NEWTONIAN PHILOSOPHY.

cities reciprocally in the subduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radi inversely; and the contrary. COR. 7. And universally, if the periodic time is as any power R" of the radius R, and therefore the velocity reciprocally as the power R of the radius the centripetal force will be reciprocally as the power R22 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 appl cation 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 uniform ly 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 fo: ce, 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 proposition 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 demonIn any circle strated in the following manner. suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given ve locity 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 um 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), in that given time, and as the length describ 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 This is the centrifugal applied to the radius. 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.

He

On these principles hangs the whole of sir
Isaac Newton's mathematical philosophy.
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 nascent arc directly, and as the square of
the time inversely, or directly as the square of
the velocity, and inversely as the chord of the
From these premises he deduces the
method of finding the centripetal force directed

nascent arc.

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 cemonstrated; and, as a necessary preliminary to this pa t, 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 intension 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.

3. The

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, 2. The same thing is likewise are in the sesquiplicate ratio of their distances from its centre. observed of the phenomena of Saturn. five primary planets, Mercury, Venus, Mars, 4. The fixed stars being supJupiter, and Saturn, with their several orbits, encompass the sun. posed at rest, the periodic times of the five primary planets, and of the earth, about the sun, are in the sesquiplicate proportion to their mean 5. The primary planets, distances from the sun. by radii drawn to the earth, describe areas no ways proportionable 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 expla ned at large under the article ASTRO

NOMY.

The mathematical demonstracions are next applied by sir Isaac Newton in the following propositions.

PROP. I. The forces by which the satellites of Jupi er are continually drawn off from rectilinear motions, and retained in their proper orbit, tends

1

are reciprocally
use satellites
vt of this pro-
and the latter
same thing we

For a very small aberration cal duplicate proportion would on of the apsides, sensible in every 100, and in many of them enor

The force by which the moon is reats orbit tends towards the earth; and segreculy as the square of the distance of its To the centre of the earth. The former thus proposition is evident from phenom. theor. the latter from phenom. 6. and r. It is also evident from the very uonon of the moon's apogee; which, in very single revolution, amounting but to 3° 3′ in et, may be neglected: and this more fully Apps from the next proposition.

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 30 Paris feet; altogether against experience.

es of Saturn. which the primary »n from rectilinear Air proper orbits, tend cly as the squares of Suds cent e. The former as manifest from phe- The demonstration of this proposition may be oned, and from theor. 2.; more diffusely explained after the following manmeson 4 and cor. 6. of ner. Suppose several moons to revolve about at the proposition is with the earth as in the system of Jupiter or Saturn, cobie from the quiescence of 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. 1. 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 deserted by its centrifugal force that carries it through its orbit, it would de cend 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.

PROP IV. The moon gravitates towards the earth, and by the force of gravity is continually dawn off from a rectilinear motion, and retained in its orbit. The mean distance of the moon from the earth in the syzigies in semidiameters of the latter, is about 04 Let us assume the mean distance of 60 semidia": eters in the syzigies; and suppose one revolution of the moon in respect of the fixed stars to be completed in 274. 7h. 43, as astronon.ers 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 toward the earth with the impulse of all that force by which it is retained in its o-bit, it will, in the space of one minute of time, describe in its fall 154 Paris feet. For the versed sine of that are 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 Paris feet; or more accurately, 15 feet 1 inch and one line 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 × 60 times greater than at the moon; a body in our regions, falling with that force, ought, in the space of one minue of time, to describe 60 x 60 x 15 Paris feet; and in the space of one second of time to describe 15 of those feet; or, more accurately, 15 feet 1 inch, 1 line 3. 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 84 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. And therefore the by which the moon is retained in its orbit

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 experi ments, 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

NEWTONIAN PHILOSOPHY.

Newton tried the thing in gold, silver, lead, glass, He sand, common salt, wood, water, and wheat. provided two wooden boxes, round and equal, filed 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 motrix) 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 certaiu, 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 susquiplicate proportion of their distances from Jupiter's centre, their accelerative gravities towards Jupiter will be reciprocally as the squares of their distance 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 And by the same argubodies do on our earth. ment, if the circumsolar 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 in equality of attraction. If, at equal distances from the sun, any satellite, in proportion to the quan tity 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 centre 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 orb from the sun would be less than the distance of the centre of Jupiter's from the san 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 Too
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 part of the whole distance;
that is, by a fifth part of the distance of the ut-
most satellite from the centre of Jupiter; an ec-
centricity of the orbit which would be very sensi-
ble. But the orbits of the satellites are concen-
tric to Jupiter; therefore the accelerative gravi-
ties 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 accu
rately 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 propor-
tion 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
Nor is it of any moment whether
less than in proportion to the quantity of matter
in the whole.
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 For if the depend upon their forms and textures. weight scould 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 3.) 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 others) there is no difference betwixt that and other 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 mat. ter, would fall nothing short of the specific gravity

Cor. 4. The smaller the planets are, they are, cæteris paribus, of so much the greater density. For so the powers of gravity on their several surfaces come nearer to equality. They are likewise, cæteris paribus, of the greater density as they are nearer to the sun. 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 COR. I. Hence we may find and compare to- far as the orb of Saturn, would be converted into gether the weights of bodies towards different ice, and in the orb of Mercury would quickly fly planets. For the weights of bodies revolving in away in vapour. For the light of the sun, to which circles about planets are as the diameters of the its heat is proportional, is seven times denser in the circles directly, and the squares of their periodic orb of Mercury than with us: and by the thermotimes reciprocally; and their weights at the sur-meter sir Isaac found, that a sevenfold heat of our