NOVEMBEB 8, 1901.] 



SCIENCE. 



713 



" To which is added an appendix contain- 

 ing notices of methods at different times 

 proposed for getting oyer the difficulty in 

 the ' Twelfth Axiom of Euclid.' " Svo, pp. 

 X + 148. This dissects most brilliantly 

 twenty-one methods of getting rid of 

 Euclid's postulate; so brilliantly that it 

 deserves to be reprinted and could scarcely 

 be improved upon. Then, nothing daunted 

 by the failure of every one else of whom he 

 has ever heard, the brave Thompson adds 

 one of his own, which perhaps he also af- 

 terward impaled upon the point of his keen 

 dissecting scalpel, for he lived long and 

 prospered. In 1865 De Morgan, whose un- 

 known letters to Sylvester I had the pleas- 

 ure of publishing in the Monist, writes : 



" With your note came an acknowledge- 

 ment from General Perronet Thompson, B. A. 

 of 1802, and Fellow of Queen's before he was 

 an ensign. And he works at acoustics as 

 hard as ever he did at the Corn Laws." 



But even in 1833, had he but known it, 

 the question of two thousand years, as to 

 whether Euclid's Parallel-Axiom could be 

 deduced, had been settled at last by the 

 creation and indeed publication, by Bolyai, 

 and also by Lobachevski, of a geometry in 

 which it is flatly contradicted. 



The newlj' created methods, which thus 

 settled this old, old question, give entirely 

 new views concerning the investigation of 

 axioms in general ; and this diamond mine 

 has been masterfully preempted by Hil- 

 bert, of Gottingen. His wonderful ' Grund- 

 lagen der Geometric ' is ablaze with gems 

 from this non-Euclidean mine. 



After Bolyai and Lobachevski, Hilbert's 

 closest forerunner is Friedrich Schur, of 

 Karlsruhe. One of the most fundamental 

 advances of this decade is the strict rigorous 

 reduction of the comparison of areas to the 

 comparison of sects. 



This was first given on January 23, 1892, 

 by Schur before the Dorpater Natur- 

 forscher-Gesellschaft. 



The account printed in Russia in the 

 society's Proceedings, a Referat given by 

 Schur, is of course almost inaccessible, nor 

 is this inaccessibility much lessened for us 

 by the fact that it has been translated into 

 Italian (Per. di Mat., VIIL, p. 153). 



The essence of the matter is the proof 

 that, a certain sect being taken as the 

 measure of the area of a triangle, the sum 

 of these sects is the same for an}' set of 

 triangles into which a given pol.ygon can 

 be cut, and so gives a sect which may be 

 taken as the measure of the area of the 

 polygon. The Referat begins as follows : 



" On the surface content of plane figures 

 with straight boundaries, by Friedrich 

 Schur. 



'' So simple a problem as the measuring 

 of plane figures with straight boundaries 

 as it seems from the literature to me ac- 

 cessible, has not yet been set forth with the 

 rigor and purity of method herein possible. 



*' Not to mention the introduction of 

 endless processes, still general magnitude- 

 axioms are used unjustifiably, which are 

 only then immediately clear when these 

 magnitudes are straight sects, their com- 

 parison therefore capable of being made by 

 superposition. 



" Such a general magnitude-theorem, 

 which is used in all text-books of elemen- 

 tar}' mathematics known to me in proving 

 the theorem of the equal area of two par- 

 allelograms with common base and equal 

 altitude, is, e. g., this, that the subtraction 

 of equal magnitudes from equal magni- 

 tudes gives again equal magnitudes. 



" If the sides of the two parallelograms 

 lying opposite the common base have a 

 piece or at least a point in common, then 

 the two parallelograms can at once be cut 

 into parts such that each part of the one 

 parallelogram corresponds to a part con- 

 gruent to it of the other parallelogram. 



" On the contrary, if those two sides have 

 no point in common, then it has been be- 



