ZENO of ELEA (c.490 – after 445 BCE)

Fifth Century BCE – Greece

‘A number of paradoxes; some of which seemed to prove the impossibility of motion’



Zeno was a member of the Eleatic school of thought founded by Parmenides. The School held the belief that the underlying nature of the Universe was unvarying and immobile.

In support of Parmenides’ argument, Zeno’s puzzles appeared to show that change and motion were paradoxical and that everything is one – and changeless.

Zeno’s paradoxes were based on the false assumption that space & time are infinitely divisible: that is, the sum of an infinite number of numbers is always infinite. Though they were based on fallacies, the paradoxes remained unsolved for two millennia.

Without the concepts of zero, infinity and the idea of limits, Greek philosophy and mathematics were not equipped to solve the puzzles.

The race between Achilles and the tortoise is not made up of a number of tiny distances but it is continuous until the end.

In the seventeenth century CE the Scottish mathematician James Gregory showed that it is sometimes possible to add infinite terms together to get a finite result – but to do so the terms being added together must approach zero. (This is a necessary, but not sufficient, condition – if the terms go to zero too slowly, then the sum of the terms does not converge to a finite number).
Such a series of numbers is called the convergence series, which occurs when the difference between each number and the one following it becomes smaller throughout the sequence.

The numbers 1, 1/2, 1/4, 1/8, 1/16… are approaching zero as their limit.

When you add up the distance that Achilles runs, with the terms becoming smaller and smaller, each term becomes closer to zero; each term is a step along a journey where the destination is zero.

Since the Greeks rejected the number zero they could not understand that this journey could ever have an end. To them, the terms aren’t approaching anything; the destination does not exist. Instead the Greeks saw the terms as simply getting smaller and smaller.

Using modern definitions, we know that the terms have a limit. The journey has a destination. We now may ask how far away is that destination and how long it will take to get there?

We sum up the distances that Achilles runs:
1 + 1/2 + 1/4 + 1/8 + 1/16… + 1/2n + …
As each step that Achilles takes gets smaller and smaller, it gets closer and closer to zero and the sum of these steps gets closer and closer to 2. How do we know this?
Starting with 2, we subtract the terms of the sum, one by one:

2-1=1 ; 1-1/2=1/2 ; 1/2-1/4=1/4 ; 1/4-1/8=1/8 ; 1/8-1/16=1/16…

We already know that series has a limit of zero, thus as we subtract the terms from 2 we have nothing left and the limit of the sum 1 + 1/2 + 1/4 + 1/8 + … is 2

The source of the trouble is infinity – Zeno had taken continuous motion and divided it into an infinite number of steps, assuming that the race would continue for ever as the steps get smaller and smaller. The race would never finish in finite time, but with the concept of zero we find a key to solving the puzzle.

Of course, Achilles wins the race.
Achilles takes two steps in catching up with the tortoise, even though he does it in an infinite number of increments.

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DEMOCRITUS of ABDERA (c.460 – c.370 BCE)

Democritus of Abdera


Fifth century BCE – Greece

‘Matter is made up of empty-space and an infinite number of tiny invisible particles called atomos or atoms’

Democritus’ atomic theory was probably based on previous ideas of other Greek philosophers. It was the first scientific attempt to explain the nature of matter; however, many of Democritus’ assumptions have now been proved wrong.

Democritus left no written record of his work and little is known about his life, but we know about his atomic theory from the second century CE Greek cynic and biographer Diogenes Laertius’ book ‘Lives of Eminent Philosophers’.

Democritus reasoned that, if he were to attempt to cut an object in half over and over, he would eventually reach a tiny grain of matter that could not be cut in half. Democritus called these hypothetical building blocks of matter “atoms”, after the Greek atomos, ‘uncuttable’ – suggesting that atoms could not be divided indefinitely into smaller parts and that it is impossible to create new matter. All that you can see is a product of packing together a myriad of miniscule atoms.
He said that atoms were always in motion and as they moved about they collided with other atoms; sometimes they interlocked and held together, sometimes they rebounded from collisions. The Roman poet Lucretius (c.94 – c.55 BCE) imagined Democritus’ atoms with hooks that fastened them together.

His remains one of the earliest attempts to explain the universe with a few simple physical and mathematical laws. For Democritus, there were only two things, space and atoms. Both had always existed and always would exist because de nihilo nihil ‘nothing could come from nothing’. Balancing the idea that the atom is the only unit of being, the Void is the single kind of not being. The higher the atom-to-void ratio, the denser the material. Atoms simply combined with other atoms in the Void; solid, impenetrable, indivisible blocks which never change; they combine to form different things, from rocks to plants to animals. When these things died or fell apart the structure disintegrated and the atoms were free to form new things by combining again in a different shape with other atoms.

Democritus reasoned that the method by which the atoms could combine was through their different shapes. While all the atoms were the same in substance, liquids were thought to have smooth round edges so they could fall over each other, while those that made up solids had toothed rough edges, which could hook on to each other.
Atoms of fire had tetragonal shape; atoms of earth, cubic; atoms of air, octahedral; and those of water were icosahedral.


Democritus’ thesis completely rejects the notion of the spiritual or the religious. The soul was explainable through a fast-moving group of atoms brought together by encasement in the body. The motion produced sensations that interacted with the mind (itself a collection of atoms) to produce thoughts and feelings. Once dead, the object that held the fast-moving atoms disintegrated and thus released they could separate and interact with other atoms to form new things. Leaving no place for abstract notions of the supernatural or an afterlife, this marked the arrival of materialism.


As with physical form, Democritus argued that other perceived differences in things, such as their taste, could be explained by the edges of the atoms. Likewise the colour of things was explained by the position of the atoms within a compound, which would result in darker or lighter shades.

Democritus had envisaged free atoms as flying about ceaselessly through empty space. What was needed therefore, was a precise picture of how atoms moved through space. Such a picture required knowledge of the laws that governed all motion.

Picture of the moon crater named after Democritus


The Greek philosopher ARISTOTLE rejected Democritus’ idea of the atom and said that matter was completely uniform and continuous. The influence of Aristotle was extraordinary. His concept of matter was at variance with modern thinking, but it was accepted for around 20 centuries until it was replaced by DALTON‘s atomic theory in 1808.



HIPPOCRATES of COS (c.470 – c.410 BCE)

Fifth/Fourth Century BCE – Greece

‘About 60 medical writings, known collectively as the Hippocratic corpus’

Before the time of Hippocrates there had been little science in medicine. Disease was believed to be the punishment of the gods. Divine intervention came not from the natural, but from the supernatural.

‘Treatment’, therefore, came also from the supernatural, through magic, witchcraft, superstition or religious ritual. Hippocrates’ approach was remarkable given the age in which he lived. ‘There are in fact two things,’ said Hippocrates, ‘science and opinion’; the former begets knowledge, the latter ignorance.’

Although little is known about him, Hippocrates (Bukarat in the Muslim world) is now remembered as the Father of Medicine. He was a contemporary of Socrates and lived on the island of Cos. It is widely acknowledged that Hippocrates himself could not have written many of the texts attributed to him. Written over a century and varying widely in style and argument, it is thought they came from the medical school library of Cos, possibly put together in the first instance by the author to whom they later became attributed.

The corpus is the oldest surviving Western scientific text. It laid the foundations of the western medical tradition. Although its remedies are now considered ‘imaginative’, the corpus speaks the language of science; it does not speak of spells, daemons or gods.

Hippocrates cast aside superstition and focused on the natural, in particular observing, recording and analysing the symptoms and passages of disease. The prognosis of an illness was central to this approach to medicine, partly with a view to being able to avoid in the future the circumstances that were perceived to have initiated the problems in the first place. The development of far-fetched cures or drugs was not important. What came from nature should be cured by nature; therefore rest, healthy diet, exercise, hygiene and air were prescribed for the treatment and prevention of illness. [Vis medicatrix naturae – the healing power of nature]. The fact that Hippocrates was prescribing such a natural solution at all was a major advancement.

Bust said to be of HIPPOCRATES ©


Hippocratic medicine was based on the balance of four elements – water (cold and moist), air (moist and hot), fire (hot and dry) and earth (cold and dry) – and four humors (bodily fluids) – phlegm, blood, yellow bile and black bile.

Whereas homeopathy attempts to treat disease by inducing in the patient symptoms similar to those displayed, according to the theory that symptoms are a manifestation of the body’s way of coping with the disease, allopathy describes a therapeutic technique that attempts a cure by inducing in the patient different symptoms. Basing treatment on symptoms, not causes, this enantiopathic technique can be fatal when applied to a system based on the humors. A person suffering from influenza is hot and thirsty (dry), so treatment with a mixture of cold and wet could result in a patient sitting a bath of cold water in a draughty room, awaiting a cure.

Personality or temperament was viewed as having four major types – sanguine, choleric, melancholic and phlegmatic.

Hippocrates regarded the body as a single entity, or whole, and the key lay in preserving the natural balance within this entity. Sickness was the sign of imbalance. When each of the factors was present in equal quantities, a healthy body would result. If one element became too dominant, then illness or disease would take over. If, for example, the sickness consisted of an excess of cold, moist humors, then the task of the physician was to restore the balance. The way to treat the problem would be by trying to undertake activities or eat foods which would stimulate the other humors, while at the same time trying to restrain the dominant one, in order to restore the balance and consequently, health.

The concept and treatment of humors endured for the next two thousand years, at least as far as the seventeenth century. The answers he prescribed for healthy living such as diet and exercise are still ‘good medicine’ and moreover, language introduced by Hippocrates still endures; an excess of black bile in Greek was ‘melancholic’, whilst someone with a too dominant phlegm humor became ‘phlegmatic’.

Physicians no longer practice Hippocratic medicine, but his name survives in the Hippocratic Oath that medical students take today. The Oath, probably penned by one of his followers, is a short passage constituting a code of conduct to which henceforth all physicians were obliged to pledge themselves. It outlines, amongst other things, the ethical responsibilities of the doctor to his patients and a commitment to patient confidentiality – an attempt to set physicians in the Hippocratic tradition apart from the spiritual and superstitious healers of their day.

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EMPEDOCLES of AGRIGENTUM (Sicily) (c.494 – c.434 BCE)

‘The four roots of all things are: AIR, WATER, FIRE and EARTH’

Two forces exist – LOVE and STRIFE.

The view of Empedocles developed the monists’ ideas that all substances are derived from a single source, into the concept of objects consisting of different compositions of these four basic elements.
The materials of the natural world being wrought from different blends of the four elemental principles, brought about through the eternal conflict between Love and Strife; their waxing and waning applied to cause mixing when Love is dominant, or separation by Strife.
Empedocles argued that this was the cause of transformation not just of the elements but also of the lives of people and cultures.

Popular mythology described how Aphrodite fashioned the human eye out of the four elements, held together by Love. She kindled the fire of the eye at the hearth fire of the universe, so that it would act like a lantern, transmitting the fire of the eye out into the world and making sight possible.
Empedocles realised that there must be more to sight than this, and that the darkness of night is caused by the body of the Earth getting in the way of light from the sun.

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ANAXAGORAS (c.500 – c.428 BCE)

‘The notion of the indivisible particle’

Anaxagoras came from Ionia but settled in Athens, where he remained for thirty years and taught both Pericles and Euripides. Charged with impiety because of his theory that the Sun is a red-hot stone (such an explanation, denying the role of Helios the sun-god, was enough to warrant prosecution) he fled Athens before the trial and settled in Asia Minor.
What we know about Anaxagoras is based on references to him by later writers.

In the cosmology of Anaxagoras, the Universe began as a homogenous sea of identical basic particles. Nous gave this sea a stir, in the knowledge that in time the particles would so combine to arrange themselves such that everything would be as it is today.

Bust said to be of ANAXAGORAS ©


Picture of a document showing a seal with a likeness of Anaxagoras ©

Nous was a vital principle akin to the life force of vitalism – the nearest English words being ‘mind’ or ‘intellect’.
The range of the word ‘nous’ is vastly greater, however, as it refers to the combination of insight and intuition which permits the apprehension of the fundamental principles of the cosmos – the concept is closer to the oriental idea of ‘seeing’ than the occidental notion of intelligence founded upon EUCLIDEAN LOGIC.

At the same time, Nous could be the creative, motive intelligence behind the cosmos, almost indistinguishable from the Christian concept of the will of God.

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THALES of MILETUS (624 – 545 BCE)

bust of Thales


‘WATER is the basis of all matter’

Thales is remembered because he disregarded the classical, mythical understanding of life and the universe in favour of a more physical understanding. Thales and his followers worked with the conviction that there exist natural laws governing the behaviour of natural processes and that future events could be predicted by understanding these laws. One did not have to examine chicken entrails in order to understand the vagaries and whims of the gods who ultimately determined these events.
The basic principle that you can understand nature by studying these natural laws and what things are made of forms the foundation of future philosophy and science.

Born in Miletus on the Aegean coast, Thales engaged in disciplines as varied as engineering and statesmanship.
He had a belief that there must be a fundamental substance or building block from which everything else is made. His conclusion was that this was water. Water was essential for life, drink it and you will grow, take it away and everything dies. Water can exist in a solid form as ice or snow; a bowl of water left exposed will vanish into the air. Water, he thought, could readily be made very fine, in which case it becomes air; alternatively it could be compacted into a slime that becomes earth. Not only was everything made of this elemental substance, but also all life was supported on it, literally. The Earth, said Thales, floated on water.
There were two compelling strands of evidence. First, it couldn’t be supported by air because air was incapable of supporting anything, but water could hold up large objects like logs or ships. Secondly, you could observe the effects of the Earth floating on water in that on occasions it rocked suddenly – an earthquake. Quite obviously this was caused by the water’s movement.

Legend has it that Thales, while traveling in Egypt observed that the ratio of the length of a shadow and the height of an object is the same for all objects if measured at the same time of day, hence when his shadow was the same length as his height, then at the same moment the height of the pyramid would be equal to the length of its shadow.

Nothing remains of Thales’ original work, and all knowledge of him is derived from later philosophers. His way of thinking marks the beginning of a way of investigating the hidden nature of natural things by investigation and systematisation.






These pre-Socratics survive only in enigmatic fragments.

Pittacus of Mitylene (c.650 – c.570 BCE)  ‘Know thine opportunity’

Solon of Athens (638 – 559 BCE)  ‘Know Thyself’

Thales of Miletus (c.642 – c.545 BCE) ‘Who hateth suretyship is sure’

Periander of Corinth (died 585 BCE)  ‘Nothing is impossible to industry’

Chilon of Sparta (sixth century BCE)  ‘Consider the end’

Bias of Prinene (sixth century BCE)  ‘Most men are bad’

Cleobus of Lindoe (sixth century BCE)  ‘The golden mean’ or ‘Avoid extremes’

In Athens of the fourth century BCE, empirical science held little appeal. Socrates had an aversion to natural philosophy.

PLATO celebrated mathematics but opposed any form of experimentation.
ARISTOTLE was more receptive to natural philosophy, but practiced little experimentation or even observation.