404 



SCIENCE. 



[N. S. Vol. XIX. No. 480. 



year, throughout the vast system of ex- 

 aminations carried on by the British gov- 

 ernment, by Oxford and by Cambridge, 

 to be accepted, no proof of a theorem in 

 geometry should infringe Euclid's sequence 

 of propositions. For two millenniums his 

 axioms remained undoubted. 



4. THE NEW IDEA. 



The break from Euclid's charmed circle 

 came not at any of the traditional centers 

 of the world's thought, but on the circum- 

 ference of civilization, at Maros-Vasarhely 

 and Temesvar, and again at Kazan on the 

 Volga, center of the old Tartar kingdom; 

 and it came as the creation of a willful, 

 wild Magyar boy of twenty-one and an 

 insuboi'.dinate young Russian, who, a poor 

 widow's son from Nijni-Novgorod, enters 

 as a charity student the new university of 

 Kazan. 



The new idea is to deny one of Euclid's 

 axioms and to replace it by its contradic- 

 tory. There results, instead of chaos, a 

 beautiful, a perfect, a marvelous new 

 geometry. 



5. HOW THE NEW DIFFERS FROM THE OLD. 



Euclid had based his geometry on cer- 

 tain axioms or postulates which had in all 

 lands and languages been systematically 

 used in treatises on geometry, so that there 

 was in all the world but one geometry. 

 The most celebrated of these axioms was 

 the so-called parallel-postulate, which, in a 

 form due to Ludlam, is simply this: 'Two 

 straight lines which cut one another can not 

 both be parallel to the same straight line.' 



Now this same Magyar, John Bolyai, and 

 this Russian, Lobachevski, made a geom- 

 etry based not on this axiom or postulate, 

 but on its direct contradiction. Wonder- 

 ful to say, this new geometry, founded on 

 the flat contradiction of what had been 

 forever accepted as axiomatic, turned out 

 to be perfectly logical, true, self-consistent 



and of marvelous beauty. In it many of 

 the good old theorems of Euclid and our 

 own college days are superseded in a sur- 

 prising way. Through any point outside 

 any given straight line can be drawn an 

 infinity of straight lines in the same plane 

 with the given line, but which nowhere 

 would meet it, however far both were pro- 

 duced. 



6. A CLUSTER OP PARADOXES. 



In Euclid, Book I., Proposition 32 is 

 that the sum of the angles in every recti- 

 lineal triangle is just exactly two right 

 angles. In this new or non-Euclidean 

 geometry, on the contrary, the sum of the 

 angles in every rectilineal triangle is less 

 than two right angles. 



In the Euclidean geometry parallels 

 never approach. In this non-Euclidean 

 geometry parallels continually approach. 



In the Euclidean geometry all points 

 equidistant from a straight line are on a 

 straight line. In this non-Euclidean geom- 

 etry all points equidistant from a straight 

 line are on a curve called the equidis- 

 tantial. 



In the Euclidean geometry the limit ap- 

 proached by a circumference as the radius 

 increases is a straight line. In the non- 

 Euclidean geometry this is a curve called 

 the oricycle. Thus the method of Kempe 's 

 book 'How to draw a straight line,' would 

 here draw not a straight line, but a curve. 



In the Euclidean geometry, if three 

 angles of a quadrilateral are right, then 

 the fourth is right, and we have a rect- 

 angle. In this non-Euclidean geometry, if 

 three angles of a quadrilateral are right, 

 then the fourth is acute, and we never can 

 have any rectangle. 



In the Euclidean geometry two perpen- 

 diculars to a line remain equidistant. In 

 this non-Euclidean geometry two perpen- 

 diculars to a line spread away from each 

 other as they go out; their points two 



