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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.)

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  • $\begingroup$ Here is a reference: from quora.com/What-is-third-proportional-of-12-and-18: "The third proportional of two numbers a and b is defined to be that number c such that a : b = b : c." $\endgroup$
    – MasB
    Jan 28, 2015 at 22:40
  • $\begingroup$ @BernardMassé: It doesn't answer my question, which is: How do we know that it is a:b=b:c and not b:a=a:c? $\endgroup$
    – user45220
    Jan 28, 2015 at 22:55
  • $\begingroup$ Obviously the value of c is a function of the ordered pair (a,b), not of the unordered pair {a,b}. So we have a function f((a,b))= $b^2/a$ and someone calls it the third proportional,I would just call it $b^2/a$. The doubled brackets "f((a,b))" are not a typo.I just wish to emphasize that the ordered pair (a,b) is an object in the domain of the function f. It is usual to write f(x) when x belongs to dom(f) but everyone writes f(a.b), not f((a,b)). Including me. $\endgroup$ Sep 2, 2015 at 5:42

2 Answers 2

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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 $b\times (b/a)^{n-2}$.

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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.

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  • $\begingroup$ Sorry, this also doesn't answer my question. However, I have figured it out on my own, thankfully. 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. $\endgroup$
    – user45220
    Jan 28, 2015 at 22:57

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