History of Mathematics - Essay Example

HISTORY OF MATHEMATICS TEST 2 Discuss the idea of number mysticism. Give concrete examples of the mysticism surrounding the numbers through 10, and any other numbers with superstitions attached to them. Which of these still exist today A certain aspect of number mysticism is older than the Pythagoreans. The number seven had been thought of as a special number for perfection in many early cultures. The idea may be from the seven stars or planets that name the days of the week: Sun, Moon, Mars, Mercury, Jupiter, Venus, and Saturn (What is the Origin of the Seven Day Week). The Pythagorean idea of the world was that natural numbers were the answer to the different secrets of humans and matter. They thought that everything was made up of numbers, the reason for what anything was could only be figured out in numbers (Early Concepts of Number and Number Mysticism). For the numbers from 1 to 10 each number has a special property. The number 1 is a monad and stands for unity because it is the cause of the other numbers and in number mysticism is the number of reason. The number 2 is called a dyad for diversity or opinion and is mystically the first female number (all the even numbers are called female in many early cultures). The number 3 is a triad and means harmony because it adds up unity (number 1) plus diversity (number 2) and is considered the first male number (all the odd numbers are male in many early cultures). Number 4 is the number for justice or the squaring of accounts (number 2 squared). Number 5 is for marriage because the number 2 (first female) plus the number 3 (first male) equals the number 5. Number 6 is for creation because the number 2 (first female) plus the number 3 (first male) plus 1 more monad equals 6. The Number 10 is for the Universe because as tetractys it is the sum of all possible dimensions in geometry. Ten is the sum of the number of points that make up: a single point (1) + a two point line (2) + a three point triangle (3) + a four point tetrahedron (4) 1 + 2 + 3 + 4 = 10. Today there is still superstition attached to the number 13, Friday the 13th, and the 13th floor is often left out of the numbering in buildings, seating in public places, etc. Today numerology is used in the occult to look for secret meanings of events telling the future. In that case numerology is a form of divination, or fortune telling such as astrology and dream interpretation (Numerology). Mathematics still uses the Pythagorean theorem that was the first rigorous geometric proof of the relationship between the length of the sides of a right-angled triangle, the Pythagorean definitions for perfect, abundant and deficient numbers and prime and composite numbers, the study of number patterns in things like Magic Squares and Sudoku, and the mathematical relationship between music and number in music theory (Early Concepts of Number and Number Mysticism). 2. Discuss Euclid's contributions to the field of mathematics. Which of his famous works has been most influential to modern mathematics and is still studied today Euclid of Alexandria wrote the oldest Greek mathematics text still in existence called the Elements, and explained the principles for two-dimensional Euclidean geometry or plane geometry, and three-dimensional Euclidean geometry known as solid geometry. Some of Euclid's works, such as the Porisms, have been lost and might have been very interesting to read today. The Euclidian Elements has been very influential in modern mathematics and is still studied even now. Instead of discovering anything new, Euclid was a very good teacher and he was really exceptional at explaining elementary or basic mathematics in the Elements on the theory of numbers, the synthetic geometry of points, lines, planes, circles and spheres and some geometric algebra and the geometry of solids. Euclid wrote the Elements in 300 BC and there were Greek copies made of it from the 10th to the 12th Centuries AD, later Arabic translations of it, then Latin and the first printed version was one of the very first mathematical text books in 1482 AD. 3. a) Explain how Archimedes' Law of the Lever works. Give an example of how this theory could be used today. According to Archimedes, objects balance each other at distances from the fulcrum that are reciprocally proportional to their weights. The law of the lever says that if two weights, w and W, are placed at either end of a straight lever balance, when w is a distance l from the fulcrum of the lever and W is a distance L from the fulcrum, then the length (distance to the fulcrum) times the width of one weight is equal to the length (distance to the fulcrum) times the width of the other weight or lw = LW, then Archimedes says that the ratio of the weights in balance is the same as the inverse ratio of their lengths from the fulcrum or w/W =L/l. The theory can be used, for instance, to enable a workman to lift an object heavier than he is by using a longer distance from the fulcrum for himself and a shorter distance for the weight. b) How did Archimedes use the idea of buoyancy to prove that King Hiero's gold crown was not pure gold The Law of Hydrostatics (buoyancy) says that when you immerse some object in a liquid, the object is buoyed up by a force equal to the weight of the liquid displaced by the object. Archimedes filled up a vessel to the top and put in King Hiero's crown. From the Law of Hydrostatics Archimedes knew that the mass of the water that overflowed the vessel was equal to the mass of the crown. When he did the same thing with what the original amount of gold had been that was supposed to be in the crown, Archimedes found that the gold displaced more water, meaning it had more mass than King Hiero's crown and concluded that not all the gold the goldsmith had been given for the crown was used to make the crown (Archimedes). 4. Identify the conic sections discussed in the reading. Why are they called conic sections The conic sections are the ellipse, the parabola and the hyperbola. Apollonius was able to show that it was not necessary to take perpendicular sections up the axis of a cone from three different cones to get the various sections, but that all three types of conic sections could be sections of the same cone, depending on the angle of inclination of the cutting plane. It was an important step to link the three types of curve as being different angles of interception sliced across the same cone. A second important aspect that Apollonius showed was that the cone did not need to be a cone with an axis perpendicular to a circular base, but could also be an oblique or scalene circular cone with the axis at an angle. The properties of a curve from a conic section are the same whether they are cut from oblique or right cones. Every oblique circular cone has an infinite number of circular sections parallel to the base, and another infinite number of subcontrary sections oriented in the opposite direction. They are called conic sections because the ellipse, parabola and hyperbola are the circumferences of all possible intersecting sections at various angles across a cone. 5. What signified the decline of Greek mathematics Why did this era come to an end The work of the Greeks was in more abstract thinking. Most of the important ancient mathematicians were philosophers who liked to speculate about the nature of the world. The main Greek strengths were in pure mathematics, geometry and abstract philosophy, but their weakness was in practical application, and in the relation between theorizing and experimenting. The Greeks used natural concepts and imagery in their reasoning. The Greeks have given mathematics many astute theories, such as the geometric principles and the atomic theory, that have continued to be important to modern mathematics. Now modern science tests the mathematical theories with empirical testing. The Greeks figured out vital things about the relation of mathematics to physics and the world and left modern science to develop the notion of mathematical physics. The Greeks had woven mathematics into the basics of their reasoning. When speculative thinking fell out of favor, and the time of mysticism and Greek theorizing was overtaken by research, Greek mathematics entered a decline. The history of mathematics shows that around 500 AD a large-scale degeneration took place. Mathematics as an intellectual interest was gradually neglected. It left the West with a mathematical Greek heritage that suddenly seemed to vanish. Some speculate also that algebra that is based almost entirely on symbolic statements took over the ordinary ways of argument in Greek mathematics. Sometime later the disintegration of the Roman empire destroyed the social stability needed in general for speculative science (Fragments of the Past). Works Cited "Archimedes." Biographies of Mathematicians Andrews University. Available at www.andrews.edu/calkins/math/biograph/bioarch.htm Boyer, Carl B. and Merzbach, Uta C. A History of Mathematics. 2nd Ed. New York: John Wiley and Sons, Inc. 1991. "Early concepts of Number and Number Mysticism." Pythagoras of Samos. Math Gym Available at: http://www.mathgym.com.au/history/pythagoras/pythnum.htm#10. "Fragments of the Past." Science News Online. Available at http://www.sciencenews.org/pages/sn_arc97/1_18_97/mathland.htm "Numerology." The Mystica. Available at: http://www.themystica.com/mystica/articles/n/numerology.html "What is the Origin of the Seven-day Week." Calendars Through the Ages Available at: http://webexhibits.org/calendars/week.html Read More