Battling about equations

Abstract

One of the most exciting episodes in the study of human heredity was the battle between two incompatible algebraic expressions to describe its properties in the beginning of the 20th Century. In one corner was Karl Pearson (1857–1936) with his 1898 formulation for ancestral heredity, which had the authority of Charles Darwin and Francis Galton for support.1 It reads: The inherited contribution of parent to offspring (y) can be formulated as: ![Formula][1] where σ/σn are the ratios of the standard deviations of measurable traits in the offspring to the standard deviations of the mid-parental generation; and k1,k2 … kn are the deviations of the mid-parental means from the mean of the offspring for each trait. He admits to reservations of how ‘mid-parent’ should be defined. The subscripts refer to each of the ancestral generations (parents, grandparents, great-grandparents, etc.). The original paper should be consulted for fuller details.2 Pearson has elaborated on earlier formulations using a convergent geometric series to represent the hereditary contribution (f(y)) by the parents, the four grand parents, eight great-grand parents, etc. to the offspring.1 This type of convergent series is of the general form: ![Formula][2] where A,B,C,D … are constants independent of x (=0.5n in Pearson's expression). The expression can be used to represent a great many biological phenomena. The formulation has no theoretical significance; all it postulates is that the phenomenon in question varies continuously. Then Maclaurin's or Taylor's theorems (depending on the number of variables involved) can be used to determine the values of the coefficients that will make the series useful to any desired degree of approximation.3 Pearson's equation certainly satisfies the requirements for continuous variation of inherited traits which Darwin believed to be true; and it reaches a limit … [1]: /embed/graphic-1.gif [2]: /embed/graphic-2.gif