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conds.

began the motion, and which con- one second, then will tv represent tinues to act on the body already the velocity at the end of sein 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 inust 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 results from the force at the first impression.

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

represent the successive velocities
4, 2, 3, 4 4, &c. to,
since the velocity v, multiplied by
at each successive instant; and,
the time in uniform motions is
equal to the space, and since we
form during any indefinite small
may consider the motions as uni-
instant of time, the above may also
be taken to represent the spaces
described at each successive in-
stant; 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

It may,

be represented by v, and thus the being the final velocity, it may sum of the series will be expressed byt (+v). But since ф repreBut since the action of the force sents the first velocity of the deis supposed still to continue, in the scending body, it is indefinitely second moment of time the body small, and may be considered as will receive a new impulse down-nothing with regard to v. wards, equal to what it received therefore, be cancelled out of the in the first; and thus its velocity whole space will be represented expression, and consequently the will be double of what it was in the first moment: in the third, it by tv; or, taking s to represent will be treble; in the fourth, quad- that is, the space described by the space, we shall have s= } tv, ruple; and so on continually: for the impression made in one mo- a body uniformly accelerated in ment, is not at all altered by what any time, is equal to half the space is made in another. The whole that would be described by the are, as it were, united into one same body, in the same time, with Hence the velocity will be a uniform velocity equal to that proportional to the time in which last acquired. it is acquired.

sum.

Thus, if a body, by means of this constant 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 v; 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

vacuum.

It has been found by experiment that a body falling freely in the latitude of London, passes through a space equal to 181 feet in the first second of time; hence we have, by representing this space by g,g=

tv, or g=v; because t=1, whence v 2g, is the velocity acquired at the end of the first second of time, and, therefore, from what has been above demonstrated, v=2g t, will represent the acquired velocity at the end of any

time t.

=

Hence, if t be made to repre- Also st; substituting, theresent the time a body has been fail-fore, for v, we have s t (2gt), ing, and the velocity acquired in or stig; and since g is constant

it follows that the spaces describ When the accelerating forces ed by falling bodies are to each are different, but constant, the other as the squares of the times spaces will be as the products of of descent, the spaces themselves the forces into the square of the being accurately expressed by the times; and the times will be as formula s tg, where g repre- the square roots of the spaces disents 16 feet, the space a body rectly, and of the forces inversely. falls through in the first second of For when the force is given or constant, the velocity (V) is as the time. time (T); and when the forces are time is given, the velocity (v) will different, but constant, and the

Hence we deduce the following general laws of motions uniformly

accelerated, viz.

1. The velocities acquired are constantly proportional to the

times.

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 (FXT).
Hence, V: v=FXT:fxt
conseq. FXT2:ƒxt2=VXT:vxt=

or,

S: s

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12=

T: t = √

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, nine times that space in a treble time, and so on. And therefore, T2: universally, if the times be in arithmetical proportion, as 1, 2, 3, 4, &c. t, the spaces described will be as 1, 4, 9, 16, &c. t. Thus a body, which falls by gravity through 10 feet in the first second, will fall through 643 feet in two seconds, and so on. And since the velocities acquired in falling are as the times, the spaces will be the square of the velocities; and both the times and velocities will be as the square roots of the

as

spaces.

3. The spaces described by a fall ing body in a series of equal moments, 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 161 feet in the first second, will fall through 3 × 161 in the second, 5 16 in the third, and so on.

These properties may be otherwise represented, thus :-Let S, V, be put for the space and velocity of a falling body in any time T; and s, v, the same for the time t; then we shall have

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S

S

F

From the properties above demonstrated, we obtain the following practical theorems: Let g denote the space passed over in the first second of time, by a body urged by any uniform force, denoted by 1; and let 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 we shall have v = 2gt, and s=gt2; and from these two equations we obtain the following general formulæ :

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Also, v=2 g t = 2 × 16 × 12 may be supposed constant during 388 feet, the last acquired velo- the indefinitely small space of city. time t, and spaces and velocities in uniform motions being proportional to the times, we shall have these two fundamental proportions, viz.

EXAM. 2. How long will a body, be in falling through a space of 1608 feet?

S 1608 162 g 10 seconds nearly. EXAM. 3. How far must a body fall to acquire a velocity of 386 feet per second?

Here s=1608, and t√

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12 seconds nearly.

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 X g; and by using this instead of g, the above formula 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 formulæ for expressing them in final determinate quantities, we subjoin those that belong to the cases of variably

accelerated motions.

Here the formula will be fluxionary expressions, the fluents of which, adapted to particular cases, will give the relation of time, space, 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 32 feet, as that variable force is to the force of gravity. Then, because the force

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These formulæ, 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 55" of mean time, or nearly 3 56; that is to say, a

star rises or sets, or passes the meridian, 3 56" 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/ 81" of a degree every day. In consequence of this, the star which passed the meridian at the same moment with the sun yesterday, is about 59/ 81 beyond the meridian to the west, when the sun arrives at it; and this distance will require about 3/ 56" 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 reai 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, acuteangled 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 founded, 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+b, 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 1 in any of them is equal to 1 in each of the others. When this is not the case, an alteration must 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.

ted, according to the nature of the Addition is variously denominanumbers or quantities that are to be added. Thus,

the sum of several simple or abSimple Addition is the finding of stract numbers; and it is thus performed,

places stand under each other. Arrange the numbers so that like Collect the sum of the units, write below the column what is over in the next column. Do this with tens, and for every ten reckon one all the columns, in order, to the left

hand.

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Addition may be proved in various ways: One may begin alternately at the top and bottom of the columns, and see if the results correspond. The excess over 9 may be noted in each line, and also in the sum, and if the sum of the former be equal to the latter, or differ from it only by one or more 9's, there can be no error but 9 in the addition: Thus in the first of the preceding examples, the remainder on the lines are 7, 6, 5, 5; the remainder in the sum 5; and

the sum of the former (23,) abating | two nines, leaves 5 also. Hence the addition is right.

Compound Addition is the collecting several quantities of different denominations into one sum.

Place the same denominations to stand directly under each other. Add up the figures in the lowest denomination, and find how many units of the next higher denomi nation are contained in their sum.

Write down the remainder, and carry these units to the next higher denomination; which add up in the same manner as before: and proceed thus through all the dif ferent denominations to the highest.

EXAMPLES.

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Z. s. d.

cwt. gr. lb.

17 18 9

11 3

17

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34.17

10

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19.143

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167.13

143.5

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2d. Exam.

.1176

.1344

.746

.1468

106 7 10 Sums 155

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363.943 Sums 1.1448

ADDITION of Intermediate Decimals, is the finding the sum of any number of circulating decimals.

1. Reduce all the decimals to

their equivalent fractions; and the sum of these will be the answer required; or,

ent denominations, they must be first reduced to the same denomination. 2. Reduce all mixed numbers to improper fractions; and all fractions, having different denominators, to the same denominator. 3. Add all the numerators into one sum; which, placed over the common denominator, will be the an-place; then add them up as in the swer required.

EXAMPLES.

2. Carry on the repetends till they all begin and end their periods of circulation in the same lines, and let the circulation of each be carried two figures beyond this

former rule, observing not to set down any thing in the first two places; only carry the proper num

1. Add together the fraction ber from them to the conterminous

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