Hegel both criticises the Newtonian doctrines, or what he states as

such; and also, not denying the truth of the laws of phenomena which he

refers to, for instance Kepler's laws, offers his own proof of these

laws. I shall make a few brief remarks on each of these portions of the

pages before me. And I would beg it to be understood that where I may

happen to put my remarks in a short, and what may seem a peremptory

form, I do so for the sake of saving time; knowing that among us, upon

subjects so familiar, a few words will suffice. For the same reason, I

shall take passages from Hegel, not in the order in which they occur,

but in the order in which they best illustrate what I have to say. I

shall do Hegel no injustice by this mode of proceeding: for I will

annex a faithful translation, so far as I can make one, of the whole of

the passages referred to, with the context.

No one will be surprised that a German, or indeed any lover of science,

should speak with admiration of the discovery of Kepler's laws, as a

great event in the history of Astronomy, and a glorious distinction to

the discoverer. But to say that the glory of the discovery of the proof

of these laws has been unjustly transferred from Kepler to Newton, is

quite another matter. This is what Hegel says (_a_)[363]. And we have

to consider the reasons which he assigns for saying so.

He says (_b_) that "it is allowed by mathematicians that the Newtonian

Formula maybe derived from the Keplerian laws," and hence he seems to

infer that the Newtonian law is not an additional truth. That is, he

does not allow that the discovery of the cause which produces a certain

phenomenal law is anything additional to the discovery of the law

itself.

"The Newtonian formula may be derived from the Keplerian law." It was

professedly so derived; but derived by introducing the Idea of _Force_,

which Idea and its consequences were not introduced and developed till

after Kepler's time.

"The Newtonian formula may be derived from the Keplerian law." And the

Keplerian law may be derived, and was derived, from the observations of

the Greek astronomers and their successors; but was not the less a new

and great discovery on that account.

But let us see what he says further of this derivation of the Newtonian

"formula" from the Keplerian Law. It is evident that by calling it a

_formula_, he means to imply, what he also asserts, that it is no new

law, but only a new form (and a bad one) of a previously known truth.

How is the Newtonian "formula," that is, the law of the inverse squares

of the central force, derived from the Keplerian law of the cubes of

the distances proportional to the squares of the times? This, says

Hegel, is the "immediate derivation." (_c_).--By Kepler's law, _A_

being the distance and _T_ the periodic time, _A_^3/_T_^2 is constant.

But Newton _calls_ _A_/_T_^2 universal gravitation; whence it easily

follows that gravitation is inversely as _A_^2.

This is Hegel's way of representing Newton's proof. Reading it, any one

who had never read the _Principia_ might suppose that Newton _defined_

gravitation to be _A_/_T_^2. We, who have read the _Principia_, know

that Newton _proves_ that in circles, the _central force_ (not the

_universal gravitation_) is as _A_/_T_^2: that he proves this, by

setting out from the idea of force, as that which deflects a body from

the tangent, and makes it describe a curved line: and that in this way,

he passes from Kepler's laws of mere motion to his own law of Force.

But Hegel does not see any value in this. Such a mode of treating the

subject he says (_i_) "offers to us a tangled web, formed of the Lines

of the mere geometrical construction, to which a physical meaning of

independent forces is given." That a _measure_ of forces is _found_ in

such lines as the sagitta of the arc described in a given time, (not

such a _meaning_ arbitrarily _given_ to them,) is certainly true, and

is very distinctly proved in Newton, and in all our elementary books.

But, says Hegel, as further showing the artificial nature of the

Newtonian formulæ, (_h_) "Analysis has long been able to derive the

Newtonian expression and the laws therewith connected out of the Form

of the Keplerian Laws;" an assertion, to verify which he refers to

Francœur's _Mécanique_. This is apparently in order to show that the

"lines" of the Newtonian construction are superfluous. We know very

well that analysis does not always refer to visible representations of

such lines: but we know too, (and Francœur would testify to this also,)

that the analytical proofs contain equivalents to the Newtonian lines.

We, in this place, are too familiar with the substitution of analytical

for geometrical proofs, to be led to suppose that such a substitution

affects the substance of the truth proved. The conversion of Newton's

geometrical proofs of his discoveries into analytical processes by

succeeding writers, has not made them cease to be discoveries: and

accordingly, those who have taken the most prominent share in such a

conversion, have been the most ardent admirers of Newton's genius and

good fortune.

So much for Newton's comparison of the Forces in different circular

orbits, and for Hegel's power of understanding and criticising it. Now

let us look at the motion in different parts of the same elliptical

orbit, as a further illustration of the value of Hegel's criticism. In

an elliptical orbit the velocity alternately increases and diminishes.

This follows necessarily from Kepler's law of the equal description

of the areas, and so Newton explains it. Hegel, however, treats of

this acceleration and retardation as a separate fact, and talks of

another explanation of it, founded upon Centripetal and Centrifugal

Force (_o_). Where he finds this explanation, I know not; certainly

not in Newton, who in the second and third section of the _Principia_

explains the variation of the velocity in a quite different manner, as

I have said; and nowhere, I think, employs centrifugal force in his

explanations. However, the notion of centrifugal as acting along with

centripetal force is introduced in some treatises, and may undoubtedly

be used with perfect truth and propriety. How far Hegel can judge

when it is so used, we may see from what he says of the confusion

produced by such an explanation, which is, he says, a maximum. In the

first place, he speaks of the motion being _uniformly_ accelerated and

retarded in an elliptical orbit, which, in any exact use of the word

_uniformly_, it is not. But passing by this, he proceeds to criticise

an explanation, not of the variable velocity of the body in its orbit,

but of the alternate access and recess of the body to and from the

center. Let us overlook this confusion also, and see what is the value

of his criticism on the explanation. He says (_p_), "according to

this explanation, in the motion of a planet from the aphelion to the

perihelion, the centrifugal is less than the centripetal force; and

in the perihelion itself the centripetal force is supposed suddenly

to become greater than the centrifugal;" and so, of course, the body

re-ascends to the aphelion.

Now I will not say that this explanation has never been given in a book

professing to be scientific; but I have never seen it given; and it

never can have been given but by a very ignorant and foolish person.

It goes upon the utterly unmechanical supposition that the approach

of a body to the center at any moment depends solely upon the excess

of the centripetal over the centrifugal force; and reversely. But the

most elementary knowledge of mechanics shows us that when a body is

moving _obliquely_ to the distance from the center, it approaches to

or recedes from the center in virtue of this obliquity, even if no

force at all act. And the total approach to the center is the approach

due to this cause, _plus_ the approach due to the centripetal force,

_minus_ the recess due to the centrifugal force. At the aphelion, the

centripetal is greater than the centrifugal force; and _hence_ the

motion becomes oblique; and _then_, the body approaches to the center

on _both_ accounts, and approaches on account of the obliquity of the

path even when the centrifugal has become greater than the centripetal

force, which it becomes before the body reaches the perihelion. This

reasoning is so elementary, that when a person who cannot see this,

writes on the subject with an air of authority, I do not see what can

be done but to point out the oversight and leave it.

But there is, says Hegel (_q_), another way of explaining the motion

by means of centripetal and centrifugal forces. The two forces are

supposed to increase and decrease gradually, according to different

laws. In this case, there must be a point where they are equal, and in

equilibrio; and this being the case, they will always continue equal,

for there will be no reason for their going out of equilibrium.

This, which is put as _another_ mode of explanation, is, in fact, the

same mode; for, as I have already said, the centrifugal force, which is

less than the centripetal at the aphelion, becomes the greater of the

two before the perihelion; and there is an intermediate position, at

which the two forces are equal. But at this point, is there no reason

why, being equal, the forces should become unequal? Reason abundant:

for the body, being there, moves in a line oblique to the distance, and

so changes its distance; and the centripetal and centrifugal force,

depending upon the distance by different laws, they forthwith become

unequal.

But these modes of explanation, by means of the centripetal and

centrifugal forces and their relation, are not necessary to Newton's

doctrine, and are nowhere used by Newton; and undoubtedly much

confusion has been produced in other minds, as well as Hegel's,

by speaking of the centrifugal force, which is a mere intrinsic

geometrical result of a body's curvilinear motion round a center, in

conjunction with centripetal force, which is an extrinsic force, acting

upon the body and urging it to the center. Neither Newton, nor any

intelligent Newtonian, ever spoke of the centripetal and centrifugal

force as two distinct forces both extrinsic to the motion, which Hegel

accuses them of doing. (_n_)

I have spoken of the third and second of Kepler's laws; of Newton's

explanations of them, and of Hegel's criticism. Let us now, in the same

manner, consider the first law, that the planets move in ellipses.

Newton's proof that this was the result of a central force varying

inversely as the square of the distance, was the solution of a problem

at which his contemporaries had laboured in vain, and is commonly

looked upon as an important step. "But," says Hegel, (_d_) "the proof

gives a conic section generally, whereas the main point which ought

to be proved is, that the path of the body is an ellipse only, not a

circle or any other conic section." Certainly if Newton _had_ proved

that a planet cannot move in a circle, (which Hegel says he ought to

have done), his system would have perplexed astronomers, since there

are planets which move in orbits hardly distinguishable from circles,

and the variation of the extremity from planet to planet shows that

there is nothing to prevent the excentricity vanishing and the orbit

becoming a circle.

"But," says Hegel again, (_e_) "the conditions which make the path to

be an ellipse rather than any other conic section, are empirical and

extraneous;--the supposed casual strength of the impulsion originally

received." Certainly the circumstances which determine the amount

of excentricity of a planet's orbit are derived from experience, or

rather, observation. It is not a part of Newton's system to determine

_à priori_ what the excentricity of a planet's orbit must be. A system

that professes to do this will undoubtedly be one very different

from his. And as our knowledge of the excentricity is derived from

observation, it is, in that sense, empirical and casual. The strength

of the original impulsion is a hypothetical and impartial way of

expressing this result of observation. And as we see no reason why the

excentricity should be of any certain magnitude, we see none why the

fraction which expresses the excentricity should not become as large

as unity, that is, why the orbit should not become a parabola; and

accordingly, some of the bodies which revolve about the same appear to

move in orbits of this form: so little is the motion in an ellipse, as

Hegel says, (_f_) "the only thing to be proved."

But Hegel himself has offered proof of Kepler's laws, to which,

considering his objections to Newton's proofs, we cannot help turning

with some curiosity.

And first, let us look at the proof of the Proposition which we have

been considering, that the path of a planet is necessarily an ellipse.

I will translate Hegel's language as well as I can; but without

answering for the correctness of my translation, since it does not

appear to me to conform to the first condition of translation, of

being intelligible. The translation however, such as it is, may help

us to form some opinion of the validity and value of Hegel's proofs as

compared with Newton's. (_r_)

"For absolutely uniform motion, the circle is the only path.... The

circle is the line returning into itself in which all the radii are

equal; there is, for it, only one determining quantity, the radius.

"But in free motion, the determination according to space and to

time come into view with differences. There must be a difference in

the spatial aspect in itself, and therefore the form requires two

determining quantities. Hence the form of the path returning into

itself is an ellipse."

Now even if we could regard this as reasoning, the conclusion does

not in the smallest degree follow. A curve returning into itself and

determined by two quantities, may have innumerable forms besides the

ellipse; for instance, any _oval_ form whatever, besides that of the

conic section.

But why must the curve be a curve returning into itself? Hegel has

professed to prove this previously (_m_) from "the determination of

particularity and individuality of the bodies in general, so that

they have partly a center in themselves, and partly at the same time

their center in another." Without seeking to find any precise meaning

in this, we may ask whether it proves the impossibility of the orbits

with moveable apses, (which do not return into themselves,) such as the

planets (affected by perturbations) really do describe, and such as

we know that bodies must describe in all cases, except when the force

varies exactly as the square of the distance? It appears to do so:

and it proves this impossibility of known facts at least as much as it

proves anything.

Let us now look at Hegel's proof of Kepler's second law, that the

elliptical sectors swept by the radius vector are proportional to the

time. It is this: (_s_).

"In the circle, the arc or angle which is included by the two radii is

independent of them. But in the motion [of a planet] as determined by

the conception, the distance from the center and the arc run over in a

certain time must be compounded in one determination, and must make out

a whole. This whole is the sector, a space of two dimensions. And hence

the arc is essentially a Function of the radius vector; and the former

(the arc) being unequal, brings with it the inequality of the radii."

As was said in the former case, if we could regard this as reasoning,

it would not prove the conclusion, but only, that the arc is _some

function or other_ of the radii.

Hegel indeed offers (_t_) a reason why there must be an arc involved.

This arises, he says, from "the determinateness [of the nature of

motion], at one while as time in the root, at another while as space in

the square. But here the quadratic character of the space is, by the

returning of the line of motion into itself, limited to a sector."

Probably my readers have had a sufficient specimen of Hegel's mode of

dealing with these matters. I will however add his proof of Kepler's

third law, that the cubes of the distances are as the squares of the

times.

Hegel's proof in this case (_u_) has a reference to a previous doctrine

concerning falling bodies, in which time and space have, he says, a

relation to each other as root and square. Falling bodies however

are the case of only _half-free_ motion, and the determination is

incomplete.

"But in the case of absolute motion, the domain of _free_ masses, the

determination attains its totality. The time as the root is a mere

empirical magnitude: but as a component of the developed Totality,

it is a Totality in itself: it produces itself, and therein has a

reference to itself. And in this process, Time, being itself the

dimensionless element, only comes to a formal identity with itself and

reaches the square: Space, on the other hand, as a positive external

relation, comes to the full dimensions of the conception of space,

that is, the cube. The Realization of the two conceptions (space and

time) preserves their original difference. This is the third Keplerian

law, the relation of the Cubes of the distances to the squares of the

times."

"And this," he adds, (_v_) with remarkable complacency, "represents

simply and immediately _the reason of the thing_:--while on the

contrary, the Newtonian Formula, by means of which the Law is changed

into a Law for the Force of Gravity, shows the distortion and inversion

of _Reflexion_, which stops half-way."

I am not able to assign any precise meaning to the _Reflexion_, which

is here used as a term of condemnation, applicable especially to the

Newtonian doctrine. It is repeatedly applied in the same manner by

Hegel. Thus he says, (_g_) "that what Kepler expresses in a simple and

sublime manner in the form of Laws of the Celestial Motions, Newton has

metamorphosed into the _Reflexion-Form_ of the Force of Gravitation."

Though Hegel thus denies Newton all merit with regard to the

explanation of Kepler's laws by means of the gravitation of the

planets to the sun, he allows that to the Keplerian Laws Newton added

the Principle of Perturbations (_k_). This Principle he accepts to a

certain extent, transforming the expression of it after his peculiar

fashion. "It lies," he says, (_l_) "in this: that matter in general

assigns a center for itself: the collective bodies of the system

recognise a reference to their sun, and all the individual bodies,

according to the relative positions into which they are brought by

their motions, form a momentary relation of their gravity towards each

other."

This must appear to us a very loose and insufficient way of stating the

Principle of Perturbations, but loose as it is, it recognises that the

Perturbations depend upon the gravity of the planets one to another,

and to the sun. And if the Perturbations depend upon these forces,

one can hardly suppose that any one who allows this will deny that

the primary undisturbed motions depend upon these forces, and must be

explained by means of them; yet this is what Hegel denies.

It is evident, on looking at Hegel's mode of reasoning on such

subjects, that his views approach towards those of Aristotle and the

Aristotelians; according to which motions were divided into _natural_

and _unnatural_;--the _celestial motions_ were circular and uniform in

their nature;--and the like. Perhaps it may be worth while to show how

completely Hegel adheres to these ancient views, by an extract from

the additions to the Articles on Celestial Motions, made in the last

edition of the _Encyclopædia_. He says (_w_),

"The motion of the heavenly bodies is not a being pulled this way

and that, as is imagined (by the Newtonians). _They go along, as the

ancients said, like blessed gods._ The celestial conformity is not

such a one as has the principle of rest or motion external to itself.

It is not right to say because a stone is inert, and the whole earth

consists of stones, and the other heavenly bodies are of the same

nature as the earth, therefore the heavenly bodies are inert. This

conclusion makes the properties of the whole the same as those of the

part. Impulse, Pressure, Resistance, Friction, Pulling, and the like,

are valid only for other than celestial matter."

There can be no doubt that this is a very different doctrine from that

of Newton.

I will only add to these specimens of Hegel's physics, a specimen of

the logic by which he refutes the Newtonian argument which has just

been adduced; namely, that the celestial bodies are matter, and that

matter, as we see in terrestrial matter, is inert. He says (_x_),

"Doubtless both are matter, as a good thought and a bad thought are

both thoughts; but the bad one is not therefore good, because it is a

thought."

APPENDIX TO THE MEMOIR ON HEGEL'S CRITICISM OF NEWTON'S PRINCIPIA.

HEGEL. _Encyclopædia_ (2nd Ed. 1827), Part XI. p. 250.

C. _Absolute Mechanics._

§ 269.

Gravitation is the true and determinate conception of material

Corporeity, which (Conception) is realized to the Idea (zur Idee).

_General_ Corporeity is separable essentially into _particular_

Bodies, and connects itself with the Element of _Individuality_ or

subjectivity, as apparent (phenomenal) presence in the _Motion_, which

by this means is immediately a system of _several Bodies_.

Universal gravitation must, as to itself, be recognised as a profound

thought, although it was principally as apprehended in the sphere

of Reflexion that it eminently attracted notice and confidence on

account of the quantitative determinations therewith connected, and

was supposed to find its confirmation in _Experiments_ (Erfahrung)

pursued from the Solar System down to the phenomena of Capillary

Tubes.--But Gravitation contradicts immediately the Law of Inertia,

for in virtue of it (Gravitation) matter tends _out of itself_ to the

other (matter).--In the _Conception of Weight_, there are, as has been

shown, involved the two elements--Self-existence, and Continuity,

which takes away self-existence. These elements of the Conception,

however, experience a fate, as particular forces, corresponding to

Attractive and Repulsive Force, and are thereby apprehended in nearer

determination, as _Centripetal_ and _Centrifugal Force_, which (Forces)

like weight, _act upon Bodies_, independent of each other, and are

supposed to come in contact accidentally in a third thing, Body. By

this means, what there is of profound in the thought of universal

weight is again reduced to nothing; and Conception and Reason cannot

make their way into the doctrine of absolute motion, so long as the

so highly-prized discoveries of Forces are dominant there. In the

conclusion which contains the _Idea_ of Weight, namely, [contains

this Idea] as the Conception which, in the case of motion, enters

into external Reality through the particularity of the Bodies, and

at the same time into this [Reality] and into their Ideality and

self-regarding Reflexion, (Reflexion-in-sich), the rational identity

and inseparability of the elements is involved, which at other times

are represented as independent. Motion itself, as such, has only its

meaning and existence in a system of _several_ bodies, and those,

such as stand in relation to each other according to different

determinations.

§ 270.

As to what concerns bodies in which the conception of gravity

(weight) is realized free by itself, we say that they have for the

determinations of their different nature the elements (momente) of

their conception. One [conception of this kind] is the _universal_

center of the abstract reference [of a body] to itself. Opposite

to this [conception] stands the immediate, extrinsic, centerless

_Individuality_, appearing as _Corporeity_ similarly independent. Those

[Bodies] however which are particular, which stand in the determination

of extrinsic, and at the same time of intrinsic relation, are centers

for themselves, and [also] have a reference to the first as to their

essential unity.

The Planetary Bodies, as the immediately concrete, are in their

existence the most complete. Men are accustomed to take the Sun

as the most excellent, inasmuch as the understanding prefers the

abstract to the concrete, and in like manner the fixed stars are

esteemed higher than the Bodies of the Solar System. Centerless

Corporeity, as belonging to externality, naturally separates itself

into the opposition of the lunar and the cometary Body. The laws

of absolutely free motion, as is well known, were discovered by

Kepler;--a discovery of immortal fame. Kepler has proved these

laws in this sense, that for the empirical data he found their

(_a_) general expression. Since then, (_a_) it has become a common way

of speaking to say that Newton first found out the proof of these

Laws. It has rarely happened that fame has been more unjustly

transferred from the first discoverer to another person. On this

subject I make the following remarks.

1. That it is allowed by Mathematicians that the Newtonian

(_b_) Formulæ may be derived from the Keplerian Laws. The

completely immediate derivation is this: In the third Keplerian

(_c_) Law, _A_^3/_T_^2 is the constant quantity. This being put

as _A.A_^2/_T_^2 and calling, with Newton, _A_/_T_^2 universal

Gravitation, his expression of the effect of gravity in the

reciprocal ratio of the square of the distances is obvious.

(_d_) 2. That the Newtonian proof of the Proposition that a body

subjected to the Law of Gravitation moves about the central body

in an _Ellipse_, gives a _Conic Section_ generally, while the

main Proposition which ought to be proved is that the fall of

such a Body is _not_ a _Circle or any other Conic Section_, but

an _Ellipse only_. Moreover, there are objections which may be

made against this proof in itself (_Princ. Math._ I. 1. Sect. II.

(_e_) Prop. 1); and although it is the foundation of the Newtonian

Theory, analysis has no longer any need of it. The conditions which

in the sequel make the path of the Body to a determinate Conic

Section, are referred to an _empirical_ circumstance, namely, a

particular position of the Body at a determined moment of time, and

(_f_) the _casual_ strength of an _impulsion_ which it is supposed

to have received originally; so that the circumstance which makes

the Curve be an Ellipse, which alone ought to be the thing proved,

is extraneous to the Formula.

3. That the Newtonian Law of the so-called Force of Gravitation is

in like manner only proved from experience by Induction.

(_g_) The sum of the difference is this, that what Kepler

expressed in a simple and sublime manner in the Form of _Laws_

_of the Celestial Motions_, Newton has metamorphosed into the

_Reflection-Form_ of the _Force of Gravitation_. If the Newtonian

Form has not only its convenience but its necessity in reference

(_h_) to the analytical method, this is only a difference of the

mathematical formulæ; Analysis has long been able to derive the

Newtonian expression, and the Propositions therewith connected,

out of the Form of the Keplerian Laws; (on this subject I refer

(_i_) to the elegant exposition in _Francœur's Traité Elém. de

Mécanique_, Liv. II. Ch. xi. n. 4.)--The old method of so-called

proof is conspicuous as offering to us a tangled web, formed of

the _Lines_ of the mere geometrical construction, to which a

physical meaning of _independent Forces_ is given; and of empty

Reflexion-determinations of the already mentioned _Accelerating

Force_ and _Vis Inertiæ_, and especially of the relation of

the so-called gravitation itself to the centripetal force and

centrifugal force, and so on.

The remarks which are here made would undoubtedly have need of

a further explication to show how well founded they are: in a

Compendium, propositions of this kind which do not agree with

that which is assumed, can only have the shape of assertions.

Indeed, since they contradict such high authorities, they must

appear as something worse, as presumptuous assertions. I will not,

on this subject, support myself by saying, by the bye, that an

interest in these subjects has occupied me for 25 years; but it

is more precisely to the purpose to remark, that the distinctions

and determinations which Mathematical Analysis introduces,

and the course which it must take according to its method, is

altogether different from that which a physical reality must

have. The Presuppositions, the Course, and the Results, which

the Analysis necessarily has and gives, remain quite extraneous

to the considerations which determine the physical value and the

signification of those determinations and of that course. To this

it is that attention should be directed. We have to do with a

consciousness relative to the deluging of physical Mechanics with

an _inconceivable_ (unsäglichen) _Metaphysic_, which--contrary to

experience and conception--has those mathematical determinations

alone for its source.

It is recognized that what Newton--besides the foundation of the

analytical treatment, the development of which, by the bye, has

of itself rendered superfluous, or indeed rejected much which

belonged to Newton's essential Principles and glory--has added

to the Keplerian Laws is the Principle of _Perturbations_,--a

Principle whose importance we may here accept thus far (hier in

(_k_) sofern anzuführen ist); namely, so far as it rests upon the

Proposition that the so-called attraction is an operation of all

(_l_) the individual parts of bodies, as being material. It lies

in this, that matter in general assigns a center for itself (sich

das centrum setzt), and the figure of the body is an element in the

determination of its place; that collective bodies of the system

recognize a reference to their Sun (sich ihre Sonne setzen), but

also the individual bodies themselves, according to the relative

position with regard to each other into which they come by their

general motion, form a momentary relation of their gravity

(schwere) _towards each other_, and are related to each other not

only in abstract spatial relations, but at the same time assign to

themselves a joint center, which however is again resolved [into

the general center] in the universal system.

(_m_) As to what concerns the features of the path, to show how

the fundamental determinations of Free Motion are connected _with

the Conception_, cannot here be undertaken in a satisfactory and

detailed manner, and must therefore be left to its fate. The proof

from reason of the quantitative determinations of free motion can

only rest upon the _determinations_ of _Conceptions_ of space and

time, the elements whose relation (intrinsic not extrinsic) motion

is.

That, _in the first place_, the motion in general is a motion

_returning into itself_, is founded on the determination of

particularity and individuality of the bodies in general (§ 269),

so that partly they have a center in themselves, and partly at the

same time their center in another. These are the determinations of

Conceptions which form the basis of the false representatives of

(_n_) Centripetal Force and Centrifugal Force, as if each of these

were self-existing, extraneous to the other, and independent of

it; and as if they only came in contact in their operations and

consequently _externally_. They are, as has already been mentioned,

the Lines which must be drawn for the mathematical determinations,

transformed into physical realities.

Further, this motion is _uniformly accelerated_, (and--as returning

into itself--in turn uniformly retarded). In motion as _free_,

Time and Space enter as _different_ things which are to make

(_o_) themselves effective in the determination of the motion

(§ 266, note). In the so-called _Explanation_ of the uniformly

accelerated and retarded motion, by means of the alternate

decrease and increase of the magnitude of the Centripetal Force

and Centrifugal Force, the _confusion_ which the assumption of

(_p_) such independent Forces produces is at its greatest height.

According to this explanation, in the motion of a Planet from the

Aphelion to the Perihelion, the centrifugal is _less_ than the

centripetal force, and on the contrary, in the Perihelion itself,

the centrifugal force is supposed to become greater than the

centripetal. For the motion from the Perihelion to the Aphelion,

this representation makes the forces pass into the opposite

relation in the same manner. It is apparent that such a sudden

conversion of the preponderance which a force has obtained over

another, into an inferiority to the other, cannot be anything taken

out of the nature of Forces. On the contrary it must be concluded,

that a preponderance which one Force has obtained over another

must not only be preserved, but must go onwards to the complete

annihilation of the other Force, and the motion must either, by

the Preponderance of the Centripetal Force, proceed till it ends

in rest, that is, in the Collision of the Planet with the Central

(_q_) Body, or till by the Preponderance of the Centrifugal Force

it ends in a straight line. But now, if in place of the suddenness

of the conversion, we suppose a gradual increase of the Force in

question, then, since rather the other Force ought to be assumed as

increasing, we lose the opposition which is assumed for the sake

of the explanation; and if the increase of the one is assumed to

be different from that of the other, (which is the case in some

representations,) then there is found at the mean distance between

the apsides a point in which the Forces are _in equilibrio_. And

the transition of the Forces out of Equilibrium is a thing just as

little without any sufficient reason as the aforesaid suddenness

of inversion. And in the whole of this kind of explanation, we see

that the mode of remedying a bad mode of dealing with a subject

leads to newer and greater confusion.--A similar confusion makes

its appearance in the explanation of the phænomenon that the

pendulum oscillates more slowly at the equator. This phænomenon is

ascribed to the Centrifugal Force, which it is asserted must then

be greater; but it is easy to see that we may just as well ascribe

it to the augmented gravity, inasmuch as that holds the pendulum

more strongly to the perpendicular line of rest.

§ 240.

(_r_) And now first, as to what concerns the _Form of the Path_,

the _Circle_ only can be conceived as the path of an _absolutely

uniform_ motion. _Conceivable_, as people express it, no doubt it

is, that an increasing and diminishing motion should take place in

a circle. But this conceivableness or possibility means only an

abstract capability of being represented, which leaves out of sight

that Determinate Thing on which the question turns.

The Circle is the line returning into itself in which all the radii

are _equal_, that is, it is completely determined by means of the

radius. There is only _one_ Determination, and that is the _whole_

Determination.

But in free motion, in which the Determinations according to

space and according to time come into view with Differences, in a

qualitative relation to each other, this Relation appears in the

spatial aspect as a _Difference_ thereof in itself, which therefore

requires two Determinations. Hereby the Form of the path returning

into itself is essentially an _Ellipse_.

(_s_) The abstract Determinations which produces the circle

appears also in this way, that the arc or angle which is included

by two Radii is independent of them, a magnitude with regard to

them completely empirical. But since in the motion as determined

by the Conception, the distance from the center, and the arc

which is run over in a certain time, must be comprehended in one

determinateness, [_and_] make out a whole, this is the sector, a

space-determination of two dimensions: in this way, the arc is

essentially a Function of the Radius Vector; and the former (the

arc) being unequal, brings with it the inequality of the Radii.

That the determination with regard to the space by means of the

(_t_) time appears as a Determination of two Dimensions,--as a

Superficies-Determination,--agrees with what was said before (§

266) respecting Falling Bodies, with regard to the exposition of

the same Determinateness, at one while as Time in the root, at

another while as Space in the Square. Here, however, the Quadratic

character of the space is, by the returning of the Line of motion

into itself, limited to a Sector. These are, as may be seen, the

general principles on which the Keplerian Law, that in equal times

equal sectors are cut off, rests.

This Law becomes, as is clear, only the relation of the arc to

the Radius Vector, and the Time enters there as the abstract

Unity, in which the different Sectors are compared, because as

Unity it is the Determining Element. But the further relation is

that of the Time, not as Unity, but as a Quantity in general,--as

the time of Revolution--to the magnitude of the Path, or, what is

the same thing, the distance from the center. As Root and Square,

we saw that Time and Space had a relation to each other, in the

case of Falling Bodies, the case of half-free motion--because

that [_motion_] is determined on one side by the conception, on

(_u_) the other by external [_conditions_]. But in the case of

absolute motion--the domain of _free_ masses--the determination

attains its Totality. The Time as the Root is a mere empirical

magnitude; but as a component (moment) of the developed Totality,

it is a Totality in itself,--it produces itself, and therein has

a reference to itself; as the Dimensionless Element in itself, it

only comes to a formal identity with itself, the Square; Space,

on the other hand, as the positive Distribution (aussereinander)

[_comes_] to the Dimension of the Conception, _the_ CUBE. Their

(_v_) Realization preserves their original difference. This is the

third Keplerian Law, the relation of the _Cubes_ of the _Distances_

to the _Squares_ of the _Times_;--a Law which is so great on this

account, that it represents so simply and immediately _Reason

as belonging to the thing_: while on the contrary the Newtonian

Formula, by means of which the Law is changed into a Law for the

Force of Gravity, shows the Distortion, Perversion and Inversion of

_Reflexion_ which stops half-way.

Additions to new Edition. § 269.

The center has no sense without the circumference, nor the

circumference without the center. This makes all physical

hypotheses vanish which sometimes proceed from the center,

sometimes from the particular bodies, and sometimes assign this,

sometimes that, as the original [cause of motion] ... It is silly

(läppisch) to suppose that the centrifugal force, as a tendency to

fly off in a Tangent, has been produced by a lateral projection, a

projectile force, an impulse which they have retained ever since

they set out on their journey (von Haus aus). Such casualty of the

motion produced by external causes belongs to inert matter; as when

a stone fastened to a thread which is thrown transversely tries to

fly from the thread. We are not to talk in this way of Forces. If

we will speak of Force, there is one Force, whose elements do not

(_w_) draw bodies to different sides as if they were two Forces.

The motion of the heavenly bodies is not a being pulled this way or

that, such as is thus imagined; it is free motion: they go along,

as the ancients said, as blessed Gods (sie gehen als selige Götter

einher). The celestial corporeity is not such a one as has the

principle of rest or motion external to itself. Because stone is

inert, and all the earth consists of stones, and the other heavenly

bodies are of the same nature,--is a conclusion which makes the

properties of the whole the same as those of the part. Impulse,

Pressure, Resistance, Friction, Pulling, and the like, are valid

(_x_) only for an existence of matter other than the celestial.

Doubtless that which is common to the two is matter, as a good

thought and a bad thought are both thoughts; but the bad one is not

therefore good, because it is a thought.

FOOTNOTES:

[Footnote 353: Cartes. _Princip._ iv. 23.]

[Footnote 354: Jac. Bernoulli, _Nouvelles Pensées sur le Système de M.

Descartes_, op. t. i. p. 239 (1686).]

[Footnote 355: _De la Cause de la Pesanteur_ (1689), p. 135.]

[Footnote 356: _Journal des Savans_, 1703. Mém. Acad. Par. 1709.

Bulfinger, in 1726 (Acad. Petrop.), conceived that by making a sphere

revolve at the same time about two axes at right angles to each other,

every particle would describe a great circle; but this is not so.]

[Footnote 357: Acad. Par. 1714, _Hist._ p. 106.]

[Footnote 358: Acad. Par. 1733.]

[Footnote 359: Acad. Sc. 1709. If we abandon the clear principles of

mechanics, the writer says, "toute la lumière que nous pouvons avoir

est éteinte, et nous voilà replongés de nouveau dans les anciennes

ténèbres du Peripatetisme, dont le Ciel nous veuille preserver!"

It was also objected to the Newtonian system, that it did not account

for the remarkable facts, that all the motions of the primary planets,

all the motions of the satellites, and all the motions of rotation,

including that of the sun, are in the same direction, and nearly in

the same plane; facts which have been urged by Laplace as so strongly

recommending the Nebular Hypothesis; and that hypothesis is, in truth,

a hypothesis of vortices respecting the _origin_ of the system of the

world.]

[Footnote 360: _Nouvelle Physique Céleste_, Op. t. iii. p. 163.

The deviation of the orbits of the planets from the plane of the

sun's equator was of course a difficulty in the system which supposed

that they were carried round by the vortices which the sun's rotation

caused, or at least rendered evident. Bernoulli's explanation consists

in supposing the planets to have a sort of _leeway_ (_dérive des

vaisseaux_) in the stream of the vortex.]

[Footnote 361: See _Hist. Sc. Ideas_, b. iii. c. ix. Art. 7.]

[Footnote 362: See Mill's _Logic_, vol. i. p. 311, 2nd ed.]

[Footnote 363: These letters refer to passages in the Translation

annexed to this Memoir.]

APPENDIX K.

DEMONSTRATION THAT ALL MATTER IS HEAVY.

(_Cam. Phil. Soc._ FEB. 22, 1841.)

The discussion of the nature of the grounds and proofs of the most

general propositions which the physical sciences include, belongs

rather to Metaphysics than to that course of experimental and

mathematical investigation by which the sciences are formed. But such

discussions seem by no means unfitted to occupy the attention of the

cultivators of physical science. The ideal, as well as the experimental

side of our knowledge must be carefully studied and scrutinized,

in order that its true import may be seen; and this province of

human speculation has been perhaps of late unjustly depreciated and

neglected by men of science. Yet it can be prosecuted in the most

advantageous manner by them only: for no one can speculate securely

and rightly respecting the nature and proofs of the truths of science

without a steady possession of some large and solid portions of such

truths. A man must be a mathematician, a mechanical philosopher, a

natural historian, in order that he may philosophize well concerning

mathematics, and mechanics, and natural history; and the mere

metaphysician who without such preparation and fitness sets himself

to determine the grounds of mathematical or mechanical truths, or the

principles of classification, will be liable to be led into error at

every step. He must speculate by means of general terms, which he

will not be able to use as instruments of discovering and conveying

philosophical truth, because he cannot, in his own mind, habitually and

familiarly, embody their import in special examples.

Acting upon such views, I have already laid before the Philosophical

Society of Cambridge essays on such subjects as I here refer to;

especially a memoir "On the Nature of the Truth of the Laws of Motion,"

which was printed by the Society in its Transactions. This memoir

appears to have excited in other places, notice of such a kind as to

show that the minds of many speculative persons are ready for and

inclined towards the discussion of such questions. I am therefore the

more willing to bring under consideration another subject of a kind

closely related to the one just mentioned.

The general questions which all such discussions suggest, are (in

the existing phase of English philosophy) whether certain proposed

scientific truths, (as the laws of motion,) be _necessary_ truths;

and if they are necessary, (which I have attempted to show that in a

certain sense they are,) _on what ground_ their necessity rests. These

questions may be discussed in a general form, as I have elsewhere

attempted to show. But it may be instructive also to follow the general

arguments into the form which they assume in special cases; and to

exhibit, in a distinct shape, the incongruities into which the opposite

false doctrine leads us, when applied to particular examples. This

accordingly is what I propose to do in the present memoir, with regard

to the proposition stated at the head of this paper, namely, that _all

matter is heavy_.

At first sight it may appear a doctrine altogether untenable to assert

that this proposition is a necessary truth: for, it may be urged, we

have no difficulty in conceiving matter which is not heavy; so that

matter without weight is a conception not inconsistent with itself;

which it must be if the reverse were a necessary truth. It may be

added, that the possibility of conceiving matter without weight

was shown in the controversy which ended in the downfall of the

phlogiston theory of chemical composition; for some of the reasoners

on this subject asserted phlogiston to be a body with positive levity

instead of gravity, which hypothesis, however false, shows that such

a supposition is possible. Again, it may be said that _weight_ and

_inertia_ are two separate properties of matter: that mathematicians

measure the quantity of matter by the inertia, and that we learn

by experiment only that the weight is proportional to the inertia;

Newton's experiments with pendulums of different materials having been

made with this very object.

I proceed to reply to these arguments. And first, as to the possibility

of conceiving matter without weight, and the argument thence

deduced, that the universal gravity of matter is not a necessary

truth, I remark, that it is indeed just, to say that we cannot even

distinctly conceive the contrary of a necessary truth to be true;

but that this impossibility can be asserted only of those perfectly

distinct conceptions which result from a complete development of

the fundamental idea and its consequences. Till we reach this stage

of development, the obscurity and indistinctness may prevent our

perceiving absolute contradictions, though they exist. We have abundant

store of examples of this, even in geometry and arithmetic; where the

truths are universally allowed to be necessary, and where the relations

which are impossible, are also inconceivable, that is, not conceivable

distinctly. Such relations, though not distinctly conceivable, still

often appear conceivable and possible, owing to the indistinctness

of our ideas. Who, at the first outset of his geometrical studies,

sees any impossibility in supposing the side and the diagonal of a

square to have a common measure? Yet they can be rigorously proved to

be incommensurable, and therefore the attempt distinctly to conceive

a common measure of them must fail. The attempts at the geometrical

duplication of the cube, and the supposed solutions, (as that of

Hobbes,) have involved absolute contradictions; yet this has not

prevented their being long and obstinately entertained by men, even of

minds acute and clear in other respects. And the same might be shewn to

be the case in arithmetic. It is plain, therefore, that we cannot, from

the supposed possibility of conceiving matter without weight, infer

that the contrary may not be a necessary truth.

Our power of judging, from the compatibility or incompatibility of our

conceptions, whether certain propositions respecting the relations of

ideas are true or not, must depend entirely, as I have said, upon the

degree of development which such ideas have undergone in our minds.

Some of the relations of our conceptions on any subject are evident

upon the first steady contemplation of the fundamental idea by a sound

mind: these are the _axioms_ of the subject. Other propositions may be

deduced from the axioms by strict logical reasoning. These propositions

are no less _necessary_ than the axioms, though to common minds their

_evidence_ is very different. Yet as we become familiar with the steps

by which these ulterior truths are deduced from the axioms, _their_

truth also becomes evident, and the contrary becomes inconceivable.

When a person has familiarized himself with the first twenty-six

propositions of Euclid, and not till then, it becomes evident to him,

that parallelograms on the same base and between the same parallels

are equal; and he cannot even conceive the contrary. When he has a

little further cultivated his geometrical powers, the equality of the

square on the hypothenuse of a right-angled triangle to the squares on

the sides, becomes also evident; the steps by which it is demonstrated

being so familiar to the mind as to be apprehended without a conscious

act. And thus, the contrary of a necessary truth cannot be distinctly

conceived; but the incapacity of forming such a conception is a

condition which depends upon cultivation, being intimately connected

with the power of rapidly and clearly perceiving the connection of the

necessary truth under consideration with the elementary principles on

which it depends. And thus, again, it may be that there is an absolute

impossibility of conceiving matter without weight; but then, this

impossibility may not be apparent, till we have traced our fundamental

conceptions of matter into some of their consequences.

The question then occurs, whether we can, by any steps of reasoning,

point out an inconsistency in the conception of matter without weight.

This I conceive we may do, and this I shall attempt to show.

The general mode of stating the argument is this:--the quantity of

matter is measured by those sensible properties of matter which undergo

quantitative addition, subtraction and division, as the matter is

added, subtracted and divided. The quantity of matter cannot be known

in any other way. But this mode of measuring the quantity of matter,

in order to be true at all, must be universally true. If it were only

partially true, the limits within which it is to be applied would be

arbitrary; and therefore the whole procedure would be arbitrary, and,

as a method of obtaining philosophical truth, altogether futile.

We may unfold this argument further. Let the contrary be supposed, of

that which we assert to be true: namely, let it be supposed that while

all other kinds of matter are heavy (and of course heavy in proportion

to the quantity of matter), there is one kind of matter which is

absolutely destitute of weight; as, for instance, phlogiston, or any

other element. Then where this _weightless_ element (as we may term

it) is mixed with _weighty_ elements, we shall have a compound, in

which the weight is no longer proportional to the quantity of matter.

If, for example, 2 measures of heavy matter unite with one measure

of phlogiston, the weight is as 2, and the quantity of matter as 3.

In all such cases, therefore, the weight ceases to be the measure of

the quantity of matter. And as the proportion of the weighty and the

weightless matter may vary in innumerable degrees in such compounds,

the weight affords no criterion at all of the quantity of matter

in them. And the smallest admixture of the weightless element is

sufficient to prevent the weight from being taken as the measure of the

quantity of matter.

But on this hypothesis, how are we to distinguish such compounds from

bodies consisting purely of heavy matter? How are we to satisfy

ourselves that there is not, in every body, some admixture, small or

great, of the weightless element? If we call this element _phlogiston_,

how shall we know that the bodies with which we have to do are, any of

them, absolutely free from phlogiston?

We cannot refer to the weight for any such assurance; for by

supposition the presence and absence of phlogiston makes no difference

in the weight. Nor can any other properties secure us at least from

a very small admixture; for to assert that a mixture of 1 in 100 or

1 in 10 of phlogiston would always manifest itself in the properties

of the body, must be an arbitrary procedure, till we have proved this

assertion by experiment: and we cannot do this till we have learnt

some mode of measuring the quantities of matter in bodies and parts of

bodies; which is exactly what we question the possibility of, in the

present hypothesis.

Thus, if we assume the existence of an element, _phlogiston_, devoid of

weight, we cannot be sure that every body does not contain some portion

of this element; while we see that if there be an admixture of such

an element, the weight is no longer any criterion of the quantity of

matter. And thus we have proved, that if there be any kind of matter

which is not heavy, the weight can no longer avail us, _in any case or

to any extent_, as a measure of the quantity of matter.

I may remark, that the same conclusion is easily extended to the case

in which phlogiston is supposed to have absolute levity; for in that

case, a certain mixture of phlogiston and of heavy matter would have

no weight, and might be substituted for phlogiston in the preceding

reasoning.

I may remark, also, that the same conclusion would follow by the same

reasoning, if any kind of matter, instead of being void of weight, were

heavy, indeed, but not _so_ heavy, in proportion to its quantity of

matter, as other kinds.

On all these hypotheses there would be no possibility of measuring

quantity of matter by weight at all, in any case, or to any extent.

But it may be urged, that we have not yet reduced the hypothesis of

matter without weight to a contradiction; for that mathematicians

measure quantity of matter, not by weight, but by the other property,

of which we have spoken, inertia.

To this I reply, that, practically speaking, quantity of matter is

always measured by weight, both by mechanicians and chemists: and as

we have proved that this procedure is utterly insecure in all cases,

on the hypothesis of weightless matter, the practice rests upon a

conviction that the hypothesis is false. And yet the practice is

universal. Every experimenter measures quantity of matter by the

balance. No one has ever thought of measuring quantity of matter by

its inertia practically: no one has constructed a measure of quantity

of matter in which the matter produces its indications of quantity by

its motion. When we have to take into account the inertia of a body,

we inquire what its weight is, and assume this as the measure of the

inertia; but we never take the contrary course, and ascertain the

inertia first in order to determine by that means the weight.

But it may be asked, Is it not then true, and an important scientific

truth, that the _quantity of matter_ is measured by the _inertia_?

Is it not true, and proved by experiment, that the _weight_ is

_proportional_ to the _inertia_? If this be not the result of Newton's

experiments mentioned above, what, it may be demanded, do they prove?

To these questions I reply: It is true that quantity of matter is

measured by the inertia, for it is true that inertia is as the quantity

of matter. This truth is indeed one of the laws of motion. That weight

is proportional to inertia is proved by experiment, as far as the laws

of motion are so proved: and Newton's experiments prove one of the laws

of motion, so far as any experiments can prove them, or are needed to

prove them.

That inertia is proportional to weight, is a law equivalent to that

law which asserts, that when pressure produces motion in a given body,

the velocity produced in a given time is as the pressure. For if the

velocity be as the pressure, when the body is given, the velocity will

be constant if the inertia also be as the pressure. For the inertia is

understood to be that property of bodies to which, _ceteris paribus_,

the velocity impressed is _inversely_ proportional. One body has

twice as much inertia as another, if, when the same force acts upon

it for the same time, it acquires but half the velocity. This is the

fundamental conception of _inertia_.

In Newton's pendulum experiments, the pressure producing motion was

a certain resolved part of the weight, and was proportional to the

weight. It appeared by the experiments, that whatever were the material

of which the pendulum was formed, the rate of oscillation was the same;

that is, the velocity acquired was the same. Hence the inertia of the

different bodies must have been in each case as the weight: and thus

this assertion is true of all different kinds of bodies.

Thus it appears that the assertion, that inertia is universally

proportional to weight, is equivalent to the law of motion, that the

velocity is as the pressure. The conception of inertia (of which,

as we have said, the fundamental conception is, that the velocity

impressed is inversely proportional to the inertia,) connects the two

propositions so as to make them identical.

Hence our argument with regard to the universal gravity of matter

brings us to the above law of motion, and is proved by Newton's

experiments in the same sense in which that law of motion is so proved.

Perhaps some persons might conceive that the identity of weight

and inertia is obvious at once; for both are merely resistance

to motion;--inertia, resistance to all motion (or change of

motion)--weight, resistance to motion upwards.

But there is a difference in these two kinds of resistance to motion.

Inertia is instantaneous, weight is continuous resistance. Any

momentary impulse which acts upon a free body overcomes its inertia,

for it changes its motion; and this change once effected, the inertia

opposes any return to the former condition, as well as any additional

change. The inertia is thus overcome by a momentary force. But the

weight can only be overcome by a continuous force like itself. If an

impulse act in opposition to the weight, it may for a moment neutralize

or overcome the weight; but if it be not continued, the weight resumes

its effect, and restores the condition which existed before the impulse

acted.

But weight not only produces rest, when it is resisted, but motion,

when it is not resisted. Weight is measured by the reaction which would

balance it; but when unbalanced, it produces motion, and the velocity

of this motion increases constantly. Now what determines the velocity

thus produced in a given time, or its rate of increase? What determines

it to have one magnitude rather than another? To this we must evidently

reply, _the inertia_. When weight produces motion, the inertia is the

reaction which makes the motion determinate. The accumulated motion

produced by the action of unbalanced weight is as determinate a

condition as the equilibrium produced by balanced weight. In both cases

the condition of the body acted on is determined by the opposition of

the action and reaction.

Hence inertia is the reaction which opposes the weight, when

unbalanced. But by the conception of action and reaction, (as mutually

determining and determined,) they are measured by each other: and hence

the inertia is necessarily proportional to the weight.

But when we have reached this conclusion, the original objection may be

again urged against it. It may be said, that there must be some fallacy

in this reasoning, for it proves a state of things to be necessary when

we can so easily conceive a contrary state of things. Is it denied,

the opponent may ask, that we can readily imagine a state of things in

which bodies have no weight? Is not the uniform tendency of all bodies

in the same direction not only not necessary, but not even true? For

they do in reality tend, not with equal forces in parallel lines, but

to a center with unequal forces, according to their position: and we

can conceive these differences of intensity and direction in the force

to be greater than they really are; and can with equal ease suppose the

force to disappear altogether.

To this I reply, that certainly we may conceive the weight of bodies

to vary in intensity and direction, and by an additional effort of

imagination, may conceive the weight to vanish: but that in all these

suppositions, even in the extreme one, we must suppose the rule to be

universal. If _any_ bodies have weight, _all_ bodies must have weight.

If the direction of weight be different in different points, this

direction must still vary according to the _law of continuity_; and the

same is true of the intensity of the weight. For if this were not so,

the rest and motion, the velocity and direction, the permanence and

change of bodies, as to their mechanical condition, would be arbitrary

and incoherent: they would not be subject to mechanical ideas; that

is, not to ideas at all: and hence these conditions of objects would

in fact be inconceivable. In order that the universe may be possible,

that is, may fall under the conditions of intelligible conceptions, we

must be able to conceive a body at rest. But the rest of bodies (except

in the absolute negation of all force) implies the equilibrium of

opposite forces. And one of these opposite forces must be a _general_

force, as weight, in order that the universe may be governed by general

conditions. And this general force, by the conception of force, may

produce motion, as well as equilibrium; and this motion again must

be determined, and determined by general conditions; which cannot

be, except the communication of motion be regulated by an inertia

proportional to the weight.

But it will be asked, Is it then pretended that Newton's experiment,

by which it was intended to prove inertia proportional to weight,

does really prove nothing but what may be demonstrated _à priori_?

Could we know, without experiment, that all bodies,--gold, iron, wood,

cork,--have inertia proportional to their weight? And to this we reply,

that experiment holds the same place in the establishment of this, as

of the other fundamental doctrines of mechanics. Intercourse with the

external world is requisite for developing our ideas; measurement of

phenomena is needed to fix our conceptions and to render them precise:

but the result of our experimental studies is, that we reach a

position in which our convictions do not rest upon experiment. We learn

by observation truths of which we afterwards see the necessity. This is

the case with the laws of motion, as I have repeatedly endeavoured to

show. The same will appear to be the case with the proposition, that

bodies of different kinds have their inertia proportional to their

weight.

For bodies _of the same kind_ have their inertia proportional to their

weight, both quantities being proportional to the quantity of matter.

And if we compress the same quantity of matter into half the space,

neither the weight nor the inertia is altered, because these depend

on the quantity of matter alone. But in this way we obtain a body

of _twice the density_; and in the same manner we obtain a body of

any other density. Therefore whatever be the density, the inertia is

proportional to the quantity of matter. But the mechanical relations

of bodies cannot depend upon any difference of _kind_, _except_ a

difference of density. For if we suppose any fundamental difference of

mechanical nature in the particles or component elements of bodies, we

are led to the same conclusion, of arbitrary, and therefore impossible,

results, which we deduced from this supposition with regard to weight.

Therefore all bodies of different density, and hence, all bodies

whatever, must have their inertia proportional to their weight.

Hence we see, that the propositions, that all bodies are heavy, and

that inertia is proportional to weight, necessarily follow from those

fundamental ideas which we unavoidably employ in all attempts to

reason concerning the mechanical relations of bodies. This conclusion

may perhaps appear the more startling to many, because they have

been accustomed to expect that fundamental ideas and their relations

should be self-evident at our first contemplation of them. This,

however, is far from being the case, as I have already shown. It is

not the _first_, but the most complete and developed condition of our

conceptions which enables us to see what are axiomatic truths in each

province of human speculation. Our fundamental ideas are necessary

conditions of knowledge, universal forms of intuition, inherent types

of mental development; they may even be termed, if any one chooses,

results of connate intellectual tendencies; but we cannot term them

_innate_ ideas, without calling up a large array of false opinions. For

innate ideas were considered as capable of composition, but by no means

of simplification: as most perfect in their original condition; as to

be found, if any where, in the most uneducated and most uncultivated

minds; as the same in all ages, nations, and stages of intellectual

culture; as capable of being referred to at once, and made the basis

of our reasonings, without any special acuteness or effort: in all

which circumstances the Fundamental Ideas of which we have spoken, are

opposed to Innate Ideas so understood.

I shall not, however, here prosecute this subject. I will only remark,

that Fundamental Ideas, as we view them, are not only not innate, in

any usual or useful sense, but they are not necessarily _ultimate_

elements of our knowledge. They are the results of our analysis so far

as we have yet prosecuted it; but they may themselves subsequently

be analysed. It may hereafter appear, that what we have treated as

different Fundamental Ideas have, in fact, a connexion, at some point

below the structure which we erect upon them. For instance, we treat of

the mechanical ideas of force, matter, and the like, as distinct from

the idea of substance. Yet the principle of measuring the quantity of

matter by its weight, which we have deduced from mechanical ideas, is

applied to determine the substances which enter into the composition

of bodies. The idea of substance supplies the axiom, that the whole

quantity of matter of a compound body is equal to the sum of the

quantities of matter of its elements. The mechanical ideas of force

and matter lead us to infer that the quantity both of the whole and

its parts must be measured by their weights. _Substance_ may, for some

purposes, be described as that to which properties belong; _matter_

in like manner may be described as that which resists force. The

former involves the Idea of permanent Being; the latter, the Idea of

Causation. There may be some elevated point of view from which these

ideas may be seen to run together. But even if this be so, it will by

no means affect the validity of reasonings founded upon these notions,

when duly determined and developed. If we once adopt a view of the

nature of knowledge which makes necessary truth possible at all, we

need be little embarrassed by finding how closely connected different

necessary truths are; and how often, in exploring towards their roots,

different branches appear to spring from the same stem.

END OF THE APPENDIX.

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Transcriber's Notes

Obvious typographical errors have been silently corrected. Other

variations in spelling, punctuation and hyphenation remain unchanged.

In the Table of Contents Chap. XV. item 3. is not listed.

In the Table of Contents Chap. XXVIII. item 5.

Italics are represented thus _italics_.

The identification of the appendices skips I and J.