Proposition 38

PROPOSITION 38. THEOREM

304. With the centripetal force present varying inversely as the distance from the centre of force C (Fig. 28) the time of descent through the whole distance AC = a π , with a .. denoting the distance AC, f the distance at which the centripetal force is equal to the force of gravity, and π : 1 the ratio of the periphery to the diameter of a circle. [p. 124]

DEMONSTRATION

Since the speed fl ay corresponds to the height the body descends from some point P (266), then the speed itself is equal to fl ay and the time to fall the distance PC is equal …

Therefore with the integral of this thus taken, in order that is vanishes

…

when y = 0, the time to pass through PC is indeed given. Whereby if y = a now is substituted in this expression, the total time to descend through AC is given.

On putting y = a z, there is obtained a . ∫ dz . Truly I have established the …

in the Commentariis Academiae Scientiarum Petropol, for the year 1730, and if z = 1 or y = a is put in place, resulting in the definition of this 1, 2, 6, 24 etc., the terminus of which, with the index equal to − 12 , is equal to π , that has been shown by another method in the same place.

From which it is understood that the total time to descend through AC is a π . Q. E. D.

Corollary

305. Therefore if many bodies are released from different distances to the same centre C, the times of descent are in proportion to the distances.

Scholium 1

306. In this proposition I have neglected the fraction 250 time, elicited from the integration of the element of distance divided by the square root of the speed corresponding to the height, is to be multiplied (222), clearly in order that the time can be inserted in seconds, if the lengths are expressed in scruples of Rhenish feet.

Also in a similar manner for the following times that I am about to define, unless the times are wanted in seconds, these will be avoided as encumbrances.

It is easily seen that nothing else is to be found by expressing the time in seconds, unless the use is forced upon us, in which case the expressions of time are divided by 250 and the lengths are shown in scruples of Rhenish feet.

Scholium 2

307. This paradox is quite apparent, since for the integral of dz , with z put equal to 1 .. [in the upper limit], it becomes equal to π . For no one is able to directly show this result by any method ; I myself only knew about this equality later, as can be seen from …

the cited paper. Therefore these two integrals give the same … value, if z is put equal to one after the integration, and yet they are not equal to each other ; indeed they cannot to be compared.

PROPOSITION 39. THEOREM

308. If the centripetal force is as the power of the exponent of the distance n and many bodies are released to fall from different distances, the times of the descents are proportional to the powers of the distances, of which the exponent is 1−2n .

DEMONSTRATION

Let AC = a be the distance of any body from the centre C and f the distance at which the centripetal force is equal to the force of gravity [p. 126]. Then when it arrives at P, CP is put equal to y and the height corresponding to the speed in this place is equal to v, then …

Therefore the time, in which the distance CP is completed, is equal to ..

Because this integral cannot be evaluated for all n, yet thus it may be compared, as the as a and y have the same dimension 1−2n for the individual terms of a and y, since in the differential they make a number of the same dimension, with dy as one dimension.

As if after integration, y is put equal to a, in which case the time for the whole descent arises, … only a will have just the same dimensions, obviously 1−2n , or it will be a multiple of a 2 . Whereby, since another factor is not included apart from f , the numbers thus retain the 1− n same value, however a is varied, and the different times of descent will be as a 2 , i. e. as the powers of the distances, the exponent of which is 1−2n . Q. E. D.

Corollary 1

309. Therefore when all the times of descent are equal to each other, it is necessary that … be a constant quantity, whatever a may be changed into, and since that happens if n = 1, or the centripetal force is directly proportional to the distance, as we have seen (283).

Corollary 2

310. In a similar manner it is at once apparent from these, that if the centripetal force varies inversely as the square of the distance or n = – 2, the times of descent to this centre are to each other as the distance raised to the power 32 , or in the three on two ratio of the distances (287).

Corollary 3

311. If there were many similar attractive centres of force, but with different strengths [or measures of effectiveness], and to these bodies are released from equal distances, then the times will be between themselves as f 2 , since a is considered as a constant, and f indeed is the variable.

Truly the strength is as the centripetal force at a given distance, for example 1, therefore fn will vary inversely with the strength, and these times are in the inverse square root ratio with the ratio of the strengths of the centres of force (285).

Corollary 4

312. If to the different centres of force of this kind bodies are released from any distances, the times of descent of these are in a ratio composed from the direct 1−2n power of the distance, and inversely as the square root of the effectiveness [or strength of the attracting source]

Scholium

313. From these propositions, which have been discussed above concerning centripetal forces, it has been made abundantly clear how the motion of bodies should be found, if the centrifugal force is substituted in place of the centripetal force, or a force repelling the body from the centre.

[It is important to note that Euler’s usage of the term ‘centrifugal force’ is different from what is now understood: in Euler’s day it was a true repulsive force, while in modern times it has come to mean an apparent or fictitious force.]

Everything remains as in the preceding discussions, except that in place of the formula expressing the centripetal force, which was .. , the negative of this must be used.

Yet neither do I judge it superfluous to report on certain other cases; for these are known from the general rules pertaining to forces for general motion, which cannot be deduced from a single calculation.

Moreover these rules pertain to the action of forces on a body at rest, to which our calculation, clearly when the increment of the speed with respect to the first is infinitely small, is not so well adapted, with the thing itself reduced to absurdity, unless the first element of the distance is traversed in an infinitely short element of time.

Moreover I make use of this axiom in order to elucidate the matter, that a body placed anywhere will always be repelled from the centre of force, even if the centrifugal force for that point is indefinitely small or zero ; and because that happens, when the power of the distances to which the centrifugal force is in proportion is a number greater than zero or positive.