March 29, 1900] 



NATURE 



51 



As might be expected, the chapter on contemporary 

 mathematics and mathematicians is meagre, and lacking 

 in proportion ; but it has the merit of emphasising some, 

 at least, of the most important lines of inquiry, and it is 

 not disfigured by any of those erroneous statements which 

 so often appear in summaries of this kind. 



One feature of the book, which will make it attractive to 

 all classes of readers, consists in the illustrations. They 

 comprise four facsimiles of manuscripts, one of the title- 

 page of the "Acta Eruditorum," two plates showing the 

 form of mathematical instruments in the seventeenth 

 century, and nineteen portraits. The portraits are fairly 

 representative, and are derived from authentic originals ; 

 one would gladly have dispensed with Mme. du Chatelet 

 and Saunderson in exchange for Gauss and Abel, who 

 are unaccountably omitted. 



The appearance of so many popular histories of mathe- 

 matics lately suggests a few remarks upon the purpose 

 which they are, or should be, designed to fulfil. Their 

 proper object is that of supplying stimulus ; and there is 

 no doubt that the interest of a mathematical student is 

 greatly encouraged by historical notes on the subject of 

 his reading. To the teacher, the history of mathematics 

 is a subject not merely of interest, but of vital import- 

 ance, because the psychological history of a race tends 

 to repeat itself, in little, in the mind of the individual ; 

 hence the proper order of teaching a subject is not 

 necessarily the most logical one ; and indeed the best 

 compromise between logical sequence and what we may 

 call " historical sequence " is precisely the golden mean 

 at which the teacher must do his best to aim. A popular 

 history does good if it awakens the teacher to some idea 

 of this : the great risk is that he may imagine that the 

 popular history contains all the information that he re- 

 quires. This is so far from being the case that he may 

 be actually in a worse position after reading his " History " 

 than before. Points of the highest importance are neces- 

 sarily ignored in a popular treatise, either from want of 

 space, or because the author is afraid of frightening his 

 readers with technicalities. For instance, it is compara- 

 tively easy to ascertain the net results of Greek geometry, 

 expressed in modern terminology ; but without some 

 acquaintance with the actual works of Euclid, Archi- 

 medes and Apollonius, as they wrote them, it is simply 

 impossible to have any correct ideas of the aims and 

 methods of Greek mathematicians. If our teachers of 

 mathematics were really familiar even with the "Ele- 

 ments " of Euclid, instead of with a garbled version of a 

 part of it, they would be far better able to discuss 

 intelligently the question of "Euclid and his Modern 

 Rivals." 



Again, the real interest of the ante-Descartes period in 

 ICurope consists in the gradual improvement of algebraic 

 notation, and of methods of arithmetical computation. 

 This question of notation is of the greatest interest from 

 every point of view. What we have now is simply the 

 survival of the fittest, and may have to submit to modifi- 

 cations more drastic than any of us at present imagine ; 

 still it is far ahead of its predecessors, and our admiration 

 of Archimedes and Fermat is greatly enhanced when we 

 realise the wretchedly inadequate notation which they 

 had to employ. To have traced, even in a general way, 

 this advance of notation and method, is far more in- 

 NO. 1587. VO I. Ol] 



structive and important than to know, for instance, that 

 logarithms were invented by Napier of Merchiston, or 

 that Newton discovered the Binomial Theorem ; yet 

 very little help in this direction is afforded by the popular 

 history. 



A really good history of mathematics in the nineteenth 

 century has yet to be written ; it would probably require 

 the combined labour of an organised body of experts, 

 such as those engaged on that invaluable work, the 

 " Encyclopadie der mathematische Wissenschaften." 

 Until such a scientific history has been composed, it is 

 idle to expect anything worth reading in a popular 

 treatise. It may even be questioned whether a popular 

 writer, however competent, could profitably deal with the 

 subject at all, unless our methods of school teaching are 

 greatly modified. For the history of modem mathe- 

 matics is not mainly that of individual discoveries, how- 

 ever brilliant ; but that of the systematic investigation of 

 mathematical notions, such as "number," "continuity," 

 "function," "limit," and the like. If these technical 

 terms are ignored, even a popular sketch of the subject 

 becomes impossible ; yet how many of these terms are 

 even approximately understood by any but mathematical 

 specialists t And what is the use of trying to explain 

 the theory of doubly periodic functions to readers who 

 are unaware that, in learning trigonometry, they were 

 studying singly periodic functions without knowing it ? 



G. B. M. 



SCIENTIFIC LENS-MAKING. 

 Theorie und Geschichte des photographischen Objectivs, 

 Von Moritz von Rohr. Pp. xx -f 435, (Berlin : 

 Springer, 1899.) 



DR. VON ROHR'S book contains much to attract 

 students interested in the theory and development 

 of photographic lenses. The author is one of the 

 scientific workers attached to Zeiss' manufactory in Jena, 

 and has a practical acquaintance with his subject. 



The book is most instructive to an English reader, 

 specially for the reason that while the debt due to the great 

 English opticians of the present day, as well as to those 

 of the past, is freely owned, and the author's appreciation 

 of the value of their work is warm and cordial, yet the 

 contrast between the methods of the English school and 

 those of the pupils of Abbe and Schott is sharply drawn, 

 and it is clear that in the opinion of the author the future 

 is with the latter. Schott and Abbe began in 1881 "to 

 study carefully, as far as possible, all chemical elements, 

 which in any form can become constituents of amorphous 

 compounds produced by fusion, with regard to their 

 influence on the refraction and dispersion of the com- 

 pounds." On this secure basis is raised the great Jena 

 glass factory, to whose work the scientific world is already 

 so deeply indebted. 



We doubt if any English manufacturer would have 

 attempted thus to improve his products ; the new 

 methods, the methods which Englishmen must adopt 

 if England is to retain her place among the manufacturing 

 peoples of the world, have not yet found a home among 

 us, and unless a change is made, England must cede the 

 place of honour, not merely in lens-making, but in every 

 branch of manufacture. 



