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

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that many of his predecessors had concerned themselves with 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 Ein 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.

do not know exactly when, and some authorities even ascribe it to Heraclides, the astronomic theory was advanced which is exactly that of Tycho Brahe. Since the matter is obscure we need not dwell on it.

But the Greek who went the whole length of Copernicus, and whose ideas were well known to Coper. nicus, was Aristarchus of Samos. Many passages in ancient literature attest his heliocentric theory. Aristarchus lived about 310-230 B.C., and his theory was held by astronomers after his death for about three-quarters of a century. One of his writings has come down to usa treatise on the size and distance of the sun and the moon. The ideas of Aristarchus, like those of Copernicus, met with a chorus of disapproval from the reactionaries of his day. Cleanthes, the Stoic, for example, wrote a tract against him, calling him a blasphemer, and demanding that he should be silenced. Eratosthenes, who followed Aristarchus in the observatory at Alexandria, an extraordinarily vital and versatile character, distinguished himself in many ways. We need mention in this place only three of his contributions to science. He gave us our so-called Julian Calendar; he took a pretty accurate measurement of the angle of the obliquity of the ecliptic-he was less than half a minute of one degree wrong—and he made a scientific measurement of the earth's circumference, which was also a close approximation.

The next and last great figure in Greek astronomy is Hipparchus, who lived in Alexandria in the second century B.C. His measurements and calculations are much more accurate than those of his predecessors. This points not only to an enlarged collection of astronomical observations, but also to more precise instru ments. As we have seen, Hipparchus measured the length of the lunar month to within one second. He discovered the precession of the equinoxes, and in general he created for himself a great reputation in his own day and later, when Ptolemy-who can hardly be called an independent astronomer-based his work on him. Unfortunately, Hipparchus did not accept the heliocentric theory, and his great reputation for accuracy did much to obliterate the system of his predecessor. But as we have said of the work done by the Pythagoreans,

and of the system of Democritus, the thing had been accomplished, and lay more or less ready to the hands of any succeeding generation of men who could use it. Copernicus acknowledges his debt to Aristarchus.

After Hipparchus the science of astronomy fell rapidly taway. The Stoics, superstition, and man's incurable anthropomorphism conquered. When Cæsar wished to reform the calendar, about fifty years B.C., he had to hunt about for any one who understood the work of Eratosthenes. I think it a mistake, therefore, to describe Greek astronomy as continuing from Anaxagoras to Ptolemy. This would be a period of six hundred years—the same period as that which separates Einstein from Roger Bacon. It is more accurate to consider the development as terminated soon after the death of Hipparchus (120 B.C.). Even then the continuous and active growth of astronomy among the Greeks covers the same period as that which separates us from the death of Copernicus!

For the sake of one of them something must be said of the Romans. For with him, except in the case of pure mathematics, Greek science may be said to end. It is easy to make game of the Romans after speaking of the Greeks. If the ancients be conceived of as a University one must say that the Greeks constituted the Faculty of Litteræ Humaniores and Mathematics, with their branches, the Faculty of Natural Science, the Faculty of Medicine, and a large part of the Faculty of Law; whereas the Romans, though they were lawyers too, avowedly and by preference devoted themselves to the Faculty of Applied Science. Look about anywhere in Western Europe and you will find a tribute to Roman engineers. A good deal of their best work was done underground, in the form of sewers and of tunnels hown through mountains to drain away malaria. It is easy to be facetious over such a contrast. It is easy, too, to be patronising and to represent Rome as an uncouth but kindly giant, protecting for centuries the fairy child, Greek Thought.

There is, however, as often happens, an exception to any such generalisation. One of the noblest pieces of European literature was written by a Roman, Lucretius, who was deeply imbued with Greek philosophy and