5 -=4 



NA TURE 



Atkil 2, 1903 



bodies, retain the property of producing scintillations on a 

 blende' screen, and are non-penetrating]. 



It seems probable that in these phenomena we are actually 

 witnessing the bombardment of the screen by the electrons' 

 hurled off by radium with a velocity of the order of that of 

 light; each scintillation rendering visible the impact of an 

 electron on the screen. Although, at present, I have nol 

 been able to form even a rough approximation to the number 

 of electrons hitting the screen in a given time, it is evident 

 that this is not of an order of magnitude inconceivably 

 great. Each electron is rendered apparent only by the 

 enormous extent of lateral disturbance produced by its im- 

 pai 1 on the sensitive surface, just as individual drops of rain 

 falling on a still pool are not seen as such, but by reason 

 of the splash they make on impact, and the ripples and 

 waves they produce in ever-widening circles. 



THE PSYCHOLOGY AND NATURAL DEVELOP- 

 MENT OF GEOMETK Y. 

 T N connection with recent endeavours to place the teaching 

 J- of geometry on the best possible basis, much interest 

 attaches to Dr. Mach's attempt to trace the order in which 

 geometrical facts first made themselves known in the 

 natural order of evolution. 



The earliest notions of space must have been suggested 

 by the relations of physical bodies to the parts of the human 

 body, the spacial behaviour of bodies towards one another 

 subsequently acquiring a mediate and indirect interest far 

 transcending that of the momentary sensations. While the 

 senses of sight and touch only give rise to sensations of 

 surface, crude physical experience soon impels us to conceive 

 the notion of volume, and the constancy of ■volume of bodies 

 would be one of the first attributes to manifest themselves 

 to our senses. Geometry, although asserted to be con- 

 cerned with ideal objects only, arose from the consideration 

 of the space relations of physical bodies. The earliest units 

 of measurement were derived from our hands and feet. But 

 the material properties of bodies rather than their spaii.il 

 properties possess the greatest interest for us, ana Dr. 

 Mach considers that the first ideas of measurement were 

 those of volume, and arose from counting the number of 

 equal identical immediately adjacent bodies which would 

 fill a given space. The notion of areas would be derived 

 from the number of food-bearing plants which a given field 

 would contain or the labour required in planting them, 

 distance would be estimated by hours of travel. The 

 measurement of lines and areas by means of solids is a 

 notion now completely estranged from our geometrical ideas, 

 but in early times we should have measured lengths and 

 areas by the number of solid bodies placed in line or dis- 

 tributed over a surface required to cover them, an idea 

 which is borne out by the remarkably elegant methods of 

 mensuration expounded in the seventeenth century by 

 Cavalieri. 



Although movable bodies present different spacial sensa- 

 tion^ to the visual sense dependent on the position and 

 distance of the observer, the notion of spacial constancy 

 becomes associated with them both by the sense of touch 

 and by combined experience. 



The earliest conceptions of purely spacial properties 

 naturally asserted themselves in the pursuit of trades and 

 arts. The property that a number of equal and similar 

 triangles of any shape can be fitted together in regular 

 order to form a pavement or mosaic naturally leads to the 

 property that the three angles of a triangle are together 

 equal to a straight angle. A consideration of the way in 

 which the triangles run in rows would lead to the notion 

 of parallels, and the property that the adjacent angles made 

 by the parallel lines with any transversal are together equal 

 to two right angles. The theorem of the Pythagoreans, 

 according to which superficial space can only be partitioned 

 into regular polygons in three ways, namely, into equilateral 

 triangles, squares, or hexagons, naturally finds its origin 

 in the same source. 



1 Radiant matter, satellites, corpuscles, nuclei ; whatever they are, they 



ike material masses. 

 - Abstract of a paper by Dr. E. Mach in the Monist. Translated by 

 I. J. McCormack. 



A stretched string furnishes the simplest visualisation of 

 a straight line, and leads to the property that a straight 

 line is the shortest distance between two points, but Dr. 

 Mach reminds us that this property cannot be regarded as 

 being established by mere visualisation. It is true that we 

 have learnt instinctively to reproduce in our imagination 

 some method of demonstrating that, for example, two sides 

 of a triangle are greater than the third side, but the source 

 of our knowledge here is physical experience derived from 

 our knowledge of material bodies. Another property of 

 straight lines, namely, that a straight line is self-congruent 

 if made to slide or rotate upon itself, is also a result of ex- 

 perience with straight and bent wires. 



The knowledge that the measures of geometry depend 

 on one another was reached in divers ways. The division 

 of a parallelogrammatic field into smaller fields gave rise 

 to the area being measured by the product of the length and 

 breadth, and the knowledge that the area of a rectangle is 

 greater than that of a parallelogram having the same sides 

 gave rise to the idea that the area also depended on the 

 angles. 



In regard to angles, Dr. Mach points out that the defini- 

 tion of an angle as the difference between two directions 

 is a physiological definition, the notion of direction being 

 a purely physiological conception. In abstract space, 

 obtained by metrical experiences with physical objects, 

 differences of direction do not exist. An angle is deter- 

 mined when the distance is assigned between two points 

 on its arm at given distances from the vertex, but, as Dr. 

 Mach points out, this measure, though closely resembling 

 those adopted in trigonometry, was not used in geometry, 

 because angles so measured would not possess additive 

 properties. The simpler measure of an angle by the arc or 

 area which it intercepts on a circle surrounding the vertex 

 thus became generally adopted. In connection with l)r 

 Mach's views on this point, it may be maintained that even 

 with our present experience of geometry an angle in- 

 stinctively suggests the idea of space, extending, no doubt, 

 indefinitely from the vertex, but possessing the remarkable 

 property of being a definite fraction of the whole space 

 surrounding that point. 



The object of geometry is to answer questions that occur 

 repeatedly in the same form, and with this object has arisen 

 the study of deductive geometry, which takes theorems and 

 proves them once for all. But it will be seen that Dr. Mach 

 strongly emphasises the physical and material origin of 

 geometry, and his studies will naturally support the view 

 that geometry is likely to be best understood when taught 

 in its early stages from the experimental side. 



NO. I744, VOL. 67] 



THE E UCA L i 'P TS. ' 



THE economic importance of the genus Eucalyptus to our 

 *- Australian Colonies accounts, no doubt, for the some- 

 what extensive official literature which has grown up there 

 on this subject. This includes numerous publications by the 

 Government botanists and forest officials of the Australian 

 colonies, and especially the classic " Eucalyptographia," 

 now, unfortunately, no longer obtainable, of the late Baron 

 von Mueller, whose enthusiasm for the genus is main)} 

 responsible for the large Eucalyptus plantations now exist- 

 ing in Italy, France, Algeria, California and other countries. 

 Messrs. Baker and Smith, in their contribution to Euca- 

 lyptus literature, give an account of the results they have 

 secured in the course of a systematic study of the Euca- 

 lypts, both from the botanical and chemical points of view, 

 and they conclude from the data so obtained that the tiers 

 belonging to this genus may be divided into a series of 

 natural groups, in which there is a striking correlation 

 between the structure of the leaves, and to a certain extent, 

 also, of the barks, and the composition of the essential oils 

 produced by the species ; thus, in Eucalyptus tesselaris, 

 which the authors regard as the primitive type, the leaves 

 have a characteristic parallel lateral venation and furnish 



1 " A Research on the Eucalypts especially in regard to their Essential 

 Oik." By R. T. Baker, F.L.S., and H. G. Smith, F.C.S. Pp. 295 ; with 

 q plates. (Technological Museum : New South Wales.) 



" Eucalypts Cultivated in the United Stales." By A. J. McClatchie. 

 M-.\ Pp. 101 ; with 91 plates. (Department of Agriculture, U.S.A.) 



