CHAPTER V[380

THE LAST AND THE LEAST THINGS: ATOMS AND SOUL-MONADS

The reaction against Aristotelianism had, as one of its results, a renascence of the atomic theory of Democritus and Lucretius; and one of the earliest adherents of the renovated doctrine was Bruno. Although a complete presentation of the theory was not given until his later works, the _De Minimo_ and the _Articuli adv. Mathematicos_, appeared, yet already in the Italian dialogues there were frequent references to it. In the _Cena_,[381] for example, it is said that in the physical division of a finite body infinite progress is impossible, and, as we shall afterwards find, in Bruno there is no distinction between physical and mathematical division. Again, in the _Cena_ an animistic atomism is suggested, which presents a curious anticipation of some of Leibniz' characteristic views. "It is more than probable, as all things partake of life, that many or innumerable individuals live not only in us, but in all composite things; when anything "dies," as is said, we must believe it to be not death, but change only; the accidental composition or concord ceases, the things that enter into it remaining always immortal; and this is truer of those things we call spiritual than of those we call corporeal or material."[382] Thus every body or organism, for all bodies are organisms to Bruno, is itself constituted by other living beings, the atoms--living atoms--being alike the origin and the end of all. So Leibniz wrote:--"Every living body has a presiding entelechy, which is the soul in the animal; but the members of this living body are full of other living beings--plants, animals,--each of which, again, has its entelechy or presiding soul."[383] In the _Infinito_ Bruno refers to the continuous changes of all composite bodies as arising from the ceaseless flux of atoms out of and into each body, even the greater "animals," the stars and planets, sending out particles, which wander through the universe from one to another.[384] Again, when discussing the four elements, he ascribes to water the power of holding together the atoms of earth, or "the dry." "If from the earth all water were to be removed, so that there remained purely dry matter, this remainder would necessarily be an incoherent, rare, loose substance, easy to be dispersed through the air, in the form of innumerable discontinuous bodies; for while the air or ether makes a continuum, that which makes a _coherent_ continuum is water or moisture."[385] These indivisible "prime bodies," of which the worlds are originally composed, are spoken of as flying throughout space from world to world, in infinite movement, entering now into this, now into that "composition."[386] Finally, in the _Spaccio_, we are reminded that "every trifle, however worthless, is of value in the order of the whole, the universe, for great things are composed of little, little things of the least, and these of the individuals (or indivisibles) or minima."[387] In its main outlines, accordingly, Bruno's atomic theory was already formed in his mind when he wrote his earlier philosophical works, and even some of his peculiar applications of it had already suggested themselves. It is hardly possible, therefore, to find any very marked development in this regard between the London and the Frankfort periods. There is elaboration and completion rather than development in any definite direction;[388] and, as we have seen, the writing of the larger works, containing the developed system, was projected in London, and even carried out to a certain extent before Bruno left England.[389] In the _Acrotismus_, which occupies a middle place between the two periods, the doctrine is equally in evidence, in reference both to the atoms and to the continuous ether in which they move. "There is a limit to the division of nature--an indivisible something; the division of nature arrives at ultimate minimal parts, unapproachable by human instruments. Of these minimal bodies every sensible body is composed, and such a body, resolved into its minima, can retain no semblance of complexity; for these are the first bodies out of which all others are made, and which are, in the truest sense, the matter of all things that have corporeal existence. Resolved into these parts, stone has no look of stone, flesh of flesh, bone of bone; in their elements, bone, stone, and flesh do not differ, but only when formed out of these, compounded, compacted, and arranged in diverse manners, do flesh, stone, and bone and other things become different one from another."[390] And Bruno describes how, between the heavenly bodies, there is a substance, "ingenerable and incorruptible, the immeasurable air, a kind of spiritual body"--the ether.[391]

[Sidenote: Object of _De Minimo_.]

[Sidenote: Atomism a metaphysical doctrine.]

Its full extension, however, the theory receives in the _De Minimo_, where the atom, or corporeal unity, is not the sole minimum discussed. The full title of the work is:--"On the threefold minimum, and measure, being the principles of the three speculative sciences and of many practical arts." We find nowhere any distinct statement as to what Bruno meant by the "threefold minimum," and the three speculative sciences to which its several members refer. It was supposed that the minima were (1) the monad or unity which is the unit of number, (2) the point, which is the unit of the line, and (3) the atom, which is the unit of body. But arithmetic and geometry can hardly be called speculative sciences, and Tocco has shown that Bruno had in view the triad of _God_, the _soul_ and the _atom_--the three kinds of simple substance, each immortal and indestructible:--_God_ as the supreme and most simple unity, Monad of Monads; _soul_ as that which lives in each composite being and holds in unity the atoms which from time to time enter into its composition; and the _atom_, the most simple of material substances, in the sum of which, with their containing ether, the material universe consists. Had Bruno carried out his subdivision of the speculative sciences, he would probably have referred God, as the substance of all reality, to a speculative theology, of Neoplatonist type; soul as the simple substance of animate beings to metaphysics proper; and the atoms, the substance of body, to a speculative physics, dealing with the metaphysical presuppositions of the general theory of nature, which was set forth in the _De Immenso_. The scheme, however, was never fully carried out,[392] the times being not yet ripe for the complete separation of the speculative and the experimental or observational sciences. In referring the atomic theory to metaphysics, Bruno showed a true instinct, for while in one sense atomism is a scientific hypothesis capable of furnishing laws which explain the interaction of bodies,--the corpuscular theory,--and as such has proved its value by the brilliant developments of recent years, on the other hand, it is also a presupposition of knowledge, a ground of the possibility of our knowledge of body, and therefore has its place in speculative theory, or metaphysics, in the widest sense. Both points of view are presented in Bruno's doctrine, but that from which he starts is the epistemological, following in this the guidance of Nicholas of Cusa.

[Sidenote: Knowledge implies the atom.]

[Sidenote: Relativity of minimum.]

Knowledge is measurement, and all measure implies a minimum in each kind of being. Were it possible to subdivide anything _ad infinitum_, the half would be potentially equal to the whole, and measurement frustrated. There must be a limit to division, an ultimate part, which itself has no parts, and which is the _substance_ of the composition into which it enters, the composition on the other hand being an "_accident_" of this minimum. As it is primarily a condition of measurement, the minimum differs in the different spheres of measure or knowledge to which the category of quantity applies. In magnitudes of one or two dimensions it is the point, in bodies the atom, in numbers the monad or unity. Thus number is accident of the monad, monad is the essence of number, as composition is accident of the atom, atom is essence of the composite. Again, the "sensible minimum" must be far greater than the natural or real minimum, for in so far as minimum is qualified by sensible, it is implied that the minimum is not absolutely such, but is a composite. The minimum of taste, touch, etc., must possess certain qualities, by which it has relation to sense, and these can derive only from some form of composition. In their primary form the minima of nature must be without difference; therefore that some are sensible, others not, must be due to some addition in the former.[393]

Thus each species of existence, as light, moisture, vital force,[394] has its own minimum, and the minimum is relative in this sense also, that there are different kinds of existence not resolvable one into another: the absolute minimum would be God, who is also the absolute maximum. The relative minimum, accordingly, is determined either by the thought and design of the observer, or by the species of existence to which the subject belongs; nature has set limits, both lower and upper, within which the individual of any species must stay, or cease to belong to that species. Accordingly, what _one_ regards as great and composite, _another_ may take as first and minimum: the unit of one science may be analysed in another into further elements. "Pythagoras in his philosophy started with the monad and numbers; Plato with atoms, lines and surfaces; Empedocles with the four elements; the physicians with the four humours, and so on; but the Pythagorean monad is prior to the placed monad (the atom), Plato's matter of bodies to the qualified bodies of Empedocles, the four simple bodies of Empedocles to the four first combinations of these, the four humours. So to the universe the whole solar system, the sun and all its planets, may be a simple unit."[395]

Here Bruno suggests two principles for the classification and systematising of the sciences, to which it would have been well had he himself and his successors faithfully adhered. The one is, that the modes of measurement, _i.e._ the methods and laws of the sciences, must differ for the different kinds of existence studied: that a biological law, for example, cannot be adopted as an explanation of mental phenomena, nor the atomic theory account for the phenomena of life. On the other hand there are _orders_ of existence, according to the complexity of the subjects involved. If we regard the science which deals with the more concrete subject as "higher," then each higher science (_e.g._ psychology) must take for granted the principles and results of each lower science (biology, physics, mathematics),--each must adopt and retain a unit for itself, which it has not further to analyse.

[Sidenote: The "minima" in the classification of sciences.]

In the same way the minima offer a ground for the distinction of the more abstract sciences one from another. The term "individual nature" (_atoma natura_) may, according to Bruno, have one of several uses. It may be applied either "negatively or privatively, and if negatively, then either accidentally or substantially." His instance of the _accidental_ use is a voice or sound, which expands spherically, is wholly wherever it is, _i.e._ the full content of the sound is heard, wherever its influence extends, not a part here, a part there, although the intensity may vary in degree. Of the _substantial_ use examples are the spirit, which is wholly in the whole body of man, or that spirit which is in the whole extent of the life of the earth, by whose life we live and in which we have our being, or, above this substantial nature or individual soul, that of the universe, and supreme above all, the mind of minds, God, one spirit completely filling all things.[396] The atom-nature is _privatively_ so-called, when it is the element and substance of a magnitude which is the same in kind with it, and may be reduced to it, and it is distinguished from the atom _negatively_ so-called, because it is not divisible, either in genus or in species, either _per se_ or _per accidens_. Examples are, (1) in discrete quantities:--unity to the mathematician, the universal proposition to the logician, the syllable to the grammarian; and (2) in continuous quantities, varying with the species of continuum:--the minimal pain, sweetness, colour, light, triangle, circle, straight line, curve; in duration, the instant; in place, the minimal space; in length and breadth, the point; in body, the least and first body.

[Sidenote: Minimum as substance.]

In the second place, the atom or minimum is also a metaphysical [Greek: pou stô]; not only is it the last result of analysis, but it is also the permanent substance of being, and again it contains all being in itself--it is essence of being. Thus such an individual nature "never comes into existence by way of generation, nor passes out of it by way of corruption or dissolution; only _per accidens_ may we say that it now is, now is not."[397] Certain of them, however, the souls, deities, God, are in their intrinsic nature eternal, immortal, indissoluble. Of these it was Bruno's intention to treat at large in a _Metaphysics_ and a _De Anima_ which he purposed to write "if God granted him time."[398] Unfortunately, it was willed otherwise.

Nothing that becomes, changes, decays, is real (_ens_). It is by meditating on this perpetual unity of nature, by conforming ourselves, and preserving ourselves in likeness to it, that we come to partake in the life of the gods, and to deserve the name of substance. That which time, movement, fate bring to us is nought; for while they are, they are not. "Let us then," cries Bruno, "supply the mind with material, in the contemplation of the _minimum_, through which it may exalt itself to the _maximum_."[399] Since the real minimum, whether atom or soul, is immortal and indestructible, we know, as Pythagoras saw, that there is no death, but only transition; death is a dissolution which can occur only to the composite, for the composite is never _substance_, but is always _adventitious_. Otherwise we should be changing our substance every moment with the continuous influx of atoms into our bodies. Only by the individual substance of the soul are we that which we are; about it as a centre, which is everywhere in its whole being (_ubique totum_), the disgregation and aggregation of atoms takes place. According to a law of the soul-world, all bodies and forces tend to the spherical form; God, as monad of monads, is the perfect or infinite sphere, of which the centre is at once nowhere and everywhere; and in Him (as in all minima, simple substances, monads) all opposites coincide, the many and the few, finite and infinite; therefore that which is _minimum_ is also _maximum_, or anything between these, each is all things, the greatest and the whole.[400] Therefore, if contemplation is to follow in the footsteps of nature, it must begin, continue, and end with the _minimum_.[401] In other words, the minimum in each sphere of being contains implicitly in itself the whole reality of that sphere. The minimum is its substance, not merely the ultimate of analysis, but the actual source, the dynamic origin of reality, as God is implicitly the whole universe and also the source of the universe as it actually exists. It is because the minimum is all reality, is the _maximum_, that the knowledge of it gives us that of the whole.

[Sidenote: Uniqueness of all things.]

[Sidenote: Sense and knowledge.]

[Sidenote: Relativity.]

[Sidenote: Judgment based upon sensation.]

In the third place the atomic theory offers an explanation of the uniqueness of each natural existence, which Bruno's philosophical theory already assumed. The ever moving atoms present a mechanism by which the infinite diversity and infinite succession of change in things may be brought about. The _appearance_ of similarity, exactness, etc., is, as we have found, an illusion. Mathematically exact figures or bodies--a true circle, for example--are unattainable by sense, even if they exist in nature; but they do not exist in nature. Sense is the primary faculty, through which the material of all others must pass, so that what has not entered through that window of the soul cannot be known at all. But a single point out of place on the circumference of a circle makes it cease to be a true circle, and our sense-apprehension is necessarily so confused and indistinct that we cannot distinguish between the true and the false, where truth depends upon so inappreciable a difference. Moreover sense-knowledge is relative to the knowing subject, or to the subject's position with regard to the object. What to the eye of one is too large is to another too small; a sound which is pleasant to one ear is not so to another; the food which to the hungry man tastes sweet, to the full man is nauseous; the ape to the ape is beautiful, but to the man is of laughter-inspiring ugliness. Hence the circumspect will not say "this has a good odour, taste, sound, this has a beautiful appearance," but will add "to me," "now," "sometimes." Nothing is good or evil, pleasant or painful, beautiful or ugly, _simply_ and _absolutely_; but the same objects in relation to individual subjects receive from the senses contrary denominations, as they in fact produce contrary effects. In deciding what is to be called good or bad, honourable or base, nature and custom have been the chief agents, and alterations have issued from the slow rise and victory of different opinions. Among the Druids and Magi certain things were performed publicly at sacrifices which now, even when committed in privacy, are regarded as execrable, and are so by way of law, and in the present condition of affairs. Philosophy, as it teaches to abstract from particulars, to bring the nature and condition of things as far as possible under an absolute judgment, must define differently the useful and good in an _absolute_ sense, from the useful and good as _contracted_ to the human species. Objectively there is no definitely good or definitely evil, definitely true or definitely false, so that from one point of view we may say that all things are good; from another that all things are evil; from a third that nothing is good or evil, as neither of the contraries is true; from a fourth that all things are both good and evil, as each of the contraries is true. No sense deceives or is deceived: each judges of its proper object according to its own measure. There is no higher tribunal to which to refer its object, nor can reason judge of colour any more than can the ear; sensible truth does not follow any general or universal rule, but one which is particular, mutable, and variable. In the working of an external sense there may be different degrees of perfection or defect, but not of truth or falsity, which consist in the reference of the subject and predicate to one another. The faculty by which we judge this or that to be _true_ colour or light, and distinguish from apparent colour or light, is not in the eye. To affirm that man is an animal, we must know both man and animal, know that animal nature is in man, and other things which, as means or circumstances, concur directly or indirectly in this knowledge. External sense can apprehend only one species or image of the object; from the colour and figure to pass to its name, its truth, its difference from other objects, belongs to a more inward faculty. Yet the latter is always based upon sense;--a deaf man can neither imagine nor dream of sounds which he has never heard, nor a blind man of colours and figures which he has never seen.[402] This digression on the relativity of knowledge, and on the different functions of sense and reason, in which Bruno follows partly the teaching of Lucretius, partly the Peripatetic doctrine of knowledge, shows that even if a true or perfectly exact geometrical figure existed in nature, none of the faculties with which we are endowed could apprehend it, since it is not given by external sense.[403]

[Sidenote: No exactness or similarity in composites.]

But in the second place[404] reason tells us that no true circle, or other figure, is possible in nature: for there is in nature no similarity except in the atoms; a true circle would imply the equality of all lines from the centre, but no two lines in nature are entirely and in all respects equal to one another. The circle or part of a circle which appears most perfect to us--the rainbow--is an illusion of the senses, due to the reflection of the light of the sun from the clouds; so the circles made by a stone falling into water cannot be perfect, for this would mean that the stone itself is perfectly spherical, that the water is everywhere of the same density, that no wind is playing upon its surface. Sound is not equally diffused owing to differences in the density and rarity of the air, nor is the horizon ever a perfect circle, owing to differences of clearness in different directions. Object and faculty alike are in continuous change; all natural things are continually altering their form or changing their position; therefore although they seem to sense to remain fixed for a time, we know that this is impossible, from the nature of things.[405] Whatsoever falls in the scope of sense-perception, even the distant sphere and stars, we judge to consist of the same elements, therefore to be subject equally to perpetual variability and vicissitude. Thus--the atoms alone being simple, and remaining ever the same--no composite thing can be the same for one moment even, as each is being altered continually in all parts and on all sides by the efflux and influx of innumerable atoms.[406] "Hence nothing is perfectly straight, nothing perfectly circular among composites, nothing absolutely solid but the atoms, nothing absolutely void but the spaces between them." The facet of a diamond appears to be a perfect plane, perfectly compact, yet in reality it is rough and porous.[407] In matter no two lines or figures are entirely equal, nor can the same figure be repeated twice.[408] No man is twice of the same weight, the very instruments by which we measure and weigh things are themselves in constant change, and the flux of atoms is never equal, but now denser, now rarer. In general no two things are of the same weight, length, sound, or number, nor are two motions or parts of motion ever the same. To say that ten trees are equal to ten others is to speak merely from a _logical_ point of view, for in fact each is _one_ in a peculiar and special sense.[409] "Equality is only in those things which are permanent and the same; changing bodies are unequal to themselves at any two instants."[410] "Nothing variable or composite consists at two moments of time wholly of the same parts and the same order of parts, since the efflux and influx of atoms is continuous, and therefore not even from the primary integrating parts will you be able to name a thing as the same twice."[411]

Number itself is not an absolute, but a relative determination: it does not touch the nature of the thing itself. Nature has no difference of number, as we have, of odd and even, tens and hundreds; nor do the gods, spirits, or other rational beings define the numbers and measures of objects by the same series of terms. Both numbers and the methods of numbering are as diverse as are the fingers, heads, and mental equipment of the numberers. That which fits in with the numbers of nature will therefore never fit in with our numbers. Thus ten horses and ten men, although determined arithmetically by one and the same number, are in nature, or physically, wholly unequal to one another.[412]

[Sidenote: The atoms.]

[Sidenote: Time and space.]

In order that men's minds may be better disposed for the reception of truth, it is necessary first to demolish the foundations of error;[413] Bruno accordingly sets himself to disprove the infinite divisibility of the continuum.[414] It was the common belief that there were no limits set to the dividing power of either nature or art, so that, however small a part might be arrived at, it was possible to divide it into yet smaller parts, on the analogy of the division of a fraction into tens of thousands of parts. Bruno denied this analogy to be justifiable, as in the latter case we are concerned not with division but with multiplication or addition, not with a continuum, but with discrete quantities, and it was part of his general theory that the addition of discretes might be carried on _ad infinitum_; the inverse process he denied. He thus held opinions directly contrary to those of Aristotle, with whom the mass of the universe was finite, limited by its enclosing sphere, the parts of the universe unlimited. Aristotle had an upper but not a lower limit; Bruno a lower but not an upper. So time and space, which Aristotle had treated as _finite_ in duration or extent, but as _infinitely divisible_, like the universe itself, are regarded by Bruno as unlimited in their dimensions, but as consisting of discrete minimal parts. "In every point of duration is beginning without end, and end without beginning"; it is the centre of two infinities. Therefore the whole of duration is one infinite instant, both beginning and end, as immeasurable space is an infinite _minimum_ or centre. "The beginning and source of all errors, both in physics and in mathematics, is the resolution of the continuous _in infinitum_. To us it is clear that the resolution both of nature and of true art, which does not advance beyond nature, descends from a finite magnitude and number to the atom, but that there is no limit to the extension of things either in nature or in thought, except in regard to the form of particular species. Everywhere and always we find the _minimum_, the _maximum_ nowhere and never. The _maximum_ and _minimum_, however, may in one sense coincide, so that we know the maximum to be everywhere, since from what has been said it is evident that the maximum consists in the minimum and the minimum in the maximum, as in the many is the one, in the one the many. Yet reason and nature may more readily separate the minimum from the maximum than the maximum from the minimum. Therefore the immeasurable universe is nothing but centre everywhere; eternity nothing but a moment always; immeasurable body an atom; immeasurable plane a point; immeasurable space the receptacle of a point or atom."[415]

The chief source of error on the part of the Peripatetics was their failure to distinguish between the minimum as a part, and the minimum a _terminus_ or limit. Hence their idea that no combination of physical minima would give a magnitude, since two or more would touch one another with their whole surface, _i.e._ would coincide:--otherwise the minimum would have parts, a part of each touching the other, and a part not touching. On their theory it would follow that magnitudes do not consist of parts, or at least not of elementary parts. This is inconsistent with nature, for existing magnitudes must have been built up out of nature's elements, and with art, for art can measure only on the assumption of first parts. It is true that what is posited as first part in one operation may be the last result in another, for the _minimum_, as we have seen, is a relative conception, but _some_ first part is always assumed in any operation. And as the operation of art is not infinite, so neither is there infinite subordination of parts.[416] When two minima touch one another, they do not do so with their whole body, _or any part of it_, but one with its terminus or limit may touch several others; no body touches another with the whole of itself or a part, but with either the whole or the part of its _limiting_ surface. The _terminus_ of a thing is therefore no part of it, and by implication not a minimal part. Hence there are two kinds of _minima_ concerned--that of the touching body, or part, and the minimum of that by which the contact is effected, the _terminus_.[417] The atom, which is the minimal sphere, touches in the absolutely minimal point, the smallest _terminus_. Other spheres do not touch in a point simply, but in more than one, or in a plane circle.[418] By adding limit to limit we never obtain a magnitude; the _terminus_ is no part, and therefore if in contact it would touch with its whole self, so that magnitude is not made up of _termini_, whether points, atoms, lines, or surfaces which are termini; and this was the false ground on which the Aristotelians denied the possibility of the atom. It remained to ask if the _termini_ were infinite, since the atoms were not; but it was clear that their number was determined by that of the atoms. For two limits do not touch one another:--"They do not cohere or make a _quantum_, but through them others in contact with one another make a _contiguum_ or _continuum_."[419] It may be added that if the parts of a divisible body were infinite in number, the parts of the whole would be equalled by the parts of the half, for in the infinite there can be no greater and less. In the infinite, as we have seen above, there is no difference between palms, digits, miles, between units and thousands, nor in the infinite time that has elapsed are there more months than years, more years than centuries. If any one set of these were less than the others it would be finite, and if one finite number may be applied to the whole, then the whole is finite.[420] The force of the Achilles dilemma was derived from the false idea that the _minimum_ of one kind had some relation to that of another kind, _e.g._ that of time to that of motion, that of impulsive force to that of the motion produced. A thing of one kind does not define or measure a thing of another, and the duration of one does not compare in the same sense with the duration of another. Parts of different things are only equivocally called parts, and _minima_ are _minima_ only according to their proper (and diverse) definitions; therefore one is not measured by another, except in a rough way, for practical purposes.[421]

[Sidenote: The vacuum. Atoms spherical.]

As the atoms come into contact with one another, not in all points of their surface, but in a definite number, it follows that there is a space between them, in the interstices; it was this thought which led Democritus to posit a vacuum.[422] The figure of the corporeal minimum must be spherical, for any mass which has projections can always be thought of as smaller, when these projections have been removed; and nature itself suggests this, by the gradual rounding off of substances through time, and the apparent roundness and smoothness of rough and jagged bodies when the observer is at a distance.[423] Diversity of forms of composite bodies results easily from spherical atoms, through differences in situation and order, differing amounts of vacuum and solid; but a simple vacuum with solid bodies is not sufficient,--there must be a certain matter through which the latter cohere together.[424] Although all other determinations may be abstracted from, figure at least must be predicated of the atoms; quantity cannot be asserted of that which is thought to be unfigured. These determinations of the minimum, though not given to sense, may nevertheless be made object of thought, by analogy or inference from the combinations of sensible _minima_ in larger composites, the same forms of aggregation being repeated in the higher which occur in the lower forms.[425]

From the consideration of mathematical figures as consisting of _minima_, Bruno attempted both to remodel and to simplify the existing mathematical theory, and, unfortunately fell foul of the new analytical mathematics, the theory of rationals and of approximations, which at that time was receiving marked extensions, and which has since justified itself so completely by results. It is true he did not entirely reject it, but he regarded it as merely an artifice for rough practical measurements. The true measure is always the _minimum_, inferred by analogy from the combinations of greater parts, which are perceived by sense. Thus the minimal circle, after the atom itself, consists of seven minima, the minimal triangle of three, and the minimal square of four, and as each figure increases not by the addition of one atom merely, but by a number determined by the original number of atoms in the figure, it follows that no one figure is ever equal to another. Thus the second triangle is of six minima, the second square of nine, the second circle of nineteen. The "squaring of the circle" is therefore impossible,[426] although it may be approximately reached through the ultimate coincidence of arc and chord, by which the circle becomes equal to a polygon with an infinite number of sides.[427] This, however, is only an approximation of sense, which fails to observe the infinitesimal differences that are caused by the existence of a few atoms, more or less, in a figure. They are visible to the eye of reason, which comprehends that no two figures in nature are ever exactly equal. In exact geometry the number of one species of figure has nothing in common with that of another. It is clear, however, that even on his own ground Bruno was in error in this regard; for example, the seventh triangle and the fifth square are each composed of thirty-six minima.[428] But it is hardly necessary to take seriously his teaching in this respect. He was wholly governed by the belief in the infinite diversity of nature, and the absolute incommensurability of any member of one species of beings with one of a different species. "Since a definite minimum exists, it is not possible either in reality or in thought for a square to be equalled by a circle, nor even a square by a pentagon, a triangle by a square, nor in fine any species of figure by a figure of another species; for difference in the number of sides implies also difference in the order and number of parts. As figures in this respect are as numbers, and one species of number cannot be equalled by another either 'formally' or fundamentally (_i.e._ either in idea or in fact), we can never make an equilateral figure of any kind equal to one of another by first parts."[429] Where this transformation is apparently carried out, as where a cube of wax is moulded to another figure, the result is due to the varying degrees of density in the different parts of the material; no solid parts are added or subtracted, but the disposition and extent of the pores or vacua are altered. But no argument can be drawn from this rough method, for the principles of practice are different from those of science.[430]

The latter principles are then applied boldly to geometrical science: thus it is shown that an angle, although it may be multiplied indefinitely, can be divided only into two parts; all its lines, it is understood, consisting of _fila_ or rows of atoms;[431] that the circle has not an infinite number of radii, for from the circumference to the centre only six such lines can be drawn;[432] that not every line can be divided into two equal parts, for the physical line or _filum_ may, naturally, consist of an odd number of atoms;[433] in any case geometrical bisection can at best be a near approximation,--though the two halves be apparently equal, they may really differ by many atoms. On this basis, in the fourth and fifth books of the _De Minimo_, Bruno offers a simplification of the geometry of Euclid. As nature itself is the highest unification of the manifold, and the monad is the unity and essence of all number, so we are taught to pass "from the infinite forms and images of art to the definite forms of nature, which the mind in harmony with nature grasps in a few forms, while the _first mind_ has at once the potentiality and the reality of all particular things in the (simple) monad."[434] In accordance with the method of simplification suggested by this doctrine, Bruno sets himself to show that the greater part of Euclid may be intuitively presented in three complicated figures, named respectively the _Atrium Appollinis_, _Atrium Palladis_, and _Atrium Veneris_. He hoped that by this means, "if not always, for the most part at any rate, without further explanation, the demonstration and the very evidence of the thing might be presented to the senses of all, without numbers,--not after the partial method of others, who in considering a statue take now the foot, now the eyes, now the forehead, now other parts separately,--but explaining all in each and each in all."[435] It is no part of the purpose of this book to go at length into the mathematics of Bruno, which unfortunately have not yet met with a competent exposition. Apart from the difficulty of the matter itself, the poetical form and setting of his theorems is an additional stumbling-block in the way of understanding. Bruno was put to many shifts in order to give a poetical colouring to the most prosaic of subjects.

We have gone thus fully into the detail of Bruno's atomic theory, more so perhaps than its intrinsic value seems to demand, because this aspect of his doctrine is the most important philosophically, and has exercised the greatest influence upon the course of speculation. It also provides most clearly an exemplification of the return which was made, or thought to be made, by the Renaissance to the older pre-Aristotelian philosophy and science. The rejection by Aristotle and his scholastic followers of the atomic theory of Leucippus and Democritus had been based upon the identification of space and body. The possibility of a vacuum in the corporeal world was denied, on the ground that discreteness was inconsistent with the continuity which was felt to be a necessary condition of space. Accordingly, the reintroduction of the atom was possible only in one of two ways--either by the distinction between body and space, or by the application of the atomic constitution of body to space itself. The former and truer solution was not open to Bruno. His time was still too much under the domination of Peripatetic thought for him to be able to take the important step of critically separating these two notions. The latter way, therefore, was that which he followed. Hence the curious attempt to remodel mathematical theory on the basis of the atom, which we have described above, and the reduction of mathematical certainty to an illusion of sense. Figure is to be found only in the combinations of atoms; and owing to the spherical form of the atom, the infinite number of them existing in any body which is presented to sense, and the space which lies between their surfaces, mathematical equality and exactness are impossible. Neither straight line, therefore, nor perfect circle are to be found in reality. Mathematics, which should be based upon, or which presupposes, continuity, is confounded with physics, which presupposes the analysis of body into discrete, impenetrable atoms. Physical atomism finds its justification in the experienced fact of resistance, which is the primary quality of body as perceived by our senses. In mathematical space, on the other hand, we abstract from all qualities except that of dimension only. Resistance would be inexplicable were it possible to proceed _ad infinitum_ in dividing matter; it implies an ultimate irreducible and indestructible unit, whether we regard this unit as a centre of force or as an inert substance merely.

The same influence of Aristotelian thought led Bruno to posit a subtle matter, the Ether, as filling up the interstices between the atoms. Space and body having been identified, it was seen that a vacuum was inconsistent with the nature of things. The Aristotelian _plenum_ was reintroduced in this form, that there might be some reality where the discrete atoms were not. The bolder step of asserting the fact, and indeed, the necessity of a vacuum as a presupposition of knowledge of the material world, was not taken until there appeared the work of Gassendi, by whom the final blow was given to the old conception of body and space, and through whom the critical separation of the one from the other was first rendered possible. It is curious that Bruno did not think of applying to the continuous ether any geometrical measure; had he done so, he would have understood the value of the new theory of infinitesimals and irrationals which he opposed so strongly. Again, had he carried out more fully the distinction which he draws between the atom and the _terminus_ or limit, the same result would have followed. Pure geometry is the geometry of the limit; for the limit is not only between atom and atom, or body and body, but also between atom and vacuum or ether. In this sense it is both continuous and figured, the compatibility of which qualities Bruno had denied; the continuous is measured, not by making it discrete, but by making the number, the measure, fluid or continuous.

[Sidenote: Metaphysical atomism.]

Lasswitz has shown that there are in Bruno's theory three distinct aspects, not, however, clearly separated one from another, of the atomic hypothesis: they may be named severally the metaphysical, the physical, and the critical aspects. From the metaphysical point of view the atom is the ultimately simple, indeterminate substance of things; its conception results from the effort to find the real substance which is outside of, and unaffected by, the change and decay apparent on the surface of things, but felt to be unreal. Simplicity, unity, substance, is that which is sought, an abiding somewhat underlying the flux of the universe, which is regarded as an illusory appearance to sense. From this aspect it is that the identity of _minimum_ and _maximum_, of the least with the greatest, is to be explained. Number, plurality, and diversity no longer apply to the absolutely simple: all are determinations of human and finite origin which are here no longer valid. In the simple all contraries coincide, for the very reason that it has no determinations in itself; even the highest qualities which men would attribute to God, for example,--justice and goodness,--are improperly predicated of him, for as in him the greatest and the least coincide, so do goodness and evil and all other contrary qualities. In this respect Bruno was following closely in the footsteps of Nicolaus of Cusa.

[Sidenote: Physical Atomism.]

[Sidenote: Critical Atomism.]

From the second point of view, that of _physical_ atomism, the atom is nothing more than a hypothesis to explain the constitution and qualities of nature as we experience it. We seek to account for the differences in material bodies and in their ways of acting upon one another by the interaction of ultimate elements of which the nature and laws may be variously interpreted. Of this point of view also there are traces in Bruno, although for it he had least regard. He does not attempt, for example, to apply the theory of the atoms to explain the four elements which had come down from Aristotle. He leaves them practically intact, and we have seen that they form a standing difficulty in the way of a consistent theory. The earth alone is atomic in its nature; water, air, and fire seem alike fluid and continuous in quality, but wherein their difference from one another consists he was unable, or did not care, to make clear. Perhaps, if we take his view at its best, we should say that all three represent strata, varying in density, of the one fluid and all-pervading ether. Had he worked out this conception, which was evidently present, on occasions, to his mind, he would have given an example of what is meant by physical atomism. But this was left for another century to fulfil. From the third or _critical_ point of view, which inquires into the presuppositions or the possibility of knowledge, Bruno may be regarded as being, to some extent, a forerunner of Kant, in the stress he lays upon the relation of the minimum to measure or knowledge, and in his doctrine of the relativity of the conception of the _minimum_. The minimum, instead of a last of division, becomes a first of composition--a ground which we must necessarily assume in order to account for the experienced fact of composition. To know a composite is to measure it, and measurement implies the _minimum_ or first part, without which quantity in any form cannot be explained. As the comparison of numbers with one another, their determination as greater or less, is only possible on the assumption of a unit, a common measure to which each may be referred, so the comparison of bodies with one another, as to quantity and quality alike, demands a corporeal minimum, to which their differences must be reduced. This relation to knowledge carries with it the relativity of the minimum according to the subject-matter with which the knower is for the time being concerned. If all knowledge is of the same type, then in each application of it--each subdivision of knowledge as a whole--there is presupposed the corresponding minimum. That which is least in one sphere may be greatest in another; that which is element of one science may be that which another seeks to analyse into lesser constituents. The celestial body, which is a highly complex combination of elements, may be the unit of astronomical science. The phrase, which is the unit of the rhetorician, is analysed by the logician and the grammarian into terms and words; these are analysed by another science into syllables and letters; these by the mathematician into lines and points. Thus every science has its own (relative) minimum. Only one _minimum_ is _absolutely_ so named,--God as the monad of monads. It is to be noted that the relativity of the monad is dependent upon the origin of its conception, in the conditions of knowledge; it is because quantity is universal that a minimum is necessary, and it is because quantity differs in kind, in each subject of knowledge,--because it is, in scholastic phrase, equivocally applied in the different cases,--that the minima differ from one another. The minimal number is no measure of the minimal body nor of the geometrical figure, and the numbers which are in use among men are not those which may be employed by other and higher rational beings. Thus, even number itself is a relative determination; ten horses, said Bruno, are not _really_ equal to ten men, but only _conventionally_.

The ancient atomism upon which Bruno founded his theory was, at any rate in its traditional rendering, frankly materialistic. It admitted nothing but atoms and the void, all things else being dependent upon the _composition_ of atoms, which itself, and all that results from it, is merely an appearance to sense, without corresponding reality in nature. All physical operations were explained by mechanical arrangement and movement of the atoms. The method which was pursued thus unscientifically, without consciousness of the extent of its validity, modern atomic theory has followed scientifically, with full comprehension of its bearings, and perhaps without due consideration of its limits. Bruno tells us that he had at one time been an adherent of Democritus' atomic theory, but on reflection had been unable to rest satisfied with his materialistic account of the nature of things. In this case also he showed himself unable to get rid of the ties which bound all the thought of his time--even that thought which most believed itself to be free.

Aristotle's distinction of form and matter in nature, of pure activity and pure passivity, had still sufficient influence to render even in Bruno's time a purely mechanical treatment of nature an impossibility. The opposing school, the Neo-Platonism which attracted so many minds of that period, because of its supposed inconsistency with Aristotle's system, was itself an offshoot, to some extent, of that system, and was still less scientific in its tendency. Mysticism, of which it was partly a cause and partly an effect, lent its weight also against any mechanical interpretation of nature. Thus even while apparently governed by scientific aspiration, Bruno gives a teleological scheme of the universe which renders any scientific explanation of it impossible. Not only, as we have seen, is the ether identified with the first substance, spirit, or soul of the universe, but also the greater and lesser organic bodies are governed each by its individual soul, which is somehow distinguished from the universal spirit, and within each of these is an infinite number of smaller living bodies. In other words, the atoms themselves are animated _virtually_, if not _actually_. This animistic interpretation is in direct conflict with the mechanical interpretation which science has followed, and which it must continue to follow if it is to produce any result. Thus, motion and the changes of composition that derive from motion are explained not by the mechanical impact of atoms and bodies upon one another, but by the action of the intrinsic soul in each being, which causes the motion of the body, in accordance with its need and desire of self-preservation. All motion, even the slightest, is thus explained by a final cause. In the whole universe also, the constantly occurring changes and transformations are due to a similar final cause--the need for each thing to become _explicitly_ that which it already is _implicitly_, _i.e._ the whole of reality. It required once more a critical separation of the spheres of validity of the respective conceptions of nature and spirit, such as Kant attempted, before full scope could be given to mechanical interpretation on the one side, and teleology restricted to the domain of spirit only on the other.