IN answer to the objection of Aristotle, that if the elements are generated by a dissolution into planes, it is absurd to suppose that all things are not generated from each other,—Proclus observes, "that we must assert the very contrary. For the phænomena do not accord with those who transmute earth, and move things immovable. For we never see earth changed into other things; but terrestrial natures are changed, so far as they are full of air or water. All earth, however, is unchangeable,
because earth alone becomes, as it were, ashes, or a calx. For in metallic operations, the whole of the moisture in metals is consumed, but the ashes remain impassive. Not that earth is entirely impassive to other things; for it is divided by them falling upon it; yet the parts of it remain, until again falling on each other, they from themselves make one body. But if it should be said that earth, on account of its qualities, is changed into other things, being itself cold and dry, earth will be more swiftly changed into fire than into water; though water, indeed, appears to be burnt, but earth, when subsisting by itself, (i.e. when it is pure earth, and earth alone,) is not burnt." He adds, "And the heaven, indeed, is neither divisible nor
imitable; but the earth existing as the most ancient of the bodies within the heaven, is divisible, but not mutable; and the intermediate natures are both divisible and mutable."
Aristotle observes, "that earth is especially an element, and is alone incorruptible, if that which is indissoluble is incorruptible, and an element. For earth alone is incapable of being dissolved into another body." The philosopher Proclus replies to this objection, yielding to what Aristotle says about earth, viz. that it is perfectly incapable of being changed into the other three elements. And he says, "that Plato, on this account, calls it the first and most ancient of the bodies within the heaven, as unchangeable into other things, and that the other elements give completion to the earth, in whose bosom they are seated, viz. water, air, and sublunary fire. But in consequence of being, after a manner, divided by the other elements, it becomes one of them; for division is a passion which exterminates continuity. If, however, it suffers being divided by the other elements, and energises on them, embracing, compressing, and thus causing them to waste away, it is very properly co-divided with those things from which it suffers, and on which it energises according to the same passion in a certain respect. For there is a division of each,
though the more attenuated are divided by the more sharp in one way, as in the arts by saws, augers, and gimlets; and the more gross in another way, by trampling and compression."
In the next place, Aristotle says, "But neither in those things which are dissolved, is the omission of triangles reasonable. This, however, takes place in the mutation of the elements into each other, because they consist of triangles unequal in multitude."
The philosopher Proclus here observes, "that in the dissolution of water into air, when fire resolves it, two parts of air are generated, and one part of fire. But when, on the contrary, water is generated from air, three parts of air being resolved, the four triangles which are mingled together from the same cause, viz. from condensation, together with two parts of air, make one part of water." He adds, "But it is not at all wonderful, that they should be moved in a certain form; for it must be granted, that in all mutations there is something without form, to a certain extent; but being vanquished by some form, they pass into the nature of that which vanquishes. For we also acknowledge, that, in the mutation of the elements with which we are conversant, certain half-generated parts frequently remain."
Aristotle adduces, as a fourth absurdity, "that
this hypothesis makes the generation of body simply, but not of some particular body. But if body is generated upon body, it was before shewn that there must necessarily be a separate vacuum, which the authors of this hypothesis do not admit. For if body is generated, it is generated from that which is incorporeal. It is necessary, therefore, that there should be some void place the recipient of the generated body. Hence, if they say that body is generated from planes, it will not be generated from body; for a plane has length and breadth alone." To this, however, Proclus replies, "that natural planes are not without depth; for if body distends the whiteness which falls upon it, it will much more distend the planes which contain it. But if the planes have depth, the generation of fire will no longer be from that which is incorporeal; but the more composite will be generated from a more simple body."
In the next place Aristotle observes, "that those who attribute a figure to each of the elements, and by this distinguish the essences of them, necessarily make them to be indivisibles. For a pyramid or a sphere being in a certain respect divided, that which remains will not be a sphere or a pyramid. Hence, either a part of fire is not fire, but there will be something prior to an element, because every body is either an element
or from elements; or not every body is divisible." Proclus, in reply to this, "blames him who makes lire to be a pyramid, and who does not abide in the Platonic hypothesis, since Plato says that a pyramid is the figure of fire; but he does not say that it is fire. For fire is a collection of pyramids, any one of which is invisible, on account of its smallness; nor will fire, so long as it is divided into fire, be divided into pyramids. One pyramid, however, is no longer fire, but the element of fire, invisible from its smallness. If, therefore, this pyramid were divided, it would neither be an element, nor composed of elements, since it would not be divided into pyramids or planes. And why is it wonderful that there should be something inordinate in sublunary bodies? For, in the mutation of the elements with which we are conversant, there is something inordinate." Proclus adds, "that certain differences also are produced, which occasion pestilential consequences in the whole genus, and turn the elements into a condition contrary to nature. But what impossibility is there," says he, "that this section of an element being taken, and fashioned into form and figure by atoms, should again become a pyramid, or some other element, in consequence of being assimilated to the natures which comprehend and compress it."
The sixth argument of Aristotle endeavours to shew, that if the elements are fashioned with the above-mentioned figures, there must necessarily be a vacuum which is not even asserted by the advocates for planes. But he spews this from there being but few figures, both in planes and solids, which are able to fill the place about one point, so as to leave no vacuum. *
Proclus observes, in reply to this argument of Aristotle, "that the elements being placed by each other, and supernally compressed by the heaven, the more attenuated are compelled into the places of the more gross. Hence, being impelled, and entering into the place about one point, they fill up the deficiency. For Plato also assigns this as the cause of no vacuum being left, viz. that less are arranged about greater things. For thus the cavities of the air have pyramids which fill up the place; those of water have dispersed octaedra; and those of earth have all the figures; and no place is empty."
In the seventh argument, Aristotle says, "that all simple bodies appear to be figured in the place which contains them, and especially water and air." He adds, "it is impossible, therefore, that the figure of an element should remain; for the whole would not on all sides touch that which contains it. But if it were changed into another figure, it would no longer be water, if it differed in figure; so that it is evident that the figures of it are not definite," &c.
Proclus, in opposition to this seventh argument, observes, "that he does not admit that the elements have a characteristic figure, since they can neither have it stably, nor abandon it." He also says, "that it is not the wholenesses of these four
bodies which are fashioned with these figures, but the elements of these, viz. those small and invisible bodies from the congress of which these sensible natures, fire, water, air, and earth, are produced. But the wholes of the elements have a spherical figure, being on all sides assimilated to the heaven. For each of them has something better than its own characteristic property, from more divine natures, just as things which approximate to the heaven have a circular motion. It is evident, therefore, that the last of the pyramids which are with the circumambient, (i.e. which are in contact with the sphere of the moon, this being the sphere in which fire is proximately contained,) though they consist of plane triangles, yet, being compressed, they become convex, in order that they may be adapted to the cavity of the heaven. But the parts existing in other things, as in vessels, and receiving configuration together with them, do not destroy the figure of the elements. For the bodies which contain others are from right-lined elements, and nothing prevents them from concurring with each other. But we, expecting to see the superficies of the containing bodies to be cylindrical or spherical, in consequence of being ignorant that they also consist of right-lined elements, are involved in doubt. All the containing natures, therefore,
were from the same things as the natures which they contain, and all are adapted to each other, according to planes."
In the eighth argument, Aristotle says, "that neither flesh nor bone, nor any other composite, can be generated from the elements themselves, because that which is continued is not generated from composition, nor from the conjunction of planes: for the elements are generated by composition, and not those things which consist of the elements."
Proclus, in objection to this, says, "that composition is not produced from air alone, nor from water alone. In these, therefore, things that have the smallest parts, being assumed between those that have great parts, fill place, and leave no void. But if this is opposition, and not union, you must not wonder; for it is necessary that they should be distant from each other. And if, when placed by each other, they are with difficulty separated, neither is this wonderful: bodies which consist of larger planes, not being naturally adapted to yield to those which consist of smaller, nor those which are composed of firmer, to those which derive their composition from easily movable planes."
Aristotle, in the ninth argument, says, "that if the earth is a cube, because it is stable and abides; and if it abides not casually, but in its proper
place, and is moved from a foreign place, if nothing impedes it; and if this, in a similar manner, happens to fire and the other elements,—it is evident that fire, and each of the elements in a foreign place, will be a sphere or a pyramid, but in its proper place a cube."
In opposition to this ninth argument, Proclus says, "that though the elements are in their proper places, yet such as consist of easily movable figures are not without motion; for pyramids are always moved from the dissimilitude of the vertex to the base. Thus also with respect to air, the elements of it, when it exists in its proper place, are assimilated to things perpetually flowing; and the elements of water love collision. For the summits are adjacent to the bases of their similars, and being impelled, they strike against the whole in the place in which each is contained. But being thus moved, they imitate the motion in a circle, neither being moved from the middle nor to the middle, but revolving about each other in their own place. The elements of earth, however, remain, because they have their summits the same with their bases. But nothing similar acts on the similar, whether they possess similitude according to figures, or according to power, or according to magnitude."
"Farther still," says Aristotle, "if fire heats
and burns through its angles, all the elements will impart heat, but one perhaps more than another; since all of them will have angles; as, for instance, the octaedron and the dodecaedron. And according to Democritus, a sphere also burns, as being a certain angle; so that they will differ by the more and the less. This, however, is evidently false."
Proclus, in opposition to this tenth argument, says, "that it is improperly assumed that an angle is calorific, and that a false conclusion is the consequence of this assumption. For Timæus assumes from sense, that sharpness and a power of dividing are certain properties of heat. But that which cuts, cuts not simply by an angle, but by the sharpness of the angle, and tenuity of the side. For thus also the arts make incisive instruments, and nature sharpens the angles of those teeth that are called incisores, and giving breadth to the grinders, has attenuated the sides. An acute angle also is subservient to rapid motion. Hence a power of this kind is not to be ascribed to an angle simply, but to the penetrating acuteness of the angle, the incisive tenuity of the side, and the celerity of the motion. It is likewise necessary that magnitude should be present, as in the pyramid, that it may forcibly enter. If, therefore, in fire alone there is acuteness of
angle, tenuity of side, and swiftness of motion, this element alone is very properly hot. This, however, is not the case with all fire, but with that alone which consists of larger pyramids; on which account, as Timæus says, there is a certain fire which illuminates indeed, but does not burn, because it is composed of the smallest elements. And according to this, fire is visible."
Aristotle adds, "at the same time also it will happen that mathematical bodies will burn and impart heat; for these likewise have angles; and atoms, cubes, spheres, and pyramids, are inherent in them, especially if, as they say, these are indivisible magnitudes. For if some of them burn, and others do not, the cause of this difference must be assigned, but not simply so as they assign it."
Proclus, well opposing what is here said, does that which Aristotle desires, viz. he assigns the difference consequent to the hypothesis according to which some bodies burn, but mathematical bodies do not burn. For Plato says, that burning bodies are material and moved figures; on which account also he says, that ϐ is added to the name, this letter being the instrument of motion. Not every thing, therefore, which is angular, is calorific, unless it is acute-angled, is attenuated in its sides, and may be easily moved.
Again, Aristotle says, "let it be reasonable, therefore, that to cut and divide should be accidents to figure; yet, that a pyramid should necessarily make pyramids, or a sphere spheres, is perfectly absurd, and is just as if some one should think that a sword may be divided into swords, or a saw into saws."
To this also Proclus replies, "that fire dissolves the elements of that which it burns, and transmutes them into itself. But a sword does not act upon the essence of that which it cuts. For it does not dissolve the essence of it, but by dividing it, makes a less from a greater quantity; since it has not its figure essentially, but from accident. If, therefore, nothing which cuts changes that which is cut into the essence of itself, nor dissolves the form of it, how can it make a division into things similar to itself? But it may be said, Let bodies which are burnt be dissolved into triangles, for instance, water and air, and the elements of them, the icosaedron and octaedron, yet what is which composes the triangles of these into the figure of fire, viz. into the pyramid, so as that many such being conjoined, fire is produced? Plato therefore says, in the Timæus, that the triangles being dissolved by fire, do not cease to pass from one body into another until they conic into another form; for instance, the triangles of
the icosaedron, which are divisible into octaedra, or rather till they pass into fire, which is of a dividing nature. For if they are composed into the nature of fire, they cease their transition; since similars neither act upon, nor suffer from each other. But it will be well to hear the most beautiful words themselves of Plato: 'When any one of the forms (says he), becoming invested by fire, is cut by the acuteness of its angles and sides, then, passing into the nature of fire, it suffers no farther discerption. For no form is ever able to produce mutation or passivity, or any kind of alteration, in that which is similar and the same with itself; but as long as it passes into something else, and the more imbecile contends with the more powerful, it will not cease to be dissolved.' It is evident, however, that the planes are not composed casually, and as it may happen, at one time in this, and at another in that figure; but that which dissolves them exterminates the aptitude which they had to that figure, for instance, to the icosaedron, this aptitude being more gross and turbulent, and transfers it to the purer aptitude of the air which is near. And in the first place, they acquire a bulk from octaedra. Afterwards being dissolved by fire, they are more purified and attenuated, and become adapted to the composition of a pyramid. But it is evident that
to whatever form they are adapted, from their figure, they easily receive this form, and on this account, from water air is first generated, and then from air fire."
In the next place, Aristotle says, "that it is ridiculous to attribute a figure to fire for the purpose of dividing alone; for fire appears rather to collect and bring boundaries together, than to separate. For it separates accidentally things which are not of a kindred nature, and collects especially those which are."
Proclus opposes this argument, and says, "that the very contrary is true. For fire essentially separates, but collects things together accidentally; since to take away things of a foreign nature from such as are similar, predisposes the concurrence of the latter into each other, and their tendencies to the same thing. For all fiery natures, according to all the senses, have a separating power. Thus, heat separates the touch, the splendid separates the sight, and the pungent the taste. And farther still, all medicines which are of a fiery nature have a diaphoretic power. Again, every thing which collects strives to surround that which is collected, at the same time compelling it; but fire does not endeavour to surround, but to penetrate through bodies." Proclus adds, "that according to those, also, who
do not give figures to the elements, fire is thought to rank among things of the most attenuated parts. But a thing of this kind is rather of a separating nature, entering into other things, than of a collective nature. That what essentially separates, however, belongs to fire, is evident from this, that it not only separates things heterogeneous from each other, but every particular thing itself. For it melts silver, and gold, and the other metals, because it separates them."
Aristotle farther observes, "in addition to these things, since the hot and the cold are contrary in capacity, it is impossible to attribute any figure to the cold, because it is necessary that the figure which is attributed should be a contrary; but nothing is contrary to figure. Hence all physiologists omit this, though it is fit either to define all things or nothing by figures."
This objection also, Proclus dissolving says, "that the argument of Aristotle very properly requires that a figure should be assigned adapted to the cold; but that it is necessary to recollect concerning heat, how it was not said that heat is a pyramid, but that it is a power affective, through sharpness of angles and tenuity of side. Cold, therefore, is not a figure, as neither is heat,
but it is the power * of a certain figure. And as heat is incisive, so cold has a connective property. And as the former subsists according to sharpness of angles and tenuity of sides, so, on the contrary, the latter subsists according to obtuseness of angles and thickness of sides. Hence, the former power is contrary to the latter, the figures themselves not being contrary, but the powers inherent in the figures. The argument, however, requires a figure, not in reality contrary, but adapted to a contrary power. Such figures, therefore, as have obtuse angles and thick sides, have powers contrary to the pyramid, and are connective of bodies. But such figures are the elements of three bodies. Hence all things that congregate, congregate through impulsion; but fire alone, as we have observed, has a separating power. †
Aristotle adds a fifteenth argument, after all that has been said, objecting to magnitude, and shewing that the Pythagoreans make the power of cold a cause, as consisting of great parts, because it compresses and does not pass through pores, as is indicated by what Plato says in the Timæus about cold. * Proclus, however, in opposition
to this, observes as follows: "We do not determine the elements of simple bodies by magnitude alone, but also by thinness and thickness, by sharpness and facility of motion, and by immobility and difficulty of motion, which give variety to forms, and cause things which have the same form, not to differ by magnitude alone. For the magnitude of planes makes the largeness or smallness of parts in bodies; since the parts of them are called elements. Thus, the pyramids of fire, of which fire consists, are the parts of fire, and octaedra are the parts of air. For the octaedron is greater than the pyramid, both being generated from an equal triangle. But the composition, together with so great a multitude, make the acute and the obtuse. For more or fewer triangles coming together, an angle, either acute or obtuse, is generated; an acute angle, indeed, from a less, but an obtuse from a greater multitude. But the characteristic property of the planes produces facility or difficulty of motion; these planes existing in a compact state, through similitude, but being prepared for tendency
through dissimilitude. Large pyramids, therefore, do not belong to things which refrigerate, but to the larger parts of fire; just as larger octaedra belong to the larger parts of air, and larger icosaedra to larger parts of water. For from this cause waters are thin and thick, and airs are attenuated and gross; since it is evident that these are determined by quantity."
11:* In order to understand what is said by Proclus in answer to the objections of Aristotle, it is requisite to relate, from Simplicius, the hypothesis of the Pythagoreans and Plato, respecting the composition of the elements from the five regular bodies. "They supposed two primogenial right-angled triangles, the one isosceles, but the other scalene, having the greater side the double in length of the less, and which they call a semi-triangle, because it is the half of the equilateral triangle, which is bisected by a perpendicular from the vertex to the base. And from the isosceles triangle, which Timæus calls a semi-square, four such having their right angles conjoined in one centre, a square is formed. But the union of six such triangles † having eight angles, forms a cube, which is the element of earth. The semi-triangle, however, constitutes the pyramid, the octaedron, and the icosaedron, which are distributed to tire, air, and water. And the pyramid, indeed, consists of four equilateral triangles, each of which composes six semi-triangles. But the octaedron consists of eight equilateral triangles, and forty-eight semi-triangles; and the icosaedron is formed from twenty equilateral triangles, but one hundred and twenty semi-triangles. Hence, these three, deriving their composition from, one element, viz. the semi triangle, are naturally adapted, according to the Pythagoreans and Plato, to be changed into each other; but earth, as deriving its composition from another triangle specifically different, can neither be resolved into the other three bodies, nor be composed from them."
11:† Viz. of six squares, or six times four isosceles triangles, whose right angles are conjoined in one centre.
17:* In planes this can only be accomplished by the equilateral triangle, the square, and the hexagon; viz. by six equilateral triangles, four squares, and three hexagons. But in solids, the pyramid and cube alone can fill the place, which is about one point. Of the first part of this admirable theorem, which is also mentioned, with the praise it deserves, by Proclus in his Commentary on the First Book of Euclid, the following demonstration is given by Tacquet.—In order that any regular figures frequently repeated may fill space, viz. may form one continued superficies, it is requisite that the angles of many figures of that species composed about one point make four right angles; for so many exist about one point as is evident from Coroll. 3. Prop. 13. of the First Book of Euclid. Thus, for instance, that equilateral triangles may fill place, it is requisite that some angles of such triangles composed about one point should make four right angles. But 6 equilateral triangles make 4 right angles; for 1 makes 2/3 of one right angle, and therefore 6 make 12/3 of 1 right, i.e. 4 right angles. The 4 angles of a square, also, as is evident, make 4 right angles; and this is likewise the case with the 3 angles of a hexagon. For one makes 4/3 of 1 right, and consequently 3 make 12/3 of 1 right, that is, again 4 right. But that no other figure can effect this, will clearly appear, if, its angle being found, it is multiplied by ally number; for the angles will always be less than, or exceed, 4 right angles.
28:* It is well observed by Simplicius, (De Cœlo, p. 142,) "that Plato and the Pythagoreans by a plane denoted something more simple than a body, atoms being evidently bodies; that they assigned commensuration and a demiurgic analogy [ i.e. active and fabricative powers] to their figures, which Democritus did not to his atoms; and that they differed from him in their arrangement of earth."
28:† Simplicius here remarks, "that it may be doubted, how the powers which are in figures, being contrary, the figures themselves will not be contrary; for powers are adapted to the things by which they are possessed. Perhaps, therefore, he H. e. Proclus] calls the four figures, the pyramid and the other regular bodies, which not being contrary, their powers are contrary; since their powers are not according to their figures. For neither the thick nor the thin, neither that which has large nor that which has small parts, neither that which is moved with difficulty nor that which is easily moved, are the differences of figure. Perhaps, too, neither are acuteness nor obtuseness of angles simply the differences of figure, since neither is an angle simply a figure. If, therefore, the dispositions of the hot and the cold, which are contrary, are effected according to these contrarieties, no absurdity will ensue. Hence the proposition which says, that things which are determined by figures are not contrary, requires a certain circumscription. For they are not contrary according to figures, yet they are not prevented from having contraries. If, however, some one should insist, that contrarieties are according to figures, it is necessary to recollect that Aristotle in this treatise says, that there is also in figures a certain contrariety."
29:* What Plato says on this subject in the Timæus, is as follows: "The moist parts of bodies larger than our humid parts, entering into our bodies, expel the smaller parts; but not being able to penetrate into their receptacles, coagulate our moisture, and cause it through equability to pass from an anomalous and agitated state, into one immovable and collected. But that which is collected together contrary to nature, naturally opposes such a condition, and endeavours by repulsion to recall itself into a contrary situation. In this contest and agitation, a trembling and numbness takes place; and all this passion, together with that which produces it, is denominated cold."
