TABLE OF SYMBOLS.

We denote

triangle by ▲; the vertices by A, B, C;
the angles at A, B, C by a, ß, r;
the opposite sides by a, b, c;

the altitudes from A, B, C by ha, hb, he;
the bisectors of a, ß, r by ta, tb, tc;
the medians to a, b, c by ma, mb, mc;

the feet of ha, hb, hc by D, E, F;
the centroid by G;

the orthocenter by H;

the in-center by I; the in-radius by r;

the ex-centers beyond a, b, c by I1, I, I,; their ex-radii

by r1, 72, 73;

the circumcenter by 0; the circumradius by R;

angle by ; angles by xs;

angle made by the rays BA and BC by ABC;

angle made by the rays a and b both from the point O by

(a, b) or ab;

bisector by bi'; circle by ; circles by Os;

circle with center C and radius r by OC(r);

congruent by =;

equal or equivalent by completion by =; for example [exempli gratia] by e...

parallelogram by g'm; perimeter [sum of sides] by p; perpendicular by 1; perpendiculars by Ls;

plus by +;

quadrilateral by quad';

right by r't;

spherical angle by &;

spherical triangle by Â;

similar by ~; symmetrical by +; therefore by ..

RATIONAL GEOMETRY.

CHAPTER I.

ASSOCIATION.

THE GEOMETRIC ELEMENTS.

1. Geometry is the science created to give understanding and mastery of the external relations of things; to make easy the explanation and description of such relations and the transmission of this mastery.

2. Convention. We think three different sorts of things. The things of the first kind we call points, and designate them by A, B, C, . . . ; the things of the second system we call straights, and designate them by a, b, c, ...; the things of the third set. we call planes, and designate them by a, ß, r, . . .

3. We think the points, straights, and planes in certain mutual relations, and we designate these relations by words such as "lie,” “between,' parallel," "congruent."

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The exact and complete description of these rela

tions is accomplished by means of the assumptions of geometry.

4. The assumptions of geometry separate into five groups. Each of these groups expresses certain connected fundamental postulates of our intuition.

I. The first group of assumptions: assumptions of association.

5. The assumptions of this group set up an association between the concepts above mentioned, points, straights, and planes. They are as follows: I 1. Two distinct points, A, B, always determine a straight, a.

Of such points besides "determine" we also employ other turns of phrase; for example, A “lies on" a, A "is a point of" a, a "goes through” A "and through" B, a “joins" A "and" or "with" B, etc.

When we say two things determine some other thing, we simply mean that if the two be given, then this third is explicitly and uniquely given.

If A lies on a and besides on another straight b we use also the expression: "the straights" a “and” b "have the point A in common."

I 2. ANY two distinct points of a straight determine THIS straight; and on every straight there are at least two points.

That is, if AB determine a and AC determine a, and B is not C, then also B and C determine a. I 3. Three points, A, B, C, not costraight, always determine a plane a.

We use also the expressions:

A, B, C "lie in" a, A, B, C, "are points of" a,

etc.

I 4. ANY three non-costraight points A, B, C of a plane a determine THIS plane a.

I 5. If two points A, B of a straight a lie in a plane a, then every point of a lies in a.

In this case we say: The straight a lies in a.

I 6. If two planes a, ẞ have a point A in common, then they have besides at least another point B in

common.

I7. In every plane there are at least three noncostraight points. There are at least four non-costraight non-coplanar points.

6. Theorem. Two distinct straights cannot have two points in common.

Proof. The two points being on the first straight determine (by I 2) that particular straight. If by hypothesis they are also on a second straight, therefore (by I 2) they determine this second straight. Therefore the first straight is identical with the second.

7. Theorem. Two straights have one or no point in common.

Proof. By 6 they cannot have two.

8. Theorem. Two planes have no point or a straight in common.

Proof. If they have one point in common, then (by I 6) they have a second point in common, and therefore (by I 5) each has in it the straight which (by I 1) is determined by these two points.

9. Corollary to 8. A point common to two planes