2. The side FB is equal to the side G C, as may thus be shown:-The whole AF is equal to the whole A G (cons.), parts of which, namely, A B and A C, are also equal (hyp.). Therefore the remainders F B and GC are equal (Ax. 3).

3. Wherefore the two sides CF, FB of the triangle FCB are equal to the two sides BG, GC of the triangle G B C, each to each.

4. And the included angle B FC has been proved to be equal to the included angle C G B (Part I.).

5. Therefore, by Prop. IV., the two triangles FCB and GBC are equal in every respect. Wherefore the remaining angles, to which the equal sides are opposite, are equal, viz. the angle FBC to the angle G C B, and the angle B C F to the angle CB G.

PART III.

1. Now the whole angle A B G was proved, in Part I., to be equal to the whole angle AC F.

2. Parts of which, viz. the angle C BG, and the angle BCF, were proved equal, in Part II.

3. Therefore the remaining angle A B C is equal to the remaining angle ACB (Ax. 3). And these are the angles at the base of the isosceles triangle. 4. Also the angle FBC has been shown (Part II.) to be equal to the angle G C B. And these are the angles on the other side of the base.

Wherefore, if a triangle is isosceles, the angles at the base are equal; and if the equal sides be produced, the angles on the other side of the base are also

equal.

Q.E.D.

1

From the above proposition may easily be deduced the Corollary that every equilateral triangle is also equiangular.

Let A B C be an equilateral triangle:

It is required to prove that it is equiangular.

A

B

Demonstration of Corollary.

1. Any two of its sides (as A. B and A C) are equal; therefore, by Prop. V. the angle A B C is equal to the angle A CB.

2. Also, any other two of its sides (as A B and B C) are equal; therefore, by Prop. V., the angle BAC is equal to the angle B C A.

3. Also the two other sides, A C, CB, are equal; therefore, again by Prop. V., the angles CAB and CBA are equal.

Therefore all the angles are equal.

Q.E.D.

[Of course, having proved that any two angles of the equilateral triangle are, each of them, equal to the third angle, as is done in 1 and 2, it might be inferred by Axiom 1 that they are equal to one another.]

1 Whatever may be obviously gathered or deduced from a proposition is called a 'corollary' to it. Corollarium meant originally a garland or wreath of thin metal given as a reward. Perhaps the idea of a wreath or garland, hanging from its support, caused the word to be used later, by philosophical writers, to express a deduction, an inference.

THE SIXTH PROPOSITION DISCUSSED.

The general enunciation of the Sixth Proposition is as follows:

It is given that two angles of a triangle are equal to each other :

It is required to prove that the sides also which subtend or are opposite to the equal angles, are equal to one another.

Particular Enunciation.

Let ABC be a triangle having the angle A B C equal to the angle A CB:

It is required to prove that the side AB (opposite to the angle A C B) is equal to the side AC (opposite to the angle A B C).

This proposition is what is called a converse proposition. It is the converse of the Fifth. By writing out the enunciations of the two propositions, the Fifth and Sixth, side by side, you will see the meaning of the word 'converse.'

In the Fifth Proposition it is said:

If the side AB is equal to the side A C:-the angle A B C is equal to the angle A CB.

In the Sixth Proposition it is said:

If the angle ABC is equal to the angle ACB :-the side A B is equal to the side A C.

What is given in the Fifth has to be proved in the Sixth, and what is given in the Sixth has to be proved in the Fifth -in other words, the premises (what is given) and the conclusion (what has to be proved) change places.

Now very few converse propositions admit of a direct proof. Euclid cannot prove directly that, if the angle ABC is equal to the angle A C B, the side A C is equal to the side A B. But he can and does prove that it is false to say that A C and AB are unequal; and hence he infers that they are equal.

This method of indirect proof is called a Reductio ad absurdum.

B

A

an

To appreciate this method of proof, let us consider Euclid as forcing conviction on an tagonist who denies what Euclid contends is true.

D

Imagine this antagonist to say: 'I deny that A B is equal to A C.'

Euclid replies: 'If it is as you say— i.e., if AB is not equal to A C-one of them must be greater than the other.'

The opponent is obliged to admit this.

'Well, then,' says Euclid, 'suppose it is A B that is the greater, then we can, by Prop. III., cut off from AB, the greater, a part equal to A C the less.'

The opponent must admit this also.

'Suppose it done,' says Euclid, that is, suppose B D to be cut off from B A,-which you say is the greater,equal to A C the less.'

"Then,' adds Euclid, we will join C D.'

The opponent must allow him to do so: it is the first postulate.

'Now then,' says Euclid, 'I have before me two triangles ACB and DB C, the bases of which are AB and DC respectively; and these triangles I can prove equal in every respect by the Fourth Proposition, as follows:

1. The side D B of the triangle DBC has been cut off from A B [which the antagonist said was the greater] so as to be equal to the side A C of the triangle AC B.

2. The side B C is common to both triangles.

3. Wherefore the two sides DB, BC of the triangle DBC are equal to the two sides A C, CB, of the triangle АСВ.

4. And the angle D B C, contained by the two sides D B, BC, is, by the hypothesis, equal to the angle A C B contained by the two sides A C, C B.

5. Therefore, by Prop. IV., the two triangles A B C and DBC are equal in every respect, and therefore in respect of area. That is, the space enclosed by the three straight lines DB, BC, CD, is equal to the space enclosed by the three straight lines A B, BC, CA. But this first space is only a part of the second, and a whole is greater than a part (Axiom 9); therefore the conclusion that they are equal is absurd.'

[It is to be hoped that no one who has followed Euclid's reasoning so far will here be disposed to make the common, but weak observation, What is the use of proving what is absurd? but will rather see that Euclid has now completely 'shut up' his opponent. The opponent had said A B and A C are unequal. Euclid answers, For the sake of argument, suppose it to be as you say, and see what comes of it. And what does come of it? Why this, that if what the opponent asserts, viz., that AB and A C are unequal, is true, a part can be proved equal to a whole. But a part cannot equal a whole, therefore what the opponent asserts, viz., that AB, AC are unequal, is false,1 and all that Euclid has to add, is:]

'Therefore A B is not unequal to A C, that is, it is equal to it. Which was to be proved.'

'mise

This illustration of Euclid's reasoning, when en scène' in the following manner, has amused, and while 1 An absurd conclusion must result, either from the hypothesis (the foundation of the reasoning) being false, or from a flaw in the reasoning itself. It is hoped that the pupil's knowledge of the Fourth Proposition, and his power of rightly applying it, are now so confirmed that he will without hesitation decide :-' It is the hypothesis that is false.'