Mathematicians From Greece

"Pappus proceeds as follows: ...If three straight lines are given in position, and if straight lines be drawn from one and the same point, making given angles with three given lines; and if there be given the ratio of the rectangle contained by two of the lines so drawn to the square of the other, the point lies on a solid locus given in position, namely, one of the three conic sections Again, if lines be drawn making given angles with four straight lines given in position, and if the rectangle of two of the lines so drawn bears a given ratio to the rectangle of the other two; then, in like manner, the point lies on a conic section given in position. It has been shown that to only two lines there corresponds a plane locus. But if there be given four lines, the point generates loci not known up to the present time (that is, impossible to determine by common methods), but merely called 'lines'. It is not clear what they are, or what their properties. One of them, not the first but the most manifest, has been examined, and this has proved to be helpful. These, however, are the propositions concerning them. If from any point straight lines be drawn making given angles with five straight lines given in position, and if the solid rectangular parallelepiped contained by three of the lines so drawn bears a given ratio to the sold rectangular parallelepiped contained by the other two and any given line whatever, the point lies on a 'line' given in position. Again, if there be six lines, and if the solid contained by three of the lines bears a given ratio to the solid contained by the other three lines, the point also lies on a 'line' given in position. But if there be more than six lines, we cannot say whether a ratio of something contained by four lines is given to that which is contained by the rest, since there is no figure of four dimensions."

"He is known in three instances to have copied theorems without giving due credit and that he may have done the same tiling in other cases in which we have no data by which to ascertain the real discoverer"

"But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis."

"Analysis... takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (ɣϵɣονός) and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (άνάπαλɩν λὐσɩν)."

"He Apollonius of Perga] spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought."

"The so called άναλυόμϵνος ('Treasury of Analysis') is... a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis."

"Ptolemy proved in the fifth book of his Syntaxis that, if the radius of the earth is taken as the unit, the greatest distance of the moon at the conjunctions is 64 10/60 of such units, the greatest distance of the sun 1210, the radius of the moon 17/60 33/602, the radius of the sun 5 30/60. Consequently, if the diameter of the moon is taken as the unit, the earth's diameter is 3 2/5 of such units, and the sun's diameter 18 4/5. That is to say, the diameter of the earth is 3 2/5 times the diameter of the moon, while the diameter of the sun is 18 4/5 times the diameter of the moon and 5 1/2 times the diameter of the earth. From these figures the ratios between the solid contents are manifest, since the cube on 1 is 1 unit, the cube on 3 2/5 is very nearly 39 1/4 of the same units, and the cube on 18 4/5 very nearly 6644 1/2, whence we infer that, if the solid magnitude of the moon is taken as a unit, that of the earth contains 39 1/4 and that of the sun 6644 1/2 of such units; therefore the solid magnitude of the sun is very nearly 170 times greater than that of the earth."

"Archimedes tells us that Aristarchus wrote a book of hypotheses, one of which was that the sun and the fixed stars remain unmoved and that the earth revolves round the sun in the circumference of a circle. Now Archimedes was a younger contemporary of Aristarchus; he must have seen the book of hypotheses in question, and we could have no better evidence for attributing to Aristarchus the first enunciation of the Copernican hypothesis."

"There is... no reason to doubt the unanimous testimony of antiquity that Aristarchus was the real originator of the Copernican hypothesis."

"Archimedes distinctly says in his Psammites or Sand-reckoner that Aristarchus was the first to discover that the apparent diameter of the sun is about 1/720th part of the complete circle described by it in the daily rotation, or, in other words, that the angular diameter is about 1/2°, which is very near the truth."

"One of the persons in the dialogue, being called to account for turning the world upside down, says that he is quite content so long as he is not accused of impiety, "like as Kleanthes held that Aristarchus of Samos ought to be accused of impiety for moving the hearth of the world as the man in order to save the phenomena supposed that the heavens stand still and the earth moves in an oblique circle at the same time as it turns round its axis.""

"The hypothesis of Aristarchus included the rotation of the earth, as might be expected."

"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."

"Apart from its [On the sizes and distances of the Sun and Moon] astronomical content, it is of the greatest interest for its geometry. Thoroughly classical in form and language, as befits the period between Euclid and Archimedes, it is the first extant specimen of pure geometry used with a trigonometrical object, and in this respect is a sort of forerunner of Archimedes' Measurement of a Circle."

"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..."

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

"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."

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

"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."

"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."

"In his book on sizes and distances Aristarchus lays down these six hypotheses: 1. That the moon receives 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. Now the first, third, and fourth of these hypotheses practically agree with the assumptions of Hipparchus and Ptolemy. For the moon is illuminated by the sun at all times except during an eclipse, when it becomes devoid of light through passing into the shadow which results from the interception of the sun's light by the earth, and which is conical in form; next the (circle) dividing the milk-white portion which owes its colour to the sun shining upon it and the portion which has the ashen colour natural to the moon itself is indistinguishable from a great circle (in the moon) when its positions in relation to the sun cause It to appear halved, at which times (a distance of) very nearly a quadrant on the circle of the zodiac is observed (to separate them); and the said dividing circle is in the direction of our eye, for this plane of the circle if produced will in fact pass through our eye in whatever position the moon is when for the first or second time it appears halved. But, as regards the remaining hypotheses, the aforesaid mathematicians have taken a different view. For according to them the earth has the relation of a point and centre, not to the sphere in which the moon moves, but to the sphere of the fixed stars, the breadth of the (earth's) shadow is not (that) of two moons, nor does the moon's diameter subtend one fifteenth part of a sign of the zodiac, that is, 2°. According to Hipparchus, on the one hand, the circle described by the moon is measured 650 times by the diameter of the moon, while the (earth's) shadow is measured by it 2½ times at its mean distance in the conjunctions; in Ptolemy's view, on the other hand, the moon's diameter subtends, when the moon is at its greatest distance, a circumference of 0° 31' 20", and when at its least distance, of 0° 35' 20", while the diameter of the circular section of the shadow is, when the moon is at its greatest distance, 0° 40' 40", and when the moon is at its least distance, 0° 46'. Hence it is that the authors named have come to different conclusions as regards the ratios both of the distances and of the sizes of the sun and moon."

"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 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 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 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."

"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 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 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 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 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 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."

"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)."

"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."

"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."

"[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."

"We are now in a position to prove the following propositions : — 1. 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); this follows from the hypothesis about the halved moon. 2. The diameter of the sun has the same ratio (as aforesaid) to the diameter of the moon. 3. The diameter of the sun has to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6; this follows from the ratio thus discovered between the distances, the hypothesis about the shadow, and the hypothesis that the moon subtends one fifteenth part of a sign of the zodiac."

"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."

"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."

"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."

"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]."

"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."

"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."

"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."

"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."