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the most simple manner. If it be not admitted, motion, and in the other case thirty-two parts of bat the solution by re-action is insisted upon, it C's motion, by an equal resistance? And how will be incumbent on the party to account for does B communicate in one case twenty-four the whole effect of communication of motion ; degrees of motion, and C thirty-two, by equal otherwise he will lie under the imputation of actions? If the actions and resistances are un rejecting a solution which is simple, obvious, equal, it is asked how the same mass can resist and perfect; for one complex, unnatural, and differently to bodies impinging upon it with incomplete. However this may be determined, equal momenta, and how bodies possessed of it will be allowed that the circumstances men- equal momenta can exert different actions, it tioned afford no ground for the inference, that being admitted that bodies resist proportional to action and re-action are equal, since appearances their masses, and that their power of overcomirg may be explained in another way.'

resistance is proportional to their momenta ?-It Thus, if there be a perfect reciprocity betwixt is incumbent on those who maintain the doctrine an impinging body and a body at rest sustaining of universal re-action to free it from these diffiits impulse, may we not at our pleasure consider culties and apparent contradictions. either body as the agent, and the other as the re- Others grant that Sir Isaac's axiom is very true sistent ? Let a moving body, A, pass from north with respect to terrestrial substances; but they to south, an equal body B at rest, which receives affirm, that, in these, both action and reaction are the stroke of A, act upon A from south to north, the effects of gravity. Substances void of graand A resist in a contrary direction, both in- vity would have no momentum; and without elastic: let the motion reciprocally communi- this they could not act; they would be moved cated be called six. Then B at rest communi- by the least force, and therefore could not resist cates to A six degrees of motion towards the or re-act. If, therefore, there is any fluid which north, and receives six degrees towards the south. is the cause of gravity, though such fluid could B, having no other motion than the six degrees it act upon terrestrial substances, yet these could communicated, will, by its equal and contrary loss not re-act upon it, because they have no force o and gain, remain in equilibrio. Let the original their own, but depend entirely upon it for their motion of A have been twelve, then A received momentum. In this manner, say they, we may a contrary action equal to six, six degrees of its conceive that the planets circulate, and all the motion will be destroyed or in equilibrio; con- operations of nature are carried on by means of sequently, a motive force as six will remain to A a subtile fuid ; which being perfectly active, towards the south, and B will be in equilibrio, and the rest of matter altogether passive, there is or at rest. A will then endeavour to move with neither resistance nor loss of motion. six degrees, or half its original motion, and B From the preceding axiom Sir Isaac draws the will remain at rest as before. A and B being following corollaries S1. A body by two forces equal masses, by the laws of communication conjoined will describe the diagonal of a parallelthree degree of motion will be communicated to ogram in the same time that it would describe B, or A with its six degrees will act with three, the sides by those forces apart. 2. Hence we and B will re-act also with three, B then will may explain the composition of any one direct act on A from south to north equal to three, force out of any two oblique ones, viz. by making while it is acted upon or resisted by A from the two oblique forces the sides of a parallelonorth 10 south, equal also to three, and B will gram, and the direct one the diagonal. 3. The remain at rest as before; A will also have its quantity of motion which is collected by taking six degrees of motion reduced to one half by the the sum of the motions directed towards the contrary action of B, and only three degrees of same parts, and the difference of those that are motion will remain to A, with which it will yet directed to contrary parts, suffers no change endeavour to move; and, finding B still at rest, "from the action of bodies among themselves; bethe same process will be repeated till the whole cause the motion which one body loses is commotion of A is reduced to an infinitely small municated to another; and, if we suppose friction quantity, B all the while remaining at rest, and and the resistance of the air to be absent, the there will be no communication of motion from motion of a number of bodies which mutually A to B, which is contrary to experience. impelled one another would be perpetual, and

Let a body, A, whose mass is twelve, at rest, its quantity always equal. 4. "The common be impinged upon first by B, having a mass as centre of gravity of two or more bodies does twelve, and a velocity as four, making a momen- not alter its state of motion or rest by the actions tum of forty-eight; and 2dly, by C, whose niass of the bodies among themselves; and therefore is six, and velocity eight, making a momentum the common centre of gravity of all bodies actof forty-eight equal to B, the three bodies being ing upon each other (excluding outward actions inelastic. In the first case, A will become pos- and impediments) is either at rest, or moves sessed of a momentum of twenty-four, and uniformly in a right line. 5. The motions of twenty-four will remain to B; and, in the 2d bodies included in a given space are the same case, A will become possessed of a momentum among themselves, whether that space is at rest, of thirty-two, and sixteen will remain to C, both or moves uniformly forward in a right line withbodies moving with equal velocities after the out any circular motion. The truth of this is shock, in both cases, by the laws of percussion, evidently shown by the experiment of a ship, It is required to know, if in both cases A resists where all motions happen after the same manner, equally, and if B and C act equally? the whether the ship is at rest, or proceeds uniformly actions and resistances are equal, how does A forward in a straight line. 6. If bodies, any in the one case destroy twenty-four parts of B's how moved among themselves, are urged in the direction of parallel lines by equal accelerative

to the method of ancient geometers. For deforces, they will all continue to move among monstrations are more contracted by the method themselves, after the same manner as if they had of indivisibles ; but because the hypothesis of been urged by no such forces,

indivisibles seems somewhat harsh, and thereThe whole of the mathematical part of the fore that method is reckoned less geometrical, I Newtonian philosophy depends on the following chose rather to reduce the demonstrations of the lemmas; of which the first is the principal. following propositions to the first and last sums

Lem. I. Quantities, and the ratios of quanti- and ratios of nascent and evanescent quantities, ties, which in any finite time converze continually that is, to the limits of those sums and ratios; to equality, and before that time approach nearer and so to premise, as short as I could, the dethe one to the other than by any given difference, monstrations of those limits. For hereby the become ultimately equal. If you deny it; sup- same this is performeil as by the method of pose them to be ultimately unequal, and let D indivisibles; and now, those principles being debe their ultimate difference. Therefore they monstrated, we may use them with more safety. cannot approach nearer to equality than by that Therefore, if hereaiver I should happen to congiven difference D; which is against the suppo- sider quantities as made up of particles, or sition.

should use little curve lines for right ones, I Concerning the meaning of this lemma philo- would not be understood to mean muivisibles, sophers are not agreed; and unhappily it is the but evanescent divisible quantities; not the sums very fundamental position on which the whole and ratios of determinate parts, but always the of the system rests. Many objections have been limits of sums and ratios; and that the force of raised to it by people who supposed themselves such demonstrations always depends on the mecapable of understanding it. They say that it thod laid down in the foregoing lemmas. is impossible we can come to an end of any in- • Perhaps it may be objected that there is no finite series, and therefore that the word ult- ultimate proportion of evanescent quantities, bemate can in this case have no meaning. In cause the proportion before the quantities have some cases the lemma is evidently false. Thus, vanished, is not the ultimate, and, when they are suppose there are two quant'ties of matter, A vanished, is none. But by the same argument and B, the one containing half a pound, and the it may be alleged that a body arriving at a cerother a third part of one. Let both be con- tain place, and there stopping, has no ultimate tinually divided by two; and though their ratio, velocity ; because the velocity before the body or the proportion of the one to the other, does comes to the place is not its ultimate velocity, not vary, yet the difference between them per- when it is arrived it has none.

But the answer petually becomes less, as well as the quantities is easy ; for by the ultimate velocity is meant themselves, until both the difference and quan- that with which the body is moved, neither betities themselves become less than any assignable fore it arrives at its place and the motion ceases, quantity; yet the difference will never totally nor after, but at the very instant it arrives; that vanish, nor the quantities become equal, as is is, that velocity with which the body arı ves at evident from the two following series :

its last place, and with which the motion ceases.

And in like manner, by the ultimate ratio of 8 1ū 32 67 128 256 512 1024 lc. evanescent quantities is to be understood the ratio

of the quantities, not before they vanish, nor

1 ,&c. afterwards, but with which they vanish. In like 0 12 24 18 90 102 334 768 1536 manner, the first ratio of nascent quantities is

that with which they begin to be. And the tirst Diff.

,&c. 6 12 2+ 48 96 192 384708 1536 3072

or last sum is that with which they begin and

cease to be (or to be augmented and diminished). Thus we see that though the difference is con- There is a limit which the velocity at the end of tinually diminishing, and that in a very large the motion may attain, but not exceed; and proportion, there is no hope of its vanishing, or this is the ultimate velocity. And there is the the quantities becoming equal. In like manner, like linit in all quantities and proportions that let us take the proportions or ratios of quan- begin and cease to be. And, since such limits tities, and we shall be equally unsuccessful. are certain and definite, to determine the same is Suppose two quantities of maiter, one containing a problem strictly geometrical. Bui whatever is eight and the other ten pounds: these quantities geometrical we may be allowed to make use of already have to each other the ratio of eight to ten, in determining and demonstrating any other thing or of fiur to five; but let us and two continually that is likewise geometrical. to each of them, and, though the ratios con

• It may also be objected, that, if the ultimate tinually come nearer to that of equality, it is in ratios of evanescent quantities are given, their vain to hope for a perfect coincidence. Thus, ultimate magnitudes will be also given ; and so

8 10 14 16 18 20 22 24, &c. all quantities will consist of indivisibles, which
10 12
20 22 24 26, &c.

is contrary to what Euclid has demonstrated

concerning incommensurables, in the tenth book Ratio

&c. of his dements. 6

But this objection is founded

on a false supposition. For those ultimate ratios For this and his other lemmas Sır Isaac makes with which quantities vanish are not truly the the following apology:-“These lemmas are pre- ratios of ultimate quantities, but limits towards mised, to avoid the jediousness of deducing pure which the ratios of quantities decreasing conplexed demonstrations ad absurdum, according tinually approach.'

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Lem. II. If in the figure A ac E, terminated are described.-For, suppose the time to be diby the right line A a, A

vided into equal parts, and in the first part of E, and the curve ac E,

that time let the body by its innate force describe there be inscribed any

the right line A B, in the following diagram

fi number of parallelo

K қ grams A6,B c, Cd, &c., comprehended under

L

a equal bases A B, BC, CD, &c., and the sides

M Bb, C c, D d, &c., parallel to one side Aa of the figure; and the parallelograms a Kbl, Lem,c Mdn, &c. are

D с FB A completed. Then if the breadth of these parallelograms be supposed to be diminished, and their number augmented in infinitum, the ultimate ratios which the

BE inscribed figure AK 6Lc Md D, the circumscribed figure Aalbmondo E, and curvilinear figure Aabcd E, will have to one another, are ratios of equality. For the difference of the inscribed and circumscribed figures is the sum of the parallelograms Kl, L , 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. In the second part of that time, the same would, But this rectangle, because its breadth A B is by law 1, if not hindered, proceed directly to c supposed diminished in infinitum, becomes less along the line Bc=AB: so that by the radii than any given space. And therefore, by lem. I., AS, BS, CS, drawn to the centre, the equal the figures inscribed and circumscribed become areas ASB, BSc, would be described. But, ultimately equal the one in the other; and much when the body is arrived at B, suppose the cenmore will the intermediate curvilinear figure be tripetal force acts at once with a great impulse, ultimately equal to either.

and, turning aside the body from the right line Lem. III. The same ultimate ratios are also Bc, compels it afterwards to continue its motion ratios of equality, when the breadths A B, BC, along the right line BC. Draw cC parallel to CD, &c., of the parallelograms are unequal, and BS, meeting BC in C; and at the end of the are all diminished in infinitum.— The demon- second part of the time, the body, by cor. 1, of stration of this differs but little from that of the the laws, will be found in C, in the same plane former.

with the triangle A SB. Join SC; and, because In his succeeding lemmas, Sir Isaac goes on S B and e C are parallel, the triangle S B C will to prove, in a manner similar to the above, that be equal to the triangle SCD, and therefore the ultimate ratios of the sine, chord, and tan- also to the triangle SA B. By the like argugent of arcs infinitely diminished, are ratios of ment, if the centripetal force acts successively in equality; and, therefore, that in all our reason- C, D, E, &c., and makes the body in each single ings about these we may safely use the one for particle of time to describe the right lines C D, the other :—that the ultimate form of evanescent D E, E F, &c., they will all lie in the same plane, triangles made by the arc, chord, and tangent, is and the triangle SC D will be equal to the trithat of similitude, and their ultimate ratio is angle SBC, and S D E to SCD, and S E F to that of equality; and hence, in reasonings about SDE. And, therefore, in equal times, equal ultimate ratios, we may safely use these triangles areas are described in one in inoveable plane; for each other, whether made with the sine, the and, by composition, any sums SADS, SAFS, arc, or the tangent.--He then shows some pro- of those areas are, one to the other, as the times perties of the ordinates of curvilinear figures; in which they are described. Now, let the numand proves that the spaces which a body de- ber of those triangles be augmented, and their scribes by any finite force urging it, whether that size diminished in infinitum; and then, by the force is determined and immutable, or is con- preceding lemmas, their ultimate perimeter tinually augmented or continuallly diminished, ADF will be a curve line : and therefore the are, in the very beginning of the motion, one to centripetal force by which the body is perpethe other in the duplicate ratio of the powers. tually drawn back from the tangent of this curve And, lastly, having added some demonstrations will act continually; and any described areas concerning the evanescence o angles of contact, SADS, S AFS, which are always proportional he proceeds to lay down the mathematical part to the times of description, will in this case also of his system, and which depends on the follow- be proportional to those times. Q. E. D. ing theorems :

Cor. 1. The velocity of a body attracted Theor. I. The areas which revolving bodies towards an immoveable centre, in spaces void of describe, by radii drawn to an immoveable cen- resistances, is reciprocally as the perpendicular tre of force, lie in the same immoveable planes, let fall from that centre on the right line which and are proportional to the times in which they touches the orbit. For the velocities in these VOL. XV.

2 T

theoi.

Cor. 2.

places 1, B, C', D, E, are as the bases AB, BC', different whether the superficies in which a body DE, EF, of equal triangles; and these bases are describes a curvilmear figure be quiescent, or reciprocally as the perpendiculars let fall upon move together with the body, the figure de

scribed, and its point S, uniformly forward in Cor. 2. If the chords AB, BC, of two arcs right lines. successively described in equal times by the same

Cor. 1. In non-resisting spaces of mediums, body, in spaces void of resistance, are completed if the areas are not proportional to the times, into a parallelogram ABC1, and the diagonal the forces are not directed to the point in which Bl of this parallelogram, in the position which the radii meet; but deviate therefrom in conseit ultimately acquires when those arcs are vi- quentia, or towards the parts to which the motion minished in infinitum, is produced both ways, it is directed, if the description of the areas is acwill pass through the centre of force.

celerated; but in antecedentia if retarded. Cor. 3. If the chords AB, BC, and DE, EF,

And even in resisting mediums, if of arcs described in equal times, in spaces void the description of the areas is accelerated, the of resistance, are completed into the parallelo- directions of the forces deviate from the point in grams ABCT, DEFZ, the forces in B and E which the radii meet, towards the parts to which are one to the other in the ultimate ratio of the the motion tends. diagonais B1, E2, when those ares are di- Scholium. A body may be urged hy a centriminished in infinitum. For the motions B C and petal force compounded of several ices. In Ef of the body (by cur. 1 of the linis), are com- which case the meaning of the proposition is, pounded of the motions Be, B l, and Ef; EZ; that the force which results out of all tends to the but Bl and E 2, which are equalto (o and Fl, point S. But if any force acts perpetually in in the demonstration of this proposition, witte the direction of lines perpendicular to the generated by the impulses of the centripetal forve described surface, this force will make the body in B and F, and are therefore proportional to to deviate from the plane of its motion, but will those impulses.

neither augment nor diminish the quantity of the Cor. 4. The forces by which bodies, in spaces described surface; and is therefore not to be negvoid of resistance, are drawn back from rectilinear lected in the composition of forces. motions, and turned into curvilinear orbits, are Theor. III. Every body that, by a radius one to another as the versed sines of arcs described drawn to the centre of another body, howsoever in equal times; which versed sines tend to the moved, describes areas about that centre proporcentre of force, and bisect the chords when these tional to the times, is urged by a force comares are diminished to intinity. For such versed pounded of the centripetal forces tending to that sines are the halves of the diagonals mentioned other body, and of all the accelerative force by in cor. 3.

which the other body is impelled.—The demonCor. 5. And therefore those forces are to the stration of this is a natural consequence of the force of gravity as the said versed sines to the theorem immediately preceding. versed sines perpendicular to the horizon of those Hence, if the one body L, by a radius drawn parabolic ares which projectiles describe in the to the other body T, describes areas proportional same time.

to the times, and from the whole force by which Cor. 6. And the same thints do all hold good the first body L is urged (whether that force is (by cor. 5 of the laws) when the planes in which simple, or, according to cor. 2 of the laws, comthe bodies are moved, together with the centres pounded of several forces) we subduct that of force, which are placed in those planes, are whole accelerative force by which the other body not at rest, but move uniformly forward in right is urged; the whole remaining force by which lines.

the first body is urged will tend to the other body Theor. II. Every body that moves in any T, as its centre. And, vice versâ, if the remaincurve line described in a plane, and, by a radius ing force tends nearly to the other body T, those drawn to a point either inmoveable or moving areas will be nearly proportional to the times. forward with uniform rectilmear motion, If the body L, by a radius drawn to the other describes about that point areas proportional to body T, describes areas which compared with the times, is urged by the centripeial force di- the times are very unequal, and that other body rected to that point.

T be either at rest or moves uniformly forward Case I. For every body that moves in a curve in a right line, the action of the centripetal force line is (by law 1) wrned aside from its rectilinear tending to that other body T is either none at all, course by the action of some force that impels it; or it is mixed and combined with very powerful and that force by which the body is turned off actions of other forces: and the whole force from its rectilinear course, and made to describe compounded of them all, if they are many, is in equal times the least equal triangles, SAB, directed to another (immoveable or moveable) SBC, SCD, &c., about the immoveable point The same thing obtains when the other S (by Prop. \L. E. 1, and law 2), acts in the body is actuated by any other motion whatever: place B according to the direction of a line par- provided that centripetal force is taken which allel to C; that is, in the direction of the line remains after subducting that whole force acting BS; and in the place C according to the direc- upon that other body T. tion of a line parallel 10 d ), that is, in the direc- Scholium. Because the equable description tion of the line CS, &c.; and therefore acts always of areas indicate that a centre is respected hy in the direction of lines tending to the immoveable that force with which the body is most affected, point S. Q. E. D.

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and by which it is drawn back from its rectilinear Case II. And (by cor. 5 of the laws) it is in- motion, and retained in its orbit, we may always

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be allowed to use the equable description of centripetal force, describes in any time, is a areas as an indication of a centre about which mean proportional between the diameter of the all circular motion is performed in free spaces. circle and the space which the same body, falling

Theor. IV. The centripetal forces of bodies, by the same given force, would descend through which by equable motions describe different cir- in the same given time. cles, tend to the centres of the same circles; and By means of the preceding proposition and are one to the other as the squares of the arcs its corollaries,' says Sir Isaac, we may discover described in equal times applied to the radii of the proportion of a centripetal force to any other circles.- For these forces tend to the centres of known force, such as that of gravity. For, if a the circles (by theor. II. and cor. 2 theor. I.), body by means of its gravity revolves in a circle and are to one another as the versed sines of the concentric to the earth, this gravity is the centrileast arcs described in equal times (by cor. 4 petal force of that body. But, from the descent theor. I.), that is, as the squares of the same arcs of heavy bodies, the time of one entire revoluapplied to the diameters of the circles by one of tion, as well as the arc described in any given the lemmas: and therefore since those arcs are time, is given (by cor. 9 of this theorem). And as arcs described in any equal times, and the by such propositions Mr. Huygens, in his exceldiameters are as the radii, the forces will be as lent book De Horologio Oscillatorio, has comthe squares of any arcs described in the same pared the force of gravity with the centrifugal time, applied to the radii of the circles. Q.E.D. forces of revolving bodies."

Cor. 1. Therefore, since those arcs are as the The preceding proposition may also be demonvelocities of the bodies, the centripetal forces are strated in the following manner:-In any circle in a ratio compounded of the duplicate ratio of suppose a polygon to be inscribed of any numthe velocities directly, and of the simple ratio of ber of sides. And if a body, moved with a given the radii inversely.

velocity along the sides of the polygon, is reCor. 2. And since the periodic times are in a flected from the circle at the several angular ratio compounded of the ratio of the radii di- points; the force with which, at every reflection, rectly, and the ratio of the velocities inversely; it strikes the circle, will be as its velocity: and the centripetal forces are in a ratio compounded therefore the sum of the forces, in a given time, of the ratio of the radii directly, and the dupli- will be as that velocity and the number of refleccate ratio of the periodic times inversely. tions conjunctly; that is (if the species of the

Cor. 3. Whence, if the periodic times are polygon be given), as the length described in equal, and the velocities therefore as the radii, that given time, and increased or diminished in the centripetal forces will be also as the radii; the ratio of the same length to the radius of the and the contrary.

circle; that is, as the square of that length apCor. 4. If the periodic times and the veloci- plied to the radius; and therefore, if the polygon, ties are both in the subduplicate ratio of the by having its sides diminished in infinitum, coinradii, the centripetal forces will be equal among cides with the circle, as the square of the arc themselves; and the contrary.

described in a given time applied to the radius. Cor. 5. If the periodic times are as the radii, This is the centrifugal force, with which the body and therefore the velocities equal, the centripetal impels the circle; and to which the contrary forces will be reciprocally as the radii; and the force, wherewith the circle continually repels the contrary.

body towards the centre, is equal. Cor. 6. If the periodic times are in the ses- On these principles bangs the whole of Sir quiplicate ratio of the radii, and therefore the Isaac Newton's mathematical philosophy. He velocities reciprocally in the subduplicate ratio now shows how to find the centre to which the of the radii, the centripetal forces will be in the forces impelling any body are directed, having duplicate ratio of the radii inverse; and the con- the velocity of the body given: and finds the trary.

centrifugal force to be always as the versed sine Cor. 7. And universally, if the periodic time of the nascent arc directly, and as the square of is as any power Rn of the radius R, and there- the time inversely; or directly as the square of fore the velocity reciprocally as the power Rn–1 the velocity, and inversely as the chord of the of the radius, the centripetal force will be re

From these premises he deduces ciprocally as the power R2n-2 of the radius; the method of finding the centripetal force diand the contrary.

rected to any given point when the body revolves Cor. 8. The same things all hold concerning in a circle; and this whether the central point the times, the velocities, and forces, by which is near or at an immense distance; so that all the bodies describe the similar parts of any similar lines drawn from it may be taken for parallels. figures, that have their centres in a similar posi- The same thing he shows with regard to bodies tion within those figures, as appears by applying revolving in spirals, ellipses, hyperbolas, or pathe demonstrations of the preceding cases to rabolas. Having the figures of the orbits given, those. And the application is easy, by only sub- he shows also how to find the velocities and stituting the equable description of areas in the moving powers; and, in short, solves all the place of equable motion, and using the distances most difficult problems relating to the celestial of the bodies from the centres instead of the bodies with an astonishing degree of mathemaradii.

tical skill. These problems and demonstrations Cor. 9. From the same demonstration it like- are all contained in the first book of the Princi. wise follows, that the arc which a body, uni- pia; to which we must refer those who wish for formly revolving in a circle by means of a given farther information.

nascent arc.

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