Introduction to Euclid's Geometry: Facts, Axioms and Postulates

Euclidean Geometry is a mathematical theory credited to Alexandrian Greek mathematician Euclid and documented in his classic ‘The Elements’. Euclid's technique involves starting with a limited collection of intuitively acceptable axioms and deducing a large number of additional propositions (theorems) from them. Euclidean Geometry, is hence, a study of different geometrical figures including planes and solids based on different postulates and theorems. Let's look at the Euclid Geometry, axioms and its postulates with a few solved examples.

Key Terms: Geometry, Measurement, Angle, Line, Circle, Algebra

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What is Euclid Geometry?

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The words ‘geometry’ and ‘metron’ are Greek words that signify ‘earth’ and ‘measurement’, respectively. Geometry arose from the necessity to measure land and was studied in various ways by many ancient civilizations, including Egypt, Babylonia, India and others. When Euclid compiled all of the notions and basics of geometry into a book called 'Elements’, Euclid's geometry came into action. The definitions, axioms, theorems, and proofs of numerous forms were discussed by Euclid in a very simplified way.

Euclid particularly addressed the shape, size, and location of solid shapes, as well as terminology such as surface, straight or curved lines, points, and so on in his geometry. The following are some of the facts postulated by him:

• There are no pieces to a point.
• A line is a length that has no width.
• A line’s terminal position is called point.
• A straight line is one that has all of its points in the same place.
• A surface has only two dimensions: length and width.
• Lines define the margins of a surface.
• A planar surface is one that has straight lines all the way around it.

Angles and circles are two classic Euclidean geometry examples. An angle is the degree of inclination between two straight lines. A circle is a planar form with all of its points at the same distance from the centre (called the radius). Euclid knew that a thorough study of geometry must begin with the fundamentals. He then defined the concepts like angles, circles, triangles, and several other polygons and figures based on these terminology.

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Postulates of Euclid Geometry

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There are a few words that we need to understand before we can examine Euclid's postulate. Solids-surface-lines-points is a three-step procedure described by Euclid from solids to points. Each step removes one dimension from the form. As a result, a solid is a three-dimensional form, a surface is two-dimensional, a line is one-dimensional, and points are dimensions. The term surface refers to something that merely has length and width.

Let's have a look at the five Euclid's postulates:

Postulate 1: A straight line can be drawn from any one point to another point

This postulate proves that at least one straight line goes through two separate points, but it does not rule out the possibility of additional. Take a look at the line below; only one line crosses through P and Q, and that line is PQ, which passes through both Q and P.

Postulate 2: A terminated line can be further produced indefinitely.

The second postulate states that a line segment can be stretched in any direction to produce a line.

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

A circle is a planar figure that may be drawn with its centre and radius and consists of a group of points that are equidistant from a reference point. The third postulate states that when the radius of a circle changes, the form of the circle does not change.

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

Every angle that is right angled or 90 degrees is the same as the one before it. Regardless of the lengths of their arms, a right-angle measures exactly 90 degrees. As a result, according to postulate 4, all right angles are equal. This only applies to right-angled triangles.

Postulate 5: If a straight line falling on two other 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 the side on which the sum of angles is less than two right angles.

When the internal angles of two lines cut by a third line total less than 180°, the two lines will intersect when extended on that side. In the illustration below,∠1 + ∠2 < 180^o ∠1 + ∠2 < 180^o. As a result, when Lines m and n are extended on the side of 1 and 2, they will meet.

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Euclid’s Axioms

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In his book Elements, Euclid included a few axioms or common notions regarding geometric forms.

Axiom 1: Things which are equal to the same thing are equal to one another.

Assume that a rectangle's area is equal to a triangle's area, which is equal to a square's area. After using the first postulate, we can say that the area of the triangle and the square are equal.

For example, if p = q and q = r, we can say p = r.

Axiom 2: If equals are added to equals, the wholes are equal.

Consider the line segment AB, where AP equals QB. When PQ is applied to both sides, AP + PQ = QB + PQ, i.e. AQ = PB, according to axiom 2.

Axiom 3: If equals are subtracted from equals, the remainder are equal.

Consider the rectangular shapes ABCD and PQRS, both of which have the same size. If the triangle XYZ is eliminated from both rectangles, the areas of the remaining sections of the two triangles are identical, according to postulate 3.

Axiom 4: Things which coincide with one another are equal to one another.

Consider the line segment AB, which has C in the middle. The line segment AB is intersected by AC + CB. As a result of axiom 4, we may deduce that AC + CB = AB.

Axiom 5: The whole is greater than the part.

AC is a portion of AB, as seen in the diagram above. As a result of axiom 5, we may conclude that AB > AC.

Axioms 6: Things which are double of the same things are equal to one another.

Axiom 7: Things which are halves of the same things are equal to one another.

Consider two identical circles with diameters and radii of and, respectively. We may assert that = and = since the circles are similar considering both axioms 6 and 7.

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Non- Euclidean Geometry

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Non-Euclidean geometry is a sub-discipline of geometry. Everything that does not come within Euclidean geometry is referred to as non-Euclidean geometry. It is frequently used to describe spherical and hyperbolic geometry. Because spherical geometry is non-euclidean, we must adjust real lengths, point locations, region area, and actual angles to convert it to euclidean or Euclid's geometry or fundamental geometry.

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Things to Remember

• Euclidean geometry, sometimes known as parabolic geometry, is a discipline of mathematics based on Euclid's five postulates and a set of assumptions.
• Euclidean geometry is divided into two types: plane geometry (two-dimensional Euclidean geometry) and solid geometry (three-dimensional Euclidean geometry).
• The most fundamental words in geometry are a point, a line, and a plane.
• A point does not have any dimensions (length or breadth), but it does have a location.
• A line is a straight line that extends forever in both directions.
• A plane is a flat, endlessly extending surface that is level.

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