doctrine of the universe, and particularly of the heavenly bodies, their laws, affections, &c. as delivered by Sir Isaac Newton.

The term Newtonian Philosophy is applied very differently; whence divers confused notions relating thereto. Some authors under this philosophy include all the corporeal philosophy, considered as its object, now stands corrected and reformed by the discoveries and improvements made in several parts thereof by Sir Isaac Newton. In which sense it is that Gravendande calls his elements of physics, Introductio ad Philosophiam Newtonianam. And in this sense the Newtonian is the same with the new philosophy; and stands distinguished from the Cartesian, the Peripatetic, and the ancient Corporeal.

Others, by Newtonian Philosophy, mean the method or order which Sir Isaac Newton observes in philosophizing; viz. the reasoning and drawing of conclusions directly from phenomena, exclusive of all previous hypotheses; the beginning from simple principles; deducing the first powers and laws of nature from a few select phenomena, and then applying those laws, &c., to account for other things. And in this sense the Newtonian philosophy is the same with the experimental philosophy, and stands opposed to the ancient Corporeal.

Others, by Newtonian philosophy, mean that wherein physical bodies are considered mathematically, and where geometry and mechanics are applied to the solution of the appearances of nature. In which sense the Newtonian is the same with the mechanical and mathematical philosophy.

Others again, by Newtonian philosophy, understand that part of physical knowledge which Sir Isaac Newton has handled, improved, and demonstrated, in his Principia.

Others, lastly, by Newtonian philosophy, mean the new principles which Sir Isaac Newton has brought into philosophy; the new system founded thereon; and the new solutions of phenomena thence deduced; Newtonian or that which characterizes and distinguishes his philosophy from all others.—Which is the sense wherein we shall chiefly consider it.

As to the history of this philosophy, we have nothing to add to what has been given in the preceding article. It was first made public in the year 1687, by the author, then a fellow of Trinity college, Cambridge; and in the year 1713, republished with considerable improvements.—Several authors have since attempted to make it plainer; by setting aside many of the more sublime mathematical researches, and substituting either more obvious reasonings or experiments in lieu thereof; particularly Whiston in his Prelect. Phys. Mathemat. Gravemane in Element. & Instit. and Dr Pemberton in his View.

The whole of the Newtonian Philosophy, as delivered by the author, is contained in his Principia, or Mathematical Principles of Natural Philosophy. He founds his system on the following definitions:

1. The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.—Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity, &c.

2. The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly. This is evident, because the motion of the whole is the motion of all its parts; and therefore in a body double in quantity, with equal velocity, the motion is double, &c.

3. The vis inertia, or innate force of matter, is a defined power of resisting, by which every body, as much as it lies, endeavours to persevere in its present state, whether it be of rest, or moving uniformly forward in a right line.—This definition is proved to be just, only by the difficulty we find in moving anything out of its place; and this difficulty is by some reckoned to proceed only from gravity. They contend, that in those cases where we can prevent the force of gravity from acting upon bodies, this power of resistance becomes insensible, and the greatest quantities of matter may be put in motion by the very least force. Thus there have been balances formed so exact, that when loaded with 200 weight in each scale, they would turn by the addition of a single drachm. In this case 400 lb. of matter was put in motion by a single drachm, i.e. by \( \frac{1}{200} \) parts of its own quantity: and even this small weight, they say, is only necessary on account of the inaccuracy of the machine; so that we have no reason to suppose, that, if the friction could be entirely removed, it would take more force to move a ton weight than a grain of sand. This objection, however, is not taken notice of by Sir Isaac; and he bestows on the resisting power above-mentioned the name of vis inertia; a phrase which is perhaps not well chosen, and with which inferior writers have endeavoured to make their readers merry at the expense of Newton. A force of inactivity, it has been said, is a forceless force; and analogous to a black white, a cold heat, and a tempestuous calm.

But objections of more importance have been made to the whole of this doctrine than those which merely respect the term vis inertia. “An endeavour to remain at rest (we are told*) is unnecessary, whilst nothing attempts to disturb the rest. It is likewise impossible to be conceived, as it implies a contradiction. Newtonian A man, by opposing force to force, may endeavour philosophy not to be moved; but this opposition is an endeavour to move, not with a design to move, but by counteracting another force to prevent being moved. An endeavour not to move therefore cannot exist in bodies, because it is absurd; and if we appeal to fact, we shall find every body in an actual and constant endeavour to move.” It has been likewise observed, and we think justly, that “if bodies could continue to move by any innate force, they might also begin to move by that force. For the same cause which can move a body with a given velocity at one time, could do it, if present, at any other time; and therefore if the force by which bodies continue in motion were innate and essential to them, they would begin to move of themselves, which is not true.” Newton indeed says that this innate force is the cause of motion under certain circumstances only, or when the body is acted upon by a force impressed ab extra. But if this impressed force do not continue as well as begin the motion, if it cease the instant that the impression is over, and the body continue to move by its vis inertia, why is the body ever stopped? “If in the beginning of the motion the body, by its innate force, overcomes a certain resistance of friction and air, in any following times, the force being undiminished, it will overcome the same resistance forever. These resistances, therefore, could never change the state of a moving body, because they cannot change the quantity of its motive force. But this is contrary to universal experience.” For these reasons we are inclined to think that bodies are wholly passive; that they endeavour nothing; and that they continue in motion not by any innate force or vis inertia, but by that force, whatever it be, which begins the motion, and which, whilst it remains with the moving body, is gradually diminished, and at last overcome by opposite forces, when the body of course ceases to move.

4. An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.—This force consists in the action only; and remains no longer in the body when the action is over. For a body maintains every new state it acquires by its vis inertia only.

It is here implied, and indeed fully expressed, that motion is not continued by the same power that produced it. Now there are two grounds on which the truth of this doctrine may be supposed to rest.

“First, On a direct proof that the impressed force does not remain in the body, either by showing the nature of the force to be transitory and incapable of more than its first action; or that it acts only on the surface, and that the body escapes from it; or that the force is somewhere else, and not remaining in the body. But none of these direct proofs are offered.

“Secondly, It may rest on an indirect proof, that there is in the nature of body a sufficient cause for the continuance of every new state acquired; and that therefore any adventitious force to continue motion, though necessary for its production, is superfluous and inadmissible. As this is the very ground on which the supposition stands, it ought to have been indubitably certain that the innate force of the body... Newtonian is sufficient to perpetuate the motion it has once acquired, before the other agent, by which the motion was communicated, had been dismissed from the office. But the innate force of body has been shown not to be that which continues its motion; and therefore the proof, that the impressed force does not remain in the body, fails. Nor indeed is it in this case desirable to support the proof, because we should then be left without any reason for the continuance of motion.* When we mention an impressed force, we mean such a force as is communicated either at the surface of the body or by being diffused through the mass.

5. A centripetal force is that by which bodies are drawn, impelled, or any way tend towards a point, as to a centre.—The quantity of any centripetal force may be considered as of three kinds, absolute, accelerative, and motive.

6. The absolute quantity of a centrifugal force is the measure of the same, proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about.

7. The accelerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time.

8. The motive quantity of a centripetal force is a measure of the same, proportional to the motion which it generates in a given time.—This is always known by the quantity of a force equal and contrary to it, that is just sufficient to hinder the descent of the body.

Scholia.

I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably, without regard to any thing external, and, by another name, is called duration. Relative, apparent, and common time, is some sensible and external measure of duration, whether accurate or not, which is commonly used instead of true time; such as an hour, a day, a month, a year, &c.

II. Absolute space, in its own nature, without regard to any thing external, remains always similar and immovable. Relative space is some moveable dimension or measure of the absolute spaces; and which is vulgarly taken for immovable space. Such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position to bodies, and which is vulgarly taken for immovable space; as the distance of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. Absolute and relative space are the same in figure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air which, relatively and in respect of the earth, remains always the same, will at one time be one part of the absolute space into which the earth passes; at another time it will be another part of the same; and so, absolutely understood, it will be perpetually mutable.

III. Place is a part of space which a body takes up; and is, according to the space, either absolute or relative. Our author says it is part of space; not the situation, nor the external surface of the body. For the places of equal solids are always equal; but their superficies, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts; that is, the translation of the whole out of its place is the same thing with the sum of the translations of the parts out of their places: and therefore the place of the whole is the same thing with the sum of the places of the parts; and for that reason it is internal, and in the whole body.

IV. Absolute motion is the translation of a body of motion from one absolute place into another, and relative motion the translation from one relative place into another. Thus, in a ship under sail, the relative place of a body is that part of the ship which the body possesses, or that part of its cavity which the body fills, and which therefore moves together with the ship; and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real absolute rest is the continuance of the body in the same part of that immovable space in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body which relatively rests in the ship will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise partly from the true motion of the earth in immovable space; partly from the relative motion of the ship on the earth; and if the body moves also relatively in the ship, its true motion will arise partly from the true motion of the earth in immovable space, and partly from the relative motions as well of the ship on the earth as of the body in the ship; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth where the ship is, was truly moved towards the east, with a velocity of 1000 parts; while the ship itself with a fresh gale is carried towards the west, with a velocity expressed by 10 of those parts; but a sailor walks in the ship towards the east with one part of the said velocity: then the sailor will be moved truly and absolutely in immovable space towards the east with a velocity of 1001 parts; and relatively on the earth towards the west, with a velocity of 9 of those parts.

Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the vulgar time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time: astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be that there is no such thing as an equable motion whereby time may be accurately measured. All motions may be accelerated or retarded; but the true or equable progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all; and therefore ought to be distinguished from what are only sensible measures thereof, and out of which we collect it by means of the astronomical equation. The necessity of which equation for determining the times of a phenomenon is evinced, as well from the experiments of the pendulum-clock as by eclipses of the satellites of Jupiter. As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if we may be allowed the expression) out of immutability of time themselves. For times and spaces are, as it were, the places of themselves as of all other things. All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places; and translations out of those places are the only absolute motions.

But because the parts of space cannot be seen, or distinguished from one another by the senses, therefore in their stead we use sensible measures of them. For, from the positions and distances of things from any body, considered as immoveable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs: but in philosophical disquisitions we ought to abstract from our senses, and consider things themselves distinct from what are only sensible measures of them. For it may be, that there is no body really at rest, to which the places and motions of others may be referred.

But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes, and effects. It is a property of rest, that bodies really at rest do rest in respect of each other. And therefore, as it is possible, that, in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest, tho' it be impossible to know from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote body; it follows, that absolute rest cannot be determined from the position of bodies in our regions.

It is a property of motion, that the parts which retain given positions to their wholes do partake of the motion of their wholes. For all parts of revolving bodies endeavour to recede from the axis of motion; and the impetus of bodies moving forwards arises from the joint impetus of all the parts. Therefore if surrounding bodies are moved, those that are relatively at rest within them will partake of their motion. Upon which account the true and absolute motion of a body cannot be determined by the translation of it from those only which seem to rest; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions; and though that translation was not made, they would not really be at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded, as the exterior part of a whole does to the interior, or as the shell does to the kernel; but if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell.

A property near akin to the preceding is, that if a place is moved, whatever is placed therein moves along with it; and therefore a body which is moved from a place in motion, partakes also of the motion of its place. Upon which account all motions from places in motion, are no other than parts of entire and absolute motions; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place; and so on, until we come to some immoveable place, as in the above-mentioned example of the sailor. Wherefore entire and absolute motions can be no otherwise determined than by immoveable places. Now, no other places are immoveable but those that from infinity to infinity do all retain the same given positions one to another; and upon this account must ever remain unmoved, and do thereby constitute what we call immoveable space.

The causes by which true and relative motions are distinguished one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved: but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that, by their giving way, that relation may be changed, in which the relative rest or motion of the other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body; but relative motion does not necessarily undergo any changes by such force. For if the same forces are likewise impressed on those other bodies with which the comparison is made, that the relative position may be preserved; then that condition will be preserved, in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true motion suffers some change. Upon which account true motion does by no means consist in such relations.

The effects which distinguish absolute from relative Absolute motion are, the forces of receding from the axis of circular motion purely relative: but, in a true and absolute circular motion, they are greater or less according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and let go, it will be whirled about the contrary way; and while the cord is untwisting itself, the surface of the water will at first be plain, as before the vessel began to move; but the vessel, by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure; and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. At first, when the relative motion in the water was greatest, it produced no endeavours. Newtonian deavour to recede from the axis; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface; and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavour to recede from the axis; and this endeavour showed the real circular motion of the water perpetually increasing, till it had acquired its greatest quantity, when the water relented relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies; nor can true circular motion be defined by such translations. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect: but relative motions in one and the same body are innumerable, according to the various relations it bears to external bodies; and, like other relations, are altogether destitute of any real effect, otherwise than they may perhaps partake of that only true motion. And therefore, in the system which supposes that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them, the several parts of those heavens and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another, which never happens to bodies truly at rest; and being carried together with the heavens, participate of their motions, and, as parts of revolving wholes, endeavour to recede from the axis of their motion.

Wherefore relative quantities are not the quantities themselves whose names they bear, but those sensible measures of them, either accurate or inaccurate, which are commonly used instead of the measured quantities themselves. And then, if the meaning of words is to be determined by their use, by the names time, space, place, and motion, their measures are properly to be understood; and the expression will be unusual and purely mathematical, if the measured quantities themselves are meant.

It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from those that are only apparent: because the parts of that immovable space in which those motions are performed, do by no means come under the observation of our senses. Yet we have some things to direct us in this intricate affair; and these arise partly from the apparent motions which are the difference of the true motions, partly from the forces which are the causes and effects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity; we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of motion, and from thence we might compute the quantity of their circular motions. And then, if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord we might infer the increment or decrement of their motions; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be Newtonian most augmented; that is, we might discover their hindrances, or those which follow in the circular motion. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible, with which the globes might be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions; we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the translation of the globes among the bodies, we should find the determination of their motions.

Having thus explained himself, Sir Isaac proposes to show how we are to collect the true motions from their causes, effects, and apparent differences; and vice versa, how, from the motions, either true or apparent, we may come to the knowledge of their causes and effects. In order to this, he lays down the following axioms or laws of motion.

1. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.—Sir Isaac's proof of this axiom is as follows: "Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts, by their cohesion, are perpetually drawn aside from rectilinear motions, does not cease its rotation otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions, both progressive and circular, for a much longer time."—Notwithstanding this demonstration, however, the axiom hath been violently disputed. It hath been argued, that bodies continue in their state of motion because they are subjected to the continual impulse of an invisible and subtile fluid, which always pours in from behind, and of which all places are full. It hath been affirmed that motion is as natural to this fluid as rest is to all other matter. It is said, moreover, that it is impossible we can know in what manner a body would be influenced by moving forces if it was entirely destitute of gravity. According to what we can observe, the momentum of a body, or its tendency to move, depends very much on its gravity. A heavy cannonball will fly to a much greater distance than a light one, though both are actuated by an equal force. It is by no means clear, therefore, that a body totally destitute of gravity would have any proper momentum of its own; and if it had no momentum, it could not continue its motion for the smallest space of time after the moving power was withdrawn. Some have imagined that matter was capable of beginning motion of itself, and consequently that the axiom was false; because we see plainly that matter in some cases hath a tendency Newtonian to change from a state of motion to a state of rest, and Philosophy from a state of rest to a state of motion. A paper appeared on this subject in the first volume of the Edinburgh Physical and Literary Essays; but the hypothesis never gained any ground.

2. The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.—Thus, if any force generates a certain quantity of motion, a double force will generate a double quantity, whether that force be impressed all at once, or in successive moments. To this law no objection of consequence has ever been made. It is founded on this self-evident truth, that every effect must be proportional to its cause. Mr Young, who seems to be very ambitious of detecting the errors of Newton, finds fault indeed with the expressions in which the law is stated; but he owns, that if thus expressed, "The alteration of motion is proportional to the actions or resistances which produce it, and is in the direction in which the actions or resistances are made," it would be unexceptionable.

3. To every action there always is opposed an equal re-action: or the mutual action of two bodies upon each other are always equal, and directed to contrary parts.—This axiom is also disputed by many. In the above-mentioned paper in the Physical Essays, the author endeavours to make a distinction between re-action and resistance; and the same attempt has been made by Mr Young. "When an action generates no motion (says he), it is certain that its effects have been destroyed by a contrary and equal action. When an action generates two contrary and equal motions, it is also evident that mutual actions were exerted, equal and contrary to each other. All cases where one of these conditions are not found, are exceptions to the truth of the law. If a finger presses against a stone, the stone, if it does not yield to the pressure, presses as much upon the finger; but if the stone yields, it reacts less than the finger acts; and if it should yield with all the momentum that the force of the pressure ought to generate, which it would do if it were not impeded by friction, or a medium, it would not react at all. So if the stone drawn by a horse, follows after the horse, it does not react so much as the horse acts; but only so much as the velocity of the stone is diminished by friction, and it is the reaction of friction only, not of the stone. The stone does not react because it does not act, it resists, but resistance is not action.

"In the loss of motion from a striking body, equal to the gain in the body struck, there is a plain solution without requiring any reaction. The motion lost, is identically that which is found in the other body; this supposition accounts for the whole phenomenon in the most simple manner. If it be not admitted, but the solution by reaction is insisted upon, it will be incumbent on the party to account for the whole effect of communication of motion; otherwise he will lie under the imputation of rejecting a solution which is simple, obvious, and perfect; for one complex, unnatural, and incomplete. However this may be determined, it will be allowed, that the circumstances mentioned, afford no ground for the inference, that action and reaction are equal, since appearances may be explained in another way" (A).

Others grant that Sir Isaac's axiom is very true in respect to terrestrial substances; but they affirm, that,

(a) If there be a perfect reciprocity betwixt an impinging body and a body at rest sustaining its impulse, may we not at our pleasure consider either body as the agent, and the other as the resisting? Let a moving body, A, pass from north to south, an equal body B at rest, which receives the stroke of A, act upon A from south to north, and A resist in a contrary direction, both inelastic: let the motion reciprocally communicated be called fix. Then B at rest communicates to A six degrees of motion towards the north, and receives six degrees towards the south. B having no other motion than the six degrees it communicated, will, by its equal and contrary loss and gain, remain in equilibrium. Let the original motion of A have been twelve, then A having received a contrary action equal to six, six degrees of its motion will be destroyed or in equilibrium; consequently, a motive force as six will remain to A towards the south, and B will be in equilibrium, or at rest. A will then endeavour to move with six degrees, or half its original motion, and B will remain at rest as before. A and B being equal masses, by the laws of communication three degrees of motion will be communicated to B, or A with its six degrees will act with three, and B will react also with three. B then will act on A from south to north equal to three, while it is acted upon or resisted by A from north to south, equal also to three, and B will remain at rest as before; A will also have its six degrees of motion reduced to one half by the contrary action of B, and only three degrees of motion will remain to A, with which it will yet endeavour to move; and finding B still at rest, the same process will be repeated till the whole motion of A is reduced to an infinitely small quantity, B all the while remaining at rest, and there will be no communication of motion from A to B, which is contrary to experience.

Let a body, A, whose mass is twelve, at rest, be impinged upon first by B, having a mass as twelve, and a velocity as four, making a momentum of 48; and secondly by C, whose mass is six, and velocity eight, making a momentum of 48 equal to B, the three bodies being inelastic. In the first case, A will become propelled of a momentum of 24, and 24 will remain to B; and, in the second case, A will become possessed of a momentum of 32, and 16 will remain to C, both bodies moving with equal velocities after the shocks, in both cases, by the laws of percussion. It is required to know, if in both cases A resists equally, and if B and C act equally? If the actions and resistances are equal, how does A in one case destroy 24 parts of B's motion, and in the other case 32 parts of C's motion, by an equal resistance? And how does B communicate in one case 24 degrees of motion, and C 32, by equal actions? If the actions and resistances are unequal, it is asked how the same mass can resist differently to bodies impinging upon it with equal momenta, and how bodies Newtonian in these, both action and re-action are the effects of Philofphy by gravity. Substances void of gravity would have no momentum; and without this they could not act; they would be moved by the least force, and therefore could not resist or react. If therefore there is any fluid which is the cause of gravity, though such fluid could act upon terrestrial substances, yet these could not react upon it; because they have no force of their own, but depend entirely upon it for their momentum. In this manner, say they, we may conceive that the planets circulate, and all the operations of nature are carried on by means of a subtile fluid; which being perfectly active, and the rest of matter altogether passive, there is neither resistance nor loss of motion. See Motion.

From the preceding axiom Sir Isaac draws the following corollaries.

1. A body by two forces conjoined will describe the diagonal of a parallelogram in the same time that it would describe the sides by those forces apart.

2. Hence we may explain the composition of any one direct force out of any two oblique ones, viz. by making the two oblique forces the sides of a parallelogram, and the direct one the diagonal.

3. The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves; because the motion which one body loses is communicated to another; and if we suppose friction and the resistance of the air to be absent, the motion of a number of bodies which mutually impelled one another would be perpetual, and its quantity always equal.

4. The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line.

5. The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forward in a right line without any circular motion. The truth of this is evidently shown by the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or proceeds uniformly forward in a straight line.

6. If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces.

The whole of the mathematical part of the Newtonian philosophy depends on the following lemmas; of which the first is the principal.

LEM. I. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before that time approach nearer the one to the other than by any given difference, become ultimately equal. If you deny it; suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is against the supposition.

Concerning the meaning of this lemma philosophers are not agreed; and unhappily it is the very fundamental position on which the whole of the system rests. Many objections have been raised to it by people who supposed themselves capable of understanding it. They say, that it is impossible we can come to an end of any infinite series, and therefore that the word ultimate can in this case have no meaning. In some cases the lemma is evidently false. Thus, suppose there are two quantities of matter A and B, the one containing half a pound, and the other a third part of one. Let both be continually divided by 2; and though their ratio, or the proportion of the one to the other, doth not vary, yet the difference between them perpetually becomes less, as well as the quantities themselves, until both the difference and quantities themselves become less than any assignable quantity; yet the difference will never totally vanish, nor the quantities become equal, as is evident from the two following series.

\[ \frac{1}{2} \frac{1}{4} \frac{1}{8} \frac{1}{16} \frac{1}{32} \frac{1}{64} \frac{1}{128} \frac{1}{256} \ldots \]

Diff. \( \frac{1}{2} \frac{1}{4} \frac{1}{8} \frac{1}{16} \frac{1}{32} \frac{1}{64} \frac{1}{128} \frac{1}{256} \ldots \)

Thus we see, that though the difference is continually diminishing, and that in a very large proportion, there is no hope of its vanishing, or the quantities becoming equal. In like manner, let us take the proportions or ratios of quantities, and we shall be equally unsuccessful. Suppose two quantities of matter, one containing 8 and the other 10 pounds; these quantities already have to each other the ratio of 8 to 10, or of 4 to 5; but let us add 2 continually to each of them, and though the ratios continually come nearer to that of equality, it is in vain to hope for a perfect coincidence. Thus,

\[ 8 \ 10 \ 12 \ 14 \ 16 \ 18 \ 20 \ 22 \ 24 \ldots \]

Ratio \( \frac{8}{10} \frac{10}{12} \frac{12}{14} \frac{14}{16} \frac{16}{18} \frac{18}{20} \frac{20}{22} \frac{22}{24} \ldots \)

For this and his other lemmas Sir Isaac makes the following apology. "These lemmas are premised, to avoid the tediousness of deducing perplexed demonstrations ad absurdum, according to the method of ancient geometers. For demonstrations are more contracted by the method of indivisibles: but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following proportions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios; and so to premise, as short as I could, the demonstrations of those limits."

For bodies possessed of equal momenta can exert different actions, it being admitted that bodies resist proportional to their masses, and that their power of overcoming resistance is proportional to their momenta?

It is incumbent on those who maintain the doctrine of universal re-action, to free it from these difficulties and apparent contradictions. For hereby the same thing is performed as by the method of indivisibles; and now those principles being demonstrated, we may use them with more safety. Therefore, if hereafter I should happen to consider quantities as made up of particles, or should use little curve lines for right ones; I would not be underlaid to mean indivisibles, but evanescent divisible quantities; not the sums and ratios of determinate parts, but always the limits of sums and ratios; and that the force of such demonstrations always depends on the method laid down in the foregoing lemmas.

"Perhaps it may be objected, that there is no ultimate proportion of evanescent quantities, because the proportion, before the quantities have vanished, is not the ultimate, and, when they are vanished, is none. But by the same argument it may be alleged, that a body arriving at a certain place, and there stopping, has no ultimate velocity; because the velocity before the body comes to the place is not its ultimate velocity; when it is arrived, it has none. But the answer is easy: for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its place and the motion ceases, nor after; but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with which they vanish. In like manner, the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented and diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed; and this is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And, since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to make use of in determining and demonstrating any other thing that is likewise geometrical.

"It may be also objected, that if the ultimate ratios of evanescent quantities are given, their ultimate magnitudes will be also given; and to all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the tenth book of his elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing continually approach."

LEM. II. If in any figure AacE (Pl. CCCXLV. n° 1.) terminated by the right line Aa, AE, and the curve acE, there be inscribed any number of parallelograms Ab, Bc, Cd, &c., comprehended under equal bases AB, BC, CD, &c., and the sides Bb, Cc, Dd, &c., parallel to one side Aa of the figure; and the parallelograms Akbl, bLcm, cMdD, &c., are completed. Then if the breadth of these parallelograms be supposed to be diminished, and their number augmented in infinitum; the ultimate ratios which the inscribed figure AkblLcMdD, the circumscribed figure AlbmcndoE, and curvilinear figure Aa

bcdE, will have to one another, are ratios of equality.

—For the difference of the inscribed and circumscribed figures is the sum of the parallelograms K/, Lm, Mn, Do; that is, (from the equality of all their bases), the rectangle under one of their bases Kb, and the sum of their altitudes Aa; that is, the rectangle Ablu. But this rectangle, because its breadth AB is supposed diminished in infinitum, becomes less than any given space. And therefore, by lem. i. the figures inscribed and circumscribed become ultimately equal the one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either.

LEM. III. The same ultimate ratios are also ratios of equality, when the breadths AB, BC, CD, &c. of the parallelograms are unequal, and are all diminished in infinitum.—The demonstration of this differs but little from that of the former.

In his succeeding lemmas, Sir Isaac goes on to prove, in a manner similar to the above, that the ultimate ratios of the sine, chord, and tangent of arcs infinitely diminished, are ratios of equality, and therefore that in all our reasonings about these we may safely use the one for the other:—that the ultimate form of evanescent triangles made by the arc, chord, and tangent, is that of finitude, and their ultimate ratio is that of equality; and hence, in reasonings about ultimate ratios, we may safely use these triangles for each other, whether made with the sine, the arc, or the tangent.—He then shows some properties of the ordinates of curvilinear figures; and proves that the spaces which a body describes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminished, are, in the very beginning of the motion, one to the other in the duplicate ratio of the powers. And, lastly, having added some demonstrations concerning the evanescence of angles of contact, he proceeds to lay down the mathematical part of his system, and which depends on the following theorems.

THEOR. I. The areas which revolving bodies describe by radii drawn to an immovable centre of force, lie in the same immovable planes, and are proportional to the times in which they are described.—For, suppose the time to be divided into equal parts, and in the first part of that time, let the body by its innate force describe the right line AB (n° 2.); in the second part of that time, the same would, by law i. if not hindered, proceed directly to c along the line Bc = AB; so that by the radii AS, BS, cS, drawn to the centre, the equal areas ASB, BSc, would be described. But, when the body is arrived at B, suppose the centripetal force acts at once with a great impulse, and, turning aside the body from the right line Bc, compels it afterwards to continue its motion along the right line BC. Draw cC parallel to BS, meeting BC in C; and at the end of the second part of the time, the body, by cor. i. of the laws, will be found in C, in the same plane with the triangle ASB. Join SC; and because SB and cC are parallel, the triangle SBC will be equal to the triangle SBC, and therefore also to the triangle SAB. By the like argument, if the centripetal force acts successively in C, D, E, &c. and makes the body in each single particle of time to describe the right lines CD, DE, EF, Newtonian EF, &c. they will all lie in the same plane; and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore, in equal times, equal areas are described in one immoveable plane; and, by composition, any sums SADS, SAFS, of those areas are, one to the other, as the times in which they are described. Now, let the number of those triangles be augmented, and their size diminished in infinitum; and then, by the preceding lemmas, their ultimate perimeter ADF will be a curve line; and therefore the centripetal force by which the body is perpetually drawn back from the tangent of this curve will act continually; and any described areas SADS, SAFS, which are always proportional to the times of description, will, in this case also, be proportional to those times Q. E. D.

Cor. 1. The velocity of a body attracted towards an immoveable centre, in spaces void of resistance, is reciprocally as the perpendicular let fall from that centre on the right line which touches the orbit. For the velocities in these places A, B, C, D, E, are as the bases AB, BC, DE, EF, of equal triangles; and these bases are reciprocally as the perpendiculars let fall upon them.

Cor. 2. If the chords AB, BC, of two arcs successively described in equal times by the same body, in spaces void of resistance, are completed into a parallelogram ABCV, and the diagonal BV of this parallelogram, in the position which it ultimately acquires when those arcs are diminished in infinitum, is produced both ways, it will pass through the centre of force.

Cor. 3. If the chords AB, BC, and DE, EF, of arcs described in equal times, in spaces void of resistance, are completed into the parallelograms ABCV, DEFZ, the forces in B and E are one to the other in the ultimate ratio of the diagonals BV, EZ, when those arcs are diminished in infinitum. For the motions BC and EF of the body (by cor. 1. of the laws), are compounded of the motions Be, BV and Ef, EZ; but BV and EZ, which are equal to Ce and Ff, in the demonstration of this proposition, were generated by the impulses of the centripetal force in B and E, and are therefore proportional to those impulses.

Cor. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed sines of arcs described in equal times; which versed sines tend to the centre of force, and bisect the chords when these arcs are diminished to infinity. For such versed sines are the halves of the diagonals mentioned in cor. 3.

Cor. 5. And therefore those forces are to the force of gravity, as the said versed sines to the versed sines perpendicular to the horizon of those parabolic arcs which projectiles describe in the same time.

Cor. 6. And the same things do all hold good (by cor. 5. of the laws) when the planes in which the bodies are moved, together with the centres of force, which are placed in those planes, are not at rest, but move uniformly forward in right lines.

Theor. II. Every body that moves in any curve line described in a plane, and, by a radius drawn to a point either immoveable or moving forward with an uniform rectilinear motion, describes about that point Newtonian areas proportional to the times, is urged by a centripetal force directed to that point.

Case I. For every body that moves in a curve line is (by law 1.) turned aside from its rectilinear course by the action of some force that impels it; and that force by which the body is turned off from its rectilinear course, and made to describe in equal times the least equal triangles SAB, SBC, SCD, &c. about the immoveable point S, (by Prop. 40. E. 1. and law 2.) acts in the place B according to the direction of a line parallel to C; that is, in the direction of the line BS; and in the place C according to the direction of a line parallel to dD, that is, in the direction of the line CS, &c.; and therefore acts always in the direction of lines tending to the immoveable point S. Q. E. D.

Case II. And (by cor. 5. of the laws) it is indifferent whether the superficies in which a body describes a curvilinear figure be quiescent, or moves together with the body, the figure described, and its point S, uniformly forward in right lines.

Cor. 1. In non-resisting spaces or mediums, if the areas are not proportional to the times, the forces are not directed to the point in which the radii meet; but deviate therefrom in consequentia, or towards the parts to which the motion is directed, if the description of the areas is accelerated; but in antecedentia if retarded.

Cor. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends.

Scholium. A body may be urged by a centripetal force compounded of several forces. In which case the meaning of the proposition is, that the force which results out of all tends to the point S. But if any force acts perpetually in the direction of lines perpendicular to the described surface, this force will make the body to deviate from the plane of its motion, but will neither augment nor diminish the quantity of the described surface; and is therefore not to be neglected in the composition of forces.

Theor. III. Every body that, by a radius drawn to the centre of another body, howsoever moved, describes areas about that centre proportional to the times, is urged by a force compounded of the centripetal forces tending to that other body, and of all the accelerative force by which that other body is impelled.—The demonstration of this is a natural consequence of the theorem immediately preceding.

Hence, if the one body L, by a radius drawn to the other body T, describes areas proportional to the times, and from the whole force by which the first body L is urged, (whether that force is simple, or, according to cor. 2. of the laws, compounded of several forces), we subduct that whole accelerative force by which the other body is urged; the whole remaining force by which the first body is urged will tend to the other body T, as its centre.

And vice versa, if the remaining force tends nearly to the other body T, those areas will be nearly proportional to the times. If the body L, by a radius drawn to the other body T, describes areas, which, compared with the times, are very unequal, and that other body T be either at rest, or moves uniformly forward in a right line, the action of the centripetal force tending to that other body T is either none at all, or it is mixed and combined with very powerful actions of other forces; and the whole force compounded of them all, if they are many, is directed to another (immoveable or moveable) centre. The same thing obtains when the other body is actuated by any other motion whatever; provided that centripetal force is taken which remains after subduing that whole force acting upon that other body T.

**Scholium.**

Because the equable description of areas indicates that a centre is reflected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit, we may always be allowed to use the equable description of areas as an indication of a centre about which all circular motion is performed in free spaces.

**Theor. IV.** The centripetal forces of bodies which by equable motions describe different circles, tend to the centres of the same circles; and are one to the other as the squares of the arcs described in equal times applied to the radii of circles.—For these forces tend to the centres of the circles, (by theor. 2 and cor. 2, theor. 1.) and are to one another as the versed sines of the least arcs described in equal times, (by cor. 4, theor. 1.) that is, as the squares of the same arcs applied to the diameters of the circles, by one of the lemmas; and therefore, since those arcs are as arcs described in any equal times, and the diameters are as the radii, the forces will be as the squares of any arcs described in the same time, applied to the radii of the circles. Q. E. D.

Cor. 1. Therefore, since those arcs are as the velocities of the bodies, the centripetal forces are in a ratio compounded of the duplicate ratio of the velocities directly, and of the simple ratio of the radii inversely.

Cor. 2. And since the periodic times are in a ratio compounded of the ratio of the radii directly, and the ratio of the velocities inversely; the centripetal forces are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inversely.

Cor. 3. Whence, if the periodic times are equal, and the velocities therefore as the radii, the centripetal forces will be also as the radii; and the contrary.

Cor. 4. If the periodic times and the velocities are both in the subduplicate ratio of the radii, the centripetal forces will be equal among themselves; and the contrary.

Cor. 5. If the periodic times are as the radii, and therefore the velocities equal, the centripetal forces will be reciprocally as the radii; and the contrary.

Cor. 6. If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radii inversely; and the contrary.

Cor. 7. And universally, if the periodic time is as any power \( R^n \) of the radius R, and therefore the velocity reciprocally as the power \( R^{n-1} \) of the radius, the centripetal force will be reciprocally as the power \( R^{n-2} \) of the radius; and the contrary.

Cor. 8. The same things all hold concerning the times, the velocities, and forces, by which bodies describe the similar parts of any similar figures, that have their centres in a similar position within those figures, as appears by applying the demonstrations of the preceding cases to those. And the application is easy, by only substituting the equable description of areas in the place of equable motion, and using the distances of the bodies from the centres instead of the radii.

Cor. 9. From the same demonstration it likewise follows, that the arc which a body uniformly revolving in a circle by means of a given centripetal force describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body, falling by the same given force, would descend through in the same given time.

"By means of the preceding proposition and its corollaries (says Sir Isaac), we may discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by cor. 9. of this theorem). And by such propositions Mr Huygens, in his excellent book *De Horologio Oscillatorio*, has compared the force of gravity with the centrifugal forces of revolving bodies.

The preceding proposition may also be demonstrated in the following manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points; the force with which, at every reflection it strikes the circle, will be as its velocity: and therefore the sum of the forces, in a given time, will be as that velocity and the number of reflections conjunctly; that is, (if the species of the polygon be given), as the length described in that given time, and increased or diminished in the ratio of the same length to the radius of the circle; that is, as the square of that length applied to the radius; and therefore, if the polygon, by having its sides diminished in infinitum, coincides with the circle, as the square of the arc described in a given time applied to the radius. This is the centrifugal force, with which the body impels the circle; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal.

On these principles hangs the whole of Sir Isaac Newton's mathematical philosophy. He now shows how to find the centre to which the forces impelling any body are directed, having the velocity of the body given: and finds the centrifugal force to be always as the versed sine of the inscendent arc directly, and as the square... square of the time inversely; or directly as the square of the velocity, and inversely as the chord of the nascent arc. From these premises he deduces the method of finding the centripetal force directed to any given point when the body revolves in a circle; and this whether the central point is near or at an immense distance; so that all the lines drawn from it may be taken for parallels. The same thing he shows with regard to bodies revolving in spirals, ellipses, hyperbolas, or parabolas—Having the figures of the orbits given, he shows also how to find the velocities and moving powers; and, in short, solves all the most difficult problems relating to the celestial bodies with an astonishing degree of mathematical skill. These problems and demonstrations are all contained in the first book of the Principia; but to give an account of them here would far exceed our limits; neither would many of them be intelligible, excepting to first-rate mathematicians.

In the second book, Sir Isaac treats of the properties of fluids, and their powers of resistance; and here he lays down such principles as entirely overthrow the doctrine of Des Cartes's vortices, which was the fashionable system in his time. In the third book, he begins particularly to treat of the natural phenomena, and apply them to the mathematical principles formerly demonstrated; and, as a necessary preliminary to this part, he lays down the following rules for reasoning in natural philosophy.

1. We are to admit no more causes of natural things than such as are both true and sufficient to explain their natural appearances.

2. Therefore to the same natural effects we must always assign, as far as possible, the same causes.

3. The qualities of bodies which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

4. In experimental philosophy, we are to look upon propositions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.

The phenomena first considered, are, 1. That the satellites of Jupiter, by radii drawn to the centre of their primary, describe areas proportional to the times of their description; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate ratio of their distances from its centre. 2. The same thing is likewise observed of the phenomena of Saturn. 3. The five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, with their several orbits, encompass the sun. 4. The fixed stars being supposed at rest, the periodic times of the five primary planets, and of the earth, about the sun, are in the sesquiplicate proportion of their mean distances from the sun. 5. The primary planets, by radii drawn to the earth, describe areas no ways proportional to the times; but the areas which they describe by radii drawn to the sun are proportional to the times of description. 6. The moon, by a radius drawn to the centre of the earth, describes an area proportional to the time of description. All these phenomena are undeniable from astronomical observations, and are explained at large under the article Astronomy. The mathematical demonstrations are next applied by Sir Isaac Newton in the following propositions.

Prop. I. The forces by which the satellites of Jupiter are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the centre of that planet; and are reciprocally as the squares of the distances of those satellites from that centre. The former part of this proposition appears from theor. 2. or 3. and the latter from cor. 6. of theor. 5.; and the same thing we are to understand of the satellites of Saturn.

Prop. II. The forces by which the primary planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the sun; and are reciprocally as the squares of the distances from the sun's centre. The former part of this proposition is manifest from phenomenon 5., just mentioned, and from theor. 2.; the latter from phenomenon 4., and cor. 6. of theor. 4. But this part of the proposition is with great accuracy deducible from the quiescence of the aphelion points. For a very small aberration from the reciprocal duplicate proportion would produce a motion of the aphides, sensible in every single revolution, and in many of them enormously great.

Prop. III. The force by which the moon is retained in its orbit, tends towards the earth; and is reciprocally as the square of the distance of its place from the centre of the earth. The former part of this proposition is evident from phenom. 5. and theor. 2.; the latter from phenom. 6. and theor. 2. or 3. It is also evident from the very slow motion of the moon's apogee; which, in every single revolution, amounting but to $3^\circ$ $3'$ in consequentia, may be neglected; and this more fully appears from the next proposition.

Prop. IV. The moon gravitates towards the earth, and by the force of gravity is continually drawn off from a rectilinear motion, and retained in its orbit.—The mean distance of the moon from the earth in the syzygies in semidiameters of the latter, is about 60. Let us assume the mean distance of 60 semidiameters in the syzygies; and suppose one revolution of the moon in respect of the fixed stars to be completed in $27\frac{1}{3}$ hours, $43'$, as astronomers have determined; and the circumference of the earth to amount to $123,249,600$ Paris feet. Now, if we imagine the moon, deprived of all motion, to be let go, so as to descend towards the earth with the impulse of all that force by which it is retained in its orbit, it will, in the space of one minute of time, describe in its fall $15\frac{1}{3}$ Paris feet. For the verified fine of that arc which the moon, in the space of one minute of time, describes by its mean motion at the distance of 60 semidiameters of the earth, is nearly $15\frac{1}{3}$ Paris feet; or more accurately, $15$ feet $1$ inch and one line $\frac{1}{3}$. Wherefore since that force, in approaching to the earth, increases in the reciprocal duplicate proportion of the distance; and, upon that account, at the surface of the earth is $60\times60$ times greater than at the moon; a body in our regions, falling with that force, ought, in the space of one minute of time, to describe $60\times60\times15\frac{1}{3}$ Paris feet; and in the space of one second of time to describe $15\frac{1}{3}$ of those feet; or, more accurately, $15$ feet $1$ inch, And with this very force we actually find that bodies here on earth do really descend.

For a pendulum oscillating seconds in the latitude of Paris, will be three Paris feet and 8½ lines in length, as Mr Huygens has observed. And the space which a heavy body describes by falling one second of time, is to half the length of the pendulum in the duplicate ratio of the circumference of the circle to its diameter; and is therefore 15 Paris feet, 1 inch, 1 line 3½.

And therefore the force by which the moon is retained in its orbit, becomes, at the very surface of the earth, equal to the force of gravity which we observe in heavy bodies there. And therefore (by rule 1. and 2.) the force by which the moon is retained in its orbit is that very same force which we commonly call gravity. For were gravity another force different from that, then bodies descending to the earth with the joint impulse of both forces, would fall with a double velocity, and, in the space of one second of time, would describe 30½ Paris feet; altogether against experience.

The demonstration of this proposition may be more distinctly explained after the following manner. Suppose several moons to revolve about the earth, as in the system of Jupiter or Saturn, the periodic times of those moons would (by the argument of induction) observe the same law which Kepler found to obtain among the planets; and therefore their centripetal forces would be reciprocally as the squares of the distances from the centre of the earth, by Prop. I. Now, if the lowest of these were very small, and were so near the earth as almost to touch the tops of the highest mountains, the centripetal force thereof, retaining it in its orbit, would be very nearly equal to the weights of any terrestrial bodies that should be found upon the tops of these mountains; as may be known from the foregoing calculation. Therefore, if the same little moon should be deflected by its centrifugal force that carries it through its orbit, it would descend to the earth; and that with the same velocity as heavy bodies do actually descend with upon the tops of those very mountains, because of the equality of forces that oblige them both to descend. And if the force by which that lowest moon would descend were different from that of gravity, and if that moon were to gravitate towards the earth, as we find terrestrial bodies do on the tops of mountains, it would then descend with twice the velocity, as being impelled by both these forces conspiring together. Therefore, since both these forces, that is, the gravity of heavy bodies, and the centripetal forces of the moons, respect the centre of the earth, and are similar and equal between themselves, they will (by rule 1. and 2.) have the same cause. And therefore the force which retains the moon in its orbit, is that very force which we commonly call gravity; because otherwise, this little moon at the top of a mountain must either be without gravity, or fall twice as swiftly as heavy bodies use to do.

Having thus demonstrated that the moon is retained in its orbit by its gravitation towards the earth, it is easy to apply the same demonstration to the motions of the other secondary planets, and of the primary planets round the sun, and thus to show that gravitation prevails throughout the whole creation; after which, Sir Isaac proceeds to show from the same principles, that the heavenly bodies gravitate towards each other, and contain different quantities of matter, or have different densities in proportion to their bulks.

Prop. V. All bodies gravitate towards every planet; and the weights of bodies towards the same planet, at equal distances from its centre, are proportional to the quantities of matter they contain.

It has been confirmed by many experiments, that all sorts of heavy bodies (allowance being made for the inequality of retardation by some small resistance of the air) descend to the earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy by the help of pendulums. Sir Isaac Newton tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. He provided two wooden boxes, round and equal, filled the one with wood, and suspended an equal weight of gold in the centre of oscillation of the other. The boxes hanging by equal threads of 11 feet, made a couple of pendulums, perfectly equal in weight and figure, and equally receiving the resistance of the air. And placing the one by the other, he observed them to play together forwards and backwards, for a long time, with equal vibrations. And therefore the quantity of matter in the gold was to the quantity of matter in the wood, as the action of the motive force (or vis matrix) upon all the gold, to the action of the same upon all the wood; that is, as the weight of the one to the weight of the other. And the like happened in the other bodies. By these experiments, in bodies of the same weight, he could manifestly have discovered a difference of matter less than the thousandth part of the whole, had any such been. But, without all doubt, the nature of gravity towards the planets, is the same as towards the earth. For, should we imagine our terrestrial bodies removed to the orb of the moon, and there, together with the moon, deprived of all motion, to be let go, so as to fall together towards the earth; it is certain, from what we have demonstrated before, that, in equal times, they would describe equal spaces with the moon, and of consequence are to the moon, in quantity of matter, as their weights to its weight. Moreover, since the satellites of Jupiter perform their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter's centre, their accelerative gravities towards Jupiter will be reciprocally as the squares of their distances from Jupiter's centre; that is, equal at equal distances. And therefore, these satellites, if supposed to fall towards Jupiter from equal heights, would describe equal spaces in equal times, in like manner as heavy bodies do on our earth. And by the same argument, if the circumfolar planets were supposed to be let fall at equal distances from the sun, they would, in their descent towards the sun, describe equal spaces in equal times. But forces, which equally accelerate unequal bodies, must be as those bodies; that is to say, the weights of the planets towards the sun must be as their quantities of matter. Further, that the weights of Jupiter and of his satellites towards the sun are proportional to the several quantities of their matter, appears from the exceeding regular motions of the satellites. For if some of those bodies were more strongly attracted to the sun in proportion to their quantity of matter than others, the motions of the satellites would be disturbed by that inequality of attraction. If, at equal distances from the sun, any satellite, in proportion to the quantity of its matter, did gravitate towards the sun, with a force greater than Jupiter in proportion to his, according to any given proportion, suppose of $d$ to $e$; then the distance between the centres of the sun and of the satellite's orbit would be always greater than the distance between the centres of the sun and of Jupiter nearly in the subduplicate of that proportion. And if the satellite gravitated towards the sun with a force less in the proportion of $e$ to $d$, the distance of the centre of the satellite's orbit from the sun would be less than the distance of the centre of Jupiter's from the sun in the subduplicate of the same proportion. Therefore, if, at equal distances from the sun, the accelerative gravity of any satellite towards the sun were greater or less than the accelerating gravity of Jupiter towards the sun but by $\frac{1}{1000}$ part of the whole gravity; the distance of the centre of the satellite's orbit from the sun would be greater or less than the distance of Jupiter from the sun by $\frac{1}{1000}$ part of the whole distance; that is, by a fifth part of the distance of the utmost satellite from the centre of Jupiter; an eccentricity of the orbit which would be very sensible. But the orbits of the satellites are concentric to Jupiter; therefore the accelerative gravities of Jupiter, and of all its satellites, towards the sun, are equal among themselves. And by the same argument, the weight of Saturn and of his satellites towards the sun, at equal distances from the sun, are as their several quantities of matter; and the weights of the moon and of the earth towards the sun, are either none, or accurately proportional to the masses of matter which they contain.

But further, the weights of all the parts of every planet towards any other planet are one to another as the matter in the several parts. For if some parts gravitated more, others less, than in proportion to the quantity of their matter; then the whole planet, according to the sort of parts with which it most abounds, would gravitate more or less than in proportion to the quantity of matter in the whole. Nor is it any moment whether these parts are external or internal. For if, as an instance, we should imagine the terrestrial bodies with us to be raised up to the orb of the moon, to be there compared with its body; if the weights of such bodies were to the weights of the external parts of the moon as the quantities of matter in the one and in the other respectively, but to the weights of the internal parts in a greater or less proportion; then likewise the weights of those bodies would be to the weight of the whole moon in a greater or less proportion; against what we have shewed above.

Cor. 1. Hence the weights of bodies do not depend upon their forms and textures. For if the weights could be altered with the forms, they would be greater or less, according to the variety of forms in equal matter; altogether against experience.

Cor. 2. Universally, all bodies about the earth gravitate towards the earth; and the weights of all, at equal distances from the earth's centre, are as the quantities of matter which they severally contain. This is the quality of all bodies within the reach of our experiments; and therefore (by rule 5.) to be affirmed of all bodies whatsoever. If ether, or any other body, were either altogether void of gravity, or were to gravitate less in proportion to its quantity of matter; then, because (according to Aristotle, Des Cartes, and Newtonian others) there is no difference betwixt that and other philosophy bodies, but in mere form of matter, by a successive change from form to form, it might be changed at last into a body of the same condition with those which gravitate most in proportion to their quantity of matter; and, on the other hand, the heaviest bodies, acquiring the first form of that body, might by degrees quite lose their gravity. And therefore the weights would depend upon the forms of bodies, and with those forms might be changed, contrary to what was proved in the preceding corollary.

Cor. 3. All spaces are not equally full. For if all spaces were equally full, then the specific gravity of the fluid which fills the region of the air, on account of the extreme density of the matter, would fall nothing short of the specific gravity of quick-silver or gold, or any other the most dense body; and therefore, neither gold, nor any other body, could descend in air. For bodies do not descend in fluids, unless they are specifically heavier than the fluids. And if the quantity of matter in a given space can by any rarefaction be diminished, what should hinder a diminution to infinity?

Cor. 4. If all the solid particles of all bodies are of the same density, nor can be rarefied without pores, a void space or vacuum must be granted. [By bodies of the same density, our author means those whose vis inertiae are in the proportion of their bulks.]

Prop. VI. That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.

That all the planets mutually gravitate one towards another, we have proved before; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence it follows, that the gravity tending towards all the planets is proportional to the matter which they contain.

Moreover, since all the parts of any planet $A$ gravitate towards any other planet $B$, and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole; and (by law 3.) to every action corresponds an equal re-action; therefore the planet $B$ will, on the other hand, gravitate towards all the parts of the planet $A$; and its gravity towards any one part will be to the gravity towards the whole, as the matter of the part to the matter of the whole. Q.E.D.

Cor. 1. Therefore the force of gravity towards any whole planet, arises from, and is compounded of, the forces of gravity towards all its parts. Magnetic and electric attractions afford us examples of this. For all attraction towards the whole arises from the attractions towards the several parts. The thing may be easily understood in gravity, if we consider a greater planet as formed of a number of lesser planets, meeting together in one globe. For hence it would appear that the force of the whole must arise from the forces of the component parts. If it be objected, that, according to this law, all bodies with us must mutually gravitate one towards another, whereas no such gravitation anywhere appears: it is answered, that, since the gravitation towards these bodies is to the gravitation towards the whole earth, as these bodies are to the whole earth, the gravitation towards them must be far less than to NEW

Cor. 2. The force of gravity towards the several equal particles of any body, is reciprocally as the square of the distance of places from the particles.

Prop. VII. In two spheres mutually gravitating each towards the other; if the matter, in places on all sides round about and equidistant from the centres, is similar; the weight of either sphere towards the other will be reciprocally as the square of the distance between their centres.

For the demonstration of this, see the Principia, book i. prop. 75. and 76.

Cor. 1. Hence we may find and compare together the weights of bodies towards different planets. For the weights of bodies revolving in circles about planets are as the diameters of the circles directedly, and the squares of their periodic times reciprocally; and their weights at the surfaces of the planets, or at any other distances from their centres, are (by this prop.) greater or less, in the reciprocal duplicate proportion of the distances. Thus from the periodic times of Venus, revolving about the sun, in 224d. 16h33m; of the utmost circumjovial satellite revolving about Jupiter, in 163d. 16h33m; of the Huygenian satellite about Saturn in 133d. 22h43m; and of the moon about the earth in 27d. 7h43m; compared with the mean distance of Venus from the sun, and with the greatest heliocentric elongations of the utmost circumjovial satellite from Jupiter's centre, 8'16''; of the Huygenian satellite from the centre of Saturn, 3'4''; and of the moon from the earth, 10'33''; by computation our author found, that the weight of equal bodies, at equal distances from the centres of the sun, of Jupiter, of Saturn, and of the earth, towards the sun, Jupiter, Saturn, and the earth, were one to another as 1/06797, 1/07971, and 1/109 respectively.

Then, because as the distances are increased or diminished, the weights are diminished or increased in a duplicate ratio; the weights of equal bodies towards the sun, Jupiter, Saturn, and the earth, at the distances 10000, 997, 791, and 109, from their centres, that is, at their very surfaces, will be as 10000, 943, 579, and 43 respectively.

Cor. 2. Hence likewise we discover the quantity of matter in the several planets. For their quantities of matter are as the forces of gravity at equal distances from their centres; that is, in the sun, Jupiter, Saturn, and the earth, as 1/06797, 1/07971, and 1/109 respectively. If the parallax of the sun be taken greater or less than 10'' 30'', the quantity of matter in the earth must be augmented or diminished in the triplicate of that proportion.

Cor. 3. Hence also we find the densities of the planets. For (by prop. 72. book 1.) the weights of equal and similar bodies towards similar spheres, are, at the surfaces of those spheres, as the diameters of the spheres. And therefore the densities of dissimilar spheres are as those weights applied to the diameters of the spheres. But the true diameters of the sun, Jupiter, Saturn, and the earth, were one to another as 10000, 997, 791, and 109; and the weights towards the same, as 10000, 943, 579, and 43 respectively; and therefore their densities are as 100, 943, 67, and 43. The density of the earth, which comes out by this computation, does not depend upon the parallax of the sun, but is determined by the parallax of the moon, and therefore is here truly defined. The sun therefore is a little denser than Jupiter, and Jupiter than Saturn, and the earth four times denser than the sun; for the sun, by its great heat, is kept in a sort of a rarefied state. The moon also is denser than the earth.

Cor. 4. The smaller the planets are, they are,憔tteris paribus, of so much the greater density. For so the powers of gravity on their several surfaces come nearer to equality. They are likewise,憔tteris paribus, of the greater density as they are nearer to the fun. So Jupiter is more dense than Saturn, and the earth than Jupiter. For the planets were to be placed at different distances from the sun, that, according to their degrees of density, they might enjoy a greater, or less proportion of the sun's heat. Our water, if it were removed as far as the orb of Saturn, would be converted into ice, and in the orb of Mercury would quickly fly away in vapour. For the light of the sun, to which its heat is proportional, is seven times denser in the orb of Mercury than with us: and by the thermometer Sir Isaac found, that a sevenfold heat of our summer-pan will make water boil. Nor are we to doubt, that the matter of Mercury is adapted to its heat, and is therefore more dense than the matter of our earth; since, in a denser matter, the operations of nature require a stronger heat.

It is shown in the scholium of prop. 22. book 2. of the Principia, that, at the height of 200 miles above the earth, the air is more rare than it is at the superficies of the earth, in the ratio of 30 to 0,000000000000003998, or as 7,300,000,000,000 to 1 nearly. And hence the planet Jupiter, revolving in a medium of the same density with such superior air, would not lose by the resistance of the medium the 1000,000th part of its motion in 100,000 years. In the spaces near the earth, the resistance is produced only by the air, exhalations, and vapours. When these are carefully exhausted by the air-pump from under the receiver, heavy bodies fall within the receiver with perfect freedom, and without the least sensible resistance; gold itself, and the lightest down, let fall together, will descend with equal velocity; and though they fall through a space of four, six, and eight feet, they will come to the bottom at the same time; as appears from experiments that have often been made. And therefore the celestial regions being perfectly void of air and exhalations, the planets and comets meeting no sensible resistance in those spaces, will continue their motions through them for an immense space of time.

Newton (Richard) D D. the founder of Hertford college, is a man of whom we regret that we can give but a superficial and rather a vague account. By one writer he is said to have been a Northamptonshire gentleman; by another, we are told that his father enjoyed at Lavendon Grange in Bucks a moderate estate, which is still in the family, though he lived in a house of Lord Northampton's in Yardley-Chace, where in 1675 our doctor was born. All agree that the family from which he sprung had long been respectable, though its fortunes had been much injured during the great rebellion.

The subject of this article was educated at Westminster school, and from that foundation elected to a scholarship of Christ-church, Oxford. At what age he was admitted into the university we have no certain information; but in the list of graduates he is thus distinguished: "Newton (Richard,) Christ-church, M.A. April 12th 1701; B.D. March 18th 1707; Hart-hall, D.D. December 7th 1710." He was appointed a tutor in Christ-church as soon as he was of the requisite standing in his college, and discharged the duties of that important office with honour to himself and advantage to the society of which he was a member. From Oxford he was called (we know not at what precise period) into Lord Pelham's family to superintend the education of the late duke of Newcastle and his brother Mr Pelham; and by both these illustrious persons he was ever remembered with the most affectionate regard. In 1710 he was by Dr Aldrich, the celebrated dean of Christ-church, inducted principal of Hart-hall, which was then an appendage to Exeter college. From this state of dependance Dr Newton wrote it against much opposition, especially from the learned Dr Conybeare, afterwards dean of Christ-church and bishop of Bristol. In no contest, it has been observed, were ever two men more equally matched; and the papers that passed between them, like Junius's letters, deserved to be collected for the energetic beauty of their style and the ingenuity of their arguments. Dr Newton, however, proved successful; and in 1743 obtained a charter, converting Hart-hall into Hertford college; of which, at a considerable expense to himself, and with great aid from his numerous friends, he was thus the founder and first head.

Though this excellent man was Mr Pelham's tutor, and, if report be true, had by him been more than once employed to furnish king's speeches, he never received the smallest preferment from his pupil when first minister; and when that statesman was asked why he did not place in a proper station the able and meritorious Dr Newton? his reply was, "How could I do it? he never asked me." He was not, however, neglected by all the great. Dr Compton, bishop of London, who had a just sense of his merits, had, at an early period of his life, collated him to the rectory of Sudbury in the county of Northampton, which he held together with the headship of Hart-hall. He resided for some years on that living, and discharged all the parts of his office with exemplary care and fidelity. Amongst other particulars he read the prayers of the liturgy in his church at seven o'clock in the evening of every week-day (hay-time and harvest excepted), for the benefit of such of his parishioners as could then assemble for public devotion. When he left the place, returning again to Oxford about 1724, he enjoined his curates to observe the same pious practice; and was fortunate enough to have three successively who trod in the steps of their worthy principal. Being always an enemy to pluralities with cure of souls, he exerted his utmost endeavours from time to time with Dr Gibbon, Bishop Compton's successor in the see of London, for leave to resign his rectory in favour of his curate. To the resignation his lordship could have no objection; but being under some kind of engagement to confer the living on another, Dr Newton retained it himself, but bestowed all the emoluments upon works of charity in the parish, and curates who so faithfully discharged their duty. Dr Sherlock, who succeeded Bishop Gibbon, being under no engagement of a like nature, very readily granted Dr Newton's request, by accepting his resignation, and collating to the rectory Mr Saunders, who was the last of his curates. Upon a vacancy of the public orator's place at Oxford, the head of Hertford college offered himself a candidate; but as the race is not always to the swift nor the battle to the strong, Dr Digby Coates carried the point against him. He was afterwards promoted to a canonry of Christ-church, but did not long enjoy it; for in April 1753 death deprived the world of this excellent man in the 78th year of his age.

He was allowed to be as polite a scholar, and as accomplished a gentleman, as almost any of the age in which he lived. In eloquence of argument, and perspicuity of style, he had no superior. Never was any private person employed in more truths, nor were truths ever discharged with greater integrity. He was a zealous friend to religion, the university, the clergy, and the poor; and such was his liberality of sentiment, that he admitted to his friendship every man, whatever might be his religious creed, who was earnestly employed in the same good works with himself—the promotion of virtue and unaffected piety. Of his works we have seen only his Theobaphes, which was published after his death; and his Pluralites Indefensibiles; but he published several other things during his life, and left a volume of sermons prepared for the press at his death.

Newton (Thomas), late lord bishop of Bristol and dean of St Paul's, London, was born on the first of January 1704. His father, John Newton, was a considerable brandy and cider merchant, who, by his industry and integrity, having acquired what he thought a competent fortune, left off trade several years before he died.

He received the first part of his education in the free school of Litchfield; a school which, the bishop observes with some kind of exultation, had at all times sent forth several persons of note and eminence; from Bishop Smaldrige and Mr Wollaston, to Dr Johnson and Mr Garrick.

From Litchfield he was removed to Westminster school, in 1717, under the care of Dr Friend and Dr Nicoll.

During the time he was at Westminster, there were, he observes, more young men who made a distinguished figure afterwards in the world, than perhaps at any other period, either before or since. He particularly mentions William Murray, the late earl of Mansfield, with whom he lived on terms of the highest friendship to the last.

He continued six years at Westminster school, five of which he passed in the college. He afterwards went to Cambridge, and entered at Trinity college. Here he constantly resided eight months at least in every year, till he had taken his Bachelor of Arts degree. Being chosen Fellow of his college, he came afterwards to settle in London. As it had been his inclination from a child, and as he was also designed for holy orders, he had sufficient time to prepare himself, and composed some sermons, that he might have a stock in hand when he entered on the ministry. His title for orders was his fellowship; and he was ordained deacon in December 1729, and priest in the February following, by Bishop Gibbon. At his first setting out in his office, he was curate at St George's, Hanover-square; and continued for several years assistant-preacher to Dr Trebeck. His first preferment was that of reader and afternoon-preacher at Grosvenor chapel, in South-Audley street.

This introduced him to the family of Lord Tyrconnel, to whose son he became tutor. He continued in this situation for many years, very much at his ease, and on terms of great intimacy and friendship with lord and lady Tyrconnel, "without so much (says he) as an unkind word or a cool look ever intervening."

In the spring of 1744, he was, through the interest of the earl of Bath (who was his great friend and patron, and whose friendship and patronage were returned by grateful acknowledgments and the warmest encomiums), presented to the rectory of St Mary le Bow; so that he was 40 years old before he obtained any living.

At the commencement of 1745, he took his doctor's degree. In the spring of 1747 he was chosen lecturer of St George's, Hanover square, by a most respectable vestry of noblemen and gentlemen of high distinction. In August following he married his first wife, the eldest daughter of Dr Trebeck; an unaffected, modest, decent young woman, with whom he lived very happy in mutual love and harmony near seven years.

In 1749 he published his edition of Milton's Paradise Lost, which (says he, very modestly) it is hoped hath not been ill received by the public, having, in 1775, gone through eight editions. After the Paradise Lost, it was judged (says he) proper that Dr Newton should also publish the Paradise Regained, and other poems of Milton; but these things he thought detained him from other more material studies, though he had the good fortune to gain by them more than Milton did by all his works put together. But his greatest gain (he says) was their first introducing him to the friendship and intimacy of two such men as Bishop Warburton and Dr Jortin, whose works will speak for them better than any private commendation.

In 1754 he lost his father, at the age of 83; and within a few days his wife, at the age of 38. This was the severest trial he ever underwent, and almost overwhelmed him. At that time he was engaged in writing his Dissertations on the Prophecies; and happy it was for him; for in any affliction he never found a better or more effectual remedy than plunging deep into study, and fixing his thoughts as intensely as he possibly could upon other subjects. The first volume was published the following winter; but the other did not appear till three years afterwards; and as a reward for his past and an incitement to future labours, he was appointed, in the mean time, to preach the Boyle's lecture. The bishop informs us, that 1250 copies of the Dissertations were taken at the first impression, and 1000 at every other edition: and "though (says he) some things have been since published upon the same subjects, yet they still hold up their head above water, and having gone through five editions, are again prepared for another." Abroad, too, their reception hath not been unfavourable, if accounts from thence may be depended upon." They were translated into the German and Danish languages; and received the warmest encomiums from persons of learning and rank.

In the spring of 1757, he was made prebendary of Westminster, in the room of Dr Green, and promoted to the deanery of Salisbury. In October following, he was made sub-almoner to his majesty. This he owed to Bishop Gilbert. He married a second wife in September 1761. She was the widow of the Rev. Mr Hand, and daughter of John Lord-Vicount Lifburn. In the same month he kissed his majesty's hand for his bishopric.

In the winter of 1764, Dr Stone, the primate of Ireland, died. Mr Grenville sent for Bishop Newton, and in the most obliging manner desired his acceptance of the primacy. Having maturely weighed the matter in his mind, he declined the offer.

In 1768 he was made dean of St Paul's. His ambition was now fully satisfied; and he firmly resolved never to ask for anything more.

From this time to his death, ill health was almost his constant companion. It was wonderful that such a poor, weak, and slender thread as the bishop's life, should be spun out to such an amazing length as it really was. In the autumn of 1781 (usually the most favourable part of the year to him) he laboured under repeated illnesses; and on Saturday the 9th of February 1782, he began to find his breath much affected by the frost. His complaints grew worse and worse till the Thursday following. He got up at five o'clock, and was placed in a chair by the fire; complained to his wife how much he had suffered in bed, and repeated to himself that portion of the Psalms, "O my God, I cry unto thee in the day-time;" &c. &c. About six o'clock he was left by his apothecary in a quiet sleep. Between seven and eight he awoke, and appeared rather more easy, and took a little refreshment. He continued dozing till near nine, when he ordered his servant to come and dress him, and help him down stairs. As soon as he was dressed, he inquired the hour, and bid his servant open the shutter and look at the dial of St Paul's. The servant answered, it was upon the stroke of nine. The bishop made an effort to take out his watch, with an intent to set it; but sunk down in his chair, and expired without a sigh or the least visible emotion, his countenance still retaining the same placid appearance which was so peculiar to him when alive. Of his numerous works, his Dissertations on the Prophecies are by much the most valuable. His learning was undoubtedly very considerable; but he seldom exhibits evidence of a very vigorous mind. On one occasion, indeed, he appears to have thought with freedom; for we believe he was the first dignitary of the church of England who avowed his belief of the final restitution of all things to harmony and happiness.