Unit Four: Quiz

A. Complete Euclid's Fifth Theorem and identify the definitions, common notions, postulates and prior theorems by number. To save time, you may omit the proof that the angles under the base are equal.


The Fifth Theorem (Heath's translation)

In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.

Let ABC be an isosceles triangle having the side AB equal to the side AC;

and let the straight lines BD, CE be produced further in a straight line with AB, AC. (________________) I say that the angle ABC is equal to the angle ACB, and the angle CBD to the angle BCE.

Let a point F be taken at random on BD; from AE the greater let AG be cut off equal to AF the less; (________________)

and let the straight lines FC, GB be joined. (________________)

Then, since AF is equal to AG and AB to AC,the two sides FA, AC are equal to the two sides GA, AB, respectively; and they contain a common angle, the angle FAG. Therefore the base FC is equal to the base GB, and the triangle AFC is equal to the triangle AGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle ACF to the angle ABG, and the angle AFC to the angle AGB (________________).

And, since the whole AF is equal to the whole AG, and in these AB is equal to AC, the remainder BF is equal to the remainder CG. (________________)

But FC was also proved equal to GB; therefore the two sides BF, FC are equal to the two sides CG, GB respectively; and the angle BFC is equal to the angle CGB, while the base BC is common to them; therefore the triangle BFC is also equal to the triangle CGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend (________________); therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CBG.

Accordingly, since the whole angle ABG was proved equal to the angle ACF, and since ...

__________________________________________________________________

______________________________________________________________________

______________________________________________________________________

(________________). Therefore, angle ABC is equal to the angle ACB.

The very thing which it was necessary to show. (o{per e[dei deivxai)

Note: This is a complicated theorem involving two pairs of congruent triangles. In the Middle Ages it was called the "pons asinorum," Latin for "bridge of asses," because it was the first really difficult problem in the Elements and stumped many beginning students. Also, the diagram looks like a bridge, doesn't it?

In the last paragraph it is worth noting that Euclid places no reliance on a notion that we commonly accept, that there are always 180 degrees in a straight line.

Definitions (None is necessary.)


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.

Prior Theorems

1. upon a given, straight, bounded line to construct an equilateral triangle
2. upon a given point to place a straight line equal to a given straight line
3. given two unequal straight lines, to cut off from the greater a straight line equal to the less
4. if two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.


B. Prove two of the historical propositions using at least two different pages from my history of The Elements as building blocks for each. You must use a total of four different pages in your essays. If you want to be really fancy, prove one first and then use it as a building block for your second proof.


  CONTENTS ARTICLES ESSAYS UNIT 1 UNIT 2 UNIT 3 UNIT 4 REFERENCES