268 



NATURE 



[July 21, 1904 



well true elliptic motion can be simulated by an 

 eccentric circle ani Ptolemy's equant. The equ'ant is 

 a point about which motion in the circle appears uni- 

 form. In elliptic motion it may be easily seen that 

 the empty focus is approximately such a point. Using 

 the equant, the maximum error in longitude is only 

 one quarter the square of the eccentricity — 8' only for 

 Mars, and for the other planets, except Mercury, less 

 than 7.1. But if any reader wants to know all the 

 equant can possibly be made to do before it must be 

 condemned, let him read this account of Kepler's 

 efforts. 



OVR BOOK SHELF. 

 The Fourth Dimension. By C. Howard Hinton, M.A. 



Pp. viii + 247; with coloured frontispiece. (London : 



Swan Sonnenschein and Co., Ltd., 1904.) Price 



4s. 6d. 

 A BOOK bearing the present title may be reasonably 

 expected to contain certain things. In the first place 

 it should have a clear exposition of Descartes 's appli- 

 cations of algebra to geometry, and conversely of 

 geometry to algebra, the logical conclusion of which 

 consists in the removal of all restrictions as to the 

 conceivable number of dimensions of space. In the 

 second place it should contain clear, concise, and 

 exactly worded statements of the peculiar and dis- 

 tinctive geometrical properties which are characteristic 

 of spaces of two, three, four, or more dimensions re- 

 spectively. Among these peculiarities might be cited, 

 as examples, the number of possible regular figures 

 corresponding to the five regular polyhedra of three- 

 dimensional space, the number of independent motions 

 of a rigid body, the properties analogous to those of 

 the shortest distance between two lines, the symmetry 

 of crystals, and, in short, any results calculated to 

 convince the reader that the study of space not only 

 of four, but of five, six, and generally n dimensions 

 leads to the discovery of geometrical theorems no less 

 interesting than those of ordinary plane and solid 

 geometry. 



Now such things as these are either entirely absent 

 from the book or else they are mixed up with such 

 a mass of irrelevant and discursive matter as to render 

 it often quite impossible to make out what the author 

 is driving at. The notion of a fourth dimension is 

 associated with the belief in a higher world with 

 electricity and magnetism, with organic life, with 

 logic and philosophy, with the nature of the human 

 soul, and with a variety of other ideas onlv calculated 

 to mislead the reader as to the real use of such in- 

 quiries. It is doubtful whether any tangible idea of 

 the " eight cell " or any other four-dimensional figure 

 can be gained by mere playing with coloured squares 

 or cubes. The proper way to realise the nature of such 

 figures is by studying their projections on pairs of 

 coordinate planes, and four-dimensional space has the 

 great advantage over three-dimensional in that anv 

 figure formed of points can be completely represented 

 by projections on two sheets of paper, whereas for a 

 three-dimensional figure one sheet is insufficient and 

 two sheets are too much. 



There is a certain class of individual, far too common 

 in this country, who busies himself in pestering his 

 mathematical friends with long and rambling letters 

 on such questions as " What is the fourth dimension? " 

 or " What is the ether? " Such people verv rarelv 

 know anything about the three dimensions of the space 

 they live in, but Mr. Hinton 's book will, it is to be 



NO. 181 2, VOL. 70] 



hoped, give them something to think about which will 

 at least amuse them and keep them occupied. The 

 great misfortune is that such books are believed by 

 the general public to be descriptions by a mathe- 

 matician of the work of other mathematicians. Con- 

 sequently, mathematicians obtain a reputation for 

 being unpractical which they certainly do not deserve. 



The Hill Towns of Italy. By Egerton J. Williams, 

 jun. Pp. xiv + 398; with illustrations from photo- 

 graphs and map. (London : Smith, Elder and Co., 

 1904.) Price los. 6d. net. 



The majority of English people who visit Italy con- 

 fine their attention to large towns such as Florence, 

 Rome, Naples and Venice. The mediaeval towns 

 of Etruria and Umbria constitute practically a 

 terra incognita to the ordinary tourist. The author 

 has done useful work in directing attention to 

 a district full of historic associations, and the 

 picturesque glimpses which he has given us both of 

 towns and country may well tempt those who have 

 the time and opportunity to go and visit the district 

 themselves. 



If there is one feature which lends itself to criticism, 

 it is that a perusal of the book does not give one a 

 mental picture so much of the towns themselves as of 

 an American traveler's impressions of them. It is 

 probably very hard for any writer to describe Italian 

 life who has not spent several of his early years in 

 Italy. So long as the writer confines himself to 

 purely descriptive matter the facts are Italian enough, 

 but where he endeavours to give colour to the scene, 

 that colour hardly feels right. We may cite such 

 sentences as " The exquisite grace and sweetness of 

 the madonna hold the onlooker like a vise " (query 

 vice) ; " One more ancient madonna greeted me as 

 I passed out by the left aisle." It would also be 

 interesting to know the author's authority for such 

 spelling as Velathri and Thrasymene. Velitrae and 

 Trasimene are certainly usual. Seeing, however, that 

 the book was written as the result of only a sojourn 

 of a spring and summer among the hill towns, we 

 can but be surprised at the amount of interesting 

 matter which it contains. 



Our Mountain Garden. By Mrs. Theodore Thomas. 

 Pp. 212. (New York: The Macmillan Company; 

 London : Macmillan and Co., Ltd., 1904.) Price 

 6s. 6d. net. 



Suburban gardeners sometimes attempt, with less or 

 more success, generally less, to establish a mountain 

 in the back garden. The author of this book has 

 adopted the converse plan of establishing a garden on 

 a New Hampshire mountain side. In this little book 

 she tells us how she did it, what patience she exercised, 

 what disappointments she experienced, what ultimate 

 success she achieved. 



The story is well told, and it is obvious that the 

 gardener was not only successful, but that she deserved 

 to be. 



Nevertheless, her sympathies seem rather to be with 

 the birds and wild animals to which she acted as 

 hostess than with the plants she used for decoration. 

 She seems to have looked on the plants as so many 

 cakes of colour, useful for producing effect, but to 

 have ignored the mental refreshment which a more 

 thorough study of their peculiarities and of their 

 manners and customs would have afforded. 



Her " practical hints " are excellent, and wiU be 

 serviceable to those disposed to follow her example 

 and make a garden for themselves according to their 

 own notions. 



