LETTER the SECOND.
Concerning the Nature of Mathematical Certainty, and the various Degrees of Moral Probability proper for Conjecture.
SIR,
Y
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:
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.
Plate I.
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 thought it would not be amiss to explain to you the Nature of those Steps, by which we arrive at moral Certainty, and where the Subject will admit of it, Mathematical Conviction, which will not a little contribute to strengthen many of the Arguments hereafter made use of, and in some Degree serve to supply the Place of Proof, where infallible Demonstration cannot from the Nature of the Thing be discovered.
But besides the indisputable Principles of Geometry, the universal Law of Analogy and Similitude of things, have a Privilege to assist us, in Conjectures relating to the heavenly Bodies, and though not of equal Force with the former, is often as conclusive as the Subject requires. This sort of probable Evidence (as Dr. Butler observes,) is essentially distinguished from "Demonstrative by this, that it admits of Degrees; and of all Variety of them, from the highest moral Certainty to the very lowest Presumption; and that which chiefly constitutes Probability, is expressed in the Word Likely, or Natural Likeness, as to State or Being." This general Way of arguing, I think, is allowed to be evidently natural, just and conclusive, and unquestionably to have its Weight in various Degrees, towards determining our Judgment: For Instance, should any ignorant Person, endowed with rational Principles, cut open a Pomegranate of the natural Growth of England, and finding it full of small Globules, or Kernels, upon being presented with an every way similar Fruit, said to be the Produce of Italy, doubt of its being of the same Nature, and composed of like globular Seeds within; here indeed would be no mathematical Evidence to assist the Judgment, the Object of Proof being invisible, but sure from the external Similitude, the strongest Probability of their being also internally the same. Again,
Is it natural to suppose, that the first Person who found a Lark's Nest, and in it several of the Female's Eggs, should have any Apprehensions of finding none in the Nightingale's, only because he had never seen one before, I believe the most illiterate Person of the earliest Ages, who had Curiosity enough for such a Search, would be greatly disappointed in such a Case, and far from concluding that the Nightingale had none. Farther, should any one who had seen several Sorts of Fish taken out of the River Thames, or out of the Nyle, have any sort of Suspicion that he should find no such Creatures in the Seine or the Ganges, though it should be allowed that he had never seen any such Creatures that were known to come from thence. Ocular Demonstration, in such a Case, would sure be unnecessary, and an Evidence of the first, I believe would be abundantly sufficient to convince us of what we ought to look for at least in the last: But then the Fishes of different Seas, and of Rivers are not of the same Species you'll say; but as it were infinitely diversified through all the aqueous World, this is, and must be granted, and alike Variety of Species must also be granted, in the former Case of the Birds: But no Objection can possibly arise from any such Diversity, since we don't pretend to say, nor is it at all necessary, that the Beings in the sidereal Planets should be every where the same with these of our solar System, a Variety must every where be admitted, and will always be admired, where the Work is Nature's, and the Design God's.
All then that I here pretend to argue for, is a Universality of rational Creatures to people Infinity, or rather such Parts of the Creation, as from the Analogy and Nature of Things, we judge to be habitable Seats for Beings, not unlike the mortal human.
Every Animal, and every Vegetable, that, as it were, naturally exists by the Virtues, Properties, or Laws of the mineral Kingdom, has something of a secondary Nature, depending upon it as a Principle; and to say that the Stars, which are a certain visible sort of Cotemporaries in Space with the Sun, have no like planetary Bodies with ours moving round them, because we cannot possibly see them, is no less absurd and ridiculous, than to argue, that we can have no Reason to expect to find, in the proper Season, Grapes upon every Vine—Figs upon every Tree—Roses upon every Bush—only because some of them are at such a Distance, that neither Rose, Fig or Grape, can be discovered by the Eye.
This sort of Reasoning, though some perhaps may neglect it, I am perswaded you will look upon as abundantly sufficient for Things out of the Reach of Science to determine; and that the collective Body of Stars have not been discovered, to be together a proper Subject for such Conjectures before, can surely only proceed from the Want of Time, necessary to compleat the Observations proper for a Foundation to build such an Hypothesis, or Theory upon. This is the great Article in which the Moderns have so much, and ever will have, an Advantage over the Antients. And hence it will appear, That
The Improvements and Discoveries of latter Ages are not at all owing to the greater Capacity of the Moderns, but from the Advantages received, or arising from the Inventions and Progress made by the Ancients. We at first in a manner walked by their Leading-strings, and though many of them now are broke, or useless, none can deny, but that formerly they were of great Advantage in promoting and directing philosophical Enquiries.
In an Assembly of the most eminent Men of all Ages, if we may suppose such a Conference amongst the illustrious Dead, on Purpose to deliver their several Sentiments familiarly together, on the most interesting Subjects of natural Knowledge, who would not lament the Disadvantages, poor old Thales, an Hipparchus, or a Ptolomy, would lie under, who had nothing but the Eye of Reason to direct them, in Opposition to the Judgment of a Brahe, or a Galilæus, who reaped so much Benefit from their compound Opticks? But on the other hand, perhaps if the solar System, was the Topic of Discourse, a [H]Pythagorean might very pertinently say to a Newtonian, "You have not gone much farther in the Light with our Direction, than we did in the Dark alone; for you are still roving round the same Circles." Much might be said upon this Head; but I believe it would be a difficult Matter to do Justice to all Parties: So here I intend to leave them, only must observe, that Posterity will always have the Advantage over their Predecessors; and that After-ages, in all Probability, will reap so great a Benefit from the Invention and Improvement of Fluxions, that scarce any thing, which is the immediate Object of such Enquiry, will long lie concealed from a true mathematical Genius.
[H] The true System of the Planets have been discovered above two thousand Years.
For this, in which he has surpassed all the Antients, and greatly advanced the philosophical Sciences, the World is indebted to Sir Isaac Newton.
But as many of his Discoveries, such as relate particularly to the Laws of the planetary System, are but as so many Confirmations of the Conjectures and Imaginations of Astronomers and Philosophers before him, it perhaps will not be amiss to acquaint you a little with the Astronomy of the Antients concerning the Universe. And before I proceed to those of my own, shew you in the first Place how far their Speculations in the visible Creation have been carried; and with these I shall conclude this preparatory Epistle.
The Universe, or mundane Space, by which the Antients comprehend all Creation, has, from time to time, according to the Progress of Science, come under a sort of Necessity of being variously modell'd agreeable to the Opinion of the several Authors, who have judged themselves wise enough to write upon it with a mathematical Foundation: And the cosmical System, by which is meant the Co-ordination of its constituent Parts has undergone almost as many Changes as its Elements are even capable of; every Age of the World, as Knowledge has increased, either from improved Imagination, or repeated Observations, producing something new concerning it.
Milton, no doubt, had all this Diversity of Opinions in View, as appears from his supposed Pre-knowledge of Raphael, in the following Passage, Book. VIII.
Hereafter, when they come to model Heaven,
And calculate the Stars, how they will weild
The mighty Frame! how build, unbuild, contrive
To save Appearances, how gird the Sphere
With centric and eccentric scribbl'd o'er;
Cycle, and Epicycle, Orb in Orb.
But the following Synopsis, I believe, will abundantly convince you that from certain Observations only, we ought to form all our Notions of it, if we either hope to arrive at Truth, or expect our Ideas should be supported by Reason.
Aristotle was of Opinion, that the Universe, or Heaven, was all one World, and St. Chrysostom, Tertullian, St. Bonaventure, Tycho Brahe, Longomontanus, Kepler, Bulialdus and Tellez, were of an united Opinion, that this one Heaven, or Universe, was all sidereal and fluid. But Aegidius, Hurtadus, Cisalpinus, and Aversa, believing the same Heaven with them to be all one World, and that sidereal, yet on the contrary held it to be solid.
Clemens, Acacius, Theodoret, Anastasius, Synaita, Procopius, Suidus, S. Bruno, and Claudianus Mamertus, supposed the universal mundane Space as divided into two Heavens, namely,
The Empyræum created the first Day,
And the Firmament created the second Day.
Two Heavens were also held by Justin Martyr, the one sidereal, and the other aerial. The first supposed by St. Gregory Nyssene, to be that of the fixed Stars, and the last, that of the Planets. But Mastrius and Bellutus, though agreeing in the Number of Heavens, calls one the *Primum Mobile, and the other, the Starry Heaven.
Farther, St. Basil, St. Ambrose, Damascene, Cassiodorus, Genebrardus, Suarez, Tannerus, Hurtadus, Oviedus, Tellez, and Borrus, distinguished the Universe as divided into three Portions, or Heavens.
| Or, as Cajetan. | Tho. Aquinas. | |
| The first called the Empyræum, | Watery, | |
| The second supposed Sidereal, | Sidereal, | Watery, |
| And the last of all, Aerial. | Aerial, | Sidereal. |
Again, St. Athanasius adds to those of the fix'd Stars, the Planets, and the Air, that of the Empyræum, and makes in all four Heavens.
But as the Number of the Heavens thus increases, and will become subdivided in the subsequent Account of them, to give you a better Idea of the Order of these celestial Portions of the mundane Space, it will not be amiss to form what remains of them into regular Sections of their proper Spheres and Systems.
See Plate III. in which Figure, the first represents a Section of the cosmical Theory of Oviedus and Ricciolus: Both consisting of five Heavens, viz.
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Fig. II. represents that of venerable Bede and Rabanus, viz. of Seven Heavens.
| And according to Bede composed of | But by Rabanus, | |
| The Air, | P | The Atmosphere, |
| The Æther, | O | The upper Air, |
| Olympus, | N | The inferior Fire, |
| The Element of Fire, | M | The superior Fire, |
| The Firmament, | A | Sphere of the fixed Stars, |
| The Angelical Region, | L | The Crystalline Heaven, |
| Realm of the Trinity. | K | The Empyræum. |
Fig. III. Represents the Hypotheses of Eudoxus, Plato, Calippus, Cicero, Riccius, Philo, Remigius, Aben-Ezra, Carthusianus, Lyranus, Tostatus, Brugensis, Orontius, Cremoninus, Philalethæus, Amicus, and Ruvius; also the Babylonians and Egyptians.
Consisting of Eight Heavens,
All Sidereal, viz. The Sphere of the fix'd Stars, and those of the Seven Planets.
Fig. IV. is that of Macrobius, Haly Alpetragius, Rabbi-Josue, Rabbi Moyses, Scotus, Abraham Zagutus, Sacroboscus, Claromontius, Avigra, and Arraiga.
All of Nine Heavens,
Comprehend a Primum Mobile Q, or, according to Arriaga, a solid Empyræum. The Sphere, of fixed Stars A, and the seven Regions of the solar Planets.
Fig. V. is that of the great Alphonsus, Fernelius, Regiomontanus, Amicus, Maurolycus and Langius; also of Azabel, Thebit, and Isaac Israelita; and likewise of Gulielmus Parisiensis, and Johannes Antonius Delphinus.
Consisting of Ten Heavens, made up of
| A Primum Mobile | S | Empyræum. |
| A Sphere of Tripidation in Longitude | R | Primum Mobile. |
| The Sphere of the fixed Stars | A | |
| And those of the seven solar Planets within. |
Note, Some Authors place the Sphere of Tripidation in Longitude below that of the Aplain, or Eighth Sphere.
Lastly, Fig. VI. is the Heaven of Petrus Alliacensis, the College of Conimbra, Martinensis, (and sometime) of Clavius; and also Johannes Warnerus, Leopoldus de Austriâ, Johannes Antonius Maginus; and lastly, of Clavius.
In all Eleven Heavens containing,
| T | A Primum Mobile, or, as others say, an Empyræum. |
| V | A Sphere of Libration in Latitude. |
| W | A Sphere of Libration in Longitude. |
| A | The Sphere of the fixed Stars, and those of the Planets. |
Plate III.
Thus you see how many various Opinions have from time to time been embraced concerning the Fabric and Formation of the visible Universe; all of which are now and have long been exploded; and although at first advanced by Men of the greatest Learning, and of the deepest Penetration in natural Knowledge, it does not appear from any one of their Opinions, that they had any the least Notion of infinite Space, but as it were confined the Divine Being to their limited Notions, as one may say in an Egg-shell. If therefore what I shall hereafter advance, extend so far without the known Creation, that you can possibly conceive no Bounds to the Works of infinite Wisdom and Power, I hope you will be in no Danger of looking upon it as more ridiculous, or absurd, than what so many of the wisest Men of every Age have thought proper to attempt, and have judged worthy of their Attention so long before me. If any thing less so, I shall think myself happy enough in having broke, or rather passed the narrow Limits to which the Creation has for so many Years been confined, in hopes of tempting Men of greater Talents to look up wards, and pursue so noble a Subject as far as the human Understanding is capable of comprehending it.
To the Opinions above might be added many more, particularly that of Johannes Baptista Turrianus, and Fracastorius, who increased the Number of Heavens to fourteen, viz. seven on each Side the Aplané.
But of this I have said enough; in my next I shall proceed to Matter better grounded,
And am, &c.
