762 



SCIENCE 



[N. S. Vol. XXXIII. No. 855 



2. By using the projective assumptions 

 of the plane and those of space. 



Therefore, the theorem could be discov- 

 ered by a two-dimensional animal, to whom 

 a third dimension would seem as incon- 

 ceivable as to us a fourth; who conse- 

 quently would not know the projective as- 

 sumptions of space, but who would have 

 seen displaced in the plane he inhabited 

 rigid figures analogous to our solid bodies, 

 and who consequently would know the 

 metric assumptions. Equally well the the- 

 orem could be discovered by a tri-dimen- 

 sional animal who should know the pro- 

 jective assumptions of space, but who, 

 having never seen solid bodies displaced, 

 would not know the metric assumptions. 



But would it be possible to establish the 

 theorem of Desargues without using either 

 the projective assumptions of space or the 

 metric assumptions, hut only the projective 

 assumptions of the plane ? It was thought 

 not, but we were not sure. Hilbert has 

 settled the question by constructing a non- 

 arguesian geometry, which is of course a 

 plane geometry. 



Non-pascalean Geometry. — Hilbert does 

 not stop there; he introduces still a new 

 conception. To understand it, we must 

 return a moment to the domain of arith- 

 metic. We have above seen the notion of 

 number enlarged by the introduction of 

 non-arcJiimedean numbers. We need a 

 classification of these new numbers, and to 

 get it we first classify the assumptions of 

 arithmetic into four groups : 



1. The laws of associativity and of com- 

 mutativity of addition, the associative law 

 for multiplication, the two laws of distribu- 

 tivity of multiplication; or, to summarize, 

 all the rules of addition and of multiplica- 

 tion, save the law of the commutativity of 

 miiltiplication. 



2. The assumptions of order, that is to 

 say, the rules of the calculus of inequalities. 



3. The law of commutativity of multipli- 

 cation according to which we can invert the 

 order of the factors without changing the 

 product. 



4. The Archimedes assumption. 



The numbers which admit the first two 

 groups are called arguesian; they may be 

 pascalean or non-pascalean, according as 

 they satisfy or do not satisfy the assump- 

 tion of the third group; they will be 

 archimedean or non-archimedean, accord- 

 ing as they satisfy or do not the assump- 

 tion of the fourth group. We soon shall 

 see the reason for these names. 



The ordinary numbers are at once ar- 

 guesian, pascalean and archimedean. We 

 can prove the law of commutativity from 

 the assumptions of the first two groups and 

 the Archimedes assumption; so there are 

 no numbers arguesian, archimedean and 

 not pascalean. 



On the other hand, it is easy to make a 

 system of numbers arguesian, non-pas- 

 calean and non-archimedean. The ele- 

 ments of this system will be series of the 

 form 



S = l\sn -f- l\sn-l -f- . . ., 



where s is a sjonbol analogous to t, n an 

 integer positive or negative, and T^, T^, ■■■ 

 numbers of the system T. If therefore we 

 replace the coefficients Tg, T^, ■■■ by the 

 corresponding series in t, we shall have a 

 series depending at the same time upon t 

 and upon s. We add these series S accord- 

 ing to the ordinary rules, and likewise for 

 the multiplication of these series, we shall 

 admit the rules of distributivity and of 

 associativity, but we shall hold that the law 

 of commutativity is not true and that, on 

 the contrary, st = — ts. 



It remains to arrange these series in an 

 order so determined as to satisfy the as- 

 sumptions of order. For that, we give to 

 the series S the sign of the first coefficient 



