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## Why was the parallel postulate controversial?

Controversy. **Because it is so non-elegant**, mathematicians for centuries have been trying to prove it. Many great thinkers such as Aristotle attempted to use non-rigorous geometrical proofs to prove it, but they always used the postulate itself in the proving.

## Who proved the parallel postulate?

Ibn al-Haytham

**Ibn al-Haytham** (Alhazen) (965-1039), an Arab mathematician, made an attempt at proving the parallel postulate using a proof by contradiction, in the course of which he introduced the concept of motion and transformation into geometry.

## What is the significance of Euclid’s parallel postulate?

It states that **through any given point not on a line there passes exactly one line parallel to that line in the same plane**. Unlike Euclid’s other four postulates, it never seemed entirely self-evident, as attested by efforts to prove it through the centuries.

## Why was mathematics important to thinking of Descartes?

According to the present interpretation, Descartes relies upon mathematical reasoning **to explicate the concept of infinity**, which is essentially mathematical.

## What is the negation of Euclidean parallel postulate?

2 The **Hyperbolic Parallel Postulate** is equivalent to the negation of the Euclidean Parallel Postulate.

## How do we know that the Euclidean parallel postulate is independent of other axioms?

We have seen that the Euclidean parallel postulate is independent of the incidence axioms by **exhibiting three-point and five-point models of incidence geometry that are not Euclidean**.

## Does Euclid’s fifth postulate imply existence of parallel lines?

**Yes.** **Euclid’s fifth postulate imply the existence of the parallel lines**. According to Euclid’s fifth postulate when a line x falls on a line y and z such that ∠1+ ∠2< 180°. Then, line y and line z on producing further will meet in the side of ∠1 arid ∠2 which is less than 180°.

## What are the consequences of the parallel postulate?

One consequence of the Euclidean Parallel Postulate is the well- known fact that **the sum of the interior angles of a triangle in Euclidean geometry is constant whatever the shape of the triangle**. 2.2. 1 Theorem. In Euclidean geometry the sum of the interior angles of any triangle is always 180°.

## What are Euclid’s postulates?

Euclid’s postulates were : **Postulate 1 : A straight line may be drawn from any one point to any other point.** Postulate 2 :A terminated line can be produced indefinitely. Postulate 3 : A circle can be drawn with any centre and any radius. Postulate 4 : All right angles are equal to one another.

## What is another name for the parallel postulate?

The parallel postulate is equivalent to the **equidistance postulate, Playfair’s axiom, Proclus’ axiom, the triangle postulate, and the Pythagorean theorem**. There is also a single parallel axiom in Hilbert’s axioms which is equivalent to Euclid’s parallel postulate.