But if the four elements naturally arrange them- selves in the stated order (counting inwards) fire, air, water, earth, how comes this order ever to be disturbed, and why is anything so out of its natural place as to manifest its trend towards it? Aristotle’s answer to this question, though explicit enough as far as it goes, is not worked out with the complete- ness and detail we could desire. Broadly, Aristotle’s answer is closely akin to that of modern science. Evaporation and condensation are continuously going on, and they are typical of all the transformations which, by changing the bulk of things, tend to change their relative yravity or levity; and so, for instance, air (vapour) and water (liquids) establish a circulation under the influence of accessions or withdrawals of heat, for it is a cardinal doctrine with Aristotle that the elements can be transmuted into each other, and that the same ultimate matier occupies more or less room according as it is heated or chilled, But here the hesitancy we have already noted as to the lxv INTRODUCTION relation of fire, as an element, to heat, as a state, obscures the connexion between the sometimes isolated data to be found up and down in Aristotle’s physical treatise, or incidentally in his other works.
Under the reserve implied in these warnings, we must attempt to reconstruct Aristotle’s system. It is certain that the celestial ‘ fifth element’ is not fire and has not its properties; and since the heavenly bodies are composed exclusively of this element (being, apparently, in some sense concentrations or intensifications of it) they are not and cannot be the direct sources of heat in the sense in which an earthly fire is. Yet it is obvious (and AristoUe lays stress on the fact) that changes of heat do accompany the presence or absence of the sun’s rays, and are affected by the directness or obliquity with which they strike things. There is only one passage in his extant works in which Aristotle gives us a direct clue to the solution of the problem thus suggested. We know that friction is a potent cause of heat, potent enough to melt a leaden bullet as it passes through the air. We may suppose, then, that the friction of the inner surface of the rotating celestial shell of the universe heats the upper air in contact with it into actual fire, more especially where its potency is highest, that is to say in its concentrated presence in the sun. The heat thus generated descends by conduction and reaches the other elemental regions with diminished force. When the sun is withdrawn cooling succeeds lo heating.* 4 It may be noted, in passing, that in his biological treatises Aristotle appears, at least, to treat the heart as an independent source or agent of heating, and the moist brain as a corresponding refrigerator. We may perhaps connect lxvi INTRODUCTION This account of the inorganic or mechanical move- ments of the elements must close with a repetition of the warning that though all the principal data on which it is based are authenticated by explicit texts, yet it is impossible not Lo have some iisgiving as to their power, in every case, to bear the weight that an attempted reconstruction of the whole system neces- sarily lays upon them, Under our next step, on the other hand, we shall find the ground perfectly finn. With inexhaustible earnestness, perseverance, and ingenuity Arislotle urges throughout the Physics that no mechanical movement 1s self-initiated by matter. Uarth falls indeed, and fire rises, in virtue of an intrinsic ‘ prin- ciple of movement.’ But that principle is a passive capacity for being moved, and not a capacity for self-initiated movement. The falling clod does not move itself, though it is in itself capable of a natural and unforced movement if suitably acted upon.
Now in the inorganic world movement in a body is always caused by some other body that is already in motion, This will land us in an ad infinitum chase for the ultimate origin of movement unless we can find some cause of movement that is not itself in motion.
The cosmic and ‘ theological ’ conclusions to which this leads us close the Physics; but our next step towards them must be the observation that living things, and especially animals, unlike inorganic things, do seem to initiate movements. This Aristotle admits, but points out with elaborate care this with the belief that some infusion of the actual celestial element was an invariable concomitant of the phenomena of life. But the links of this connexion (if it exists) are lost.
Ixvii INTRODUCTION the danger of expressing this fact by saying that animals ‘ move themselves.’ An animal’s free move- ments are probably prompted in every case by some change in its physical organism or environment, but it is always some kind of desire (however prompted or suggested) that actually makes the animal move. Now a desive is not a material body or substance that can impinge upon another body and cleave or thrust it. Yet we see that as a fact—a primary datum of observation and experience that we must accept whether we can or cannot ‘ explain’ it—an animal’s ‘ desire ’ can and dues move its body. And since a desire is not capable, directly and on its own account, of moving locally (for only a ‘ body ’ can so move), we seem already to have reached our ‘ un- moving source of motion.’
But this conclusion would be too hasty. With the sole exception of the human intelligence, with its natural ‘ desire to know ’ for the sake of knowing, all the desires of animals are prompted by physical and sense-perceived changes, and therefore postulate them as antecedents and cannot be their prime source. And even intellectual desires, though Aristotle thought they could rise into independence of all sense organs, can draw the materials on which they so rise only from material objects through the ae of sense.
ut let us look from earth to heaven. Aristotle’s astronomy will require our attention in due course, but for our present purpose we need only know that he believed the whole ethereal shell of the universe to consist of a number of concentric spheres, one inside another, the inner surface of each being in frictionless contact with the outer surface of the next lxviti INTRODUCTION within iL, until the inner surface of the inmost rubs against the outmost edge of the sublunary or ‘ four- elemental ’ world.
Perhaps the homely illustration of the coats of an onion will best serve us here. Replace the core of the onion by some imaginary ‘stuffing’ of unlike substance; Aristotle’s concentric spheres will then be represented by the remaining coats of the onion, and the imagination will easily rise to the idea that they can slip about within each other in immediate but frictionless contact. This is what Aristotle's concentric ethereal shells do. Fach of them has a ‘proper’ movement of its own, but, with the ex- ception of the outmost, cach of them also sym- pathetically obeys the movement of some one or more of the spheres above it. It is thus that the spiral course of the sun throughout the year, of the moon throughout the month, of the planets (with their retrocessions) throughout their periods, and the simple circular movement of the fixed stars (in Aristotle’s time the ‘ precession of the equinoxes’ had not been observed) were accounted for by a single circular movement in one case, ov a‘ proper ’ circular movement imposed upon one or more derivative movements (all circular) in the other cases. The fundamental principle of this analysis and synthesis still survives as the basis of our modern astronomy.
But our immediate concern is with the ‘ proper ’ movement of each several ‘ heaven,’ or ethereal coat, shell, or envelope. Each such heaven was seen by Aristotle as ‘ moving itself,’ just as an animal does, in addition to any movement it may receive from other moving spheres. Such a heaven, then, mani- fests the phenomenon which Aristotle regarded as lxix INTRODUCTION peculiar to animated beings, the power, namely, of initiating bodily movement in virtue of an inherent vital principle, not by impact but by the impulse of desire.
Bach heaven, then, is an animated being, But the desires of sublunary animals are stirred, directly or indirectly, by sense-impressions or physical changes in the organs of sense, and (except in the case of the human intelligence) are directed towards, or at least accompanied by, actual ‘ sensations ’ of a material order; whereas the heavens or celestial animals have no organs of sense, and the substance that constitutes their bodies is incapable of any modification except that of (rotatory) movement. What, then, can be the nature and the object of the ‘ desire ’ that their motion implies?
The highest desire of the highest of sublunary animals and the least dependent on material sensa- tions is the human ‘ desire to know.’ Lift that into a higher potency, and make it continuous instead of intermittent, and regard it as continuously gratifying itself by fruition, and you will have the best concep- tion that man can form of the blessed and eternal joy of the divine animals.
But what are the objects of their desires? These must needs be higher than themselves. Moreover, since they all move, and we have seen that the ultimate source of movement must itself be motion- less, and we now know that it must be higher than even the celestial animals with celestial bodies, we must take it that the object of each desire that originates the proper movement of a celestial animal or ‘heaven’ must be an immaterial consciousness free from movement, in blessed possession of ‘ truth ’ Ixx INTRODUCTION its very self, wherein desire is quenched in eternal fruition.
Such, in the supreme degree, is the life of the supreme being, described by Aristotle as the ‘ prin- ciple on which all heaven and nature depend.’ Such is the direct object to which the desire of the supreme heaven, expressed in 1ts continuous and simple move- ment, is directed. The proper movement of cach of the other heavens is exercised in independence, indeed, but in a certain subordimation to that of the supreme or oulmost heaven; and the divine and immaterial object of the desire that inspires it must be conceived (though Aristotle docs not expressly develop this line of inference) as standing in a corresponding relation of subordination to the Supreme Principle ‘on which heaven and nature depend.’
The Poet Laureate may well call the attention of his readers to the ‘ extreme interest’ of the chef passage in which this ‘theological’ doctrine is expounded. ΤῸΣ it is, as he declares, ‘the one original foundation of the Christian doctrine on the subject.’* More especially it is the basis on which all the elaborate Angelology of the Schoolmen is built, and that because it gave the Neoplatonists, Christian and ethnic alike, and the Arabian Aristo- telians of the tenth to twelfth centuries, the point of attachment for their synthesis of the Platonic and the Aristotelian philosophies.
Indeed it must be evident to the most careless observer that in all this theological construction Aristotle betrays himself as still dominated in his highest spiritual life by Plato’s teaching. While he 9 Robert Bridges, The Spirit of Alan, extract 39. Ixxi INTRODUCTION is honestly persuaded that he is building up his system towards a conclusion that only reveals itself as we approach and reach it, it is obvious that in reality his biological predispositions, his reverence for popular instinctive and traditional beliefs, and his own instruments and methods of thought, are all directed into the channel they take in this argument by the influence of the pre-established goal to which they lead us.
Plato had taught Aristotle—a willing pupil—that the supreme life is a life of direcl perception of immaterial and eternal realities, in which truth and beauty are one. And though Aristotle on his own ground broke away from his master, yet his ultimate conception of the divine and ideal life not only remained Platonic but continued to influence him in all his thought.
We shall never understand the under-currents of Aristotle’s thought till we understand this subtle influence. It is at the root of his ‘ teleology,’ that is to say his doctrine that the perfect and the actuahzed is presupposed in the potential and the undeveloped and at once directs and explains its course. It is that that makes him regard even the up-and-down-ness that observation establishes as a fact in the movement of the four elements as finding its reason and explanation in the consideration that it is the nearest approximation that the in- complete linear movement can make to the ‘ circum- lation’ of the complete and self-re-entrant circular movement.
On the same principle he explains why animals and plants that are condemned to mortality by the divergent trends of their component elements Ixxii INTRODUCTION nevertheless strive after immortality by leaving beings like themselves behind them to make the collective life of their kind eternal.
It is the same impulse that makes him—though he defines Nature as the objective sum of things, atiributes, and funclionings that are subject to change—nevertheless treat her and feel towards her as a divine (impersonal, it is true) agent of impulse, purposeful and aspiring.
But in one respect Anstotle has the advantage of Plato. It must be particularly obvious to the reader who compares the Uimaeus with Aristotle’s physical and metaphysical treatises. Plato, starting with the perfect and absolute, but unable to get rid of the world in which we live, gives us a saddening, some- times a hormfying, description of the badness of the ‘second best ’ which was all that a material creation could possibly aspire to, and our prevailing impression may well be of the degradation imvolved in our material nature. Aristotle, starting from the world in which we live, finds it full of intensest interest in itself, and ennobled, from centre to circumference, by upliftings and aspirations towards that which is above itself. Even its failures are not-yet-nesses, or not-quite-nesses. There is no degradation or confinement in our having bodies, but there is a divine glory in our having intelligence, and joy in our having disinterested affections. And if there is a region of divine life to which we mortals cannot rise, it is inspiring to push out the limits of our thinking to the utmost of our power and to dwell upon the thought that there are diviner lives in which those limits are broken.
[Ὁ remains that the common, if not the universal, lxxiii INTRODUCTION verdict pronounces that Plato’s power of effectually bringing us into an ideal world transcends Aristotle’s. But if to the mystic and the metaphysician Aristotle can never be more than the philusopher of the plain man, yet 1o the plain man himself he may bring moments in which he forgets that he is not a mystic.
III, Nore on tue J/erroronousres As to the Aristotelian canon, I am content in the main to accept the authority of Ross (dAmstoile, pp. 9 sqq.) as to the spuria (such as the De coloribus, the Mechanica, the Problemata, parts of the Iistoria animahum, ig Fees the dubia. But Iam compelled to reject the Meteorologica in its entirety, although it is classed by Ross as‘ undoubtedly genuine’ (p. 11); and as it might naturally be expected that its data would be taken into account in expounding Aristotle’s cosmography, I must at least indicate the grounds for rejecting it, though I cannot here enter upon a detailed argument.
To begin with, the Meteorologica is written in conspicuously pleasanter and more flowing Greek than is at all frequently to be met with in Aristotle’s genuine works. This was noticed by earlier com- mentators. On the other hand the argumentation is indefinitely looser than Aristotle tolerates even when he is least severe, as, for instance, in parts of his biological treatises. Even where Anistotle’s reasoning lags farthest behind his observation, and where he is rcadiest to adopt any explanation that seems to serve him for the moment, he never ap- proaches the random and facile opportunism of the author of the Meteorologica.
lxxiv INTRODUCTION Apart from these general characteristics, the whole system of the Meteorologica differs from that of the Physica, the De caelo, and the De generatione et corruptione. In these genuine treatises, for mstance, there is no trace of the doctrine of ‘ dry and moist exhalations’ which in the Meteorologica is so pro- iminent as almost to oust the doctrine of the ‘ four elements,’ nor is this doctrine of exhalations itself anywhere clearly and fundamentally expounded.
Again, in the Meteorologica the revolving heaven earries the sphere of fire and the sphere of air with it in its circulation, down to the region in which the mountains, by thei barriers, make as it were pools of still air; whereas in the De caelo it seems to be the friclion of the heavenly sphere against the stationary atmosphere that heats the upper air into fire; and in any case the idea of the rotating atmo- sphere would bring confusion into the whole mechan- ism of the natural up-and-down-ness of the motion of the elements.
1 may add that the traits in medicval science drawn specifically from the Meteorologica have never appeared to me either to connect up with the general system or to have any definiteness and consistency of their own. I cannot but think that other students must have shared this impression, IV. Nore on toe Princietes or ΤΒΑΝΒΠΑΤΊΟΝ HERE ADOPTED @ “The frankly paraphrastic and expository character of the translation of the Physics here submitted to 4. Compiled by ἢ, W. The sentences between inverted commas were written as they stand hy the translator; the rest is from his notes, VOL, 1 Ft Ixxv INTRODUCTION students was adopted under the conviction that a so-ealled literal translation would be wholly un- intelliguble m many passages to the English reader, and would be of very little value to the student of the Greek text.
“The writer’s object was primarily to make what he understood Aristotle to mean clear to the English reader who knows no Greek, but is not afraid of hard thinking; and secondarily (if he could) to make what Aristotle says intelligible to the reader of Greek who has made no previous study of Lhe book or of the Greek and mediaeval commentators who have thrown so much light upon it.”
The translation, therefore, must not be regarded as an introduction to Aristotle’s technical terms, but as an attempt to express his thoughts in terms with which the Enghsh reader is familiar. “ Such an attempt must necessarily be tentative and must at best be open to legitimate challenge at many points, at worst be convictable of positive and frequent error. I should not have had the audacity to enter upon so high and perilous an emprise, had I not been able to lean at every step on the ample Greek and Mediaeval commentaries, without which the Physics must have remained to me a sealed book almost from the first page to the last, and which the reader to whom I hope to be of service cannot be supposed to have studied.”
A ‘literal translation’ ‘‘ would have necessitated the constant use of English words in meanings that they cannot bear in Finglish writing, a practice which is particularly dangerous and misleading when the words in question carry strong and definite associa- tions of their own. No insistency of warning can Ixxvi INTRODUCTION suffice to prevent such words from drawing the reader after them in their natural and customary suggestions.” Therefore the principle of never using a well-known English word in a sense alien from that in which it is used in current literary English speech oy writing has been carried as far as possible.* “Vor instance, to translate ψυχή by ‘soul’ (as distinguished from ζωή as ‘ life’) would raise false but inexpugnable associations from which no effort, however firm and vigilant, could protect the reader.” Both ψυγή and ζωή mean ‘hfe’ in exactly the same sense—but ζωή is ‘life,’ ψυχή is ‘ a life.’ There is no indefinite article in the classical languages, but ψυχὴ has the individualized quality of the indcfinite article already in itself? Therefore ψυχή will be found 4 Some of his readers will think that he has carried it too far, a few may think that he has not earned it far enough, as, for instance, when he uses ‘air’ where we should use ‘vapour.’ But what we mean by ‘air’ was to Aristotle the elemental vapour, as ‘ water’ waa the elemental liquid, and to substitute ‘vapour’ for ‘ air’ here would be sub- stituting not merely a current English word, but a modern conception.
> “Tn the middle ages there is an Unum quod (unus qui) convertilur cum ente; it 1s the hypercategorical unum, and it pluys an enormous and very perplexing part in Mediaeval physics. It is an unum which is not numerical and therefore is not opposed to plurality, and there is no contradiction in saying that things which are numerically distinct may be hypercategorically wnum. This «num is simply the indefinite article. There is, so far as I am aware, no one of the Western group of the Indo-European languages, except English and Anglo-Saxon, that have a distinctive form for this unum quod convertitur cum ente.”
Even in English, though we have an indefinite article, ‘ one’ is sometimes used in its place, eg., “I am now wanting a Ixxvii INTRODUCTION translated as ‘ a life’ instead of ‘ a soul,’ for we may speak with propriety of a celestial spirit and a plant as each having a life (ψυχή) of sts own, but it would be quite another thing to say that a celestial spirit and a plant each had a soul of its own.
There is, however, no one English word which will always be the equivalent of ψυχή, and in order to make clear what Aristotle is thinking of at the moment, it has becn translated variously by—a life principle, a terminable life (which may be cut short by an accident), a reproductive or digestive function, a thought, mentality or mind—but it is always an individual something, never (as ζωή is) just life, or thought in general.
Again ‘ substance ’ has been rejected in favour of ‘ being ’ or ‘ conercte entity ’ as the English equiva- lent of οὐσία “ chiefly because ‘ substance ’ irresistibly suggests materiality; and to raise the question whether the soul is a substance, or to assert that the immaterial and non-spatial, but conscious beings of Aristotle’s theology on the one hand, or in- dividual men, stones and trees on the other, are ‘ substances ’ is too great a violation of English usage to be permissible except in professedly technical compositions.” ‘ Concrete ’ means compositum in the Latin sense, ἐ.6, the complex of body and form, which everything is that has a primary existence of its own: everything that would be there whether we existed or not, and this constitutes the first category, οὐυσία.
_ on book on such and such a subject; can you recommend one?” means just the same as “ Can you recommend a book on such and such a subject? I am now wanting one.” And there is no intention here to set the limit at a single one, Ixxviti INTRODUCTION Yet again: shortage is substituted for privation (the established translation of στέρησις), for privaton implies thal something has been taken away, and shortage merely that something has not been attained For instance, the seals that might under suitable circumstances have developed the full form of quad- ruped, or the water that might have become ‘ air,’ though they are both short of something which they might have had, are neither of them deprived of anything which they have had.
“ Whustrations could of course be multiplied, but these will suffice as apologies to the students of Greek and of philosophy, and as warnings to the Hnglish reader that he must never suppose himself to be in possession of Aristotle’s terminology in an English dress, however faithfully I may hope to convey his meaning, “On the other hand I have to crave indulgence for the use of uncouth words, some of them ignored and others expressly disallowed by the dictionaries, such as ‘ aliveness,’ ‘ awareness,’ ‘ dimensionality,’ ‘ quan- titive ’ and ‘qualitive’; and of innovations or archaisms such as ‘station’ as the correlative of ‘motion’ or ‘continent’ as the correlative of ὁ content,’ * and of some frank barbarisms in the way of hyphened composites which are in no danger of causing confusion of thought however much they may offend propriety.”
4 T say ‘a motion interrupted by stations’; * a motion is not made up of stations, any more than a line is made up of points.’ And I call a vessel that contains ‘ the vessel continent’ or even ‘the continent’ in contradistinction to the ‘ content,’ as meaning in fact the vessel that contains the content.”
Ixxix INTRODUCTION V. Nore on certain MatuematicaL Concertrions The following is an attempt to give the un- mathematical reader some idea of the mathematical conceptions which permeate the Physics. No detailed knowledge of inathematics is necessary, but the meaning of certain abstractions such as magnitude, dimension, divisibility, surface, line, point and so forth must be firmly grasped in order fully to under- stand the significance of Aristotle's reasoning.
Divisibility —The question of divisibiity is a crucial one, and we will begin by considering that.
We see that a collection of individuals can be broken up into groups, and these can again be broken into smaller groups, and so on and so on; all these divisions will, roughly speaking, be alike, and this can go on until each individual stands alone. The process of division can still be carried on, but if an individu-al is broken up it thereby loses itséndevidu-alit Vor instance, if a chair is broken up, the parts wil not be small chairs but pieces of wood, cane, and so on. The divisions of an orange are not small oranges, nor the parts of a man small men. The divisions of a homogeneous mass on the other hand are all exactly alike. But things that in everyday life we call homo- geneous may not really be so, and experiment may show that if they are divided and subdivided again and again the parts will be homogeneous only up to a certain point, after which they too are reduced to individuals, and if the process is continued they will lose their individuality.
* Observe that the term ‘ division ’ 1s here, as elsewhere, used in different senses.
(a) The Process of dividing a limited quantum into parts 18 called davesion.
Ixxx INTRODUCTION Tf a lump of gold is cut in two there will be a certain amount of wastage and distortion, the whole of the gold will not remain in the two parts, and they will not fit together perfectly. If the process is con- tinued, the wastage will tend to exhaust the gold (it does not destroy it, but it makes 2t unavailable); but this is due to the limitations of actual conditions. Wastage and distortion are imperfections, noi frne- tions of dividing. The perfect ‘cut’ would nerther waste nor distort: the two parts would comprise the whole of the gold, and they would fit together so as to leave no spaces between them and would take up no more room than the uncut lump. There would then be no trace of the ‘cut,’ but it would still ‘ mark off’ the parts, and if the pressure which held them together were removed it would determine where they fell apart and how much of the gold was in each part. So though the cut takes up no room, ‘and is no part either of the gold it divides or of any- thing else, it has a definite position and extension.
(b) The Parts into which a limited quantum is divided are called divisions, These divisions when added together make up the whole of the quantum of which they are parts, and if they are successively removed there will be less and less of the original quantum left.
‘There is yet another sense in which the term is used, viz.: (ce) The Boundaries whose function it is to mark of the parts are also called divisions, In practice these divisions ‘ use up’ some of the quantum the divide. For instance, the hedges which mark out fields, oceupy some of the ground, and if there were enough of them they would eventually fill up the whole of the ground; but this is not part of the function of a boundary; it is an imperfection due to physical conditions. The perfect boundary would tuke up no ruom, and removing it would neither increase nor diminish the available space.
Ixxxi INTRODUCTION Its existence, though not material, is none the less real; it as in fact what is meant by a mathematical surface.
Now suppose a lump of gold to be divided (by ‘perfect ’ cuts) into layers. These layers may be again divided into thinne: layers, and so on and so on. The layers are parts of the gold and when added together build up the whole of it. The ‘cuts’ or ‘surfaces,’ on the other hand, are not parts of the gold; and as we have seen they have no material existence.
‘There is no ἔπη, to the conceptual possibility of thus slicing up a solid continuum, for however long we go on there must always be a layer between one ‘ent’ and the next, which layer, however thin, must oceupy some room and is conceptually divisible into yet thinner layers. Thus we can never say that any two cuts (surfaces) are the nearest conceivable to each other, for they are always separated by a con- ceptually divisible layer.
A surface is conceptually divisible into less exten- sive surfaces, which exactly fit into each other, and together make up the whole of the original surface. The conceptual ‘ boundaries’ by which it is divided up are not themselves parts of the surface, and there- fore do not tend to exhaust it; they are in fact mathematical lines which are no more paris of a surface than a surface is part of a solid.
Two lines running side by side may be as close together as we choose to imagine them, but they will always be ‘spaced’: there is always a con- ceptually divisible strip of surface between them, and a strip, however narrow, is still a part of the surface, and the surface can be built up from and divided into these strips, but the dividing lines are no part of the surface; they simply mark off its parts.
Ixxxii INTRODUCTION Ycl a line, though it has no material existence, has length and position, 10 therefore does exist, though not inaterially. Its length can be divided into shorter lengths, the sum of which is identical with ihe whole length. The ‘ boundaries ᾿ of a line are not parts of jt; they simply serve to mark it off into parts. They are in fact mathematical points, which are na more parts of a line than a line is part of a surface or a surface part of a solid.
Two points may be as close together as we choose to imagine them, but they must always be spaced; there is always a conceptually divisible length of line between them, and the line can be built up from, and divided into these lengths. But the dividing points are no part of the line; they simply mark off its parts.
Yo sum up: A solid body which has a material existence is con- ceptually divisible by surfaces.
A surface has no material existence; it exists in virtue of having extension and position, and it is conceptually divisible by lines.
A line exists in virtue of having length and position, and it is conceptually divisible by points, A point exists in virtue of having position; it is indivisible.
Magnitude can be conceived as ‘that which is divisible’; that which is indivisible is not a mag- nitude. The terms ‘ divisible’ and ‘ indivisible’ belong to the class of opposites which he calls con- tradictories; they are mutually exclusive and there can be no intermediate state.* 5. Some opposites are not contradictories; hot and cold, for instance, or wel and dry, are opposites but they are nat contradictories. They arc extremes between which there are lxxxiii INTRODUCTION Points are indivisible and so are not magnitudes, Lines, surfaces, and solids are all divisible and are therefore all magnitudes. But they do not all belong to the same ‘ order ’ of maguitude.
The ‘order’ or ‘dimensions’ of a magnitude depends on the further divisibility of the boundaries by which it can be divided up.
A line is a one-dimensional magnitude, being divisible by in-divisible boundaries (points).
A surface is a two-dimensional magnitude, being divisible by one-dimensional boundaries (ines).
A solid is a three-dimensional magnitude, being divisible by two-dimensional boundaries (surfaces).
Now as there is no bridge between divisibles and indivisibles, nerther can there be any bridge between magnitudes of different dimensions. For if a one- dimensional magnitude could become two-dimen- sional, its indivisible boundaries would have to become divisible, which is impossible, as we have seen; neither can a two-dimensional be built up from one-dimensional magnitudes, for adding magnitudes together does not alter the dimensions of thei boundaries. It is easy to see that transformation is equally impossible the other way round.
The same reasoning will demonstrate that there is no bridge between two-dimensional and_ three- dimensional magnitudes.
Now, owing to our inability to visualize abstractions different degrees of hotness and coldness, or of wetness and dryness, ‘The same thing may vary from time to time between the opposite extremes, tending now to become hotter now colder. ‘The actuality of one extreme implies the poten- tiality of the other in the same subject. But with con- tradictory opposites the actuality of the one is the absolute negation both of the actuality and of the potentiality of the other, mm the same subject. See Bk. V. ch. iif.
Ixxxiv INTRODUCTION