On the application of the Binomial Law to Statistical Enquiries, illustrated by the Law of the growth of Man at different ages. By SAMUEL BROWN, F.S.S.

[Read before the Institute, 24 February 1873.]

IN recent years, the writer who has done the most good service in

bringing scientific methods to bear on the collection and comparison of statistics, is M. Quetelet, the Director of the Royal Observatory in Bruxelles, President of the Central Commission of Statistics, and the Perpetual Secretary of the Academy of Sciences. Nearly all the learned Academies of Europe claim the credit of enrolling him as a member. He was amongst the small number of illustrious men, including Professor Babbage, Whewell, Malthus, Drinkwater, Jones, Col. Sykes, &c., who at the third meeting of the British Association for the advancement of Science, at Cambridge in 1833, succeeded in establishing the special Section F on Economic Science and Statistics, and which afterwards led to the formation of the Statistical Society of London and various provincial Societies. He was also the originator of the International Statistical Congresses, which have been so useful in improving the methods of collection and publication of facts in all branches of statistical enquiry by every Government in Europe.

As far back as 1835, M. Quetelet, in his work entitled "Sur l'homme et sur le développement de ses facultés, ou Essai de Physique sociale," published a series of researches of a most interesting character into the physical and moral condition of man. This excellent work, enriched with additional observations and Tables, the results of 35 years' further experience, and the zealous aid of scientific friends, was republished in 1869, in compliment to the Delegates of the Continental Governments about to assemble at the Hague, who were then and are now engaged in preparing a systematic comparison of international Statistics in every branch which affects the condition or progress of the people.

In treating of the physical qualities of man, the first and most essential part of the enquiry is as to his growth, and the relative proportion of the various parts of the body at different ages until his complete maturity. The last work of M. Quetelet, entitled

Anthropométrie, ou mesure des différentes facultés de l'homme,” published about the end of 1870, comprizes the results of many years of observations, in which, by the assistance of scientific friends, artists, and medical men, he has succeeded in collecting sufficient

trustworthy facts to trace the law of growth in every portion of the human body at all periods of life.

The methods formerly employed to ascertain the true proportions which constitute the typical man, were not satisfactory. Artists, and sculptors especially, whilst endeavouring to obtain. models of perfection for the object they had in view, found it necessary to study nature with great care, and have left various observations on the proportions of the human body at different periods of life and in both sexes. But it does not in general appear clear how the measures were taken, nor whether the models chosen would fairly represent the form and proportions of man in general. If some typical form of the human race could be supposed to exist, the relative proportions of which were so fixed that any deviations from it in excess or defect could only arise from accidental causes, the observations recorded as to the measurements of any part of the body may be divided into groups, and according to the theory of probability the specific number which ought to be found in each group may be predicted beforehand with a very near approach to accuracy. The greater the number of observations, the more certainly will the observed number in each group agree with the number calculated by the theory. If the measures are divided into equal spaces, the group which approaches nearest to the mean will be the most numerous; and the other groups will be found to contain numbers as they differ from the mean in excess or defect in exact proportion to the coefficients of the terms of the binomial theorem.

This remarkable law, called for shortness the Binomial law, applies not only to the measurement of the height and growth of man, but to his weight, his strength, all his physical, moral, and intellectual actions, so far as they can be expressed in numbers,—to animals, to plants,-in fact, to all classes of facts capable of being classed by groups in numbers, and containing some typical form or character from which there are only what may be called accidental deviations.

In accordance with this law, when the extremes are known, dwarfs and giants are not casual monstrosities. If we obtain from a sufficient number of observations the height of certain groups of the population in any country, and out of these observations the giants and dwarfs had been purposely excluded, we should not only be able to state the numbers which had been omitted, but to assign to them their relative statures as compared with the rest of the people.

M. Quetelet devotes the second part of his work to a very interesting enquiry as to the measurement of man, as estimated by the proportions which artists in different countries and in ancient or modern times have given to the human form, male and female.

The work of Audran, published in Paris in 1683, entitled "Les proportions du corps humain mesurées sur les plus belles statues de l'antiquité," contains the proportions of Laocoon and his sons, of Poetus, of the Egyptian Termes, of Antinous, of the Greek statue of Peace, of the Pythian Apollo, an antique fragment, and of two female figures. M. Quetelet has compared the proportions with those of the mean proportions obtained by the measurement of adult man in Belgium. The coincidences are striking, and the differences not greater than would be caused by errors of observation. The hand and the foot are slightly smaller amongst the Greeks than the Belgian model; but both differ from the common rule of the schools that the head and the foot are of equal length; for the Belgian model gives the former 135 and the latter 154 of the total height, and the Greek statues 130 and 149 respectively. All the measures tend to establish the fact that the proportions of the human form of the present day are almost identical with those deduced by observation from the most regular statues of Grecian art.

The proportions amongst the Romans are deduced from the rules laid down by Vitruvius, many of which have been since adopted by the moderns without naming the authority from which they proceed.

Pliny, after citing examples of extremes of stature, lays down the rule that the height of a man is the same as the distance between the tips of the middle fingers when the arms are stretched out in a strait line; and that at three years of age man will have reached half the height which he will eventually attain.

Amongst the Italian artists, Leone Battista Alberti, who was born in 1398, and lived 77 years, ascertained the proportions of the human body by selecting a great number of models reputed of fine and regular shape; and obtained a mean for the measurement of certain parts by rejecting what were in excess or defect, thus constructing as it were a type of the human form in theoretical proportions.

Various other Italian writers and artists of the highest celebrity have left their rules for the true proportions of man, but none appear to have taken the same simple and more natural method of Alberti, whereby the mean proportions from a considerable number

of measurements of each part are taken to represent the type of the human form. The German artists have in Albert Durer (born 1471, died 1528) the credit of possessing one who, amidst his celebrated works on mathematics and the arts, paid special attention to the present subject. Taking the height of man as the unit, he measured off the proportions of the various parts in length or diameter, in fractions of his unit. The female he makes th part shorter than the man. M. Quetelet, by a great number of measurements, obtained 1.684 metres for the height of man, and 1.579 of the female, or 1th part less. Albert Durer's measures of an infant about one year old, in which artists frequently fail as to the correct proportions, seem to have been carefully obtained from

nature.

Schadow gave, from 1788 to 1848, frequent memoirs on the subject; but his principal work was "Polyclete, or theory of the measure of man according to sex and age." He claims the credit of having been the first to show the difference in the sexes of the head and the face, and the profile in maturity and old age, the measure of the infant just born, the proportions at various stages of growth in males, &c.

In none of the authors alluded to in these brief historical notices do we find the full application of those scientific principles on which M. Quetelet has himself established the harmony of proportions which constitute the type of man. The law of the growth of man is expressed by the binomial law of Newton, which, simple and general as it is, applies equally to his stature, his weight, and his physical and other powers. In the 40 years which have elapsed since M. Quetelet first pointed out its adaptation to the researches as to man, many facts have been added to strengthen his conclusions.

A remarkable confirmation is afforded in a paper presented by Mr. E. B. Elliott to the International Statistical Congress at Berlin in 1865, showing the measurement of the height of 25,878 volunteers in the United States, taken during the civil war. The intervals of height are taken as every 25 millimetres. The mean height is 1.75 metre, at the interval comprizing which the largest number of the men are found, 157 out of every 1000, when the calculated number would give 153 out of every 1000. At every other interval the calculated numbers correspond very closely to the observed numbers, the total deviations in 18 intervals being only -28 and +28 in every 1000 cases.

It is the result of observations that the extremes in the height

of men, if the human race were all measured, would be about 2.60 metres in one direction and 0.70 metre in the other. Between these limits, if all were grouped at intervals of 1 decimetre, the numbers at each group would be found not to be arbitrary or uncertain, but in a most remarkable manner to range themselves at equidistance from the mean, with a regularity and precision which may be calculated beforehand with the greatest accuracy from a few terms.

Two laws relating to the growth of man may be represented by

two curves.

1. The ordinate, representing by its length men of mean height at each age, in passing along what may be called the line of life, will be found to touch in its upper point a descending hyperbolic curve, which may be called "the curve of mean height."

2. At any point in this curved line may be seen, on the right and left, the number of men of the same age, taller or shorter than those of the mean height, distributed with the greatest regularity in a plane, perpendicular to the line of mean height, and ranged according to a second line, which may be named the binomial line.

The two extremes of the last line would be the giants and the dwarfs at each age.

The expression for the curve representing by its abscissa and ordinate the individual height and the number of individuals of a given age, is in its most simple analytical form,

The simplicity of this formula rests upon the fact that every observed phenomenon must depend upon causes which are favourable or unfavourable to its happening. In one case these