Explanation of Mathematics


Elements Organization

It is believed that of the thirteen books collected, Elements would have been composed from earlier mathematicians. Books I and II likely originated from Pythagoras, Book III probably was the work Hippocrates, and Book V was assumed to be Eudoxus. There is evidence that many of the other books came from Pythagorean and Athenian sources.(22) So in other words Elements was not a collection of Euclid's ideas but mostly a collection of the known geometric works of his day. Each book is divided into definitions, postulates, axioms (common notions), and propositions that could be built from previous statements. Definitions, postulates, and axioms were not proofs but clear ideas and logical steps needed to prove propositions.

The first six books dealt with plane geometry which deals with lines, polygons, circles and other two-dimensional objects. Books seven through ten are considered arithmetic and number theory proofs. In other words this is looking into relationships of the known numbers and the operations of addition, congruence, division, factorization, multiplicand, root extraction and subtraction. In this section of the book you will find Euclid's own discoveries, such as the first known proof that there are infinitely many prime numbers (Book IX Proposition 20). Books eleven through thirteen discuss solid geometry or three-dimensional solids. Although some of the wording can be confusing. With the help of videos and applets much of Euclid's meaning can seem simple and even obvious. It is important to remember that this was the point. Euclid was building a base for the structure of geometry and he wanted them to be reasonable enough that you wouldn't doubt them. He was finding the most elementary ways of describing the normal space around him as he understood it. He was not looking into the astronomical or infinity of the unknown.(23)


Explanation of Inductive and Deductive Principles

Further explanation of the processes used by Greek mathematicians, philosophers, and scientist to find truth in the world around them may be needed. The two fundamental ways in which they looked for answers was through inductive and deductive procedures. Inductive reasoning is the process of looking for strong evidence or probability. It was not intended to be an absolute proof or intended to create structural understanding. The experimentation allowed for theoretical science and even religious ideology be formed around the evidence they found. The deductive process that was formalized by Aristotle and formally structured by Euclid was the process of starting with simple "self-evident" truths and building mathematical knowledge from there. The Greeks came to believe that this was the only acceptable way of obtaining knowledge. The potential logical fallacies that come from the deductive process are usually found in the starting point or the initial thoughts of the philosopher. Aristotle, having been a student of Plato, admitted the importance of induction while he built his systematic approach of deduction and logic.(24) As a side note, it is important to recognize that inductive reasoning should not be confused with the mathematical proof technique known as Mathematical Induction. The second step in Mathematical Induction is known as the inductive step and is actually using a deductive principle. This method is just taking a given statement and going one logical step in the natural numbers to imply that the next statement is true. This step is using well established and agreed upon statements and ideas and taking the next logical step. Therefore, the mathematical induction proof is actually a form of deductive reasoning.


A Few of the Well Known Proofs

The following proofs provide three different locations for visualizing Euclid's proofs. The first two are different YouTube stations which have extensive video libraries that explain many of the propositions. These stations are great resource when looking for a detailed explanation or trying to review the structure of Euclid's proofs. The last proof Proposition 47 provides a link that gives all of Book I's propositions explanations in applet form. These are effective ways of visualizing how the axiomatic process is used to build a clear proof.

Various Euclid Proofs
Book I - Proposition 4 (Angle Side Angle)(j)
Book I - Proposition 32 (Angle Sum Postulate)(k)
Book I - Proposition 47 (Pythagorean Theorem)(l)


To find more technology assistance on other postulates within Elements the following links are the video libraries in which the two first videos were found

mathematicsonline (m)
Sandy Bultena (n)

This last proof in the table above is the method Euclid used to prove the Pythagorean principle. It may seem confusing, but by taking each of the steps in the applet you can recognize how Euclid went about using any number of definitions, postulates, and axioms. There are other simplified ways of looking at this proof using modern technology. Here are some other examples to help visualize and conceptualize deductive proofs:

Pythagorean Animation (o)

Pythagorean Water GIF (p)

(q)



Section Description
History and Background This section of the wepage introduces Euclid, who influenced him, Elements, and questions that arise from Elements.
Significance and Application This section addresses the influnce of Elements and applications into other fields of study.
Explanation of Mathematics This section looks at how Elements is organized, explains inductive and deductive principles, and provides a few examples of proofs
References This section provides the references cited in this webpage.


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