208 



NATURE 



[April 13, 191 1 



on a line and provislonaHy labt'l them P(o), P(i). 

 ( 00 ), and if P(.v). r(y) are provisional labels of any 

 other two point;:, »iien there are projective construc- 

 tions defining P(* + y) and P(.vy), such that the laws 

 for adding and multiplying labels obey the ordinary 

 laws of algebra, e.j^. P(jf + y) = P(y + «), including the 

 limiting cases when * or y, or both, have one of the 

 special values o, i, oo. Hence it follows that from 

 the base points P (o, i, «) we can construct a 

 rational scale of points V(r), where r is any rational, 

 positive or negative number. It may he added that 

 these are the only points on the line which can be 

 reached from the base-points by projective construc- 

 tions, and it would be a good exercise for the student 

 to prove what is not absolutely demonstrated in the 

 book, that this deduction is free from ambiguity; for 

 instance, suppose P{x), P(y), Pix + y) have each been 

 deduced from the base-points by a chain of projective 

 construction, it is required to prove that the same 

 point, P(x + y), is derivable from P{x) and P(y) by 

 the single operation of addition. 



The net result of these considerations may be put 

 (among other ways) in the following form. Suppose 

 we have a tetrahedral frame of lines with a point 

 given on each line distinct from the two vertices of 

 the tetrahedron which it contains. Then on each of 

 the six lines we can construct a rational scale, and 

 hence, by projective constructions alone, arrive at all 

 |X)ints which can be defined by four rational homo- 

 geneous coordinates. This rational projective space 

 is not continuous : to fill up the lacuna, it will be 

 necessary to assume the existence of one linear con- 

 tinuum of points, and apparently this will be also 

 sufficient. 



Staudt, on the other hand, gives his constructions 

 as the definitions of the addition and multiplication of 

 throws; and because the laws of algebra are satisfied, 

 he deduces the possibility of assigning numerical 

 values to throws. On the whole, Staudt's procedure 

 seems the more scientific, but it is not a matter of 

 much importance. 



A propos of involutions, attention may be directed 

 to the proposition on p. 223 : " Any projectivity in a 

 one-dimensional form may be obtained as the product 

 of two involutions." This is very interesting, because 

 it shows that although involution is, in the first in- 

 stance, a derivative idea (as a special case of projec- 

 tive correspondence) it may ultimately be regarded as 

 elementary. 



To indicate how far this volume proceeds, it will be 

 sufficient to say that chapter x. deals with pairs and 

 pencils of conies in a plane, and gives the typical 

 algebraic forms according to the elementary divisors 

 of the discriminant ; while the next, and final, chapter 

 is on families of lines, and treats briefly of ruled 

 quadrics, }'\ne coordinates, and linear congruences and 

 complexes. It might, by the way, interest applied 

 mathematicians to point out that if we suppose a 

 unit force acting along a given line, the six homo- 

 geneous coordinates of the line may be taken to be 

 proportional to the moments of the force about the 

 edges of the tetrahedron of reference. 



G. B. M. 

 NO. 2163, VOL. 86] 



THE SEW PSYCHOLOGy. 

 Manual of Mental and Physical Tests : a Book < f 

 Directions compiled with special Reference to llh 

 Experimental Study of School Children m tin 

 Laboratory or Class-room. By Prof. G. M 

 Whipple. Pp. xix + 534. (Baltimore T "^^ ' 

 Warwick and York, inc., 1910.) 



IS psychology to rank among the exact sl: 

 This is the question which is at once , 

 when we look into Prof. Whipple's volume. Wc 

 are reminded of Kant's famous pronouncement that 

 psychology never could be a science, because it was 

 impossible either to apply mathematics to its problems 

 or to perform experiments upon the minds of others. 

 Kant's dictum is a classical instance of the danger 

 of prophesying the impossible. In the book before us 

 the mathematical treatment of mental measurements 

 is discussed in the third chapter, and the rest of th« 

 volume is made up of more or less happily devistd 

 experiments upon the minds and bodies of oth< r 

 people. 



Yet nobody knows better than the author himstit 

 how relatively slight actual accomplishment has been. 

 The contrast between the position of the psychologist 

 and that of the doctor, for example, is verj- great. 

 An insurance company will decide quite serious 

 financial questions on the report of a medical man 

 whose tests give sufficiently good average results for 

 their purpose. But the psychologist has not yet 

 achieved a position of such confidence. He deals with 

 far subtler problems, and it is no fault of his that 

 we have not yet begun to consult him on the future 

 of our children. He frankly confesses that he is not 

 ready to render such positive service ; and whilst on 

 one hand we may fairly congratulate ourselves 

 on the fact that a serious attempt is being made to 

 arrive at a better understanding of mental phenomena, 

 it would, on the other hand, in Dr. Whipple's words, 

 be wrong to speak 



" as if a science of mental tests had already been 

 achieved. . • . To make such an assertion is surely 

 misleading, for . . . there is, at the present time, 

 scarcely a single mental list that can be applied un- 

 equivocally as a pyschical measuring-rod." 

 There is no general agreement about procedure, and 

 in many cases psychologists do not know exactly what 

 it is they are measuring, such is the "astounding com- 

 plexity, variet} , and delicacy of form of our psychical 

 nature." 



It would, nevertheless, ill become professional 

 students of the older sciences to speak in contemptuous 

 terms of a younger brother who is so conscious of his 

 own shortcomings. Psychology may have long to 

 wait for its Newton, but in the meantime the effort 

 to collect facts in a scientific way should surely meet 

 with every possible encouragement. The " man in the 

 street" has long recognised the existence of ditTer- 

 ences in the mental characteristics of his friends. He 

 accepts them in much the same spirit as folk re- 

 garded the weather, until the meteorologist began that 

 painstaking collection of data which is just beginning 

 to bear scientific fruit. We have more words to 

 describe these personal differences than we have for 



