MAT-351 Lesson 004 – The Unit of Measure
In the last lesson we learned that a quantity can be measured only by a quantity of its own kind, and that to measure is to find how many times one quantity is contained in another. We spoke there of the known quantity we lay beside the unknown one, but we did not name it. It has a name, and we shall learn it now.
Ray gives the definition: one of the single parts of which a quantity of multitude is composed is called the unit of measure. His example is the one he used before. Take the quantity three cents. It is composed of single parts, and each of those parts is one cent. Therefore one cent is the unit of measure of the quantity three cents.
We should notice at once that the definition speaks of a quantity of multitude, and we should ask why. The reason takes us back to what we learned in Lesson 002. A multitude is a quantity made up of individual and distinct parts. Those parts are already there before we come to them, and any one of them may serve as our unit of measure. We do not have to make it or choose it; we have only to see it. The flock of sheep is composed of single sheep, and one sheep is the unit of measure of the flock. The shelf of books is composed of single books, and one book is the unit of measure. In every quantity of multitude the unit of measure is given to us along with the quantity.
Now we may state the principle of measuring, which Ray sets down immediately after the definition: the value or measure of any quantity is the number of times it contains its unit of measure.
Let us see this worked. We wish to measure the quantity three cents. We have its unit of measure, one cent. We ask how many times one cent is contained in three cents, and we find that it is contained three times. The measure of the quantity is therefore three. In the same way, if a flock contains one sheep forty times over, the measure of the flock is forty; and we say there are forty sheep.
This may seem a laborious way of saying something a child already knows how to do. But we should see what has been made plain by it. Counting and measuring are not two different operations. When we count sheep we are measuring the flock, and the unit of measure is one sheep. Every act of measuring, however it is carried out, comes to this same thing: taking the unit and finding how many times it is contained in the whole.
We should also see how this joins to the rule of Lesson 003. There we said that the known quantity must be of the same kind as the one measured. We can now say something stronger and more exact. The unit of measure is not merely a quantity of the same kind; it is a part of the very quantity we are measuring. One cent is not simply something that resembles three cents. It is one of the three. This is why the rule of Lesson 003 cannot be broken. A quantity that is not of the same kind could not be a part of the quantity we are measuring, and so could not be its unit of measure at all.
One caution before we leave the matter. The unit of measure is not fixed by nature for all cases, even in a multitude. We may measure a quantity of eggs by one egg, and say there are twenty-four; or we may measure it by one dozen, and say there are two. Both are true, and neither is an error. We shall take up what follows from this in Lesson 009. For now we need only hold that the measure of a quantity is always the number of times it contains whatever its unit of measure is.
There remains a difficulty which the definition does not answer. It tells us what the unit of measure is in a quantity of multitude, where the parts are found. But we learned in Lesson 002 that a magnitude has no parts by nature. What then shall we do when we wish to measure water, or land, or time? That is the subject of the next lesson.
Memory Work:
- One of the single parts of which a quantity of multitude is composed is called the unit of measure.
- One cent is the unit of measure of the quantity three cents.
- The value or measure of any quantity is the number of times it contains its unit of measure.
- The unit of measure is not merely a quantity of the same kind as the quantity measured; it is one of its parts.
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