MAT-351 Lesson 002 – Magnitude and Multitude
In the last lesson we learned that quantity is anything capable of increase or decrease. That definition covers a great many different things — water and money, rope and time — and we shall now see that all of them fall into two sorts, according to the way the quantity is considered. Ray gives us the two names.
Quantity is called magnitude when it is considered in an undivided form. His example is a quantity of water. When we speak of the water in a cistern, we do not think of it as so many separate waters. We think of it as one whole thing, larger or smaller. So considered, it is a magnitude.
Quantity is called multitude when it is made up of individual and distinct parts. His example is three cents. Here the quantity does not come to us as one undivided whole. It comes as three separate coins, each one complete in itself, and the quantity is what they make when taken together. So considered, it is a multitude.
Let us try some further examples. The length of a rope is a magnitude, for the rope is one continuous thing and we do not find separate lengths lying along it. The water in a river, the ground in a field, the time in an hour, the weight of a stone — all of these are magnitudes. But the sheep in a flock, the men in a company, the books on a shelf, and the dollars in a purse are multitudes, for each of these is made up of parts that are already separate and distinct before we come to them.
Here we must clear away a difficulty of words, or it will trouble us later. Euler, whom we quoted in the last lesson, uses the word magnitude in a wider sense than Ray does. Euler writes that whatever is capable of increase or diminution is called magnitude, or quantity, treating the two words as meaning the same thing. Ray uses magnitude in a narrower sense, for one of the two ways in which quantity is considered. Both authors are careful, and neither is in error; they have simply chosen differently. In this course we shall follow Ray, and we shall use magnitude in the narrower sense: not as another name for quantity, but as the name of quantity considered in an undivided form.
Now we should ask why this distinction is worth making at all. It is not a matter of putting things into two boxes for the pleasure of sorting them. The distinction matters because it decides how the quantity can be measured.
Consider three cents. The parts are given to us already. One cent is a part of the quantity, and it is a part that nature, or rather the mint, has already marked off. To say how much money we have, we need only say how many of these ready-made parts there are. The multitude carries its own part with it.
Now consider the water in a cistern. Where are its parts? There are none. The water does not arrive divided into pints; we may draw off a pint, or a quart, or a spoonful, and the water submits to any of these equally. A magnitude has no parts by nature. If we mean to say how much water there is, we must first fix upon a part ourselves, and only then can we count.
This is the whole difference, and it is the reason the two names are worth learning. In a multitude, the part is found; in a magnitude, the part must be chosen. Everything we shall say about measuring quantity turns upon this.
We should add one caution. Because Ray says that magnitude is quantity considered in an undivided form, the same thing may sometimes be considered either way. The water in a cistern is a magnitude; but if it has been poured into twenty jars, we may consider it a multitude of twenty jarfuls. What has changed is not the water but the way we are regarding it. So when we ask whether a quantity is a magnitude or a multitude, we are asking how it is being considered in the case before us.
In the next lesson we shall learn a rule that follows from all this: that a quantity can be measured only by a quantity of its own kind.
Memory Work:
- Quantity is called magnitude when it is considered in an undivided form.
- Quantity is called multitude when it is made up of individual and distinct parts.
- A quantity of water is a magnitude; three cents is a multitude.
- In a multitude the part is found; in a magnitude the part must be chosen.
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