Where did all the Darrochs go?
My name is Darroch. Darroch is a Highland Scottish
name derived from the Gaelic word for 'oak ' which is spelled
'darrach '. It is officially a Highland Clan of which there are
fewer than a hundred. The name has been known in Scotland in its
present form for at least four hundred years, probably five hundred
years. But there are so few Darrochs listed in the phone books
of Britain and North America that it must be the smallest of all
the clans in numbers. How many members should a clan or family,
that has been multiplying for five hundred years, have?
My great grandfather, Archibald Darroch born in
1788, had, living and dead, over 1200 descendants. My grandfather,John
Darroch born in 1822, had fifteen children. Two boys died early.
One girl did not marry. The other twelve, six boys and six girls,
all married and had children, some of them large families. Altogether
they produced 74 grandchildren for John Darroch. Of these 37 were
boys and 37 were girls. The girls all married out of the name.
Of the boys 14 were the sons of daughters, hence did not carry
the name. That left 23 sons of the sons who still Darrochs. If
that last number had been 19 out of 74, the ratio would have been
19:57 or 1:3, which is the Mendelian ratio for recessive against
dominant characteristics or genes in the breeding of plants and
animals. There may be no significant connection but what happens
to a surname in a theoretically perfect descent with a minimum
number of offspring? The smallest number would be a family where
each member of succeeding generations had one boy and one girl.
A man ( and his wife. Unfortunately this only refers
to men as we are only concerned with the transmission of the surname.
) has two children, a boy and a girl. The girl marries and loses
her surname. The boy retains his surname and the ratio is 1:1
or one-half. Each of the children has a boy and a girl, providing
four grandchildren for the man. The daughter's children, both
boy and girl, carry another name. The son's children marry and
the grand-daughter changes her name while the grandson keeps his.
The ratio here is 1:3 or the fraction retaining the surname is
The four grandchildren all marry and have a boy
and a girl each. There are eight great-grandchildren of whom only
the son of the grandchild who could retain his born name, can
pass it along to his children. The plan would appear as follows:-
S is a son carrying the surname: d is a daughter
who must relinquish her name on marriage: s is a son, grandson,
etc bearing another name.
Therefore, out of sixteen great-great-grandchildren
only one or a fraction of 1/16th, could bear the name of the great-great-grandfather.
And so on - ad infinitum.
This theoretical family would double each generation.
In the sixteenth generation a man could have more than 32,000
descendants of whom only one male would carry his name. In the
whole sixteen generations, over about four hundred years, he's
have over 64,000 descendants, of whom only sixteen would have
been males to bear his name. This theoretical family does not
allow for sterility, non-marriage, early deaths, disease, or congenital
defects. No does it make any provision for ' runs ' of boys or
girls. More boys are born then girls, but more boys die at birth
or in early age, and, of course, until recently large families
were the rule.
Let us go back and compare our theoretical descendants
with our actual ancestors. I had a father who was a Darroch. One
grandfather out of four grandparents was a Darroch. One great-grandfather
out of eight great-grandparents was a Darroch, and one great-great-grandfather
out of sixteen great-great-grandparents was a Darroch. Thus the
fraction of my ancestors who bore my name in each generation is
the same as the fraction of my descendants who would bear the
name if I and my children, etc. had an equal number of boys and
girls in every family.
Therefore this statement appears to be true. In
families large enough to be considered random ( that is equal
boys and girls ) and under optimum conditions ( that is barring
sterility, non-marriage, accidents, disease, or congenital defects
) the fraction of a man's descendants that bear his surname will
be the same in each succeeding generation as it was in each corresponding
generation among his ancestors. In other words - in inverse geometrical
The larger the family, of course, the greater that
fraction will amount to in actual numbers. Now I know where all
the Darrochs went. They went into other families.
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