oblique to the ordinates, and where its curvature is sharp, the ordinates must be closer together. Much time is saved in calculation by the use of subdivided intervals, as follows: In those parts of the figure where close ordinates are required, the ordinary intervals may be subdivided into half-intervals, quarter-intervals, or smaller subdivisions if necessary; each ordinate belonging to a set of subdivided intervals having its multiplier diminished in the same proportion in which the intervals are subdivided. Thus, the series of multipliers for ordinates at whole intervals being 1, 4, 2, 4, 2, &c., the series of multipliers for ordinates at half-intervals will be— 1, 2, 1, 2, 1, &c., and at quarter-intervals, 1, 1, 1, 1, 1, &c. 2 When an ordinate stands between a larger and a smaller interval its multiplier will be the sum of the two multipliers which it would have had as an end-ordinate for each interval. Thus, for an ordinate between a whole-interval and a halfinterval, the multiplier is 1+1=1; for an ordinate between a whole-interval and a quarter-interval, 1+1=11; for an ordinate between a half-interval and a quarter-interval, 1+1=1, &c. The ordinates having been multiplied by their proper multipliers, and the products added together, the sum is to be multiplied by one-third of a whole-interval, to find the area. In the following Table, those rules are applied to the calculation of the approximate area of a quadrant of a circle of 12 feet radius (Fig. 9). This figure is chosen, because its true area to the 100th of a square foot is otherwise known to be 12×12 × .7854=113.10 square feet, and this affords the means of testing the degree of approximation attained by the rules. The ordinates are computed, to the accuracy of the 100th of a foot, by the formula, Ordinate√ (radius2--abscissa3). In the first division of the Table, the base of the figure is divided into six intervals of two feet each. This gives an error of nearly of the whole area. In the second division of the Table, the last two intervals, where the curve becomes very oblique to the ordinates, are subdivided into four half-intervals, 100 and the error is reduced to about of the whole area. 1 250 In the third division of the Table, the last two half-intervals are further subdivided into four quarter-intervals; and the error in the area becomes only of the whole. o Approximate Area of the Quadrant of a Circle of 12 feet CHAPTER IX. Volumes of Solid Figures-Direct Measurement of Volumes-Measurement of Volumes in Layers or in Rectangular Divisions. THE Volumes of Solid Figures are computed as follows:Conceive the figure to be traversed in a convenient direction by a straight line, as base-line or axis of abscissæ, on which line divide the length of the solid into a sufficient number of equal intervals, and let the solid be conceived to be traversed by a series of plane cross-sections at the points of division of the base-line. If the solid figure has flat ends perpendicular to the base-line, those ends themselves will be the endmost sections. If it is pointed, wedge-shaped, or rounded at the ends, each of the endmost sections will be 0. Measure and compute the areas of the cross-sections by the rules applicable to plane figures. Then conceive the areas of the sections to represent the ordinates of a plane curve of the same length with the solid figure; compute the area of that ideal curve by the rules applicable to plane curves: the area so computed will be equal to the volume of the solid figure. The curve whose ordinates represent areas of sections is sometimes drawn on paper, and is then called the "curve of areas." If drawn with sufficient accuracy, it obviously enables the volume of a figure to be found by means of the platometer. In determining the closeness of the cross-sections, and subdividing, if necessary, the intervals between them, the same rules are to be followed as those which are applicable to the ordinates of plane figures. Direct Measurement of Volumes.-A solid, standing on a rectangular plane base, may have its volume computed directly, without the intermediate process of finding sectional areas, by the following rule, which is founded on Simpson's first rule:— Divide each side of the rectangular base into an even number of equal intervals, and through the points of division draw a network of lines, so as to divide the base into a number of equal rectangular subdivisions. At the angles of those subdivisions measure ordinates, which multiply by the factors shown in the following table, and add the products together. Multiply the sum by one-ninth of the product of the longitudinal and transverse intervals; the product will be the volume required. Measurement of Volumes in Layers, or in Rectangular Divi sions. The volume of a solid may be computed in separate layers; the thickness of each layer being, if the trapezoidal rule is employed, one interval of the length; if Simpson's first rule, two intervals; if Simpson's second rule, three intervals. The trapezoidal rule is not always sufficiently exact; Simpson's first rule is the most generally useful. Supposing, then, that this rule is to be adopted, each layer will be bounded by two of the plane sections of the body to be measured, and will have a third plane section at the middle of its thickness. Then, I. Add together the two outer sections and four times the middle section: one-third of the sum multiplied by the interval of the sections (or half-thickness of the layer) will be the volume of the layer:-Or otherwise, one-sixth of the sum will be the mean sectional area of the layer, which multiplied by its thickness, will give its volume. The volume of a layer whose thickness is one interval, may be computed by the following rule, viz.: To eight times the middle ordinate add five times the near end ordinate, and subtract the far end ordinate: multiply the remainder by one-twelfth of the common interval; the product will be the area required. The volume of a solid standing on a rectangular base may also be computed in separate rectangular prisms, each standing on one or more rectangular subdivisions of the base. If Simpson's first rule be taken as the foundation of the method, each prism will stand on four subdivisions of the base, measuring two longitudinal intervals lengthwise by two transverse intervals breadthwise, and will have its curved boundary defined by nine ordinates; one in the centre, one in the middle of each of the four sides, and four at the corners. Then, II. Add together, the corner ordinates, four times the side ordinates, and sixteen times the middle ordinate; one-ninth of the sum, multiplied by the longitudinal and transverse intervals, will be the volume:-Or otherwise, one thirty-sixth part of the sum will be the mean ordinate, which multiplied by the area of the base of the prism, will give its volume. The rectangular divisions at the edges of the base may sometimes become wedge-shaped instead of prismatic, by their outer ordinates becoming = 0. When the volume of a solid has thus been computed in rectangular divisions, these may be added together so as to give either longitudinal or transverse layers. |