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which is also again, by art. 135, the same as by descending through G. 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 ABCD, &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 ang 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.
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 V AC to Nac.
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, which, by the supposition, are as ac to Vac, the roots of the whole curves. Therefore, the whole times are in the same ratio of NAC to vac.
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 v DC to DC,
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..
A pendulum consists of a ball, or any other heavy body B, hung by a fine string or thread, moveable about a centre A, and describing the arc CBD; 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. If the 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 co, or by a body falling freely through ER. This velocity will be sufficient to cause the ball to ascend through an equal arc BD, to the same height p from whence it fell at c; having there lost all its motion, it will again be. gin 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 ci 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 cor p, but the arcs described con
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.
DEFINITION. 146. If the cir. cumference of a
B circle be rolled on
E a right line, beginning at any point A, and continued till the same point
А A arrive at the line 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 circular arc AD;
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 ABa 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
E ball descend from F, and,
KAK N M in vibrating, describe the arc Fbf. Divide fB into innumerable small parts, one of which is Gg; draw FEL, GM, gm, perpendicular to
th H h Arn
DB. On LB describe the semicircle LMB, whose cene tre is o; draw mp parallel to DB; also draw the chords BE, BH, EH, and the radius
L Now the triangles BEH,
HAK NM BHK, are equiangular; therefore BK:BH::BH:BE, or BH? = BK, BE, or BH = (BK.BE). And the equiangular triangles Mmp, Mon, give MP : Mm :: MN: MO. Also, by the nature of the cycloid, Hh is equal to Gg.
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 kk 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 hh to kk, or by to bk by similar triangles, or (BK. BE) to BK, or ✓ Be to VBK, or as ✓ BL to BN by similar triangles.
That is, the time in yg : time in kk :: VBL : VBN.
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 kk, is to the velocity at Ry 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
EB time in kk: time in EB :::
VIN' ŽVLB VIN *V LB (by sim. tri.) :: Nn or mp: 2V(BL.LN.) That is, the time in Kk : time in EB :: mp: 2N (BL . LN) But it was, time in gg : time in kk :: VBL: VBN; theref. by comp. time in gg: time in EB :: Mp:2w(BN.NL) or 2NM. But, by sim. tri. Mm: 20m or bL :: mp : 2NM. Theref. time in gg : time in EB :: Mm : FL.
Consequently the sumn of all the times in all the gg's, is to the time in EB, or the uime in DB, which is the same thing, as the sum of all the mn's, is to LB; that is, the time in Fg : time in DB :: and the time in FB : time in DB :: LMB : LB, or the time in FBf : time in DB :: 21MB : LB. That is, the time of one whole vibration,
is to the time of falling through half cb,
149. Corol. 1. Hence all the vibrations of a pendulum in a cycloid, whether great or small, are performed in the same time, which time is to the time of falling through the axis, or half the length of the pendulum, as 3:1416 to 1, the ratio of the circumference to its diameter; and hence that time is easily found thus. Put p = 3:1416, and I the length of the pendulum, also g the space fallen by a heavy body in 1" of time.
2 then vg: 11:: 1": the time of falling through jl,
28 theref. 1:8::
which therefore is the time
2g of one vibration of the pendulum.
150. And if the pendulum vibrate in a small arc of a circle; because that small arc nearly coincides with the small cycloidal arc at the vertex B; therefore the time of vibration in the small arc of a circle, is nearly equal to the time of vibration in the cycloidal arc; consequently the time of vibration
1 in a small circular arc, is equal to R v where l is the radius of the circle.
151. So that, if one of these, g or l, be found by experi ment, this theorem will give the other. Thus, if g, or the space fallen through by a heavy body in 1" of time, be found, then this theorem will give the length of the second pendulum. Or, if the length of the second pendulum be observed by experiment, which is the easier way, this theorem will give g the descent of gravity in 1'. Now, in the latitude of London, the length of a pendulum which vibrates seconds, has been found to be 39 inches; and this being
994 written for l in the theorem, it gives PV =l": hence is
28 found g = 'p?l= p X 394 = 193.07 inches = 16 feet, for the descent of gravity in 1"; which it has also been found to be, very nearly, by many accurate experiments.
152. Hence is found the length of a pendulum that shall make
any number of vibrations in a given time. Or, the number of vibrations that shall be made by a pendulum of. a given length. Thus, suppose it were required to find the length of a half-seconds pendulum, or a quarter-seconds pendulum; that is, a pendulum to vibrate twice in a second, or 4 times in a second. Then, since the tin.e of vibration is as the square root of the length,