March 4, 1886] 



NATURE 



413 



circular and hyperbolic functions in both differentiation and 

 integration. The gain is for mathematicians ; its use to 

 practical men may be doubted, as the numerical calcula- 

 tion of these functions is (at present) best done by the 

 familiar logarithms. In the older treatises the applica- 

 tions were chiefly algebraic and geometric ; the author's 

 system is to introduce the student at once to a wide scope 

 of applications in both geometry and physics, including 

 some of the higher branches {e.g. central orbits, harmonic 

 vibration, Fourrier's and Green's theorems, &c.). It is 

 clear that the account of each must be very brief In 

 some cases (<'.^'". the article on " Curve-Tracing," Art. 127) 

 it amounts to merely a sketch of procedure and results 

 with scarcely any proof In an " introductory " work this 

 seems a defect. It is, however, a masterly introduction 

 to the subject, and the wide scope of the applications is 

 well fitted to interest the student. 



It remains to notice some defects (in our judgment). 

 About ten pages are devoted to ordinary trigonometric 

 relations and tables of mere trigonometric formulas. This 

 seems too much space (being 4 per cent, of the whole) to 

 such elements. No definition is given of a maximum or 

 minimum, and the treatment of maxima and minima is 

 made to depend wholly on geometry. 



On p. 189 it is stated that Taylor's theorem is one "by 

 means of which any function whatever can be expanded " 

 — an obvious slip, corrected lower down (pp. 193, 201). 

 The necessity for the subject-functions, and in many 

 cases also their differential coefficients, being continuous 

 and generally also finite within the limits of any question 

 is not stated. This is, unfortunately, a not uncommon 

 omission in elementary works. Allan Cunningham 



Elementary Algebra. By Charles Smith, M.A., Fellow 

 and Tutor of Sidney Sussex College, Cambridge. 

 (London : Macmillan and Co., 1886.) 

 It is a pleasure to come across an algebra-book which has 

 manifestly not been written in order merely to prepare 

 students to pass an examination. Not that we think Mr. 

 Smith's book unsuitable for this purpose; indeed, with its 

 carefully-worked examples, graduated sets of exercises, 

 and regularly-recurring miscellaneous examination-papers, 

 it compares favourably with the most approved "grinders' " 

 books. The real want of the present day is a text-book 

 logically arranged and logically written. Apparently no 

 author cares to risk the chance of the financial ruin of his 

 book by going thoroughly to the root of the evil. A policy 

 of "safety" is the most we can expect. This is Mr. Smith's 

 policy, and although we think he might have made fewer 

 concessions to custom and yet have been safe, we wel- 

 come his effort very cordially, trusting that, when his 

 book has gained the success which it well deserves, he 

 will see his way to introduce further improvements. He 

 shows to great advantage as a teacher, his style of exposi- 

 tion being most lucid : the average student ought to find 

 the book easy and pleasant readmg. The second set of 

 exercises on the binomial theorem is worth specially 

 noting ; in many other mathematical books the sets of 

 exercises proposed to the student might well be, as in 

 this instance, collectioits of really useful theorems. 



LETTERS TO THE EDITOR 



[ The Editor does not hold himself responsiblefor opinioiisexpressed 

 hy his correspondents. Neither can he undertake to return, 

 or to correspond with the writers of, rejected manuscripts. 

 No notice is taken of anonymous communications. 

 [ The Editor urgently requests correspondents to keep their letters 

 as short as possible. The pressure on his space is so great 

 that it is impossible otherwise to insure the appearance even 

 of communications containinginteresting and no7>el facts.'] 

 The Pleomorphism of the Schizophyta 

 Some students of natural history are content, when the ex- 

 planations of phenomen.i which they have advanced and the 



arguments by which they have supported those explanations are 

 appropriated by other observers, to remain silent, trusting to the 

 justice of future generations for the vindication of their claims. 

 So far as my own experisnce goes, an active observer who shoukl 

 trouble himself to obtain honest treatment from all his con- 

 tetnporaries in regard to the significance of his published 

 wiitings, might abundantly employ the latter half of his Hfe in 

 struggling with new writers for that just recognition of his efforts 

 in earlier years in advancing the knowledge of this or that sub- 

 ject, which is the one reward desired above all others by most 

 men who have not attained to the heights of philosophic con- 

 tempt for the regard and sympathy of fellow-labourer.':. I do 

 not intend to largely employ my leisure in this pursuit, but there 

 is one subject on which I am anxious once for all to establish 

 the significance of my observations and reasonings published 

 twelve years ago in relation to similar views .idvanced and 

 accepted .it this moment. 



That subject is what is now spoken of as the pleomorphism of 

 the Schi/.ophyta or Bacteria. 



The view that the genera then recently est.iblished by Cohn, 

 viz. Micrococcus, Bacterium, Bacillus, Vibrio, Spirillum, and 

 Leptothrix, are form-phases, or variations of growth of a 

 number of " Protean" species of Bacteria, each of which may 

 exhibit, according to undetermined conditions, all or some of 

 these forms, was definitely and precisely formulated by me in 

 my memoir on " A Peach-coloured Bacterium," published in the 

 Quart. Journ. of Microscop. Science in 1873. I distinctly recog- 

 nised the existence of true species of Bacteria or Schizophyta, 

 but I pointed out that these must be characterised, not by the 

 simple form-fealures used by Cohn, but by the ensemble of their 

 morphological and physiological properties as exhibited in their 

 complete life-histories. I illustrated my conception of the Protean 

 or pleomorphic character of Bacterian species by a reference to the 

 similar character of the species of Calcareous Sponges, and I had 

 in my mind also the closely pai-allel facts established by 

 Carpenter in relation to the endless variety of forms of the 

 Prolozoic Foraminifera. 



My view was no merely speculative suggestion, but was 

 based upon a careful study of a remarkable peach-coloured 

 Bacterium, which exhibited a wide range of forms, con- 

 nected by intermediate forms, growing together in the same 

 vessel, and linked to one another most unmistak.ably by the 

 fact that they all were coloured by a special pigment which I 

 studied with the spectroscope, and to which I gave the name 

 " Bacterio-purpurin." I observed this organism on many different 

 occasions from various localities ; I figured and described its 

 various form-phases ; I obtained some modifications of form by 

 cultivation, but chiefly depended upon the association of the 

 different forms, the presence of completely transitional forms, 

 and the common bond of the pigment, for the view as to their 

 nature which I put forward. I gave the name Bacterium 

 rubescens to this pleomorphic, or, as I termed it, "Protean," 

 species. I gave an account of further observations on this 

 organism in the Quart. Journ. Mir, Sci., 1876, pp. 27-40. 



Cohn opposed my view as to the genetic connection of the 

 various forms associated by me under this name, and, contrary 

 to the established laws of nomenclature, substituted a manu- 

 script name in one of Rabenhorst's collections (viz. " roseo- 

 fersicina "), for the duly-published name applied by me to this 

 organism. fie further described some of its form-phases, 

 already figured by me, as Monas olceni, Alonas viiwsa, and 

 Rhabdohionas warmingii. 



On the other hand, two years later, Dr. Warming of Copen- 

 hagen ( Vidensk. Meddelelser. naturhist. For. i. Kjobenhavn, 

 1875), after studying the same organism and figuring many of 

 its form-phases, .adopted my view as to their nature, and the 

 extension of that view to the Schizophyta generally. He says : 

 " Les bacteries sont douees en realite d une plasticite illimitee, et 

 je crois qu'il faudra renoncer au systime de M. Cohn." In 

 1883 Dr. Neelsen, in his " Studien iiber die blaue Milch" 

 (Cohn's Beitriige, vol. iii. p. 241) cites my views and their 

 confirmation by Warming, and rightly contrasts them with the 

 later views of Nageli and Billroth, and with that of Lister, who 

 conceived that certain Bacteria were deA'eloped from a filamentous 

 fungus (Dcmatiitm fiscispoi um). As the result of his investiga- 

 tion of the Baclirium cyatiogenutn of blue milk, Neelsen says : 

 " Viel eherwiirde fiir unsern Fall der Ausspruch Lankesters 

 zutreffend ersclieinen, von dem Proteus-ahnlichen Organismus, 

 dessen einzeinc F.rscheinungs-formen eineSerievon Adaptationen 

 vorstellen." 



In 1884 Prof de Bary of Strasburg, in his " Vergleichende 



