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 ½00 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