376 Mr. W. H. Walenn on Unitation. 



for November 1868^ p. 346, for December (Supplement) 1875, 

 pp. 522, 523, 524, and for November 1876, pp. 345-350 inclu- 

 sive. In other words, the remainder to a division is dependent 

 for its value upon the quotient, and is a function of an expres- 

 sion containing the quotient ; whereas a unitate is quite inde- 

 pendent of the quotient, and is a function of an expression that 

 does not contain the quotient. The expression of which it is 

 the function contains only the number to be unitated and the 

 base of the unitation ; the way in which the base affects the 

 unitate is as the determinator of the extent of a series of in- 

 tegral values, commencing with unity and finishing with the 

 base itself. The unitate may therefore be said to express the 

 place of its corresponding number in a recurring series whose 

 period is the base of the system of unitation. 



25. The numeration of unitation is therefore limited by the 

 base, and, with certain exceptions, always consists of integer 

 numbers ranging between unity and the base. The exceptions 

 arise from an extension of the meaning of the word unitate, or 

 rather fromincludinor in the list of unitates numbers not oripi- 

 lially comprehended in the definition, but which, for certain pur- 

 poses, fulfil their conditions, — just as, by analogy, the decimal 



1*414213 is properly made to do duty for V2 ; whereas 



by the construction of numbers, which makes all numbers 

 functions of unity, v/2 is not a function of unity, but is in- 

 commensurable therewith. The exceptions occur in the case 

 of fractional unitates, as shown in the Philosophical Magazine 

 for July 1873, p. 38 et seq., and in other cases. For instance, 



strictly speaking, there is no unitate (to the base 9) of k, 



o 



nor any number whose unitate is ^. Taking the unitate of ^ 



and of U^M Q ) ^s equal to ^ has, however, an advantage 



the decimal equivalent of a surd quantity, inasmuch as it com- 

 pletely fulfils the conditions of the principles of unitation (see 

 Phil. Mag. July 1873, p. 40) ; whereas the interminable 

 decimal 1-414213.... can only approximate to the result, 

 although the degree of approximation is practically unlimited, 

 or limited only by the labour of calculation. 



26. If the application of the theory of functions to unitation 

 had not yielded such definite and such original results, and if 

 the interpretation of symbols had not been brought to bear 

 upon the whole s ibject with especial rigidity and care, unita- 

 tion would have been merely a name for something that existed 

 already, namely the obtaining remainders to divisors ; but the 



over 



