Plate 1. Fig. 11.

Plate 1.
Fig. 12.

Plate 1.

Fi. 12.

PROP. XXXIV. A body acquires the same velocity in falling down an inclined plane, which it would acquire by falling freely through the perpendicular elevation of the plane.

The square of the velocity which a body acquires by falling to D, is (by Prop. XXVI. com. pared with Prop. XXIX. Cor.) to the square of the velocity it acquires by falling to B, as the space AD is to the space AB, that is (El. VI. 8. Cor.) as the square of AD is to the square of AC; and consequently the velocity at D is to the velocity at B, as AD is to AC.

But, be

cause AD and AC (Prop. XXXI. Cor.) are passed over in the same time, the velocity acquired at D is (by Prop. XXXII.) to that which is acquired at C, as AD to AC. Since then the velocity at D has the same ratio to the velocities at B, and at C, namely, the ratio of AD to AC, the velocities at B and C (EI: V. 9.) are equal.

COR. I.

Hence the velocities acquired by bodies falling down planes differently inclined are equal, where the heights of the planes are equal The velocities acquired in falling from A to C, and from A to G, are each equal to the velocity acquired in falling from A to B, and therefore equal to one another.

COR. 2. Hence if bodies descend upon inclined planes, whose heights are different, the velocities will be as the square roots of their heights. For (ig. 8. and 9) the velocity in D is equal to that in A, and the velocity in D is equal to that in G. Therefore the velocity in D (Fig. 8.) is to that in D (Fig. 9.) as AB is to FG (by Cor. Prop. XXVI.)

PROP. XXXV. A body falls perpendicularly through the diameter, and obliquely through any chord of a circle, in the same time.

In the circle ADB, let AB be a diameter, and AD any chord; draw BC a tangent to the circle at B; produce AD to C, and join DB. Because ADB (El. III. 31 ) is a right angle, a body (by Prop. XXXI Cor.) will fall from A to D on the inclined plane in the same time in which it will fall from A to B perpendicularly. In like manner let the chord A be produced to G; and because AEB is a right angle, a body will fall from A to E in the inclined plane in the same time in which it would fall from A to B.

COR. I.

COR. 2.

chords.

Hence all the chords of a circle are described in equal times.

Hence also the velocities, and accelerating forces, will be as the lengths of the

PROP. XXXVI. If a body descends along several contiguous planes, the velocity which it acquires by the whole descent, provided it lost no motion in going from one to another, is the same which it would acquire,

if it fell from the same perpendicular height along one continued plane; and this velocity will be the same with that which would be acquired by the pendicular fall from the elevation of the planes.

per

Let AB, BC, CD, be several contiguous planes; through the points A and D, draw HE, DF, parallel to the horizon, and produce the contiguous planes CB, CD, to G and E. By Prop. XXXIV. Cor. the same velocity is acquired at the point B, whether the body descends from A to B, or from G to B. Therefore, the line BC being the same in both cases, the velocity acquired at C must be the same, whether the body descends through AB, BC, or along GC. In like manner, it will have the same velocity at D, whether it falls through AB, BC, CD, or along ED, that is, (by Prop. XXXIV.) its velocity will be equal to the velocity acquired by the perpendicular fall from H to D.

COR. Hence, if a body descends along any arc of a circle, or any other curve, the velocity acquired at the end of the descent is equal to the velocity acquired by falling down the perpendicular height of the arc; for such a curve may be considered as consisting of indefinitely small right lines, representing contiguous inclined planes.

SCHOL. The velocity of a body, passing from one inclined plane to another, is diminished in the ratio of radius, to the co-sine of the angle between the directions of the planes. Let BC, or B m (Fig. 20.) represent the velocity acquired at B, and resolve BC into B n and C n, by letting fall the perpendicular Cn: m n will be the velocity lost, therefore the velocity at B is to the velocity diminished by passing from AB to BD as BC to B n, or as radius to the cosine of the angle between the directions of the planes.

PROP. XXXVII. If two bodies fall down two or more planes equally inclined, and proportional, the times of falling down these planes will be as the square roots of their lengths.

Plate 1.

Fig. 13.

Let the inclined planes be AB, BC, DE. EF: let AG, DH, be lines drawn parallel to the Plate 1. horizon; let B, DE, be equally inclined to the plane of the horizon, and also BC, EF; let AB Fig. 17, be to DE as AG to DH and as BC to EF, and draw GB, HE.

Because ABG, DEH are similar triangles, AB is to DE (El. VI. 4.) as BG to EH, and ✔AB to ✔DE as ✔BG to ✅EH: also AB is to DE as BG + BC is to HE + EF, and ✔AB to ✅DE as ✔BG + BC, or GC is to ✔HE + EF, or ✔HF.

And since by construction) AB is to DE as BC to EF, AB is to DE as AB + BC is to DE + EF, and ✔AB to ✔DE, as ✅AB + BC to ✔DE + EF. But AB, DE, being planes equally inclined, the accelerating force of gravitation will be the same upon each, and the bodies descending upon them may be considered as falling down different parts of the same plane. Hence, (Prop. XXVI. Cor. 1. and XXIX. Cor.) the time of descent along AB is to that along ᎠᎬ,

9

DE, as AB to DE; and the time of descent along GC is to that along HF, as GC is
HF, that is, as AB to ✔DE. Again, the time of descent along GB is to that along HE as
✔BG is EH, that is, as ✔AB to DE. Since, therefore, the time of descent along the
whole plane GC is to that along the whole plane HF, as AB to DE, and that of the part
GB is also to that of the part HE. as AB to DE, the time of descent al ng the remainder
BC is to that along the remainder EF (El. V. 19.) as ✔AB to ✔DE. Consequently, the time
of descent down AB + BC is to that down DE + EF, as AB to DE, that is, as
ABBC to DE + EF.

COR. Hence, if bodies descend through arcs of circles, the times of describing similar arcs will be as the square roots of the arcs. For such similar arcs may be considered as composed of an equal number of proportional sides, or planes, having the same inclination to each other, and their elevations equal: whence, by this proposition, the times of descent will be as the square roots of the lengths of the arcs.

PROP. XXXVIII. If a body be thrown up along an inclined plane, or the arc of a curve, it will, in the same time, rise to the same height, from which, with equal force, it would have descended; and any velocity will be lost in the same time in which it would, in descending, have been acquired.

For the same force of gravitation has, in every respect, the same efficacy to retard the motion of bodies ascending, as to accelerate them descending on an inclined plane or curve.

Plate 1.
Fig. 15.

SECT. III.

Of the Pendulum and Cycloid.

DEF. V. A pendulum is a heavy body, hanging by a cord or wire, and moveable with it upon a centre.

PROP. XXXIX. The vibrations of a pendulum are produced by the force of gravitation.

Let the ball A, suspended from the centre B by the cord BA, be drawn up to C and let fall from thence it will descend by the force of gravitation to A, from whence (being prevented from falling farther by the cord) it will proceed (by Prop. XXXVI. Cor.) with a velocity equal to that which it would have acquired in falling perpendicularly from E to A, which

will carry it on the opposite side to the height from which it fell. Being brought back again towards A by the force of gravitation, it will acquire a new velocity which will carry it towards C and in this manner it will vibrate by the force of gravitation, till the resistance of the air, and the friction of the string, stop its motion,

PROP. XL. The same pendulum, vibrating in small unequal arcs, performs its vibrations nearly in equal times.

In the circle CGA, the small arcs CA, EA, will differ little from their respective chords in Plate L Fig. 16. length or declivity. But (by Prop. XXXVI. Cor.) the times in which the chords are passed and 17. over are equal; therefore the times of describing the arcs CA, EA, and also (by Prop. XXXVIII.) of describing their doubles CAD, EAF, will be nearly equal.

EXP. Two equal pendulums, vibrating in small, but unequal arcs, will, for a long time, keep pace in their vibrations.

PROP. XLI. If a pendulum vibrate through small arcs of circles of different lengths, the velocity, it acquires at the lowest point, is as the chord of the arc which it describes in its descent.

Let BA be the pendulum, and CAD, EAF, the arcs through which it vibrates; and draw the plate 1. horizontal lines EK, CH The velocity acquired in falling from H to A is (by Prop. XXVI. Fig. 16. Cor.) to that acquired by falling from G to A, as ✔HA, to ✔G, that is, (by El. VI. 8. Cor.) and 17. as CA to GA. For the same reason, the velocity acquired in falling from G to A, is to that acquired in failing from K to A, as GA to EA. Consequently, ex aquali, the velocity acquired in fa ling from H to A, is to that acquired by falling from K to A, as CA to EA. But (by Prop. XXXVI. Cor.) the velocity acquired in falling from H to A is equal to that from C to A; and the velocity acquired in falling from K to A is equal to that from E to A. Therefore the velocity acquired in descending through the arc CA, is to that through EA, as the chord CA to the chord EA: and the same may be shewn concerning the remaining half of the vibrations, AF, AD.

COR. Hence the lengths of the chords of arcs through which pendulums move, are mea- Fig. 17. sures of velocity.

PROP. XLII. The time of the descent and ascent of a pendulum, supposing it to vibrate in the chord of a circle, is equal to the time in which a body, falling freely, would descend through eight times the length of the pendulum.

For

Plate 1. Fig. 18.

Fig. 19.

For the time of the descent of a body upon the chord, is (by Prop. XXXV.) equal to that of the fall through the diameter of the circle, which is twice the length of the pendulum; but in double that time, that is, in the descent and ascent, or whole vibration, the body would fall (by Prop. XXVII.) through four times the space, that is, through eight times the length of the pendulum.

PROP. XIII. The times in which pendulums of different lengths perform their vibrations, are as the square roots of their lengths.

Let the two pendulums, AB, CD, be of different lengths. The time in which the first, AB, vibrates through a chord, is equal to that in which a body (Prop. XXXV) would fall freely through twice AB, the diameter of the circle of which AB is radius: in like manner, the time in which CD vibrates. is equal to that in which a body would fall through twice CD. But the times in which a body would fall through these different spaces are (Prop. XXVI. Cor 1) as the square roots of the spaces, that is, as the square roots of AB and CD, the lengths of the pendulums: therefore the vibrations are in the same ratio.

COR. The times in which pendulums of unequal lengths vibrate, are as the square roots of the similar arcs through which they move. Let BA, BC, be pendulums of different lengths vibrating in the similar arcs FG, DE. Since the times of vibration are as the square roots of the lengths BA, BC, and that similar arcs are as the diameters, the times of vibration, are as the square roots of the arcs, FA, DC, or of their doubles, FG, DE.

EXP. Two pendulums, the lengths of which are as 1 to 4, will perform their vibrations in times as 1 to 2, that is, the shorter pendulum will make two vibrations, whilst the longer makes one for T:t::√ √ T.

PROP. XLIV. The squares of the times, in which a pendulum of a of a giv en length performs its vibrations, are inversely as the accelerating forces, or gravities.

By Prop. XXVI where the accelerating force is given, the space described is as the square of the time in which it is described. And since, in any given moving body, the velocity is as the accelerating force (Prop. A. p. 14.) where the square of the time, or the time itself, is given (by Prop. II.) the space described will be as the accelerating force. Consequently, where neither the accelerating force, nor the square of the time, is given, the space described will be in the ratio compounded of both. If then the space described be called S, the accelerating force A, and the square of the time T, S will be as TA, T&A S whence For Te is as A

Ꭺ

But, when the spaces are equal, S is a given quantity:

whence