November 1, 1912] 



SCIENCE 



597 



Never believe that Gauss ever doubted the actual 

 truth of the parallel assumption for our space. 



Yet just now, 1912, this space of ours is 

 being proved non-euclidean by the principle 

 of relativity. Says Vladimir Varicak in a 

 wonderful lecture, " TJeber die nichteuklidische 

 Interpretation der Eelativtheorie," ^ 



I postulated that the phenomena happened in a 

 Lobachevski space, and reached by very simple 

 geometric deduction the formulas of the relativity 

 theory. Assuming noneuclidean terminology, the 

 formulas of the relativity theory become not only 

 essentially simplified, but capable of a geometric 

 interpretation wholly analogous to the interpreta- 

 tion of the classic theory in the euclidean geometry. 

 And this analogy often goes so far, that the very 

 wording of the theorems of the classic theory may 

 be left unchanged. 



To see that this will prove our space 

 Bolyaian, we have only to remember Poincare's 

 dictum : 



These two propositions, the earth turns round, 

 and, it is more convenient to suppose that the earth 

 turns round, have one and the same meaning.* 



The first man to so bring forth the non- 

 euclidean geometry that it was not stillborn, 

 but lived and grew, was the Frenchman 

 Hoiiel, by his translations of Lobachevski in 

 1866 and John Bolyai in 1867. Thirty years 

 later, in my translator's preface, I said: 



No part of Lobachevski 's largest work, ' ' New 

 Elements, ' ' has ever before been published in any 

 language but the original Russian. 



I gave an account of it in 1893 at the Mathe- 

 matical Congress of the World's Columbian 

 Exposition, and promised then the publication 

 of my translation.* This promise was delayed 

 for a personal visit to Kazan, the home of 

 Lobachevski, and Maros-Vasarhely, the home 

 of Bolyai. Only through his little book " Geo- 

 metrical Researches," ^ have Lobachevski's 

 ideas been heretofore accessible to the world 

 in general. 



But it is preeminently in his ' ' New Elements ' ' 

 'Jahresher. D. Math. Ver., 21, 103-127. 



* ' ' The Value of Science, ' ' Halsted 's translation, 

 p. 140. 



* See ' ' Mathematical Papers of Chicago Con- 

 gress, " pp. 92-9.5. 



= Hoiiel, 1866; Halsted, 1891. 



that the great Bussian allows free expression to 

 his profound philosophic insight, which on the 

 one hand shatters forever Kant's doctrine of our 

 absolute a priori knowledge of all fundamental 

 spatial properties, while on the other hand em- 

 phasizing the essential relativity of space. 



The realities which with the aid of the 

 euclidean space form we understand under 

 motion and position, may, with the coming of 

 more accurate experience, refuse to fit in that 

 form. Our mathematical reason may decide 

 that they would be fitted better by a non- 

 euclidean space form. Space is presupposed 

 in all human notions of motion or position. 

 We may drop out such specifications from our 

 space form as render it specifically euclidean. 

 Euclidean space is a creation of that part of 

 mind which has worked and works yet uncon- 

 sciously. 



It is not the shape of the straight lines which 

 makes the angle-sum of a rectilineal triangle two 

 right angles. 



With straight lines of precisely such shape but 

 in a non-euclidean space, this sum may be greater 

 or less. In non-euclidean spaces, if one edge of a 

 flat ruler is a straight line the other edge is a 

 curve, if the ruler be everywhere equally broad. 

 In any sense in which it can be properly said that 

 we live in space, it is probable that we really live 

 in such a space. 



And now fifteen years later comes the rela- 

 tivity theory to prove all this, and to make 

 non-euclidean geometry a powerful machine 

 for advance in physics. 



George Bruce Halsted 

 Gkeeley, Colo. 



Allen's Commercial Organic Analysis. Vol- 

 ume V. Tannins, Dyes and Coloring Mat- 

 ters, Inks. Edited by W. A. Davis and 

 Samuel S. Sadtler. P. Blakiston's Son 

 and Co. Philadelphia, 1911. Price $5.00. 

 This volume contains the following chap- 

 ters : Tannins by W. P. Dreaper. Analysis of 

 Leather by W. P. Dreaper. Dyes and Color- 

 ing Matters by W. P. Dreaper and E. Feil- 

 mann. Dyestuffs of Groups 6 to 12 by J. T. 

 Hewitt. Coloring Matters of Natural Origin 

 by W. M. Gardner. Analysis of Coloring 

 Materials by W. P. Dreaper and E. Feihnann. 



