PROPOSITION XXVII.

134. A Body acquires the Same Velocity in descending down ang Inclined Plane BA, as by falling perpendicular through the Height of the Plane BC.

FOR, the velocities generated by any constant forces, are in the compound ratio of the forces and times of acting. But if we put

F to denote the whole force of gravity in BC, f the force on the plane AB,

t the time of describing BC, and

T the time of descending down AB; then by art. 119, F :ƒ :: BA : BC; and by art. 132, t T: BC: BA; theref. by comp. Ft: fr:: 1 : 1.

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E

That is, the compound ratio of the forces and times, or the ratio of the velocities, is a ratio of equality.

135. Corol. 1. Hence the velocities acquired, by bodies descending down any planes, from the same height, to the same horizontal line, are equal.

136. Corol. 2. If the velocities be equal, at any two equal altitudes, D, E; they will be equal at all other equal altitudes A, C.

137. Corol. 3. Hence also, the velocities acquired by descending down any planes, are as the square roots of the heights.

POPOSITION XXVIII.

138. If a Body descend down any Number of Contiguous Planes, AB, BC, CD; it will at last acquire the Same Velocity, as a Body falling perpendicularly through the Same Height ED, supposing the Velocity not altered by changing from one Plane to

another.

PRODUCE the planes DC, CB, to meet the horizontal line EA produced in F and G. Then, by art. 135, the velocity at B is the same, whether the body descend through AB or FB. And therefore the velocity at c will be the same,

whether the body descend through ABC or through FC,

which is also again, by art. 135, the same as by descending through GC. Consequently it will have the same velocity at D, by descending through the planes AB, BC, CD, as by descending through the plane GD; supposing no obstruction to the motion by the body impinging on the planes at B and c: and this again, is the same velocity as by descending through the same perpendicular height ED.

139. Corol. 1. If the lines AECD, &c, be supposed indefinitely small, they will form a curve line, which will be the path of the body; from which it appears that a body acquires also the same velocity in descending along any curve, as in falling perpendicularly through the same height.

140. Corol. 2. Hence also, bodies acquire the same velocity by descending from the same height, whether they descend perpendicularly, or down any planes, or down any curve or curves. And if their velocities be equal, at any one height, they will be equal at all other equal heights. Therefore the velocity acquired by descending down any lines or curves, are as the square roots of the perpendicular heights.

141. Corol. 3. And a body, after its descent through any curve, will acquire a velocity which will carry it to the same height through an equal curve, or through any other curve, either by running up the smooth concave side, or by being retained in the curve by a string, and vibrating like a pendulum: Also, the velocities will be equal, at all equal altitudes; and the ascent and descent will be performed in the same time, if the curves be the same.

PROPOSITION XXIX.

142. The Times in which Bodies descend through Similar Parts of Similar Curves, ABC, abc, placed alike, are as the Square Roots of their Lengths.

THAT is, the time in AC is to the time in ac, as VAC to Vac.

D

C

For, as the curves are similar, they may be considered as made up of an equal number of corresponding parts, which are every where, each to each, proportional to the whole. And as they are placed alike, the corresponding small similar parts will also be parallel to each other. But the time of describing each of these pairs of corresponding parallel parts, by art. 128, are as the square roots of their

lengths,

lengths, which, by the supposition, are as AC to ac, the roots of the whole curves. Therefore, the whole times are in the same ratio of AC to ac..

143. Corol. 1. Because the axes DC, DC, of similar curves, are as the lengths of the similar parts AC, ac; therefore the times of descent in the curves AC, ac, are as DC to De, or the square roots of their axes.

144. Corol. 2. As it is the same thing, whether the bodies run down the smooth concave side of the curves, or be made to describe those curves by vibrating like a pendulum, the lengths being DC, DC; therefore the times of the vibration of pendulums, in similar arcs of any curves, are as the square roots of the lengths of the pendulums.

SCHOLIUM.

145. Having, in the last corollary, mentioned the pendulum, it may not be improper here to add some remarks concerning it.

B

If the

A pendulum consists of a ball, or any other heavy body в, hung by a fine string or thread, moveable about a centre A, and describing the arc Cân; by which vibration the same motions happen to this heavy body, as would happen to any body descending by its gravity along the spherical superficies CBD, if that superficies were perfectly hard and smooth. pendulum be carried to the situation AC, and then let fall, the ball in descending will describe the arc CB; and in the point B it will have that velocity which is acquired by descending through CB, or by a body falling freely through EB. This velocity will be sufficient to cause the ball to ascend through an equal arc BD, to the same height D from whence it fell at c; having there lost all its motion, it will again begin to descend by its own gravity; and in the lowest point B. it will acquire the same velocity as before; which will cause it to re-ascend to c: and thus, by ascending and descending, It will perform continual vibrations in the circumference CBD. And if the motions of pendulums met with no resistance from the air, and if there were no friction at the centre of motion A, the vibrations of pendulums would never cease. But from these obstructions, though small, it happens, that the velocity of the ball in the point B is a little diminished. in every vibration; and consequently it does not return precisely to the same points C or D, but the arcs described con

tinually

tinually become shorter and shorter, till at length they are insensible; unless the motion be assisted by a mechanical contrivance, as in clocks, called a maintaining power.

146. If the circumference of a circle be rolled on a right line, beginning at any point A, and continued till the same point A arrive at the line

DEFINITION.

A

E

D

again, making just one revolution, and thereby measuring out a straight line ABA equal to the circumference of the circle, while the point a in the circumference traces out a curve line ACAGA; then this curve is called a cycloid; and some of its properties are contained in the following lemma.

LEMMA.

147. If the generating or revolving circle be placed in the middle of the cycloid, its diameter coinciding with the axis AB, and from any point there be drawn the tangent CF, the ordinate CDE perp. to the axis, and the chord of the circle AD: Then the chief properties are these:

The right line CD
The cycloidal arc AC

The semi-cycloid ACA

the circular arc AD;
double the chord AD;

double the diameter AB, and

The tangent CF is parallel to the chord AD.

PROPOSITION XXX.

148. When a Pendulum vibrates in a Cycloid; the Time of one Vibration, is to the Time in which a Body falls through Half the Length of the Pendulum, as the Circumference of a Circle

is to its Diameter.

LET ABа be the cycloid; DB its axis, or the diameter. of the generating semicircle DEB; CB=2DB the length of the pendulum, or radius of curvature at B. Let the ball descend from F, and, in vibrating, describe the arc FBf. Divide FB into innumerable small parts, one of which is Gg; draw FEL, GM, gm, perpendicular to

Mp Mm :: MN: MO.

нh is equal to Gg.

Also, by the nature of the cycloid,

1

If another body descend down the chord EB, it will have the same velocity as the ball in the cycloid has at the same height. So that кk and Gg are passed over with the same velocity, and consequently the time in passing them will be as their lengths Gg, Kk, or as нh to кk, or вH to BK by similar triangles, or (BK. BE) to BK, or BE to BK, or as

BL to BN by similar triangles. That is, the time in Gg: time in Kk:: BL: √BN. Again, the time of describing any space with a uniform motion, is directly as the space, and reciprocally as the velocity; also, the velocity in K or кk, is to the velocity at B, as EK to EB, or as LN to LB; and the uniform velocity for EB is equal to half that at the point B, therefore the

(by sim. tri.):: Nn or мp: 2√(BL. LN.)

That is, the time in Kk: time in EB :: MP: 2 (BL. LN.) But it was, time in Gg: time in Kk :: √BL: √BN; theref. by comp. time in Gg: time in EB :: Mp:2√(BN. NL) or 2NM. But, by sim. tri. Mm: 20м or BL:: Mp: 2NM.

Theref. time in Gg : time in EB :: Mm: BL.

Consequently the sum of all the times in all the Gg's, is to the time in EB, or the time in DB, which is the same thing, as the sum of all the mm's, is to LB;

that is, the time in Fg: time in DB::

and the time in

or the time in

FB: time in DB ::

Lm : LB,

LMB: LB,
FBf time in DB:: 2LMB: lb.

That is, the time of one whole vibration,

is to the time of falling through half CB,
as the circumference of any circle,

is to its diameter.

149. Coral.