Up to Mendel no one proposed to answer this question in any other way than by reference to the intensity of the character in the progenitors, and _primarily_ in the parents, _A_ and _a_, in whose bodies the gametes had been developed. It was well known that such a reference gave a very poor indication of what _Aa_ would be. Both _A_ and _a_ may come from a population consisting of individuals manifesting the same character in various intensities. In the pedigree of either _A_ or _a_ these various intensities may have occurred few or many times. Common experience leads us to expect the probability in regard to _Aa_ to be influenced by this history. The next step is that which Galton took. He extended the reference beyond the immediate parents of _Aa_, to its grandparents, great-grandparents, and so on, and in the cases he studied he found that from a knowledge of the intensity in which the given character was manifested in each progenitor, even for some few generations back, a fairly accurate prediction could be made, not as to the character of any individual _Aa_, but as to the average character of _Aa_’s of similar parentage, in general. But suppose that instead of individuals presenting one character in differing intensities, two individuals breed together distinguished by characters which we know to be mutually exclusive, such as _A_ and _B_. Here again we may speak of the individuals producing the gametes as _A_ and _B_, and the resulting zygote as _AB_. What will _AB_ be like? The population here again may consist of many like _A_ and like _B_. These two forms may have been breeding together indiscriminately, and there may have been many or few of either type in the pedigree of either _A_ or _B_. Here again Galton applied his method with remarkable success. Referring to the progenitors of _A_ and _B_, determining how many of each type there were in the direct pedigree of _A_ and of _B_, he arrived at the same formula as before, with the simple difference that instead of expressing the probable average intensity of one character in several individuals, the prediction is given in terms of the probable number of _A_’s and _B_’s that would result on an average when particular _A_’s and _B_’s of known pedigree breed together. The law as Galton gives it is as follows:-- “It is that the two parents contribute between them on the average one-half, or (0·5) of the total heritage of the offspring; the four grandparents, one-quarter, or (0·5)^2; the eight great-grandparents, one-eighth, or (0·5)^3, and so on. Then the sum of the ancestral contributions is expressed by the series {(0·5) + (0·5)^2 + (0·5)^3, &c.}, which, being equal to 1, accounts for the whole heritage.”