December 15. 1898] 



NA TURE 



>47 



'forests, no unmistakable spocinu-n of that date lias yet 

 been discoveicd. 



The concluding rhaplcis arc devoted to vascular 



cryptogams, over 140 pages being assigned to the 



E(|uisetales, chiefly to the remarkable group of Cala- 



imir-;, which must have been so conspicuous an element 



(il ,irl)oniferous vegetation. Though cryptogamic, they 



loiMied large trees forty or fifty feet high, with woody 



trunks of exogenous growth. I'or this reason a section 



of the Calaniites named Camelodendron have been 



! and are even yet regarded as Gymnosperms by some 



'French writers. The genera and species of this group 



''are peculiarly difilirult to diagnose, every organ being 



!' detached and preserved in a different manner. Internal 



casts of pith cavities in sandstone are the most familiar 



: objects, but the more valuable specimens are those which 



preserve their internal structure, so ably deciphered by 



' Williamson and others. The foliary organs are found 



'separately in the shales and ironstone nodules ; and the 



strobili in various conditions, which have permitted their 



internal and external structure to be examined. The 



roots of several kinds are also found detached from the 



i; stems. The author, without attempting to unite these 



' scattered organs into specific wholes, has grouped the 



' facts in the clearest manner. The variety presented 



I prove that several distinct generic types existed, and as 



each variety of each separate organ was first described 



I in ignorance of its probable relationship to the other, a 



I complicated nomenclature has resulted. The Calaniites, 



I well represented in the IJevonian, did not survive the 



I Permian, though represented in the newer rocks by the 



' closely related Kquisetujn. 



Tin- second important carboniferous group, Spheno- 

 I phyllum, is also placed in a separate class, the Spheno- 

 phyllalcs, as a type that cannot be assigned to any 

 existing group. Its leaves are wedge-shaped, with one 

 or several veins and disposed in whorls, the strobili long 

 and narrow, and the stem slender and woody. It was 

 possibly a climbing plant, and is regarded as linking the 

 Calaniites and Lycopods. 



In so brief a notice it is difficult to do justice to a 

 work so full of matter and observation. I'.otaiiists and 

 geologists must equally congratulate themselves on 

 having so obscure and difficult a subject put before them 

 for the first time in a really lucid and comprehensive 

 manner. J. ,S. V,. 



INFINITESIMAL CALCULUS. 

 Injinilcsimal Analysis. By William Henjamin .Smith. 



Vol. i. I'p. xvi-t-352. (London: Macmillan and Co., 



Ltd., 1898.) 

 T T may be assumed thai the contents of this volume 

 •*• represent, on the whole, the author's conception of 

 a reasonable first course for the average University 

 student. Judged from this point of view, the work 

 certainly deserves approval, and is a favourable specimen 

 of the class to which it belongs. 



In the first two chapters the processes of differenti- 



ati',111 and integration are explained, with a|)propriate 



graphical illustrations. No attempt is made to discuss 



all the subtleties which modern function-theory has 



NO. 1520, VOL. 59] 



shown to be involved in the assumption of the possibility 

 of differentiation and integration, but the analysis, so far 

 as it goes, is sound, and something is done to guard the 

 student from making false generalisations. 



The next four chapters deal mainly with applications. 

 These have been judiciously selected, and are of prac- 

 tical importance as well as theoretical interest. Kine- 

 mntical applications might have been advantageously 

 included ; in fact, consitlering the general character of 

 the book, it is strange that kinematical considerations 

 have been almost entirely ignored. 



Chapter vii., on partial integration, concludes with 

 Creen's theorem ; it is a pity that Stokes's theorem was 

 not also included. A short but useful chapter on 

 definite integrals, and another on curve-tracing, con- 

 clude the volume. 



On pp. 18-20 there arc some remarks about velocity 

 with which we profoundly disagree. After allowing 

 that "according to the most familiar notions" a.s/a/ "is 

 the ai'cragc speed (or velocity) during the time A/,'' and 

 that "if the space be a function of the time" (it is 

 difficult to see how any other .issumption could be made) 

 then in general a.s/a^ has a definite limit dsldt when at 

 becomes infinitesimal, I'rof Smith proceeds : 



" Mechanically, however, this limit is not itself an 

 average speed at all, it is not of the same nature as the 

 variable difference-quotient A,f/A/. For this quotient 

 ?ieTer assumes this limiting value, no matter how small 

 A/ be made. And this is quite what we should expect 

 and what the nature of the case demands. For motion 

 implies duration, however small, of time, and change, 

 however small, of place. When there is no lapse of 

 time and no displacement there is no motion, and hence 

 no s|)eed (or velocity). In all strictness, there can be no 

 iiio/ioii at an itistaiit and hence no speed (or velocity) at 

 an instant. The concept of speed (or velocity) or motion 

 will not combine with tlie concept of instant (or point of 

 time) to form a compound concept." 



Surely I'rof Smith has here confounded the concepts 

 of motion and displacement. If we allow that motion 

 at an instant is impossible, how are we to escape Zeno's 

 paradoxical conclusion that all motion is impossible? 

 I low can I move from one place to another during a 

 minute, say, if at enery instant of that interval motion is 

 impossible? The remark, later on, that "this limit of 

 the average velocity, characterises not the action but the 

 state of the body, and is itself not a velocity though 

 everywhere named so," does not improve matters, and is 

 really irrelevant. The definition of velocity is ([uite 

 independent of such question-begging terms as "action" 

 and " state." Each of these terms, as applied to velocity, 

 is just as good and just as bad as the other ; it is when 

 we add the words "of the body" that the metaphysical 

 difficulty comes in, on account of the relativity of motion. 

 But assuming that we can form a clear concept of a con- 

 tinuous displacement expressed by a law s—/{l), there is 

 neither a logical nor a metaphysical difficulty in proceed- 

 ing to .;■ =/'(/) and saying that this is the velocity at 

 time /, if we have already agreed that when ,f at + />, the 

 velocity is a (a, b being constants) : that is, in whatever 

 sense a measures the velocity for the law s = at+b, then 

 in precisely tlic same sense /'{t) measiucs the velocity at 

 time / for the law s=/{t). 



Another passage to which we feel bound to call 



