This law was proposed by Gay-Lussac in $1808$.
He observed that when gases combine or are produced in a chemical reaction,they do so in a simple ratio by volume,provided all gases are at the same temperature and pressure.
For example,$100 \ mL$ of hydrogen combines with $50 \ mL$ of oxygen to give $100 \ mL$ of water vapour.
$\text{Hydrogen} + \text{Oxygen} \rightarrow \text{Water vapour}$
$100 \ mL$ $50 \ mL$ $100 \ mL$
Thus,the volumes of hydrogen and oxygen which combine together (i.e.,$100 \ mL$ and $50 \ mL$) bear a simple ratio of $2:1$.
Gay-Lussac's discovery of an integer ratio in volume relationships is essentially the law of definite proportions by volume,whereas the earlier law was with respect to mass.