THALES OF MILETUS
J. J. O'CONNOR AND E. F. ROBERTSON
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Thales of Miletus Born: about 624 BC in Miletus,
Asia Minor (now Turkey) Died: about 547 BC
in Miletus. Article by: J J O'Connor and
E F Robertson Thales of Miletus was the son
of Examyes and Cleobuline. His parents are
said by some to be from Miletus, but others
report that they were Phoenicians. But the
majority opinion considered him a true Milesian
by descent, and of a distinguished family.
Thales seems to be the first known Greek
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 (see
[8]):-
[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 in [11] that the saros
was an invention of Halley. Neugebauer wrote
[11]:-
... 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.
Longrigg in [1] even doubts 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 [6]
(or see [8]):-
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
(see [8]):-
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]
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 in [12]:-
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 B L van der Waerden [16]
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:-
A circle is bisected by any diameter.
The base angles of an isosceles triangle
are equal.
The angles between two intersecting straight
lines are equal.
Two triangles are congruent if they have
two angles and one side equal.
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 fifth 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 [6]:- 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 (see [8]):-
[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.
Heath in [8] gives 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 points 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's
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'.
This, as Brumbaugh writes [5]:-
... 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.
Sambursky writes in [15]:-
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 in the West!
The bust of Thales shown above 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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