RULE.-(2) Find a diameter half way between the middle and end, as at O P, multiply it by 2, square the product and add the square to the square of a diameter taken in the middle, multiply the sum by the length, and the product again by 1309, the last product is the solidity, very nearly.

Questions.-1. Describe an elliptic spindle.-2. How do you find the solidity of an elliptic spindle?

EXAMPLES.

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

Diam. O P. 4×2=8×8=64 sq. of twice O P.
Diam. C D 5×5=25 sq. of C D add.

89 sum.

Sum 89X15X1309-174-7515 Ans.

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

CASE XV.

Ans.

1636-25.

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

RULE. (1) To the square of twice a diameter taken half way between the middle and end, add the squares of the middle end diameters; multiply the sum by the length, and again by 1309, the last product is the solidity required.

NOTE. E F G H in the preceding figure represents the middle frustum of an elliptic spindle.

Question.-1. Give the rule for finding the solidity of the middle frustum of an elliptic spindle?

EXAMPLES.

1. What is the solidity of the middle frustum E G F H, 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.
Diam. C D
6x 6 36 sq. of

C D.

Diam. E For G H 4× 4 16 sq. of E F or G H.

152 sum.

Sum 152X10X 1809-198.968 Ans.

CASE XVI.

The common, but erroneous method of finding the solidity of square timber contained in a round stick.*

RULE. In order to ascertain the quantity of hewn tim ber contained in a round stick, it has been the common practice to (1) girt the stick in the middle with a line, (after taking off the bark) and then to double the line into four equal parts, and one of these parts is considered equal to a side of square timber, that can be hewn from the stick; which side is multiplied into itself in inches, and the product by the length in feet; the last product divided by 144 is considered the solidity in solid feet.

Question.-1. What is the common method of finding the solidity of square timber contained in a round stick?

EXAMPLES.

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

Girt 48-4-12×12=144×21-8024÷144-21 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? Ans 7 ft. 3. What is the solidity of a stick of timber that girts 56 inches, and its length being 11 ft. 9 inches? Ans. 15 ft. 11' 11", or 1543 ft. 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 34

CASE XVII.

1868 ft.

6912

A 2nd and more accurate 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 square.

RULE.—(1) Girt the stick (after taking off the bark,) and annex a cypher to the girt in inches and divide the whole by 4.4 the quotient is a side of the greatest square in inches that can be hewn from such a stick; multiply this side into

* This method gives too much for the solidity, if it is meant to measure the solidity of square timber only; and not enough if it is meant to include the four slabs, or segments.

itself, and this product again by the length in feet, and the last product divided by 144 will be the solidity of the stick in feet after it is hewn.

Question.-1. What is the 2nd and more accurate method of finding the solidity of square timber contained in a stick of round timber?

Illustration of the preceding rule.

Make a circle whose diameter is 20, and of course its circumference would be 62.8t (which may represent the girt line of a stick of timber;) annex a cypher to the circumference and divide by 44, the quotient will be 14:3 nearly, 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; but if the same circumference were divided by, (which would represent the length of the girt line in case sixteenth, after it was doubled into four equal parts) you would find that the quotient would be 15.7 for a side of the square timber.

The following figure will further illustrate the preceding rule.

Diameter A B or C D 20, and the sides of the square A D, or D B, or B C, or CA 143, 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 inches?

Girt 62-80 4 4 14.3 nearly x 14.3=204.49 X by length 20-4089-80-144-28.4† solidity, Ans.

The same question solved by case sixteenth.

Girt 62-8-15·7×15·7-246-49 X length 20-4929-80÷ 144-34-2† solidity Ans.

NOTE.-By comparing the two preceding answers, it is evident that the ⚫ last method of operation gives too much for the solidity of square timber in such a stick.

Solidity found by using the circumference
True solidity found by case 17th

difference 5.8

2. What is the solidity of square timber in a stick of round timber, that is 21 feet long, and 48 inches in circumference?

Girt 48.0÷4-4=10·9† a side of square timber in the stick. 10-9×10-9×21=2495·01÷144-17:32† solidity, Ans. The same question solved by using of the circumference, Girt 48-4=12×12×21=3024-144-21 feet, Ans.

Difference 3'68 ft. too much. 3. What is the solidity of square timber in a stick of round timber, that is 11 ft. 9' long, and girts 56 inches? Girt 56 0-4-4=1 ft. 0' 8"† a side x 1 ft. 0' 8"-1 ft. 1' 4" 5' 4" mul. by the length 11 ft. 9=18 ft. 1' 1'' 2''' 8'''' Ans. 4. What solidity of square timber is in a round stick, that is 31 ft. 7 inches long, and girts 50 inches?

50·0-4-4-11tin. a side. Ans. 26 ft. 6' 5" 7" NOTE. Having shown by the two first questions in this case, that using of the circumference for a side of square timber, gives too much for the solidity, or more than the stick will measure after it is hewn square; it now remains to show that of the circumference multiplied by itself, and then by the length, does not produce the solidity of the stick, if the four segments or slabs are to be included.

EXAMPLES.

1. What is the solidity of a stick of timber whose length is 20 ft. circumference 62.8 in. and diameter, 20 in. measured by using of the circumference? Ans. 34.2 ft.

The same stick measured as a cylinder.

Ans. 48.5 ft.

NOTE. It is obvious from the preceding examples that the square of the circumference, multipied by the length, gives too much for the solidity of square timber in the stick, (or more than it would measure if it were hewn square:) and not enough if the four slabs, or segments are to be included; It ought to be a matter of consideration by those concerned in buying and selling timber, how it should be measured; and if it is agreed by the parties to measare only the square timber, (or what the stick would measure if it were hewn) then apply the rule in case 17th; but if it is agreed to measure the whole solidity of the stick (including the segments, or four slabs,) then apply the rule in the following case.

CASE XVIII.

To find the solidity of a round stick of timber, including the four slabs, or segments.

RULE.-(1) Girt the stick in the middle (after taking off the bark,) annex two cyphers to the girt, or circumference, and divide by 3.14 the quotient is the diameter nearly: multiply the girt, or circumference and diameter together; and one fourth part of the product multiplied into the length will be the solidity required.

A

Question.-1. How do you find the solidity of a round stick of timber, including the four slabs or segments?

EXAMPLES.

1. What is the solidity of a stick of timber, that girts 94.2 in. and length 20 feet?

94.20 3.14-30 diam. 94.2×30÷1=706·5×20÷144=98 T4 ft. Ans.

144

2. What is the solidity of a stick of timber, that is 22 ft. long; and the girt line measuring 31-4 in.? Ans. 114.

NOTE. In all the preceding examples in timber measure, the timber has been considered of equal bigness from end to end; it now remains to treat of tapering timber, both round and hewn.

CASE XIX.

To find the solidity of hewn timber in a round stick, when the stick is tapering from end to end.

RULE (1) Girt the stick at both ends, annex a cypher to the girts, or circumferences, and divide each girt by 4·4 the quotients will be the sides of square timber, multiply the two sides together; find the difference between the two sides; square the difference, and add one third of its square to the product of the two sides, and multiply this sum by the length, the last product is the solidity required.