unite these points, is obviously mutual to the two points; that is, the distance of the point B from the point A is the same as the distance of the point A from the point B. The direction of the point A from the point B is opposite to the direction of the point B from the point A. THE STRAIGHT LINE. 5. Of all geometrical lines, the simplest is the straight line. Although the idea of a straight line, is the first to which we are conducted by our experience, and the use of our senses, still it is very difficult to define it. The definition usually given is as follows: A straight line is the shortest distance between two points. A more general definition is as follows: A straight line is an indefinite line, such, that any limited portion whatever, is the shortest distance between the points which fix this limit. In any line the direction of one of its points, B, from another point, A, as has already been noticed, is opposite the direction of A from B; so that a line has two different directions exactly opposite, either of which may be regarded as the direction of the line. A Straight line is one which has the same direction throughout its whole extent. A Curved line is one which changes its direction at every point. Two straight lines are evidently capable of superposition, that is, of being placed the one on the other, so as to coincide. Hence, two straight lines coincide throughout their whole extent when they have two points common. Or, in other words, two points determine the position of a straight line. We also infer that two distinct straight lines can intersect or meet each other in only one point. THE PLANE. 6. The plane surface, or, as usually expressed, the Plane, is the simplest of all surfaces. It may be defined as follows: A plane is an indefinite surface, on which we may conceive that through each of its points a straight line may be made exactly to coincide with it throughout its whole extent. It immediately results from this definition, and the nature of a straight line, That any line, having two of its points common with the plane, lies wholly in this plane. Hence, a straight line cannot be partly in a plane and partly out of it. When a straight line has only one point in common with a plane, it is said to meet or pierce the plane, and the plane is said to cut the line, and the segments or portions of the line thus separated will be on different sides of this plane. THE CIRCLE. 7. When a line is not a straight line, or made up of finite portions of straight lines, it is called a curved line. Any When The simplest of all curved lines is the circumference of a circle, which may be thus defined: The circumference of a circle is a plane curve returning into itself, every point of which is equally distant from a certain point in its plane, which point is called the centre of the circumference. The portion of the plane limited by this circumference is called a circle. portion of the circumference of a circle is called an arc. the arc is equal to one-fourth of the circumference it is called a quadrant. Each of the straight lines drawn from the centre to the circumference is called a radius. Hence, all radii of the same circle are equal. B F A E D A line passing through the centre and terminating in both directions by the circumference, is called a diameter. All diameters of the same circle are equal, since each is twice the radius. THE ANGLE. C 8. When two straight lines meet, the opening between them is called a plane angle, or simply an Angle. The magnitude of the angle does not depend upon the lengths of these lines, but only upon the difference of their directions. If in any circle we draw two radii, the distance between their extremities which terminate in the circumference, will embrace an When the arc between the two radii is equal to a quadrant, these radii form with each arc. B A F C B E D other an angle, which we call a right angle. And the radii are When the arc said to be the one perpendicular to the other. between the radii is less than a quadrant, the angle is called acute. When the arc is greater than a quadrant, the angle is called obtuse. The magnitude of an angle may be estimated or measured by means of any particular angle, taken as the unit angle. The right angle is generally the angle chosen as the unit angle. THE RULER AND THE COMPASS. 9. The straight line and the circumference of the circle, which are the only lines treated of in Elementary Geometry, are respectively traced or drawn upon a plane, by the aid of the Ruler and of the Compass. These instruments are so simple, and of such general use, as to need no description in this place. With the Ruler we can draw a straight line on a plane from any one point to any other point. With the Compass we can describe on a plane the circumference of a circle having any given point as a centre, and for its radius any given line. METHODS OF DEMONSTRATION. 10. There are two distinct methods employed in Geometrical demonstration: The Direct Method, and the Indirect Method. The most simple process of direct demonstration is the principle of superposition, which consists in being able to make two figures exactly coincide, by applying the one upon the other. The demonstration is also direct when we employ, by a direct course of reasoning, axioms, definitions, and principles already established. The indirect method, known under the name of Reducing to an absurdity, consists in first supposing the proposition not to be true; afterwards, by certain deductions, to draw, from truths already recognized as rigorously exact, a result contradictory to some one of these truths, or to the proposition itself. We will terminate this subject by noticing two kinds of false reasoning, very common with beginners, and against which they should be constantly on their guard. The first is called, Reasoning in a circle. The second is called, Begging the question. We are said to reason in a circle when, in the demonstration of a proposition, we employ, either implicitly or explicitly, a second proposition, which cannot, itself, be established without the aid of the first. We are said to beg the question, when, in order to establish a proposition, we employ the proposition itself. GEOMETRY. FIRST BOOK. THE PRINCIPLES. DEFINITIONS. I. GEOMETRY is the science of Position and Extension. II. A Point has merely position, without any extension. III. Extension has three dimensions; Length, Breadth, and Thickness. IV. A Line has only one dimension; length. V. A Surface has two dimensions; length and breadth. VI. A Solid has three dimensions; length, breadth, and thickness. VII. A Straight line is one which has the same direction through its whole extent. In reality a line has two directions, the one exactly opposite the other; either of which may be considered as its direction. VIII. A Broken line is one which is made up of two or more straight lines. IX. A Curved line is one which changes its direction at every point. X. Parallel lines are those which have the same direction. XI. An Angle is the difference in direction of two straight lines meeting or crossing each other. The Vertex of the angle is the point where its sides meet. XII. When one straight line meets or crosses another, so as to make the adjacent angles equal, each of these angles is called a Right angle, and the lines are said to be perpendicular to each other. |