546 HEREDITY 
in the Ff, generation were of many kinds. The combinations 
in this case also agreed quite well with the mathematical laws 
of combination, when a large number of the F, individuals 
were taken into account. The kinds of combinations and their 
proportions follow quite well the general algebraic formula 
(a + b)", in which n represents the number of characters in- 
volved. Thus (a + 6)? expanded gives a? + 2 ab + b? which is 
in accord with the 1: 2:1 ratio, the ratio expressing the inheri- 
tance of two contrasting characters. The formula (a + b)* gives 
the combinations when plants are crossed that have two pairs 
of contrasting characters. Of course the results obtained scarcely 
ever exactly agree with the mathematical formula, and the more 
individuals taken into account, the closer the agreement. 
As a result of his work with a number of pairs of characters, 
Mendel showed that by means of repeated artificial fertilization, 
the constant characters of different varieties of plants may be 
obtained in all of the associations which are possible according 
to the mathematical laws of combination. This means that, by 
crossing in a certain way, the desirable characters of different 
varieties may be brought together, and thereby plants of a more 
desirable type produced. 
Mendel’s Law. — Mendcl’s discoveries and conclusions con- 
cerning the way parental characters appear in hybrids and their 
offspring constitute Mendel’s law, and his discoveries may be sum- 
marized as follows: (1) characters are of two kinds, dominant and 
recessive; (2) characters do not blend but behave as units and 
separate completely from one another; (3) gametes are, there- 
fore, pure, never containing genes for both of a pair of contrast- 
ing characters; (4) the offspring of a hybrid consist of dominants 
and recessives in the ratio of three dominants to one recessive, 
and the recessives and one-third of the dominants breed truce, 
while two-thirds of the dominants breed as hybrids, producing 
offspring consisting of dominants and recessives and again in 
the ratio 3:1; (5) when any number of pairs of characters 
are considered, each pair behaves independently, and all com- 
binations of characters according to the mathematical laws of 
combination can be obtained. 
After eight years of work, Mendel published an account of 
his remarkable discoveries, but unfortunately his publication 
remained unnoticed until 1900, thirty-five years after its pub- 
