figure of their bases; round, square, oblong, triangular, &c.; the point at the top is called the vertex, and a line drawn from the vertex, perpendicular to the base, is called the height of the pyramid.

RULE. Find the area of the base, whether round, square, oblong, or triangular, by some one of the foregoing rules, as the case may be; then multiply this area by one-third of the height, and the product will be the solid content of the pyramid.

EXAMPLES.

1. What is the solid content of a true-tapered round stick of timber, 24 feet perpendicular length, 15 inches diameter at one end, and a point at the other?

15x15x,7854×8

144

-=9,8175 solid feet. Ans.

2. What is the solid content of a square stick of timber of a true taper, 30 feet perpendicular length, 18 inches square at one end, and a point at the other? Ans. 22 feet.

3. What is the solid content of a triangular tapering stick of timber, 21 feet long, 10 inches each side of the triangle, 8 inches the perpendicular of the triangle at the large end, and the other end a point? Half perpendicular-4,33 and 4,33×10×7

144

-=2,1ft.+Ans.

NOTE. If a stick of timber be hewn three square, and be equal from end to end, you find the area of the base as in the last question, in inches, multiply that area by the whole length, and divide the product by 144, to obtain the solid content.

4. If a stick of timber be hewn three square, be 12 feet long, and each side of the base 10 inches, the perpendicular of the base being 83 inches, what is its solidity? Ans. 3,6+feet.

CASE 9.-To find the solidity of the frustum of a cone or pyramid.

Definition. The frustum of a cone is what remains after the top is cut off by a plane parallel to the base, and is in the form of a log greater at one end than the other, whether round, or hewn three or four square, &c.

RULE. If it be the frustum of a square pyramid, multiply the side of the greater base by the side of the less; to this product add one third of the square of the difference of the sides, and the sum will be the mean area between the bases; then multiply this sum by the height, and it will give the content of the frustum. Or, if it be a tapering square stick of timber, take the girth of it in the middle; square of the girth, (or multiply it by itself in inches ;) then say, as 144 inches to that product, so is the length, taken in feet, to the content in feet.

EXAMPLE.

What is the content of a tapering square stick of timber, whose side of the largest end is 12 inches, of the least end, 8 inches, and whose length is thirty feet, calculat-. ing it by both rules?

By the first Rule: 12×8-96. 12-S=4

And 96+5x30

144

4x4

=51

3

=214ft. Ans.

10in. of the girth in

Then 10x10=100-area in the middle of the stick. And, as 144: 100 :: 30ft.: 20, 83+feet. Ans. RULE. If it be a triangular pyramid, or a tapering three square stick of timber, multiply the sum of the mean area, as found in the first rule, by ,433-and that product by the height or length. Or, multiply the area in the middle, as found in the second rule, by,433-and then state the proportion as before.

EXAMPLE.

What is the content of a tapering three-square stick of timber, whose side of the largest end is 15 inches, of the least end, 6 inches, and whose length is 40 feet, calculating it by both rules?

9×9

By the first Rule: 15×6=90. 15—6—9. =27.

And 90+27x,433×40

3

144

1

-=-14,0725ft. Ans.

15+6

By the second Rule:

2

10,5 in. of the girth

=

in the middle, if it were four-square. Then 10,5×10,5×,433=47,73825in. And, as 144: 47,73825 :: 40 feet: 13,260025ft. Ans.

area in middle.

RULE.-If it be a circular pyramid or cone, multiply the diameters of the two bases together, and to the product add one third of the square of the difference of the diameters; then multiply this sum by ,7854-and it will be the mean area between the two bases; multiply this area by the length of the frustum, and it will give the solid content.

Ör multiply each diameter into itself; multiply one diameter by the other; multiply the sum of these products by the length; annex two ciphers to the product, and divide it by 382; the quotient will be the content, which divide by 144 for feet as in other cases.

EXAMPLE.

What is the solid content of a tapering round stick of timber, whose greatest diameter is 13 inches, the least 6 inches, and whose length is 24 feet, calculating it by both rules?

13×6,5=84,5

13—6,5=6,5

And 84,5+14,083×,7854×24

144

By the second Rule :*

6,5×6,5

≈14,083+

3

-12,904+feet. Ans.

13x13+6,5x6,5+13×6,5×2400

382

-1858,115+

And 1858,115÷÷144=12,903+ft. Ans.

*To find the content of timber in the tree, multiply the square of 1-5 of the circumference at the middle of the tree, in inches, by twice the length in feet, and the product divided by 144 will be the content, extremely near the truth. In oak an allowance of 1-10 or1-12 must be made for the bark if on the tree; in other wood less. Trees of irregular growth must be measured in parts. T

CASE 10.-To find the solid content of a Sphere or Globe.

NOTE. For definition of a Globe, see Case 9 of Superficies.

RULE. Find the superficial content by Case 9 of Superficies; multiply this surface by one-sixth of the diameter, and it will give the solidity.

Or, multiply the cube of the diameter by ,5236—and the product will be the solidity.*

EXAMPLE.

What is the solidity of our earth, if its diameter be 7957 miles, nearly, and its circumference at the equator be just 25000 miles?

7957,75×25000x7957,75÷6-263857106187,5+solid

miles. Ans.

CASE 11.-To find the solid content of a frustum or segment of a Globe.

Definition. The frustum of a globe is any part cut off by a plane.

RULE. To three times the square of the semidiameter of the base, add the square of the height; multiply this sum by the height, and the product again by ,5236; the last product will be the solid content.

EXAMPLE.

If the height of a coal-pit, at the chimney, be 9 feet, and the diameter at the bottom be 24 feet, how many cords of wood does it contain, allowing nothing for the chimney?

24÷2=12=semidiam.

And 432+81×9×,5236

12×12×3=432. 9×9=81

128

-18,886+ cords. Ans.

*If the diameter of a sphere be 1, its solidity will be,5236; and if its circumference be 1, its solidity will be ,016887.

SECTION III.

OF CASK GAUGING.

Definition-Gauging is the finding of the content of any Cask, Box, Tub, or other Vessel.

Among the many different rules for gauging, the following is as exact as any.

RULE. Take the diameter at the bung and head, and length of the cask; subtract the head diameter from the bung diameter, and note the difference.

If the staves of the cask be much curved or bulging between the bung and head, multiply the difference of diameters by,7; if not quite so much curved, by ,65; if they bulge yet less, by ,6; and if they are almost or quite straight, by ,55-and add the product to the head diameter; the sum will be a mean diameter.

Square the mean diameter, thus found, and multiply the square by the length; divide the product by 359 for ale or beer gallons, and by 294 for wine gallons.

NOTE.-I. To measure the length of the cask, take the length of the stave; then take the depth of the chimes, which, with the thickness of the heads, (that are 1 inch, 1 inch, or 2 inches, according to the size of the cask) being subtracted from the length of the stave, leaves the length within.

2. In taking the bung diameter observe by moving the rod backward and forward whether the stave opposite to the bung, be thicker or thiner than the rest, and if it be, make allowance accordingly.

EXAMPLE.

How many ale and wine gallons will a cask contain, whose bung diameter is 30 inches, head diameter 25 inches and length 40 inches?

5x,7=3,5 25+3,5-28,5 mean

30-25=5.

diam.

28,5x28,5×40

=90,5+ ale gal. Ans.

359

28,5x28,5×40

=110,51+wine gal. Ans.