¹ I owe the term ‘genetic linguistics’ to my colleague Yakov Malkiel, who employed it in a paper read to the Linguistic Group at the University of California, Berkeley, on 10 February 1965. It is an accurate and convenient surrogate for the more cumbersome ‘historical and comparative linguistics’.

² F. A. Ficken, ‘Mathematics and the layman’, American scientist 52.423 (1964); Ficken adds that ‘the popular meaning of the term may be helpful but often ... it may be completely misleading’.

³ Y. R. Chao, ‘Models in linguistics and models in general’, in E. Nagel et al., edd., Logic, methodology and philosophy of science 558–66 (Stanford, Calif., 1962), lists 39 occurrences of the term ‘model’, mostly from linguists. ‘As far as I am aware, the earliest mention of models in linguistics was by [Z. S.] Harris ... in 1944‘ (559). The entire paper is a valuable contribution to an understanding of models in linguistics and models in general.

⁴ See Z. S. Harris, ‘The transformational model of language structure’, Anthropological linguistics 1:1.27 (1959): ‘in an untechnical sense’ a model is ‘any framework in respect to which language is described’. See also J. H. Rowe, ‘The Renaissance foundations of anthropology’, AA 67.1 (1965) : ‘Renaissance grammars of Classical Latin and Greek became models for the description of spoken languages in all parts of the world.‘

⁵ ‘The definition of explanation is ... one of the unsolved problems in the philosophy of science’: Leo Apostel, ‘Towards the formal study of models in the non-formal sciences’, in H. Freudenthal, ed., The concept and the rôle of the model in mathematics and natural and social sciences 14 (Dordrecht, 1961).

⁶ Some writers do not sharply distinguish Theory and Model: R. B. Braithwaite, ‘Models in the empirical sciences’ in Nagel 224, states that ‘in psychology and the social sciences the word “model” is frequently used merely as a synonym for ... theory’; A. P. Elkin, ‘Rethinking anthropology: A review’, Oceania 34.84 (1963), complains that at the 1951 International Symposium on Anthropology ‘the term model was used in several senses’, and cites approvingly (85) the remark of Robert Redfield, in Sol Tax, ed., An appraisal of anthropology today 120 (Chicago, 1953): 'I shall not meddle in this muddle of models.' P. Suppes, 'Models of data', in Nagel 252, keeps the two sharply separate, speaking of ‘a model of a theory’, the model being as it were two degrees removed from the data. On the other hand, in an interesting nontechnical discussion of models Hugh Skilling, ‘An operational view’, American scientist 52.388A (1963), essentially eliminates theory and relates models directly to data: 'models, of course, are based on observations. They are generalizations of a number of observations.' The view taken in the present paper reflects that of H. Cramér, The elements of probability theory 22 (New York, 1955), who speaks of a mathematical theory's serving as a model; the question to be considered is whether probability mathematics can serve as a model for genetic linguistic phenomena. It should be added here that 'theory' and ‘model’ are not only often confused with each other; they are often confused with ‘method’. This is especially true in descriptive linguistics, where ‘theory’ may be equivalent to the method used in describing a language; so also a model may serve as a method, as with Latin grammar. An excellent example of a theory in the strict sense of the term is to be found in J. H. Greenberg, ‘An axiomatization of the phonologic aspect of language’, in L. Gross, ed., Symposium on sociological theory 437–80 (Evanston, 1959).

⁷ Indeed, more than one acceptable theory may be devised for the same phenomena; the corpuscular and wave theories of light are a classical illustration.

⁸ This is even, or rather I should say especially, true of mathematical theories and models of phenomena; cf. Cramér 21: 'Any attempt at mathematical description of observed phenomena must involve a certain idealization of really observed facts. The formulas of the mathematician can only provide ... a kind of idealized picture of the characteristic features of the phenomenon under investigation.' See also H. Izui, 'Comments', Current anthropology 3.141 (1962). L. Hogben, Mathematics for the million 637 (New York, 1937), puts the matter bluntly: ‘the form of a mathematical statement does not necessarily tell you anything about the process it describes. All it does is to tell you how to calculate a result.‘

⁹ Apart from statements about the probability that an event is due to chance; these are cited under ‘chance’.

¹⁰ S. M. Lamb, ‘Some proposals for linguistic taxonomy’, Anthropological linguistics 1:2.34 (1959).

¹¹ J. H. Greenberg, ‘The measurement of linguistic diversity’, Lg. 32.109 (1956). A fruitful development of Greenberg's work is to be found in S. Lieberson, ‘An extension of Greenberg's linguistic diversity measures’, Lg. 40.526–31 (1964).

¹² J. H. Greenberg, ‘The nature and uses of linguistic typologies’, IJAL 23.76 (1957).

¹³ R. B. Lees, ‘The basis of glottochronology’, Lg. 29.114 ('basic-root-morpheme inventory ... of several thousand items'), 115 ('the random sample'). There is some confusion here because 'the random sample' is also ‘some arbitrarily chosen sample subset’; these two are incompatible.

¹⁴ Ibid. 119.

¹⁵ Ibid. 119.

¹⁶ Ibid. 124.

¹⁷ J. E. Grimes, ‘Measures of linguistic divergence’, Proceedings of the Ninth International Congress of Linguists, Cambridge, Mass., 1962 45, 46 ('s-Gravenhage, 1964).

¹⁸ D. H. Hymes, ‘Na-Déné and the positional analysis of categories’, AA 58.627–8 (1956).

¹⁹ J. H. Greenberg, ‘Genetic relationship among languages’, Essays in linguistics 37 (Chicago, 1957).

²⁰ Ibid. 37.

²¹ D. H. Hymes, ‘Genetic classification: Retrospect and prospect’, Anthropological linguistics 1:2.55 (1959).

²² Ibid. 54. What constitutes a rigorous method for identifying resemblances is not discussed.

²³ Ibid. 55.

²⁴ J. H. Greenberg, ‘Historical linguistics and unwritten languages’, in A. L. Kroeber, ed., Anthropology today 269 (Chicago, 1953).

²⁵ Ibid. 269.

²⁶ Hymes (1956) 627.

²⁷ Hymes (1959) 55.

²⁸ Henri Poincaré, Science and method 75–6 (New York, n.d.).

²⁹ C. I. Lewis, An analysis of knowledge and valuation 276–9 (La Salle, Illinois, 1962), raises inconvenient questions about such a limit.

³⁰ The story is delightfully told by Warren Weaver, Lady Luck, the theory of probability 45–51 (Garden City, New York, 1963).

³¹ Greenberg, Essays 36.

³² Ibid. 37.

³³ It is interesting that L. Bloomfield, Language 297 (New York, 1933), discusses chance resemblances without using the word ‘chance’ itself (‘Other resemblances between languages bear no significance whatever‘). What constitutes resemblance, however, is a question that neither Bloomfield nor Greenberg answered precisely. Yet it plagues every comparativist. R. L. Oswalt, in a paper read to the Linguistic Group at the University of California, Berkeley, on 14 November 1963, defined forms as phonologically resemblant if they have in common one consonant of similar articulation. This seems extreme.

³⁴ This practical consequence has its limitations. If both players have equally large resources and are equally well versed in probability theory it is clear, from W. A. Whitworth, Choice and chance 220–30 (New York, 1934), that the best thing for them to do is not to play with each other.

³⁵ Aristotle, Metaphysics VI.3 (1027bl) tells a story similar to Poincaré's; it is commented upon by Jacques Maritain, A preface to metaphysics 133–4 (New York, 1962). Aristotle distinguished sharply between what always or usually happens and what happens by chance ; cf. Physics II.5 (196bl0, 197al2), De caelo I.12 (283b3), De generatione et corruptione II.6 (333b7), Magna moralia II.8 (1206b40), Ethica endemia VII.13 (1247a6), Rhetoric I.10 (1369a31).

³⁶ According to Mario Bunge, Causality 99 (Cleveland and New York, 1963); I have not been able to run this down. Chrysippus was a Stoic who lived a century after Aristotle.

³⁷ Anicius Manlius Severinus Boethius, In librum de interpretatione editto secunda in J. P. Migne, Patrologiae cursus completus series Latina 64.491C9–492A11. I owe this reference to Bunge 99 and to A. A. Cournot, An essay on the foundations of our knowledge 44 (New York, 1956); the latter was first published (in French) in 1851. Boethius lived from 480 to 524 a.d.

³⁸ Summa theologica I.Q115 art 6 in A. C. Pegis, The basic writings of Saint Thomas Aquinas 1.1067 (New York, 1945). St. Thomas's views are treated in some detail by Maritain 135–9.

³⁹ Cournot 40–4 (1956) ; also Considérations sur la marche des idées et des événements dans les temps modernes 1.1 (Paris, 1872).

⁴⁰ J. B. Bury, ‘Cleopatra's nose’, in H. Temperly, ed., Selected essays of J. B. Bury 61 (Cambridge, 1930). This essay was first published in 1916; the title is taken from Blaise Pascal, Pensées §ii.l62: 'Le nez de Cléopâtre s'il eût été plus court, toute la face de la terre aurait changé.' See also another essay of Bury's entitled 'Darwinism and history', where chance coincidence is defined as ‘the collision of two unconnected sequences’ (Temperly 36; first published in 1909); and The idea of progress 304, 368 (London, 1920).

⁴¹ E. H. Carr, What is history 129 (New York, 1964).

⁴² Hymes (1956) 627.

⁴³ Interestingly enough neither Hymes nor Greenberg suggests an interpretation for an observed number of resemblant forms less than the ‘chance’ number, yet in many types of statistical problem an observation considerably less than the ‘chance’ value may have importance. Perhaps they expect the ‘chance’ value to be so small that any value smaller would be close to zero.

⁴⁴ I assume this, but I am not sure that the authors cited intend it. It may be that they accept a greater value as demonstrating historical connection.

⁴⁵ Hymes (1959) 54, 55.

⁴⁶ Greenberg, Essays 37.

⁴⁷ Greenberg (1953) 270–1.

⁴⁸ Bloomfield 297.

⁴⁹ L. J. Savage, The foundations of statistics 1 (New York, 1954).

⁵⁰ M. R. Cohen, A preface to logic 115 (Cleveland, 1956).

⁵¹ Ibid. 127.

⁵² C. S. Peirce, ‘On the doctrine of chances, with later reflections’, in J. Buchler, ed., Philosophical writings of Peirce 164 (New York, 1955).

⁵³ Savage 3.

⁵⁴ F. Mosteller, R. E. K. Rourke, and G. B. Thomas Jr., Probability with statistical applications 3 (Reading, Mass., 1961). For a dispassionate discussion of the two views see M. G. Kendall, ‘On the reconciliation of theories of probability’, Biometrika 36.101–16 (1949).

⁵⁵ Max Born, Natural philosophy of cause and chance 122 (Oxford, 1949).

⁵⁶ I say ‘two languages’ rather than ‘two unrelated languages’ because relationship is irrelevant to the question. If such a number can be determined it can be determined for any two languages.

⁵⁷ The sensible remarks of Bloomfield 297 are in point here.

⁵⁸ J. L. Hodges and E. L. Lehmann, Basic concepts of probability and statistics 4 (San Francisco, Calif., 1964) : ‘We shall refer to experiments that ... do not always yield the same result when repeated under the same conditions as random experiments.‘

⁵⁹ I have omitted to discuss the concept of ‘independence’, which is mentioned in connection with ‘chance’ near the beginning above, and which underlies most of my own statistical work. My views on this subject have undergone considerable change over the past twenty-five years; what I now believe to be a reasonable position is suggested in my paper, ‘A classification of twenty-one Philippine languages’, Philippine journal of science 91.487–90 (1962), and further developed and corrected in my ‘Statistical structure of the Proto-Austronesian morph’, Lingua 14.243–70 (1965).

⁶⁰ Quoted, but with no reference, in J. M. Keynes, A treatise on probability 333 (London, 1929).