March i6, 191 i] 



NATURE 



75 



He had travelled extensively in remote parts of Ger- 

 manv, had discovered many more plants than had 

 been made known since the revival of botanical study 

 in Germany, and had described these carefully in a 

 work, " Historia Plantarum," left in manuscript. 

 This was published some years after his death, edited 

 by Conrad Gasner, who, by desire of the publisher, 

 employed illustrations (prepared to accompany Tragus's 

 work) to illustrate the descriptions of Cordus, to which 

 they were occasionally incorrectly fitted. From a 

 careful study of the descriptions. Dr. Greene shows 

 cause to regard Valerius Cordus as of rare ability and 

 insight, and esteems him to have been immeasurably 

 the greatest of the "German fathers of botany." 

 Among the services to botany ascribed to him we are 

 told that "he is the inventor of the art of phyto- 

 graphy " ; that in all descriptions "attention is given 

 to the morphology and life-history of the plant in as 

 far as is known to him"; that new terms are em- 

 ployed e.Kpressing new ideas and points of view in the 

 science, and that new conceptions appear in regard 

 to inflorescences, flowers, fruits, and seeds. In 

 taxonomy he shows clearer views with regard to 

 species, and his groups were more often based on 

 relationships than were those of his predecessors. A 

 number of his groups of generic rank stand good, 

 though in most cases the names given by him were 

 needlessly changed by Linnaeus. He paid heed to 

 internal structure (so far as that could be determined 

 by him, that is, by the unaided eye), and to physio- 

 logy, as regards prefloration, modes of climbing, and 

 similar features of plant-life. He also gave attention 

 to the varieties of cultivated fruits, of which excellent 

 descriptions are extant by him. What he succeeded 

 in doing suffices to show how grievous a loss botany 

 sustained in his earlv death. 



VECTOR ANALYSIS. 



Elements de Calcul vectoriel, avec de nombreuses 

 Applications a la Geometrie, a. la Mdcanique, et 

 a la Physique mathdmatique. By Prof. C. Burali- 

 Forti and Prof. R. Marcolongo. Edition fran^aise 

 traduite de I'ltalien et augment^e d'un Supplement 

 l^ar S. Lattes. Pp. vi + 229. (Paris: A. Hermann 

 et Fils, 1910.) Price 8 francs. 



'X'HE variety of matter contained in this small book 

 J- shows the condensing power of vector notation, 

 especially when combined with a concise literary style. 

 The theoretical part includes the elements of the 

 barycentric calculus, as well as a vector analysis in 

 , which vectors are written either in single letters, or 

 in the form B-A, where A, B are points. Scalar and 

 vector products are treated separately, so that 

 quaternions do not come in. Special points to notice 

 are that a scalar product has given to it the sign 

 opposite to that assigned by Hamilton ; the effect of 

 this is that if o, 0, y are three orthogonal unit-vectors, 

 «* = 0* = 7^=i, and versors have to be treated by intro- 

 ducing a symbol i, such that i' = — i, and is not a 

 vector. There is a good deal to be said for this ; but 

 it is most unfortunate that the authors take the clock- 

 wise sense of rotation for the positive one, especially 1 

 NO. 2159, VOL. 86] 



considering the use of vectors and vector products in 

 physics. 



The applications include geometrical, mechanical, 

 hydrodynamical and electrical formulae. Specially to be 

 noted are the proofs of Green's theorem and its con- 

 geners, Stokes's theorem of circulation, and Hertz's 

 formula for variation of flux. 



There is an appendix, partly historical, partlv 

 critical and even polemic. Probably everv reader will 

 find something here with which he cordially disagrees; 

 but there is one statement that deserves special atten- 

 tion. We believe that the authors are right in think- 

 ing that the final notation of the vector calculus will 

 be based on Grassmann's " Ausdehnungslehre," as im- 

 proved and modified by subsequent writers. The 

 Hamiltonians will have nothing more than a senti- 

 mental grievance if this proves to be the case. Nothing 

 can upset, or even modify, the quaternion calculus, be- 

 cause it is a definite type of linear algebra ; the main 

 question now is whether this algebra is the best for 

 the treatment of physical, and especially electrical, 

 problems. Judging by the attitude of Gibbs, Heavi- 

 side, and Lorenz (to name only these), the answer 

 appears to be no. 



There is very little fear that a really convenient 

 notation will not be ultimately agreed upon ; it will 

 probably be invented by a physicist. Meanwhile, dis- 

 passionate observers will derive some amusement, as 

 well as much instruction, from the lively controversies 

 of the champions of this or that particular symbol, as 

 if its retention or rejection were of vital importance in 

 itself. For instance, our authors seriously object to a 

 symbol such as [a/3] for a scalar product, on the ground 

 that functional symbols are invariably placed on one 



side of the operand ! The example of / ydx, where 

 / Qdx is practically a functional symbol, shows that the 

 statement is barely true, except in an artificial sense; 

 but even if it were strictly true, this would be no 

 reason for regarding it as a necessary law of mathe- 

 matical notatiort. G. B. M. 



MAP-MAKING. 

 The Theory of Map-Projections, with special refer- 

 ence to the Projections used in the Survey Depart- 

 ment. By J. I. Craig. Pp. iv + 80. (Cairo: 

 National Printing Department, 1910; Ministry of 

 Finance, Egypt Survey Department.) Price 200 

 milliemes. 



THE subject of map projections is one in which 

 the English language is strangely deficient, a 

 deficiency the more apparent when contrasted with the 

 wealth of Continental literature on the subject. Those 

 interested in the higher theory of map-making will, 

 therefore, welcome the appearance of this little 

 treatise, which seems to give in a compact and prac- 

 tical shape all the essentials of this attractive branch 

 of the geometry of surfaces. Starting with a state- 

 ment of the problem to be solved, and an allusion to 

 possible improvements in nomenclature; the term 

 projection itself, in the meaning of a representation 

 in accordance with any law, for instance, is not a 

 particularly happy one ; a history is given of the adop- 



