328 Professor Hamitron on Conjugate Functions, 
manner, if we admit in arithmetic the idea of the cardinal number none, as one of the 
possible answers to the fundamental question How many, the rules of the arith- 
metical addition of this number to others, and of others to it, and the properties 
of the arithmetical sums thus composed, agree with the rules and properties 
of such combinations as 0 +p, &+v+0, explained by the equations (109.), 
when the whole numbers, p», v, £ are positive; we shall, therefore, not clash 
in our enlarged phraseology with the language of elementary arithmetic, respecting 
the addition of numbers regarded as answers to the question How many, if we now 
establish, as a definition, in the more extensive Science of Pure Time, that any com- 
bination of whole numbers up v &, of the form v + p, or E+v+ p, interpreted so as to 
satisfy the equations (109.), is the swm of those whole numbers, and is composed by 
adding them together, whether they be positive, or contra-positive, or null, But as a 
mark that these words swm and adding are used in AuGEpra (as the general Science 
of Pure Time), in a more extensive sense than that in which -drithmetic (as the 
science of counting) employs them, we may, more fully, call v + u the algebraic sum 
of the whole numbers » and v, and say that it is formed by the operation of algebrai- 
cally adding them together, v to p. 
In general, we may extend the arithmetical names of sum and addition to every 
algebraical combination of the class marked by the sign +, and may give to that 
combining sign the arithmetical name of Plus ; although in Algebra the idea of 
more, (originally implied by plus,) is only occasionally and accidentally myolved in 
the conception of such combinations. or example, the written symbol b +a, by 
which we haye already denoted the compound step formed by compounding the step b 
as a successive step with the step a, may be expressed in words by the phrase 
‘*a plus b,”’ (such written algebraic expressions as these being read from right to left, ) 
or ‘the algebraic sum of the steps a and b ;”? and this algebraic sum or compound 
step b + a may be said to be formed by “algebraically adding » to a :” although this 
compound step is only occasionally and accidentally greater in length than its com- 
ponents, being necessarily shorter than one of them, when they are both effective 
steps with directions opposite to each other. Even the application of a step a to a 
moment A, so as to generate another moment a + A, may not improperly be called 
(by the same analogy of language) the algebraic addition of the step to the moment, 
and the moment generated thereby may be called their algebraic sum, or ‘the original 
moment plus the step ;” though in this sort of combination the moment and the step 
to be combined are not even homogeneous with each other. 
With respect to the process of calculation of an algebraic sum of whole numbers, 
the following rules are evident consequences of what has been already shown respect- 
