Sent from Karl Pearson
16 Sep 1895
Description:

‘Dear Professor Petrie,

I have been so pressed with work the last few days, that I have had no time to sit down & think over your scheme until today. Nor does it seem to me one that admits of discussion by way of a few odd notes on your memorandum. It involves an immense round of highly complex matters, which have been discussed and fought over mathematically and otherwise. I will try and put my points as concisely as possible.

(1) Suppose we have two ‘pure’ races A & B – whatever ‘pure’ may mean. We may take a sample of n of one and m of the other and form a new group without hybridization of the two. I will call this as I have always done a mixture of races. If the two groups conjugate together we have a ‘mixed race’. It is most vital in all these matters to distinguish between a ‘mixture of races’ & a “mixed race”. Illustration. The late Professor Weldon took fawn & white waltzing mice, & albino white mice. If he had killed m of the latter & n of the former & measured say an index character of the femur, he would have had a mixture of races. He crossed these two and got ‘hybrids’. These ‘hybrid’ were again crossed inter se & with the parent races. The children of the hybrids were again crossed inter se & so on down to the 9^th generation. Each generation of hybrids or “mixed races” presented special features, and certain points of this I may indicate to you, although the results are not yet published.

(2) Let us consider first a “mixture of races.” If the characters dealt with in both A & B followed a normal curve, then the character in the mixture would follow a normal curve. The solution in this case was given in my first seminar on evolution. There is as far as I am aware no graphical method by which this resolution can be made with even approximate correctness. We tried it very fully in 1895 & the graphical resolution gave nothing at all close to the true components. Further, the probable errors of the resolution are very large & 500 to 1000 individuals must be used to get good results. You will find that in the Phil. Trans. memoir & in a subsidiary memoir in the Phil. Mag. it was applied to a considerable number of cases, excluding the resolution of Reibengräber[?] & Roman’s[?] British Skulls.

But, the method suffers from an a priori hypothesis which is that the frequency distribution of the original races is Gaussian. Now nearly all the chief skull characters, if you take a sufficiently large sample can be demonstrated to diverge sensibly from the Gaussian distribution & what is more the simpler the measurement the more frequently it appears to be non-Gaussian; a compound measurement like stature, because it is compound appears to follow more closely the Gaussian chance distribution than a single bone measurement, or skull measurement such as you describe as having “a single element of growth”.

(2) Your plan was actually used by Professor Weldon & myself on Naples crabs; we found one measurement markedly skew, & I resolved it into two Gaussian components, we naturally suspected that other measurements would now break up into two groups containing the same numbers, but we found that there was no such solution at all, a single Gaussian curve being the answer. In other words a proportion may be dimorphic with regard to one character and not with regard to other characters.

It was perfectly possible to get graphically the two Gaussian curves to describe other series of measurements, e.g. the skew frequency curves for characters in herrings, but mathematical analysis showed that not only these solutions were invalid, but no such solutions at all were valid. That investigation led up to the general recognition that skewness in variation is not necessarily a mark of polymorphism. In the years that have gone by since we have established for organ after organ that with sufficient numbers, there will be found sensible evidence of want of Gaussian distribution. The only reason that many cranial measurements can be represented by a Gaussian distribution is that the probable errors of the constants are so large that you cannot say it differs from the [illeg.]. But I can show that a triangle or bit of various other curves will give equally good results.

(3) Thus even for the case of a “mixture of races” your a priori hypothesis of the German curves is not established. Further assuming it to be true & your material not being a priori known to be a mixture, you may find, as we have done, it is dimorphic with regard to one character & not with regard to the others.

We tried some years ago on the only long series of skulls that we had then available the problem; divide the series into two components – one the males & the other the females – by the very measure you suggest. We have at present a very long series of human femurs & in despair of sexing them, we are going to try this very process again, but I am anything but hopeful of the result, because I hold that other sources of polymorphism quite obscure the dimorphism of sex. And I think this will be so with your crania. There is (i) the doubtful sexing, (ii) the great variety of ages, (iii) the lumping or polymorphism due to whole families buried together. On these grounds I feel sure you will hardly do better than we have done (& the labour of any rigid method is great) & find some of your groups of characters will not break up at all, others will show dimorphism, but will not give the same size of component groups.

As to your idea of getting rid of peakings by large sub-ranges, I do not think it helps matters. Whatever range you take the constants if corrected by Sheppard’s method will be identical & it is from the constants that the final result must be deduced. All we have to note is the size of their probable errors, the exact amount of peakiness does not trouble one.

(4) I now pass to the second problem, which I believe is the one you have at heart, a mixture of races or a hybridisation. Or, what is most probable in your case, a part of the population is a ‘mixed race’ and a part a “mixture of races”. Now surely here before you apply any resolution you must settle how the character you are dealing with is inherited? It may be a case of ‘alternate inheritance’ or of ‘blending’ or partly of one & partly of the other. Or again it may be something entirely different.

Let me illustrate my point. When you cross albino white mice with Japanese parti-coloured fawn both pink eyed, the offspring are all black eyed & in bulk are parti-coloured wild grey with occasional parti-yellow, black, etc. The hybrids have characters wholly unknown to both pure races, which have bred pure for generations! Is it recession or atavism or what? You could not possibly break up such a mixed race into two components, & if you did with regard to coat colour, you could not with regard to eye colour. But to go further, you can cross the hybrids together, and what happens? A new generation arrives – what is termed the “segregation” generation – pink eyes come back, but a large proportion of black eyes remain[,] fawns, blacks, lilacs, chocolates, whole & particoloured [sic] come in, which existed in neither pure race & one of the pure races, albinos, returns also. The hybridisers are teaching us a great deal, much that we know already with regard to human hair & eye colour inheritance bears also on this point. Or, again, take something, I have been dealing with myself. Your pure races are shorthorn cattle (horned, red, white & [illeg.]) & Galloways (polled & black). You cross these two. The hybrids are all black nearly all polled, but a few are “scarred” & have loose horns hanging from the skin, small & short. These can be removed & the whole series of hybrids are (by the dishonest) or might be sold for pure Galloways. So long as you breed them with Galloways you get black polled cattle. Now it is clear that if your character was length of horn, this mixture of races would give you no dimension at all, possibly some bone of the skeleton would give you dimorphism. Thus the amount of dimorphism & the size of your component groups would vary from character to character & your work of resolution would give you impossible results.

Now cross these Galloway skeleton hybrids (a) among themselves, reds & particulars come back, longside blacks; scurs[?] occur more frequently & more rarely horns. (b) cross them with shorthorns, scurs & rarely horns appear again & there are reds & particolours. But (c), as we have seen, if crossed with Galloways the result is the bulk are hornless & black.

Now, I think the results are sufficient to show that when we have a mixed race, the results at first may show polymorphism & not dimorphism, that we have reversion to characters which did not exist in either “pure” race inbreeding, & that the extent of any component of the polymorphic result depends (a) on the particular character with which we are dealing and (b) on the extent to which (i) the cross breeding has been carried and (ii) the amount to which any hybrid generation has been mated with either parent race.

Even if your material proved dimorphic, you must nor expect your individuals for different characters to fall into the same groups, or those groups to be of the same size, for each character. That somewhat similar laws hold not only for such character as I have described, but for bone-characters is, I think, probable. No doubt after continuous hybridisation for many generations, you do settle down for measurable characters to an approximately smooth frequency possible Gaussian in character, but the variation from the double Gaussian of the mixture of the races to the simple Gaussian of a much “mixed race” one whose members have interbred for generations is one we are only now learning something about, and it appears to me highly improbable that at any stage from the first crossing to the complete mingling after so many generations that a Gaussian determination would be the rule.

At any rate you ought before attempting it to know the nature of your inheritance of each character. It seems to me that for a mixed race you are dogmatically assuming that the characters are all (i) alternative & (ii) that if there be a prepotency of one individual for one character, he will be prepotent for all his characters. This is directly contrary to all experience. But without these two assumptions your process of resolution is illegitimate.

To sum up:

Your method might apply to a “mixture of races”, I think it has no application to a “mixed race”. There is no gain in large sub-ranges, if Sheppard’s corrections are made in finding the constants. The reduction would be laborious, especially, as if you found the two components n,1 & n2 for one character & two n1’ & n2’ for another character, you require the probable error of the difference n1-n1’ to determine whether the differences are significant or not.

In some races certain characters are different, others alike, you would not therefore get the same sized components necessarily from each character. When true characters are nearly alike in mean & variability & the latter is nearly the same for all races, any attempt to separate a mixture leads to wild results, the influence of small irregularities being very great.

The method has been attempted before, but has not hitherto been successful; marked dimorphism in one character being found not to be accompanied by any sensible dimorphism in a second.

I am afraid this is very lengthy, but it is the only way I can express my views & I fear will not be helped.

Yours very sincerely,

Karl Pearson.’