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tion KL, and so on, and S, the last section, BD; also let CL be represented by l.

Then, according to the prismoidal formula,

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Adding all these, we have for the contents of the whole

excavation, the expression

(S1+4S+ 2S+4S..... S.).

Hence the following rule:

"Divide the length of the cutting into an even number of equal parts, and find the areas of the transverse sections at the points of division.

"Add together the areas of the extreme sections, twice the sum of the areas of all intermediate sections of an odd order, and four times the areas of all intermediate sections of an even order, and multiply by one third the distance between two consecutive sections."

On account of the irregularity of the ground, neither of these rules will give precisely correct results. The nearer we take the sections to each other, the more accurate the result will be.

5. GAUGING OF CASKS.-In order to calculate the capacity of a cask, it is necessary to know the following internal dimensions:

2R = the diameter at the bung,
2r = the diameter at the head,
1= the length of the cask.

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II. It has been found by an examination of a great number of casks that the curvature is confined to the middle third of the length, and is an arc of a parabola, while the end thirds are frustums of cones. The investigation is too abstruse for insertion here, but the following formula has been deduced, and is found to give correct results in most cases:

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or, in case we use diameters, the formula is

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D and d being the bung and head diameters. This is known as Hutton's Rule.

6. LUMBER AND LOG MEASURE. - Sawed boards are measured by the square feet of flat surface, supposing the boards to be one inch thick. Thus, a board 12 feet long, 1 feet wide, and 1 inch thick, would contain 12 feet. A board 12 feet long, 1 foot wide, and 2 inches thick, would contain 24 feet. A board 12 feet long, 14 feet wide, and 14 inches thick, would contain 27 feet.

It is frequently required to ascertain the number of feet of boards which may be obtained from a given log. The slabs, AFB, etc., are considered worthless; the remaining rectangle, AE, is supposed capable of making boards without any loss from sawing. Thus, if AE were the end

A

C

F

D

B

E

of a log 12 feet long, and AB, AC were each 1 foot, the log would be considered as able to make 144 feet of boards.

If AE be a square, its area is equal to the product of CB and AD. This product multiplied by the length will give the cubical contents of the squared log; and this multiplied by 12 will give the feet of boards. It is most convenient to measure the diameter in inches and the length in feet. In this case the result must be divided by 144 to obtain the cubical feet. But as this must be afterward multiplied by 12 to reduce it to board measure, the two operations may be condensed by simply dividing by 12. Hence the following rule:

Multiply the diameter in inches by half the diameter in inches, and the product by the length in feet, and divide by 12. The result will be the feet of boards.

If the log taper, the mean diameter is taken. The bark must be removed before measuring it.

The following rule, which is evidently an approximation, is given as producing the most satisfactory results:

Add together the two extreme diameters in inches, and divide by 2 for the mean diameter. Subtract one third for the side of the square the log will make when hewn. Square the side thus obtained, and multiply the result by the length of the log in feet. Divide by 144. The quotient will be the cubical contents in feet and twelfths of a foot.

This rule gives less than the preceding, and, as trees do not grow exactly round or regular, is probably, as a general rule,

more accurate.

EXAMPLES.

1. The base of a wedge is 8 inches by 2, the edge 6 inches, the altitude 10 inches. What is the volume?

Ans. 73.

2. A level road 10 feet wide is to be laid out from A to B; it is re

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3. The staves of a cask are in the form of an arc of a circle. The bung diameter is 3 feet, the head diameter 2 feet, and the length 4 feet. How many gallons will it contain?

π

V= l(r2+2R2) = 4(+2) = 25.3946 cu. ft. = 189.95 gals.

4. What would be the capacity of the cask by Hutton's Rule ?

πι

V=

=

π

360(39D2+26Dd+25d2) 90(351+195+156.25) = 24.5045 cu. ft.

183.29 gals.

5. How many cubic feet and how many feet of boards in a squared log 20 feet long, which when round measured 24 inches in diameter at

one end and 18 at the other, by the two rules ?

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2d rule,

Cu. ft., 27.222; ft. boards, 326.664.

PART III.

MODERN GEOMETRY.

SECTION I

SYMMETRY.

I. With respect to an axis. Two points are symmetrical with respect to a line, when the line joining them is bisected at right angles by the given line; thus, A and B are symmetrical with respect to the line OP when AB is bisected at right angles by OP. The line OP is called the axis of symmetry.

Two lines, surfaces or solids are symmetrical with respect to a line, when any point in one has a symmetrical point in the other.

The following propositions are easily deduced. The student should prove them:

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1. If one of two symmetrical lines intersect the axis of symmetry, the other intersects it in the same point.

2. They make equal angles with it.

20*

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