Euclid’s first two axioms (of five), to review, defined the very notion of a straight line. While this may not seem like something altogether in need of definition, Euclid took two full steps to do it, in order to be provided a basis upon which to build the rest of his geometry (and thus, upon which future generations, including the present one, could understand geometry).
First, he said that any two points, no matter where they are situated, can technically be connected with a straight line.
Second, he said that the line connecting these two points (and any points which happen to lie in between) can be extended to infinity in either direction.
And that’s all there is to say about lines (though the principles that can be derived from these two axioms is rather astounding).
The Third Axiom
Euclid’s third axiom leaves the definition of lines behind, but not the subject altogether. He uses the idea of a line in order to define the nature of a common circle (which explains why the definition of the line came first).
Simply stated, the axiom says that, given a line segment of arbitrary length, a circle can be drawn using the endpoint of the segment as a center, and the length of the segment as the radius.
Want proof of this (as if it was necessary)?
Tie a string around a nail or pin stuck into a surface, and then tie a pencil to the other end. Pull the string tight and draw a line following the outstretched string as it moves around the circle. What should result is a nearly perfect circle (if performed right).
It is important to remember, also, that all of Euclid’s axioms and later proofs were constructed using a total of two tools – a compass and an unmarked straight edge. Now, using just these tools, Euclid has been able to construct any possible line, and any possible circle.
Using the Third Axiom
The fact that a line segment is utilized as a crucial component in the construction of the circle would be of practically unlimited value to Euclid in his later proofs. He would use his simple circle construction throughout book three (of thirteen) in Elements, providing proof after proof regarding the nature of circles and their relationship to other lines and figures.
In addition, a rather simple definition regarding the total area of a circle can be written using the idea of a line segment as the radius:
The area of a circle is the total area swept over by the circle’s radius as it rotates around the circle’s center.
This is easy enough to state, but slightly more difficult to actually calculate (this is where that pesky “π” enters into the picture, with the equation A = πr²).
And now, in the space of just three short, simple sentences, Euclid has fully defined the nature of a straight line, and of circles.
Just a couple more axioms, and he’ll have a complete, working set of axioms upon which to build his very own mathematics.
See Also:
References:
Dunham, William. “Journey Through Genius: The Great Theorems of Mathematics.” John Wiley and Sons. 1990.
Euclid. “Elements.”
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