(21) But how deep or how large the place of this world is, no man knoweth, though some physicists or astrologers have undertaken to measure the deep with their measures of circles; their measuring is but conjectural, or a measuring of somewhat that is comprehensible or palpable; as if a man would grasp the wind in his fist.

(71) Magnitude is divided into two parts--magnitude which is stationary and magnitude which is movable, the stationary pare having priority. Multitude is also divided into two parts, for it is related both to itself and to other things, the first relationship having priority. Pythagoras assigned the science of arithmetic to multitude related to itself, and the art of music to multitude related to other things. Geometry likewise was assigned to stationary magnitude, and spherics (used partly in the sense of astronomy) to movable magnitude. Both multitude and magnitude were circumscribed by the circumference of mind. The atomic theory has proved size to be the result of number, for a mass is made up of minute units though mistaken by the uninformed for a single simple substance.

(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.

(3) A: Assuredly. H: Of what size, then, must be the space in which it's moved, and of what kind [must be] the nature [of that space]? Must it not be far vaster [than the cosmos], in order that it may be able to find room for its continued course, so that the moved may not be cramped for want of room and lose its motion? A: Something, Thrice-greatest one, it needs must be, immensely vast.

(11) But how can the unextended reach over the defined extension of the corporeal? How can it, so, maintain itself as a unity, an identity?

This is a problem often raised and reason calls vehemently for a solution of the difficulties involved. The fact stands abundantly evident, but there is still the need of intellectual satisfaction.

We have, of course, no slight aid to conviction, indeed the very strongest, in the exposition of the character of that principle. It is not like a stone, some vast block lying where it lies, covering the space of its own extension, held within its own limits, having a fixed quantity of mass and of assigned stone-power. It is a First Principle, measureless, not bounded within determined size- such measurement belongs to another order- and therefore it is all-power, nowhere under limit. Being so, it is outside of Time.

Time in its ceaseless onward sliding produces parted interval; Eternity stands in identity, pre-eminent, vaster by unending power than Time with all the vastness of its seeming progress; Time is like a radial line running out apparently to infinity but dependent upon that, its centre, which is the pivot of all its movement; as it goes it tells of that centre, but the centre itself is the unmoving principle of all the movement.

Time stands, thus, in analogy with the principle which holds fast in unchanging identity of essence: but that principle is infinite not only in duration but also in power: this infinity of power must also have its counterpart, a principle springing from that infinite power and dependent upon it; this counterpart will, after its own mode, run a course- corresponding to the course of Time- in keeping with that stationary power which is its greater as being its source: and in this too the source is present throughout the full extension of its lower correspondent.

This secondary of Power, participating as far as it may in that higher, must be identified.

Now the higher power is present integrally but, in the weakness of the recipient material, is not discerned as every point; it is present as an identity everywhere not in the mode of the material triangle- identical though, in many representations, numerically multiple, but in the mode of the immaterial, ideal triangle which is the source of the material figures. If we are asked why the omnipresence of the immaterial triangle does not entail that of the material figure, we answer that not all Matter enters into the participation necessary; Matter accepts various forms and not all Matter is apt for all form; the First Matter, for example, does not lend itself to all but is for the First Kinds first and for the others in due order, though these, too, are omnipresent.

(18) Things seen have as their property manifestation, action, inertia. They form the basis of the elements and the sense-powers. They make for experience and for liberation.

(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.

(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.

(15) How far is it true that equality and inequality are characteristic of Quantity?

Triangles, it is significant, are said to be similar rather than equal. But we also refer to magnitudes as similar, and the accepted connotation of similarity does not exclude similarity or dissimilarity in Quantity. It may, of course, be the case that the term "similarity" has a different sense here from that understood in reference to Quality.

Furthermore, if we are told that equality and inequality are characteristic of Quantity, that is not to deny that similarity also may be predicated of certain quantities. If, on the contrary, similarity and dissimilarity are to be confined to Quality, the terms as applied to Quantity must, as we have said, bear a different meaning.

But suppose similarity to be identical in both genera; Quantity and Quality must then be expected to reveal other properties held in common.

May the truth be this: that similarity is predicable of Quantity only in so far as Quantity possesses differences? But as a general rule differences are grouped with that of which they are differences, especially when the difference is a difference of that thing alone. If in one case the difference completes the substance and not in another, we inevitably class it with that which it completes, and only consider it as independent when it is not complementary: when we say "completes the substance," we refer not to Subtance as such but to the differentiated substance; the particular object is to be thought of as receiving an accession which is non-substantial.

We must not however fad to observe that we predicate equality of triangles, rectangles, and figures generally, whether plane or solid: this may be given as a ground for regarding equality and inequality as characteristic of Quantity.

It remains to enquire whether similarity and dissimilarity are characteristic of Quality.

We have spoken of Quality as combining with other entities, Matter and Quantity, to form the complete Sensible Substance; this Substance, so called, may be supposed to constitute the manifold world of Sense, which is not so much an essence as a quale. Thus, for the essence of fire we must look to the Reason-Principle; what produces the visible aspect is, properly speaking, a quale.

Man's essence will lie in his Reason-Principle; that which is perfected in the corporeal nature is a mere image of the Reason-Principle a quale rather than an essence.

Consider: the visible Socrates is a man, yet we give the name of Socrates to that likeness of him in a portrait, which consists of mere colours, mere pigments: similarly, it is a Reason-Principle which constitutes Socrates, but we apply the name Socrates to the Socrates we see: in truth, however, the colours and shapes which make up the visible Socrates are but reproductions of those in the Reason-Principle, while this Reason-Principle itself bears a corresponding relation to the truest Reason-Principle of Man. But we need not elaborate this point.

(11) "But, given Magnitude and the properties we know, what else can be necessary to the existence of body?"

Some base to be the container of all the rest.

"A certain mass then; and if mass, then Magnitude? Obviously if your Base has no Magnitude it offers no footing to any entrant. And suppose it sizeless; then, what end does it serve? It never helped Idea or quality; now it ceases to account for differentiation or for magnitude, though the last, wheresoever it resides, seems to find its way into embodied entities by way of Matter."

"Or, taking a larger view, observe that actions, productive operations, periods of time, movements, none of these have any such substratum and yet are real things; in the same way the most elementary body has no need of Matter; things may be, all, what they are, each after its own kind, in their great variety, deriving the coherence of their being from the blending of the various Ideal-Forms. This Matter with its sizelessness seems, then, to be a name without a content."

Now, to begin with: extension is not an imperative condition of being a recipient; it is necessary only where it happens to be a property inherent to the recipient's peculiar mode of being. The Soul, for example, contains all things but holds them all in an unextended unity; if magnitude were one of its attributes it would contain things in extension. Matter does actually contain in spatial extension what it takes in; but this is because itself is a potential recipient of spatial extension: animals and plants, in the same way, as they increase in size, take quality in parallel development with quantity, and they lose in the one as the other lessens.

No doubt in the case of things as we know them there is a certain mass lying ready beforehand to the shaping power: but that is no reason for expecting bulk in Matter strictly so called; for in such cases Matter is not the absolute; it is that of some definite object; the Absolute Matter must take its magnitude, as every other property, from outside itself.

A thing then need not have magnitude in order to receive form: it may receive mass with everything else that comes to it at the moment of becoming what it is to be: a phantasm of mass is enough, a primary aptness for extension, a magnitude of no content- whence the identification that has been made of Matter with The Void.

But I prefer to use the word phantasm as hinting the indefiniteness into which the Soul spills itself when it seeks to communicate with Matter, finding no possibility of delimiting it, neither encompassing it nor able to penetrate to any fixed point of it, either of which achievements would be an act of delimitation.

In other words, we have something which is to be described not as small or great but as the great-and-small: for it is at once a mass and a thing without magnitude, in the sense that it is the Matter on which Mass is based and that, as it changes from great to small and small to great, it traverses magnitude. Its very undeterminateness is a mass in the same sense that of being a recipient of Magnitude- though of course only in the visible object.

In the order of things without Mass, all that is Ideal-Principle possesses delimitation, each entity for itself, so that the conception of Mass has no place in them: Matter, not delimited, having in its own nature no stability, swept into any or every form by turns, ready to go here, there and everywhere, becomes a thing of multiplicity: driven into all shapes, becoming all things, it has that much of the character of mass.

(35) of the sense perceptions. Not only is there a great deal more to light than anyone has ever seen but there are also unknown forms of light which no optical equipment will ever register. There are unnumbered colors which cannot be seen, as well as sounds which cannot be heard, odors which cannot be smelt, flavors which cannot be tasted, and substances which cannot be felt. Man is thus surrounded by a supersensible universe of which he knows nothing because the centers of sense perception within himself have not been developed sufficiently to respond to the subtler rates of vibration of which that universe is composed.

(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.

(2) Side by side exist the Authentic All and its counterpart, the visible universe. The Authentic is contained in nothing, since nothing existed before it; of necessity anything coming after it must, as a first condition of existence, be contained by this All, especially since it depends upon the Authentic and without that could have neither stability nor movement.

We may be reminded that the universe cannot be contained in the Authentic as in a place, where place would mean the boundaries of some surrounding extension considered as an envelope, or some space formerly a part of the Void and still remaining unoccupied even after the emergence of the universe, that it can only support itself, as it were, upon the Authentic and rest in the embrace of its omnipresence; but this objection is merely verbal and will disappear if our meaning is grasped; we mention it for another purpose; it goes to enforce our real assertion that the Authentic All, at once primal and veritable, needs no place and is in no way contained. The All, as being an integral, cannot fall short of itself; it must ever have fulfilled its own totality, ever reached to its own equivalence; as far as the sum of entities extends, there this is; for this is the All.

Inevitably, also, anything other than this All that may be stationed therein must have part in the All, merge into it, and hold by its strength; it is not that the thing detaches a portion of the All but that within itself it finds the All which has entered into it while still unbrokenly self-abiding, since Being cannot lodge in non-Being, but, if anything, non-Being within Being.

Being, then, is present to all Being; an identity cannot tear itself asunder; the omnipresence asserted of it must be presence within the realm of Being; that is, it must be a self-presence. And it is in no way strange that the omnipresence should be at once self-abiding and universal; this is merely saying omnipresence within a unity.

It is our way to limit Being to the sense-known and therefore to think of omnipresence in terms of the concrete; in our overestimate of the sensible, we question how that other Nature can reach over such vastness; but our great is small, and this, small to us, is great; it reaches integrally to every point of our universe- or, better, our universe, moving from every side and in all its members towards this, meets it everywhere as the omnipresent All ever stretching beyond.

The universe in all its reach can attain nothing further- that would mean overpassing the total of Being- and therefore is content to circle about it; not able to encompass or even to fill the All, it is content to accept place and subordination, for thus it preserves itself in neighbouring the higher present to it- present and yet absent; self-holding, whatever may seek its presence.

Wherever the body of the universe may touch, there it finds this All; it strives for no further advance, willing to revolve in that one circle, since to it that is the All and in that movement its every part embraces the All.

If that higher were itself in place there would be the need of seeking that precise place by a certain right path; part of seeker must touch part of sought, and there would be far and near. But since there is no far and near there must be, if presence at all, presence entire. And presence there indubitably is; this highest is present to every being of those that, free of far and near, are of power to receive.

(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.

(10) We shall understand the mode of purification by confession, and that of contemplation by analysis, advancing by analysis to the first notion, beginning with the properties underlying it; abstracting from the body its physical properties, taking away the dimension of depth, then that of breadth, and then that of length. For the point which remains is a unit, so to speak, having position; from which if we abstract position, there is the conception of unity.

(3) Discern what size the water is, that it is immeasurable (and) incomprehensible, both its beginning and its end. It supports the earth; it blows in the air where the gods and the angels are. But in him who is exalted above all these there is the fear and the light, and in him are my writings revealed. I have given them as a service to the creation of the physical things, for it is not possible for anyone to stand without that One, nor is it possible for the aeon to live without him. It is he who possesses what is in him by discerning (it) in purity.

(8) The light of our world can be allocated because it springs from a corporeal mass of known position, but conceive an immaterial entity, independent of body as being of earlier nature than all body, a nature firmly self-based or, better, without need of base: such a principle, incorporeal, autonomous, having no source for its rising, coming from no place, attached to no material mass, this cannot be allotted part here and part there: that would be to give it both a previous position and a present attachment. Finally, anything participating in such a principle can participate only as entirety with entirety; there can be no allotment and no partition.

A principle attached to body might be exposed, at least by way of accident, to such partition and so be definable as passive and partible in view of its close relationship with the body of which it is so to speak a state or a Form; but that which is not inbound with body, which on the contrary body must seek, will of necessity go utterly free of every bodily modification and especially of the very possibility of partition which is entirely a phenomenon of body, belonging to its very essence. As partibility goes with body, so impartibility with the bodiless: what partition is possible where there is no magnitude? If a thing of magnitude participates to any degree in what has no magnitude, it must be by a participation without division; divisibility implies magnitude.

When we affirm unity in multiplicity, we do not mean that the unity has become the multiples; we link the variety in the multiples with the unity which we discern, undivided, in them; and the unity must be understood as for ever distinct from them, from separate item and from total; that unity remains true to itself, remains itself, and so long as it remains itself cannot fail within its own scope , yet it is not to be thought of as coextensive with the material universe or with any member of the All; utterly outside of the quantitative, it cannot be coextensive with anything.

Extension is of body; what is not of body, but of the opposed order, must be kept free of extension; but where there is no extension there is no spatial distinction, nothing of the here and there which would end its freedom of presence. Since, then, partition goes with place- each part occupying a place of its own- how can the placeless be parted? The unity must remain self-concentrated, immune from part, however much the multiple aspire or attain to contact with it. This means that any movement towards it is movement towards its entirety, and any participation attained is participation in its entirety. Its participants, then, link with it as with something unparticipated, something never appropriated: thus only can it remain intact within itself and within the multiples in which it is manifested. And if it did not remain thus intact, it would cease to be itself; any participation, then, would not be in the object of quest but in something never quested.

(27) For it is the contraction, compacting or imaging of a thing, and therefore it is the first quality, and a beginning of the angelical creatures, and of all images or likenesses which are in heaven, and all which are in this world, and all whatsoever that can be named or expressed.

(6) This uncertainty shared by all true students of the subject proves conclusively that it is unwise to make definite statements founded on the indefinite and fragmentary information available concerning the Pythagorean system of mathematical philosophy. The material which follows represents an effort to collect a few salient points from the scattered records preserved by disciples of Pythagoras and others who have since contacted his philosophy.

(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.