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PROBLEM II.

Given the dimensions of any vessel, to find the dimensions of another similar vessel, that shall be any number of times greater or less than the given vessel.

EXAM. 1.

Here 2163 10077696, the cube of the given length; and 10077696 x 8 (the ratio) 80621568; hence 80621568 432 inches, the length of the required vessel.

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Also, 1253 = 1953125, the cube of the given breadth; and 1953125 x 8 = 15625000; hence 15625000 = 250 inches, the breadth of the required vessel.

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Lastly, 643 = 262144, the cube of the given depth; and 262144 x 8 = 2097152; hence/2097152 = 128 inches, the depth of the required vessel.

EXAM. 2.

Here 643 262144, the cube of the given depth; and 262144 × 27 (the ratio) = 7077888; hence

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7077888192 inches, the depth of the required

vessel.

Also, 273 = 19683, the cube of the given diameter; and 19683 × 27 = 531441; hence ✓531441 = 81 inches, the diameter of the required vessel.

EXAM. 3.

Here 3433— 40353607, the cube of the given depth; and 40353607 ÷ (the ratio) = 5044200.875; hence 5044200.875 = 171.5 inches, the depth of the required vessel.

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Also, 1253 = 1953125, the cube of the given diameter; and 1953125 ÷ = 244140.625; hence

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244140.623 = 62.5 inches, the diameter of the required vessel

EXAM. 4.

Here 30o = 2,000, the cube of the given length: and 27000 × 3 (the ratio) = 81000; hence 81000 = 43.267 inches, the length of the required cask.

Also, 24 = 13824, the cube of the given bung diameter; and 13824 x 3 = 41472; hence 41472 = 34.614 inches, the bung diameter of the required cask.

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Lastly, 185 5832, the cube of the given head diameter; and 5832 x 3 = 17496; hence 17496 =25.96 inches, the head diameter of the required cask.

PROBLEM III.

Given the dimensions and content of any vessel, to find the dimensions of a similar vessel of a given

content.

EXAM. 1.

Here, as 2625: 253 :: 21000; 125000, the cube of the length; and 125000 = 50 inches, the length of the required vessel.

Also, as 2625: 153 :: 21000

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the breadth; and 27000 = 30 of the required vessel.

Lastly, as 2625

the depth; and, the required vessel.

27000, the cube of

inches, the breadth

75: 21000 2744, the cube of

2744 14 inches, the depth of

EXAM. 2.

Here 125.33 × 282 = 35343.06 cubic inches, the

content of the given vessel.

Also, 68.5 x 282 = 19317 cubic inches, the content of the vessel, of which the dimensions are required.

Then, as 35343.06 303 :: 19317 14757.041410, the cube of the diameter of the said vessel; hence

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14757.041410 24.52 inches, its diameter.

Lastly, as 35343.06 503:19317 68319.636160, the cube of the depth of the said vessel; hence 68319.636160 = 40.88 inches, the depth required.

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EXAM. 3.

Here 599.45 × 231 = 138472.95 cubic inches, the content of the given vessel.

Also, 946 × 231 = 218526 cubic inches, the content of the vessel, of which the dimensions are required.

Then, as 138472.95 463 :: 218526: 153607.233 297, the cube of the bottom diameter of the said ves

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sel; hence 153607.233297 = 53.55 inches, its bottom diameter.

And, as 138472.95: 623 :: 218526: 376108.500 731, the cube of the top diameter of the said vessel; hence/376108.50073172.18 inches, its top dia

meter.

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Lastly, as 138472.95 603 218526: 340872.466 427 the cube of the depth of the said vessel; hence 340872.466427 69.85 inches, the depth re

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quired.

EXAM. 4.

Here 38.75 x 282 = 10927.5 cubic inches, the content of the given vessel.

Also, 55.5 x 282 = 15651 cubic inches, the content of the vessel, whose dimensions are required.

Then, as 10927.5: 303 :: 15651 38670.967741 the cube of the length, of the said vessel; hence 38670.967741 = 33.82 inches, its length.

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Again, as 10927.5: 243 :: 15651: 19799.535483
E

the cube of the bung diameter, of the said vessel;

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hence 19799.535483 = 27.05 inches, its bung diameter.

Lastly, as 10927.5 : 183 :: 15651: 8352.929123 the cube of the head diameter, of the said 2028 inches, the head

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vessel; and/8352.929123

diameter required.

PART II.

THE USE OF THE SLIDING RULE.

PROBLEM I.

Multiplication by the lines A and B.

EXAM. 2.

As 1 on A: 8 on B:: 15 on A: 120 on B.

EXAM. 3.

As 1 on B 24 on A:: 64 on B: 1536 on A.

EXAM. 4.

As 1 on A: 128 on B :: 265 on A: 33920 on B.

EXAM. 5.

As 1 on B ; 5.3 on A :: 8.5 on B ; 45.05 on A.

EXAM. 6.

As 1 on A ; 3.5 on B :: 4.7 on A ; 16.45 on B.

EXAM. 7.

:

As 1 on B: 1.7 on A :: 3.5 on B: 5.95 on A.

EXAM. 8.

As 1 on A ; 2.4 on B ;; 6.2 on A ; 14.88 on B.

EXAM. 9.

As 1 on B.9 on A: 1.8 on B 1.62 on A.

EXAM. 10.

:

As 1 on A 32.8 on B:: 64.7 on A: 2122.16 on B.

EXAM. 11.

As 1 on B.32 on A :: 86.3 on B: 27.616 on A.

EXAM. 12.

As 1 on A: .238 on B:: .562 on A: .133756

on B.

PROBLEM II.

Division by the lines A and B.

EXAM. 2.

As 7 on A: 1 on B: 42 on A: 6 on B.

EXAM. 3.

As 8 on B 1 on A :: 96 on B: 12 on A.

EXAM. 4.

As 24 on A: 1 on B:: 1536 on A: 64 on B.

EXAM. 5.

As 128 on B: I on A:: 33920 on B: 265 on A.

EXAM. 6.

As 5.3 on A : 1 on B:: 45.05 on A: 8.5 on B.

EXAM. 7.

As 4.7 on B 1 on A:: 16.45 on B: 3.5 on A.

EXAM. 8.

As 2.4 on A 1 on B:: 14.88 on A: 6.2 on B.

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