2l8 



NA TURE 



[July 7, 1892 



in the case of common minerals, where the number of 

 published analyses is very great, a judicious selection of 

 the best and most recent analyses has been made. 



The statement of the optical constants and the physical 

 characters of minerals has been treated in much the same 

 fashion as the chemical data. The best and most 

 trustworthy determinations have been selected, while 

 measurements of doubtful value have been omitted. 



It is on the crystallographic portion of the work, how- 

 ever, that Prof. E. S. Dana has expended the greatest 

 amount of labour. We are informed in the preface 

 that "an attempt has been made to trace back to 

 the original observer the fundamental angles for each 

 species, then the axes hav^ been recalculated from 

 them, and finally the important angles of all common 

 forms have been calculated from these axes." The 

 author is able to state that in every case this recalcula- 

 tion of the angles of all the forms of a mineral has been 

 undertaken, and that no pains has been spared in the 

 verification and correction of the results. The crystal 

 forms are indicated by letters, and the symbols em- 

 ployed are in the first instance those of Miller, and in the 

 second instance the modified form of Naumann's symbols 

 familiar to all who have used the earlier editions of the 

 work. The author gives it as his opinion that the former 

 should eventually supplant the latter altogether. In 

 the hexagonal and rhombohedral system, however, the 

 Bravais-Miller system is adopted in preference to that of 

 Miller. 



With few exceptions, the figures of crystals (1400 in 

 number) are new. Many have been drawn from original 

 data, and those taken from other works have been re- 

 drawn so as to secure uniformity of projection ; the habits 

 of each species and the types of twinning in crystals have 

 been very fully illustrated. 



While the general account of the mode of occurrence 

 and association of mineral species has been very carefully 

 attended to, there has been no attempt to make this part 

 of the work exhaustive, for to have done so would have 

 greatly increased the bulk of the volume. The account 

 of American localities — which has always been an 

 important feature of Dana's work, and has made it for 

 North America what the treatises of Kokscharov and 

 Zepharovitch are for the Russian and Austrian Empires 

 respectively— has been greatly added to. The works of 

 Roth and Hintze, with the numerous books and memoirs 

 devoted to the geology of particular regions, now supply 

 all the information that is needed in respect to 

 mineralogical distribution in other areas. 



We have tested the volume in many ways as to the 

 completeness and recent nature of the information 

 given with respect to particular species, and always 

 with satisfactory results. To pass such a voluminous 

 mass of information through the press has required 

 eighteen months of labour, and notices of important con- 

 tributions to our knowledge that have appeared since the 

 earlier pages of the book were printed off have been 

 relegated to a supplement. This supplement, which 

 extends to 28 pages, also contains brief accounts 

 of minerals of unknown composition, and of doubtful 

 species having little or no claim to recognition. 



In conclusion, we must congratulate both the original 

 author of the " System," and the writer of the volume 

 NO. 1 184, VOL. 46] 



in its present form, on the completion of their useful 

 labours. It is not too much to say that the publication 

 of each successive edition of this work has constituted 

 an epoch in the history of mineralogical science ; and the 

 present edition, coming from the hands of a new author, 

 completely maintains the prestige of former ones. 



J. W. J. 



MODERN INFINITESIMAL CALCULUS. 

 An Introduction to the Study of the Elements of the 

 Differential and Integral Calculus. From the German 

 of the late Axel Harnack, Professor of Mathematics 

 at the Polytechnicum, Dresden. (London and Edin- 

 burgh : Williams and Norgate, 1891.) 



MR. G. L. CATHCART'S translation forms a hand- 

 some volume, and will prove acceptable to those 

 engaged in mathematical teaching, as a storehouse of 

 suggestive methods and ideas for analytical exegesis. 



But let us examine the work from the standpoint of 

 the student approaching the subject of the Calculus for 

 the first time, supposing this book to be put into his 

 hands to acquire his first acquaintance with the method 

 and reasoning. 



Until very recently the Classics, Greek and Latin, as 

 taught at school, were looked upon chiefly as collections 

 of grammatical examples, and the subject-matter was 

 lost sight of in the careful parsing and analysis of the 

 sentences. Boys were taught on a system which implied 

 that they were all, in their turn, to become schoolmasters 

 and instructors ; and the interests of the majority, who 

 would profit intellectually from the literary study of the 

 ancient masterpieces, were completely neglected. 



So, too, in Mathematics : the ordinary text-books give 

 an excellent schoolmaster's training in the subject ; but 

 the large and increasing class of students, brought into 

 existence recently by the commercial developments of 

 scientific application, who are required to put into imme- 

 diate practice the theory which they find indispensable, 

 cannot afford the time to be dragged the whole length of 

 the quagmire of the Convergency of Series, of Inequali- 

 ties, of Discontinuity, and of the so-called Failure of 

 Taylor's Theorem. These are the quagmires in which the 

 mere mathematician delights to lose himself, and also to 

 lure in others after him. 



To one who is already very familiar with the notation 

 and operations of the Calculus the present treatise will 

 prove, not repellent, but even fascinating to minds who 

 pursue the subject for its purely analytical interest. 

 Having been over the road before, they will be prepared 

 to appreciate the strictly logical order in which the 

 theorems are developed, starting in Chapter I. with the 

 fundamental conceptions of Rational Numbers, of their 

 Addition, Subtraction, Multiplication, and Division — the 

 subject of Arithmetic in short ; and passing on in Chap- 

 ter II. to Radicals and Irrational Numbers in general. 

 The next three chapters treat of the Conceptions of 

 Variable Quantities, of Functions of a Variable, their 

 Geometric Representation and Continuity ; and it is not 

 till the sixth chapter that the Differential Coefficient is 

 introduced and determined for the simplest functions. 



But the beginner, who has had the courage to read 

 thus far, will wonder what on earth the subject is all 



