Quotient of Rationally Expressible Numbers is Rational

Proof

 * Euclid-X-20.png

Let the rational area $AC$ be applied to the rational straight line $AB$ such that $BC$ is its breadth.

Let the square $AD$ be described on $AB$.

From, $AD$ is rational.

But $AC$ is also rational.

Therefore $DA$ is commensurable in length with $AC$.

From Areas of Triangles and Parallelograms Proportional to Base:
 * $BD : BC = DA : AC$

Therefore from Commensurability of Elements of Proportional Magnitudes $DB$ is also commensurable in length with $BC$.

As $DB = BA$ it follows that $AB$ is also commensurable in length with $BC$.

But $AB$ is rational.

Therefore $BC$ is rational and commensurable in length with $AB$.