Does the notation $r^3\propto t^2$ mean the same as $t^2\propto r^3$?

Question

I was recently reading about Kepler's third law of planetary motion. There in two books I saw two different things. In one place it is written $r^3\propto t^2$ and in the other book it as written $t^2\propto r^3$.

Now to see if both was the same thing I did this with an example:

Thus we see that $B=2A$ and thus $B\propto A$

Also we see A=$\frac12$B and thus $A\propto B$. Thus we see that A $\propto B$ is the same as $B \propto A$.

But how can this be? If $A\propto B$ then A=kB and K= $\frac AB$ and if $B\propto A$ then B=kA and $k= \frac {B}{A}$. Thus the value of the constant changes.

I am confused about this. Please help and tell me whether $r^3\propto t^2$ is same as $t^2\propto r^3$.

Answer 1

$A\propto B$ tells us that as $A$ varies, $B$ also varies in such a way that the ratio of $A$ to $B$ never changes. But $A\propto B$ says nothing about the value of that ratio. For example, suppose $x = 2y$ and $y = 3z$. The statement $x \propto y$ says the ratio of $x$ to $y$ is fixed and unchanging. This is true. The ratio of $y$ to $z$ also is fixed and unchanging. It is fixed at a different value than the ratio of $x$ to $y$, but it is still fixed. Therefore $y \propto z$.

This works just as well when the two ratios are $2$ and $\frac12$ as it does when the ratios are $2$ and $3$.

Another way to look at this: let $A\propto B$, so that $A$ and $B$ vary in such a way that the ratio of $A$ to $B$ never changes. If the ratio of $A$ to $B$ never changes, the ratio of $B$ to $A$ cannot change. Therefore $B \propto A$.

So in fact it really does not matter whether you write $A\propto B$ or $B \propto A$. They mean exactly the same thing.

Answer 2

$A\propto B$ means that there exists a nonzero constant $k$ with $A=kB$. $B\propto A$ means that there exists a nonzero constant $k'$ with $B=k'A$ (we are forced to call it $k'$ this time because it certainly is allowed to be a different constant than the $k$ from the first sentence; after all the name of the constant does not matter). But (using $k\ne 0$ and $k'\ne 0$) by simple reordering of the equation we have $A=kB\iff B=\frac1k A$ as well as $B=k'a\iff A=\frac1{k'}B$. Hence, as we can choose $k'=\frac 1k$ if we are givren $k$, or vice versa pick $k=\frac1{k'}$ if we are given $k'$, we conclude $$ A\propto B\iff B\propto A.$$

Maybe compare with this: An integer $n$ is called even if there eixts an integer $k$ with $n=2k$. So is $42$ even? Yes, because we can pick $k=21$ and then have $42=2\cdot 31$. And is $100$ even? Yes, because we can pick $k=50$ and then have $100=2\cdot 50$. Here, too, we have "Thus the value of the constant changes." without that it matters.