The Movement of Comets

Proposition 40 Theorem 20: The comets move in some of the conic sections, having their foci as the sun. They trace an orbit to the sun by radii drawn to the sun’s center.

centre of the in the a r eas proportional to the times.

This proposition appears from Cor. 1, Prop. 13, compared with Prop. 8, 12, and 13, Book 3.

Corollary 1

Hence if comets are revolved in orbits returning into themselves, those orbits will be ellipses; and their periodic times be to the periodic times of the planets in the sesquiplicate proportion of their principal axes. And therefore the comets, which for the most part of their course are higher than the planets, and upon that account describe orbits with greater axes, will require a longer time to finish their revolutions. Thus if the axis of a comet’s orbit was four times greater than the axis of the orbit of Saturn, the time of the revolution of the comet would be to the time of the revolution of Saturn, that is, to 30 years, as 4 4 {\displaystyle \scriptstyle {\sqrt {4}}} (or 8) to 1, and would therefore be 240 years.

Corollary 2

But their orbits will be so near to parabolas, that parabolas may be used for them without sensible error.

Corollary 3

Therefore, by Cor. 7, Prop. XVI, Book 1, the velocity of every comet will always be to the velocity of any planet, supposed to be revolved at the same distance in a circle about the sun, nearly in the subduplicate proportion of double the distance of the planet from the centre of the sun to the distance of the comet from the sun’s centre, very nearly. Let us suppose the radius of the orbis magnus, or the greatest semidiameter of the ellipsis which the earth describes, to consist of 100000000 parts; and then the earth by its mean diurnal motion will describe 1720212 of those parts, and 71675½ by its horary motion. And therefore the comet, at the same mean distance of the earth from the sun, with a velocity which is to the velocity of the earth as 2 {\displaystyle \scriptstyle {\sqrt {2}}} to 1, would by its diurnal motion describe 2432747 parts, and 101364½ parts by its horary motion. But at greater or less distances both the diurnal and horary motion will be to this diurnal and horary motion in the reciprocal subduplicate proportion of the distances, and is therefore given.

Corollary 4

Wherefore if the latus rectum of the parabola is quadruple of the radius of the orbis magnus, and the square of that radius is supposed to consist of 100000000 parts, the area which the comet will daily describe by a radius drawn to the sun will be 1216373½ parts, and the horary area will be 50682¼ parts. But, if the latus rectum is greater or less in any proportion, the diurnal and horary area will be less or greater in the subduplicate of the same proportion reciprocally.