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 η άπαντα γαρ χωρεί)
: ”Place 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. Within this extension, things have a
position. Points, lines, planes and
solids related by
distances and
angles follow from this presumption.
Some have argued that his geometry was simply a lucky happenstance resulting from empirical method worked out by the
Babylonians or
Egyptians and that he had no understanding of the basic principles involved. This overskeptical view neglects Thales own predilection for insight and also human nature. The
mathematics of the times was not especially difficult or obscure and we have a convincing story from DL that when he had inscribed a
right triangle in a
circle he sacrificed an ox. According to Lonergan in his noted study called “Insight”, such behavior is a typical of insights, or sudden realizations of the truth. Better known is
Archimedes’ shouting, eureka! (“I have found it!”) with reference to Archimedes’ Principle, into which he had just had an insight. Less dramatically, most of us just evidence the behavior associated with being startled.
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 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.
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 93.33 cubits. These figures sufficed for the Egyptians and Thales. We would go on to calculate the cotangent as 70 divided by 93.33 or.75003 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. 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.)