Suppose you had accidentally found a very small Part of a visibly broken Medallion, with nothing more express upon it, than what is represented at _Fig._ 1. _Plate_ I. a Person totally unacquainted with the mathematical Sciences, we may naturally conclude, would not be able to make any thing of it, or in the least comprehend what it originally was, or meant; but if an Astronomer should chance to see it, who of course we are to suppose knew the Order and Proportion of the planetary Orbits, he would immediately conclude, and with great Probability, on the Side of his Conjectures, that it might be Part of a Medal representing the Solar System. In such a Case may we not very naturally suppose he would reason thus? The Arches A and B seem to be Portions of the respective Orbits of _Saturn_ and _Jupiter_, and what may lead us to believe, that they are really so, and Part of the Solar System, is the oblique Curve C, which looks not unlike the Trajectory of a Comet. This surely would be far from an irrational Conjecture, and consequently in some Degree probable: But this is not sufficient you'll say; To prove it we must have farther recourse to the Mathematicks, and a Mathematician would immediately thus demonstrate it to be true. [Illustration: Plate I.] [Illustration: Plate II.] First, by compleating the Circles geometrically from the fourth Book of _Euclid_, by the Assistance of any three Points E. F. G. the original Figure will be restored, as at _Fig. 2._ And secondly, by assuming any two Points, as F, E in the Curve C, if admitted a Parabola, by a well-known Problem in Conic Sections the Heliocentric Portion X. Y. Z. will easily be projected and shewn, as in _Fig. 3._ Lastly, join this in Position to the former, and it will justly supply the Orbit, or Path of some one of the Comets; and if required, even what Comet may be discovered by comparing the Perihelion Distance Y. S. with their general Elements or Theories, in Dr. Hally's _Synopsis_ of the Motion of these Bodies. And if a farther Confirmation of the Truth of these Conjectures were wanting, the small concentric Circles at D would now be allowed beyond a Contradiction, to represent the secondary Orbits of _Saturn_; and thus the first Presumption being carried thro' several corroborating Degrees of Probability, almost past a Dispute, would become a mathematical Certainty; and the above imperfect Piece of Medallion, would evidently appear beyond a Contradiction to be Part of a Representation of the said solar System, and such as is shewn in _Plate_ II. Q. E. D. Thus in many Cases, it often happens, that from a very small Part of _orbicular Things_, we are able to determine the Form and Direction of the Whole: And hence you may conceive it no very difficult Task to a Mathematician, to describe the Orbits of all the Planets in the solar System, though he had never observed them but in one and the same Sign of the _Zodiack_; thus far I have