(13) We turn to ask why Quantity is not included among the primary genera, and Quality also.

Quantity is not among the primaries, because these are permanently associated with Being. Motion is bound up with Actual Being , since it is its life; with Motion, Stability too gained its foothold in Reality; with these are associated Difference and Identity, so that they also are seen in conjunction with Being. But number is a posterior. It is posterior not only with regard to these genera but also within itself; in number the posterior is divided from the prior; this is a sequence in which the posteriors are latent in the priors . Number therefore cannot be included among the primary genera; whether it constitutes a genus at all remains to be examined.

Magnitude is in a still higher degree posterior and composite, for it contains within itself number, line and surface. Now if continuous magnitude derives its quantity from number, and number is not a genus, how can magnitude hold that status? Besides, magnitudes, like numbers, admit of priority and posteriority.

If, then, Quantity be constituted by a common element in both number and magnitude, we must ascertain the nature of this common element, and consider it, once discovered, as a posterior genus, not as one of the Primaries: thus failing of primary status, it must be related, directly or indirectly, to one of the Primaries.

We may take it as clear that it is the nature of Quantity to indicate a certain quantum, and to measure the quantum of the particular; Quantity is moreover, in a sense, itself a quantum. But if the quantum is the common element in number and magnitude, either we have number as a primary with magnitude derived from it, or else number must consist of a blending of Motion and Stability, while magnitude will be a form of Motion or will originate in Motion, Motion going forth to infinity and Stability creating the unit by checking that advance.

But the problem of the origin of number and magnitude, or rather of how they subsist and are conceived, must be held over. It may, thus, be found that number is among the primary genera, while magnitude is posterior and composite; or that number belongs to the genus Stability, while magnitude must be consigned to Motion. But we propose to discuss all this at a later stage.

(18) The Ideal Principle possessing the Intellection of Magnitude- assuming that this Intellection is of such power as not merely to subsist within itself but to be urged outward as it were by the intensity of its life- will necessarily realize itself in a Kind not having its being in the Intellective Principle, not previously possessing the Idea of Magnitude or any trace of that Idea or any other.

What then will it produce by virtue of that power?

Not horse or cow: these are the product of other Ideas.

No: this Principle comes from the source of Magnitude and therefore Matter can have no extension, in which to harbour the Magnitude of the Principle, but can take in only its reflected appearance.

To the thing which does not enjoy Magnitude in the sense of having mass-extension in its own substance and parts, the only possibility is that it present some partial semblance of Magnitude, such as being continuous, not here and there and everywhere, that its parts be related within it and ungapped. An adequate reflection of a great mass cannot be produced in a small space- mere size prevents- but the greater, pursuing the hope of that full self-presentment, makes progress towards it and brings about a nearer approach to adequate mirroring in the parallel from which it can never withhold its radiation: thus it confers Magnitude upon that which has none and cannot even muster up the appearance of having any, and the visible resultant exhibits the Magnitude of mass.

Matter, then, wears Magnitude as a dress thrown about it by its association with that Absolute Magnitude to whose movement it must answer; but it does not, for that, change its Kind; if the Idea which has clothed it were to withdraw, it would once again be what it permanently is, what it is by its own strength, or it would have precisely the Magnitude lent to it by any other form that happens to be present in it.

The Soul- containing the Ideal Principles of Real-Beings, and itself an Ideal Principle- includes all in concentration within itself, just as the Ideal Principle of each particular entity is complete and self-contained: it, therefore, sees these principles of sensible things because they are turned, as it were, towards it and advancing to it: but it cannot harbour them in their plurality, for it cannot depart from its Kind; it sees them, therefore, stripped of Mass. Matter, on the contrary, destitute of resisting power since it has no Act of its own and is a mere shadow, can but accept all that an active power may choose to send. In what is thus sent, from the Reason-Principle in the Intellectual Realm, there is already contained a degree of the partial object that is to be formed: in the image-making impulse within the Reason-Principle there is already a step or we may put it that the downward movement from the Reason-Principle is a first form of the partial: utter absence of partition would mean no movement but repose. Matter cannot be the home of all things in concentration as the Soul is: if it were so, it would belong to the Intellective Sphere. It must be all-recipient but not in that partless mode. It is to be the Place of all things, and it must therefore extend universally, offer itself to all things, serve to all interval: thus it will be a thing unconfined to any moment but laid out in submission to all that is to be.

But would we not expect that some one particularized form should occupy Matter and so exclude such others as are not able to enter into combination?

No: for there is no first Idea except the Ideal Principle of the Universe- and, by this Idea, Matter is all things at once and of the particular thing in its parts- for the Matter of a living being is disparted according to the specific parts of the organism: if there were no such partition nothing would exist but the Reason-Principle.

(13) 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 polyhedANSWER: 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.

(17) Nor can we, on the other hand, think that matter is simply Absolute Magnitude.

Magnitude is not, like Matter, a receptacle; it is an Ideal-Principle: it is a thing standing apart to itself, not some definite Mass. The fact is that the self-gathered content of the Intellectual Principle or of the All-Soul, desires expansion : in its images- aspiring and moving towards it and eagerly imitating its act- is vested a similar power of reproducing their states in their own derivatives. The Magnitude latent in the expansive tendency of the Image-making phase runs forth into the Absolute Magnitude of the Universe; this in turn enlists into the process the spurious magnitude of Matter: the content of the Supreme, thus, in virtue of its own prior extension enables Matter- which never possesses a content- to exhibit the appearance of Magnitude. It must be understood that spurious Magnitude consists in the fact that a thing not possessing actual Magnitude strains towards it and has the extension of that straining. All that is Real Being gives forth a reflection of itself upon all else; every Reality, therefore, has Magnitude which by this process is communicated to the Universe.

The Magnitude inherent in each Ideal-Principle- that of a horse or of anything else- combines with Magnitude the Absolute with the result that, irradiated by that Absolute, Matter entire takes Magnitude and every particle of it becomes a mass; in this way, by virtue at once of the totality of Idea with its inherent magnitude and of each several specific Idea, all things appear under mass; Matter takes on what we conceive as extension; it is compelled to assume a relation to the All and, gathered under this Idea and under Mass, to be all things- in the degree in which the operating power can lead the really nothing to become all.

By the conditions of Manifestation, colour rises from non-colour . Quality, known by the one name with its parallel in the sphere of Primals, rises, similarly, from non-quality: in precisely the same mode, the Magnitude appearing upon Matter rises from non-Magnitude or from that Primal which is known to us by the same name; so that material things become visible through standing midway between bare underlie and Pure Idea. All is perceptible by virtue of this origin in the Intellectual Sphere but all is falsity since the base in which the manifestation takes place is a non-existent.

Particular entities thus attain their Magnitude through being drawn out by the power of the Existents which mirror themselves and make space for themselves in them. And no violence is required to draw them into all the diversity of Shapes and Kinds because the phenomenal All exists by Matter and because each several Idea, moreover, draws Matter its own way by the power stored within itself, the power it holds from the Intellectual Realm. Matter is manifested in this sphere as Mass by the fact that it mirrors the Absolute Magnitude; Magnitude here is the reflection in the mirror. The Ideas meet all of necessity in Matter : and Matter, both as one total thing and in its entire scope, must submit itself, since it is the Material of the entire Here, not of any one determined thing: what is, in its own character, no determined thing may become determined by an outside force- though, in becoming thus determined, it does not become the definite thing in question, for thus it would lose its own characteristic indetermination.

(9) But how can we conceive a thing having existence without having magnitude?

We have only to think of things whose identity does not depend on their quantity- for certainly magnitude can be distinguished from existence as can many other forms and attributes.

In a word, every unembodied Kind must be classed as without quantity, and Matter is unembodied.

Besides quantitativeness itself does not possess quantity, which belongs only to things participating in it, a consideration which shows that Quantitativeness is an Idea-Principle. A white object becomes white by the presence of whiteness; what makes an organism white or of any other variety of colour is not itself a specific colour but, so to speak, a specific Reason-Principle: in the same way what gives an organism a certain bulk is not itself a thing of magnitude but is Magnitude itself, the abstract Absolute, or the Reason-Principle.

This Magnitude-Absolute, then, enters and beats the Matter out into Magnitude?

Not at all: the Matter was not previously shrunken small: there was no littleness or bigness: the Idea gives Magnitude exactly as it gives every quality not previously present.

(1) Seen from a distance, objects appear reduced and close together, however far apart they be: within easy range, their sizes and the distances that separate them are observed correctly.

Distant objects show in this reduction because they must be drawn together for vision and the light must be concentrated to suit the size of the pupil; besides, as we are placed farther and farther away from the material mass under observation, it is more and more the bare form that reaches us, stripped, so to speak, of magnitude as of all other quality.

Or it may be that we appreciate the magnitude of an object by observing the salience and recession of its several parts, so that to perceive its true size we must have it close at hand.

Or again, it may be that magnitude is known incidentally from the observation of colour. With an object at hand we know how much space is covered by the colour; at a distance, only that something is coloured, for the parts, quantitatively distinct among themselves, do not give us the precise knowledge of that quantity, the colours themselves reaching us only in a blurred impression.

What wonder, then, if size be like sound- reduced when the form reaches us but faintly- for in sound the hearing is concerned only about the form; magnitude is not discerned except incidentally.

Well, in hearing magnitude is known incidentally; but how? Touch conveys a direct impression of a visible object; what gives us the same direct impression of an object of hearing?

The magnitude of a sound is known not by actual quantity but by degree of impact, by intensity- and this in no indirect knowledge; the ear appreciates a certain degree of force, exactly as the palate perceives by no indirect knowledge, a certain degree of sweetness. But the true magnitude of a sound is its extension; this the hearing may define to itself incidentally by deduction from the degree of intensity but not to the point of precision. The intensity is merely the definite effect at a particular spot; the magnitude is a matter of totality, the sum of space occupied.

Still the colours seen from a distance are faint; but they are not small as the masses are.

True; but there is the common fact of diminution. There is colour with its diminution, faintness; there is magnitude with its diminution, smallness; and magnitude follows colour diminishing stage by stage with it.

But, the phenomenon is more easily explained by the example of things of wide variety. Take mountains dotted with houses, woods and other land-marks; the observation of each detail gives us the means of calculating, by the single objects noted, the total extent covered: but, where no such detail of form reaches us, our vision, which deals with detail, has not the means towards the knowledge of the whole by measurement of any one clearly discerned magnitude. This applies even to objects of vision close at hand: where there is variety and the eye sweeps over all at one glance so that the forms are not all caught, the total appears the less in proportion to the detail which has escaped the eye; observe each single point and then you can estimate the volume precisely. Again, magnitudes of one colour and unbroken form trick the sense of quantity: the vision can no longer estimate by the particular; it slips away, not finding the stand-by of the difference between part and part.

It was the detail that prevented a near object deceiving our sense of magnitude: in the case of the distant object, because the eye does not pass stage by stage through the stretch of intervening space so as to note its forms, therefore it cannot report the magnitude of that space.

(4) "Dialecticians of the day," replied the Spirit of the River, "all say that the infinitesimally small has no form, and that the infinitesimally great is beyond all measurement. Is that so?" "If we regard greatness as compared with that which is small," said the Spirit of the Ocean, "there is no limit to it; and if we regard smallness as compared with that which is great, it eludes our sight. The infinitesimal is a subdivision of the small; the colossal is an extension of the great. In this sense the two fall into different categories. "Both small and great things must equally possess form. The mind cannot picture to itself a thing without form, nor conceive a form of unlimited dimensions. The greatness of anything may be a topic of discussion, or the smallness of anything may be mentally realized. But that which can be neither a topic of discussion nor be realized mentally, can be neither great nor small.

(12) It is the corporeal, then, that demands magnitude: the Ideal-Forms of body are Ideas installed in Mass.

But these Ideas enter, not into Magnitude itself but into some subject that has been brought to Magnitude. For to suppose them entering into Magnitude and not into Matter- is to represent them as being either without Magnitude and without Real-Existence or not Ideal-Forms but Reason-Principles whose sphere could only be Soul; at this, there would be no such thing as body

The multiplicity here must be based upon some unity which, since it has been brought to Magnitude, must be, itself, distinct from Magnitude. Matter is the base of Identity to all that is composite: once each of the constituents comes bringing its own Matter with it, there is no need of any other base. No doubt there must be a container, as it were a place, to receive what is to enter, but Matter and even body precede place and space; the primal necessity, in order to the existence of body, is Matter.

There is no force in the suggestion that, since production and act are immaterial, corporeal entities also must be immaterial.

Bodies are compound, actions not. Further, Matter does in some sense underlie action; it supplies the substratum to the doer: it is permanently within him though it does not enter as a constituent into the act where, indeed, it would be a hindrance. Doubtless, one act does not change into another- as would be the case if there were a specific Matter of actions- but the doer directs himself from one act to another so that he is the Matter, himself, to his varying actions.

Matter, in sum, is necessary to quality and to quantity, and, therefore, to body.

It is, thus, no name void of content; we know there is such a base, invisible and without bulk though it be.

If we reject it, we must by the same reasoning reject qualities and mass: for quality, or mass, or any such entity, taken by itself apart, might be said not to exist. But these do exist, though in an obscure existence: there is much less ground for rejecting Matter, however it lurk, discerned by none of the senses.

It eludes the eye, for it is utterly outside of colour: it is not heard, for it is no sound: it is no flavour or savour for nostrils or palate: can it, perhaps, be known to touch? No: for neither is it corporeal; and touch deals with body, which is known by being solid, fragile, soft, hard, moist, dry- all properties utterly lacking in Matter.

It is grasped only by a mental process, though that not an act of the intellective mind but a reasoning that finds no subject; and so it stands revealed as the spurious thing it has been called. No bodiliness belongs to it; bodiliness is itself a phase of Reason-Principle and so is something different from Matter, as Matter, therefore, from it: bodiliness already operative and so to speak made concrete would be body manifest and not Matter unelaborated.

(16) But here we may be questioned about these numbers which we describe as the primal and authentic:

"Where do you place these numbers, in what genus among Beings? To everyone they seem to come under Quantity and you have certainly brought Quantity in, where you say that discrete Quantity equally with the continuous holds place among Beings; but you go on to say that there are the numbers belonging to the Firsts and then talk of other numbers quite distinct, those of reckoning; tell us how you arrange all this, for there is difficulty here. And then, the unity in sense-things- is that a quantity or is quantity here just so many units brought together, the unity being the starting-point of quantity but not quantity itself? And, if the starting-point, is it a kindred thing or of another genus? All this you owe it to us to make clear."

Be it so; we begin by pointing out a distinction:

You take one thing with another- for we must first deal with objects of sense- a dog and a man, or two men; or you take a group and affirm ten, a decad of men: in this case the number affirmed is not a Reality, even as Reality goes in the sphere of sense, but is purely Quantity: similarly when you resolve into units, breaking up the decad, those units are your principle of Quantity since the single individual is not a unity absolute.

But the case is different when you consider one man in himself and affirm a certain number, duality, for example, in that he is at once living and reasoning.

By this analysis and totalling, you get quantity; but there are two objects under consideration and each of these is one; each of the unities contributes to the complete being and the oneness is inherent in each; this is another kind of number; number essential; even the duality so formed is no posterior; it does not signify a quantity apart from the thing but the quantity in the essence which holds the thing together. The number here is no mere result of your detailing; the things exist of themselves and are not brought together by your reckoning, but what has it to do with essential reality that you count one man in with another? There is here no resultant unity such as that of a choir- the decad is real only to you who count the ten; in the ten of your reckoning there cannot be a decad without a unitary basis; it is you that make the ten by your counting, by fixing that tenness down to quantity; in choir and army there is something more than that, something not of your placing.

But how do you come to have a number to place?

The Number inherent apart from any enumeration has its own manner of being, but the other, that resulting upon the appearance of an external to be appraised by the Number within yourself, is either an Act of these inherent numbers or an Act in accordance with them; in counting we produce number and so bring quantity into being just as in walking we bring a certain movement into being.

But what of that "Number within us having its own manner of being"?

It is the Number of our essence. "Our essence" we read "partakes of Number and harmony and, also, is Number and harmony." "Neither body nor magnitude," someone says: soul, then, is Number since it is essence. The number belonging to body is an essence of the order of body; the number belonging to soul constitutes the essences of souls.

In the Intellectuals, all, if the Absolute Living-Form, there is a multiple- a triad, let us say- that Triad of the Living-Form is of the nature of essence: and the Triad prior to any living thing, Triad in the realm of Being, is a principle of essence.

When you enumerate two things- say, animal and beauty- each of these remains one thing; the number is your production; it lay within yourself; it is you that elaborate quantity, here the dyad. But when you declare virtue to be a Tetrad, you are affirming a Tetrad which does actually exist; the parts, so to speak, make one thing; you are taking as the object of your act a Unity- Tetrad to which you accommodate the Tetrad within yourself.

(4) We are told that number is Quantity in the primary sense, number together with all continuous magnitude, space and time: these are the standards to which all else that is considered as Quantity is referred, including motion which is Quantity because its time is quantitative- though perhaps, conversely, the time takes its continuity from the motion.

If it is maintained that the continuous is a Quantity by the fact of its continuity, then the discrete will not be a Quantity. If, on the contrary, the continuous possesses Quantity as an accident, what is there common to both continuous and discrete to make them quantities?

Suppose we concede that numbers are quantities: we are merely allowing them the name of quantity; the principle which gives them this name remains obscure.

On the other hand, line and surface and body are not called quantities; they are called magnitudes: they become known as quantities only when they are rated by number-two yards, three yards. Even the natural body becomes a quantity when measured, as does the space which it occupies; but this is quantity accidental, not quantity essential; what we seek to grasp is not accidental quantity but Quantity independent and essential, Quantity-Absolute. Three oxen is not a quantity; it is their number, the three, that is Quantity; for in three oxen we are dealing with two categories. So too with a line of a stated length, a surface of a given area; the area will be a quantity but not the surface, which only comes under that category when it constitutes a definite geometric figure.

Are we then to consider numbers, and numbers only, as constituting the category of Quantity? If we mean numbers in themselves, they are substances, for the very good reason that they exist independently. If we mean numbers displayed in the objects participant in number, the numbers which give the count of the objects- ten horses or ten oxen, and not ten units- then we have a paradoxical result: first, the numbers in themselves, it would appear, are substances but the numbers in objects are not; and secondly, the numbers inhere in the objects as measures , yet as standing outside the objects they have no measuring power, as do rulers and scales. If however their existence is independent, and they do not inhere in the objects, but are simply called in for the purpose of measurement, the objects will be quantities only to the extent of participating in Quantity.

So with the numbers themselves: how can they constitute the category of Quantity? They are measures; but how do measures come to be quantities or Quantity? Doubtless in that, existing as they do among the Existents and not being adapted to any of the other categories, they find their place under the influence of verbal suggestion and so are referred to the so-called category of Quantity. We see the unit mark off one measurement and then proceed to another; and number thus reveals the amount of a thing, and the mind measures by availing itself of the total figure.

It follows that in measuring it is not measuring essence; it pronounces its "one" or "two," whatever the character of the objects, even summing contraries. It does not take count of condition- hot, handsome; it simply notes how many.

Number then, whether regarded in itself or in the participant objects, belongs to the category of Quantity, but the participant objects do not. "Three yards long" does not fall under the category of Quantity, but only the three.

Why then are magnitudes classed as quantities? Not because they are so in the strict sense, but because they approximate to Quantity, and because objects in which magnitudes inhere are themselves designated as quantities. We call a thing great or small from its participation in a high number or a low. True, greatness and smallness are not claimed to be quantities, but relations: but it is by their apparent possession of quantity that they are thought of as relations. All this, however, needs more careful examination.

In sum, we hold that there is no single genus of Quantity. Only number is Quantity, the rest quantities only in a secondary degree. We have therefore not strictly one genus, but one category grouping the approximate with the primary and the secondary.

We have however to enquire in what sense the abstract numbers are substances. Can it be that they are also in a manner quantitative? Into whatever category they fall, the other numbers can have nothing in common with them but the name. 5. Speech, time, motion- in what sense are these quantities?

Let us begin with speech. It is subject to measurement, but only in so far as it is sound; it is not a quantity in its essential nature, which nature is that it be significant, as noun and verb are significant. The air is its Matter, as it is Matter to verb and noun, the components of speech.

To be more precise, we may define speech as an impact , though it is not so much the impact as the impression which the impact produces and which, as it were, imposes Form . Speech, thus, is rather an action than a quantity- an action with a significance. Though perhaps it would be truer to say that while this motion, this impact, is an action, the counter-motion is an experience ; or each may be from different points of view either an action or an experience: or we may think of speech as action upon a substrate and experience within that substrate.

If however voice is not characteristically impact, but is simply air, two categories will be involved: voice is significant, and the one category will not be sufficient to account for this significance without associating with a second.

With regard to time, if it is to be thought of as a measure, we must determine what it is that applies this measure. It must clearly be either Soul or the Present Moment. If on the contrary we take time to be something measured and regard it as being of such and such extension- a year, for example- then we may consider it as a quantity: essentially however time is of a different nature; the very fact that we can attribute this or that length to it shows us that it is not length: in other words, time is not Quantity. Quantity in the strict sense is the Quantity not inbound with things; if things became quantities by mere participation in Quantity, then Substance itself would be identical with Quantity.

Equality and inequality must be regarded as properties of Quantity-Absolute, not of the participants, or of them not essentially but only accidentally: such participants as "three yards' length," which becomes a quantity, not as belonging to a single genus of Quantity, but by being subsumed under the one head, the one category.

(525) how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply 4 , taking care that one shall continue one and not become lost in fractions. That is very true. Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible,—what would they answer? They would answer, as I should conceive, that they were speaking of those numbers which can only be realized in thought. Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth? Yes; that is a marked characteristic of it. And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been. Very true, he said.

(11) 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.

(13) So, the student of Hermetics need not lay aside any of his cherished scientific views regarding the Universe. All he is asked to do is to grasp the underlying principle of "THE ALL is Mind; the Universe is Mental--held in the mind of THE ALL." He will find that the other six of the Seven Principles will "fit into" his scientific knowledge, and will serve to bring out obscure points and to throw light in dark corners. This is not to be wondered at, when we realize the influence of the Hermetic thought of the early philosophers of Greece, upon whose foundations of thought the theories of modern science largely rest. The acceptance of the First Hermetic Principle (Mentalism) is the only great point of difference between Modern Science and Hermetic students, and Science is gradually moving toward the Hermetic position in its groping in the dark for a way out of the Labyrinth into which it has wandered in its search for Reality.

(5) What then is the veritable nature of Number?

Is it an accompaniment upon each substance, something seen in the things as in a man we see one man, in a being one being and in the total of presentations the total of number?

But how explain the dyad and triad? How comes the total to be unitary and any particular number to be brought under unity? The theory offers a multiplicity of units, and no number is reducible to unity but the simple "one." It might be suggested that a dyad is that thing- or rather what is observed upon that thing- which has two powers combined, a compound thing related to a unity: or numbers might be what the Pythagoreans seem to hold them in their symbolic system in which Justice, for example, is a Tetrad: but this is rather to add the number, a number of manifold unity like the decad, to the multiplicity of the thing which yet is one thing. Now it is not so that we treat the ten things; we bring them together and apply the figure ten to the several items. Or rather in that case we say ten, but when the several items form a unity we say decad. This would apply in the Intellectual as in the sensible.

But how then can number, observed upon things, rank among Real Beings?

One answer might be that whiteness is similarly observed upon things and yet is real, just as movement is observed upon things and there is still a real existence of movement. But movement is not on a par with number: it is because movement is an entity that unity can be observed upon it. Besides, the kind of real existence thus implied annuls the reality of number, making it no more than an attribute; but that cannot be since an attribute must exist before it can be attributed; it may be inseparable from the subject but still must in itself be something, some entity as whiteness is; to be a predicate it must be that which is to be predicated. Thus if unity is observed in every subject, and "one man" says more than "man's oneness being different from the manness and common to all things- then this oneness must be something prior to man and to all the rest: only so can the unity come to apply to each and to all: it must therefore be prior also to even movement, prior to Being, since without unity these could not be each one thing: of course what is here meant is not the unity postulated as transcending Being but the unity predicable of the Ideas which constitute each several thing. So too there is a decad prior to the subject in which we affirm it; this prior would be the decad absolute, for certainly the thing in which the decad is observed is not that absolute.

Is this unity, then, connate and coexistent to the Beings? Suppose it coexistent merely as an accidental, like health in man, it still must exist of itself; suppose it present as an element in a compound, there must first exist unity and the unity absolute that can thus enter into composition; moreover if it were compounded with an object brought into being by its agency it would make that object only spuriously a unity; its entry would produce a duality.

But what of the decad? Where lies the need of decad to a thing which, by totalling to that power, is decad already?

The need may be like that of Form to Matter; ten and decad may exist by its virtue; and, once more, the decad must previously exist of its own existence, decad unattached.

(56) Timaeus: when taken singly each in its several kind, is seen by us, but when many are collected together their masses are seen. And, moreover, as regards the numerical proportions which govern their masses and motions and their other qualities, we must conceive that God realized these everywhere with exactness, in so far as the nature of Necessity submitted voluntarily or under persuasion, and thus ordered all in harmonious proportion. From all that we have hitherto said about these Kinds,

(13) But how account, at this, for its extension over all the heavens and all living beings?

There is no such extension. Sense-perception, by insistence upon which we doubt, tells of Here and There; but reason certifies that the Here and There do not attach to that principle; the extended has participated in that kosmos of life which itself has no extension.

Clearly no participant can participate in itself; self-participation would be merely identity. Body, then, as participant does not participate in body; body it has; its participation must be in what is not body. So too magnitude does not participate in magnitude; it has it: not even in addition of quantity does the initial magnitude participate in magnitude: the two cubits do not themselves become three cubits; what occurs is that an object totalling to a certain quantity now totals to another: for magnitude to participate in magnitude the actual two cubits must themselves become the new three .

If, then, the divided and quantitatively extended is to participate in another Kind, is to have any sort of participation, it can participate only in something undivided, unextended, wholly outside of quantity. Therefore, that which is to be introduced by the participation must enter as itself an omnipresent indivisible.

This indivisibility must, of course, not be taken in any sense of littleness: littleness would be still divisible, could not cover the extension of the participant and could not maintain integral presence against that expansion. Nor is it the indivisibility of a geometric point: the participant mass is no single point but includes an infinity of points; so that on the theory this principle must be an infinity of points, not a simultaneous entire, and so, again, will fail to cover the participant.

If, then, the participant mass in its entirety is to contain that principle entire, the universe must hold that one soul present at its every point.

(530) other proportion. No, he replied, such an idea would be ridiculous. And will not a true astronomer have the same feeling when he looks at the movements of the stars? Will he not think that heaven and the things in heaven are framed by the Creator of them in the most perfect manner? But he will never imagine that the proportions of night and day, or of both to the month, or of the month to the year, or of the stars to these and to one another, and any other things that are material and visible can also be eternal and subject to no deviation—that would be absurd; and it is equally absurd to take so much pains in investigating their exact truth. I quite agree, though I never thought of this before. Then, I said, in astronomy, as in geometry, we should employ problems, and let the heavens alone if we would approach the subject in the right way and so make the natural gift of reason to be of any real use. That, he said, is a work infinitely beyond our present astronomers. Yes, I said; and there are many other things which must also have a similar extension given to them, if our legislation is to be of any value. But can you tell me of any other suitable study? No, he said, not without thinking. Motion, I said, has many forms, and not one only; two of

(12) 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 , 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 be referred to Motion, as must activity also.

(1) Since, however, we are narrating the wisdom employed by Pythagoras in instructing his disciples, it will not be unappropriate to relate that which is proximate in a following order to this, viz. how he invented the harmonic science, and harmonic ratios. But for this purpose we must begin a little higher. Intently considering once, and reasoning with himself, whether it would be possible to devise a certain instrumental assistance to the hearing, which should be firm and unerring, such as the sight obtains through the compass and the rule, or, by Jupiter, through a dioptric instrument; or such as the touch obtains through the balance, or the contrivance of measures;—thus considering, as he was walking near a brazier’s shop, he heard from a certain divine casualty the hammers beating out a piece of iron on an anvil, and producing sounds that accorded with each other, one combination only excepted.

But he recognized in those sounds, the diapason, the diapente, and the diatessaron, harmony. He saw, however, that the sound which was between the diatessaron and the diapente was itself by itself dissonant, yet, nevertheless, gave completion to that which was the greater sound among them. Being delighted, therefore, to find that the thing which he was anxious to discover had succeeded to his wishes by divine assistance, he went into the brazier’s shop, and found by various experiments, that the difference of sound arose from the magnitude of the hammers, but not from the force of the strokes, nor from the figure of the hammers, nor from the transposition of the iron which was beaten. When, therefore, he had accurately examined the weights and the equal counterpoise of the hammers, he returned home, and fixed one stake diagonally to the walls, lest if there were many, a certain difference should arise from this circumstance, or in short, lest the peculiar nature of each of the stakes should cause a suspicion of mutation.

Afterwards, from this stake he suspended four chords consisting of the same materials, and of the same magnitude and thickness, and likewise equally twisted. To the extremity of each chord also he tied a weight. And when he had so contrived, that the chords were perfectly equal to each other in length, he afterwards alternately struck two chords at once, and found the before-mentioned symphonies, viz. a different symphony in a different combination. For he discovered that the chord which was stretched by the greatest weight, produced, when compared with that which was stretched by the smallest, the symphony diapason. But the former of these weights was twelve pounds, and the latter six. And, therefore, being in a duple ratio, it exhibited the consonance diapason; which the weights themselves rendered apparent.

But again, he found that the chord from which the greatest weight was suspended compared with that from which the weight next to the smallest depended, and which weight was eight pounds, produced the symphony diapente. Hence he discovered that this symphony is in a sesquialter ratio, in which ratio also the weights were to each other. And he found that the chord which was stretched by the greatest weight, produced, when compared with that which was next to it in weight, and was nine pounds, the symphony diatessaron, analogously to the weights. This ratio, therefore, he discovered to be sesquitertian; but that of the chord from which a weight of nine pounds was suspended, to the chord which had the smallest weight [or six pounds,] to be sesquialter.

For 9 is to 6 in a sesquialter ratio. In like manner, the chord next to that from which the smallest weight depended, was to that which had the smallest weight, in a sesquitertian ratio, [for it was the ratio of 8 to 6,] but to the chord which had the greatest weight, in a sesquialter ratio [for such is the ratio of 12 to 8.] Hence, that which is between the diapente and the diatessaron, and by which the diapente exceeds the diatessaron, is proved to be in an epogdoan ratio, or that of 9 to 8. But either way it may be proved that the diapason is a system consisting of the diapente in conjunction with the diatessaron, just as the duple ratio consists of the sesquialter and sesquitertian, as for instance, 12, 8, and 6; or conversely, of the diatessaron and the diapente, as in the duple ratio of the sesquitertian and sesquialter ratios, as for instance 12, 9, and 6.

After this manner, therefore, and in this order, having conformed both his hand and his hearing to the suspended weights, and having established according to them the ratio of the habitudes, he transferred by an easy artifice the common suspension of the chords from the diagonal stake to the limen of the instrument, which he called chordotonon . But he produced by the aid of pegs a tension of the chords analogous to that effected by the weights.

(15) The same holds also of astronomy. For treating of the description of the celestial objects, about the form of the universe, and the revolution of the heavens, and the motion of the stars, leading the soul nearer to the creative power, it teaches to quickness in perceiving the seasons of the year, the changes of the air, and the appearance of the stars; since also navigation and husbandry derive from this much benefit, as architecture and building from geometry. This branch of learning, too, makes the soul in the highest degree observant, capable of perceiving the true and detecting the false, of discovering correspondences and proportions, so as to hunt out for similarity in things dissimilar; and conducts us to the discovery of length without breadth, and superficial extent without thickness, and an indivisible point, and transports to intellectual objects from those of sense.