Saturday, April 01, 2006

Euclid of Alexandria

Euclid of Alexandria is perhaps the greatest math teacher of all time. His textbook The Elements is the perhaps the most influential math book ever published.

Much of what we know about Euclid must be deduced from the various clues that are available in the writings of others. We know for example that he must have lived around the time of Ptolemy I in Alexandria from 325 - 265 BC from Proclus who writes:
He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry.
(from David Joyce's Web Site)

It is believed that Euclid must have attended Plato's Academy in Athens because of his deep knowledge about the works Eudoxus and Theaetetus. This is also clear since the organization of the Elements is an attempt to understand the Platonic ideal shapes. The Greek philosopher Proclus writes:
In his aim he [Euclid] was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures.
(from Euclid, MacTutor Biography)

It is probable that Euclid ran a school of mathematics in Alexandria. Pappus mentions that the Greek mathematician Apollonius learned geometry from the students of Euclid in Alexandria (Donald Lancon, here). There is a story told by Stobaeus in the fifth century:
... someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid "What shall I get by learning these things?" Euclid called his slave and said "Give him threepence since he must make gain out of what he learns".
(from Heath, a History of Greek Mathematics)

Euclid had a reputation for being fair-minded, polite, and a serious scholar. Pappus writes that Euclid was:
... most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself.
(from Euclid, Mac Tutor Biography)

For some reason, Euclid did not write a preface to his great work as did many of the other Greek mathematicians. As a result of this, we are forced to rely on writings that came hundreds of years after his death.

It is believed that when Euclid was writing the Elements, he borrowed content from other textbooks that came earlier. For example, he presents definitions for oblong, rhombus, and rhomboid which are never used (Euclid, MacTutor).

The Elements consists of 13 books. The first two books focus on triangles, parallel lines, parallelograms, rectangles and squares. Included in these book is the parallel postulate, which is the starting point for non-Euclidean geometry, the famous Pythagorean Theorem, a construction for squaring a rectangle.

Books three and four deal with the properties of circles. Book five presents Eudoxus's theory of proportions. Book six deals with similar triangles.

Books seven to nine deal with number theory including Euclid's algorithm for greatest common divisor. Books ten deals with irrational numbers which is taken from Theaetetus's theory.

Books eleven through thirteen cover three-dimensional geometry. Book twelve ends with the proof that circles are to each other as the square of their diameters and spheres to each other as the cubes of their diameters. Book thirteen ends with the proof that there can be only 5 types of regular polyhedra (three-dimensional shapes with equal sides) which are the: tetrahedon (4-sided), cube (6-sided), octahedron (8-sided), dodecahedron (12-sided), and icosahedron (20-sided). These are of course the Platonic solids (see here for details):

1. Tetrahedron

2. Cube

3. Octahedron

4. Dodecahedron

5. Icosahedron

Euclid wrote other works besides the Elements including a work on perspective called Optics. Many of his works are lost including a work he did on conics which predates Apollonius.

The Elements was first published in book form in 1482. Since then, there have been over 1000 editions of this classic math book. It set the standard for rigor and clarity of mathematics and one of the chief themes in the history of mathematics is the effort to place other areas of mathematics on the same firm foundations at Euclid's Elements. Today, Euclid is known as the father of geometry.


Friday, March 31, 2006

More on Euler's Identity and Roots of Unity

When Leonhard Euler came up with his Formula and his Identity, he stood on the shoulders of many giants. In the next few blogs, I plan to focus a bit on some of the giants that Euler stood upon including: Euclid, Archimedes, Hipparchus, Ptolemy, Napier and Bernoulli.

Thinking about Euler's identity in the context of Fermat's Last Theorem raises some questions which I think need to be answered:
  • Pi, Sine, and Cosine are based on Euclid's plane geometry. What validity can it have for number theory which is independent of Euclidean or non-Euclidean geometry?
  • How is it possible for a number to be put to the power of an imaginary number? How can this construction possibly have any meaning?
It turns out that trigonometric functions can be defined independently of Euclid (see here). It also turns out that it is possible to use the Maclaurin Series to define a exponents so that they can include any complex power including i (see here)

One of the goals of this blog is to provide a complete set of proofs for each of the propositions that I present or to state those propositions as postulates. Implicit in the use of sine, cosine, and pi is a set of assumptions that are often not thought about:
  • How can we be sure that all right triangles regardless of their size have the same ratio between their sides so that for a given angle, there is one and only one sine or cosine value? (See here for the answer)
  • How can we be sure that pi is really constant? It may be obvious but where's the proof? How can we be sure that for all circles, the ratio of circumference-to-diameter is the same? (See here for the answer)
While these questions are very elementary, I think that it is still important to ask them.

In understanding Roots of Unity and Cyclotomic Integers, I think it is worthwhile to review the achievements of DeMoivre, Taylor, and Maclaurin.

I think it is important and valuable to understand all of these details before exploring Kummer's proof.