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 ca.n.a.ls 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 a.s.sertions, 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 surpa.s.ses 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 a.s.sault 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 mult.i.tude 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 pa.s.s 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 sh.o.r.e 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 a.s.sault and defend, in all manner of sieges and a.s.saults. 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 a.s.saulted 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 a.s.sault 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 a.s.sault 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 embarra.s.sed," 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 a.s.sociated 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 ma.s.sy 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-gla.s.ses 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 a.s.sured, 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 ma.s.s 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 circ.u.mscribing 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 circ.u.mference of the circle and for its alt.i.tude the radius. Archimedes solved also the problem of the relation of the diameter of the circle to its circ.u.mference; 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...o...b..t, making certain hypothetical a.s.sumptions 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 a.s.sumptions as to celestial mechanics. Thus Archimedes starts out with the preliminary a.s.sumption that the circ.u.mference of the earth is less than three million stadia.

It must be understood that this a.s.sumption 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 a.s.serts that the sun is larger than the earth, and the earth larger than the moon. In this a.s.sumption, he says, he is following the opinion of the majority of astronomers. In the third place, Archimedes a.s.sumes 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 a.s.sumption 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 circ.u.mference of the sun is greater than one side of the thousand-faced figure inscribed in its...o...b..t. The measurement, it is expressly stated, is based on the measurements of Aristarchus, who makes the diameter of the sun 1/170 of its...o...b..t. 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 doc.u.ment 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 pa.s.sing 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 s.p.a.ce so rapidly as to make the circuit of its...o...b..t 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.

It must be understood that in following out the steps of reasoning by which we suppose Aristarchus to have reached so remarkable a conclusion, we have to some extent guessed at the processes of thought-development; for no line of explication written by the astronomer himself on this particular point has come down to us. There does exist, however, as we have already stated, a very remarkable treatise by Aristarchus on the Size and Distance of the Sun and the Moon, which so clearly suggests the methods of reasoning of the great astronomer, and so explicitly cites the results of his measurements, that we cannot well pa.s.s it by without quoting from it at some length. It is certainly one of the most remarkable scientific doc.u.ments of antiquity. As already noted, the heliocentric doctrine is not expressly stated here. It seems to be tacitly implied throughout, but it is not a necessary consequence of any of the propositions expressly stated. These propositions have to do with certain observations and measurements and what Aristarchus believes to be inevitable deductions from them, and he perhaps did not wish to have these deductions challenged through a.s.sociating them with a theory which his contemporaries did not accept. In a word, the paper of Aristarchus is a rigidly scientific doc.u.ment unvitiated by a.s.sociation with any theorizings that are not directly germane to its central theme. The treatise opens with certain hypotheses as follows: