May 19, 1911] 



SCIENCE 



763 



r,,; we shall say that a series is less than 

 another, when if taken away from this, it 

 gives a positive difference. This, therefore, 

 is always the same rule: t is looked upon 

 as very great with regard to any ordinary 

 real number, and s is looked upon as very 

 great with regard to any number of the 

 system T. 



The law of eommutativity not being true, 

 these now are non-pasealean numbers. 



Before going farther I recall that Ham- 

 ilton long ago introduced a system of com- 

 plex numbers where the multiplication is 

 not commutative; these are the quater- 

 nions, which the English so often use in 

 mathematical physics. But, for quater- 

 nions the assumptions of order are not 

 true; what therefore is original in Hu- 

 bert's conception is that his new numbers 

 satisfy the assumptions of order without 

 satisfying the rule of eommutativity. 



To return to geometry. Admit the as- 

 sumptions of [the first] three groups, that is 

 to say, the projective assumptions of the 

 plane and of space, the assumptions of 

 order, and Euclid's postulate; the theorem 

 of Desargues will follow from them since 

 it is a consequence of the projective as- 

 sumptions of space. 



We wish to establish our geometry with- 

 out using metric assumptions; the word 

 length therefore has now for us no mean- 

 ing ; we have no right to use the compasses ; 

 on the other hand, we may use the ruler, 

 since we admit that we may pass a straight 

 through two points, in virtue of one of the 

 projective assumptions; equally we know 

 how through a point to draw a parallel to a 

 given straight, since we admit Euclid's 

 postulate. Let us see what we can do with 

 these resources. 



We can define the homothety (perspec- 

 tive similarity) of two figures ; and through 

 it proportion. We can also define equality 

 in a certain measure. 



The two opposite sides of a parallelo- 

 gram shall be equal iy definition; thus we 

 know how to recognize whether two sects 

 are equal to one another, provided they ie 

 parallel. 



Thanks to these conventions, we now are 

 prepared to compare the lengths of two 

 sects, but with the proviso that these sects 

 ie parallel. 



The comparison as to length of two sects 

 differing in direction has no meaning, and 

 there would be needed, so to speak, a dif- 

 ferent unit of length for each direction. 

 It is unnecessary to add that the word 

 angle has no meaning. 



Sects will thus be expressed by numbers ; 

 but necessarily these will not be ordinary 

 numbers. All we can say is that if the 

 theorem of Desargues is true, as we sup- 

 pose, these numbers will belong to an ar- 

 guesian system. 



Inversely, having given any system S of 

 argTiesian numbers, we can make a geom- 

 etry such that the lengths of the sects of a 

 straight may be exactly expressed by these 

 numbers. 



The equation of the plane will be a linear 

 equation as in the ordinary analytic geom- 

 etry ; but since in the system S multiplica- 

 tion will not be commutative, in general it 

 is needful to make a distinction and to say 

 that in each of the terms of this linear 

 equation the coordinate will play the role 

 of multiplicand, and the constant coeffi- 

 cient the role of multiplier. 



Thus to each system of arguesian num- 

 bers will correspond a new geometry satis- 

 fying the projective assumptions and those 

 of order, the theorem of Desargues and 

 Euclid's postulate. What now is the geo- 

 metric meaning of the arithmetical assump- 

 tion of the third group, that is to say, of 

 the rule of eommutativity of multiplica- 

 tion? 



The geometric translation of this rule is 



