P l a n e G e o m e t r y
An Adventure in Language and Logic
We explained in the Introduction that it is not possible to prove every statement. Nevertheless, we should prove as many statements as possible. Which is to say, the statements on which the proofs are based should be as few as possible. They are called the First Principles. There are three categories of them: Definitions, Postulates, and Axioms or Common Notions. We will follow each with a brief commentary.
11. An angle is the inclination to one another of two straight lines that meet.
12. The point at which two lines meet is called the vertex of the angle.
13. If a straight line that stands on another straight line makes the adjacent angles equal, then each of those angles is called a right angle; and the straight line that stands on the other is called a perpendicular to it.
14. An acute angle is less than a right angle. An obtuse angle is greater than a right angle.
16. Rectilinear figures are figures bounded by straight lines. A triangle is bounded by three straight lines, a quadrilateral by four, and a polygon by more than four straight lines.
17. A square is a quadrilateral in which all the sides are equal, and all the angles are right angles.
17. A regular polygon has equal sides and equal angles.
18. An equilateral triangle has three equal sides. An isosceles triangle has two equal sides. A scalene triangle has three unequal sides.
19. The vertex angle of a triangle is the angle opposite the base.
10. The height of a triangle is the straight line drawn from the vertex perpendicular to the base.
11. A right triangle is a triangle that has a right angle.
12. Figures are congruent when, if one of them were placed on the other, they would exactly coincide. (Congruent figures are thus equal to one another in all respects.)
14. A parallelogram is a quadrilateral whose opposite sides are parallel
15. A circle is a plane figure bounded by one line, called the circumference, such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
16. And that point is called the center of the circle.
17. A diameter of a circle is a straight line through the center and terminating in both directions on the circumference. A straight line from the center to the circumference is called a radius; plural, radii.
1. Grant the following:
1. To draw a straight line from any point to any point.
2. To extend a straight line for as far as we please in a straight line.
3. To draw a circle whose center is the extremity of any straight line, and whose radius is the straight line itself.
4. All right angles are equal to one another.
5. If a straight line that meets two straight lines makes the interior angles on the same side less than two right angles, then those two straight lines, if extended, will meet on that same side.
(That is, if angles 1 and 2 together are less than two right angles, then the straight lines AB, CD, if extended far enough, will meet on that same side; which is to say, AB, CD are not parallel.)
Axioms or Common Notions
1. Things equal to the same thing are equal to one another.
2. If equals are joined to equals, the wholes will be equal.
3. If equals are taken from equals, what remains will be equal.
4. Things that coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Equal magnitudes have equal parts; equal halves, equal thirds,
2. and so on.
Commentary on the Definitions
A definition describes what is being defined, and gives it a name. A definition has the character of a postulate, because we must agree to use the name in precisely that way. A definition however does not assert that what has that name exists -- and we may not assume it does. For, just as the postulates and axioms -- the statements we do not prove -- must be as few as possible, so must be the assumptions that things exist. We may assume, for example, that points exist, but very little else. A definition is required only to be understood.
And so an "equilateral triangle" is defined. But the very first proposition shows that what has that name exists. How does it show it? By presenting the logical steps that allow us to construct a figure that satisfies the definition. The definition of an equilateral triangle correctly describes something we can actually draw.
Now, an equilateral triangle exists as an idea, for we have understood the definition. But for a defined term to exist for mathematics, it must mean more than that. It must mean that we are able to bring it into this world. With any definition, we must either postulate the existence of what has been defined (that is done in the case of a circle, Postulate 3), or we must prove it.
By maintaining the logical separation of the definition of a thing and its existence, mathematics becomes a science in the same way physics is a science. Physics must show that the things of which it speaks
A definition is reversible. That means that when the conditions of the definition are satisfied, then we may use that word. And conversely, if we use that word, that implies those conditions have been satisfied. A definition is equivalent to an if and only if sentence.
Note that the definition of a right angle says nothing about measurement, about 90°. Plane geometry is not the study of how to apply arithmetic to figures. In geometry we are concerned only with what we can see and reason directly, not through computation. A most basic form of knowledge is that two magnitudes are simply equal -- not that they are both 90° or 9 meters.
How can we know when things are equal? That is one of the main questions of geometry. The definition (and existence) of a circle provides our first way of knowing that two straight lines could be equal. Because if we know that a figure is a circle, then we would know that any two radii are equal. (Definitions 15 and 17.)
We have not formally defined a point, although Euclid does. ("A point is that which has no part." That is, it is indivisible. Most significantly, Euclid adds, "The extremities of a line are points.") And we have not defined a "line," although again Euclid does. ("A line is length without breadth.") Euclid defines them because they are elementary objects of geometry, and their existence is assumed. (Postulate 1, however, guarantees the existence of a straight line.) But since there is never occasion to prove that something is a point or a line, a definition of one is not logically required.
Commentary on the Postulates
We require that the figures of geometry -- the triangles, squares, circles -- be more than ideas. We must be able to draw them. The fact that we can draw a figure is what permits us to say that it exists. For, as we have noted in the Commentary on the Definitions, we may not assume that what we have called a "triangle" or a "circle" actually exists.
The first three Postulates narrowly set down what we are permitted to draw. Everything else we must prove. Each of those Postulates is therefore a "problem" -- a construction -- that we are asked to consider solved: "Grant the following."
The instruments of construction are straightedge and compass. Postulate 1, in effect, asks us to grant that what we draw with a straightedge is a straight line. Postulate 3 asks us to grant that the figure we draw with a compass is a circle. And so we will then have an actual figure that refers to the idea of a circle, rather than just the word "circle."
Note, finally, that the word all, as in "all right angles" or "all straight lines," refer to all that exist, that is, all that we have actually drawn. Geometry -- at any rate Euclid's -- is never just in our mind.
Commentary on the Axioms or Common Notions
The distinction between a postulate and an axiom is that a postulate is about the specific subject at hand, in this case, geometry; while an axiom is a statment we acknowledge to be more generally true; it is in fact a common notion. Yet each has the same logical function, which is to authorize statements in the proofs that follow.
Implicit in these Axioms is our very understanding of equal versus unequal, which is: Two magnitudes of the same kind are either equal or one of them is greater.
So, these Axioms, together with the Definitions and Postulates, are the first principles from which our theory of figures will be deduced.
Please "turn" the page and do some Problems.
Continue on to Proposition 1.
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Copyright © 2012 Lawrence Spector
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