(ii.) To continue the division we may take as our new unit a **submultiple** of Q, such as Q/r, where r is an integer, and repeat the process.

**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,..

The relation of **submultiple** is the converse of that of multiple; thus if a is I 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.

Dividing 12 by 2, we get a **submultiple** 6, which again has a prime 2 as a factor.

A fraction of a quantity is a **submultiple**, or a multiple of a **submultiple**, of that quantity.

Subdivision of **Submultiple**.-By 7 of A we mean 5 times the unit, 7 times which is A.

The relation expressed by proportion includes the relations expressed by multiple, **submultiple**, fraction and ratio.

(iii) Finally, there are the cases of linear measurement, where it is theoretically possible to find, by geometrical methods, an exact **submultiple** of a given unit, but both the unit and the **submultiple** are not really concrete objects, but are spatial relations embodied in objects.

The former would be an exact **submultiple** of the 30-day month, but the exact relation of seven days to the month is not very clear.