History of Math

How did Greek mathematics develop?

Why is Euclid important?


      Greek mathematics developed from reason and logic. Egyptians would not ask the question why? where Greeks were interested in how something works down to the core. Egyptians might of said it works because it does but Greeks wanted to dig deeper. This was best illustrated in the first chapter of the book I am reading "Journey Through Genius" by William Dunham. This type of thinking became prominent in the Greek culture at that time influencing many great mathematicians, inventors, astronomers, engineers, and many others. Greek mathematics developed from this non-rigorous, taken for granted, idea to a much more extreme proof-orientated way of mathematics. It was around this time that this free thinking or 'awakening' happened and many great individuals conjectured, proved, and invented things that we heavily rely on to this day.This awakening was responsible for great individuals like Thales, Pythagoras, and Euclid who showed that humans were starting to think outside of the box. We will briefly examine each individual.



























      Thales (624-547) is regarded to be the first Greek mathematician. He was the founder of the Ionian school of philosophy in Miletus, and the teacher of Anaximander. Thales went to Egypt and studied with the priests where he learned mathematics and brought it back to Greece. Upon studying in Egypt Thales did geometrical research which allowed him with the use of triangles to calculate the distance from the shore of ships at sea. Using the same technique Thales was also able to determine the height of pyramids in Egypt. Thales is credited with the following five theorems of geometry:

1. 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.
4. If one triangle has two angles and one side equal to another triangle, the two triangles are equal in all respects. (See Congruence)
5. Any angle inscribed in a semicircle is a right angle. This is known as Thales' Theorem.

     Pythagoras (569-500) is considered the first pure mathematician of our time. Pythagoras was interested in the study of numbers along with angles, triangles, areas, proportions, polygons, and polyhedra. Pythagoras also played a seven string lyre which allowed him to study the relationship between music and mathematics. He noted that the vibrating strings of the lyre sounded when the lengths of the strings were proportional to whole numbers like 2:1, 3:2, 4:3, and so on. His most famous theorem, the Pythagorean Theorem is what he is most remembered by. He is credited with the following six results:

1. The sum of the angles of a triangle is equal to two right angles.
2. The theorem of Pythagoras - for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians understood this 1000 years earlier, but Pythagoras proved it.
3. Constructing figures of a given area and geometrical algebra. For example they solved various equations by geometrical means.
4. The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely to have been the idea of Pythagoras because it does not align with his philosophy the all things are numbers, since number to him meant the ratio of two whole numbers.
5. The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two.
6. Pythagoras taught that Earth was a sphere in the center of the Kosmos (Universe), that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus.

     Euclid (330-260) is often referred to as the "Father of Geometry". He taught mathematics in Alexandria, Egypt at the Alexandria library. He also wrote the most enduring mathematical work of all time which is Elements. This is a thirteen volume work that is a comprehensive compilation of all geometric knowledge known up to that point in time. The book is based off the works of Thales, Pythagoras, Plato, Eudoxus, Aristotle, Menaechmus, and others. It was used for over 2000 years. Euclid wrote Elements at the Alexandria library which covered plane geometry, solid geometry, arithmetic, and number theory. The organization of Elements is interesting because Euclid organized the known geometrical ideas, starting with simple definitions, axioms, formed statements called theorems, and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates, and demonstrated logically 467 propositions in plane and solid geometry. Elements would be the most widely used textbook of all time and would appear in classrooms until the twentieth century. It has sold more copies than any other book besides the bible.

     Personally with this research of Greek mathematics, it makes me wonder what were the main factors that made these people such great thinkers. Besides this idea of free thinking, what else were influencing these people that made them achieve this greatness. The Greeks made major developments in philosophy, politics, mathematics, physics, and many other fields. These ideas are commonly used today like the democracy in our government or Euclid's axioms in geometry.
Sources:

http://www.mathopenref.com/thales.html

http://www.mathopenref.com/pythagoras.html

http://www.mathopenref.com/euclid.html

What Is Math?

Mathematics is a generalized term that encompasses many different branches within. To name a few there is Geometry that deals with shapes, Calculus that deals with rates of change, Arithmetic that deals with counting and many more. To ask what is mathematics is like asking what is life. There is no definitive answer.

One of the significant discoveries within the history of mathematics would have to be the discovery of zero. The Mayans independently discovered the concept of zero without any communication from the outside world. Islamic civilizations discovered this concept as well and many other important mathematical findings.

Off of the top of my head and doing no research, the next important discovery would have to go to Issac Newton and Gottfried Leibniz for their work in the development of infinitesimal calculus.

Mathematical tools are also very important to consider. Starting with an early counting object called an abacus and progressing to the TI-84 calculator took not only a lot of time, but a lot of mathematics.

There are many more of these discoveries and yet there are still more to be found. To rank them in significance seems odd but chronologically might make more sense. Mathematics builds on itself. We first learn Arithmetic and then move on to basic Geometry and Algebra. We do not start with Calculus and work backwards.