January 18, 1906] 



NA TURE 



>.6j 



farmer regard the investigator as his necessary helper 

 in the conduct of his business. 



In matters connected with the physics of the soil 

 and its bearing upon the operations of cultivation the 

 American workers have accumulated much novel in- 

 formation, and to this some of the chapters of Prof. 

 Snyder's book form a good introduction. The re- 

 quirements of the crop are treated from a sound 

 general standpoint which never forgets that water 

 and air, soil texture, and cultivation are perhaps the 

 prime factors in plant production. In this country 

 students are a little too apt to fancy that farming 

 begins and ends with the application of artificial 

 manures; we can recommend this book to them for 

 the truer point of view, even though the conditions 

 which regulate our use of manures are not quite the 

 same as in America. 



RECENT ASPECTS OF ELEMENTARY 

 GEOMETRY. 

 The First Book of Geometry. By Grace Chisholm 

 Young, Ph.D., and W. H. Y'oung, M.A., Sc.D. 

 Pp. xvi + 222. (London: J. M. Dent and Co., 

 1905.) Price ix. 6d. net. 



OF late years a very remarkable change has been 

 made in the theory of elementary geometry, the 

 general effect of which has been to make it more 

 abstract, and to reduce a great deal of it to the 

 application of logic without any appeal to intuition. 

 It has been realised that geometry must be based on 

 the assumption of certain undefinable entities, of 

 elementary relations between them, and a complete 

 system of independent axioms. For the purposes of 

 ordinary Euclidean geometry, it is probably the 

 simplest way to assume the straight line as the one 

 undefinable entity, and intersection as the elementary 

 relation from which the notions of point and plane 

 may be derived. What system of axioms we adopt 

 will partly depend upon the nature of the geometry 

 we study; for instance, the axioms which are 

 necessary and sufficient for the purposes of projective 

 geometry require supplementing when we discuss the 

 theory of measurement. . 



It is the theory of measurement which presents the 

 greatest difficulty at the present time. If we assume 

 all the results of projective geometry, we may proceed 

 as follows: — Taking any three points O, I, X on a 

 line, we may associate them with the numbers (or 

 indices) o, I, co (where =o is the vague infinity of 

 ordinary arithmetical algebra). We can then give a 

 purely projective rule for finding a point on the line 

 to be associated with any given rational number pjq ; 

 we thus get on the line a set of points corresponding 

 to the whole field of rational numbers, and, more- 

 over, the arrangement of the points corresponds to the 

 arrangement of the numbers according to their 

 magnitude; that is, if m>n>p, the point N lies on 

 that segment MP which does not contain X. If we 

 like, we can define the distance AB as being measured 

 bv b — a, where a, b are the indices of A, B. This 

 satisfies the relation AB + BC = AC, but equal seg- 

 ments as thus defined are not intuilionally equal, 



no. 1890, VOL. 7$^ 



except when X is " the " point at infinity on the 

 line; and even then we cannot prove, but must assume 

 the intuitional equality. Moreover, there are points 

 on the line which do not have rational indices, unless, 

 in spite of common sense, we assume that the points 

 on the line form a discrete aggregate. Now in 

 arithmetic we have a perfect continuous aggregate, 

 where each irrational element separates all the rational 

 ones into two complementary parts, respectively 

 greater or less than itself. If we assume that all the 

 points in the line which have not rational indices 

 behave in a similar way, we have a complete corre- 

 spondence between the succession of points on a line 

 and the elements of the arithmetical continuum. So 

 far as appears at present, this is a pure assumption ; 

 but if it is not made, anything like the ordinary 

 theory of measurement seems to be impossible, for 

 two distinct points ought to have a measurable dis- 

 tance, and the measure must be a number ; if the two 

 distinct points cannot be associated with two distinct 

 numbers, how is their distance to be defined as a 

 measurable quantity? Other difficulties arise in con- 

 nection with transfinite numbers and their represent- 

 ation by point aggregates ; but these are compara- 

 tively unimportant, it it is remembered that the 

 assumption of the correspondence of points on a line 

 with the arithmetical continuum involves a similar 

 correspondence between the arithmetical continuum 

 and the points on any finite segment. 



It is very interesting to see how this recent theory 

 has reacted on the question of teaching elementary 

 geometry. Instead of tending to make it more 

 abstract and more logical, it has done exactly the 

 reverse ; and the reason for this is not difficult to find. 

 The notions of geometry, so far as it is distinct from 

 logic, are derived from concepts, and these, again, 

 from experience. There must be an intuitional basis 

 for geometry; and although, from a logical point of 

 view, it is desirable, for any given species of 

 geometry, to reduce its necessary assumptions to a 

 minimum, progress in geometrical invention is to be 

 expected from those who cultivate their powers of 

 observation as well as their logical faculties. One 

 result of recent research has been to explode, once 

 for all, the pretence that the " Elements " of Euclid 

 present geometry in its most logical form ; on the 

 other hand, to try to teach beginners the subject in 

 what would now be considered the most rigorous 

 way would be certain to end in failure. 



The book which has been written by Dr. and Mrs. 

 Young illustrates very well what has just been said. 

 Its main object is to awaken the pupil's mind to the 

 ideas by which we classify the properties of space; 

 this is done by directions in paper-folding, in dis- 

 section of areas, in the construction of solid models, 

 and the like. At the same time, various theorems are 

 stated and proved, so that the beginner may learn 

 the difference between experimental and deductive 

 geometry. As in the case of other text-books with a 

 similar aim, the teacher will have to be careful to see 

 that hi~. pupil distinguishes proofs from verifications; 

 e.g., on p. 173 we have a proof that the angles of 

 a triangle make up two right angles, while on p. 121 



we have a verification in a special case. 



