Quantity
In the previous chapter we pointed out
that lengths are measurable in terms of some unit length, areas in term of a unit area, and volumes in terms of a unit volume.
When we have a set of things such as lengths which are measurable in terms of any one of them, we say that they are quantities of the same kind. Thus lengths are quantities of the same kind, so are areas, and so are volumes. But an area is not a quantity of the same kind as a length, nor is it of the same kind as a volume. Let us think a little more on what is meant by being measurable, taking lengths as an example.
Lengths are measured by the foot-rule. By transporting the foot-rule from place to place we judge of the equality of lengths. Again, three adjacent lengths, each of one foot, form one whole length of three feet. Thus to measure lengths we have to determine the equality of lengths and the addition of lengths. When some test has been applied, such as the transporting of a foot-rule, we say that the lengths are equal; and when some process
has been applied, so as to secure lengths being contiguous and not overlapping, we say that the lengths have been added to form one whole length. But we cannot arbitrarily take any test as the test of equality and any process as the process of addition. The results of operations of addition and of judgments of equality must be in accordance with certain preconceived conditions. For example, the addition of two greater lengths must yield a length greater than that yielded by the addition of two smaller lengths. These preconceived conditions when accurately formulated may be called axioms of quantity. The only question as to their truth or falsehood
which can arise is whether, when the axioms are satisfied, we necessarily get what ordinary people call quantities. If we do not, then the name "axioms of quantity" is ill-judged–-that is all.
These axioms of quantity are entirely abstract, just as are the mathematical properties of space. They are the same for all quantities, and they presuppose no special mode of perception. The ideas associated with the notion of quantity are the means by which a continuum like a line, an area, or a volume can be split up into definite parts. Then these parts are counted; so that numbers can be used to determine the exact properties of a continuous whole.
Our perception of the flow of time and of
the succession of events is a chief example of the application of these ideas of quantity. We measure time (as has been said in considering periodicity) by the repetition of similar events–-the burning of successive inches of a uniform candle, the rotation of the earth relatively to the fixed stars, the rotation of the hands of a clock are all examples of such repetitions. Events of these types take the place of the foot-rule in relation to lengths. It is not necessary to assume that events of any one of these types are exactly equal in duration at each recurrence. What is necessary is that a rule should be known which will enable us to express the relative