December 3, 1914J 



NATURE 



363 



tion as to the reality of the facts described. For 

 instance, both a map and a photograph are gfiven 

 of Churchdown, the conspicuous outlier that rises 

 above the Lias plain near Gloucester ; while the 

 description of the successive layers in the Victoria 

 Cave furnishes an impressive picture of the history 

 of British man. The statement of the loss of land 

 along the Yorkshire coast demonstrates the effec- 

 tiveness of forces now in action, and two succes- 

 sive views are given of the destruction of a chalk 

 headland near Swanage. 



The illustrations are practically all new, includ- 

 ing a fine view into Lulworth Cove, taken from 

 the hill at its west end. The printers have twice 

 gone astray on the difficulties of Rhynchonella 

 (pp. 60 and 79), and a bracket needs extension on 

 p. xi to include the Cainozoic systems ; but the 

 mode of production shows how generously pub- 

 lishers are prepared to meet the demand for read- 

 able works on nature-study. G. A. J. C. 



Catalogue of the Ungulate Mammals in the British 

 Museum (Xatural History). Vol. iii. Artio- 

 dactyla. Families Bovidae, Subfamilies -^^py- 

 cerotinae to Tragelaphinae (Pala, Siaga, Gaz- 

 elles, Oryx Group, Bushbucks, Kudus, Elands, 

 etc.), Antilocapridae (Prongbuck), and Giraffidae 

 (Giraffes and Okapi). By R. Lydekker, assisted 

 by G. Blaine. Pp. xv + 283. (London: British 

 Museum (Xatural History), 1914.) Price 75. 6d. 

 The third volume of the British Mus»*Tim catalogue 

 of L'ngulates completes the account of the ante- 

 lopes (saiga, gazelles, oryx group, bushbucks, 

 kudus, elands, etc.) and deals also with prongbucks 

 and giraffes. Like its predecessors it is a fine 

 piece of work with terse descriptions and scholarly 

 synonymy. Its usefulness has been notably in- 

 creased by the inclusion of fifty excellent figures, 

 mostly of heads. As a work of reference it will 

 be of great value and interest to those who have 

 collected trophies of this sort. The prongbuck, 

 the position of which has been the subject of dis- 

 cussion, is ranked by Mr. Lydekker as the only 

 living representative of a separate family, Antilo- 

 capridae. Another interesting type, the Okapi, 

 represents a genus along with the giraffe in the 

 family Giraffidae. Among the many other inter- 

 esting forms may be mentioned, Ammodorcas, 

 ^pyceros (the Pala), the Saiga, and the Chiru. 



LETTERS TO THE EDITOR. 



l^The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manusctipts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.^ 



The Age of a Herring. 



To my doubts as to the truth of Dr. Hjort's 

 theories of the herring's age, Dr. Hjort and Mr. 

 Einar Lea have now replied (Natcre, November 5) 

 by recapitulating their main arguments. I can do 

 little more than reiterate my own unaltered in- 

 credulity-. My position is simply this, that a theory 

 has been put forward which seems to me, a priori, 



NO. 2.^^.^, VOL. Q4l 



extremely improbable, and that the statistical data 

 on which it is based are, to my mind, not strong 

 enough to support it. The large assertion that the 

 Norwegian spring herring have consisted, in pre- 

 ponderating or overwhelming proportion, year by year 

 from 1908 to 19 13, of fish spawned in 1904, is based 

 (so far as I can discover) on only nineteen samples, or 

 little more than two a year, averaging somewhat 

 above 300 herring each. Dr. Hjort and Mr. Lea have 

 no ditficuity in showing that their 1913 herring, 

 grouped according to scale-rings or alleged years of 

 growth, form a curve which is a bad fit to a proba- 

 bility curve; but they do not remind us that this 

 so-called sample is no sample at all, but is a con- 

 glomeration of three separate samples from far-distant 

 localities. Dr. Hjort's _full data have not been pub- 

 lished, so far as I know, for the j-ears since 1909 ; let 

 us try to discover how, for the year 1908, he arrives 

 at the figure 348, which he gives (cf. Nature, 

 August 27, p. 672) as the percentage proportion, in 

 the entire stock of Norwegian spring herring, of fish 

 spawned (according to the evidence of their scale- 

 rings) in 1904. I find that this determination was 

 based on two samples only,' from different localities, 

 one consisting of 881 fish, the other of 549. In the 

 former sample the percentage of 1904 fish is given 

 as 15-9, in the latter as 652. The former deter- 

 mination, by Mr. K. Dahl, seems a little shak\- : for 

 an independent determination of a part of the same 

 sample, by Mr. Einar Lea,* gave the alternative value 

 of IO-2. But be that as it may, I find that Dr. Hjort's 

 result is obtained by the simple method of adding 

 together the two samples, one of 881 fish showing 

 159 per cent., the other of 549 fish showing 65-2 

 per cent., and so the resulting mean value of 348 per 

 cent., decimal and all, is straightway arrived at. It 

 does not need the eye of a mathematician to see that, 

 in a problem of biological statistics, such a method 

 of calculation is inadequate; and that it neither proves 

 nor even renders probable the conclusion drawn from 

 it, namely, that four-ringed herring (whatever those 

 four rings may mean) constituted 348 per cent, of all 

 the spring herring in Norway, in the year in question. 



D'Arcy W. Thompson. 



The Cross X as a Symbol for Multiplication. 



Historians of mathematics attribute the first use of 

 the cross x as a symbol for multiplication to William 

 Oughtred ("Clavis Mathematicae," London, 1631). 

 See \V. W. R. Ball's "Short Account of the History 

 of Mathematics," fifth edition, 1912, p. 239; M. Can- 

 tor's "Geschichte der Mathematik," Bd. ii., 1892, 

 p. 658; J. Tropfke's "Geschichte der Elementar Mathe- 

 matik," Bd. i., 1902, p. 135. In some places, as, for 

 instance, in Oughtred's " Circles of Proportion " (Lon- 

 don, 1632, p. 38), the two bars of the cross are not 

 quite straight, giving the symbol the appearance of 

 the small letter x. In some of John Wallis's writr 

 ings, as, for example, his " Elenchus geometriae Hob- 

 bianae," etc. (Oxford, 1655, p. 23). the symbol is 

 not the usual cross, but is plainly the capital letter 

 X turned on its side. In a paper by Lord Viscount 

 Brouncker in the Philosophical Transactions (vol. ii., 

 1668, p. 646), the capital letter X occurs regularly 

 as the symbol for multiplication. These and similar 

 cases lead to the inference that the cross and the letter 

 X were considered practically one and the same symbol 

 for multiplication. 



In this connection we desire to point out that the 

 small letter x, and also the capital letter X, occur as 

 symbols for multiplication before Oughtred (1631) in 



1 Hjort, " Rapports et Proces-Verbaux, vol. xx., p. 29. (Copenhagen, 

 1914.) 

 - "Public, de Circonstance," Xo. 53, p. 94. (1910.) 



