The birth of Greek astronomy has been attributed to Thales of Miletus. Thales brought from Egypt a number of fundamental geometric principles. He was able to take what he learned, develop upon it, and put it to practical use for the Greeks. Another important contributor to the foundation of Greek geometry was Pythagoras. Pythagoras is credited with the discovery of the famous Pythagorean theorem which equates the sides of a right triangle. Pythagoras and his followers, the Pythagoreans, developed and proved a few significant theorems and may have discovered the existence of irrational numbers. Plato also played a crucial role in laying out the beginnings of Greek geometry. His main contribution was not the in the content of his discoveries, but in his contribution to the philosophy of mathematics.
Thales, an Ionian who was active near the start of the sixth century B.C.,(Herodotis I, 74) has been credited with completing a number of tasks that imply he must have had a basic knowledge of the underlying geometric theorems. Thales was able to determine the height of a pyramid by measuring the length of its shadow at a particular time of day (Heath pp. 128-139). He may have been able to do this in a couple ways. The simplest way would be to measure the shadow of the pyramid at the time of day when an objects shadow was the same length as the height of the object. Thales may have been able to observe that at a certain position of the sun an objects height is equal to the length of its shadow. This is true for all objects at the same time of day, irregardless of their size.
Thales may have been able to calculate the height of the pyramids by applying a similar triangle theorem. An object and it's shadow will form two legs of a right triangle. At any fixed time of day all of the right triangles formed in this manner will be similar triangles. Hence given an object of known height and the size of its shadow, the height of an unknown object can be calculated by measuring the length of its shadow at the same time of day.
The following are the general theorems attributed to Thales:
1. He was the first to demonstrate that a circle is bisected by its diameter.
2. Angles at the base of any isosceles triangle are equal.
3. If two straight lines intersect the opposite angles formed are equal (note this was discovered but not proved).
4. If two triangles have two angles and one side respectively equal, the triangles are equal in all respects.
Thales most likely did not demonstrate that a circle is bisected by its diameter. Probably Thales observed this phenomenon in inscriptions on Egyptian monuments. There is no evidence to support the idea that he was able to prove this observation. Theorems two and three were both observed by Thales, but he was unable to prove theorem three.
Thales was presumed to be familiar with theorem 4. Eudemus believed Thales had to know this theorem to be able to predict the distance of a ship from shore. Thales, observing a ship from a tower could have determined its distance from shore in the following manner. Using two sticks and a plum line fastened together Thales would have sighted the angle at which the ship was visible, then looking toward shore with the same angle Thales would have been able to target an object that was the same distance from the tower as the boat. By keeping two angles constant, the right angle at the base of the tower and the angle of his eye sight, and one side constant, the height of the tower, Thales would have constructed two identical triangles according to the theorem. Instead, Thales may have been able to use a similar triangle theorem to calculate the distance of a ship from shore, but this is unlikely.
Pythagoras of Samos made a number of significant breakthroughs toward the development of ancient Greek geometry (Heath pp. 141-162). Pythagoras did not just found a school of thought, but for his followers it was a way of life. (Rep. 600 A-B)Pythagoras or at least the Pythagoreans are credited with the following basic ideas:
1. All of the angles in a triangle add up to the sum of two right angles.
2. The theorem of Pythagoras, a^2 + b^2 = c^2.
3. Discovery of irrational numbers
4. Defining the five regular solids (polyhedrons)
5. Developing and using geometrical algebra
Pythagoras not only observed that all the angles in a triangle summed to the value of two right angles, but he was able to prove it. By far the most famous and important of Pythagoras's discoveries was the Pythagorean theorem. It is fabled that this discovery was so astounding that Pythagoras went out and sacrificed an ox to celebrate. Equating the sides of a right triangle has had a dramatic effect on the study of geometry even up to the present. However, the Pythagorean theorem did lead to the inevitable discovery of irrational numbers, numbers that can not be described in terms of a ratio. The first irrational number discovered was the square root of 2 (diagonal of a unit square). The discovery of irrational numbers was so disturbing to the Greeks that it was rumored that the first Pythagorean to discover their existence met with an unfortunate boating "accident."
Another significant contribution of the Pythagoreans was the discovery of the fifth regular solid. Regular solids are any polyhedron that can be inscribed in (have its vertices on) or circumscribed about (have its faces tangent to) a sphere. The five regular polyhedrons are the regular tetrahedron, which has four equilateral triangles as faces; the cube, with six squares as faces; the regular octahedron, with eight equilateral triangles as faces; the regular icosahedron, with 20 equilateral triangles as faces; and the regular dodecahedron; with 12 regular pentagons as faces. Discovery of the fifth regular polyhedron, the dodecahedron, required the discovery of the equilateral pentagon. This discovery is credited to the Pythagoreans for this reason. No one has ever identified a sixth regular polyhedron.
The Pythagoreans also worked with applying areas to develop geometric algebra. Geometric algebra could be used to solve quadratic equations (x^2 + x + 1 = 0) in a similar way that modern algebra is used today. However, the complicated nature of geometric algebra prevented it from being used to solve more complex polynomials (x^3 + x^2 + x + 1 = 0).
Plato's attitude towards the study of geometry is best stated by an inscription above the door of his school, "let no one destitute of geometry enter my doors." Many people of Plato's time learned enough math to get by in trading and so forth, but it was Plato's firm belief that a complete pursuit of geometry and mathematics was essential. Plato taught that geometry should be pursued for the sake of knowledge, not just for the basic practicalities of life (Rep. vii 525 C,D). Plato believed that understanding these two sciences was essential to understand the true direction of philosophy (Rep. 526 D, E). He thought that these sciences could construct a new foundation of the mind that would bring to light ideas and observations that would have otherwise gone unnoticed .
Plato argued for the importance of geometry and mathematics as essential tools of life. A good example of this was Plato's arguments about the science of astronomy. In Plato's writings he emphasized that the future astronomers would have to reach beyond the traditional star gazing and use mathematics and geometry to discover the true nature of the science astronomy. When asked his opinion on who will be the future astronomers Plato states, "... he who is truly an astronomer must be wisest..." By wisest Plato infers an individual who is well trained in both mathematics and geometry. In this passage Plato discusses that the old times of star gazing and observing the risings and settings are over. Astronomers will have to move forward and begin an attempt to study and describe the motion of the sun, moon, and five planets in a more scientific manor. In a passage from Plato's republic Plato compares and contrasts star gazing to the examination of a fine piece of art. Simply gazing at the stars for their beauty is like wasting a valuable tool.
Plato was also unique in his approach to geometry. He was very careful to define everything in an unequivocal way. Some of his definitions paralleled the Pythagoreans, but many of them are original. The most common and well known of Plato's definitions was dividing the numbers into two groups. He defined numbers as being either even or odd.
Plato also described two different intellectual methods toward the development of geometrical and mathematical thought. Both methods deal with proving a hypothesis. In the first method you start with a hypothesis and do not add any additional hypotheses, but instead build upon it with the aid of diagrams and images until you are able to prove or disprove the hypothesis. In the second method the original hypothesis is built upon with a number of additional hypotheses until a principle is reached about which there is nothing hypothetical. When this is reached it is possible to descend back through all the previous steps and prove the original hypothesis. The first intellectual method has been more typically used in the development of mathematical and geometrical thought.
Greek geometry was founded upon the original discoveries of the Egyptians. Thales played an important role by integrating these ideas along with some of his own advances into the Greek culture. It is unclear exactly how much Thales really knew and was able to apply towards real life scenarios like measuring the height of a pyramid, or determining the distance of a boat from shore. But, the achievements attributed to Thales do prove that he had a fundamental understanding of several basic geometrical principles. Not much is known about what advances in geometry were made between the times of Thales and the Pythagoreans. The Pythagoreans advanced the understanding of not only triangles, but a number of geometric shapes. They were the first to relate relative areas and use the concepts to formulate a primitive algebra. Plato did not make any famous contributions to geometry. However, he made important advances toward the acceptance of geometry in the society. Plato was the first person to stress the importance of everyone learning geometry. Thales, Pythagoras, and Plato were not the only Greek philosophers to contribute to the development of ancient Greek geometry. However, their achievements do outline the development and acceptance of geometry in ancient Greek society.
Heath, T. A History of Greek Mathematics. Dover Publications, Inc. New York 1981.
