2:165 ATTEMPT TO INTRODUCE THE CONCEPTION OF NEGATIVE QUANTITIES INTO PHILOSOPHY

Things contrary to one another means that one cancels what the other effects. This contrariety is twofold, either logical by way of contradiction, or actual—i.e., without a démenti.

The former, the logical, is that which so far alone has claimed attention. It involves both assertion and denial of one and the same thing. The conclusion of this logical relationship is nothing at all (nihil negativum irrepræsentabile), as the Principle of Contradiction expresses it. A body in motion is something, a body not in motion is something (cogitabile); but a body which was in motion and again in precisely the same sense which was not in motion is a sheer impossibility.

The second, the actual, is where two predicates of one thing conflict, but not by the principle of contradiction. Here also the one cancels what the other causes, but the result is something (cogitabile). The power of a body to move in one direction and an 118equal effort of the same body in the opposite do not contradict each other. As predicates they are at once possible in the same body. The consequence of this is rest, which is something (repræsentabile). None the less, this rest is a true opposition. For what of itself the one tendency effects the other cancels, both tendencies, which meet in it together, being true of one 2:172and the same thing. The consequence of this certainly is also nothing, but in another sense than in the case of the contradiction (nihil privatum repræsentabile). For the future we shall call this actual nothing, Zero = 0. Its significance is identical with that of a negation, loss, absence, which otherwise is in use by philosophers, subject only to a closer definition which will be stated later on.

In the case of the former, the logical démenti, only that relation will be noticed by which the predicates of a thing cancel themselves and their consequences by contradiction. Which of the two in truth affirms and which denies need concern no one here—e.g., to be at once and in an identical manner dark and not dark involves a contradiction in the very same subject. The predicate dark affirms, the predicate not-dark denies; but metaphysically the former itself is a negation. In actual opposites the cancellation also rests on a relation towards each other of two predicates of the very same thing, but in this case of quite another kind. What the one affirms the other does not deny, this being impossible, for both predicates A and B are affirmative; only, the results a and b from each apart are neither a nor b when both are conjoined in the same subject, where the result is Zero. Given one person owes another £100, on this ground the sum payable is £100. But if the person to whom this amount is owing, himself owes £100—this is a ground 119for so much disbursement. Together the debts form a ground of Zero—i.e., money is neither to be paid out nor received. It is easily discoverable that this Zero is a relative nothing. As a certain movement did not transpire, in the above cited case, so a certain asset falls away in this case; but in the case of the démenti, on the other hand, there is simply nothing. The nihil negativum is not therefore to be expressed by Zero, because Zero involves no démenti. One can think that a certain movement does not occur, but that it together occurs and does not occur allows of no thinking.

Mathematicians employ the conception of this actual opposition for their quantities, denoting them by the signs + and −. This opposition being two-sided, it is easy to see, without on this account 2:173involving any difference in the quantity before which + stands, and the quantity before which − stands, that the one either partially or wholly cancels the other. A ship sails from Portugal to Brazil. + denotes the distances it makes by the east wind, and − those it loses by the west. The figures themselves represent miles. In seven days thus the voyage is + 12 + 7 − 3 − 5 − 1 − 8 = 19 miles to the west. These quantities before which − stands possess this sign in the contrary sense only so far as they are taken together with the quantities before which + stands. Should they stand in conjunction with others also denoted by −, there is no actual opposition, for opposition is a counter-position only occurring between + and −. Over and above, since subtraction is a cancellation occurring when opposite quantities are taken together, it is clear that the − cannot properly be a sign of subtraction, as it is usually represented to be, first together only the + and the − betraying a deduction. Consequently − 4 − 5120 = − 9 forms no subtraction, but an actual increase and addition of like quantities. But + 9 − 5 = 4 indicates a deduction, because the contrary signs imply that the one to the extent of its amount cancels the other. Similarly, the sign + by itself signifies no addition, but only so to the extent the quantity before which it stands is taken together with another before which also + stands, or is supposed to stand. Shall it, however, be taken together with one before which − stands, this cannot otherwise happen than by means of opposition, and then the sign + indicates subtraction as well as the sign −; that is to say, one quantity cancels to the extent of its value another, as for instance − 9 + 4 = − 5. This shows that the sign − in the example − 9 − 4 = − 13 is no subtraction, but just as much an addition as the sign + in the example + 9 + 4 = + 13. For in any case, in so far as the signs are identical the units must clearly be added, but in so far as they are different they can only be included by opposition—i.e., by means of subtraction. In mathematics and geometry, therefore, these two signs serve only to distinguish what is mutally opposed; in other words, what, either wholly or partially, in the conjunction cancels one the other. The purpose of this is two-fold: (1) that the conjunction makes the counterrelation evident; and (2) that it may be known, the one 2:174having been subtracted from the other, to which of the two quantities the resultant belongs. From the above-mentioned case of the sailing ship, if − denoted the progress with the east wind and + the retardation by the west, the forthcoming result would be identical, only in this case the resultant would require the sign −.

It is from this that the mathematical conception of negative quantities arises. One quantity in respect 121of another is negative in so far as it cannot otherwise than by opposition be taken in conjunction—i.e., that one to the extent of its value cancels the other. This is now unquestionably a counter-position, and quantities so opposed to each other mutually cancel in each other an equal amount. No quantity may thus be properly called absolutely negative. It must merely be said of + a and − a that one is the negative quantity of the other. This, however, can always be kept in mind; so mathematicians have long since been accustomed to call the quantities before which − stands negative, in which case let it not be forgotten that such a signification designates no exclusive class of things, according to their inner character. It denotes this counter-position with certain other things, before which + stands, to be taken in conjunction in one opposition.

In order, distinct from any particular regard for the quantity, to extract from this conception what forms the object proper of Philosophy, above all see that it contains the opposition which has just been termed actual. Let assets amount to + 8 and debts to − 8, no contradiction hampers this occurring in the one person. None the less, the one cancels what the other effected, and, the result being Zero, I call the debts negative assets. But by this I do not understand them to be mere negations or denials of assets, for in that case they by themselves would require the sign Zero, and this conjoined asset and debt would produce an estate of the value of 8 + 0 = 8, which is false. The meaning is that the debts are positive grounds for the reduction of the assets. Such denominations, uniformly denoting the correspondence of things only in their mutual relation, apart from which this conception immediately ceases, it would be unreasonable on 122that account 2:175to assume a class of things in order to call it exclusively negative, for which even the mathematical expression of negative quantities is not exact enough. For negative things would all round imply negations, which, however, is by no means the conception that we seek to establish. Sufficient that we have already explained the counter-relation comprising the whole c'onception, and contained in the actual opposition. In order, then, at once to let the expressions themselves announce that one opposite is not a démenti of the other, and that when the counter-position is something positive it is not its mere negation, but, as we shall see presently, something affirmative opposed to it, pursuing the method of mathematicians, we shall call sinking a negative rising, descent a negative ascent, retreat a negative advance. From this will emerge that descent is not so different from ascent as non a and a, but something equally as positive as ascent. The one only comes to contain the ground of a negation in conjunction with the other. It is also quite obvious, since here only the counter-position is taken into consideration, that inversely I may with equal justice call rising a negative sinking, as sinking a negative rising. Similarly, assets are as much negative debts as debts negative assets. But if one seeks to denote its actual opposite, it is more appropriate to bestow the name negative on that to which in each case the appellation is preferably meant to apply. For example, it is handier to call debts negative assets than vice versa, although the difference does not lie in the counterposition itself, but in the bearing of its result on a wider issue. I should just like to recall that, at times, I shall employ the expression that one thing is the negative (case) of the other—e.g., the negative of rising 123is sinking, which shall not convey that it is a démenti of the other, but a case of its actual opposition.

The following principle is to be regarded as a fundamental rule in such actual opposition. Actual contrariety only occurs when of two things as positive grounds one cancels the effect of the other. Let motive power be a positive ground, then an actual conflict can only occur when in mutual conjunction with it another 2:176motive power cancels the effect. The sequel may generally serve to prove this. Ends conflicting with each other must, in the first instance, be found in one and the same subject. For no actual opposition springs from an end in one thing and from another end in another thing.¹ In the second instance, one of the contrary ends, where actual opposition is concerned, cannot form a démenti of the other, for then the conflict would be logical, and, as shown above, impossible. In the third instance, one end cannot negate anything else than what the other effects, in negation there being no opposition. Fourthly, in so far as the two ends oppose each other, they cannot both be negations, for were they so nothing would have been effected for the other to cancel. Accordingly, the predicates in each actual opposition must both be positive, but in the sense that conjointly, in the same subject, the effects on each side cancel themselves. By these means, things of which one is regarded as the negative of the other, by themselves are both regarded as positive, but conjoined in one subject the effect is Zero. The course westward is just as positive as the course eastward, only in the same ship the distances covered cancel each other partially or wholly.

124 By this I do not mean that, otherwise related, this actual opposition of two things does not involve many negations. A ship carried westward does not make progress east or south, etc. Nor is it at once in other places. There are many negations which attach to its motion. But that which among all other exclusions remains positive in both eastward and westward motion is what alone permits of actual conflict within itself, and of which the effect is Zero.

This can be illustrated by general signs, on the following method. All genuinely possible negations (for it is impossible to negate the very thing that simultaneously is posited in the subject) can be expressed by Zero = 0, and the affirmative by a corresponding positive sign; but the conjunction in the 2:177same subject by both + and −. One sees here that A + 0 = A, A − 0 = A, 0 + 0 = 0, 0 − 0 = 0¹ are one and all no oppositions, and that in none of them is there anything cancelled that was effected. Similarly, A + A implies no cancellation, and the only remaining case is this: A − A = 0—i.e., that of things where one is the negative of the other both are A, and therefore genuinely positive, yet so that the one cancels what the other effects, which is here indicated by the sign −.

The second rule, which really is the converse of the other, runs: On every side, where a positive ground exists, and the result notwithstanding being Zero, there is actual opposition—i.e., this ground 125stands in conjunction with another positive ground, which is the negative of the former. If a ship in the open sea, actually driven along by the east wind, does not move from its position, at least not to the extent the wind gives occasion for, a current must run against it. In a general sense this is as much as to say that the cancellation of the effect of one positive ground invariably demands another positive ground. Take at pleasure a ground of the result b; in that case there can never be a result o, except in so far as a ground to − b exists—i.e., a ground to something positive which is opposed to the first; b − b = 0. If someone is left a legacy of £1,000, the actual inheritance cannot be £600, except 1,000 − 400 = 600—i.e., except that £400 debts, or other expenditure, is connected with it. The sequel will greatly help to explain this.

I conclude this section by the following note. Negation, in so far as it is the result of a real opposition, I will call dispossession; but every negation which does not come from actual opposition 2:178will here be termed a want (defectus, absentia). The latter calls for no positive ground, but only for the absence of one. The former, however, possesses not only a true ground of position, but an equally true one of opposition. Rest in a body is either a mere loss—i.e., a negation of motion, in so far as no motive-power exists—or a dispossession in so far as there is indeed motive-power to be met with; but the effect—i.e., the motion to which an opposite force gives rise—is annulled.

WHEREIN EXAMPLES ARE ADDUCED FROM PHILOSOPHY IN WHICH THE CONCEPTION OF NEGATIVE QUANTITIES OCCURS

Every body, by impenetrability, withstands the motive force of another to penetrate the space occupied by it. Since the body, the motive force of the other notwithstanding, is a ground of its rest, from the above it follows that impenetrability presupposes in the parts of the body, through whose cohesion a space is filled, as true a force as that wherewith another is impelled to occupy this space.

Conceive, explanatory of this, two springs which exert pressure on each other. Beyond question, equality of force maintains them at rest. Place between them a spring of equal powers of expansion, the pressure will not alter the effect; and, according to the law that action and reaction are equal and opposite, both springs will be maintained at rest. Instead of this spring, now insert a solid body; the result will still be similar, the first-mentioned springs being maintained at rest by impenetrability. The principle of impenetrability is, accordingly, a true force, because it effects the very same as an actual force. Call attraction a cause, whatever it may be, of one body compelling another to press, or be moved, towards the space which the former occupies (it suffices here only to suppose this attraction), then impenetrability is a negative attraction. 2:180This also proclaims it to be just as positive a ground as any other 127natural motive power; and since negative attraction is in reality a true repulsion, occasion is offered in the powers of the elements—in virtue of which they occupy a space in the sense, however, of putting limitations to this space by the conflict of two forces which oppose each other—for many explanations which I believe have afforded me knowledge both clear and trustworthy, to be disclosed by me in a separate treatise.

Take an example from Psychology. The question concerns: Is pain a mere absence of pleasure? or is it of itself a positive ground of the deprivation of pleasure; its actual contrary, not solely its logical démenti; whether, in short, pain can be called a negative pleasure?

To commence with; inner sensation tells us at once that pain is more than a mere negation. For, to whatever extent one may enjoy, still, so long as we are limited beings, a possible pleasure is lacking. He who imbibes a medicine tasting like pure water possibly experiences another pleasure than the expectant health, but still he feels no pleasure in the taste; a want such as this, however, is not productive of pain. Give him a drug composed of wormwood, and the sensation is very positive. In this case, it is no mere want of pleasure, but something which is a true ground of feeling, called unpleasantness.

One can, at any rate, gather from this illustration that pain is no mere negation, but a positive sensation. The fact, however, of its being as much something positive as it is the actual contrary of pleasure, comes clearest to light in the following 128way. A Spartan mother is brought the news of her son’s heroic fight in the cause of his country. Feelings of delight pervade her soul. Then the news is added, he died a hero’s death. This detracts from the delight considerably, and reduces its degree. Call the degree of delight from the first news alone 4a, and the pain nothing else than a negation = 0; then, weighed together, the value of pleasure is 4a + 0 = 4a. In that case, as is wrong, there would have been no diminution 2:181of pleasure by the news of death. Let, however, the pleasure from the heroic conduct be = 4a, and what remains over after the effect of the cause of pain = 3a; then the pain is = a, and it is the negative of pleasure, i.e., − a, the consequence being that 4a − a = 3a.

Very absurd, too, were displeasure a mere negation and the equivalent of Zero, would an estimate, under mixed conditions, of the total value of all pleasure be. An estate has been bought, the income from which is £2,000. Let the degree of pleasure derived from this income, so far as it is unencumbered, be expressed by 2,000. Whatever has to be deducted from the enjoyment of this provides a ground for displeasure. Ground rent £200, wages £100, depreciation £150 a year. Were the displeasure a mere negation = 0, then, all reckoned together, the pleasure in the purchase would be 2,000 + 0 + 0 + 0 = 2,000—i.e., just as much as if the income were enjoyable without deductions! But it is obvious the enjoyment must be reckoned according to the receipts after deduction of expenditure. Thus the degree of pleasure is 2,000 − 200 − 100 − 150 = 1,550. According to this, displeasure is not merely an absence of pleasure, but a positive ground to cancel, partially or wholly, that pleasure which owes itself to another ground. Therefore I 129call it a negative pleasure. The absence of both pleasure and displeasure, so far as it is derivable from an absence of grounds thereto, is called indifference. The absence of both pleasure and displeasure, so far as it results from an actual clash of similar grounds, is called equilibrium. Both are Zero; the first, however, a pure and simple negation, the second a dispossession. The state of mind prevailing from one of these two sensations, on pleasure and pain being unequally opposed, is the preponderance of either pleasure or pain. Herr v. Maupertis sought, in his essay on Moral Philosophy, to estimate the sum of human happiness from conceptions such as these. It certainly cannot be otherwise estimated; but because only similar sensations may be added (feeling, however, in the very intricate state of life, where the emotions are so varied, appearing very diverse) the problem is insoluble 2:182for mankind. The estimate of this scholar was negative, but I do not agree with him here.

Aversion, on these grounds, may be called a negative desire, hate a negative love, ugliness a negative beauty, blame a negative fame, etc. One may think this is all a desultory hair-splitting of words. But only those will judge it so who are unaware of the benefit accruing when expressions, as by a flash, reveal the ratio between already known conceptions, the truth of which the slightest mathematical experience attests to every one. The errors are obvious to which many philosophers fall victim on account of this neglect. They frequently are found treating evil as mere negations, although on our showing it is at once apparent that there are two kinds of evil—that of absence (mala defectus), and that of dispossession (mala privationis). The former are negations for 130whose contrary position no ground exists; the latter presupposes positive grounds to cancel that good for which another ground actually exists. Therefore they form a negative good. This latter evil is far greater than the former. Not to give to him who is needy constitutes an evil; but to take, to extort, to steal from him, is much worse, and taking is a negative giving. In logical affairs a similar process is observable. Errors are negative truths (avoid confusing this with the truth of negative principles); a refutation is a negative proof. But I am taking unnecessary trouble to dwell on this. I want only to circulate these ideas—practice will show their use, and in the Third Part I will suggest some of the avenues that open up.

These ideas of an actual opposition have their useful application in practical Philosophy as well. Vice (demeritum) is not solely a negation, but a negative virtue (meritum negativum). For vice can only occur in so far as within a being an inner law exists (either merely conscience, or it may be the consciousness of a positive law) which is rebelled against. This inner law is the positive ground of a good action, and solely on account of the cancellation of that act which alone would flow from consciousness of the law can the result be Zero. It is 2:183here, therefore, a dispossession, an actual opposition, not merely an absence. Do not imagine that this applies solely to sins of commission, and not equally to sins of omission. An unreasoning beast exercises no virtue; but, on account of no inner law being contravened, this omission is not vice. The unreasoning beast was not prompted by an inner moral feeling to 131a good act; and thus, as its act was not withstood, or counteracted, the Zero, or the omission, was not determined as an effect. In this case, owing to the absence of a positive ground, it is simply a negation, and no deprivation. Contrast with this a man who renders no help to another he sees in distress, whom, too, he might easily assist. In his heart, as in that of everyone else, a positive law of charity exists. This has to be quenched. To provide for the omission an actual inner motive is called for. This Zero is the result of an actual opposition. It actually costs some men in early life a marked effort to leave a trifling kindness undone, in which case they are not blind to the positive incentives. Habit makes all easy, and eventually this incentive becomes less taken to heart. Morally, then, sins of commission do not differ in kind from sins of omission, but only in degree. Physically—i.e., in respect of the external consequences, no doubt, they also differ in kind. He who gets nothing suffers from the evil of penury, and he from whom is taken, from the evil of dispossession. But in respect of the moral state of him who practises the sin of omission, only a greater degree of incentive is required for the sin of commission; just as the counter-weight employed on a beam exerts a true force to keep the burden at rest, and only a little increase to move it on the other side. In the same way, he who does not pay what he owes will, in certain circumstances, cheat in order to gain; and he who does not help when he can will ruin another as soon as the incentives multiply. Charity and uncharitableness are contradictory counterparts of each other. Uncharitableness is a true negation; but, in respect of a conscious obligation to be charitable, this negation, being only possible by an actual 132opposition, is 2:184a dispossession. In such a case, the act of loving and the act of hating differ only in degree. On the other hand, ail omissions, which are certainly a want of greater moral perfection, but not sins of omission, are simply nothing but negations of a definite virtue, and not plunder or vice. The shortcomings of saints and the failings of devoted souls are of this nature. A certain surer ground of perfection is lacking, and the failing does not come about on account of an incentive to act evilly. It would be possible to apply such conceptions much more extensively to the objects of practical Philosophy. Prohibitions are negative commands, punishments negative rewards, etc. But my purpose is achieved for the present if only the application of this conception is generally understood. I recognise that to readers with minds enlightened there is more repetition about these explanations than the occasion demands. But he who for a moment considers that there exists a very unenlightened class of judges, who are confined to knowing only the one book that they spend their lives in reading, will forgive me. With that class in mind indefinite prolixity is not superfluous.

Let us take another case from science. In nature many dispossessions occur from the conflict of two active causes, of which one cancels the effect of the other by actual opposition. But whether this may not merely be a negation—i.e., the absence of a positive cause—or whether it is the outcome of two active forces in opposition, is often as uncertain as whether, in the same sense, Zero is attributable to the absence of motive, or to the conflict of two mutually suspending forces. For example, it is a 133perennial question whether cold demands a positive cause, or whether it simply is attributable to absence of warmth as cause. I will dwell on this for a little, so far as it assists me to my purpose. Beyond question, of itself cold is only a negation of warmth, and it is easily seen that without a positive ground it is possible of itself. On the other hand, it is as easy to understand that cold can originate from a positive ground; that, indeed, however the origin of warmth is to be accounted for, it does at times actually so originate. No absolute cold is known throughout nature, relativity being meant whenever it is mentioned. Experience and reason 2:185agree in confirming Professor von Musschenbroek’s idea, that calefaction does not consist of internal commotion, but in the actual passage of elemental heat from one material to another, although this passage presumably is associated with some internal shock, in the same way as the shock promotes the liberation of the elemental heat. On this footing, when among bodies the elemental heat in a certain space is held in equilibration, they are relative to each other neither cold nor warm. Suspend the equilibration, then that material is cold into which the elemental heat passes, relative to the body which thereby is deprived of it, which terms this body, on the other hand, in so far as it has warmth to part with to the other, in respect of it, warm. The state in this change is called in the former increase of temperature, in the latter decrease, until all is again in equilibration.

Nothing is now more natural to suppose than that the attractive forces of matter set this subtle and elastic fluid so long in motion and fill the physical mass with it, till it everywhere is in equilibrium, in which case the spaces, proportionate to the attractions 134operating within them, are filled with it. And here it is evident that one mass which cools the other in contact appropriates by actual force (attraction) from this other the elemental heat With which the mass of the other was filled, and that the cold of the other body may be called a negative warmth, because the negation which in the warmer body follows on this is a deprivation. But it would be useless, and not much better than a dallying with words, to introduce this signification here. My purpose is directed, in this instance, only to what follows.

It has long been known that magnetic bodies have two ends called poles, standing in opposition to each other, one of which repulses at the negative point, while the other attracts at the positive. However, in a treatise on the similarity of electric and magnetic force, Professor Aepinus showed that electrified bodies, under certain treatment, also of themselves displayed two poles, of which he called the one positive, the other negative, and of which the one attracted what the other repelled. This phenomenon is most clearly in evidence when a rod 2:186is brought sufficiently near an electric body without, however, drawing any sparks from it. I assert now that in heating or in cooling processes—i.e., in the case of all changes of warmth or cold—chiefly the rapid ones, occurring in a continuous intermediate space or in extension, there are invariably two poles of warmth to be found, of which the one becomes positive—i.e., above the past degree of the body in question warm—the other negative, under this degree—i.e., cold. It is known that various caves are subject in their interiors to a severer degree of frost the more the exterior earth and air are warmed by the sun. Matthias Bel, who describes the caves of the 135Carpathian Mountains, adds that it is a custom of the peasants in Siebenbürgen to cool their drinks by burying them in the earth and lighting a quickburning fire over them. It appears that within the time the upper surface of the earth cannot become positively warm without at a somewhat greater depth becoming the negative of this. Boerhaave gives also an example of the fire at a forge causing cold at a certain distance. This counteraction seems to obtain as well in the upper air, specially where temperatures change quickly. Jacobi gives an instance, somewhere in the Hamburger Magazin, that in cases of severe cold, such as occur in very long stretches of plain, it is quite usual to be temperate and mild in lengthy intermediate patches. Similarly, Professor Aepinus found in the case of the electricised rod, of which I was speaking, that from the positive pole of one end to the negative of the other the positive and negative electric parts alternated for some lengths. The evidence goes to show that the air cannot be heated in any one region without in the other bringing into play the action of a negative pole—i.e., without causing cold. On this footing, conversely, the rapid increase of cold in one district serves to promote warmth in another, just as a metal rod heated at one end, when plunged into cold water, sends the warmth to the other end.[*] The difference in the 2:187temperature, accordingly, ceases whenever the communication, or the deprivation, has had sufficient time to distribute itself uniformly throughout the whole mass, just as Professor Aepinus’s rod displayed only one kind of electricity as soon as it had drawn sparks. Perhaps, too, this explains how the marked cold of the lofty atmosphere is not solely attributable to a lack of means of warmth, but to a 136positive cause—i.e., that in respect of the degree of warmth it becomes negative as the lower atmosphere and earth are positive. In any case, magnetic force, electricity, and heat appear to owe their occurrence to one and the same means. Friction stirs them all into activity, and I suspect, were the subject cleverly treated, that the difference of the poles 2:188and the opposition of positive and negative activity might become just as noticeable in the phenomena of warmth.

Galilei’s inclined plane, Huygen’s pendulum, Torricelli’s tube, Guericke’s air-pump, and Newton’s glass prism have unlocked for us many great secrets of nature. The negative and positive activity of masses, particularly in the case of electricity, conceal in all likelihood important disclosures; and it is to be hoped a more fortunate posterity, whose fair days we can discern, will learn general laws of that which for us still appears in doubtful harmony.

CONTAINING SOME CONSIDERATIONS PREPARATORY TO THE APPLICATION OF THE ABOVE CONCEPTIONS TO PHILOSOPHICAL CONCERNS.

The glances I have so far cast on this important, but no less obscure, matter are only preliminary. When one mounts from instances which, as those above, are simple of comprehension to general principles, one shrinks from responsibility for errors that, possibly, on an untrod field, may come to light only in the course of time. What I have still to say on the 137subject I submit, then, as only a very imperfect venture, although, certainly, I do promise myself various good results from the interest it may excite. I am well aware that a confession of this sort is a very poor bid for the favour of those who like the overbearing, dogmatic tone, which wheedles them into assenting to anything. But, wholly indifferent to the loss of favour gained by this means, I find, in so precarious a pursuit as that of metaphysics, instead of ostentatiously giving one’s thoughts out ex cathedra, that to submit them first to a public test is far the better plan, because, in the other case, one being apt to take offence at corrections, every error that creeps in becomes past mending.

Everyone easily understands why, without a positive ground, a thing does not exist; but how that which does exist ceases to exist is less easy of comprehension. For example, at this moment a picture of the sun exists in my mind through the force of my imagination. A moment later I cease to entertain the object. The image which was ceases to be in me, and the next state is the zero of the former. Were I to urge that the thought’s cessation was caused by my no longer entertaining it, the answer would not differ from the question; for here it concerns how an action which actually occurs can be discontinued—i.e., cease to be.

I declare, then, every passing out of existence is a negative coming into existence—i.e., there is just as true an actual ground required in order to cancel something positively existing as, previous to its existence, there is one required to call it into existence. The reason for this is contained in the above. Take as given A, then only does A − A = 0, 138which is equivalent to saying that A can be annulled only in so far as with the ground of A there is an equal, but opposite, actual ground conjoined. Physical nature teems with examples of this. A movement never ceases, wholly or partially, without a motive force, equal to that capable of causing the loss of motion, being conjoined in opposition with it. Inner experience also very clearly testifies to the suspension of ideas and desires which activity of mind has made actual. The truly unusual amount of energy it costs to suppress a grief-laden thought is a matter of keen personal experience. Genuine effort is demanded to dismiss a mirth-provoking idea, if one wishes to attune the mind to seriousness. Every abstraction is nothing but a holding of certain discrete ideas in abeyance, which one usually puts into operation so that the generalisation be more fully grasped. No one but knows well the amount of energy that this requires; and thus the abstraction may be called a negative mindfulness. That means, it is a genuine labour and exercise, which is opposed to the operation whereby the discrete idea is kept clear, whose zero, or eclipse, is caused by comprehension 2:191under the generalisation. For otherwise, were the abstraction simply a negation and disjunction, it would put just as little strain on the mind as might be required not to know something, on account of there never being a ground to it. Precisely the same necessity of a positive ground to annul an inner mental effort is displayed in the conquering of desires, in which case the above-cited examples stand at service. Generally speaking, however, apart from those cases in which one is actually conscious of this contrary activity, examples of which have been given, the fact that one does not clearly perceive it inwardly affords no 139sufficient occasion for incredulity. For example, at this moment I entertain the idea of a tiger. This thought vanishes, and a jackal occurs to me. Certainly, during the change of ideas, no marked mental effort to annul the one is discernible. But is it not a marvellous activity that lies concealed in the depths of our mind, which in the thick of its execution we do not remark, because of the complexity of operations, of which each separate one is but very dimly perceptible? The evidences of this are known to everyone. Let one only take into consideration what goes on unnoticed in us when reading. The conclusion is, that the play of ideas, and generally of all our psychological activity, in so far as the effects, after actually existing, again cease, presupposes opposed actions, of which the one is the negative of the other. Although inner experience cannot always at once inform us of them, the reason of this is as above.

If the grounds on which the rule advanced rests are taken into consideration, one will be immediately aware that in respect of the annulment of an existing fact mental results cannot differ from those of active forces in the physical world. That is to say, nothing else than the strictly opposed motive-power of another mental power can annul the one; no single inner occurrence, a mental thought, can cease to be without a genuinely active force of precisely the same thinking subject. The distinction concerns here only the 2:192different laws to which these two kinds of natures are subject, the material state being only changed by an external cause, the mental one, however, additionally by an inner. The necessity of the actual opposition remains, none the less, always the same with this distinction.

Once again, I repeat, it is a delusion to think of the 140annulment of the positive results of our mental activity as discontinuances. It is quite remarkable that the more one probes his most every-day and surest beliefs, the more of such sophistries one discovers, the reason being that we are contented with words, without understanding anything of the facts. That I do not now entertain a certain thought, even if it had not previously existed, is certainly conceivable enough when I say I omit to think this, for in that case omission implies the want of the effect. But when it is asked, “Wherefore Is there no longer a thought in me that shortly before existed?” the former answer is useless. For this non-existence is now a dispossession, and in this instance the omission has a wholly different meaning[*]—i.e., the annulling of an energy that shortly before existed. This is now just the question I raise, and in whose case I will not be easily put off by a word. A favourite propensity in applying the rule to the various natural cases is to take the démenti for positive, so one must be careful. For our principle applies to the origination and expiry of something that positively exists. For example, a flame! Its extinction, on exhaustion of its fuel, is no negative origination. That is to say, the extinction is not caused by an actual motive-force, the opposite of that which originated the flame. For so long as the flame endures it does not correspond with an already permanent motion, but with the constant generation of motions renewed by molecular combustion.¹ Thus the extinction of a flame is not the annulment of an actual motion, but the loss of fresh 2:193fuel and of various 141dissolutions. The cause is that the ground fails—namely, further combustible material—and this is not to be regarded as the annihilation of an existing thing, but as the lack of ground to a possible occurrence (further resolution into its elements). But of this enough. I write it to stimulate further experiments in science of this kind. The inexperienced would certainly be justified in demanding more enlightenment.

The principles that I undertake to expound in this section appear to me of extreme importance. Previously, however, I have to add a definition to the general conception of negative quantities, which I had left aside, with a view to avoid burdening an already strained attention with many details. So far I have weighed only the grounds of actual opposition, inasmuch as they set in one and the same object determinations of which one is the negative of the other—e.g., motive-powers of one and the same body in directions opposite to each other, where the grounds of the respective results (i.e., the motions) actually cancel each other. This opposition I will call the actual (oppositio actualis). On the other hand, those, too, are rightly called predicates which pertain, no doubt, to distinct things, of which one, although it is the negative of the other, yet does not cancel the effect of the other. This applies to the extent each is constituted to annul either the effect of the other, or at least of something else as definitely determined as this effect and its equivalent. This opposition may be termed potential (oppositio potentialis). Both are actual—i.e., different from the démenti; both are constantly employed in mathematics, and both also 142deserve to be employed in Philosophy. Two bodies move towards each other in the same straight line, with equal forces. Owing to their forces mutually communicating themselves by impact to both bodies, these two may be called one the negative of the other. In the former sense this occurs by actual opposition. In the case of two bodies in the same straight line, which with equal forces move in directions contrary from each other, one is the negative of the other; but, since here they do not mutually communicate their forces, they stand only in potential opposition, because were one—either 2:194of them—to collide with another body moving in the same direction as its opposite, each would annul in it just the same amount of force as is in the other body. In the immediately following, this is the sense in which I also shall understand it of all grounds of actual opposition throughout the world, and not merely of those to which motive-powers belong. To give an example of such grounds, it might be said that the pleasure of one individual and the suffering of another stand in potential opposition, for incidentally the one does annul the consequences of the other. If the two conflict, the suffering of one individual often destroys what the pleasure of another procures. In taking now quite generally the grounds which in both senses really stand in opposition, let it not be demanded of me that I, in every instance, furnish concrete examples of the conceptions. For so clear and comprehensive as is everything that can be held up to view, so hard and obscure is it for us to grasp the non-mechanical actual grounds, so as to make, either in opposition or in agreement, their ratios and their effects intelligible. I content myself, therefore, with expounding the general purport of the following principles.

143The first principle is this: in all of the world’s natural changes the sum of the positive, in so far as it is estimated by adding consonant (not opposed) occurrences, and subtracting actually opposed ones from each other, is neither increased nor diminished.

All change consists in this: either something positive, that had not existed takes place, or what was in existence is annulled; but the change, in so far as both its ground and the effect pertain to the world, is natural. Accordingly, in the first case, on what did not exist taking place, the change is an origination. Previous to this change, in respect of this occurrence, the state, universally, equals Zero = 0, and the effect of the origination = A. What I assert, however, is that on A originating in any physical change, − A must also originate. That is to say, there cannot be any natural ground of an actual effect without there being equally a ground for another effect, its negative. Because (apart from 2:195the ground being given), when the effect is nothing = 0, the sum of the occurrence contains no more in the effect than the universal condition contained, in so far as the ground lay there.¹ This universal state, however, included the Zero of what takes place in the effect; in other words, in the previous state is not to be found what occurs in the effect. The outcome is: neither can the change issuing from the previous state, regarded universally according to its actual or potential results, be other 144than the equivalent of Zero. Since now, on the one hand, the effect is positive and equals A (notwithstanding that the universal condition, as previously in respect of the change, shall be Zero = 0, which, however, unless A − A is taken in conjunction, is impossible), the outcome is: never, on natural lines, can there occur any change whose effect does not consist in a self-cancelling opposition, either actual or potential. This sum, however, produces Zero = 0; and, since previous to the change it was also = 0, it neither increases nor diminishes by this means.

In the second case, since the change consists in annulling something positive, the effect is = 0. But, according to the earlier argument, the total ground did not merely condition = A, but A − A = 0. Therefore, keeping to the method of estimating with which I here set out, the world’s position is neither increased nor diminished.

I will try to elucidate this principle, which seems to me of considerable importance. In physical changes it is a long-established rule, expressed thus: If forces tending in one direction are added, and those tending in another are subtracted, their reciprocal activity (collision, pressure, attraction) does not alter the quantity of motion. But the fact that this rule, in pure mechanics, is not directly derived from the metaphysical ground from which our general principle is deduced does not disturb the validity of its resting on this ground. For the law of inertia, underlying the ordinary proof, borrows its truth solely from the above chain of reasoning, as I could very quickly prove were I inclined to be discursive.

2:196 The elucidation of the rule with which we are occupying ourselves is naturally difficult where it concerns non-mechanical psychological changes. Generally, 145too, those dependent on the mind are not so comprehensible as those of the physical world, nor can they be made conspicuous by way of ocular demonstration. None the less, so far as it is possible, I will try to throw some light on the subject.

Aversion is just as positive as desire. The former is an effect from a positive displeasure, the latter a positive effect of pleasure.¹ Desires and aversions find themselves actually opposed only when we experience them in one and the same object. But since one and the same ground which excites displeasure in one object is at once the ground of genuine pleasure in another, the grounds of desires are identical with those of aversions. Thus the ground of a desire is equally the ground of something standing towards it in actual opposition, although this may, for the time being, be only potential. It is as with the movements of bodies, which, in the same straight line, take opposite directions to each other, and, although remote from mutually destroying each other’s motions, may none the less be regarded as negatives of each other, because they are potentially opposed. In accordance with this, a marked degree of ambition creates, in respect of the opposite, an equally marked degree of aversion, which certainly is only potential until the circumstances in respect of the ambition actually stand in opposition. Still, it is precisely the same cause of the ambition that establishes a positive ground of an equal degree of mental displeasure, in so far as external circumstances, which favour the 146former, might provide occasion for those opposed. We shall soon see that this is not the case with the most perfect being, and that, indeed, the ground of its highest pleasure excludes all possibility of displeasure.

As a matter of fact, we find in the affairs of the understanding that the more pronounced the clearness or comprehensibility of a certain idea, the 2:197more the clearness and comprehensibility of the others diminish; so that the positive, which becomes marked in the case of such a change, is conjoined with an actual opposition, which, when estimated all together, according to the adopted plan, neither diminishes nor increases the degree of the positive by the change.

The second principle is as follows: all the actual grounds of the universe, those identical being added, and those contrary to one another being subtracted from each other, produce a resultant which is the equivalent of zero.

The world, as a whole, is of itself nothing, except in so far as it is something existing by the will of another whole. Therefore, regarded by itself, the sum of all existing actuality, in so far as it is rooted in the world, is equal to Zero. The fact of all possible actuality in position to the divine will producing a positive resultant does not annul the nature of a world. Necessarily, however, this nature gives rise to the fact, that of and by itself the existence of a world rooted in it is the equal of Zero. So the sum of what exists in the world, in position to that ground which is apart from it,¹ is 147positive, but in position to the internal actual grounds in opposition it is equal to Zero. Since, now, in the first position, no opposition of the actual grounds of the world to the divine will can ever take place, in this respect there being no annulment, the sum is positive. But because in the second position the resultant is Zero, it follows that the positive grounds must stand in opposition. Viewed, and summed up, then, in opposition, they produce Zero.

I have stated these two principles with the intention of inviting the reader to reflect on this matter. I also confess that for myself they afford neither sufficient enlightenment, nor is their obviousness quite apparent from their grounds. None the less, I am thoroughly convinced that imperfect ventures in abstract knowledge, problematically stated, can foster the higher knowledge in a marked degree. The reason is, that very often another will more quickly find the answer to a deeply hidden question than the one who suggests it to him, one’s own efforts possibly only half surmounting 2:198the difficulties. The subject-matter of these principles appears to me of itself to possess a certain dignity—a fact that, provided their sense has been well apprehended, which in such knowledge is by no means easy, should stimulate to their more thorough testing.

In the meantime I will try to forestall some more misconceptions. People would fairly misunderstand me if, in the former of these two principles, they thought I had sought to assert that changes in the world never, under any circumstances, increased or lessened the sum of the actually existing. This is so wholly foreign to my meaning that even the 148mechanical law, which I stated as an example, asserts the very contrary. For the clash of the objects now¹ increases and now diminishes the sum of the motions, if it is to be regarded of itself; but the resultant, reckoned according to the method subjoined, remains identical. For in many instances the oppositions are only potential, when, the motive powers not actually cancelling each other, an increase does take place. But, according to whatever reckoning is once adopted as a standard, even potential oppositions also must be deducted from each other.

One must judge in precisely the same way when applying the principle to non-mechanical changes. A similar misunderstanding would arise if one allowed himself to fancy that by the same principle the perfection of the world could not grow at all.² For this principle does not in any way deny that generally the sum of reality cannot by natural means be increased. Moreover, the perfection of the world generally lies very definitely rooted in this conflict of the opposed actual grounds, just as the material portion of it is palpably maintained in a regular course by the conflict of forces. And to identify the sum of actuality with the greatness of perfection always remains a serious mistake. We say above that pain is just as positive as pleasure. Who, however, would call pain a perfection?

We have already noticed how difficult it often is to decide whether certain natural negations are mere 149wants on account of the ground lacking, or dispossessions, resulting from the counter-action of two positive grounds. Examples of this come frequently to light in the material world. Cohesive particles in any body adhere with true forces (attraction), and were 2:199it not that forces just as effective counteracted it in equal degree by repulsion of the elements, whose ground creates the effect of impenetrability, the tendency would diminish the spatial displacement. Rest occurs here, not because there is a lack of motive-forces, but because they counteract each other. Similarly, the weights in two scales are at rest, when by the laws of equilibrium they hold each other in check. This idea can be extended far beyond the bounds of the material world. It does not at all follow that the sum of the actual grounds of thinking and desiring is lesser when we believe our minds to be wholly inert than when we are aware of a small measure of activity. When the greatest savant is idle and reposing, propose to him to tell something of his impressions! Not being cognisant of anything in this state, you will find his mind vacant, without definite deliberations or opinions. But give him occasion by a question, or by offering an opinion of your own, and then his learning flows uninterruptedly in the direction of unburdening to himself and to you what lay in his consciousness.

Unquestionably, long previously the actual grounds of this were discoverable in him; but since the effect, in respect of consciousness, was Zero, they must, in so far, have cancelled each other. Ina similar way, stored in the armoury of a ruler for future warfare, there lies that ingenuously conceived engine of destruction, ominously quiescent, till a diabolic bit of tinder is applied to it; then, with a flash, it explodes, and devastates all around it. A big idea, it appears 150to me, and very right, lies concealed in the thought of Leibniz. The mind comprehends the entire universe in its powers of presentation, although but an indefinitely small part of these perceptions is clear. In fact, all kinds of conceptions can only rest, as on their ground, on the inner activity of our minds. External things may well contain the conditions under which, by one means or another, they put themselves forth, but not the power actually to produce them. The mind’s power for thought must contain all its own actual grounds to the extent, by natural means, they take their rise in it; and the manifestation of ideas that originate and expire 2:200can, to all appearance, only be attributed to the agreement, or opposition, of all this activity. These considerations may be looked on as explanatory to the first of the principles in Section III.

Similarly, Zero, in moral affairs, is not invariably to be regarded as a negation of want; and the positive effect of an increased mass is not invariably a proof of greater activity, which can be directly related to this effect. Let a miser have ten degrees of passion, which in a certain instance conflicts with the laws of duty, and let him exercise twelve degrees of charitable impulse. The effect is two degrees to the advantage of benevolence and helpfulness. Take the case of another man with three degrees of miserliness and seven of a capacity to act according to the principles of obligation. The act will be four degrees great, after his desires to render others assistance have conflicted. It is, however, indisputable, in so far as the said passion can be regarded as natural and undeliberate, that the moral value of the former act is greater than that of the latter, although it is true, if they are to be estimated by the living force, the effect 151in the latter case outbids the other. On this account it is impossible for men to decide from their actions with certainty the degree of moral intention in others; and, moreover, the power which sees into the innermost heart reserves judgment to itself.

If one dared to apply these conceptions to so important a knowledge as that which man harbours of the eternal God, what difficulties do not then surround our remotest thoughts! Since we can derive the basis of these thoughts only from ourselves, it remains doubtful, in most cases, whether we ought strictly to utilise the conception for such an incomprehensible theme at all, or at the best only by way of analogy. Simonides is no less a sage because, after much faltering and protraction, he replied to his Prince in these terms: “The more I revolve the problem of God, the less capable am I to understand it.” But the erudite clique does not express itself thus. It knows nothing but talking on every subject; it puts its trust in what it talks.

Grounds neither of dispossession nor of actual opposition can find a place in the Supreme Being. For in it, and by it, everything is given; so the fact that it is the exclusive disposer of all destinies renders there an inner annulment impossible. 2:201Therefore the sentiment of displeasure is not a becoming predicate of the Godhead. Never does man, however, possess a desire for an object without possessing a positive aversion for the opposite, not merely in the sense that his will is opposed to the desire as a negation, but as its actual counterpart (aversion)—i.e., the effect of positive displeasure. In the case of every desire, diligent in training its pupils to acquire excellence in 152practice, because it is not to its taste, every effect that is positively contrary to it affords a ground of displeasure. Objects related to the divine will are of quite another kind. Strictly, no external thing finds a ground within itself of either pleasure or of displeasure, the ground not being in the least dependent on something else; and it is not because the good exists apart from the ground that pure delight dwells in the divine Being; but this good exists because the eternal idea of its potentiality, and the delight associated with this, are a ground of consummate desire. If this notion is compared with that of the material yearning, natural to every created being, one will be aware how little the will of omnipotence can have in common with it; a comparison which also, in respect of the minor destinies, will not appear strange to him who has thoroughly grasped the immeasurable difference in quality existing between things that are nothing by themselves and that thing-in-itself in virtue of which alone everything has its being.

Since the grounded philosophers, as they term themselves—who see so deeply into all matters that nothing whatever comes amiss to them to explain and comprehend—daily grow in number, I can already, very clearly in advance, detect that this conception of an actual opposition, set forth by me as a basis at the commencement of this treatise, will appear to them very shallow, while the conception of the negative quantities, which has been built tip on it, will not be sufficiently thorough. Making no secret of the fallibility of my judgment—in view of which I usually understand last of all that which the rest so glibly persuade themselves to believe—I feel 153proud of the right to support from these great minds, which my inferior capacity gives me. It is their superior wisdom 2:202that may fill the gap which my poor judgment must leave behind.

I understand quite well how a conclusion flows from a premiss in accordance with the law of identity, for this reason: it is found by analysing the conception contained in the premiss. Thus necessity is a premiss of unchangeableness; synthesis a premiss of divisibility, infinity a premiss of omniscience, etc., and I can clearly see this conjoining of the ground with the conclusion, because the conclusion is really identical with a minor premiss of the ground, and, by being already contained in it, is predicated, in virtue of it, by the law of identity. But how, not according to the law of identity, one thing results from something else, that is a problem I should very much like to have explained to me. I call the first class the logical ground, because its position to the conclusion can be logically understood—i.e., is clearly in accordance with the law of identity; but the ground of the second class I call actual, because, although this position may well belong to my indubitable conceptions, its process cannot be discerned in any possible way.

So far, now, as it concerns this actual ground, and its position to the effect, my question allows of being put in this simple form: How am I to understand that, on account of something existing, something else exists? A logical conclusion can only come about because it is identical with the premiss. A man can fail; the ground of this fallibility lies in the finality of his nature; for, if I resolve the conception of a finite mind, I see that the fallibility lies here, that this is identical with what is contained in the conception of a finite mind, The will of God alone contains 154the real ground of the world’s existence. The divine will is something; the existing world is something wholly different. None the less, one posits the world because of the Divine will! The state in which I hear the name Stagyrite mentioned is something by means of which something else—namely, my thought of a philosopher—is called into being. A body A moves; another B in its direct line is at rest. The motion A is something, that of B is something different, and yet the one is caused by the other. You may now analyse the conception of the Divine will as much as it pleases you, never will you find there an existing world, as if the world was contained in it, and by the law of identity posited by it; and so in the 2:203rest of the cases. Not by such words as cause and effect, force and act, will I allow myself to be put off. For although I am ready to admit a thing as a cause, or if I attribute to it the notion of a force, then I have thought into it the position of the actual ground to the effect, and after that it is a simple matter to see the relation of the consequence according to the rule of identity. For example, one can quite clearly understand the existence of the world from the almighty will of God. But here the might signifies that something in God by means of which some other things are posited. This word “might,” however, already posits an actual ground to the effect, which I should like very much to have explained to me. I take the opportunity of noting that the cleavage of Herrn Crusius into real and ideal ground wholly differs from my cleavage. For his ideal ground is identical with the logical ground, and in that case one sees immediately that if I already admit something as a ground I can from that conclude the effect. Therefore, according to his principles, the 155south wind is not only an actual ground of rain-clouds, but also an ideal, because I can recognise and presuppose rain-clouds from the south wind. But, according to our conceptions, the actual ground is never a logical ground; therefore the wind is not the cause of the rain in virtue of the law of identity. The distinction between logical and actual opposites which has been propounded above by us is parallel to the logical and actual ground now under consideration.

The logical case I clearly understand by means of the principle of contradiction, and I conceive how, if I posit the infinity of God, thereby the predicate ot mortality is suspended, because the predicate contradicts the premiss. But how the motion of one body annuls that of another, since the one is not in contradiction with the other, is a different question. If I presuppose impenetrability, which actually opposes every power that strives to thrust itself into the space occupied by a body, I can already understand the annulment of the motions; but then I have set one actual opposition against another. Let it now be attempted clearly to give a general explanation of this actual opposition—how, because something is, something else is cancelled, and see whether more can be said than I have said—namely this, and this alone: that it does not happen by the principle of contradiction. I have revolved the question of the nature of our knowledge in respect of our judgment of grounds and effects, and 2:204I shall soon publish the complete results of these considerations. From them it will be gathered that the position of a real ground to something, which, because of it, is either posited or suspended, cannot be expressed by a judgment, but solely by an idea. One can easily reduce this by resolution to simpler conceptions of actual grounds; 156in such a way, however, that finally all our knowledge of this connection resolves itself into simple and insoluble conceptions of the real ground, whose relation to the effect cannot be made intelligible at all. Till then those whose self-assumed wisdom is illimitable may try the methods of their philosophy to see how far it can penetrate to the core of such problems.

2:377 The eminent Leibniz was the author of many sound theories with which he enriched science, but of many still greater schemes for whose completion the world has looked to him in vain. Whether the cause is attributable to his speculations appearing to him too immature—habitually a scruple with men of merit, and of old a source of withholding many valuable fragments from science; or, as Boerhaave suspects of great chemists, that often they gave out inventions as if they had already mastered them, trusting their ability to avoid failure whenever it occurred to them to work at their solution; that I will not here decide. To all appearance, at any rate, a certain mathematical problem which Leibniz had entitled in advance, Analysis Situs, the lack of which, among others, Buffon regretted when studying the doubling in nature of the embryos,¹ was never more than a passing thought. I scarcely know to what extent the problem that I undertake here to consider is allied to that which passed through the mind of Leibniz; but, 158judging by the significance of the terms, I philosophically seek the first ground of the possibility of that whose quantities he was concerned to establish mathematically. For the positions of the divisions of space in relation to each other pre-suppose the environment in virtue of which they find place in such an order. In the most abstract sense environment is not the relation of one object in space to another, this being properly the conception of position, but the relation of the array of these positions to absolute space. The relative position of the parts, in the case of all extended objects, is sufficiently explained from itself. But the environs, to which this array of parts lies open, refers to space apart, not, 2:378indeed, as localised, because this would only be the position of identically the same divisions externally related, but to universal space as a synthesis, from which every extension must be regarded as a part. No wonder if the reader finds these notions, only to be explained in the sequel, still incomprehensible. So, in respect of this, I only add that my intention in this treatise is to seek whether in the intuitive perceptions of extension, such as Geometry contains, plain proof is to be found of absolute space, independent of the existence of all matter, and itself forming the first ground of the possibility of its composition, possessing a reality of its own. Everyone knows how vainly philosophers have tried to settle this question definitely by means of the most abstract metaphysical tenets. Apart from a treatise of the renowned Euler (senior) in the Historie der Königl. Academie der Wissenschaften zu Berlin (1748), I know of no à posteriori attempt to establish this—i.e., by means of certified principles which truly are not metaphysical themselves, but which by analogy 159could still afford a test of its correctness. Euler does not wholly attain his purpose, because he only raises the difficulties that lie in the way of assigning a definite significance to the most general laws of motion, if, without broaching the no lesser remaining difficulties, no other conception is accepted for space than that arising from the relationship ot actual things among themselves, should the said principles be applied when one tries to present them to the mind according to the conception of absolute space in the concrete. The proof I seek shall not, as lay in Professor Euler’s intention, afford mechanicians, but geometricians themselves, an invincible ground to assert, on the usual evidence, the reality of their absolute space. For this end I make the following preparations.

In corporeal space, on account of its three dimensions, one may conceive three planes which intersect each other at right angles. Since we know whatever is external to us through the senses only in so far as we ourselves form the medium, it is no wonder that we derive the first ground to construct the conception of environs in space from the relationship of these planes of intersection to our 2:379body. The plane on which the body stands upright is called, in respect of us, horizontal, and this gives occasion to distinguish the environs denoted by above and below. Two other vertical planes can stand on this horizontal plane, each intersecting the other at right angles, so that the length of the human body is taken in the line of intersection. One of these divides the body into two halves, externally alike, which provide the ground for distinguishing the left from the right side. The other division gives us the conception of back and front. A printed paper, for example, sets us first to 160distinguish between the top and bottom; then we notice the difference between the front of the page and the back of the page; and finally we regard the setting of the type from left to right, or reversed. In this case the printed letters on the surface are always in the same position to one another; and, turn the paper as one likes, the form remains in all particulars unaltered. But the diversity of the environs has to be so much considered, and it so exactly conditions the impression which the viewed object makes, that the very same writing reversed—i.e., regarded from right to left, instead of, as formerly, from left to right—becomes unintelligible.

Indeed, we judge of the cosmic environs, subordinate to the general notion we entertain of environment, in so far as they take their directions from the sides of our body. What we otherwise observe in the skies and on the earth, independent of this ground conception of relationships, is only the positions of objects among themselves. Were I even more sure than I am of the order of the sections of the horizon, none the less by them I can still only determine the environs by being conscious of on what hand this order runs its length; and the most exact chart of the skies would never place me in the position to know from one section—say the north—on which side of the horizon I should have to seek the sunrise, unless (apart from the positions of the stars among themselves) the setting of the chart towards my hands determined the direction.

It is just the same with the geographical—nay, with our most ordinary—knowledge of the position of localities, which in no wise assists us if we cannot refer the objects thus ordered, and the whole array of the 2:380interchangeable positions, to the sides of our 161bodies verging on the environs. A very striking characteristic of natural products, offering even occasion to distinguish species, consists in the definite direction which the formation of their parts assumes, and by whose means two entities can be distinguished, although completely agreeing in respect of size, of proportion, and even, among themselves, of the allocation of their parts. The hairs on the crowns of all men’s heads are twisted from left to right. All hops wind themselves round their sticks in this direction, while beans take the opposite. Regarded from the tapered end to the orifice, the spiral of almost all shells, excepting perhaps only those of three species, winds from left to right. This definite feature pertains without exception to the identically same species, apart from any reference to the hemisphere where it nourishes or to the direction of the diurnal solar and lunar motion, whose course to us is from left to right, but inversely to the dwellers at our antipodes. The reason of this is that where, as in the natural products adduced, the source of the winding lies in the seeds themselves, on the other hand, where it is possible to attribute a certain rotation to the course of these astronomical bodies, this circulation at the southern hemisphere must go the other way. Such a law Mariotte asserts to have observed with the winds, which from new to full moon have a preference throughout the whole compass for blowing from left to right. Don Ulloa claims, by his observations in the Southern seas, that this is confirmed.

Since the diverse feeling of left and right is so markedly necessary in judging environs, nature has automatically attached it to the organisation of the human body, whereby, in point of skill—maybe also of strength—one side possesses an undoubted advan162tage over the other. Thus people are universally dextral (isolated exceptions being left out of account, which, as in the case of squinting, do not affect nature’s rule). On mounting a horse, or striding over a ditch, one impels his body with greater facility from the right to the left than reversed. It is with the right hand that one writes and performs 2:381whatever demands skill and strength. But just as the right side, in impelling power, to all appearance has an advantage over the left, in respect of sensibility the left has it over the right—at least, if some naturalists are to be trusted—e.g., Borelli, who asserts of the left eye, and Bonnet of the left ear, that the sense in them is stronger than in the same organs on the right. So the two sides of the human body, despite their great external resemblance, are clearly enough distinguished by a marked sensibility, if, at the same time, the different situations of the internal organs, also the beating of the heart—which, because of its muscular contraction, causes an oblique encounter of its point with the breast—are left out of account.

We shall demonstrate, therefore, that the be-all and end-all of a corporeal form does not solely lie with the relation and the position of its parts towards each other, but, over and above, with a reference to absolute space generally, as conceived by geometricians; yet in such a way that this relation cannot be directly apprehended, as may be done in the case of these physical diversities of left and right side, which rest solely and alone on this ground of difference.

If two figures drawn on one plane are equal and similar to one another, then they cover each other. But with corporeal extension, as also with the lines and levels which do not lie in one plane, a very 163different state of affairs often exists. They can be completely equal and similar, and yet of themselves so different, that the boundaries of the one cannot, on the same space, form the boundaries of the other. The thread of a male screw, which winds round its shank from left to right, will never fit into a female whose thread goes from right to left, be the thickness of the shank and the number of the windings identical. One spherical triangle can be completely equal and similar to another without covering it. But it is from the limbs of the human body, symmetrically arrayed against its vertical plane, that we get the clearest and commonest example. The right hand is similar and equal to the left; and were one of the two regarded alone, in respect of the proportions and the positions of the parts relative to themselves and to the total dimensions, a complete description of the one would do in all particulars as well for the other.

2:382 I call a body which is completely equal and similar to another, although it cannot be contained within precisely the same boundaries, its incongruous counterpart. In order now to show the feasibility of this, let one take an entity not consisting of two halves, which are symmetrically arrayed against a single plane of intersection, but something of the nature of a human hand. To a panel placed opposite such an object, draw from all points of its surface straight lines, and then extend them as far on the other side as they are on this. In that case, the extremities of such extended lines, when conjoined in one object, comprise the plane of a corporeal figure which is the incongruous counterpart of one placed over and against it. That is to say, if the given hand is a right one, its counterpart is a left one. The mirrored image of an object rests on the same grounds. For always appearing 164just as far behind the pane of the mirror as in front of it, the picture of a right hand in the mirror is invariably a left one. Should the object itself, such as a human body, be made up of two incongruous counterparts, if its back is parted from its front by vertical intersection, then its own image is congruent. This is easily seen, if one imagines it to have made a halfturn, for the reverse of the reverse of an object is necessarily congruent to its image.

The above may suffice to make the feasibility of completely similar and equal, yet incongruous, spaces comprehensible. We now proceed to the philosophic application of these conceptions. The ordinary example of the two hands already makes it sufficiently clear that the figure of one body may completely resemble, and the quantitative extent wholly equal, that of another, so that none the less an internal difference remains—namely, this: the surface containing the one can impossibly enclose the other. Such a difference must have an internal source, because, turn the surface as one likes, that which bounds the corporeal space of the one cannot serve as the boundary of the other. But this inner source of diversity cannot be attributed to the distinct manner of the connection of the parts of the body relative to one another, because, as one sees from the adduced example, in this respect everything can be completely identical. Nevertheless, if one conceives the first act of creation shall be a human hand, then it must necessarily be 2:383either a left or a right hand, and in order to produce the one, another deed of the creator was necessary than that by which its counterpart could be made.

Let one now take the notion of many modern, preferably German, philosophers—namely, that 165space merely consists of the external relationship of juxtaposed parts of matter, then in the adduced example all actual space would be confined to that which this first-created hand occupies. But let it be a left or a right hand, because no difference whatever exists in the relationship of the parts among themselves, this hand, in respect of such a characteristic, would be absolutely indefinite—that is to say, it would fit both sides of the body, which is impossible.

From this it is clear that directions in space are not the outcome of positions of material parts, but these positions are the outcome of the directions. Furthermore, differences in the constitution of bodies, actual differences, may be found, which are solely referable to absolute and primeval space, because only in virtue of it is the relativity of corporeal things possible; and, again, because absolute space is no object of external sensibility, but a fundamental conception which first renders all objects possible. Solely by the counter-attitude to other bodies can we apprehend what in the figure of a body concerns alone the relationship to pure space.

A thoughtful reader will not, therefore, consider the conception of space, as geometricians think it, and as also acute philosophers have accepted it in the doctrine of natural science, a mere hallucination, although difficulties do not fail to harass our conception if one tries by abstract reason to grasp its reality, which is sufficiently perceptible to the inner sense. But the toil of abstract reason presents itself everywhere, if one will persist in philosophising on the first data of our knowledge; although its difficulty is never so formidable as that which comes to light when the conclusions of a theory, taken for granted, contradict the most obvious experience.

2:13 A NEW THEORY OF MOTION AND REST, AND THE CONSEQUENT RESULTS FOR THE RUDIMENTS OF NATURAL SCIENCE

2:15 If, in a matter concerning Philosophy, the unanimous verdict of professors formed a wall, the scaling of which would be considered a crime as punishable as the act of Remus, then I might well allow myself the pleasantry of again appealing to the decisive opinion of the venerable multitude for that liberty which is justified by nothing else than by sound reason. If it occurred to me to dispute a law which for centuries had maintained undisputed possession in the textbooks of philosophers by hereditary right, for my part I should readily agree that either I ought to have been earlier on the scene, or now ought to refrain from the controversy. However, seeing myself in the midst of so many innovators, who are little inclined to abide by the conventional law, and who are at least treated with this consideration—that their opinions are tried and weighed in the balance; trusting to equal favour, I venture to mix myself among them, and to examine and challenge the conceptions of motion and rest, as well as the law of inertia associated with the latter. But I well know that these gentlemen, who are accustomed to dismiss as worthless all thoughts that are not offered up on the altar of the Wolffian or some other temple of learning, will at 167first sight declare that no test is necessary, and that the whole theory is incorrect.

I wish that for a moment my readers could translate themselves into that frame of mind which Descartes considered so indispensable for the attainment of right thinking, and in which, so long as this inquiry lasts, I am to find myself—namely, to forget all acquired theories, and voluntarily to tread the path towards truth, without any other guide than frank, wholesome reason.

In this frame of mind, I understand by motion a change of place. But at once I conceive the place of a thing is known by its position, or external relation, towards other things surrounding me. I can now observe a body relative to other external objects, which in close proximity surround it; and in that case, should it not alter its relationship to them, I shall say it rests. But from the moment I regard it in relationship to a wider sphere it is possible that the very same body, together with its near objects, alters its position in respect of that wider sphere; and from this point of view I shall impute motion to it. It now stands open to me to widen my intellectual horizon as I choose, and to regard this body in relation to ever remoter surroundings; and I understand that my judgment as to its motions and rest is never constant, but may always alter with novel outlooks. Granted, for example, I find myself on a ship, which is moored at the river mouth. Before me, on the table, is lying a ball, which, regarded in relation to the table, the walls, and the other parts of the ship, I say rests. Soon after this I look to the bank from 168the ship, and remark that, having 2:17cast off its moorings, it is slowly floating clown the stream. Forthwith I say the ball is in motion from east to west in the direction the river flows. But some one informs me that the earth daily revolves at a far higher speed from west to east. Forthwith I am of a different mind, and attribute to the ball a wholly contrary motion, with a velocity easily determined by astronomical science. However, I am reminded that the whole ball of the earth, in respect of the planetary system, moves from west to east with still greater velocity. Being compelled to attribute the same motion to the ball on the table, I alter the velocity formerly given it. Finally, Bradley teaches me that the entire planetary system, inclusive of the sun, is very likely experiencing a displacement in respect of the fixed stars.¹ I inquire in what direction and with what velocity. I am not answered, so now I get bewildered. I know no more whether my ball rests, or in what direction and with what velocity it moves. Beginning now to see that something is wanting in the expression of motion and rest, I shall never use the term in an absolute, but only in a relative, sense. I shall never say a body rests without stating in relation to what thing it rests, and never speak of its moving without simultaneously mentioning the objects in viecv of which it alters its position. Were I to imagine a mathematical space, devoid of all creation, as a mere receptacle for objects, even this would not in the least 169help me. For how am I to distinguish its divisions and separate, corporeally unoccupied, localities?

Now I take two bodies, of which one, B, rests in respect of all visible objects in close proximity to me; the other, A, however, approaches B with a definite velocity. The ball B may now never by so much as an inch alter its position in respect of any other external objects; yet, viewed in respect of the approaching ball A, it is not stationary. For, if their relationship is reciprocal, then their change of position is also reciprocal. The ball B, which is said to be at rest in respect of external objects, takes an equal share in respect of its reciprocal relations to the ball A. Both approach each other. Why, then, in spite of the intractability of language, 2:18shall I not say the ball B, which, it is true, is at rest in respect of other external objects, none the less finds itself in equal motion in respect of the approaching ball A?

You will allow, if we discuss the effect produced by a collision of the two bodies, that the external relationship with other things has here no concern. Diverting thought, then, from all other extraneous objects, and solely regarding the transaction between A and B, tell me this: while one rests and only the other moves, would it be possible to say which rests and which moves? Will not one be compelled to attribute motion to both, and, as a matter of fact, to both in equal measure? Their reciprocal approach concerns one just as much as the other. Given that a ball, A, 3lbs. in weight, approaches another, B, 2lbs. in weight, the latter, in respect of the surrounding space, being at rest, the distance of five feet between them is covered in one second. If I only notice the change occurring between the two bodies, all I can assert is that a mass of 3lbs. and one of 1702lbs. approach each other by five feet in one second. Since I have not the slightest occasion to attribute a greater share in the change to one of the bodies in preference to the other, so as to preserve perfect equality to both sides, I shall have to apportion the velocity of five feet per second in inverse proportion to the masses. That is to say, the body weighing 3lbs. will have two degrees of velocity, but that weighing 2lbs. three degrees, and on clashing they will actually take effect on each other with these forces. Disregardful, then, of whatever rest the body B may have in respect of other surrounding objects in space, it none the less possesses a true motion in respect of any other body approaching it; a motion, too, that is the equal of the other. So long, therefore, as one considers B to be absolutely at rest, the sum of both motions equals that which is supposed to be only in the body A.

If, notwithstanding this, one allows himself to be troubled about the intractable language, then I have no longer scruples whether he also will use exactly the same expression. If a 12lb. cannon-ball were fired 2:19against a wall in the direction east to west, the philosopher himself says it moves at a velocity of 600 feet in one second, although he immediately observes that, because from west to east the earth, within the same distance, has almost the same motion, the force of the powder properly suffices only to annul this motion of the cannon-ball. None the less, and without being beguiled by the diurnal or annular motion of the earth, one tacitly acquiesces in the fact of the relationship of ball and wall, in respect of the near and far surroundings, having nothing to do with the case, which merely concerns the reciprocal relations of these two bodies. But acquiescing in 171this, to which of the two, respectively, would one attribute rest? The manifestation of change affords no other clue than that the two are brought together; if one, indeed, is not more likely to admit that both move towards each other—the ball towards the Avail and the wall towards the ball, the one with the same force as the other.

Regard the distance which is covered between the two bodies, divided by the time, as the sum of the reciprocal velocities. Express it thus: as the sum of the masses A and B stands to the mass of the body A, so the given velocity stands to the velocity of the body B, which, if deducted from the said total velocity, leaves as remainder the velocity of A. In that case one will have apportioned the whole preceding change equally between the two bodies, and so with these equal forces will they also clash. From this I, for my argument, deduce only the two following corollaries.

(1) Every body, in respect of which another moves, is also, in respect of it, in motion itself. Therefore, it is impossible that a body shall approach another which is absolutely at rest.

Had not experience declared that, in a state generally taken to be one of rest, a body, in all cases of action, reacts on the other with equal force, it would probably never have occurred to anyone to suggest that, in order to cancel an opposite force in the oncomer, a body—wholly resting, so long as it is untouched by one approaching it, or, if it is preferred, whose force remains in equilibrium—suddenly, on being struck, 172assumes a 2:20motion contrary to the other, or is charged with an excess of gravity. Having, however, proved that what is falsely taken for rest in relation to the striking body is really in respect of it a motion, it is self-evident that this force of inertia is an unnecessary fiction, in every case of clash the motion of one body against another being met with an equal degree of contrary motion. Without the need of devising an exceptional kind of natural force, this quite simply and clearly explains the equality of action and reaction, as it also admirably serves the purpose of simply and clearly accounting for all the laws of motion. In this respect, however—similar to the Newtonian law of attraction explanatory of the grand motions of the cosmic system—it only serves as a law of a general manifestation known by experience, of which one does not know the cause. Consequently, there must be no undue haste to elevate this law into an inner natural force claiming to be that unknown cause. In this sense, without yielding an iota of the correctness of my theory, I can quite well admit that all bodies, in respect of those acting on them, possess a force of inertia—i.e., a force which equally opposes the action, for this is a mere law of experience. But as an inner force they only appear to possess inertia of themselves when they are at complete rest; while the fact is they have it solely on account of their being in actual and equal motion towards the approaching body—never, in so far as they rest, in respect of this body. On other grounds it is not difficult to refute the accepted notions of the force of inertia.

In the first place, a body may have never so many forces, yet, when it is at rest, these, in every case, must be in equilibrium. How, then, can it come about that, whenever the striking body touches the 173one at rest, suddenly, of itself, the latter, in order to annul a portion of the other’s force, shall transform itself into a motion, or preponderating effect, opposed to that of the oncomer? For were its inner force, 2:21even at the moment of contact, still to remain in equilibrium, nothing in the way of resistance with it would be accomplished.

In the second place, allowing that this suddenly aroused effort were possible, the passive body would itself receive no motion from the blow; for it and the counter-effect would annul each other, and the sole result would be that the two bodies would cease to act on each other; but not that, on receiving the blow, the struck one should move. And, apart from this, since the force of inertia is natural, even if equilibrium is suspended by the clash, the inertia would need of itself to be restored the instant after—i.e., the struck body must immediately again come to rest. I refrain from adducing many more grounds which I have ready to oppose the notion of the force of inertia. I could just as well throw light on the obvious metaphysical evidences. It is not, however, a book, but only a pamphlet that I write at present, to whose limits the copious material must be adapted.

Concerning the Law of Continuity in so far as it is Inseparable from the Force of Inertia.

What must, in the highest degree, embarrass the advocates of the ordinary conception of motion is this: if they seek to explain the laws of motion according to their theory, in spite of their wishes, they cannot prevent another law unsolicited asserting itself. This resourceful hypothesis is the law of 174continuity, of which probably the scantiest number of mechanicians have noticed that, no matter how they disagree with it, none the less, if they would explain the clash of bodies from the accepted conceptions of motion, they must tacitly accept it. Under this, however, I only understand the physical law of continuity, which never admits of being proved, although it may well permit of refutation; for what concerns the logical¹ sense it is 2:22a very excellent and correct method for forming judgments, but in no way does it affect the present objection. In Leibniz’s sense, then, the law of continuity would run: one body imparts no force to another instantaneously, but in such a way as to convey it through all infinitely minute intermediate degrees, from rest to the precise velocity. Listen, now, to the way all those who are keen to explain the laws of impact according to the accepted ideas of motion must consistently make use of this rule of Leibniz. Why does not a completely rigid body, on coming into contact with another equally rigid, impart its full force?—why only always the half, as statistics make plain? One answers, this happens because the striking body keeps pressing and compelling what encounters it till the velocity ol both is equal; that is to say, if the masses are equal, till each possesses half the velocity of the striker. Following on this, the struck body shuns all further action of the striker. But in such a case has not one 175assumed that the total effect of the striking body-passes by degrees into the body at rest, through a succession of infinitely many minute moments of pressure? For if the striker acted with all its force instantaneousljq it would impart its entire motion to the other, and itself remain at rest, which conflicts with the law of the clash of perfectly rigid bodies. The body at rest intercepts the whole motion of the striker; therefore, if this can act instantaneously with all its force, it will certainly do it; and what is true of the whole force is true of the half, of the quarter, etc. So it will not act instantaneously with any infinite force at all, but only bit by bit, through all infinitesimal moments, which the law of continuity attests.

Seeing from this that, if one does not repudiate the ordinary conceptions of motion and rest, the law of continuity must absolutely be accepted, I will only curtly show why nevertheless the most renowned physicists are not keen to admit its validity even as a hypothesis; and, since it cannot be proved, better than this no one will pretend it to be.

2:23 If I plead that one body, without previously going through all possible intermediate minute degrees, can never act instantaneously on another with one degree of force, then I assert it will not be able to act on it at all. For the moment may never be so infinitely small with which it instantaneously acts, and which, in a definite fraction of time, amounts to a precise velocity, this moment is always a sudden effect, which, according to the law of continuity, ought to have first, and could only first, have passed through all infinite degrees of lesser moments; for of any precise moment a yet smaller can always be thought, from whose summation the other has grown. For example, the moment of gravity is unquestionably infinitely 176slower than that of action in the case of bodies clashing, because, while the clash can generate a high degree of velocity in a wholly insensible time, gravity can do so only in a much longer. Therefore, even the moment of action in the case of impact is instantaneous and contrary to the law of continuity. One may not offer as an objection that in nature there are no perfectly hard'bodies. For here it is sufficient merely to conjecture them, and to determine their laws of motion, because only by means of them are the laws, according to which flexible bodies clash, to be found. In any case, every soft body has a certain degree of cohesion, so that, in respect of the equal or smaller moments of the striking body, it can be regarded as a hard body. And if only in respect of this a sudden effect is possible, then it will also be able to take place in respect of a greater degree.

Key to the Explanation of the Laws of Impact, according to the New Conceptions of Motion and Rest.

Our theory has now made clear what occurs from the clash of two reciprocally active bodies. It merely consists in action and re-action being on each side equal, and—provided that, devoid of any elastic or spring power, they have fairly encountered each other—in both bodies, relative to each other, coming to rest after the impact. Only, under the term “laws of motion” one does not merely understand the respective order given to the clashing bodies, one in relation to the other, but 2:24essentially also the change of their external state in respect of the space in which they find themselves. Strictly speaking, this is so far only the outer manifestation of that which has directly come to pass between them, and what this is one demands to know.

177 To this end let one in the first instance take two bodies, A and B, the former weighing 3lbs., and the latter 2lbs. In respect of the space in which they find themselves the latter rests, while the former moves directly towards the body B with a velocity of five degrees. Since now our principles compel us to attribute to the body B, relative to A, a velocity of three degrees, to A, however relative to B, one of two degrees, the consequence is that these two like forces annul each other, and both bodies, respective to each other, will come to rest. But since B, which relative to the surrounding objects was at rest, incurs by this a velocity of two degrees towards A, the consequence is that this motion also must be recognised parallel to the surrounding space, and with a velocity similar to that of the body B. Now the blow from A annuls this motion of two degrees in the body B, but not in the surrounding space in which no effect takes place. The consequence is A will continue to move in the former direction towards the body B; or, what is the same thing, the body B, after the blow, will advance in the opposite direction—i.e., in the direction of the striker A, with a velocity, relative to the surrounding space, of two degrees, consequently also the body A in the same direction and with the same velocity, on account of its now resting in respect of B. Both bodies will therefore advance after the clash with two degrees of velocity. One sees from this that one velocity, annulled in the one body, which is only transferred to the struck body in respect of the oncomer (a velocity which it did not have in respect of the space), produces really in it an equal degree of motion, with regard to the space, in the direction of the impact.

Let now two bodies, A and B, with the former 178weight, be taken; but they approach each other from opposite directions, A with a velocity of three degrees and B with one of two degrees. In this case, if one regards only the reciprocal relationship of the motion of these bodies towards each other, the velocities three and two must be added, and by the above theory the sum must be divided among them in inverse relationship to the masses, so that while A receives two degrees velocity B receives three degrees, the consequence being that relative to 2:25each other they are placed at respective rest by the equality of the opposed forces. Since now B is accredited with a velocity of three degrees owing to the respective motion of both bodies towards each other, a motion of which B, relative to the surrounding space, possesses, not the whole, but only two degrees, following what has just been remarked above, the consequence is that the annulment of a velocity, which is not discoverable in the body relative to the space, will establish a motion in the opposite direction in relation to precisely the same space.¹ That is to say, B will be advanced in the direction in which A struck the blow, with one degree velocity, and A also with this degree, because respective to B it rests.

It would be a simple matter to derive from these rudimentary conceptions, not only the laws of motion, on bodies clashing, which advance in the same direction with unequal velocities, but also the rules of the shock of elastic bodies. It would also be needful 179to place the argument in a more favourable light by a variety of illustrations. This could all happen if it were possible, with material so copious and conditions so restricted, to be complete as regards content, and yet lavish as regards expression.[*]

We now give the extracts from Spencer and Descartes mentioned in the footnote on p. 168.

“A body impelled by the hand is perceived to move, and to move in a definite direction: doubt about its motion seems impossible. Yet we not only may be, but usually are, quite wrong in both these judgments. Here, for instance, is a ship which we will suppose to be anchored at the equator with her head to the West. When the captain walks from stem to stern, in what direction does he move? East, is the obvious answer—an answer which for the moment may pass without criticism. But now the anchor is heaved, and the vessel sails to the West, with a velocity equal to that at which the captain walks. In what direction does he now move when he goes from stem to stern? You cannot say East, for the vessel is carrying him äs fast towards the West as he walks to the East; and you cannot say West for the converse reason. In respect to things outside the vessel he is stationary, though to all on board he seems to be moving. But now are we quite sure of this conclusion?—Is he really stationary? On taking into account the Earth’s motion round its axis, we find that he is travelling at the rate of 1,000 miles per hour to the East; so that neither the perception of one who looks at him, nor the inference of one who allows for the ship’s motion, is anything like right. Nor, indeed, on further consideration, do we find this revised conclusion to be much better. For we have not allowed for the Earth’s motion in its orbit. This being some 68,000 miles per hour, it follows that, assuming the 180time to be midday, he is moving, not at the rate of 1,000 miles per hour to the East, but at the rate of 67,000 miles per hour to the East. Nay, not even now have we discovered the true rate and the true direction of his movement. With the Earth’s progress in its orbit we have to join that of the whole Solar system towards the constellation Hercules. When we do this, we perceive that he is moving neither East nor West, but in a line inclined to the plane of the Ecliptic, and at a velocity greater or less (according to the time of the year) than that above named. And were the constitution of our Sidereal System fully known, we should probably discover the direction and rate of his actual movement to differ considerably even from these. Thus we are taught that what we are conscious of is not the real motion of any object, either in its rate or direction, but merely its motion as measured from an assigned position—either our own or some other. Yet in this very process of concluding that the motions we perceive are not the real motions, we tacitly assume that there are real motions. We take for granted that there is an absolute course and an absolute velocity, and we find it impossible to rid ourselves of this idea. Nevertheless, absolute motion cannot even be imagined, much less known. Apart from those marks in space which we habitually associate with it, motion is unthinkable. For motion is change of place; but in space without marks, change of place is inconceivable, because place itself is inconceivable. Place can be conceived only by reference to other places; and, in the absence of objects dispersed through space, a place could be conceived only in relation to the limits of space; whence it follows that in unlimited space place cannot be conceived—all places must be equi-distant from boundaries which do not exist. Thus, while obliged to think that there is an absolute motion, we find absolute motion cannot be represented in thought.

“Another insuperable difficulty presents itself when we contemplate the transfer of Motion. Habit blinds us to the marvellousness of this phenomenon. Familiar with the fact from childhood, we see nothing remarkable in the ability of a moving thing to generate movement in a thing that is stationary. It is, however, impossible to understand it. In what respect does a body after impact differ 181from itself before impact? What is this added to it which does not sensibly affect any of its properties, and yet enables it to traverse space? Here is an object at rest and here is the same object moving. In the one state it has no tendency to change its place, but in the other it is obliged at each instant to assume a new position. What is it which will for ever go on producing this effect without being exhausted? And how does it dwell in the object? The motion, you say, has been communicated. But how? What has been communicated? The striking body has not transferred a thing to the body struck; and it is equally out of the question to say that it has transferred an attribute. What, then, has it transferred?

“Once more there is the old puzzle concerning the connexion between Motion and Rest. A body travelling at a given velocity cannot be brought to a state of rest, or no velocity, without passing through all intermediate velocities. It is quite possible to think of its motion as diminishing insensibly until it becomes infinitesimal; and many will think equally possible to pass in thought from infinitesimal motion to no motion. But this is an error. Mentally follow out the decreasing velocity as long as you please, and there still remains some velocity; and the smallest movement is separated by an impassable gap from no movement. As something, however minute, is infinitely great in comparison with nothing, so is even the least conceivable motion infinite as compared with rest.

“Thus, neither when considered in connection with Space, nor when considered in connection with Matter, nor when considered in connection with Rest, do we find that Motion is truly cognisable. All efforts to understand its essential nature do but bring us to alternative impossibilities of thought.”

Special attention is drawn to Descartes having enounced that no more energy is required for a state of motion than of rest.

Abelard, 218

Adamson’s Kant quoted, 209 n.

Ali Baba, 178 n.

Amfortas, 266

Analogy, Kant’s distinction between mathematical and philosophical, 44 n., 256; reasoning by, 254-57; Kant’s definition of, 256

Angelus Silesius, quoted, 218

Anglicanism, its non-spiritual founders, 47 n.

Anthropology, Kant’s, quoted on symbolic expression, 257

Antigone, Liberal type, 56

Apathy. See Moral Apathy

Apollo, moral type, 109

Apollonius of Tyana, on value of Philosophy, xiii

Argyll, Duke of, claims immunity from criticism for the Church, 51-2

Arnold, Matthew, 96, 97

Art, xiii-xiv, xlii-xliii, 94-5

Asquith, Mr., his liability to refutation, xxvi

Bacon, quoted, 41 n.

Baldwin’s Dictionary of Philosophy and Psychology quoted, 235

Balfour, Arthur James: Begbie’s accusation of, xxiv-xxv; his liability to refutation, xxvi

Barbey d’Aurevilly quoted, lxii

Bax, Belfort, 223 n., 241

Begbie, Harold, on Balfour, xxiv, xxv n.

Behmen, definition of God, 219

Belloc, Hilaire, xxiv, 114 n.

Bergson, 80

Bishop, Anglican, association with Greek Church, 68 n.

Bishop of Exeter, on Bishops in the House of Lords, xxxvi

Bishop of St. Albans, on nonpolitical nature of Church, xxxvi

Blood, Benjamin Paul, 238

Boehm, Paul, on Negative Quantities essay, 62; on difference of Kant and Leibniz, 224

Bottomley, Horatio, 114 n.

Bradlaugh, Charles, possessed of the religious spirit, 43-59; asked by a friend, “Why trouble about God?”, 77

Bradley, F., xxvi

Brandes', Georg, reply to question on belief in God, 71 n.

British Weekly, 10 n., 54 n.

Brynhild, type of Liberalism, 56

Buddha, 95, 99, 109

Buddhism, 1, 278

Buffon’s Works quoted, 157 n.

Bunyan, John, patronised by Anglican parson, xlv, xlvii-xlviii

Burns, Robert, his mental absorption, 272

Butler, Samuel, 68 n.

Caiaphas, lii n., 55, 108

Caird, Prof. Ed., on Negative Quantities essay, xvi-xvii, 63; on Conceptions and Perceptions, 79 n.

Campbell, Rev. R. J., 10 n., 205

Carnegie, Mr., and education, liii

Carus, Dr. Paul, 241, 245

Cecil, Lord Hugh, on the Church’s privileges, xxxv-xxxvii; analogy with Mons. Jourdain, xliii; his invitation to Non-conformists, 216

284 Chamberlain, Houston Stewart, on Negative Quantities and First Ground essays, 78

Chateaubriand, lix

Chesterton, G. K., on journalistic corruption, xiv-xv n.

Christ, type of non-officialism, 26, 55; not an Optimist, 95; as Moral Apathist, 107-11 s and present-day Protestants, 265. See Jesus.

Christianity: its meaning obscured by qualifying adjectives, 1; as myth, 21 n.; Wagner on, 21 n.; Huxley and Newman on, 34 n.; and ecclesiasticism, 68

Church, the: its mediocrity and formalism, xxx-xxxiii; its politicalism, xxxvii-xxxviii; its patronage of Bunyan, xliv-xlix; Laurence Oliphants criticism of, 49; claims immunity from criticism, 52

Churchill, Lord Randolph, criticised by W. T. Stead, 43-59

Clemens’s definition of God, 219

Coincidence distinguished from causality, 5

Coleridge on Kant’s earliest essay, 61

Consciousness: Problem of Rest and Moral Apathy culminates in, 103; origin of, 103-4

Conservatism, shirks criticism, 19; à priori grasp of morality, 21, 65; by impeding intellectual progress gives rise to Liberal consciousness, 105-6; orders its mind according to Theism, 210, 216; is materialistic, 262; subordinates Art, 265; delights in external pomp, 276; pays Liberalism a compliment by its theft, 277; accuses Liberalism of raising class-strife, 278-9. See Liberalism.

Controverted Questions, Huxley’s, quoted, 50, 52

Copernicus, 78

Courtney, W. L., edits misrepresentation of Wagner, lvi

Creighton, Bishop, insisted Anglicanism was founded on scholarship, 47 n.; contrast with Oliphant’s saying, 49 n.

Creon, type of Conservative, 56

Critique of Judgment, quoted, 44 n., 243; on analogy, 257

Critique of Practical Reason, 209 n.

Critique of Pure Reason, 15, 57, 62, 77, 223 n., 238; quoted on necessity for criticism, 17, 19; on analogy, 44 n., 256; on our prescription of laws to Nature, 78 n.; on conceptions and perceptions, 79 n., 238

Daily Mail’s paganism, 68 n.

Daily News, 49 n.

Daily Telegraph, quoted, see Bishop of Exeter; its clientele, xli

Dante, 189

Darwin, 252, 267; on causes and effects in Nature, 258-9

Davidson, Dr., translator of Schlosser’s Literary History, 60, 61

De Quincey, on Kant’s Negative Quantities essay, 60, 61

Descartes, 167, 168 n., 179, 249; on motion, 181-83

Diogenes’ witty reply, 22

Dooley, 71 n.

Drews, Arthur, on Negative Quantities essay, 62; on Von Hartmann and consciousness, 105

Eckhart’s definition of God, 73, 219

Elizabeth, Queen, founder of Anglicanism, 47 n.

Ellis, William Ashton, 98 n.; his treatment of Schopenhauer, 3

Elsmere, Robert, 242, 246, 249

Emerson, 7, 94, 98, 111, 279; on Fable, xlii; on Christianity as Mythus, 21 n.; on Compensation, 184-92

Fairbairn, Prof. A. M.: his nauseating rhetoric, 200-19

285 Farrar, Archdeacon, denounces formalism, xxxii-xxxiii

Farrer, J. A., his Paganism and Christianity quoted, xiii n.

Fichte on grain of sand, 208 n.

Fielding’s Tom Jones, a new vein of knowledge, xv, xvi n., 64

Fricka, Conservative type, 56

Galilei, 136

Gessler, Conservative type, 56

God, Schopenhauer’s definition of, Ixiv-lxv, 89 n.·, objectification of, 16; a definition of, 29; Goldschmidt's question to Ibsen, 71 n.; the point of indifference, 73; responsibility of, 204; mystics’ definition of, 217 sq.

Gods of the Vulgar. See Bacon.

Godwin, 12 n., 208 n.

Goethe on Protestants, lvii, lxi, 23, 253

Goldschmidt’s question to Ibsen and Brandes, 71 n.

Gournemanz, 80

Grace, Dr. W. G., 270

Graham, Prof. William, criticises Spencer, 231

Green, John Richard, on non-spirituality of Queen Elizabeth, 47 n.

Hamilton, Sir William, 226, 227, 230, 232, 236, 237

Hamlet, 95

Hartmann, Ed. von, 7; on the origin of consciousness, 104-5; on brain-destroying effect of professorial philosophy, 105 n.; on evolution, 196 n., 199

Hatch, Dr., quoted, xxxiv

Hegel, on God, lxi, 16; his eternal process, 196 n.

Heine on Kant, 15, 17

Henry VIII., founder of Anglicanism, 47 n.

Hibbert Journal, the, article on “Pragmatism,” xxv, xxvii-xxviii; article on “Pluralistic Mysticism,” 238

Holmes, Edmond, his book on education, liii

Horne, Rev. John, on Spurgeon and clerical dress, 54 n.

Hugo, Victor, claimed by Roman Catholics, 33

Hunding, conservative type, 56

Huxley, 20, 55, 84; on Mr. Balfour’s reasoning, xxvi; refutation of Newman, 34-35 n.; controversy with Dr. Wace, 50; reply to the Duke of Argyll, 52

Huysmans, lxii

Ibsen’s embarrassment, 71 n.

Impenetrability a true force, Negative Quantities essay, 126

Jacobi quoted by Kant, 135

James, Prof. William, 238

Jesus, the historical, 20; Emerson on, 21 n.; Matthew Arnold on, 96-97; employment of parable, 99. See Christ.

John the Baptist, 91

Judaism Christianity's millstone, 39. See Theism.

Juvenal quoted, 22

Kant, xv, 4, 7, 8, 15, 17, 19, 22, 24, 32, 49, 51, 57, 60, 61, 64, 65, 70, 72, 73, 80, 83, 94 n., 112, 178 n., 214, 215, 217, 226, 227, 236, 241, 242, 256, 273, 277, 278, 281; on place of the definition in philosophy, xviii; gave Protestantism a creed, 16, 90, 91; cannot be appropriated by Theists, 33; on analogy, 44 n., 256-57; contrast with Wolff and Leibniz, 73-74; on the erudite clique, 76, 219; on the prescription of laws to nature, 78; on conceptions and perceptions, 79, 238 n.; on moral apathy, 99-111; contrast with Spencer, 195, 199, 200, 220-251; distinction from Leibniz on negation, 224; on universale and particulars, 238 n.

Kelvin, Lord, on drop of water, 208 n.

King officially theistic, 24; his Army—the National Debt, 28

Knox, behind Scottish Presbyterianism, 47 n.

Kundry, 98

286 Lamb, Charles, quoted, 272

Laurier, Sir Wilfrid, quoted by Asquith, xxvi

Leibniz, 150, 215, 273; propounder of Optimism, 12; his rationalism theistic, 63 n.; contrasted with Kant, 73; his consideration of spatial problems, 157; on motion, 174; on negation, 224

Leonardo, xxvi

Liberalism, its need of introspection, xvii; discouraging indifference of Liberals to, li-liv, 8,9; feared by Liberals themselves, 15; must seek criticism, 19, 26; allows Conservatism first voice in morality, 21; naturally antagonistic to officialism, 25; must break its association with present-day Non-conformists, 38; what it may learn from Kant’s Critique, 57; a male God objectionable for it, 226-27; accused of raising class-strife, 278-79. See Conservatism.

Liddon, Canon, 52

Literary Guide, 49 n.

Locke, 244

Luther, founder of German Protestantism, 47

Malebranche on God, 219

Mallock, W. H., his reasoning compared favourably with Fairbairn's, 203

Maudsley quoted, 272

Metaphysical Rudiments of Natural Science, Kant’s, quoted on motion, 249-51

Milton quoted, 93, 110

Mind on self-defeating argument and Balfour's dealing with Kant, xxvi; on Kant’s synthesis, 247

Molière’s satire, xlii-xliv

Moore, George, 28 n.

Moral Apathy, 107-11; described by Kant, 99-100

Morgiana, 178 n.

Morning Post on revolutions, 279

Motion and Rest: rest as important a rôle as motion, 101; motion only possible by means of rest, 102; Spencer on, 179; Descartes on, 181

Mozart, xxvi

Müller, Max, attempt to dissuade him from translating the Critique of Pure Reason , 77

Murray, David Christie, on coincidence, 5

Mystics’ definition of God, 217-19

Myth. See Christianity.

Napoleon, 93

New Age, the, 49 n.

Newman, Cardinal, on value of a soul, liv-lv, Ivi; on historical Christianity, 35 n.

Newman, Colonel, on humble-bees, 259

Newman, Ernest, misrepresentation of Wagner, lvi n.

New Paralipomena, Schopenhauer’s quoted, on God, lxiv-lxv, 89 n.

New Testament compared with Kant’s essay, 273

Newton, Sir Isaac, 136; misconstrued meaning of negation, 61

Nicol, Sir William Robertson, 10 n.

Nietzsche, 253; Nietzscheites, 1, 84, 100, 108, 109

Nirvana, positive, 214

Norm, possibility of raising, 84-88, 92

North, Sir “Kit,” Editor of Blackwood, 60

Old Testament compared with Leibniz, 273

Oliphant, Laurence, on the Church’s monopoly, lvii, 49; on the root of moral disease, 99 n.

Optimism, the bane of Liberalism, xxiv; hopelessness of, 7; in relation to intellectual restraint, 10; a modern craze, 12; in relation to moral apathy, 100-1. See Pessimism.

Origin of Species, quoted, 258-60

Owen, Rev. J., quoted, 218 n.

Pain as positive as pleasure, 287Negative Quantities essay, 127-29

Parsifal, 80, 266; in relation to Schopenhauer’s philosophy, 97-99

Paulsen, on Kant as Protestant philosopher, 16 n.; on Negative Quanitities essay, 63

Peall, the billiard-player, 270

Pearson, Karl, on drop of water, 208 n.

Pessimism, definition of, 7 n.; and moral apathy, 103; what shocks the Pessimist into, 201; sets forth no absolute nonexistence, 214. See Optimism.

Philosophy, reasons for its lack of popularity, ix-xii; provisional explanation of, 5; seeks criticism, 19; by intuition forestalled science, 21; its task, 83; ready to guide Liberalism, 216; simplifies the moral problem, 275

Poetry in intuitional agreement with philosophy, 94-99

Pope, Alexander, 96-97

Pragmatists, the, xxv, xxvii-xxviii

Pratt, J., on Pragmatism, xxv

Professorial philosophers withhold philosophy from the people, x, 76; are servants of Theism, 71 n.; destroy original powers of thought, 105 n. See Hartmann.

Prolegomena, Kant's, quoted, on dogmatic twaddle, 8; on prescription of laws to nature, 78 n., 223 n.; on space, 239-41; on analogy, 256

Prudhomme on property, 75

Rationalism needs to come to an agreement with philosophy, 21, 22, 32, 34; modern distinguished from that of Wolff and Leibniz, 63 n.; must associate itself with art, 94

Rationalists must draw distinction between Christianity and adjectived-Christianity, l-li

Religion within the Bounds of Pure Reason, Kant’s, quoted, 32 n.

Renan, 14 n.

Ridge, Pett, quoted, 51

Robertson, J. M., tells anecdote of Bradlaugh, 77

Roman Catholicism, innocent of toleration, xxxviii-xli; provides the creed for Liberals, 11; not optimistic, 12; dissent from the original simplicity of Christianity, 30; compared advantageously with Non-conformism, 36

Rosencrantz on Negative Quantities, 62

Schelling on art, xiv; on God, lx

Schiller’s reason for belonging to no religion, 36 n.; his Gessler and Tell as types of Conservatism and Liberalism, 56

Schlosser’s Literary History quoted by De Quincey, 60, 61

Schopenhauer, xvi, xlviii, lv, lvi, lxii, 7, 12, 13, 16, 55, 61 n., 96, 97, 205, 209 n., 227, 238, 254, 281; on God, lxiii-lxv, 89 n.; and W. Ashton Ellis, 3; his pessimism, 7; and Shakespeare, 95; was Wagner’s philosopher, 97; distinguished Judaism from Christianity, 110; contrasted with Spencer, 200, 229, 242, 243, 246, 248; on order and optimism, 212; and the mystics, 215, 217; on Will, 229; distinguished Religion from Theism, 230; on the Absolute, 231; on revolution, 280

Scotus Erigena, 218

Shakespeare, xxvi, 95, 189, 208 n.

Shaw, Bernard, 85, 112, 264

Siegmund, Liberal type, 56

Simonides’ reply to question on God, 76

Smith, Goldwin, on Judaism, 38

Sophocles’ Creon and Antigone as types, 56

Southey on an essay of Kant, 60

Space more sublime than time, 80

288 Spectator, The, 47 n.; as a moral influence, 49 n.; its theistical caution, 54 n.

Spencer, Herbert, 20, 84, 168 n., 219, 230; on present-day education, 105 n.; extract from First Principles, 179-181; contrasted with Kant, 195, 199, 200, 220-51; contrasted with Schopenhauer, 200, 242, 246, 248; on, the Absolute, 225-54; on Space, 241-49

Spurgeon, on clerical dress, 54 n.

State, the, assumes guardianship of Christianity and suborns professors of philosophy, x-xi; cannot initiate, 9; shirks criticism, 22

Stead, W. T., his inconsistency, 43-59

Summun bonum, philosophic conception of God, lxiii-lxiv, 42, 227

Superstition, 5

Symonds on Christ as myth, 21 n.

Tell (Schiller’s), as Liberal type, 56

Tennyson: a senseless qualification, 50

Theism, the enemy in the Liberal camp, xx; belongs to Conservatism and Officialism, 24, 210; essentially political, 33 n.; a moral thief, 75-6. See Theology.

Theology usurps legitimate field of Philosophy, xi-xii; masks as Philosophy, xii, 10; shirks criticism, 19; opposes science, 20

Theologians not artists, 14; ignore Kant and Schopenhauer, 16. See Theism.

Time, the great illusion, 7; less sublime than Space, 80

Times, the, articles quoted: on Mr. Asquith’s reasoning, xxvi-xxvii; Lord Hugh Cecil on the Church’s privileges, xxxv; Bishop of St. Albans on Church’s freedom from politics, xxxvi; Father Vaughan quotes Chateaubriand, lix; M. Montague Villiers and definition of Protestant faith, lxi; Lord Hugh Cecil's invitation, 216 n.; on the demagogic temper, 279

Tolstoy, as type, 55; on Art, 273

Tom Jones. See Fielding. Tyndall, 19

Understanding does not draw its laws from nature, 78

Vaughan, Father, quotes Chateaubriand, lix

Villiers, H. Montague, asks definition of Protestant faith, lxi

Wace, Dr., 55, 56; controversy with Huxley, 50

Wagner, xlii, lv, 55, 242, 254; Newman's misrepresentation of, lvi n.; Life of, 3; on Christian myth, 21 n.; his characters as types, 56; on time and space in Parsifal, 80; on Schopenhauer, 96; Parsifal, 98-9, 266; on the Sacrament, 265 n.

Watson, John, 241; criticises Balfour, xxvi

Wesendonck, Matilda, Wagner’s Letters to, quoted, 98

Wilde, Oscar, 112

Wolff, 213, 224; his Rationalism distinguished from modern, 63 n.; his procedure contrasted with Kant’s, 73-4

World as Will, 61 n.; quoted on order in Nature, 212; on Will and Thing-in-Itself, 230; on Space, 248-49

Chap.	I.	Taking up an Attitude	3
„	II.	Locating the Adversary	24
„	III.	A Searching Blow	40
„	IV.	Preparing to Smite the Philistines	60

	Attempt to Introduce the Conception of Negative Quantities into Philosophy. By Immanuel Kant			117
	Sec.	I.	General Explanation of the Conception of Negative Quantities	117
	„	II.	Wherein Examples are Adduced from Philosophy in which the Conception of Negative Quantities Occurs	126
	„	III.	Containing some Considerations Preparatory to the Application of the above Conceptions to Philosophical Concerns	136
vi	Concerning the First Ground of the Difference of Environs in Space. By Immanuel Kant			157
	New Theory of Motion and Rest, and the Consequent Results for the Rudiments of Natural Science. By Immanuel Kant			166
	Extracts from Emerson’s Essay on Compensation			184

Chap.	I.	Theological Scalp	195
„	II.	Where Science Fizzles Out	220
„	III.	Finis Fundat Opus	252

Index			283

			page
Preface			vii
Introduction			ix