Finding the orbits from the focus given

PROPOSITION 43 Problem 30

Make a body move in a trajectory that revolves around the center of force in the same way as another body in the same trajectory at rest

In the orbit VPK, given by position, let the body P revolve, proceeding from V towards K.

From the centre C let there be continually drawn Cp, equal to CP.

This makes:

• the angle VCp proportional to the angle VCP
• the area which the line Cp describes will be to the area VCP, which the line CP describes at the same time, as the velocity of the describing line Cp to the velocity of the describing line CP; that is, as the angle VCp to the angle VCP.

Therefore, it is in a given ratio, and therefore proportional to the time.

Since, then, the area described by the line Cp in an immovable plane is proportional to the time, it is manifest that a body, being acted upon by a just quantity of centripetal force may revolve with the point p in the curve line which the same point p, by the method just now explained, may be made to describe an immovable plane. Make the angle VCu equal to the angle PCp, and the line Cu equal to CV, and the figure uCp equal to the figure VCP, and the body being always in the point p, will move in the perimeter of the revolving figure uCp, and will describe its (revolving) arc up in the same time that the other body P describes the similar and equal arc VP in the quiescent figure VPK. Find, then, by Cor. 5, Prop. VI., the centripetal force by which the body may be made to revolve in the curve line which the point p describes in an immovable plane, and the Problem will be solved. Q.E.F.