Geometry & Euclid’s definitions axioms and postulates

Geometry & Euclid’s definitions axioms and postulates

Geometry

The word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Geometry appears to have originated from the need for measuring land. This branch of mathematics was studied in various forms in every ancient civilization, be it in Egypt, Babylonia, China, India, Greece, the Incas, etc. The people of these civilizations faced several practical problems which required the development of geometry in various ways. For example, whenever the river Nile overflowed, it wiped out the boundaries between the adjoining fields of different land owners. After such flooding, these boundaries had to be redrawn. For this purpose, the Egyptians developed a number of geometric techniques and rules for calculating simple areas and also for doing simple constructions. The knowledge of geometry was also used by them for computing volumes of granaries, and for constructing canals and pyramids. They also knew the correct formula to find the volume of a truncated pyramid. You know that a pyramid is a solid figure, the base of which is a triangle, or square, or some other polygon, and its side faces are triangles converging to a point at the top.

Truncated Pyramid

In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc. show that the Indus Valley Civilization (about 3000 BC) made extensive use of geometry. It was a highly organized society. The cities were highly developed and very well planned. For example, the roads were parallel to each other and there was an underground drainage system. The houses had many rooms of different types. This shows that the town dwellers were skilled in mensuration and practical arithmetic. The bricks used for constructions were kiln fired and the ratio length : breadth : thickness, of the bricks was found to be 4 : 2 : 1.

Thales 
Thales 

In ancient India, the Sulbasutras (800 BC to 500 BC) were the manuals of geometrical constructions. The geometry of the Vedic period originated with the construction of altars (or vedas) and fireplaces for performing Vedic rites. The location of the sacred fires had to be in accordance to the clearly laid down instructions about their shapes and areas, if they were to be effective instruments. Square and circular altars were used for household rituals, while altars whose shapes were combinations of rectangles, triangles and trapeziums were required for public worship. The sriyantra (given in the Atharvaveda) consists of nine interwoven isosceles triangles. These triangles are arranged in such a way that they produce 43 subsidiary triangles. Though accurate geometric methods were used for the constructions of altars, the principles behind them were not discussed.
These examples show that geometry was being developed and applied everywhere in the world. But this was happening in an unsystematic manner. What is interesting about these developments of geometry in the ancient world is that they were passed on from one generation to the next, either orally or through palm leaf messages, or by other ways.

Also, we find that in some civilizations like Babylonia, geometry remained a very practical oriented discipline,

Pythagoras
Pythagoras

as was the case in India and Rome. The geometry developed by Egyptians mainly consisted of the statements of results. There were no general rules of the procedure. In fact, Babylonians and Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science. But in civilizations like Greece, the emphasis was on the reasoning behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reason

ing. A Greek mathematician, Thales is credited with giving the first known proof. This proof was of the statement that a c

ircle is bisected (i.e., cut into two equal parts) by its diameter. One of Thales’ most famous pupils was Pythagoras (572 BC). Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued till 300 BC. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise, called ‘Elements’. He divided the ‘Elements’ into thirteen chapters, each called a book. These books influenced the whole world’s understanding of geometry for generations to come.

Euclid’s definitions axioms and postulates

Definitions

The Greek mathematicians of Euclid’s time thought of geometry as an abstract model of the world in which they lived. The notions of point, line, plane (or surface) and so on were derived from what was seen around them. From studies of the space and solids in the space around them, an abstract geometrical notion of a solid object was developed. A solid has shape, size, position, and can be moved from one place to another. Its boundaries are called surfaces. They separate one part of the space from another, and are said to have no thickness. The boundaries of the surfaces are curves or straight lines.

These lines end in points. Consider the three steps from solids to points (solids-surfaces-lines-points). In each step we lose one extension, also called a dimension. So, a solid has three dimensions, a surface has two, a line has one and a point has none. Euclid summarized these statements as definitions. He began his exposition by listing 23 definitions in Book 1 of the ‘Elements’.

A few of them are given below :

1. A point is that which has no part.

2. A line is breathless length.

3. The ends of a line are points.

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

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

6. The edges of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

We should note certain things.

1. Euclid seems to define a point twice (definitions 1 and 3) and a line twice (definitions 2 and 4). This is rather strange.
2. Euclid never makes use of the definitions and never refers to them in his further definations.
3. Some concepts are never defined. For example there is no notion of ordering the points on a line, so the idea that one point is between two others is never defined, but of course it is used.
4. In Book V of The Elements considers magnitudes and the theory of proportion of magnitudes. However Euclid leaves the concept of magnitude undefined and this appears to modern readers as though Euclid has failed to set up magnitudes with the rigour for which he is famed.
5. When Euclid introduces magnitudes and numbers he gives some definitions but no postulates or common notions. For example one might expect Euclid to postulate a + b = b + a, (a + b) + c = a + (b + c), etc., but he does not.
6. When Euclid introduces numbers in Book VII he does make a definition rather similar to the basic ones at the beginning of Book I: A unit is that by virtue of which each of the things that exist are called one.

Axioms

Starting with his definitions, Euclid assumed certain properties, which were not to be proved. These assumptions are actually obvious universal truths. He divided them into two types: axioms and postulates. He used the term ‘postulate’ for the assumptions that were specific to geometry. Common notions (often called axioms), on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry.
“Axiom” is from Greek axíôma, “worthy.” An axiom is in some sense thought to be strongly self-evident.
Euclid’s axioms are:
(1)Things which are equal to the same thing are equal to one another.

(2) If equals are added to equals, the wholes are equal.

(3) If equals are 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.

(6) Things which are double of the same things are equal to one another.

(7) Things which are halves of the same things are equal to one another.

These ‘common notions’ refer to magnitudes of some kind. The first common notion could be applied to plane figures. For example, if an area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square. Magnitudes of the same kind can be compared and added, but magnitudes of different kinds cannot be compared. For example, a line cannot be added to a rectangle, nor can an angle be compared to a pentagon.
The 4th axiom given above seems to say that if two things are identical (that is, they are the same), then they are equal. In other words, everything equals itself. It is the justification of the principle of superposition.
Axiom (5) gives us the definition of ‘greater than’. For example, if a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C. Symbolically, A > B means that there is some C such that A = B + C.

Postulates

Euclid’s five postulates are:
Postulate 1 : A straight line may be drawn from any one point to any other point.                                                                                                  *Note- This postulate tells us that at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line. However, in his work, Euclid has frequently assumed, without mentioning, that there is a unique line joining two distinct point.

Postulate 2 : A terminated line can be produced indefinitely.         *Note- What we call a line segment now-a-days is what Euclid called a terminated line. So, according to the present day terms, the second postulate says that a line segment can be extended on either side to form a line.

Postulate 3: A circle can be drawn with any center and any radius.

Postulate 4: All right angles are equal to one another.

Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Modern version of Euclid’s geometry

In modern terminology, angles are normally measured in degrees or radians.
Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length). Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as “if the line is extended to a sufficient length,” although he occasionally referred to “infinite lines.” A “line” in Euclid could be either straight or curved, and he used the more specific term “straight line” when necessary.

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