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Ex. 1. What is the area of a segment whose height is 16, the diameter of the circle being 48?
2. What is the area of a segment whose height is 32, the diameter being 48? Ans. 1281.55.
The following rules may also be used for a circular seg
1. To the chord of the whole arc, add of the chord of half the arc, and multiply the sum by 3 of the height.
If C and c=the two chords, and h=the height;
The segment (C+c)h nearly.
2. If h=the height of the segment, and d-the diameter of the circle;
The term solidity is used here in the customary sense, to express the magnitude of any geometrical quantity of three dimensions, length, breadth, and thickness; whether it be a solid body, or a fluid, or even a portion of empty space. This use of the word, however, is not altogether free from objection. The same term is applied to one of the general properties of matter; and also to that peculiar quality by which certain substances are distinguished from fluids. There seems to be an impropriety in speaking of the solidity of a body of water, or of a vessel which is empty. Some writers have therefore substituted the word volume for solidity. But the latter term, if it be properly defined, may be retained without danger of leading to mistake.
NOTE F. p. 35.
The geometrical demonstration of the rule for finding the solidity of a frustum of a pyramid, depends on the following proposition:
A frustum of a triangular pyramid is equal to three pyramids; the greatest and least of which are equal in height to the frustum, and have the two ends of the frustum for their bases; and the third is a mean proportional between the other two.
Let ABCDFG (Fig. 34.) be a frustum of a triangular pyramid. If a plane be supposed to pass through the points AFC, it will cut off the pyramid ABCF. The height of this is evidently equal to the height of the frustum, and its base is ACB, the greater end of the frustum.
Let another plane pass through the points AFD. This will divide the remaining part of the figure into two triangular pyramids AFDG and AFDC. The height of the former is equal to the height of the frustum, and its base is DFG, the smaller end of the frustum.
To find the magnitude of the third pyramid AFDC, let F be now considered as the vertex of this, and of the second pyramid AFDG. Their bases will then be the triangles ADC and ADG. As these are in the same plane, the two pyramids have the same altitude, and are to each other as their bases. But these triangular bases, being between the same parallels, are as the lines AC and DG. Therefore, the pyramid AFDC is to the pyramid AFDG as AC to DG; and AFDC AFDG2:: AC2: DG2. (Alg. 391.) But the pyramids ABCF and AFDG, having the same altitude, are as their bases ABC and DFG, that is, as AC2 and DG2. (Euc. 19, 6.) We have then
AFDC AFDG2:: AC: DG2
ABCF AFDG:: AC2: DG2
Therefore, AFDC2: AFDG2:: ABCF : AFDG.
That is, the pyramid AFDC is a mean proportional between AFDG and ABCF.
Hence, the solidity of a frustum of a triangular pyramid is equal to of the height, multiplied into the sum of the areas of the two ends and the square root of the product of these areas. This is true also of a frustum of any other pyramid. (Sup. Euc. 12, 3. Cor. 2.)
If the smaller end of a frustum of a pyramid be enlarged,
till it is made equal to the other end; the frustum will become a prism, which may be divided into three equal pyramids. (Sup. Euc. 15 3.)
NOTE G. p. 59.
The following simple rule for the solidity of round timber, or of any cylinder, is nearly exact:
Multiply the length into twice the square of of the circum
It is common to measure hewn timber, by multiplying the length into the square of the quarter-girt. This gives exactly the solidity of a parallelopiped, if the ends are squares. But if the ends are parallelograms, the area of each is less than the square of the quarter-girt. (Euc. 27. 6.)
Timber which is tapering may be exactly measured by the rule for the frustum of a pyramid or cone (Art. 50, 68.); or, if the ends are not similar figures, by the rule for a prismoid. (Art. 55.) But for common purposes, it will be sufficient to multiply the length by the area of a section in the middle between the two ends.
OF THE SEGMENTS OF A CIRCLE, WHOSE DIAMETER IS 1, AND IS SUPPOSED TO BE DIVIDED INTO 1000 EQUAL PARTS.