Geometry, moreover, embraced logic, grammar and rhetoric, because it was approached purely verbally. There was no algebra, no symbols for "angle" or "equals." What the student saw, he explained. For geometry is based on looking, and the sensitivity it develops is the essence of science.
In the 4th century B.C., Alexandria in Egypt was the center of culture and learning, and it was there that the Greek mathematician Euclid assembled the most remarkable textbook the world has ever seen: the Elements of geometry and arithmetic. Written in simple, straightforward language, the Elements has been translated the world over, and through the centuries it has been the model for clear and eloquent reasoning. It was the first written work to introduce what is called rigor into mathematics. That same rigor -- What gives us the right to say that we really know? -- is part of the culture of mathematics today, and it is the model followed in theoretical physics. Anyone truly interested in what mathematics is, can have no firmer foundation than Euclid.
angles at the base are also equal. Or, when two straight lines intersect, the angles labeled 1 and 2 are equal. But what distinguished the Greeks is that they wanted to explain how they knew that what they saw was true. They were the first to present those facts in the framework of a logical science. And after introducing some terms, we will see precisely what that means.
Knowledge of figures
Geometry, which literally means land measurement, is the study of
figures. A circle is a figure, a triangle is a figure. What we aspire to is knowledge of figures.
A figure is whatever has a boundary. In plane geometry, we study figures that are flat, and their boundaries are called lines, or, in the case of a circle, a single line. We say that a line, which may be either straight or curved, is a length. We do not mean length as opposed to width; we mean any actual or potential boundary of a plane figure.
A line, too, may have its boundaries, or extremities, and we call these points. "Point" is a convenient word, when we need it, to call attention to a specific place. There.
We name a line by naming its extremities, its endpoints, with capital letters, thus we speak of the line AB.
Or we could call it BA, it does not matter, except if we want to emphasize that it extends in one direction, we would call it AB; if in the other direction, BA.
The space enclosed by the boundary -- the figure itself -- is the
magnitude area.
Magnitudes
A magnitude is whatever has size; it could be larger or smaller. In plane geometry we study three kinds of magnitudes: length, area, and angle. We compare magnitudes of the same kind -- two lengths, two areas, or two angles -- and we try to decide how they are related. Either two magnitudes of the same kind will be equal to one another, or one of them will be larger.
Therefore if we say that these two triangles are equal --
-- we mean that they are equal areas. Because that is what kind of magnitude a triangle is. The space enclosed by each boundary will be exactly the same.
A magnitude is not like a natural number, because a magnitude is not composed of indivisible units. Since a natural number is composed of indivisible units, it will have only certain parts. 10 people can be divided only in half (which is 5), into fifths (2's), or into tenths (1's). But a magnitude -- such as the line AB -- can be cut into any parts. Halves, thirds, fourths, fifths, millionths! We say therefore that a magnitude is continuous. This is in contrast to a natural number, whose units we say are discrete.
The words length and area are often used to mean the measure of those magnitudes -- a number. But in plane geometry, the length and the area are the magnitudes themselves, not numbers.
Straight lines
Each straight line that forms the boundary of a square,
for example, will have two extremities; obviously. But we imagine that we could extend a straight line for as far as we please, and here we touch on the question of infinity. Which is something more than "for as far as we please." It is the idea of something with no
endpoints at all.
For as long as there have been mathematicians, there have been quarrels about
"infinity." What does it mean to say that something is infinite? Not that it could be, or that it will be, but that, at this moment, it is. Does what can never be whole exist?
In plane geometry, however, we have no need of the idea of infinite lines (for they are only ideas); in what is called analytic geometry, they think they do. It is only finite lines -- the actual or potential boundaries of a figure -- that we ever require. Hence, when we speak of a "straight line," we mean what we can actually experience or draw: a line with two extremities. A straight line will be potentially infinite, but it is never actually infinite.
These quarrels about actual versus potential infinities arise only when straight lines are abstracted from the boundaries of figures. But in plane geometry it is the figures that concern us, and in the ultimate theorems, straight lines appear only as the boundaries of figures.
Also, just because we can define something (such as an actually infinite line) does not guarantee that it exists. (We can define a unicorn. But does a unicorn exist?) As we shall see, the mathematical existence of what has been defined, requires that we be able to produce it.
This completes the preliminary description, this is what plane geometry is about; we are now ready to study it as a logical science.
Logic
The method of logic is to know or prove something through reasoning, and it provides one answer to the question, "How do I know?" One can reply, "I deduced it." (Latin de, away from + ducere, to lead.) Hence when we deduce or prove something, we are led, through reasoning, from what we know to what we can conclude.
Now, it is not possible to prove every statement. If we had to do that, there would be no end. Rather, to prove or explain anything is to do so in terms of something simpler, something that we already know and accept.
Similarly, it is not possible to define every word, because we must already understand the words in which a definition is framed. An example from arithmetic is the impossibility of defining one. (Try
) That does not mean that we do not understand one. An example from geometry is the impossibility of defining a straight line -- because nothing is known better, or is more fundamental, than the idea of a straight line.
Logic, then, which is the science of reasoning, is not based on reasoning. It is based on irreducible understanding, on what we call first principles. They provide the basis for proving. The first principles of Euclid's geometry are in three categories:
1) Definitions
2) Postulates
3) Axioms or Common Notions
The statements that are found in these first principles will justify the statements we will make in the proofs, or propositions, that follow.
Before presenting the first principles, let us become familiar with the vocabulary of logic.
Introduction to Logic
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