38 Mr, W. H. Walenn on Negative and Fractional Unitates. 



tions, such as subtraction and division, when regarded as opera- 

 tions upon unitates, involve mathematical investigation and a 

 rigid comparison of results. These results used for comparison, 

 may either consist of the separate determination of unitates by 

 totally distinct processes, or of individual examples obtained by 

 the application of unitation to checking calculations that are 

 known to be correct. 



Applying the principles and modes of thought sketched out 

 above to the examination of the meaning of a negative unitate, 

 the following results are arrived at. It is important to bear in 

 mind certain properties of unitates. 1st. By definition, the 

 imitate of a given number to a given base, say 9, is the remainder 

 obtained by dividing the given number by the base, 9 ; it must 

 therefore always be a plus or positive quantity, equal to or less 

 than 9. 2nd. Nought (or 0) never occurs in unitation; when 

 there is no remainder the unitate of the number is 9. The 

 three cases into which subtraction of unitates resolves itself are :— 

 (I.) When the unitate of the subtrahend, or number to be sub- 

 tracted, is less than the unitate of the minuend or quantity to 

 be diminished. In this case the subtraction of the former uni- 

 tate from the latter leads to a plus or positive unitate^ which is 

 the unitate of the remainder. (II.) When the unitate of the sub- 

 trahend is equal to that of the minuend. The subtraction leads to 

 Oasa unitate; this may at once be written 9. (III.) When the 

 unitate of the subtrahend is greater than that of the minuend. 

 Regarded algebraically, this subtraction leads to a negative uni- 

 tate, and, to obtain its real value, it must be subtracted from 9. 

 Thus when, in unitation, the expression 6 — 8 occurs, it may be 

 written — 2, but should be at once reduced to a positive quantity 

 by adding 9 to it, thus 9—2 = 7, 7 is the unitate required; if it 

 be desired to give the —2a name, it may be called the co-uni- 

 tate of 7. For arithmeticians only, it may in practice be the 

 best plan to add 9 to the unitate of the minuend in the last two 

 cases before performing the subtraction of the unitate of the sub- 

 trahend therefrom. 



The performance of the operation of division upon unitates is 

 derived from the performance of the same operation upon ordi- 

 nary numbers ; but, unlike that operation upon ordinary num- 

 bers of which it is the analogue, it always has a remainder, or 

 may be presumed to have one. This is a consequence of no 



25 



unitate being equal to ; for instance, in — - = 5, the unitate 



o 



25 7 9 



of — = - ; the unitate of the quotient 5 may be written 5 + -, 



which is (for the purpose of checking the division of 25 by 5) 



