OCTOBEE 5, 1906.] 



SCIENCE. 



441 



ical. But this is a poor proof of the impossi- 

 bility of establishing ^ the Euclidian postulate,] 

 since the non-Euclidian systems have to deal with 

 a difi'erent class of phenomena; such are the 

 metrical relations upon the sphere and the pseudo- 

 sphere in two-dimensional point-space, and those 

 holding in three-dimensional curved manifolds 

 contained in w-dimensional space, or in space 

 whose element is changed from that of a point in 

 the ordinary Euclidian sense to some other geo- 

 metrical entity depending on n coordinates, like 

 Pliicker's four-dimensional line-space. I should 

 refer you for the elucidation of this point of view 

 to pp. 27-32 of the dissertation, especially to p. 

 29 and sequel, where a quotation from Bianchi 

 is discussed and refuted. 



The difference between my position and yours 

 is, it seems, as follows: while you maintain that 

 external space is either Euclidian or non- 

 Euclidian, and there is no possibility of ever 

 finding out which, for the Euclidian postulate 

 can neither be proved nor disproved, I assert that 

 external space is both Euclidian and non- 

 Euclidian, according to the point of view. [If 

 space is regarded as a point-manifold, it is 

 Euclidian, and the postulate can be proved, as 

 soon as we are allowed to look for its establish- 

 ment in three-dimensional geometry,] of which 

 two-dimensional geometry ig only a part. If 

 space, however, is regarded as a line-manifold, 

 say, then certain two- and three-dimensional mani- 

 folds contained in it are non-Euclidian. So, for 

 instance, all lines passing through a point repre- 

 sent [the two-dimensional elliptic geometry dis- 

 cussed by Klein, Lindemann and Killing], which, 

 [according to my opinion, is an absurdity for a 

 point-space in the ordinary sense of the term]. 

 As to [Poincare], he seems to stand on a very 

 similar basis — namely, in that he does not op- 

 pose the non-Euclidian to the Euclidian geom- 

 etry and [says that all depends upon convention] 

 as to what we understand by distance, straight 

 line, angle, etc. [But still he deduces from this 

 the perfectly gratuitous conclusion that therefore 

 the parallel-postulate can not be proved.] It is 

 gratuitous, according to my opinion, because, as 

 the simultaneous existence of both the Euclidian 

 and the non-Euclidian groups of motion have 

 been proved beyond a_ shadow of doubt, they must 

 evidently refer to different classes of phenomena, 

 and hence there must exist a Euclidian space and 

 a non-Euclidian space. And as the actual space 

 is only one, all must depend upon the point of 

 view ( the entity taken as the space element ) . 

 Therefore, for point-space the postulate may be 



a necessity, without involving its necessity for 

 other three-dimensional manifolds, like certain 

 line-complexes, for instance, — just as plane geom- 

 etry, even if it were admittedly Euclidian, would 

 not have to hold for the geometry of the sphere 

 or the pseudosphere. 



You will observe that the groups of motion 

 in Lie's treatment are deduced from the assump- 

 tion of an analytical point, that is some entity 

 depending upon a certain number of coordinates 

 a?!, x^, • • ■ Xn, and, evidently, the entity in this 

 case is indeterminate. You may call it point, but 

 it may actually correspond to something quite 

 different from what we understand by this name 

 in elementary geometry. 



I trust that, according to the maxim that 

 curiosity is the mother of all knowledge, the 

 perusal of my treatise, in pursuance of the grati- 

 fication of this laudable feeling, may change your 

 attitude upon this question, and will convince you 

 that, instead of the different systems of geometry 

 warring with each other, they are actually in 

 peace, — ^the non-Euclidian systems, however, still 

 needing interpretation in many particulars — an 

 interpretation realizable in our space, in the 

 space in which all of us live and think and work 

 and strive for perfection. 



I. E. Eabinovitch. 



SPECIAL ARTICLES. 



INHERITANCE OF COLOR COAT IN SWINE. 



Mr. Q. I. Simpson, tlie well-known swine 

 breeder of Palmer, 111., is conducting several 

 series of crosses between different breeds of 

 swine, the breeds thus far used being Tam- 

 worth (red), Yorkshire (white), Poland China 

 (black with white points), the wild boar of 

 Europe and Duroc- Jersey (red). 



He bred a wild boar to a Tamworth sow, 

 securing a large litter all much resembling 

 the wild boar, having his color, snout, eyes, 

 ears, length and size of legs, tail, shape of 

 body, size, wildness and characteristic move- 

 ments, Erom two of these hybrid pigs and a 

 Tamworth boar he has secured three litters, 

 each containing four pigs. What the usual 

 litter of wild pigs is I do not know, but the 

 Tamworth litter is usually eight or more pigs. 

 The body color of these three litters is as 

 follows : 



