¹ Since 1950, when Swadesh published the first paper in glottochronology (M. Swadesh, Salish internal relationships, IJAL 16.157–67), the subject has acquired a considerable bibliography, which was reviewed by D. H. Hymes in 1960 (Lexicostatistics so far, Current anthropology 1.3–44, with additional comments, 338–44; not all of the references, 41–44, 345, are to glottochronological articles, e.g. Kroeber and Chrétien, Ray, Whitney).

² R. B. Lees, The basis of glottochronology, Lg. 29.113–27 (1953).

^2a After I had written this, my colleague S. M. Lamb called my attention to the fact that Swadesh had already used the term; see Towards a satisfactory genetic classification of Amerindian languages, Anais do XXXI Congr. Internacional de Americanistas 1002 (São Paulo, 1955).

³ Some scholars regard the statement as an hypothesis still to be verified, others as an established law. I have chosen ‘postulate’ as a neutral term.

⁴ Swadesh 1950, 158.

⁵ Hymes 4. The derivation is as follows:

(4)

(6)

⁶ In theory this could be done by employing the definition of r given in (1) above, but it is usually difficult or impossible to find empirical data at thousand-year intervals.

⁷ When applied to more than two cognate languages, it takes the languages by pairs and computes t for each pair.

⁸ The discarded words are replaced; Lees 116 states this, but neither he nor anyone else seems to have been aware of the implications of replacement. I consider these implications in Section III of this paper.

⁹ This is a rewording of Lees 116. I speak of discard rather than replacement because the model which he develops at this point (116–7) is based on the concept of discard, as I shall show in Section III. I also use the term ‘language’ instead of ‘dialect’.

¹⁰ Lees 116.

¹¹ By (12) c = r ^2t, log c = log r ^2t = 2t log r, whence t = (log c)/(2 log r).

¹² I do not include the product law among the postulates because the latter are statements about language whereas the product law is a statement about probability.

¹³ M. Swadesh, Towards greater accuracy in lexicostatistic dating, IJAL 21.121–37 (1955).

¹⁴ The use of any other values (e.g. N = 165, r = 0.85 of Swadesh's original study) would change our numerical results a little but would not affect the argument.

¹⁵ This is the usual definition of a probability found in the standard textbooks, e.g. J. Neyman, First course in probability and statistics 16 (New York, 1950).

¹⁶ I.e. S_a = S_b = S.

¹⁷ In this notation the upper figure represents the total number of objects, the lower the number of these objects in a combination.

¹⁸ The computation of these probabilities, while laborious, is not the burden that might be supposed, since tables of the logarithms of combinations are available, e.g. A. Hald, Statistical tables and formulas 76–81 (New York, 1951). The use of a previously prepared form expedites the work, as does a table of antilogarithms.

¹⁹ I take the term from Neyman 15. W. Feller, An introduction to probability theory and its applications 10 (New York, 1950), speaks of sample space, each probability being the probability of a sample point; and L. J. Savage, The foundations of statistics 45 (New York, 1954), of a probability space.

²⁰ I do not intend a specious appearance of accuracy by using four decimal places; I merely happened to use four-place tables of logarithms and antilogarithms. Nothing important would have been lost had I rounded off my figures to three places, or even to two.

²¹ This same observation holds for those few instances where the highest probability and C' do not coincide, as may be seen by inspection of Tables II and III.

²² The definition of joint sets and their computation are treated in connection with Table II.

²³ The fact that the t'_c point is occasionally one step away from the point of maximum probability in the range arises, I believe, from the fact that the k- and c-functions are continuous functions whereas the probabilities are computed for integral values of K and C. Hence occasional discrepancies arise.

²⁴ Swadesh 1950, 160.

²⁵ Swadesh 1950, 160.

²⁶ I use c or C as seems appropriate to the context. C = 100 c, and is usually the value we are interested in, rather than c.

²⁷ Hymes 19–22.

²⁸ The k-function is a continuous function within the positive range of t (which is all we are interested in, since a negative t has no meaning within our frame of reference); therefore all positive values of t may be employed, not merely integral values.

²⁹ There is an apparent discrepancy here between saying that c does not become zero until t is infinite and saying that C becomes zero when t = 36. What happens is that by the k-function k = .01 and hence K = 1 when t = 30.54; k = .005 and hence K = .5 when t = 35.13. But since fractional values of K are meaningless, I have rounded off to the nearest whole number. The value of c for t = 36 gives a C which is less than .5; I therefore round off to zero.

³⁰ And this in turn makes the c-function an exponential function, since c = K ² = r ^2t.

³¹ Swadesh 1955, 124.

³² M. Swadesh, Lexicostatistic dating of prehistoric ethnic contacts, Proceedings of the American Philosophical Society 96.457 (1952).

³³ Swadesh 1952, 457. I am not sure that these criteria are entirely consistent with each other.

³⁴ Hymes 9: ‘The method does stress a certain rate of retention, but it also, necessarily, stresses a complementary rate of replacement.‘

³⁵ It is to be remembered that I am criticizing the k-function, not Postulate (I). It seems reasonable to suppose that when a word is replaced the process usually begins with the appearance of a new word which competes with it. Gradually the frequency of the new word increases and that of the old word decreases. At some point the new word becomes the customary term; we may now speak of loss and replacement, even though it may take two or three generations more for the frequency of the old word to reach zero. It also seems reasonable to suppose that this process may be operating on more than one word at the same time.

³⁶ Swadesh 1955, 126–37.

^36a He saw clearly that varying persistence would affect calculated time depth; this he called skewing. See Swadesh 1955, 128.

³⁷ Quoted from a private communication in Hymes 10.

³⁸ Hymes 10, in the same quotation.

³⁹ Quoted in Hymes 10.

⁴⁰ Strictly speaking they are extraneous and the conclusions which glottochronology draws from them are invalid, but I believe that no harm is done by using the term as I have done.

⁴¹ It may be objected that I am guilty of an inconsistency in that in Section III assumed, though I did not so state, that each item of the diagnostic list was equally exposed to risk, i.e. had equal persistence, whereas in Section IV I accepted Swadesh's observed differences in persistence. I did not state in Section II that each item had equal persistence because I there made no assumption about individual persistences. I was concerned to establish the defects of the c-function. These defects hold even if a new k-function is devised which takes individual persistences into account. For a given value of r, no matter what form the k-function may take in any future revision, there will be a range of C's for a particular t and a tier of t's for a particular C. It is inherent in the c-function that it determines a single value of t for a given C, and vice versa; it is inherent in the postulates that C determines a tier of t's and t a range of C's.