## Euclid’s First Postulate: a Line Segment between Points

Euclid’s first postulate states that any two points can be joined by a straight line segment. It does *not* say that there is only one such line; it merely says that a straight line can be drawn between any two points.

## Euclid’s Second Postulate: Extend a Straight Line

Euclid’s second postulate allows that line segment to be extended farther in that same direction, so that it can reach any required distance. This could result in an infinitely long line.

## Euclid’s Third Postulate is Central to Circles

The third postulate starts with an arbitrary line segment, and an arbitrary point, which is not necessarily on the line segment. First, use the compass to note the end points of the line segment, then, put the sharp spike of the compass on the arbitrary point, and finally, draw the circle with the same radius as the line segment.

## Euclid’s Fourth Postulate: All Right Angles are Equal

Euclid was probably thinking of right angles as made by constructing one line perpendicular to another. Any two such right angles are “equal” to one another.

Euclid’s fourth postulate states that, “if x and y are both right angles, then x=y.”

This may be more profound if the angles are oriented differently: opening to the left or right, up or down, or towards some other direction.

Euclid did not measure angles in degrees or radians, and he did not use a protractor. Instead, he usually discusses “how many angles in a diagram add up to some number of right angles.” For example, in Book I, Proposition 13 basically states that when a straight line “stands on” another straight base line, the sum of the two angles on the base line adds up to two right angles.

## Euclid’s Fifth Postulate: the Parallel Postulate

Euclid’s fifth postulate is the longest, and is now called the “parallel postulate.”

The fifth postulate has been the subject of much debate and labour over the centuries -can it be proven from the other postulates and axioms? Eventually mathematicians realized that the fifth posulate defines plane geometry, the geometry for a flat surface, and it cannot be derived from the other Euclidean axioms.

This postulate’s explanation needs diagrams.

Consider this parallogram, with the interior angles on the base marked in green. The sum of those two interior angles is 180 degrees. The two blue slanted lines are parallel; they will never meet even if extended infinitely far.

Anshuman says

It was really helpfull.

5th postulate was hard to understant

Constantine Roussos says

I think we can and should remove the word “infinite” from explanations in plane Geometry.. For example, instead of saying “they will never meet even if extended infinitely far” we should say “they will never meet no matter how far they are extended”. The concept of infinitely is not needed for plane geometry and its definition (if it exists) is beyond the scope of plane geometry.

dfggf says

that is really clever

ravi raj says

i think 5 postulate is not easy to understand

rita patel says

i think that greg is absoulutely correct , no doubt!

Greg Grant says

I don’t think what you’re saying is correct regarding Postulate 3. If Euclid allowed such an operation then he would not have to prove Proposition 2 in such a complicated manner. Prop 2 states “To place a straight line equal to a given straight line with one end at a given point.” If he could just measure the length of the line and draw a circle of that radius around the point then this would be trivial, but he doesn’t do it that way, he in fact invokes a quite complicated argument to prove this Proposition.

Therefore I think the right way to interpret Postulate 3 is that given any two points, one can draw a circle with center equal to one of the points and passing through the other point. Prior to proving a length can be duplicated, it is exactly this way that he draws circles – there are always two points.

rita patel says

yes, greg is right!!!!!