ON THE ALGEBRAIC COUPLE. 191 
angle is the same function of that angle which has at once every possible 
value, or which is the same thing, every possible value within one complete 
period. We should therefore be inclined to say, that, speaking geometrically, 
cos OC or sin OC has no particular individual value, but that it possesses at 
once every value between —1 and +1, both inclusive. We shall not, I ap- 
prehend, arrive by any process of abstract reasoning at an analytical equiva- 
lent for these and similar expressions. We may, however, interpret them by 
reference to particular problems, the solution of which involves principles not 
purely algebraical; and the question will then arise, how far it is safe, having 
regard to the nature of these principles, to consider the interpretation as uni- 
yersal. 
_ Suppose a point placed upon a sphere whose equator is horizontal, and such 
point descending by the force of gravity, and that a person is entitled to re- 
ceive, (or obliged to pay, as the case may be,) such a fraction of a pound as is 
denoted by the cosine of the longitude of the place at which the point tra- 
verses the plane of the equator. For every position of the point up to, but 
exclusive of the pole, the value of this person’s interest is simply the fraction 
cos 6 of a pound, @ being the longitude of the original position. If the point 
be actually on the pole, the problem considered as a physical one fails; and 
the answer assumes the form of a trigonometrical function of an angle which 
has no one value in particular more than another. If we bring this limiting 
ease within the scope of the problem by proposing it in this form :—a material 
point rolls from the pole of a sphere to the equator down a meridian, de- 
termined by some impulse extraneous to the problem; what is the value of 
_ the interest of a person who is to receive, (or pay, if the result be negative,) 
the fraction of a pound above indicated ?—the algebraical answer is the same 
as before, viz. the cosine of an angle which has any or every value, no one in 
particular more than another; but the problem is not now purely algebraic, 
but belongs to the Theory of Probabilities, which tells us that the answer is 
(cos n-+-cos (2 2) +... + cos (m n)) +m, where z diminishes without limit and 
mn=2 7, that is, f, *" cos Od 0; and this, which is the mean of all the possi- 
ble values of cos 6, is therefore, in this problem at least, the interpretation of 
the cosine of the indeterminate angle which denotes the longitude of the pole. 
Perhaps this doctrine of the interpretation of indeterminate values may be 
stated as follows :—If a problem when treated analytically give an indetermi- 
nate result of which all the individual values are calculable, and if the same 
problem when treated by the principles of the science to which it belongs 
give a specific result, we are warranted in saying that the latter is guoad sub- 
jectam materiam the interpretation of the former, and may be treated as its 
analytical equivalent ; this doctrine being subject to the implied condition 
that the science to which the problem belongs, and which gives us the spe- 
cific result, is a science whose fundamental principles do not rest on induc- 
tion in any other sense than the axioms of algebra do. 
A and B engage in a game in which they will win alternately, A winning 
first: what is the present value of A’s ultimate winnings when they rise ? 
—/{— Cy 
The only answer given by algebra is kad eh where 2 is the number of 
“games about to be played; but the real answer is evidently 1, so that 3 ought 
(=): 
to be the arithmetical interpretation of ead or the equivalent of 
(—1) is 0 if x be indeterminate and integer ; and that of (—1)* is the same, 
provided the infinity used be the limit of the ordinal series 1, 2, 3, 4 
and not the limit of any partial series, as 1, 3, 5, 7.... or 2,4, 6, 8 
eebesey 
