ACCELERATED Motion
, is that which receives fresh accessions of velocity; and the acceleration may be either equably or unequably: if the accessions of velocity be always equal in equal times, the motion is said to be equably or uniformly accelerated; but if the accessions, in equal times, either increase or decrease, then the motion is unequably or variably accelerated.
Acceleration is directly opposite to retardation, which denotes a diminution of velocity.
Acceleration comes chiefly under consideration in physics, in the descent of heavy bodies, tending or falling towards the centre of the earth, by the force of gravity.
That bodies are accelerated in their natural descent, is evident both to the sight, and from observing that the greater height they fall from, the greater force they strike with, and the deeper impressions they make in soft substances.
The acceleration of falling bodies has been ascribed to various causes, by different philosophers. Some have attributed it to the pressure of the air downwards: the more a body descends, the longer and heavier, say they, must be the column of atmosphere incumbent upon it; to which they add, that the whole mass of fluid pressing by an infinity of right-lines all ultimately meeting in the earth's centre, such central point must support, as it were, the pressure of the whole mass; and that consequently the nearer a body approaches to it, the more must it receive of the pressure of a multitude of lines tending to unite in the central point.
Mr. Hobbes endeavours to account for this acceleration from a new impression of the cause which makes bodies fall; in which he is so far right. But then he as far mistakes, as to the cause of the fall, which he thinks is the air: at the same time, says he, that one particle of air ascends, another descends; for in consequence of the earth's motion being two-fold, that is circular and progressive, the air must at once both ascend and circulate; whence it follows, that a body falling in this medium, and receiving a new pressure every instant, must have its motion accelerated.
But to both these systems it may be answered, that the air is quite out of the question; for it is very evident that bodies fall, and in falling have their motion accelerated, in vacuo, as in open air, and even more than in the air, in as much as this opposes and somewhat retards their fall.
The Gassendists assign another reason for the acceleration: they pretend that there are continually issuing out of the earth certain attractive corpuscles, directed in an infinite number of rays; those, say they, afcend and then descend, in such sort that the nearer a body approaches to the earth's centre, the more of these attractive rays press upon it, in consequence of which its motion becomes more accelerated.
The peripatetics endeavour to explain the matter thus: the motion of heavy bodies downward, arises, say they, out of an intrinsic principle that causes a tendency in them to the centre, as the place appropriated to their element; where, when they can once arrive, they will be at perfect rest; and therefore, continue they, the nearer bodies approach to it, the more the velocity of their motion is increased: a notion too idle to merit confutation.
The Cartesians account for acceleration, by reiterated impulses of their materia subtilis, acting continually | on falling bodies, and propelling them downwards: a conceit equally unintelligible and absurd with the former.
But, leaving all such visionary causes of acceleration, and only admitting the existence of such a force as gravity, so evidently inherent in all bodies, without regard to what may be the cause of it, the whole mystery of acceleration will be cleared up. Consider gravity then, with Galileo, only as a cause or force which acts continually on heavy bodies; and it will be easy to conceive that the principle of gravitation, which determines bodies to descend, must by a necessary consequence accelerate them in falling.
A body then having once begun to descend, through the impulse of gravity; the state of descending is now, by Newton's first law of nature, become as it were natural to it; insomuch that, were it left to itself, it would for ever continue to descend, even though the first cause of its descent should cease. But besides this determination to descend, impressed upon it by the first cause of motion, which would be sufficient to continue to infinity the degree of motion already begun, new impulses are continually superadded by the same cause; which continues to act upon the body already in motion, in the same manner as if it had remained at rest. There being then two causes of motion, acting both in the same direction; it necessarily follows, that the motion which they unitedly produce, must be more considerable than what either could produce separately. And as long as the velocity is thus continued, the same cause still subsisting to increase it more, the descent must of necessity be continually accelerated.
Supposing then that gravity, from whatever principle it arises, acts uniformly upon all bodies at the same distance from the centre of the earth: dividing the time which the heavy body takes up in falling to the earth, into indefinitely small equal parts; gravity will impel the body toward the centre of the earth, in the first indefinitely short instant of the descent. If after this we suppose the action of gravity to cease, the body will continue perpetually to advance uniformly toward the earth's centre, with an indefinitely small velocity, equal to that which resulted from the first impulse.
But then if we suppose that the action of gravity still continues the same after the first impulse; in the second instant, the body will receive a new impulse toward the earth, equal to that which it received in the first instant; and consequently its velocity will be doubled; in the third instant, it will be tripled; in the fourth, quadrupled; in the fifth, quintupled; and so on continually: for the impulse made in any preceding instant, is no ways altered by that which is made in the following one; but they are, on the contrary, always accumulated on each other.
So that the instants of time being supposed indefinitely small, and all equal, the velocity acquired by the falling body, will be, in every instant, proportional to the times from the beginning of the descent; and consequently the velocity will be proportional to the time in which it is produced. So that if a body, by this constant force, acquire a velocity of 16 1/12 feet suppose in one second of time; it will acquire a velocity of 32 1/6 feet in two seconds, 48 1/4 feet in 3 seconds, 64 1/3 in 4 seconds, and so on. Nor ought it to seem strange that all bodies, small or large, acquire, by the force of gravity, the same velocity in the same time. For every equal particle of matter being endued with an equal impelling force, namely its gravity or weight, the sum of all the forces, in any compound mass of matter, will be proportional to the sum of all the weights, or quantities of matter to be moved; consequently, the forces and masses moved, being thus constantly increased in the same proportion, the velocities generated will be the same in all bodies, great or small. That is, a double force moves a double mass of matter, with the same velocity that the single force moves the single mass; and so on. Or otherwise, the whole compound mass falls all together with the same velocity, and in the same manner, as if its particles were not united, but as if each fell by itself, separated all from one another. And thus all being let go at once, they would fall together, just as if they were united into one mass.
The foregoing law of the descent of falling bodies, namely that the velocities are always proportional to the times of descent, as well as the following laws concerning the spaces passed over, &c, were first discovered and taught by the great Galileo, and that nearly in the following manner.
Because the constant velocity with which any body moves, or the space it passes over in a given time, as suppose one second, being multiplied by the time, or number of seconds it is in motion, expresses the space passed over in that time; and the area or space of a rectangular figure being denoted by the length multiplied by the breadth; therefore the space so run over, may be considered as a rectangle compounded of the time and velocity, that is a rectangle of which the time denotes the length, and the velocity the breadth. Suppose then A to be the heavy body which descends, and AB to denote the whole time of any descent; which let be divided into a certain number of equal parts, denoting intervals or portions of the given time, as AC, CD, DE, &c. Imagine the body to descend, during the time expressed by the first of the divisions AC, with a certain uniform velocity arising from the force of gravity acting on it, which let be denoted by AF, the breadth of the rectangle CF; then the space run through during the time denoted by AC, with the velocity denoted by AF, will be expressed by the rectangular space CF.
Now the action of gravity having produced, in the first moment, the velocity AF, in the body, before at rest; in the first two moments it will produce the velocity CG, the double of the former; in the third moment, to the velocity CG will be added one degree | more, by which means will be produced the velocity DH, triple of the first; and so of the rest; so that during the whole time AB, the body will have acquired the velocity BK. Hence, taking the divisions of the line AB at pleasure; for example, the divisions AC, CD, &c, for the times; the spaces run through during those times, will be as the areas or rectangles CF, DG, &c; and so the space described by the moving body during the whole time AB, will be equal to all the rectangles, that is, equal to the whole indented space ABKIHGF. And thus it will happen if the increments of velocity be produced, as we may say, all at once, at the end of certain portions of finite time; for instance at C, at D, &c; so that the degree of motion remains the same to the instant that a new acceleration takes place.
By conceiving the divisions of time to be shorter, for example but half as long as the former, the indentures of the figure will be proportionably more contracted, and it will approach nearer to a triangle; and so much the nearer as the divisions of time are shorter: and if these be supposed infinitely small; that is, if increments of the velocity be supposed to be acquired continually, and at each indivisible particle of time, which is really the case, the rectangles so successively produced, will form a true triangle, as ABC; the whole time AB consisting of minute portions A 1, 12, 23, &c; and the area of the triangle ABC, of all the minute surfaces, or minute trapeziums, which answer to the divisions of the times; the area of the whole triangle ABC, denoting the space run through during the whole time AB; and the area of any smaller triangle A 7 g, denoting the space run through during the corresponding time A 7. Bnt the triangles A 1 a, A 7 g, &c, being similar, have their areas to each other as the squares of their like sides A 1, A 7, &c; and consequently the spaces gone through, in any times counted from the beginning, are to each other as the squares of the times.
Hence, in any right-angled triangle, as ABC, the one side AB represents the time, the other side BC the velocity acquired in that time, and the area of the triangle the space described by the falling body.
From the preceding demonstration is also drawn this other general theorem in motions that are uniformly accelerated; namely, that a body descending with a uniformly accelerated motion, describes in the whole time of its descent, a space, which is exactly the half of that which it would describe uniformly in the same time, with the velocity it has acquired at the end of its accelerated fall. For it has been shewn that the whole space which the falling body has run through in the time AB, is represented by the triangle ABC, the last velocity being BC; and the space which the body would run through uniformly in the same time AB, constantly with the said greatest velocity BC, is represented by the rectangle ABCD: but it is well known that the rectangle ABCD is double the triangle ABC; and therefore the latter space run through, is double the former; that is, the space run through by the accelerated motion, is just half of that which the body would describe in the same time, moving uniformly with the velocity acquired at the end of its accelerated fall.
Hence then, from the foregoing considerations are deduced the following general laws of uniformly accelerated motions, namely,
1st. That the velocities acquired, are constantly proportional to the times; in a double time a double velocity, &c.
2d. That the spaces described in the whole times, each counted from the commencement of the motion, are proportional to the squares of the times, or to the squares of the velocities; that is, in twice the time, the body will describe 4 times the space; in thrice the time, it will describe 9 times the space; in quadruple the time, 16 times the space; and so on. In short, if the times are proportional to the numbers 1, 2, 3, 4, 5, &c, the spaces will be as 1, 4, 9, 16, 25, &c, which are the squares of the former. So that if a body, by the natural force of gravity, fall through the space of 16 1/12 feet in the first second of time; then in the first two seconds of time it will fall through four times as much, or 64 1/3 feet; in the first three seconds it will fall nine times as much, or 144 3/4 feet; and so on. And as the spaces fallen through are as the squares of the times, or of the velocities; therefore the times, or the velocities, are proportional to the square roots of the spaces.
3d. The spaces described by falling bodies, in a series of equal instants or intervals of time, will be as the odd numbers 1, 3, 5, 7, 9, &c, 1, 4, 9, 16, 25, &c, which are the differences of the squares or whole spaces that is, the body which has run through 16 1/12 feet in the firft second, will in the next second run through 48 1/3 feet, in the third second 80 3/12, and so on.
4th. If the body fall through any space in any time, it acquires a velocity equal to double that space; that is, in an equal time, with the last velocity acquired, if uniformly continued, it would pass over just double the space. So if a body fall through 16 1/12 feet in the first second of time, then it has acquired a velocity of 32 1/6 feet in a second; that is, if the body move uniformly for one second, with the velocity acquired, it will pass over 32 1/6 feet in this one second: and if in any time the body fall through 100 feet; then in another equal time, if it move uniformly with the velocity last acquired, it will pass over 200 feet. And so on.
But, as the method of demonstration used by Galileo, by means of infinitely small parts forming a regular triangle, is not approved of by many persons, the same laws may be otherwise demonstrated thus: let the whole time of a body's free descent be divided into any number of parts, calling each of these parts 1; and let a denote the velocity acquired at the end of the first | part of time; then will 2a, 3a, 4a, &c, represent the velocities at the end of the 2d, 3d, 4th, &c, part of time, because the velocities are as the times; and for the same reason 1/2a, 3/2a, 5/2a, &c, will be the velocities at the middle point of the first, second, third, &c, part of time. But now as the velocities increase uniformly, the space described in any one of these parts of time, may be considered as uniformly deseribed with its middle velocity, or the velocity in the middle of that part of time; and therefore multiplying those mean velocities each by their common time 1, we have the same fractions 1/2a, 3/2a, 5/2a, &c, for the spaces passed over in the successive parts of the time; that is, the space 1/2a in the first time, 3/2a in the second, 5/2a in the third, and so on: then add these spaces successively to one another, and we obtain 1/2a, 4/2a, 9/2a, 16/2a, &c, for the whole spaces described from the beginning of the motion to the end of the first, second, third, &c, portion of time; namely 1/2a space in one time, 4/2a in 2 times, 9/2a in 3 times, and so on: and it is evident that these spaces are as the numbers 1, 4, 9, 16, &c, which are as the squares of the times.
And from this mode of demonstration, all the properties above mentioned evidently flow: such as that the whole spaces 1/2a, 4/2a, 9/2a, &c, are as the squares of the times 1, 2, 3, &c, that the separate spaces 1/2a, 3/2a, 5/2a, &c, 1, 3, 5, &c, described in the successive times, are as the odd numbers and that the velocity a acquired in any time 1, is double the space 1/2a described in the same time.
As the laws of acceleration are very important, I shall here insert the two following propositions, sent me by my learned friend Mr. Abram Robertson, of Christ Church College Oxford, in which those laws are demonstrated in a manner somewhat different.
If from the point P in the straight line AB, the points M, N begin to move at the same time, namely, M towards A with a motion, uniformly retarded, and N from rest towards B with a motion uniformly accelerated; and if the velocity of M decreases as much as the velocity of N increases in the same time; then the space MN is generated by an uniform motion, equal to the velocity with which M begins to move.
For, by hypothesis, whatever is lost in the velocity of M by retardation, is added to the velocity of N by acceleration: the joint velocities, therefore, of M and N must always be equal. But it is by the joint velocities of M and N that the space MN is generated. Consequently MN is generated by an uniform motion, which is evidently equal to the velocity with which M begins to move.
If a point begins to move in the direction of a straight line, and continues to move in the same di- rection with a velocity uniformly aocelerated; the space passed over in any given time, will be equal to half the space passed over in the same time with the velocity with which the acceleration ends.
Let the point D begin to move from A towards B, along the straight line AB, with a motion unisormly accelerated; the space AD passed over, is equal to half the space which the point would pass over, in the same time with the acquired velocity at D.
Let the points M, N begin to move in the straight line GH, at the same time, with equal velocities uniformly accelerated; M beginning to move from G, and N from P; and at the same time that M comes to the point P, let N come to H. Then as M and N move with equal velocities, uniformly accelerated, it is evident that the spaces, which they pass over in the same time, are equal to one another; consequently the space GP is equal to the space PH. Now as M begins to move from G with a velocity uniformly accelerated, it will arrive at P with an acquired velocity. Hence it is evident, if it be supposed to begin to move from P with this acquired velocity, and proceed toward G with a velocity uniformly retarded in the same degree that it was accelerated when it began to move from G, that it will pass over the same space GP in the same time. Wherefore, supposing the two points M, N to begin to move from P at the same time, namely the point M beginning to move with the acquired velocity mentioned above, and proceeding towards G with the velocity uniformly retarded, described above; and the point N as before with the velocity uniformly accelerated: then as the acceleration and retardation are uniform, they will be equal in equal spaces of time. Again, as M is retarded in the same degree that it was accelerated when it began to move from G, that is, in the same degree that N is accelerated, by the former prop. MN is generated by an uniform velocity. But when the point M arrives at G, its velocity becomes equal to o or nothing; and at the time that M arrives at G, N arrives at H with the acquired velocity. Wherefore, as the velocities of M and N taken jointly are equal, and consequently uniform, the space GH is passed over with the velocity of N at H, in the same time that PH is passed over by N beginning to move from P with a velocity uniformly accelerated to H. But PH is half of GH. “Hence the prop. is manifest.”
And hence the other laws of the spaces, before<*> mentioned, easily follow.
Since the spaces descended are as the squares of the times, and the abscisses of a parabola are as the squares of the ordinates, therefore the relation of the times and spaces descended may be very well represented by the ordinates and abscisses of that figure. Thus if AB be the axis of the parabola Abdfh, and AC a tangent | at the vertex perpendicular to the axis, divided into any number of equal parts Aa, ac, ce, &c, for the times; and if there be drawn ab, cd, ef, &c, parallel to the axis: hence if ab be the space descended in the time Aa, then cd will be the space descended in the time Ac, and ef the space defcended in the time Ae, and so on continually.
From the properties above-demonstrated, are derived the following practical formulas or theorems for use. Namely, if g denote the space passed over in the first second of time, by a body urged by any constant force, denoted by 1, and t denote the time or number of seconds in which the body passes over any other space s, and v the velocity acquired at the end of that time; then from the foregoing laws we have v = 2gt, and s = gt^{2}; and from these two equations result the following general formulas:
And here, when the constant force 1, is the natural force of gravity, then the distance g descended in the first second, in the latitude of London, is 16 1/12 feet: but if it be any other constant force, the value of g will be different, in proportion as the force is more or less.
The motion of an ascending body, or of one that is impelled upwards, is diminished or retarded by the same principle of gravity, acting in a contrary direction, after the same manner that a falling body is accelerated.
A body projected upwards, ascends until it has lost all its motion; which it does in the same space of time, that the body would have taken up in acquiring, by falling, a velocity equal to that with which the falling body began to be projected upwards. And consequently the heights to which bodies ascend, when projected upwards with different velocities, are to each other as the squares of those velocities.
Accelerated Motion of Bodies on Inclined Planes. The same general laws obtain here, as in bodies falling freely, or perpendicularly; namely, that the velocities are as the times, and the spaces descended down the planes as the squares of the times, or of the velocities. But those velocities are less, according to the sine of the plane's inclination; and the spaces less, according to the square of the sine. See Inclined Plane.
Accelerated Motion of Pendulums. See PENDULUM.
Accelerated Motion of Projectiles. See PROJECTILE.
Accelerated Motion of Compressed Bodies, in ex- panding or restoring themselves. See Dilatation, Compression, and Elasticity.
Accelerating Force, in Physics, is the force that accelerates the motion or velocity of bodies; and it is equal to, or expressed by, the quotientarising from the motive or absolute force, divided by the mass or weight of the body that is moved. In treating of physical considerations respecting forces, velocities, times, and spaces gone over, the first inquiry is the accelerating or accelerative force. This force is greater or less in proportion to the velocity it generates in the same time, and by this velocity it is measured. All accelerating forces are equal, and generate equal velocities, that have the motive forces directly proportional to the quantities of matter: so a double motive force will move a double quantity of matter with the same velocity, as also a triple motive force a triple quantity, a quadruple force a quadruple quantity, &c, all with the same velocity. And this is the reason why all bodies fall equally swift by the force of gravity; for the motive force is exactly proportional to their weight or mass. In general, the accelerating force is in the direct ratio of the motive force, and inverse ratio of the quantity of matter. When a body is let fall freely, to descend by the force of its natural gravity, it has been found by experiment that it falls through 16 1/12 feet in one second of time, and requires a velocity of 32 1/6 feet in that time: but if the quantity of matter be doubled, and the motive force remain the same as before, by connecting the falling body to another of equal weight by means of a thread, this other body being laid on a horizontal plane, and the falling body hanging down off the plane, and drawing the other equal body along the plane after it; then the accelerating force will be only half of what it was before, and the space fallen in one second will be only 8 1/24 feet, and the velocity acquired 16 1/12: and if the quantity of matter be tripled, or the body drawn along the plane doubled; then the accelerating force will be only one-third of what it was at first, and the space descended in one second, and velocity acquired, each one-third of the sirst: and so on.
But accelerating forces are sometimes variable, as well as sometimes constant; and the variation may be either increasing or decreasing.
The nature of constant and variable accelerating forces, may be illuftrated in the following manner. Let two weights W, w, be connected by a thread passing over a pully at A, B, or C; and let the weight W descend perpendicularly down, while it draws the smaller weight w up the line AD, or BE, or CF, the | first being a straight inclined plane, and the other two curves, the one convex and the other concave to the perpendicular. Then the small weight w will always make some certain resistance to the free descent of the large weight W, and that resistance will be constantly the same in every part of the plane AD, the difficulty to draw it up being the same in every point of it, because every part of it has the same inclination to the horizon, or to the perpendicular; and consequently the accessions to the velocity of the descending weight W, will be always equal in equal times; that is, in this case W descends by a uniformly accelerating force. But in the two curves BE, CF, the resistance or opposition of the small weight w will be constantly altering as it is drawn up the curves, because every part of them has a different inclination to the horizon, or to the perpendicular: in the former curve, the direction becomes more and more upright, or nearer perpendicular, as the small weight w ascends, and the opposition it makes to the descent of W, becomes more and more; and consequently the accessions to the velocity of W will be always less and less in equal times; that is, W descends by a decreasing accelerating force: but in the latter curve CF, as w ascends, the direction of the curve becomes less and less upright, and the opposition it makes to the descent of W, becomes always less and less; and consequently the accessions to the velocity of W will be always more and more in equal times; that is, W descends by an increasing accelerating force. So that although the velocity continually increases in all these cases, yet whilst it increases in a constant ratio to the times of motion, in the plane AD; the velocity increases in a less ratio than the time it ascended up BE, and in a greater ratio than the time increases in the other curve CF.
Now the relations between the times and velocities in all these cases, may be very well represented by the relations between the abscisses and ordinates of certain lines. Thus let AB and AC be two straight lines, making any angle BAC; and AD, AE two curves, the former concave, and the latter convex towards AB: divide AB into any parts Aa, Ab, &c, representing the times of motion; and draw the perpendiculars acde, bfgh, &c, representing the velocities. Then in the right line AC, the ordinates ad, bg, being as the abscisses Aa, Ab, this represents the case of uniformly accelerated motion, in which the velocities are always as the times: but in the curve AD, the ordinates ac, bf increase in a less ratio than the abscisses Aa, Ab; and therefore this represents the case of decreasing acceleration, in which the velocities increase in a less ratio than the times: and in the other curve AE, the ordinates ae, bh increase in a greater ratio than the abscisses; and therefore this represents the case of increasing acceleration, in which the velocities increase in a greater ratio than the times.
The several algebraic formulas or theorems, respecting the time, velocity, space, for constant accelerating forces, are delivered above, at the article Accelerated Motion, where the value of each circumstance is expressed in finite determinate quantities. But in the cases of variably accelerated motions, the formulas will require the help of the method of fluxions to express, not those general relations themselves, but the fluxions of them; and consequently, taking the fluents of those expressions, in particular cases, the relations of time, space, velocity, &c, are obtained.
Now if t denote the time in 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 the force would generate in one second of time, if it should continue invariable, like the force of gravity, during that one second; and therefore the value of this velocity 2g, will be in proportion to 32 1/6 feet, as that variable force, is to 1 the force of gravity. Then because the force may be supposed constant during the indefinitely small time t, and that in uniform motions the spaces and velocities are proportional to the times, we from thence obtain these two general fundamental porportions,
From which are derived the four formulas below, in which the value of each quantity is expressed in terms of the rest.
And these theorems equally hold good for the destruction of motion and velocity, by means of retarding forces, as for the generation of the same by means of accelerating forces.
Acceleration, in Mechanics, the increase of velocity in a moving body.
Acceleration. Astron. The Diurnal Acceleration of the fixed stars, is the time which the stars, in one diurnal revolution, anticipate the mean diurnal revolution of the sun; which is 3^{m} 55^{s} 9/10 of mean time, or nearly 3^{m} 56^{s}: that is, a star rises, or sets, or passes the meridian, about 3^{m} 56^{s} sooner each day. This acceleration of the stars, which is only apparent in them, arises from the real retardation of the sun, owing to his appa- | rent motion in his orbit towards the east, which is about 59′ 8″ 2/10 of a degree every day. So that the star which passed the meridian yesterday at the same moment with the sun, is to-day about 59′ 8″ past the meridian to the west, when the sun arrives at it; which will take him up about 3^{m} 56^{s} of time to pass over; and therefore the star passes by 3^{m} 56^{s} sooner than the sun each day, or anticipates his motion at that rate. The true quantity of this anticipation, or acceleration, is found by this proportion, 360° 59′ 8″ 1/5 :: 24 hours: 3^{m} 55^{s} 9/10, the fourth term of which is the acceleration.
The diurnal acceleration serves to regulate the lengths or vibration of pendulums. If I observe a fixed star set or pass behind a hill, steeple, or such like, when the pendulum marks for instance 8^{h} 10^{m}; and the next day, the eye being in the same place as before, the passage be at 8^{h} 6^{m} 4^{s}; I thence conclude that the pendulum is well regulated, or truly measures mean time.
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, on the other hand, the planet is said to be retarded in its motion, when the mean exceeds the real 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 nearer the sun, and slower when farther off.
Acceleration of the Moon, is a term used to express the increase of the moon's mean motion from the sun, compared with the diurnal motion of the earth; by which it appears that, from some uncertain cause, it is now a little quicker than it was formerly. Dr. Halley was led to the discovery, or suspicion, of this acceleration, by comparing the ancient eclipses observed at Babylon, &c, and those observed by Albategnius in the ninth century, with some of his own time; as may be seen in N 218 of the Philosophical Transactions. He could not however ascertain the quantity of the acceleration, because the longitudes of Bagdat, Alexandria, and Aleppo, where the observations were made, had not been accurately determined. But since his time the longitude of Alexandria has been ascertained by Chazelles; and Babylon, according to Ptolemy's account, lies 50′ east of Alexandria. From these data, Mr. Dunthorne, vol. 46 Philos. Transactions, compared the recorded times of several ancient and modern eclipses, with the calculations of them by his own tables, and thereby verified the suspicion that had been started by Dr. Halley; for he found that the same tables gave the moon's place more backward than her true place in ancient eclipses, and more forward than her true place in later eclipses; and thence he justly inferred that her motion in ancient times was slower, and in later times quicker, than the tables give it.
Not content however with barely ascertaining the fact, he proceeded to determine, as well as the observations would allow, the quantity of the acceleration; and by means of the most authentic eclipse, of which any good account remains, observed at Babylon in the year 721 before Christ, he found that the observed beginning of this eclipse was about an hour and three quarters sooner than the beginning by the tables; and that therefore the moon's true place preceded her place by computation by about 50′ of a degree at that time. Then admitting the acceleration to be uniform, and the aggregate of it as the square of the time, it will be at the rate of about 10″ in 100 years.
Dr. Long, vol. ii. p. 436 of his Astronomy, enumerates the following causes from some one or more of which the acceleration may arise. Either 1st, the annual and diurnal motion of the earth continuing the same, the moon is really carried about the earth with a greater velocity than formerly: or, 2dly, the diurnal motion of the earth, and the periodical revolution of the moon, continuing the same, the annual motion of the earth about the sun is retarded; which makes the sun's apparent motion in the ecliptic a little slower than formerly; and consequently the moon, in passing from any conjunction with the sun, takes up a less time before she again overtakes the sun, and forms a subsequent conjunction: in both these cases, the motion of the moon from the sun is really accelerated, and the synodical month actually shortened: or, 3dly, the annual motion of the earth, and the periodical revolution of the moon, continuing the same, the rotation of the earth upon its axis is a little retarded; in this case, days, hours, minutes, &c, by which all periods of time must be measured, appear of a longer duration; and consequently the synodical month will appear to be shortened, though it really contain the same quantity of absolute time as it always did. If the quantity of matter in the body of the sun be lessened, by the particles of light continually streaming from it, the motion of the earth about the sun may become slower: if the earth increases in bulk, the motion of the moon about the earth may thereby be quickened.