How Mnay Straight Lines Can Pass Through Two Distinct Points

Introduction

Geometry is a field of study in mathematics that deals with the effects and relations of lines, points, solids, surfaces and other dimensional analogues. A greek mathematician named Elucid is largely considered the father of geometry. His approach to geometry is now known as Elucid geometry. In Euclid's method, numerous more propositions (theorems) are derived from several important conceptual axioms (postulates). Many of Euclid's conclusions were already known, but Euclid was the first one to arrange them into a logical framework where each conclusion is supported by axioms and previously established theorems. In mathematics or logic, a generalisation is not self-evident but is demonstrated through a series of arguments; a truth that is established using established truths is called a theorem whereas an axiom is an unprovable rule or first principle accepted as true because it is self-evident or particularly useful.

Content

Now let us look at some of Elucid’s postulates which he wrote in his book titled “ELEMENTS”:

Postulate 1: We can draw a straight line from any one point to another point

According to this postulate, at minimum one straight line connects two separate places, however, he omitted to clarify that there can only be one such line. Even though the fact that he has assumed throughout his work that there is only one line that passes through two places.

Postulate 2: Any terminated line can be further reproduced indefinitely.

To put it simply, Euclid described a line segment as a terminated line described a line segment as terminated line, to put it simply. Since we can extend a finished line or a line segment in either direction to produce a line, this postulate implies that we can.

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

Any circle can be made starting from its end or beginning, and its diameter will be equal to the length of the line segment.

Postulate 4: All ninety-degree angles are equal

No matter how long the sides are or which way they are facing, all right angles (angles with a measure of 90°) are equal and always congruent with one another.

Postulate 5: If the interior angles on one side of a straight line, when added together, are less than two right angles, then the two straight lines, if created indefinitely, will intersect on the side where the sum of the interior angles is less than two right angles.

Conclusion

From the first postulate, an axiom can be derived that “given two points there is a unique line that passes through them”.This is a self-evident statement and has not been proven wrong to date.

Hence studying Elucid’s postulate and the axiom derived from it can be intuitively interrupted that only a single line can pass through two distinct points. The universality and timelessness of these ideas are the reason why many of his theories are still taught in high schools all over the world as the basics of geometry.