91. _Correspondence of Series of Numbers._--In SS 33-42 we have dealt
with the parallelism of the original number-series with a series
consisting of the corresponding multiples of some unit, whether a number
or a numerical quantity; and the relations arising out of
multiplication, division, &c., have been exhibited by diagrams
comprising pairs of corresponding terms of the two series. This,
however, is only a particular case of the correspondence of two series.
In considering addition, for instance, we have introduced two parallel
series, each being the original number-series, but the two being placed
in different positions. If we add 1, 2, 3, ... to 6, we obtain a series
7, 8, 9, ... , the terms of which correspond with those of the original
series 1, 2, 3,...
Again, in SS 61-75 and 84-88 we have considered various kinds of numbers
other than those in the original number-series. In general, these have
involved two of the original numbers, e.g. 5^3 involves 5 and 3, and
log2 8 involves 2 and 8. In some cases, however, e.g. in the case of
negative numbers and reciprocals, only one is involved; and there might
be three or more, as in the case of a number expressed by (a + b)^n. If
all but one of these constituent elements are settled beforehand, e.g.
if we take the numbers 5, 5^2, 5^3, ..., or the numbers [root 3]1, [root
3]2, [root 3]3, ... or log10 1.001, log10 1.002, log10 1.003 ... we
obtain a series in which each term corresponds with a term of the
original number-series.
A B C
+---+-----+ +---+----+ +---+---------+
| n |6 + n| | n | 4n | | n | [root]n |
+---+-----+ +---+----+ +---+---------+
| 0 | 6 | | 0 | 0 | | 0 | .000 |
| 1 | 7 | | 1 | 4 | | 1 | 1.000 |
| 2 | 8 | | 2 | 8 | | 2 | 1.414 |
| 3 | 9 | | 3 | 12 | | 3 | 1.732 |
| . | . | | . | . | | . | . |
| . | . | | . | . | | . | . |
| . | . | | . | . | | . | . |
+---+-----+ +---+----+ +---+---------+
This correspondence is usually shown by _tabulation_, i.e. by the
formation of a table in which the original series is shown in one
column, and each term of the second series is placed in a second column
opposite the corresponding term of the first series, each column being
headed by a description of its contents. It is sometimes convenient to
begin the first series with 0, and even to give the series of negative
numbers; in most cases, however, these latter are regarded as belonging
to a different series, and they need not be considered here. The
diagrams, A, B, C are simple forms of tables; A giving a sum-series, B a
multiple-series, and C a series of square roots, calculated
approximately.
92. _Correspondence of Numerical Quantities._--Again, in S 89, we have
considered cases of multiple-tables of numerical quantities, where each
quantity in one series is _equivalent_ to the corresponding quantity in
the other series. We might extend this principle to cases in which the
terms of two series, whether of numbers or of numerical quantities,
merely _correspond_ with each other, the correspondence being the result
of some relation. The volume of a cube, for instance, bears a certain
relation to the length of an edge of the cube. This relation is not one
of proportion; but it may nevertheless be expressed by tabulation, as
shown at D.
D
+-----------+-------------+
| Length of | Volume |
| edge in | of |
| inches. | cube. |
+-----------+-------------+
| 0 | Nil. |
| 1 | 1 cub. in. |
| 2 | 8 cub. in. |
| 3 | 27 cub. in. |
| . | . |
| . | . |
| . | . |
+-----------+-------------+
93. _Interpolation._--In most cases the quantity in the second column
may be regarded as increasing or decreasing continuously as the number
in the first column increases, and it has intermediate values
corresponding to intermediate (i.e. fractional or decimal) numbers not
shown in the table. The table in such cases is not, and cannot be,
complete, even up to the number to which it goes. For instance, a cube
whose edge is 1-1/2 in. has a definite volume, viz. 3(3/8) cub. in. The
determination of any such intermediate value is performed by
_Interpolation_ (q.v.).
In treating a fractional number, or the corresponding value of the
quantity in the second column, as intermediate, we are in effect
regarding the numbers 1, 2, 3, ..., and the corresponding numbers in the
second column, as denoting points between which other numbers lie, i.e.
we are regarding the numbers as _ordinal_, not cardinal. The transition
is similar to that which arises in the case of geometrical measurement
(S 26), and it is an essential feature of all reasoning with regard to
continuous quantity, such as we have to deal with in real life.
94. _Nature of Arithmetical Reasoning._--The simplest form of
arithmetical reasoning consists in the determination of the term in one
series corresponding to a given term in another series, when the
relation between the two series is given; and it implies, though it does
not necessarily involve, the establishment of each series as a whole by
determination of its unit. A method involving the determination of the
unit is called a _unitary_ method. When the unit is not determined, the
reasoning is algebraical rather than arithmetical. If, for instance,
three terms of a proportion are given, the fourth can be obtained by the
relation given at the end of S 57, this relation being then called the
_Rule of Three_; but this is equivalent to the use of an algebraical
formula.
More complicated forms of arithmetical reasoning involve the use of
series, each term in which corresponds to particular terms in two or
more series jointly; and cases of this kind are usually dealt with by
special methods, or by means of algebraical formulae. The old-fashioned
problems about the amount of work done by particular numbers of men,
women and boys, are of this kind, and really involve the solution of
simultaneous equations. They are not suitable for elementary purposes,
as the arithmetical relations involved are complicated and difficult to
grasp.