224 
PROFESSOR KARL PEARSOK, MATHEMATICAL 
length, and considerably under the real stature. A height, for example, of 185 
centims., 6 feet 2 inches, say, would hardly entitle a man, in England at any rate, 
to rank as a giant. 
In the next place, no notice whatever was taken of the dwarfs. I felt that, if the 
curves were determined from the giant data only, the test that they gave good results 
for dwarfs would be the most satisfactory one conceivable. As it is, I have been able, 
on the basis of the long-bone stature relations for giants, to predict the stature of 
dwarfs to within 2‘5 centims. average error. Manouvrier’s ‘'coefficients moyens 
ultimes ” give a mean error for these dwarfs of 7'25 centims., or 2'9 times as great. 
The actual fitting of the curves was conducted in the following manner. Remember¬ 
ing that the curve gives the value of the mean stature for the whole series of loug- 
bones of one size, i.e., the mean of the array of statures for a long bone of given type 
or size, I recognised that the curve, and accordingly its asymptote, must pass fairly 
centrally through the group of plotted points. An approximate value of the asymptote 
constant h was accordingly selected, and the value of c calculated from the mean of 
the observational values of y and x. If this form of the curve gave, as it generally 
did, not very satisfactory results, h was modified, and the nev/ c calculated. In this 
manner, for example, three approximations were made in the case of the radius. The 
method of least squares was not readily applicable to the data (which were at best 
not very trustworthy), for it involves the calculation of such expressions as S {x^y') 
and S which, owing to the large values of x involved, give far too great import¬ 
ance to the largest giants. 
The curves ultimately determined were the following :—^ 
For the femur : 
y — y~)- 
For the tibia : 
2/ = T-7lTo^M22-5625 -y^). 
For the humerus : 
y = (20'25 ~ ?y“). 
For the radius : 
y = T(5F8T-'»'(20*25 ~ y"). 
Here the unit for both y and x is equal to tAvo centims. of stature, or of long- 
bone. Thus the distances 7, 475, 4*5 and 4*5 centims, of the asymptotes from the 
lines of regression of the normal population are really distances of 14, 9'5, 9 and 
9 centims. in actual stature or long-bone length. 
O c* 
* The mathematical reader Avill bear in mind that it is only the “snake” arid not the otlier tvo 
branches of the quiutic curve which we require. 
