Chapter 2f

Proposition 18

Motion is the translation of a body from the place it occupies to another place. True rest is a body remaining at the same place.

Euler
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PROPOSITION 18. Problem.

  1. With a uniform force present acting in the downwards direction, to determine the time of the ascent or the descent through any arc of a circle EA (Fig.23), ending at the lowest point A.

Solution.

Let C be the centre of the circle, CA is the radius of the vertical or the line parallel to the direction of the force g. Putting AC = a and the arc AE equal to the height AG = b, the speed at the lowest point A corresponds to the height gb, since the body descending from E has such a speed when it arrives at A.

And the body must have such a speed at A, in order that it can rise as far as E. Some element Mm of the arc AE is considered and calling AP = x ; then PM = ( 2ax − x 2 ) and Mm = adx.

Now the speed at M ( 2 ax − x 2 ) … corresponds to the height g .GP = gb − gx (93). Therefore the time in which the element Mm is traversed either in the ascent or in the descent is equal to

…

Which, since it cannot be integrated, we express the integral by a series. Moreover, with putting 2a = c …

Hence this is multiplied by adx and the integration gives the time, in which the arc AM is .. completed, to equal : Now the time in which the whole arc EA is traversed can be produced, if we put x = b and the ratio of the periphery to the diameter = π : 1 , with which in place there is obtained : .. etc are the squares of the coefficients 1, 1 , 3 , which is Where the coefficients 1, 14 , 64 2 8 produced if ( 1 − z )− 2 in resolved into a series. Now the time can therefore be found approximately from this series. Q.E.I.

Corollary 1

  1. Therefore where the arc EA is made larger, then the time too is greater, in which it is traversed. Indeed on putting b = 2a = c , the time is infinite, since the body in descending is by no means able to complete the semicircle.

Corollary 2

  1. Therefore if the body in an oscillatory motion is moving in the arc EAF of the circle, then the time of one to or fro motion is twice as great as the time of one ascent or one descent, since the time to pass through ANF is equal to the time to pass through AME. Whereby the time of one to or fro motion, or the time for half an oscillation, is equal to Truly the time for one oscillation to be completed is twice as great.

Scholium 1

  1. The series expressing this time can at once be found in this way. An element of time can be resolved into these factors : and of these only the latter should be converted into a series, clearly this : with 2a = c. Moreover, because after the integration, on placing x = b, then

…

From which the whole descent time can be gathered together to be equal to :

Scholium 2

  1. From which it is apparent that the summation of the series depends on the construction of the equation

…

and the sum of the series is equal to e ∫ t with e denoting the number of which the log is equal to1. With these in place, the series is to be summed by my method explained in Comment. Acad. Petrop. Tom. VII [1740, p. 123; Opera Omnia series I, vol. 14; E41 is translated in this series; however, this appears to be a misquote by Paul Stackel, as this paper does not present the method used to sum the present series. One should look instead in E025 perhaps], and the following equation is found from the exposition:

..

From which equation, if it can be solved, q is found in terms of t and hence the sum itself is found in terms of t or bc . Moreover since the construction of the equation does not follow from inspection, it is yet apparent that it can be done, since the sum of the series for the time can be assigned with the help of quadrature. Indeed the given sum of the series is found to follow from the construction of that equation.

Corollary 3

  1. If the arc AE, in which the descent or the ascent is completed, is put infinitely small, yet the time for that motion is not infinitely small. For in the expression for the time, only b vanishes, and the time in which a vanishing arc AE is completed is equal to π 2a . 2 g [i. e. the radius of the circle a remains unchanged, while the distance fallen b tends towards zero.]

Corollary 4

  1. With the other part AF of the circle joined with AE the oscillations through the arc EAF can be made indefinitely small; still with a finite completion time. Clearly the time for one ’to’ or ‘fro’ motion, or the time for half an oscillation, is equal to π 2a .

Corollary 5

  1. Therefore the times of this kind of infinitely small oscillations are in the square root ratio directly as the radius and inversely as the force [of gravity] acting.

Corollary 6.

  1. These same formulae prevail, if the force acting should not be uniform. For whatever variable force is put in place, yet while the body driven along an infinitely small arc, it has the same constant value.

Corollary 7

  1. It is to be understood that even if the curve EAF is not a circle, but any curve, then also these results reported here pertain to infinitely small oscillations on this curve. Then indeed in place of the radius the radius of osculation of this curve is to be taken at the lowest point A.

Corollary 8

  1. Oscillations upon an infinitely small arc of the curve EAF are effected with the aid of a pendulum, the length of which is the radius AC. [p. 74]Therefore the times of indefinitely small oscillations of the pendulum vary directly as the square root of the length of the pendulum and inversely as the square root of the force acting.

Corollary 9

  1. If the curve ANF is not equal to the curve AME, [i. e. no longer circular arcs, and each with its own radius of curvature] it is still sufficient to consider the radius of osculation at the point A for infinitely small oscillations. Let this length be equal to α, then the ascent time through the indefinitely small arc AF is equal to π 2α , and since 2 g the descent time through the vanishing arc EMA is π 2a , then the time for one journey … or half an oscillation on the composite curve EAF π( a+ α ) 2 g

Corollary 10

  1. If the oscillations are not indefinitely small on the circle BAD, the oscillation times are greater, as the arcs of the oscillations are greater. And if the oscillations are yet definitely small, the time of such an oscillation to the time of an indefinitely small oscillation to is as the square of the diameter of the circle increased by the versed sine of the arc traversed to the square of the diameter itself.

Corollary 11

  1. The height, from which a body descends in the same time by the same force g acting, as it descends along an indefinitely small arc EMA , is equal to π8a , or is to the eighth part of the radius as the square of the circumference to the square of the diameter ; this height is hence approximately equal to 54 a .

Corollary 12

  1. Moreover the body descends along the chord of the arc EMA in the same time that it descends along the diameter of the circle (102). Whereby the descent time along an indefinitely small arc is to the descent time along the corresponding arc is as 2 2a to … i. e. as the diameter to the fourth part of the circumference.

The descent time .. from the diameter or from twice the length of the pendulum is to the time of one whole indefinitely small oscillation composed from a to and fro motion is as the diameter to the circumference.

Scholium 3

  1. If two circular arcs AE and FA (Fig. 24), upon which connected oscillations are carried out, are not equal, these oscillations can be made with the aid of a pendulum, if in the centre of K of the arc AF a nail is driven in, in order that the thread CA, after it has described the arc EA about the centre, is retained at K, and describes the arc AF about the centre K.

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