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 C 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. 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 arc. When the arc between the two radii is equal to a quadrant, these radii form with each B A E 25 B A 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. く XIII. An Acute angle is one which is less than a right angle. XIV. An Obtuse angle is one which is greater than a right angle. XV. When the sum of two angles is equal to a right angle, they are called Complementary angles; each being the complement of the other. XVI. When the sum of two angles is equal to two right angles, they are called Supplementary angles; each being the supplement of the other. XVII. A Plane is a surface which is straight in every direction, or one with which a straight line, joining any two of its points, will coincide. XVIII. When a surface is not a plane surface it is called a Curved surface. XIX. A plane figure is a limited portion of a plane. When it is limited by straight lines, the figure is called a rectilineal figure, or a polygon ; and the limiting lines, taken together, form the contour or perimeter of the polygon. XX. The simplest kind of polygon is one having only three sides, and is called a triangle. A polygon of four sides is called a quadrilateral; that of five sides is called a pentagon; that of six sides is called a hexagon; one of seven sides is called a heptagon; one of eight sides an octagon; one of nine sides a nonagon, and so on for figures of a greater number of sides. XXI. A triangle having the three sides equal, is called an equilateral triangle; one having two sides equal, is called an isosceles triangle; and one having no two sides equal, is called a scalene triangle. XXII. A triangle having a right angle, is called a right-angled triangle. The side opposite the right angle is called the hypotenuse. XXIII. A triangle having its three angles acute, is called an acute-angled triangle. |