Wf he os SB ftom atch aah ee nena a Os ln thnticchn A Dinallp tn oe Pa tenth tty RAD en DO, a A Al NaN i “ee Ay AUG ae i y iy 4 r ! i 1 , i ot i : 1 ‘ , n a a CF ihe H Ay { rt 1 4 : * poe ' ’ fi o ‘ i . ' ‘ ‘ ‘ T Hi ‘ =i ni A - j i ’ ‘ ' +, ' 1 = ‘ ‘ * + ‘ ' i ) WA nl POV aR KA A JOURNAL FOR THE STATISTICAL STUDY OF BIOLOGICAL PROBLEMS EDITED IN CONSULTATION WITH FRANCIS GALTON BY W. F. R. WELDON KARL PEARSON AND C. B. DAVENPORT VOLUME IV JUNE 1905 to Marcu 1906 CAMBRIDGE AT THE UNIVERSITY PRESS LONDON: CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, FETTER LANE (C. F. CLAY, Manacer) anp H. K. LEWIS, GOWER STREET NEW YORK: THE MACMILLAN COMPANY LEIPSIC : BROCKHAUS BOMBAY AND CALCUTTA: MACMILLAN AND CO., LIMITED [All Rights reserved] 194767 Cambridge : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. VII. XIII. XIV. DV. XVI. —~C? oy COIN THINGS: Memoirs. Results of Crossing Grey eogues) Mice with Albinos. By E. H. J. Scuuster, M.A. ; ; Biometrical Studies on Man. I. Variation and Correlation in Brain-weight. By RAYMOND PEarRL, Ph.D. ; ; A Study of the Relations of the Brain to the Size of the seal - REGINALD J. GLADSTONE, M.D. On the Biometric Constants of English Brain-we siite By J. BLAKE- MAN, M.Sc., assisted by ALICE LEE, D.Sc.,and Karu PEARSON, F.R.S. A Quantitative Study of the Resemblance between Man and Woman. By E. TscHEPOURKOWSKY ; ; : : : “Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson.” A Rejoinder. By Kart PEARSON, F.R.S. . Variation and Correlation in the Earthworm. By RAYMOND PEARL, Ph.D., and WiILBuR N. FULLER : : ; Data for the Problem of Evolution in Man. On Fertility, Duration of Life and Reproductive Selection. By A. O. Powys . Contributions to the Study of Interracial Correlation. By E. TSCHEPOURKOWSKY : : : : : : ; Statistical Studies in Immunity. Smallpox and Vaccination. By JOHN Brown.LEE, M.A., M.D. ; ; : : On Tests for Linearity of Regression in Frequency Distributions. By Joun BLAKEMAN, M.Sc., B. A. The Long Barrow and Round Barrow Skulls in the eee of ihe Department of Comparative Anatomy, The Museum, Oxford. By K. H. J. Scuuster, M.A. j : : : : The Egg of Cuculus Canorus. An Attempt to ascertain from the Dimensions of the Cuckoo’s Egg if the Species is tending to break up into Sub-species, each exhibiting a Preference for some one Foster-parent. Second Memoir. By OswaLp H,. LATTER Tables for Facilitating the Computation of Probable Errors. By WINIFRED GIBSON, B.Sc. p ; . 3 : Cooperative Investigations on Plants. III. On Inheritance in the Shirley Poppy. Second Memoir ‘ : : : ; On the Inheritance of Coat-Colour in Cattle. Part I. Shorthorn Crosses and Pure Shorthorns. By Amy Barrineron and KARL Pearson, F.RS. PAGE 161 169 lv Contents PAGE XVII. On Hereditary Deafness. A Discussion of the Data collected by Dr E. A. Fay in America. By E. Scnuster, M.A. . , . 465 XVIII. On the Correlation between Vaccination and Smallpox in the London Epidemic, 1901-2. By F. M. Turner, M.D. . : . 483 Miscellanea. (i) Per la risoluzione delle curve dimorfiche. Nota del Dr FERNANDO DE HELGUERO : : : é : . 230 (i) Albinism in Sicily. By W. Biaeson PRS. : : : : . 231 (iii) Notes on Statistical Processes. By W. PALIN ELDERTON . : - ote (iv) On the Distribution of aa of Attack in Cases of ee By Karu Pearson, F.R.S. ; : 505 (v) On the Mean Duration of Life of Tae ae within a eve after Birth. By RaymMonp PgaRL, Ph.D. : 4 : : - old Plates. Parts I. anv II. (i) Coat-Colour in Mice. : : ; to follow p. 6 (11) Longitudinal and Horizontal Sartions of Crania . - 116 (iii) Skull and Skeleton of a Man of 25 years with hydrocephalus. ‘ : : : : : . x Part III. (i) Long Barrow Crania. N. verticalis . : : - 3852 (ii) Round Barrow Crania. N. verticalis . : ; , . (ii) Long Barrow Crania. N. lateralis. : : ‘5 4 (iv) Round Barrow Crania. N. lateralis. : : is f (v) Long Barrow Crania. N. facialis* . ‘ ; (vi) Round Barrow Crania. N. facialis : ; ; F; F Parr IV. (i) Petal Colours in the Shirley Poppy —. ; . between pp. 394, 395 Tables. PART OUT. Three folding Tables of Measurements of Long and Round Barrow Crania . ’ : : : . to follow p. 354 Subject and Name Indices to the first five volumes will appear with the neat volume. * By an unfortunate slip of the engraver, this plate is erroneously entitled Round Barrow Crania. N. facialis. he a ‘BIOMETRIKA NAL FOR ‘THE ‘STATISTICAL STUDY OF _ BIOLOGICAL PROBLEMS tant Z = = Rea «ACES “NDITED | ” “CONSULTATION. WITH FRANCIS GALTON Ms hod : Pie es . | ge het ok KART PRARSON. oa ee ial ©. B. DAVENPORT CAMBRIDGE na. AT THE UNIVERSITY PRESS: ; Pm. | cannes ES ERESTTZ PRESS WAREHOUSE, FETTER LANE Sen se (C.F. CLAY, Manacer) j i et _ ano H, K, LEWIS, GOWER STREET é eee NEW YORE! THE MAOMILLAN COMPANY _ USIPSic: BROCKHAUS- “BoMar AND ee asa MACMILLAN AND i, LIMITED ~ ~~ buckram, 25s. net. 4 Ww. pie REDERTON : - WINIFRED GIBSON: Probable Errors of Mean BE. J, Scuuster: The ‘Long Barrow and Ed: SCHUSTER : Theat ane: of Sees in White ies, Part. race - Oswatp H. Larter: The Egg of Cuculus Canorus. Second Mem : Drapers’ Company Research Memoirs, a ‘Aitecee Methods for from 1 to 1000.. Oxford. E. TscHEPOURKOWSEY : Interracial Coefficients of Correlation i in Man. A. O. Powys: Data for the Problem of Evolution in Man.” aR J. BLAKEMAN: On Tests for Linearity gf Regression in Frequeney Distribution eS JOHN. BROWNLEE : Smallpox and Vaccination. — ete Cooperative Investigations on n Plants, Ht.. On Inheritance of the Shi oe Poppy. Part IT. eye Cooperative Investigations on. Planta Iv. _Nariation and Correlation in Lesser ae Celandine from divers localities. Part IL. 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Bs, met each, Neb ig. r (1 904). upites 216. net, bound ir ae - LONDON: ‘CAMBRIDGE, UNIVERSITY PRESS WAREHOUS! 7 C, F. CLAY, Manaaer, \ ee ae _ FETTER BANE WiC, os fies Maen VoLuME IV JUNE, 1905 Nos. 1 AND 2 BIOMETRIKA. RESULTS OF CROSSING GREY (HOUSE) MICE WITH ALBINOS. By E. H. J. SCHUSTER, M.A., New College, Oxford. THE experiments here recorded were commenced in the department of comparative anatomy in Oxford in the autumn of 1902, but various personal circumstances have delayed the publication of any account of them for more than a year after their completion. They were made at the suggestion and with the advice and help of Professor Weldon, and I have therefore numerous reasons for thanking him. Source of Material. The original stock of white mice was obtained from various dealers, among whom Jeunet, of Paris, figures most largely. Some also were given to me by Mr Darbishire. The grey mice were caught in numerous houses in Oxford, and were brought to me by their captors in answer to an advertisement in The Oxford Times. In order to avoid, as much as possible, any possibility of using grey mice with “fancy” blood in their veins, those caught in the precincts of the Museum, where Mr Darbishire’s experiments had been going on for some months, were in all cases rejected. Previous Experiments of a Similar Nature. Breeding experiments similar to mine have been made by Cuénot (’02 and ’03) and Allen (04), and the fact that my results do not seem to be in exact accordance with theirs must serve for an apology for the appearance of an other- wise superfluous publication. That such differences should have arisen at first seemed surprising ; but an ingenious theory of Cuénot’s explains the possibility of their occurrence, and I should like, before going on to the consideration of results, to give a short account of this theory. Biometrika rv 1 \portiaa NStityg ‘A os AUG15 1905 Vane ional Museu™: 2 Results of Crossing Grey (House) Mice with Albinos Cuénot (04) suggests that all mice, including albinos, have some definite colour in them, but that it needs a special faculty to make this latent colour apparent on their coats, and that the difference between all coloured mice and all albino mice lies in the possession or non-possession of this faculty. Coat-colour, then, is a compound character which must be split up primarily into two component parts: (1) nature of the colour; (2) power, or lack of power of producing colour. He represents the constitution of the gametes of a mouse with regard to these two things by two letters, the first indicating their condition with regard to (2). When they are colour-producing the letter C (chromogene) is used, when non-colour-producing A (albino). The second letter denotes the nature of the colour. Thus G grey, B black, and Y yellow. For example: CG represents an ordinary house mouse. AG represents an albino mouse whose gametes would produce the grey colour if possessed of the colour-producing faculty. CB =a pure black mouse. AB a mouse similar to AG, only with Black for the latent colour. CY a pure yellow mouse. AY albino with yellow latent. The six symbols described above denote six pure races of mice. When any two of these are mated together hybrids will of course result whose gamete formula is a combination of the gamete formula of the two parents. The colour of the hybrids is determined by the relative dominance : (1) of colour productiveness and albinism ; (2) of the nature of the two colours latent in the two parental gametes. As in all cases colour dominates albinism, it only remains to be considered which of the three colours employed is dominant over which of the others. Here we find G dominant over B, Y dominant over B, Y dominant over G. Thus a grey CG crossed with a white. AY will produce yellow mice whose formula is CG-AY. A black CB crossed with either a grey CG@ or a white AG will produce grey mice, though in the first case the formula will be CB-CG, and in the second it will be CB-AG. | All this is not a modification nor an extension of Mendel’s laws, but merely an application of them to a special case ; Cuénot’s theory as to the two factors which determine coat colour renders such an application possible. The further behaviour of the hybrids when crossed among themselves or with pure mice can therefore be predicted by any one conversant with these laws and possessed of a knowledge of the gametic formula of the hybrids used. A third element is introduced by the fact that the colour of a mouse’s coateis not always uniform, but may be marked with white. This marking may occur with mice of any colour; it is transmitted independently of the colour. and may be latent in albinos. A uniform coat dominates a piebald coat. Thus if one mates a piebald yellow mouse with an albino with uniform greyness EK. H. J. Scuvuster 5} latent in it one will obtain a uniform yellow hybrid, for yellow dominates grey, uniformity piebaldness, and colour-productiveness albinism. If such yellow hybrids are paired together and a sufficient number of young obtained they should be in the following proportion: 16 albinos, 9 piebald yellow, 27 uniform yellow, 3 piebald grey, and 9 uniform grey: see Appendix II, p. 12. Eight of these 64 mice would be pure, all of different constitutions, and there would be two of each of 12 different kinds of hybrids, four of 6 different kinds, and eight of one kind. Thus a reasonable degree of complication is obtained when only two colours are employed; when more than two colours are introduced the compli- cation is largely increased. I have brought forward this example to explain my reasons for not examining in detail the occurrence of each of the various types of mice produced in my experiments. For in addition to the inherent difficulties of such an examination, I am in total ignorance as to colours latent in the original stock of white mice. Types of mice produced. Bateson (03) gives a list of a number of types of Fancy mice with a statement of the colour of the actual pigments contained in each, and almost all the mice produced in these experiments can be referred to one or other of the groups in this list; I cannot, however, guarantee that I have referred them rightly, as no microscopic examination of them has been made, so that the colour as it appears to the naked eye is the only basis of the classification. After the name of each colour is given the abbreviation employed as a symbol for it. In giving instances of Cuénot’s theory, for the sake of uniformity I have used my own abbreviations and not those employed in his original paper. (1) Grey (G) (Cinnamon or Agouti) figure 4. (2) Golden agouti (GA). (3) Yellow (Y). In this are included both yellow and cream, as these shade insensibly into one another and form indeed a highly variable group of colours. Figures 5, 7, 8 represent three conditions of yellowness, but the lightest yellow of all obtained was almost completely white, with but the faintest suggestion of pigment. It had, however, perfectly black eyes like all the coloured mice produced. (4) Sooty Yellow (SY). (5) Chocolate (Ch.) figure 6. (6) Chinchilla (Cc.) figure 2. I cannot find this anywhere on Bateson’s list. In the best specimens it is a beautiful silvery grey in colour but it shades insensibly into the ordinary grey coat, figure 3 being one of the intermediate forms. Allen (04) records having obtained some light grey mice in crossing house mice with albinos, these may possibly have been of the same nature as my Chinchillas. His apparently died in childhood so that he had no opportunity of breeding from them. Intermediate forms between pairs of these colours did occur and are represented by the symbols of the two colours joined by a hyphen. Thus figure 1 represents 1—2 4 Results of Crossing Grey (House) Mice with Albinos a mouse intermediate in colour between a black and a grey, and would therefore be denoted by B-G. Each of these colours occurred both uniform and marked with white. The white marking, as Cuénot observes, is subject to continuous variation. The parts first affected are the tail and the paws; then a small white mark may be found on the belly and on the top of the head. The white mark on the belly may grow forwards and backwards, so as to cover the whole ventral surface ; laterally it sends out arms of whiteness immediately in front of the hind legs, these grow dorsal- wards and may meet and form a complete white belt. The mark in the head spreads forwards till it forms a good white blaze on the forehead, or further still till the front of the muzzle is completely white, and eventually it becomes continuous with the ventral white mark. The letter P in conjunction with the symbol for a colour means that the mouse is piebald and of this colour; no attempt is made to indicate the degree to which the white marking has spread, though of course a record of this has been kept. Results. 106 white mice were mated with grey mice and of these crosses 74 proved fertile. 70 families contained only grey mice. Of the remaining four 2 contained mice of varying shades of yellow in addition to grey, and the original chinchillas appeared in the other two. Thus the first generation of hybrids consisted of 342 mice, of which 329 were grey, seven yellow, and six chinchilla. The appearance of yellow in this generation is easily understood, if we assume, in the absence of any definite information on the subject, that the albino parents in each case were hybrids containing yellow and some other colour latent. Such mice would according to Cuénot’s hypothesis produce some yellow and some grey when mated with pure grey mice, as yellow is dominant over grey and grey over any other colour which they might contain. Having no sort of evidence either as to the nature of the chinchilla coat, or as to its behaviour when pure and mated with other colours, it would be futile to indulge in speculation on the possible reasons for its occurrence here. The yellow mice when mated together produced yellow and grey and white. Thirteen young from four families were obtained, and of these five were grey, six yellow and two white. These numbers are of course too small for the proportions to have any significance, but the theoretical expectation for sixteen mice would be four albino, nine yellow and three grey. From the original seven yellow mice all the yellows and sooty yellows occurring in the experiments were produced. The original pairs are labelled A and B in the Appendix, the As having the ¥ white and the Bs the . The offspring of the A and B crosses were mated with white mice and with one another. E. H. J. Scuuster 5 In the former case the crosses produced are labelled C and D, C having the # a product of an A or B cross and the § an albino, and D being the reciprocal of this. The Os produced 282 young of which 130 were albino and 152 coloured, the Ds produced 255 young of which 131 were albino and 124 coloured. Thus together we have 537 young of which 261 are albinos and 276 coloured. Con- sidering merely the characters, colour productiveness and albinism, the Mendelian expectation would be 268°5 of each, so that the actual result obtained is in fairly close agreement with it. If we go on to an enumeration of the actual colours obtained, we find among the Cs 25 black, 102 grey, 2 intermediate between grey and chinchilla, 2 golden agouti, 16 grey marked with white, and 7 black marked with white. Among the Ds we have 26 black, 77 grey, 1 golden agouti, 1 yellow, 1 sooty yellow, 1 chocolate, 12 grey marked with white and 7 black marked with white. Speculation as to the proportion of the various colours seems to be useless, but if one assumes that some of the albinos employed both in the A and B and in the Cand D crosses had any other characters but uniformity and greyness latent in them, then one would expect that these characters would become apparent in the coats of some of the mice produced. This remark applies with equal force to the consideration of the results of the various other crosses described below, but it will not be repeated again. The offspring of the A and B crosses were paired together and these pairs are labelled £. The Hs produced 119 albinos, 27 blacks, 17 intermediate between black and grey, 201 greys, 9 intermediate between chinchilla and grey, 9 chinchilla, 8 golden agouti, 9 yellow, 3 sooty yellow, 1 chocolate, 19 grey marked with white, 3 black marked with white, 1 chocolate marked with white, and 1 chinchilla marked with white. That is to say 119 albinos and 308 coloured. 119 is at any rate a possible approximation to the Mendelian quarter. The 308 coloured mice should be composed of } pure colour producers and 2 hybrids between albinos and colour producers. The F and G crosses were made to test the truth of this. The Fs have coloured offspring of the Hs for 2s and albinos for ¥‘s. The Gs are reciprocal. Out of 87 of these crosses 32 produced no albino young and 55 produced some albino young. 82 is a fair approximation to 1 of 87. The colours of the young contained in these 32 families is as follows: 18 black, 1 intermediate between black and grey, 95 grey, 2 intermediate between chinchilla and grey, 1 yellow, 8 sooty yellow, 1 golden agouti, 6 chocolate, 20 grey marked with white, 13 black marked with white. The 55 families produced 141 albinos and 137 coloured mice. If these are to be regarded as hybrids equal numbers of each should be expected; and a fair degree of equality is obtained. If we add these numbers to the results of the C and D crosses with which they are to be associated, we get out of 815 mice 402 albino and 413 coloured. 6 Results of Crossing Grey (House) Mice with Albinos The actual colours obtained in the 55 F and G@ families are as follows :— 23 black, 74 grey, 2 golden agouti, 1 chinchilla, 1 yellow, 6 sooty yellow, 2 chocolate, 1 yellow marked with white, 24 grey marked with white and 3 black marked with white. Cross L was made by breeding together those coloured mice which had been used in the #’s and Gs and which had produced only coloured young. Their offspring were all coloured, consisting of 57 grey, 3 grey marked with white, 3 black and 5 sooty yellow. In cross M the remaining coloured mice which had been used for the Fs and Gs were mated together and produced 30 albinos and 71 coloured mice, the colours being as follows:—52 grey, 5 grey marked with white, 7 black, 1 golden agouti, 6 sooty yellow. The white offspring of the Hs were also mated together, and in all cases bred true. In conclusion, it is hardly necessary to point out that with regard to characters, colour-productiveness and albinism the mice under consideration here behave in complete accordance with Mendel’s laws, both with regard to dominance and to segregation. If one considers the nature of the colour and the question as to whether it is distributed uniformly or is marked with white there is nothing in my results incompatible with similar behaviour with regard to these characters also. Certain minor difficulties do indeed suggest themselves, for instance the occurrence of intermediate forms and the tendency of yellow mice when crossed either with white or grey hybrids to produce sooty yellow young; this colour figuring somewhat conspicuously in some of the latter generations. It is only to be regretted that the true nature of the albinos was not known and it is to be hoped that in the future the experiments may be repeated without this very serious drawback. LIST OF LITERATURE REFERRED TO. 02. L. Cutnor. “La Loi de Mendel et VHérédité de la Pigmentation chez les Souris.” Arch. Zool. Exp. et Gén. Notes et Revue, 1902. 03. ——. Ibid. 1903. 704. ——. Ibid. 1904. 03. W. Batrson. “Colour Heredity in Fancy Mice and Rats.” P.Z.S. 1903, Vol. 1. 04. Gtover M. Attun. “The Heredity of Coat-Colour in Mice.” Proc. Amer. Acad. Art. and Sci. Vol. x1. DESCRIPTION OF FIGURES. These are taken from water-colour drawings ad nat. by Miss Adeline Ritchie. (1) Intermediate between grey and black. 2) Chinchilla. 3) Intermediate between chinchilla and grey. ) Grey. ), (7), (8) Three shades of yellow. ) Chocolate. : r f ° + ss —s cf. - : ' i, _- - i} q = . P - A Fe ‘ ee - Pye ees = nal a 4 = = . ~s { i nN 7 : = : ‘ CS < : 1 { — : 7 \ JAD ) 4 Asano AOR ‘T ALWId 1 “AI ‘PA ‘VSHRILANOIW Vil. Vit. COAT-COLOURS IN MICE. : Bi Vil. COaAT-COLOURS IN MICE. BIOMETRIKA, Vol. 1V., Part I. PLATE 1. VIII. E. H. J. ScHustEr 7 APPENDIX I. A List of all the Mice Produced. The following abbreviations are used in describing the mice :— A, Albino. Cc. Chinchilla. SY, Sooty Yellow. G, Grey. Ch. Chocolate. Y, Yellow (including Cream). GA, Golden Agouti. Intermediate colours are represented by the symbols of the colours between which they lie placed together. P joined to the symbol for a colour means that the mouse was marked with white. Crosses A and B. A, Father Albino, MotherGrey. B, Mother Albino, Father Grey. 70 A and B pairs produced only G mice. Family. A 32, 3Y and 2G. BAT, 4Cc. B 26, 4Y, 2G. B 43, 2Cc., 2G. Cross C. Products of A and B Crosses mated to Albino Does. Results Results C i 2A, 1B, 14 C 41 34, 1G CO 2 3A, 2B C 42 2A, 2G ORS 1A, 1G'A, 1BP C 43 2A C 4 3A, 2B, 2G°P C 45 2A, 3G'P C 5 QA, 2G C 46 3A, 1BP C 6 4A, 1G C 48 3A, 1B, 1G CO 7, 3A, 3B C49 2A, 1G C 8 1G, 3G C 47 14, 14 C 9 3A, 2B, 1G C50 QA, 2G, 1G'P C 10 6G C 51 3A, 1B C13 2A, 2B, 1G, 1BP C 52 2A, 2B C12 1A, 54 C 58 3G C14 1A, 5G C 54 4A, 4G C15 YA, 1B, 1G, 1GP C 57 2A, 1B C 16 2A, 1G C 59 3A, 3B C19 5G C 60 1A C 20 5G C 63 4A, 2G, 2G4P C 21 3A, 4G C 62 3A, 2G, 24 P, 1BP C 22 2A, 2G, 2Ch. G C 61 3A, 4G, 24P C 23 5A C 64 14, 2G, 1GP C 25 2A, 2B, 2G C 65 2A, 2B C 26 1A, 5G C 66 2A, 2G C27 3d, 2G C 67 1A, 34 C 28 1A, 2BP C68 4A, 1G C 29 4A, 1G C69 3A C 30 3A, 1B, 3G, 1¢P C71 2A, 2G C 32 3A O72 1G C 37 2A, 1B, 3G C78 4A, 1B, 14 C 40 3A, 3G, 1G4P ORM 14, 1G, 1G'A C 39 34, 1G Total 1304 , , Fi of Cs , 25B, 102G, 2Ch. G, 2G'A, 164P, 7BP 8 Results of Crossing Grey (House) Mice with Albinos Cross D. Products of A and B Crosses mated with Albino Bucks. Results Results D 2 | 3G, 24 D33 | 2A, 24, 1@A D8 | 14 1GP D3 | 24. 1B D4 | 14.1G,1GP,1BP|| D37 | 24 94 D6 | 9414 D3s | 2A, 1B, 24 D7 | 44 1B D40 | 34.16 D : 14, 4G Dit | 44, 3G D9 | 24) 4G, 1GP,1BP|| D42 | 34) 1B, 2G D12 | 24, 1B Di | 24, 294 1GP D13 | 24) 3G, 1GP Dis | 24 1GP Dis | 24. 1B,26,1BP || Die | 34 D15 | 24, 9G 1BP Diz | 2A, 4B, 1G D6 | 14, 2B, 2GP Dis | A Div | 34. 2B D52 | 3A, 2BP D18 | 24, 1B, 2G D50 | 24, 3G D19 | 24, 24 D538 | 24, 1G, 1GP D20 | 54,14 D5, | 14, 1G, 16P D21 | 34.14 Ds | 5A, 14 D2 | 2A) 3G D5? | 4A, 1B, 1BP D23 | 3A D58 | 2A, 1B,.1G D2% | 24, 34 D51 | 24,1B D26 | 14,34 D59 | 4A Dey | 3A, 1B, 14 D60 | 34, 1¥-G D28 | 34, 1B, 14 D61 | 24, 1G, 1¥, 1@P D29 | 1B, 34 D62 | 24,14 D30 | 34. 3G D6, | 34.14 Ds2 | 44, 3B D65 | 14, 1B D66 | 34, 10h. D67 | 14, 1B, 1@P, 14 D68s | 4A Total | 1314, 26B, 7BP, 77G, 124 7 18Y, 10h cre A, 26B, 7BP, 77G, 12GP, 1GA, 1Y, 18Y, 1Ch. ee 2614, 51B, 14BP, 179G, 28GP, 2Cc-G, 3G, 1Y, = 1SY, 1Ch. and Ds ? 6a oe Cross LE. E. H. J. Scuuster Products of A and B Crosses mated with One another. Results Results E 2 1A, 1B, 26 E 61 1d, 2G, 1GA E 3 2A, 2G E 62 1B, 2G, 2G'A, 1G¢P EES 2B, 2G KH 63 3A, 1B, 24 E 8 2G GY 1A, 2B, 2G E 9 2A, 14 E66 1B, 3G E10 1A, 4G E 67 1A, 3G E12 LA, 28, 2G E G68 2A, 2G E il 4G EH 69 1A, 34 E 14 1A, 2B, 1G E 70 1A, 4G E15 4G E 71 1A, 28, 4G E16 4G eae 14, 1B, 2G E17 14, 4G 133 1A, 1CcG E18 3d, 3G E TY 2G, 1G P, 2CcP E19 14, 1B, 14 13,45) 1A, 10cG, 1Cc E 20 1A, 3G, 2G'P E 6 14, 3G E 21 2A, 2G, 14P E v7 2A, 3G E 22 3A E 78 1A, 4G E 23 38G¢ E 79 3A, 2G E24 QA, 2G, 1GP E 80 1A, 3G, 1GP E 25 3A, 2G E 82 1A, 3G E 26 1A, 2G E 83 2A, 3G E27 1A, 14 E84 1A, 3B, 1GP E 28 2A, 24 H 85 2A, 1G, 1Ch., 14P E 29 1B-G, 3G EH 86 2A, 2B-G, 1G EE 30 1A, 3G'A, 1Ch. P i 87 QA, 3G E 31 1A, 3G, 1G'A EL 88 14, 2G E 33 3A, 1G, 2V E 89 2A, 3G E 84 1d, 44 E 90 2A, 3G E 35 4A, 1G E 94 3B-G Lf 40 Why, x6 E968 1A, 1B-G, 1GP E 41 3G E 96 14, 14P E 42 2G, 1SG E 97 1A, 1B, 14 E 44 2A, 2G | # 98 YA, 3G EAS 1A, 3G | EH 99 ILA, BG; E 4G 2A, 2G | #100 14, 1B, 2G E 47 1A, 1G, 1SG | #101 1A, 1B, 3G E48 2A, 1G, 1SG L102 | 1A, 2G E 49 1A, 2G Hi08 \ 1A, 2G, IGP E 50 2G ae eAswaliove E104 | 3BG E51 4G¢ E108 5G E 52 1d, 1B, 1GP E105 5BG, 14a E 54 2A, 24P E109 1BG, 1G, 14P, 2BP EI 54% 1d, 1G¢P E110 2G E55 3A, Ub, 1G. WGP E& iii 1A, 2YV E56 1A, 244 EH LIB 1A, 3Ce aT 3A, 1G E113 2G E 58 QA, 2GP Ef 113* 1A, 20cG, 1Cc E 59 3d, 1G El1l4 | 1A, 1G, 2Cc E60 5G Ef 114* | 1A, 2CcG', 2Ce | E 36 3G E115 2G 2a | £37 QA | FE 116 1A, 10cG’, 1¥ E 38 1A, 1B, 2G || #117 WEE We E 39 1B, 2G Total 1194, 27B, 17B-G, 201G, 90cG, 9Ce, 8G‘A, oY, BSF, one 1Ch., 19GP, 3BP, 10h.P, 1 Cce.P=119 Albino, 308 : coloured. Biometrika Iv bo 10 Results of Crossing Grey (House) Mice with Albinos Cross F. Being coloured ¢ Products of E Crosses mated with Albino $s. Those which produced Albinos among their young are taken first. Results | Results | ‘i | | HP a 34, 2G F 23 34, 1GP | F 2 4A, 2G | F 24 1A, 44 i 3) 2A, 2G | SEES, 34, 2G F 4 3A F 32 14, 1B, 144 Eos | 9A, 2GP F 33 3A F6 | 2A, 2BP F 34 14, 2G Fi0 | 34, 3G || #738 2A, 3G Fig | 2A, 1GP, 1BP || £36 14, 14 Foie aA F 39 1A, 1Cc, 24P Figx| 4A, 1B F 43 14, 1B, 14, 14P | Fi6é | 44,14 F 44 2A, 1G haer22 34 | Total of #8 containing albinos. 534, 3B, 24G‘, 1Cc, 1G‘A, TGP, 3BP. Cross G. Being coloured $ Products of E Crosses mated with Albino #s. Those which produced Albinos among their Young. | Results Results | aa ] G1 | 4d, 14 G3 2A, 1B, 14 | G 2 | 4A, 1G, 144 G 82 14, 4G G 3 | 3A, 24 G 385 2A, 8GP | G@ 6 | 8A, 1G, 104. || G36 2A, 2B, 3G | G8 | 44, 4B || G37 | 4A, 2G, 1GP r 9 5A, 2G || G38 2A, 2B, 2G G 11 44, 1B G 39 3d, 20c, 3GP G 12 2A, 3B G 41 5A, 2B, 1G G 13 34, 3G G 42 5A | G15 1A, 44 G 45 2A, 1G G 17 1A, 1B, 24 G 46 2A, 1B G 18 2A, 4G G 50 4A, 1G, 10h. G 19 24, 5G | G51 4A, 2G, 1G4'P G 22 14, 2G, 5SY G 53 14, 1B, 3G, 1GP G25 4A, 1¥, 1GP || G62 34, 1G, 1GP G28 | 9A,1G,1SY,1GP,1YP || G47 14, 2B, 14 | Total 884, 20B, 50G, 1G‘A, 1Y, 6SY, 2Ch., 17GP, 1Y¥P E. H. J. Scuuster 11 Fs and Gs which have produced no Albino Young. | | { Results | Results | | | | | —_—_—__—— ——— | F 9 7G G 4 2d 7 4G G 5 5B, 3G Fil 2G, 1GA Gy 104 | Fis 1B, 3G G 10 2B, 3G F19 2G, 2GP Gat 3B, 1G. 3G. F 20 2G | G20 8G, 1GP F 26 1BP 1 PS | 1B, 1G, 3BP, 2GP F 27 3G, 2GP G24 | 6BP F 29 2B, 2G Gt 26 5G F320 2B, 2G | @a7 | 34, 4@P TEE 1G, 1C¢.G, 1Ch. G 29 2G, 4SV F 38 4G, 1BP Y BO 3G, 1GP F4l 2G, 1Ch., 1BP G 33 3G, 14P, 1BP F 42 1G G 34 5G, 2GP F 4S 2G, 3Ch., 1GP G48 3G, 1Y, 48V F 46 1B-G, 1Ch., 1GP | G49 2B, 4G! | 5B, 1B-G, 37G, 2Cc-G, 1G‘A, Rp sway acvayene terns : Total 6Ch., 6GP, 3BP Total | 13B, 56G,1¥,8SY, 14G'P, 10BP Cross K. Pairs of Albino Young from Cross E. These gave all Albino Young, Cross L. Pairs made of the Coloured Offspring of E pairs which have given all Coloured Young when Crossed with Whites. i} Results Results Ia 4G L9 4G¢ L2 | 96 Zid | 1B, 44 L3 | 66 Lil 3G LA | 5E Sb tkss 1B, 4G Ui Sie a Gan Gere IL IVE: 3G, AGP Liv 2G, 5S VY L115 Beg L8 LB oG 57G, 3GP, 3B, 5SY. 12 Results of Crossing Grey (House) Mice with Albinos Cross M. Pairs made of Coloured Offspring of E pairs which have given some White Young when Crossed with White. Results | Results M 1 3A, 3S Y | M 11 1A, 1G M 2 1GA, 2G, 1GP || M12 2A, 2G M 3 | 3A, 1GP, 24 | Mis | 54 M 4 1A, 64 | Mis | 14, 4B, 24 M 5 34, 34 \ WATS WAG: M 6 2A, 2G, 1B Mi6 | 54 M 7 QA, 3G | M17 1A, 2G M 8 QA, 2GP, 14 || J 18 3G M 9 1A, 5G, 1GP || M21 | 1A, 3G, 1ISV M 10 34, 2SY, 14 | M 20* 3A, 1B, 14 | Total 30A, 52G, 5G°P, 7B, 1G‘A, 6SY. APPENDIX II. (See p. 3.) The eight pure mice would have the following gametic formulae CGU, CGP, CYU, CYP, AGU, AGP, AYU, AYP. The last four being white, and the first four coloured respectively, uniform grey, piebald grey, uniform yellow and piebald yellow. Of the 56 hybrids, there are 24 which are hybrids with regard to only one pair of characters;—two of each of the following 12 different kinds: CGU-CGP, CGU-AGU, uniform grey in colour; CGU-CYU, CYU-CYP, CYU-AYU, uniform yellow; CGP-AGP, piebald grey; CGP-CYP, CYP-AYP, piebald yellow; AGU-AGP, AGU-AYU, AGP-AYP, AYU-AYP, albino. There are 24 which are hybrids with regard to two pairs of characters, 4 of each of the following six different kinds, CGU-CYP, AYU-CGU, AYP-CYU, uniform yellow; AGP-CGP, uniform grey; AYP-CGP, piebald yellow; AGU-AYP, albino. Also 8 hybrids with regard to all three pairs of characters; of these there can be only one kind, which is uniform yellow in colour. BIOMETRICAL STUDIES ON MAN. I. VARIATION AND CORRELATION IN BRAIN-WEIGHT*. By RAYMOND PEARL, Ph.D. CONTENTS. PAGE 1. Introduction . ‘ ‘ : : ; : : ; : é : 13 2. Data ; . : 3 F ; : : ; : : : 16 3. Fundamental Goristanta . : ‘ : : : : ‘ : : 21 4, Brain-weight Types : : : ; : : : : : : 22 5 eee in Brain-weight . : : : : : F ; : 29 ‘ (Homogeneity of Material . : : : F : 29 6. On the Nature of the Frequency ingoributions é : F : : 37 7. The Correlation of Brain-weight with other Characters. : ‘ 42 8. Brain-weight and Sex. : ‘ : E : : : : 45 9. Brain-weight, Age and Stature. : : : : : : : 51 10. Brain-weight and Body-weight . : : : : : 2 . 64 11. Brain-weight and Skull Characters : é : : : ; : 67 12. The Weight of the Cerebrum : : : : : ; . - 74 13. Concluding Remarks . : : ; : ; : : : : 76 14. Summary : d : : ; : : : : 2 82 Appendix. Correlation Tables ‘ : 2 : : : ; : . 83-104 Ll. Introduction. THE subject of brain-weight in man has for a long time been given considerable attention by anatomists and anthropologists. The reason for this is obvious. Since the brain is the organ of the mind it appeared to earlier workers that size of brain ought to be an index of intellectual capacity. The substance of the brain is of such physical homogeneity that its weight is a fairly accurate, as well as simple, “size” measure of the organ. Consequently, by earlier workers, brain- weights were collected, studied and discussed to a considerable extent as measures * Contributions from the Zoological Laboratory of the University of Michigan. No. 86. 14 Variation and Correlation in Brain- Weight or indices of brain power. It became evident after a time, however, that there was not the close and definite relationship between brain-weight and intellectual capacity which had been supposed to exist. Individuals of marked intellectual power were found in not a few instances to have brain-weights below the average, while on the other hand it was not at all difficult to find individuals of very mediocre intellectual attainment who possessed brains of unusually large size and weight. These results still held even after rough corrections were made for bodily size, age, etc. While it is thus evident that brain-weight cannot be taken as a close index of intellectual power, holding for individual instances, yet it is clear that, considered from the phylogenetic standpoint, increase in brain-weight and in psychic capacity have in general gone hand in hand, and the weight of the brain increases quite regularly as we go up the taxonomic scale*. The present trend of investigation in this subject, is rather from the point of view of anthropology than of psychology. Evidently brain-weight is an important | and interesting anthropological character, and in this field we may expect significant results. The method of investigation which has been almost universally followed in brain-weight work has been to tabulate large masses of statistics of weighings, compute means for various groupings, and draw the conclusions which appeared to follow from the tabulations and averages. In other words the only statistical methods which in most cases have been applied to the data have been those of the sociological statistician. Such methods serve fairly well, of course, when only the “types” are wanted, but they are quite inadequate for some of the work which many neurologists have wished to do in this field. Practically all students of the subject have attempted to determine in one way or another the degree of correlation which exists between brain-weight and other physical characters and also age. Knowledge of these correlations is of course much to be desired. The human brain is justly to be regarded as the highest product of organic evolution. Any contribution to a knowledge of the laws governing its variation and correlation cannot fail to be of the greatest interest. Again, only through a knowledge of the degree of the correlation of brain-weight with other characters of the body is it possible to make scientifically such suitable corrections for bodily differences as will make fair any comparison of the brain-weights of different races, or of different groups of the same race. Now, as everyone knows who has even an elementary knowledge of statistics, it is possible to make the same statistical material lead to quite different conclusions, by grouping it in different ways, when the tabulations and averages are the only sources from which conclusions may be drawn. As a matter of fact this has happened in work on brain-weights. Different investigators, working in different ways, have arrived at * Cf. Ziehen, Th.: ‘Das Gehirn, Massverhiltnisse,” Bardeleben’s Handbuch der Anatomie des Menschen, Ba. 1v. 1—8 Abtheilung, pp. 353—386, 1899, p. 362 et seq.; and more recently Spitzka, H. A.: “ Brain-weight of Animals with special reference to the Weight of the Brain in the Macaque Monkey,”’ Jour. Comp, Neurol. Vol. x11. pp. 9—17, 1903. ‘etinl, Raymonp PEARL 15 quite different numerical appreciations of the relation existing between brain- weight and other characters. The very interesting question of variation in brain-weight has been scientifically investigated only by Pearson*, so far as is known to the writer. The reason which led me to undertake the present piece of work was primarily a desire to apply adequate statistical methods to a biometrical problem of peculiar intrinsic interest, and for which fairly large collections of reliable data were available. It was desired to determine as a part of a general plan of work outlined as exactly as possible from the data at hand, what were the actual conditions of variation and correlation in brain-weight. It was hoped that by such work some light might be thrown on the question of the evolution of man’s brain. The work was originally planned to include the analysis of only one large collection of brain-weight data (see below p. 16) but when the constants for this material had been determined certain of the results were seen to be rather peculiar. Not being satisfied that these indicated the true state of affairs, but thinking that they might be caused by some hidden anomalies of the raw material, it was decided to extend the work to other data in order to test the first results. The specific problems to which attention has been directed in this work are: 1. The amount of variation in the weight of adult male and female brains. 2. The relation of brain-weight to age in the adult. 3. The relation of brain-weight to stature and body-weight in the adult. 4, The relation of brain-weight to skull characters. 5. The nature of the regression of brain-weight on these other characters. 6. The effect of natural selection on brain-weight. 7. The value of brain-weight statistics from the anthropological standpoint. At this point I wish to acknowledge my indebtedness and express my thanks to the officials of the Carnegie Institution for a grant through which I obtained a large Brunsviga arithmometer and other necessary apparatus for statistical work. This aid enabled me to extend this work far beyond what would have been possible had I been restricted to “long-hand” methods of computation. The amount of arithmetical calculation necessary in any extensive biometrical in- vestigation is so great that it may almost be said that such work cannot profitably be undertaken on any large scale unless one has at command the various mechanical aids which reduce to a minimum the time necessary to carry out the computations. To my wife I am especially indebted for much aid in the computing, and for the diagrams which accompany the paper. To Prof. H. H. Donaldsou I am indebted for the loan of some unique and valuable manuscript records of brain-weighings, as well as for numerous helpful suggestions regarding the work as a whole. * Pearson, K.: ‘‘ Variation in Man and Woman,” The Chances of Death, Vol. 1. pp. 319—323. 16 Variation and Correlation in Brain-Weight 2. Data. The statistical data for this study were obtained from several sources, as has been mentioned above. The work was begun on Marchand’s statistics, then extended to include the old, but still valuable collection of brain-weights by Bischoff. With the results from these two series in hand, I was still not satisfied that I had analyzed sufficient material to be certain of the conclusions. So I determined to include the following material in the discussion: (1) the admirable series of brain- weights of Swedes collected by Retzius. (2) Matiegka’s Bohemian series. (3) The Boyd-Marshall data observed at the St Marylebone Infirmary. As will appear later the last mentioned series was used only for a special purpose. The general results of this paper are based on the analysis of the Marchand, Bischoff, Retzius and Matiegka series. Marchand’s* data were obtained from a series of brain-weighings made between the years 1885-1900 in the Pathological Institute at Marburg. The total number of brains weighed was 1234, Of this number 389 were from individuals under 15 years of age, and hence were unavailable for adult brain-weight studies. The data recorded from each individual in Marchand’s work included, in addition to brain-weight, sex, age, and stature. It was not possible, however, to determine all of these points in every case, so that omissions of one or more records for an individual occur not infrequently in these statistics. The same is of course true of all extensive series of brain-weight data ever collected. After all such incomplete records have been thrown out there remain in Marchand’s tables 475 male and 281 female complete records, or a total of 756 of both sexes. Marchand did not include body-weight in his determination, because he considered that: “das Kérpergewicht bei dem Leichenmaterial der Krankenhiuser ein so wechselnder Faktor ist, dass bestimmte Beziehungen zum Gehirngewichte sich daraus kaum ableiten lassen wiirden.” The brains were weighed in the fresh condition, usually immediately after removal from the cranial cavity. The membranes were not removed before weighing. The weighings were made in the majority of cases to the nearest 5 grams, greater accuracy not being possible on account of such unavoidable sources of error as varying amounts of blood in the vessels of the brain, amount of fluid in the ventricles and membranes, etc. These sources of error cause variations in the total weight outside the limits of 5 grams. Marchand’s material comes from the “hessische Bevélkerung.” Bischoff’s+ data comprise the results of the weighing of something over 800 brains of individuals between the ages of 17 and 85. The attempt was to determine for each individual the following characters in addition to brain-weight : sex, age, cause of death, stature and body-weight. Omissions in the records were * Marchand, F.: ‘‘ Ueber das Hirngewicht des Menschen,” Abhandl. d. math.-phys. Cl. d. Kénigl. Sachs, Gesellsch. d, Wiss. Bd. xxyut. No. tv. pp. 393 —482, 1902. + Bischoff, T. L. W. v.: Das Hirngewicht des Menschen, Bonn, 1880, Pp. vi. and 171, plus Tables. RAyMOND PEARL 17 fairly numerous, so it was only possible to extract full data on sex, stature, body- weight and brain-weight for 365 males and 241 females. This material I have referred to in this paper as the “short” Bavarian series. For determining the correlation between brain-weight and age I was able to extract 529 male and 323 female records from Bischoff’s tables. This larger group I have called the “long” Bavarian series. The brains were weighed in the fresh condition, soon after removal and with the membranes. Nearly all the individuals had died in hospitals and a considerable number of them were convicts. Practically all were members of the middle and lower classes of society. The causes of death for this and Marchand’s series were very varied, including a wide range of diseases and accidents, so that the material is not vitiated by the great preponderance of some one cause of death which might have a specific effect on brain-weight. The great majority of the individuals included in Bischoff’s series were Bavarians. The Swedish data used were collected by Retzius*. The source of the material was in the main the autopsies at the great Sabbatsberg Krankenhaus in Stockholm, although some of the returns were from autopsies at the Maria Krankenhaus. This series of brain-weights is undoubtedly one of the best which has ever been made with reference to accuracy in the individual weighings and uniformity in the conditions and method of weighing. It is only to be regretted that the series is not more extensive. Regarding the racial homogeneity of the material Retzius says (loc. cit. p. 55): “ich iiberzeugte mich aber in den allermeisten Fallen davon, dass sie eine echt schwedishe Herkunft anzeigten.” The method of weighing was as follows (loc. cit. p. 55): “Die Gehirne wurden ohne Abnahme der weichen Hirnhaute, in der Regel kurz nach der Herausnahme, d. h. nur nach dem Verlust einer dabei stets ausfliessenden geringen Menge Blut und Cerebrospinalfliissigkeit, direct auf die Waageschale gelegt.” The series as published by Retzius gives the weights of 450 male brains, and 250 female. In addition to brain-weight, age, stature, and cause of death are recorded. After throwing out individuals incom- pletely recorded, and those falling outside the age limits 20—80, there was left available for this study 416 male records and 233 female. Matiegkat has furnished some very valuable data for the student of brain- weight. His material came from the autopsies at two institutions in Prague, the Institut fiir gerichtliche Medizin, and the Pathologisch-Anatomische Institut. The series from the former source is the larger and more complete, and is the only one which is used in the present discussion. This includes the brain-weights of 372 males and 197 females, together with records of age, stature, skull-length and skull-breadth. The individuals were all adults between the ages of 20 and 80. The great majority of them were Czechs. The method of weighing the brain * Retzius, A.: ‘‘ Ueber das Hirngewicht der Schweden,” Biol. Untersuchungen, N. F. Bd. 1x., Cap. Iv. pp. 51—68, 1900. + Matiegka, H.: ‘Uber das Hirngewicht, die Schidelkapacitiit und die Kopfform, sowie deren Beziehungen zur psychischen Thiitigkeit des Menschen,” Sitzber. des kin. bihmischen Gesellsch, d. Wissensch, Mathem.-Naturwiss, Classe, Jahrg, 1902, No. xx. pp. 1—75. Biometrika 1v iS 18 Variation and Correlation in Brain- Weight was that usually, followed: the pia and arachnoid were not removed before weighing. The special value of this material of Matiegka’s les in the fact that it is the most extensive series of brain-weights in which skull characters are so recorded as to make it possible to determine their correlation with the weight of the brain. In addition, data are given for the determination of the correlation between brain-weight and stature, and brain-weight and age. Unfortunately Matiegka’s material is open to criticism in certain important respects. In the first place detailed tables showing the exact records for each individual are not given, but instead the material is thrown into the form of correlation tables. In making up these tables the units of grouping were not altogether wisely chosen. This results in introducing a certain error into the absolute values of the constants deduced from these tables. The exact cause, nature and amount of this error will be pointed out later in the paper. Again, the skull-length and skull-breadth were measured in a very unreliable manner. Regarding the method of making these measurements Matiegka says (loc. cit. p. 46): “Bei dem von mir verar- beiteten Materiale und zwar in beiden Instituten wurde gewéhnlich auch die Linge und Breite des Schideldaches d. i. an dem behufs der Hirnentnahme vorgenommenen Horizontalschnitte gemessen. Das so erlangte Lingenmass ist daher bedeutend hiirzer als die grésste Schddelldnge, die Breite wohl haufig etwas kleiner als die grésste Schiidelbreite*.” The effect of this procedure on the biometric constants is discussed later in this paper. My chief reason for including this somewhat questionable material was for the sake of comparison, and because reasonably long series of brain weighings are not so plentiful as to allow one who would study the subject a great range of choice in material. The data on the brain-weight of the English were very kindly placed at my disposal by Prof. H. H. Donaldson. He was able to obtain, some years ago, a copy of the original detailed manuscript tables which the English anatomist Marshall had compiled from Boyd’s original data collected at the St Marylebone Infirmary and the Somerset County Lunatic Asylum. These detailed tables were never published, though it was on them that Marshall’s well-known paper, “ On the Relations between the Weight of the Brain and its Parts and Stature and Mass of the Body in Man +,” was based. For the privilege of examining and using this unique and valuable material I am very grateful to Prof. Donaldson. From the biometrical standpoint the raw material available for a statistical study of the weight of the brain is peculiarly complex. The great bulk of the statistics consists of material gathered at the autopsies performed in large general hospitals or other public institutions of similar character. The result of this is that the only large collections of brain-weight data available are not representative “random” samples of the general population, Instead they represent a group of the population which has been subjected to a rather stringent selective process of a peculiar kind. It would be very difficult, if not impossible, to formulate all * Italics Matiegka’s. + Jour. Anat. and Physiol. Vol. xxv1. pp. 445—500, 1892. RAYMOND PEARL 19 the factors which contribute to the differentiation of the “general hospital popu- lation” from the “general population” of the same locality. Of some of them, however, we may be fairly certain. In the first place, there are many classes (social) of the general population which will never be represented in any significant proportion in the general hospital population. Furthermore, the nature of the injury or disease from which an individual is suffering in many cases determines whether the individual shall be in a hospital. On this point, Greenwood*, in an able discussion of the matter, says: “ Evidently the population of a general hospital will chiefly consist of: (i) persons acutely ill, (11) those suffering from surgical injuries or diseases, (iii) sufferers from medical affections requiring special treat- ment. Chronic maladies of old age, such as bronchitis, indeed, any highly chronic disease, will be under-represented in comparison with the general death-rate. Similarly, the number of cases of valvular heart disease and rarer disorders, such as Diabetes Mellitus or Insular Sclerosis and other nervous lesions, will be above the general average.” In addition to the selection which occurs in the formation of the general hospital population there is a still farther weeding out when we come to deal with autopsy records, for the reason that not every individual dying in a hospital is subjected to a post-mortem examination. Whether there shall be an autopsy or not in a given case depends on several factors, one of which is the cause of death. The individuals whose last illnesses have given doubtful or rare clinical features are more likely to be subjected to a post-mortem examination, other things being equal, than the individuals in whose case the fatal illness has run a_ perfectly typical clearly cut course of some common disease. Consequently there is a tendency for the statistics to be unduly weighted with deaths from some of the rarer maladies. There can be no doubt that the statistical constants relating to any character of the body which is in any way affected by the disease causing death will be different, in material coming from hospital and pathological laboratory autopsy returns, from those which would be obtained could we get a perfectly random sample of the general population dying outside a hospital, and subjected to post-mortem examination regardless of the cause of death. These peculiarities attendant upon hospital autopsy records are of particular importance when the character studied is the weight of the brain, for the reason that this character is definitely affected both by the age of the individual, and, in certain cases, by the nature of the disease causing death. It is evident that there will be a more or less definite relation between the nature of the cause which brings an individual into a general hospital and the age of the individual. This combined relation of age and disease is a factor of importance in analysing brain-weight data. In what manner will be apparent from a moment’s con- sideration. Suppose we take the first of Greenwood’s classes of the general * Greenwood, M.: ‘A First Study of the Weight, Variability and Correlation of the Human Viscera, with special reference to the Healthy and Diseased Heart.” Biometrika, 111. pp. 63—83, 1904, p. 65. 3—2 20 Variation and Correlation in Brain-Weight hospital population mentioned above, namely, “ persons acutely ill.” It is at once clear that this will include two sub-groups. First in number and importance will be the group of individuals suffering from diseases medically classified as “acute ” in the strict sense of the term, for example, typhoid fever. Second will be the group of individuals acutely ill at the termination of a long-standing chronic disease. These individuals in most instances have been cared for at home as long as the disease remained in the chronic or sub-acute form, and only go into the hospital when it becomes acute. A good example of such cases is afforded by Bright’s disease. There can be little doubt on d@ priort grounds that the mean age of the individuals will be lower in the first as compared with the second of these groups. This is shown to be the case by the actual statistics in the brain- weight series where “cause of death” is tabulated. In the lower age groups the causes of death returned are preponderantly “acute” in the strict sense. In the higher age groups we get a preponderance of the chronic affections. Now it is altogether likely that the acute illness which leads to death in a comparatively short time has much less effect on the weight of the brain than the chronic “wasting” sickness. So, then, the general result is that in the higher age classes where the brain-weight has decreased as the natural result of senescence there is a further artificial lowering on account of the preponderance of individuals who have been afflicted with “ wasting” diseases. On the other hand the returns during early adult life give mean values for the brain-weight which are probably nearer the true normal value for the general population. In the present paper it was desired to consider only adult brain-weights and consequently only material which fell in age between 20 and 80 years was usually used. It is the general opinion of anatomists (cf. for example, Ziehen, loc. cit. p. 359, and Marchand, loc. cit. p. 404), that after the age period 15—20 there is very little increase of weight in the brain with advancing age. In other words the brain is considered to attain practically its complete growth in about the first 20 years of life. According to Marchand (loc. cit. p. 402 et seq.) the maximum brain-weight remains practically constant till the individual is about 50 years old. After that age he believes that senile degeneration begins. Accordingly, in the discussion of the Marchand data, I have considered the period from 15—80 years to constitute the “adult period,” so far as brain-weight is-concerned. As a matter of fact, the result would have been but little different had the period 20—80 years been used in this case (as it was with all the other data), because there are only 36 males and 17 females falling in the age class 15—19, and these individuals are fairly evenly distributed among the brain-weight classes as will be seen by consulting Tables 17 and 18 of the Appendix. When the work was begun I had, of course, no means of knowing how it would turn out, so it was necessary to decide how to treat the material on the basis simply of a careful general inspection of the statistics and of the opinion of other workers on the subject. As one point which I wished especially to investigate was the change of brain-weight with age, it became a nice problem as to how to handle the RayMonpd PEARL 21 material so as to best get at this. I finally decided after a good deal of con- sideration to adopt provisionally Marchand’s view stated above, viz., that the weight of the brain reaches a maximum between the ages 15 and 20, remains constant till about age 50, then declines through the old age period. Acting on this plan I separated the material in each case into what I have called a “ young” and the “total” group. The “young” series included the individuals falling between the ages 20 to 50 (in the Hessian material, 15—50, and in the Bohemian 20—59 were the limits, in the former case from choice, in the latter from necessity). The “total” series included all the individuals between ages 20 and 80 (in the Hessian series 15 was the lower limit). The results show that on the whole this method of handling the material was adequate, considering the ends to be gained and the amount of material available. 3. Fundamental Constants. As a preliminary to the discussion of the correlation of brain-weight with other characters it 1s necessary to exhibit and discuss the constants, measuring type and variability of the characters considered. Certain problems which are of consider- able interest on their own account present themselves here. Of first importance perhaps is the question as to the amount of differentiation in respect of brain- weight which exists between different sub-races of men, both in type and variability. Further, the rather extensive material worked over makes it possible to settle the problem of the relative variability of the sexes with respect to the weight of the brain. On this point Pearson (loc. cit.) has made a brief communication based on an analysis of four short series of English data, Bischoff’s Bavarian series, and a French series. His general conclusion is that in respect of brain-weight the sexes are sensibly equally variable. Another interesting problem which merits discussion here is as to how the relative variability in brain-weight compares with the varia- bility of other organs and characters of the body. In Table I are exhibited the values of the Means, Standard Deviations and Coefficients of Variation together with their probable errors for each of the characters studied in this work. While, of course, the primary object of study is the weight of the brain, yet it is necessary for the correlation determinations that we have the fundamental constants for the other characters, age, stature, ete. To effect economy of space the constants for these other characters have been included in this first table. The frequency distributions from which the constants have been deduced will be found in the fundamental tables, numbered in Arabic numerals from 1 to 44, collected at the end of this paper. Regarding the calcula- tion of the constants the following explanations should be made. In preparing the correlation tables from the raw statistical material the following values for the units of grouping were chosen, and used uniformly throughout the work, except in certain cases where it was impossible to keep to them. 22 Variation and Correlation in Brain- Weight Character Sener UAE O rouping Brain-weight ... 50 grams Age os ob 5 years Stature ... 3 centimetres Skull length 5 millimetres Skull breadth 5 millimetres Body-weight 5 kilograms These standard units were chosen after considerable study and experimenting and are believed to be the best values for the data discussed. However, it has been shown recently by other workers* that “the system of grouping adopted is within wide limits immaterial.” An examination of those cases recorded in the present paper where it was necessary to use different units of grouping—notably in the Matiegka data where the brain-weights were grouped into 100 gram classes and the ages into 10 year classes—leads to the same conclusion. The standard deviation was evaluated according to the equation ¢=Vu,. In obtaining mw, Sheppard’s correction was used throughout for the characters brain- weight, stature, skull length, skull breadth aud body-weight. The uncorrected “rough” moment was used in getting the standard deviation in age, as in this case there is no approach to high contact at either end of the range. The decimals have been retained to three places in the tabulated values in order that the significant value of the first place may be seen. It will be under- stood, of course, that in the computations the decimals were retained to a larger number of places for arithmetical reasons. . 4. Brain-weight Types. In brain-weight work hitherto it has not been possible to make accurate com- parisons of the mean weight of the brain in different races. There were two reasons for this: one that the probable errors of the means were not determined, and the other that no method was available by which allowance could be made for the differences in the mean stature and age of the samples of material to be com- pared. Obviously if brain-weight is correlated with stature and age, we should expect to get a considerable difference between the mean brain-weight of two groups which differed widely with respect to these other characters. Inasmuch as a knowledge of the degree of correlation between these characters affords a method whereby the material can be reduced to a “standard stature-age base,” it will be worth while to examine the actual differences in brain-weight types in the four racial groups here discussed. * “ Assortative Mating in Man.” A Cooperative Study. Biometrika, Vol. 11. pp. 481—498, 1903. 23 RAYMOND PEARL "11 ‘dO « FOS: + SLL-FS CFL ¥ G18-€ GEL. F LES-€ PEL-+9L0-F 881: + LES-F @L1- + LLLP GEL: + OOF-F O8T- + GLE-F EEL. + SES-F OLG- + LIL-8 O€%- + 66E-4 €GE- + OFE-8 ELE: FOLT-L G6G- + 960-2 CGS. + B8E-L 06%: + EG6-L GES: + GSL-8 Poe: FPLE-8 EGG. + EF0 8 809: + FBE-1Z eZL- ¥ LPP-F CLL ¥ €LL-F 960: + 8&8-€ LGL- + &6E-F 6LL- + GG3-F F60- + 966-7 LIL + 796-8 F60- + 600-F €GE- + 6F8-8 00Z- + GP6-4 GGI- F8IL-8 612. + L88-4 1%. + GE8-4 L61- + 608-2 GGG. + 616-4 SLI- + 960-8 666 F PILL 6L1- + Z6G-L ‘a MONBIIVA JO YWATOTJO0D 9ZE- F109-01 | 996. 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CO8-€ FLZG-E6 | GFE F C6E-FII ECF-C FOLG-ETEL | SET I€L-F $ OGL-O9FT ""* (Buno x) . ‘ 88E-€ FZLL-96 | E1S-6F 809-E1T 099-6 F FIG-OLEL | L6T €L6-€ $ 6E8-FEPL ae (1eq0,) a suvlumeyog €69-€ ¥ 1P8-LOT | €LL-€ F 88-11 | €G6-4 F ZOG-O8SL | ELT ZOP-F + 9GL-GOPL * (Bun0 x) a < ZH9-Z F 89E-ZOL | 99F-G F GLO-ZLT 6LLF F GL8-6G21 | 18z LOP-€F LEL-L6EL ue (T2401) i‘ SUBISSOH] LOF-P FEPS-GOL | L1Z-€ ¥ O8T-6OL LI€-9F 88PF-69ZT | LET 09¢-F ¥ L9Z-S1FL + (sunoZ) . " SFL-€FLEL-OOL | 98F-3 + 6ZE-90T P-F F G89-ZGSL | EES 91G-€ ¥ 18F-00FT es (Te}OL) JYSTOM-ULeAg | **" Sepearg 6 ? é “ON ? aos —_ TOPORIVU) DRY WOTJBIAVG PABpURYY uve “SIUDISUD) JOJUGUDPUNT "T ATIDVIL 24 Variation and Correlation in Brain-Weight Taking the gross values given in Table I the following Table II has been prepared to facilitate comparison. TABLE IL. Brain-weight Types. Hessians Bavarians Swedes Bohemians Total Males Mean brain-weight 1391-737 1363°185 1400°481 1454839 Mean age... 42-889 43°171 45024 45°699 Mean stature 167°359 166°549 169°789 — Young Males Mean brain-weight 1405°756 1369°110 1415:267 1460°150 Mean age... 31°572 35°582 36°317 40°545* Mean stature 167°294 — 170°176 169°417 Total Females Mean brain-weight 1259°875 1220°356 1252°682 1310°914 Mean age... 44:795 40°333 47°639 41447 Mean stature 156°688 154°712 158°710 — Young Females Mean brain-weight 1280202 1235°504 1269-488 1313-910 Mean age... 34-061 32°731 35°689 35°613t Mean stature 156°980 a 159°673 157°650 It is apparent at once that, considering the magnitude of the probable errors, there are significant differences between the Bavarians, Swedes, and Bohemians, with respect to brain-weight, in both sexes and age classes. Between the Swedish and Hessian means the differences are smaller, and as the following table shows, are not significant when their probable errors are taken into account : Total Males: Total Females: Hessian mean— Swedish mean = 7:193+6:065. Swedish mean—Hessian mean = 8°744+4:952. Young Males: Swedish mean—Hessian mean = 9°511+46°331. Young Females: Hessian mean— Swedish mean=10°714+8°197, In no case is the difference as great as even twice its probable error, hence we must conclude that the differences exhibited might, so far as the extent of material allows us to judge, be due to random sampling. * This value was calculated by taking the first four age classes of the ‘‘total’’ series. It includes more individuals than the “ young ” series for the other characters, This was the only way in which the mean age of the ‘‘ young” series in this case could be even approximated to, as the raw material given by Matiegka is incomplete in this respect. The age as given is probably a little too high, but the error cannot be great. RAYMOND PEARL 25 Is this agreement between the Hessian and Swedish peoples in the matter of brain-weight real, or only apparent and the result of compensating differences in the other correlated characters? An answer to this question can be reached in two ways; either by reducing one set to the same “stature-age base” as the other and then comparing results, or by analyzing the effects on the brain-weight of the observed stature and age differences in the two sets of material. Both of these methods depend on the use of characteristic equations, the derivation of which will be described further on in the paper. For the present I shall make practical use of such equations without further discussion of their derivation or validity. Table II shows that in the samples with which we are dealing the Hessians, both male and female, average younger and shorter than the Swedes. Reducing the Hessians to the same “stature-age base” as the Swedes, by means of the appro- priate equations given on p. 63 infra, the following results are obtained : TABLE III. The probable brain-weight of a group of Hessian ¢ ¢ having the Equation same age and stature as the Swedish ¢ ¢ (Total) ce .. =1396:480 (23) Swedish brain-weight observed ... ae ies its ae .. =1400°481 Swedish heavier. Difference . = 4001 The probable brain-weight of a group of Hessian 9 ? having the same age and stature as the Swedish 9 9 (Total) Be .. 1258232 (25) Swedish brain-weight observed ... ae He ee v= = 1252°682 Hessian heavier. Difference .. = 5°550 The probable brain-weight of a group of Hessian ¢ ¢ having the same age and stature as the Young Swedish ¢ g aus are =1407°457 (24) Swedish brain-weight observed ... a Bee Bae age w= = 1415-267 Swedish heavier. Difference sas eco lO The probable brain-weight of a group of Hessian 9 2 having the same age and stature as the Young Swedish 9 ? Sno aes =1283°540 (26) Swedish brain-weight observed ... On 56 oe bad .. =1269-488 . Hessian heavier. Difference eee | 14:052 These differences after the stature-age corrections are made are extremely small. This is particularly well shown if they are reduced to a relative basis, by expressing the differences as percentages of the observed Swedish brain-weight : The difference between Total Swedish ¢ ¢ and Hessians (calculated) =0°29°/, of the observed Swedish brain-weight. The difference between Total Swedish @ @ and Hessians (calculated) =0-44°/, of the observed Swedish brain-weight. The difference between Young Swedish ¢ ¢ and Hessians (calculated) =0°55°/, of the observed Swedish brain-weight. The difference between Young Swedish ? 2 and Hessians (calculated) =1:11°/, of the observed Swedish brain-weight. Biometrika 1v 4 26 Variation and Correlation in Brain-Weight It appears then that, so far as the available material may be considered valid as representing the whole population, the conclusion is justified that the Hessian and Swedish peoples are sensibly alike in respect to their brain-weight. The agreement would be closer than that actually observed if the assumption made in the course of reasoning here followed, that the regression of brain-weight on age and stature is strictly linear, were exactly true. As will be shown later these regressions are not strictly linear but they approach linearity with sufficient closeness to serve for all practical purposes. This, together with the difference in age distribution of the “total” and “ young” series, also accounts for the fact that while in the “total” series and male “ young” series the gross differences between Swedes and Hessians are lowered when we reduce to a common “stature-age base,” these differences are slightly increased, on the other hand, when we deal in the same way with the shorter female “ young” series. This is in agreement with the general fact that these peoples are probably the most closely related ethnically of any with which we are dealing in the present paper. The Swedes may be considered to be among the purest representatives of the original blonde, dolichocephalic Teutonic race (Ripley* and Denikert+). In the case of the Hessians some intermixture of this Teutonic with the brachy- cephalic, characteristically brunette Alpine type has occurred. The differentiation from the Swedes in such important characters as skull form and stature is not great howevert. Turning now to the other racial groups discussed, viz. the Bavarians and Bohemians (Czechs), we find as would be expected, that the differences in brain- weight are greater. Using the Hessian data as a basis for comparison, the gross differences with their probable errors are exhibited in the following table : Total Males : Hessian mean—Bavarian mean =28°552+4°763. Total Females: Hessian mean—Bavarian mean =39°519+5'618. Young Males: Hessian mean—Bavarian mean =36°646 +5842. Young Females: Hessian mean—Bavarian mean =44°698 + 6°577. Total Males : Bohemian mean—Hessian mean =53'102 + 5°286. Total Females: Bohemian mean—Hessian mean =51:039+6:212. Young Males: Bohemian mean—Hessian mean =54°394 + 6°462. Young Females: Bohemian mean—Hessian mean = 33°708 + 7551. In all cases the differences are seen to be well above what might arise from errors in statistical sampling. Reducing to a common “stature-age base” by the method followed above the following results are obtained : * Ripley, W. Z.: The Races of Europe, New York, 1899, pp. xxxii, 624. + Deniker: The Races of Man, London and New York, 1900, pp. xxiii, 328 and 611. + This is well shown graphically in the maps indicating the distribution of stature and cephalic index in Europe, given by Ripley (loc. cit. pp. 96 and 53). Cf. also Deniker, loc. cit. pp. 328 and 329. RayMonD PEARL 27 TABLE IV. Probable Brain- Ree Difference | Difference 4 weight of a group aac panage expressed as | as per cent. puee of Hessians of ses in excess or | of observed ot Libeces the same sex, B; served defect of mean rrranae se age and stature Gees Or! Hessian |(Bavarianor| © ace 1 a as the Se) values Bohemian) ane Bavarian ¢ g¢, Total 1388 ‘984 1363°185 —25°799 —1:9°/, | No. 23, p. 63 5 2 2, Total 1265°137 1220°356 —44°781 =3°7 7], ee lOsess “ 66d, Young 1397°157* 1369-110 — 28°047 — 20) ie nm EE Sp 5 2 92, Young 1277°317* 1235°504 —41°813 —3°4°/, 5» 26, 45 Bohemian ¢ ¢, Total 1394°570+ 1454°839 +60°269 +4:1°/, my 2h oo: . 2 9, Total 1259°310+ 1310°914 +51°604 +3°'9°/, sap Os 55 5 36 6, Young 1399 527+ 1460°150 + 60°623 4+4°2°/. oy 5 2 9, Young 1278°935t 1313°910 +34°975 2 fe 5 Ash. ep This table brings out several points of considerable interest. In the first instance is to be noted the general effect of reducing the Hessians to the same “stature-age base” as the other races, upon the interracial differences in mean brain-weight. In the case of the Bavarians the deviations from the Hessian means are reduced slightly in both male series, and the “young” female series, when stature and age differences are eliminated. In the “ total” female series the difference is increased over the original gross difference. The explanation for this discrepancy in the female “total” series is to be found in a peculiar abnormality which this series shows in its elemental frequency distribution, and which will be discussed later (p. 40). In the case of the Bohemians we get the somewhat remark- able result that a reduction to a common “stature-age base” actually increases the differences of this racial group in brain-weight as compared with the Hessians. Or in other words, in the samples with which we are dealing the stature and age differences act in a compensatory way and bring the mean brain-weights closer together than they would be if we dealt with selected samples of the populations, each sample having the same mean age and stature. The fact already noted (p. 18) that the mean brain-weights deduced from Matiegka’s tables for the Bohemians are somewhat too large, may now be examined in detail. This Bohemian material was published by Matiegka in the form of correlation tables with unduly large units of grouping (cf. Tables 25 to 28, Appendix). The base unit for brain-weight was 100 gr. Now in calculating the general population mean from these tables one assumes that the individuals in each elemental frequency group centre in brain-weight at the mid-point of that group. Thus, the individuals recorded as having a brain-weight of between 1300 and 1400 grams are assumed to centre at 1350 gr. But evidently this assumption will not be true except at the middle of the whole range. For example, the brain-weights recorded * This assumes that the mean stature of the Bavarians would be the same in the “‘ young” group as it is in the “total” group. The error introduced by this procedure is practically negligible. + Here again, on account of lack of data, the mean stature is assumed to be the same in ‘‘ total” and “ young” groups. + See footnote on p. 24. 28 Variation and Correlation in Brain-Weight as between 1500 and 1600 will centre somewhat below 1550. Now, of course, if the distribution is exactly or approximately normal and the number of individuals is sufficiently large, the errors from this cause on one side of the general population mean will balance those on the other, and we shall still be able to get a very close value for this mean from the frequency distribution. Unfortunately, in Matiegka’s series, however, the total numbers are not sufficiently large to overcome entirely this error. So we have the following differences: by direct calculation from the individual observations Matiegka finds for the mean brain-weight of the “ young” (20—59) series in the male 1450-4 gr., in the female 1305°5 gr. Our corresponding values are 1460°150 gr. and 1313°910 gr. or there is an excess of 9'7 gr. and 8-4 gr. respectively. For the “total” series (20—80) I have calculated from regression values which Matiegka gives (p. 7), based on the individual observations, general population means which gives values as follows: males 1441-4 gr., females 12948 gr. The corresponding values from the tables are: males 14548 gr., females 1310°9 gr., or the excess of the table means is 13°4 gr. for the males and 161 for the females. These differences are small and for practical purposes negligible. The general conclusion may be drawn that, apart from all differences in stature, and in the mean age of the samples studied, the Bavarian mean brain-weight is lower than the Hessian (roughly about 2°/,), and the Bohemian mean brain-weight is higher than the Hessian (roughly about 3°5 °/,). These results seem to be of some importance as indicating a method whereby brain-weight may have some scientific validity as an anthropological character. This can hardly be said to be the case when nothing more is done than to tabulate means without probable errors, and no attempt is made to get rid of disturbing stature and age effects. Interracial differences in brain-weight may mean something or nothing. With modern biometrical methods it is possible to measure exactly these differences, and when such methods are used it is the belief of the writer that brain-weight can become a really significant anthropological character. In the cases discussed here we see ethnic affinities and differences clearly reflected in the brain-weight. Of the four racial groups studied the two most closely related in origin and general anthropological characters, the Swedes and the Hessians, have mean brain-weights sensibly identical. The Bavarians and Bohemians, though close neighbours geographically, arise from totally distinct ethnic stocks (‘‘ Alpine” and Slav respectively) and associated with this we find a wide difference in the weight of the brain. On the question of the homogeneity of brain-weight statistics, some evidence is furnished in the third and fourth columns of Table IV. If the inclusion of aged persons (50 years and over) introduced any considerable element of heterogeneity so far as brain-weight constants are concerned it would be expected that the interracial differences in the means would not be equal in the “total” and in the “young” series. As a matter of fact in the material here discussed they are sensibly equal, or perhaps better, they are not significantly different. The whole RAYMOND PEARL 29 question of the homogeneity of this material will be treated fully in connection with the discussion of the variability in brain-weight. The interesting questions brought up by this table of means, regarding sexual differences, association of brain-weight with other characters, etc., will be discussed in later sections of the paper. 5. Variation in Brain-weight. Homogeneity of Material. Variability in the weight of the brain has been subjected to exact investigation by only one worker, Pearson (loc. cit.), so far as is known to the writer. He deduced from his material (cf. p. 15 supra) coefficients of variation ranging in value from 7°93 °/, to 10°64°/,, the higher values being from admittedly heterogeneous series. The numerical values for the variation constants and their probable errors found in the present work are given in Table I. The question of homogeneity of material should be first discussed. For reasons which have been set forth above it would be hopeless to look for any high degree of homogeneity in any collection of human brain-weighings at present available. The best we can hope for is a fair degree of homogeneity, and reasonably the same degree in different series which are to be compared. Unfortunately there is not available here, as in the case of craniometrical investigations, series in which fair homogeneity can be inferred with high probability, so that it is not possible to make a direct estimate by comparing variabilities with such a “known base.” In- stead resort must be had to indirect methods. The best of such indirect methods is based on the fact that if a random sample be taken from a homogeneous collec- tion of material the variation constants for the sample and the whole collection will not significantly differ. On the contrary if the material is non-homogeneous such a sampling will give different values for the constants. If the sample be selected, i.e. not random, the variation constants for the character selected will of course be lowered. Now in the material as treated here we have in the “young” series for each racial group a selection from the “total” series, but a selection based on age, not on brain-weight directly. If there were no correla- tion between brain-weight and age such a selection on an age basis would be, of course, a random sample so far as brain-weight is concerned. Unfortunately for the argument, there is, as will be shown later, a sensible though in general low correlation between age and brain-weight. This being the case it would be expected that such a selection as has been made in separating out the “ young” from the “total” series would result in a lowering of the standard deviation and coefficient of variation in brain-weight. Now, as a matter of fact, as the following table shows, the correlation between brain-weight and age is so low that in the relatively small series under discussion, the “young” series forms practically a random sample of the “total” series, within the limits of error, with respect to brain-weight. 30 Variation and Correlation in Brain-Weight TABLE V. Difference in Variation Constants between “ Young” and “ Total” Series. 3 ? 8. D. _ C. of V. 8. D. C. of V. Swedes... eee | $2°85144:066 | +4+°122+°291 + 4°78645°465 | +°271+4°435 Hessians ... we. | —13474+3°971 — ‘1774 '285 — '627+4°541 —'170+°372 Bavarians .. | —1°8884+3°556 —'173 +234 — 10°357 +3°909 — 941 + °320 Bohemians Bee | ae WisWard Sil + 026 + °304 — 3°545.4+5:067 — ‘287+ °388 . (Skull)* | +1:832+4-230 | +-078+:295 | — 1:93244:866 | —-2064-372 Difference + when “Young” is greater. 5 = * " less. In only one case—the Bavarian females—is the difference as large as its probable error, and in this case the difference is not three times the probable error. The conclusion seems justified that the series are reasonably homogeneous in other respects than age. The age distribution is such as to exclude growth effects, and, as the differences indicate, the effects of senescence on brain-weight are so insig- nificant that for practical purposes the material may be considered sufficiently homogeneous to warrant further biometrical study. From the character of the differences it would appear that the most homogeneous series are, on the whole, the Swedish and the Bohemian. A direct examination of the standard deviations and coefficients of variation confirms the conclusion. In the following table I have arranged in order according to ascending value, the variation constants for the “total” series, using as the basis for the grouping the coefficient of variation: TABLE VI. Comparison of Different Races in Respect to Variation in Brain-weight and Skull Capacity. Males Females 8. D. |C. of V. S. D. C. of V. Swedes ... e's ... | 106°329 | 7°592 || Bohemians sas . | 96°772 | 7-382 Bohemians ee ... | 113°608 | 7°809 || Swedes ... ise ... | 100°757 | 8:043 Hessians ar ... | 112°675 | 8-096 || Hessians ee ... | 102°368 | 8°125 Bavarians ae ... | 110°664 | 8118 || Bavarians ies ... | 101°776 | 8°340 English t... niko ... | 124°48 | 9°20 English t ado son | LLSS7, 9°72 Skull Capacity, English} — 8°28 Skull Capacity, English } — 8°68 a 5 Germans § — 7°74 3 » Germans§ — 8°19 * See p. 17 and Table I. for the series here referred to. t+ Pearson: The Chances of Death, Vol. 1. p. 321. + W. R. Macdonell: Biometrika, Vol. m1. p. 221. § Pearson: The Chances of Death, Vol. 1. p. 333. RayMonpd PEARL 31 All of the four series used in this paper are in good agreement, both among themselves and with the skull capacity values. They are distinctly lower than the figures from the admittedly heterogeneous, Reid, Peacock, Sims and Clendinning English series, worked over by Pearson. Taking all the evidence together we are driven to the conclusion that the series are fairly homogeneous. If any one of our series is markedly heterogeneous all the others must be equally so, which would be a very improbable result. That the series cannot be very heterogeneous in their make-up is also shown by the values of the coefficients of variation for skull capacity from two admittedly homogeneous series, the Whitechapel skulls studied by Macdonell, and Ranke’s Bavarian series. A comparison of the variability in brain-weight with that shown by other organs and characters of the human body may next be undertaken. I have arranged in the following table (VII) the coefficients of variation for a variety of characters which have been studied by biometrical workers. The arrangement 1s in general that of descending order of values in the male series. The attempt has been made in the table to include representatives of all the different classes of organs and characters for which we have biometric data available. The most noticeable and remarkable fact brought out by the foregoing table is that with the exception of capacity, all skull characters are roughly only about half as variable as brain-weight. Some such a relation as this might have been predicted, on the general ground that brain-weight measurements and statistics are on many accounts rather “loose,” and would indicate a higher variability than would exact measurements on skulls, even though it had no real existence. Such reasoning, however, takes no account of the agreement, which is really remarkably close when we recall the numerous sources of error in brain-weight returns, between skull capacity and brain-weight in their variability. All will admit that capacity is the most difficult skull character to measure accurately, yet no one would maintain that the difference in variability between cephalic index, for example, and skull capacity was entirely, or even in any considerable part, due to the element of error in the measurement of the latter. The agreement in variability between skull capacity and brain-weight is, of course, to be expected on theoretical grounds. Thatit should turn out in practice to be so close is a first-rate guarantee of the general trustworthiness of brain-weight statistics. As to the explanation of the great variability in brain-weight and skull capacity as compared with the other skull characters we may tentatively reason about the matter in the following way. If the list of organs and characters given in the foregoing table be examined a natural division into three groups almost im- mediately suggests itself. First, we have at the bottom of the list the “bone” measurements, in general including all those characters which depend primarily for their values on the dimensions of various parts of the skeleton. These characters give values for the coefficient of variation up to from 5 to 7, certain of the mandibular variabilities exceeding this limit. Next comes the group giving values for the coefficients of from 7 to 10, with the limits fairly sharply marked off. 32 Variation and Correlation in Brain-Weight TABLE VII. Coefficients of Variation for Man. 3 ? Weight of Spleen (General oe pbulemens ie wis w. | 50°58 — ” ” ” (Healthy) iI 5 Ba6 eats AAG 38°21 — Dermal Sensitivity? ... ae ie ... | 85°70 | 45°70 Weight of Heart (General Hospital Population) * Bae a .. | 32°39 — Keenness of Sight{t ... aes So8 ... > | 28°68 | 32°21 Weight of Kidneys (General Hospital Population)# bir tes .. | 24°63 — Weight of Body (Bavarians).. : os ae .. | 21°32 | 24-715 Weight of Liver (General Hospital Population)... ee OE wae [eee — Swiftness of Blow{ ... ne abe is seo) | LOR 17-1 Weight of Heart (Healthy) + +56 ast see ee anh on 17°71 — re » Kidneys Magan: Pee ae ae ee ae3 sa, | 16:80) a Breathing Capacity ... eas Sa 2 ss ais Si 7 GEG 20°4 Strength of Pull} as cae 500 ave oa aes 68 15:0 19°3 Weight of Liver (Healthy) +. os as AME wee | LAESO = Height of Mandible (et ‘both sexes) § ane ies ast .. | 11°73 | 11°78 Weight of Body (English){ ... ae ana se we | LOSE | Siar Skull Capacity (Etruscan)|| ae ei Ae 586 — nae 9°58 8°54 Brain-weight (French) t ae aes oe ah) fen ays 9:16 9:14 Skull Capacity (Modern Italian)|| aA; a oe Ae Rae Be 8°34 8:99 Re 55 (English)7 - ... aa ae ace ae ane 8°28 8°68 (Egyptian Mummies) | wae sae oi ee ves 8:13 8°29 Brain- weight (Bavarian)... a er “ares me 8-118 | 8-340 - 5 (Hessian) ae ae oe une peel wee ae 8-096 | 8-125 5 (Bohemian) . : EE a Bee aes a 7:809 | 7:382 Skull Capacity (Modern German)|| .. ey 714 8-19 Be (Naqada)|| ... oa v3 ze Be ae at 7°72 6-92 Brain- -weight (Swedish) = cee oe abe ae se 7592 | 8-043 Skull Capacity (Parisian French)| : ee ote aoe ae 7°36 7°10 (Aino)|| : ar aes we 7°07 6°90 Mandible, Distance between Foramina mentalia (English, both sexes)§ 6°23 6°23 Length of Forearm**... an a 5°24 5:21 - » Femur (French)... 5:05 5:04 », Tibia elias 4975 | 5365 3 », Humerus (French) t 4°89 5°61 », Radius 4°87 5:23 Skull, Height to Breadth Index (English) T 4°86 4:16 3 Breadth to Height _,, all ; 4°83 4:17 Length of Finger (English Criminals) ++ 3 4°74 — Skull, Ratio of Height to Horizontal Length (English) 1 4°61 4°10 Length of Foot (English) ++ .. 4:59 — Skull, Cephalic Index for Horizontal Length (English) 7 4°38 3°99 Length of Cubit (English Criminals)tt ... re ee 4°36 = Skull, Least Breadth of Forehead Grpehen ys 4:29 4°55 » Height (English) 4:21 3:96 . Length of Base (English) 1 ar 4:07 4:11 5 Cephalic Index for Greatest Length (English) .. 3°95 4:03 Stature (English)** ... 3:99 3°83 Skull, Ratio of Height to Greatest Length by English)... 3°80 4:21 5 Greatest Breadth (English) 17 Pe 3°75 3°54 . ee Height (English) ae 3°73 4°12 Face Breadth (English Criminals) ++ 3°707 — Skull, Cross Circumference (English) 7 3°70 3°97 » Sagittal ¥ 3°63 3°90 Head Breadth (English Criminals) ++ 3°333 — Skull, Length (English)7 ... 3°31 3°45 Head Length (English Criminals) + 3°154 — Skull, Horizontal Circumference (English) 1 2°87 2°92 * Greenwood, M.: Biometrika, Vol. 111. p. 66. + Ibid, p. 67. t Pearson, K.: The Chances of Death, Vol. 1. pp. 293—377. § Macdonell, W. R.: Biometrika, Vol. 111. p. 225, || Ibid. p. 221 (after Pearson). WT Ibid. p. 222. ** Pearson and Lee: Biometrika, Vol. 11. p. 370. +t Macdonell, W. R.: Biometrika, Vol. 1. p. 202. Raymond PEARL 33 Finally we have all the characters giving values above 10. Now as one passes from one end of the series to the other a definite biological relationship is plain. At the upper end of the series (coefficient of variation 10 and over) the organs and characters tabulated are such as depend in a very considerable degree for their values as determined by measurement, on the general metabolic condition of the organism as a whole at the time the measurements are made or immediately before. This will be freely admitted, I think, for such characters as “ Dermal sensitivity,” “Keenness of sight,” “Weight of body,” “Swiftness of blow,” “Strength of pull,” “ Breathing capacity ” and the like. The same thing, though less apparent perhaps, is, I believe, true when the characters are the weights of viscera. Indeed Greenwood’s own results show this to be the case, when we get such different results for means, variabilities and correlation accordingly as we deal with the “general hospital population,” “healthy” organs, or the same viscera in diseased conditions of different characters. Wynn* has only recently shown that in rabbits there is a considerable degree of probability that continued administration of digitalis will in a short time raise the mean weight of the heart appreciably, while at the same time the body weight is lowered. It is to be regretted that his series of experiments included so (statistically) few individuals. Furthermore, it seems reasonable to assume that in this upper group the thing measured in the majority, if not in all cases, is not the thing natural selection has acted upon directly, allowing that it has acted at all. I think it may be fairly assumed that so far as natural selection has acted at all on these organs and characters, the selection has been in the direction of ability to function properly so as best to conserve the physiological economy of the organism as a whole, rather than in the direction of absolute size of organ or character. The physiologically balanced functioning with reference to the needs of the organism as a whole is the important thing in such organs as the spleen, pancreas, liver, etc. Absolute size of organ can hardly be a very close direct measure of ability to function well. So then, I am inclined to attribute the high variability observed in those organs and characters falling together in the uppermost part of the table to these two factors: (a) the value obtained by measurement depends to a considerable degree on the general metabolic condition of the individual at the time, and (b) the thing measured is not the thing with which natural selection, so far as it has acted at all, has had directly to do. On the other hand, if we turn to the lowest group in the above table (coefficients of variation under 5—7 for the males) we find the two factors just mentioned almost exactly reversed. Here the characters are either of the skeleton directly, or in the case of the anthropometrical data are characters which closely depend for their measured values on the size of portions of the skeleton. Obviously the general metabolic condition of the organism has, within wide limits, little effect on the length of the femur or the skull, for example, and if there is an effect it manifests itself very slowly. Further it is to be presumed that the absolute * Journal of the American Medical Association, Vol. x11. pp. 164, 165, 1904. Biometrika Iv 34. Variation and Correlation in Brain-Weight size is of greater direct selective value in the case of parts of the skeleton than in the case of organs like the liver, ete. In other words, size is much more directly related to proper functioning in the former than in the latter case. The skull may at first sight appear to form an exception here. It must be kept in mind, however, that the skull serves two very important—perhaps equally important— functions; viz., (1) the enclosing and protecting of the brain, and (2) the serving as a basis for the attachment of the complexes of muscles which actuate the masticatory apparatus and support and move the head. With reference to each of these factors natural selection may act on the skull. So far as the second factor at least 1s concerned, size of skull (in a broad sense) will be of selective value. So that here again, since size and function are interrelated, we should expect to find the results of selective action reflected in the size measurements of characters. In general, I think it can safely be maintained that the low variability shown in the group of characters under discussion is what might be expected to result from the operation of the factors just mentioned. In the middle group (coefficients of variation ranging from 7 to 10 in the male) including brain-weight and skull capacity we seem to have also an intermediate condition, with respect to the two factors which have been mentioned as among the causes which contribute to the observed variability. In the first place, there can be no doubt in face of the evidence both from the biometrical and anatomical standpoints, that the cranial capacity is quite highly correlated with volume of the brain*. Admitting this, the discussion may be restricted to the brain. Now it has been known from the time of the earliest collection of brain-weight statistics that the weight of the brain is influenced to a certain extent by the general metabolic condition of the individual preceding death. I am inclined to think however that the extent of this influence has been over-estimated. My primary reason for this view comes from the analysis of the statistics themselves. If the metabolic condition of the individual preceding death influenced brain-weight to a marked degree, one would hardly expect to find the agreement shown in the constants of variation and correlation tabulated in this paper, when different groups of individuals with different distribution of ante-mortem affections were compared. We might expect by chance to get two groups in agreement, but on this supposition the odds would be great against getting four groups to agree so closely as they actually do. Furthermore, there is a certain amount of evidence from other sources indicating that the influence of ante-mortem general metabolic conditions on brain-weight, while always present as one factor, does not produce so marked an effect on the weight of the brain as on the size measurements of some of the organs in our most variable group. For example, Miithlmann+ studied microscopically the brains of 24 individuals of different ages from birth to 90 years to determine the amount of pigment degeneration in the ganglion cells at the different ages. In the adults the causes of death included such divergent con- * On the anatomical side, cf. Symington, Nature, Vol. 68, pp. 589—544. t+ Verhandl. d. deutsch. Pathol. Gesellsch, Bd. 11. pp. 148—157. RayMonpd PEARL 35 ditions as endocarditis, pneumonia cruposa, and nephritis chronica, Yet the differences found in the appearance of the brain in different cases were clearly associated with differences in age, and not with causes of death. On this whole question more data are needed. Exact measurements of the correlation between skull capacity and brain-weight are much to be desired. We also lack definite biometrical evidence regarding the effect of morbid conditions on the weight of the brain. The evidence at present available scems to lead to the conclusion here indicated. On the other hand, it can hardly be maintained that natural selection has acted on the size of the brain to the degree that it has on skeletal structures. The brain, in this respect, comes nearer to the organs included in our most variable group. So that, on the one hand with a less marked dependence of the measure- ment on the previous metabolic condition of the organism than is found in the most variable group of organs and characters, and on the other hand, with a less stringent selection with reference to size as compared with the least variable group, an intermediate condition in variability for the brain is to be expected. Of course, too much stress cannot be laid on this argument as to the relative variability of the characters of the human body in view of the meagreness of the data at present available. The data at hand, however, plainly suggest some such an explanation, and it is only as a tentative suggestion worthy of being kept in mind as further data are available that the matter has been developed here. To summarise; when a series of the various organs and characters of the body whose variation has been measured are arranged in the order of their relative variability, those characters falling at the most variable end of the series are those whose measured value is affected markedly by the previous general metabolic condition of the organism, and in which ability to function properly is not closely related to size of organ, and in which natural selection, if it has acted at all on the character, has not acted directly upon the thing measured. The opposite end of the series includes the least variable organs and characters, in which both the above mentioned conditions are reversed. Brain-weight and skull capacity occupy an intermediate condition both with reference to variability and the conditioning factors discussed. Attention may be turned next to the question of the relative variability of the different races in respect to brain-weight. In the following scheme the races are arranged in descending order of mean brain-weight in the left-hand columns, and of variability (measured by the coefficient of variation) in the right-hand column. The lists are based on “ total” series. From these lists it would appear that as a general rule the higher the mean brain-weight is, the lower will be the variability. But it must be kept in mind that with our present series none of the differences in the variability columns is significant. Taking the extremes, Bavarians and Swedes for the males, and Bavarians and Bohemians for the females, the differences between the coefficients of variability with their probable errors are respectively *526 +°217 and ‘958 +°337. 5—2 36 Variation and Correlation in Brain- Weight In these cases the difference is less than three times its probable error and cannot be considered as certainly significant. The lists are introduced merely to show the | Mean C. of V. Mean C. of V. | Bohemians Bavarians Bohemians Bavarians | Swedes Hessians Hessians Hessians | Hessians Bohemians Swedes Swedes | Bavarians Swedes Bavarians Bohemians general trend of the results. It is possible that with much larger series of brain- weight statistics and consequently reduced probable errors some such relation as that just stated might be definitely proven. For the sake of comparison and reference I include here tabular arrangements of the means and variabilities for the characters stature and age, made in the same way as that for brain-weight. Stature. Mean C. of V. Mean C. of V. Swedes Bohemians Swedes Bohemians Bohemians Hessians Bohemians Hessians Hessians Swedes Hessians Swedes Bavarians Bavarians Bavarians Bavarians Age. Mean S. D. Mean S. D. Bohemians Hessians Swedes Bohemians Swedes Bohemians Hessians Hessians Bavarians Swedes Bavarians Swedes Hessians Bavarians Bohemians Bavarians Comparing the different characters the means seem to be entirely chaotic, but there is more regularity as regards the variabilities. The relative variabilities in stature and age for the different samples run parallel, with the exception of the Hessians and Bohemians in the male series. The samples which show the greatest variability in age of individuals included, also show the greatest variability in stature, and vice versd. This relation is, of course, to be expected, but the relation between RayMonD PEARL 37 the brain-weight series and the other two is rather curious. There is no parallelism between the variability columns of the brain-weight series and the other two, but the variability columns of the stature series together with the females of the age series exactly agree (with the exception of the transposition of the Hessians and Swedes) with the columns of means of brain-weight. In other words in those series showing the greatest variability in stature (and in the females of age also) we have the highest mean brain-weight, and vice versd. No reason for this curious parallelism is apparent, and it may be purely accidental, but it seems worth noting. The questions of the relative variability of the sexes in brain-weight and the racial differences as affected by sex will be discussed in a later section of the paper. 6. On the Nature of the Frequency Distribution. In this section the variability in brain-weight will be discussed analytically, according to the methods and nomenclature of Pearson’s fundamental memoir on Skew Variation* and its supplement}. My purpose in considering these curves analytically is not primarily that data may be furnished so that the material may be fitted with appropriate curves, but rather in order that definite knowledge may be had as to whether the variation in this character obeys the “normal” law of the deviation of errors. Both Miss Fawcettt and Macdonell§ have reached the conclu- sion that, for practical purposes at least, the majority of skull characters may be considered to conform to this law in their variation. It is of prime importance to determine in how far the same is true of the weight of the brain. Since this was my chief object in analyzing the data, I shall not at this time deal graphically with the curves, but instead shall merely present the chief analytical constants arranged in tabular form. Furthermore, from considerations of the time involved in computation, I have not determined the analytical constants for all the brain- weight frequency distributions given in this paper. Instead, after examining all the data, I decided to confine myself to the “total” series for both sexes of the four races, The analytical constants for the selected series are exhibited in Table VIII. In the second column is given the number of brains on which the calculation in each case is based. The third column gives the unit in terms of which the second, third and fourth moments about the mean (2, “3, and 4) are calculated: the next two columns give 8, and 4/@,, and the two following, 8, and 3—8,. Following this are given in order the “criterion” (x,= 28, —38,—6)||, the mean, mode and skewness. The skewness was calculated from the moments directly by the formula 1 NBit3) « h= 258, —68,—9" | and from this the distance from mean to mode was obtained by multiplying by co. * Phil. Trans. Vol. 186, A, pp. 343—414. + Ibid. Vol. 197, A, pp. 448—459. ~ Biometrika, Vol. 1. p. 448. § Ibid. Vol. 11, p. 227. || Phil. Trans. Vol. 197, A, p. 444. {| Pearson, K.: “‘On the Mathematical Theory of Errors of Judgment and on the Personal Equation,” Phil. Trans. Vol. 198, A, pp. 235—299. Page 277. if n Brain-Weight Variation and Correlation i 38 69LT- | PEE-GLGEL | POS-GEST | GESP-S | OZ8O-L —| 0Z8O0-F | GIGF- | GIFS. | LZZ9-GP | 6ZEO-E | LEPE-€ 8080. | [GG-LIGT | 9S€-0G6T | FHPO-€ | 80PS-L—| 80PE-F | FILI- | FELO- | GI84-L44 | 16E6-— | SEPI-F 900I- | LP0-GGEL | G8T-E9ET | LOPF- 9G0E- — | 9EOE-E | 86GZ- | 8EGO-. | Y1LES-64 | 616F-G | 9868-F 9690. | GOL-POET | VL6-OLEL | ShOE- — | 6ZEL- TL98-G | 6ELT- | S6GLO- | FFIG-S | EOL. GOE6- I91G- | G6Z-O€PTL | 6E8-PEFL | SLOG. — | 96ET-— | OBEL-E | GZOF- | ES9L- | OOES-E | 806. LO66- L GPGO-. | P6G-PSEL | GL48-69EL | LES. — | SZ9T.- GLE8-S | 8960. | G600- | PFLE8-6F | LO9S-— | LIGI-F Ge9l- | IL&-ELET | LEL-16EL | Scre- GB9E-— | G89E-E | EEE. | GIEL- | 8E98-98 | 8EFL-F—| E8L0-¢ LOLI: | 98G-IPSI | Z89-BSZT | 6E0. IOI. — | LEOI-€ | VGGe- | 80GO-. | SOLI-I¢ | LhP8-L | 8090-F IGOL- | L69-68€L | [8h-OOFT | VEGF. — | 9E0E. PIGL-SG | POT. | L8ZO- | G6BST-LG | G6Z9-L | ESSG-F SSOUMBTD|) IpoT[ uvayy =juomoyg| “d-—e &g Igy Tg Fr Srl Gr ‘30g | seo 4 ,.sunoz,, ae SOG eee lion | ss Bog |ezgiP “ UCLICACT BOOT /461/6 =“ es ‘SOOL|sze P “ — uermmeyog 30¢ |1ss|6 “ ‘ Bog |ouprlP ge UVISSOFT B0¢ jees/h “ ‘3O0¢ |9TF P «IPIOL, cpomg qa) | “oN Selleg pue soe "{Y Sto M-ULeIg ‘saauny fo squmsuog ywouhpouy THA dId Vib RayMonpd PEARL 39 Considering first the skewness, it is seen to be in all the series positive, or the mean is greater than the mode. Further we note that in all cases the value is low. Whether the values can be considered significant however can only be determined by an examination of the probable errors. The formula for the probable error of the skewness is ‘67449 NE 2 *, The limiting values for this probable error for values of n ranging from 197 to 529 as in the present case are respectively 0589 and ‘0359. Having regard to the number of cases on which the calculations are based it appears that in six out of the nine cases tabulated the skewness can be regarded as certainly or probably insignificant. In all of these six cases the skewness is less than thrice its probable error, in two cases it about equals its probable error, and in one is less. The remaining three cases out of the total (skewness, ‘1635, ‘2161 and ‘1769) are very probably or certainly significant. In general we may safely conclude, I think, that, in the case of the weight of the brain, the distance from the mean to the mode will be very small. If the mean and mode do not coincide the mean will be greater than the mode. This agrees with Miss Faweett’s+ conclusion for the most important skull characters in the Naqada race. Macdonellt finds, however, that in the case of the English, considering the same skull characters, “if we were to draw the curves, the mean would be found in half the number of the curves to be less, and in the other half to be greater than the mode.” In neither brain-weight nor skull series does there appear to be any definite preponderance in the value of the skewness of one sex over the other. We may turn now to the other constants, which are of most significance in determining whether the distribution may be considered normal within the limits of error; viz., /8,, 8, and the criterion, The probable error of 4/8, (= “67449 ee) ranges in value for our series between ‘0718 (n=529), and ‘1177 (n=197), that of B, (= 67449 Ja) between ‘1437 (n= 529) and ‘2354 (n= 197); and that of the criterion (= ‘67449 s/o) between ‘2873 (n=529) and ‘4708 (n=197). Considering the probable errors of /8,,1t is seen at once that of the eight “total” series three give certainly insignificant values (0958, ‘1139, ‘1113) for /8,; two others give values which are probably insignificant (1694, +2254). One (2298) is probably significant ; and the two remaining values (‘3623 and 4029) are certainly significant. The Bavarian “young” § series gives a certainly significant value for /8, Taking next the deviation of 8, from 3 in comparison with the probable error of 8, we see that in four cases (3— 8, =— 1031, ‘1628, —1396, 1329) @, differs from 3 by an insignificant amount. In one case (2036) the difference is less than twice the probable error and hence may be considered * The formulae for the probable errors of the analytical constants are given on p. 278 of Pearson’s memoir on ‘‘ The Mathematical Theory of Errors of Judgment,” loc. cit. supra. + Loc. cit. p. 443. + Biometrika, Vol. 11. p, 227. 40 Variation and Correlation in Brain-Weight as very probably insignificant. In two cases (3682 and °3026) the difference is less than thrice the probable error and hence may be considered possibly, or even perhaps, probably insignificant. The Bavarian § series, both “ total” and “ young,” give certainly significant values for 3—,. Considering finally the criterion it is seen that in all cases except the two Bavarian ¢ the criterion differs from zero by a certainly or very probably insignificant amount. These two Bavarian ¢ series differed so greatly from the normal curve in most of the analytical constants that it was thought desirable to determine their position precisely by means of another constant «,*. For the “ total” series I found «, = ‘0032, and for the “ young ” series K,= "0816. By the scheme given by Pearson (loc. cit. p. 445) we see that the “total” series, when the probable errors of the constants are considered, comes very close to the condition demanding a curve of Type II («,=0, 8, =0, 8, not = 8). The “young” series clearly demands a curve of Type IV («,>0 and <1). The deviation of these Bavarian female curves from the normal type I believe to be due to an undue accumulation of individuals in one brain-weight class ; viz., that from 1250—1300 gr. It seems altogether probable that some of the individuals which should have gone into the next higher class (represented in the “total” series by a frequency of only 26 as against 69 in the class next lower) have by some error been entered in Bischoff’s lists with too low brain-weights. What the source of error was it is, of course, impossible now to determine. The abnormality of the Bavarian females has already been noted in the discussion of the means and variabilities. Leaving these two series out of account I think that on the whole we may safely conclude, as Miss Fawcett and Macdonell (loc. cit. p. 443, and p. 227 resp.) have for skull characters, that : With series of brain weighings such as are considered in this paper we shall reach for most practical purposes adequate graphical representations of the frequency by using the normal curve of deviation: y= yy e~*?. It should always be kept in mind, however, that our series, both on the brain- weight and skull sides, are too small to fix absolutely the normality or non-normality of the variation in these characters. Some of the distributions certainly differ from normality. The conclusion stated above is to be considered simply as a practical result, rather than as a theoretical generalization. This result seems to be of considerable importance as indicating the worth of brain-weight statistics. It shows that such statistics justify careful study and analysis, and that, contrary to the statements of certain recent writers on the subject, there is no general fallacy inherent in the data themselves which renders abortive any attempt to reach through them the truth regarding the mass relations of the brain. In order to test exactly how well the normal curve represents the data in a single case I have fitted the Swedish male “total ” series with a normal curve. The frequency histogram and its fitted curve are shown graphically in the * By (B2 +3)? : hey ; -, Pearson, Phil. Trans. Vol. 197, A, p. 444. “4 (8, — 3B;) (28, — 38, — 6) RAYMOND PEARL 41 accompanying diagram. The unit of # is 50 grams. The scale of frequency at the left of the diagram gives y. Diacram I, Showing the distribution of variation in the brain-weight of adult Swedish Males. Rees 80 ——+ + tee | ee || | ors 60 sa a ae Fan 50 : ae ot te a 30 i sail 20 st] x es | 10 i Se HL - ied) Lanes ccd a i : ws ca ee 1075 1125 1175 1225 1275-1825 1975 1425 1475 1525 1575 1626 1675 1725 1775 oe and o = 2'1266, becomes y = 78-0401 e~ 011062", with the origin at 1400°481. Applying Pearson’s test of the goodness of fit*, and comparing areas instead of mid-ordinates, as is necessary where the number of frequency groups is so small, we get the results shown in the accompanying table. The table gives the observed frequencies (m,’) The general equation y = y,e-*”?”, where in this case yy = b) Roce Observed Calculated mot Under 1100 0) ‘981 ‘981 1100—1150 1 2°9 1°24 1150—1200 10 8°5 26 1200—1250 21 20°3 02 1250—1300 44 39°0 64 1300—1350 53 60°4 “91 1350—1400 86 i De2 leno 1400—1450 72 75°3 14 1450—1500 60 60°8 ‘O1 1500—1550 28 39°4 3°29 1550—1600 25 20°6 “94 1600—1650 12 8:7 119245) 1650—1700 3} 2°9 003 1700 —1750 1 8 05 1750 and over 0) ‘036 036 Totals 416 415°817 11°320 * Phil. Mag. Vol. u. pp. 157—175. Biometrika tv 6 42 Variation and Correlation in Brain-Weight the calculated frequencies (m,), and the ratio of the squared difference between the ay: two to the calculated (amy), . Here n’ the number of frequency groups is 15 and y*= 11°32. From Elderton’s Tables* the value of P may be obtained. With n’=15 and y?=11 the value of P as given in the table is ‘686036. Or, expressed in words, if the brain-weight of Swedish males obeyed the “normal” distribution of frequencies we should expect to get a greater divergence between theory and observation in roughly 65 out of every 100 trials when the trials were based on random samples of 416 indi- viduals each. In other words, the fit may be considered very fair, and certainly entirely satisfactory for all practical purposes. 7. The Correlation of Brain-weight with other Characters. In the introductory portion of this paper it was stated that one of the chief objects for which the work was undertaken was to measure exactly the degree and nature of the association between brain-weight and other characters in man. All workers on the subject have realized the importance of such determinations and numerous attempts have been made to arrive at them. The methods used, however, have not been such as to lead to definite and sound conclusions. These methods have in the main consisted in tabulating the mean brain-weights for various groupings of the other characters under consideration. In this way it can of course be determined whether there is any shift of the mean as the other character changes. Further than this one cannot go. Besides, in this method there lurk numerous pitfalls unless one uses it with a clear understanding of some of the fundamentals of statistical science. Different results can be obtained as the material is differently grouped. Hence, it is not strange that we find conservative anatomists making only very general statements as to the correlation of brain-weight with other characters, and, in those cases where an attempt at greater precision of statement is made, considerable difference of opinion as to what conclusions shall be drawn from the data. What are wanted in the case are not simply inspections and general “appreciations” of tabular lists of brain- weighings, or of ratios and indices, or even of correlation tables or regression lines, but instead definitely determined coefficients of correlation with their probable errors. With these we can get in given cases precise statements of the degree of correlation between different characters. It would of course be highly desirable to measure the correlation between the weight of the brain and a great variety of other organs and characters of the body. Such characters will at once suggest themselves to anyone interested in the problems of correlation. For instance, aside from age, stature and body-weight for which material is available, there are the whole range of skull characters; the * Biometrika, Vol. 1. p. 161. RAYMOND PEARL 43 weights of the various viscera; the weight and volume of the spinal cord; characters of the muscular system, etc. A study of the correlation between any or all of these characters and the weight of the brain would be most interesting. But, unfortunately, material for such study cannot be had. Brain-weight statistics usually provide data for but few other characters; in the majority of cases only sex, age and stature. In two of the series used in this work, viz., the Swedish and Hessian, this was the case. For these two groups I have determined all possible correlations for both sexes, and for two age groupings (“ young” and “total”). This gives as the pairs of characters, (a) brain-weight and stature, (b) brain-weight and age, (c) stature and age. Bischoff’s Bavarian material furnished in addition to the above, data on the body-weight. The following correlations have been determined for this material: (a) brain-weight and age (“total” and “young”), (b) brain-weight and stature (“total”), (c) brain-weight and body-weight (“total”), (d) body-weight and stature (“total”). In the case of the Bohemian material data were furnished from which the correlation between skull length and skull breadth could be determined so that for this group I have been able to calculate coefficients of correlation for the following pairs of characters : (a) brain-weight and stature (“young”), (b) brain-weight and age (“total”), (c) brain-weight and skull length (“young”), (d) brain-weight and skull breadth (“young”), (e) skull length and skull breadth (“young”). In Table IX are exhibited the coefficients of correlation together with their probable errors. The correlation tables from which the coefficients were calculated are given in the appendix to this paper, Tables 1 to 44. References are given in Table IX telling for each coefficient the original table from which it was calculated. The coefficients of correlation were determined from the usual Bravais formula according to which the coefficient of correlation ay) ~ Noy,’ where # and y are deviations from the means of the two correlated characters and o, and o, are the respective standard deviations; the usual* method of evaluating r from the above equation was used. The most striking fact of a general nature shown by this table is the generally low degree of correlation which exists between the weight of the brain and other characters. The coefficients run noticeably lower even than those of skull characters (cf. Macdonell’s Table V+) and very markedly below those between different characters of the long bonest. On the other hand our brain-weight correlations give values of the same general order of magnitude as those found by Greenwood§ for various abdominal and thoracic viscera. Attention may be called * Yule: Jour, Roy. Stat. Soe. Vol. ux. Part 1v. pp. 1—44 (Reprint). + Biometrika, Vol. ut. p. 232. { Lee and Pearson: Phil. Trans. Vol. 196, A, pp. 228, 229. § Loc. cit. p. 19 above. Weight rain- Variation and Correlation in B 44 ee eee, | PP | G80. F PEEP. | LFS _ &P | 860. +Z96E- | G9E - e pes ([e{O.L) oangeyg pues yYysIEM- Apog. late"! f PE | PIGO.+PSEI. | 691] €& | LZE0.+900F- | 66% | (SuUNOA) Tapeerg [[MAg puY 4y8Ue7 [MAG | Gh | ZIFO-F09GS- | LFS) Ih | PEO. +1L9T- | G9E = * ({eIJOJ,) YStem-Apog pue Ystoa- urerg | | | BE | 660. F IPOS. | 691) L€ | 9080.FGG9r. | 66% | (SUNoZ) YIpeorg [NYG “ = & | OF | G9FO-F FOIE. 6E1| 62 | €LZ0.F SSPE. | 66S] (SuNOA) YYSuET [[NYG pue qysTom-uyerg OP |ZIFO.FFIIO. | 8€%) 6E | E20. FOOTO. —| 298 | Hy fe (Sano ae 8E | FSE0.FGOPE-— ESE LE 06Z0. FEET. —|6BS_ 8B_ | PHO. F9GGs.— | L6T) LB | Geso.FGrOS.—|sLe)" ** (TeIOL,) @BY pur zyS:om-urerg a | 96 | L990. 891. | €T| os | L6g0.F FEO |993]} °° = (SunoX) 9€ | €1FO-+9EZZ. 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ON | 91481 ‘ON fe) e fs) ‘9 sallag pue sejoBreyO SUBISSOH] sapong “SU01ZD]ALL0() ‘qybram-uag ‘XI ATAVE Raymond PEARL 45 here to the general uniformity of the correlations for the same characters in different series. All the brain-weight and stature correlations are seen to be positive, while with a single exception (Bavarian female “young” series) all the correlations of brain-weight with age are negative. The coefficients are generally lower for the “young” than for the “total” series, as is to be expected. In the case of the correlation of brain-weight with age, some of the coefficients for the “young” series are evidently insignificant in comparison with their probable errors. The correlations between stature and age are negative with the exception of the Hessian male series. The positive sign in these series arises from the fact that in this material the age class 15 to 20 was included. All growth in stature has not stopped at age 20, and as a consequence there are included in the tables (21 and 23) five individuals of unusually small stature and low age. These serve to change the sign of the coefficients. Their greater effect in the “ young” series is apparent. The detailed discussion of the various brain-weight correlations and the regressions based on them I propose to take up in separate sections of the paper now to follow. 8. Brain-weight and Sex. All brain-weight statistics show that the brain of the male is absolutely heavier than that of the female. In the series here discussed the absolute differences in mean brain-weight between males and females are as follows : Male mean—Female mean | | “Total” series “Young” series | Se as | Swedes oe 147°8 | 145°8 | Bohemians ... | 1439 | 1462 Bavarians ... 142°8 133°6 Hessians het 131°9 125°6 From this table the following points are to be noted: (a) Considering the size of the probable errors involved it is evident at once that the absolute difference is sensibly the same for all four races. Taking the extremes of the “total” column the difference between the Swede and Hessian sex differences is 15°9 with a probable error of + 7:9, or in other words the difference is almost exactly twice its probable error and cannot be considered certainly significant. The “young” series points to the same conclusion. So_ then, Weisbach’s law for stature that the greatest sex differences occur in those races having the highest mean stature, does not appear to hold for brain-weight, so far as absolute differences are concerned. 46 Variation and Correlation in Brain-Weight (b) The absolute sex differences are sensibly the same for both age groupings. We may conclude then, that whatever changes occur in the weight of the brain with advancing age act in such a way as to leave the absolute sex difference unchanged. These absolute differences may be converted into relative differences by expressing them as percentages of the { and $ means. When so expressed they take the following form : Sea Differences in Relation to Means. “Total ” series “Young ” series | Race | 9 8 in eee ee a Percentage of | Percentage of | Percentage of | Percentage of | ¢ mean ? mean ¢ mean ? mean Swedes 10°6 °/, eG Sigie 10:3 °/, 11°5 °/, Bavarians .... | = -10°5°/, Were 9°8 °°], 10°8 °/, Bohemians ... | 9:9 °/, 10°97 °/, 10:0 °/, Lye Hessians 9:5 °/, Osi 8:9 °/, 98°). The relative differences confirm the conclusions reached trom the examination of absolute differences. The result that the changes which take place in the weight of the brain during adult life affect both sexes to an equal degree relatively seems to be of some importance. It implies that the factors which bring about the gradual lowering in the weight of the brain with advancing age are organic in a strict sense, and not dependent on environmental conditions. The same thing is shown if the relative differences expressed in another way, namely by taking the ratio of male to female mean, are examined. For our series these ratios are as follows: Sex Ratios. Means. Brain-weight. Race “Total” series | ‘‘ Young”’ series Swedes 1:118 1115 Bavarians 7 1:108 Bohemians 1:110 1111 Hessians 1:105 1:098 The sex ratios for a number of other characters are given in the following table for comparison. The values in this table are taken from a table in Pearson’s Chances of Death*. * Vol. 1. p. 374. RAYMOND PEARL 47 Sex Ratios. Means. Various Physical Characters. Organ or Character | Sex Ratio Body weight, Babies see .. | 1:034 <5 4 Children... ba 1-038 - . Adults ae oe 1193 Weight of Vital Organs .. ea 1°130 Stature, Children ... wae an 1:007 5 Adults... va a 1-077 Height, Sitting ... wee ies 1-032 Long Bones cos a .. | 1086 Chest Girth ee ses ae 1-024 Squeeze of Hands asd cae) E207 Keenness of Sight and Touch ... | 1:061 Skull Capacity ... es .. | 1124 », Circumference Aer .. | 1:042 Cephalic Index... a | ‘997 Head Index sei see sewed 995 Profile Angle ire “8 cee ‘994 Alveolar Angle... x aaa | 994 Nose and Palate | 1013 We may turn next to the question of the relative variability of the sexes in respect to brain-weight. Pearson* found that in the case of the English, the female showed “slightly more” variability than males; the French data which he worked over gave sensibly equal variabilities for the sexes, as did also Bischoff’s Bavarian data. In the following table are shown the differences between the male and female coefficients of variation with the probable errors of the differences. The values tabulated are the male minus the female constant in each case. Relative Variability of the Seves. Male — Female. Coefficient of Variation Race “Total” series ‘* Young ” series 7 ; | Bohemians... IDM ap BAO) | "740 + 375 Hessians ane — 029 + 293 — ‘036 + °366 Bavarians es — 222+ °254 “546 + 305 Swedes a —'451+4°310 — ‘600 + °422 In three cases out of the eight the male has a larger coefficient of variation, but in no instance can the differences between male and female coefficients be considered significant when compared with the probable errors. In only one case (Bohemian “young” series) does the difference approach closely to a value even twice as great as its probable error. Therefore we must conclude that, so far as the series here considered are concerned, there is no significant difference between * Chances of Death, Vol. 1. pp. 321, 322. 48 Variation and Correlation in Brain-Weight males and females in respect to variability in the weight of the brain. In view of the general reliability, from a statistical standpoint, which has been previously demonstrated to exist in the material on which this work is based, it seems not unreasonable to predict that it will be found that equal variability of the sexes in brain-weight holds generally. Another problem which presents itself in this connection is as to how much of the observed difference between males and females in brain-weight is to be accounted for by the fact that the male body is in general larger than the female. In other words, would a group of females of the same stature and age, say, as a given group of males have the same brain-weight? It is possible to reach a general solution of this problem by the use of characteristic equations based on the correlations and regressions. I propose now to discuss this question by this method. Of course, the prediction may be made by characteristic equations in either direction; i.e, we may predict the probable brain-weight of a group of males having other characters the same as in a given group of females, or we can predict the probable brain-weight of a group of females having the characters the same aS in a given group of males. In the former case characteristic equations based on the male data would be used, in the latter the equations would be those deduced from female data. We may consider first the characters, age, and stature. In the following tables are given (a) the probable brain-weights of a group of individuals of the sex indicated, as calculated from the characteristic equation referred to in the last column of the table; (b) the observed brain-weights of a group of the opposite sex having for the other characters the mean values used in the characteristic equation in calculating (a); (c) the difference between (a) and (b); (d) a reference to the equation from which (q@) is calculated. TABLE X. Probable Female Brain-weight with Stature and Age equal to Observed Male. Predicted female Observed male oe On ; Race brain-weight brain-weight Difference Equation | Hessian (Total) re) 1287-963 | 1391°737 | 103-774 25 » (Young)... 1308-941 —«:1405-756 96-815 26 Swede (Total) 460 1307 °340 1400°481 . 93°141 21 , (Young) ... 1319-783 1415-267 95-484 22 The mean difference here is 97°304. The mean observed difference between male and female without any allowance for stature and age is 137°8 for the same races. So then the differences in stature and age between the males and females in our Swedish and Hessian samples account for only 40°5 gr. or about 29 per cent. of the observed difference in mean brain-weight between the sexes. Let us now proceed to the reversed prediction, and find the probable brain- weights of a group of males having the same stature and age as the means for RAYMOND PEARL 49 these characters in the Hessian and Swedish females. The results are shown in Table XI, which is arranged in the same manner as the preceding table. TABLE XI. Probable Male Brain-weight with Stature and Age equal to Observed Female. Predicted male Observed female aa : Race brain-weight brain- weight Difference | Equation Hessian (Total) ios 1357°591 1259°875 97°716 23 .5 (Young)... 1369°431 1280°202 89-229 24 Swede (Total) aes 1370°543 1252-682 117°861 19 . (Young)... 1391-624 1269°488 122°136 20 The mean difference here is 106°736, or in other words we reach essentially the same result as before. The general conclusion up to this point then is that there is a difference in the mean brain-weight between the sexes of roughly 100 gr. after allowance has been made for stature and age differences by taking the mean brain-weight of an array of individuals of one sex having for type of stature and age the respective means of these characters observed in the other sex. Differences in stature and age account for less than one-third of the observed sex difference in brain-weight. We may next determine what part of the observed sex difference in brain- weight is to be accounted for by differences in skull length and breadth. Here only one race, the Bohemian, can be used on account of lack of material. Let us examine the effect on the mean brain-weight of selecting a group of individuals of one sex having the same mean skull length and skull breadth as the opposite sex. Making the designated selection first from the males and then from the females we have the results shown in Table XII. TABLE XII. Probable Male Brain-weight with Mean Skull Length and Skull Breadth equal to the Observed Female. Predicted male Observed female ieee : Race brain-weight brain-weight {DISMESIEEIMBE | Equation _ x | | | 35 | Bohemian ... 1395°631 | 1310°914 | 84°717 | Probable Female Brain-weight with Mean Skull Length and Skull Breadth equal to the Observed Male. | R Predicted female Observed male | aCe brain-weight brain-weight Difference Equation | Biometrika 1v u | Bohemian ... 1389-736 1454°839 65°103 36 50 Variation and Correlation in Brain- Weight The mean of the two differences is 74°910, while the observed difference between male and female mean brain-weights in the Bohemian “skull” series is 142-014, or in other words by selecting from either sex individuals with mean skull length and skull breadth equal to those found in the opposite sex, we are able to reduce the sex difference in mean brain-weight by about 67 grams, or 47 °/,. This result is of considerable interest. We can come nearer to the brain-weight of one sex by selecting skull length and breadth to the means of that sex than we can by selecting stature and age, but still not a great deal nearer. We should on general grounds expect that aside from skull capacity, skull length and skull breadth would be the two characters whose selection would indirectly modify brain-weight the most. Admitting this to be the case there still remains a considerable difference to be accounted for. It is interesting to note that the sexes are brought nearer together in brain-weight when males are selected than when the selection is from the females. This is explained by the fact that the brain-weight is some- what more highly correlated with the skull characters in the males in our series. Finally, we may examine the effect of a selection of stature and body-weight on the sex difference in brain-weight. The material here comes from the Bavarian series, and the prediction is made in the same way as in the other cases; viz. by selecting males with mean stature and body-weight equal to the female means and vice versd, and determining the mean brain-weight of the selected groups. The results are shown in Table XIII. TABLE XIII. Probable Male Brain-weight with Mean Stature and Body-weight equal to the Observed Female. Predicted male Observed female . : Race brain-weight brain-weight | Dinerence Equation | Bavarian ... 1321°496 1218°776 | 102°720 29 Probable Female Brain-weight with Mean Stature and Body-weight equal to the Observed Male. Predicted female Observed male brain-weight brain-weight Difference Equation Race | | | Bavarian ... 1261:°298 1357 °466 | 96°168 30 Here the better result is obtained by selecting from the females on account of the higher correlation between brain-weight and body-weight and brain-weight and stature in this sex. The mean of the two differences is 99°444, and the observed sex difference in the case is 138°690 gr. So then the differences between Raymond PEARL 51 the sexes in body-weight and stature account for 39°246 gr. or 28 °/, of the observed difference in mean brain-weights. In other words, by selecting individuals on the basis of body-weight and stature alone we shall produce about the same degree of change in the mean brain-weight as by selecting on the basis of stature and age alone. Taking all the results together we reach the general conclusion that the difference between the sexes in mean brain-weight is only in part to be accounted for by differences in other bodily characters. While in general it is true that a small body has a brain of low weight, yet in order for the observed difference in mean brain-weight in men and women to be due to this factor alone either the women ought to have very much smaller bodies than they actually possess, or the men ought to have larger bodies, or both sexes ought to be ditferent from what they actually are in size of body in the directions indicated. Of course it is very easy to say by way of speculation that the smaller brain of woman is due to the fact that the female human organism is subjected to less strenuous demands along the lines of motor and intellectual activity than the male. But such speculation leads to nothing on account of our lack of definite scientific evidence as to the degree of correlation between the weight of the brain and amount and intensity of psychic activity. One other question remains to be considered in connection with the relation of brain-weight to sex. Does either sex show any considerably higher degree of correlation between brain-weight and other characters than the other? Table IX furnishes data on this point. With the exception of the Swedish brain-weight and age correlations and the braim-weight and skull-length correlations it is seen that the female coefficients are uniformly larger than the male. The differences are in many of the cases not significant in comparison with their probable errors but the general tendency towards higher correlations in the females is clearly evident. The meaning of this tendency is in this case difficult to conjecture. It falls in line with the result which has been pointed out by Lee and Pearson in several papers’, viz., that for physical characters generally women usually show a higher degree of correlation than men, in civilized races. 9. Brain-weight, Age and Stature. The coefficients of correlation between brain-weight and age and stature respectively have already been given in Table IX, but before proceeding further in the discussion of their significance it is necessary to determine whether the regressions are linear. One might on general grounds expect the brain-weight and age correlations to be skew, since this is generally the case with growth correlations where one of the variables is age and the other the absolute magnitude of some organ or character. If in any case the correlation turns out to be skew, it of course greatly complicates the problem. The method of analysis which I have followed in determining the degree of * H.g., ‘*On the Relative Variability and Correlation in Civilized and Uncivilized Races,” Roy, Soc. Proc. Vol. txt. pp. 3483—857. 7—2 52 Variation and Correlation in Brain-Weight approach to linearity of the regressions is that given by Pearson*. The two constants which are of the greatest significance here are, (a) the mean square deviation of the means of the arrays from the regression line, ¥,,*, and (b) a con- stant , called the correlation ratio, giving the mean reduction in variability of an array as compared with the whole population. Evidently are = ont ad ro? eee a eee eee cee e eens seal); and He stdin ae (ii), Oo whence by simple substitution %j;2=(n? — 7°) o? where in these equations o is, as usual, the standard deviation of the variates about the mean for the whole population, r is the coefficient of correlation between the two variables concerned, and oy, is the standard deviation of the means of the arrays about the mean of these means. The deviation of ¥j, from zero, and of n from 7, measure the deviation of the system from linearity f. I propose to discuss the relation of brain-weight first to age alone, then to stature alone, and finally to both age and stature together. In Table XIV are exhibited the values of =, and 7 for the correlations between brain-weight and age discussed in this paper. In calculating cy, from which to obtain » according to the relation given above, in this and all other cases the means of the arrays were weighted with the number of cases on which they were based. This procedure of course gave the mean of the means of the arrays the same value as the general population mean calculated from the elemental frequency distribution. TABLE XIV. Analytical Constants for Linearity of Regression. Brain-weight and Age. r n =u Race and Series 3 e re) ? 3g ? Swedish (Total) ... | —°2493+ -0310 |— -2336 + ‘0418 | -2876 | -2770 | 143840 | 14890 (Young)... |—*1705 + 0405 |— +1512 + 0585 | 2251 | 2143 | 146960 | -15190 | ” Hessian (Total) ... |— +1673 + °0300 | — °3598 + ‘0350 | +2002 | 3864 | "10996 a | -1409¢ * (Young)... | — °0750 + :0393 |— +1650 + -0499 | 1411 | ‘1961 | -1195¢@ | -10600 | Bavarian (Total) ... |— °1225 + 0290 |— +2405 + 0354 | +1962 | -3481 | *15330 | ‘25170 Pr (Young)... |— 010040353 | -0114+ 0412 | -0676 | -1958 | -0669a0 | °19550 Bohemian (Total) ... | — 2045 + °0335 | — -2558+ -0449 | 2441 | 3033 +1333¢0 | °1628¢ * Roy. Soc. Proc. Vol. Lxx1. pp. 303—313, especially the footnote, pp. 303, 304. Since the above was written a very full treatment of the whole subject of skew correlation and non-linear regression by Prof. Pearson has appeared as ‘‘ Mathematical Contributions to the Theory of Evolution, XIV.” Drapers’ Company Research Memoirs, Biometric Series II. 1905. + Of course the deviation of 7 from r must be considered numerically simply, because 7 is necessarily a positive quantity from (ii) above, since neither o), nor o can be negative. If the difference between 7 and r were taken with reference to sign in cases where r is minus, an altogether false notion of the degree of departure of the regression from strict linearity would be obtained. The degree of this departure will always, of course, be immediately given by 2), whatever the sign of r. Raymond PEARL 53 It is seen at once that 7 differs from r and &,, from zero in all cases. This, of course, implies departure of the system from linearity. But evidently deviations of a system from linearity may be due to either one or both of two causes. Either, on the one hand, the system may be truly non-linear, in which event the means of the arrays will be fitted better by some curve than by a straight line, or, on the other hand, the points fixed by the means of the arrays may not lie exactly on a straight line and still no curve will represent the relationship between the two variables concerned better than a straight line so drawn that the mean square of the deviations of the points from the line is a minimum. In the first case we have true non-linearity of the regression, while in the second the deviation from linearity is due to the errors of random sampling, and it might reasonably be expected that if the whole population could be studied the regression would become strictly linear. Now evidently in both these cases 7 will differ from r and &,, from zero, so that recourse must be had to some further method in order to determine into which class a given case falls. Two such methods immediately suggest themselves: one, to examine the probable errors involved, the other, to inspect the fitted regression line. An examination of Table XIV makes it immediately evident that the differ- ences between 7 and r (without regard to sign) give values of the same order of magnitude as the probable errors of 7. In only one case is this difference as great as three times the probable error of 7, and in the great majority of cases it does not approach such a value*. So then it seems probable that in our series the regression of brain-weight on age is linear within the errors arising from random sampling. The approach to linearity is sensibly the same both for the whole period of adult life and for the younger half of this period. In order to bring out the facts graphically I have had prepared a series of diagrams showing the regression lines for brain-weight on age. For the sake of economizing space, and since there is essential agreement between the different series with respect to , it was decided not to publish all the regression diagrams. I have chosen for representation here the regression lines for the Swedish and the Hessian data. These are shown in Diagrams II to IX. Diacram II. Probable Brain-weight for given Age. Swedish ¢ Total. W) in grams. Brain-weight ( 22-5 275 325 375 425 475 525 575 625 675 72:5 77:5 Age (=A) in years. [* The proper test is the probable error of 7 ~7, which we hope will shortly be published. Ep.] 54 Brain-weight (=W) in grams. 135 130 W) Brain-weight ( ie) le) in grams. W) in grams. Brain-weight ( Variation and Correlation in Brain-Weight Diagram III. Probable Brain-weight for given Age. Swedish ¢ Total. 22:5 27°5 32:5 87-5 42:5. 47:5 52:5 57:5 62:5 67:5 72:5 "77-5 Age (=A) in years. Diacram IV. Probable Brain-weight for given Age. Hessian ¢ Total. 1450 r cs ° ° 17-5 22:5 275 325 375 42:5 47:5 525 57:5 625 675 72:5 77:5 Driacram V. Probable Brain-weight with given Age. Hessian ? Total. 175 225 27:5 325 375 425 475 52:5 57:5 625 675 725 775 Age (=A) in years. =W) Brain-weig ht ( in grams. =W) in grams. Brain-weight ( Brain-weight (=W) in grams. RAYMOND PEARL Diacram VI. Probable Brain-weight with given Age. Swedish ¢ Young. 1450 1400 1350 1300 22°55 27°5 32:5 375 42:5 47-5 Age (=a) in years. Diagram VII. Probable Brain-weight with given Age. Swedish ¢ Young. 1250 1200 225 275 32:5 375 425 47:5 Age (=a) in years. Diagram VIII. Probable Brain-weight for given Age. Hessian ¢ Young. 17-5 225 275 32:5 375 42:5 47-5 Age (=a) in years. 55 56 Variation and Correlation in Brain-Weight Dricram IX. Probable Brain-weight for given Age. Hessian ? Young. =W) Brain-weight ( 175 225 27:5 325 37:5 42:5 47:5 Age (=a) in years. These diagrams show clearly, I think, that the regression of brain-weight on age in these cases is sensibly linear within the ages 15 to 80, so far as can be judged on the material available. The regression diagrams for the other two racial groups show the same relation. Obviously no simple curve will represent the systems of points shown in these diagrams IT to IX better than the straight line does. So then, until we have very much larger samples of material to work with, we shall be justified in assuming for practical purposes linearity of regression between brain-weight and age, and in considering the deviations of the means of the arrays from the regression line to be due to errors incident to sampling in lots of less than 500 individuals. The equations to the regression lines for these diagrams may now be given. For convenience in practical use I have put these equations in the form of “characteristic” equations. The significance of the letters used is as follows: W denotes the probable brain-weight in grams of an array of individuals of age type A ora. The ages included by A are 15 or 20 to 80; by a 15 or 20 to 50, or in other words the equations containing A are deduced from the “ total” series and those containing a from the “young” series. = is the standard deviation of the array having the probable W as its mean. (1) Swedish ¢ W=1487-783-1:939.4, 32=102-972 (2) » ¢ W=1501-411-2372a, 3%=107-582 (3) » 9 W=1326-475-1:549 4, S= 97-969 (4) » § W=1340'438-1:988a, 2=104:329 (5) Hessian ¢ W=1439°'734-1:119 A, S=111:087 (6) » ¢ W=1432'497- -847a, 3=111-014 (7) 33 @ W=1361-:067 —2:°259 A, S= 95°512 (8) » § W=1339-298-1°735a, %=100-445 The foregoing constants bring out a number of interesting and important facts regarding the relation of brain-weight to age. In the first place it is clear that diminution in brain-weight with advancing years takes place very slowly. In 10 years the Swedish males lose 19°39 grs. and the Hessian males 11:19 grs.* * In order to economize space I have not tabulated the regression coefficients separately. It will be understood of course by anyone wishing to refer to or use these regression coefficients that the number prefixing a letter in the second member of any of the characteristic equations 1 to 36 is the regression coefficient (either gross or partial) of brain-weight on the character indicated by the letter. RaymonD PEARL 57 These are comparatively speaking slight changes, amounting respectively to 1:3 °/, and -8°/, of the mean brain-weight for the whole adult period. In the same length of time (10 years) males lose an average of ‘34” in stature or approximately *5°/, of the mean stature*. The diminution in brain-weight seems thus to be relatively somewhat more rapid than the diminution in stature. The fact that the regression is linear implies that this diminution of brain- weight with advancing age so far as can be judged from the material here discussed is sensibly equally distributed over the whole adult period. To be sure the slope of the lines is in all but one case (the Swedes) less when the individuals between 20 and 50 are treated separately, but the difference is a small one when the probable error is considered (cf. Table IX or XIV). Splitting the material up in this way of course greatly reduces the already statistically small series, so that the probable errors of the coefficients of correlation for the “ young” series become large. It is significant, however, that with a single exception (the otherwise abnormal Bavarian female series) all the coefficients of correlation between brain- weight and age for the “ young” series are negative. The evidence presented by each of the four series here discussed, which are representative and have been shown to be statistically trustworthy, and which if not as large as might be desired by the statistician are among the largest ever collected by the neurologist, is mutually accordant, and leads to but one conclusion: namely, that after age 15—20 there is a steady though very gradual diminution in the weight of the brain with advancing age. Thisconclusion I have emphasized because it is not in agree- ment with that reached by some eminent neurological authorities. I would further emphasize the fact that the conclusion is by no means final in the mind of the biometrician. It simply marks the point to which the material at present available leads. Very much larger series of brain-weighings are needed before the precise form of the regression can be finally determined. But the fact that four distinct series of fair length taken by different observers on different races lead to such closely accordant results as those shown in Table XIV may be taken as very strong presumptive evidence of the essential correctness of the conclusion reached. It is noteworthy that there is ne tendency towards more rapid diminution on brain-weight after age 50 to 60. The rate of diminution shows no such marked changes at this time as has been maintained to occur by many of the workers on the subject. With the exception of the Swedish series the females in all cases show a higher correlation between brain-weight and age than the males. This difference seems in the case of the Hessians and Bavarians (total series) to be probably significant. According to Powys’ (loc. cit.) figures the slope of the regression line between stature and age is slightly steeper for the females. It is difficult to assign any reason for this greater correlation between brain-weight and age in * Powys, A. O.: ‘‘Data for the Problem of Evolution in Man. Anthropometric Data from Australia,” Biometrika, Vol. 1. pp. 30—49. Biometrika 1v 8 58 Variation and Correlation in Brain-Weight the female. It has been shown above that there is in general no significant difference between the sexes in respect to variability in brain-weight, so that the usual formula of “low variation and high correlation” connoting stability of type does not hold in this case. It is possible that we have here an expression in a particular case of a greater general “evenness” in the females of the bodily changes accompanying increasing age, which in turn might be due to the generally more even environmental conditions to which women are subjected. It is note- worthy in this connection that the correlation in respect to duration of life is generally higher between pairs of female relatives than between pairs of male relatives *, The linearity of the regression of brain-weight on age is of interest as possibly indicating a fundamental difference in the modes of action of the biological processes of growth on the one hand and senescence on the other. When growth in absolute magnitude of a character is plotted on a base line of age the result is usually a curved line (cf. for example the 12—25 portion of Powys’ stature curve, loc. cit.), which implies, of course, that the amount of increment in the character in question is not the same for each unit of time. On the other hand in the case of decrease of brain-weight and stature with advancing age the decrement seems to be practically uniform for each unit of time. If these relations should prove to be generally true they would furnish a very interesting field for further study and analysis. Turning now to the relation of brain-weight to stature, we have in Table XV the values of 7, 7 and },,, for all the brain-weight and stature correlations. TABLE XV. Linearity of Regression. Brain-weight and Stature. 7 U] Zu Race and Series 3 2 3 ? g ? Swedish (Total) ... | *1830+°0320 | -3490+:°0388 | :2439 | °3847 | ‘1612¢ | ‘16186 3 (Young)... | *1796+°0403 | °3390+:0530 | ‘2837 | °4738 | -2196¢ | °3310¢ Hessian (Total) ... | *1823-+°0299 "1828+ ‘0389 | *2864 | 3215 | -2209¢0 | 26450 5 (Young)... | *1741+°0383 | -1809+:0496 | ‘2714 | °3807 | ‘20820 | °3350¢ Bavarian (Total) ... °1664+ °0343 22364 °0413 | -2262 | °3270 | °153260 | ‘23866 Bohemian(Young)... | *2084+:0397 | *2168+°0557 | °2419 | °3591 | +1809¢ | -2863¢ Again, the values of 7 and =, are seen to diverge considerably from r and 0 respectively. It is clear that on the whole the regression of brain-weight on stature approaches less closely to linearity than does the regression of brain- weight on age. This is, I think, a somewhat remarkable result, and one not likely to have been foreseen. * Beeton and Pearson: Biometrika, Vol. 1. p. 60. =W) Brain-weight ( Brain-weight ( in grams. RAayMonD PEARL 59 In order to determine whether the departure of the regression from strict linearity is due simply to errors of sampling or to a fundamentally different law of relation between the two variables it is again necessary to have recourse to regression diagrams. I have selected eight diagrams which are fairly repre- sentative. Diagram X. Probable Brain-weight for given Stature. Swedish ¢ Total. 4 s S & S g ‘= 1455 148-5 1515 154-5 157-5 1605 163-5 1665 1695 1725 1755 1785 1815 1845 187-5 190°5 Stature (=S) in centimetres. Diacram XI. Probable Brain-weight for given Stature. Swedish ¢ Total. . 1350 WD S S S 1300 g ‘= <> 1250 = II 1100 Brain-weight ( I oa a oO 1335 1365 139°5 1425 145-5 1485 151-5 154-5 1575 160-5 163-5 1665 169-5 172:5 1755 Stature (=S) in centimetres. Diacram XII. Probable Brain-weight for given Stature. Swedish ¢ Young. v cS {o) (oe; 1350 1455 148-5 151-5 154-5 1575 160-5 163-5 1665 169'5 1725 175°5 178-5 181-5 1845 1875 190°5 Stature (=s) in centimetres. 8—2 60 W) in grams. Brain-weight ( W) in grams. Brain-weight ( 1450 1400 1850 1300 1250 1400 1350 1300 1250 1200 1150 1100 1050 Variation and Correlation in Brain-Weight Diacram XIII. Probable Brain-weight for given Stature. . Swedish ¢ Young. % 1350 iS 3 ~ = 1300 2 ~ [1250 = Il —~ 1200 ~ 2 gS ‘S 1150 re) = As = 83 1100 ice} 1395 1425 1455 1485 1515 154-5 157-5 1605 163:5 1665 1695 1725 175:5 Stature (=s) in centimetres. Diacram XIV. Probable Brain-weight for given Stature. Hessian ¢ Total. 1395 142-5 145-5 148-5 151-5 154-5 1575 1€0-5 163-5 166°5 1695 172°5 175°5 178-5 181-5 1845 187°5 190°5 Stature (=8S) in centimetres. Discram XV. Probable Brain-weight for given Stature. Hessian ? Total. 127'5 1305 1335 136'5 1395 1425 1455 1485 1515 1545 1575 1605 1635 1665 1695 1725 1755 1785 1815 Stature (=S) in centimetres. RAYMOND PEARL 61 Diagram XVI. Probable Brain-weight for given Stature. Hessian ¢ Young. 1450 1400 W) in grams. 1350 1300 Brain-weight ( 1395 1425 1455 1485 1515 154-5 157-5 1605 1635 1665 1695 172°5 175-5 1785 181°5 184-5 Stature (=s) in centimetres. Diagram XVII. Probable Brain-weight for given Stature. Hessian ¢ Young. W) in grams. Brain-weight ( 127'5 1305 1335 136°5 139'5 1425 1455 148-5 1515 1545 157-5 1605 163-5 166°5 1695 172-5 1755 1785 1815 Stature (=s) in centimetres. It is evident from these diagrams that a straight line represents the regression relation between brain-weight and stature better than any simple curve would. So that we are justified in concluding, as in the case of age, that until we have very much larger collections of data than are at present available we can do no better than proceed on the assumption that the regression of brain-weight on stature is linear. In fact, in these samples there is no evidence that there is any tendency towards anything but a linear relation between the two variables. The deviations of the means of the arrays from the regression line are only such as would reasonably be expected to arise when we deal with relatively small samples. In connection with all of the regression diagrams in this paper it should be kept in mind that the outlying points are based on single individuals, or at most only on comparatively few individuals. Hence deviations of these outlying points from the regression line have very little or no significance as regards the general trend of the results. In fact it would really be better if 62 Variation and Correlation in Brain-Weight the outlyimg points were left out entirely in the graphical representations. The non-biometrical reader should understand that each plotted point carries weight in proportion to the fraction which the number of individuals in the array on which it is based is of the total number of individuals. With this caution in mind the substantial linearity of the regression through the bulk of the observations, which alone are of importance, becomes evident. The regression equations showing numerically the relation between brain- weight and stature, may now be given. In these equations W denotes the probable brain-weight of an array of individuals of mean stature S or s in centi- metres. As before, the equations involving S are based on the “total” series, and those involving s on the “ young” series. & is the standard deviation of the array having the probable mean brain-weight W. (9) Swedish g¢ W=915:054+2°859 S, 3=104'533 (10) e g W=919-37442'914s, 3=107-405 (11) , Q W=421:994+4+5-9348, S= 94-421 (12 ‘ 9 W=451-643+5:121s, S= 99-294 (13) Hessian ¢ W=913'59242'857S, 2=110°787 (14 3 6 W=950'21442'723s, F=109°628 (15) » 9° W=834:62442-'7148, 2=100°643 (16 3 9 W=894:031+42'460s, S=100°161 (17) Bavarian ¢ W=836°6674+3:127S, 5=118°452 (18) 5 Q W=627:157+3'824s, S=103°562 We may now proceed to the general conclusions to be drawn from the data given above regarding the relation of brain-weight to stature. In the first place it is to be noted that the correlation coefficients between these two variables are for each series of the same general order of magnitude as those between brain- weight and age, though of course positive instead of negative, as they are in that case. Or it may be concluded that during adult life, at least, brain-weight is only a very little more closely related to general size of body, in so far as we may take stature as a measure of this, than to age. When we turn to the regression coefficients, however, it is seen that a unit of stature connotes a somewhat larger change of brain-weight than a unit of age. Thus, in the Swedish males, an increase of 10cm. in stature connotes an increase in brain-weight of 28°59 gr., while an increase of 10 years in age connotes a decrease of 19°39 gr. in the brain-weight. As in the correlation of brain-weight with age, the females have uniformly a ° higher degree of correlation between stature and brain-weight than the males. I am inclined to think that the same general sort of explanation may be given in this case as in that of the brain-weight and age correlations. The correlation between brain-weight and stature and the regression of brain-weight on stature are sensibly the same for the early adult period of life as for the total adult period. Without any question, I think, this correlation is to be regarded in its origin as a growth correlation, the relation between the two variables being Raymond PEARL 63 practically fixed at the end of the growth period. After that time the relation undergoes no marked change throughout the remainder of life, since both stature and brain-weight regress on age at not far from the same rate, Finally, the combined effect of stature and age on the weight of the brain may be examined. This relationship can best be examined through the medium of multiple regression equations of the form: a = Bre By + Bis V3, in which £,, and @,; are the partial regression coefficients of x, on its associated variables z, and «;, it being understood that #,, 7, and «, here stand for deviations from the means of the respective variables. For our material these equations, when reduced to absolute magnitudes, take the following forms, the significance of the letters being the same as in the regression equations given before. (19) Swedish ¢ W=+1091:021+2°288 S—1°755 d, 3=101°819 (20) » & W=1080-715+2°362s —1:856a, 3=106'477 (21) » § W= 56143344679 S-1:078 4, S= 93-066 (22) - Q W= 533-407+4+4'854s -1:092a, 3= 98-923 (23) Hessian ¢ W= 942:1544+2:989S-1:181 4, 2=108-992 (24) » ob W= 926586431075 -1:318a, 3=108-889 (25) » W=1005-607 +2:244S8—2:173.4, S= 94-261 (26) » 9 W= 974:382+2'345s -1:8294, T= 98883 These equations give the probable mean brain-weight of an array of adult individuals of either sex, having any given age (in years) and mean stature (in centimetres). They are, I believe, the first multiple regression equations relating brain-weight to stature, age and sex, to be published. They are of interest from several points of view. Examples of their practical use have been given earlier in the paper (pp. 25 and 48) in connection with the discussion of racial differences in mean brain-weight and later in connection with the sex differences. They afford a means whereby it is possible to make scientific comparisons of the mean brain- weight of different races, since by their use the probable brain-weight of a group of Swedes or Hessians having the same mean stature and age as the sample of the race with which comparison is to be made can be ascertained. We can in effect reduce both races to the same stature-age base and then examine the brain-weight differences. _It would seem that these equations ought to prove very useful to anthropologists, who frequently wish to make comparisons of this kind. They are based on the two sets of brain-weighings which I believe to be on the whole the most reliable and trustworthy at present available. These equations also make it possible to determine the effect of indirect selection of stature and age on brain- weight. As a practical example of this last use of these equations let us consider the following problem: By how much would it be necessary to modify the stature of a group of Swedish males, having a mean age of 45 years, in order that they might have the same mean brain-weight as a group of Bohemian males of the same mean 64 Variation and Correlation in Brain- Weight age? The mean brain-weight of the Bohemian males with a mean age of 45°699 is 1454839 er. From equation (19) we have then 1454-839 = 1091-021 + 2:288 S— (1°'755 x 45°699) in which S is the mean stature of the selected group of Swedes. Solving we find S= 194-06, or, in other words, our group of Swedes would have a mean height of 6 ft. 4:4 in. ! Now as a matter of fact the Swedish males with a mean age of 45°02 years have a mean stature of 169'79cm. In order then to get from the adult male Swedish population a group having the same mean brain-weight as the adult male Bohemian population, by a selection of stature alone, it would be necessary to raise the mean stature of the Swedes 24°3.cm. or roughly 94 inches. Of course it is not for a moment to be supposed that evolution in brain-weight proceeds by the indirect selection of stature alone. Indeed when it is remembered that the mean stature of Bohemian males is only 169'4cm. the example just cited shows how little the indirect selection of stature has to do with the matter. In the discussion of evolutionary problems, however, it is important to know just how much or how little effect each factor involved is capable of producing. It is in this direction of measuring the relative effect of different combinations of factors that regression equations such as those given above have great value. It is apparent from equations (19) to (26) that stature and age in a measure compensate one another so far as their effect on brain-weight is concerned. The influence of stature is however in all cases greater. It is of some interest to know in the particular cases how great an increase in stature compensates a given increase in age and vice versd. In order to show this I have prepared the following table. An increase of 10cm. in stature is compensated in its effect on brain-weight by an increase in age in the Swedish males of 13:0 years Swedish females of 43:4 Hessian males of 25°3 ” + Total series only. > Hessian females of 103 ,, j 10. Brain-weight and Body-weight. In only one of the four series of brain-weighings studied (the Bavarian) was body-weight tabulated in the returns. Recent workers on the subject have left this character out of their data on the general ground that it is of very doubtful significance or reliability in post-mortem returns from a general hospital popula- tion. Certainly it is true that it is influenced to a very marked degree by the Raymond PEARL 65 conditions preceding death. Yet as Donaldson* has pointed out if we could get this character free of disturbing ante-mortem effects it would furnish the best measure of the general “size” of the body. Since “size” of the body is one of the things to which brain-weight should first of all be related I decided to include in this study a consideration of the correlation of brain-weight with body-weight. I was fully aware of the weakness of the material and expected much worse results than those actually obtained. A glance at Table I will recall the facts as to the distribution of the frequency of body-weight. they would be for the living population of the same race and age. The means are of course lower than This is to be expected. The variation is large but there is very good accord between males and females in this respect. Turning now to the correlations I have arranged in Table XVI the values of 7, 7 and &,, for the correlation between brain-weight and body-weight. Linearity of Regression. TABLE XVI. Brain-weight on Body-weight. Race and Series 3 Bavarian (Total) 1671 + '0343 2260 + -0412 1845 | 3189 The result for the males is very satisfactory, mdicating a quite reasonable Ay > approach to linearity of regression. The female series does not give so good a result, but it will be recalled that throughout the work this Bavarian female series has been found to be somewhat abnormal. i Diagrams XVIII and XIX. Diacram XVIII. W) in grams. Brain-weight ( 27°35 32-5 Biometrika tv 375 Bavarian ¢ Total. Probable Brain-weight for given Body-weight. The regression lines are shown in 42:5 47:5 52:5) Sito 62:5 67°5 72:5 Body-weight (=W,) in kilograms. * H. H. Donaldson: The Growth of the Brain. 77-5 982°'5 87°5 92:5 9 66 Variation and Correlation in Brain-Weight Dracram XIX. Probable Brain-weight for given Body-weight. Bavarian ¢ Total. 1300 1200 Brain-weight (=W) in grams. 175 22:5 27:5 32:5 37:5 42:5 47:5 52:5 575 62:5 67:5 72:5 77-5 82:5 87:5 Body-weight (=W,) in kilograms. Through the great bulk of the observations the regression is evidently linear*. The male series comes very close to linearity. The equations to the regression lines may now be given. In addition to the equation showing the regression of brain-weight on body-weight I have included multiple regression equations giving the probable brain-weight for given types of body-weight and stature together. The significance of the letters is as follows: W denotes the probable brain-weight of an array having a mean body-weight in kilograms W;, or body-weight Wy, and stature S. = is the standard deviation of the array. (27) Bavarian ¢ W=1263'308 +1-'886 W,, ==118°439 (28) . Q W=1121621 +2:265 W,, 3 =103:495 (29) . g¢ W= 917-7484+2-234 +1355 W,, 3=117-711 (30) me @ W= 741:078+2:646 S+1°593 W,, 3=102-432 The correlation of brain-weight with body-weight is of about the same degree as the correlation of the former with the stature, but the regression approaches more closely to linearity in the case of body-weight. As in the other cases the female shows the higher correlation. The absolute increase of brain-weight with every increase of 10 kilos. in body- weight is for the males 18°86 gr. and for the females 22°65 gr. Equations (29) and (30) show that in each case stature contributes absolutely more to an increase in brain-weight than does body-weight. The exact proportions are as follows: in order to bring about a change in brain-weight equal to that produced by an increase of stature of 10 cm. the body-weight would have to be increased 165 kilos. in the males, and 16°6 kilos. in the females. The greater dependence of the brain-weight on bodily characters in the female as compared with the male is very well shown in equations (29) and (30). * Cf. pp. 61, 62. RAYMOND PEARL 67 11. Brain-weight and Skull Characters. As was to be expected the correlation of brain-weight with the skull length and breadth gives the highest values for the coefficients of any of the correlations dealt with. In fact one would predict on general grounds, I think, a somewhat closer degree of correlation between these characters than is actually found in the Bohemian statistics. It is likely, however, that the values here obtained (Table XVII) are lower than they should be on account of the methods used in measuring skull length and breadth. As has been fully described above (p. 18) these measurements were taken on the horizontal cut surfaces after the dome of the skull had been removed by sawing around it. This is clearly not a very precise way in which to get skull measurements. In order to show so far as may be what has been the effect of this procedure I have prepared the following tables com- paring the various constants of the Bohemian skull length and breadth series with other well-known cranial series. Taking first the means we have: Skull Length Skull Breadth Race | 3 se re) ? Bohemians : : 176°547 1707142 149°841 144°953 Whitechapel English * 189-06 180°36 140-67 134°68 Altbayerischt ... ae 180°58 173°45 150°47 143-98 Wiirtemburgert AGG 179°48 172°74 147°88 142°90 The means give reasonably accordant values considering that the Bohemians are a markedly brachycephalic race. ‘Turning next to the coefficients of variation we have: Skull Length Skull Breadth Race 3 % 3 ? Bohemians : ee 4173 3233 4:44] 3°815 Whitechapel English t bee |) BNL I Bee 3°75 Sia! Bavarian § “is oso 3°57 3°89 3°39 French|| .. ao sae 3°97 3°65 4°21 3°67 Naqada... tos nee 3°17 3°14 3°29 3°45 Aino$... 500 is 3°20 3°08 2°76 2°68 * Macdonell, Biometrika, Vol. 111. pp. 208, 209. + Alice Lee, Phil. Trans. Vol. 196, A, pp. 225—264. + Macdonell, loc. cit. p. 220. § Lee, loc. cit. || Unpublished reduction of Measurements in Broca’s MS. by C. D. Fawcett. {| Miss Faweett, Biometrika, Vol. 1. pp. 438 and 456. 68 Variation and Correlation in Brain- Weight The Bohemian females are fairly accordant in variability with the other series in both length and breadth, but the males are apparently somewhat more variable. This greater variability in the Bohemian male series is due I believe not to less homogeneous material but to the manner in which the measurements are made. The coefficients of correlation between length and breadth may next be considered. Race 3 f°] Bohemians ... aie ‘4006 1954 Whitechapel English* ta ‘240 350 Frencht ... ee “089 — 042 German}... os "286 488 English Criminals & : 402 — English Middle Classes. 345 = Ainot a be "432 376 Naqadat at a aes 344 143 The results here are somewhat chaotic, but clearly the difference in degree of correlation between Bohemian males and females is too great to be entirely normal. It is probable that the male coefficient is somewhat too high and the female too low. The agreement between the Bohemian male coefficient and Macdonell’s value for measurements on the heads of living English criminals is striking. We should expect measurements made as the Bohemian were, to agree better with measurements on the living head than with those from the skull. The practical result to which we come is that conclusions from the Bohemian data as to the correlation between brain-weight and skull length and breadth must be only tentative until they can be tested by other less objectionable material. We may now examine in more detail the results as to the correlation of brain- weight with these skull characters. In Table XVII are given the values of 7, 9, ANG Saar. TABLE XVII. Linearity of Regression. Brain-weight and Skull Characters. r UT] 2a Race and Series a = 3 * 3 " 3g ° Bohemians, Skull Length | :5482 + -0273 | 36044-0465) -5660 | 4032 | 1408 | -1808¢ 9 » Breadth | -4655 + 0306 | ‘5041 +:0399) °5498 | -5368 | 292660 | 1845¢ The four regression diagrams concerned are Diagrams XX to XXIII. * Macdonell, loc. cit. p. 233, + C. D. Faweett, loc. cit. + Lee, loc. cit. § Macdonell, Biometrika, Vol. 1. pp. 181 and 188, ‘‘ Measurements on living head and not skull.” Brain-weight (=W) in grams. 1600 1550 1500 1350 1300 RayMonpD PEARL Diagram XX. Probable Brain-weight for given Skull Length. 157°5 Bohemian g¢ Young. 20—59 years. 162°5 167°5 172:°5 1775 182°5 Skull Length (=L) in millimetres. 187°5 192:5 197-5 69 70 =W) in grams. Brain-weight ( 1650 1600 1550 1400 1350 1300 1250 1200 Brain-weight (=W) in grams. Variation and Correlation in Brain-Weight Diacram XXI. Probable Brain-weight for given Skull Breadth. Bohemian ¢ Young. 20—59 years. Gq Git po = 122°5 1275 132:5 1375 142-5 147°'5 152-5 1575 162°5 167:5 172°5 177:5 Skull Breadth (=B) in millimetres. Diacram XXII. Probable Brain-weight for given Skull Length. Bohemian ? Young. 20—59 years. 1400 1350 1300 1250 1200 1150 152°5 157-5 162°5 167:5 172:5 177-5 182:5 Skull Length (=L) in millimetres. RAYMOND PEARL 71 Diacram XXIII. Probable Brain-weight for given Skull Breadth. Bohemian ¢ Young. 20—59 years. 1400 W) in grams, 1350 1300 Brain-weight ( 1250 1382°5 1387°'5 142-5 147°5 1525 157-5 162°5 Skull Breadth (=B) in millimetres. Through the bulk of the observations the regression is very evidently linear. Diagram XXI shows that the considerable deviations of » from r and &,, from zero in the case of the male brain-weight and skull breadth regression are really due to the inclusion in the material of five individuals which are to be regarded as either erroneously measured or recorded, or else abnormal or extremely rare normal cases (cf. Table 31, Appendix). The regression equations follow. In these equations W denotes the probable brain-weight in grams of an array of individuals having a mean skull length Z and skull breadth B in millimetres. (31) Bohemian ¢ W=8'589 L— 52-650, = 96548 (32) , g¢ W=8-076 B+ 253-596, 3=102°170 (33) 53 Q W=6:215 L + 264265, S= 88-466 (34) * 9 W=8646 B+ 68°434, S= 81-908 (35) A g W=6751L+ 5-082 B—489-649, s= 99°745 (36) 55 9 W=4696L+ 7°766 B—602:994, S= 77°893 One rather curious point may be noted here in passing. As the figures stand a smaller probable error will be made in estimating the mean brain-weight of males from a knowledge of skull length alone (equation 31) than if both length and breadth of skull are used. This apparent paradox arises for the following reason*, The standard deviation of the array in the regression of 2, on its associated variables x, and 2; is given by NM S=0,V¥1—R;, Ti Tig — 20 ial sla: where R= 12 13 au 13123 1 — To3" When the regression is of x, on 2 alone the standard deviation of the array is L=o,V1—7,2. * Cf, Yule, loc. cit. 72 Variation and Correlation in Brain-Weight Now the condition under which we shall make a smaller standard error in estimating a, from the two variables 2, and a, than in estimating it from 2, only, is evidently that (1 — RY) < (1 — 12°) or that dpe eS ye Now in the case noted above R,?< 7,,.°, the figures being '2534 < °3005. Returning to the main line of the discussion, the results as to the correlation of brain-weight with skull characters are not altogether satisfactory, as they are apparently not entirely in accord with themselves. One of the points here on which information is most desired is whether brain-weight is more closely correlated with length or breadth of skull. According to our results brain-weight is more closely correlated with length of skull in the male and breadth of skull in the female. This seems a somewhat anomalous result as there is no apparent reason why one should not expect the sexes to show the same relation in the matter. However, Macdonell’s (loc. cit.) figures for the correlation of skull capacity with length and breadth show a similar relation but in the opposite direction. He finds for the Whitechapel English skulls that the correlation between capacity and length is lower than the correlation between capacity and breadth in the males, and higher in the females. Other skull series do not, however, show this reversed relationship between the sexes. With the data at present available one cannot reach any general conclusion as to the relative closeness of correlation between brain-weight and skull length and breadth. It is very much to be hoped that future collectors of brain-weight statistics will include in their returns such head measurements as length, breadth, height and circumference taken before the cranial cavity is opened, and by standard anthropometrical methods. A comparison of the values of the coefficients of correlation between brain- weight and skull length and breadth and between skull capacity and these characters is of interest. I have adapted the following table from Macdonell’s* Table XIII. Brain-weight and Skull Length Brain-weight and Skull Breadth Race 3 # 3 2 Bohemians ... *56482 + 0273 3604+ °0465 | °4655 + ‘0806 5041 + :0899 Skull Capacity and Length Skull Capacity and Breadth English* ... ‘597+ °051 “691+ °040 631 + 048 | 646 + 044 Naqadat ... 501 + 054 599+ 039 | -484+-058 532 + 044 German] ... 515 + :050 “687+ °037 | 672 + 037 "706 + ‘034 Aino eas *893 + 016 663 + °053 561 + 053 502 + ‘070 * Macdonell, loc. cit. p. 235. + Fawcett, loc. cit. + Lee, loc. cit. RayMonDd PEARL 73 The brain-weight correlations with two exceptions are smaller than the corresponding skull capacity correlations. A portion at least of this difference is probably to be accounted for on the ground that brain-weight fluctuates as a result of causes which do not at all atfect skull capacity. Possibly the whole of the difference is to be accounted for in this way, but until there is available a more reliable collection of material on the brain-weight side the question cannot be definitely settled. Until such material is available we shall be justified, I think, in assuming that the values of the coefficients for the correlation between brain- weight and skull characters given by the male series probably represent more nearly the normal relation than do those of the female series. Making all due allowance for the shortcomings of the material the following points are clearly brought out: (a) Brain-weight is positively correlated with both skull length and skull breadth. (b) The correlations here are higher than any of the other correlations of brain-weight with bodily characters. (c) The regression of brain-weight on these skull characters is approximately linear. (d) Indirect selection of skull characters is capable of modifying brain-weight to a considerable extent. The use of equations (31) to (36) in selection problems has been illustrated above (p. 49). One other example may be given because it bears very directly on a question of perennial interest in certain quarters. The problem to which I have reference may be stated for our present purpose in this form: Does dolichocephaly imply a greater brain-weight than brachycephaly or vice versd* ? Mere inspection of equations (35) and (36) shows that neither dolichocephaly nor brachycephaly will have a great advantage over the other in respect to brain-weight, but the matter can be made somewhat more anschaulich by taking a particular example. Con- sequently I have chosen the followmg figures. In Macdonell’s Tables I and II (loc. cit.) are given the values for cephalic index, skull-length and skull-breadth for a dolichocephalic race (the Whitechapel English series) on the one hand and a brachycephalic race (the Altbayerisch) on the other. Those values are as follows: rs) ? English 3B Cephalic index= 74°34 English re Cephalic index= 74°73 Altbayerisch ... 5 = 0820 Altbayerisch ... » eB Il(0) English aor Length = 189-06 English ve Length =180°36 3 Bee Breadth. = 140°67 x ae Breadth = 13468 Altbayerisch ... Length =180°58 Altbayerisch ... Length =173°45 aan Breadth =150°47 A ta Breadth = 143-98 * Evidently the real essence of the question, so far as it has what might be called ‘“‘ humanistic ”’ interest, rests on the assumption of a reasonably high positive correlation between brain-weight and intellectual ability. ‘‘ Brains” not brain-weights are the things! Up to this point I have carefully avoided any discussion of the validity of such an assumption as that mentioned, deeming it entirely irrelevant to the main purpose of the paper. The point is discussed in the concluding section. Biometrika tv 10 74 Variation and Correlation in Brain-Weight Now using equations (35) and (36) for males and females respectively we have for the probable mean brain-weights of arrays having the types of length and breadth of the English and Altbayerisch series respectively, the following values: 3 “ English B.W. = 1501-580 English B.W. = 1289-901 Altbayerisch B.W.=1494:135 Altbayerisch B.W.=1329°676 In the males the dolichocephalic and brachycephalic arrays have practically equal brain-weights, while in the females the brachycephalic array gives a brain- weight about 40 grs. higher than the dolichocephalic. But, as has been pointed out above, the male equations probably come much nearer representing the normal relations than do the female. On the whole, then, there seems to be little or no evidence that either dolichocephaly or brachycephaly is associated with large brain-weight. It should perhaps be noted that no significance is to be attached to the absolute values of the brain-weights calculated from the English and Altbayerisch skull types, as indicating the probable brain-weight of those races. The differences only are of significance. The measurements of these skulls were taken only for the sake of convenience ; hypothetical dolichocephalic and brachycephalic skull means would have served the purpose as well. The brain-weights obtained are of course the probable mean brain-weight of arrays of Bohemians having the specified skull characters. Intra-racial regression equations can of course only be used for inter- racial prediction when the regression of all characters on one another is the same in both races, which is a condition never likely to be exactly fulfilled in practice. Of the races and characters discussed in this paper the Hessians and Swedes in brain-weight, stature, and age come nearest to fulfilling this condition. 12. The Weight of the Cerebrum. Up to this point the discussion has had to do entirely with the variation and correlation of the whole brain or encephalon. In the manuscript material placed at my disposal by Prof. Donaldson (vide supra, p. 15) data were given from which it was possible to determine the variation in the weight of the cerebral hemispheres, and the correlation of this character with stature and age for English males. The results obtained from this material may now be examined. It should be stated that the records of weight were in ounces, and of stature in inches in the original material. The units of grouping and the distribution of the frequencies may be seen in Tables 45 and 46 of the Appendix. After the constants had been calculated in ounces and inches they were transferred to the metric system in order to facilitate comparison with the other parts of the paper. The values of the constants are exhibited in Table XVIII. RayMond PEARL 75 TABLE XVIII. Variation in the Weight of the Cerebrum. English Males. Boyd-Marshall Data. Mean S. D. C. of V. Character No. | Table Weight of cerebrum | 308 | 1184°940+4°859 gr. | 45 126-438 + 3°436 er. 10°670 + ‘293 Stature Mae ... | 308 | 171°602+ °268cm.| 45 6976+ ‘190 cm. 4:065+°111 Age... Ags ... | 308 49°481+ ‘645 yr. | 46 16-7704 °456 yr. — For the correlation of weight of cerebrum with age I find from Table 46 r=— 1412 + 0377, and for the correlation of the weight of cerebrum with stature from Table 45 Te L202 0809: These values bring out the following points. The cerebral hemispheres are markedly more variable in weight as judged by the coefficient of variation than is the entire encephalon. This greater variation may denote a really greater variability of this part of the brain, or it may be due to the variable element which enters as a result of the separating of these organs from the rest of the brain. It seems to me likely that it is in part at least real, for the following reasons: first, the cerebral hemispheres have attained relatively enormous develop- ment late in their phylogenetic history ; and again from the functional standpoint the cerebrum is the most variable part of the brain. The correlations of the weight of the cerebrum with age and stature are both somewhat lower than the corresponding values for the other series of males falling within the ages 20 to 80. The differences can hardly be considered as significant however in view of the probable errors. For the coefficient of regression of brain-weight on age I find b,=— 1-065, and on stature b;=2178. By comparing these with the regression coefficients for the entire encephalon on the same characters in the series previously discussed it becomes evident at once to what a great extent changes in the total brain-weight are due to changes in the weight of the cerebrum. Unfortunately it is not possible to get relative figures here on account of lack of knowledge of the regression of the total brain-weight on stature and age for the English series. Of course the weight of the cerebrum forms a large part (ca. 87 per cent. for the English series) of the total brain-weight. The problem here is to determine whether the changes in the weight of the cerebruin cause less or more than their proportional part (say 87 per cent.) of the changes in total brain-weight due to stature and age changes, I have tried a number of approximate methods 10—2 76 Variation and Correlation in Brain-Weight of getting at this question with the data available, but none of them led to any result. sufficiently well grounded to warrant taking space for its publication. Putting all the results so obtained together they seemed to indicate the following conclusion, which is of course subject to modification when further material is available: viz., that changes in stature affect all parts of the brain equally (ie., in the proportion of their absolute masses) while changes in age have a relatively greater effect on the cerebral hemispheres than on the remainder of the brain (cerebellum, pons, and medulla). The final conclusions regarding variation and correlation in the weight of the cerebrum may be summed up as follows: the cerebrum is somewhat more variable in weight than is the entire encephalon, and this character is slightly less closely correlated with age and stature. The values of the variation and correlation constants are, however, as is to be expected, of the same general order of magni- tude as those for the total brain-weight. 13. Concluding Remarks. In concluding this paper I wish to call attention to what seem to me to be some of the broader aspects of the work. There are at least two general results of the work which alone justify its being done, I think. The first and most important is that the paper contributes reduced material to the existing collection of biometric data on man. In the nature of the case such a collection grows slowly, but every increase in it means a definite, although it may be small, step in advance in our knowledge regarding the fundamental problems of anthropology. I regard as the second most important result the fact that it may now fairly be said that the essential trustworthiness of the most important of the existing collections of brain-weight statistics has been demonstrated, and that consequently reasoning by statistical methods on the problems involved is not of necessity lacking in validity. The agreement in the statistical constants from four series of data so divergent in their origin as those treated in this paper cannot reasonably be held to be fortuitous. It can mean, I think, but one thing, namely, that the same kind of general lawfulness underlies the variation and correlation of brain- weight and the variation and correlation in other characters of the organism With this conclusion presumably no student of brain-weight would disagree, but some of the most eminent students have disagreed with the converse proposition that it is possible to gain a knowledge of the nature of this lawfulness by statistical methods applied to large masses of material when brain-weight is the thing concerned. This sceptical attitude owes its origin, I think, to the nature of the statistical methods which have hitherto been applied to the problem. Brain- weight statisticians have erred in two directions in the handling of their material. On the one hand, following the much-to-be-condemned practice in other fields of anthropology, entirely heterogeneous material has been grouped together to attain large total numbers. The culminating example here is, perhaps, Topinard’s series of the brain-weights of 13,000 Europeans. Secondly, certain brain-weight workers RAYMOND PEARL Yar have attempted to apply to the individual results gained by somewhat doubtful statistical analysis of mass data, and vice versd, what is much worse, have argued that the relations found to hold in a single individual might be considered to hold for the mass. As an example here might be cited Marshall’s* attempt to correct the brain-weights of De Morgan, Thackeray, and others for stature and age. The results detailed in this paper will indicate to the biometrician at least how dangerous both of these methods of procedure are with such material as brain- weight statistics. Of course both disregard fundamental canons of general statistical theory, but the chance of grave error becomes very much greater when the character under consideration shows on the one hand definite and clearly marked types in different races, and on the other hand low correlations+. With the application of adequate statistical methods it is possible to reach definite and significant results regarding the weight of the brain. When we have biometric constants for a considerable number of long series of brain-weight statistics from different races we shall be able to advance our knowledge greatly with reference to the general problems of the evolution of man’s brain. While the discussion of such problems can at the present time admittedly lead to only tentative results, still I think it will be worth while to examine for a moment the bearing of our results on a single evolution problem. Progressive evolution through natural selection may be brought about in any character either by the direct selection of that particular character or by the indirect selection of other characters correlated with it. As we proceed in the analysis of the problem the proportional effect of each of these two sets of factors must be quantitatively determined. A first approximation to such a determination has been here made * Loe. cit., supra, p. 18. + In this connection Weigner’s recent paper ‘“‘Hin Beitrag zur Bedeutung des Gehirngewichts beim Menschen,”’ Anatomische Hefte, Bd. 23, pp. 69—109 is decidedly open to criticism. But when one, after very briefly analyzing the weighings of 69 male and 66 female brains (sic) from individuals ranging in age from 14 to 82 years, by the most superficial of statistical methods, feels justified in drawing the following sweeping conclusions inter alia, what can biometry offer that will be in any way effective? Weigner’s three principal conclusions as to fact are (loc. cit. p. 108): 1. ‘*Das Gehirngewicht steht in keinem direkten Zusammenhang mit dem Alter, und wenn auch die angegebenen Zahlen sich zu Gunsten dieses Zusammenhanges verwerten einen, verlieren dieselben an der Bedeutung, da durch dieselben nicht der Beweis gefiithrt werden kann, wie sich Gehirngewicht bei einem und demselben Individuum in verschiedenen Altersperioden verhalt ; 2. zwischen dem Gehirngewichte und der Koérperlinge lisst sich keine bestimmte Proportion feststellen ; 3. die Durchschnittszahl hat bloss einen relativen Wert, und zwar deshalb, weil die das Gehirngewicht reprasentierenden Zahlen durch die Wigung eines zufiillig uns gerade zur Verfiigung stehenden Materials gewonnen sind und das aus demselben bestimmte arithmetische Mittel keine konstante Grdsse, sondern eine frei bewegliche Reihe von Zahlen uns darstellt.” It should be said that the material on which these conclusions are based is apparently within the limits of its probable errors entirely normal, and shows an approximately linear regression of brain-weight on age during adult life. This being the case the very positive conclusions seem to take on the character of extreme dogmatism. If 1. were true, by the same token no life insurance company could fix upon a premium rate for the insurance of lives of individuals, aged 21 say, which would enable it to carry on its business justly and fairly, because no life insurance company knows when any one individual aged 21 will die. Yet life insurance companies seem to be struggling on! 78 Variation and Correlation in Brain-Weight for brain-weight and “size of body.” It has been shown (p. 66) that of a given change in mean brain-weight between 25 and 30 per cent. can be brought about by a selection of “size of body,” measuring this both by stature and weight of body. As it is fair to assume that natural selection has acted on “size of body” we thus reach some idea of how much of an effect such selection has had on the weight of the brain. A vital point at issue is as to what are the underlying causes of brain-weight differences. We have seen that there are definite racial types in brain-weight apart from differences in the other bodily characters which could be studied in this connection. To take a concrete example, we find that groups of male and female Bohemians having the same mean length and breadth of skull have probable mean brain-weights differing by about 69 grams, the female mean being lower than the male, of course. To what is this difference to be ascribed? Evidently there are a considerable number of possible factors which may enter into the matter. In the first place these males and females would still differ in stature by a certain amount so that a part of the remaining brain-weight difference would be accounted for in this way, but probably less than 25 per cent. Selecting too the same skull size would undoubtedly reduce the gross sexual difference in stature to some extent. The mean age of the arrays is essentially the same, so that nothing is gained from this source. To account for the still remaining difference there are at least the following possible factors: 1. The specific gravity of the female brain may be lower. 2. A unit increase in skull length and breadth may be associated with a greater change in skull height in the male than in the female. 3. The walls of the female skull may be thicker than in the male. 4. The female brain may fill the skull cavity less closely than does the male. (The opposite has been stated to be the case. Cf. Donaldson, loc. cit. p. 118.) 5. The ventricles of the brain may be proportionately larger in the female than in the male brain. With the material at present available it is impossible to determine which of these possible factors really play the important réle. It is my purpose merely to call attention to the kind of problem that confronts one at this point in the analysis of brain-weight constants. It at least will serve to show some of the directions in which more definite information is desired. Donaldson* has well said: “The search for correlation between the size and form of the brain and the degree of the intelligence has interested all who have worked on this organ, and although it might be designated as the psychologist’s standpoint, it has, from the very first, been in some measure before the minds of all.” This point has so far been avoided in this paper for the reason that it seemed to me desirable to settle the question of the purely physical variations and * Loe. cit. p. 85. Raymonp PEARL 79 correlations before making any attempt on the much more difficult matter of brain-weight and intelligence correlation. I have, however, one point to add to the discussion of this already much debated subject. It follows rather simply from the theory of multiple correlation and regression that if the correlation coefficients 7. and 7; measuring the correlation between characters «, and a, and a, and z;, are known, then it is possible to determine the limits between which the value of the coefficient of correlation 7,, between 2, and a, must lie. Without going into the details of the theory involved, which is given in convenient form by Yule*, it may be said that since (123 = Nels)" Is not > 1 + 2%? — 1? — Ns", 73 Must lie between the limits MyNig £ (/1 + e113" — Ny? — 113"). Now, very fortunately, Pearson and his co-workers+ at University College have CC determined for Cambridge undergraduates the correlation between “intelligence” as denoted by place in degree examinations and certain physical characters whose correlation with brain-weight we also know. Evidently then we have the material for getting at an approximation of the limits of the correlation between brain- weight and intelligence except that the two correlations from which the limits of the third are to be determined are not from the same material. This of course introduces an element of error, but as the coefficients of correlation of brain- weight with other characters seem to be fairly constant from race to race, a very serious error will probably not be made in assuming that the values of these correlations would not differ greatly in the English from what they are for continental races or from one another in two different groups of English. Leaving for the moment this question of possible error, let us examine the figures, first taking this problem: Assuming that Cambridge undergraduates have the weight of the cerebrum correlated with stature to roughly the same degree as is shown in Boyd’s English data, what will be the limits of the correlation between weight of cerebrum and intelligence? The necessary data are as follows: Yj. = — ‘0056 correlation coefficient between stature and intelligence. m= 12 ° is P . , weight of cerebrum. The limits of 7,;, or of the coefficient of correlation between brain-weight and intelligence, are roughly +98 and —-99. Now mr, is really insignificant, the conclusion the authors draw being that (loc. cit. p. 107), “stature is not correlated with place in degree examinations.” Putting r,=0 the limits of 7, are + V1 — 73% Or the practical conclusion we reach is that, so far as the known correlations of weight of cerebrum and intelligence respectively with stature give evidence, the correlation between weight of cerebrum and intelligence may have any value * Loe, cit. + Lee, A. Lewenz, M.A., and Pearson, K.: ‘‘On the Correlation of the Mental and Physical Characters in Man, Part 1.” Roy. Soc. Proc. Vol, uxx1. pp. 106—114, 1902. 80 Variation and Correlation in Brain-Weight between 0 and +1. There is no evidence from this source as to whether the correlation is positive or negative. Suppose we take still another case, stating the problem this time in this way: Assuming that the Cambridge undergraduates have brain-weight correlated with skull length to roughly the same degree as is shown by Bohemian males, what will be the limits of the correlation between brain-weight and intelligence? The necessary coefficients are: 71, = ‘0861 correlation coefficient between skull length and intelligence. Lin 5482 ” ” ” ” » and brain-weight. The limits of 7, or the correlation between brain-weight and intelligence, are + °7 and —°6, or practically we reach much the same result as before. Now the Bohemian coefficient, as we have seen, probably is in error in the direction of being too small, or the probability is that the limits of 7, are still wider than the figures indicate. As to the reliability of these limits this may be said: in order to fix correctly the sign of the correlation between brain-weight and intelligence, both the coefficients between intelligence and physical characters and brain-weight and the same physical characters would have to be greater than /°5 or ‘707. Now I think it is reasonably evident from what we already know, that for civilized man at least there is no likelihood whatever of ever finding values as high as ‘707 for the coefficients concerned. Or in other words the degree of the correlation between brain-weight and intelligence is indeterminate, with the probability that it is sensibly equal to zero. That is, brain-weight and intelligence in the sense of mental capacity are probably not sensibly correlated. All the inferential evidence when subjected to careful scrutiny leads to the same conclusion, I think. In closing I may perhaps be permitted to point out what seem to me to be the most promising lines for future investigation of the problems regarding the weight of the brain. In the first instance it seems to me highly desirable to transfer the problem for a time at least from man, where in the nature of the case all the fundamental records must be vitiated from the fact that we cannot weigh the brains of any considerable number of normal men instantly killed. As a conse- quence of this impairment of the data resulting from complex ante-mortem con- ditions the absolute values of certain of the biometric constants concerned cannot in the case of man be proven to be sensibly identical with what we should consider necessarily normal values. ‘lo be sure, such a sensible identity cannot, on the other hand, be disproven, but a state in which one can neither prove nor disprove is scientifically not altogether satisfactory. By working with animals it is possible to weigh any number of brains of instantly killed normal individuals. From records so obtained the values of the important biometric constants like the coefficients of correlation can be satisfactorily determined. In the hands of Prof. Donaldson this line of work is yielding results of the greatest significance and value. For continuing the work on the human side the primary need is for more and larger collections of brain-weights in which close attention is paid to the racial homogeneity of the material and to the recording of other subsidiary facts RayMond PEARL 81 as well as brain-weight. In comparison with the present series of brain-weighings a Series which should record the following facts would be almost ideal. 1. Brain-weight, by the standard method. 2. Race. In this the more essential detail given the better. 3. Country of residence during adult life. 4. Occupation (as an index of social status). 5. Immediate cause of death. 6. Chronic diseases of adult life. 7. Stature. 8. Body-weight. 9. Head length. \ Measurements all made be- 10. Head breadth. | fore skull is opened, and 11. Head height. with hair removed at points 12. Maximum horizontal circumference of head.) — of contact. 13. Age. 14. Sex. Such a list at first sight appears formidable from the practical standpoint, but evidently the things which would give the greatest difficulty are 2 to 6 inclusive. The head measurements could be made with little trouble, and a relatively small expenditure of time. This list is presented with the hope that any future workers who may be about to undertake the great labour involved in obtaining a large mass of human brain-weight statistics will at least consider the points raised. The few additional facts would take but little more time in the collection, and they would greatly enhance the value of the completed series. I would especially call attention to the need for larger series of brain-weight statistics than those we now have. In order to do really “close” statistical work on the subject it is desirable that we have larger arrays of individuals of given age and stature types in order that the regressions may be smoothed and the brain-weights in more absolutely homogeneous material may be investigated. Finally it should be said that the present paper is, from the biometrical stand- point, only a “ first” study of the problems of brain-weight correlation. Much yet remains to be done, and as will have been apparent to the reader could be done with the present material. One thing especially which might be done is to separate the material into a third “old” group, comprising the individuals falling in age between 50 and 80, and treat this group separately. I very much doubt, however, in view of the apparent substantial linearity of the regression of brain- weight on age through the whole of adult life, and considering the statistically small number of individuals which in each case would have been available, whether the value of the results so obtained would have been in any way commensurate with the labour involved. Even a biometrician must stop somewhere. When we have much larger collections of brain-weight statistics to work with an old age division of the material will be at least experimentally justified. Biometrika 1v 11 82 Variation and Correlation in Brain-Weight 14. Summary. Some of the more important general conclusions of a study of five series of brain-weighings, representing Swedish, Hessian, Bavarian, Bohemian, and English sub-races of man and including altogether the weights of 2100 adult male and 1034 adult female brains, may be summarised as follows: 1. There are definite racial types in brain-weight. The differences between racial groups in this character are only in part to be accounted for by differences in other characters of the body. 2. The series studied exhibit a very fair degree of homogeneity. 3. In respect to variability the characters, brain-weight and skull capacity are sensibly equal. The coefficients of variation for brain-weight are intermediate in value between those which have been determined for skeletal characters in man on the one hand, and those for the weights of the viscera and for various physio- logical characters on the other hand. 4. Having regard to the size of the present series we conclude that variation in weight of the brain may for practical purposes be considered to follow the “normal” law of the distribution of errors. If the mean and the mode do not exactly coincide the mean will be shghtly greater than the mode in brain-weight frequency distributions. 5. The correlation of brain-weight with age, stature and body-weight, is in all cases low. 6. The correlation of brain-weight with skull length and skull breadth is, in comparison with the other characters studied, fairly high, and is for both skull length and breadth positive. 7. The sexual differences in mean brain-weight are practically constant in all the races studied, whether considered absolutely or relatively. 8. Only a part of the sexual difference is to be accounted for by differences between the sexes in other bodily characters. 9. The sexes are equally variable in respect to brain-weight. 10. The weight of the brain tends to be more highly correlated with other characters in the female than in the male. 11. The correlation of brain-weight with age is negative. The regression of brain-weight on age is, so far as can be determined from our present series, linear throughout the period of life comprised between the ages 20 to 80. In other words, there is a steady decline in the weight of the brain with advancing age, beginning at about the twentieth year and continuing throughout adult life. RAYMOND PEARL 83 12. The correlation of brain-weight with stature and with body-weight is positive and in each case the regression is linear. 13. The weight of the cerebrum follows the same laws of variation and cor- relation as does the weight of the entire brain, but exhibits a somewhat greater relative variability. 14. The results as to variation and correlation in brain-weight are closely accordant for all the races studied. 15. The indirect selection of bodily characters can modify the mean weight of the brain to a considerable extent. The amount of this effect has been measured in certain cases. 16. There is no evidence that brain-weight is sensibly correlated with intel- lectual ability. The limits of this correlation have been shown to be not closer than 0 and +°6. 17. So far as can be determined from present material dolichocephaly has associated with it no advantage in mean brain-weight over brachycephaly and vice verst.. 18. There is great need for further large and homogeneous collections of brain-weight statistics. When these are available for a considerable number of races it will be possible to pass from intra-racial to inter-racial problems. APPENDIX OF MEASUREMENTS. Correlation Tables. The fundamental correlation tables on which the results of this paper are based have been placed together here at the end for convenience in reference. The tables are numbered consecutively in Arabic numerals, and each bears its own explanation. A word may be said as to the plan of arrangement. The racial order is Swedes Tables 1 to 12 inclusive. Hessians P 13, 24 , Bohemians 5 25 ,, 34 Fe Bavarians Fr 35, 44 se English a 45 and 46 4 Under each race the order of arrangement is in each case for the tables involving brain-weight as one of the variables to precede the subsidiary tables involving only the related characters. The tables of males precede those for females, and the “total” series precede the “young.” 11—2 84 Variation and Correlation in Brain-Weight TABLE 1. Swedes. Brain-weight and Stature. “Total” Series. Males. Stature (centimetres). 156—158 M1173 1838—-185 186—188 | 1100—1149 1150—1199 1200—1249 1250—1299 1800—1349 1550—1399 L4O0—1449 L450—1499 | 1500—1549 | 1550—1599 | 1600—1649 | 1650—1699 | 1700—1749 —_ OW oro w - bw j EwWwWons ©Or-10 OF lo Brain-weight (grams). PP CrOWWOOe Nr eee OP Oe Ord) We Totals TABLE 2. Swedes. Brain-weight and Stature. “Total” Series. Females. Stature (centimetres). = lrilolea |e] a] e |]. io |] x i | | slat elise el silo }e] ats | ~ 4S ~ tl N ~ i) | Let eR el lee ea ale ey elena Totals » Weg od ee (fetes = ~ S Sa) ve} fon) sls | ni ele] se] oe foe 16 1 oe nN I NOUN 4 n x _ 4 4S “a | 900— 949 ] 8} 950— 999 — S | 1000—1049 i 7 1 | 1050 1099m| sn elena 1} 1] 1} 1} 1;/—{—]— 9 aa | 1200—1159') 4 = 4) OE We | 8 oe) ies) a4 sa ie Sonal ee eos = | 1150—1199 i 2 3 8 Cit Oat tO 2 eer 30 Say) Segue ses) |p | | 3 6 9) 1 Si ak, 4/2 );—|— 53 = | 1250—1299 33 ipa Be 9) Sa 7) 28 ae 40 4 | 1300-1349 | 4 8 6 6 5} 1 | —|-—Y 30 "3 | 1850--1399 1 Sa 5} 38) 3/ 5] 2; 2) 1 |—f] 23 ca | 1400-1449 : =) 2) 4) 8) oe) tS eeeo 1450—1499 1 | — 2 2;/—|—j|—]1 6 1500—1549 | 1 =e 1 1550—1599 rie ee 71 Totals RAYMOND PEARL 85 TABLE 3. Swedes. Brain-weight and Stature. “ Young” Series. Males. 20—50 years. Stature (centimetres). Brain-weight (grams). 1500—1549 1550—1599 1600—1649 1650—1699 1700—1749 Ne D> ISS] Xs} ee) baal a ~~ S Sa) Ne) D> ISN] wD iv.) im = ty KS) WD iS) Ne) Ne} Rey aS N mS ~N se) ice) i) Dd ™~ ial ~ a) ~ il ~ ~ al al ial ™~ ™~ ~ ~ ™ Sa ee a hea uliee eed ee eda eeell gall cek cleave ah ie mse || Sse 1) ks) 1D wD wD So) Rs} es) ~ |e i Se) 6 Se) Se) i ~ ~ ~ ~ ~ al ~ ~ ™ ial ™ ~ a ~ aa 1100—1149 > 1150—1199 il 2) — 1200—1249 2 1 4 1 1250—1299 1 2 5 — 1300—13849 4 4 5 6 1850—1399 4 / 11 2 6 1400—1449 1 3.16 5 1450—1499 2 5 Wall 5 — 1 1 7 3 | — 3 2 2 — il — Totals TABLE 4. Swedes. Brain-weight and Stature. ‘“ Young” Series. Females. 20—50 years. Stature (centimetres). S | 99 Oo | Q SY eS) | ~ | ™m eon an ot seen (UE: Sa) R SSP aot S| ast WD Mey || ASS xe) OS | © ~ a NS ~ ™~ ~ Fe ol ‘eal ||) oa 2 aaa: ~ ™ ™ Pei | TA ee Ae al ete) ol}. [ ihocals SistisiSl/Sl eles Sl ss; sl sl RI s at ™ he | ™ | om = ~ ™ | “ne 900— 949 | S| 950— 999 | S 1000—1049 ee | in| ©2| 1050—1099 | — a= 1 a 2 ® | 1100—1149 | 1 Veit Ona. eo) Tel oo 12 | 1150-1199] —| —| 1 |—|2]3}]5]—]|1] 2 16 sv 11200-12494 —| —| 9 )2)/1)/2)5)/ 3°16) 1 24 = | 1250—1299 ee Oe eeprom or abe He a3) |, cb 26 4 | 1300—1349 O Wes, ike oa | go 16 SOO == S59 Wi Nee Wee Sy 8) | 2S.) 0 14 ea te ee a ie 8 8 EO 1 OO ee ine Nee ee | ee eo. | ae 6 f500—1549) | — | —= | — 2429 Se a ats | SS ES ee ee 1 Totals | 67LI—OOLT Totals 6691 — 09ST Males. 669I—OG9T Females. 6%SI1— 0061 649I—0091 | 6691—O0SS9T MAMA HMANA 667I—0S4T cue Rescu AN A lelea| 2a 674 I— 007 T afore] | | | | Brain- Weight ae 67SI—00&1 AGA Hid Hess 661 —0G&T [OO GORI Pe— hor) ot | | | “ Total” Series. 664I—OG4 I rHODMOAADOMMNN 67EI1—008T pebien! iareochen Mousse) sitexk | | “ Total” Serves. 697 I—009T NAOODOOMrI- OM a TABLE 6661096 DOWD ADMOOANAA ae Lal 6%EI1—008L TABLE 6, OA WMHOOHMAINM ioe) Brain-weight (grams). vation and Correlation Brain-weight and Age. 6661 —0G6T Brain-weight (grams). OOtmArANAN 6661—0G61 6401 —006T eo) fe aero) ral Fey Sees NAMWAIDMm WOO OHM HA 64@I—006T Arona MAN 6611 —OSTT Brain-weight and Age. Var Swedes. 64 II—O01IT Swedes. 86 CGT OGTT ease }| sesame ga ea 6tIT—oor1 [eae > GN Sacre NUS soon |} |a]l laanw | ade COL =O00F ba Wl) fet ei elie ceo—ove PIL IIT I ILE Cf6=006> Palle | ey el | te 87 RAYMOND PEARL ie TABLE Males. ” Series. “ Young Brain-weight and Age. Swedes. Brain-weight (grams). 64L1—O0LT 669T-—OG9T 649I—009T 6691—0SST 61SI1— 001 6641—0S4T 6441I—0047T 6661—O0¢ET 64%EI—O008L 6661—OG6T 6461 —002T 6611 —OgSTT 64II—O01T ‘(savak) o8V TABLE 8. Females. Series. ” “ Young Brain-weight and Age. Swedes. Brain-weight (grams). | | 674I—O0TT | | | | | 666I—0G61 679I1—009T | 6741—00S1 | 66YI—OS4T 66I11—OSTT 6691—O0S&41 6661—OGE&T 1d seo ot 64EI—00FL 64EI—008T 67II—O0IT Ke) | celta Ge) NON 2D P20 —— | Hoo 6601 —O90I — a Ne) pat) Ne} nN wt N we) a OANA N re nN 4 640I—000T 666 —O006 88 Variation and Correlation in Brain- Weight TABLE 9. Swedes. Stature and Age. “Total” Series. Males. Stature (centimetres). Age (years). ejajejelslsisisle|sielelaisiels Aa ~ ~ ~ ™~ ~ TS SE ~ | ™~ ™~ ~ ~ ~ mm Total Wig este Oe fin aig nes Ps Pi a TE | | | | es tS) S15} 6 1] oe] el] StS] Ole y_yR |] | 6 1 6 | ww 1 & N 4s 4S nS S 5 4 | 4 a 4 [4 ms ny 20—24 | — = f—|. 3) a) oe) 4) 7). 81 20) Ss eon | 25—29 | — — Ta) 7 4 6 6; 10; 1 1 | = 37 30—34 1 1 2 5 9 | 12 6 6 een ea | 48 35—89 | 1 | —| — | T=) Sh 42") 99:1) 6 8. | 6a on) eee LO i 1) 6] 8) 49) 910 45 | 5) 3) Saal 45—49 | 1 —|{—|{— 1 4 8| 5] 6 5 | 5 4) — 1 — | — 40 50—64 | — | 2 | = |= | 1) 8] 9) 8h) 10) 6) 93) 43235207) eG 55—59 | — as ee 2 FANS G3 ll, SC 6 9; 5 2| 2 1 }—}] — 39 60—64 | 1 3] 3.447 178.) Sat) a Leese 65—69 | 1; 2 1 1 5 3 | 2 3 18 | Y0—74 PP} —) 2] 2) Ve bl] |= Selec | 7—79f—|—|—]}1]}—; 1] 1] 2] 1] 1] 1) 1}]—j;—}|—|}—] 9» Totals g g : 10 | 29 | 47 TABLE 10. Swedes. Stature and Age. “Total” Series. Females. Stature (centimetres). xslxrlelalelalel»seioin SS tense Wes: | Re ny SH | >|] ~s => = wD ite WwW | Ne) Ne} ~~ 2~ iS ~ S| ey ~ ~ ia ~ ~ LH | ~ ~ ™~ ~ ™~ Pe Ras a ea a ee ee Pe ese lech | | fittoias RN > | sel Seal tet lee > 6S | D R wD | % a > Sa) 9 | at ie => WwW Mw) WwW WwW Ke) Ks) <<) ~~ i al ™~ | ~ ™ i inl ~ ~ ~ Lal ™~ ~ ™~ mm 20—24 —/|—|1 4 9) 1 2, | 25—29 —|— 3 1 2) 4 2 | 30—34 | 23ers 2 rel S | 35—39 2| 2) 41 6 241 > | 40—44 ees 4 Sale Se i= 19 tl ied | eset bp | 50—54 | OD!) i aa 3 | 3-| -2 ) RX 1 | 99 ™ So ss : $9 So) => a) Ris lola} SiS sis louol_eloelols |] ale [xr] x]TX | o | | ™~ ™ = hae Fath adie | =| An | La | ™ ™ | tem Le | | im ™ ™~ ™ Oe ee are ee ee fee te ele les) yl [Moves S1ISIS | SiS i Slslslele lee) Sse eae SIS{SISISlSfSlSiS ISIS iS ls sysysye {sls | g(—|—) sl slabs) a) 22 SS eS 20—24 = |e i aw) 3 4;—} 1}]—) 1 }|—]|— 22 25—29 | L)/—| a) —|—] Po) 8) el 6) a) el ee ec 30—S4 1);—j)} 1 2 2; 5 2 2 1} —j}2)—)—}] — 18 35—3BI =| 2 2 1 4 1 AM a oe 4}/—]—]} 1 |]—|—]|— 32 40—44 1 ZO BY ST ay a BP a jp Se Se | 31 ASAIO | 1 2 1 4 4 7 4 2 2)/—]—]—-|—|— 27 50—54 Lo al eid Vega aales eee |) a7 55—59 2 |—}— 6 6 6 1 2); 2 )—}—)|]—]— 25 60—64 of De We) B84) aBoletge ll = 2 ist ae eS oe 65—69 —|;—|1 1 | — 3 3 5 2) — 1 | - 16 70—74 2) — 3 4+ 4 2 3 2 —|— 20 V58—79 | 1 — i) Be eS 4 Totals 95 RayMond PEARL 23. TABLE Males. Series. ”? Stature and Age. “ Young Hessians. 15—50 years. Stature (centimetres). | 68I—O8T GS8I—E€8T || lalaa|+ Sete ea i=) co AN OO NOD SH o> = ISLE Wer nouneax 2 oL1—89r 12 DW AH I~ O So keyi—Choye || SOP no seo ls ‘t91—e9r tonomat | Sas st © < Ajram |S sor—oer | |e paoe |S eee ‘261-067 - i | mee | Jes a er ie ott—tt1 oad I! as | [a eto—rtr | > | | i OfI—ssr i | | | ae 24. TABLE Females. Series. ” “ Young Stature and Age. Hessians. 15—50 years. Stature (centimetres). Bo ; | $ | CAAA Pe = fal E8I—08T = CLT aL outa i (iil ee OLI—89T + L9T—G9T TON fe 9I—@9L A H10 = ae a = T9I-—66I ae aes 2 ee oO I~ a 5 I~ i~l~ eA GGT —eG, eS Ave sto) SH x | ger—oor | @ [AAA [eg | 64I—24T | pote- = oft 41 | | [ana = Sheik Jana | ~ OvI—86E1 2 LG1--GEL = Folie Gop | 1&I—66éT | 86I—96T — & SSSssss la ‘(steat) aSy Stature (centimetres), Age (years). 96 Bohemians. Variation and Correlation in Braim-Weight TABLE 25. Brain-weight and Stature. “ Young” Series. Brain-weight (grams). Males. 20—59 years. “ Young ” Series. Bohemians. TABLE 26. Brain-weight and Stature. 20—59 years. Females. Brain-weight (grams). o> D> | =r) D >) | > Sd imp) > jen) a) oS Sd Sd Q isp) S o> | jer) ion) Sin [TR] _ Ri sluelolas Styles ~ | ~ ~ = | ™~ = ian ™~ ™ | | | all | | | | | Totals ela Totals Sisisssgsr7rsisissis al s/s | Ti] St | ~ ™ | el ™~ | ~ ® | ™~ ~ 4 il jie - a OV 180134, —}— 1 | 250-—154) — |)“ | 3) = 5 = | 185—189 1 1 155—-159 ey Ze a aN ee) Se | 140—1hh 2) — 2 160—164 On el Be22 9 2; 1)— 54 S| 145—149 fi 5 2 16 | 165—169| — 3 | 15 | 24 | 17 i. 66 —~| 150—154 1 6 4 22 170—174 Seely ele | on 69 2 | 155—159 9/14] 2 42 175—179} — 3 ui 9} 10 5 || 36 = 160—164 6 | 14 7 32 180—184f — | 1}/—] 7] 4] 2]—] 1 15 S | 165—169 | by) a 1 | 185—189} — | — 2/— 2);—/;/—)— 4 RN | 170—174 2 2 1 5 190—194 | 1}; —)|— 1 | 175—179 1 1 | | Totals 19 | 63 | 88 | 68 | 90 | 5 2 | 266 Totals 133 eee ee cals se aes TABLE 27. TABLE 28. Bohemians. Brain-weight and Age. Bohemians. Brain-weight and Age. “Total” Series. Males. “Total” Series. Females. Brain-weight (grams). Brain-weight (grams). D> D Dd D> | D> D> Dd 2 fon) | ale lo > D> Dd S3 > Sd > D> op) is) S o> S > ior) Dd St} msm ]_RQT_HAl_ sels | oe Sse ecoriesee se | BS ™ ™ | ™~ ~ ™~ ™~ ™~ ™ ™~ Trae ~ | ™ ee ee ee eee 0) de ee S =) S =) S S S S SSeS) S S S S 1S S Ss =) S S =) S S S S S S S | Sy) ny Se WwW Re) aN : | Sy), iSy3 Se) PSs WwW in| et ™ ™ i! ™~ ™~ ™ Fe rsh) ta = ™ ™m Lae 20—-29 | ¢ @ |:2O— 29 1 3|' 21) 28") 26n Seme2 30—89 | Z| 30—39| — | 2 ap ell Bh 4O—49 2: @.| JO 99 |) 44 8) 10 eae 50—59 Dd fon) Dd on) > 2) S > S Sli BIS S) Sd S ™ | & 3D tS | 6 S BR cS La I ~ ~ ~ ~ ~ ~ = | ie Te] We i yal potas S S S > = S o S S Ss S S S S S S S S | Ss a SY 99 S| Rs) ~ nn ~ ~ ae al a ~ ~m ~ 180—184] 195—199 Sy = i 1 | 2 |] TT GE ee ok | ak e— ON SOn Ts) Ss) a) | =) ao | ee as hh) ee 1/68 —|—| 4] 19/29] 23} 4|—|—J] 79 aa LO 9) SS Sel | —4) el) | —|—|-—]| 1 Oe e405 | h 19 | el eel OR ao ee le 1 10 Se Sa ele aie 4 TABLE 31. Brain-weight and Skull Breadth. Males. 20—59 years. Brain-weight (grams). | ) | G os 1500-—1599 1100—1199 1200—1299 1800—1399 1400—1499 1600-—1699 1700—1799 | Skull breadth (millimetres). 110—114 tp—119 120—124 eo 129, 1380—134 30159 L40—144 15—149 150—154 155—159 160—164 165—169 70-175 175—179 Totals 21 Biometrika 1v ® | my hos fe dp hee’ i) ™ LON 4 | | | | | | | |Totals © Se sue ais Abo eles | S S/A/Rlaxls{s =a ny al N il nN S505 alate) | en 2 | 155—159| — | 2} 1}—}]—]— 3 1 160—164) 1 | 1) 18 | 4) 3) — | 27 bo| 165-169] —| 4]16| 9] 3] 1] 33 Ol 21s 33 lial ol 7 eo 1-7 9 |e 4 7 17 = 180—184 | ee ae 6 “4 i a “| Totals | 2 | 10 | 59 | 54 | 28| 6 [ 159 | | TABLE 382. Bohemians. Brain-weight and Skull Breadth. Females. 20—59 years. Brain-weight (grams). Salalaslalealsa SiS (SASSI Ais S/H ;Rl]_ua |] Ss] s 4 ~ il ~ ia) il | | | | | | }Totals ae ee eee , =i|2 2 ]/s Ss = (Ss |S Ss les ls Sia /QRQ/ eH |S] ~ il il ~ ~ ~ Skull breadth (millimetres). 130—13J oo 9 140—1hs 15—149 150—15) 155d—159 160—164 Total 13 Variation Bohemians. and Correlation in Brain-Weight TABLE 33. Skull Length and Skull Breadth. Males. 20--59 years. Skull length (millimetres). Totals a do ee @ | 110—114 1 | | 115—119 | 120—124 "EB 125—129 Z | 1380—184 — —- 3 _q | 195—189 3 | 8 3 | 140-145 14 1 58 S| 145—149 17 5 78 - q | 150—154 25 6 95 © | 155—159 | 5 5 40 = | 160—164 3 — ul & | 165—169 2 | 4 TD | 170-174 | 175—179 1 | Totals | 2 | 14] 42 | 68 | 79] 61] 19] 10| 4 |. 299 TABLE 34. Bohemians. Skull Length and Skull Breadth. Females. 20—59 years. Skull length (millimetres). 130—1384 185—139 140-—144 145—149 150—154 155—159 160—164 Skull breadth (millimetres). Totals Brain-weight (grams). RAYMOND PEARL 99 TABLE 35. Bavarians. Brain-weight and Stature. “ Total,’ Short Series. Males. Stature (centimetres). 5 Totals ‘ th as ‘ 1 1 156—158 159—161 elle: |S 1—1 180—182 1S86—189 1838—18. L44—146 0 1000—1049 1050—1099 1100—1149 1150—1199 1200—-1249 1250—1299 1300—1349 1350—1399 1400—1449 1450—1499 1500—1549 1550—1599 1600—1649 1650—-1699 it oN Ww | ARIA | pe phe wR | Hr RDA woe | —_ Ll ulae bo | | — Totals We | 12 TABLE 36. Bavarians. Brain-weight and Stature. ‘“ Total,” Short Series. Females. Stature (centimetres). Brain-weight (grams). oY ‘ 6—158 16% — 168—170 15 16: 800— 849 850— 899 900— 949 950— 999 1000—1049 1050—1099 1100-—1149 1150—1199 1200—1249 1250—1299 1300—1349 1350—1399 1400—1449 1450—1499 1500—1549 1550—1599 — wo ~119 + | PEDO D&W = rere w-l me = = bd bOD HH oaunokonn | [4 — Hoaw-~10 8 wr | Totals Age (years). 100 Variation and Correlation in Brain-Weight TABLE 37. Bavarians. Brain-weight and Age. “Total,” Long Series. Males. Brain-weight (grams). 49 | PQQ on — 3 — > | D> S dS > o> jon) ier) Se S| Se eS Sear Se Os aa Oo seal = S ™ ™ 2 2 B) p = SS 9D Ko) 5 Ne} | ~ ~ ™~ ~ = ~ ~ ~ al ™ | ri ™~ ~ ie | Ps eee eee ee te te I || Bees SilmelsilselSielsisls |e 1S | 48 eS ales S Ss ™~ = RN RN % | 9d Sree eS Ww’ > Ss >| ~ al i ~ ~ ™~ ~ > al > ~ ~ ~ ee) | | | 20—24 joes 6 | basal =| =p | 25—29 —]| 6 8 — 2 | 30-—34 2 a | 1) 2 3 ® | 35—39 4/110 22 | 8 1 | YS | 40-44 -- ii 4 2 1 - 45—AQ — 5 9 3 il on S D> D> oy D> oy Dd Sp g Ss S> ~> yp ~> dy ~ Dd = d Dp oe fe») > s/S/S/SliSI/SIHlIsierlR—i sl sl s S S S = S > SS Ss S re = Ig S Ag S wD S 9 | S wD S wD S 1D Oo |s S Dd S S i —s | & NR | 9 | => > 3 SS | > > mm | al ™~ ~ ~ m ~ ~ ™ 20—24 || 9 1 47 25—29 6 9 3 | 51 30—S4 6 15 vi 3 51 35—89 1 | 9 3 3 | 45 4O—A4 2 | 5 4 2 21 45—AI 5 | 7 6 | 23 50—54 6 | 6| 4 2 28 55—59 1} 2 5 ao 10 60—C4 4 8) 2 2 17 | 65—69 3} 1 3 1 15 | TO—T4 ] 2 1 9 75—T79 4 = —_— 6 SoS (ell Totals 323 RayMonpD PEARL TABLE 39. Bavarians. Brain-weight and Age. “ Young,” Long Series. Males. Brain-weight (grams). | Sa aa a ee eo la lo | S 2/3 + 1 Dd Si al se] os S DB | st | od Sin lH larel/alal l/s |si/sio}sef/_o]s ~ ™ LOM esl ~ ~ ~ ™ ™ ™~ ™ ~ ™~ ; ey er | | tetas Sia | SS) SoS eis SSS Ss wD S WD Saale) — | & S wD S wD S wD 4 SPH lH / Rl Rls] ses Psi sige /s]s a | ™~ mm LON ~ | ns s m ™~ ~ ™~ ™~ ~ 2 7 1] 3 — | 1 9s 50 1a ii 2 | < | 14 Tale 5 Sa? 6 6] 3 | | TABLE 40. Bavarians. Brain-weight and Age. “ Young,’ Long Series. Females. Age (years). 40—4h j5—49 Totals S> | ir) QD | D > S ier) S> 3) | Sea S3 | oe |S t+ | D St | SD ory S =) ™ al N RN 89 Sr) a] ~ ~ ~ ~ ial ae al | eae elie ae yt be le S S S > S S ) S wD S WwW S wD S| ES S iS) =) S S san ~ MN Y Sa) Sa) ~ ~ ~ ~ ™ ~ al ™~ — 20—24 25—29 30—34 35—39 101 102 Brain-weight (grams). Brain-weight (grams). Bavarians. Brain-weight and Body-weight. TABLE 41. Variation and Correlation in Brain-Weight “ Total,’ Short Series. Body-weight (kilograms). Males. 1000—1049 1050—1099 1100—1149 1150—1199 1200—1249 1250—1299 1300—1349 1350—1399 1400—1449 | 1450—1499 1500—1549 1550—1599 1600—1649 | 1650—1699 Totals 1 De it le eal Ph taal Sat ofp as = As | oa ales tlle : Al ele eey eo) HF OP are || ee | Sees 1 Ten elOg ise | gba leaalpead 9.118.) 104) 11) 95) orl e3 1 B21 5 | a 7a Aa To G.G Obes Gal att aul = 25 Ae Sel Oe see BS Ta ere eh aa a 2 =P il : ASS or 31 Bavarians. TABLE 42. Brain-weight and Body-weight. Body-weight (kilograms). “ Total,” Short Series. Females. 1500—1549 800— 849 850— 899 900— 949 950— 999 1000—1049 1050-—1099 1100—1149 1150—1199 1200—1249 1250—1299 1800—1349 1350—1899 1L400—1449 1450—1499 | > | is | | 5 1550—1599 Totals | Ml aelerneslirerel | cs koe | t 1 | Lee Rc aamel! | | | trae Seste | — | Ll lle ie WO Dd Ow wn! | | eo | [Pilinorasrerine con i | | mHrreocrco | » | De eK bo | | . Body-weight (kilograms). Stature (centimetres). Raymond PEARL 103 TABLE 43. Bavarians. Body-weight and Stature. ‘ Total,’ Short Series. Males. Stature (centimetres). © | o | R Ney co sam st] & S 99 R D> | | s+) SS | wd WwW WD Ko} a) Ks) ~~ tan =~ ~~ ™ ™ | a | ~ ~ deat | onl ~ ™ ™ ™~ 2 | eS heal et OP Pk tT FS in Cam de a Potals | SISISISlIS/S/SlPSislelsfE ae a ~ S| MN |S ~ ~ ri ~ 25—29 1 | 1 30—34 Bi eae 18 poo NT SB a ee 6: | BB -| 43 a ae eee eS. On 5a! LON Te) 4a ee) 69 4I—4I | — | — 1 1 5 D (lie) U7 | Lo 3 6 6 | - —- 76 50—54 hea Mal onl toe eae) delete eT 55—59 | 1 2 33 7) 16 6 OMA 2 | — 1 45 60—64 | — | — | — 1 1 | — | 4 4 8 4 Dal) a3 1 = 31 65—69 1 1 3 3 j 14 1O—F4 1| 2 10 45—79 3 | 80—84 2 | 85—89 | 90—94 2 Totals 365 | TABLE 44. Bavarians. Body-weight and Stature. “ Total,’ Short Series. Females. slalwsflalwalials)/alseialsia | S i] iS) 89 9 | “~S Se im) S ~~ | 2 S | | | | | Totals’ 132—1384 2 135— 137 1 138—140 1 141—143 — 1 1 2 Me wep—| 1) 8) 4) 5) 1138 =| 17 147—149 | — ere 5 6 4 6 31 50 152e—=) | 3 OA On We | OF 5 41 Hos — 100 ee ON TO tb 5 36 156—158} — | — 2 oy) 1d LE 54 159—161 1 1 fy 8 8 37 162—164 2); — 2, 1 ” 11 165—167 ii - | - oO: | 168—170 | 2 I71—178 3 ETF 3 241 Age (years). 104 Variation and Correlation in Brain-Weight TABLE 45. English. Weight of Cerebrum and Stature. Males. Stature (inches). | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 Totals| (ae ae | 28 | | al 29 | } 1 1} — 2 30 1 - 1) a) 2) eee 31 | = ; p= 1 ei 882. 2 3 n | 33 f : o| % |—;—|—}1)—] 1] 2}/—]} 1;/-—|;—]/—]|] 1]-| 2 8 = | 385 [Df =) 4) 4.) ==) 0 | ae eS iar S| ia | 2}/—| 2} 2) 2; —j|«3}]—] 17 12 ali ae : 1} — 4) 2° 32): oe a eae ee B | ees | — |e |e |e QV | 5, 6 ulee4 | 23 eB ee ee &| 89 }—/;—]|1/--/ 1] 1 L}/ 2) 5;—] 2] 3) 2] —)-Sa aes) >| 40 | - 1]; 3) 4-| 4) 6) 95) SOF) = seeieenoo: Beall ag ee leh leg (alte ee 3) 2) Blea} 3) 938) Sh ene aiesO Ol 42 | Li4) 4) 6) 6] 9) S50 1c) Seiler, ‘| 43 | 1) @| 2/4/96) 4) 2) 27) =i eaainee >| 44 : }—=j| 1] 2) @) 4) %2.) a) 60) 5 os a eees ee 9) | ee eee |e | : 64) 2) Ol We er Soe eee ales 20H Cm | | a | 2) 4) V4 93 | ==") os eon iaty te Ai 1 ec an ieee Fel eI Na Seal De ds) | 1 48 }—|—|—|]—/|—|]1]—| 2/—] 2] 5) 1) 1/—] 14-18 os oma ese ema eg are | eae [Wo 2 2 a ea) a 6 50 1 1 | = 3 gt || | 1 eros | | ee 4 Totals | TABLE 46. English. Weight of Cerebrum and Age. Males. Weight of Cerebrum (ounces). | | | 35 | 36 37 | 88 39 40 | 41 | 42| 43 44 | 45 | 48 | 47 Totals 20—29) 12) 1 | 2 |—j— | al |) | 25) 2) 2) a6: ees sl oume online 2] 49 30—39 ea 3/ 2) 3] 2) 3] 3) 4] 4] 3] 5) 4) 5) 4 4) 2)3/—|—] 55 40—494—|—|211)1|—| 2) 21—| 3] 6] 6! 4] 4) 3) 3) 6) 3) 1) ois RomeaonaaS 50—59 = 4| 2| 2) 4\°8| 4) 6] 42] 4)-9)) 4) 4) nee eee 60—69 1 1 2] 3/ 2| 2) 6] 1) 3} 8] 6) 4] 3] 5] 1) 1]|—|—/—/—] 49 7079) — | 1 4) 2) 3] 8) 2] 1] 4] 8} 6; 4; 1}).1/ Lj-1/—/1|]—] 48 28 | 26 | 16 |17 | 14 i3| 6 | 3 | 4 | 308 | I i A STUDY OF THE RELATIONS OF THE BRAIN TO THE SIZE OF THE HEAD. By REGINALD J. GLADSTONE, M.D. OnE of the main objects of the investigation which forms the subject of this paper, has been to obtain a series of reconstruction formula, by which it will be possible, when in possession of certain chief measurements of the head, to predict within the limits of normal variation, the approximate weight of the brain. The collection of data for this purpose, which has occupied a period of over three years, has been carried out with the permission of the Resident Medical Officer, in the ‘ post-mortem’ room of the Middlesex Hospital, whereas the mathe- matical treatment, in the following paper, is chiefly the work of Mr J. Blakeman, M.Sce., of University College, London. Briefly, the method which we have adopted has been to measure the heads of ‘ post-mortem’ room subjects, and afterwards in each case to take out the brain and weigh it. In each case, therefore, a comparison can be made between the out- side measurements of the head, and the weight of the brain. The individuals measured have been a few presumably healthy subjects, who have died an accidental death ; the ordinary subjects of post-mortem examination at the Middlesex Hospital, London ; and a fairly large proportion of subjects who have died from malignant disease in the Cancer Department of the same Hospital. They may be regarded, therefore, as forming a fairly typical sample of what has been termed a ‘hospital population’; containing, however, owing to the inclusion of ‘cancer cases,’ a rather larger proportion of individuals who have died from wasting diseases than is ordinary. A comparison, however, of the ratio that the ' brain-weight bears to the outside measurements of the head in a series of cases in which death took place from accident, or acute illness, with cases of death from wasting diseases, has shown that the diminution of brain-weight in the latter, although measurable, is small ; and that their inclusion, therefore, with the acute cases does not seriously affect the general result. Biometrika 1v 4 106 Brain-weight and Head-size No case has been included in which the brain showed a distinctly pathological condition which would have obviously affected its weight; nor have individuals with foreign or Jewish names been included. For permission to make use of this valuable material, and for assistance in many ways, I am indebted to Dr R. A. Young, Pathologist, and Lecturer on Morbid Anatomy at the Middlesex Hospital, and to Mr W. T. Hillier, Pathological Assistant in the Cancer Department of the Middlesex Hospital. In each case the following items have been recorded: (a) name, (b) age, (c) sex, (d) cause of death, (e) remarks: on general condition, degree of emaciation, &c., (f) stature, (g) measurements of head : These were : L, the length of the head from the glabella to the occipital point. B, the maximum transverse diameter of the head above the level of the zygo- matic arches. H, the height of the cranium as indicated by the vertical distance from the biauricular line * to the bregma. U, the horizontal circumference, taken in a plane passing in front through a point just above the glabella, and behind through the occipital point. S, the longitudinal or sagittal arc, measured from the glabella, over the vertex to the external occipital protuberance. Q, the transverse or coronal arc, measured from the tragus of one side, over the vertex to the tragus of the opposite side. These measurements of the head having been obtained, the scalp was reflected, and (3) the length, (28) the breadth, ({) the height, and (G1) the circumference were measured on the bared skull. The vault of the skull was then removed, and the brain taken out and weighed in the usual manner, without removing the pia and arachnoid membranest, the weight being recorded in grammesf. The longitudinal and transverse diameters of the head have been taken with a Flower’s craniometer, made by Aston and Mander, of 25 Old Compton Street, * Taken at the eentre of the auricular orifices. + The pia and arachnoid membranes vary considerably in their total weight and an increase in their weight accompanies an increase in age; thus according to Broca, the weight of the ‘pia’ at different ages in males, averages 20—30 years ... ... 45 grammes 31—40_ ,, ... ... 50 as 60 aes ... 60 9 The variations ranged between 38 and 130 grammes. The mean weight of the ‘pia’ in the case of 133 females was 48°7 grammes, and of 273 males 55:8 grammes. The term ‘pia’ as used by Broca - obviously includes both the pia and arachnoid membranes. t The weight of the brain in ounces was in a large number of cases also ascertained, and recorded in the Hospital Reports; and a reference to these reports proved to be a useful means of verifying the recorded weight in grammes, of certain cases in which there was a considerable deviation from the usual relation between the brain-weight, and the outside measurements of the head. REGINALD J. GLADSTONE 107 London, and the vertical diameter by an instrument, Fig. 1, similar to the one which I have previously described and figured in the Report of the Proceedings of the Anatomical Society, November, 1901; the instrument which we have used for the post-mortem measurements has, however, been improved by the adoption of a suggestion made by Mr J. Gray, namely, the substitution of a vertical screw to act upon the indicator, for the rack and pinion with which the first instrument was fitted. Any slipping of the measuring rod, from the instrument working loose, is thus avoided. Fic. 1. Instrument for measuring the vertical height of the head, from the biauricular line to the vertex. The circumference and the longitudinal and transverse arcs have been taken with a Chesterman’s steel tape-measure. Although the longitudinal and transverse arcs were measured, and are pre- served in each case, we have not made use of them in constructing any of the tables or formule. The diameters being in our opinion the more trustworthy measurements, since they are less influenced by variations in the amount and thickness of the hair, and because the points between which the measurements are 14—2 108 Brain-weight and Head-size taken are in the case of the diameters much more precise. In many individuals the external occipital protuberance, even when the muscles of the neck are thoroughly relaxed, is difficult to localize. This is the case in the living subject, and still more in the cadaver, so much so that in some subjects it is impossible to determine its exact position. Moreover, when prominent and easily felt, there may be as much as 1 cm. difference between the measurement taken by one individual and that by another, according to whether the top or bottom of the projection has been taken as the starting point. The small tubercle on the tragus from which the measurement is taken in recording the transverse arc is also relatively to the centre of the external auditory meatus, a somewhat variable point. I regard these measurements, therefore, as less reliable than the diameters. The measurement of the ares and circumference has this important advantage, however—they can HW =] =] — 155mm: Fic. 2. be taken by a tape measure, which can be carried in the waistcoat pocket, and they can thus be more readily ascertained by travellers who would be unable to carry a more bulky instrument. Moreover the measurement of the longitudinal and transverse arcs, and of the horizontal circumference, will in some cases give a truer estimate of size than the greatest diameters taken through the principal axes, as will be seen by the accompanying Fig. 2, which shows tracings of the horizontal circumference of two heads, one of which, represented by a continuous line, was narrow in the frontal region, but wide in the parietal, whereas the other, represented REGINALD J. GLADSTONE 109 by broken lines, although having a wider frontal region, had a less maximum transverse or interparietal diameter. The total area of the latter is greater than the former, as is indicated by the circumference, which is 543 mm. in the broken tracing as compared with 530mm. in the continuous. The product of the two principal diameters of the continuous tracing (1=181 x B= 155) is 28,055, whereas that of the broken curve (1 = 182 x B= 150) is 27,300, or 755 less than the continuous. The product of the diameters thus gives in this case an erroneous indication of the area enclosed by the tracings*. We have accordingly made use of the horizontal circumference, although discarding the longitudinal and transverse ares for the reasons stated above. The correlation between brain-weight and the outside measurements of the head is naturally not nearly so close as that which exists between the outside measurements of the skull and the capacity of the skull. For in addition to varia- tions in the thickness and shape of the skull, we have to deal with variations in the thickness of the scalp and hair, and also with the very considerable variations in the amount of space which exists between the surface of the brain and the internal surface of the skull. Moreover the weight of the brain is modified by variations in densityt+, the size of the cerebral ventricles}, and by the amount of blood contained in its vessels. The weight of the brain also, as is well known, varies considerably with age, a progressive decrease of weight and most probably also of size taking place after middle age. This decrease, however, appears according to both Boyd’s§ and Vierordt’s|| statistics, to commence at a much earlier age, the highest average brain- weights for different ages occurring between 14 and 20 in both sexes. The brain in children and in youth is both actually and relatively to the body weight and stature very large. According to Vierordt’s statistics, the average brain-weight of 35 boys, between 12 and 16 years of age inclusive, was as much as 1423 grammes; and in a table published by the same author, showing the average brain-weight for each year from birth up to the age of 25, the highest average brain-weights, 1490 grm. ¥, and 1345 grm. $, occur at the ages of 15 and J4 respectively ; whereas, according to Boyd’s statistics, the average brain-weights of adult British subjects (mostly of the lower classes), are : Males between 20 and 40 ..... . 1360 grm. or 48 oz. Females ,, 20 and 40 ...... 1230 grm. or 435 02. {* The problem really is which gives the better average result. It may be worth noting that the volume of an ellipsoid is proportional to the diametral product, but not to the product of the three circumferences. Ep.] + According to Professor Donaldson’s statistics, ‘‘ the average specific gravity of the entire encephalon should be for the adult male 1:0363, and for the adult female 1:0360.” H. H. Donaldson: Growth of the Brain, p. 95. t “The cast of the ventricles as made by Welcker displaces 26 cm.* of water so that the fluid filling such a cavity would weigh a trifle over 26 grammes.” H. H. Donaldson: Growth of the Brain, p. 87. The specific gravity of the cerebro-spinal fluid is about 1-009. § See table compiled from the observations of R. Boyd, Phil. Trans. 1860, in Quain’s Anatomy, Vol. ur. Part 1. p. 178. || Table 17, p. 104, H. H. Donaldson: The Growth of the Brain. 110 Brain-weight and Head-size The maximum, average and minimum weights of the brain, in four groups, arranged according to age and sex, of the ‘ post-mortem’ room subjects of the Middlesex Hospital, may be seen in the following table: The weight of the brain is expressed in grammes, and each group contains 50 subjects. | Males Females | Max. | Aver. Min. Max. Aver. Min. | 20—46 | 1685 1370°5 1207 1520 1223°5 1027 | 46 upwards ... 1588 131671 1120 1408 1195°7 955 | It will be noticed that in passing from the younger to the older groups there is a decrease in the average brain-weight of 544 grm. in the male, and of 27'S grm. in the female, the mean decrease for both male and female being approximately 40 grammes. This result corresponds very closely with the table compiled from the observa- tions of R. Boyd, and published in Quain’s Anatomy, Vol. 11. Part 1. p. 178, in which it is shown that “the brain is absolutely heavier between 14 and 20 years of age than at any other period of life, and that at the age of 80 it has lost about 90 grammes, or rather more than 3 ozs., 1.e., 35 of its total weight.” Not only is there a marked change in brain-weight corresponding with different periods of life, but there are also considerable variations in the thickness of the skull and its coverings. The scalp is thin in infants and young children, and a sensible attenuation of the scalp occurs in old age, which is probably largely due to atrophy of the hair follicles following the loss of hair. The amount, although small, is measurable, and is sufficient to affect the general result of a statistical investigation. The amount of atrophy may be seen by a comparison of the figures in the following tables, which show the average ditferences between the diameters of the head and the diameters of the skull as measured before and after the scalp has been turned down in order to remove the ‘skull cap.’ 18 Males between 20 and 46 years of age. mm. Height of head minus height of skull = 4:05. Length 5 » length ye he Breadth _,, » breadth ,, =8'27, 27 Males from 46 years upward. mm. Height of head minus height of skull=3-79. Length S » Jength =e Breadth __,, » breadth ,, =7-40. REGINALD J. GLADSTONE 111 17 Females from 20—46 years of age. mm. Height of head minus height of skull=3:82. Length _,, 35 length ne Breadth _,, ; breadth ,, =7°'88. 27 Females from 46 years upward. mm. Height of head minus height of skull=3°50. Length _,, - length alos Breadth _,, A breadth ,, =6°98. It will be observed that the scalp is thinner in female subjects than in males and that there is a diminution in the thickness of the scalp, which amounts to : J : 026 mm. § : 032 mm. at the vertex. 1:22 mm. 0:40 mm. at the glabella and occipital point together. 0°87 mm. 0:90 mm. on the two sides. There is thus a diminution in the length of the principal diameters of the head, attributable to the atrophy attendant on old age, which amounts if the mean of both sexes be taken to: 0:29 mm. in the vertical diameter. 081mm. _,, longitudinal 0885 mm. _,, transverse ” 2) The thickness of the skull also varies with age, as will be noted on comparing the cut surface of the vault of a skull shown in Plate II. Fig. D, which is a photograph of the vault of the skull of a child about four years old, with that in Fig. C, which is the photograph of the vault of a normal adult skull. The difference in thickness of the skull of a child, and that of an adult will also be seen on comparing Fig. A, Plate II. with Fig. B on the same plate. Fig. A represents the right half of the skull of a child about five or six years of age; Fig. B the right half of a thick, adult skull. Note also the absence of frontal sinuses in Figs. A and C. The extent to which the thickness of the skull may be affected by disease, may be seen in Fig. 3, which represents part of the vault of a skull, from a case of ‘osteitis deformans*, the walls of which averaged about one inch in thickness. The increase in thickness of the skull in these cases takes place partly on the outer and partly on the inner surface of the skull, so that although the size of the head is enlarged the capacity of the skull is diminished, and compression exerted on its contents. The reverse condition is met with in cases of chronic hydrocephalus in which the size of the head is increased by an augmentation in the volume of its contents. * Specimen 1239. Museum of the Royal College of Surgeons, England. 112 Brain-weight and Head-size An extreme case of this disease is seen in Figs. KE and F in Plate III., which are photographs of specimen 3878 inthe Hunterian Museum of the Royal College of Surgeons, England. It is described in the catalogue as “The skeleton of a man who died with hydrocephalus at the age of 25 years. Its greatest horizontal circumference is 914 cms.! Besides the enlargement of the frontal and _ parietal bones by which the greater part of the enlarged cranial cavity is formed rows of Wormian bones from 2°5 to 3°8 ems. in breadth are placed in the whole course of the lambdoid and sagittal sutures, and in great part of the length of the squamous sutures.” Fie. 3. The enlargement of the head in hydrocephalus is associated with an increase in the amount of the cerebro-spinal fluid, which is found either in the space between the arachnoid and pia mater, outside the brain, within the cerebral ventricles, or in both these situations. In the case of James Cardinal, the skeleton of whom is preserved in the Museum of Guy’s Hospital, seven pints of cerebro-spinal fluid were found between the membranes, whereas the ventricles contained one pint. In this case “ it appeared that the fluid had been originally contained within the ventricles, but had burst through an opening in the corpus callosum *.” It is obvious that both in the case of osteitis deformans, and hydrocephalus the size of the brain cannot be gauged by the size of the head. The enlargement of the head in hydrocephalus is also of interest, as it indicates one means by which the size of the cranium is increased in the course of normal growth, namely by an expanding force acting from within. In the early stages of development the foetal brain has a smooth even surface, at a later stage from the 8th week of intrauterine life to the 4th month, what are called the ‘ transitory fissures’ make their appearance ; these are infoldings of the thin walls of the * See description of case in Tumours, Innocent and Malignant, p. 448, J. Bland Sutton. REGINALD J. GLADSTONE 113 hemisphere vesicles, which appear to be occasioned by the growth of the brain being more rapid than the capsule which confines it. At a still later stage following a more rapid enlargement of the skull many of the early fissures disappear, but they are afterwards replaced by the permanent fissures or sulci of the adult brain. The appearance and disappearance of these fissures seem to indicate that there are two counteracting forces concerned in the growth of the brain, namely an expanding force due to increase in brain matter, and cerebro- spinal fluid, and a passive restraining force exerted by the skull. In anencephalic monsters the brain is imperfectly developed ; the hemisphere vesicles appear at an early stage of development to have ruptured and collapsed, and the cerebro-spinal fluid to have escaped through a triangular aperture, which may often be recognized at the time of birth in the position of the anterior fontanelle. The cerebro-spinal fluid thus passes into the cavity of the amnion, and the amount of fluid within this sac becomes thereby much increased. In these cases the expanding force concerned in the growth of the skull is absent, and consequently the bones forming the vault of the skull, if they are developed at all, grow only to a sufficient size to cover over the base of the skull, and rudiments of the brain, whereas the bones of the face, and of the skeleton generally, grow to their normal size. In rickets on the other hand the bones of the skull are abnormally soft and yielding, and apparently do not afford the usual resistance to the expanding pressure from within, and as a result the cranial portion of the skull enlarges, and the well-known square overhanging forehead of a typical rickety child is produced, while the face remains of the normal size and owing to the enlargement of the head above, appears unusually small. In the course of normal growth the two forces continue to counteract one another until the brain has attained its maximuin size. Later a progressive diminution in the weight and probably also of the size of the brain takes place, and the sutures of the skull afterwards, gradually consolidate. Not only is there the ditference mentioned above in the general thickness of the skull at different ages, but there is also a considerable difference in the frontal region, due to the development after puberty of the frontal air sinuses, which tend to increase in size with the advance of age, and which are usually considerably larger in the male than in the female. Their position and the size which they may attain are well seen in Fig. 4, D, and Plate II., Figs. B and C. It is obvious, therefore, that in any attempt to calculate the weight of the brain age must be taken into account, since it affects not only the brain itself, but also the thickness of the scalp and of the skull. It may be noted, however, that after middle age the variations attributable to age are so far as the thickness of the scalp and the weight of the brain are concerned, in the same direction, namely diminution, so that the proportion between the outside measurements of the head and the brain-weight is not very materially altered. The diminution in brain- Biometrika 1v 15 114 Brain-weight and Head-size —— Sam Lc oes . a ‘3 ’ GM APR SSS 88 ee er toe al Pye Pe rhe a eterna ee Seas ~ peewee Se os *rl8 Me, Fic. 4. Four horizontal sections through the frontal bone, showing the condition of the bone at various ages. A. Male, aged 9. Bone thin, no frontal sinuses. Bs 5, 14. Bone considerably thicker, no frontal sinuses. Osea. ,, 82. Illustrating the occasional absence of the frontal sinuses in an adult. 1D », 54. Unusually large frontal sinuses. From specimens in the Museum of the Royal College of Surgeons, London. REGINALD J. GLADSTONE 115 weight follows very closely the diminution in the size of the head, as is shown by the following table, which gives the proportion between these two quantities in 4 groups of 50, arranged according to age and sex. The size of the head is here indicated by a figure obtained from the product of the three principal diameters (L x Bx H) expressed in millimetres, the first four figures of this number represent the number of cubie centimetres which would be contained in a rect- angular block having the same diameters as the cranial portion of the head, and I shall refer to it subsequently as the ‘index of size’ of the head, thus : (196) x (156) x (138) = 4219488. 4219, the ‘index of size’ of the head, represents cubic centimetres and may therefore be conveniently compared with the weight of the brain expressed in grammes. Index of | Brain- | Index of Size | Number in Males H L B size weight | Brain-weight | each group | Age, 20—46 ... | 184°8 | 190°8 | 149°5 3876 1370°5 2-806 50 » 46 and upwards | 132°4 | 189°8 | 148-7 3736 1316°1 2°838 50 Females Age, 20—46 ... | 129°8 | 1838°4 | 144°9 3449 HOO Br5 2°818 50 » 46 and upwards | 128°2 | 182°1 | 143°5 3350 WS yor 2-801 50 By combining the male with the female groups, we obtain the following ratios between the size of the head and the brain-weight: Age, 20-46. : : 2812 46 and upwards. 2°819 In comparing the extreme ends of the series, however, it appears that the diminution in brain-weight, with the advance of age, is more rapid than the diminution in the size of the head ; thus taking the mean of 4 ( and 3 cases over 70 years of age, the ratio between size of head and brain-weight is 2°821, whereas an equal number of cases between 20 and 25 years of age give 2°702. ” The influence of stature upon the ratio between the outside measurements of the head and the brain-weight is apparently very small; but it has a marked influence on the actual size of the head as compared with the general mean, and a less but measurable influence on the proportion of the size of the head to the body in the individual. In other words, tall men in the aggregate have larger heads than short men, but proportionally to the size of their bodies their heads are con- siderably smaller than those of short men. The formula which I have employed to express this relationship of the size of the head to the stature is the following: BYAG an ny Index of size ; : 100x & ua i = Capitulo-statural index. Stature in centimetres 4/3921 em? x 100 15°77em.x100 1577 Ex. = —— Se Ob 170 cm. 170 em. 170 116 Brain-weight and Head.-size It will be seen on referring to the table given below that a considerable : diminution of the capitulo-statural index takes place with increase in stature ; and also that the size of the head increases with the stature, but that the enlargement of the head does not keep pace with the increase in stature. Table showing the proportion that the size of the head bears to the stature in groups of individuals arranged according to their height, the degrees of which are expressed in intervals of 3 inches. l | Navieallyse @e Stature ae: ee Capitulo-statural index cases in =|) yee oe | re Mind eReotesiae Index of size x 100 each group | (L x Bx H) SESE TIEC = % can ee Inches cm. em.? cm, tature 37 74—72 | 185:3 4135 16-05 8-66 120 71—69 | 177-7 4071 15:97 8:98 124 68—66 | 170-0 3993 15°87 | 9-00 67 | 65—63 | 162-4 3902 15°75 9°69 15 62—60 | 154'8 3778 15°56 10°05 This table is based upon the measurements of 363 male subjects of all classes in England above 20 years of age, the larger number of these belong to the professional class, so that both the mean stature and mean size of the head are slightly above the general average. The increase in size of the head, accompanying increase in stature, is shown also in the following chart, which is planned from the same figures as the preceding table: ze Index of si Number in each group 67 124 120 4007 Mean Index of size Nie. nuit. 1548 1624 nue. non. 1700 1777 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 Inches Fic. 5. Chart showing the increase in the size of the head which takes place with increase of stature. The chart is based on the measurements of 363 adult male subjects, above 20 years of age. viduals have been grouped according to their stature, which is expressed in intervals of 3 inches. figures indicating the size of the head are shown in the column on the left. The indi- The Plate Il Biometrika, Vol. IV, Parts | and II Biometrika, Vol. IV, Parts | and Il Plate Ill REGINALD J. GLADSTONE lye The following table is also of interest, as exemplifying the same law in certain extreme cases, in giants and dwarfs”. . a = = = = ) Capitulo-statural index inde. ——<<—$————— ea tae ee es De Stature of size J Index of size TR, ds Se EY O~N 67 14l | 200 | 160 4512 | 1530 | 574 | 80°0 | Chronic nephritis. Cardiac degeneration 68 137 | 182°5 | 149° 3738 | 1297 | 550 | 81°9 | Accident. Fractures of bones 71 139 | 201: | 152° 4261 1335 | 569 | 75:9 | Pulmonary tuberculosis 70 134°5 | 193 | 145: 38777 |: 1282 | 542 | 75-4 | Malignant growth affecting mouth 68 140 |195 | 153 4177 | 1590 | 576 Accident. Fracture of skull 65 132 | 186 | 146 3585 | 1300 | 547 Cerebral haemorrhage 68 137°5 | 192°5 | 143 3785 | 1400 | 549 Pulmonary tuberculosis, Skull thin 34°5 3559 | 1255 | 535 Carcinoma of stomach 68 132°5 | 185°5 | 147 3613 | 1355 | 555 Pneumonia. Pleural effusion Cr Or or 00) We) = iS S 64 135 | 196 |150°5| 3982 | 1375 | 565 Cerebral haemorrhage 66 1327) 1859 a 3443 | 13840 | 5387 Meningitis. Mastoid disease 63 136°5| 197 =| 148°5 | 3993 1380 | 570 ‘75 | 136 188°5 | 142 3640 1355 | 558 66°5 | 138 198 154 4208 1522 | 576 71 132 191 152 3832 1208 | 561 Cellulitis of upper lip. Septiczemia Basal meningitis. Pulmonary tuberculosis Fracture of skull. “Large pear-shaped head” 60°5 | 135 | 194 | 148 3876 | 1405 | 569 Fracture of skull [hemorrhage 63°5 | 1380 | 183 | 147 3497 | 1358 | 551 | “Run over.” Rupture of liver. Cerebral 65 130 | 184°5 | 144°5| 3466 | 1292 | 541 | Malignant growth. R. ext. auditory meatus 71:5 | 126 178 138 3095 1340 | 517 68 144 192 160 4424 1400 | 580 68°5 | 137 190, 149 3878 1357 | 568 68°5 | 130 197 158 4046 1287 | 582 68°5 | 137°5 | 189°5 | 146 3804 1275 | 554 68°5 | 135 182 151 3710 1270 | 541 71 142 199 168 4747 1635 | 602 Burns. Congestion of brain Fatty liver and kidneys. Delirium tremens Tubercular nephritis. Thin skull Fatty heart. Congestion of lungs Morbus cordis Carcinoma of cesophagus, and stomach Impaction of food in pharynx 72°5 |1385 | 195 | 168 4493 | 1505 | 577 Inguinal hernia. Pulmonary embolism 69°25 | 135 | 198 | 151 4036 | 1490 | 564 Appendicitis. Thick skull. Foreign 70°5 | 138 193 151 4022 1485 | 570 69°5 | 1385 | 184 | 139 3454 | 1310 | 537 Appendicitis. General peritonitis 69 142 200 147 4175 1420 | 574 5 | 136 182 153 3787 1318 | 567 69°75 | 135 190 148 3796 1432 | 564 5 | 138 187 159 4103 1364 | 574 71 137 SH 159 4161 1405 | 584 5 4158 1432 | 574 61 134 191 149 3814 1207 | 548 65 135 | 184 | 142 3527 1375 | 530 Cerebral tumour 61°5 | 138i. | 187 | 158 3748 | 1350 | 560 “ Run over.” Fractures of ribs and sternum 61 129 182 | 142 3334 1236 | 528 | 78°0 | Pulmonary tuberculosis 5 | 129 | 192 | 141 3492 | 1250 | 540 |] 73 Malignant growth in floor of mouth 61°5 | 134 | 194°5 | 152 3962 | 1350 | 558 | 78: Ulceration of larynx be | 29 190 143 3505 | 1320 | 5389 | 75 Carcinoma of omentum. Emaciation 66 138°5 | 201 | 155 4315 | 1525 | 587 | 77 Jerebral abscess following empyema 67°5 {130 |190 | 154 3804 | 1570 | 550] 81 Pulmonary embolism. Inguinal hernia 69 131°5 | 192 153 3863 1340 | 563 | 79 Diabetes 69 134 | 193 | 156 4034 | 1422 | 578 | 80 Cirrhosis of liver. Morbus cordis 5 |142 | 192 | 158 4308 | 1506 | 569 | 82° Chronic nephritis. Gidema of lung. Foreign 68°5 | 124 /|181 | 141 3165 | 1215 Infective endocarditis. Skull very thin | 67 134 |190 | 143 3641 1311 | 545 | 75: Renal calculus 32°5 | 191 | 144 3644 | 1300 | 555 | 75: Pneumonia. Well nourished 68°5 | 130°5| 193 | 154°5| 3891 1224 | 570 | 80° Cirrhosis of liver. Well nourished 68°5 | 132 | 193°5| 148-5 | 3793 | 1350 | 560 | 76 Fractures of sternum and ribs OF) 136 4270 | 1335 | 558 | 82 Aortic regurgitation. Well nourished 69 132 | 190 | 162 4063 | 1390 | 557 | 85 Meningitis. Bullet in brain 14 years 69 135 | 193 | 154 4012 | 1400 | 569 | 79 Fall down lift. Well nourished 66 125 |182 | 152 3458 1225 | 588 | 83 Fibroid degeneration of heart 60°5 | 134 | 191 | 152 3890 | 1310 | 549 Bronchitis and emphysema. Thick scalp Purulent meningitis. Fracture of nasal bones Malignant growth of kidney Lymphatic leukemia. Punctiform hem. c’blm. Lobar pneumonia. Enlgemt. of liver. Foreign Chronic laryngitis. Thin skull. Scalp very thick Malignant growth of lip. Metastasis Fracture of tibia. Delirium tremens [on] ~I ra ww We) = we) Or an e Or oe WSOWONMEWOTKAAKWATODWADWDODONTUADOL HOC ST SI AT 00 WT OO ~T ~T AT TO OO ~T 1 0 ~T ~3 OH ~T TT WT TT TY TT I SDN SKNHOCSEAUNHNWNHKROCONKYAWHWHKEADWANYANE NA [ee) ea I = a We ror —_ < ref ae [ep on = oO ~I -~I HOWATERWOWOTEE WEE +l ive} lon) B. Male Subjects from 46 and Upwards. Middlesex Hospital. REGINALD J. GLADSTONE 119 Acute or Chronic (Millimetres) (Cm.*) AxLxB | Sl) | ASS Sse] | GQQapRaGhaqeqabres | Soqqn | |] Keak | KA] | ab | QQRaar = te bo — CO ie) wb) § _— i a 126 184°5 | 146 131 183°5 | 148 a oe) > [fluid 35 | 63°5 | 1382°6 | 187 | 158 38791 | 1260 558 Cellulitis and cdema of pharynx. Thick 40 60 126 | 182 | 144 | 3302 | 1165 | 532 Carcinoma uteri. Thin skull [skull | 30?) 51 «| 126 =| 1785/1388 | 3104 | 1080 | 498°5 Ureemia Valvular disease of heart. Emboli in brain Pneumonia. Pericarditis Fatty heart Rheumatism. Bronchitis Raynaud’s disease y Carcinoma. Cervix uteri Gun-shot wound in L. breast | 33?/ 58 |199-5/181 |143 | 3171 | 1127 | 531 | 64 |137 |175 |149 | 3572 | 1270 | 532 42 | 62 |132 |187 |143 | 3530 | 1959 | 552 135 | 66 |196 |180 |140 | 3175 | 1200 |521 | 3438 | 1290 | 539 158 | 3903 | 1334 |554 93 | 69 |142 |176 |156 | 3899 | 1380 |555 ww hae lop) (on) i ww pa es CO — a) a or 38 | 65°5 | 129 185 142°5 | 3401 1140 | 542 Burns of neck and chest | 23 | 67°5 | 128 181 14] 3267 | 1243 | 540 | Septiceemia By) lh a 133 184 141 3451 1340 | 532 Pneumonia. Bronchitis 34 | 58 126 179 137 | 3090 | 1168 | 505 | Carcinoma mammee. Skull thin on | 60) | 32 186 139 S403) Ti lB22 538 Carcinoma mamme. Skull and scalp thin 39 «64. TAS |) heey | EP | BRR 1249 | 521 | Broncho-pneumonia 34, 63 1345 |190 |144 | 3680 | 1321 | 552 Pneumonia. Skull very thick 41 | 66°5 | 127°5 | 183°5 | 147) | 3439 | 1192 | 533 Morbus cordis. Skull very thick, scalp thin | 43°) 66°5 | 136 187 151°5 | 3853 1373 | 565 Be} | g : 3156 1170 | 530 28 | 64 129 179 =| 142 3279 | 1265 | 528 33 67 132 186 |} 151 3707 1235 | 560 34. 66 138 191 152 4006 1302 | 546 388 65'S | 131 177 141 3269 | 1241 Carcinoma mamme. Skull and scalp thick Carcinoma of stomach. Skull thick behind Intestinal ulceration probably syphilitic Sarcoma of frontal bone Pulmonary embolism. Fibromyoma of uterus Carcinoma uteri. Scalp thin. Foreign TJIwsMOT1THOOsaTTINANN TN WOOO AHWdONN No ST OSHORDHONARDOT~1ODEP ON aOAOCHOPOD ASN KHEKSHATGATABCADOSHARUHOKHHMOOADN REGINALD J. GLADSTONE VAL C. Female Subjects from 20 to 46 Years of Age. Middlesex Hospital. (Continued.) 82 Bites (Millimetres) (cm. | ST | wom. | Cephati D 5 nches) | m,’ Brain m. ephalic Cause of Death, etc. No. Be Age Stature | &xLxB weight Cire. | Index 85 | L B 45 68 125 178 138 3071 1078 | 514 | 77% Carcinoma uteri. Thickscalp, Thick bushy hair 5 162 25 67 _ 136 181 153°5 | 3779 | 1520 | 550 |. 84:8 | Unresolved lobar pneum, Cerebral congestion 163 39 62 1133 | 184 145 3548 | 1460 | 535 | 78:8 | Carcinoma uteri. Thick hair. Thin scalp 164 36 | 61 | 131 | 179° | 140 3292 | 1075 | 519 | 78:0 | Carcinoma uteri. Skull thick, except at sides. | 165 36 | 62°5 | 130 | 183 | 147 3497 | 1280 | 535 | 80°3 | Carcinoma mammi [Foreign | 166 40 60 123%5 ) 74-5") 143 3082 1180 | 513 | 81°9 | Malignant dis. R. side of face. Emaciated 167 22 |58°5 |126 | 179 | 144 3248 | 1250 | 513 | 80-4 | Diabetic coma. Scalp rather thin 168 40 | 63°5 | 125 | 184 146 3358 | 1190 | 528 | 79°3 | Carcinoma uteri. Scalp thin 169 40 61°5 — 130°5 | 193 151 3803 | 1874 | 548 | 78:2 | Addison’s disease. Skull rather thick [thick 170 40 |65 | 128°5|185 | 150 3566 1806 | 5385 | 81:1 | Tubercular nephritis. Skull very thin. Scalp | 171 S063, (126 | 177 | 141 3145 1202 | 502 | 79°7 | Malignant disease of R. ear. Skull very thin 172 42 64°5 | 1285] 188 | 145 3503 | 1240 | 5389 | 771 | Carcinoma mamme. Skull thick 173 34 63°5 | 131 188 | 145 B571 1316 | 540 | 7771 | Pernicious anemia. Well nourished 174 42 | 62°5 | 180°5}189 | 151 3724 | 1280 | 543 | 79°9 | Carcinoma uteri, Skull thin ; 3°5 | 126 190 151 3615 | 1350 | 555 | 79°5 | Cirrhosis of liver. Ascites 176 39 62° | 128 |175 | 143 3203 1180 | 515 | 81-7 | Carcinoma uteri. Skull thin. Scalp thick | 177 24 64 132 | 186 | 147 3609 | 1210 | 560 | 79°0 | Pyonephrosis. Foreign Mm S RAAQEQNQONANagggggagageaaggana! aagalasc w eo} > Ww 21 | 66°5 | 128°5 | 191 | 148 3632 | 1163 | 541 °5 | Sarcoma of maxilla. Emaciated. Scalp thin 178 44 64°5 | 128°5 | 188°5 | 147 356L | 1127 | 541 | 78:0 | Softening and growth in brain | | 179 24 6475 | 136 | 190 | 154 3979 = 1324 | 552 | 81:1 — Bronchitis. Empyema. Dilated heart | 180 30 63°5 | 134 | 187 | 141 3533 1210 | 540 | 75-4 | Diabetic coma. Wasting. Skull thick 181 37 63°75| 137 | 187 | 144 3689 1290 | 530 | 77°0 | Carcinoma mamme. Wasted | 182 33 55 120. 179 | 147 | 8158 = 1100 | 490 82°1_ ~~ Carcinoma uteri. Emaciated. Alcoholic | 133 38 | 67 132 |197 | 154 4005 | 1280 | 578 78:2 | Calcareous tumour of cerebellum 184 26 | 64:5 | 128 |175 | 142 B18L | 1175 | 545° 81-1 | Chorea. Well nourished 185 25 61°56 | 129 | 186 | 145 3479 | 1160 542) 78:0 | Tuberculosis of lungs and intestine 186 35 63 129°5 | 190 | 148 3642 | 1205 | 547 | 77°9 | Pulmonary tuberculosis. Emaciated | 187 77 | D. Female Subjects from 46 Years of Age and Upwards. Middlesex Hospital. | 52 (Millimetres) | Grms. ; No. | $5 | Age Gnehey ieee), | Heats | Mi | Celie eS | a6 H L B weight . 3 | 188 | C | 46 | 66 116 7-185 |: 143 3069 = 1022 } 77°3. | Carcinoma uteri 189 | C | 50 | 60 132 | 187 | 187°5 | 83894 | 1248 | 528 | 73°5 | Carcinoma of breast, and pleura 190 | C | 46 | 66 139 | 185 | 144 3703-1350 | 5382.) 77°8 | Malignant growths in peritoneum 191 | C | 50 | 63 124 | 181 141 3165 1287 | 522.) 77°9 | Epithelioma of scalp and face 192 | C | 46 | 67% |130 | 183 | 141 3354 1204 | 522 | 77 Carcinoma of urethra 193 | C | 49 | 60°5 | 123 | 173 | 141 3000 1090 | 514.) 81°5 | Carcinoma uteri. Scalp and hair thin 194 | C | 49 | 655 |132 | 185 | 151 3687 =—-:1355 | 545 | 81°6 | Cystic kidneys. Ureemia. Skull thin | 195 | C | 48 | 64 129 | 185 | 149 3556 1250 | 530 | 80°56 | Carcinoma uteri. Skull thin | 196 | € | 47 | 66°5 | 120°5| 177 | 130 2773-1076 | 516 | 734 | Carcinoma mamme. Skull very thick | 197 | C | 53 | 64 130 | 168 | 140 3058-1120 | 515 | 83°3 | Carcinoma uteri 198 | C | 56 | 60 131 187 | 13865 | 3344 = 1220 | 5381 | 73°0 | Carcinoma mammie 199 | — | 64 | 61°5 | 132 | 182°5 | 145 3493 1240 | 540 | 79°7 | Diabetes. Gangrene of foot 200 | C | 69 | 62°5 |125°5)185 | 142 3297 | 1220 | 525 | 76°8 Morbus cordis 201 | C | 60 | 61 130 | 182 | 142 3360 1095 | 515 | 78:0 | Carcinoma uteri, Skull thick 202 | C | 55 | 61°5 |129 |180 | 1389 3228 | 1235 | 518 | 77:2 | Chronic bronchitis. Skull thin 208 | C | 55 | 59 129 174 146 3277 | 1105 | 506 | 83-9 Bronchitis. Alcoholism. Skull soft and thick | 204 | — | 53 | 61 133 | 188 | 154 3851 1405 | 577 | 81:9 | Cerebral hemorrhage. Thick skull and scalp | | 205 | C | 60 | 60°5 |126 | 179 136 3067 1150 | 518 | 76:0 | Multiple sarcomata. Very thin skull & scalp | | | | Biometrika 1v 16 122 Brain-weight and Head-size D. Female Subjects from 46 Years of Age and Upwards. Middlesex Hospital. (Continued.) | Sg ae (Millimetres) (cms. | Cn iat lGepnete No. | SE | Ase | Stature | | HXLXB | werent Cin || eandex Cause of Death, etc. 2d EL L B : 206 | C | 54 | 60°5 | 129°5 | 181 57°5 | 3692 | 1305 | 548 | 87:0 | Carcinoma. Metastases in L. hip. Thin skull 207 | C | 60 | 64:5 | 182 1179 | 144 3402 | 1220 | 523 | 80°4 | Chronic interstitial nephritis. Scalp thin 208 | — | 62 | 62°5 | 1386 | 192 | 153 3995 | 1296 | 561 | 79°7 | Chr. nephritis. Fatty degen. of Heart. Very | thick skull 209 | C | 53 | 61 128 | 180 | 144 3318 | 1175 | 523 80°0 | Carcinoma mamme. Skull thick. Scalp thin 210 | C | 68 | 61 120 | 173 | 181 2720 | 955 | 502) 75°7 | Carcinoma uteri. Scalp thin 211 | C | 63) 62°5 | 1265} 172 1185 | 2937 | 1070 | 520 | 78:5 | Sarcoma of nose 212 | — | 51 66 | 134 | 183 | 146 3580 | 1320 | 532) 79°8 | Appendicitis. Hair and skull thin hss |) OL NW vksh |) ctayshetoy || PAL 176 138 2939 | 1060 | 509 | 78°4 | Carcinoma recti. Skull very thin 214 | C | 57 | 61 123 |180 | 135 2989 | 1130 | 525 | 75:0 | Carcinoma mammez. Emaciated 215 | — | 68 | 64 128 |188 | 149 3586 1250 | 540 | 79°3 | Carcinoma of pancreas. Well nourished 216 | C | 58 | 605 | 124 | 178 | 1438 3156 | 1225 | 515 | 80°3 | Carcinoma mamme. Cerebral abscess 217 | C | 59 | 65 127 180 142 3246 | 1180 | 524) 78°9 | Carcinoma mamme. Skull thin 218 | © | 49 | 57 126 | 181 139 3170 | 1178 | 512 | 76:8 | Carcinoma uteri. Fairly nourished 219 | C | 56/ 60 |122 |186 | 144 3268 | 1142 | 541 | 77°4 | Carcinoma. Sigmoid flexure of colon 220 | C | 58 | 59 18h} 182 | 140 3389 1130 | 520 | 76°9 | Carcinoma uteri 221 | C | 62 | 62°5 | 1285] 184 | 148 3381 | 1185 | 528 | 77°7 | Carcinoma uteri. Wasted 222 | C | 67 | 63 124 175 | 182 2864 | 1012 | 509 | 75:4 | Carcinoma uteri. Well nourished 223 | — | 47) 64 133 185 52 3740 | 1280 | 547 | 82:2 224 | C | 64 | 59 131°5) 180 | 147 3479 | 1103 | 5382 | 81°7 | Carcinoma mamme 225 | C | 47 | 65 131 | 192 | 145 3647 | 1408 | 565 | 75°5 | Carcinoma mamme. Skull rather thick 226 | A | 74} 68 |128 | 191 | 152 3716 | 1300 | 564 | 79°6 | Strang. femoral hernia. Well nourished 227 | C | 67 | 62 125 | 185 | 142 3284 | 1246 | 528 | 76°8 | Cerebral hem. Interstitial nephritis. Emac. 228; A |} 50| — |142 | 191 | 155 4204 | 1380 | 559 | 81:2 | Fall. Rupture ofaorta. Fracture of ribs. Fat 229 | C | 63] 638 |1380 {189 | 152 3735 | 1850 | 548 | 80°4 | Malignant dis. of intestines. Very fat. Skull 2380 | A | 74 | 62 121 | 186 | 148 3218 | 1060 | 542 | 76:9 | Fractured ribs. Skull and scalp thick [thin 281 | C | 49 | 65°5 | 137 | 185°5 | 145 3685 | 1350 | 5385 | 78:2 | Carcinomaofomentum. Very fat. Skull thin 2382 | C | 46 | 67 135 | 196 | 140 3704 | 1220 | 521, 714 | Carcinoma uteri. Emaciated. Skull thick 233 | C | 46 | 62°5 | 126°5]174 | 146 3214 | 1110 | 532) 83:9 | Pulmonary tuberculosis. Skull thick & spongy 234 | C | 50 | 64:5 | 125 |181 | 150 3394 | 1215 | 542 82°9 Carcinoma uteri. Emaciated. Scalp thin 235 | C | 63 | 58:5 |127 |178 | 148 3233-1104 | 522.) 80°3 | Carcinoma mamme. Emaciated. Skull thin 236 | OC | 56 64:5 126°5/184 | 144 3352 | 1170 | 533) 78°3 | Carcinoma of stomach. Slight wasting 2387 | C | 63 | 62 128 |179 | 148 3391 | 1120 | 587 | 82:7 | Carcinoma uteri. Emaciated. Scalp thin E. Children &. HO “11: No. Py Age (Inches) RS (Cms.*) Bae Mm. | Cephalic Cause of Death, etc. Ba Stature H L B AxLxB | weight | Cire. | Index ay 288 | C | 25 ms. | 22 89°5| 125 95 1063 450 361 | 76:0 | Broncho-pneumonia 239 | A 9ms. | 28 114 155 | 117 2067 — — | 75°5 | Peritonitis 240 | — | 1ly.-8m. | 37°5 | 119 168 | 125 2499 | 1052 | 481 | 74:4 | Tubercular meningitis 241 | — |2y.-6m. | 31°75 | 111 160 | 131 2327 | 1015 474 | 81:9 | Acute general tuberculosis 242 | — |\2y.-6m.| 33: | 121 172 | 135°5 | 2820 | 1120 | 486 | 78°8 | Broncho-pneumonia. Foreign 243 | — | 2y.-8m. | 38 116 162 | 139 2612 | 1240 | 483 | 85°8 244 | A 3 37 124°5| 181 | 1387 3087 | 1310 | 514 | 75°7 | Meningitis 245 | C | 8y.-6m. | 355 | 121 170 | 129°5| 2664 | 1095 | 476 | 76:2 | Septic broncho-pneumonia. Emaciated 246 | CO |By.-10 m.| 34 127 174 | 135 2983 | 1080 | 501 | 77°6 | Otitis media. Emaciated 247 | — 4 39 127 173 | 134 2944 | 1202 | 501 | 77°5 | —. Skull very thin 248 | — 5 87°5 | 125 170 | 132 2805 | 1200 | 490 | 77°7 | Death by suffocation. Foreign 249 | C 7 46 127 175 | 140 3112 | 1375 | 504 | 80°0 | Tubercular cervical glands 250 | — 8 43 131°5] 192 | 140 3535 | 1450 | 580 72°9 | Pericarditis. Head large and long REGINALD J. GLADSTONE 123 E. Children §. (Continued.) (Millimetres) | 62 | Grms. 22 Age eee ee Brain pin. Cause of Death, ete. qo L B == 10 52 137°5 | 187 149°5 | 3844 | 1417 | 5389 | 79°9 | Pericarditis. Foreign C 11 55 181 145 3412 | 1412 | 532 | 80°71 | Morbus cordis [spleen A 12 65 183 | 141 3509 | 1379 | 540 | 7771 | “Run over.” Fracture of ribs. Rupture of C 12 48°5 | 1225) 183 | 142 3183) 13822 | 528 | 77°6 | Broncho-pneumonia. Pulm. tuberculosis — 12-13 56°5 184 143 3526 | 1454 | 580 | 77:7 C 15 62°5 19093 | 151 | 4056 | 1618 | 547 | 79°2 | Diabetes. Coma C 17 67 (189 | 1505 | 3669 | 1095 | 536 | 79°6 | Morbus cordis. Wasted. Foreign C 18 71 172 «|: 142 3200 | 1280 | 520 | 82°6 | Tubercular meningitis. Foreign C 19 67 187 | 146 3904 | 1460 | 556 | 78:1 | Psoas abscess. Tubercular meningitis A; 19 67°5 | 191 157 4048 | 1278 | 558 | 82:2 | Bullet wound in head A. 19 65°5 | 188°5 | 147 3698 | 1412 | 540 | 78:2 | Pneumonia —/; 19 69°5 | 189 | 149 3886 | 1450 | 547 | 78:8 | Cerebral abscess. Meningitis F. Children $. Reo sy7s he: | 38 Age Inches) anaes | (Cms.3) | arain Mm. | Cause of Death, etc. Bs tature L acd | HxLxB > weight | Cire. 26 | — | 3 wks. 21°5 5 | 114 88 933 385 | 324 | 77°6 | Bronchitis. “ Brought in dead ” C | 2°5 ms. | 22 5|129°5| 99 1212 490 | 369 | 76°4 | Gastro-enteritis. Foreign C | 5 ms. 22°5 5 | 129 95 1085 512 | 368 | 73°6 | Convulsions C | 6ms. 24°5 132°5 | 104°5 | 1385 503 | 885 | 78°8 Malgnant growth of orbit (? sarcoma) C | 8ms. | 25 144 | 126 2141 960 | 437 | 87:5 | Rickets. Gastro-enteriti. Foreign C | 2y.-6ms. | 33°5 172 | 132 2815 995 | 496 | 76:7 | Tuberculosis. Emaciated A |3y.-6ms.) 38°5 164 | 126 2356 | 1100 | 475 | 76:8 | Fracture of base of skull 24 7 47 169 | 131 2679 | 1120 | 493 | 77°5 | Pulmonary tuberculosis. Meningitis C 7 48 174 | 1338 2731 | 1205 | 485 | 76°4 C 15 61°5 185 | 139 3292 1205 | 518 | 75-1 | Otitis media. Thrombosis of R. lateral sinus C 17 61 5/178 | 143°5 | 3206 | 1180 | 524 | 80° | Tubercular meningitis. Skull thick. Emac. C 18 53°5 179 =| 135 3021 | 1147 | 507 | 75:4 Mitral regurgitation. Gangrene of hands C 18 66 186 | 151 3904 | 1419 | 554 | 81:2 | Chronic otitis media. Cerebellar abscess — 19 65 179 141 3306 | 1245 | 512 | 78°8 16 —2 A STUDY OF THE BIOMETRIC CONSTANTS OF ENGLISH BRAIN-WEIGHTS, AND THEIR RELATIONSHIPS TO EXTERNAL PHYSICAL MEASUREMENTS. By J. BLAKEMAN, B.A., M.Sc., assisted by ALICE LEE, D.Sc., and KARL PEARSON, F.R.S. (1) Introductory. THE purpose of this paper is to present a biometric analysis of the measure- ments provided by Dr R. J. Gladstone and published in this volume. The conclusions reached are therefore of the same order of validity as the data upon which they are based. An attempt has been made to compare them with the fuller material reduced by Dr Raymond Pearl, and in many points where comparison was possible general confirmation of his conclusions has been obtained. Gladstone’s statistical material differs from that used by Pearl in two essential points. It is in the first place more meagre, but in the second place it provides additional measuremeuts which enable us to predict with a moderate degree of accuracy brain-weight from external measurements on the living subject. The characters recorded by Gladstone were the age (A), stature (S), cause of death, the maximum head-length (L), the maximum head-breadth (5), the auricular height (/), the horizontal circumference (U), and the brain-weight (w). The sagittal arc, nasion to inion, and the biauricular transverse arc (Q of our notation) were also measured, but are not considered, as the difficulty of determining the inion, and the impediment formed by the hair, especially in the case of the female subjects, was found to be so great as to make the measurements unreliable. The product P=Z~x Bx H was formed, and the present study is concerned with the biometric relations of w, A, S, L, B, H, P, and U. The distribution of the work has been as follows: The whole of the labour of obtaining the multiple regression formula, p. 146, for the males fell to J. Blakeman, the similar formula for the females to A. Lee. All the other statistical and calculating work throughout the paper as well as the preparation of Figures 8—6 is further due to J. Blakeman. K. Pearson is responsible only for suggesting the general lines the paper should take, and preparing the draft of its final form. In dealing with Gladstone’s material a reduction was first made in its amount by (a) the exclusion of a small number of foreigners ; (b) the exclusion of those individuals for whom all the seven characters referred to above were not available ; J. BLAKEMAN 125 (c) the exclusion of all males under 24 years of age and all females under 20 years of age. The reason for adopting this restriction was that from considerable experience of anthropometric measurements we have found that these ages roughly mark in man and woman a turning-point in the nature of the growth curves. Up to these ages there is a rather rapid increase, after these ages in most characters a slight but continuous decrease. With the above exclusions we had 117 male and 94 female cases. These numbers are small, but the material is fairly homogeneous and the probable errors have in each case been given. The publication of the actual data (pp. 118—123) has been undertaken largely with the hope that other medical schools may collect similar material so that ultimately proper numbers will be available, large enough for due allowance to be made for the cause of death and the condition at death. No such separate classifications are possible at present. (2) On the Special Character of the Present Material. The present material is what has been described in this journal* as a “ General Hospital Population,’—with perhaps in this case a rather large proportion of cancer cases. It cannot be too often insisted upon that such a population is not a fair sample of the “general population” of a given district. There is a larger amount of what it would be convenient to call “shrinkage” due to illness and defective nourishment. The well-nourished, physically well-developed middle classes are largely absent, and taking the brain-weights of these hospital returns as typical of the “general population ”—especially applying them to problems in the relationship of intelligence to physical measurements—is lable to lead to very erroneous conclusions. An illustration of this may be given from Gladstone’s classification into acute and chronic cases—the A and C of the second column of the tables. We find for average brain-weights: TABLE I. Ages | Cause of Death | No. Male No. | Female | 20 to 46... | Acute | 16 1430 5 1303 Chronic | 27 1331 32 1227 | Over 46 ... Acute 22 1318 3 1248 | | Chronic 34 1354 40 1178 It will thus be seen that, with the exception of the males over 46, the cause of death makes a very substantial difference in the average brain-weight. The numbers are however far too few to allow us to make any separate classification, but the point is a vital one and must undoubtedly be properly dealt with when greater numbers are forthcoming. The reader will find as he proceeds further that we have other grounds for suspecting that the general hospital population differs in some essential features from the general population. * Greenwood: Biometrika, Vol. 11. p. 65. 126 Biometric Constants of English Brain-weights (3) On the Means and Variabilities of the Anthropometric Characters of the Present Material. Table II. contains the means, standard deviations, and coefficients of variation of all the characters dealt with. We notice that the woman is absolutely more variable than man in four out of the nine characters considered, and relatively more variable in eight out of the nine. In the ninth, stature, the difference in variability between the two sexes is well within the probable error of the difference. This is a further illustration of the point that man is not essentially move variable than woman. We can now make some comparisons between this general hospital population and other English groups. In the first place let us take the head-measurements : TABLE II. Means and Variabilities. Mean Standard Deviation Coefficient of Variation Character Les 3 ? 3 "3 3 2 Head Length ... 190°36 183°20 5°62 5°59 2°95 3°05 (Z) in mm. ae + °35 +°39 2 D0 +28 +13 +:15 Head Breadth ... 149°34 144°55 5°41 5:76 3°6 3°99 (B) in mm. ae +°34 +40 +24 + °28 +1 +°20 Auricular Height 132°90 129°21 4°51 4°35 3°39 3°37 (#) in mm. + °28 + °30 + °20 +°21 +#°15 anally Cephalic Index ... 78°48 78°93 2°76 2°87 3°52 3°64 (100 B/L) re Spoil) + °20 ap by +:'14 +:16 +:18 Horizontal Cir- cumference ... 555°79 533°25 15°84 16°83 2°85 3°16 (VU) in mm. ee +°99 +1:17 +:°70 +°83 +:'13 +°16 Product ... | 3782°91 | 3427°03 | 290°47 282 °94 7°68 8:26 (P=Lx Bx #) in cm.3 cen | eS +19°68 | +12°81 +13:°92 + °34 +°41 Stature wae 67°16 63°05 3°05 2°79 4°55 4:42 GS) in inches... +19 +:19 + 13 +:14 + °20 +29 Age Aas 49°26 45°90 12°11 13°69 24°60 29°83 (A) in years ass +°76 +°95 +°53 + °67 +1°15 +1°59 Brain-Weight ... | 1827°69 | 1224:90 | 104:03 100°16 7°84 8:18 (w) In grs. nie +6°49 +6°70 +4:°59 + 4°93 +35 +°40 N.B. In working out all further biometric constants, regression formulae, etc., not the above values but the values as actually calculated to four decimal places were used. J. BLAKEMAN 127 TABLE III. Results for Cephalic Indew. | Group Male Female | General Hospital Population ... 78°48 78°93 3000 Criminals* Sie 78°54 —- 1000 Cambridge Graduatest ... 79°56 _ | 2000 School Children ... 78°92 78°29) | 100 to 200 British Association § 77°75 | 78°75 We see that as far as the cephalic index is concerned there is nothing to markedly differentiate our material from other groups of the general population that have been dealt with by other observers. Turning next to the diameters we have TABLE IV. Male Female Group ] | L B H L B | | : | General Hospital Population .... | 190°4 | 149°3 | 132-9 | 183-2 | 144-6 | 129-2 3000 Criminals* .. | 191°7 | 150°4 | — I. < 1000 Cambridge Graduatest ... | 193°5 | 154-0 British Association ||... se 198°1 | 155-0 | 130°9 | 185°6 | 147°3 128-4 Congress of Anatomists|| .» | 198-4 || 157-2 | 133-1 | — — — University College Staff|| .. | 196°4 | 153°5 | 134°8 Bedford College Students ||... — — — 189°7 | 146°8 | 132°7 These results show very clearly that the general hospital population has a smaller head than the middle class English population. The only exceptions are the male and female auricular heights for the British Association members. These were probably not measured with an ear-plug craniometer. The nearest approach is the criminal population. Before commenting on this let us consider the stature returns. We have * Macdonell: Biometrika, Vol. 1. p. 177 et seq. + Pearson: The Chances of Death, Vol. 1. p. 351, + Pearson: Biometrika, Vol. ur. p. 140. § Pearson: The Chances of Death, Vol. 1. p. 351. || Lee and Pearson: Phil. Trans. Vol. 196 A, p. 251 et seq. 128 Biometric Constants of English Brain-weights TABLE V. Group Male Stature | Female Stature | | yeneral Hospital Population a aes 672 63'"1 Galton’s South Kensington Returns* 67'"°9 633 / 3000 Criminals + ie ae eae aes 655 | — | Australian Criminals } ies +63 se 669 | 62"°2 Cambridge Students § e | 689 638 Pearson’s Family Measurements \|, ‘Offspring 68'°7 63'°9 | 3 3 ‘ Parents | Cigar | 625 Thus we see that the general hospital population is shorter than the average of the upper middle classes, approaching closer to the lower middle classes, who would chiefly supply the South Kensington Exhibition returns. It is taller than the criminal classes either in England or New South Wales, and approaches closely to the average of Pearson’s “ Parent” Group, who must all have been over 40 years of age. It would accordingly seem that our general hospital population belongs anthropometrically to a class intermediate between the classes from which criminals are drawn and the lower middle class. Its reduced head-measurements are probably due in part to this fact, in part to chronic illness, and in part to a concomitant of class differentiation, namely want of good nourishment and exercise in the period of growth. We might reasonably attribute a great deal to this want of nurture, and it seems as hopeless to settle the problem of correlation between size of head and intelligence by measuring different social classes, as to settle the correlation between stature and intelligence by the same process. If we turn to age at death we shall get much confirmation of the view that we are dealing with a badly nourished, physically degenerate class. We have TABLE VI. Mean Age at Death in Years. | Group | Male | Female | | General Hospital sore 49°3 45-9 : | Society of Friends %.. | 60°9 | 60:8 Peerage and Landed Gentry 4 62:9 | — * Galton’s Natural Inheritance, pp. 200—1; Pearson: The Chances of Death, Vol. 1. p. 311, + Macdonell: Biometrika, Vol. 1. p. 191. + Powys: Biometrika, Vol. 1. p. 44. § Biometrika, Vol. 1. p. 191. || Pearson and Lee: Biometrika, Vol. 1. p. 370. 4 Pearson and Beeton: Biometrika, Vol. 1. p. 64. J. BLAKEMAN 129 Here in the latter two groups adults, 21 years and over, have been taken, while in the hospital group among the males 24 was the minimum admitted age. Clearly we have to deal, even allowing for a larger percentage of accidents and an absence of some of the diseases of old age, for a population of lesser constitutional vitality as measured by expectation of life. It is to be noted that the female is shorter- lived than the male*. For the horizontal circumference we have no reliable data for comparison. Dr Beddoe gives itt for 30 groups, containing various numbers from two to a hundred, and also for a hundred individuals, but in not one single case of the groups or of the individuals has he obtained a result as low as our mean of 558 mm. On the other hand it is almost exactly the value 560 Lewenz and Pearson} have found for Jeremy Bentham’s head, which in other respects is close to the English mean type. There appears to be little doubt accordingly that Dr Beddoe has an individual method of measuring this circumference. Lastly, turning to the brain-weights, we have, pooling the results of Reid and Peacock, Clendenning, and Sims, the following mean values. TABLE VII. English Brain-weights in grms. 7 = 7 | tne || Group | 3 je) | | | Gladstone’s Middlesex Hospital Data =... | 1328 1224 | Reid and Peacock, Clendenning, Sims$ ... | 1335 1235 | Thus the average brain-weight of an English general hospital population appears to be fairly constant when we compare Gladstone’s data with the means of other material. But this is of course no evidence that the general hospital population is at all representative of the general population. On the contrary, the fact that for the head diameters, for stature, and for constitutional robustness, as measured by duration of life, the hospital population falls below the general population, would lead us to believe that it probably does so in brain-weight ; and we should attribute the probable source of this differentia- tion to want of nourishment and “shrinkage” due to chronic disease. (4) Variation Constants. Turning first to the head measurements we may compare our results with those for other English groups. This is done in Table VIIL. * On this point see “ Assortative Mating in Man,” Biometrika, Vol. 1. p. 488. + L’Anthropologie, Vol. x1v. pp. 284, 291—4. + Biometrika, Vol. m1. p. 394, § Pearson: The Chances of Death, Vol. 1. p. 321. Of course the pooling of the results of these (lifferent observers is open to considerable criticism. Biometrika Iv 17 130 Biometric Constants of English Brain-weights TABLE VIII. Variability of Head Measurements. aaa « Length Breadth Auricular Height Group S. D. C. of V. | S. D. C. of V. 8. D. C. of V. | — = = | | ec mec emiecm i |) 6 | 2 =e | | | General Hospital Population | 5°62 | 5°59 | 2°95 | 3°05) 5-41 | 5°76 | 3°62 | 3°99 | 4:51 | 4°35 | 3-39 | 3°37 English Criminals* wes 6205 | — 8e6) |=) | 5-01 | ==. N3:338 : Cambridge Graduates + 616] — |318} — | 5:06) — | 3-28 | | | : — Ae Sj | = English Skulls 6:27 | 6°22 | 3°31 | 3-45 | 5°28 | 4°77 | 3°75 | 3°54 | 4°28 | 4°50 | 3°73 | 4°12 Very little of a definite kind can be deduced from these rather sparse results. In length the general hospital population seems less variable than either the English criminals or Cambridge graduates, but it is more variable in breadth. we compare the variability of the diameters of the living head with those of the skull, we see that the head is clearly not definitely more variable than the skull, and accordingly the assumption often made that owing to the presence of the flesh it must be so, is erroneous. If The paradox is explicable on the basis of a negative correlation between skull diameters and amount of covering fleshf, i.e., individuals growing much bone must on the average have less flesh. Taking next cephalic index we have for English groups: TABLE VIII bis. Cephalic Index * Macdonell: Biometrika, Vol. 1. p. 202. Group Standard Deviation | Coefficient of Variation 3 ? 3 ? General Hospital Population 2°76 2°87 3°52 3°64 English Criminals § 2°74 — 355 | — Cambridge Graduates|| 2:90 — 3°70 — School Children 7 3°31 3°99 4:20 5:09 British Association ** 2°52 2°96 3°24 3°76 English Skulls t+ ... 3°26 2°98 4°38 3°99 + Faweett: R. S. Proc. t Let o, be s.v. of skull length, o,, of head length, oc, of flesh thickness; then : OL = {oy +07? + 2oyaerp}h or o, must be greater than c, unless 77, the correlation between amount of flesh and bone size, be negative. § Macdonell: Biometrika, Vol. 1. p. 191. 4] Pearson: Biometrika, Vol. 11. p. 140. || Idem, p. 191. ++ Macdonell: Biometrika, Vol. ut. p. 222. ** Pearson: The Chances of Death, Vol. 1. p. 351. J. BLAKEMAN 131 We see at once that a general hospital population is about equally variable with other English groups, whether of lower grade, like the criminal, or of higher grade, like the middle class, chiefly drawn upon for Cambridge graduates or B.A. members. School children are more variable—probably a phenomenon of growth. But again we see that our skull series is more variable than the head series. For the circumference our comparable data are nil; we can only compare with the English skull. This provides: TABLE IX. Coefficients of Variation for English Horizontal Circumference. | | Group 3 fo) | : General Hospital Popuanice | 2°85 3°16 English Skull* ... was ah 2°87 2°92 | In the matter of the circumference we should therefore conclude that there is equal variability for head and skull measurements. Stature provides us with more material. We find: TABLE X, Variation in Stature in Inches. Standard Deviation | Coefficient of Variation Group ves 9 g 9 | | General Hospital Population .. pee ie 305 2°79 4:55 4:42 Galton’s South oe Recordst ... | 255 | 2:40 3°95 379 | English Criminals? ... 3 ey 2°59 _ 388 | — Cambridge Students t eae |} 251 2°42 364 | 3°79 Pearson’s s Family Measurements, Offspring § 2°71 2°61 3°95 4:09 3 5 - Parents § 270 | 2:89 3:99 3°83 It will be seen that both absolutely and relatively the general hospital population is, in the case of stature, more variable than any of the other English groups. We believe this increased variability in stature to be due to the measure- ments being taken on the corpse, not on the living subject. As a rule we may say that the coefficient of variation of the living stature of the adults of any race is under 4. Pearl gives twelve cases of this coefficient for corpse measurements in his Table I, p. 23, and in ten of these cases the coefficient of variation is over 4. * Macdonell: Biometrika, Vol. 11. p. 222. + Pearson : The Chances of Death, Vol. 1. pp. 313 and 3811. + Macdonell: Biometrika, Vol. 1. p. 181. § Pearson and Lee: Biometrika, Vol. 1. p. 370. 17—2 132 Biometric Constants of English Brain-weights In fact the mean values are ¢‘s 411, 2s 439; and we may, we hold, safely conclude that the increased variability does not make a class difference, but is peculiar to measurements made under some difficulties*. Lastly, we may turn to the variation in age at death: TABLE XI. Variation in Duration of Life. | Standard Deviation Coefficient of Variation Group iz = 3 " | =é 3 —— _ — | — General Hospital Population ... | 1211 | = 13°69 24°59 29°83 | Society of Friends + os nee 1713 | 1873 | 28:13 30°80 | Peerage and Landed Gentry +t .... | 16°84 — | 27°61 — Yorkshire Tombstones } re 12°97 16-40 | 18:97 24°53 Oxfordshire Tombstones ¢ en |) —beerto) 14°67 18°55 21°68 London Tombstones? ... Soe eee eile 14-06 19°56 21°22 Thus the general hospital population, while more variable in duration of life than the groups taken from tombstone records, which were all of married adults, is less variable than the pedigree records of either the Society of Friends or the Peerage and Landed Gentry, which were simply for persons over 21. We think, therefore, that the general hospital population, while sensibly differentiated from the other classes in actual duration of life, is not sensibly differentiated in the variability of that duration. We may sum up then the result of the last two sections as follows : A general hospital population is sensibly differentiated in its average anthropometric characters from other groups of the same population. But there appears to be no markedly significant difference in variation. It would seem, therefore, that observations made on a hospital population may with due precautions be used for problems in variation, but they should not be used for drawing inferences as to the type of the surrounding general population without extreme care. (5) On the Relative Magnitude and Variability of English Brain-weights compared with those of other Races. Professor Raymond Pearl’s reductions enable us to form a fairly extensive table of comparative brain-weights. We have added a few results for stature and skull capacity. In the bulk of cases these are dependent upon few individuals, and are far from reliable, so that they must be treated as merely suggestions until further data are available. * See on difference between stature and corpse length Pearson: Phil. Trans. Vol. 192 A, p. 191. + Pearson and Beeton: Biometrika, Vol. 1. pp. 60 and 64. t Biometrika, Vol. 11. p. 488. 133 OST “AU ‘SV ZG6L “TOA “SUD “2% _ ‘WOsavag puv vary Aq paonpar vywp S.yoT[OIY, | ‘urtuayeltg Aq poonpoat puev poxooT[oo wnio1up49 snunvsayT, START PABUIV UL VIURIO YOUSIA MI VW ‘eee ‘d "720 ‘007 ‘uosavag AQ paonped anSopeyeg Yorunyy Ul viuvso Touatq $$§ "Eze “d ‘220 ‘0072 ‘uosavag Aq paonpat sruvg ap arbopodowyUP .p 239100F JO BIBT un~ ‘usuteye[g Aq poonpart vyep s,auoyspryyy || || GGG pure gos ‘dd ‘mr ‘JoA ‘py2ajzaworg ‘uy Aq paonpatr [[ouopory Jo vyep jedeyooqty Ay S$ "Tag ‘d +729 -907 ‘uosaveg Aq poonpert puv peulIqMod suIIg puv SUIUUApPUETD ‘yoooveg ‘prey jo eByeq 44 ‘uBmayetg ‘¢ Aq paonpar sansoreyeo [vorso[odoryqyuy UBUIIELD) BY} WO] U9YVY VIULIO PoYBOST MIJ VW xx ‘egg “d +729 +00] ‘uosaveg Aq poonpor vyep sayurry |, ‘EGG pur cez ‘dd ‘1 ‘oq ‘ygnaqr fo saouvyD ayy, :Uosreag Aq paonpet vyep s.yoyostg § "eZ “tal “JOA ‘vYyRYQaWOIg Bas SUOIJONpaL SJIvag Og ‘jAvag Aq poonpart vyep s,puvyorryy || J. BLAKEMAN ‘[twag Aq poonpar eyep snizjyoy “9 + ‘Tavag Aq poonpar Bep s.eysoneyy ,. il cae ie = ; ) = im | in | j = 7 ar j | y — se - | eRe. = — | ¥9L | bos || 824 | TOL 9661 1-6 COL PrLT ee: “ 006- aa a 991 | ibbOe |] €4L | LOL LPL Z.6 IéL SEL 2S 1 ever aa | | | | 38 O01 GBI Aaa : 668- | PP I-4 | ogt Eee) a Men ooet | §$og BL FOL 8zEL a See C06: | 9-F ue TAL €8 BEL LLPL | §83L]|| 2-6 611 CEL 7 6 +06- i z-6 FEL CEEL “PT Ustsug €16- EF 9-9 PEL § 99% Z-8 601 9GE1 ,00L F-8 €OL OZEI Sure |S ¥ 906- 0-F L9 | 99 § 06€ Lek OIL FOGL ,OOT P-8 FIL Z9ET Segg |" P suvrreang alo : een Py iy z | aera (ame | : alae = ¥-F 6-9 LOT || 18% = = = = ji 18 Z0L O9I llTs@ | 7" fo) = €-F BL 191 lon = = = ae |: €11 B6ET lqnp | 7" P uvissoH — oa _ = = <= | — — eI —s ar &F L9 | 6a 469 = aa ee — || 08 LOL ECZL $83 | °° Oreos a= | 0-7 8-9 OLT 491F = = = 9-2 901 OOFT +91P | 77 P YSIpang — GF ZL 8oL EET = ae = ri PL 16 11 *L6L |" 3 (i) F00- T EF Cafe) SGOT #996 Se Wile 02 SFFL | xx(i)OT 8-L PIT CCFL xELE | 7° P suerueyog Sypovdeo-yays 0, | “AJO'O) “A'S Brey ‘ON «|/"AJO"O | “A'S | Bee ‘on |i-agoo| ‘a's | avon ‘ON qSIOM-UlBIg < == = <7. = = = = dnoiy jo one ‘sumo UT a1njeqyg asd109 ‘smo ur Ayroedeg [yg ‘SI9 UT JUSIOM-UIRIG | ‘ja ‘sqyhiam-unug eaynundwon = "TTX ATAVL 134 Biometric Constants of English Brain-weights Now, although this table is very incomplete, and impresses upon one the need for the further collection of material, we can safely, we think, draw one or two conclusions : (i) The order of racial average brain-weights is very far from the order of average racial intelligence. Nor is the order bettered if we allow in any manner for stature. We think that even allowing for differences in racial stature, it is quite impossible to compare the brain-weights of men of different races, and any consideration of intelligence and brain-weight which proceeds by “ pooling” the brain-weights of distinguished men of a great variety of races is fundamentally fallacious *. (ii) The variability, as measured by the coefficient of variation, of brain-weight is for the three races for which we possess fairly reliable data greater than that of skull capacity. Roughly, the average brain-weight variability is 9, the average skull capacity 8. On the whole the female shows very slightly greater variability in both cases. (iii) The ratio of brain-weight to skull capacity has an average value of about ‘9, and this is almost exactly the value of the ratio of the variation in brain-weight to that in skull capacity. It would appear that the cranial chamber is more closely packed with brain in the case of the female than in that of the male, but the evidence, which is based upon heterogeneous material, is not convincing t. (iv) The standard deviation of brain-weight in grammes is fairly close to the standard deviation of capacity in cubic centimetres. (v) Gladstone’s results are, as far as the means are concerned, in close agreement with the results obtained by Pearson from mixing the data of a variety of observers. The increased variability, however, of the English Series I shows its real hetero- geneity, and not too much stress ought accordingly to be laid on the sensible identity of the mean values. The means of the component groups, indeed, vary widely. (6) On the Correlations of the various Characters measured. Table XIII gives the values of the correlation coefficients and their probable errors. They are tabled to four decimal places, not because they are worth more than one or two, but because these values must be used if the regression coefficients are to be tested. We propose to consider the separate entries in this table, and test them against such scant data as are available at present from other sources. As there are nine observed characters, each sex would provide 36 coefficients of correlation, but only 17 have been calculated for each case, partly because of the * Especially when the next stage is to compare this heterogeneous group with the fairly homo- geneous results for a local general hospital population ! + Data are being collected on more homogeneous material, and it is further hoped to test more completely than has hitherto been done the relative densities of different parts of the male and female brain. 135 J. 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LELE- — 6EF0-F, __ | 9990-F - = ae = a a I = ae = PLO9- 690% — gq _ |GLr0-F} __ | 8190-F) __ io Sas _ oe — = I = = CRSP. 0160: — 6870-+| _ | 6190. + = er 1 a OGPE. 9ECT- — si 7 _ | 10F0-F) «| 0@90-F | ae = & pots at _ _ = == I Og6¢- I¢L0- — é 2 ) io ) P. ) 2 ) P ) 2 ) 2 3} 2 ) | 2 m V sg Hx@*T=d a Tid OOLT=I H I a ; ‘suappinyg .ayjo pun qybram-umug fo suoynpe440og “FJIX ATAWL 136 Biometric Constants of English Brain-weights great labour involved (each coefficient being found without grouping), and partly because we realized at once that the product of the diameters and the horizontal circumference had the highest correlations with brain-weight, and that accordingly these, and not the individual diameters or cephalic index, were the appropriate characters to use in the multiple regression equations, if the latter were not to be too unmanageable. The relationship of the head diameters to each other and to stature has already been determined for more copious and better material, and was accordingly hardly worth repeating here*. (a) We will consider first the influence of sex on the observed characters. We note that with two exceptions, the correlations of length of head with brain- weight and the product of the diameters with the horizontal circumference, the female has higher correlations than the male. Even in these cases the difference is within the probable error of the difference. This confirms the result already reached by Pearson and Lee that in civilised races the organs of the women are more highly correlated than those of the men+. It will be remembered that this was attributed to more intense selection in the case of the male. (b) We turn next to the consideration of the intluence of age on the other characters. We note at once that all twelve coefficients of correlation are negative, or the average value of each physical character dealt with decreases with age. We conclude that at any rate for a general hospital population, the physical characters all shrink with age. While the brain-weight shrinkage is most marked the auricular height is also very significant, then follows the product of the diameters, while the shrinkage of stature only comes fourth on the list. If age be measured in years the following are the regression lines for Gladstone’s data for stature, together with those for Swedes deduced from the reduction of Retzius’ data by Pearl f. Shrinkage of Stature unth Age. Malest English : ee — 0378-4 + 69:02 (unit of stature=inch) oa suet 1 = — 0960.4 +175°31 (unit of stature= cm. ) Swedes: S=—-0807A+173°42 (_,, i ms ) Tae 1, [S=—0655 A+ 66:05 (unit of stature =inch) Romie. neo ae ee — 1662 A+167°78 (unit of stature= cm. ) Swedes: S=—-1008 4+163°51 (_,, RS - ys From these we conclude that the English shrink in stature more than the Swedes with age, and that women of the General Hospital Population in both races shrink more than the men. The lines are represented in Fig. 1 below, which gives a fair idea of the effect of age on stature. We now inquire whether * See for example Macdonell’s results for 3000 criminals, Biometrika, Vol. 1. p. 202, and Fawcett and Pearson’s results for 1000 Cambridge graduates, R. S. Proc., Vol. 66, p. 23. + R. S. Proc., Vol. 61, pp. 365. + Biometrika, Vol. 1v. pp. 53, 54. J. BLAKEMAN 137 this shrinkage is peculiar to the General Hospital Population, or extends to the normal population of a district. The only comparative results that we know Fig. 1. ——— SArrinkage tn SlCature with Age so ———— Salure tn Centimcelers 7 = Li tree eg, Pras ~~ Sex i ee or = | re =I saan ores oo | ee Ze. | Std test =) | __— Sth aloe SS 1 ~~ SS —_— ees ee — A —— SS Z Le = =| + = Sac— | en | | | | | | | 130 | il | 20 25 30 35 ao cr} oo ao os 7o 7o 8&0 so Age of are the regression lines calculated for A. O. Powys’ experience of New South Wales criminals by K. Pearson*. We have: English Males, Hospital Population : S=— ‘0378.4 + 69-02 New South Wales Males, Criminal Population: S=— ‘0337. + 68°34 English Females, Hospital Population : S=—'0655A + 66:05 New South Wales Females, Criminal Population: S=—-03614 + 63°61 In all cases the unit of stature is an inch. Hence it would appear that the hospital population of males shrinks slightly more rapidly than the criminal population, and the hospital population of females very sensibly more. This is probably due to the larger proportion of male acute eases and of female chronic cases. We do not think the present results do more than confirm the conclusion of Powys’ paper that there is for normally healthy individuals a shrinkage of a little more than 4” per ten years. (c) If we take in the next place head diameters we are not able to compare our General Hospital Population with Matiegka’s data, for his papert+ gives no clue to the ages of the individuals whose head measurements he has taken, and further those measurements are scarcely comparable with external measurements * Biometrika, Vol. 1. p. 49. + Sitzberichte des k. béhmischen Gesellschaft d. Wissenschaften, Math. Naturw. Classe, Jahrg. 1902, pp. 1—75. Biometrika 1v 18 138 Biometric Constants of English Brain-weights on the head. Our results therefore must at present stand unsupported. They show a sensible shrinkage of the head diameters, greater in the woman and greatest in the auricular height for both sexes. The amount of it is represented graphically in Fig. 2. The equations to the regression lines are as follows, age being measured in years and the diameters in centimetres : Maximum Head Length, Males: =— 03484 + 192:07, ie oe Females: [=—°0623A + 186-06, Maximum Head Breadth, Males: B=—-0406A 4 151°34, 7 - Females: B=—‘0867A + 148°53, Auricular Height, Males: H =—‘0799A + 136°84, . x Females: H =—'1059A + 13407. Fic. 2. — Srrinkage of stxe f ead wtlhr Age SL | a= ann eee Pe =e a a : === == | | [ ie ees | leans eal | | — t— a ii - t } Se =o Se ae Senos eee ae | 4 [ wae ss ab | } aeaee eae a HASAN CN ee ee I : 4 = ol ements eae Te ere eae } a a a : cae a SS = oe SS a a “ — poaeal pasnhewter tele T Ze == = Z Saar ee —_ 20 26 30 fv te} go a3 Pine 3s 60 65 7a Vie} a0 It will thus be seen that the woman shrinks with age about twice as fast as the man in head length and breadth, but only about & as fast in auricular height. The very sensible shrinkage for both sexes in head height is peculiarly noteworthy, as it of course contributes to the shrinkage in stature. It seemed worth while investigating whether these results are peculiar to a hospital population. Accordingly 1306 criminals were taken, whose age, head length and head breadth but (unfortunately) not head height were recorded, and the correlations worked out for age and these two diameters. The correlation tables are given as J. BLAKEMAN 139 Tables XIV. and XV. below. The actual values obtained are, ages running from 21 to 70: Mean Age: 33°32 4°19. Standard Deviation: 10°2219 yrs. +°1349. Maximum Length: 191:44+:11. Standard Deviation: 6°0025 ems. + 0792. Maximum Breadth: 15030 +°10. Standard Deviation: 5'1329 cms. + 0677. Correlation of Age and Head Length +1138 + ‘0184. Correlation of Age and Head Breadth: + 0349 + 0186. TABLE XIV. 1306 Criminals. Correlation between Head Length and Age. > | Totals ig = 1 1 72} .— 2s 1 17g 1 _ 2 i = = 175 1 as 1 3 vo |, — 2 1 3 177 1 4 5 78 3 3 = 1 7 79 5 2 3 — 1 12 180 4 3 2 3 a Sa 13 181 i 5 3 1 == 1 7 182 Y fi 6 3 3 1 28 183 6 6 2 6 2 2 24 184 2 4 1 i = 43 185 6 5 4 1 i 53 ey | 8S Yi 5 7 3 = 57 |) Ee 3 7 5 3 1 55 a | 188 ii 6 2 = 68 3) 189 15 4 6 3 83 : 190 13 5 4 as 85 Bee 197 3 9 3 2 96 ® | 192 7 7 i] 102 | 103 3 3 2 79 194 9 3 4 83 195 7 8 3 66 196 4 6 — 66 197 6 6 8 1 56 198 6 2 1 es 43 199 10 5 1 1 35 200 9 2 1 = 30 201 4 = 4 2 20 202 9 1 a3 2 24 2038 4 1 whe = 14 204 7 ze oz 13 205 2 a 1 8 206 1 = 3 207 1 am 6 208 = 209 — 1 1 210 1 1) ) Totals 1306 140 Biometric Constants of English Brain-weights TABLE XV. 1306 Criminals. Correlation between Head Breadth and Age. Totals | | m bo - | = COON D oP Ww sT Re Orb oe Ov OO [bw rsgren Head Breadth. | ro bo | DPN NOKWATOOAAMWOAMOOK OD eH | 1 1 | romero as | es | don fb ® or ed | ro | ene RES aey ren er | wWPRwWhe ele | | tee | eel tee ed cole | ae | — | eo | row | DO EDPOOKR RTO Pe Oe A comparison of Table II. shows that the criminal is younger and less variable in age than the General Hospital Population. To test the effect of the more youthful population, we omitted the criminals less than 26 years of age. The results were as follows for head length of 916 cms.: Mean Age: 37°46 +°21. Standard Deviation: 9°5254 + 1498. Maximum Length: 19173 4°14. Standard Deviation: 61534 + 0968. Correlation of Age and Head Length: +1062 + 0220. Clearly the correlation is not sensibly affected, while the age is sensibly and the mean head length slightly raised. We think we must conclude from this result J. BLAKEMAN 141 that in the criminal population there is no shrinkage of head breadth, perhaps a slight increase, with age, and a sensible increase, not a shrinkage, of head length with age. Unfortunately we could not test the auricular height change. It follows from this that the shrinkage of head size with age appears to ditferentiate the General Hospital from the criminal and very possibly from the normally healthy general population. Hence in determining the probable brain-weight from size of the head in a normal population probably little stress ought to be laid on any change of the diameters and therefore their product with age. It is possibly in our case a result of the increasing amount of chronic illness in the older members of the General Hospital Population. The possible shrinkage or the head with age in the general population would form an interesting subject for investigation. Of course when the three diameters have negative correlation with age, we should expect to find, as we do find, that the product would shrink with age. We have no data to compare with the observed shrinkage of the horizontal circumference with age. (d) The correlation of the horizontal circumference with stature, ‘33, is we believe new, but is very much what we might have anticipated. So also is the correlation, ‘31, of stature with the diametral product. Thus Macdonell found for 3000 criminals the correlation of stature and maximum head length, “34, of stature and maximum head breadth, ‘18, and of stature and face breadth, °35*. With regard to the correlation between the horizontal circumference U and the product P of the three diameters L, B, H, we note that if V be a coefficient of variation we must have: rey =(Virurt+ Verve+ Varun) Ve => (2:95ryz, + 3°62ryup + 3°39r yq)/7-68 for English males from Table II. Now we are unacquainted at present with any determinations of ryz, ryg and ryy, but in round numbers ry, and rygz cannot differ much from ‘7 and ryy from ‘5. Substituting we find py ="82; so that the observed value ‘83 for males appears quite probable. A similar investigation gives ‘79 as against the observed ‘80 for females. (e) We turn lastly to the correlations between brain-weight and the other characters. We see in the first place that cephalic index is the least important of these correlated characters, and further that its correlation with brain-weight varies markedly from male to female. It is within the limits of random sampling zero for men, and has the quite sensible value ‘2 for women. We have no similar data to compare our results with. All we can do is to consider the correlation of skull capacity with cephalic index, which has been determined in a number of cases. The irregular and divergent values of this correlation were first pointed out by * Biometrika, Vol, 1. p. 202. 142 Biometric Constants of English Brain-weights Dr A. Lee*. Thus she found it sensibly zero for $ Theban mummies and —‘15 for male Thebans. A list of values is given by Lewenz and Pearson+ in which it is shown that § English crania give a practically zero value for the correlation between cephalic index and capacity. That value was, however, obtained indirectly, i.e. from the ZL and B correlations with each other and the capacity, and accordingly it seemed worth while to investigate the matter directly on the Whitechapel crania. Accordingly we extracted from Macdonell’s tables{ the capacities and cephalic indices of 70 ¢ and 80 ¢ crania and worked out directly the correlation between the two characters. We found: Correlation of Cranial Capacity and Cephalic Index for Whitechapel Crania : Males: +:°093 +:080. Females: — ‘099 + ‘075. Neither value is really significant having regard to the probable error, and we might therefore conclude that this correlation is sensibly zero for English crania. Such a result would be in good accordance with the sensibly zero correlation between brain-weight and cephalic index for males, but not in accordance with the result for English females. Assuming that there might be an error somewhere in the arithmetic the correlation between brain-weight (w) and cephalic index J was tested indirectly by the formula Tiw = (Vz Y Bw — Vi Tr) V; sieietetolereiciotekeleisickereieteketetetelereleter= (2), with the following results : TABLE XVI. Cephalic Index and Brain-weight. “| | Sex Direct Calculation | By Formula (2) | Males ase +0014 +0373 | Females... +°2110 + °2084 | L ale The degree of closeness is as great as we might expect from the two methods, the square of V/100 being neglected in obtaining formula (e). There is, therefore, we think, no doubt that, while the correlation between cephalic index and capacity is sensibly zero for English crania of both sexes, the females of a General Hospital Population have a sensible correlation between brain-weight and cephalic index. It is conceivable that this is due to some exaggeration of breadth in the measurement ° of the heads of the younger females, for whom the brain-weight would be larger and as a rule the hair interfere more than would be the case for older women or for the males§. On the whole we think we have very adequate grounds for omitting the cephalic index entirely from our prediction equations for the brain-weight. * Phil. Trans. Vol. 196 A, p. 232. + Biometrika, Vol. 11. p. 374. + Biometrika, Vol. 11. Tables after p. 244. § The hair difficulty occurs twice in the case of measuring B and only once in measuring L. Hence B would tend to be more exaggerated than L. J. BUAKEMAN 1438 Looking generally at the brain-weight correlations we see that predictions based on the diametral product and the circumference will give us the best results. Age and stature have formed such a large part of the anatomical discussions on the subject that we feel compelled to retain them, but clearly a knowledge of any diameter of the head is of more importance in predicting brain-weight than either stature or age. After careful consideration we determined to make our most complex prediction formula depend upon a knowledge of the diametral product, the horizontal circumference, the stature and the age of the subject. Before considering prediction formulae, however, it is well to compare our correlations with any available material. In the first place we take the English skull. TABLE XVII. Comparison of Skull and Head Correlations. | General Hospital Population Whitechapel Crania * Character -| 3 ? 3 “ | | L with w 598 + -040 545+ :°049 | C 597 + 051 691+ °040 | B with w 489 + 048 6074044 | C ‘631 + '048 646 + 044 Af with w 544 + 044 673 + ‘038 C | [501+ -059 600 + 049] | O with w 623 + °032 627 + 042 C "820 + °026 848+ °021 | P with w “703 + °032 “749 + ‘031 C | [°750+ -035 "823 + 025] | We notice that the correlations of capacity and external measurements are all greater than the corresponding correlations for the brain-weight. The apparent exception (C and ZL, ¥ is within the limits of probable error), namely C and H for both sexes, arises, we think, from the fact that H is here the basio-bregmatic and not the auricular height of the skull, and no stress can thus be laid upon this exceptiont. The conclusion, which we might have anticipated, is therefore forced upon us that it is easier to predict capacity from external measurements on the skull than brain-weight from external measurements on the living head. The values of the correlation between C and P, although obtained indirectly, are probably very close: thus A. Lee} gives for the corresponding values in Bavarian crania: ¥, ‘700 + 034 and 9, ‘814 + :023, the differences for the two races being well within the limits of random sampling. * Cited from Macdonell, Biometrika, Vol. 111. p. 232. Except those for P and C which are deduced from the formula : rpp=(Viprer + Vetcent+ Vurcu)|Vr. All these constants are known for the Whitechapel crania except 7,,, the correlation of capacity and auricular height, we have used for this the known cor- relation of capacity and basio-bregmatic height, cited in brackets under C and H. + If the auricular height be more highly correlated with capacity than the basio-bregmatic height, the result is very satisfactory, because the former is all that is obtainable on the living head. The point should be investigated as ample data exist. + Phil. Trans. Vol. 196 A, p. 237. 144 Biometric Constants of English Brain-weights If we pass from intra-racial to inter-racial comparisons, Pearl’s memoir offers valuable material. Thus he has calculated the correlation of w with Z and B for Bohemians, based, it is true, on Matiegka’s rather doubtful skull measurements. Again, he provides us with the brain-weight and stature and brain-weight and age correlations for Swedes and Hessians. These are reproduced in the accompanying table. Considering how widely correlation results differ from race to race the general agreement is not without significance. It increases our confidence in the possibility of good results being obtained from careful post-mortem room measurements. TABLE XVIII. Comparison of English and other European Brain-weight Correlations. Middlesex Hospital Data Other European Data Correlation 3 g Group 3 f°) wand L 598 + 040 545 + 049 Bohemian 548 + 027 360 + 047 wand B "489 + 048 607 + 044 Bohemian ‘466 +031 504 + 040 wand S 289 + ‘057 367 + 060 Swedes 183 + 032 349 + ‘039 "i 33 . “ 3 i Hessians 182 + 030 183 + 039 wand A | —'256+°'058 | —°372+:060 Swedes —'249+:'031 | —:234+-042 . * - ) ra 3 Hessians —'167+°080 | —°360+°035 The general agreement is, we have said, not bad considering the differences in method of measuring and probably in heterogeneity of material. The Swedish are quite fairly concordant with the English. So are the Bohemians (considering the poor method used by Matiegka for determining the breadth and length of the skull), except for the correlation of weight and length for women, where there is a bad and inexplicable fall, although a slight fall might have been expected from the English results. The Hessian values appear to be irregular, and one is inclined to doubt the accuracy with which the corpse length has been determined, or the relative homogeneity of the series; individuals under 20 ought not to have been included in the age and stature correlations. (7) On the Regression Equations of Brain-weight on other Characters. (a) The following table will allow the reader to see at a glance the average changes in brain-weight wand product P involved in unit change from the mean in each of the observed characters. As illustration, take the following case, which is closely that of Jeremy Bentham : Stature: Mean (?), U=560, L=192, B=153, H* =127, and Age = 85 yrs. * Measured from top of auricular orifice H is 121 (see Biometrika, Vol. 11. p. 394), we must add, say, 6mm. to compare with Gladstone’s data from centre of orifice. J. BLAKEMAN 145 TABLE XIX. Increase of Brain-weight in grs. and Diametral Product in cm. above their means for each unit increase in the other observed Characters. Regression Coefficient of Character with Probable Error. | One mm. One mm. | One mm. One mm. | One Year|} One inch | in L inB | inH | inU | ind in § | é 19036 | 149°34 | 13220 555-79 49:26 | 67-16 | | Mean of Character g 183°20 | 144°55 129°21 | 533°25_ | 45°90 63°05 Mean | | | | aeov.co .., | 110728 | 9:3936/ 12:5628| 4:0895 | —2-1974| 9-827 eee csant 3 | 1327°69 ers. |S .g954 | 41-0463 | +1-2074| +°3204 | +-5245| 42-0338 | 8 | | Increase in |— ices |_ | | oie 9°7599 | 10°5593 15°5037 | 3°7344 2°7245 | 13-1934 | Sha 2 59s 5593 | 15°5087| 3°7344 | —2°724: 31934 | P | 1224°90 ers. | 47-0446] +-9607| +1°1841, +°3224 | +-4721| +2-3271 | = | 151711 | —4:1097 | 29°6394 ; 3782-91 cm.3 =e = = Diametral Saya fee lene | +6424 | +1-4730| +5-6361 roduct | Increase in | cms. 13-5656 | —5-9478| 40-5999 AQT + 3 == _ = ‘ He eseeO8cern. | +6913 | +1°3769 | +6-4775 A man of his age would have lost 78°5 grs. of his original mean age brain-weight, or have on this account only 1249 grs, instead of 1328 grs., the average at age 49°26. He has an excess of 1°64 mm. in head length, an excess of 3°66 mm. in head breadth, and a defect of 5°20 mm. in auricular height. Also an excess of 4°21 mm. in horizontal circumference. Thus on the first count his brain-weight would be 18:15 grs. above the average, on the second 34°38 grs. above the average, on the third 65°33 below the average, and on the fourth 17:22 grs. above the average. But it would not be proper to’club all these together and say that on account of his head measurements we should expect Bentham to be 4°42 grs. above the average when at the mean age. For all these characters are correlated, and we have seen that they, especially the auricular height, are influenced by age. Hence it is absolutely needful to use multiple regression formulae. These will be applied to Bentham’s measurements later, but the above will suffice to show how a single character may have its influence appreciated. (b) We now pass to the multiple regression equations. These were found in the usual way, namely, by the calculation in this case of 25 constituent determinants and the minors of their first row of constituents. The work is straightforward, but laborious. For purposes of comparison the prediction formulae based on diametral Biometrika Iv 19 146 Biometric Constants of English Brain-weights product and horizontal circumference together (without age and stature), and on each of these alone, are also added. Prediction Formulae. (i)® ds w='1987P+:8644U— 119104 + 175088 + 368559. Probable error of prediction: 48°5 grs. Mean error: 57:4 grs. (i)> QO: w='2195P+ 5067 U— 1:2395.4 + 1:07218 + 1918298. Probable error of prediction: 48:2 grs. Mean error: 51:0 grs. (ii) S: w=-2136P +8486 U + 47-9443. Probable error of prediction: 49°6 grs. Mean error: 58°7 grs. (ii)> 9: w=2466P +3893 U + 172-2420. Probable error of prediction: 447 grs. Mean error: 52°8 grs. (iii)? os w=2519P +374°7628. Probable error of prediction: 49°9 grs. Mean error: 59-0 grs. (iii)> Q: w=-2653P +315-8281. Probable error of prediction: 44°7 grs. Mean error: 52°9 grs. (iv)? &: w=4:0895U —945°1785. Probable error of prediction: 54°9 grs. Mean error: 64:9 grs. (iv)> 9: w=3°7344U — 766-4563. Probable error of prediction: 53:0 grs. Mean error: 62'8 grs. We see from these results: first, that owing to the higher correlations it is possible to predict female brain-weights somewhat more accurately than male brain-weights from external measurements of the head. Secondly, that the funda- mental character is the product of the three diameters. A comparison of (iii) and (iv) shows that it is worth a good deal more than the circumference, and further comparing (111) with (1) we see that if we predict from the diametral product alone, we shall only make 2 grs. worse mean error than if we predict from all four characters. From this standpoint it is clear that age and stature allowances are not nearly as important matters as head diameters if we wish to predict brain-weights from physical measurements. The formulae were then applied to 24 and 24 cases selected at random, being every fourth or fifth case in the observation book. It does not seem needful to reproduce the individual results, but the general conclusions are given in Table XX. Our actual results are therefore somewhat in defect of the mean errors to be expected by theory, especially in the case of (1), (ii), and (111) for the males. In the case of the females the defect is very little, and the theoretical and observed mean errors are close. We think it quite safe to say that the brain-weight of a living individual can be predicted with a mean error of not more than 50 grs. It is J. BLAKEMAN 147 noteworthy that Dr Lee’s diametral formula* gave a mean error in predicting skulJ capacity also of about 50cm.* It seems to us that no linear formula based upon external measurements is likely to give any better result. For all such measure- ments are more or less closely correlated, and, as has been shown elsewhere, increasing the number of correlated variables does not much narrow the range of variation of the predicate. TABLE XX. Prediction of the Brain-weights of 48 Individuals from External Measurements. Mean Error, in each instance of 24 cases. | Formula go Sex | ? sex 2 | (i) 47°50 ers. 50°65 grs. | (11) 48°83 ers. 50°65 grs. | (iii) 52-63 ers. | 49:10 ers. (iv) 60°21 gers. | 60°15 gers. (c) It may be of interest to try these formulae on one or two actual heads. In the Société d Anthropologie de Paris, Bulletins 1866, I. Série, T. 1. p. 209 are given the measurements of a skull reputed to be that of Dante. Unfortunately all the measurements we require are not given. We find for cranial lengths: L=178 mm., B=140 mm., U=525 mm. The bi-auricular transverse arc, doubtless measured from centre to centre of auricular orifice, was 310 mm., and the total height of the skull, i.e. the basio-bregmatic height, 140 mm. By examining crania with a similar transverse are and basio-bregmatic height, we consider that 118 mm. would be a close value of the auricular height of Dante’s skull. The height of the corresponding skeleton was 155 cms. The mean stature of the Tuscans is 165 cms., and of the more northern Piedmontese 162 cms. We shall almost certainly exaggerate Dante’s stature if we put it at 164 cms. We have next to convert skull measurements into head measurements, and in doing this we have rather magnified the values in the light of recent investigations on the thickness of the flesh. Dante died at 56 years of age. Alongside Dante we can take the measure- ments of Jeremy Bentham’s headt, and for further illustration the head measure- ments of one of ourselves. We have then: * Phil. Trans, Vol. 196 A, p. 238 et seq. Biometrika, Vol. 11. p. 371. + Phil. Trans. Vol. 190 A, p. 466. t+ Biometrika, Vol. 11. p. 394. 19—2 148 Biometric Constants of Engtish Brain-weights TABLE XXI. Special Individual Heads. Characters | Dante Bentham 15 | | ae | a tier le ve 188 192 191 B 150 | 153 150 H uae 124 126* 135 U tea dae 525 560 555 Age... sail 56 85 47 Stature = 164 cms. Mean (?) lig At 56 years | At 85 years | At 49 years | At 47 years Probable Brain-weight (1) 1244 1272 1351 1353 —\_ —~-—————"" - ty (ii) 1240 1314 1328 3 “ (iil) 1256 1307 1329 of » Uv) 1202 1345 1324 The average English brain-weight for a “General Hospital Population ” of the mean age of 49 years is 1328 ers. It is therefore clear that P.’s brain-weight is essentially mediocre. The great Bentham could have had a brain-weight which at the mean age was at most only a few grammes above the normal. This corresponds exactly to the general mediocrity which has been already noticed in Bentham’s physical measurementst. As for Dante his probable brain-weight was 80 to 90 grs. below that of the mean of the Enghsh General Hospital Population, and this corresponds well with the observed skull capacity. If the reputed skull were really Dante’s we are forced to conclude either (i) that there are most striking individual exceptions to any rule that extreme ability is associated with large brain-weight, or (11) that it is impossible to apply the formulae deduced from one race to determine the brain-weight of a member of a second. As the present paper and Pearl’s memoir both show that at least (11) is true we may at once reject (a) any intra- racial reconstruction formula used inter-racially, and (b) any demonstration of the association of great intellectual power with large brain-weight, which is based on lists of the brain-weights of distinguished men of all nations, clubbed together without any regard to age, stature, or race, and compared with the mean brain-weight of a General Hospital Population. We think, therefore, that the above formulae should be confined to English persons. That if they are to be apphed to any other race the mean values of w, P, U, A, and S, say 7, P, U, A, and 8S, should be adopted for that race. The fundamental formulae (1) then become : * Formulae are calculated for H measured from centre of auricular orifices, and OH in loc. cit. was from the top. + Biometrika, Vol. 1. p. 394. J. BLAKEMAN 149 (i) ff: w—W="1987 (P — P) +°8644 (U — U) +6893 (S — 8) — 11910 (A — A), Gi)’ 2: w—w='2195 (P— P) +5067 (U- U7) +4221 (S — 8) — 1:2395 (A — A), where w and W are in grs., P and P in cm’, U and UJ in mm., S and S in centi- metres, no longer inches, and A and A in years. These formulae may enable the reader to obtain approximations for other groups, for which some of the mean characters are determinable. For example: a knowledge of L, B, and F for the Swedes would enable us to reach an approximate formula for Swedes from Retzius’ data for @, S and A. The assumption made is that the multiple regression coefficients change far less from race to race than the “type *.” (8) On the Graphic Representation of “Scatter” of Brain-weight about the Probable Value. In the course of the present investigation we have given the probable errors and mean errors of brain-weight prediction as deduced from the arithmetical deduction of our data. It seems desirable, with a view of impressing on anatomists the real character of variation in physical measurements, and how hopeless is the hunting for anything like an exact formula for brain-weight, to put the relationship between brain-weights and diametral products and brain-weights and horizontal circumferences in a graphical form for the present English data. In these diagrams each dot represents an actual observed case, and the best fitting straight line to the system of dots, calenlated by the usual correlation method, is drawn slanting across the diagram. Figs. 38 and 4 give the results for males and females deducible from the diametral products. Figs. 5 and 6 give similar results for the horizontal circumferences. We see that the scatter is less for the diametral product than for the horizontal circumference, but we see further how hopeless it would be to attempt from these intra-racial data to achieve more by any curve than can be done by a straight line. When we pass, as in formula (11), to double regression results we have merely such a system of dots in space of three dimensions, and multiple regression data, as in formula (1), only represent a more complicated system of dots in many dimensioned space. So soon as this conception of the actual conditions of variation is realized, we believe that anatomists will once for all abandon any attempt to represent by mathematical formulae, other than correlation results, this “scatter” which marks all the inter-relationships of anthropometric characters. We see in a general way also that increasing the number of dimensions of our space does not in the least necessitate any sensible reduction in the extent of the scatter—for if it did it would involve the principle that a plane must fit a system of points in space more closely than a line a system of points in a plane. : | 8 FS 750 Ik a Ls 500 $50 ir 475 ; 500 525 550 575 60Q Horizontal Circumference in mm. Biometrika 1v 20 154 Biometric Constants of English Brain-weights (9) On the Human Prime in the Case of Brain-weight. Pearl cites* Marchand as stating that the human brain-weight reaches its maximum between 15 and 20 years, and remains practically constant until 50, when senile degeneration begins. Gladstone cites Boyd and Vierordt to the effect that the maximum brain-weights are to be found in youth. Gladstone’s own data certainly show this maximum in almost extreme youth, but the degeneration begins immediately the prime is reached. It is true the data are very scanty, but such as they are, they are suggestive. We have: Males 8—19, Brain-weight 1407 grs. » 20—25, io 1394 grs. » 20—46, . 1380 grs. » over 46, . 1320 grs. Even among the 8 to 19 group the group 12 to 15 has the largest average brain-weight, i.e. 1443 grs. The numbers are very small, and of course the cases of chronic illness in the older groups are more numerous. But Table I, p. 125 shows that even among the acute cases the fall in brain-weight owing to age alone is very substantial. Accordingly the human prime in brain-weight seems to fall before 20 years, and after 20 years there is, on the whole, a continuous fall in brain- weight. This result must be steadily kept in view when any attempt is made to correlate brain-weight and intelligence. It can hardly be argued that the human intellectual prime corresponds to the human brain-weight prime. And while brain-weight steadily changes, first rising and then falling with age, intelligence will be shown in a forthcoming memoir by one of our number to be almost stationary during the rise of brain-weight. Further, it is clear that the correlations and regressions will be much modified if we include in our results the rising period of brain-weight with the falling period. For this reason we have limited the material of this paper to cases in which the individual was 20 years and upwards. In this way our regressions will be much more nearly linear. A glance, for example, at Powys’s diagram for change of stature with age+ would, we think, convince the reader that without the use of skew correlation curves it is not possible to mingle observations on a physical character for ages before and after the “prime.” We think that the source of the irregularity in the Hessian correlations for weight, stature, and age may be due to including individuals between 15 and 20 years of age f. The determination of the “prime” in every anthropometric measurement is a matter of very great importance and interest. Our own experience is that up to the prime we get a curved line of growth not widely divergent from the logarithmic in character. After the prime the degeneration is very gradual, and for practical purposes almost linear. It would be simple, therefore, to allow for this by age * Biometrika, Vol. tv. p. 20. + Biometrika, Vol. 1. p. 47. + Stature has a later prime than brain-weight, and this might well account for the irregular Hessian results in Pearl’s Table IX. See p. 44 of this number of the Journal. J. BLAKEMAN 155 correlations, were it not that the prime appears to vary sensibly from one character to another. In other words, the “human prime” is purely relative to the special organ considered. There appears to be also no period of prime, but rather on the average an instant of prime, the time-graph of the character not being horizontal for any very sensible period. In the determination of the prime we ought to proceed by repeated measurements of the same series of individuals, and then reducing the individual time-graphs to a mean time-graph. This, of course, is absolutely impossible in the case of brain-weight and other internal organs. We are in these cases thrown back on the measurements of populations at different ages, the population at one age not consisting of the same individuals in whole or even in part as the population of another age. This method is admittedly de- fective, for it tacitly dismisses the whole question of natural selection*, and may lead us into strange pitfallst; but it is often the only possible, and occasionally —since anthropometers will hardly work entirely for the future—the only practical method of approximating to the prime. (10) We are able to a certain extent now to answer a problem which has much exercised anatomists. Has the woman relatively to the man less brain- weight, or is the absolute difference in brain-weight merely the result of the different sizes of the two sexes? There is, we think, only one proper method of answering this question. We must ask: What would be the brain-weight of a man having all his physical characters identical with that of the mean woman ? Would he have more or less average brain-weight ? Conversely, given a woman with all her physical characters identical with those of the mean man, would she have more or less brain-weight? Now unfortunately we do not know all the physical characters in either case, but we can judge from age, stature and the diametral product. Using the multiple regression equation (i) we find that: The Englishman of the same age, stature and diametral product as the mean woman has 1235 grs. brain-weight, or only 10 grs. more than the average woman. Further using (1)? we find that : The Englishwoman of the same age, stature and diametral product as the mean man has 1315 grs. brain-weight, or only 13 grs. less than the average man. But we are not certain that this even is the limit of difference because the woman reaches her prime rather sooner than the man and declines more rapidly. It would be really more reasonable to compare the man physically like the woman, when at his prime, and the woman physically like the man when at her prime. Brain-weight primes are so little determined at present that it seems impossible to do this. But we think that there is probably as in other physical * The selective death-rate in man is probably 60 to 80 p.c. See Biometrika, Vol. 1. p. 74. + Pfitzner’s conclusions drawn from the dead are probably to be largely discounted for this reason: see Biometrika, Vol. 11. p. 465. 20-—2 156 Biometric Constants of English Brain-weights characters several years’ interval between the two primes. If we change the problem and ask, what is the brain-weight of a man of male mean age equal to the mean woman in physical characters, we find it is 1230 grs. or only 5 grs. more than the brain-weight of the mean woman. On the other hand the brain- weight of a woman of female mean age equal to the mean man in physical characters is 1319 grs. or only 9 grs. less than the brain-weight of the mean man. These differences are extremely slight, and would probably disappear entirely if we took another important cranial character like frontal breadth into con- sideration and gave the woman equal frontal breadth with the mean man and vice versd. On the whole as far as present evidence goes, we can safely conclude that there is no sensible relative difference in the brain-weights of man and woman, the absolute differences observed are quite compatible with the differences which result from the relative sizes of the two sexes. In other words any argument as to the relative intellectual capacity of men and women based upon their absolute brain-weights applies equally to the thesis that the average big man has more intellectual capacity than the average small man, because he has on the average greater absolute brain-weight. While our results thus apparently contradict those of Pearl on p. 51 of this Journal, the contradiction is only on the surface, for we have been able to use a far more complete system of physical measurements; and it is clear that the more measurements taken the closer is the approach to relative equality in the brain-weights of man and woman. (11) On the Capitulo-Statural Indez. Gladstone has introduced a quantity equal to 100 x VDiametral Product/Stature or to 100P3/S in our notation and termed it the capitulo-statural index*. He shows that it decreases with stature—an almost necessary result of its algebraic form. As he has not discussed it with special relation to the Middlesex Hospital measurements, it appears worth while to discuss some of its properties and correlations. We will represent it by the letter y and take x = PHS. Clearly, since it involves no new measurement, all its properties can be deduced from the biometric constants already tabled. We have at once dy =1P-36P/S — P8S/8?, whence squaring, summing and dividing by the number of individuals, we find: of = {Sop + Page —3PSopasr ps}/S*P?, or Re = (1000,)?/x? = 4 V3 + Ve a ar pg Vs V> acer cee rereeeveces (1). Again, multiply the value of dy above by 68, sum and divide by number of individuals, and we have: fg = rg V p= Vaan eee eee (ii). * Biometrika, Vol. tv. p. 115. J. BLAKEMAN 157 Here V, denotes the coefficient of variation of any character J, or V;=100a,/J, and J is a mean value. For the practical purposes of the present inquiry we may take ¥=P#/S although this is not absolutely exact*. As P is in cm.* we must reduce S to em. We then find: TABLE XXII. Capitulo-Statural Index, Coefficient of | = Standard Deviation 100 Correlation | Regression a a =100c, Variation =—_"* | with Stature | with Stature | ¢ 9°1340 “4095 4°4828 — 8366 — 0442 + °0255 + ‘0180 + 1977 +0187 + ‘0018 9°4136 3924 4°1686 — °7957 — 0441 ? + 0273 + 0198 + 2051 + 0255 + 0024 From the regression coefficients we at once find: For ‘s: Capitulo-Statural Index = 16°6674 — 044168, Hor {)s% Capitulo-Statural Index = 16°4812 — 044138, S being measured in centimetres. Comparing this with Gladstone’s Table on p. 116 of this volume of the Journal we have: TABLE XXIII. Capitulo-Statural Index for Different Statures. hatane Gladstone’s Observed | Values deduced from Values on Males Post-mortem data 185°3 cms. ... 8°66 8:48 ide CIN Saunas 8:98 8°82 170‘0 cms... 9°00 9°16 162°4 cms... 9°69 9°50 154°8 ems... 10°05 9°83 Probably the fair homogeneity of the post-mortem material gives sensibly better results, especially as it is smoothed by the correlation process, than the * See R. S. Proc. Vol. 60, pp. 491, 492. 158 Biometric Constants of English Brain-weights averages of Gladstone’s groups, which were composed of post-mortem, wérk-house, and professional class measurements *. Generally we see that the woman has a rather higher capitulo-statural index than the man; absolutely it is about equally variable, but its coefficient of variation is less. These coefficients, however, have very much the values which have been found for other bone measurements on man. The drop of this index per centimetre of stature is almost identical in male and female. For anthropometric purposes the index appears to be of little value, for the change in the diametral product with stature is given at once by the regression coefficient tabled in Table XIX., where the reader will also find the influence of age and horizontal circumference on this product. In fact, the theory of linear correlation shows that we should expect P! for a given S to be closely of the form Pi =C,8+C,, and accordingly if C, is positive we might not unnaturally anticipate that P!/S would decrease with S. Such an index would only be of value if C, were zero, and up to the present a wide experience shows that even a sensible approach to zero in C,, if it ever does occur, is an occurrence of the most marked variety ft. (12) On the Indea = Ratio of Diametral Product to Brain-weight. What we have said at the conclusion of the previous article applies with a good deal of force to the index: Diametral Product/Brain-weight, which we will represent by the letter z. All the properties of it are involved in the correlations already tabled between the diametral product, the brain-weight, and the other characters. Still there is a physical side to this index which may be of some interest. It measures in a rough sort of way the ratio of volume to weight of enclosed brain-matter. We will investigate how far this index changes with age. All the requisite constants are given in Table XXIY. The results were deduced, as in the case of the capitulo-statural index, from the formulae : Ve — (100¢;/2) = V2 + V, —= 2 Vip Viel rp Cree eesccnretvccce (111), TA = (Verp, a Viel wa) ve COC eee rer eesereeereneseseseseseeseos (iv), * Tf we apply the above formula to male giants and dwarfs (see Gladstone, p. 117), we find: Individual and Stature Gladstone’s Value Formula | Trish Giant, O’Brian, 231-1 ems. 712 6-46 | American Giant, Freeman, 205-7 ems. 777 7:59 Dwarf from Kiel, 121-9 cms. bn 11:15 11-29 Dwarf from Holstein, 97:8 cms. ae 13°78 12°35 but quite different values have been given by other authorities for these statures and accordingly no great stress can be laid on such divergences as occur between the formula and ‘‘ observed ” values. t+ Cf. Lewenz and Pearson, Biometrika, Vol. m1. pp. 373 and 374. J. BLAKEMAN 159 by aid of the known values of the coefficients of variation Vp, V,, (see Table IL.), and the known correlations rp,,, 7p4, and 7,4 (see Table XIII.)*. TABLE XXIV. Index: Diametral Product/ Brain-weight. Sg M Standard | Coefficient of | Correlation Regression 2s siete Deviation | Variation with Age Coefficient on Age | ¢ 2°8492 “1703 59771 +1153 +0016 +°0106 + 0075 + °2636 +0615 + ‘0008 9 2°7978 “1628 5°8189 +1150 + 0014 +°0113 + -0080 + °2863 + 0696 + ‘0008 | The mean value of the index is closely the same, 2°8 for both sexes. The difference of the variations is also within the difference due to random sampling. The influence of age is, within the limits of the probable errors, the same for both sexes, and for the sixty years covered by the data would not alter the index by as much as ‘1 or by sensibly less than four per cent. We have thus the interesting result that while age tends to shrink diametral product and brain-weight its effect on their ratio is comparatively insignificant. It tends slightly to increase it, the shrinkage of the brain-weight being somewhat more than the shrinkage of the diameters. Turning to the regression coefficients in Table XIX., we see that the shrinkage of brain-weight in 60 years is 132 grs. on 1328 grs. average, or roughly 10 per cent., and the shrinkage in diametral product 247 cm.’ on 3783 cm.’ average, or roughly 6°5 per cent. We have thus the source of the 4 per cent. change in the index. It is, therefore, not surprising that the index increases and does not shrink like the simple characters with increasing age. Supposing the index were absolutely constant, then equations (111) p. 158 show us that it would only be a very inexact method of determining brain-weight from diametral product; for the constant terms are here 25 per cent. of the total probable brain-weights. (13) General Conclusions. While the writers of the present paper would admit that many of its conclusions are only tentative as the data are too sparse, they yet consider that they will be useful as suggesting lines for further research and as confirming and supplementing some of the results of Pearl’s more elaborate study. They hold that: (i) The biometric constants determined fit in reasonably well with the biometric results hitherto obtained for other brain-weight material. * See footnote, however, to Table II. 160 Biometric Constants of English Brain-weights (11) They are, allowing for the differentiation of a general hospital population, consonant with what we know of other English anthropometric and craniometric material. (iui) They show that a general hospital population differs in mean from other groups of the general population, but that the difference in variability and correlation 1s not very marked. (iv) Important results would be reached if the change in physical characters with age were more completely studied ; especially if the human prime for various physical measurements were more adequately determined. (v) Allowance can be made for the effect of age and stature on brain-weight, but neither of these factors is nearly as important for brain-weight prediction as the knowledge of diametral product or horizontal circumference. (vi) The probable brain-weight of an individual can be determined from stature, age, diametral product, and horizontal circumference with a mean error of about 50 grs. The mean deviation in brain-weight from the mean of the group is about 80 grs. In other words, we improve our prediction in the ratio of 8 to 5 by taking into account these characters. Nothing better can probably be achieved (see Figures 3 to 6) by introducing further external characters, or by considering regression as curved instead of plane. (vi) Generally Pearl’s conclusion that brain-weights for homogeneous material, uniformly measured, can be dealt with as satisfactorily as any other anthropometric characters is confirmed. We would add that the material ought to be restricted so that the highest prime of any character dealt with is the minimum age included in the data, (vill) The apparent shrinkage of the diameters of the head with age is probably peculiar to general hospital material. Its absence in criminal data is significant. The subject, however, is at present being directly investigated by one of the present writers. (ix) There is no sensible relative difference between the brain-weights of man and woman, when proper allowance is made for the relative difference in size of man and woman. A QUANTITATIVE STUDY OF THE RESEMBLANCE BETWEEN MAN AND WOMAN. By E. TSCHEPOURKOWSKY, Moscow. In my studies on the cephalic index, which are partly published in Russian and partly in German journals, I affirm that sexual selection plays a very important rdéle in the development of its varieties. Since then I have collected from different sources a certain amount of data for the study of interracial and sexual correlation, which can possibly give an answer to the following question: “If different characters of the human frame are inherited and if we admit that some of them are the result of natural selection, then the environment—in a wider sense—not being the same for both sexes we may expect that some characters are characteristic for one sex only. How far are they inherited in comparison with others by the opposite sex ?” We may suppose @ priort that the cephalic or nasal index has not the same importance in the struggle for life as for instance the arm length. If all kinds of characters are inherited equally by both sexes, the coefficients of correlation of their male and female values in different races must be equally high for them all. In order to investigate this point I have used the data of various Russian authorities for living subjects; firstly, because they are collected in a very uniform manner, and secondly, because the diversity of races within the large territory of Russia is itself sufficient. The work of collecting the data was very much facilitated by the new compilation of Iwanowsky: “ Anthropological Constituents of the Population of Russia.” The raw material is represented in the tables given below in which the number of cases and the authors’ names are also indicated. The numerical values for crania which I have used are partly extracted from different authors but mostly deduced from my own measurements made in the various anthropological museums of Europe. The cross-correlations can also be studied from these tables by anyone who wishes to do so*, * See the remark on the next page. Biometrika 1v a 162 Study of the Resemblance between Man and Woman The number of races is not very high, but the persistently high value of the coefficients of correlation and the fact that the measurement of men and women of the same race was made by the same investigator make the results very valuable. TABLE I. Constants for Interracial Variation and Correlation between Man and Woman*. Standard | Coefficient of M Sexual Deviation Variation hel a Ratio ; ' : as orre Source No. Character inne ie di |S eel eo ac oa * Iwanowsky | 31 | Stature ue 3°59 | 3°11 | 2°20 | 2:04 | 163°16 | 152°42 | +°842 93°4 | a 33 | Head Length 3°61 | 3°17 | 1°92 | 1°76 | 187-09 | 179°76 | +844 96°0 | Rs 45 | Cephalic Index 2°59 | 2°54 / 3°16 | 3°10 | 81°71) 81°93) +°875} 100-2 35 22 | Face Indext... 1°81 | 1°89 | 2°39 | 2°43 77°91 | 77°73 | +°726 99°7 is 25 | Nasal Index ... .. | 4°43 4°60 6°51 | 6°75 | 68-00] 68°08) +°836 | 100°1 . 22 | Relative Arm Length} ; 1°30 | 1°50 | 2°86 | 3°32 | 45°41| 45°18) +°731 99°4 Crania 49 | Cephalic Index 4°53 | 4:00 | 5°92 | 5°18 | 76°51) 77:16) +°929}| 100-8 Means — — 2°73 | 2°60 | 3°12 | 3:07 — = + °826 — | TABLE Ia.§ , an Coefficient of Mean Mean Standard Deviation Vaniatinn Character Correlation | No. 3 2 oa “ 3 2 Stature see ... | 164°00 + °42 | 152°28+ °39 | 4°10+°21 | 3°764'19 | *836+ 023 | 49 | 2°50 | 2°47 Head Length ... | 187°11 4°37 | 178°364°33 | 3°76+4 °26 | 3°36 + °23 | ‘909+ :009 | 49 | 2°01 | 1°88 Relative Arm Length} 45°82+°29| 45°414°27 | 2°00+°20 | 1'85+:°19 | -832+°033 | 22 | 4:36 | 4:09 Cephalic Index 81:°944°24 | 82°374°24 | 2°534°12 | 2°50+'12 | :869+°017 | 43 | 3-08 | 3:03 Facial Index 77°35+4°29 | 77°944°29 | 2°04+4°29 | 2°06+ °29 | -687+°057 | 23 | 2°67 | 2°61 Nasal Index 68°00 +°64 | 68°01+°60 4:°96+4°36 | 4°66+°40 °688+°054 | 27 | 7:28 | 6:84 * In forming the correlation tables of this memoir as well as those of a second to be shortly published in Biometrika, Ihave sometimes used not all the given data but have chosen from the means of the same nationality only a few for groups which are geographically isolated. + The relation of Bizygomatic Breadth to the total Face Length (from the top of the forehead to the chin). + Ratio of the absolute arm length to stature. § I owe this table to the kindness of my friend M. Gekanowsky (Ziirich-Warshawa), who formed (using my material quite independently from me) the correlation tables and calculated the coefficients with their probable errors. The number of cases is different, because he had not seen my tables—(and I excluded the means deduced from a small number), but the constants calculated by him are not very different from mine. between man and woman is close to °8. This table can be used as control table. The mean resemblance (interracial) EK. TscHeEPOURKOWSKY 163 We see firstly, that the coefficients lie between °726 and ‘929, and in the mean they are equal to ‘826 (Table I.). Secondly, that in sexual resemblance there does not exist a large difference between characters which are apparently indifferent and those which can be presumed to play an important rdle in the struggle for life. This is shown also by the means of indices and relative lengths, which are nearly the same for both sexes. Two explanations of this fact are I think possible: 1. The racial proportions of the human frame have no special value in the struggle with environment, or they have the same value for both sexes. 2. The characters acquired in this struggle under the influence of natural selection by one sex are inherited by another. If we admit that the racial characters of the human frame are evolved under the influence of an environment which is different for each sex, then why are they not characteristic for one sex only, i.e. why in the species Homo is the male not ditferent from the female, as in many other species of animals and plants? It shows I think that in the formation of the different races of man many characters are probably the result of homogamic and endogamic instincts. From this point of view a race is very clearly defined by Karl Pearson as a population which by geographical, religious, or other causes has been isolated during a time sufficient for thorough intercrossing and for the work of selection. If that be so, the degree of resemblance between male and female of the same race is the product of this selection. Within the limits of one race the degree of resemblance between the closest relatives of the opposite sex is nearly ‘4. If a part of the members of this race with a special deviation of any one character become isolated and form a new race, then the resemblance between their male and female elements increases to ‘8. The difference ‘4 is perhaps the measure of the work of selection, and the sexual correlation coefficient is the criterion of the formation of a new race. From the Table I. it follows, that judged by coefficients of interracial variation woman is in the mean less variable than man; but the difference is very small. In three characters out of seven she is more variable than man. But in at least five of the seven cases the difference is not sensible, having regard to the probable error of the difference. 21—2 164 Study of the Resemblance between Man and Woman APPENDIX I. TABLE II. Cephalic Index-Skulls. Nationality re Nationality 3 ie) | Authority = - | Whitechapel Been || Seth |) 7s) | Fijians Be ... | 69°6 | 68°9 | Flower, etc. French Paris, City | 79°2 | 78:0 || Duke of York Islanders | 70°9 | 71°8 | Godefroi Collection Italian ... | 80°8 | 80°0 || Australans ... ... | 71°0| 72°7 Different authors Ancient Romans ... | 78°1 | 78°7 Micronesians ... 74°4 | 75:0 Dutch 80°0 | 79°4 | Lenftemberg ... | 84:2 | 85:1 | Niederle Bavarians ... ... | 83°4 | 83:1 | Australians 68°8 | 71°4) Krause Waichenfeld Graves | 83-9 | 84°6 | Czechs | 83°2 | 83°6| Matiegka Badeners 83°8 | 83°4 | Australians 69°0 | 72°0| Challenger Expedition Swiss : 78°9 | 80°5 | Wurtembergers .. | 82°6 | 82-7) German Catalogue Row-graves 73°7 | 74:1 | Kalmuek Astrackan ... | 81°6 | 81:0 | Iwanowsky Ancient Gaulish ... | 78°4 | 75-4 Minousinsk 79°6 | 75:9 | Goroshenko Bretons 77°3 | 76°8 | Thebans | 75°1 | 76°5 | Pearson Romans 77°3 | 79:0 Bashkirs 81°0 | 81:0 | Nikolsky Friesians TET | TSO | Hottentots | 71:4 | 74:9 | Broca (registers) Anglo-Saxons 75:0 | 75:0 | Eskimo : 71°4 | 71°8 Fe S Swedes 77°9 | 78°1 | New Caledonians 71°3 | 74:4 | Crania Ethnica Etruscans ... secs || fiesta) |} vase | Hawaiians 75°5 | 78-4 50 5 Ancient Egyptians | 75°1 | 76°5 _ Loyalty I. 69°8 | 71:8 - 3 Naqadas ... | 73°2 | 74°6 Dzungarians ... . | 78:2 | 80°7 m1 » Ainos 76°5 | 77°7 Taithi 75'1 | 77°4 re aA Negroes 74°3 | 74°8 Tasmanians 76°1 | 74:8 | Broca Panjabis 70°7 | 72°3 Chinese 77-9 | 76°5 as Kanakas 79°3 | 80°3 Javanese 81°5 | 80°9 3; Andamanese 80°6 | 82°7 _Marquesans ... 75:0 | 78°2 5 Torres Straits 68°3 | 70°1 | * Extracted from ‘‘ Variation in Man and Woman.” Chances of Death, Vol. 1. pp. 349 —372. 165 E. TscHEPOURKOWSKY "SUBISSTAP-OUT A x Aysmoyseztay | (S) GL | (9%) LLP | | | (9*) e368 (gz | a azole in sig qn | AasMoys 1 ea abe (26 “egg (0%) [PCT on aoe ee wee ") SMISBT, moureyy | (02x) Top | (8) 2gr (4-2) c6L| (#9) 682 |(42) T6L| (#9) 981] — — (402) 138) (#9)608 |("2£) ATOT |(F°*) 60ST | °°* a an oa “ sqnye TOTUTUIOG — = — = = = = — | (8§)F6L] (94)Z08; — — mse oo atc eee “ Hourly a a — | (s*)pgr| ()e8t} — = (8062) (S202) (7) GTOT) Ge eter| ~~" 7 8s es SSTUTA.LO YL) SNIZIOYY — — (2) p62] (°£) 62) (2) 98T | GE)I8T| = — — | ()eze| (7£)eee] 4) 6291] C*) aaeT| *° a3 “* gsnosysemuy, AMOIODOUT[OY — -— (0St) 89), (001) TAL, (0S1) L6T (0°r) [81 \(0S1) 21g (oor 199 (51) 08 (7) 618 ($1) 9991 (001) CFGT eee arate ahd pete BaIOqU A, ‘SUT arqUeyy) _ — — (22) ZF | (8*) BBO (174) GDA | (st) 192 | — — age aise “* ursteqaazy SULTYSNLG AA a =a (soz) 9g7,| (9€) FLL |(40%) 981 | (48) 6L1 id =a wee | (4£) 319 |(9%) GFOL| (48) TST] °° aes ase “ uezey ‘Srey ey, oURYsSo.10r) —= —_ (a | . = zl) QeQ (02) 9Zg | pass af nee ree eee wee eee $}OIOG aarmmmog | (1%) ech | (°t) 6FF = Se OS DI Gee = as (8£) 68] (rs)OGeT| (9*)OSFT] Hes fa Ap sparourrg ZIMOUTII[() = za — = POD TOES LR Ra 1 (7268) MOG Pai ea ee Oe sq uyquy ZIMOUYIO[C) — = _ — . |@£x)ggt} (6)6LT| — ((t£r)o¢gg| (65) 238 \(#£*)L2T9T| (6°) Lest] °° HE a “* mopey “ Sysurzay, | (0%) egh | (05) 84% [(ooregn| (S8)ZL4 \(97) GBT | ($8) BLT [(CO*)6BL) ($8) FEL (977) 228) (£8) 9€8 | iene (£8) LeGT| °"° Be cies “ urqny puryyay = = (40z) 997, (S71) E84 (f2) ¢gT (6+) 217 |(922) €e9 |(9F) G09 |(97) 808 |(F1) ETS | (761) OF OT (vr) GEG T] °° im ue “ neypore a ‘Sood qaTTUAMOg — — — — | (6)e6T} est] — — | ($6)z6L] (°£)062| (8) ¢9¢T| (44) IFFT] 7° a aa ie “* SHRISO ourysuy | (02) BFF | (09) epP [(-zJeTs| (09) LL4 |(0°8)06T | (09) GBT — |(°£) ¢T8} (°9)6T8} (°£)O69T| (09) ELST] °° sss ee oH “+ SUIZOSSC) moureyy | (297) Lr | (4) 687 | — — |(29*) L8T | (£4) 18T |(#92) 299 ($7) 989 — | — | epot| (S4)orpst}] tt BMVRLOTAT AMOTPNYSOY = — — |(z0r) gan] (8*)98z |(2°x) 48 | (876 a (zor) 118| (st) cog |(2o?) 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(9S) pss] — je si “*eIssNy JO YING - GopMozULty)-OF[Y TL, Sa 4 = = (3€) 668 |(9°2)0€8| cates 2 ve mery oyuomoyer | (42) gpp | (&) ope [(s'*)zox| (8) epz [(*) EBT | (08) 8L1| (04) 009] (98) E89 | (98) 418 /(9°r) 208 |() LTOT |(0°) LOST] *** ie “ motgopy “ Ran — — — |(4er) pep |(00r) 297 \(002) FET | (S2) GAT (474) O49 |(2°#) LOD |(0°#) B18 |(S4*) 6ZB | (CO) OTOT |(£22) 90GT | °° ee Wegore Meso: purlraz — — -- Se Ge PASH Casi) are (5)¢e1g (9) eee} (8) TL9T (4©)69¢T (vitoqig) ysueyoounmeg =“ auryory | (S€2) gop | (05) pep |(St2)on2| (0%) Pee |(S*) G87 | (S)84T) — |(s)1T9} )org (S2)0F9T| (SeseT| tt ys . Leper | (002) F9% | (90) FOF |(o0%) 9F4 |(7°7)OGL |(CO%) BLT |(00%) SLT (°°%) L69 |(°7F) FL9 |(0°) 088 (Cox) CEB (0%) 099T (cor) gzeq | *** ap EOC, wen “ MOTGO.IO AL — — {(€£) ths |(0t) ens |(S2®) BRT (COP LAT| = — i(#£)eT8 (°°) 0F8 (££) Tg9T |(9°*)O€GT | *** ie “uezey = ss AYSUIA\YSIPSOY — — = (4 ») PFOT (7*) 86o1 a aes VMYSOT ‘SUBISSNY FRITH, moyorlog | (9%) cpp | (%) spp | — — | (6) get] (or) eet |(cot) eae | (°F) 989 |(0o#) FB | (°*) 108 Pete ERS Wee oe STELINOT, kysuomysopsoy | (45) 267 | (0%) prP | 45)6GL) 4PEL| — — | (4)og9} (4*)ee9| (45) 918) (41) ez8 (S)gF9T (1) 6g01 oe 2 Sul | Ayztmoarpeys — — |r) egy |(S42) egy |(5%*) OBT [(£42) GL \(2**) F69 (S42) TTL |(S**) BOB |(542) Gag |(o7) SGOT |(S42) FEAT | °° + ysuapoUrg «,, SUPISSNIO[OIgG atqueyy) = = < — —|(z6z)egt | (6') 11 \(262) 9F9| (8*) BF9 |(26) gag | (6) Tre |(f)es9T| (St)EZeT] suvIuetty roursoy | (5) 09% | (99) ter | (€4) 182] ()082] (S8)F6T] (*4)FeT| (42) L289] (*)EL9| (F)eLL| (4) PBL) (76) LOGT] (C9) TLPT] OUT 2 } “g 4 2 ) 2 } ~ 4 2 | fr) Agro qn y AGTTEUOTB N Yysuery] Wry oATYBlAy | ~xXopuy [eIoryT yysueTy prox xopuy [VseN xXopuy peoyy arnyeyg (sanyoa wnap) vissny fo samy Sumy "TIT ATAVL TI XIGNUddV 166 Study of the Resemblance between Man and Woman APPENDIX III. CORRELATION TABLES. I. Stature. Female. 158 | Totals Male. ed a | ee ie 1a | (ea "a | 1 mHwrotrrcowar | Heo Totals ie id II. Cephalic Index-Skulls. Male. Totals Je | wr | Be rorw | RwWaranaproree Male. EK. TscHEPOURKOWSKY 167 Ill. Head Index, Living Subjects. Female. Totals 1 2 2 3 33 : 8 3s 12 s 7 = 2 if 34 1 1 45 | IV. Nasal Indea. Female. 67 | 68 yf ¢ UBM i 1K 76 Y Totals) RS WWRe NNR EN NF W& Male. 168 Study of the Resemblance between Man and Woman V. Head Length. Female. 174.) 175 | 176 | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | Totals | E : 1 | shee aes | Snel alee api} | = es | j00\ Seah Steel see eee eee 3 ie eee i 3 itl i ge ee = =| — | 1 2 — 1 4 2 — 1 - - 3 il - 1 D) ea 1 3 fe Teo a) 1 ee 4 = 1 —— | 1 — it — 3 1 1 — | — D 1 -—— | — | — 1 Totals VI. Facial Index. Female. 78 Totals nv 19 Male. Pe PDO BR Re VIL. Relatiwe Arm Length. Female. | g : 4 | 45 | 46 | 47 | 48 | 49 | Totals Male. = ie 22 | Totals “DAS FEHLERGESETZ UND SEINE VERALLGEMEINER- UNGEN DURCH FECHNER UND PEARSON*.” A REJOIN DER. By KARL PEARSON, F.R.S. THERE is not much profit as a rule in complaining of the treatment one receives at the hands of critics, but still I think the needlessly hostile tone of Dr K. E. Ranke and Dr Greiner’s review of my memoir on skew variation requires some protest on my part. As an illustration of the want of courtesy of which I complain I would cite for example the following statement (p. 323): Er (Pearson) weist aber einmal darauf hin, dass zwar seine eingeschriebenen Kurven teilweise unbegrenzt werden kénnen, das Gesetz selbst, die hypergeometrische Reihe aber nicht, und vertréstet auf eine spatere Arbeit, in der an Stelle der eingeschriebenen Kurven die Anpassung der Reihen selbst gegeben werden solle. Diese spatere Arbeit ist nie geschrieben worden. This is not an isolated instance of the manner in which the authors criticise my work. It was quite open to them to have examined the customary sources of bibliographical information, or even to have written to me and asked if the memoir in question had been published. But here as elsewhere they assumed, without making proper investigation, that I could say nothing further and therefore had said nothing further. The memoir in question appeared so long ago as 1899 in a well known British scientific journal} from which the authors actually cite another paper of mine. Although my memoir is me geschrieben * The criticism of my work appeared in the Archiv fiir Anthropologie, Bd. 11. pp. 295—331, 1904. The proper place to reply to an attack of the kind would be in the Archiv itself. Professor J. Ranke accepted a rejoinder and asked that it should be in German and not exceed 40 pp. I have heartily to thank Miss M. Lewenz for the labour of a translation, which I much regret I cannot make use of, because the Editors of the Archiv do not now see fit to publish this reply to K. E. Ranke’s attack. As the reply was an endeavour to give an historico-critical account of the theory of skew variation it may interest readers of Biometrika, and will possibly reach in the course of time some readers of the Archiv fiir Anthropologie. + ‘*On certain Properties of the Hypergeometrical Series, and on the fitting of such series to Observation Polygons in the Theory of Chance,” Phil. Mag. 1899. Biometrika tv 29 170 Shew Variation, a Rejoinder worden according to my critics, it was actually written in 1895 and its substance given in academic lectures of the same or the following year; it was not published till some years afterwards owing to that want of leisure for preparing matter for press which every teacher who has to lecture four hours a day will appreciate. Dr Ranke says that had the paper been written, it would not have influenced his judgment. That is quite possible, and I only cite the matter here to indicate the tone adopted by my critics. A very similar instance occurs in Dr K. E. Ranke’s treatment of my fitting of Professor J. Ranke’s data for 900 Altbaierisch crania. I used this example purposely because it had already been used by Stieda. I should not myself have mixed, even for the cephalic index, # and ? data as Stieda did, but I wished to compare the results reached by the generalised curve with those reached by the Gaussian curve. I actually spoke of the resulting curves in my memoir, whether the generalised curve or the Gaussian, as being “quite good for this type of statistics*.” My object of course was to show that the generalised method did not fail where the Gaussian succeeded, but surpassed it. Now how does Dr Ranke treat this instance? He cites an example (actually inserted+ by me!) in a memoir by Palin Elderton giving the Ranke’sche Messungen as an illustration of my method of testing goodness of fit in the case of the normal curve. Undoubtedly as I said in 1894 the Gaussian curve is quite good for J. Ranke’s data, but it does not follow that the Type IV. frequency curve does not give a better fit, and is not significant for constants which the Gaussian process cannot deal with. Now it was open to Dr Ranke to test the values given for the distance from mode to mean, the skewness and the other constants in the case of the Altbaierisch crania. I have given the probable errors of these constants in my memoir: On the Mathe- matical Theory of Errors of Judgment, etc., Philosophical Transactions, Vol. 198 A, see p. 278. Had Dr Ranke fairly tested my results he would have found that the asymmetry was not significant, and that the mode sensibly coincided with the mean, but that the constant 8,, which should equal 3 for the normal curve, has a value 3°65 with a probable error of only about ‘11. Now this constant and its probable error have no relation at all to any particular theory of variation. They follow quite easily from the general Gaussian theory. It is accordingly extremely improbable that Ranke’s measurements are truly given by a Gaussian distribution in all their features. The fact that 8, is >3 points to an emphasis * p. 389. + The same remark applies to the illustrations of goodness of fit given by Faweett, cited in footnote Ranke u. Greiner, p. 326. The reference to Powys is inexact ; his paper shows that in at least three cases the Gaussian curve is quite impossible. Dr Macdonell’s work on the English skull shows that at least in 4 out of 13 cases the asymmetry is significant. ‘ Die englische Schule,” by which Dr Ranke refers to workers in my Biometric Laboratory has not discovered a truth which had escaped me; they: have shown that the Gaussian curve is of wide applicability, but not of universal truth in anthropometric measurements. This result was reached with a view to testing whether the theory of inheritance, so far ag it is based on the Gaussian theory, might be safely applied to human characters. In testing this validity of the Gaussian theory, it was of course needful to have a more general theory from which to determine the chief physical constants involved in non-Gaussian distributions. K. PEARSON 171 of the modal frequency and to a reduction of the extreme frequencies which are inconsistent with the Gaussian curve. This is actually shown in my Plate 11 and referred to in my text, and corresponds to a sensible deviation from the Gaussian law. It was open to Dr Ranke to attribute this exaggeration of the modal at the expense of the extreme variates to heterogeneity in the material. But he had no right when the material for a judgment was before him in my memoirs to conclude that, because the general distribution of frequency was not on the average incompatible with the Gaussian law, the deviation of a particular constant of the distribution from its Gaussian value might not be most significant. This case of the deviation of the Ranke’s measurements from the Gaussian form is of special interest, for it is not one of asymmetry, but of a non-Gaussian type of symmetry. Dr Ranke suggests that because my test of goodness of fit shows that the Gaussian curve is “quite a good fit,” my generalised method of dealing with frequency is idle. Assisted by a mathematician he ought to have recognised that the expression 8,—3 (which measures whether the frequency towards the mean is emphasised more or less than that required by the Gaussian law) had a sensible value, and that my method not only led to the discovery of this deviation but provided a method of allowing for it in the description of the frequency. Had Dr Ranke read my memoir on errors of observation (Phil. Trans., Vol. 198 A, pp. 274—286), he would have recognised that the two tests (a) whether special physical constants of the distribution satisfy the Gaussian law, and (b) whether the general distribution of frequency satisfies within reasonable limits the Gaussian law, are not necessarily identical. Finally had he concluded that (b) for the Altbaierisch crania was satisfied, but not (a), and that there was thus no necessary discrepancy between my memoir of 1894 and the statement in Palin Elderton’s paper of 1900, he might indeed have fallen back on his customary assumption that when frequency is not Gaussian it is heterogeneous. But at any rate had he adopted this course he would have avoided the appearance of criticising his author without endeavouring to understand what the meaning of his investi- gations was, or striving to elucidate them by a study of his other memoirs on the same subject. I do not wish to say anything further on this point. I want merely to indicate by these two out of several cases that the reader must not look for a really impartial statement of my position from Drs Ranke and Greiner. I am quite unable to account for the peculiar tone they have at times given to their criticism. (2) With this preamble I should like to divide my reply under these headings: (A) The need for generalised frequency curves, even in anthropological science. (B) The nature of the assumptions made in the Gaussian theory and their insufficiency. 22—2 172 Skew Variation, a Rejoinder (C) The hypotheses made to generalise the Gaussian law. (1) Poisson, Laplace and the early writers. Gi) Francis Galton and D. McAlister. Git) Fechner. (iv) Edgeworth and Kapteyn. (v) General Results for Asymmetry *. (D) The criticisms of my theory by Dr Ranke and Dr Greiner and the reply to be made to them. A. The need for Generalised Frequency Curves, even in Anthropological Science. I have already pointed out that even Prof. Ranke’s measurements are not fully in accordance with the Gaussian theory—for the odds are great against a quantity exceeding its probable error more than five times. It is perfectly true that the English School have found that many characters, especially craniological characters, are for practical purposes sufficiently described by the Gaussian curve. But it is equally true that they have found other cases in which the deviation from the Gaussian curve is significant, and that they have only been able to measure this significance because they had a wider theory to base their researches upon. Dr Ranke entirely disregards the statements of Miss Fawcett and Dr Macdonell on this point. Both find a definite number of cases, the one in Egyptian skulls, the other in English skulls, in which the deviation from the Gaussian law is definitely significant. Both conclude as I have done that in the case of many characters for a variety of practical purposes the Gaussian curve is sufficient ; this is, however, not a theoretical justification of the Gaussian curve, but an argument in favour of its empirical use in a certain definite number of cases. Dr Ranke may of course say that the exceptions that we have found are due to heterogeneity of our material. If so he must face the difficulty that the same set of crania can be homogeneous and give the Gaussian curve for their length and be heterogeneous for their breadth, deviating therein largely from the Gaussian curve. If he asserts that this is quite possible then he must meet the further difficulty that they can be homogeneous for their cephalic indices, which are based upon the ratio of the supposed heterogeneous to the homogeneous material! The fact is that no unprejudiced observer can examine the constants by which we have defined the deviations from the Gaussian law without seeing that they present every variety of value, starting from the values to be expected on the Gaussian theory and rising to values which are absolutely incompatible with any Gaussian theory at all. In fact he must come to the conclusion that some theory is absolutely needful, which will provide a curve or series of curves capable of representing the * T have left out of consideration the general method of Thiele, followed in Germany by Lipps, because I have dealt with these authors in a recent memoir. + Biometrika, Vol. 1. p. 443, and Biometrika, Vol. 111. p. 227. K.: PEARSON 173 fundamental deviations of any distribution from the Gaussian curve and deter- mining whether these deviations are significant or not. Looked at.solely from this standpoint—which I am very far from accepting—my curves provide an empirical series which accurately measures the deviations from the Gaussian law and enables the enquirer to determine how far that law is applicable. Hach one of them passes into the Gaussian curve if that curve is the better fit to the observations. This is not true of many of the other remedies which have been proposed to supplement what I venture to call the universally recognised inadequacy of the Gaussian law. They cannot as we shall see in the sequel effectively describe the chief deviations from the Gaussian distribution. The chief physical differences between actual frequency distributions and the Gaussian theoretical distribution are : (i) The significant separation between the mode or position of maximum frequency and the average or mean character. (ii) The ratio of this separation between mean and mode to the variability of the character—a quantity I have termed the skewness. (iii) A degree of flat-toppedness which is greater or less than that of the normal curve. Given two frequency distributions which have the same variability as measured by the standard deviation, they may be relatively more or less flat-topped than the normal curve. If more flat-topped I term them platykurtic, if less flat-topped leptokurtic, and if equally flat-topped mesokurtic. A frequency distribution may be symmetrical, satisfying both the first two conditions for normality, but it may fail to be mesokurtic, and thus the Gaussian curve cannot describe it. The Gaussian curve is usually fitted from the mean’ square deviation, but it may also be fitted from the probable error, or the mean error, or again from the mean fourth power of the deviations—p, in my notation. Whichever method is adopted we ought to get the same result within the errors of random sampling. When I first began to describe frequency data by the normal curve, I was startled to find the very large number of cases in which these different processes led to Gaussian curves, differing widely from one another, i.e. beyond all the limits of probable error. I was soon led to see that in actual statistics two distributions might have equal total frequency, be sensibly symmetrical, and have the same standard deviation and yet differ largely in their flat-toppedness. The mesokurtosis of the Gaussian curve is not a universal characteristic of frequency distributions. When we test a theoretical distribution of frequency against observation, we may find an excellent fit for the total distribution and yet the distinction between mode and mean, the skewness, and the deviation from mesokurtosis may be most significant. The reason for this is that the test for goodness leaves a margin of variation which may be due to random sampling, or to the non-normal character of an important constant of the distribution. For example, 10 coins are tossed a hundred times, and the proportion of cases with five and more heads is somewhat 174 Skew Variation, a Rejoinder in excess of the theoretical distribution 100(4+ 4)", but within the limits of a random sample. It is quite conceivable that if the returns for each individual coin were analysed it would be found that those of one exceeded in proportion of heads the limits of random sampling, and that the coin proved to be loaded when delicately tested. Thus as I have shown in my memoir on errors of observation, we have not only to test for general goodness of fit, but also to consider the probable errors of the fundamental constants of the distribution. Because the general distribution of frequency is given within the limits of random sampling by a normal curve it does not follow that the system will be mesokurtic. Consider for example the two curves: 2a? (m, + 4) V2ara (am, + $) T (am) and ar f s oy aes ah we N _ Pm, +1) . F 2a*(m— Pb = Pima Win BP (4 +B They are both symmetrical, they both for any value of m, or m, which is moderately large are indistinguishable in appearance from the Gaussian curve. If they represented actual observations, we should try to fit them (i) by finding the area, (ii) by finding the standard deviation. The former for both curves is N and the latter for both curves is ¢. Hence we should fit them with But this in both cases would be incorrect. Both cases would only pass into the Gaussian curve when m, and m, are theoretically infinite, practically large. No Gaussian fitting could distinguish one of these curves from the other. Why ?— Because it does not proceed further than the standard deviation. 'To measure the difference between either of the above distributions and the Gaussian curve we must proceed to higher moments. Let Vy, be the nth moment about the mean, i.e. if % be the mean value of 2, Nyy = ty (7 a x)" dx, where the limits of the integral are those of the range. Then if B,=,/p,°, we easily find : my ee and m= 2 fe = , 6 6 es Bi~3= 5g 8 Bo Thus we reach one of the conditions for the Gaussian curve, i.e. 8. =3, in either case when m, and m, are considerable, but if 8, be >3, m, will be positive and if : see _ m—1. MAT oe 8, < 3, m, will be positive. Now since ~*~ is always less than —*—_ , it is easy m,+4 m,— 4 K. PEARSON 175 to show that in the neighbourhood of the origin y/y, is always greater than y/y, for the same value of a. In other words, the first curve is flatter topped than the second, and both lie on different sides of the corresponding Gaussian curve. The first curve type is platykurtic and the second leptokurtic. Now there is nothing to prevent us fitting curves of the above types to any series of frequency observations. Supposing those observations are truly normal, then m, or m, will be so large that 8,=3 within the error of random sampling. Now the probable error of 8, for a Gaussian distribution of total frequency V*: and if £8, differs from 3 by several times this probable error, it is absolutely impossible to treat the system as mesokurtic. In any such case one or other of the above curves must give a truer representation than the Gaussian curve. It is easy to show that for leptokurtic distributions the maximum frequency is greater than that given by the normal curve and for platykurtic distributions it is less. The Gaussian curve compels us to assert that the product of the maximum frequency into the standard deviation is a constant (ie. yo=N/V27). This condition of mesokurtosis is unfulfilled—within the limits of random sampling— for a great variety of frequency distributions. Further it is absolutely certain that divergencies from the Gaussian or normal curve are not exclusively in the direction of either platykurtic or leptokurtic distributions. Thus the symmetrical binomial is essentially leptokurtic, ie. 8, < 3, and therefore cannot be used for a great variety of distributions. In general all skew binomials with p>°2113 and <‘7887 are leptokurtic; outside these limits they are platykurtic. The test whether a curve satisfies the mesokurtic condition has nothing to do with my particular views on frequency, it is merely deduced from the general principles of probability and is a test of normal distribution. Of course there are many other conditions to be satisfied, e.g. p., should equal (2n—1) py». But as I have shown elsewhere the probable errors of the high moments increase so rapidly, that it becomes easier and easier to satisfy such conditions within the errors of random sampling, and without very large numbers they are of little practical value. The following are significantly platykurtic distributions : The Maximum Breadth in English ¥ skulls, The Nasal Breadth in English ¥ skulls, The Cephalic Index of Altbaierisch skulls, The Auricular Height in ? Nagada skulls. * Pearson: Phil. Trans. Vol. 198 A, p. 278. 176 Skew Variation, a Rejoinder As a rule the data available in craniological investigations are too sparse to give any real test of mesokurtosis, and this is the true reason why we must content ourselves with the Gaussian curve. Again Mr Powys found out of twelve frequency distributions for the ‘stature of men and women that 11 were leptokurtic and the twelfth essentially mesokurtic. This tendency to leptokurtic distributions—which can hardly be due to chance— is actually given by Rauke and Greiner as a case in favour of the Gaussian curve ! (Anmerkung S. 326). They further cite Fawcett and Lee in the following manner : In der an letzter Stelle zitierten Arbeit ist der Nachweis einer bestimmt gerichteten Asymmetrie fiir die Mehrzahl der Masse und zwar in der nach Fechner zu erwartenden Richtung besonders beachtenswerth. They do not say that of the 24 curves given by Fawcett and Lee 14 are leptokurtic and that Fechner’s curve can only represent platykurtic distributions. They do not draw attention to the fact that the Fechner curve would be impossible for the whole of Powys’ stature data, and for 12 out of Macdonell’s 26 curves for the English skull! In other words, if the Abweichungen of Fawcett and Macdonell and Powys’ data are to be used as an argument at all, 38 out of these 62 distributions diverge from the normal curve in a manner which cannot possibly be represented by Fechner’s theory ! If we turn from the condition for mesokurtosis to those for differentiation of mode and mean and for skewness we meet other considerations. So far we are not dependent for anything we have said on any theory of frequency other than the Gaussian. On that theory 8,=3. and if the difference 8,—3 be significant the distribution cannot be Gaussian. If we want to distinguish between the mode and the mean, we cannot start from the Gaussian theory, because that theory supposes the two values absolutely the same. On the other hand if we consider asymmetry, we ought to have, within the limits of random sampling, all the odd moments zero, 1.€. eo: Ms = Ps = M7 = +. = Mon = 0. Now it is of very little practical value testing the high moments because their probable errors are excessive. The probable error of pu; for the normal curve = 67449 fee and of pu; = "67449 Rye o, or in terms of o as our unit is thirteen times as large. These are the gross errors; the percentage probable errors are of course infinite. Asa rule it is hardly worth testing these conditions beyond p;. We determine whether the third moment is zero within the limits of random sampling. If we wish a relative magnitude: we can take 8,=4y,"/u,°, a quantity which occurs over and over again in frequency discussions. The probable error of 2, is obviously zero for the normal curve, because £, is of the square of the order of small quantities. The probable error of VB, = 67449 Vo and K. PEARSON digg VB, for all truly normal distributions ought not to differ by more than two or three times the above expression from zero. We can form other expressions involving 8, and 8, and ask what their value is for the Gaussian curve. We can calculate their probable errors, and determine whether the given distribution satisfies the Gaussian value within the limits of random sampling. Thus I take the expression : ae VB,(8,+3) _1 VB,(6+8,-8) a5 5B,—68,-9 2 SE(Crssjyaes, 0 Clearly this expression vanishes for the normal curve, and =4¥V@, nearly when VB, and 8,—3 are not very large, i.e, when we have not a very wide deviation from normality. The probable error of this expression, if the distribution be really normal, is 67449 Ve 2N Again, consider the expression : ase VB, (B, +3) = 40 59 Gg 9 cites This is a length which vanishes, if the distribution be truly normal. Its probable error is ‘67449 NE ay 7 the case of the Gaussian curve, and accordingly d should not differ from o by more than two or three times the above probable error. Now let us write 7=8,-3. Then it is absolutely impossible for any distribution to be looked upon as Gaussian unless y, d and 7 are zero within the limits of random sampling. These limits being deduced from their known probable errors. Now it will be asked why choose such an expression as y instead of the simpler VB,? The answer is quite simple. We want to determine whether the mode coincides with the mean or not, and we cannot do this on the basis of the Gaussian curve where no distinction is made between the two. We must take some curve which is not Gaussian to determine this important quantity from. Now the equation to the Gaussian curve is _(a-myP y=ye ra, where m is the mean value of w and a, the standard deviation, and we have for its differential equation : Biometrika Iv 23 178 Skew Variation, a Rejoinder Now if we assume that the actually observed character is not w but X, and that X is some function of a, we shall not in plotting the frequencies to X obtain a normal curve, but we ought if Y be the ordinate of this curve to have Yr r r di ; YdX =yde or } = de Taking logarithmic differentials 1dY lidyd« , d« | dx Vax ydedX * dX*/ dX — 1 gm % 4 Fe [te Sea re ie) a Assume «—m=f(X) and we find: UPD xX : VaX ogP(Q) 0 ee (iv), where F(X) = Xf (Xf F(X) (f' (XP af" (X)}- The form has been so chosen that the origin is the mode, i.e. dY/dX vanishes with XY. The proposal to thus generally transform the Gaussian curve is due in a quite ditferent form to Edgeworth*. Kapteyn following Edgeworth and without any acknowledgment takes: S(X) = B(X +)! where @, « and gq are constants to be determined. He therefore puts: cae X (X +«) PAY= G3(q— 1) = mgB(X +0) — gk to This is a somewhat complex expression. The resulting frequency curve is g§ A) ake ((X4 «)!—m}? V=¥,(X+)oe re ee ee (v), and has been suggested by Kapteyn as a general form of the skew frequency curve. We shall consider it later. Galton and McAlister as early as 1879 took f(X)=blog* — m, where 6 and a@ are constants. Ranke and Greiner, without apparently knowing the history of research in this field, take the same value and attribute to Fechner the well-known Galton-McAlister curve of the geometric mean which results. We find 72 F(X)=—_y, A a; + b? log 7A * He has developed it in a long series of papers published in the R. Statistical Society’s Journal, K. PEARSON 179 j ce See (i i ar a whence Van oe Fc aoe (toe X\? : and Y= Vy @7 oot (Ws =) A AC ae (v1). Edgeworth himself has made other suggestions as to suitable values for f(X) and accordingly of /(X). Now it is quite clear that assuming the character to be a definite function of another character which really obeys the normal law, there 1s no more reason for assuming one form of #/(X) than another, because we are in absolute ignorance of the nature of this function. Kapteyn’s, or Edgeworth’s, or Galton’s are equally valid, and the only test of their relative suitability les in the extent to which the resulting curves fit actual data. Clearly to assume «=f(X) is to assume the actual frequency distribution to follow any law whatever. It is only screening the generality of the assumption y= $ (2); where ¢ is unknown, by an appeal to the supposed universality of the Gaussian curve and by a perfectly arbitrary selection of the subsidiary function /. But there is another manner of looking at this proposal. Returning to the equation eh xX YdX ~~ ofF(X)’ and writing 0? =o,’ F(X), we see that it becomes identical in form with the normal equation, Le. LdY . xX VdxX o@" In other words the distribution of any frequency may be looked upon as given in the neighbourhood of any point by a normal curve of standard deviation o VF (a). Hence the conception arises that if the causes which produced variation in the immediate neighbourhood of any value a, of the character, were constants for the whole range of variation, we should have a normal curve of standard deviation o,VF(a,)*. In reality there is a continuous and gradual change of the tendency to variability as we pass from one value of the character to a secondt. Analytically * This method of looking at the matter throws light on another point. If a curve be of limited range, it signifies that c=0 at certain points, or the curve stops because we have reached the limits of local variation. In a curve of unlimited range it is not the capacity for local variation but the absence of individuals to vary, which is the special feature. + The matter is of such importance relative to some of Ranke’s criticisms that I give another proof of equation (iv) here, based on the conception of an infinite number of infinitely small cause groups which Ranke considers can only lead to the normal curve. Let y,.,, be the (r+1)th term of a binomial, skew or symmetrical, say for simplicity the latter, i.e. ($+4)". Then Yrs Yr w+ 1-2r : (Yr tYr) 2 (m+1)° Now let c, be the distance between y, and y,.,, used in plotting these ordinates to obtain a curve, and let it be related to some small value c, by the relation c,=c, x function of r=c, x ¢ (r). Let X,. be measured 23—2 180 Skew Variation, a Rejoinder we may look at it in this way: If da, be the variation in the neighbourhood of 2, then 82, is not independent of a, but correlated with it. We may have a perfectly continuous population from dwarfs to giants, but it does not follow that the actual tendency to vary of dwarfs and of giants is identical. All proofs that I have seen of the normal curve fail in this respect. They assume that the character « is due to a number of increments, which are due to an indefinitely large number of independent cause groups. They assume that 6x, is not correlated with the already accrued value x. All processes like those of Edgeworth, Galton, Kapteyn and Fechner are really devices for getting over this gradual change in the tendency to vary from point to point of the range. It appears to me best to directly acknowledge and face this difficulty by selecting a fitting function for F(X). If we drop now the distinction between X and # as unnecessary we reach as our frequency equation : or if we use Maclaurin’s theorem for F' (): aes i ee ydx o2(1 +a" + aa? + asa 4+...) Now I have shown* how to determine the successive constants ¢,7, 0,7, O22, ete. Further all these constants but o,? are zero, when the distribution is normal, and the series will be found to converge rapidly, when the distribution is from the largest term of the binomial, then X,=c, ./(n-+1) x function of r=c, Jnl) f(r), say, and conversely r=a function of X,./{cy./(u+1)}. Divide both sides of the above equation by c,, which may be written on the left AX, and we obtain : AY. — ¢ ei Dees XAX™ F (e+) 62/0) (r+) — By Put op=3/n+1c, and F(r) for the expression f(r) ¢ (r)/{r/(n+1)—4} which does not become infinite with 2r=n+1, because XY, and therefore f(r) vanishes for this value of r, and accordingly f (r) contains 27 —(n+1) as a factor. We then have: ne x, yAXx oy" F(X,/o9) © Now make n infinite and c, vanishingly small, then we have if ¢,=4 ./(n +1) ¢, be still finite dy | Xx YdX a ZF (X/o,)’ a result in agreement with the above investigation. In other words this, and not the Gaussian curve, is the generalised frequency curve we reach if we directly abrogate the third Gaussian principle, that contributory increments of the variate are independent. Of course the first two Gaussian principles simultaneously disappear. This view of the matter occurred to me many years ago, when considering Hagen and Crofton’s proofs of the Gaussian law. It was expressed in my memoir of 1894 by the statement that we require curves produced by conditions in which the contributory cause groups are not independent, i.e. in which an increment 6.x to the variate « depends upon the value of 2, or is cor- related with it. My method of reaching such curves, however, was a direct appeal to discrete series in which such a condition was fulfilled. * «Mathematical Contributions to the Theory of Evolution, XIV.” p. 6. Dulau and Co., London. K. PEARSON 181 in the least approximately normal. Accordingly if we wish to get a good interpolation curve to determine the distance between the mode and the mean, we may assume [ikl 3 ae (vii) bis. ydx o7f(1+a27+a,2") In this case we discover with the previous notation, that ga! NBAB.+3) 2 5B» ~~ en —J9 4 VB, (8, + 3) 5B, — 68,:—9 of this distance to the variability, or what I term the skewness, or the asymmetry relative to the variability. is the distance between mean and mode, and that y = is the ratio If we leave out a, we find the skewness given by y¥=18, and the distance between mean and mode d=4V£,c. In practice these give fairly closely the same values as the fuller expressions above, and the fuller expressions are not numerically much modified if we include a;. Shortly we have got a very fair mathematical process of determining the position of the mode and the degree of asymmetry. Now the constants of such a curve as (vii) bis are absolutely determined by a knowledge of o, 6,, and 8,; or looked at inversely they suffice to fix o, B,, and £,,. In other words the degree of kurtosis (8,—3), the skewness y and the distance between mean and mode—all most definite physical constants—are at once fixed by a knowledge of the constants of the curve, or on the other hand, being known, they fix those constants. It is of course allowable to replace any one of the three by the variability of the system. The actual position of the mode and the total magnitude of frequency suffice to fix the position and size of the curve. I have already called 8,-3= 7 the degree of kurtosis; I call d the modal divergence. Then unless n, the degree of kurtosis be zero, subject to probable error, 67449 V24/N, x, the skewness be zero, subject to probable error, 67449 V15/N, d, the modal divergence be zero, subject to probable error, ‘674490 V1:5/N, * no distribution can be legitimately described as normal or Gaussian. It would be of interest to know how far Ranke and Greiner have applied such tests to any series containing a large number WV of individuals. I think if they had done so, they must have come to the same conclusion as the majority of statisticians that the normal curve has only a limited range of application. Of course if V be small, as in most craniological series, we find our probable errors so large, that it is not possible to say more than that for short series the Gaussian curve may roughly describe the result. But for long series in economics, sociology, zoology, botany and anthropometry the Gaussian curve over and over again fails. If in all these cases Ranke and Greiner assert that the material is 182 Skew Variation, a Rejoinder heterogeneous they are merely arguing in a circle. The distributions are as continuous and smooth as those which occur in the case of the Gaussian curve, and they occur for characters in the same group of individuals which present for other characters the normal distribution. Thus the length of meropodite of mght claw in Gelasimus pugilator* is quite sensibly normal, but the length of the carpodite of the right claw is almost as certainly platykurtic and skew. If two characters are normal, a third character which is their difference, whether they be correlated or not, should have a normal distribution, yet in the case of Gelasimus pugilator for the whole series of measurements the difference distributions are essentially platykurtic +. The size of the dise in Ophicoma ngrat has a modal difference in the distribution of 1000 cases of °271 mm. and the probable error on the basis of a normal curve is ‘056. The deviation is thus five times its probable error and the asymmetry undoubtedly significant. The outer diameter of Arcella vulgaris§ in 504 cases gives a modal difference of 3:226 mikrons, and the probable error of this difference 1s only ‘211. The asymmetry is therefore undoubtedly significant. The distance between the mean and mode in the case of the length of shell of Nassa obsoleta|| from Lloyd Pomt, Long Island, U.S. was 68 mm. for 368 individuals, the probable error of this modal difference was ‘08. The asymmetry is therefore significant. Other characters of IV. obsoleta were as definitely asymmetrical, while some from exactly the same individuals were sensibly normal. The transverse arc in ~ Naqada skulls has for 115 individuals a modal ditference of 2°34mm. and the probable error of this difference is ‘78 mm., or it is probably significant. Yet the breadth of the male Naqada skulls is significantly symmetrical. The height/length index of 117 English skulls has a modal difference of °85 and a probable error of only ‘22, the skewness is therefore significant. The same is true of the distribution of many internal organs in man. For example, if we exclude recognised diseased hearts, we obtain a markedly skew distribution such as is given in the broken line of Fig. 1, This is for 1382 heart- weights. If this be supposed to be due to the great variety of ages, we have only to look at the continuous curve for hearts of 358 young adults, 25 to 35, to see the same asymmetry. This is drawn for four times the scale. * G. Duncker: Biometrika, Vol. mu. p. 313. + There is another point to which I will only refer briefly here. If characters were always distributed according to the Gaussian law the regression curves must be straight lines, The generalised Mendelian theory of determinants I have developed makes them, however, hyperbolas, and I have given instances in a recent memoir of a variety of curved regression lines + McIntosh : Biometrika, Vol. 11. p. 470. g R. Pearl and F. J. Dunbar ; Biometrika, Vol. 11. p. 327. | A. C. Dimon : Biometrika, Vol. u. p. 29. Frequency. K. PEARSON 183 Lastly if we consider that the question of health determines the skewness, we have in the dotted curve the weight-distribution of 699 hearts stated to be “healthy.” We see that there is still the same essential skewness; the pathologist has merely cut off a small portion of the tail on the left and far too much of the tail on the right, i.e. unusually big hearts were discarded as necessarily “ unhealthy.” The form of the curve undoubtedly indicates that many of these large hearts are abnormal, but any continuous curve fitted to the remainder, the “healthy hearts,” would not only be significantly skew, but would project a long way into the portion of the tail discarded as “unhealthy.” The list of asymmetrical distributions might be indefinitely extended, but these must suffice to indicate that asymmetry cannot be lightly put on one side in the manner adopted by Ranke and Greiner. fees i eeareeeeeS sexe coe Weight of Heart in English ozs. pa ae |e cei ad RRS ees se sige e eae oe zSReERSS Sa PE a Teese aes ees es SS a ee 2k Sees Se LLL ptt | Ee a Ld ~ = > = = = Fic. 1. Frequency Polygons of Weight of Heart in Males (Greenwood*). [geeks = A= = * 25 to 55 years. Without specific disease of heart. Number of cases 1382. Il O——o 25 to 35 years. Without specific disease of heart. Number of cases 358. IT. oe ----- e 25 to 55 years. Definitely sound hearts. Number of cases 669. The scale is four times as great for II. and twice as great for III. as for I. If we pass to discrete variates, we find as large a number, if not a larger number of distributions in which skewness is well marked, for example, fertility in the Aphis Hyalopterus Trirhodust, fertility in man}, fecundity in race-horses§, and fertility and fecundity in mammals generally. I illustrate this with an example of fertility in English mothers in Fig. 2. It will be seen at once that no normal curve could be used to describe this distribution. It is equally * Biometrika, Vol, 111. p. 45 et seq. + Warren: Biometrika, Vol. 1. p. 127. t~ Pearson: Phil. Trans. Vol, 192 A, p. 257, and The Chances of Death, Vol. 1. p. 63. § Pearson: Biometrika, Vol. 1. p. 292. 35-125 86-125 Skew Variation, a Rejoinder 184 ‘uaippyo fo oe Il Ol 6 Perse ee ee eer one fr a eee eee ee ee a eae de Sees a eS eee See ee Pree Nese ese a pee ea vel ee ee SOSA V ORE ee Bee Mies) et ee ee ee Hoe sees Nae eee Seen e eee aS Hee SREB Reese ie 2a PEER EEE CHAE ‘SOTIWIVA YSU Url uatpliyO jo Jequinn °“% ‘Ply ‘saying OOT (ad howanbasyy K. PEARSON 185 impossible also for a curve of the McAlister-Galton type, for the simple reason that that curve has high contact at both ends of the range. Now I contend that the anthropologist who either neglects such matters as human fertility, or confesses that he has no means of succinctly describing their distributions—and as long as he sticks to the Gaussian curve he certainly will not have—is simply putting on one side a fundamental factor in the science of man. Ranke lays great stress on homogeneity. He does not, however, clearly define what he means by the term. Apparently any series which follows a Gaussian curve is to him homogeneous, any other series is not. I should be glad if he would then consider any craniological series, say, of adult crania. This will involve crania of adults from perhaps 25 or 30 years of age to 50, and these are rather narrow limits considering our paucity of material. Now our data show that the correlation between head measurements and age may be of the order of about —‘1 to—‘2. After about 25 to 28 years of age in man, there is a continual shrinkage not only of stature but of skull capacity, brain-weight, circumferences and diameters of the skull. Under the circumstances, what right have we because the Gaussian curve is obtained to call this material “homogeneous”? I will go further; suppose we could, and we can, obtain the measurements on one or two thousand individuals of the same age; are these to be considered as a homogeneous distribution? My reply will be, in man the order of birth is an essential feature in determining the dimensions of the physical characters. My investigations show that physique and health are sensibly correlated with the position of a member in his own family. In mammals others have shown that the physique of an individual is sensibly correlated with the number of members born in that individual’s litter. Now with these facts before us what stress can be laid on Ranke’s conceptions of homogeneity? The practical anthropologist requires curves which will successfully graduate his data. Only on the basis of such graduations can he allow for the influence of disturbing factors like age, order of birth, season or special position of production in the organism. I take this very case: How is it possible to allow for the influence of order of birth, unless you know the size of families or the distribution of births within the community? It can only be achieved provided the distribution can be represented by a few simple constants which allow of definite mathematical handling. Ranke and Greiner say with considerable asperity that my method of determining the range from given data can be of no service. Yet take this very case of size of families. In the English middle classes for 4390 instances, I find that the observed limit is 17, but fitting a skew curve the range is determined as 22 children running practically from just before 0 to over 21 births (Fig. 2). For Denmark by the same process in 34,000 cases the theoretical range is 26 and the observed range 22*. I then proceeded to take statistics for the Argentine Republic, and found for the town of Buenos Ayres, 27,510 births, that the range of the curve was from ‘25 to 36°61 births, or 87 possible births. The maximum observed in these 27,510 births was 23. But among the South- and Mid-American * The Chances of Death, Vol. 1. Reproduction Selection. Biometrika tv 24 186 Skew Variation, a Rejoinder populations cases of 24 (Trinidad), 26 (Cuba), 27 (Nicaragua) and 34 (Colombia), the women beginning to bear at 13 and continuing to 50, have been recorded! It will, I think, be realised by the impartial reader that Ranke’s statements : Fiir den Anthropologen ist also nur das Gaussche Gesetz von Wichtigkeit. Fiir seine Probleme beansprucht es aber auch vollige Giiltigkeit (S. 327) and So haben anderseits auch die Werte von Variationsumfiingen die bislang aufgefunden sind, keinerlei weitere Erkenntniss gebracht (8. 324) fall wide of the mark. Ranke has either very much circumscribed the field of the anthropologist, or he has not in this, as in other cases noted in his paper, studied the literature of the subject, or finally he has disregarded the results reached. It is quite true that the range cannot always be determined and being determined does not always give a very good result. The reasons for this are not far to seek. For example, If a discrete quantity has for its minimum 0 units the start of the curve must naturally fall on the negative side of the origin—since its area measures frequency between —‘5 and +°5. Ranke would probably find something mysterious in this “reichen oft ins Negative.” Actually it is to be expected, especially if due allowance be given for the probable error of the range. In most biometric statistics, we cannot as in the case of births deal with 20,000 to 30,000 cases and get small errors for our constants. We have only perhaps 500 to 2000 cases and even less than this in craniology. This may denote an error of 14 to 17 p.c. in the calculated range, and it is quite possible that the range may “reichen ins Negative.” Take the case worked out by me* of the number of Miillerian glands in the forelegs of $ swine. The range theoretically calculated is 18 glands with a probable error of + 2°54; the start of this range ought not to have exceeded —°5. It is actually —°82 with a probable error of "16. The actual skewness of this distribution is ‘81 with a probable error of ‘02. The distribution is accordingly significantly asymmetrical. I have cited these cases as sufficient for our present purpose, but there are many other cases in which the discovery of the range has been of biological or special anthropological interest, e.g. the earliest appearance of certain diseases in childhood, the range of cancer attacks, the first occurrence of signs of puberty, ete. It has been apphed also effectively to a number of zoological and botanical data. A more striking case, perhaps, of usefulness is the limit to high barometric pressure obtained by dealing with the frequency statistics of barometric height at series of stationst+. Throughout the whole of the stations of the British Isles dealt in, there is sensible skewness of distribution, and with one Irish exception, which is sensibly mesokurtic, the whole series of curves are platykurtic, and this deviation from normality cannot be chance, but is a significant character of the frequency distributions. In these cases the limit to high pressure has been found, and appears to be a constant of considerable physical importance for the local climate. * Pearson and Filon: Phil. Trans. Vol. 191 A, p. 289. + Pearson and Lee: Phil. Trans. Vol. 190 A, p. 423 et seq. K. PEARSON 187 I again am forced to consider that Ranke has not been aware of what has been published, still less what has been done in this matter. He appears to base his conclusions chiefly on my first paper on skew-variation, and he has not noticed the fact that being the fist paper much has to be corrected in the light of more recent work in the past ten years. Ranke speaks of the writer's: Andwendung von allerlei grésseren und kleineren Anderungen in seiner Methode ad hoc in eine fiir den vorliegenden Zweck nicht zu unwahrscheinliche Form bringen (S. 324). Now I contend that this gives a grossly unjust description of the paper in question. Had Ranke read recent literature, he would have been aware that the great difficulty with frequency distributions is to obtain the true values of the “moments” from records which merely give data for arbitrary “Spielraume,” often far too large and usually selected by the observer without any regard to the needs of the computator. My method is one based on the method of moments, but to deduce the moments from given data is the real difficulty which Ranke never for an instant seems to grasp or at any rate refer to. The standard deviation (which he appears to consider sufficient for anthropologists) will vary, and often very sensibly, with the nature of the grouping of the data. This difficulty was very present in my mind in 1894, and is constantly referred to in my memoir, the “allerlei grdssere und kleinere Anderungen in seiner Methode” are no changes in method at all but attempts to obtain some approximation to the true moments of the data. It was not till 1898 that Sheppard showed the correct manner of calculating the moments from the raw data in his important memoir on frequency constants*, for one type and one type only of frequency distribution. The curves calculated by Sheppard’s method, now in general use, would give better results undoubtedly than are to be found in my memoir of 1894. Further, however, Sheppard's method applies only to curves with high contact with the horizontal axis at both ends. It leaves us still in doubt as to how to find the moments of curves, which cut the axis at the end of the range or are asymptotic at one or both ends to the vertical axis. At such ends of the range, the real solution lies in recording the frequency for very small elements, but this was not provided in any of the statistics which were then before me. It is just these cases of limited range at one or both ends which present difficulty in the determination of the moments. The difficulty will be familiar to all statisticians, if it has escaped Ranke. To some extent it is met in my memoir on the systematic fitting of curves issued in April, 1902+. Yet granting all these difficulties what do we find in my memoir of 1894? An analysis of the cases in which range is dealt with seems justified by the charges made: Example I, Range determined of Cambridge Barometric Heights. There is nothing physically improbable in the result. Example VI. Range found for enteric fever runs from — 1°35 years to about 385 years. The probable error of the range is not given, but the whole difficulty * Proc. London Math. Society, Vol. xxx. p. 353 et seq. + Biometrika, Vol. 1. p. 265. 24—2 188 Skew Variation, a Rejoinder turns upon the great changes introduced into the range by different methods of calculating the moments. More recent investigations, in which the sexes are separated, the moments more accurately determined, and larger numbers dealt with, give far better results for zymotic diseases. I presume that one character being age, however, Ranke and Greiner would dismiss these data from consideration under any circumstances. Example VII. Guesses at 9 tints. Possible range 1 to 9, i.e. curve to run from ‘5 to 95. Observed guesses run from 1 to 8. Theoretical range of 11 instead of 9. The paucity of the observations gives a probable error of at least 20 to 30 per cent. in the determination of the range,.and the result is rather better than might have been anticipated. Example VIII. Ratio of forehead to body length in Carcinus moenas, observed range 30, calculated range 51. This range is probably not very close but it is not in any way that I can see impossible. The material is probably dimorphic. Example XI. H. de Vries’s data for Ranunculus bulbosus. Actually observed range 5 to 10 petals. Calculated range 5 to 11 petals. Example XII. H. de Vries’s data for a race of Trifolium repens. Actually observed range 0 to 10 high blossoms. Theoretical range in complete agreement. Example XIV. Pauperism percentages for 632 cases. Observed range 18 for the year 1891 dealt with. Calculated range 31. This range gives 2 units of negative pauperism. Its prcbable error is, perhaps, 14 per cent. It will be seen that out of the seven examples in which range is calculated only three reichen ins Negative, and that this reichen is well within the limits of the errors arising on the one hand from random sampling and on the other from the defective methods of determining the moments, which were alone available in 1894. While quite appreciating the honour done me when other workers use my methods, I must decline ‘to be responsible in any way for their application of my formulae. I have so often found that their failure to fit my curves is due to a misapprehension of my methods or to actual errors in arithmetic, that I have long given up any attempt to set such matters right. The frequent assumption made that statistical methods can be applied without adequate mathematical training is the source of most of the slips in this matter*. So far then I think we may conclude that Ranke is completely unjustified both in his statement that the Gaussian curve fully describes all the frequency that is of importance to the biologist, and in his attempt to discredit any result of scientific value which flows from endeavouring to measure such differences from the Gaussian law as we find in the distance between mode and mean, the skewness, the kurtosis and range of many actual frequency distributions. * A good illustration, by no means unique, of this is F. Reinéhl: Die Variation im Andriécewm der Stellaria Media, 1903. He finds it impossible to fit certain distributions with my curves, owing to ignorance of the full literature and to faulty determination of the moments. He then argues from this want of fit to biological conclusions. K. PEARSON 189 All the leading statisticians, from Poisson to Quetelet, Galton, Edgeworth, and Fechner, with botanists like de Vries, zoologists like Weldon have realised that asymmetry must be in some way described before we can advance in our theory of variation. In innumerable cases the important quantities measured by y, x and d actually exist; these have each their physical significance and they must be found. It is perfectly open to Ranke and Greiner to criticise my method of determining these quantities, but that they should shut their eyes to their existence appears to me only compatible with a very small acquaintance with the data of variation. Let us see now how various authorities have met this difficulty of skewness. B. The Gaussian Curve*. Gauss proceeds from the axiom that: The arithmetical mean of a series of observations gives their most probable value, i.e. the mean is the value of maximum frequency. This result is not axiomatic. It can only be a result of experience, and if it were true it would make the normal curve as much a result of experiment, ie. an empirical result, as any other proposed curve of frequency. Gauss’s proof demands, however, something more than this first statement. It involves (i) the equal probability of errors in excess of the mean and of errors in defect, (ii) the continuity of magnitude in the errors, and (iii) the independence of all the small contributions to the total error. Experience shows that Gauss’s fundamental axiom as to the mode and mean coinciding is not universally true. It is not true of errors of observations, it is not true of variations in living forms. Gauss reaches a differential equation which leads to the normal curve. His proof seems to me, as it has done to many others, quite invalid, because the equal probability of errors in defect and excess of the mean is not demonstrated, the possible dependence of contributory elements is not discussed, and the question of continuity of errors is not considered. C. (1) Laplace and Poisson. Laplace and after him Poisson took, I venture to think, much firmer ground. They did not assume (i) and (ii), but they did not realise the importance of (iti). They proceeded by evaluating the terms of the binomial : (p+q)”. * In writing for Germans I naturally spoke of the Gaussian curve. But I am not clear that precedence is to be given to Gauss. Gauss first gave a proof of the well-known equation y=y,e— 27/7" in his Theoria Motus Corporum Coelestium of 1809. This was three years before the publication of Laplace’s Théorie Analytique des Probabilités of 1812. But to give absolute priority to Gauss is to disregard Laplace’s earlier memoirs, particularly those of 1782, ‘‘ Sur les approximations des Formules qui sont fonctions des trés-grands nombres,” and its Suite du Mémoire of 1783. On p. 433 of the latter memoir Laplace actually suggests the importance of forming a table of the probability integral je rat. The Théorie des Probabilités reproduces the substance of this memoir, and on this account some writers have post-dated Laplace’s work, Gauss stated that he had used the method of least squares in 1795, but this does not necessarily involve a knowledge of the probability integral, and if it did, it is ten years after Laplace. On the whole my custom of terming the curve the Gauss-Laplacian or normal curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians. 190 Shew Variation, a Rejoinder Using Stirling’s theorem they showed that if the mth be the largest term in the binomial, then the sum p, of all the terms from m—r to m+~r is very nearly given by: 2 Pg 1 papas Pr = Nore e@ 20° da + ern OOM) olcigis ciciastiote eiaistere rele ere (vill), where ¢ = V Npgq. Here we have the first appearance of the probability integral as representing a series of discontinuous binomial terms. In fact when WV is fairly large Laplace and Poisson show that sums of terms of the binomial are closely given by the areas of the probability curve. It is an approximate result based upon Stirling’s theorem, and it does not for a moment involve making NV infinitely large, or the spacing apart of the binomial terms very small. This representation of a number of finite terms by the probability integral seems to be unfamiliar to Ranke and Greiner, but no practical statistician would calculate the sum of 7 terms in the binomial (p + q)* for even moderate values of VN. He would simply calculate the standard deviation ¢ = Npq of the binomial and turn up tables of the probability integral. This fundamental property of the normal curve, ie. that it closely represents a discontinuous series, is passed over in silence by my critics. It is the very purpose for which the probability integral was originally introduced by Laplace. In other words it arises without any consideration of (1) continuity of variation, or (11) equal probability of negative and positive deviations. It will be observed that the above approximation to the binomial, ie. to (p+q)* is symmetrical, but we can easily allow for some degree of asymmetry. Still writing o = Vv Npg, and for the binomial Bi =(1—4pq)(Npq), 1 = 8B: —3 = (1 — 6pq)/(N pg). I have shown*, y, being the maximum term in the binomial, that the rth term from the maximum is given by: i fT —r ~ 5¢3 (1 Bi + 4m) - NB, + ANB, & (1 - $6, + 4) ete. Yr=ye 7 xe ee Ee a2 (1x). The term in 7/o was, I believe, first added by Poisson, and expresses his attempt to allow for asymmetrical variation. Edgeworth expanding the exponential has adopted for his asymmetrical curve, a form easily deduced from (ix), Fe ee {1 - We, (2 =; = sr ataciel aa fe It will thus be seen that the normal function and the probability integral arise naturally from the expression for a single term or a series of terms of the binomial polygon. ‘This is their historical origin and the historical origin of the conception of asymmetrical variation. Instead of the complex form given above resulting from Stirling’s theorem, I approached the subject by looking at the relation of the * Phil. Trans. Vol. 186 A, p. 348, footnote. Frequency. 5 5 S °o K. PEARSON 191 normal curve to a symmetrical binomial in a totally different manner. I succeeded in showing that the ordinates and areas of the normal curve gave exceedingly closely the terms and sums of terms of the symmetrical binomial even for relatively small values of n. This had already been done by Laplace. The reader will realise this if he looks at the closeness of the normal curve with SSW N ag = aN, and the binomial ($+4)" with Y=10 in the accompanying Figure 3. But my method enabled me to give a simpler expression to the asymmetrical binomial Pele SS RSeaee fe SaaS aes INES EES Seaee 4 | EA s FPS Se aaeis 7 y SRE SS oe a eS a ea (eS Pap | ht FEN) se Ree eee ee es TER aE SA SESSReeeSs ae a ARR eRe Cia eaaeee Sa eaeees ee ; = 9 10 ~ = ~ 8 Number of Binomial Terms. Fic. 3. Comparison of Point Binomial 1024 (4+4)” with the Gaussian Curve. S.D. = /Npq=1'5811. Maximum Ordinate 258-35. than had been obtained by Poisson or Edgeworth using Stirling’s theorem. Figure 4 shows how closely the terms of the asymmetrical binomial 5000 (4 + 5)” and the sums of terms are reproduced by my curve of Type III, ie. “ 14 y = 153654 (1 iE 5) en, I had no higher ambition—nor could I have had one higher—than Laplace had when he discovered the normal curve. I wanted to find a close mathematical expression for the terms of the asymmetrical binomial for relatively small values of N. Now Laplace and Poisson had both retained the last of Gauss’s limiting conditions, i.e. they had by adopting the binomial supposed each increment of the deviation to be independent of previous increments. It seemed needful to me to get rid of this condition, and I therefore introduced instead of the binomial the hypergeometrical series. Here the successive increments are correlated. In order 192 Shew Variation, a Rejoinder to place this new representation on the same footing as the symmetrical binomial to which Laplace approximated with the normal curve, I deduced as I had done for the symmetrical and the asymmetrical binomials, curves which gave the hyper- geometrical series and the sum of its terms as closely as Laplace’s normal curve gave the symmetrical binomial. This is the complete history of the development of my skew curves. Before I proceed to discuss Ranke and Greiner's criticisms, I must remark that their attack on this point does not concern me only. Every practical statistician uses Laplace’s representation of the point binomial by the CE ae CER EEE EEEEt CEPA 1700 1600 1500 Frequency. o So ° =2 —1 (0) 1 2 3 4 6 6 qT 8 9 Number of Dice showing 5 and 6 pips. Fic. 4. 5000 Throws with 12 Dice. fe) © Points of the Binomial (4+ §)! x 5000. \l —. Curve y=1536'54 (1+) OE 5 Origin at the mean 2. Mode of curve at x= —-3, Frequency of each number of 5 and 6 pips -1 0 1 2 3 4 5 6 7 8 9 Ordinates of Binomial ae ae 0 561 1346 1480 987 444 142 33 6 1 °0 Areas of Curve a isd 557 1340 1493 982 443 139 37 8 1 0 K. PEARSON 193 probability integral, when he is discussing the probable errors of death rates, class indices and a multitude of other problems, and this when the binomial is skew and N relatively small! It is the whole theory of current statistics which Ranke and Greiner are tilting at when they object to the use of what is equivalent to the Euler-Maclaurin theorem, 1.e. the mathematical representation of a finite sum of terms by a definite integral *. There is another point also which may be noted here before we leave the binomial. The quantity 8, is a measure of the asymmetry. Now consider the ratio §,/n, or the ratio of this measure of asymmetry to the kurtosis. For the asymmetrical binomial we have (,/n =(1 — 4pq)/(1 — 6pq). Since p and gq if positive must give a product lying between 0 and }, this ratio cannot take any value between 0 and +1. Hence any curve which gives for 8,/y a value less than unity cannot possibly diverge from the normal curve in the direction of a binomial series. We shall see the application of this later. C. (ii) The Galton-McAlister Curve. I have already referred to the attempt of Poisson to give the skew binomial by an extra term applied to the Gauss-Laplacian probability integral. Quetelet endeavoured to meet the asymmetry of frequency distributions by placing graphically skew binomials on top of the frequency polygon—a very rough and somewhat deceptive process. The next step in the advance was taken by Francis Galton, who in 1879 suggested that the geometrical mean and not the arithmetical mean is likely to give the most probable result in many vital phenomena. Galton refers to Gauss’s assumption that errors in excess or in defect are equally probable, and says “this assumption cannot be justified in vital phenomena.” He cites especially the cases of errors in human judgment, guessing at temperatures, tints, pitch, etc. He appeals to Fechner’s law in its simplest form as evidence to the contrary, and placing the matter in the hands of D. McAlistert, the law of frequency = Ae 4 h (Joe 4) was deduced, and methods for fitting this curve were discussed. The curve is well-known in England and also on the continent §. It is therefore curious to find Ranke and Greiner attributing this curve to Fechner’s work which was not published till 18 years later. It was not till I had made a fairly complete set of experimental determinations of the kind supposed to give this curve, that I finally discarded it. Thus I asked audiences of 100 to 800 persons to match tints in several ways, I asked them to guess heights, to determine mid-lengths, to state which figures in randomly distributed series were most closely circles, h POO Sera * See Lacroix : Traité du Calcul différentiel et intégral, Tom. ut. p. 136. + R.S. Proc. Vol. 29, p. 365 et seq. “ The Geometric Mean in Vital and Social Statistics.” + R. 8. Proc. Vol. 29, p. 367 et seg. ‘‘ The Law of the Geometric Mean.” § It is cited by Kapteyn, for example. Biometrika 1v 25 194 Skew Variation, a Rejoinder squares, equilateral triangles, ete. etc. In all these results I found the distribution asymmetrical, but the most probable value was not the geometrical mean, nor the distribution the Galton-McAlister curve. One of the striking defects of the curve was its high contact at both ends. The distributions clearly often corresponded to curves in which the contour cut the axis at a finite angle. Another point was that the skewness was in the opposite direction to that presupposed by the reasoning from which the curve is deduced. It was precisely this experience which showed me that putting #=@log(&/a) in the Gaussian curve is not a sufficient generalisation. Ranke and Greiner, to say nothing of Fechner himself, are remarkably vague as to the accurate determination of the position and constants of the Galton- McAlister curve. McAlister gives no clear description of how the curve is to be placed if neither the mode nor start of the range is known. I think it desirable therefore, having regard to the inferences I wish to draw, to give the fitting by my method of moments. I write the curve: 1 ( e\2 Yo -—=— (log : y= Yo € 2 A) Ree OE Apa Sanhde 0006 (x1). Differentiation shows us at once that the distance «,,. of the mode from the origin is given by: nig = 0-8 he neue ese ee ee (xil). Integrating the expression Vy,/ =| yada we tind if NV = total frequency: 0 and generally : Thus the distance from the origin to the mean, @me., is given by Cig SIGS Weeds. ee eee (xv). Now write e* =) and we have if ~, be a moment coefficient about mean : Pa = be — py? = WA (A — 1) pls = ps) — Spy py’ + 2p? = a®V/A (At — BAZ + 2d) .».(XVi). fos = pes — Ayes’ py! + Ope’ py? — Buy! = at A? (AS — 4A? + GA — 38) Forming the usual constants of frequency we have: Br) ee = OSI 2)? eee eee (xvi1), n= 82-38 =(A—1) (4374+ 604 6) oe. (xvill), where By = p/p," va @'= Bina — Lng = O(N NN) eee (xix), Stas X= VR Fe (xx). K. PrARSON 195 We see then that the mean and second and third moments must be found to determine this curve. From the second and third moments we have 8,, whence by equation (xvil) \ is determined. The first equation of (xvi) then gives a=a/VrX(A—1). Then equation (xv) gives the distance of the start of the curve from the known mean. Further since X=e”, we determine c¢ and finally y, is determined by equation (xiii). There is no obscurity or difficulty with the fitting, if we use the method of moments. The cubic equation (xvil) is solvable at once either by Lill’s, Mehmke’s or Reuschle’s mechanisms. But what are the objections ? (i) The curve touches the axis at the end of the range. Skew curves extremely often cut it at a finite angle. (11) The skewness has a definite direction, which to be logically consistent we ought not to neglect, i.e. since A is always > 1, d remains always of one sign. (iii) Since X>1, y, the kurtosis, is always positive and the curve can only represent platykurtic distributions. It can never give a curve which deviates from the Gaussian curve in the direction of the Laplace-Poisson skew binomial for p >'2118 < ‘7887, because this is essentially leptokurtic. (iv) The range of skewness given by y is very limited. Differentiating » we find it is a maximum for ) = 1°7200 and this gives y ="2075. The Galton-McAlister curve cannot therefore describe any curve whose skewness does not lie between 0 and ‘2. A cursory examination of the observational results reached, shows that the skewness in all kinds of data over and over again exceeds °2. (v) 8, and 7 are both functions of X only*. Hence there is a relation between them or between 7 and y. That is to say the kurtosis is determined by the skewness. The kurtosis must vanish with the skewness. But experience shows that many distributions are sensibly symmetrical and yet have far from zero kurtosis, e.g. nasal breadth in English women, ete. ete. Finally consider the ratio §,/n. If we approach the normal curve as the limit to a point binomial (p+ q)* we have seen that Bi/n = (1 — 4pq)/(1 — 6g)... eee ccecevssceesceees (Xx): and this equals nothing if we take the symmetrical binomial. Otherwise it has a finite value depending upon the particular binomial along which we reach the Gaussian curve. The Galton-McAlister curve, if we make aV) infinite, but aVAVX—1 finite, approaches the Gaussian curve. * Actually it is By4 — 128,? + 1568,? + 648, — 4° + 124? - 36 + 188,27 — 68,n? -— 1178,n=0. I have to thank my assistant, Mr J. Blakeman, for much aid in the analysis of this section and the following section of this memoir, 25—2 196 Skew Variation, a Rejoinder Now for the Galton-McAlister curve (A+ 2)? M+ 3+ 6X4 6’ Bin i and this approaches the limit 9/16, its maximum value, when \ approaches unity. If we take (1 — 4pq)/(1 — 6pq) = 9/16, we get imaginary values for pandq. Thus while the normal curve itself gives an indeterminate value for 8,/n =0/0, and as Laplace has shown describes with fair accuracy any slightly skew binomial with large power, the Galton-McAlister curve cannot describe even approximately any skew binomial, however near to a normal distribution. On all these grounds we see that the “law of the geometric mean” fails to supply the fundamental need of describing the modal difference, the kurtosis and the skewness of actual frequency distributions. It cannot describe these physical characteristics of the frequency. C. (au) Pechner’s Double Gaussian Curve*. We have noted that the Gaussian curve was first deduced by Laplace to represent a finite number of the terms of a binomial expression, and that Gauss deduced it on hypotheses which amount to the following: (i) The arithmetic mean is the most probable value. (ii) Deviations in excess and defect of the mean are equally probable if of the same magnitude. (ii) The facility of an increment is the same for all values of the character. Now every one of these assumptions is negatived when the double Gaussian curve is used, and yet the Gaussian curve which ts only deduced by aid of them is adopted to describe what conflicts with its fundamental axioms. This proceeding is the reverse of logical. However, if the double Gaussian curve be adopted, there is absolutely no reason why we should adopt the rough process by which Fechner determines the mode and obtains the constants of the distribution. The fitting by my method of moments is perfectly straightforward, and as it leads to the points we have to consider it will be indicated here. Let the two half curves be: Ny, epi e Yi 7 =aantG) 12 o;", v> 0 Vr o, ra Po ceeeeteeseeeeeeeneees (xxil) Ns a; ere Yo € 20%, 2<0| ~ \/2ara, Then, since the modal value is common, o,/n,=0./n.. Further, the total frequency N=4(u4+). Now write «=V2/r and uw=0,—o, v=a,0,. Then taking moments round the mode we easily find: * Here again it is historically incorrect to attribute these curves to Fechner. They had been proposed by De Vries in 1894, and termed ‘‘ half-Galton curves,” and Galton was certainly using them in 1897. See the discussion in Yule’s memoir, R. Statist. Soc. Jour, Vol. ux. p. 45 et seq. K. PEARSON 197 fy = KU, pl =U, py = Qeu(ur+ Qv), py =3 {u? (Ww + 3v) + VI. Transferring to the mean we deduce: Mig UU ELE ce oes e's ecidtncanies sionals casa Mlea e anieae wars Poanien ea (xxi), Oily OU CUE tle to CICAU)) ass retort Bes peat ne Pa ach es Abe (xxiv), py = 8 (a? (u? + 80) + 07} — eu? (20? + 100 4+ Baru} oe. (xxv). fs, 3, @, Will be known quantities as soon as the frequency distribution is known. Now determine ®, = p2/u.2 and write = u/V ms, we easily find: 6 Jo } Ny . 29) v8 = = ee : ( 2) P+R 9 clei We cyan gure enee ee ee (xxvl). This cubic* gives by its real root the value of w=o,—o,. We then easily deduce o = EV po ({4 + (40? — 3) X7}F + 0’) ce . a naan neeeacs (xxvll) Oo = EV wy ({[4 + (4? — 3) 07}2 — 0) These determine the different variabilities of the two halves. Then ¥ 2No, 2Noaz Gpo n= =, Ny = FIN re (XXv111) 0,+ 0, O,+ 0% give the frequencies in each Gaussian curve, while PG TON TIN a saat iel dst educa Sans (xxix) fixes the position of the origin relative to the known mean value of the system. Thus the complete solution depends on a knowledge of the mean, and the second and third moment coefficients. As before the cubic is readily solved by Lill, Reuschle or Mehmke’s mechanisms. The analysis is now a little more complex than in the case of the Galton- McAlister curve. Write ¢=v/w=o,0,(0,—0,). Then we have: B= (e—14 2x21 — x2 +6) i Bo = {3B (1+ 3e+ &) — «(24 84? + 10e)}/(1 — + e)J Thus again we see that 8, and @, are both functions of € only, or the skewness is not independent of the kurtosis}. Whenever the skewness is zero, the kurtosis must also be zero or the curve be normal. Now consider the expression 1 — «?+ ¢ which we will write y, or, y = 36338 + o,0,/(0, — 2)”. The last term is positive or y must be >'36338. * This cubic was, I believe, first given by Edgeworth. + The actual relation is: 295217? + 625008,2 — 1105068, + 13468n? — 113458, + 15925n=0, which, as in the Galton-McAlister case, has no obvious physical significance. 198 Skew Variation, a Rejoinder Now: > 45352 05008 = fy 3= PE Y a and this with the above limitation to the value of y can never become negative. Hence the double Gaussian curve is, like the Galton-McAlister curve, invariably platykurtic. Now consider the value of ‘63662 °11477 00518 A= ——— +. 7 , my Hence the ratio e tends as y increases to take the value 140374. Equating this to (1 — 4pq)/(1 — Gpq) we see that the double Gaussian curve approaches the normal curve along the particular platykurtic binomial p=°8985, ¢=‘1015, or it cannot in the neighbourhood of the normal curve represent any skew binomial but this. Lastly it may be shown that £, has its maximum value when y = 36338 or its minimum value. Thus we find that the maximum possible value of 6, 1s about ‘99. In the same way the maximum skewness is 13236. These values are sufficiently high to cover the great bulk of cases, but I have found 6, = 4071 for scarlet fever incidence, = 1:9396 for age of brides who marry men in their 24th year aud =41683 for the distribution of lips in the medusa P. pentata. These exceptions suffice to show that the curve is not general enough. Summing up we conclude that the double Gaussian curve is not satisfactory because theoretically (i) It starts by denying the very axioms from which alone we can reach the Gaussian curve ; and empirically because (ii) It can describe no frequency distribution which cuts the axis at a finite angle, and such distributions constantly occur. (i) It is essentially platykurtic. Therefore it is not available for leptokurtic curves, nor even for any but very special skew binomials, i.e. those in which p does not lie between ‘2113 and ‘7887. As we approach close to the normal curve we get nearer and nearer to one definite point binomial, 1.e. that in which p =‘8985. (iv) There is always a relation between the skewness and the kurtosis, or these important physical constants are not independent. In particular we cannot have any form of symmetry but the mesokurtic. (v) The range of @, and of the skewness is fairly large, but frequency distributions actually occur markedly outside this range. (vi) Lastly, and of much import, the kurtosis can never exceed ‘8692, or the maximum value of 8,=8'8692. This degree of kurtosis is exceeded in a great number of distributions. Thus in the lips of P. pentata, in tint guessing, in the breadth of male English skulls, in the nasal breadth of female English skulls, in K. PEARSON 199 no less than eight of Duncker’s series in the case of Gelasimus pugilator and in various other distributions. In all these cases the platykurtosis is significant, and the double Gaussian curve fails us hopelessly. C. (iv) The Edgeworth-Kapteyn Curves. Kapteyn*, without recognizing Edgeworth’s priority, has proceeded in the manner indicated on p. 178 above. He assumes that some quantity « obeys the normal distribution y = ee J /Ino He then takes «= 4 (X)— M and reaches the frequency distribution : = ae ER OXGY Get A) Is Soe crerresrccnuca ean. aes art (XXx1). V2 Y Thus far (as we have already shown) nothing has been achieved, because this equation may by a proper choice of #’(X) represent any curve whatever. As Kapteyn himself says, following Edgeworth, “as #’(v) may represent any function, we see that the equation may be made to represent any curve whatever. Therefore it must be the most general form of frequency curve possible” (p. 17). There is, however, one point to be raised here. What is # of which the observed character X is a function? Is it, as in the explanatory illustrations cited by Kapteyn, another characteristic of the organism? If so we ought in some cases to be able to determine it. What is the character which obeys the normal law? For example, sagittal are in English women is almost exactly normal in its distribution, and nasal breadth is very asymmetrical. Shall we take w=sagittal are and X =nasal breadth and make c= EF (X)—-M? Now every biologist knuws that such a relation is not in the least true. No two characters in an organism are in any way connected by a mathematical function, such that when one is given the other is determined. The relation is always of the loose kind that we term association or correlation. X does not fix w, and a multitude of z’s with varying degree of probability are associated with a given w This correlation is often of a very low order. Between any two characters of a given organism, no such relation of perfect correlation as that involved in Edgeworth or Kapteyn’s relation has ever been discovered. Very imperfect correlation or at any rate all degrees of correlation have been invariably demonstrated to exist. The function # has no real existence as a biological entity. It is only a mechanism for introducing the normal curve, and is not a true character of the organism at all. Supposing, as in English female crania, nasal breadth is asymmetrical, what is the quantity which is symmetrically distributed of which nasal breadth is a function? It has no reality in the organism at all, and Kapteyn proceeds to make it still more impossible in the following manner. If « has * Skew Frequency Curves in Biology and Statistics, Groningen, 1903. 200 Skew Variation, a Rejoinder existence at all, its limits lie between — » and + 0, ie. the whole range of the normal curve. But in order to get a range limited at one end, not the whole series of values of X corresponding to # are taken. A value is selected for X, such that X becomes impossible after a certain value of «. In other words, w is a character which although following the normal curve is abruptly terminated as far as X is concerned at a value with a finite frequency! Kapteyn takes* v=(X +x«)1— M, where «x, g and M are to be determined from the data. For example, in Professor Weldon’s data for the measurement of foreheads of Carcinus moenas, Kapteyn (p. 39) finds that with our notation a= 002,204, Hj b 002,561, K=— 05781, C= eZ Thus when X =—x, we have «=— M=-—o roughly, or the Gaussian frequency curve for x is to be abruptly cut off, and about 15 per cent. of its tail discarded. If it be said that this could be achieved by natural selection of foreheads, the reply is the simple question: Please show what physical character in a crab is given by an abruptly truncated normal curve! The fact is no such character has ever been met with, and it must be recognised that w represents a wholly fictitious variable having no physiological relation to the character X at all, but introduced solely to reduce the frequency by hook or by crook to that fetish distribution the Gaussian curve. We can now sum up the objections to Kapteyn’s method. Theoretical: (i) There is no justification whatever for assuming that some character « actually exists which obeys the normal law of distribution, and that the observed character is a function of this. Some characters are found as a rule in any organism which obey the normal law, but no two characters in an organism have ever been found to be the one a mathematical function of a second, they are always imperfectly correlated. (ii) Kapteyn’s hypothesis involves if his normal character were a physiological entity, that distributions of organic characters should occur which would be represented by fragments of Gaussian curves, or such curves abruptly curtailed. We have no experience of such distributions in actual vital statistics. If they did exist they would contradict the first two axioms on which the Gaussian law itself is based, and would thus deprive that law of the sole justification for its application. As it cannot be supposed that all skewly distributed characters X in an organism are functions of one and the same a, for in this case they would be * This becomes the Galton-McAlister curve for the limit ¢=0. K. PEARSON 201 perfectly correlated with each other, which is contrary to experience, it must follow that if Kapteyn’s hypothesis were correct large quantities of characters distributed in truncated Gaussian curves ought to appear when we deal with variation. The total absence of such characters is evidence that the «-characters are shadow variables and of no biological import. (ii) The previous statements reduce Kapteyn’s special choice of F(X)=(X +0) to a mere artifice adopted to get an empirical curve of variation by Edgeworth’s hypothesis, Many other functions are @ priori equally valuable, and might be adopted to get curves of limited range, e.g. F(X) =(X + «)1(X —«#')”. The hypothesis gains nothing in logical consistency by its appeal to the Gaussian curve; that appeal is one adopted for convenience of fitting, and the sole test of Kapteyn’s curve is empirical goodness of fit. Practical: (iv) HKvery frequency curve should be a graduation formula. Kapteyn’s method of fitting is by equating certain total frequencies in order to determine his four constants. They thus fail to successfully smooth any special causes tending to exaggerate any particular frequency group. Such screening of special causes of frequency deviation is far less likely to occur when we use the method of moments, which is a true method of graduation*. (v) We ought in every law of frequency distribution to be able to judge of the effect of the wnit of grouping on the values of the constants. This has been satisfactorily achieved for, perhaps, the bulk of cases, when the method of moments is used by Sheppard’s corrections}. In Kapteyn’s process we have no means of ascertaining the extent to which the size of the unit of grouping influences the constants of his distribution. In his Example II., for instance, he takes his curve to accurately reproduce the total area of the group of houses under £10 annual value. What difference would * Thus Kapteyn deals with some statistics of the values of house property in England fitted by me (Phil. Trans. Vol. 186 A, p. 396). I specially state that £20 was the limit to taxable value, and that accordingly the frequency of houses immediately below this value will be exaggerated. Kapteyn’s method fails to indicate such a source of @ priori recognised irregularity. For example, one of his conditions is that the houses of value less than £10, i.e. more than half the total frequency, shall be identical in his result with the observed frequency. He thus cuts away at once any possibility of smoothing this group or allowing for the large probable error in it due to random sampling even. His method leads to a limiting house value of £2. 2s., while mine leads to £4. 4s. Mine corresponds to a weekly rent of about 2s.; his to a weekly rent of 1s. The latter rent hardly occurs in England unless the house is given in part payment of wages, or in charity. Kapteyn says that his distribution starts with a zero frequency, and mine with an infinite ordinate. ‘It seems hardly admissible that the latter solution can be in accordance with nature (sic) in this particular.” Why not? An infinite ordinate may and does in my case give a finite frequency. + See reference, p. 187. Biometrika tv 26 202 Skew Variation, a Rejoinder be made if the first group had included only houses under £5? We are unable to answer this question. (vi) very frequency curve should be determined by constants of which the probable errors are easily deducible. The method of moments admits of the probable errors of the moments being easily determined (see Biometrika, Vol. 11. pp: 273 et seq.). My system of skew curves gives all the constants in terms of moments whose probable errors are known. The moments in Kapteyn’s theory depend on the integration of : ioe) n | (a + Mya e-*e? da, -M and there is no means of readily evaluating this integral. Jn fact the arithmetical mean*, the standard deviation, the skewness and the kurtosis, and the modal divergence are unobtainable from the constants of Kapteyn’s theory. This seems to me sufficient to deprive the method of any practical significance even as an empirical representation. It has further been shown by Sheppard that the probable errors of constants determined by class frequencies (partial areas) are higher than when these constants are determined by the method of moments. We may give the above statement a separate paragraph as: (vu) The fundamental physical constants of the frequency distribution are not determinable from Kapteyn’s empirical curve. To illustrate the results of this want of a knowledge of the probable errors, I turn to the three illustrations given by Kapteyn. Example (1). Observations on the Threshold of Sensation. Kapteyn himself shows that his solution is hardly less satisfactory if he uses the Galton-McAlister curve (our equation (xi) p. 194). He does not therefore know whether g =:00 and q=— 04 differ within the probable error of q. Example (ii). Valuation of House Property. Kapteyn fits this with a Galton-McAlister curve for his q comes out ‘00. Owing to the difficulty in calculating moments, we cannot do more than approximate to the value of y the skewness in these data. I make it 18. It is certainly well over unity. We have already seen that it is impossible for a Galton-McAlister curve to give a skewness above *21. The apparent agreement Kapteyn finds for the frequencies is not therefore sufficient evidence that the fundamental constants of the distribution will be really given by reasonable values. Example (iii). Foreheads of Carcinus moenas. Kapteyn fits these first with qg=221. “The agreement seems satisfactory.” Then with q=0, or a Galton- McAlister curve, “The representation is hardly less satisfactory.” Then with * «The arithmetic mean of all the X’s cannot be generally found in a simple and rigorous way,” Kapteyn, p. 44. K. PEARSON 203 q=«, “The representation, though sensibly less satisfactory than that by the [previous] solutions, is still pretty close.” A method by which a fundamental constant of the distribution may take any value between 0 and and still give a “pretty close” representation, must I think condemn itself. Such a statement demonstrates effectively that the author has not yet determined numerically or even approximately in his own mind the probable errors of the constants he uses. It will be seen that as far as the three illustrations Kapteyn himself gives go he has not advanced the matter beyond the Galton-McAlister curve. That curve fits reasonably (according to Kapteyn) all his three series. But the skewnesses of the three series are respectively ‘72, 1°8 and 32. I have calculated these roughly, but I think there can be no doubt of their approximate correctness. In every one of these cases: the skewness sensibly exceeds the maximum limit of skewness, ie. ‘21, possible for the Galton-McAlister curve which Kapteyn applies to them *. C. (v) The General Results which flow without the Third Gaussian Aaiom. It seems to me accordingly that very grave objections can be raised not only from the theoretical but from the practical standpoint to the methods I have discussed which attempt to allow for asymmetry, Le. (i) The Galton-McAlister Geometrical Mean Law (ii) The Galton-Fechner use of Half Gaussian Curves, (iii) The Edgeworth-Kapteyn use of transformed Gaussian Curves. All these experienced statisticians differ in toto from the opinion of Ranke and Greiner—that we need not trouble about descriptive curves for asymmetrical distributions—but their methods seem to me unsatisfactory theoretically and insufficient practically, because they still make a fetish of the Gaussian axioms. They do not return to the Laplace-Poisson method of replacing those fundamental axioms by more general conceptions. If a Gaussian curve does not fit, they will consent to deduce their own curves from a truncated Gaussian curve, which some shadow variable of the mathematician is supposed to follow, and of which we have no experience in any organic characters hitherto measured. Indeed if we had such experience, it would at once negative the very axioms on which the Gaussian curve is based. Now it seems to me that all these attempts, whether embodied in the general method of Edgeworth or in the special hypotheses of Galton-McAlister or Kapteyn, amount to abolishing the third of the Gaussian assumptions, namely that small increments of the variable or the character are independent of the total already reached. That is to say that they amount to saying that increments of the * Tam unable to say how far the general form of Kapteyn allows for the requisite range of skewness and kurtosis, because neither the modal difference, nor the standard deviation, to say nothing of the higher moments, can in general be evaluated. 26—2 204. Shew Variation, a Rejoinder variate are correlated with the value of the variate already reached*. Galton and Fechner made the increment proportional to the variate. But in our ignorance of the actual nature of variation in organisms, we have no reason at all for making such a narrow assumption. We can to please our critics put the matter as I have already indicated in the Gaussian form. We simply assume that if the causes of variation in the immediate neighbourhood of the character #, remained the same for the whole range they would give a normal curve, hence we should have a relation of the form: j They do not, however, remain the same; the tendency to vary at x is a function of x, in other words ¢=o,V F(z), where F(x) is an arbitrary function. We have then 1dy _ x ydx =o 2 (a) This is a result as general as Edgeworth’s and more so than Kapteyn’s or Fechner’s. We now take the simplest possible functional series for F'(«), ie. =Ap tet A,u7+.... The coefficients a), @,, dz... @, can be found at once in terms of the moments}, and my special curves result if we stop at a. Against going to higher powers are the objections I have raised in my memoir on skew correlation+, namely (i) that the higher powers involve moments of the 5th and higher orders and their probable errors are very large, (ii) that it has not yet been shown that going to a, does not suffice to describe all the types of frequency which occur in common practice. The above is the simplest and most general form into which I would put my theory of asymmetrical frequency for those who feel compelled to approach all frequency from the Gaussian standpoint. D. Specific Criticisms of Ranke and Greiner on ny Theory. I think these may be summed up as follows: (a) That all distributions of variates are continuous, and that accordingly no curves, however closely they may approximate to finite discontinuous series like the binomial and the hypergeometrical series, can be applicable to variation in nature. * Suppose we draw r cards from a pack, and wish to consider the chance of s being of one suit, we may do so by drawing one card at a time, observing it and returning it, and then drawing again. Here there is not correlation between the successive contributions to r. Or we may draw the r cards, without replacing the individual; here the successive contributions are correlated with the previous contributions, and the third Gaussian principle is upset. + “Mathematical Contributions to the Theory of Evolution, XIV. On the General Theory of Skew Correlation and Non-Linear Regression,” Drapers’ Company Research Memoirs. Biometric Series, 1. (Dulau and Co. 1904) p. 6. t+ Ibid. p. 7. K. PEARSON 205 (b) That what I term the “number of cause-groups” must be infinite in number, for without such infinity it is impossible to reach continuity. (c) That looked at from the standpoint of binomial or hypergeometrical series the constants of some one or more of my curves may become unintelligible. Now not one of these objections has any application to the method which has been used in this paper to deduce the differential equation to my curves. But I still think there are very grave objections to every one of the above statements. To begin with (a). We meet in an immense variety of living forms with discrete variates. For example, the number of teeth on the rostrum of a prawn, the number of lips of a medusa, the number of veins in a leaf, the number of glands in a swine’s foot, the number of tentaculocysts in Hphyra, the number of individuals in a litter, the number of bands on snails’ shells, the number of somites in the body of an earthworm, the number of petals or sepals in a flower, ete. ete. Are we to put all these distributions of variation on one side because Ranke and Greiner hold that all distributions of variates are continuous? We have in these cases probably continuous causes producing discontinuous distributions. Are we not to use the areas of a continuous curve to give the frequency of such discrete variates ? Consider for example the function given by: en vr pr =a oa * ln—r it V2are, n Ve NxS |p" Cea - rie . 0 This is compounded of n+1 normal curves, the area of the (7 +1)th normal : n ; ee curve being Np"-"q” ie - , Le. the (r+1)th term of the binomial V (p+ q)”, and |n—r |p this (r+ 1)th normal curve has a, for its standard deviation. The origin of the system is at the mode of the normal curve corresponding to r=0, and the means of these normal curves are spaced equal distances c apart. When every o, =0 we have discrete variation. When c, is small, less say than $c, it would probably be difficult to distinguish the result from discrete variation. Enlarging o, we pass on till we get a system which it would be practically impossible to distinguish from continuous variation, even if n were only moderate in magnitude. I lay no stress whatever on the above expression because I am in no sense pledged to any Gaussian curve, but it illustrates well what I want to express: namely, in actual nature the frequency might fundamentally fall on certain values of the character, but that the effect of nurture, environment, and growth may well scatter the values of the variable round the fundamental value, so that continuity of variation is all that can be actually observed. The number of somites in an annulose animal is discrete and probably inherited, but the length of the body may appear as a continuous variate. I do not think for a moment that the distinction made by Ranke between discrete and continuous variation, 206 Skew Variation, a Rejoinder and the further statement that variation in man is essentially continuous, is at all valid. Our units of grouping for the numbers available are not very fine, we can hardly successfully classify a few hundred observations into more than 20 groups, and with this unit of grouping it would be practically impossible to distinguish between the apparently continuous distribution and a discrete distribution of a similar number of classes with the variates modified by growth, nurture or any other scattering tendencies. Ranke appears entirely to have overlooked the current biological theory of inheritance summed up in the words inheritance by deter- minants. Such theories, whether they be those of Weismann or Mendel, lead us directly to discrete variation*., The discreteness of the variation will be more or less, in many cases probably entirely, obscured by the environmental influence. In such cases the number of fundamental cause-groups is not infinitely great, and Ranke is overlooking current biological views when he asserts that we must take “die Anzahl der Elementarursachen selbst als unendlich gross und die Grésse der Wirkung der einzelnen Ursache als unendlich klein.” If the number of determinants which fix a character is finite, that character would correspond to a discrete variation of limited range. If the number of determinants be very large, the distribution would by Laplace’s theorem be represented more and more closely by the normal curve. I toss ten coins into the air and for every head in the result I pay a gramme of gold-dust, the frequency distribution of gold-dust would closely be given by the terms of the binomial (4 + 4)” as in the points of our Fig. 1. But suppose instead of weighing my gramme of gold-dust accurately, I give a “handful” of sugar. If 6 heads turn up I give six handfuls of sugar, but each of these will not be exactly my standard mean handful. I am unlikely to give either.five or seven standard handfuls as my six approximate handfuls, but in some cases even these might be possible ; we pass in fact from discrete to continuous variation, and the multimodal character of the discrete variates will disappear with the roughness of the handfuls, or have the petky appearance of random sampling. The total area up to any midpoint between two discrete groups s and s+1 will be given by the continuous integral which represents the first s terms of the binomial. If we have two such total areas, one up to the midpoint between groups s and s+1 and the other up to the midpoint between groups s+1 and s+2, then an interpolated area between these values as given by the continuous integral will be sensibly the same as if, c being the unit of discrete difference, we determined a curve corresponding to the mean binomial frequency in the Spielraum c, Le. by simply fractionising 1, i.e. we replaced the factorials by Stirling’s theorem or used I’ functions, and supposed 7 to change continuously from s tos+1. This is * Thus I have shown that a generalised Mendelian theory leads directly to skew binomial distributions of characters in the general population. Phil. Trans. Vol. 203 A, pp. 53—86. K. PEARSON 207 merely another way of looking at the change from discrete to continuous variation, due to the influence of a multitude of causes on the discreteness of the variates which fall into a given Spielrawm. I still find nothing absurd in the statement that the actual effect of the scatter is sensibly equivalent to a fractionising of the indices. It is simply equivalent to the statements, (1) that the ordinates of the Gaussian curve closely give, even for small values of n, the terms of the binomial, (ii) that the ordinate of the Gaussian curve between two terms of the binomial closely gives a fractionised binomial term (owing to Stirling’s theorem being true for fractionised factorials or I’ functions), (111) that we have no knowledge of how the “scatter” within the Spielrawm may be distributed so as to give a continuous effect*. Now these points are not in the least needful for the deduction of my skew curves, they are merely given here because in our complete ignorance of the nature of the causes, hereditary and environmental, which produce continuous variation, I think we have no warranty for saying that a limited number of cause- groups is impossible, or that no such limited number of fundamental cause-groups could give a continuous variation. In the present state of our knowledge we cannot agree with Ranke in sweeping away as impossible all the discreteness which follows from determinantal theories of inheritance. We cannot afford to be dogmatic as to the continuous or discontinuous character of the ultimate sources of variation and any effective theory must like the Laplace probability integral be equally applicable to the sum of a discontinuous series as well as to the areas of a continuous curve. (b) Any finite series of cause-groups, Ranke tells us, must lead to dis- continuity. I have endeavoured to show above that the discontinuity may be as real and yet as undetectable as the distribution of lengths, say, of the vertebral columns of sharks which yet depends on the number of discrete vertebrae, with a scatter of their individual sizes. But Ranke’s argument in itself is a false one, many discontinuous systems lead at once to continuous distributions. In our ignorance of the exact sources of variation, all we can do is to show that a limited number of cause-groups can quite well lead to continuous variation. To take a perfectly arbitrary illustration, suppose that a character can only take values lying between a, and a, and that this character is to be settled by the determinants derived from s+1 ancestors, 1.e. suppose all but s+1 to be cast out in the successive divisions of the germ-cell. Then it by no means follows that the character will be a blend of these s+ 1 determinants, one or other of them may be dominant. It does not follow that the dominant one represents either the one with the least or the greatest value of the character. It might be the one with s—vr determinants below and r above it. * For example I have a variety of Binomial machines or ‘‘Quicunxes”’ like that figured in my memoir, Phil. Trans. Vol. 186 A, Plate 1. Fig. 2, p. 414. It is quite possible to arrange a quicunx in which there are only a limited number of compartments, but in which the top of the seed in these compartments is not horizontal, but gives a continuous curve, e.g. the greater air draft of the greater frequency might be used to pile up the material in any receptacle on the side of the greater frequency. 208 Skew Variation, a Rejoinder In this case the frequency of mdividuals with character w would be given by the curve Y = Yo (4% — My)" (Ay — )". This is a perfectly continuous curve, and one of my series of skew curves. Thus it is quite conceivable that a finite number of fundamental cause-groups should lead to an absolutely continuous distribution *. Now how does Ranke treat this illustration? He first states that all continuity must involve an infinite number of cause-groups, or variation in man _ being continuous, must be associated with an infinite number of cause-groups. He had this very case before him, and yet he writes: Die Analyse der Elementarursachen ergibt uns also unweigerlich die bisher immer angenom- mene unendliche Anzahl derselben, die unendliche Kleinheit der Wirkung jeder einzelnen Ursache und die Kontinuitit der moglichen Wirkungserade. Sie ergibt also wirklich die Verhialtnisse, die wir zum Verstiindniss der kontinuerlichen Variationskurve ganz unumgiinglich nétig haben. Denn wie soll eine kontinuerliche Kurve sich aus der Kombination endlicher Bausteine ergeben (S. 321) ? Ranke only gets out of the difficulty by asserting that since the number of causes is finite, but must be infinite for variation, my continuous curve based on a finite number of cause-groups cannot represent variation! A more remarkable specimen of circular reasoning can hardly be conceived. The fact is that Ranke suffers from the old third Gaussian axiom, 1.e. the supposition that the increments that go to build up the variate are independent of each other. The fundamental cause-groups are by no means Bausteine in the sense that the total variate is the sum of these Bausteine placed on top of each other! The causes determine the magnitude of the variate, but not at all necessarily by their sum. (c) Ranke asserts that some of my curves have constants which if we endeavour to interpret them from the standpoint of the binomial give impossible or improbable values for the constants. The answer to this is that the series were only the scaffolding to deduce the curves. The differential equation to the curves contains the limit to a good many other frequency systems which directly diverge from the fundamental axioms of Gauss. I used the original series as a means of dispensing with the Gaussian axioms in familiar cases, but the result reached involves a good deal more than can be interpreted by the original series. Ranke can only see absurdity in a binomial with a negative p or g. But the nature of the sources of variation is so little known to us that we cannot possibly assert the absurdity of such values. We may not indeed be able to directly interpret them in the case of man, say, but they occur and recur in chance investigations. I will illustrate this in one case only, but such will demonstrate the required possibility and dispose at once of Ranke’s argument as to absurdity. * Making r and a, infinite, but s—7 finite, we get the curve I have deduced as the limit to a binomial of finite power. In other words, that curve is also shown to correspond to a possible continuity. K. PEARSON 209 Suppose an organ to require the conjunction of exactly n determinants of one kind to fix it, but that the size of the organ depends on how soon this conjunction takes place. Let D be the necessary kind of determinant and 7 the chance that it is left in the right position after each operation, say a cell-division or cell-fusion. Let D’ be any other kind of determinant and « its chance of appearing, so that «+a=1. Then if D appears n times in the first 2 operations, we have a certain size for the organ, if in the first n+1 operations another size, and if only in the first n +7 operations a third size. But the chances of these respective appearances are the terms of the series marc + OY) ied 4. bob aE ie aE NOD) one (n a ie Le eae eee Ee ie =m'(l—x«)y"=([-— -— — = —q)-”. se a (1—«) (= “) (p— 4), say, where p—q=1. Here p and q have lost the condition that they are both to be less than unity. I do not for a moment suggest that this is the real interpretation of a binomial with a negative g. I only assert that because we are as yet unable to certainly interpret such expressions, the absence of interpretation does not involve the absurdity which Ranke postulates. There are many other matters to which I might justifiably take exception in Ranke and Greiner criticism*, but I think I shall have said sufficient to convince the impartial reader of the following points: (i) The great bulk of modern statisticians are agreed that the Gaussian law is absolutely insufficient to describe observed facts. They may disagree as to the method of supplementing it. I do not think that the opinion of Ranke and Greiner can possibly weigh against those of Poisson, Quetelet, Galton, Edgeworth, Fechner and Kapteyn—all authorities who have had to deal for years with statistical data. (ii) The original use of the probability integral (the areas of the Laplace- Gaussian curve) as introduced by Laplace was to represent the sum of terms of a discontinuous series. To the mathematical mind there is no absurdity in this replacement of discontinuity by continuity; it is the basis of the Euler-Maclaurin theorem. (iii) The dogmatic assertion of Ranke that variation in man is due to an infinite number of infinitely small fundamental cause-groups, simply neglects the * For example, all the discrete variates mentioned on p. 205 have been dealt with by biometric writers, and many others besides, yet Ranke speaks as if such writers had not dealt with discrete variation. He speaks of Ludwig’s multimodal curves for flowers as if there had been no controversy as to the actuality of the ‘ Fibonaccizahlen’”’ modes, when due regard is paid to homogeneity of season and environment. He speaks as if Johannsen had demonstrated normal variation in his ‘‘Erbsenspopulation,” when he has really applied no valid criterion whatever to test for asymmetry, etc. In short he seems to me to have neglected a great deal of the modern literature of the subject, and, if I may venture to say so, to write over-dogmatically on what he has read, Biometrika Iv 27 210 Shew Variation, a Rejoinder whole determinantal theory of inheritance. A complete theory of asymmetrical frequency must describe in the manner of Laplace’s probability integral either continuous or discontinuous variation. (iv) The apparent or practical continuity of many variation data may be due either to real continuity or to discontinuity effectually masked by: (a) the relative paucity of material and roughness of our measurements, which compels us to divide it into groups of the same order of number as the number of determinants, (b) the influence of age, nurture and environment superposed upon the pure hereditary results, or (c) the fact that many of the characters measured by us are built up of a larger or smaller, but not necessarily an infinite, number of simple organs or characters, which may possibly individually have discontinuous variation *, : (v) The assertion of Ranke that a finite number of fundamental cause-groups must lead to discontinuity 1s disposed of by illustration. It is quite possible to invent a great variety of determinantal systems—the number of the determinants being finite—which lead to continuous variation. In our present ignorance of the sources of variation, especially of the mechanism of inheritance, it would be idle to lay weight on any special interpretation of this kind. It is important, however, to observe that continuity or discontinuity of variation are not, as Ranke asserts them to be, associated with the finite or infinite number of the cause-groups. (vi) The absurdity which Ranke finds in the values taken by some of the constants of my curves, exists only when a very narrow view is taken of the sources of organic variation. A binomial series with negative power or with negative p or q is capable, as is shown in this paper, of perfectly rational imterpretation. But in the present state of our knowledge it would be idle to specify any particular interpretation as the correct one. (vii) The problem of variation can be looked at in the following manner without the least loss of generality. Modify Gauss by replacing his third axiom, the independence of contributory increments to the variate, by the postulate that the increments are correlated with previous increments}. Start with any binomial and we reach the generalised probability curve for an infinite number of cause- groups: RT yda of (alor)’ where f is an arbitrary function. This theory covers Galton, Edgeworth, Kapteyn and Fechner. Expanding / (x/c,) in a series of ascending powers of /o, we have * Ranke has quite overlooked the work by Galton and myself on the discontinuity of the series of individuals even when the population obeys the Gaussian law. See Biometrika, Vol. 1. pp. 289—299. + Ranke apparently considers that (p+q)" with p, q and n positive is interpretable. A little philosophical consideration will show that it is merely ‘‘ familiar,’’ not really intelligible. There is no physiological meaning in p, q, n, and we cannot as yet associate them with any true organic mechanism. + This postulate of course abrogates the first two axioms of the Gaussian theory as well. K. PEARSON PA my generalised probability curves*. A very few terms of the expansion, however, suffice for describing practical frequency distributions. If we keep only three terms, we see that the same system of curves suffices to describe continuous and discrete variates is because Ranke insists on an infinity of cause-groups and supposes no continuity can arise without them— an important point. If I lay stress upon this method here, it “a truth Looks freshest in the fashion of the day.” (viii) The important physical constants of a frequency distribution are those which can be determined with the least probable errors. The probable errors of the moment coefficients increase rapidly with the moments. Hence the important physical constants are those which depend on the low moment coefficients, 1.e. on the early terms of the expansion of f(a#/c). Now these physical constants are (a) the mean, (b) the modal difference or distance of mean from mode, (c) the skewness, and (d) the kurtosis. We may replace either (0) or (c) by the standard deviation. Experience shows that these four physical constants are certainly independent. The constants of my skew curves directly give them and we are able to determine by their probable errors whether they are significant or not. (ix) With regard to the other theories discussed I have shown: (a) That the Galton-McAlister curve, ascribed by Ranke to Fechner, is not applicable to a great number of cases, for its kurtosis is a function of its skewness and its skewness cannot exceed ‘21. (b) That the double Gaussian curves due to Galton and Fechner are illogical, because they reach a Gaussian result by rendering invalid every one of the Gaussian principles. Further, the skewness is always a function of the kurtosis and the kurtosis cannot exceed ‘87, a degree which is exceeded in a great variety of data. (c) That the Edgeworth curves as developed by Kapteyn fail from the logical standpoint, for they appeal to a truncated Gaussian distribution which has never been observed in experience. They are not true graduation formulae, and are obtained in such manner that it is not possible to determine any one of the chief physical constants or evaluate their probable errors. Further in the examples given by Kapteyn they all sensibly reduce to the Galton-McAlister curve. But this curve has in every one of the cases dealt with by Kapteyn a skewness significantly less at a maximum than is required by every one of the statistical series involved. Finally it seems to me that all discussion of asymmetrical frequency must turn in one form or another on the proper form to be given to F(z) in the equation 1 dy een ee a yd“a a7 F(a)’ See ‘‘ Mathematical Contributions to the Theory of Evolution, XIV.” Dulau and Co., London. 27—2 212 Skew Variation, a Rejoinder If we assume it to be F(a) => (a,a”) as I have done, we fall back on the normal curve when a, for r>0 is zero within the limits due to its probable error. Ranke, if he wishes to demonstrate the Gaussian law as general, must show this to be the case. It has been over and over again demonstrated that a, a,, etc. differ significantly from zero for a great variety of series. Another advantage of the form F (x)= (a,a") is that it covers as I have shown discrete as well as continuous variation. Considering =o, F(a) as the standard deviation of the “instantaneous Gaussian curve,” we see that the “instantaneous Gaussian curve” varies from one position to a second, like the “instantaneous ellipse” of the astronomers. A reasonable first hypothesis to make is that the local mean square deviation o? is independent of w, we obtain the Gaussian curve. A next assumption is that it is a linear function of «—perhaps it would be better to say that its mean local value is a linear function of , i.e. the mean square of the local variability o? is correlated lineally with 2 This gives my curve of Type III. The next easiest assumption is to suppose the regression line of o on x to be parabolic. In this case we obtain the remainder of the curves treated in my II. and XI. memoirs. If we stop at ag we have what I have termed the skew frequency curves of the qth order*, and we . see that this involves a regression curve between the square of the mean local variability and the character of the gth order*. I see, however, at present no practical necessity for proceeding beyond skew curves of the 2nd order, although I propose shortly to publish a discussion of skew curves of the 3rd order illustrating some theoretical points which arise in their discussion. To sum up I think Ranke’s criticism fails (a) because he has disregarded the universally recognised need of modern statistical science for asymmetrical frequency curves, (8) because he has not appreciated the mathematical transformation by which a number of finite terms are replaced by an integral expression, (y) because he has not realised that modern theories of heredity lead directly to discontinuous skew distributions, (6) because continuity does not depend upon infinity of fundamental cause-groups, and lastly (e) because, and this may be due to my fault in the first deduction of my curves, he has quite failed to see either their scope or their real generality. * «Mathematical Contributions to the Theory of Evolution, XIV. On the Theory of Skew Regression.” Dulau and Co. VARIATION AND CORRELATION IN THE EKARTHWORM™. By RAYMOND PEARL anp WILBUR N. FULLER. 1. Introductory. THE purpose of this paper is to present the results of a study of the variation and correlation exhibited in certain specific characters of the common garden earth- worms. The earthworm which occurs most abundantly in the region about Ann Arbor and with which we have dealt in the present work, has usually been identified as Lumbricus herculeus, Sav. or L. agricola, Hoffm. The counting and measuring and a portion of the computing were done by W. N. Fuller in connection with certain class-work in the University. The values of the constants were twice independently computed, and the paper written by R. Pearl. There were two principal reasons which led to the undertaking of this work. In the first place we wished to determine if possible more exactly than had previously been done the typical condition of the form of earthworm commonly found in this portion of the United States. The earthworm is such a generally used type form, both for laboratory teaching and for investigation in a number of fields, notably regeneration, that an exact determination of the existing type and variation with reference to the several specific characters seemed decidedly desirable. Secondly, the writer hoped to get from this form some light on certain general biometrical problems on which he is collecting data. The material for the work consisted of sexually adult specimens only. The criterion used to determine whether an earthworm was sexually adult, was the presence of a fully developed clitellum. That this is a justifiable criterion has been shown by Cole. * Contributions from the Zoological Laboratory, University of Michigan, No. 89. + Zool. Anz. Bd. xvi. pp. 440 and 453, 214 Variation and Correlation in the Earthworm The worms were collected on rainy nights, in gardens and lawns near the campus of the University of Michigan. On such nights the worms leave their burrows, and it is comparatively easy, after some experience, to pick up a large number in a reasonably short time. The worms were brought to the laboratory, stupefied in weak alcohol, straight- ened to their full length, injected with weak chromic acid or formalin, and hardened and preserved in either alcohol or formalin. As the worms are stupefied in alcohol the muscles become relaxed, and it is easily possible to straighten them to their full length. They are hardened in this straightened position. The specimens used in this investigation were subsequently used by students for laboratory work. The measurements and counts were made during the first half of the fall semester of the college year, 1902-03. Quantitative determinations of the following characters were made on each worm : 1. Total number of somites in the body. 2. Length of body in centimetres. 3. The number of somites from the anterior end of the worm to the most anterior somite of the clitellum. 4, The number of somites included in the clitellum. 5. The position of the genital openings. The following points regarding the methods used in making the determinations may be noted. In all counts of somites which began at the anterior end of the worm the prostomium was not counted, but the first somite which was complete on the ventral side of the worm was counted as 1. All the other somites in the body were included in the count. All worms having spiral segments or other segmental anomalies were excluded in this work. Such worms were preserved for use in connection with a general study of teratology in the earthworm. In making the counts it was found very helpful to use a “tally register,” such as is used in counting telegraph poles, and by gate keepers at places of amusement for counting people. This instrument has been found very useful in collecting biometrical data on integral variates for a variety of characters (e.g., scale rows and gill-rakers in fish, etc.). Anyone who has ever attempted to make a series of counts of repeated like characters knows how difficult it becomes in a short time to “ keep the place,” both mentally and on the specimen. All this difficulty is overcome by the use of a tally register, and its use makes it possible to work for a long time without becoming unduly fatigued. The only thing necessary to secure both speed and accuracy is that the operator have some preliminary practice. With regard to the character “length of body” considerable care had to be exercised in order to get results in any way reliable. Of course, it would be nearly or quite impossible to get any reliable measure of the length of an earthworm RAYMOND PEARL AND WiLBur N. FULLER 215 while it was alive, on account of the varying degrees of contraction of the muscles. This source of error is present to a certain degree in worms fixed and hardened in the way described above, but it is very greatly reduced im the fixed as compared with the living worm. Since, however, there was present in all cases a source of error on account of different degrees of contraction, no attempt was made to obtain close measurements of length. What was desired was to get all the worms classified into a comparatively few, rather large length groups, for purposes of correlation. This end was attained by using as our smallest unit of measurement a distance of 2°5 centimetres. A worm was laid on a measuring stick divided into 2°5 cm. spaces, its length observed, and the record made by writing on the record sheet a letter with an algebraic sign, denoting the division of the measuring stick within which the end of the worm fell. Thus a record of C— indicated that that particular worm fell, in length, somewhere between 10 cm. and 125 cm. In making the computations all worms falling within the class C'— were considered to have been 11°25 cm. long, or in other words it was assumed that within a group the lengths were arranged symmetrically about the centre of that group. The assumption is, of course, the same as is usually made in dealing with the frequency distribution of graduated variates. In the counts for the position and number of somites in the clitellum the following arbitrary rules were followed. The number of somites from the anterior end of the worm back to the fist somite on which any chitellar tissue was present, was recorded as “Number of somites to clitellum.” Al somites on which any clitellar tissue was present were recorded as “Number of somites in clitellum.” Some such arbitrary rules were necessary in dealing with the cases in which the clitellum included only a part of one or both of its terminal somites. A record was kept, according to a plan which will be described later, of the exact extent and form of the clitellum in each particular worm. The only variations which were found in the collection of worms with reference to the position of the genital openings were regarded as teratological and the specimens exhibiting such variations were excluded from our tabulations. Abnor- mality in position of the genital pores in the very few cases in which it appeared was found associated with other abnormalities. The position of these openings is perfectly constant in normal worms. There is no “fluctuating” or continuous variation. On this account these characters will not be discussed further in this paper. It should be stated that the collection was made entirely at random. No selection on the basis of any characteristic of the worm, with the exception of the presence or absence of the clitellum as noted above, was exercised in forming the collection. The question of the homogeneity of the material will be considered farther on in the paper. We may turn now to the results. 216 Variation and Correlation in the Earthworm 2. Total Number of Somites and Length of Body. In Table I. is shown the frequency distribution for the total number of somites in the body. TABLE J. Total Somites. | NBomites, | Frequency || *ymver ot | Frequeney 7 1 130 i 80—88 = 181 4 89 2 132 7 90—92 — Tat: 4 93 1 134 3 94 — 135 5 95 1 136 6 I6—98 — 137 9 99 2 138 10 100 1 139 6 101 = 140 5 oe 2 14 17 103 es 142 27 104 3 143 21 1 1 144 24 pue 1 145 37 107 2 146 32 108—112 =e 1pY 3] oe l 148 29 114 3 149 26 115 3 150 5] 116 _. 151 15 L117 ae 152 15 118 2 153 12 119 = 154 9 120 5 155 9 121 aie 156 10 122 1 157 4 123 2 158 ml 124 2 159 2 125 5 160, 161 a 126 2 162 2 127 1 163 eat 128 5 164 1 129 | 4 Total 487 This distribution is shown graphically in Fig. 1. There is hardly any doubt that the apparently multimodal condition of this histogram is due simply to the fact that we have a limited amount of material and a very wide range of variation. Our sample is not sufficiently large to cover smoothly the whole of the range. RAYMOND PEARL AND WILBUR N. FULLER 217 The values for the mean, standard deviation (with Sheppard’s correction), and coefficient of variation deduced from this frequency distribution are as follows : Mean = 142°715 + ‘362 somites. Standard Deviation = 11°850 + ‘256 somites. Coefficient of Variation = 8:303 + °181°/.. Fig. 1, Variation in number of Somites in body of Earthworm. Number of Worms. LI 79 = 105 110 “Tis 20 125 130 135 140 145 150 155 160 164 Number of Somites. Turning to the character “length of body ” we have the frequency distribution given in Table II. TABLE II. Length of Worms. Class unit = 2°5 cms. Length in F centimetres cit ace 10 —12°5 | 7 12°5—15 50 15 —175 | 75 175—20 | 153 20 —22°5 | 137 22°5—25 55 25 —27°5 9 27'5—30 1 Total 487 Biometrika ty ; 28 218 Variation and Correlation in the Earthworm The constants deduced from this distribution have the following values: Mean = 19°171 + 094 cms. Standard Deviation = 3:077 +067 cms. Coefficient of Variation = 16:049 + °356 °/.. From these values the following points are to be noted: (a) The range of variation in respect to number of somites in the body is very great. The explanation of this fact is simple. The sample of worms under con- sideration is not homogeneous with respect to age. Up to a certain age at least, Lumbricus adds somites at the posterior end of the body with growth. It is clear from either Table I. or Figure 1, that the great extent of the total range in the variation in this character arises from the presence of a comparatively small number of individuals with few somites which unduly extend the lower end of the range. It seems reasonably certain from what we know of the method of growth in the earthworm that if this sample of worms could have been allowed to go on growing the range of variation in total number of somites would have steadily decreased with lapse of time. The fact that there is no apparent tendency towards extreme extension of the range at the upper end indicates that the bulk of the worms included in the sample have either completed the process of adding somites or at least that the process is going on very slowly if at all. So then the sample serves to define the existing typical condition of the worms in the region with respect to this character with a reasonable degree of accuracy. It is to be regretted that an entirely homo- geneous sample with respect to age cannot be obtained, but in the nature of the case this is practically impossible. It would involve following each individual worm from the time that its development began. (b) The earthworm is apparently more variable in respect to length of body than in respect to number of somites. This would indicate that the observed variation in length is not due primarily to variation in number of somites, but rather to variations in the lengths of the individual somites in different worms. Of course, it immediately occurs to one that possibly some of the greater variation in length is due to varying degrees of contraction in individual worms. As has been stated above, however, this source of error has been largely eliminated by taking large length classes. The range of variation in length due to degree of contraction, when the worms are fixed in the way described, is very probably considerably less than one length class unit of 25 cm. The conclusion, that the earthworm is more variable in length than in number of somites, may seem at first sight paradoxical, but it is really no more so than to conclude, what is very obviously the fact, that man is more variable in respect to sitting height than in respect to number of vertebrae. It simply indicates that in the earthworm we have two kinds of growth occurring together: one the addition of somites, the other increase in size of the individual somites. That these two kinds of growth are acting together will be clear when the correlation surface between total number of somites and length of body is examined. RAYMOND PEARL AND WILBUR N. FULLER 219 (c), The average length of the individual somite in this sample of worms is evidently 191°71 142-715 = 1:34 mm. This value so obtained accords very well with observation on the single somites. Its significance from a taxonomic standpoint will be considered later. We may turn next to the question of the correlation between total number of somites and length of body. The correlation surface is exhibited in Table III. Computing the value of the coefficient of correlation from this table by the usual equation Say) No,o,’ we get r= '260 + 028. This is a rather low coefficient and indicates that, so far as can be judged from this sample, there is no very close relationship between length and number of somites in the earthworm. It is seen at once by mere inspection that the relationship between the two characters is not the same in all parts of the correlation table. Thus the worms in the two lowest length classes (i.e. from 10 to 15cm. in length) are not, in general, individuals having few somites, but instead these arrays centre well towards the right end of the table. The five worms having the smallest number of somites all fall in or above the third length class. . This apparently paradoxical result of low correlation between length and number of somites is evidently to be explained as a consequence of the fact that the length of a worm depends not only on the number of somites in the body, but also on the size of the individual somites. It is the same sort of result as would be expected if the sitting height and number of vertebrae in man were correlated. This factor of increase in length of the individual somite can now be analysed somewhat more precisely, through the medium of the regression equations. For the regression of length on number of somites we have the coefficient of regression: S00 ae b, = 260 x Fy gx 9 = 0675, and for the regression of number of somites on length the regression coefficient The characteristic equations, referred to the means as origin, are G) £=9538 + 0675 S, and (ii) S= 123525 +1001 L, 28—2 tion and Correlation in the Earthworm. Varia 220 L8P | I € STPqOL I |- S| SS |e | ec es dime Se eid aa a lh a ad Ne 6 GasleT | Me lags Se (ee eel ee | ee ee oe cg ag ete Ite a @jy jefe }/—|—|—|1|/—|—|—] sees LEI 9 |e |9 eS |v Ons (SoS sis it | % | 1 Wiel eee— o¢ egl oe Pee L | 8 SoOePMRIEOr le Caley Onl ete ymin Oo Gear cL =H it ARS Cer dee i ole ee a0 Ws eee or 0g feeleee lt ry |F (ASN GE Oe ea ear Ala | era ! —|—|— —|— —|]—]t }—}—}|—}1}—!—|—|—]} eer or S[eIOL| LOT | SOT | 6ST STL | £41 srcaadore panuyuoy S[e}OL, 084-18 Oop RG GE—F-BE C= 6 == =) 6 ||" We eee Tj-|]-— oe ae | eee aie Wa | GI—G.61 | | | Lt ralt | SS eri ONG SO.1}0UI1}U80 > f, Gor | for mt yy8uer] SEI | ser | TST | OST | aT | SBI | 4eT | 9ST | GT \ eT | Ger | BET | OGT | SIT | SIT | TIT| SIT | LOT | 90T GOT |00T| 66 | S6 | 86 | 68 | BL | | ‘SOYLMIOG JO Toquin yy ‘sayuuog fo waqunyy pun hpog fo ybuaT uaamjaq wo0rynpa110y TI] Wilavia Raymonp PEARL AND Wiipur N. FULLER 21 where Z denotes the length of the body in centimetres, and S the total number of somites. The probable error made in estimating length from number of somites is 2:004 cms., and in estimating somite number from length the probable error is 7718 somites. From (i) it is seen that an increase of one somite is associated, on the average, with an increase in length of the worm of ‘6 mm. But, as has been shown above, the average length of an individual somite is more than twice this amount (1:34 mm.). Now if the individual somites did not increase in size after they are added the value given in equation (i) and the average value obtained by dividing the total length by the number of somites should agree. The values actually obtained lead us to conclude that the individual somite after being laid down about doubles in length. This result is of interest as showing the quantitative relation of the two factors considered in the growth of this simple segmented animal. The regressions are sensibly linear. 3. Variation and Correlation in the Clitellum. In the discussion of this subject the term “ position of the clitellum” will be used to indicate the number of somites from the anterior end of the body back to the first somite which bears any clitellar tissue. It will be recalled that “number of somites in the clitellum” is taken to include all somites on which any clitellar tissue is present. The frequency distributions for these two characters are given in Tables IV. and V. respectively. TABLE IV. TABLE V. Position of Clitellum. Somites in Clitellum. | | | Number of F Number of F | Somites SSC ya Somites Deny, 29 1 6 394 30 117 if 84 ol 376 8 17 32 1 Total 495 Total 495 ae The very small range of variation in both cases is noteworthy. In the case of the position of the clitellum practically all the individuals are included within a range of two classes. The number of somites in the clitellum with a total range of variation of only three somites has approximately four-fifths of the individuals in a single class. 999 Variation and Correlation in the Earthworm The variation constants for position of clitellum deduced from Table IV. have the following values: Mean = 30°762 + ‘013 somites. Standard Deviation = °436 + ‘009 somites. Coefficient of Variation = 1:416 + 030 °/.. For the number of somites in the clitellum the values are: Mean = 6238 + ‘015 somites. Standard Deviation = °500 + ‘011 somites. Coefficient of Variation = 8:019 + 173 °/ It is at once evident that the variation in respect to number of somites in the clitellum is significantly greater, as indicated both by the standard deviation and the coefficient of variation, than the variation in the position of the clitellum. The difference between the coefficients of variation in the two cases is very large (6°608 °/,). The difference between the two standard deviations is absolutely small (064), but relatively it is large enough to be significant, being more than three times its probable error (which is + 014). The probable reason for the difference in variability between these two characters will be discussed later. Turning now to the correlation between position and number of somites in the clitellum, we have the following table : TABLE VI. Correlation between Number of Somites in Clitellum and Position of that Organ. Number of Somites in Clitellum. 6 7 8 Totals [i os | 29 1 le ae aes 39 | .62 | 16 5D 31 354 21 1 i ot 32 1 = = = Totals | 394 84 17 The value of the coefficient of correlation deduced from this table is r= — ‘629 + 018. This rather high negative value for the coefficient is of considerable interest from the morphogenetic standpoint. It shows that as the number of somites anterior to the clitellum increases, the number included in the clitellum itself tends to decrease, and vice versd. Or in other words, it means that the posterior end of the clitellum is a definitely fixed morphological point, subject to very little variation. Raymond PEARL AND WiuBur N. FULLER 223 That the posterior end of the clitellum shows the least variation of any point in connection with this organ is further shown by direct measurement. The frequency distribution concerned is as follows: Number of Somites — to posterior end of | Frequency Clitellum | 36 | 40 37 | 416 3 38 39 1 Total | 495 i} \\= The mean deduced from this is 37 somites and the standard deviation *407, giving a coefficient of variation of 1:100 °/,, which is significantly lower than that for the position of the anterior end of the clitellum. This relatively fixed position of the posterior end of the clitellum is very probably associated with the fixed position of the openings of the male ducts and the other parts of the reproductive system in this form. The clitellum is to be regarded as an accessory organ, at least, of the reproductive system. It has been shown by teratological work that the various portions of the reproductive system proper vary, so far as position is concerned, as a unit*. Whenever one part develops in another somite than that in which it normally occurs all the other parts appear in abnormal positions such that the normal relations between them are preserved. Beddard+ has shown that in the highly variable form Perionyx the same sort of relationship holds between the position of the clitellum and the other organs of the reproductive system proper. We may now turn to the consideration of the exact form which the clitellum takes in different cases. It was found during the collection of the data, that the clitellum did not, in all cases, completely include all the somites which it touched. In other words, it was rather frequently observed that clitellar tissue would only partially cover a somite at the anterior or the posterior end of the organ, or, in rare cases at both ends. It is of interest to know how frequently this condition occurs, and accordingly we have prepared a table to bring out this point. The following method was used in recording clitellum form: in case the clitellar tissue ended ‘at both extremities of the organ, precisely at intersegmental grooves, two parallel, unbroken vertical lines were made, with a figure indicating the number of somites in the clitellum between them. Thus |6| indicates that the clitellum * Cf. Bateson, Materials for the Study of Variation, pp. 159—167, and a paper by the present writer in the Anat. Anz. Bd. xvi11. pp. 123—127 describing a case in which this close interconnection of the parts in respect to position is very clearly shown. + Proc. Zool. Soc. 1886, pp. 298—314. 224 Variation and Correlation in the Earthworm contains six somites and ends anteriorly and posteriorly in an intersegmental groove. Whenever the clitellar tissue overlapped so as to cover only a part of a somite at one end or the other, either the left or right vertical line was dotted, according as the overlapping occurred at the anterior or posterior end of the clitellum. Thus :6| indicates that the clitellar tissue is present on six somites, and that it only covers a portion of the most anterior one of these six somites. | 6: signifies that it is the most posterior somite which is only partially covered, and :6: indi- cates that overlapping occurs at both ends. In Table VII. are given the results of a determination of clitellum form in 495 worms, the form classes being arranged in the descending order of the frequency of their occurrence. TABLE VII. Clitellum Form. TereeOneee nicer | i7| | | 8 | | :7| | :6| | 7: | | Frequency | 380| 72 | 14| 9 | 4|2/2/|1 [1/1 495 | In 95°9 °/. of all the worms the clitellum ends exactly in an intersegmental groove at both ends. In the remaining 4°1 °/, it will be seen that overlapping occurs much more frequently at the anterior than at the posterior end of the clitellum. Thus in only 1:4 °/, of all the worms is there any overlapping at the posterior end, including the three cases, or ‘6 °/, of the whole number, in which overlapping occurs at both ends of the clitellum. So far as may be judged from the present sample of worms, then, it appears that (1) clitellar tissue does not extend over more than eight whole somites; (2) in a majority of cases it extends over exactly six whole somites; (3) in the great majority of cases clitellar tissue ends exactly on intersegmental grooves, both anteriorly and posteriorly, and (4) when overlapping does occur, it is more apt to be at the anterior than at the posterior end of the clitellum. Now this spreading out or “overlapping” of the clitellum beyond the limits set by intersegmental grooves would apparently indicate a growth in longitudinal extent of the clitellum after it is first laid down. It seemed very desirable to determine exactly whether any such growth actually does occur, because it is usually supposed that the clitellum is an organ of definite extent which does not change so far as size is concerned. The matter can be definitely settled by deter- mining whether large worms have extensive clitella, or, in other words, whether extent of clitellum and length of worm are sensibly correlated. In order to test this matter we have made use of Pearson’s recently published Method of Contingency*. This extension of the theory of correlation makes it * « Math. Cont. to the Theory of Evolution, XIII.” Drapers’ Company Research Memoirs, Biometric Series I. pp. 35, Two plates, RAYMOND PEARL AND WiLBurR N. FULLER 225 possible to deal with a variety of problems which could not before be attacked by biometric methods on account of inherent peculiarities of the data. At the outstart the following contingency table (Table VIII.) between length of worm and extent of clitellum was formed : TABLE VIIL Length. Class unit 2°5 cm. C- O+ D- | D+} E- 15 0 F+ | Totals 379 g 3 = 1 oO a 72 + 8 A 2 s 1 S| 14 col es 1 1 Totals 7 49 73 155 135 53 9 ] 482 This gives a table of 80 sub-contingencies, a comparatively large number for this method. Following the notation of Pearson in the memoir referred to, we find for the square contingency from this table y? = 101°5385, whence the mean square contingency go? = 2107. The mean square contingency coefficient hence equals 4172. From the same grouping ap = 1167, whence, interpolating by the diagram in Pearson’s Table I., the mean contingency coefficient C, = ‘34, It is at once evident that these values do not approach very closely the relation C,;=C,, which should be the case if the proper grouping is used in the Biometrika Iv 29 226 Variation and Correlation in the Earthworm table and the correlation is normal. It seemed probable that the discrepancy was due to too fine grouping in the table. In this particular case with the grouping adopted a considerable number of compartments have no observed occurrences falling in them. With the hope of bettering the results a somewhat coarser grouping was tried, the table then taking the form shown in Table IX. TABLE IX. Length. Class unit 2°5 cm. a C- | C+ | D— | D+ | HE- | E+ | F- | F+ | Totals 3 |-———_— 2 |6| and :6| Ss |6: and |7| i :7| and |7: | $7: and |8| Oo - |8: and :8: 2 om Totals This gives only 40 sub-contingencies and a correspondingly larger number of the compartments are filled with observations. With this grouping we find the following values: x? = 710°615, ¢ge= 1465, C,= ‘3575. For the mean contingency from the same grouping we find : = 1093, whence C, = '33. This evidently comes fairly close to satisfying the relation C, = C,, and further manipulation of the grouping seems hardly worth while. We may conclude, then, that there is a very sensible degree of correlation between the size of the worm and the extent of the clitellum, the value of r being about ‘357 with a probable error of somewhat less than + ‘054. This result implies that growth occurs in the clitellum after it is formed. Putting all the results together we are able to form a fairly complete and connected picture of the morphogenetic phenomena connected with the position and extent of this organ. These phenomena we conceive to have probably the following relations: (a) the clitellum develops at such a point on the body that its posterior end lies a definite and, except in teratological cases, a fixed number of somites behind the openings of the genital ducts. (b) The clitellum when formed has a Raymond PEARL AND WILBUR N. FULLER 227. variable number of somites but as a consequence of (a) any increase in the number of somites in this organ must necessarily entail a decrease in the number of somites between the anterior end of the worm and the anterior end of the clitellum. Hence a negative correlation between these two characters arises. (c) There is a tendency for the clitellum after being formed to increase its extent by growth. This growth tends to extend the clitellum both anteriorly and posteriorly. In a sample of worms non-homogeneous with respect to age any extension in the posterior direction tends to increase the variation in the number of somites in the clitellum without correspondingly affecting the amount of variation in the number of somites lying in front of this organ. (d) The increase in extent of the clitellum by growth is more marked at the anterior than at the posterior end of the organ. If this analysis be correct the factors concerned in the form taken by the clitellum after it is first laid down are fairly simple and understandable. The problem as to what causes the clitellum to originally appear in the precise region where it does is not so simple however. It is a particular case falling under the general problem of the “localization of morphogenetic phenomena.” While natural selection is capable of accounting for the facts when the worm is entirely normal, this explanation fails to account for such cases as those decribed by Bateson for Perionyx and referred to above, where the position of the clitellum bears the same relation to the other organs of the reproductive system, whatever its absolute position. 4. The Taxonomic Position of the Form Studied. It was supposed that the sample of worms studied in connection with this work was sufficiently large to enable a very precise determination of the systematic position of the form to be made. The constants obtained for the various characters from this collection were compared with the values given by Beddard* for the same characters in the various species of Lwmbricus. Somewhat to our surprise it was found that our specimens differed widely in some points from the condition found in any described species of the genus. The worms come nearest to L. hercu- leus, Sav., and should evidently be classed with, or at least very close to this species. The points of distinction and agreement are shown in the following table: a ee xs —_ L. herculeus Sav. Character Ann Arbor form (as given by Beddard) Number of Somites ... Mean 142°715 180 Length ... se ee 3) LOle7 mm: 360 mm. Somites to Clitellum ... is 30°76 31 Somites in Clitellum ... 3 6:24 6 (Other diagnostic characters in agreement in the two cases.) * A Monograph of the Order of Oligochaeta. Oxford, 1895, pp. 721—724. 29—2 228 Variation and Correlation in the HKarthworm In position and extent of the clitellum there is essential agreement between the Ann Arbor form and the typical L. herculeus. In size and number of somites, however, there is a striking discrepancy. It would appear that the earthworms typical for this region approach towards a dwarf condition of the type of the species, being on the average only a little more than half as long and having about 40 fewer somites. The typical L. herculeus, according to the above figures, has for the single somite an average length of 2 mm., as against 1:34 mm. for our sample. Applying equations (1) and (11) (p. 219) we find the mean length of an array of worms each having 180 somites, 216°9 mm. instead of 360 mm., as given for L. herculeus, and for the mean number of somites for an array having a length type of 360 mm., 159°6 as against 180. Is this Ann Arbor form to be regarded as a true dwarf variety, or is the small size to be explained in some other way? Several possible explanations occur to one but none of them seems adequate. In the first place it might be maintained that the specimens in our sample were young, and had not completed their growth. This is, undoubtedly, true for some of the specimens in the sample. The earth- worm continues to add somites for a long period, possibly throughout its life. If such addition does occur throughout the life of the worm it must, after a time, be at a very slow rate. This is evidenced by the fact that in the present sample there is no tendency towards extreme extension of the range at the upper end. The form of the frequency polygon for these worms, as has been pointed out above, indicates that the majority have either stopped growing or are growing at a very slow rate. Another possibility is that in the collection of these worms an unconscious and unavoidable selection was made on the basis of size. This might arise in the following way. Suppose it were a habit of very large (old) worms not to leave the burrows on rainy nights. Then it would necessarily result that any collection made in the way the present one was made, would contain no or few very large worms. Now it is possible that this is the case, but it seems to me hardly probable, and for the following reason: in the large number of earthworms which have been collected for class-work in this laboratory during the last five years, including several thousand individuals at least, specimens have not been found of twice the size of the average of the present collection. Furthermore worms collected by digging are of no larger size than those collected in the way described in this paper*. Finally no student of earthworm habits has recorded, so far as I am able to find, any tendency towards a habit of the sort mentioned. Again it might be thought that the relatively small size of the worms in this sample as compared with the type of the species was due to the method of killing and fixing. That this cannot be the case is shown, first by comparison with living * These last two facts also argue strongly against the first point raised; viz., that the worms in the present*sample are all young specimens. Surely in five years collecting a fair number at least of old worms would be found. RaymMonp PEARL AND Wiupur N. FULLER 229 worms from this same locality, and secondly because such an explanation would not in any way account for the smaller number of somites. Altogether, it must be concluded that the typical form of Lumbricus of this locality is very considerably smaller and contains fewer somites than the type of the species to which it is most closely related; viz. L. herculeus*. There is no evidence available as to what is the cause of this dwarfing, but it is not unlikely that it is due to direct environmental influence. 5. Summary. A study of variation and correlation in several characters of the earthworm Lumbricus herculeus, Sav., leads to the following conclusions : 1. The earthworm is more variable in length of body than in number of somites. 2. There is a rather low degree of correlation (7 =*260) between these two characters. 3. The results just stated (1 and 2) arise from the fact that the organism increases in length as a consequence of the combined action of two processes, (a) the addition of new somites, and (b) the growth of somites already added. 4. There is greater variability in the number of somites in the clitellum than in the number of somites lying in front of that organ. The factors concerned in this greater variability are analysed. 5. The number of somites included by the clitellum is negatively correlated to a relatively high degree with the number of somites making up the portion of the body anterior to that organ. 6. This implies that the posterior end of the clitellum is a relatively fixed point. This conclusion is confirmed by direct measurement. 7. There is a strongly pronounced tendency for the clitellum to end both anteriorly and posteriorly exactly at intersegmental grooves. Overlapping occurs more frequently at the anterior than at the posterior end of the organ. 8. The clitellum increases in extent as the worm grows in length. 9. The form of Lwmbricus found at Ann Arbor is shorter and contains fewer somites than the typical L. herculeus, Sav. In conclusion I wish to acknowledge my indebtedness to the officials of the Carnegie Institution for a grant to aid in the carrying on of certain biometrical work of which this paper forms a part. * There remains, of course, still another possibility, namely, that the values for the type of the species are based on a few unusually large specimens, but if this is the case it is difficult to understand why some one has not called attention to the matter. Especially does this seem remarkable when we remember that there is no organism more universally used as a type in zoological laboratories. MISCELLANEA. I. Per la risoluzione delle curve dimorfiche. Nota pet Dk FERNANDO DE HELGUERO, Roma. Il Prof. Karl Pearson ha trattato il problema della decompozizione di una curva dimorfica risultante da componenti normali in una memoria inserita nelle Phzl. Trans. Vol. 185 a, pp. 71-110. Egli riduce tutta la difficolta del problema alla risoluzione del sistema: ie — 4 pig 03 — 2s" — Ngo O72 =O \saacie caus cueracisecueescosecsteesuecerens (24) pg" — 25° + 4 p05 po? — 20 pts Po? — Az Po7=O es eeee deadvan here seseer (2D) dove pz, Ay, A; Sono dedotti dai dati empirici e p,, p; sono le incognite. Ovvero eliminando p, si trova l unica equazione risolvente : 24p,? — 28D po? — 36 ps"p0" — (24 gh; — 104”) pro? — (148574 + 20;”) po? + (288 p13! — L245 wy — Ay?) Po? + (24 Mg9As — THs?Ay?) Po? + 82g! Ag po — Wg =0....(29) Studiando questo problema io sono giunto a formule un po’ pitt semplici di quelle sopra scritte, che da quelle possono direttamente dedursi colle posizioni a= Bus? M4, Ags=— Bus! 5) P3= M3 P's P2=bs! Pay: Le mie formule risolventi sono: o il sistema: 1— 4p,’ — 2p, + 6p,'? + 8p9'vy=0 5p’ — 20p,'3 — 2p, + dps pr’? + 3py?v5=0, o l unica equazione: 24p,9 + 8404 po" + 36p./° + 8 (Av; +5y4”) poo’? +6 (744 + 395”) po +9 (32 —12yyv, + 3y48) po’? — 9 (8v5-+7r4) Po? — 964 po’ — 24=0. ...(B) I] modo di usare praticamente di queste formule é il segnente : Siano n= —e 2%? , Y= le equazioni delle due componenti prendendo ! asse delle ordinate passante per il baricentro della curva complessa; si vogliono determinare ¢,, C2, 01, 02; 01, dy. Miscellanea 231 Siano ps, 3, fy, fs 1 Momenti della curva complessa rispetto all’ asse delle ordinate, a I’ area. afty ~ Byy” apts ~ LO pg pg ry ) 6 ae & a p38 a® ps3" a’ 3 Si trovino p,’, ps, per mezzo del sistema (A), 0 p,’ per mezzo della nonica (B) e p,’ per mezzo della Si calcolino: Va — 894'9 + 399/24 + p09" vs) +2 es 2p + By, py +4 meats sae tien oe (CO) Noti p.', p,' si calcoli p,’ =3 3, poi y; € yg che sono le radici della 2 Y— plyt pi =0. I parametri delle curve normali componenti sono allora date dalle formule : ie Ce / a= B He nan /', b=» ao aye : ayy ’ y, 2 YI Ml V1 ¥2 o_ Me 1 ae o_ Hy 1 -*)- } Gi Ore lay, (2: =) i}, of=— + by? {5 (27 F Lee L’ equazione (B) pud ammettere pitt radici reali, cos) il sistema (A). Per il nostro problema interessano solo le soluzioni con p,’ negativo. A ciascuna soluzione corrisponde una coppia di curve normali componenti. Riguardo al numero di queste soluzioni si osservi che esso ¢ sempre pari: se vy>—7 si hanno o nessuna o due soluzioni; se vy<—7 esistono, al pill, quattro soluzioni, due sono certo reali se v4< —5°94587. La dimostrazione di questi risultati, come pure dei metodi per la risoluzione del sistema (A), si troveranno in una speciale memoria che verr’ pubblicata altrove. [In this Vote p, stands for the nth moment about the centroid of the frequency system—the notation of my memoir of 1893—and not ap, the more usual notation now. K.P.] II. Albinism in Sicily. A further Correction. In Biometrika, Vol. 111, Pt. tv. I pointed out that Dr Arcoleo states explicitly that he never met with any case of an albino child being born to an albino parent, and consequently that Professor Weldon’s use of this authority to prove the production of two pigmented children (+3 albinos) by albino parents was erroneous. Professor Pearson replied that the meaning of the explicit statement I quoted from Arcoleo was not clear to him, and that “Dr Arcoleo states explicitly that the mother of the four albinos in Family No, 6 was ‘una albina di belle forme,’ &c.” Happily we need not examine the various discrepancies and self-contradictions which Professor Pearson’s version would presuppose in Arcoleo’s table and text. It is merely a 232 Miscellanea mis-translation. The beautiful “albéna” was not the mother of Family No. 6, but the friend of the mother. The text reads :— Bese “Tl Cay. N. N. senti grande simpatia per una albina di belle forme e desideré avere una figlia che le rassomigliasse. Con tale scopo indusse la moglie a farlese amica, e gia ambedue vagheggiarono la medesima idea. I] desiderio fu soddisfatto colla nascita di una bella bambina; ma moriva a due anni, &c.” The words “ indusse...... amica” mean that he induced la moglie, his wife to make friends with the “albina”; literally, far, to make, Ze, her [the albina], amica, a friend, sz, to herself. Arcoleo evidently attributes the albino births to “sympathy,” or other mysterious influence, in which case he might well call the occurrence a “fatto curiosissimo.” Nevertheless his memoir is perfectly consistent, save for a numerical slip in one place, and when he wrote that he knew no instance of hereditary albinism he meant no more or less than he said, doubtful as this has seemed to Professor Pearson. W. BATESON. CAMBRIDGE, 27 Feb. 1905. [I am sorry that I overhastily read Arcoleo’s paragraph and thus stated that Cav. N. N. made an amica of the “fair albino.” It is clear that he caused his wife to make a friend of her with the object that his wife and not the albino lady should provide the albinotic daughter. Arcoleo’s consistency is thus established, and I can only hope that his powers of scientific observation are not detrimentally influenced by the capacity thus exhibited for archaic belief. K.P.] Journal of Anatomy and Physiology NORMAL AND PATHOLOGICAL, HUMAN AND COMPARATIVE. CONDUCTED BY PRINCIPAL SIR WILLIAM TURNER, K.C.B., AND Provessors D. J. CUNNINGHAM, G. 8S. HUNTINGTON, A. MACALISTER, anv J. G. M‘KENDRICK. i VOL. XXXIX. se NEW SERIES—VOLUME XIX. PART IV.—JULY 1905. NOW READY—PRICE 6/-, 3 ‘ANNUAL SUBSCRIPTION 21/- POST FREE. With Plates and Figures in Text. | CONTENTS. I. Wauter H, Gasxeuu, M.D., LL.D., F.R.S. Origin of Vertebrates: deduced. from the Study of * Ammocetes. I]. F.G. Parsons, #.R.C.S8. Pressure Epiphyses. III. Waurer Kip. Imbrication of the Papillary Ridges in Man. IVY. Ww. Wricut, M.B., D.Sc., F.R.C.8. Skulls from the Round Barrows of Hast Yorkshire. V. Gituman Moorueap, M.D., M.R:C.P.I. The Anatomy of a Sirenomelian Monster. VI. G. J. Jenxins, M.D., F.R.C.S.. Cranio-cerebral Topography. VII. Joun CAmmron, M.D. (Edin.), D.Sc. Development of the Retina in Amphibia: an Embryological and - Cytological Study. Part III. VIII. D. Macran Arprun, F.R.C.S.E. Variations of the Tibia and Astragalus. IX. M.Brrraetor. On the Limits of the Sensibility to Odours and of the Methods of detecting Emanations. X. CHarres R: Warrraxer, L.R.C.S8,, L.R.C.P., ete. Abnormality of * the Renal Arteries. XI. Epwarp Fawcett, M.B. Edin. © Ossification of the Lower Jaw of Man. XII. Bestram C. A. Winpie, M.D., Sc.D., F.R.S. Recent Teratological Literature. XIII. Notice of New Book. XIV. Index. XV. Proceedings of Anatomical Society of Great Britain and Ireland. ' LONDON: CHARLES GRIFFIN ayp COMPANY, Lrp., Exeter Street, Strand. Now Ready. JOURNAL OF THE ANTHROPOLOGICAL INSTITUTE. Vol. XXXIV. Part II. July—December, 1904. Contents:— = - ; Les Six Races composant la Population actuelle de Hurope (the Huxley Lecture)—Dr J. Dentzer. A Sea Dyak Love Philtre—Rev. W. Howrtn and R. Sannrorp. The Magic Origin of Moorish Designs —Dy E. Westermarcs, ‘Native Stories from Santa Cruz and Reef Islands—Rey. W. O’Frrranu. Notes . on some Ancient Chilian Skulls and other remains—R. HE. Larcnam. Further Notes on the Kikuyu Tribes of British East Africa—H. R. Tarn. A Method of Estimating Skull Capacity from Peripheral Measures—Dr J. Beppon. The Wiradyuri and other Languages of New South Wales—R. H. Maruews. Further Excavations on.a Palsolithic Site in Ipswich—N. F. Layarp. Ethnological Report on the Strseélis and Sk’atlits Tribes of British Columbia—C. Hinx Tovr. WITH FULL-PAGE PLATES AND ILLUSTRATIONS IN TEXT, PRICE 10s. NET, Published by THE ANTHROPOLOGICAL INSTITUTE of GREAT BRITAIN and IRELAND, 8, Hanover Square, London, W. And to be obtained through all Booksellers. a A MONTHLY RECORD OF ANTHROPOLOGICAL SCIENCE. Published under the direction of the Anthropological Institute of Great Britain and Ireland. — Bach number of MAN consists of 16 Imp. 8vo. pages, with illustrations in the text together with one full-page plate ; and includes Original Articles, Notes, and Correspondence; Reviews and Summaries ; Reports of Meetings; and Descriptive Notices of the Acquisitions of Museums and Private Collections. Price, 1s. Monthly or 10s. per Annum prepaid. : TO BE_OBTAINED FROM THE 7 ANTHROPOLOGICAL INSTITUTE, 3, Hanover Square, London, W.— AND THROUGH ALL BOOKSELLERS, ) ; CONTENTS (All Rights reserved) I. Results of Crossing Grey (House) Mice with Albinos. By E. H. J. SonusTer, M.A. (With coloured Platel) - . 7. 5. «. h™ II. Biometrical Studies on Man. I. Variation and Correlation in Bratt weight. By RayMonD PEar, Ph.D. oe ge sige Aes in the text.) . : ; 4 13 III. A Study of the Relations of the Brain to the Size of the Head. By: REGINALD J. Griapstone, M.D. (With Plates II and III and five ~ Figures in the text.) 0.9% "000 ae) oc" ie) rr IV. On the Biometric Constants of English Brain-weights. By J. BLaKE- ad MAN, M.Se., assisted by ALIcE Lx, D.Sc, and KARL PEARSON, F.R.S. (With six Figures in the text.) . : : : ik ae 124 V. -A Quantitative Study of the Resemblance between Man gad Woman. as Su By E. TSCHEPOURKOWSKY. . : : ; . : . wey | sp E VI. “Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und f Pearson.” A Rejoinder. By Kari Pearson, F.R.S. (With four ‘ Figures in the text.) ; ‘ : re ast : ii Ala Se Oe fc VII. Variation and Correlation in the Karthworm. By RAYMOND PEARL, ry Ph.D., and Witpur N. Futter. (With one Figure in the text.) . 218 = y Miscellanea. Gd) Per la. risoluzione delle curve dimorfiche. Nota del paw hehe las Dr FERNANDO DE HELGUERO . . 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The publication of | a paper in 1 Blom eiriecAMnatl that in 1 the Editors it contains either {n method or material something of interest to biom But the Editors desire it to ne ae! understood that such Theses of Sees a its Relation Ms Ressoation ne 1 Norinal Corre-- oe lation. By Karu Pearson, F. RS. Issued. Price 4s. II. Mathematical Contributions to the Theory of Hivolufion ey : Theory of Skew Correlation and Non-linear re pe : _PEARSON, F.RS. Issued. Price DES typosis in the Animal lenedo By moka Wanien, D. So. bee : Tee D.Sc, Epna’ LEA-SMITH, Marion SEO and Kant’ Bee Eprrep BY GEORGE BP NUTTALL, M.D, ak ! Lecturer in Bacteriology and Preventive Medicine in the University of Camb dge, Late. Associate in Hygiene in the Johns Hopkins be can Siee og IN CONJUNCTION WITH “ee JOHN s, HALDANE, MD., F.RS., "Lecturer in Physiology in the University of. Oxford. ss - Medical Officer of Health of Brighto n : CHARLES J. MARTIN, M. B., D.Sc., F.R.S., Director of the Lister Tustitute of, Preventive Medicine, London, ek 7 Q ‘> 6 § 5 3 z * %} S Se = 1 oL_L |_| 1 012 3 4 5 6 7 8 9 10 1112 13 1415 16 17 Size of Family. Equations to Regression Straight Lines: aa’, y= 9°6209 +-18187r bb’, y=11:1936 — -147822 | Origin at 0 children. dd’, y= 2:1903+4:058442 | Unit of r=1 child. ee’, y= 3°0170 — -09788x Equations to Cubic Parabolas : ec’, y=10°465864 — °099753x — 0330962? — -00035223) Origin at 8-5 children. Sf 5 y= 2°245851 — -089609x — 0079002? + -000422.3 Unit of c=1 child. 31—2 244 Duration of Life and Number of Offspring In these deferred expectations the correlation is very high and within very small limits of error, and again it is positive in both series for mothers of small families and negative for mothers of large families. We may therefore safely conclude that the advantage in longevity is distinctly in favour of mothers of moderate sized families of about six children, and that in New South Wales at all events, maximum fertility is extremely unfavourable to extreme longevity. I think the reasons are sufficiently obvious—and what applies to New South Wales must apply with equal force elsewhere—the incessant strain upon the physique of women who bear large families during the periods of gestation, parturition and lactation must be very prejudicial to longevity, whilst the mental strain involved in the rearing of such families cannot be regarded as other than detrimental. But we are also presented with the fact that childless women and mothers of extremely small families have shorter expectation of life than mothers of moderate sized families. This I consider must arise partly from the inferior physique of unprolific women generally, and partly—perhaps chiefly— in consequence of the prejudicial effects of the efforts to prevent and to limit families. But whatever may be the explanation, the shortened life thus occasioned is by no means equal to that occasioned by unrestrained excessive fertility. In New South Wales there is but little Malthusian restraint when compared with the old countries of the world—notwithstanding that the Government of that State, alarmed at the decline in the birthrate, appointed a Royal Commission to investigate the causes operating to produce such decline. In its Report, relying upon medical evidence and on statistics which I think were wrongly interpreted, it declared that there had been a serious decline and that this was chiefly to be attributed to artificial restraint. That there has been a decline in the birthrate, whether measured as per 1000 of the population, or per 1000 married women of the reproductive period, or in age groups within that period, is obvious from the figures produced in evidence, but it was entirely overlooked when dealing with the latter method—the only correct one—that there had been a considerable change in the physique of the population. In the earlier days the community consisted chiefly of specially selected immigrants drawn for the most part from the most prolific classes of the British community, viz. the agri- cultural and artisan. Now that immigration has practically ceased, the birthrate has declined from an abnormal condition due to a differentiated community to that of a more normal population. That there is restriction to some extent amongst certain classes of the community is doubtless trae—although it is equally probable that this has always been the case—but, as we shall subsequently see in another part of this paper*, the extent of this practice is very limited. It was previously remarked that the expectation or mean duration of life used throughout the investigation should not be mistaken for the actuarial expectation of life at the ages referred to. Thus according to the English Registrar-General’s life table}, the expectation of females aged 46 is 23°34 years, * See “ Distribution of Fertility,” p. 247. + Supplement to Fifty-fifth Annual Report, Part 1. pp. xviii. and xix, A. O. Powys 245 so that the expectation beyond 45 of those surviving age 46 is 24°34 years; the expectation at age 60 is 14°10 and at age 75, 6°68 years*. The complete expecta- tions at similar ages computed by the present writer from the New South Wales census experience of 1891 are 26°17, 14:37, and 7:02 years respectively. The average durations of life determined from the experience of married women in the present investigations are 23°05, 13°82, and 7:22 years respectively. It is seen that at the first two ages the averages for married women are less than the expectations according to both the English and New South Wales life tables for the general population of females, although in excess at the later period. At first sight it might appear that the married condition was somewhat unfavourable to longevity; but this is not so. We are able to compare, not the actuarial expectation, but the mean duration of life of married and celibate men and women upon exactly the same basis. The ages at death of each sex are tabulated in quinquennial age groups, and by subtracting the married, the numbers of bachelors and spinsters have been obtained as shown in Table XII. TABLE XII. Number of Deaths of each Sex, Single and Married. Deaths of Females (1898 to 1902) Deaths of Males (1901 and 1902) Age at Death = Geen aT Tea All Females} Married Single All Males Married Single 45 and under 50 1181 1050 131 939 616 323 50 . 55 1093 972 121 851 588 263 55 “5 60 1218 1126 92 931 652 279 60 Fi 65 1484 1362 122 1083 750 333 65 a 70 1609 1491 118 1382 937 445 70 5 ve 1513: 1423 90 1276 871 405 75 i 8&0 1356 1281 75 830 606 224 80 - 85 1132 1080 52 589 433 156 85 90 579 551 28 310 271 39 90 ri 95 219 206 13 129 98 31 95 » 100 69 65 4 41 33 8 100 a LOS! 14 14 — 13 9 4 Totals ine 11,467 10,621 846 8374 5864 2510 Note:—The numbers given in the columns headed “Married” do not agree with those given in Tables I. and IL., as in this table are included those mothers whose ages at death were known but the numbers of whose offspring were unknown. The latter were of course excluded from Tables I. and II. From the data in this table the average durations of married and celibate life were determined, the central age of each age group being adopted as the mean age of that group. The results for females were as followst : * T cannot tell from the table referred to whether the expectations given are complete or curtate. + It will be noticed that the averages determined by the adoption of central ages of a quinquennial group differ but slightly from those determined by the adoption of single years. 246 Duration of Life and Number of Offspring Average duration of life after 45 years of age: Married Women 22°60 years. Single do. 18:98 _,, Average duration of life after 60 years of age : Married Women 13°91 years. Single do. 1223 Average duration of life after 75 years of age: Married Women 7:40 years. Single do. (24 3e At each period there is exhibited a sensible superiority of the married over the unmarried state, and this difference being undoubtedly real, the natural inference is that the exercise of the functions incidental to the married state is at all events after the reproductive period. It can hardly conducive to longevity be urged that an unmarried woman who survives the age of 45 suffers generally from any constitutional infirmity which, from motives of prudence or otherwise, deterred her from contracting marriage or which physically incapacitated her. The greatest enemy of mankind—phthisis—has at that age practically exhausted itself. A life at that age may be regarded as a selected life, but more and more undoubtedly so at ages 60 and 75. For males the results were* : Average duration of life after 45 years of age: Married Men 21°59 years. Single do. NOS aes Average duration of life after 60 years of age: Married Men 13:07 years. Single do. 168 3; Average duration of life after 75 years of age: Married Men 7-49 years. Single do. 6°60 __,, Again we find the differences consistently in favour of the married state—not so pronounced at the first two periods as amongst females, but somewhat more so at the latest. In view of the results obtained for both sexes, we are warranted in concluding that marriage, at all events so far as longevity is concerned, is not a failure. It may be remarked that the material we have been dealing with may be said to be homogeneous—the deaths are those of persons almost entirely of British stock or origin. Of the 11,396 deaths of females which occurred during the years 1898 to 1902 aged 45 and upwards, 2572 or 22°5 per cent. were of Australasian birth ; 8406 or 73°8 per cent. were born in the United Kingdom, whilst only 418 or 3°7 * Tt will be noticed that the averages determined by the adoption of central ages of a quinquennial group differ but slightly from those determined by the adoption of single years. A. O. Powys 247 per cent. were born elsewhere. Of the 6457 who died at age 65 or over these figures are 665 or 10°3 per cent., 5545 or 85°9 per cent., and 247 or 3:8 per cent. respectively. From the results of this section of the investigation we are amply justified in concluding : (a) That in females extreme fecundity is unfavourable to extreme longevity. (b) That mothers of moderate sized families of about six live on the average longer than those with smaller or larger families. (c) That although in males increased fertility is apparently associated with increased longevity throughout the whole of life, such association in part at least is due to second and subsequent marriages. (d) That the married of both sexes have a greater expectation of life than the unmarried, and that this superiority is not all due to physical infirmities which prevented marriage. B. Distribution of Fertility. Professor Pearson, in his essay “Reproductive Selection*,” has dealt with this subject rather exhaustively, using the data derived from certain sections of the Anglo-Saxon community and from the upper and middle classes of the city of Copenhagen tabulated by Messrs Rubin and Westergaard. The statistics now to be presented are for the whole state of New South Wales—urban and rural— and will exhibit the distribution of fertility under varying conditions of ages of women and durations of marriage. Taking first of all the women used in the last investigation, i.e. those who died beyond the reproductive period irre- spective of the age at marriage or of the duration of marriage, the distribution is as Shown in Table XIII. From this table we see that nearly 11 per cent. of the women were sterile, and it would appear from Fig. 5 (broken lines) that this is unduly high. The excess might at first sight be attributed to absolute prevention of family, but as these statistics relate to married women dying beyond the reproductive period, they comprise widows—whether widowed before or after the reproductive period— and of women who may have married late in that period and whose fertility had but little opportunity to display itselft, and of women who may have married beyond that critical period and consequently no opportunity whatever. Therefore although this percentage of barren women agrees fairly closely with that obtained by Rubin and Westergaard for marriages of duration of 15 years and upwards with both husband and wife alive, viz. 12°87, this ratio is distinctly too high for New South Wales. For marriages under the last-mentioned conditions, as we shall subsequently see, the barren marriages are but 5563 per cent., which * Chances of Death, pp. 63 et seq. + Vide Biometrika, Part 1. Vol. 1. p. 33, see curve showing initial fertility in deferred marriages. 248 Distribution of Fertility TABLE XIII. Distribution of Fertility. Women dying beyond the Reproductive Period. Eaperience of New South Wales. Size of ane Per Cent. of Sum of Number of | Per Cent. of Sum of Family Women Total Percentages Children Total Percentages 0 1110 10°802 10°802 — — — 1 533 5°187 15°989 533 837 837 2 581 5°654 21°643 1162 1°826 2°663 3 644 6°267 27°910 1932 3°035 5°698 4 702 6°831 34°741 2808 4-411 10°109 a) 813 7912 42°653 4065 6°230 16°339 6 855 8°320 50'973 5130 8°216 24°555 7 976 9°498 60°471 6832 10°733 35°288 8 963 9°371 69°842 7704 12°103 47°391 9 847 8°243 78°085 7623 11°976 59°367 10 786 7649 85°734 7860 12°348 71°715 11 568 5°527 91°261 6248 9°816 81°531 12 422 4°107 95°368 5064 7956 89°487 13 226 2°199 97567 2938 4°615 94°102 1h 129 1°255 98°822 1806 2°838 96°940 15 57 555 99°377 855 1°343 98°283 16 39 380 99°757 624 980 99263 17 12 116 99°873 204 *320 99°583 18 5 049 99°922 90 142 99°725 19 2 020 99-942 38 “060 99°785 20 2 ‘019 99°961 40 063 99°848 a1 1 ‘010 99°971 21 033 99°881 22 1 010 99°981 22 034 99°915 2 1 010 99-991 24 038 99°953 30 1 009 100000 30 047 100°000 Totals 10276 100000 — 63653 100-000 — Median 5°8831 8°2179 Mean Family =6:1943 children agrees remarkably close with that computed by Professor Pearson for the Danish statistics, viz. 5°717. This is of course for all women at whatever age married, but whose marriage had existed 15 years or more at the time of the New South Wales census of 1901. From the returns of that census I have been enabled to compile not only the durations of marriage but the ages of the women at the time of the census together with the number of childless women. The results are shown in Table XIV. This table shows that under suitable conditions of age at marriage and duration of marriage, the naturally sterile women do not exceed 24 per cent. of the total, and even of that small percentage some part is probably due to impotent husbands. We see also from this table that barrenness increases with the age at marriage, A. O. Powys 249 135 130 125 120) 115 110) 105 — 10 = 95 | $e Fic. 5. Distribution of Fertility. Married 90) Ht | Women, New South Wales. c / = 85 ao Broken Lines—Deaths over 46 years of Age. | 3 80 ; ; x Firm Lines—All married Women at Census. as) : \ / \ 2 = \ Ss 65 £ me I \ \ | ih S 60) 1 a 2 Se aa sala) Fy z \ E 50 4 . = oo ee eee 45 = 40 4 35 a \ 30 25 : | | ; 4 \ 15 10 \ 5 ~ 012 34 5 6 7 8 9 10 1112 13 141516 171819 20 21 29 23 24 25 26 27 28 2930 Size of Family. TABLE XIV. Duration of Marriage at Census 1901 Age of Wife at 10 Years and Over 15 Years and Over 20 Years and Over Census 1901 Total Without Issue Total Without Issue Total Without Issue Wi ce : oe ae Number Per Cent. pine Number | Per Cent. Wives Number | Per Cent. | - a | 20 and under 25 6 1 16°67 — —_— — — — _— 25 a 30 | 3712 119 321 5 = — — — | 30 , 35 | 18000 726 4°08 3438 89 2°59 a = ; o— So} 5 40 | 26520 1281 4°83 16458 536 3°26 2799 68 2°43 40 35 45 | 23904 1183 4:95 20681 740 3°58 12672 309° | 244 4S 3 50 | 17169 1165 6:79 16137 | 878 5°44 13771 598 | 5:98 50 55 | 35294 35038 9°93 34338 2818 8°21 32848 2127 6°47 Totals ... | 124605 | '7978 6°40 91057 5061 5°56 62097 3102 5:00 Biometrika 1v 250 Distribution of Fertility and that for the same age at marriage it decreases with its duration. Thus of women who married at age under 20, and whose duration of marriage was 10 years and upwards, 3°21 per cent. were barren; of those whose duration was 15 years and upwards, 2°59 per cent., whilst for a duration of 20 years and upwards the percentage was only 2°43. These differences are I think really due to first births occurring from 10 to 15 and from 15 to 20 years after marriage, rather than to increased restraint in the more modern marriages, for on reference to the “Vital Statistics” of New South Wales for the 10 years 1893—1902, I find that the average annual first births from 10 to 15 years after marriage were 24, from 15 to 20 years after marriage 5, and from 20 to 25 years ‘5. When the number of marriages and the proportion of those marrying at such ages as to be able to bear children at the intervals referred to are considered, as also the ruptured marriages caused by the death of husband or wife, there is little or nothing left to be accounted for by restraint in the more modern marriages. It seems safe to assume then that the naturally sterile women cannot exceed 24 per cent. So far as I am aware these statistics are the first from which an accurate determination of the value of this ratio can be made. Again referring to Table XIIL, it is seen that the condition of things existing after the close of the reproductive period is that the average family is 6°19 children, the median 5°88, the mode 7, and that 28°36 per cent. of such women produce 50 per cent. of the children. If, however, the married women at all ages at death be taken, it is found that the average family is 5°31 children, the median 4°55, and that 25°61 per cent. of the women produce 50 per cent. of the children. This latter result—based on what I regard as the true method, being on deaths at all ages—is almost in exact accordance with Prof. Pearson’s conclusion from the Anglo-Saxon and Danish statistics, viz. “that one quarter of the popula- TABLE XV. Distribution of Fertility. Duration of Marriage 15 years and upwards. Experience of New South Wales Census 1901. Families | Families Families { Size of | Size of Size of | Family || Family Family Number | Per Cent. || Number | Per Cent. Number | Per Cent. | 0 5074 5°563 9 8420 9:231 18 67 073 il 3309 3°628 10 7260 7959 19 32 035 2 4079 4°472 fat 5076 5°565 20 10 ‘O11 3. 5416 5938 || 12 3550 3°892 21 9 ‘010 4 6909 7575 -|| 13 2078 2°278 22 4 “004 5) 8273 9:070 Lh 1171 1°284 23 3 ‘003 6 9551 10°471 || 16 591 648 26 1 ‘001 tf 10022 10°987 16 283 310 8 9907 10°861 17 119 131 Renee a... | Totals | 91214 100-000 A. O. Powys 251 ? tion produce one half of the next generation*.” This additional corroboration affords strong evidence of its universal application. Next taking the marriages, the duration of which is 15 years and upwards, the distribution of fertility is as shown in Table XV. A comparison of these statistics with the similar statistics of Messrs Rubin and Westergaard’s quoted by Prof. Pearson will show an. immense superiority of fertility in favour of the New South Wales woman, but it must be borne in mind that the Danish statistics refer only to the professional and middle classes of the city of Copenhagen, and which necessarily exclude the most prolific sections (at least it is so in New South Wales, as will be subsequently shown in the section “Reproductive Selection”) of the community, viz. the artisan and the agricultural and pastoral. Here we find 5:56 of the marriages are childless, and as this appears unduly high I have, in calculating the fertility curve, altered the value to 2 per cent., and the computed curve justified the alteration. The following were found to be the constants: fo= 11°32501 Mean Family = 695103 fs= 488272 Modal Family = 568098 p44 = 350°82375 Median Family = 8:03020 (Observed) Sa 016414 Range = 2951447 B,= 2°735343 Standard Deviation= 3°3653 ead Yo = N09;0507 and the equation to the curve of fertility 3 e 6°6225 / we \9"2039 Base DPN ¢ OREO ie Tee) In this series it is found that 31:16 of the women produce one half of the children. The statistics and curve will be found plotted on Fig. 6. On comparing the constants of this curve with those of the Danish or Anglo- Saxon curves, it is seen that we are dealing either with a naturally more prolific community or one in which there is little resort to Malthusian restraint; for although the artisan section of the community is not included in the former statistics, yet the mean family of the New South Wales woman married 15 years and upwards, viz. 6951, is far in excess of the mean Danish artisan family for marriages of 25 years and upwards duration, viz. 526+. As the Scandinavian race is un- questionably a very prolific race, and the Australian mothers are, as shown in Section A, of essentially British stock—which, as Prof. Pearson remarks}, is markedly less fertile than the Danish—it must be concluded that but. little Malthusian restraint is practised in New South Wales: yet, as previously stated, the Government of that State deemed it necessary to appoint a Royal Commission to enquire into the causes of the so-called decline in the birthrate. That Com- * Chances of Death, p. 92. + Loe. cit. p. 98. t Loe, cit. p. 89. 32—2 252 Distribution of Fertility mission concluded “That there is a diminution of fecundity and fertility in recent years which is due to: (a) Deliberate prevention of conception and destruction of embryonic life ; (b) Pathological causes consequent upon the means used and the practices involved therein ; “and from the evidence of medical witnesses, police officers and others, we are led to the conclusion that: (a) The practice of preventing conception by artificial means is common among all classes of the community and in all parts of the State ; (b) This practice has greatly increased during the last 15 years; and (c) The extension of the deliberate and intentional avoidance of procreation by means used to prevent conception is a main factor in producing the decline in birthrate, which we have ascertained to have taken place in New South Wales since the year 1888.” Fic. 6. Curve of Fertility. Duration of Marriage 15 years and upwards. [Experience of New South Wales Census 1901. 110, 105 10 | a 95 90 85 80 | 76 Frequency. a re) a 3 | | 012 3 4 5 67 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Number of Children in Family. 4 x 6°6225 x 9°2039 Equation to Curve y=109°9507 (1+ :53500) (1-s7a55) é Mean Family =6°95103. Modal ,, =6°68020. A. O. Powys 253 It is not my intention to deal with this subject from the ethical standpoint, but a thorough statistical analysis compels me to disagree not only with the conclusions of the Commission but the unscientific reasons in support thereof. It is not necessary here to traverse the evidence given, but it will suffice to produce the actual facts as recorded at the census. It is clearly seen from the distribution of fertility as revealed in Table XV. that we are dealing with a naturally most fertile race. It may be reasonably argued that the distribution shown in that table refers only to marriages which were contracted at least fifteen years ago, and not to present marriages. But we can well judge of the effect of the alleged fifteen years’ excessive restriction if we examine the women who were married from 15 to 20 years at the time of the census and who were then under 40 years of age. The results I find to be as follow : Size of Family Number of Families Size of Family Number of Families 0 553 6 2662 1 637 7 2570 2 948 & 2063 3 1402 9 1125 4 1967 10 539 5 2235 11 and over* 369 Here it is seen that (allowing only 11 children to the 369 women who had 11 and over) the mean family is 5°61 children and only 3:24 per cent. who had so far proved sterile, and consequently these figures utterly refute the Commission’s conclusions. In view of these figures, we can assert that there cannot have been much restraint practised in New South Wales during the last fifteen years. I wish to emphasise this point in order that it may be recognised that the curves of fertility for various durations of marriage in New South Wales are as nearly as possible those free from restraint. That restraint is practised by certain sections of the community I am well aware, as also that it always has been so in some form or other, but in view of the figures which have been produced and those to be produced, I think it is altogether erroneous to assert that the practice is common to all sections of the community and to all parts of the State. Now let us ascertain the distribution for durations of marriage of less than 15 years. I have taken those women who were under 20 years of age at marriage, and have divided the duration of marriage into three periods—from 4 to 5 years, from 5 to 10 years, and from 10 to 15 years—the ages at the census being under 30 for the first two, and under 35 for the latter period. The statistics and percentages are shown in Table XVI. * The returns do not show the distribution beyond 10 children. 254 Distribution of Fertility TABLE XVI. Duration of Marriage and Number of Offspring. Women’s Age at Marriage under 25. Eaperience of New South Wales Census 1901. Duration 4 to 5 Years Duration 5 to 10 Years Duration 10 to 15 Years Number | of ! Offspring | ies Percentage Vere Percentage ae Percentage 0 541 9°47 | 1094. 6°05 757 4:14 1 1386 24°27 | 2230 12°33 1039 5°68 2 2731 47°81 4811 26°60 1874 10°25 ol 954 16°70 5567 30°78 2931 16°03 4 96 1°68 3124 17-27 3603 19°71 5 4 ‘O07 996 5°51 3715 20°32 6 — — 226 1:25 2542 13°91 tif —_ — 36 *20 1233 6°75 8 = — 2 ‘Ol 422 2°31 9 — _ — — 118 65 10 = = — — 38 21 11 — — = = if 04 Totals 5712 100°00 18086 100°00 18279 100°00 These percentages have been plotted on Fig. 7, and the fertility curves computed. The constants were : Duration of Marriage. 4 to 5 years 5 to 10 years 10 to 15 years He *72604 1°6271 3°6399 ite — ‘11092 + +1283 — ‘5179 Ms 1°51477 77554 36°6377 By 032146 ‘003821 005562 Bs 2°873581 2929381 2°765345 Range 10:0683 22°5456 182893 Mean Family 1°7706 2°6348 4°1229 Modal Family 1°863] 2°5932 4°2086 Standard Deviation "8521 1:2756 1°9078 Yo 45-9164 30°9904 20°2118 It is worthy of notice that for marriages of an average duration of 4°5 years the mean family is 1:77, and that for an average duration of 7°5 years it is 263— the difference of 3 years’ duration being responsible for ‘86 child or ‘288 annually. The difference in mean families for marriages of 74 and 124 years’ mean duration is 1488 child or 298 annually. It is thus seen that the average annual rate of increase is very nearly uniform, viz. about 290 or 300 children per 1000 married women. It would thus appear that the initial rate of fertility is maintained at all events for 15 years of marriage by women marrying at under 25 years of age. Whilst such a rate as that prevails in a community I do not think Per 100 Married Women. A. O. Powys 255 Fic. 7. Distribution of Fertility amongst Women—Married at Age Under 25. Experience of New South Wales Census 1901. A, Duration of Marriage 4 to 5 Years. 50 40 4p 5 | 5 = 30 1 RS vo = “4 | a) S20 Z| Ss S x s re | \ s 10 : { (oR Ee 7 fe) 1 2 3 Number of Offspring. Mean at 1°7706 Children. Mode at 1-8631 Children (Origin of Curve). E ti toc 45-9164 (1 x \ 18°85225 x 10°78065 oats Rrra sae ( + ¢-a95%0 ) ( Sccse) : B, Duration of Marriage 5 to 10 Years. 30 i) {eo} Per 100 Married Women. _ ° ° Oo 1 2 3 4 5 6 7 8 9 Number of Offspring. Mean at 2°6348 Children. Mode at 2°5932-Children (Origin of Curve). : a x 31-6924 a 41°9672 Equation to Curve y=30-9904 (1 ae 7708) (2 = ae - C, Duration of Marriage 10 to 15 years. 30 20 l I oS x i l 4 5 6 7 8 9 10.1 12 Number of Offspring. Mean at 4:1229 Children. Mode at 4:2086 Children (Origin of Curve). 3 i 1086664 "85926 Equation to Curve y=20-2118 (14 z ) : (1-a5 y 4 a oO 1 2 3 10-0752 82141 256 Age at Marriage and Duration of Life it can be seriously held that there has been very much Malthusian restraint; especially is this so when it is borne in mind that the potential embryonic life has not of course been taken into account. It may therefore I think be held that the distributions of fertility as set forth in Tables XV. and XVI. are as nearly as possible those of a non-restricting population. C. On the Correlation between Age at Marriage and Duration of Life beyond Age 46. Although in Section A of this investigation we found that the number of offspring born to a woman is highly correlated with the duration of life beyond the reproductive period, irrespective of the age at marriage, yet it is possible that the fact that maximum fertility is not favourable to extreme longevity may be due to the generally early contraction of marriages, which is of course generally productive of large families—before the proper development of the reproductive organs. The period of mature or maximum development has been shown else- where to be about age 25*. Although it is clear that early marriage has not the effect of arresting development (perhaps it tends to too rapid development), this may be accomplished to the subsequent prejudice of life. ‘Had it been possible to have examined each age at marriage with each sized family and subsequent life incidental to each sized family, this could readily have been ascertained, but the voluminous statistics involved present an almost insuperable obstacle to its accomplishment. However, statistics are published in the “ Vital Statistics” of New South Wales for the two years 1901 and 1902 which show the age at marriage, the number of offspring, and the duration of life after marriage, but we are unable to connect mothers of each sized family with the duration of marriage. One table in the “Vital Statistics” shows the ages at marriage in quinquennial groupings, together with the durations of marriage by single years and the total offspring. In order to obtain from this table the average number of children born to women marrying in each quinquennial group, and who survived the reproductive period, I have taken the earliest duration such that if added to the latest year of such group the sum would be 49. Thus for age at marriage under 20 the duration of marriage was 29 years and upwards; for age at marriage 20—25 years the duration was 24 years and upwards; and so on. From the table referred to I have compiled the information shown in columns 2, 3 and 4 in Table XVII. Another table in the “Vital Statistics” shows the ages at marriage in single years and the duration of marriage also in single years, but not the number of offspring, and from this table I have computed for each single year of age at marriage the mean duration of life beyond age 46, and the results appear in Table XVIII. The means of the quinquennial groups are shown in the fifth column of Table XVII. Although the two sets of women shown in the * Biometrika, Vol, 1. Part 1 p. 33. A. O. Powys 257 two tables are not necessarily the same, it will be noticed that there is but little difference in the numbers within the groups. It must also be noted that all the figures refer to age at first marriage, if more than one has been contracted. TABLE XVII. | . Average Life beyond Age at Marriage N Eas of ae a soe Age 46 by Survivors eaths hildren nildren ofthat Age Under 20 years 1226 9892 8:06 20°66 years 20 and under 25 years 1661 11462 6°90 20°70 ,, 5 eagle 843 4399 5°22 23-b4,, 30 . 35 4, 326 1096 3°36 23°44 ,, 19) 3 LO 121 253 2°09 25°20 ,, 40 53 45 ,, 92 60 0°65 TABLE XVIII. 5 || r | Age at | Deaths after an eee Mean Duration || Ageat | Deaths after beh ses a Mean Duration Marriage Age 46 alae after Age 46 | Marriage Age 46 ae eine ee after Age 46 Age 46 Age 46 15 years Die 1277 22°40 years 28 years 156 3723 23°87 years LG 3, 162 3401 20°99 __,, gs 93 2013 21°65, IY op 238 4795 20°15 ,, || 30, 143 3487 24°39, 137), 399 8203 20°56 ,, ol, 44 902 20°50, YG 378 7793 20°62, ei Ty 56 1383 24°70, AD see 417 8616 20°66 _,, 33 yy 42 915 ATA il ee 395 8246 20:8350 5, B44, 39 907 23°26 ,, ies 319 6855 21°49, 35 4, 38 928 24°42 ,, 23s, 272 5224 19°21, || 36 ,, 29 760 26°21, cp , 251 5298 PAIL || 37, 24 593 24°71 —,, BB oe 269 6339 23°57 ,, || 38 ,, 21 514 24°48, 26! es 183 4425 24°18, i) os 10 279 27°90, CE os 132 3107 23°54, From the statistics contained in Table XVIIL. the correlation between the age at marriage and the duration of life after age 46 was determined. The following were the constants: Mean Avera: Marmages...0.m: msc: eascommcns. sequins 22°6806 years Mean Duration of Life after Age 46 ............... 21°5942 ; bs ae. a Mammater cartes: 45835 _ ,, = deme eld Poel Duration of Life ............ 16004 — ,, Gomme ablomie camecnciveveseqnuens ier eakaeens + °70679 + 00523 Coefficient of Regression ...............055 + ‘23163 Regression Straight Line (Origin at Age at Marriage 14 years. Unit of e=1 year), y = 19°5835 + °23163 a. Biometrika Iv 33 258 Age at Marriage and Duration of Life The statistics and regression straight lines are shown on Fig. 8. The correlation is high and within very small limits of probable error, and we are therefore justified in concluding that early marriage, whether in consequence of generally larger families (see Section A) or in consequence of immature develop- ment, is distinctly prejudicial to longevity beyond the reproductive period. If any weight is to be attached to the results of the comparatively meagre ob- servations of the marriages contracted between ages 35 and 40, we would be Fic. 8. Age at Marriage and Duration of Life beyond Age 46 by Survivors of that Age. Females. (ere)/ (ea a ee WE Soe aa a a OO a EP Duration of Life beyond Age 46. 11 “SRR ReR ees o 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Age at Marriage. Equations to Regression Straight Lines : aa’ (Origin at 14 years of Age. Unit of x=1 year) y =19°5835 + 231630. bb’ (Origin at 14 years of Age. Unit of x=1 year) y =20°8118 — -0228x. ce’ (Origin at 24 years of Age. Unit of x=1 year) y = 23°4057 + 056292. further justified in concluding that the longer marriage is deferred the greater the probable lifetime after the reproductive period is passed. If, however, we examine Fig. 8, we see that the statistics divide into two fairly well defined sections, viz. from age 15 at marriage to age 24, and from age 25 upwards. The dividing age (24—25) is remarkable as that found to be that of maximum initial A. O. Powys 259 fertility and also that of greatest physical development, at least so far as stature is concerned*, I have therefore determined the correlation for each of these sections with the following results : Age at Marriage 15 to 24 years 25 to 39 years Mean Age at Marriage F sis 20°0104 years 28°7099 years Mean Duration of life beyond Age 46 20°6745_,, 23°6708 ,, ae Age at Marriage — 2°3557 3°5277 Standard Deviat ei: oy a ” si eae ee of Life "6221 = ,, 1:0769_,, Correlation ... a ea ee — 0865 + 0124 +:1844+ 01822 Coefficient of Regression ae — 02284 + 05629 Origin at Age at Marriage (Origin at Age at Marriage Regression Straight Line 14 years 24 years y= 20'8118 — 02282 y = 23°4057 + 05629x In these two series we see that the correlation is low, though sensibly within the limits of probable error. The coefficient of regression shows that there is little difference in the average duration of life beyond the reproductive period by marriage at any age under 25 years—the mean being 20°01 years; also that there is but little difference by marriage at any age between 25 and 389—the mean of which is 23°67 years. But there is a most decided advantage in longevity by deferring marriage until after age 25—a difference of 32 years—and as these are the results of over 4000 observations, they must be accepted as reliable and not merely as accidental. In comparing these results with those of Section A, it must be remembered that in these observations the average lifetime is curtate, Le. only completed years have been counted—the year of death is not considered— and that the age 46 is the origin. To bring these into line, therefore, with those of Section A, 14 years must be added to the present results. We see from Table XVII. that the average number of offspring to marriages contracted under age 25, and in which the wife at least survives to age 46, is 7°40, and that of marriages contracted between ages 25 and 40, 446. If now we turn to Fig. 3 of Section A we find from the parabola that the average durations of life after age 45 to mothers of 7:40 and 4°46 children are almost identical, viz. 23°4 years, whilst (adding the corrections referred to above, viz. 1} years) the average durations of those who marry before and after 25 years of age are 22°2 and 25°2 years respectively. These differences I think show that the age at marriage, as well as the number of offspring, is a factor in connection with the duration of life beyond the reproductive period. During the reproductive period it is of course obvious that as each successive birth has its risk, mothers of large families imperil their lives more than those of smaller ones—not simply in proportion to the number of births, but after the first confinement in a constantly increasing ratio. From the “ Vital Statistics” of New South Wales, I have been enabled to deter- mine the risk incidental to each confinement. In one table is to be found the order of confinement—first, second, etc., whilst in another are to be found the * Biometrika, Vol. 1. Part 1. p. 33. 33—2 260 Age at Marriage and Duration of Life deaths in childbirth in order of confinement. This information is available for the nine years 1894 to 1902 and, together with the crude probability of death . in each confinement, is shown in Table XIX. TABLE XIX. Probability of Death in Confinement. yaperience of New South Wales, 1894-—1902. | Order of | é Probability |, rte. a Probability of | Confine- | Births Deaths 11 | of Death in CG ries of t Births eens me A Death in ment | Confinement Confinement | rau eieya Onions) Confinement 1 65778 552 ‘00839 11 4698 46 ‘00979 2 52818 258 ‘00488 12 2716 34 "01252 oe | 43462 240 *00552 13 1427 24 "01682 4 35510 213 | ‘06600 14 703 9 ‘01280 5 28920 204 00705 15 338 6 01775 6 | 23527 161 ‘00684 | 16 129 1 ‘00775 tf 18729 130 ‘00694 17 68 | oo il 01471 \. 8 | 14659 132 “00900 18 aif 288| 9 | 4 | -ogsa4 f OL625 9 | 10580 114 “01078 Beyond 18 18 — — 10 7230 61 *00844 Rejecting the statistics of the sixteenth and subsequent confinements as being too limited to yield satisfactory results, the probabilities of death were graduated by means of a parabola of the fourth order, the equation to which was found to be y = 8384571 + 846415 & + 040384 a — 002525 a* + 000568 a%, where y=the probability of death, and # the number of confinement—the origin of the curve being at the eighth confinement. There are no points of inflexion, and the curve is convex throughout. The theoretical minimum occurs at the 3:0954 confinement. The statistics and curve will be found plotted on Fig. 9. Obtaining the probabilities of death from the parabola, we are enabled to ascertain the number of survivors of any given number of fertile married women at each confinement—presuming the mortality of childbirth to be the only operative cause of death. Thus adopting the symbols used in life tables*, calling x the number of confinement, gq, the probability of death in the xth confinement, p, the probability of surviving it, and l,, we have (taking 10,000 fertile married women): x Vee Px by a Vx Px l, 1 “00667 99333 9933 ig, 00927 *99073 9387 2 “00604 ‘99396 9873 10 01027 ‘98973 9291 3 00583 *99417 9815 11 01127 ‘98873 9186 4 00595 “99405 9757 12 01240 ‘98760 9072 5) 00632 *99368 9695 13 01367 98633 8948 6 00691 99309 9628 14 ‘O1511 “98489 8831 7 ‘00758 “99242 9555 15 ‘01679 98321 8665 & 00838 99162 9475 * Institute of Actuaries Text Book, Part 1, p. xxi. A. O. Powys 261 Thus if no other causes of death were in operation, we see that no less than 134 out of every 1000 women would perish from accidents of childbirth in accomplishing the fifteenth confinement, and each subsequent confinement would be accompanied by an ever-increasing risk. It thus appears that granting that economic reasons (such as in sparsely populated territories as Australia, where there is but little more than one individual to the square mile) demand that there Fic. 9. Probability of Death in Confinement. Experience of New South Wales, 1894—1902. cle | 17}—4 16+— 15 14 13 _ (S) Ss a Ht 4 —t 12 3 4 5 6 7 8 9 10 11 1213 14 15 Number of Confinements. Mortality per 1000 Confinements. orp wk GDN wD O Equation to Curve (Origin at 8 Children): y =8°384571 + 8464152 + 04038427 — 00252527 + 00056824. should be but little if any restriction in families, it is distinctly prejudicial to the individual, both during and beyond the reproductive period, to bear a large family. It may of course be urged that it is prejudicial within the reproductive period even to bear but one, but it will be seen from the last table that to bear a family of six the risk cannot be said to be large—over 96 per cent. of the women will survive this—whilst the reproductive period passed almost a maximum expectation is attained. Turning now to married men marrying at various ages, I have computed from similar tables to those from which the information was obtained for married women, the mean duration of life beyond age 46 by married men surviving that age and marrying at the various ages. The age of 46 is not of course a critical age in man as it is in woman, but has been adopted so that comparison 262 Age at Marriage and Duration of Life might be effected with the duration of life in women. The results are shown in Table XX. TABLE XX. Mean Duration of Life of Married Men beyond Age 46 by survivors of that age. Experience of New South Wales, 1901 and 1902. eae Deaths | Total years | Mean Duration | ioe ak Deaths | Total years) Mean Duration M ge. after | lived after | of life after | M et after | lived after of life after arriage! Ace 46| Age 46 | Age 46 jetnace! Age 46) Age 46 Age 46 = | 20 222 4290 19°32 years | 33 137 3075 22°45 years 21 342 6595 OE2S se. 3 158 3152 19°95 _,, 22 358 7021 19:61 —,, 35 136 3407 25°05 ,, 23 336 6510 19°38, 36 109 2409 22°10 ,, 2 342 6712 19°63, 37 91 1937 21°29, 25 410 7958 1954; | 38 86 1803 22°13" ,, 26 351 6777 19°31, 39 | 69 1390 20°14 ,, 27 300 5894 9:65"; 40 | 102 2504 24°55, 28 339 6489 19°14, 41 | 39 821 21°05, 29 241 4990 | 20°71 ,, | 2 52 957 18°40, 30 285 5864 | 20°58 ,, | Tis Nie 47 | 483 LEO 31 158 3223 20°40 ,, 4A | 835 824 23°54 ,, 82 180 4075 22°64 ,, I] | The correlation between age at marriage and duration of life beyond age 46 was determined, and also the regression straight line. The following are the general results: Mean Age at Marriage: aie ccascsncreue star eine: 27°94414 years. Mean Duration of Life beyond Age 46....%....... 20°2365 - : ae ee (Ape at Marrmlagetrn je scene (le Mley -y, Suen Deviation} tion ofplite. aus ceeee 1°4623 @onrelationy 2csdcesiee eae eee + 51973 + 007038 Coefficient of Regression .....3......... + 10642 Regression Straight Line (Origin at Age at Marriage 19 years. Unit of a=1 year), y= 19-2847 + 10642 a. The statistics and regression straight line will be found plotted on Fig. 10. Here we see that early marriage is again apparently unfavourable to longevity in man, though not so pronounced as in woman, as perhaps a priori might have been suspected. It has been suggested that two causes operate against longevity in excessively fertile women, viz. the physical strain in bearing and the mental strain in rearing large families. The latter factor only is operative in the case of males, and probably contributes to the shortening of the duration of life in men marrying at the early ages. Again we see that the statistics divide into two well-defined series, the demarcation being at age 28 at marriage. Up to and inclusive of that age the results are remarkably uniform, the mean of the mean durations of life beyond age 46 being 19:41 years. For marriages contracted beyond age 28 the results are rather irregular, but distinctly exhibit an increase A. O. Powys 263 in longevity, the mean duration being 21°53 years—an advance of over 2 years— and there can be no doubt that the increase is a real one, whatever may be the cause. It of course may be urged, and with much force, that amongst those following hazardous occupations, such as the industrial classes, marriage is generally Fic. 10, Age at Marriage and Duration of Life beyond Age 46 by Survivors of that Age. Males. AEE Duration of Life beyond Age 46. Age at Marriage. Equation to Regression Straight Line: aa’ (Origin at 19 years of Age. Unit of c=1 year) y = 19-2847 + 106422, contracted early in life, whilst amongst those followmg the least hazardous— the professional—marriage is contracted much later. This is true, as will be seen from Section K, though the difference in age at marriage does not exceed 12 years in the two classes. The adults’ death rates are also shown in the same section, that of the professional class being 15°73 and that of the industrial 22°72 per 1000 exposed to risk. But notwithstanding this great ditference in the mortality in view of the slight difference in the age at marriage, it is doubtful whether if all allowances be made the difference in the duration of life can be thus accounted for. Some portion is probably due to the struggle to maintain the large family generally connected with early marriage *. [* It will be seen from Biometrika, Vol. 1. p. 46 et seq., that a man’s prime is about 28 in the same sense as a woman’s is about 24. May not therefore the lessened expectation of life in the case of man as in the case of woman be due to exercise of sex-functions before the prime? Further is not the association of extreme fertility with shorter duration of life in women due in part to the fact that very large families mean as a rule early marriages? K. P.] 264 Duration of Marriage and Number of Offspring D. On the Correlation between Duration of Marriage of Females and Number of Offspring—at various ages at Marriage. There are two methods by which the necessary information may be obtained : first, by noting at death the date of and age at marriage and the total offspring ; second, at a census obtaining the information direct from the living married women. In New South Wales the materials requisite for both methods are available, but the information for the first method refers to women who at the time of death were either married or widows, or who had been separated from their husbands. This information will be found in the “ Vital Statistics” of TABLE XXI. Deaths of Females, Age at Marriage, Duration of Life after Marriage, and Number of Offspring. Age at Marriage Duration oe Under 20 20 and under 25 25 and under 30 30 and under 35 after Marriage (years) Offspring Offspring Offspring Offspring Deaths| | Deaths |__| Deaths|__ >| Deaths! |= Total | Average Total | Average Total | Average Total | Average 0 11 7 64 26 13 50 13 8 62 5 4 80 1 17 13 76 66 53 80 41 34 83 10 2 20 2 9 14 1°56 37 43 1:16 19 20 1:05 10 13 1°30 3 16 26 | 1°62 29 44 1°52 20 37 1°61 8 13 1°62 4 16 31 | 1°94 48 90 1°87 18 24 1°33 8 14 1°75 5 11 35 3°18 37 1 1°95 15 29 1:93 6 8 1:33 6 22) 50 POAT 51 125 2°45 26 66 2°54 4 9 2°25 7 15 38 2°53 27 ee 2°85 Iiy/ 37 2°18 7 13 1°86 8 18 49 2°72 32 92 2°88 22, 55 2°50 6 19 3°17 9 27 87 3°22 34 98 2°88 14 49 3°50 9 24 2°67 10 27 99 3°67 42 153 3°64 15 41 2°73 9 31 3°44 11 oT, 103 3°82 37 141 3°81 27 79 2°93 6 14 2°33 12 17 65 | 3°82 46 192 4°17 23 104 4°52 3 17 5°67 13 31 120 3°87 29 130 4:48 20 85 4:25 6 13 2°17 14 23 99 4°30 41 179 4°36 13 48 3°69 4 8 2°00 15 28 122 4°36 51 230 4°5] i, 82 4°82 16 23 111 4°83 42 234 5°57 20 104 5°20 17 29 136 4°69 39 239 6°13 9 38 4:29 18 34 184 5°41 44 244 5°55 16 72 4°50 — — = 19 26 124 4°77 55 281 5°11 19 87 4°58 — _ — 20 37 206 5°57 40 217 5°43 21 34 208 6°12 36 233 6°47 22 93° || 186) a 8 09n ey 44s oad 5°B2 — 23 28 185 6°61 41 281 6°85 24 28 204 7°29 41 230 5°61 eo 32 243 | 7°59 26 34 244 | 7°18 = = 27 23 al |e e43 — = 28 18 143 7°94 — | — 29 26 187 7:19 - — — — —- A. O. Powys 265 New South Wales for the two years 1901 and 1902, whilst that relating to married women living with their husbands at the time of the census of 1901 will be found in Part “Conjugal Condition and Families” of the Census returns. The former tables show for married women dying in those years the age at marriage (in quinquennial groups), the duration of marriage in single years, and total offspring. Limiting the information to the reproductive period, the statistics are shown in Table XXI. The correlation and regression straight lines were determined for each of these series with the results set forth in Tables XXII. and XXIII. TABLE XXII. General Results. Mean Duration Standard Deviation Correlation. of Life after Mean Duration of Life Age at Marriage | Marriage(within| Size of | Duration of | Size | after Marriage | Regression Reproductive | Family | Life after of and Size of Period) Marriage | Family Family | Under 20 16°7648 years | 49155 | 7-9165 years | 2°0168 | 97328 + :00133 "24796 20 and under 25 | 12°6961_,, 3°8670 | 7°2585 ,, | 1°8226 | 96554 + -00143 *24245 25 3 30| 9°3217 ,, 2°8398 | 5°7459 4, | 174459 | 94568 + 00362 ‘23797 30 i$ 35 | 6:8762 ,, 2°0000 | 4°1622 ,, 1:0963 | °65700 + ‘03812 17304 TABLE XXIII. Regression Formule. z=Duration of Life after Marriage (within Reproductive Period). y=Number of Offspring. Age at Marriage, Under 20 years, y = "7585 + °24796x ms 20 and under 25 years, y="7888+4'242454 H 3 25 . 30s, y= 621542379747 Fe 5) 30 e 35s, y= 8101 +°17304x Note:—Origin at marriage and unit of v=1 year. The statistics and regression straight lines will be found plotted on Figs. 11, 12, 13 and 14; the average number of offspring for each year of life after marriage is shown by small circles, and that for quinquennial periods by triangles. It is seen that in each series although the statistics are somewhat irregular they are well represented by the straight lines, and thus it would appear that the initial fertility incidental to the age at marriage is preserved almost unchanged through- out the reproductive period. It must be clearly understood that the foregoing are the results of women dying who had been married and whose husbands may or may not have survived them. Although it is seen that the initial fertility is preserved throughout the whole of the reproductive period, yet the age at marriage is the determinant of that fertility. Up to age 30 there would seem to be but little difference in this initial fertility, although it is slightly less at Biometrika Iv 34 266 Duration of Marriage and Number of Offspring Age at Marriage, Duration of Marriage and Number of Offspring. Eaperience of New South Wales, Deaths of Married Women, 1901 and 1902. Fic. 11. Age at Marriage, Under 20 years. 8 am eT ieee PEAR AY | - BaSESaE Poppet ts th Bec BRREPRY aaa cease FEE vat ‘2 5 oO ‘> 4 33 ae 1 a te) 5 10 30 Duration of Marriage. Fic. 12. Age at Marriage, 20 to 25 years. irae Slag > = 5 & 4 we 3 = 2 a 1 a = Py le oh eee) We be 0) 5 10 15 20 25 Duration of Marriage. Fie. 13. Age at Marriage, 25 to 30 years. Fic. 14. Age at Marriage, 30 to 35 years. pa a Tatty? =; ang | = a7 = ~ S = 8 > = & g2 3 3 S = 4 a 2 4 0 5 10 i 20 0 Duration of Marriage. Duration of Marriage. Regression Straight Lines aa’. Averages of Quinquennial Period shown thus Aves A A. O. Powys 267 each advance in the age at marriage. At age 30—35 there is a most decided fall in the initial fertility, and the extent of the fall when marriage is deferred to age 35—40 may be judged from the fact that 20 women who married at that age, and whose duration of marriage was under 5 years, bore only 14 children, and 7 whose duration was 5 to 10 years bore 10 children. After age 40 at marriage—by unprejudiced* spinsters—the initial fertility may be said to have reached zero. Calculating the probable average number of children at the end of the reproductive period (end of forty-fifth year) by means of the regression formule, we obtain For Age at Marriage under 20 ..............008. 755 Soe er > 20-and under 25. v.02... 6°51 Sha hie ‘5 PASTE oe (AOU Paes ae isa ib. » ee 3 Sil). fo. DOwaaeaies 3°21 —results which are somewhat less than the actual results of those women who survived the reproductive period, and which are shown in Table XVII. Turning now to the census results of New South Wales for 1901, we are able to obtain—for wives who were living with their husbands on the night of the census—the age, the duration of marriage, and the number of offspring. From the nature of the tabulation we are unable to obtain this information for all wives, but this will best be understood from the following specimen of the tabulation: — SPECIMEN OF TABULATION. Number of Wwes without Issue. Age of Wife Duration of Marriage Total Wives Under 1 year 1 year | 2 years 3 years 4 years 5 and under 10 years | 10 and under 15 years 15 and under 20 years And so on in quinquennial periods 21 and under 25 14 years LON, 5 HO SCN 555 20 » 30 o 5 35 These tabulations appear for each sized family up to 10 children, 11 and over being amalgamated. Thus if we wish to trace women married, say, at age under 20, and with duration of marriage under 5 years, we can only make use of those who were specifically stated at the census to be under 20 and who had been married less than 5 years. This of course does not give us all such women, since * Those for whom conception was not antenuptial. 34—2 268 Duration of Marriage and Number of Offspring some who had been married under 1 year, and between 1 and 2 years, and were aged 20 at census, belong to the age at marriage group under 20. So also do some women aged 21 to 25 years, and who had been married 2 and under 5 years. If we take duration of marriage 5 to 10 years with age at census 20—25, we see that the whole of these women must have been married at under age 20. But so too do some who were aged 25—30 with the same duration of marriage, but we are unable to determine how many. I have therefore been obliged to confine the investigation to such groups as are absolutely certain. Another difficulty has to be faced when we reach the duration of marriage sufficient to yield 11 or more children. To effect the distribution of families beyond 10 TABLE XXIV. Distribution of Fertility amongst Wives Married at Age under 20 Years. Huperience of New South Wales Census, 1901. Duration of Marriage Number = _ annals : of : | : _j,., | 9 and under 10 | 10 and under 15 | 15 and under 20 | 20 and under 25 | 25 and under 30 see Under 5 years years years years years years ees | Total i Total Total Total Total Total : ota 5 Pota : Tota P Tota a Tota Bats. ota Wives | children | V4¥¢8 | Children | Wives | Children | Wives | Children | Wi¥€s | Children | ¥#¥€*| Children 0 1171 -— 166 — 120 — 89 — 68 — il 1166 1166 438 438 228 228 158 158 77 77 2 188 376 1026 2052 376 752 195 390 88 176 3 22 66 1034 3102 624 1872 289 867 152 456 4 = = 422 1688 812 3248 429 1716 170 680 5 — = 83 415 864 4320 490 2450 231 1155 6 = — 17 102 484 2904 583 3498 312 1872 of — — 4 28 155 1085 520 3640 330 2310 8 — — = — 39 312 398 3184 396 3168 9 —- — = = 9 81 170 1530 367 3303 10 = — = = 1 10 72 720 314 | 3140 11 — — | = 36 396 260 | 2860 12 — — =r | 34 408 13 = — _ eis 14 = = = _ = 16 ~- — = — _ — — — = = 16 = = = = : 2 = = re = = = — 18 = = = zits = = i) = — = —_ a — —_ 20 = = — — Totals | 2547 1608 3190 7825 3712 | 14812 3429 | 18549 2799 | 19605 Fonte 63 2-45 3-99 5-41 7-00 Per cent without 46°0 5°20 3°23 2°60 2°43 1:90 issue A. O. Powys 269 children, we are obliged to call to our assistance another table in the census which shows the number of wives with each sized family and the ages at marriage, but not the duration of marriage. In the manner thus shown—having regard to the age at marriage—the distribution of families of 11 and more children has been effected, and to this extent the columns of the last three durations in Table XXIV. are vitiated, though the averages obtained cannot possibly be affected to any appreciable extent. ‘This table shows the distribution of fertility to wives married at under 20 years for various durations of marriage within the reproductive period. The average issue for under 5 years’ duration of marriage is evidently too small and the percentage of women without issue too large, owing to the large’ number of women married under 1 year to be fair criteria of the results of a mean duration of 24 years’ marriage. The results of this duration were obtained from the census and were found to be 422 women with 468 children (56 had 0, 270 had 1, 90 had 2, and 6 had 3), or an average of 1:11 each with 13°3 per cent. without issue. As, however, the result was 1°32 in the case of deaths of married women, which I think is more probably the truth in view of the fact that so many of those marrying under 20 being undetermined in the ages 20 and 21—25, I adopted this value in the calculation of the correlation and regression straight line, with the following general results : MeameA erat Marriage: aici assault wtaeonsieieeewees 14-6415 years. Pe MOUZCPOL TH AIIWLY © seasieiincioes gee uo asians ads 4:6967 children. : .,. _ (Duration of Marriage ...... 79585 years. Sa cmanion ee OL WAM beans ncores 2°3841 children. Orne AbiG meyer nec cates Arcee cs Ga saa sooo ‘99790 + 00002 Cocticient of Repression) ..sec.sc.sescdevcescessce scar 29894 Regression Straight Line (Origin at Marriage. Unit of «= 1 year) y = 3198 + 29894 2, where x = duration of marriage, and y the number of offspring. The statistics and regression straight line are plotted on Fig. 15, on which is also plotted for the purpose of comparison the regression straight line as determined by deaths. The methods of tabulation of the census do not admit of tracing the marriages contracted between intervals of 5 years, say, between 20 and 25, but does admit of them being dealt with as under 25 years at marriage*. The information for * It might at first sight be thought possible to have effected the desired computations for quin- quennial periods. For the results are ascertainable for marriages contracted as under each quinquennial age: as for instance, under age 20, under age 25, under age 30, &c., and the difference between any two succeeding ages would appear to give the necessary results. Thus, take the duration of marriage 5 and under 10 years, marrying under age 20 there were 3190 such women who had 7825 children ; marrying under age 25 there were 18,086 such women who had 47,653 children; but it is not true that the difference between these two series, viz. 14,896 women haying 39,828 children married between the ages 20 and 25, for many (and it is impossible to say how many) of the latter were married at ages under 20. The number shown as marrying under 20 are those we are certain of from the returns but are not all. The same applies to all the groups. 270 Duration of Marriage and Number of Offspring the latter period I have also taken from the census returns, and the results are shown in Table XXV. TABLE XXV. Distribution of Fertility amongst Wives Married at Age under 25 Years. Experience of New South Wales Census, 1901. Duration of Marriage Number _ of Wilde 5 and under 10 | 10 and under 15 | 15 and under 20 | 20 and under 25 SHShee Under 5 years years years years years Tee Total ime 1 1 Total Total Total : Tota : ota : ota : Tota Q ota Wives | Ghildren | W#¥e8 | Children | Wi¥€S | Children | ¥i¥e | Children | ¥i¥e8 | children @) 6152 —_ 1094 | — TEE — 553 = 332 — if 8133 8133 2230 2230 1039 1039 637 637 344 344 2 3668 7336 4811 9622 1874 3748 928 1856 477 954 34 597 1791 5567 | 16701 2931 8793 1402 4206 669 2007 4 43 72 3124 | 12496 3603 | 14412 1967 7868 858 3432 5 2 10 996 4980 3715 | 18575 2235 | 11175 1122 5610 6 — = 226 1356 2542 | 15252 2662 | 15972 1383 8298 7 = — 36 252 1233 8631 2570 17990 1548 | 10836 8 — — 2 | 16 422 3376 | 2063 | 16504 1735 | 13880 9 118 | 1062 | 1125 10125 | 1670 | 15030 10 eae 38 380 539 | 5390 1368 | 13680 il 7 OU 200* 2200 652+} 7172 12 - 90*) 1080 454+) 5448 13 = — = = — = 50* ~~ 650 264+} 3432 14 = = = = = = 29* 406 149+] 2086 15 = =: = — — = — — 74+} 1110 16 = == = = 36t 576 7 — = — = 15+ 255 18 : = a =e 8t| 144 Totals | 18595| 17442 |18086) 47653 | 18279| 75345 | 17050) 96059 | 13158 | 94294 Average | ‘94 2°64 412 5°64 717 Family Per cent. without Son 6:05 4:14 3°24 2°52 issue Owing to the number of wives (4114) who had been married under 1 year, the statistics of under 5 years’ duration of marriage were not used to determine the average number of offspring for the mean of that duration, but the actual result of 24 years’ duration, viz. 3968 wives (563 without issue, 2570 with 1 child, 814 with 2, and 21 with 3) having 4261 children, or an average of 107. The per- centage of childless women was 142. With this substitution the correlation * Estimated—Total 11 and over, 369. + Estimated—Total 11 and over, 1652. A. O. Powys 271 Average Offspring. Duration of Marriage (Years). Regression Straight Lines: aa’ (Census), a A ar at Marriage. bb’ (Deaths), y=°7585+°247962 } Unit of «=1 year. between duration of marriage and number of offspring was determined from the figures in the above table with the following results : Mean Duration of Marriage ........0..0-sssseeeeees 13:7295 years. PRMD ZEN Ol HANAN s eucRisvestiadiucuilacs acy tes ¥aee 4°5031 children. ._. (Duration of Marriage...... 59345 years. Buandare Deviation See Of Mamrly:sisscs: csaenn 1:7972 children. Wore a GlOM ead celal ctan vaculiteaiean eeaneuetaeakanir 1:0000 Coefficient of Regression ...........csceeeeeeseee ees 30286 Regression Straight Line (Origin at Marriage. Unit of « = 1 year) y = 3450 + °30286 a, where w= duration of marriage, and y the number of offspring. For the purpose of comparison the regression straight line was deterinined from the deaths on a similar basis (age at marriage under 25 years, and duration of marriage in quinquennial groupings), and was found to be y=°7360 + °2484 a. On comparing the results from the census figures of those marrying at under 20 with those marrying under 25, we see that in both cases, as in the results obtained from the deaths, that the initial fertility is preserved throughout the reproductive period, and that it is slightly greater when marriage is contracted under age 25 than when contracted under age 20; that is, that it is greater at age 20 to 25 than under 20. This is in accordance with results arrived at else- where* in connection with first births. It is also seen that the mean annual * Biometrika, Vol. 1. Part 1. p. 33. 272 Duration of Marriage and Number of Offspring births are about 300 per 1000 married women marrying under 25 years. But if marriage be delayed beyond that age, not only is the initial fertility reduced, but it apparently diminishes with the duration of marriage. As has already been shown, it is impossible to obtain from the census returns the results of marriages within quinquennial groups of age at marriage, but it can be obtained in decennial groupings, though, for reasons previously set forth, not for all such marriages. Thus, those women whose ages at the census were between 30 and 35 years, and whose duration of marriage was under 5 years, must have been married between the ages of 25 and 35, as also those who were aged 35 to 40, and had been married 5 to 10 years, and so on. In this way was obtained the information relating to women marrying at ages 25 to 35, and also upon the same principle that of women marrying at ages 35 to 45. The results are shown in Table XXVI. Fic. 16. Women Married under 25. Average Offspring. 0 5 10 15 20 25 Duration of Marriage (Years). Regression Straight Lines : aa’ (Census), y =*3450 + °30286x Origin at Marriage. bb’ (Deaths), y=*7360 + °24840x § Unit of w=1 year. It would now be well to show side by side the average families and proportion of wives without issue for each age at marriage and each duration of marriage. This is done in Table XXVII. The second column in that table shows the mean age at marriage of those marrying within the groups shown in the first column according to the experience of the years 1897 to 1902, when the ages were shown in single years. These results are shown graphically in Figure 17—except for marriages under 20. The table shows in a striking manner the effects of delayed marriage and how the generative organs in females tend to become atrophied through non-use. For marriages contracted under age 25 we see that only 24 per cent. are sterile at the end of the reproductive period, whilst for marriages contracted between 25 and 35 or at a mean age of 28-4 years the "percentage is nearly 12, and for ages 35 to 45 Distribution of Fertility amongst Wives Married at Ages 25 to 35, and 35 to 45. A. O. Powys TABLE XXVI. Experience of New South Wales Census, 1901. 273 Age at Marriage 25 to 35 years ete Mee TOES, Size of Family Duration of Marriage Duration of Marriage 24 Under | 5 to 10 | 10 to 15 | 15 to 20 24 Under | 5 to 10 years | 5 years years years years years 5 years years 0 312 1879 654 445 281 1383 619 345 1 614 1938 548 293 162 41 193 100 2 139 1220 879 400 232 2 59 81 3 1 217 953 510 277 1 16 30 4 2 30 644 517 311 — — 14 5 — — 261 448 328 = = 5 6 — — 80 323 302 — — 2 7 — — 20 198 216 — —- = 8 — — 2 65 139 = = = 9 — — — 20 72 -— — — 10 — — — 3 25 11 — — — 2 11 — = = 12 — — — 1 6 — = — 13 — = — 1 3 = == = 1h — — — — 1 — _- — 15 = = — — 1 — = — Total Wives 1068 5284 4041 3226 2367 177 887 577 Total Children 903 5149 9682 | 11032 9936 48 359 445 Average Family) 85 con 2°39 3°42 4°20 27 “40 “tl Per cent. with-) | ,. : . . ; ; : i Bate asia 29°21 35°56 16°18 13°79 11°87 75°14 69°79 59°79 a. TABLE XXVII. Average Families and Issueless Wives. Duration of Marriage Age at ee 24 years | Under 5 years| 5 to 10 years | 10 to 15 years] 15 to 20 years | 20 to 25 years | 25 to 30 years Marriage | yiarriage 4 (Years) (Years) #0] 5G || Fy 2 o | 2 o| o =|) Gy) eo} #2 _o ep || Soe i celey oo oe ee Bion go Se 502 Siam tp > Bion ap 2 So7 faia2|se|ae| 88/22] 88] 4s | 28/28] 82) 82] $8 | Bs a Relenn) 3s Riel mm ae n= ay e\E 2/25/28) 48/62) 48/52) 28/24 | <6 | 28 | 28 | 58 Under 20 | 18°6 | 1:10] 12°7| 1:03 | 46-0 | 2:45 | 5:20] 3:99 | 3-23) 5:41 2°60 | 7:00 | 2°43 | 8°83 | 1°90 - 5 45] 21°4 |1:07| 14-2} -94 | 33-1 | 2°64 | 6°05] 4:12 | 4°14] 5°64 | 3:24) 7°17 | 2°52 ——- — 25 to 35 | 28°4 85 | 29°2| ‘97 | 35°6 | 2°39 | 16°18 | 3:42 | 13°79 | 4:20 | 11°87 85 to 45 | 38°4 '27| 75:1] °40 | 69°8 77 {59°79 | — — = 35 Biometrika Iv 274 Duration of Marriage and Number of Offspring or a mean of 384 years nearly 60 per cent. are sterile. Although it is noticeable that for marriages contracted under 25 years the average offspring is greater for each duration of marriage than those contracted under 20—except for the duration 21 years—yet the percentage of issueless wives is also greater. This I think more an apparent than a real physiological fact and is due to the larger proportion of Fic. 17. Average Issue and Proportion of Issueless Wives. Average Issue shown by Firm Lines. Issueless Wives shown by Broken Lines. 80 Average Issue. Percentages of Issueless Wives. O 5 10 15 20 25 Duration of Marriage (Years). Age at Marriage, Under 25, Lines 4d’ and aa’. * . 25—35 » BB’ and bb’. 38 5 35—45 smc. C eandace.. prejudiced marriages in the younger group—and to this is also due the slightly larger average offspring in the 2} years’ duration of marriage. Taking the six years 1897 to 1902 there were 9490 marriages and 3910 antenuptial conceptions, or 41:2 per cent. of prejudiced marriages in the age group under 20, whilst of the 26,453 marriages contracted between 20 and 25 there were 7899 antenuptial con- ceptions or 29°9 per cent. Nor does this represent the full difference, for some women who were aged 19 at marriage, were 20 at the birth of the child and consequently were included in the later group. If adjustment be made the result will of course intensify the difference. This excess of antenuptial conception in the lower age group must amply account for the slightly higher proportion of issueless wives in the group under 25, but the fact that the average offspring is greater in the latter group than in the former—notwithstanding the greater proportion of sterile women—proves conclusively that the maximum fertility— initial and continuous—lies somewhere between age at marriage 20 and 25. Of course the test of fertility is not in the number of offspring produced during the reproductive period—for that will be greatest at the earliest age at marriage—but the rapidity with which they are produced. A favourite method amongst statisticians and others of comparing birthrates (at least amongst those who have been able to comprehend the fallacies in the a A. O. Powys 275 popular method of estimating the births per 1000 of the total population) has been to show the proportion of legitimate births per 1000 married women under 45 years of age. But it is evident from Table XX VII. that this method is liable to yield misleading comparisons, for the kinetic fertility is dependent not only upon the age of the woman but also upon the age at marriage. This objection of course also applies even when the more accurate measure of determining the movements in the birthrate is adopted, viz. when the rate is determined not in one group under age 45, but in quinquennial groups within that group. In New South Wales there has been a decline in each quinquennial group except at age under 20 since 1881, as will be seen from the following statement* : A f Wi 1871 Birthrate | 1881 Birthrate | 1891 Birthrate | 1901 Birthrate ghee Me per cent. per cent. per cent. per cent. 15—19 years 50°10 51°60 47°91 56°28 20—24 ,, 44°15 45°79 41°63 39°70 25—29 ,, 40°75 40°52 soso 29°87 30—384 ,, 33°67 33°86 29°23 22°68 35—39 ,, 27°04 27°36 23°63 17°25 40-5 5, 13°41 12°89 11°84 8°81 The decline since 1881 is a large one and although there has been an un- doubted rise in the age at marriage since that period, the decline cannot all be attributed to that rise—though probably much is so due. Prior to 1897 the ages at marriages were not tabulated—even if they had been it would not have helped us much since the marriages to which the births in 1871 and 1881 were due were for the most part contracted in Great Britain—the great majority of the adult population then being of British birth. But as has been said the decline is not wholly due to the advance in the age at marriage—it is chiefly due I think to the Law of Regression so ably formulated and demonstrated by Mr Francis Galton in “ Natural Inheritance.” Whilst the systems of free and assisted immigration were adopted in New South Wales the immigrants were of special physique and chiefly drawn from the two most fertile classes of the British community, viz. the agri- cultural and artisan. Medical examination was necessary before departure from Great Britain and thus a differentiated community was obtained. This immigra- tion ceased about 1885, and this was followed by the smaller decline in 1891 and the larger in 1901—the standard of the fertility of the offspring of the immigrants regressing towards that of the population from which they originally sprang. What difference there will be in the average issue of women of past generations and the ultimate families of the present generation—the age at marriage being the same—it is of course impossible to say, but I think a very close estimate may be made. Take for instance age at marriage under 20. We see from Table XVII. the average family of women of past generations marrying at this age was 8°06. This is for women whose husbands may have predeceased them before the end of * Vide Report of Decline in the Birth-rate Commission, p. 7. 35—2 276 Duration of Marriage and Number of Offspring the reproductive period or who may have been separated from them otherwise. For women of the present generation marrying at the same age and under the same conditions we may make use of the regression formula from Table XXIII.— limiting the reproductive period to age 46 and adopting the average age at marriage as 18°6 years (see Table XX VII.) we obtain 7°55. This is a reduction of about half a child per family. Adopting the same principle the averages for the present generation are for age at marriage 20—25, 6°51, 25—30, 5:12, 30—385, 3°21, as against 6°90, 5°22 and 3°36 respectively for past generations. These differences are probably the extent of the regression. I have calculated from the information contained in the English Registrar- General’s Reports the average age of spinsters marrying at ages under 45 for the years 1861-62 and 1901-2. The meau age at the former period was 24°3 years ; at the latter 25-1—an advance of nearly 1 year. This however hardly conveys an adequate idea of the change that has taken place in England and Wales during the last forty years in the age constitution at marriage. At the former period only 30°9 per cent. were married after what may be regarded as the critical age (so far as fertility is concerned) of 25, whilst at the latter period the percentage had increased to 39:5. Having in view the rapid fall in the initial and subsequent fertility of marriages contracted beyond age 25 it 1s unquestionable that some not inconsiderable portion of the decline in the birthrate in England and Wales is due to the increasing age of spinsters at marriage. In Australian age at marriage statistics distinction is not made between first and subsequent marriages. In Victoria—which may be regarded as typical of Australia—the average age of all women marrying at ages under 45 was 24'1 years in 1875-6; in 1901-2 this had advanced to 25°6 years. At the former period 33°8 per cent. were over 25 years of age, whilst at the latter the percentage had increased to 47:3. It would thus appear that there is a general tendency to postpone marriage both in England and Australia (probably more pronounced in the latter) and thus not only diminishing the average family but as we have seen lengthening the intervals between the offspring. So long as the average age of the female at marriage is in the vicinity of 25 and there are no further restrictions of the family than at present the community will not be a loss for natural increase. The average family will—at the present Australian rate—be between 5 and 6 children, and although this is the gross family it must be borne in mind that the infantile—as well as the subsequent—mortality is very hght when compared with Great Britain and European countries. Indeed the deaths of infants form such a small percentage of the births—about 10 per cent.— that the birthrate is considerably diminished thereby. And here (although not directly connected with this section of the investigation) it may not be out of place to refer to the correlation between the birthrate and the infantile mortality—the latter being the deaths under 1 year of age per 100 births. It will readily be seen from the next table of birthrates and infantile mortalities in various countries that A. O. Powys ar | there is such correlation, for it can scarcely be regarded as fortuitous that high birth- rates are associated with high mortalities and low birthrates with low mortalities. Although a higher rate of mortality might probably have been anticipated where the births are numerous, yet I think the high mortality is more the cause than the effect of the high birthrate. Now according to the English and Australian statistics of those infants who die under 12 months of age 50 per cent. die within 3 months of birth, and 70 per cent. within 6 months. If infants be reared in the natural way (which is a fairly general practice in Australia) an almost complete check is given to further conception until the completion of the period of lactation. Therefore where the mortality is heavy, even though the children have been naturally fed (falling as we have seen heaviest in the first few months of life) the impediment to further conception is removed. Where artificial feeding is resorted to of course there is no obstacle to immediate conception. Therefore in countries or communities in which artificial feeding may be the rule but where through ignorance or carelessness, or a like indifference to the production of life as to its subsequent preservation prevail, a high mortality and a high birthrate should be expected ; as also in those countries where artificial feeding is the general practice. Thus we find that in Sweden, where it is stated* that almost every child is nursed by its own mother, both the infantile mortality and the birthrate are the lowest in Europe (except the birthrates in France and Ireland—both of which are abnormal). The same authority asserts that in Lower Bavaria where natural nursing has become the exception the infantile mortality reaches the extraordinary high proportion of 50 per cent. The following table shows the birthrates and infantile mortalities in various countries so far as I have been able to obtain them. France and Ireland have been intentionally omitted—the former because the low birthrate (associated with high mortality) is well known to be due to restrictive measures, and the latter because the low birthrate (associated in this case with low mortality) is due to the constant emigration of adults of reproductive ages. Russia has also been omitted in consequence of the want of reliable figures, but it is generally understood that both the birthrate and infantile mortality are the highest in the civilized world: Country Birthrate Wranaiits | Country Birthrate i - Be {| Saxony ... 39°9 28°3 || Japan ee foe | 29°8 15°2 Hungary... 38°9 25°6 Scotland ... 004 29°8 12°2 Bavaria ... 37°0 27°9 _ England and Wales 29°0 Tbe7 Austria ... 37°0 24-7 Belgium... oe 28°9 16°3 Prussia ... 36°4 20°8 | Switzerland ae 28°6 5-9 Italv aE 33°5 19°0 || Australia ... re 27°1 11°0 Holland ... 32°0 17°5 || Sweden... aia 26°8 10°7 Norway ... 30°2 9°6 | New Zealand... 257, 81 Denmark . 29:9 13°6 | * Hugh R. Jones, M.A., M.D., B.Sc., Journal of the Royal Statistical Society, March 1894. 9 = 78 Duration of Marriage and Number of Offspring From these figures the correlation between infantile mortality and birthrate was determined. The general results were: Mean Birthrate 31:79 » Mortality mee Dials ope Birthrate 4°3328 Standard Deviation ier 61795 ‘Correlation 93369 + 00210 Coefficient of Regression ‘6547 Regression Straight Line (#=Infantile Mortality, y= Birthrate, Origin of 2, Inf. Mort. = 0) y=°6547a + 205423, the probable error of y being + 1-046, The next table shows the number of survivors of age 5 per 1000 births in various countries *: Country Survivors Country Survivors Country | Survivors || Country | Survivors New Zealand ... 889 || Ireland 837 Belgium ... 756 Italy ... 632 New South Wales 850 || Sweden 783 || Denmark... 755 Austria 614 Victoria 844 | Scotland . 780 France 751 Hungary| 598 Norway.. 838 |; England and Wales 762 || Switzerland 748 | Spain... 571 So great is the mortality generally in high birthrate countries that the numerical advantage to the community almost entirely disappears by the end of the fifth year, reckoning per 1000 of the total population. Reducing the birthrates in the proportions given in the last table, the net birthrate at the end of the fifth year is shown in the third columns of the following table: Birth- | Survivin Birth- | Survivin Birth-| Survivin Country || rates| 56h eae Country rate | 5th as Country rate | 5th cast Hungary | 38°9 23°3 Denmark .. 29°9 22°6 Sweden 26°8 20°9 Austria | 37-0 22°7 England and Wales | 29-2 22°1 New Zealand | 25:7 22°8 Spain ... | 34°8 199 || Belgium ds pon || asi) 21°9 Victoria 25°7 21°7 Italy ... | 335 222 | Switzerland ... | 28°6 21°4 France 22°0 16°5 Norway | 302 | 253 | | New South Wales O7°4 2353 || It may be urged that whilst the birthrate has fallen in England and Wales during the last 15 or 20 years the infantile mortality has practically remained unchanged. Even if artificial limitation of families has become more generally the practice during that period an investigation of the birthrates and infantile mortalities of the various registration districts shows that the association between the two rates does exist and that those districts which include large cities and towns (where I should imagine the knowledge of restrictive measures would be * Tam responsible for the New Zealand, New South Wales and Victorian figures; the others have been obtained from Mulhall. A. O. Powys 279 ‘best known and the practice reach its zenith) still maintain a high birthrate associated with a high infantile mortality, whilst the essentially rural districts have generally a low birthrate and a low mortality. These rates I have computed from the data in the Registrar-General’s Report for 1902, and are shown in the following table: : : Hee Birth- | Infantile Registration | Birth-| Infantile Registration | Birth-| Infantile Begistration District rate | Mortality District rate | Mortality District rate | Mortality Durham 35°9 14:1 London 28°6 14:0 Cambridge 24°2 11°0 Monmouth 35°5 13°5 Cumberland | 28°0 11:0 Huntingdon 24:1 9°4 Stafford ... 33°7 14:3 Worcester 27°8 11-7 Berkshire 23°9 10°1 South Wales 33°1 14:6 Cheshire 27°5 12°7 Surrey 23°7 11-2 Northumberland 32°7 13:1 Lincolnshire | 26°7 12°3 Hertford 23°6 9°2 Nottingham 31°6 14:7 North Wales | 26°6 12°3 Oxford 23°5 10°9 Warwick 30°3 14:1 Shropshire 26°6 10°4 Devonshire 23°4 12°4 Derbyshire 30°3 12°7 Suffoll 25°8 10°7 || Hereford 23°4 10°8 Essex : 29°9 11°9 Northampton | 25:7 11°2 Bedford 23°3 10°7 Lancashire we we. [2958 15-0 Buckingham | 25°6 10°4 Cornwall 23°2 13°9 North Riding, Yorkshire | 29°5 13°5 Gloucester 25°6 12°1 Somerset 230 10:0 East Riding i 29°2 12°9 Norfolk 25°5 12°8 Dorset 229 iS147/ Middlesex 28°9 12°4 Kent 25-4 11°6 Sussex 22°1 9°8 Leicester Bee ann || teil 13°6 Hampshire 24°7 11:4 Westmoreland | 21:9 82 West Riding, Yorkshire | 28°5 14:4 Wiltshire 24°3 9°8 Rutland 21°9 71 Thus we see that notwithstanding a few irregularities birthrates and infantile mortalities are closely associated in the registration districts of England and Wales—the irregularities being probably due to the differing ages, conjugal habits, sex proportions and other conditions. Where these factors are alike I think the infantile mortality becomes the chief determinant of the birthrate. As we have seen from a previous table, although natural selection is responsible for a larger birthrate its further operation tends to restore the balance. It might be inferred from the table showing the survivors at 5 years of age in various countries that by the operation of natural selection in countries where the mortality was heavy during that period the survivors could be regarded as specially selected lives, and that their subsequent expectation of life would be greater than in those countries where the mortality was comparatively light. But this is not always the case, as will be seen from the following table, which is extracted from life tables given by Mulhall from the experiences of the censuses of 1880 and 1881. I have added the results of New South Wales, which I have computed for the census of 1891. Survivors at various ages of 1000 born. l ] | | | [Rene Age | England) France | Prussia| Austria} Italy | Spain | Sweden| Norway| Belgium | Switzerland ae | : | 20 706 685 616 532 554 496 711 760 699 | 689 798 40 597 566 514 423 462 412 621 644 589 | 578 696 60 405 408 351 267 328 292 473 494 419 393 500 80 96 104 65 44 65 69 | 139 161 101 | 69 116 280 Reproductive Selection From these figures may be derived the numbers of survivors to age 80 of every 1000 who attain age 5 and which are as follows: England 126 Sweden 178 France 138 Norway 192 Prussia 95 Belgium 134 Austria 72 Switzerland 92 Italy 103 New South Wales 137 Spain 121 From this it is seen that those countries with one exception—France—with heavy mortality under 5 years of age do not yield as favourable a result at age 80 even after such a severe weeding out, as those countries where the mortality under 5 1s light. E. Reproductive Selection. Professor Pearson says* with regard to the question “ Does Society ultimately recruit itself from above or below ?” “(a) The fertility of one class must be shown to be sensibly greater than that of another class. Now the causes of fertility for two classes in the same com- munity differ comparatively slightly. Hence the small but sensible effect of natural selection must be carefully estimated and allowed for, i.e., we must deal with net fertilities. “(b) But the net fertility of one class may still be greater than that of a second and yet the second increase at a greater rate. We have to determine how far the greater fertility is merely potential. We must ascertain what propor- tion of the two classes remain unmated. “(c) Not only may the net fertility be greater—ze., the number of children per family who survive infancy, say live to 15 years, be greater—but also the per- centage of mating may be greater in one class than another and yet the same class may yet have no reproductive advantage. We ought further to take into account the adult death rates in the two classes. Natural selection may not only be effective in modifying the gross fertility into a net fertility but also as an adult death rate.” The “Vital Statistics” of New South Wales for the years 1895 to 1899 inclusive throw much light upon this question of reproductive selection. In them tables will be found showing the occupations of married men who died during that period together with the total offspring—distinguishing the living and dead. The ages of the living offspring are not given, so that the net fertility must be regarded as the number of offspring living at the time of the death of the father and irrespective of their ages. Thus the conditions contained in (a) and (c) above except that the net fertility is as just stated are fairly complied with, and as the * The Chances of Death, Vol 1. pp. 96 and 97. A. O. Powys 281 results can be modified in accordance with the rates of marriage in the various classes of occupations, condition (b) will also be complied with. The occupations are divided into six classes, viz., I. Professional (mainly en- gaged in the government and defence of the country, and in satisfying the moral, intellectual and social wants of its inhabitants). II. Domestic (persons engaged in the supply of board and lodging and in rendering personal service for which remuneration is usually paid). III. Commercial (persons directly connected with the hire, sale, transfer, distribution, storage, and security of property and materials). IV. Industrial (persons principally engaged in various works of utility, &c., Le. artisans). V. Pastoral, Agricultural, and Mining. VI. Indefinite. Table XXVIII. shows the number of deaths in the various classes during the years 1895 to 1899, together with the living and dead offspring, the average to each class, and the proportion of living to total offspring. TABLE XXVIII. ; Offspring Average Offspring | Percentage : Number of of Living Occupation Déesthis to Total Living| Dead | Total [Living| Dead | Total | Offspring I. Professional ee eet are 886 2896 | 972) 3868 | 3-269 | 1:097 | 4°366 74:9 II. Domestic ... oes els nes 707 2429| 847) 3276/ 3°436 | 17198 | 4°634 74°1 III. Commercial ee ws Aer 3116 | 10929 3706 14635 | 3-507 | 1°189 | 4:696 74:7 IV. Industrial ... ae we aes 5074 | 19176) 7143) 26319) 3°779 | 1:408 | 5°187 72°9 V. Pastoral, Agricultural and Mining 5564 | 26730) 7484 | 34214) 4°804 | 1°345 | 6-149 78:1 VI. Indefinite ... aes 8 toi 454 1821| 727 | 2548 | 4°011 | 1°601 | 5°612 71°5 Total aoe Sug ... | 15801 | 63981 20879 | 84860 4:049 | 1°321 | 5°370 75°4 Taking the results as they stand and disregarding the Indefinite class (the members of which should really have been distributed amongst the other classes) we find that the essentially Rural class—Pastoral, Agricultural, and Mining—yields the largest gross and largest net results and the Professional the lowest in both. The order of gross fertility which is also that of net fertility is Pastoral &c., Industrial, Commercial, Domestic, Professional. We also see with regard to the last four classes which comprise for the most part the urban occupations that the greatest mortality occurs in the largest families, with the slight exception in the cases of the Domestic and Commercial classes. This is in agreement with the results obtained by Professor Pearson in dealing with Rubin and Westergaard’s Copenhagen statistics. But in this instance natural selection has not disturbed the order of gross fertility. But the rural class with by far the greatest fertility has also the least infantile and juvenile mortality, and it would seem likely that this class would ultimately recruit society. But the results shown in the sixth column of Table XXVIII, have yet to be modified by the marriage rates of each Biometrika 1v 36 282 Reproductive Selection class, and fortunately I have been able to obtain these with a great degree of accuracy. The total number of males over 20 years of age in each class was obtained at the census of 1901, and the occupations of the bridegrooms have also been obtained for the years 1901 and 1902, and which are published in the “ Vital Statistics” of New South Wales for those years. Hence the marriage rates per 1000 adults of each can readily be ascertained. The statistics and results are shown in Table XXIX. TABLE XXIX. Marriage Rates of Various Classes of Occupations. | a ; Number of Adults Ei | Marriage Rate Tades Noe ass of Occupation (20 upwards) at M ty per 1000 M ar Census 1901 Siete oni a Adults Geule= 1901 and 1902 Professional oe see = 23300 764°5 32°81 1:181 Domestic ... See se ae 17277 Qi L572, “566 Commercial &, i238 ee 91939 2759°5 29°99 1:079 Industrial ... Ay os ner 102071 4130°5 40°46 1°456 Agricultural, Pastoral, and Mining 139947 2545°5 18:18 654 Indefinite ... a s ee 3575 39°5 Total 2. He oe 378109 10511°5 27°79 By applying the index number shown in the last column to the net fertility shown in Table XXVIII. we obtain finally as the comparative net fertility : 1st, Industrial Class 5°502, 2nd, Professional - 3861, 3rd, Commercial " 3°784, 4th, Pastoral, Agricultural, &c._,, 3142, 5th, Domestic ae 1:945. Thus although natural selection failed to change the position of the classes in the net fertility from those occupied in the gross yet the class marriage rates have completely done so. The rural class with the highest gross and net fertilities, with the least mortality amongst the offspring, and also as we shall presently see with the lowest adult mortality, now falls to nearly the lowest position, whilst the industrial class with both these mortalities the highest occupies the highest position through the frequency of mating in that class. We have seen that the average gross fertility of the agricultural and pastoral class is 6°149, and the industrial 5°187 —merely a difference of one child per father; but this difference is probably not altogether due to a difference in actual fertility but to the heavier adult mortality in the latter class and consequently shortened life. This will be seen from the following A. O. Powys 283 statement showing the adult deaths per 1000 of each class, and which have been calculated in the same way as the marriage rates: Mortality Rate per Class of Occupation 1000 Males over 20 years Agricultural, Pastoral, &c .... bee 12°69 Commercial... sive ses oF 13°29 Professional... er ase den 15°73 Domestic oe ee nt vee 16°59 Industrial dials sieve be 2a 22°72, Total... “ie sine se ive 16°47 Thus notwithstanding an adult mortality far in excess of the other occupations and with the highest infantile and juvenile mortalities the frequency of marriage is also so much in excess that as Professor Pearson puts it, “after all society recruits itself from below,” that is from the artisans—although as his statistics referred only to the City of Copenhagen he was not in possession of information regarding the rural occupations. This class it is seen with the highest fertility and the lowest mortality at every age absolutely fails to contribute its fair quota to the community in consequence of its low marriage rate. When we examine the gross fertilities of the various classes (remembering that these are the results of all durations of marriage and of widowers, as well as of married men whose wives survive them) and compare with the Anglo-Saxon and Danish statistics, we again see that up to the present there is but little Malthusian restraint upon the population in New South Wales—what little there may be apparently being confined to the Professional, Domestic, and Commercial classes. The ages of the brides at marriage in these classes have not been tabulated, but it may be of interest to show the average age at marriage of the bridegrooms, which according to the marriages of the years 1901 and 1902 were as follows : Class Average Age at Marriage Professional... ie cs eee 30°22 years Domestic ah ees we Aas 30°97 =, Commercial ... sh ee ay 28°88 Industrial sae ca ee ae 28°57, Agricultural, Pastoral, &c. ... sor 29°18, As the classes of occupations are rather comprehensive I append Table XXX. which shows similar information to Table XX VIII. for some of the sub-classes. The agricultural class is unquestionably the most prolific in the community accompanied by an exceedingly low mortality among its offspring. The medical profession would appear to be conspicuously below the average of the whole com- munity, whilst ministers of religion just reach that average. The comparatively heavy mortality amongst the offspring of the artisan classes is significant. 36—2 284 Reproductive Selection TABLE XXX. Occupation Total Offspring Average Offspring Percentage of Living PENS | to Total | Class Sub-Class Living) Dead | Total | Living) Dead | Total | Offspring | | | Religion* 68 264. 101 365 | 3°882 | 1°485 | 5°367 72°3 I. | Health + pon sts 151 475 148 623 | 3°146 | -980 | 4°126 76°2 IV. | Houses and Buildings{) 1173 | 4614) 1822 | 6436 | 3:934 | 1°553 | 5-487 717 IV. | Labourers, &e. 1488 | 5527) 1933 | 7460/| 3°715 | 1-299 | 5-014 74:1 V. | Agricultural ... 3149 | 16613 4877 | 20990 | 5-275 1390 | 6-665 79°71 Vv Pastoral 1024 | 4712) 1249 | 5970| 4°610 | 1°220 | 5°830 79°71 Distribution of Fertility in Occupations. TABLE XXXI. 1895—1902. . Deaths of Married Men, New South Wales, Class of Occupation Number oH Agricultural Children Professional Domestic Commercial Industrial Pes fee res Allg in B , Family Deaths ne Deaths | en Deaths cn Deaths oe Deaths ee Deaths ete 0) 238 15°276| 240 |19°818] 756 14°237 | 1136 12°900 | 899 10°103| 3372 | 12°719 1 171 10°976 116 | 9°579) 539 10°151 667 7575 602 6°766 | 2151 8°113 2 161 10°334 110 9083 | 529 9°962 |} 819 9°300] 606 6°811 2267 | 8°551 3 155 9°949 108 8°918 510 9°605 | 742 8°427 580 6°518 | 2163 | 8:159 4 138 8°858 92 7597 501 9°435 734 8°335 | 648 7:283 | 2169 | 8181 5 135 8°665 84 6°9386 | 476 8°965 722 8:199 | 645 7°249} 2103 | 7°932 6 110 7060 94 7°762 | 384 TeBy 720 8176 | 631 7-091 | - 1999 7°540 7 90 Oeil 69 5°698 | 350 6°592 | 673 7°642 721 8°103 1953 7366 8 96 67162 70 5°781 336 6°328 | 666 7°563 | 750 8°429 1984 77483 9 88 5°648 88 7°267 283 5°330 | 522 5°928 | 680 7°642 1720 | 6°488 10 Te 4°685 41 3°386 | 212 3°992 497 | 5644] 684 7687 1564 | 5:°899 ila 35 2°246 40 37303 182 3°427 Bhoyey |) LEILA 510 5°732 1159 | 4°372 IE 2 1-412 22 1°817 105 1977 SD) 2°669 | 421 4°731 830 | 3:131 Ih} 19 1°220 20 1°652 vl seh) 144 1°635 228 2°562 488 1°841 14 il *706 9 743 34 “640 84. "954 139 1°562 278 1°049 15 6 *385 4 330 13 *Q45 oy 363 61 “686 120 "452 16 2; "128 2) "165 14 264 24 ile 49 “BSI 93 351 ili 3 193 2 "165 8 ‘151 9 “102 10 sali2, 34 "128 18 2 128 = — 4 ‘O75 8 ‘091 15 “169 30 113 19 2, 128 = — 1 ‘O19 4 046 ll 124 18 ‘068 20 1 ‘064 “000 1 ‘O1l 5 056 rt 026 21 2 ‘038 3 034 2 022 7 "026 22 1 OE — — Hf “004 Cite — = il ‘O11 1 004 29 — — oo 1 ‘O11 = a 1 ‘004 qi | Totals 1558 |100:000 1211 100-000} 5310 | 100-000} 8806 /|100-000| 8898 |100:000] 26512 | 100-000 * Ministers of Religion, &e. + Carpenters, Bricklayers, Masons, Plumbers, &e. + Doctors, Chemists, &c. § Including the Indefinite class not shown in Table. A. O. Powys 2865 In Table XXVIII. the results were obtained from the deaths during 1895-9 when the tabulation distinguished between the living and dead offspring. Since that date the offspring have not been so distinguished, but only the total or gross shown. As it may be of some value to show these for the longer period 1895-1902 and to show the range of families in each class I have compiled Table XXXI. The mean family, median family, median children and percentage of married men producing half the children in each class are shown in Table XXXII. TABLE XXXII. eaten Mean | Median | Median | Percentage producing | pauor Family | Family | Children | Ghilaven | via Ne | Professional 4°55 3°39 7°09 22°58 | Domestic wae ses .. | 4°52 3°34 7°43 23°12 | Commercial Ste a .. | 4°70 3°64 7°08 23°30 | Industrial ... Pees FP reall Ds26 4°42 7°57 PAA Agricultural, Pastoral,and Mining | 6°21 5°74 8:56 27°37 All... ane nar Res ae 5°39 4°54 7°85 25°07 It will be noticed that the averages—with one slight exception—are sensibly greater in each class for the period 1895-1902 than for the period 1895-9. This is not due to increasing fertility during the additional period but to the fact that New South Wales is annually increasing its proportion of old people which has not yet reached that of a normal community—in consequence of its comparatively recent settlement. For instance, the proportions of the population over 60 years of age in New South Wales were at the censuses of 1861, 1871, 1881, 1891, and 1901, 3:1, 40, 4°3, 4°4, and 5°6 per cent. respectively, whilst the proportion in England and Wales at the last census was 74 per cent. But as this increase in age is common to all sections of the community the comparison between the fertilities of the various occupations is in no way vitiated. Tables XXXI.and XXXII. need but little comment. It is seen that Professor Pearson’s conclusion that generally 25 per cent. of the married contribute 50 per cent. of the next generation is corroborated, though the percentage varies in the various classes, being largest in the rural and smallest in the professional classes. Further that his view that society is recruited from below is also sustained— that it is the comparatively short-lived artisan class, notwithstanding the highest mortality among its offspring, that is contributing the largest quota to the population. CONTRIBUTIONS TO THE STUDY OF INTERRACIAL CORRELATION. By E. TSCHEPOURKOWSKY, Moscow. Many authors have tried to explain mechanically several of the distinctive characters of the races of man, especially the different forms of the skull. I may mention Nystrom’s hypothesis: he asserted that the head-form may be the product of the work of muscles on the one hand, and of the inner pressure of blood on the other. We have also similar theories due to Ranke and Thomson, but the evidence on which they are based is very slight. I propose in this paper to state some considerations based on biometrical methods. Ina paper read before the Congress of Naturalists and Physicians assembled in St Petersburg in 1900 I made the following statements based on the measurements of 700 skulls*. (1) The form of the skull is connected with the form of its base, and the variation of the first produces a series of the variations in the second, some of which resemble embryonic growth. (2) We do not find between the lengths and breadths of skulls in the same race a relation which is in accordance with the principle spoken of as that of “compensational growth.” If one increases, the other also increases. But if we pass from one race to another we find that with the growth of length there is a decrease of the breadth, and vice versa. This arises from the fact that the most brachycephalic races have the greatest breadths and the smallest lengths. The effect would be the same if, instead of different races, we took dolichocephalic and brachycephalic groups in the same race. (8) As the decrease of the skull length is connected with the decrease of the length of skull-base and of the face-length, we should expect that lower races with powerful jaws and small cranial capacity must have a tendency to dolichocephaly, and again that the decrease of the facial parts and the development of the brain are influences which are favourable to brachy- cephaly. (4) In the geographical distribution of the cephalic index we have two well-marked regions: generally speaking southern continents are dolichocephalic and northern are brachycephalic. We can perhaps distinguish on the earth two * Extracted from the Report of the Congress. E. TscHEPOURKOWSKY 287 centres isolated by the highest mountains of the globe, from which only the northern centre was favourable to the development of brachycephaly. The southern remained in a primitive state. In the same report and in that read by me before the Anthropological Congress in Worms I made the following deductions, based on the measurements of 1,300 women and their new-born children*: (1) that the head-form of the mother is inherited by the new-born child; and (2) that not only is the form of the skull, but also the form of the base determined in this very early period of life. As the base is then relatively very small we can hardly speak about its influence on the form of the skull: both are equally inherited. These last facts brought me more and more to the conclusion that sexual selection has played an important rdéle in the variation of the cephalic index. Ripley denies this on the ground that it is not a prominent feature of the human body. But selection may be at work if the index be simply correlated with selected characters. In order to ascertain this, and throw light on many other problems which I hope shortly to treat in a special anthropological essay, I have collected material for the study of interracial correlation. This material is partly extracted from Ivanovsky’s Anthropological Constituents of the Population of Russia, partly from Deniker’s Races of Man, and as far as regards crania from different authors and from my own measurements. From the data given below, upon which the cross correlations can also be studied, the correlation tables (Appendix, Tables 1 to 28) were formed and the constants calculated, which are represented in the following table. Many conclusions can be drawn from this. I will mention some of the results. 1. About the correlation between stature and head form there exist the most contradictory opinions between anthropologists. Interracially for the races of Russia I found a very sensible negative correlation. The control table for other races of the earth was formed on Deniker’s data, all Russian races of the preceding correlation table being excluded, and I obtained the same negative result. As the total number of races in both tables is 176, this result is most probably correct. I had very few data for women—only 28 cases—and they give me a very slight positive correlation. The material is not large enough to draw conclusions concerning the sexual difference. It follows from the masculine correlation that the head-form changes when stature is selected. 2. The correlation between the cephalic index and the shape of face, i.e. the relation of bizygomatic breadth to the total height of the face, measured from the top of the forehead to the chin, is positive and very sensible. It proves that the head-form can be influenced by sexual selection, because the shape of face is a prominent feature of racial “habitus.” But as we pass to the upper face index, Le. to the relation upper face height to bizygomatic breadth, which is calculated * “Ueber die Vererbung der Kopfform von Seiten der Mutter,” Correspondenzblatt der Deutschen Anthrop. Ges. 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TscHEPOURKOWSKY 289 mostly from my own measurements, we find a very interesting fact: the correlation of the cephalic index and upper face index is positive, ie. the upper face height increases more rapidly with brachycephaly than the bizygomatic breadth, so that mongoloid races having very prominent jaws have, however, the upper face longer than negroid races. It depends, I think, upon the diminution of prognatism, and represents morphologically a more advanced state of phylogenetic growth. Perhaps it stands in connection with the more advanced development of the base in the case of brachycephals, which was first noticed by Schaffer. Intraracially I obtained the results represented in the Table II. TABLE II. Skulls ~. Intraracial Variation and Correlation of Cephalic Index and Upper Face Form. Means Standard Deviations No. of Coefticient of | Race | = . oa Cephalic | Upper Face Cephalic | Upper Face Comelation Index Index Index Index Australians... Ae 66 71:08 49°35 2°85 2°93 + °089 Negroes — wee an 74°34 tag SPAT 3°60 3°95 +182 _Duke of York Islanders 53 70°90 48°70 2°91 3°22 — ‘093 Malays ... oS: re 60 80°20 50°23 4°70 2°80 —'185 Fijians Nae ae 32 69°59 49°47 4°58 Soul +°:217 Papuans ae pee || oa) 72°36 49°51 4°41 3°40 — °255 Polynesians... Bas 44 77°68 50°23 5:20 3°70 + 002 Alfourous ace ae 19 80°73 50°89 5°71 3°82 — 302 Micronesians ... Me 32 74°41 51°15 3°69 3°71 — 251 Copts ... ae A 34 77°44 54°53 4°83 3°10 —'147 Etruscans nae me 47 78°15 p4elo 2°93 O75 — ‘021 Europeans ses wise 80 80°24 53°95 3°80 3°44 — 198 Ancient Thebans ses |) lay 75°05 53°52 3°17 | 3°54 — 067 It follows from them that intraracially the correlation is mostly negative. For nasal and cephalic indexes I found a slight positive correlation for the races of Russia. 3. The interracial correlation of the diameters of the head with stature shows that the height—measured from tragus to vertex—and the length slightly increase, but the breadth decreases. In accordance with these correlations we find that with the increase of stature not only the breadth of skull, but also the bizygomatic breadth of the face decreases. It is perhaps a general rule that horizontal diameters of the head decrease with increase of stature while vertical and longitudinal diameters increase. 4, Passing now to the cranial correlations we find interracially the negative correlation between length and breadth, which isin accordance with Dr Alice Lee’s researches. The correlation of length with height is large and positive, of height Biometrika tv 37 290 Contributions to the Study of Interracial Correlation with breadth negative. In the diameters of the face we find a high correlation between upper face height and bizygomatic breadth. Very interesting is the fact that the upper face height is more closely correlated with breadth than with length. The bizygomatic breadth on the contrary is more correlated with length than with breadth. Both diameters of the face are very slightly correlated with the height. The minimum frontal breadth is positively correlated with length and breadth and negatively with height. It follows from the values of the above constants that interracially with the increase of the length of the skull the breadth diminishes; the height and basal length on the contrary increase as well as the stature. 5. For interracial variation we have the chief results presented in the Table III. below. We conclude from these constants that interracially the length of the TABLE III. Coefficients of Interracial Variation. | Coefficients of Variation Character | Livin Bubiects Suanlis Stature... ae bbe 2°02 = Cephalic Index... ass 3°58 6°34 Face Index noo aoe 3°29 — Upper Face Index sie ~- 3°87 Nasal Index oes Ae 10°40 = Cephalic Length ... ae 1:49 2°57 Cephalic Breadth ... 6 2°93 4°71 Cephalic Height ... Fee 4:95 3°30 Bizygomatic Breadth _... 3°40 314 Minimum Frontal Breadth — 2°76 Upper Face Heigh eee — 5°29 Skull Basis ae nee — 3°32 skull is less variable than the breadth. The height in the case of skulls is also less variable than the breadth. The variation constants for length and breadth of living subjects are in accordance with those for the skulls, but tbe height is not to be compared in the two cases: in living subjects it is measured as the distance between tragus and vertex, and its great variability is, I think, partly owing to the impossibility of perfect measurement, partly to inconstancy of tragus. Very interesting also is the great variability of the upper face height as compared with that of the bizygomatic breadth. From the above given constants we can deduce, I think, the following statement: The varieties of the human head-form are mostly owing to the variation of the breadth. EK. TscHEPOURKOWSKY 291 In concluding this study I wish to state some theoretical considerations which, although perhaps far from the absolute truth may, I hope, yet show how important for anthropology the biometrician’s work will be in the future when aided by extended anatomical studies. I call first the reader’s attention to the fact that the several diameters of the head in new-born children do not stand in equal relation to those of adults. In other words they need not accomplish the same growth to reach the adult stage. The most complete table of these differences will be found in the work of Papillaut. From my own measurements of the base of the skulls of new-born infants I have reached, as a general result, the conclusion that the breadths of the skull and of its base are far less developed in the embryonic stage than the lengths when we compare their ratios with those of adults. I give here some numerical data extracted from the work of Papillaut. Gi - | i Mini oka ee Upper Face | Bizygomatic | Breadth of | Length of ingen Character Height Breadth Skul | Skull aie | readth | New-Born | ion : —__.— ABE oe 44° 48° 56°5° 0° 60°/, Relation delt Hh ie 56°5°/, | 60°/, / Coefficient of Interracial Variation 5°29 3:14 4°71 2°57 276, Compared with the coefficients of interracial variation they show parallelism with only one exception—that of bizygomatic breadth, I cannot affirm that racial differences in the skull are simply differences of growth, but the above- mentioned parallelism seems to me not without interest when compared with the other point which is also connected with growth. We have seen that interracially and partly intraracially—probably as far as we can eliminate the influence of mixture—the cephalic index diminishes with stature, because the length and height increase and the breadth decreases. I have also shown that the length of the basis is positively related with the length and height and negatively with the breadth of the skull. Now the length of the basis consists of two parts, which are very different embryologically. The distance between sutura spheno-basilaris and the basion is the end of the primordial skeleton-axis, i.e. of the chorda dorsualis. The second part, i.e. the distance between nasion and sutura, is connected with the development of the brain and partly influenced by sinus frontales. Manouvrier has shown that only the first part of the base-length depends on stature. It has much more to grow in order to reach the adult stage than the second, which is developed very early. The reason is the great development of the embryonic brain owing perhaps to its physiological faculties. It follows from these considerations that the negative correlation between stature and cephalic index can be explained by the growth of the length from basion to sutura basio-sphenoidalis, which increases much more rapidly with the stature than others do because it is a part of the primordial body-skeleton. With the increase of this diameter the length of the skull and the basio-bregmatic height increase also, but the breadth compensa- tionally decreases. As it follows from the tables, all these correlations are 37—2 292 Contributions to the Study of Interracial Correlation observed only as averages, and separate races may present many exceptions. The cause of the negative correlation between stature and cephalic index is thus only one of many factors which can influence the head-form. We must, therefore, conclude that the diameters of the head and the cephalic index respectively depend on many causes. The anthropometric measurements are often artificial simplifications, and will, with the progress of anthropology, be made to approach more closely to those regions which are physiologically and embryologically individualized. The organic influences which exist in the individual, or within the limits of a race, can be masked, and correlations between numerical values of measurements inverted as we pass to interracial correlation because other factors come into play. The one of these factors is sexual selection, which is very powerful in the species Homo. Admit for instance that the general growth of an organism produces an elongation of the head. Then if the selection is connected with stature, and tall individuals survive, the new race thus formed must be long-headed. But if round-faced individuals become a type of racial beauty, the sexual selection works in the opposite direction, i.e. produces a round-headed tall race, so that in a particular case the organic influence can be masked. The progress of the evolution of a race is the result of a number of organic correlations, and intellectual or instinctive impulses play an important part in it. Many correlations which are preserved interracially are organic, i.e. physiological correlations. Perhaps we may define the characteristics of a given race as its deviations from the most probable values, calculated from the coefficients of organic correlation. These deviations are the result of processes which were not the same in the evolution of all races, and which have been quite special for each individual race, 293 E. TscHEPOURKOWSKY ‘solos OY} UL S[VNPIAIPUL Jo JaqtanuU ay} 09 Iajar sjayovaq Ul StoquINU IT], y = — | = (41) 1.29| GPT (71) 6.92) (4) P81 == = erl (€F) $81 = (£F) Gt, moyonqyurg | 7" “ eisvonvosuvay, “ os | (8) 0.FE| LOET 7-29| Gel | ()1-92| (eS) est] eet | (es) 299] Tot | (8) eet] (S)9z9t} (8) 868] °° _Aysmouvmp fe 7 BSSEPQ | 6L ==) || "Goal = a — = IZE_ | (%) T-99} TEL | (cox) est | (62) eFOT | (07) F.3g | Stequessieyy | BESS yNOg | (041) 6.69] SET | (8h) 92] (981) Tet] — = OG RC est Ce) TOT oe) Leas SOeROE is wottqoyy — | (49) ISL 9IT (49) Vara) FL C 9) L8T aie! (49) & BBQ aiele ann Bpelyg d ee an YSury 74 | “el | (#5) 8.9L] (9) 6 = = oct | (*) eet] (*§)ezor | (2) 0-88 | *: wemyooyg f° 2 ouMON = | on (9) 6191 | (69) LTB] ee viaengary |“ | — aay —— (8€ ) eZOT (8&?) 6-28 eee eee 3 oe eee MOTI ; ce gg |(ot)¢ee| eset |(4)0.¢9| oer |G) L-64|/(48) pet] per | ©) 1-29) LOT | (2) PST | (2) oTOT | (0%) 6-18) 7° “PUR |! 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Agro (-J22) Ayo xapuy mupeere xoput | aon jo qysuery is 3 I 5 aH YOUNT |yyprerg| yysuey xopuy ‘ eo poe | P80] TPSPN | cake | TOBA | WAT MEPH | sumeq | peo | peo | OS | peo ay pe ear = youduy unssnmy ayn 02 quaovlpy so fo saomyy "TT ATAVEL SHHLOVaVHO NVAW TVIOVYE T XIGNdddV Contributions to the Study of Interracial Correlation 294 &6 = CPST | (St) 1-69] OFT | (5) 9.92) (S)82T| oet | (*)0-99) zor | (=) 26] (S) LF9L| () seg] moyonqyueg | ce “* suvrpeaSuryy 6 = BIST | ($4) ¢-e9| — = (*)68L| OFT | () LLL) vot | (*) eet | (5%) cor] (*) Te] °° * yeasoz |" ae “* SHBIIOZSe cg | (%) F-98] Z6el = Pel | (2) LbL| *)6LT| eet | 4) or) ert | ()oer| (4) erat] (2) ¢.92 | 7 “mopar | a “* guese[soyy — — _ 9-69} TFT | (79)9.64) (9) 241) — == SGT | (79) egt | (9) 1OLT | (9) Lee} oyuRy[rD |" ueqnoy Mi — = — | (8)g99] zet | (st) Zon] (8) set] — am 6rT | (8) 981} (8) LGOT |] (8*) 8.62] 77° ee “ TITOAA, ie — — _ = FEL | (20t) 22] (22) LBL] = — = SEL | (202) 2eT | (°F) SeOT | (oF) LTA] 7 AMoyansoy | **"asatuod0 ss — — — |()91L] TPT | (24) 9.42) (2) gt] — == 9c1_ | (°t) gL | (°") 999T | (922) 9.¢8 | °° perpoperg | “-mosiareyy 5: —- = — | (°8)8.69] SFI | (68) $08] (°8) 941 — (68) 6-8 | °° “mousey |" MoyeyO 68 — ELEL | (9%) G79] — — (9) LT] Tet | (0%) #14} eer | (0%) FET | (022) B99T | (9%) G.7g | °° “* pjoqerq | *"* MOLY ‘SUBISSNY 194] 98 | (%%)6LE] 9681 = — = ($2) 181] SL | (#7) 2.99] pet | (r)egt| (2) erat | ('2) 0.48 JOUMIOG |" tt te se gedory = = = = 6EI | (94) o:ez | (4) eoT LY — = eet | (4) PEt] (°%4) 9G9T | (4) Lee] ot ssvuoreg | *** 35 se - 6L — 9Tel | (°9)6-64) — = (99) est] LIT | (9)%19] get | (99) ter} (69) s99T | (9) 9-18 | °° uyosuuorg | *** sine “" suvruenyyry = — = = = — [Fst] SIL | ()T29] zer | (ex) o6T | (0) 9e2T | (1) 6.6L] °° raneqpread | *° a * SUvIUOATT 68 | (68) 2.98] FOST = GFL | (68) 6.18] (68) 221] 6eL | (68)g.¢2| eet | (68)egt| (68) O89T | (62) 2.99] °° MOPANOL) | *** : ss * ulszerT 9b = 16ZL | (99) 8-9] — — | ©9)¢gT} SIL | (9)F09] est | (9)0gT| (°9) cot | (09) ¢.08 | °° TaqoeAy | °° me ay * Ysiqqery (42) $49; — CMF | ae = = e9t | (4) 61 = (Ze );qe))| 27: “ eaqgaryy | sae a azery == MGS rays) = = | LPT | (6) 681 | (6) SPOT | (Ot) 2.22) °° "sopra | + “" suvisteg =“ = a — | (8) 9.32, — = = = == 6st | (S£) e6r = (SE) Cre eal ects ee ei ; rif suvfg “ = = — | (et) 2.19 SET | (ht) e6T | (09) LLOT | (oft) c.gy | oo * oaqueyy | “* suvmmoyiny, “4 €8 | (0%) Fee] LOFT | (0%) €-9] BET | (f) zen | (Of) est] get | (Of) LoL} LET | (oof) Bgr | (c¥€) GR9T | (of) g.np |v AySmouvay | wisvonvosuvdy, ‘Spanyy cg — e6el |(4°1)¢.c9| FFT | (€)¢.62) (&) 181} est | (or) F-1L] zet | (or) et | (922) BROT | (4or) O19 | AYSIOp Ung | +** at se yrumoy Hg | (*)¢:¢e| cert | (8) eg] — = Z6L| PHT | (%&) LLL] OFT of) 981 | (fF) OL9T| (§) F-8L] °° « Aysmorzepp] st eS[NOY] esouttyy) 68 | (%*)0.98| ZerL | (°f) Fee] — = = 6ST 1 CCGG) zen 9) egt | (°f) Te9T | (of) zag] ot AYsMowzeyy | “" esptnoy =“ = 191 #S) 91 | (#5) GPOT | (5) 6-18 AYSMOTSL] | 77° “s unredg-aig —“ 88 (©) L-9¢| Tgp1 | (©) 8.12} spr | (6) F.08} (6) zer} Tet | (6)9.221 eo | (66) tet] (66) Teg1| (65) t.6@ | Aysmouray | + “" Vpdo Tupedg = ($6) $8L} — = ZOT | (S€t) 29 | (t#2) BZOT | (S*™) G98] “°° uISnaeyy) eplo-MoaIyNog ‘ZIqSIry = = — —= 6el | (4) 281) (2) 681} — = IST 4z) 981 | (82) 06L1| (44) 3-18] °° snizgeyy | ee ae SUVITOIE — — = —= = = Z9L Zz) TEL | (9% GLOT | (2) ppg | 0° TOTUAULOG | ++ ag sas leqoeley 88 (+) 0.98] FOFL | (0%) Z92) — _ () est] set | (*)0.2} tor | (or) FeL| () eco) (oh) eg] “* puepeag | : * ZIYSILY-eIe yy = = — Varas) — — | 9c (9%) BET | (2) GFOT | (02) QRZ | Saoquessto ay | #8 oA SUIIBIV YY] €8 (%) Fe] Leet | (f) egg} — — | (%)9gt} 2et | (£)e¢02) eet o£) o8T | (°f) Te9T | (ef) g.eg] °° ss ee “UBIO, i 88 (°£) 2.68] PZPL | (£) rg] — — (f) 781] O8t | (£)869| get | (°€) g9gT} (£) az9L, (ef) 9.¢9 | °° Aysmorzeyy | “ Treypeyg = = 69 | ect | 7) 68T | (9%) FOL | (2) 0.68 | SeroyNY Bid}: s . 8 (36) ¢.c¢| erel | (46) F-e2]} LPL | (5%) @.6L) (55)egt] Ist | (6) 279! eer | (06) BRT (£6) gF9T | (29) Lig} 7 “** MOTOIO YT | *** " URyeaysy ‘synuIpey 18 = GLrt | (£)0.c9| ert | (4£)¢.08! (6) ezT] get | (er) 622! ger (oF) 9971 (Oh) S297) (ON. eon <= potosyst My | me oO spreqe yy €8 8-9¢| OLET = 6El | (£)944) Cf) 98T] LET (2) L-bL| Sbt | C®) pet} CG) se9r| (*) 08] 7° “ mopraeg | ne Re UIZIVS14ST = = — | ()629] OFT | (S¢) #982) (S2)98T] — — | Let | CS)06t] (9) etat | (5) ¢-%98 | moyonqzurg | +" 3 ae Tysnosuy 98 (°4) 9.98] QOPI | (8°) 4-49] LET | (86) %92| (86) Z6r| get | (96)9.92 Het | (86) 227] (46) QzoT) (8%) eng | MOUTITE NT ie ay “URL 16 = O8PL 9-19] — — — TPL | () ope) srt | (f) ort] Cz) egot| (f) 622) °° oxueysorox | -° ; oe “" apisay Ago (-Tea) A410 xopuy Uipeesey xepuy | 908g jo qyyduery ee Suc xo A } ce ‘ acess qsuery pul é ‘(panuyuoo\— JT ATV 295 E. TscHEPOURKOWSKY 16 — ELPL = = = (9%) e6T| FET | (*%)GoL] 9st | (9%) 06T | (9) Lo9T| (9) e.g) °° Aysmoqoegt Ay | ot : ef a 16 = SLPL = SPI | (402) g.6L | (47) 281] set | (42) 69} Let | (42) T6T | (402) peOT | (42) egg jr oo ourey | * ; ss SINCL — 0 CCE CT se = PST | (°°) PET | (8) SFOT | (OX) E62) °° 9 aqnag | “et suermoq4ysg i} = 6EFL = = _ = PEL | ()614} est | (Ht) LST] ()099T] (4) BIg] **aagnsyo |” ae tpi rong — - = ShT | (8) Pet | (8?) GI9T | (9) 0.62 |“ “* MOTTeY | °° ce “* SsTUTatoy) — | G)poe| — | G8) 14g] TST | (£)9.92,| G2) z6r) — = GcT | (9) 68L | (9%) TIOT | (9) 6-18 | Yormezutay-oF[e J, bases seqorey — — |(S)%18] OFT | (St) g.92| (8) est] — a ESL | (051) O6T | (05#) 999T | (9S!) F.08 | “°° — ModosotUTOY | * SI0qiM = = = = OPE | @*) #62) (2)9LT| — == ect | ()9st| (4) 6291] ()ezeg]° * snimgey fo snyoysemey, SuUty _— OFT | (96) p24 | (99) O61] — = BST | (95) 28L | (96) BLOT | (9%) 9.98 | °° —- ASMorseTT | ; " s8aqz 68 | (6) 0.8] O9FT LLG| PPT | (St) 9.621 (08) I8L] PET | (2) 1-24} O9T | (8) est | (6) PFOT | (052) 6.98 | °° mouTyMay | ts suTpNg €8 = SOFT = a rae (88) GST} Bel | (85)9-89] OFT | (85) 6] (85) 969T | (85) 9.62 | °° ASTON] | *** “* suemoyan J, — = = |(&)@9¢) <= = aa SGT | (*) est} (%) 0991] (%) Lg] moxonqueg | * gee “* syany, gg | (77) 6-Ge| SZhI | (6) 6-99] ToT | (S8)@.18| (98)e8T} Est | (°8)%c9] Sct | (98) Z6T | (98) Teg] (48)ee8) °° 9 tt Moureyg | °° ; 5 snsun fF, 6g | (°£)0.ce| errT | (°f)G-.2¢) — = (8) JST) S81 | (E812) Let | (&)ost| (Cf) esol! ()¢7g|/°* Aysmomey | "* °~* elnoy “ 6g | (*)0.¢e| ert | (€4)¢.09] 6GL | (&4)s.c8| (4) 98t| oet | (4)e69] 6ct | (4) se | (st) ee9T| (€4)GpR | sysmoueng | ** reyedeqiey, ‘SyNS1OJ, gg | (41) §.FE) SEPT | (77) 1-69] IGT | (2%) g.08| (°%) SBT} OsT | (8) 6.69} O9T | (072) 98T | (602) 9E9T | (OF) T-98 | °° oyueznoy | “* synSueyay, _— = = | Gero) = — ial 8€) g8T | (22) TOA | (8®) 9-82 | °° “ oaqueyy | 7" ot este 08 (¥€) ¢.9@| eceL | (69) 3-29] OFT | (24)6.¢2] (84) F8L| set | (£9)6.02| PFT | (40%) BT | (42) egoT | (402) 9.27 Sooqiny Bid | "* urlleqiazy = PET | Co). 2 23 (2) P20 = Bet 28) g81 | (9) e99T | (°5) 9.08 | °° ail | ef = = (0S) g.¢¢ — (68) Ts ee = LET | (88) 9ST | (98) L99T | (49) Gp, | sSergoyyNY ‘Biq | °° vould ~ : = = = = 6FT | (19) Get | (19) O9T | (79) 6-08 | WorMezup-oyfeL | QS €8 == g9eT | (f)6-€9}) — -- = Set | ()¢69] SEI °f) PBT | (Of) SFOT | (°f) 8.ce | °° Jesuesueg | ** uuiseyy +8 -_ LLET | (66) L-0L] IPT | (S02) 9.82] (40%) GEL] LBL | (’2) 1-89] eet | (402) 98T | (992) GFOT | (9°) Teg]? ourygsnae yy | 7 : uezey ‘sieyey, 18 | (°9) 1-98] 8zET | (9°f) ¢-99] SFL | (Pf) g.22/ (0%) GST] 6BL | (99%) LBL] Gt | (0°) BAT | (4°€) QPOT | (o£) 6.98 | °° 2 Tesskeq | ; zh TpouRe J, — = — = SPL | (9S£) gen] (95) e6L] — = 9ST | (95) PST | (99€) TE9T | (95£) 8-FB | 77 AYSMOTSLTL | *7* : Oe ) sxiley, 98 ($f) ¢.Ge| SPF | (£) 8.1¢ — (£) 661} LeT | ()SPpL| Bet | (£) PET | (°£) 899T} (£) Fag | °° x is ig “t SUIQTYG-OqIg zg | (°£) Se] g6El | (£) F.7g] — = (f) est} Let | (£)T-92] ost | (%€) 621] (Pf) 9697] (£)6eg) °° Aysmorzaqeyy | : eu “ saqaeg — — — = O9T | (89) 68L | (29) OFET | (09) FFB | 7 qarmmog | ‘ 06 =| () 8-88] OgFT = = — (2) G61] pet | ()LeL| est | (2) est} (=) 6LcT] (2) 6.38 | °° "7 Feasog | °° : tpefouteg — — — — — = — — GGT | (*€) 98T | (7£") ATOT | (7£1) 0.8 | °°" ZIMOUYOeTO | °° “wuopey — |(%) 2.78] — |(0x)e.6L} OFT | (02) e.G4) (7) 98T]| — = PSL | (92) OBL | (9%) eeQT | (9%) ag] ot © AYsutztezq | °° os =“ uqug 9 — |(&)epe] — |(9%)¢-¢9| get | (42) 9.92 | (4%) 621) — = OST | (r2) cet | (#6) OFT | (92) 8.08 | °"° puny” | 7 ze MBSTE AA selec 98 (s+) T.98| esrt | (2) 8-99] xet | (6) 6-e4/ (%)e8T| get | (9) zen} LET | (9*) e8L | (9) GOT | (9%) Ber | * woplueg | us Gg = ZEPl = OFT | (2) ¢-pL/(77) 881] Get | (9)¢-1L] GFL | (£2) BBL | (1) LLOT | (fF) 1.82 samroyyny “BIC | ae SULISIOT — OFT | (9%) 0-18 | (9%) E41] — = SPL | (2) O8T | (cet) SLOT | (0°) Beg | °° “" AOTTEHT | * ee SYR T +6 = O6FL | (1) T-P4} FPL | (0%) 9.6L] (0%) IST] get | (8)0-TL] Bat | (6°) BEI | (°°r) CECT | (0°) EL | °° mounsnyy | 1" : see SYVIYSC) gg | (674) 9-8 | ZEFT | (C°€) €-99| CFT (*1) €-18 (92x) S41] 9EL | (%4)614] Set | (6°) O6T | (°°) OBOT | (°°£) ¢-1g | °° “* 9uLysny | 3 a “** 897 9SS() = ae = (42)1L| — — = = ect | (4) pet} ()erer| (4) 908) = moqraeSzeyy | °° oe suoTooI() = es ==! | (92) 00) | ee = OIG = 9ST | (#9") LET | (292) ePOT | (292) Beg | °° * MOUTYPT a VIARLOT, = | £410 (121) | gy | xopay |WPPE| xo 3u0 : I pul | sox jo Wqeue'T lyypeerg| qysuery xoepu peaee eee -edeg | yeseyy | see reoeg | qaduer, | W8PH USO, oe el ornyeig pe e Aqrr0yny dnory 10 aoRyy ‘(panurjuoo)s—T ATA VL Contributions to the Study of Interracial Correlation TABLE IL. Living Races from Deniker’s Races of Mankind. Race or Group Stature eae Race or Group Stature a ae Aetas ... A 146 85 Korumbas 156 77 Andamanese ... 148 81 Kulu-Lahulis... 161 75 Annamese 157 83 Laotians 159 84 Arabs ... 166 76 Lapps ... 153 87 Australians 167 74 Malé 156 75 Badagas 166 72 Magyars “s 165 84 Baluchi 166 80 Mal-Paharias ... 158 76 Baltis ... 161 74 Mandingans 170 75 Batekes 164 74 Maricopas 172 83 Battas ... 160 81 Moquis = 163 85 Bashilange 168 77 Mushikongos ... 166 72 Belgians ust 165 82 Nicobars 163 80 Bhumi (Bhils) 159 75 Omahas 173 82 Bilkula Indians 166 84. Pawnees sae 171 80 New Britain 162 " Piedmontese ... 165 86 Burmese 165 83 Pimas ... 170 78 Caribs ... 157 80 Polynesians 173 80 Chakamas ; 160 84 Tahitians 173 85 Southern Chinese 161 81 Punjabis sas 168 74 Comanches 168 85 Rotti Islanders 160 Ua Corsicans : 163 He Sardinians 162 77 Crow Indians ... 3) 80 Santals 161 76 Dalmatians 171 87 Sicilians 163 79 Danakils 167 74 Singhalese 162 79 Dards ... 161 76 Siouans 7/3} 80 Dravidians 163 74 Solorese Br 158 83 Eskimos, Alaska a: 163 79 Solomon Islanders 162 ai 3 Greenland ... 162 Gi Basques, Spanish 164 79 French 165 84 Spaniards 164 79 Basques 166 83 Sundanese 159 86 Fulahs ... 174 75 Tamils... ae 165 76 Gypsies 165 80 » ». India 167 al Hindus 2 163 73 Tenggerese 160 80 5, Behar an 163 76 Tipperahs 161 80 Troquois 173 79 Toucouleurs 172 7 Italians 164 83 Trao Mois 158 ah Javanese 162 85 Uru-Kurubas ... 164 76 Jakuns lds 81 Ute Indians 166 79 Kharvars 162 76 Veddahs 155 75 Kothas 163 74 Wolofs Ae 75 Kols 165 72 ZUDIS: 2. we 162 83 Kurmis 161 76 Zandehs, W. ... 172 78 E. TScHEPOURKOWSKY TABLE III. Cranial Indices. 297 Race or Group elt ba | Race or Group eis 9D re Negroes 74:3 51°3 Hottentots 72°4 52°1 Australians 71:0 79°3 Tasmanians 75°8 46°3 Javanese 81°5 49°7 Torres Straits 68°6 52°6 Marquisans 75'8 51°6 Viti-Levu 66°8 48°2 Auvergnats 86°3 49°8 Maoris ws 74°0 49°4 Esthonians ... 78°6 53:1 New-Caledonians 71°6 48°6 Dawson Straits 74:2 49°5 Eskimos oe ae 71°8 51°7 Wiirttembergians 82°6 53:0 Re Greenland ... 71°4 56:1 Bavarians 82°7 52°6 Veddahs 69°9 49°9 Chinese 79°0 54:0 Andamanese ... 81°5 48°7 Ainos 76°5 50:0 Malays 80°2 50°2 Kalmuks 81°6 53°2 Polynesians ita 50:2 Minusinsk 73°2 531 Alfurous 80°7 50°9 3 74°6 52°5 Micronesians ... 73°4 51°1 French 79°5 51°9 Papuans 72°4 49°5 Bashkirs os teen POLO 52-1 Czechs 83°2 51°4: Duke of York Islanders | 70:4 48°7 Telenguts 86°1 51:0 Kanakas Tee Sen) ihsks! 49°8 Soiots ... 86°3 53-2 Celebes 82°1 51°8 Samoyeds wae 82°4 53°2 Naqada 72°9 53°8 Tarbagatai-Torguts ... 83°3 52°2 Thebans 751 53°5 Abyssinians ... oss 75°7 52°3 Copts littl 54°5 Senftenberg Crania ... 84°2 49°5 Etruscans 781 54°1 || 9 Rumanians 80°6 52°3 Kalmuks 81°2 DOs2 ey iil When a race is repeated, two series have been used. Biometrika Iv 38 Contributions to the Study of Interracial Correlation 298 8-281 | 1-89 9-96 1-961 | €-POT | 9-681 | T-L8T | 8-681 | 0-4 6 ie 8-9€L | 0-04 0-66 €-G3L | ¢-POL | 6-LEL | L-IPL | 6-E81 = ge “* STTMAG e89T[09 “yz &-GeL | ¢-1L €-86 —= | 7-SOE | F-SEE | 0.081 | 9-061 | 89: ei ss SBULOY I, €-1€L | 9-09 1-°6 9-1ZT | ¢-OOL | 0-€€1 | €-68I | €-S8T = re oe "* o7o “ToMOLT 1&1 G9 16 cag 16 ISL | PPT GLI = es ae ui O€L 89 6 = GOL 9€1 Lel I8l — ome e a jourls0 yy Z-8ZL | 8-L9 a 9-61 | 8-GOL | 9-9€T | 8-EET | 0-081 = ae "+ sgornog yuaTORIC, 8-€€L | 1-99 L-€6 6-0ZL | 6-66 | €&-PEL | L-1ZL | 9-641 = Mie oi ‘oq ‘toajopon) 1-821 | ¢-99 = — | &O0L | 9-GT | €-O€1 | 8-28L | FTL ee a - ae ae = — | &66 | 9-181 | €-3rI | 8-G8T = ay 2° as see == ies i = 6-16 | O-[€T | 0-PST | T-84T 7 a ia ie * B001g cg aa c6 == 901 61 CEL 88I | 8-14 i “" BolUgyy BlUety) 9-C€L | GCL 6-16 — | P-O0L | G-OPT | G-OFT | T-08T | €-82 ae ai es loUvsO yf cel BL ¥6 = 06 aaa SPL ALL | @eg | oct Aysmoueay 9B a 66 = €0L Gel GEL GST | 0-84 ipa re = 2 LET aad L6 eae FOL 6E1 GEL GST | €Ts ay a a a CEL = 96 = FOL OLT 1g1 SB! 22e6Oa 1" goss = eorayy” eluedy 6-1ZL | 1-09 9-06 €-O1L | 4:86 | 9-1 | 8-FZT | P-841 | 6-69 — oh ie . 9-€ZL | 0-09 0-€6 VEIL | 6-86 | &-6EI | G-9ET | €-L9T | &18 ai a he JOMOL VEU 46 = SOL | IFT 6EL Se | eae 2s Ged Oa ae S tS lh a 96 = FOL 6&1 Bel CS ies: OO al eeu a eee ers . ESL |) = 6 = GOL er OF1 G20 6-82 | ee 7 % €€1 = 6 a 901 raat 6ET GOTT M-Geer eo eh et “ 0-661 — 96 = 101 8é1 9€1 PSI | 6-€L as “ woruyyy eruedC Z-1él | 6-04 =i €-2Z1 | O-GOL | 6-61 | 8-GFT | 8-28L | 8-62 es ce ypraayog €-LET | 8-69 Z-96 — | GOI | $-6EL | Z-I7T | 8-€8L | ¢-9L = ore sf Toue50 yy 0-Z€L | 8-89 r-46 — |0-L46 | &-631 | 8-OFL | 7-FLT |] 0-18 a aa nt AYSTOOTNT €-1é€L | &89 Z-96 — | 266 | 0O-TEl | T-FPT | F-08T = ie: aa ie JouIa0 yy 0-GEL | 8-04 L-€0L — | €00L | 8-€81 | ¢-OGL | 9-081 | 3&8 ve a ne ayuey 8-GZL | 1-89 L-16 — | 6-O0L | GLET | 8-981 | O-LLE | @LL ue a ig i €-82L | 8-89 8-66 — | 9-001 | 0-98T | 9-9€T | 6-181 | T-Gd ae a oe ty 9-31 | 9-29 1-16 — | €66 | &GET | 8-FEL | 9-F8L | 6-L ag ae as 409M J Z-1€L | 8-49 €-F6 I-GIL | @ZOL | LOFT | 9-L6T | 6-061 | 8-99 aa HT es IOMOT G-9EL | €-89 G.16 L-G6L | &FOL | 0-8EL | LOFT | 9-€8T ‘049 ‘TodJopoy) ‘stAR(T ‘1OMOT YT G.rel | L-L9 Z-16 6-FEL | 8-€OL | 8-1FL | 8-GPL | 6-18T = a a ee “** STAR v-€€L | ¢-89 0-26 T-8I1 | 1-001 | O-L€L | 0-G€L | T-O8T a i se ts Toajapox 8-9€L | 8-PL L-ZOL — | 8-1OL | LG€L | €&8EL | 3-981 | &€L ia a “** OXMOYSOIOF) 0-8EL | 0-€L 0-96 — | 0-96 | 0-681 | 0-9F1 | 0-64T | 9-18 ast i s+ AYsMouvMT L-PEl | T-1h Z-86 — | 9-86 | &1€L | FLPL | 8621 | 9.88 re ~ aS x 9-861 | 1-89 L-16 = — | % PET | OTST | 8-LAT = at oe ae “151g €-PSL | &L9 6-6 G-1GL | &-66 | P-SFL | 6 IPT | G-PLL | G18 ae ‘oqo “ToMOpT STAR, Gel | 1-99 €-96 L-8IL | 1-GOL | GET | GZEL | 2-481 | 0-TL ‘Oyo “TOMOTY ‘Torypurg “Gpraig €-O€L | 8-29 1-46 6-L1T | 6-LOL | 6-€€1 | GEL | &-€8l | €-3L ca SolpoyNY JUsLeYI yIprerg | yUsw_ | Uprorq | siseq | yYSIEH xapuy o1yeUL aoe | [eyuolg jo aovg | IqSIeH | UYpeerg | qISueryT otpeqdag Ayrioyyny -osdzig | geddq | wnurturpy | yysuery | soddq : sa0RTy suvsonbie yl SPTVIYG SorI0 J, SUVIUVUIS? J, (Spuv[1oyyON ) Jopur[p10 Ny yas are suviuissiq y SUBISOUOIOTTL eee eee eee suendeg $30719}40 sonbseq syeUsIOANW a6 oa “" SOUILASTT asoulyy Leyeseq ie J, Sqvly “ “epeuey “ =: “"SURIUOpE[RA MON 00 66 “= sueppa A, ssouvulepuy eee eee see Sk@MOLST ci ““srapuesy A4[esorT a ay suvldvsang “ suURIyIyRy oo dot “+ graqaagi “* suROSnIy cite aes ee souty SaLyyseq Youad jy “* SUBIIVACT ose eee eee sqydon sueqay J, epeben EAN suetsoudyog pod 300 seyeaey “** SdopuURIST YO A Jo ony in ee SSUISNOUTPL $4ns.10 J, be *SURISIOUIO} JAN A, ca “* sqreaqyg uosMecy ae ee “ gsomRAre SURT[RIYSNY eee eee eee $0139 NT dnoiy 10 9081 SaODY Snowy, fo SJUBULAANS Da JT 1pUDLAQ “AT WIDVL Upper Face Index. EK. TScHEPOURKOWSKY 299 APPENDIX II. CORRELATION TABLES. TABLE I. Russian Races, cf. Table II. p. 300. Head Index. 78 81 | 82 | 83 86 | 87 | 88 | 89 | Totals 154 aes 155 = et ee eel oe lel NO) a Stature. my (er) nN HR OWwWoomoowr O@On | | He ror] | | ete | DrFNWONNE BW =| aa ee bp Totals 20 | 11 92 (For Table II. see p. 300.) TABLE III. Head Index. 3 89 | Totals &1 | &2 Face Index. eos eeo en | AA — = Dw OMO Phe oa WwbDOe 13 TABLE IV. Cranial Cephalic Index. TON ON TN TON eo 2 3 86 |Totals | =) me OH oO~TO i ~] Contributions to the Study of Interracial Correlation 300 AAOAMANOOMOOMH n a fas} ft) jo) iS) ‘xopuy [eseN ‘A UTaVL (66% ‘d 908 “AT 9[qGeJ, 10,7) [elas Ja jaa [a T9T | O9T STRIOLI TAT | & $ ‘ § G 91 | 891 | @9T “aI yeyG ‘soony woiwssny-UuoNT ‘TT AIAVL “XOpuy peof, ‘xopuy peof, Stature. Stature. E. TscHEPOURKOWSKY 301 TABLE VI. Length of the Head. 183 | 184 186 | 157 | 188 | 189 ] oul eel | Ili tel ig XR S rs ee lptortoa|) [etor S ~© — Totals} 2 |—|3 | 2 3 | 4 | 10] 9 4 | 12 | & | — | me 096 are ND OAT | eget ” TABLE VIL. Head Breadth. 146 147 | 148 149 | 150 | 151} 152) 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163|Totals 154 155 156 157 1 158 1 159 1 160 a 161 2 4 162 | 1 7 D ile = 8 —_— 2, 12 2 | — 13 a 7 1 1 8 =) al 9 1 1 33 —}— 4 1 1 1 es Contributions to the Study of Interracial Correlation 302 OO P29 ei hee nel — s[rqoL| 97 OTT | 681 £67 9&T eer | ret O&T | 66T ‘WYSIeH prey THA AAVL “a.INyRyg Stature. 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 Totals E. TscHEPOURKOWSKY TABLE IX. Bizygomatic Breadth. 303 136 | 137 138 139 140 WH 142 143 | 144 146 iy 148 151 | 152 — Py HE HWP RH Ow 3 Stature. TABLE X. Face Index. 163 165 166 167 Totals ~ NH es on = J 78 “) 50 85 | Totals _ mele | ee | we) _ HEH wWoakRoOo mA | & py ao (S) Skull Breadth. Skull Height. 304 Contributions to the Study of Interracial Correlation TABLE XI. Skull Length. TABLE XII. Skull Length. 167 | — | 174 | 175 176 | 177 | 178 | 179 | 180] 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191} Totals 305 KE. TscHEPOURKOWSKY s[eqO J, G Gd TL I GL =, i! e |s i Id ® G OL = € 69 | & 6 89 ior I 49 . iG 99 os | tS G9 go I 79 > ae st I 19 j 09 |ST@IOL] OST | 67E | 87 | LTT | OFT | STL | TTT | fT | xT rr | ort 6ET | 881 | LET | 961 | GET | VET | 8ST | SST | TET | OST | BGT | 8T G6I | — | 6@T “Gapeoetg [[NAS TAX WIAVI (90¢ ‘d 008 “ATX 9qRI, 104) &P € I T G € S[eqIOL, I SYT a GYT G STT if BqT I IVT F OVT g 681 g 8ST € LST g 9ST g GST 9 TET z SST I 6ST - IST — OsL € 66I I G6T ‘s7eq0,L] 7ST OST \6VT | STL | ATT ore | ote rrr | ert OTT rrr | ort 681 | 8&1 | 461 | 981 | SET | TSI | G&L | SST | T&L | OGL | GET QO ae UME eS (MES) THX ATAVI ‘4qySlOH [RAG Biometrika Iv 39 Upper Face Height. 306 Contributions to the Study of Interracial Correlation TABLE XIV. Upper Face Height. 66 | 67 | 68 | 69 | 70 Bizygomatic Breadth. Totals TABLE XV. Skull Length. 167 | — \174) 175 | 176 | 177 | 178 179 | 180 | 181 | 182| 183 184,| 185 | 186 | — | 190 | 191} Totals (For Table XVI. see p. 305.) Bizygomatic Breadth. EK. TscHEPOURKOWSKY 307 TABLE XVII. Skull Height. 129 | 180 | 131 | 182 | 183 | 184| 185 | 186 | 137 | 138 | 139 | 140 | 141 | 142) — | 145 Totals Upper Face Height. TABLE XVIII. Skull Length. 179 | 180 Z : 3: 191 | Totals w NDNRWORWHONNWW 41 Contributions to the Study of Interracial Correlation 308 ) PSKOHALSS TH} Ge) Ss 8S) Ge) Sy! © ih ES s[eqo| ofr er | 981 G&T | TST. | &&T s[B1O J, 88T LET 9ET GET TEL 8ST 6ST IST OST 661 8é6T LOT 961 G6I Yer 86I Ger ‘QYSloAT [MAS XX WIAVL OVALS | ‘Yypeoig o1ewMo ian H MANOA M Hint sA belies, Roo) S[RqYOT, OST 881 | LET | 981 | SST | TE Sa) 83 ~ Agieaoehes Fidei “XIX ATAVE “yyprRaig oyjeuoSkzig VM WH RQ MN VN Skull Length. Skull Breadth. EK. TSscHEPOURKOWSKY 104} Totals TABLE XXI. Minimum Frontal Breadth. 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | — | 102) 103 176 _ If — 1 —- 2. | =e) $e | oan 178 Ik —}—|]—|]— ROM Pela dre [a a 4 180 — vy) — 1 — 1 = 1 181 — | — | — | — 1 1 182 —|]—-—/]— 7) ee ey) ab ee tees 183 - 1 184} — | — | — | 2 }— | 1 oe ty el Te TRA. | Se ed SEG as yo oe fer ee So i ee 187 — | — aw pose 188 — | — | — | — 1 1 ee Hl ee = TON es Nis ee ee el 190 1 VAG en (cama | ARNE See | | ee een Totals | 2 4 1 10 5 8 o y 1 — 1 — 1 w | won| seer one | oe _ TABLE XXII. Minimum Frontal Breadth. 104 | Totals een ae | | to | ro | | ee ree Rr we mow | O— _ 148 | 149 150 Totals} 2 4 309 310 Contributions to the Study of Interracial Correlation TABLE XXIII. Minimum Frontal Breadth. 91 | 92| 93 | 94 | 95 | 96 | 97 | 98 | 99 | — | 102) 103 | 104} Totals Skull Height. 8 Totals TABLE XXIV. Skull Basis. 90 | — | 94 | 95 | 96 | 97 | 9S | 99 | 100 | 101| 102 | 103 | 104 | 105 | 106 Totals, Naso-Alveolar Height. | Totals} 1 | —|] 1} —]| 1 2/;—|4 7 2 5 1 4 2 1 31 Skull Breadth. Bizygomatic Breadth. K. TscHEPoURKOWSKY TABLE XXV. Skull Basis. 311 100 | 101 2| 103 | 104 106 | Totals Totals 42 TABLE XXVI. Skull Basis. 125 126 127 106 | Totals ro | NRW RWRE ON Ww 312 Contributions to the Study of Interracial Correlation TABLE XXVIII. Skull Basis. — 90 | — | 94 | 95 | 96 | 97 | 98 | 99 | 100 | 101 | 102) 103 | 104) 105 | 106 | Totals i aS) rcs) e \ I | | | | | | | | | | Ne iS) SS) | | toro] | | Skull Height. & | | | [peti [Al eal ciel | = | | . tear Fil afe| pw ee He Totals Jt “I we [op) bo I wo w 1{—|1]}—|1]3|— TABLE XXVIII. Skull Basis. 96 | 97 | 98 | 99 | 100] 102 | 102| 103 | 104| 105 | 106 {Totals Skull Length. z eee | eee | | 4 Totals STATISTICAL STUDIES IN IMMUNITY. SMALLPOX AND VACCINATION. By JOHN BROWNLEE, M.A., M.D., Glasgow. IN writing this paper on smallpox and vaccination I have no intention of entering into any controversy upon the subject. The fact that vaccination protects against smallpox is assumed, and consideration is alone given here to the relationship between vaccination, revaccination, and smallpox, in so far as the facts throw light upon the growth, decline, or establishment of immunity in persons who have passed through one or other of these infections. I would rather have chosen for this investigation some other disease, but as an inquiry of this kind is essentially statistical there was no other series of figures available. The disadvantage of choosing vaccination and smallpox arises from the fact that the former is to a considerable extent different from the latter, and though there seems good reason for believing that vaccination is a disease caused by the parasite of smallpox so modified as to have lost one stage in its life history, yet the fact that it protects chiefly against the second stage of smallpox causes the immunity relationship to be of a more complex character than is desirable. Before proceeding, however, to consider the relationship of smallpox to vacci- nation one fact demands special preliminary notice: that is the change which has taken place in the age at which vaccinated persons are attacked by smallpox. All over the country from the earliest period of last century for which statistics are available there has been a steady increase in the mean age at which such persons have been attacked by smallpox. This is quite distinct from the rise in the mean age at death, shown by the combined mortality statistics of the vaccinated and unvaccinated which is com- monly and correctly looked upon as largely due to the protective influence of vaccination upon the younger members of the community. What I refer to is a process which applies specially to the vaccinated, and which does not seem to admit of any complete explanation from the point of view of alteration in the character of the population. In London, for instance, between the years 1836 Biometrika 1v 40 314 Smallpox and Vaccination and 1851, of 3,094 cases 2,719 were under the age of thirty, or more than seven vaccinated persons under thirty were affected with smallpox for each individual above that age. In Dr Gayton’s statistics relating to the period 1870 to 1883 the corresponding proportion is 54 to 1. In Dr Sweeting’s referring to 1880 to 1885 the proportion is 3 to 1, while in the epidemics of the last ten years two TABLE I, Age of Smallpox Attack in Case of Vaccination. Age London, 1836-51 London, 1870-72 London, 1880-85 London, 1892-95 London, 1902 Periods Cases Deaths Cases Deaths Cases Deaths Cases Deaths Cases Deaths 0—5 7 2 195 38 6 — Toh? © == 16h 5—10 56ST 786 60 1082 101 = 98 2 10-15 206 10) 256 8 1882 265 3 76 20) (866 & 40 ee 405 10 53300 65417 20-25 1058 93 man _ 639 16 104945 25-30 526 55 OE eee Soe ee 454 16 1017 70 30-385 210. 21 see 370 «14 827 91 35—40 102 20 peak =e! 934 (14 616 111 40—50 61 10 435 110 jee es 2938 726 154 50—60 2 1 134 30 a 89 8 954 =i 60—70 - 2) 62 19 24 7 105 27 10— \ 1 1 366 0-30 2719 216 9246 756 1427 79 1950 39 3099 137 30— 375 «2 1928 389 455 68 1020 52 2564 441 Mean Age 22°6 24°5 219 © 26-4 264 B34 286 38-2 persons under thirty years only are affected for each one over that age. In every town where there are statistics for a series of outbreaks of smallpox during the last thirty years exactly the same fact is observed. To mention Glasgow ; in 1872 this proportion was 6 to 1, in 1892 to 1895 24 to 1, and during the epidemic of 1901 the ratio reached unity. Now there can be no doubt that a part of this is due to the gradual disappearance of adult persons protected by previous attacks of smallpox. In addition during the years immediately succeeding Jenner’s discovery vaccination was performed promiscuously on young and old alike, and persons vaccinated in adult years must have constituted during the first sixty years of the century a very considerable proportion of the total vaccinated persons at higher ages, as it can easily be seen that any person in 1836 whose age was greater than thirty-seven must have been vaccinated at the earliest at an age corresponding to the difference of the actual age and of thirty-seven years. By 1870, however, both these influences must have to a considerable extent disappeared, and yet we see that the change referred to has been going on just as markedly since that date. Against these two factors must be set one which undoubtedly acts in the opposite direction, and that is the gradual better enforce- ment of vaccination during infancy, which, by continually providing a larger number of vaccinated persons at lower ages, ought if no other change were taking place to supply a greater number of cases at these ages. The influence of this should at least be sufficient, at any rate since 1870, to neutralise that due JOHN BROWNLEE 315 to the factors before mentioned. Unfortunately there is a complete absence of statistics referring to this matter, so that an examination of the relative influences of these different factors is impossible. The effect of revaccination on the age incidence of persons suffering from smallpox can be more readily estimated. When performed in the absence of an epidemic, or when slight outbreaks only of this disease are present, there is not a sufficient amount done to influence statistics much one way or another. In Sheffield, for instance, when the great epidemic of 1887 occurred, about five to seven per cent. of the population at each age period above fifteen years were found to be revaccinated. When, how- ever, as in this epidemic in Sheffield and in the late epidemics in Glasgow revaccination is performed on a large scale, it is found that a much larger propor- tion of persons between ten and twenty years of age avail themselves of this means of protection than at higher ages. The effects of this on later epidemics are obviously, (1), if the second epidemic occurs within a few years of the first to raise the mean age; and (2), if the later epidemic occur about fifteen years there- after to lower the mean age of attack. Revaccination, however, as a rule only begins to have an important modifying effect on the epidemic concerned when it is naturally approaching its end. The change in the age incidence of smallpox just discussed is one which not only applies to the vaccinated as a class, but equally to the different groups comprehended in this class. Persons suffering from smallpox are for the purpose of classification commonly divided into groups as they present on their arms one, two, three, and four or more vaccinal cicatrices. In each of these groups the mean age has steadily risen. Now with regard to the group of those who present one cicatrix this is to be expected. Since 1870 persons vaccinated in this manner have been becoming progressively fewer and fewer, and consequently in London the bulk of those presenting one vaccinal cicatrix must consist of persons at the higher ages. Exactly the same change in the mean age, however, is found to be among the members of the group presenting one cicatrix in Glasgow, although no such change in the method of vaccination has taken place there. With regard to the group, however, of persons with four or more scars, year by year since 1870 a larger and larger number of such have come into existence at lower ages and consequently it would be expected that the mean age among this group would be at any rate stationary if not lower. This is not found to be the case. The annexed table exhibits the change in the mean age which has taken place in London since 1870. The rise noted in the mean age of persons with four marks TABLE II. Mean Age of Cases and Deaths in the Epidemics in London, 1870—1902. One Scar Two Sears Three Scars Four Scars Cases Deaths Cases Deaths Cases Deaths Cases Deaths 1870-83 21°8 26°2 21°0 25°1 18°1 24°0 170 24°8 1892-95 26°8 ta 13053 29°5 43°9 26°5 S17 Oe Siey/ 1902 34°2 39°6 34°0 41'3 31°0 By (Or 272 34°5 40— 2 316 Smallpox and Vaccination in view of the fact just mentioned seems to establish that this phenomenon is not due to alteration in the character of the population, but presents a biological as distinct from a statistical change. The explanations which can be offered from the biological point of view are three. Firstly, there may be some secular change taking place in the susceptibility of the population towards smallpox itself; secondly, there may be a like change taking place in the smallpox organism; and thirdly, it is possible that the passage of the organism of vaccinia through several thousand generations in the human subject has gradually increased the protective value of lymph; here again, however, there may be secular change. This increase in protective value, while it cannot be held proved, is in striking contradiction to what has been hitherto considered as probable. Plants propagated by cuttings, by tubers, and such like means, are well known to deteriorate in quality, and frequent recourse must be had to propagation by seeds if the strength of the species is to be maintained. To this rule, however, there are certain exceptions. Though it cannot be distinctly averred, however, that the vaccinal lymph has increased in potency, it can with all certainty be maintained that there is no evidence in favour of its deterioration. The evidence on which the argument regarding the deterioration of lymph was based cannot be regarded as complete. The chief points were, firstly, that as the century went on it became increasingly more possible to revaccinate successfully ; and secondly, that in making attenuated forms of sheep-pox corresponding, as was thought, to vaccination, it was found that after a few generations these could not be relied on to give protection. In view of our fuller knowledge of the probable lite history of the smallpox organism and its relations to vaccinia this latter argument ceases to present a reasonable analogy, while the former is more explicable on the view of the increasing potency of the lymph than on the other hypothesis. The change just described is one which affects vaccinated persons as a whole and bears no special relationship to the question of the growth and decay of immunity. It is then to be noted that in all further discussions of smallpox statistics throughout the course of this paper the comparisons which are drawn, as far as possible concern persons in the same epidemic in the same place, because with different age constants at different places and at different times it can be seen what contradictory inferences could be made. The first fact with regard to the immunity conferred by vaccination is one which seems unexpected. It is found that persons who are well protected have for all epidemics at all places a lower mean age than those who are badly protected. Exactly the same remarks apply to the groups which are based on the size or character of the cicatrix. Thus in each epidemic persons with four good marks have a lower mean age than those with four indifferent marks, a relationship which also applies to the group of persons presenting one, two, or three cicatrices. Taking the statistics of the large epidemics in Glasgow, of which more than half the cases consist of persons with one mark, it is found that the mean age in each of the three outbreaks falls steadily as the size of the mark JOHN BROWNLEE 317 increases; that this is not due merely to the fact that larger marks are now put on than were formerly is shown moderately conclusively when it is considered further in detail. The degree of foveation of a cicatrix has been regarded as an indication of its protective value. Foveation, as a result of vaccination, no TABLE III. Mean Age of Persons with one Scar of different sizes, Glasgow 1901, 1902, and 1904. 1901 SS Size of Scar Foveated Not Foveated 1902* 1904 O—-25 sq. in. 32°7 33°7 38°3 37°3 5) = 5) 55 31°6 32°1 34°8 33°3 *5—1 ss 28°1 29°3 33°4 32°0 1— ‘s 26'1 27°5 28°3 32°9 physician can foretell. The number, and, to a certain extent, the size of the cicatrices, is in the hands of the operator, but as to whether the scar will turn out to be foveated or not depends upon a number of factors beyond his control, and yet the mean age of those with unfoveated scars is for each group higher than that of those in which the scars are foveated. This relationship also applies uniformly as far as I have been able to observe, and in the large epidemics where there is a sufficient number of cases to allow of finer gradation the same relation- ship holds. For instance, taking those cases in the epidemic in London in 1902 in which there are four cicatrices of collective area greater than 4 square inch it is found that among those in which more than half of this area is foveated the mean age is lower than that of those in whom less than half is foveated, and that the mean age of the latter is in defect of that of those patients in whom there is no foveation, the cicatrix being glazed or puckered. I do not think that this can be explained as may be suggested on the ground that foveation is a feature of the scar which tends to disappear with age, for it must be noted that the relationship just described bears a constantly present inverse proportion to that observed to exist between the same subdivisions and the mortality: that is to say, many marks are a better protection against death than few, larger than smaller, and foveated than non-foveated, and this to a greater extent than the mere difference of age incidence would imply. The susceptibility to smallpox of those with different numbers of marks is unfortunately a matter which cannot be determined directly, and consequently its full significance cannot be expressed in figures. It however can be investigated for London to a certain extent. At intervals of ten years since 1870 large out- breaks of smallpox have occurred in London, The number of persons affected at each age period with the number of their vaccinal scars is therefore known at four epochs separated by almost equal intervals of time, and consequently the way * The incidence of mean age is here due to the fact that the revaccination in 1901 was performed more largely in the earlier than in the later ages. 318 Smallpox and Vaccination in which persons are protected by different kinds of vaccination can be judged by a comparison of the proportions in each group at the same age period at the different epochs. Persons thus of the age five to ten years in 1872 will be fifteen to twenty in 1882, and will be thirty-five to forty-five in 1902. These proportions are recorded in the accompanying table. From this it may be seen that in general after forty years all vaccinated persons are apparently equally liable to smallpox. The proportions of persons attacked with smallpox who were from forty to sixty in 1872, sixty to eighty in 1892, and above seventy in 1902 with one, two, three, and four scars respectively, are sensibly constant considering the small number of TABLE IV. Table showing the Percentage of Persons attacked by Smallpox, with the different numbers of Vaccinal Cicatrices at each age period, in the Epidemics 1870—72, 1880—85, 1892—95, and 1902 in London. Group I. Persons 0—10 years, 1870—72. Epidemic Age Periods One Scar Two Sears Three Scars Four Scars 1S7O—72 0—10 24°6 25°5 22°6 27°3 1880—85 10—20 20°7 25°7 31°3 22-2 1892—95 20—30 14:3 23°7 28°3 35:1 1902 30—40 7%) 14°2 24°8 51i'1 Group II. Persons 10—20 years, 1870—72. Epidemic Age Periods One Scar Two Scars Three Scars Four Scars 1870—72 10—20 22°2 36°8 23°3 17°5 1880—85 20—30 21°3 32°3 26°0 19°9 1892—95 30—40 20 27°0 29°0 23 1902 40—50 18°4 28°4 27°71 25°9 Group III. Persons 20—30 years, 1870—72. Epidemic Age Periods One Scar Two Scars Three Scars Four Scars 1870—72 20—30 24°6 39°6 19°4 16°7 1880—85 30—40 = — — — 1892—-95 40—50 28°6 28°4 15 22°8 1902 50—60 16°7 35°6 23 22°5 Group IV. Persons 30—40 years, 1870—72. Epidemic Age Periods One Scar Two Scars Three Scars Four Scars 1870—72 30—40 29°8 41°6 17°8 16°6 1880—85 40—50 — = — =e 1892—95 50—60 24°7 30°3 21°3 23°6 1902 60—70 26°4 31°8 18°7 23°1 Group V. Persons 40—60 years, 1870—72. Epidemic Age Periods One Sear Two Scars Three Scars Four Sears 1870—72 40—60 32°7 39°1 16°6 10°4 1880—85 50—70 — — — — 1892—95 60—80 50 32°3 59 11°8 1902 70— 39°9 371 11°2 11°2 JOHN BROWNLEE 319 attacks at these ages. Persons with four marks are relatively well protected under five years, which is to a certain extent explained on the ground that the bulk of the patients at that age period belong to the class with very imperfect marks. Between twenty and forty years the advantage seems greatest, and after forty it has apparently entirely disappeared. Exactly the opposite would appear to be the case with persons with one mark. The cause of this in the early years, however, has just been explained, and the apparent decrease in later years is accounted for largely by the fact that a steady equalisation of susceptibility among the four groups takes place, and the number of persons with several cicatrices increases in proportion. It is to be noted that it is apparently between twenty and forty that persons with four scars have most advantage over persons with one. It may be added that the group of those with three marks resembles in general that with four, and the group with two that with one. This question of relative susceptibility at different ages is closely allied to that just discussed regarding the lowering of the mean age in those groups where vaccination is most perfectly performed. It certainly cannot be inferred from the latter that the more incomplete the vaccinal protection the more likely is the attack of smallpox to be delayed, provided such an attack is going to come. It appears much more likely that the decay of protective power is longer delayed, and that consequently while for the first twenty years or so any kind of fair mark indicates much the same protective power, there is a period in the middle ages of life when those who have good cicatrices are much better protected than those with inferior, and again after this period of relative protection has decayed all again become equally susceptible *. The only statistics which are available to check this conclusion are those of Warrington in the epidemic 1892-3, where a census as to the state of * How this produces a higher mean age among those badly vaccinated than among those well vaccinated can be seen if the figures of Table V. are examined, though it is perhaps not immediately apparent. Here the divisions do not lend themselves to any accurate calculation of mean ages. In the approximations however the mean age of all those found to be badly vaccinated is in much the same excess over that of all found to be well vaccinated, as the mean age of those in the former class who were attacked is in excess of the mean age of those in the latter. But this high mean age among the badly vaccinated attacked by smallpox is not necessarily a consequence of this grouping of the population in the houses invaded by smallpox. As high ages are more susceptible among the badly vaccinated the population of the houses invaded will necessarily contain more old persons as there must be some correlation of age among persons living in the same houses, young families being of course more common among young parents. The high mean age referred to is independent of this, as will be seen if the percentage attacks in both are calculated. These show that in equal populations the mean age of those badly vaccinated attacked by smallpox would be in considerable excess of that of the well vaccinated. Proportions Attacked. Age Periods Well Vaccinated Badly Vaccinated 0—10 37 6:8 10—20 22°6 19°5 20—30 48°3 47-1 380—40 27°5 54°9 40—50 16:2 19°7 50— 13°4 19°5 320 Smallpox and Vaccination vaccination was taken for the different age periods of all those in infected houses. Unfortunately the vaccinal scars are simply divided into two classes according as they exhibit good or imperfect characters. It will be seen that under ten years those well vaccinated have a certain measure of protection probably for the reason TABLE V. Correlation between Exposed and Attacked, Vaccinated Well and Badly, Warrington, 1893. Well Vaccinated Badly Vaccinated Age Periods Not Attacked Attacked Not Attacked Attacked Correlation 0—10 466 18 124 9 +21 — 10—20 346 114 136 32 — 13 20—30 144 135 74 66 — 03 30—40 95 36 46 56 +°43 40—50 98 19 61 15 +:°09 50— 45 7 68 14 +:14 already indicated, from ten to thirty years there is practically no difference between the classes. Between the ages of thirty and forty the advantage however is very definitely in favour of those with good marks, and after the age of forty this is still present, though not nearly so marked. These figures so far as they go confirm the explanation just given. Before proceeding further to discuss the character of the immunity conferred by vaccination it is necessary to have as far as possible an accurate knowledge of the susceptibility to smallpox exhibited at different ages among persons who have no acquired protection at all either from vaccination or previous attack of the disease. With regard to the age susceptibility to smallpox in an unvaccinated community not a great deal can be said. In prevaccination days statistics were rarely kept with any accuracy, and though records of death in age periods exist relating to considerable numbers of persons and considerable periods of time, yet these afford a quite inadequate basis on which to compute the susceptibility at different ages. The most complete tables calculated from the number of deaths are those published by Duvillard with the authority of the French Academy of Sciences in 1806. He deduces the result that persons at all ayes are about equally susceptible to smallpox, but that the maximum susceptibility is between the ages of ten and eleven years, though the excess is small. His conclusions lead him, however, into the difficulty of requiring that a much larger number of persons at a higher age should be attacked by smallpox than is actually observed. He explains this on the ground that the great majority of these attacks are so mild as to escape notice. Now of this there was no evidence at the time at which Duvillard wrote, nor is there any at the present date, when the diagnosis of modified smallpox has been brought to as great a pitch of perfection as any clinical diagnosis can. Consequently his conclusions afford an example of the untrustworthiness of calculating the susceptibility at different ages from the number of deaths alone without any knowledge of the actual case mortalities. For the determination of the susceptibility at different ages among persons who JOHN BROWNLEE 321 are unvaccinated at the present day certain statistics exist. The chief of these are contained in the account of the epidemic in Sheffield in 1887 presented in Dr Barry’s report to the Local Government Board. In this case a census was taken of nearly the whole population. The facts are tabulated so as to give the number of cases at each period who were unvaccinated, vaccinated, or revacci- nated, and also those who had previously had smallpox. The state as regards the vaccination of the inhabitants of those houses actually invaded by smallpox is also tabulated separately. On account of the size of the epidemic this census must be held as the most important group of statistics at our disposal. It seems better to consider fully in the first instance the conclusions which may be based on these figures, and afterwards to compare the results with the statistics affecting naturally much smaller numbers which have been published with regard to other towns. Considering, in the first place, the alterations in susceptibility to smallpox among the unvaccinated which takes place as the age increases, and expressing the relationship in terms of the correlation coefficients between those exposed to infection who escape, and those exposed who are attacked for each two succeeding age periods, it is found that these coefficients vary in the following TABLE VI. Correlation between those who were Attacked, and those who Escaped in Vaccinated and Unvaccinated respectively, at each age and that immediately succeeding.— Sheffield, 1887—S. Age Periods Vaccinated Unvaccinated Age Periods Vaccinated Age Periods Unvaccinated O=> 1 ee = —'10 — 28 — = = i Ge = By — 28 FOC O10 —"2] —'17 — — 1— 5& 5—10 — 47 5 10 & L015 — 34 —°538 — — 5—10 & 10—15 — 84 10—15 & 15—20 — 29 “72 — = HOKU 8k WIS HG — 20 15—20 & 20—25 04 04 Vo 20 G0 —— 25 O04 15—20 & 20—25 —'18 20—25 & 25—30 Oil 38 20—25 & 25—80 “O04 20—25 & 25—30 24 PI— HO) CA BOSS ‘09 14 25—30 & 30—35 13 25—30' & 380—85 *B4 30—35 & 35—40 "26 924 30—35 & 35—40 “ol 380—385 & 35—40 54 35—h0 & 4O—45 ‘47 75 35—40 & 4O—45 53 35—40 & 40—45 73 40—15 & J5—50 "48 — 25 LO—45 ®& J5—5O 67 4O—4E ®& 45—5O “AT Vip—=50) & 50—55 ‘Ol “80 45I—50 & 50—55 68 4I—5O & 50—55 95 50—55 & 55—60 “5 —'17 50—55 & 55—60 etd GOL) (85 (10) 93 55—60 & 60— 02 “O04 55—60 & 60— 79 55—60 & GO— 92 way. From birth till between ten and fifteen years they are negative, indicating that the susceptibility steadily increases to this age, as was before found by Duvillard, and that thereafter each succeeding coefficient is positive of greater or less amount. If the susceptibility under one year is chosen for comparison with that of each succeeding age period it is seen that the strength of resistance which was less marked between ten and fifteen years has between twenty and thirty again increased so as to be equivalent to that in the first year of life, and thereafter steadily becomes greater. The period of maximum. susceptibility between ten and fifteen years is one which is well marked in a large number of epidemics, although incapable of any very accurate measurement. The number Biometrika tv 41 322 Smallpox and Vaccination of children successfully vaccinated is published as a Government Return for every year since 1872, so that for epidemics occurring since 1890 for any definite place TABLE VII. Table showing the rate per Million of Attack by Smallpox among the Unvaccinated calculated as described in the teat with the number of Attacks in an epidemic of Smallpox in 1726 at Aynho in Northamptonshire for comparison. London, Leicester, Gloucester, Aynho, Age Periods 1902 1892-93 1895-96 1726 0— 5 — 255 727 13 5—10 315 607 909 15 10—15 543 700 955 33 15—20 584 554 774 14 20—25 339 396 434 16 25—30 225 386 446 9 30=— 35) 150 aa 351 32* a rough approximation can be made of the respective number of vaccinated and unvaccinated, it being always borne in mind that the number of unvaccinated thus calculated is a maximum, and that many persons included have probably been vaccinated at some period subsequently during the existence of smallpox scares. The occurrence of such vaccination obviously becomes more probable the older the patient is, so that if a maximum is found between ten and fifteen years it is probably, as thus measured, to be in defect of the reality. The relative insusceptibility of old persons is also in statistics of epidemics an almost constant feature but the small numbers of unvaccinated persons attacked with smallpox from twenty years upwards makes the tracing of the development of insuscep- tibility in many cases less definite than in the case of Sheffield. This represents the main course of the increase and the decrease of susceptibility towards smallpox presented by the unvaccinated person. It is in general agreement with the few figures which exist regarding the age incidence of smallpox in the eighteenth century. This age incidence of course depends in a very large measure in any locality on the frequency with which epidemics occur. If epidemics break out every three or four years, as is known to have been the case in many places, then it is obvious that the most frequent age of attack will be that between two and three years, a characteristic well shown for example in the statistics of some German towns. The ages of two to three years indeed seem to constitute a period of greater relative susceptibility, as is already indicated in the figures for Sheffield, where the correlation coefficient between the age periods 0-1 and 1-5 years is greater than that between the periods of 1-5 and 5-10, though the numbers are too small to make this absolute. If, however, epidemics occur at greater intervals the natural period of greatest susceptibility will tend to become more evident. A maximum between ten and fifteen years is shown markedly in the statistics of Aynho in Northamptonshire, given by Dr Jurin, and, though in his record there is no statement to indicate the intervals between the succeeding epidemics, it is probable that this represents an epidemic OBIE of smallpox among a comparatively virgin community. * This is number of cases above 30. JOHN BROWNLEE 323 Still confining our consideration to the Sheffield figures we proceed to inves- tigate the susceptibility among the vaccinated. Among these, as we pass from age period to age period an increase of susceptibility occurs from the period of vaccination till between the ages of fifteen to twenty years. At this age the period of greatest susceptibility is reached, and thereafter immunity is gradually regained, increasing with succeeding years, but never actually reaching that existing immediately after the operation was performed. It is to be noticed that the growth of immunity with old age proceeds from the age of twenty upwards in practically the same manner as among the unvaccinated. The curve is much more continuous, as is to be expected from the larger numbers which are available for its construction. The correlation coefficients fall slightly short of the similar ones for the unvaccinated, which is to be expected, as the old age protection starts originally from a higher level, the acquired immunity not being wholly lost at the age chosen for comparison. It is a matter of great interest from the point of view of the theory of immunity that with this large mass of statistics suitable for the calculation of the effect which one form of immunity produces upon another, namely, the effect of the presence of acquired immunity upon the growth of natural immunity, it is found that the latter develops exactly as if the former were completely absent. This is a result which could not have been anticipated, and its meaning will be further discussed in another paper. The relative susceptibilities of the vaccinated and unvaccinated now falls to be examined, and here the Sheffield census is exceptionally valuable, as it allows the relative susceptibilities of vaccinated and unvaccinated to be determined and TABLE VIII. Table showing Correlation between Vaccinated and Unvaccinated Persons as regards Attack by Smallpox and escape from Attack when the numbers are considered of all persons (a) in the town itself, (b) in infected houses. (a) All persons in (b) Those in the infected houses the town considered alone considered Sheffield, Sheffield, Dewsbury, Leicester, Age Periods 1887-88 1887-88 1892* 1892-93 0— 1 84 “94 “90 1 ys) 63 94 °93 1 5—10 63 96 84 93 10—15 67 “99 56 TH SO) "45 72, ‘73 20—25 34 ‘74 66 } 50 25—30 “12 “40 380—385 “Aliz¢ BD ag 385—40 16 39 17 4O—45 sat "28 30 J5—5O 19 ‘49 50—55 — ‘05 —'ll | 55—60 “05 ‘O07 — 03 60— aT ‘05 J * Tn this instance the number of revaccinated persons and of those who have had previous attacks of smallpox are excluded in the statistics, so that the correlation coefficients are as free from error due to the admittance of such persons as is possible. The whole numbers affected are however small. ae) 324 Smallpox and Vaccination compared, not only when the risk of infection is comparatively slight, as in the general population, but when it is much more considerable, as in infected houses. In the annexed table are given in parallel columns a series of coefficients expressing the relative susceptibilities under each of these conditions for all age periods. It will be seen that the protection which vaccination affords against attack is much less strong, considered in relation to the total numbers of such persons in the town generally, than to the numbers in houses actually infected. In the former it ceases to be considerable after twenty-five years of age, and is practically negligible after the age of fifty. In infected houses, however, this decline is not nearly so marked, but after the age of fifty it also becomes practically negligible. This is an example of a property which at times seems to ensure a greater value to action on a mass than on a small section of that mass which differs from the mass in being apparently more vulnerable. It is not a case of the poison acting equally on the two classes as may be taken for granted is likely to be the case in invaded houses, but that the relative sparseness of unvaccinated persons affords them a protection beyond what random selection would give. It is conceivable that some such factor plays an adjuvant part in natural selection. There are no other statistics which enable comparison to be made between those in infected houses and those of the general population, but the relative susceptibility between the two classes of vaccinated and unvacci- nated in infected houses has been calculated for some other epidemics, Leicester, etc., which show local differences, but exhibit on the whole the same kind of relation from age period to age period which has been noted in the case of Sheffield. All the figures agree in making a rapid fall in the protective power of vaccination after the age of fifteen years. It is, however, to be noticed that the protection against a severe attack of smallpox or death afforded by vaccination is of a much more permanent nature than that against an attack of the disease. This is to be expected when the clinical resemblance between vaccination and the second stage of the course of smallpox both as regards local appearances and the course of the pyrexia is recalled, and when it is noted that the clinical course of smallpox suggests the presence of two very different classes of toxins in the two different stages of the disease, so that the immunity against the first stage is much more readily lost than that agamst the second. That this is the explanation will be apparent when the close relationship between the protection against severe attack, that is an attack with a severe second stage of the fever, and a fatal attack is noted. There are no very ample data to determine this comparison, but in the annexed table will be seen in parallel columns the correlation betwixt recoveries and deaths, and mild cases and severe for the Gloucester epidemic of 1896, and it will be seen that the correspondence, considering the small numbers involved, is sufficiently close. The figures which Macdonell has already given are quoted for comparison, when it will be readily seen that the deaths give a very fair measure indeed of the severity of the disease, closer than might be expected when it is considered how large a factor the personal equation plays in the tabula- tion of degrees of severity. This is important, as the great majority of statistics JOHN BROWNLEE 325 as published refer only to cases and deaths, and give no detail of the character of the illness. TABLE IX. Correlation, Vaccinated and Unvaccinated. Recoveries and Deaths, and Cases Mild and Severe, for Epidemic in Gloucester, 1896. Recoveries Cases Mild Age Periods and Deaths and Severe 0O— 5 — — 5—10 — “94 L0—15 94 93 15—20 aiid, "75 20—30 63 *88 30-—40 "73 | 63 68 4O—50 434 0° 63 50—60 (eis 60 60— eee uae TABLE IX bis. Correlation between Cases Mild and Severe and Recoveries and Deaths. Vaccinated and Unvaccinuted (Macdonell). Recoveries and Deaths | Mild and Severe Sheffield... se “hel 79° Glasgow, 1892—5... 78 “91 Glasgow, 1901 tas 63 51 The way in which the protection against death from smallpox afforded by vaccination is gradually lost is of considerable interest. As shown in the appended table it gradually decreases from the date of vaccination up to the age of twenty- TABLE X. Correlation, Vaccinated and Unvaccinated, Recoveries and Deaths in Various Epidemics. Age Periods London London Sheffield | Gloucester London 1856-61 1870-72 1887 1896 1902 5 = a 1 1 1 5—10 34 74 79 1 17 10—15 oN een 78 ‘94 93 15—20 nn 76 17 Td 20-25 66 67 Rs 61 25 —30 65 Y 49 oe 67 30—85 . ‘55 51 35—0 i ae 53 i a 43 5015 ete melee: 38 es W550 ( 76 ( 31 G4 74 50—55 48 ee 55—60 {42 Le ce eames 49 62 60—65 ‘AG 65—70 ( 326 Smallpox and Vaccination five, after which it is moderately constant. One point, however, of interest becomes manifest here, and that is the way in which the correlation coefficients for the different ages between vaccinated and unvaccinated tend to assume higher values, at the ages of 10 to 20 years*, as the century has gone on. Taking the first set of statistics which exist, namely those of Dr Marson, passing from these to the great epidemic of 1870-72 in London up to the last epidemic of 1902, we see the tendency for those coefficients to become larger continuously manifest, and the corresponding figures for the other epidemics for which they have been calculated seem to fit into place much as might be expected from their date of occurrence. This affords some confirmation of what has been suggested earlier in this paper in connection with the rise in age at which vaccinated persons are attacked with smallpox, namely, that vaccinal lymph has gradually gained potency through being passed for many generations through the human subject fF. The question now arises as to whether in different epidemics the extra protec- tion which good scars are known to afford to their possessors compared with bad scars assumes greater relative importance in mild or severe epidemics. There seems, however, to be little difference. In the accompanying table will be seen TABLE XI. Correlation between Recoveries and Deaths in the Epidemics 1902 and 1892—95, and 1902 and 1870—83 for each Vaccinal Group. Nature of Group 1902 & 1892-95 1902 & 1870-83 One Scar 22, 43, Two Scars 18 37 Three Scars ‘24 43 Four Scars 21 HQ the correlation between cases and deaths for the epidemic in 1902 and those in 1892-95 and 1870-83. These coefficients indicate that there is no great differ- ence in a severe epidemic or in a mild between the manner in which each different group is affected in comparison with the same group in a severe or in a mild epidemic. The figures, it is true, show some little difference, but not much when it is considered that the change in the age distribution of the different groups has been proceeding not by any means equally in them all. Another point which may be noted is the manner in which the relative protection against death between those with good and those with bad marks varies. It is found to differ not very considerably below the age of twenty, to * The whole correlation coefficients are given in the table, but only the early years are referred to in the text, as adult primary vaccination was very common in the first half of the century. + It is this change in the correlation coefficients which serves to differentiate the two interpretations which one puts on the rise of the mean age: (1) that due to the lymph gaining potency which is the cause of the rise of the general mean age of vaccinated persons attacked by smallpox, here the corre- lation between vaccinated and unvaccinated has increased as the century has gone on: and (2) the higher mean age of indifferently vaccinated persons taking smallpox than of well vaccinated, here the correlation between recoveries and deaths is in favour of the well vaccinated. It is thus seen that the mean age may be raised in two different ways. (Cf. footnote, p. 319.) JOHN BROWNLEE 327 be at a maximum between twenty and thirty years, and then to gradually disappear towards old age. The figures here, however, are not large enough to base any definite conclusions. It now falls to be considered at what rate immunity develops after vaccina- tion, that is to say, how long must vaccination be performed prior to the onset of smallpox to prevent the attack? The incubation period of smallpox is from twelve to fourteen days, and it is found to be practically an absolute rule that no person develops smallpox more than fourteen days after a successful vaccination has been performed. If vaccination be performed during incubation period of smallpox the attack is seen often to be much modified. Here again the statistics refer to small numbers. In the first place, taking all ages together and estimating the correlation between the attacks and deaths in unvaccinated persons and in TABLE XII. Correlation between Recoveries and Deaths in Unvaccinated Persons, and those Vaccinated at various Dates prior to onset of illness. Time before Gloucester London London onset (all ages) (all ages) (ages above 1 year) Days 1896 1902 1902 0— 3 13 ‘06 “40 3— 6 16 12 “40 6— 9 65 aly) 2 I-14 14 2, 45 14— 1:00 1:00 1:00 those vaccinated after infection, we find that even among those vaccinated within three days of the onset of the illness there is a distinct positive correlation in favour of the process. Among those vaccinated from three to six days before the onset of attack the correlation is slightly higher, from six to nine days it is very considerable, from nine to fourteen days, however, it is again very much less, apparently due to the fact, as will be shown later, that only a few persons who are really very susceptible develop the disease after nine days. After fourteen days the protection against death is absolute. If children under one year be eliminated from the numbers considered the protection is seen to be considerably higher, but still to exhibit a maximum at the period of six to nine days. Though the subtraction of those cases under one year makes such a difference in the correlation coefficients, yet when those who have been vaccinated more than six days before the onset of the illness are compared for the varying age TABLE XIII. Correlation between Recoveries and Deaths of those Unvaccinated, and those Vaccinated 6—14 days prior to onset of illness. London, 1902. Age Periods 0— 1 61 ats 63 5—10 73), 10—15 Boge 15—20 49 20—30 61 328 Smallpox and Vaccination periods with those who are unvaccinated at the same age period it is found that the correlation is practically constant from birth onwards, and that the smaller correlation coefficients obtained when all ages are grouped together is brought about by the fact that a large proportion of the total cases are under one year, and that among these there is so special a mortality as to vitiate seriously the value of the correlation coefficients of the statistics when all ages are combined. As to the actual susceptibility of persons vaccinated after exposure to small- pox there are no actual figures in existence. Some, however, can be inferred in the Gloucester epidemic; out of 653 persons vaccinated after exposure only 85 developed the disease as compared with 604 out of 832 among the unvaccinated who were exposed to infection and were not vaccinated. Out of the 85, 23 were vaccinated on the day of onset of the disease or on the day after, at a time when, therefore, the process could have no preventive action. The distribution of the remaining 62 persons is as follows: The number of those vaccinated from 0-8 days before the onset of attack was 19, from 3-6 days also 19, from 6-9 days 18, while for 9 days and upwards only 6. Now the great majority of persons exposed to infection are vaccinated within four days of the exposure, so that it becomes evident that if the operation be performed up to the ninth day before that on which onset of the attack could be expected there is a very high proba- bility that no further developments will take place. This is a question on which statistics cannot be compiled, because, though a person may be exposed to infec- tion for three to four days there is nothing to indicate on which day he may be infected, if at all, unless he subsequently develops the disease, so that the above figures are probably as satisfactory as any can be. The similar relation which exists between persons revaccinated more than six days before the onset of the TABLE XIV. Correlation between Recoveries and Deaths of those Vaccinated in Infancy only, and those Revaccinated 6—14 days prior to onset of illness. Age Periods London, 1902 20—40 "46 40—50 35 illness shows a correlation of a similar kind, but not so great as in the former class, which is to be expected as before remarked, as the level of the protection from which the improvement is made is originally much higher. The smallness of the figures does not permit it to be given for more than two age groups. The effect which an attack of smallpox exercises in establishing immunity is much more difficult to ascertain than that produced by vaccination. In general the great bulk of persons who are vaccinated have been vaccinated in infancy. With regard, however, to those who have had smallpox there are no data as to the ages at which they were attacked. In London between 1836 and 1851 out of 2654 unvaccinated persons suffermg from smallpox treated in hospital only JOHN BROWNLEE 329 690 were under the age of ten years; between 1870 and 1883 out of 2169 like persons 1197 were under the age of ten years. In both of these sets of statistics the distribution of the numbers from birth till ten years is fairly uniform. So that it may be taken for granted that towards the end of last century among persons over twenty years of age who had had smallpox previously one half was the greatest proportion of these who had the attack during the first ten years of life. In the accompanying table I have collected the details of fifty-one cases TABLE XV. Table of Second Attack of Smallpoz. Age at Second Attack. Salpesr | solte | Sal Silas fas ls |S i= |S ls ) Te 2 Lid Gd ig lade] ™ ra Sse It ase Ss A See eee Se | ts asa ce Wes. [eee | ee lcs fas] S Cmeret seO N aNeae p Io oe aa oy fp yet a BeeOe We eel ho i | D 10—15 {—|—]/ 1 Te te 9 | a= | 1 1 Bet) 75==20 : a | ea || 20-25 | Beem marge ee ca eae ag | Ron s00. P| | = 1 ree | eee ee eee ee | 2 aoe 2p) ys J—|—}—};—}—|—]—-]1 —|—|}— —|- Totals 4 9 3 vi 6 3 5 6 1 Y, 33 iy | <= it | where persons have had a second attack of smallpox between the years 1887 and 1902, and of these fifty-one it is seen at once that thirty suffered from their first attack of smallpox below the age of five years, nine between the age of five and ten years, leaving a total of only twelve who had their first attack of smallpox above this age. This would seem to show that the protection afforded by an attack of smallpox in childhood is much less complete than that afforded by an attack after ten years. A study of the figures in the table seems to show that this is not due simply to the natural lessening in numbers which occurs in the population at each age period from childhood onwards. A certain amount of support is lent by this to the suggestion advanced earlier in this paper that vaccination performed in early adult life afforded a longer period of protection than when performed in infancy, and in both instances it may be surmised that as the natural susceptibility to smallpox increases from birth onwards till about twelve years there may be a greater loss of acquired immunity during the same period of life than is likely to occur when the natural protection against smallpox has become more definitely established. With regard to the protection which an attack of smallpox provides against death even less can be said. Of these fifty- one cases seven died. Five of these deaths, however, occurred in one epidemic out of twenty-three cases, and, although one of these five died from apoplexy, yet even four deaths out of twenty-three cases represents a mortality so much in Biometrika 1v 42 330 Smallpox and Vaccination excess of what might be expected as to make it probable that these figures are not sufficiently large to be taken as trustworthy. The actual protection, however, which one attack of smallpox affords against a second is much larger than that conferred by vaccination; this protection, judging from the Sheffield census, is very high, the correlation coefficient being ‘6, and this high correlation of protec- tion against attack suggests the probability of some error in the death rate as given above. The protection given by revaccination is still higher than that given by a previous attack of smallpox and by recent revaccination practically absolute. Of the 400,000 persons revaccinated in Glasgow in 1901 six were admitted during the epidemic of 1904; three of these stated that there had been no local reaction, they were therefore not effectively revaccinated ; one stated that there had been a local reaction, which, however, left no scar. In the other two the condition of the arm indicated successful revaccination. That is out of 400,000 persons three at the most were affected with smallpox after a lapse of three years, while cf the other 250,000 inhabitants of Glasgow more than 800 succumbed. In the statistics published in the London epidemic of 1902 the mean period which has elapsed between revaccination and the attack of smallpox (23-9 years) falls considerably short of the mean age of those attacked by smallpox in the same epidemic (28°6 years) with the protection alone of imfantile vaccination. These persons, however, must represent a more susceptible class, for, judging from the Sheffield epidemic, the correlation between the revaccinated who escape and those who were attacked and the like classes of the vaccinated is very considerably in favour of the former. TABLE XVI. Table of Revaccinated Persons Attacked by Smallpox. London, 1902. Age at Attack. a ee a ee ee re ee ee rae el! ees tet |e See eS3 9) SEE Cea ese Set Beye ees || Rss esy |p rss A 05 | —| — le FO: a oe ae a ilk es Please eee ee 9 S| 10-16 Fi |—| 2) 519) 4) FB) ye eae ‘el 16-20 - —) —| 9 6 1 9 tot) 85) elke alk enh hale or, 2 | 2025 1 Sea lero 1 12 = | 25—30 toe cat) sol eeea Ate |e ees ae 5 2 | 30—85 : 1) | PS ly eoelee Nl ie pee 7 , | 8—)0 casas) a 1 i DA este fa ee aaa 1 »| 45—50 Talo = 1 50—55 el 1a) | a 1 Ss epeeryolee| | || me hy REN ang a ee a GO— 65 Mf Nn ee ae ee 1 SS SS a EEE EE Votals (179) = |-3elc@ ja] 9 al ganl een on Neer ies | 1 | 92 JOHN BROWNLEE 331 In the accompanying table those cases of revaccinated persons attacked by smallpox during the London epidemic of 1902 are given; the number is ninety- two in all, and among these two deaths occurred, in both of which the period of the latest vaccination had been at least twenty-four years previously. Few inferences can be drawn from this table; it is, however, to be noted that more than a third of the cases occur in persons revaccinated before the age of fifteen years, who, as far as the statistics of this hospital go, constitute a very much smaller proportion of the total revaccinated persons, indicating again that the operation performed at this age affords less permanent protection than when done later. In conclusion, it is necessary to state that all the correlation coefficients in this paper were calculated by the short method given at the end of Prof. Pearson’s memoir on the correlation among attributes not quantitatively measurable * by means of the formula r=sine — where x? = dabod.N* : Beas ieere ~ (ad — be)? (a + d)(b +e)’ If the values of r corresponding to log «* from 1 to 4 be first calculated, as is easily done with the help of a table of Gaussian addition logarithms, then it is amatter of only a few minutes to obtain a correlation coefficient. Prof. Pearson has found this formula true in most instances within the probable error; I myself have checked it with about twenty examples and only once found it give a result outside that error, so for the accuracy required in this paper it may be considered sufficient. When the numbers are in nearly all cases small the probable error is large, and little stress can be placed on the accuracy of an individual coefficient. A far better test in a subject like this, when very large numbers of coefficients have to be calculated, is their general concordance. In many cases they form part of a series which is @ priori continuous, and consequently each affords a mutual check on the other of much greater value considering the number of cases involved than the calculation of the probable error. Considering the series tabulated in this paper, many instances will be seen where the coefficients fall so out of line with their immediate neighbours that it is evident that some large error is present. REFERENCES TO ORIGINAL SOURCES OF THE FIGURES ON WHICH THIS PAPER IS BASED. Douvittarp. Analyse et tableaux de linfluence de la petite vérole sur la mortalité 4 chaque Age. Paris, 1806. MacponEtt. Biometrika, Vol. 1. p. 375, Vol. 1m. p. 135. General Board of Health. History and Practice of Vaccination, 1851. Local Government Board. Epidemic of Smallpox at Sheffield, 1887-8. Royal Commission on Vaccination: Appendices on Epidemics in Warrington, Dewsbury, Gloucester and Leicester. Report of the Metropolitan Asylums Board Hospitals, 1870-1902. * Phil. Trans. Vol. 195, A, p. 16. ON TESTS FOR LINEARITY OF REGRESSION IN FREQUENCY DISTRIBUTIONS. By JOHN BLAKEMAN, M.Sc. B.A. In considering any frequency distribution of two characters, say # and y, one of the things to be determined is whether the regression curve of y on & is linear or non-linear. In his memoir “On the general theory of skew correlation and non-linear regression *” Karl Pearson has given the test for linearity of regression expressed in terms of the correlation coefficient 7 and the correlation ratio 7. It is there shown that, while is by definition an essentially positive quantity r may be either positive or negative; that ? and 7? are both quantities which can only vary between the limits 0 and 1 and moreover 7? is never less than r°; finally, that the necessary and sufficient condition for linear regression 1s 7? — 7? = 0. It has been remarked that r may be positive or negative, but it 1s obvious that whenever 7 works out negative we can, by changing the direction of either one of the axes of reference, make r positive. Thus, in all frequency distributions, the axes may be so drawn that 7 shall be positive. There is therefore no loss of generality in assuming 7 positive and, for simplicity of statement, in what follows r will always be taken as positive. With this convention we see that, if we write C=7?-7, S=n-7, 7= log. 5 the test for lmear regression may be written indiscriminately as €=0, or $=0, or @ =(, and in many other equivalent ways. Now, in any actual distribution, we shall not expect to find the quantities €, 3, w absolutely zero but, even when the regression of y on « is linear, we shall expect the values of ¢,, @ to be influenced in the usual way by the fact that our material is only a random sample of the general population. Thus, as in all statistical tests, having found ¢, or 8, or ow, to have certain positive values, we want to know if these values are such as might arise from * Drapers’ Company Research Memoirs. Biometric Series u. Dulau and Co. JoHN BLAKEMAN 333 errors due to random sampling from a general population whose regression is linear, or whether the values are such as to indicate significant non-linearity. For this purpose we compare the calculated value of € or 3, or @ with its probable error, where, if the standard deviations of ¢,93, @ are written X., 5, Le the probable errors, say H;, H,, Lg, are defined by the equation # ='67449%. While Karl Pearson has given in his memoir a formula for %,, and the formula for &, is well-known, no formula has yet been given for calculating any such quantity as =~, ,, 2 for the testing of linear regression. It is the object of the present investigation to determine such formulae; the obvious thing to do would be to determine X,, as € is the quantity occurring naturally in Karl Pearson’s work; but the formula for =, is so very much simpler that it has been made the basis of this investigation; it is then shown how to determine >, and , from &,, and other known quantities. Thus it is at the worker’s choice to test linearity of regression by comparing any one of the quantities €, 3, @ with its probable error. Following Karl Pearson’s notation Ne, will denote the number of individuals ‘ ‘ : ee ae te n : In the a,-array of 4’s, ny, the number of individuals in the y;-array of z’s and Nay the number of individuals in the sub-group common to these two arrays. JV is the total number of individuals in the table; %, 7 are the means of the two characters, while o,, o, are their standard deviations ; Yar, is the mean of the x,-array of y’s while on, is the standard deviation of that array and pyy, Am, oa 7,7 ) are quantities defined by the following equations, S’ denoting a double summation: Now =p {Nxy, (ap a (Ys oad yy", Nm =8 (ax, (Yu, yy", Nox? = 8 {nz, (Yu, — Y)"} (Lencu —We), ne Cy TxFy’ and further we shall define a quantity X,» by the equation Nw = 5" {ne, (@p — 2)" (Ye, — Dh The fundamental results on which our work is based may be stated as follows:— Let n,n’ be the number of individuals falling within any two mutually exclusive yi roups, then if &,,, », be the standard deviations of n,n’ and R,,,, the correlation n ) ny between deviations in n and n’ due to random sampling, we have 2 -_ Ly en! = ni : S2=0 (1 oF Swien (1 See (i), Sl Ban = LN aon eee ON (11) 334 On Linearity of Regression General Theorem. Let mn, n2,... be the number of individuals in any number of mutually exclusive groups and let quantities X, Y be defined by the equations X = Ayn, + Agn, + Agngt+..., Y= Bn, + Bn, + Byun, +..., where the A’s and B’s are quantities which remain constant as the frequency distribution is changed. Then &X = A,dn, + A,dn, + A367, + ..., OY = Bon, + B,6n, + Bon, +... Multiply these two expressions together, sum for all random samples and divide by the number of such samples; then using (1) and (11) we get immediately > XY o0t YyltyRyy=- Se + A, Byn, + ApByng + AgBytts +... ceseeceveres (111). Problem I. To determine the correlation between the deviations due to random sampling in the values of nz,y, and Yy,,. We have Ney Yay =S {Nay yYs}- Hence Ne OYxy =S {Ona u,Yst — Nay Yaeger ONx yy = ONx ye Multiply these two expressions together, sum for all random samples and divide by the number of such samples; then (a) p and p’ different Magus ay Yay ae Mays ay Yay 0 Ned > =— = i Noe sane Ys “Yat Beng ity N oe ee (iv). (8) p equal to p’ s >) = = = Pa Naty ~My 4 Vary Beg ye, = Nz yes Nz ys Ya, Nays (Ys Yay) ascites (v). We shall need, besides, the results proved by Karl Pearson as Propositions ITI, IV, VIII in the memoir previously cited, which we will state here for reference. Proposition III. Proposition IV. Ry, ng, =0(p’ not equal to p) ............ (vii). Proposition VIII. pa, Kay — 1) Son 4Nd, a N where y; is a quantity defined by the equation S (May Tne (Ye, a y)’} = No, (1 = °) om’ Xx xX and is obviously such that, in the case of normal distributions when ony is constant p and r=, we get y= 1. JOHN BLAKEMAN 335 Problem II. To find the correlation between the deviations due to random sampling in the values of oz and oj,. We have Nias = (Nx, (x, = x}, J. 2Nozdaq = 8 {Snz, (Xp — X)} — WHS (ne, (% — x). But S {nz, (&p— %)} =0. Hence 2Noxp8o,=S [Snz, (Xp — BY}. Again 2Noydsoy=S {Snx, (Yx, — 7} +28 {SYx,, Nx, (Yr. — yt. Multiply these two expressions together, sum for all random samples, and divide by the number of such samples; then using (iii), (vi) and (vii) we find ; No 2 oa? aa men AN? 620 yo, 20 yoyo y = ae aes ar S (nx, (%p a z)? (Yay _ yy} =— WM Oy oy” + IN Nese Hence 4020 920,04 Ro 01 = wan gens DOU DOU DOOROSOUONOCUD DUUOnGoOUDdO. Go (ix). Problem III. To find the correlation between the deviations due to random sampling in the values of p, and oy. We have Np = 8" |Nx,y,(% — ©) (Ys — YI, N8py = S' {Srx,y5 (% — 2%) (Ys — y)} — d@8" {Maeaug (Ys — y)} — dy’ [Neg (p — 2)}. Hence Nop = S' [6nx,y, (Xp — Z) (Ys — Y)}- But 2No dom = 8 (nx, (Ye, — Y)"} + 2S [Syx,Nx, (Yx, — Y)}- Multiply these two expressions together, sum for all random samples and divide by the number of such samples ; then using (111), (iv) and (v) we find N*pnow’ , : S = QIN Grp p. oad soar es ee a S {Mx (ap ~~ @) (Ys —_ y) (Ya, ae yy} + 28’ (Regus (@p — ©) (Ys — Y) (Ys — Yu) (Ye, = 2) But s {Maus (tp —@) (Ys — ¥) (Yx, —yya=S {Mays (@p — @) (Ys — Ya) (Ya, — yy} rs {Maes (%p — @) (Ya, = Op AD while S {ney, (Ys — Yx,)} = 9. Summing with respect to s and keeping p constant. Hence S" {na gy, (% — ®) (Ys — 9) (Yu, — Y)} = Ns. Again S' (Mz y, (Lo — ©) (Ys — Y) (Ys — Yup) (Yan — Y)} =S" (Mayu, (ep —F) (Ys — Yu)” Yn — Y)} +S" (Mays (% — B(Ys— Yury) (Yu, — Y)}- Sum with respect to s, keeping p constant. Since S Maays Ys — Yax,)} = 9, and S [ay (Ys— Yx,)”} = Ne, Fry» 336 On Linearity of Regression we get . oF [Rays (a, — 2) (Ys —Y) Ys — Ya) (Yo aa ”)} =S (Mii, ng * (x, — x) (Yr, = Y)}. We may write S (Mary Fry” (&p — 2) (Yx, = yD} = No; (lis n°) Pu xX V3, where y; is a purely numerical constant, which is obviously equal to unity in the case of normal correlation, when op, ?=o,?(1—1?) for all values of p and n =r. Pei Hence finally 1 2o4 De oapltn tan aaa N {Aas — Pu om” ap 2pn oy” (al — ”) Xs} aisvelsisteto tele Problem IV. To find the standard deviation of the values of @ defined by the equation a = log, — a7) We have o@ = log, a 5 or Hence See es sues etrsint tts satiet eee (x1) n 7 But n= od Oy r) r) $a, “ Hence scald = oo Yul x, eens een ee (x11), n om oy and r= Pu } OnGy . 6a, 3 8 = Opn = Lz End aN ye. (xiii). r Pu Ox Oy Squaring equations (x1), (x11), and (xiii), and forming the product of equations (xii) and (xiii), then on summing for all random samples and dividing by the number of such samples we get in succession 2 SS 2 s ate 5 _Babrby (xiv) pe . ———— ainjoisin'n'sle ei nisiein 0 aleinsn eal clarelouicisiateree ; gy Y nr 2 YY 2 ss ae = You mat) = 227 my Rey cy (xv) yoy oy oucy 2, =p ; Lon Xo, 2=, Yo ttp ox = WW ab. a Yas SA a) 2 (One Ox oP Pulte oy SS 22, Roy tty oy , 20, may Roxey (xv1) Poa rcistoitiere eee ; Puy Ox0y s SS Sys ahi) >, Roy - Xp, “oy Ry ox 5 ee Loy Pogo ay _ Yoy Xo yoyo nr Purtu OxOM Oyo Mu Zoy =p, Rep Yay roy Royo Xow ee — —2 uu Ser ee (xvi). Oyoy Oy CyPu JOHN BLAKEMAN 337 Adding together equations (xiv), (xv), (xvi) and (xvi) multiplied by — 2, we get ee mak + s a Dla a 2 2 2 Oomu Pur Ox Pula S 2p, Zener, ne ae Peg Pen Rego (xviii) Pulmu CxO M But we have as well-known results » — Px» — Pu » _ P10 — Pro" . Xp, Figen Ye, = Trai Poses a (Gab-8) — . y — Pa— PoPn xx). 22, Xp, Rox, igen ee (xx) Hence, substituting in equation (xviii) the values given by equations (vil), (ix), (x), ae and (xx), we get — A? 1-7 | Px- Pu? , Pw — Po Pa —PwPu , Aw—Fa'Ae 2— i 2 Ea ee, hate. ie a aa +e Np * 4Npat Npopr | 2Nog ry 1 , : = Np X {Nig — Pure + Que, (1 — 7’) x3} aeeitele (xx), ee which may obviously be re-written s 2 ue 1 (= *) Ps =" Poo” _ Pas BPoPu +(e 5 a Opie = 1 aml + N4 —— Nee n° n aN Ne AN Woo? Np Pu Np, Nr? ANKE is — Bands, Noy — Boa" Aa i NryAs 2No77r. This is the complete formula for =, to come which has been the first object of the present investigation. The formula has been thrown into the form given as equation (xxii) because each term as there exhibited vanishes for a normal distribution, since for normal surfaces we have {(%1 — 1) -2(y%3—-1)} «ee. (xxii). — 3 ac 1 — v2 Ps = Spo", Pa = 3PoPu> ee Me = i . Pu” fe Again for a normal distribution ee Ya, ¥ ad on, (&p @) ne ro," i Hence Nr, =S8 {nz, (Yx,— Y)'} = = S (nz, (%» — @)4 ded . ="°" 3Noz! = BNA, Cx Nis = 8 {10x (@p — B) (Yu, — Y)} = BOW 3 No =O . 10. X Ton6, — Ol. Ag. - ro Nn = 8 {nz (Xp — 2) (Ya, — YY} = -. 3No;z! == DIV TON fg — GIN Ng Oa Further y,=y,=1, while 7 =r. Biometrika 1y 43 338 On Linearity of Regression ia): Now the ay mula for ~,? consists of a term a involving r only together with a number of terms involving product moments which are just such terms as the product moment terms of equation (xxii)*. Again the formula for &,? consists i of a term involving 7 only together with a number of terms of exactly the same nature as these moment termst. In practical statistical work the simple _ 72/2 | ay seuss . formulae = peat W ; ee are used and the nature of the justification for these simple formulae can only be appreciated by a reference to the numerical work of Karl Pearson’s memoir where he shows that, as a matter of experience, in actual frequency distributions which are very far from normal, although the separate product terms occurring in the formula for &,? may be significant, yet, these product terms occur in such a manner in the formula as to be insignificant in the aggregate. This work suggests that we should consider if anything of a like nature occurs in the case of the formula for {,,. Equation (xxii) gives &,,’as 1 LN ; the sum of a term =, (=- =) involving » and 7 only and a number of product Un V moment terms of exactly the same type as those occurring in the formulae for x7, =,°. Isuggest therefore the formula eral Ci ie eS NW {3 - | sieiwiuitnns fete. eee (xxiii), as being exactly analogous to the simple formulae for ¥,?, 5,?; the complete justifi- cation for the formula (xxii) will be found in the practical statistical work given later in the paper. Before proceeding to the numerical work I will give some formulae which enable us to determine other quantities of importance when &,,? is known. Problem V. To find the correlation between the deviations due to random sampling in 7 and r. a ee We have a = =n aa ea -_ 23, 27m y pe nr Xn >, ' vipa s ee) ear (1—R&,,), o> > > > 2 or os (1 — By) = Ye? — @ = =) eer (xxiv). This equation gives the numerical value of R,, as soon as 7, 7, >,, Zn» Le have been determined, no matter in what way. Assuming the simple formulae for ¥,, %, and the formula (xxiii) for =, we get a corresponding formula 2nr (1 — 7?) (Ll — 7) (1 — By) = 9? — 2? —(n—ry (1 t+ qr)... (xxv). * Phil. Trans. A, Vol. 191, p. 245. t Loc, cit. p. 20. JoHN BLAKEMAN 339 Problem VI. To find the standard deviation of the values of ¢ defined by the equation C= — 7 We have 18 = bn — ror. Hence 1 2=9'D,? + P32 — Anrd, Brae But Se Bains Bet Lie w n r? nr ee ep ae NT Dy Sp lve Hence fo =P =P) SP — SPAT eg cevdsssevveeves (xxvi). As in equation (xxiv) this is a formula always applicable. The formula for =, which corresponds to equation (xxiii) for &,, is therefore sb) ms ae Le dee ee 0 See ee (xxvii). N Problem VII. To find the standard deviation of the values of S defined by the equation S=n-7. Proceeding exactly as before, we get Ag =™, 4D = 22, Selb ye giving HP (YT) (Te — dette cise een een stars (xxviil), applicable in all cases. The corresponding simple formula is NV s? = a a i es Oe) cet ene ee eee (xxix). Statistical Illustrations. I proceed to discuss, by appeals to actual statistical experience, the degree of closeness to which the simple formulae (xxiii), (xxvii) and (xxix), may be used instead of the complete formulae (xxii), (xxvi), and (xxviii). If, as regards any frequency distribution, we have tabulated for each #-array of y’s the Nx,» Yx, aNd oy, , We may proceed to calculate 2, from the formula (xxii) @, : without further reference to the sub-groups of the correlation table. This follows at once since we have, in general, Nomi =) {Ne os (@p — nym (Ys — . (x, yy (Bp — £)" (Ys — Pe) +8 [Ne os (Gaye (Yap — =) [since S {nx,4, (Ys — Yx,)} = 0), hence Dink tise Nani Bee reden da es nese con evemesnnesws vite (xxx), 43—2 340 On Linearity of Regression Also we have Npmz=8 {rx,4 (,—@)™ (ys— WY} =S {nay (Wp — 2)” (Ys — Yay + Yn — YY} = 8 (May, (%p—F) (Ys — Yu,)} +8 (Magy, (Mp —£)™ (Ya, — YY} +28 {ragy, Yo 2)" (Yr D) (Ys Ya) =§ (Mai, Fng, (a) — £)™} + Nine, It i é hence Dmna= Mme apd {Nx,on el Cee 9) Fao se concise bas oseroacsocnc Coca)! p N Thus the formula for Y,, contains only X’s and y,, y; which, by their definition, are calculable when Rugs Ytpr Tng, ae known for every array, together with product moments of the form Pm, Pm2 Which are also calculable if we use equations (xxx) and (xxx1). In his memoir Karl Pearson discusses four actual frequency distributions which exhibit skew correlation and which he names illustrations A, B, C, D. For these distributions he has given the numerical values of many of the statistical constants and has also tabled the Naas Yap» Ong? for each array, to four decimal places in the cases of illustrations A, B, C, and to three decimal places in the case of illustration D. It is thus relatively easy to calculate &,, from the formula (xxii) for these four cases. Starting with the tabled values of Nazar Yargs Try » I recalculated ‘p all the constants necessary, keeping six decimal places throughout the arithmetical processes and using Sheppard’s corrections or not, according as Karl Pearson did or did not use them in his work. I have tabled all the constants to four decimal places, but, whenever any constant was used to obtain a new quantity, its value as obtained by me to six decimal places was used. Although Karl Pearson obtained his constants by a different arithmetical process, and his results are necessarily more accurate, since he has kept six decimal places throughout, yet, wherever the values of constants given by my work may be compared with his, they agree to at least four decimal places excepting in the case of illustration C,in which the general agreement is only as far as the third decimal figure. This illustration has given throughout more irregular results than the others, due probably to the disturbing influence of the ecdyses. Having obtained the necessary constants I proceeded to calculate 2, for each case from the formula (xxii). It remained to test the closeness to which the simple formulae (xxvil), (xxix) might be used instead of the formulae (xxvi), (xxviil) for X¢, 2,. For this purpose Karl Pearson’s four illustrations are again extremely convenient since these are the only published frequency distributions for which &, has been calculated from the complex formula. In each of the four cases I substituted in the formulae (xxvi) and (xxvill) for =, =, the values found from the complex formulae, and JOHN BLAKEMAN 341 : 1-r’y for =, the value found from the usual approximate formula ¥,? = i Thus values for =, 2g, 2 were found which are certainly true values excepting in so far as the formula for ,? is only approximate. Writing £,, H:, Hy for the probable errors thence deduced we have EE, ='674492,,, H,='674492,, HL, ='67449%,. The standard deviations and probable errors were now found from the simple formulae (xxiii), (xxvii) and (xxix) and the resulting values denoted by 2’,, ¥’¢, &’s, Ge 6 Sy, Dies +f 7 ie in or non-significance of the deviations from linearity judged by the values of these ratios. EK’ 4, E';, E’,. Finally the ratios were calculated and the significance Exhibiting the values of the constants thus deduced; we have Illustration A. On the Skew Correlation between Number of Branches to the Whorl and Position of the Whorl on the Spray in the case of Asperula odorata. The position of the whorl is the #-character and the number of branches to the whorl is the y-character. Taking the values of Migs Yap» Trg, AS given by Pearson, we have Ngo Yap rep ay 150 6°7800 "8553 a 150 6°8133 8437 L3 150 6°8133 9047 v4 142 6-4859 ‘8780 Xs 87 6°1724 "8605 We find, the quantities being determined without the use of Sheppard’s corrections, VON, eo =28027, y = 66554, oz, = 1:3369, oy = ‘8978, Te = Ilse, #, = °8061. vy =5'8417, oy =r, = ‘0503, Pu = ‘2492, hy = “0075, Ps = '8964, As = 0342, Pu = 1°4906. he wlono oy = 2244, n = ‘2499, 7 = 2076, Xi — ‘9909, X3 = ‘O771. 342 On Linearity of Regression Substituting these values in formula (xxii) we may exhibit the result so as to show the numerical Saar of each term in the form 2 = 1 iy: 1970 — 2928 +°9870 — 1:1980 HW — ‘0129 + 2704 — 5673 + 5492} =; {71970 — -5088 — 3099 + 5492} 1 aa ee 1970 — 2544} = = 5, (6:9426} Srastgatumeene (xxxil). This gives > — Lom i ="0682: Pearson gives E, = 0242. Hence >, = 0359. F : 1 ; ae (1 = ry rom the formula meer re >, — OOlS: Hence from formulae (xxvi) and (xxviii) we get 2-= ‘0108, 2, = 021% Hence, finally #, =:0682, E, = 00738, E, = 0147, while, from the simple formulae (xxiii), (xxvii), (xxix) we get as = 0694, £’,=°0071, f= “0150: F g J oreo: These give Br =O (PASy, E’,7 = 2°7389, E,= 2°8236 Illustration B. On the Correlation between Age and Head Height in Girls. Age is taken as the #-character and height of head as the y-character. We have in this case Ly Nz, Ly Cia oa) 1 115-2500 00000 y6 7 116-9643 5°7706 ben 18 117-4722 58552 Can 40 119-1000 59282 (es 76 120°3026 59764 —_ 125 121-6340 52732 9—10 7 121°7246 6°7754 1013 235 122-8160 59306 11—12 261 123°1427 64178 12—18 309 123-8908 6-4122 13—1h 263 124-8622 6°7178 ihpaaae is 198 125°7146 7°1730 15—16 214 126°1565 6-9326 16—17 162 126-5340 7°7392 17—18 95 126-9132 6-3358 e719 61 127-0205 6-2470 19—20 13 129°5577 9-6812 20-—21 7 123-8214 50622 21—22 8 126°5000 8-2828 22-93 2 125-2500 1°9148 JOHN BLAKEMAN 343 We find, using Sheppard’s corrections throughout, NV = 2272, Cal 00K, y = 124-0467, o, = 3:0648, oy= 69083, by — NS Naille fa = 47-7239. V4 = 259-050 1 owf=r, = 43822, n= 62274, Nie 62:3991 Da = 1488952, Xi = 80°4155, Dum = 5101883. eae 10 T3209, oy = 2:0934, 7 = °3030, r = ‘2941, See MER, Ys, 0 0De. Hence we deduce from formula (xx11) Lia =, (6688 — 0724 + 4546 - -4035 + 0623 + 0533 — 1964 +0867} =5 {6688 — 0213 — 0808 + :0867} i, aed, ete a al 6688 — 0154} = Ww! Goaw take wesaeseceanes (xxxill). This gives pe — 010) FE, = ‘0114. Pearson gives E, = 0129. Hence > — O19 I. From the formula f= a8) ; N = = 0004. Hence from formulae (xxvi) and (xxviii) we get = = 0030, Se UDI Hence, finally he —AQU A. E, = +0020, EL, = 0034, while, from the simple formulae (xxiii), (xxvii) and (xxix) we get EF’, = 0116, EB’. = 0021, E’, = 0034. These give ae —— Ae 52 ae By 9-597 RE’. 2 Diag EY 2°5879, E’, 5973 344 On Linearity of Regression Illustration OC. On the Skew Correlation between Size of Cell and Size of Body in Daphnia magna. Length of body is taken for the #-character and size of cell for the y-character. We have a, Mage Yary "re, | 1 20 53000 0:9092 | 2 150 5°8333 16620. | 3 100 7°7900 1°6860 4 80 80500 21539 5 150 94733 26694 6 140 84357 1°7087 7 230 85957 1°9384 8 120 10-2667 1°9439 9 180 10°7611 24469 10 150 11-0267 26949 11 150 10-9533 2:4470 | 12 140 91000 1°7947 18 120 90000 1°8619 lh 220 10-0364 1°8894 15 60 10°3167 1°5165 We find, using Sheppard’s corrections for the cell sizes, but not for the body lengths, N = 2010, & = 85025, 7 = 92687, ao, = 3'8648, Cy= \2DAlT, V, = 149366, fo = 64604. vs = 432-7695, Pals Ale: pn= 38929, Ma = 151375, Pu = 127°3508, Ay= 27-8071, ‘Don = 108'1612, hog = BO3594, Base) n = ‘5720, r = 3963, x, = 8918, Vs— coud Substituting in formula (xxil) we get Sq? = 7 (BB108 — 2651 +8098 — 12303 + ‘0971 — 3796 — 6074 +°7752} _ = (33108 — “6855 — 8898 + °7752} = {33108 — 8001} = ad AZ TOLOG): “eeectencice savers (xxxiv) W wy 25106}... This gives P0308, Lg O20: Pearson gives E,, = 0097. Hence Dy = 0144. JOHN BLAKEMAN 345 , dl-rp From the formula ye a NW ) ; > 0004, Hence from formulae (xxvi) and (xxvii) we get de = 0125, >, = 0138. Hence, finally Eg = 0238, Ei = 0084, E, = 0093, while, from the simple formulae (xxiii), (xxvii), (xx1x) we get Lae 2 As E’-=°0107, i’, = 0114. These give aa = 13-4052 £ § 15°8934 * _ 153972, Ee : i’. , ve Illustration D. On the Skew Correlation between Number of Branches to the Whorl and Position of the Whorl on the stem in Hguisetum arvense. Position of whorl is taken as the a#-character and number of branches to the whorl as y-character. We have v er Yay ‘Ria | _ | 1 126 7-619 2360 2 126 9294 1/273 3 126 9°627 1°187 y 126 9°730 1151 5 126 9-643 1°158 6 124 9-427 1°375 7 123 8732 | 1°78 8 121 7297 | 2-291 9 119 5555 | = 2558 10 110 3964 2199 il 97 2-443 1°506 12 67 1°866 0-960 13 39 1-462 0-746 | i 12 1°333 0-471 | 15 4 1-250 0-433 16 2 1-000 0-000 | Not using Sheppard’s corrections for either character, we find N = 1448, ze = 64033, y = £2169, Oz, = 3'5426, C—O, Ye = 125500, fig = 10°7486, Vy = 3195158, Cie — he oa, Dy = 32256, ry = 140°4409, psx = 2050840, yg = 152°2416, Poo = 2052746. Ae = 171:1280, oe = 27899, n = 8510, r = "7082, Xi = 6437, xs = "1992. Biometrika 1y 44 346 On Linearity of Regression Hence from formula (xxii) we have Sj? = + (6128 — 2428 + 1:0134 — 9598 — +1705 + 9708 — 6241 +0173} = 5 (6128 — 1893 +1762 +0173} = + §.19814:°0042} =~ (6170) cee (xxxv). N N This gives = — 0206, Ey = 0139. Pearson gives E, = 0054. Hence >, =, 0080! From the formula ye a NV > =:0002. Hence from formulae (xxvi) and (xxvill) we get De 0229) Ds = 0153. Hence, finally Eg = 0139, E,= 0154, E, = 0103, while, from the simple formulae (xxiii), (xxvil) and (xxix) we get i = 0139: E’; =:0152, B= 0102; These give 7 2132341, ~ =146291 * 13-9785. Ee 3 EL’, , E’s, In order to appreciate the significance of the results in these four cases they are collected and tabulated as follows: [ Note : @w=log, 7 logy Ty log, 10] ip Uk iS aS E E E al 7 77 or ¢ % E E ¢ E mn E' 0694 | -0071 | -0150 A "1856 | °0194 | -0423 | 2°6728 | 2°7389 | 2:8236 E 0682 | -0073 | -0147 E'~E | :0012 | 0002 | -0003 £' "0116 | 0021 | 0034 B 0298 | -0053 | -0089 2°5752 | 2°5879 | 2°5973 E 0114 | -0020 | -0034 E'~E | :0002 | -0001 | -0000 yo 0238 | -0107 | -0114 C 3669 | *1701 | °1757 | 13°4052 | 15°8934 | 15°3972 EL 0274 | :0084 | -0093 E' ~E | :0036 | -0028 | -0021 Et’ 70139 | °0152 | 0102 D 1836 | *2226 | +1438 | 13°2341 | 14°6291 | 13-9785 E 0139 | 0154 | -0103 E'~E | :0000 | :0002 | -0001 JoHNnN BLAKEMAN 347 We conclude from this table: I. As regards the linearity of the regression in the statistical examples chosen, illustrations C and D are certainly skew. The skewness of these examples is obvious either from a glance at the diagrams accompanying Karl Pearson’s memoir or from his discussion of the cubic and quartic regression curves for these cases. Illustrations A and B give oa = 2 values between 2 and 3. The w $ > limiting value of these ratios which should denote significance is a matter which can only be decided as a result of large statistical experience. When the ratio gets as low as 2 we are getting on the border line and for cases A and B we can only say that the skewness is probably significant. This is the kind of result we should expect from the corresponding diagrams of Karl Pearson’s memoir. He finds, however, that he obtains so much better fits by curves of higher order that he concludes that the regression of A is parabolic and of B cubic. II. If we refer to equations (xxxii), (xxxili), (xxxiv), (xxxv) we see exhibited the numerical contribution of each term of formula (xxii) for =”. As regards the numerical values of the quantities which vanish for normal correlation we see that they by no means become zero in the examples chosen. Also in these equations I have grouped together the three p terms and the three X terms, but the numerical values of the separate terms are still considerable. When the whole expression for =,” is summed however we find that the last three terms of these numerical equations so nearly cancel that the difference between E, and EL’, is at the most ‘004, a quantity which is of no significance in the value of a probable error. This is exactly the kind of result Karl Pearson found in comparing the values of #, as found from the complex formula and the simple formula. There is thus exactly the same justification for the use of formula (xxiii) as for the formula >,? = eae Comparing the values of Z;, H, with E’;, #’, we see that the agreement is closer still, the difference being at the most of order ‘002; and if we except illustration C the agreement in the values of H,, #, is remarkably close, being of the order 0003 at the most. Thus our numerical work leads us to the conclusion that we have the same justification for the use of the simple formulae (xxiii), (xxv), and (xxix) for E,, E,, Ey as for the use of the accepted simple formulae for H’,, ,. These simple formulae have been obtained by making approximations of a statistical nature, i.e. approximations suggested entirely by statistical experience apart from arithmetic. I proceed to examine the formulae thus obtained to see if any simplification can be made by arithmetical approximation. 442 348 On Linearity of Regression Accepting the simple formula (xxiii) for 2, we get flew x [ve +9] yso7 lied as 1 r : (r +3) aay : ae : Assuming ~ <1, we can expand this expression in a series of ascending powers a of S, and, neglecting cubes and higher powers, we get Proceeding in exactly the same way from equations (xxvii) and (xxix), we get o ry SF & A WSe7 pot ae a Rte) gal Ol less gs) RPC a onror acts o> (xxxvil), S27 Se = aby Vs2" 2 ee 6 (Le Be) CL 7) coats ce (xxxvill). Sis : : When = is small these approximate formulae will be very nearly accurate, and 7 they exhibit the arithmetical reason why We —_ = w ¢ 9 and, moreover, we see in what manner they will differ when they are not equal. work out nearly equal oe a: In our statistical examples = is small and we should expect from these second 7 order approximations that Loy ee would be greater than , which will be coy K,’ Ee 5 Ee 3 ees ; seen to be the case on referring to the table of values. Again, —- = acd according 5 2 K Re i ¢ 2 ; Cee . 1 & : as 143r° = 8rz,ie.as1= 5r?; if rv? 1s nearly =, =, will be nearly equal, but = < oO iE > us BY; we cannot say which will be the greater as the third order terms are then likely to become important in determining the difference. These conclusions are seen to be borne out by the table of values, and other peculiarities in the values might be explained in the same way. When 7s small, equations (xxxvi), (xxxvu), and (xxxviii), obviously suggest the arithmetical approximation since aa JOHN BLAKEMAN 349 I append the values given by (xxxix), for the four statistical illustrations : A B C D S = 2039 0302 4433, 2016 rp VN og me Seas | 2-688: 2°5 3°7078 | 13°3086 coun aaa 6883 2°5711 || 13°707 Even when E is not very small re 1 EK’, 67449 V1 —2 (72-1) 4+7'-1 and it is obvious that if there be any doubt as to linearity of regression, when E See EL’. 67449" good arithmetical approximation for equation (xxvii). 14/¢ will still be a is not small, then r and 7 must be both small and General Conclusion. Thus to test whether the regression of any frequency distribution is linear or not we have three possible methods, viz.:—to compare the value of either a, €, or S, with its probable error. As regards the best quantity to choose for this we may first say that @ has no advantages whatever. In favour of the use of S we may say that the terms of S are just 7 and r, which are the quantities in terms of which we naturally judge our distribution; we must be careful however in using formula (xxix) for ~, to remember that 7 is supposed positive ; i.e. if r is negative for any case we inust change its sign before substituting in the formula and we then get the probable error of the difference between 7 and the absolute magnitude of r. € however is the term occurring naturally in the work and is the one quantity of the three with a direct physical meaning; i.e. gives the mean square deviation of the distribution from the regression line. This will probably be sufficient to assure that tests for linearity will be conducted in terms of €. Finally, (i) A simple test for linearity of regression which will be sufficient in very many cases is 350 On Linearity of Regression (ii) For more exact work cpus should be calculated from the formulae Ee’ by ees WE 1 EH, 67449" * ‘W140 —7P-G—ry’ S VN ie Vdegr 1 . EE, 67449" 7 >’ n +r a rdl-7P—7nd-ryp T+ Mast h Each of these results gives an approximate formula for the same physical quantity (i.e. the number to be looked up in the table of the probability integral if we want the probability in favour of the distribution being linear). If these ratios work out sufficiently nearly equal this constitutes some Jjustifi- cation for the statistical approximations made in obtaining the formulae, but if on comparing the values of the ratios the agreement is unsatisfactory, and no mistake in the arithmetic can be detected, then the probable errors of Hz, EH, must be calculated from the complete formulae (xxii), (xxvi), (xxvill) In terms of the subsidiary quantity » and substituting for &,? its value as determined by the complete formula given by Karl Pearson. THE LONG BARROW AND ROUND BARROW SKULLS IN THE COLLECTION OF THE DEPARTMENT OF COMPARATIVE ANATOMY, THE MUSEUM, OXFORD. By E. H. J. SCHUSTER, M.A., New College, Oxford. THE following work was undertaken at the desire of Professor Weldon. It is the first part of a descriptive catalogue of the British skulls in the collection of the Department of Comparative Anatomy, Oxford, and is an attempt to lay before the public in an accessible and partially digested form the material in this collection. Owing to the small number of the measurements of each group which it has been possible to take, the statistical constants deduced from them are not of very much value as they stand, but it is hoped that similar catalogues of other collections may be published which will enable one to deal with larger groups. Professor Weldon and Professor Karl Pearson have written a descriptive catalogue of the skulls, which has been based to a certain extent on the manuscript catalogue prepared some years ago for the Department of Comparative Anatomy by Mr Hatchett Jackson. The account of the anatomical peculiarities was most kindly provided by Professor Thane. Professor Thane’s determinations of the sex have also been adhered to throughout. The following measurements were taken to such extent as was possible in each individual case. A. CRANIAL. LT’, Horwzontal length. This was measured with instruments described in Miss Fawcett’s memoir (Biometrika, Vol. 1. p. 408). L. Greatest length measured from the glabella to occiput. L’. Greatest length measured from the nasion. F. Ophryo-occipital length. Flower. Greatest length measured from the middle point between the frontal prominences. 352 Long Barrow and Round Barrow Skulls B. Greatest breadth measured outside the squamosals. B”’. Auricular breadth of Virchow—distance between the upper borders of the auditory canal. B’. Smallest frontal breadth. H’. Height. From the anterior border of the foramen magnum to the point on the surface of the skull, in a line vertical to the horizontal plane. H. Ausiliary height. From the anterior border of the foramen magnum to the bregma, i.e. the juncture of the sagittal and coronary sutures. OH. Auricular height, measured with the vertical scale and sliding rod of the craniophor (Biometrika, Vol. 1. p. 414). LB. Basicranial length, measured from the anterior border of the foramen magnum to the nasion. fml. Greatest length of foramen magnum. fmb. Greatest breadth of foramen magnum. U. Horizontal circumference, measured with the steel band measure over the superciliary ridges in front and the most projecting part of the skull behind. S. Sagittal circumference, measured from the nasofrontal suture to the posterior border of the foramen magnum over the sagittal suture. Q. Vertical transverse circumference, measured from the upper border of the ear opening of one side to that of the other, in a plane vertical to the horizontal plane. The foregoing measurements are described in the Archiv fiir Anthropologie, Band xv., Braunschweig, 1884. B. Factat MEASUREMENTS. GH. Upper face height, measured from the nasion to the middle of the central process of the upper jaw between the middle incisor teeth, (ie. the alveolar point). GB. Face breadth, being the distance from one zygomatic maxillary suture to the other. The measurement must be taken from the lower end of these sutures, from the lower front rim of one cheek-bone to that of the other. J. Bizygomatic breadth, from the outermost point of one zygomatic arch to the like point on the other. NH. Nasal height, measured from the nasion to the lowest edge of the pyriform aperture. ; NB. Nasal breadth, greatest breadth of the nasal aperture, wherever it may be. O,. Breadth of orbit for both left (LZ) and right (A) eyes, measured from the “point lacrymal” on the inner side to the inner margin of the orbit on the other side, along a line parallel to the general slope of the orbit. Plate i. Vol. IV. Part III. Biometrika. Song Barrow Crania. N. verticalis. Vol. IV. Part III. Biometrika. Round Barrow Crania. N. verticalis. v a SE . S1pesaze| “N ‘eves MoLeg Huoy Il eteld IM Weg “AL 1OA “e4l4zowWoIg Vol. IV.) Part Ill. Biometrika. Round Barrow Crania. N. lateralis. Biometrika. Vol. IV. Part III. Plate v. Round Barrow Crania. N. facialis. Biometrika. Vol. IV. Part iii. Plate vi. Round Barrow Crania. N. facialis. *j E. H. J. Scuustrer 353 “Point lacrymal” is the point at which the hinder edge of the lacrymal groove meets the lacrymo-frontal suture. O,. Greatest height of orbit, for both (L) and (R) eyes, taken as closely as possible perpendicular to O,. G,. Length of palate, measured from the point of the Spina nasalis posterior to the inner wall of the alveolar rim between the middle incisors. G,. Breadth of palate between the alveolar walls at the second molars. GL. Profile length from the basion to the alveolar point. W,. Condylar width. Greatest width of mandibles at condyles, from the outside of one condyle to the outside of the other. W,. Angle width. Greatest width of mandibles at angles, from the outside of one angle to the outside of the other. h,. Greatest height of mandible, measured from the lowest median projection to the top of process between middle incisors. Jf. Distance between foramina mentalia. C. INDICEs. B/L’. Cephalic index for horizontal length. B/L. Cephalic index for greatest length measured from the glabella. H'/L’. The ratio of height to length, for horizontal length and the whole height. H/L. The ratio of height to length, for the greatest length measured from the glabella and for the auxiliary height. G’H/GB. The upper face height to face breadth index. NB/NH. The nasal breadth to nasal height index. 0,/0, (R), 0/0; (LZ). The orbital height to orbital breadth index for both right and left eyes. G,/G,. The palate breadth to palate length index. P. The profile angle found with Ranke’s Goniometer. The description of the facial measurements, and of the indices calculated, is taken from Miss Faweett’s paper on the Naqada Crania (Biometrika, Vol. 1.), and the lettering used in the tables of that memoir to denote these various measure- ments and indices has been retained. But it will be noticed that certain of the measurements have been omitted, and others slightly modified. Thus owing to the imperfect state of the teeth, in many cases, the face height was not measured. Also the orbital breadths have been measured from a different point on the inner side as it was found more convenient to do them in this way. Owing to the few cases in which these determinations could have been made, it was not considered worth while to calculate the angles A 2, NZ, BZ, 0, 0. The measurement of the capacity was not attempted owing to the fragmentary nature of the majority of the skulls*. * The following six were found fit for the measurement of the capacity which is given in cm.°; e 24, 1606. 26, 1496. 388, 1592. 83, 1667, 173, 1437. 180, 1502. Biometrika 1y 45 304 Long Barrow and Round Barrow Skulls Table I. shews the means of the measurements thus taken, with the number of measurements in each case. Tables II. to IV. give the actual measurements. Plates I. to VI. give characteristic specimens of Long and Round Barrow Crania with a few abnormal cases. TABLE I. Showing means for each group of skulls and each measurement, with number of observations in each case. Long Barrow Round Barrow | Measure- aT : j ment Number | Number Number Number Mean ¢ |of obser-| Mean? jofobser-| Mean g |of obser-| Mean? _ | of obser- vations g vations ? vations ¢ vations ¢ L 191°88 8 185°33 33 ite fa9e 75} 8 179°45 ll L 190°56 16 182°61 13 188°61 36 177°97 34 JB 188°44 16 181-50 12 185°038 32 175°88 30 fF 187715 ANZ 183°96 12 187°01 37 178°63 35 3 142°44 18 138 °62 12 144°77 31 136°56 34 Bt PANES Yi 16 116°94 9 121°68 25 113°64 22 B 98°94 16 94°09 | ll 98°27 32 95°98 29 HH’ 138°44 9 138°83 | 33 138 3 136:°00 2 Jab 137°83 12 135711 | 9 135°94 16 130°80 15 OH 120°67 9 118-00 3 123 °62 & 115°91 11 LB 101°95 ll 96°81 | 8 103°94 16 99°33 12 tml 35°68 Wil 34°50 | 6 36°97 15 35°44 9 Jmb 27°73 1! 30°25 6 31-00 15 28°79 7 U 534°87 16 alsiazal a 587°85 27 506°43 30 S 384°77 13 382°00 8 376°53 15 365°30 15 Q 321°78 9 312-00 3 333 °38 8 313°54 ll G" HH 69°92 13 66°75 4 70°44 16 66°20 10 GB 95°88 12 92°75 4 97°42 12 90°29 a if 134-00 3 132°5 ] 133°50 3 _ — NH 49°43 15 47°00 7 50°18 14 48°22 9 | WB 24°07 1b) 22°79 if 24°61 14 23°50 9 0 (LZ 38°12, 8 36°87 4 39°03 15 38°78 9 Oe. 38°28 9 37°87 4 40-00 21 39°35 10 0 (L 31°81 8 30°62 4 32°46 14 32°67 9 Ne 31°28 9 30°87 4 31°79 21 32°19 8 GL 95°33 9 92°62 4 96-00 8 93°43 7 Gy 55°31 8 57°00 1 55°59 11 50°69 8 Gy 43°70 5 36°00 1 40°73 15 39°11 9 Wy 121°21 uf 113°83 3 123°38 13 117°35 10 W, 104°90 10 92°92 6 105°04 24 94°85 13 hy 34°13 15 31°80 | 5 33°83 21 30°89 17 af 45°67 15 8535 | 3 45°23 22 43°69 16 BI’ 74°38 8 14°33 5 3 77°88 8 76°36 11 BIL 74°93 16 76°33 | i2 76°70 30 77°73 33 H'/L' (2°25 8 74°66 | 3 73°67 3 74:50 2 A/L eS 11 74:00 | 8 72°81 16 74°85 13 GH/GB 71°44 9 81:00 1 72°83 12 73°50 6 NB/NH 49-00 15 49°10 | 6 49°14 14 48-89 9 0,/0,:L 83°37 8 81°66 33 82°87 15 84°22 9 O,/0,:R 81°89 9 80°66 3 79°60 20 82°12 8 G's/G, 78°40 5 oo — (3°82 = | 11 76°75 8 P 83°00° 5 — — 84°14° Ul 84:00° 6 { TABLE II. S | a ela atoll Wel — 7O — —— 74 74 71 7i — 76 = 72 — 76 os 68 — 86 76 == 81 75 | | i | | W,| m | ¢ | BL’ | BL | wy/L'| BL G'H|GB| NB/NH | O21) Osf01 G/G,| P | eae Sie. | | | OAK) eee el eee ae | | al = eee some os pe, 69 — ee = ee osea aoe le | | TUS SS (5m [4 05ie| = = = = | | — = Sara ae oat Oz y |) "71, 69 ee al = 44 ee hee | ma 71 71 72, 72 = 43 88 89 oe = 72 2 71 71 65 | 45 82 | 8&3 86° = = =a = = = = jf || HK) 80 | 78 — | 84° 73 72 71 70 = | es 89 | 87 = = — | 38 42 = = | 45 Enea sees = | ala, eee hese SP eileen Sete fee kl ei | eae OSE) | 37°5° | 45 agp eA Ot emma ia 75) 0 245 Ec ed | Borsa Ste Mi AOL || == |) 206 | sae ||). 33 G45 73% 1 R73) #6974, | 189 (0p) wat 85 of == | OT DEZES WEGOsG i AO Swi 72. Nee a) 7208) 973 71 55 PM OS | ee LOPS SO Su SL Gn eyy® | vere |. 71 71 78 48 Sete dea eS 97°5 | 32°5 | 43 Sat Roce. eae te eee Se ae eae TOS AG) SES!) 45 See pil, LA eed ge Le es 77 | 78 |. 84. | = 99°5 35 46 80 81 78 79 67 sé 76 82 83° LOOM Nese S Sila | Odea a ar | 72 79.9 1 253 a Mesee nee 102°5 | 32°5 | 48 77 ae TOs cyan) 10S 53 82 wel CC at ie ears | | ; Ge a 50 == 79 aad a 81 48 81 80 — LONG BARROW. d. TABLE I. | Number MEASUREMENTS OH { LB sme | fad NH W, 197°5 192°5 184 185 181 188 176'5 | H'|L', H/L G'H|GB| NB|NH | 0,/0,| 04,0, Lb | # ody 190 182°5 183 175 188'5 1715 184°5 175'5 SUuFSl lt | Sal 358 399 | 325 321 333 346 Wet ty 55 oe & eA <3 | x o Sil on LONG BARROW. @. 2B | — 75 ‘izs| 130°5 101 | - | | 502 | | | | = 81 = 24187 187 "187°5 138 | 119 | 92 | agt's|} rg | 118 | r00'5 | 34 | 31 530 | 389 | 311 | — | 92°5 | 1325) 49 | 24'5 | 365 | 37°5 | 31 | 31 os 74 | 73 | 78 | 74 = 50 | Bae] |) = 26 181 186°5 | 136 | 113, | S9°5 |) 142 | 140"5) 121 | 98 | 33°5 | 30°5 | 508 | 387 | 3:4 | — Se Nee HASISH 123 bs P3035 29 | paZS) 1] m7 oe pzSann | na 27) = SI ee MNS || = Bey) — 1 132 | — | — | — |139°5| — | 96:5} — | — | — | 382 | — | 645] — | — | 46° |r =) = | Ss} = 8535 1 | 75 79 4 | 28 = | | = | — | 90] — Juans} — | 98° | 365 | 29's Se Nh sell es NOSES E92 Se eS | 365 }3t | 31 | 98) | 57 | 36 | | = = = ao | — 182°5 136 | r1375| 85:3] — - | ===] |*= | | | | | 75 | re 185°5 | 131 | 115°5 |} 87-5) — =| — = = sol 5270377: | | | i JO, -— = | a | ace 188 139°5 115 93° | 133 | 31's] 115 | 99 | 34 | 20°5 | S24 | 384 | arr | — | 98 — |44 |2 38 | 30 | 117 86°5 | 30 47°5 | 74 74 7 7 = so | — 79 =| = Ss leon aos aes (2 (= = =| ee ae an = BR At 194°5 | 145°5| 122, | 97°95] — | 1355) — | 89°35} — | — | S40 | 405 | | | | = | 96°5 | 37°5 | 47°5 76 | 72 — 78 = | 178 | 174°5 | 180°5 | 135 | 113 98 See! =) 11193) = =) || $$10) 1/362"), — |°6575:1)/ — — | 4475 | 22'5 | | 93°5 89° | 30 37 76 68 50 79 = | 169 | — | 170°5 | 146 | 12075) — | = [n28ist) — | = 135 (25 ~ | | | | — | s6 | — | 76 BB) a= *)|/182/5)1 80) war} 1005 | — 138° | — |100 | 34 | 305] — | 370 | — | 71°5 | 88:5 | — | 50 | 24 391395519} ates Fares 11 93:50 ha |e 4031119315 |42i5 91a Ce ea zs RT 48 | 8 | 8 | — | — if if i rods | we a Romer tos —— ee ene nee ttt nanan ae one ed to CaM a tnt tye | a) aOR) oat ei |e Spb e°IOI | 3°e1 oeL Yer ee ‘as 3 53°5 rae 20 ert oRt 103 gs “S \ ga vel CO ep - —s 6e 5 1 gk s , roa 20 sis] der | yer 6& } 35 34°5 { — —- | jo : = Os 23 - art |; Ser /iarg8a) Y8r teroen| Se ec | 325 1 ae 2e0! ee6l | e per -— LHe : gsi} ask OT) POLY Ber SONe | 2st | esr | ssr 638 : yE'5 } 7 —~ |) =. gt 28 FET | Gpr CSOY, ors Spr eeQh | EOE | exer del yest 9 |< eed | yee eye lenges Op sé OPE | esol Qt 68 rox Ops @ est Set Yr: — -— [8 Seat = C eger| Spr 2081, esas fOL) - gs: 29 Sg On) | OBL Te EEL | Opi O81 eal cor O81 88 3 re ys OFT |) = oo asi £21 eu - 1AL ~ $38 i Op} Ups ae Ht) SM 2'88r 2 $81 8dr OR! 08 3 = oO | 2: USE! Py VyT se eNl eOv : V8 «ere ye es fe pega “ 1 AO AT ENTE NT NOM cei si 12, 30 | Ne ¢ = se | 2001 | - OeL gOeT! ef c os 228! le yo SSi | Pept | Vapor essr | oes = ay | 60 el aeb e@osr ve pyr |) 8x - By 93 o 2° OS ORL | 2OTI 00} = ey 27001 1g} Tht | 2°S8R). O81 | e° SST - 68 5 15 |} 95 “~~ ——-s TABLE III. [| M11 | os Dien) a, | w,| ww, |», | ¢ | Be | ae | aye'| aye |enjes| NBINH “ See G./G, | | a7 eae. | 1985 Zi ame lle 2-2 | Ohare iad Naar Bie cael es See 2°5 38 = | — -— _ 66 == 1-68) 82 46 — — _ Remnants) 9 93137 | 4215) — | = | =| 75 65 51 ha erfelall ie oe Seite ale || 729'22| 44 SC aan |e eT 4 a is real a Aare —_— — | 122 107 30°5 | 49°5 — 74 ME ECOROTeOSroN| ee Sx Se Neer = 96 | 95 = eae) 3o5 | «-— | 109 = EE! Sr PH) oa = 55 yl 73m A02 — —_ == | — = — 69 54 | 42°5 | 119 | TON’ | 35°5 | 43 a abl eodiens| Boorse B74 75 49 86 | 82 | 79 See) |e | 37. 4755 | = We = 72 50 i Wieeeere sa 219) 57 46°5 |129 | 107 | 34 | 465 | — of — 72 75 45 83 78 82 _ eat en aaa ange ieee S) 1 i aE ailk [aes cee ier Sa rates Wes = = — | — = — 80 — 63 = = as == 67 om ous sos mele | ee, 98°. | 36 See dire 10 e NeeTOn Nae 74 51 190 | es mel ae eoo yFOTs5.| 32 MOMS Wiles a ae aes = 55°5 | 44 mL 6) 8285 Wi On NGM.) 77 11178 73 45 Se i COMM lero he fa = a as zt Pp = 71 = 81 80 — 57°5 | 42 | 130 | 119 | 36°5 | 49 Seat eS = — 73 47 88 81 73 eee ern!) FO7°5.132°5) (0 — | — 9 | Sr | — | 74 = = aged eae le nee aee | ase) ss | 77) | 68 seis Te eae Ona eer se | FT5°5:/''35 51°5 = matin Vila Pace pee 82295. | FIO'S | 34 | 44. = = peek aad — | 46 = eR eS ees | ciges 81 83 — — 68 52 78 | 8 | — ==) | — | 12675 | 105°5 | 35°5 | 42 ee _ = Tig tern ee meee os LETS | 34° 145 — }—}— ] — = a a ae eas 30 43°5 55°5 | 41 Lian T0250 30-5 |) — 77 77 74 74 73 = i= i 74 —— — — | eg — a 9 — Semin 29) |, 98°95 | 32°5 1-435) 72 | 72 | — BOR 79h al ve 54°5 | 37 120°5 89 SS eh 73 74 70 71 78 42 = = 68 ee eee 7 51/1. 99'5-/) — | 49°5 | — SHON eee Mite oy GUS ie el a CLS 53 85 | 82 | 86 — | — | — |122 | 35°5 | 45 83 82 — | — — = 78 79 = = = 9/28) 1) Se ee 83 = Ss = = = 81 = oma a a) Sek S| — | oe) — oy) 66 54 S30 OOM eras 53 36 82 82 — 77 — 48 90 | — 68 — — ie [1O255 | — 485 81 eae ROUND BARROW. ¢. TABLE III. MEASUREMENTS Number (ef ee _~ js (<2 2 See ee eee = = = - sku |, | 5 vl foal ell, | | 0, 0. | | | 04/0, | 0,0, | 1 ete Wican ath de ela | oor low | ae | a | on) be | gut | foo | u | s | Q jem) en] os | ye) we] pp | pp | GE] Ge) Gr] ma | We] |e | Bin) oye | aye’) je \ourcn| Nant au H al 1] GG] P ieee di ee SE iE i el is = 2 Ih ee eee (Ee |) © 904 | 535 | 387 82 7. S ne tS ligce 77 | 94 | 130 | 56 66 a 82 46 — — = = eattta | 305 fs 369 — | 66 | 101 = 50 eS |r) S| 65 5I 72) 708) 72 | ae = 545 ha) | - | 67 74 = Wj) = 520 | oe 4 74 == | n7 | = 29 «| 524 | 367 sr 7. : | 4 9 | 95 | — | — ae ea 558) | 385) |) = 2 = | 132 48 = fe ai — = 55 = a 620 || — ib : Bs 540 — = = 123: | — = 8, ae : ella : 534 | 382 75°5 | 100°5 515 SS NA | Ne 7 49 86 | 82 = : i | = See lee SE eee ke = aa = 34 | 543, | 3765) — 75. jroo | — | s7's = Sra |i= ye es 45 | 83 | 78 | 82 = 1 a = 350 390 80 63 | 3 6: — — alse a is = — | 520 |372 | 347 | 70 Joss | — | 46 Ge eral alae GSI} = || = 138 | 182 | 181 | 178 | 18. 143°5 | 1 Ss re ae. | Se <7 z = = aaa eae aee © i's iagesi|ttees |-on:si|| Sean pe | 22) Sess RSS Rs oe 330 | 67 | 92_ | 1490'S | 50 ED (a3 eS | = 8 | 79 | 82 — | 18775] 187 | 186°5 | 148 | 12 97°51) | a2'5| — “5 | 3 x = 23 an a = fas 153 — | 1915 | 187°5| 190 | 158°5 8 Tae eS rs a Ba eee 378 = 100 = 0 79 a et ee es > 156 — ji 180 | 183°5 | 146 | 12 103 133'5 115°5 534 73 se = 3 = a 73 47 88 ar 2B = wep] = aot) Jerk |ea8 fag | | fide*| = |"S9) ass Jars | 88 [ass Sea Se = un ait | LE beh! 177, | 147) | 1215] 97 = 138 = lee a 235 515 | 370 84 79 = ¢ ; 5 A te = | — 8 = |= Hg. | 287 18H | 18 8° | tse reBiS cars ae | 228 een all soa Lesisi| GaN ees PIS lalalel eilale | lle WY | — 182 | 80 | 181°5 = J | ee | | = 7 | 76 | — | 84 190 | 191 1915 1865} 190 | 147 | 1285) 99°5| ryt | 141 120 | 1065] — | — 545. | 382 28 | 715 | 98: S|) ee || ame ipa 4 191 | sp tay | tg | = | “a lic==e|ks88 llha5 SPY Keel pack | it ras = || he Were |) Za =|) S ie 74 | 85° 18g 189 1845 | 185°5 | 137 100°5 128 Hl 195 | 18 |as 385 : 7 - | 39. | 3275 | 532 335 =| S|) = = 25 = || = | 198 asd pS P85 | aBe 28:5, rues || GOnsi sz bae | x20 | 9855] 375 | 385 | 527 [ators Ser | 70°5 Joos | — | st Grete Woes (lies niles | eae ae 197 | — | 183 | 177°5| 183 a i fy NS | AUT. | 183. 285'5'| 80 ify fasts [x26 | 99 | = |S | cae | "25 sass | 205 Se | ae Slelalsl sf 2 tae els y is | -— [ioe] 205) es (tsas asses | = |S elle : = SoS evs fete) el=|= 217 — | 191°5| 189°5 | 185 | 150 — | 97 = =, > Q18 | 8: | 181°) 177 | 180 | 5| — SS | ase ssn L0Cr | OOK5| || a8 =| & 83))|| 86) |) 222 | al. pes (EE |p) BBP | 2255 TOR) || Se | || 152281372) |) (340) /(69) al apis ge lee =| a) = as Moll = tes lhe | = eal ee | — — — — — — — 81 a — " 3: of rf moe as oo » _ soda to ao | x Vlog Mawr 4 Hes 2 eL | esau Xvi} 1per 0e 261 hol TO1 | ros oe 2 ONL | atene | 20x! ae EOL 66r | 2°20! LOL bEL } anvil | a-8or ! cos i f= PE OF 1 | 2208 = pai SE Spr | erpar | 2081 | ser. = Vil Thl | 5001 | sor | arsos | ic Sil a*sel | egos | e881 | ser) = efL 2°OnT E01 O81 OVI : R&T TEI | exG@1 | e2per | etder |) 74 oct @ED | ecxar ; teen | ever 7sa— |} HOST @SEI | SROL) ‘sel’) pen] |= ogi Par, soar | sey) p81 TSC | bel | 27701 101 él eset! cod: | 8811 2101 Wis eten lai eREL| SO@t | edetd oor) = bel) | QEr| O81 growr | ess p81 cbL | 0 | -- —_ ven SERL | ger! Syr | S¥6r) Br totes ath | eoor | — | der — | tpt Spr | 27081 Ror | ays - ,; OSS eiber| “ORI | avai ares éel OSU) ciger| (oar; ‘o8ny erent Ber) coer | aoe rer) = |, tol — | = -- + GOL, | SKI 4 5XNI cjovr | pie), = 6YL - {es - _ SY L:, sei) 5931 rer | Aber! RSs 6gl His - ~- ions — jeder; o8: | 8! = Yel a | -- ~ Bel TAL!) soQi | p:0d1 | atze1 «pron, 10eL = s81 2 etn — Let NGL} aer, apex) se8r) edi |) COL @SEr |) SNE | Poss.) Anbu 7681 VOL _ —_— = - - | el _ eer | oy | ei | be=o 4! WOE aig) “s81)! -o81 | aasr | per YOx 2°EcE 2981) 2t0Cl | geet | saa PIR = eel ski off ogi! 97 = 34 | 265 | 499 | 364 | — | 63 | 885 | — | gb6r5 | 23°5 | 36° 6" 2 86 | = | = 168 | = ae | ce 5 | 13. | aad ee | 11g | = SS) fsgan| a =alliega ea » | 5 | 365 | 3 3 49° | 36 | 92 | 30 | 45 = 5 — | 79 au sr 88 | 85 73) V71 | 178 | 1785] 176° | 17°51 132 119_-| 99> ian 333 { | | = =| SPER: | i | AS } 77 | 184 | 184° | 1 183 138°5 116°5 | 95 | 8 ae 20K GB IBD e Ne ASS I ahora tat Sei SESH INS2N Wee tie ase osu lraze neal i7a iihe7dy |i ai || st ="\577.1o— | — } 180 | — | 175 | 17g's| 172-5) 1425 | 11g | 100 | — | argos} — | 100 | 37°5 | 28 5 Sallie = 40 | 33 — | — | — | — Jizz | o9:5| 32 | 43 7 | 75 | — = 820 Se 192 | 178 (178 | 17675 1785 138-5 110 | 99 a $ jo | 367 | — | 65'5 | 90 — | 43 | 22°5 | 39°5 | 40°5 | 32°5 | — | 98:5 149 | ars | 115 | 86° | 365 | 435 | — | 8x | — | 80 2 8 8 ° 199 | — | 167166 | 169 =} 14175) — | 92 al 372 324 ar j42 .|32 |33 | — | = 123°5| — | 29° 3 | 78 | 78 us seallcaael eee ees | 200 | 176 | 175 | 174° fs > | 490 = ee Ele al 3°5 95} 415 | 7! 7 _ = | ee78le7on |e | Sor 76 | 175 | 174°5 | 176°5 | 136 | 120 121 | 500 a | | 43 84 = eg TS | 202 ) — }rz3sl azz | 175 | =| 136°5 | | : =|=]= cae |) = So licig eee, |\a9¢ laa) ||P a =e = 04 | — | 1835) — | 188 | 128 sal 4985) (302530051 eM | a em LOSES | SeSOFseee(e el | e | eo EE oe : Sy S| oS pS ale ao8 | — |182°| — | i845] 149 | we = | 513 SS "5 ae | = SIE SES NSS oe oxi = pe mee | ae a tl Ta SOU erg ee a I= gi | fis ges ts SS ay s 2/242) | aa | bape | == A! °5| 197°5 | 139°5 113, | 100 | — | — | 120 | — | — | — oy bse = hse | —/—-/]—/]—)]—] & = Nee 221 182 | 18175 | 180" | 143°5 | 118'5| 93 | — | 123 |.— | 95 | 34 SUT) (358) 1324 Cal ieee yh = 385 | =" (335) a c= 51 | 38:5 | 115'5| 100 | 30 | 445 | 77 | 78 | — s = = EPS B= | = 79 =f Sy S15 —_— a. oar’ CISC Pe a PO ra Vaates baa Sa Lacs baa alae Le >» | | i ~ os ~ co isc) me BET "% sa Pres fa Wt eee on e ue a Bs a as ly a8 ec es efei! | e te! ape ipo Rls! BEE ~ i enn = a a PERIST MI UBISI Cen SiR Biss ara oo =i ol ao cas bd ict i cht Farle ¥ | —— : ey | — —- ae Kadi, — _e . am E. H. J. Scuuster 355 DESCRIPTIVE CATALOGUE. In the accompanying remarks : , Jalvaria = Dome =Schiideldach. Cranium= Braincase. Skull=Braincase and face bones. ad.= Adult. B. B.=Greenwell’s British Barrows. G. C.=Greenwell Collection in Oxford Museum. The page of the British Barrows, together with the line on the page in which the particular skull is referred to are, when possible, cited. The Roman numerals give the number of the special excavation in the same work. LONG BARROWS (MALE). No. 3.—Calvaria, ad. Mandible without ascending portion of left ramus. Alveolar and palatal portions of maxillae, also left occipital condyle in box. Posterior sagittal depression. B. B. p. 484, ecxxi. Ebberston, N. R. Yorkshire. G. C. No. 4.—Calvaria, old ad. Torus occipitalis. B. B. p. 484, cexxi. Ebberston, N. R. Yorkshire. G. C. Cf. also MS. Catalogue, Oxford. No. 9.—Imperfect Calvaria, ad. Mandible without ascending portion of right ramus ; the left coronary process broken. Maxillae detached (query from same individual ?). Supra-inial depression. B. B. p. 509, cexxvii. Over Silton, N. R. Yorkshire. G.C. Cf. MS. Catalogue, Oxford. No. 18.—Skull, ad. The right face-bones and the basi-cranial axis wanting, the mandible imperfect. Distorted by pressure. B. B. pp. 497—501, p. 501, 1. 1, cexxiv. Rudstone, E. R. Yorkshire. G. C. No. 19.—Skull, old ad. Imperfect: basis cranii missing: mandible with most of the right ramus missing. B. B. p, 501, lL 6, and pp. 612 e¢ seg. (figured and measured). Rudstone, EK. R. Yorkshire. G. C. No. 21.—Skull, old ad. Right parietal and squamosal imperfect. Thurnam, J/em. Anthrop. Soc, 1. p. 132, p. 478. Dinnington, near Rotherham, W. R. Yorkshire. No, 22.—Skull, old ad. Occipital imperfect. Metopic suture. Thurnam, Joc. cit. Dinnington, near Rotherham, W. R. Yorkshire. No. 25.—Skull, adolescent. Frontal defective (from abscess ?); left palate broken. Paired precondylar eminences, short torus palatinus. Thurnam, loc. cit. Dinnington, near Rotherham, W. R. Yorkshire. No. 38.—Skull, ad. Ossicles in lambdoid ; vestiges of transverse occipital suture ; left infra- orbital suture in face. Crania britannica, No. Ixviii. Vol. 1. pp. 244, 245, Vol. 11 pp. 59 (3). Rodmarton, Gloucestershire. Thurnam Collection, No. 163. No. 40.—Skull, young ad. Very defective ; left temporal and basis cranii absent. Metopic suture. Crania britannica, No. \xx. Vol. 1. pp. 244, 245, figured, Vol. 11. 59 (4). Rodmarton, Gloucestershire. Thurnam Collection, No. 166. No. 48.—Skull with mandible, ad. Sphenoid, palatine region, much of right frontal and parietal absent. Mandible without angle of right ramus. Rolleston, Journ. Anthrop. Inst. v. p- 151 (1875). Nether Swell, Gloucestershire. No. 59.—Skull with mandible, ad. Basis cranii wanting: other extensive defects. Shght torus occipitalis ; minute ossicles in lambdoid. B. B. p. 521, ecxxx. Rolleston, loc. evt. p. 169. Swell, Gloucestershire. 45—2 356 Long Barrow and Round Barrow Skulls No. 62.—Skull with mandible, young ad. Very fragmentary. Mandible wanting both condyles, and not belonging to skull. B. B. pp. 524—527, 1. 37, cexxxii. Upper Swell, Gloucestershire. No. 65.—Skull with mandible, adolescent. Mandible with greater part of right ramus wanting. Metopic suture ; right infra-orbital suture on face, left m 3 just appearing. B. B. p. 529, 1. 5. Upper Swell, Gloucestershire. No. 72.—Imperfect calvaria, ad. Part of right side of face and temporal bone attached. B. B. p. 531, 1. 11. Upper Swell, Gloucestershire. No. 77.—Skull with mandible, ad. Much broken. G. Y. Ackerman, Archaeologia, xxxvii. p. 482. Thurnam, Archaeologia, xlii. Crawley, near Witney, Oxfordshire. No. 80.—Skull with mandible, young ad. Mandible broken at symphysis. Ossicles in lambdoid : slight bathrocephaly. Ackerman and Thurnam, Joc. cit. sub No. 77. Crawley, near Witney, Oxfordshire. No. 81.—Calvaria, with separate maxillae and mandible, ad. Mandible with right condyle broken. Metopic suture. Trace of torus occipitalis, Ackerman and Thurnam, Joc. cit. sub No. 77. Crawley, near Witney, Oxfordshire. No. 82.—Skull with mandible, ad. Defects. Right canine displaced in front of lateral incisor ; mastoids very small. Ackerman and Thurnam, Joc. cit. sub No. 77. Crawley, near Witney, Oxfordshire. No. 83.—Skull with mandible, ad. (young ?). Right zygoma broken. Ossicles of pterion ; the right 30x10 mm., the left 20x7 mm. Two ossicles in left coronal suture, the larger 20x20 mm. Ossicles of lambdoid, two on right, three on left side, the largest 20x15 mm. Ackerman and Thurnam, Joc. cit. sub No. 77. Crawley, near Witney, Oxfordshire. No. 86.—Skull with mandible, ad. Skull much broken. Ackerman and Thurnam, loc. cit. sub No. 77. Crawley, near Witney, Oxfordshire. No. 87.—Skull, ad. Basis cranii and most of face wanting. Ackerman and Thurnam, loc. cit. sub No. 77. Crawley, near Witney, Oxfordshire. LONG BARROW (IMMATURE, OR SEX DOUBTFUL). No. 7.—Calvaria (? %). Very imperfect. B. B. p. 484, ccxxi. Ebberston, N. R. Yorkshire. G. C. No. 23.—Skull (sex ?). Thurnam, Joc. cit. No. 21. Dinnington near Rotherham, W. R. Yorkshire. No. 29.—Skull (¢ ?), ad. Left temporal and left side of face wanting. Very long mastoid. Thurnam, loc. c’t. No. 21. Dinnington, near Rotherham, W. R. Yorkshire. No. 51.—Calvaria with mandible. Child 6 years. Metopic suture. Rolleston, oc. cit. sub No. 48, p. 156, 1. 7, and p. 158, 1. 21. Swell, Gloucestershire. No. 54.—Skull with mandible. Face and basis cranii very imperfect. Rolleston, loc. cit. sub No. 48, p. 161. Swell, Gloucestershire. No. 66.—Calvaria with left temporal (g ?), ad. Slight sagittal ridge ; torus occipitalis ; ossicles of lambdoid. B. B. p. 530. Upper Swell, Gloucestershire. LONG BARROW (FEMALE). No. 13.—Calvaria, ad. Imperfect : additional fragments. B. B. pp. 491, 492, 1. 8, cexxiii. Weston, E. R. Yorkshire. No. 24.—Skull, old ad. Ossicle of lambdoid 37x25 mm.: faint post-coronal depression : small pterygo-spinous ridge. Thurnam, loc. cit. sub No. 21. Dinnington, near Rotherham, W. R. Yorkshire. KE. H. J. Scuuster 357 No. 26.—Skull, old ad. Right malar wanting. Teeth lost. Thurnam, Joc. cit. sub No. 21. Dinnington, near Rotherham, W. R. Yorkshire. No. 27.—Skull, ad. Both zygomata, the right temporal, and part of the occipital wanting. Thurnam, loc. cit. sub No. 21. Dinnington, near Rotherham, W. R. Yorkshire. No. 28.—Skull, ad. Left temporal wanting, occipital and left parietal injured. Posterior sagittal and lambdoid ossicles. Thurnam, Joc. cit. sub No. 21. Dinnington, near Rotherham, W. R. Yorkshire. No. 30.—Cranium, old ad. Occipital condyles wanting. Sagittal groove. Thurnam, loe. cit. sub No. 21. Dinnington, near Rotherham, W. R. Yorkshire. No. 31.—Calvaria, ad. Thurnam, loc. cit. sub 21. Dinnington, near Rotherham, W. R. Yorkshire. No. 39.—Skull with mandible, ad. Right infra-orbital suture on face : perforation in anterior wall of left external auditory meatus. Crania britannica, Vol. 1. pp. 244, 245, evii. Rodmarton, Gloucestershire. No. 43.—-Calvaria with temporals, detached maxillae, and mandible, young ad. I I a if 8 G a4 oa I v a — I i € =< Li G Se et |e ee | ee eG I — ae I a cam a I aon = os G I T il a I G I = a — = 8 v SE eee ee al oe a ee ORE aan tame r I aa 9 ee é I 9T € | coe ee z z — a — a I aad — mai I I = : = G I = = in I = sy I I I = =e L I T-61 wn | 2 | = a nk a Aver ileal lex Es 2 | do iS Ee| § | fe | se Pelee |Poleeles | & | 2) eelee| Se) & | 2 | ge e8 |) @ | Eh | eB ee |S4|/8e/ 2 | ee | = | § | EB| Se] F] & | F | & | BS alge se es: ies Ne: ar ae ee 4, 8* : B SAIN) SOVY, 368 ‘Soq jo yysuey ‘shigq sooyong fo squawamsvayy “TIT ATAVL 369 O. H. Larrer Té | 8G | 8h | 99 | 8 | GIT| Loe} Le | Le | ge | er | 79 | Fo | 98 | T6 | BLA | Zagt| 214] ¢eg Isqerog, —}|— Se (Mires | hee) re Wea I I I aah | eT i tS) ee —|-—}|-—- I I = | i G98 a =e ae era = i = a, a as rox neal I I Sa FCG = ae —/}—-—}]—/]—-]—]—|]-—-F]1 Sit LG = |¢ se i == we ee | ee eee z L I 9 | 0-&@ I ees = = ft I — | 6% I I —}|}—/]/—/—|]—|—]e € == 01 9:7G SS | = —-{—/]—|]— | ¢ z € L4G save ene anl cil lc I — | 9.6% z wl I I I 8 I i C.F SS eee | ee ee OT — |—T|e € — | £96 Ear ee ra —/|/—/]1 Sel af r I € | &bZ a ees | ae com ie ee en yee Seen IL L I CFG ieee We = a ho nk = ae eas ee le I I L-F =e 7 7 € 7 9 I I I Abe arte ee! ye wy | LT | 48 | 0-78 aT ete Sia I a ace Le WL hes Gree = z € 2 VAT on ae z — | T I I Gl rt | 1 8-83 = 14% eal eG G I —/1 nial cI | ¢ OL | 2-8 he = heat z z => Vie — | = —|— z Lh | bp | 9-€% I I I 9 I I L Se a a a 7 z I = 6 og | 8 GS | G-€% en eal == lal cane I = |— | ee G I F-€3 z I r == € L I Pe I — a ig =i ge | 83 | OL | &E z I z I € I 9 a eae I € Se hese he ar eso a elie a crt = ne I z 7 € ¢ z age ett ica ee I Saori gk if eo 76 Hort | 16 z ¢ I 9 r HL | We a SS 3 z 9 Sls € Th Ny SLs 668 a) 7k 210-6 I = ee i z P 9 z z = | i ee ale © P € ey | 6s 41 pt | 6:66 be SA wre 6 I ole 1 = 138 I j v c= les 6S | €@ | 93° | 8-8 z € z I z 8 91 | & z =) li == % Se z j ya | 60 | Se | 266 I I Se 0 D | te = |e j € sé | 61 | 61 | 9-8 z L € ig 7 6 Civea| al I 9 j 7 g G GIs uae z6 | 63 | €9 | 9.2% I ena a= ee ¢ 8 we calsnG SG SG rd rd p He arcinl eon Mirce I z rasa arc v 7 Ie Nee € I I rd I 9 L I LOM eee, Wael cece I eee! € € ¢ Sl a1 I j € z = |g € I 69 | “63 | 08 | %-2z ae lt 9 I z I j OU | 1 I SG 1 € j z j Lp | 66 | SL | 1-8 Biometrika Iv 47 370 The Egg of Cuculus Canorus TABLE III.—(continued). Breadth of Egg. GENTES a 2 . 3 +2 ra o cal ‘ot el oe mw eB ta : eae e(Zlel#l|2|2/2l3|Slalels = ai Sie )als/81e8)8 | a 12 | | oS eee E S/F (ES ISIE IE |S 1e |S 2 le See 2 Be ps Ee |e) oe | ele 1a |e PS) a eee 3 eis Slele le | 8) 433°) 8) ogee a ase ae eos lye a= 3 Cues ml ry ae U | 2) SS SS eee 1 ee ee |=] =| 2) 2 |) oe | 29)) a) Se ey Peete (ae a ei el at 2 ae = — |) — a ee al 2 a se 1/—|—| —h 9) — | — | | 2) eee 1) 4)/—/ 5S5—] Gt @l a) -v) Bo) =) a ee 1jJ—| ilj—l— | eth ar ae | SaaS aS —| 3| 3) 4/ 2) 3) a1 94-1) 7) ae ee 1/ 1/ 2) 2/1} 2) 2) 34-3 & ee) 204s eee eee 91 1) 2) 3)/—41,2) 9) — | oF @:) 23) Saale eee ee 10°] 24) 2/10/10! 4)/°4] 81) 37) 45) 13: se5\eron ionomers Te ee eee oe ae 91 6) -1 |2So) shale = ieoulee 31 6] 81 4) -4)— | 9] 91 42)49 | 4)) TOURS Seiesalmes 31—] 4])—] 1/29) 7) 2) 3 18), Qc ioaler een eo 3) 3.) 61,2) 2] 2) =) 8 [nas 1) eco iis tN eee 7) 8| 4) 31) 94) 7) 6) 4) 99) 44) 166) 7a oe emcee 66 2) 4] 2] 1| 1) 1) e219 ) 10) Saiesh onos oes 6°7 21 5/ 31 1) 2] 1) @}]—) 3) 20) (5 1695.) oa ee ene 6:8 3/ 6/10) 3| @1—| 1) =) 9354) 1) 3 Seon ae 6:9 4/—) 6)/—] 1] PT) 14) Ss) 9 196 Weal) sen aa eee eee ee 7-0 71412113) 4/4717 | 4) 31 9 84 | 940) aCe ere ie ‘] 4/9) 4| 9) W.) 34°11 —1) 9)) a9) 9 Ks) Ssaon ae 2 9) 1) 8] 1) 1)-2) a) tl sa) a) Salsa ae eee 3 4) 4/4) 1)/—}) tly aw) a2) eee oe oe ae ‘4 Silica eee) all eset eee | nee tee) = = 5 5) 21 2) 1) 2) —) Vi ty 2) scl 26) oe ee eee | ii Wp) ee | ea reg ieee |. oes ie Meee ee |e Ti DN ee 2) Gil tte || a aoelit See |p es 6 cl) aT | Slo pel pre ea ee ee eee 90 | | =e ee a ee ey a ae een eg es a roi |) | — | 31) —) 1) | a eae a SS See 1 wee eee ee Ue ee ee ee eee |e 1 ay as in (ee ee ee | 78 | 91 | 86 | 54 | 64 | 48 | 35 | 37 | 37 |307/115] 85 | 66 | 48 | 58 |'31 Of the remaining 332 Cuckoo’s eggs the measurements of which are given in the totals: 46 were not classed, 11 were in Reed-Bunting’s nests, 17 in Wood-Warbler’s, 17 in Linnet’s, 14 in Chaftfinch’s, 4 in Bullfinch’s, 14 in various Lark’s, 14 in Yellow-Wagtail’s, 15 in Blueheaded-Yellow-Wagtail’s, 10 in Spotted- Flycatcher’s, 15 in Willow-Wren’s, 5 in Nightingale’s, 15 in Greenfinch’s, 20 in Blackeap’s, 7 in Whinchat’s, 14 in Lesser-Whitethroat’s, 6 in Wheatear’s nests, and the remainder in numbers of 5 and under in the nests of various other species. O. H. Larrer 371 TABLE IV. Cuckoo’s Eggs in mm. Foster-Parent’s Egg . a au, Approximate Length of Length Breadth Length Breadth SPE Robin 5a0 22°8 16°8 20°2 15°4 53—6 Garden-Warbler 21°9 16°4 *20°3 16:0 5s Meadow-Pipit 22°3 16°6 19°7 14°5 57—6 Wren 21°4 16°2 17-7 127 By Redstart... 22°4 16°6 *19-0 14:0 55 Sedge- Warbler 22°1 16-4 gallone 14:0 43 Barred-Warbler 21°8 16°3 *29°8 15°5 6t Whitethroat 21°8 16°2 *19'0 =| 142 5s Marsh-Warbler. 22°1 16°3 *19°0 14:0 54 Red-backed-Shrike.. 22°3 16°5 *29-8 16°7 7s White-Wagtail 22°4 16°7 *20°3 15°5 Ue Pied-Wagtail 223 16°5 *20°3 15:2 73 Reed-Warbler 226 16°6 sll) 12°7 54 Tree-Pipit ... 22°6 16°6 20°0 15°1 6 —6} Hedge-Sparrow 22°6 16°8 20°0 14:7 54—6 Yellow-Ammer 222 16°5 21°5 16:0 63—7 Cuckoo 22°3 16°5 — — 14 TABLE V. Frequency Distributions of Cuckoo's Eggs. A. With 5 mm. unit of grouping. Number Number Length Breadth Rey + 0. H.1L.} 0. H. L. only Rey + 0. H.L.' O. H. L. only | 18°75—19°25 1 1 13°75—14°25 uf 1 19°25—19°75 3 2 || 14°25—14°75 2 1 19°75— 20°25 33 14 || 14°75—-15°25 19 7 20°25—20°75 39 8 lb s25 == lou 102 40 20°75—21°25 156 69 15°75—16°25 443 227 21°25-—21°75 152 54 16°25—16°75 395 152 21°75 —22°25 392 179 16°75—17°25 456 222 22°25—22°75 288 121 17°25—17°75 115 47 22°75—23°25 286 148 17°75—18°25 37 18 23°25—23°75 100 53 18°25—18°75 0) 0) 23°75—24°25 86 53 18°75—19°25 2 2 24°25—24°75 21 8 | 24°75—25°25 12 5 25°25—25 "75 2 1 Totals 1572 717 25°75—26°25 0) 0 26 °25—26°75 1 1 Totals 1572 lay * Computed from measurements given in inches by W. J. Gordon in Our Country’s Birds. 47—2 372 The Egg of Cuculus Canorus B. With 1 mm. unit of grouping. Number Number Length = ; = Breadth | Rey |0.H.L. Rey |0.H.L. Rey +0. H.L. only only Rey +0.H. L. only only | | 18°75—19°75 4 1 3 13°75—14°75 3 1 2 19°75—20°75 72 | 50 22 14°75—15°75 121 74 47 20°75—21°75 308 | 185 123 15°75—16°75 838 459 379 21°75—22°75 680 | 380 300 16°75-—17°75 571 302 269 22°75—23°75 386 | 185 201 17°75—18°75 37 19 18 23°75—24°75 107 46 61 18°75—19°75 2 0) 2 24°75 —25°75 14 8 6 25°75—26°75 1 | 0 1 Fees: Totals 1572 855 717 Totals 1572 855 717 | prejudice in favour of the half and whole millimetres may have tended to obscure some of the points involved and renders it difficult to deal satisfactorily with the distribution of frequency. The multimodal character of the previous distributions (see Biometrika, Vol. 1. pp. 175 and 176) was chiefly due to the manner in which this emphasis of certain values upset the grouping. Admitting the defect the only legitimate grouping is that which takes an entire millimetre as its unit, but this is very coarse, especially in the case of the breadths, for the purpose of drawing any conclusions. In Table V. B the frequencies are given with 1 mm. unit of grouping and we see that the ‘5 mm. modes have disappeared, but the grouping is very coarse. Working, however, from these we find : TABLE VI. Moan Standard Deviation an in mm. EN a Authority Length Breadth Length Breadth Rey ... Sas 22266 16°559 “9450 5955 OME. Ws 22-486 | 16°613 9546 “6021 Total we 22366 16°583 9557 5992 Av examination of this table seems to indicate that the English Cuckoo’s egg is for both dimensions slightly larger and more variable than the German. The difference, however, is not such that we can lay any stress upon it, and in round numbers we may certainly take : Length of egg 22°4, Breadth 16°6, standard deviation of Length =-96 and of Breadth = °60. The following results were obtained by calculating the normal curves for four sets of frequencies* : * The determination of P, the goodness of fit, from x? has been made by aid of Palin Elderton’s Tables, Biometrika, Vol. 1. p. 159. O. H. Larrer 373 Length Breadth Rey | Latter Rey Latter Obs. Cale Obs. Cale. Obs. Cale. Obs. Cale. 1 3°32 3 1:49 _ — a — 50 43°14 22 23°27 1 1:01 2 0-71 185 203°66 123 133°24 74 73°43 47 53°44 380 | 344°74 300 278°73 459 460 °36 379 369-01 185 210739 201 213°77 302 300°75 269 272°68 46 46°06 61 59°16 19 19°35 18 21°02 8 3°57 6 711 0) 10 2 0-14 0 10 1 0°23 — ae _ = =r x?=16°36 x7=8:09 x? = 62 x7 = 28°59 P< :025 P = 325 R99 P< 00004 The odds are thus more than 40 to 1 against Rey’s lengths of eggs fitting a normal distribution. But only once in 100 random samples of 855 eggs should we expect a better result than we have got for the breadths, if the distribution be really normal. Turning to the 717 measurements we find the length distribution reason- able on the basis of a normal curve; once in every three random samples of 717 eggs we should get a worse result, but the breadths are quite impossible. This impossibility arises, however, entirely from the two giant breadths of 18°8 and 19:2. They are undoubtedly abnormalities and if they be excluded, the fit is a good one, Le. about 55 samples in 100 would give a worse result. With regard to Rey’s length distribution the sources of improbability are seen on analysis to be (a) the crowding up of seven eggs on 25:0 mm., (b) of 131 eggs on 22:0, and (c) of 63 on 225mm. These contribute more than half the value of x?! In other words there is little reason to doubt that the variability in length of Cuckoo’s eggs would follow a normal curve, were it not for Rey’s tendency to heap up observations on the °5 and ‘0 millimetre groups*. It does not therefore seem worth while pursuing the distribution of variability further on the present observations. We see that it is quite possible that the normal curve would really suffice to describe the frequency. On the other hand, if the gentes theory be considered as established, we should naturally expect the heterogeneity to show itself in some deviations from any smooth curve of frequency. In conclusion I must gratefully acknowledge the generous assistance of Prof. Karl Pearson in the preparation of this paper and in the statistical exami- nation of the measurements obtained: indeed without his aid I should have been quite unable to perceive their significance, nor would this paper have been written. * The evil of this heaping up may be easily seen if we take the first group 19°05—20-05 instead of 18:75—19°75 still grouping by mm. The mean will now be found to be 22°10 and the standard deviation 1:0014 instead of 22:27 and -9450 respectively ! MISCELLANEA. Notes on Statistical Processes. By W. PALIN ELDERTON, Actuary. I. An Alternative Method of calculating the Rough Moments from the Actual Statistics. Let s be the position of any term in the series and f(s) the frequency corresponding, then the following table (page 375) explains itself. Consider the line opposite the first term and we notice that the sum of the series is given by the first summation S; and that the second which we will call S, gives the first moment of the whole distribution about a vertical through a point situated at unit distance before the point corresponding to f(1). Still considering only the first line we see that S; gives each n (n+1) nen function multiplied by 9 or yo? Le. (mm +m,')+2 where m’ is written for the rough Pa a 34 3n242 eee 4 341172 moment. Similarly S; gives a and SS; gives c Ee , and hence So=my, (Ee mz +3mg + 2m = gees : sg _ my + 6s! +1 Ling’ +6 5 24 ; A numerical example is as follows. TABLE II. Function | Istsum | 2ndsum 3rd sum 4th sum 239 524 978 1648 2587 157 285 454 670 939 93 128 169 216 269 29 35 41 47 53 6 6 6 6 6 524 978 1648 2587 3854 375 Miscellanea WL (ufot(-wf (u)sort(1-“)sfo+(6-wS iP (- Sera euyalice)* +(e)fe+(at iv (g+u) (¢+u) (T+u) a0 tere Ce) Ja+ (1s (wf (u)f (u)f+(1-OL (w) fol+(1-“)fe+(o-u)f hats he acces u({—4) +(e) fot (2s | v i€ (g+u) (T+) u +(e) fe +(DS (u) + (wins 4339) U OF $ THOTT uuinjoo snotaead jo ung (wins 43.07) WU 07 $ WOT uuinjoo snotaeid Jo wing | (u)f (wf ws (ws| a (w)fer(1- wf (fot (1- wf (Tw) f+ (uf) (I-wW)f | 1-u (wso+(1-Wfet(e-wSL] (wet (L-WSE+(6- WSL] (Z-WS4(1-WL4(L (B-u)f | g—u -f_at “Het OF) MLI-WM+' +(e) £64 (BL (Wft"+@f @L] & OEerenrn ras | +(8)fe+ ( fer" +@ fers, MSE THOLADSL) DSL] 1 (rms pt) Pee ees eee uuinyoo snotAead jo ung uuIn[oo snotaerd jo ung al ae 2a ak TOT OUn eT |30"ON T dav 376 Miscellanea It is unnecessary to form the fifth sum in detail as the total of the fourth gives the only term required. The totals check the work throughout. The rough moments (not reduced to unit frequency) from the equations given above are 978, 2318, 6612 and 21458 which agree with the result we should obtain by direct calculation. If the rough moments about the mean are required the following is a convenient form : a= 283/S,—d (1+d), v3 = 68,/S, — 3v2(d+1)—d(1+d) (2+d), y= 2485/8) — 2v3 {1 +2 (d+1)}—v {6 (14d) (2+d)—-1}-d(1+d)(2+d)(3+4+d), where d is the distance between the mean and the point used for calculating the untransferred rough moments. Obviously d=S}/ Sj. I find that if there are few terms to be dealt with the direct method is quicker but for longer series the above has a distinct advantage. The summation method is really due to Mr G. F. Hardy, but though he has remarked that it is practically the same as the method of moments, I have not traced any previous note of the actual connection between the two methods as statistical processes, though Professor Pearson tells me he believes the summation method of reaching moments is given by Coradi in a paper on the Abdank-Abanowitz integrator, and J. Massau, “Mémoire sur lintégration graphique et ses applications,” Paris, 1887, gives the connection between successive integrations and the moments. I understand that the integraph does not give very satisfactory results for the higher moments, but though the method is like the one with which we are dealing, the latter is clearly not open to the same objection. The method can be extended to enable us to deal with correlation tables. In order to find the coefficient of correlation we require the means and standard deviations of the a’s and y’s and the zy-moment. The mean and s. D.’s can be found by treating the totals of the w-columns or y-rows in the same way as we did the entries in Table II. but a quicker method can be devised, which however will be easier to explain with the help of an example. TABLE III. Correlation Table. Totals Totals We now obtain from this table another (Table IV.) in the same form giving the y-sum of it by summing each column continuously, and then obtain Table V. by summing Table IV. across continuously. TABLE IV. y-sum of Table IIT. Totals 8 | 43 | 198 | 192 | 68 | 509 Miscellanea 377 TABLE V. a-sum of Table IV., ve. a table giving all cases for x, y group and over in Table ILI. —~ ty Os MN | Totals The totals in the right-hand column of Table IV. give the first sum of the total in the right- hand column of Table III. and are the same as the column #=1 in Table V. The total of Table ITI. or the sum of col. 1 in Table IV. gives the mean of the y’s (509+ 200) ; similarly the sum of the row y=1 gives the mean of the 2's (691 +200). The total of Table V. gives the vy-moment (1796) and the x-standard deviation is found by forming from the first row the series 691, 491, 295, 119, 22 and summing it, i.e. 1618. The second moment about the mean can then be found ; the numerical working being as follows : #-mean = 691 + 200=3'455, vy = 283/S,—ad (1+d) by formula given above a — 3°455 x 4:455 ="788. Similarly with the y-moments y-mean = 509 + 200 = 2545, . (509 +309 + 150 +38) x 2 2 300 — 2°545 x 3°545 = 1°088. The ay-moment= 0 _ 2°545 x 3°455 ='1870, r="1870+ /°788 x 1-038 = 207. In this example the moments have not been adjusted as the totals of the rows did not seem to warrant the assumption of high contact, Table V. can be formed directly from Table III. but it is a little troublesome to do so at first. The method can best be understood by seeing how any item is made up; thus 63=30+17+26-—10, 97=20+74+4+22-19, 102=28+463+37-26 and so on. It is not difficult to see how Table V. succeeds in giving the xy-moment ; we have merely to remember the numbers by which the various terms in a correlation table have to be multiplied to produce the required result. The final item in Table III. for instance has to be multiplied by 20. It occurs in every item in the e=5 column of Table IV. and therefore in every item of Table V. ; there are 20 items. Similarly the entry 28 in Table III. has to be multiplied by 9; Biometrika 1v 48 378 Miscellanea it occurs in #=3 and y=1, 2 and 3 in Table IV. and consequently in all the first 9 groups of Table V. and it is therefore correctly treated. I have found when dealing with correlation tables that the summation method saves time when the table is fairly large. The method might be extended to simplify some of the calculations which are required when non-linear regression is investigated (see Professor Pearson’s Drapers’ Research Memoir, Biometric Series, II]. On Skew Correlation). II. The Application of Certain Quadrature Formulae. The usual quadrature formulae express an area in terms of the ordinates at the beginning and end of the base and a varying number of intermediate ordinates, but in some statistical work it is convenient to have values for the areas in terms of ordinates both within and without + the base on which the area stands. Symbolically we have to express | ; ydx in terms of Yo, Yr Y-1» Yr, Y-a, ete. Let Yr= at bx +ca* + dai + ext, } 2 ce then | ete-etg te and Y= 4, 5 ¥-1+-Y41=2(a+er+e), Y 2+ Y 49=2 (a+ 4ce+4 16e). Now assume the required integral can be put in the form hy tk (nity-)+l (yoty -2), substitute the values given just above and equate coefficients of a, ¢ and e respectively to 1 1 : 1, Dp and 30 and we obtain 3 1 tl, i da = =e {51784 +308 (1 +y-1)- 17 (Yoty—o)}. II. If y=a+be+ cx, 3 2 1 | yytem gg lyst 22 ty. Ill. If Yu=atba+cu? + dx + ext, k ie yall = an {6463 y — 2092. y; + 2298 y2— 113273 + 223 y4}. Iv. If Yn=at bat+cxu? + dx, 3 1 | _ Ysde= F727 Yo +1741 + 5y2— ys}: =] Vv. If Y= a+ ba + cx”, : 1 I _ Ye = 57 {25 Yo — 2 + Y}- VI. If Yu = a+ ba + cx, 3 1 | ynto=qq 849) -U stan) Miscellanea 379 VII. If Yu = a+ ba + ca? + da3 + ex, 5 1 iS Yule = 55 (802 (y_;+Y43)—93 Y_g +4) +11 (Y_3+y;)}- VIII. If Yo= a+ bat cu? +da3 + ext, a i | nee=% {29y_,+124y,+24y, +4y, —Y;}. IX. If Yn = a+ bx +cx?, z ie ; _ = B (5y_,+8y,—Ys}- The most obvious purpose for which these formulae can be used in biometric work is for calculating the areas from the values of ordinates when the equation to the curve has been determined. It is clearly convenient to calculate as few ordinates as possible and I have found Formulae I. and II. especially convenient when expressed in the form of differences; thus II. can be written 2 1 [yen Hy {Ayo— Ay -1}; and the adjustment from the mid-ordinate is easily calculated. Formula I. can be written 4 291 17 ydx=Yor reo AYo— AY 1} — 5760 {Ay — Ay—2} -} and the fractions are ‘0505 and -00295. Formulae VI. and VII. are useful when the ordinates at the beginning and end of each group have been calculated. A numerical example of the use of these formulae in a case where their accuracy can be tested will be of interest. We will assume that the ordinates of the normal curve have been calculated for the equidistant values of x and the areas corresponding to these as mid-ordinates are required. TABLE VI. Value of Area b Area b - ordinate A Ronnie 1b Peenaiaon True value “80 2896916 0785785 | “40 *3682701 ‘0306722 366223 *366274 | ‘366218 ‘00 3989423 — ‘0306722 *396307 *396386 *396298 ‘40 | °3682701 — 0785785 366223 *366274 *366218 *80 | +2896916 — 0955055 "288992 "288986 “288994 1°20 | ‘1941861 — 0832653 "194740 "194696 "194746 1°60 *1109208 — 0569298 2°00 0539910 The second and the last columns of this table were found from W. F. Sheppard’s Tables, Biometrika, Vol. 1. pp. 182 et seg. For frequency curve work we do not generally require a large number of figures, and as the total frequency does not often exceed 5000, Formula II. would be sufficiently accurate while Formula I. would give a satisfactory answer in almost all possible cases, 48—2 380 Miscellanea MOG Adjustment of Moments. The quadrature formulae given above can be used very conveniently for adjusting the statistical moments. We may first take the calculation of moments when the ordinates at equidistant points are n—-t known. Let Yo, ¥1, Yo... Yn—1 be given and we require | ; y,dx. Now 1 n-% % 3 n-% | ; yatte= | yedie | | ' Yn A. 3 n If we now apply Formula I. we see that we can use it for all the integrals on the right- hand side of this equation except the first two and the last two and the value of these can be taken from Formula IV. Summing the values we have n-% | an Yue = —_ {6463 yp) + 4371 4; + 6669 yy + 5537 ys +5760 (Yq aia ann + Yn —5) +5537 Yn — mn +6669 Yn —34+4371 Yy 2+ 6463 Yy 1} ceseceececeeceees xe which means that we can multiply the first and last ordinates by aes (or 1:122205), the second and last but one by a (or *758854), the third and last but two by san (or 1157813), the inIRC fourth and last but three by een (or *961285), leave all the other ordinates unaltered and work out the moments in the usual way from the modified series of ordinates which will now give the proper values for equating to the moments from the formula for the curve. If there be high contact at each end of the curve—and ordinates are known—there is no reason why Formula I. should not be used even for the end groups, and this tells us that the rough statistical moments require no modification in such a case. The above rules have been applied in actual cases, thus Formula VI. was found when tested, da o 142’ though it is greater than the best formulae given by Professor Pearson in Biometrika, Vol. 1. pp. 278, 279, is sufficiently accurate for almost all conceivable purposes. The rule in cases of high contact was tested by adding 12 ordinates of the normal curve calculated to 5 decimal places, which gave 1:24998 instead of 1:25000. A type III. curve which had high contact gave 24473 with 9 ordinates instead of 24475, so that in each case the result was practically exact. A further test can however be applied, for if we assume that areas and not ordinates are known we can apply the formulae and reach Sheppard’s adjustments. For if areas are known we are given as the rough ¢t4 moment the expression by using 12 ordinates to approximate to to give an error of —-000000205 which, 2 3 3 nz | Yanda xt | Yada (X+ y+ | Yn (X +2)EL eee +] Yanda (Xx +n—1)t, 72 2 : n-3 n-4 and we require | _ (X+2)'y,dx. The series of integrals can be written by the help of Formula I. as : 1 5760 where X is neglected in order to simplify, and working out this general coefficient we have {...-+[5178 hi +308 {(h—1)§+(h+1)% —17 (h-2)'+ (L429 ] yp to}, 1 5760 {5760 A! + 240¢ (¢— 1) ht 24. 38¢ (¢— 1) (¢ - 2) (¢— 8) A~4 ete.}. Miscellanea 381 If ¢=2 this becomes 4?+12 and if ¢=4, eg leg a We have however already noticed n- that if there is high contact the value of Vey ; ye xv)'y,dx is given by the sum of the 2 ordinates, i.e. the second moment is given by a series of which the general term is /?y, and the fourth by one whose general term is A*ty,; hence if » be written for the true and »y for the unadjusted moment we have 1 batTg=a» and a es Hat 5 Ba 35 Mas 1 or ¥2~ Jo He 1 and nog Mat gon M4. The odd moments are seen to require no modification. It is interesting to note that Formula II. will give the adjustment for the second moment but that for the fourth it gives Ya 5 2 vg+ =~ =py which only differs from the true result by = : 1 24 To demonstrate Sheppard’s oes for the nth moment a parabola of at least the nth order must be used. There is another method of a very simple character by which adjustments when areas are given can be reached ; it consists of finding ordinates from the given areas and then working on the values obtained. It is easy to obtain formulae suitable for our purpose in the following way : Let y,=a+br+cxu?+dx° + ex, then 2c fe Yn da = Ag easy aoe 80 242 64 -% | Yd =A _y=a—b +50 0= D d+ ane = : 26 i Yygda=A, =at+b taqet 64 160 ° 3 z eA = 98 544 2882 i Y_dt = Ay =G+26+57 e+ 64 d+ 160 ° -3 | 5 Yam A 90 B40 Tale dei 4°” 64 G0) These equations can easily be solved and we obtain the following result : a= 7559 (2184 do— 116 (A144 41) 49 (4-2 +Ays)} ee eae ees Nile b= ree5{ — 1860(A_1— A 41) +200(A 9-44 2)} o= a0 {- 2640 Ay +1440 (A_1+441)— 120(A_9 +A yo), d=reg{ -880(A_1—4y1)- 160 (A244), e= Zag (480-4o—320(A_1 +4 41) +80(4 -2+A,42)}. 382 Miscellanea But y=a, and if there be high contact we can form all the ordinates Yo, 7/1, 2... Pei Gai 005 by using the expression given for a, hence, if Y be the distance of y) from the vertical about which moments are calculated we have as the general term for the nth moment [2134.X"—116 {(X—1)"+ (X41) +9 (X—2)" 4 (X+42)9], which after reduction becomes n—1)(n—2)(n—-3) n—1) eee 80. A7- 24 ra [ 1920 fae 28. anal and this will be found to give Sheppard’s adjustments. The values of 6, c, d and e are given above as they may be useful in some cases other than those with which we are now dealing, e.g. for some work when parabolas are being used. They enable us to find, for instance, 1 J-2=T990 {1689 A_»+684A_ — 746A)+ 3644, coal 71Ag}, which, by the way, if applied to all terms of the series also give Sheppard’s adjustments when high contact is assumed. The method in a slightly modified form might be applied to enable us to deal with the cases in which the curve rises sharply from zero, but the difficulty is that in all such cases the actual starting point makes a very great difference to the result we obtain. In fact it is possible if we assume the curve to start at the beginning of one of the groups to obtain a negative ordinate at the middle point of the group and the result is in consequence extremely unsatis- factory, though it gives some idea of where the curve really starts. r+} Using the same notation as above, viz. | ydx=A,, and also the same method, the py following formulae can be obtained : If Yn=at ba+cx?+dx>+ex', Yo= 7999 (2134 40- 116 (A_,+A,)-49 (A_2+4s)} as above, th= p99 (— 7142+ 2044 Ay ~ 26 dy — 36.4, +943} Re Stl, —— {16804 «+ 684746 Ags 364. Au a7 eee eee XII, If y=atbe + cx, 1 Yorggt— 4-14-26 Ao Ars 0:80.00 le 0106.40 cineie vieiss 20.00 oo cleacce XIV., 1 n= gq (23A1 + 249— Aa} cisiaieleleleisla elaiajale slole'n sjalelaisisleleiaialeivelelajatete XV. In order to apply these formulae to the problem of correcting moments we first find the ordinates corresponding to the given areas by means of Formulae XI. to XV. and then use Formula X. or some similar simpler expression such as rn—k 1 | ; Yxdu= sy {264% + 21y, + 25y2 +24y3 + 24y4 +.. a eenccesccecccce XVL., met Miscellanea 383 which is formed from equations II. and V. in order to find the terms to be used for giving the corrected moments. The exact area may not be reproduced, for if we find the first term yo by XV. and the other terms by XIV. and then use XVI. we have the following result : n-% | ‘ yall =o {26 yo + 214, +25y24 24y3+...} Il eye 24 x 54 126 x 23 Ay +26 x 2A,—26 A, —21 Ay +21 x 26.4, —21 Ap — 25 A,+25 x 26A,—25A3 —etc.} 1 v 576 {577 Ay +573.A, +579 Ao +575 A3 +576 Agt...}, or the area is overstated by 1 576 lo 8A +342 — Ay}. When a parabolic curve is being fitted these formulae can be applied as they stand, for in such cases we can generally choose our range to some extent, that is the formula itself does not fix an actual starting place for our curve ; but in much frequency curve work the difficulty of adjusting the moments is, as has been already remarked, considerably increased owing to the rapid rise of the curve from zero. In such cases, especially when the terms are few in number or there seems a likelihood from the original statistics that the curve does not start at the beginning of the base of the first group, I think the formulae should be applied neglecting the first group, which should be examined separately. . Taking the following unpromising series (col. 1) from Phil. Trans. Vol. 197 a, pp. 454—456, the following table was made. TABLE VII. Frequency | Ordinate by | Modified ordinate XIV. & XV. by XVI. 47 (neglected) (neglected) 762 742°7 804°6 160°5 141°2 123°6 20 14°8 15:4 5° 4°5 4°5 1°5 1°3 1°3 996 Tf the curve starts at the middle of the base of the 47 group its mid-distance from the middle point of the base of the 762 group is really ; and this should be taken into account in calculating the moments. Now it is easy to see that the first group probably relates to a small base because the mid-ordinate calculated from 3 (28d9+24, ~ Ax) is —8. If in order to 384 Miscellanea simplify matters we assume that half a base is the only alternative to a whole one the moments can be calculated as follows : TABLE VIII. : Distance d Modified : ; F ordinate f poten 6 fd fa fa fat | 47°0 —*75 —35°2 +26'4 | —19°8 +14°8 804°6 — = — | — — 123°6 1 123°6 123°6 123°6 123°6 15°4 2 30°8 61°6 123 °2 246°4 4°5 3 13°5 40°5 121°5 364°5 1°3 4 52 20°8 83°2 332°8 996°4 137°9 272°9 431°7 1082°1 The series of numbers from which Professor Pearson found the graduated figures given in col. (1) of Table VII. was 1, 49, 798, 123, 18, 5,2; but owing almost entirely to the differences of 2°6 between 18 and 15:4 and ‘7 between 2 and 1°3 the totals that are obtained are 996, 131, 325, 473, 1393, which differ widely from those given above. The mean is very close in comparison to that obtained by using the figures as they stand in col. (1) Table VII. but the other moments are little better. I give the result of the attempt to find adjustments in this awkward case because it shows that the method can give a series of numbers very like that required to give a true result, but the small differences make the result unsatisfactory. Journal of Anatomy and Physiology. CONDUCTED BY PRINCIPAL SIR WILLIAM TURNER, K.(C.B., : ; AND : : Prorgssors D. J. CUNNINGHAM, G. 8, HUNTINGTON, A. MACALISTER, anp J. G. M‘KENDRICK. VOL. XL. NEW SERIES—VOLUME I. PART I, NOW READY—PRICE 6}.-. With Plates and Illustrations in the Text. ANNUAL SUBSCRIPTION 21/- POST FREE, CONTENTS OF PART I.—OCTOBER 1905. I. O. Onannook Brapizy, M.B., D.Sc., F.R.S.E. On the Development of the Hind-Brain of the Pig. (Plates 1.—x1.) II. E. Wacr Carer, M.D., F.R.S.E., etc. Note on the Elastic Tissue in the Hye of Birds. (Plates xu, x1.) III. Wu. Sr C. Symmers, M.B. Aberdeen. Pigmentation of the Pia Mater, with Special Reference to the Brain of Modern Egyptians.. IV. Joun McCraz, M.B,, L,R.C.P. Lond. 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The Breeding Industry “Its Value to the Country, and its Needs By WALTER HEAPE, M.A. TRINITY COLLEGE, CAMBRIDGE rn: ‘ This book isan attempt to estimate the value of the stock breeding industry, — to show the nature of the work required to foster its growth, and to consider how far the existing attitude and methods of Government help to that end, . Nee ew 3 se “e hy Le MMe RES eRe a ee Pere ere ee REN OP eT ene es | Peete = ; eT aie te es et ag LONDON: CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, hg OR C.F. CLAY, Manacer, FETTER LANE, E.C. Hie Pi Ba e Lass i anes Mie ee eee Se Votums IV MARCH, 1906 No. 4 TABLES FOR FACILITATING THE COMPUTATION OF PROBABLE ERRORS. By WINIFRED GIBSON, B.Sc., Jessel Student, University College, London. ‘\ Tue following tables have been prepared with the view of rendering the computation of the probable errors of the mean, the standard deviation and the coefficient of variation as brief as possible. It is well known that if o be the standard deviation, V the coefficient of variation, and n the number of observa- tions, then: (i) The Probable Error of the Mean = ‘6744898 o/,/n. (ii) The Probable Error of the Standard Deviation = °6744898 o/,/(2n). (iii) The Probable Error of the Coefficient of Variation = 6744898 V {1 +2 (sta) | / J/(2n). It is clear, therefore, that the tabling of the expressions °6744898/\/n, 7 \2)4 6744898/,/(2n), and Vile (aoa) would make the computations of these probable errors much easier. The question then arose whether the actual values or the logarithms of these expressions should be given. To those who work with a calculator, the latter would be of little service, while the former would still be helpful to those who have not this advantage. As nearly all biometricians now use a Tate, Brunsviga, or other mechanical computator, it was settled that the actual values should be given, With the numbers usually available in biometric work, exclusive of human vital statistics, n rarely exceeds 1000. Hence ¥, = 6744898//n and x» ="6744898//(2n) Biometrika Iv 49 IQA RAQ 386 Tables for Facilitating the Computation of Probable Errors have been tabled for n=1 to 1000. Clearly for any even value of n between 1000 and 2000 the value of y, can be found from the value of y, for jn. Thus x, for n=1594 is x» for n=797, and is found to be ‘01689. . As the first difference for x2 > 500 does not exceed ‘00002, and the second difference is zero to five figures, it will be clear that y, can for odd numbers between 1000 and 2000 be found by the simple process of halving the sum of the adjacent values of y, for 3(n—1) and L(n+1). Further, for any even value of n between 1000 and 2000, x. may be found by halving the value of x, for 4n, and for any odd value of n between 1000 and 2000 x. may be found by quartering the sum of the values of y, for $(m—1) and $(n+1). Thus suppose n=1731; we have $(n—1)=865 and 4(n+1)= 866. Hence y, =4$(01622 + °01621) =-016215, and y, =}(-02293 + 02292) = 0114625. The actual values are y, = "016212 and y, =-011463. Such differences will hardly | ever have any statistical importance. To determine the probable error of the coefficient of variation we use Table III. Here we find ‘pe vi 42 (a0) | tabled for values of V from 0 to 50. The probable error of V must therefore be found by taking out the value of x, corresponding to the given value of n,and multiplying it by the W found for the given value of V by interpolation from this table. Thus suppose V = 23°65 for 583 observations. We have by the usual advancing difference formula (Biometrika, Vol. 1. p. 175) p = 2418612 +-65 x 115861 — 82 * 33 ,.o1g16 4 © Ser 139 00045 which evaluated by the Brunsviga = 24°93774*. Further, from Table IT., y, = -01975. Thus the probable error = Wy, =°49252. The value actually found by direct calculation is “49259, the ditference, which is of no importance for practical statistics, depending upon cutting off x, at the fifth figure. While Tables I. and II. are the work of the author, Table III. is due to Dr Raymond Pearl and J. Blakeman. All the values were found to seven figures, but it seemed sufficient for practical work to register them to five. * Actual value 24°937,739, so that the y table is amply sufficient. TABLES I. AND ILI. wv Xi Xy vi 67449 47694 Z “47694 33724 3 38942 27536 i 33724 23847 5 30164 *21329 6 27536 19471 7 25493 18026 8 23847 16862 &) "22483 15898 10 21329 “15082 vial 20337 14380 12 19471 13768 3 “18707 13228 th “18026 12747 15 17415 12314 16 16862 11923 7 16359 11567 18 15898 “11241 19 “15474 “10942 20 15082 “10665 21 14719 10408 22 14380 10168 23 14064 09945 2h 13768 09735 25 13490 09539 26 13228 09353 27 “12981 ‘09179 28 12747 09013 29 12525 08856 30 12314 08708 31 12114 08566 32 11923 08431 33 ‘11741 08302 4 11567 ‘08179 35 11401 08062 36 11241 ‘07949 37 “11088 ‘07841 38 10942 07737 39 “10800 076387 40 “10665 07541 41 “10534 ‘07448 42 “10408 07359 43 "10286 ‘07273 dh ‘10168 ‘07190 45 10055 ‘07110 46 09945 ‘07032 47 09838 ‘06957 48 09735 06884 49 09636 06813 50 09539 06745 WINIFRED GIBSON 387 Probable Errors of Means and Standard Deviations. AMMAN Os O59 Or a & ds MN NN MAW Ss WN SVVenra OOnwrwew 09445 09353 “09265 ‘O91L79 “09095 09013, 08934 08856 ‘08781 ‘08708 ‘08636 08566 ‘08498 08431 ‘08366 08302 “08240 ‘08179 “08120 08062 “08005 07949 07894 ‘07841 ‘07788 OT 737 ‘07687 07637 ‘07589 ‘O7541 “O7494 07448 07403 07359 ‘07316 07273 ‘07231 ‘07190 07150 ‘O7T110 ‘O7071 ‘07082 “06994. “06957 “06920 ‘06884 “06848 06813, ‘06779 06745 ‘06678 “06614. ‘06551 ‘06490 06431 06373 06317 06262 *06209 ‘06157 ‘O6 107 ‘06057 “06009 705962 “05916 “05871 05827 05784 05742 05700 “05660 05621 "05582 05544 ‘05507 ‘05471 05435 “05400 05366 05332 "05299 *05267 05235 *05204 ‘05173 05143 05113 05084 05056 05027 “05000 "04972 04946 “04919 04893 04868 04843 “04818 ‘04793, 04769 Li a eee Ge Os Ce w WK os Co Mt Cg mM re re ee re et te Mists 9 xX ‘06711 ‘06678 06646 ‘06614 06582 06551 06521 °06490 06460 0643 1 “06402 ‘06373 ‘06345 ‘06317 “06290 06262 06236 06209 06183 06157 06132 ‘06107 “06082 ‘06057 06033 ‘06009 ‘05985 05962 *05939 ‘05916 05893 05871 “05849 *05827 “05805 ‘05784 ‘05763 05742 ‘05721 ‘O5700 ‘05680 “05660 “05640 ‘05621 ‘05601 05582 *05563 05544 *05526 "05507 03947 03934. 03920 "03907 “OB894 04746 04722 ‘04699 *O4677 04654 04632 ‘04611 “04589 04568 *O4547 "04527 “04507 “04487 ‘04467 “04447 “04428 “04409 04391 04372 04354 (04336 04318 04300 04283 04266 “04249 04232 04216 “04199 04183 “04167 0415] ‘04136 04120 “O41L05 “04090 ‘04075 “04060 “04045 ‘04031 ‘OAO1L7 “04002 “03988 ‘03974 ‘03961 388 Tables for Facilitating the Computation of Probable Errors TABLES I. anp Il.—(continued). 05489 05471 05453 05435 05418 05400 05383 05366 05349 05332 05316 05299 05283 05267 05251 05235 05219 05204 05188 05173 ‘05158 05143 05128 05113 "05099 05084 “05070 05056 ‘05041 05027 05013 “05000 04986 ‘04972 04959 04946 04932 04919 04906 04893 ‘04880 ‘04868 04855 04843 04830 04818 04806 04793 04781 04769 03724 03713 03702 03691 03680 03669 03658 03647 03637 ‘03626 03616 03605 03595 03585 08575 03565 03555 03545 03535 03526 03516 03507 08497 03488 03478 03469 03460 03451 03442 03433 03424 038415 03407 03398 03389 03381 03372 n XG Xe 201 ‘04757 03364 202 | 04746 ‘03356 203 | 04734 ‘03347 204 | 04722 ‘03339 205 ‘04711 ‘03331 206 | -04699 ‘03323 207 | -04688 ‘03315 208 04677 ‘03307 209 | -04666 “03299 210 | 04654 03291 211 04643 ‘03283 212 | -04632 ‘03276 213 | -04622 ‘03268 214 | 04611 ‘03260 215 ‘04600 (03253 216 | -04589 ‘03245 aty | -04579 ‘03238 218 | -04568 ‘03230 219 04558 | 03223 220 ‘04547 ‘03216 221 ‘04537 ‘03208 222 | -04527 ‘03201 223 | -04517 ‘03194 224 ‘04507 ‘03187 225 | -04497 ‘03180 226 | -04487 ‘03173 207 | -04477 ‘03166 228 | -04467 ‘03159 229 ‘04457 ‘03152 230 | -04447 ‘03145 231 04438 ‘03138 232 | -04428 ‘03131 233 | -04419 ‘03125 234 “04409 ‘03118 235 | -04400 | -O3111 236 | 04391 ‘03105 237 | -04381 ‘03098 238 | -04372 “03092 239 04363 ‘03085 240 | -04354 ‘03079 Q41 04345 ‘03172 242 | -04336 | -03066 243 | -04327 ‘03060 244 | -04318 ‘03053 245 | -04309 ‘03047 246 04300 | 03041 247 | 04292 03035 248 | -04283 ‘03029 249 | 04274 03022 250 | -04266 03016 nN x Xe 251 04257 03010 252 04249 03004 258 04240 02998 254 04232 02993 255 04224 02987 256 04216 02981 257 04207 02975 258 04199 02969 259 04191 02964 260 04183 02958 261 04175 02952 262 04167 02947 263 04159 02941 264 04151 02935 265 04148 02930 266 04136 02924 267 04128 “02919 268 04120 02913 269 04112 02908 27 04105 02903 271 04097 02897 272 04090 02892 273 04082 02887 274 04075 02881 275 04067 02876 276 04060 02871 277 04053 02866 278 04045 02860 27 04038 02855 280 04031 02850 281 04024 02845 282 04017 02840 283 04009 02835 284 “04002 02830 285 03995 02825 286 03988 02820 287 03981 02815 288 03974 02810 289 03968 02806 290 03961 02801 291 03954 02796 292 03947 02791 293 03940 02786 294 -03934 ‘02782 295 03927 02777 296 03920 02772 297 03913 02767 298 03907 02763 299 03901 02758 300 03894 02754 nv X, Xx 301 ‘03888 02749 302 “OB88 1 "02744 303 *OB875 ‘02740 304 ‘03868 02735 B05 *03862 02731 306 03856 02726 307 “03850 02722 808 03843 02718 809 ‘03837 02713 310 ‘03831 “02709 311 “03825 "02704 Sue, ‘OB819 02700 313 ‘03812 ‘02696 314 ‘03806 02692 B19) “03800 ‘02687 316 ‘OB794 02683 Bir. ‘03788 ‘02679 318 03782 "02675 319 ‘03776 ‘02670 320 ‘03771 -02666 821 ‘03765 “02662 822 *03759 02658 B23 ‘03753 *02654 B24 ‘O3747 02650 O25) ‘O8741 02646 326 ‘03736 “02642 Bair ‘03730 02637 328 ‘03724 02633 329 ‘OB719 *02629 330 ‘03713 02625 331 ‘03707 ‘02621 832 “03702 ‘02618 B33 ‘03696 ‘02614 334 ‘03691 “02610 335 "03685 02606 336 “03680 *02602 3837 ‘03674 ‘02598 338 03669 °02594 B39 ‘03663 “02590 340 *03658 ‘02587 341 03653 *02583 B42 ‘03647 02579 B43 ‘03642 “02575 BAA ‘03637 02571 B45 03631 “02568 346 ‘03626 02564 BLT ‘03621 “02560 B48 03616 02557 BAI ‘03610 702553 350 “03605 "02549 WINIFRED GIBSON TABLES I. ann II.—(continued). | 351 ‘03600 02546 352 ‘03595 ‘02542 353 03590 02538 B54 03585 “02535 855 03580 02531 356 ‘03575 "02528 BOM 03570 *02524 358 °03565 02521 3AQ *03560 ‘O2517 360 03555 02514 361 "03550 02510 362 03545 *02507 363 “03540 02503 364 “03535 *02500 365 03530 02496 366 03526 02493 307 03521 “02490 368 “03516 ‘02486 369 ‘03511 02483 3870 03507 ‘02479 ond “03502 ‘02476 B72 *03497 02473 BIS. 03492 ‘02469 BTA ‘03488 ‘02466 BID ‘03483 ‘02463 376 *03478 “02460 OW ‘03474 *02456 378 ‘03469 *02453 Cue, *03465 "02450 380 ‘03460 "02447 3S 1 ‘05456 02443 382 *03451 “02440 383 03446 02437 ISA “03442 02434 38H *03488 02431 386 03433 ‘02428 387 08429 | °02424 388 (03424 | -02421 389 03420 02418 5390 ‘08415 ‘O2415 | 391 | 03411 02412 392 “03407 “02409 393 03402 *02406 394 03398 | -02403 395 ‘08394 | “02400 396 ‘03389 °02397 397 03385 | -02394 398 03381 | -02391 | 899 ‘03377 ‘02388 400 03372 ‘02385 7 xX 4O1 03368 402 03364 AOS 03360 JOY 03356 AOS 03352 AUG 03847 407 03343 JOS 03339 409 03335 410 03331 411 03327 412 ‘03323 413 03319 Ald 03315 415 03311 416 03307 417 03303 ALS 03299 AID 03295 420 03291 421 ‘03287 422 03283 423 ‘03279 AZ 03276 425 03272 426 03268 427z 03264 428 03260 429 03256 430 03253 431 03249 432 03245 ABB 03241 M4 03238 ABS 03234 IBC 03230 437 03227 438 0322: ABI 03219 440 ‘03216 AAA 03212 442 03208 AAS 03205 AAA 03201 AAS ‘03197 JAG 03194 yyy ‘03190 448 ‘03187 AAI 03183 450 03180 389 Xe 02382 °02379 ‘02376 5) iD *02370 ‘02367 02364 02361 02358 *02355 "02353 *02350 “02347 “02344 02341 02338 02336 02333 02330 02327 02324 02322 “02319 02316 02313 02311 02308 *02305 02303 02300 "02297 °02295 *02292 ‘02289 ‘02287 *02284 02281 02279 02276 02274 ‘02271 ‘02269 *02266 02263 ‘02261 *02258 ‘02256 02253 02251 ‘02248 390 Tables for Facilitating the Computation of Probable Errors TABLES I. anp II.—(continued). n x, Xe 451 ‘03176 02246 452 03173 02243 458 03169 02241 4 03166 02238 455 03162 02236 456 03159 02233 457 03155) | °02231 458 703152 | *02229 459 03148 02226 AGO 03145 "02224 461 ‘03141 02221 462 03138 02219 ACS 03135 02217 44 03131 "02214 AOS 03128 02212 466 03125 | °02209 ACT "03121. | °02207 G8 O8118 02205 469 OB115 02202 470 ‘OB111 02200 471 03108 02198 472 03105 02195 4i3 ‘03101 02193 44 03098 02191 45 03095 02188 AVG 03092 02186 Air 03088 02184 4s 03085 02181 479 03082 02179 480 03079 ‘02177 ASI ‘08075 ‘02175 4S2 03072 02172 ASB 03069 02170 4S) 03066 02168 ASS *03063 ‘02166 480 03060 02163 487 03056 02161 488 03053 02159 ASI 03050 02157 490 03047 02155 AOL “08044 02152 492 ‘03041 02150 493 03088 02148 AOS 03035 02146 495 03032 02144 496 03029 02142 AIT ‘03026 02139 AIS “08022 ‘02137 499 03019 02135 500 03016 02133 nv Xy Xo nN Xt Xz 501 03013 02131 551 02873 02032 502 03010 02129 552 02871 02030 503 03007 02127 553 “02868 02028 504 “03004 02124 554 “02866 02026 505 “03001 02122 555 02863 02024 506 02998 02120 556 02860 02023 507 02996 02118 557 02858 "02021 508 ‘02993 02116 558 02855 02019 509 “02990 02114 559 02853 02017 510 ‘02987 02112 560 02850 02015 | 511 “02984 02110 561 02848 02014 | 512 ‘02981 02108 562 "02845 02012 513 ‘02978 “02106 563 02843 “02010 514 02975 02104 564 "02840 02008 515 02972 02102 565 02838 02006 | 216 02969 02100 566 02835 02005 | Lil 02966 02098 567 02833 02003 518 02964 02096 568 02830 02001 519 02961 02094 569 02828 ‘01999 3.20 02958 02092 570 02825 01998 521 02955 702089 571 02823 01996 522 02952 02087 572 02820 01994 523 "02949 “02085 573 02818 “01992 524 02947 02084 oT 4 "02815 ‘01991 525 02944 02082 5V5 02813 01990 326 02941 02080 576 02810 01987 527 02938 02078 a77 02808 01986 528 02935 02076 57. 02806 01984 529 02933 “02074 579 02803 01982 330 02930 02072 580 ‘02801 01980 531 02927 02070 581 02798 01978 532 02924 02068 582 02796 01977 Hote) 02922 02066 583 02793 01975 IBS 02919 02064 584 02791 01974 i) 02916 02062 585 02789 01972 536 02913 02060 586 02786 01970 5387 02911 02058 587 02784 ‘01969 538 02908 02056 588 02782 01967 539 *02905 02054 589 02779 01965 | 540 02903 “02052 590 ‘02777 01964 | SAT 02900 02051 591 ‘02774 01962 542 02897 02049 592 02772 01960 OS 02895 02047 593 ‘02770 01959 544 “02892 “02045 594 ‘02767 ‘01957 54S 02889 02043 595 02765 01955 546 02887 02041 596 02763 01954 IAT 02884 02039 597 ‘02761 01952 148 02881 02037 598 02758 ‘01950 549 02879 02036 599 02756 01949 950 ‘02876 "02034 600 02754 01947 WINIFRED GIBSON 391 TABLES I. ano [J.—(continued). Xy Xo nu Xy Xo t x Xy 02751 =| = +01945 651 02644 ‘01869 | TOL 02548 ‘01801 02749 01944 652 02642 01868 102 “02546 ‘01800 02747 01942 | 653 02639 01866 03 02544 | 01799 02744 O1941 | 654 ‘02637 01865 TO4 “02542 ‘01798 02742 01939 655 “02635 01864 TOS 02540 01796 02740 01937 656 02633 01862 706 ‘02538 01795 02738 01936 657 02631 ‘01861 707 02537 | °01794 02735 01934 658 "02629 “01859 TO8 02535 | -01792 02733 01933 659 02627 01858 709 02533 | ‘01791 02731 01931 660 02625 01856 710 02531 01790 ‘02729 01929 661 02623 01855 711 ‘02530 01789 ‘02726 01928 662 02621 01854 Ge ‘02528 01787 02724 01926 663 ‘02620 01852 Gls} 02526 ‘01786 02722 01925 664 02618 01851 TL4 02524 01785 02720 01923 665 02616 01849 715 02522 01784 02718 01922 666 02614 01848 716 02521 ‘01782 02715 01920 667 02612 ‘01847 717 02519 ‘01781 02713 01919 668 “02610 01845 718 02517 01780 ‘02711 ‘01917 669 “02608 01844 719 02515 ‘O1779 02709 01915 670 02606 01843 720 02514 ‘O1777 ‘02707 01914 671 02604 01841 721 02512 ‘01776 "02704 01912 672 "02602 01840 722 02510 01775 “02702 01911 673 “02600 01838 EBs “02508 01774 “02700 01909 OVA 02598 01837 TQ 02507 ‘01773 02698 01908 675 02596 01836 726 “02505 01771 02696 ‘01906 676 "02594 01834 726 “02503 01770 02694 01905 67 "02592 01833 727 02502 ‘01769 02692 01903 6S “02590 01832 728 “02500 ‘01768 02689 01902 679 02588 01830 729 ‘02498 01766 ‘02687 “01900 680 02587 01829 7380 02496 ‘01765 “02685 ‘01899 681 ‘02585 ‘01828 | 781 “02495 ‘01764 02683 01897 682 ‘02583 01826 | 732 02493 ‘01763 02681 01896 683 02581 01825 83 02491 01762 “02679 01894 OSS 02579 01824 184 “02490 ‘01760 02677 01893 685 02577 01822 1386 "02488 01759 02675 01891 686 02575 01821 736 ‘02486 01758 "02672 01890 687 02573 01820 (eu 02485 “O1707 02670 01888 688 02571 ‘01818 738 “02483 01756 “02668 ‘01887 689 ‘02570 ‘O1817 7389 ‘02481 ‘O1754 02666 01885 690 02568 01816 T4O 02479 01753 02664 ‘01884 GOT 02566 “O1814 T4HL 02478 ‘O1752 02662 01822 692 “02564 01813 742 02476 | -01751 ‘02660 ‘01881 693 "02562 01812 743 02474 | 01750 ‘02658 01879 694 -02560 ‘01810 THA ‘02473 ‘01749 02656 01878 695 “01809 | 145 02471 01747 02654 ‘01876 696 01808 T46 ‘02469 01746 02652 01875 697 01807 TH 02468 | °01745 02650 | 01874 | 698 “01805 748 02466 01744 02648 = 01872 699 i ‘01804 THO 02465 | -01743 02646 01871 700 "02549 01803 750 02463 01742 392 Tables for Facilitating the Computation of Probable Errors TABLES I. anp II.—(continued). | i X | Xe n Xe) taco ea oe mG % GOL 02461 | :01740 SO] ‘02383 ‘01685 | | S851 02312 ‘01635 V52 “02460 ‘O1L739 S02 ‘02382 01684 | | 852 ‘02311 ‘01634 753 “02458 ‘01738 S803 “02380 ‘01683 | 858 *02309 ‘01633 T54 ‘02456 01737 SO4 02379 01682 S54 “02308 01632 755 02455 ‘01736 895 02377 ‘01681 | 855 02307 ‘01631 756 02453 01735 806 ‘02376 “01680 856 ‘02305 ‘01630 "57 ‘O2451 ‘01733 Soy 02374 ‘01679 857 02304 ‘01629 158 “02450 ‘OL732 808 ‘02373 ‘O1678 858 *02303 ‘01628 759 “02448 ‘01731 809 02371 ‘O1677 859 02301 01627 760 02447 | 01730 810 02370 01676 860 02300 01626 761 02445 | ‘01729 811 “02368 ‘01675 SO1 ‘02299 01625 762 02443 | -01728 812 ‘02367 ‘01674 | S62 ‘02297 ‘01624 768 “0244957 ) 01727 813 02366 01673 | 863 “02296 *01624 TO4 02440 | :01725 S14 ‘02364 ‘01672 SOL "02295 01623 T65 "02439 | 01724 815 ‘02363 ‘O1671 | 865 02293 ‘01622 | 766 *02437 | 01723 S16 02361 ‘01670 S66 "02292, ‘01621 767 02435 | 701722 817 *02360 ‘01669 SOT “02291 01620 768 “02434 | ‘01721 S18 *02358 “O1658 SOS "02289 ‘01619 169 *02432 | -01720 819 "02357 ‘OL667 S869 ‘02288 ‘01618 770 02431 | °01719 820 "02355 “01666 870 ‘02287 ‘01617 (el "02429 ‘01718 821 "02354 ‘01665 871 "02285 ‘01616 W712 "02428 ‘OLT17 822 *02353 -01664 872 "02284 01615 1S 02426 ‘O1715 823 02351 “01662 873 02283 ‘01614 TT4 "02494 701714 S24 *02350 ‘O1661 ST) *02281 ‘01613 V75 02423 ‘01713 825 +02348 ‘01660 875 “02280 01612 176 “02421 “OTL 826 *02347 ‘01659 STO *02279 “O1611 ah 02420 | ‘01711 827 "02345 ‘01658 877 ‘02278 “01610 718 O2418e ie O17 10 828 “02344 -01657 878 ‘02276 ‘01610 Cpe) "02417 ‘O1709 829 *02343 ‘01656 879 ‘02275 -01609 TSO ‘O2415 | -01708 830 ‘02341 °01655 880 ‘022974 *01608 TSI ‘02414 ‘O1707 83. *02340 “01654 S81 "02272 *01607 782 "02412 ‘O1706 832 *02338 *01653 882 02271 ‘01606 TSS. “02410 ‘01704 833 02337 “01652 883 ‘02270 “01605 784 02409 | -O1L703 834 *02336 01651 884 02269 01604 T85 ‘02407 | ‘01702 835 *02334 ‘O1L651 885 °02267 ‘01603 7S6 ‘02406 ‘O1701 836 02333 *01650 886 *02266 -01602 787 02404. | -01700 837 02331 ‘01649 887 02265 -01601 788 *02403 “01699 838 *02330 °01648 888 “02263 “01600 789 ‘02401 01698 839 *02329 ‘01647 S89 02262 “01600 790 “02400 “O1697 S40 02327 ‘01646 890 02261 “01599 791 *02398 “01696 | S41 02326 *O1645 SI] ‘02260 °01598 792 *02397 “O1695 852 02324 01644 892 02258 *O1597 TIS "02395 ‘01694 S43 102323 ‘01643 S938 °O2257 ‘01596 794 *02394 -01693 S44 *02322 *01642 SIA "02256 *01595 795 ‘02392 "01692 S45 ‘02320 ‘O1641 S95 *02255 “01594 796 “02391 “01690 S46 *02319 ‘01640 SIC 02253 ‘01593 197 "02389 ‘O1689 Shir *02318 *01639 soy "02252 *01592 798 02388 “01688 SAS ‘02316 ‘01638 S98 02251 “01592 T99 02386 O1687 | SAI 02315 ‘01637 899 02250 01591 SOO *02385 | ‘O1686 S5O 02313 ‘016386 900 “02248 ‘01590 WINIFRED GIBSON 393 TABLE III. TABLES I. anp II.—(continued). Probable Errors of Coefficients of Variation. a = aw i = | _ | a= a F - — 7" | | | : A | a2 A X, Xe | | a X, X» v + + | + oe SS |- —| eel 02247 | 01589 | 951 | -02187 | -01547 Sy eats | 1-00010 sore taee 02246 01588 952 | ones | ois | | 2 fF Sve? | 100070 | ey | 60 02245 “01587 953 | 02185 01545 Pal? speed) aicootgon |) tek ead 02243-01586 | 954 | 09184 | -o1544 POE Sehnseteen| OOS 70: sn yen, co 02242 | “01585 955 | -02183 | 01543 ee acer iee |) 1e0OG09, «I ea heed 02241 | 01585 | 956 | 02181 | -01543_ | ele Berean |) etoog08 Nea woo 02240 | 01584 gsr | 02180-01542 | Oe eco \iatcolzee- |) “ie |nibd 02238 | 01583 958 | -02179 | -O1541 Gl rere. |p We O1e8o I, Gee Gos 02237 | 01582 959 | -02178 | “01540 | Sue cco We weOStoy 0!) tae al 108 02936 ‘O15 | q 0217 ‘O15! ; Ee 0266 : 5 02236 | -OL581 960 | 02177 | 01539 BAUM r sevaene petconeeo’: |) heen, Wer a 02235 | 01580 961 | -02176 | -01539 ba, 1-08280 E N8y 02233-0157! 962 | 02175 | 01538 | pe eee | rosea |) CF eeleue 02232 | -01578 | | 963 | 02174 | 01537 | | ve | is-a1787 |- 104680 759 | 08 02231 | 01578 | =| «964 | 02172 | 01536 A) se eaten We tense) Naan eee "092: | . Lyd | BH N917 4 523) 4 A 202 5 02230-01577 965 | -o2171 | -O1535 Fea re ag, 10620 ea] ay! 02229-01576 |= 66 «| 02170 ‘| 01535 Don aeorur0 Sea Moe 02227 | “01575 | 96r | -02169 | -01584° FO rade, a) oreot || ea. oe 02226 | O1574 968 | 02168 | -01533 bo Neaise ge |) Osseo. |) aaa eee 02225 | “01573 969 | 02167 | -01532 Pe aeeeesce’ || 100090" |) Ghana Reg “()2 Z . we ry, PY ays a © o fh Or - r. 02224 01572 970 | 02166 | -O1531 Pema SG cuiey, sli uetlOSG ||| Sige le oo 02223 | ‘01572 ort | 02165 | -01531 : epana | 112192 wx, | 49 02221 | -O1571 972 | -02163 | -01530 ol Rese oy eek MBE a ieee 02220 01570 gv3 | -o2162 | -o1529 ool ives Wk LOW | Cage eed 02219 | 01569 97h | 02161 | -01528 Seal seers sles I ara ince . q . YP ‘ “A197 Zh 20°'e : sai oie 5 2 02218 | -01568 975 | -o2i60 | -01597 A |e eee TS rm eee ne 02217 | 01567 976 | 02159 | 01597 genes 1718539 44 02215 | -01566 ovr | -02158 | -01526 eee eres (10045 | ore ll ae 02214 | 01566 978 | 02157 | 01525 Soule etearce. | el S1S90 Bh aaah lee 02213 | 01565 | 979 | 02156 | -o1524 Eel eiteuene | 228688 | tees. || a 02212 | 01564 930 | 02155 | “01524 oa Wee eta pace a era ee 02211 | -01563 931 | 02153 | -01523 rele reece | ensolnh ano rse 02209 | -01562 932 | -02152 | -01522 ease ee) AavGO als aacn. Naas 02208 | -O1561 933 | 02151 | “01521 Poh se agen |) 128203 Gear, Wee 02207 01561 98h 02150 01520 oy 37-79621 130940 1723 36 . ‘O15 “05 O15; 34 37°7262 296 2 ate 02206 | -01560 935 | “02149 | -01520 See suapeas jiaisseece ||| Gites age 02205 | -01559 936 | -02148 | -o1519 oe Pete cer OL a meee 34 02203 | 01558 93y | 02147 | -01518 a eeroee |e tsn05: || aeons |Laae 02202 | -01557 938 | 02146 | “01517 el ees ale ee0al all ieee) pase 02201 | 01556 989 | 02145 | 01517 ee legrreice ge W) o0800" | Geno le! 02200 | - | Monae 2 | 3s 5 ie ( 02200 | -01556 990 | -02144 | -o1516 yo | 45-5650 | 142789 | 999 | 30 02199 | 01555 991 | 02143 | -O1515 - 1-43709 ee) 02198 | 01554 992 | -02142 | -01514 HE MN ates, eed. .\| lagea (ez 02196 01553 993 | 02140 | ‘01514 PMU ee cece | el tr038. | “aayeo ees ‘02195 ‘01552 994 ‘02139 01513 | 6 ; 182296 149642 2033 27 “—5 * = “5 . (5 A+ ‘ re! Als lolx cated > 02194 | -01551 995 | 02138 | 01512 etl eereae aay || Gta al = Sons) (mee 02193 | -01551 996 | 02137 | -O1511 es 153734 | , 25 02192 |» -01550 997 | -02136 | -01510 Pee een atopic hares. tame 02191-01549 998 | -02135 | -O1510 eA Becigciey | one ll Asyas dl 22 02189 | ‘01548 999 | 02134 | -01509 Peel ee ren, || 160089 IL gine, lines -02188 01547 | 1000 02133 ‘01508 4 diesel ints Men ape sain 22 50 61 “23724 | 2177 | | | | | Biometrika ty 50 COOPERATIVE INVESTIGATIONS ON PLANTS. Ill. ON INHERITANCE IN THE SHIRLEY POPPY *. SECOND MEMOIR. CONTENTS. PAGE (1) Material and Manner of Fertilisation : ; ; : ; : : : 394 (2) Nature of the Observations : : : ; ; ; ; ; , . 398 Influence of Environment on the Shirley Characters : ; : ; : 401 (4) Influence of Selection of Maternal Plants ; : : 3 : : : 403 (5) Influence of Variability of like Organs—Homotyposis—On Coefficients of Heredity . ; ; : : ; ‘ : : : : : : d 409 (6) Parental Correlation . : : ; : : 2 : : : , 9 410 (7) Fraternal Correlation. — . : ‘ : : : : : : ; ; 413 (8) General Conclusions . : : : : : F : : : ‘ : 415 Appendix: Correlation Tables . 5 : : : ‘ : : é ‘ 416 (1) Material and Manner of Fertilisation. A first memoir on the subject of inheritance in the Shirley Poppy was published in November, 1902+. In summing up our results in that paper we stated that: “Most plant organs being multiple in appearance, we have to apply special methods to deduce the intensity of heredity from multiple observations on the individual. It will probably be better in future experiments to confine attention to the first or principal flower, instead of using the indirect method of homotyposis; but this will involve the observation (previous to harvesting) of individual plants in large series—500 to 1000—and much increase the labour of superintendence and observationt.” We continued : “ Notwithstanding the difficulties referred to above we find that for a variety of plant characters in the Shirley Poppy the values of hereditary influence found are on the whole in fair agreement with the like values in man.” * These experiments have been materially assisted by the Grant made to the Department of Applied Mathematics at University College, London, by the Worshipful Company of Drapers. The establish- ment of a paid calculating staff has in addition largely accelerated the labour of reduction. + Biometrika, Vol. 11. pp. 56-100. t Ibid., p. 83. _ BIOMETRIKA,VOL IV. PART IV. __ TYPES OF SHIRLEY POPPY PETALS PAINTED FROM NATURE BY MISS VA BIGGS FOR THE MEMOIR ON INHERITANCE IN THE SHIRLEY POPPY. BIOMETRIKA,VOL IV. PART IV. 41 Ww : 49 L 33 Re orn, fF] TYPES OF SHIRLEY POPPY PETALS PAINTED FROM NATURE BY MISS EVA BIGGS FOR THE MEMOIR ON INHERITANCE IN THE SHIRLEY POPPY. { ar a ; as ; n . F y Til. /nheritance in Shirley Poppy 395 The object of the first series of experiments, like those which have followed it, was to ascertain the intensity of heredity in a free plant population, and to com- pare such heredity with that of similar populations in other types of life. If we have to deal with evolution in actual living forms in which isolation of individuals is not the rule, it is the law of inheritance in populations with which we are concerned. We desire to know the statistics of sub-classes, and the proportions in which those sub-classes reproduce themselves. A statistical investigation of this kind cannot be opposed to any ultimate analysis of the result of crossing indi- viduals; it can only indeed serve as a criterion of the truth of any hypothesis based upon individual crossings. At the same time it avoids the dangers which arise from neglecting what we may term the “law of small numbers.” It is too often forgotten that an imposing display of numbers, obtained by dealing with the numerous offspring of a number of crossings which can be counted on the fingers, must on any hypothesis of parental gametic constancy be given the weight of the number of crossings and not of the number of offspring, when the results are applied to the race as a whole. In our earlier experiments we were not certain to what extent Shirley Poppies grown in masses were cross or self-fertilised *. This doubt left a double interpre- tation open when we came to deal with the resemblance of siblings. Our results in the previous investigation were in general accordance with those for man, provided we assumed that the bulk of the plants were cross-fertilised. Another point of considerable importance was the question of whether any really well-defined colour sub-groups could be asserted to exist in the case of the Shirley Poppy. In order to determine this point water-colour sketches of the petals of a long series of the poppies were kindly made for us by Miss Eva Biggs immediately after gathering. It was soon seen that in distribution of colour, amount of colour, and shade of pigmentation there was practical continuity. We found it impossible to draw a rigid line between scarlets, reds, orange-reds, terra- cottas, rose-pinks, pinks, and pink-whites, although such terminology served fairly well for rough classification. There appeared always to be transitional individuals, although such individuals might be numerically few. The ocular demonstration of this was difficult even in Miss Biggs’ paintings, because the colour of the poppy is very difficult to match in water-colours, and being matched is very evanescent. We believe that Miss Biggs’ careful work fully achieves what is possible in this direction. But the task becomes harder still when the colour printer has to match the artist’s tints. If honest, he frankly tells you that he can approximately catch the artist’s shades, but that he can only do so by using non-durable printer's inks. The alternative before the Editors was therefore to produce something like the artist’s work which will not be permanent, or something unlike it which will be permanent. The latter seems to fail entirely in its object, for while it might show continuity, 1t would give colours which we should be told were not even approxi- nately those of the Shirley. We have therefore determined on the second course, * Loc. cit., p. 62. 396 Cooperative Investigations on Plants namely, to allow the colour-printer to reproduce in possibly evanescent colours, which will show for some time at least the apparent continuity of pigmentation in the Shirley Poppy. It will be seen from our plate that while white-pinks, pinks, rose-pinks, terra-cottas*, and reds are fairly safe broad classifications, there is no absolute discontinuity when we come to deal with individual instances. In the present experiments the observations were all made on the first or apical flower. They had to be made while the plant was in blossom. The seed of the apical capsule observed was taken and marked with the new pedigree number. A second and sometimes a third bud on the same plant was enclosed in a bag of very fine bolting cloth, or for about half the experiments in oiled paper bags obtained from Cologne. ‘These were invariably used in the second year’s experi- ments. The advantages of the bolting cloth bag are the free admission of air and hight, but it becomes heavy in rainy weather and much damage was done in the wet season of 1903. It is possible that very minute insects may occasionally penetrate it, but the risk here is not much greater than with paper bags, however carefully they are tied round the stem. In the spring of 1903 series of 100 packets of pedigree seed were issued to a number of cooperators. Each of these packets contained seed from one plant only, and the ultimate origin of the seed was the 1899 Hampden crop of Pearson, although there had been several inter- vening generations. In the spring of 1904 series of 100 packets of apical capsule seeds of 1903 and of 50 packets of bagged capsule seeds of 1903 were issued, and the same process was followed of collecting the seed of the apical capsules and of a secondary bagged capsule. To test fertility the seed in the free and bagged capsules was merely examined after gathering and not again sown. The following are the 1903 crops: (A) Parkstone Crop. This was grown and observed by H. Blanchard. The crop was sown on May 13th, the first seedlings were observed on the 20th, and the thinning took place between June 13 and 15. Of the 100 packets of single capsule seed provided all produced plants and in most cases more than sufficient plants to get 10 siblings. In one case only one seed and in a second three seeds alone germinated+. But the observer was able to deal with nearly 1000 individual plants. The weather was extremely unfavourable during the harvesting and much harm was done to the bagged capsules. Tabulation and calculations by Alice Lee, D.Sc. (B) Arnclife Crop. This was grown and observed by the Rev. W. A. Shuffry, and 50 packets of seed were sown on May 30; first seedlings observed June 10, thinned July 13 and 20. Second series of 50 sown June 18, first seed- * In the original Hampden Shirley Poppies the terra-cotta was extremely rare, if it occurred at all. It is still scarce, but is creeping up in numbers, and may ultimately need a special class. As meanwhile it is being introduced commercially, we cannot say whether it is a natural evolution or has been introduced by cross-fertilisation into our crops. + One of these plants also gave a distorted apical capsule with a ‘stigmatic labyrinth” instead of bands. Ill. Jnheritance in Shirley Poppy 397 lings observed June 25, thinned July 29-30. Out of the earlier sowing, none ; out of the later sowing four sets failed to germinate. There were, however, a number of series in which less than ten plants failed to appear or be observed. The lateness of the Dale* season and the badness of the weather were much against this crop. Tabulation and calculation by J. Clerk-Maxwell Garnett. (C) Streatham Crop. This was grown and observed by W. Palin Elderton. Sown on April 30, thinned at end of July. There was no case in which none of the seed of the 95 packets sown failed to germinate, but in some cases very few plants were grown. The weather was very wet during the whole of the flowering season and also very windy; many of the plants were much damaged. The bagged capsules suffered very much, and the seed had to be gathered earlier than was desirable. The tabulation is due to Mary Beeton, and the calculations to Alice Lee and her conjointly. (D) Boar's Hill Crop. This was grown and observed by A. D. Darbishire. Sown April 23-25, first seedlings observed May 7, thinned June 25 to July 1. Out of 42 packets of seed sown at least 40 germinated. But unfortunately the observations were not made on the principal flower, but on the 2nd, 3rd, 4th, and even 5th flowers, so that the material is not comparable with the other observa- tions of this series. Not yet tabled or calculated +. (E) Kidderminster Crop. This was grown and observed by John Notcutt. Seed sown April 17, first seedlings observed April 29, thinned June 25. Of the 100 packets sown all germinated, and there were nearly 1000 plants observed. The tabulation and calculations are by Mary Beeton. With regard to the seed provided for crops (A) to (E), it must be remarked that in making up the packets fertile capsules were directly chosen and those capsules which provided unfertilised seed deliberately rejected. This method could not be followed in the seed for the 1904 crops, because the harvest showed that the bagged capsules contained apparently no fertile seed. Accordingly the capsules for the seed-sowing in 1904 were taken from the 1903 (A) crop at random, each capsule having been at harvest placed in a bag whether or no it contained fertile seed. The 1904 crops were: (F) Crewe Crop. This was grown and observed by H. E. Ward. Sown May 18, first seedlings observed May 27, thinned June 30. Out of 129 packets of free apical capsule seed 92 germinated. This may be taken to represent the reduction in fertility due to no selection of fertile seed capsules. In 1903 there was practically no failure to germinate. Out of 50 bagged capsule packets 46 failed entirely to germinate, and 4 germinated, giving respectively 5, 2, 1, and 1 * Arncliffe is in Upper Wharfdale. + It is hoped that enough material may ultimately be accumulated to test the difference between sibling resemblance’ measured on apical and on other flowers. 398 Cooperative Investigations on Plants actually developed plants; the 92 which germinated of the free capsules gave 400 actually observed plants*. The tabulation and calculations are due to Dr A. Lee. (G) Bushey Crop. Grown by H. Payne, but not recorded owing to his appointment to the South African College. The bagged capsules, however, failed to give seed which germinated, while the free capsules gave a very good crop. (H) Rushden Crop. Grown by G. Abbott. Mr Abbott’s time was so occupied by other matters during the flowering season that few records were made of the flowers, but the same result occurred in this sowing as at Bushey—the bagged capsules gave no germinating seed, the free capsules a plentiful supply. We may therefore conclude that while a plentiful supply of fertile seed is obtained when free capsules growing in the midst of a poppy crop are collected, practically no seed is obtained when we deal with bagged capsules, at any rate in seasons like 1903 and 1904. It seems reasonable therefore to infer that the Shirley Poppy is in the bulk cross and not self-fertilised+, and we should accord- ingly expect in the resemblance of sister plants the intensity to approach what we are familiar with in other types of life as that of half and not whole siblings. We may remark that the observers noted continual visits of wasps, bees, and other insects to the crops. As far as could be determined from a careful examination of the unused bagged capsules of 1903, they presented no fertile seed whatever. (2) Nature of the Observations, A full crop consisted of 100 to 150 plots each sown with seed from a single pedigree individual plant. In each plot at a maximum ten and of necessity sometimes fewer individual plants were selected and the characters of their apical flowers observed. The characters were as follows: (i) Petals. (a) The number of petals. (b) The number of petaloid stamens. (i) Number of stigmatic bands. These being counted at the periphery and not at the centre. (ii) Colour of the “middle third” of the petals. This was recorded for one petal of the flower only, unless the petals were unlike, when the colours of both petal pairs were given. The colour categories used were: Red (7). Pink-white (pw). White (2). Red-pink (ip). Veined-white (vw). Red-white (7w). Pink (p). * Tf these beds had not been thinned, and if all the plants in some of the beds had been observed, there would have been far more, i.e. 400 represents the minimum, 9 the maximum fertility of the two groups. + Another, we consider fairly strong, argument in favour of wide cross-fertilisation can be found in the occurrence of poppies with dark pollen or dark basal patches. Such ‘‘rogues’’ occur where there is no trace of black in the maternal plant, but they generally appear as isolated units ; some- times, but rarely, two or three cases appear in a fraternity. If the whole fraternity were due to one pollen parent, they could hardly fail to be more frequent in the group. IIL. /nheritance in Shirley Poppy 399 Scarlet was tabled under red, the few terra-cottas and salmon-pink under red- pink, and intermediate colours by intermediate values (e.g. p or pw). If the petal pairs were of different colours, they were recorded as p x w, say, and tabled under intermediate values. Veined-white was used for white faintly pink or for white with distinct pink veins. (iv) Breadth of margin. The classes used were : Broad (6). Slight (s). None (7). Broad to slight (bs). Slight to none (sn). “None” signifies that the border of the petal was of the same colour as its body or middle-third. F (v) Colour of the margin, if it existed. This was only completely noted in the Parkstone crop, and it enabled us to form a classification of the flowers by colour at the border of the petal; this was the same as in the middle-third, if there were no separate margin. Colour scale used as in (iii). (vi) Intensity of the wrinkling of the petal. This was observed some time, 12 to 24 hours, after the opening of the flower. But while frilling is a marked feature of the Shirley, this character was not a satisfactory one to observe, and but little weight can be given to the results. The following classes were adopted: Very much wrinkled or frilled ( ). Slightly wrinkled (sz). Wrinkled (w). Not wrinkled (nw). (vii) Base of petal. (a) If the petal base differ in colour from the middle-third it may be said to have a basal patch, and the following classification of basal patches was adopted : None (7). Well defined (d). None to slight (ns). Well defined to large (dl). Slight (s). Large (/). Slight to well defined (sd). (8) If there were a basal patch its colour was noted; white, black and lilac basal patches were recorded. (viii) The shape of the capsule cover, whether flat or conical, the colour of the pollen, and all abnormalities were further noted in a separate column. Actual shape of capsules was only fully noted in the Crewe crop. The following are samples of entries from the records: B Petals | | wel | Pedigree Bands | Colour | Margin |} Wrinkling | Remarks (a) | @) | —@ | @) 144 | | 30°16 4 0 12 | p nm | sw | @ w Pollen brown 826 23°3 C. F. 400 Cooperative Investigations on Plants The first number (144) in the first column refers to the original Hampden great-grandmaternal plant, the second (30°16) to the Kidderminster I. grand- maternal capsule, the third (826) to the Parkstone maternal capsule, and the last number is the key number of the Crewe crop. C. F. signifies that it was from a free bud, or 826 capsule was in all probability cross-fertilised. Again : = = - ee ee oe ae Sierra re . Petals ae | eee Base Pedigree | Bands | Colour | Margin | Wrinkling Remarks (4) | ©) | (@) |) = -|— = | 149 6 | 3 | 13 | r bs n | di | w | Conical topped capsule | 26°15 D. 47:1 | | | | | This gives a grand-daughter capsule from Arncliffe of an original Hampden plant. It will be seen that with the crop-books before one it is possible to rapidly look up the pedigree of any plant, although the pollen ancestry must remain unknown. By observing characters on the apical capsules only we at first considered that we should get rid of the difficulty of having to allow for homotyposis, i.e. the variability of character within the individual plant. This variability, as was pointed out in the first memoir, is very considerable. But as the work went on the question was again raised as to whether the apical flower is as true a measure of individuality as the totality of flowers on the plant. This point has been to some extent discussed in the first memoir on the Shirley*, and is by no means easy to answer. We found in that paper that offspring from seed of all the capsules of a plant had a greater average resemblance to the plant than offspring from seed taken from a single capsule, while the offspring were more like each other when their seed was taken from a single capsule. If the arrangement of buds on a plant were as definite as the digits of a man, we should have no hesita- tion in comparing the characters of the apical flowers of mother and daughter plants. But this is certainly not the case, and when we deal with characters which are by no means peculiar to the apical flower it becomes a very difficult point to determine how far the variation within the individual plant can or must be allowed for. Characters like “wrinkling,” number of petals and number of stigmata have a very considerable variation within the individual. Frequently when the general tendency of the flowers of the plant is to a high number of stigmata, or to an abnormal number of petals, this tendency may fail to be repre- sented on the first flower. Other characters like colour and margin may possibly vary less from flower to flower of the same plant; but are nevertheless sensibly influenced by environment. Lastly, a very considerable selection takes place of the capsules which give any germinating seed. The mother plants which provide germinating seed are far from a random sample of the general Shirley population. * Biometrika, Vol. 11. pp. 73, 74, and p. 82. Ill. Jnheritance in Shirley Poppy 401 We have then the following points which must always be borne in mind and which render in our experience the problem of inheritance in plants extremely complex. (a) The wide-reaching effect of a changed environment, whether this be a change of climate, soil, or season. (b) The effect of selection in modifying the intensity of inheritance; the variability of the actual mother plants is for a number of characters far less than the variability in the general population. (c) The difficulty as to the multiplicity of lke organs in the individual and the choice as to which of those organs is to represent the individual. We propose to consider these points in the following sections. (3) On the Influence of Environment on the Shirley Characters. In the first place let us consider the type and variability in number of stigmatic bands—a character which we can compare with the previous crops. Of course this comparison can only be partial, because in the earlier crops this charac- ter was recorded for every flower on the plant, while in 1903-4 only the first or apical flowers were dealt with. Further, for the earlier crops the mean and variability are those found for the homotyposis tables, while in the 1903-4 crops they are taken from the fraternal correlation tables. TABLE IL. Change of Type and Variability with Environment. Mean No. of Crop Stigmata Sab: C. of V. Parkstone, 1903 a 14:08 1°562 11°13 Enfield J, 1900 ... see 13°78 1518 11°02 Kidderminster IT, 1903 ... 13°58 1-880 13°84 Crockham, 1900... na 13°39 2°055 15°35 Kidderminster I, 1900 ... 1BeLS 1°702 12°91 Bookham, 1900 ... BBP 13°14 2-330 17°73 Crewe, 1904 Bee ec 13°06 1°687 12°92 Enfield ITI, 1900 ... are 12°72 1‘717 13°50 Hampden, 1899 ... ae 12°61 1°885 14°95 Chelsea, 1899... ate 12°37 1°680 13°58 Arncliffe, 1903... aen IPD PALE) 17°48 Chelsea IT, 1900... “ee 12°04 1694 14:07 Highgate, 1900... ... 12-03 1°890 15°71 Oxford, 1900 tate sale 11°39 2°173 19:08 Streatham, 1903... ate 11°39 1°507 13°20 Considering that the probable error in these cases is a small quantity in the first or second place of decimals, we reach a very wide range of difference in type and variability occurring in different localities and seasons. We see at once that plants growing under diversities so great as are evidenced above are not directly comparable. Without laying too much stress on this table we may remark that the mean variation of the first seven entries is 1°82, and of the last eight 1:83. In other Biometrika ty 51 402 Cooperative Investigations on Plants words, it does not appear that the circumstances which lower the mean of the stigmata are the same as those which lower variability. If we classify in order of variability we find the seven most variable stations give a mean of 12°6 bands, and the eight least variable a mean of 12°8 bands, two values, the difference of which is only doubtfully significant. In fact, it will be found that very fertile crops like Kidderminster I., Enfield I., and Parkstone rank in the same class of lesser variability with starveling crops like those of Streatham and Chelsea; while the high variability group embraces low types lke those of Highgate, Arncliffe, and Oxford. The high situation, cold and late season of Arncliffe reduced the number of stigmata, but left the plants highly variable, the potting at Oxford produced the same result, but the causes which produced the starveling crop at Streatham were as influential in reducing its variability as in lowering its type. The general inference to be drawn is that both in variability and type the influence of environment, local or seasonal, on plant life is extremely large, and that careful investigations of the correlation of locality, soil, and seasonal meteorological conditions with plant type and variability would give valuable results. Meanwhile we can only adopt methods of studying resemblance, which as far as possible eliminate this influence. TABLE II. Percentages of Certain Characters in Different Crops of same Season. Character Crop Percentage With more than 13 Stigmata ies er Parkstone 63-0 » oF os bes au Kidderminster IT. 53°3 ” ” ” goo one Arnclifte 24°83 ” ” » ae a Streatham 88 73 > ‘ ahha ale , 1 With more than four Petals or ae i Pantavene 176 some petaloid stamens ” » ” p ee Kidderminster II. 16°8 ” ” ” ” eee Arnclifte 16°2 ” ” ” ” state Streatham 13°8 Percentages of Flowers with some red... Kidderminster IT. 35:0 ” ” ” % Arncliffe 33°8 »” ” ” ” eee Parkstone 25'5 ” ” ” as Pe Streatham 25-0 Without margins... ee aC ee Kidderminster IT. 72°6 s 5 ise ies cae ae Parkstone 71°3 e * ae ee aie aes Arnclifte 63°9 ” 97 tae vas wae ies Streatham 61:0 With slight or no basal patches ... Bud Streatham 52°4 H 0 5 ees wee Kidderminster IT. 36°4 ”» » ” Bae a Parkstone 30°7 ” ” ” Arnclifte 18°4 With slight or no wrinkling ies oe Parkstone 92°9 ” » 5 ae ae Kidderminster IT. 78°7 ” ” » ox es Streatham 78°6 ” 9 % se ts Arncliffe 65°6 Ill. Jnheritance in Shirley Poppy 403 Another suggestive manner of examining the influence of environment is to consider the percentage of certain characters observed in the different crops of one and the same season. We first place all the characters together without endeavouring to analyse how far the results may be due to differences of standard in different observers. Now examining this table we see that although the seed for these four crops was all taken from a single local crop by sampling fertile individuals at random, yet the environment influenced widely the results. This is clear, because in counting the number of stigmata or noting if margins existed there was no room for sensible influence of personal equation. Luxuriant crops like those at Parkstone and Kidderminster had more stigmata, petals, and petaloid stamens than a poor crop like that at Arncliffe, or a starveling one like that at Streatham. Further we see that nearly the same order is preserved for the percentages without margins, or we suspect that poverty in a crop increases the number of individuals in whom the pigment does not reach the edge of the petal. With the other characters the results are not so congruous. Arncliffe and Streatham, which agree in having petals well margined, are different in basal patch frequency, Streatham showing more flowers without basal patch than any other crop and Arncliffe fewer. In wrinkling the results are not very definite, but indicate that the more luxuriant crops have less wrinkling ; while reds appear to be favoured by the bleaker north. The chief point to be noticed is that there is immense diversity from one crop to a second, and that this diversity is hardly greater in characters like colour and base, where personal equation may be influential, than it is in stigmata and number of petals, where it must be of small or no account. All the above percentages have been deduced from the results weighted with the number of sister plants, so as to bring out the influence of change of environment on the results we have to work with in measuring sibling resemblance. These great differences between local crops of the same season forcibly suggest that individual plant-environment within the local crop, depending on subtle and inappreciable differences in soil, aspect, and amount of moisture, will in the case of material thus susceptible produce very sensible differentiations. We believe that a good deal of the irregularity, and possibly the lessened intensity of the inheritance coefficients for plant life, is due to this marked susceptibility of plants to small differences of environment. But until crops can be grown year by year in the same place, and under nearly similar conditions, it will be impossible to measure these conditions, or accurately deter- mine their influence on the intensity of inheritance. (4) The Influence of Selection in Modifying the Intensity of Inheritance. If p be the actually observed correlation, and r the correlation if all members of a race produce offspring, then it is well known that any selection of parents, natural or artificial, tends to reduce p below the value of r. If the selection be stringent, then p may be so much lower than r that it may appear to have no relation to it at all, and fallacious conclusions be drawn as to the variability of inheritance for different species or for different characters in the same species, 51—2 404 Cooperative Investigations on Plants the real fact of the case being that these species or characters are being subjected to very different intensities of selection. If the distribution of the character be approximately normal, and if wu be the catio of the variability of selected parents to the variability of the general popula- tion for a given character, then it is known that the true value of inheritance r is closely given by: = P ‘ v1-(—#) (1 —p') Thus with anything like stringency of selection p may be quite small, and the value of 7 completely screened. We may illustrate this on various characters. (a) Stigmata. In the case of a crop under very favourable conditions, or where every plant is separately grown as at Oxford, we have a high variability, amounting to 2 or over. But at Parkstone the actual variability was only 1°56, and on a random selection of 129 seed capsules from this crop only 92 sets germinated at Crewe, and of these sets the maternal plants had a stigmatic band variability of 14590. Thus ~=°7295 for the stringency of selection of the maternal plants at Crewe, as compared with Shirley Poppies at large. (b) Katent of Margin. If we take the 971 plants at Parkstone and the 408 offspring at Crewe from certain of them we note the following distributions (see p. 417): b bs s sn n Totals Parkstone General Population es 52 73 84 71 691 971 Parkstone Mothers of Crewe Daughters ... 1 58 64 10 275 408 | Or, dividing into three groups, we have the percentages: Broad Margins Slight Margins Insignificant General Population... 129 87 78:4 Mother Plants via 14°5 15°7 69°8 We see that there has been a considerable alteration in the distribution of margins, between the actual mothers and the general population from which they were selected at random. Let us treat the distribution of margin as approximately normal to get some idea of the stringency of selection. In this case if d be the range of margins classed by this one observer as slight we find, if o be the standard deviation of the general Parkstone population and a’ that of the selected mothers : d = 34400 and d='5497o". Hence j= jo = "62589: Ill. Lnheritance in Shirley Poppy 405 Thus a fairly stringent selection of margins took place between the Parkstone general population and those members of it who actually provided germinating seed, and so daughter-plants at Crewe. But this is not the whole selection. Parkstone has for stigmatic bands a low variability, as compared with what we know of Shirleys in general. It is extremely unlikely that it has not also a reduced variability in extent of margin. How much, however, it is impossible to say, for we cannot compare Mr Blanchard’s estimate of what is classed as “ slight ” in margin with estimates on other crops by other observers. We have no know- ledge of the relative personal equations. Thus the above stringency of selection can only be considered as a minimum. vw in reality is probably considerably lower. (c) Number of Petals. If we consider number of petals we find : Parkstone General Parkstone Mothers of Kidderminster IT. Population Crewe Crop General Population 4 Petals 758 333 798 4 Petals and oe 89 21 47 petaloid stamens More than 4 Petals 125 with and without 41 122 petaloid stamens aS _ = 972 395 967 Or, reduced to percentages : 4 Petals 78:0 84:°3 82°5 4 Petals and sont 9:9 5:3 re petaloid stamens More than 4 Petals with and without 12°8 10°4 12°6 petaloid stamens Taking d as the range from normal four petals to the end of the range of normal four petals accompanied by petaloid stamens we have, if op, o’, and ox be the respective standard deviations : d = 36140 p = 26130’ = 20900. We see that in this case the Parkstone maternal plants are an “inverse” selection of the general population, being more variable than the general Parkstone crop, but they are considerably less variable than a typical crop like the Kidder- minster. Thus on the general population of Shirleys there is a stringency of selection in the maternal plants of the Crewe crop at least equal to w=°8 and possibly more intense. At Streatham even the percentages being 861, 2°3, and 116 respectively, we have d='11030 5, and the stringency of selection of the Crewe maternal plants would give w as low as °53. But there are all degrees of petaloid stamens, and all observers may not have noted equally every slight deviation from normality in the stamens of their flowers. 406 Cooperative Investigations on Plants (d) Colour of Middle Third of Petal. We have the following results : Parkstone General Parkstone Mothers Population of Crewe Crop Some red in Petal* 240 153 All Pink ae 481 161 Some White fay 252 94 973 408 Or, in percentages : Some red in Petal 24:7 37°5 All Pink siaia 49°4 39°5 Some White ats 259 23°0 Taking the all pink as our reference group we have d = 1:05620’ = 13315, or p=a'/o = 1:2606. In other words, there has been such a selection of red mothers that the mothers are more variable than the general population at Parkstone. The mean poppy, which in the general population is about half-way + up the pink range, has in the maternal plants crept nearly up to 1/8 from the beginning of the red groupf. If we turn to the Crewe offspring we find the following values: Mothers ; Daughter Plants 153 37°5 p.c. 139 34:1 p.c. 161 39°5 p.c. 199 48°8 p.c. 94 23°0 p.c. 70 17°1 p.c. 408 408 Thus we see regression towards the pinks. In terms of d the pink range we have for o” the standard deviation of offspring o” x 13586 = d, or o/c” = 1:02 nearly. Thus the variability of the general population at Crewe is sensibly the same as that of the general population at Parkstone, but the mothers who provided germinating seed at Crewe were redder than the average Parkstone poppy, and the resulting offspring, though still regressing slightly towards the red, are yet about + off the boundary of the red§, instead of half-way across the pink range as in the Parkstone crop. The exact interpretation of this inverted selection is difficult, and has not yet been theoretically worked out. The formula on p. 404 involves approximate normality of the distribution, and becomes meaningless when the selection curve does not theoretically fall wholly inside that of the population from which selection takes place. The distributions of colour percentages at Kidderminster and Arncliffe are : Some Red All Pink Some White Kidderminster ... 34°5 40:0 25°5 Arncliffe He 33°8 40°4 25°8 * The two groups include under red 14 and 2 flowers respectively, which were noted to have a certain amount of white with the red or red-pink. + °6851 to 1:3315. + °3186 to 1:0562. § *4106 to 1:3586. Hl. Jnheritance in Shirley Poppy 407 and thus much alike, and very close to the Parkstone mothers of the Crewe crop. It seems therefore impossible to assert that the maternal plants at Crewe, although certainly an “inverse” selection from the Parkstone crop, are really a selection from the general population of Shirleys. We can only consider them as an average sample. (e) The Basal Patch. With the sole exception of wrinkling the basal patch is the most unsatisfactory of the characters observed. Not only do the flowers showing no basal patch vary very much in percentage with the environment, but there is undoubtedly a considerable amount of divergence in the estimations made by different observers of the categories. We have the following results : Parkstone General Parkstone Mothers Crewe Crop Population of Crewe Crop n ns |= or none 295 134 53 s sd . i } = Definite ~ 342 128 214 i } = Large 332 137 132 969 399 399 Or, in percentages : Parkstone General Parkstone Mothers Crewe Crop Population of Crewe Crop Slight or none 30°4 33°6 13°3 Definite one 35°3 32°1 53°6 Large 506 34°3 34°3 33°1 Thus we see that while the maternal plants of the Crewe crop were a quite reasonably random sample of the Parkstone population, the distribution of Crewe offspring is totally different. How far is this due to environment, and how far to personal equation? The corresponding percentages at Kidderminster, Arncliffe, and Streatham are: Kidderminster General Arncliffe General Streatham General Population Population Population Slight or none 36°5 18°4 52°4 Definite ae 50°4 47°7 30°9 Large wee 13°1 33°9 16°7 If these greater differences are not due to personal equation, which series can be taken as a typical crop of Shirleys? We confess that there does not seem to us any means of separating out the environmental influence from the personal equation in this case. All we can say is that there is no evidence that the maternal plants at Crewe were a selection from the general population of Shirleys, and if they were, we have no means of measuring the stringency of that selection. (f) Wrinkling of Petals. This, which is a marked feature of the beauty of the Shirley, is the most unsuccessful of all the characters recorded. The instruction to observe it between 12 and 24 hours after opening of the bud could not always 408 Cooperative Investigations on Plants be carried out, and there is little doubt that it is not only a feature much modified by season and environment but even by the interval between opening of the bud and the time of record. a. The following table gives our results: Parkstone General Parkstone Mothers Crewe General Population of Crewe Crop Population Frilled and Wrinkled ... 70 12 137 Slight Wrinkling eee 346 112 234 No Wrinkling a 553 275 28 969 399. 399 Or, as percentages : Frilled and Wrinkled 72 3°0 34:3 Shght Wrinkling eee 35°7 28°1 58°7 No Wrinkling 500 57°1 68°9 70 Now it will be clear that while the maternal plants of the Crewe crop are a selected group of the Parkstone crop, yet there is no relation between the per- centages in these mothers and in the offspring. The 70 per cent. of not wrinkled maternal plants actually produced four times as many wrinkled and eight times as many slightly wrinkled as not wrinkled plants, and the 3 per cent. wrinkled produced twice as many not wrinkled and slightly wrinkled as wrinkled plants !* In fact while the Parkstone crop had more non-wrinkled poppies than any other and the maternal selection 12 per cent. more than the general Parkstone popula- tion, the Crewe offspring had only 7:0 per cent. of non-wrinkled poppies, less than half that shown by the most wrinkled crop, Arncliffe, which showed about 15 per cent. Whatever changes took place in environment and season between Parkstone and Crewe they were sufficient to totally upset wrinkling as an inherited character ! Some possibly, but far from all, of this difference we believe may be due to personal equation. The main feature to be noticed is that whereas the southern crops, Parkstone and Streatham, have more than 45 per cent. of unwrinkled flowers, the northern crops have less than 17 per cent. (Arncliffe 14°9, Kidderminster 16:4), sinking to 7 per cent. at Crewe. If we confine our attention to the Parkstone crop and take d for the range of slightly wrinkled we have: d = 1:3860 x o’ =1:2812 6, Or the stringency of maternal selection =p=a'lo= "9244. If we take the Oxford crop of 1900 as one in which the wrinkling was very carefully observed, we find d = 1:1946 o, or we have p= o/c ='8620; so that it is quite possible that the stringency may be greater than that indicated by the Parkstone crop. * See Appendix I., Table F. Ill. Lnheritance in Shirley Poppy 409 (5) The Influence of Variability of like Organs in the Plant on Coefficients of Heredity. Homotyposis. We have already seen that if all the flowers of a plant were identical, or if the first flower were as markedly differentiated from other flowers as one digit in a man is from a second (i.e. if there were a very marked correlation between order of flowering and character), the problem of inheritance in plants would be much simpler. But the variation of like organs in a plant is very considerable, and in many cases the degree of variation is not highly correlated with the position or order of production. Now if there were no such correlation, it has been shown elsewhere that the extent of the variation of the individual plant—its homotyposis —would be properly measured by the coefficient of correlation of like organs. Hence our selection for record of apical flowers will mean that we get closer to the true values of the inheritance coefficients than by using all or any flowers on the plant, provided 1st there be marked differentiation of the apical flower from other flowers (i.e. if the first flowers of two related plants are much more alike than the first flower of one and any other flower of the second), or again 2nd there be no variation of the character observed in the plant at all. For example, colour and margin in the broad classification of our records do not exhibit much variation in the plant. Number of stigmata, size of local patch and wrinkling, do exhibit much variation, and further this variation is by no means perfectly correlated with position or order of budding. In order therefore to allow for the neglected homotyposis, we ought to have obtained measurements of (i) the correlation between the order of flowering and the characters observed, and (ii) the homotyposis of these characters. The first would have enabled us to measure the differentiation of the first or apical flower from other flowers, and by the aid of the second result we could have made allowance for the neglected homotyposis. But to record some half-dozen characters on all the flowers of each plant would have multiplied the work of recording tenfold, and the task seemed at. present beyond the volunteer labour, which was all that was available. Accordingly we must simply state that our inheritance coefficients are too low because we have at present no adequate measure of the differentiation of the apical flowers, nor of the homotyposis of most of the characters dealt with. But it is satisfactory to note that those characters in which mere inspection has shown least differentiation with position and most variation in the individual, such as number of stigmata, size of basal patch and wrinkling, are precisely those in which the coefficients of inheritance come out lowest. We hope that investigations into character differentiations with position and into homotyposis will be started this season. Meanwhile the results in sections (3) to (5) will, we hope, suffice to indicate to the reader that in our opinion the study of inheritance in plants is not a simple, straightforward matter. We have not in Shirleys at any rate found the definite and exclusive categories which the Mendelians appear to encounter at every turn. We have found our characters widely modified by environment and season; we have found selection acting on the seed sown, and both type and variability Biometrika 1y 52 410 Cooperative Investigations on Plants susceptible of much alteration. Lastly, although we have classified each character into more sub-groups than we have met with in Mendelian writings, we find many cases in which the category is hard to determine, and we are convinced that personal equation must be given its due weight. (6) Parental Correlation. As we stated at the beginning of this paper, we intended to base the 1903 experiments on examinations of apical flowers only. We were not therefore able to compare in the bulk of characters the crops of 1903 with the maternal plants. Ample material should, however, have been provided for parental correlation in the three crops of 1904. Unfortunately two of our observers were unable to complete their tasks. We are therefore thrown back on a comparison of the Crewe daughter plants with the Parkstone maternal plants. The fact that the Parkstone and Crewe crops were grown in very different environments is against very good results, but the number of bagged capsules at Parkstone and their comparatively good state led to our selecting that crop as the chief source for the seed supply of 1903. The tables are formed for the apical flowers of the maternal plants at Parkstone and the daughter plants at Crewe, and this was done with six characters. The work of tabling and reduction is due to Dr A. Lee. The coefficients of correlation, except in the case of the stigmata where the method of moments was used, were found by the fourfold table method. We should, if the investigations were now to be started de novo, probably use the method of mean square contingency, but it was considered desirable to preserve as far as possible uniformity of treatment throughout the whole round of poppy investigations, and the fourfold table method had been adopted in the first memoir. We refer briefly to the characters dealt with. Stigmata. There was probably a stringency of selection here of the maternal plants of about ‘7295. It is an open question to what extent the homotyposis of about *5 ought also to be allowed for, at any rate the value found for parental inheritance is a minimum value. Extent of Margin. This was determined by a fourfold table, the divisions being between ‘no’ and ‘some’ margins. The factor of selection of the maternal plants is here 6259. The homotypic influence is, as in the case of colour, probably very small, the flowers on a plant being usually all whole coloured or all margined. Number of Petals. This was determined by a fourfold table, division into normal four petals and non-normal flowers. There was inverse selection of maternal plants compared to the general Parkstone population, Le. they were more variable than that population; but if we compare the maternal group with the Streatham crop as a Shirley population of maximum petal variability there would have been a stringency of selection of °5278. It is not clear, however, how great a selection we ought to allow for, because the observers may not have been equally definite as to what they reckoned as an incipient petaloid II. Inheritance in Shirley Poppy 411 stamen. Further, we have at present no measure of the homotyposis of plants in relation to supernumerary petals and petaloid stamens. Colour of Middle Third of Petal. This was worked by a fourfold table, the division being into poppies showing some and those showing no red. We cannot do more than consider the maternal plants as an average random selection of Shirleys (see p. 406). In the matter of body colour, within the classifications red, pink, pink-white, etc., there appears to be small variation within the plant, or we may look upon this broad colour classification as individual to the plant and not to the bud. The Basal Patch. This was dealt with by a fourfold table, the division being, none to slight, and slight-definite to large. No evidence is available as to whether there was or was not a selection of maternal plants. The extent of basal patch varies considerably on flowers of the same plant, and is probably a good deal influenced by personal equation and homotyposis. Wrinkling. The division here was by a fourfold table. Mr Blanchard’s esti- mates of wrinkling were so low as compared with Mr Ward’s that the division of the maternal plants was made into not wrinkled and some wrinkling, while the Crewe population was divided into wrinkled and slight or no wrinkling. There may be a selection stringency of ‘8 to ‘9, and probably normal homotyposis, as wrinkling varies considerably on the same plant. But we have no definite knowledge on these points, and we consider wrinkling, although very characteristic of the Shirley, as very unsuitable for record. We sum up our results in the following table: TABLE III. Parental Correlation, Parkstone Maternal and Crewe Daughter Plants. Character Raw | Selection Corrected Correlation Stringency Correlation | Stigmata.. vat iL r/Ur/ "7295 "2324 Extent of Mar ein ss 3175 "6259 4718 Number of Petals oe: "2740 5278 (2) "4750 (2) Colour of Middle Third 4008 0 “4008 | Basal Patch ... "1305 “6900 (7?) ‘1864 (2) | Wrinkling ae aes 1595 8620 (27) 1842 (7) Mean ... ae 24 69 33 We have not the least hesitation in admitting the full inconclusiveness of these results. All we consider that they show is, that if we could accurately allow for selection of parentage, for the variability of character in a plant with its multiplicity of organs, and for the extreme response to seasonal and environ- mental changes, and, further, eliminate the effects of personal equation, there is no reason to suppose that the parental resemblances for different characters would 52—2 412 Cooperative Investigations on Plants not group themselves between ‘4 and ‘5, the values found for characters in other types of life. We do not believe that experiments on plant heredity, which employ no measurement and allowance for these modifying factors, can lead to conclusive results. In our opinion these factors are all potent, and modify widely the limits of any category which may be selected for classification of simple characters in plants. (7) On Collateral Heredity. The following table gives the results obtained for the 1903 crops with a com- parison of such characters as are comparable in the Oxford crop of 1900. In all cases the apical capsules in the sibling plants are compared. In the Appendix are given the actual tables from which the results are deduced. It is desirable to point out the methods adopted in the individual cases: (a) Resemblance in number of Stigmatic Bands. Found by usual correlation table. (b) Number of Petals. Fourfold table, the grouping being into the normal four petals and the abnormal more than four petals, including in the latter class semi-double, double flowers, and flowers with one or more petaloid stamens. (c) Colour of Middle Third. Fourfold table, the grouping being into flowers showing some and flowers showing no red. (d) Breadth of Margin. Fourfold table, the grouping being into flowers showing some and flowers showing no margin. (e) Colour of Flower at Margin of Petal. Fourfold table, the grouping being into flowers showing some and flowers showing no white there. (f) «tent of Basal Patch. Fourfold table, the divisions being into none or slight, and large or definite patches. (g) Wrinkling. Fourfold table, the divisions being into marked wrinkling (‘frilled’ or ‘ wrinkled’) and insignificant wrinkling (slight or no wrinkling)*. Several points are brought out by this table. In the first place we see that the starveling Streatham crop shows a great reduction in the fraternal correlations. Now if we endeavour to interpret this result we should reasonably suppose that much selection took place at Streatham; the mean of the stigmatic bands and their variability has been much lowered. There is further no reason for supposing a correlated selection in brother plants. Now if we suppose two organs A and B with standard deviations o, and o, and correlation 7,., to be selected in a non-correlated manner so that their standard deviations become s, and 8, it is easy to show, on the hypothesis of nearly normal distribution, that the correlation after selection will be 818912 Pir = ny eee eet ao} n a3 2 o 3 oo Totals NM q NN. Arneliffe. First Sister Plant. Second Sister Plant. 3092 1485 pww , Totals 49 47 384 1727 1181 3092 1485 477 18 384 8364 Sister Plants’ Apical Flowers. Breadth of Margin. bs s 51 44. 69 44 69 78 33 55 57 650 sn n Totals 279 453 55 431 650 57 456 704 82 450 677 450 4542 6158 6158 8642 Sister Plants’ Apical Flowers. Breadth of Margin. Breadth of Margin. OO. Streatham. Sister Plants’ Apical Flowers. Second Sister Plant. Breadth of Margin. First Sister Plant. Totals 1159 | 621 | 1638 Breadth of Margin. Totals 1159 621 1638 1129 7117 1129 | 7117 ‘| 11664 423 4 4 Second Sister Plant Extent of Basal Patch. Second Sister Plant. Extent of Basal Patch. Cooperative Investigations on Plants PP. Kidderminster. Sister Plants’ Apical Flowers. First Sister Plant. Breadth of Margin. Second Sister Plant. Breadth of Margin QQ. Parkstone. First Sister Plant. Totals 573 686 Totals 573 0 1035 686 6070 Sister Plants’ Apical Flowers. Extent of Basal Patch. sd dl Totals n 1765 ns 330 s 543 sd 918 d 2103 al 1603 l 1326 Totals | 1765 330 543, 918 2103 1603 1326 8588 RR. Arneliffe. Sister Plants’ Apical Flowers. First Sister Plant. Extent of Basal Patch. | dl Totals n 647 ns 56 s 194 sd 312 d 2016 dl 1257 t 398 Totals 4880 Il. Lnheritance in Shirley Poppy SS. Streatham. Sister Plants’ Apical Flowers. First Sister Plant. Extent of Basal Patch. a5 a nN ns 8 sd d dl i Totals 26 =o fax} bas Ay Ay n 1259 r 676 cae 420] 8 PACE i GA] sd 1432 MQ. | a 2182 7a 0° dl 419 Sse] 2 1533 2 2 —_— cea | Totals 11702 | TT. Kidderminster. Sister Plants’ Apical Flowers. 7 First Sister Plant. Extent of Basal Patch. 3 el n ek ae sad | a | dl 1 Totals 3) | | AL Ay n 826 99 124 517 426 1 326 2319 | ons 99 | 92 | 93 | 6 | 59 | — 42 | 310 Oe Ss 124 a | 64 96 61 = 51 419 2 faa sd 517 65 96 680 494 6 255 2113 Mo d 426 59 | 61 494 868 197 2105 gohan dl 1 a ee in = 2 9 A 5 l 326 42 | 51 255 | 197 2 216 1089 (5) Olas ; ; NS Totals] 2319 | 310 | 419 | 2113 | 2105 | 9 | 1089 | 8364 E, a | UU. Parkstone. Sister Plants’ Apical Flowers. First Sister Plant. Colour at Margin of Petal. ue s r rp Pp | TW pw Vw | WwW Totals fae re 274 | 106 | 426 3) 73 21 | 118 Se | ap 106 90 295 3 85 12 | 101 mi ep Pp 426 295 1860 7 423 150 646 a a Tw 3 3 7 — 1 — | 3 Ds | pw ie 85 423 1 196 45 | 243 oe vw 21 12 150 = 45 20 | 93 g s w 118 101 646 3 243 93 476 os | ns Totals | 1021 692 3807 7 1066 341 | 1680 @) | Z : | 7 o Oo. TS + n -=) SO) Tce, SS om a D S| nM 6s ice Biometrika ty VV. Crewe. First Sister Plant. Form of Capsule. Sister Plants’ Apical Flowers. Round Conical Flat Totals | Round 314 258 ils}5) 727 Conical 258 342 179 779 | Flat 155 179 92 426 Totals 727 779 426 1932 | 425 54 426 Cooperative Investigations on Plants WW. Parkstone. Sister Plants’ Apical Flowers. First Sister Plant. Wrinkling of Petal. =r & a if w sw nw Totals Oy Be aw Bi an w M Ss sw Conny nw as Q.= | Totals | 81 528 3065 | 4920 | 8594 oe | : XX. Arneliffe. Sister Plants’ Apical Flowers. First Sister Plant. Wrinkling of Petal. eee = aeons 8 & i | w sw nw Totals me ii 78 131 252 57 518 ae w 131 314 596 111 1152 ps sw 252 596 1310 296 2454 a ry nw 57 111 296 260 724 a ————— es 9 | Totals | 518 | 1152 | 2454 724 4848 ne YY. Streatham. Sister Plants’ Apical Flowers. First Sister Plant. Wrinkling of Petals. sw 113 742 1380 1704 3939 nw 149 936 1704 2472 5261 Totals 345 2157 3939 5261 11702 Second Sister Plant Wrinkling of Petal. ZZ. Kidderminster. Sister Plants’ Apical Flowers. First Sister Plant. Wrinkling of Petals. vi w sw nw Totals t 2 | 28 59 17 106 w 28 | 444 1013 194 1679 3356 779 5207 nw 17 | 194 779 382 1372 8364 Second Sister Plant Wrinkling of Petal. or Ne} 5 w ON THE INHERITANCE OF COAT-COLOUR IN CATTLE. PART I. SHORTHORN CROSSES AND PURE SHORTHORNS. By A. BARRINGTON anp K. PEARSON. CONTENTS. Page (1) Introductory . 427 (2) Material 430 (3) Colour Inheritance in Blac: Gren Cattle 431 (4) Horn Inheritance in Blue-Grey Cattle 435 436 (5) Colour History of Shorthorns : (6) Possible Results of Various Colour Matings and Mendelian Tntenpee ticis : 440 (7) Assortative Mating in the Shorthorn Population : : : : : 444 (8) On Direct Inheritance. First Degree . : p 445 (9) On Direct Inheritance. Second and Higher bepreees ‘ ; : : 448 (10) On Collateral Inheritance. Whole and Half Siblings ; ; : : 453 “ : a : e 454 (11) General Conclusions (1) Introductory. As far as we are aware no statistical investigation into the inheritance of coat- colour in cattle has hitherto been published. An attempt will be made in the present memoir to cover some of this ground in a manner similar to that in which investigations have already been made in the case of the coat-colours of the thoroughbred horse* and of the greyhound. It is, perhaps, idle to assert that we have throughout desired to give as complete and accurate an account as lay within our power of the mass of material available, and this without prejudice in favour of any theory. . Those acquainted with the spirit which animates the work of biometricians will scarcely doubt this; those who have throughout misrepresented the purport of mathematical investigations into heredity will doubtless continue to do so. At the same time it may not be without service to reiterate what has * Phil. Trans. Vol. 195 A, pp. 92-102 ; Biometrika, Vol. 11. pp. 229-240. + Biometrika, Vol. 111. pp. 245-298. 428 On the Inheritance of Coat-Colour in Cattle been previously stated as to the scope of biometric work in this field. Our object is to state first the proportion in which sub-classes exist in a “ population,” to measure next the degree in which these sub-classes are mated at random, and then to determine the extent to which these sub-classes reproduce themselves. We believe that a statistical record of this kind is of fundamental importance whatever physiological theory of heredity be adopted. In treating cattle we have done precisely what was done in the case of thoroughbred horses and grey- hounds, namely, the latest available volumes of the studbook have been taken*, and the colour pedigree of individuals entered therein worked out. In the work on cattle this was done on standard pedigree slips, of which a sample is given on the following page. These were filled in where possible as far as the great- great-grandparents and these schedules, representing a random sample of the existing general population, form the basis of our tables. As in the case of the greyhounds, supplementary investigations were made by collecting special pedi- grees as those of white individuals. These special pedigrees were not embraced in the general sample except in so far as they fell into the period there dealt with, but they served to illustrate special points which it seemed desirable to elucidate. On the basis of the above random sample} we first determined the extent to which random mating occurs in the general population. To what extent is there a fashion or taste in mating cattle? We next enquired how far our tabulated results admitted of a Mendelian interpretation. Let us suppose this question answered in the positive or negative sense, as the case may be, it leaves the tabulated data absolutely where they were as to the further question we have to ask : What is the average degree of resemblance between relatives in a population ? To what extent do the features of the ancestry impress themselves upon the offspring? This is not a matter of theory, it is simply a statistical enquiry as to the degree of independence of two distributions. It resolves itself mathematically into determining the degree of independence of probability in two “chance” distributions. In order to dispose of all doubts as to quantitative scale and as to “normal” distributions of characters, we have adopted throughout the contingency method of measuring the deviation from independent probability, and in particular that form of it spoken of as mean square contingency. This makes all our results for one race strictly comparable inter se. But the colour classifications being different for different species, it is not so legitimate to compare these results together. In order to throw light on this point other methods have been adopted occasionally for comparison and control; these will be referred to in their proper places. * In the second paper on coat-colour in horses, we began with the earliest studbooks in order to test the influence of the far more frequent greys and blacks. + The word “random” is used with reference to coat-colour. When one or both parents had appeared several times, no further offspring of such individuals were taken, in order to avoid overweighting certain individuals. This affected the cows more than the bulls, because the former are classified in herds with sometimes close inbreeding. Thus 1000 consecutive cows might possibly have given only 50 to 60 sires. 429 A. BARRINGTON AND K. 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And without knowledge of the amount of mating of sub-groups no determinantal system can be applied to wild populations. If proven a theory of determinants would immensely assist breeders, but it is extremely difficult to see how it could be applied to wild populations and to the Darwinian theory of evolution by inheritance and selection. Hence, if we lay special stress on finding directly the coefticients which fix the intensity of inheritance in populations, it is not because we should in the least undervalue the discovery of a well-demonstrated determinantal theory of inheritance, but because we should then have, before applying it to populations and seeing its full bearing on the problem of evolution, to actually go through the theoretical determination of constants, which can without any such hereditary theory at all be determined from the material at our disposal. When, however, we have found the degree of resemblance between relatives quantitatively, that is to say, the empirical numbers which for any population mark the divergence from independent probability between any relational grades, we can next ask: Is there any connection between these numbers within the same species ? and are the coefficients for different species, or for different characters in the same species alike? When we answer these questions we are determining whether heredity is the same or not for all characters and species, and to what extent ancestry and stock are of importance. It may be said that a valid determinantal theory would answer these questions also. So it undoubtedly would, but no such theory is, in our opinion, even approximately established as yet ; and meanwhile there is and will remain every justification for attacking the problem by a method, which starts from no hereditary hypothesis, but merely endeavours to interpret the observed numbers of each class. So much may be said of the methods adopted in this paper. It is a repetition of what has already been written in this Journal, but it is desirable to restate it here in order to emphasize the aim of the present investigation. (2) Material. When we determined some years ago to consider the inheritance of coat-colour in cattle we directly sought for a simple case which might possibly enable us to convince ourselves of the efficiency of a quite brief Mendelian formula. The Guernsey cattle have a very considerable variety of coat-colour. With shorthorns a briefer classification is sufficient, but still it is wide enough to allow of the possibility of very complex allelomorphs. We were aware that the white short- horn breeds nearly true, and that the black Galloway does the same. Hence the eross of the white shorthorn and the black Galloway seemed to offer a possibility of testing the simplest Mendelian formula. Unfortunately the resulting blue-grey A. Barrinaton AND K. PEARSON 431 cattle, although of great value for trade purposes, do not form the subject of any herdbook record. Still it seemed possible to obtain some simple results by an appeal to large breeders of blue-greys. Their returns form the subject of the first part of the present paper. We next endeavoured to obtain some general idea of the colour inheritance of British cattle, but the colours of the original progenitors of existing types seem very hypothetical, and our researches in this direction, embodied in a later part of this paper, are not very helpful. Attention was then devoted to two types: the shorthorn and the Guernsey. In the case of the former Coates’ Shorthorn Herdbook, issued by the Shorthorn Society of Great Britain and Ireland, has now reached the fiftieth volume, and we have to heartily thank the Society for placing a copy of it at our disposal for the present investigation. This herdbook is very complete in character and deals with very large numbers of beasts. At least some breeders insert the coat-colour of calves sent to the butcher, but we doubt whether such entries are universal. In the case of the Guernseys, the English Guernsey Herdbook has appeared for eighteen years, the pedigrees lead fairly soon back to the Guernsey Guernsey Studbook, of which ten volumes had appeared up to date. We have to heartily thank the English Guernsey Society for placing a copy of their herdbook at our disposal. We supplemented it by the purchase of the Guernsey Guernsey Studbook. The latter seems to us a less reliable work, as cattle are included with many coat-colours, which would not be admitted now into the English Guernsey record. But a full discussion of this matter is postponed until we come to deal with Guernseys in the second half of this paper. Pedigrees with the statement of coat-colour were filled in for shorthorn and Guernsey cattle from these herdbooks, and to obtain an adequate supply of such pedigrees has demanded a large amount of very laborious work. We shall now take our subdivisions in order : (3) Blue-Grey Cattle. Colour Inheritance. In this case the blue-grey cattle are as a rule produced by crossing a black polled Galloway cow with a white shorthorn bull. The white shorthorn (with black eyes) and the black cattle breed relatively true, so that if we term one B and the other W we might test a simple Mendelian formula (BB) x (WW)=(BW), and then further investigate the crosses (BW)x(WW), (BW) x (BW) and (BW) x (BB). As in the case of Japanese waltzing mice crossed by Albino white mice we do not meet with any simple phenomenon of dominance, the general coat-colour of the offspring being blue-grey. The problem therefore turns on the action of these hybrids. Our first enquiry was as to the nature of the shorthorn bull purchased by the blue-grey cattle breeders. Mr H. L. Fife, of the Raby Estates Office, speaking of experience with the sales of produce from the Raby herd, writes: “Breeders of blue-grey cattle in the North-West of England, who produce the same by crossing a white shorthorn bull with a black polled cow, are always most particular when buying a white sire to satisfy themselves if possible, that such sire is the produce of two white parents. If he is not, that is to say, if the parent on one side only is white, their theory is that his stock when crossed with black cows do not get the blue-grey colour they like.” 432 On the Inheritance of Coat-Colour in Cattle This statement led to the framing of the following questions, which were issued in a circular form to the breeders of blue-grey catitle*: We are told that breeders of blue-grey cattle object to white shorthorn bulls, whose parents were not both pure white as giving bad results. (i) Can you from your own experience tell us whether you have bred from such a shorthorn bull, and if so whether you found it to give bad colour results and what these results were? (ii) If a blue-grey cow be crossed with a white shorthorn what is the resulting coat-colour ? Could you give us actual instances of this cross? In particular do pure white cattle ever result from this cross? (iii) If the blue-grey cross be covered by a Galloway bull, what is the resulting coat-colour ? Do black as well as blue-grey offspring result? Can you give us instances in your own experience? (iv) If the blue-grey cross be mated with itself, blue-grey cow with blue-grey bull, what is the resulting coat-colour? Do you know instances of black cattle or pure white cattle resulting from this cross? Are the majority of the offspring blue-grey ? It will be observed that questions (ii), (iii), and (iv) deal respectively with the crosses (BW) x (WW), (BW) x (BB) and (BW) x (BW) and were intended to elicit information with regard to these matings. Question (1) was framed so as to draw forth either facts or impressions of the breeder as to the need for purity in the white ancestry. These questions produced a number of answers—the experience of all breeders being by no means alike—which we have summarised below. We have most heartily to thank these gentlemen for their kindness and invariable patience in replying to our queries. There appears to be general assent to the proposition that a pure bred Galloway cow and a white shorthorn bull of white parentage give the desired blue-grey cross. When the white shorthorn bull is not from white parentage, Messrs Hyslop state that besides greys, there appear reds, blacks, particolour black and white, and occasionally a white with dark nose and insides of ears. Mr de Vere Irving prefers to use a shorthorn white bull of white parentage on both sides. He finds on using a white bull of roan parentage, that the offspring are liable to come dark, some almost black in colour, and others very dark blue-grey. Mr G. Hodgson always tries if possible to get a bull from white parents, which he finds makes the progeny the desired light blue-grey. If a white bull of white parentage of a satisfactory type is not to be had, then he takes one from roan parents, but this in his experience makes the colour of a darker nature. Mr R. Tinniswood has bred from a pure bred Galloway and a white shorthorn bull not of white parentage and has had occasional coloured calves from this cross. He considers that it is very desirable that the white shorthorn bull should be from white parents to get the best results. Bulls not from white parents mated with pure bred Galloways in the majority of cases give the right colour, but may give reds, roans+ or nearly blacks. Mr Tinnis- * We have to thank Messrs Harrison and Son of Carlisle for kindly providing us with a list of blue-grey cattle breeders. + The red and roan hair we have examined is exactly alike in appearance to that of red and roan shorthorns. A. BARRINGTON AND K. PEARSON 433 wood considers that the ancestry of the Galloway cow is even more important than that of the shorthorn bull; and states that the true pedigree black Galloway crossed with the white shorthorn of non-white parentage will give a blue-grey, though some of the offspring will be very dark blue-grey. The whole of this evidence seems to show that black Galloway x white shorthorn* does not give invariably the desired light blue-grey blend, but that the deviations from this colour in the opinions of experienced breeders depend on the ancestry of the (BB) and the (WW). The hybrid generation is not uniform unless the parents come of selected stock. Such appears to be the best answer we can get to our first problem. Our second question relates to the crossing of the hybrid (BW), the blue-grey cattle with (WW) the white shorthorn. This ought to give half blue-grey (BW) and half white shorthorns (WW). The experience of breeders is not uniform, but most probably it is based on white shorthorn bulls with varying colour pedigree. Mr Tinniswood finds that the cross is invariably a white calf with a black nose and ear linings. He has been careful to use shorthorn bulls which as to ear and nose are perfectly white. Mr Hodgson has frequently bred this cross and finds the results always white, occasionally with a black nose. He says this cross gives white cattle which are not, however, to be reckoned as pure white shorthorns. Thus in the experience of both these breeders (BW) has not occurred as a result of (BW) x (WW), nor in the experience of at least one of them (WW), but white with black markings. Other breeders, however, have had different experiences. Mr G. Richardson finds all colours to arise from the cross. Messrs Hyslop find like Messrs Tinniswood and Hodgson that white with coloured nose and ears was the resultant coat-colour, but some light blue-grey, some “grizzled” (i.e. red mixed with white hairs), and some black generally at base of coat. Mr de Vere Irving considers the coat-colour uncertain, and cites as illustrations, a blue-grey cow x white shorthorn bull giving a white calf with black nose and blue-grey ears. Another blue-grey cow with the same bull giving last year a blue-grey calf and this year a roan. He has also had pure white cattle from this cross. We think we may conclude from these results that the essential feature to which Mendel drew attention, 1.e. segregation in the second crosses, certainly does occur, but its nature is not compatible with any simple Mendelian formula. We have not only (WW) and (BW) occurring from (BW) x (WW), but also red and roan appearing in the hair, as well as white with black markings which did not exist in either the pure bred Galloway or the shorthorn parents. Further, this segregation although possibly more marked than in the hybrid generation is to be found in the hybrid generation itself, if the cow and bull while presenting the external features of pure Galloway and pure shorthorn, have not pure black and pure white ancestry. We now turn to the cross (BW) x (BB). This should give (BB) and (BW) in equal proportions. Mr de Vere Irving finds the resulting offspring black or dark * The usual cross is 9 (BB) x ¢(WW), but the reciprocal cross is sometimes made. The produce of this cross is said, however, not to be uniform in colour. It is adopted because shorthorn heifers can be bought on better terms than pure bred Galloway heifers ; while the shorthorn heifer as a cow makes a better price than a Galloway for towns’ dairying, Biometrika 1v 55 434 On the Inheritance of Coat-Colour in Cattle blue-grey. He has at present a black heifer from this cross and has personally known instances of the dark blue-greys. Mr Richardson’s experience is the same as Mr Irving’s, ie. black and blue-grey. Mr Tinniswood’s experience is that the offspring take black, but occasionally, especially if from light blue-grey cows, the black is accompanied by white markings. Mr Hodgson finds the coat-colour not very certain, generally black; but from a light blue-grev cow he has had blue- greys and roans; from a dark blue-grey cow, generally black with only an occa- sional roan. ‘This is interesting because we have seen that the light blue-grey results from the purity of the Galloway cow and the white parentage of the short- horn bull. It would thus appear that the grandparental ancestry influences the nature of the cross (BW) x (BB), through the character of the (BW) itself. Messrs Hyslop have had still more varied results, namely, dark blue-grey, some “ orizzled” (red and white hairs), black, and white with black nose and ears. In other words if white with black nose and ears be taken to represent the (WW) arising from (WW) x (BW), it cannot represent the (BW) in (BB) x (BW). We now turn to the last problem: what does (BW)x(BW) give? Messrs Tinniswood and de Vere Irving have never tried the cross. In fact it is clearly not one that blue-grey cattle breeders are likely to make*. Mr Hodgson has had no personal experience of this cross, but has seen instances in which a blue-grey bull being used on blue-grey cows, blue-grey and roan offspring resulted. Mr Richardson has known instances in which this cross being made the offspring were nearly all blue-grey, but the following season with the same cows and bull, red, roan, white, black, with only a few blue-grey calves appeared. Messrs Hyslop state that the cross gives blue-grey, black, red, “ grizzled” (red with white hairs) and an occasional white. They know no breeder, however, who makes at present this cross although it used to be followed. So far as this experience goes therefore (BW) x (BW) appears to give (BW), (BB) and possibly (WW), although it is not white with black markings, such as occurs in the (BW) x (WW) and(BW) x (BB) crosses. But on the other hand red hair has appeared when the original pure bred Galloway and shorthorn did not patently contain this colour. To sum up, we may conclude, we think: (a) That when pure black and pure white parents of races which breed nearly true are crossed, the hybrids are not homogeneous, there is a certain amount of segregation even in this generation, and colour may appear which was latent in the parents. It seems to us a difference in words only, whether we attribute this appearance to a latent determinant, or say that it is reversion to an ancestor. The pedigree is, in fact, the only @ priori guide the breeder has to the possibility * Mr Tinniswood breeds about 100 blue-grey calves a year. As he remarks, it is possible to get blue-greys from crossing a blue-grey cow with either a Galloway or shorthorn bull, but the colour ‘ very seldom occurs.” Hence the breeder naturally turns to the white shorthorn and Galloway first cross. Mr Tinniswood has used both crosses, generally the shorthorn bull, but also the pure shorthorn white cow and the Galloway bull, a cross he has never known to give red or roan. He has known red or roan from the reciprocal cross, and attributes it to want of pedigree in the Galloway cow. A. BarriInaton AND K. PEARSON 435 of such reversion. As he puts it, “it is desirable that the white shorthorn should be of white parentage,” and “ the Galloway a pure Galloway.” (b) More segregation takes place when the hybrid generation is crossed with either original stock. Red hair appears, the white shorthorn is replaced by white with black nose and ears, and this or pure white appears from the cross (BW) x (BB). Our hope therefore that a simple Mendelian formula might be tested on this case—the only one in which we knew of two whole colour cattle of different races being habitually crossed—has failed. Our enquiries were, we believe, justified, however, by the apparent simplicity of black x white = blue- grey*. We did not think actual colour statistics in this case worth collecting, for if we must deal with more complex colour categories, it is at once best to turn to breeds like the shorthorn and Guernsey, for which ample colour statistics are already on record. So far as it is possible to make a comparison between rough experience of the present kind on blue-grey cattle and more definite quantitative experience on mice, the colour results of crossing are in many points strikingly like the Oxford experience with Japanese waltzing and albino white mice. It will need a complex allelomorph to describe these colour changes, if indeed the phenomena can be thus described at all. (4) Blue-Grey Cattle. Horn Inheritance. It seemed worth considering whether a second character, the horns, would admit of a simple Mendelian analysis. The black Galloways are polled cattlet, and the white shorthorns, horned. We can represent these respectively by (PP) and (HH), and we have to enquire as to the following results: (PP) x (HH) =(PH), Carat: ye Ui ix (HH) and. CPA) x (PH). First, (PP) x (HH) gives theoretically the blue-grey (PH). Actually Messrs Harrison say (Pf) are “chiefly” polled, and Mr de Vere Irving finds that about one in twenty of (PH) is horned. Thus “polled” is dominant, but it is not absolutely dominant. Mr Irving has not found that exceptions are to be associated with want of pedigree in the Galloway, and he has found no difference with regard to horns between §(PP)x ¢ (HH) and the more usual cross 2 (PP) x § (HH). Mr Robert Tinniswood agrees on this point, and finds the result polled if both sire and dam are pure bred. Mr Hodgson finds not more than about 1 in 30 to 40 horned; he considers that this is usually due to impurity in the Galloway pedigree, but horns may occur with exceptionally well bred sire and dam. (PH)x(PP). This cross should give (PP). Mr Irving finds the offspring are polled. Myr Richardson says that they are all polled. Mr Tinniswood has not * As a matter of fact this blue-grey is, to judge by hair samples, simply a blend of white and black hairs, varying in proportion from almost pure black to a large majority of white. + Our English informants state that the Galloway is in their experience polled. If the breed now contained (PH)’s, (HH)’s would appear. Boyd Dawkins, Quart. Journal Geol. Soc. 1867, p. 177, says that in the 18th century Galloways were not polled, but his evidence for universality of horns seems far from conclusive. Further details as to the crossing of polled and horned breeds will be given in the next number of this Journal. 55—2 436 On the Inheritance of Coat-Colour in Cattle largely bred this cross but all the calves have been polled. Mr Hodgson on the other hand has found 1 out of 8 to 10 horned. (PH) x (HH). This should equal 2(PH) +2 (H#) and give half horned and half polled. Messrs Hyslop find the majority horned. Mr Richardson finds three- quarters horned, and Mr Irving that the horned are in a majority of three to one. Mr Irving says that it sometimes happens that the horns do not grow to much size but in the majority of cases they are there, and he does not consider that such cattle can be described as polled. Mr Tinniswood finds the calves bred this way generally horned, perhaps 2 or 3 in twenty would come polled. Mr Hodgson on the other hand finds in this case only 1 in 4 to 6 horned and illustrates by three blue-grey $s put to a white shorthorn bull last year giving all polled calves. (PH)x (PH). This should give (PP)+2(PH)+(H#), which being inter- preted is polled in a majority of three to one. Few breeders make this cross, and Messrs Irving, Richardson and Tinniswood have had no experience of it. Mr R. Hyslop finds six polled to one horned, and Mr Hodgson says one out of 6 to 8 horned. These numbers are far from the Mendelian ratio. Thus, as in other cases which it has been our misfortune to come across, we find approximations to Mendelism, but we do not find Mendelism an accurate account of the records provided. The polled (PH) might really be a (PP), it cannot be an (Hf), and therefore we could account for less than a half being horned in the cross (PH) x (H#), but not for more than a half being horned. The only explanation could be that polled in a number of cases are latently horned, le. we cannot distinguish (PP) from (PH). This would account for the horned cattle appearing in (PP) x (Hf), but not for the usual absence of horns in (PH) x (PP), for some reputed (PP)’s would be (PH)’s. Anyhow the further pursuit of a somewhat elusive Mendelism in blue-grey cattle did not seem to us likely to be profitable. (5) Colour History of the Shorthorn. The classification of shorthorns into broad categories is extremely simple as compared with some other types of domestic cattle. The recognised classes are Red (Rf), this is supposed to contain whole reds; Red with a little White (RIW) or red with occasional white spots or with white markings; Red and White (RW), Le. particolour; Roan (Ro), a mixture of red and white hairs; and White (W). The white are not albinos*, having black eyes and occasionally black noses. Attempts more or less successful to get rid of the black noses have been ener- getically made by breeders. The existence of black pigment in the nose is probably not always recorded in the herdbook records. We have seen that breeders of blue- grey cattle strive to get shorthorn bulls without black noses, but as there is always black pigment in the eye, the question of absence of black pigment im the nose is perhaps not of first-class importance in the problem of crossing. It appears perfectly certain that eye and nose pigmentation in intercrossing white shorthorns * A herd of true albino white cattle (whole white and pink eyes) has been reported to us as existing in Sweden, but we have not succeeded in locating it. A. BARRINGTON AND K. PEARSON 437 does not now introduce black into the coat-colour, though how far it may appear as red may be open to discussion. Red in the shorthorn appears in different shades roughly classified by breeders as dark and light, and we have corresponding dark and light roan, according to the character of the red hair in the roan. hese finer distinctions are not recorded in the herdbook *. From the standpoint of our present investigation, it would be of very great suggestiveness if we could determine in number and colour the original factors of the modern shorthorn. So far as we have been able to gather from writings on the subject there seems to be a good deal of diversity of opinion among authorities as to the origin of the domestic breeds of British cattle. There are three some- what divergent main views. First that of Boyd Dawkins}. He identifies the Bos Urus of the prehistoric and historic periods with the Bos primigenius of the Pleistocene period, but he does not consider that either survived in this country as source of the wildpark cattle. He holds that the small dark Welsh and Scotch mountain cattle represent an original Celtic shorthorn, and that the Romans used this Celtic cattle and did not import larger breeds of their own. This whole coloured Celtic shorthorn was driven out not by the Romans, but by the Anglo- Saxons. The Chillingham wild cattle on this view are living representatives of the breed brought by the Anglo-Saxons. They are not survivors of the Bos Urus in this country, but imported Anglo-Saxon cattle run wildf. At first sight there does not appear the same reason for the disappearance of the Anglo-Saxon cattle, that there was for the expulsion of the Celtic cattle, and Boyd Dawkins’ citations from Welsh Laws may be read in more ways than one, for fines have generally become customary many generations before they are codified. It is also unlikely that all the place-names in Ireland referring to white cattle are later than the Saxon invasion. Another view has been taken by McKenny Hughes§. He considers that the characteristics of the Urus nowhere appear in the Romano-British cattle. The Kerry cattle are typical of the Celtic shorthorn, the Chillingham cattle are the nearest representatives of the breed introduced by the Romans||. The Highland and Welsh cattle are derived largely from the Roman breed with admixture of the Celtic shorthorn. These are all whole coloured or shaded. The Longhorns, frequently particoloured or sheeted, are the offspring of breeds introduced from * The range and continuity of the reds seem fairly obvious from the samples of hair colour in our possession. t+ Quarterly Journal of the Geological Society, Vol. xx11. 1866, pp. 391-401. ‘Fossil British Oxen,” Pleistocene Mammalia, Palaeontographical Society, Boyd Dawkins and Sandford, 1866-1886, in parts. + B. A. Report on Herds of Wild Cattle, Report, 1887, p. 135, appears to leave the question of importation open. Riitimayer (‘‘ Ueber Art und Race des zahmen europiischen Rindes,” Archiv fiir Anthropologie, Bd. 1. S. 219-250) holds that the English wild cattle, the large horned Hungarian and Italian cattle belong to the Primigenius race. § Archaeologia, Vol. ty. Part 1. pp. 125-158, 1896. ‘On the more Important Breeds of Cattle which have been recognised in the British Isles, etc.” || A comparison of the skulls of Chillingham and Italian bulls with that of the Bos primigenius can be easily made at the Natural History Museum, South Kensington. 438 On the Inheritance of Coat-Colour in Cattle the Low Countries and Holstein in later mediaeval times. McKenny Hughes says that the Durham and modern shorthorns belong to a much later period and fall outside his enquiry. Thus writings on the subject do not give us very definite results for our present purposes. We further appealed to Professor James Wilson, of the Royal College of Science, Dublin, who is an authority on the matter, and who most kindly provided us with the following brief statement : (a) The first cattle in Britain were of a large type, Bos primigenius, which became extinct here in the stone age, although possibly surviving later on the Continent. If they had anything to do with existing British cattle, which is extremely doubtful, the relationship is very remote and is only to be traced through later importations from the Continent. (b) The next type to be noted in Britain is the Celtic shorthorn, Bos longi- frons, a much smaller animal. Patches of hair have been found in the caves, and these are said to be of a dark colour. Professor Wilson is strongly of opinion that B. longifrons was mainly black. It has been said to be red or black. (c) The next invasion, that of the Romans, brought the Roman white cattle, which probably had black markings. The Romans formed joint settlements with the British, and a Romano-British cattle resulted in the neighbourhood of Roman settlements, from crossing the Roman and Celtic types. (d) In the next place we have a Saxon invasion, bringing with it Saxon cattle. There was probably some mixture of breeds on the Celtic and Saxon frontiers, but generally speaking Saxon cattle occupied the parts from which the Romano-British population had been expelled. These cattle were red. (e) Following on the Anglo-Saxon was a Dutch invasion of cattle into the east of England. By Dutch we understand from the “Low Countries.” This invasion began in the middle ages and continued till a century ago, and again involved a certain amount of mixture. The Dutch cattle were probably “flecked,” the flecking being white with black, red, or brown. The Englishman developing a prejudice against black, we may suppose the reddish mixture to remain, and to have become the basis of roan. Before, therefore, the beginning of pedigree cattle-breeding, or earlier in the sixteenth century say, we should expect a distribution of cattle races much as follows: Wales and Scotland, Celtic or black cattle, with some of Roman descent and perhaps some whole reds. Midland England, very mixed cattle, compounded of the following races in order: Celtic, Roman, Saxon aud Dutch. The colours were black, white with black or brown points, red and flecked. Round the edges of the country the breeds were comparatively pure, but in the Midlands, where they all met, they were a mixture ; the Romano-Celtic element probably prevailing. Southern England, Anglo-Saxon, red. Eastern England, Dutch cattle, flecked or roan, A. BARRINGTON AND K. PEARSON 439 The purest British cattle would thus be found in Wales, Scotland (and West Treland), where black is the prevailing colour. The Romano-British were up the west and middle of England, where they still survived to about a century ago. They were the old longhorn breeds. The wild white cattle of Chillingham, Chartley, and elsewhere, are Roman cattle run wild owing to the unrest of the Anglo-Saxon, and later and more especially the Danish invasions. Traces of the Roman cross are still to be seen among Scottish, Welsh, and Irish cattle. The wild white cattle have black muzzles and black or brown points; that is their hair is brown, or dark at the ears, round the eyes.and muzzle and often at and below the knees. The Anglo-Saxon red cattle, which were once spread over the south of England, are still to be found in the red cattle of Norfolk, Sussex, Devon, and Hereford. The Romano-British or longhorns were some black, some white, but mainly flecked, and were largely driven out by the flecked Dutch, although some mixture probably took place. It will be seen that Professor Wilson’s account differs to some extent from both those of Boyd Dawkins and McKenny Hughes. Upon the ingredients just referred to the breeders had to work when pedigree cattle-breeding, which is scarcely more than a century old, came into vogue. The shorthorn has possibly arisen from four races, the Celtic, a Romano-British, an Anglo-Saxon, and the “ Dutch*”, and even some of these are mixtures. Confining our attention to coat-colour, the black Celtic, if it contributed any- thing, must have been bred out early. But it is doubtful whether there was not a red Celtic, and a red Galloway heifer (1760-1770) appears to have been connected with the beginnings of the breed. Thus the shorthorn red may have had three sources, an Anglo-Saxon red, the red of the Dutch flecking, and the supposed Celtic red. The white may have come through the Romano-British, through an Anglo-Saxon white or possibly through the white in the Dutch. The particolours and the roans are of equally doubtful origin, although it probably is safe to assert that they are due to the breeds of latest importation ; and it thus seems fairly impossible to determine @ priori how many distinct red, roan, parti- colour, or white types may really exist in the case of the shorthorn. The importance of this statement for any Mendelian interpretation must be obvious. We may have reds which are dominant, recessive or even heterozygous to white or even to other reds, and the search for a Mendelian formula becomes very elusive. If we turn from the possible ingredients of the shorthorn breed to the history of its origin we find matters still more difficult to disentangle. There appears, by the will of John Percy of Harum, to have been a breed of shorthorns in Yorkshire as early as 1400, and these were not whole colour. In * Their nearest Continental cousins to-day are the German Fleek-Vich. 440 On the Inheritance of Coat-Colour in Cattle 1640 we again hear of a shorthorn breed; and a native race with short horns, large bodies, and black or red in colour, existed in Holderness and on the banks of the Tees in the middle of the eighteenth century. A new breed of shorthorns was introduced from Holland in the eighteenth century, and much improvement was attributed to the introduction of Dutch bulls, but it is said that these were the offspring of English shorthorn cows sent by James II to William of Orange. In the early shorthorn pedigrees we find some black, flecked, and spotted beasts. However debateable the origin of the shorthorn may be, it appears to have existed as a distinct breed by the middle of the eighteenth century, and possibly 50 or 160 years previously. Its source is probably Yorkshire and Durham with Dutch reinforcements. The early breeders seem to have had much the same colours, red, roan, white, and their mixtures. One noteworthy early breeder introduced a cross which is supposed to have become a potent factor with a red Galloway polled cow. The blacks and fleckeds rapidly disappeared. The proportion of colours was, however, somewhat different. Thus from pedigrees before 1850 we find: Red R. W. Roans Whites 14:5 42 35 5°5 per cent. and from recent statistics : Red R. L. W. R. W. Roan White 30 6°5 45 D4 5 per cent. Thus the whites have remained about stationary in number, but the whole reds and roans have increased at the expense of the particoloured, and this even if we throw the red with white markings into the particolour. It cannot be said that this brief examination* casts much light on the possible components of the modern shorthorn. We do not know the actual constituents of the early Yorkshire shorthorn. It suggests that the white may have come from Holland as a whole colour and not be a contribution of Saxon or Romano- British factors. We see a definite Dutch contribution and a possible Celtic red through Collings’ Galloway cross. We do not know any more certainly whether an Anglo-Saxon red also contributed. Failing this, we are thrown back on actual breeding experiments to disentangle possible Mendelian factors. (6) Possible results of the various colour matings and Mendelian interpre- tations. There are fifteen colour crossings possible with our five shorthorn colour categories, and since each crossing might give offspring of one of the five cate- gories, 75 possibilities arise. * See: Thomas Bates and the Kirklevington Shorthorns, by C. J. Bates, 1897. Bell’s History of Improved Shorthorn Cattle, 1871. B. Rudd: An Account of some of the Stock of Shorthorned Cattle of Ch. and R. Collings, 1821; Origin and Pedigree of the Sockburn Shorthorns, 1822. Lewis F. Allen: The American Herdbook, to which is prefixed a Concise History of English and American Shorthorns, 1856, and History of American Cattle, 1868. A. Barrinaton AND K. PrARSON 441 These categories are arranged in the following table: TABLE I. Possible Offspring Mating R. R. L. W. R. W. Ro. Ww. R. x R. x | x x x wp R.x R. L. W. x x | x Ss 2 R.x R. W. x x | x x wp R. x Ro. x | x x * x R. x W. ae x | x | x x wp | R.L. W. xR. L. W. x x x x wp | RL. W.xR. W. x | x | x x wp Waetinlan Wax iRos © i.e) Nl “x | x x wp } nts LD AW es ANE | wp 2 | wp x wp | R.W.xR. W, x x | «x |-x AHB | BR. W.x Ro. x | x | x | x wp Ra Wiex Wie x | wp | x | x wp InKOy Se RKO nt eee fil eX x | x x x Ro.x W.... as |X x x | x x WeSC We: ie ai | wp | ? | AHB | wp x All the categories marked x were found in the first random sample of 2172 calves taken out of the Herdbook. Some of course are infrequent, but their relative frequency will be found at once by examining Tables II and III. Those marked wp came to light in following up the pedigrees of white cattle. They are taken from vols. 37 to 49, in the bulk of cases from vols. 47 to 49; they are not exhaustive, but in most cases several schedules with each type occur in our record. The two cases marked AHB were noted in a single volume of the American Herdbook. No stress is laid on them, Thus out of the 75 possible arrangements all but three have been found, and these three are all concerned with the comparatively scarce category of red cattle with white markings. Many of the 75 groups are rare, some excessively rare. It is always possible to assert that the rarest of them are due to misprints or to breeders’ mistakes. We have made enquiries of breeders and in some cases additional instances have been provided, but we have kept to the cases actually on Herdbook record*. Take as an illustrative case WxW. This is a comparatively rare cross; because some breeders think white cattle delicate, others do not breed it because the produce have less sale and at lower prices in the export market. It is either unfashionable or unprofitable. Most breeders assert that WxW always gives W; it generally does, but their experience when you come to actual numbers must be very limited. In fourteen volumes of the Herdbook containing about 57,000 calves, 91 cases only of W xW were found, though quite possibly * Mr T. Milne tells us, for example, of an all red heifer turned down with two young bulls, one all red and the other RL, producing from one or other a white calf. Biometrika ty 56 442 On the Inheritance of Coat-Colour in Cattle some few were overlooked. So that the cross is one of extreme rarity. In these 91 cases the calves were in 86 cases white, in four cases roan*, and in one case red ft. The American Herdbook examined for other matters produced incidentally a case of WxW=RW. Thus Coates’ Herdbook indicates that in 5} per cent. of cases white does not breed true to white. We are therefore driven to the conclusion that, if the records are to be trusted, there are latent determinants in white short- horns or in some white shorthorns which under certain circumstances can be called into play. This view is confirmed by another consideration. Some wild white cattle, like the late Chartley Herd with its representatives at the Zoological Gardens, have black muzzles and black ear-linings. They are reputed to drop occasionally black or red calves. Other herds of wild white cattle have black noses and red inside the ears, and have bred absolutely true to this type for at least a century. As we have seen, the white shorthorns breed generally true to their type. The cross wild white cow x shorthorn white bull gives a hybrid pure white cow, but such pure white cows, if crossed with a shorthorn white bull, may produce a considerable percentage of both roan and red as well as white calves. In other words, while wild white cattle of the type considered breed absolutely and the white shorthorns sensibly true to type, and while the hybrid is pure white, the hybrids’ offspring by white shorthorns segregate into colours occurring in the near ancestry of the shorthorn, and possibly but not probably in the distant ancestry of the wild white. These facts, coupled with the experience that white shorthorns if crossed will, although rarely, give roans and reds seem to show the existence of latent colour determinants in the whites depending upon their ancestral history. The phenomena described are similar to those noted in albino white mice, which carry in latent form traces of their colour ancestry, traces which can become patent under various crossings. The simplest Mendelian formula would be based on considering the white as recessive and the red as dominant, and the roan as hybrid, ie. (RR) x (WW )=(Ro), and this is roughly an expression of some of the facts of the case. Thus(WW)x( WW) gives (WW) in all but 54 per cent. of cases. In 196 crosses of (RR) x (RR) we find 156 give (RR), or in about 79°5 per cent. of cases. But, besides about 3 per cent. of roans, we have 12 per cent. of calves with white markings and 5:5 per cent. of particoloured. We are thus compelled to consider all red, as we have just con- sidered all white, as either consisting of several subclasses only to be discovered by breeding, some all reds being really heterozygous, or else as containing latent white marking and particolour determinants. But the difficulty does not stop here, there appear to be authentic cases of (RR) x (RR) giving (WW). Thus we find Favourite Duke (£) got out of Clara 3rd (R) not only a normal red bull calf, but a red and white cow calf and the white cow calf Clara 5th} and the (R) Heydon * Bull calves: Unnamed, Vol. xxxvi. p. 277; Shaftesbury, Vol. xxxvu1. p. 292. Cow calves: Crystal Valley, Vol. xu1v. p. 273; Seam, Vol. xtvirt. p. 809. Other cases in American Herdbook. + Cow calf, Bright Anna 2nd, Vol. xuirt. p. 318. + Coates’ Herdbook, Vol. x11. p. 623. A. BARRINGTON AND K. PEARSON 443 Rose crossed by Earl Clarence 2nd (R) gave the white cow calf Minnette*. Such cases may be very rare indeed but, if authentic, reduce the Mendelian formula to a rough empirical statement of a statistical ratio; they are inconsistent with any theory of a pure gamete. If we pass from the question of purity in the supposed homozygous elements to the first hybrid generation, we find in the Tables II. and III. of our random samples 90 cases of (RR)x(WW), and the result is (Ro) the assumed hybrid in 85 cases or in 945 per cent. of cases, but particoloured (RW) has occurred in two cases, white marking in one case and whole red in a third; while travelling beyond the field of our working sample (RR)x (WW) is actually recorded to have given(WW). Crossing the hybrid generation together, we should expect: (Ro) x (Ro) = (RR) + 2 (Ro) +( WW). Our actual numbers on the random samples are: 514 (Ro) x (Ro) = 86 (RR)+ 4+ 31 (RLW) +35 (RW) + 278 (Ro) + 84( WW), as against the Mendelian formula: =128°5(RR) + 257 (Ro) + 128:5(WW). The (Ro) is thus seen to be, like (RR) and (WW), a non-homogeneous group ; we are compelled to suppose that some (fo) have white marking and others parti- colour determinants latent in them. If we cross the hybrid (Ro) by the original stocks, we have: 656 (Ro) x (RR) = 243 (RR) +46 (RLW) + 39 (RW) +324(Ro)+4( WW), as against the Mendelian formula: = 328 (RR) +328 (Ro). The close approximation to the Mendelian number of (Ro) is noteworthy, but the appearance of 4(WW) is again impossible unless some of the reds are to be treated as heterozygous. The latent white marking and particolour determinants are again manifestly needed. Lastly, we take the case of (Ro) x (WW): 74(Ro) x (WW) =1(RR)+1(RLW)+ 1 (RW) + 47 (Ro) + 24( WW), as against the Mendelian formula: = 37 (Ro) + 37(WW). Here again we are compelled to assert a heterozygous white, or if we treat the individual instances of (RR), (RLW), and (RW) as heterozygotes and club them with the (Ro), the proportion 50 to 24 is very far from the Mendelian equality. Of course the same process could not then be applied to explain the anomalies in the crosses (Ro) x(Ro) and (Ro)x(RR). The anomalous cases in these crosses can * Coates’ Herdbook, Vol. xxxvuit. p. 231. + No attempt is made to express anything but (RR) and (JVI) in Mendelian form, the constitution of the other individuals being unknown. 56—2 444 On the Inheritance of Coat-Colour in Cattle be multiplied by seeking for them and, not doing as we are here doing, dealing with the material provided in the pedigrees of a random sample from a few volumes of the Herdbook. It would thus seem that no simple Mendelian formula can possibly fit the shorthorn cases. Roughly, such a formula approaches the data in one or two points, but the roughness appears inconsistent with a theory of Mendelism being due to the purity of gametes. It is of course clear that the introduction of a complex allelomorph may improve matters, or the differentiation of whites and reds into different classes, homozygous and heterozygous. Increase in the number of available variables usually does give better fits. But the attempt to differentiate cows by the results of breeding from them must shatter on the paucity of their offspring. A cow will frequently give no more than 8 to 10 calves, and, bred even to the same bull, or to the same type bulls, it may not be till the 9th or 10th calf that she diverges from uniformity in her offspring. Thus one cow may throw 8 white calves and only the 9th be red or roan. This type of experience is not uncommon, and breeders not infrequently attribute it to the effect of age on the cow. To those who believe in individual prepotency—a view hardly consistent with Mendelism—this influence of age may not appear impossible. In the next place, the prejudice against white crosses on the part of breeders, indeed the dislike to white shorthorns in general, renders it almost impossible to effectively test the purity of any (RR) or (WW) bull, although his offspring may be fairly numerous. Further, the bulls are often mated solely within their particular owner’s strain, and it is not so evident what would happen if they were mated outside it. This tendency to assortative mating among shorthorn breeders will be discussed more fully in another section. We have seen that there is historically a possibility of two strains of red and two strains of white having been mingled in the shorthorn. Determinants representing parti- colour and white markings can undoubtedly be introduced also—we confess to having made an attempt from this standpoint, which shattered with further examination of Table I—but the introduction needs a wider practice than we can boast of in inventing Mendelian formulae, and until we are more convinced than we are at present of the soundness of such formulae we should prefer to leave the invention to those who have had it. Coates’ Herdbook presents a wide range of material, and whatever we may think of the categories selected, the record has been made by persons in absolute ignorance of recent controversies about heredity. It is therefore really impartial material for Mendelians to unravel. For the remainder of this paper we shall confine ourselves to the biometric treatment of the problem, which, as we have already noted, is independent of the existence or non-existence of any Mendelian formula. We shall compare the results with other investigations of a like kind and endeavour to interpret the results in biometric language. (7) Assortative Mating in the Shorthorn Population. Three sets of matings were worked out, namely: (a) the very recent matings of the parents of calves entered on the pedigree schedules, (b) the matings of the A. Barrineton AND K. PEARSON 445 great-great-grandparents of the calves on these schedules, and (c) a special series of matings taken from the early volumes of the Herdbook. Thus we have measures of the extent to which breeders now, 15 or 20 years ago, and perhaps 50 years ago, have mated like with like. The Tables each contain about 825 matings, and are given as Tables A, B, C of the Appendix. The general measure of the deviation from random mating was obtained by calculating the mean square con- tingency coefficient. The results were as follows: Table of Assortative Mating Coefficients. Recent Times : ‘20380 15—20 Years ago: ‘1912 50 Years ago: 1219 It will thus be seen that the tendency of breeders to mate like with like has been very sensible for many years, and has now reached a value comparable with the unconscious assortative mating in the case of man*, or with that which breeders of fancy mice consciously or unconsciously produce}. This selective action ought to be taken into account when we endeavour to estimate the resemblance of parent to offspring. (8) On Direct Inheritance. First Degree. The usual four tables, sires and dams with bull and cow calves, were formed. These tables will be found as D, E, F, and G of the Appendix. They were worked out by mean square contingency, and, for the sake of comparison and control, also by mean contingency and by fourfold tables. In order to provide material, on which it might be possible to consider Mendelian formulae, Tables II. and III. are provided, giving both sire and dam for over 1000 bull and 1000 cow calves respectively. These tables have been already used in the previous section. They differ from the mating Table A, in that the same cow and bull are repeated not more than three times, while in the mating tables they are entered as often as they occur on the schedules. A word must be said here about the contingency method as applied to these tables. The coefficient of mean square contingency is an absolute measure of the dependence of the offspring upon the parents, as long as we stick to the five groups now dealt with, but it would most probably be increased in value if we subdivided these groups—for example, divided the reds and roans into light and dark reds and roans. It has been pointed out elsewhere that this sub- division, without we immensely increase the number of individuals dealt with, must be limited, or we shall obtain very large contributions to the contingency from isolated units, which cannot of course be subdivided. Experience in com- paring tables reduced by different methods seems to indicate that 25- to 49-fold contingency tables approach closely in average value to the results reached by 4-fold table methods}. It would have been a great advantage in the case * Biometrika, Vol. 111. pp. 373 and 487, + Biometrika, Vol. ur. p. 20. + Biometrika, Vol. ut. p. 253. 446 On the Inheritance of Coat-Colour in Cattle of the shorthorn tables to have been able to compare the 25-fold table results with those for 36- and 49-fold tables; and thus determine how far we were approximating to a limit. This would be the more important if we had any method of differentiating the large classes of reds and roans, which we have seen may quite possibly have sprung from two original races. As no sub-classification is provided by the Herdbook, this cannot be attempted, and we are left with the TABLE II. Distribution to Sires and Dams of £ Offspring. Coat-Colour of Sire. Red | 4 Red, : | | Offspring Red Little White | and White | 022 White | Totals | eae | | | R. 75 | 10 5 68 1 159 R. L. W 10 4 3 13 — 30 Red | R. W. 5 3 — 4 1 13 | Ro. 4 1 1 91 27 124 W. — — — 2 — 2 | Group Total 94 18 9 178 29 328 R. ss 6 | 2 _- | 8 — 16 | Red, | R.L. W.... 9 1 | 1 5 — 16 Little | R. W. 3 ] | 2 3 | 9 White | Ro. — I — 15 6 22 W. — — oa — — | S Group Total 18 5 31 6 63 S i Pee z R. AG 14 2 3 10 1 30 iS Red | R. L. W.... 8 1 1 8 — 18 Ab and R. W. Hy 2 9 10 = 26 = | White | Ro. 3 2 _ 40 9 54 Te W. — | — — — | = — Oo | a 3 Group Total 30 | 7 13 68 10 128 = - - | = R. 4 56 7 6 45 _ 114 R. L. W.... 12 | 5 — 18 —— 35 Roan. | R:W. aj) 14; | 3 -- 13 — 30 Ro. S00 79 | 25 14 142 7 267 Wie aso = | — — 53 5 58 Group Total] 161 | 40 20 271 12 504 Bess cate ge | 1 a 1 Re Wir. 1 — — 1 — he | White | R. W. — | a = we Se Ro. ae 26 4 3 18 — 51 Wi. _ — 8 — 8 Group Total Pa 4 3 28 — 62 eae a Grand Total) 330 | 74 48 576 57 1085 A. Barrineron AND K. PEARSON 447 TABLE IIL. Distribution to Sires and Dams of 2 Offspring. Coat-Colour of Sire. = - | Offspring Red gh ee pee Roan White Totals | R. aie 81 13 5 58 — 157 | R. L. W.... 13 8 33 7 — 31 | Red R. W. 6 = 3 11 = 20 Ro. 2 — 4 76 13 95 W. = = 2 == a | Group Total | 102 21 15 154 13. | 3805 R. . | 18 6 a 14 — | 39 Red, Rega Wises 4 5 ot 5 = 14 Little | R. W.... 2 3 — 5 a 10 White | Ro. 1 — 1 28 4. 34 W. ae ie ae ee ca De: g | Group Total 19 14 1 52 4 90 eB R. sate 17 2 3 10 — By to) Red | R. L. W.... 4 3 = ih — | le u and R. W. 18 8 9 ll = | 46 5 | White | Ro. 1 3 1 41 9 bd wo) W. — — =A and ae aa Sy | ee a Group Total 40 16 13 69 9 147 ©) 3 iN R. sai al 12 8 41 = 122 Re Li W.ce. 14 4 3 13 — 34 Roan | R. W.... 10 6 5 22 = 43 Ro. 78 17 17 136 7 | 255 | W. — — — 31 4 35 | Group Total] 163 39 33 243 ll 489 R. ee = = = = ee et Weee | 2 ial - i = = White | R. W. 2 = 1 1 = 4 Ro. 19 3 5 15 — 42 W. — — — 1 3 10 Group Total 21 3 6 2o 33 56 | Grand Total) 345 93 68 541 40 1087 somewhat overweighted groups of red and roan—amounting to nearly + and 4 of the total offspring—to deal with. These large sub-groups are likely to reduce the mean square contingency coefficients, and we have no means of really testing how far we have already approached a limiting value of the mean square contingency coefficient. In dealing with the material by the 4-fold table method, we considered that the best division we could make was between beasts with some whole red and 448 On the Inheritance of Coat-Colour in Cattle beasts with no whole red, the latter class including roans and whites. Table IV. gives our results: TABLE IV. Parental Inheritance in Shorthorns. ; . Mean Square Mean Fourfold Nature of Percentage Contingency | Contingency Table Sire and Bull Calf ... 37 ‘43 | “46 | Sire and Cow Calf ... | “40 “50 52 | Dam and Bull Calf ... 40 “44 “44 | Dam and Cow Calf ... 42 “46 43 Mean sae see “40 46 “46 The result seems to us to indicate that the values by mean square contingency have come out somewhat too small. The general effect of an assortative mating if there were no selection of parents would be to increase the apparent parental correlation. If 7 were the real correlation, p the assortative mating, and r the observed parental correlation, we should have Bye eaep ie 2p?" If r='5 and p= ‘2, we should expect the correlation altered by about 35 increment. But this assumes that there has been no selection of parents, whereas in the case of the bulls at least there is obviously very considerable selection. The tables show that the red whites of both kinds are rejected, and that red and roan bulls are the most fashionable. But we know that any stringency in parental selection is more effective in reducing parental correlation than assortative mating can be in raising it*. The amount of this cannot be determined in the present case, but it very probably contributes to the reduction observed in the above table below the values determined for horse and grey- hound. The general conclusion that we can draw is that the parental resemblance to offspring in shorthorn cattle exceeds ‘4, and may quite reasonably equal ‘46, or even a higher value. It lies therefore well within the range of values ‘4 to ‘5 found already for man, horse, and dog. (9) On Direct Inheritance. Second and Higher Degrees. We now turn to the degree of resemblance between the offspring and more distant ancestry. The following results were obtainedT: * ¢¢Mathematical Contributions to the Theory of Evolution,” Phil. Trans. Vol. 200 A, p. 39, + The tabulated material will be found in the Appendix as Tables H, I, J, K, L, M, N, and O. No individual grandparent was taken more than six times, and on the average no individual occurs more than three times. A. BARRINGTON AND K. PEARSON 449 TABLE V. Grandparental Inheritance in Shorthorns. Sex change — —| Paternal Grandsire and Bull Calf... 19 0) Paternal Grandsire and Cow Calf... 25 1 Maternal Grandsire and Bull Calf... 18 2 Maternal Grandsire and Cow Calf... en, 1 Paternal Grandam and Bull Calf ... 5 1 Paternal Grandam and Cow Calf... 16 2 Maternal Grandam and Bull Calf ... "24 1 Maternal Grandam and Cow Calf... 25 0 Mean = “20 It will be seen at once that this is lower than the results obtained for man, horse, and dog; i.e. about *2 instead of 3. This is what we might expect, as the parental correlations found for the shorthorn by mean square contingency are about ‘4 instead of the 5 for pigmentation in man, horse, and dog. We are not prepared to assert, however, that it connotes an actual lessening of the intensity of inheritance, because other methods in the parental case seem to indicate that the resemblance is really greater than is shown by mean square contingency with only a classification into five very unequal groups. The values are clearly somewhat irregular. If we group them according to the number of sex changes we find: Grandparental resemblance with no sex change =‘22 5 % one 5 ='20 3 two sex changes=‘17 It would thus appear that, as in the case of eye-colour in man, there is a weakening of the intensity of resemblance with each change of sex. But if this be more than an appearance, it must be screened by other factors in the case of parentage, for in that case the values are ‘40 for no change of sex and ‘40 for one change of sex. This result is remarkably consonant with what we have previously observed in the case of greyhounds* where we found no sensible influence of change of sex in the parental resemblance, but an apparent influence on the grand- parental relation. In eye-colour in the case of man it was found throughout all relationshipst. We failed to demonstrate a change of sex influence in the case of coat-colour in horses}, and it seems worth considering the matter further in short- horns. Accordingly in attempting to estimate the great-grandparental influence, when we gave up the excessive labour of forming the 16 great-grandparental tables which would have meant dealing with upwards of 16,000 entries, and a far * Biometrika, Vol. 111. p. 258. + Pearson: Phil. Trans. Vol. 195 A, pp. 114-117. Lutz: Biometrika, Vol. 1. pp. 237-240. t Ibid. Vol. 1. p. 232. Biometrika 1v 57 450 On the Inheritance of Coat-Colour in Cattle larger number than this of pedigree great-grandparental relations *, we have simply classified into great-grandparents and (i) male, (il) female descendants, with 0, 1, 2, and 3 changes of sex. This required the preparation of 8 instead of the full 16 tables. If we write the pedigree in the following manner: 8g =99 10g =119 12¢ =139 14g =159 ees ee | 4g = 59 63 = 79 eae aren 2g = 39 ae eee eee 0g 19 then: 8 with 0 gave Table P of great-grandsire and ¢@ offspring, no sex change 15 with 1 - Q = grandam and @ - % ¥ 8, 12, 14 with 1 - R ; grandsire and 9? " one 9, 11, 15 with 0 i S ; grandam and ¢ e 5 5 10, 12, 14 with 0 55 T my grandsire and ¢ ss two changes 9, 11, 13 with 1 . U § grandam and 9? ms ; > 10 with 1 e Vv ie grandsire and 9 53 three ,, 13 with 0 53 We 5 grandam and ¢ " 5 55 The results obtained, still by the method of mean square contingency, are given in the table below: > TABLE VI. Great-Grandparental Inheritance in Shorthorns. Sex change | Resemblance Great-Grandsires and ¢ Offspring 0) “20 ” ” ” ” 2 “12 ” ” g ”? 1 18 ” ” ” Eh | 3 whe! Great-Grandams and ¢ Offspring 1 15 ” ” ” 9) 3 "15 ” ” 2 ” 0 26 ” ” ” ” 2 ‘12 Mean =°17 These results are irregular, and not very conclusive; probably large enough numbers were not taken in several of the cases, but the tabulation of great-grand- parents in large numbers is by no means an easy task, as in a pedigree stock high ancestors tend to be repeated many times. But in both grandsires and grandams, if we do not club together ~ and $ offspring we find that no change of sex is * The process throughout has been the same in all the ancestral tables; the produce is taken once only, and the same ancestor occurs on an average not more than three times, although individuals occur much more frequently, because as we go back there is much repeated blood owing to the fashion in bulls, and it would be almost impossible labour to reach 1000 independent great-great-grandsires. This method was adopted to avoid overweighting the tables with any particular individuals. The same principle was also used in the horse tables. A. BARRINGTON AND K. PEARSON 451 more potent than two changes (and than either one or three). There does not, however, seem to be much marked difference between one and three changes. On the whole it seems probable that a change of sex does produce some slight lessening in the ancestral resemblance beyond the parents; it is marked in eye-colour in man, and just appears in the coat-colours of greyhound and shorthorn, although if it exists in coat-colour of horses it has been screened by the probable errors of the results. The mean result for eye-colour in man for great-grandparental resemblance is ‘19, and for horse’s coat-colour °20, both results sensibly higher again than that for shorthorns. Still one further ancestral resemblance has been worked out, namely, that between great-great-grandparents and the offspring. But the complete set of 32 tables has been replaced by four dealing only with each sex in the ancestry and the offspring regardless of the exact line of descent. The results are given below*: TABLE VII. Great-Great-Grandparent and Offspring. | Great-Great-Grandsire and ¢ Offspring ... 09 Great-Great-Grandsire and 9 Offspring ... 14 Great-Great-Grandam and ? Offspring ... 13 Great-Great-Grandam and ¢ Offspring ... a3 Mean = ‘12 The only comparable result is the combined table for all sexes provided by Dr Alice Leet in the case of coat-colour in horses, and this gave for the great- great-grandparent and offspring the value ‘15, thus agreeing with the previous cases in giving a higher value than the shorthorn results do. We can now put together our results for ancestry in shorthorns and place alongside them those for other pigmentation researches: TABLE VIII. Ancestral Resemblance. Species Mant | Horset ae Greyhound|| |} Shorthorn See Character Colour | Colour | Colour | Coat-Colour | Coat-Colour | *ifion” Method lowed Fourfold | Fourfold | Fourfold 36-fold 25-fold Vari Se hag tet Table Table Table | Contingency | Contingency scene Parent... Ae wa “49 “52 BS 53 “40 “49 Grandparent... one 32 "30 22 33 “20 "27 Great-Grandparent rol 19 19 — — len 18 Great-Great-Grandparent — 15 — — 12 13 * Given as Tables X, Y, Z,Q,in the Appendix. + Biometrika, Vol. 1. p, 235. + Ibid. Vol. 11. p. 222. § R. S. Proc., Vol. uxvr. p. 157, omitting the cases of inheritance through the sire, as there can be little doubt that the sires are unreliable. || Biometrika, Vol ut. pp. 254-8. 57—2 452 On the Inheritance of Coat-Colour in Cattle But we do not consider these results as they stand as strictly comparable. We believe the lower values in the case of the shorthorns are not due to a lesser intensity of inheritance, but to the fact that the only possible contingency table is not finely enough divided. This we hold is indicated by the fourfold and mean contingency methods giving sensibly higher values than the mean square con- tingency method (see p. 448). It appeared desirable to determine the best geometrical series: a, ar, ar’, ar’, to fit the shorthorn results. Accordingly we made: (40 — a)? + (20 — ar? + C17 — ar’)? + (12 — ar’? aminimum. This led to the least square type equations : ‘40 + °20r +177? +127? =a (1 +7? 4+ rt + 1°), ‘20 + 347 + 367? =a(r + 2r? + 87°). Eliminating a, 7 is given by: ‘L777 + °28r8 + 1:20r* — -04r4 + 6373 — 367? + ‘06r — 20 = 0. The appropriate root of this is r=64 nearly, giving a=°382, and the best fitting geometrical series is ‘38, ‘24, *16, ‘10, as against the observed °40, ‘20, ‘17, ‘12. But the remarkable result of the investigation is that the factor of reduction for the geometrical series in the case of eye-colour in man was found * to be ‘62, and for coat-colour in horses ‘66+, and the reducing factor for shorthorns is ‘64, exactly their mean. The reducing factor for greyhounds is seen to be about ‘62¢. Thus disregarding absolute values we see that the manner in which ancestral resemblance decreases with ascent is very closely the same for such different species as man, horse, greyhound, and shorthorn, and for most purposes may be taken equal to the round number 2/3. If we assume r = 2/3, we find the best value of a for the shorthorn is almost exactly ‘37, giving the series ‘37, ‘25, ‘16, and ‘11, against the observed °40, °20, ‘17, and +12. It would thus appear that the general effect of applying the 25-fold mean contingency method has not been to lower the relative intensity of the successive grades of ancestry, but to weaken in about the same proportions the absolute values. The factor of reduction is about 38/50, or we should have for shorthorns the best fitting series, 50, 32.) 2 weal: as against an observed series, 53, ‘26, para “16. Either or both of these are in quite reasonable agreement with what has been observed for man and horse. Generally we may conclude that ancestral influence in the shorthorn diminishes at the same rate sensibly as in man, horse, and dog, and that absolute values are * Biometrika, Vol. 1. p. 222. }+ Ibid. Vol. 11. p. 222. {+ The mean of Dr Warren’s rather irregular results for Hyalopterus and Daphnia is again -61. Biometrika, Vol, 1. p. 189. A. Barrinaron AND K. PEARSON 453 probably closely alike in the four cases, although the categories in the case of the shorthorn render the comparison harder. (10) On Collateral Inheritance. The work of tabling pairs of whole siblings is somewhat lengthy in the case of shorthorns, the breeding life of a cow is short, and there is a good deal of changing in the case of the bull used. Accordingly we have contented ourselves with the cases of whole brothers and whole sisters*. It is much easier to table cases of half siblings, and six tables of half siblings—brother-brother, sister-sister, brother- sister, the bull parent being common, and again the cow parent being common— have been constructed+. These half-sibling tables are of peculiar interest, because there is a good deal of obscurity at present about the degrees of likeness we should expect to find between half siblings. A priori we should expect the degree of likeness of half siblings to be half that of whole siblings, but this relationship has not. been verified on any of the material hitherto considered: half siblings are relatively more alike than whole siblings, the addition of another common parent does not double the resemblance. In Table IX below we have put together the results with regard to whole and half siblings as far as pigmentation results have yet been worked out. Several conclusions may be drawn from this table. In the first place we see that half siblings through the bull are more alike than half siblings through the cow, and this is so for all three combinations. It would thus appear that the bull is more influential than the cow, but there is no prepotency of the bull over the cow in the parental resemblances. We should thus have the paradox that the bull caused the offspring to be more alike than the cow does, but not by causing them to be more like himself! Notwithstanding that the whole-sibling resemblances in shorthorns are of the same general value as those for eye-colour in man or coat- colour in Basset Hounds or Greyhounds of different litters, our opinion is that they are too low. Our experience with the parental tables leads us to believe that the grade of the resemblance is nearer that for hair-colour in man or coat- colour in horses. But we should not expect the reduction to be as great in this as in the parental case, owing to the special emphasis breeders lay on certain classes of parents. It is quite clear that neither in horses nor in shorthorns is the degree of resemblance for half siblings half that between whole siblings. The ratio of resemblance of maternal half siblings (the only case worked out for horses) to whole siblings is for horses ‘57, and for shorthorns is ‘56. These results are closely in accord, and our impression that a constant ratio exists between half and whole sibling resemblance, and that this ratio is greater than °5, is not shaken by the values in the case of Basset Hounds. We think that a very profitable line of investigation would deal with the relation of half to whole brethren in a variety of characters, and, if possible, with measurable characters as well as pigments. So far as we can see in this, as in other features of collateral inheritance, short- horns range themselves well within the limits already found for other species. * Tables a and B of the Appendix. + Tables y, 6, «, & 7, 0, of the Appendix. 454 On the Inheritance of Coat-Colour in Cattle = TABLE IX. Collateral Inheritance. Species | Man Horse} Shorthorn pee Greyhound Mean = | ee ee lan | zi | gu Seales asrite teat Coat Coat-Col Pi | ; air-| Coat- | Qoat-Colour oat- oat-Colour, igmen- | Character I qq Colour | Colour | ea Colour | Studbook Returns | tation '36-fold Contingency Reduction 4-fold | 4-fold | 4-fold | 4-fold 25-fold 4-fold = ere Method Table | Table | Table | Table | Contingency Table Same | Different | *8"04S Litter Litter Whole Brothers “52 54 62 62 52 "52 52 “55 Whole Sisters ... | °45 52 57 69 54 53 "60 “56 "56 Whole Brother ) | ee pe : : : i f and Sisters ¢{| 48 | 88 | “85: | °58 = 56 51 53 Half Brothers )|) __ eo . =e through Father § | < | = Be 5 ai af Half Brothers ii es _ i 5 through Mother ie B6 2 % ie =, we Half Sisters 7 mat 36 6 Be ean -36 through Father 4 Half Sisters 1 eae a. 9 ma "22% ; through Mother § [re 3 ae a an ca 36 Half Sister and \ | 2 Brotherthrough = “BQ a = = “32 Father J 3 Half Sister and = Brother through 28 30 a —_— = 29 Mother | Mean result fon W246 |e 2538 ensiieeGa 53 53 56 53 BB whole siblings Mean result for om | ; : | eye ; half siblings at all vee a2 22 5 az BS (11) General Conclusions. (a) With the exception of blue-grey cattle we are hardly likely to meet with a simpler colour combination in cattle than that to be found in the case of the shorthorn. If we consider the case of the cross of the polled black Galloway and the white shorthorn, then whether we deal with coat-colour or horns we find certain phases which suggest Mendelian ratios, but no simple Mendelian formula applies rigidly ; we find ourselves neglecting sensible percentages of occurrences incom- patible with a theory of the pure gamete. The hybrids of two races which at any rate breed approximately true, like the white shorthorn and white wild cattle, throw a considerable range of colours when they are mated. Thus to describe the colour action of shorthorn crosses will need a differentiation of the primary colour categories into homozygous and heterozygous groups, or the introduction of a * Not included in the general means of half siblings, as of doubtful validity. A. BARRINGTON AND K. PEARSON 455 somewhat complex system of latent characters, because practically every colour group will be found—with larger or smaller frequencies—to reproduce every other colour group. The only alternative is to reject the records in bulk as patently in error, because they do not fit rigidly a simple Mendelian scheme. (b) When we turn to the statistical treatment of the data we are at once met by a further difficulty. The method of contingency enables us to free ourselves from any question of colour order or scale, and thus we are not troubled with any hypothesis as to a normal or Gaussian distribution. But while thus taking its place as the rational statistical process for dealing with colour data, we are dependent on breeders’ colour classifications which are not made with a view to statistical treatment, and provide the statistician, as in this case of shorthorns, with groupings like those of red and roan not fine enough for his purposes. He obtains measures of intensity of resemblance comparable within the species—he can investigate the relative intensity of ancestral and collateral likenesses, but it is not so easy to compare the results for one species with a second. It would be a great advantage if red and roan could be recorded in light and dark shades, for the range in these colours is very wide and needs differentiation. With a 36- or 49-fold table we should undoubtedly get results better comparable with those for horses and greyhounds. (c) Notwithstanding these difficulties we actually find, however, that the shorthorns fit quite reasonably into the general range of results for man, horse, and dog. The “decay” of ancestral resemblance was found to be ‘64, a value lying between that found for man and dog, and expressible in round numbers by saying that the relationship decreases 2/3 at each upward grade. Thus the geometrical series feature of the ancestral law is maintained in the case of short- horns. The parental relationship lies between ‘4 and ‘5, and is probably nearer the latter than the former number; the whole sibling resemblance is a little over ‘5 at a minimum, and appears to be very similar in range of value for that found for man and dog. There is a substantial assortative mating produced by breeders mating like with like, but while this would raise the apparent values of the con- tingency coefficients, the fact that red and roan are especially fashionable tends by its selective action to produce more than an equivalent lowering. Generally the shorthorn data strengthen our view that—whether it be Mendelian, or some other yet to be found—there is a single basal law of inheritance, manifesting itself notwithstanding assortative mating, selection, and environment in the clustering of the inheritance constants for all species of populations about the value 45 to ‘5 for parental inheritance, and about ‘5 to °55 for fraternal inheritance, with a geometrical series of a ratio roughly about 2/3 for ancestral grades. Such an ancestral law will, we are convinced, be found to flow from the physiological basal law if it be discovered. Meanwhile each fresh series as it is worked out biometrically tends to strengthen the view that apparent variations in the in- tensity of hereditary resemblance are not real variations in nature. ‘he hereditary process is one, producing in a population definite degrees of average resemblance 456 On the Inheritance of Coat-Colour in Cattle between grades of kinship, and the apparent variation in intensity of mheritance is due to environmental influence, the selective action of breeders, assortative mating, and the difficulty of applying a uniform method of reduction to categories so varying in their extent and frequency as those we meet with in dealing with the records of quantitatively undetermined characters like pigmentation. APPENDIX. TABLES OF SHORTHORN MATING AND INHERITANCE. I. ASSORTATIVE MATING PRODUCED BY BREEDERS’ SELECTIONS. A. Modern Matings. Bull. R. LR. Tews ERewe |) Rowell mows Totals | RB. 246 ~ | B LW. 76 5 | RW. 86 te) 369 48 B. Matings of 15 to 20 Years ago. Bull. R. |. TW. | Ro Wall Groce eee otals 63 10 Wea peat she as 190 re 11 4 1 19 5 40 z 39 8 13 62 4 126 S| 142 30 40 | 188 18 418 | 29 3 7 2] a 60 284 BB 70 | 395 30 834 C. Matings of 50 Years ago. Bull. R. R.L.W.*| R. w. | Ro. | W. } Totals R. an 10 = 20 54 it 95 2) aves Ue ce ap ||) ae al ae RSW es a ee heya aiaS Pt 239 |) “Ro. $8 | = |) 6a Sooo ay 435 are i es 15 | 34 7 59 Totals 68 = | 146 | 517 97 828 * Formerly recorded under R. W. A. BARRINGTON AND K. Parson Il PARENTS AND OFFSPRING. D. Sires and Bull Calves. Sires. R. Rk. L. W. | RB. W. Ro. W. Totals S 151 | 21 14 | 132 2 320 = 40 i 5 | 45 aa 101 6) 27 9 11 30 1 78 =| Ro. 112 33 18 306 49 518 3s | W. ae ae 63 5 68 pQ ae Totals 330 "4 48 576 | 57 1085 | E. Sires and Cow Calves. Sires. R. |R.L.W.| BR. W.| Ro. W. | Totals > 343 3 93 O 123 > 481 CS) & 47 1087 F. Dams and Bull Calves. Dams. R. |RL.W.| RB. W.| Ro. W. | Totals | | $ 159 | 16 30 | 114 1 320 =a) Rel. W. 30 | 16 18 35 2 101 Gon ME. We 13 9 26 30 ae 78 — | Ro. 124-22 54 267 51 518 | 3 W. Q | —_ a 58 8 68 | ia) Totals 328 | 63 128 | 504 62 1085 | G. Dams and Cow Calves. Dams. Cow Calves. Totals Biometrika ty R. L. W. R. W. Totals 343 93 123 481 47 457 58 458 On the Inheritance of Coat-Colour in Cattle III. GRANDPARENTS AND OFFSPRING. H. Paternal Grandsire and Male Offspring. Paternal Grandsire. Rk. L. W. | R. W. Ro. : Totals bo bo OR ATR & Offspring. I. Paternal Grandsire and Female Offspring. Paternal Grandsire. R. L, W.| B. W.|'Ro. | w. | Totals on a 19 14 | 115 7 315 Ss. 8 4 54 1 95 & 7 ul 42 2 101 BS 30 96 | 261 | 15 456 } 4 4 sa) 4 57 O+ | | 65 59 | 506 | 29 | 1024 J. Maternal Grandsire and Male Offspring. Maternal Grandsire. rR. |RL.w.|R W.| Ro. | w. | Totals | on a 5 151 Dill 20 130) 5 342 ES L. W.. 39 6 10 49 2 106 Be R. W. Fe Alt 9 6 48 » 86 O Ro. ae 180 34 38 285 26 563 W. ae 19 1 2 34 2 58 5 1155 K. Maternal Grandsire and Female Offspring. Maternal Grandsire. R. R. L. W.| R. W. Ro. W. Totals a0 a R. 126 28 23 161 13 351 5, R. L. W. 39 8 7 37 4 95 Ag RW. . 29 10 15 40 1 95 ) Ro. 117 39 43 245 20 464 ep : 5 5 ‘ 61 & Offspring, ¢ Offspring. 2 Offspring. Jd Offspring. A. BARRINGTON AND K. PEARSON L. Paternal Grandam and Male Offspring. Paternal Grandam. R. L. W.| R. W. | Ro. M. Paternal Grandam and Female Offspring. Paternal Grandam. R. R. L. W.| R. W. Ro. W. Totals R. 123 i | ove |) “is 359 R. L. W. 36 3 16 47 3 105 R.W. . 22 4 12 51 5) 94 Ro. 145 17 53 306 40 561 W. 14 — a 28 7 53 1172 N. Maternal Grandam and Male Offspring. Maternal Grandam. R |RLW.|R.W.| Ro. | W. | Totals R. 131 32 43 125 11 342 Rip WWE 29 6 18 | 34 3 90 | B. W. 16 7 11 | 42 2 78 Ro. 127 33 72 | 287 32 551 | W. ov 2 2 46 6 63 | Totals 310 80 146 | 5384 | 654 1124 O. Maternal Grandam and Female Offspring. Maternal Grandam. R. L. W.| BR. W. Ro. W. Totals | R. 133 22 | 46 | 130 | 10 341 | Rel aWe..: 28 8 8 | 33 3 80 | R. W. 25 6 26 43 8 108 Ro. 94 32 67 | 235 32 460 W. 6 4 — 32 5 47 Totals 286 72 147 473 58 1036 | 0 460 On the Inheritance of Coat-Colowr in Cattle IV. GREAT-GRANDPARENTS AND OFFSPRING. P. Great-Grandsire and Male Offspring. No sex change. Great-Grandsire. BR |R. hk. Wo) Row. | Ro. | weanlemecsen oD | = ee Bl oR GW... @| RW. oO | Ro. W. So | Totals Q. Great-Grandam and Female Offspring. No sex change. Great-Grandam. R. R. L. W.| R. W.} Ro. W. Totals ? Offspring. R. Great-Grandsire and Female Offspring. One sex change. Great-Grandsire. chai : —- ' R. BR. L. W.| R. W. | Ro. W. Totals op | ] a R. 220 30 | 32 207 10 499 a Rea Were 57 11 9 70 5 152 4 R. W. 47 7 10 82 4 150 ) Ro. we 228 50 43 468 25 814 W. ane 21 6 10 60 6 103 O+ Totals 573 104 104 887 50 1718 8S. Great-Grandam and Male Offspring. One sex change. Great-Grandam. |R. L. W.| RB. W. | Ro. Jv Offspring. & Offspring. ¢ Offspring. V. Great-Grandsire and Female Offspring. ¢ Offspring. W. Great-Grandam and Male Offspring. Three sex changes. J Offspring. A. BARRINGTON AND K. PEARSON T. Great-Grandsire and Male Offspring. Great-Grandsire. Two sex changes Ro. W. Totals R. | RL W. | R.W.| Ro. | W. 263 36: itm 33 230 | 15 60 LF a 71 5 47 10 | 8 | 78 4 287 41 | 56 400 29 30 8 | 5 42 4 Totals Great-Grandam and Female Offspring. Great-Grandam. Ro. W. Totals R. L. W.| R. W. Great-Grandsire. Totals RW. R. W. Great-Grandam. Re Th, W. 461 462 On the Inheritance of Coat-Colour in Cattle V. GREAT-GREAT-GRANDPARENTS AND OFFSPRING. X. Great-Great-Grandsire and Male Offspring. Great-Great-Grandsire. R. L. W.| R. W. | Ro. W. | Totals or ike 609 2B] RLW 136 a Re We 188 = | Ro. 806 | W. 95 Ro | Totals Y. Great-Great-Grandsire and Female Offspring. Great-Great-Grandsire. | R. |R.LW.|R.W.| Ro. | W. | Totals rota) 5 : a | R. 215 20 29 177 27 468 = |; RLW 43 3 11 61 7 125 ap Pl ie 35 4 8 39 5 91 Ss Ro: 250 16 63 318 39 686 W. 26 4 8 | 59 6 103 Or ) | Totals 569 47 119 | 654 84 1473 | Z. Great-Great-Grandam and Male Offspring. Great-Great-Grandam. R. | RL. W.| BR. W. | Ro. 2 236 25 105 | 364 a 64 6 7 79 Z 46 5 36 || 116 5 256 33 133) | 505 29 9 23 50 So 631 78 304 | 1114 Q. Great-Great-Grandam and Female Offspring. Great-Great-Grandam. | BR. | Be. Wei) Wen Ro W. | Totals oO | = 169 19°) 5168 258 27 541 a 36 3 19 69 4 131 Z 19 5 17 58 15 114 S 214 25 98 437 54 828 9 oF | 24 3 18 53 11 109 1723 VAD: A. COLLATERAL INHERITANCE. BARRINGTON AND K. PEARSON a. Whole Brothers. First Brother. Second Brother. =e) = 290 29 69 155 27 29 44 8 22 1 69 8 108 109 12 155 22 109 | 494 75 27 1 12 75 | 50 Totals | 570 104 306 855 165 | 2000 R. | R. LW.) RB. W.} Ro. W. Totals WHOLE AND HALF-SIBLINGS. 570 104 306 855 165 B. Whole Sisters. First Sister. R. L. W.| R. W. Totals w 2 d 549 aA _L. W. 100 | BW. 284 q 0. 911 3 : 156 RD Totals y. Half Brothers through Sire. First Half Brother. 5 = s R. W. Totals ~ e R 719 aeRe lL Wee, 191 Pa ReeWer’ ds 151 It | Ro. 857 aot W. 122 (=| ° ® | Totals 857 M | 6. Half Sisters through Sire. First Half Sister. . ts i ia aaa a | 3 R. |R. L. W.| R. W. |] Ro. W. Totals nm M R. . 266 41 41 246 17 611 $ inch . Totals Deaths 1695 | Totals | 5163 6858 | r= —'2997. F. M. Turner 487 TABLE VIII. Vaccinated cases only, 1901—2. Number of Vaccination Scars and Severity. Recoveries | Deaths Totals Less than 4 scars 3955 4 scars or more 2930 | | Totals 6885 r= — ‘2631. TABLE IX. Vaccinated cases only, 1901—2. Foveation of Scars and Severity. Recoveries | Deaths Totals Scars less than half foveated ... 5203 | 602 5805 Scars more than half foveated... 980 58 1038 Totals 2. te 6193 || 660 r= —'1664, These figures agree well with those given by Macdonell for the Glasgow epidemic of 1900-1, less closely for the epidemic of 1892-5 in the same town. The slight differences in the total numbers observed in the three tables are due to the measurement of area or the counting of the number of scars having presented special difficulty in certain cases. An important practical point to settle, if possible, is whether these three correlations are independent or two of them secondary to the third. It seems probable @ priori that the protection afforded by vaccination is proportional to the area of the vaccine vesicles, which may be regarded as laboratories for the produc- tion of an antitoxin. If this were so the correlation between number of scars and severity would be secondary to that between number and area of scars. TABLE X. Vaccinated cases only, 1901—2. Number and Area of Scars. Totals Area < } inch | Area >} inch 3924 2926 Less than 4 scars 4 scars or more | Totals... 1691 | 5159 r=+°6648. Denoting by ry, Yys, Tay, the total correlations between area of scars, number of scars and severity of disease taken in pairs, and pjs, pys, the partial correlations 488 Correlation between Vaccination and Smallpox between area and number of scars with severity of disease, we have, when (i) area is constant and (ii) number is constant respectively, trys = —'2631, tys=—°2997, ray =+ 6648. Tus — Tas’ an : ies : — =— 0897, ‘i V(1 = 1245) (1 = 724y) yh Has Unslan A oo) V0=ry5)(U= Fa) Dealing in the same manner with area and foveation of scar, we have the following table: TABLE XI. Vaccinated cases only, 1901—2. Area and Foveation of Scars. | : Area <$ inch | Area >4 inch | Totals | Scars less than half foveated ... | Scars more than half foveated... | | Totals r= +2386. And Me=— 2991, Tr=— 1664, 1,,— +12386; Trs — Vas’ ar = = =— 1012, ae vd = fas) Uae) i, v1 — rps) Ci) It seems therefore not improbable that among cases with equal area of vac- cination marks, there is no difference in the amount of protection afforded by different numbers of scars. The apparent protection afforded by foveation of the scars, though not very great in itself, can with less probability be attributed to secondary association of foveation with large scars. It would be a good thing if these two points could be definitely settled, for a great deal of time is spent in smallpox hospitals in observing and recording these data, some of which may possibly be unnecessary. The fact that pys as above calculated is not equal to 0 may be due to the fact that some scars grow considerably between infancy and adult life. This of course would tend to lessen ry, (the area being measured many years after the operation), but would leave ry, unaltered unless, as is probably not common, one or more scars disappeared entirely. The above formulae also apply only to linear correla- tion, while the correlation in these cases may be skew. F. M. TurRNER 489 I have paid much attention to the subject of age in these statistics, as the subject seems to me important from two points of view. It is known that the protection afforded by vaccination against attack is not permanent, but diminishes with lapse of time. A quantitative estimate of the amount of protection soon after vaccination and after the lapse of different intervals of time would be a useful guide as to the proper period at which to perform revaccination. Since no statistics are available as to the protection against attack, it seems reasonable to enquire into the time-decrement of the protection against death and take this as a guide. The second reason for enquiry as to age of the patients is the hetero- geneity of the figures given above as regards age. If we take the correlation coefficient between vaccination and severity to be a measure of the protection afforded by the operation, it seems reasonable to demand that the two groups, vaccinated and unvaccinated, should be similar in all respects, so far as that is possible of attainment. But this is far from the case in respect to age in the above tables. Table IV. shews that the ages of the vaccinated and unvaccinated differed greatly, and this would tend to alter the value of the correlation. Using the language of multiple correlation there is a very high correlation, among the patients treated in hospital, between age and vaccination. That being so, we cannot affirm d@ priori a close correspondence in magnitude, nor even in sign, between the total correlation 7, between vaccination and severity, and the partial correlation py; between the same factors. To ascertain the time-decrement of protection I have divided up the cases into five age groups and worked out the value of rin each. Nearly all the cases who were vaccinated at all were vaccinated in infancy, and the number of these who had been revaccinated was small. The assumption that the age of the patient measures the number of years that had elapsed between the last operation of vaccination and the attack of smallpox is in most cases correct. The following shews the number of revaccinated cases admitted : TABLE XII. Number of Revaccinated Persons Admitted in 1901 and 1902. Interval in Years since Revaccination. 0—10 | 10—20 20 and over | Uncertain Totals 1901 12 240 45 | 1 82 | 1902 34 79 161 g 76 Totals | The figures for 1902 are not complete in that no record was kept of revaccina- tion at Gore Farm Hospital. As the cases admitted to that hospital numbered 565 only, as against 7316 to the two hospitals where the record was kept, the omitted cases cannot have been numerous. Biometrika Iv 62 490 Correlation between Vaccination and Smallpox A complete list of the 276 cases recorded in 1902 is published in the report From this I have compiled the following details as to age: TABLE XIII. Ages of Revaccinated Persons Admitted in 1902. | Under 15 | 15 20. 20 25 | 25 25——30 | 30 —40 = 40—50 | 50—60 | Over 60 | Totals | | | No. of cases 3 | 103 | 44 20 | 276 No. of deaths —— v | 8 2 27 From these figures it will be seen that Tables XVI. and XVII. below only contain 3 revaccinated cases, and the high value of r at this period is not due to prevalent revaccination. TABLE XIV. Patients under 10 years of age, 1901—2. Recoveries Deaths Totals Unvaccinated Vaccinated or doubtful Totals r= — ‘5567 TABLE XV. Patients under 10 years of age, 1901—2. Toe | Deaths Totals Unvaccinated or doubtful ... 859 448 Vaccinated... ae see 132 2 Totals 991 | 450 7 = — ‘6636. TABLE XVI. Patients from 10 to 19 years of age, 1901—2. Recoveries Deaths Totals Unvaccinated Vaccinated or doubtful Totals r= — ‘6414. F. M. Turner TABLE XVII. Patients from 10 to 19 years of age, 1901—2. Recoveries Deaths Totals Unvaccinated or doubtful ... 527 165 692 Vaccinated 1140 23 1163 Totals 1667 188 r= — 6933. TABLE XVIII. Patients from 20 to 29 years of age, 1901—2. Totals Recoveries | Deaths Unvaccinated 244 Vaccinated or doubtful Totals TABLE XIX. Patients from 20 to 29 years of age, 1901—2. Recoveries | Deaths Totals Unvaccinated or doubtful ... Vaccinated Totals TABLE XX. Patients from 30 to 39 years of age, 1901—2. Recoveries Deaths Totals 90 1832 Unvaccinated = Vaccinated or doubtful Totals 1597 r= —°4093. TABLE XXI. Patients from 30 to 39 years of age, 1901—2. Recoveries | Deaths Totals Unvaccinated or doubtful ... 90 | wi 167 Vaccinated 1507 248 1755 Totals 1597 | 325 1922 | i ae r= —:4790 _—_ 491 TABLE XXII. Patients 40 years of age and over, 1901—2. Correlation between Vaccination and Smallpox 1189 Totals Recoveries Deaths | Unvaccinated es 17 34 | Vaccinated or doubtful Wi 359 1531 1582 Totals... tee | 393 r= —‘4771. TABLE XXIII. Patients 40 years of age and over, 1901—2. Recoveries Totals Unvaccinated or doubtful ... Vaccinated 98 1091 Totals 1189 r= —'4440. Of the two methods of dealing with the doubtful cases 1 have given my reasons above for preferring their inclusion with the vaccinated rather than the unvaccinated. As between Tables XIV. and XV., however, I think it possible that a majority of the “doubtful cases” under 10 were unvaccinated, in which case the value given by Table XV. would be the more accurate. On comparing the values of either series, however, we find that the value of r does not vary very much in the three periods under 30 years of age, whereas after the age of 30 there is a moderate fall in value. The maximum occurs in the period from 10 to 20. This result is in striking contrast to the opinion of the members of the medical profession as to the duration of immunity against attack ; which is that the protection afforded by vaccination is highest for the first few years after the operation and then talls off rapidly, so that at 10 or 12 years of age revaccination becomes desirable. So long as no statistics are available of the number of persons exposed to risk of infection by smallpox, it is impossible to calculate by mathematical methods the protection afforded by vaccination against attack. But the above opinion as to the duration of vaccination-immunity is really arrived at by methods identical in principle to the above fourfold tables, that is by recording the numbers attacked of vaccinated and unvaccinated persons at different ages, and contrasting these with the very roughly known proportion existing between vaccinated and unvaccinated persons in the community at large. Inside a smallpox hospital there is found a very high correlation between age and vaccination, say ‘9 or ‘8 (see Tables XXIV. XXVI. and XXVIII). If there existed an equally high correlation among the whole community between age and vaccination, we might assume that vaccination either failed to protect, or protected with equal force throughout hfe. But if there is only slight correlation between F. M. TurnER 493 these two conditions among the population outside, which is most probably the case, the protection gained by the vaccinated in infancy must diminish as age advances. We seem therefore driven to the conclusion that protection against attack and protection against death, though both produced by the same operation, disappear at widely different rates. In a later part of this paper, however, I put forward a suggestion which I think accounts for the discrepancy in the figures, and allows us to suppose that the protections against attack and against death produced by vaccination are the same, not only in cause, but also quantitatively as measured by correlation coefficients. The effect of the age of the patients upon the value of r in Table I. is perhaps best shewn by comparing it with the values derived from Tables XIV.—X XIII. The mean value of 7 from the five tables, in which the doubtful cases have been included with the vaccinated, is — ‘5234, against —°4246 of Table I. And the mean value from the other five tables is — 5795 against —"4922 of Table III. The effect has therefore been to diminish r. It is interesting, however, to use the formulae for multiple correlation to test the result. Taking the division of age at 10 years, we have the following table: TABLE XXIV. All Cases, 1901—2*. Under 10 Over 10 Totals | Unvaccinated Vaccinated Totals 1408 y=+°912. TABLE XXV. All Cases, 1901—2. Recoveries Deaths Totals Under 10 Over 10 Totals 8030 9659 r= —'3150. * T have inadvertently omitted the doubtful cases in calculating this table. As these high corre- lations are laborious to work out I have not thought it necessary to recalculate the value with the doubtful cases included. In all the tables of age and vaccination I have carried the series in r to twelve terms, and the terms beyond would certainly influence the third figure, probably the second also. The Tables XIV. and XV. also give series in + which converge slowly. In these two cases also I have stopped at twelve terms. 494 Correlation between Vaccination and Smallpox: From Tables I., XXIV. and XXV. we get Pre= = 4QAG, Typ= 4-912, Fy = ooRinO: lys—Taylas ore Zu — 3530. P ro = — = _ v1 — 1 ay) (l= Tags) If we take the age division at 20 or 80 years we get the four Tables XXVI.— XXIX. TABLE XXVI. All Cases, 1901—2. Under 20 | Over 20 Totals | Unvaccinated.., 1893 2278 | Vaccinated or doubtful 1403 7381 | Totals 9659 7 ; : = + 821. ; } TABLE XXVII. All Cases, 1901--2. _ ; | Recoveries | Deaths Totals Under 20 3296 Over 20 6363 Totals... 8030 | 1629 9659 Whence we deduce Tyy = — 4246, ryp=t'821, ry4,=—°0904, Pvs =— 6178. TABLE XXVIII. All Cases, 1901—2. Under 30 Unvaccinated nae Vaccinated or doubtful F. M. TurNER 495 TABLE XXIX. All Cases, 1901—2. Recoveries Deaths Totals Under 30 Over 30 Totals 8030 r= +°1360. Whence we deduce Pyg = — "4246, ryp=+ 671, ryg = +1860, pys=— 7022. The discrepancy between the above three values of p;, is largely due to the want of linear correlation between age and severity of disease as measured by death or recovery. Another possible cause has occurred to me, which may have contributed to this result; and as this cause would also account for the small value shewn above for the diminution of vaccination protection with time, I am inclined to lay stress upon it. Let us take any individual out of a population exposed to smallpox. Three things may happen to him, either he may escape the disease altogether, he may contract smallpox im a mild form and eventually recover, or he may contract it in a severe form and die. Into which of the three classes any individual falls, depends on his power of resistance to the disease or his immunity, and on the amount of virulence of the infection to which he is exposed. In some diseases, as diphtheria and tetanus, we are acquainted with immunity produced by a chemical substance, and which may therefore be quantitatively measured. In diphtheria and tetanus the above three classes correspond to definite quantitative amounts of antitoxin. In smallpox we have not this knowledge, but the analogy makes it justifiable to regard immunity as a quantity, though not as yet measurable. The immunity may be natural, or acquired by vaccination, or the result of the two causes combined. If in such a population we desire to examine the correlation between severity of disease, which is the observed fact rendering the immunity visible to us*, with any other factor as vaccination, age, &c., we may make a sixfold table, thus: TABLE C. e State as regards Disease. ee 7 Eee NouDidens Disease and Disease and 0 Disea: q Sate pn Recovery Death Totals Orr acs af! Unvaccinated as} = . 8 Vaccinated os ~ ar Totals (op) = : * The influence of exposure to different degrees of virulence of infection is probably also an important factor, but is ignored, because at present beyond our observation. In dealing with a large population there will probably be no marked differences in this respect between the vaccinated and the unvaccinated classes. 496 Correlation between Vacemation and Smallpox where g=atd, l=bt+e, m=c+f, s=atb+eo, t=dt+et+f, N=s+t=g+l+4+m. This sixfold table can be turned into a fourfold table by adding the middle column to either the first or second. If the distribution of the two characters among the population is a normal one, the same value of r will be obtained by applying the “fourfold table” formulae to either of these. This, however, will not be the same as the value calculated by the same formulae from the fourfold table obtained by omission of the first column, thus: TABLE D. Recovered Died Totals Unvaccinated ... b c wu Vaccinated 2s e f v Totals... U m n where u=b+e, v=et+f, n=l+m=utv. We have no accurate evidence on the distribution of immunity among the population at large, but the following reasons point to the assumption of normal distribution being more correct for the former sixfold than the latter fourfold table. (1) The same agent, vaccination, is known to protect against attack and also against death. On the former supposition this fact is intelligibly represented in the table, for if an array representing a group of persons all vaccinated to an equal degree were to be vaccinated further, the severity of disease in each individual case would be reduced. It follows that some who would have con- tracted a fatal attack would now recover, i.e. some cases would be transferred from column 3 to column 2. Similarly others would be transferred from column 2 to column 1, and would not suffer from smallpox after exposure. In the fourfold table the first transfer is similarly represented, but the second is not. Whatever unit may ultimately be used to measure severity, the hypothesis of normal distri- bution applied to the latter table cannot explain a reduction of severity to 0 or a negative quantity. (2) Among vaccinated persons the severe cases are much less frequent than the mild. The hypothesis of normal distribution amongst those affected with smallpox requires that in the total, and also in each array, the cases shall cluster round a certain mean severity and that cases of very great or very slight severity shall be extremely rare. If, however, normal distribution of immunity applies to the whole population, we can readily see how grades of slight severity can be the commonest. F. M. Turner 497 Thus in Fig. 1 the abscissae represent degrees of immunity, which in the case of those who contract the infection are represented by the degree of severity Fig. 1. of the resulting disease. In those who escape infection the degree of immunity is greater than that required to prevent the disease appearing. It may be much more than this, but the excess is latent, and we have no means of observing it. The ordinates represent frequencies, the line ab divides those who escape infection from those who do not; the line cd divides the deaths from the recoveries. The area abdc represents the number of recovered cases, and acd the number of deaths. It is obvious that, if those who escape infection exceed half of those who are exposed, the frequencies of the different degrees of severity diminish without exception, as the degree increases. Probably the actual frequency curve among the vaccinated is of this type. Assuming for the above reasons that the normal distribution applies to Table C we see that the same cannot hold also for Table D, though all the above calculations have been based on the latter assumption. It is therefore important to examine what is the effect of neglecting one column in a sixfold normal table and treating the remaining figures as if they conformed to a normal distribution. Such a table I shall call a “curtailed” normal distribution, and the value of r obtained from the full table I shall call the true value. That which is obtained from the curtailed table by the use of any formula I shall call the “apparent” value and shall denote by 7’. As a first example I take the table on p. 214 of Biometrika, Vol. 1., which is shewn by Macdonell to be a close fit to the normal correlation table. Drawing a line under “head breadth 15:0” I have kept all figures above the line and discarded all those below. The resulting table is therefore a curtailed correlation table, analogous to our smallpox tables drawn from the inmates of a smallpox hospital, all those who remained unaffected by the disease having escaped observation. A curtailed table in measurable characters would result if we collected statistics of height from soldiers, a population from whom all below a certain standard height had been already rejected. Using product moments, I have found in the above case 7’ = ‘2446, whereas the true value r= 4016 + ‘010. Biometrika ty 63 498 Correlation between Vaccination and Smallpox Using fourfold tables the value of r’ varies very much according to the point chosen for division. Thus I have calculated the following three tables: TABLE XXX. Head Length. = | 19:0 or less Over 15:0 Totals = 3 © | Less thando:0) 7 701 599 1300 aa) | 16:0 AP wae 109 119 228 rs | ———— v | Totals Wes 810 718 1528 a8 7 ='0837. TABLE XXXI. Head Length. a 19:0 or less Over 19:0 Totals So S _——— 2 Less than 14°9 ... 619 458 1077 P| i90r150— ... 191 260 451. | oS | Se © | Totals... 810 718 1528 | a2 i 7’ = "2266. TABLE XXXII. Head Length. < 19-0 or less Over 19:0 Totals as) a 2 Less than 14°8 ... aa 14°8 to 15:0 cS ® Totals on 7 = "2595. It is seen that in every case 7’ is less than r. If we use product moments a mathematical formula can be obtained. N cart (5-244) Z>= = mG 2(1 - r) Oe Tx Fy oy” 27 V1 —P ogy represent the equation to the correlation surface, and «=h’=ha, be the equation to the bounding line. Let Let @, y be the coordinates of the centroid, r’ the correlation, both the centroid and the correlation applying to the curtailed surface only. 2 f+o [| @- mya zdedy Then tT a co oe (IL) if | (a — zy eddy x [ i (y- yy eee ay hJ -o# hj -o@ F. M. Turner 499 and 0 | i zdudy Since the centroid of every y array lies on the regression line, the centroid of the figure must also lie upon this line. Consequently All the integrals in (IL) and (III.) can be reduced to the probability integral or directly integrated. Putting e 8" \/2ap and I ede = V2arq, +o 1 x? h | ie led =—{* et a2de we ave Pe ZAXL Y Woe se P o we = Noq, a [+00 N rr) = 1 and | | xzdady = —— | e onda h/J -«© \/ Qer Ould h From (IV.) gar. a! A eee eee (VI). Substituting the values from (V.) and (VI.) we obtain the following values for the three integrals in (IL): wo r+o 2 __ a2 | | (w —2)zdady = No? WS hj-a qY 2 fo by OD 2 I, Ie (y —j'zdudy = Noy "4 i ae ie i (a —2x)(y — 9) zdady = Nrogoy PLEO. Substituting these values in (IL) we obtain: pT (hpq + ¢ — Pp’) Le icc ec eee ncncceeeenenes VA "8 (hpqg— Pp) +@ ee or ie Ee Ps A EP ta (VIL). q+ (1 — r*)(hpg — p*) From (V.) we see that the distance between the centroids of the curtailed and the whole figures, resolved along the axis of a, is rae The distance of the limit of the curtailed figure from the latter centroid resolved in the same way is h’ =hoz; so that ? me h is the expression for the distance of the curtailed centroid from the limiting straight line. It follows that 2 a h is always positive. 63—2 500 Correlation between Vaccination and Smallpox Since p and q are essentially positive and r? < 1 we see from (VIII.) that it is impossible for 7’ to be numerically greater than r. If, however, r =0 or 1, then r also =0 or 1. The same can be shewn by differentiating p —h. The equation (VIIT.) can be written thus: r!2 a2 u 1 2 pees) ae @ In the case of smallpox correlations 7’ is the calculated apparent correlation. r is unknown, and cannot be calculated since h is also unknown. But it will be seen that when 7’ is constant 7 is greater when p? = hpg _ (2) - h (2) un q q is less. Calling this quantity z, the following gives values of z for a range of values of h: h Zz h z ~ 2-0 +119 +01 +658 15 "227 0-2 662 1-0 371 03 -700 0-5 514 0:4 ‘715 0-4 540 | 05 “729 03 565 1:0 ‘796 0-2 592 15 “848 0-1 ‘613 | 2-0 “854 ~0°0 +637 | +3:0 1:30 It will be seen that, as ) increases, z and consequently the discrepancy between 7 and 2” also increase*, * A simple geometrical interpretation of the quantity z is obtained thus. Let C, Fig. 2, represent the centroid of the whole figure, CX, CY the axes, PN the dividing line, and O the centroid of the curtailed figure. Then CN=ho,. Draw OM perpendicular to CX. P Y O CG NIM x Fic. 2, cM=!.c,, Gq q 5 C— z\ -h (2) ; \q q “2X 0,2=CM?-CM.CN=CM. MN. F. M. Turner 501 In the numerical example worked out above r ='4016 + 010, Lo23he from the tables, h = — 023 398, p= +399 0257, whence , © = 2570. This agrees closely enough with the value ‘2446 obtained by direct calculation considering the probable error of 7. In a curtailed normal distribution the use of fourfold tables and formulae is not strictly applicable, i.e. the result obtained from them is not the true correlation of the figure, for the formulae only apply to truly normal figures. Since, however, the formulae give a numerical result to whatever fourfold table they are applied, it is useful to discover the relation between the value so obtained, and the correla- tion coefficient of the whole figure. I have not succeeded in finding a solution of the general problem. I have, however, calculated a fair number of curtailed normal tables by the fourfold formulae either chosen froin the tables in Macdonell’s paper on “ Criminal Anthropometry,” or formed by first using the fourfold formulae to get the frequency of a normal table with the value of 7 = ‘5 at various points of division ; and then using these frequencies to form curtailed tables and calculating 7’. It is rash to generalise from a few instances, but I think the following statements are true. (1) The apparent correlation 7” is always less than the true correlation 1, The difference increases as the unobserved portion g increases, or as h decreases, (2) In the same distribution, when r and h are constant, 7’ varies with the point of division. If this is close to the bounding line 7” becomes very small. An interesting problem, which may prove of practical application, is to solve a table of the form of Table C., when @ and d are unknown, but g=a-+d is known. This may be done by the tedious process of trial and error. Give any value to a and from the sixfold tables deduce two fourfold tables. If the normal distribution holds good, the values of 7 obtained from the two tables will be identical. As an example let us suppose that the number of persons exposed to infection in London was double the number who developed the disease. From this we get the following sixfold table. TABLE E. All Exposed to Infection. ¥ Correlation between Vaccination and Smallpox Unvaccinated Vaccinated Totals Not Diseased |- Affected with Smallpox Recovered Died Totals Unvaccinated Vaccinated Totals Not Diseased If a= 400 we find 9659 9659 19318 r= — 5373. | Alive Dead Totals Unvaccinated 1975 753 2728 Vaccinated 15714 876 16590 Totals 17689 1629 19318 fo= — 5429. 7, =— 5611, 1, = — 5496. Whence by interpolation we get the correct solution : If, however, the number exposed to risk is not much larger than the number a= 434, r=—'5451, d = 9225, who contract the disease the values of 7 and 7’ more nearly coincide. exceeded —°54. Thus if the number exposed who do not contract the disease is equal to only one-tenth of those who do, we have g = 966, and I have calculated a= 39, r=—°4492. On the other hand if the number of persons exposed to infection was several times larger than that of those who developed the disease, the true value of 7 considerably F. M. Turner 503 TABLE F. Diseased Not Diseased Totals Recoveries | Deaths Unvaccinated ... 39 | 1525 | 753 2317 Vaccinated ans 927 6505 876 8308 Motalsse 966 8030 | 1629 10625 r= —'4492 I think the above figures afford some explanation of the apparent small diminution of vaccination protection as age advances. If my contention be correct that a normal distribution in immunity may more correctly be assumed for the whole population exposed to attack, than for that part only which actually contracts the disease, and appears in smallpox hospital statistics, then the values given above (Tables IT. to IV. and XIV.— XXIII.) are apparent values of r corresponding to curtailed distributions. If any time-diminution occurs, then the number of persons sufficiently protected to escape the disease altogether will be much larger at the ages 0-10 than at higher ages, say 20—30. Hence, using the above notation, in Table XIV. the value of g is much greater than in Table XIX. Hence r—7’ will be greater in the former case than in the latter. Although, therefore, the values of 7’ in the two cases are nearly equal, the values of r probably differ considerably. The above investigation of curtailed distributions also gives a possible explana- tion of the small correlation between severity of disease and number, size and foveation [i.e. depth] of the scars, For we have seen that if in any normal distribution, if one of the factors is divided into three classes, then leaving out either of the extreme classes from consideration will reduce the sixfold to a fourfold table. The resulting apparent correlation, however, will be less than the true value. Now in arranging our Tables I.—III. we have recognised only two degrees of vaccination, “ vaccinated ” and “unvaccinated.” But if we split the vaccinated group into two, “ four scars or more” and “less than four scars,” we may arrange the whole as a sixfold table. Hence Table VIII. is seen to be acurtailed form of Table I; and similar arguments apply to Tables VII. and IX. Macdonell, who discovered the lower correlation in these instances, remarks*: “It is obvious that in dealing with the last four tables we have descended to a much lower plane of correlation, and the results may possibly somewhat modify medical opinion as to the degree of significance of foveation, number of scars and scar area.” If, however, the normal distribution held good for Table I., then a “lower plane of correlation” would necessarily follow in Tables VIT.—IX., not as a medical, but as a mathematical phenomenon. * Biometrika, Vol. % p. 383. 504 Correlation between Vaccination and Smallpox Summary. 1. The correlation coefficient between severity of disease and previous vaccination in the late London epidemic was lower than in any of the previously investigated cases. 2. There seem stronger reasons for assuming normal distribution for the whole population exposed to risk of infection, than for the population attacked by the disease. 3. If the former assumption is correct the smallpox correlation tables of Pearson, Macdonell, and of this paper, correspond only to a portion of the normal correlation surface. 4. Values of r calculated from such “curtailed” normal distributions are lower than the true value, whether product moments are used or fourfold formulae, or contingency*. As to the first method a complete investigation is given, as to the other two I have failed to find a mathematical proof, and base the statement upon calculated instances, unfortunately not numerous. 5. Values of 7 have been calculated for each age group from the figures of the London epidemic. These appear to shew that the protection of vaccination lasts without perceptible diminution for about 30 years. This result is opposed to the opinion of most of those who have had experience of smallpox. My hypothesis given under (2) would explain the discrepancy, for the real values of the correla- tion at early ages would be considerably higher than the apparent values drawn from smallpox cases only. 6. The correlations between area of scars and severity, number of scars and severity, foveation of scars and severity in the London epidemic agree closely with those calculated by Macdonell from Glasgow statistics. They are all considerably lower than the correlation between vaccination and severity. The difference would be expected, if the normal distribution held for the total population admitted to hospital. For the tables from which the three former correlations have been calculated are “curtailed” by the omission of the unvaccinated group. 7. The figures from the London epidemic do not enable us to decide whether the correlations of severity with number and foveation of scars are secondary to that between severity and area of scars; though they may be so. 8. The gross correlation between vaccination and severity is lower than the mean net correlation at various ages, owing to the very high correlation between age and vaccination among inmates of smallpox hospitals. In conclusion, I have to thank Prof. Pearson, of University College, and Dr Ricketts, Superintendent of the Smallpox Hospitals of the Metropolitan Asylums Board, for much help in the preparation of this paper. * No examples of contingency formulae are given in the paper, as none of the tables were adapted to this method. MISCELLANEA. I. On the Distribution of Severity of Attack in Cases of Smallpox. By KARL PEARSON, F.R.S. Dr Turner finds that the protection provided by vaccination does not diminish as rapidly with age as he would expect it to do. His expectation arises from the fact that current medical opinion considers that the immunity provided by vaccination diminishes rather rapidly with the increase of the period which has elapsed since vaccination. I do not propose to consider whether protection against death when disease is incurred is really of the same character as immunity against an attack. But I should wish to point out that Dr Brownlee has reached a very similar conclusion to Dr Macdonell on this point by a very different process, and further that it follows, if we simply use the coefficient of association, which makes no appeal to normality of distribu- tion. On the other hand, while I am distinctly interested in Dr Turner’s theory of a “curtailed normal curve,” I am compelled to say that I consider it an extremely improbable hypothesis in the present application. The group of persons who catch smallpox are a selected group of the general population, selected because (i) they have come in contact with the disease, (ii) they were at the time of such contact not sufficiently immune against it. This sufficiency of im- munity must depend not only on prior vaccination, but on a host of other causes, the virulence of the poison they encountered, their particular state of health at the time in question, their conduct before and after the risk was run, ete., ete. It is impossible to suppose a rigid line drawn at a certain grade and say all below this grade escape this disease, all above will contract it. I cannot understand how those ‘selected’ to incur the disease differ from any other naturally selected group with which we are acquainted. Now biometricians are dealing with selected groups every time they measure the variation of a character in a species, but no such truncated normal distributions have yet exhibited themselves*. Dr Turner says “A curtailed table in measurable characters would result if we collected statistics of height in soldiers, a population from whom all below a certain standard height had already been rejected” (p. 497 above). Now this is a case which can be well tested. For example, height standards exist for both the American and Italian armies. Yet what do we find? That the distribution of the statures of the accepted recruits in both countries, so far from forming a curtailed normal distribution is in some cases as close to a complete Gaussian * American trotting horses give the nearest approach, there being a time limit to entry in the record of trotters. But this is a perfectly arbitrary line drawn across the trotting population by the hand of man, and not a selection due to a complex of natural causes, Biometrika Iv 64 506 Miscellanea curve as anything with which we are acquainted*. The fact is that there are so many other factors on which selection depends that we get no “curtailing” of the distribution at all. I should imagine that it would be precisely the same with those who are selected to incur small- pox. The selection depends on so very much else than a certain absolutely fixed grade of immunity. This point is of such importance that I think it worth illustrating by an example. It is one which Professor Weldon has most kindly provided me with out of the extensive reductions he has made of Italian conscripts and recruits. In the Province of Verona in the five years between 1875 and 1879 there were 16,203 conscriptst and from these were selected 3810 recruits}. The following table contains the two distributions : lg fa Sy an] Sa) wD = D> dae s | 9 ~~ | Q ~ Sa) Xs) = Dd baal Kian Sfal sf} else | alse lisl| sels slau lefelse] se a) Stature in 7 i | 7 Fi ij ii 7 i 7 ) 7 qt 7 meek a i 7 7 7 Centimetres Ale e Ay | F = A Li a SIN IN il Tet a) a) al my | ~ il al = ~N ~ Conscripts 4}—| 2 1 1 i 7 7 | 22 | 33 | 55 | 117] 225 | 626 | 725 | 1027 | 1503 Selected Recruits | — | —| — | —— . —|—|—|—] 16,148] 239| 354 -_ © coal ae eae Vee Sate = care, , i as jl Stature in | | l l | | | | | | | | | | | | | Totals Centimetres a > 3S x So | & slo }/aol[ otlTrRrisiolaolalsleae cS SS) 6 2) mY A Bs eS ||| BS 4] Go 1! co. |) co! Tool coma Sosaal lines a a ~ 1 ~™ | ON N | i 4 n n nn — — Conscripts | 1869 | 2065 | 2125 | 1703] 1525 1058 | 699 3a! 205 | 131 45 | 24 197 14 “ee aigens | Selected Recruits | 471} 542) 538] 425] 414) 256/170/133) 55| 35} 9| 3/—j/ 2 | —]|—J 3810 | | | | In the diagram the two distributions with their corresponding curves are given, and we see that in the selected recruits there is not the slightest approach to a curtailed normal curve. In fact, if we examine the fundamental constants of the distribution § we find: For the Conscripts For the Recruits s/8; ="1658 + 0130 _A/B, = "3409 + 0268 By,=3'7516 £0260 By=2°7989 + ‘0536. The probable errors are those which would arise if the distribution were truly normal. The deviations from symmetry are for the two cases 12°6 and 12’7 times their probable errors respec- tively. It is impossible therefore to say more than that the two curves are both markedly skew and deviate equally from normality in this respect. In the next place the deviations from meso- kurtosis are in the two cases 289 and 3°8 times their probable errors ; in other words the curve for the conscripts diverges indefinitely more from the normal curve than that for the selected recruits. This is only one instance out of many which emphasise the same points, i.e. that there is no approach in the selected curve to curtailment and it differs on the whole less from normality than is the case with the unselected material. * See, for example, Baxter’s statistics for U.S. recruits discussed, Phil. Trans. Vol, 186, A, p. 385. + R. Livi, ‘‘Classificazione delle stature...... dei coscritti delle leve di terra negli anni 1875-79.” Annali di Statistica, Vol. vu. 1883, pp. 144—9. + BR. Livi, Antropometria Militare, Parte 1., p. 260, Roma, 1898. § Phil. Trans. Vol. 186, A, p. 368. Biometrika, Vol. 1v., pp. 174—7. 4) Phil. Trans, Vol. 198, A, p. 278. S-Z61_G-06! S-881_ 9-981 Sv8l Svel G-O8l G- "sw U2 aungnig BLL G-OLL Syst GULL S-OLl G-B91 G-991 G+r9L GZ9L G-O9L GBS GIG S-pSL S-ZSl GOSL S-Bbl G-Opl G-rbl GZpL SOL G-8El G-9E1 S-rEL GEL S-OEL G81 G-9Zl G-bZl_G-ZoI 1 =e | = — 00% -- ils OOF aoe — — ———— 4 009 008 (oyeyer 00% | +ttb F — oom oosi AMEE SMR SOR RRReeeeAee i 002% ‘spINdooy pezopeg pur sydrsosuog vuoseA Jo suonqiaysiq Aouenberg jo uostreduto~ *houanbaig fo a)no9 508 Miscellanea I have next endeavoured to form an estimate of the distribution of severity in the cases of smallpox, and the only method of doing this seemed to be to form a frequency distribution for the intervals which elapse between either (a) onset or (0) eruption and the first bath. This may be taken in a rough sort of way as a measure of the severity of the attack. -Unluckily any character which depends on time usually gives a skew frequency distribution and the present case is no exception to the rule. But I do not think that the resulting distributions can be in any way described as curtailed normal curves. Dr John Brownlee kindly provided me with the particulars of between 800 and 900 cases. I could distinguish no sensible differences in the male and female distributions. Further, the vaccinated cases formed a very large proportion of the total, for example 779, as against 55 unvaccinated and 21 doubtful. Hence, taking into account that what Dr Turner is dealing with is the distribution of cases in all classes, vaccinated or unvaccinated, I have clubbed all groups together to get my distribution of severity. There were 57 deaths, which must of course be excluded, from a time to bath test of severity, they occurred with rather an erratic distribution at a mean interval of 10:4 days after onset or 7:6 days after beginning of the eruption. Thus the interval between onset and eruption in the fatal cases is about 2°8 days, while in the recoveries it is 3:1 days, so that there may, when more material is forthcoming, be found to be a sensible difference in this interval for the two classes of cases. The following table gives the distribution of severity of attack as measured by the two tests of days (i) from onset to bath and (ii) from eruption to bath. Be ee RONUOG BM MNICORUD ODOR GRU Gna hahcghd bd hgh ays | : | | s/e/e/ole la isielajalals (s/s) sis icing Onset to Bath ...| 2/13] 40/131/192/152]99|73|40/24]13/17/10/ 6] 4/6|/2/1/—/1 | 826 Eruption to Bath mee 164} 174/156} 96) 71/39)26/13/21) 6) 8) 2)7)3]3)]1)j—|/— Neither of these distributions can be considered as a curtailed normal curve. They must, I think, be looked upon as significantly skew distributions of the usual type such as in practice almost invariably arise, when time is the variate to which we plot frequency. They do not appear to me to give any support to Dr Turner’s view expressed on p. 496 above as (2), or to the sugges- tion on p. 497, that the mildest cases are more frequent than those more severe ; the different degrees of severity do not diminish without exception as the degree increases, and since the bulk of our cases are all vaccinated, it is not possible to suppose that the actual frequency curve among the vaccinated is of the type suggested by Dr Turner. The modal frequency corresponds to a sensible degree of severity of attack, i.e. to about 13 days from onset to first bath, while the mildest cases correspond to only 4 days. It seems therefore quite impossible to suppose, as Dr Turner does on p. 502, that the severity frequency distribution is half a normal curve of all exposed to risk. The distribution of the attacked very considerably passes the mode, and on the assumption made by Dr Turner, those exposed to risk and escaping must be a very small fraction indeed of the total population exposed to risk. Indeed the above distributions show that the severity of the attack rises from zero to a maximum, and then falls, in the usual skew frequency fashion, at a slower rate to zero. The constants of the above distributions have been worked out ; it will suffice here to give, however : Onset to Bath By =2'3229, B,=6°2466. Eruption to Bath £,=2°606, 6,=6'3664. Thus both distributions are sensibly non-Gaussian. Miscellanea 509 It may be said that this skewness might itself account for the results which Dr Turner considers opposed to current medical opinion and which he would account for by a curtailed normal distribution. Possibly it may, but I hold that this is very unlikely, and for the following reasons : (i) The severity is here measured by a time test, which invariably gives a skew distribution ; but if severity could be measured in some other quantitative manner, for example by intensity of eruption, mean rise of temperature or in some such fashion, we should most probably reach a more normal distribution. (ii) The effect of even the large amount of skewness indicated above on the correlation found by a fourfold division is, I think, not likely to be very large. It is too often assumed that the distribution must be very closely normal to give a good result by the fourfold table method. As a matter of fact, if the fourfold division falls on the “long tailed side” of the mode, as it does in these vaccination statistics, we get close values to the actual correlation even by the fourfold table method. To illustrate this 1 took the only correlation which was available on the data, namely the correlation between onset to eruption and eruption to bath. This correlation was found by the ordinary product moment process, which is independent of any hypothesis of normal distribution and came out = —‘174+°023. In other words, if severity of the disease be measured by a long period from eruption to bath, then a short period between onset and eruption is associated with severity. This is in accordance with the previous indication that the fatal cases have a short period from onset to eruption. I now formed a fourfold table from my exceedingly skew distribution, dividing between 14 and 15 days from eruption to bath and between 3 and 4 days from onset to eruption. This gave: Onset to Eruption. & 3 days and less | 4 days and more | Totals eles ote) ae ‘Ss 3 HE devs aoe less mas days and more... = a Totals whence the correlation —'166 was deduced. It is clear that the skewness has not sensibly influenced the value of the correlation as determined from this fourfold table. The divergence of the fourfold dividing lines from the median is certainly not quite as great in this case as in the vaccination problem, but it is difficult to go nearer the tails and get anything like a reliable result. Dividing between 4 and 5 days from onset to eruption, and 16 and 17 days from eruption to bath, I get: Onset to Eruption. 2 4 days and less | 5 days and more | Totals ae —— os 16 days and less 699 as 17 days and more... 120 E co Totals 819 | In this case the probable error of the frequency 8 is no less than 2, but a change of 2 in the value of this frequency would change the value of 7 by about 20 p.c. Such a division is 510 Miscellanea therefore very unreliable*. We find, however, that the correlation =—-238 with a probable error of about +°05. In other words within the limits defined by the probable errors, we get a correlation sensibly equal to the actual value —‘174. I think accordingly that we may assume that the skewness of our severity test would not materially alter the condition of affairs. T conclude as follows : (i) There is no reason @ priort for supposing that the distribution of severity of attack in smallpox follows a curtailed normal curve. Such curves are contrary to any existing experience of the distribution of frequency in selected groups. In particular they do not occur in the case of soldiers suggested by Dr Turner. (ii) The only tests I have been able to apply to the frequency distribution of severity is that of length of period from onset or eruption to bath. These both give a continuous and not a curtailed curve. The curves are skew, as we are accustomed almost invariably to find them when time is the variate. (iii) This skewness, due to the time as variate, may not really be characteristic of the distribution of smallpox severity ; this could only be determined if we chose a variate other than time to measure severity. But if it were characteristic, it does not appear that a fourfold division taken so far towards the tail as occurs when we divide severity at death would seriously affect the result. It is shewn that within the limits of probable error the product moment method of calculating the coefficient of correlation and the fourfold division method lead to sensibly the same results, even when we have a skewness in the distribution as great as is indicated in the eruption to bath test of severity. II. On the Mean Duration of Life of Individuals Dying within a Year after Birth}. By RAYMOND PEARL, Ph.D. The usual custom in tabulating census returns of mortality at different ages is to use a five year base unit for ages above 5, and below that age a one-year unit. This method of tabulation makes the finding of the moments of the frequency distribution somewhat less simple than would be the case if all the base elements were equal. Furthermore, the age distribution of the heavy mortality of the first year of life is not given at all. It becomes a very important matter to know, at least approximately, this first year distribution when one attempts to find the moments for the whole material, because the frequency in this element must be centred at some point before one can proceed with the calculations. The mean age at death of those dying under one year must be known. In connection with some work on vital statistics which is being carried on in the Zoological Laboratory of the University of Michigan, it became necessary to have as exact a determination of the centering point of this first year mortality as it was possible to get. It is the purpose of this note to present the results obtained. * Dr Turner’s Tables XIV. and XV. contain only 8 and 2 cases in their fourth quadrants and the probable errors of the resulting coefficients are very considerable. I think Dr Brownlee’s figures (Biometrika, Vol. 1v. p. 325; Journal of Hygiene, Vol. v. pp. 533—4) may be safely taken as giving a reasonable measure of the reduction with time of vaccination protection. + Contributions from the Zoological Laboratory, University of Michigan, No. 88. Miscellanea 511 TABLE I. Deaths (excluding still-born) at the designated age per 1000 legitimate or ulegitimate children born. LEGITIMATE Age in days and 1877—1881 average | 1882—1890 average | 1891—1895 average months 3 = 3 ye ) 2 0 to 1 day 9:32 7°16 10:27 7:89 10°83 8°12 ieee) days 5°41 4°12 5°60 4°30 5°52 4715 eee, Ser. 3520") 2580" ||" 3:60 2°77 3°45 2°67 Ge. Ie 2-33) im leSia | eee 1°85 2°24 1°73 rs ea 1:93) | 1:51 9) 7 Wer | 1-60 1°73 1°38 Pett Gon 2°32 1°76 J 1°64 1:86 1°40 (A oie 2°87 DB 2°54 1°89 2°13 1°61 mee Be: 271 | 2°08 2°37 1°88 203 1°59 Slap) ic 2°31 | 1°84 2°05 1°62 1°76 1°41 D) Sagres (Oars 186 | 1°51 1°75 1:38 1°56 1°24 TIO) de A i 190 | 1°54 | 1°81 1°46 1°70 1°33 TT ie 1-89 1°47 1°82 1°43 1°67 1°39 HOW eS: he. 1-96 1°61 1:91 1°54 1°83 1:46 TID yao 1:93. | 1°60 1°86 1°54 ei 1-49 ee, £ month*| 21-73 |~ 18:90 21°97 18-98 22°14 18°80 1 ,, 2months*| 22:59 | 19°76 22°70 19°79 23°20 19°78 Yee GREE hey 18°58 | 15°86 18-92 16°44 19°72 16°62 6} J. seu hee ae 15:96 13°59 16-45 14:18 16°90 14°40 pe hire, 13°30 11:26 13-99 11-93 14°15 12°13 Diag aoe | 11-51 9°85 12-07 10°46 12°24 10°55 Gera re 10°61 9°05 | 11:05 9-60 11:01 9°53 i 9°30 8°33 | 9-91 8°87 | 9-72 8°77 a oe 8°74 7:96 9-22 8°47 9:18 8:20 Oy oO ee 8:29 7°83 8-69 8:20 | 8:30 7°85 ORR T= «5 751 7°22 7°80 751 7°44 6°89 ike ase 6°94 6°92 717 | 6:98 6°69 6°49 197-32 | 169°62 | 201°89 | 174-20 | 200°77 | 170°98 ILLEGITIMATE 0 to 1 day 11°48 9°37 12°18 9°62 12°82 10°27 1 , 2 days 8-10 6°37 8°05 6°23 8:06 | 6:24 Dag Pres 5°28 4-06 5°50 4:32 5°54 4-25 Been. as 3°54 2-96 3°79 314 | 369 | 2:76 IAEA Nae he 3°32 2°77 Bey 2°85 3:08 | 2°48 a Cae. 4-09 2:97 3°65 2°85 3°35 | 2°48 Cie ede 4°54. 3°58 4:03 | 3-29 3°51 2°65 mee 8: 4-06 3°43 3.5988 3517 3°26 2°98 Sapo 5. 3°62 3:12 SAT | 9°64 3°08 2°41 Dies. Os. 3°28 2°60 310 | 92°54 2°91 2°43 OD aloes 3°53 2°67 3°32 2°67 3°04 2°51 Hh ape 3°30 2°67 3°32 2°94 3°BD O71 HORT Oe be, 3°43 3-11 3°50 2°85 3°48 2°86 Gi SR Be 3°33 3712 3°37 2°84 3°75 | 2:95 14 ,, Zmonth*| 44°83 39°59 46°19 40:05 | 48°11 42-00 1 ,, 2months*) 53-39 46°48 54°03 47-77 | 55:03 48:96 a alee 40-95 37:06 40°93 37°86 43°36 38°63 So ls, 31-98 29°74 33°01 30°15 33°84 30°66 pacman ia be 24-91 23°04 95°77 23°81 26°49 25°12 Dee 6. 19°61 17:97 20°12 19-22 20°49 19°96 Gi a ae 16-03 15°38 1704 16°45 17°45 16°84 aS ko. 12°91 13-00 13°93 | 13°54 14-04 13°36 ee a T3904) bila NIB yizf 12-02 1-51 11:98 a y 9°35 10-00 10°55 10-49 10°33 10-06 HO gp, dik one 8°50 8°63 8°56 8°87 | 8:27 8:29 Teen One yield aor Tol 8-11 | 7°39 7°08 345°16 | 31217 | 35365 =| 320-29 | 359-23 322,92 | * It is assumed that these are calendar months. is Miscellanea Now it is evident that the larger the number of frequency elements for the mortality of the first year for which we can get data, the more accurate will be the determination of the mean, for the reason that the rate of infantile mortality changes rapidly with lapse of time after birth. It is fairly easy to get data giving the number of deaths occurring in each month of the first year, but a still finer division is desirable. Especially is this true for the first month of life. Roughly from 25 to 45 per cent. of all the deaths falling within the first year occur in the first month. Of these the larger proportion occur early in the month. Fortunately the German statistics, which are in many respects among the best vital statistics available, give a rather detailed age grouping of the deaths under one year. I have used in this work the Prussian statistics* for the years 1877 to 1895, split up into three groups as follows: 1877-81, 1882-90, 1891-95. The data are given in the form of death-rates per 1000 born of the same class, in the given period. Separate returns are given for male and female, and legitimate and illegitimate mortality. Up to the fourteenth day after birth the mortality for each day is recorded. The mortality of the remainder of the first month is grouped together in a single class, and from the beginning of the second month to the end of the first year after birth the rates are given by months. Still-births are, of course, excluded.