ELEMENTS

OF

GEOMETRY.

CHAPTER I.-PRELIMINARY.

Article 1. BEFORE the student begins the study of geometry, he should know certain principles and definitions, which are of frequent use, though they are not peculiar to this science. They are very briefly presented in this chapter.

LOGICAL TERMS.

2. Every statement of a principle is called a PROPO

SITION.

Every proposition contains the subject of which the assertion is made, and the property or circumstance asserted.

When the subject has some condition attached to it, the proposition is said to be conditional.

The subject, with its condition, if it have any, is the HYPOTHESIS of the proposition, and the thing asserted is the CONCLUSION.

Each of two propositions is the CONVERSE of the other, when the two are such that the hypothesis of either is the conclusion of the other.

3. A proposition is either theoretical, that is, it declares that a certain property belongs to a certain thing; or it is practical, that is, it declares that something can be done.

Propositions are either demonstrable, that is, they may be established by the aid of reason; or they are indemonstrable, that is, so simple and evident that they can not be made more so by any course of reasoning.

A THEOREM is a demonstrable, theoretical proposition. A PROBLEM is a demonstrable, practical proposition. An AXIOM is an indemonstrable, theoretical proposition.

A POSTULATE is an indemonstrable, practical proposition.

A proposition which flows, without additional reasoning, from previous principles, is called a COROLLARY. This term is also frequently applied to propositions, the demonstration of which is very brief and simple.

4. The reasoning by which a proposition is proven is called the DEMONSTRATION.

The explanation how a thing is done constitutes the SOLUTION of a problem.

A DIRECT DEMONSTRATION proceeds from the premises by a regular deduction.

An INDIRECT DEMONSTRATION attains its object by showing that any other hypothesis or supposition than the one advanced would involve a contradiction, or lead to an impossible conclusion. Such a conclusion may be called absurd, and hence the Latin name of this method of reasoning-reductio ad absurdum.

A work on geometry consists of definitions, propositions, demonstrations, and solutions, with introductory or explanatory remarks. Such remarks sometimes have the name of scholia.

5. REMARK.-The student should learn each proposition, so as to state separately the hypothesis and the conclusion, also the condition, if any. He should also learn, at each demonstration, whether it is direct or indirect; and if indirect, then what is the false hypothesis and what is the absurd conclusion. It is a good exercise to state the converse of a proposition.

In this work the propositions are first enounced in general terms. This general enunciation is usually followed by a particular statement of the principle, as a fact, referring to a diagram. Then follows the demonstration or solution. In the latter part of the work these steps are frequently shortened.

The student is advised to conclude every demonstration with the general proposition which he has proven.

The student meeting a reference, should be certain that he can state and apply the principle referred to.

GENERAL AXIOMS.

6. Quantities which are each equal to the same quantity, are equal to each other.

7. If the same operation be performed upon equal quantities, the results will be equal.

For example, if the same quantity be separately added to two equal quantities, the sums will be equal.

8. If the same operation be performed upon unequal quantities, the results will be unequal.

Thus, if the same quantity be subtracted from two unequal quantities, the remainder of the greater will exceed the remainder of the less.

9. The whole is equal to the sum of all the parts. 10. The whole is greater than a part.

EXERCISE.

11. What is the hypothesis of the first axiom? Ans. If several quantities are each equal to the same quantity.

What is the subject of the first axiom? Ans. Several quantities.

What is the condition of the first axiom? Ans. That they are each equal to the same quantity.

What is the conclusion of the first axiom? Ans. Such quantities are equal to each other.

Give an example of this axiom.

RATIO AND PROPORTION

12. All mathematical investigations are conducted by comparing quantities, for we can form no conception of any quantity except by comparison.

13. In the comparison of one quantity with another, the relation may be noted in two ways: either, first, how much one exceeds the other, or, second, how many times one contains the other.

The result of the first method is the difference between the two quantities; the result of the second is the RATIO of one to the other.

Every ratio, as it expresses "how many times" one quantity contains another, is a number. That a ratio and a number are quantities of the same kind, is further shown by comparing them, for we can find their sum, their difference, or the ratio of one to the other.

When the division can be exactly performed, the ratio is a whole number; but it may be a fraction, or a radical, or some other number incommensurable with unity.

14. The symbols of the quantities from whose comparison a ratio is derived, are frequently retained in its expression. Thus,

The ratio of a quantity represented by a to another represented by b, may be written 7.

α

b

A ratio is usually written a: b, and is read, a is to b.

This retaining of the symbols is merely for convenience, and to show the derivation of the ratio; for a ratio may be expressed by a single figure, or by any other symbol, as 2, m, 1/3, or . But since every ratio is a number, therefore, when a ratio is thus expressed by means of two terms, they must be understood to represent two numbers having the same relation as the given quantities.

The second term is the standard or unit with which the first is compared.

So, when the ratio is expressed in the form of a fraction, the first term, or ANTECEDENT, becomes the numerator, and the second, or CONSEQUENT, is the denominator.

15. A PROPORTION is the equality of two ratios, and is generally written,

ab:: cd,

and is read, a is to b as c is to d,

but it is sometimes written,

all of which express the same thing: that a contains b exactly as often as e contains d.

The first and last terms are the EXTREMES, and the second and third are the MEANS of a proportion.

The fourth term is called the FOURTH PROPORTIONAL of the other three.

A series of equal ratios is written,

abcdef, etc.

When a series of quantities is such that the ratio of each to the next following is the same, they are written, abcd, etc.