December 23, 1898.] 



SCIENCE. 



909 



standard set, e. g., III., or 1 + 1 + 1; and finally 

 the written symbol, 3. Such symbols serve to 

 represent and convey the numeric quality. 

 The word ' number ' is applied indiscriminately 

 to the quality or idea and to its symbol. 



Schubert tells us that in antiquity the Romans 

 represented the numbers from one to nine by 

 rows of strokes, as 4 is still represented on our 

 watches ; while the Aztecs used to put together 

 single circles for the numbers from one to nine- 

 teen. I have seen Japanese use columns of cir- 

 cles in the same way. Thus, also, our striking 

 clocks convey a numeric quality by a group 

 possessing it. But the number pertaining to a 

 group or artificial individual is far from being 

 the simple notion it seems. If numbers are 

 used to express exactly this definite attribute of 

 finite systems they are called cardinal numbers. 



Schubert's first sentence is: "Dinge zahlen 

 heisst, sie als gleichartig ansehen, zusammen 

 aufiassen, und ihnen einzeln andere Dinge 

 zuordnen, die man auch als gleichartig ansieht." 

 This maybe rendered : "To count things meaus 

 to consider them as alike, to take them together, 

 and to associate them singly to other things 

 which one also considers as alike." I would 

 prefer to say: "To count distinct things means 

 to make of them an artificial individual or 

 group, and then to identify its elements with 

 those of a familiar group." When the mind 

 of man made these artificial individuals they 

 were found to possess a sort of property or 

 quality which was independent of the distinctive 

 marks of the natural individuals composing 

 them, also independent of the order or sub-as- 

 sociation of these natural individuals. Whether 

 the artificial individual were made of a church, 

 a noise and a pain, or made of three peas, or 

 composed of two eyes and a nose, it had one 

 certain quality — it was a triplet. 



I see no necessity for Schubert to consider the 

 church as like the noise and the pain. Again, 

 the individuals of the familiar group vised in the 

 count need not be alike. Even the individuals 

 used by a clock in counting differ ordinally, and 

 when we follow the count of the clock we use 

 words all different. The primitive written 

 number is such a picture of a group of individ- 

 uals as represents their individual existence and 

 nothing more, e. g., III.; so, however different 



they may be, this number is independent of the 

 order in which they are associated with its ele- 

 ments. 



Schubert wastes three sentences on the so- 

 called concrete number, benannte Zahl. Three 

 quails is not a number, but is a particular bevy. 

 His Section 2, Addition, he begins thus: "If 

 one has two groups of units such that not only 

 all units of each group are alike, but that also 

 each unit of the one group is like each unit of 

 the other group," etc. All this likeness and 

 alikeness seems unnecessary. Any two groups 

 may be thought into one group. Any two 

 primitive numbers may be added. 



In Section 5, Peacock's Principle of Perma- 

 nence is given in Hankel's general form : The 

 combination of two numbers by any defined 

 operation is a number, such that the combina- 

 tion may be handled as if it gave one of the pre- 

 viouslj' defined numbers. New kinds of num- 

 bers, like all numbers, are defined by the 

 operations from which they result. Thus are 

 introduced zero and negative numbers, and 

 later the fraction. After this all is easy to the 

 end of Schubert's contribution. 



It only remains to point out, as of especial 

 importance, that from beginning to end not the 

 slightest mention is made of measurement. Not 

 a word is wasted on people who do not clearly 

 see that number is long prior to measurement. 



The second section of the Encyklopaedie is 

 ■•Kombinatorik,' by E. Netto. This is a part 

 of mathematics which never fulfilled the hopes 

 of the school which was lost in it during the 

 early part of this century. Of the most com- 

 prehensive monographs the last two are in 

 1826 and 1837. For us it has gone over into 

 determinants, and more than half of Netto's 

 article is devoted to determinants. This article 

 is particularly valuable from a bibliographic and 

 historic point of view. 



The third section is ' Irrationalzahlen und 

 KonvergenzunendlicherProzesse,'by A. Prings- 

 heim. It begins on page 47, and goes past the 

 end of the Heft. This is a modern subject, of 

 intense living interest. How entirely modern 

 it is might not be suspected by readers of such 

 sentences in Cajori's excellent history of mathe- 

 matics as those on page 70 : " The first incom- 

 mensurable ratio known seems to have been 



