Mathematical Contributions to the Theory of Evolution. 419 



(a) No constancy of the ratio stature to long bone is theoretically 

 to be expected, but the ratio of deviation from mean stature to 

 deviation from mean long bone, i.e., the regression coefficient is the 

 quantity, the constancy of which might be expected. 



(6) No method of predicting individual stature from the indi- 

 vidual long bones, whether one or all are used, can give a result 

 with a less probable error than 2 cm. 



(c) For the same length of femur, tibia, and humerus, the stature 

 is shorter the longer the radius. This result has considerable bear- 

 ing on the relationship of man to the anthropomorphous apes. 



6. Formulae are then obtained for the reconstruction of probable 

 stature, as measured : 



(a) On corpse, from the lengths of the long bones containing 

 animal matter, and with the cartilages attached. These will possibly 

 be of service for purposes of criminal investigation. 



(6) In life, from the lengths of the long bones without cartilages,, 

 and free of all animal matter. 



Corrections are given for cases in which the femur is measured in 

 the oblique position ; the tibia is measured with the spine ; and the 

 left, instead of the right, hand members are known. 



7. While the formulae are tested on twenty cases, taken at random 

 from Reliefs data, with fairly satisfactory results, the step from 

 variation within the local race to racial variations is not made without 

 a consideration of the circumstances under which natural selection, 

 will modify a regression formula. It is pointed out that the diver- 

 gence between such regression formulae really enables us to predict 

 to some extent the nature of the differential selection which has 

 taken place between two local races. To test how far we may safely 

 apply our formulas to other than French measurements, the stature 

 of the Ainos <$ and ? is reconstructed by means of them from 

 Koganei's measurements of the long bones, and the result is found 

 to be very satisfactory. If French regression formulas give good 

 results for the Aino, they will give, in all probability, good results- 

 for prehistoric European races. At the same time it is most import- 

 ant that material should be obtained for an independent investiga- 

 tion of the regression formulas for another European race. With 

 a view of illustrating the change in the regression formulae owing to 

 selection, the anthropomorphous apes are considered, and it is shown 

 that the gorilla, in the regression formulas for femur and tibia 

 stands much closer to man than either the chimpanzee or orang. 



8. The formulas are applied to reconstruct the stature of Palaeolithic 

 man, Neolithic man in France and Britain, of the Dolmen-builders 

 and Guanches, of Bound Barrow British, and Bow Grave Germans^ 

 of Romano-British and Bomano-Gauls, of Anglo-Saxons, Franks, 

 mediaeval French, and of the Naqada race discovered by Professor 



