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PROBLEM XI.

To find the Convex Surface and Solid Content of a

Cylindric Ring. RULE 1.–Multiply the thickness of the ring added to the inner diameter, by the thickness and by 9.8698, and the product will be the convex surface.

RULE 2.—To the thickness of the ring, add the inner diameter.—Multiply that sum by the square of the thickness and by 2.4674, and the product will be the solid content.

EXAMPLE 1.—The thickness of a cylindric ring AC or D B = 2 inches, and inner diameter = 18, re quired the convex superficies.

18 + 2 x 2 x 9.8698 = 394.792 square inches and = 144 = 2.741 superficial feet nearly

D B в

EXAMPLE 2.-Required the solid content of the ring as above.

18 + 2 x 22 x 2.4674 = 197.392 cubic inches and = 1728 = .114 cubic feet nearly,

OF TIMBER MEASURE.

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Timber is chiefly estimated by the square or superficial foot of 144 inches, or cubic foot of 1728; the calculation of which is performed by duodecimals, that is, the foot or inch, &c. divided into 12 parts or divisions, thus :12 fourths make..

I third, 12 thirds

1 second, 12 seconds

1 inch,
12 inches

1 foot.
And the several values arising are:
Feet multiplied by feet give feet,
Feet multiplied by inches give inches,
Feet multiplied by seconds give seconds,
Inches multiplied by inches give seconds,
Inches multiplied by seconds give thirds,
Seconds multiplied by seconds give fourths, &c.

But this rule is more commonly called Cross Mul. tiplication, on account of commencing with the left hand figure of the multiplier.

Rule 1.-Place the multiplier under the multiplicand, feet under feet, inches under inches, seconds under seconds, &c.

2.-Multiply each denomination of the length by the feet of the breadth, beginning at the lowest, and place each product under that denomination of the multiplicand from which it arises, always carrying one for every 12.

3.—Multiply by the inches, and set each product one place farther to the right hand.

4.-Then multiply by the seconds, and set each pro. duct another place toward the right hand, &c.

Thus proced in like manner with all the other denominations, and their sum will be the content.

EXAMPLE 1.-Required the superficial content of a board 12 feet 6 inches long, and I foot 54 inches broad.

F. I.

12 6 Multiplied by 1 5 6

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Feet 18 2 9 When the two ends of a board or plank are of different breadths, add the two breadths together, and multiply the length by half the sum.

Example 2.-A plank that is 1 foot 4 inches broad at one end, 11inches broad at the other, and 18 feet 9 inches long, what is its superficial content ? 16 + 114 = 27! 2 = 131 inches.

F. I.
Then

18 9
131 inches = 1 1 9

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Superficial Measure by the Engineer's Slide Rule.

When the length is given in feet, and the breadth in inches, the gauge point is 12; but if the dimensions are all inches, the gauge point is 144.

Rule.-Set the breadth upon B to the gauge point upon A, and against the length upon A is the content in square

B. Example 1.-Required the number of square feet contained in a board 114 inches broad and 18 feet long Set 11.5 upon B to 12 upon A; and against A is 17.3 feet upon

B.

feet upon

18 upon

The content of one board being found, the content of any number of the same dimensions may

be found by setting 1 upon B to the content of the one found upon A ; and against any number of boards upon B is the whole content upon A.

Find the content of 8 boards, each being 17.3 square

feet.

Set 1 upon B to 17.3 upon A ; and against 8 upon B is 138.4 feet upon A. EXAMPLE 2.-If a board is 10 inches broad at one end, and 7 at the other, what must be its length to make a square foot ?

10 +7= 17 = 2 = 84 inches. Set 8.5 upon B to 144 upon A; and against 1 upon B is 16.9 inches long upon A.

To find the Solidity of Timber. The solid content of timber (according to custom) is found by multiplying the length by the square of the girth.

EXAMPLE.—Required the content of a tree, in cubic feet, whose girth in the middle is 84 inches and length 25 feet 6 inches. 84 · 4 = 21 inches 4 girth.

F.
and 21 inches = 1
Multiplied by 1

1 9
1 3 9

I. 9 9

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F. 1.
Then 25 6
Multiplied by 3 0 9

6

1 7 1 6 Feet

1 1 6

76

78

But a more expeditious method is obtained by means of the following

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19
192
20
203
21
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22
222

75 7 7$

23

.250 .272 .294 .317 .340 .364 .390 .417 .444 .472 .501 .531 .562 .594 .626 .659 .694 .730 .766 .803 .840 .878 .918 .959 1.000

124 12 12 13 134 13 13% 14 145 142 144 15 154 15 152 16 164 164 16 17 174 177 174 18 185

1.042
1.085
1.129
1.174
1.219
1.265
1.313
1.361
1.410
1.460
1.511
1.562
1.615
1.668
1.722
1.777
1.833
1.890
1.948
2.006
2.066
2.126
2.187
2.250
2.376

231
24
244
25
254
26
262
27
271
28
282
29
294
30

2.506 2.640 2.777 2.917 3.062 3.209 3.362 3.516 3.673 3.835 4.000 4.168 4.340 4.516 4.694 4.876 5.062 5.252 5.444 5.640 5.840 6.044 6.250

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Rule.-Multiply the area corresponding to the 1 girth in inches by the length of the timber in feet; and the product is the solidity in feet and decimal parts.

Example.--A piece of tim ber 18 feet long, and 14 inches square, how many cubic feet does it contain ?

1.361 x 18 = 24.498 cubic feet.

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