Mark Humphrys (amongst others) asks whether we can estimate the number of descendants of King Edward III. A slightly different but related question is the probability that a present-day English person descends from Edward III (1312-1377). This is my attempt to calculate this probability.
First we can consider the number of ancestors that we have. Wachter (1978), in considering the number of ancestors at the time of the Norman conquest for a child born in England in 1947, suggested that in 1587 (13 generations at 30 years) the child would have 7938 ancestors, a figure that allows for some cousin-cousin marriages, based on the observation that Wachter himself had 63 rather than 64 ancestors six generations back. Smith (2001) has examined 626 birth-briefs at the Society of Genealogists, and reviewed other literature on cousin-cousin marriages, showing that although the rate is difficult to estimate, the estimates which can be considered reliable range from 0.28% to 0.86% of marriages. Over many generations, this would mean that on average the number of ancestors increased not twofold per generation, but between 1.9828 and 1.9944 per generation, leading to slightly lower estimates than Wachter's, of 7322 to 7899 ancestors at the 13th generation (and 14518 to 15753 at the 14th generation).
Marriages between more distant cousins, also have an impact, but these are ignored here, following the original argument of Wachter. Wachter (1978) asserts, and Smith (2001) provides some limited evidence, that second cousin marriages are much less common than first cousin marriages, and third or more distant cousin marriages occur no more frequently than would be expected from the number of third, etc., cousins in the population. Third, etc., cousin marriages are therefore very rare. Sturges and Haggett (1987) estimated the average growth rate of the English population c.1350-c.1994 was 1.14 times per generation, implying 2.28 surviving children per couple. With an average 2.28 children per couple then on average each individual has 28 third-cousins, and less than 9000 tenth-cousins in a population of millions. Thus marriages between very distant cousins are more common, but not as common as first cousin marriages until one reaches about 12th cousins, which is more distant than is possible in the 13 generations considered here. Second, etc., cousin marriages also have less impact on the number of ancestors than first cousin marriages. The child of first cousins has 24 rather than 32 distinct 3x-great grandparents, but the child of second cousins has 28, and the child of third cousins has 30. It is therefore safe to follow Wachter in ignoring marriages between cousins other than first cousins, as including them is a lot of calculation which will only reduce the average number of ancestors in 1587 by a fraction of a percent, and this is probably within the margin of error derived from the uncertainty in the rate of first-cousin marriages.
In line with Wachter we can assume "wide diffusion of ancestors throughout society and the country by 1600." Taking those numbers as a starting point, then if we knew the number of descendants of Edward III born in England in the 30-year generation centred on 1587, we could estimate the probability of at least one of them being one of those ancestors.
Leo van de Pas in his website "Genealogics" gives as comprehensive a list of the descendants of Edward III as can probably be compiled at present. In the fifth generation he lists 321 descendants, of whom I count about 245 as 'English' (though distinguishing English from Welsh and Irish nobles in this list is difficult). This gives a rate of increase of 3.00 times in each generation within 'English' descendants or 3.17 per generation for all descendants. This is high in comparison to Sturges and Haggett's (1987) estimate of 2.28 surviving children per couple on average over c.1350-c.1994, but accords with evidence that richer people had more surviving children than poorer people (Clark and Hamilton ). Predicting forward at the rate of 3.00 times for a further four generations to c.1587, Edward III is predicted to have about 20000 descendants. This estimate is somewhat sensitive to the count in the fifth generation: if there were only 150 3x-great-grandchildren the growth would be 2.72 times per generation, and the total only 8300. If a population of 245 grew at 2.28 times per generation there would be only 6600 descendants.
Wrigley and Schofield (1981:208) estimated the population of England to be 3.8 million in 1586, and this is probably the best estimate that there is of the population at that date. Edward III's 20000 ninth-generation descendants therefore constituted about 0.52% of the population. Using Wrigley and Schofield's estimated total births for the 30 years centred on 1587 to express them as the fraction of a generation gives a very similar figure of 0.54%. The probability of a specific one of the 7938 ancestors of the 1947 baby being descended from Edward III is therefore 0.52% and the chance of that ancestor not being descended from Edward III is 99.48%. However with 7938 ancestors the probability that NONE of them is descended from Edward III is 0.9948 raised to the power 7938: a miniscule 1.0x10-18. Even if the number of ancestors in England in 1587 is over estimated by nearly a factor of 2 (so really 4000) and the descendants of Edward III by a factor of four (so really 5000), which seems about the limits of what is possible, the chances of the 1947 baby not descending from Edward III are still only 0.5%. For the generation born around 1977, the estimated number of ancestors in 1587 is 15833 according to Wachter, and the calculated probabilty of no descent drops to 1.4x10-26 (or 0.003% in the limiting case).
Conclusion: there is an extremely high probabilty that a modern English person with predominantly English ancestry descends from Edward III, at a very minimum over 99%, and more likely very close to 100%. The number of descendants of Edward III must therefore include nearly all of the population of England, and probably much of the populations of the rest of the UK and Eire, as well as many millions in the USA, former British colonies and Europe, so 100 million seems a conservative estimate. Documenting one's own descent from Edward III is, however, another matter!
Clark G & Hamilton G (2006) Survival of the richest: The Malthusian mechanism in pre-industrial England. Journal of Economic History 66 707-736.
Sturges CM & Haggett BC (1987) Inheritance of English Surnames. London: Hawgood Computing.
Wachter KW (1978) Ancestors at the Norman Conquest, in Wachter KW, Hammel EA, &: Laslett P (eds.) Statistical Studies of Historical Social Structure. London: Academic Press.
Wrigley EA & Schofield RS (1989) The population history of England 1541-1871: a reconstruction . 2nd ed. Cambridge: Cambridge University Press.(p.208)
Created: 18 July 2008. Updated: 14 August 2010.
Thanks to Mark Humphrys for reading this and spotting various errors, and to Edward Hubbard for prompting me to include detail about distant cousin marriages