The word Geometry comes from the Greek word geometrin meaning earth-to measure. In fact, early geometry was a method of surveying, or measuring land. Early geometry was also done with approximations rather that fact. Not until the Greek, Thales of Miletus, was this system changed from trial and error to deductive reasoning. In the 300s B.C., a man named Euclid organized the teachings of Pythagoras, his disciples and other Greek thinkers, into his great work, The Elements. In fact, this book is a synthesis of past teachings. Euclid, thought to be a student of Plato's disciples, organized the epic, Elements, from centuries of Greek geometry, and refined lots of it. Plato, the founder of a major academy in Athens, emphasized geometry in his teachings. The Elements is one of the most widely read books ever, and his approach has dominated mathematics for the last two millenniums. I have traveled back in time to 370 B.C. to interview Euclid right before his death:
Q: When and where were you born?
A: I was born in 300 B.C. in Greece.
Q: What turned you on the most about geometry?
A: What turned me on the most about mathematics in general was much the same as what turns people on to music and the arts--there was a beauty and mystery to it. What turned me on to geometry was its power to expand our knowledge of the world and how things work.
Q: How did you compile The Elements?
A: I synthesized this work out of hundreds of years of the teachings of mathematicians.
Q: What is included in this great work?
A: The Elements is made up of 13 volumes dealing with such subjects as plane geometry, proportion, the properties of numbers, incomprehensible magnitudes, and solid geometry.
Q: Besides writing and studying, what do you do with yourself?
A: I teach at a school in Alexandria, Egypt.
Q: Why would you consider yourself important?
A: I think I'm important because I am the first person who organized geometry and I made many great discoveries.
Q: What kind of discoveries did you make?
A: One of the discoveries that I'm most proud of, that most people don't credit me for, is in the theory of numbers, or Number Theory.
Q: What do you think is the basis of geometry?
A: Well I based my geometry on five fundamental assumptions, called axioms or postulates.
Q: Can you go into detail on one or two of these postulates?
A: Well my first postulate goes like this:
Postulate I: For every point P and for every point Q not equal to P there is a unique line l that passes through P and Q.3 - or - two points P and Q determine an exclusive line, PQ.
Definition: Given two point A and B. The
segment AB is the set including A and B and all the points that
lie on the line AB but in between A and B. The points A and B are
therefore the endpoints of he line segment AB.
Postulate II: For every segment, such as AB and for every segment, such as CD, there's a unique point such as E so that B is between A and E and the segment CD is congruent to segment BE. So those are two of the five main postulates in Elements.
Q: Can you quickly go over the other three?
Postulate III: If you have a point O and a point A not equal to O, there exist a circle that's center is O and radius is OA.
Postulate IV: All right angles are congruent (equal) to each other.
Postulate V: Otherwise known as the Parallel Postulate. For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l. Parallel: lying or moving in the same direction but always the same distance apart.
Q: Can you give me an example of what you would call deductive reasoning?
A: Well deductive reasoning is basically deducing facts from previously known facts. Here's an example: Things that are equal to the same thing, are equal to each other.
Or: if a=b
Q: The Elements is basically a compilation of hundreds of years of unorganized Geometry; whom would you credit to a lot of the discoveries featured in your book?
A: Many of the theorems in the book I would credit to great mathematicians, Eudoxus, and Theaetetus.
Q: Ptolemy once asked you, In geometry, is there a shorter way than The Elements?
A: I replied, There is no royal path to geometry.
Q: Can you give me an example of geometrical algebra?
A: My book includes 10 algebraic propositions. Here's one: (a + b) a = ab+a2
Q: Did you write any more books besides The Elements?
A: Yes, although The Elements is considered my greatest work, I did write many other books.
Q: How do you feel that The Elements has been translated and published in more edition than any other book other than the bible?
A: Well I'm very proud of that fact. When I was compiling this book I had no idea it was going to be that influential.
The Elements were in fact so extremely influential that up until about 100 years ago, that book was the principle source and textbook on geometry throughout the world. Euclid is thought to have been so influential on that brand of geometry that is called, Euclidean Geometry. Euclidean Geometry, even today, is what most kids learn in elementary and middle school. Though little is known of his life, his influence on not just geometry, but modern mathematics cannot be understated.
Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries- Development and History, 3rd ed. New York: W.H. Freeman and Company, 1993
Heath, Sir Thomas. A History of Greek Mathematics- Volume I: From Thales to Euclid. Canada: General Publishing Company, 1981.
Euclid, Merriam-Webster, Inc. and Encyclopedia Britannica, Inc., 1991. http://acnet.pratt.edu/~arch543p/help/euclidean_geometry.html
Euclid (mathematician), Microsoft® Encarta® Online Encyclopedia 2001 http://encarta.msn.com 1997-2000 Microsoft Corporation. All Rights Reserved.
Jacobs, Harold R. Geometry. New York: W.H. Freeman and Company, 1987.