Mostly by weighting reactants and reaction products. Let’s say you burn hydrogen (H2) in oxygen (O2) two get water: 2 H2 + O2 = 2 H2O (definitely not the reaction they used, but it’s simpler). You carefully weigh all three of them (hydrogen and oxygen before the reaction, water after), and notice that you need to burn 4 grams of hydrogen with 32 grams of oxygen to get 36 grams of water. If you already know the equation for this particular reaction (the one I’ve written a couple sentences back), then it’s quite straightforward to deduce that oxygen atomic mass is 16 times that of hydrogen.
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