October 21, 1920] 



NATURE 



239 



court and Esson and of Wilhelmi are readily 

 accessible, and well known, at least to English 

 chemists, the impression on the reader is far from 

 satisfactory. Other instances of like nature could 

 be quoted. J. R. P. 



Our Bookshelf. 



The British Charophyta. By James Groves and 

 Canon George Russell Bullock-Webster. Vol. i. 

 Nitellea;. With Introduction. Pp. xiv+i4i + 

 XX plates. (London : The Ray Society, 1920.) 

 Price 25s. 



This monograph of the British Charophyta is a 

 valuable addition to the literature of British 

 botany. It has also a personal interest for many 

 British botanists as representing the work on this 

 group, embracing much of the leisure of forty 

 years, of the brothers Henry and James Groves, 

 to the former of whom the volume is fittingly dedi- 

 cated. In 1880 Messrs. H. and J. Groves pub- 

 lished in the Journal of Botany a "Review of 

 the British Characeae," in which an attempt was 

 made to give an account of all the then-known 

 British species, with illustrations and some par- 

 ticulars as to their variation and distribution. This 

 was the first of a series of papers by the same 

 authors, in which have been included descriptions 

 and figures of fresh species added from time to 

 time to the British list, records of distribution, and 

 other notes. The present monograph, in which 

 Canon Bullock-Webster co-operates, is the care- 

 fully considered outcome of these years of work. 

 The systematic portion, which includes the first of 

 the two subdivisions (Nitellea; and Charese) of the 

 group, is preceded by an introductory section 

 dealing with the growth and structure of the 

 Charophyta generally, and their distribution and 

 affinities ; this is well illustrated by numerous text- 

 figures from various sources, and several plates. 

 Each of the species is beautifully represented in 

 a lithographed plate, mainly from drawings by 

 Miss ^Iary Groves. The authors recognise six 

 genera of Charophyta, five of which, Nitella and 

 Tolypella, comprising the Nitellea;, and Nitellopsis, 

 Lamprothamnium, and Chara, included in the 

 Charea, are represented in Britain. The key to all 

 the British species, which precedes the general 

 .systematic account, includes thirty-two species, in 

 several of which distinct varieties are recognised. 

 Under each species there is a complete account of 

 the synonymy with reference to previous publica- 

 tions, a full description in English, and an account 

 of the distribution ; notes on variation, affinities, 

 and nomenclature are also added. 



Monograjia de I'Ordrc dels Rafididpters (Ins.). 

 By R. P. Liongl Navas, S.J. (Publicacions de 

 I'institut de Citncies.) Pp. 93, (n.irr«|fina : 

 Institut d'Estudis Catalans, 1918.) 



I'ATMER Navas is well known as a student of the 

 taxonomy of that miscellaneous assemblage of 

 insects formerly included in the old Linncan order 

 NO. 2660, VOL. 106] 



o( the Neuroptera. In the monograph before us 

 he deals with the curious and remarkable "snake- 

 flies." Their position in any scheme of classifica- 

 tion has long been a difficulty, and opinions 

 thereon are very diverse. Father Navas prefers 

 to follow Handlirsch and to regard them as con- 

 stituting an order of their own — the Raphidio- 

 ptera. Others merge them along with the " alder- 

 flies " (Sialida;) to form the order Megaloptera, 

 while a third alternative is that followed bv some 

 entomologists of combining the Megaloptera with 

 the Plannipennia into a single order, Neuroptera. 

 We are inclined to follow the intermediate course, 

 as there is little doubt that the Raphidiida have 

 their nearest allies in the Sialida'. although they 

 are more highly specialised than the latter. 



The present monograph is exclusively system- 

 atic — only eight lines are devoted to the larval 

 stages, for example — and the sole observations on 

 structure deal entirely with those characters of 

 the external anatomy which are utilised by the 

 systematist. Twa families are recognised, com- 

 prising thirteen genera and seventy-one species. 

 The greater number 'of genera occur in Europe 

 and North America ; only one genus is .African, 

 and four are Asiatic, but none are peculiar to 

 either of those continents. In Britain we have 

 four species comprised within three genera, but 

 the group has been hitherto so little collected that 

 in the next decade we shall probably totally rpvise 

 our views on its geographical distribution.- The 

 author has done a service in bringing together the 

 various species within a single memoir, and his 

 keys and descriptions will enable tH6 different 

 forms to be identified. Of the forty odd figures, 

 many are sketchv and rather deficient in dctailv 



A. D. I, , , 



Some Famous Problems of the Theory of' Numbers 

 and in particidar IVaring's Problem: An Inau- 

 gural Lecture delivered before the University of 

 Oxford. By Prof. G. H. Hrtrdy. Pp.. 34. 

 (Oxford : .-\t the Clarendon Press, 1920.) , Price 

 15. 6d. net. 



The theory of the integral numbers is a subject in 

 which it is frequently easy to conjecture new 

 results and extremely difficult to prove them. An 

 example of a result which must have been based 

 on conjecture is known as \\'aring's theorem, that 

 every positive integer is the sum of nine (or fewtfr) 

 positive cubes, of nineteen (or fewer) biquadrates, 

 and so on. .\ proof of this result, asserted in 1782, 

 was first approached by Prof. HillK-rt, of Gftttin- 

 gen, who showed in 1909 that every integer n is 

 the sum of a finite number not exceeding g{k), 

 independent of n, of exact feth powers. It has 

 been established, by transcendental analysis de- 

 veloped long since the days of Waring, that 

 ;r(3) = 9 as asserted by him, but whether ^(4) = 19 

 is still uncertain, though tliis number has been 

 shown not to exceed 37. The only positive 

 integers known to be inexpressible as a sum of 

 eight cubes (at most) are J3 and J39. 



Prof. Hardy and Mr. Littlewood have recently 

 developed a new method of applying properties of 



