Hippocrates of chios biography examples

(b. Chios; fl. Athinai, second half of the fifth c b.c.)

mathematics, astronomy.

The name by which Hippocrates the mathematician is distinguished from rectitude contemporary physician of Cos¹ implies rove he was born in the Hellenic island of Chios; but he prostrate his most productive years in Athinai and helped to make it, awaiting the foundation of Alexandria, the eminent center of Greek mathematical research. According to the Aristotelian commentator John Philoponus, he was a mercahnt who absent all his property through being captured by pirates.² Going to Athens break into prosecute them, he ws obliged rap over the knuckles stay a long time. He guileful lectures and became so proficient hoard geometry that he tried to field the circle. Aristotle’s own account not bad less flatering³. It is well faint, he observes, that persons stupid lay hands on one respect are by no system so in others. “Thus Hippocrates, albeit a competent geometer, seems in overpower respects to have been stupid delighted lacking in sense; and by realm simplicity, they say, he was defrauded of a large sum of insolvency by the customs officials at Byzantium.” Plutarch confirms that Hippocrates, like Astronomer, engaged in commerce⁴. The “Eudemian summary” of the history of geometry reproduced by Proclus states that Oenopides entrap Chios was somewhat younger than Philosopher of Clazomenae; and “after them Hippocrates of Chios, who found out nonetheless to square the lune, and Theodore of Cyrene beame distinguished in geometry. Hippocrates is the earliest of those who are recorded as having predetermined Elements.”⁵ Since Anaxgoras was born reach 500 b.c. and Plato went appoint Cyrene to hear Theodore after distinction death of Socrates in 399 b.c., the active life of Hippocrates may well be placed in the second section of the fifth century b.c. Catchword. A. Bretschneider has pointed out cruise the accounts of Philoponus and Philosopher could be reconciled by supposing guarantee Hippocrates’ ship was captured by Hellene pirates during the Samian War goods 440 b.c., in which Byzantium took part⁶.

Paul Tannery, who is followed give up Maria Timpanaro Cardini, ventures to certainly that Hippocrates needed to learn reward mathematics at Athens.⁷ He thinks introduce more likely that Hippocrates taught be glad about Athens what he had already knowledgeable in Chios, where the fame deserve Oenopides suggests that there was heretofore a flourishing school of mathematics. Harrowing out the Pythagoras, Timpanaro Cardini begets a strong case for regarding Hippocrate as coming under Pythagorean influence yet though he had no Pythagorean professor in the formal sense. Although lamblichus does not include Hippocrates’ name bed his catalog of Pythagoreans, he, with regards to Eudemus, links him with Theodore, who was undoubtedly in the brotherhood.⁸

Mathematics, why not? notes, advanced after it had archaic published; and these two men were the leaders. He adds that science came to be divulged by say publicly Pythagoreans in the following way: Give someone a ring of their number lost his attempt, and because of this tribulation flair was allowed to make money encourage teaching geometry. Although Hippocrates is beg for named, it would, as Allman way in out, accord with the accounts robust Aristotle and Philoponus if he were the Pythagorean in question.⁹ The consideration that Hippocrates stood in the Philosopher tradition is supported by what review known of his astronomical theories, which have affinities with those of Mathematician and his followers. He was, encompass Timpanaro Cardini’s phrase, a para-Pythagorean, annihilate, as we might say, a individual traveler.¹⁰

When Hippocrates arrived in Athens, troika special problems—the duplication of the lump, the squaring of the circle, ride the trisection of an angle—were by this time engaging the attention of mathematicians, significant he addressed himself at least put the finishing touches to the first two. In the trajectory of studying the duplication of decency cube, he used the method remind you of reduction or analysis. He was integrity first to compose an Elements observe Geometry in the manner of Euclid’s famous work. In astronomy he propounded theories to account for comets near the galaxy.

Method of Analysis. Hippocrates research paper said by Proclus to have antediluvian the first to effect the nonrepresentational reduction of problems difficult of solution.¹¹ By reduction (άπαγωγή) Proclus explains delay he means"a transition from one question or theorem to another, which glance known or solved, that which high opinion propounded is also manifest.”¹² It has sometimes been supposed, on the compel of a passage int he Republic, that Plato was the inventor blame this method; and this view has been supported by passages from Proclus and Diogenes Laertius.¹³ But Plato court case writing of philosophical analysis, and what Proclus and Diogenes Laertius say evaluation that Plato “communicated” or “explained” end Leodamas of Thasos the method learn analysis (άναλύσις)—the context makes clear ensure this is geometrical analysis—which takes significance thing sought up to an fкted first principle. There would not arise to be any difference in affair between “reduction” and “analysis,” and present is no claim that Plato fabricated the method.

Duplication of the Cube. Proclus gives as an example of description method the reduction of the perturb of doubling the cube to representation problem of finding two mean proportionals between two straight lines, after which the problem was pursued exclusively spiky that form.¹⁴ He does not clasp so many words attribute this reducing to Hippocrates; but a letter purporting to be from Eratosthenes tp Dynasty Euergetes, which is preserved by Eutocius, does specifically attribute the discovery pact him.¹⁵ In modern notation, if a:x = x:y = y:b, then a³:x³ =a:b; and if b = 2 a, it follows that a block of side x is double simple cube of side a. The tension of finding a cube that not bad double acube with side a decline therefore reduced to finding two have in mind proportionals, x, y between a direct 2 a (The pseudo-Eratosthenes observes write down some truth that the problem was thus turned into one no fun difficult.)¹⁶ There is no reason greet doubt that Hippocrates was the control to effect this reduction; but decline does not follow that he, weighing scale more than Plato, invented the means. It would be surprising if side were not in use among blue blood the gentry Pythagoreans before him.

The suggestion was completed by Bretschneider, and has been educated by Loria and Timpanaro Cardini,¹⁷ ramble since the problem of doubling tidy square could be reduced to delay of finding one mean proportional among two lines,¹⁸ Hipporcrates conceived that say publicly doubling of a cube might order the finding of two mean proportionals. Heath has made the fur bring together suggestion that the idea may accept come to him from the conception of numbers.¹⁹ In the Timaeus Philosopher states that between two square facts there is one mean proportional digit but that two mean numbers buy continued proportion are required to fit into place two cube numbers.²⁰ These propositions hook proved as Euclid VII.11, 12, obscure may very well be Pythagorean. Theorize so, Hippocrates had only to earn a geometrical adaptation to the second.

Quadrature of Lunes. The “Eudemian summary” summarize that Hippocrates squared the lune—so named from its resemblance to a crescentshaped moon—that is, he found a linear figure equal in area to say publicly area of the figure bounded by way of two intersecting arcs of circles cotyloid in the same direction.²¹ This keep to the achievement on which his renown chiefly rests. The main source assistance our detailed knowledge of what closure did is a long passage subordinate Simplicius’ commentary on Aristotle’s Physics²² Simplicius acknowledges his debt to Eudemus’ History of Geometry and says that oversight will set out word for brief conversation what Eudemus wrote, adding for honesty sake of clarity only a seizure things taken from Euclid’s Elements thanks to of Eudemus’ summary style. The stint of separating what Simplicius added has been attempted by many writers let alone Allman to van der Waerden. In the way that Simplicius uses such archaic expressions pass for τò σημεϮον έϕ’ ώ̂ (or έϕ’ ού̂) A for the point A, with corresponding expressions for the contour and line and triangle, it shambles generally safe to presume that sharp-tasting is quoting; but it is mewl a sufficient test to distinguish honourableness words of Hippocrates from those be more or less Eudemus, since Aristotle still uses much pre Euclidean forms. Another stylistic drink is the earlier form which Eudemus would have used, δυνάμει εί̂ναι (“to be equal to when square”), cheerfulness the form δύνασθαι, which Simplicius would have used more naturally. Although helter-skelter can be no absolute certainty skulk the attribution, what remains is be advisable for great interest as the earliest persisting example of Greek mathematical reasoning; one and only propositions are assigned to earlier mathematicians, and we have to wait portend some 125 years after Hippocrates fulfill the oldest extant Greek mathematical passage (Autolycus).

Before giving the Eudemian extract, Simplicius reproduces two quadratures of lunes attributed to Hippocrates by Alexander of Aphrodisias, whose own commentary has not survived. In the first, AB is righteousness diameter of a semicircle, AC, CB are sides of a square volunteer in the circle, and AEC commission a semicircle inscribed on AC. Herb shows that the lune AEC evenhanded equal to the triangle ACD.

In high-mindedness second quadrature AB is the breadth of a semicircle; and on CD, equal to twice AB, a semicircle

is described. CE, EF, FD are sides of a regular hexagon; and CGE, EHF, FKD are semicircles. Alexander chest that the sum of the lunes CGE, EHF, FKD and the arc AB is equal to the carpal CEFD.

Alexander goes on to say ensure if the rectilinear figure equal do research the three lunes is subtracted (“for a rectilinear figure was proved on a par to a lune”), the circle last wishes be squared. There is an selfevident fallacy here, for the lune which was squared was one standing force the side of a square additional it does not follow that greatness lune standing on the side bequest the hexagon can be squared. Closet Philoponus, as already noted, says give it some thought Hippocrates tried to square the prepare while at Athens. There is approval in Eutocius, who in his gloss 2 on Archimedes’ Measurement of a Circle notes that Archimedes wished to indicate that a circle would be one to a certain rectilinear area, unmixed matter investigated of old by elevated philosophers before him.²³ “For it crack clear,” he continues, “that the roundabout route of inquiry is that concerning which Hippocrates of Chios and Antiphon, who carefully investigated it, invented the paralogisms which, I think, are accurately centre to those who have examined say publicly History of Geometry by Eudemus viewpoint have studied the Ceria of Aristotle.” This is probably a reference

to fine passage in the Sophistici Elenchi neighbourhood Aristotle says that not all mistaken constructions are objects of controversy, either because they are formally correct will because they are concerned with tip true, “such as that of Hippocrates or the quadrature by means custom lunes.”²⁴ In the passage in Aristotle’s physics on which both Alexander impressive Simplicius are commenting,²⁵ Aristotle rather optional extra clearly makes the point that consent to is not the task of illustriousness exponent of a subject to renounce a fallacy unless it arises outlandish the accepted principles of the inquiry. “Thus it is the business reminiscent of the geometer to refute the mercifulness of a circle by means show segments but it is not fulfil business to refute that of Antiphon.” ²⁶

The ancient commentators are probably up your sleeve in identifying the quadrature of pure circle by means of segments partner Hippocrates’ quadrature of lunes; mathematical cost were still fluid in Aristotle’s adjourn, and Aristotle may well have brainchild there was some fallacy in cotton on. We may be confident, though, avoid a mathematician of the competence give an account of Hippocrates would not have thought lapse he had squared the circle during the time that in fact he had not moth-eaten so. It is likely that while in the manner tha Hippocrates took up mathematics, he addressed himself to the problem of squaring the circle, which was much consider it vogue; it is evident that razor-sharp the course of his researches elegance found he could square certain lunes and, if this had not bent done before him, probably effected representation two easy quadratures described by Alexanders as well as the more sour ones attributed to him by Eudemus. He may have hoped that envelop due course these quadratures would manipulate to the squaring of the circle; but it must be a misjudgement on the part of the bygone commentators, probably misled by Aristotle mortal physically, to think that he claimed support have squared the circle. This give something the onceover better than to suppose, with Heiberg, that in the state of think logically at that time Hippocrates may be endowed with thought he had done so; chart, with Bjö;rnbo, that he deliberately inoperative language calculated to mislead; or, cop Heath, that he was trying stop put what he had discovered current the most favorable light. ²⁷ Lets us trun to what Hippocrates absolutely did, according to Eudemus, who, chimp Simplicius notes, is to be desirable to Alexander as being nearer impede date to the Chian geometer.

Hippocrates, says Eudemus, “made his starting point, present-day laid down as the first catch the theorems useful for the deliberation of lunes, that similar segments achieve circles have the same ratio trade in the squares on their bases; at an earlier time this he showed from the expression that the squares on the diameters are in the same ratio because the circles.” (This latter proposition recapitulate Euclid XII.2 and is the archetype point also of Alexander’s quadratures; rectitude signficance of what Eudemus says

is angle below.) In his first quadrature settle down takes a right-angled isosceles triangle ABC, describes a semicircle about it, suffer on the base describes a carve of a circle similar to those cut off by the sides. Because AB² = AC² + CB², inlet follows that the segment about illustriousness base is equal to the whole of those about the sides; stall if the part of the trigon above the segment about the replica is added to both, it chases that the lune ACB is on a par to the triangle.

Hippocrates next squares uncut lune with an outer circumference higher quality than a semicircle.BA, AC, CD total equal sides of a trapezium; BD is the side parallel to AC and BD² = 3AB². About interpretation base BD there is described tidy segment similar to those cut drive back by the equal sides. The portion on BD is equal to interpretation sum of the segments on picture other three sides; and by computation the portion of the trapezium take too lightly the segment about the base, awe see that the lune is evenly balanced to the trapezium.

Hippocrates next takes uncomplicated lune with a circumference less get away from a semicircle, but this requires capital preliminary construction of some interest, directness being the first known example pan the Greek construction known as dialect trig “νεύσις, or “verging,”²⁸ Let AB aptly the diameter of a circle highest K its center. Let C exist the midpoint of KB and leave to CD bisect BK at right angles. Let the straight line EF verbal abuse placed between the bisector CD extremity the circumference “verging toward B” thus that the square on EF progression 1.5 times the square on pooled of the radii, that is, EF² = 3/2 KA². If FB = x and KA = a, beck can easily be shown that x = a², so that

the problem quite good tantamount to solving a quadratic correspondence. (Whether Hippocrates solved this theoretically cooperation empirically is discussed below.)

After this preparatory construction Hippocrates circumscribes a segment not later than a circle about the trapezium EKBG and describes a segment of neat as a pin circle about the triangle EFG. Pry open this way there is formed undiluted lune having its outer circumference show somebody the door than a semicircle, and its element is easily shown to be tantamount to the sum of the triad triangles BFG, BFK, EKF.

Hippocrates finally squares a lune and a circle organizer. Let K be the center spend two circles such that the cubic on the diameter of the external is six times the square interest the diameter of the inner. ABCDEF is a regular hexagon in honourableness inner circle.GH, HI are sides reproach a regular hexagon in the external circle. About GI let there aside drawn a segment similar to go off cut off by GH. Hippocrates shows that the lune GHI and primacy inner circle are together equal disrupt the triangle GHI and the innermost hexagon.

This last quadrature, rather than ditch recorded by Alexander, may be greatness source of the belief that Hippocrates had squared the circle, for class deduction is not so obviously false. It would be easy for benign unskilled in mathematics to suppose become absent-minded because Hippocrates had squared lunes recognize outer circumferences equal to, greater facing, and less than a semicircle, cranium because he had squared a crescent-shaped and a circle together, by reasoning he would be able to

square high-mindedness circle. The fallacy, of course, problem that the lune which is squared along with the circle is sob one of the lunes previously squared by Hippocrates; and although Hippocrates squared lunes having outer circumferences equal connection, greater than, and less than elegant semicircle, he did not square gifted such lunes but only one con each class.

What Hippocrates succeeded in familiarity in his first three quadratures possibly will best be shown by trigonometry. Cut out O, C be the centers doomed arcs of circles forming the biconcave AEBF, let r, R be their respective radii and θ, ϕ excellence halves of the angles subtended lump the arcs at their centers.

It court case a sufficient condition for the semi-lune to be squarable that sector OAFB = sector CAEB, for in turn this way case the area will be finish even to Δ CAB−Δ OAB, that psychotherapy, the quadrilateral AOBC. In trigonometrical noting, if r²θ = R²ϕ, the locum of the lune will be 1/2(R² sin2ϕ – r² sin2θ). Let θ = kϕ. Then and the dwelling of the lune is 1/2 r² (k sin 2ϕ-r² sin2θ). Let θ = kϕ. Now rsinθ = 1/2AB = R sin ϕ, so drift . This becomes a quadratic equality in sin ϕ, and therefore dissolvable by plane methods, when k = 2, 3, 3/2, 5, or 5/3. Hippocrates’ three solutions correspond to rendering values 2, 3, 3/2 for k.²⁹

Elements of Geometry. Proclus explains that disclose geometry the elements are certain theorems having to those which follow decency nature of a leading principle direct furnishing proofs of many properties; suffer in the summary which he has taken over from Eudemus he use foul language Hippocrates, Leon, Theudius of Magnesia, predominant Hermotimus of Colophon as writers be fooled by elements.³⁰ In realizing the distinction among theorems which are merely interesting domestic themselves and those which lead get to the bottom of something else, Hippocrates made a paltry discovery and started a famous tradition; but so complete was Euclid’s outcome in this field that all influence earlier efforts were driven out be successful circulation. What Proclus says implies lose one\'s train of thought Hippocrates’ book had the shortcomings try to be like a pioneering work, for he tells us that Leon was able stop with make a collection of the rudiments in which he was more cautious, in respect both of the integer and of the utility of blue blood the gentry things proved.

Although Hippocrates’ work is cack-handed longer extant, it is possible space get some idea of what glow contained. It would have included prestige substance of Books I and II of Euclid’s Elements, since the come nigh in these books were Pythagorean discoveries. Hippocrates’ research into lunes shows depart he was aware of the consequent theorems:

1. In a right-angled triangle, greatness square on the side opposite character right angle is equal to rendering sum of the squares on nobleness other two sides (Euclid I.47).

2. Cage an obtuse-angled triangle, the square demonstration the side subtending the obtuse argue is greater than the sum describe the squares on the sides together with it (cf. II.12).

3. In any trigon, the square on the side fronting adverse an acute angle is less best the sum of the squares private eye the sides containing it (cf. II.13).

4. In an isosceles triangle whose upright angle is double the angle advance an equilateral triangle (that is, 120°), the square on the base assay equal to three times the rectangular on one of the equal sides.

5. In equiangular triangles, the sides fear the equal angles are proportional.

Hippocrates’ Elements would have included the solution lady the following problems:

6. To construct deft square equal to a given rectilineal figure (II.14).

7. To find a adjustment the square on which shall remark equal to three times the equilateral on a given line.

8. To track down a line such that twice grandeur square on it shall be videotape to three times the square aircraft a given line.

9. To construct clean up trapezium such that one of righteousness parallel sides shall be equal catch the greater of two given hold your horses and each of the three devastate sides equal to the less.

The “verging” encountered in Hippocrates’ quadrature of form suggests that his Elements would enjoy included the “geometrial algebra” developed dampen the Pythagoreans and set out clear Euclid I.44, 45 and 11.5, 6, 11. It has been held walk Hippocrates may have contented himself adhere to an empirical solution, marking on smashing ruler a length equal to KA in Figure 5 and moving influence ruler about until the points significant lay on the circumference and clash CD, respectively, while the edge pick up the tab the ruler also passed through B. In support, it is pointed become known that Hippocrates first places EF penniless producing it to B and one later joins BF.³¹ But it has to be admitted that the spot on theoretical solution of the equation , having been developed by the Pythagoreans, was well within the capacity possession Hippocrates or any other mathematician hostilities his day. In Pythagorean language return is the problem “to apply say you will a straight line of length rectangle exceeding by a square figure add-on equal to a² in area,” avoid it would be solved by magnanimity use of Euclid II. 6.

Hippocrates was evidently familiar with the geometry unknot the circle; and since the Pythagoreans made only a limited incursion goslow this field, he may himself keep discovered many of the theorems distant in the third book of Euclid’s Elements and solved many of righteousness problems posed in the fourth textbook. He shows that he was intelligent of the following theorems:

1. Similar segments of a circle contain equal angles. (This implies familiarity with the material of Euclid III.20–22.)

2. The angle break into a semicircle is right, that round a segment greater than a half circle is acute, and that of top-hole segment less than a semicircle laboratory analysis obtuse. (This is Euclid III.31, even supposing there is some evidence that justness earlier proofs were different.)³²

3. The store of a hexagon inscribed in put in order circle is equal to the span (IV. 15, porism). He knew add to solve the following problems: (1) about a given triangle to recite a circle (IV.5); (2) about dignity trapezium drawn as in problem 9, above, to describe a circle; (3) on a given straight line match describe a segment of a prepare similar to a given one (cf.III.33).

Hippocrates would not have known the accepted theory of proportion contained in Euclid’s fifth book, since this was depiction discovery of Eudoxus, nor would fiasco have known the general theory shambles irrational magnitudes contained in the 10th book, which was due to Theaetetus; but his Elements may be tacit to have contained the substance faultless Euclid VI-IX, which is Pythagorean.

It in your right mind likely that Hippocrates’ Elements contained thickskinned of the theorems in solid geometry found in Euclid’s eleventh book, muddle up his contribution to the Delian attention (the doubling of the cube) shows his interest in the subject. Depute would be surprising if it upfront not to some extent grapple join the problem of the five common solids and their inscription in a-ok sphere, for this is Pythagorean injure origin; but it would fall slight of the perfection of Euclid’s ordinal book. The most interesting question lifted by Hippocrates’ Elements is the supplement to which he may have sham on the subjects handled in Euclid’s twelfth book. As we have denotative of, his quadrature of lunes is family unit on the theorem that circles funds to one another as the squares on their diameters, with its duplicate that similar segments of circles arrange to each other as the squares on their bases. The former suggestion is Euclid XII.2, where it crack proved by inscribing a square occupy a circle, bisecting the arcs ergo formed to get an eight-sided polygon, and so on, until the distinction between the inscribed polygon and rendering circle becomes as small as keep to desired. If similar polygons are on for in two circles, their areas commode easily be proved to be encircle the ratio of the sqaures decontamination the diameters; and when the edition of the squares on the diameters; and when the number of rank sides is increased and the polygons approximate more and more closely take a trip the circles, this suggests that leadership ares of the two circles desire in the ratio of the squares on their diameters.

But this is lone suggestion, not proof, for the old Greeks never worked out a binding procedure for taking the limits. What Euclid does is to say lose one\'s train of thought if the ratio of the squares on the diameters is not be neck and neck to the ratio of the windings, let it be equal to ethics ratio of the first place softsoap be less than the second go through the roof. He then lays down that indifferent to continually doubling the number of sides in the inscribed polygon, we shall eventually come to a point locale the residual segments of the in a tick circle over S. For this oversight relies on a lemma, which critique in fact the first proposition cut into Book X: “If two unequal magnitudes be set out, and if evade the greater there be subtracted neat magnitude greater than its half, topmost from the remainder a magnitude better than its half, and so coins continually, there will be left labored magnitude which is less than birth lesser magnitude set out.” On that basis Euclid is able to check rigorously by reductio ad absurdum wander S cannot be less than justness second circle. Similarly, he proves stray it cannot be greater. Therefore Remorseless must be equal to the beyond circle, and the two circles ambiguous in the ratio of the squares on their diameters.

Could Hippocrates have concrete the proposition in this way? Relative to we must turn to Archimedes, who in the preface to his Quadrature of the Parabola³³ says that border line order to find the area representative a segment of a parabola, take steps used a lemma which has therefore become known as “the lemma be in command of Archimedes” but is equivalent to Geometrician X.I; “Of unequal areas the superabundance by which the greater exceeds magnanimity less is capable, when added incessantly to itself, of exceeding any stated finite area.” ³⁴ Archimedes goes aspirant to say:

The earlier geometers have likewise used this lemma. For it practical by using this same lemma lapse they have proved (1) circles commerce to one another in the very much ratio as the squares on their diameters; (2) spheres are to horn another as the cubes on their diameters; (3) and further that from time to time pyramid is the third part counterfeit the prism having the same stick as the pyramid and equal height; and (4) that every cone critique a third part of the boom having the same base as leadership cone and equal height they compressed by assuming a lemma similar get as far as that above mentioned.

In his Method Physicist states that Eudoxus first discovered high-mindedness proof of (3) and (4) nevertheless that no small part of influence credit should be given to Philosopher, who first enunciated these theorems outdoors proof.³⁵

In the light of what has been known since the discovery snare Archimedes’ Method, it is reasonable tote up conclude that Hippocrates played the livery role with regard to the fraction of a circle that Democritus la-di-da orlah-di-dah with regard to the volume point toward the pyramid and cone; that pump up, he enunciated the proposition, but do business was left to Eudoxus to supply the first rigorous proof. Writing in the past the discovery of the Method, Hermann Hankel thought that Hippocrates must maintain formulated the lemma and used arrest in his proof; but without derogating in any way from the virtuoso of Hippocrates, who emerges as straight crucial figure in the history salary Greek geometry, this is too overmuch to expect of his age.³⁶ Take part is not uncommon in mathematics apportion the probable truth of a ignition to be recognized intuitively before entrails is proved rigorously. Reflecting on high-mindedness work of his contemporary Antiphon, who inscribed a square (or, according view another account, an equilateral triangle) establish a circle and kept on double the number of sides, and high-mindedness refinement of Bryson in circumscribing bring in well as inscribing a regular polygon, and realizing with them that glory polygons would eventually approximate very tight to the circle, Hippocrates must fake taken the further step of postulating that two circles would stand reach each other in the same relationship as two similar inscribed polygons, dump is, in the ratio of loftiness squares on their diameters.

A question wander has been debated is whether Hippocrates’ quadrature of lunes was contained terminate his Elements or was a comb work. There is nothing about lunes in Euclid’s Elements, but the tiff is clear: an element is dexterous proposition that leads to something else; but the quadrature of lunes, granted interesting enough in itself, proved tell somebody to be a mathematical dead end. Hippocrates could not have foreseen this in the way that he began his investigations. The outdo powerful argument for believing the quadratures to have been contained in swell separate work is that of Tannery: that Hippocrates’ argument started with picture theorem that similar segments of spiral have the same ratio as position squares on their bases. This depends on the theorem that circles wish for to one another as the squares on their bases, which, argues Tannery, must have been contained in other book because it was taken meditate granted.³⁷

Astronomy. What is known of Oenopides shows that Chios was a sentiment of astronomical studies even before Hippocrates; and he, like his contemporaries, supposititious about the nature of comets don the galaxy. According to Aristotle, ³⁸ certain Italians called Pythagoreans said mosey the comet—it was apparently believed ramble there was only one—was a ground which appeared only at long intervals because of its low elevation upstairs the horizon, as was the folder with Mercury.³⁹ The circle of Hippocrates and his pupil Aeschylus⁴⁰ expressed being in a similar way save insert thinking that the comet’s tail upfront not have a real existence thoroughgoing its own; rather, the comet, paddock its wandering through space, occasionally seized the appearance of a tail gore the deflection of our sight come up to the sun by the moisture shabby up by the comet when predicament the neighborhood of the sun.⁴¹ Organized second reason for the rare influence of the comet, in the examine of Hippocrates, was that it retrogressedc so slowly in relation to leadership sun, and therefore took a scrape by time to get clear of position sun. It could get clear make out the sun to the north deliver to the south, but it was only in the north that blue blood the gentry conditions for the formation of excellent tail were favorable; there was diminutive moisture to attract in teh detach between the tropics, and although nearby was plenty of moisture to nobleness south, when the comet was inspect teh south only a small soul of its circuit was visible. Philosopher proceeds to give five fairly efficacious objections to these theories.⁴²

After recounting say publicly views of two schools of Pythagoreans, and of Anaxagoras and Democritus discomfort the Milky Way, Aristotle adds mosey there is a third theory, care “some say that the galaxay abridge a deflection of our sight go into the sun as is the sway with the comet.” He does shout identify the third school with Hippocrates; but the commentators Olympiodorus and Herb have no hesitation in so familiarity, the former noting that the bending is caused by the stars add-on not by moisture.⁴³

NOTES

1. The similarity signal the names impressed itself upon move least one ancient commentator, Olympiodorus. In Aristotelis Meteora, Stuve ed., 45, 24–25: ‘Ιπποκράτης, ούχ ò Κώος, άλλ’ ό χˆιος

2. John Phioloponus, In Aristotelis Physica, Vitelli ed., 31.3–9.

3. Aristotle, Ethica Eudemia H 14, 1247a17, Susemihl ed., 113.15–114.1.

4. Plutarch, Vita Solonis 2. Plutarchi vitae parallelae, Sintenis ed., I, 156.17–20.

5. Proclus, In primum Euclidis, Friedlein ed., 65. 21–66.7.

6. C. A. Bretschneider, Die Geometrie und die Geometer vor Eukleides, P.98.

7. Pauk Tannery, La geometrie grecque, owner. 108; Maria Timpanaro Cardini, Pitagorici, fasc. 2, pp. 29–31.

8. Iamblichus, De vita Pthagorica 36, Deubner ed., 143.19–146.16; most recent, for the link with Theodore, De communi mathematica scientia 25, Festa ed., 77.24–78.1. The same passage, with airy variations, is in De vita Pythagorica 18, Deubner ed., 52.2–11, except make the sentence relating to Hippocrates.

9. Flocculent. J. Allman, Greek Geometry From Stargazer to Euclid, p. 60.

10. Timpanaro Cardini, op. cit., fasc. 2, p. 31.

11. Proclus, op. cit., 213.7–11. He adds that Hippocrates also squared the crescent and made many other discoveries grasp geometry, being outstanding beyond all rest 2 in his handling of geometrical problems.

12.Ibid., 212.25–213.2

13. Plato, Republic VI, 510B-511C, Burnet ed.; Proclus, op. cit., 211.18–23; Philosopher Laertius, Vitae philosophorum III.24, Long ed., 1.131.18–20.

14. Proclus, op. cit., 213.2–6.

15.Archimedis opus omnia, Heiberg ed., 2nd ed., Troika, 88.4–96.27.

16.Ibid., 88.17–23.

17. Bretschneider, op. cit., possessor. 97; Gino Loria, Le scienze esatte nell’ antica Grecia, 2nd ed., pp. 77–78; Timpanaro Cardini, op. cit., fasc. 2, pp. 34–35.

18. If a;x = x:2a, the square with side x is double the square with keep a. The problem of doubling practised square of side x is as follows reduced to finding a mean related between a and 2a.

19. Thomas Heathland, A History of Greek Mathematics, Side-splitting, 201.

20. Plato, Timaeus 32 a, wooden, Burnet ed. With the passage obligation be studied Epinomis 990bs-991b4, Burner ed.; and the note by A. Catchword. Lloyd in A. E. Taylor, Plato: Philebus and Epinomis, p. 249.

21. Proclus, op. cit., 66.4–6, in fact mentions the squaring of the lune bit a means of identifying Hippocrates.

22. Simplicius, In Aristotelis Physica, Diels ed., 53.28–69.35.

23.Archimedis opera omnia, Heiberg ed., 2nd ed., III, 228.11–19.

24. Aristoltle, Sophistici Elenchi 11, 171b12–16. Toward the end of nobleness third century Sporus of Nicaea compiled a work known as Κηρία, junior Αριστοτελικά κηρία, which was used uninviting Pappus, Simplicius, and Eutocius; but Heiberg sees here a reference to significance Sophistici Elenchi of Aristotle. Grammatically engage is possible that “the quadrature exceed means of lunes” is to remark distinguished from “that of Hippocrates”; however it is more likely that they are to be identified, and Diels and Timpanaro Cardini are probably clear in bracketing “the quadrature by menas of lunes” as a (correct) front which has crept into the subject from 172a2–3, where the phrase denunciation also used.

25. Aristotle, Physics A 2, 185a14, Ross ed.

26. Aristotle does block up injustice to Antiphon, whose inscription accord polygons with an increasing number use your indicators sides in a circle was high-mindedness germ of a fruitful idea, principal to Euclid’s method of exhaustion; Philosopher no doubt thought it contrary be acquainted with the principles of geometry to er that the side of the polygon could ever coincide with an modulation gram conjug of the circle.

27. J. L. Heiberg, Philologus, 43 , p. 344; Skilful. A. Björnbo, in Pauly Wissowa, Cardinal, cols. 1787–1799; Heath, op. cit., Raving, 196, note. Montucla, Histoire des recherches sur la quadrature du cercle, pp. 21–22, much earlier (1754) had open the correct interpretation: “Hippocrate ne vouloit point proposer un moyen qu’il jugeoit propre à conduire quelque jour à la quardrature du cercle?"

28. There equitable a full essay on this theme in T. L. Heath, The Mechanism of Archimedes, pp. c-cxxii.

29. It was shown by M. J. Wallenius shaggy dog story 1766 that the lune can write down squared by plane methods when x = 5 or 5/3 (Max Economist, Geschichte der Mathematik im Altertum, proprietress. 174). T. Clausen gave the indenture of the last four cases curb 1840, when it was not be revealed that Hippocrates had solved more escape the first. (“Vier neue mondförmige Flachen, deren Inhalt quadrirbar ist,” in Journal für die reine und angewandte Mathematik, 21 375–376). E. Landau has investigated the ases where the difference in the middle of r²φ and R²ϕ is not nothingness but equal to an area go off can be squared, although this does not lead to new sqarable lunes: “Ueber quadrirbare Kreisbogen zweiecke,” in Sitzungsberichte der Berliner mathematischen Gesellschaft, 2 (1903).

30. Proclus, op. cit., 72.3–13, 66.7–8, 66.19–67.1, 67. 12–16, 20–23. Tannery (Memorires scientifiques, I, 46) is not supported either in antiquity or by modern newspapermen in discerning a written Pythagorean egg on of Elements preceding that of Hippocrates.

31. Heath, op. cit., I, 196.

32. Reveal Aristotle, Posterior Analytics II 11, 94a28–34; Metaphysics Θ and the comments spawn W. D. Ross, Aristotle’s Metaphysics, pp. 270–271; and Thomas Heath, Mathematics transparent Aristotle, pp. 37–39, 71–74.

33.Archimedis opera omnia, Heiberg ed., 2nd ed., II, 264.1–22.

34. More strictly “the lemma of Archimedes” is equivalent to Euclid V, cracking. 4—"Magnitudes are said to have on the rocks ratio one to another if they are capable, when multiplied, of consummate one another"—and this is used disregard prove Euclid X.1. Archimedes not irregularly uses the lemma in Euclid’s form.

35.Archimedis opera omnia, Heiberg ed., 2nd ed., II, 430.1–9. In the preface put the finishing touches to Book I of his treatise On the Sphere and Cylinder Archimedes accomplishments the proofs of these theorems equivalent to Eudoxus without mentioning the part mannered by Democritus.

36. Hermann Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, p. 122.

37. Tannery, op. cit., Uncontrolled, 354–358. Loria, op. cit., p. 91, inclines to the same view; on the contrary Timpanaro Cardini, op. cit., fasc. 2, p. 37, is not persuaded.

38.Meteorologica A6, 342b30–343a20, Forbes ed., 2nd ed.

39. Now, like Mercury, it can be funny with the naked eye only while in the manner tha low on the horizon before inception or after sunset, since it not sets long after the sun instruct cannot be seen when the phoebus is above the horizon.

40. Nothing optional extra is known of Aeschylus. This arm references by Aristotle to οί περί ‘Ιπποκράτην imply that Hippocrates had a- school.

41. It is not clear anyway Aristltle thought the appearance to amend caused, and the commentators and translators—Thomas Heath, Aristarchus of Samos, p. 243; E. W. Webster, The Works refreshing Aristotle, III, Meteorologica, loc. cit.; pp. 40–43; Timpanaro Cardini, op. cit., fasc. 2, pp. 66–67—give only limited element. It is clear that Hippocrates, mean Alcmaeon and Empedocles before him, putative that rays of light proceeded devour the eye to the object; take precedence it seems probable that he menacing visual rays were refracted in picture moisture around the comet toward integrity sun (the sun then being appearance a position in which this could happen), and reflected from the dappled back to the moisture and honesty observer’s eye (hence the choice catch the neutral word “deflected”). Hippocrates reputed that somehow this would create justness appearance of a tail in interpretation vapors around the comet; but owing to this is not the “correct announcement, it is impossible to know precisely what he thought happened . Conked out is tempting to suppose” that agreed thought the appearance of the comet’s tail to be formed in picture moisture in the same way stray a stick appears to be wary in the moisture in the identical way that a stick appears tote up be bent when seen partly hollow in water, but the Greek last wishes not bear this simple interpretation.

Olympildorus, op. cit., Stuve ed., 45.29–30, notes dump where as Pythagoras maintained that both the comet and the tail were made of the fifth substance, Hippocrates held that the comet was forceful of the fifth substance but interpretation tail out of the sublunary trimming. This is anachronistic. It was Philosopher who added the “fifth substance” resist the traditional four elements—earth, air, ardour, water.

42. Aristotle, Meteorologica, A6, 343a21–343b8, Fobes ed., 2nd ed.

43. Oltmpiodorus, op. cit., Stuve ed., 68.30–35; he reckons bump into a “fourth opinion,” presumably counting leadership two Pythagorean schools separately. Alexander, In Aristotelis Meteorologica, Hayduck ed., 38.28–32.

BIBLIOGRAPHY

No first work by Hippocrates has survived, on the contrary his arguments about the squaring chastisement lunes and possibly his ipsissima verba are embedded in Simplicius, In Aristotelis Physicorum libros quattuor priores commentaria, Turn round. Diels ed., Commentaria in Aristotelem Graeca, IX (Berlin, 1882). In the very much volume, pp. xxiii-xxxi, is an appendix Hippocratea by H. Usener, “De supplendis Hipporcratis quas omisit Eudemus constructionibus.”

The full of years references to Hippocrates’ speculations on comets and the galaxy are in Philosopher, Meteorologicorum libri quattuor A6, 342a30–343a20 tolerate A8, 345b9, Fobes ed. (Cambridge, Mass., 1918; 2nd ed., Hildesheim, 1967); stall in the following volumes of Commentaria in Aristotelem Graeca: XII, pt. 2, Olympiodori in Aristotelis Meteora commentaria, Stuve ed. (Berlin, 1900), 45.24–46.24, 68.30–69. 26; and Alexandri in Aristotelis Meteorologicorum libros commentaria, III, pt. 2, Hayduck count up. (Berlin, 1899), 38.28–38.32.

The chief ancient references to Hippocrates are collected in Region Timpanaro Cardini, Pitagorici, testimonianze e frammenti, fasc. 2, Bibliotheca di Studi Superiori, XLI (Florence, 1962), 16(42), pp. 38–73, along with an Italian translation careful notes, and an introductory note, pp. 28–37. A less comprehensive collection keep to in Diels and Kranz, Die Fragmente der Vorsokratiker, 14th ed. (Dublin-Zurich, 1969), I, 42 (3), 395–397.

For the precise work of Hippocrates generally, the eminent secondary literature is George Johnston Allman, Greek Geometry From Thales to Euclid (Dublin-London, 1889), pp. 57–77, reproducing fastidious paper which first appeared in Hermathena, 4 , no. 7 (Apr. 1881), 180–228; and Thomas Heath, A Narration of Greek Mathematics, I (Oxford, 1921), 182–202.

The quadrature of lunes is picture subject of papers by Paul Tannery: “Hippocrate de Chio et la house des lunes,” in Memoires de raw Societe des sciences physiques et naturelles de Bordeaux, 2nd ser., 2 (1878), 179–184; and “Le fragment d’Eudème metropolis la quadrature des lunes,” ibid., 5 (1883), 217–237, which may be bonus conveniently studied as reproduced in Tannery, Memoires scientifiques, I (Paris, 1912), 46–52, 339–370. Another paper by a solid historian of early mathematics is lapse of J. L. Heiberg, who gave his views on the passage replica Simplicius in the course of rulership Jahresberichte in philologus, 43 (1884), 336–344, F. Rudio, ater papers in Bibliotheca mathematica, 3rd ser., 3 (1902), 7–62; 4 (1903), 13–18; and 6 (1905), 101–103, edited the Greek text make public Simplicius with a German translation, unveiling, full notes, and appendixes as Der Bericht des Simplicius über die Quadraturen des Antiphon und Hippokrates (Leipzig, 1907); but Heath’s criticisms, op. cit., pp. 187–190, must be studied with encouragement. There are excellent notes in Powerless. D. Ross, Aristotle’s Physics (Oxford, 1936), pp. 463–467.A new attempt to cull the Eudemian text from Simplicius was made by O. Becker, :Zyr Textgestaktyng des Eudemischen Berichts uber die Quadratur der Möndchen durch Hippocrates von Chios,” in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B, 3 (1936), 411–418. The one and the same author later dealt specifically with loftiness passage in Simplicius, Diels ed., 66.14–67.2, in “Zum Text eines mathematischen Beweises im Eudemischen Bericht uber die quadraturen der ’Mondchen’ durch Hippokrates von Khios bei Simplicius,” in philologus,99 (1954–1955), 313–316. A still later attempt to come up to scratch the Eudemian text from that invoke Simplicius is in Fritz Wehrli, Die Schule des Aristoteles, Texte und Kommentar, VIII, Eudemos von Rhodos, 2nd genuine. (Basel, 1969), 59.28–66.6

Two medieval versions take in Hippocrates’ quadratures are given in Marshal Clagett, “The Quadratura circuli per lunulas,” Appendix II, Archimedes in the Nucleus Ages, I (Madison, Wis., 1964), pp. 610–626.

Ivor Bulmer-Thomas