I cannot understand how the Law of Multiple Proportions is significant, or how does it further improve over Law of Conservation of Mass or Law of Definite Proportions. From Wikipedia:
If two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will be ratios of small whole numbers.
Now I must say that I don't understand anything from the "small whole numbers" part, since it is not rigorously defined. For example, $31454315/43546546$ may be a ratio of small whole numbers for me but not for you.
Nevertheless the following is the algebraic interpretation of the law, with substituting the "ratios of small whole numbers" part with: "a real constant", since the ratios of two whole numbers is always a real constant.
Compound 1: Has $m$ mass of component $A$ and $x$ mass of component $B$.
Compound 2: Has $m$ mass of component $A$ and $y$ mass of component $B$.
By the Law of Definite Proportions, the following are correct:
$x/m = c_1$
$y/m = c_2$
where $c_1$ and $c_2$ are real constants. This is because Law of Definite Proportions states that "a chemical compound always contains exactly the same proportion of elements by mass".
Dividing $x/m$ by $y/m$ gives:
$x/y = c_1/c_2$
Since the ratio of two real constants is another real constant, we can express this as:
$x/y = c_3$
Since a real constant can always be expressed as the ratio of two whole numbers, this gave us the Law of Multiple Proportions.
As I have shown above, the Law of Multiple Proportions is a corollary of the Law of Definite Proportions.
So again: What is the significance of the Law of Multiple Proportions? Isn't it just an application of the Law of Definite Proportions? How does Law of Multiple Proportions improve over Law of Definite Proportions and Law of Conservation of Mass? How does it contribute to the first scientific atomic theory?