### Hippocrates of Chios

Born: about 470 BC in Chios (now Khios), Greece

Died: about 410 BC

Hippocrates of Chios taught in Athens and worked on the classical problems of squaring the circle and duplicating the cube. Little is known of his life but he is reported to have been an excellent geometer who, in other respects, was stupid and lacking in sense. Some claim that he was defrauded of a large sum of money because of his naivetÃ©. Iamblichus [4] writes:-

One of the Pythagoreans [Hippocrates] lost his property, and when this misfortune befell him he was allowed to make money by teaching geometry.

Heath [6] recounts two versions of this story:-

One version of the story is that [Hippocrates] was a merchant, but lost all his property through being captured by a pirate vessel. He then came to Athens to persecute the offenders and, during a long stay, attended lectures, finally attaining such proficiency in geometry that he tried to square the circle.

Heath also recounts a different version of the story as told by Aristotle:-

... he allowed himself to be defrauded of a large sum by custom-house officers at Byzantium, thereby proving, in Aristotle's opinion, that, though a good geometer, he was stupid and incompetent in the business of ordinary life.

The suggestion is that this 'long stay' in Athens was between about 450 BC and 430 BC.

In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. We describe this impressive achievement more fully below.

Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double. This had a major influence on attempts to duplicate the cube, all efforts after this being directed towards the mean proportionals problem.

He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements. Proclus, the last major Greek philosopher, who lived around 450 AD wrote:-

Hippocrates of Chios, the discoverer of the quadrature of the lune, ... was the first of whom it is recorded that he actually compiled "Elements".

Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.

Eudemus of Rhodes, who was a pupil of Aristotle, wrote History of Geometry in which he described the contribution of Hippocrates on lunes. This work has not survived but Simplicius of Cilicia, writing in around 530, had access to Eudemus's work and he quoted the passage about the lunes of Hippocrates 'word for word except for a few additions' taken from Euclid's Elements to make the description clearer.

We will first quote part of the passage of Eudemus about the lunes of Hippocrates, following the historians of mathematics who have disentangled the additions from Euclid's Elements which Simplicius added. See [6] both for the translation which we give and for a discussion of which parts are due to Eudemus:-

The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation of lunes to the circle, were first investigated by Hippocrates, and his exposition was thought to be correct; we will therefore deal with them at length and describe them. He started with, and laid down as the first of the theorems useful for the purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases. And this he proved by first showing that the squares on the diameters have the same ratio as the circles.

Before continuing with the quote we should note that Hippocrates is trying to 'square a lune' by which he means to construct a square equal in area to the lune. This is precisely what the problem of 'squaring the circle' means, namely to construct a square whose area is equal to the area of the circle. Again following Heath's translation in [6]:-

After proving this, he proceeded to show in what way it was possible to square a lune the outer circumference of which is that of a semicircle. This he affected by circumscribing a semicircle about an isosceles right-angled triangle and a segment of a circle similar to those cut off by the sides. Then, since the segment about the base is equal to the sum of those about the sides, it follows that, when the part of the triangle above the segment about the base is added to both alike, the lune will be equal to the triangle. Therefore the lune, having been proved equal to the triangle, can be squared.

To follow Hippocrates' argument here, look at the diagram.

ABCD is a square and O is its centre. The two circles in the diagram are the circle with centre O through A, B, C and D, and the circle with centre D through A and C.

Notice first that the segment marked 1 on AB subtends a right angle at the centre of the circle (the angle AOB) while the segment 2 on AC also subtends a right angle at the centre (the angle ADC).

Therefore the segment 1 on AB and the segment 2 on AC are similar. Now

segment 1/segment 2 = AB2/AC2 = 1/2 since AB2+ BC2= AC2 by Pythagoras's theorem, and AB = BC so AC2= 2AB2.

Now since segment 2 is twice segment 1, the segment 2 is equal to the sum of the two segments marked 1.

Then Hippocrates argues that the semicircle ABC with the two segments 1 removed is the triangle ABC which can be squared (it was well known how to construct a square equal to a triangle).

However, if we subtract the segment 2 from the semicircle ABC we get the lune shown in the second diagram. Thus Hippocrates has proved that the lune can be squared.

However, Hippocrates went further than this in studying lunes. The proof we have examined in detail is one where the outer circumference of the lune is the arc of a semicircle. He also studied the cases where the outer arc was less than that of a semicircle and also the case where the outer arc was greater than a semicircle, showing in each case that the lune could be squared. This was a remarkable achievement and a major step in attempts to square the circle. As Heath writes in [6]:-

... he wished to show that, if circles could not be squared by these methods, they could be employed to find the area of some figures bounded by arcs of circles, namely certain lunes, and even of the sum of a certain circle and a certain lune.

There is one further remarkable achievement which historians of mathematics believe that Hippocrates achieved, although we do not have a direct proof since his works have not survived. In Hippocrates' study of lunes, as described by Eudemus, he uses the theorem that circles are to one another as the squares on their diameters. This theorem is proved by Euclid in the Elements and it is proved there by the method of exhaustion due to Eudoxus. However, Eudoxus was born within a few years of the death of Hippocrates, and so there follows the intriguing question of how Hippocrates proved this theorem. Since Eudemus seems entirely satisfied that Hippocrates does indeed have a correct proof, it seems almost certain from this circumstantial evidence that we can deduce that Hippocrates himself developed at least a variant of the method of exhaustion.

### Antiphon the Sophist

Born: 480 BC in (possibly) Athens, Greece

Died: 411 BC in Athens, Greece

Antiphon was an orator and statesman who took up rhetoric as a profession. He was a Sophist and a contemporary of Socrates. These definite assertions are, however, disputed by some historians. The problem seems to revolve round whether there was one Sophist philosopher named Antiphon who lived around this time or whether there are two, or as some experts claim, three distinct Antiphons.

In what follows we shall assume that at least the orator named Antiphon was the same person as the Sophist who made the mathematical advances. This is the same line as taken in [1] while in [2] only Antiphon as an orator is discussed without reference to the philosophical or mathematical works. In [7] the hypothesis that Antiphon is one, or several different men is discussed without any definite view being preferred either way.

A number of speeches which were written by Antiphon have been preserved. Three of these speeches were real speeches made by Antiphon as the prosecutor in murder trials. Twelve speeches are specimen speeches written by Antiphon for use in teaching students the skills of prosecuting and defending clients in cases. The speeches come as three collections of four; two prosecution speeches and two defence speeches for each of three different cases.

Antiphon published a number of works on philosophy which have been lost except for a small number of fragments which have been discovered together with some quotations from the works in the writings of other authors. These works include On Truth, On Concord, The Statesman, and On Interpretation of Dreams. The work On Truth is written to support the views of Parmenides who believed that there was a single sole reality and that the apparent world of many things was unreal. In this work Antiphon is defending the same philosophical ideas which Zeno of Elea supported with his paradoxes.

In On Concord Antiphon [1]:-

... defends the authority of the community as a safeguard against anarchy and recommends the ideals of concord and self-restraint both within communities and within the individual soul. Most probably he was only concerned to criticise the laws of a city by asking whether or not they satisfy the "natural" needs of the individual.

Hobbs in [7] notes that:-

... some have doubted whether the same man could have written "On Truth" and the conventional gnomic utterances of "On Concord".

In [7] three reasons are given to support at least the same author for these two philosophical works:-

(1) "On Truth" is not as radical as it appears, but simply a plea for legal reform;

(2) its doctrines, although radical, are not endorsed by Antiphon;

(3) Antiphon changed his mind.

Died: 411 BC in Athens, Greece

Antiphon was an orator and statesman who took up rhetoric as a profession. He was a Sophist and a contemporary of Socrates. These definite assertions are, however, disputed by some historians. The problem seems to revolve round whether there was one Sophist philosopher named Antiphon who lived around this time or whether there are two, or as some experts claim, three distinct Antiphons.

In what follows we shall assume that at least the orator named Antiphon was the same person as the Sophist who made the mathematical advances. This is the same line as taken in [1] while in [2] only Antiphon as an orator is discussed without reference to the philosophical or mathematical works. In [7] the hypothesis that Antiphon is one, or several different men is discussed without any definite view being preferred either way.

A number of speeches which were written by Antiphon have been preserved. Three of these speeches were real speeches made by Antiphon as the prosecutor in murder trials. Twelve speeches are specimen speeches written by Antiphon for use in teaching students the skills of prosecuting and defending clients in cases. The speeches come as three collections of four; two prosecution speeches and two defence speeches for each of three different cases.

Antiphon published a number of works on philosophy which have been lost except for a small number of fragments which have been discovered together with some quotations from the works in the writings of other authors. These works include On Truth, On Concord, The Statesman, and On Interpretation of Dreams. The work On Truth is written to support the views of Parmenides who believed that there was a single sole reality and that the apparent world of many things was unreal. In this work Antiphon is defending the same philosophical ideas which Zeno of Elea supported with his paradoxes.

In On Concord Antiphon [1]:-

... defends the authority of the community as a safeguard against anarchy and recommends the ideals of concord and self-restraint both within communities and within the individual soul. Most probably he was only concerned to criticise the laws of a city by asking whether or not they satisfy the "natural" needs of the individual.

Hobbs in [7] notes that:-

... some have doubted whether the same man could have written "On Truth" and the conventional gnomic utterances of "On Concord".

In [7] three reasons are given to support at least the same author for these two philosophical works:-

(1) "On Truth" is not as radical as it appears, but simply a plea for legal reform;

(2) its doctrines, although radical, are not endorsed by Antiphon;

(3) Antiphon changed his mind.

### Leucippus of Miletus

Born: about 480 BC in (possibly) Miletus, Asia Minor

Died: about 420 BC

Leucippus of Miletus carried on the scientific philosophy which had begun to become associated with Miletus. We know little of his life but it is thought that he founder the School at Abdera on the coast of Thrace near the mouth of the Nestos River. Today the town is in Greece and is called Avdhira. At the time that Leucippus would have lived in Abdera it was a prosperous town which politically was a member of the Delian League.

The philosopher Protagoras was born in Abdera and he was a contemporary of Leucippus bu

Died: about 420 BC

Leucippus of Miletus carried on the scientific philosophy which had begun to become associated with Miletus. We know little of his life but it is thought that he founder the School at Abdera on the coast of Thrace near the mouth of the Nestos River. Today the town is in Greece and is called Avdhira. At the time that Leucippus would have lived in Abdera it was a prosperous town which politically was a member of the Delian League.

The philosopher Protagoras was born in Abdera and he was a contemporary of Leucippus bu

### Oenopides of Chios

Born: about 490 BC in Chios (now Khios), Greece

Died: about 420 BC

Very little is known about the life of Oenopides of Chios except that his place of birth was the island of Chios. We believe that Oenopides was in Athens when a young man but there is only circumstantial evidence for this.

In Plato's Erastae Oenopides is described as (see for example [1]):-

... having acquired a reputation for mathematics...

and Plato also describes a scene where Socrates comes across two young men in the school of Dionysius who was Plato's teacher. The young men were discussing a question in mathematical astronomy which had been tackled by Oenopides and Anaxagoras. This question was certainly that of the angle that the ecliptic makes with the celestial equator. Bulmer-Thomas writes in [1]:-

... it was probably [Oenopides] who settled on the value of 24, which was accepted in Greece until refined by Eratosthenes. Indeed, if Oenopides did not fix on this or some other figure it is difficult to know in what his achievement consisted, for the Babylonians no less than the Pythagoreans and Egyptians must have realised from early days that the apparent path of the sun was inclined to the celestial equator.

However, in contrast to these claims, Heath writes [2]:-

It does not appear that Oenopides made any measurement of the obliquity of the ecliptic.

Another major contribution to mathematical astronomy made by Oenopides was his discovery of the period of the Great Year. Originally the "Great Year" was the period after which the motions of the sun and moon came to repeat themselves. Later it came to mean the period after which the motions of the sun, moon and planets all repeated themselves so in the period of one Great Year all should have returned to their positions at the beginning of the Great Year.

Oenopides gave a value of the Great Year as 59 years. Heath writes [2]:-

His Great Year clearly had reference to the sun and moon only; he merely sought to find the least integral number of complete years which would contain an exact number of lunar months.

Based on this Paul Tannery [5] showed that Oenopides' result leads to a lunar month of 29.53013 days which is remarkably close to the modern value of 29.53059 days. However many historians doubt whether Oenopides could have collected sufficient good quality data to enable him to obtain a value as accurate as this. To collect the data for even one period requires 59 years and this makes it almost impossible for someone to gather the data in their own lifetime.

Toomer believed that in fact despite Oenopides' Great Year of 59 years, he did not have this accurate value for the length of the month, and later calculations were made using better data than would have been available to Oenopides to give this very accurate value for the length of the month, more accurate than Oenopides could ever have known.

Paul Tannery in [5] makes another claim however when he states that Oenopides considered some of the planets as well as the sun and moon as part of his 59 year Great Year. The data works well for some of the planets, for example Saturn is only 2 from its starting position at the end of the 59 year cycle. Tannery is forced to conclude that not all the planets could have been taken into account by Oenopides, however, as some of the planets would be in the wrong sign of the Zodiac after the period ended.

Proclus attributes two theorems which appear in Euclid's Elements to Oenopides. These are to draw a perpendicular to a line from a given point not on the line, and to construct on a line from a given point a line at a given angle to the first line. These are elementary results but Heath believes that their significance might be that Oenopides set out for the first time the explicit 'ruler and compass' type of allowable construction. He writes [2]:-

... [Oenopides] may have been the first to lay down the restriction of the means permissible in constructions t the ruler and compasses which became a canon of Greek geometry for all plane constructions...

Oenopides also developed a theory to account for the Nile floods. He suggested that heat stored in the ground during the winter dries up the underground water so that the river shrinks. In the summer the heat disappears, as testing the temperature of deep wells suggests, and water flows up into the river so causing floods. This theory, which of course is false, did not prove popular as other rivers in Libya were subject to similar conditions but did not behave in the same way.

We have some other indications of the philosophy of Oenopides. He is said to have believed in fire and air as basic elements and thought of the world as a living being with God as its soul.

Died: about 420 BC

Very little is known about the life of Oenopides of Chios except that his place of birth was the island of Chios. We believe that Oenopides was in Athens when a young man but there is only circumstantial evidence for this.

In Plato's Erastae Oenopides is described as (see for example [1]):-

... having acquired a reputation for mathematics...

and Plato also describes a scene where Socrates comes across two young men in the school of Dionysius who was Plato's teacher. The young men were discussing a question in mathematical astronomy which had been tackled by Oenopides and Anaxagoras. This question was certainly that of the angle that the ecliptic makes with the celestial equator. Bulmer-Thomas writes in [1]:-

... it was probably [Oenopides] who settled on the value of 24, which was accepted in Greece until refined by Eratosthenes. Indeed, if Oenopides did not fix on this or some other figure it is difficult to know in what his achievement consisted, for the Babylonians no less than the Pythagoreans and Egyptians must have realised from early days that the apparent path of the sun was inclined to the celestial equator.

However, in contrast to these claims, Heath writes [2]:-

It does not appear that Oenopides made any measurement of the obliquity of the ecliptic.

Another major contribution to mathematical astronomy made by Oenopides was his discovery of the period of the Great Year. Originally the "Great Year" was the period after which the motions of the sun and moon came to repeat themselves. Later it came to mean the period after which the motions of the sun, moon and planets all repeated themselves so in the period of one Great Year all should have returned to their positions at the beginning of the Great Year.

Oenopides gave a value of the Great Year as 59 years. Heath writes [2]:-

His Great Year clearly had reference to the sun and moon only; he merely sought to find the least integral number of complete years which would contain an exact number of lunar months.

Based on this Paul Tannery [5] showed that Oenopides' result leads to a lunar month of 29.53013 days which is remarkably close to the modern value of 29.53059 days. However many historians doubt whether Oenopides could have collected sufficient good quality data to enable him to obtain a value as accurate as this. To collect the data for even one period requires 59 years and this makes it almost impossible for someone to gather the data in their own lifetime.

Toomer believed that in fact despite Oenopides' Great Year of 59 years, he did not have this accurate value for the length of the month, and later calculations were made using better data than would have been available to Oenopides to give this very accurate value for the length of the month, more accurate than Oenopides could ever have known.

Paul Tannery in [5] makes another claim however when he states that Oenopides considered some of the planets as well as the sun and moon as part of his 59 year Great Year. The data works well for some of the planets, for example Saturn is only 2 from its starting position at the end of the 59 year cycle. Tannery is forced to conclude that not all the planets could have been taken into account by Oenopides, however, as some of the planets would be in the wrong sign of the Zodiac after the period ended.

Proclus attributes two theorems which appear in Euclid's Elements to Oenopides. These are to draw a perpendicular to a line from a given point not on the line, and to construct on a line from a given point a line at a given angle to the first line. These are elementary results but Heath believes that their significance might be that Oenopides set out for the first time the explicit 'ruler and compass' type of allowable construction. He writes [2]:-

... [Oenopides] may have been the first to lay down the restriction of the means permissible in constructions t the ruler and compasses which became a canon of Greek geometry for all plane constructions...

Oenopides also developed a theory to account for the Nile floods. He suggested that heat stored in the ground during the winter dries up the underground water so that the river shrinks. In the summer the heat disappears, as testing the temperature of deep wells suggests, and water flows up into the river so causing floods. This theory, which of course is false, did not prove popular as other rivers in Libya were subject to similar conditions but did not behave in the same way.

We have some other indications of the philosophy of Oenopides. He is said to have believed in fire and air as basic elements and thought of the world as a living being with God as its soul.

### Zeno of Elea

Born: about 490 BC in Elea, Lucania (now southern Italy)

Died: about 425 BC in Elea, Lucania (now southern Italy)

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Very little is known of the life of Zeno of Elea. We certainly know that he was a philosopher, and he is said to have been the son of Teleutagoras. The main source of our knowledge of Zeno comes from the dialogue Parmenides written by Plato.

Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around 450 BC.

Despite Plato's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place. However, Plato tells us that Socrates, who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20, Zeno about 40, and Parmenides about 65 years of age at the time, so Plato's claim is certainly possible.

Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there. Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus, contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, were to have a profound influence on the development of mathematics.

Diogenes Laertius [10] gives further details of Zeno's life which are generally thought to be unreliable. Zeno returned to Elea after the visit to Athens and Diogenes Laertius claims that he met his death in a heroic attempt to remove a tyrant from the city of Elea. The stories of his heroic deeds and torture at the hands of the tyrant may well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology and again there is no supporting evidence regarding this, but we shall give some indication below of the details.

Zeno's book of forty paradoxes was, according to Plato [8]:-

... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.

Proclus also described the work and confirms that [1]:-

... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum.

In his arguments against the idea that the world contains more than one thing, Zeno derived his paradoxes from the assumption that if a magnitude can be divided then it can be divided infinitely often. Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved many fragments of earlier authors including Parmenides and Zeno. Writing in the first half of the sixth century he explained Zeno's argument why something without magnitude could not exist [1]:-

For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing.

Although Zeno's argument is not totally convincing at least, as Makin writes in [25]:-

Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense.

The paradoxes that Zeno gave regarding motion are more perplexing. Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium. For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [8]):-

There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.

In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the 1/4 point, to do this one must reach the 1/8 point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum

1/2 + 1/4 + 1/8 + ... = 1

On the one hand Zeno can argue that the sum 1/2 + 1/4 + 1/8 + ... never actually reaches 1, but more perplexing to the human mind is the attempts to sum 1/2 + 1/4 + 1/8 + ... backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get 1/4 of the way, but before we get 1/4 of the way we must reach 1/8 of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today.

Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements. However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):-

If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.

The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance). Aristotle argues against the paradox by claiming:-

... for time is not composed of indivisible 'nows', no more than is any other magnitude.

However, this is considered by some to be irrelevant to Zeno's argument. Moreover to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel writes in [20]:-

The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.

Vlastos (see [32]) points out that if we use the standard mathematical formula for velocity we have v = s/t, where s is the distance travelled and t is the time taken. If we look at the velocity at an instant we obtain v = 0/0, which is meaningless. So it is fair to say that Zeno here is pointing out a mathematical difficulty which would not be tackled properly until limits and the differential calculus were studied and put on a proper footing.

As can be seen from the above discussion, Zeno's paradoxes are important in the development of the notion of infinitesimals. In fact some authors claim that Zeno directed his paradoxes against those who were introducing infinitesimals. Anaxagoras and the followers of Pythagoras, with their development of incommensurables, are also thought by some to be the targets of Zeno's arguments (see for example [10]). Certainly it appears unlikely that the reason given by Plato, namely to defend Parmenides' philosophical position, is the whole explanation of why Zeno wrote his famous work on paradoxes.

The most famous of Zeno's arguments is undoubtedly the Achilles. Heath's translation from Aristotle's Physics is:-

... the slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.

Most authors, starting with Aristotle, see this paradox to be essentially the same as the Dichotomy. For example Makin [25] writes:-

... as long as the Dichotomy can be resolved, the Achilles can be resolved. The resolutions will be parallel.

As with most statements about Zeno's paradoxes, there is not complete agreement about any particular position. For example Toth [29] disputes the similarity of the two paradoxes, claiming that Aristotle's remarks leave much to be desired and suggests that the two arguments have entirely different structures.

Both Plato and Aristotle did not fully appreciate the significance of Zeno's arguments. As Heath says [8]:-

Aristotle called them 'fallacies', without being able to refute them.

Russell certainly did not underrate Zeno's significance when he wrote in [13]:-

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ....

Here Russell is thinking of the work of Cantor, Frege and himself on the infinite and particularly of Weierstrass on the calculus. In [2] the relation of the paradoxes to mathematics is also discussed, and the author comes to a conclusion similar to Frankel in the above quote:-

Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable.

It is difficult to tell precisely what effect the paradoxes of Zeno had on the development of Greek mathematics. B L van der Waerden (see [31]) argues that the mathematical theories which were developed in the second half of the fifth century BC suggest that Zeno's work had little influence. Heath however seems to detect a greater influence [8]:-

Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please.

We commented above that Diogenes Laertius in [10] describes a cosmology that he believes is due to Zeno. According to his description, Zeno proposed a universe consisting of several worlds, composed of "warm" and "cold, "dry" and "wet" but no void or empty space. Because this appears to have nothing in common with his paradoxes, it is usual to take the line that Diogenes Laertius is in error. However, there is some evidence that this type of belief was around in the fifth century BC, particularly associated with medical theory, and it could easily have been Zeno's version of a belief held by the Eleatic School.

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List of References (35 books/articles)

A Poster of Zeno of Elea Mathematicians born in the same country

Cross-references in MacTutor

History Topics: An overview of the history of mathematics

History Topics: The rise of the calculus

History Topics: Infinity

History Topics: A history of time: Classical time

Chronology: 500BC to 1AD

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Other Web sites

Astroseti (A Spanish translation of this biography)

The Catholic Encyclopedia

Kevin Brown

The big view

Internet Encyclopedia of Philosophy

S M Cohen (Zeno's paradoxes)

Stanford Encyclopedia of Philosophy (Zeno's paradoxes)

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### Empedocles of Acragas

Born: about 492 BC in Acragas (now Agrigento, Sicily,Italy)

Died: about 432 BC in Peloponnese, Greece

Empedocles was born in Acragas on the south coast of Sicily. The name Acragas is Greek, while the Latin name for the town was Agrigentum. Later the town was called Girgenti and more recently it became known by its present name of Agrigento. It was one of the most beautiful cities of the ancient world up to the time it was destroyed by the Carthaginians in 406 BC. It was, in Empedocles time, a rich city containing the finest Greek culture. Some of the Pythagoreans had come there after being attacked in their centre at Croton.

Empedocles was born into a rich aristocratic family. He travelled throughout the Greek world participating fully in the extraordinary desire for learning and understanding which gripped that part of the world. He is described as follows by Sarton :-

He was not only a philosopher but a poet, a seer, a physicist, a social reformer, a man of so much enthusiasm that he would easily be considered a charlatan by some people, or become a legendary hero in the eyes of others.

There are many legends regarding Empedocles life. He wrote poetry and 450 lines of such had been preserved by later writers such as Simplicius, Aristotle, Plutarch and others. It is not difficult to see the source of most of the legends about Empedocles for these are built on the poems that he wrote himself. In these he claims god-like powers, but whether this was simply a poetic style or whether he really did believe that he had such powers it is hard to say. Certainly his poems were much appreciated, for example Lucretius admired his hexametric poetry.

If we are to gather anything about the character of the man then it will come from the lines of poetry which have been preserved: 400 lines from his poem Peri physeos (On Nature) and the remainder from his poem Katharmoi (Purifications). These :-

... reveal a man of fervid imagination, versatility, and eloquence, with a touch of theatricality.

Some details of his travels appear accurate. He went to Italy and was in the town of Thurii, Lucania shortly after 445 BC. From there he went to the Peloponnese and he was in Olympia in 440 BC. His songs were sung at the Olympic games in that year. He had a young friend, Pausanias the son of Anchitos, who went with him on his travels. Of the many legends regarding his death, the most likely would appear to be that he died following a feast in the Peloponnese. Sarton writes :-

Empedocles was so great and rare a man that he left no school; none of his admirers or disciples, not even the faithful Pausanias, was able to continue the master's work.

Certainly Empedocles was attributed with many "firsts". Aristotle is said to have considered him the inventor of rhetoric while Galen regarded him as the founder of the science of medicine in Italy. He is best known, however, for his belief that all matter was composed of four elements: fire, air, water, and earth.

The reason for his four element theory was to argue a modification of the belief of the Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, which had been founded by Parmenides in Elea in southern Italy. The philosophy of this school, which included Zeno of Elea, was the claim that the many things which appear to exist are merely a single eternal reality. Empedocles did not go for the "all is one" version, but his "all is composed of the four elements" is extremely important in the development of science since it was adopted by Plato and Aristotle. As Sarton writes :-

In spite of its arbitrariness, that hypothesis had a singular fortune, for it dominated Western thought in one form or another almost until the eighteenth century.

We should also note an important feature of the hypothesis. It, like the ideas of Pythagoras, tried to explain the multitude of complexity seen in the world as being the consequence of a small number of simple underlying properties. Although we no longer believe in Empedocles' four element theory, we do still look for simple mathematics which will explain the complex phenomena that surround us.

Empedocles did not base his four element hypothesis on any experimental evidence. He did base some other scientific ideas on experiment, however, and he showed by experiment that air existed and was not empty space. He did this with a clepsydra, a vessel with a hole in the bottom and one in the top. Placing the bottom hole of the vessel under water, Empedocles observed that the vessel filled up with water. If, however, he put his finger over the top hole, then the water did not enter the hole at the bottom but it did once he removed his finger. Empedocles correctly deduced that the air in the container prevented the water entering.

Empedocles believed that light travelled with a finite velocity, not through any experimental evidence, of course, but simply through reasoning. Aristotle writes in De sensu :-

Empedocles says that the light from the Sun arrives first in the intervening space before it comes to the eye, or reaches the Earth. This might plausibly seem to be the case. For whatever is moved through space, is moved from one place to another; hence, there must be a corresponding interval of time also in which it is moved from the one place to the other. But any given time is divisible into parts; so that we should assume a time when the sun's ray was not as yet seen, but was still travelling in the middle space.

It is remarkable how many of Empedocles' ideas have turned out to be correct. In addition to his belief in the finite velocity of light he also developed a crude evolutionary theory based on the survival of the fittest. He also had a form of the law of conservation of energy and had a theory of constant proportions in chemical reactions. His ideas, although they had little influence on the development of science, can be seen in the light of our current scientific knowledge to be quite incredible. If we have to explain how such prophetically correct ideas could have such little influence we have to agree with the philosopher Hans Reichenbach who, in a book published in 1957, said:-

... a good idea stated within an insufficient theoretical frame loses its explanatory power and is forgotten.

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