*Michael Fowler, UVa Physics,
8/23/08*

The first recorded important
contributions to Greek science are from the city of **Miletus****Thales** in about **585 B.C**., followed by **Anaximander**
about **555 B.C.**, then **Anaximenes** in **535 B.C.** We shall argue
below that these Milesians were the first to do real science, immediately
recognizable as such to a modern scientist, as opposed to developing new
technologies.

The crucial contribution of Thales to
scientific thought was the **discovery of nature**. By this, we mean the idea that the natural
phenomena we see around us are explicable in terms of matter interacting by
natural laws, and are not the results of arbitrary acts by gods.

An example is Thales’ theory of earthquakes, which was that the (presumed flat) earth is actually floating on a vast ocean, and disturbances in that ocean occasionally cause the earth to shake or even crack, just as they would a large boat. (Recall the Greeks were a seafaring nation.) The common Greek belief at the time was that the earthquakes were caused by the anger of Poseidon, god of the sea. Lightning was similarly the anger of Zeus. Later, Anaximander suggested lightning was caused by clouds being split up by the wind, which in fact is not far from the truth.

The main point here is that *the gods are just not mentioned* in
analyzing these phenomena. The
Milesians’ view is that nature is a dynamic entity evolving in accordance with
some admittedly not fully understood laws, but not being micromanaged by a
bunch of gods using it to vent their anger or whatever on hapless humanity.

An essential part of the Milesians’ success in developing a picture of nature was that they engaged in open, rational, critical debate about each others ideas. It was tacitly assumed that all the theories and explanations were directly competitive with one another, and all should be open to public scrutiny, so that they could be debated and judged. This is still the way scientists work. Each contribution, even that of an Einstein, depends heavily on what has gone before.

The theories of the Milesians fall into two groups:

(1) theories regarding particular phenomena or problems, of the type discussed above,

(2) speculations about the nature of the universe, and human life.

Concerning the universe, *Anaximander suggested that the earth was a
cylinder*, and the sun, moon and stars were located on concentric rotating
cylinders: the first recorded attempt at a *mechanical
model*. He further postulated that
the stars themselves were rings of fire.
Again, a very bold conjecture—all heavenly bodies had previously been
regarded as *living gods*.

He also considered the problem of the origin of life, which is of course more difficult to explain if you don’t believe in gods! He suggested that the lower forms of life might be generated by the action of sunlight on moist earth. He also realized that a human baby is not self-sufficient for quite a long time, so postulated that the first humans were born from a certain type of fish.

All three of these Milesians struggled with the puzzle of the origin of the universe, what was here at the beginning, and what things are made of. Thales suggested that in the beginning there was only water, so somehow everything was made of it. Anaximander supposed that initially there was a boundless chaos, and the universe grew from this as from a seed. Anaximenes had a more sophisticated approach, to modern eyes. His suggestion was that originally there was only air (really meaning a gas) and the liquids and solids we see around us were formed by condensation. Notice that this means a simple initial state develops into our world using physical processes which were already familiar. Of course this leaves a lot to explain, but it’s quite similar to the modern view.

One of the most important
contributions of the Greeks was their development of geometry, culminating in
Euclid’s *Elements*, a giant textbook
containing all the known geometric theorems at that time (about 300 BC),
presented in an elegant logical fashion.

Notice first that the word
“geometry” is made up of “geo”, meaning the earth, and “metry” meaning
measurement of, in Greek. (The same
literal translations from the Greek give geography as *picturing* the earth and geology as *knowledge* about the earth. Of
course, the precise meanings of all these words have changed somewhat since
they were first introduced.)

The first account we have of the beginnings of geometry is from the Greek historian Herodotus, writing (in 440 B.C. or so) about the Egyptian king Sesotris (1300 B.C.):

*“This king moreover (so they
said) divided the country among all the Egyptians by giving each an equal
square parcel of land, and made this the source of his revenue, appointing the
payment of a yearly tax. And any man who
was robbed by the river of a part of his land would come to Sesotris and
declare what had befallen him; then the king would send men to look into it and
measure the space by which the land was diminished, so that thereafter it
should pay the appointed tax in proportion to the loss. From this, to my thinking, the Greeks learnt
the art of measuring land...” *

On the other hand Aristotle, writing a century later, had a more academic, and perhaps less plausible, theory of the rise of geometry:

“.*.the sciences which do not aim
at giving pleasure or at the necessities of life were discovered, and first in
the places where men first began to have leisure. That is why the mathematical arts were
founded in *.”

However, as Thomas Heath points out
in *A History of Greek Mathematics*, page 122, one might
imagine that if this (that is, Aristotle’s theory) were true, Egyptian geometry
“would have advanced beyond the purely practical stage to something more like a
theory or science of geometry. But the
documents which have survived do not give any grounds for this supposition; the
art of geometry in the hands of the priests never seems to have advanced beyond
mere routine. The most important available source of information about Egyptian
mathematics is the Papyrus Rhind
written probably about 1700 BC, but copied from an original of the time of King
Amenemhat III (Twelfth Dynasty), say 2200 BC.”

Heath
goes on to give details of what appears in this document: areas of rectangles,
trapezia and triangles, areas of circles given as (8*d*/9)^{2}, where *d*
is the diameter, corresponding to pi equal to 3.16 or so, about 1% off. There
are approximate volume measures for hemispherical containers, and volumes for
pyramids.

Another important Egyptian source is the Moscow Papyrus, which includes the very practical problem of calculating the volume of a pyramid! (Actually with a flat top: look at the figure, from Wikipedia.)

A brief overview of the early
history of geometry, up to

Pythagoras was born
about **570 B.C.** on the **Samos
**(on the map above), less than a hundred miles from **Croton**, a Greek
town in southern

Pythagoras founded what we would nowadays call a cult, a religious group with strict rules about behavior, including diet (no beans), and a belief in the immortality of the soul and reincarnation in different creatures. This of course contrasts with the Milesians’ approach to life.

The Pythagoreans believed strongly
that numbers, by which they meant the positive integers 1,2,3, ..., had a
fundamental, mystical significance. The
numbers were a kind of eternal truth, perceived by the soul, and not subject to
the uncertainties of perception by the ordinary senses. In fact, they thought that the numbers had a
physical existence, and that the universe was somehow constructed from them. In support of this, they pointed out that
different musical notes differing by an octave or a fifth, could be produced by
pipes (like a flute), whose lengths were in the ratios of whole numbers, 1:2
and 2:3 respectively. Note that this is
an *experimental* verification of an hypothesis.

They felt that the motion of the heavenly bodies must somehow be a perfect harmony, giving out a music we could not hear since it had been with us since birth. Interestingly, they did not consider the earth to be at rest at the center of the universe. They thought it was round, and orbited about a central point daily, to account for the motion of the stars. Much was wrong with their picture of the universe, but it was not geocentric, for religious reasons. They felt the earth was not noble enough to be the center of everything, where they supposed there was a central fire. (Actually there is some debate about precisely what their picture was, but there is no doubt they saw the earth as round, and accounted for the stars’ motion by the earth’s rotation.)

To return to their preoccupation with numbers, they coined the term “square” number, for 4,9, etc., drawing square patterns of evenly spaced dots to illustrate this idea. The first square number, 4, they equated with justice. 5 represented marriage, of man (3) and woman (2). 7 was a mystical number. Later Greeks, like Aristotle, made fun of all this.

Pythagoras is of course most famous for the theorem about right angled triangles, that the sum of the squares of the two sides enclosing the right angle is equal to the square of the long side, called the hypotenuse.

This
is easily proved by drawing *two*
diagrams, one having four copies of the triangle arranged so that their
hypotenuses form a square, and their right angles are all pointing outward,
forming a larger overall square, in the other this larger square is divided
differently - the four triangles are formed into two rectangles, set into
corners of the square, leaving over two other square areas which are seen to be
the squares on the other two sides.

You can prove it yourself by clicking here!

Actually, it seems very probable that this result was known to the Babylonians a thousand years earlier (see the discussion in the lecture on Babylon), and to the Egyptians, who, for example, used lengths of rope 3, 4 and 5 units long to set up a large right-angle for building and surveying purposes.

As we discussed above, the Pythagoreans greatly revered the integers, the whole numbers 1, 2, 3, …, and felt that somehow they were the key to the universe. One property of the integers we’ll need is the distinction between prime numbers and the rest: prime numbers have no divisors. So, no even number is prime, because all even numbers divide exactly by 2. You can map out the primes by writing down all the integers, say up to 100, cross out all those divisible by 2 (not counting 2 itself), then cross out those divisible by 3, then 5, etc. The numbers surviving this process have no divisors, they are the primes: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, … Now, any integer can be written as a product of primes: just divide it systematically first by 2, then if it divides, by 2 again, until you get something that doesn’t divide by 2 (and give a whole number). Then redo the process with 3, then 5, until you’re done. You can then write, for example, 12 = 2x2x3, 70 = 2x5x7 and so on.

Notice now that if you express a
number as a product of its prime factors in this way, then the square of that
number is the product of the same factors, but each factor appears twice as
often: (70)^{2} = 2x2x5x5x7x7.
And, in particular, note that the square of an even number has 2
appearing at least twice in its list of factors, but the square of an odd
number must still be odd: if 2 wasn’t on the list of factors of the number,
then it won’t be on the list for its square, since this is the same list with
the factors just appearing twice as often.

Of course, from the earliest times,
from

Let’s think about all possible numbers between one and ten, say, including all those with fractional parts, such as 3/2 or 4567/891, to choose a number at random. Suppose we take a piece of paper, mark on it points for the whole numbers 1, 2, 3, ...,10. Then we put marks for the halves, then the quarters and three quarters. Next we put marks at the thirds, 4/3, 5/3, 7/3, up to 29/3. Then we do the fifths, then the sevenths, ... Then we buy a supercomputer with a great graphics program to put in the higher fractions one after the other at lightening speed!

The question is: is this list of
fractions *all the numbers there are* between one and ten?

In other words, can we prove that there’s a number you could never ever reach by this method, no matter how fast your computer?

Two thousand five hundred years ago, the Pythagoreans figured out the answer to this question.

The answer is ** yes**:
there

This discovery greatly upset the Pythagoreans, since they revered the integers as the mystical foundation of the universe, and now apparently they were not even sufficient foundation for the numbers. Ironically, this unnerving discovery followed from applying their very own theorem—Pythagoras’ theorem—to the simplest possible right-angled triangle: half a square, a triangle with its two shorter sides both equal to one.

*This means its long side-the
hypotenuse—has a length whose square is two*.

We shall now go through their argument showing that the length of this longest side cannot be written as a ratio of two integers, no matter how large you choose the integers to be.

*The basic strategy of the proof
is to assume it *can* be written
as a ratio of integers, then prove this leads to a contradiction.*

So, we assume we *can* write
this number—the length of the longest side—as a ratio of two whole numbers, in
other words a fraction *m*/*n*. This is the length whose square is 2, so *m*²/*n*²
= 2, from which *m*² = 2*n*².

Now all we have to do is to find two whole numbers such that the square of one is exactly twice the square of the other. How difficult can this be? To get some idea, let’s write down the squares of some numbers and look:

1^{2} = 1, 2^{2} =
4, 3^{2} = 9, 4^{2} = 16, 5^{2} = 25, 6^{2} =
36, 7^{2} = 49, 8^{2} = 64, 9^{2} = 81, 10^{2}
= 100, 11^{2} = 121, 12^{2} = 144, 13^{2} = 169, 14^{2}
= 196, 15^{2} = 225, 16^{2} = 256, 17^{2} = 289, … .

On perusing this table, you will
see we have some near misses: 3^{2}
is only one more than twice 2^{2}, 7^{2} is only one less than
twice 5^{2}, and 17^{2} is only one more than twice 12^{2}.
It’s difficult to believe that if we
keep at it, we’re not going to find a direct hit eventually.

In fact, though, it turns out *this never happens*, and that’s what the
Pythagoreans proved. Here’s how they did
it.

**First, assume we canceled any common factors between numerator
and denominator.**

This means that** m and n can’t
both be even**.

Next, notice that** the square
of an even number is even**. This
is easy to check: if

On the other hand, ** the square
of an odd number is always odd**. If a number doesn’t have 2 as a factor,
multiplying it by itself won’t give a number that has 2 as a factor.

Now, back to the length of the
square’s diagonal, *m*/*n, *with ** m² = 2n²**.

Evidently, ** m² must be even**,
because it equals 2

Therefore, from what we have just
said above about squares of even and odd numbers,** m must itself be even**.

This means, though, that ** m²
must be divisible by 4**.

*This* means that 2*n*²
must be divisible by 4, since *m*² = 2*n*² -- but in this case, *n²
must be divisible by 2!*

It follows that ** n must itself
be even** --- BUT we stated at the beginning that we had

Thus a watertight logical argument has led to a contradiction.

The only possible conclusion is: **the
original assumption is incorrect**.

*This means that the diagonal
length of a square of side 1 cannot be written as the ratio of two
integers*, no matter how large we are willing to let them be.

This was the first example of an*
irrational* number—one that is not a *ratio* of integers.

Legend has it that the Pythagoreans who made this discovery public died in a shipwreck.

The historical significance of the
above proof is that it establishes something new in mathematics, which couldn’t
have been guessed, and, in fact, *something the discoverers didn’t want to be
true*. Although fractions very close
to the square root of 2 had been found by the Babylonians and Egyptians, there
is no hint that they considered the possibility that *no fraction would ever
be found* representing the square root of 2 exactly.

The kind of abstract argument here is far removed from practical considerations where geometry is used for measurement. In fact, it is irrelevant to measurement - one can easily find approximations better than any possible measuring apparatus. The reason the Pythagoreans worked on this problem is because they thought they were investigating the fundamental structure of the universe.

Abstract arguments of this type, and the beautiful geometric arguments the Greeks constructed during this period and slightly later, seemed at the time to be merely mental games, valuable for developing the mind, as Plato emphasized. In fact, these arguments have turned out, rather surprisingly, to be on the right track to modern science, as we shall see.

Over the next century or so, 500
B.C.- 400 B.C., the main preoccupation of philosophers in the Greek world was
that when we look around us, we see things changing all the time. How is this to be reconciled with the feeling
that the universe must have some constant, eternal qualities? **Heraclitus**,
from **“everything flows”**, and
even objects which appeared static had some inner tension or dynamism. **Parminedes**,
an Italian Greek, came to the opposite conclusion, that **nothing ever changes**, and apparent change is just an illusion, a
result of our poor perception of the world.

This may not sound like a very
promising debate, but in fact it is, because, as we shall see, trying to
analyze **what is changing and what isn’t**
in the physical world leads to the ideas of **elements**, **atoms** and **conservation laws**, like the
conservation of matter.

The first physicist to give a clear
formulation of a possible resolution of the problem of change was **Empedocles**
around **450 B.C**., who stated that
everything was made up of f**our elements**:
**earth, water, air and fire**. He asserted that **the elements themselves were eternal and unchanging**. Different substances were made up of the
elements in different proportions, just as all colors can be created by mixing
three primary colors in appropriate proportions. Forces of attraction and repulsion (referred
to as love and strife) between these elements cause coming together and
separation, and thus apparent change in substances. Another physicist, **Anaxogoras**, argued
that no natural substance can be more elementary than any other, so there were
an infinite number of elements, and everything had a little bit of everything
else in it. He was particularly
interested in nutrition, and argued that food contained small amounts of hair,
teeth, etc., which our bodies are able to extract and use.

The most famous and influential of
the fifth century B.C. physicists, though, were the **atomists**, **Leucippus** of **Democritus**** **of Abdera. They claimed that the physical world consisted
of atoms in constant motion in a void, rebounding or cohering as they collide
with each other. Change of all sorts is
thus accounted for on a basic level by the atoms separating and recombining to
form different materials. The atoms
themselves do not change. This sounds amazingly like our modern picture, but of
course it was all conjecture, and when they got down to relating the atoms to
physical properties, Democritus suggested, for example, that things made of
sharp, pointed atoms tasted acidic, those of large round atoms tasted sweet. There was also some confusion between the idea
of physical indivisibility and that of mathematical indivisibility, meaning
something that only exists at a point. The
atoms of Democritus had shapes, but it is not clear if he realized this implied
they could, at least conceptually, be divided. This caused real problems later on, especially
since at that time there was no experimental backing for an atomic theory, and
it was totally rejected by Aristotle and others.

It is also worth mentioning that at
this same time, on the **Kos** (see map) just a few miles from **Hippocrates**. He and his followers adopted the Milesian
point of view, applied to disease, that it was not caused by the gods, even
epilepsy, which was called the sacred disease, but there was some rational
explanation, such as infection, which could perhaps be treated.

Here’s a quote from one of Hippocrates’ followers, writing about epilepsy in about 400 B.C.:

*It seems to me that the disease
called sacred … has a natural cause, just as other diseases have. Men think it divine merely because they do not
understand it. But if they called
everything divine that they did not understand, there would be no end of divine
things! … If you watch these fellows treating the disease, you see them use all
kinds of incantations and magic—but they are also very careful in regulating
diet. Now if food makes the disease
better or worse, how can they say it is the gods who do this? … It does not
really matter whether you call such things divine or not. In Nature, all things are alike in this, in
that they can be traced to preceding causes.*

The Hippocratic doctors criticized the philosophers for being too ready with postulates and hypotheses, and not putting enough effort into careful observation. These doctors insisted on careful, systematic observation in diagnosing disease, and a careful sorting out of what was relevant and what was merely coincidental. Of course, this approach is the right one in all sciences.

In the fourth century B.C., Greek
intellectual life centered increasingly in

Actually, this all began somewhat
earlier with Socrates, Plato’s teacher, who, however, was not a scientist, and
so not central to our discussion here. One of Socrates’ main concerns was how to get
the best people to run the state, and what were the ideal qualities to be
looked for in such leaders. He believed
in free and open discussion of this and other political questions, and managed
to make very clear to everybody that he thought the current leaders of

This had a profound effect on his
pupil Plato, a Greek aristocrat, who had originally intended to involve himself
in politics. Instead, he became an
academic-in fact, he invented the term! He,
too, pondered the question of what is the ideal society, and his famous book *The Republic* is his suggested answer. He was disillusioned with Athenian democracy
after what had happened to Socrates, and impressed with

Plato, then, had a rather abstract view of science, reminiscent of the Pythagoreans. In particular, he felt that the world we apprehend with our senses is less important than the underlying world of pure eternal forms we perceive with our reason or intellect, as opposed to our physical senses. This naturally led him to downgrade the importance of careful observation, for instance in astronomy, and to emphasize the analytical, mathematical approach.

Plato believed the universe was created by a rational god, who took chaotic matter and ordered it, but he also believed that because of the inherent properties of the matter itself, his god was not omnipotent, in the sense that there were limits as to how good the universe could be: one of his examples was that smart people have large brains (he thought), but if you make the brain too large by having a very thin skull, they won’t last long! He felt this need to compromise was the explanation of the presence of evil in a universe created by a beneficent god.

Plato’s concentration on perfect underlying forms did in fact lead to a major contribution to astronomy, despite his own lack of interest in observation. He stated that the main problem in astronomy was to account for the observed rather irregular motion of the planets by some combination of perfect motions, that is, circular motions. This turned out to be a very fruitful way of formulating the problem.

Plato’s theory of matter was based on Empedocles’ four elements, fire, air, water and earth. However, he did not stop there. He identified each of these elements with a perfect form, one of the regular solids, fire with the tetrahedron, air with the octahedron, water with the icosahedron and earth with the cube. He divided each face of these solids into elementary triangles (45 45 90 and 30 60 90) which he regarded as the basic units of matter. He suggested that water could be decomposed into fire and air by the icosahedron breaking down to two octahedra and a tetrahedron. This looks like a kind of atomic or molecular theory, but his strong conviction that all properties of matter could eventually be deduced by pure thought, without resort to experiment, proved counterproductive to the further development of scientific understanding for centuries. It should perhaps be mentioned, though, that the latest theory in elementary particle physics, string theory, known modestly as the theory of everything, also claims that all physical phenomena should be deducible from a very basic mathematical model having in its formulation no adjustable parameters—a perfect form.

Lloyd, G. E. R. (1970). *Early
Greek Science: Thales to Aristotle*, Norton.

Heath, Sir Thomas (1921, 1981). *A
History of Greek Mathematics*, *Volume
I*. Dover.