# Definition:That which produces Medial Whole with Rational Area

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## Definition

Let $a, b \in \R_{>0}$ be in the forms:

- $a = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} + k}$
- $b = \dfrac \rho {\sqrt {2 \left({1 + k^2}\right)} } \sqrt{\sqrt {1 + k^2} - k}$

where:

- $\rho$ is a rational number
- $k$ is a rational number whose square root is irrational
- $\lambda$ is a rational number whose square root is irrational.

Then $a - b$ is **that which produces a medial whole with a rational area**.

In the words of Euclid:

*If from a straight line there be subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational; and let it be called***that which produces with a rational area a medial whole***.*

(*The Elements*: Book $\text{X}$: Proposition $77$)

## Terms

The **terms** of $a - b$ are the elements $a$ and $b$.

### Whole

The real number $a$ is called the **whole** of the straight line which produces with a rational area a medial whole.

### Annex

The real number $b$ is called the **annex** of the straight line which produces with a rational area a medial whole.

## Also known as

This can also be described as **that which produces with a rational area a medial whole**.

And in answer to your next question: no, there isn't.