What is the difference between postulates, common notions, propositions, and definitions in Euclid’s Elements?

To drawing the straight line between the two point

In straight line to producing the limited straight line constantly

Within any radius and center to describing the circle

All the right angels are equal to one another.

Common notions

It includes all the things which are equal to the many other same thing and equal to other as well.

By adding the equals in the equals than all the item will be equal.

By subtracting the equal from equal then the reminders will be equal

 All the things which coinciding with one another it will equal to one another.

Propositions

It is used for constructing an equilateral triangle for given limited straight line

For the given straight line to placing a straight line equal at the one end of the given point.

To cutting off from the greater of two provided an unequal straight lines and this is equal to the less

Definitions

It is that point which has not any part

This line has the length of the breadth less

The ends of these lines are known as the points.

What is the method of exhaustion? How is it used in the Elements?

         The method of exhaustion is very important technique which is used in the classical Greek mathematicians for proving the results that can be dealt with means of the limits. It is considered as the one of the initial form of the integral calculus. It is also known as the methods for finding an area of the shape by scratching it inside the sequence of the polygons and its areas are converging for the area of the containing shapes.

            This method is required for proving the contradiction which is normally known as the reductio ad absurdum. This method is used for finding the area of the region by comparing it first with the area of the second region. This method is used by the Euclid exhaustion for proving the various proposition which are given in the book 12 of his Elements.  The way is provided by the method of the Exhaustion to proving it rigorously which are the volume A must be equal to volume B. By assuming this the work will proved. It will also use for the ascribing simpler areas outside the A and then deriving contradiction (Math Ubc Ca).  

What is the Euclidean algorithm? How does it work? What does it have to do with commensurability?

Euclidean algorithm is the good technique which used for quick findings of the GCD of two integers (Khan Academy Org).

For finding GCD (A,B) the Euclidean Algorithm will be follow as

This algorithm will worked as shown in the below given figure

Source: https://stackoverflow.com/questions/6005582/how-does-the-euclidean-algorithm-work

When b will be divided in to a, r will considered as the remainder, Then b will be assigned to a and the remainder will assigned to b and a will be returned.

Source: https://stackoverflow.com/questions/6005582/how-does-the-euclidean-algorithm-work