P l a n e G e o m e t r y
An Adventure in Language and Logic
I N T R O D U C T I O N
Geometry: The Study of Figures
GEOMETRY, which literally means land measurement, is the study
of figures. A circle is a figure, a triangle is a figure. A figure is whatever has a boundary. What we aspire to is knowledge of figures: their properties, their relationships, and how to construct them. It is with the ideas of figures already in mind that geometry proceeds.
In these pages, we present an English version of the very first textbook on geometry, a book assembled by the Greek scholar Euclid in the 4th century B.C. It is 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 work to introduce what is called rigor into mathematics. That same rigor ("What gives us the right to say that we know?") is part of the culture of mathematics today, and it is the model followed by theoretical physics. Anyone truly interested in what mathematics is, can have no firmer foundation than Euclid.
Efforts have always been made to express the Elements in the language of each time and place. The pages that follow are adapted from the translation by Sir Thomas Heath (Dover) as well as the edition of Isaac Todhunter (Elibron Classics).
What distinguishes Euclid's text from today's, is that it is completely verbal. There is no algebra, no symbols for "angle" or "equals." And there are no two-column proofs Geometry in this way embraces logic, grammar and rhetoric, which at one time were the essential liberal arts. What students see, they put into words. For geometry is based on looking, and the sensitivity it develops is the essence of science.
Now, we can see that when two sides of a triangle are equal, then the
angles at the base are also equal. Or, when two straight lines meet, the angles labeled 1 and 2 are equal. But what distinguished the Greeks is that they wanted to explain why that 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.
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 them points. "Point" is the word we use, when we need it, to call attention to a specific place, such as where a line ends, or where two lines meet. Points exist potentially. When we indicate a point, we may then say that it "exists." Anything more than that is unnecessary.
(See The mathematical existence of numbers, "Is a line really composed of points?")
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 called area.
An angle is formed when two straight lines meet. The point --
the place -- where they meet is called the vertex. We name an angle with three letters -- "angle ABC" -- and we place the vertex in the center. When there is no doubt as to which point is the vertex, we may say "the angle at B," or simply "angle B."
A magnitude is whatever has size: it could be larger or smaller. Length, area, and angle are the three kinds of magnitudes we study in plane geometry. We compare magnitudes of the same kind, and we try to decide how they are related. Two lengths, two areas, or two angles either will be equal to one another, or one of them will be larger.
If we say, then, 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 continuous, while a natural number, which is a collection of indivisible units, is discrete.
Now in algebra, the words length and area are sometimes used to mean the measure of those magnitudes -- a number. But in plane geometry, length and area are the magnitudes themselves, not numbers.
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 therefore we say that a straight line is potentially infinite. To be potentially infinite is in marked contrast to being actually infinite -- which would mean that the straight line has no endpoints at all.
In plane geometry we are concerned only with what we can actually observe, and so we have no need even for the idea of actually infinite lines. (For they are only ideas; we cannot draw them, and they serve no practical purpose.) In what is called analytic geometry, however, they think they do need such lines. But in plane geometry, 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" in these pages, we mean what we can actually experience or draw: a line with two extremities.
Those who think that even in plane geometry they need actually infinite lines, imagine that any finite line, such as a side of a square, is a segment -- a part -- of an actually infinite line. (Strange. No?) Hence they call a side of a square a "line segment." The idea in Euclidean geometry is quite the reverse, namely that we can extend any straight line for as far as we please.
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 their boundaries.
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 existence for mathematics of whatever we define, 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.
The method of logic is to know or prove something through reasoning. 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, there would be no end. Rather, to prove or explain anything is to do so in terms of something simpler, something that we already acknowledge and accept.
Similarly, it is not possible to give a verbal definition of every word, because we must already understand the words in which a definition is framed. An example from arithmetic is he impossibility of defining one. (Try) That does not mean that you 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 founded on reasoning. It is founded 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:
3) Axioms or Common Notions
The statements found there will justify the statements we will make in proving the Theorems and Problems that follow.
Before presenting the first principles, let us become familiar with the vocabulary of logic.
Copyright © 2012 Lawrence Spector
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