Chapter 2g

Proposition 19

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 19. Problem

  1. For a given force acting, to find the length of the pendulum making indefinitely small oscillations, which completes a to and fro motion in a time of one second. Solution. [p. 76] With the length of the pendulum a sought and the force acting g, with the force of gravity denoted by one, the time of one indefinitely small oscillation is equal to π 2a . ..

This has to be expressed in seconds of minutes, with the length a expressed in thousandth parts of Rhenish feet and the formula π 2a is to be divided by 250, as is … apparent from the first book (221). On account of which the time of one half oscillation is obtained π 2a seconds. Whereby, since the time has to be one second, that is … thousandth parts of Rhenish feet.

Therefore, this is the length of the pendulum completing a semi-oscillation in a time of one second. Q.E.I.

Corollary 1

  1. Hence the lengths of the pendulums executing oscillations in the same time, but with different forces acting, are in the ratio of the forces.

Corollary 2

  1. If the force acting g is equal to the force of gravity 1, which case agrees with oscillations on the surface of the earth, the length of the pendulum which makes a single to and fro journey [i. e. half an oscillation in one second] is equal to 3.16625 Rhenish feet, or three and one sixth feet. [p. 77]

Scholium 1

  1. This length agrees extremely well with that found by Huygens from experiment; from which it is apparent that we have assumed correctly the number in the preceding book (220) of 15625 scruples of Rhenish feet that a body falls, acted on by the force of gravity, for a time of one second from rest; for indeed this number departs from the number 250, by which the expressions for the time must be divided, in order that a time of one second is presented. [Recall that the number 250 was just a useful number introduced by Euler as a memory aid, and so was only approximately correct.] Therefore since it may be generally wished to have the sixth part of a foot for the Huygens length of the pendulum of 3.166, clearly the length of this must be determined by observations

…

everywhere on the surface of the earth, from which it generally consists of 1055 thousandth parts of a Rhenish foot.

Scholion 2

  1. Now from observations the universal foot can be determined in the following manner. A pendulum of length f is taken, which is set in motion to make the smallest oscillations, and let the number of these counted in a time of one hour be n, thus in order that a single semi-oscillation is completed in a time of 3600 seconds. Now let the length n of the pendulum completing semi-oscillations in one second be z. Whereby, since the times of oscillations of different pendulums acted on by the same force are as the square root ratio of the lengths of the pendulums (171), then the ratio is 3600

…

thus z = 12960000 , [p. 78] and consequently the universal foot is equal to 38880000 .

Corollary 3

  1. Therefore a pendulum four times longer than 3166 14 scruples of Rhenish feet completes semi-oscillations in two seconds, since the times of the oscillations are in the square root ratio of the lengths of the pendulums.

Corollary 4

  1. Since the radius of the earth is 20382230 Rhenish feet., if a pendulum of such a length is conceived, a single semi-oscillation of this will last for 2536 sec. Whereby in 24 hours almost 17 whole oscillations are completed.

[There is no section 184.]

Corollary 5

  1. Since the time of half an oscillation is π 2a , the time of .. whole oscillations is 2π 2a . But this time is equal to the time of the revolution .. performed on the periphery of a circle of radius a by a body in free motion, which is drawn towards the centre by a force equal to g, as from the preceding book is apparent (612).

On this account the time of a whole oscillation of the pendulum equal to the radius of the earth is equal to the time that a body projected on the surface carries out a complete revolution.

Huygens also showed that a body completes almost 17 revolutions in a time of 24 hours in performing this motion.

Corollary 6

  1. Since the force of gravity shall be to the force that a body on the surface of the sun is urged towards the centre of the sun, as 41 to 1000, the length of the pendulum which on the surface of the sun performs a semi-oscillation in a time of one second is equal to 77.226 Rhenish feet. In a similar manner on account of gravity on the surface of Jupiter , for such a pendulum the length is 6.448 feet. And on the surface of being equal to 167

…

Saturn on account of gravity equal to 105 , the length of such a pendulum is 4.054 feet.

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