MAT-351 Lesson 003 – Quantity Is Measured Only by Quantity of Its Own Kind
We have learned what quantity is, and we have learned that quantity is considered either as magnitude or as multitude. We come now to the first rule of measuring, and Euler states it plainly: we cannot measure or determine any quantity except by considering some other quantity of the same kind as known, and pointing out their mutual relation.
Let us take his examples one at a time, for the rule is easier to see in a particular case than in the abstract.
Suppose we wish to say how much money a man has. We take some known piece of money — Euler names a crown, and we might as well say a dollar — and we show how many of these are contained in the sum before us. We do not measure his money by pounds of weight, nor by feet of length. We measure money by money.
Suppose we wish to say how heavy a thing is. We take some known weight, a pound or an ounce, and we show how many times that weight is contained in the thing we are weighing. We measure weight by weight.
Suppose we wish to say how long a thing is. We take some known length, such as a foot, and we find how many times it is contained in the length before us. We measure length by length.
In every case the same thing happens. We hold up a known quantity beside the unknown one and ask how the two stand to each other. And in every case the known quantity is of the same kind as the one we are measuring.
Now we should ask why this must be so, for it is not enough to observe that it is always done this way. The reason is that measuring is comparing, and only things of the same kind can be compared as greater and less.
Consider what it would mean to measure a rope by an hour. We should have to ask how many hours long the rope is. But the question has no answer, and not because we lack an instrument. It has no answer because there is nothing in the rope that an hour could be laid against. An hour is not longer or shorter than a rope; it is not related to the rope in that way at all. The question is not hard, it is empty.
This shows us something about the rule. It is not a rule of practice that a clever man might find a way around. It follows from what measuring is. To measure is to say how many times one quantity is contained in another, and one quantity can be contained in another only if the two are of one kind.
We should now connect this with what we learned in the last lesson, for the two go together. We saw that in a multitude the part is found, and in a magnitude the part must be chosen. We may now say why the choice, when we must make it, is not free in every respect. We may choose whether to measure a rope by feet or by inches or by paces, and no one of these is the only right choice. But we may not choose to measure it by pounds. What is left open to us is which quantity of the right kind to use; what is settled beforehand is that it must be of the right kind.
One thing more deserves notice, because a student may otherwise think the rule broken. We often speak of quantities of different kinds in one breath — we say that a wagon travels thirty miles in one hour. But we are not measuring miles by hours here. We are measuring miles by miles, and hours by hours, and then setting the two results side by side. The rule stands. We shall have more to say of such joined quantities later in this chapter.
In the next lesson we shall give a name to the known quantity we hold up beside the unknown one. It is called the unit of measure.
Memory Work:
- We cannot measure any quantity except by considering some other quantity of the same kind as known, and pointing out their mutual relation.
- We measure money by money, weight by weight, and length by length.
- To measure is to say how many times one quantity is contained in another, and one quantity can be contained in another only if the two are of the same kind.
- What we may choose is which quantity of the right kind to measure by; that it must be of the right kind is not a matter of choice.
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