= Solution
\b[<Atomic theory> explains fixed composition by fixed numbers of <chemical atoms>, and <multiple proportions> by different small whole-number combinations of the same <chemical atoms>.] The two laws are empirical regularities which the theory explains; they are not themselves direct observations of individual <chemical atoms>.
For the <law of definite proportions>, suppose a pure <chemical compound> has composition $\mathrm A_x\mathrm B_y$. If the <atomic masses> are $m_A$ and $m_B$, every formula unit contains masses $xm_A$ and $ym_B$. Enlarging the sample multiplies both by the same number, so
$$
\boxed{\frac{\text{mass of B}}{\text{mass of A}}=\frac{ym_B}{xm_A}.}
$$
The <mass ratio> is therefore independent of sample size or preparation route, provided the same pure stoichiometric <chemical compound> is obtained. Ordinary <water> supplies an example: its <hydrogen>-to-<oxygen> <mass ratio> is approximately $1:8$.
For the <law of multiple proportions>, compare $\mathrm A_x\mathrm B_y$ and $\mathrm A_u\mathrm B_v$. At a fixed <mass> of A, the masses of B are in the ratio
$$
\boxed{\frac{y/x}{v/u}=\frac{yu}{vx}.}
$$
This is a ratio of integers, and is often a ratio of small integers for simple <chemical compounds>. With <carbon monoxide> and <carbon dioxide>, about $12$ <mass> units of <carbon> combine with about $16$ and $32$ of <oxygen>: the <oxygen> masses are in the ratio $1:2$. The <chemical formulas> explain why adding a second <oxygen> <chemical atom> doubles the <oxygen> contribution without changing the <carbon> contribution.
The laws alone do not uniquely determine the correct <chemical formulas> or the <atomic weights>; independent information is needed to distinguish, for example, one formula unit from a multiple of it. Modern qualifications include variations in <isotopic composition> and <non-stoichiometric compounds>. These refine the simple nineteenth-century formulation without removing its usefulness for pure compounds of fixed composition.
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