In the former year the ratios were 40.2 and 41.5, in the latter 41-6 and 427%.
In the same preface is included (a) the famous problem known by Pappus's name, often enunciated thus: Having given a number of straight lines, to find the geometric locus of a point such that the lengths of the perpendiculars upon, or (more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to the product of the remaining ones; (Pappus does not express it in this form but by means of composition of ratios, saying that if the ratio is given which is compounded of the ratios of pairs - one of one set and one of another - of the lines so drawn, and of the ratio of the odd one, if any, to a given straight line, the point will lie on a curve given in position), (b) the theorems which were rediscovered by and named after Paul Guldin, but appear to have been discovered by Pappus himself.
He discovered that gases always combined in volumes having simple ratios, and that the volume of the product had a simple ratio to the volumes of the reacting gases.
Equal changes in temperature and pressure occasion equal changes in equal volumes of all gases and vapours - Avogadro deduced the law: Under the same conditions of temperature and pressure, equal volumes of gases contain equal numbers of molecules; and he showed that the relative weights of the molecules are determined as the ratios of the weights of equal volumes, or densities.
He maintained that, under varying conditions, two substances could combine in an indefinitely large number of different ratios, that there could in fact be a continuous variation in the combining ratio.