386 



NATURE 



[August 25, i, 



of a homographic range, then the two rows are congruent, 

 it is necessary and sufficient that the distance between 

 two elements a,b\?,K\o% {abi'j), where k is a constant and 

 {abij) is the cross-ratio of a, b, z, /, the last two being two 

 fixed elements on the "line" ab, the so-called absolute 

 point-pair of the line. This leads to Cayley's theory of 

 the absolute quadric, and the classification of metrical 

 geometry into the three kinds, elliptic, parabolic, and 

 hyperbolic. The theory of angles between lines or 

 planes, the theory of parallels, and the general definition 

 of perpendicularity follow in due course. In all this 

 there is no hocus-pocus whatever ; we have an analytical 

 theory, based upon precise definitions, which is quite 

 independent -of any appeal to the senses. But the 

 question is bound to arise: " What is the relation of 

 this to real geometry? What has it to do with the 

 space of which we have experience, with the practical 

 measurements which we are making every day?" To 

 answer this inquiry in anything like a satisfactory way 

 it is necessary to clear our mind of prejudices and mis- 

 conceptions which obscure the whole matter until they 

 are removed. 



First of all it must be remembered that we cannot 

 distinguish between real and imaginary space in the 

 same sense as we do, for instance, between a real 

 experience and an hallucination, or between a photo- 

 graph and a landscape composition. Space is essentially 

 an ideal conception, and strictly speaking we have no 

 experience of space at all ; we evolve, each of us probably 

 with his own degree of precision or vagueness, a scheme 

 to which we relate certain aspects of our sense-impres- 

 sions. To attempt to define real space as the space in which 

 real things exist is, of course, mere playing with words 

 and avoiding the true issue : when we say that a thing 

 " exists in space," we refer an actual (or imagined) objec- 

 tive experience to an ideal scheme, and our statement 

 has a meaning for us simply so far as the scheme is 

 clearly developed in subjecto. Again, to say that real 

 space is of three dimensions, as contrasted with the 

 ^-dimensional space of abstract analytical geometry, 

 merely means that, hitherto, a three-dimensional scheme 

 has proved sufficient for the classification of those sense- 

 impressions which admit of a spatial interpretation. It 

 is a very interesting experiment to walk along a street 

 and attend exclusively to one's visual impressions ; this 

 gives a consistent experience of a /■rw-dimensional space 

 with a time-series of continuous projective transform- 

 ations. The exhibitions of "animated photographs" 

 affiDrd a similar experience ; the conclusion seems 

 obvious that the properties of "real" space are con- 

 ditioned by the range of sensations that we refer to 

 it. Supposing that we could develop a new sense, it 

 is quite possible that we might experience a " real " 

 space of four dimensions. 



From the purely mathematical side these discussions 

 are more or less irrelevant. The definition of a posi- 

 tional manifold of n dimensions is perfectly clear and 

 intelligible ; and it is quite legitimate to assume such 

 postulates of construction as will make the corresponding 

 geometry just as much a true geometry as the elements 

 of Euclid. Of course, if n>i, we lose the help of 

 " intuition," that is, the suggestions of sense-impressions ; 

 but these suggestions are not essential, and the modern 

 NO 1504, VOL. 58] 



development of geometrical theory is, in fact, chiefly due 

 to a sceptical criticism of the crude results of merely 

 objective experience. 



Then, again, as to the metrical properties of space. 

 The analytical theory leads, as we have seen, to three 

 distinct varieties. No conceivable experiment can de- 

 cide whether "real" space is elliptic, hyperbolic, or 

 parabolic : one sufficient reason is that it is pure assump- 

 tion to suppose that we can move a ruler about without 

 altering its length. It is enough for all practical purposes 

 to know that the hypothesis of parabolic space is com- 

 paratively simple, and serves nearly enough for the 

 interpretation of physical measurements. In this con- 

 nection, special attention may be directed to Mr. 

 Whitehead's notes on pp. 499 and 451. The last is 

 particularly important, as pointing out that a space of 

 one type may be a locus in a space of one more dimen- 

 sion and of a different type : thus ordinary Euclidean 

 space of three dimensions may be regarded as a limit- 

 surface in a hyperbolic space of four dimensions. 



On p. 369 will be found a very useful bibliography of 

 treatises and memoirs dealing with the general theory of 

 metrics ; one omission that may be noted is that no 

 reference is given to Lie's large treatise on transform- 

 ation-groups, which contains a section on this subject, 

 with detailed criticism of the theories of Riemann, 

 Helmholtz and others. 



It would not be right to conclude this notice without 

 saying a word or two in appreciation of the spirit of 

 thoroughness and of independence in which Mr. White- 

 head's valuable book has been written. It possesses a 

 unity of design which is really remarkable, considering 

 the variety of its themes ; and the author's own contribu- 

 tions, not only in illustrative detail, but in additions to 

 the general theory, are well worthy of attention. AH 

 who are interested in the comparative study of algebra 

 will look forward with pleasurable anticipation to the 

 appearance of the second volume, and wish the author all 

 success in bringing his formidable task to a conclusion. 



G. B. M. 



EARLY GREEK ASTRONOMY. 

 The First Philosophers of Greece. An edition and trans- 

 lation of the remaining fragments of the pre-Sokratic 

 philosophers, together with a translation of the ipore 

 important accounts of their opinions contained in the 

 early epitomes of their works. By Arthur Fairbanks. 

 Pp. vii 4- 300. (London : Kegan Paul, Trench, and 

 Co., Ltd., 1898.) 



THE histories which we possess and to which we 

 readily turn for information concerning the early 

 science of the ancients have been prepared mainly by 

 two kinds of writers, having in view two different objects. 

 We have on the one hand, works like those of Delambre, 

 or in later times of Mr. Narrien, authors possessing a 

 comprehensive knowledge of mathematical analysis, and 

 who, writing for the benefit of physicists, are most in- 

 terested in exhibiting the scientific connection existing 

 between the older philosophers and modern science. As 

 an example of the other kind, we may refer to such works 

 as that by Sir G. C. Lewis, whose classical attainments 

 were probably in advance of his knowledge of physics, 



