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661 i A segment of a sphere is a part of it cut off by any plane; as the figure A C B D. The plane is the base of the segment; the per- A pendicular distance from the center of the base to the convex surface is the height of the segment; as C D.
662i A spherical zone is a part of the surface of a sphere included between two parallel planes, which form its bases; and the height of a spherical zone is the perpendicular distance between the planes forming its bases.
663i A cylindrical ring is a figure formed by bending a cylinder uniformly till the two ends meet; as A C D B.
664i A spheroid is a figure resembling a sphere, and which may be formed by the revolution of an ellipse about one of its axes; as A E C B D F A.
If the ellipse revolves about its longer or A transverse axis or diameter, the spheroid is prolate, or oblong; if about its shorter or conjugate diameter, the spheroid is oblate, or flattened.
665. A segment of a spheroid is a part cut off" by any plane, as F A E F.
666. To find the surface of a sphere. Multiply the diameter by the circumference.
Ex. 1. Required the convex surface of a globe, whose diameter is 24 inches. Ans. 1809.55+ inches.
2. Required the surface of the earth, its diameter being 7957J miles, and its circumference 25,000 miles. Ans. 198943750 square miles.
667i To find the solidity of a sphere. Multiply the cube of the diameter by .523598.
Ex. 1. What is the solidity of a sphere, whose diameter is 12 inches? Ans. 904.78+inches.
2. Required the solidity of the earth, supposing its circumference to be 25,000 miles. "Ans. 263858149120.06886875 miles.
668. To find the convex surface of a segment or of a zone of a sphere.
Multiply the height of the segment or zone by the circumference of the sphere of which it is a part.
Ex. 1. If the diameter of a sphere is 12J feet, what is the convex surface of a segment cut off from it, whose height is 2 feet?
Ans. 78.54 sq. ft
2. If the diameter of the earth, considered as a perfect sphere, is 7970 miles, and the height of each temperate zone is taken at 2143.623553 miles, what is the surface of each temperate zone?
Ans. 53673229.81+ sq. m.
669. To find the solidity of a segment of a sphere.
Multiply the square of the height plus three times the square of the radius of the base, by the height, and this product by .5236.
Ex. 1. Required the solidity of a spherical segment, whose height is 3 feet, and the radius of whose base is 4^ feet. Ans. 109.56 cu. ft.
2. Required the solidity of the segment of a sphere, whose height is 9 feet, and the diameter of whose base is 20 feet .
Ans. 1795.42 cu. ft.
670. To find the surface of a cylindrical ring.
Multiply the sum of the thickness and the inner diameter by the thickness, and that product by 9.8696.
Ex. 1. Required the surface of a cylindrical ring, whose inner diameter is 21 inches, and whose thickness is 4 inches.
Ans. 986.96 sq. in.
671. To find the solidity of a cylindrical ring.
Multiply the sum of the thickness and the inner diameter by the square of the thickness, and that product by 2.4674.
Ex. 1. Required the solidity of a cylindrical ring, whose inner diameter is 25 inches, and whose thickness is 5 inches.
Ans. 1850.55 cu. in.
672. To find the solidity of a spheroid.
Multiply the square of the revolving axis by the fixed axis, and that product by .523598.
Ex. 1. If the fixed axis of a spheroid is 32 inches, and the revolving axis 20 inches, what is the solidity? Ans. 6702.08 cu. in.
2. Required the contents of a balloon in the form of a prolate spheroid, having its longer diameter 48 feet, and its shorter 38 feet.
Ans. 36291.76 cu. ft
MENSURATION OF LUMBER.
673. Boards are usually measured by the square foot.
Planks, joists, beams, &c. are usually measured by board measure, the board being considered to be 1 inch in thickness.
Round timber is sometimes measured by the ton, and sometimes by board measure.
674i To find the number of square feet in a board.
Multiply the length of the board, taken in feet, by its width, taken in inches; and the product divided by 12 will give the contents in square feet. Or,
Take both the length and width in feet, and their product will be the contents in feet.
Note. — If the board is tapering, take half the sum of the width of its ends for the width.
Ex. 1. What are the contents of a board 24 feet long, and 8 inches wide? Ans. 16 feet .
2. What are the contents of a board 30 feet long, and 16 inches wide? Ans. 40 feet.
3. What are the contents of a tapering board, 30 feet long, whose ends are, the one 26 inches, and the other 14 inches wide?
675i To find the number of feet, board measure, in a plank, joist, beam, &c.
Multiply the width taken in inches by the thickness in inches, and this product by the length, in feet: and the last product divided by 12 will give the contents in feet, board measure.
Note. —If the plank, joist, &c. is tapering in width, take half the sum of the width of the ends for the width; and if the taper be both of the width and the thickness, the common rule of obtaining the contents in cubic feet is, to multiply half the mm of the areas of the two ends by the lengtji, and divide the product by 144.
Ex. 1. How many feet are there in 3 joists, which are 15 feet long, 5 inches wide, and 3 inches thick? Ans. 56^- feet.
2. How many feet in 20 joists, 10 feet long, 6 inches wide, and 2 inches thick? Ans. 200 feet.
3. How many feet in a beam 20 feet long, 10 inches thick, whose width tapers from 18 to 16 inches? Ans. 283^ feet.
676i To find the contents of round timber.
Multiply the length, taken in feet, by the square of one fourth of the mean girth, taken in inches; and this product divided by 144 will give the contents in cubic feet.
Note. — The girth of tapering timber is usually taken about J the distance from the larger to the smaller end.
The rule is that in common use, though very far from giving the actual number of cubic feet in round lumber measured by it. 40 cubic feet, as given by the rule, are in fact equal to 50j9^ true cubic feet. The following rule gives results more nearly accurate, requiring to be diminished by only one foot in 190, to give exact contents. Multiply the square of one fflh of the mean girth, taken in inches, by twice the length, in feet; and divide by 144.
Ex. 1. How many cubic feet in a stick of timber which is 30 feet long, and whose girth is 40 inches? Ans. 20£ feet.
2. If a stick of timber is 50 feet long, and its girth is 56 inches, what number of cubic feet does it contain? Ans. 68^ feet.
3. What are the contents of a log 90 feet long, and whose circumference is 120 inches? Ans. 5G2^ feet.
GAUGING OF CASKS.
677 i Gauging is the process of finding the capacities of casks or other vessels.
Casks are generally considered to be of four varieties: 1. Having the staves nearly straight; 2 Having the staves very little curved; :) Having the staves of a medium curve ; 4. Having the staves considerably curved.
Note. — Casks of the first variety approach very nearly the form of a cylinder; those of the third variety are ot the shape of a molasses hogshead; those of the second variety have a curvature of stave between that of the first and third; and the fourth have a greater curvature than that of the third.
678i In gauging casks, it is necessary first to find the mean diameter. This is found by taking the end and middle diameters, and the length in inches; and then adding to the end diameter the product of the difference between the end and middle diameters by .55, .C0, .65, or .70, as the cask may be of the first, second, third, or fourth variety.
679i To find the capacity of a cask in gallons
Multiply the square of the mean diameter, in inches, by the length, in inches; and the product multiplied by .U0;i4 will give the capacity in liquid or wine gallons.
Note 1. —If the capacity is required in ale or beer gallons, use for a multiplier .0028 instead of .0034. If imperial gallons are required, multiply the liquid or wine gallon, as found by the rule, by .833.
Note 2. — The contents of any vessel being known in cubic inches, its capacity in liquid gallons may be found by dividing by 231; in ale or beer gallons, by dividing by 282; and in bushels, by dividing by 2150.42.
Ex. 1. Required the capacity in gallons of a cask of the fourth variety, whose middle diameter is 35 inches, head diameter 27 inches, and length 45 inches. Ans. 102.6.
2. What is the capacity in gallons of a cask of the third variety, whose middle diameter is 38 inches, head diameter 30 inches, and length 42 inches'!
3 What are the contents in liquid measure of a tub 40 inches in diameter at the top, 30 inches at the bottom, and whose height is 50 inches? Ans. 209.66gal.
4. How many wine gallons will a cubical box contain, that is 10 feet long, 5 feet wide, and 4 feet high.! Ans. 149(iT8»gal.
5. How many ale gallons will a trough contain, that is 1 2 feet long, 6 feet wide, and 2 feet high V Ans. 882^gal.
6. How many bushels of grain will a box contain that is 15 feet long, 5 ieet wide, and 7 feet high.! Ans. 421.8bu.
TONNAGE OF VESSELS.
680? The tonnage of a ship is the number of tons burden it will carry, with safety, under the ordinary circumstances of navigation.
The light-loaded water-line of a vessel is the line made by the water upon the outside of the hull as it floats without load; and the deeploaded water-line is that made in like manner when it is fully laden.
The number of cubic feet of the hull between these two waterlines, divided by 35, the number of cubic feet of sea-water which must be taken to weigh a ton, represents the weight of water displaced in sinking the vessel from the light to the deep-loaded waterline, and thereibre its true tonnage.
681? The present government rule, or that adopted 1864, requires too many data properly to have a place in arithmetic. The former rule, although it does not always give the actual tonnage, yet as builders and others often make their estimates by it, is here given.
For Single-decked Vessels. Take the length on deck from the forward side of the main stem to the after side of the stern post, and the breadth at the broadest part abooe the main wales; take the depth from the under side of the deck plank to the ceiling of the hold; and deduct from the length three .fifths of the breadth; multiply the remainder by the breadth, and the product by the depth; and divide the last product by 95.
For Double-decked Vessels. Proceed as with single-decked vessels, except for the depth take half the breadth.'
Note — The rule is differently construed. The length is usually taken in a line with the deck; the depth at the main hatch. But with regard to the breadth, there is a great want of uniformity among measurers; most take the breadth about 45 inches below the plank-sheer at the broadest part; some consider the upper wales, and others the lower, at the main wales, thus making a considerable difference in their results.
The rule for single-decked vessels operates very well, but the rule for double-decked vessels, which is also intended to include all vessels of more than one deck, often fails to give the true tonnage. A more accurate method would, for Double-decked Vessels, take the breadth 5 feet below the upper deck, at the broadest part, and for Three-pecked Vessels 7 feet below the upper deck; and in each case for depth of hold three fifths of the breadth.
Ex. 1. A. & G. T. Sampson, of East Boston, have contracted to build a clipper ship 191A feet long, 36 feet wide, 22^- feet deep; what is the tonnage of the ship? Ans. 1184?^j7j tons.
2. What is the tonnage of the ship Meridian, whose length is 184T\, width 38ii, and depth 28 feet? Ans. 1284fff!^ tons.
3. The ship Mattakeeset is 195T% feet long, 39^ wide, and 27T^ deep; whit is the tonnage of the same? Ans. 1397T^|^; tons.
4. Required the tonnage of a single-decked vessel, whose length is 78 feet, width 21 feet, and depth 9 feet. Ans. 130-t% tons.
5. What is the tonnage oe a double-decked vessel, whose length is 159 feet, and width 30 fret? Ans. 667U tons.
6. What is the tonnage of Noah's ark, admitting its length t6 Aave been 479 feet, its breadth 80 feet, and its depth 48 feet?
Ans. 14517}$ tons.