ter taken in the middle, multiply the sum by the length, and the product again by 1309, the last product is the solidity, very nearly.

Examples.

1. What is the solidity of the elliptic spindle V B CD; length 15; diameter C D 5; and diameter taken half way between the middle and end equal to 4?

Least diam. 4X28X864 sq. of twice of the diam. Diam. C-D 5X5=25 sq. of C D add.

89 sum.

Sum. 89 X 15 X 1309—174·7515 Ans.

2. What is the solidity of an elliptic spindle that is 20 in length; greatest diameter 15; and the diameter half way between the middle and end, equal to 10? Ans. 1636 25

To find the solidity of the middle frustum of an
elliptic spindle.

RULE. To the square of twice the diameter taken half way between the middle and the end, add the squares of the middle and end diameters; multiply the sum by the length, and again by 1309, the last product is the solidity required.

Examples.

1. What is the solidity of the middle frustum E G FH length 10; diameter at the end 4: diameter C D 6, and diameter taken at O P, equal to 5?

Diam. O P 5X2=10X10=100 sq. of twice O P. 6X 6: 36 sq.

of

C D.

Diam. C D

Diam. E F or G H 4× 4=

16 sq. of E For GH.

152 sum.

Sum. 152 X 10 X 1309198.968 Ans.

CASE XVI.

Timber measure; the common method.

RULE. The common method of measuring round timber, (in order to ascertain the quantity of hewn timber, that is contained in a round stick) has been to girt the stick in the middle with a line, and then to double the line into four equal parts, (and one of these parts is considered equal to one side of the square timber that can be hewn from such a stick); and one of these parts, (or one fourth of the circumference) is multiplied into itself and again by the length of the stick, the last product is called the solidity.

NOTE. If of the girt in inches, is multiplied into itself and again by the length in inches, the last product must be divided by 1728, the quotient is the solidity in feet: if the girt in inches is squared and multiplied into the length in feet, the product must be divided by 144; the quotient will be the solidity in feet.

NOTE. This method of measuring round timber is very erroneous; it gives too much for the solidity of square timber in the stick, and not enough if the four slabs, or segments are to be included; as I shall prove in case 17th.

Examples.

1. What is the solidity of square timber, that can be hewn from a round stick that is 21 ft. long, and its circumferente, or girt line 48 inches?

Girt 4812 × 12=144×21=3024144=21 ft. Ans.

2. What is the solidity of square timber in a round stick that is 30 ft. long, and its circumference or girt measures 24 inches? 7 ft. Ans.

3. What is the solidity of that stick of timber that girts 56 inches, and its length being 11 ft. 9 inches? Duodecimally. Girt 561 ft. 2 in. quarter girt.

4. What is the solidity of square timber in a stick of round timber, that girts 50 in. and is 31 ft. 7 in. long?

Ans. 34 ft. 3' 2" 10" 9""" or 3481

NOTE. In all the preceding examples the stick of timber has been considered all of its length, of equal bigness, or a cylinder. If the stick is tapering (as it always is) the common method is to girt it in the middle; and sometimes it is girted in two, or three places, and the sum of the girts is divided by the number of them, and the quotient is taken for the mean girt of the stick.

CASE XVII.

A new method of finding the solidity of square timber that is contained in a stick of round timber, or to find how much the stick will measure after

it is hewn.

RULE. Girt the stick (after taking off the bark,) and annex a cypher to the girt in inches and divide the whole by 44 the quotient is a side of the greatest square that can be hewn from such a stick; multiply this side into itself, and this product again by the length, and the last product will be the solidity of the stick after it is hewn.

The preceding rule illustrated.

Make a circle whose diameter is 20, and of course its circumference will be 62.8t, (which will represent the girt of a stick of timber); annex a cypher to the circumference and divide by 4.4, the quotient is 14 and nearly 3 tenths; now make the largest square in this circle that can be made in it, and you will find that the sides of the square will be 14.3, very nearly. If the above circumference were divided by the quotient would be 15.7 for a side of square timber that could be hewn from such a stick.

15-7-14-3-1-4 difference.

Q

The following figure will serve to illustrate the rule.

Diameter A B or C D 20, and the sides of the square A D, or D B, or B C, or C A 14.3, and are sides of the greatest square that can be made in the circle.

Examples.

1. What solidity of square timber can be hewn from a stick that is 20 feet long, and girts 62.8 in.?

Girt 62.8044-143 nearly X 14.3-204-49 X by length 20=4089 80144-28 4† solidity Ans.

What is the solidity of the above stick of timber allowing of the circumference to be a side of the square timber, as directed in case 16th.

Girt 62-8=15-7 X15-7-246-49 X length 20 4929-8014434.2 solidity Ans.

NOTE. By comparing the two preceding answers, it is evident that the last method of operation gives