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same weight; those in K being the heaviest, L the next heaviest, and so on down to Q, which was the lightest. The precise weights are given in Table V, together with the corresponding diameter, which I ascertained by laying 100 peas of the same sort in a row. The weights run in an arithmetic series, having a common average difference of 0.172 grain. I do not of course profess to work to thousandths of a grain, though I did to less than tenths of a grain ; therefore the third decimal place represents no more than an arithmetical working value, which has to be regarded in multiplications, lest an error of sensible importance should be introduced by its neglect. Curiously enough, the diameters were found to run approximately in an arithmetic series also, owing, I suppose, to the misshape and corrugations of the smaller seeds, which gave them a larger diameter than if they had been plumped out into spheres. The results are given in Table V, which show that I was justified in sorting the seeds by the convenient method of the balance and weights, and of accepting the weights as directly proportional to the mean diameters, which can hardly be measured satisfactorily except in spherical seeds.

In each experiment seven beds were prepared in parallel rows; each was 1 feet wide and 5 feet long. Ten holes of 1 inch deep were dibbled at equal distances apart along each bed, and one seed was put into each hole. They were then bushed over to keep off the birds. Minute instructions were given and followed to ensure uniformity, which I need not repeat here. The end of all was that the seeds as they became ripe were collected from time to time in bags that I sent, lettered from K to Q, the same letters being stuck at the ends of the beds, and when the crop was coming to an end the whole foliage of each bed was torn up, tied together, labelled, and sent to me. I measured the foliage and the pods, both of which gave results confirmatory of those of the peas, which will be found in Table VI, the first and last columns of which are those that especially interest us; the remaining columns showing clearly enough how these two were obtained. It will be seen that for each increase of one unit on the part of the parent seed, there is a mean increase of only one-third of a unit in the filial seed; and again that the mean filial seed resembles the parental when the latter is about 15.5 hundredths of an inch in diameter. Taking then 15:5 as the point towards which filial regression points, whatever may be the parental deviation (within the tabular limits) from that point, the mean filial deviation will be in the same direction, but only one-third as much.

This point of regression is so low that I possessed less evidence than I desired to prove the bettering of the produce of very small seeds. The seeds smaller than Q were such a miserable set that I could hardly deal with them. Moreover, they were very infertile. It did, however, happen that in a few of the sets some of the Q seeds turned out very well.

If I desired to lay much stress on these experiments, I could make my case considerably stronger by going minutely into the

details of the several experiments, foliage and length of pod included, but I do not care to do so.

TABLE V.

WEIGHTS AND DIAMETERS OF SEEDS (SWEET PEA).

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PARENT SEEDS AND THEIR PRODUCE.

Table showing the proportionate number of seeds (sweet peas) of different sizes, produced by parent seeds also of different sizes. The measurements are those of mean diameter, in hundredths of an inch.

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II.-Separate Contribution of each Ancestor to the Heritage of the Offspring.

When we say that the mid-parent contributes two-thirds of his peculiarity of height to the offspring, it is supposed that nothing is known about the previous ancestor. We now see that though nothing is known, something is implied, and that something must be eliminated if we desire to know what the parental bequest, pure and simple, may amount to. Let the deviate of the mid-parent be a, then the implied deviate of the mid-grandparent will be a, of

the mid-ancestor in the next generation a, and so on.

Hence the

sum of the deviates of all the mid-generations that contribute to the heritage of the offspring is a(1+} +}+&c.) =a.

Do they contribute on equal terms, or otherwise? I am not prepared as yet with sufficient data to yield a direct reply, therefore we must try the effects of limiting suppositions. First, suppose

they contribute equally; then as an accumulation of ancestral deviates whose sum amounts to a, yields an effective heritage of only a, it follows that each piece of property, as it were, must be reduced by a sucession tax to of its original amount, because

Another supposition is that of successive diminution, the property being taxed afresh in each transmission, so that the effective heritage would be

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and this must, as before, be equal to a 3, whence

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The third limiting supposition of a mid-ancestral deviate in any one remote generation contributing more than a mid-parental deviate, is notoriously incorrect. Thus the descendants of "pedigree-wheat" in the (say) twentieth generation show no sign of their mid-ancestral magnitude, but those in the first generation do so most unmistakably.

The results of our two valid limiting suppositions are therefore (1) that the mid-parental deviate, pure and simple, influences the offspring to of its amount; (2) that it influences it to the of its amount. These values differ but slightly from, and their mean is closely, so we may fairly accept that result. Hence the influence, pure and simple, of the mid-parent may be taken as 1, of the mid-grandparent, of the mid-great-grandparent, and so on. That of the individual parent would therefore be, of the individual grandparent, of an individual in the next generation , and so on.

Explanation of Plates IX and X.

Plate IX, fig. a. Rate of Regression in Hereditary Stature. The short horizontal lines refer to the stature of the mid-parents as given on the scale to the left. These are the same values as those in the left hand column of Table I.

The small circles, one below each of the above, show the mean stature of the children of each of those mid-parents. These are the values in the right hand column of Table I, headed “Medians.” [The Median is the value that half the cases exceed, and the other fall short of it. It is practically the same as the mean, but is a more convenient value to find, in the way of working adopted throughout in the present instance.]

The sloping line AB passes through all possible mid-parental heights.

The sloping line CD passes through all the corresponding mean heights of their children. It gives the "smoothed" results of the actual observations.

The ratio of CM to AM is as 2 to 3, and this same ratio connects the deviate of every mid-parental value with the mean deviate of its offspring.

The point of convergence is at the level of mediocrity, which is 681 inches.

The above data are derived from the 928 adult children of 205 mid-parents, female statures having in every case been converted to their male equivalents by multiplying each of them by 1.08.

Fig. b. Forecasts of stature. This is a diagram of the mechanism by which the most probable heights of the sons and daughters can be foretold, from the data of the heights of each of their parents.

The weights M and F have to be set opposite to the heights of the mother and father on their respective scales; then the weight sd will show the most probable heights of a son and daughter on the corresponding scales. In every one of these cases it is the fiducial mark in the middle of each weight by which the reading is to be made. But, in addition to this, the length of the weight sd is so arranged that it is an equal chance (an even bet) that the height of each son or each daughter will lie within the range defined by the upper and lower edge of the weight, on their respective scales. The length of sd is 3 inches 2f; that is, 2 × 1.50 inch.

=

A, B, and C are three thin wheels with grooves round their edges. They are screwed together so as to form a single piece that turns easily on its axis. The weights M and F are attached to either end of a thread that passes over the movable pulley D. The pulley itself hangs from a thread which is wrapped two or three times round the groove of B and is then secured to the wheel. The weight sd hangs from a thread that is wrapped in the same direction two or three times round the groove of A, and is then secured to the wheel. The diameter of A is to that of B as 2 to 3. Lastly, a thread wrapped in the opposite direction round the wheel C, which may have any convenient diameter, is attached to a coun

terpoise.

It is obvious that raising M will cause F to fall, and vice versâ, without affecting the wheels AB, and therefore without affecting sd; that is to say, the parental differences may be varied indefinitely without affecting the stature of the children, so long as the midparental height is unchanged. But if the mid-parental height is changed, then that of sd will be changed to of the amount.

The scale of female heights differs from that of the males, each female height being laid down in the position which would be occupied by its male equivalent. Thus 56 is written in the position of 60 48 inches, which is equal to 56 × 108. Similarly, 60 is written in the position of 6480, which is equal to 60 × 1·08.

In the actual machine the weights run in grooves. It is also

taller and has a longer scale than is shown in the figure, which is somewhat shortened for want of space.

Plate X. This is a diagram based on Table I. The figures in it were first "smoothed" as described in the memoir, then lines were drawn through points corresponding to the same values, just as isobars or isotherms are drawn. These lines, as already stated, formed ellipses. I have also explained how calculation showed that they were true ellipses, and verified the values I had obtained of the relation of their major to their minor axes, of the inclination of these to the coordinates passing through their common centre, and so forth. The ellipse in the figure is one of these. The numerals are not directly derived from the smoothed results just spoken of, but are rough interpolations so as to suit their present positions. It will be noticed that each horizontal line grows to a maximum and then symmetrically diminishes, and that the same is true of each vertical line. It will also be seen that the loci of maxima in these follow the lines ON and OM, which are respectively inclined to their adjacent coordinates at the gradients of 2 to 3, and of 1 to 3. If there had been no regression, but if like bred like, then OM and ON would both have coincided with the diagonal OL, in fig. a, as shown by the dotted lines.

I annex a comparison between calculated and observed results. The latter are inclosed in brackets.

Given

"Probable error" of each system of mid-parentages = 1.22. Ratio of mean filial regression = 3

"Probable error of each system of regressed values = 1·50.

Sections of surface of frequency parallel to XY are true ellipses.

[Obs. Apparently true ellipses.]

MX YO 6: 175, or nearly 1: 3.

[Obs.-1: 3.]

Major axes to minor axes = 7: √ 2 = 10 : 5:35.

[Obs.-10 5.1.]

Inclination of major axes to OX = 26° 36'.

[Obs.-25°.]

Section of surface parallel to XY is a true curve of frequency. [Obs. Apparently so.]

"Probable error" of that curve = 1·07.

[Obs.-10 or a little more.]

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