A stout, robust child will weigh less, in an absolute sense, than an adult man who is extremely thin and emaciated; but relatively to the mass of his body, he will weigh more. Now this relative weight or index of weight, the ponderal index, gives us precisely this idea of relative embonpoint, of the more or less flourishing state of nutrition that any given individual is enjoying. Hence it is a relation of great physiological importance, especially when we are dealing with children.
The calculation of the ponderal index ought to be analogous to that of other indexes; what has to be found is its relation to the stature reduced to a scale of 100. In this case, however, we find ourselves facing a mathematical difficulty, because volumetric measurements are not comparable to linear measurements. Consequently it is necessary to reduce the measurement of weight by extracting its cube root, and to establish the following equation:
St:∛(W) = 100:X
whence
Pi = 100(∛(W))/S
The application of this formula necessitates a troublesomely complicated calculation, which it would be impracticable to work out in the case of a large number of subjects. But as it happens, tables of calculations in relation to the ponderal index already exist, thanks to the labours of Livi[35] and it remains only to consult them, as one would a table of logarithms, by finding the figure corresponding to the required stature, as indicated above in the horizontal line, and the weight as indicated in the vertical column.
Some authors have thought that they were greatly simplifying the relation between weight and stature by calculating the proportional weight of a single centimetre of stature and assuming that they had thus reduced the relation itself to a ratio based upon a single linear measurement (one centimetre), analogous to the ratio established by the reduction of the total stature to a scale of 100. But evidently such a calculation is based upon two fundamental errors, namely: first, no comparison is ever possible between a linear measure and a measure of volume; and secondly, the relation which we are trying to determine is that between synthetic measurements, i.e., measurements of the whole, and not of parts.
Fig. 37.
In the aforesaid method of computing (which is accepted by such weighty authorities as Godin and Niceforo), the number expressing the weight in grams is divided by the stature expressed in centimetres, and the quotient gives the average weight of one centimetre of stature expressed in grams. This method, which sounds plausible, may easily be proved to be fallacious, by the following illustration, given by Livi in his treatise already cited (Fig. 37). The two rectangles A and B represent longitudinal sections of two cylinders, which are supposed to represent respectively (in A) the body of a child so fat that he is as broad as he is long (the rectangle A is very nearly square), and (in B) that of a man of tall stature and so extremely thin that he very slightly surpasses the child in the dimensions of width and thickness (note the length and narrowness of rectangle B). Evidently the ponderal index of A is very high and that of B is very low. But if we calculate the proportional weight of one centimetre of stature, it will always be greater in the man than in the child, and consequently we obtain a relation contrary to that of the ponderal index.