Yet if our analysis does not lead us astray, the theory of Democritus was not truly monistic; his indestructible atoms, differing from one another in size and shape, utterly incapable of being changed from the form which they had maintained from the beginning, were in reality as truly and primordially different as are the primordial elements of Anaxagoras. In other words, the atom of Democritus is nothing less than the primordial seed of Anaxagoras, a little more tangibly visualized and given a distinctive name. Anaxagoras explicitly conceived his elements as invisibly small, as infinite in number, and as made up of an indefinite number of kinds—one for each distinctive substance in the world. But precisely the same postulates are made of the atom of Democritus. These also are invisibly small; these also are infinite in number; these also are made up of an indefinite number of kinds, corresponding with the observed difference of substances in the world. "Primitive seeds," or "atoms," were alike conceived to be primordial, un-changeable, and indestructible. Wherein then lies the difference? We answer, chiefly in a name; almost solely in the fact that Anaxagoras did not attempt to postulate the physical properties of the elements beyond stating that each has a distinctive personality, while Democritus did attempt to postulate these properties. He, too, admitted that each kind of element has its distinctive personality, and he attempted to visualize and describe the characteristics of the personality.

Thus while Anaxagoras tells us nothing of his elements except that they differ from one another, Democritus postulates a difference in size, imagines some elements as heavier and some as lighter, and conceives even that the elements may be provided with projecting hooks, with the aid of which they link themselves one with another. No one to-day takes these crude visualizings seriously as to their details. The sole element of truth which these dreamings contain, as distinguishing them from the dreamings of Anaxagoras, is in the conception that the various atoms differ in size and weight. Here, indeed, is a vague fore-shadowing of that chemistry of form which began to come into prominence towards the close of the nineteenth century. To have forecast even dimly this newest phase of chemical knowledge, across the abyss of centuries, is indeed a feat to put Democritus in the front rank of thinkers. But this estimate should not blind us to the fact that the pre-vision of Democritus was but a slight elaboration of a theory which had its origin with another thinker. The association between Anaxagoras and Democritus cannot be directly traced, but it is an association which the historian of ideas should never for a moment forget. If we are not to be misled by mere word-jugglery, we shall recognize the founder of the atomic theory of matter in Anaxagoras; its expositors along slightly different lines in Leucippus and Democritus; its re-discoverer of the nineteenth century in Dalton. All in all, then, just as Anaxagoras preceded Democritus in time, so must he take precedence over him also as an inductive thinker, who carried the use of the scientific imagination to its farthest reach.

An analysis of the theories of the two men leads to somewhat the same conclusion that might be reached from a comparison of their lives. Anaxagoras was a sceptical, experimental scientist, gifted also with the prophetic imagination. He reasoned always from the particular to the general, after the manner of true induction, and he scarcely took a step beyond the confines of secure induction. True scientist that he was, he could content himself with postulating different qualities for his elements, without pretending to know how these qualities could be defined. His elements were by hypothesis invisible, hence he would not attempt to visualize them. Democritus, on the other hand, refused to recognize this barrier. Where he could not know, he still did not hesitate to guess. Just as he conceived his atom of a definite form with a definite structure, even so he conceived that the atmosphere about him was full of invisible spirits; he accepted the current superstitions of his time. Like the average Greeks of his day, he even believed in such omens as those furnished by inspecting the entrails of a fowl. These chance bits of biography are weather-vanes of the mind of Democritus. They tend to substantiate our conviction that Democritus must rank below Anaxagoras as a devotee of pure science. But, after all, such comparisons and estimates as this are utterly futile. The essential fact for us is that here, in the fifth century before our era, we find put forward the most penetrating guess as to the constitution of matter that the history of ancient thought has to present to us. In one direction, the avenue of progress is barred; there will be no farther step that way till we come down the centuries to the time of Dalton.

HIPPOCRATES AND GREEK MEDICINE

These studies of the constitution of matter have carried us to the limits of the field of scientific imagination in antiquity; let us now turn sharply and consider a department of science in which theory joins hands with practicality. Let us witness the beginnings of scientific therapeutics.

Medicine among the early Greeks, before the time of Hippocrates, was a crude mixture of religion, necromancy, and mysticism. Temples were erected to the god of medicine, aesculapius, and sick persons made their way, or were carried, to these temples, where they sought to gain the favor of the god by suitable offerings, and learn the way to regain their health through remedies or methods revealed to them in dreams by the god. When the patient had been thus cured, he placed a tablet in the temple describing his sickness, and telling by what method the god had cured him. He again made suitable offerings at the temple, which were sometimes in the form of gold or silver representations of the diseased organ—a gold or silver model of a heart, hand, foot, etc.

Nevertheless, despite this belief in the supernatural, many drugs and healing lotions were employed, and the Greek physicians possessed considerable skill in dressing wounds and bandaging. But they did not depend upon these surgical dressings alone, using with them certain appropriate prayers and incantations, recited over the injured member at the time of applying the dressings.

Even the very early Greeks had learned something of anatomy. The daily contact with wounds and broken bones must of necessity lead to a crude understanding of anatomy in general. The first Greek anatomist, however, who is recognized as such, is said to have been Alcmaeon. He is said to have made extensive dissections of the lower animals, and to have described many hitherto unknown structures, such as the optic nerve and the Eustachian canal—the small tube leading into the throat from the ear. He is credited with many unique explanations of natural phenomena, such as, for example, the explanation that "hearing is produced by the hollow bone behind the ear; for all hollow things are sonorous." He was a rationalist, and he taught that the brain is the organ of mind. The sources of our information about his work, however, are unreliable.

Democedes, who lived in the sixth century B.C., is the first physician of whom we have any trustworthy history. We learn from Herodotus that he came from Croton to aegina, where, in recognition of his skill, he was appointed medical officer of the city. From aegina he was called to Athens at an increased salary, and later was in charge of medical affairs in several other Greek cities. He was finally called to Samos by the tyrant Polycrates, who reigned there from about 536 to 522 B.C. But on the death of Polycrates, who was murdered by the Persians, Democedes became a slave. His fame as a physician, however, had reached the ears of the Persian monarch, and shortly after his capture he was permitted to show his skill upon King Darius himself. The Persian monarch was suffering from a sprained ankle, which his Egyptian surgeons had been unable to cure. Democedes not only cured the injured member but used his influence in saving the lives of his Egyptian rivals, who had been condemned to death by the king.

At another time he showed his skill by curing the queen, who was suffering from a chronic abscess of long standing. This so pleased the monarch that he offered him as a reward anything he might desire, except his liberty. But the costly gifts of Darius did not satisfy him so long as he remained a slave; and determined to secure his freedom at any cost, he volunteered to lead some Persian spies into his native country, promising to use his influence in converting some of the leading men of his nation to the Persian cause. Laden with the wealth that had been heaped upon him by Darius, he set forth upon his mission, but upon reaching his native city of Croton he threw off his mask, renounced his Persian mission, and became once more a free Greek.

While the story of Democedes throws little light upon the medical practices of the time, it shows that paid city medical officers existed in Greece as early as the fifth and sixth centuries B.C. Even then there were different "schools" of medicine, whose disciples disagreed radically in their methods of treating diseases; and there were also specialists in certain diseases, quacks, and charlatans. Some physicians depended entirely upon external lotions for healing all disorders; others were "hydrotherapeutists" or "bath-physicians"; while there were a host of physicians who administered a great variety of herbs and drugs. There were also magicians who pretended to heal by sorcery, and great numbers of bone-setters, oculists, and dentists.

Many of the wealthy physicians had hospitals, or clinics, where patients were operated upon and treated. They were not hospitals in our modern understanding of the term, but were more like dispensaries, where patients were treated temporarily, but were not allowed to remain for any length of time. Certain communities established and supported these dispensaries for the care of the poor.

But anything approaching a rational system of medicine was not established, until Hippocrates of Cos, the "father of medicine," came upon the scene. In an age that produced Phidias, Lysias, Herodotus, Sophocles, and Pericles, it seems but natural that the medical art should find an exponent who would rise above superstitious dogmas and lay the foundation for a medical science. His rejection of the supernatural alone stamps the greatness of his genius. But, besides this, he introduced more detailed observation of diseases, and demonstrated the importance that attaches to prognosis.

Hippocrates was born at Cos, about 460 B.C., but spent most of his life at Larissa, in Thessaly. He was educated as a physician by his father, and travelled extensively as an itinerant practitioner for several years. His travels in different climates and among many different people undoubtedly tended to sharpen his keen sense of observation. He was a practical physician as well as a theorist, and, withal, a clear and concise writer. "Life is short," he says, "opportunity fleeting, judgment difficult, treatment easy, but treatment after thought is proper and profitable."

His knowledge of anatomy was necessarily very imperfect, and was gained largely from his predecessors, to whom he gave full credit. Dissections of the human body were forbidden him, and he was obliged to confine his experimental researches to operations on the lower animals. His knowledge of the structure and arrangement of the bones, however, was fairly accurate, but the anatomy of the softer tissues, as he conceived it, was a queer jumbling together of blood-vessels, muscles, and tendons. He does refer to "nerves," to be sure, but apparently the structures referred to are the tendons and ligaments, rather than the nerves themselves. He was better acquainted with the principal organs in the cavities of the body, and knew, for example, that the heart is divided into four cavities, two of which he supposed to contain blood, and the other two air.

His most revolutionary step was his divorcing of the supernatural from the natural, and establishing the fact that disease is due to natural causes and should be treated accordingly. The effect of such an attitude can hardly be over-estimated. The establishment of such a theory was naturally followed by a close observation as to the course of diseases and the effects of treatment. To facilitate this, he introduced the custom of writing down his observations as he made them—the "clinical history" of the case. Such clinical records are in use all over the world to-day, and their importance is so obvious that it is almost incomprehensible that they should have fallen into disuse shortly after the time of Hippocrates, and not brought into general use again until almost two thousand years later.

But scarcely less important than his recognition of disease as a natural phenomenon was the importance he attributed to prognosis. Prognosis, in the sense of prophecy, was common before the time of Hippocrates. But prognosis, as he practised it and as we understand it to-day, is prophecy based on careful observation of the course of diseases—something more than superstitious conjecture.

Although Hippocratic medicine rested on the belief in natural causes, nevertheless, dogma and theory held an important place. The humoral theory of disease was an all-important one, and so fully was this theory accepted that it influenced the science of medicine all through succeeding centuries. According to this celebrated theory there are four humors in the body—blood, phlegm, yellow bile, and black bile. When these humors are mixed in exact proportions they constitute health; but any deviations from these proportions produce disease. In treating diseases the aim of the physician was to discover which of these humors were out of proportion and to restore them to their natural equilibrium. It was in the methods employed in this restitution, rather than a disagreement about the humors themselves, that resulted in the various "schools" of medicine.

In many ways the surgery of Hippocrates showed a better understanding of the structure of the organs than of their functions. Some of the surgical procedures as described by him are followed, with slight modifications, to-day. Many of his methods were entirely lost sight of until modern times, and one, the treatment of dislocation of the outer end of the collar-bone, was not revived until some time in the eighteenth century.

Hippocrates, it seems, like modern physicians, sometimes suffered from the ingratitude of his patients. "The physician visits a patient suffering from fever or a wound, and prescribes for him," he says; "on the next day, if the patient feels worse the blame is laid upon the physician; if, on the other hand, he feels better, nature is extolled, and the physician reaps no praise." The essence of this has been repeated in rhyme and prose by writers in every age and country, but the "father of medicine" cautions physicians against allowing it to influence their attitude towards their profession.



VIII. POST-SOCRATIC SCIENCE AT ATHENS—PLATO, ARISTOTLE, AND THEOPHRASTUS

Doubtless it has been noticed that our earlier scientists were as far removed as possible from the limitations of specialism. In point of fact, in this early day, knowledge had not been classified as it came to be later on. The philosopher was, as his name implied, a lover of knowledge, and he did not find it beyond the reach of his capacity to apply himself to all departments of the field of human investigation. It is nothing strange to discover that Anaximander and the Pythagoreans and Anaxagoras have propounded theories regarding the structure of the cosmos, the origin and development of animals and man, and the nature of matter itself. Nowadays, so enormously involved has become the mass of mere facts regarding each of these departments of knowledge that no one man has the temerity to attempt to master them all. But it was different in those days of beginnings. Then the methods of observation were still crude, and it was quite the custom for a thinker of forceful personality to find an eager following among disciples who never thought of putting his theories to the test of experiment. The great lesson that true science in the last resort depends upon observation and measurement, upon compass and balance, had not yet been learned, though here and there a thinker like Anaxagoras had gained an inkling of it.

For the moment, indeed, there in Attica, which was now, thanks to that outburst of Periclean culture, the centre of the world's civilization, the trend of thought was to take quite another direction. The very year which saw the birth of Democritus at Abdera, and of Hippocrates, marked also the birth, at Athens, of another remarkable man, whose influence it would scarcely be possible to over-estimate. This man was Socrates. The main facts of his history are familiar to every one. It will be recalled that Socrates spent his entire life in Athens, mingling everywhere with the populace; haranguing, so the tradition goes, every one who would listen; inculcating moral lessons, and finally incurring the disapprobation of at least a voting majority of his fellow-citizens. He gathered about him a company of remarkable men with Plato at their head, but this could not save him from the disapprobation of the multitudes, at whose hands he suffered death, legally administered after a public trial. The facts at command as to certain customs of the Greeks at this period make it possible to raise a question as to whether the alleged "corruption of youth," with which Socrates was charged, may not have had a different implication from what posterity has preferred to ascribe to it. But this thought, almost shocking to the modern mind and seeming altogether sacrilegious to most students of Greek philosophy, need not here detain us; neither have we much concern in the present connection with any part of the teaching of the martyred philosopher. For the historian of metaphysics, Socrates marks an epoch, but for the historian of science he is a much less consequential figure.

Similarly regarding Plato, the aristocratic Athenian who sat at the feet of Socrates, and through whose writings the teachings of the master found widest currency. Some students of philosophy find in Plato "the greatest thinker and writer of all time."(1) The student of science must recognize in him a thinker whose point of view was essentially non-scientific; one who tended always to reason from the general to the particular rather than from the particular to the general. Plato's writings covered almost the entire field of thought, and his ideas were presented with such literary charm that successive generations of readers turned to them with unflagging interest, and gave them wide currency through copies that finally preserved them to our own time. Thus we are not obliged in his case, as we are in the case of every other Greek philosopher, to estimate his teachings largely from hearsay evidence. Plato himself speaks to us directly. It is true, the literary form which he always adopted, namely, the dialogue, does not give quite the same certainty as to when he is expressing his own opinions that a more direct narrative would have given; yet, in the main, there is little doubt as to the tenor of his own opinions—except, indeed, such doubt as always attaches to the philosophical reasoning of the abstract thinker.

What is chiefly significant from our present standpoint is that the great ethical teacher had no significant message to give the world regarding the physical sciences. He apparently had no sharply defined opinions as to the mechanism of the universe; no clear conception as to the origin or development of organic beings; no tangible ideas as to the problems of physics; no favorite dreams as to the nature of matter. Virtually his back was turned on this entire field of thought. He was under the sway of those innate ideas which, as we have urged, were among the earliest inductions of science. But he never for a moment suspected such an origin for these ideas. He supposed his conceptions of being, his standards of ethics, to lie back of all experience; for him they were the most fundamental and most dependable of facts. He criticised Anaxagoras for having tended to deduce general laws from observation. As we moderns see it, such criticism is the highest possible praise. It is a criticism that marks the distinction between the scientist who is also a philosopher and the philosopher who has but a vague notion of physical science. Plato seemed, indeed, to realize the value of scientific investigation; he referred to the astronomical studies of the Egyptians and Chaldeans, and spoke hopefully of the results that might accrue were such studies to be taken up by that Greek mind which, as he justly conceived, had the power to vitalize and enrich all that it touched. But he told here of what he would have others do, not of what he himself thought of doing. His voice was prophetic, but it stimulated no worker of his own time.

Plato himself had travelled widely. It is a familiar legend that he lived for years in Egypt, endeavoring there to penetrate the mysteries of Egyptian science. It is said even that the rudiments of geometry which he acquired there influenced all his later teachings. But be that as it may, the historian of science must recognize in the founder of the Academy a moral teacher and metaphysical dreamer and sociologist, but not, in the modern acceptance of the term, a scientist. Those wider phases of biological science which find their expression in metaphysics, in ethics, in political economy, lie without our present scope; and for the development of those subjects with which we are more directly concerned, Plato, like his master, has a negative significance.

ARISTOTLE (384-322 B.C.)

When we pass to that third great Athenian teacher, Aristotle, the case is far different. Here was a man whose name was to be received as almost a synonym for Greek science for more than a thousand years after his death. All through the Middle Ages his writings were to be accepted as virtually the last word regarding the problems of nature. We shall see that his followers actually preferred his mandate to the testimony of their own senses. We shall see, further, that modern science progressed somewhat in proportion as it overthrew the Aristotelian dogmas. But the traditions of seventeen or eighteen centuries are not easily set aside, and it is perhaps not too much to say that the name of Aristotle stands, even in our own time, as vaguely representative in the popular mind of all that was highest and best in the science of antiquity. Yet, perhaps, it would not be going too far to assert that something like a reversal of this judgment would be nearer the truth. Aristotle did, indeed, bring together a great mass of facts regarding animals in his work on natural history, which, being preserved, has been deemed to entitle its author to be called the "father of zoology." But there is no reason to suppose that any considerable portion of this work contained matter that was novel, or recorded observations that were original with Aristotle; and the classifications there outlined are at best but a vague foreshadowing of the elaboration of the science. Such as it is, however, the natural history stands to the credit of the Stagirite. He must be credited, too, with a clear enunciation of one most important scientific doctrine—namely, the doctrine of the spherical figure of the earth. We have already seen that this theory originated with the Pythagorean philosophers out in Italy. We have seen, too, that the doctrine had not made its way in Attica in the time of Anaxagoras. But in the intervening century it had gained wide currency, else so essentially conservative a thinker as Aristotle would scarcely have accepted it. He did accept it, however, and gave the doctrine clearest and most precise expression. Here are his words:(2)

"As to the figure of the earth it must necessarily be spherical.... If it were not so, the eclipses of the moon would not have such sections as they have. For in the configurations in the course of a month the deficient part takes all different shapes; it is straight, and concave, and convex; but in eclipses it always has the line of divisions convex; wherefore, since the moon is eclipsed in consequence of the interposition of the earth, the periphery of the earth must be the cause of this by having a spherical form. And again, from the appearance of the stars it is clear, not only that the earth is round, but that its size is not very large; for when we make a small removal to the south or the north, the circle of the horizon becomes palpably different, so that the stars overhead undergo a great change, and are not the same to those that travel in the north and to the south. For some stars are seen in Egypt or at Cyprus, but are not seen in the countries to the north of these; and the stars that in the north are visible while they make a complete circuit, there undergo a setting. So that from this it is manifest, not only that the form of the earth is round, but also that it is a part of a not very large sphere; for otherwise the difference would not be so obvious to persons making so small a change of place. Wherefore we may judge that those persons who connect the region in the neighborhood of the pillars of Hercules with that towards India, and who assert that in this way the sea is one, do not assert things very improbable. They confirm this conjecture moreover by the elephants, which are said to be of the same species towards each extreme; as if this circumstance was a consequence of the conjunction of the extremes. The mathematicians who try to calculate the measure of the circumference, make it amount to four hundred thousand stadia; whence we collect that the earth is not only spherical, but is not large compared with the magnitude of the other stars."

But in giving full meed of praise to Aristotle for the promulgation of this doctrine of the sphericity of the earth, it must unfortunately be added that the conservative philosopher paused without taking one other important step. He could not accept, but, on the contrary, he expressly repudiated, the doctrine of the earth's motion. We have seen that this idea also was a part of the Pythagorean doctrine, and we shall have occasion to dwell more at length on this point in a succeeding chapter. It has even been contended by some critics that it was the adverse conviction of the Peripatetic philosopher which, more than any other single influence, tended to retard the progress of the true doctrine regarding the mechanism of the heavens. Aristotle accepted the sphericity of the earth, and that doctrine became a commonplace of scientific knowledge, and so continued throughout classical antiquity. But Aristotle rejected the doctrine of the earth's motion, and that doctrine, though promulgated actively by a few contemporaries and immediate successors of the Stagirite, was then doomed to sink out of view for more than a thousand years. If it be a correct assumption that the influence of Aristotle was, in a large measure, responsible for this result, then we shall perhaps not be far astray in assuming that the great founder of the Peripatetic school was, on the whole, more instrumental in retarding the progress of astronomical science that any other one man that ever lived.

The field of science in which Aristotle was pre-eminently a pathfinder is zoology. His writings on natural history have largely been preserved, and they constitute by far the most important contribution to the subject that has come down to us from antiquity. They show us that Aristotle had gained possession of the widest range of facts regarding the animal kingdom, and, what is far more important, had attempted to classify these facts. In so doing he became the founder of systematic zoology. Aristotle's classification of the animal kingdom was known and studied throughout the Middle Ages, and, in fact, remained in vogue until superseded by that of Cuvier in the nineteenth century. It is not to be supposed that all the terms of Aristotle's classification originated with him. Some of the divisions are too patent to have escaped the observation of his predecessors. Thus, for example, the distinction between birds and fishes as separate classes of animals is so obvious that it must appeal to a child or to a savage. But the efforts of Aristotle extended, as we shall see, to less patent generalizations. At the very outset, his grand division of the animal kingdom into blood-bearing and bloodless animals implies a very broad and philosophical conception of the entire animal kingdom. The modern physiologist does not accept the classification, inasmuch as it is now known that colorless fluids perform the functions of blood for all the lower organisms. But the fact remains that Aristotle's grand divisions correspond to the grand divisions of the Lamarckian system—vertebrates and invertebrates—which every one now accepts. Aristotle, as we have said, based his classification upon observation of the blood; Lamarck was guided by a study of the skeleton. The fact that such diverse points of view could direct the observer towards the same result gives, inferentially, a suggestive lesson in what the modern physiologist calls the homologies of parts of the organism.

Aristotle divides his so-called blood-bearing animals into five classes: (1) Four-footed animals that bring forth their young alive; (2) birds; (3) egg-laying four-footed animals (including what modern naturalists call reptiles and amphibians); (4) whales and their allies; (5) fishes. This classification, as will be observed, is not so very far afield from the modern divisions into mammals, birds, reptiles, amphibians, and fishes. That Aristotle should have recognized the fundamental distinction between fishes and the fish-like whales, dolphins, and porpoises proves the far from superficial character of his studies. Aristotle knew that these animals breathe by means of lungs and that they produce living young. He recognized, therefore, their affinity with his first class of animals, even if he did not, like the modern naturalist, consider these affinities close enough to justify bringing the two types together into a single class.

The bloodless animals were also divided by Aristotle into five classes—namely: (1) Cephalopoda (the octopus, cuttle-fish, etc.); (2) weak-shelled animals (crabs, etc.); (3) insects and their allies (including various forms, such as spiders and centipedes, which the modern classifier prefers to place by themselves); (4) hard-shelled animals (clams, oysters, snails, etc.); (5) a conglomerate group of marine forms, including star-fish, sea-urchins, and various anomalous forms that were regarded as linking the animal to the vegetable worlds. This classification of the lower forms of animal life continued in vogue until Cuvier substituted for it his famous grouping into articulates, mollusks, and radiates; which grouping in turn was in part superseded later in the nineteenth century.

What Aristotle did for the animal kingdom his pupil, Theophrastus, did in some measure for the vegetable kingdom. Theophrastus, however, was much less a classifier than his master, and his work on botany, called The Natural History of Development, pays comparatively slight attention to theoretical questions. It deals largely with such practicalities as the making of charcoal, of pitch, and of resin, and the effects of various plants on the animal organism when taken as foods or as medicines. In this regard the work of Theophrastus, is more nearly akin to the natural history of the famous Roman compiler, Pliny. It remained, however, throughout antiquity as the most important work on its subject, and it entitles Theophrastus to be called the "father of botany." Theophrastus deals also with the mineral kingdom after much the same fashion, and here again his work is the most notable that was produced in antiquity.



IX. GREEK SCIENCE OF THE ALEXANDRIAN OR HELLENISTIC PERIOD

We are entering now upon the most important scientific epoch of antiquity. When Aristotle and Theophrastus passed from the scene, Athens ceased to be in any sense the scientific centre of the world. That city still retained its reminiscent glory, and cannot be ignored in the history of culture, but no great scientific leader was ever again to be born or to take up his permanent abode within the confines of Greece proper. With almost cataclysmic suddenness, a new intellectual centre appeared on the south shore of the Mediterranean. This was the city of Alexandria, a city which Alexander the Great had founded during his brief visit to Egypt, and which became the capital of Ptolemy Soter when he chose Egypt as his portion of the dismembered empire of the great Macedonian. Ptolemy had been with his master in the East, and was with him in Babylonia when he died. He had therefore come personally in contact with Babylonian civilization, and we cannot doubt that this had a most important influence upon his life, and through him upon the new civilization of the West. In point of culture, Alexandria must be regarded as the successor of Babylon, scarcely less directly than of Greece. Following the Babylonian model, Ptolemy erected a great museum and began collecting a library. Before his death it was said that he had collected no fewer than two hundred thousand manuscripts. He had gathered also a company of great teachers and founded a school of science which, as has just been said, made Alexandria the culture-centre of the world.

Athens in the day of her prime had known nothing quite like this. Such private citizens as Aristotle are known to have had libraries, but there were no great public collections of books in Athens, or in any other part of the Greek domain, until Ptolemy founded his famous library. As is well known, such libraries had existed in Babylonia for thousands of years. The character which the Ptolemaic epoch took on was no doubt due to Babylonian influence, but quite as much to the personal experience of Ptolemy himself as an explorer in the Far East. The marvellous conquering journey of Alexander had enormously widened the horizon of the Greek geographer, and stimulated the imagination of all ranks of the people, It was but natural, then, that geography and its parent science astronomy should occupy the attention of the best minds in this succeeding epoch. In point of fact, such a company of star-gazers and earth-measurers came upon the scene in this third century B.C. as had never before existed anywhere in the world. The whole trend of the time was towards mechanics. It was as if the greatest thinkers had squarely faced about from the attitude of the mystical philosophers of the preceding century, and had set themselves the task of solving all the mechanical riddles of the universe, They no longer troubled themselves about problems of "being" and "becoming"; they gave but little heed to metaphysical subtleties; they demanded that their thoughts should be gauged by objective realities. Hence there arose a succession of great geometers, and their conceptions were applied to the construction of new mechanical contrivances on the one hand, and to the elaboration of theories of sidereal mechanics on the other.

The wonderful company of men who performed the feats that are about to be recorded did not all find their home in Alexandria, to be sure; but they all came more or less under the Alexandrian influence. We shall see that there are two other important centres; one out in Sicily, almost at the confines of the Greek territory in the west; the other in Asia Minor, notably on the island of Samos—the island which, it will be recalled, was at an earlier day the birthplace of Pythagoras. But whereas in the previous century colonists from the confines of the civilized world came to Athens, now all eyes turned towards Alexandria, and so improved were the facilities for communication that no doubt the discoveries of one coterie of workers were known to all the others much more quickly than had ever been possible before. We learn, for example, that the studies of Aristarchus of Samos were definitely known to Archimedes of Syracuse, out in Sicily. Indeed, as we shall see, it is through a chance reference preserved in one of the writings of Archimedes that one of the most important speculations of Aristarchus is made known to us. This illustrates sufficiently the intercommunication through which the thought of the Alexandrian epoch was brought into a single channel. We no longer, as in the day of the earlier schools of Greek philosophy, have isolated groups of thinkers. The scientific drama is now played out upon a single stage; and if we pass, as we shall in the present chapter, from Alexandria to Syracuse and from Syracuse to Samos, the shift of scenes does no violence to the dramatic unities.

Notwithstanding the number of great workers who were not properly Alexandrians, none the less the epoch is with propriety termed Alexandrian. Not merely in the third century B.C., but throughout the lapse of at least four succeeding centuries, the city of Alexander and the Ptolemies continued to hold its place as the undisputed culture-centre of the world. During that period Rome rose to its pinnacle of glory and began to decline, without ever challenging the intellectual supremacy of the Egyptian city. We shall see, in a later chapter, that the Alexandrian influences were passed on to the Mohammedan conquerors, and every one is aware that when Alexandria was finally overthrown its place was taken by another Greek city, Byzantium or Constantinople. But that transfer did not occur until Alexandria had enjoyed a longer period of supremacy as an intellectual centre than had perhaps ever before been granted to any city, with the possible exception of Babylon.

EUCLID (ABOUT 300 B.C.)

Our present concern is with that first wonderful development of scientific activity which began under the first Ptolemy, and which presents, in the course of the first century of Alexandrian influence, the most remarkable coterie of scientific workers and thinkers that antiquity produced. The earliest group of these new leaders in science had at its head a man whose name has been a household word ever since. This was Euclid, the father of systematic geometry. Tradition has preserved to us but little of the personality of this remarkable teacher; but, on the other hand, his most important work has come down to us in its entirety. The Elements of Geometry, with which the name of Euclid is associated in the mind of every school-boy, presented the chief propositions of its subject in so simple and logical a form that the work remained a textbook everywhere for more than two thousand years. Indeed it is only now beginning to be superseded. It is not twenty years since English mathematicians could deplore the fact that, despite certain rather obvious defects of the work of Euclid, no better textbook than this was available. Euclid's work, of course, gives expression to much knowledge that did not originate with him. We have already seen that several important propositions of geometry had been developed by Thales, and one by Pythagoras, and that the rudiments of the subject were at least as old as Egyptian civilization. Precisely how much Euclid added through his own investigations cannot be ascertained. It seems probable that he was a diffuser of knowledge rather than an originator, but as a great teacher his fame is secure. He is credited with an epigram which in itself might insure him perpetuity of fame: "There is no royal road to geometry," was his answer to Ptolemy when that ruler had questioned whether the Elements might not be simplified. Doubtless this, like most similar good sayings, is apocryphal; but whoever invented it has made the world his debtor.

HEROPHILUS AND ERASISTRATUS

The catholicity of Ptolemy's tastes led him, naturally enough, to cultivate the biological no less than the physical sciences. In particular his influence permitted an epochal advance in the field of medicine. Two anatomists became famous through the investigations they were permitted to make under the patronage of the enlightened ruler. These earliest of really scientific investigators of the mechanism of the human body were named Herophilus and Erasistratus. These two anatomists gained their knowledge by the dissection of human bodies (theirs are the first records that we have of such practices), and King Ptolemy himself is said to have been present at some of these dissections. They were the first to discover that the nerve-trunks have their origin in the brain and spinal cord, and they are credited also with the discovery that these nerve-trunks are of two different kinds—one to convey motor, and the other sensory impulses. They discovered, described, and named the coverings of the brain. The name of Herophilus is still applied by anatomists, in honor of the discoverer, to one of the sinuses or large canals that convey the venous blood from the head. Herophilus also noticed and described four cavities or ventricles in the brain, and reached the conclusion that one of these ventricles was the seat of the soul—a belief shared until comparatively recent times by many physiologists. He made also a careful and fairly accurate study of the anatomy of the eye, a greatly improved the old operation for cataract.

With the increased knowledge of anatomy came also corresponding advances in surgery, and many experimental operations are said to have been performed upon condemned criminals who were handed over to the surgeons by the Ptolemies. While many modern writers have attempted to discredit these assertions, it is not improbable that such operations were performed. In an age when human life was held so cheap, and among a people accustomed to torturing condemned prisoners for comparatively slight offences, it is not unlikely that the surgeons were allowed to inflict perhaps less painful tortures in the cause of science. Furthermore, we know that condemned criminals were sometimes handed over to the medical profession to be "operated upon and killed in whatever way they thought best" even as late as the sixteenth century. Tertullian(1) probably exaggerates, however, when he puts the number of such victims in Alexandria at six hundred.

Had Herophilus and Erasistratus been as happy in their deductions as to the functions of the organs as they were in their knowledge of anatomy, the science of medicine would have been placed upon a very high plane even in their time. Unfortunately, however, they not only drew erroneous inferences as to the functions of the organs, but also disagreed radically as to what functions certain organs performed, and how diseases should be treated, even when agreeing perfectly on the subject of anatomy itself. Their contribution to the knowledge of the scientific treatment of diseases holds no such place, therefore, as their anatomical investigations.

Half a century after the time of Herophilus there appeared a Greek physician, Heraclides, whose reputation in the use of drugs far surpasses that of the anatomists of the Alexandrian school. His reputation has been handed down through the centuries as that of a physician, rather than a surgeon, although in his own time he was considered one of the great surgeons of the period. Heraclides belonged to the "Empiric" school, which rejected anatomy as useless, depending entirely on the use of drugs. He is thought to have been the first physician to point out the value of opium in certain painful diseases. His prescription of this drug for certain cases of "sleeplessness, spasm, cholera, and colic," shows that his use of it was not unlike that of the modern physician in certain cases; and his treatment of fevers, by keeping the patient's head cool and facilitating the secretions of the body, is still recognized as "good practice." He advocated a free use of liquids in quenching the fever patient's thirst—a recognized therapeutic measure to-day, but one that was widely condemned a century ago.

ARCHIMEDES OF SYRACUSE AND THE FOUNDATION OF MECHANICS

We do not know just when Euclid died, but as he was at the height of his fame in the time of Ptolemy I., whose reign ended in the year 285 B.C., it is hardly probable that he was still living when a young man named Archimedes came to Alexandria to study. Archimedes was born in the Greek colony of Syracuse, on the island of Sicily, in the year 287 B.C. When he visited Alexandria he probably found Apollonius of Perga, the pupil of Euclid, at the head of the mathematical school there. Just how long Archimedes remained at Alexandria is not known. When he had satisfied his curiosity or completed his studies, he returned to Syracuse and spent his life there, chiefly under the patronage of King Hiero, who seems fully to have appreciated his abilities.

Archimedes was primarily a mathematician. Left to his own devices, he would probably have devoted his entire time to the study of geometrical problems. But King Hiero had discovered that his protege had wonderful mechanical ingenuity, and he made good use of this discovery. Under stress of the king's urgings, the philosopher was led to invent a great variety of mechanical contrivances, some of them most curious ones. Antiquity credited him with the invention of more than forty machines, and it is these, rather than his purely mathematical discoveries, that gave his name popular vogue both among his contemporaries and with posterity. Every one has heard of the screw of Archimedes, through which the paradoxical effect was produced of making water seem to flow up hill. The best idea of this curious mechanism is obtained if one will take in hand an ordinary corkscrew, and imagine this instrument to be changed into a hollow tube, retaining precisely the same shape but increased to some feet in length and to a proportionate diameter. If one will hold the corkscrew in a slanting direction and turn it slowly to the right, supposing that the point dips up a portion of water each time it revolves, one can in imagination follow the flow of that portion of water from spiral to spiral, the water always running downward, of course, yet paradoxically being lifted higher and higher towards the base of the corkscrew, until finally it pours out (in the actual Archimedes' tube) at the top. There is another form of the screw in which a revolving spiral blade operates within a cylinder, but the principle is precisely the same. With either form water may be lifted, by the mere turning of the screw, to any desired height. The ingenious mechanism excited the wonder of the contemporaries of Archimedes, as well it might. More efficient devices have superseded it in modern times, but it still excites the admiration of all who examine it, and its effects seem as paradoxical as ever.

Some other of the mechanisms of Archimedes have been made known to successive generations of readers through the pages of Polybius and Plutarch. These are the devices through which Archimedes aided King Hiero to ward off the attacks of the Roman general Marcellus, who in the course of the second Punic war laid siege to Syracuse.

Plutarch, in his life of Marcellus, describes the Roman's attack and Archimedes' defence in much detail. Incidentally he tells us also how Archimedes came to make the devices that rendered the siege so famous:

"Marcellus himself, with threescore galleys of five rowers at every bank, well armed and full of all sorts of artillery and fireworks, did assault by sea, and rowed hard to the wall, having made a great engine and device of battery, upon eight galleys chained together, to batter the wall: trusting in the great multitude of his engines of battery, and to all such other necessary provision as he had for wars, as also in his own reputation. But Archimedes made light account of all his devices, as indeed they were nothing comparable to the engines himself had invented. This inventive art to frame instruments and engines (which are called mechanical, or organical, so highly commended and esteemed of all sorts of people) was first set forth by Architas, and by Eudoxus: partly to beautify a little the science of geometry by this fineness, and partly to prove and confirm by material examples and sensible instruments, certain geometrical conclusions, where of a man cannot find out the conceivable demonstrations by enforced reasons and proofs. As that conclusion which instructeth one to search out two lines mean proportional, which cannot be proved by reason demonstrative, and yet notwithstanding is a principle and an accepted ground for many things which are contained in the art of portraiture. Both of them have fashioned it to the workmanship of certain instruments, called mesolabes or mesographs, which serve to find these mean lines proportional, by drawing certain curve lines, and overthwart and oblique sections. But after that Plato was offended with them, and maintained against them, that they did utterly corrupt and disgrace, the worthiness and excellence of geometry, making it to descend from things not comprehensible and without body, unto things sensible and material, and to bring it to a palpable substance, where the vile and base handiwork of man is to be employed: since that time, I say, handicraft, or the art of engines, came to be separated from geometry, and being long time despised by the philosophers, it came to be one of the warlike arts.

"But Archimedes having told King Hiero, his kinsman and friend, that it was possible to remove as great a weight as he would, with as little strength as he listed to put to it: and boasting himself thus (as they report of him) and trusting to the force of his reasons, wherewith he proved this conclusion, that if there were another globe of earth, he was able to remove this of ours, and pass it over to the other: King Hiero wondering to hear him, required him to put his device in execution, and to make him see by experience, some great or heavy weight removed, by little force. So Archimedes caught hold with a book of one of the greatest carects, or hulks of the king (that to draw it to the shore out of the water required a marvellous number of people to go about it, and was hardly to be done so) and put a great number of men more into her, than her ordinary burden: and he himself sitting alone at his ease far off, without any straining at all, drawing the end of an engine with many wheels and pulleys, fair and softly with his hand, made it come as gently and smoothly to him, as it had floated in the sea. The king wondering to see the sight, and knowing by proof the greatness of his art; be prayed him to make him some engines, both to assault and defend, in all manner of sieges and assaults. So Archimedes made him many engines, but King Hiero never occupied any of them, because he reigned the most part of his time in peace without any wars. But this provision and munition of engines, served the Syracusan's turn marvellously at that time: and not only the provision of the engines ready made, but also the engineer and work-master himself, that had invented them.

"Now the Syracusans, seeing themselves assaulted by the Romans, both by sea and by land, were marvellously perplexed, and could not tell what to say, they were so afraid: imagining it was impossible for them to withstand so great an army. But when Archimedes fell to handling his engines, and to set them at liberty, there flew in the air infinite kinds of shot, and marvellous great stones, with an incredible noise and force on the sudden, upon the footmen that came to assault the city by land, bearing down, and tearing in pieces all those which came against them, or in what place soever they lighted, no earthly body being able to resist the violence of so heavy a weight: so that all their ranks were marvellously disordered. And as for the galleys that gave assault by sea, some were sunk with long pieces of timber like unto the yards of ships, whereto they fasten their sails, which were suddenly blown over the walls with force of their engines into their galleys, and so sunk them by their over great weight."

Polybius describes what was perhaps the most important of these contrivances, which was, he tells us, "a band of iron, hanging by a chain from the beak of a machine, which was used in the following manner. The person who, like a pilot, guided the beak, having let fall the hand, and catched hold of the prow of any vessel, drew down the opposite end of the machine that was on the inside of the walls. And when the vessel was thus raised erect upon its stem, the machine itself was held immovable; but, the chain being suddenly loosened from the beak by the means of pulleys, some of the vessels were thrown upon their sides, others turned with the bottom upwards; and the greatest part, as the prows were plunged from a considerable height into the sea, were filled with water, and all that were on board thrown into tumult and disorder.

"Marcellus was in no small degree embarrassed," Polybius continues, "when he found himself encountered in every attempt by such resistance. He perceived that all his efforts were defeated with loss; and were even derided by the enemy. But, amidst all the anxiety that he suffered, he could not help jesting upon the inventions of Archimedes. This man, said he, employs our ships as buckets to draw water: and boxing about our sackbuts, as if they were unworthy to be associated with him, drives them from his company with disgrace. Such was the success of the siege on the side of the sea."

Subsequently, however, Marcellus took the city by strategy, and Archimedes was killed, contrary, it is said, to the express orders of Marcellus. "Syracuse being taken," says Plutarch, "nothing grieved Marcellus more than the loss of Archimedes. Who, being in his study when the city was taken, busily seeking out by himself the demonstration of some geometrical proposition which he had drawn in figure, and so earnestly occupied therein, as he neither saw nor heard any noise of enemies that ran up and down the city, and much less knew it was taken: he wondered when he saw a soldier by him, that bade him go with him to Marcellus. Notwithstanding, he spake to the soldier, and bade him tarry until he had done his conclusion, and brought it to demonstration: but the soldier being angry with his answer, drew out his sword and killed him. Others say, that the Roman soldier when he came, offered the sword's point to him, to kill him: and that Archimedes when he saw him, prayed him to hold his hand a little, that he might not leave the matter he looked for imperfect, without demonstration. But the soldier making no reckoning of his speculation, killed him presently. It is reported a third way also, saying that certain soldiers met him in the streets going to Marcellus, carrying certain mathematical instruments in a little pretty coffer, as dials for the sun, spheres, and angles, wherewith they measure the greatness of the body of the sun by view: and they supposing he had carried some gold or silver, or other precious jewels in that little coffer, slew him for it. But it is most certain that Marcellus was marvellously sorry for his death, and ever after hated the villain that slew him, as a cursed and execrable person: and how he had made also marvellous much afterwards of Archimedes' kinsmen for his sake."

We are further indebted to Plutarch for a summary of the character and influence of Archimedes, and for an interesting suggestion as to the estimate which the great philosopher put upon the relative importance of his own discoveries. "Notwithstanding Archimedes had such a great mind, and was so profoundly learned, having hidden in him the only treasure and secrets of geometrical inventions: as he would never set forth any book how to make all these warlike engines, which won him at that time the fame and glory, not of man's knowledge, but rather of divine wisdom. But he esteeming all kind of handicraft and invention to make engines, and generally all manner of sciences bringing common commodity by the use of them, to be but vile, beggarly, and mercenary dross: employed his wit and study only to write things, the beauty and subtlety whereof were not mingled anything at all with necessity. For all that he hath written, are geometrical propositions, which are without comparison of any other writings whatsoever: because the subject where of they treat, doth appear by demonstration, the maker gives them the grace and the greatness, and the demonstration proving it so exquisitely, with wonderful reason and facility, as it is not repugnable. For in all geometry are not to be found more profound and difficult matters written, in more plain and simple terms, and by more easy principles, than those which he hath invented. Now some do impute this, to the sharpness of his wit and understanding, which was a natural gift in him: others do refer it to the extreme pains he took, which made these things come so easily from him, that they seemed as if they had been no trouble to him at all. For no man living of himself can devise the demonstration of his propositions, what pains soever he take to seek it: and yet straight so soon as he cometh to declare and open it, every man then imagineth with himself he could have found it out well enough, he can then so plainly make demonstration of the thing he meaneth to show. And therefore that methinks is likely to be true, which they write of him: that he was so ravished and drunk with the sweet enticements of this siren, which as it were lay continually with him, as he forgot his meat and drink, and was careless otherwise of himself, that oftentimes his servants got him against his will to the baths to wash and anoint him: and yet being there, he would ever be drawing out of the geometrical figures, even in the very imbers of the chimney. And while they were anointing of him with oils and sweet savours, with his finger he did draw lines upon his naked body: so far was he taken from himself, and brought into an ecstasy or trance, with the delight he had in the study of geometry, and truly ravished with the love of the Muses. But amongst many notable things he devised, it appeareth, that he most esteemed the demonstration of the proportion between the cylinder (to wit, the round column) and the sphere or globe contained in the same: for he prayed his kinsmen and friends, that after his death they would put a cylinder upon his tomb, containing a massy sphere, with an inscription of the proportion, whereof the continent exceedeth the thing contained."(2)

It should be observed that neither Polybius nor Plutarch mentions the use of burning-glasses in connection with the siege of Syracuse, nor indeed are these referred to by any other ancient writer of authority. Nevertheless, a story gained credence down to a late day to the effect that Archimedes had set fire to the fleet of the enemy with the aid of concave mirrors. An experiment was made by Sir Isaac Newton to show the possibility of a phenomenon so well in accord with the genius of Archimedes, but the silence of all the early authorities makes it more than doubtful whether any such expedient was really adopted.

It will be observed that the chief principle involved in all these mechanisms was a capacity to transmit great power through levers and pulleys, and this brings us to the most important field of the Syracusan philosopher's activity. It was as a student of the lever and the pulley that Archimedes was led to some of his greatest mechanical discoveries. He is even credited with being the discoverer of the compound pulley. More likely he was its developer only, since the principle of the pulley was known to the old Babylonians, as their sculptures testify. But there is no reason to doubt the general outlines of the story that Archimedes astounded King Hiero by proving that, with the aid of multiple pulleys, the strength of one man could suffice to drag the largest ship from its moorings.

The property of the lever, from its fundamental principle, was studied by him, beginning with the self-evident fact that "equal bodies at the ends of the equal arms of a rod, supported on its middle point, will balance each other"; or, what amounts to the same thing stated in another way, a regular cylinder of uniform matter will balance at its middle point. From this starting-point he elaborated the subject on such clear and satisfactory principles that they stand to-day practically unchanged and with few additions. From all his studies and experiments he finally formulated the principle that "bodies will be in equilibrio when their distance from the fulcrum or point of support is inversely as their weight." He is credited with having summed up his estimate of the capabilities of the lever with the well-known expression, "Give me a fulcrum on which to rest or a place on which to stand, and I will move the earth."

But perhaps the feat of all others that most appealed to the imagination of his contemporaries, and possibly also the one that had the greatest bearing upon the position of Archimedes as a scientific discoverer, was the one made familiar through the tale of the crown of Hiero. This crown, so the story goes, was supposed to be made of solid gold, but King Hiero for some reason suspected the honesty of the jeweller, and desired to know if Archimedes could devise a way of testing the question without injuring the crown. Greek imagination seldom spoiled a story in the telling, and in this case the tale was allowed to take on the most picturesque of phases. The philosopher, we are assured, pondered the problem for a long time without succeeding, but one day as he stepped into a bath, his attention was attracted by the overflow of water. A new train of ideas was started in his ever-receptive brain. Wild with enthusiasm he sprang from the bath, and, forgetting his robe, dashed along the streets of Syracuse, shouting: "Eureka! Eureka!" (I have found it!) The thought that had come into his mind was this: That any heavy substance must have a bulk proportionate to its weight; that gold and silver differ in weight, bulk for bulk, and that the way to test the bulk of such an irregular object as a crown was to immerse it in water. The experiment was made. A lump of pure gold of the weight of the crown was immersed in a certain receptacle filled with water, and the overflow noted. Then a lump of pure silver of the same weight was similarly immersed; lastly the crown itself was immersed, and of course—for the story must not lack its dramatic sequel—was found bulkier than its weight of pure gold. Thus the genius that could balk warriors and armies could also foil the wiles of the silversmith.

Whatever the truth of this picturesque narrative, the fact remains that some, such experiments as these must have paved the way for perhaps the greatest of all the studies of Archimedes—those that relate to the buoyancy of water. Leaving the field of fable, we must now examine these with some precision. Fortunately, the writings of Archimedes himself are still extant, in which the results of his remarkable experiments are related, so we may present the results in the words of the discoverer.

Here they are: "First: The surface of every coherent liquid in a state of rest is spherical, and the centre of the sphere coincides with the centre of the earth. Second: A solid body which, bulk for bulk, is of the same weight as a liquid, if immersed in the liquid will sink so that the surface of the body is even with the surface of the liquid, but will not sink deeper. Third: Any solid body which is lighter, bulk for bulk, than a liquid, if placed in the liquid will sink so deep as to displace the mass of liquid equal in weight to another body. Fourth: If a body which is lighter than a liquid is forcibly immersed in the liquid, it will be pressed upward with a force corresponding to the weight of a like volume of water, less the weight of the body itself. Fifth: Solid bodies which, bulk for bulk, are heavier than a liquid, when immersed in the liquid sink to the bottom, but become in the liquid as much lighter as the weight of the displaced water itself differs from the weight of the solid." These propositions are not difficult to demonstrate, once they are conceived, but their discovery, combined with the discovery of the laws of statics already referred to, may justly be considered as proving Archimedes the most inventive experimenter of antiquity.

Curiously enough, the discovery which Archimedes himself is said to have considered the most important of all his innovations is one that seems much less striking. It is the answer to the question, What is the relation in bulk between a sphere and its circumscribing cylinder? Archimedes finds that the ratio is simply two to three. We are not informed as to how he reached his conclusion, but an obvious method would be to immerse a ball in a cylindrical cup. The experiment is one which any one can make for himself, with approximate accuracy, with the aid of a tumbler and a solid rubber ball or a billiard-ball of just the right size. Another geometrical problem which Archimedes solved was the problem as to the size of a triangle which has equal area with a circle; the answer being, a triangle having for its base the circumference of the circle and for its altitude the radius. Archimedes solved also the problem of the relation of the diameter of the circle to its circumference; his answer being a close approximation to the familiar 3.1416, which every tyro in geometry will recall as the equivalent of pi.

Numerous other of the studies of Archimedes having reference to conic sections, properties of curves and spirals, and the like, are too technical to be detailed here. The extent of his mathematical knowledge, however, is suggested by the fact that he computed in great detail the number of grains of sand that would be required to cover the sphere of the sun's orbit, making certain hypothetical assumptions as to the size of the earth and the distance of the sun for the purposes of argument. Mathematicians find his computation peculiarly interesting because it evidences a crude conception of the idea of logarithms. From our present stand-point, the paper in which this calculation is contained has considerable interest because of its assumptions as to celestial mechanics. Thus Archimedes starts out with the preliminary assumption that the circumference of the earth is less than three million stadia. It must be understood that this assumption is purely for the sake of argument. Archimedes expressly states that he takes this number because it is "ten times as large as the earth has been supposed to be by certain investigators." Here, perhaps, the reference is to Eratosthenes, whose measurement of the earth we shall have occasion to revert to in a moment. Continuing, Archimedes asserts that the sun is larger than the earth, and the earth larger than the moon. In this assumption, he says, he is following the opinion of the majority of astronomers. In the third place, Archimedes assumes that the diameter of the sun is not more than thirty times greater than that of the moon. Here he is probably basing his argument upon another set of measurements of Aristarchus, to which, also, we shall presently refer more at length. In reality, his assumption is very far from the truth, since the actual diameter of the sun, as we now know, is something like four hundred times that of the moon. Fourth, the circumference of the sun is greater than one side of the thousand-faced figure inscribed in its orbit. The measurement, it is expressly stated, is based on the measurements of Aristarchus, who makes the diameter of the sun 1/170 of its orbit. Archimedes adds, however, that he himself has measured the angle and that it appears to him to be less than 1/164, and greater than 1/200 part of the orbit. That is to say, reduced to modern terminology, he places the limit of the sun's apparent size between thirty-three minutes and twenty-seven minutes of arc. As the real diameter is thirty-two minutes, this calculation is surprisingly exact, considering the implements then at command. But the honor of first making it must be given to Aristarchus and not to Archimedes.

We need not follow Archimedes to the limits of his incomprehensible numbers of sand-grains. The calculation is chiefly remarkable because it was made before the introduction of the so-called Arabic numerals had simplified mathematical calculations. It will be recalled that the Greeks used letters for numerals, and, having no cipher, they soon found themselves in difficulties when large numbers were involved. The Roman system of numerals simplified the matter somewhat, but the beautiful simplicity of the decimal system did not come into vogue until the Middle Ages, as we shall see. Notwithstanding the difficulties, however, Archimedes followed out his calculations to the piling up of bewildering numbers, which the modern mathematician finds to be the consistent outcome of the problem he had set himself.

But it remains to notice the most interesting feature of this document in which the calculation of the sand-grains is contained. "It was known to me," says Archimedes, "that most astronomers understand by the expression 'world' (universe) a ball of which the centre is the middle point of the earth, and of which the radius is a straight line between the centre of the earth and the sun." Archimedes himself appears to accept this opinion of the majority,—it at least serves as well as the contrary hypothesis for the purpose of his calculation,—but he goes on to say: "Aristarchus of Samos, in his writing against the astronomers, seeks to establish the fact that the world is really very different from this. He holds the opinion that the fixed stars and the sun are immovable and that the earth revolves in a circular line about the sun, the sun being at the centre of this circle." This remarkable bit of testimony establishes beyond question the position of Aristarchus of Samos as the Copernicus of antiquity. We must make further inquiry as to the teachings of the man who had gained such a remarkable insight into the true system of the heavens.

ARISTARCHUS OF SAMOS, THE COPERNICUS OF ANTIQUITY

It appears that Aristarchus was a contemporary of Archimedes, but the exact dates of his life are not known. He was actively engaged in making astronomical observations in Samos somewhat before the middle of the third century B.C.; in other words, just at the time when the activities of the Alexandrian school were at their height. Hipparchus, at a later day, was enabled to compare his own observations with those made by Aristarchus, and, as we have just seen, his work was well known to so distant a contemporary as Archimedes. Yet the facts of his life are almost a blank for us, and of his writings only a single one has been preserved. That one, however, is a most important and interesting paper on the measurements of the sun and the moon. Unfortunately, this paper gives us no direct clew as to the opinions of Aristarchus concerning the relative positions of the earth and sun. But the testimony of Archimedes as to this is unequivocal, and this testimony is supported by other rumors in themselves less authoritative.

In contemplating this astronomer of Samos, then, we are in the presence of a man who had solved in its essentials the problem of the mechanism of the solar system. It appears from the words of Archimedes that Aristarchus; had propounded his theory in explicit writings. Unquestionably, then, he held to it as a positive doctrine, not as a mere vague guess. We shall show, in a moment, on what grounds he based his opinion. Had his teaching found vogue, the story of science would be very different from what it is. We should then have no tale to tell of a Copernicus coming upon the scene fully seventeen hundred years later with the revolutionary doctrine that our world is not the centre of the universe. We should not have to tell of the persecution of a Bruno or of a Galileo for teaching this doctrine in the seventeenth century of an era which did not begin till two hundred years after the death of Aristarchus. But, as we know, the teaching of the astronomer of Samos did not win its way. The old conservative geocentric doctrine, seemingly so much more in accordance with the every-day observations of mankind, supported by the majority of astronomers with the Peripatetic philosophers at their head, held its place. It found fresh supporters presently among the later Alexandrians, and so fully eclipsed the heliocentric view that we should scarcely know that view had even found an advocate were it not for here and there such a chance record as the phrases we have just quoted from Archimedes. Yet, as we now see, the heliocentric doctrine, which we know to be true, had been thought out and advocated as the correct theory of celestial mechanics by at least one worker of the third century B.C. Such an idea, we may be sure, did not spring into the mind of its originator except as the culmination of a long series of observations and inferences. The precise character of the evolution we perhaps cannot trace, but its broader outlines are open to our observation, and we may not leave so important a topic without at least briefly noting them.

Fully to understand the theory of Aristarchus, we must go back a century or two and recall that as long ago as the time of that other great native of Samos, Pythagoras, the conception had been reached that the earth is in motion. We saw, in dealing with Pythagoras, that we could not be sure as to precisely what he himself taught, but there is no question that the idea of the world's motion became from an early day a so-called Pythagorean doctrine. While all the other philosophers, so far as we know, still believed that the world was flat, the Pythagoreans out in Italy taught that the world is a sphere and that the apparent motions of the heavenly bodies are really due to the actual motion of the earth itself. They did not, however, vault to the conclusion that this true motion of the earth takes place in the form of a circuit about the sun. Instead of that, they conceived the central body of the universe to be a great fire, invisible from the earth, because the inhabited side of the terrestrial ball was turned away from it. The sun, it was held, is but a great mirror, which reflects the light from the central fire. Sun and earth alike revolve about this great fire, each in its own orbit. Between the earth and the central fire there was, curiously enough, supposed to be an invisible earthlike body which was given the name of Anticthon, or counter-earth. This body, itself revolving about the central fire, was supposed to shut off the central light now and again from the sun or from the moon, and thus to account for certain eclipses for which the shadow of the earth did not seem responsible. It was, perhaps, largely to account for such eclipses that the counter-earth was invented. But it is supposed that there was another reason. The Pythagoreans held that there is a peculiar sacredness in the number ten. Just as the Babylonians of the early day and the Hegelian philosophers of a more recent epoch saw a sacred connection between the number seven and the number of planetary bodies, so the Pythagoreans thought that the universe must be arranged in accordance with the number ten. Their count of the heavenly bodies, including the sphere of the fixed stars, seemed to show nine, and the counter-earth supplied the missing body.

The precise genesis and development of this idea cannot now be followed, but that it was prevalent about the fifth century B.C. as a Pythagorean doctrine cannot be questioned. Anaxagoras also is said to have taken account of the hypothetical counter-earth in his explanation of eclipses; though, as we have seen, he probably did not accept that part of the doctrine which held the earth to be a sphere. The names of Philolaus and Heraclides have been linked with certain of these Pythagorean doctrines. Eudoxus, too, who, like the others, lived in Asia Minor in the fourth century B.C., was held to have made special studies of the heavenly spheres and perhaps to have taught that the earth moves. So, too, Nicetas must be named among those whom rumor credited with having taught that the world is in motion. In a word, the evidence, so far as we can garner it from the remaining fragments, tends to show that all along, from the time of the early Pythagoreans, there had been an undercurrent of opinion in the philosophical world which questioned the fixity of the earth; and it would seem that the school of thinkers who tended to accept the revolutionary view centred in Asia Minor, not far from the early home of the founder of the Pythagorean doctrines. It was not strange, then, that the man who was finally to carry these new opinions to their logical conclusion should hail from Samos.

But what was the support which observation could give to this new, strange conception that the heavenly bodies do not in reality move as they seem to move, but that their apparent motion is due to the actual revolution of the earth? It is extremely difficult for any one nowadays to put himself in a mental position to answer this question. We are so accustomed to conceive the solar system as we know it to be, that we are wont to forget how very different it is from what it seems. Yet one needs but to glance up at the sky, and then to glance about one at the solid earth, to grant, on a moment's reflection, that the geocentric idea is of all others the most natural; and that to conceive the sun as the actual Centre of the solar system is an idea which must look for support to some other evidence than that which ordinary observation can give. Such was the view of most of the ancient philosophers, and such continued to be the opinion of the majority of mankind long after the time of Copernicus. We must not forget that even so great an observing astronomer as Tycho Brahe, so late as the seventeenth century, declined to accept the heliocentric theory, though admitting that all the planets except the earth revolve about the sun. We shall see that before the Alexandrian school lost its influence a geocentric scheme had been evolved which fully explained all the apparent motions of the heavenly bodies. All this, then, makes us but wonder the more that the genius of an Aristarchus could give precedence to scientific induction as against the seemingly clear evidence of the senses.

What, then, was the line of scientific induction that led Aristarchus to this wonderful goal? Fortunately, we are able to answer that query, at least in part. Aristarchus gained his evidence through some wonderful measurements. First, he measured the disks of the sun and the moon. This, of course, could in itself give him no clew to the distance of these bodies, and therefore no clew as to their relative size; but in attempting to obtain such a clew he hit upon a wonderful yet altogether simple experiment. It occurred to him that when the moon is precisely dichotomized—that is to say, precisely at the half-the line of vision from the earth to the moon must be precisely at right angles with the line of light passing from the sun to the moon. At this moment, then, the imaginary lines joining the sun, the moon, and the earth, make a right angle triangle. But the properties of the right-angle triangle had long been studied and were well under stood. One acute angle of such a triangle determines the figure of the triangle itself. We have already seen that Thales, the very earliest of the Greek philosophers, measured the distance of a ship at sea by the application of this principle. Now Aristarchus sights the sun in place of Thales' ship, and, sighting the moon at the same time, measures the angle and establishes the shape of his right-angle triangle. This does not tell him the distance of the sun, to be sure, for he does not know the length of his base-line—that is to say, of the line between the moon and the earth. But it does establish the relation of that base-line to the other lines of the triangle; in other words, it tells him the distance of the sun in terms of the moon's distance. As Aristarchus strikes the angle, it shows that the sun is eighteen times as distant as the moon. Now, by comparing the apparent size of the sun with the apparent size of the moon—which, as we have seen, Aristarchus has already measured—he is able to tell us that, the sun is "more than 5832 times, and less than 8000" times larger than the moon; though his measurements, taken by themselves, give no clew to the actual bulk of either body. These conclusions, be it understood, are absolutely valid inferences—nay, demonstrations—from the measurements involved, provided only that these measurements have been correct. Unfortunately, the angle of the triangle we have just seen measured is exceedingly difficult to determine with accuracy, while at the same time, as a moment's reflection will show, it is so large an angle that a very slight deviation from the truth will greatly affect the distance at which its line joins the other side of the triangle. Then again, it is virtually impossible to tell the precise moment when the moon is at half, as the line it gives is not so sharp that we can fix it with absolute accuracy. There is, moreover, another element of error due to the refraction of light by the earth's atmosphere. The experiment was probably made when the sun was near the horizon, at which time, as we now know, but as Aristarchus probably did not suspect, the apparent displacement of the sun's position is considerable; and this displacement, it will be observed, is in the direction to lessen the angle in question.

In point of fact, Aristarchus estimated the angle at eighty-seven degrees. Had his instrument been more precise, and had he been able to take account of all the elements of error, he would have found it eighty-seven degrees and fifty-two minutes. The difference of measurement seems slight; but it sufficed to make the computations differ absurdly from the truth. The sun is really not merely eighteen times but more than two hundred times the distance of the moon, as Wendelein discovered on repeating the experiment of Aristarchus about two thousand years later. Yet this discrepancy does not in the least take away from the validity of the method which Aristarchus employed. Moreover, his conclusion, stated in general terms, was perfectly correct: the sun is many times more distant than the moon and vastly larger than that body. Granted, then, that the moon is, as Aristarchus correctly believed, considerably less in size than the earth, the sun must be enormously larger than the earth; and this is the vital inference which, more than any other, must have seemed to Aristarchus to confirm the suspicion that the sun and not the earth is the centre of the planetary system. It seemed to him inherently improbable that an enormously large body like the sun should revolve about a small one such as the earth. And again, it seemed inconceivable that a body so distant as the sun should whirl through space so rapidly as to make the circuit of its orbit in twenty-four hours. But, on the other hand, that a small body like the earth should revolve about the gigantic sun seemed inherently probable. This proposition granted, the rotation of the earth on its axis follows as a necessary consequence in explanation of the seeming motion of the stars. Here, then, was the heliocentric doctrine reduced to a virtual demonstration by Aristarchus of Samos, somewhere about the middle of the third century B.C.