The above principle, that the pressure on the support is

equal to the sum of the bodies supported, is often stated as

an axiom in the outset of books on Mechanics. And this

appears to be the true place and character of this

principle, in accordance with the reasonings which we have

already urged. The axiom depends upon our conception of

action and reaction. That the two weights are supported,

implies that the supporting force must be equal to the force

or weight supported.

In order further to show the foundation of this principle,

we may ask the question:--If it be not an axiom, deriving

its truth from the fundamental conception of equal action

and reaction, which equilibrium always implies, what is the

origin of its certainty? The principle is never for an

instant denied or questioned: it is taken for granted, even

before it is stated. No one will doubt that it is not only

true, but true with the same rigour and universality as the

axioms of Geometry. Will it be said, that it is borrowed

from experience? Experience could never prove a principle to

be universally and rigorously true. Moreover, when from

experience we prove a proposition to possess great exactness

and generality, we approach by degrees to this proof: the

conviction becomes stronger, the truth more secure, as we

accumulate trials. But nothing of this kind is the case in

the instance before us. There is no gradation from less to

greater certainty;--no hesitation which precedes confidence.

From the first, we know that the axiom is exactly and

certainly true. In order to be convinced of it, we do not

require many trials, but merely a clear understanding of the

assertion itself.

But in fact, not only are trials not necessary to the proof,

but they do not strengthen it. Probably no one ever made a

trial for the purpose of showing that the pressure upon the

support is equal to the sum of the two weights. Certainly no

person with clear mechanical conceptions ever wanted such a

trial to convince him of the truth; or thought the truth

clearer after the trial had been made. If to such a person,

an {218} experiment were shown which seemed to contradict

the principle, his conclusion would be, not that the

principle was doubtful, but that the apparatus was out of

order. Nothing can be less like collecting truth from

experience than this.

We maintain, then, that this equality of mechanical action

and reaction, is one of the principles which do not flow

from, but regulate our experience. To this principle, the

facts which we observe must conform; and we cannot help

interpreting them in such a manner that they shall be

exemplifications of the principle. A mechanical pressure not

accompanied by an equal and opposite pressure, can no more

be given by experience, than two unequal right angles. With

the supposition of such inequalities, space ceases to be

space, force ceases to be force, matter ceases to be matter.

And this equality of action and reaction, considered in the

case in which two bodies are connected so as to act on a

single support, leads to the axiom which we have stated

above, and which is one of the main foundations of the

science of Mechanics.

[2d ed.] [To the doctrine that mechanical principles, such

as the one here under consideration (that the pressure on

the point of support is equal to the sum of the weights),

are derived from our Ideas, and do not flow from but

regulate our experience, objections are naturally made by

those who assert all our knowledge to be derived from

experience. How, they ask, can we know the properties of

pressures, levers and the like, except from experience? What

but experience can possibly inform us that a force applied

transversely to a lever will have any tendency to turn the

lever on its center? This cannot be, except we suppose in

the lever tenacity, rigidity and the like, which are

qualities known only by experience. And it is obvious that

this line of argument might be carried on through the whole

subject.

My answer to this objection is a remark of the same kind as

one which I have made respecting the Ideas of Space, Time,

and Number, in the last Book. The mind, in apprehending

events as causes {219} and effects, is governed by Laws of

its own Activity; and these Laws govern the results of the

mind's action; and make these results conform to the Axioms

of Causation. But this activity of the mind is awakened and

developed by being exercised; and in dealing with the

examples of cause and effect here spoken of, (namely,

pressure and resistance, force and motion,) the mind's

activity is necessarily governed also by the bodily powers

of perception and action. We are human beings only in so far

as we have existed in space and time; and of our human

faculties, developed by our existence in space and time,

space and time are necessary conditions. In like manner, we

are human beings only in so far as we have bodies, and

bodily organs; and our bodies necessarily imply material

objects external to us. And hence our human faculties,

developed by our bodily existence in a material world, have

the conditions of matter for their necessary Laws. I have

already said (chap. v.) that our conception of Force arises

with our consciousness of our own muscular exertions;--that

Force cannot be conceived without Resistance to exercise

itself upon;--and that this resistance is supplied by

Matter. And thus the conception of Matter, and of the most

general modes in which Matter receives, resists, and

transmits force, are parts of our constitution which, though

awakened and unfolded by our being in a material world, are

not distinguishable from the original structure of the mind.

I do not ascribe to the mind _innate_ Ideas--Ideas which it

would have, even if it had no intercourse with the world of

space, time, and matter; because we cannot imagine a mind in

such a state. But I attempt to point out and classify those

Conditions of all Experience, to which the intercourse of

all minds with the material world has necessarily given rise

in all. Truths _thus_ necessarily acquired in the course of

all experience, cannot be said to be learnt _from

experience_, in the same sense in which particular facts, at

definite times, are learnt from experience--learnt by some

persons and not by others--learnt with more or less of

certainty. These latter _special truths of_ {220}

_experience_ will be very important subjects of our

consideration; but our whole chance of discussing them with

any profit depends upon our keeping them distinct from the

_necessary and universal conditions of experience_. Here, as

everywhere, we must keep in view the fundamental antithesis

of Ideas and Facts.]

6. _Oblique Forces._--By the aid of the above axiom and a

few others, the Greeks made some progress in the science of

Statics. But after a short advance, they arrived at another

difficulty, that of Oblique Forces, which they never

overcame; and which no mathematician mastered till modern

times. The unpublished manuscripts of Leonardo da Vinci,

written in the fifteenth century, and the works of Stevinus

and Galileo, in the sixteenth, are the places in which we

find the first solid grounds of reasoning on the subject of

forces acting obliquely to each other. And from that period,

mathematicians, having thus become possessed of all the

mechanical principles which are requisite in problems

respecting equilibrium, soon framed a complete science of

Statics. Succeeding writers presented this science in forms

variously modified; for it was found, in Mechanics as in

Geometry, that various propositions might be taken as the

starting points; and that the collection of truths which it

was the mechanician's business to include in his course,

might thus be traversed by various routes, each path

offering a series of satisfactory demonstrations. The

fundamental conceptions of force and resistance, like those

of space and number, could be contemplated under different

aspects, each of which might be made the basis of axioms, or

of principles employed as axioms. Hence the grounds of the

truth of Statics may be stated in various ways; and it would

be a task of some length to examine all these completely,

and to trace them to their Fundamental Ideas. This I shall

not undertake here to do; but the philosophical importance

of the subject makes it proper to offer a few remarks on

some of the main principles involved in the different modes

of presenting Statics as a rigorously demonstrated science.

{221}

7. _A Force may be supposed to act at any Point of its

Direction._--It has been stated in the history of

Mechanics[12\3], that Leonardo da Vinci and Galileo obtained

the true measure of the effect of oblique forces, by

reasonings which were, in substance, the same. The principle

of these reasonings is that expressed at the head of this

paragraph; and when we have a little accustomed ourselves to

contemplate our conceptions of force, and its action on

matter, in an abstract manner, we shall have no difficulty

in assenting to the principle in this general form. But it

may, perhaps, be more obvious at first in a special case.

[Note 12\3: _Hist. Ind. Sc._ b. vi. c. i. sect. 2.]

If we suppose a wheel, moveable about its axis, and carrying

with it in its motion a weight, (as, for example, one of the

wheels by means of which the large bells of a church are

rung,) this weight may be supported by means of a rope (not

passing along the circumference of the wheel, as is usual in

the case of bells,) but fastened to one of the spokes of the

wheel. Now the principle which is enunciated above asserts,

that if the rope pass in a straight line across several of

the spokes of the wheel, it makes no difference in the

mechanical effect of the force applied, for the purpose of

putting the bell in motion, to _which_ of these spokes the

rope is _fastened_. In each case, the fastening of the rope

to the wheel merely serves to enable the force to produce

motion about the center; and so long as the force acts in

the same line, the effect is the same, at whatever point of

the rope the line of action finishes.

This axiom very readily aids us in estimating the effect of

oblique forces. For when a force acts on one of the arms of

a lever at any oblique angle, we suppose another arm

projecting from the center of motion, like another spoke of

the same wheel, so situated that it is perpendicular to the

force. This arm we may, with Leonardo, call the _virtual

lever_; for, by the axiom, we may suppose the force to act

where the line of its direction meets this arm; and thus we

reduce the case {222} to that in which the force acts

perpendicularly on the arm.

The ground of this axiom is, that matter, in Statics, is

necessarily conceived as _transmitting_ force. That force

can be transmitted from one place to another, by means of

matter;--that we can push with a rod, pull with a rope,--are

suppositions implied in our conceptions of force and matter.

Matter is, as we have said, that which receives the

impression of force, and the modes just mentioned, are the

simplest ways in which that impression operates. And since,

in any of these cases, the force might be resisted by a

reaction equal to the force itself, the reaction in each

case would be equal, and, therefore, the action in each case

is necessarily equal; and thus the forces must be

transmitted, from one point to another, without increase or

diminution.

This property of matter, of transmitting the action of

force, is of various kinds. We have the coherence of a rope

which enables us to pull, and the rigidity of a staff, which

enables us to push with it in the direction of its length;

and again, the same staff has a rigidity of another kind, in

virtue of which we can use it as a lever; that is, a

rigidity to resist flexure, and to transmit the force which

turns a body round a fulcrum. There is, further, the

rigidity by which a solid body resists _twisting_. Of these

kinds of rigidity, the first is that to which our axiom

refers; but in order to complete the list of the elementary

principles of Statics, we ought also to lay down axioms

respecting the other kinds of rigidity[13\3]. These,

however, I shall not here state, as they do not involve any

new principle. Like the one just considered, they form part

of our fundamental conception of matter; they are not the

results of any experience, but are the hypotheses to which

we are irresistibly led, when we would liberate our

reasonings concerning force and matter from a dependence on

the special results of experience. We cannot even {223}

conceive (that is, if we have any clear mechanical

conceptions at all) the force exerted by the point of a

staff and resisting the force which we steadily impress on

the head of it, to be different from the impressed force.

[Note 13\3: Such axioms are given in a little work (_The

Mechanical Euclid_) which I published on the Elements of

Mechanics.]

8. _Forces may have equivalent Forces substituted for them.

The Parallelogram of Forces._--It has already been observed,

that in order to prove the doctrines of Statics, we may take

various principles as our starting points, and may still

find a course of demonstration by which the leading

propositions belonging to the subject may be established.

Thus, instead of beginning our reasonings, as in the last

section we supposed them to commence, with the case in which

forces act upon different points of the same body in the

same line of force, and counteract each other in virtue of

the intervening matter by which the effect of force is

transferred from one point to another; we may suppose

different forces to act at the same point, and may thus

commence our reasonings with a case in which we have to

contemplate force, without having to take into our account

the resistance or rigidity of matter. Two statical forces,

thus acting at a mathematical point, are equivalent, in all

respects, to some single force acting at the same point; and

would be kept in equilibrium by a force equal and opposite

to that single force. And the rule by which the single force

is derived from the two, is commonly termed _the

parallelogram of forces_; the proposition being this,--That

if the two forces be represented in magnitude and direction

by the two sides of a parallelogram, the resulting force

will be represented in the same manner by the diagonal of

the parallelogram. This proposition has very frequently been

made, by modern writers, the commencement of the science of

Mechanics: a position for which, by its simplicity, it is

well suited; although, in order to deduce from it the other

elementary propositions of the science, as, for instance,

those respecting the lever, we require the axiom stated in

the last section.

9. _The Parallelogram of Forces is a necessary Truth._--In

the series of discussions in which we are {224} here

engaged, our main business is to ascertain the nature and

grounds of the certainty of scientific truths. We have,

therefore, to ask whether this proposition, the

parallelogram of forces, be a necessary truth; and if so, on

what grounds its necessity ultimately rests. We shall find

that this, like the other fundamental doctrines of Statics,

justly claim a demonstrative certainty. Daniel Bernoulli, in

1726, gave the first proof of this important proposition on

pure statical principles; and thus, as he says[14\3],

'proved that statical theorems are not less necessarily true

than geometrical are.' If we examine this proof of

Bernoulli, in order to discover what are the principles on

which it rests, we shall find that the reasoning employs in

its progress such axioms as this;--That if from forces which

are in equilibrium at a point be taken away other forces

which are in equilibrium at the same point, the remainder

will be in equilibrium; and generally;--That if forces can

be resolved into other equivalent forces, these may be

separated, grouped, and recombined, in any new manner, and

the result will still be identical with what it was at

first. Thus in Bernoulli's proof, the two forces to be

compounded are represented by P and Q; P is resolved into

two other forces, X and U; and Q into two others, Y and V,

under certain conditions. It is then assumed that these

forces may be grouped into the pairs X, Y, and U, V: and

when it has been shown that X and Y are in equilibrium, they

may, by what has been said, be removed, and the forces, P,

Q, are equivalent to U, V; which, being in the same

direction by the course of the construction, have a result

equal to their sum.

[Note 14\3: _Comm. Petrop._ vol. i.]

It is clear that the principles here assumed are genuine

axioms, depending upon our conception of the nature of

equivalence of forces, and upon their being capable of

addition and composition. If the forces, P, Q, be

_equivalent_ to forces X, U, Y, V, they are equivalent to

these forces added and compounded in any order; just as a

geometrical figure is, by our conception of {225} space,

equivalent to its parts added together in any order. The

apprehension of forces as having magnitude, as made up of

parts, as capable of composition, leads to such axioms in

Statics, in the same manner as the like apprehension of

space leads to the axioms of Geometry. And thus the truths

of Statics, resting upon such foundations, are independent

of experience in the same manner in which geometrical truths

are so.

The proof of the parallelogram of forces thus given by

Daniel Bernoulli, as it was the first, is also one of the

most simple proofs of that proposition which have been

devised up to the present day. Many other demonstrations,

however, have been given of the same proposition. Jacobi, a

German mathematician, has collected and examined eighteen of

these[15\3]. They all depend either upon such principles as

have just been stated; That forces may in every way be

replaced by those which are equivalent to them;--or else

upon those previously stated, the doctrine of the lever, and

the transfer of a force from one point to another of its

direction. In either case, they are necessary results of our

statical conceptions, independent of any observed laws of

motion, and indeed, of the conception of actual motion

altogether.

[Note 15\3: These are by the following mathematicians; D.

Bernoulli (1726); Lambert (1771); Scarella (1756); Venini

(1764); Araldi (1806); Wachter (1815); Kaestner; Marini;

Eytelwein; Salimbeni; Duchayla; two different proofs by

Foncenex (1760); three by D'Alembert; and those of Laplace

and M. Poisson.]

There is another class of alleged proofs of the

parallelogram of forces, which involve the consideration of

the motion produced by the forces. But such reasonings are,

in fact, altogether irrelevant to the subject of Statics. In

that science, forces are not measured by the motion which

they produce, but by the forces which they will balance, as

we have already seen. The combination of two forces employed

in producing motion in the same body, either simultaneously

or successively, {226} belongs to that part of Mechanics

which has motion for its subject, and is to be considered in

treating of the laws of motion. The composition of motion,

(as when a man moves in a ship while the ship moves through

the water,) has constantly been confounded with the

composition of force. But though it has been done by very

eminent mathematicians, it is quite necessary for us to keep

the two subjects distinct, in order to see the real nature

of the evidence of truth in either case. The conditions of

equilibrium of two forces on a lever, or of three forces at

a point, can be established without any reference whatever

to any motions which the forces might, under _other_

circumstances, produce. And because this can be done, to do

so is the only scientific procedure. To prove such

propositions by any other course, would be to support truth

by extraneous and inconclusive reasons; which would be

foreign to our purpose, since we seek not only knowledge,

but the grounds of our knowledge.

10. _The Center of gravity seeks the lowest place._--The

principles which we have already mentioned afford a

sufficient basis for the science of Statics in its most

extensive and varied applications; and the conditions of

equilibrium of the most complex combinations of machinery

may be deduced from these principles with a rigour not

inferior to that of geometry. But in some of the more

complex cases, the results of long trains of reasoning may

be foreseen, in virtue of certain maxims which appear to us

self-evident, although it may not be easy to trace the exact

dependence of these maxims upon our fundamental conceptions

of force and matter. Of this nature is the maxim now

stated;--That in any combination of matter any how

supported, the Center of Gravity will descend into the

lowest position which the connexion of the parts allows it

to assume by descending. It is easily seen that this maxim

carries to a much greater extent the principle which the

Greek mathematicians assumed, that every body has a Center

of Gravity, that is, a point in which, if the whole matter

of the body be collected, the effect will remain unchanged.

For the Greeks asserted this of a {227} single rigid mass

only; whereas, in the maxim now under our notice, it is

asserted of any masses, connected by strings, rods, joints,

or in any manner. We have already seen that more modern

writers on mechanics, desirous of assuming as fundamental no

wider principles than are absolutely necessary, have not

adopted the Greek axiom in all its generality, but have only

asserted that two _equal_ weights have a center of gravity

midway between them. Yet the principle that every body,

however irregular, has a center of gravity, and will be

supported if that center is supported, and not otherwise, is

so far evident, that it might be employed as a fundamental

truth, if we could not resolve it into any simpler truths:

and, historically speaking, it was assumed as evident by the

Greeks. In like manner the still wider principle, that a

collection of bodies, as, for instance, a flexible chain

hanging upon one or more supports, has a center of gravity;

and that this point will descend to the lowest possible

situation, as a single body would do, has been adopted at

various periods in the history of mechanics; and especially

at conjunctures when mathematical philosophers have had new

and difficult problems to contend with. For in almost every

instance it has only been by repeated struggles that

philosophers have reduced the solution of such problems to a

clear dependence upon the most simple axioms.

11. _Stevinus's Proof for Oblique Forces._--We have an

example of this mode of dealing with problems, in Stevinus's

mode of reasoning concerning the Inclined Plane; which, as

we have stated in the History of Mechanics, was the first

correct published solution of that problem. Stevinus

supposes a loop of chain, or a loop of string loaded with a

series of equal balls at equal distances, to hang over the

Inclined Plane; and his reasoning proceeds upon this

assumption,--That such a loop so hanging will find a certain

position in which it will rest: for otherwise, says

he[16\3], its motion must go on for ever, which is absurd.

It may be asked how {228} this absurdity of a perpetual

motion appears; and it will perhaps be added, that although

the impossibility of a machine with such a condition may be

proved as a remote result of mechanical principles, this

impossibility can hardly be itself recognized as a

self-evident truth. But to this we may reply, that the

impossibility is really evident in the case contemplated by

Stevinus; for we cannot conceive a loop of chain to go on

through all eternity, sliding round and round upon its

support, by the effect of its own weight. And the ground of

our conviction that this cannot be, seems to be this

consideration; that when the chain moves by the effect of

its weight, we consider its motion as the result of an

effort to reach some certain position, in which it can rest;

just as a single ball in a bowl moves till it comes to rest

at the lowest point of the bowl. Such an effect of weight in

the chain, we may represent to ourselves by conceiving all

the matter of the chain to be collected in one single point,

and this single heavy point to hang from the support in some

way or other, so as fitly to represent the mode of support

of the chain. In whatever manner this heavy point (the

center of gravity of the chain) be supported and controlled

in its movements, there will still be some position of rest

which it will seek and find. And thus there will be some

corresponding position of rest for the chain; and the

interminable shifting from one position to another, with no

disposition to rest in any position, cannot exist.

[Note 16\3: Stevin. _Statique_, livre i. prop. 19.]

Thus the demonstration of the property of the Inclined Plane

by Stevinus, depends upon a principle which, though far from

being the simplest of those to which the case can be

reduced, is still both true and evident: and the evidence of

this principle, depending upon the assumption of a center of

gravity, is of the same nature as the evidence of the Greek

statical demonstrations, the earliest real advances in the science.

12. _Principle of Virtual Velocities._--We have referred

above to an assertion often made, that we may, from the

simple principles of Mechanics, demonstrate the

impossibility of a perpetual motion. In reality, {229}

however, the simplest proof of that impossibility, in a

machine acted upon by weight only, arises from the very

maxim above stated, that the center of gravity seeks and

finds the lowest place; or from some similar proposition.

For if, as is done by many writers, we profess to prove the

impossibility of a perpetual motion by means of that

proposition which includes the conditions of equilibrium,

and is called the _Principle of Virtual Velocities_[17\3],

we are under the necessity of first proving in a general

manner that principle. And if this be done by a mere

enumeration of cases, (as by taking those five cases which

are called the _Mechanical Powers_,) there may remain some

doubts whether the enumeration of possible mechanical

combinations be complete. Accordingly, some writers have

attempted independent and general proofs of the Principle of

Virtual Velocities; and these proofs rest upon assumptions

of the same nature as that now under notice. This is, for

example, the case with Lagrange's proof, which depends upon

what he calls the _Principle of Pulleys_. For this principle

is,--That a weight any how supported, as by a string passing

round any number of pulleys any how placed, will be at rest

then only, when it cannot get lower by any small motion of

the pulleys. And thus the maxim that a weight will descend

if it can, is assumed as the basis of this proof.

[Note 17\3: See _Hist. Ind Sc._ b. vi. c. ii. sect. 4.]

There is, as we have said, no need to assume such principles

as these for the foundation of our mechanical science. But

it is, on various accounts, useful to direct our attention

to those cases in which truths, apprehended at first in a

complex and derivative form, have afterwards been reduced to

their simpler elements;--in which, also, sagacious and

inventive men have fixed upon those truths as self-evident,

which now appear to us only certain in virtue of

demonstration. In these cases we can hardly doubt that such

men were led to assert the doctrines which they discovered,

not by any capricious conjecture of arbitrary selection, but

by having a keener and deeper insight than other persons

{230} into the relations which were the object of their

contemplation; and in the science now spoken of, they were

led to their assumptions by possessing clearly and

distinctly the conceptions of mechanical cause and

effect,--action and reaction,--force, and the nature of its

operation.

13. _Fluids press Equally in all Directions._--The doctrines

which concern the equilibrium of fluids depend on principles

no less certain and simple than those which refer to the

equilibrium of solid bodies; and the Greeks, who, as we have

seen, obtained a clear view of some of the principles of

Statics, also made a beginning in the kindred subject of

Hydrostatics. We still possess a treatise of Archimedes _On

Floating Bodies_, which contains correct solutions of

several problems belonging to this subject, and of some

which are by no means easy. In this treatise, the

fundamental assumption is of this kind: 'Let it be assumed

that the nature of a fluid is such, that the parts which are

less pressed yield to those which are more pressed.' In this

assumption or axiom it is implied that a pressure exerted

upon a fluid in one direction produces a pressure in another

direction; thus, the weight of the fluid which arises from a

downward force produces a lateral pressure against the sides

of the containing vessel. Not only does the pressure thus

diverge from its original direction into all other

directions, but the pressure is in all directions exactly

equal, an equal extent of the fluid being taken. This

principle, which was involved in the reasoning of

Archimedes, is still to the present day the basis of all

hydrostatical treatises, and is expressed, as above, by

saying that _fluids press equally in all directions_.

Concerning this, as concerning previously-noticed

principles, we have to ask whether it can rightly be said to

be derived from experience. And to this the answer must

still be, as in the former cases, that the proposition is

not one borrowed from experience in any usual or exact sense

of the phrase. I will endeavour to illustrate this. There

are many elementary propositions in physics, our knowledge

of which {231} indisputably depends upon experience; and in

these cases there is no difficulty in seeing the evidence of

this dependence. In such cases, the _experiments_ which

prove the law are prominently stated in treatises upon the

subject: they are given with exact measures, and with an

account of the means by which errours were avoided: the

experiments of more recent times have either rendered more

certain the law originally asserted, or have pointed out

some correction of it as requisite: and the names, both of

the discoverers of the law and of its subsequent reformers,

are well known. For instance, the proposition that 'The

elastic force of air varies as the density,' was first

proved by Boyle, by means of operations of which the detail

is given in his _Defence_ of his _Pneumatical

Experiments_[18\3]; and by **Mariotte in his _Traité de

l'Équilibre des Liquides_, from whom it has generally been

termed Mariotte's law. After being confirmed by many other

experimenters, this law was suspected to be slightly

inaccurate, and a commission of the French Academy of

Sciences was appointed, consisting of several distinguished

philosophers[19\3], to ascertain the truth or falsehood of

this suspicion. The result of their investigations appeared

to be, that the law is exact, as nearly as the inevitable

inaccuracies of machinery and measures will allow us to

judge. Here we have an example of a law which is of the

simplest kind and form; and which yet is not allowed to rest

upon its simplicity or apparent probability, but is

rigorously tested by experience. In this case, the

assertion, that the law depends upon experience, contains a

reference to plain and notorious passages in the history of

science.

[Note 18\3: Shaw's _Boyle_, Vol. ii. p. 671.]

[Note 19\3: The members were Prony, Arago, Ampère, Girard,

and Dulong. The experiments were extended to a pressure of

twenty-seven atmospheres; and in no instance did the

difference between the observed and calculated elasticity

amount to one-hundredth of the whole; nor did the difference

appear to increase with the increase of pressure.--Fechner,

_Repertorium_, i. 110.]

Now with regard to the principle that fluids press equally

in all directions, the case is altogether different. {232}

It is, indeed, often asserted in works on hydrostatics, that

the principle is collected from experience, and sometimes a

few experiments are described as exhibiting its effect; but

these are such as to illustrate and explain, rather than to

prove, the truth of the principle: they are never related to

have been made with that exactness of precaution and

measurement, or that frequency of repetition, which are

necessary to establish a purely experimental truth. Nor did

such experiments occur as important steps in the history of

science. It does not appear that Archimedes thought

experiment necessary to confirm the truth of the law as he

employed it: on the contrary, he states it in exactly the

same shape as the axioms which he employs in statics, and

even in geometry; namely, as an assumption. Nor does any

intelligent student of the subject find any difficulty in

assenting to this fundamental principle of hydrostatics as

soon as it is propounded to him. Experiment was not

requisite for its discovery; experiment is not necessary for

its proof at present; and we may add, that experiment,

though it may make the proposition the more readily

intelligible, can add nothing to our conviction of its truth

when it is once understood.

14. _Foundation of the above Axiom._--But it will naturally

be asked, What then is the ground of our conviction of this

doctrine of the equal pressure of a fluid in all directions?

And to this I reply, that the reasons of this conviction are

involved in our idea of a fluid, which is considered as

matter, and therefore as capable of receiving, resisting,

and transmitting force according to the general conception

of matter; and which is also considered as matter which has

its parts perfectly moveable among one another. For it

follows from these suppositions, that if the fluid be

confined, a pressure which thrusts in one side of the

containing vessel, may cause any other side to bulge

outwards, if there be a part of the surface which has not

strength to resist this pressure from within. And that this

pressure, when thus transferred into a direction different

from the original one, is not altered in intensity, {233}

depends upon this consideration; that any difference in the

two pressures would be considered as a defect of _perfect_

fluidity, since the fluidity would be still more complete,

if this entire and undiminished transmission of pressure in

all directions were supposed. If, for instance, the lateral

pressure were less than the vertical, this could be

conceived no other way than as indicating some rigidity or

adhesion of the parts of the fluid. When the fluidity is

perfect, the two pressures which act in the two different

parts of the fluid exactly balance each other: they are the

action and the reaction; and must hence be equal by the same

necessity as two directly opposite forces in statics.

But it may be urged, that even if we grant that this

conception of a perfect fluid, as a body which has its parts

perfectly moveable among each other, leads us necessarily to

the principle of the equality of hydrostatic pressure in all

directions, still this conception itself is obtained from

experience, or suggested by observation. And to this we may

reply, that the conception of a fluid, as contemplated in

mechanical theory, cannot be said to be derived from

experience, except in the same manner as the conception of a

solid and rigid body may be said to be acquired by

experience. For if we imagine a vessel full of small, smooth

spherical balls, such a collection of balls would approach

to the nature of a fluid, in having its parts moveable among

each other; and would approach to perfect fluidity, as the

balls became smoother and smaller. And such a collection of

balls would also possess the statical properties of a fluid;

for it would transmit pressure out of a vertical into a

lateral (or any other) direction, in the same manner as a

fluid would do. And thus a collection of solid bodies has

the same property which a fluid has; and the science of

Hydrostatics borrows from experience no principles beyond

those which are involved in the science of Statics

respecting solids. And since in this latter portion of

science, as we have already seen, none of the principles

depend for their evidence upon any special experience, the

doctrines of Hydrostatics also are not {234} proved by

experience, but have a necessary truth borrowed from the

relations of our ideas.

It is hardly to be expected that the above reasoning will,

at first sight, produce conviction in the mind of the

reader, except he have, to a certain extent, acquainted

himself with the elementary doctrines of the science of

Hydrostatics as usually delivered; and have followed, with

clear and steady apprehension, some of the trains of

reasoning by which the pressures of fluids are determined;

as, for instance, the explanation of what is called _the

Hydrostatic Paradox_. The necessity of such a discipline in

order that the reader may enter fully into this part of our

speculations, naturally renders them less popular; but this

disadvantage is inevitable in our plan. We cannot expect to

throw light upon philosophy by means of the advances which

have been made in the mathematical and physical sciences,

except we really understand the doctrines which have been

firmly established in those sciences. This preparation for

philosophizing may be somewhat laborious; but such labour is

necessary if we would pursue speculative truth with all the

advantages which the present condition of human knowledge

places within our reach.

We may add, that the consequences to which we are directed

by the preceding opinions, are of very great importance in

their bearing upon our general views respecting human

knowledge. I trust to be able to show, that some important

distinctions are illustrated, some perplexing paradoxes

solved, and some large anticipations of the future extension

of our knowledge suggested, by means of the conclusions to

which the preceding discussions have conducted us. But

before I proceed to these general topics, I must consider

the foundations of some of the remaining portions of the

science of Mechanics.

{{235}}

CHAPTER VII.

OF THE ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS.

1. IN the History of Mechanics, I have traced the steps by

which the three Laws of Motion and the other principles of

mechanics were discovered, established, and extended to the

widest generality of form and application. We have, in these

laws, examples of principles which were, historically

speaking, obtained by reference to experience. Bearing in

mind the object and the result of the preceding discussions,

we cannot but turn with much interest to examine these

portions of science; to inquire whether there be any real

difference in the grounds and nature between the knowledge

thus obtained, and those truths which we have already

contemplated; and which, as we have seen, contain their own

evidence, and do not require proof from experiment.

2. _The First Law of Motion._--The first law of motion is,

that _When a body moves not acted upon by any force, it will

go on perpetually in a straight line, and with a uniform

velocity._ Now what is the real ground of our assent to this

proposition? That it is not at first sight a self-evident

truth, appears to be clear; since from the time of Aristotle

to that of Galileo the opposite assertion was held to be

true; and it was believed that all bodies in motion had, by

their own nature, a constant tendency to move more and more

slowly, so as to stop at last. This belief, indeed, is

probably even now entertained by most persons, till their

attention is fixed upon the arguments by which the first law

of motion is established. It is, however, not difficult to

lead any person of a speculative habit {236} of thought to

see that the retardation which constantly takes place in the

motion of all bodies when left to themselves, is, in

reality, the effect of extraneous forces which destroy the

velocity. A top ceases to spin because the friction against

the ground and the resistance of the air gradually diminish

its motion, and not because its motion has any internal

principle of decay or fatigue. This may be shown, and was,

in fact, shown by Hooke before the Royal Society, at the

time when the laws of motion were still under discussion, by

means of experiments in which the weight of the top is

increased, and the resistance to motion offered by its

support, is diminished; for by such contrivances its motion

is made to continue much longer than it would otherwise do.

And by experiments of this nature, although we can never

remove the whole of the external impediments to continued

motion, and although, consequently, there will always be

some retardation; and an end of the motion of a body left to

itself, however long it may be delayed, must at last come;

yet we can establish a conviction that if all resistance

could be removed, there would be no diminution of velocity,

and thus the motion would go on for ever.

If we call to mind the axioms which we formerly stated, as

containing the most important conditions involved in the

idea of Cause, it will be seen that our conviction in this

case depends upon the first axiom of Causation, that nothing

can happen without a cause. Every change in the velocity of

the moving body must have a cause; and if the change can, in

any manner, be referred to the presence of other bodies,

these are said to exert _force_ upon the moving body: and

the conception of force is thus evolved from the general

idea of cause. _Force is any cause which has motion, or

change of motion, for its effect_; and thus, all the change

of velocity of a body which can be referred to extraneous

bodies,--as the air which surrounds it, or the support on

which it rests,--is considered as the effect of forces; and

this consideration is looked upon as explaining the

difference between the motion which really takes places in

the experiment, and that motion {237} which, as the law

asserts, would take place if the body were not acted on by

any forces.

Thus the truth of the first law of motion depends upon the

axiom that no change can take place without a cause; and

follows from the definition of force, if we suppose that

there can be none but an _external_ cause of change. But in

order to establish the law, it was necessary further to be

assured that there is no _internal_ cause of change of

velocity belonging to all matter whatever, and operating in

such a manner that the mere progress of time is sufficient

to produce a diminution of velocity in all moving bodies. It

appears from the history of mechanical science, that this

latter step required a reference to observation and

experiment; and that the first law of motion is so far,

historically at least, dependent upon our experience.

But notwithstanding this historical evidence of the need

which we have of a reference to observed facts, in order to

place this first law of motion out of doubt, it has been

maintained by very eminent mathematicians and philosophers,

that the law is, in truth, evident of itself, and does not

really rest upon experimental proof. Such, for example, is

the opinion of d'Alembert[20\3], who offers what is called

an _à priori_ proof of this law; that is, a demonstration

derived from our ideas alone. When a body is put in motion,

either, he says, the cause which puts it in motion at first,

suffices to make it move one foot, or the continued action

of the cause during this foot is requisite for the motion.

In the first case, the same reason which made the body

proceed to the end of the first foot will hold for its going

on through a second, a third, a fourth foot, and so on for

any number. In the second case, the same reason which made

the force continue to act during the first foot, will hold

for its acting, and therefore for the body moving during

each succeeding foot. And thus the body, once beginning to

move, must go on moving for ever.

[Note 20\3: _Dynamique._]

{238} It is obvious that we might reply to this argument,

that the reasons for the body proceeding during each

succeeding foot may not necessarily be all the same; for

among these reasons may be the time which has elapsed; and

thus the velocity may undergo a change as the time proceeds:

and we require observation to inform us that it does not do so.

Professor Playfair has presented nearly the same argument,

although in a different and more mathematical form[21\3]. If

the velocity change, says he, it must change according to

some expression of calculation depending upon the time, or,

in mathematical language, must be a _function_ of the time.

If the velocity diminish as the time increases, this may be

expressed by stating the velocity in each case as a certain

number, from which another quantity, or _term_, increasing

as the time increases, is subtracted. But, Playfair adds,

there is no condition involved in the nature of the case, by

which the _coefficients_, or numbers which are to be

employed, along with the number representing the time, in

calculating this second term, can be determined to be of one

magnitude rather than of any other. Therefore he infers

there can be no such coefficients, and that the velocity is

in each case equal to some constant number, independent of

the time; and is therefore the same for all times.

[Note 21\3: _Outlines of Natural Philosophy_, p. 26.]

In reply to this we may observe, that the circumstance of

_our not seeing_ in the nature of the case anything which

determines for us the coefficients above spoken of, cannot

prove that they have not some certain value _in nature_. We

do not see in the nature of the case anything which should

determine a body to fall sixteen feet in a second of time,

rather than one foot or one hundred feet: yet in fact the

space thus run through by falling bodies is determined to a

certain magnitude. It would be easy to assign a mathematical

expression for the velocity of a body, implying that

one-hundredth of the velocity, or any other {239} fraction,

is lost in each second[22\3]: and where is the absurdity of

supposing such an expression really to represent the

velocity?

[Note 22\3: This would be the case, if, _t_ being the

number of seconds elapsed, and _C_ some constant quantity,

the velocity were expressed by this mathematical formula,

_C_(99/100)^_t_.]

Most modern writers on mechanics have embraced the opposite

opinion, and have ascribed our knowledge of this first law

of motion to experience. Thus M. Poisson, one of the most

eminent of the mathematicians who have written on this

subject, says[23\3], "We cannot affirm _à priori_ that the

velocity communicated to a body will not become slower and

slower of itself, and end by being entirely extinguished. It

is only by experience and induction that this question can

be decided."

[Note 23\3: Poisson, _Dynamique_, ed. 2, art. 113.]

Yet it cannot be denied that there is much force in those

arguments by which it is attempted to show that the First

Law of Motion, such as we find it, is more consonant to our

conceptions than any other would be. The Law, as it exists,

is the most simple that we can conceive. Instead of having

to determine by experiments what is the law of the natural

change of velocity, we find the Law to be that it does not

change at all. To a certain extent, the Law depends upon the

evident axiom, that no change can take place without a

cause. But the question further occurs, whether the mere

lapse of time may not be a cause of change of velocity. In

order to ensure this, we have recourse to experiment; and

the result is that time alone does not produce any such

change. In addition to the conditions of change which we

collect from our own Ideas, we ask of Experience what other

conditions and circumstances she has to offer; and the

answer is, that she can point out none; When we have removed

the alterations which external causes, in our very

conception of them, occasion, there are no longer any

alterations. Instead of having to guide ourselves {240} by

experience, we learn that on this subject she has nothing to

tell us. Instead of having to take into account a number of

circumstances, we find that we have only to reject all

circumstances. The velocity of a body remains unaltered by

time alone, of whatever kind the body itself be.

But the doctrine that time alone is not a cause of change of

velocity in any body is further recommended to us by this

consideration;--that time is conceived by us not as a cause,

but only as a condition of other causes producing their

effects. Causes operate in time; but it is only when the

cause exists, that the lapse of time can give rise to

alterations. When therefore all external causes of change of

velocity are supposed to be removed, the velocity must

continue identical with itself, whatever the time which

elapses. An eternity of negation can produce no positive

result.

Thus, though the discovery of the First Law of Motion was

made, historically speaking, by means of experiment, we have

now attained a point of view in which we see that it might

have been certainly known to be true independently of

experience. This law in its ultimate form, when completely

simplified and steadily contemplated, assumes the character

of a self-evident truth. We shall find the same process to

take place in other instances. And this feature in the

progress of science will hereafter be found to suggest very

important views with regard both to the nature and prospects

of our knowledge.

3. _Gravity is a Uniform Force._--We shall find observations

of the same kind offering themselves in a manner more or

less obvious, with regard to the other principles of

Dynamics. The determination of the laws according to which

bodies fall downwards by the common action of gravity, has

already been noticed in the History of Mechanics[24\3], as

one of the earliest positive advances in the doctrine of

motion. These laws were first rightly stated by Galileo, and

{241} established by reasoning and by experiment, not

without dissent and controversy. The amount of these

doctrines is this: That gravity is a uniform accelerating

force; such a _uniform force_ having this for its character,

that it _makes the velocity increase in exact proportion to

the time of motion_. The relation which the spaces described

by the body bear to the times in which they are described,

is obtained by mathematical deduction from this definition

of the force.

[Note 24\3: _Hist. Ind. Sc._ b. vi. c. ii. sect. 2.]

The clear Definition of a uniform accelerating force, and

the Proposition that gravity is such a force, were

co-ordinate and contemporary steps in this discovery. In

defining accelerating force, reference, tacit or express,

was necessarily made to the second of the general axioms

respecting causation,--That causes are measured by their

effects. Force, in the cases now under our notice, is

conceived to be, as we have already stated, (p. 236,) any

cause which, acting from without, changes the motion of a

body. It must, therefore, in this acceptation, be measured

by the magnitude of the changes which are produced. But in

what manner the changes of motion are to be employed as the

measures of force, is learnt from observation of the facts

which we see taking place in the world. Experience

_interprets_ the axiom of causation, from which otherwise we

could not deduce any real knowledge. We may assume, in

virtue of our general conceptions of force, that under the

same circumstances, a greater change of motion implies a

greater force producing it; but what are we to expect when

the circumstances change? The weight of a body makes it fall

from rest at first, and causes it to move more quickly as it

descends lower. We may express this by saying, that gravity,

the universal force which makes all terrestrial bodies fall

when not supported, by its continuous action first _gives_

velocity to the body when it has none, and afterwards _adds_

velocity to that which the body already has. But how is the

velocity added proportioned to the velocity which already

exists? Force acting on a body at rest, and on a body in

motion, appears under very different {242} conditions;--how

are the effects related? Let the force be conceived to be in

both cases the same, since force is conceived to depend upon

the extraneous bodies, and not upon the condition of the

moving mass itself. But the force being the same, the

effects may still be different. It is at first sight

conceivable that the body, acted upon by the same gravity,

may receive a less addition of velocity when it is already

moving in the direction in which this gravity impels it; for

if we ourselves push a body forwards, we can produce little

additional effect upon it when it is already moving rapidly

away from us. May it not be true, in like manner, that

although gravity be always the same force, its effect

depends upon the velocity which the body under its influence

already possesses?

Observation and reasoning combined, as we have said, enabled

Galileo to answer these questions. He asserted and proved

that we may consistently and properly measure a force by the

velocity which is by it generated in a body, in some certain

time, as one second; and further, that if we adopt this

measure, gravity will be a force of the same value under all

circumstances of the body which it affects; since it

appeared that, in fact, a falling body does receive equal

increments of velocity in equal times from first to last.

If it be asked whether we could have known, anterior to, or

independent of, experiment, that gravity is a uniform force

in the sense thus imposed upon the term; it appears clear

that we must reply, that we could not have attained to such

knowledge, since other laws of the motion of bodies

downwards are easily conceivable, and nothing but

observation could inform us that one of these laws does not

prevail in fact. Indeed, we may add, that the assertion that

the force of gravity is uniform, is so far from being

self-evident, that it is not even true; for gravity varies

according to the distance from the center of the earth; and

although this variation is so small as to be, in the case of

falling bodies, imperceptible, it negatives the rigorous

uniformity of the force as completely, though {243} not to

the same extent, as if the weight of a body diminished in a

marked degree, when it was carried from the lower to the

upper room of a house. It cannot, then, be a truth

independent of experience, that gravity is uniform.

Yet, in fact, the assertion that gravity is uniform was

assented to, not only before it was proved, but even before

it was clearly understood. It was readily granted by all,

that bodies which fall freely are _uniformly_ accelerated;

but while some held the opinion just stated, that uniformly

accelerated motion is that in which the velocity increases

in proportion to the _time_, others maintained, that _that_

is uniformly accelerated motion, in which the velocity

increases in proportion to the _space_; so that, for

example, a body in falling vertically through twenty feet

should acquire twice as great a velocity as one which falls

through ten feet.

These two opinions are both put forward by the interlocutors

of Galileo's Dialogue on this subject[25\3]. And the latter

supposition is rejected, the author showing, not that it is

inconsistent with experience, but that it is impossible in

itself: inasmuch as it would inevitably lead to the

conclusion, that the fall through a large and a small

vertical space would occupy exactly the same time.

[Note 25\3: _Dialogo_, iii. p. 95.]

Indeed, Galileo assumes his definition of uniformly

accelerated motion as one which is sufficiently recommended

by its own simplicity. 'If we attend carefully,' he says,

'we shall find that no mode of increase of velocity is more

simple than that which adds equal increments in equal times.

Which we may easily understand if we consider the close

affinity of time and motion: for as the uniformity of motion

is defined by the equality of spaces described in equal

times, so we may conceive the uniformity of acceleration to

exist when equal velocities are added in equal times.'

Galileo's mode of supporting his opinion, that bodies

falling by the action of gravity are thus uniformly {244}

accelerated, consists, in the first place, in adducing the

maxim that nature always employs the most simple

means[26\3]. But he is far from considering this a decisive

argument. 'I,' says one of his speakers, 'as it would be

very unreasonable in me to gainsay this or any other

definition which any author may please to make, since they

are all arbitrary, may still, without offence, doubt whether

such a definition, conceived and admitted in the abstract,

fits, agrees, and is verified in that kind of accelerated

motion which bodies have when they descend naturally.'

[Note 26\3: _Dialogo_, iii. p. 91.]

The experimental proof that bodies, when they fall

downwards, are uniformly accelerated, is (by Galileo)

derived from the inclined plane; and therefore assumes the

proposition, that if such uniform acceleration prevail in

vertical motion, it will also hold when a body is compelled

to describe an oblique rectilinear path. This proposition

may be shown to be true, if (assuming by anticipation the

Third Law of Motion, of which we shall shortly have to

speak,) we introduce the conception of a uniform statical

force as the cause of uniform acceleration. For the force on

the inclined plane bears a constant proportion to the

vertical force, and this proportion is known from statical

considerations. But in the work of which we are speaking,

Galileo does not introduce this abstract conception of force

as the foundation of his doctrines. Instead of this, he

proposes, as a postulate sufficiently evident to be made the

basis of his reasonings, That bodies which descend down

inclined planes of different inclinations, but of the same

vertical height, all acquire the same velocity[27\3]. But

when this postulate has been propounded by one of the

persons of the dialogue, another interlocutor says, 'You

discourse very probably; but besides this likelihood, I wish

to augment the probability so far, that it shall be almost

as complete as a necessary demonstration.' He then proceeds

to describe a very ingenious and simple experiment, which

shows that when a body is made to swing upwards at the end

of {245} a string, it attains to the same height, whatever

is the path it follows, so long as it starts from the lowest

point with the same velocity. And thus Galileo's postulate

is experimentally confirmed, so far as the force of gravity

can be taken as an example of the forces which the postulate

contemplates: and conversely, gravity is proved to be a

uniform force, so far as it can be considered clear that the

postulate is true of uniform forces.

[Note 276\3: _Dialogo_, iii. p. 36.]

When we have introduced the conception and definition of

accelerating force, Galileo's postulate, that bodies

descending down inclined planes of the same vertical height,

acquire the same velocity, may, by a few steps of reasoning,

be demonstrated to be true of uniform forces: and thus the

proof that gravity, either in vertical or oblique motion, is

a uniform force, is confirmed by the experiment above

mentioned; as it also is, on like grounds, by many other

experiments, made upon inclined planes and pendulums.

Thus the propriety of Galileo's conception of a uniform

force, and the doctrine that gravity is a uniform force,

were confirmed by the same reasonings and experiments. We

may make here two remarks; _First_, that the conception,

when established and rightly stated, appears so simple as

hardly to require experimental proof; a remark which we have

already made with regard to the First Law of Motion: and

_Second_, that the discovery of the real law of nature was

made by assuming propositions which, without further proof,

we should consider as very precarious, and as far less

obvious, as well as less evident, than the law of nature in

its simple form.

4. _The Second Law of Motion._--When a body, instead of

falling downwards from rest, is thrown in any direction, it

describes a curve line, till its motion is stopped. In this,

and in all other cases in which a body describes a curved

path in free space, its motion is determined by the Second

Law of Motion. The law, in its general form, is as

follows:--When a body is thus cast forth and acted upon by a

force in a direction {246} transverse to its motion, the

result is, That _there is combined with the motion with

which the body is thrown, another motion, exactly the same

as that which the same force would have communicated to a

body at rest_.

It will readily be understood that the basis of this law is

the axiom already stated, that effects are measured by their

causes. In virtue of this axiom, the effect of gravity

acting upon a body in a direction transverse to its motion,

must measure the accelerative or deflective force of gravity

under those circumstances. If this effect vary with the

varying velocity and direction of the body thus acted upon,

the deflective force of gravity also will vary with those

circumstances. The more simple supposition is, that the

deflective force of gravity is the same, whatever be the

velocity and direction of the body which is subjected to its

influence: and this is the supposition which we find to be

verified by facts. For example, a ball let fall from the top

of a ship's upright mast, when she is sailing steadily

forward, will fall at the foot of the mast, just as if it

were let fall while the ship were at rest; thus showing that

the motion which gravity gives to the ball is compounded

with the horizontal motion which the ball shares with the

ship from the first. This general and simple conception of

motions as _compounded_ with one another, represents, it is

proved, the manner in which the motion produced by gravity

modifies any other motion which the body may previously have had.

The discussions which terminated in the general reception of

this Second Law of Motion among mechanical writers, were

much mixed up with the arguments for and against the

Copernican system, which system represented the earth as

revolving upon its axis. For the obvious argument against

this system was, that if each point of the earth's surface

were thus in motion from west to east, a stone dropt from

the top of a tower would be left behind, the tower moving

away from it: and the answer was, that by this law of

motion, the stone would have the earth's motion impressed

upon it, as well as that motion which would {247} arise from

its gravity to the earth; and that the motion of the stone

relative to the tower would thus be the same as if both

earth and tower were at rest. Galileo further urged, as a

presumption in favour of the opinion that the two

motions,--the circular motion arising from the rotation of

the earth, and the downward motion arising from the gravity

of the stone, would be compounded in the way we have

described, (neither of them disturbing or diminishing the

other,)--that the first motion was in its own nature not

liable to any change or diminution[28\3], as we learn from

the First Law of Motion. Nor was the subject lightly

dismissed. The experiment of the stone let fall from the top

of the mast was made in various forms by Gassendi; and in

his Epistle, _De Motu impresso a Motore translato_, the rule

now in question is supported by reference to these

experiments. In this manner, the general truth, the Second

Law of Motion, was established completely and beyond

dispute.

[Note 28\3: _Dialogo_, ii. p. 114.]

But when this law had been proved to be true in a general

sense, with such accuracy as rude experiments, like those of

Galileo and Gassendi, would admit, it still remained to be

ascertained (supposing our knowledge of the law to be the

result of experience alone,) whether it were true with that

precise and rigorous exactness which more refined modes of

experimenting could test. We so willingly believe in the

simplicity of laws of nature, that the rigorous accuracy of

such a law, known to be at least approximately true, was

taken for granted, till some ground for suspecting the

contrary should appear. Yet calculations have not been

wanting which might confirm the law as true to the last

degree of accuracy. Laplace relates (_Syst. du Monde_, livre

iv. chap. 16,) that at one time he had conceived it possible

that the effect of gravity upon the moon might be slightly

modified by the moon's direction and velocity; and that in

this way an explanation might be found for the moon's

_acceleration_ (a deviation of her observed from her

calculated place, which long {248} perplexed

mathematicians). But it was after some time discovered that

this feature in the moon's motion arose from another cause;

and the second law of motion was confirmed as true in the

most rigorous sense.

Thus we see that although there were arguments which might

be urged in favour of this law, founded upon the necessary

relations of ideas, men became convinced of its truth only

when it was verified and confirmed by actual experiment. But

yet in this case again, as in the former ones, when the law

had been established beyond doubt or question, men were very

ready to believe that it was not a mere result of

observation,--that the truth which it contained was not

derived from experience,--that it might have been assumed as

true in virtue of reasonings anterior to experience,--and

that experiments served only to make the law more plain and

intelligible, as visible diagrams in geometry serve to

illustrate geometrical truths; our knowledge not being (they

deemed) in mechanics, any more than in geometry, borrowed

from the senses. It was thought by many to be self-evident,

that the effect of a force in any direction cannot be

increased or diminished by any motion transverse to the

direction of the force which the body may have at the same

time: or, to express it otherwise, that if the motion of the

body be compounded of a horizontal and vertical motion, the

vertical motion alone will be affected by the vertical

force. This principle, indeed, not only has appeared evident

to many persons, but even at the present day is assumed as

an axiom by many of the most eminent mathematicians. It is,

for example, so employed in the _Mécanique Céleste_ of

Laplace, which may be looked upon as the standard of

mathematical mechanics in our time; and in the _Mécanique

Analytique_ of Lagrange, the most consummate example which

has appeared of subtilty of thought on such subjects, as

well as of power of mathematical generalization[29\3]. And

{249} thus we have here another example of that circumstance

which we have already noticed in speaking of the First Law

of Motion, (Art. 2 of this chapter,) and of the Law that

Gravity is a uniform Force, (Art. 3); namely, that the law,

though historically established by experiments, appears,

when once discovered and reduced to its most simple and

general form, to be self-evident. I am the more desirous of

drawing attention to this feature in various portions of the

history of science, inasmuch as it will be found to lead to

some very extensive and important views, hereafter to be

considered.

[Note 29\3: I may observe that the rule that we may

_compound_ motions, as the Law supposes, is involved in the

step of _resolving_ them; which is done in the passage to

which I refer. (_Méc. Analyt._ ptie. i. sect. i. art. 3. p.

225.) 'Si on conçoit que le mouvement d'un corps et les

forces qui le sollicitent soient _decomposées_ suivant trois

lignes droites perpendiculaires entre elles, on pourra

considérer séparément les mouvemens et les forces relatives

à chacun de ces trois directions. Car à cause de la

perpendicularité des directions il est visible que chacun de

ces mouvemens partiels peut être regardé comme indépendant

des deux autres, et qu'il ne peut recevoir d'altération que

de la part de la force qui agit dans la direction de ce

mouvement; l'on peut conclure que ces trois mouvemens

doivent suivre, chacun en particulier, les lois des

mouvemens rectilignes accélérés ou retardés par les forces

données.' Laplace makes the same assumption in effect,

(_Méc. Cél._ p. i. liv. i. art. 7), by resolving the forces

which act upon a point in three rectangular directions, and

reasoning separately concerning each direction. But in his

mode of treating the subject is involved a principle which

belongs to the Third Law of Motion, namely, the doctrine

that the velocity is as the force, of which we shall have to

speak elsewhere.]

5. _The Third Law of Motion._--We have, in the definition of

Accelerating Force, a measure of Forces, so far as they are

concerned in producing motion. We had before, in speaking of

the principles of statics, defined the measure of Forces or

Pressures, so far as they are employed in producing

equilibrium. But these two aspects of Force are closely

connected; and we require a law which shall lay down the

rule of their connexion. By the same kind of muscular

exertion by which we can support a heavy stone, we can also

put it in motion. The question then occurs, how is the rate

and manner of its motion determined? The answer to this

question is contained in the Third Law {250} of Motion, and

it is to this effect: that the _Momentum_ which any pressure

produces in the mass in a given time is proportional to the

pressure. By _Momentum_ is meant the product of the numbers

which express the velocity and the mass of the body: and

hence, if the mass of the body be the same in the instances

which we compare, the rule is,--That _the velocity is as the

force which produces it_; and this is one of the simplest

ways of expressing the Third Law of Motion.

In agreement with our general plan, we have to ask, What is

the ground of this rule? What is the simplest and most

satisfactory form to which we can reduce the proof of it?

Or, to take an instance; if a double pressure be exerted

against a given mass, so disposed as to be capable of

motion, why must it produce twice the velocity in the same time?

To answer this question, suppose the double pressure to be

resolved into two single pressures: one of these will

produce a certain velocity; and the question is, why an

equal pressure, acting upon the same mass, will produce an

equal velocity _in addition_ to the former? Or, stating the

matter otherwise, the question is, why each of the two

forces will produce its separate effect, unaltered by the

simultaneous action of the other force?

This statement of the case makes it seem to approach very

near to such cases as are included in the Second Law of

Motion, and therefore it might appear that this Third Law

has no grounds distinct from the Second. But it must be

recollected that the word _force_ has a different meaning in

this case and in that; in this place it signifies

_pressure_; in the statement of the Second Law its import

was _accelerative_ or _deflective force_, measured by the

velocity or deflexion generated. And thus the Third Law of

Motion, so far as our reasonings yet go, appears to rest on

a foundation different from the Second.

Accordingly, that part of the Third Law of Motion which we

are now considering, that the velocity generated is as the

force, was obtained, in fact, by a separate train of

research. The first exemplification of this {251} law which

was studied by mathematicians, was the motion of bodies upon

inclined planes: for the force which urges a body down an

inclined plane is known by statics, and hence the velocity

of its descent was to be determined. Galileo

originally[30\3] in his attempts to solve this problem of

the descent of a body down an inclined plane, did not

proceed from the principle which we have stated, (the

determination of the force which acts down the inclined

plane from statical considerations,) obvious as it may seem;

but assumed, as we have already seen, a proposition

apparently far more precarious;--namely, that a body sliding

down a smooth inclined plane acquires always the same

velocity, so long as the _vertical_ height fallen through is

the same. And this conjecture (for at first it was nothing

more than a conjecture) he confirmed by an ingenious

experiment; in which bodies acquired or lost the same

velocity by descending or ascending through the same height,

although their paths were different in other respects.

[Note 30\3: _Dial. della Sc. Nuov._ iii. p. 96. See _Hist.

Ind. Sci._ b. vi. c. ii. sect. 5.]

This was the form in which the doctrine of the motion of

bodies down inclined planes was at first presented in

Galileo's _Dialogues_ on the Science of Motion. But his

disciple Viviani was dissatisfied with the assumption thus

introduced; and in succeeding editions of the _Dialogues_,

the apparent chasm in the reasoning was much narrowed, by

making the proof depend upon a principle nearly identical

with the third law of motion as we have just stated it. In

the proof thus added, 'We are agreed,' says the

interlocutor[31\3], 'that in a moving body the impetus,

energy, momentum, or propension to motion, is as great as is

the force or least resistance which suffices to sustain it;'

and the impetus or momentum, in the course of the proof,

being taken to be as the velocity produced in a given time,

it is manifest that the principle so stated amounts to this;

that the velocity produced is as the statical force. And

thus this law of motion appears, {252} in the school of

Galileo, to have been suggested and established at first by

experiment, but afterwards confirmed and demonstrated by _à

priori_ considerations.

[Note 31\3: _Dialogo_, p. 104.]

We see, in the above reasoning, a number of abstract terms

introduced which are not, at first at least, very distinctly

defined, as _impetus_, _momentum_, &c. Of these, _momentum_

has been selected, to express that quantity which, in a

moving body, measures the statical force impressed upon the

body. This quantity is, as we have just seen, proportional

to the velocity in a given body. It is also, in different

bodies, proportional to the mass of the body. This part of

the third law of motion follows from our conception of

matter in general as consisting of parts capable of

addition. A double pressure must be required to produce the

same velocity in a double mass; for if the mass be halved,

each half will require an equal pressure; and the addition,

both of the pressures and of the masses, will take place

without disturbing the effects.

The measure of the quantity of matter of a body considered

as affecting the velocity which pressure produces in the

body, is termed its _inertia_, as we have already stated (c.

v.). Inertia is the property by which a large mass of matter

requires a greater force than a small mass, to give it an

equal velocity. It belongs to each portion of matter; and

portions of inertia are added whenever portions of matter

are added. Hence _inertia is as the quantity of matter_;

which is only another way of expressing this third law of

motion, so far as quantity of matter is concerned.

But how do we know the quantity of matter of a body? We may

reply, that we take the weight as the measure of the

quantity of matter: but we may then be again asked, how it

appears that the weight is proportional to the inertia;

which it must be, in order that the quantity of matter may

be proportional to both one and the other. We answer, that

this appears to be true experimentally, because all bodies

fall with equal velocities by gravity, when the known causes

of difference are removed. The observations of falling {253}

bodies, indeed, are not susceptible of much exactness: but

experiments leading to the same result, and capable of great

precision, were made upon pendulums by Newton; as he relates

in his _Principia_, Book III. prop. 6. They all agreed, he

says, with perfect accuracy: and thus the weight and the

inertia are proportional in all cases, and therefore each

proportional to the quantity of matter as measured by the other.

The conception of inertia, as we have already seen in

chapter V., involves the notion of action and reaction; and

thus the laws which involve inertia depend upon the idea of

mutual causation. The rule, that the velocity is as the

force, depends upon the principle of causation, that the

effect is proportional to the cause; the effect being here

so estimated as to be consistent both with the other laws of

motion and with experiment.

But here, as in other cases, the question occurs again; Is

experiment really requisite for the proof of this law? If we

look to authorities, we shall be not a little embarrassed to

decide. D'Alembert is against the necessity of experimental

proof. 'Why,' says he[32\3], 'should we have recourse to

this principle employed, at the present day, by everybody,

that the force is proportional to the velocity? ... a

principle resting solely upon this vague and obscure axiom,

that the effect is proportional to the cause. We shall not

examine here,' he adds, 'if this principle is necessarily

true; we shall only avow that the proofs which have hitherto

been adduced do not appear to us unexceptionable: nor shall

we, with some geometers, adopt it as a purely contingent

truth; which would be to ruin the certainty of mechanics,

and to reduce it to be nothing more than an experimental

science. We shall content ourselves with observing,' he

proceeds, 'that certain or doubtful, clear or obscure, it is

useless in mechanics, and consequently ought to be banished

from the science.' Though D'Alembert rejects the third law

of motion in this form, he accepts one of {254} equivalent

import, which appears to him to possess axiomatic certainty;

and this procedure is in consistence with the course which

he takes, of claiming for the science of mechanics more than

mere experimental truth. On the contrary, Laplace considers

this third law as established by experiment. 'Is the force,'

he says'[33\3], 'proportioned to the velocity? This,' he

replies, 'we cannot know _à priori_, seeing that we are in

ignorance of the nature of moving force: we must therefore,

for this purpose, recur to experience; for all which is not

a necessary consequence of the few data we have respecting

the nature of things, is, for us, only a result of

observation.' And again he says[34\3], 'Here, then, we have

two laws of motion,--the law of inertia [the first law of

motion], and the law of the force proportional to the

velocity,--which are given by observation. They are the most

natural and the most simple laws which we can imagine, and

without doubt they flow from the very nature of matter; but

this nature being unknown, they are, for us, only observed

facts: the only ones, however, which Mechanics borrows from

experience.'

[Note 32\3: _Dynamique_, Pref. p. x.]

[Note 33\3: _Méc. Cél._ p. 15.]

[Note 34\3: p. 18.]

It will appear, I think, from the views given in this and

several other parts of the present work, that we cannot with

justice say that we have very 'few data respecting the

nature of things,' in speculating concerning the laws of the

universe; since all the consequences which flow from the

relations of our fundamental ideas, necessarily regulate our

knowledge of things, so far as we have any such knowledge.

Nor can we say that the nature of matter is unknown to us,

in any sense in which we can conceive knowledge as possible.

The nature of matter is no more unknown than the nature of

space or of number. In our conception of matter, as of space

and of number, are involved certain relations, which are the

necessary groundwork of our knowledge; and anything which is

independent of these relations, is not unknown, but

inconceivable. {255}

It must be already clear to the reader, from the phraseology

employed by these two eminent mathematicians, that the

question respecting the formation of the third law of motion

can only be solved by a careful consideration of what we

mean by observation and experience, nature and matter. But

it will probably be generally allowed, that, taking into

account the explanations already offered of the necessary

conditions of experience and of the conception of inertia,

this law of motion, that the inertia is as the quantity of

matter, is almost or altogether self-evident.

6. _Action and Reaction are Equal in Moving Bodies._--When

we have to consider bodies as acting upon one another, and

influencing each other's motions, the third law of motion is

still applied; but along with this, we also employ the

general principle that action and reaction are equal and

opposite. Action and reaction are here to be understood as

momentum produced and destroyed, according to the measure of

action established by the Third Law of Motion: and the cases

in which this principle is thus employed form so large a

portion of those in which the third law of motion is used,

that some writers (Newton at the head of them) have

stated the equality of action and reaction as the third law

of motion.

The third law of motion being once established, the equality

of action and reaction, in the sense of momentum gained and

lost, necessarily follows. Thus, if a weight hanging by a

string over the edge of a smooth level table draw another

weight along the table, the hanging weight moves more slowly

than it would do if not so connected, and thus loses

velocity by the connexion; while the other weight gains by

the connexion all the velocity which it has, for if left to

itself it would rest. And the pressures which restrain the

descent of the first body and accelerate the motion of the

second, are equal at all instants of time, for each of these

pressures is the tension of the string: and hence, by the

third law of motion, the momentum gained by the one body,

and the momentum lost by the other in virtue of the action

of this string, are equal. And similar {256} reasoning may

be employed in any other case where bodies are connected.

The case where one body does not push or draw, but _strikes_

another, appeared at first to mechanical reasoners to be of

a different nature from the others; but a little

consideration was sufficient to show that a blow is, in

fact, only a short and violent pressure; and that,

therefore, the general rule of the equality of momentum lost

and gained applies to this as well as to the other cases.

Thus, in order to determine the case of the direct action of

bodies upon one another, we require no new law of motion.

The equality of action and reaction, which enters

necessarily into every conception of mechanical operation,

combined with the measure of action as given by the third

law of motion, enables us to trace the consequences of every

case, whether of pressure or of impact.

7. _D'Alembert's Principle._--But what will be the result

when bodies do not act directly upon each other, but are

_indirectly_ connected in any way by levers, strings,

pulleys, or in any other manner, so that one part of the

system has a mechanical advantage over another? The result

must still be determined by the principle that action and

reaction balance each other. The action and reaction, being

pressures in one sense, must balance each other by the laws

of statics, for these laws determine the equilibrium of

pressure. Now action and reaction, according to their

measures in the Third Law of Motion, are momentum gained and

lost, when the action is direct; and except the indirect

action introduce some modification of the law, they must

have the same measure still. But, in fact, we cannot well

conceive any modification of the law to take place in this

case; for direct action is only one (the ultimate) case of

indirect action. Thus if two heavy bodies act at different

points of a lever, the action of each on the other is

indirect; but if the two points come together, the action

becomes direct. Hence the rule must be that which we have

already stated; for if the rule were false for indirect

action, it would {257} also be false for direct action, for

which case we have shown it to be true. And thus we obtain

the general principle, that in any system of bodies which

act on each other, action and reaction, estimated by

momentum gained and lost, balance each other according to

the laws of equilibrium. This principle, which is so general

as to supply a key to the solution of all possible

mechanical problems, is commonly called _D'Alembert's

Principle_. The experimental proofs which convinced men of

the truth of the Third Law of Motion were, many or most of

them, proofs of the law in this extended sense. And thus the

proof of D'Alembert's Principle, both from the idea of

mechanical action and from experience, is included in the

proof of the law already stated.

8. _Connexion of Dynamical and Statical Principles._--The

principle of equilibrium of D'Alembert just stated, is the

law which he would substitute for the Third Law of Motion;

and he would thus remove the necessity for an independent

proof of that law. In like manner, the Second Law of Motion

is by some writers derived from the principle of the

composition of statical forces; and they would thus

supersede the necessity of a reference to experiment in that

case. Laplace takes this course, and thus, as we have seen,

rests only the First and Third Law of Motion upon

experience. Newton, on the other hand, recognizes the same

connexion of propositions, but for a different purpose; for

he derives the composition of statical forces from the

Second Law of Motion.

The close connexion of these three principles, the

composition of (statical) forces, the composition of

(accelerating) forces with velocities, and the measure of

(moving) forces by velocities, cannot be denied; yet it

appears to be by no means easy to supersede the necessity of

independent proofs of the last two of these principles. Both

may be proved or illustrated by experiment: and the

experiments which prove the one are different from those

which establish the other. For example, it appears by easy

calculations, that when we apply our principles to the

oscillations of a pendulum, {258} the Second Law is proved

by the fact, that the oscillations take place at the same

rate in an east and west, and in a north and south

direction: under the same circumstances, the Third Law is

proved by our finding that the time of a small oscillation

is proportional to the square root of the length of a

pendulum; and similar differences might be pointed out in

other experiments, as to their bearing upon the one law or

the other.

9. _Mechanical Principles become gradually more simple and

more evident._--I will again point out in general two

circumstances which I have already noticed in particular

cases of the laws of motion.--Truths are often at first

assumed in a form which is far from being the most obvious

or simple;--and truths once discovered are gradually

simplified, so as to assume the appearance of self-evident

truths.

The former circumstance is exemplified in several of the

instances which we have had to consider. The assumption,

that a perpetual motion is impossible, preceded the

knowledge of the first law of motion. The assumed equality

of the velocities acquired down two inclined planes of the

same height, was afterwards reduced to the third law of

motion by Galileo himself. In the History[35\3], we have

noted Huyghens's assumption of the equality of the actual

descent and potential ascent of the center of gravity: this

was afterwards reduced by Herman and the Bernoullis, to the

statical equivalence of the solicitations of gravity and the

vicarious solicitations of the effective forces which act on

each point; and finally to the principle of D'Alembert,

which asserts that the motions gained and lost balance each

other.

[Note 35\3: B. vi. c. v. sect. 2.]

This early assertion of principles which now appear neither

obvious nor self-evident, is not to be considered as a

groundless assumption on the part of the discoverers by whom

it was made. On the contrary, it is evidence of the deep

sagacity and clear thought which were {259} requisite in

order to make such discoveries. For these results are really

rigorous consequences of the laws of motion in their

simplest form: and the evidence of them was probably

present, though undeveloped, in the minds of the

discoverers. We are told of geometrical students, who, by a

peculiar aptitude of mind, perceived the evidence of some of

the more advanced propositions of geometry without going

through the introductory steps. We must suppose a similar

aptitude for mechanical reasonings, which, existing in the

minds of Stevinus, Galileo, Newton, and Huyghens, led them

to make those assumptions which finally resolved themselves

into the laws of motion.

We may observe further, that the simplicity and evidence

which the laws of mechanics have at length assumed, are much

favoured by the usage of words among the best writers on

such subjects. Terms which originally, and before the laws

of motion were fully known, were used in a very vague and

fluctuating sense, were afterwards limited and rendered

precise, so that assertions which at first appear identical

propositions become distinct and important principles. Thus

_force_, _motion_, _momentum_, are terms which were

employed, though in a loose manner, from the very outset of

mechanical speculation. And so long as these words retained

the vagueness of common language, it would have been a

useless and barren truism to say that 'the momentum is

proportional to the force,' or that 'a body loses as much

motion as it communicates to another.' But when 'momentum'

and 'quantity of motion' are defined to mean the product of

mass and velocity, these two propositions immediately become

distinct statements of the third law of motion and its

consequences. In like manner, the assertion that 'gravity is

a uniform force' was assented to, before it was settled what

a uniform force was; but this assertion only became

significant and useful when that point had been properly

determined. The statement that 'when different motions are

communicated to the same body their effects are {260}

compounded,' becomes the second law of motion, when we

define what composition of motions is. And the same process

may be observed in other cases.

And thus we see how well the form which science ultimately

assumes is adapted to simplify knowledge. The definitions

which are adopted, and the terms which become current in

precise senses, produce a complete harmony between the

matter and the form of our knowledge; so that truths which

were at first unexpected and recondite, became familiar

phrases, and after a few generations sound, even to common

ears, like identical propositions.

10. _Controversy of the Measure of Force._--In the History

of Mechanics[36\3], we have given an account of the

controversy which, for some time, occupied the

mathematicians of Europe, whether the forces of bodies in

motion should be reckoned proportional to the velocity, or

to the square of the velocity. We need not here recall the

events of this dispute; but we may remark, that its history,

as a metaphysical controversy, is remarkable in this

respect, that it has been finally and completely settled;

for it is now agreed among mathematicians that both sides

were right, and that the results of mechanical action may be

expressed with equal correctness by means of _momentum_ and

of _vis viva_. It is, in one sense, as D'Alembert has

said[37\3], a dispute about words; but we are not to infer

that, on that account, it was frivolous or useless; for such

disputes are one principal means of reducing the principles

of our {261} knowledge to their utmost simplicity and

clearness. The terms which are employed in the science of

mechanics are now liberated for ever, in the minds of

mathematicians, from that ambiguity which was the

battleground in the war of the _vis viva_.

[Note 36\3: B. vi. c. v. sect. 2.]

[Note 37\3: D'Alembert has also remarked (_Dynamique_, Pref.

xxi.) that this controversy 'shows how little justice and

precision there is in the pretended axiom that causes are

proportional to their effects.' But this reflection is by no

means well founded. For since both measures are true, it

appears that causes may be _justly_ measured by their

effects, even when very different kinds of effects are

taken. That the axiom does not point out one _precise_

measure, till illustrated by experience or by other

considerations, we grant: but the same thing occurs in the

application of other axioms also.]

But we may observe that the real reason of this controversy

was exactly that tendency which we have been noticing;--the

disposition of man to assume in his speculations certain

general propositions as true, and to fix the sense of terms

so that they shall fall in with this truth. It was agreed,

on all hands, that in the mutual action of bodies the same

quantity of force is always preserved; and the question was,

by which of the two measures this rule could best be

verified. We see, therefore, that the dispute was not

concerning a definition merely, but concerning a definition

combined with a general proposition. Such a question may be

readily conceived to have been by no means unimportant; and

we may remark, in passing, that such controversies, although

they are commonly afterwards stigmatized as quarrels about

words and definitions, are, in reality, events of

considerable consequence in the history of science; since

they dissipate all ambiguity and vagueness in the use of

terms, and bring into view the conditions under which the

fundamental principles of our knowledge can be most clearly

and simply presented.

It is worth our while to pause for a moment on the prospect

that we have thus obtained, of the advance of knowledge, as

exemplified in the history of Mechanics. The general

transformation of our views from vague to definite, from

complex to simple, from unexpected discoveries to

self-evident truths, from seeming contradictions to

identical propositions, is very remarkable, but it is by no

means peculiar to our subject. The same circumstances, more

or less prominent, more or less developed, appear in the

history of other sciences, according to the point of advance

which each has reached. They bear upon very important

doctrines respecting the prospects, the {262} limits, and

the very nature of our knowledge. And though these doctrines

require to be considered with reference to the whole body of

science, yet the peculiar manner in which they are

illustrated by the survey of the history of Mechanics, on

which we have just been engaged, appears to make this a

convenient place for introducing them to the reader.

{{263}}

CHAPTER VIII.

OF THE PARADOX OF UNIVERSAL PROPOSITIONS OBTAINED FROM

EXPERIENCE.

1. IT was formerly stated[38\3] that experience cannot

establish any universal or necessary truths. The number of

trials which we can make of any proposition is necessarily

limited, and observation alone cannot give us any ground of

extending the inference to untried cases. Observed facts

have no visible bond of necessary connexion, and no exercise

of our senses can enable us to discover such connexion. We

can never acquire from a mere observation of facts, the

right to assert that a proposition is true in all cases, and

that it could not be otherwise than we find it to be.

[Note 38\3: B. i. c. iv. Of Experience]

Yet, as we have just seen in the history of the laws of

motion, we may go on collecting our knowledge from

observation, and enlarging and simplifying it, till it

approaches or attains to complete universality and seeming

necessity. Whether the laws of motion, as we now know them,

can be rigorously traced to an absolute necessity in the

nature of things, we have not ventured absolutely to

pronounce. But we have seen that some of the most acute and

profound mathematicians have believed that, for these laws

of motion, or some of them, there was such a demonstrable

necessity compelling them to be such as they are, and no

other. Most of those who have carefully studied the

principles of Mechanics will allow that some at least of the

primary laws of motion approach very near to this character

of necessary truth; and will confess that it would be

difficult to imagine any other consistent {264} scheme of

fundamental principles. And almost all mathematicians will

allow to these laws an absolute universality; so that we may

apply them without scruple or misgiving, in cases the most

remote from those to which our experience has extended. What

astronomer would fear to refer to the known laws of motion,

in reasoning concerning the double stars; although these

objects are at an immeasurably remote distance from that

solar system which has been the only field of our

observation of mechanical facts? What philosopher, in

speculating respecting a magnetic fluid, or a luminiferous

ether, would hesitate to apply to it the mechanical

principles which are applicable to fluids of known

mechanical properties? When we assert that the quantity of

motion in the world cannot be increased or diminished by the

mutual actions of bodies, does not every mathematician feel

convinced that it would be an unphilosophical restriction to

limit this proposition to such modes of action as we have tried?

Yet no one can doubt that, in historical fact, these laws

were collected from experience. That such is the case, is no

matter of conjecture. We know the time, the persons, the

circumstances, belonging to each step of each discovery. I

have, in the History, given an account of these discoveries;

and in the previous chapters of the present work, I have

further examined the nature and the import of the principles

which were thus brought to light.

Here, then, is an apparent contradiction. Experience, it

would seem, has done that which we had proved that she

cannot do. She has led men to propositions, universal at

least, and to principles which appear to some persons

necessary. What is the explanation of this contradiction,

the solution of this paradox? Is it true that Experience can

reveal to us universal and necessary truths? Does she

possess some secret virtue, some unsuspected power, by which

she can detect connexions and consequences which we have

declared to be out of her sphere? Can she see more than mere

appearances, and observe more than mere facts? Can {265} she

penetrate, in some way, to the nature of things?--descend

below the surface of phenomena to their causes and origins,

so as to be able to say what can and what can not be;--what

occurrences are partial, and what universal? If this be so,

we have indeed mistaken her character and powers; and the

whole course of our reasoning becomes precarious and

obscure. But, then, when we return upon our path we cannot

find the point at which we deviated, we cannot detect the

false step in our deduction. It still seems that by

experience, strictly so called, we cannot discover necessary

and universal truths. Our senses can give us no evidence of

a necessary connexion in phenomena. Our observation must be

limited, and cannot testify concerning anything which is

beyond its limits. A general view of our faculties appears

to prove it to be impossible that men should do what the

history of the science of mechanics shows that they have done.

2. But in order to try to solve this Paradox, let us again

refer to the History of Mechanics. In the cases belonging to

that science, in which propositions of the most

unquestionable universality, and most approaching to the

character of necessary truths, (as, for instance, the laws

of motion,) have been arrived at, what is the source of the

axiomatic character which the propositions thus assume? The

answer to this question will, we may hope, throw some light

on the perplexity in which we appear to be involved.

Now the answer to this inquiry is, that the laws of motion

borrow their axiomatic character from their being merely

_interpretations_ of the Axioms of Causation. Those axioms,

being exhibitions of the Idea of Cause under various

aspects, are of the most rigorous universality and

necessity. And so far as the laws of motion are

exemplifications of those axioms, these laws must be no less

universal and necessary. How these axioms are to be

understood;--in what sense _cause_ and _effect_, _action_

and _reaction_, are to be taken, experience and observation

did, in fact, teach inquirers on this subject; and without

this teaching, the laws of motion could never have been

distinctly known. If two forces {266} act together, each

must produce its effect, by the axiom of causation; and,

therefore, the effects of the separate forces must be

_compounded_. But a long course of discussion and experiment

must instruct men of what kind this _composition_ of forces

is. Again; action and reaction must be equal; but much

thought and some trial were needed to show what _action_ and

_reaction_ are. Those metaphysicians who enunciated Laws of

motion without reference to experience, propounded only such

laws as were vague and inapplicable. But yet these persons

manifested the indestructible conviction, belonging to man's

speculative nature, that there exist Laws of motion, that

is, universal formulæ, connecting the causes and effects

when motion takes place. Those mechanicians, again, who,

observed facts involving equilibrium and motion, and stated

some narrow rules, without attempting to ascend to any

universal and simple principle, obtained laws no less barren

and useless than the metaphysicians; for they could not tell

in what new cases, or whether in any, their laws would be

verified;--they needed a more general rule, to show them the

limits of the rule they had discovered. They went wrong in

each attempt to solve a new problem, because their

interpretation of the terms of the axioms, though true,

perhaps, in certain cases, was not right in general.

Thus Pappus erred in attempting to interpret as a case of

the lever, the problem of supporting a weight upon an

inclined plane; thus Aristotle erred in interpreting the

doctrine that the weight of bodies is the cause of their

fall; thus Kepler erred in interpreting the rule that the

velocity of bodies depends upon the force; thus

Bernoulli[39\3] erred in interpreting the equality of action

and reaction upon a lever in motion. In each of these

instances, true doctrines, already established, (whether by

experiment or otherwise,) were erroneously applied. And the

error was corrected by further reflection, which pointed out

that another mode of interpretation was requisite, in order

that the axiom {267} which, was appealed to in each case

might retain its force in the most general sense. And in the

reasonings which avoided or corrected such errors, and which

led to substantial general truths, the object of the

speculator always was to give to the acknowledged maxims

which the Idea of Cause suggested, such a signification as

should be consistent with their universal validity. The rule

was not accepted as particular at the outset, and afterwards

generalized more and more widely; but from the very first,

the universality of the rule was assumed, and the question

was, how it should be understood so as to be universally

true. At every stage of speculation, the law was regarded as

a general law. This was not an aspect which it gradually

acquired, by the accumulating contributions of experience,

but a feature of its original and native character. _What_

should happen universally, experience might be needed to

show: but that what happened should happen _universally_,

was implied in the nature of knowledge. The universality of

the laws of motion was not gathered from experience, however

much the laws themselves might be so.

[Note 39\3: _Hist. Ind. Sc._ b. vi. c. v. sect. 2.]

3. Thus we obtain the solution of our Paradox, so far as the

case before us is concerned. The laws of motion borrow their

_form_ from the Idea of Causation, though their _matter_ may

be given by experience: and hence they possess a

universality which experience cannot give. They are

certainly and universally valid; and the only question for

observation to decide is, how they are to be understood.

They are like general mathematical formulæ, which are known

to be true, even while we are ignorant what are the unknown

quantities which they involve. It must be allowed, on the

other hand, that so long as these formulæ are not

interpreted by a real study of nature, they are not only

useless but prejudicial; filling men's minds with vague

general terms, empty maxims, and unintelligible

abstractions, which they mistake for knowledge. Of such

perversion of the speculative propensities of man's nature,

the world has seen too much in all ages. Yet we must not, on

that account, despise these forms of {268} truth, since

without them, no general knowledge is possible. Without

general terms, and maxims, and abstractions, we can have no

science, no speculation; hardly, indeed, consistent thought

or the exercise of reason. The course of real knowledge is,

to obtain from thought and experience the right

interpretation of our general terms, the real import of our

maxims, the true generalizations which our abstractions

involve.

4. If it be asked, How Experience is able to teach us to

interpret aright the general terms which the Axioms of

Causation involve;--whence she derives the light which she

is to throw on these general notions; the answer is

obvious;--namely, that the relations of causation are the

_conditions_ of Experience;--that the general notions are

_exemplified_ in the particular cases of which she takes

cognizance. The events which take place about us, and which

are the objects of our observation, we cannot conceive

otherwise than as subject to the laws of cause and effect.

Every event must have a cause;--Every effect must be

determined by its cause;--these maxims are true of the

phenomena which form the materials of our experience. It is

precisely to them, that these truths apply. It is in the

world which we have before our eyes, that these propositions

are universally verified; and it is therefore by the

observation of what we see, that we must learn how these

propositions are to be understood. Every fact, every

experiment, is an example of these statements; and it is

therefore by attention to and familiarity with facts and

experiments, that we learn the signification of the

expressions in which the statements are made; just as in any

other case we learn the import of language by observing the

manner in which it is applied in known cases. Experience is

the interpreter of nature; it being understood that she is

to make her interpretation in that comprehensive phraseology

which is the genuine language of science.

5. We may return for an instant to the objection, that

experience cannot give us general truths, since, after any

number of trials confirming a rule, we may for aught we can

foresee, have one which violates the {269} rule. When we

have seen a thousand stones fall to the ground, we may see

one which does not fall under the same apparent

circumstances. How then, it is asked, can experience teach

us that _all_ stones, rigorously speaking, will fall if

unsupported? And to this we reply, that it is not true that

we can conceive one stone to be suspended in the air, while

a thousand others fall, without believing some peculiar

cause to support it; and that, therefore, such a supposition

forms no exception to the law, that gravity is a force by

which _all_ bodies are urged downwards. Undoubtedly we can

conceive a body, when dropt or thrown, to move in a line

quite different from other bodies: thus a certain

missile[40\3] used by the natives of Australia, and lately

brought to this country, when thrown from the hand in a

proper manner, describes a curve, and returns to the place

from whence it was thrown. But did any one, therefore, even

for an instant suppose that the laws of motion are different

for this and for other bodies? On the contrary, was not

every person of a speculative turn immediately led to

inquire how it was that the known causes which modify

motion, the resistance of the air and the other causes,

produced in this instance so peculiar an effect? And if the

motion had been still more unaccountable, it would not have

occasioned any uncertainty whether it were consistent with

the agency of gravity and the laws of motion. If a body

suddenly alter its direction, or move in any other

unexpected manner, we never doubt that there is a cause of

the change. We may continue quite ignorant of the nature of

this cause, but this ignorance never occasions a moment's

doubt that the cause exists and is exactly suited to the

effect. And thus experience can prove or discover to us

general rules, but she can never prove that general rules do

not exist. Anomalies, exceptions, unexplained phenomena, may

remind us that we have much still to learn, but they can

never make us suppose that truths are not universal. We may

observe facts that show us we have not fully {270}

understood the meaning of our general laws, but we can never

find facts which show our laws to have no meaning. Our

experience is bound in by the limits of cause and effect,

and can give us no information concerning any region where

that relation does not prevail. The whole series of external

occurrences and objects, through all time and space, exists

only, and is conceived only, as subject to this relation;

and therefore we endeavour in vain to imagine to ourselves

when and where and how exceptions to this relation may

occur. The assumption of the connexion of cause and effect

is essential to our experience, as the recognition of the

maxims which express this connexion is essential to our

knowledge.

[Note 40\3: Called the Bo-me-rang.]

6. I have thus endeavoured to explain in some measure how,

at least in the field of our mechanical knowledge,

experience can discover universal truths, though she cannot

give them their universality; and how such truths, though

borrowing their form from our ideas, cannot be understood

except by the actual study of external nature. And thus with

regard to the laws of motion, and other fundamental

principles of Mechanics, the analysis of our ideas and the

history of the progress of the science well illustrate each

other.

If the paradox of the discovery of universal truths by

experience be thus solved in one instance, a much wider

question offers itself to us;--How far the difficulty, and

how far the solution, are applicable to other subjects. It

is easy to see that this question involves most grave and

extensive doctrines with regard to the whole compass of

human knowledge: and the views to which we have been led in

the present Book of this work are, we trust, fitted to throw

much light upon the general aspect of the subject. But after

discussions so abstract, and perhaps obscure, as those in

which we have been engaged for some chapters, I willingly

postpone to a future occasion an investigation which may

perhaps appear to most readers more recondite and difficult

still. And we have, in fact, many other special fields of

knowledge to survey, before we are led by the order of our

subject, to {271} those general questions and doctrines,

those antitheses brought into view and again resolved, which

a view of the whole territory of human knowledge suggests,

and by which the nature and conditions of knowledge are

exhibited.

Before we quit the subject of mechanical science we shall

make a few remarks on another doctrine which forms part of

the established truths of the science, namely, the doctrine

of universal gravitation.

{{272}}

CHAPTER IX.

OF THE ESTABLISHMENT OF THE LAW OF UNIVERSAL GRAVITATION.

THE doctrine of universal gravitation is a feature of so

much importance in the history of science that we shall not

pass it by without a few remarks on the nature and evidence

of the doctrine.

1. To a certain extent the doctrine of the attraction of

bodies according to the law of the inverse square of the

distance, exhibits in its progress among men the same

general features which we have noticed in the history of the

laws of motion. This doctrine was maintained _à priori_ on

the ground of its simplicity, and was asserted positively,

even before it was clearly understood:--notwithstanding this

anticipation, its establishment on the ground of facts was a

task of vast labour and sagacity:--when it had been so

established in a general way, there occurred at later

periods, an occasional suspicion that it might be

approximately true only:--these suspicions led to further

researches, which showed the rule to be rigorously

exact:--and at present there are mathematicians who

maintain, not only that it is true, but that it is a

necessary property of matter. A very few words on each of

these points will suffice.

2. I have shown in the _History of Science_[41\3], that the

attraction of the sun according to the inverse square of the

distance, had been divined by Bullialdus, Hooke, Halley, and

others, before it was proved by Newton. Probably the reason

which suggested this conjecture was, that gravity might be

considered {273} as a sort of emanation; and that thus, like

light or any other effect diffused from a center, it must

follow the law just stated, the efficacy of the force being

weakened in receding from the center, exactly in proportion

to the space through which it is diffused. It cannot be

denied that such a view appears to be strongly recommended

by analogy.