4. If two lines cut each other, their common intersection is called a point. Thus, if the lines AB and CD cut each other, their common intersection, O, is a point. B 5. A solid has extension in every direction; but this is not true of surfaces and lines. A point has extension in no direction, but simply position in space. 6. A surface may be conceived as existing independently in space, without reference to the solid whose boundary it forms. In like manner, we may conceive of lines and points as having an independent existence in space. 7. A straight line is a line which has the same direction throughout its length; as AB. A straight line is also called a right line. NOTE. The word "line" will be used hereafter as signifying a straight line. A curved line, or simply a curve, is a line no portion of which is straight; as CD. A broken line is a line which is composed of different successive straight lines; as EFGH 8. A plane surface, or simply a plane, is a surface such that the straight line joining any two of its points lies entirely in the surface. Thus, the surface MN is a plane if the straight line PQ joining any two of its points lies entirely in the surface. M P N 9. A curved surface is a surface no portion of which is plane. 10. We may conceive of a straight line as being of unlimited extent in regard to length; and in like manner we may conceive of a plane as being of unlimited extent in regard to length and breadth. 11. A geometrical figure is any combination of points, lines, surfaces, or solids. 12. A plane figure is a figure formed by points and lines all lying in the same plane. 13. A geometrical figure is called rectilinear, or rightlined, when it is composed of straight lines only. 14. Geometry treats of the properties, construction, and measurement of geometrical figures. 15. Plane Geometry treats of plane figures only; Solid Geometry, also called Geometry of Space, or Geometry of Three Dimensions, treats of figures which are not plane. 16. An Axiom is a truth assumed as self-evident. A Theorem is a truth which requires demonstration. A Proposition is a general term for either a theorem or a problem. A Postulate is an assumption of the possibility of solving a certain problem. A Corollary is a secondary theorem, which is an immediate consequence of the proposition which it follows. A Scholium is a remark or note. An Hypothesis is a supposition made either in the statement or the demonstration of a proposition. One proposition is said to be the Converse of another when the hypothesis and conclusion of the first are respectively the conclusion and hypothesis of the second, 17. POSTULATES. 1. A straight line can be drawn between any two points. 2. A straight line can be produced indefinitely in either direction. 18. AXIOMS. 1. Things which are equal to the same thing, or to equals, are equal to each other. 2. If the same operation be performed upon equals, the results will be equal. 3. The whole is equal to the sum of all its parts. 4. The whole is greater than any of its parts.. 5. But one straight line can be drawn between two points. 6. A straight line is the shortest line between two points. 19. A straight line is said to be determined by any two of its points. SYMBOLS. 20. The following symbols will be used in the work: In addition to these, the following are useful in writing demonstrations on the blackboard, or in exercise books: PLANE GEOMETRY. BOOK I. RECTILINEAR FIGURES. DEFINITIONS AND GENERAL PRINCIPLES. 21. If two straight lines be drawn from the same point in different directions, the figure formed is called an Angle. Thus, if the straight lines OA and OB be drawn from the same point O in different directions, the figure AOB is an angle. A B The point O is called the vertex of the angle, and the lines OA and OB are called its sides. The symbol is used for the word "angle." 22. If there is but one angle at a given vertex, it may be designated by the letter at that vertex; but if two or more angles have the same vertex, it is necessary, in order to avoid ambiguity, to name also the letters at the extremities of the sides, placing the letter at the vertex between the others. Thus, we should call the angle of the preceding article "the angle 0"; but if there were other angles having the same vertex, we should read it either AOB or BOA. Another method of designating an angle is by means of a letter placed between its sides; an example of this will be found in § 71. 5 23. Two geometrical figures are said to be equal when one can be applied to the other so that they shall coincide throughout. 24. To prove that two angles are equal, it is not necessary to consider the lengths of their DE and EF, respectively, the angles are equal, even if the sides AB and BC are not equal in length to DE and EF, respectively. 25. Two angles are called adjacent when they have the same vertex, and a common side between them; as AOB and BOC. 26. Two angles are called vertical, or opposite, when the sides of one are the prolongations of the sides of the other; as DHF and GHE. PERPENDICULAR LINES. B XX E 27. If through a given point in a straight line a line be drawn meeting the given line in such a way as to make the adjacent angles equal, each of the equal angles is called a right angle, and the lines are said to be perpendicular to each other. Thus, if A be any point in the line CD, and the line AB be drawn in such a way as to make the angles BAC and BAD equal, each of these angles is a right angle, and the lines AB and CD are perpendicular to each other. B C D A |