For Greek Science
Professor Crane
May 9, 1995
Look at the comments on this paper.
Although present day society is technically dimensions ahead of the ancient Greeks, we owe much of our progress to them. What is amazing is not so much that we are more scientifically advanced today, but that we are so similar. Granted some of ancient science is far from accurate, ironically some of what we laughed at for centuries has turned out to be true! click here for more! Incredibly, today we not only practice, teach and respect the same ideas, we think about them in much of the same manner. A great example of the living link between ancient science and current science is the study of geometry. The purpose of this project is to show that geometry is not only useful, but can be fun to learn by taking its roots into account.
TO BEGIN...
To begin, we should know just what geometry means:
"In ancient Egypt, from which Greece inherited this study, the Nile would flood its banks each year, covering the land and obliterating the orderly making of plot and farm areas. This yearly flood symbolized to the Egyptian the cyclic return of the primal watery chaos, and when the waters receded the work began of redefining and re-establishing the boundaries. This work was called geometry and was seen as a re-establishment of the principle of order and law on earth....This activity of 'measuring the earth' became the basis for a science of natural law as it embodied in the archetypal forms of the circle, square, and triangle"[1]
Geometry is still applied today in Egypt as is shown in this picture of an Egyptian field. This panaroma shows a patchwork of squares crisscrossed by irrigation canals and ditches. The years of applying geometry in order to redraw property lines after the Nile's annual flooding, led the Egyptians to become masters of their land, "Thus the needs of agriculture led the ancient Egyptians to become more than good farmers: mastering both the desert and the river, they not only achieved new skills but expanded their intellectual horizons in the process."[2]
HOW DID THE GREEKS VIEW GEOMETRY?
The Greeks viewed geometry as important on both the practical and more abstract, philosophical level. Plato considered geometry to be the ideal philosophical language since it is the most reduced and essential.[3] Plato believed that geometry was an incredible mode of immersing oneself into philosophical contemplation. In fact, the notice above Plato's porch read, "Let no one unversed in geometry enter my doors."[4] We can see the connection between the practical and philosophical aspects of geometry in Margarita philosophica (Basle 1583). Lawlor describes this image:
,
"Geometry as a contemplative practice is personified by an elegant and refined woman, for geometry functions as an intuitive, synthesizing, creative, yet exact activity of mind associated with the feminine principle. But when these geometric laws come to be applied in the technology of daily life they are represented by the rational, masculine principle: contemplative geometry is transformed into practical geometry."[5]
HOW DID AND HOW DO STUDENTS VIEW GEOMETRY?
Students in ancient Greece and students today both tend(ed) to gripe about the uselessness of geometry. As Sir Thomas Heath notices, "notwithstanding the influence of Plato, the attitude of cultivated people in general towards mathematics was not different in Plato's time from what it is to-day."[6] In Plato'sRepublic, we learn that students at that time doubted the importance of studying geometry. The ancient Greek philosopher Teles also mentions geometry as one of the plagues of the lad.[7] Likewise, Isocrates says that most people think studies of geometry idle "since they are of no use in private or public affairs; moreover they are forgotten directly because they do not go with us in our daily life and action, nay, they are altogether outside everyday needs."[8]
Isocrates, however, believed that, as Heath writes, "the study of these subjects up to the proper point trains a boy to keep his attention fixed and not to allow his mind to wander; so, being practiced in this way and having his wits sharpened, he will be capable of learning more important matters with greater ease and speed."[9]
Students today feel much the same way. They wonder what good geometry could possible do them, and when asked, are usually answered something like we have just seen -- studying geometry allows one to think more logically and opens the mind to a new level of thinking and reasoning ability. While students in Plato's time most probably aired their gripes about geometry orally, students today turn to the internet. One such cyberspace discussion focuses on whether students today need to take one year of geometry. Another discussion talks about what the math curriculum in American high schools should be like.
DO GEOMETRY STUDENTS TODAY REALLY LABOR OVER THE SAME PROBLEMS AS STUDENTS DID IN ANCIENT GREECE?
Absolutely! One such problem is squaring the circle. This problem was quite popular in 414 B.C.E., and thus appears in Aristophanes'Birds 1001-1005. Plutarch writes that Anaxagorus worked on the problem while in prison (Plutarch, On Exile 17, 607E, F). Many ancient Greek philosophers including Aristotle, Themistius, Philoponus and Simplicius also worked or commented on the problem. Today, the discussion continues. An internet geometry graphics.
WHAT IS ONE EXAMPLE OF FUNDAMENTAL GEOMETRY THAT WE OWE TO THE GREEKS?
The Greeks began the system of lettering of geometric figures. For instance, "Cantor points out that the letters spelling the word 'health' were placed at the vertices of the pentagram, the Pythagorean emblem."[10]
Not surprisingly, Aristotle also noted the importance of lettering geometric symbols. Since Aristotle created the first system of formal logic, it is no wonder that he also understood the importance of lettering geometric figures. Aristotle is credited with saying that "much time and trouble is saved by a general symbolism."[11] So, when students today letter their geometric figures, they can keep in mind that they are working just as the Pythagoreans did in the sixth century B.C.E..
WHO WAS PYTHAGORAS?
Pythagoras (c.585-c.500 B.C.E.) profoundly influenced geometry. Students today probably recognize his name from the Pythagorian Theorem. Although the Egyptians before Pythagoras knew that in a triangle whose sides had the ratio 3 :4 :5, the square on the hypotenuse equals the sum of the squares on the other sides, it was Pythagoras who realized that this theorem is true in any right triangle.
Not too unlike mathematicians today, Pythagoras wanted to teach mathematics because he understood that it was important for people to study. Proclus describes the Pythagoreans, "'Herein,' says Proclus, 'I emulate the Pythagoreans who even had a conventional phrase to express what I mean, 'a figure and a platform, not a figure and a sixpence,' by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among sensible objects and so become subservient to the common needs of this moral life."[12]
This passage shows us that the Pythagoreans were anxious to teach mathematics so that people would be able to think on a higher level. Unfortunately, not unlike contemporary society, it was difficult to find interested students. The Pythagorean motto, 'a figure and a platform, not a figure and a sixpence' is literal and refers to an episode when Pythagoras had to bribe a boy to study geometry.
Heath relates this story: "He [Pythagoras] adopted therefore this plan of communicating his arithmetic and geometry, so that it might not perish with him. Selecting a young man who from his behaviour in gymnastic exercises seemed adaptable and was withal poor, he promised him that, if he would learn arithmetic and geometry systematically, he would give him sixpence for each 'figure' (proposition) that he mastered. This went on until the youth got interested in the subject, when Pythagoras rightly judged that he would gladly go on without the sixpence. He therefore hinted that he himself was poor and must try to earn his daily bread instead of doing mathematics; whereupon the youth, rather than give up the study, volunteered to pay sixpence himself to Pythagoras for each proposition."[13]
The progress of arithmetic is depicted in a 1503 woodcut in which Arithmetic is personified as a woman.
On her thighs are geometric progressions which relate both to Pythagoreans and to Plato. On her left thigh is a series of even numbers which the Pythagoreans associated with being passive and feminine. On her right side is a series of odd numbers which the Pythagoreans associated with being masculine and active.[14]
In the Timeaus, Plato uses these series of numbers to describe the world soul which we will see (a bit further down in the section on music) also relates to music. The man with the abacus is Pythagoras who still relies on spatial arrangement for calculating numbers. The other man is Boethius who is more advanced since he uses Arabic numerals in his calculations. His system is more abstract, and less connected to the spatial, geometric system that Pythagoras uses. However, the more archaic system is the origin of the more modern system.
HOW DID PLATO INFLUENCE GEOMETRY?
Plato and his school began to emphasize the importance of solid geometry. You can view a great on-line depiction of the platonic solids by clicking here.
In The Republic,, we learn that Plato believes we need a science of solid objects in order to consider issues of objects in motion, such as astronomy.
Lawlor (p.8) gives us a great example of how ancient Greek astronomers relied
on geometry. 
GEOMETRY IS ALSO FUNDAMENTAL TO MUSIC:
Another connection the the Greeks recognized between geometry and other disciplines is the relationship between geometry and music. You can click to a cool image and description of harmony and the sequence of numbers click here.
In the Timaeus, Plato explains the connection between multiplying numbers in geometric series to obtain musical proportions which construct universal harmonies. This diagram (from Lawlor, p.83) describes what Plato is talking about in the Timaeus.
As Lawlor explains, "This diagram by Giorgi shows the two progressions of 2 and 3, as given by Plato in the Timaeus, placed in association with the musical proportion of 6, 8, 9, 12. It uses the musical proportion as a basis for generating number for a succession of musical octaves, fourths, and fifths, thus constructing an harmonic system which could be used as a model for architecture, painting, and other arts."[15]
CONCLUSIONS:
We have seen that not only do students view geometry today much like students viewed it ancient Greece, but also that this view (generally that geometry is tedious, and unimportant) is unwarranted. Studying geometry today can be fun. Already on the internet are discussions and images of geometry. Geometry is a tool to 1)deep philosophical contemplation in its harmony and universality as well as to 2)practical studies such as its applicability to other disciplines including agriculture, astronomy and music.
Geometry is also more fun to study by recognizing its links to ancient Egypt and Greece. Geometry should be taught by connecting what we do today to its ancient roots. For instance, learning the Pythagorean Theorem without learning who Pythagoras was and what it was like to learn during his time sacrifices a depth of understanding and knowledge which is available to make the study of geometry richer and more rewarding.