1 described by any forces, in the same time, are as the forces, or as the velocities. 130. Corol. 3. Consequently the velocities and spaces descended by bodies down different inclined planes, are as the sines of elevation of the planes. 131. Corol. 4. If CD be drawn perpendicular to AB; then, while a body falls freely through the perpendicular space BC, another body will, in the same time, descend down the part of the plane BD. For by similar triangles, BC BD BA BC, that is, as the space descended, by corol. 2, Or, in any right-angled triangle BDC, having its hypothenuse BC perpendicular to the horizon, a body will descend down any of its three sides BD, BC, DC, in the same time. And therefore, if on the diameter BC a circle be described, the time of descending down any chords BD, BE, BF, DC, EC, FC, &c, will be all equal, and each equal to the time of falling freely through the perpendicular diameter BC. PROPOSITION XXVI. 132. The Time of descending down the Inclined Plane BA, is to the Time of falling through the Height of the Plane BC, as the Length BA is to the Height BC. DRAW CD perpendicular to AB. Then the times of describing BD and BC are equal, by the last corol. Call that time t, and the time of describing BA call T. Now, because the spaces described by constant forces, are as the squares of the times; therefore t: T2 :: BD : BA. But the three BD, BC, BA, are in continual proportion ; therefore BD: BA :: BC2 :: BA2; hence, by equality, ť: T2 :: BC2: BA2, 133. Corol. Hence the times of descending down different planes, of the same height, are to one another as the lengthsof the planes. PROPOSITION PROPOSITION XXVII. 134. A Body acquires the Same Velocity in descending down any 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; BC; T: BC: BA; : 1. 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, to 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. PROPOSITION 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 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. curve or curves. 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 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 alti tudes; 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 Ac to /ac. 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 a B 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 DC, 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. 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 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 OB; and in the point в it will have that velocity which is acquired by descending through cp, 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 в is a little diminished in every vibration; and consequently it does not return precisely to the same points cor D, but D, but the arcs described continually 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 B E again, making just one revolution, and thereby measuring out a straight line ABA equal to the circumference of the cir cle, 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: CD The right line the circular arc AD; double the chord AD; The semi-cycloid ACA = double the diameter AB, and 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. |