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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 1/4.

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 progression.

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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