and on Algebra as the Science of Pure Time. 333 
which new remainder © » + has still a quotity less than that of », but lies between 
0 and 6x, instead of lying (like ») between 0 and », in the general series of whole 
numbers (103.), and is therefore contra-positive if « be positive, or positive if « be 
contra-positive. With respect to the actual process of calculation, for discovering 
whether a proposed algebraical division (or measuring), of one whole number by 
another, conducts to an accurate integer quotient, or only to two approximate integer 
quotients, a next preceding and a next succeeding, with positive and contra-positive 
remainders ; and for actually finding the names of these several quotients and re- 
mainders, or their several special places in the general series of whole numbers : this 
algebraical process differs only by some slight and obvious modifications (on which it 
is unnecessary here to dwell,) from the elementary arithmetical operation of di- 
viding one quotity by another ; that is, the operation of determining what multiple 
the one is of the other, or between what two successive multiples it is contaimed. 
Thus, having decomposed by arithmetical division the quotity 8 into the arithmetical 
sum of 1 x 5 and 3, and having found that it falls short by 2 of the arithmetical pro- 
duct 2 x 5, we may easily infer from hence that the algebraic whole number contra- 
positive eight can be only approximately measured (in whole numbers), as a mensur- 
and, by the measurer positive five ; 
? 
the next succeeding integer quotient or measure 
being contra-positive one, with contra-positive three for remainder, and the next pre- 
ceding integer quotient or measure being contra-positive two, with positive two as the 
remainder. It is easy also to see that this algebraic measuring of one whole number 
by another, corresponds to the accurate or approximate measuring of one step by 
another. And in like manner may all other arithmetical operations and reasonings 
upon quotities be generalised in Algebra, by the consideration of multiple steps, and 
of their connected positive and contra-positive and null whole numbers. 
On the Sub-multiples and Fractions of any given Step in the Progression of Time ; 
on the Algebraic Addition, Subtraction, Multiplication, and Division, of Re- 
' ciprocal and Fractional Numbers, positive and contra-positive ; and on the 
impossible or indeterminate act of sub-multipling or dividing by zero. 
16. We have seen that from the thought of any one step a, as a base or unit- 
step, we can pass to the thought of a series or system of multiples of that base, 
namely, the series (98.) or (99.) or (104.), having each a certain relation of its own 
