History of Geometry
Geometry was thoroughly organized in about 300 BC, when the Greek mathematician Euclid gathered what was known at the time, added original work of his own, and arranged 465 propositions into 13 books, called 'Elements'. The books covered not only plane and solid geometry but also much of what is now known as algebra, trigonometry, and advanced arithmetic. Through the ages, the propositions have been rearranged, and many of the proofs are different, but the basic idea presented in the 'Elements' has not changed. In the work facts are not just cataloged but are developed in a fashionable way. Even in 300 BC, geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which is how to follow lines of reasoning, how to say precisely what is intended, and especially how to prove basic concepts by following these lines of reasoning. Taking a course in geometry is beneficial for all students, who will find that learning to reason and prove convincingly is necessary for every profession. It is true that not everyone must prove things, but everyone is exposed to proof. Politicians, advertisers, and many other people try to offer convincing arguments. Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction. Geometry provides a simplified universe, where points and lines obey believable rules and where conclusions are easily verified. By first studying how to reason in this simplified universe, people can eventually, through practice and experience, learn how to reason in a complicated world. Geometry in ancient times was recognized as part of everyone's education. Early Greek philosophers asked that no one come to their schools that had not learned the Elements' of Euclid. There were, and still are, many who resisted this kind of education. Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards. However not all of the knowledge of the more learned peoples of the past was false. In fact without people like Euclid or Plato we may not have been as advanced in this age as we are. Mathematics is an adventure in ideas. Within the history of mathematics, one finds the ideas and lives of some of the most brilliant people in the history of mankind’s’ populace upon Earth. First man created a number system of base 10. Certainly, it is not just coincidence that man just so happens to have ten fingers or ten toes, for when our primitive ancestors first discovered the need to count they definitely would have used their fingers to help them along just like a child today. When primitive man learned to count up to ten he somehow differentiated him from other animals. As an object of a higher thinking, man invented ten number-sounds. The needs and possessions of primitive man were not many. When the need to count over ten aroused, he simply combined the number-sounds related with his fingers. So, if he wished to define one more than ten, he simply said one-ten. Thus our word eleven is simply a modern form of the Teutonic ein-lifon. Since those first sounds were created, man has only added five new basic number-sounds to the ten primary ones. They are “hundred,” “thousand,” “million,” “billion” (a thousand millions in America, a million millions in England), “trillion” (a million millions in America, a million-million millions in England). Because primitive man invented the same number of number-sounds as he had fingers, our number system is a decimal one, or a scale based on ten, consisting of limitless repetitions of the first ten number sounds. Undoubtedly, if nature had given man thirteen fingers instead of ten, our number system would be much changed. For instance, with a base thirteen number system we would call fifteen, two-thirteen’s. While some intelligent and well-schooled scholars might argue whether or not base ten is the most adequate number system, base ten is the irreversible favorite among all the nations. Of course, primitive man most certainly did not realize the concept of the number system he had just created. Man simply used the number-sounds loosely as adjectives. So an amount of ten fish was ten fish, whereas ten is an adjective describing the noun fish. Soon the need to keep tally on one’s counting raised. The simple solution was to make a vertical mark. Thus, on many caves we see a number of marks that the resident used to keep track of his possessions such a fish or knives. This way of record keeping is still taught today in our schools under the name of tally marks.
The earliest continuous record of mathematical activity is from the second millennium BC when one of the few wonders of the world was created mathematics was necessary. Even the earliest Egyptian pyramid proved that the makers had a fundamental knowledge of geometry and surveying skills. The approximate time period was 2900 BC The first proof of mathematical activity in written form came about one thousand years later. The best-known sources of ancient Egyptian mathematics in written format are the Rhind Papyrus and the Moscow Papyrus. The sources provide undeniable proof that the later Egyptians had intermediate knowledge of the following mathematical problems, applications to surveying, salary distribution, calculation of area of simple geometric figures' surfaces and volumes, simple solutions for first and second degree equations. Egyptians used a base ten number system most likely because of biologic reasons (ten fingers as explained above). They used the Natural Numbers (1,2,3,4,5,6, etc.) also known as the counting numbers. The word digit, which is Latin for finger, is also another name for numbers that explains the influence of fingers upon numbers once again. The Egyptians produced a more complex system then the tally system for recording amounts. Hieroglyphs stood for groups of tens, hundreds, and thousands. The higher powers of ten made it much easier for the Egyptians to calculate into numbers as large as one million. Our number system which is both decimal and positional (52 is not the same value as 25) differed from the Egyptian, which was additive, but not positional. The Egyptians also knew more of pi then its mere existence. They found pi to equal C/D or 4(8/9)ª whereas a equals 2. The method for ancient peoples arriving at this numerical equation was fairly easy. They simply counted how many times a string that fit the circumference of the circle fitted into the diameter, thus the rough approximation of 3. The biblical value of pi can be found in the Old Testament (I Kings vii.23 and 2 Chronicles iv.2)in the following verse “Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.” The molten sea, as we are told is round, and measures thirty cubits round about (in circumference) and ten cubits from brim to brim (in diameter). Thus the biblical value for pi is 30/10 = 3. Now we travel to ancient Mesopotamia, home of the early Babylonians. Unlike the Egyptians, the Babylonians developed a flexible technique for dealing with fractions. The Babylonians also succeeded in developing more sophisticated base ten arithmetic that were positional and they also stored mathematical records on clay tablets. Despite all this, the greatest and most remarkable feature of Babylonian Mathematics was their complex usage of a sexagesimal place-valued system in addition a decimal system much like our own modern one. The Babylonians counted in both groups of ten and sixty. Because of the flexibility of a sexagismal system with fractions, the Babylonians were strong in both algebra and number theory. Remaining clay tablets from the Babylonian records show solutions to first, second, and third degree equations. Also the calculations of compound interest, squares and square roots were apparent in the tablets. The sexagismal system of the Babylonians is still commonly in usage today. Our system for telling time revolves around a sexagesimal system. The same system for telling time that is used today was also used by the Babylonians. Also, we use base sixty with circles (360 degrees to a circle). Usage of the sexagesimal system was principally for economic reasons. Being, the main units of weight and money were mina,(60 shekels) and talent (60 mina). This sexagesimal arithmetic was used in commerce and in astronomy. The Babylonians used many of the more common cases of the Pythagorean Theorem for right triangles. They also used accurate formulas for solving the areas, volumes and other measurements of the easier geometric shapes as well as trapezoids. The Babylonian value for pi was a very rounded off three. Because of this crude approximation of pi, the Babylonians achieved only rough estimates of the areas of circles and other spherical, geometric objects. The real birth of modern math was in the era of Greece and Rome. Not only did the philosophers ask the question “how” of previous cultures, but they also asked the modern question of “why.” The goal of this new thinking was to discover and understand the reason for mans’ existence in the universe and also to find his place. The philosophers of Greece used mathematical formulas to prove propositions of mathematical properties. Some of who, like Aristotle, engaged in the theoretical study of logic and the analysis of correct reasoning. Up until this point in time, no previous culture had dealt with the negated abstract side of mathematics, of with the concept of the mathematical proof. The Greeks were interested not only in the application of mathematics but also in its philosophical significance, which was especially appreciated by Plato (429-348 BC). Plato was of the richer class of gentlemen of leisure. He, like others of his class, looked down upon the work of slaves and crafts worker. He sought relief, for the tiresome worries of life, in the study of philosophy and personal ethics. Within the walls of Plato’s academy at least three great mathematicians were taught, Theaetetus, known for the theory of irrational, Eodoxus, the theory of proportions, and also Archytas (I couldn’t find what made him great, but three books mentioned him so I will too). Indeed the motto of Plato’s academy “Let no one ignorant of geometry enter within these walls” was fitting for the scene of the great minds who gathered here. Another great mathematician of the Greeks was Pythagoras who provided one of the first mathematical proofs and discovered incommensurable magnitudes, or irrational numbers. The Pythagorean theorem relates the sides of a right triangle with their corresponding squares. The discovery of irrational magnitudes had another consequence for the Greeks since the length of diagonals of squares could not be expressed by rational numbers in the form of A over B, the Greek number system was inadequate for describing them. As, you might have realized, without the great minds of the past our mathematical experiences would be quite different from the way they are today.