6th Ennead | Plotinus | Book 3-1
Book 3
On the Kinds of Being (3)
We have now explained our conception of Reality [True Being] and considered how far it agrees with the teaching of Plato. We have still to investigate the opposed principle [the principle of Becoming].
There is the possibility that the genera posited for the Intellectual sphere will suffice for the lower also; possibly with these genera others will be required; again, the two series may differ entirely; or perhaps some of the sensible genera will be identical with their intellectual prototypes, and others different — “identical,” however, being understood to mean only analogous and in possession of a common name, as our results will make dear.
We must begin on these lines:
The subject of our discussion is the Sensible realm: Sensible Existence is entirely embraced by what we know as the Universe: our duty, then, would seem to be clear enough — to take this Universe and analyse its nature, classifying its constituent parts and arranging them by species. Suppose that we were making a division of speech: we should reduce its infinity to finite terms, and from the identity appearing in many instances evolve a unity, then another and another, until we arrived at some definite number; each such unit we should call a species if imposed upon individuals, a genus if imposed upon species. Thus, every species of speech — and similarly all phenomena — might be referred to a unity; speech — or element — might be predicated of them all.
This procedure however is as we have already shown, impossible in dealing with the subject of our present enquiry. New genera must be sought for this Universe-genera distinct from those of the Intellectual, inasmuch as this realm is different from that, analogous indeed but never identical, a mere image of the higher. True, it involves the parallel existence of Body and Soul, for the Universe is a living form: essentially however Soul is of the Intellectual and does not enter into the structure of what is called Sensible Being.
Remembering this fact, we must — however great the difficulty — exclude Soul from the present investigation, just as in a census of citizens, taken in the interests of commerce and taxation, we should ignore the alien population. As for the experiences to which Soul is indirectly subject in its conjunction with Body and by reason of Body’s presence, their classification must be attempted at a later stage, when we enquire into the details of Sensible Existence.
Our first observations must be directed to what passes in the Sensible realm for Substance. It is, we shall agree, only by analogy that the nature manifested in bodies is designated as Substance, and by no means because such terms as Substance or Being tally with the notion of bodies in flux; the proper term would be Becoming.
But Becoming is not a uniform nature; bodies comprise under the single head simples and composites, together with accidentals or consequents, these last themselves capable of separate classification.
Alternatively, Becoming may be divided into Matter and the Form imposed upon Matter. These may be regarded each as a separate genus, or else both may be brought under a single category and receive alike the name of Substance.
But what, we may ask, have Matter and Form in common? In what sense can Matter be conceived as a genus, and what will be its species? What is the differentia of Matter? In which genus, Matter or Form, are we to rank the composite of both? It may be this very composite which constitutes the Substance manifested in bodies, neither of the components by itself answering to the conception of Body: how, then, can we rank them in one and the same genus as the composite? How can the elements of a thing be brought within the same genus as the thing itself? Yet if we begin with bodies, our first-principles will be compounds.
Why not resort to analogy? Admitted that the classification of the Sensible cannot proceed along the identical lines marked out for the Intellectual: is there any reason why we should not for Intellectual-Being substitute Matter, and for Intellectual Motion substitute Sensible Form, which is in a sense the life and consummation of Matter? The inertia of Matter would correspond with Stability, while the Identity and Difference of the Intellectual would find their counterparts in the similarity and diversity which obtain in the Sensible realm.
But, in the first place, Matter does not possess or acquire Form as its life or its Act; Form enters it from without, and remains foreign to its nature. Secondly, Form in the Intellectual is an Act and a motion; in the Sensible Motion is different from Form and accidental to it: Form in relation to Matter approximates rather to Stability than to Motion; for by determining Matter’s indetermination it confers upon it a sort of repose.
In the higher realm Identity and Difference presuppose a unity at once identical and different: a thing in the lower is different only by participation in Difference and in relation to some other thing; Identity and Difference are here predicated of the particular, which is not, as in that realm, a posterior.
As for Stability, how can it belong to Matter, which is distorted into every variety of mass, receiving its forms from without, and even with the aid of these forms incapable of offspring.
This mode of division must accordingly be abandoned.
How then do we go to work?
Let us begin by distinguishing Matter, Form, the Mixture of both, and the Attributes of the Mixture. The Attributes may be subdivided into those which are mere predicates, and those serving also as accidents. The accidents may be either inclusive or included; they may, further, be classified as activities, experiences, consequents.
Matter will be found common to all substances, not however as a genus, since it has no differentiae — unless indeed differentiae be ascribed to it on the ground of its taking such various forms as fire and air.
It may be held that Matter is sufficiently constituted a genus by the fact that the things in which it appears hold it in common, or in that it presents itself as a whole of parts. In this sense Matter will indeed be a genus, though not in the accepted sense of the term. Matter, we may remark, is also a single element, if the element as such is able to constitute a genus.
Further, if to a Form be added the qualification “bound up with, involved in Matter,” Matter separates that Form from other Forms: it does not however embrace the whole of Substantial Form [as, to be the genus of Form, it must].
We may, again, regard Form as the creator of Substance and make the Reason-Principle of Substance dependent upon Form: yet we do not come thereby to an understanding of the nature of Substance.
We may, also, restrict Substance to the Composite. Matter and Form then cease to be substances. If they are Substance equally with the Composite, it remains to enquire what there is common to all three.
The “mere predicates” fall under the category of Relation: such are cause and element. The accidents included in the composite substances ire found to be either Quality or Quantity; those which are inclusive are of the nature of Space and Time. Activities and experiences comprise Motions; consequents Space and Time, which are consequents respectively of the Composites and of Motion.
The first three entities [Matter, Form, Composite] go, as we have discovered, to make a single common genus, the Sensible counterpart of Substance. Then follow in order Relation, Quantity, Quality, Time-during-which, Place-in-which, Motion; though, with Time and Space already included [under Relation], Time-during-which and Place-in-which become superfluous.
Thus we have five genera, counting the first three entities as one. If the first three are not massed into a unity, the series will be Matter, Form, Composite, Relation, Quantity, Quality, Motion. The last three may, again, be included in Relation, which is capable of bearing this wider extension.
What, then, we have to ask, is the constant element in the first three entities? What is it that identifies them with their inherent Substance?
Is it the capacity to serve as a base? But Matter, we maintain, serves as the base and seat of Form: Form, thus, will be excluded from the category of Substance. Again, the Composite is the base and seat of attributes: hence, Form combined with Matter will be the basic ground of Composites, or at any rate of all posteriors of the Composite — Quantity, Quality, Motion, and the rest.
But perhaps we may think Substance validly defined as that which is not predicated of anything else. White and black are predicated of an object having one or other of these qualities; double presupposes something distinct from itself — we refer not to the half, but to the length of wood of which doubleness is affirmed. father qua father is a predicate; knowledge is predicated of the subject in whom the knowledge exists; space is the limit of something, time the measure of something. Fire, on the other hand, is predicated of nothing; wood as such is predicated of nothing; and so with man, Socrates, and the composite substance in general.
Equally the Substantial Form is never a predicate, since it never acts as a modification of anything. Form is not an attribute of Matter hence, is not predicable of Matter it is simply a constituent of the Couplement. On the other hand, the Form of a man is not different from the man himself [and so does not “modify” the Couplement].
Matter, similarly, is part of a whole, and belongs to something else only as to a whole and not as to a separate thing of which it is predicated. White, on the contrary, essentially belongs to something distinct from itself.
We conclude that nothing belonging to something else and predicated of it can be Substance. Substance is that which belongs essentially to itself, or, in so far as it is a part of the differentiated object, serves only to complete the Composite. Each or either part of the Composite belongs to itself, and is only affirmed of the Composite in a special sense: only qua part of the whole is it predicated of something else; qua individual it is never in its essential nature predicated of an external.
It may be claimed as a common element in Matter, Form and the Couplement that they are all substrates. But the mode in which Matter is the substrate of Form is different from that in which Form and the Couplement are substrates of their modifications.
And is it strictly true to say that Matter is the substrate of Form? Form is rather the completion which Matter’s nature as pure potentiality demands.
Moreover, Form cannot be said to reside in Matter [as in a substrate]. When one thing combines with another to form a unity, the one does not reside in the other; both alike are substrates of a third: thus, Man [the Form] and a man [the Composite] are substrates of their experiences, and are prior to their activities and consequents.
Substance, then, is that from which all other things proceed and to which they owe their existence; it is the centre of passivity and the source of action.
These are incontrovertible facts in regard to the pseudo-substance of the Sensible realm: if they apply also in some degree to the True Substance of the Intellectual, the coincidence is, doubtless, to be attributed to analogy and ambiguity of terms.
We are aware that “the first” is so called only in relation to the things which come after it: “first” has no absolute significance; the first of one series is subsequent to the last of another. “Substrate,” similarly, varies in meaning [as applied to the higher and to the lower], while as for passivity its very existence in the Intellectual is questionable; if it does exist there, it is not the passivity of the Sensible.
It follows that the fact of “not being present in a subject [or substrate] is not universally true of Substance, unless presence in a subject be stipulated as not including the case of the part present in the whole or of one thing combining with another to form a distinct unity; a thing will not be present as in a subject in that with which it co-operates in the information of a composite substance. Form, therefore, is not present in Matter as in a subject, nor is Man so present in Socrates, since Man is part of Socrates.
Substance, then, is that which is not present in a subject. But if we adopt the definition “neither present in a subject nor predicated of a subject,” we must add to the second “subject” the qualification “distinct,” in order that we may not exclude the case of Man predicated of a particular man. When I predicate Man of Socrates, it is as though I affirmed, not that a piece of wood is white, but that whiteness is white; for in asserting that Socrates is a man, I predicate Man [the universal] of a particular man, I affirm Man of the manhood in Socrates; I am really saying only that Socrates is Socrates, or that this particular rational animal is an animal.
It may be objected that non-presence in a subject is not peculiar to Substance, inasmuch as the differentia of a substance is no more present in a subject than the substance itself; but this objection results from taking a part of the whole substance, such as “two-footed” in our example, and asserting that this part is not present in a subject: if we take, not “two-footed” which is merely an aspect of Substance, but “two-footedness” by which we signify not Substance but Quality, we shall find that this “two-footedness” is indeed present in a subject.
We may be told that neither Time nor Place is present in a subject. But if the definition of Time as the measure of Motion be regarded as denoting something measured, the “measure” will be present in Motion as in a subject, while Motion will be present in the moved: if, on the contrary, it be supposed to signify a principle of measurement, the “measure” will be present in the measurer.
Place is the limit of the surrounding space, and thus is present in that space.
The truth is, however, that the “Substance” of our enquiry may be apprehended in directly opposite ways: it may be determined by one of the properties we have been discussing, by more than one, by all at once, according as they answer to the notions of Matter, Form and the Couplement.
Granted, it may be urged, that these observations upon the nature of Substance are sound, we have not yet arrived at a statement of its essence. Our critic doubtless expects to see this “Sensible”: but its essence, its characteristic being, cannot be seen.
Do we infer that fire and water are not Substance? They certainly are not Substance because they are visible. Why, then? Because they possess Matter? No. Or Form? No. Nor because they involve a Couplement of Matter and Form. Then why are they Substance? By existing. But does not Quantity exist, and Quality? This anomaly is to be explained by an equivocation in the term “existence.”
What, then, is the meaning of “existence” as applied to fire, earth and the other elements? What is the difference between this existence and existence in the other categories? It is the difference between being simply — that which merely is — and being white. But surely the being qualified by “white” is the same as that having no qualification? It is not the same: the latter is Being in the primary sense, the former is Being only by participation and in a secondary degree. Whiteness added to Being produces a being white; Being added to whiteness produces a white being: thus, whiteness becomes an accident of Being, and Being an accident of whiteness.
The case is not equivalent to predicating white of Socrates and Socrates of white: for Socrates remains the same, though white would appear to have a different meaning in the two propositions, since in predicating Socrates of white we include Socrates in the [whole] sphere of whiteness, whereas in the proposition “Socrates is white” whiteness is plainly an attribute of Socrates.
“Being is white” implies, similarly, that Being possesses whiteness as an attribute, while in the proposition “whiteness is Being [or, is a being]” Being is regarded as comprising whiteness in its own extension.
In sum, whiteness has existence because it is bound up with Being and present in it: Being is, thus, the source of its existence. Being is Being on its own account, but the white is due to whiteness — not because it is “present in” whiteness, but because whiteness is present in it.
The Being of the Sensible resembles the white in not originating in itself. It must therefore be regarded as dependent for its being upon the Authentic Being, as white is dependent upon the Authentic Whiteness, and the Authentic Whiteness dependent for its whiteness upon participation in that Supreme Being whose existence is underived.
But Matter, it may be contended, is the source of existence to the Sensible things implanted in it. From what source, then, we retort, does Matter itself derive existence and being?
That Matter is not a Primary we have established elsewhere. If it be urged that other things can have no subsistence without being implanted in Matter, we admit the claim for Sensible things. But though Matter be prior to these, it is not thereby precluded from being posterior to many things-posterior, in fact, to all the beings of the Intellectual sphere. Its existence is but a pale reflection, and less complete than that of the things implanted in it. These are Reason-Principles and more directly derived from Being: Matter has of itself no Reason-Principle whatever; it is but a shadow of a Principle, a vain attempt to achieve a Principle.
But, our critic may pursue, Matter gives existence to the things implanted in it, just as Socrates gives existence to the whiteness implanted in himself? We reply that the higher being gives existence to the lower, the lower to the higher never.
But once concede that Form is higher in the scale of Being than Matter, and Matter can no longer be regarded as a common ground of both, nor Substance as a genus embracing Matter, Form and the Couplement. True, these will have many common properties, to which we have already referred, but their being [or existence] will nonetheless be different. When a higher being comes into contact with a lower, the lower, though first in the natural order, is yet posterior in the scale of Reality: consequently, if Being does not belong in equal degrees to Matter, to Form and to the Couplement, Substance can no longer be common to all three in the sense of being their genus: to their posteriors it will bear a still different relation, serving them as a common base by being bound up with all alike. Substance, thus, resembles life, dim here, clearer there, or portraits of which one is an outline, another more minutely worked. By measuring Being by its dim manifestation and neglecting a fuller revelation elsewhere, we may come to regard this dim existence as a common ground.
But this procedure is scarcely permissible. Every being is a distinct whole. The dim manifestation is in no sense a common ground, just as there is no common ground in the vegetal, the sensory and the intellectual forms of life.
We conclude that the term “Being” must have different connotations as applied to Matter, to Form and to both conjointly, in spite of the single source pouring into the different streams.
Take a second derived from a first and a third from the second: it is not merely that the one will rank higher and its successor be poorer and of lower worth; there is also the consideration that, even deriving from the same source, one thing, subjected in a certain degree to fire, will give us an earthen jar, while another, taking less of the heat, does not produce the jar.
Perhaps we cannot even maintain that Matter and Form are derived from a single source; they are clearly in some sense different.
The division into elements must, in short, be abandoned, especially in regard to Sensible Substance, known necessarily by sense rather than by reason. We must no longer look for help in constituent parts, since such parts will not be substances, or at any rate not sensible substances.
Our plan must be to apprehend what is constant in stone, earth, water and the entities which they compose — the vegetal and animal forms, considered purely as sensibles — and to confine this constant within a single genus. Neither Matter nor Form will thus be overlooked, for Sensible Substance comports them; fire and earth and the two intermediaries consist of Matter and Form, while composite things are actually many substances in one. They all, moreover, have that common property which distinguishes them from other things:
serving as subjects to these others, they are never themselves present in a subject nor predicated of any other thing. Similarly, all the characteristics which we have ascribed to Substance find a place in this classification.
But Sensible Substance is never found apart from magnitude and quality: how then do we proceed to separate these accidents? If we subtract them — magnitude, figure, colour, dryness, moistness — what is there left to be regarded as Substance itself? All the substances under consideration are, of course, qualified.
There is, however, something in relation to which whatever turns Substance into qualified Substance is accidental: thus, the whole of fire is not Substance, but only a part of it — if the term “part” be allowed.
What then can this “part” be? Matter may be suggested. But are we actually to maintain that the particular sensible substance consists of a conglomeration of qualities and Matter, while Sensible Substance as a whole is merely the sum of these coagulations in the uniform Matter, each one separately forming a quale or a quantum or else a thing of many qualities? Is it true to say that everything whose absence leaves subsistence incomplete is a part of the particular substance, while all that is accidental to the substance already existent takes independent rank and is not submerged in the mixture which constitutes this so-called substance?
I decline to allow that whatever combines in this way with anything else is Substance if it helps to produce a single mass having quantity and quality, whereas taken by itself and divorced from this complementary function it is a quality: not everything which composes the amalgam is Substance, but only the amalgam as a whole.
And let no one take exception on the ground that we produce Sensible Substance from non-substances. The whole amalgam itself is not True Substance; it is merely an imitation of that True Substance which has Being apart from its concomitants, these indeed being derived from it as the possessor of True Being. In the lower realm the case is different: the underlying ground is sterile, and from its inability to produce fails to attain to the status of Being; it remains a shadow, and on this shadow is traced a sketch — the world of Appearance.
So much for one of the genera — the “Substance,” so called, of the Sensible realm.
But what are we to posit as its species? how divide this genus?
The genus as a whole must be identified with body. Bodies may be divided into the characteristically material and the organic: the material bodies comprise fire, earth, water, air; the organic the bodies of plants and animals, these in turn admitting of formal differentiation.
The next step is to find the species of earth and of the other elements, and in the case of organic bodies to distinguish plants according to their forms, and the bodies of animals either by their habitations — on the earth, in the earth, and similarly for the other elements — or else as light, heavy and intermediate. Some bodies, we shall observe, stand in the middle of the universe, others circumscribe it from above, others occupy the middle sphere: in each case we shall find bodies different in shape, so that the bodies of the living beings of the heavens may be differentiated from those of the other elements.
Once we have classified bodies into the four species, we are ready to combine them on a different principle, at the same time intermingling their differences of place, form and constitution; the resultant combinations will be known as fiery or earthy on the basis of the excess or predominance of some one element.
The distinction between First and Second Substances, between Fire and a given example of fire, entails a difference of a peculiar kind — the difference between universal and particular. This however is not a difference characteristic of Substance; there is also in Quality the distinction between whiteness and the white object, between grammar and some particular grammar.
The question may here be asked: “What deficiency has grammar compared with a particular grammar, and science as a whole in comparison with a science?” Grammar is certainly not posterior to the particular grammar: on the contrary, the grammar as in you depends upon the prior existence of grammar as such: the grammar as in you becomes a particular by the fact of being in you; it is otherwise identical with grammar the universal.
Turn to the case of Socrates: it is not Socrates who bestows manhood upon what previously was not Man, but Man upon Socrates; the individual man exists by participation in the universal.
Besides, Socrates is merely a particular instance of Man; this particularity can have no effect whatever in adding to his essential manhood.
We may be told that Man [the universal] is Form alone, Socrates Form in Matter. But on this very ground Socrates will be less fully Man than the universal; for the Reason-Principle will be less effectual in Matter. If, on the contrary, Man is not determined by Form alone, but presupposes Matter, what deficiency has Man in comparison with the material manifestation of Man, or the Reason-Principle in isolation as compared with its embodiment in a unit of Matter?
Besides, the more general is by nature prior; hence, the Form-Idea is prior to the individual: but what is prior by nature is prior unconditionally. How then can the Form take a lower rank? The individual, it is true, is prior in the sense of being more readily accessible to our cognisance; this fact, however, entails no objective difference.
Moreover, such a difference, if established, would be incompatible with a single Reason-Principle of Substance; First and Second Substance could not have the same Principle, nor be brought under a single genus.
Another method of division is possible: substances may be classed as hot-dry, dry-cold, cold-moist, or however we choose to make the coupling. We may then proceed to the combination and blending of these couples, either halting at that point and going no further than the compound, or else subdividing by habitation — on the earth, in the earth — or by form and by the differences exhibited by living beings, not qua living, but in their bodies viewed as instruments of life.
Differentiation by form or shape is no more out of place than a division based on qualities — heat, cold and the like. If it be objected that qualities go to make bodies what they are, then, we reply, so do blendings, colours, shapes. Since our discussion is concerned with Sensible Substance, it is not strange that it should turn upon distinctions related to sense-perception: this Substance is not Being pure and simple, but the Sensible Being which we call the Universe.
We have remarked that its apparent subsistence is in fact an assemblage of Sensibles, their existence guaranteed to us by sense-perception. But since their combination is unlimited, our division must be guided by the Form-Ideas of living beings, as for example the Form-Idea of Man implanted in Body; the particular Form acts as a qualification of Body, but there is nothing unreasonable in using qualities as a basis of division.
We may be told that we have distinguished between simple and composite bodies, even ranking them as opposites. But our distinction, we reply, was between material and organic bodies and raised no question of the composite. In fact, there exists no means of opposing the composite to the simple; it is necessary to determine the simples in the first stage of division, and then, combining them on the basis of a distinct underlying principle, to differentiate the composites in virtue of their places and shapes, distinguishing for example the heavenly from the earthly.
These observations will suffice for the Being [Substance], or rather the Becoming, which obtains in the Sensible realm.
Passing to Quantity and the quantum, we have to consider the view which identifies them with number and magnitude on the ground that everything quantitative is numbered among Sensible things or rated by the extension of its substrate: we are here, of course, discussing not Quantity in isolation, but that which causes a piece of wood to be three yards long and gives the five in “five horses,”
Now we have often maintained that number and magnitude are to be regarded as the only true quantities, and that Space and Time have no right to be conceived as quantitative: Time as the measure of Motion should be assigned to Relation, while Space, being that which circumscribes Body, is also a relative and falls under the same category; though continuous, it is, like Motion, not included in Quantity.
On the other hand, why do we not find in the category of Quantity “great” and “small”? It is some kind of Quantity which gives greatness to the great; greatness is not a relative, though greater and smaller are relatives, since these, like doubleness, imply an external correlative.
What is it, then, which makes a mountain small and a grain of millet large? Surely, in the first place, “small” is equivalent to “smaller.” It is admitted that the term is applied only to things of the same kind, and from this admission we may infer that the mountain is “smaller” rather than “small,” and that the grain of millet is not large in any absolute sense but large for a grain of millet. In other words, since the comparison is between things of the same kind, the natural predicate would be a comparative.
Again, why is not beauty classed as a relative? Beauty, unlike greatness, we regard as absolute and as a quality; “more beautiful” is the relative. Yet even the term “beautiful” may be attached to something which in a given relation may appear ugly: the beauty of man, for example, is ugliness when compared with that of the gods; “the most beautiful of monkeys,” we may quote, “is ugly in comparison with any other type.” Nonetheless, a thing is beautiful in itself; as related to something else it is either more or less beautiful.
Similarly, an object is great in itself, and its greatness is due, not to any external, but to its own participation in the Absolute Great.
Are we actually to eliminate the beautiful on the pretext that there is a more beautiful? No more then must we eliminate the great because of the greater: the greater can obviously have no existence whatever apart from the great, just as the more beautiful can have no existence without the beautiful.
It follows that we must allow contrariety to Quantity: whenever we speak of great and small, our notions acknowledge this contrariety by evolving opposite images, as also when we refer to many and few; indeed, “few” and “many” call for similar treatment to “small” and “great.”
“Many,” predicated of the inhabitants of a house, does duty for “more”: “few” people are said to be in the theatre instead of “less.”
“Many,” again, necessarily involves a large numerical plurality. This plurality can scarcely be a relative; it is simply an expansion of number, its contrary being a contraction.
The same applies to the continuous [magnitude], the notion of which entails prolongation to a distant point.
Quantity, then, appears whenever there is a progression from the unit or the point: if either progression comes to a rapid halt, we have respectively “few” and “small”; if it goes forward and does not quickly cease, “many” and “great.”
What, we may be asked, is the limit of this progression? What, we retort, is the limit of beauty, or of heat? Whatever limit you impose, there is always a “hotter”; yet “hotter” is accounted a relative, “hot” a pure quality.
In sum, just as there is a Reason-Principle of Beauty, so there must be a Reason-Principle of greatness, participation in which makes a thing great, as the Principle of beauty makes it beautiful.
To judge from these instances, there is contrariety in Quantity. Place we may neglect as not strictly coming under the category of Quantity; if it were admitted, “above” could only be a contrary if there were something in the universe which was “below”: as referring to the partial, the terms “above” and “below” are used in a purely relative sense, and must go with “right” and “left” into the category of Relation.
Syllable and discourse are only indirectly quantities or substrates of Quantity; it is voice that is quantitative: but voice is a kind of Motion; it must accordingly in any case [quantity or no quantity] be referred to Motion, as must activity also.
It has been remarked that the continuous is effectually distinguished from the discrete by their possessing the one a common, the other a separate, limit.
The same principle gives rise to the numerical distinction between odd and even; and it holds good that if there are differentiae found in both contraries, they are either to be abandoned to the objects numbered, or else to be considered as differentiae of the abstract numbers, and not of the numbers manifested in the sensible objects. If the numbers are logically separable from the objects, that is no reason why we should not think of them as sharing the same differentiae.
But how are we to differentiate the continuous, comprising as it does line, surface and solid? The line may be rated as of one dimension, the surface as of two dimensions, the solid as of three, if we are only making a calculation and do not suppose that we are dividing the continuous into its species; for it is an invariable rule that numbers, thus grouped as prior and posterior, cannot be brought into a common genus; there is no common basis in first, second and third dimensions. Yet there is a sense in which they would appear to be equal — namely, as pure measures of Quantity: of higher and lower dimensions, they are not however more or less quantitative.
Numbers have similarly a common property in their being numbers all; and the truth may well be, not that One creates two, and two creates three, but that all have a common source.
Suppose, however, that they are not derived from any source whatever, but merely exist; we at any rate conceive them as being derived, and so may be assumed to regard the smaller as taking priority over the greater: yet, even so, by the mere fact of their being numbers they are reducible to a single type.
What applies to numbers is equally true of magnitudes; though here we have to distinguish between line, surface and solid — the last also referred to as “body” — in the ground that, while all are magnitudes, they differ specifically.
It remains to enquire whether these species are themselves to be divided: the line into straight, circular, spiral; the surface into rectilinear and circular figures; the solid into the various solid figures — sphere and polyhedra: whether these last should be subdivided, as by the geometers, into those contained by triangular and quadrilateral planes: and whether a further division of the latter should be performed.
How are we to classify the straight line? Shall we deny that it is a magnitude?
The suggestion may be made that it is a qualified magnitude. May we not, then, consider straightness as a differentia of “line”? We at any rate draw on Quality for differentiae of Substance.
The straight line is, thus, a quantity plus a differentia; but it is not on that account a composite made up of straightness and line: if it be a composite, the composite possesses a differentiae of its own.
But [if the line is a quantity] why is not the product of three lines included in Quantity? The answer is that a triangle consists not merely of three lines but of three lines in a particular disposition, a quadrilateral of four lines in a particular disposition: even the straight line involves disposition as well as quantity.
Holding that the straight line is not mere quantity, we should naturally proceed to assert that the line as limited is not mere quantity, but for the fact that the limit of a line is a point, which is in the same category, Quantity. Similarly, the limited surface will be a quantity, since lines, which have a far better right than itself to this category, constitute its limits. With the introduction of the limited surface — rectangle, hexagon, polygon — into the category of Quantity, this category will be brought to include every figure whatsoever.
If however by classing the triangle and the rectangle as qualia we propose to bring figures under Quality, we are not thereby precluded from assigning the same object to more categories than one: in so far as it is a magnitude — a magnitude of such and such a size — it will belong to Quantity; in so far as it presents a particular shape, to Quality.
It may be urged that the triangle is essentially a particular shape. Then what prevents our ranking the sphere also as a quality?
To proceed on these lines would lead us to the conclusion that geometry is concerned not with magnitudes but with Quality. But this conclusion is untenable; geometry is the study of magnitudes. The differences of magnitudes do not eliminate the existence of magnitudes as such, any more than the differences of substances annihilate the substances themselves.
Moreover, every surface is limited; it is impossible for any surface to be infinite in extent.
Again, when I find Quality bound up with Substance, I regard it as substantial quality: I am not less, but far more, disposed to see in figures or shapes [qualitative] varieties of Quantity. Besides, if we are not to regard them as varieties of magnitude, to what genus are we to assign them?
Suppose, then, that we allow differences of magnitude; we commit ourselves to a specific classification of the magnitudes so differentiated.