Investigations into the supposed number of the phanerogamic species of plants, which have hitherto been described or are preserved in herbariums. Numerical ratios of plant-forms. Discovered laws of the geographical distribution of the families. Ratios of the great divisions: of the Cryptogamia to the Cotyledons, and of the Monocotyledons to the Dicotyledons, in the torrid, temperate, and frigid zones. Outlines of arithmetical botany. Number of the individuals, predominance of social plants. The forms of organic beings stand in mutual dependence on each other. If once the number of species in one of the great families of the Glumaceæ, Leguminosæ, or Compositæ, on any one point of the earth, be known, an approximative conclusion may be arrived at not only as to the number of all the phanerogamia, but also of the species of all remaining plant-families growing there. Connection of the numerical ratios here treated on in the geographical distribution of the families, with the direction of the isothermal lines. Primitive mystery in the distribution of types. Absence of Roses in the southern, and of Calceolarias in the northern zone. Why has our heath (Calluna vulgaris), and why have our Oaks not progressed eastwards across the Ural into Asia? The vegetation-cycle of each species requires a certain minimum heat for its due organic development—pp. 273–287. Analogy with the numeric laws in the distribution of animal forms. If more than 35,000 species of phanerogamia are now cultivated in Europe, and if from 160,000 to 212,000 phanerogamia are now contained, described and undescribed, in our herbariums; it is probable that the number of collected insects scarcely equals that number of phanerogamia; whereas in individual European districts the insects collected preponderate in a threefold ratio over the phanerogamia—pp. 287–291. Considerations on the proportion borne by the number of the phanerogamia actually ascertained, to the entire number existing on the globe—pp. 291–295. Influence of the pressure of atmospheric strata on the form and life of plants, with reference to Alpine vegetation—pp. 295–296.