taken up in § 10. The following examples will show that they are not always true:

"An apple tree has leaves." It does not follow that (1) If it is not an apple tree, it has not leaves.

(2) If it has leaves, it is an apple tree.

"This desk is made of wood." It does not follow that (1) If it is not this desk, it is not made of wood. (2) If it is made of wood, it is this desk.

"A donkey has a head." It does not follow that because you have a head you are a donkey.

WARNING. Never assume that the negative or the converse of ONE true statement is also true.

10. Law of Converse. If conditional statements such that their conditions cover all possibilities, and no two conclusions can be true at once, are true, then the converses of those statements are also true.

NOTE. This law is discussed more fully in the Appendix, § 340.

This law will be understood when it is applied to definite cases, but the following examples make its meaning somewhat clearer.

(1) If A is true, x is true.

If A is not true, X is not true.

These two conditions (true and untrue) cover all possibilities, and the two conclusions (true and untrue) are such that they cannot both be true at the same time, so the converses are also true; i.e.

These three conditions (>,=, <) cover all possibilities,

=

and but one of the conclusions (>,: once, so the converses are also true; i.e.

If X> Y, A > B.

If x = Y, A = B.

If x < Y, A> B.

11. General and Special Cases.

<) can be true at

NOTE. This need not be read until the pupil is ready to begin the theorems.

Anything known of mankind is known of each man separately; but anything known of certain men only, would not necessarily be true of all men. If it is true for each man in existence, it is true for all mankind. In other words, anything known of a class as a whole, or of all members of the class, is known to be a characteristic of that class, both as a whole and by individuals. On the other hand, anything known of part of a class is not known of the class as a whole, or of other members of that class.

So in Geometry, proofs should be made for the general case whenever that is possible, and when that does not seem possible, the proof should be worked for each of the different cases separately. For example, in working with triangles, the triangle used should always be one about which no assumption (other than the given of the theorem) is made. The triangle should neither be isosceles, nor be assumed to have any certain sized angle, such as a right or acute angle. It is better to draw the triangle in the figure so that it does not even appear to have any special characteristic, or the one studying the figure may carelessly assume that the characteristic which the figure. appears to have really belongs to it.

In theorems where it does not seem possible to find a

proof for the general case (and these are comparatively rare), it is necessary to prove enough cases to cover all possibilities in order that the general case may be known; for example, a proof for right, acute, and obtuse-angled triangles would be true for all triangles. In the same way, a proof for triangles having three equal sides, two equal sides, and no equal sides, would be true for all triangles.

Sometimes a proof is true only for a special case on account of points or lines that are added to the figure and are assumed to lie in certain positions, when as a matter of fact they can equally well lie in other positions. If points or lines are added to a figure, the proof must hold for all possible positions in which they can lie. This is discussed in more detail in § 110.

SECTION II. POINTS, LINES, AND SURFACES

12. Geometry. This subject studies points, lines, and figures formed by them. It proves facts about the figures, and uses as a basis for the reasoning definitions and axioms.

13. Definitions. A definition is such a description of the thing defined as will distinguish it from all other things; it might be said to be an agreement as to what a term shall be used to indicate. Some things are of such simple nature that it is difficult, if not impossible, to define them in terms still simpler, and in such cases an explanation in regard to them may well take the place of a definition.

14. Axioms. A truth that is taken as one of the foundation facts of a subject is called an axiom. It is often defined as a truth so simple that it cannot be derived from truths still simpler; but for Elementary Geometry this is not strictly true (Appendix, § 343). It will be found that the axioms of Geometry are facts so self-evident that there is no doubt as to their truth.

15. Space. The space in which everything exists is, as far as experience shows, unlimited. At any rate, the space studied in Elementary Geometry (sometimes called Euclidean Space) is unlimited. Space is evidently divisible, for all bodies occupy portions of space.

16. Solids. Any limited portion of space, such as the space occupied by any body, is called-irrespective of the nature of the body which may occupy it--a geometric

solid, or simply a solid. Solids are said to have three dimensions: length, breadth, and thickness.

17. Surfaces. That which separates one portion of space from an adjoining portion is called a surface. If two adjoining lots are considered as extending down into the ground so that any distance down there is still a boundary between the lots, that boundary is a surface. It evidently does not occupy space, for any particle of soil belongs to one lot or to the other, yet there is a distinct boundary such that all on one side of it belongs to one lot, while all on the other side of it belongs to the other lot.

NOTE. The terms "side," "between," "within," "outside," will be used in this syllabus in their ordinary meaning without any attempt to define them geometrically.

Another example of a surface is the outside of any object, as a box. It separates the space occupied by the object from the space outside the object, but itself occupies no

space.

A surface may be limited or unlimited in extent, and may have limited portions; it is said to be two dimensional, having length and breadth.

18. Lines. That which separates one portion of a surface from an adjoining portion is called a line. The surface boundary between two lots is a line. The line occupies no space, yet definitely divides one lot from the other.

A line may be indefinite in extent, but has limited portions; it is said to be one dimensional, having length only.

19. Points. That which separates one portion of a line from an adjoining portion is called a point. If four lots come together in what is ordinarily called a corner,