* For comments on the draft of a different paper, out of which the present article has grown, I should like to express my thanks to Prof. D.W. Arnott, Prof. J.H. Greenberg, Prof. J. Vansina, Prof. W.E. Welmers, and Prof. E.O.J. Westphal. However, since they have not seen the article in anything like its present form, I cannot permit myself to hint (in the usual way, by saying exactly the opposite) that they somehow share the responsibility. For all deficiencies, without reservation, I am solely to blame.

1. Greenberg, J.H., “Studies in African Linguistic Classification I-VII,” Southwestern Journal of Anthropology, 5(1949), 79–100, 190–98, 309–17CrossRefGoogle Scholar; 6(1950), 47-63, 143-60, 223-37, 388-98. An eighth article was published in the same journal in 1954 (see note 11), and the whole series was then reprinted as a book: Greenberg, J.H., Studies in African Linguistic Classification (New Haven, 1955).Google Scholar This was later rewritten as Greenberg, J.H., Languages of Africa (Bloomington, 1963)Google Scholar; the second edition (Bloomington, 1966) includes two pages of additions and corrections.

2. The early steps in his academic career can be summarized as follows: first degree from Columbia University (1936), a postgraduate training in anthropology and linguistics, a year's fieldwork in northern Nigeria, and a Ph.D. from Northwestern (1940), supervised by Herskovits. After five years' service with the U.S. army, Greenberg joined the Department of Anthropology at the University of Minnesota. In 1948 he was invited back to Columbia, and he remained there for the next fourteen years, reaching the rank of full professor in 1956. For further details see the introduction in Dil, A.S., ed., Language, Culture, and Communication: Essays by Joseph H. Greenberg (Stanford., 1971), xi–xiv.Google Scholar

3. For another example of the way in which his African experience fed wider, more abstract, interests see Greenberg, J.H., “The Logical Analysis of Kinship,” Philosophy of Science, 16(1949), 58–64.CrossRefGoogle Scholar

4. A paper by Greenberg on “The Genetic Classification of Australian Languages” was read at the annual meeting of the Linguistic Society of America, New York, December 1951 Google Scholar, but apparently was never published. Also Greenberg, J.H., “The General Classification of Central and South American Languages” in Wallace, A.F.C., ed., Men and Cultures (Philadelphia, 1960), 791–94.CrossRefGoogle Scholar

5. Greenberg, J.H., “The Classification of African Languages,” American Anthropologist, 50(1948), 24–30.CrossRefGoogle Scholar

6. According to the statement of editorial policy printed on the inside front cover of every issue, the journal was “designed to include articles in all branches of anthropology… relating to peoples and cultures, past and present, in any region.” It was be be “a vehicle of expression for anthropologists in all parts of the world.”

7. I am obliged to Professor Greenberg for explaining the circumstances which led to the publication of the original series of articles. It is not the case--as reported by Vansina, J., “Bantu in the Crystal Ball I,” HA, 6(1979), 315 Google Scholar--that he had “the greatest difficulty” in getting these articles published.

8. Herskovits, M.J., “Foreword” in Greenberg, , Studies, v.Google Scholar

9. Greenberg, , “Studies I,” 82 Google Scholar; idem, Studies, 4; idem, Languages of Africa, 2.

10. Greenberg, , “Studies I,” 79 Google Scholar; idem, Studies, 1.

11. Greenberg, J.H., “Studies in African Linguistic Classification VIII,” Southwestern Journal of Anthrology, 10(1954), 405–15CrossRefGoogle Scholar; idem, “Historical Linguistics and Unwritten Languages” in A.L. Kroeber, ed., Anthropology Today: An Encyclopedic Inventory (Chicago, 1953), 265-86; idem, Essays in Linguistics (Chicago, 1957), chapters 3-4; idem, “Africa as a Linguistic Area” in W.R. Bascom and M.J. Herskovits, eds., Continuity and Change in African Cultures (Chicago, 1959), 15–27, possibly written as early as 1953. See also “Central and South American Languages” as cited in note 4.

12. Greenberg, , Languages of Africa, 1.Google Scholar

13. Greenberg, , Essays, 35.Google Scholar

14. Greenberg, J.H., “Linguistic Evidence Regarding Bantu Origins,” JAH, 13(1972), 190–92.CrossRefGoogle Scholar On the practicalities of data compilation see also Greenberg, , “Central and South American Languages,” 792–93.Google Scholar

15. Greenberg, , “Studies VIII,” 407–08Google Scholar; idem, Studies, 108-09; idem, Languages of Africa, 4.

16. As recognized by Winston, F.D.D., “Greenberg's Classification of African Languages,” African Language Studies, 7(1966), 160–70, esp. 168.Google Scholar

17. The point can be made a little clearer, perhaps, by using the notation developed later in the text. We define an equivalence relation on each separate column, writing a↑b to mean that an item from language A links with an item from language B. We also define an equivalence relation on the matrix as a whole, writing A↑B to mean that language A links with language B. Initially these statements may all be very tentative, and some may well be wrong; but the method consists of adjusting each set of statements alternatively, to make it more consistent with the other. By definition, I suppose, the best classification is the one which maximizes the number of resemblances which can be explained on the hypothesis that a↑b and A↑B are both true, over each column separately and over the whole matrix.

18. “Reflexive” means that A must link with itself; “symmetric” means that if A links with B then B must also link with A. There are many binary relations which do not possess these properties, and one will turn up in the later part of this paper.

19. Greenberg, J.H. “Nilo-Saharan and Meroitic” in Sebeok, T.A., ed., Current Trends in Linguistics, 7: Linguistics in Sub-Saharan Africa (The Hague, 1971), 421–42, esp. 431–33, 436–37.Google Scholar

20. Comparing successive versions of Greenberg's classification, we can see that Nilo-Saharan was mainly assembled from the bottom upwards, whereas for Niger-Congo the reverse is largely true.

21. Greenberg, , Essays, 42–44.Google Scholar

22. Greenberg, , “Studies VIII,” 405 Google Scholar; idem, Studies, 106.

23. Greenberg, , “Historical Linguistics and Unwritten Languages,” 275.Google Scholar

24. For a recent description with references to some of the earlier literature, see Tucker, A.N. and Bryan, M.A., “The Mbugu Anomaly,” BSOAS, 37(1974), 188–207.CrossRefGoogle Scholar On the problem of its classification see Welmers, W.E., African Language Structures (Berkeley, 1973), 8–9 Google Scholar, and Bynon, T., Historical Linguistics (Cambridge, 1977), 253–55.CrossRefGoogle Scholar

25. Greenberg, , Essays, 54.Google Scholar For an acknowledgement of his help in the development of the technique see Swadesh, M., “Lexicostatistic Dating of Prehistoric Ethnic Contacts: With Special Reference to North American Indians and Eskimos,” Proceedings of the American Philosophical Society, 96(1952), 455.Google Scholar

26. Sapir, J.D., “West Atlantic: An Inventory of the Languages, Their Noun Class Systems, and Consonant Alternation” in Sebeok, , Current Trends, 7: 45–112.Google Scholar The example is not a completely fair one, however, because Sapir (quite rightly) has already filtered out some of the distortion which seems to be due to borrowing.

27. Heine, B., “Historical Linguistics and Lexicostatistics in Africa,” Journal of African Languages, 12(1974), 7–20.Google Scholar

28. Henrici, A., “Numerical Classification of Bantu Languages,” African Language Studies, 14(1973), 82–104.Google Scholar

29. N. Jardine, quoted by Henrici, “Numerical Classification,” 91n6.

30. There is a consequential point which also deserves some stress. In biological data, distortions of the pattern arise merely from ‘noise.’ In dealing with languages, however, we ought to find information in the distortion of the pattern, as well as in the pattern itself. This is certainly the case with Henrici's data. Even in the range of “fundamental” vocabulary, relationships between languages are never perfectly hierarchical. Borrowing does impinge--enough to skew the pattern systematically, but not enough to distort it beyond recognition. It is generally true, I suspect, that lexicostatistical data will reveal something of the pattern of borrowing between languages, as well as the pattern of genetic relationships. For the implications of this see Sankoff, D., “Reconstructing the History and Geography of an Evolutionary Tree,” American Mathematical Monthly, 79(1972), 596–603, 1100.Google Scholar

31. In the latter sense, a “complete” classification is one in which every group in divided into exactly two sub-groups.

32. Greenberg, , “Studies I,” 81, 83 Google Scholar; idem, Studies, 3, 5.

33. As will doubtless be recognized, I borrow the terminology (but not much else) from Kuhn, T.S., The Structure of Scientific Revolutions (2d ed., Chicago, 1970).Google Scholar

34. Meeussen, A.E., “Review of Greenberg, Languages of Africa ,” Journal of African Languages, 2(1963), 170–71.Google Scholar

35. In Britain, for example, there were only five libraries which subscribed to the early issues of the Southwestern Journal of Anthropology--three in London, one each in Cambridge and Edinburgh. However, Greenberg did submit summaries of his articles to the IAI, for publication in African Abstracts (beginning with African Abstracts, 1(1950), 137–38.Google Scholar

36. Greenberg, , “Studies III,” 309 Google Scholar; idem, Studies, 33; idem, Languages of Africa, 31. The compliment was quoted by Westermann himself, in his review of Greenberg's original seven articles (note 38). In British circles, however, it seems to have caused some amusement (note 70).

37. Greenberg, , “Historical Linguistics and Unwritten Languages,” 278–79Google Scholar, citing a personal communication from Westermann.

38. Westermann, D., “African Linguistic Classification,” Africa, 22(1952), 250–56.CrossRefGoogle Scholar

39. Westermann, D. and Bryan, M.A., The Languages of West Africa (London, 1952).Google Scholar

40. Language, 30(1954), 302–09.CrossRefGoogle Scholar

41. Westermann, D., “Foreword” in Westermann, and Bryan, , Languages of West Africa, 3.Google Scholar He is reported to have visited London in 1947 and 1948 to help with the preparation of this book: see Forde, D., “Obituary: Diedrich Westermann,” Africa, 26(1956), 329–31.CrossRefGoogle Scholar Bryan had been appointed as a research assistant to work on the Handbook in 1944.

42. Ward, I.C., et al, “A Handbook of African Languages,” Africa, 16(1946), 156–59.Google Scholar This document was drafted by the Handbook Sub-committee, the membership of which was not recorded in print. I assume that it included Ward and Tucker; from the final paragraph, justifying the use of geographical “zones,” I infer that Guthrie was also involved.

43. The first draft is said to have been seen by Ward, who died in October 1949; the text was being “finally reviewed by members of the Linguistic Advisory Committee” early in 1950: Africa, 20(1950), 243.Google Scholar

44. Africa, 18(1948), 129.Google Scholar

45. For instance no mention was made of Greenberg's work, nor even of his name, in the publications eventuating from the linguistic survey of the “Northern Bantu Borderland” (note 61). At one point, however, Richardson did remark that the problems of classification encountered by Jacquot and himself would “merit examination by those who deny the existence of mixed languages:” Jacquot, et al, Linguistic Survey, 1:53.Google Scholar

46. Tucker, A.N. and Bryan, M.A., The Non-Bantu Languages of North-Eastern Africa (London, 1956), 139–57, with “Addenda,” xviGoogle Scholar; Huntingford, G.W.B., “The ‘Nilo-Hamitic’ Languages,” Southwestern Journal of Anthropology, 12(1956), 200–22CrossRefGoogle Scholar; Hohenberger, J., “Comparative Masai Word List,” Africa, 26(1956), 281–87.CrossRefGoogle Scholar The book by Tucker and Bryan--forming part 3 of the Handbook--was reviewed by Huntingford: Africa, 27(1957), 297–300.CrossRefGoogle Scholar From the way in which these publications are cross-referenced, it is clear that they were co-ordinated in advance--presumably by Tucker. Huntingford was a colleague of his at SOAS; Hohenberger was a Lutheran pastor who had served in Tanganyika as a missionary before the War. See also Tucker, “Philology and Africa” as cited in note 48.

47. For a lexicostatistical treatment of the ‘Nilo-Hamitic’ problem, see Heine, , “Historical Linguistics,” 15–16.Google Scholar

48. Tucker, A.N., “Philology and Africa,” BSOAS, 20(1957), 554.CrossRefGoogle Scholar

49. For the term “genealogical,” perhaps not quite synonymous with “genetic,” see Ward, and others, “Handbook,” 157.Google Scholar

50. Bryan, M.A., “The T/K Languages: A New Substratum,” Africa, 29(1959), 1–21.CrossRefGoogle Scholar Her theoretical remarks (1-2), were paraphrased from Westphal's article (see note 54).

51. A slightly more positive response than that of the linguists came from Daryll Forde in a lecture delivered to the Anthropology section of the New York Academy of Sciences in March 1953. There was even a cautious suggestion that Greenberg's view of Bantu might be worthy of “further study.” Forde, , “The Cultural Map of West Africa: Successive Adaptations to Tropical Forests and Grasslands,” Transactions of the New York Academy of Sciences, 2d. Ser., 15(1953), 206–19, esp. 208.CrossRefGoogle ScholarPubMed

52. Greenberg, , Languages of Africa, v.Google Scholar

53. As cited in note 46.

54. Westphal, E.O.J., “On Linguistic Relationship,” Zaïre, 11 (1957), 513–24.Google Scholar This was written in reply to an article by Meeussen (note 56), at the latter's suggestion. The concept of “mixed languages” was also defended by Whiteley, W.H., “Linguistic Hybrids,” African Studies, 19(1960), 95–97.Google Scholar Though Whiteley did not join the staff at SOAS until 1959 (when he was appointed directly to a readership), he had been in close contact with the Department of Africa throughout the 1950s. The article in question was provoked by a review of Hohenberger's book on Maasai, written by an American linguist, Hans Wolff, and published in a South African periodical.

55. Cole, D.T., “African Linguistic Studies, 1943-1960,” African Studies, 19(1960), 219–29CrossRefGoogle Scholar; Meeussen, A.E., “Hamietisch en Nilotisch,” Zaïre, 11(1957), 263–72.Google Scholar

56. For reviews of Greenberg, Studies, see Welmers, W.E., Language, 32(1956), 556–63CrossRefGoogle Scholar, and Gleason, H.A., American Anthropologist, 58(1956), 948–49.Google Scholar See also Gleason, H.A., An Introduction to Descriptive Linguistics (New York, 1955), esp. 344–46.Google Scholar

57. Welmers, W.E., “Note on the Classification of African Languages,” Linguistic Reporter, 1(1959), supplement 1, 2–6.Google Scholar For a recent review of the controversy see Welmers, , African Language Structures, 2–14.Google Scholar

58. Dalby, D., “Levels of Relationship in the Comparative Study of African Languages,” African Language Studies, 7(1966), 171–79Google Scholar; idem, “Reflections on the Classification of African Languages, With Special Reference to the Work of Sigismund Wilhelm Koelle and Malcolm Guthrie,” African Language Studies, 11(1970), 147-71.

59. Meeussen, , “Review of Greenberg,” 170.Google Scholar

60. As a spinoff from the Handbook, the IAI conceived an ambitious scheme for mapping the northern boundary of Bantu. Funds were procured from the British, French, and Belgian governments, and four “research workers” were hired to carry out the survey. They were trained in London, then sent into the field in teams of two. The work of the “western team” (Jacquot and Richardson) was supervised by Guthrie; the “eastern team” (Van Bulck and Hackett), was under Tucker's direction. Progress reports can be found in Africa: see especially Africa, 21(1951), 146–47.CrossRefGoogle Scholar

61. Jacquot, A. et al, Linguistic Survey of the Northern Bantu Borderland, I (London, 1956)Google Scholar; Richardson, I., Linguistic Survey…, II (London, 1957)Google Scholar; Tucker, A.N. and Bryan, M.A., Linguistic Survey…, IV (London, 1957).Google Scholar The first volume is a “summary report;” the second gives details of the languages encountered by the “western team;” the third was never published; the fourth deals with the “far eastern section,” surveyed separately by Tucker in 1951.

62. Jacquot, and Richardson, , Linguistic Survey, I, 22.Google Scholar There is no reason for singling this passage out except that it has already been quoted by Welmers (see note 67 below). See also Richardson, I., “Some Problems of Language Classification With Particular Reference to the North-West Bantu Borderland,” Africa, 25(1955), 161–69.CrossRefGoogle Scholar

63. In other words, unlike “links with,” “links to” is not an equivalence relation. It is non-reflexive, in that A→A must be forbidden by definition; it is non-symmetric, in that A→B does not imply B→A (though B→A may also be true); and it is non-transitive, in that A→B and B→C do not imply A→C. It might even be described as “anti-transitive,” in that A→B by itself implies the negation of A→C.

64. Westphal, , “On Linguistic Relationship,” 514–15Google Scholar, followed by Bryan, , “The T/K Languages,” 1–2.Google Scholar

65. The number of available vertices will normally be six, except towards the periphery of the lattice. We are free to draw the edge in any position, as long as it does not complete a circuit of length greater than two.

66. In general, if we allow the circuit in the center to be of any length, we can define a trap as a (finite) connected digraph in which every vertex is of out-degree one. I do not know whether graphs of this type have been studied or given a name by mathematicians: they are not mentioned in such elementary textbooks as I have seen--for example, Wilson, R.J., Introduction to Graph Theory (2d ed., London, 1979).Google Scholar

67. Welmers, , African Language Structures, 6 Google Scholar, alleging that “it is difficult if not impossible to assign a linguistically rational interpretation to a statement such as “the one cited above in note 62.

68. Guthrie, Malcolm, The Classification of the Bantu Languages (London, 1948), 27.Google Scholar

69. Ward, et al, “Handbook,” 158.Google Scholar

70. For an expression of the opinion that Westermann was “over-optimistic” see Tucker, “Philology and Africa,” 544n4.

71. Ward, et al, “Handbook,” 159.Google Scholar

72. Nonetheless, it would not be fair to suggest that “mixed languages” were accepted as a facile solution to problems which would have yielded to more rigorous investigation, along Greenbergian lines. The belief that “mixed languages” existed, had existed, or might have existed, served precisely as a constraint on undue simplification. From the British point of view, it was Greenberg who had taken the easy way out.

73. After the event it may be possible to find some individual “isogloss” which coincides with the boundary between one second-order group and the next. It may also be convenient to define the boundaries formally in such terms. But there is no way in which the small number of “isoglosses” needed for this purpose can be distinguished from all the others in advance.

74. Ward, et al, “Handbook,” 159 Google Scholar, admitting that this would be true of “zones.” I think it must also be true of second-order “groups.”

75. In case I have failed to make this clear, I should like to suggest the following experiment: (1) trace the boundaries from Fig. 2; (2) turn the page upside down and trace them again, so as to simulate a different first-order pattern; (3) superimpose the two; (4) distinguish the boundaries which are “well-defined” (i.e., which show up in both patterns) from those which are not; (5) now construct a classification to satisfy these conditions: (a) except in the last resort, a second-order boundary can only be drawn in places where we have found a first-order boundary; (b) as far as possible, a second-order boundary ought to be drawn in places where we have found a well-defined first-order boundary; (c) as far as possible, the second-order boundaries ought to be drawn in such a way that the groups they define are roughly equal in size. Note that some of the second-order groups are fairly obvious, whereas others are just residual (compare Guthrie, , Classification, 40 Google Scholar). A solution is available on request, but I do not guarantee that it is optimal.

76. Flight, C., “Malcolm Guthrie and the Reconstruction of Bantu Prehistory,” HA, 7(1980), 84.Google Scholar

77. Heine, B., “Zur genetischen Gliederung der Bantu-Sprachen,” Afrika und Ubersee, 56(1973), 164–85.Google Scholar

78. Ward, et al, “Handbook,” 159 Google Scholar, to be followed by a long list of other references.

79. In addition, a summary version of Guthrie's classification was published by the British Standards Institution, for the guidance of librarians.

80. Dalby, David, Language Map of Africa and the Adjacent Islands: Provisional Edition (London, 1977), 13–14.Google Scholar The term “referential” was first introduced by Dalby, , “Reflections,” 161–63.Google Scholar

81. Tucker, , “Philology and Africa,” 550–51, 551n1.Google Scholar His conception of African history is exemplified at length in Tucker, A.N., The Eastern Sudanic Languages (London, 1940), chapters 2-3.Google Scholar Relying entirely on earlier writers, Tucker envisaged the history of the Wele basin as a succession of “invasions,” which only began “towards the end of the Neolithic Age, in the sixteenth century [A.D.]” (23).

82. Westphal, , “On Linguistic Relationship,” 518.Google Scholar

83. The most useful sources are the inaugural lectures given by the first two British professors of African history: Oliver, R.A., African History For the Outside World (London, 1964)Google Scholar, and Fage, J.D., On the Nature of African History (Birmingham, 1965).Google Scholar See also Fage, J.D., “History” in Lystad, R.A., ed., The African World: A Survey of Social Research (New York, 1965), 40–56.Google Scholar