Thales was the first known philosopher, scientist and mathematician
although his occupation was that of an engineer. He is believed
to have been the teacher of Anaximander (611 BC - 545 BC) and
he was the first natural philosopher in the Milesian School.
However, none of his writing survives so it is difficult to determine
his views or to be certain about his mathematical discoveries.
Indeed it is unclear whether he wrote any works at all and if
he did they were certainly lost by the time of Aristotle who
did not have access to any writings of Thales. On the other hand
there are claims that he wrote a book on navigation but these
are based on little evidence. In the book on navigation it is
suggested that he used the constellation Ursa Minor, which he
defined, as an important feature in his navigation techniques.
Even if the book is fictitious, it is quite probable that Thales
did indeed define the constellation Ursa Minor.

Proclus, the last major Greek philosopher, who lived around
450 AD, wrote:-

*[Thales] first went to Egypt and thence introduced this
study [geometry] into Greece. He discovered many propositions
himself, and instructed his successors in the principles underlying
many others, his method of attacking problems had greater generality
in some cases and was more in the nature of simple inspection
and observation in other cases.*

There is a difficulty in writing about Thales and others from
a similar period. Although there are numerous references to Thales
which would enable us to reconstruct quite a number of details,
the sources must be treated with care since it was the habit
of the time to credit famous men with discoveries they did not
make. Partly this was as a result of the legendary status that
men like Thales achieved, and partly it was the result of scientists
with relatively little history behind their subjects trying to
increase the status of their topic with giving it an historical
background.

Certainly Thales was a figure of enormous prestige, being
the only philosopher before Socrates to be among the Seven Sages.
Plutarch, writing of these Seven Sages, says that:

*[Thales] was apparently the only one of these whose
wisdom stepped, in speculation, beyond the limits of practical
utility, the rest acquired the reputation of wisdom in politics.*

This comment by Plutarch should not be seen as saying that
Thales did not function as a politician. Indeed he did. He persuaded
the separate states of Ionia to form a federation with a capital
at Teos. He dissuaded his compatriots from accepting an alliance
with Croesus and, as a result, saved the city.

It is reported that Thales predicted an eclipse of the Sun
in 585 BC. The cycle of about 19 years for eclipses of the Moon
was well known at this time but the cycle for eclipses of the
Sun was harder to spot since eclipses were visible at different
places on Earth. Thales's prediction of the 585 BC eclipse was
probably a guess based on the knowledge that an eclipse around
that time was possible. The claims that Thales used the Babylonian
saros, a cycle of length 18 years 10 days 8 hours, to predict
the eclipse has been shown by Neugebauer to be highly unlikely
since Neugebauer shows that the saros was an invention of Halley.
Neugebauer wrote:

*... there exists no cycle for solar eclipses visible
at a given place: all modern cycles concern the earth as a whole.
No Babylonian theory for predicting a solar eclipse existed at
600 BC, as one can see from the very unsatisfactory situation
400 years later, nor did the Babylonians ever develop any theory
which took the influence of geographical latitude into account.*

After the eclipse on 28 May, 585 BC Herodotus wrote:

*... day was all of a sudden changed into night. This event had been
**foretold by
Thales, the Milesian, who forewarned the Ionians of it, fixing for it the
very year in which it took place. The Medes and Lydians, when they observed
the change, ceased fighting, and were alike anxious to have terms of peace
agreed on.*

Some doubt that Thales predicted the eclipse by guessing writing:

*... a more likely explanation seems to be simply that
Thales happened to be the savant around at the time when this
striking astronomical phenomenon occurred and the assumption
was made that as a savant he must have been able to predict it.*

There are several accounts of how Thales measured the height
of pyramids. Diogenes Laertius writing in the second century
AD quotes Hieronymus, a pupil of Aristotle:

*Hieronymus says that [Thales] even succeeded in measuring
the pyramids by observation of the length of their shadow at
the moment when our shadows are equal to our own height.*

This appears to contain no subtle geometrical knowledge, merely
an empirical observation that at the instant when the length
of the shadow of one object coincides with its height, then the
same will be true for all other objects. A similar statement
is made by Pliny:

*Thales discovered how to obtain the height of pyramids
and all other similar objects, namely, by measuring the shadow
of the object at the time when a body and its shadow are equal
in length.*

Plutarch however recounts the story in a form which, if accurate,
would mean that Thales was getting close to the idea of similar
triangles:

*... without trouble or the assistance of any instrument
[he] merely set up a stick at the extremity of the shadow cast
by the pyramid and, having thus made two triangles by the impact
of the sun's rays, ... showed that the pyramid has to the stick
the same ratio which the shadow [of the pyramid] has to the shadow
[of the stick].*

Now of course Thales could have used these geometrical methods
for solving practical problems having merely observed the properties
and having no appreciation of what it means to prove a geometrical
theorem. This is in line with the views of Russell who writes
of Thales contributions to mathematics:

*Thales is said to have travelled in Egypt, and to have
thence brought to the Greeks the science of geometry. What Egyptians
knew of geometry was mainly rules of thumb, and there is no reason
to believe that Thales arrived at deductive proofs, such as later
Greeks discovered.*

On the other hand, there are claims that Thales put geometry
on a logical footing and was well aware of the notion of proving
a geometrical theorem. However, although there is much evidence
to suggest that Thales made some fundamental contributions to
geometry, it is easy to interpret his contributions in the light
of our own knowledge, thereby believing that Thales had a fuller
appreciation of geometry than he could possibly have achieved.
In many textbooks on the history of mathematics Thales is credited
with five theorems of elementary geometry:

*(i) A circle is bisected by any diameter.* *(ii)
The base angles of an isosceles triangle are equal.* *(iii)
The angles between two intersecting straight lines are equal.*
*(iv) Two triangles are congruent if they have two angles and
one side equal.* *(v) An angle in a semicircle is a right
angle.*

What is the basis for these claims? Proclus, writing around
450 AD, is the basis for the first four of these claims, in the
third and fourth cases quoting the work *History of Geometry*
by Eudemus of Rhodes, who was a pupil of Aristotle, as his source.
The *History of Geometry* by Eudemus is now lost but there
is no reason to doubt Proclus. The firth theorem is believed
to be due to Thales because of a passage from Diogenes Laertius
book *Lives of eminent philosophers* written in the second
century AD:

*Pamphile says that Thales, who learnt geometry from
the Egyptians, was the first to describe on a circle a triangle
which shall be right-angled, and that he sacrificed an ox (on
the strength of the discovery). Others, however, including Apollodorus
the calculator, say that it was Pythagoras.*

A deeper examination of the sources,
however, shows that, even if they are accurate, we may be crediting Thales
with too
much. For example Proclus uses a word meaning something closer
to 'similar' rather than 'equal- in describing (ii). It is quite
likely that Thales did not even have a way of measuring angles
so 'equal- angles would have not been a concept he would have
understood precisely. He may have claimed no more than "The
base angles of an isosceles triangle look similar". The
theorem (iv) was attributed to Thales by Eudemus for less than
completely convincing reasons. Proclus writes:

*[Eudemus] says that the method by which Thales showed
how to find the distances of ships from the shore necessarily
involves the use of this theorem.*

Other scholar give three different methods which Thales might
have used to calculate the distance to a ship at sea. The method
which he thinks it most likely that Thales used was to have an
instrument consisting of two sticks nailed into a cross so that
they could be rotated about the nail. An observer then went to
the top of a tower, positioned one stick vertically (using say
a plumb line) and then rotating the second stick about the nail
until it point at the ship. Then the observer rotates the instrument,
keeping it fixed and vertical, until the movable stick point
at a suitable point on the land. The distance of this point from
the base of the tower is equal to the distance to the ship.

Although theorem (iv) underlies this application, it would
have been quite possible for Thales to devise such a method without
appreciating anything of 'congruent triangles'.

As a final comment on these five theorems, there are conflicting
stories regarding theorem (iv) as Diogenes Laertius himself is
aware. Also even Pamphile cannot be taken as an authority since
she lived in the first century AD, long after the time of Thales.
Others have attributed the story about the sacrifice of an ox
to Pythagoras on discovering Pythagoras' theorem. Certainly there
is much confusion, and little certainty.

Our knowledge of the philosophy of Thales is due to Aristotle
who wrote in his *Metaphysics:*

*Thales of Miletus taught that 'all things are water':*

*...may seem an unpromising beginning for science and
philosophy as we know them today; but, against the background
of mythology from which it arose, it was revolutionary.*

Others suggest:

*It was Thales who first conceived the principle of explaining
the multitude of phenomena by a small number of hypotheses for
all the various manifestations of matter.*

Thales believed that the Earth floats on water and all things
come to be from water. For him the Earth was a flat disc floating
on an infinite ocean. It has also been claimed that Thales explained
earthquakes from the fact that the Earth floats on water. Again
the importance of Thales' idea is that he is the first recorded
person who tried to explain such phenomena by rational rather
than by supernatural means.

It is interesting that Thales
has both stories told about his great practical skills and also about him
being an unworldly
dreamer. Aristotle, for example, relates a story of how Thales
used his skills to deduce that the next season's olive crop would
be a very large one. He therefore bought all the olive presses
and then was able to make a fortune when the bumper olive crop
did indeed arrive. On the other hand Plato tells a story of how
one night Thales was gazing at the sky as he walked and fell
into a ditch. A pretty servant girl lifted him out and said to
him "How do you expect to understand what is going on up
in the sky if you do not even see what is at your feet".
As Brumbaugh says, perhaps this is the first absent-minded professor
joke!

The bust of Thales is in the Capitoline Museum in Rome, but
is not contemporary with Thales and is unlikely to bear any resemblance
to him.

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