Euclid's View (2)

 

In Euclid's View (1) it was noted that Euclid's text (ELEMENTS) set out a coherent and logical framework for Geometry as he understood it. The ELEMENTS is itself based on the works of other mathematicians and philosophers - Plato, Aristotle, Eudoxus, Thales, Hippocrates and Pythagoras.

 

*** Euclid's Axioms ***

(or Common notions)

Statements accepted as true and apply to all branches of mathematics

 

Common notion 1.

 

Things which equal the same thing also equal one another. 

 

If A = B and B = C then A = C 

This is a transitive property

Common notion 2.

 

If equals are added to equals, then the wholes are equal. 

If A = B then A + C = B + C

Common notion 3.

 

If equals are subtracted from equals, then the remainders are equal. 

If A = B then A - C = B - C

Common notion 4.

 

Things which coincide with one another equal one another. 

 This is a reflexive property

Common notion 5.

 

The whole is greater than the part. 

 

The Common Notions are common to all branches of mathematics and are used to regulate all forms of logical reasoning. This is what Euclid’s word common means - common to all the branches of science and mathematics. For further discussion on this see Euclid's Common Notions

 

*** Definitions ***

23 definitions and these are set out fully at Euclid's Elements Book 1. 

The first 4 are:

Definition 1.

 

A point is that which has no part. Essentially, this means that a point is simply a position.

Definition 2.

 

A line is breadthless length (or, a line has length but no width)

Definition 3.

 

The ends of a line are points.

Definition 4.

 

A straight line is a line which lies evenly with the points on itself. 

 

 

*** POSTULATES ***

Postulate 1.

Given any two points such as A and B, there is a line AB which has them as endpoints. This is one of the constructions that may be done with a straightedge (the other being described in the next postulate).

Postulate 2.

 

Here we have the second ability of a straightedge, namely, to extend a given line AB. 

 

Postulate 3.

 

To describe a circle with any center and radius. This can be done with a straightedge and compass.

Postulate 4.

 

That all right angles equal one another.

Postulate 5.

 

That, if a straight line falling on two straight lines makes the interior angles on the 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.

Postulate 5 is the "Parallel Postulate" and has presented problems which led to the development of non-Euclidean geometries.

The 5th postulate was re-stated by John Playfair -

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point

(John Playfair). 

The Five Postulates apply only to Euclidean geometry. In a sense, they define Euclidean geometry—which for Euclid was the only geometry.

*** PROPOSITIONS ***

Various propositions were derived from the axioms, definitions and postulates. These are set out at Euclid's Elements Book 1. Here are the first 4 

Proposition 1.

 

To construct an equilateral triangle on a given finite straight line.

Proposition 2.

 

To place a straight line equal to a given straight line with one end at a given point.

Proposition 3.

 

To cut off from the greater of two given unequal straight lines a straight line equal to the less.

Proposition 4.

 

If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. (Essentially, this is one situation in which 2 triangles are congruent).

 

 

*** EUCLIDEAN PLANE *** 

Postulate 1 (above) tells us - Given any two points such as A and B, there is a line AB which has them as endpoints. If that line is straight (i.e. of constant direction) then the line AB lies in a plane but, at this stage, the plane is not uniquely defined. If there is a third point (C), NOT on line AB, then points A, B and C define a unique plane. 

Such a plane is referred to as a Euclidean plane. It has infinite size and contains an infinite number of points. Such a plane has dimension two (denoted denoted or ) and real numbers are required to determine the position of each point. The Euclidean plane includes the concept of parallel lines. Furthermore, the plane has metrical properties so that, for example, distance and angle measurement are possible.

 

Acknowledgment

This post fully acknowledges the excellent publication Euclid's Elements Book 1 by David E. Joyce - Department of Mathematics and Computer Science - Clark University. 

Other material

 BYJUS.COM - Euclid Elements

Euclidean Geometry - Wikipedia 

The Euclidean Plane (Wikipedia) 

Euclidean Geometry (Libre Texts) examines points, lines, angles, planes.