GEOMETRY: ANCIENT AND MODERN. 265 



trovertible. The denial of the parallel-postulate leaves Lobatchewsky 

 to face the fact that under the conditions given in the postulate the 

 two lines, if continually produced, may never meet on that side of the 

 transversal on which the sum of the interior angles is less than two 

 right angles. In other words, through a given point we may draw in 

 a plane any number of distinct lines which will never meet a given line 

 in the same plane. A result of this is that the sum of the angles of a 

 triangle is variable (depending on the size of the triangle), but is always 

 less than two right angles. Notwithstanding the shock to our precon- 

 ceived notions which such a statement gives, the geometry of 

 Lobatchewsky is thoroughly logical and consistent. What, then, does 

 it mean? Simply this: We must seek the true explanation of the 

 parallel-postulate in the characteristics of the space with which we are 

 dealing. The Euclidean geometry remains just as true as it ever was, 

 but it is seen to be limited to a particular kind of space, space of zero- 

 curvature the mathematicians call it; that is, for two dimensions, space 

 which conforms to our common notion of a plane. Lobatchewsky's 

 geometry, on the other hand, is the geometry of a surface of uniform 

 negative curvature, while ordinary spherical geometry is geometry of 

 a surface of uniform positive curvature. The Lobatchewskian geometry 

 is sometimes spoken of as geometry on the pseudo-sphere. 



The 'absolute geometry' of the Bolyais (Wolfgang Bolyai de Bolya, 

 1775-1856, and his son, Johann Bolyai, 1802-1860) is similar to that of 

 Lobatchewsky. 'The Science Absolute of Space,' by the younger Bol- 

 yai, published as an appendix to the first volume of his father's work, 

 has immortalized his name. 



The work of Lobatchewsky and the Bolyais has been rendered ac- 

 cessible to English readers by the translations and contributions of 

 Prof. George Bruce Halsted, of the University of Texas. 



If we proceed beyond the domain of two-dimensional geometry we 

 merge the ideas of non-Euclidean and hyper-space. The ordinary 

 triply-extended space of our experience is purely Euclidean; and if we 

 approach the conception of curvature in such a space it must be curva- 

 ture in a fourth dimension, and here the mind refuses to follow, 

 although by pure reasoning we can show what must take place in such 

 a space. 



H. Grassman, Blemann and Beltrami have written profoundly on 

 these questions, and it is to the last that is due the discovery that the 

 theorems of the non-Euclidean or Lobatchewskian geometry find their 

 realization in a space of constant negative curvature. 



We naturally ask the question: Is there any reason to suppose that 

 the space which we inhabit is other than Euclidean? To this a nega- 

 tive reply must be returned. We may have suspicions, but we have no 

 evidence. If we could discover a triangle the sum of whose angles by 



