Chapter X: Arithmetical Reasoning

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.