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and then, though the sun's motion should be really uniforni, it would not appear to be so when seen from the earth; and in this case there would be an optic equation, but not an eccentric one. • Again, let us imagine the sun's orbit not to be circular, but elliptical, and the earth to be in its focus, then it is evident that the sun cannot appear to have a uniform motion in such ellipse, and, therefore, his motion will be subject to two equations, viz. the Optic and Eccentric Equations, the sum of which is the Absolute Equation. ABSOLUTE Term, or Number, in Algebra, is that which is completely known, and to which all the other parts of the equation is made equal. A B S T R A CT Mathematics, or Pure Mathematics, is that which treats of the properties of magnitude, figure, or quantity, absolutely and generally considered, without restriction to auy species in particular, such as Arithmetic and Geometry. . It is thus distinguished from Mia!ed Mathematics, in which simple and abstract quantities, rinatively considered in Pure athematics, are applied to sensible objects, as in Astronomy, Mechanics, Optics, &c. ACCELERATION is principally used in Physics, to denote the increasing lapidity of bodies in falling towards the centre of the earth, by a force called gravity, whether a property of matter, or an effect of the earth's motions. That matural bodies are accelerated in their descent, is evident from various considerations, both a priori and posteriori. Thus we actually find, that the greater height a body descends from, the more rapidly it descends, the greater impression it makes, and the more intense is the blow wi.h which it strikes the obstacle upon which it impinges. Some have attributed this acceleration to the pressure of the air; others to an inherent principle in matter, by which all bodies tend to the centre of the earth as their proper seat or element, where they would be at rest; and hence, say they, the nearer that bodies ap
proach to it, the more is their motion accelerated. Another class held, that the earth emitted a sort of attractive effluvia, innumerable threads of which continually ascend and descend, proceeding, like radii, from a common centre, and diverging the more the further they go; so that the nearer a heavy body is to the centre, the more of these magnetic threads it receives; and hence the more its motion is accelerated. But leaving all such visionary theories, and only admitting the existence of some such force as gravity, inherent in all bodies, without regard to what may be the cause of it, the whole mystery of acceleration will be cleared up, and the theory of it established on the most obvious principles. Suppose a body iet fall from any height,” and that the primary cause of its beginning to descend is the power called gravity; then, when once the descent is cosmomenced, motion becomes, in some measure, matural to the body; so that, if left to itself, it would persevere in it for ever; as we see in a stone cast from the hand, which continues to move after it is left by the cause that first gave it motion ; and which motion would continue for ever, was it not destroyed by resistance and gravity, which cause it to fall to the earth. But beside this ten dency, which of itself is sufficient to continue the same degree of motion, in finitum, there is a constant accession of subsequent efforts of the same principle, which
* Sir Richard Phillips, in his Essays, maintains that there is no such force inherent in matter as the attraction of gravitation, and that the cause of a body’s descend. ing to the earth, as well as all the other phenomena usually ascribed to the action of this force, are the natural and necessary results of the two inotions of the earth. (See articles Attraction and Motion.)—This explanation of the true cause of the phenomena does not, however, alter the law of acceleration, or, indeed, any law of the earth or the planetary system; though it varies our reasoning.
began the motion, and which continues to act on the body already in motion, in the same manner as if it were at rest. Here, then, being two causes of motion, and both acting in the same direction, the motion they jointly produce must necessarily be greater than that of one of them; and the same cause of increase acting still on the body, the descent of it must, of course, be continually accelerated. For, supposing gravity, or the earth’s motion, to act uniformly on all bodies, at equal distances from the earth's centre ; and that the time in which a heavy body falls to the earth to be divided into equal parts, indefinitely small. Let this force incline the body towards the earth's centre, while it moves in the first indefinitely small space of time of its descent; if after this the force be supposed to cease, the body would proceed uniformly towards the centre of the earth, with
a velocity equal to that which re
sults from the force at the first impression. But since the action of the force is supposed still to continue, in the second moment of time the body will receive a new impulse downwards, equal to what it received in the first; and thus its velocity will be double of what it was in the first moment: in the third, it will be treble; in the fourth, quad. ruple; and so on continually : for the impression made in one moment, is not at all altered by what is made in another. The whole are, as it were, united into one sum. Hence the velocity will be proportional to the time in which it is acquired. Thus, if a body, by means of this coustant force, acquire a velocity v, in one second of time, it will, in two seconds, acquire a velocity 2 v; in three seconds, it will acquire a 3 w; and so on ; and all bodies, whatever be their quantity of matter, will acquire, by the force of gravity, the same velocity in the same time; that is, supposing no resistance from the atmosphere, or any other obstacle in a perfect vacuumHence, if t be made to represent the time a body has been failint, and v the velocity acquired in 6
one second, then will t w represent the velocity at the end of t seconds.
And, now if we represent by + the velocity of a body when it first begins to fall, or its velocity acquired in the first instant of its descent, then will the terms of the series
4', 2 *, 3 p., 4 p., &c. top,
represent the successive velocities at each successive instant ; and, since the velocity, v, multiplied by the time in uniform motions is equal to the space, and since we may consider the motions as uniform during any indefinite small instant of time, the above may also be taken to represent the spaces described at each successive instant; and hence the sum of them will be the whole space described in the time t. Now the number of terms in this series being t, and t $ being the final velocity, it may be represented by v, and thus the sum of the series will be expressed by t ($ 4 v). But since + represents the first velocity of the descending body, it is indefinitely small, and may be considered as nothing with regard to w. It may, therefore, be cancelled out of the expression, and consequently the whole space will be represented by , t w = or, taking s to represent the space, we shall have s = | tv, that is, – the space described by a body uniformly accelerated in any time, is equal to half the space that would be described by the same body, in the same time, with a uniform velocity equal to that last acquired.
It has been found by experiment that a body falling freely in the latitude of London, passes through a space equal to 10% feet in the
MATHE MATICAL AND PHYSICAL SCIENCE.
it follows that the spaces described by falling bodies are to each other as the squares of the times of descent, the spaces themselves being accurately expressed by the formula s = to g, where g represents 16, feet, the space a body falls through in the first second of time. iience we deduce the following general laws of motions uniformly accelerated, viz. 1. The velocities acquired are constantly proportional to the times. 2. That the spaces are proportional to the squares of the time; so that if a body describe any given space in a given time, it will describe four times that space in a double time, mine times that space in a treble time, and so on. And universally, if the times be in arithmetical proportion, as 1, 2, 3, 3, &c. t, the spaces described will be as 1, 4, 9, 16, &c. *. Thus a body, which falls by gravity through 16, seet in the first second, will fall through 64% feet in two seconds, and so on. And since the velocities acquired in falling are as the times, the spaces will be as the square of the velocities; and both the times and velocities will be as the square roots of the spaces. 3. The spaces described by a falling body in a series of equal mo: ments, or intervals of time, will be as the odd numbers 1, 3, 5, 7, 9, &c. which are the differences of the squares or whole spaces ; that is, a body which falls through 16, feet in the first second, will
When the accelerating forces are different, but constant, the spaces will be as the products of the forces into the square of the times; and the times will be as the square roots of the spaces directly, and of the forces inversely. For when the force is given or constant, the velocity (V) is as the time (T); and when the forces are different, but constant, and the time is given, the velocity (v) will be as the force (F). But when neither the force nor the time is given, the velocity (v) will be as the product of the force into the time, that is as (FX T). Hence, V : v = FXT : foxt “go. FxTa: foto = WXT : vxt=
N. : S.
The same law of acceleration obtains equally in the descents of bodies down inclined planes, except that the force of gravity will in that case vary as the sines of the angles of inclination of the planes, that is, the force down the inclined plane is to the whole force of gravity as the sine of the angle of inclination of the plane to radius. If, therefore, the angle of inclination of the plane be a, the force down the plane will be sin. a × g; and by using this instead of g, the above formulae will be equally applicable to the descents of heavy bodies down inclined planes.
Of variably accelerated motion. Having illustrated the laws of accelerated motion, when the accelerating forces are constant, and deduced the formulae for expressing them in final determinate quantities, we subjoin those that belong to the cases of variably accelerated motions.
Here the formulae will be fluxiomary expressions, the fluents of which, adapted to particular cases, will give the relation of time, i. velocity, &c. Let t denote the time of motion; v the velocity generated by any force; s the space passed over; and 2g the variable force at any part of the motion, or the velocity that the force would generate in one second of time, if it should continue inva. riably like the force of gravity during that one second, and the value of this veiocity, 2g, will be in proportion to 324 feet, as that
variable force is to the force of gravity. Then, because the force
These formulae, like those in the preceding part of this article, are equally applicable to the destruction of motion and velocity, by means of retarding forces, as to the generation of them by means of accelerating forces. The motion of a body ascending, or impelled upwards, is diminished or retarded from the same principle acting in a contrary direction, in the same manner as a falling body is accelerated. A body projected upwards, rises till it has lost all its motion; which it does in the same time that a falling body would have acquired a velocity equal to that with which the body was thrown up. Hence the same body projected up, will rise to the same height, from which, if it fell, it would have acquired the velocity with which it was projected upwards. And hence the heights to which bodies thrown upwards with different velocities, ascend, are to each other as the squares of those velocities. ACCELERATION, in Astronomy, is a term applied to the fixed stars. The diurnal acceleration is the time by which the stars, in one diurnal revolution, anticipate the mean diurnal revolution of the sun,
which is 3, 551%;" of mean time, or nearly 3, 56; ; that is to say, a star rises or sets, or passes the meridian, 31 500 sooner each day. This apparent acceleration of the stars is owing to the real retardation of the sun, which depends upon his apparent motion towards the east, at the rate of about 59. sy of a degree every day. In
consequence of this, the star which passed the meridian at the same imoment with the sun yesterday, is about 59 by beyond the meridian
to the west, when the sun arrives at it; and this distance will require about 3/ 561 of time for the sun to pass over, and therefore the star will anticipate the motion of the sun at this rate every day. ACCELERATION of a Planet. A planet is said to be accelerated in its motion, when its real diurnal motion exceeds its mean diurnal motion ; and retarded in its motion, when the mean exceeds the rea; diurnal motion. This inequality arises from the change in the distance of the planet from the sun, which is continually varying; the planet moving always, quicker in its orbit when nearest the sun, and slower when furthest off. ACCELERATION of the Moon, is the increase of the moon’s mean motion from the sun, compared with the diurnal motion of the earth; and it appears that, from a certain cause, it is now a little quicker than it was formerly. La Place has shown this acceleration of the moon’s motion to arise from the action of the sun upon the moon, combined with the variation of the eccentricity of the earth's orbit. . By the present diminution of the eccentricity, the moon’s mean motion is accelerated; but when the eccentricity is arrived at the minimum, the acceleration will cease; after this the eccentricity will increase, and the moon’s mean motion will be retarded. ACUTE, or SHARP, a term opposed to obtuse; thus we say, acutegood triangle, acute-angled cone, C. ADDITION, one of the fundamental operations in Arithmetic and Algebra, denotes the finding of one number or quantity, equal to two or more given numbers or
quantities. The general principle upon which it is i. is that the whole is equal to all the parts. Addition is indicated or expressed by writing the sign +, which is called the sign of addition between the numbers or quantities; thus 7+5, and a-Hb, indicate the addition of 7 and 5, and of a and b. The result of the addition is called the sum. It may be expressed in one number, when the proposed numbers are all of the same kind, and when I in any of them is equai to 1 in each of the others. When this is not the case, an alteration naust be made in the numbers, and if such an alteration cannot be made, then the sum cannot be obtained in one number. In Algebra, quantities cannot be collected into one quantity, unless they consist of the same power of the same letters. Addition is variously denominated, according to the nature of the numbers or quantities that are to be added. Thus, Simple Addition is the finding of the sum of several simple or Tabstract numbers; and it is thus performed, Arrange the numbers so that like places stand under each other. Collect the sum of the units, write below the column what is over tens, and for every ten reckon one in the next column. Do this with all the columns, in order, to the left hand.