astronomers-from-greece

"‘Yet some few of such investigations we have in the five first propositions of Euclid’s thirteenth book … seems to be the work of Theo, […] rather than of Euclid himself.’"

"All our Greek texts of the Elements up to a century ago…purport in their titles to be either ‘from the edition of Theon’…or ‘from the lectures of Theon... [Greek commentaries] commonly speak of the writer of the Elements instead of using his name."

"that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements..."

"In speech articulate and logical, in her actions prudent and public-spirited. The city gave her suitable welcome and accorded her special respect."

"Most people, both Christian and non-Christian, saw Hypatia’s killing as a brutal, unprovoked murder that exploded out of a toxic set of circumstances for which Hypatia bore little responsibility."

"Hypatia was a University lecturer denounced by Church dignitaries and torn to pieces by Christians. Such will probably be the fate of this book: therefore it bears her name. What I have written here I believe and shall not retract or change for similar episcopal denunciations."

"I have composed a book on the length of the year in which I show that the tropical year contains 365 days plus a fraction of a day which is not exactly 1/4 days as the mathematicians-astronomers suppose, but which is less than 1/4 by about 1/300."

"In comparison with the great size of the earth the protrusion of mountains is not sufficient to deprive it of its spherical shape or to invalidate measurements based on its spherical shape. For Eratosthenes shows that the perpendicular distance from the highest mountain tops to the lowest regions is ten stades [c.5,000-5,500 feet]. This he shows with the help of dioptras which measure magnitudes at a distance."

"Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces."

"[Eratosthenes] ... is a mathematician among geographers, and yet a geographer among mathematicians; and consequently on both sides he offers his opponents occasions for contradiction."

"Eratosthenes declares that it is no longer necessary to inquire as to the cause of the overflow of the Nile, since we know definitely that men have come to the sources of the Nile and have observed the rains there."

"Eratosthenes... knew that the Sun was straight overhead in... Syene at noon on the , but that it was 7.2 degrees south of straight overhead in , located 794 kilometers farther north. He concluded... 794 kilometers corresponded to 7.2 degrees out of the 360 degrees... around Earth's circumference, so that the circumference must be 794 km x 360°/7.2°≈39,700 km... remarkably close to the modern value of 40,000 km. Amusingly Christopher Columbus totally bungled this... confusing Arabic miles with Italian miles..."

"The most important name from the point of view of this chapter is Hippocrates of Chios. He is indeed the first person of whom it is recorded that compiled a book of Elements. This is lost, but Simplicius has preserved in his commentary on the Physics of Aristotle a fragment from Eudemus's History of Geometry giving an account of Hippocrates's quadratures of certain lunules or lunes."

"One would suppose that the relation between the pseudo-didactic and the didactic syllogism, was the same as that between the pseudo-dialectic and the dialectic; so that, if the pseudo-dialectic deserved to be called sophistic or , the pseudo-didactic would deserve these appellations also; especially, since the formal conditions of the syllogism are alike for both. This Aristotle does not admit, but draws instead a remarkable distinction. The (he says) is a dishonest man, making it his professional purpose to deceive; the pseudo-graphic man of science is honest always, though sometimes mistaken. So long as the pseudo-graphic syllogism keeps within the limits belonging to its own special science, it may be false, since the geometer may be deceived even in his own science [of] geometry, but it cannot be sophistic or eristic; yet whenever it transgresses those limits, even though it be true and though it solves the problem proposed, it deserves to be called by those two epithets. Thus, there were two distinct methods proposed for the quadrature of the circle—one by Hippokrates, on geometrical principles, the other by Bryson, upon principles extra-geometrical. Both demonstrations were false and unsuccessful; yet that of Hippokrates was not sophistic or eristic, because he kept within the sphere of geometry; while that of Bryson was so, because it travelled out of geometry. Nay more, this last would have been equally sophistic and eristic, and on the same ground, even if it had succeeded in solving the problem. If indeed the pseudo-graphic syllogism be invalid in form, it must be considered as sophistic, even though within the proper scientific limits as to [the] matter; but, if it be correct in form and within these same limits, then however untrue its premisses may be, it is to be regarded as not sophistic or eristic."

"[Hippocrates] elaborated the geometry of the circle: proving, among other propositions, that similar segments of a circle contain equal angles; that the angle subtended by the chord of a circle is greater than, equal to, or less than a right angle as the segment of the circle containing it is less than, equal to, or greater than a semicircle (Euc. III, 31); and probably several other of the propositions in the third book of Euclid. It is most likely that he also established the propositions that [similar] circles are to one another as the squares of the diameters (Euc. XII, 2), and that similar segments are as the squares of their chords. The proof given in Euclid of the first of these theorems is believed to be due to Hippocrates."

"Of original writings... we have only a fragment concerning the lunes of Hippocrates, quoted by Simplicius... and taken from Eudemus's lost History of Geometry..."

"Hippocrates... denoted the square on a line by the word... power which it still retains in algebra."

"Hippocrates of Chios who lived in Athens in the second half of the fifth century B.C., and wrote the first text book on Geometry, was the first to give examples of curvilinear areas which admit of exact quadrature. These figures are the menisci or lunulae of Hippocrates."

"It would appear that Hippocrates was in Athens during a considerable portion of the second half of the fifth century, perhaps from 450 to 430 B.C. We have quoted the story that what brought him there was a suit to recover a large sum which he had lost, in the course of his trading operations, through falling in with pirates; he is said to have remained in Athens on this account a long time, during which he consorted with the philosophers and reached such a degree of proficiency in geometry that he tried to discover a method of squaring the circle. This is of course an allusion to the quadratures of lunes."

"If on the sides of a right-angled triangle ACB semi-circles are described on the same side, the sum of the areas of the two lunes AEC, BDC is equal to that of the triangle ACB. If the right-angled triangle is isosceles, the two lunes are equal, and each of them is half the area of the triangle. Thus the area of the lunula is found."

"He was the first to observe that the problem of doubling the cube is reducible to that of finding two mean proportionals in continued proportion between two straight lines. The effect of this was, as Proclus says, that thenceforward people addressed themselves (exclusively) to the equivalent problem of finding two mean proportionals between two straight lines."

"If AC = CD = DB = radius OA (see Fig. 3), the semi-circle ACE is ¼ of the semi-circle ACDB. We have now◐AB - 3◐AC = [trapezium] ACDB - 3 · meniscus ACE, [where the meniscus is the lunulae, i.e., lune] and each of these expressions is ¼◐AB or half the circle on ½AB as diameter. If then the meniscus AEC were quadrable so also would be the circle on ½AB as diameter. Hippocrates recognized the fact that the meniscus is not quadrable, and he made attempts to find other quadrable lunulae in order to make the quadrature of the circle depend on that of such quadrable lunulae."

"He commenced by finding the area of a lune contained between a semicircle and a quadrilateral arc standing on the same chord... as follows. Let ABC be an isosceles right-angled triangle inscribed in the semicircle ABOC, whose centre is O. On AB and AC as diameters describe semicircles as in the figure. Then, since by Euc. I, 47,sq. on\,BC = sq. on\,AC + sq. on\,AB,therefore, by Euc. XII, 2,area\;\frac{1}{2} \bigodot on\,BC = area\;\frac{1}{2} \bigodot on\,AC + area\;\frac{1}{2} \bigodot on\,ABTake away the common parts.\therefore area\,\triangle ABC = sum\;of\;areas\;of\;lunes\;AECD\;and\;AFBG.Hence the area if the lune AECD is equal to half that of the triangle ABC."

"In [his] textbook Hippocrates introduced the method of "reducing" one theorem to another, which being proved, the thing proposed necessarily follows; of this method the ' is an illustration. No doubt the principle had been used occasionally before, but he drew attention to it as a legitimate mode of proof which was capable of numerous applications."

"He next inscribed half a regular hexagon ABCD in a semicircle whose centre was O, and on OA, AB, BC, and CD as diameters described semicircles of which those on OA and AB are drawn in the figure [2]. Then AD [by equilateral triangles within the half-hexagon] is double any of the lines OA, AB, BC, and CD,\therefore\;square\;on\;AD = sum\;of\;sqs.\;on\;OA, AB, BC, and\;CD,\therefore\;area\;\frac{1}{2} \bigodot on\,ABCD = sum\;of\;areas\;of\;\frac{1}{2} \bigodot s\;on\;OA, AB, BC, and\;CD.Take away the common parts\therefore\;area\;trapezium\;ABCD = 3\;lune\;AEBF + \frac{1}{2} \bigodot on\;OA.If therefore the area of this latter lune be known, so is that of the semicircle described on OA as diameter."

"Hippocrates of Chios... was one of the greatest of the Greek geometricians. He... began life as a merchant. The accounts differ as to whether he was swindled by the Athenian custom-house officials who were stationed at the Chersonese, or whether one of his vessels was captured by an Athenian pirate near ... somewhere about 430 B.C. he came to Athens to try to recover his property in the law courts. ...the Athenians seem only to have laughed at him for his simplicity, first in allowing himself to be cheated, and then in hoping to recover his money. While prosecuting his cause he attended the lectures of various philosophers, and finally (in all probability to earn a living) opened a school of geometry himself. He seems to have been well acquainted with the Pythagorean philosophy, though there is no sufficient authority that he was ever initiated as a Pythagorean."

"The circle being after rectilineal figures, the most simple in appearance, geometricians very naturally soon began to seek for its measure. Thus we find that the philosopher Anaxagoras occupied himself with the question in prison. Then Hippocrates of Chios tried the same problem, and it led him to the discovery of what is called the lune, a surface in the shape of a crescent, bounded by two arcs and exactly equal to a given square. He also found two unequal lines which were together equal to a rectilineal figure, so that if their relation could have been found the solution of the problem would have been obtained. But this no one has yet been able to do, nor is it likely ever to be done."

"[Hippocrates] wrote the first elementary text-book of geometry... on which probably Eudlid's Elements was founded; and therefore he may be said to have sketched out the lines on which geometry is still taught in English schools."

"The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation (of lunes) to the circle, were first investigated by Hippocrates, and his exposition was thought to be in correct form... He started with, and laid down as the first of the theorems useful for his purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases have... And this he proved by first showing that the squares on the diameters have the same ratio as the circles. For, as the circles are to one another, so also are similar segments of them. For similar segments are those which are the same part of the circles respectively, as for instance a semicircle is similar to a semicircle, and a third part of a circle to a third part... It is for this reason also... that similar segments contain equal angles...'"

"Hippocrates also enunciated various other theorems connected with lunes... of which the theorem last given is a typical example. I believe that they are the earliest examples in which areas bounded by curves were determined by geometry."

"The most celebrated discoveries of Hippocrates were... in connection with the quadrature of the circle and the duplication of the cube, and owing to his influence these problems played a prominent part in the history of the Athenian school."

"Hippocrates of Chios... the most famous mathematician of his century... is credited with the idea of arranging theorems so that later ones can be proven on the basis of earlier ones, in the manner familiar to us from... Euclid. He is also credited with introducing the indirect method of proof into mathematics. His text on geometry, called the Elements, is lost."

"Hippocrates of Chios... attempted the solution [for squaring the circle] and was the first actually to square a curvilinear figure. He constructed semicircles on the three sides of an isosceles right-angled triangle and showed that the sum of the two lunes thus formed is equal to the area of the triangle itself. ...His proof involves the proposition that the areas of circles are proportional to the squares of their diameters,—a proposition which Eudemus... tells us that Hippocrates proved. To the quadrature problem as such, however, his contribution was not important."

"The history of the Athenian school begins with the teaching of Hippocrates about 420 B.C."

"It is supposed that the use of letters in diagrams to describe a figure was made by him or introduced about this time, as he employs expressions such as "the point on which letter A stands" and "the line on which AB is marked.""

"The sun is pure fire: so Posidonius in the seventh book of his Celestial Phenomena. And it is larger than the earth, as the same author says in the sixth book of his Physical Discourse. Moreover it is spherical in shape like the world itself according to this same author and his school."

"There are never any occasions when you need think yourself safe because you wield the weapons of Fortune; fight with your own! Fortune does not furnish arms against herself; hence men equipped against their foes are unarmed against Fortune herself."

"Riches are a cause of evil, not because, of themselves, they do any evil, but because they goad men on so that they are ready to do evil."

"A single day among the learned lasts longer than the longest life of the ignorant."

"Things which bestow upon the soul no greatness or confidence or freedom from care are not goods. But riches and health and similar conditions do none of these things; therefore, riches and health are not goods. Things which bestow upon the soul no greatness or confidence or freedom from care, but on the other hand create in it arrogance, vanity, and insolence, are evils. But things which are the gift of Fortune drive us into these evil ways. Therefore these things are not goods."

"When men were scattered over the earth, protected by eaves or by the dug-out shelter of a cliff or by the trunk of a hollow tree, it was philosophy that taught them to build houses."

"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 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."

"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..."

"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."

"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."

"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."