512 



NA TURE 



[April 2, 1903 



In addition, wood as well as brick was found to be 

 stronglv active under the conditions employed. Metals ex- 

 posed for some time outside the buildings showed a marked 

 increase of activity over the metal which had been carefully 

 cleaned. E- Rutherford. 



McGill University, Montreal, March 12. 



Mendel's Principles of Heredity in Mice. 



The points raised by Mr. Bateson in Nature of March 19 

 cannot be discussed within the limits of a short letter ; a full 

 discussion will be published in an early number of Bio- 

 nictrika. In the meantime I would ask Mr. Bateson one 

 question : — 



He represents the mice used by Mr. Darbishire as differ- 

 ing in two characters ; one (pinkness of eye with white coat) 

 he calls G ; the other (pinkness of eye with some colour in 

 the coat) he calls G'. The hybrids produced by crossing 

 these mice he calls GG' ; and by reference to the mysterious 

 properties of " heterozygotes " any difficulties presented by 

 their eye-colour are avoided. But when these hybrids are 

 paired inter se t they are said to produce offspring of three 

 kinds, in the proportions 



GG + 2GG' + G'G'. 



Now the mice G'G' are of the same constitution in 

 respect of all the characters concerned as their pure-bred 

 grand-parent G'. Mr. Darbishire has shown (Biometrika, 

 vol. ii. part ii.) that they do not always resemble their grand- 

 parent, or either of their parents, in one of the characters 

 (coat-colour) denoted by G'. They may show a new colour, 

 *' lilac," not present in any of their near ancestors. Six 

 out of eighteen mice of this category, at present old enough 

 for study, show lilac colour. 



I would ask Mr. Bateson 's explanation of this fact and 

 of the coat-colour of the first hybrids GG'. 



Oxford, March 24. W. F. R. • Weldon. 



Historical Note in regard to Determinants. 



In the last-issued part of the American Journal of Mathe- 

 matics, vol. xxv. pp. 97-106, there is a short paper by Mr. 

 E. D. Roe entitled " Note on Symmetric Functions " which 

 in my opinion should not pass unnoticed. It concerns two 

 fundamental theorems regarding alternants which it 

 appears Mr. Roe had previously dealt with in the American 

 Mathematical Monthly, vol. vi. (1S99) p. 25, and had been 

 there attributed by him to Prof. Gordan. In a footnote 

 he now says : — 



" Prof. Metzler has kindly called the writer's attention 

 to the reference to Muir (' Determinants,' p. 176, § 129), 

 from which it appears that Muir has the priority of publi- 

 cation, as far, at least, as theorem i. is concerned. It may, 

 however, be added that in a recent letter Prof. Gordan states 

 that he has used the two theorems for thirty years." 



From this it might possibly be inferred that my publi- 

 cation of the said theorem twenty years ago, and Gordan 's 

 alleged private use of it thirty years ago, are matters of 

 moment in connection with its history. This would be a 

 fatal error, as the theorem has been in print for at least 

 seventy-eight years, having been exhaustively dealt with 

 by Schweins in his " Theorie der Differenzen und Differen- 

 tiale, . . ." published at Heidelberg in 1825. ' 



The part of my connection with it which gives me most 

 satisfaction is not the fact that I discovered it for myself, 

 but that I discovered an earlier and neglected discoverer of 

 it, Schwrins, and have since tried my best to do justice to 

 his merits. His treatise had been absolutely lost sight of, 

 «ven in Germany, until the appearance of my paper, " An 

 Overlooked Discoverer in the Theory of Determinants," 

 which was published in the Philosophical Magazine for 

 November, 18S4. In this paper was given a brief account 

 of that portion of his work which concerned general deter- 

 minants, and at the same time it was indicated that this 

 was but a small fraction of the whole contents, several 

 special determinants being equally familiar to him. In 1888 

 the subject was returned to, and entered into more fully in 

 the Proceedings Roy. Soc. Edinburgh, vol. xv. pp. 526-542, 



1 V. the second Abtheilung (pp. 369-398) and the second chapter of it in 

 particular. 



NO. I744, VOL. 67] 



the account there given being afterwards republished in the 

 first volume of my " History of Determinants," pp. 157- 

 173. At a later date Schweins's chapter on alternants, ex- 

 tending to about thirty pages, was dealt with in a similar 

 in. inner, the account appearing in a paper in the Proc. 

 Roy. Soc. Edinburgh, vol. xxiii. pp. 93-132. On pp. 98- 

 103 of this the theorem will be found, accompanied by con- 

 siderable detail. To the present day, nevertheless, Schweins 

 has not received his due from any of his own countrymen. 



Speaking generally, I would urge that the greatest 

 possible caution should be exercised by everyone who finds 

 it necessary to attach to a theorem the name of an author, 

 not merely when the theorem concerns alternants, but when 

 it belongs to any part of the general subject of determinants. 

 As a second example, let us take a case where the mathe- 

 matician who is unfairly dealt with is not German but 

 English. No fact ought to be better known than that the 

 first discoverer of continuants was Sylvester, his paper con- 

 taining the discovery having been published in the Philo- 

 sophical Magazine for June, 1853. In the early part of 

 1875, however, S. Giinther published a text-book which 

 assigned the credit to the Danish mathematician, C. Ramus, 

 and notwithstanding the fact that an effort was made in 

 the Philosophical Magazine for February, 1877 (vol. iii. pp. 

 137-138), and still more pointedly in the American Journal 

 of Mathematics for 1878 (vol. i. p. 344) to rectify the error, 

 it has lingered on in Germany and the Continent of Europe 

 to the present day. The details of the story are instructive. 

 Giinther's statement was : — 



" Die Moglichkeit einer solchen Darstellung scheint 

 zuerst von Ramus [Kjobenhavn, Vid. Selsk. Overs. 1855, 

 pp. 106-119) bemerkt worden zu sein : auch Spottiswoode 

 (Crelle's Jour 11., Ii. p. 374) und Heine (Crellc's Journ., hi. 

 p. 97) wurden im Verlaufe anderweitiger L'ntersuchungen 

 auf dieselbe gef iihrt. " 



This was republished in 1877 without alteration. In 

 opposition to it the following are the facts 



(1) As above stated, Sylvester's discovery was published 

 in June, 1853. 



(2) Spottiswoode, writing in August of the same year, and 

 having just become familiar with Sylvester's discovery, re- 

 produced the substance of the latter's remarks in the second 

 edition of his " Elementary Theorems Relating to Deter- 

 minants," which appeared in Crelle's Journal in 1856. 



(;) In September, 1853, Sylvester returned to the subject 

 (v. I'lul. Mag. [4] vi. pp. 297-299). 



(4) In August, 1854, a result of Sylvester's on the subject 

 appeared in the Nouv. Annates de Math., xiii. p. 305, under 

 the significant title " Theoreme sur les Determinants de M. 

 S3 Ivester." 



(5) In 1855, as Giinther states, Ramus made his com- 

 munication. 



These five assertions have always been easily verifiable; 

 and since the claim made publicly in 1S77 and 1878, ought 

 to have been verified by any writer who had to refer to the 

 subject. Strange to say, this has never been done, the most 

 recent text-book, Pascal's, having only got as far as the 

 Following sentence indicates: — 



" I primi che si sono occupati dell' argomento sono stati 

 Ramus, Sylvester, Spottiswoode, Heine, Thiele, e Giinther." 



If we turn for aid on such matters to the Encyklopadie 

 Jer math. II "issensch., which is now in course of publica- 

 tion, and aims at being a standard work of reference, 

 there is nought for- us but disappointment. In connection 

 with alternants, therein called " Vandermonde'sche " or 

 " Potenzdeterminanten, " the name of Schweins is not men- 

 tioned, and as for the early history of continuants, we find 

 In .ill] confusion worse confounded. Ramus's paper, i( is 

 true, does not appear, but unfortunately we are referred to 

 one of still later date (1858), by Painvin, and to a note 

 which is attributed to Sylvester, but which Sylvester never 

 wrote. The name " continuant," too, is wrongly attributed, 

 and when in connection with the application to continued 

 fractions Sylvester's name is again mentioned, the first date 

 attai bed thereto is 1859 ! This may be a misprint for 1853, 

 but if so there is a further error in the specification of the 

 page. Heine's name is still to the fore; unluckily, how- 

 ever, it is not attached to the right paper. Something of 

 Giinther's is referred to, but the title is left out. 



Cape Town, S.A., Februarv 28. Thomas Muir. 



