class men has sunk to 294, and the percentage becomes progressively lower as we go to the lower classes; thus it is 247 among the fathers of the fourth class men, 13.8 among those of the pass men and 12.8 in those of the no degree class.

The two points to be noticed about these figures are firstly that the percentage of first and second class fathers diminishes as one passes down through the various classes of sons, and secondly that it does so in a very orderly way. In order to show both these points quite clearly Diagram I. has been constructed. In this diagram the height of the six vertical lines is proportional to the percentage of fathers in the first or second class of each of the six groups of sons respectively. The upper end of the left hand upright, which represents this percentage among the fathers of first class sons, and the upper ends of the five other uprights, which indicate the same fact concerning the fathers of the five other classes of sons, lie almost on a straight line, shown in the diagram, which slopes down steeply from left to right, making an angle of 45 degrees approximately with the horizontal.

One more fact must be referred to concerning Table IA. before leaving it for a time to consider Table II A. It may give the impression that exactly the same number of fathers are included in it as sons; that is to say that it deals with only one son of each 2459 fathers; but this is not the case. 2459 men of the younger generation are included, but as in some cases two or three or even more of these are sons of the same father he may be included two or three or even more times, being counted once over for each of his sons that are dealt with in the table.

We will now pass on to the consideration of Table II A., which has been constructed in exactly the same way as Table I A., but out of entirely independent material, for in this case all the sons took their degrees or should have taken them prior to 1860, but since no fathers prior to 1800 could be included it happens that almost all the sons took their degrees between 1830-1860, and all the fathers between 1800-1830.

As will be remembered the fourth class of honours was not introduced till 1830, so that those fathers who took honours are only divided into three classes. If we examine this table in the same way as we have examined the previous one we shall find a remarkable agreement, for 41.9 per cent. of the sons in the first class had fathers in either the first or second class, and the percentages of the remainder whose fathers achieved this distinction are as follows: 407 per cent. of the second class, 33.3 per cent. of the third, 28.1 per cent. of the fourth, 20.1 per cent. of the pass degree men, and 12.9 per cent. of those who took no degree. These numbers are shown graphically in Diagram II. As in Diagram I. the upper ends of the vertical lines which represent the percentages lie almost on a straight line which slopes steeply downwards from left to right.

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10

Enough has now been said to show not only that signs of hereditary influences are well marked, but-although nothing certain can be predicted in individual cases-there is a considerable regularity when we deal with large numbers, a regularity which becomes more marked as the numbers are increased. However we proposed to go further than this and make a definite measure of the intensity of this force, and we will now go on to describe the results of this attempt.

15

20

25

Sons with first class honours

30

35

Sons with second class honours

DIAGRAM II. (date of degree of sons, 1830-60).

The heights of the vertical lines show in what percentage of cases the fathers have taken either first class or second class honours. The diagram is intended to show that the percentage of fathers who obtained this degree of distinction diminishes with some regularity, as one passes downwards from the sons with first class honours to those with no degree.

Two methods have been used in doing this, namely that of "Contingency" and that of "Fourfold Correlation." The former of these will be found in the paper by Professor Karl Pearson "On the Theory of Contingency and its relation to Association and Normal Correlation," which forms No. 1 of the Biometric Series of the Drapers' Company Research Memoirs, published by Dulau and Co. in 1904. The latter will be found fully described in the paper by the same author "On the Correlation of Characters not quantitatively measurable" (Phil. Trans.

Vol. cxcv. A. pp. 1–47). It will be interesting to see how far the results obtained by the two methods used are in accordance with one another. With regard to the first named or "Contingency" method two slightly different variations of it are employed, the result of one of these, called "the Mean Square Contingency Coefficient," and represented by the symbol C1, was found for Table I A. to be 26, while for Table II A. it was 29. The result of the other, called "the Mean Contingency Coefficient," and represented by the symbol C1, gave 29 for Table I A. and 31 for Table II A.

In order to obtain the correlation coefficients by the fourfold method from Tables IA. and II A. it is necessary to reduce these tables to their simplest form. As this simplification can be done in many slightly different ways, slightly different values of the correlation coefficient may be found. Thus Tables IB., IC., and ID. are simplifications of Table I A. Table IB. tells one that of the 2459 sons, 478 obtained honours in classes I. and II.; whereas 1981 were placed in the third or fourth classes or obtained either a pass degree or no degree; and that of the 478, 160 had fathers of equal distinction to themselves and 318 who were inferior to them, having merely obtained third or fourth classes or pass degrees, or having obtained no degree at all; of the 1981 composing the inferior class of sons, 348 had fathers in the upper class and 1633 in the lower. From this table the correlation coefficient, for which the symbol ↑ is used, r was calculated and found to be 29. Table IC. was made in exactly the same way as IB. except that the division between the sheep and the goats was made below the third class instead of below the second, so the former category is larger than in the previous case, both with regard to the fathers and to the sons; it is unnecessary to repeat the numbers here, as they are set out in the table in question. The correlation coefficient calculated from this table = 31. In making Table ID. the same process was repeated except that all those who obtained honours were separated from all those who did not do so. The value of the correlation coefficient obtained was 28. Thus it will be seen that the value of this coefficient is in this case very much the same whether the division between the upper and lower groups of fathers and between the upper and lower groups of sons, is made beneath the second class, the third class, or the fourth class. The mean of the three values obtained is 293.

Tables II B. and II C. correspond with Tables I B. and IC. and are made from the earlier material classified in Table II A. in precisely the same way as these tables are made from the more modern material included in Table I A. The correlation coefficients obtained from them are 34 and 33 respectively, the mean of which is 335, a somewhat higher value than the 293 mentioned above. Table III. summarizes all the coefficients calculated hitherto; it shows the degree of similarity in the results which may be expected in treating two quite independent sets of material each by three different methods. Thus if we consider

unity to be the measure of perfect resemblance and nothing of no resemblance, by disregarding minor differences and speaking somewhat broadly, we may say that degree of intellectual similarity between father and son, as indicated by the degrees which each took, is 3 or nearly 3.

C. Resemblance between Brother and Brother.

We will now pass on to the consideration of fraternal resemblance. In Table IV A. the material used consisted of those families of brothers in which the majority of the brothers took their degrees between the years 1860-1892; it is constructed much on the same principle as Tables I A. and II A., except that instead of considering each son in relation to his father, every man included is considered in relation to each of his brothers taken in turn. Thus in dealing with a family of two brothers, X and Y, who may have taken a first and a fourth class respectively, in the first place X is taken and is entered in the fourth square from the left of the top row as being a first class man with a fourth class brother, and then Y is entered in the left hand square of the fourth row as being a fourth class man with a first class brother, so that each family of two contributed two to the total number of 4266 pairs. Now let us consider a family of three brothers, X, Y and Z, who were placed respectively in the first, second and third classes; X is in this case entered twice-once in the second square from the left of the top row, as being a first class man with a second class brother, and once in the third square of the same row as having a third class brother; but if both Y and Z had been in the second class he would have been entered twice in the second square. Similarly Y is entered once in the left hand square and once in the third square of the second row, and Z is entered once in the left hand square and once in the second square of the third row. Thus the family of three contributes 3 x 2 = 6 to the total number, a family of four, 4×3=12, of five, 5×4=20, and so on; so that the total 4266 does not represent 8532 men taken together two by two as might possibly have been supposed, but 4266 pairs made from a smaller number of men. It will be noticed as a result of this method of construction, the table is symmetrical about an axis which runs from the left hand top corner to the right hand bottom corner. If to begin with we examine this table in the same way in which the examination of Tables I A. and II A. was begun, we shall see that out of the 339 first class men reckoned in the way described above, 154 or 45.5 per cent. had brothers in the first and second classes; of the 668 second class men, 254 or 38 per cent. had brothers attaining this standard; and that the percentages diminish steadily and fairly rapidly as we pass down our scale of classes to 276 per cent. for the third class men, 21-2 per cent. for the fourth class men, 15.3 per cent. for the pass

G. M.

2

men and 12.6 per cent. for those who obtained no degrees. If we compare this series with the corresponding one obtained from Table I A. we shall see a wellmarked general similarity between the two, but, although the percentage of first and second classes among the brothers of the men who failed to take degrees is practically identical with that among the fathers, it is considerably higher among the brothers than among the fathers of first class men; so that we can conclude from this alone that the resemblance between brother and brother is more marked than that between father and son; and it will be seen when we come to treat the contingency and correlation coefficients that the same result is obtained from them also.

It will be unnecessary to consider separately the Tables V A. and VI A.; the former is made from those families in which the majority of brothers took their degrees between 1830 and 1860, and in the latter those between 1800-1830 were used; a few individual brothers in this case did actually take their degrees later than 1830 and thus were capable of obtaining fourth class honours, but very few were so placed, and as the number of these was so small, in making the table it was not thought worth while to treat them separately and they were included among the pass degree men.

For all the three sets of material included in Tables IV A., VA. and VI A. respectively, the mean square contingency coefficient and the mean contingency coefficient were calculated, and in order to obtain various values for the correlation coefficient the original tables were simplified in different ways. Thus in constructing Tables IV B., V B., and VI B., divisions both vertical and horizontal were made in IV A., V A., and VI A., between the second and third classes and all the numbers in each of the four divisions of these tables thus defined were added together; to make Tables IV C., V C., and VIC. the original tables were divided between the third and fourth classes; and Tables IV D. and VD. were made by simply separating honours men from the rest. The results of all these variations in method are summarized in Table VII. From this it will be seen that all the correlation coefficients agree tolerably well together, and all the mean square contingency coefficients agree together exceedingly well, but that the latter do not agree with the former, being in each case considerably less, and it will be remembered that a difference in the same direction but not nearly so well marked was noticed with regard to the tables dealing with the fathers and sons, the results of which are summarized in Table III.

D.

The Influence of Family Traditions on the Results obtained.

The differences that are pointed out in the preceding paragraph between the mean square contingency coefficients and the correlation coefficients are probably due to the fact that the material is not homogeneous, that is to say that each