MAT-351 Lesson 001 – What Quantity Is
Before we can study algebra, we must settle what algebra is about. Every science has its own subject. The physician studies the body, the grammarian studies speech, and the mathematician studies quantity. So we begin by asking what quantity is, and we shall find that the answer, once we have it, governs everything that follows.
Euler opens his Elements of Algebra with this definition: whatever is capable of increase or diminution is called magnitude, or quantity. Ray gives the same definition in his own words: quantity is anything that is capable of increase or decrease. Two authors, writing in different countries and different centuries, arrive at the same answer. We shall take it as ours.
Let us try the definition on some ordinary things.
A sum of money is a quantity. A man has ten dollars. He may be given more, and then he has more; he may spend some, and then he has less. The sum admits of being made greater and of being made less, and so it is a quantity.
The weight of a sack of grain is a quantity. Grain may be poured in or taken out, and the weight rises or falls as we do so.
The same holds of the length of a rope, of the ground enclosed by a field, of the hours a journey takes, and of number itself. Each of these can be made greater and can be made less. Each is a quantity.
Now we should ask why the definition is framed as it is. Euler and Ray might have answered our question by giving a list — quantity is number, line, space, time, weight, and money. Why did they not? Because any list would be incomplete. The kinds of quantity are more numerous than we could set down. A merchant measures money, a farmer measures land, a physician measures the pulse, a builder measures stone, and each of these is measuring a quantity. No list could hold them all, and a definition that is incomplete is of no use to us. So instead of naming the kinds, Euler and Ray name the mark by which any kind whatever may be known: that it admits of more and of less. This is why the definition is short and yet leaves nothing out.
We should also notice how little the definition asks. It does not ask that the thing be seen or handled. Time cannot be handled, yet an hour is longer than a minute, and time is a quantity. It does not ask that the thing come in separate pieces that can be counted. A rope may be cut anywhere along its length, and its length is a quantity still. It asks one thing only, and we must hold to that one thing: that the thing admit of more and of less.
This gives us a test we may apply to anything whatever. We ask of it: can it be greater, and can it be less? If both answers are yes, the thing is a quantity, and mathematics can treat of it. If not, it is not a quantity, and mathematics has nothing to say of it as such. The name of a city is not a quantity, for one name is not greater or less than another; but the number of people living in that city is a quantity, for that number may rise and fall. Notice that the same city gives us both, which shows us that we must ask the question of the particular thing we mean to measure, and not of the object it belongs to.
We must hold this definition exactly as it stands, for everything that follows in this course depends on it. Quantity is the subject of mathematics, and algebra is the part of mathematics which treats of quantity in general, by means of letters and signs. A student who is uncertain what quantity is will therefore be uncertain of everything built upon it. Why letters and signs are used, and how algebra differs from arithmetic, we shall learn in Lessons 022 through 026, when we have the words needed to state it fully.
In the next lesson we shall learn that quantity is considered in two ways, which are called magnitude and multitude, and we shall learn what distinguishes the one from the other.
Memory Work:
- Quantity is anything that is capable of increase or decrease.
- To know whether a thing is a quantity, we ask whether it can be made greater and whether it can be made less.
- Among the kinds of quantity are numbers, lines, space, time, weight, and sums of money.
- The definition of quantity names no kinds, because the kinds are too many to be numbered.
- Algebra is the part of mathematics which treats of quantity in general, by means of letters and signs.
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