124 



NA TURE 



[December i r, T902 



The introduction by the latter contains a brief statement of 

 the uses and purposes, not only of the differential calculus, 

 but also of the integral calculus and of differential equa- 

 tions. The present volume, however, deals exclusively 

 with the differential calculus, and that only so far as it 

 refers to functions of one and two variables. A notable 

 and important exception to this limitation, however, occurs 

 in the chapter on successive differentiation, where the 

 notion of D"y naturally leads to that of D~"y, in other 

 words, the ;/th integral of y. Here, however, the nota- 

 tion D - " is alone used, the familiar " F-hole of a violin " 

 being conspicuous by its absence. Probably the latter 

 symbol might advantageously be eliminated from our 

 mathematical notation altogether, were it not for the 

 important difference between differentiation and integra- 

 tion introduced by the appearance of the inevitable 

 ''constant of integration" which leads to the further 

 notion of "definite integrals." 



The amount of attention given to graphs will be wel- 

 comed by the great majority of teachers, and chapter vi., 

 which deals with the application of graphic methods to 

 the approximate solution of equations, is an important 

 feature which ought certainly to occur at some stage or 

 other of an ordinary mathematical curriculum, and may 

 probably be inserted here quite as well as elsewhere. 

 The feature which is most calculated to arouse criticism 

 is the adoption of the method of differentials as the basis 

 of the whole work. The author states that he has found 

 this method most useful and helpful to the student of 

 physics and mechanics, but it has the great disadvantage 

 of throwing into the background something which is very 

 important, namely, the notion of a limit. In examin- 

 ations there has recently been a tendency on the part of 

 candidates, when asked to find from first principles the 

 differential coefficient of sin x, to send up the following 

 answer : — 



d (sin x) sin (x + dx) - sin x sin x + dx cos x - sin x 

 dx dx dx 



dx cos x 

 = — = cos X. 



dx 



Even this might be excused if the candidates showed 

 an intelligent appreciation of the meaning of what they 

 were writing down, but as soon as they are asked to 

 differentiate x x , log sin x, or anything which is not in the 

 book, they exhibit hopeless ignorance, thereby proving 

 conclusively that the stock differentiations have been 

 merely written down by rote. 



There is no doubt a tendency on the part of another 

 class of writer to rush to the opposite extreme by making 

 the student read long discussions on continuity before 

 introducing him to the notation of the calculus. But 

 cannot a happy mean be found by introducing the notions 

 implied in the relation dy = f'(x)dx immediately after 

 the principal algebraic and transcendental functions have 

 been differentiated by means of the method of limits ? 

 Apart from this matter of opinion the book appears to 

 be excellent. 



We are glad to see the author does not relegate 

 Taylor's and Maclaurin's theorems to the end of the 

 book. In a logical treatment, that might possibly be 

 their proper position, but the postponement would prevent 

 many readers from acquiring an intimate familiarity of 

 what are probably the most important theorems in the 

 whole of the calculus. G. H. B. 



NO. 1728, VOL. 67] 



GALL-INSECTS. 



Monographie des Cynipides d' Europe et d ' Algtrie. By 

 TAbb^ J. J. Kieffer. Vol. I. Ibalynae et Cynipinae. 

 With 27 plates. Pp. vii + 687. (Paris : Hermann, 

 1 897-1901.) Price fr. 40. 



I "HE present work is a portion of the great series of 

 J- monographs commenced by the late E. Andre, under 

 the title of "Species des Hymenopteres d'Europe et 

 d'Algerie," by himself, his brother and other specialists, 

 among whom are the Rev. T. A. Marshall and the Abbe 

 Kieffer. The character of the work is well known to all 

 hymenopterists, and in this regard we need only say that 

 another volume will complete the Cynipidae, including 

 the parasitic subfamilies Allotriinae, Eucoilina; and 

 Figitina^, and will also include the families Evaniida;, 

 Stephanidae and Trigonalidae, and full systematic and 

 synonymic catalogues and indices to both volumes, the 

 first volume containing only an index of families and 

 genera, and a table of contents. 



In addition to the systematic portion of the work, the 

 structure, metamorphoses, broods, galls, parasites, biology, 

 bibliography, classification, geographical distribution, 

 &c, of the Cynipidae are discussed at considerable length - y 

 and the author mentions in his preface that though, when 

 he undertook this work in 1S96, few or no Cynipid.x were 

 known from any of the more southern countries of 

 Europe or from Algeria, he has now obtained, through 

 the kindness of various contributors whom he mentions, 

 considerable information on these countries, though much 

 of it reached him too late for the first volume and will 

 have to be deferred to the supplement in vol. ii. 



Notwithstanding the insignificant appearance of the 

 Cynipidae, on which the Abbe remarks, they are of ex- 

 treme scientific interest on account of the alternations 

 which the various broods present of winged and wingless, 

 and sexual and sexless, individuals at different times of 

 the year, in which respect they have much resemblance 

 to the Aphidae, though the Cynipidae, unlike the latter, 

 are seldom or never to be regarded as destructive insects, 

 one reason for which may be that the Cynipidae (or at 

 least certain species) are liable to the attacks of an in- 

 ordinate number of small parasitic Hymenoptera, chiefly 

 belonging to the Chalcididae, so that an entomologist may 

 breed a great variety of Hymenoptera from (say) a large 

 quantity of galls of Cynips kollari, without obtaining a 

 single specimen of the original species which formed the 

 galls. 



Although the insects themselves are inconspicuous, their 

 galls are conspicuous enough, and some of the large 

 fleshy eastern galls on oaks, such as the Apple of Sodom,, 

 resemble brightly-coloured fruit ; while the moss-like 

 galls, such as the bedeguar on the wild rose, are likewise 

 very pretty objects. One peculiarity of these insects is that 

 a considerable number of the species are attached either 

 to oaks or roses, though some few are met with on other 

 trees. They are also among the few insects which yield 

 products of great value to the human race, the most im- 

 portant of which, of course, is ink ; but various galls have 

 been, or are still, used for illumination, for tanning, in 

 medicine, for chemical purposes, and in the case of a few 

 species, even for food. 



Though some of our earlier hymenopterists, such as- 



