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Example. Suppose the length or longest diameter, TR, of the accompanying figure is ascertained to be 34 inches, and the breadth, C O, drawn at right angles to and across the middle of T R, to be 25 inches. Also suppose from the place of their intersection, five equal distances of 3 inches to be set off both ways on the line TR, and through the points 1, 2, 3, 4, &c., double ordinates to be drawn parallel to C 0; it is required to find the area of the surface TCR O, the lengths of the ordinates being as follows :-(1) = 11.76; (2) = 17.71; (3) = 21.21; (4) = 23.39; (6) = 24.61 ; (6) or CO= 25; (7) = 24.61 ; (8) = 23.39; (9) = 21.21; 10 = 17.71; (11) = 11.76 ; Extreme ordinates, Even ordinates.
Odd ordinates, (1) 11.76 (2) 17.71
(3) 21-21 (11) 11.76
(5) 24.61 Sum 23.52
(7) 24-61 (8) 23.39
635.60 Area of space a C c d 06. In order to complete the area of the surface TCR O, it is necessary to determine also the area of each of the segments a T ba and c R dc. Assuming, for convenience of calculation, that the curves a T 6 and c R d are arcs of a parabola, the areas of the corresponding segments will be found by multiplying the ordinates or bases a b, c d, by their respective heights, T1, R 11, and taking two-thirds of the products. Accordingly, since the length of the entire line TR= 34, and the sum of the 10 equal distances of three inches measured from the central point (6) = 30, T 1 and R 11 are together equal to 4, or each of them to 2 inches, and we have a l x T 1 X 2= 11.76 x 2 X 2= 15.68, which answers also for the other segment, as the extreme ordinates in this case are of the same length. The sum of the segments, therefore, = 15.68 x 2 = 31.36 square inches; and
Area of space a Ccd0b = 635.60
* As the common distance happens in the present example to be the same number as the constant divisor 3, there is, of course, no necessity to go through the operations shown in the text, but the entire process has been exhibited for the sake of illustrating the application of the Rule.
A shorter way of computing the areas of the two segments is as follows:-Since in every instance from the mode of measuring equal distances along the diameter T R from its middle point, there can be no difference between the heights of the segments, T 1, R 11, we may add together the extreme ordinates and multiply the sum by the sum of the heights. One-third of the product will then give the space included in both segments. Thus,
Sum of extreme ordinates ............... 23-52
31-36 sq. inches, as before. This method also possesses the advantage of simplifying the computation of the area of the entire figure; for, if to the result obtained in the first part of the process by multiplying by the common distance of the ordinates, we add the product of the sum of the extreme ordinates multiplied by the sum of the heights of the segments, it will only be necessary to divide by 3, to arrive at the magnitude of the whole curvilinear surface, e.g.,*
666-96 The figure represented in the drawing is that of a true ellipse, as no doubt will have been inferred from the character of the ordinates assigned to it, and was 80 chosen for the purpese of illustrating the correctness of the Rule.
These ordinates were calculated as follows:-In a circle, the rectangles under the segments of all intersecting chords are equal (Euclid III., 35), that is, the rectangles under the segments of a diameter are equal to the squares of the corresponding ordinates; while in an ellipse, the rectangles under the segments of a diameter are proportional to the squares of the ordinates. Now, in the ellipse TCR O, the
he ordinat crosses the transverse diameter at 2 inches from one extremity, and 32 inches from the other. The product (rectangle) of the segments of the diameter is, therefore, 32 x 2 = 64. Again, the ordinate (2) makes a section at 5 inches from one end, and 29 from the other; 29 x 5 = 145; and, similarly, for the remaining ordinates up to the conjugate diameter. Then, as the square of half the transverse diameter is to the product of the segments T 1 and R 1, so ist
the segments T 1 and R 1, so is the square of the conjugate diameter (regarded as a double ordinate) to the square of the ordinate (1), that is, As 172 : 32 X 2 :: 252 : 138.41, and extracting the square-root of 138-41, we obtain 11.76 as the length of ordinate (1).'
On the same principle, for the length of the next ordinate, we say, As 17° : 29 x 5 :: 252 : 313.58. V 313-58 = 17.71. And so on, until we arrive at the conjugate diameter, after which the successive ordinates can only be a repetition in an inverse order of those which have preceded.
Regarding T R as the transverse diameter, and C O as the conjugate diameter, and retaining the previous dimensions, we have for the area of TORO, calculated as that of an ellipse :
34 x 25 x 7854 = 667.59
Difference .... .63 This error, small as it is, would be considerably reduced if the areas of the segments a T ba, c R d c, were computed by the Rule for elliptical segments, as given on page 196, instead of that for parabolic arcs. We should then find the segmental spaces equal together to 31.90 square inches, which being substituted for the value obtained by treating the segments as parabolas, brings up the area of the whole figure to 667.50, a result only .09 of a square inch less than the exact area. As a general practice, however, it will be found best, even when the curve is known to depart but slightly from the boundary of a true ellipse, to consider the terminal segments as parabolas, and to determine their areas accord
* See the note on pages 66 and 67 of the present Distillery Instructions.
ingly. A sufficiently close approximation will be had in this manner, whatever be the form of the curve, and much labor of calculation saved.
It is advisable to place the ordinates at such distances apart as will leave as. small a space as possible at either end to be estimated as segments of a parabola, for it is chiefly in the determination of the area of these terminal portions that the greatest error is committed. No general rule can be laid down as to the number of ordinates that should be taken. That must be decided in each case by the extent of the surface, and the character of the curved boundary. Figures of tolerably regular outline, which approach to the elliptical form, will not, of course, require so many ordinates as where the curvature presents at particular points abrupt turns or deflections from its previous course. But cases of this kind can hardly occur in the experience of an excise oflicer, as the vessels used by traders are almost always constructed with a certain uniformity of plan, although they may not entirely conform to any recognized geometrical model.
So universally applicable and trustworthy is the system of equi-distant ordinates, that the separate estimation of the terminal segments may even be dispensed with, the ordinates being so taken that the first and last shall coincide with the curve itself, and therefore be reckoned as 0, and the area computed as in the case of the figure represented on page 198, without introducing any great error into the result. If, for instance, the diameter TR (last fig.) be divided into 10 equal portions of 3.4 inches each, and 9 double ordinates drawn through the points of section, the area might be calculated on the assumption that there were altogether 11 ordinates, of which the first and last were equal to 0. The values of the intermediate ordinates then being determined in the manner just exemplified, and the Rule on page 199 applied, the surface of the ellipse would be found by this method to be 658.78 square inches, which differs from the more accurate area otherwise computed by only 8.18 square inches, an error of not much more than 1 per cent.
It does not appear necessary to furnish any additional examples of the mode of Mensuration by equi-distant ordinates, as the general process is the same for all classes of plane surfaces bounded by irregular lines of continuous curvature, and also for those enclosed by crooked boundaries, or mixtures of straight lines and curves; but in the latter case it is important that ordinates be drawn from the points of junction of the component lines, and that other ordinates be interposed at equal distances.
A few remarks on the practical application of the system of equi-distant ordinates will be given in the succeeding chapter.
(11.) MENSURATION OF SOLIDS.— DEFINITIONS.—A solid, geometrically considered, is that species of magnitude or portion of space which extends in the three directions, and has the three dimensions of length, breadth, and depth or thickness. The boundaries of every solid consist of one or more surfaces, and the quantity of space included by such surface or surfaces constitutes that which is termed indifferently the bulk, volume, solidity, or content of the figure.
It is important to observe that the enclosing surfaces must not be regarded as making up any part of the volume of the solid. They serve merely to define the limits and determine the shape or configuration of the body to which they belong. In a mathematical point of view or for purposes of Mensuration, no account is taken of the particular kind of matter of which a «solid” may consist. The
only requisite is, that a certain object should exist, possessing the positive attributes of length, breadth, and depth. A sphere, for example, fulfils all the conditions imposed by the terms used to define it, without any reference to what may be within the space enclosed by its surface. That space may be occupied by wood, iron, a gas, a liquid, or may even be a vacuum, and yet the nature of the sphere, as a geometrical magnitude, remain unaffected. In this sense, therefore, a drop of water resting of itself on a plate of glass, may with propriety be called a « solid.” Hollow bodies or vessels come under the general denomination of « solids," when there is occasion to consider them with regard either to their external or internal dimensions. It is usual to apply either the term content or capacity to the quantity of space included by the inner surfaces of such bodies.
Solids are divided into three classes :-st. Those bounded by planes. 2nd. Those bounded by plane and curved surfaces. 3rd. Those bounded by curved surfaces only.
The first-class comprehends the pyramid and prism, and their several varieties; the second class embraces the cylinder and cone; and the third class, the sphere or globe, together with other allied figures, which cannot be adequately treated of in Elementary Mensuration.
It must be obvious that solids, like surfaces, may exist under all conceivable varieties of form. It is with reference to the character of their external surfaces that solids are arranged in distinct classes or denominations. This classification, however, as in the case of plane figures, cannot be carried beyond a very limited extent, the general terms employed for the purpose being applicable to such bodies only as present a certain uniformity of aspect, in the entire absence of which, the body is simply called an irregular solid; and by this term it is to be understood that the body is wanting in the characteristics which would bring it under one or other of the classes of solid figures, the more important of which have now to be considered.
As the length and breadth in plane figures are measures always perpendicular to each other, so in solids, the three dimensions, length, breadth, and thickness, are to be regarded as taken not in obedience to the outline of the figure, but in accordance with the principle that such dimensions must invariably be at right angles to one another.
(12.) METHODS OF FINDING THE CONTENTS or SOLIDS.—The quantity of space occupied by any given solid, whatever be its shape, is estimated numerically by comparing the figure with another solid of known magnitude and regular form, taken as the standard measuring unit. It is first necessary to ascertain the length, breadth, and thickness, or equivalent dimensions, of the proposed figure, in some denomination of linear measure, such as inches, feet, &c. From these data, we are enabled by the help of rules or processes derived from geometry, to determine either exactly or approximately, how many times the space included between the surfaces of the body in question would contain the portion of space known to be enclosed by the unit of solid measure. The result thus obtained expresses the relative volume or content of the proposed figure. That which is employed as the measuring unit of solids in general, is the solid termed a cube. A cube is contained by six equal square surfaces, the edges* of which are at right angles to one
• By the edges of a solid are meant the boundary lines of its surfaces. Thus, the sides of a rectangle are the edges of a rectangular solid.
another. The dice used in games of chance are almost perfect cubes. Whatever be the linear unit adopted, the cube constructed upon this as an edge, or upon the corresponding superficial unit as a base, supplies the measuring unit of solidity.
Thus, a cubic inch represents the space enclosed by six equal perpendicular surfaces, each one inch long and one inch broad, or one inch square. Similarly, a cubic foot has six equal limiting surfaces, each one foot square, and so of a cubio yard, &c.
Suppose it required to compute the content of the solid 4 figure A D, all the sides or surfaces of which are equal squares, 4 inches long by 4 inches broad. If the three adjacent edges, A B, BC, and C D, be each divided into 4 equal parts, and parallel lines be drawn from the points of section, as in the diagram, it will be seen how the solid might thus be resolved into 4 equal blocks or slices of which A E is one. Now it is evident that the solid A E contains the same number of cubic inches as the surface A C contains square inches, viz., 4 X 4 = 16 cubic inches. Hence, the entire solid being equal to 4 times the portion A E, must consist of 4 times 16, that is, 64 cubic inches. Again, let the edges A B, BC, CD, bo supposed to be divided into 31 inches, instead of 4 inches each. Then the surface A C will contain 31 x 37 = 421 square inches, (see page 173) and the block A E will contain the same number of cubic inches. But as the depth C D is 31 linear inches there will manifestly be in the entire solid 31 x 421 = 342 1 cubic inches. In both cases, therefore, the content of the solid is obtained by multiplying together the three equal dimensions expressing the length, breadth, and depth, that is, cubing the length of one of the edges of the figure.
A Prism is a solid, the side-faces of which, as ACDB, A CEF, &c., whatever their number, are alway parallelograms, and the ends or bases, as A A BF, C D E, any parallel and rectilinear plane figures, similar and equal to each other.
Prisms are distinguished by the forms of the ends or bases. The name triangular, rectangular, pentagonal, &c., is given to a prism according as its base is a triangle, a rectangle, a pentagon, &c. c Thus, in the first of the accompanying figures, there is depicted a triangular prism ; in the second, a pentagonal or polygonal prism. If the base be an irregular figure, the prism is called an irregular prism ; in every instance, however, the side-faces must be parallelograms.
A right prism is one which has its side-faces perpendicular to its ends ; otherwise, it is an oblique prism.
A prism, the ends and side-faces of which are all parallelograms, is sometimes termed a parallelopiped. When these parallelograms are rectangles, it is called a « rectangular parallelopiped.” When they are equal squares, the solid is usually styled a cube. Thus it is evident, that the parallelopiped is only a particular kind of prism, and the cube a particular kind of parallelopiped, just as amongst plane figures, the rectangle is a variety of the parallelogram and the square a variety of the rectangle.