Pedagogical Anthropology

Cephalic Index.—This is the proportion between the maximum transverse and longitudinal diameters. It is obtained by applying the familiar formula:

Ci = (100×d)/(D)

in which d represents the transverse diameter and D the longitudinal. The index represents the percentual relation between the two diameters, and is obtained from the formula by reducing the greater diameter to a scale of 100, as follows:

D:100 = d:X, whence X = 100×d/D

Instead of working out the calculations, we may find the required index in the tables already compiled.

Volume.—The volume of the cranium cannot be taken directly, except in the case of a skull. After the various osseous foramina have been closed, the cranial cavity is filled through the occipital foramen with any one of a number of substances (millet, shot, water, etc.), which is afterward measured. The method of taking this measurement is practised on a facsimile of a cranium already calculated, and usually made of metal.

But in the case of a living person the direct calculation of the volume is impossible. Nevertheless various empirical methods have been sought for obtaining this measurement, even though imperfect and approximate. Recently renewed use has been made, especially in France, of an approximate calculation made by means of Broca's cubic index. The volume of the cranium is equal to half the product of the three diameters, divided by an index which varies according to age.

This index is as follows:

An index of cranial development is afforded by the maximum circumference. The average volume of the normal adult cranium is about 1,500 cubic centimetres: mesocephalic cranium.

When the cranium is much inferior in volume, it is called microcephalic (from 1,200 down to 700 cubic centimetres). When on the contrary it is much superior (from 1,900 up to 2,200 cubic centimetres), it is called macrocephalic or megalocephalic.

For the face, the following craniometric points should be noted:

Along a longitudinal line:

1. The nasion (point of meeting of the nasal and frontal bones).
2. Subnasal point (meeting of nasal septum with upper maxilla).
3. Upper alveolar point (between the two upper incisors at their point of insertion).
4. Lower alveolar point (point corresponding to the above, in the lower maxilla).
5. Mental point (middle point of the chin).
6. The following craniometric points are situated laterally.
7. Auricular point (corresponding to the auricular foramen; in living persons it is situated on the tragus).
8. Malar point (on the malar bones).
9. Zygomatic point (corresponding to the zygomatic arches).
10. Gonion or goniac point (angle of mandible).

The face also may be studied by inspection—prosoposcopy; and by measurement—prosopometry.

Prosoposcopy.—We proceed to inspection according to two norms: A. facial norm; B. lateral norm or norm of profile.

A. Facial Norm.—If it is a question of a living person, we make complete inspection of the visage, from the roots of the hair to the chin. First of all we direct attention to the forehead, which will give us an index of the development of the anterior region of the brain; next, we observe whether a plane passing longitudinally through the median line would divide the face into two equal halves (facial symmetry).

From an æsthetic point of view, the three following vertical distances ought to correspond in length:

Height of forehead (from the roots of the hair to the nasion).

Length of nose (from the nasion to the subnasal point).

Labio-mental height (from the subnasal point to the point of the chin). And in regard to width the three following horizontal distances ought, according to the æsthetic laws of art, very nearly to correspond (especially in the female face):

Width of forehead, between the two external orbital points.

Bimalar width, between the two malar points.

Bigoniac width, between the two gonia.

It should be remembered that the standards of beauty do not necessarily coincide with those of normality.

B. Lateral Norm.—In observing the face according to this norm, three facts should be chiefly noted:

1. The relative volumetric development between facial and cerebral cranium.
2. The direction of the forehead, which, in the normal profile, ought to be vertical.
3. Whether the facial profile protrudes or not beyond the extreme anterior limit of the forehead.

Prosopometry.—-Many forms of measurements are taken on the skeleton of the face and many total and partial indices are obtained, such, for instance, as the facial index, the orbital index, the nasal index, etc.

Measurements of diameters and angles are also taken on the face of the living subject and indices are obtained.

We, however, shall limit ourselves to indicating only those measurements which are taken most frequently in our special field of application.

The diameters and the height of the face are obtained by the craniometric calipers and Mathieu's compass with sliding branches; the facial angle is measured in projection by means of the double square; and directly, by the goniometer.

One mode of measuring the facial angle in projection is that of drawing the facial profile with the help of special instruments; or else of taking a photograph in perfect profile and tracing and measuring the facial angle on the picture.

Principal Linear Measurements:

Total length of visage: from line of hair root to point of chin.

Total length of face: from the nasion to the point of the chin.

Length of the nose: from the nasion to the subnasal point.

Height of mandible: from the upper edge of the lower incisors to the lower edge of mandible.

Subnase-mental height: from the subnasal point to the point of the chin.

Bizygomatic diameter: between the two bizygomatic arches.

Bimalar diameter: between the two malar points.

Bigoniac diameter: between the two gonia.

Biorbital diameter: between the two external borders of the orbits.

Gonio-mental distance: from the goniac point to the point of the chin.

Auriculo-frontal radius: from the tragus or from the auricular point to the ophryon.

Auriculo-subnasal radius.

Auriculo-mental radius.

(The last four measurements, if compared right and left, give an index of facial symmetry; the radii when compared together serve as an indirect measure of prognathism.)

Width of nose between the external borders of the nostrils (the branches of Mathieu's compass are placed tangent to the nostrils).

(The index of the nose is obtained from the length and breadth, by applying the well-known formula of indices; the nose thereupon receives various names—leptorrhine, mesorrhine, platyrrhine).

Width of orbit: from the inner extremity of the ocular rima (eye-slit) to the external border of the orbit.

Width of the ocular rima: between the two extremities of the rima.

Width of the labial rima: between the two extremities of the rima.

Length of the ear: from the highest upper edge of the auricle to the lower extremity of the lobule.

Index of the ear: this is obtained, by the well-known formula, from the length and breadth. The normal index is 50; the types of ear above 50 are low types.

Anthropologists obtain the facial index from the skeleton, especially for the purpose of determining the proportion of the face in human remains found in the geological strata. In such crania the mandible is wanting, and the teeth are wanting. Consequently, there are several ways of computing the facial index, because, while the transverse or bizygomatic diameter, which is considered as the lesser diameter, always remains constant, the longitudinal, which is considered as the greater, varies. The longitudinal diameter is calculated sometimes from the ophryon to the chin, at others from the ophryon to the point of insertion of the two upper middle incisors. In the first case it is now less, and again greater than the bizygomatic diameter; in the second case, it is always less, and the resulting facial index is notably greater than 100.

The most usual formula for the facial index is the following:

Fi = (bizygomatic diameter×100)/(ophryo-mental diameter)

on the basis of which Pruner Bey gives the following mean averages according to race, for the general facial index:

This index is not exact and constant, like that for the cranium; in fact, in case a person loses his teeth the index is altered. At the present day, especially in the French school, the anterior or total facial index is taken into consideration, in which the vertical diameter is measured from the vertex of the head to the chin (Collignon), and, consequently, the index is always less than 100. The following is the nomenclature that results for the anterior facial index:

If we take for the measure of length that of the visage, i.e., the distance between the middle point of the frontal line of roots of the hair and the chin, we obtain indices that are higher by 5 than those of the French school, namely:

In many cases this index differs in the individual by as much as 10 from the cranial index, as I proved in my work on the population of Latium. Consequently, anyone who has a cranial index of 81 ought to have a visage index of 71, etc.

Contrary to what happens in the case of the cranium, the index of the face varies according to the age, the face being very short in childhood, and much longer in the adult.

Angles.—The angles distinguished by anthropologists are so numerous that it is impossible for us to take them all under consideration.

In the case of a living person, the angles may be measured directly with the aid of Broca's goniometer; the transverse branch passes across the subnasal point; the two antero-posterior branches are inserted, with the buttons with which they terminate, into the external auricular canals; the vertical branch, swinging on a hinge, is adjusted in such a way that the little rod which it carries at the end rests upon the ophryon.

This complicated instrument resembles an instrument of torture and could not be applied to children; furthermore, it is difficult to adjust, and consequently the angles that it gives are inexact: every muscular contraction causes the angle to vary. For this reason the goniometer is impracticable.

If, by means of an instrument we trace the projection of the facial profile, the facial angle may be taken on such a drawing; it may also be traced and calculated on a photograph taken in profile.

Broca's angle is that included between the auricular foramen, the subnasal point and the ophryon.

Camper's angle is that included between the auricular foramen, the point of insertion of the upper incisors and the metopic point.

We, on the contrary, in judging of the facial angle, or rather of the existence and degree of prognathism, have resorted to inspection, aided by certain facial lines, namely (Fig. 104):

a. Vertical Facial Line.—If the subject holds his head level, with the occipital point in contact with a vertical rod, and his gaze fixed straight before him, then what we call the vertical line is the line perpendicular to the horizontal direction of the gaze, and tangent to the extreme anterior limit of the brain. This line, in the perfect human face, is perpendicular to the horizontal line uniting the auricular point with the subnasal point, and hence forms a right angle with it.

b. Line of Facial Profile.—This is the line uniting the nasal point with the subnasal point. This line is never vertical, and therefore cannot form a right angle with the auriculo-subnasal line, but forms an angle that approximates more or less nearly to a right angle (85°): this is the facial angle.

Transversely there is only one line for us to consider, and it has already been noted:

c. The auriculo-subnasal line, or line of orientation.

Facial Norm.—Our attention should be directed, as we have already said:

1. To the forehead.

This, if anomalous, may be:

• Broad (if greater than 133 mm.).
• Narrow (if less than 100 mm.).
• High (if over 60 mm.).
• Low (if under 50 mm.).

2. To the Symmetry of the Face.—If the face is notably asymmetrical, in respect to a plane dividing it longitudinally, the fact is at once perceptible. But a slight asymmetry may fail to be detected either by measurements (trago-mental diameters) or by inspection. Consequently, it will be well to follow certain practical rules in making this observation.

Observe first of all the median line of the face: the bridge of the nose, the nasal septum, the upper labial furrow and the point of the chin ought all to lie in the same vertical line; very often a slight deviation of the nasal septum above the upper labial furrow will betray the asymmetry; furthermore, the two naso-labial plicæ or folds should be noted, for they ought to be symmetrical in direction and in depth; lastly, we must observe the symmetry of the zygomatic prominences. We shall often discover three concurrent facts: a slight deviation in the median line of the face usually corresponding to the nasal septum; a greater depth of one of the naso-labial plicæ; and a greater prominence of the zygoma and the cheek on the same side.

Our attention should next be turned to the correspondence required by æsthetics between the following three diameters:

• Minimum frontal.
• Bizygomatic.
• Bigoniac.

A very notable difference between these distances may also lead to the discovery of anomalies.

Sometimes we may discover, even by inspection alone, a notable narrowness of the frontal diameter, as compared with the other two.

The bizygomatic diameter may show an exaggerated development, and this is frequently accompanied by a hollowness in the temporal and upper maxillary regions and by a beak-like prognathism (prominence of the middle portion of the upper maxilla); at other times this degenerative sign calls our attention to the mongoloid type.

The bigoniac diameter may also show an exaggerated development due to the enormous volume of the mandible (criminaloid type—Lombroso's assassin type). It is necessary to supplement our observation with the measurement of these three diameters, because it may very often appear to the eye that the minimum frontal diameter is below the normal, merely by comparison with the other two diameters which are overdeveloped; while when measured, it may turn out to be normal. Or, conversely, the other diameters, the bizygomatic or bigoniac, although actually normal, may appear overdeveloped, because of the shortness of the minimum frontal diameter (see "Faces of Inferior Type.")

Meanwhile we must not forget that the following are signs of grave degeneration:

a. The minimum frontal diameter less than 100 mm. (the gravity of this is increased if at the same time the other two diameters are found as described in b).

b. The other two diameters greater than 110 mm. (Lombroso's born delinquents, assassin type).

Lateral Norm, or Norm of Profile.—Our attention ought to be directed, as we have already said:

1. To the direction of the forehead. If abnormal, this may be:

• a. Receding;
• b. Bombé.

The receding forehead is an indication of an incomplete or defective development of the frontal lobe of the brain; we find the forehead notably receding in the microcephalic type.

The bombé forehead is characteristic of hydrocephaly, but may occur also in the scaphoid cranium. When the forehead is bombé, the facial angle becomes equal to or greater than a right angle, because the face recedes beneath the extreme anterior boundary of the brain; in this case we have the opposite case to prothognathism, namely, orthognathism.

2. Our attention should next be directed to the facial profile, in order to observe the form and degree of prognathism.

The authorities distinguish three principal forms of prognathism:

a. Prognathism properly so-called: prominence of the upper maxilla as a whole.

b. Prophatnia.—Prominence of the alveoli.

c. Progeneism.—Prominence of the mandible—the lower dental arch projects in front of the upper.

Measurements of the Thorax

Principal anthropometric points: acromial point; sternal fossa; xiphoid point; mammillary points.

Measurements.—Thoracic Circumference.—Already described among the measurements of the form.

Recording instruments are now made that are exceedingly complicated and quite costly, that register the movements of respiration; they are used in medical clinics, but would be of little practical use in our schools.

Axillary and Submammary Circumference.—Taken as above, but at different levels.

Biacromial Diameter.—This is taken by means of special calipers called a thoracimeter or pelvimeter, because it is used to obtain the big measurements of the body (thorax and pelvis). The two buttons at the ends of the branches are applied to the acromial points, while the measurer occupies a position in front of the subject to be measured.

Transverse Thoracic Diameter.—The buttons of the thoracimeter are applied on a level with the mammary papillæ, along the axillary lines (vertical lines descending from the centre of the arm-pits).

Antero-posterior Thoracic Diameter.—This is also taken at the level of the nipples: the branches are applied anteriorly on the sternum and posteriorly on the vertebral channel.

These two diameters serve to furnish the thoracic index:

Ti = (100×d (antero-posterior))/(D (transverse))

Spirometer.—The subject takes a maximum inspiration and retains his breath until he has exactly fitted his mouth to the apparatus; then he emits all his breath in a forced expiration. This causes the index to rise, and the amount may be read upon it.

Sternal Length.—From the xiphoid point to the sternal fossa.

Bimammillary Diameter.—Distance between the two nipples.

Abdomen.—It would be really difficult to take measurements of the abdomen in the school. The principal anthropometric points to remember are the umbilical point, the two antero-superior iliac points, the pubis.

The distances which it would be useful to take are the following: xipho-umbilical and umbilico-pubic distances, which give an idea of the upper development (liver) and lower development (intestines) of the abdomen, and the biacromial diameter which measures the width of the pelvis.

Limbs.—In the case of the limbs also it is by no means easy or practicable to take many measurements. Consequently it should be sufficient to indicate that there are a great number of different measurements for every different segment of the limbs.

There are two principal instruments needed for this: a large compass with adjustable branches, for the long segments, and a small compass for the short segments. With the large compass we measure the length of the upper arm and forearm, the length of the thigh and shin, the length of the foot. With the small compass we measure the total length of the hand, its width, the length of the fingers and of the digital segments, etc.

The circumference of the limbs is taken with the ordinary metallic tape.

In order to fulfil the present-day scope of pedagogic anthropology, it is sufficient to take only a few measurements (the form and the head), but it is necessary to take them with great accuracy, and above all, to verify one's personal ability as a measurer, so that everyone who wishes to try the experiment may have a reliable method of testing himself. To this end it is necessary to know how to calculate one's own special personal error.

The Personal Error

In anthropometry, a knowledge of the anthropometric points, the instruments to apply to them, their use and their interpretation, is not sufficient. There is need of prolonged experience in accordance with the accepted method and under a practical guide.

As a matter of fact, the degree of accuracy with which a measurement is taken is always relative, no matter who takes it, but in the case of a person who has had no practice this relativity may present so wide a margin as to be practically useless.

To obtain an approximate figure of a measurement means nothing, unless the figure is supplemented not only by a statement as to which of the accepted methods was used in taking it, but also by a minute description of the manner in which this method was carried out.

It is necessary to bear in mind:

1. That the ability to find the anthropometric points implies a certain knowledge of anatomy; it is a practical research, to be made under the guidance of a teacher, while the actual finding of the points as well as the taking of the measurements, should be left to the learner.
2. That the manner of applying the instruments is not without effect upon the resulting figure: for example, if the compass is held horizontally in measuring the frontal diameter, the result is different from what it would be if the instrument were held vertically. If the compass is held by the extremities of the branches, the diameter is slightly different from what it would be if the compass was held by the handle. Accordingly, it is necessary to describe minutely how we are accustomed to hold the instruments.
3. That the resulting figure differs according to whether or not the screw has been turned, or whether it has been read in position, or by approaching the instrument to the eye.
4. That when an instrument is old, it registers different results from those it gave when new; consequently, it is necessary to verify it, before proceeding to take a series of measurements. Hence it is proper to state not only precisely what instrument is used, but also that the precaution has been taken to verify it.

But what is still more important is to find out one's own personal data.

If the same measurement is taken twice under precisely similar conditions, the same figure is hardly ever obtained both times; everyone, even the most experienced, has his own personal error. By practice the amount of this error may be steadily lowered, but cannot be eliminated. Constant figures are an evidence of dishonesty, of mere copying; they are almost certainly not authentic.

It is important to know one's own average error.

It is calculated as follows:

Let us suppose that successive attempts have resulted in the following figures relative to the same measurement:

9, 10, 11, 12, 8

The mean average of these numbers is

(9+10+11+12+8)/5 = 10

Let us see how the values obtained differ in respect to 10:

9 10 11 12 8 10

-1, 0, +1, +2, -2 = differences from the mean average figure. We now take the average of these differences, disregarding the plus and minus signs:

(1+0+1+2+2)/5 = 6/5 = 1.2 = mean average error

The personal mean error is a datum that it is necessary to know in order to give value to any measurements that we may wish to give forth.

In taking the various test measurements for the purpose of calculating one's personal error, it is well to use the precaution of not taking them twice at the same sitting, but after an interval of time, not only so that all marks will have disappeared that may have been left upon the skin by the instrument in the act of measuring, but also that the preceding figure will have faded from our memory. Accordingly, the measurements should be repeated on successive days and if possible under the same conditions of time and place.

It is well to make a careful choice of the time and place, because these also have their effect upon the figures.

It will be observed that if the measurements are made in a well-appointed place, with a steady light, without noises, in short, without disturbing causes, the personal error is much more easily decreased, i.e., the measurements are more exact, because the measurer can better concentrate his attention.

Even the hour of the day has an influence upon the figures. It is known that none of us has the same ability to perform our various tasks at all the different hours of the day; for instance, it is not a matter of indifference whether we ask the pupils in a school to solve a problem at one hour of the day rather than at another. This is true of all occupations, and hence also of anthropometry; there are certain hours of the day at which fewer errors in measurement will be made, independently of the state of fatigue.

Consequently, it is well to know this individual datum, and to tell at what hour and in what environment the measures have been taken.

The figures are of more value if they have been compared with the results of other observers; it is necessary, after we have found our own average error, to select, for the purpose of verifying our results, some other observer, of similar experience to our own, and whose personal error is also known.

Here it is necessary to take into consideration still another factor—one's personal susceptibility to suggestion. If we have confidence in the person through whom we verify our figures, we are inclined to obtain figures equal to his own. We have only to compare our earlier figures with those since we began to use him as a test, in order to see whether, and to what extent we are influenced by suggestion. Hence, to obviate this danger it is necessary to obtain our respective figures without communicating them to each other.

It will also be necessary to take precautions not to be influenced by suggestion under any other circumstances. For instance, we are in hopes, while taking a series of measurements of school children, that we shall be able to prove that the heads of the more intelligent are larger than those of the less intelligent. In order that the figures shall be free from alterations due to suggestion, it is necessary that the measurer, while actually taking the measurements, shall be unaware which children are better and which are worse, from the intellectual point of view.

The personal error cannot be calculated in regard to a single measurement and then applied to all the others, but it must be worked out anew for every separate measurement; it oscillates variously, as a matter of fact, in relation to the longer and shorter diameters, the cranial measurements, and the measurements of the trunk and the limbs.

We are sufficiently skilled to take measurements when we have attained for measurements of cranial diameters a mean error of from 1 to 2 mm., for the vertical cranial diameter one of 4 mm., and for the stature, one of from 5 to 6 mm.

Finally, in anthropometry, theory is of no value without a long and intelligent practice, constituting an actual and personal education in anthropometric technique.

All anthropometric figures have a relative value dependent upon the extent of this education in the individual investigator.

This is a case in which it may be said that the figures are worthless without the signature.

CHAPTER VIII
STATISTICAL METHODOLOGY

Having taken measurements with the rigorous technical precision that is to-day demanded by anthropometry, we should know how to extract from these figures certain laws, or at least certain statistical conclusions.

There are two principal methods of regrouping the figures:—mean averages and seriations.

Mean Averages.—Averages are obtained, as is a matter of common knowledge and practice, by taking the sum of all the figures and dividing the result by the number of data. The general formula is as follows:

(a+b+c+d)/(1+1+1+1)

When comparative figures are given, as, for example, those recorded by Quetélét for the stature, the diameters of the head, etc., such figures are always mean averages.

Such averages may be more or less general. We might, for example, obtain a mean average of the stature of Italians, and this would be more general than the mean stature for a single region of Italy, and this again more general than the mean stature for a city, or for some specified social class, etc.

It is interesting to know how the mean will be affected, according to the number of individuals examined, because it is obvious that the mean stature of Italians cannot be based upon measurements of all Italians, but upon a larger or smaller number of individuals. Now, if we take various different numbers of individuals, shall we obtain different mean statures? And if so, what number of subjects must we have at our disposal in order to obtain a constant medial figure, and hence the one that represents the real mean average? It has been determined that a relatively small number will suffice to give the mean, if the measurements are taken with uniform method and from the same class of subjects (sex, age, race, etc.); for the cranium, 25 subjects are sufficient, and for the stature, 100 subjects.

This method furnishes us with an abstract number, insofar as it does not correspond to any real individual, but it serves to give us the synthetic idea of an entirety. In anthropology we need this sort of fundamental synthesis before proceeding to individual analysis for the purpose of interpreting a specified person.

Now, it is evident that the figures representing the mean stature for each region in Italy give us a basis for judging of the distribution of this important datum, while an accumulation of a hundred thousand individual figures would lead to nothing more profitable than confusion and weariness.

The following table, however, is quite clear and instructive:

MEAN STATURE IN ITALY

(According to Departments)

Yet the interpretation of such a table is not simple; it is necessary to read the numbers, to remember them in their reciprocal relation; and it demands effort and time to acquire a clear and synthetic idea of the distribution in Italy of this one datum, stature.

On the other hand, we must lose as little time and spare our forces as far as possible. The value of positive methodology lies in the extent to which it accomplishes these two subjects.

Geographical charts serve the purpose of this desired simplification. Let us take an outline map of Italy, divide it into regions, and colour these different regions darker or lighter, in proportion as the stature is higher or lower.

The gradations and shadings in colour will tell us at a single glance, and without any fatigue on our part, what the table of figures reveals at the cost of a very perceptible effort. Little squares must be added on the margin of the chart, corresponding to the gradations in colour, and opposite them the figures which they respectively indicate—after the fashion in which the scale of reduction is given in every geographical map. In this way we may study these charts, and their examination is pleasant and interesting, while it successfully associates the two ideas of an "anthropometric datum" and of a "region," a result which a series of figures, pure and simple, could not achieve.

We have seen Livi's charts of Italy, both for stature and for the cephalic index. Analogous charts may be constructed for all the different data, for example, the colour of the hair, the shape of the nose, the facial index, etc. In the same manner we may proceed to a still more analytical distribution of anthropometric data among the different provinces of a single region. For example, I myself prepared charts of this sort for the stature, the cephalic index and the pigmentation of the population of Latium.

Sometimes we want to see in one single, comprehensive glance, the progress of some anthropological datum; for instance, in its development through different ages. Quétélet's series of figures for growth in stature, in weight, in the diameters of the head, the cranial circumference, etc., offer when read the same difficulty as the similar tables of distribution according to regions. On the contrary, we get a synthetic, sweeping glance in diagrams, such as the one which shows the growth of stature in the two sexes. The method of constructing such diagrams is very simple, and is widely employed. When we wish to represent in physics certain phenomena and laws; or in hygiene, the progress of mortality through successive years, etc., we make use of the method of diagrams.

Let us draw two fundamental lines meeting in a right angle at A (Fig. 151): AS is known as the axis of the abscissæ; AO, the axis of the ordinates. We divide each of these lines into equal parts. Let us assume that the divisions of AS represent the years of age, and those of AO the measurements of stature in centimetres; and since the new-born child has an average height of 50 cm., we may place 50 as the initial figure. From the figure O (age) and from 50 cm. (measure), we erect perpendiculars meeting at a, where we mark the point. At the age of one year the average stature is about 70 cm., accordingly we erect perpendiculars from 1 (age) and from 70 (measure), obtaining the point c. Since the stature at two years is about 80 cm. the same procedure gives us the point e. Since the stature at the age of three is about 86 cm., I erect the perpendicular from a level slightly higher than half-way between 80 and 90, obtaining the point i; and so on, for the rest. Meanwhile we begin to be able to see at a glance that the stature increases greatly in the first year and that thereafter the intensity of its growth steadily diminishes.

If we unite the points thus constructed, the line of representation is completed.

The verticals 0a, 1c, 2e, etc., are the ordinates, and the horizontals 50a, 70c, etc., are the abscissæ of the line of representation; and since it is constructed along the intersections of these lines, they are for that reason collectively called coordinates. It is usual in constructing these diagrams to mark the coordinates in such a way that they will not be apparent, instead of which only the axes and the line representing the development of the phenomenon are shown (Fig. 152).

Sometimes a different method of representing the phenomenon graphically is followed, namely, by tracing the successive series of distances developed on the ordinates (Fig. 153); in which case the characteristic arrangement of the lines causes this to be known as the organ-pipe method.