PART II. OF THE IDEAS OF SPACE AND TIME.

SECT. I. OF THE INFINITE DIVISIBILITY OF OUR IDEAS OF SPACE AND TIME.

Whatever has the air of a paradox, and is contrary to the first and

most unprejudiced notions of mankind, is often greedily embraced by

philosophers, as shewing the superiority of their science, which could

discover opinions so remote from vulgar conception. On the other hand,

anything proposed to us, which causes surprize and admiration, gives

such a satisfaction to the mind, that it indulges itself in those

agreeable emotions, and will never be persuaded that its pleasure is

entirely without foundation. From these dispositions in philosophers and

their disciples arises that mutual complaisance betwixt them; while the

former furnish such plenty of strange and unaccountable opinions, and

the latter so readily believe them. Of this mutual complaisance I

cannot give a more evident instance than in the doctrine of infinite

divisibility, with the examination of which I shall begin this subject

of the ideas of space and time.

It is universally allowed, that the capacity of the mind is limited, and

can never attain a full and adequate conception of infinity: And though

it were not allowed, it would be sufficiently evident from the plainest

observation and experience. It is also obvious, that whatever is capable

of being divided in infinitum, must consist of an infinite number of

parts, and that it is impossible to set any bounds to the number of

parts, without setting bounds at the same time to the division. It

requires scarce any, induction to conclude from hence, that the idea,

which we form of any finite quality, is not infinitely divisible, but

that by proper distinctions and separations we may run up this idea

to inferior ones, which will be perfectly simple and indivisible. In

rejecting the infinite capacity of the mind, we suppose it may arrive at

an end in the division of its ideas; nor are there any possible means of

evading the evidence of this conclusion.

It is therefore certain, that the imagination reaches a minimum, and

may raise up to itself an idea, of which it cannot conceive any

sub-division, and which cannot be diminished without a total

annihilation. When you tell me of the thousandth and ten thousandth

part of a grain of sand, I have a distinct idea of these numbers and of

their different proportions; but the images, which I form in my mind to

represent the things themselves, are nothing different from each other,

nor inferior to that image, by which I represent the grain of sand

itself, which is supposed so vastly to exceed them. What consists of

parts is distinguishable into them, and what is distinguishable is

separable. But whatever we may imagine of the thing, the idea of a grain

of sand is not distinguishable, nor separable into twenty, much less

into a thousand, ten thousand, or an infinite number of different ideas.

It is the same case with the impressions of the senses as with the ideas

of the imagination. Put a spot of ink upon paper, fix your eye upon that

spot, and retire to such a distance, that, at last you lose sight of it;

it is plain, that the moment before it vanished the image or impression

was perfectly indivisible. It is not for want of rays of light striking

on our eyes, that the minute parts of distant bodies convey not any

sensible impression; but because they are removed beyond that distance,

at which their impressions were reduced to a minimum, and were incapable

of any farther diminution. A microscope or telescope, which renders them

visible, produces not any new rays of light, but only spreads those,

which always flowed from them; and by that means both gives parts to

impressions, which to the naked eye appear simple and uncompounded, and

advances to a minimum, what was formerly imperceptible.

We may hence discover the error of the common opinion, that the capacity

of the mind is limited on both sides, and that it is impossible for

the imagination to form an adequate idea, of what goes beyond a certain

degree of minuteness as well as of greatness. Nothing can be more

minute, than some ideas, which we form in the fancy; and images, which

appear to the senses; since there are ideas and images perfectly simple

and indivisible. The only defect of our senses is, that they give

us disproportioned images of things, and represent as minute and

uncompounded what is really great and composed of a vast number of

parts. This mistake we are not sensible of: but taking the impressions

of those minute objects, which appear to the senses, to be equal or

nearly equal to the objects, and finding by reason, that there are other

objects vastly more minute, we too hastily conclude, that these are

inferior to any idea of our imagination or impression of our senses.

This however is certain, that we can form ideas, which shall be no

greater than the smallest atom of the animal spirits of an insect a

thousand times less than a mite: And we ought rather to conclude, that

the difficulty lies in enlarging our conceptions so much as to form a

just notion of a mite, or even of an insect a thousand times less than a

mite. For in order to form a just notion of these animals, we must have

a distinct idea representing every part of them, which, according to the

system of infinite divisibility, is utterly impossible, and, recording

to that of indivisible parts or atoms, is extremely difficult, by reason

of the vast number and multiplicity of these parts.

SECT. II. OF THE INFINITE DIVISIBILITY OF SPACE AND TIME.

Wherever ideas are adequate representations of objects, the relations,

contradictions and agreements of the ideas are all applicable to the

objects; and this we may in general observe to be the foundation of all

human knowledge. But our ideas are adequate representations of the

most minute parts of extension; and through whatever divisions and

subdivisions we may suppose these parts to be arrived at, they can never

become inferior to some ideas, which we form. The plain consequence is,

that whatever appears impossible and contradictory upon the comparison

of these ideas, must be really impossible and contradictory, without any

farther excuse or evasion.

Every thing capable of being infinitely divided contains an infinite

number of parts; otherwise the division would be stopt short by the

indivisible parts, which we should immediately arrive at. If therefore

any finite extension be infinitely divisible, it can be no contradiction

to suppose, that a finite extension contains an infinite number of

parts: And vice versa, if it be a contradiction to suppose, that

a finite extension contains an infinite number of parts, no finite

extension can be infinitely divisible. But that this latter supposition

is absurd, I easily convince myself by the consideration of my clear

ideas. I first take the least idea I can form of a part of extension,

and being certain that there is nothing more minute than this idea, I

conclude, that whatever I discover by its means must be a real quality

of extension. I then repeat this idea once, twice, thrice, &c., and find

the compound idea of extension, arising from its repetition, always

to augment, and become double, triple, quadruple, &c., till at last it

swells up to a considerable bulk, greater or smaller, in proportion as I

repeat more or less the same idea. When I stop in the addition of parts,

the idea of extension ceases to augment; and were I to carry on the

addition in infinitum, I clearly perceive, that the idea of extension

must also become infinite. Upon the whole, I conclude, that the idea of

all infinite number of parts is individually the same idea with that of

an infinite extension; that no finite extension is capable of containing

an infinite number of parts; and consequently that no finite extension

is infinitely divisible.[1]

[1] It has been objected to me, that infinite divisibility supposes

only an infinite number of PROPORTIONAL not of ALIQIOT parts, and

that an infinite number of proportional parts does not form an

infinite extension. But this distinction is entirely frivolous.

Whether these parts be calld ALIQUOT or PROPORTIONAL, they cannot

be inferior to those minute parts we conceive; and therefore cannot

form a less extension by their conjunction.

I may subjoin another argument proposed by a noted author,[2] which

seems to me very strong and beautiful. It is evident, that existence in

itself belongs only to unity, and is never applicable to number, but on

account of the unites, of which the number is composed. Twenty men may

be said to exist; but it is only because one, two, three, four, &c. are

existent, and if you deny the existence of the latter, that of the

former falls of course. It is therefore utterly absurd to suppose any

number to exist, and yet deny the existence of unites; and as extension

is always a number, according to the common sentiment of

metaphysicians, and never resolves itself into any unite or indivisible

quantity, it follows, that extension can never at all exist. It is in

vain to reply, that any determinate quantity of extension is an unite;

but such-a-one as admits of an infinite number of fractions, and is

inexhaustible in its sub-divisions. For by the same rule these twenty

men may be considered as a unit. The whole globe of the earth, nay the

whole universe, may be considered as a unit. That term of unity is

merely a fictitious denomination, which the mind may apply to any

quantity of objects it collects together; nor can such an unity any

more exist alone than number can, as being in reality a true number.

But the unity, which can exist alone, and whose existence is necessary

to that of all number, is of another kind, and must be perfectly

indivisible, and incapable of being resolved into any lesser unity.

[2] Mons. MALEZIEU

All this reasoning takes place with regard to time; along with an

additional argument, which it may be proper to take notice of. It is a

property inseparable from time, and which in a manner constitutes its

essence, that each of its parts succeeds another, and that none of them,

however contiguous, can ever be co-existent. For the same reason, that

the year 1737 cannot concur with the present year 1738 every moment must

be distinct from, and posterior or antecedent to another. It is certain

then, that time, as it exists, must be composed of indivisible moments.

For if in time we could never arrive at an end of division, and if

each moment, as it succeeds another, were not perfectly single and

indivisible, there would be an infinite number of co-existent moments,

or parts of time; which I believe will be allowed to be an arrant

contradiction.

The infinite divisibility of space implies that of time, as is evident

from the nature of motion. If the latter, therefore, be impossible, the

former must be equally so.

I doubt not but, it will readily be allowed by the most obstinate

defender of the doctrine of infinite divisibility, that these arguments

are difficulties, and that it is impossible to give any answer to them

which will be perfectly clear and satisfactory. But here we may

observe, that nothing can be more absurd, than this custom of calling a

difficulty what pretends to be a demonstration, and endeavouring by that

means to elude its force and evidence. It is not in demonstrations as

in probabilities, that difficulties can take place, and one argument

counter-ballance another, and diminish its authority. A demonstration,

if just, admits of no opposite difficulty; and if not just, it is a

mere sophism, and consequently can never be a difficulty. It is either

irresistible, or has no manner of force. To talk therefore of objections

and replies, and ballancing of arguments in such a question as this, is

to confess, either that human reason is nothing but a play of words, or

that the person himself, who talks so, has not a Capacity equal to such

subjects. Demonstrations may be difficult to be comprehended, because of

abstractedness of the subject; but can never have such difficulties as

will weaken their authority, when once they are comprehended.

It is true, mathematicians are wont to say, that there are here equally

strong arguments on the other side of the question, and that the

doctrine of indivisible points is also liable to unanswerable

objections. Before I examine these arguments and objections in detail,

I will here take them in a body, and endeavour by a short and decisive

reason to prove at once, that it is utterly impossible they can have any

just foundation.

It is an established maxim in metaphysics, That whatever the mind

clearly conceives, includes the idea of possible existence, or in other

words, that nothing we imagine is absolutely impossible. We can form the

idea of a golden mountain, and from thence conclude that such a mountain

may actually exist. We can form no idea of a mountain without a valley,

and therefore regard it as impossible.

Now it is certain we have an idea of extension; for otherwise why do we

talk and reason concerning it? It is likewise certain that this idea,

as conceived by the imagination, though divisible into parts or inferior

ideas, is not infinitely divisible, nor consists of an infinite number

of parts: For that exceeds the comprehension of our limited capacities.

Here then is an idea of extension, which consists of parts or inferior

ideas, that are perfectly, indivisible: consequently this idea implies

no contradiction: consequently it is possible for extension really to

exist conformable to it: and consequently all the arguments employed

against the possibility of mathematical points are mere scholastick

quibbles, and unworthy of our attention.

These consequences we may carry one step farther, and conclude that all

the pretended demonstrations for the infinite divisibility of extension

are equally sophistical; since it is certain these demonstrations cannot

be just without proving the impossibility of mathematical points; which

it is an evident absurdity to pretend to.

SECT. III. OF THE OTHER QUALITIES OF OUR IDEA OF SPACE AND TIME.

No discovery could have been made more happily for deciding all

controversies concerning ideas, than that abovementioned, that

impressions always take the precedency of them, and that every idea,

with which the imagination is furnished, first makes its appearance in a

correspondent impression. These latter perceptions are all so clear and

evident, that they admit of no controversy; though many of our ideas are

so obscure, that it is almost impossible even for the mind, which forms

them, to tell exactly their nature and composition. Let us apply this

principle, in order to discover farther the nature of our ideas of space

and time.

Upon opening my eyes, and turning them to the surrounding objects,

I perceive many visible bodies; and upon shutting them again, and

considering the distance betwixt these bodies, I acquire the idea of

extension. As every idea is derived from some impression, which

is exactly similar to it, the impressions similar to this idea of

extension, must either be some sensations derived from the sight, or

some internal impressions arising from these sensations.

Our internal impressions are our passions, emotions, desires and

aversions; none of which, I believe, will ever be asserted to be the

model, from which the idea of space is derived. There remains therefore

nothing but the senses, which can convey to us this original impression.

Now what impression do oar senses here convey to us? This is the

principal question, and decides without appeal concerning the nature of

the idea.

The table before me is alone sufficient by its view to give me the idea

of extension. This idea, then, is borrowed from, and represents some

impression, which this moment appears to the senses. But my senses

convey to me only the impressions of coloured points, disposed in a

certain manner. If the eye is sensible of any thing farther, I desire

it may be pointed out to me. But if it be impossible to shew any thing

farther, we may conclude with certainty, that the idea of extension is

nothing but a copy of these coloured points, and of the manner of their

appearance.

Suppose that in the extended object, or composition of coloured points,

from which we first received the idea of extension, the points were of

a purple colour; it follows, that in every repetition of that idea we

would not only place the points in the same order with respect to each

other, but also bestow on them that precise colour, with which alone we

are acquainted. But afterwards having experience of the other colours of

violet, green, red, white, black, and of all the different compositions

of these, and finding a resemblance in the disposition of coloured

points, of which they are composed, we omit the peculiarities of

colour, as far as possible, and found an abstract idea merely on that

disposition of points, or manner of appearance, in which they agree. Nay

even when the resemblance is carryed beyond the objects of one sense,

and the impressions of touch are found to be Similar to those of sight

in the disposition of their parts; this does not hinder the abstract

idea from representing both, upon account of their resemblance. All

abstract ideas are really nothing but particular ones, considered in

a certain light; but being annexed to general terms, they are able to

represent a vast variety, and to comprehend objects, which, as they are

alike in some particulars, are in others vastly wide of each other.

The idea of time, being derived from the succession of our perceptions

of every kind, ideas as well as impressions, and impressions of

reflection as well as of sensations will afford us an instance of an

abstract idea, which comprehends a still greater variety than that of

space, and yet is represented in the fancy by some particular individual

idea of a determinate quantity and quality.

As it is from the disposition of visible and tangible objects we receive

the idea of space, so from the succession of ideas and impressions we

form the idea of time, nor is it possible for time alone ever to make

its appearance, or be taken notice of by the mind. A man in a sound

sleep, or strongly occupyed with one thought, is insensible of time;

and according as his perceptions succeed each other with greater or

less rapidity, the same duration appears longer or shorter to his

imagination. It has been remarked by a great philosopher, that our

perceptions have certain bounds in this particular, which are fixed by

the original nature and constitution of the mind, and beyond which no

influence of external objects on the senses is ever able to hasten or

retard our thought. If you wheel about a burning coal with rapidity, it

will present to the senses an image of a circle of fire; nor will there

seem to be any interval of time betwixt its revolutions; meerly because

it is impossible for our perceptions to succeed each other with the same

rapidity, that motion may be communicated to external objects. Wherever

we have no successive perceptions, we have no notion of time, even

though there be a real succession in the objects. From these phenomena,

as well as from many others, we may conclude, that time cannot make

its appearance to the mind, either alone, or attended with a steady

unchangeable object, but is always discovered some PERCEIVABLE

succession of changeable objects.

To confirm this we may add the following argument, which to me seems

perfectly decisive and convincing. It is evident, that time or duration

consists of different parts: For otherwise we could not conceive a

longer or shorter duration. It is also evident, that these parts are not

co-existent: For that quality of the co-existence of parts belongs to

extension, and is what distinguishes it from duration. Now as time is

composed of parts, that are not coexistent: an unchangeable object,

since it produces none but coexistent impressions, produces none that

can give us the idea of time; and consequently that idea must be

derived from a succession of changeable objects, and time in its first

appearance can never be severed from such a succession.

Having therefore found, that time in its first appearance to the mind

is always conjoined with a succession of changeable objects, and that

otherwise it can never fall under our notice, we must now examine

whether it can be conceived without our conceiving any succession

of objects, and whether it can alone form a distinct idea in the

imagination.

In order to know whether any objects, which are joined in impression,

be inseparable in idea, we need only consider, if they be different

from each other; in which case, it is plain they may be conceived apart.

Every thing, that is different is distinguishable: and everything,

that is distinguishable, may be separated, according to the maxims

above-explained. If on the contrary they be not different, they are

not distinguishable: and if they be not distinguishable, they cannot be

separated. But this is precisely the case with respect to time, compared

with our successive perceptions. The idea of time is not derived from a

particular impression mixed up with others, and plainly distinguishable

from them; but arises altogether from the manner, in which impressions

appear to the mind, without making one of the number. Five notes played

on a flute give us the impression and idea of time; though time be not

a sixth impression, which presents itself to the hearing or any other of

the senses. Nor is it a sixth impression, which the mind by reflection

finds in itself. These five sounds making their appearance in this

particular manner, excite no emotion in the mind, nor produce an

affection of any kind, which being observed by it can give rise to a new

idea. For that is necessary to produce a new idea of reflection, nor can

the mind, by revolving over a thousand times all its ideas of sensation,

ever extract from them any new original idea, unless nature has so

framed its faculties, that it feels some new original impression arise

from such a contemplation. But here it only takes notice of the manner,

in which the different sounds make their appearance; and that it may

afterwards consider without considering these particular sounds, but

may conjoin it with any other objects. The ideas of some objects it

certainly must have, nor is it possible for it without these ideas ever

to arrive at any conception of time; which since it, appears not as any

primary distinct impression, can plainly be nothing but different

ideas, or impressions, or objects disposed in a certain manner, that is,

succeeding each other.

I know there are some who pretend, that the idea of duration

is applicable in a proper sense to objects, which are perfectly

unchangeable; and this I take to be the common opinion of philosophers

as well as of the vulgar. But to be convinced of its falsehood we need

but reflect on the foregoing conclusion, that the idea of duration is

always derived from a succession of changeable objects, and can never

be conveyed to the mind by any thing stedfast and unchangeable. For it

inevitably follows from thence, that since the idea of duration cannot

be derived from such an object, it can never-in any propriety or

exactness be applied to it, nor can any thing unchangeable be ever said

to have duration. Ideas always represent the Objects or impressions,

from which they are derived, and can never without a fiction represent

or be applied to any other. By what fiction we apply the idea of time,

even to what is unchangeable, and suppose, as is common, that duration

is a measure of rest as well as of motion, we shall consider

afterwards.[3]

[3] Sect 5.

There is another very decisive argument, which establishes the present

doctrine concerning our ideas of space and time, and is founded only on

that simple principle, that our ideas of them are compounded of parts,

which are indivisible. This argument may be worth the examining.

Every idea, that is distinguishable, being also separable, let us take

one of those simple indivisible ideas, of which the compound one of

extension is formed, and separating it from all others, and considering

it apart, let us form a judgment of its nature and qualities.

It is plain it is not the idea of extension. For the idea of extension

consists of parts; and this idea, according to the supposition, is

perfectly simple and indivisible. Is it therefore nothing? That is

absolutely impossible. For as the compound idea of extension, which is

real, is composed of such ideas; were these so many non-entities, there

would be a real existence composed of non-entities; which is absurd.

Here therefore I must ask, What is our idea of a simple and indivisible

point? No wonder if my answer appear somewhat new, since the question

itself has scarce ever yet been thought of. We are wont to dispute

concerning the nature of mathematical points, but seldom concerning the

nature of their ideas.

The idea of space is conveyed to the mind by two senses, the sight

and touch; nor does anything ever appear extended, that is not either

visible or tangible. That compound impression, which represents

extension, consists of several lesser impressions, that are indivisible

to the eye or feeling, and may be called impressions of atoms or

corpuscles endowed with colour and solidity. But this is not all. It is

not only requisite, that these atoms should be coloured or tangible,

in order to discover themselves to our senses; it is also necessary

we should preserve the idea of their colour or tangibility in order to

comprehend them by our imagination. There is nothing but the idea of

their colour or tangibility, which can render them conceivable by the

mind. Upon the removal of the ideas of these sensible qualities, they

are utterly annihilated to the thought or imagination.

Now such as the parts are, such is the whole. If a point be not

considered as coloured or tangible, it can convey to us no idea; and

consequently the idea of extension, which is composed of the ideas of

these points, can never possibly exist. But if the idea of extension

really can exist, as we are conscious it does, its parts must also

exist; and in order to that, must be considered as coloured or tangible.

We have therefore no idea of space or extension, but when we regard it

as an object either of our sight or feeling.

The same reasoning will prove, that the indivisible moments of time must

be filled with some real object or existence, whose succession forms the

duration, and makes it be conceivable by the mind.

SECT. IV. OBJECTIONS ANSWERED.

Our system concerning space and time consists of two parts, which

are intimately connected together. The first depends on this chain of

reasoning. The capacity of the mind is not infinite; consequently no

idea of extension or duration consists of an infinite number of parts

or inferior ideas, but of a finite number, and these simple and

indivisible: It is therefore possible for space and time to exist

conformable to this idea: And if it be possible, it is certain they

actually do exist conformable to it; since their infinite divisibility

is utterly impossible and contradictory.

The other part of our system is a consequence of this. The parts, into

which the ideas of space and time resolve themselves, become at last

indivisible; and these indivisible parts, being nothing in themselves,

are inconceivable when not filled with something real and existent. The

ideas of space and time are therefore no separate or distinct ideas, but

merely those of the manner or order, in which objects exist: Or in

other words, it is impossible to conceive either a vacuum and extension

without matter, or a time, when there was no succession or change in any

real existence. The intimate connexion betwixt these parts of our system

is the reason why we shall examine together the objections, which have

been urged against both of them, beginning with those against the finite

divisibility of extension.

I. The first of these objections, which I shall take notice of, is more

proper to prove this connexion and dependence of the one part upon the

other, than to destroy either of them. It has often been maintained in

the schools, that extension must be divisible, in infinitum, because

the system of mathematical points is absurd; and that system is absurd,

because a mathematical point is a non-entity, and consequently can never

by its conjunction with others form a real existence. This would

be perfectly decisive, were there no medium betwixt the infinite

divisibility of matter, and the non-entity of mathematical points. But

there is evidently a medium, viz. the bestowing a colour or solidity on

these points; and the absurdity of both the extremes is a demonstration

of the truth and reality of this medium. The system of physical points,

which is another medium, is too absurd to need a refutation. A real

extension, such as a physical point is supposed to be, can never exist

without parts, different from each other; and wherever objects are

different, they are distinguishable and separable by the imagination.

II. The second objection is derived from the necessity there would be of

PENETRATION, if extension consisted of mathematical points. A simple and

indivisible atom, that touches another, must necessarily penetrate it;

for it is impossible it can touch it by its external parts, from the

very supposition of its perfect simplicity, which excludes all parts. It

must therefore touch it intimately, and in its whole essence, SECUNDUM

SE, TOTA, ET TOTALITER; which is the very definition of penetration.

But penetration is impossible: Mathematical points are of consequence

equally impossible.

I answer this objection by substituting a juster idea of penetration.

Suppose two bodies containing no void within their circumference, to

approach each other, and to unite in such a manner that the body, which

results from their union, is no more extended than either of them; it

is this we must mean when we talk of penetration. But it is evident this

penetration is nothing but the annihilation of one of these bodies, and

the preservation of the other, without our being able to distinguish

particularly which is preserved and which annihilated. Before the

approach we have the idea of two bodies. After it we have the idea

only of one. It is impossible for the mind to preserve any notion of

difference betwixt two bodies of the same nature existing in the same

place at the same time.

Taking then penetration in this sense, for the annihilation of one body

upon its approach to another, I ask any one, if he sees a necessity,

that a coloured or tangible point should be annihilated upon the

approach of another coloured or tangible point? On the contrary, does

he not evidently perceive, that from the union of these points there

results an object, which is compounded and divisible, and may be

distinguished into two parts, of which each preserves its existence

distinct and separate, notwithstanding its contiguity to the other? Let

him aid his fancy by conceiving these points to be of different colours,

the better to prevent their coalition and confusion. A blue and a red

point may surely lie contiguous without any penetration or annihilation.

For if they cannot, what possibly can become of them? Whether shall the

red or the blue be annihilated? Or if these colours unite into one, what

new colour will they produce by their union?

What chiefly gives rise to these objections, and at the same time

renders it so difficult to give a satisfactory answer to them, is the

natural infirmity and unsteadiness both of our imagination and senses,

when employed on such minute objects. Put a spot of ink upon paper, and

retire to such a distance, that the spot becomes altogether invisible;

you will find, that upon your return and nearer approach the spot

first becomes visible by short intervals; and afterwards becomes always

visible; and afterwards acquires only a new force in its colouring

without augmenting its bulk; and afterwards, when it has encreased to

such a degree as to be really extended, it is still difficult for

the imagination to break it into its component parts, because of the

uneasiness it finds in the conception of such a minute object as a

single point. This infirmity affects most of our reasonings on the

present subject, and makes it almost impossible to answer in an

intelligible manner, and in proper expressions, many questions which may

arise concerning it.

III. There have been many objections drawn from the mathematics against

the indivisibility of the parts of extension: though at first sight that

science seems rather favourable to the present doctrine; and if it

be contrary in its DEMONSTRATIONS, it is perfectly conformable in its

definitions. My present business then must be to defend the definitions,

and refute the demonstrations.

A surface is DEFINed to be length and breadth without depth: A line

to be length without breadth or depth: A point to be what has neither

length, breadth nor depth. It is evident that all this is perfectly

unintelligible upon any other supposition than that of the composition

of extension by indivisible points or atoms. How else could any thing

exist without length, without breadth, or without depth?

Two different answers, I find, have been made to this argument; neither

of which is in my opinion satisfactory. The first is, that the objects

of geometry, those surfaces, lines and points, whose proportions and

positions it examines, are mere ideas in the mind; and not only never

did, but never can exist in nature. They never did exist; for no one

will pretend to draw a line or make a surface entirely conformable to

the definition: They never can exist; for we may produce demonstrations

from these very ideas to prove, that they are impossible.

But can anything be imagined more absurd and contradictory than this

reasoning? Whatever can be conceived by a clear and distinct idea

necessarily implies the possibility of existence; and he who pretends

to prove the impossibility of its existence by any argument derived from

the clear idea, in reality asserts, that we have no clear idea of

it, because we have a clear idea. It is in vain to search for a

contradiction in any thing that is distinctly conceived by the mind. Did

it imply any contradiction, it is impossible it could ever be conceived.

There is therefore no medium betwixt allowing at least the possibility

of indivisible points, and denying their idea; and it is on this latter

principle, that the second answer to the foregoing argument is founded.

It has been[4] pretended, that though it be impossible to conceive a

length without any breadth, yet by an abstraction without a separation,

we can consider the one without regarding the other; in the same manner

as we may think of the length of the way betwixt two towns, and

overlook its breadth. The length is inseparable from the breadth both

in nature and in our minds; but this excludes not a partial

consideration, and a distinction of reason, after the manner above

explained.

[4] L'Art de penser.

In refuting this answer I shall not insist on the argument, which I have

already sufficiently explained, that if it be impossible for the mind

to arrive at a minimum in its ideas, its capacity must be infinite, in

order to comprehend the infinite number of parts, of which its idea of

any extension would be composed. I shall here endeavour to find some new

absurdities in this reasoning.

A surface terminates a solid; a line terminates a surface; a point

terminates a line; but I assert, that if the ideas of a point, line or

surface were not indivisible, it is impossible we should ever conceive

these terminations: For let these ideas be supposed infinitely

divisible; and then let the fancy endeavour to fix itself on the idea of

the last surface, line or point; it immediately finds this idea to break

into parts; and upon its seizing the last of these parts, it loses its

hold by a new division, and so on in infinitum, without any possibility

of its arriving at a concluding idea. The number of fractions bring

it no nearer the last division, than the first idea it formed. Every

particle eludes the grasp by a new fraction; like quicksilver, when we

endeavour to seize it. But as in fact there must be something, which

terminates the idea of every finite quantity; and as this terminating

idea cannot itself consist of parts or inferior ideas; otherwise it

would be the last of its parts, which finished the idea, and so on; this

is a clear proof, that the ideas of surfaces, lines and points admit

not of any division; those of surfaces in depth; of lines in breadth and

depth; and of points in any dimension.

The school were so sensible of the force of this argument, that some of

them maintained, that nature has mixed among those particles of matter,

which are divisible in infinitum, a number of mathematical points, in

order to give a termination to bodies; and others eluded the force of

this reasoning by a heap of unintelligible cavils and distinctions. Both

these adversaries equally yield the victory. A man who hides himself,

confesses as evidently the superiority of his enemy, as another, who

fairly delivers his arms.

Thus it appears, that the definitions of mathematics destroy the

pretended demonstrations; and that if we have the idea of indivisible

points, lines and surfaces conformable to the definition, their

existence is certainly possible: but if we have no such idea, it is

impossible we can ever conceive the termination of any figure; without

which conception there can be no geometrical demonstration.

But I go farther, and maintain, that none of these demonstrations

can have sufficient weight to establish such a principle, as this of

infinite divisibility; and that because with regard to such minute

objects, they are not properly demonstrations, being built on ideas,

which are not exact, and maxims, which are not precisely true. When

geometry decides anything concerning the proportions of quantity, we

ought not to look for the utmost precision and exactness. None of its

proofs extend so far. It takes the dimensions and proportions of

figures justly; but roughly, and with some liberty. Its errors are never

considerable; nor would it err at all, did it not aspire to such an

absolute perfection.

I first ask mathematicians, what they mean when they say one line or

surface is EQUAL to, or GREATER or LESS than another? Let any of them

give an answer, to whatever sect he belongs, and whether he maintains

the composition of extension by indivisible points, or by quantities

divisible in infinitum. This question will embarrass both of them.

There are few or no mathematicians, who defend the hypothesis of

indivisible points; and yet these have the readiest and justest answer

to the present question. They need only reply, that lines or surfaces

are equal, when the numbers of points in each are equal; and that as

the proportion of the numbers varies, the proportion of the lines and

surfaces is also varyed. But though this answer be just, as well as

obvious; yet I may affirm, that this standard of equality is entirely

useless, and that it never is from such a comparison we determine

objects to be equal or unequal with respect to each other. For as the

points, which enter into the composition of any line or surface, whether

perceived by the sight or touch, are so minute and so confounded with

each other, that it is utterly impossible for the mind to compute their

number, such a computation will Never afford us a standard by which we

may judge of proportions. No one will ever be able to determine by an

exact numeration, that an inch has fewer points than a foot, or a foot

fewer than an ell or any greater measure: for which reason we seldom or

never consider this as the standard of equality or inequality.

As to those, who imagine, that extension is divisible in infinitum, it

is impossible they can make use of this answer, or fix the equality of

any line or surface by a numeration of its component parts. For since,

according to their hypothesis, the least as well as greatest figures

contain an infinite number of parts; and since infinite numbers,

properly speaking, can neither be equal nor unequal with respect to each

other; the equality or inequality of any portions of space can never

depend on any proportion in the number of their parts. It is true, it

may be said, that the inequality of an ell and a yard consists in the

different numbers of the feet, of which they are composed; and that of

a foot and a yard in the number of the inches. But as that quantity we

call an inch in the one is supposed equal to what we call an inch in

the other, and as it is impossible for the mind to find this equality by

proceeding in infinitum with these references to inferior quantities: it

is evident, that at last we must fix some standard of equality different

from an enumeration of the parts.

There are some[5], who pretend, that equality is best defined by

congruity, and that any two figures are equal, when upon the placing of

one upon the other, all their parts correspond to and touch each other.

In order to judge of this definition let us consider, that since

equality is a relation, it is not, strictly speaking, a property in the

figures themselves, but arises merely from the comparison, which the

mind makes betwixt them. If it consists, therefore, in this imaginary

application and mutual contact of parts, we must at least have a

distinct notion of these parts, and must conceive their contact. Now it

is plain, that in this conception we would run up these parts to the

greatest minuteness, which can possibly be conceived; since the contact

of large parts would never render the figures equal. But the minutest

parts we can conceive are mathematical points; and consequently this

standard of equality is the same with that derived from the equality of

the number of points; which we have already determined to be a just but

an useless standard. We must therefore look to some other quarter for a

solution of the present difficulty.

[5] See Dr. Barrow's mathematical lectures.

There are many philosophers, who refuse to assign any standard of

equality, but assert, that it is sufficient to present two objects, that

are equal, in order to give us a just notion of this proportion. All

definitions, say they, are fruitless, without the perception of such

objects; and where we perceive such objects, we no longer stand in need

of any definition. To this reasoning, I entirely agree; and assert, that

the only useful notion of equality, or inequality, is derived from the

whole united appearance and the comparison of particular objects.

It is evident, that the eye, or rather the mind is often able at one

view to determine the proportions of bodies, and pronounce them equal

to, or greater or less than each other, without examining or comparing

the number of their minute parts. Such judgments are not only common,

but in many cases certain and infallible. When the measure of a yard and

that of a foot are presented, the mind can no more question, that the

first is longer than the second, than it can doubt of those principles,

which are the most clear and self-evident.

There are therefore three proportions, which the mind distinguishes

in the general appearance of its objects, and calls by the names of

greater, less and equal. But though its decisions concerning these

proportions be sometimes infallible, they are not always so; nor are our

judgments of this kind more exempt from doubt and error than those on

any other subject. We frequently correct our first opinion by a review

and reflection; and pronounce those objects to be equal, which at first

we esteemed unequal; and regard an object as less, though before it

appeared greater than another. Nor is this the only correction, which

these judgments of our senses undergo; but we often discover our error

by a juxtaposition of the objects; or where that is impracticable, by

the use of some common and invariable measure, which being successively

applied to each, informs us of their different proportions. And even

this correction is susceptible of a new correction, and of different

degrees of exactness, according to the nature of the instrument,

by which we measure the bodies, and the care which we employ in the

comparison.

When therefore the mind is accustomed to these judgments and their

corrections, and finds that the same proportion which makes two figures

have in the eye that appearance, which we call equality, makes them also

correspond to each other, and to any common measure, with which they

are compared, we form a mixed notion of equality derived both from the

looser and stricter methods of comparison. But we are not content with

this. For as sound reason convinces us that there are bodies vastly more

minute than those, which appear to the senses; and as a false reason

would perswade us, that there are bodies infinitely more minute; we

clearly perceive, that we are not possessed of any instrument or art of

measuring, which can secure us from ill error and uncertainty. We are

sensible, that the addition or removal of one of these minute parts,

is not discernible either in the appearance or measuring; and as we

imagine, that two figures, which were equal before, cannot be equal

after this removal or addition, we therefore suppose some imaginary

standard of equality, by which the appearances and measuring are exactly

corrected, and the figures reduced entirely to that proportion. This

standard is plainly imaginary. For as the very idea of equality is that

of such a particular appearance corrected by juxtaposition or a common

measure. The notion of any correction beyond what we have instruments

and art to make, is a mere fiction of the mind, and useless as well

as incomprehensible. But though this standard be only imaginary, the

fiction however is very natural; nor is anything more usual, than for

the mind to proceed after this manner with any action, even after the

reason has ceased, which first determined it to begin. This appears very

conspicuously with regard to time; where though it is evident we have no

exact method of determining the proportions of parts, not even so exact

as in extension, yet the various corrections of our measures, and their

different degrees of exactness, have given as an obscure and implicit

notion of a perfect and entire equality. The case is the same in many

other subjects. A musician finding his ear becoming every day more

delicate, and correcting himself by reflection and attention, proceeds

with the same act of the mind, even when the subject fails him, and

entertains a notion of a compleat TIERCE or OCTAVE, without being able

to tell whence he derives his standard. A painter forms the same fiction

with regard to colours. A mechanic with regard to motion. To the one

light and shade; to the other swift and slow are imagined to be capable

of an exact comparison and equality beyond the judgments of the senses.

We may apply the same reasoning to CURVE and RIGHT lines. Nothing is

more apparent to the senses, than the distinction betwixt a curve and a

right line; nor are there any ideas we more easily form than the ideas

of these objects. But however easily we may form these ideas, it is

impossible to produce any definition of them, which will fix the precise

boundaries betwixt them. When we draw lines upon paper, or any continued

surface, there is a certain order, by which the lines run along from one

point to another, that they may produce the entire impression of a

curve or right line; but this order is perfectly unknown, and nothing

is observed but the united appearance. Thus even upon the system of

indivisible points, we can only form a distant notion of some unknown

standard to these objects. Upon that of infinite divisibility we cannot

go even this length; but are reduced meerly to the general appearance,

as the rule by which we determine lines to be either curve or right

ones. But though we can give no perfect definition of these lines, nor

produce any very exact method of distinguishing the one from the other;

yet this hinders us not from correcting the first appearance by a more

accurate consideration, and by a comparison with some rule, of whose

rectitude from repeated trials we have a greater assurance. And it is

from these corrections, and by carrying on the same action of the mind,

even when its reason fails us, that we form the loose idea of a perfect

standard to these figures, without being able to explain or comprehend

it.

It is true, mathematicians pretend they give an exact definition of a

right line, when they say, it is the shortest way betwixt two points.

But in the first place I observe, that this is more properly the

discovery of one of the properties of a right line, than a just

deflation of it. For I ask any one, if upon mention of a right line he

thinks not immediately on such a particular appearance, and if it is not

by accident only that he considers this property? A right line can be

comprehended alone; but this definition is unintelligible without a

comparison with other lines, which we conceive to be more extended. In

common life it is established as a maxim, that the straightest way is

always the shortest; which would be as absurd as to say, the shortest

way is always the shortest, if our idea of a right line was not

different from that of the shortest way betwixt two points.

Secondly, I repeat what I have already established, that we have no

precise idea of equality and inequality, shorter and longer, more than

of a right line or a curve; and consequently that the one can never

afford us a perfect standard for the other. An exact idea can never be

built on such as are loose and undetermined.

The idea of a plain surface is as little susceptible of a precise

standard as that of a right line; nor have we any other means of

distinguishing such a surface, than its general appearance. It is in

vain, that mathematicians represent a plain surface as produced by the

flowing of a right line. It will immediately be objected, that our idea

of a surface is as independent of this method of forming a surface, as

our idea of an ellipse is of that of a cone; that the idea of a right

line is no more precise than that of a plain surface; that a right line

may flow irregularly, and by that means form a figure quite different

from a plane; and that therefore we must suppose it to flow along two

right lines, parallel to each other, and on the same plane; which is a

description, that explains a thing by itself, and returns in a circle.

It appears, then, that the ideas which are most essential to geometry,

viz. those of equality and inequality, of a right line and a plain

surface, are far from being exact and determinate, according to our

common method of conceiving them. Not only we are incapable of telling,

if the case be in any degree doubtful, when such particular figures are

equal; when such a line is a right one, and such a surface a plain one;

but we can form no idea of that proportion, or of these figures, which

is firm and invariable. Our appeal is still to the weak and fallible

judgment, which we make from the appearance of the objects, and correct

by a compass or common measure; and if we join the supposition of

any farther correction, it is of such-a-one as is either useless or

imaginary. In vain should we have recourse to the common topic, and

employ the supposition of a deity, whose omnipotence may enable him to

form a perfect geometrical figure, and describe a right line without any

curve or inflexion. As the ultimate standard of these figures is derived

from nothing but the senses and imagination, it is absurd to talk of

any perfection beyond what these faculties can judge of; since the true

perfection of any thing consists in its conformity to its standard.

Now since these ideas are so loose and uncertain, I would fain ask any

mathematician what infallible assurance he has, not only of the more

intricate, and obscure propositions of his science, but of the most

vulgar and obvious principles? How can he prove to me, for instance,

that two right lines cannot have one common segment? Or that it is

impossible to draw more than one right line betwixt any two points?

should he tell me, that these opinions are obviously absurd, and

repugnant to our clear ideas; I would answer, that I do not deny, where

two right lines incline upon each other with a sensible angle, but it is

absurd to imagine them to have a common segment. But supposing these two

lines to approach at the rate of an inch in twenty leagues, I perceive

no absurdity in asserting, that upon their contact they become one. For,

I beseech you, by what rule or standard do you judge, when you assert,

that the line, in which I have supposed them to concur, cannot make

the same right line with those two, that form so small an angle betwixt

them? You must surely have some idea of a right line, to which this line

does not agree. Do you therefore mean that it takes not the points in

the same order and by the same rule, as is peculiar and essential to a

right line? If so, I must inform you, that besides that in judging after

this manner you allow, that extension is composed of indivisible points

(which, perhaps, is more than you intend) besides this, I say, I must

inform you, that neither is this the standard from which we form the

idea of a right line; nor, if it were, is there any such firmness in our

senses or imagination, as to determine when such an order is violated or

preserved. The original standard of a right line is in reality nothing

but a certain general appearance; and it is evident right lines may be

made to concur with each other, and yet correspond to this standard,

though corrected by all the means either practicable or imaginable.

To whatever side mathematicians turn, this dilemma still meets them.

If they judge of equality, or any other proportion, by the accurate and

exact standard, viz. the enumeration of the minute indivisible parts,

they both employ a standard, which is useless in practice, and actually

establish the indivisibility of extension, which they endeavour to

explode. Or if they employ, as is usual, the inaccurate standard,

derived from a comparison of objects, upon their general appearance,

corrected by measuring and juxtaposition; their first principles,

though certain and infallible, are too coarse to afford any such subtile

inferences as they commonly draw from them. The first principles are

founded on the imagination and senses: The conclusion, therefore, can

never go beyond, much less contradict these faculties.

This may open our eyes a little, and let us see, that no geometrical

demonstration for the infinite divisibility of extension can have so

much force as what we naturally attribute to every argument, which is

supported by such magnificent pretensions. At the same time we may learn

the reason, why geometry falls of evidence in this single point, while

all its other reasonings command our fullest assent and approbation.

And indeed it seems more requisite to give the reason of this exception,

than to shew, that we really must make such an exception, and regard

all the mathematical arguments for infinite divisibility as utterly

sophistical. For it is evident, that as no idea of quantity is

infinitely divisible, there cannot be imagined a more glaring absurdity,

than to endeavour to prove, that quantity itself admits of such a

division; and to prove this by means of ideas, which are directly

opposite in that particular. And as this absurdity is very glaring in

itself, so there is no argument founded on it which is not attended

with a new absurdity, and involves not an evident contradiction.

I might give as instances those arguments for infinite divisibility,

which are derived from the point of contact. I know there is no

mathematician, who will not refuse to be judged by the diagrams he

describes upon paper, these being loose draughts, as he will tell us,

and serving only to convey with greater facility certain ideas, which

are the true foundation of all our reasoning. This I am satisfyed with,

and am willing to rest the controversy merely upon these ideas. I desire

therefore our mathematician to form, as accurately as possible,

the ideas of a circle and a right line; and I then ask, if upon the

conception of their contact he can conceive them as touching in a

mathematical point, or if he must necessarily imagine them to concur

for some space. Whichever side he chuses, he runs himself into equal

difficulties. If he affirms, that in tracing these figures in his

imagination, he can imagine them to touch only in a point, he allows

the possibility of that idea, and consequently of the thing. If he says,

that in his conception of the contact of those lines he must make

them concur, he thereby acknowledges the fallacy of geometrical

demonstrations, when carryed beyond a certain degree of minuteness;

since it is certain he has such demonstrations against the concurrence

of a circle and a right line; that is, in other words, he can prove an

idea, viz. that of concurrence, to be INCOMPATIBLE with two other

ideas, those of a circle and right line; though at the same time he

acknowledges these ideas to be inseparable.

SECT. V. THE SAME SUBJECT CONTINUED.

If the second part of my system be true, that the idea of space

or extension is nothing but the idea of visible or tangible points

distributed in a certain order; it follows, that we can form no idea

of a vacuum, or space, where there is nothing visible or tangible. This

gives rise to three objections, which I shall examine together, because

the answer I shall give to one is a consequence of that which I shall

make use of for the others.

First, It may be said, that men have disputed for many ages concerning

a vacuum and a plenum, without being able to bring the affair to a

final decision; and philosophers, even at this day, think themselves

at liberty to take part on either side, as their fancy leads them. But

whatever foundation there may be for a controversy concerning the things

themselves, it may be pretended, that the very dispute is decisive

concerning the idea, and that it is impossible men could so long reason

about a vacuum, and either refute or defend it, without having a notion

of what they refuted or defended.

Secondly, If this argument should be contested, the reality or at least

the possibility of the idea of a vacuum may be proved by the following

reasoning. Every idea is possible, which is a necessary and infallible

consequence of such as are possible. Now though we allow the world to be

at present a plenum, we may easily conceive it to be deprived of motion;

and this idea will certainly be allowed possible. It must also be

allowed possible, to conceive the annihilation of any part of matter by

the omnipotence of the deity, while the other parts remain at rest. For

as every idea, that is distinguishable, is separable by the imagination;

and as every idea, that is separable by the imagination, may be

conceived to be separately existent; it is evident, that the existence

of one particle of matter, no more implies the existence of another,

than a square figure in one body implies a square figure in every one.

This being granted, I now demand what results from the concurrence of

these two possible ideas of rest and annihilation, and what must we

conceive to follow upon the annihilation of all the air and subtile

matter in the chamber, supposing the walls to remain the same, without

any motion or alteration? There are some metaphysicians, who answer,

that since matter and extension are the same, the annihilation of one

necessarily implies that of the other; and there being now no distance

betwixt the walls of the chamber, they touch each other; in the same

manner as my hand touches the paper, which is immediately before me.

But though this answer be very common, I defy these metaphysicians to

conceive the matter according to their hypothesis, or imagine the floor

and roof, with all the opposite sides of the chamber, to touch each

other, while they continue in rest, and preserve the same position. For

how can the two walls, that run from south to north, touch each other,

while they touch the opposite ends of two walls, that run from east

to west? And how can the floor and roof ever meet, while they are

separated by the four walls, that lie in a contrary position? If you

change their position, you suppose a motion. If you conceive any thing

betwixt them, you suppose a new creation. But keeping strictly to the

two ideas of rest and annihilation, it is evident, that the idea, which

results from them, is not that of a contact of parts, but something

else; which is concluded to be the idea of a vacuum.

The third objection carries the matter still farther, and not only

asserts, that the idea of a vacuum is real and possible, but also

necessary and unavoidable. This assertion is founded on the motion we

observe in bodies, which, it is maintained, would be impossible and

inconceivable without a vacuum, into which one body must move in order

to make way for another.. I shall not enlarge upon this objection,

because it principally belongs to natural philosophy, which lies without

our present sphere.

In order to answer these objections, we must take the matter pretty

deep, and consider the nature and origin of several ideas, lest we

dispute without understanding perfectly the subject of the controversy.

It is evident the idea of darkness is no positive idea, but merely the

negation of light, or more properly speaking, of coloured and visible

objects. A man, who enjoys his sight, receives no other perception from

turning his eyes on every side, when entirely deprived of light, than

what is common to him with one born blind; and it is certain such-a-one

has no idea either of light or darkness. The consequence of this is,

that it is not from the mere removal of visible objects we receive

the impression of extension without matter; and that the idea of utter

darkness can never be the same with that of vacuum.

Suppose again a man to be supported in the air, and to be softly

conveyed along by some invisible power; it is evident he is sensible of

nothing, and never receives the idea of extension, nor indeed any idea,

from this invariable motion. Even supposing he moves his limbs to

and fro, this cannot convey to him that idea. He feels in that case a

certain sensation or impression, the parts of which are successive to

each other, and may give him the idea of time: But certainly are not

disposed in such a manner, as is necessary to convey the idea of space

or the idea of space or extension.

Since then it appears, that darkness and motion, with the utter removal

of every thing visible and tangible, can never give us the idea of

extension without matter, or of a vacuum; the next question is, whether

they can convey this idea, when mixed with something visible and

tangible?

It is commonly allowed by philosophers, that all bodies, which discover

themselves to the eye, appear as if painted on a plain surface, and that

their different degrees of remoteness from ourselves are discovered

more by reason than by the senses. When I hold up my hand before me, and

spread my fingers, they are separated as perfectly by the blue colour

of the firmament, as they could be by any visible object, which I could

place betwixt them. In order, therefore, to know whether the sight can

convey the impression and idea of a vacuum, we must suppose, that amidst

an entire darkness, there are luminous bodies presented to us, whose

light discovers only these bodies themselves, without giving us any

impression of the surrounding objects.

We must form a parallel supposition concerning the objects of our

feeling. It is not proper to suppose a perfect removal of all tangible

objects: we must allow something to be perceived by the feeling; and

after an interval and motion of the hand or other organ of sensation,

another object of the touch to be met with; and upon leaving that,

another; and so on, as often as we please. The question is, whether

these intervals do not afford us the idea of extension without body?

To begin with the first case; it is evident, that when only two luminous

bodies appear to the eye, we can perceive, whether they be conjoined or

separate: whether they be separated by a great or small distance; and if

this distance varies, we can perceive its increase or diminution, with

the motion of the bodies. But as the distance is not in this case any

thing coloured or visible, it may be thought that there is here a vacuum

or pure extension, not only intelligible to the mind, but obvious to the

very senses.

This is our natural and most familiar way of thinking; but which we

shall learn to correct by a little reflection. We may observe, that

when two bodies present themselves, where there was formerly an entire

darkness, the only change, that is discoverable, is in the appearance

of these two objects, and that all the rest continues to be as before, a

perfect negation of light, and of every coloured or visible object. This

is not only true of what may be said to be remote from these bodies, but

also of the very distance; which is interposed betwixt them; that being

nothing but darkness, or the negation of light; without parts, without

composition, invariable and indivisible. Now since this distance causes

no perception different from what a blind man receives from his eyes, or

what is conveyed to us in the darkest night, it must partake of the

same properties: And as blindness and darkness afford us no ideas of

extension, it is impossible that the dark and undistinguishable distance

betwixt two bodies can ever produce that idea.

The sole difference betwixt an absolute darkness and the appearance of

two or more visible luminous objects consists, as I said, in the objects

themselves, and in the manner they affect our senses. The angles, which

the rays of light flowing from them, form with each other; the motion

that is required in the eye, in its passage from one to the other; and

the different parts of the organs, which are affected by them; these

produce the only perceptions, from which we can judge of the distance.

But as these perceptions are each of them simple and indivisible, they

can never give us the idea of extension.

We may illustrate this by considering the sense of feeling, and the

imaginary distance or interval interposed betwixt tangible or solid

objects. I suppose two cases, viz. that of a man supported in the air,

and moving his limbs to and fro, without meeting any thing tangible; and

that of a man, who feeling something tangible, leaves it, and after a

motion, of which he is sensible, perceives another tangible object; and

I then ask, wherein consists the difference betwixt these two cases?

No one will make any scruple to affirm, that it consists meerly in the

perceiving those objects, and that the sensation, which arises from the

motion, is in both cases the same: And as that sensation is not capable

of conveying to us an idea of extension, when unaccompanyed with some

other perception, it can no more give us that idea, when mixed with

the impressions of tangible objects; since that mixture produces no

alteration upon it.

But though motion and darkness, either alone, or attended with tangible

and visible objects, convey no idea of a vacuum or extension without

matter, yet they are the causes why we falsly imagine we can form such

an idea. For there is a close relation betwixt that motion and darkness,

and a real extension, or composition of visible and tangible objects.

First, We may observe, that two visible objects appearing in the midst

of utter darkness, affect the senses in the same manner, and form the

same angle by the rays, which flow from them, and meet in the eye, as if

the distance betwixt them were find with visible objects, that give us

a true idea of extension. The sensation of motion is likewise the same,

when there is nothing tangible interposed betwixt two bodies, as when

we feel a compounded body, whose different parts are placed beyond each

other.

Secondly, We find by experience, that two bodies, which are so placed

as to affect the senses in the same manner with two others, that have a

certain extent of visible objects interposed betwixt them, are

capable of receiving the same extent, without any sensible impulse or

penetration, and without any change on that angle, under which they

appear to the senses. In like manner, where there is one object, which

we cannot feel after another without an interval, and the perceiving

of that sensation we call motion in our hand or organ of sensation;

experience shews us, that it is possible the same object may be felt

with the same sensation of motion, along with the interposed impression

of solid and tangible objects, attending the sensation. That is, in

other words, an invisible and intangible distance may be converted into

a visible and tangible one, without any change on the distant objects.

Thirdly, We may observe, as another relation betwixt these two kinds

of distance, that they have nearly the same effects on every natural

phænomenon. For as all qualities, such as heat, cold, light,

attraction, &c. diminish in proportion to the distance; there is but

little difference observed, whether this distance be marled out by

compounded and sensible objects, or be known only by the manner, in

which the distant objects affect the senses.

Here then are three relations betwixt that distance, which conveys the

idea of extension, and that other, which is not filled with any coloured

or solid object. The distant objects affect the senses in the same

manner, whether separated by the one distance or the other; the second

species of distance is found capable of receiving the first; and they

both equally diminish the force of every quality.

These relations betwixt the two kinds of distance will afford us an easy

reason, why the one has so often been taken for the other, and why we

imagine we have an idea of extension without the idea of any object

either of the sight or feeling. For we may establish it as a general

maxim in this science of human nature, that wherever there is a close

relation betwixt two ideas, the mind is very apt to mistake them, and

in all its discourses and reasonings to use the one for the other. This

phænomenon occurs on so many occasions, and is of such consequence,

that I cannot forbear stopping a moment to examine its causes. I shall

only premise, that we must distinguish exactly betwixt the phænomenon

itself, and the causes, which I shall assign for it; and must not

imagine from any uncertainty in the latter, that the former is also

uncertain. The phænomenon may be real, though my explication be

chimerical. The falshood of the one is no consequence of that of the

other; though at the same time we may observe, that it is very natural

for us to draw such a consequence; which is an evident instance of that

very principle, which I endeavour to explain.

When I received the relations of resemblance, contiguity and causation,

as principles of union among ideas, without examining into their causes,

it was more in prosecution of my first maxim, that we must in the end

rest contented with experience, than for want of something specious and

plausible, which I might have displayed on that subject. It would have

been easy to have made an imaginary dissection of the brain, and have

shewn, why upon our conception of any idea, the animal spirits run

into all the contiguous traces, and rouze up the other ideas, that are

related to it. But though I have neglected any advantage, which I might

have drawn from this topic in explaining the relations of ideas, I am

afraid I must here have recourse to it, in order to account for the

mistakes that arise from these relations. I shall therefore observe,

that as the mind is endowed with a power of exciting any idea it

pleases; whenever it dispatches the spirits into that region of the

brain, in which the idea is placed; these spirits always excite the

idea, when they run precisely into the proper traces, and rummage that

cell, which belongs to the idea. But as their motion is seldom direct,

and naturally turns a little to the one side or the other; for this

reason the animal spirits, falling into the contiguous traces, present

other related ideas in lieu of that, which the mind desired at first to

survey. This change we are not always sensible of; but continuing

still the same train of thought, make use of the related idea, which is

presented to us, and employ it in our reasoning, as if it were the same

with what we demanded. This is the cause of many mistakes and sophisms

in philosophy; as will naturally be imagined, and as it would be easy to

show, if there was occasion.

Of the three relations above-mentioned that of resemblance is the most

fertile source of error; and indeed there are few mistakes in reasoning,

which do not borrow largely from that origin. Resembling ideas are not

only related together, but the actions of the mind, which we employ

in considering them, are so little different, that we are not able to

distinguish them. This last circumstance is of great consequence, and we

may in general observe, that wherever the actions of the mind in forming

any two ideas are the same or resembling, we are very apt to confound

these ideas, and take the one for the other. Of this we shall see many

instances in the progress of this treatise. But though resemblance be

the relation, which most readily produces a mistake in ideas, yet

the others of causation and contiguity may also concur in the same

influence. We might produce the figures of poets and orators, as

sufficient proofs of this, were it as usual, as it is reasonable, in

metaphysical subjects to draw our arguments from that quarter. But lest

metaphysicians should esteem this below their dignity, I shall borrow

a proof from an observation, which may be made on most of their own

discourses, viz. that it is usual for men to use words for ideas, and

to talk instead of thinking in their reasonings. We use words for ideas,

because they are commonly so closely connected that the mind easily

mistakes them. And this likewise is the reason, why we substitute

the idea of a distance, which is not considered either as visible or

tangible, in the room of extension, which is nothing but a composition

of visible or tangible points disposed in a certain order. In

causing this mistake there concur both the relations of causation and

resemblance. As the first species of distance is found to be convertible

into the second, it is in this respect a kind of cause; and the

similarity of their manner of affecting the senses, and diminishing

every quality, forms the relation of resemblance.

After this chain of reasoning and explication of my principles, I am now

prepared to answer all the objections that have been offered, whether

derived from metaphysics or mechanics. The frequent disputes concerning

a vacuum, or extension without matter prove not the reality of the idea,

upon which the dispute turns; there being nothing more common, than to

see men deceive themselves in this particular; especially when by means

of any close relation, there is another idea presented, which may be the

occasion of their mistake.

We may make almost the same answer to the second objection, derived from

the conjunction of the ideas of rest and annihilation. When every thing

is annihilated in the chamber, and the walls continue immoveable, the

chamber must be conceived much in the same manner as at present, when

the air that fills it, is not an object of the senses. This annihilation

leaves to the eye, that fictitious distance, which is discovered by the

different parts of the organ, that are affected, and by the degrees of

light and shade;--and to the feeling, that which consists in a sensation

of motion in the hand, or other member of the body. In vain should we.

search any farther. On whichever side we turn this subject, we shall

find that these are the only impressions such an object can produce

after the supposed annihilation; and it has already been remarked, that

impressions can give rise to no ideas, but to such as resemble them.

Since a body interposed betwixt two others may be supposed to be

annihilated, without producing any change upon such as lie on each

hand of it, it is easily conceived, how it may be created anew, and yet

produce as little alteration. Now the motion of a body has much the same

effect as its creation. The distant bodies are no more affected in the

one case, than in the other. This suffices to satisfy the imagination,

and proves there is no repugnance in such a motion. Afterwards

experience comes in play to persuade us that two bodies, situated in the

manner above-described, have really such a capacity of receiving body

betwixt them, and that there is no obstacle to the conversion of the

invisible and intangible distance into one that is visible and tangible.

However natural that conversion may seem, we cannot be sure it is

practicable, before we have had experience of it.

Thus I seem to have answered the three objections above-mentioned;

though at the same time I am sensible, that few will be satisfyed

with these answers, but will immediately propose new objections and

difficulties. It will probably be said, that my reasoning makes nothing

to the matter in hands and that I explain only the manner in which

objects affect the senses, without endeavouring to account for their

real nature and operations. Though there be nothing visible or tangible

interposed betwixt two bodies, yet we find BY EXPERIENCE, that the

bodies may be placed in the same manner, with regard to the eye, and

require the same motion of the hand in passing from one to the other,

as if divided by something visible and tangible. This invisible and

intangible distance is also found by experience to contain a capacity of

receiving body, or of becoming visible and tangible. Here is the whole

of my system; and in no part of it have I endeavoured to explain the

cause, which separates bodies after this manner, and gives them a

capacity of receiving others betwixt them, without any impulse or

penetration.

I answer this objection, by pleading guilty, and by confessing that my

intention never was to penetrate into the nature of bodies, or explain

the secret causes of their operations. For besides that this belongs not

to my present purpose, I am afraid, that such an enterprise is beyond

the reach of human understanding, and that we can never pretend to

know body otherwise than by those external properties, which discover

themselves to the senses. As to those who attempt any thing farther, I

cannot approve of their ambition, till I see, in some one instance at

least, that they have met with success. But at present I content myself

with knowing perfectly the manner in which objects affect my senses, and

their connections with each other, as far as experience informs me of

them. This suffices for the conduct of life; and this also suffices for

my philosophy, which pretends only to explain the nature and causes of

our perceptions, or impressions and ideas.[6]

[6] As long as we confine our speculations to the appearances of

objects to our senses, without entering into disquisitions

concerning their real nature and operations, we are safe from all

difficulties, and can never be embarrassed by any question. Thus,

if it be asked, if the invisible and intangible distance,

interposed betwixt two objects, be something or nothing: It is easy

to answer, that it is SOMETHING, VIZ. a property of the objects,

which affect the SENSES after such a particular manner. If it be

asked whether two objects, having such a distance betwixt them,

touch or not: it may be answered, that this depends upon the

definition of the word, TOUCH. If objects be said to touch, when

there is nothing SENSIBLE interposed betwixt them, these objects

touch: it objects be said to touch, when their IMAGES strike

contiguous parts of the eye, and when the hand FEELS both objects

successively, without any interposed motion, these objects do not

touch. The appearances of objects to our senses are all consistent;

and no difficulties can ever arise, but from the obscurity of the

terms we make use of.

If we carry our enquiry beyond the appearances of objects to the

senses, I am afraid, that most of our conclusions will be full of

scepticism and uncertainty. Thus if it be asked, whether or not the

invisible and intangible distance be always full of body, or of

something that by an improvement of our organs might become visible

or tangible, I must acknowledge, that I find no very decisive

arguments on either side; though I am inclined to the contrary

opinion, as being more suitable to vulgar and popular notions. If

THE NEWTONIAN philosophy be rightly understood, it will be found to

mean no more. A vacuum is asserted: That is, bodies are said to be

placed after such a manner, is to receive bodies betwixt them,

without impulsion or penetration. The real nature of this position

of bodies is unknown. We are only acquainted with its effects on

the senses, and its power of receiving body. Nothing is more

suitable to that philosophy, than a modest scepticism to a certain

degree, and a fair confession of ignorance in subjects, that exceed

all human capacity.

I shall conclude this subject of extension with a paradox, which will

easily be explained from the foregoing reasoning. This paradox is, that

if you are pleased to give to the in-visible and intangible distance,

or in other words, to the capacity of becoming a visible and tangible

distance, the name of a vacuum, extension and matter are the same, and

yet there is a vacuum. If you will not give it that name, motion

is possible in a plenum, without any impulse in infinitum, without

returning in a circle, and without penetration. But however we may

express ourselves, we must always confess, that we have no idea of any

real extension without filling it with sensible objects, and conceiving

its parts as visible or tangible.

As to the doctrine, that time is nothing but the manner, in which

some real objects exist; we may observe, that it is liable to the same

objections as the similar doctrine with regard to extension. If it be a

sufficient proof, that we have the idea of a vacuum, because we dispute

and reason concerning it; we must for the same reason have the idea

of time without any changeable existence; since there is no subject of

dispute more frequent and common. But that we really have no such idea,

is certain. For whence should it be derived? Does it arise from an

impression of sensation or of reflection? Point it out distinctly to us,

that we may know its nature and qualities. But if you cannot point

out any such impression, you may be certain you are mistaken, when you

imagine you have any such idea.

But though it be impossible to shew the impression, from which the idea

of time without a changeable existence is derived; yet we can easily

point out those appearances, which make us fancy we have that idea. For

we may observe, that there is a continual succession of perceptions in

our mind; so that the idea of time being for ever present with us; when

we consider a stedfast object at five-a-clock, and regard the same at

six; we are apt to apply to it that idea in the same manner as if every

moment were distinguished by a different position, or an alteration

of the object. The first and second appearances of the object, being

compared with the succession of our perceptions, seem equally removed as

if the object had really changed. To which we may add, what experience

shews us, that the object was susceptible of such a number of changes

betwixt these appearances; as also that the unchangeable or rather

fictitious duration has the same effect upon every quality, by

encreasing or diminishing it, as that succession, which is obvious to

the senses. From these three relations we are apt to confound our ideas,

and imagine we can form the idea of a time and duration, without any

change or succession.

SECT. VI. OF THE IDEA OF EXISTENCE, AND OF EXTERNAL EXISTENCE.

It may not be amiss, before we leave this subject, to explain the ideas

of existence and of external existence; which have their difficulties,

as well as the ideas of space and time. By this means we shall be the

better prepared for the examination of knowledge and probability, when

we understand perfectly all those particular ideas, which may enter into

our reasoning.

There is no impression nor idea of any kind, of which we have any

consciousness or memory, that is not conceived as existent; and it

is evident, that from this consciousness the most perfect idea and

assurance of being is derived. From hence we may form a dilemma, the

most clear and conclusive that can be imagined, viz. that since we never

remember any idea or impression without attributing existence to it,

the idea of existence must either be derived from a distinct impression,

conjoined with every perception or object of our thought, or must be the

very same with the idea of the perception or object.

As this dilemma is an evident consequence of the principle, that every

idea arises from a similar impression, so our decision betwixt the

propositions of the dilemma is no more doubtful. So far from there being

any distinct impression, attending every impression and every idea,

that I do not think there are any two distinct impressions, which are

inseparably conjoined. Though certain sensations may at one time be

united, we quickly find they admit of a separation, and may be presented

apart. And thus, though every impression and idea we remember be

considered as existent, the idea of existence is not derived from any

particular impression.

The idea of existence, then, is the very same with the idea of what we

conceive to be existent. To reflect on any thing simply, and to reflect

on it as existent, are nothing different from each other. That idea,

when conjoined with the idea of any object, makes no addition to it.

Whatever we conceive, we conceive to be existent. Any idea we please

to form is the idea of a being; and the idea of a being is any idea we

please to form.

Whoever opposes this, must necessarily point out that distinct

impression, from which the idea of entity is derived, and must prove,

that this impression is inseparable from every perception we believe to

be existent. This we may without hesitation conclude to be impossible.

Our foregoing reasoning[7] concerning the distinction of ideas without

any real difference will not here serve us in any stead. That kind of

distinction is founded on the different resemblances, which the same

simple idea may have to several different ideas. But no object can be

presented resembling some object with respect to its existence, and

different from others in the same particular; since every object, that

is presented, must necessarily be existent.

[7] Part I. Sect. 7.

A like reasoning will account for the idea of external existence. We may

observe, that it is universally allowed by philosophers, and is besides

pretty obvious of itself, that nothing is ever really present with the

mind but its perceptions or impressions and ideas, and that external

objects become known to us only by those perceptions they occasion. To

hate, to love, to think, to feel, to see; all this is nothing but to

perceive.

Now since nothing is ever present to the mind but perceptions, and since

all ideas are derived from something antecedently present to the mind;

it follows, that it is impossible for us so much as to conceive or form

an idea of any thing specifically different from ideas and impressions.

Let us fix our attention out of ourselves as much as possible: Let us

chase our imagination to the heavens, or to the utmost limits of the

universe; we never really advance a step beyond ourselves, nor can

conceive any kind of existence, but those perceptions, which

have appeared in that narrow compass. This is the universe of the

imagination, nor have we any idea but what is there produced.

The farthest we can go towards a conception of external objects, when

supposed SPECIFICALLY different from our perceptions, is to form a

relative idea of them, without pretending to comprehend the related

objects. Generally speaking we do not suppose them specifically

different; but only attribute to them different relations, connections

and durations. But of this more fully hereafter.[8]

[8] Part IV, Sect. 2.