“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.”
In the former case where _A_ and _a_ are characters which can be
denoted by reference to a common scale, the law assumes of course that
the inheritance will be, to use Galton’s term, _blended_, namely that
the zygote resulting from the union of _A_ with _a_ will on the average
be more like _a_ than if _A_ had been united with _A_; and conversely
that an _Aa_ zygote will on the average _be more like A than an aa
zygote would be_.
But in the case of _A_’s and _B_’s, which are assumed to be mutually
exclusive characters, we cannot speak of blending, but rather, to use
Galton’s term, of _alternative_ inheritance.
Pearson, finding that the law whether formulated thus, or in the
modified form in which he restated it[16], did not express the
phenomena of alternative inheritance known to him with sufficient
accuracy to justify its strict application to them, and also on general
grounds, proposed that the phenomena of blended and alternative
inheritance should be treated apart--a suggestion[17] the wisdom of
which can scarcely be questioned.
[16] In Pearson’s modification the parents contribute 0·3, the
grandparents 0·15, the great-grandparents ·075.
[17] See the works referred to above.
Now the law thus imperfectly set forth and every modification of it is
incomplete in one respect. It deals only with the characters of the
resulting zygotes and predicates nothing in regard to the gametes which
go to form them. A good prediction may be made as to any given group of
zygotes, but the various possible constitutions of the gametes are not
explicitly treated.
Nevertheless a definite assumption is implicitly made regarding the
gametes. It is not in question that differences between these gametes
may occur in respect of the heritage they bear; yet it is assumed
that these differences will be distributed among the gametes of any
individual zygote in such a way that each gamete remains capable,
on fertilisation, of transmitting _all_ the characters (both of the
parent-zygote and of its progenitors) to the zygote which it then
contributes to form (and to the posterity of that zygote) in the
intensity indicated by the law. Hence the gametes of any individual
are taken as collectively a fair sample of all the racial characters
in their appropriate intensities, and this theory demands that there
shall have been no qualitative redistribution of characters among the
gametes of any zygote in such a way that some gametes shall be finally
excluded from partaking of and transmitting any specific part of
the heritage. The theory further demands--and by the analogy of what
we know otherwise not only of animals and plants, but of physical or
chemical laws, perhaps this is the most serious assumption of all--that
the structure of the gametes shall admit of their being capable of
transmitting any character in any intensity varying from zero to
totality with equal ease; and that gametes of each intensity are all
equally likely to occur, given a pedigree of appropriate arithmetical
composition.
Such an assumption appears so improbable that even in cases where
the facts seem as yet to point to this conclusion with exceptional
clearness, as in the case of human stature, I cannot but feel there is
still room for reserve of judgment.
However this may be, the Law of Ancestral Heredity, and all
modifications of it yet proposed, falls short in the respect specified
above, that _it does not directly attempt to give any account of the
distribution of the heritage among the gametes_ of any one individual.
Mendel’s conception differs fundamentally from that involved in the Law
of Ancestral Heredity. The relation of his hypothesis to the foregoing
may be most easily shown if we consider it first in application to the
phenomena resulting from the cross-breeding of two pure varieties.
Let us again consider the case of two varieties each displaying the
same character, but in the respective intensities _A_ and _a_. Each
gamete of the _A_ variety bears _A_, and each gamete of the _a_ variety
bears _a_. When they unite in fertilisation they form the zygote _Aa_.
What will be its characters? The Mendelian teaching would reply that
this can only be known by direct experiment with the two forms _A_ and
_a_, and that the characters _A_ and _a_ perceived in those two forms
or varieties need not give any indication as to the character of the
zygote _Aa_. It may display the character _A_, or _a_, or a character
half way between the two, or a character beyond _A_ or below _a_. The
character of _Aa_ is not regarded as a _heritage_ transmitted to it by
_A_ and by _a_, but as a character special and peculiar to _Aa_, just
as NaCl is not a body half way between sodium and chlorine, or such
that its properties can be predicted from or easily stated in terms of
theirs.
If a concrete case may help, a tall pea _A_ crossed with a dwarf _a_
often produces, not a plant having the height of either _A_ or _a_, but
something _taller_ than the pure tall variety _A_.
But if the case obeys the Mendelian principles--as does that here
quoted--then it can be declared _first_ that the gametes of _Aa_
will not be bearers of the character proper to _Aa_; but, generally
speaking, each gamete will either bear the pure _A_ character or the
pure _a_ character. There will in fact be a redistribution of the
characters brought in by the gametes which united to form the zygote
_Aa_, such that each gamete of _Aa_ is pure, as the parental gametes
were. _Secondly_ this redistribution will occur in such a way that, of
the gametes produced by such _Aa_’s, on an average there will be equal
numbers of _A_ gametes and of _a_ gametes.
Consequently if _Aa_’s breed together, the new _A_ gametes may meet
each other in fertilisation, forming a zygote _AA_, namely, the pure
_A_ variety again; similarly two _a_ gametes may meet and form _aa_,
or the pure _a_ variety again. But if an _A_ gamete meets an _a_ it
will once more form _Aa_, with its special character. This _Aa_ is
the hybrid, or “mule” form, or as I have elsewhere called it, the
_heterozygote_, as distinguished from _AA_ or _aa_ the _homozygotes_.
Similarly if the two gametes of two varieties distinguished by
characters, _A_ and _B_, which cannot be described in terms of any
common scale (such as for example the “rose” and “single” combs of
fowls) unite in fertilisation, again the character of the mule form
cannot be predicted. Before the experiment is made the “mule” may
present _any_ form. Its character or properties can as yet be no more
predicted than could those of the compounds of unknown elements before
the discovery of the periodic law.
But again--if the case be Mendelian--the gametes borne by _AB_ will be
either _A_’s or _B_’s[18], and the cross-bred _AB_’s breeding together
will form _AA_’s, _AB_’s and _ BB_’s. Moreover, if as in the normal
Mendelian case, _AB_’s bear on an average equal numbers of _A_ gametes
and _B_ gametes, the numerical ratio of these resulting zygotes to each
other will be