Euclid and the “elements”

Euclid (300 BC, 265 BC (?) ) was a Greek mathematician,

often referred to as the "Father of Geometry”. Of course this is not a true Euclid’s portrait, as some textbook wants you to believe, but it’s quite suggestive

Euclid was active in Alexandria (modern Egypt)  during the reign of Ptolemy I (323–283 BC). There, he was a scholar and a preceptor, and wrote a textbook of geometry, the  Elements, which  is one of the most influential works in Mathematics;

it served as the main textbook  for teaching  geometry from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms, or postulates

• An Axiom, or postulate, is a proposition that we assume as “true”. All other proposition can be deduced from the axioms and the definitions using the rules of logic.

• Many of the proofs in the Elements are by contradiction

A proof by contradiction …

It’s a lie being exposed.

The son tells the father “ “I did not set break grandma’s vase. It was the cat …”

And the father tells him “ Son, you should not lie…”

“ if the cat had broken the vase, he would have knocked it off the table. But since the vase is still on the table, the cat can’t have broken it. “

So the father has proved his theorem…

•Euclidean Geometry deals with points, lines and planes and how they interact to make more complex figures.

•Euclid’s Postulates define how the points, lines, and planes interact with each other. The “Elements” start with 23 definitions.

1. What is Euclidean Geometry?

http://aleph0.clarku.edu/~djoyce/java/elements/aboutText.html

• Definition 1.– A point is that which has no part.

• Definition 2.– A line is breadthless length.

• Definition 4.– A straight line is a line which lies evenly with the

points on itself.

• Definition 5.– A surface is that which has length and breadth only.

• Definition 7.A plane surface is a surface which lies evenly with the

straight lines on itself.

• Probably some of these

definitions are not Euclid’s but have been added by other scholars. Keep in mind that the oldest copy of the “Elements” that we have was written 800 years after Euclid’s death. Can you imagine how many copies (with changes) of the book have been made in 800 years??

• A fragment of the Elements’ oldest copy

Definition 10.When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

This is Euclid’s last definition.

• Definition 23– Parallel straight lines

are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

• If the lines are not in the same plane, they may not have any point in common even if they are not parallel. Two lines that are not on the same plane and do not intersect are called Skew

After the definitions, Euclid states 5 postulates. Here are the first 4.

• 1. A straight line segment can be drawn joining any two points.

• 2. Any straight line segment can be extended indefinitely in a straight line.

• 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

• 4. All right angles are equal.

… And here is the 5th.

• 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines must intersect  each other on that side if extended far enough.

• 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines must intersect  each other on that side if extended far enough.

The 5th postulate is known as “the parallel postulate”. In fact is equivalent to the following: (the Mayfair axiom). Given a line and a point not on that line,

there is exactly one line through the point that is parallel to the line

But do parallel line really exist? How to verify that two lines never

ever meet?

For 2000 years people

were uncertain of what

to make of Euclid’s

fifth postulate.

Euclid himself had doubts

about it.

Remarks about the 5th postulate:

• It was very hard to understand. It was not as simplistic as the first four postulates.

• The parallel postulate does not say that parallel lines exist; it shows the properties of lines that are not parallel.

• Euclid proved 28 propositions before he utilized the 5th postulate, but once he started utilizing this proposition, he did so with power: he used the 5th postulate to prove well-known results such as the Pythagorean theorem and that the sum of the angles of a triangle equals 180.

• If the parallel postulate is not true, that means that given a line and a point not on the line, there is

A) either more than one line, or B) no line at all,

that pass through the given point and are parallel to the given line.

• How can this be possible? • How would a world without parallel lines

would look like?

Remember that points, lines, and planes are undefined terms. Their meaning comes only from postulates. So if you change the postulates you can change the meaning of points, lines, and planes, and how they interact with each other… and a proposition that is “false” from the point of vies of a set of axioms becomes “true” is the point of view is changed

• That is to say that a “true statement” always depend on the context! To me, that thing in the picture may look like a broken vase, ready for the garbage can…

• … but instead it is a piece of art at the Brooklyn museum.

• For example, is this sentence true or false?

• “Today is September 11, 2015”

“Today is Sept. 11, 2015”This sentence is false in

our calendar (the Gregorian calendar, the system in use from 1582 till the present) because today is Sept. 24, but is true according to the Julian calendar (the system in use up till 1582) (See the calendar’s converter)

So to summarize…

what is “true” and what is “false” depends on our perception, or our assumptions. In Mathematics, “true propositions” are: a set of axioms or postulates, that we assume to be “true”, and their logical consequences.

So, we cannot ask whether the 5th postulate is true or false. Instead we should ask:

*Is the parallel postulate really necessary in geometry? That is, can we develop a geometry without parallel?

*Do Euclid’s axioms describe the structure of the universe? * Do parallel line really exist in nature?

The first question was answered only in the 18th century AD, the when non-Euclidean geometries took shape.

The other two questions were answered in the 20th century AD, with the theory of relativity

The first question was answered only in the 18th century AD, the when non-Euclidean geometries took shape.

The other two questions were answered in the 20th century AD, with the theory of relativity