Math symbols
About us
Contact us
Site map
Search The Site
   Program of Lessons
Study Guide
Topics of problems
Tests & exams
www.bymath.com Study Guide - Arithmetic Study Guide - Algebra Study Guide - Geometry Study Guide - Trigonometry Study Guide - Functions & Graphs Study Guide - Principles of Analysis Study Guide - Sets Study Guide - Probability Study Guide - Analytic Geometry Select topic of problems Select test & exam


Thales of Miletus (c. 624 BC – c. 546 BC) was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition. According to Bertrand Russell, "Western philosophy begins with Thales. "Thales attempted to explain natural phenomena without reference to mythology and was tremendously influential in this respect. Almost all of the other pre-Socratic philosophers follow him in attempting to provide an explanation of ultimate substance, change, and the existence of the world—without reference to mythology. Those philosophers were also influential, and eventually Thales' rejection of mythological explanations became an essential idea for the scientific revolution. He was also the first to define general principles and set forth hypotheses, and as a result has been dubbed the "Father of Science", though it is argued that Democritus is actually more deserving of this title.
In mathematics, Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and is the first known individual to whom a mathematical discovery has been attributed. Also, Thales was the first person known to have studied electricity.
Thales was born in the city of Miletus around the mid 620s BC. Miletus was an ancient Greek Ionian city on the western coast of Asia Minor (in what is today Aydin Province of Turkey), near the mouth of the Maeander River.
The dates of Thales' life are not known. However, the time of his life is roughly established by a few dateable events mentioned in the sources and an estimate of his length of life. According to Herodotus, Thales once predicted a solar eclipse which has been determined by modern methods to have been on May 28, 585 BC. Diogenes Laertius quotes the chronicle of Apollodorus of Athens as saying that Thales died at 78 in the 58th Olympiad (548–545 BC).
Diogenes Laertius states that ("according to Herodotus and Douris and Democritus") Thales' parents were Examyes and Cleobuline, both Phoenician nobles. Giving another opinion, he ultimately connects Thales' family line back to Phoenician prince Cadmus. Diogenes also reports two other stories, one that he married and had a son, Cybisthus or Cybisthon, or adopted his nephew of the same name. The second is that he never married, telling his mother as a young man that it was too early to marry, and as an older man that it was too late. A much earlier source - Plutarch - tells the following story: Solon who visited Thales asked him the reason which kept him single. Thales answered that he did not like the idea of having to worry about children. Nevertheless, several years later Thales, anxious for family, adopted his nephew Cybisthus.
Thales involved himself in many activities, taking the role of an innovator. Some say that he left no writings, others that he wrote "On the Solstice" and "On the Equinox". Neither has survived. Diogenes Laertius quotes letters of Thales to Pherecydes and Solon, offering to review the book of the former on religion, and offering to keep company with the latter on his sojourn from Athens. Thales identifies the Milesians as Athenians.
Several anecdotes suggest that Thales was not solely a thinker but was also involved in business and politics. One story recounts that he bought all the olive presses in Miletus after predicting the weather and a good harvest for a particular year. In another version of the same story, Aristotle explains that Thales reserved presses ahead of time at a discount only to rent them out at a high price when demand peaked, following his predictions of a particular good harvest. Aristotle explains that Thales' objective in doing this was not to enrich himself but to prove his fellow Milesians that philosophy could be useful, contrary to what they thought.
Thales’ political life had mainly to do with the involvement of the Ionians in the defense of Anatolia against the growing power of the Persians, who were then new to the region. A king had come to power in neighboring Lydia, Croesus, who was somewhat too aggressive for the size of his army. He had conquered most of the states of coastal Anatolia, including the cities of the Ionians. The story is told in Herodotus.
The Lydians were at war with the Medes, a remnant of the first wave of Iranians in the region, over the issue of refuge the Lydians had given to some Scythian soldiers of fortune inimical to the Medes. The war endured for five years, but in the sixth an eclipse of the Sun (mentioned above) spontaneously halted a battle in progress (the Battle of Halys).
It seems that Thales had predicted this solar eclipse. The Seven Sages were most likely already in existence, as Croesus was also heavily influenced by Solon of Athens, another sage. Whether Thales was present at the battle is not known, nor are the exact terms of the prediction, but based on it the Lydians and Medes made peace immediately, swearing a blood oath.
The Medes were dependencies of the Persians under Cyrus. Croesus now sided with the Medes against the Persians and marched in the direction of Iran (with far fewer men than he needed). He was stopped by the river Halys, and then unbridged. This time he had Thales with him, perhaps by invitation. Whatever his status, the king gave the problem to him, and he got the army across by digging a diversion upstream so as to reduce the flow, making it possible to ford the river. The channels ran around both sides of the camp.
The two armies engaged at Pteria in Cappadocia. As the battle was indecisive but paralyzing to both sides, Croesus marched home, dismissed his mercenaries and sent emissaries to his dependents and allies to ask them to dispatch fresh troops to Sardis. The issue became more pressing when the Persian army showed up at Sardis. Diogenes Laertius tells us that Thales gained fame as a counsellor when he advised the Milesians not to engage in a symmachia, a "fighting together", with the Lydians. This has sometimes been interpreted as an alliance, but a ruler does not ally with his subjects.
Croesus was defeated before the city of Sardis by Cyrus, who subsequently spared Miletus because it had taken no action. Cyrus was so impressed by Croesus’ wisdom and his connection with the sages that he spared him and took his advice on various matters.
The Ionians were now free. Herodotus says that Thales advised them to form an Ionian state; that is, a bouleuterion ("deliberative body") to be located at Teos in the center of Ionia. The Ionian cities should be demoi, or "districts". Miletus, however, received favorable terms from Cyrus. The others remained in an Ionian League of 12 cities (excluding Miletus now), and were subjugated by the Persians.
Diogenes Laertius tells us that the Seven Sages were created in the archonship of Damasius at Athens about 582 BC and that Thales was the first sage. The same story, however, asserts that Thales emigrated to Miletus. There is also a report that he did not become a student of nature until after his political career. Much as we would like to have a date on the seven sages, we must reject these stories and the tempting date if we are to believe that Thales was a native of Miletus, predicted the eclipse, and was with Croesus in the campaign against Cyrus.
Thales had instruction from Egyptian priests, we are told. It was fairly certain that he came from a wealthy and established family, and the wealthy customarily educated their children. Moreover, the ordinary citizen, unless he was a seafaring man or a merchant, could not afford the grand tour in Egypt, and in any case did not consort with noble lawmakers such as Solon.
He did participate in some games, most likely Panhellenic, at which he won a bowl twice. He dedicated it to Apollo at Delphi. As he was not known to have been athletic, his event was probably declamation, and it may have been victory in some specific phase of this event that led to his being designated sage.
The Greeks often invoked idiosyncratic explanations of natural phenomena by reference to the will of anthropomorphic gods and heroes. Thales, however, aimed to explain natural phenomena via a rational explanation that referenced natural processes themselves. For example, Thales attempted to explain earthquakes by hypothesizing that the Earth floats on water, and that earthquakes occur when the Earth is rocked by waves, rather than assuming that earthquakes were the result of supernatural processes. Thales was a Hylozoist (those who think matter is alive). It is unclear whether the interpretation that he treated matter as being alive might have been mistaken for his thinking the properties of nature arise directly from material processes, more consistent with modern ideas of how properties arise as emergent characteristics of complex systems involved in the processes of evolution and developmental change.
Thales, according to Aristotle, asked what was the nature (Greek Arche) of the object so that it would behave in its characteristic way. Physis comes from phyein, "to grow", related to our word "be". (G) nature is the way a thing is "born", again with the stamp of what it is in itself.
Aristotle characterizes most of the philosophers "at first" as thinking that the "principles in the form of matter were the only principles of all things", where "principle" isarche, "matter" is hyle ("wood" or "matter", "material") and "form" is eidos.
Arche is translated as "principle", but the two words do not have precisely the same meaning. A principle of something is merely prior (related to pro-) to it either chronologically or logically. An arche ("to rule") dominates an object in some way. If the arche is taken to be an origin, then specific causality is implied; that is, B is supposed to be characteristically B just because it comes from A, which dominates it.
The archaic that Aristotle had in mind in his well-known passage on the first Greek scientists are not necessarily chronologically prior to their objects, but are constituents of it. For example, in pluralism objects are composed of earth, air, fire and water, but those elements do not disappear with the production of the object. They remain as archaic within it, as do the atoms of the atomists.
What Aristotle is really saying is that the first philosophers were trying to define the substance(s) of which all material objects are composed. As a matter of fact, that is exactly what modern scientists are attempting to accomplish in nuclear physics, which is a second reason why Thales is described as the first western scientist.
Thales' most famous belief was his cosmological thesis, which held that the world started from water. Aristotle considered this belief roughly equivalent to the later ideas of Anaximenes, who held that everything in the world was composed of air.
The best explanation of Thales' view is the following passage from Aristotle's Metaphysics. The passage contains words from the theory of matter and form that were adopted by science with quite different meanings.
"That from which is everything that exists and from which it first becomes and into which it is rendered at last, its substance remaining under it, but transforming in qualities, that they say is the element and principle of things that are."
And again: "For it is necessary that there be some nature, either one or more than one, from which become the other things of the object being saved... Thales the founder of this type of philosophy says that it is water."
Aristotle's depiction of the problem of change and the definition of substance is clear. If an object changes, is it the same or different? In either case how can there be a change from one to the other? The answer is that the substance "is saved", but acquires or loses different qualities (the things you "experience").
A deeper dip into the waters of the theory of matter and form is properly reserved to other articles. The question for this article is, how far does Aristotle reflect Thales? He was probably not far off, and Thales was probably an incipient matter-and-formist.
The essentially non-philosophic Diogenes Laertius states that Thales taught as follows: "Water constituted ('stood under') the principle of all things."
Heraclitus Homericus states that Thales drew his conclusion from seeing moist substance turn into air, slime and earth. It seems likely that Thales viewed the Earth as solidifying from the water on which it floated and which surrounded Ocean.
Thales was known for his innovative use of geometry. His understanding was theoretical as well as practical. For example, he said: "Space is the greatest thing, as it contains all things".
Tops is in Newtonian-style space, since the verb, chorea, has the connotation of yielding before things, or spreading out to make room for them, which is extension. Within this extension, things have a position. Points, lines, planes and solids related by distances and angles follow from this presumption.
Thales understood similar triangles and right triangles, and what is more, used that knowledge in practical ways. The story is told in DL (loc. cit.) that he measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The length of the pyramid’s shadow measured from the center of the pyramid at that moment must have been equal to its height.
This story indicates that he was familiar with the Egyptian seked, or seqed - the ratio of the run to the rise of a slope (cotangent). The seked is at the base of problems 56, 57, 58, 59 and 60 of the Rhind papyrus - an ancient Egyptian mathematics document.
In present day trigonometry, cotangents require the same units for run and rise (base and perpendicular), but the papyrus uses cubits for rise and palms for run, resulting in different (but still characteristic) numbers. Since there were 7 palms in a cubit, the seked was 7 times the cotangent.
To use an example often quoted in modern reference works, suppose the base of a pyramid is 140 cubits and the angle of rise 5.25 seked. The Egyptians expressed their fractions as the sum of fractions, but the decimals are sufficient for the example. What is the rise in cubits? The run is 70 cubits, 490 palms. X, the rise, is 490 divided by 5.25 or 931?3 cubits. These figures sufficed for the Egyptians and Thales. We would go on to calculate the cotangent as 70 divided by 931?3 to get 3/4 or .75 and looking that up in a table of cotangents find that the angle of rise is a few minutes over 53 degrees.
Whether the ability to use the seked, which preceded Thales by about 1000 years, means that he was the first to define trigonometry is a matter of opinion. More practically Thales used the same method to measure the distances of ships at sea, said Eudemus as reported by Proclus ("in Euclidem"). According to Kirk & Raven (reference cited below), all you need for this feat is three straight sticks pinned at one end and knowledge of your altitude. One stick goes vertically into the ground. A second is made level. With the third you sight the ship and calculate the seked from the height of the stick and its distance from the point of insertion to the line of sight.
The seked is a measure of the angle. Knowledge of two angles (the seked and a right angle) and an enclosed leg (the altitude) allows you to determine by similar triangles the second leg, which is the distance. Thales probably had his own equipment rigged and recorded his own sekeds, but that is only a guess.
Thales has learned to define distance from coast up to the ship for what used similarity of triangles. In a basis of this way the theorem named subsequently by Thales’ Theorem lays: if the parallel straight lines crossing sides of angle, cut equal pieces on its one side they cut equal pieces and on its other side. Thales’ Theorem is stated in another article. (Actually there are two theorems called Theorem of Thales, one having to do with a triangle inscribed in a circle and having the circle's diameter as one leg, the other theorem being also called the intercept theorem.) In addition Eudemus attributed to him the discovery that a circle is bisected by its diameter, that the base angles of an isosceles triangle are equal and that vertical angles are equal. According to a historical Note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal. It would be hard to imagine civilization without these theorems.
It is possible, of course, to question whether Thales really did discover these principles. On the other hand, it is not possible to answer such doubts definitively. The sources are all that we have, even though they sometimes contradict each other.
Ever since, interested persons have been asking what that new something is. Answers fall into (at least) two categories, the theory and the method. Once an answer has been arrived at, the next logical step is to ask how Thales compares to other philosophers, which leads to his classification (rightly or wrongly).
The most natural epithets of Thales are "materialist" and "naturalist", which are based on ousia and physis. The Catholic Encyclopedia notes that Aristotle called him a physiologist, with the meaning "student of nature." On the other hand, he would have qualified as an early physicist, as did Aristotle. They studied corpora, "bodies", the medieval descendants of substances.
Most philosophic analyses of Thales’ philosophy come from Aristotle, a professional philosopher, tutor of Alexander the Great, who wrote 200 years after Thales death. Aristotle, judging from his surviving books, does not seem to have access to any works by Thales, although he probably had access to works of other authors about Thales, such as Herodotus, Hecataeus, Plato etc., as well as others whose work is now extinct. It was Aristotle's express goal to present Thales work not because it was significant in itself, but as a prelude to his own work in natural philosophy.
Aristotle's philosophy had a distinct stamp: it professed the theory of matter and form, which modern scholastics have dubbed hylomorphism. Though once very widespread, it was not generally adopted by rationalist and modern science, as it mainly is useful in metaphysical analyses, but does not lend itself to the detail that is of interest to modern science. It is not clear that the theory of matter and form existed as early as Thales, and if it did, whether Thales espoused it.

| Math symbols | Jokes | Responses About us | Links | Contact us | Site map |

Copyright © 2002-2012 Dr. Yury Berengard.  All rights reserved.
Last updated: July 29, 2012