Here is Lavoisier's description of the use of the big round flask in the portrait by David (from Traité élémentaire de Chémie Part Three, Chapter II). Note how clearly he describes the procedure, and how he looks out for possible errors (red highlights).

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How to determine the absolute weight of different gases.

 In everything I have written on the method of measuring gas volumes and correcting them for pressure and temperature, I have supposed that one knows their specific weights and from these can derive their absolute weights : now I must give an idea of how one can arrive at this knowledge. One has a large bulb A, plate V, fig. 10, with a capacity of at least half a cubic foot, that is 17 to 18 pints; one glues to it a copper cap bcde to which is screwed a plate de with a stopcock fg. This piece is screwed using a threaded union figure 12 to a bell jar BCD the capacity of which must be several pints greater than that of the bulb. This bell jar is open on top and its neck is fitted with a copper cap hi and a stopcock lm; one of these stopcocks is illustrated separately, figure 11. The first task is to determine the volume of the bulb ; one manages this by filling it with water and weighing to measure the quantity. Then one empties the water and dries the flask by running a cloth through the opening de ; the last traces of moisture disappear spontaneously when one evacuates the bulb once or twice. When one wants to determine the weight of a gas, one screws the bulb A to the platform of the vacuum pump through the stopcock fg. One opens the stopcock and evacuates as much as possible, being very careful to observe how far the test barometer descends. Having evacuated, one closes the stopcock and weighs the bulb with scrupulous precision, after which one screws it back to the bell jar BCD, which is placed on the table with a depression ABCD, same Plate, fig. I [not shown here. Notice that he assumes you know that the gas is collected in BCD by displacing water, i.e. a tube is inserted under the lip of the bell jar to suck out the air until the jar is filled with water, then another tube bubbles the gas into the jar]. One passes into this bell jar the gas that one wishes to weigh ; then opening the stopcocks fg and lm the gas in the bell jar passes into bulb A : at the same time water climbs back into the bell jar BCD. To avoid a troublesome correction, it is necessary to sink the bell jar into the depression until the exterior level of the water matches that of the water inside the bell jar. Then one closes the stopcocks, unscrews the bulb and reweighs. The weight, reduced by that of the empty bulb, gives the weight of the volume of air or gas that it contains. Multiplying this weight by 1728 [cubic] inches and dividing the product by a number of cubic inches equal to the capacity of the bulb one has the weight of a cubic foot of the gas studied. In these determinations it is necessary to take account of the barometric pressure and temperature ; after which nothing is easier than to correct the weight of the cubic foot that one has found to what the same gas would give at 28 inches of pressure and 10 degrees of temperature. I have given details of the required calculations in the preceeding paragraph. One must not neglect the small portion of residual air in the bulb after evacuation ; the size of which is easy to reckon according to the height of the test barometer. If this height were, for example, one one-hundredth of the total barometric pressure, one must conclude that one percent of air remained in the bulb and the volume of gas introduced would be only 99/100 of the total volume of the bulb.