F Now if ċ the number 3•1416, then 2c. X FG is the circumference of the circle EFG E G whose radius is FG; therefore ¢X FGʻis the area of that circle. B But, by cor. theor. 1, Parabola, P x AF = FGʻ, where p denotes the parameter of the parabola; consequently pc x AF till also express the same circular section Eg, and therefore pc x the sum of all the Af's will be the sum of all those circular sections, or the whole content of the solid paraboloid. But all the Af's form an arithmetical progression, beginning at 0 or nothing, and having the greatest term and the sum of all the terms each expressed by the whole axis ad. And since the sum of all the terms of such a progression, is equal to AD X AD or ( AD’, half the product of the greatest term and the number of terms; therefore Ad’ is equal to the sum of all the Af's, and consequently pc x į AÐ?, or c x P x AD’, is the sum of all the circular sections, or the content of the paraboloid. DCZ But, by the parabola, p:dc:: DC : AD, or p = sequently {c x P x ADbecomes {c X ADÏX Dc for the solid content of the paraboloid. But c X AD X DG* is equal to the cylinder BCIH; consequently the paraboloid is the half of its circumscribing cylinder. ; con AD Q. E. D. THEOREM XIX. The Solidity of the Frustum BEGC of the Paraboloid, is equal to a Cylinder whose Height is DF, and its Base Half the Sum of the two Circular Bases eG, BC. For, by the last theor. } pc X AD' = the solid abc, and, by the same, ż pc X AF? = the solid AEG, theref. the diff. ž pc x (AD’— AF)=the frust. BEGĆ. But AD? AF? = DE X (AD + AF), theref. į pc X DF X (AD + AF) = the frust. BEGC. But, by the parab. p X AD = Dc, and p X AF = FG?; theref, jc X DF X (DC? + FG) = the frust. BEGC. Q. E. D. OF OF MOTION, FORCES, &c DEFINITIONS. Art. 1. BODY is the mass, or quantity of matter, in any material substance; and it is always proportional to its weight or gravity, whatever its figure may be. 2. Body is either Hard, Soft, or Elastic. A Hard Body is that whose parts do not yield to any stroke or percussion, but retains its figure unaltered. A Soft Body is that whose parts yield to any stroke or impression, without restoring themselves again; the figure of the body remaining altered. And an Elastic Body is that whose parts yield to any stroke, but which presently restore themselves again, and the body regains the same figure as before the stroke. We know of no bodies that are absolutely, or perfectly, either hard, soft, or elastic; but all partaking these properties, more or less, in some intermediate degree. 3. Bodies are also either Solid or Fluid. A Solid Body, is that whose parts are not easily moved among one another, and which retains any figure given to it. But a Fluid Body is that whose parts yield to the slightest impression, being easily moved among one another; and its surface, when left to itself, is always observed to settle in a smooth plane at the top. 4. Density is the proportional weight or quantity of matter in any body. So, in two spheres, or cubes, &c, of equal size or magnitude; if the one weigh only one pound, but the other 2 pounds; then the density of the latter is double the density of the former; if it weigh 3 pounds, its density is triple; and so on. 5. Motion is a continual and suecessive change of place. If the body move equally, or pass over equal spaces in equal times, it is called Equable or Uniform Motion. But if it increase or decrease, it is Variable Motion; and it is called Accelerated Motion in the former case, and Retarded Motion in the latter.-Also, when the moving body is considered VOL. II. K with SeeIserna zaezizicia; consequently pe X AF as sarske ure EG, and therefore :he sun ar be the son of all those SICHI SLos 12 v cca: o he solid paraex Zural's Art z sedice progression, begin. ving ! CANT. Ne zezres term and the sun is stressed by the whole axis ad. BESSIES of such a progression, is Direzte of the greatest en nie uczess; az is equal to e sun fals i espex AD, or sans ie SE se crezia sections; or the II Iait. DC AD erueni' X secoces e X AD X DC for the si meme ce se zarzdcioni But. X AD X DG is equal weiter rs cocsequerely the paraboloid is the half asserising concer. ; con Q. E.d. TEORIN II Crner viese Height is DF, and its Base Half FUR, ville at ther. x = the solid A MA, e, X AF = the solid seref be : pex (AD - AF*)=tBus av - AF = PE X (AD + AF), X CE X AD + AF) = the Bury i zde, ar X AD = pc, and p Vera DE X, DC + FG) = the with respect to some other body at rest, it is said to be Ab. solute Motion. But when compared with others in motion, it is called Relative Motion. 6. Velocity, or Celerity, is an affection of motion, by which a body passes over a certain space in a certain time. Thus, if a body in motion pass uniformly over 40 feet in 4 seconds of time, it is said to move with the velocity of 10 feet per second ; and so on. 7. Momentum, or Quantity of Motion, is the 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. 8. Force is a power exerted on a body to move it, or to stop it. If the force act constantly, or incessantly, it is a Permanent Force: like pressure or the force of gravity. But if it act instantaneously, or but for an imperceptibly small time, it is called. Impulse, or Percussion: like the smart blow of a hammer. 9. Forces are also distinguished into Motive, and Accelerative or Retarding. A Motive or Moving Force, is the power of an agent to produce motion; and it is equal or proportional to the momentum it will generate in any body, when acting, either by percussion, or for a certain time as a permanent force. 10. Accelerative, or Retardive Force, is commonly understood to be that which affects the velocity only: or it is that by which the velocity is accelerated or retarded; and it is equal or proportional to the motive force directly, and to the mass or body moved inversely. So, if a body of 2 pounds weight, be acted on by a motive force of 40.; then the accelerating force is 20. But if the same force of 40 act on another body of 4 pounds weight; then the accelerating force in this latter case is only 10; and so is but half the former, and will produce only half the velocity. 11. Gravity, or Weight, is that force by which a body endeavours to fall downwards. It is called Absolute Gravity, when the body is in empty space; and Relative Gravity, when immersed in a fluid. 12. Specific Gravity is the proportion of the weights of different bodies of equal magnitude; and so is proportional to the density of the body. |