Proposition 2

PROPOSITION 2. Theorem.

20. While the body is moving uniformly along the curve AM (Fig. 1), it pressed the curve normally with a force at the individual points M with a force which is to the force of gravity as the height corresponding to that speed is to half the radius of osculation.

Demonstration

If the body on the curve AM must be moving freely with a uniform motion, then it is necessary for the force to be present everywhere only acts normally along MO, which has the ratio to itself to the force of gravity as the height corresponding to the speed of the body to half the radius of osculation MO, as appears from the demonstrations of the preceding book (165, 209, 552).

For unless such a force is present then the body travels in a straight line. Moreover in this case the channel AM, in which the body is considered to be enclosed, impedes the motion, in which the body progresses less in a straight line. On account of which the body presses the channel normally with so great a force following the direction Mn. If indeed the body is pressed by such a normal force present, then it can move freely in the channel AM ; and neither might it press the channel; but truly with this force missing, as we put here, it is necessary in order that the body presses the channel itself with such a force. [Newton’s third law.]Q.E.D.

Corollary 1.

21. Therefore if the height corresponding to the speed of the body is put as v and the radius of osculating MO is equal to r and the gravity on the body is equal to 1, as clearly it has if it has been put on the surface of the earth, the force will be by which the body pressing on the channel at M along Mn, is equal to 2rv .[p. 10]

Corollary 2. 22. If the body moves with a greater or smaller speed along the curve AM, then the force pressing at M is greater or less in the square ratio of the speed, since the height v is proportional to the square of the speed.

Corollary 3

23. The direction of this force is normal to the curve and is in the opposite direction to the position of the radius of osculation MO. Whereby the radius of osculation in the other part of the curve produced gives the direction of this force.

Corollary 4. 24. If the body is moving in a straight line, this force is zero on account of the infinite radius of osculation. This is also evident from the nature of the motion. For the body moves in a straight line uniformly spontaneously and on this account is not pressed by the channel.

Corollary 5.

25. If the curve AM is a circle, the force is the same everywhere. Truly with that to be greater as the radius of the circle is made less. Indeed with the same speed present, the force varies inversely as the radius of the circle.

Scholium 1.

26. Where the body is able to move freely along the curve AM uniformly, it is necessary that [p. 11] it is drawn along the normal MO by a force equal to 2rv . From which it is to be understood that the body struggles with such a force in the opposite region, otherwise the body cannot be kept on the curve by that force. Therefore while the body is forced to move along the channel AM it is being carried along by the struggle with this force, and this force is exercised by the channel itself. On account of which the channel must have such firmness, in order that it is able to sustain such force.

Corollary 6.

27. Therefore the body is able to carry out the motion without any expenditure of the speed, which clearly is consistent with the definition of the force.

Corollary 7.

28. Therefore the force arises from the motion alone. On account of which, just as motion is generated from forces, thus forces can arise from the motion.

Scholium 2.

29. Hence it is understood, as now in the first book above we have agreed with the notion (102), that it is unclear whether motion is owing to forces or whether forces to motion. For we see each in the world, truly forces and motion to arise; therefore one is the cause of the other, the question is to be decided from reasoning as well as from observation. Indeed there seems to be hardly any agreement about forces that arise on bodies that remain at rest, with much less forces being decided to arise form these. Besides truly, that everything can be shown to arise from motion, is considered to be the natural cause to be given of all phenomena. [p. 12] For motion once in existence must always to be conserved we have clearly shown above (63); this we have truly elaborated upon, as forces arise from motion. As truly forces without motion that are able either to be present or to be conserved cannot be conceived. On account of which we can conclude that all forces, which are observed in the world, arise from motion; and it falls upon the diligent investigator, that for each and every motion of bodies from any kind of force in the world their origin has to be observed.

Scholium 3

30. Since it is difficult to understand how such an effect clearly of a continuous force can act on a body without any change arising in the speed, it is a worthwhile task to inquire about the cause of this effect. We have seen in the preceding proposition that the motion of the body was not exactly uniform, but the speed is actually allowed to decrease, while the body is moving through singular elements in the curve. Truly these decrements are equivalent to second order differentials, as also only infinitely small changes in the speed repeated infinitely often are able to diminish the speed. Therefore I declare that the force acting is ascribed to an infinitely small decrement in the speed; and I become more confirmed in this belief, because when the decrement in the speed becomes greater, so also does the force present increase. Since the force at M is equal to 2rv and while it is being traversed, the whole element Mm is acted upon by this force [p. 13], it is permitted to express the effect of this force on the element Mm = ds by 2vds .

Truly the above r decrement in the speed, while the element Mm has been traversed, is found to be cds 2 (12). But because this is equal to c there, which here we have as 2r 2 2 equation arises : − dc = cds2 , and it becomes 2r v ; hence the Therefore we have − 4vdv = 4v ds2 equal to the square of the force that supports the 2 2 2r element Mm.

Corollary 8.

31. Therefore the square of the force acting on Mm is equivalent to the decrement of 2v 2 . And if this decrement is equal to ds 2 , then the force is equal to the force of gravity, from which the comparison of these forces is known.

Corollary 9

32. Therefore it is conceded that an infinite number of infinitely small decrements in the speed suffices in the production of a finite force. For as long as the decrement v 2 itself is the homogeneous ds 2 , the force is finite; but truly if that infinite number of infinities becomes greater than ds 2 , then the force also becomes infinitely large.

Definition 2. 33. This force, which the body exercises on the body in the line of the curve is called the centrifugal force, since the direction of this pulls from the centre of osculation O. [p. 14] Corollary 1. 34. The centrifugal force is therefore to the force of gravity as the height corresponding to the speed to half the radius of osculation. Corollary 2. 35. Therefore when the body is forced to move along the line of the curve the centrifugal force presses against the curve, even if no external force is acting. Scholium. 36. Therefore when the body is acted on by some external forces, a force also arises in the channel from these forces as well as from the channel itself, both pressing in a two- fold ratio, truly partially from the external forces and partially from the centrifugal force. Now therefore, what the force shall be that prevails on the constrained body is to be found.