What is the significance of Law of Multiple Proportions? 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?
These laws all date back from the very early days of chemistry and are only taught at school to give pupils a way to grasp parts of the subject more easily. Except for the law of conservation of mass, which some professors/supervisors love stating frequently, they are hardly mentioned in universities outside of science history classes.

That said, as I understood them, they refer to slightly different cases. The law of definite proportions should be understood to read:


  No matter how much mass of two elements will react with each other, the product will contain the masses of its elements in a certain ratio and this ratio only.


Taking an easy example, if I reacted $46~\mathrm{g}$ of sodium and $16~\mathrm{g}$ of oxygen, I would get $62~\mathrm{g}$ of $\ce{Na2O}$ where the ratio of sodium to oxygen is $46:16$. If I had instead reacted $92~\mathrm{g}$ of sodium with an unlimited amount of oxygen, I would have gotten $124~\mathrm{g}$ of the same product which contains sodium and oxygen in a $46:16$ ratio. No matter how much sodium I let react with how much oxygen, the product will contain the definite proportion of sodium to oxygen $46:16$.*

The law of multiple proportions now states that some elements are special. For example, iron can react with chlorine to give two different chlorides: $\ce{FeCl2}$ and $\ce{FeCl3}$. For each of the two you can, as you realised set up a law of definite proportions which often gives you a rather weird rational ratio. However, when comparing those two with each other, you find that the mass of chlorine in $\ce{FeCl3}$ is 1.5 times as much as the mass of chlorine in $\ce{FeCl2}$ — corresponding to a $3:2$ ratio. And it is these very small (usually single digit) whole numbers that are actually meant.

Taking a different example, the relative atomic masses of sulphur and chlorine are $32.07$ and $35.45$, respectively. So a sulphur dichloride would have mass ratios of $32.07:70.9$ — far from any small number, but still a rational ratio. Sulphur tetrachloride exists, too, with another pretty unclean ratio. However, the ratio of the chloride contents is $1:2$ — isn’t that a nice simple ratio?



*: This example is not technically correct. Burning sodium gives $\ce{Na2O2}$, sodium peroxide. But I wanted nice and whole numbers and oxygen and sodium have nice and whole mass numbers.