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 preconceived 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 Bolyai, 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 accessible 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 curvature 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 negative 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
