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 [See Dr. Barrow's mathematical lectures.], 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.
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 coued 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. Whe