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Hence it follows, that any one of these narrow parts, as
IG, is X
× AI2; hence, AD and DC being given or constant quantities, it appears that the said parts IG, &c, are proportional to Ar2, &c, or proportional to a series of square numbers, whose roots are in arithmetical progression, and the
area ADCGA equal to drawn into the sum of such a series of arithmeticals, the number of which is expressed by AD.
Now, by the remark at pag. 233, vol. i, the sum of the squares of such a series of arithmeticals, is expressed by zn. n + 1. 2n+1, where n denotes the number of them. In the present case, represents an infinite number, and then the two factors n + 1, 2n + 1, become only n and 2n, ommitting the 1 as inconsiderable in respect of the infinite number 7: hence the expression above becomes barely žn.n. 2n = n3.
To apply this to the case above: n will denote AD or BC; and the sum of all the Ar's becomes AD3 or BC3; conse
quently the sum of all the × AI's, is × AD3 =
AD. DC = BD, which is the area of the exterior part ADCGA. That is, the said exterior part ADCGA, is of the parallelogram ABCD; and consequently the interior part ABCGA is of the same parallelogram.
Q. E. D.
Corol. The part AFCGA, inclosed between the curve and the right line AFC, is of the same parallelogram, being the difference between ABCGA and the triangle ABCFA, that is between and of the parallelogram.
The Solid Content of a Paraboloid (or Solid generated by the Rotation of a Parabola about its Axis), is equal to Half its Circumscribing Cylinder.
LET ABC be a paraboloid, generated by the rotation of the parabola AC about its axis AD. Suppose the axis ad be divided into an infinite number of equal parts, through which let circular planes pass, as EFG, all those circles making up the whole solid paraboloid.
But, by cor. theor. 1, Parabola, p ́× AF = FG2, where p denotes the parameter of the parabola; consequently pc × AF will also express the same circular section EG, and therefore pcx 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 O 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 ADX AD or AD2, 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 AD, or X px AD, is the sum of all the circular sections, or the content of the paraboloid.
But, by the parabola, p: DC :: DC: AD or p = -;
consequently x px AD2 becomes ex AD X DC2 for the solid content of the paraboloid. But cx AD X re2 is equal to the cylinder BCIH; consequently the paraboloid is the half of its circumscribing cylinder.
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, EC.
OF MOTION, FORCES, &c.
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 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 successive 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
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 undertood to be that which affects the volocity 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, it 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.
13. EVERY body naturally endeavours to continue in its present state, whether it be at rest, or moving uniformly in a right line.
14. The Change or Alteration of Motion, by any external force, is always proportional to that force, and in the direction of the right line in which it acts.
15. Action and Re-action, between any two bodies, are equal and contrary. That is, by Action and Re-action, equal changes of motion are produced in bodies acting on each other; and these changes are directed towards opposite or contrary parts.
GENERAL LAWS OF MOTION, &c.
16. The Quantity of Matter, in all Bodies, is in the Compound Ratio of their Magnitudes and Densities.
THAT is, b is as md; where b denotes the body or quantity of matter, m its magnitude, and d its density.
For, by art. 4, in bodies of equal magnitude, the mass or quantity of matter is as the density. But, the densities remaining, the mass is as the magnitude: that is, a double magnitude contains a double quantity of matter, a triple magnitude a triple quantity, and so on. Therefore the mass. is in the compound ratio of the magnitude and density.
17. Corol. 1. In similar bodies, the masses are as the densities and cubes of the diameters, or of any like linear dimensions. For the magnitudes of bodies are as the cubes of the diameters, &c.
18. Corol. 2. The masses are as the magnitudes and specific gravities.-For, by art. 4 and 12, the densities of bodies are as the specific gravities.
19. Scholium. Hence, if b denote any body, or the quantity of matter in it, m its magnitude, d its density, g its