ATTEMPTS AT ARITHMETISATION 227 



the rule for finding the "ultimate" or " ultimator " 

 of a" \ this ultimator is ua'^'^a^; he also gives the 

 rule for writing down ' ' the subject of every Ulti- 

 mator"; the subject of the ultimator na'^'^a^ h^ing 

 a" + c. He applies these rules when the exponents 

 are fractional. 



The ** ultimator " of the product of two variables, 

 ae, is found thus. **Put ae — bee^ and ae = caa. 

 Whence a — be, and e = ca^ and ae = \ bee + \ caa. 

 The Ultimator of which last is . . . bee^-\-caa^, and 

 consequently is equal to the Ultimator of ne. But 

 be = (i^ and ca = e. Therefore these substituted for 

 their Equals in that Ultimator give ae^-^ea^ for the 

 Ultimator of ae'' (p. 43). It will, of course, be 

 noticed that special limitations are placed upon the 

 variables a and e, when the coefficients b and e are 

 tacitly assumed to be constants. Kirkby proceeds 

 to the derivation of the ultimators of fractions and 

 logarithms. He explains the necessity of retaining 

 in the Ultimator each variant (variable) under its 

 Power. ** Without this we cou'd bave no Means 

 from the Nature of the thing itself, whereby to 

 distinguish an Ultimator from a Subject." The 

 functions of a^, e^ are more than simply to represent 

 unity ; just what they are is not very clear, although 

 to the author " it is evident then, as often as any 

 Subject consists of different Variants Ex gr. ,i', j, ^, 

 that the Expressions x^^ j®, s^,- in the Ultimator 

 have the same Difference in Power with the 

 same Variants under any other common Exponent 

 x'\ y\ ::*".... Therefore the Expression x^, f^, 5*^, 



