GEOMETRY INTRODUCTION FUNDAMENTAL CONCEPTIONS 1. Def.-Geometry is the science of space. 2. Every one has a notion of space extending indefinitely in all directions. Every material body, as a rock, a tree, or a house, occupies a limited portion of space. The portion of space which a body occupies, considered separately from the matter of which it is composed, is a geometrical solid. The material body is a physical solid. Only geometrical solids are here considered, and they are called simply solids. Def.-A solid is, then, a limited portion of space. 3. Def.-The boundaries of a solid are surfaces (that is, the surfaces separate it from the surrounding space). A surface is no part of a solid. 4. Def.-The boundaries of a surface are lines. A line is no part of a surface. 5. Def.-The boundaries (or ends) of a line are points. A point is no part of a line. 6. The solid, surface, line, and point are the four fundamental conceptions of geometry. They may also be considered in the reverse order, thus: (1.) A point has position but no magnitude. (2.) If a point moves, it generates (traces) a line. This motion gives to the line its only magnitude, length. (4.) If a surface moves (not along itself), it generates a solid. This motion gives to the solid, besides length and breadth, thickness. Def.-A figure is any combination of points, lines, surfaces, or solids. 7. Def.-A straight line is a line which is the shortest path between any two of its points. 8. Def. A plane surface (or simply a plane) is a surface such that, if any two points in it are taken, the straight line passing through them lies wholly in the surface. 9. Def.-Two straight lines are parallel which lie in the same plane and never meet, however far produced. GEOMETRIC AXIOMS 10. All the truths of geometry rest upon three fundamental axioms, viz.: (a.) Straight line axiom.-Through every two points in space there is one and only one straight line. This is sometimes expressed as follows: Two points determine a straight line. ↓ (b.) Parallel axiom.-Through a given point there is one and only one straight line parallel to a given straight line. (c.) Superposition axiom.-Any figure in a plane may be freely moved about in that plane without change of size or shape. Likewise, any figure in space may be freely moved about in space without change of size or shape. GENERAL AXIOMS 11. In reasoning from one geometric truth to another the following general axioms are also employed, viz. : (1.) Things equal to the same thing are equal to each other. 2 (3.) If equals be taken from equals, the remainders are equal. (4.) If equals be added to unequals, the wholes are unequal in the same order. (5.) If equals be taken from unequals, the remainders are unequal in the same order. (6.) If unequals be taken from equals, the remainders are unequal in the opposite order. (7.) If equals be multiplied by equals, the products are equal; and if unequals be multiplied by equals, the products are unequal in the same order. (8.) If equals be divided by equals, the quotients are equal; and if unequals be divided by equals, the quotients are unequal in the same order. (9.) If unequals be added to unequals, the lesser to the lesser and the greater to the greater, the wholes will be unequal in the same order. (10.) The whole is greater than any of its parts. (11.) The whole is equal to the sum of all its parts. (12.) If of two unequal quantities the lesser increases (continuously and indefinitely) while the greater decreases; they must become equal once and but once. (13.) If of three quantities the first is greater than the second and the second greater than the third, then the first is greater than the third. 12. Def.-Plane Geometry treats of figures in the same plane. 13. Def.-Solid Geometry, or the geometry of space, treats of figures not wholly in the same plane. PLANE GEOMETRY BOOK I FIGURES FORMED BY STRAIGHT LINES 14. Defs.—An angle is a figure formed by two straight lines diverging from the same point. This point is the vertex of the angle, and the lines are its sides. A clear notion of an angle may be obtained by observing the hands of a clock, which form a continually varying angle. & A FIG. I B B' FIG. 2 We may designate an angle by a letter placed within as a and b in Fig. 1, and in Fig. 2. Three letters may be used, viz.: one letter on each of its sides, together with one at the vertex, which must be written between the other two, as AOC, BOC, and AOB in Fig. 1, and A'O'B′ in Fig. 2. If there is but one angle at a point, it may be denoted by a single letter at that point, as O' in Fig. 2. Angles with a common vertex and side, as a and b, are said to be adjacent. 15. Def.—Two angles are equal if they can be made to coincide. Also, in general, any two figures are equal which can be made to coincide. Thus, suppose we place the angle AOB on the angle A'O'B' so that O shall fall at O', and the side OA along O'A'; then, if the side OB also falls along O'B', the angles are equal, whatever may be the length of each of their sides. 16. Def.-When one straight line is drawn from a point in another, so that the two adjacent angles are equal, each of these angles is a right angle, and the lines are perpendicular. A RIGHT ANGLES ACUTE ANGLE OCTUSE ANGLE Thus, if the angles AOC and COB are equal, they are right angles, and CO is perpendicular to AB. When a straight line is perpendicular to another straight line, its point of intersection with the second line is called the foot of the perpendicular. 17. Def.-An acute angle is an angle less than a right angle; an obtuse angle, greater. The term oblique angle may be applied to any angle which is not a right angle. |