Duhem and atomism

I need help understanding Duhem's comments on the "law of multiple proportions". His statement is this:

"The masses of bodies A, B, and C combining to form the compound M are to one another as the three numbers a, b, and c. Then the masses of the elements A, B, and C combining to form the compound M' will be to one another as the numbers xa, yb, and zc (x, y, and z being three whole numbers). ... Now, in whatever relations the elements A, B, and C are combined within the compound M', we can always represent these relations with as close an approximation as you please, by the mutual relations of three products xa, yb, and zc ... in other words, whatever the results given by the chemical analysis of the compound M', we are always sure to find three integers x, y, and z."

What he claims is that this does not verify the law of multiple proportions. I'm missing something. I understand what he's going after in a similar argument about the law of gravitation, but he lost me here.

He makes the experiment sound arbitrary, as if we can just pick any integers at random and get the answer we expect, but I don't see how that can be.

Is anyone familiar enough with this argument that they can explain it to me? Mechanics is my thing, not chemistry. So, I'm probably being a bit thick. I'd appreciate any help.