10. An hypothesis is a proposition assumed to be true in order to argue from. The following expressions show the meaning of signs used through the book: A =B means that A is equal to B. A is taken twice, or A+A. A is taken twice as a factor, or A× A. AABC means the triangle ABC, as distinguished from the angle ABC. ... means therefore. Figures in parenthesis through a demonstration, thus, (I. 15), (VI. 2), refer to the previous proposition in which the statement was proved, meaning (Book I. Prop. 15), (Book VI. Prop. 2). PART I. PLANE GEOMETRY. BOOK I. LINES, ANGLES, TRIANGLES, PARALLELOGRAMS. DEFINITIONS. Point.-Line.-Surface. 1. A point is that which has position, but not magnitude. 2. A line is length without breadth. The extremities of lines are points. The intersection of one line with another is also a point. 3. If a line preserve the same direction throughout, it is a straight line. If it change its direction at every point, it is a curved line or curve. Corollary. Two straight lines cannot enclose space; neither can they coincide in any two points without coinciding altogether. 4. Parallel straight lines are such as are in the same plane, and being pro duced ever so far both ways do not meet. 5. A surface is that which has length and breadth only. 6. A plane surface is one in which, if any two points be taken, the straight line between them lies wholly in that surface. 企 Angle. 7. A plane rectilineal angle is the inclination to each other of two straight lines which meet. Α B D E If a number of angles be at one point, as B, they are designated by three letters, of which the middle one is at the vertex of the angle. Thus, DBC means the angle formed by the two lines DB, BC. If there be but one angle at a point, as at E, it may be designated simply as the angle E. 8. When a straight line standing on another straight line makes the adjacent angles equal to each other, each of them is called a right angle, and the straight line which stands on the other is called a perpendicular. 9. An obtuse' angle is one that is greater than a right angle. 10. An acute2 angle is one that is less than a right angle. Figure. 11. A figure is that which is enclosed by one or more boundaries. 12. Rectilineal3 figures are those contained by straight lines; 13. Triangles by three straight lines; 1 Obtusus, blunt. 3 Rectus, right; linea, a line. 2 Acutus, sharp-pointed. 4 Tres, tria, three; angulus, a corner. 14. Quadrilaterals' by four straight lines; Triangle. 16. Considering only the sides of a triangle, an equilateral3 triangle is one which has three equal sides. 17. An isosceles triangle is one which has two equal sides. 18. A scalene' triangle is one which has three unequal sides. 19. Considering only the angles of a triangle, a right-angled triangle is one which contains a right angle. 20. An obtuse-angled triangle is one which contains an obtuse angle. 21. An acute-angled triangle is one which contains three acute angles. Quadrilateral. 22. Of quadrilaterals, a parallelogram is one which has the opposite sides parallel to each other. |