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CASE XIII.

To find the solidity of a sp heroid, or ellipsoid; and also to find the solidity of the middle frustim of a spheroid.

DEFINITION.- A spheroid is a solid generated by the revolution of an ellipse, or oval, about the transverse or conjugate diameter. [See the figure A B C D.]

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Po find the solidity of the spheroid A B C D. RULE.-Multiply the revolving diameter B D, into itself, and the product by the fixed diameter A C, and the product again by 5236, the last product is the solidity.

Examples. 1. What is the solidity of a spheroid whose revolving diameter B D is 20, and fixed diameter AC is 30? Revolv. diameter 20x20=400X30X•5236=6283.2 Ans.

2. What is the solidity of a spheroid whose revolving diameter is 30, and fixed diameter 50 ? Ans. 23562.

To find the solidity of the middle frustum of a spheroid.

RULE.-To the square of the end diam. add twice the square of the middle diameter, multiply this sum by the length, and the product again by 2618 the last product is the solidity

Examples. 1. What is the solidity of the frustum E F G H length 40, end diameters E G, or F H 24, and middle diameter BD 32. ?

End diam. 24x24= 576
Mid. diam. 32X32X2=2048 add.

2624x40X.2618=27478-5280 Ans. 2. What is the solidity of the middle frustum of a spheroid whose length is 30; middle diameter 25; and end diameter 20 inches?

Ans. 12959.1 CASE XIV. To find the solidity of an elliptic spindle. DEFINITION.-An elliptic spindle is formed by any of the three conic sections revolving about a double ordi. nate; the following figure represents an elliptic spindle.

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Rule.-Find a diameter half way between the middle and end, multiply it by two, 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.

Erainples. 1. What is the solidity of the elliptic spindle V BCD; length 15; diameter C D 5; and diameter taken half way between the middle and end equal to 4 ? Diam. O P.4x2=8X8=64 sq. of twice O P. Diam. C D 5X5=25 sq. of C D add.

89 sum.

Sum, 89X15X.13091747515 Ans. 2. What is the solidity of an elliptic spindle thar is 22 long ; greatest diameter 15; and the diameter his way between the middle and end, equal to 10?

P

Ans. 18

To find the solidity of the middle frustum of an

elliptic spindle. RULE.-To the square of twice a diametere 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.

Note.-EF G H in the preceding figure represents the Toiddle frustum, of an elliptic spindle.

Examples. 1. What is the solidity of the middle frustum E GF 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 0 P.
Diam, CD

6X 6 =
= 36 s

sq.

of CD. Diam. E F or G H 4X 4= 16 sq. of E For G H.

152 sum. Sun, 152 X10 X 13093198.968 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. 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,

Nore.-This method of measuring round timber is very erTokeous ; it gives two much for the solidity of square timber in the stiak. and not enoligh if the four slabs, or segments are to be includea ; mi I shall prove in case 17th.

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+1=12x12=144x21–3024; 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 measúres 24 inches ?

71 ft. Ans. 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 1514 ft. 4. What is the solidity of square timber in a stick of l'ound timber, that girts 50 in. and is 31 ft. 7 in, long

Ans. 34 ft. 3' 2'' 10" 21" or 34151

8 68 6912

CASE XVII.

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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 square.

Rule--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 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.

Illustration of the preceding rule. 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 circum. ference 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 same circum. ference were divided by the quotient would be 15.7 for a side of square tiiber that could be hewn from such a stick.

15.7 14.3.1.4 difference.

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The following figure will further illustrate the

preceding rule.

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D Diameter AB or CD 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 62.80-4414.3 nearly x14-3204:49 x by length 20=4089.80---144-28°47 solidity, Ans.

The same question solved by case sixteenth.

Girt 62-8-1=1507x157246:49 xlength 204929-80 14434.27 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 circum. 34.27
True solidity found by case 17th. 28 41

difference 5.8

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