Unit Three

     1.  Discussion of historical propositions
     2.  Preparation for tomorrow's graded exercise

FIRST you will be asked to identify the building blocks of a given Euclidean proposition from a list of definitions, postulates, and prior propositions. MAKE SURE that you are very familiar with these building blocks before the quiz so that you can find them easily. (The partial list given below includes all that you will need and some that you will not need but which are of particular interest.) You will then have to complete the last step in the proof.

Here is an example:

Euclid's First Proposition

upon a given, straight, bounded line to construct an equilateral triangle

Let there be the given, straight, bounded line, AB. It is necessary, then, upon the straight line, AB, to construct an equilateral triangle. With center A, then, and with radius AB, let a circle be drawn, the circle BCD. (Postulate 3) And again, with center B, then, and radius BA, let a circle be drawn (Postulate 3), the circle ACE. And from point C, through which the circles cut one another, to the points A, B let there be joined straight lines, the lines CA, CB (Postulate 1.) And since the point A is the center of the circle CDB, the line AC is equal to the line AB (Definition 15). Again, since the point B is the center of Circle CAE, the line BC is equal to BA (Definition 15). It has been shown that CA is also equal to AB. For indeed, both of the lines CA and CB are equal to the line AB. Things, then, that are equal to the same thing are also equal to one another (Common Notion 1). And the line CA is surely equal to the line CB. The three lines, then, CA, AB, BC are equal to one another. Equilateral, then, is the triangle ABC. And it has been constructed upon the given, straight, bounded line, AB. [Upon the given, straight, bounded line, then, an equilateral triangle has been constructed.] The very thing it was necessary to make.

Definitions

1.

A point is that of which there is no part.

2.

A line is a widthless length.

3.

A line's ends are points.

4.

A straight line is one which lies evenly with the points on itself.

5.

A surface is that which has only length and width.

6.

The ends of a surface are lines.

7.

A plane surface is one which lies evenly with the lines on it.

10.

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called perpendicular to that on which it stands.

15.

A circle is a plane figure contained by one line [which is called the circumference], and all the straight lines coming from one point of those lying within the figure and falling [upon the circumference of the circle] are equal to one another.

Note: An interpolation is suspected in the brackets. Manuscripts have the words, but the other ancient sources, namely the commentators, do not. Nor does papyrus Herculanensis No. 1061., the oldest source on The Elements. Heath notes (The Thirteen Books 184), "The words were doubtless added in view of the occurrence of the word "circumference" in Deff. 17, 18 immediately following, without any explanation. But no explanation was needed ... Euclid was perfectly justified in employing the word if Deff. 17, 18 and elsewhere, but leaving it undefined as being a word universally understood and not involving in itself any mathematical conception."

23.

Parallel lines are whatever straight lines which, being in the same plane and being shot out to infinity in both directions, fall together with one another in neither direction.

Euclid's 5 Postulates

1.

Let it be demanded that from every point to every point a straight line be drawn,

2.

and that a limited straight line extend continuously upon a straight line,

3.

and that to every point and every distance a circle be drawn,

4.

and that all right angles are equal to one another,

5.

and that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Euclid's 5 Common Notions

1.

Things equal to the same thing are also equal to one another.

2.

And if equal things should be added to equal things, the wholes are equal.

3.

And if from equal things equal things should be taken away, the remaining things are equal.

4.

and things fitted to one another are equal to one another.

5.

And the whole is greater than the part.

Note on common notion 4: What Euclid apparently means by "fitting" one thing to another is imaginarily picking up, for instance, a triangle and placing it down upon a corresponding triangle to see if all the points correspond to one another. Many mathematicians feel that this "fitting" belongs to the physical sciences and as such is incompatible with the metaphysical, hypothetical science of geometry. Euclid uses it in his fourth theorem.

SECOND you will be asked to use your history of The Elements to "prove," or at least argue, two of the historical "propositions" or your own "theorem." Before you begin the body of your argument, remember to define the terms of the argument, i.e. define what you are going to prove in your own words just as Euclid did.


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