November 8, 1901.] 



SCIENCE. 



715 



Island, has followed this up by actually- 

 constructing in her doctor's dissertation 

 at Gottingen (1900), under Hilbert, a sect- 

 calculus independent of the parallel-axiom. 



This is a beautiful piece of non-Euclid- 

 ean geometry, and is, so far as I know, the 

 first feminine contribution to our fascinat- 

 ing subject. 



In 1899 appeared Hilberb's ' Grundlagen 

 der Geometric,' in which the remarkable 

 contributions of Schur are all retouched by 

 a master hand. 



In Schur' s proof of the Pascal theorem 

 the space axioms are used. Hilbert re- 

 places them by the parallel -axiom, thus 

 proving Pascal as a theorem of plane Eu- 

 clidean geometry. 



Schur makes a sect-calculus, and shows 

 that the theory of proportion can be founded 

 without the introduction of the difficult idea 

 of the irrational number. He indicates that 

 this could be done without the Archimedes 

 axiom. 



Hilbert actually does it. 



Schur proves for the first time the funda- 

 mental theorem for a rigorous treatment of 

 area. 



Hilbert simplifies this proof, and proceeds 

 to treat this whole subject without the 

 Archimedes axiom, making here the new 

 distinction between flachengleich and in- 

 haltsgleich. 



Two polygons are said to have equal sur- 

 face when they can be resolved into a finite 

 number of triangles congruent in pairs. 



Two polygons are said to have equal con- 

 tent if it is possible to add to them polygons 

 of equal surface, so that the two new com- 

 pound polygons have equal surface. 



Thus Euclid only tried to treat equal con- 

 tent, and Hilbert is here a return to the 

 great Greek. 



The intense interest in all these unex- 

 pected developments is voiced in a hand- 

 some volume : ' Questioni riguardanti la 

 geometria elementare ' (Bologna, 1900, 8vo, 



pp. vii+532), edited by Federigo En- 

 riques, who has been chosen to contribute 

 the part on the foundations of geometry to 

 the great German Encyclopsedia of the 

 Mathematical Sciences, and who contrib- 

 utes the first article (28 pages) to this 

 Italian work. It is entitled ' On the Scien- 

 tific and Didactic Importance of the Ques- 

 tions which Relate to the Principles of 

 Geometry.' 



The whole book may be properly de- 

 scribed as an outcome of the non-Euclidean 

 geometry, but more specifically, the longest 

 of the fourteen articles which make it up 

 is by Bonola : ' On the Theory of Parallels 

 and on the non-Euclidean Geometries ' (80 

 pages, 26 figures). 



The first fifty of his eighty pages are de- 

 voted to an historico-critical exposition ; 

 the last thirty to general theory, hyperbolic 

 geometry, elliptic geometry. Though the 

 article was published only last year, it is in 

 certain respects antiquated. The proofs 

 freely use the Archimedes postulate, with- 

 out saying anything more about it than I 

 did in 1885, that is, nothing at all. His 

 § 7 is headed ' Postulates Equivalent to 

 the Postulate of Euclid,' and gives those 

 adopted by Proclos, Wallis, Bolyai Farkas, 

 Carnot, Legendre, Laplace, Gauss. But 

 now we know that all these men failed in 

 attempting to rival the choice of Euclid. 

 Their axioms are not the equivalent of his 

 immortal postulate. 



In this section the name Legendre is mis- 

 spelled, and in § 5 Bonola says, " The at- 

 tempts of Legendre for the demonstration 

 of the Euclidean hypothesis, published in 

 the various editions of the ' Elements ' of 

 Euclid, which appear under his name," 

 etc. 



Of course Legendre never published any 

 edition of Euclid. lb was on the contrary 

 Legendre's geometry which cursed the sub- 

 ject with that definition, " A straight line 

 is the shortest distance between two points," 



