LEM

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

equal and contrary loss and gain, remain in equilibrio. 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 equilibrio; consequently, a motive force as six will remain to $A$ towards the south, and $B$ will be in equilibrio, 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 re-act 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 possessed 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 shock, 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 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.

Newtonian Philosophy 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.

Newtonian Philosophy. In his succeeding lemmas, Sir Isaac goes on to prove, in a manner similar to the above, that the ultimate ratios of the fine, 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 similitude, 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 fine, 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 determinate 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 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 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. 1. 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, &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, SAES, 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, SAES, 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 Newtonian locities in these places, A, B, C, D, E, are as the Newtonian Philosophy. 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 Bc, BV and Ef, EZ; but BV and EZ, which are equal to Cc 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 fines of arcs described in equal times; which versed fines tend to the centre of force, and bisect the chords when these arcs are diminished to infinity. For such versed fines 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 fines to the versed fines 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 areas proportional to the times, is urged by a centripetal force directed to that point.

CASE 1. 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.

Newtonian Philosophy 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.

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 accelerated by any other motion whatever; provided that centripetal force is taken which remains after subducting that whole force acting upon that other body T.

Because the equable description of areas indicates that a centre is respected 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. Newtonian Philosophy.

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 equal among themselves; 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.

Newtonian Philosophy. "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 conjointly; 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 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 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. Newtonian 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 intention 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 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 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 apsidies, sensible in every single revolution, and in many of them enormously great.

PROP. III. The force by which the moon is retained in

Newtonian in its orbit, tends towards the earth; and is reciprocally
Philosophy. 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 $3^\{\backslash circ\}\; 3\text{'}$ in con-
sequentia, may be neglected: and this more fully ap-
pears 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
syzgies in semidiameters of the latter, is about $60\backslash frac\{1\}\{2\}$.
Let us assume the mean distance of 60 semidiameters in
the syzgies; and suppose one revolution of the moon in
respect of the fixed stars to be completed in $27^d\; 7^h\; 43\text{'}$,
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 or-
bit, it will, in the space of one minute of time describe
in its fall $15\backslash frac\{1\}\{2\}$ Paris feet. For the versed sine of that arc
which the moon, in the space of one minute of time,
describes by its mean motion at the distance of 60 se-
midiameters of the earth, is nearly $15\backslash frac\{1\}\{2\}$ Paris feet;
or more accurately, 15 feet 1 inch and one line $\backslash frac\{1\}\{2\}$.
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 $60\; \backslash times\; 60$ times greater than that at the moon;
a body in our regions, falling with that force ought, in
the space of one minute of time, to describe $60\; \backslash times\; 60\; \backslash times$
$15\backslash frac\{1\}\{2\}$ Paris feet; and in the space of one second of time
to describe $15\backslash frac\{1\}\{2\}$ of those feet; or, more accurately, 15
feet 1 inch, 1 line $\backslash frac\{1\}\{2\}$. And with this very force we ac-
tually 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 $8\backslash frac\{1\}\{2\}$ 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 diam-
eter; and is therefore 15 Paris feet, 1 inch 1 line $\backslash frac\{1\}\{2\}$.
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 bod-
ies 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 $\backslash frac\{1\}\{2\}$
Paris feet; altogether against experience.

The demonstration of this proposition may be more
diffusely explained after the following manner: Sup-
pose 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 for-
ces would be reciprocally as the squares of the distan-

ces from the centre of the earth, by Prop. I. Now, if Newtonian
Philosophy. 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 descend to the
earth; and that with the same velocity as heavy bod-
ies 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 for-
ces 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 gra-
vity; 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 con-
tain 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 di-
stinguish to a great accuracy by the help of pendu-
lums. Sir Isaac Newton tried the thing in gold, sil-
ver, 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 ob-
served them to play together forwards and backward,
for a long time, with equal vibrations. And there-
fore 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