FATHER OF GEOMETRY: EUCLID

 Euclid of Alexandria (lived around 300 BC) organized Ancient Greek and Near Eastern mathematics and geometry. He wrote the Elements, the most widely used mathematics and geometry textbook in history, though older books sometimes confuse him with Euclid of Megara. Modern economics is referred to as a series of footnotes to Adam Smith, author of The Wealth of Nations (1776 AD). Similarly, much of Western mathematics is a series of footnotes either developing Euclid's ideas or testing them.

Life of Euclid

Almost nothing is known about Euclid's life. He led his own school in Alexandria, Egypt, around 300 BC. We do not know the places or years of his birth and death. Euclid seems to have written about a dozen books, many of which are now lost.
The Athenian philosopher Proclus (412-485 AD), who lived seven hundred years after Euclid, states that Euclid "brought together the elements, collected most of the theorems of Eudoxus, perfected many of those of Theaetetus, and gave undeniable proofs of what his predecessors had only cursorily proved." The scholar Stobaeus lived at roughly the same time as Proclus. Stobaeus collected Greek manuscripts that were in danger of being lost. He tells a story about Euclid that seems to be true.


Someone who started studying geometry said to Euclid. 'What can I achieve by learning?' he asked. Euclid he called his slave and said to him, 'Give him some money give because he has to get something out of what he learns.'
(Health, 1981, loc.8625)

Pre-Euclidean Geometry

In the Elements, Euclid collected, organized and proved the ideas of geometry that were already being used as applied techniques. With the exception of Euclid and some Greek pioneers such as Thales (624-548 BC), Hippocrates (470-410 BC), Theaetetus (417-369 BC), Eudoxus (408-355 BC), almost no one tried to work out why the ideas were correct or whether they applied generally. Thales even gained fame in Egypt for seeing the mathematical principles behind the rules of some specific problems and then applying them to other problems, such as determining the length of pyramids.
The ancient Egyptians had a serious knowledge of geometry, but this knowledge consisted of practical methods based on testing and experimentation. For example, to calculate the area of a circle, they made a square whose length was eight times nine times the diameter of the circle. The area of this square was close enough to the area of the circle that they could detect no difference. Their method took the value of pi to be 3.16, a bit far from the real value of 3.14... but close enough for simple engineering. Much of what we know about ancient Egyptian mathematics comes from the Rhind papyrus, discovered in the middle of the 19th century and now held in the British Museum.
The ancient Babylonians also knew a lot about applied mathematics, including the Pythagorean theorem. During archaeological excavations in Nineveh, clay tablets have been discovered that contain triples of 3-4-5, 5-12-13 and relatively large numbers, all of which obey the Pythagorean theorem. As of 2006, some 960 tablets have been analyzed.

Elements

Euclid was not the first to produce many of the ideas in the Elements. His contributions were fourfold:

• He collected important mathematical and geometrical information in a single book. Elements was a textbook rather than a reference book, so it does not contain everything known.
• Euclid gave definitions, postulates and axioms. He called axioms "common ideas".
• He presented geometry as an axiomatic system: Every statement was either an axiom, a postulate, or a statement proved in clear logical steps from axioms and postulates.
• He presented his own original discoveries, such as the first known proof of the infinite number of prime numbers.

The Elements consists of 13 sub-chapters divided into 3 main parts: (often each part is called a "book").

• Section 1-6: Plane geometry
• Chapters 7-10: Arithmetic and number theory
• Chapters 11-13: Solid geometry

Each chapter starts with definitions. Section 1 specifically includes postulates and "common ideas" (axioms). Examples are as follows:

Definition: "A point is something that has no parts."
Postulate: "Draw a straight line from any point to any other point" (This is Euclid's way of saying that a straight line exists).
Common idea: "Things that are equal to the same thing are also equal to each other."


If the ideas seem self-evident, this is what must be done. Euclid wanted to build his geometry on ideas so obvious that no one could logically doubt them. From his definitions, postulates and common ideas, Euclid deduces the rest of geometry. His geometry defines the normal space we see around us. Modern 'non-Euclidean' geometry describes space at speeds close to the speed of light or at astronomical distances bent by gravity.

Other Works of Euclid

About half of Euclid's works are lost. The only reason we know about them is that ancient authors refer to them. His missing works include books on conic geometry, logical errors and porisms. We are not sure what porisms are. Euclid's extant works are Elements, Data, Division of Geometric Figures, Phenomena and Optics. In his book on Optics, Euclid defends the same theory of vision as the Christian philosopher St. Augustine.

Euclid's Effect

From ancient times until the late 19th century, people recognized the book of Elements as a perfect example of correct reasoning. The Elements went through more than a thousand editions, making it one of the most popular books after the Bible. In the 17th century, the Dutch philosopher Baruch de Spinoza wrote his Ethics, using the same form of definitions, postulates, axioms and proofs from the Elements as his model. In the 20th century Austrian economist Ludwig von Mises adopted Euclid's axiomatic method in his book on economics, Human Action.