November 12, 1908J 



NA TURE 



35 



tinuous space hold unchanged for such a discontinuous 

 space. Some operations, however, such as rotation 

 and transformation of coordinates to new axes in- 

 clined to the original ones, are possible only in certain 

 cases. The study of the conditions under which 

 such operations are possible, and of the effect of these 

 operations when the conditions are satisfied, forms 

 the main drift of the book. 



The most interesting chapters are those which deal 

 with what the author calls modular spaces. .A modular 

 plane space of modulus m is a square of in- points, 

 a point {a, b) of this square representing all points 

 (a+pm, b-i-qm) of the unlimited arithmetical space, 

 p, q being arbitrary integers. Geometrical properties 

 of the complete unlimited space yield corresponding 

 properties of the fundamental modular square, the 

 coordinates of the original points being replaced by 

 their congruent numbers of modulus m. 



Transformations of coordinates in such modular 

 spaces lead to the construction of magic squares and 

 abacs. 



Graphical methods are given for the solution of 

 diophantine equations, and the last chapter deals with 

 a number of problems, among them the following, 

 originally proposed by Euler ; from each of six dif- 

 ferent regiments six officers of different rank 

 :ire taken. The problem is to arrange them 

 in a solid square so that in each row and in 

 each column there shall not be two ofificers of the 

 same rank or of the same regiment. This problem, 

 which was shown to be insoluble by MM. G. and 

 H. Tarrv in the case of thirty-six officers, is soluble 

 when there are sixteen, and the reasons for this are 

 here discussed. 



Strangely enough, this branch of mathematics, 

 although it might well be classed amongst the purest 

 of the pure, is not without its industrial applications, 

 notablv in the weaving of tissues and fabrics. 



.-Mtogether, w-e commend M. .^rnoux's book to 

 those interested in the mathematical curiosities of 

 the theorv of numbers. 



L. X. G. FiLON. 



Contributions to the Stuily of the Eiirly Development 



and Imbedding of the Human Ovum. Bv Dr. T. H. 



Bryce, Dr. J. H. Teacher, and J. M. M. Kerr. 



Pp. viii-)-93; lo plates. (Glasgow: J. MacLehose 



and Sons, 1908.) Price 12s. 6d. net. 

 It will be a glad day for the science of embryology 

 when all the details of the sequence of the develop- 

 ment of man are described from successive stages of 

 the human ovum and embryo. The chick has, to 

 a great measure, passed from the position that once 

 it occupied, and even the lower mammals cannot be 

 taken as substitutes for human material, when human 

 development is to be rightly studied. Much that is 

 confusing in embryology to-day is the outcome of 

 reading whole pages of the embryonic life-histories 

 of other creatures into the early chapters of human 

 development. 



In certain special directions the primates form a 

 group distinguished developmentally from other 

 mammals, and man and the anthropoids differ in 

 some details from the other primates. Our know- 

 ledge of the development of man will, therefore, not 

 be ideal until all our st;iges are accurately described 

 from purelv human material. Towards the attain- 

 ing of this ideal, the description of the Teacher- 

 Bryce ovum materially helps; at the same time, it 

 probablv holds out a guarantee for the further ex- 

 tension of our knowledge of the earliest stages of 

 human development, for the material so carefully 

 treated in this case is material that is often neglected. 



The Teacher-Brvce ovum is the earliest human 

 NO. 2037, VOL. 79] 



ovum yet described — its age is computed at thirteen 

 to fourteen days — and, owing to the care taken in 

 ascertaining the details of its history, this computa- 

 tion may be taken as final. 



It is younger, by probably a day, than the well- 

 known ovum of Hubert Peters, described in 1899, 

 although that ovum was originally considered to be 

 no more than three to four days old. 



Great care and a wealth of detail have been used 

 in making the account of this ovum as complete as 

 possible, and in order to render the material of more 

 value, a table of all the recorded early human ova 

 has been incorporated for comparative purposes. 



The volume in which this ovum is described also 

 contains the description of an early ovarian preg- 

 nancy, and this — like the uterine ovum — is the earliest 

 stage that has yet been described. 



It is but natural that, in dealing with such material, 

 many new details should come to light, and all the 

 man)' points of novelty receive very ample discussion 

 and illustration. The whole technic of the work, and 

 especially the many fine illustrations, mark a distinct 

 advance on the ordinary run of English scientific 

 publications, and towards this perfection the authors 

 have to thank the Carnegie Trust for assistance. 

 Dr. Bryce has already demonstrated his specimens 

 at the meetings of scientific societies, and the general 

 features of his earh- ovum are now well known to- 

 embryologists, but the book in which he describes 

 it contains, apart from the mere description, a vast 

 amount of well-assorted detail, got together and 

 presented in most workmanlike fashion. 



Graphic Algebra. By Dr. .Arthur Schultze. Pp. viii + 

 93. (New York : The Macmillan Co. ; London : 

 iVIacmillan and Co., Ltd., 1908.) Price 45. bd. 

 In this text-book the author first gives examples of 

 plotting from physical and statistical data, and the 

 graphing of simple functions of one and two variables. 

 He then proceeds to the main purpose of the book, 

 which is that of solving algebraical equations by the 

 use of squared paper and a few standard curves. 

 Equations up to the fourth degree are fully dealt 

 with, and, in order to facilitate the work, a method 

 is cleverly developed in which the direct graph is 

 replaced by "two loci of a simpler nature, the inter- 

 sections of which give the required roots. Thus a 

 quadratic equation is solved by reading off the inter- 

 section of a standard parabola and a straight line ; 

 the same parabola is used for all quadratics, and 

 it is only the scale and the position of the line which 

 vary. Instead of the parabola, a rectangular hyper- 

 bola may be used. Cubics are dealt with by means 

 of the curve y = x^ and a suitable straight line. Bi- 

 quadratic equations are solved by the intersection of 

 a circle and the standard parabola or standard hyper- 

 bola. In all cases it is shown how to find the 

 imaginary or complex roots, if such exist. 



The whole subject is treated in a very concise 

 and interesting manner, and the reader should become 

 fully conversant with the principles of graphing and 

 the nature of algebraical equations. But the special 

 methods, however ingenious, must be regarded rather 

 in the nature of mathematical exercises than as 

 having any very useful practical applications, for such 

 equations occur so seldom outside text-books that 

 when an actual case does arise, simple direct methods 

 of solution are usually to be preferred. This admir- 

 able manual concludes with an appendix containing 

 some " statistical data suitable for graphic representa- 

 tion," a short table of squares, cubes, square roots, 

 and reciprocals of numbers, and a collection of answer? 

 to the man\- exercises which are provided throughout 

 the text. 



