True Weight from False Balance/Unequal Arms

Theorem
Let $B$ be a body whose weight is $W$.

Let $B$ be weighed in a false balance with unequal arms.

Let the readings of the weight of $B$ be $a$ and $b$ when placed in opposite pans.

Then:
 * $W = \sqrt {a b}$

Proof
We have that the false balance has arms of different lengths.

Let the lengths of the arms of the false balance be $x$ and $y$.

, placing $B$ in the pan at the end of $x$ gives:
 * $W x = a y$

and placing $B$ in the pan at the end of $y$ gives:
 * $W y = b x$

Then: