period of petty nationalism and exclusiveness choked or soured it; and all this was going on, as I have said at the beginning, over a period of hundreds of years.

The earliest science of which we have any record among the Greeks is that of medicine and surgery, and about the time of Pythagoras, or soon afterward, this became a very scientific study indeed. In Sicily, Empedocles enjoyed a considerable reputation as a physician. Philosopher and poet as well, he was a sort of Sicilian Goethe in the influence he wielded both on philosophy and science. By this time, however, the physicians were a well-established guild all over the Greek world, and in the writings of Hippocrates, still extant, there is a corpus of genuinely scientific literature which is remarkably free from dross of any kind. Modern physicians still exclaim at Hippocrates' accurate diagnosis of certain types of fever, of pneumonia and phthisis; at his skill in dealing with empyema and stone, in his surgical operations on the skull, and in all types of dislocations. The use of certain drugs, the use of wine for washing wounds, the insistence on cleanliness, quiet, simple regimen, and above all on the duty of the physician to assist the curative powers of Nature-nothing could be more scientific. The surgeon, Francis Adams, who translated Hippocrates seventy-five years ago, wrote that in his day there existed nothing so good as Hippocrates' work on dislocations. To-day Mr Charles Singer, of the University of London, says that the change in medical practices from the 19th to the 20th centuries is very largely in the direction of Hippocrates.

In the meantime, physics and astronomy had not been neglected. The earlier records are hazy, but by the fifth century B.C. the evidence is clearer. We know, for example, a good deal about Anaxagoras, who gave discourses on physics and astronomy when Socrates was a young man, and who was later banished from Athens because some of his sayings were considered impious. He was an acute observer; he explained accurately enough the inundations of the Nile; he described in modern fashion the nature of the sun, the phases of the moon and the source of its light, as well as the rotations of the universe, and he hinted at something like the atomic theory. He was soon followed by Leucippus,

and then by the great Democritus, whose atomic theory was most comprehensive. It will not do to dismiss Democritus as a mere maker of guesses. He was a trained mathematician, as many accounts of him show; he was a much-travelled observer and thinker. Besides completing a theory which lay ready for any generation of men who could use it, he had a profound influence on the mathematicians and astronomers who followed him in these subjects, though they did not go to his distance in physics.

Let us return to the progress of mathematical study, confining ourselves at first roughly to the two centuries before Euclid, say from 500 to 300 B.C. During this period the Pythagorean guild, Pythagoras himself being dead, collected a corpus of geometric lore which amounted practically to the first four and the sixth books of Euclid. They had also a very considerable literature on arithmetical series, square and cube root, rational and irrational numbers, and on the properties of number in general. They had also a pretty complete theory of sound, and particularly of musical sounds, overtones, and harmonies.

Once more, there was no sudden development about this. Pythagoras himself taught the secret of the octave and something about harmony, and had a proof of Euclid, 1, 47, at least as early as 525 B.C. His own mathematical teaching preceded Euclid, that is to say, by a period as long as the time between Newton and Einstein. All through this period mathematics was progressing. Not only so, but there were, in the Pythagoreans, a band of men devoting their lives to mathematical investigation.

Now I am not one of those dogmatists who undertake to weigh periods of history in a grocer's scale, and roundly declare that, but for A, B would never have happened. But I think it would not be exaggerated or fanciful to say that few things in the last twenty-five centuries of human striving and aspiration have endured as has that mathematical study of the Pythagoreans. Not only did they accomplish a task which had not to be done again, not only in their zeal for astronomy, physics, and optics and geometry did they prepare the way for Euclid and Democritus, for Archimedes and

Aristarchus, and so directly inspired Copernicus and Galileo; but they set an example of hard abstract thinking, of science for its own sake, and established a discipline for the human intellect to which nothing superior has ever been evolved.

It is perhaps more useful to plot out the field in this way than to deal with single mathematicians and say what they individually accomplished. Certainly this is the truer historical method. The common belief of the man who does not read Greek, and who has not studied this period attentively, is that there were a few eminent and lonely thinkers, who somehow hit upon certain bits of scattered truth, but who lived before their time and hence accomplished very little. But we know the names of a great many Pythagoreans, all of whom collaborated in the work which Euclid collected and added to. We know the names of correspondents of Archimedes, to whom he sent his demonstrations, and from whom he took hints and suggestions. Similarly, Apollonius, the inventor of conic sections, acknowledges his indebtedness not only to his predecessors but also to mathematical friends. But, of course, Greek mathematics did not end with Euclid. Aristarchus, the inventor of plane and spherical trigonometry, outlived Euclid; and Diophantus, who is sometimes called the inventor of algebra, but who like Euclid really collected the work of his predecessors to a large extent, came more than five centuries after Euclid. The whole period covered by Greek mathematics, including in that written records only, amounts to about nine hundred years.

It is sometimes said that for all their mathematical lore the Greeks were unable to use their knowledge in a mechanical way. This is a flat contradiction of history. Their use of the compound pulley, of the siphon, and dozens of other mechanical appliances is well attested, and their skilful engineering, for purposes of war and peace, can be learned from reading their literature and from their archæological remains. They applied their knowledge of optics to elaborate uses of mirrors, and there are references, other than the story of Archimedes' exploit, to burning glasses. The Greeks drove tunnels accurately. Archimedes made practical use of his knowledge of statics; at the same time Ctesibius in Alexandria Vol. 249.-No. 493,

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was making practical use of his study of pneumatics, in water-engines, and so forth. The skill of Greek clockmakers is, comparatively speaking, well known.

Just

as Greek military engineers used torsion in making engines of destruction, so Greek physicians used it in reducing difficult dislocations. What more practical use could be made of knowledge than to give Europe an accurate calendar? About 125 B.C. Hipparchus calculated the lunar month as 29 days, 12 hours, 44 minutes, 2 seconds. Sir Thomas Heath says that this differs from the present accepted length by less than one second!

Let us deal with Greek biology just sufficiently to show how perverse 'historians' can be. When I was a youth, and before I had read much Greek, I stumbled on a Latin translation of Aristotle's 'De Animalibus.' I was then dabbling in botany and engaged in a very elementary study of zoology. It may be judged, therefore, how deep was my prejudice about the indispensable aid rendered to science by the microscope'! But having had impressed upon me by my masters the importance of scientific classification, I marvelled and held my breath when I found that not only had Aristotle said the same thing but had classed whales among the mammals! That was my first lesson in the principles of science. It remained with me because in a small way it was something of an original discovery. Still, for years after that I was told, and one may still read in alleged histories of science, that the Greeks made practically no contribution to natural history. Truth may be gained, however, from learned monographs on the subject. In particular, there is a brilliant little essay on Aristotle as a biologist, by D'Arcy Wentworth Thompson, himself a biologist and the son of a Professor of Greek. The same writer has translated into English Aristotle's work, 'On Animals.'

It must not be imagined, because I mention here only Aristotle, that the period of Greek biology was short. One of the first investigators we mentioned was Anaximander, who interested himself in geology and shell-fish, and laid down a theory of evolution and variation of the species. It is not, therefore, an accident, or even an original thing, in Aristotle to conceive of Nature as having an ascending scale, and of creatures adapting themselves to their environment. His writings show

that many of his predecessors had concerned themselves with these these questions, particularly Anaxagoras and Empedocles. We have already compared the scientific influence of Empedocles with that of Goethe. In biology he seems to have come pretty close to Lamarck's Theory of Use and Disuse. There is the same evolutionary idea in Theophrastus, who followed Aristotle. Now between Anaximander and Theophrastus, with whom Greek biology may be said to close, there is a period of two and a half centuries. Within two and a half centuries they will accept Evolution in Tennessee.

Meanwhile astronomy proceeded apace. From mere observation the Greeks held certain very creditable opinions about the cosmos as early as the fifth century B.C. Both Empedocles and Anaxagoras taught that the light of the moon was borrowed, and the latter asserted that the sun was a molten mass and that very probably the stars had been detached from it. He held also that the whole universe was in rotating motion. The Pythagoreans held that all these separate worlds were spherical in shape; Democritus taught that there were an infinite number of such worlds, all in constant motion, and each of them consisting of an infinite number of infinitely small atoms. Enopides of Chios is credited with two astronomical discoveries: first, the obliquity of the ecliptic; and, second, the existence of a Great Year, so far as earth, sun, and moon were concerned. He flourished about the close of the fifth century.

In the following century Plato, undoubtedly, gave a great impetus to astronomy. Any one who reads only his Republic' will see his zeal for the subject, fanciful, exaggerated, and literary as many of his references are. It is not to be thought that he was an astronomer; even his mathematical knowledge is a little doubtful. His pupil Aristotle added very slightly to astronomy; he argued from the shadow cast by the earth in lunar eclipses that it must be spherical in shape. But another pupil of Plato, Heraclides of Pontus, actually took two great steps forward. He discovered that the earth rotates daily on its axis, also that Venus and Mercury revolve around the sun as a centre. Soon after this, we

* Vide especially 'De Partibus Animalium,' iv, 10.