing how frequently one or another of them crops up even in the most recent work, and in places where one would least expect it. Even Sir Thomas Heath, the learned editor of Aristarchus, Archimedes, Diophantus, and Apollonius, and author of a monumental history of Greek mathematics, says in one place that science was short-lived among the Greeks.

We begin to see a good deal of the fantastic in the ordinary account of Greek science if, basing ourselves on the story of Syracuse and Archimedes, we ponder it with any sort of historic judgment and with any sort of discerning knowledge of the history of mankind. We reflect, almost at once, that the continuous, free, and yet unadulterate development of Syracuse was not without parallel in the Greek world; that the city of Cumæ, for one, enjoyed an even longer period under the same conditions. As we have said, it will not do to think of all Syracusan history as belonging to the history of science. Yet we remember that Hipparchus and Diophantus continued the scientific period a long while after the death of Archimedes. And as for the basis of the culture out of which science grew, we remember that Corinth and the fine culture of Samos existed before Syracuse began. Again, is it conceivable that a people sufficiently gifted to produce Archimedes, whom Sir Thomas Heath calls probably the greatest mathematical genius of all time, should not have produced scientific genius of another kind? Later we shall return to the connexion between mathematics and science. For men to undertake to write about the history of science and to be themselves so unscientific as to suppose that mathematics and other sciences dwell apart, and that a people can go on deducing, and never experimenting, for centuries--this is a thing only to be explained, I suppose, by the curse of specialisation which weighs on us. As to mechanical equipment, any boy who had read Plutarch's Life of Marcellus' would know that Archimedes at least was not stinted in that respect. A more serious student, reading Hippocrates on trepanning the skull, and his references to elaborate surgical instruments, or seeing the collection of such instruments in the museum in Naples, would be still more enlightened. But any one who attempts to work through the written

remains of Archimedes with the help of Heiberg or Heath, will understand that he was an endlessly precise inventor of laboratory equipment; and, further, that he and other Greeks had a numerical notation which, for the subtlest calculations, in no way fell short of our own.

As to religious persecution: should it not be sufficient to ask whether any people could escape this exceedingly human trait for six or seven or eight centuries? If the Greeks were so unlike ourselves as that, could we understand them, could we study them with profit? The truth is that many a Greek sighed, with Faust: Und leider auch Theologie.' Let one illustration do. So far as we know, Anaximander of Miletus, born about four centuries before the death of Archimedes, was the first Greek to teach evolution. He taught, as clearly as any modern, the possibility of change from species to species, and used some of the arguments afterwards used by Charles Darwin and A. R. Wallace to support his thesis. Like Darwin he had his T. H. Huxley to confute a certain type of theologian in his day. This was his pupil, Xenophanes, who like Huxley had a ready and somewhat satiric tongue, and whose favourite argument was one which Huxley often used-that every age is incurably anthropomorphic. Some of his fragmentary sayings are remarkably like bits of Huxley's 'Life of Hume.' Huxley did distinguished work on crustaceans. It is on record that Xenophanes examined fossilised shellfish at Paros, Malta, and Syracuse, and that he recognised an extinct sea-beach when he saw one.

The story of Thales of Miletus and Pythagoras of Samos has often been told; in the main, it must be said, with relatively too much attention to the former. Thales and many Greeks after him puzzled over the question which every child asks, Of what is the world made?' Thales answered, 'Water.' It has been pointed out that he lived on a large gulf, which since his time has become an alluvial plain, and that he had seen the Nile. If we credit him with greater subtlety we may imagine that he took the three forms of Water-solid, liquid, and invisible vapour-as a type merely of all kinds of chemical change, growth, and decay.

In Thales himself a ferment of thought was working. In two fields he had found treasure. In Babylon exceed

ingly careful records of astronomic phenomena had been kept; and this, with the Babylonian theory of the cycle of things, enabled their 'wise men' to predict eclipses. So Thales himself predicted an eclipse in 585 B.C. He may have done nothing original to achieve this. We cannot say anything about that, for his treatise on astronomy has not come down to us. But with the lore which Thales collected in Egypt it was certainly different. The Egyptians were at one time credited with profound mathematical lore; but recent writers, whose authority must be esteemed-Heiberg and Heath, for example-say that Egyptian mathematics was really a very limited and practical mensuration, connected with building and land surveying. From this Thales went off into real mathematics. He is credited, very generally, with the proof of five of the propositions occurring in the first six books of Euclid. He knew how to take the distance of a ship at sea from the height of a tower. Thales certainly founded a scientific school at Miletus and may be said to have provided a basis for European science. He, and his associate, Anaximander, in astronomy, biology, and physics, held that all that had been must have moved by the same processes as they then saw at work; that the past might be understood and described, and the future predicted in accordance with law. Above all, they suggested, and to a certain extent demonstrated, the mathematical relationship of some of these laws.

We must, however, not fail to see that Pythagoras, about sixty years younger than Thales, made a far greater bound forward. Pythagoras was born in the highly cultured civilisation of Samos, 580 B.C. Whereas Thales still boggled over material unity, Pythagoras left all that aside and looked for a unity in structure. He did not attempt explanation, he was satisfied with description. Moreover, though Pythagoras, like Thales, was a geometer-a much more advanced geometer, as we should expect he gave geometry a numerical content which was of the very greatest importance. He investigated the physics of sound and formulated the laws of musical harmony. It is easy to say that the numerical basis of the octave and of the right-angled triangle is a simple thing; but as we moderns have

recently discovered, the things that men really come to understand in chemistry, electricity, and everything else, have a numerical basis, and a beautiful simplicity.

It was the peculiar greatness of the school which Pythagoras founded in the west of Hellas-and as his authority was so reverently quoted we cannot doubt that he set the example himself—that they went on patiently observing, deducing, and proving in an abstract, scientific way. A Pythagorean proverb ran: σχῆμα καὶ βᾶμα, ἀλλ' οὐ σχῆμα καὶ τριώβολον. What a motto for Science! The above-drawn distinction between Pythagoras and his predecessors is not based on mere hints or insufficient records. The record is plain; and in it we see, first of all, that Pythagoras definitely directed men away from the question, 'What is matter?' and turned their attention to the question, 'What is matter like?' That, again, may seem to be a small thing. But it is worth remembering that it all had to be done over again for Europeans in the days of Leibnitz and Newton. Indeed, in many works on the history of philosophy it is said that one or other of these men first drew this very necessary distinction. Certainly the vehemence with which Leibnitz maintained that all description of nature would in the future be mechanical, whereas explanation might still continue to be spiritual, shows that in his generation the idea was unusual. And many will remember how Newton insists over and over again that he uses terms not as an explanation, but merely as a mathematical description.

There is, however, something even more modern about the Pythagoreans than this. It is their doctrine that number, though not indeed the explanation of the matter of the universe, might well prove to be the key to its structure. I know that many Pythagoreans went off into mystical nonsense at this point, men of sufficient authority to draw even Plato in their train. But that does not discredit the main theory, any more than the science of medicine is to-day discredited because some individuals who profess it, talk like primitive medicinemen about magical vaccines and mumble-jumble of one sort or another. The Pythagorean principle shone

A new diagram, that means a step forward; but we do not draw it to make a threepence.

clear throughout. And the glory and honour of the Pythagoreans increase as scientific knowledge increases. Think of Lavoisier with his balance; think of Coulomb with his passionate striving for the numerical formulæ of electricity; think of chemical equivalents, of Mendeléeff's law, of Gauss, of the goniometer and the science of crystallography! Listen to Liebig:

'To investigate theessence of a natural phenomenon three conditions are necessary: We must first study the phenomenon itself from all sides, we must then determine in what relation it stands to other phenomena, and lastly... we have to solve the problem of measuring these relations ..that is of expressing them in number.'

Once it was said that while this might be true of chemistry, and certain other sciences, its truth was limited to them, and that it could not be true of biology, for example. For a long time, however, metabolistic theories of life have pointed biology in the same direction. And the life-work of Sir Jagadis Bose, crowned by his discovery of the pulse of trees, can hardly be of isolated importance.

Before passing on to examine the purely mathematical development of Pythagoreanism, and in order to be chronological, let us turn aside from mathematics proper, and take a general survey of the condition of science after Thales and Pythagoras, or rather before the death of the latter. We come now to a period of Greek development which abounds in written records. We need not speculate as to what the Greeks may have meant or thought. We can follow suggestions made by certain individuals, criticised by many others, and finally wrought out, often by collaboration, to a scientific law which still stands. We know how political disturbances made one part of the Greek world cease, for a while, to be the home of scientific thought. We can see science rise and fall from period to period according as one generation continued to be rationalistic or gave itself up to superstition. The Greeks, too, had their superstitious moods, and had always some intolerant folk among them, being human like ourselves. We can see too how freedom of trade and freedom of intercourse made for the growth of scientific thought; whereas a