THE ELEMENTS OF GEOMETRY.

BOOK I.

CHAPTER I.

ON LOGIC.

I. Definitions. - Statements.

1. A statement is any combination of words that is either true or false; e.g. (exempli gratia, "for example "),

All x's are y's.

2. To pass from one statement to another, with a consciousness that belief in the first warrants belief in the second, is to infer.

3. Two statements are equivalent when one asserts just as much as the other, neither more nor less; e.g., A equals B, and B equals A.

4. A statement is implied in a previous statement when its truth follows of necessity from the truth of the previous statement; e.g.,

All x's are y's implies some y's are x's.

I

5. Any declarative sentence can be reduced to one or more simple statements, each consisting of three parts, namely, two terms or classes, and a copula, or relation, connecting them.

A equals B, x is y, are simple examples of this typical form of all statements. Here x and y each stand for any word or group of words that may have the force of a substantive in naming a class; e.g., x is y may mean all the x's are y's; man is mortal, means, all men are mortals.

6. A sentence containing only one such statement is a logically simple sentence; e.g., Man is the only picture-making animal.

7. A sentence that contains more than one such statement is a logically composite sentence; e.g., A and B are respectively equal to C and D, is composite, containing the two statements A equals C, and B equals D.

8. Statements that are expressly conditional, such as, if A is B, then C is D, reduce to the typical form as soon as we see that they mean

(The consequence of M) is N.

9. In the typical form, bird is biped, x is y, we call x and y the terms of the statement; the first being called the subject, and the last the predicate.

II. Definitions. — Classes.

10. Terms that name a single object are called individual term; e.g., Newton.

11. Terms that name any one of a group of objects are called class terms; e.g., man, crystal. The name stands for any object that has certain properties, and these properties are possessed in common by the whole group for which the name stands.

12. A class is defined by stating enough properties to decide whether a thing belongs to it or not.

Thus, "rational animal" was given as a definition of "man." 13. If we denote by x the class possessing any given property, all things not possessing this property form another class, which is called the contradictory of the first, and is denoted by non-r, meaning "not x;" e.g., the contradictory of animate is inanimate.

14. Any one thing belongs either to the class x or to the class non-r, but no thing belongs to both.

It follows that x is just as much the contradictory of non-r as non-x is of x. So any class y and the class non-y are mutually the contradictories of each other, and both together include all things in the universe; e.g., unconscious and conscious.

III. The Universe of Discourse.

15. In most investigations, we are not really considering all things in the world, but only the collection of all objects which are contemplated as objects about which assertion or denial may take place in the particular discourse. This collection we call our universe of discourse, leaving out of consideration, for the time, every thing not belonging to it.

Thus, in talking of geometry, our terms have no reference to perfumes.

16. Within the universe of discourse, whether large or small, the classes x and non-x are still mutually contradictory, and every thing is in one or the other; e.g., within the universe mammals, every thing is man or brute.

17. The exhaustive division into x and non-x is applicable to any universe, and so is of particular importance in logic. But a special universe of discourse may be capable of some entirely different division into contradictories, equally exhaustive. Thus, with reference to any particular magnitude, all

magnitudes of that kind may be exhaustively divided into the contradictories,

Greater than, equal to, less than.

IV. Contranominal, Converse, Inverse, Obverse.

18. If x and y are classes, our typical statement x is y means, if a thing belongs to class x, then it also belongs to class y; e.g., Man is mortal, means, to be in the class men, is to be in the class mortals.

If the typical statement is true, then every individual belongs to the class y: hence no belongs to the class non-y, or no thing not y is a thing ; that is, every non-y is non-x:

e.g., the immortals are not-human.

The statements x is y, and non-y is non-x, are called each the contranominal form of the other.

Though both forms express the same fact, it is, nevertheless, often of importance to consider both. One form may more naturally connect the fact with others already in our mind, and so show us an unexpected depth and importance of meaning.

19. Since x is y means all the x's are y's, the class Y thus contains all the individuals of the class x, and may contain others, besides. Some of the y's, then, must be x's. Thus, from "a crystal is solid" we infer "some solids are crystals."

This guarded statement, some y is x, is called the logical converse of x is y. It is of no importance in geometry.

20. If, in the true statement x is y, we simply interchange the subject and predicate, without any restriction, we get the inverse statement y is x, which may be false.

In geometry it often happens that inverses are true and important. When the inverse is not true, this arises from the circumstance that the subject of the direct statement has been more closely limited than was requisite for the truth of the

statement.

21. The contranominal of the inverse, namely, non-x is non-y, is called the obverse of the original proposition.

Of course, if the inverse is true, the obverse is true, and vice versa. To prove the obverse, amounts to the same thing as proving the inverse. They are the same statement, but may put the meaning expressed, in a different light to our minds.

22. If the original statement is x is y, its contranominal is non-y is non-x, its inverse is y is x, its obverse is non-x is non-y. The first two are equivalent, and the last two are equivalent.

Thus, of four such associated theorems it will never be necessary to demonstrate more than two, care being taken that the two selected are not contranominal.

23. From the truth of either of two inverses, that of the other cannot be inferred. If, however, we can prove them both, then the classes x and y are identical. A perfect definition is always invertable.

V. On Theorems.

24. A theorem is a statement usually capable of being inferred from other statements previously accepted as true.

25. The process by which we show that it may be so inferred is called the proof or the demonstration of the theorem.

26. A corollary to a theorem is a statement whose truth follows at once from that of the theorem, or from what has been given in the demonstration of the theorem.

27. A theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom.

28. A geometric theorem usually relates to some figure, and says that a figure which has a certain property has of necessity also another property; or, stating it in our typical form, x is y, "a figure which has a certain property" is "a figure having another specified property."