Is a true Projection of the System of the known Comets; in which are represented nine of the chief Trajectories, from their _Aphelii_ to their _Perihelii_, all in just Proportion and Position to the Orbits of _Saturn_ and _Jupiter_, which are also represented by the two concentric Circles, supposed to be drawn round the Sun as their Center. The Ellipsis, or Trajectory, marked A, shews the Position and Path of the Comet which appeared in the Year 1684, whose Period is supposed to be about fifty Years, and has been observed within the Region of the Planets once. That mark'd B, is the Way of the Comet of 1682; The Period conjectured to be about seventy-five Years and a half, and has been observed thrice. C, Way of the Comet of 1337; The Period about 100 Years, observed once. D, That of the Comet of 1661; The Period about 129 Years, observed twice. E, Tract of the Comet of 1618; The Period about 160 Years, observed once. F, Way of the Comet of 1677; The Period about 200 Years, observed once. G, Way of the Comet of 1744; The Period about 300 Years, observed once. H, Way of the Comet of 1665; The Period about 400 Years, observed once. I, Way of the Comet of 1680; The Period about 575 Years, observed thrice. [Illustration: Plate IV.] [Illustration: Plate V.] The Scale of this System is equal to one Third of the former. Here I must observe to you, as a Thing I judge may prove of great Consequence with regard to the System of Comets, which is as yet very imperfect: That I am strongly of Opinion, that the Comets in general, through all their respective Orbits, describe one common Area, that is to say, all their Orbits with regard to the Magnitude of their proper Planes, are mathematically equal to one another; which, if it once could be proved, and confirmed by Observation, the Theories of all the Comets that have been justly observed, might easily be perfected, and their Periods at once determined, which now we can only guess at, or may wait whole Ages for more Certainty of. What leads me to believe, that this may prove to be really the Case is this. I find by Calculation, that the Orbits of the two last Comets, whose Elements have been most corrected by Sir _Isaac Newton_ and Dr. _Hally_, are to one another, according to their Numbers, nearly as [J]13 to [K]17, notwithstanding one of them is one of the most erratick that ever came under our Observation; and the other one of the most neighbouring to the Sun. [Footnote J: 1316539,968282 Comet of 1680.] [Footnote K: 1708155,4644 Comet of 1682.]