PROBLEM II.

TO FIND THE SOLIDITY OF A CIRCULAR HAY STACK WHEN THE SIDES ARE CURVED.

Rule.-Reduce the curved sides to the figures of a cone and the frustum of a cone, by taking and giving equal quantities, and proceed as by the last rule.

Note. Buyers and sellers of hay generally measure a mean circumference of the stack, and for the height they measure from the ground to about of the top part or cone, and by so doing reduce the whole into a cylinder. A good judge may sometimes approximate near the truth by this method, but this method is not to be recommended to a learner, as it is neither scientific nor correct.

1.

Examples.

The circumferences A-B and C-D are 34 and 40 feet, and the perpendicular heights E-D 15, and D-F 15 feet; what is the solidity of the following hay stack?

2. What is the solidity of the above stack according to the common method, the mean circumference being 37 feet, and the mean height 20 feet?

Answer, 80.70 yards.

Note. If a middle circumference be taken exactly between the two, the content will be much nearer; but when accuracy is required, the bottom part should be measured as the frustum of a cone, and the top part as a cone.

PROBLEM III.

Rule. If the stack be straight from the bottom to the eaves, and from the eaves to the top, measure the lower part

as a prismoid, and the top part as a prism; but if the sides are curved, reduce them to a prism and a prismoid of equal solidity. See prism and prismoid.

Note. Some measurers find the area of the end of the stack, and multiply by the mean length for the solidity; and when the end of the stack is perpendicular with the ground, this method is correct; but it is very rarely that stacks are of this figure.

Example.

1. Required the solidity of the following stack, the dimensions at the bottom are 50 and 20 feet, and the length and breadth at the eaves, 63 and 25; and the perpendicular height 16 feet, and from the eaves to the top 14 feet*

TO MEASURE THE EXCAVATIONS OF DRAINS, CANALS, NEW ROADS, AND RAILWAYS.

Divide the length of the drain, &c., into a number of equal parts, the more the better, then measure perpendicular sections at those parts; if the drain, &c., is over level ground each section will form a trapezoid, the top and bottom the two parallel sides, and depth of the drain, &c., the perpendicular distance between them; but if the drain, &c., passes obliquely through a hill, the section will form two triangles.

Note 1. The contents of drains are generally returned in solid yards, or by the floor.

2. The common method practised is to measure the width and depth of the drain in different places for a mean width and depth. See rule III., page 197.

TO FIND THE SOLIDITY.

Rule. To the sum of the areas of the first and last sections, add four times the sum of the areas of the even ones, and twice the sum of the areas of the odd ones, rejecting the first and last; multiply the sum of the areas by one-third of the common distance of the sections, and the product will be the solidity. Many irregular solids are measured as above when they can not be divided into regular figures. See rule II., page 197.

Examples.

1. Fig. 1, represents part of a canal through level ground measured in three parallel trapezoidal sections, the common distance being 24 feet, and the dimensions as below.

Feet. 391.62

372.00

K-L-41.0)

M-N=21.0area 372.00

Depth=12.0

}

sum of the first and last sections.

1416.00 four times the even section.

2179.62

8=of the common distance.

17436.96 feet, the solidity.

2. What is the solidity of part of a new road, cut obliquely through the side of a hill, from the following dimensions, the sections being right-angled triangles, the common distance of the sections being 60 feet? See fig. 2.

The section of an embankment is either a triangle or a trapezoid, and the embankment should be divided into equidistant transverse sections, similarly to the last problem, and the perpendicular height and breadth of the base found as below stick poles at A and B, and erect perpendiculars therefrom just to touch the top of the embankment, then you will have the perpendicular height B-C, and C-D which is equal to the breadth of the base.

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DESCRIPTION OF THE CARPENTER'S RULE.

This rule is generally used in measuring timber and arti ficers' works; not only in taking the dimensions, but is particularly useful in casting up the contents. It consists of two pieces of box, each a foot in length, which are connected together by a folding joint.

On one side of the rule the whole length is divided into inches, and half quarters or eights, for the purpose of taking dimensions; and on the same face there are several plane scales, divided into twelfth parts, which are designed for planning such dimensions as are taken in feet and inches. On one part of the other face is a slider, and four lines marked A, B, C, and D, the two middle ones, B and C, being upon the slide.

Three of these lines, viz., A, B, C, are called double lines, because they proceed from 1 to 10 twice over; these three lines are all exactly alike, both in number and division, and the fourth line D is a single one, proceeding from 4 to 40, and is called the girt-line.

The use of the double lines A and B, is for working proportions, and finding the areas of plane figures; and the use of the girt-line D, and the other double line C, is for measuring solids.

When 1 at the beginning of any line is accounted 1, then the 1 in the middle will be 10, and the ten at the end 100; and all the small divisions are altered in value accordingly.