June 12, 1913] 



NATURE 



569 



the successive periods as applicable to England, 

 and concludes that the Chellean is not pre-Glacial, 

 but is referable to the middle of the Pleistocene 

 age. He discusses the human skeleton embedded 

 in ochre found by Buckland in the Paviland cave, 

 Glamorganshire, and 'points out that it is paral- 

 leled by similar burials of a late Palaeolithic 

 period in France and Moravia. He considers that 

 the importance of the Engis skull has been exag- 

 gerated, and concludes that the Tilbury man 

 cannot be later than the beginning of the Neolithic 

 period. He gives reasons for supposing that the 

 early Neolithic immigrants to western Europe 

 may have lived for some time with the Palaeolithic 

 peoples whom they found there. 



The description of the Terramara settlements, 

 which occupies nearly 200 pages, forms an exhaus- 

 tive work of reference on a subject on which Dr. 

 Munro speaks with special authority. More than 

 100 of these large flat mounds in the valley of 

 the Po have now been examined, and it seems 

 clear that they are all referable to the Bronze 

 age. They are generally quadrangular in shape, 

 and their average superficial area is about seven 

 acres. Implements, seeds and fruits, and bones 

 of the animals used for food are so numerous that 

 a very fair idea of the life of the inhabitants can 

 be obtained, and it is evident that they were 

 occupied with many industries. They made their 

 own bronze implements and ornaments, worked 

 also in bone and terracotta, and must have woven 

 much cloth, judging by the extraordinary variety 

 and abundance of spindle-whorls and loom- 

 weights. Their pottery is especially artistic, and 

 the peculiar horned appendages fixed to the tops 

 of the handles have not been found outside the 

 area of the Terramare and certain districts influ- 

 enced by the civilisation of their inhabitants. 



MATHEMATICAL TEXT-BOOKS. 



(1) Matematica Dilcttevole e Curiosa. By Ing. 

 Italo Ghersi. Pp. viii + 730. (Milano : Ulrico 

 Hoepli, 1913.) Price 9.50 lire. 



(2) New Analytic Geometry. By Prof. P. F. 

 Smith and Prof. A. S. Gale. Pp. x + 342. 

 (Boston and London : Ginn and Co., n.d.) 

 Price 6j\ 6d. 



(3) Experimental Mensuration: An Elementary 

 Text-Book of Inductive Geometry. By H. 

 Stanley Redgrove. Pp. xvii + 328. (London: 

 W. Heinemann, 1912.) Price 2s. 6d. net. 



(4) Geometrical Optics. By A. S. Percival. Pp. 

 vii+132. (London: Longmans, Green and 

 Co., 1913.) Price 4s. 6d. net. 



(5) Problemes d'Analysr Mathematique. By Prof. 

 E. Fabry. Pp. 460. (Paris : A. Hermann et 

 Fils, 1913.) Price 12 francs. 



NO. 2276, VOL. 91] 



(6) Lccons sur I'Intdgration des Equations Diffd- 

 rentielles aux Derivees Partielles. By Prof. M. 

 V. Volterra. Pp. ii -|- S3. (Paris: A. Hermann 

 et Fils, 1912.) Price 6 francs. 

 (1) ' I "'HIS collection of mathematical paradoxes 

 J. should be of interest to a large circle 

 of readers. It is remarkably comprehensive, 

 occupying more than seven hundred closely printed 

 pages, is profusely illustrated, and is published 

 at a very reasonable price. Those who are ac- 

 quainted with Mr. Rouse Ball's work will find 

 much in these pages with which they are already 

 familiar, but there will be very few who will not 

 discover something that is new and surprising. 

 In addition to the standard problems of antiquity 

 — the squaring of the circle, the trisection of an 

 angle, and the duplication of the cube — there are 

 elaborate sections on arithmetic, algebra, geo- 

 metry, and mechanics. There are problems on 

 shunting, map-making, perpetual motion, tram- 

 mells, constructions with limited instruments, 

 probability, magic squares, boomerangs, draughts, 

 chess, and many other themes of a popular 

 character. It is, however, impossible to give any 

 detailed account of so miscellaneous a work ; 

 enough has, perhaps, been said to show that there 

 is abundant material to while away any number 

 of idle hours. 



(2) It is gradually becoming recognised that the 

 practice in elementary, text-books of restricting 

 cartesian geometry to straight lines and curves 

 of the second degree gives the student a narrow 

 idea of the power and generality of analytic 

 methods. For ordinary purposes a course of the 

 character set out in the volume before us is, in 

 our view, far more valuable than that given in 

 most English text-books. Intricate properties of 

 conies and systems of conies, homogeneous co- 

 ordinates, invariants, &c. , are omitted, and space 

 is thus obtained for curves of higher degree and 

 the elements of three-dimensional geometry. 

 There is an excellent collection of examples, few 

 of which demand from the student any high degree 

 of analytic skill. 



(3) The author gives us, in the present work, 

 a distinctly original text-book on geometry. He 

 is in sympathy with the principles laid down in 

 the circular issued by the Board of Education, and, 

 in addition, contends that this subject loses most 

 of its educational value if it is not united from 

 the first with mensuration and trigonometry. The 

 trigonometrical ratios are accordingly introduced 

 in the second chapter, and applications are made 

 at every convenient opportunity. With the ex- 

 ception of a few of the chief geometrical theorems, 

 which are mainly derived by inductive processes, 

 there is no formal geometry, and practically all 



