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Er. 2. What is the content of a conical cask, the length being 20, and the bung and head diameters 16 and 12 inches ?

910-98752 ale gallons,

Answer { 13:41866 wine gallons

Prob. 16. To find the content of a cask by four dirnensions.

Add together, the squares of the bung and head diameters, and the square of double the diameter taken in the middle between the bung and head; then multiply the sum by the length of the cask, and the product again by '00043 for ale gallons, or by .00053 for wine gallons.

Er. 1. Required the content of any cask whose length is 40, the bung diameter being 32, the head diameter 24, and the iniddle diameter between the bung and head 289 inches.

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Set 40 on C, to 420 on D; then against

24 on D, stands 13.0
32 on D, stands 23.2
574 on D, stands 75.0

sum 111.2 wine gallons.

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Ex. 2. What is the content of a cask, whose length is 20, the bung diameter being 16, the head diameter 12, and the diameter in the middle between them 143?

5 11:4479 ale gallons,

113.9010 wine gallons. Prob. 17. To find the content of any cask from three dimensions only.

Add into one sum, 39 times the square of the bung diameter, 25 times the square of the head diameter, and 26 times the product of the two diameters : then multiply the sum by the length, and the *00034

00034 product again by

for wide gallons, or by or '000037's 9

Il for ale gallons.

Ex. 1. Required the content of a cask, whose length is 40, and the bung and head diameters 32 and 24 ? 32

24 32 24

24

32

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2972160
00034

2972160

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11888640
8916480

8916480

270196 9)1010°53440

112-2816 wine gallons 91.86676 ale gallons

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Er. 2. What is the content of a cask, whose length is 20, and the bung and head diameters 16 and 12 ?

Answer {

{140352 wine gallons.

4833 ale , Note.--This is the most exact rule of any, for three dimensions only; and agrees nearly with the diagonal rou.

Of the Ullage of Casks. The ullage of a cask, is what it contains when only partly filled. And it is considered in two positions, namely, as standing on its end with the axis perpendicular to the horizon, or as lying on its side with the axis parallel to the horizon.

Prob. 18. To find the ullage by the sliding rule.

By one of the preceding problems find the whole content of the cask Then set the length on N, to 100 on SS, for a segment standing, or set the bung diameter on N, to 100 on SL, for a segment lying ; then against the wet inches on N, is a nunber on SS or SL, to be reserved.

Next, set 100 on B, to the reserved number on A; then, against the whole content on B, will be found the ullage on A.

Ex. 1. Required the ullage answering to 10 wet inches of a standng cask, the whole content of which is 92 gallons, and length 40

hes.
Having set 40 on N, to 100 on SS; then

against 10 on N, is 23 on SS, the reserved number. Then set 100 on B, to 23 on A ; and

against 92 on B, is 2162 on A, the ullage required. 2. What is the ullage of a standing cask whose whole length is 20 inches, and content 11į gallons; the wet inches being 5 ?

Ans. 2 65 gallons. 3. The content of a cask being 92 gallons, and the bung diameter 32, required the ullage of the segment lying when the wet inches are

Ans. 17:6 gallons. Prob. 19. To ullage a standing cask by the pen.

Add all together, the square of the diameter at the surface of the liquor, the square of the diameter of the nearest end, and the square of double the diameter taken in the middle between the other two; then multiply the sum by the length between the surface and nearest end, and the product again by *00043 for ale gallons, or by 0005; for wine gallons, in the less part of the cask, whether empty or filled.

Er. The three diameters being 24, 27, and 29 inches, required the ullage for 10 wet inches.

Here 243=576, 29°=84), and 542=2916.

Now 2916+ 841 +576=4333;
whence 4333 x 10 x .0005}=24:5535 wine gallons ;

and 4333 X 10 X.0004;=20-22:05 ale gallons.

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MECHANICS.

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1. Mechanics is that Science which treats of the laws of Equilibrium and Motion.

As it relates to solid bodies, it is divided into Statics and Dynamics ; the former regards the theory of equilibrium, and the latter that of motion. ON MATTER, MOTION, FORCES, &c.

Definitions. 2. Matter, or Substance, of which bodies are composed, is, in respect of its essence, wholly unknown. All that we know of matter relates to the various properties and qualities which present themselves to our senses.

Of these properties, the following are the principal, as they are always united to matter : viz. extension, figure, solidity, mobility, gravity, and inertia or natural inactivity.

3. Extension may be considered in three points of view :

Ist. As simply denoting that part of space which lies between two points, in which case it is called distance.

2dly. As implying both lengib and breadth, when it is denominated surface, or area.

3dly. As comprising three dimensions, length, breadth, and thickness ; in which case it may be called bulk, capacity, or content.

4. Figure is the boundary of extension. Thus all portions of matter, from which we receive our ideas of this substance, are bounded; that is, they have figure.

5. Solidity is that property of matter by which it fills space ; or, by which any portion of matter excludes every other portion from that part of space

which it occupies. 6. Mobility, or the capacity of being transferred from one place to another, is a quality which experience teaches to belong to all bodies that fall under our cognizance, and we therefore conclude it to be an inherent property of matter.

7. Divisibility signifies a capacity of being separated into parts. That matter is thus divisible, daily experience assures us ; nor does there

appear to be any limit to this division. We know that many bodies may be reduced 10 a very fine powder by trituration ; and by chemical solution, the parts of a body may be so far divided as not to be sensible to the sight; and, again, by the help of the microscope, we discover myriads of organized bodies, which were totally unknown before such instruments were invented. From these and other considerations we are naturally led to suppose, that this capacity of divisiún is without limit.

8. Gravity is the tendency which all bodies have towards the centre of the earth.

We are convinced of the existence of this tendency by observing, that whenever a body is sustained, its pressure is exerted in a direction perpendicular to the horizon; and that when the impediment is removed, the body always descends in that direction. The weight of any body is its tendency to the earth, compared with the like tendency of some other body, considered as a standard.

Gravity is not an accidental property of matter, arising from the figure or disposition of the parts of a body; for, if this were the case, by changing its shape, or altering the arrangement of the particles which compose it, the gravitation of the mass would be aliered. But we find that no separation of the particles, no change of structure, which the human power can effect, produces any alteration in the weight. Hence the gravity of the whole is the aggregate of the gravities of all the parts; and hence the weight of a body can be altered only by an increase or diminution of the number of particles of which it is composed.

9. Inactivity inay be considered in two lights:

Ist. As an inability in matter to change its state of rest, or of uniform rectilinear motion.

2dly. As that quality by which it resists any such change. In this latter sense it is usually called inertia, inertness, or vis inertiæ, or the force of inactivity.

That a body resists any change in its state of rest, or of uniform rectilinear motion, we know from constant experience ; for we can. not move the least particle of matter without some exertion ; nor can we destroy any motion without perceiving some resistance. Hence we conclude that inertia is a property inherent in all bodies with which we are concerned ; differing in quantity, indeed, in different cases, but existing in a greater or less degree in all.

The preceding properties are always found to exist together in the same substance, and they are, by some, considered as essential and inseparable from matter.

10. By the Quantity of Matter in a body, we understand the aggregate of its particles, each of which has a certain degree of weight and of inertia. Now the oniy properties of matter which admit of exact comparison, and which depend upon the number, and not upon the arrangement of the particles, are weight and inertia, and either of these may properly be made use of as a measure of the quantity of matter. Thus, if b represent the quantity of matter or mass, and w the weight of any body, then 6 is always proportional to w.

The Density of a body is measured by the quantity of matter in a given bulk.

11. By Motion, we understand a change of place : it is of two kinds, absolute and relative.

A body is said to be in absolute motion, when it is actually trans. ferred from one point in fixed space to another; and io be relatively in motion, when its situation is changed with respect to the surrouuding objects.

When a body alwavs passes over equal parts of space in equal

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