Thales of Miletus one of the Seven Sages

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Thales of Miletus (Θαλής,Thales,Thalês (pronounced /ˈθeɪliːz/ or "THEH-leez") , ca. 624 BCca. 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 was known for his innovative use of geometry. His understanding was theoretical as well as practical. For example, he said: Megiston topos: hapanta gar chorei (Μέγιστον τόπος· άπαντα γαρ χωρεί) Space is the greatest thing, as it contains all things Topos is in Newtonian-style space, since the verb, chorei, has the connotation of yielding before things, or spreading out to make room for them, which is extension [disambiguation needed]. 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 pyramids shadow measured from the center of the pyramid at that moment must have been equal to its height. This story reveals that he was familiar with the Egyptian seqt, or seked, defined by Problem 57 of the Rhind papyrus as the ratio of the run to the rise of a slope, which is currently the cotangent function of trigonometry. It characterizes the angle of rise. Our cotangents require the same units for run and rise, 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 seqt was 7 times the cotangent. Thales' Theorem : \textstyle \frac{DE}{BC} = \frac{AE}{AC } = \frac{AD}{AB} 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 seqt. 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 93 1/3 cubits. These figures sufficed for the Egyptians and Thales. We would go on to calculate the cotangent as 70 divided by 93 1/3 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 seqt, 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 seqt from the height of the stick and its distance from the point of insertion to the line of sight. The seqt is a measure of the angle. Knowledge of two angles (the seqt 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 seqts, but that is only a guess. 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. 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. (The most we can say is that Thales knew these principles. There is no evidence for Thales discovering these principles, and, based on the evidence, we cannot say that Thales discovered these principles.)