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NEWTONIAN PHILOSOPHY.

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 For if the same forces any changes by such force. are likewise impressed on those 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 therefor 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 rela

tions.

The effects which distinguish absolute from relative moti n are, the forces of receding from the axis of circular motion. For there are no such forces in a 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 and may be to the relative, discovers itself, At first, when measured by this endeavour. the relative motion in the water was greatest, it produced no⚫ndeavour 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 decreas ed, 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 rested relatively in the vessels. And therefore this endeayour does not depend upon any translation of the water in respect of the ambient bodies; nor can VOL VIIL

true circular motion be defined by such transla-
tions. 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 mo-
tion, 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 exter-
nal bodies; and, like other relations, are altogether
destitute of any real effect, otherwise than they may
perhaps participate 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 seve-
ral parts of those heavens and the planets, which
are indeed relatively at rest in their heavens, do yet
For they change their position one to
really move.
another, which never happens to bodies truly at rest;
and being carried together with the heavens, parti-
cipate 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 immoveable 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 the 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 er ular 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 most augmented; that is, we might discover their hindermost faces, 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 au 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 EE 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 mict on to be in the globes, and the bodies to be at rest; and then, lastly, from the transation 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 then 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 compeiled to change that state by forces impressed upon it.

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.

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.

From the preceding axioms 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 moyed 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.

For this and his other lemmas sir Isaac makes the following apology. "These lemmas are premised, to avoid the tediousness of deducing per plexed 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 in. divisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the follow. ing propositions 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 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 understood 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, be cause 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 ve locity 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 so all quantities will consist of indivisibles, which is

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contrary to what Euclid has demonstrated con- radii AS, BS, c S, drawn to the centre, the equal cerning incommensurables, in the 10th book of his areas ASB, BS c, would be described. But when Elements. But this objection is founded on a false the body is arrived at B, suppose the centripetal supposition. For those ultimate ratios with which force acts at once with a great impulse, and, turnquantities vanish are not truly the ratios of ulti-ing aside the body from the right line Bc, compels mate quantities, but limits towards which the ratios of quantities decreasing continually approach,"

LEM. II. If in any figure A ac E (Pl. 129, fig. 1.) terminated by the right line Aa, A E, and the curve ac E, there be inscribed any number of parallelo grams Ab, Be, Cd, &c. comprehended under equal bases, AB, BC, CD, &c. and the sides Bb, Cc, Dd, &c. paralel to one side Aa of the figure; and the parallelograms a K bl, b Lcm, c M dn, &c. are completed. Then if the breadth of these paralJelograms be supposed to be diminished, and their number augmented in infinitum; the ultimate ratios which the inscribed figure A K 6Lc M dD, the circumscribed figure A alb mendo E, and curvilinear figure A a b c d E, will have to one another, are ratios of equality.-For the difference of the inscribed and circumscribed figures is the sur of the parallelograms K . L m, Mn, Do; that is, (from the equality of all their bases), the rectangle under one of their bases K b, and the sum of their altitudes A a, that is, the rectangle A Bla. But this rectangle, because its breadth A B is supposed diminished in infinitum, becomes less than any given space. And therefore by lem, 1. the figures inscrived 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:- hat the ultimate form of evanescent triangles made by the arc, chord, and tangent, is that of similitude, and their ultimate ratio is that of equality; and hence, in reasonings about ultimate ratios, we may safely ase the triangles for each other, whether made with the sine, the arc, or the tangent.-He then shows Some properties of the ordmates 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 contionally 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 immoveable centre of force, lie in the same immoveable 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 (fig. 2.); in the second part of that time, the same would, by law. 1. if not hindered, proceed directly to calong the line BcAB; so that by the

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it afterwards to continue its motion along the right line BC. Draw c C parallel to BS, meeting BC in C; and at the end of the second part of the time, the body, by cor. 1. of the laws, will be found in C, in the same plane with the triangle ASB. SC; and because SB and c C 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 particie of time to describe the right lines CD, DE, 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 tiines, equal areas are described in one immoveable plane; and, by composition, any sums SADS, SAFS, of those areas are, ope 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. F. D.

COR. I. 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 cqual times, in spaces void of resistance, are completed into the parallelograins ABCV, DEFZ, the forces in B and E are one to the other in the ultimate ratio of the diagonals BV, EZ, when those ares are diminished in infini tum. For the motions BC and EF of the body (by cor. 1. of the laws), are compounded of the motions Br, BV and E, 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 ares 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

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