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As a check upon the correctness of this answer, let the content of each pair of successive slices be computed from the same dimensions, and the results added together.
In practice it would be found most convenient and accurate to effect the mensuration of such a body as A D, where there is no marked irregularity of contour, by measuring ordinates on one of the side-faces, and multiplying the area of this surface by the depth of the figure,—that is, on the supposition that the solid consists of a pile of infinitely thin plates similar and equal to either of its side-faces. But in those cases in which there is much protuberance or unevenness of the top and bottom, it is best to employ the method prescribed in the rule above laid down.
Example (2.) A vessel with irregularly protuberant or bulging sides, has its horizontal sections all true circles. The length of the vessel is 96 inches, and nine equi-distant sections, inclusive of the two ends, measured from the bottom upwards, have the following diameters,—80, 83, 86, 88, 90, 89, 87, 84 and 81 inches. What is the cubical content ? Squares of diams. Squares of diams. of odd
Squares of diams. of even of end sections.
7396 = 7744
= 6100 812
Prismoidal Formula.-As respects any solid, or a frustum or zone of any solid, generated by the revolution of one of the conic sections about its axis, it is only necessary to ascertain the dimensions of the two ends and of a section exactly mid-way between the ends, in addition to the length of the solid. The common distance between the sections will here be half the length of the body, and as there is but a single intermediate section, the rule will become
To four times the area of the middle or intermediate section, add the area of each end, and multiply the sum by one-sixth of the length or height.
This rule, it will be seen, is identical with that otherwise derived, and given on page 210, in the mensuration of prismoids.
Example (3.) The ends of a pyramidal frustum are regular hexagons ; a side of the greater end is 13, and a side of the less, 8 inches; the frustum is 24 inches long. What is its solid content ?
In the case of all regularly tapering bodies, such as the pyramid, cone, &c., the side or perimeter of a section taken mid-way between the top and bottom must evidently be equal to half the sum, or the arithmetical mean of the sides or perimeters of the extreme sections,
• The proof of this may easily be inferred from Euclid VI., 2.
= 10.5 = a side of the middle hexagonal section. We have therefore by the rule and table given on page 181, and by the present rule, 13 x .8660 x 13 x
439.062 Area of greater end. 8 X 8660 X
166.272 Area of less end. 10.5 x 8660 x 10.5 x x 4= 1145.716 4 times area of mid. sec.
1751.050 One-sixth of length ....
4 7004.200 Answer. 7004.2 cub, inches.
8 x 1
The shortest way of performing this calculation would be, of course, to add together the squares of 13, 8, and 4 times the square of 10.5, and to multiply the sum by •8660, 3, and 4 successively. But the principle of the process would not thus be so obvious.
It will be seen on trial, that this result agrees exactly with that which would be obtained by an application of the special rule for pyramidal frustums on page 207.
The contents of frustums of cones may be determined in a similar manner, the diameter of the middle section being half the sum of the diameters of the two ends.
Example (4.) A segment of a sphere—that is, any portion cut off by a plane -measures 40 inches across its base.
Its perpendicular depth is 18 inches : required the solidity.
The diameter of a section at the middle of the depth is found as follows:-If the segment be divided by a plane passing vertically through the centre of the base, the section so made will be a segment of a circle, having the given diameter 40 as its chord and tho given depth 18 as its height. From these data the diameter of the circle is computed on the principle laid down in Euclid III., 35, to be 40-222 inches. Again, the diameter of a horizontal section through the middle point of the depth, regarded as a chord of the same circle, will be found by a similar calculation to be 33.526. Accordingly we have 402
Square of diam. of base.
0 Square of diam. of top of segment.
6095-96 1-6th of depth 3
14363-3 cubic inches.
Example (5.) The diameter of a sphere is 30 feet. What is its solidity ?
14137.2 cubio inches. Answer. Example (6.) The longer axis of a prolate spheroid is 55 and the shorter 33 inches. What is the solid content of the spheroid ?
Every transverse section of a spheroid is evidently a circle, and the middle section of a prolate spheroid is a circle of which the shorter axis is the diameter. The diameter of each end = 0. We have consequently, by the general rule,
31361.022 cubic inches. Answer. This is the same result as was otherwise obtained for a spheroid of equal dimensions on page 214.
Example (7.) The middle zone or frustum* of a prolate spheroid has the following dimensions :-Diameter of each circular base, 18; diameter of middle section, 30; and length of height of frustum, 40 inches. Required the solid content. 18° x 2
648 Sum of squares of diams. of ends.
22242-528 cubic inches. Answer. All casks are, in the practice of the revenuo departments, gauged on the assumption that the figure corresponds to that of the middle frustum of a prolate spheroid.
NOTE.--It will be unnecessary in the present place to furnish additional examples of the applicability of this rule to all classes of regular solids and the frustums and segments of such solids, as the mode of proceeding is the samo in every instance. Hence there is no occasion to introduce, as is usually done, special rules for the mensuration of such bodies as paraboloids, hyperboloids, the various forms of spindles, &c., and their frustums respectively, since the content or capacity can be readily found by the one general rule, where an intermediate diameter can be measured, or computed from the other dimensions. It must be borne in mind, however, that in the case of solids which are protuberant about the centre and run off on cither hand towards a point, what is meant by a middle or intermediate diameter is not the diameter of the middle section of the entire solid, but a diameter taken half way between the greater and less diameters. The middle diameter of a cask, for instance, is not in this sense, what is called the “bung diameter," but a diameter mid-way between the bung and head diameters.
So, as regards a spindle, an intermediate diameter means one taken at one-fourth of the distance from point to point of the solid, all such solids being viewed as consisting of two similar solids joined together at their bascs, and having therefore two intermediate diameters.
When a given solid is seen to be very irregular, in some parts increasing and in others decreasing in breadth, diameters should be taken, one at the commencement and one at the end of each irregular piece, and every such piece should be treated by itself, and its content computed from an intermediate diameter of its own.
(13.) RATIOS OF SIMILAR FIGURES.-Two figures, surfaces or solids, aro said to be similar, when, though unequal in point of size or absolute magnitude, tho relative proportion of the corresponding parts of each is the same. In other
* By the middle frustum is meant the part remaining after an equal piece has been cut off from each end.
words, similar figures are those which have the same shape or form, but may be constructed on a different scale. The sides and all the corresponding dimensions of such figures must have the same proportion one to the other, and their corresponding angles must be equal. Thus, if any one side of one of the figures be double or triple the corresponding side in the other figure, then every side in the one must be double or triple the corresponding side in the other, and the angle formed by each pair of sides in the one must be equal to the angle formed by the corresponding sides in the other. This relation between magnitudes is that with which we are most familiar in the arts and in the ordinary affairs of life. The delineation of maps and plans consists in expressing on a small scale, but without altering their proportions, the shape of tracts of country, parts of buildings, &c., that is, in drawing a similar figure with shorter sides.
It should be clearly understood that the precise conditions under which two plane geometrical figures will be similar, are as follows:-1st, that they shall have the same number of angles. 2nd, that these angles shall be equal respectively each to each. 3rd, that the sides containing the angles which are equal shall bear to one another the same proportion.*
For example, equilateral triangles, however much they may vary in size, must, from their construction, always have their three angles equal and their sides all proportional to each other. Such figures, accordingly, are always similar ; so also are squares, regular polygons, and circles, which can never undergo any change of their proportions.
In solid bodies, the test of similarity is, that they shall be enclosed by the same number of similar planes, similarly situated, and having like inclinations to one another. Spheres, cubes, and the other regular solids are invariable in their proportions, and consequently are always similar.
Similar triangles have to each other the same proportion as the squares of their corresponding sides (Euclid VI., 19); and from this it follows, that as every plano rectilinear figure may be divided into triangles, all similar rectilinear figures are to one another in the same ratio as the squares of their corresponding sides, or as the squares of their perimeters. Thus, the area of any given triangle is to the area of any similar triangle, as the area of a square described on a side of the first is to the area of a square described on the corresponding side of the second, and the same proportion holds good as regards all other plane figures whatever. It is obvious, therefore, that the area of a triangle increases or decreases at a much quicker rate than the sum of the lengths of its sides ; for if we double each side of a triangle, the area will be four times as great as before ; if we treble each side, the area will be nine times as great as before. Similarly, if we divide each side by two, the area becomes diminished four times, and so on. Again, since all circles are similar figures, the area of one circle is to the area of another circle as a square described on the diameter of the first is to a square described on the diameter of the second. Accordingly, if we double the diameter of a circle, the area of the circle whose diameter is so increased will be four times as great as at first, and if we reduce the diameter by one-half, the area will be only one-fourth of its previous amount.
The chief practical conclusion to be drawn from these principles is, that if we have a certain plane figure of known dimensions, and wish to construct another
• Treatise on Geometry. Cab. Cycl.
similar figure which shall have, say, nine times the area of the former, we must draw the sides or outline three times as long as the corresponding sides of the given figure, and similarly for other proportions. A triangle whose three sides measure respectively 8, 7, and 5 inches has only one-ninth of the area of a triangle, the sides of which measure 24, 21, and 15 inches, that is, three times the lengths of the sides of the first triangle. Again, a regular decagon the side of which measures 12 inches, has four times the area of a similar polygon, of which the side is but the half of 12, that is, 6 inches. These statements may be easily verified by calculation.
Rectangular surfaces, it is evident, must vary in the ratio of the product of their lengths and breadths, since the area in each case is found by multiplying together the dimensions of length and breadth. The area of a parallelogram which measures 12 feet by 7, is to the area of a parallelogram which measures 15 feet by 11, as 12 x 7 : 15 x 11.
It should be observed, that if one only of the dimensions of each of two surfaces varies while all the other dimensions remain unchanged, then these surfaces will vary only in the ratio of the altered dimension. Thus, let the base of a triangle be 13, and its altitude 9, and let the base of another triangle be also 13 inches, but the altitude only 7, then the area of the first is to the area of the second triangle as 9 : 7.
Similar solids of unequal dimensions vary in the ratio of the products of their lengths, breadths, and depths respectively, because the volume of each is found by multiplying these dimensions together.
It follows, therefore, that when such bodies vary proportionally, that is, when the length, breadth, and depth increase or diminish in the same ratio, the solidities will vary as the cubes of their like dimensions.
A square prism, a side of whose base is 5 and whose height is 7 inches, has a solidity of 175 cubic inches ; and a square prism a side of whose base is 12.5 and whose height is 17.5 inches has a solidity of 2734-375 cubic inches. But 175 : 2734.375 :: : 12.5", or as 78 : 17.5', that is, the solidities are in the same ratio as the cubes of the sides of the base or the cubes of the heights.
Again, if the diameters of two spheres be respectively 8 and 10, their solidities will be 268.0832 and 523.6; and these two numbers are to one another in the same proportion as the cubes of the diameters, or as 88 : 109. That the same relation holds good as regards all other pairs of similar solids may be readily established by computing the actual solidities according to the proper rules, and comparing the results so obtained with the cubes of any simple dimension of either.
If one dimension only of a solid be doubled, tripled, halved, &c., the bulk will be increased or reduced only to that extent. If the body be increased or diminished superficially in the same ratio, that is, in the dimensions of length and breadth, the total alteration of bulk will be as the squares of these dimensions; and if an equal change be made in the depth also, the new solidity will be to the former as the cubes of any two like dimensions. For example, let the length of a parallelopiped be 4, its breadth 3, and its height 2. Then if the length be increased to 6, the solidity will increase from 24 to 36, that is, in the ratio of 6 to 4. If the breadth be also increased in the ratio of 6 to 4, it will become 4.5, and the solidity will become 54; and 24 : 54
3: 4-5?. If,