are the different figures produced by different positions of the plane cutting the solid sphere? How is a triangle formed by the cutting plane? How is a circle formed by the intersecting plane? How is an ellipse formed by the cutting plane? How is a parabola formed by the cutting plane? How is the hyperbola produced by the intersecting plane? What are the vertices of any section? What is the axis, or transverse diameter of an ellipse? What is the centre of an ellipse, and where are the centres of the hyperbola and parabola? What is the diameter of an ellipse? How may a parabola be described? What is the focus of the parabola? What is the vertex of the diameter? What is the diameter of the parabola? What is the ordinate of the parabola? What is a conjugate to a diameter? What is an absciss? What is a parameter? What is a tangent? What is the ordinate to a diameter of an ellipse? How many foci have the ellipse and hyperbola? What foci has the parabola? CHAP. VIII. . MENSURATION. — LAND SURVEYING. MENSURATION, in its most extensive signification, comprehends, as particular branches, geometry, trigonometry, algebra, conic sections, and even fluxions; but the term is more frequently used in a more confined sense, and is then applied to a system of rules and methods, by which numerical measures of geometrical quantities are obtained. In all practical applications of mathematics, it is necessary to express magnitudes, of every kind, by numbers. For this purpose, a line of some determinate length, as one inch, one foot, and so on, is assumed as the measuring unit, of lines; and the number expressing how often this unit is contained in any line is the numerical value, or measure, of that line. A surface, of some determinate magnitude, is assumed as the measuring unit of surfaces; and the number of units contained in any surface, is the numerical measure of that surface, and is called its area. It is usual to assume, as the measuring unit of surfaces, a square, whose side is the measuring unit of lines. A solid, of a determinate figure and magnitude, is, in like manner, assumed as the measuring unit of solids; and the number of units contained in any solid is its solidity, or content. The unit of solids is a cube, each of whose edges is the measuring unit of lines; and, consequently, each of its faces is the measuring unit of surfaces. A right angle is conceived to be divided into ninety equal angles; and one of these, called an angle of one degree, is assumed as the measuring unit of angles. The measures generally employed in the application of mensuration to the common business of life, and their proportions to each other, are as follows: TABLE OF LINEAL MEASURES. 12 Inches = 1 Foot. 3 Feet = 1 Yard. 6 Yards = Fathom. 5 Yards =1 Pole, Rod, or Perch. 40 Poles = 1 Furlong. 8 Furlongs = 1 Mile. 691 Miles = 1 Degree. Degrees =3 360 The Earth's Circumf. ན TABLE OF SQUARE MEASURES. 144 Square inches 9 Square feet 30 Square yards 1607 10 Square chains or 100,000 sqr. links, 640 Square Acres The Scotch acre is to 1 Foot square. = 1 Yard square. = 1 Rood. = 1 Acre. S 100,000 to 78,694. MENSURATION OF HEIGHTS AND DISTANCES. By the application of geometry, the measurement of lines, which, on account of their position, or other circumstances, are inaccessible, is reduced to the determination of angles and of lines, which are accessible, and admit of being measured by well known methods. A line, considered as traced upon the ground, may be measured with rods, or an instrument, called Gunter's chain of sixty-six feet; but still 1 Winchester bushel. 1 Scotch pint. Scotch pints. more expeditiously, by measuring tapes of fifty, or a hundred feet. By these, if the ground be tolerably even, and the direction of the line be traced pretty correctly, a distance may, by using proper care, be measured within about three inches of the truth, in every fifty feet; so that the error may not exceed the two hundredth part of the whole line. Vertical angles may be measured with an instrument called a quadrant, furnished with a plummet and sights. If an angle of elevation is to be measured, as the angle contained by a horizontal line AC, (FIG. 1.) and a line drawn from A to B the top of a tower or hill; or to a celestial body; the centre of the quadrant must be fixed at A, and the instrument moved about A in the vertical plane, till to an eye placed at G, the object B, be seen through the two sights D, d. Then will the arch EF, cut off by the plumb line AF, be the measure of the angle CAB. An angle of depression CAB (FIG. 2.) is to be measured in the same manner, except that here the eye is to be placed at A, the centre of the quadrant; and the measure of the angle is the arch EF. The most convenient instrument for measuring angles, whether vertical or horizontal, is the theodolite. This instrument consists of a telescope and its level; a vertical arc; a horizontal limb, or plate, with a compass; which limb is generally about seven inches in diameter; and a staff with parallel plates. In the focus of the eye glass of the telescope, are two very fine wires or hairs, at right angles to each other, whose intersection is in the plane of the vertical arc. The vertical arc is firmly fixed to a long axis, which is at right angles to the plane of the arc. This axis, sustained by and moveable upon, two supporters which are fixed firmly on the horizontal plate. On the upper part of the vertical arc are two brackets for holding the telescope; the inner sides of which brackets are so framed as to be tangents to the cylindric rings of the telescope, and, therefore, bear only on one part. One side of the vertical arc is graduated to half degrees, which are subdivided to every minute of a degree. On the other side of the vertical arc, are two ranges of divisions; one for taking the upright height of timber, in one hundredth parts of the distance between the instrument and the tree whose height is to be measured; and the other for reducing hypothenusal lines to such as are horizontal. The vertical arc is cut with teeth, or a rack, and be moved regu may |