the general arrangement of the diagram being as shown in E or E′:—

Multiplication is therefore equivalent to completion of the diagram by entry of the product.

36. Multiple-Tables.—The diagram C or D of § 35 is part of a complete table giving the successive multiples of the particular unit. If we take several different units, and write down their successive multiples in parallel columns, preceded by the number-series, we obtain a multiple-table such as the following:—

11291s.   5d.3 yds. 2 ft.17359 ...
224182s. 10d.7 yds. 1 ft.34718 ...
336274s.   3d.11 yds. 0 ft.52077 ...
448365s.   8d.14 yds. 2 ft.69436 ...
5510457s.   1d.16 yds. 1 ft.86795 ...
....... ...
....... ...
....... ...
....... ...

It is to be considered that each column may extend downwards indefinitely.

37. Successive Multiplication.—In multiplication by repetition the unit is itself usually a multiple of some other unit, i.e. it is a product which is taken as a new unit. When this new unit has been multiplied by a number, we can again take the product as a unit for the purpose of another multiplication; and so on indefinitely. Similarly where multiplication has arisen out of the subdivision of a unit into smaller units, we can again subdivide these smaller units. Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of £3 by 20, 12 and 4, and B the successive reduction of £3 to shillings, pence and farthings. The principle on which the diagrams are constructed is obvious from § 35. It should be noticed that in multiplying £3 by 20 we find the value of 20·3, but that in reducing £3 to shillings, since each £ becomes 20s., we find the value of 3·20.

38. Submultiples.—The relation of a unit to its successive multiples as shown in a multiple-table is expressed by saying that it is a submultiple of the multiples, the successive submultiples being one-half, one-third, one-fourth, ... Thus, in the diagram of § 36, 1s. 5d. is one-half of 2s. 10d., one-third of 4s. 3d., one-fourth of 5s. 8d., ...; these being written “½ of 2s. 10d.,” “1⁄3 of 4s. 3d.,” “¼ of 5s. 8d,”...

The relation of submultiple is the converse of that of multiple; thus if a is 1⁄5 of b, then b is 5 times a. The determination of a submultiple is therefore equivalent to completion of the diagram E or E′ of § 35 by entry of the unit, when the number of times it is taken, and the product, are given. The operation is the converse of repetition; it is usually called partition, as representing division into a number of equal shares.