 File XII. 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 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 diversions, 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 principle 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 colored points disposed in a certain manner. If the eye is sensible of anything further, I desire it may be pointed out to me. But if it be impossible to shoe anything further, we may conclude with certainty that the idea of extension is nothing but a copy of these colored points and of the manner of their appearance. Suppose that in the extended object or composition of colored points from which we first received the idea of extension, the points were of a purple color. 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 color with which alone we are acquainted. But afterwards, having experience of the other colors of violet, green, red, white, black, and of all the different compositions of these and finding a resemblance in the disposition of colored points of which they are composed, we omit the peculiarities of color as far as possible and found an abstract idea merely on that disposition of points or a 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 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 determinant, 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 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 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 merely 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 by 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 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 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. Everything 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 anything steadfast and unchangeable. For it inevitably follows from this 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 anything unchangeable be ever said to have duration. Ideas always represent 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 in section five afterwards. 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 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 a part 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 appears somewhat new since the question itself has scarce ever yet been thought of. We are want 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 color and solidity but this is not all it is not only requisite that these atoms should be colored or tangible in order to discover themselves to our senses it is also necessary we should preserve the idea of their color or tangibility in order to comprehend them by our imagination there is nothing but the idea of their color 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 as the whole if a point be not considered as colored 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 colored 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 conceivable by the mind end of file 12 file 13 of a treatise of human nature by David Hume volume one this LibriVox recording is in the public domain read by George Yeager book one part two section four 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 connection 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 one the first of these objections which i shall take notice of is more proper to prove this connection 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 where there are no medium betwixt the infinite divisibility of matter and the non-entity of mathematical points but there is evidently a medium that is the bestowing a color 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 two 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 say 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 adjuster 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 anyone if he sees a necessity that a colored or tangible point should be annihilated upon the approach of another colored or tangible point on the contrary does he not evidently perceive that from the union of these points their 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 colors 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 colors unite into one what new color 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 retired 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 coloring without augmenting its bulk and afterwards when it has increased 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 three 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 favorable 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 of this is perfectly unintelligible upon any other supposition than that of the composition of extension by indivisible points or atoms how else could anything 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 anything 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 pretended lord de panse 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 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 endeavor 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 endeavor 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 endeavor 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 finish 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 schoolmen 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 cavals 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 extends 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 air at all did it not aspire to such an absolute perfection i first asked 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 readiness and justice 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 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 L 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 enumeration 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 L 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 adjust 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 i 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 persuade 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 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 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 matter 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 an 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 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 complete terse or octave without being able to tell once he derives his standard a painter forms the same fiction with regard to colors 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 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 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 definition of it for I ask anyone 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 maximum 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 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 that is 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 infallible 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 further 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 inflection 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 anything consists in its conformity to its standard now since these ideas are so loose and uncertain i would feign 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 20 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 that is 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 endeavor to explode or if they employ as as 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 subtle 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 ascent and approbation and indeed it seems more requisite to give the reason of this exception than to shoo that we really must make such an exception and regard all the mathematical arguments for infinite divisibility as utterly sophisticated for it is evident that as no idea of quantity is infinitely divisible there cannot be imagined a more glaring absurdity than to endeavor 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 drafts 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 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 that is that of concurrence to be incompatible with two other ideas that is those of a circle and right line though at the same time he acknowledges these ideas to be inseparable end of file 13 file 14 of a treatise of human nature by david hume volume one this libra vox recording is in the public domain read by george yeager book one part two section five 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 the 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 the 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 subtle 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 anything 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 colored 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 extension since then it appears that darkness and motion with the utter removal of everything 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 color of the firmament as they could be by any visible object which i could place betwixt them in order therefore to know whether the site 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 anything colored 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 colored 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 that is that of a man supported in the air and moving his limbs to and fro without meeting anything 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 merely 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 unaccompanied 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 falsely 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 filled 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 appeared 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 shoes 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 phenomenon for as all qualities such as heat cold light attraction etc diminish in proportion to the distance there is but little difference observed whether this distance be marked out by compounded and sensible objects or be known only by the manner in which the distant objects affect the senses here then our three relations betwixt that distance which conveys the idea of extension and that other which is not filled with any colored 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 to ideas the mind is very apt to mistake them and in all its discourses and reasonings to use the one for the other this phenomenon 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 phenomenon 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 phenomenon may be real though my explication be chimerical the falsehood 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 endeavor 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 what 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 shun why upon our conception of any idea the animal spirits run into all the contiguous traces and rouse 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 shoo 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 informing 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 where it as usual as it is reasonable in metaphysical subjects to draw our arguments from that quarter but less 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 that is 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 proved 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 everything is annihilated in the chamber and the walls continue immovable 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 the 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 to 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 emotion 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 satisfied 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 hand and that i explain only the manner in which objects affect the senses without endeavoring 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 endeavored 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 anything 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 footnote four as long as we can find 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 between two objects be something or nothing it is easy to answer that it is something that is a property of the objects which affect the senses after such a particular matter 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 if 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 inquiry beyond the appearances of objects to the senses i am afraid that most of our conclusions will be full of skepticism 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 as 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 skepticism to a certain degree and a fair confession of ignorance in subjects that exceed all human capacity end of footnote four 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 invisible 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 pointed 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 shoot 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 forever present with us when we consider a steadfast object at five o'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 shoes 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 increasing 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 end of file 14 file 15 of a treatise of human nature by David Hume volume one this LibriVox recording is in the public domain read by George Yeager book one part two section six 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 that is 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 those 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 anything 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 in part one section seven 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 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 anything 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 in part four section two end of file 15 file 16 of a treatise of human nature by david hume volume one this libra vox recording is in the public domain recording by george yeager book one part three of knowledge and probability section one of knowledge there are seven see part one section five different kinds of philosophical relation that is resemblance identity relations of time and place proportion in quantity or number degrees in any quality contrarity and causation these relations may be divided into two classes into such as depend entirely on the ideas which we compare together and such as may be changed without any change in the ideas it is from the idea of a triangle that we discover the relation of equality which it's three angles bear to two right ones and this relation is invariable as long as our idea remains the same on the contrary the relations of contiguity and distance betwixt two objects may be changed merely by an alteration of their place without any change on the objects themselves or on their ideas and the place depends on a hundred different accidents which cannot be foreseen by the mind it is the same case with identity and causation two objects though perfectly resembling each other and even appearing in the same place at different times may be numerically different and as the power by which one object produces another is never discoverable merely from their idea it is evident cause and effect are relations of which we receive information from experience and not from any abstract reasoning or reflection there is no single phenomenon even the most simple which can be accounted for from the qualities of the objects as they appear to us or which we could foresee without the help of our memory and experience it appears therefore that of these seven philosophical relations there remain only four which depending solely upon ideas can be the objects of knowledge and certainty these four are resemblance contrarity degrees in quality and proportions in quantity or number three of these relations are discoverable at first sight and fall more properly under the province of intuition than demonstration when any objects resemble each other the resemblance will at first strike the eye or rather the mind and seldom requires a second examination the case is the same with contrarity and with the degrees of inequality no one can once doubt but existence and non-existence destroy each other and are perfectly incompatible and contrary and though it be impossible to judge exactly of the degrees of inequality such as color taste heat cold when the difference betwixt them is very small yet it is easy to decide that any of them is superior or inferior to another when their difference is considerable and this decision we always pronounce at first sight without any inquiry or reasoning we might proceed after the same manner in fixing the proportions of quantity or number and might at one view observe a superiority or inferiority betwixt any numbers or figures especially where the difference is very great and remarkable as to equality or any exact proportion we can only guess at it from a single consideration except in very short numbers or very limited portions of extension which are comprehended in an instant and where we perceive an impossibility of falling into any considerable error in all other cases we must settle the proportions with some liberty or proceed in a more artificial manner i have already observed that geometry or the art by which we fix the proportions of figures though it much excels both in universality and exactness the loose judgments of the senses and imagination yet never attains a perfect precision and exactness its first principles are still drawn from the general appearance of the objects and that appearance can never afford us any security when we examine the prodigious minuteness of which nature is susceptible our ideas seem to give a perfect assurance that no two right lines can have a common segment but if we consider these ideas we shall find that they always suppose a sensible inclination of the two lines and that where the angle they form is extremely small we have no standard of a right line so precise as to assure us of the truth of this proposition it is the same case with most of the primary decisions of the mathematics they remain therefore algebra and arithmetic as the only sciences in which we can carry on a chain of reasoning to any degree of intricacy and yet preserve a perfect exactness and certainty we are possessed of a precise standard by which we can judge of the equality and proportion of numbers and according as they correspond or not to that standard we determine their relations without any possibility of error when two numbers are so combined as that the one has always a unit answering to every unit of the other we pronounce them equal and it is for want of such a standard of equality in extension that geometry can scarce be esteemed a perfect and infallible science but here it may not be a miss to obviate a difficulty which may arise from my asserting that though geometry falls short of that perfect precision and certainty which are peculiar to arithmetic and algebra yet it excels the imperfect judgments of our senses and imagination the reason why i impute any defect to geometry is because its original and fundamental principles are derived merely from appearances and it may perhaps be imagined that this defect must always attend it and keep it from ever reaching a greater exactness in the comparison of objects or ideas than what our eye or imagination alone is able to attain i own that this defect so far attends it as to keep it from ever aspiring to a full certainty but since these fundamental principles depend on the easiest and least deceitful appearances they bestow on their consequences a degree of exactness of which these consequences are singly incapable it is impossible for the eye to determine the angles of a kilogon to be equal to 1,996 right angles or make any conjecture that approaches this proportion but when it determines that right lines cannot concur that we cannot draw more than one right line between two given points its mistakes can never be of any consequence and this is the nature and use of geometry to run us up to such appearances as by reason of their simplicity cannot lead us into any considerable error i shall here take occasion to propose a second observation concerning our demonstrative reasonings which is suggested by the same subject of the mathematics it is usual with mathematicians to pretend that those ideas which are their objects are of so refined and spiritual a nature that they fall not under the conception of the fancy but must be comprehended by a pure and intellectual view of which the superior faculties of the soul are alone capable the same notion runs through most parts of philosophy and is principally made use of to explain our abstract ideas and to shoe how we can form an idea of a triangle for instance which shall neither be an isosceles nor scale numb nor be confined to any particular length and proportion of sides it is easy to see why philosophers are so fond of this notion of some spiritual and refined perceptions since by that means they cover many of their absurdities and may refuse to submit to the decisions of clear ideas by appealing to such as our obscure and uncertain but to destroy this artifice we need but reflect on that principle so often insisted on that all our ideas are copied from our impressions for from thence we may immediately conclude that since all impressions are clear and precise the ideas which are copied from them must be of the same nature and can never but from our fault contain anything so dark and intricate an idea is by its very nature weaker and fainter than an impression but being in every other respect the same cannot imply any very great mystery if its weakness render it obscure it is our business to remedy that defect as much as possible by keeping the idea steady and precise until we have done so it is in vain to pretend to reasoning and philosophy end of file 16