Proposition 16

PROPOSITION 16 THEOREM 13

If the density of the medium in each of the places is reciprocally as the distance of the places from the immoveable centre, and the centripetal force be reciprocally as any power of the same distance, that the body may revolve in a spiral intersecting all the radii drawn from that centre in a given angle.

This is demonstrated in the same manner as the foregoing Proposition.

If the centripetal force in P is reciprocally as any power SPn+1 of the distance SP whose index is n + 1; it will be collected, as above, that the time in which the body describes any arc PQ, will be as PQ × PS 1/2n

The resistance in P as Rr / .. or as .. and therefore as

that is (because

is a given quantity), reciprocally as SPn+1. And therefore, since the velocity is reciprocally as SP½n, the density in P will be reciprocally as SP.

Cor. 1. The resistance is to the centripetal force as .. to OP.

Cor. 2. If the centripetal force be reciprocally as SP³, 1 - ½n will be = 0; and therefore the resistance and density of the medium will be nothing, as in Prop. IX, Book I.

Cor. 3. If the centripetal force be reciprocally as any power of the radius SP, whose index is greater than the number 3, the affirmative resistance will be changed into a negative.

SCHOLIUM

This Proposition and the former, which relate to mediums of unequal density, are to be understood of the motion of bodies that are so small, that the greater density of the medium on one side of the body above that on the other is not to be considered. I suppose also the resistance, cæteris paribus, to be proportional to its density. Whence, in mediums whose force of resistance is not as the density, the density must be so much augmented or diminished, that either the excess of the resistance may be taken away, or the defect supplied.

PROPOSITION 17 PROBLEM 4

To find the centripetal force and the resisting force of the medium, by which a body, the law of the velocity being given, shall revolve in a given spiral.

Let that spiral be PQR. From the velocity, with which the body goes over the very small arc PQ, the time will be given; and from the altitude TQ, which is as the centripetal force, and the square of the time, that force will be given. Then from the difference RSr of the areas PSQ and QSR described in equal particles of time, the retardation of the body will be given; and from the retardation will be found the resisting force and density of the medium.

PROPOSITION 18 Problem 5

The law of centripetal force being given, to find the density of the medium in each of the places thereof, by which a body may describe a given spiral. From the centripetal force the velocity in each place must be found; then from the retardation of the velocity the density of the medium is found, as in the foregoing Proposition.

But I have explained the method of managing these Problems in the tenth Proposition and second Lemma of this Book; and will no longer detain the reader in these perplexed disquisitions. I shall now add some things relating to the forces of progressive bodies, and to the density and resistance of those mediums in which the motions hitherto treated of, and those akin to them, are performed.