Chapter 6

[3]Dr Berkeley.

[4]'Tis evident, that even different simple ideas may have a similarity or resemblance to each other; nor is it necessary, that the point or circumstance of resemblance should be distinct or separable from that in which they differ.Blueandgreenare different simple ideas, but are more resembling thanblueandscarlet; though their perfect simplicity excludes all possibility of separation or distinction. 'Tis the same case with particular sounds, and tastes, and smells. These admit of infinite resemblances upon the general appearance and comparison, without having any common circumstance the same. And of this we may be certain, even from the very abstract termssimple idea. They comprehend all simple ideas under them. These resemble each other in their simplicity. And yet from their very nature, which excludes all composition, this circumstance, in which they resemble, is not distinguishable or separable from the rest. 'Tis the same case with all the degrees in any quality. They are all resembling, and yet the quality, in any individual, is not distinct from the degree.

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 showing the superiority of their science, which could discover opinions so remote from vulgar conception. On the other hand, any thing proposed to us, which causes surprise 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.

'Tis 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, 'twould be sufficiently evident from the plainest observation and experience. 'Tis also obvious, that whatever is capable of being dividedin infinitum, must consist of an infinite number of parts, and that 'tis 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 theidea, 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.

'Tis therefore certain, that the imagination reaches aminimum, and may raise up to itself an idea, of which it cannot conceive any subdivision, 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.

'Tis 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; 'tis plain, that the moment before it vanished, the image, or impression, was perfectly indivisible. 'Tis 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 aminimum, 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 aminimumwhat 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 'tis 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 areinferior 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 according to that of indivisible parts or atoms, is extremely difficult, by reason of the vast number and multiplicity of these parts.

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 whateverappearsimpossible and contradictory upon the comparison of these ideas, must bereallyimpossible 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 stopped 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: andvice 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 additionin infinitum, I clearly perceive, that the idea of extension must also become infinite. Upon the whole, I conclude, that the idea of an 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]

I may subjoin another argument proposed by a noted author,[2]which seems to me very strong and beautiful. 'Tis 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 'tis 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. 'Tis 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. 'Tis 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 subdivisions. For by the same rule, these twenty menmay be considered as an unite. The whole globe of the earth, nay, the whole universemay be considered as an unite. 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.

All this reasoning takes place with regard to time; along with an additional argument, which it may be proper to take notice of. 'Tis 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 coexistent. 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. 'Tis 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 'tis 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 adifficultywhat pretends to be ademonstration, and endeavouring by that means to elude its force and evidence. 'Tis not in demonstrations, as in probabilities, that difficulties can take place, and one argument counterbalance another, and diminish its authority. A demonstration, if just, admits of no opposite difficulty; and if not just, 'tis a mere sophism, and consequently can never be a difficulty. 'Tis either irresistible, or has no manner of force. To talk therefore of objections and replies, and balancing of arguments in such a question as this, is to confess, eitherthat 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 the abstractedness of the subject; but can never have any such difficulties as will weaken their authority, when once they are comprehended.

'Tis 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 'tis utterly impossible they can have any just foundation.

'Tis 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 'tis certain we have an idea of extension; for otherwise, why do we talk and reason concerning it? 'Tis 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 'tis possible for extension really to exist conformable to it: and consequently,all the arguments employed against the possibility of mathematical points are mere scholastic 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 'tis certain these demonstrations cannot be just without proving the impossibility of mathematical points; which 'tis an evident absurdity to pretend to.

[1]It has been objected to me, that infinite divisibility supposes only an infinite number ofproportionalnot ofaliquotparts, and that an infinite number of proportional parts does not form an infinite extension. But this distinction is entirely frivolous. Whether these parts be calledaliquotorproportional, they cannot be inferior to those minute parts we conceive; and therefore, cannot form a less extension by their conjunction.

[2]Mons. Malezieu.

No discovery could have been made more happily for deciding all controversies concerning ideas, than that above mentioned, 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 'tis 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 distancebetwixt 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 our 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 coloursof 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 carried 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 sensation, 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 determined quantity and quality.

As 'tis 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 occupied 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 tohis imagination. It has been remarked by a great philosopher,[3]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; merely because 'tis 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 by someperceivablesuccession of changeable objects.

To confirm this we may add the following argument, which to me seems perfectly decisive and convincing. 'Tis evident, that time or duration consists of different parts: for otherwise, we could not conceive a longer or shorter duration. 'Tis also evident, that these parts are not co-existent: for that quality of the coexistence of parts belongs to extension, and is what distinguishes it from duration. Now as time is composed of parts that are not co-existent, an unchangeable object, since it produces none but co-existent impressions, produces none that can give us the idea of time; and, consequently, that idea must be derived from a successionof 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 beconceivedwithout 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 separable in idea, we need only consider if they be different from each other; in which case, 'tis plain they may be conceived apart. Every thing that is different is distinguishable, and every thing 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. Forthatis necessary to produce a new idea of reflection; nor canthe 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 our themannerin 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 commonthat duration is a measure of rest as well as of motion, we shall consider afterwards.[4]

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 ofextensionis formed, and separating it from all others, and considering it apart, let us form a judgment of its nature and qualities.

'Tis 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 nonentities there would be a real existence composed of nonentities, 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 any thing ever appear extended, that is not either visible or tangible. That compound impression, which represents extension, consists of several lesser impressions, that are indivisibleto the eye or feeling, and may be called impressions of atoms or corpuscles endowed with colour and solidity. But this is not all. 'Tis not only requisite that these atoms should be coloured or tangible, in order to discover themselves to our senses, 'tis 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.

[3]Mr Locke.

[4]Sect. 5.

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: 'tis therefore possible for space and time to exist conformable to this idea: and if it be possible, 'tis 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, 'tis 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 nonentity, 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 nonentity 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 ofphysicalpoints, 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 ofpenetration, if extension consisted of mathematical points. A simple and indivisible atom that touches another must necessarily penetrate it; for 'tis 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 novoid 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; 'tis this we must mean when we talk of penetration. But 'tis 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. 'Tis 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 unsteadinessboth 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 increased to such a degree as to be really extended, 'tis 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 themathematicsagainst 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 itsdemonstrations,'tis perfectly conformable in itsdefinitions. My present business then must be, to defend the definitions and refute the demonstrations.

A surface isdefinedto 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. 'Tis 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 tothis 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 any thing 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. 'Tis in vain to search for a contradiction in any thing that is distinctly conceived by the mind. Did it imply any contradiction, 'tis 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 'tis on this latter principle that the second answer to the foregoing argument is founded. It has been pretended,[5]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 ourminds; but this excludes not a partial consideration, and adistinction of reason, after the manner above explained.

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 aminimumin 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 theideasof a point, line, or surface, were not indivisible, 'tis 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 onin 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.

Theschoolmenwere so sensible of the force of this argument, that some of them maintained that nature has mixed among those particles of matter, which are divisiblein 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, 'tis 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 any thing concerning the proportions of quantity, we ought not to look for the utmostprecisionand 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 isequalto, orgreater, orlessthan 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 divisiblein 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 varied. But though this answer bejustas well as obvious, yet I may affirm, that this standard of equality is entirelyuseless, 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 'tis 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 enumeration, 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 divisiblein infinitum, 'tis 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 beequalnorunequal 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. 'Tis 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 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 'tis impossible for the mind to find this equality by proceedingin infinitumwith these references to inferior quantities, 'tis evident that at last we must fix some standard of equality different from an enumeration of the parts.

There are some who pretend,[6]that equality is best defined bycongruity, 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 'tis 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 determinedto 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 ofequality, but assert, that 'tis 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.

'Tis 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 ofgreater, less, andequal. 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 juxta-position 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 callequality, 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 bodiesvastlymore minute than those which appear to the senses; and as a false reason would persuade us, that there are bodiesinfinitelymore minute, we clearly perceive that we are not possessed of any instrument or art of measuring which can secure us from all 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 measuringare 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 juxta-position 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 any thing 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 'tis 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 us 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 become 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 completetierceoroctave, 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 onelightandshade, to the otherswiftandslow, are imagined to be capable of an exact comparison and equality beyond the judgments of the senses.

We may apply the same reasoning tocurveandrightlines. 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 theideas of these objects. But however easily we may form these ideas, 'tis 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 merely 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 'tis 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.

'Tis 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 definition 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 'tis not by accident only thathe 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 'tis 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 undeterminate.

The idea of aplain surfaceis 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. 'Tis in vain that mathematicians represent a plain surface as produced by the flowing of a right line. 'Twill 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 essentialto 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, 'tis 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 inflection. As the ultimate standard of these figures is derived from nothing but the senses and imagination, 'tis 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 'tis 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 'tis 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 'tis 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 fails 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 show that we really must make such an exception, and regard all the mathematical arguments for infinite divisibility as utterly sophistical. For 'tis 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 thepoint of contact. I know there is no mathematician, who willnot 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 satisfied 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 chooses, 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 carried beyond a certain degree of minuteness; since, 'tis 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 beincompatiblewith two other ideas, viz. those of a circle and right line; though at the same time he acknowledges these ideas to beinseparable.

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