Fig. 154.
This law may be represented graphically by arranging the anthropologic data on the abscissæ (e.g., those of stature), and the number of individuals on the ordinates.
In such cases we have a curve with a maximum central height and a symmetrical bilateral diminution (Fig. 121): this is the curve of Quétélet.
Or better yet, it is known as Quétlét's binomial curve, because this anthropologist was the first to represent the law graphically and to perceive that its development was the same as that so well known in mathematics for the coefficients in Newton's binomial theorem.
Newton's binomial theorem is the law for raising any binomial to the nth power, and is expanded in algebra as follows:
(a+b)n = an+ na(n-1)b+ (n(n-1)/2)a(n-2)b2+ ((n(n-1)(n-2))/(2.3))a(n-3)b3+ ((n(n-1)(n-2)(n-3))/(2.3.4))a(n-4)b4+ ((n(n-1)(n-2)(n-3)(n-4))/(2.3.4.5))a(n-5)b5+ ... +bn
substituting for n some determined coefficient, for example, 10, the binomial would develop, in regard to its coefficients, after the following fashion:
(a+b)10 = a10+10×a9b+ ((10.9)/2)a8b2+ ((10.9.8)/(2.3))a7b3+ ((10.9.8.7)/(2.3.4))a6b4+ ((10.9.8.7.6)/(2.3.4.5))a5b5+ ((10.9.8.7.6.5)/(2.3.4.5.6))a4b6+ ((10.9.8.7.6.5.4)/(2.3.4.5.6.7))a3b7+ ((10.9.8.7.6.5.4.3)/(2.3.4.5.6.7.8))a2b8+ ((10.9.8.7.6.5.4.3.2)/(2.3.4.5.6.7.8.9))ab9+ b10.
Whence it appears that, after performing the necessary reductions, the coefficients following the central one diminish symmetrically in the same manner as they increased: that is, according to the selfsame law that we meet in the anthropological statistics of seriations.