Mathematicians 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.’"

"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..."

"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."

"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."

"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."

"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."

"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... 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..."

"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] ... 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."

"Such... being the middle nature of the soul, Plato, with great propriety, in the Phædrus, and in his tenth book of laws, defines it to be number moving itself; which definition he received from Philolaus, and Philolaus from Pythagoras."

"Plato was instructed in their nature by Philolaus his preceptor, and the disciple of Pythagoras."

"Pythagoras, in the sacred discourse, calls number "the ruler of forms and ideas." But Philolaus, "the commanding and self-begotten container of the eternal duration of mundane concerns." And and all those... under the probation of the quinquennial silence... "the first exemplar of the mundane fabric, and the judiciary instrument of its artificer.""

"The importance of the decuple system in relation to the Pythagoreans is much greater. For as they considered numbers over ten to be only the repetition of the first ten numbers, all numbers and all powers of numbers appeared to them to be comprehended in the decad, which is therefore called by Philolaus, great all-powerful and all-producing, the beginning and the guide of the divine and heavenly, as of the terrestrial life."

"Fragment 2. All things, at least those we know, contain number; for it is evident that nothing whatever can either be thought or known, without number. Number has two distinct kinds: the odd, and the even, and a third, derived from a mingling of the other two kinds, the even-odd. Each of its subspecies is susceptible of many very numerous varieties; which each manifests individually."

"According to the Pythagorean Philolaus, "the Dekad, the full and perfect number, was of supreme and universal efficacy as the guide and principle of life, both to the Kosmos and to man. The nature of number was imperative and lawgiving, affording the only solution of all that was perplexing or unknown; without number, all would be indeterminate and unknowable.""

"Philolaus divided the world into three parts—viz. Olympus, which holds within itself the purity of the elements, i. e. probably the central fire and the fire outwardly embracing the world; Kosmus, or the world in a limited sense, i. e. the perfectly ordered world, which comprises all mundane bodies except the earth; and Uranus, i. e. the part of the universe which belongs to the terrestrial sphere."

"Fragment 4. This is the state of affairs about nature and harmony. The essence of things is eternal; it is a unique and divine nature, the knowledge of which does not belong to man. Still it would not not be possible that any of the things that are, and are known by us, should arrive to our knowledge, if this essence was not the internal foundation of the principles of which the world was founded, that is, of the finite and infinite elements. Now since these principles are not mutually similar, neither of similar nature, it would be impossible that the order of the world should have been formed by them, unless the harmony had intervened... the dissimilar things, which have neither a similar nature, nor an equivalent function, must be organized by the harmony, if they are to take their place in the connected totality of the world."

"There is a fire in the middle at the centre, which is the Vesta of the universe, the house of Jupiter, the mother of the Gods, and the basis coherence and measure of nature."

"The ancient theologists and priests... testify that the soul is united with the body as if for the sake of punishment; and so is buried in body as in a sepulchre."

"Fragment 1. (Stob.21.7; Diog.#.8.85) The world's nature is a harmonious compound of infinite and finite elements; similar is the totality of the world in itself, and of all it contains. b. All beings are necessarily finite or infinite, or simultaneously finite and infinite; but they could not all be infinite only."

"[Number is] the commanding and self-begotten container of the eternal duration of mundane concerns."

"In the old Pythagorean representation of the celestial system, according to Philolaus, the five planets were mentioned... among the ten deified bodies which revolve round the central fire (the focus of the universe έστἱα) "immediately beneath the region of fixed stars;" these were succeeded by the Sun, Moon, Earth, and... the anti-Earth..."

"In regard to Philolaus, we are told... that he derived geometrical determinations (the point, the line, the surface, the solid) from the first four numbers, so he derived physical qualities from five, the soul from six; reason, health, and light, from seven; love, friendship, prudence, and inventive faculty from eight. Herein (apart from the number schematism) is contained the thought that things represent a graduated scale of increasing perfection; but we hear nothing of any attempt to prove this in detail, or to seek out the characteristics proper to each particular region."

"Philolaus, in a fragment preserved by (Eclog. Phys. p. 51), says, "that there is a fire in the middle at the centre, which is the Vesta of the universe, the house of Jupiter, the mother of the Gods, and the basis, coherence, and measure of nature." Hence... they are greatly mistaken who suppose the Pythagoreans meant the Sun by the fire at the centre..."

"Fragment 3. The harmony is generally the result of contraries; for it is the unity of multiplicity, and the agreement of discordances. (Nicom.Arith.2:509)."

"The first publication of the Pythagorean doctrines is pretty uniformly attributed to Philolaus. He composed a work on the Pythagorean philosophy in three books, which Plato is said to have procured at the cost of 100 minae through , who purchased it from Philolaus, who was at the time in deep poverty. ...Out of the materials which he derived from these books Plato is said to have composed the Timaeus. But in the age of Plato the leading features of the Pythagorean doctrines had long ceased to be secret; and if Philolaus taught the Pythagorean doctrines at Thebes, he was hardly likely to feel much reluctance in publishing them... little more can be regarded as trustworthy, except that Philolaus was the first who published a book on the Pythagorean doctrines, and that Plato read and made use of it."

"[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."

"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."

"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."

"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...'"

"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.""

"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."

"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."

"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."

"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 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."

"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."

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