III. "Results derived from the Natality Table of Kőrösi by employing the Method of Contours or Isogens." By FRANCIS GALTON, F.R.S. Received January 12, 1894. There are three variables in the statistics of natality. The age of the father is one, that of the mother is another, and the percental offspring of parents of those ages is the third. These three variables may be coordinated in the same way as that which is daily followed at meteorological offices in dealing with (1) the longitudes of the various stations, (2) their latitudes, and (3) the barometric height at each. After these data have been entered on a chart in their proper places, contours, known by the name of isobars, are drawn to show the lines of equal barometric pressure. In natality tables, the ages of the father and the mother take the place of the longitudes and latitudes in weather charts, and lines of similar birth rates, or as I would call them, " isogens," take the place of isobars.

Table I contains the means of each set of four adjacent entries as shown by the arrangement below, the left-hand diagram showing the four entries, and the right-hand one showing their mean.

The

entries themselves were copied to the nearest integer from Kőrösi's tables. The means are recorded in Table I to the nearest integer only, subject to an allowance of correction not exceeding 0·30 for the sake of slight smoothing; thus 24-25, which would otherwise have been entered as 24, might be treated as if it were 24·25+0·30 : =24.55 and be entered as 25. Similarly 24-75 might be entered either as 25 or as 24. It will be seen by the right-hand diagram that the position of the mean corresponds to the first moment of the years shown at the side and top; therefore the interval to which the annual birth rate corresponds is made up of the half year before and after that epoch.

The means that are enclosed in brackets are those in which one or

Table I.--Annual Percentage of Births according to the Ages of the Father and Mother, derived from Körösi's Table

of Natality at Budapest.

The tabular values refer to the half-year before and after the beginning of the year entered at the top and side.

more of the four squares from which they were derived was blank. They are, of course, less trustworthy than the rest; moreover, they may depend on less than 100 families.

The ages of married couples are distributed over only about one-half of the squares of Table I, as there are too few examples of other ages to be statistically available. This partial distribution is well seen in the diagram of isogens, where a dotted outline encloses all the material that can be used with safety. The broken line AB corresponds to the instances in which both parents are of the same age. The chart is practically limited to marriages in which the wife is less than five years older, and less than seventeen years younger, than her husband.

It will be noticed that the isogens run in nearly straight, diagonal, and equidistant lines across the greater part of the chart. If we omit six squares in the upper left-hand corner where there is no room for an isogen, we shall find these diagonal lines to cross 89 of the total number of 118 entries, or between eight and nine tenths of them. This indicates the existence of a very curious and unexpected law of natality, which is well brought out by Table II, which shows the values measured from the dots marked on the isogens. They have been taken at convenient places to serve as examples, one at the beginning, one at the end of the straight portion of each, and at some other intervening places.

In Table II are given the ages of the father and mother that correspond to each of these dots.

As a consequence of the straightness of the isogens, the sums of the

ages of the parents to which each point in the straight portion of the same isogen refers are constant. The difference between their ages is of no account whatever in eight or nine tenths of the total number of marriages; it is only when the wife is older than the husband or when she approaches the limit of the child-bearing age, that this curious

law ceases to hold true. The connexion between it and the straightness of the isobar is easily understood from the equation to a straight line of x+y= constant, for if æ represent the age of the father, f, and if y represent that of the mother, m, then ƒ+m = constant. That this is a fact is conspicuously evident from the columns headed B+C in Table II. This is the first curious law.

Again, through a coincidence between the increasing age of either parent, and the decrease of fertility, it happens that the sum of the three elements of (1) father's age, (2) mother's age, (3) percental birth-rate in a year has a value that is itself approximately constant, as is seen in the column headed A+B+C. Its lowest limit is 90 and its highest up to the isogen of 10 per cent. is 96, but it has increased to 98 at the isogen of 5 per cent. If we accept for it a constant value of 93 or 94 we shall never be far wrong in the larger part of the chart.

From this follows the second curious law that if we wish to calculate the percental birth-rate per annum for a married couple within the limits of the chart where the isogens run straight and parallel, we have only to add the ages of the father and mother and subtract the total from 93 or 94, in order to obtain it with considerable pre cision. The approximate limits within which this law obtains are : (1) the wife is not to be older than her husband; (2) she is not to be less than twenty-three years of age, nor (3) more than forty.

Example. In any large number of husbands and wives living under like conditions to the inbabitants of Budapest, whose respective ages at their nearest birthdays, to 21st June, 1892, were: that of the father, thirty-five, that of the mother, twenty-seven; then the number of children born to them during the year 1892 would be at the rate of 93-(35+27) per cent. 31 per cent.; the isogen makes it about 32 per cent.*

=

I shall not now enter into the other salient peculiarities of the isogens further than to allude to the curious change in their course which occurs when the wife is older than the husband. When she is from thirty to thirty-eight she certainly seems to be appreciably more fertile with a husband of her own age or somewhat older than she is with one who is younger. I should hesitate to ascribe this to physiological causes without corroborative evidence derived from breeders of stock. It is very possible that indifference on the part of young husbands to ageing wives may have something to do with it.

It is almost needless to say that if it be desired to obtain the observed birth-rates for a mother of any specified age and for fathers of

* A rough mechanical arrangement was exhibited by which isogens may be drawn. It consists of three sliding pieces connected by a string. A coloured patch is pasted on the back board to show the limits within which the isogens drawn by it are trustworthy.