GEOMETRY. PRELIMINARY DEFINITIONS. 1. Space has extension in all directions, and so far as our experience can teach us it is limitless. 2. A material or physical body occupies a definite portion of space, and this space freed from the body is called a geometrical solid, which for brevity will be known as a Solid. 3. The limits, or boundaries, of a solid are Surfaces. The limits, or boundaries, of a surface are Lines. The intersection of two lines is a Point. 4. Therefore a solid has three dimensions: Length, Breadth, and Thickness. A surface has only two dimensions: length and breadth. A line has only one dimension: length. A point is without dimension, having simply position. 5. In drawings and diagrams material figures are employed for purposes of demonstration, but they are merely the representatives of mathematical figures. 6. A Straight Line, or Right Line, is the shortest line between two points; as AB A B F 9. A Plane Surface, or simply a Plane, is one such that the straight line which joins any two of its points lies entirely in the surface. 10. A Carved Surface is one no portion of which is plane. 11. A Geometrical Figure is any combination of points, lines, surfaces, or solids formed under specific conditions. Plane Figures are formed by points and lines in a plane; Rectilinear, or Right-lined Figures, are formed of straight lines. 12. Geometry is that branch of mathematics which treats of the construction of figures, of their measurement, and of their properties. Plane Geometry treats of plane figures. Solid Geometry, sometimes called Geometry of Space and Geometry of Three Dimensions, treats of solids, of curved surfaces, and of all figures that are not represented on a plane. 13. A Theorem is a truth requiring demonstration. 14. A Problem is a question proposed for solution. 15. A Postulate assumes the possibility of the solution of some problem. 16. An Axiom is a truth assumed to be true, or a truth verified by intuition or our experience with material things. 17. A Proposition is a general term for theorem, axiom, problem, and postulate. 18. A Demonstration is the course of reasoning by which the truth of a theorem is established. 19. A Corollary is a conclusion which follows immediately from a theorem, but this conclusion may at times demand demonstration. 20. A Lemma is an auxiliary theorem required in the demonstration of a principal theorem. 21. A Scholium is a remark upon one or more propositions. 22. An Hypothesis is a supposition made either in the enunciation of a proposition or in the course of a demonstration. 23. A Solution of a problem is the method of construction which accomplishes the required end. 24. A Construction is the drawing of such lines and curves as may be required to prove the truth of a theorem, or to solve a problem. 25. The Enunciation of a theorem consists of two parts: the Hypothesis, or that which is assumed; and the Conclusion, or that which is asserted to follow therefrom. 27. Given POSTULATES. 26. 1. A straight line can be drawn between any two points. 2. A straight line can be produced, indefinitely in either direction. Prove Theorem That something is true Problem That something can be done AXIOMS. 28. 1. Things which are equal to the same thing are equal to each other. 2. If equals be multiplied or divided by equals, the results will be equal. 3. If equals be added to or subtracted from equals, the results will be equal. 4. If equals are added to or subtracted from unequals, the results will be unequal. 5. The whole is equal to the sum of its parts. 6. The whole is greater than any of its parts. ABBREVIATIONS. 29. The following is a list of the symbols which will be used as abbreviations: [R, right angle. In addition to these, the following may be used for writing demonstrations on the board or in exercise books, but no use is made of them in the present work. 1, perpendicular. Il, parallel. lls, parallels. ▲, triangle. >, is greater than. <, is less than. .*., therefore. <, angle. A, triangles. O, parallelogram. , circles. , arc. PLANE GEOMETRY BOOK I. RECTILINEAR FIGURES. 30. An Angle is the difference in direction of two lines; if the two lines meet, the point of meeting is called the Vertex, and the lines are called its Sides. Thus, in the angle formed by AB and BC, B is the vertex, and AB and BC are the sides. A B с 31. An isolated angle may be designated by the letter at its vertex, as "the angle 0"; but when several angles are formed at the same point by different lines, as OA, OB, OC, we designate the angle intended by three letters; namely, by one letter on each of its sides, together with the one at its vertex, which must be written between the other two. Thus, with these lines there are formed three different angles, which are distinguished as AOB, BOC, and AOC. B A 32. Two angles, such as AOB, BOC, which have the same vertex O and a common side OB between them, are called Adjacent. 33. The magnitude of an angle depends wholly upon the amount of divergence of its sides, and is independent of their length. |