By analyzing the weight ratios of components in chemical reactions. Like H2 and O2 give you H2O, but it’s also possible to make H2O2 from those same components, given that you supply twice as much oxygen for the same amount of hydrogen. Why could that be? There aren’t that many possibilities to consider before you arrive at the corresponding chemical formulas for these compounds. And comparing the weight of H2 and O2 participating in the reaction you can figure out the ratio of the atomic weights of these elements. Piece by piece you’d be accumulating such information, until one day you figure out the periodic table. And after that it gets even more straightforward.
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