The quadrilateral may then, by a similar process, be converted into an equivalent triangle, and thus a polygon of any number of sides may be gradually converted into an equivalent triangle. The area of this triangle may then be found. III. The third method is chiefly employed in practice by Surveyors. с D M N Let ABCDEFG be an irregular polygon. Draw AE, the longest diagonal, and drop perpendiculars on AE from the other angular points of the polygon. The polygon is thus divided into figures which are either right-angled triangles, rectangles, or trapeziums; and the areas of each of these figures may be readily calculated. NOTE 7. On Projections. a The projection of a point B, on a straight line of unlimited length AE, is the point M at the foot of the perpendicular dropped from B on AE. The projection of a straight line BC, on a straight line of unlimited length AE, is MN,—the part of AE intercepted between perpendiculars drawn from B and C. When two lines, as AB and A.E, form an angle, the projection of AB on AE is AM. We might employ the term projection with advantage to shorten and make clearer the enunciations of Props. XII. and XIII. of Book II. Thus the enunciation of Prop. XII. might be : "In oblique-angled triangles, the square on the side subtending the obtuse angle is greater than the squares on the sides containing that angle, by twice the rectangle contained by one of these sides and the projection of the other on it.” The enunciation of Prop. XIII. might be altered in a similar a manner, NOTE 8. On Loci. Suppose we have to determine the position of a point, which is equidistant from the extremities of a given straight line BC. There is an infinite number of points satisfying this condition, for the vertex of any isosceles triangle, described on BC as its base, is equidistant from B and C. Let ABC be one of the isosceles triangles described on BC. If BC be bisected in D, MN, a perpendicular to BC drawn through D, will pass through A. It is easy to shew that any point in MN, or MN produced in either direction, is equidistant from B and C. It may also be proved that no point out of MN is equidistant from B and C. The line MN is called the Locus of all the points, infinite in number, which are equidistant from B and C. DEF. In plane Geometry Locus is the name given to a line, straight or curved, all of whose points satisfy a certain geometrical condition (or have a common property), to the exclusion of all other points. Next, suppose we have to determine the position of a point, which is equidistant from three given points A, B, C, not in the same straight line. E À B If we join A and B, we know that all points equidistant from A and B lie in the line PD, which bisects AB at right angles. If we join B and C, we know that all points equidistant from B and C lie in the line QE, which bisects BC at right angles. Hence 0, the point of intersection of PD and QE, is the only point equidistant from A, B and C. PD is the Locus of points equidistant from A and B, B and C, and the Intersection of these Loci determines the point, which is equidistant from A, B and C. Examples of Loci. Find the loci of (1) Points at a given distance from a given point. (2) Points at a given distance from a given straight line. (3) The middle points of straight lines drawn from a given point to a given straight line. (4) Points equidistant from the arms of an angle. (5) Points equidistant from a given circle. (6) Points equally distant from two straight lines which intersect. NOTE 9. On the Methods employed in the solution of Problems. In the solution of Geometrical Exercises, certain methods may be applied with success to particular classes of questions. We propose to make a few remarks on these methods, so far as they are applicable to the first two books of Euclid's Elements. The Method of Synthesis. In the Exercises, attached to the Propositions in the preceding pages, the construction of the diagram, necessary for the solution of each question, has usually been fully described, or sufficiently suggested. The student has in most cases been required simply to apply the geometrical fact, proved in the Proposition preceding the exercise, in order to arrive at the conclusion demanded in the question. This way of proceeding is called Synthesis (oúvbeois=composition), because in it we proceed by a regular chain of reasoning from what is given to what is sought. This being the method employed by Euclid throughout the Elements, we have no need to exemplify it here. The Method of Analysis. The solution of many Problems is rendered more easy by supposing the problem solved and the diagram constructed. It is then often possible to observe relations between lines, angles and figures in the diagram, which are suggestive of the steps by which the necessary construction might have been effected. This is called the Method of Analysis (åvávols=resolution). It is a method of discovering truth by reasoning concerning things unknown or propositions merely supposed, as if the one were given or the other were really true. The process can best be explained by the following examples. Our first example of the Analytical process shall be the 31st Proposition of Euclid's First Book. |