GEOMETRY. Q. What is Geometry? A. It is a science which treats of all kinds of bodies, delineates their shape or figure, and teaches to measure their superficial and solid contents; as, to measure boards, timber, casks, boxes, cisterns, land, &c. &c. Q. What is the first thing to be attended to, in the study of geometry? A. The first thing, necessary to be learned, in obtaining a competent knowledge of geometry, is the definition of those terms which are of most frequent occurrence; such as points, lines, angles, squares, circles, and the various kinds of geometrical figures. As the object here is, merely to give the scholar so much knowledge of geometry as to enable him to measure superficies and solids, with facility and accuracy, such definitions only will be given, as are necessary for the accomplishment of that object. GEOMETRICAL DEFINITIONS. Q. What is a point, in geometry, or mathematics? A. A point is considered without regard to dimensions, and is merely a mark, or dot, at which to commence the measure. ment of any surface or distance. Q. What is a line? A. A line is considered as having length, but neither breadth nor thickness. Q. What is a superficies or plane? A. It is a surface, having length and breadth, without regard to thickness. Q. What is a solid ? A. It is a body having length, breadth and thickness, as a solid block of wood or stone. Q. How many kinds of lines are there? U A. There are two kinds of lines, straight, and crooked or curved. Q. What is a straight line? A. It is a direct line between two points, and is the shortest line that can be drawn between those points, and is also called a right line. Q. What is a crooked, or curved line? A. It is a line, whose ends verge or incline towards each other, and if continued would meet and form a circle. Q. In what other particulars are lines distinguished from each other? A. All lines, in regard to each other, are either parallel, perpendicular, oblique, or diagonal; as compared with other bodies or surfaces, they are distinguished by horizontal and vertical lines. Q. What are parallel lines? A. They are two lines, whose ends are exactly at an equal distance from each other, and if extended to any length, would never meet. Q. When are lines perpendicular to each other? A. One line is perpendicular to another, when it does not lean towards either end, or when they meet in such a manner as to make a square corner. Q. When are lines oblique to each other? A. When they are drawn so as to be neither parallel nor perpendicular to each other, but slant either towards or from each other. Q. What is a diagonal line? A. It is a line drawn from one corner of a plane or superficies, to the other corner. Q. What is a horizontal line? A. It is a line drawn parallel to the horizon, or on a level with the surface of the earth. Q. What is a vertical line? A. It is a line drawn perpendicular to a horizontal line. Any body that stands upright, as a tree, or a steeple, is said to be vertical. Q. What is a square? A. It is a figure, having four equal sides, meeting ther perpendicularly, which make the corners square. Q. What is a parallelogram, or long square? A. It is a four sided figure, having its opposite sides equal, and meeting each other perpendicularly, like a square. Q. What other regular four sided figures are there? A. Rhombs, Rhomboids, and Trapezoids, besides various others of an irregular form. Q. What general name will comprehend all other four sided figures? A. They are all included under the general name of Trape zium. Q. What general name will include all figures of more than four sides? A. They are all comprehended under the name of Polygon. If they are of equal sides, they are called regular polygons. If their sides are unequal, they are called irregular polygons. Q. What is an angle? A. It is a point where two lines meet in such a manner as to include a space between them. Q. How many kinds of angles are there? A. Three; right angles, acute angles, and obtuse angles. Q. What is a right angle? A. It is an angle, made by two lines meeting each other perpendicularly, and making a square corner. Q. How is the magnitude of an angle determined? A. Angles are measured or determined by the number of degrees contained between the lines that form them. Q. How many degrees are contained in a right angle? A. Every right angle contains 90 degrees, or one fourth part of a circle, whether the space included between the sides is great or small. Q. What is an acute angle? A. It is an angle less than a right angle; that is, containing less than 90 degrees. 30 40 50 60 70 80 90 Q. What is an obtuse angle? A. It is an angle larger than a right angle; that is, containing more than 90 degrees. Q. What is meant by an oblique angle? A. All angles, either greater or less than a right angle, are called oblique angles. Q. What is the least number of right lines, necessary to enclose a space, and what is the figure, formed by them, called ? A. Three lines are necessary, and the figure, formed by them, is called a triangle. Q. How many kinds of triangles are there? A. There are three kinds, viz: Equilateral Triangles, Isosceles Triangles, and Scalene Triangles. Q. What is an Equilateral Triangle? Q. What is an Isosceles Triangle? A. It is an angle of two equal sides, and having the other longer or shorter. Q. What is a Scalene Triangle. -A. It is an angle, having all three of its sides unequal. Q. How many angles are contained in every triangle? A. Three, viz.: One right angle, and two acute; or one ob tuse angle, and two acute; or three acute angles. Q. How many degrees are contained in the three angles of every triangle? A. One hundred and eighty degrees, or half a circle. Q. What is a circle? A. It is a round figure or ring, bounded on all sides by a line, equally distant from a point within, called the centre. Q. Into how many equal parts is every circle supposed to be divided? A. Every circle, whether great or small, is supposed to be divided into 360 equal parts, called degrees. Each degree is also subdivided into 60 minutes, and each minute into 60 seconds. 245 MENSURATION OF SUPERFICIES. Q. What is meant by Superficial Measure? A. The measurement of the surface or outside of any body, as of a piece of land, the outside of a building, of painting, paving, &c. Q. How are superficies or surfaces measured? A. By the superficial foot, yard, rod, or acre, according to the measure of different artificers. Q. How are the superficial contents of every surface found? A. By the proper rule of its figure, whether square, triangular, or circular. ARTICLE I. TO MEASURE A SQUARE AND PARALLELOGRAM. Q. What is the RULE for finding the superficial contents of a square of equal sides? A. Multiply the length of one side into itself, and the product will be the superficial contents or area, in the same denomination of the given sides, which must be brought to the denomination required. Q. How do you find the superficial area of a parallelogram, or long square? A. Multiply the length by the breadth, and the product will be the superficial area as before. EXAMPLES. 1. There is a garden, laid out in an exact square, measuring, on each side, 160 feet; how many square yards does it contain? Operation. 160×160-25600÷9—28444 yds. Ans. 2. There is a farm, lying in the form of a parallelogram, measuring 240,5 rods in length, and 87,5 rods in width; how many acres does it contain? Ans. 131 acres, 2 roods, 3 rods. 3. How many feet of boards will lay the floor of a room, measuring 16 by 18 feet? Ans. 288 feet. 4. A man bought a tract of land, measuring 320 rods in length, and 175 rods in breadth, at $12 an acre; how many acres did it contain, and what was the amount of his purchase? Ans. 350 acres. Amount, $4375. ARTICLE II. TO MEASURE A RHOMBUS AND RHOMBOID. Q. What is a Rhombus ? A. It is a figure in the form of a diamond, having four equal sides, its opposite angles being equal; two of which are greater, and two less than the angles of a square. U* |