GEOMETRY. DEFINITIONS. 1. Geometry is the science which treats of position, and of the forms, measurements, mutual relations, and properties of limited portions of space. SPACE extends without limit in all directions, and contains all bodies. 2. A Point is mere position, and has no magnitude. 3. Extension is a term employed to denote that property of bodies by virtue of which they occupy definite portions of space. The dimensions of extension are length, breadth, and thickness. 4. A Line is that which has extension in length only. The extremities of a line are points. 5. A Right or Straight Line is one all of whose parts lie in the same direction. 6. A Curved Line is one whose consecutive parts, however small, do not lie in the same direction. 7. A Broken or Crooked Line is composed of several straight lines, joined one to another successively, and extending in different directions. When the word line is used, a straight line is to be understood, unless otherwise expressed. 8. A Surface or Superficies is that which has extension in length and breadth only. 9. A Plane Surface, or a Plane, is a surface such that (9) 15109 if any two of its points be joined by a straight line, every point of this line will lie in the surface. 10. A Curved Surface is one which is neither a plane, nor composed of plane surfaces. 11. A Plane Angle, or simply an Angle, is the difference in the direction of two lines proceeding from the same point. The other angles treated of in geometry will be named and defined in their proper connections. 12. A Volume, Solid, or Body, is that which has extension in length, breadth, and thickness. These terms are used in a sense purely abstract, to denote mere space-whether occupied by matter or not, being a question with which geometry is not concerned. Lines, Surfaces, Angles, and Volumes constitute the different kinds of quantity called geometrical magnitudes. 13. Parallel Lines are lines which have the same direction. Hence parallel lines can never meet, however far they may be produced; for two lines taking the same direction cannot approach or recede from each other. Two parallel lines cannot be drawn from the same point; for if parallel, they must coincide and form one line. PLANE ANGLES. To make an angle apparent, the two lines must meet in a point, as AB and AC, which meet in the point A, Angles are measured by degrees. B 14. A Degree is one of the three hundred and sixty equal parts of the space about a point in a plane. If, in the above figure, we suppose AC to coincide with AB, there will be but one line, and no angle; but if AB retain its posi tion, and AC begin to revolve about the point A, an angle will be formed, and its magnitude will be expressed by that number of the 360 equal spaces about the point A, which is contained between AB and AC. Angles are distinguished in respect to magnitude by the terms Right, Acute, and Obtuse Angles. 15. A Right Angle is that formed by one line meeting another, so as to make equal angles with that other. The lines forming a right angle are perpendicular to each other. 16. An Acute Angle is less than a right angle. 17. An Obtuse Angle is greater than a right angle. Obtuse and acute angles are also called oblique angles; and lines which are neither parallel nor perpendicular to each other are called oblique lines. 18. The Vertex or Apex of an angle is the point in which the including lines meet. 19. An angle is commonly designated by a letter at its vertex; but when two or more angles have their vertices. at the same point, they cannot be thus distinguished. For example, when the three lines AB, AC, and AD meet in the common point A, we designate either of the angles formed, by three letters, placing that at the vertex between those at the opposite extremities of the including lines. Thus, we say, the angle BAC, etc. B 20. Complements. - Two angles are said to be complements of each other, when their sum is equal to one right angle. 21. Supplements. - Two angles are said to be supplements of each other, when their sum is equal to two right angles. PLANE FIGURES. 22. A Plane Figure, in geometry, is a portion of a plane bounded by straight or curved lines, or by both combined. 23. A Polygon is a plane figure bounded by straight lines, called the sides of the polygon. The least number of sides that can bound a polygon is three, and by the figure thus bounded all other polygons are analyzed. FIGURES OF THREE SIDES. 24. A Triangle is a polygon having three sides and three angles. Tri is a Latin prefix signifying three; hence a Triangle is literally a figure containing three angles. Triangles are denominated from the relations both of their sides and angles. 25. A Scalene Triangle is one in which no two sides are equal. 26. An Isosceles Triangle is one in which two of the sides are equal. 27. An Equilateral Triangle is one in which the three sides are equal. 28. A Right-Angled Triangle is one which has one of the angles a right angle. 29. An Obtuse-Angled Triangle is one having an obtuse angle. |