January 19, 1893] 



NATURE 



267 



please make explicit use of the Principle of the Perman- 

 ence of Equivalent Forms, which, after having been ex- 

 pounded at length and defended by Peacock (appendix 

 to '"Algebra"), has been summarily rejected as misleading 

 and unmeaning by many recent authors. To the formal 

 reintroduction of this principle Mr. Hayward's language 

 exhibits a tendency to return. Outside the domain of 

 elementary Algebra, its strict employment in the prolon- 

 gation of an analytical function into a new region is 

 indeed of common occurrence in Analysis ; while its ten- 

 tative application in unrestricted form, as an instrument 

 of suggestion and discovery in the Theory of Operations, 

 is fundamental. To the effort to widen the limits of 

 interpretation in connexion with it, has been due most of 

 the advances in Analysis. 



It is a fundamental question in mathematical logic how 

 far, after having carried the stream of our analysis through 

 regions of uninterpreted symbols, and having at length 

 arrived at a stage in which these symbols have disap- 

 peared, we are entitled to claim this procedure to be 

 demonstrative. It is of course of the very essence of 

 Algebra that the intermediate steps of its analysis remain 

 uninterpreted ; though in the Algebra of real quantities we 

 have a tacit assurance that an interpretation can be sup- 

 plied if necessary. Why then was there an objection 

 to a similar procedure in the Algebra of complex quanti- 

 ties ; and what is the source of the timidity and doubt 

 which characterized the use of complex analysis before 

 its geometrical interpretation was developed .'' Simply 

 that complex quantities turned out to be multiple-valued, 

 and that the selection of the proper value under given 

 circumstances had to be settled by tracing the continuity 

 of the quantities in a way that was to the mind practically 

 impossible until a visual geometric representation was 

 discovered. The Argand diagram is not essential to the 

 logic of the matter ; it rather makes Analysis possible by 

 bringing its scope within our grasp. It simply forms a 

 more extensive and systematic example of the method 

 which has been in use since the time of Descartes for 

 studying functions and approximating to their roots, by 

 aid of their graphical representation. 



The Principle of Permanence of Equivalent Forms thus 

 lies at the very root of Algebra, but it is rendered ineffective 

 by indeterminateness of interpretation. Its strict use, 

 when most needed, is subject also to another hitch. It 

 requires that the forms be expressed in exact terms ; an 

 infinite series must be expressed in the sum of n terms 

 together with a residue R. These residues must be 

 retained throughout the analysis until we arrive at a point 

 where interpretation comes in ; and it must then be 

 settled how far they can be neglected in the circumstances 

 of the actual interpretation. In the language of Mr. 

 Hayward, it cannot be asserted about series that are not 

 absolutely convergent, that the fundamental laws of 

 Algebra hold without limitation. 



It is perhaps a question how far the idea, thus restricted 

 and safeguarded, is worth being expressly retained as a 

 working principle of ordinary Algebra. In subjects like 

 the Calculus of Operations and Finite Differences, which 

 are still in an unsytematized stage, it cannot be dispensed 

 with ; and the extent to which its use is boldly pushed, 

 by De Morgan and Boole, even to the discussion in an 

 operational manner of divergent series without their 

 NO. 12 I 2, VOL. 47] 



residues, contrasts with the more exact processes of recent 

 Analysis. How far this boldness arises from the profound 

 logical studies of these writers, and their appreciation of 

 the imperfect character of inference at the best, may be 

 a subject open to discussion. 



In connexion with the doctrine of convergence of 

 series, the author gives a very clear account, from Sir G. 

 Stokes, of how it is that, on approaching certain critical 

 points, the convergence may gradually fall off and finally 

 disappear. The illustrations employed are algebraic 

 series of an exceptional character ; and the whole cir- 

 cumstances may possibly suggest to the uninitiated that 

 it is a phenomenon of exceptional rarity. The most 

 natural context is, of course, in connexion with the 

 wonderful and far-reaching theory initiated by Fourier, 

 by means of which functions arbitrarily discontinuous are 

 expressed by seemingly continuous series. In that con- 

 nexion, the necessity of explanation is so obvious that it 

 is interesting to examine the previous attempts at eluci- 

 dation. Thus De Morgan, in 1839, is able to conclude 

 (" Diff. and Int. Calc," pp. 233, 239) that such discon- 

 tinuity cannot occur in series proceeding by powers of a 

 real variable ; that in other cases it occurs only through 

 the series becoming divergent at the point of discontinuity. 

 It is, however, an important question how far it would be 

 allowable to avoid burdening an elementary exposition 

 by complete precautions against the existence of anomalies 

 like this, which would hardly have originally occurred to 

 any one in that early stage. 



The book ends with a wider survey, including a clear 

 and interesting account of Cauchy's theory of the radical 

 points of a rational function. The graphs of the cubic 

 2^ + az, which are given as an illustration, would also 

 form excellent and rapid examples of the Rankine-Max- 

 well method of graphical addition, applied to the separate 

 graphs of z^ and az. J. L. 



FOSSIL PLANTS AS TESTS OF CLIMA TE. 

 Fossil Plants as Tests of Clitnate, being the Sedgwick 

 Prize Essay for the Year 1892. By A. C. Seward, 

 M.A., F.G.S., Lecturer in Botany in the University of 

 Cambridge. (London : C. J. Clay and Sons, Cam- 

 bridge University Press, 1892.) 



nPHIS admirable essay is really a digest of the opinions 

 -■- of the principal writers on fossil plants, so far as 

 they throw light on geological climates, and a critical 

 rhume of the subject up to date. It should be read by 

 all who prefer to deduce the relative temperatures 

 of various latitudes in the past from such solid data as 

 assemblages of ferns, cycads, and conifers, the ancestors 

 of living genera and species, rather than from utterly 

 extinct belemnites, ammonites, and saurians, of whose 

 habits little can ever be known, and which might have 

 drifted far out of their temperature zones by warm and 

 cold sea-currents. 



Perhaps if any criticism can be made, it is that 

 too little has been said by the author as to what is 

 known of the Mesozoic floras, which, if scanty, are 

 extremely interesting. In fact only the widely-sepa- 

 rated Palaeozoic and Cenozoic floras are fully dealt 

 with. Owing to the magnitude, difficulty, and freshness 



