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A GEOMETRICAL SYLLOGISM.
Here we have a
straight line. If we
want to speak of it, we might point to it and say, that straight line. But if we want to speak of it without pointing to it, we must give it a name. Just as we give names to things, that we may be able to talk about them; so we give names to points and lines, and circles and triangles, that we may be able to talk about them. Now we might call this line the line Fchn, or by any other such name. But we find it more convenient to call geometrical figures by the letters of the alphabet.
Thus we may call this line AB; placing the letter A at one end, and the letter B at the other. So that when we speak of the line AB, the letters A B are simply the name by which we distinguish that particular line from all other lines.
You know, I daresay, that a point marks position; and that the extremities (or ends) of lines are points. Now let us make a syllogism about the line AB. (Refer to general syllogism on page 10). Here is one:
(a.) Every line has two points.
(b.) The figure A B is a line.
(c.) Therefore the figure A B has two points. It is not always necessary to write out the whole of a syllogism in this way. Sometimes one of the premisses is so evident that it is left out, and only one premiss and the conclusion given. But, whether expressed or understood, two premisses are absolutely necessary to form a conclusion. To distinguish one premiss from the other, the first is called the major premiss, and the second the minor premiss.
In the above syllogism for instance, we might leave out the major premiss (a), and the argument would stand thus:
The figure A B is a line.
Therefore the figure A B has two points.
Here the major premiss (every line has two points) is left to be understood, but is none the less necessary to the conclusion.
Put in the Premiss which has been omitted in each of the following Syllogisms.
Conclusion... A B and CD if produced will never
Minor Premiss. A B and CD are E each equal to E F
Conclusion... AB is equal to CD. C
A proposition in geometry is something proposed to be done or proved. If something is to be done, the proposition is called a problem; if something is to be proved, the proposition is called a theorem. But in both cases a proof is required.
Thus in a problem, it is not enough to do what is required to be done. We must also prove, by syllogisms, that it is done.
Every proposition, whether problem or theorem, consists of two parts. That which is given, and that which is required.
Here is a proposition:
If two circles have equal radii, those circles are equal.
Here we are told that two circles have equal radii. That is said to be given. We are then asked to
prove that those circles are equal. This is said to be required.
Here is another proposition:
On a given straight line to describe an equilateral triangle.
NOTE.-To describe means to draw, to construct.
In this case we are given-a straight line; and we are required to describe an equilateral triangle on it. EXERCISE.-Rule your exercise paper in two columns, thus:
and place each of the two parts of the following propositions in its proper column.
1. From a given point to draw a straight line equal to a given straight line.
2. From the greater of two given straight lines to cut off a part equal to the less.
3. If two triangles have two sides and the included angle of one, equal to two sides and the included angle of the other each to each, the bases shall be equal.
NOTE.-When two sides of a triangle have been mentioned, the third side is called the base.
4. The angles at the base of an isosceles triangle are equal.
5. If two straight lines cut one another, the vertical (or opposite) angles are equal.
An axiom is a self-evident truth. That is to say, an assertion so evidently true that it need not (or indeed cannot) be proved.
The following are the axioms of Euclid:
1. Things which are equal to the same thing are equal to one another.
2. If equals be added to equals, the wholes are equal.
2a. (Not given by Euclid, but assumed by him). If the same thing be added to equals the wholes are equal. 3. If equals be taken from equals, the remainders are equal.
3a. (Not given by Euclid, but assumed by him.) If the same thing be taken from equals, the remainders are equal.
4. If equals be added to unequals, the wholes are unequal.
4a. (Not given by Euclid, but assumed by him.) It the same thing be added to unequals the wholes are unequal.
5. If equals be taken from unequals the remainders are unequal.
6. Things which are doubles of the same thing are equal to one another.
6a. (Not given by Euclid, but assumed by him). Things which are doubles of equal things are equal to one another.
7. Things which are halves of the same thing are equal to one another.
8. Magnitudes which coincide with one another are equal to one another.