358 Professor J. Arthur Thomson [March 30, 



exact, we may fancy a youth inheriting an estate, in regard to which 

 it might be said that half of it had belonged to his father, and half 

 of it to his mother, yet one with a full knowledge of the family 

 history and the gradual acquisition of the property, might be able to 

 make the story of the heritage much more interesting by showing how 

 this meadow was due to a grandmother and that forest to a great- 

 grandfather. 



To appreciate the possible complexity of our mosaic inheritance 

 we must recall the number of our ancestors. We have two parents, 

 four grandparents, eight great-grandparents, about sixteen great-great- 

 grandparents, and so on. " If," as Prof. Milnes Marshall said, " we 

 allow three generations to a century, there will have been twenty-five 

 since the Norman Invasion, and a man may be descended not merely 

 from one ancestor who came over in 1066, but directly and equally 

 from over sixteen million ancestors who lived at or about that date." 

 But on these theoretical lines the existence of one man to-day would 

 involve the existence of nearly seventy thousand millions of millions 

 of ancestors at the commencement of the Christian era. Which is 

 absurd, for it overlooks the frequent occurrence of close inter- 

 marriage, of cousins for instance. 



The problem of reduction in the number of ancestors has been 

 very carefully discussed by genealogists like Prof. Lorenz and Dr. F. 

 T. Richter, but we should soon lose ourselves in the discussion. We 

 must be content to take one example. Theoretically, Kaiser Wilhelm 

 II. might have had in the direct line the number of ancestors indicated 

 in the upper row of the following scheme ; the second row indicates 

 the number actually known, on to the twelfth generation ; the third 

 row gives the number of those possible ancestors of whose existence 

 there is deficient record ; and the fourth row gives the probable total. 



Generations. I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. 



(1) Theoreti- j 2 4 8 16 32 64 128 256 512 1024 2048 4096 

 cal Number J 



(2) Actual ) 



number J 2 4 8 14 24 44 74 111 162 206 225 275 

 known j 



(3) Inadequately known 5 15 50 117 258 



(4) Probable total 116 177 256 342 533 



According to Galton's law, " the two parents between them con- 

 tribute on the average one-half of each inherited faculty, each of 

 them contributing one-quarter of it. The four grandparents contri- 

 bute between them one-quarter, or each of them one-sixteenth ; and 

 so on, the sum of the series, i + 4 + J + tV + etc -> heing equal to 

 1, as it should be. It is a property of this infinite series that each 

 term is equal to the sum of all those that follow : thus \ = \ + % + 

 y 1 -^ + etc. ; 5 = -g- + tV "f~ etc., aD ^ so on " T ne prepotencies or 

 subpotencies of particular ancestors, in any given pedigree, are 

 eliminated by a law that deals only with average contributions, and 



