486 



NATURE 



[March 22, 1900 



book full of information, and illustrated by examples 

 excellently chosen and ably elaborated. 



A little adverse criticism may be devoted to the 

 following points: — Proof should surely be given of the 

 important problem (5) on p. 82, for which the reader is 

 referred to works on spherical trigonometry, where he 

 may not find it, or to Reusch's treatise on stereographic 

 projection, which is probably not accessible to him ; 

 tetrahedral is a misleading name for the class to which 

 sodium chlorate belongs ; r, being used to indicate 

 tetartohedral classes (p. 149), should scarcely be applied 

 to the trigonal bipyramidal class considered as belonging 

 to the rhombohedral system ; the nature of this class 

 and of some others would be much simplified by the 

 modern conception of the simultaneous action of an 

 axis and plane known as " composite symmetry," as 

 one of the general elements of crystal symmetry ; this is 

 only alluded to on p. 274, but its introduction as a mode 

 of crystal symmetry would render possible a definition 

 of the tetragonal system by means of its tetragonal axis 

 instead of the somewhat awkward definition on p. 139. 

 Similarly, the joint action of an axis and plane of 

 twinning has to be taken into account to explain certain 

 twins of sodium periodate mentioned on p. 359, and is 

 overlooked in the discussion on p. 463. Most readers 

 will find the argument on pp. 258-9 that the conception 

 of merohedrism leads to inconsistencies far from con- 

 vincing. 



In describing the stereographic projection, it is really 

 confusing to the student, and unnecessary, to speak of his 

 eye as being situated on the surface of the sphere. 

 Mention might have been made of the convenient device 

 for crystal drawing described by Maskelyne, under the 

 name crystallograph ; and the method of finding the 

 edge between two faces in a perspective drawing by re- 

 ducing them to a common intercept on an axis, and 

 finding their trace on the other two, might have been 

 introduced into Chapter vi. 



If the above be some defects of the book, many are 

 the features in which it is superior to its predecessors. 



Among new or specially instructive propositions may 

 be noted the proof relating to tetrad axes on pp. 276- 

 278, and the discussion of indices on pp. 288-295 ; the 

 proof of the relation between a face and its inverse on p. 

 356 ; the propositions in the rhombohedral system re- 

 lating to Millerian and Naumannian symbols, to indices 1 

 referred to three and four axes, and to the drawing of the 

 rhombohedron (p. 376). The useful proposition relating 

 to a small circle (p. 83), and its application, are not gener- 

 ally found in text-books. Especially to be commended are 

 the examples illustrative of the drawing of twin crystals. 

 Among the new terms introduced, "stereogram" will 

 doubtless be found serviceable. Finally, as evidence of 

 the up-to-date character of the book, we may note the 

 adoption of Cesaro's proof of the anharmonic ratio, the 

 discussion of Wellsite, and the description of Mr. 

 Smith's three-circle goniometer. 



Owing to the author's desire to avoid analytical 

 methods and spherical trigonometry, many of the proofs 

 are somewhat tedious ; but Chapter xix. contains ana- 

 lytical proofs and much suggestive material for the more 

 mathematical reader, particularly some propositions 

 NO. 1586, VOL. 6l] 



relating to the rhombohedral system ; ^._^, the expression 

 for the length of a trapezohedron edge (p. 578). 



The book is an eloquent witness to the scientific 

 method of the teaching which Prof. Lewis has carried 

 on at Cambridge for nearly twenty years — teaching to 

 which the present writer is glad to acknowledge his own 

 indebtedness. 



The author and the University Press may be congratu- 

 lated on the completion of a treatise worthy of the subject 

 and of the University. H. .\. Miers. 



THE CORRESPONDENCE OF OLBERS AND 



GA USS. 

 Wilhelni Olbers, sein Leben unci seine Werke. Im 



Auftrage der Nachkommen herausgegeben von Dr. 



C. Schilling. Zweiter Band. Briefwechsel zwischen 



Olbers und Gauss, Erste Abtheilung. Pp. viii -|- 767. 



8vo. (Berlin : Springer, 1900.) 



THE first volume of this work, published in 1894' 

 (Nature, li. p. 74), contained the collected 

 scientific papers of Olbers ; the present one gives the 

 first half (1802-19) of his correspondence with Gauss, 

 These old letters will nearly all be read with great 

 attention by any one interested in the history of astro- 

 nomy during the early part of this century, as the two 

 correspondents were equally devoted to theoretical and 

 practical astronomy, and discussed new publications and 

 new discoveries in all their bearings. Many readers will 

 perhaps think with the reviewer that here and there 

 some parts of the letters might with advantage have 

 been omitted, and that the editor when leaving out 

 ephemerides of comets and minor planets might have 

 gone further, and have omitted many results of ob- 

 servations, &c., which have been published elsewhere. 



The correspondence began in January 1802, when 

 Olbers had just succeeded in recovering the lost planet 

 Ceres by means of the elliptic elements calculated by 

 the young mathematician Gauss by a new method de- 

 vised by himself. The great sensation which Piazzi's 

 discovery had produced was kept up for some years by the 

 discovery of Pallas, Juno and Vesta, the first and last of 

 these minor planets being found by Olbers, and Juno by 

 Harding, so that (as Gauss remarks) of five planets 

 found in the years 1781 to 1807, the four were found by 

 natives of Hanover. The great respect in which the 

 wonderful success of the computations of Gauss with 

 regard to Ceres were held by astronomers, naturally led 

 to his being left to compute orbits and ephemerides 

 of all the four minor planets, and they consequently 

 occupy a very large part of the letters for the first seven 

 or eight years, until Gauss gradually handed over this 

 work to his pupils. Among many interesting matters 

 connected with the minor planets, which are touched on 

 in the letters, we may mention Olbers' well-known 

 hypothesis as to the origin of these bodies, which 

 directly led him to the discovery of Vesta, also the 

 annoyance of Bode at the discovery of a second planet 

 between Mars and Jupiter, whereby his ideas about the 

 harmony in the solar system were upset. That these 

 new bodies were not followed with the same attention 

 outside Germany is evident from the fact that Vidal, of 



