Astronomers From Greece

"Archimedes... does not attempt to argue for or against this hypothesis, he merely objects to the unmathematical idea of there being a certain ratio between a point, which has no magnitude, and the surface of a sphere. Of course the meaning of Aristarchus is clear enough, that if we suppose the earth to move round the sun in a large orbit, the distance of the fixed stars must be immensely great as compared with that of the sun, as our motion round the latter would otherwise produce apparent displacements parallax] among the stars, if they are at different distances from the centre of the world, or at any rate, if they are on the surface of a sphere, make the stars in the neighbourhood of the ecliptic appear to close up or spread out according as the earth is at the part of its orbit farthest from them or nearest to them. This is indeed a very startling hypothesis to meet with so far back as the third century before our era... It would almost seem that there was nothing more to say; that Aristarchus had merely thrown out this suggestion or hypothesis without devoting a book or essay to its discussion, and the fact, that his book on the distance of the sun does not contain anything on the subject, tends to confirm this impression."

"You know that according to most astronomers the world is the sphere, of which the centre is the centre of the earth, and whose radius is a line from the centre of the earth to the centre of the sun. But Aristarchus of Samos has published in outline certain hypotheses, from which it follows that the world is many times larger than that. For he supposes that the fixed stars and the sun are immovable, but that the earth is carried round the sun in a circle which is in the middle of the course; but the sphere of the fixed stars, lying with the sun round the same centre, is of such a size that the circle, in which he supposes the earth to move, has the same ratio to the distance of the fixed stars as the centre of the sphere has to the surface. But this is evidently impossible, for as the centre of the sphere has no magnitude, it follows that it has no ratio to the surface. It is therefore to be supposed that Aristarchus meant that as we consider the earth as the centre of the world, then the earth has the same ratio to that which we call the world, as the sphere in which is the circle, described by the earth according to him, has to the sphere of the fixed stars."

"We have to depend on statements of subsequent writers when we endeavor to give Aristarchus his proper place in the history of cosmical systems."

"The only book of his which has been preserved is a treatise "On the dimensions and distances of the sun and moon," in which we find the results of the first serious attempt to determine these quantities by observation. He observed the angular distance between the sun and the moon at the time when the latter is half illuminated (when the angle at the moon in the triangle earth-moon-sun is a right angle) and found it equal to twenty-nine thirtieths of a right angle or 87°. From this he deduced the result that the distance of the sun was between eighteen and twenty times as great as the distance of the moon. Although this result is exceedingly erroneous, we see at any rate that Aristarchus was more than a mere speculative philosopher, but that he must have been an observer as well as a mathematician. This treatise does not contain the slightest allusion to any hypothesis on the planetary system..."

"As related by Archimedes in the "sand-counter", Aristarchus advanced the bold hypothesis that the earth rotates in a circle about the sun. Most astronomers rejected this... as Archimedes tells us also. [I]n view of the status of mechanics at the time, there are weighty arguments against the motion of the earth... already found in Aristotle and, developed more fully, in Ptolemy. If the earth had such an enormously rapid motion, says Ptolemy, then everything that was not clinched to and riveted to the earth, would fall behind and would therefore appear to fly off in the opposite direction. Clouds... would be overtaken by the rotation of the earth and would lag behind. ...[T]here is nothing to be said against this since the Greeks did not know the law of inertia and required a force to account for every motion. If the earth does not drag the clouds along, they have to lag behind. We do not know how Aristarchus met these arguments."

"Of the two mathematicians Aristarchus of Samos and Seleucus of Babylon, whose systems came most nearly to his own, he Copernicus] mentions only the first, making no reference to the second. It has often been asserted that he was not acquainted with the views of Aristarchus of Samos regarding the central sun and the condition of the earth as a planet, because the Arenarius, and all the other works of Archimedes, appeared only one year after his death, and a whole century after the invention of the art of printing; but it is forgotten that Copernicus, in his dedication to Pope Paul III., quotes a long passage on Philolaüs, Ecphantus, and Heraclides of Pontus, from Plutarch's work on The Opinions of Philosophers (III., 13), and therefore that he might have read in the same work (II., 24) that Aristarchus of Samos regards the sun as one of the fixed stars."

"[Aristarchus] is so obscure that Wallis was obliged to annotate him from one end to the other, in the effort to make him intelligible."

"The theory of Copernicus not only founded modern system of astronomy, but made men to examine other articles of their creed, after were thus convinced that they had taught and believed the earth to be stationary 6000 years. All the opinions of the ancients respecting the motion of the earth were speculative hypotheses, arising from the Pythagorean school, which, as we know, considered fire the centre of the world, round which all was moving. Thus we ought explain the passage of Aristarchus of Samos, mentioned by Aristotle in his Arenario. Aristarchus, a Pythagorean, held the idea that the earth revolves round its axis, and at the same time, in an oblique circle round the sun; and that the distance of the stars is so great, that this circle is but a point in comparison with their orbits, and therefore the motion of the earth produces no apparent motion in them. Every Pythagorean might have entertained this idea, who considered the sun or fire as the centre of the world, and who was, at the same time, so correct a thinker, and so good an astronomer, as Aristarchus of Samos. But this was not the Copernican system of the world. It was the motions the planets, their stations, and their retrogradations, which astronomers could not explain, and which led them to the complicated motions of the epicycles, in which the planets moved in cycloids round the earth."

"Aristarchus... was a contemporary of Euclid. His fame rests on his heliocentric theory... Perhaps "theory" is too strong a word, for his proofs were weak; yet it was a great idea, an idea redeveloped centuries later by Copernicus. ...an observer on Earth sees a half-moon only when ∠EMS [the angle between the Earth and Sun, from the persective of the Moon] is a right angle. ...sometimes both Sun and Moon are visible when the phase of the Moon is half-moon. So? Measure ∠MES [the angle between the Moon and Sun, from the persective of earth]... This is what Aristarchus did. ...the third angle is determined [since the sum of the three angles of any triangle is 180º], so the shape but not the size of ⃤ EMS [triangle formed by Earth-Moon-Sun] is known. Consequently, although the actual length of any side is not determinate, the ratio of any pair is. ...[Moon-Earth distance/Sun-Earth distance] ME/SE = cos α [where ∠MES = α]."

"It was the ancient opinion of not a few in the earliest ages of philosophy, That the fixed stars stood immovable in the highest parts of the world; that under the Fixed Stars the Planets were carried about the Sun; that the Earth, as one of the Planets, described the annual course about the Sun, while a diurnal motion it was in the mean time revolved about its own axe; and that the Sun, as the common fire which served to warm the whole, was fixed in the center of the Universe. This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans. And this was the judgment of Anaximander, more ancient than any of them, and of that wise king of the Romans, Numa Pompilius; who, as a symbol of the figure of the World with the Sun in the center, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it."

"Copernicus, Kepler and Galileo were ‘revisionists’ in rejecting the geocentric system of Ptolemy (which held sway for some 1500 years) and, against an oppressive and repressive mainstream opinion (and officialdom), reinstated—with improvements—the heliocentric system of Aristarchos of Samos (3rd cent BCE)."

"You (King Gelon) are aware the 'universe' is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the Floor, and that the sphere of the fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface."

"Proposition 18. The earth is to the moon in a [volume] ratio greater than [15.8] that which 1259712 has to 79507, but less than [31.5] that which 216000 has to 6859."

"Proposition 17. The diameter of the earth is to the diameter of the moon in a ratio greater than [2.51] that which 108 has to 43, but less than [3.16] that which 60 has to 19."

"Proposition 16. The sun has to the earth a [volume] ratio greater than [254] that which 6859 has to 27, but less than [368] that which 79507 has to 216."

"Proposition 15. The diameter of the sun has, to the diameter of the earth a ratio greater than [6.3] that which 19 has to 3, but less than [7.2] that which 43 has to 6."

"Proposition 14. The straight line joined from the centre of the earth to the centre of the moon has to the straight line cut off from the axis towards the centre of the moon by the straight line subtending the (circumference) within the earth's shadow a ratio greater than that which 675 has to 1."

"Proposition 13. The straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the moon move is less than double of the diameter of the moon, but has to it a ratio greater than [2.0] that which 88 has to 45; and it is less than 1/9th part of the diameter of the sun, but has to it a ratio greater than [1/10th] that which 22 has to 225. But it has to the straight line drawn from the centre of the sun at right angles to the axis and meeting the sides of the cone a ratio greater than [0.097] that which 979 has to 10125."

"Proposition 12. The diameter of the circle which divides the dark and the bright portions in the moon is less than the diameter of the moon, but has to it a ratio greater than [0.99] that which 89 has to 90."

"Proposition 11. The diameter of the moon is less than 2/45ths [0.04], but greater than 1/30th [0.03] of the distance of the centre of the moon from our eye."

"Proposition 10. The sun has to the moon a [volume] ratio greater than that which 5832 has to 1, but less than that which 8000 has to 1."

"Proposition 9. The diameter of the sun is greater than 18 times, but less than 20 times, the diameter of the moon."

"Proposition 8. When the sun is totally eclipsed, the sun and the moon are then comprehended by one and the same cone which has its vertex at our eye."

"Proposition 7. The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth."

"Proposition 6. The moon moves (in an orbit) lower than (that of) the sun, and, when it is halved, is distant less than a quadrant from the sun."

"Proposition 5. When the moon appears to us halved, the great circle parallel to the circle which divides the dark and the bright portions in the moon is then in the direction of our eye; that is to say, the great circle parallel to the dividing circle and our eye are in one plane."

"Proposition 4. The circle which divides the dark and the bright portions in the moon is not perceptibly different from a great circle in the moon."

"Proposition 3. The circle in the moon which divides the dark and the bright portions is least when the cone comprehending both the sun and the moon has its vertex at our eye."

"Proposition 2. If a sphere be illuminated by a sphere greater than itself, the illuminated portion of the former sphere will be greater than a hemisphere."

"Proposition 1. Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres."

"[Hypotheses] 1. That the Moon receives its light from the sun. 2. That the earth is in the relation of a point and centre to the sphere in which the moon moves. 3. That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye. 4. That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant. 5. That the breadth of the (earth's) shadow is (that) of two moons. 6. That the moon subtends one fifteenth part of a sign of the zodiac."

"Such was the celebrity of this philosopher, that many illustrious names appear in the train of his disciples, particularly Philolaus, Eudoxus, and Plato. ...He excelled, not only in speculative philosophy, but in geometry and mechanics, He is said to have invented a kind of winged automaton, and several curious hydraulic machines. He was in such high reputation for moral and political wisdom, that, contrary to the usual custom, he was appointed seven different times to the supreme magistracy in Tarentum."

"That tho' a Man were admitted into Heaven to view the wonderful Fabrick of the World, and the Beauty of the stars, yet what would otherwise be Rapture and Extasie, would be but melancholy Amazement if he had not a Friend to communicate it to."

"The unwritten laws of the gods were promulgated against depraved manners, inflicting a severe destiny and penalty on the disobedient; and these unwritten laws are the fathers and leaders of those that are written, and of the dogmas established by men."

"Simplicius attributes to him a work on Opposites, to which he says that Aristotle was indebted for what he says on this subject in his Categories. His Harmonicon is quoted by Nicomachus in his Arithmetic. ...Fragments of the works attributed to him "On the Good and Happy Man," and "On Wisdom," are... extant. ...A letter of Archytas to Plato and Plato's reply are preserved by Laertius."

"Among the mathematical problems Archytas solved or attempted to solve the duplication of the cube, for which purpose he attempted to find two mean proportionals between the two right lines formed by the section of a cylinder, as Laertius expresses it. ...Among his mechanical inventions is mentioned a wooden pigeon that could fly, of which Gellius speaks particularly. The invention of a rattle, perhaps a child's toy, is also attributed to him."

"The fragments of the works "On the Good and Happy Man," and "On Wisdom," were published by T. Gale in 1670, and were given again with other things in his "Opuscula Mythologica," Cambridge, 1671, 8vo.; Amsterdam, 1688, 8vo. The fragment of the Greek text of the work on the "Nature of the Universe," was published at Venice, 1561, 8vo.; with a Latin version by Dom. Pizimentius, under the title "Architæ Tar. X. Prædicamenta." ...A complete collection of the fragments was published by I. Cn. Orelli, Leipzig, 1821, 8vo. The "Political Fragments of Archytas, Charondas, &c., translated from the Greek by Thomas Taylor, was published at London, 1822, 8vo. There is a work by Nic. T. Reimer intitled "Archytas, eiusque Solutio Problematis Cubi Duplicationis," Göttingen, 1798, 8vo."

"Of his writings none remain except a metaphysical work, "On the Nature of the Universe," in which he has explained the predicaments; and sundry fragments "On Wisdom," and "On the Good and Happy Man," preserved by Stobaeus and edited from him by Gale."

"His death, which is said to have been occasioned by a shipwreck, is made a subject of poetical description by Horace, who celebrates him as an eminent geographer and astronomer."

"Concerning the philosophical tenets of Archytas... Aristotle, who was an industrious collector from the Pythagoreans, borrowed from him the general arrangements which are usually called his ten categories."

"The sum of his moral doctrine is; that virtue is to be pursued for its own sake in every condition of life; that all excess is inconsistent with virtue; that the mind is more injured by prosperity than by adversity; and that there is no pestilence so destructive to human happiness as pleasure."

"It is probable that Aristotle was indebted to Archytas for many of his moral ideas; particularly for the notion which runs through his ethical pieces, that virtue consists in avoiding extremes."

"Even thee, thou measurer of earth and sea, thou counter of the sands, Archytas, how small a portion of earth contains thee now! It profits thee not to have searched the air and traversed the heavens since thou wert to die. So Tantalus, Tithonus, and Minos have died, and Pythagoras too with all his learning hath gone down once more to the grave. But so it is: all must die alike; some to make sport for Mars, some swallowed up in the deep: old and young go crowding to the grave: none escape: I too have perished in the waters. But grudge me not, thou mariner, a handful of earth: so may the storm spend itself on the woods while thou art safe and thy merchandize increases. Is it a small matter with thee to bring ruin on thy children? Yea, perhaps retribution awaits thyself: my curses will be heard, and then no atonement shall deliver thee. 'Tis but the work of a moment—thrice cast earth upon me and hasten on."

"Te maris et terrae numeroque carentis arenae Mensorem cohibent, Archyta, Pulveris exiqui prope litus parva Matinum Munera, nec quidquam tibi prodest The sea, the earth, the innumerable sand, Archytas, thou coulds't measure; now, alas! A little dust on Matine shore has spann'd That soaring spirit; vain it was to pass The gates of heaven, and send thy soul in quest O'er air's wide realms; for thou hadst yet to die."

"There cannot be a single, simple body which is infinite, either, as some hold, one distinct from the elements, which they then derive from it, nor without this qualification. For there are some who make this (i.e. a body distinct from the elements) the infinite, and not air or water, in order that the other things may not be destroyed by their infinity. They are in opposition one to another — air is cold, water moist, and fire hot—and therefore, if any one of them were infinite, the rest would have ceased to be by this time. Accordingly they say that what is infinite is something other than the elements, and from it the elements arise."

"All things must in equity again decline into that whence they have their origin for they must give satisfaction and atonement for injustice each in the order of time."

"In Antiquity, Anaximander understood that the sky continues beneath our feet long before ships had circumnavigated the Earth. ...Only one small original fragment of his writings has survived... Things are transformed one into the other according to necessity, and render justice to one another according to the order of time. From one of the crucial, initial moments of natural science there remains nothing but... this appeal to "the order of time.""

"Anaximandros... pupil and companion of Thales, was like him an astronomer, geographer, and physicist, seeking for a first principle (for which he invented the name); affirming an infinite material cause, without beginning and indestructible, with an infinite number of worlds;—and still showing the Chaldean impulse—speculating curiously on the descent of man from something aquatic, as well as on the form and motion of the earth (figured by him as a cylinder), the nature and motions of the solar system, and thunder and lightning. It seems doubtful whether, as affirmed by Eudemus, he taught the doctrine of the earth's motion; but that this doctrine was derived from the Babylonian schools of astronomy is so probable that it may have been accepted in Miletos in his day."

"Anaximander the Milesian, affirmed the infinite to be the first principle, and that all things are generated out of it, and corrupted again into it. His infinite is nothing else but matter."

"Anaximander displays all the symptoms of the intellectual fever spreading through Greece. His universe is no longer a closed box, but infinite in extension and duration. The raw material is none of the familiar forms of matter but a substance without definite properties except for being indestructible and everlasting. Out of this stuff all things are developed, and into it they return... infinite multitudes of other universes have already existed, and been dissolved again into the amorphous mass. The earth is a cylindrical column, surrounded by air; it floats upright... without support or anything to stand on, yet it does not fall because, being in the centre, it has no preferred direction... if it did, this would disturb the symmetry and balance of the whole. The spherical heavens enclose the atmosphere "like the bark of a tree", and there are several layers... to accommodate the various stellar objects. ...The sun is merely a hole... the moon... it phases... due to recurrent partial stoppages of the puncture, and so are the eclipses. The stars are pin-holes in a dark fabric through which we glimpse the cosmic fire filling the space between two layers of "bark". ...it is the first approach to a mechanical model of the universe. ...yet the machinery looks like it had been dreamed up by a surrealist painter... closer to Picasso than to Newton."