September 27, 1^94] 



NA TURE 



laid down in the Introduction. The artifice of using two 

 units, the finite or Euclid'5 unit, and the infinitely great 

 or Riemann's unit, is an essential part of the theory, and 

 is referred to in the title of the book. 



Let us now look a little more closely at the abstract 

 development of Geometry as treated by Prof. Veronese. 

 I Such notions as "point" and "line" are suggested by 

 ! simple intuitions ; abstractly considered what are they t 

 The point is simply the fundamental element of geo- 

 j metrical forms. It is an axiom that there are dif- 

 I ferent points, but all points are identical. .A. straight 

 ( line is a continuous point-system of one dimension, 

 identical in the position of its parts, and determined by 

 two of its points. Here it is to be observed that the 

 straight line is not necessarily determined by miy two of 

 its points. It is an a.xiom that any point on the line and 

 any point off the line determine only a single straight 

 line. Hence if there are two or more points on a straight 

 line by which it is not determined, any straight line 

 through one goes through all. So far the Euclidean and 

 non-Euclidean systems are not in any way discriminated. 

 The choice of a system, excluding the Hyperbolic Geo- 

 metry, is made by means of an hypothesis concerning 

 different units. Let straight lines be drawn from a point, 

 and any length in one of them chosen as a unit. On 

 each of them there will be a "range of the scale" with 

 that line as unit, and, as in the Introduction, there will be 

 points outside the range of the scale. The points within 

 the range of the scale form the finite domain about the 

 point. The points at an actually infinite distance of the 

 tir~t order form the domain of the infinitely great of the 

 iJKt order. There can thus be a number of domains of 

 infinitely great or infinitely small order of the space 

 .fijout a point. Suppose a point A taken on a straight 

 line, and a point R outside it, and let the distance between 

 them be chosen as a unit. Then if we join R to a point 

 M on the line at a finite distance from A, the lines K B, 

 \ 1! are different relatively to the unit ; if the distrnce 

 V B is infinitely great in comparison with A R, they coin- 

 cide relatively to the unit. The hypothesis which excludes 

 the Hyperbolic Geometry is that two straight lines going 

 from a point which in any domain are different relatively 

 to the unit of that domain will not in any other domain 

 coincide relatively to the unit of that domain, and it is 

 proved that, on this hypothesis, two straight lines joining 

 a point R to points at infinity in opposite directions on a 

 line A B lie in the same straight line through R. 



A point-system, defined as a straight line is defined, 

 may be closed or open. In the former case starting 

 from one point, and going through the system continu- 

 ously in one direction, the point of starting will be ulti- 

 mately arrived at ; in the latter it will never be arrived 

 at. If we assume the straight line open, and make the 

 hypothesis just now described to exclude the Hyperbolic 

 Geometry, we shall come to an absolute Euclidean 

 Geometry. If we assume the straight line closed, but its 

 entire length actually infinite in comparison with a per- 

 ceivable unit, we shall come to a .Spherical or Elliptic 

 Geometry which coincides with the Euclidean in the 

 domain of the perceivable unit about any point. This is 

 the assumption chosen by Prof. Veronese. But there is 

 still a choice open between the Spherical and the pro- 

 perly Elliptic Form. .\s mentioned above, the former 



: C . 1300, VOL. 50] 



is chosen by means of the hypothesis that a straight line 

 contains pairs of points by which it is not determined. 

 This hypothesis is adopted to avoid the kind of compli- 

 cation which occurs in the Elliptic Geometry, and which 

 may be associated with the statement that the plane of 

 the Elliptic Geometry is "unifacial " in the sense in which 

 that word is used in Geometry of Position ; but it is pointed 

 out that for the purpose of obtaining a system including 

 Euclid's as a limit, the hypothesis is a pure convention. 



We have described the foundations of Prof. \'eronese's 

 system at considerable length, because it is by these 

 that his system must be judged. For the subsequent 

 developments'it will be almost sufficient to say that they 

 are clear and orderly, and, in places, very interesting. 

 The construction of the plane by means of a pencil of 

 rays meeting a straight line, leads to the essential pro- 

 perties of the plane and of plane figures. The like 

 method by means of the " star" of rays from a point out- 

 side a plane to points on the plane, leads to the proper- 

 ties of figures in space of three dimensions. The word 

 Star {Stern) is introduced in place of the older Sheaf of 

 rays [Strahlen-Bitndel). The construction of space of 

 four dimensions is made by means of a star of rays from 

 a point outside a space of three dimensions to the points 

 within it, and so on for a space of any number of 

 dimensions. Abstractly considered there cannot be in 

 the nature of the case any restriction of Geometric 

 Forms to space of three dimensions. .\11 the forms^ 

 the straight line, plane, &c. — are treated first as Euclidean 

 and afterwards as "complete" in the sense of the 

 Spherical Geometry above described. The Euclidean 

 forms first considered are regarded as the parts of the 

 complete forms in the domain of the perceivable unit. 



The use of more than one unit precludes the applica- 

 tion to geometric magnitude of the axiom V of Archi- 

 medes ; but there is another principle which has fre- 

 quently been supposed to lie at the basis of Geometry with 

 which our author also dispenses, we refer to the Prin- 

 ciple of Superposition, or Motion without Deformation. 

 He points out that, although this principle has been very 

 extensively used as the test of equality, it yet involves in 

 its statement the notion of equality, albeit in a limited 

 form, and, as a test of equality, it is thus without mean- 

 ing in an abstract sense. By placing the notion of equality 

 of geometric magnitudes, or, as he says, identity of 

 figures, on a different footing, he is enabled to prove the 

 equality of congruent and symmetric figures, and to 

 establish the idea of motion without deformation by 

 means of continuous systems of identical figures. 



The reader will see that the purpose of the book is not 

 didactic, but the author hopes to produce a book adapted 

 for learners, founded on the principles laid down, but 

 limited to the Euclidean domain of a single unit. We 

 shall look forward with much interest to its appearance. 

 The indictment of Euclid is perhaps not yet complete, as 

 almost every advance in Geometry throws light on some 

 weakness in his logic, or defect in his method : but it is 

 not too much to say that no well-reasoned didactic 

 treatise on Elementary Geometry has yet appeared. In 

 the meantime those who have studied the subject in the 

 existing defective works will do well to clear their ideas 

 by reading at least some parts of Prof. Veronese's. 



A. E. H. L. 



