In 1804, Dalton explained his atomic theory to his friend and fellow chemist Thomas Thomson, who published an explanation of Daltons theory in his book A System of Chemistry in 1807. According to Thomson, Daltons idea first occurred to him when experimenting with "olefiant gas" (ethylene) and "carburetted hydrogen gas" (methane). Dalton found that "carburetted hydrogen gas" contains twice as much hydrogen per measure of carbon as "olefiant gas", and concluded that a molecule of "olefiant gas" is one carbon atom and one hydrogen atom, and a molecule of "carburetted hydrogen gas" is one carbon atom and two hydrogen atoms. In reality, an ethylene molecule has two carbon atoms and four hydrogen atoms (CH), and a methane molecule has one carbon atom and four hydrogen atoms (CH). In this particular case, Dalton was mistaken about the formulas of these compounds, and it wasnt his only mistake. But in other cases, he got their formulas right. The following examples come from Daltons own books A New System of Chemical Philosophy (in two volumes, 1808 and 1817):
Example 1 — tin oxides: Dalton identified two types of tin oxide. One is a grey powder that Dalton referred to as "the protoxide of tin", which is 88.1% tin and 11.9% oxygen. The other is a white powder which Dalton referred to as "the deutoxide of tin", which is 78.7% tin and 21.3% oxygen. Adjusting these figures, in the grey powder there is about 13.5 g of oxygen for every 100 g of tin, and in the white powder there is about 27 g of oxygen for every 100 g of tin. 13.5 and 27 form a ratio of 1:2. These compounds are known today tin(II) oxide (SnO) and tin(IV) oxide (SnO). In Daltons terminology, a "protoxide" is a molecule containing a single oxygen atom, and a "deutoxide" molecule has two. Tin oxides are actually crystals, they dont exist in molecular form.
Example 2 — iron oxides: Dalton identified two oxides of iron. There is one type of iron oxide that is a black powder which Dalton referred to as "the protoxide of iron", which is 78.1% iron and 21.9% oxygen. The other iron oxide is a red powder, which Dalton referred to as "the intermediate or red oxide of iron" which is 70.4% iron and 29.6% oxygen. Adjusting these figures, in the black powder there is about 28 g of oxygen for every 100 g of iron, and in the red powder there is about 42 g of oxygen for every 100 g of iron. 28 and 42 form a ratio of 2:3. These compounds are iron(II) oxide (FeO) and iron(III) oxide (FeO). Dalton described the "intermediate oxide" as being "2 atoms protoxide and 1 of oxygen", which adds up to two atoms of iron and three of oxygen. That averages to one and a half atoms of oxygen for every iron atom, putting it midway between a "protoxide" and a "deutoxide". As with tin oxides, iron oxides are crystals. 
Example 3 — nitrogen oxides: Dalton was aware of three oxides of nitrogen: "nitrous oxide", "nitrous gas", and "nitric acid". These compounds are known today as nitrous oxide, nitric oxide, and nitrogen dioxide respectively.  "Nitrous oxide" is 63.3% nitrogen and 36.7% oxygen, which means it has 80 g of oxygen for every 140 g of nitrogen. "Nitrous gas" is 44.05% nitrogen and 55.95% oxygen, which means there are 160 g of oxygen for every 140 g of nitrogen. "Nitric acid" is 29.5% nitrogen and 70.5% oxygen, which means it has 320 g of oxygen for every 140 g of nitrogen. 80 g, 160 g, and 320 g form a ratio of 1:2:4. The formulas for these compounds are NO, NO, and NO.
The earliest definition of Dalton's observation appears in an 1807 chemistry encyclopedia:
Dalton's atomic theory garnered widespread interest but not universal acceptance shortly after he published it because the law of multiple proportions by itself was not complete proof of the existence of atoms. Over the course of the 19th century, other discoveries in the fields of chemistry and physics would give atomic theory more credence, such that by the end of the 19th century it had found universal acceptance.