132 THE RATE OF GROWTH [ch. 



Gaussian curve extends, in theory, to infinity at either end; and 

 this infinite extension, or asymptotism, has its biological significance. 

 We know that this or that athletic record is lowered, slowly but 

 continually, as the years go by. This is due in part, doubtless, to 

 increasing skill and improved technique ; but quite apart from these 

 the record would slowly fall as more and more races are run, owing 

 to the indefinite extension of the Gaussian curve*. 



On the other hand, while the Gaussian curve extends in theory 

 to infinity, the fact that variation is always limited and that extreme 

 v3,riations are infinitely rare is one of the chief lessons of the law 

 of frequency. If, in a population of 100,000 men, 170 cm. be the 

 mean height and 6 cm. the standard deviation, only 11 per cenl^., 

 or say 130 men, will exceed 188 cm., only 10 men will be over 

 191 cm., and only one over 193 cm., or 13 J per cent, above the 

 average. The chance is negligible of a single one being found over 

 210 cm., or 7 ft. high, or 24 per cent, above the average. 



Yet, widely as the law holds good, it is hardly safe to count it 

 as a universal law. Old Parr at 150 years old, or the giant Chang 

 at more than eight feet high, are not so much extreme instances of 

 a law of probabihty, as exceptional cases due to some peculiar cause 

 or influence coming inf. In a somewhat analogous way, one or two 

 species in a group grow far beyond the average size ; the Atlas moth, 

 the Gohath beetle, the ostrich and the elephant, are far-off outhers 

 from the groups to which they belong. A reason is not easy to. find. 

 It looks as though variations came at last to be in proportion to the 

 size attained, and so to go on by compound interest or geometrical 

 progression. There may be nothing surprising in this ; nevertheless, 

 it is in contradistinction to that summation of small fortuitous 

 differences which lies at the root of the law of error. If size vary in 

 proportion to the magnitude of the variant individuals, not only 



* This is true up to a certain extent, but would become a mathematical fiction 

 later on. There will be physical limitations (as there are in quantum mechanics) 

 both to record-breaking, and to the measurement of minute extensions of the 

 record. 



t We may indeed treat old Parr's case on the ordinary lines of actuarial 

 probability, but it is "without much actuarial importance." The chance of his 

 record being broken by a modern centenarian is reckoned at (5)^°, by Major 

 Greenwood and J. C. Irwin, writing on Senility, in Human Biology, xi, pp. 1-23, 

 1939. 



