What is a third proportional?

Question

I searched online, couldn't find anything clear.

If I had two numbers, $a,b$, what is their third proportional? Apparently it can be either $c$ such that $a/b=b/c$ or $b/a=a/c$, but obviously these $c$'s aren't equal in general. So is a third proportional indeterminate?

For example, given $5,12$, what is their third proportional. How do I know which one of $5/12=12/c$ and $12/5=5/c$ is the correct one?

This is so frustrating, I can't believe online sources can be so unclear. How can I check like 20 websites on third proportionals and still not find an answer to my question? Ugh. (Sorry.)

Answer 1

A continued proportion is a pair of equations like:

$$\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$$

A fourth proportional is a solution of the equation

$$\frac{a}{b}=\frac{c}{d}$$

where three of the numbers are known and the other is the unknown. There are several cases.

As mean proportional is a solution of either

$$\frac{a}{b}=\frac{b}{c}\ \ \ \ \text{ or }\ \ \ \ \frac{b}{c}=\frac{c}{d}$$

where the unknown is the repeated variable.

A third proportional is a solution to either

$$\frac{a}{b}=\frac{b}{c}\ \ \ \ \text{ or }\ \ \ \ \frac{b}{c}=\frac{c}{d}$$ where the unknown is the numerator of the left-hand side or the denominator of the right-hand side.

Answer 2

I figured it out on my own, and since none of the answers answer my question I will just post mine.

The third proportion to $a$ and $b$ is defined as the number $x$ such that $a,b,x$ are in a geometric progression. If the question was instead the third proportion to $b,a$, then it would be the number $x$ such that $b,a,x$ are in geometric progression. Then it's easy to find in both cases, e.g. in the first one, since the geometric ratio is constant, it must be $b/a$, then we simply multiply this by b to get $x=b^2/a$.

In general, the $n$th proportional of $a,b$ is defined as $b\times (b/a)^{n-2}$.