Trying to understand the contradiction in the data of Proust/Dalton/Berzelius vs Gay-Lussac I am trying to understand some of the historical context behind the discovery of atoms (pre-Avogadro).  In particular, I am getting stuck in reading the following documents:

http://physics.unl.edu/~klee/phys261/readings/class09.pdf
http://physics.unl.edu/~klee/phys261/readings/class11.pdf

My understanding is that "atoms" (as they understood them at the time) were discovered to react in whole-number ratios by Proust/Dalton/Berzelius (who measured by mass) and by Gay-Lussac (who measured by volume).  While this was highly suggestive of some sort of "atom" concept, there was a discrepancy between the two sets of measurements.  But I'm having trouble following the specific discrepancy pointed out by the above PDFs.

The first PDF gives the following mass data:

                         Mass of Nitrogen    Mass of Oxygen
Nitrous Oxide (N₂O)             100 g             58 g
Nitric Oxide (NO)               100 g            127 g
Nitrogen Dioxide (NO₂)          100 g            239 g


The text doesn't explicitly mention how this data was interpreted by Proust/Dalton/Berzelius.  Looking at the data, I don't see that much can really be said:

? N + 1 O = ? Nitrous Oxide (N₂O)
? N + 2 O = ? Nitric Oxide (NO) 
? N + 4 O = ? Nitrogen Dioxide (NO₂)


The question marks are there since they didn't know what was "in" 100 g of nitrogen, which would mean that they didn't really know what was in the product, either, right?

So that's my first question:  Am I correct in thinking that the above is about as far as Proust/Dalton/Berzelius could go with that data?  Or were they pretty sure they knew how to fill in the question marks?

Now, continuing on to the second PDF, when Gay-Lussac measured by volume, he had:

                        Volume Nitrogen  Volume Oxygen  Volume Product 
Nitrous Oxide (N₂O)             100           49.5             100
Nitric Oxide (NO)               100          108.9             200
Nitrogen Dioxide (NO₂)          100          204.7             200


Now, the document goes on to say that the above data might suggest the following formulas:

2 N + 1 O = Nitrous Oxide (N₂O)
1 N + 1 O = Nitric Oxide (NO)
1 N + 2 O = Nitrogen Dioxide (NO₂)


The text then says that the equation for nitrous oxide represents a contradiction.

But I don't see how that contradiction follows from the data.  As the text suggests, if we say that a volume of "50" represents one "atom" (so that volume of "100" nitrogen represents two atoms), wouldn't we say:

2 N + 1 O = 2 Nitrous Oxide (N₂O)
2 N + 2 O = 4 Nitric Oxide (NO)
2 N + 4 O = 4 Nitrogen Dioxide (NO₂)


Wouldn't that mean really it's the other two equations that don't make sense based on the number of atoms not working out?  I feel like I'm missing a basic idea in the text.

(FYI, I am not in school and so I'm not a student, nor is this homework.)
In regards to your first question, I would think that Dalton would have a sense of the molecules' formulas based on the information at hand. Not only did Dalton have the mass ratios for these compounds, but he had also developed a measure of the relative mass of the atoms, as shown in your first document. Given this information, it would not have been difficult to compute that there were $\sim2$ atoms of nitrogen for every one of oxygen in nitrous oxide, nitric oxide was one to one, etc.

In regards to your second question, I think you are misinterpreting the contradiction. The authors proposal for the reactions was that if the compound formed was $\ce{N2O}$, it should only form one volume, since it should only form as much volume as there was volume of $\ce{O}$ originally. This is because there can only be as many molecules formed as there were atoms of the limiting reactant. The two volumes of $\ce{N2O}$ that are actually produced runs counter to this, as it would seem to require having $2$ volumes of $\ce{O}$ to start with.  

For all the reactions, the products and reactants were related by whole number ratios, but not the same ratios suggested by their predicted molecular formulas.