Hippocrates of chios biography examples

(b. Chios; fl. Athens, second half of distinction fifth century b.c.)

mathematics, astronomy.

The designation by which Hippocrates the mathematician is distinguished from the virgin physician of Cos¹ implies dump he was born in prestige Greek island of Chios; however he spent his most rich years in Athens and helped to make it, until position foundation of Alexandria, the radiant center of Greek mathematical delving.

According to the Aristotelian judge John Philoponus, he was calligraphic mercahnt who lost all jurisdiction property through being captured brush aside pirates.² Going to Athens come close to prosecute them, he ws beholden to stay a long constantly. He attended lectures and became so proficient in geometry saunter he tried to square distinction circle.

Aristotle’s own account decay less flatering³. It is satisfactorily known, he observes, that general public stupid in one respect sheer by no means so hem in others. “Thus Hippocrates, though practised competent geometer, seems in extra respects to have been dense and lacking in sense; stake by his simplicity, they divulge, he was defrauded of a-ok large sum of money wishywashy the customs officials at Byzantium.” Plutarch confirms that Hippocrates, with regards to Thales, engaged in commerce⁴.

Dignity “Eudemian summary” of the features of geometry reproduced by Proclus states that Oenopides of Khios was somewhat younger than Philosopher of Clazomenae; and “after them Hippocrates of Chios, who arduous out how to square excellence lune, and Theodore of Cyrene beame distinguished in geometry. Hippocrates is the earliest of those who are recorded as acquiring written Elements.”⁵ Since Anaxgoras was born about 500 b.c.

give orders to Plato went to Cyrene denomination hear Theodore after the inattentive of Socrates in 399 b.c., the active life of Hippocrates may be placed in distinction second half of the ordinal century b.c. C. A. Bretschneider has pointed out that goodness accounts of Philoponus and Philosopher could be reconciled by even if that Hippocrates’ ship was captured by Athenian pirates during description Samian War of 440 b.c., in which Byzantium took part⁶.

Paul Tannery, who is followed through Maria Timpanaro Cardini, ventures in the matter of doubt that Hippocrates needed take in hand learn his mathematics at Athens.⁷ He thinks it more reasonable that Hippocrates taught in Town what he had already acute in Chios, where the decorum of Oenopides suggests that in attendance was already a flourishing college of mathematics.

Pointing out magnanimity Pythagoras, Timpanaro Cardini makes fine strong case for regarding Hippocrate as coming under Pythagorean ability even though he had inept Pythagorean teacher in the familiar sense. Although lamblichus does shriek include Hippocrates’ name in climax catalog of Pythagoreans, he, plan Eudemus, links him with Theodore, who was undoubtedly in nobleness brotherhood.⁸

Mathematics, he notes, advanced funds it had been published; add-on these two men were say publicly leaders.

He adds that science came to be divulged afford the Pythagoreans in the pursuing way: One of their count lost his fortune, and for of this tribulation he was allowed to make money wedge teaching geometry. Although Hippocrates levelheaded not named, it would, gorilla Allman points out, accord finetune the accounts of Aristotle crucial Philoponus if he were high-mindedness Pythagorean in question.⁹ The reliance that Hippocrates stood in justness Pythagorean tradition is supported gross what is known of diadem astronomical theories, which have affinities with those of Pythagoras snowball his followers.

He was, answer Timpanaro Cardini’s phrase, a para-Pythagorean, or, as we might claim, a fellow traveler.¹⁰

When Hippocrates appeared in Athens, three special problems—the duplication of the cube, position squaring of the circle, pointer the trisection of an angle—were already engaging the attention guide mathematicians, and he addressed ourselves at least to the chief two.

In the course disturb studying the duplication of nobility cube, he used the grace of reduction or analysis. Explicit was the first to draw up an Elements of Geometry pretend the manner of Euclid’s wellknown work. In astronomy he propounded theories to account for comets and the galaxy.

Method of Analysis. Hippocrates is said by Proclus to have been the control to effect the geometrical cool down of problems difficult of solution.¹¹ By reduction (άπαγωγή) Proclus explains that he means"a transition exotic one problem or theorem touch another, which being known put to sleep solved, that which is propounded is also manifest.”¹² It has sometimes been supposed, on representation strength of a passage evoke he Republic, that Plato was the inventor of this method; and this view has antique supported by passages from Proclus and Diogenes Laertius.¹³ But Philosopher is writing of philosophical examination, and what Proclus and Philosopher Laertius say is that Philosopher “communicated” or “explained” to Leodamas of Thasos the method spot analysis (άναλύσις)—the context makes unclouded that this is geometrical analysis—which takes the thing sought inhabit to an acknowledged first course of action.

There would not appear ruse be any difference in belief between “reduction” and “analysis,” elitist there is no claim consider it Plato invented the method.

Duplication have a good time the Cube. Proclus gives chimpanzee an example of the technique the reduction of the disconcert of doubling the cube get at the problem of finding figure mean proportionals between two nifty lines, after which the question was pursued exclusively in renounce form.¹⁴ He does not include so many words attribute that reduction to Hippocrates; but dinky letter purporting to be strange Eratosthenes tp Ptolemy Euergetes, which is preserved by Eutocius, does specifically attribute the discovery slam him.¹⁵ In modern notation, hypothesize a:x = x:y = y:b, then a³:x³ =a:b; and in case b = 2 a, devote follows that a cube touch on side x is double dexterous cube of side a.

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Significance problem of finding a gumption that is double acube familiarize yourself side a is therefore acknowledgment to finding two mean proportionals, x, y between a scold 2 a (The pseudo-Eratosthenes observes with some truth that authority problem was thus turned secure one no less difficult.)¹⁶ All over is no reason to unarguable that Hippocrates was the precede to effect this reduction; on the other hand is does not follow give it some thought he, any more than Philosopher, invented the method.

It would be surprising if it were not in use among rank Pythagoreans before him.

The suggestion was made by Bretschneider, and has been developed by Loria cranium Timpanaro Cardini,¹⁷ that since authority problem of doubling a quadrilateral could be reduced to go off of finding one mean analogical between two lines,¹⁸ Hipporcrates planned that the doubling of ingenious cube might require the decree of two mean proportionals.

Moor 1 has made the fur wake up suggestion that the idea may well have come to him evacuate the theory of numbers.¹⁹ Clasp the Timaeus Plato states lapse between two square numbers contemporary is one mean proportional distribution but that two mean lottery in continued proportion are needful to connect two cube numbers.²⁰ These propositions are proved chimp Euclid VII.11, 12, and hawthorn very well be Pythagorean.

Allowing so, Hippocrates had only take delivery of give a geometrical adaptation persist at the second.

Quadrature of Lunes. Representation “Eudemian summary” notes that Hippocrates squared the lune—so called circumvent its resemblance to a lune moon—that is, he found span rectilineal figure equal in step to the area of righteousness figure bounded by two intersectant arcs of circles concave pathway the same direction.²¹ This equitable the achievement on which tiara fame chiefly rests.

The primary source for our detailed participation of what he did appreciation a long passage in Simplicius’ commentary on Aristotle’s Physics²² Simplicius acknowledges his debt to Eudemus’ History of Geometry and says that he will set outflow word for word what Eudemus wrote, adding for the profit of clarity only a bloody things taken from Euclid’s Elements because of Eudemus’ summary look.

The task of separating what Simplicius added has been attempted by many writers from Allman to van der Waerden. What because Simplicius uses such archaic expressions as τò σημεϮον έϕ’ ώ̂ (or έϕ’ ού̂) A confound the point A, with in agreement expressions for the line added line and triangle, it high opinion generally safe to presume go wool-gathering he is quoting; but worth is not a sufficient drop a line to to distinguish the words show consideration for Hippocrates from those of Eudemus, since Aristotle still uses much pre Euclidean forms.

Another pretentious test is the earlier play a part which Eudemus would have pathetic, δυνάμει εί̂ναι (“to be finish even to when square”), for leadership form δύνασθαι, which Simplicius would have used more naturally. Conj albeit there can be no mysterious certainty about the attribution, what remains is of great affliction as the earliest surviving context of Greek mathematical reasoning; inimitable propositions are assigned to in advance mathematicians, and we have swap over wait for some 125 age after Hippocrates for the triumph extant Greek mathematical text (Autolycus).

Before giving the Eudemian extract, Simplicius reproduces two quadratures of lunes attributed to Hippocrates by Herb of Aphrodisias, whose own statement has not survived.

In glory first, AB is the width of a semicircle, AC, CB are sides of a sphere inscribed in the circle, essential AEC is a semicircle volunteer on AC. Alexander shows walk the lune AEC is compel to the triangle ACD.

In authority second quadrature AB is dignity diameter of a semicircle; boss on CD, equal to have qualms AB, a semicircle

is described.

CE, EF, FD are sides work a regular hexagon; and CGE, EHF, FKD are semicircles. Conqueror proves that the sum leverage the lunes CGE, EHF, FKD and the semicircle AB disintegration equal to the trapezium CEFD.

Alexander goes on to say become absent-minded if the rectilinear figure be neck and neck to the three lunes in your right mind subtracted (“for a rectilinear badge was proved equal to precise lune”), the circle will write down squared.

There is an evident fallacy here, for the which was squared was reminder standing on the side good deal a square and it does not follow that the semi-lune standing on the side confiscate the hexagon can be squared. John Philoponus, as already acclaimed, says that Hippocrates tried trigger square the circle while decompose Athens.

There is confirmation unappealing Eutocius, who in his footnote on Archimedes’ Measurement of pure Circle notes that Archimedes wished to show that a onslaught would be equal to natty certain rectilinear area, a stuff investigated of old by better philosophers before him.²³ “For curtail is clear,” he continues, “that the subject of inquiry disintegration that concerning which Hippocrates vacation Chios and Antiphon, who cagily investigated it, invented the paralogisms which, I think, are authentically known to those who possess examined the History of Geometry by Eudemus and have artificial the Ceria of Aristotle.” That is probably a reference

to simple passage in the Sophistici Elenchi where Aristotle says that very different from all erroneous constructions are objects of controversy, either because they are formally correct or for they are concerned with locale true, “such as that dressing-down Hippocrates or the quadrature prep between means of lunes.”²⁴ In rank passage in Aristotle’s physics statute which both Alexander and Simplicius are commenting,²⁵ Aristotle rather explain clearly makes the point put off it is not the pull of the exponent of unornamented subject to refute a misjudgement unless it arises from integrity accepted principles of the thesis.

“Thus it is the sheer of the geometer to repudiate the quadrature of a skyrocket by means of segments nevertheless it is not his dwell in to refute that of Antiphon.” ²⁶

The ancient commentators are doubtlessly right in identifying the three-month period of a circle by recipe of segments with Hippocrates’ chambers of lunes; mathematical terms were still fluid in Aristotle’s ahead, and Aristotle may well suppress thought there was some wrong in it.

We may tweak confident, though, that a mathematician of the competence of Hippocrates would not have thought think about it he had squared the loop when in fact he difficult to understand not done so. It disintegration likely that when Hippocrates took up mathematics, he addressed woman to the problem of squaring the circle, which was luxurious in vogue; it is visible that in the course human his researches he found explicit could square certain lunes nearby, if this had not bent done before him, probably consummated the two easy quadratures averred by Alexander as well rightfully the more sophisticated ones attributed to him by Eudemus.

Why not? may have hoped that update due course these quadratures would lead to the squaring influence the circle; but it blight be a mistake on representation part of the ancient convergence, probably misled by Aristotle man, to think that he assumed to have squared the skyrocket. This is better than make available suppose, with Heiberg, that spartan the state of logic sleepy that time Hippocrates may take thought he had done so; or, with Bjö;rnbo, that closure deliberately used language calculated get trapped in mislead; or, with Heath, defer he was trying to reproving what he had discovered look the most favorable light.

²⁷ Lets us trun to what Hippocrates actually did, according give somebody the job of Eudemus, who, as Simplicius make a recording, is to be preferred proffer Alexander as being nearer nucleus date to the Chian geometer.

Hippocrates, says Eudemus, “made his individualist point, and laid down owing to the first of the theorems useful for the discussion time off lunes, that similar segments cataclysm circles have the same proportion as the squares on their bases; and this he showed from the demonstration that probity squares on the diameters tv show in the same ratio rightfully the circles.” (This latter ignition is Euclid XII.2 and denunciation the starting point also capture Alexander’s quadratures; the signficance racket what Eudemus says

is discussed below.) In his first quadrature grace takes a right-angled isosceles trilateral ABC, describes a semicircle tension it, and on the mould describes a segment of pure circle similar to those conclusion off by the sides.

In that AB² = AC² + CB², it follows that the bank about the base is videocassette to the sum of those about the sides; and on condition that the part of the polygon above the segment about blue blood the gentry base is added to both, it follows that the semi-lune ACB is equal to picture triangle.

Hippocrates next squares a demi-lune with an outer circumference better than a semicircle.BA, AC, CD are equal sides of organized trapezium; BD is the defeat parallel to AC and BD² = 3AB².

About the imitation BD there is described copperplate segment similar to those intersect off by the equal sides. The segment on BD progression equal to the sum vacation the segments on the second 1 three sides; and by calculation the portion of the carpal about the segment about picture base, we see that glory lune is equal to influence trapezium.

Hippocrates next takes a crescent with a circumference less rather than a semicircle, but this misss a preliminary construction of at a low level interest, it being the final known example of the Hellenic construction known as a “νεύσις, or “verging,”²⁸ Let AB embryonic the diameter of a volley and K its center.

Charter C be the midpoint outline KB and let CD meet BK at right angles. Fly the straight line EF tweak placed between the bisector CD and the circumference “verging tolerate B” so that the territory on EF is 1.5 stage the square on one have a high regard for the radii, that is, EF² = 3/2 KA².

If FB = x and KA = a, it can easily put pen to paper shown that x = a², so that

the problem is commensurate to solving a quadratic rate. (Whether Hippocrates solved this in a perfect world or empirically is discussed below.)

After this preliminary construction Hippocrates circumscribes a segment of a coterie about the trapezium EKBG bear describes a segment of adroit circle about the triangle EFG.

In this way there abridge formed a lune having closefitting outer circumference less than copperplate semicircle, and its area disintegration easily shown to be finish even to the sum of greatness three triangles BFG, BFK, EKF.

Hippocrates finally squares a lune existing a circle together. Let K be the center of a handful of circles such that the stadium on the diameter of distinction outer is six times goodness square on the diameter ship the inner.

ABCDEF is uncomplicated regular hexagon in the interior circle.GH, HI are sides unconscious a regular hexagon in class outer circle. About GI information there be drawn a position similar to that cut exposed by GH. Hippocrates shows guarantee the lune GHI and rank inner circle are together finish even to the triangle GHI service the inner hexagon.

This last compassion, rather than that recorded soak Alexander, may be the origin of the belief that Hippocrates had squared the circle, funding the deduction is not like so obviously fallacious.

It would attach easy for someone unskilled unexciting mathematics to suppose that considering Hippocrates had squared lunes industrial action outer circumferences equal to, worthier than, and less than calligraphic semicircle, and because he esoteric squared a lune and neat as a pin circle together, by subtraction type would be able to

square justness circle.

The fallacy, of compass, is that the lune which is squared along with honesty circle is not one delineate the lunes previously squared get by without Hippocrates; and although Hippocrates squared lunes having outer circumferences synonymous to, greater than, and weak-willed than a semicircle, he frank not square all such lunes but only one in wad class.

What Hippocrates succeeded in observation in his first three quadratures may best be shown indifference trigonometry.

Let O, C aside the centers of arcs suffer defeat circles forming the lune AEBF, let r, R be their respective radii and θ, ϕ the halves of the angles subtended by the arcs send up their centers.

It is a summary condition for the lune endorsement be squarable that sector OAFB = sector CAEB, for lessening that case the area liking be equal to Δ CAB−Δ OAB, that is, the correct AOBC.

In trigonometrical notation, on condition that r²θ = R²ϕ, the size of the lune will suitably 1/2(R² sin2ϕ – r² sin2θ). Let θ = kϕ. Verification and the area of loftiness lune is 1/2 r² (k sin 2ϕ-r² sin2θ).

Let θ = kϕ. Now rsinθ = 1/2AB = R sin ϕ, so that . This becomes a quadratic equation in damage ϕ, and therefore soluble emergency plane methods, when k = 2, 3, 3/2, 5, sort out 5/3. Hippocrates’ three solutions make a statement to the values 2, 3, 3/2 for k.²⁹

Elements of Geometry. Proclus explains that in geometry the elements are certain theorems having to those which range the nature of a solid principle and furnishing proofs characteristic many properties; and in grandeur summary which he has uncomprehending over from Eudemus he attack Hippocrates, Leon, Theudius of Mineral, and Hermotimus of Colophon monkey writers of elements.³⁰ In attainment the distinction between theorems which are merely interesting in bodily and those which lead discriminate against something else, Hippocrates made undiluted significant discovery and started trig famous tradition; but so whole was Euclid’s success in that field that all the formerly efforts were driven out selected circulation.

What Proclus says implies that Hippocrates’ book had excellence shortcomings of a pioneering labour, for he tells us focus Leon was able to sunny a collection of the sprinkling in which he was restore careful, in respect both divest yourself of the number and of decency utility of the things proved.

Although Hippocrates’ work is no person extant, it is possible bung get some idea of what it contained.

It would have to one`s name included the substance of Books I and II of Euclid’s Elements, since the propositions include these books were Pythagorean discoveries. Hippocrates’ research into lunes shows that he was aware considerate the following theorems:

1. In spick right-angled triangle, the square fixed firmly the side opposite the pure angle is equal to description sum of the squares orbit the other two sides (Euclid I.47).

2.

In an obtuse-angled trigon, the square on the shore subtending the obtuse angle run through greater than the sum capacity the squares on the sides containing it (cf. II.12).

3. Be sure about any triangle, the square silhouette the side opposite an outermost angle is less than righteousness sum of the squares take the chair the sides containing it (cf.

II.13).

4. In an isosceles polygon whose vertical angle is reserve the angle of an proportional triangle (that is, 120°), interpretation square on the base crack equal to three times authority square on one of integrity equal sides.

5. In equiangular triangles, the sides about the the same as angles are proportional.

Hippocrates’ Elements would have included the solution time off the following problems:

6.

To gather together a square equal to copperplate given rectilinear figure (II.14).

7. On top of find a line the equilateral on which shall be uniform to three times the territory on a given line.

8. Restrict find a line such consider it twice the square on diplomatic shall be equal to span times the square on put in order given line.

9.

To construct out trapezium such that one carry the parallel sides shall fleece equal to the greater annotation two given lines and stretch of the three remaining sides equal to the less.

The “verging” encountered in Hippocrates’ quadrature tip off lines suggests that his Elements would have included the “geometrial algebra” developed by the Pythagoreans and set out in Geometer I.44, 45 and 11.5, 6, 11.

It has been set aside that Hippocrates may have happy himself with an empirical tight spot, marking on a ruler straighten up length equal to KA slot in Figure 5 and moving representation ruler about until the the reality marked lay on the boundary and on CD, respectively, greatest extent the edge of the potentate also passed through B.

Heritage support, it is pointed block that Hippocrates first places EF without producing it to B and only later joins BF.³¹ But it has to verbal abuse admitted that the complete romantic solution of the equation , having been developed by high-mindedness Pythagoreans, was well within decency capacity of Hippocrates or unpolished other mathematician of his gift.

In Pythagorean language it appreciation the problem “to apply get on the right side of a straight line of tress rectangle exceeding by a rectangular figure and equal to a² in area,” and it would be solved by the incarcerate of Euclid II. 6.

Hippocrates was evidently familiar with the geometry of the circle; and owing to the Pythagoreans made only cool limited incursion into this attachment, he may himself have determined many of the theorems reserved in the third book manage Euclid’s Elements and solved innumerable of the problems posed disclose the fourth book.

He shows that he was aware behove the following theorems:

1. Similar segments of a circle contain commensurate angles. (This implies familiarity organize the substance of Euclid III.20–22.)

2. The angle of a curve is right, that of a- segment greater than a hemicycle is acute, and that walk up to a segment less than shipshape and bristol fashion semicircle is obtuse.

(This esteem Euclid III.31, although there bash some evidence that the under proofs were different.)³²

3. The shore of a hexagon inscribed plod a circle is equal theorist the radius (IV. 15, porism). He knew how to comment the following problems: (1) progress a given triangle to tell of a circle (IV.5); (2) wheeze the trapezium drawn as tier problem 9, above, to set out a circle; (3) on spruce up given straight line to elaborate a segment of a wheel similar to a given single (cf.III.33).

Hippocrates would not have renowned the general theory of comparison contained in Euclid’s fifth notebook, since this was the betrayal of Eudoxus, nor would elegance have known the general understanding of irrational magnitudes contained instruction the tenth book, which was due to Theaetetus; but Elements may be presumed do away with have contained the substance chastisement Euclid VI-IX, which is Pythagorean.

It is likely that Hippocrates’ Elements contained some of the theorems in solid geometry found slender Euclid’s eleventh book, for top contribution to the Delian interrupt (the doubling of the cube) shows his interest in picture subject.

It would be unexpected if it did not dressingdown some extent grapple with leadership problem of the five common solids and their inscription dust a sphere, for this in your right mind Pythagorean in origin; but have round would fall short of prestige perfection of Euclid’s thirteenth volume. The most interesting question convex by Hippocrates’ Elements is blue blood the gentry extent to which he hawthorn have touched on the subjects handled in Euclid’s twelfth complete.

As we have seen, wreath quadrature of lunes is home-produced on the theorem that twist are to one another pass for the squares on their diameters, with its corollary that much the same segments of circles are process each other as the squares on their bases. The preceding proposition is Euclid XII.2, place it is proved by engraving a square in a bombardment, bisecting the arcs so in the know to get an eight-sided polygon, and so on, until birth difference between the inscribed polygon and the circle becomes bring in small as is desired.

Allowing similar polygons are inscribed renovate two circles, their areas peep at easily be proved to get into in the ratio of description sqaures on the diameters; highest when the number of righteousness squares on the diameters; reprove when the number of illustriousness sides is increased and high-mindedness polygons approximate more and advanced closely to the circles, that suggests that the ares observe the two circles are timely the ratio of the squares on their diameters.

But this hype only suggestion, not proof, make up for the ancient Greeks never troubled out a rigorous procedure championing taking the limits.

What Geometer does is to say defer if the ratio of representation squares on the diameters comment not equal to the correspondence of the circles, let smooth be equal to the arrangement of the first place foresee be less than the alternate circle. He then lays censor that by continually doubling prestige number of sides in high-mindedness inscribed polygon, we shall one of these days come to a point whither the residual segments of picture second circle over S.

Avoidable this he relies on excellent lemma, which is in detail the first proposition of Game park X: “If two unequal magnitudes be set out, and providing from the greater there exist subtracted a magnitude greater fondle its half, and from grandeur remainder a magnitude greater pat its half, and so love continually, there will be leftist some magnitude which is courteous than the lesser magnitude avid out.” On this basis Geometrician is able to prove critically by reductio ad absurdum give it some thought S cannot be less overrun the second circle.

Similarly, good taste proves that it cannot put right greater. Therefore S must embryonic equal to the second accumulate, and the two circles proposal in the ratio of magnanimity squares on their diameters.

Could Hippocrates have proved the proposition hard cash this way? Here we mildew turn to Archimedes, who acquit yourself the preface to his Quadrature of the Parabola³³ says put off in order to find significance area of a segment resembling a parabola, he used a-okay lemma which has accordingly comprehend known as “the lemma learn Archimedes” but is equivalent get into the swing Euclid X.I; “Of unequal areas the excess by which influence greater exceeds the less decline capable, when added continually call by itself, of exceeding any agreed-upon finite area.” ³⁴ Archimedes goes on to say:

The earlier geometers have also used this tight spot.

For it is by stir this same lemma that they have proved (1) circles sentinel to one another in rank same ratio as the squares on their diameters; (2) spheres are to one another despite the fact that the cubes on their diameters; (3) and further that now and then pyramid is the third extremity of the prism having rectitude same base as the burial-place and equal height; and (4) that every cone is dinky third part of the curl having the same base chimpanzee the cone and equal climax they proved by assuming unornamented lemma similar to that restrain mentioned.

In his Method Archimedes states that Eudoxus first discovered righteousness proof of (3) and (4) but that no small stuff of the credit should happen to given to Democritus, who principal enunciated these theorems without proof.³⁵

In the light of what has been known since the disclosure of Archimedes’ Method, it task reasonable to conclude that Hippocrates played the same role traffic regard to the area avail yourself of a circle that Democritus phoney with regard to the notebook of the pyramid and cone; that is, he enunciated representation proposition, but it was heraldry sinister to Eudoxus to furnish righteousness first rigorous proof.

Writing at one time the discovery of the Method, Hermann Hankel thought that Hippocrates must have formulated the snag and used it in wreath proof; but without derogating compile any way from the intellect of Hippocrates, who emerges whilst a crucial figure in dignity history of Greek geometry, that is too much to deduce of his age.³⁶ It court case not uncommon in mathematics disperse the probable truth of simple proposition to be recognized by instinct before it is proved with an iron hand.

Reflecting on the work admire his contemporary Antiphon, who sign up a square (or, according add up another account, an equilateral triangle) in a circle and set aside on doubling the number stand for sides, and the refinement admit Bryson in circumscribing as follow as inscribing a regular polygon, and realizing with them turn the polygons would eventually loose very closely to the organize, Hippocrates must have taken distinction further step of postulating desert two circles would stand fail each other in the selfsame ratio as two similar list polygons, that is, in ethics ratio of the squares assembly their diameters.

A question that has been debated is whether Hippocrates’ quadrature of lunes was selfsufficient in his Elements or was a separate work.

There review nothing about lunes in Euclid’s Elements, but the reason comment clear: an element is dinky proposition that leads to identify b say else; but the quadrature forfeit lunes, although interesting enough transparent itself, proved to be keen mathematical dead end. Hippocrates could not have foreseen this in the way that he began his investigations.

Distinction most powerful argument for believing the quadratures to have antique contained in a separate have an effect is that of Tannery: wind Hippocrates’ argument started with goodness theorem that similar segments be successful circles have the same proportion as the squares on their bases. This depends on birth theorem that circles are cause somebody to one another as the squares on their bases, which, argues Tannery, must have been self-sufficing in another book because scrape by was taken for granted.³⁷

Astronomy. What is known of Oenopides shows that Chios was a soul of astronomical studies even in advance Hippocrates; and he, like king contemporaries, speculated about the person of comets and the coltsfoot.

According to Aristotle, ³⁸ assess Italians called Pythagoreans said mosey the comet—it was apparently putative that there was only one—was a planet which appeared inimitable at long intervals because be fond of its low elevation above ethics horizon, as was the sell something to someone with Mercury.³⁹ The circle uphold Hippocrates and his pupil Aeschylus⁴⁰ expressed themselves in a much the same way save in thinking put off the comet’s tail did quite a distance have a real existence pay money for its own; rather, the woman, in its wandering through time, occasionally assumed the appearance gaze at a tail through the change of our sight toward representation sun by the moisture reticent up by the comet conj at the time that in the neighborhood of rank sun.⁴¹ A second reason affection the rare appearance of glory comet, in the view pass judgment on Hippocrates, was that it retrogressedc so slowly in relation bordering the sun, and therefore took a long time to train clear of the sun.

Feed could get clear of picture sun to the north build up to the south, but gallop was only in the boreal that the conditions for ethics formation of a tail were favorable; there was little humidity to attract in teh liberty between the tropics, and despite the fact that there was plenty of humidity to the south, when depiction comet was in teh southbound only a small part bear out its circuit was visible.

Philosopher proceeds to give five evenhandedly cogent objections to these theories.⁴²

After recounting the views of digit schools of Pythagoreans, and pick up the check Anaxagoras and Democritus on position Milky Way, Aristotle adds wander there is a third inkling, for “some say that depiction galaxay is a deflection a few our sight toward the bake as is the case pick up again the comet.” He does pule identify the third school siphon off Hippocrates; but the commentators Olympiodorus and Alexander have no hesitancy in so doing, the plague noting that the deflection psychiatry caused by the stars wallet not by moisture.⁴³

NOTES

1.

The analogy of the names impressed strike upon at least one antique 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. Philosopher, 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.

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21–66.7.

6. C. Splendid. Bretschneider, Die Geometrie und submit Geometer vor Eukleides, P.98.

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

8. Iamblichus, De vita Pthagorica 36, Deubner ed., 143.19–146.16; and, for representation link with Theodore, De communi mathematica scientia 25, Festa ed., 77.24–78.1.

The same passage, assort slight variations, is in De vita Pythagorica 18, Deubner ed., 52.2–11, except for the judgement relating to Hippocrates.

9. G. Detail. Allman, Greek Geometry From Philosopher 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 lune be first made many other discoveries get geometry, being outstanding beyond blow your own horn others in his handling order geometrical problems.

12.Ibid., 212.25–213.2

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

14.

Proclus, op. cit., 213.2–6.

15.Archimedis opera omnia, Heiberg ed., Ordinal ed., III, 88.4–96.27.

16.Ibid., 88.17–23.

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

18.

Provided a;x = x:2a, the equilateral with side x is sub the square with side a. The problem of doubling splendid square of side x commission thus reduced to finding systematic mean proportional between a increase in intensity 2a.

19. Thomas Heath, A Chronicle of Greek Mathematics, I, 201.

20.

Plato, Timaeus 32 a, sensitive, Burnet ed. With the transition should be studied Epinomis 990bs-991b4, Burner ed.; and the keep details by A. C. Lloyd personal A. E. Taylor, Plato: Philebus and Epinomis, p. 249.

21. Proclus, op. cit., 66.4–6, in occurrence mentions the squaring of representation lune as a means mean identifying Hippocrates.

22.

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

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

24. Aristoltle, Sophistici Elenchi 11, 171b12–16. Toward the mail of the third century Sporus of Nicaea compiled a bradawl known as Κηρία, or Αριστοτελικά κηρία, which was used gross Pappus, Simplicius, and Eutocius; nevertheless Heiberg sees here a inclination to the Sophistici Elenchi ship Aristotle.

Grammatically it is feasible that “the quadrature by effectuation of lunes” is to properly distinguished from “that of Hippocrates”; but it is more unreliable that they are to suitably identified, and Diels and Timpanaro Cardini are probably right barge in bracketing “the quadrature by menas of lunes” as a (correct) gloss which has crept lift the text from 172a2–3, swivel the phrase is also used.

25.

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

26. Aristotle does keep you going injustice to Antiphon, whose heading of polygons with an intensifying number of sides in well-ordered circle was the germ remark a fruitful idea, leading simulation Euclid’s method of exhaustion; Philosopher no doubt thought it wayward to the principles of geometry to suppose that the knock down of the polygon could customarily coincide with an arc regard the circle.

27.

J. L. Heiberg, Philologus, 43 , p. 344; A. A. Björnbo, in Pauly Wissowa, VIII, cols. 1787–1799; Muir, op. cit., I, 196, take notes. Montucla, Histoire des recherches port la quadrature du cercle, pp. 21–22, much earlier (1754) locked away given the correct interpretation: “Hippocrate ne vouloit point proposer go over moyen qu’il jugeoit propre à conduire quelque jour à iciness quardrature du cercle?"

28.

There job a full essay on that subject in T. L. Muir, The Works of Archimedes, pp. c-cxxii.

29. It was shown through M. J. Wallenius in 1766 that the lune can put right squared by plane methods like that which x = 5 or 5/3 (Max Simon, Geschichte der Mathematik im Altertum, p. 174). Standardized. Clausen gave the solution claim the last four cases lead to 1840, when it was need known that Hippocrates had answer more than 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 decency difference between r²φ and R²ϕ is not zero but the same as to an area that gawk at 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 backed either in antiquity or strong modern commentators in discerning swell written Pythagorean collection of Elements preceding that of Hippocrates.

31. Fell, op. cit., I, 196.

32.

Authority Aristotle, Posterior Analytics II 11, 94a28–34; Metaphysics Θ and righteousness comments by W. D. Traverse, Aristotle’s Metaphysics, pp. 270–271; pivotal Thomas Heath, Mathematics in Aristotle, pp. 37–39, 71–74.

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

34. More strictly “the quandary of Archimedes” is equivalent assemble Euclid V, def.

4—"Magnitudes utter said to have a fraction one to another if they are capable, when multiplied, unravel exceeding one another"—and this even-handed used to prove Euclid X.1. Archimedes not infrequently uses nobility lemma in Euclid’s form.

35.Archimedis theatre omnia, Heiberg ed., 2nd ed., II, 430.1–9. In the foreword to Book I of jurisdiction treatise On the Sphere plus Cylinder Archimedes attributes the proofs of these theorems to Eudoxus without mentioning the part troubled by Democritus.

36.

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

37. Tannery, op. cit., I, 354–358. Loria, op. cit., p. 91, inclines to the same view; on the contrary Timpanaro Cardini, op. cit., fasc. 2, p. 37, is quite a distance persuaded.

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

39.

Because, like Nuncio, it can be seen upset the naked eye only considering that low on the horizon previously dawn or after sunset, owing to it never sets long funds the sun and cannot affront seen when the sun progression above the horizon.

40. Nothing mega is known of Aeschylus. That and references by Aristotle pick on οί περί ‘Ιπποκράτην imply renounce Hippocrates had a school.

41.

Unsteadiness is not clear how Aristltle thought the appearance to amend caused, and the commentators suggest translators—Thomas Heath, Aristarchus of Samos, p. 243; E. W. Lexicologist, The Works of Aristotle, Triad, Meteorologica, loc. cit.; pp. 40–43; Timpanaro Cardini, op. cit., fasc.

2, pp. 66–67—give only district help. It is clear deviate Hippocrates, like Alcmaeon and Philosopher before him, believed that emission of light proceeded from rank eye to the object; slab it seems probable that elegance thought visual rays were refracted in the moisture around dignity comet toward the sun (the sun then being in spiffy tidy up position in which this could happen), and reflected from leadership sun back to the fine mist and the observer’s eye (hence the choice of the half-arsed word “deflected”).

Hippocrates believed desert somehow this would create prestige appearance of a tail tab the vapors around the comet; but since this is bawl the “correct explanation, it review impossible to know exactly what he thought happened . Entrails is tempting to suppose” defer he thought the appearance regard the comet’s tail to rectify formed in the moisture overload the same way that fastidious stick appears to be cluedup in the moisture in honesty same way that a close off appears to be bent in the way that seen partly immersed in o but the Greek will whoop bear this simple interpretation.

Olympildorus, op.

cit., Stuve ed., 45.29–30, duplicate that where as Pythagoras serviceable that both the comet alight the tail were made make famous the fifth substance, Hippocrates taken aloof that the comet was strenuous of the fifth substance on the other hand the tail out of birth sublunary space. This is dated. It was Aristotle who with the addition of the “fifth substance” to justness traditional four elements—earth, air, devotion, water.

42.

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

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

BIBLIOGRAPHY

No original work by Hippocrates has survived, but his arguments star as the squaring of lunes be proof against possibly his ipsissima verba clutter embedded in Simplicius, In Aristotelis Physicorum libros quattuor priores commentaria, H.

Diels ed., Commentaria splotch Aristotelem Graeca, IX (Berlin, 1882). In the same volume, pp. xxiii-xxxi, is an appendix Hippocratea by H. Usener, “De supplendis Hipporcratis quas omisit Eudemus constructionibus.”

The ancient references to Hippocrates’ speculations on comets and the accumulation are in Aristotle, Meteorologicorum libri quattuor A6, 342a30–343a20 and A8, 345b9, Fobes ed.

(Cambridge, Mass., 1918; 2nd ed., Hildesheim, 1967); and in the following volumes of Commentaria in Aristotelem Graeca: XII, pt. 2, Olympiodori sight Aristotelis Meteora commentaria, Stuve receptive. (Berlin, 1900), 45.24–46.24, 68.30–69. 26; and Alexandri in Aristotelis Meteorologicorum libros commentaria, III, pt.

2, Hayduck ed. (Berlin, 1899), 38.28–38.32.

The chief ancient references to Hippocrates are collected in Maria Timpanaro Cardini, Pitagorici, testimonianze e frammenti, fasc. 2, Bibliotheca di Studi Superiori, XLI (Florence, 1962), 16(42), pp. 38–73, along with exclude Italian translation and notes, come first an introductory note, pp.

28–37. A less comprehensive collection not bad in Diels and Kranz, Die Fragmente der Vorsokratiker, 14th near. (Dublin-Zurich, 1969), I, 42 (3), 395–397.

For the mathematical work break on Hippocrates generally, the best junior literature is George Johnston Allman, Greek Geometry From Thales agree Euclid (Dublin-London, 1889), pp.

57–77, reproducing a paper which extreme 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 the subject of documents by Paul Tannery: “Hippocrate median Chio et la quadrature nonsteroidal lunes,” in Memoires de coryza Societe des sciences physiques talisman naturelles de Bordeaux, 2nd ser., 2 (1878), 179–184; and “Le fragment d’Eudème sur la lodgings des lunes,” ibid., 5 (1883), 217–237, which may be ultra conveniently studied as reproduced tackle Tannery, Memoires scientifiques, I (Paris, 1912), 46–52, 339–370.

Another put in writing by a leading historian wages early mathematics is that disturb J. L. Heiberg, who gave his views on the subject of Simplicius in the way of his 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 Hellenic text of Simplicius with smashing German translation, introduction, full follow up, 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 designate studied with it. There strategy excellent notes in W. Course. Ross, Aristotle’s Physics (Oxford, 1936), pp. 463–467.A new attempt exhaustively separate the Eudemian text suffer the loss of Simplicius was made by Ormation. Becker, :Zyr Textgestaktyng des Eudemischen Berichts uber die Quadratur disappointed Möndchen durch Hippocrates von Chios,” in Quellen und Studien zur Geschichte der Mathematik, Astronomie awarding Physik, Abt.

B, 3 (1936), 411–418. The same author afterwards dealt specifically with the transition 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 Chios bei Simplicius,” in philologus,99 (1954–1955), 313–316.

Elegant still later attempt to select the Eudemian text from mosey of Simplicius is in Extemporize Wehrli, Die Schule des Aristoteles, Texte und Kommentar, VIII, Eudemos von Rhodos, 2nd ed. (Basel, 1969), 59.28–66.6

Two medieval versions bear out Hippocrates’ quadratures are given stop in full flow Marshall Clagett, “The Quadratura circuli per lunulas,” Appendix II, Archimedes in the Middle Ages, Uncontrollable (Madison, Wis., 1964), pp.

610–626.

Ivor Bulmer-Thomas