Download Euclid and his influence on mathematics

Euclid and his influence on mathematics
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Euclid lived in Alexandria, a Greek city near the mouth of the Nile River, around 300
B.C.
He wrote the world’s most famous axiomatic system in a book called the Elements.
Euclid’s work was studied by Greek and Roman scholars for a thousand years. It was
then translated to Arabic around 800 A.D. and studied by Arab scholars too.
It was then studied all throughout Europe and is still a commonly studied book today.
This book is the second most published book of all time. Can you guess the first?
In the book, the Elements, mathematics is studied from an axiomatic perspective.
This means that a set number of mathematical statements were assumed at the beginning
of the book (these are called axioms, or postulates) and then everything else in the book
was derived logically from those statements.
Prior to Euclid
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Geometry as a logical discipline actually began with Thales, around 600 B.C.
Thales was the first known person to start thinking about geometry as a deductive study.
Even though many of the statements that Thales proved to be true were already known,
his way of logically linking them together was new.
The Pythagoreans and other Greek mathematicians continued to use this method of
logical deduction in mathematics.
By the time of Euclid, Plato’s philosophy and Aristotle’s logic were firmly established;
scholars understood the need for deductive forms of justification.
At the time, however, proofs were disorganized and each one started from its own set of
assumptions.
Euclid’s Elements fixed that.
Euclid’s Elements
The Elements consisted of 13 books.
Book 1 contains 23 Definitions, 5 Postulates, 5 Common Notions (Axioms) and 48 Propositions
(Theorems). Each Proposition is stated and proven.
Books 2-13 simply contain more Definitions and Propositions (Theorems), each according to the
theme of that book.
The Elements contain 465 Propositions amongst the 13 books. Most books contain between 15
and 40 Propositions, but Book 10 has 115 Propositions.
Not all of the Elements are about geometry. Books 1-4 and 6 are about plane geometry. Books
11-13 are about solid geometry.
Books 5 and 10 are about magnitudes and ratios.
Books 7-9 are about whole numbers.
Euclid’s Common Notions and Postulates
Common Notions:
1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
Postulates:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines make the interior angles on that same side
less than two right angles, the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.
Why the Elements have lasted the test of time
The Elements are not only about geometry, but also about how to think logically.
• In the 17th century, Descartes based part of his philosophical method on the “long chains
of reasoning” used in Euclid to move from simple principles to complex conclusions.
• Isaac Newton and Baruch Spinoza used the form of Euclid’s Elements to present their
ideas.
• Abraham Lincoln carried a copy of Euclid with him and studied it in order to become a
better lawyer.
• The format of the Declaration of Independence is based on opening with a set of axioms
and using these axioms and deductive reasoning to justify the need for breaking away
from Great Britain and forming an independent country.
• E.T. Bell, a mathematician from the early 1900s once said, “Euclid taught me that without
assumptions there is no proof. Therefore, in any argument, examine the assumptions.”
Constructions
Proposition 9: To bisect a rectilineal angle.
Let the angle BAC be the given rectilineal angle.
Thus it is required to bisect it.
Let a point D be taken at random on AB;
let AE be cut off from AC equal to AD;
let DE be joined, and on DE let the equilateral triangle DEF be constructed;
let AF be joined.
I say that the angle BAC has been bisected by the straight line AF.
For, since AD is equal to AE, and AF is common,
the two sides DA, AF are equal to the two sides EA, AF respectively.
And the base DF is equal to the base EF;
therefore the angle DAF is equal to the angle EAF.
Therefore, the given rectilineal angle BAC has been bisected by the straight line AF.