There are methods to add two lines of arbitrary lengths or multiply them together known since Greek times; and more advanced methods based on the concepts of bases and units.
But, I have not been able to find a way to exponentiate a number geometrically without using algebra. I would love if someone could somehow illustrate the concept.
Basically I am asking is it possible to draw the graph of a^x geometrically.
On questions raised by Aretino and RickyDemer I want to clarify that: I am talking about Euclidean geometry (so a collapsible compass,straight-edge are allowed); although, Cartesian geometry is fine, too.
Also, is there a book that can teach a basic concept as this? You know, a book on Euclidean geometry that teaches exponentiation, multiplication etc.
Let $AB = 1$, $AD=a$. We draw another line at any angle with $AB$, mark a point C on it such that $AC=a$. Let a line through $D$ parallel to $BC$ meet this line at $E$
, then $AE=a^2$, continuing this way we can raise it to any integer power.
Using this you can only calculate negative integral powers as well. Calculation of square roots is simple so we can construct all numbers of form $a^\frac{n}{2}$. But we cannot go cube roots or anything as using scale and compass we are limited to square roots.