_Concerning the Nature of Mathematical Certainty, and the various Degrees of Moral Probability proper for Conjecture._ _SIR_, You know how much I am an Enemy to the taking of any thing for granted, merely because a Person of reputed Judgment, has been heard to say, _it absolutely is so_; an _Ipse dixit_, and implicit Faith in some Cases, may be both necessary and useful; but here, in Astronomy, I mean, every Man's Reason, by the Help of a very little Mathematicks, is able to bring wonderful Truths to Light without them; and Truths not only of the highest Importance to every Individual, but of a great and common Consequence to all Mankind: And as such, in all Ages of the World, have been judged worthy to be enquired into, by the best and wisest of Philosophers. You are likewise very sensible how far the human Understanding is even at the best, from being infallible, and don't want to be told, how difficult it is in a Subject of this Nature to arrive at any tolerable Degree of Certainty, which before the Days of the sagacious _Euclid_, and the penetrating _Archimedes_, was a Thing not to be expected. And many things which were then but barely Objects of Conjecture and Probability, have since been demonstrated to be infallibly true. Time and Observation will undoubtedly, at last, discover every thing to us necessary to our Natures, and proper for us to know. As a Proof of which, we see human Wisdom daily increases; and while a Capacity continues to make ourselves still more acquainted with the manifest Wisdom and Power of God in the Works of his Creation, who is to tell us where to stop our Enquiries? Or who is so impious to set Bounds to a Science, which so evidently spreads through all Infinity, the Attributes of God, and an eternal Basis for future Hope? This Branch, or rather Body of Astronomy, I believe you will find to be quite new; and though evident Truths, are the principal Thing to be regarded in it, yet as being in its infant State, where lineal Demonstration fails, as in some Cases it cannot be otherwise, I hope you will give me Leave to make use of a weaker Way of Reasoning, to convince you of the Point in Dispute, I mean of that by the Analogy of known and natural Things. I shall be extremely unwilling to affirm any thing for a _Fact_, or Truth, without hearing, if not the real Evidence, at least a plausible Reason, next to a Conviction, or moral Certainty, along with it; and therefore I will here endeavour to explain to you what I mean by moral Certainty and also by mathematical Proof. Mathematical Proof, or Certainty, proper for Conjectures, may, to almost every Capacity, be illustrated as follows: