Er. 2. What is the content of a cask, whose length is 20, a:id the bung and head diameters 16 and 12? 11 4833 ale gallons, Answer { 14:0352 wine galvis. Note. This is the most exact rule of any, for three dimensions only; and agrees nearly with the diagonal rod. 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 stanu. ing, or set the bung diameter on N, to 100 on SL, for a segment lying ; then against the wet inches on N, is a number 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. Er. 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. 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 21'2 on A, the ullage required. 2. Wbat is the ullage of a standing cask whose whole length is 20 inches, and content 114 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 8? 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 '00044 for ale gallons, or by *00051 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 24*=576, 29°=84), and 54°=2916. Now 2916+841 +576=4333 ; and 4333 X 10 X .0004;=2022:05 ale gallons. MECHANICS. 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 : 1st. 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 au 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 division 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 parti. cles which compose it, the gravitation of the niass would be altered. 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 may 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 cannot 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 to be relatively in motion, when its situation is changed with respect to the surrounding objects. When a body alwavs passes over equal parts of space in equal successive portions of time, its motion is said to be uniform. When the successive portions of space, described in equal times, continually increase, the motion is said to be accelerated ; and retarded, when those spaces continually decrease. Again, motion is said to be uniformly accelerated or retarded, when the increments or decrements of the spaces, described in equal and successive portions of time, are always equal. 12. The Velocity of a body, or the rate of its motion, is measured by the space, uniformly described in a given time. The given time, taken as a standard, is usually one second ; and the space described is measured in feet. Thus, when v represents the velocity of a body, v is the number of feet which the body would uniformly describe in one second. If the motion of a body be accelerated or retarded, the velocity at any point is not measured by the space actually described in a given time, but by the space which would have been described in the given time, if the motion bad continued uniform from that point.- 13. The Quantity of Motion, or Momentum, is that power or force in moving bodies, by which they continually tend from their present places, or with which they strike any obstacle that opposes their motion. It is measured by he velocity and quantity of matter jointly. 14. Whatever changes, or tends to change, the state of rest, or of uniform rectilinear motion of a body, is called force, or power. Thus pressure, iinpact, gravity, &c. are called forces. When a force produces iis effect instantaneously, it is said to be impulsive. When it acts incessantly, it is called a constant force. Constant forces are of two kinds, uniform and variable. A force is said to be uniform, when it always produces equal effects in equal successive portions of time ; and variable, when the effects produced in equal time are unequal. Forces which are known to us only by their effects, must be compared by estimating those effects under the sanje circumstances. Thus, impulsive forces must be nieasured by the whole effects produced ; uniform forces, by the effects produced in equal times; and variable forces, by the effects which would be produced in equal times, were they to become and continue uniform during those times. The effects produced by the actions of forces are of two kinds, velocity and momentum ; and thus we have two methods of comparing thein, according as we conceive them to be the causes of velocity or moinentum. Under this point of view, forces are distinguished into motive, and accelerative, or retarding. 15. A motive, or moving force, is the power of an agent to produce motion ; and it is measured by the momentum uniformly generated in a given time. If the munienta thus generated, in any two cases, be as 14 to 15, the moving forces are said to be in that ratio. 16. An accelerative, or retarding force, is understood to affect the velocity only, and is measured by the velocity uniformly generated or destri yed in a given time, without regard to the quantity of ma:ter moved. Thus, if the velocities uniformly generated in two cases, in equal times, be as 6 to 7, the accelerating forces are said to be in that ratio. Other qualities and properties of bodies, as Solidity, Fluidity, Hardness, Softness, Elasticity, &c. will be defined in the succeeding articles. The relations between the quantities of matter, magnitudes, or bulks, densities of bodies, &c. may be expressed as follows. 17. The quantity of matter in all bodies is in the compound ratio of their magnitude and densities. For when their magnitudes are equal, the quantity of matter which they contain is evidently as their densities, and when their densitiesare the same, the quantity of matter which they contain is as their magnitudes. When, therefore, neither their magnitudes por densities are equal, the quantity of matter which they contain is in a ratio compounded of both. Cor. 1. When the bodies are similar, then masses, or quantities of matter, are as their densities and the cubes of their diameters. For the magnitudes of bodies is shown, in Geometry, to be as the cubes of their diameters, or other homologous dimensions. 2. The specific grarity of a body, is the weight of a certain magnitode of that substance, such as a cubic foot, a cubic inch, &c. Hence the specific gravities of bodies are proportional to their densities; an. hence, again, the masses are as the magnitudes of the bodies and their specific gravities. Scholium. Let b denote any body, or quantity of matter m its magnitude, a its diameter. Then, from the last Articles and Corollaries, we obtain the following general Table of Proportions. bamda msaa'd aa' maga sa b aagama az 6 ald ada m As an example of the application of this Table :-Required the proportion of the diameters of two spheres, in which the quantities of matter are as 6 to 5, and the densities as 4:3. Let A, a represent the respective diameters; B, 6 the quantities we B 6 6 5 shall have A' : QS :: D d :: 18 : 20 :: 9:10. Hence 3 the cubes of the diameters are as 9 : 10; and the diameters-themselves as 19 : 10. of matter ; and D, d the densities ; then, since by the Table acca |