FRIZELL: FOUNDATIONS OF ARITHMETIC. 411 



primes, and so on, and fill these sets with the correspond- 

 ing sets of numbers permuted in any way we please. 



100. We have seen that every possible arrangement 

 of all the natural numbers in a series is of type lower 

 than W. Therefore the quotient places of our frame- 

 work m the order in which we propose to fill them (or 

 rather state how they shall be filled ) constitute an or- 

 dinal type of the second class. That is, the numbered 

 places in a permutation of all the natural numbers, ar- 

 ranged, however, in the order in which they are to be 

 filled, form a series obtainable by the process of ^ 97.* 

 Hence there can be no permutation of all the natural 

 numbers not obtainable by this process, since such a 

 permutation would be eo ipso different from all of 

 those hitherto obtained and well ordered in some defi- 

 nite numbered places. In other words, this would mean 

 that the series of the places as they are filled, or the 

 series of numbers with which each place is to be filled, 

 was not obtainable by the process of § 97. Therefore 

 all possible permutations of the natural numbers form 

 a series closely related to the series of type W. 



BIOGRAPHY. 



I, Arthur Bowes Frizell, member of the Protestant Episcopal Church, 

 was born in Boston, July 14, 1865. My parents were Joseph Palmer Fes- 

 senden Frizell, civil and hydraulic engineer, and Julia Anna (Bowes) Frizelh 

 My early education was chiefly at home and in a private school in Dorches- 

 ter, Mass. I graduated from the high school in St. Paul, Minn., and spent 

 three years at the Massachusetts Institute of Technology, where I after- 

 wards served as assistant instructor in mathematics, 1888-'91. I received 

 the degree of Bachelor of Arts from Harvard College in 1893 and that 

 of Master of Arts from Harvard University in 1900. I served as instructor 

 in mathematics at New York University 1895-'96, and at Harvard 1897-1906, 

 when I resigned to study abroad. After three semesters at Gottingen, I 

 returned to America, and was appointed, 1908, Professor of Mathematics 

 in Midland College, Atchison, which position I resigned, 1909, to accept an 

 instructorship in the University of Kansas. 



*And the preceding statements apply verbatim to every series of values 

 admissible in a given numbered place. 



