ELEMENTS OF GEOMETRY. BOOK I. DEFINITIONS.* 1. EXTENSION has three dimensions, length, breadth, and thickness. 2. GEOMETRY is the science which has for its object: 1st. The measurement of extension; and 2dly. To discover, by means of such measurement, the properties and relations of geometrical magnitudes. 3. A POINT is that which has place, or position, but not magnitude. 4. A LINE is length, without breadth or thickness. 5. A STRAIGHT LINE is one which lies in the same direction between any two of its points. 6. A BROKEN LINE is one made up of straight lines, not lying in the same direction. 7. A CURVE LINE is one which changes its direction at every point. The word line when used alone, will designate a straight line; and the word curve, a curve line. 8. A SURFACE is that which has length and breadth without thickness. * See Davies' Logic and Utility of Mathematics. §1. 9. A PLANE is a surface, such, that if any two of its points be joined by a straight line, such line will be wholly in the surface. 10. Every surface, which is not a plane surface, or composed of plane surfaces, is a curved surface. 11. A SOLID, or BODY is that which has length, breadth, and thickness: it therefore combines the three dimensions of extension. 12. A plane ANGLE is the portion of a plane included between two straight lines meeting at a common point. The two straight lines are called the sides of the angle, and the common point of intersection, the vertex. Thus, the part of the plane includ ed between AB and AC is called an angle: AB and AC are its sides, and A its vertex. An angle is sometimes designated A4 simply by a letter placed at the vertex, C -B as, the angle A; but generally, by three letters, as, the angle BAC or CAB,—the letter at the vertex being always placed in the middle. 13. When a straight line meets another straight line, so as to make the adjacent angles equal to each other, each angle is called a right angle; and the first line is said to be perpendicu lar to the second. 14. An ACUTE ANGLE is an angle ess than a right angle. 15. An OBTUSE ANGLE is an angle greater than a right angle. 16. Two straight lines are said to be parallel, when being situated in the same plane, they cannot meet, how far soever, either way, both of them be produced. 17. A PLANE FIGURE is a portion of a plane terminat ed on all sides by lines, either straight or curved. 18. A POLYGON, or rectilineal figure, is a portion of a plane terminated on all sides by straight lines. The broken line that bounds a polygon is called its perimeter. 19. The polygon of three sides, the simplest of all, is called a triangle; that of four sides, a quadrilateral; that of five, a pentagon; that of six, a hexagon; that of seven, a heptagon; that of eight, an octagon; that of nine, an nonagon; that of ten, a decagon; and that of twelve, a dodecagon. 20. An EQUILATERAL polygon is one which has all its sides equal; an equiangular polygon, is one which has all its angles equal. 21. Two polygons are equilateral, or mutually equilateral when they have their sides equal each to each, and placed in the same order: that is to say, when following their bounding lines in the same direction, the first side of the one is equal to the first side of the other, the second to the second, the third to the third, and so on. 22. Two polygons are equiangular, or mutually equiangu lar, when every angle of the one is equal to a correspond. ing angle of the other, each to each. 23. Triangles are divided into classes with reference both to their sides and angles. 1. An equilateral triangle is one which has its three sides equal. 2. An isosceles triangle is one which has two of its sides equal. 3. A scalene triangle is one which has its three sides unequal. 4. An acute-angled triangle is one which has its three angles acute. 5. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypothenuse, and the other two sides, the base and perpendicular. 6. An obtuse-angled triangle is one which has an obtuse angle. 24. There are three kinds of QUADRILATERALS: 1. The trapezium, which has no two of its sides parallel. 2. The trapezoid, which has only two of its sides parallel. 3. The parallelogram, which has its opposite sides parallel. 25. There are four varieties of PARALLELOGRAMS: 1. The rhomboid, which has no right angle. 2. The rhombus, or lozenge, which is an equilateral rhomboid. 3. The rectangle, which is an equiangular parallelogram. 4. The square, which is both equilateral and equiangular. 26. A DIAGONAL of a figure is a line which joins the vertices of two angles not adjacent. 27. A base of a plane figure is one of its sides on which it may be supposed to stand. DEFINITIONS OF TERMS. 1. An axiom is a self-evident truth. 2. A demonstration is a train of logical arguments brought to a conclusion. 3. A theorem is a truth which becomes evident by means of a demonstration. 4. A problem is a question proposed, which requires a solution. 5. A lemma is a subsidiary truth, employed for the demonstration of a theorem, or the solution of a problem. |