THEOREM 59. [Euc. III. 37.] If from a point outside a circle two straight lines are drawn, one of which cuts the circle, and the other meets it; and if the rectangle contained by the whole line which cuts the circle and the part of it outside the circle is equal to the square on the line which meets the circle, PROBLEM 33. To divide a given straight line so that the GEOMETRY. PART I. AXIOMS. ALL mathematical reasoning is founded on certain simple principles, the truth of which is so evident that they are accepted without proof. These self-evident truths are called Axioms. For instance: Things which are equal to the same thing are equal to one another. The following axioms, corresponding to the first four Rules of Arithmetic, are among those most commonly used in geometrical reasoning. Addition. If equals are added to equals, the sums are equal. Subtraction. If equals are taken from equals, the remainders ure equal. Multiplication. Things which are the same multiples of equals are equal to one another. For instance: Doubles of equals are equal to one another. Division. Things which are the same parts of equals are equal to one another. For instance: Halves of equals are equal to one another. The above Axioms are given as instances, and not as a complete list, of those which will be used. They are said to be general, because they apply equally to magnitudes of all kinds. Certain special axioms relating to geometrical magnitudes only will be stated from time to time as they are required. DEFINITIONS AND FIRST PRINCIPLES. Every beginner knows in a general way what is meant by a point, a line, and a surface. But in geometry these terms are used in a strict sense which needs some explanation. 1. A point has position, but is said to have no magnitude. This means that we are to attach to a point no idea of size either as to length or breadth, but to think only where it is situated. A dot made with a sharp pencil may be taken as roughly representing a point; but small as such a dot may be, it still has some length and breadth, and is therefore not actually a geometrical point. The smaller the dot however, the more nearly it represents a point. 2. A line has length, but is said to have no breadth. A line is traced out by a moving point. If the point of a pencil is moved over a sheet of paper, the trace left represents a line. But such a trace, however finely drawn, has some degree of breadth, and is therefore not itself a true geometrical line. The finer the trace left by the moving pencil-point, the more nearly will it represent a line. 3. Proceeding in a similar manner from the idea of a line to the idea of a surface, we say that A surface has length and breadth, but no thickness. And finally, A solid has length, breadth, and thickness. Solids, surfaces, lines and points are thus related to one another : (i) A solid is bounded by surfaces. (ii) A surface is bounded by lines; and surfaces meet in lines. (iii) A line is bounded (or terminated) by points; and lines meet in points. 4. A line may be straight or curved. A straight line has the same direction from point to point throughout its whole length. A curved line changes its direction continually from point to point. AXIOM. There can be only one straight line joining two given points: that is, Two straight lines cannot enclose a space. 5. A plane is a flat surface, the test of flatness being that if any two points are taken in the surface, the straight line between them lies wholly in that surface. 6. When two straight lines meet at a point, they are said to form an angle. B The straight lines are called the arms of the angle; the point at which they meet is its vertex. The magnitude of the angle may be thus ō explained: A Suppose that the arm OA is fixed, and that OB turns about the point (as shewn by the arrow). Suppose also that OB began its turning from the position OA. Then the size of the angle AOB is measured by the amount of turning required to bring the revolving arm from its first position OA into its subsequent position OB. Observe that the size of an angle does not in any way depend on the length of its arms. Angles which lie on either side of C a common arm are said to be ad jacent. For example, the angles AOB, BOC, which have the common arm OB, are adjacent. When two straight lines such as AB, CD cross one another at O, the angles COA, BOD are said to be vertically opposite. The B angles AOD, COB are also vertically opposite to one another. B A |