**Starting in the 6 ^{th} century BC, a group of Greek
philosophers, now known as the pre-Socratics, began to produce an avalanche of
new ideas that would change the world forever. **

**Dissatisfied with invoking
the Olympic gods to explain the world, they began to look for naturalistic
explanations. Some hypothesized that the world was made of various elements,
while others claimed it was composed of bouncing atoms. Pythagoras famously
discovered music had a mathematical foundation, giving him the idea that
perhaps mathematics itself was at the root of the universe.**

**Over a number of
centuries, they produced some of the greatest intellectual achievements of
humanity. Thales became the first to provide a formal proof of a mathematical
theorem, Aristotle discovered formal logic and gave various proofs for the
spherical shape of the Earth, Eratosthenes measured the circumference of the
Earth with remarkable accuracy, and Ptolemy created an accurate model of the
motion of the planets. **

Starting in the 6^{th} century BC, a number of
philosophers began to seek a **physical explanation of the universe**.
Today, these philosophers are called the **pre-Socratic** **philosophers**,
but Aristotle referred to them as **physiologoi** (“natural philosophers”),
distinguishing them from the earlier **theologoi **(“theologians”) and**
mythologoi** (“storytellers”), who both attributed natural phenomena to the
gods. The earliest known Greek thinker to inquire into the physical workings of
the universe was **Thales** (624–545 BC), a merchant from the Greek colony **Miletus**,
on the west coast of Turkey. As a merchant in this multicultural city on the
edge of Asia, he must have had contacts introducing him to the knowledge of the
civilizations of the Middle East and Egypt. As with many early Greek thinkers,
we only know about his work through summaries in secondary sources. Both
Aristotle and Herodotus agree that Thales used **logic to** **understand
the** **physical universe**. He developed a belief still central to modern
science, namely that despite the wide variety of phenomena around us, the
universe is, in essence, an **orderly place **that operates on a few **basic
naturalistic mechanisms**. This attempt to reduce the complexity of the world
is now called **reductionism**. This idea turned the study of the universe
from an attempt to understand the will of the gods into a rudimentary form of
science. Aristotle also credited him with the **pursuit of knowledge** **for
its own sake**. He wrote:

They were pursuing science **in
order to know**, and **not for any utilitarian end**. […] It was when
almost all the necessities of life […] had been secured, that such knowledge
began to be sought. […] It is owing to their wonder that men […] began to
philosophize. They wondered […] about the phenomena of the moon and those of
the sun and of the stars, and about the genesis of the universe. [109]

By presenting **logic **as the ultimate tool to understand
the universe—as opposed to receiving knowledge through divine revelation—Thales
inadvertently placed humanity on a pedestal, placing **great trust** in the **capacity
of the human mind**. This became the central idea of Greek philosophy.
Aristotle made it explicit by stating that **reason** was man’s most **defining
characteristic**:

**All men by nature desire to
know**.

Thales’s execution of these principles, however, was rather simplistic. He observed that water existed as a solid, a liquid, and a gas. He also noticed that water was essential to life. From these observations, he erroneously concluded that “everything is made of water.”

Perhaps surprisingly, his materialistic worldview did not make Thales an atheist. In fact, according to Aristotle, “Thales thought all things are full of gods.” In fact, most pre-Socratics retained some form of divinity, although their gods were generally more abstract than the traditional gods of Greece and they rarely required worship.

Following in the
footsteps of Thales, many other philosophers attempted to discover the true building
blocks of the universe. **Heraclitus** (535–475 BC) concluded the primary
building block must be fire because, like everything in this world, it is constantly
subject to **change**. Take, for instance, a river. Every time you step into
a river, it is filled with different water. Therefore, he famously concluded:

**No man ever steps into
the same river twice**.

Two other early influential naturalistic thinkers were **Anaximander**
(c. 610–546 BC), a student of Thales, and his own student **Anaximenes **(c.
586–526 BC), both also from the city of Miletus. Whereas traditionally, Greeks
believed **thunder **was caused by Zeus, Anaximander theorized it might be
caused by **wind colliding with clouds**. He even claimed life originated in
water and later evolved to land. Anaximenes explained the **seasons** based
on the **position of the sun **in the sky, **rainbows **in terms of the
effects of **sunlight on clouds**, and stars as burning pieces of material
pushed upward on evaporated moisture. He explained **earthquakes** as caused
by the **drying of land**, requiring no intervention by the “earth-shaker”
Poseidon. As with Thales, Anaximander and Anaximenes did believe in gods. For
Anaximander God was **infinity** and for Anaximenes it was the element **air**
(which comes dangerously close to atheism).

Another naturalistic thinker
was **Anaxagoras **(c. 510–428 BC). Anaxagoras suggested
that earthquakes were caused by air escaping from cavities below the earth. He
also called the **sun** a “**glowing stone**,” **denying its divinity**,
for which he was put on trial for impiety and exiled. He was accused of being
an atheist, although in reality he believed in a divinity called **nous **(“mind”),
which he believed was unchanging and infinite and was behind every motion in
the universe. Plato would later criticize him, stating that although he spoke
of the divine nous, when it came down to it he only used materialistic
explanations to explain the universe.

**Empedocles**
(495–435 BC) tried to unify a number of earlier theories and concluded that the
universe was built up out of **four elements**: **water**, **fire**, **earth**,
and **wind**. Mixing these elements was believed to produce the complexity
of phenomena we see in the world around us.

Around the year 500 BC,
a man named** Parmenides**, from **Elea**, a Greek colony in Southern
Italy,** **took reductionism to the extreme. Whereas Heraclitus had claimed
that the world was in constant flux, Parmenides thought that **change **and **motion
**were **illusions**. He believed that although we see things go in and
out of existence, in reality, there is only one eternal and complete Being.
This Being, according to Parmenides, is not subject to change and, as a
consequence, the world cannot be subject to change either. Since our **senses**
do perceive change, they must be **unreliable **and therefore **cannot be
trusted**. Instead, he believed, we can only get to the truth through **abstract
reasoning**.

Parmenides’s student **Zeno
**(c. 490–430 BC) set out to prove that his master was right. In his first “proof,”
he argued that an arrow in flight is, in fact, motionless. He reasoned that
since, at each moment, the arrow occupies a space that is exactly equal to
itself, it must, therefore, be at rest. Since the arrow is motionless at every
instant and since time is entirely composed of instants, this must mean that motion
is impossible.

In a second “proof,” Zeno argued that the fast Achilles could not outrun a turtle given a head start. This is why. When Achilles reaches the starting point of the turtle, the turtle has moved a little bit forward. It will then take Achilles some additional time to run that extra distance, by which time the turtle will have moved a bit forward once more. This reasoning can be continued ad infinitum, and as a consequence, Achilles will never overtake the turtle (see Fig. 283). Zeno thus concluded that it is impossible to talk sensibly about motion, so it was better to drop the concept altogether.

**Fig. ****283**
– One of Zeno’s paradoxes. Achilles can never overtake a turtle, because each
time Achilles reaches the previous position of the turtle, the turtle has moved
a little to the right. This can be repeated ad infinitum. (Martin Grandjean, CC
BY-SA 4.0)

Despite the flaws in these arguments, it is remarkable that both Parmenides and Zeno were willing to follow logic and reason, even when it ran counter to both conventional wisdom and basic experience. This, too, became a hallmark of science.

Another radical new theory about the building blocks of the
universe came from the philosophers **Leucippus** (5^{th} century
BC) and his pupil **Democritus** (c. 460–370 BC). They believed that
everything in the world is made up of indivisible and indestructible particles,
which they called **atoms** (meaning “uncuttable”). Democritus wrote:

Bitter and sweet are opinions. Color is an opinion. In truth, there are only atoms and the void.

These atoms had different shapes, could bounce off each
other and get stuck together in different combinations, explaining the
different physical phenomena in the world. For instance, Democritus believed
that bitterness was caused by small, angular atoms passing across the tongue,
whereas sweetness was caused by larger, smoother atoms. Water atoms are smooth
and slippery, while salt atoms are sharp and pointed, and iron atoms have
strong hooks that lock them into a solid. Democritus also believed we can see
the world because sheets of atoms, which he called **eidola**, are released
by objects. These sheets then pass through the air and into our eyes.

Since he believed there were
an infinite amount of atoms, there must as a consequence also be an infinite
amount of worlds. He then added that it was a matter of **chance** (“**tykhe**”)
whether any of them could sustain life. He seems to have been the first
philosopher to give an important role to chance.

Again, it is easy to
assume that Democritus was an atheist, but this was not the case. In fact, he
believed the gods were made from atoms as well. He claimed they could be seen
during sleep, when **divine eidola** penetrate our bodies. Peculiarly, he
seems to have believed that these eidola were not emitted by gods (as in the
case of ordinary objects), but actually were gods themselves, although not much
is known about his thoughts on the matter.

No matter how inventive
and creative the theories of the pre-Socratics were, they still had no method
to discover which of them (if any) was correct. Overstating their faith in
reason, they rarely bothered to corroborate their philosophical claims with **empirical
evidence**. Without evidence, however, these theories are little more than
fantasies of the human mind. The theory of atoms, however, turned out correct. Today
we can see them individually with scanning tunneling microscopes (see Fig.
284).

**Fig. ****284**
– Atoms as seen through a scanning tunneling microscope.

Thales is also considered to be the first true **mathematician**.
Since the early days of Sumer, simple arithmetic had been used for practical
purposes, such as construction projects, and in India, some remarkable
mathematical reasoning was used for the construction of altars. In Greece,
however, mathematicians started to study math **for its own sake**. They
also constructed **formal proofs** to demonstrate the validity of
mathematical theorems, instead of only giving numerical examples to make their
theorems plausible.

Later Greek writers
attributed to Thales the discovery of a **proof **for what is now called **Thales’s
theorem**. In modern vocabulary, it reads:

If points A, B, and C are points on a circle and the line AC is the diameter of the circle, then the angle at point B is always 90 degrees, no matter where point B is located on the circle (see Fig. 285).

**Fig. ****285**
– Thales’ theorem.

The Babylonians had already empirically discovered this
theorem, but Thales set out to prove it with **absolute certainty**. Let’s
go through his reasoning.

We first have to show that
the combined angles of a triangle are 180 degrees. To do this, we take a
triangle of arbitrary shape and call its angles a, b, and c (see Fig. 286). Next, we draw
two additional triangles of the exact same shape and place them as shown in the
image. At the place where the three triangles intersect, we see that the angles
a, b, and c add up to a straight angle of 180 degrees. Since the initial
triangle also has angles a, b, and c, we have now found that these angles
indeed add up to **exactly** 180 degrees. Notice that we started with an
arbitrary triangle. We could do the same procedure for **every** triangle
imaginable. So, we have discovered that **all** **triangles** have angles
that add up to exactly 180 degrees.

**Fig. ****286**
– Diagram to proof the combined angles of a triangle are 180 degrees

**Fig. ****287**
– Geometric construction used to proof Thales’ theorem

Next, we look at Fig. 287, where Thales drew an extra line, OB. Since the lines AO, BO, and CO are all radii of the circle, they are of equal length. Since the triangle ABO has two equal sides, it must mean that the angles α must also be equal. The same goes for triangle BCO and the angles β.

Since the combined angles of a triangle are 180 degrees, we find for triangle ABC:

By dividing both sides by two, it must be true that:

Since the angle at point B consists of angles α and β, it must be equal to 90 degrees. This proves Thales’s theorem.

The certainty of this
type of reasoning is unique to mathematics. Since every step of the argument is
**logically indisputable**, it must mean that the conclusion is **not merely
an opinion**, but **necessarily true**. The fact that the human mind is
able to derive such abstract truths with **absolute certainty** mesmerized
the Greeks. In fact, for some, these results were considered **sacred**. For
instance, Ptolemy, who used mathematics to obtain a “certain and steadfast”
model of the solar system (which we will discuss near the end of this chapter),

believed that the “constancy, order, symmetry, and calm” of mathematics is “associated with the divine” and “makes its followers lovers of divine beauty.”

**Pythagoras** (580–500 BC), the most famous early
mathematician, was born on the Island of Samos, close to Miletus. He was
credited by later Greek authors with a whole list of inventions, but little is
actually known about his teachings. Greek sources do agree that Pythagoras was
an ascetic who founded a sectarian school where students lived together and
were sworn to secrecy. New initiates
of the school were not permitted to see Pythagoras until after they had
completed a five-year initiation period. Instead, they could listen to him
lecture from behind a veil while they remained completely silent. The sources
also claim that Pythagoras was a strict vegetarian, believed sex weakened the
soul, and did not approve of writing. He also believed in **reincarnation**
and even claimed to remember his past lives. Based on his belief in this
“Indian” concept, a 3^{rd} century AD Greek writer even claimed Pythagoras
had studied under Hindu sages in India, but most historians do not consider this likely.

Later Greek authors
attributed to Pythagoras the discovery of a proof for what is now known as the **Pythagorean
theorem**. The theorem itself (without the proof) had been known earlier. The
Babylonians and the Egyptians had used the Pythagorean rule for over a thousand
years, and the Indian mathematician Baudhayana, between 800 and 600 BC, was the
first to explicitly
state the theorem (but saw no need to prove it). The Chinese book *Zhoubi
Suanjing* seems to offer a proof of the Pythagorean theorem for a triangle
with lengths 3, 4, and 5 (if we are not reading too much into some unclear
passages). The text is attributed to **Shang Gao**, the astronomer of the **Duke
of Zhou **(11^{th} century BC), which predates Pythagoras by about
500 years, yet the date of the proof is unsure as various parts of the book have
been updated over later centuries, some even in the first two centuries AD. A
commentator from the 3^{rd} century AD added Fig. 289 to the text, which
is a geometric figure that can be used to prove the theorem and seems to be the
figure described in words in the older text.

**Fig. ****288** – Pythagoras (Paolo Monti,
CC BY-SA 4.0)

**Fig. ****289**
– A diagram Zhao Shuang added in the 3^{rd} century AD to the *Zhoubi
Suanjing* that can be used to prove the Pythagorean theorem

The most accessible proof of the Pythagorean theorem is
depicted in Fig. 290. On the left, we see an arbitrary triangle with one angle
of 90 degrees. In the second image, four of these triangles are arranged to
form a square. The white square in the middle now has an area equal to **c ^{2}**.
In the third image, we have rearranged the triangles, revealing two white
squares with areas

**Fig. ****290**
– Geometric construction used to prove the Pythagorean theorem

Pythagoras is also credited with the discovery of the **mathematical
basis of music**. According to legend, Pythagoras made his discovery by
listening to the different sounds of blacksmith hammers. He noticed that some
hammers, when struck together, created a harmonious sound, while others didn’t.
Pythagoras discovered the hammers sounded well together if their weight **ratios
**could be described by **small integers**. For instance, the ratio of 2:1
sounded good, corresponding to a difference of an **octave**. A ratio of 3:2
gives us a difference of a **perfect fifth** and a ratio of 9:8 corresponds
to a difference of one **whole tone**. He therefore concluded that **musical
harmony** could be described by **mathematical ratios. **Curiously,
actually performing this experiment with hammers does not yield these results.
We do obtain these ratios, however, when using strings. If we cut the length of
a string in half, for instance, we get double the frequency, which corresponds
to an increase of one octave.

The discovery of the
mathematical nature of music gave Pythagoras the idea that perhaps the **entire
universe** is **governed by mathematics**. Aristotle later wrote that “the
Pythagoreans constructed the whole universe out of numbers.” Pythagoras himself
concluded:

**World is number**.

Numbers, according to Pythagoras, were the fundamental
building blocks of the universe and were therefore deemed divine. Pythagoras
was specifically impressed by the **tetractys**, which consists of ten points
arranged in rows of one, two, three, and four points and which together form a
triangle (see Fig. 291).

**Fig. ****291**
– The tetractys

A prayer of the Pythagoreans shows the importance of the tetractys:

Bless us, divine number, you who generated gods and men! O holy, holy Tetractys, you that contains the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all. [110]

Pythagoras also applied his theory of numbers to the **motion
of the planets**, which he assumed could be explained by the same **mathematical
ratios **as his musical instruments. He even took the analogy further. He
claimed the planets move in spherical concentric shells, which created
harmonious sounds, inaudible to the human ear. He called this the **harmony of
the spheres**. According to Aristotle, this was the first **physical model
of the universe** ever proposed. Anaximenes added to this model the idea that
the stars were all stuck to a **crystalline sphere**, which rotated once
every 24 hours. This idea seems ludicrous from a modern standpoint, but sounded
reasonable at the time, given that all stars seem to move in unison around the
Earth every 24 hours, with the distances between the stars kept constant. In
reality, of course, the daily motion of the stars is only apparent, and is
instead caused by the Earth rotating around its own axis.

Pythagoras’s theorem that
the whole universe can be explained in terms of (whole) numbers came under
threat when** Hippasus **(c. 530–450 BC), one of his students, discovered **irrational
numbers **(although some sources credit Pythagoras himself). An irrational
number is a number that cannot be expressed as a fraction of integers (such as
3/8 or 1/10). The first irrational number they stumbled upon was √2. When
taking a right triangle where both **a** and **b** have a length of 1,
the resulting side **c** has to be equal to √2. Horrified by the
realization that this number cannot be described in whole numbers, Pythagoras
wanted to keep the discovery a secret. One day, however, Hippasus revealed the
secret and was murdered by the sect as a result.

The most accessible proof of
the existence of irrational numbers uses the famous method of **proof by
contradiction [4]**.
In trying to prove that √2 is an irrational number, we set out

If √2 is a **rational
number**, we must be able to express it as a fraction of the form **a**/**b**,
where **a** and **b** are integers:

Squaring both sides of the equation, we find:

^{}

This can be rearranged algebraically into:

Since a number multiplied by 2 always gives an even number,
this must mean **a**^{2} is necessarily even. Since the square of an
even number is always even, this must mean that **a** is also **necessarily
even**. Since **a** is even and since the smallest even number is 2, this
must mean that **a**^{2} is always a multiple of 4. According to the
formula **a**^{2} = 2**b**^{2}, this must mean that **b**^{2}
is a multiple of 2, and this makes **b necessarily even**. And here comes
the trick. It is always possible to simplify a fraction so that **at least a
or b will be odd** (for instance, we can simplify the fraction 4/10 to 2/5).
We have now found that **a **and **b **are **both even** and that **one
of them is necessarily odd**. This **cannot both be true**. Therefore, we
must conclude that our assumption that √2 is a rational number must be
false. Therefore, √2 is an irrational number.

The classical Greeks were captivated by their mathematical discoveries,
which they also expressed in their architecture. Take, for instance, the famous
**Temple of Hera** from around 500 BC, which, to this day, is a symbol of **symmetry**,
**mathematical harmony**, and **beauty **(see Fig.
292).

The finest example of Greek
architecture is the temple of **Athena Parthenos **(“Athena the Virgin”),
also known as** the Parthenon****, **which was built between 447 and 438
BC on the top of the **Acropolis** in Athens (see Fig. 293 and Fig. 294). Here, the
obsession with mathematical perfection was pushed to the extreme. A
mathematical ratio of 9:4 was used throughout the temple. This ratio is found
in the relationship between the width and height of the temple, between the
width and length of the temple, and in the space between the columns as
compared to their diameters. To the Greeks, these exact mathematical
relationships signified beauty and harmony.

**Fig. ****292**
– The Temple of Hera (c. 500 BC) (Norbert Nagel, CC BY-SA 3.0)

**Fig. ****293**
– The Parthenon (447 – 438 BC) (Steve Swayne, CC BY-SA 2.0)

**Fig. ****294**
– The Parthenon on top of the Acropolis (447 – 438 BC). On the inside, **Pheidias
**(c. 480 – 430 BC), the most famous sculptor of Greece, added a
colossal sculpture of the goddess Athena wrought in gold and ivory. (© Shutterstock)

But there is more. Some experts believe that various sides of the Parthenon are curved to correct for the curvature caused by perspective. In Fig. 295, this effect is highly exaggerated to make it more visible. The base and the roof arch upwards, the columns tilt inwards, and the diameter of the columns and the spaces between the columns are shorter near the corners. As a result, the building appears even more geometrically perfect.

**Fig. ****295**
– Exaggerated view of the deliberate curvature of the Parthenon (Napolean Vier,
CC BY-SA 3.0)

Around 300 BC, the Greek mathematician **Euclid**
systematized mathematics in his thirteen-part masterpiece called *The
Elements*. Euclid took all the mathematical proofs that had been discovered
so far and connected them together into a coherent system. In this process, he
discovered that all known statements in geometry could be** logically** **derived**
from just five **axioms **(“things we can take for granted”). These axioms,
in turn, were so **self-evident** that no right-thinking person would doubt
their validity. The axioms are:

I. Two points can be connected by a straight line.

II. A finite line can be lengthened into an infinitely long line.

III. A finite line can be the radius of a circle.

IV. All right angles are equal to one another.

V. Only parallel lines never intersect.

Surprisingly, the whole of geometry can be derived from
these five simple statements. Let’s discuss a simple example. In Euclid’s first
proposition, he showed how to create an **equilateral triangle** (a triangle
with equal sides) from his axioms. On the left side of Fig.
296, we see a line drawn from A to B (axiom I). With a compass, we can make
this line the radius of two circles of equal size (axiom III). On the right
side of the figure, we have added point C, where the two circles intersect. Since
lines AB, BC, and CA are all radii of one or both of the circles, they must be
of equal size, and triangle ABC is, therefore, exactly equilateral.

**Fig. ****296** – Drawing used to proof Euclid’s first proposition

**Fig. ****297** – Archimedes found pi up to five decimal places, by approximating a circle by
a polygon of progressively more sides.

In proposition 47, Euclid famously showed how to prove the Pythagorean theorem from his axioms.

The greatest mathematician
of the ancient world was **Archimedes** (c. 287–212 BC). His inventions are
many. He managed to find pi to five decimal places, by approximating a circle
by a polygon of progressively more sides (see Fig. 297). He then used pi
to calculate the area of a circle, an ellipse, and a parabola, and the volume
of a sphere, a cylinder, and a cone. In a work called *The Sand Reckoner*,
he made an estimate of the amount of sand it would take to fill the universe.
At the time, the assumption was that the universe was about 10,000 Earth
diameters wide, which, Archimedes calculated, can be filled with 10^{64}
grains of sand.

Archimedes also had great
practical skills. It is said he made machines that aided the Greeks at war,
including a pulley system to lift enemy ships out of the water. He also
understood the principle of the **lever**. The longer the arm of a lever,
the lower the force required to lift an object. If long enough, a lever can be
used to lift objects of any weight. He therefore claimed, “give me a place to
stand, and I will move the Earth.” He was also asked by the king of Syracuse to
figure out whether his crown was made of pure gold or whether it was a mixture.
It is said that he figured out how to solve the problem when stepping into an
overfilled bath. The water that overflowed, he realized, was equal to the
volume of his body. This experiment could be repeated to find the volume of the
crown. Comparing the weight of the crown with a piece of pure gold of equal
volume, he should be able to find out if the crown was made of pure gold. It is
said that when he found the answer, he jumped from his bath and ran naked
through the streets shouting “eureka” (“I found it”).

Both Euclid and Archimedes
studied in the great **library of Alexandria**, which was founded around 300
BC by the Macedonian king of Egypt, **Ptolemy I Soter** (c. 367–282 BC), a
former general of **Alexander the Great** (356–323 BC). The purpose of the
library was to collect all the important texts in the known world. The library
housed an estimated 100,000 scrolls. Associated with the library was a
community of scholars, who resided in the **Musaeum**, named after the
Muses, the goddesses who inspire the arts and the sciences. In one of the
greatest traumatic events ever to hit philosophy, the library unintentionally
went up in flames when **Caesar** invaded in 48 BC. According to legend, the
documents that survived were finally burned by **Caliph Omar** in 640 AD,
who claimed their content heretical. It was said he used the scrolls to heat
public baths for nearly six months.

The philosopher **Plato** (c. 425–348 BC) was highly
influenced by Pythagoras. He was inspired by his belief that numbers were the
foundation of the universe, and he was attracted to the absolute certainty inherent
in mathematics. It was for this reason that the sign above the entrance of his
school, the **Akademia**, read:

Let no one ignorant of geometry enter here.

**Fig. ****298** – Mosaic of Plato’s Academy from
Pompeii (100 BC – 100 AD) (Jebulon; Museo Nazionale Archeologico, Naples)

Plato’s most important contribution to philosophy was his **doctrine
of the Forms** (also called the **doctrine of** **Ideas**). According
to Plato, the **world of appearances** (the world we see around us) was
imperfect and subject to change, yet at the base of every object, there existed
an ideal, eternal, and unchanging **Form**. These Forms, he believed,
existed in an objective reality in a dimension higher than the material world. He
called this reality the **world of Forms**.

In his *Timaeus*, Plato
hypothesized that the world was created by a **Divine Craftsman** (**demiurge**),
who created the universe by giving shape to already existing **unformed matter**
using **geometry**. He believed that the Divine Craftsman had fashioned
matter first into regular triangles and then combined them into the **five
Platonic solids** (see Fig. 299). A Platonic
solid is a polyhedron consisting of identical polygonal faces with equal sides
and angles. Plato was impressed by the fact that only five of such solids
exist, and he believed that they were the building blocks of **five elements**:
water, earth, fire, air, and a heavenly element.

**Fig. ****299**
– The Platonic solids by Kepler (1597) (Harmonices Mundi)

Fire, Plato believed, must be made of the pointy
tetrahedrons since it feels sharp. Air must be made of the octahedrons since it
is so smooth that one can barely feel it. Water must be made of icosahedrons
since it easily flows out of our hands when we try to pick it up. Earth must be
made out of hexahedrons since earth is tough and clumsy. The fifth Platonic
solid, the dodecahedron, Plato claimed, “God used for arranging the
constellations on the whole heaven.” His student Aristotle would later also
identify five elements (calling the fifth one “**ether**”), although he had
no interest in connecting them to the Platonic solids.

From these elements, the
demiurge created material objects based on the Forms, in the same way a builder
uses a blueprint. Matter, however, can never match the Forms perfectly, which
explains the difference between the world of Forms and the world of
appearances. Yet, since objects still resemble the Forms by approximation, we
can use **reason** to deduce what the underlying Forms must be like. A great
example is the triangle. Nowhere in the universe will we ever find a perfect
mathematical triangle, yet our mind is capable of deducing this concept after
seeing many imperfect triangles on Earth.

Surprisingly, some mathematicians and physicists to this day call themselves Platonists, as they believe that the everyday world of appearances can be fully explained by an unseen world of mathematical relationships and numerical constants.

According to Plato,
understanding these Forms comes so naturally to us that knowledge of them must
already be **inside of us**. As a believer in reincarnation, he claimed that
we must have known about them in a previous life but have since forgotten about
them. To know the Forms, therefore, we don’t need to discover them but only **recollect
**them. We read:

[The soul has] seen all things here and in the underworld. There is nothing which it has not learned, so it is in no way surprising that it can recollect the things it knew before. [111]

For instance, how do we distinguish between a river and,
say, a lake? According to Plato, this is possible because we inherently know
the Form of a river in our minds. For his student Aristotle, the concept of a
river forms by making **generalizations **after seeing many rivers, but for
Plato, it was more than that. He believed that the Forms had an **independent
existence**.

The **soul**, Plato
claimed, also preexisted in the world of Forms, and must therefore be **immortal**.
In a complete reversal from the shades in the Homeric underworld, who were
nothing without their body, Plato stated that “what gives each one of us his
being is **nothing but his soul**, whereas the **body** is no more than a
**shadow **which keeps us company.” It is now “the real man—the undying
thing called the soul” that departs after death. Plato then lets Socrates
explain that after death,** true** **sinners** go to **Tartarus**
either for a year or forever, those with **limited sins** go for **purification**
to the **Acherusian Lake**, while those who have “**purified**
themselves sufficiently **by philosophy** live thereafter altogether **without
bodies**.” [112]

In his **Myth of Er**, in
his *Republic*, Plato recounted a myth of a slain warrior named **Er**,
who after death sees divine judges send souls either through a hole in the sky
or down in a hole in the earth, depending on whether they were just or unjust.
In **heaven**, their **good deeds were rewarded** and in **hell **their
**sins were punished**. After a thousand years, the souls return to Earth for
**new incarnations**, either as humans or as animals. Before entering their
new bodies, they would drink from the **River of Forgetfulness**, causing
them to forget their previous lives.

When discussing these
stories, Plato made sure to tell us they are **just myths**, not to be taken
too literally. But in his view, they can’t be too far off:

Of course, no reasonable person ought to insist that the facts are exactly as I have described them, but that either this or something very like it is a true account of our souls and their future habitations. [112]

This idea of an **eternal soul** would later appeal to Jews
and Christians, as did the idea of a **perfect spiritual reality**
completely **disjoint from the material realm**. In a text called the *Wisdom
of Solomon* (1^{st} century BC), the idea of an eternal soul made
its way into Judaism, which before this point had no real notion of an
afterlife (ironically, the same text then rails against the influence of Greek
culture on Israel). **Augustine** (354–430), the influential Christian
saint, explicitly recognized the similarities between Christianity and Plato’s teachings,
stating the Platonists come “**closest to us**”.

**Fig. ****300** – Copy of a bust of Plato
by Silanion (c. 370 BC) (Marie-Lan Nguyen, CC BY 2.5; Capitoline Museum, Italy)

**Fig. ****301** – Roman copy of a bust of Aristotle
by Lysippos (c. 330 BC) (Jastrow; Museo nazionale romano di palazzo Altemps,
Italy)

The most influential thinker of ancient Greece was Plato’s
greatest student, **Aristotle **(384–322 BC), founder of the **Lyceum**,
another Athenian school of philosophy. Aristotle was a pioneer of great genius,
on the level of Newton and Einstein. He made world-changing contributions in
many fields, which we will discuss in this and various later chapters. The
works of Aristotle that remain are the instruction books for his students,
which can be quite dry, as academic works often are. Aristotle also wrote
public works in dialogue form, which were praised in ancient times for their
eloquence, but they have since been lost.

Unlike any other
thinker in the ancient world, Aristotle valued **empirical evidence**,
making him the grandfather of **science**. Aristotle claimed that Plato,
with his abstract speculation about an unmeasurable reality of Forms, had **neglected
the study of** **nature**. True knowledge, Aristotle claimed, was gained
by **observing similarities** in the phenomena around us and making **generalizations
**based on these observations. According to Aristotle, this was all there was
to it. He saw no need for a metaphysical world of Forms in his description of
the world. In the *Nicomachean Ethics*, he writes how he dealt with this
crucial disagreement with his friend and teacher, Plato:

But perhaps it is desirable that
we should examine the notion of a Universal Good [the Form of the Good], and
review the difficulties that it involves, although such an inquiry goes against
the grain because of our friendship for the authors of the Theory of Ideas [Forms].
Still perhaps it would appear desirable, and indeed it would seem to be
obligatory, especially for a philosopher, to sacrifice even one’s closest
personal ties in defense of the truth. Both are dear to us, yet it is our duty
to** prefer the truth**. [113]

According to Aristotle, objects did have a “form” with a
small “f” (their shape), but since this form was never observed **apart from
the object itself**, he felt little reason to assume that forms had a
separate existence in a more abstract world. This was even true for the **soul**,
which he claimed **could not exist without the body** and therefore could **not
**be **immortal **(yet, in one
curious passage he deviates from this opinion and states that part of the soul
was immortal after all).

Aristotle exposed the theory of the Forms as absurd, as it runs into problems in abstract cases. Take for instance a walk from point A to point B. Does the line of the walk exist as a Form separate from the walk? And if it exists, where would that line be located? According to Aristotle, these questions are nonsensical.

Finally, Plato had claimed
that the Forms were eternal and unchanging, but could not explain how they
introduced change into the world. Plato tried to address this problem by
stating that objects in the real world were not identical to the forms, but
only “**participated**” in them. Aristotle rightly countered that this
explained nothing, calling Plato’s explanation mere “empty words and poetical
metaphors.” [109]

Instead of focusing on the world of forms, Aristotle spent
most of his time concentrating on the world that lies within our grasp. For
example, he became the first **biologist**. He dissected animals and plants
to understand how they functioned and broke eggs at various levels of
development to chart the growth of the chick embryo.

Although these
biological studies made him the grandfather of science, Aristotle did not do **experiments**
in the modern sense. The word he used instead was **observation **(**pepeiramenoi**).
He simply observed the world but generally **did not design experiments** with
the sole purpose of isolating a phenomenon and manipulating it in order to
discover how it works.[5]
His lack of experimentation famously showed when he wrote that **heavy objects
fall faster than light objects**. This might sound recognizable from our
day-to-day experience (a lead ball falls faster than a sheet of paper), but a
simple experiment can show both objects **hit the ground at the same time**
when **air friction is small** (crumple the paper into a ball to reduce air
friction and both objects will fall at the same rate).

The Greek who came closest
to being a modern scientist was likely **Ptolemy** (c. 100–170 AD). In his *Optics*,
he described the use of a piece of equipment containing a bronze disk, metal mirrors
of various shapes, and a sighting tube. With this **experimental setup**, he
demonstrated very accurately that the **angle of incidence** of a light ray hitting
a mirror is equal to the **angle of reflection**. He also conducted an
experiment on **refraction**, which involves a change in the direction of
light when it propagates from one substance to another (for instance from air
to glass), but here he found an erroneous result, likely because he was fudging
with his own data to make it fit his hypothesis.

Aristotle also created the most sophisticated **theory of
the elements** of the ancient world, which he described in his *On
Generation and Corruption*. Aristotle hypothesized that the foundational
substance of the universe was **prime matter**, which was a substance
without any qualities. Combined with the qualities **hot**,** cold**, **wet**,
and** dry**, prime matter turned into the four elements (see Fig.
302). Although his theory later turned out to be wrong, it did have remarkable explanatory
power. For instance, if we give an object the qualities hot and dry, the object
will catch fire. When water (with the qualities cold and wet) is heated by
fire, it evaporates into air (having the qualities hot and wet).

**Fig. ****302**
– Aristotle’s theorem of the four elements.

Aristotle also set out to classify what was necessary to **fully
understand** a phenomenon. In his estimation, it required knowing **four
causes** (perhaps better translated as “four explanations”). The **material
cause** refers to the material the object is made of, the **efficient cause**
refers to the agent that produced the object, the **formal cause** refers to
the shape or **essence **of an object, and the **final cause** refers to
the **purpose** (“**telos**”) of the object. The final cause is of
particular interest here. Aristotle believed that all objects have an**
inherent ****purpose** or **goal**. If that purpose was not yet met, it
existed **inherent in the object** as a **potentiality**. For instance,
oil has the potential to burn. This potentiality, Aristotle claimed, exists in
the oil even when it is not burning.

Aristotle likely got this
idea from his study of biology. His many years of dissecting and describing
animals and plants taught him that different organs all have their own distinct
function. Muscles and tendons, for example, have the purpose of moving parts of
the body. The stomach has the purpose of digesting food. Seeds have the purpose
(or potential) of becoming trees. This idea was finally challenged by **Darwin’s**
**theory of evolution**, which showed convincingly that the **perception of
purpose** in biology was only an **epiphenomenon **of **natural selection
**(we have hearts not in order to pump blood around, but because creatures
without hearts were less successful at survival).

While talking about
purpose still has its place in biology, Aristotle also applied this type of
thinking to material objects. For instance, he believed that the **purpose**
of** fire** was to **point upwards toward the stars**. This is why the
flame on a candle always points upwards, even when the candle is turned upside
down. The purpose of the element earth was to **fall down** in a straight
line **towards the center of the universe**, which he believed was located
at the **center of the Earth**. It took 2000 years for people to realize it
is not the stone that inherently “wants” to fall downward, but that the earth
instead exerts a **gravitational force** on the stone. Aristotle’s theory,
nevertheless, had remarkable explanatory power. In *On the Heavens*,* *he
used it to explain the **spherical shape of the earth**. If matter all wants
to get as close to the center of the universe as possible, you automatically
end up with a spherical earth.

Aristotle wasn’t the first to believe in a spherical Earth. Plato had also written about it, but without providing evidence:

My conviction is that the earth is a round body in the center of the heavens, and therefore has no need of air or of any similar force to be a support. [114]

Aristotle did provide **evidence**. When the shadow of
the earth falls over the moon during a lunar eclipse, the shadow is always
circular. Since only a sphere creates a circular shadow from all sides, he
concluded that the earth must be a sphere. Aristotle also noted that when a
person travels north or south, he sees unknown stars on the horizon, which
would not make sense if the earth was flat (see Fig.
303). Similarly, when a person travels either east or west, he will find that
the sun rises at a different time. Finally, Aristotle noted that when a large
ship sails off to the horizon, its bottom disappears first, and only later does
its mast. This could be explained by assuming that the bottom of the boat
becomes concealed behind the curvature of the earth (see Fig.
304).

**Fig. ****303** – People on different parts of a spherical Earth see different stars (S. P.
Dinkgreve, worldhistorybook.com)

**Fig. ****304** – On a spherical Earth, the bottom of a boat can hide behind the curvature of
the Earth. From a 1550 copy of *On the Sphere of the World*, the original
dating to the 13^{th} century.

Aristotle’s theory could also explain the motion of a **falling
rock**. The natural motion of a stone was to move towards the center of the
universe, making it fall. However, the stone can be stopped by **external
forces**, for instance the force exerted on the rock by the floor. Aristotle
saw this as an **artificial** or **unnatural **stop, which prevented the
natural motion of the stone. As a result, the rock will finally lay still on
the surface of the Earth.

These external forces can
also explain why a **rock thrown upwards** **continues its upward motion**
after leaving the hand before finally falling down again. When a rock moves
upward, it leaves behind a **vacuum**. Since Aristotle claimed that **nature**
“**abhors a vacuum**,” the surrounding air rushes in, giving the stone an
extra push, propelling it upward. Eventually, though, the natural motion
prevails, and the rock will fall down again.

Although incorrect, his theory was definitely consistent and the most comprehensive theory of the world thus far.

Aristotle’s four elements explained motion on Earth, but could
not explain the **eternal circular motion **of the **heavenly bodies**.
Why did they not fall to the center of the universe? And why did they not slow
down and come to rest as objects on earth do? And why was there “no trace of
change in the whole of the outermost heaven or in any one of its proper parts”?
To circumvent these problems, Aristotle postulated a **fifth element**,
called **ether**, from which all heavenly bodies were made, characterized by
endless and unchanging circular motion.

Since the physics on earth
seemed so different from the physics that governed heaven, Aristotle divided
the universe into the **world below the sphere of the moon** and the **world
above the sphere of the moon**. This idea of a distinct terrestrial and
celestial physics was finally overthrown some 2000 years later by **Isaac Newton**
(as will be discussed in a later chapter). Newton showed that the **same
gravitational force** could explain both the **falling of objects **and
the **orbits of the heavenly bodies**.

With his theory, Aristotle also reasoned there could not exist “more than one world,” for if there was another Earth (necessarily made of the four elements), it too would fall to the center of the universe. In the Middle Ages, this was used as an argument against aliens on other planets.

In his *Metaphysics*, Aristotle set out to investigate
the **fundamental nature of reality**, which he equated with finding “the
principles and causes of **substances**.” This investigation into substance,
he claimed, went even deeper than physics. With physics we describe the natural
world, but with **metaphysics** we want to know what it means for an object **to
exist** in the first place. He called it the study of “**being as being**”.
Aristotle recognized this as the **First Philosophy**, the study of the most
foundational knowledge.

Aristotle’s metaphysics
rides on the idea that every **causal chain** (or, perhaps more accurately:
every explanation) should be **finite**. However, when thinking about the
world, we often run into **infinite regression **problems. This can happen
for each of the four causes:

It is impossible that one thing should come from something else as from matter [the material cause] in an infinite regress, for example, flesh from earth, earth from air, air from fire, and so on to infinity. Nor can the causes from which motion originates [the efficient cause] proceed to infinity, as though man were moved by the air, the air by the sun, the sun by strife, and so on to infinity. Neither can there be an infinite regress in the case of the reasons for which something is done [the final cause], as though walking were for the sake of health, health for the sake of happiness, and happiness for the sake of something else, so that one thing is always being done for the sake of something else. The same is true in the case of the essence [the formal cause]. [115]

To solve this problem, Aristotle began to wonder about the
cause of the orbital motion of the heavenly bodies. Just like all motion,
Aristotle believed, it was in need of a **mover** (an efficient cause). This
mover, however, had to “move without being moved,” for otherwise, it would
itself also be in need of a mover (creating infinite regression). He therefore
called it the “**unmoved mover**,” the “**uncaused cause**,” or the “**first
cause**.” But how can something move something else without itself moving?
According to Aristotle, the only solution was to assume that the unmoved mover
was an **object of desire**, “producing motion by **being loved**.” Just
as a bottle of water can “cause” a thirsty person to move towards it, so too
are the heavenly bodies compelled into orbit by the unmoved mover. It is the **final
cause** (or purpose) of the heavenly bodies to move as the unmoved mover
desires. To avoid infinite regression, Aristotle then claimed that **every
motion on earth** is (somehow) **causally linked** in a **finite number
of steps** to the **motion of the heavenly bodies**, which in turn was **caused
by the unmoved mover**.

Aristotle equated his
unmoved mover with **God**, making his metaphysics a **theological endeavor**.
He then set out to use **reason** to **derive the characteristics** of this
**god**. First of all, since all matter can be brought into motion, but the
unmoved mover cannot, it follows that the unmoved mover must be **immaterial**.
With reason being the highest quality in the universe, he claimed God must be
the **supreme thinker**, and being a thinker, he claimed his god must be **alive**.
But what does the supreme thinker think about? Being the perfect thinker, it has
to think about itself. Finally, Aristotle had to figure out whether there was
just one unmoved mover, or whether there were more of them. He had reason to
believe there were more, as **Eudoxus**—another student of Plato—had
identified 27 different circular motions in the heavens, perhaps requiring 27
gods (more on this later). Aristotle finally settled on one unmoved mover,
believing God was indivisible.

One last clarification is
necessary. In modern times, we often envision a **first cause** to occur at
the beginning of the universe and for everything to unfold from there.
Aristotle, however, believed in an **eternal universe**, since he believed
the heavenly bodies had existed unchanged for all eternity. So, his unmoved
mover was not the first cause at the beginning of time, but instead compelled
the heavenly bodies forward at every moment in time. Without the unmoved mover,
he believed, the planets would stop.

Aristotle also laid the foundation of **formal logic **in
a work called the* Organon*, which consists of six books on the topic. The
goal of this pursuit was to figure out how we can attain **reliable knowledge**.
In his book *Prior Analytics*, he discussed what **logical inferences**
could be made with **absolute certainty** from various simple statements. Take,
for instance, the following sentence: “All ravens are black.” If we regard this
statement as true, then we can with absolute certainty infer that the following
statement must also be true: “There are no white ravens.” Aristotle called this
**deductive reasoning**. He defined **deduction **(**sullogismos**) as
follows:

A deduction is an argument
in which, certain **things having been supposed**, something other than what
was laid down **results by necessity** because these things are so. By “because
these things are so” I mean that they result through these, and by “resulting
through these” I mean that no term is required from outside for the necessity
to come about. [116]

The “things supposed” are the **premises **(**protasis**)
of the argument, while what “results by necessity” are the **conclusions **(**sumperas-ma**).

To systematize his thinking, he created four generalized statements:

I. All X are Y

II. No X are Y

III. Some X are Y

IV. Some X are not Y

To reason correctly with these statements, Aristotle defined three self-evident axioms:

I.
**The law of identity**

Words either mean the same thing or they don’t.

II.
**The law of contradiction
**A statement can’t both be true and untrue at the same time.

III.
**The law of the excluded middle**

A statement is always either true or not true. There is no third option.

These simple ideas, often called the **three laws of
thought**, are still used in philosophy, mathematics, and computer
programming.

Aristotle then showed
how to combine different statements to find new conclusions. He called such a
construction a **syllogism**. Let’s study an example:

Statement 1 All X are Y

Statement 2 All Z are X

Conclusion Therefore, all Z are Y

That this conclusion follows is often easier to understand by filling in words for X, Y, and Z. For example:

Statement 1 All Greeks are mortal

Statement 2 All Athenians are Greek

Conclusion Therefore, all Athenians are mortal

When trying all combinations, he found there are 256 possible syllogisms, of which only 24 turned out true. Here we find two other examples of correct syllogisms:

Statement 1 All X are Y

Statement 2 Some Z are X

Conclusion Therefore, some Z are Y

Statement 1 No X is Y

Statement 2 All Z are X

Conclusion Therefore, no Z are Y

In his *Posterior Analytics*, Aristotle discussed a
crucial problem with syllogistic reasoning. In themselves, syllogisms tell us
nothing about the world. The **conclusions are** **only true** **if the premises
are shown to be true** in the first place. Yet, in most cases, we cannot prove
the premises with absolute certainty. Take, for instance, the premise “All
ravens are black.” Although all ravens discovered thus far have been black,
there is no way to know for sure whether they are all black. To prove that “all
ravens are black,” we therefore cannot rely on deduction and have to rely on
the next best thing: **induction **(**epagoge**). Induction works by
making **generalizations** about the world based on a **finite number of
measurements**. Aristotle called it “**an argument from the particular to
the universal**.” If we find enough particular ravens and they are all black,
we can convince ourselves of the universal statement “All ravens are black,”
but we can never be completely certain.

Aristotle then went
further, claiming that these two methods—deduction and induction—are the **only
reliable paths to knowledge**. When we are born, he claimed, our mind is like
an “**unscribed tablet**.” At this point, our mind contains **no innate
knowledge**, but merely the **capacity to know**. Anticipating the
scientific revolution that would occur two millennia after his death—making Aristotle
the greatest genius of all time—he claimed that **all knowledge had to enter
the mind through the senses**:

We conclude that […] knowledge [is]
**neither innate** in a determinate form, nor
developed from other higher states of knowledge, but from
**sense-perception**. [117]

In the Middle Ages, Saint **Thomas Aquinas **(1225–1274)
summarized his position very concisely:

**Nothing is in the intellect
that was not first in the senses**.

Through sensory observation, Aristotle claimed, we can in
turn apply induction to discover the **first principles** of science. The
rest of science should then follow by syllogistic deduction.

**Eratosthenes** (c. 276–195 BC) found a way to determine
the** circumference of the earth**. He noticed that at a certain longitude
and at a certain time of the year, the sun would be directly overhead. As a
result, sunlight would shine directly into a well, leaving no shadow at the
bottom. At this exact same time, a lighthouse situated hundreds of kilometers north
did create a shadow (see Fig. 305).

**Fig. ****305**
– The method used by Eratosthenes to calculate the circumference of the earth (S.
P. Dinkgreve, worldhistorybook.com)

The figure shows that the angle of the shadow is equal to
the angle depicted at the center of the earth (in mathematics, we call these alternate
angles). Eratosthenes measured the angle of the shadow to be 7.2 degrees, which
corresponds to 1/50^{th} of the earth’s circumference. Multiplying the
distance between the well and the lighthouse by 50, he found a circumference
only one percentage point removed from the modern value of 40,000 kilometers. Eratosthenes
also calculated the angle of the Earth’s axis at 23.3 degrees and introduced
the idea of a leap day to even out the calendar.

**Aristarchus** (c. 310–230 BC) used the circumference of the earth and
the size of the earth’s shadow on the moon to calculate the **distance to the
moon**. With the distance to the moon, he also found a way to calculate the **distance
to the sun**. Although both his techniques were mathematically correct, his
measuring equipment was not accurate enough to find the correct answers. A few
centuries later, Ptolemy managed to find out that the moon was 59 earth radii away
from the earth (the modern value is 60 earth radii).

Aristarchus was also the
first to propose a **heliocentric model of the solar system**. He showed how
the rotation of the Earth’s axis could explain the apparent daily rotation of
stars around the Earth and the rotation of the Earth around the sun could
explain the apparent yearly rotation of the zodiac. According to **Plutarch **(46–119
AD) the astronomer **Seleucus** (2^{nd} century BC) even had a proof
of the heliocentric model, although we are not told what this proof was. Although
the theory did not become widely accepted among the Greeks, Plutarch also
mentioned that “Plato, when he grew old, repented that he had placed the Earth
in the middle of the universe, which was not its place.”

The Greek philosophers
also attempted to improve the Mesopotamian model of the solar system. It had
been known since early Mesopotamian days that the planets didn’t move in simple
circles across the night sky. Instead, they periodically seemed to stop, move
back a little, stop again, and then move onwards again (see Fig.
306). This backward motion is called **retrograde motion**. The mechanism
behind this motion was not understood.

**Fig. ****306**
– The retrograde motion of Mars (S. P. Dinkgreve, worldhistorybook.com)

Plato’s faith in the geometric regularity of the world convinced
him that the motion of the planets could still be understood in terms of **uniform
circular motion**. To prove this, he famously challenged the astronomers of
his day to “**save the phenomena**,” meaning to bring his theory of circular
celestial motion in line with observed reality. This, however, turned out to be
a difficult task because of the complexity of the motion of the heavenly
bodies. First there was the **diurnal motion**, which lets all celestial
bodies rise in the east and set in the west once every 24 hours (we now know
this is caused by the rotation of the Earth around its axis). Then there was
the **annual motion**, which makes the heavenly bodies rise a little earlier
every day, making different constellations visible throughout the year (as
caused by the motion of the Earth around the sun). Then there was the **motion
of the planets** themselves. Both the velocity and the brightness of the
planets change over time, and they also make a wavy up-and-down motion along
their path within the band of the zodiac. And then, finally, there was
retrograde motion.

Several people tried to
solve the problem, among them **Eudoxus **(c. 408–355 BC), one of Plato’s
students and the second-best mathematician of Greece. His mathematical works
have not survived first hand, as his major proofs were incorporated into the
work of Euclid.[6] Eudoxus
took up the Pythagorean idea that the **fixed stars** were attached to a **sphere**,
which rotates around its axis every 24 hours. The **planets**, in his
theory, were each attached to **four spheres**, the first moving in the same
manner as the fixed stars, the second moving the planet along the band of the
zodiac, and the third and fourth spheres cleverly working together to create
retrograde motion. This model, with a total of **27 spheres**, brilliantly
reduced the complex motion of the planets to a composite of just circular
motion, in line with Plato’s challenge, but it was finally discarded because it
did not account for the changes in brightness of the planets (which is caused
by a change in distance).

The Greek astronomer**
Ptolemy**, also working in the library of Alexandria, was the first to fully
solve the problem. In his work called *Mathematike Syntaxis*, which is now
known by its Arab title *The Almagest* (from “Al-Majisti,” meaning “the
greatest”)—which is considered to be among the most influential scientific
texts of all time—he produced the first **accurate mathematical model**
describing the **motion of the heavenly bodies**.

**Fig. ****307** – Paths of the planets and
the sun as seen from the earth, by James Ferguson (18^{th} century)

**Fig. ****308**
– Ptolemy’s model of the solar system (S. P. Dinkgreve, worldhistorybook.com)

To describe retrograde motion, Ptolemy took up an idea by
the astronomer **Apollonius **(c. 240–190 BC). Apollonius had realized that
retrograde motion could be mechanically described by assuming that the planets moved
on small circular orbits, called **epicycles**, that were attached to a
bigger orbit around the earth (see Fig. 308). He then added an idea by another
great astronomer, named **Hipparchus **(c. 190–120 BC), who had discovered
that the seasons were not exactly of equal length, suggesting the sun sped up
and slowed down along its orbit. As a result, Hipparchus had claimed that the earth
was not in the center of the orbit of the sun, but 1/24^{th} the radius
of the Sun’s orbit off-center. Changes in the velocity of the planets indicated
that the same was true about the orbits of the planets. Finally, Ptolemy added
his own invention, named the **equant**, which was a point in space also
located slightly off center (see Fig. 308). Ptolemy masterfully showed that the
center of the epicycles moved with constant velocity as **seen from this point**,
saving Plato’s wish for uniform motion.

This model could describe
all observed motions of the heavenly bodies with precision and could thus be
used to make very accurate predictions of their future positions. It would
remain the dominant model of the universe for 1200 years, until it was finally
replaced by the **heliocentric model **of **Nicolaus Copernicus **(1473–1543
AD).

Some of the data in the
*Almagest* seems strangely coincidental in the geocentric model but can be
easily explained in the heliocentric model. For instance, the inner planets
Mercury and Venus always stay close to the sun, which, we now know, is because
they closely orbit the sun. Also, the outer planets all have epicycles with
periods equal to one Earth year, which, we now know, is because epicycle motion
is caused by the motion of the Earth around the sun. Ptolemy knew about these
curious coincidences, but saw no need to explain them.

Ptolemy also wrote two
other influential astronomical works. The first is the *Tetrabiblos *(“Four
Books”), which applies his model of the universe from the *Almagest* to
predict **horoscopes**. Another is the *Planetary Hypotheses*, in which
he attempts to measure the **size of the universe** based on his model. To
do this, he assumed that the spheres of the planets were stacked directly on
top of one another and then equated the distance to the sphere of the fixed
stars with the “radius of the cosmos.” His result was obviously way off, but
his attempt was ambitious nonetheless.

The same Hipparchus we
just mentioned also found that the stars slowly shift position, one degree
every 85 years (the modern value is 72 years). He discovered this by comparing
his own data with star measurements made by his predecessors about a hundred
years earlier. This effect is known as **precession**, which occurs because
the axis of the Earth is spinning like a top with a period of 26,000 years.

The Greeks also
produced at least one mechanical model of the motion of the heavenly bodies, called
the **Antikythera mechanism**, which was found in a shipwreck. The device
contains 37 bronze gears enabling the device to track the movements of
the moon and
the sun through the
zodiac (including their variable velocity). As this motion was first studied in
the 2^{nd} century BC by Hipparchus and the shipwreck dates to 70-60
BC, the device must have been built somewhere between these dates. Machines
with similar complexity did not appear again until the 14^{th} century.
The device also contains some inscriptions, which mention the **Saros cycle**
of 223 months, used to **predict eclipses**. All gears in the device are
accounted for except one, which, according to some historians, can be used to
describe the epicyclical motion of Jupiter. As it takes only a small number of
gears to describe the motion of all the planets known at the time, it is also
hypothesized that part of the device is missing.

Fig. 309 – The Antikythera mechanism (2nd-1st century BC) (Marsyas, CC BY-SA 2.5)

Ptolemy’s model was great at describing the motion of the
planets, but there was still one problem left. Although it could **describe **the
motions, it could not** explain **them. What mechanism could possibly make
the planets move as circles within circles? And why was the earth not at the
center as Aristotle had theorized? It would take the genius of Copernicus and
Newton to finally resolve these problems.

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