We often see the value of the fractions increase by the decrease of the denominator, whilst the absolute number of the species is reduced. In the fractional method which I have followed as the most applicable to questions relating to the geography of plants, there are two variable quantities; for in passing from one isothermal line to another, we do not find the sum total of the phanerogamia change in the same proportion as the number of the species of one particular family. In proceeding from the consideration of these species to that of the divisions established in the natural system according to an ideal series of abstractions, we may direct our attention to genera or races, to families, or even to still higher classes of division. There are some genera, and even whole families, which exclusively belong to certain zones; not merely because they can only thrive under a special combination of climatic relations, but also because they first sprang up within very circumscribed localities, and have been checked in their migrations. The larger number of genera and families have, however, their representatives in all regions of the earth, and at all elevations. The earliest inquiries into the distribution of vegetable forms had reference to genera alone, and are to be found in the valuable work of Treviranus.[NJ] This method is, however, less appropriate for yielding general results, than that which compares the number of the species of each family, or the great leading divisions (acotyledons, monocotyledons, and dicotyledons), with the sum total of the phanerogamia. In the frigid zone, the variety of forms, or the number of the genera, does not decrease in an equal degree with that of the species, there being in these regions relatively more genera and fewer species.[NK] The case is almost the same on the summits of high mountain-chains, where are sheltered individual members of many different genera which one would be disposed to regard as belonging exclusively to the vegetation of the plain.