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P l a n e G e o m e t r y
An Adventure in Language and Logic
based on
First Principles
Here are the first principles of plane geometry -- the Definitions, Postulates, and Axioms or Common Notions -- whose necessity we pointed out in the Introduction. They will be followed by a brief commentary.
Definitions
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.
| 5. Angles are complementary (or complements of one another) if, together, they equal a right angle. Angles are supplementary (or supplements of one another) if together they equal two right angles. |  |
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.)
| 13. Parallel lines are straight lines that are in the same plane and do not meet, no matter how far extended in either direction. |  |
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.
Postulates
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 that are equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes (the "sums") will be
2. equal.
3. If equals are subtracted 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 states the conditions under which a word or a name will be used; in that way it regulates the use of that word, and we must agree to accept that. It is never a question of whether a definition is true or false. A definition is required only to be understood.
Moreover, a definition does not assert that the thing defined exists -- and we may not assume it does. The existence of what has been defined is a separate logical issue. We say therefore that definitions are nominal. "Triangle" is merely the name -- the sound -- that we will use to refer to a figure bounded by three straight lines. And we should not assume that what we have chosen to call a "triangle" exists, any more than we should assume that any sound refers to something that exists. At some point, however, we are required to show that this thing we have called a "triangle" does exist. And we do that by showing that we are able to construct -- actually draw -- one. 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 demonstrate that we can construct what has been defined. Proposition 1, as we will see, solves the problem of constructing an "equilateral triangle."
By maintaining the logical separation of the definition of a thing and its existence, mathematics becomes a science in the same way that physics is a science. Physics must show that the things of which it speaks
-- "electrons," "protons," "neutrinos" -- actually exist. And physics does that by showing that it is possible to experience them. It was geometry that led the way. Geometry was the first science. Also, by the requirement that a definition does not imply the existence of what is defined, mathematics avoids dealing in fantasies and the possibility of any contradiction. For example, by a "hemigon" I mean a regular polygon that has half as many sides as angles. Do you understand? Good.
Definitions find their greatest importance in proofs, and that really is their function. To prove, for example, that a triangle is isosceles, we must prove that the triangle satisfies the definition of isosceles.
A definition is reversible. That means that when certain conditions are satisfied, then we may use that word. And conversely, if we use that word, that implies that 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 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 chosen not to define a "point," although Euclid does. ("A point is that which has no part." That is, it cannot be divided. 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.") 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 on paper. The fact that we can draw a figure is what permits us to say that it exists. For, as we have noted, we may not assume that what we have defined, such as 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, because they are the only instruments that will satisfy the Postulates
Postulate 1, in effect, asks us to grant that whatever 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.
As for Postulate 5, we will have more to say about it when we come to Proposition 29.
Note, finally, that the word all, as in "all right angles" or "all straight lines," refer to all that exist, that is, that have actually been 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 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.
Each of the Axioms, as well as Postulate 4, gives a criterion for things being equal (or, Axiom 5, unequal).
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.
or
Continue on to Proposition 1.
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