I should like to repeat here that these results have not been obtained with absolute certainty, and that it seems improbable that the laws of heredity in regard to various ancestral traits are the same. I do not enter into a discussion of the question of in how far these traits follow the laws of Mendelian inheritance,—a question that cannot be answered definitely at the present time (Davenport). These problems have a fundamental importance for a clearer interpretation of the conditions which prevail in the form of local types of man. In a large population which is as little stable in its habits as that of modern Europe and modern America, the number of ancestors of a single person increases very rapidly, the number of parents being two; of grandparents, four; of great-grandparents, eight; the theoretical number of ancestors twenty generations back would be over a million, or, more accurately, 1,048,576. Twenty generations represent, according to the rate of increase of modern times, about seven hundred years; according to the rate of increase of older times, about four hundred years as a minimum. These figures would apply to the series of generations represented by first-born males; for first-born females the respective numbers would be about five hundred years and three hundred and fifty years. If we consider, however, the actual descent of families, including individuals later born, we might perhaps assume that twenty generations in Europe would represent from eight hundred to nine hundred years, and among primitive peoples perhaps only little less, since in former times the differences between the rapidity of successive generations in Europe and among primitive peoples was not very great. This makes it obvious that it is entirely impossible that as great a number of ancestors as the theory requires can have contributed to the development of the individuals of the present generation. The reason for this is plain. Owing to intermarriages between the same families, large numbers of ancestors will be duplicated in different paternal and maternal lines; and in this way the real ancestry of each individual appears to be much more complex than the purely arithmetical treatment would suggest. The calculation for the ancestor table of the German Emperor, for instance, is instructive. According to O. Lorenz, the numbers of his ancestors in successive generations were as follows:—