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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.

Indeed, here is the binomial theorem with the reductions made:

(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)/(2.3.4))a4b6+ ((10.9.8)/(2.3))a2b7+ ((10.9)/(2))a2b8+10×ab9+b10.

And after calculating the coefficients, we obtain the following numbers in a symmetrical series:

10 45 120 210 252 210 120 45 10

This is why the curve of Quétélet is called binomial.

Let us assume that we wish to represent by means of Quétélet's curves, two seriations, for instance in regard to the stature of children of the same race, sex and age, but of opposite social conditions: the poor and the rich.

These two curves of Quétélet's, provided that they are based upon an equal and very large number of individuals, will be identical, because the law itself is universal. Only, the curve for the rich children will be shifted along toward the figures for high statures, and that for the poor children toward the low statures.

↣ Statures ↣ (Ascending Series)

Fig. 155.

At a certain point A the two curves meet and intersect, each invading the field of the other: so that within the space ABC there are individual rich children who are shorter than some of the poor, and individual poor children who are taller than some of the rich: i.e., the conditions are contrary to those generally established by the curve as a whole. This rule also, of the intersection of binomial curves, is of broad application; whenever a general principle is stated, e.g. that the rich are taller than the poor, it is necessary to understand it in a liberal sense, knowing that wherever we should descend to details, the opposite conditions could be found (superimposed area ABC). For all that, the principle as a whole does not alter its characteristic, which is a differentiation of diverse types (for example, the tall rich and the short poor). The same would hold true if we made a comparison of the stature of men and women; the curve for men would be shifted toward the higher figures and that for women toward the lower, but there would be a point where the two curves would intersect, and in the triangle ABC there would be women taller than some of the men, and men shorter than some of the women. The differences have reference to the numerical majority (the high portions of the curves) which are clearly separated from each other, like the tops of cypress trees which have roots interlacing in the earth. Now, it is the numerical prevalence of individuals, in any mixed community, that gives that community its distinctive type, whether of class or of race. If we see gathered together in a socialistic assemblage a proletarian crowd, suffering from the effects of pauperism, the majority of the individuals have stooping shoulders, ugly faces and pallid complexions; all this gives to the crowd a general aspect, one might say, of physical inferiority. And we say that this is the type of the labouring class of our epoch in which labour is proletarian—a type of caste. On the other hand, if we go to a court ball, what strikes us is the numerical prevalence of tall, distinguished persons, finely shaped, with velvety skin and delicate and beautiful facial lineaments, so that we recognise that the assemblage is composed of privileged persons, constituting the type of the aristocratic class. But this does not alter the fact that among the proletariat there may be some handsome persons, well developed, robust and quite worthy of being confounded with the privileged class; and conversely, among the aristocrats, certain undersized individuals, sad and emaciated, with stooping shoulders and features of inferior type, who seem to belong to the lower social classes.

For this same reason it is difficult to give clear-cut limits to any law and any distinction that we meet in our study of life. This is why it is difficult in zoology and in botany to establish a system, because although every species differs from the others, in the salience of its characteristics and the numerical prevalence of individuals very much alike, none the less every species grades off so insensibly into others, through individuals of intermediate characteristics, that it is difficult to separate the various species sharply from one another. It is only the treetops that are separate, but at their bases life is intertwined; and in the roots there is an inseparable unity. The same may be said when we wish to differentiate normality from pathology and degeneration. The man who is clearly sane differs beyond doubt from the one who is profoundly ill or degenerate; but certain individuals exist whose state it would be impossible to define.

Now, while seriations analyse certain particularities of the individual distribution, by studying the actual truth, mean averages give us only an abstraction, which nevertheless renders distinct what was previously nebulous and confused in its true particulars. The synthesis of the mean average brings home to us forcibly the true nature of the characteristics in their general effect. The analysis of the seriation brings home to us forcibly the truth regarding this effect when we observe it in the actuality of individual cases.

"When, from the topmost pinnacle of the Duomo of Milan or from the hill of the Superga," says Levi in felicitous comparison, "we contemplate the magnificent panorama of the Alpine chain, we see the zone of snow distinguished from that free from snow by a line that is visibly horizontal and that stretches evenly throughout the length of the chain. But if we enter into the Alpine valleys and try to reach and to touch the point at which the zone of snow begins, that regularity which we previously admired disappears before our eyes; we see, at one moment, a snow-clad peak, and at the next another free from snow that either is or seems to be higher than the former."

Now, through the statistics of mean averages, we are able to see the general progress of phenomena, like the spectator who gazes from a distance at the Alpine chain and concludes that the zone of snow is above and the open ground is below; while, by means of seriation, we are in the position of the person who has entered the valley and discovers the actuality of the particular details which go to make up the uniform aspect of the scene as a whole. Both aspects are true—just as both of those statistical methods are useful—for they reciprocally complete each other, concurring in revealing to us the laws and the phenomena of anthropology.

CHAPTER IX
BIOGRAPHICAL HISTORY OF THE PUPIL AND HIS ANTECEDENTS

The child, like every other individual, represents an effect of multifold causes: he is a product of heredity (biological product) and a product of society (social product). The characteristics of his ancestors, their maladies, their vices, their degeneration, live again in the result of the conception which has produced a new individual: and this individual, whether stronger or weaker, must pass through various obstacles in the course of his intrauterine life and his external life. The sufferings and the mistakes of his mother are reflected in him. The maladies which attack him may leave upon him permanent traces. Finally, the social environment receives the child at birth, either as a favoured son or as an unfortunate, and leads him through paths that certainly must influence his complex development.

All of the preceding and theoretic parts of this volume which took up each characteristic for separate consideration, have already explained all that it is necessary to know in order to interpret the characteristics present in a given individual, and the more or less remote causes which contributed to them.

We may now apply our acquired knowledge to individual study, by making investigations into the antecedents of the child and recording his biographic history. It forms a parallel to the clinical history which is recorded in medicine: and it leads to a diagnosis, or at least to a scientific judgment regarding the child.

Although this biographic part is eminently practical, certain principal points of research may be indicated for the purpose of guiding the student. But no one will ever make a successful study of medical pedagogy unless he will follow the practical lessons dedicated to the individual study of the scholar, and make a practice of personal observation. In the Pedagogical School of Rome, we provide subjects, taken from the elementary schools or from the Asylum School of De Sanctis for defective children. And we read their biographical history in regard to their antecedents, and then make an objective examination of them, frequently extending it to an examination of their sensibility and their psychic conditions and enquiring into their standard of scholarship. From these lessons based upon theory, profitable discussions often result; and they certainly are the most profitable lessons in the course.

A biographical history is essentially composed of three parts: the antecedents, which comprises an investigation of the facts antedating the individual in question; the objective examination, which studies the individual personally; and the diaries, i.e., the continued observation of the same individual who has already been studied in regard to himself and his antecedents.

The objective examination and the diaries cannot be considered solely in the light of anthropology, because they chiefly require the aid of psychology. But even anthropology makes an ample and important contribution, first, in the form of an objective morphological examination, the vast importance of which has already been shown; secondly, because it gives us a picture of the biologico-social personality which it is necessary to compare with the reactions of the subject in question, with his psychic manifestations, his degree of culture, etc.; and upon this comparison depends the chief importance of the individual study of the pupil.

Accordingly, in addition to an examination of the individual, anthropology ought to concern itself also with the conditions antedating the individual; therefore, it traces back to the origins (antecedents), while psychology reserves for itself the principal task of following the psychological development of the subject in his school life (diaries); a task in which it will nevertheless go hand in hand with anthropology since the latter must follow at the same time the physio-morphological development of the subject himself.

Accordingly, the gathering of antecedent statistics is the task of anthropology. The antecedent statistics may be called the history of the genesis of the individual; the manner of collecting them is by means of enquiries that are generally made of the child's nearest relations (the mother) or of the teachers who have superintended his previous education. The enquiries are conducted under the guidance of a certain system of which we give the following outline:

anamnesis biopathological remote ascendant collateral near mother conception pregnancy delivery lactation child first development of dentition locomotion speech maladies incurred maternal opinions of child character intelligence etc. sociological vocation of parents their morality their culture their care of their children school record opinions of teachers, history of previous schooling.

We may distinguish biopathological antecedents, which have regard to the organism of the child as a living individual; sociological antecedents, having regard to the social environment in which the child has

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