Duhem and atomism

[Resha Caner]

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.

[Resha Caner]

I'm surprised. Duhem was new to me, but I expected someone out there had already thought through this. I ventured out to other forums to ask the question, and finally got an answer ... from one guy. Even then, he had never heard of Duhem, but just helped me figure out what was going on.

OK, here is an example of the answer in a form much simplified from Duhem's argument.

Pick a random number, say 2.84. I can always find a set of integers that will represent that within the accuracy of my measurement system. In this case, 71 and 25 work very nicely.

Now suppose I have a compound where the ratio of the elements A & B is 2.84. That does not prove the compound has 71 atoms of A and 25 atoms of B.

Of course there is other evidence to support the atomic model. I wasn't trying to refute it, so don't argue with me about it. I just wanted to understand what Duhem was saying. I get it now.