CHAPTER II.

THE SCIENCE OF GEOMETRY.

1. THE science of geometry consists of two things: the forms of geometrical quantity and the relations that exist between them.

2. The forms of geometrical quantity give rise to our geometrical ideas, and their relations give rise to geometrical truths.

3. The description of the various forms of geometrical quantity gives rise to the definitions of geometry.

I. DEFINITIONS.-The Definitions of geometry are the descriptions of geometrical quantities.

4. The definitions of geometry are usually formed by placing the thing defined in the next higher class, and then stating the difference between it and others of the class.

5. Thus, in the definition "A triangle is a polygon of three sides," polygon is the next higher class, and three sides is the distinguishing characteristic.

6. This next higher class is called the genus, and the specific difference is called the differentia. Thus, in the above definition polygon is the genus and three sides is the differentia.

7. Such definitions are called logical definitions. A logical definition is one that defines by genus and differentia. Logical definitions are the best specimens of perfect definitions.

II. TRUTHS OF GEOMETRY.-The truths of geometry are of two kinds: those which are self-evident, and those which are proved by a process of reasoning.

8. The self-evident truths are called axioms; as, “Things which are equal to the same thing are equal to one another."

9. The derived truths are called theorems or propositions; as, "The sum of the angles of a plane triangle equals two right angles."

10. A theorem is often stated so that it consists of two parts the hypothesis, or that which is assumed; and the conclusion, or that which follows from the hypothesis. Thus, "If A is equal to B, C is equal to D.”

11. The converse of a theorem is formed by reversing the hypothesis and conclusion. Thus,

"If A is equal to B, C is equal to D;" this is direct. "If C is equal to D, A is equal to B;" this is the converse. 12. The opposite of a proposition is formed by stating the negative of its hypothesis and conclusion. Thus,

"If A is equal to B, C is equal to D," is direct.

"If A is not equal to B, C is not equal to D," is opposite.

13. The converse of a proposition is not necessarily true. Thus, while "All birds are bipeds" is true, its converse, "All bipeds are birds," is not true.

14. The relation of these three forms of the proposition is indicated in the following principles:

1. If a direct proposition and its converse are true, the opposite proposition is true.

2. If a direct proposition and its opposite are true, the converse proposition is true.

III. REASONING.-The process by which we discover and prove the theorems of geometry is called reasoning.

15. Reasoning may be defined as the process of deriving

a truth from one or more truths. Thus, if we know that "AB" and "B= C," we can immediately infer that "A= = C."

16. Reasoning in geometry begins with the self-evident truths, but afterward often makes use of derived truths in order to prove a new truth.

17. The simplest form of geometrical reasoning is that in which we discover the relation of two quantities on account of their relation to the same quantity.

Thus, if we know that angle A= angle B, and angle C = angle B, we can infer that angle A= angle C. The basis of this inference is the axiom that things that are equal to the same thing are equal to one another.

18. Another form of reasoning in frequent use is that in which if two quantities are equal the results of increasing or diminishing them equally are equal.

Thus, if we have A + B = C+ D, and BC, by subtracting B from the first expression and C from the second, we have A = D.

19. Another form of reasoning of frequent use is that in which one of two equal quantities is substituted in two equal expressions of quantities.

Thus, if A = B + C, and B = C, we may substitute B for C, and have A B + B, or A = 2B.

=

20. There are still other forms of comparison embraced under the general form of reasoning given above, to which the student's attention may be called as he meets them.

21. Methods of Reasoning.-There are two distinct methods of geometrical reasoning, which have been distinguished as the synthetic and analytic methods.

22. The Synthetic Method begins with the self-evident

truths or truths already proved, and passes, step by step, to the truth to be proved. This method is usually called a demonstration.

23. There are also two distinct methods of demonstration. The simplest form is that in which figures are directly compared by applying one to the other. This is called the method of superposition.

24. The more general form of demonstration is that in which truths are proved by comparing the different quantities or elements by a reference to the axioms or the truths previously proved.

25. The Analytic Method begins with an assumption, and traces the relation backward until we arrive at some knowu truth. If the result attained is true, we infer that our

assumption is true, and vice versa.

26. The synthetic method is adapted to proving a truth already known; the analytic method is adapted to the discovery of truth.

27. Reductio ad Absurdum.-The method called reductio ad absurdum consists in supposing that the proposition to be proved is not true, and then showing that such a supposition leads to a contradiction of some known truth. This proves a theorem to be true by simply showing that it cannot be false.

28. The method is frequently used in proving the converse of a proposition when there is no good direct method; it is also often used in treating incommensurable quantities.

29. This method of reasoning is known as the Indirect Method, to distinguish it from the other, which is called the Direct Method. It is not considered so satisfactory as the

direct method, and should never be used except when no good direct method can be found.

30. Errors in Reasoning.-There are two errors in reasoning into which young geometricians are liable to fall, called Reasoning in a Circle and Begging the Question.

31. We reason in a circle when, in demonstrating a truth, we employ a second truth which cannot be proved without the aid of the truth we are trying to demonstrate.

32. We are said to beg the question when in order to establish a proposition we employ the proposition itself.

CHAPTER III.

GEOMETRICAL LANGUAGE.

1. LANGUAGE is the means of expressing our ideas and thoughts.

2. We think by means of language, and hence language may be considered as an instrument of thought as well as a medium of expression.

3. The language of mathematics differs somewhat from the language of ordinary usage in being more concise and definite in its meaning. Much of the language of mathematics is symbolical; that is, a symbol is used in place of the written word.

4. The Symbols of Geometry are of three classes: symbols of Quantity, symbols of Operation, and symbols of Relation. 5. The SYMBOLS OF QUANTITY are usually pictured representations of the quantities considered. These quantities are also expressed by the letters of the alphabet. Symbols