March 27, 1903.] 



SCIENCE. 



499 



historian, Moritz Cantor, expressed the opin- 

 ion that the use of zero was probably due 

 to the Babylonians 1700 b.c. However, 

 it has not been definitely established that 

 zero was in use any earlier than 400 a.d. 

 About this time it was used in India, and 

 several centuries later the Arabs began to 

 employ it. Through the Arabs its use be- 

 came known to Europeans during the 

 twelfth century. It was not generally 

 adopted in Europe until several centuries 

 later, notwithstanding its great advantages. 

 For a considerable time there were two 

 parties among the European educators- 

 one party, known as the algorists, favored 

 the adoption of the Hindu system of nota- 

 tion (falsely called Arabic) with its posi- 

 tion values, while the other, known as the 

 abacists, favored the Roman notation with- 

 out zero or position value. 



The general adoption of the Plindu sys- 

 tem was greatly facilitated by the facts that 

 it was explained in most of the calendars 

 for more than a century beginning with 

 1300, and that the medieval universities 

 frequently offered courses devoted to the 

 use of this notation. With the opening of 

 the medieval universities we approach some 

 of the fundamental discoveries in more 

 modern mathematics. As we considered 

 these on a similar occasion,* we shall 

 merely add a few thoughts on the concept 

 of dimensions which are due to Pliieker. 



The idea of more than three dimen- 

 sions can be partially explained in a very 

 simple manner. If the total number of 

 points on a straight line is denoted by oo 

 (the symbol for infinity), it is clear that 

 there are oo - points in a plane, since 

 through each point of the given line we 

 may draw a line at right angles to this line. 

 Each of these oo' points of the plane may 

 be taken as the center of an infinite num- 

 T)er of circles, and all the circles which have 

 one point as center are distinct from those 



* Science, Vol. XL. (1900), p. 528. 



which have any other point as center. 

 Hence there are oo^ circles in a plane, 

 while there are only oo^ points in it. 



We arrive at the same result by observ- 

 ing that an infinite number of lines may 

 be drawn through each point of a plane 

 and that each of these lines is tangent to 

 an infinite number of circles going 

 through this point. Hence ooj circles pass 

 through each point of a plane and lie en- 

 tirely in the plane. As the number of 

 points on a circle is infinite, the number of 

 circles is obtained by multiplying the num- 

 ber of points by oo . Hence we say that the 

 plane is two-dimensional when the point 

 is considered as the element, but it is three- 

 dimensional if the circle is considered as 

 element. If the ellipse were taken as ele- 

 ment it could be readily shown that the 

 plane would be five-dimensional. 



Similarly space is three-dimensional if 

 the point is taken as element but it is four- 

 dimensional if the sphere is taken as ele- 

 ment. Since there are co" pairs of points 

 in space and oo^ pairs of points on a line 

 there are oo* lines in space, that is there 

 is a 1, 1 correspondence between the lines 

 and spheres of space. This is frequently 

 expressed by saying there are just as many 

 spheres in our space as there are lines, 

 while the number of each of these is in- 

 finitely larger than the number of points. 

 From this standpoint there is no limit to 

 the number of dimensions of ordinary 

 space. 



G. A. MhjLee. 



SCIENTIFIC BOOKS. 



The Yuccew. By William Trelease. From 

 the Thirteenth Annual Report of the Mis- 

 souri Botanical Garden. . Issued July 30, 

 1902. St. Louis, Mo. Published by the 

 Board of Trustees. 1902. 8vo. Pp. 107. 

 The Spanish bayonets are shrubby or tree- 

 like plants, principally of the genus Yucca, 

 and represented in gardens by short-stemmed 



