PART IV — COMMENTARIES ON THE PHYSICS AND ON THE HEAVENS¶
On the Mathematician and the Physicist, Modes of Being, Place, the Continuum Not Composed of Indivisibles, and Geometric Motion
Source Work: In octo libros Physicorum & In libros De caelo et mundo expositio (c. 1268–1273) · Switch to Diplomatic Latin · Summary · GEOMETOR Notes
Physics Book II, Lecture 3 — How the Mathematician Differs from the Natural Philosopher and Why There Is No Falsehood in Those Who Abstract¶
Latin Locus: Phys. II, lect. 3 · Commentary: Notes on Phys. II, lect. 3
[lib. 2 l. 3 n. 1] After the Philosopher has shown what nature is and in how many ways it is spoken of, here he consequently intends to show what things natural science considers. And it is divided into two parts: in the first, he shows how the natural philosopher differs from the mathematician; in the second, he shows how far the consideration of natural science extends, at Since, however, nature, etc. Concerning the first he does three things: first, he raises the question; second, he sets forth arguments bearing on the question, at For indeed planes, etc.; third, he resolves the question, at Concerning these things, therefore, he treats, etc. He says first, therefore, that after it has been determined in how many ways nature is spoken of, one must consider wherein the mathematician differs from the natural philosopher.
[lib. 2 l. 3 n. 2] Then when he says, For indeed planes, etc., he sets forth arguments bearing on the question. The first of these is as follows. Whichever sciences consider the same subjects are either the same science, or one is a part of the other; now the mathematical philosopher considers points, lines, surfaces, and bodies, and similarly so does the natural philosopher (which he proves from the fact that natural bodies have planes, that is, surfaces, and solids, that is, three-dimensional solidities, and lengths and points; and it is necessary that the natural philosopher consider all things that belong to natural bodies); therefore it seems that natural science and mathematics are either the same, or one is a part of the other. The second argument he sets forth at Moreover, astronomy, etc. And concerning this argument he first raises the question whether astronomy is altogether other than natural philosophy, or is a part of it. For it is manifest that it is a part of mathematics: hence, if it is also a part of natural philosophy, it follows that mathematics and physics agree at least in this part. Now that astronomy is a part of physics he proves in two ways. First, indeed, by the following argument. To whomever it pertains to know the substances and natures of certain things, to him also it pertains to consider their accidents; now it pertains to the natural philosopher to consider the nature and substance of the sun and the moon, since they are certain natural bodies; therefore it pertains to him also to consider their per se accidents. He proves this as well from the custom of philosophers: for natural philosophers are found to have made determinations concerning the shape of the sun and the moon and the earth and the whole world, upon which the intention of astronomers also labors. Thus, therefore, astronomy and natural science agree not only in the same subjects, but also in the consideration of the same accidents and in the demonstration of the same conclusions. Hence it seems that astronomy is a part of physics; and consequently physics does not wholly differ from mathematics.
[lib. 2 l. 3 n. 3] Then when he says, Concerning these things, therefore, indeed, etc., he resolves the aforesaid question. And concerning this he does two things: first, he sets forth the solution; second, he confirms it, at Now this will surely become manifest, etc. Concerning the first he does three things: first, he resolves the question; second, he draws a certain corollary from what has been said, at Whence also he abstracts, etc.; third, he excludes an error, at Yet this escapes notice, etc.
[lib. 2 l. 3 n. 4] He says first, therefore, that the mathematician and the natural philosopher make determinations concerning the same things—namely points, lines, surfaces, and things of this sort—yet not in the same way. For the mathematician does not treat of them insofar as each of them is the boundary of a natural body, nor does he consider the things that belong to them insofar as they are boundaries of a natural body; yet that is the way in which natural science considers them. Nor is there any incongruity in the same thing falling under the consideration of diverse sciences according to diverse modes of consideration.
[lib. 2 l. 3 n. 5] Then when he says, Whence also he abstracts, etc., he draws a certain corollary from what has been said. For because the mathematician considers lines and points and surfaces and things of this sort, together with their accidents, not insofar as they are boundaries of a natural body, he is therefore said to abstract from sensible and natural matter. And the cause why he is able to abstract is this: because according to the intellect they are abstracted from motion. To make this cause evident, one must consider that many things are conjoined in reality without the one belonging to the understanding of the other: just as the white and the musical are conjoined in some subject, and yet the one does not belong to the understanding of the other, and therefore the one can be separately understood without the other. And this is for one object of understanding to be abstracted from another. Now it is manifest that posterior things do not belong to the understanding of prior things, whereas the converse holds: hence prior things can be understood without posterior things, and not conversely. Thus it is clear that animal is prior to man, and man is prior to this man (for man stands by addition to animal, and this man by addition to man); and on account of this, man does not belong to the understanding of animal, nor Socrates to the understanding of man: hence animal can be understood without man, and man without Socrates and other individuals. And this is to abstract the universal from the particular. In a similar way, among all the accidents that accrue to substance, quantity accrues to it first, and thereafter sensible qualities and actions and passions and the motions consequent upon sensible qualities. Thus, therefore, quantity does not enclose in its understanding sensible qualities or passions or motions; yet it does enclose substance in its understanding. Quantity can therefore be understood without matter subject to motion and sensible qualities, yet not without substance. And therefore quantities of this sort, and the things that belong to them, are according to the intellect abstracted from motion and from sensible matter, though not from intelligible matter, as is stated in Metaphysics VII. Because, therefore, they are thus abstracted from motion according to the intellect, inasmuch as they do not enclose in their understanding \ sensible matter subject to motion, the mathematician is able to abstract them from sensible matter. And it makes no difference with respect to the truth of the consideration whether they are considered in this way or in that. For although they are not abstract according to being, nevertheless mathematicians who abstract them according to the intellect commit no falsehood—for there is no falsehood in those who abstract—because they do not assert that they exist outside sensible matter (for this would be a falsehood), but consider them without any consideration of sensible matter, which can be done without falsehood: just as someone can consider whiteness without music, and truly so, even though they concur in the same subject; yet it would not be a true consideration if he were to assert that the white is not musical.
[lib. 2 l. 3 n. 6] Then when he says, Yet this escapes the notice of those who do this, etc., he excludes from what has been said the error of \ Plato. For because it escaped his notice how the intellect could truly abstract things that are not abstract according to being, he posited that all things that are abstract according to the intellect are abstract in reality. Hence he not only posited that mathematical entities are separate, on account of the fact that the mathematician abstracts from sensible matter, but he even posited that natural things themselves are separate, on account of the fact that natural science is concerned with universals and not with singulars. Hence he posited that man exists separately, and horse and stone and other things of this sort; and these separate realities he called Ideas—even though natural things are less abstract than mathematical ones. For mathematical entities are altogether abstracted from sensible matter according to the intellect, because sensible matter is not included in the understanding of mathematical entities, either in universal or in particular; whereas in the understanding of natural species sensible matter is indeed included, though not individual matter; for in the understanding of man flesh and bone are included, though not this flesh and this bone.
[lib. 2 l. 3 n. 7] Then when he says, Now this will surely become manifest, etc., he makes clear the solution set down in two ways: first, indeed, through the difference of the definitions assigned by the mathematician and the natural philosopher; second, through the intermediate sciences, at This is also demonstrated by the more natural, etc. He says first, therefore, that what has been stated concerning the diverse manner of consideration of the mathematician and the physicist will become manifest if anyone attempts to state the definitions of natural and mathematical entities and of their accidents: because mathematical entities, such as the even and the odd, the straight and the curved, and number and line and figure, are defined without motion and matter; whereas flesh and bone and man are not so defined, for the definition of these is like the definition of the snub, in whose definition a sensible subject is placed, namely the nose, and not like the definition of the curved, in whose definition no sensible subject is placed. And thus from the very definitions of natural and mathematical things there appears what was stated above concerning the difference between the mathematician and the natural philosopher.
[lib. 2 l. 3 n. 8] Then when he says, This is also demonstrated, etc., he proves the same thing through the sciences intermediate between mathematics and physics. Now those sciences are called sciences intermediate between mathematics and physics which take abstract principles from the purely mathematical sciences and apply them to sensible matter: just as optics applies to the visual line the things demonstrated by geometry concerning the abstract line; and harmonics, that is, music, applies to sounds the things that the arithmetician considers concerning the proportions of numbers; and astronomy applies the consideration of geometry and arithmetic to the heaven and its parts. Yet sciences of this sort, although they are intermediate between natural science and mathematics, are nevertheless said here by the Philosopher to be more natural than mathematical, because each thing is denominated and takes its species from its term: hence, because the consideration of these sciences terminates in natural matter, even though they proceed through mathematical principles, they are more natural than mathematical. He says, therefore, concerning sciences of this sort that they stand in a way contrary to the sciences that are purely mathematical, such as geometry or arithmetic. For geometry indeed considers the line that has being in sensible matter, which is the natural line; yet it does not consider it insofar as it exists in sensible matter, according as it is natural, but abstractly, as has been said. Optics, conversely, takes the abstract line according as it stands in the consideration of the mathematician, and applies it to sensible matter; and thus it treats of it not insofar as it is mathematical, but insofar as it is physical. From the very difference, therefore, of these intermediate sciences from the purely mathematical sciences, what was said above becomes clear. For if intermediate sciences of this sort apply abstract things to sensible matter, it is manifest that the mathematical sciences conversely abstract the things that exist in sensible matter.
[lib. 2 l. 3 n. 9] And through this the reply to the objection raised above concerning astronomy is also clear. Hence astronomy is more natural than mathematical. Thus it is no wonder if it shares in conclusions with natural science. Yet because it is not purely natural, it demonstrates the same conclusion through a different middle term. Thus, that the earth is spherical is demonstrated by the natural philosopher through a natural middle term, for instance because its parts from every side converge equally toward the center; whereas by the astronomer it is demonstrated from the shape of a lunar eclipse, or from the fact that the same stars are not beheld from every part of the earth.
Physics Book III, Lecture 5 — On Action and Passion and the Division of Being into Ten Categories Proportional to Modes of Predicating¶
Latin Locus: Phys. III, lect. 5 · Commentary: Notes on Phys. III, lect. 5
[lib. 3 l. 5 n. 1] After the Philosopher has shown that motion is the act of the mobile and of the mover, he now raises a certain doubt concerning what has gone before. And first, he raises the doubt; second, he resolves it, at Yet neither is it absurd that the act, etc. Concerning the first he does two things: first, he premises certain points toward the doubt; second, he pursues the doubt, at Since, therefore, both, etc.
[lib. 3 l. 5 n. 2] He says first, therefore, that what has been stated involves a difficulty, that is, a doubt, that is reasonable, that is, dialectical: for there are probable arguments on either side. Toward this doubt, moreover, he premises that there is a certain act of the active principle and a certain act of the passive principle, just as it was said above that there is a certain act both of the mover and of the moved. And the act of the active principle is called action, whereas the act of the passive principle is called passion. And he proves this because that which is the work and end of each thing is its act and perfection: hence, since the work and end of the agent is action, while that of the patient is passion, as is manifest through itself, what has been stated follows—namely, that action is the act of the agent and passion the act of the patient.
[lib. 3 l. 5 n. 3] Then when he says, Since, therefore, both, etc., he pursues the doubt. For it is manifest that both action and passion are motion: for each is the same as motion. Either, therefore, action and passion are the same motion, or they are diverse motions. If they are diverse, it is necessary that each of them be in some subject: either, therefore, both are in the patient and the thing moved; or one of them is in the agent, namely action, and the other in the patient, namely passion. Whereas if someone should say conversely that what is in the agent is passion and what is in the patient is action, it is manifest that he speaks equivocally: for what is passion he will call action, and conversely. Now he seems to omit a fourth member, namely that both are in the agent; yet he passes over this because it has already been shown that motion is in the mobile, whereby this member—that neither is in the patient, while both are in the agent—is excluded.
[lib. 3 l. 5 n. 4] Of these two members that he touches upon, therefore, he first pursues the second, at Yet if one says this, etc. For if someone should say that action is in the agent and passion in the patient, while action is a certain motion, as has been said, it follows that motion is in the mover. Now the same account must hold concerning the mover and the moved, namely that in whichever of them motion exists, that thing is moved—or the same account holds concerning the mover and the moved just as concerning the patient and the agent. Yet in whatever thing motion exists, that thing is moved; wherefore it follows either that every mover is moved, or that something possesses motion and is not moved—each of which seems incongruous.
[lib. 3 l. 5 n. 5] Then when he says, If, however, both, etc., he pursues the other member, saying that if someone asserts that both—namely action and passion, being two motions—are in the patient and the thing moved, just as teaching, which is on the part of the teacher, and learning, which is on the part of the learner, are in the learner, two incongruities follow. The first of these is that action was said to be the act of the agent: if, therefore, action is not in the agent but in the patient, it will follow that the proper act of each thing is not in that of which it is the act. Afterward another incongruity will follow, namely that one and the same thing is moved according to two motions. For action and passion are now supposed to be two motions, and in whatever thing motion exists, it is moved according to that motion; if, therefore, action and passion are in the mobile, it follows that the mobile is moved according to two motions. And this would be the same as if there were two alterations of a single subject terminating in one species, as though one subject were moved by two whitenings—which is impossible. That the same subject should be moved simultaneously by two alterations terminating in diverse species, namely by whitening and heating, is not incongruous. Yet it is manifest that action and passion terminate in the same species: for what the agent enacts and what the patient undergoes are the same.
[lib. 3 l. 5 n. 6] Then when he says, Yet will there be one act? etc., he pursues the other main alternative. For someone may say that action and passion are not two motions, but one. And from this he draws four incongruities. The first of these is that the same thing would be the act of things diverse in species. For it was said that action is the act of the agent and passion the act of the patient, which are diverse in species: if, therefore, action and passion are the same motion, it follows that the same act belongs to things diverse in species. The second incongruity is that if action and passion are one motion, action would be the same as passion, and teaching, which is on the part of the teacher, would be the same as learning insofar as it stands on the part of the learner. The third incongruity is that to act would be the same as to undergo, and to teach the same as to learn. The fourth, which follows from this, is that every teacher would learn, and every agent would undergo.
[lib. 3 l. 5 n. 7] Then when he says, Yet neither is it absurd that the act of one, etc., he resolves the aforesaid doubt. Now it is manifest from what was determined above that action and passion are not two motions, but one and the same motion: for according as it is from the agent it is called action, whereas according as it is in the patient it is called passion.
[lib. 3 l. 5 n. 8] Hence there is no need to resolve the incongruities that follow upon the first part, in which it was supposed that action and passion were two motions, save one that remains to be resolved even when it is supposed that action and passion are one motion: for since action is the act of the agent, as was said above, if action and passion are one motion, it follows that the act of the agent is in a certain manner in the patient, and thus the act of one thing will be in another. Moreover, four incongruities followed from the other side; and thus five incongruities remain to be resolved.
[lib. 3 l. 5 n. 9] He says first, therefore, that it is not incongruous for the act of one thing to be in another, because teaching is the act of the teacher, yet tending from him into another continuously and without any interruption: hence the same act belongs to this one, that is, the agent, as that from which it proceeds, and yet exists in the patient as received in it. It would be incongruous, however, if the act of one thing existed in another in the very same way in which it is its own act.
[lib. 3 l. 5 n. 10] Then when he says, Nor is anything prevented from being one act for two, etc., he resolves the second incongruity, namely that the same act would belong to two things. And he says that nothing prevents one act from belonging to two things, provided it is not one and the same according to intelligible notion, though one in reality—just as it was stated above that the distance of two to one and of one to two is the same, and so too of what is in potency to the agent and conversely. For in this way the same act in reality belongs to two things according to diverse intelligible notions: to the agent, indeed, according as it is from it, and to the patient according as it is in it.
[lib. 3 l. 5 n. 11] To the other three incongruities, of which one was deduced from another, he replies in reverse order—first, namely, to that which was introduced last as the greater incongruity. Thus, therefore, in the third place he replies to the fifth incongruity. And he says that it is not necessary that the teacher learn, or that the agent undergo, even if to act and to undergo were the same, provided we say that they are not the same like things whose definition is one, such as tunic and garment, but like things that are the same in subject and diverse according to intelligible notion, such as the road from Thebes to Athens and from Athens to Thebes, as was said earlier. For it is not necessary that all the same attributes belong to things that are in just any way the same, but only to those that are the same both in subject or reality and in intelligible notion. And therefore, even granting that to act and to undergo were the same, since they are not the same in intelligible notion, as has been said, it does not follow that to whatever thing acting belongs, undergoing also belongs.
[lib. 3 l. 5 n. 12] Then when he says, Yet neither if teaching, etc., he replies to the fourth incongruity. And he says that even if teaching and the learning of the learner were the same, it does not follow that to teach and to learn would be the same: because teaching and learning are spoken of in the abstract, whereas to teach and to learn are spoken of in the concrete. Hence they are applied to the ends or terms according to which the diverse intelligible notions of action and passion are taken. Just as, although we may say that the interval between certain distant things is the same when taken abstractly, nevertheless if we apply it to the terms of the interval, as when we speak of being distant from here to there and from there to here, it is not one and the same.
[lib. 3 l. 5 n. 13] Then when he says, To state the matter altogether, etc., he replies to the third incongruity by overturning the inference whereby it was concluded that if action and passion are one motion, action and passion are the same. And he says that finally it must be stated that it does not follow that action and passion are the same, or teaching and learning, but rather that the motion in which both of them are found is the same. And this motion, according to one intelligible notion, is action, and according to another intelligible notion is passion. For it is one thing according to intelligible notion to be the act of this as in this, and another to be the act of this as from this. Now motion is called action according as it is the act of the agent as from this, whereas it is called passion according as it is the act of the patient as in this. And thus it is clear that although the motion of the mover and of the moved is the same, on account of the fact that it abstracts from either intelligible notion, nevertheless action and passion differ on account of the fact that they include these diverse notions in their signification. From this it also appears that, since motion abstracts from the intelligible notion of action and passion, it is contained neither in the category of action nor in the category of passion, as certain thinkers asserted.
[lib. 3 l. 5 n. 14] Yet concerning this a twofold doubt remains. The first, indeed, is that if action and passion are one motion, and do not differ save according to intelligible notion, as has been said, it seems that they ought not to be two categories, since the categories are genera of things. Second, if motion is either action or passion, motion will not be found in substance, quality, quantity, and where, as was stated above, but will be contained solely in action and passion.
[lib. 3 l. 5 n. 15] To make these points evident, therefore, one must know that being is divided into the ten categories not univocally, as a genus into species, but according to diverse modes of being. Now modes of being are proportional to modes of predicating. For in predicating something of some other thing, we say that this is that: hence too the ten genera of being are called the ten categories. Now all predication takes place in three ways. One mode, indeed, is when there is predicated of some subject that which pertains to its essence, as when I say Socrates is a man, or man is an animal; and according to this the category of substance is taken. A second mode is that whereby there is predicated of something what does not belong to its essence, yet inheres in it. And this either stands on the side of the matter of the subject, and according to this is the category of quantity (for quantity properly follows upon matter: whence \ Plato also placed the great on the side of matter); or it follows upon form, and thus is the category of quality (whence qualities are also founded upon quantity, just as color is in a surface, and figure in lines or in surfaces); or it stands by way of a respect toward another, and thus is the category of relation (for when I say the man is a father, nothing absolute is predicated of the man, but a respect inherent in him toward something extrinsic). The third mode of predicating, however, is when something extrinsic is predicated of something by way of a certain denomination: for in this way extrinsic accidents also are predicated of substances; yet we do not say that a man is whiteness, but that a man is white. Now to be denominated from something extrinsic is found in one way commonly in all things, and in another way specially in the things that pertain to human beings alone. Commonly, indeed, something is found to be denominated from something extrinsic either according to the character of cause or according to the character of measure; for something is denominated as caused and as measured from something exterior. Now since there are four genera of causes, two of these are parts of the essence, namely matter and form: hence the predication that could be made according to these two pertains to the category of substance, as for instance if we say that man is rational and man is corporeal. The final cause, however, does not cause anything apart from the agent: for the end has the character of cause insofar as it moves the agent. There remains, therefore, the agent cause alone from which something can be denominated as from something exterior. Thus, therefore, according as something is denominated from the agent cause, there is the category of passion, for to undergo is nothing other than to receive something from an agent; whereas conversely, according as the agent cause is denominated from the effect, there is the category of action, for action is an act proceeding from the agent into another, as was said above. Of measure, moreover, one kind is extrinsic and another intrinsic. Intrinsic measure is like the proper length of each thing and its breadth and depth: from these, therefore, something is denominated as from something intrinsically inhering, whence it pertains to the category of quantity. Exterior measures, however, are time and place: according, therefore, as something is denominated from time, there is the category of when; whereas according as it is denominated from place, there are the categories of where and of position, which adds above where the order of parts in place. Now it was not necessary to add this on the side of time, since the order of parts in time is imported in the very notion of time: for time is the number of motion according to before and after. Thus, therefore, something is said to be when or where by denomination from time or from place. There is, moreover, something special in human beings. For in other animals nature has sufficiently provided the things that pertain to the preservation of life, such as horns for defense, a thick and hairy hide for covering, and hooves or something of the sort for walking without injury. And thus when such animals are said to be armed or clothed or shod, in a certain way they are denominated not from something extrinsic, but from certain parts of their own. Hence in them this is referred to the category of substance: just as if it were said that man is handed or footed. Yet things of this sort could not be given to man by nature, both because they did not suit the delicacy of his bodily constitution, and on account of the manifold variety of works that befit man insofar as he possesses reason, to which determinate instruments could not be fitted by nature; instead of all these, reason is present in man, whereby he prepares exterior things for himself in place of those things that are intrinsic to other animals. Hence when a man is said to be armed or clothed or shod, he is denominated from something extrinsic that has the character neither of cause nor of measure: whence it is a special category, and is called having (habitus). Yet one must observe that this category is attributed to other animals as well, not according as they are considered in their own nature, but according as they come into human use: as if we call a horse caparisoned or saddled or armed.
[lib. 3 l. 5 n. 16] Thus, therefore, it is clear that although motion is one, nevertheless the categories taken according to motion are two, according as categorial denominations are made from diverse exterior things. For the agent, from which as from something exterior the category of passion is taken by way of denomination, is one thing; and the patient, from which the agent is denominated, is another thing. And thus the solution to the first doubt is clear.
[lib. 3 l. 5 n. 17] The second doubt, moreover, is easily resolved. For the intelligible notion of motion is completed not only through what belongs to motion in the nature of things, but also through what reason apprehends. For in the nature of things, motion is nothing other than an imperfect act, which is a certain inchoation of a perfect act in that which is moved: just as in that which is being whitened, something of whiteness already begins to be. Yet in order that this imperfect reality may possess the intelligible notion of motion, it is further required that we understand it as a kind of medium between two terms: of which the preceding is compared to it as potency to act, whence motion is called an act; whereas the subsequent is compared to it as the perfect to the imperfect or as act to potency, on account of which it is called the act of a being in potency, as was said above. Hence whatever imperfect state is taken as not tending into something further that is perfect is called the term of a motion, and will not be a motion according to which something is moved: as for instance if something begins to be whitened and the alteration is immediately interrupted. With respect, therefore, to what belongs to motion in the nature of things, motion is placed by reduction in that genus which terminates the motion, just as the imperfect is reduced to the perfect, as was said above. With respect, however, to what reason apprehends concerning motion—namely its being a certain medium between two terms—the notion of cause and effect is thereby already implied: for something is reduced from potency into act only by some agent cause. And according to this, motion pertains to the categories of action and passion: for these two categories are taken according to the notion of agent cause and effect, as has been said.
[lib. 3 l. 5 n. 18] Then when he says, What motion is, therefore, indeed, etc., he defines motion in particular: and he says that it has been stated what motion is both in universal and in particular, because from what has been stated concerning the definition of motion in universal, it can become manifest how it is defined in particular. For if motion is the act of the mobile insofar as it is such, it follows that alteration is the act of the alterable insofar as it is such; and similarly with the others. And because it was placed in doubt whether motion is the act of the mover or of the mobile, and it was shown that it is the act of the active principle as from this and of the passive principle as in this, in order to remove every doubt let us say somewhat more clearly that motion is the act of the potentially active and of the potentially passive. And thus we shall also be able to say in particular that building is the act of the builder and of the buildable insofar as it is such; and the case is similar with healing and other motions.
Physics Book IV, Lecture 7 — On the Definition of Immobile Place from Local Translation and Themistius’s Solution¶
Latin Locus: Phys. IV, lect. 7 · Commentary: Notes on Phys. IV, lect. 7
[lib. 4 l. 7 n. 1] After the Philosopher has defined place, here he shows how something is in place. And concerning this he does two things: first, he shows how something is simply in place, and how not; second, he shows how that which is not simply in place is in place in a qualified respect, at Whence also if water were made, etc.
[lib. 4 l. 7 n. 2] He concludes first from the premises, therefore, that since place is the boundary of the containing body, whatever body has adjacent to it some body containing it externally is in place simply and per se; whereas whatever body does not have adjacent to it some body externally containing it is in no way in place. Now in the world there is only one such body, namely the outermost sphere, whichever sphere that may be. Hence according to this determination it follows that the outermost sphere is not in place.
[lib. 4 l. 7 n. 3] Yet this seems impossible: because the outermost sphere is moved in place, and nothing is moved in place that is not in place. Now the difficulty of this doubt does not arise for those who hold the opinion concerning space. For there is no necessity for them to say that, in order for the outermost sphere to be in place, it must have a containing body; rather, the space that is understood to penetrate the whole world and all its parts is the place of the whole world and of each of its parts, according to them. Yet this position is impossible: because one must say either that place is nothing over and above the located body, or that there are certain dimensions of space existing per se and nevertheless interpenetrating the dimensions of sensible bodies—both of which are impossible.
[lib. 4 l. 7 n. 4] Hence Alexander said that the outermost sphere is in no way in place: for not every body is of necessity in place, since place does not fall within the definition of body. And on account of this he said that the outermost sphere is not moved in place, either according to the whole or according to its parts. Yet because every motion must be placed in some genus of motion, \ Avicenna, following him, said that the motion of the outermost sphere is not motion in place, but motion in position (situs)—contrary to \ Aristotle, who states in Book V of this work that motion exists only in three genera, namely in quantity, quality, and where. Yet this cannot stand: for it is impossible, speaking per se, for motion to exist in any genus the intelligible notion of whose species consists in an indivisible. For on account of this there is no motion in substance, because the intelligible notion of every species of substance consists in an indivisible, inasmuch as species of substance are not predicated according to more and less; and on account of this, since motion possesses succession, a substantial form is brought into being not through motion, but through generation, which is the term of motion. It is otherwise, however, with whiteness and similar things, which are participated according to more and less. Now every species of position has an intelligible notion consisting in an indivisible, so that if anything is added or subtracted, it is no longer the same species of position. Hence it is impossible that there be motion in the genus of position. And besides, the very same difficulty remains. For position, according as it is posited as a category, imports the order of parts in place, even though according as it is posited as a difference of quantity it imports nothing save the order of parts in the whole. Everything that is moved according to position, therefore, must be moved according to place.
[lib. 4 l. 7 n. 5] Certain others, however—namely Avempace—said that place is to be assigned in one way to a body moved circularly and in another way to a body moved with rectilinear motion. For because a straight line is imperfect, admitting of addition, a body moved with rectilinear motion requires an externally containing place; whereas because a circular line is complete in itself, a body moved circularly does not require an externally containing place, but a place around which it revolves: whence circular motion too is said to be motion around the center. Thus, therefore, they say that the convex surface of the contained sphere is the place of the first sphere. Yet this is contrary to the common suppositions previously laid down concerning place: namely, that place is what contains, and that place is equal to the located body.
[lib. 4 l. 7 n. 6] And therefore \ Averroes said that the outermost sphere is in place per accidens. To make this evident, one must consider that everything that has fixity through another is said to be in place per accidens from the fact that that through which it is fixed is in place, as is clear in the case of a nail fastened in a ship and of a man resting in a ship. Now it is manifest that bodies moved circularly possess fixity through the immobility of the center: hence the outermost sphere is said to be in place per accidens insofar as the center around which it revolves has being in place. That the other lower spheres have per se a place in which they are contained happens accidentally, and does not belong to the necessity of a body moved circularly. Against this, however, it is objected that if the outermost sphere is in place per accidens, it follows that it is moved in place per accidens, and thus motion per accidens is prior to motion per se. Yet to this the reply is made that circular motion does not require that what is moved per se circularly be per se in place, whereas rectilinear motion does require this. Even so, this seems to conflict with the definition that Aristotle laid down above concerning what is in place per accidens. For he said that certain things are or are moved in place per accidens from the fact that that in which they exist is moved; whereas something is not said to be in place per accidens from the fact that something altogether extrinsic to it is in place. Since, therefore, the center is altogether extrinsic to the outermost sphere, it seems ridiculous to say that the outermost sphere is in place per accidens from the fact that the center is in place.
[lib. 4 l. 7 n. 7] And therefore I more approve the judgment of \ Themistius, who said that the outermost sphere is in place through its parts. To make this evident, one must consider that, as Aristotle said above, place would not be sought except on account of motion, which demonstrates place from the fact that bodies succeed one another in one place. Hence, although place does not belong to the necessity of a body, it nevertheless belongs to the necessity of a body that is moved according to place. Thus, therefore, to any body moved locally it is necessary to assign a place according as in that motion there is considered a succession of diverse bodies in the same place. Now in things that are moved with rectilinear motion, it is manifest that two bodies succeed one another in place according to the whole: because one whole body leaves the whole place, and into that whole place another body enters. Hence it is necessary that a body moved with rectilinear motion be in place according to its whole self. In circular motion, however, although the whole comes to be in diverse places according to intelligible notion, nevertheless the whole does not change place in subject: for the place remains always the same in subject, and is diversified only according to intelligible notion, as will be stated in Book VI of this work. Yet the parts change place not only according to intelligible notion, but also in subject. In circular motion, therefore, succession in the same place is observed not of whole bodies, but of parts of the same body. Hence to a body that is moved circularly, place is owed of necessity not according to the whole, but according to its parts.
[lib. 4 l. 7 n. 8] Yet against this there seems to stand the fact that the parts of a continuous body are neither in place nor moved according to place; rather, the whole is moved, and the whole is in place. Now it is manifest that the outermost sphere is a continuous body: therefore its parts are neither in place nor moved according to place. And thus it does not seem true that place is owed to the outermost sphere by reason of its parts. To this, however, it must be said that the parts of a continuous whole, although they are not in place in act, are nevertheless in place in potency, according as the \ continuum is divisible. For a part, if it be divided off, will exist in the whole as in a place: hence it is in this manner that the parts of a continuum are moved in place. And this appears above all in fluid continua, which are of easy division, as in water, whose parts are found to be moved within the whole mass of water. Thus, therefore, because something is predicated of a whole by reason of its parts, insofar as the parts of the outermost sphere are in place in potency, the whole outermost sphere is in place per accidens by reason of its parts; and being in place in this way suffices for circular motion.
[lib. 4 l. 7 n. 9] If someone should object that what is in act is prior to what is in potency, so that it seems incongruous that the first local motion should belong to a body existing in place through parts that are in place in potency, it must be said that this accords most fittingly with the first motion. For it is necessary to descend by degrees from the one immobile principle to the diversity found in mobile things. Now the variation that takes place according to parts existing in place in potency is lesser than that which takes place according to wholes existing in place in act. Hence the first motion, which is circular, has less difformity and retains more uniformity, standing closer to immobile substances. Moreover, it is much more fitting to say that the outermost sphere is in place on account of its own intrinsic parts than on account of the center, which lies altogether outside its substance; and this harmonizes better with the mind of Aristotle, as is clear to anyone inspecting what follows, wherein the Philosopher makes plain how the heaven is in place, at Whence also if water were made, etc.
[lib. 4 l. 7 n. 10] Concerning this he does two things: first, indeed, he makes plain how the outermost sphere is in place; second, he draws a conclusion from what has been said, at Whence it is moved circularly, etc. Concerning the first he does three things: first, he makes plain that the outermost sphere is in place through its parts; second, how its parts are in place, at Now just as was said, some things indeed, etc.; third, how from its parts being in place belongs to the whole, at And some things indeed are per se, etc.
[lib. 4 l. 7 n. 11] Because, therefore, he had stated that whatever body has nothing outside containing it is not in place per se, he concludes that if some body of this sort not contained by another—such as the outermost sphere—were water (in which what is being said appears more clearly on account of the easy division of its parts), its parts will be moved insofar as they are contained by one another, thus existing in a certain manner in place. Yet the whole water will in one way be moved, and in another way not. For it will not be moved in such a way that the whole simultaneously changes place, as though translated into another place diverse in subject; rather, it will be moved circularly, and this motion requires a place of the parts, and not of the whole. And it will not be moved upward and downward, but circularly; whereas certain other bodies will be moved upward and downward, changing place according to the whole—namely rare and dense, or heavy and light, bodies.
[lib. 4 l. 7 n. 12] Then when he says, Now just as was said, etc., he shows how the parts of the outermost sphere are in place. And he says that, just as was stated above, some things are in place in act, and others according to potency. Hence when something is a continuum of homogeneous parts, its parts are in place according to potency, just as is the case in the outermost sphere; whereas when the parts are separated and merely contiguous, as happens in a heap of stones, then the parts are in place according to act.
[lib. 4 l. 7 n. 13] Then when he says, And some things indeed are per se, etc., he shows how from this it follows that the whole sphere is in place. And he says that some things are per se in place, such as every body that is per se moved in place, whether according to change of place or according to growth, as was said above. Yet the heaven, that is, the outermost sphere, is not in place in this way, as has been said, since no body contains it; rather, according as it is moved circularly, with its parts succeeding one another, place is owed to its parts in potency, as has been said, insofar namely as one part of it is adjoining, that is, consecutive to another. Certain things, on the other hand, are in place per accidens, such as the soul and all forms: and in this way too the heaven, that is, the outermost sphere, is in place, insofar as all its parts are in place because each part of it is contained by another around the circle. For in a non-circular body an extreme part remains uncontained and containing only; whereas in a circular body every part is both containing and contained, though in potency. Hence by reason of all its parts a circular body is in place. And this it takes as being per accidens—namely through its parts—just as above, when he said that the parts of a body are moved per accidens in place.
[lib. 4 l. 7 n. 14] Then when he says, Whence it is moved circularly, etc., he draws a certain conclusion from what has been said. For because he had stated that a body moved circularly need not be in place according to the whole, save only per accidens by reason of its parts, he concludes that the supreme body is moved solely in a circle, on account of the fact that the whole itself is not anywhere: because what is somewhere is itself something, and has something outside itself by which it is contained, whereas outside the whole there is nothing. And on account of this all things are said to be in the heaven as in the ultimate container, because the heaven is perhaps that which contains the whole. Now he says perhaps because it has not yet been proved that nothing exists outside the heaven. Nor is this to be understood in such a way that the very body of the heaven is place; rather, it is a certain innermost surface of it facing us, which is like a boundary touching the mobile bodies that exist within it. And on account of this we say that the earth is in water, which is in air, which is in the aether, that is, fire, which is in the heaven, while the heaven is no further in another.
[lib. 4 l. 7 n. 15] According to the intention of Averroes, however, this text must be expounded otherwise. For the example of water that he first introduces is not, according to him, to be referred to the outermost sphere, but to the whole universe, which indeed is moved insofar as its parts are moved—some circularly, such as the celestial bodies, and others by upward or downward motion, such as the lower bodies. What is introduced afterward, namely that some things are in place in act and others in potency, is not to be referred to what was said before, but must be taken as stated on its own account. For because he had said that some things are in place according to their parts and others according to the whole, he consequently adds that some things are in place according to act and others according to potency; and further, that some things are in place per se and others per accidens. Here it is to be noted that heaven, according to him, is taken here in two ways: for first heaven is taken for the totality of bodies, and chiefly the celestial ones; second, for the outermost sphere. He says, therefore, that those things are per se in place which are moved according to place, whether according to the whole or according to their parts, such as the heaven, that is, the universe; whereas those things are in place per accidens such as the soul and the heaven, that is, the outermost sphere. For one must say that all parts of the universe are in some manner in place: the outermost sphere indeed per accidens, and the other bodies per se, insofar as they are contained by an exterior body. And this he sets forth right to the end.
Physics Book VI, Lecture 1 — That No Continuum (Line, Time, or Motion) Can Be Composed of Indivisible Points¶
Latin Locus: Phys. VI, lect. 1 · Commentary: Notes on Phys. VI, lect. 1
[lib. 6 l. 1 n. 1] After the Philosopher has made determinations concerning the division of motion into its species, and concerning the unity and contrariety of motions and rests, in this sixth book he intends to determine the things that pertain to the division of motion according as it is divided into quantitative parts. And it is divided into two parts. In the first, he shows that motion, like every \ continuum, is divisible; in the second, he shows in what manner motion is divided, at Now it is necessary that the now itself, etc. The first part is again divided into two: in the first, he shows that no continuum is composed of indivisibles; in the second, he shows that no continuum is indivisible, at It is manifest from what has been said, therefore, etc. And this first part is again divided into two: in the first, he shows that no continuum is composed of indivisibles; in the second part (because the proofs premised seem to pertain rather to magnitude), he shows that the same account holds concerning magnitude, motion, and time, at Now it belongs to the same account, etc. Concerning the first he does two things: first, he resumes certain definitions laid down above, which he now employs to demonstrate what is proposed; second, he proves what is proposed, at For neither are their extremes one, etc.
[lib. 6 l. 1 n. 2] He says first, therefore, that if the definitions previously laid down of the continuous, of the contiguous (that which is in contact), and of the consecutive are fitting—namely that continuous things are those whose extremes are one; contiguous things (things in contact) are those whose extremes are together; and consecutive things are those between which there is nothing intermediate of their own genus—from these it follows that it is impossible for any continuum to be composed of indivisibles, such as a line out of points, provided that a line is called something continuous and a point something indivisible. And he adds this lest anyone should use the name of line and of point otherwise.
[lib. 6 l. 1 n. 3] Then when he says, For neither are their extremes one, etc., he proves what is proposed. And first, he introduces two main arguments to prove what is proposed; second, he makes plain certain points that could be doubtful in his proofs, at Nor can any other genus, etc. Concerning the first main argument he does two things: first, he shows that no continuum is composed of indivisibles either by way of continuity or by way of contact; second, that neither is it composed by way of consecutive things, at Yet neither can a point be consecutive, etc. Concerning the first he sets forth two arguments, of which the first is as follows. Out of whatever things something one is composed, whether by way of continuity or by way of contact, it is necessary that they possess extremes that are one or that are together. Yet the extremes of points cannot be one: because an extreme is spoken of with respect to some part, whereas in an indivisible one cannot take something that is an extreme and something else that is some other part. Similarly, it cannot be said that the extremes of points are together: because nothing can be the extreme of a partless reality, since an extreme and that of which it is the extreme are always distinct, whereas in a partless reality one cannot take one thing and another. It remains, therefore, that a line cannot be composed of points either by way of continuity or by way of contact.
[lib. 6 l. 1 n. 4] The second argument he sets forth at Moreover, it would be necessary, etc., which is as follows. If any continuum is constituted out of points, it is necessary either that they be continuous with one another, or that they touch one another: and the same account holds concerning all other indivisibles, namely that no continuum is composed out of them. Now to prove that indivisibles cannot be continuous with one another, let the first argument suffice. Yet to prove that they cannot touch one another, another argument is introduced, which is as follows. Everything that touches another touches either whole to whole, or part of the one to part of the other, or part of the one to the whole of the other. Yet since an indivisible has no part, it cannot be said that a part of the one touches a part of the other, or a part the whole; and thus it is necessary, if two points touch one another, that whole touch whole. Yet out of two things of which the whole of one touches the whole of the other, no continuum can be composed: because every continuum has distinct parts, such that this is one part and this another, and is divided into parts diverse and distinct in place, that is, in position, among things that possess position; whereas things that touch one another according to the whole are not distinguished in place or position. It remains, therefore, that a line cannot be composed out of points by way of contact.
[lib. 6 l. 1 n. 5] Then when he says, Yet neither, etc., he proves that a continuum is not composed of indivisibles by way of the consecutive. For a point will not be consecutive to another point in such a way that a length, that is, a line, could be constituted out of them; nor will one now be consecutive to another now in such a way that time could be composed out of them: because one thing is consecutive to another when there is nothing intermediate of the same genus between them, as was expounded above. Yet between any two points there is always an intermediate line: and thus if a line is composed of points, as you grant, it follows that between any two points there is always another intermediate point. And similarly between any two nows there is an intermediate time. Neither, therefore, is a line composed of points, nor time of nows, as of things standing consecutively to one another.
[lib. 6 l. 1 n. 6] The second principal argument he sets forth at Moreover, they would be divided, etc., which is taken from the other definition of the continuum laid down above at the beginning of Book III—namely, that a continuum is that which is divisible into infinity: and the argument is as follows. Out of whatever things either a line or time is composed, into those same things it is divided: if, therefore, each of these is composed of indivisibles, it follows that it is divided into indivisibles. Yet this is false, since none of the continua is divisible into partless things; for if it were so, it would not be divisible into infinity. No continuum, therefore, is composed of indivisibles.
[lib. 6 l. 1 n. 7] Then when he says, Nor can any other, etc., he makes plain two things that he had stated above. The first of these was that between two points there is an intermediate line, and between two nows, time. And he makes this plain as follows. If there are two points, they must differ according to position: otherwise they would not be two, but one. Yet they cannot be contiguous with one another, as was shown above: hence it remains that they stand apart, and that there is something intermediate between them. Now nothing else can be intermediate between them save a line between points, and time between nows. And he proves this as follows: because if between points there were some medium other than a line, it is manifest that that medium would be either indivisible or divisible. If it were indivisible, it must be distinct in position from both extremes; and since it does not touch them, there must again be some other medium between the indivisible posited as intermediate and the extremes, and so on into infinity, unless a divisible medium be posited. If, on the other hand, the medium between the two points is divisible, it will be divisible either into indivisibles or into things always further divisible. Yet it cannot be said that it is divided into indivisibles, because then the very same difficulty will return—namely how something divisible could be composed out of indivisibles. It remains, therefore, that that medium is divisible into things always further divisible. Yet this is the very definition of a continuum: therefore that medium will be a certain continuum. Now no continuum other than a line can be intermediate between two points: therefore between any two points there is an intermediate line. And by the same reasoning, between any two nows there is time; and similarly in other continua.
[lib. 6 l. 1 n. 8] Then when he says, Now it is manifest, etc., he makes plain the second point that he had supposed, namely that every continuum is divisible into divisibles. For if it were granted that a continuum were divisible into indivisibles, it would follow that two indivisibles would be contiguous with one another so as to be able to constitute a continuum. For it is necessary that continuous things have a single extreme, as appears from the definition of the continuous, and that the parts of a continuum touch one another: because if their extremes are one, it follows that they are together, as was stated in Book V. Since, therefore, it is impossible for two indivisibles to be contiguous with one another, it is impossible that a continuum be divided into indivisibles.
On the Heavens Book I, Lecture 2 — On the Perfection of Three-Dimensional Body and the Geometers’ Mode of Speaking of a Moving Point¶
Latin Locus: De Caelo I, lect. 2 · Commentary: Notes on De Caelo I, lect. 2
[lib. 1 l. 2 n. 1] After the Philosopher has shown by way of prologue that bodies and magnitudes must be treated in natural science, here he begins to pursue his principal purpose. And because, as was said above, in this work \ Aristotle principally intends to make determinations concerning the corporeal universe and its principal parts, which are the simple bodies—chief among which is the celestial body—this work is divided into three parts: in the first, he treats of the corporeal universe; in the second, he treats of the celestial body, and this in Book II, at That the whole heaven, therefore, was neither generated, etc.; in the third part, he treats of the other simple bodies, namely the heavy and the light, in Book III, at Concerning the first heaven, therefore, indeed, etc. Concerning the first he does two things: first, he shows the perfection of the universe; second, he determines certain conditions or properties of it, at Yet since it is manifest concerning these things, etc. Concerning the first he again does two things: first, he shows the perfection of the universe; second, he shows out of what parts its perfection is integrated, at Concerning the nature of the whole, therefore, indeed, etc. Concerning the first he does two things: first, he shows the perfection that the universe possesses according to the common notion of its genus, namely insofar as it is a body; second, he proves its own proper perfection, at Of partial bodies, therefore, indeed, etc. Concerning the first he does three things: first, he makes plain the definition of body, which he uses to show what is proposed; second, he proves what is proposed, at Therefore, since all and whole, etc.; third, he shows what can be manifest from the premises, at Whichever magnitudes, therefore, indeed, etc. Concerning the first he does two things: first, he defines the \ continuum, which is the genus of body; second, he makes plain the definition of body, at A body, however, etc.
[lib. 1 l. 2 n. 2] Concerning the first, one must consider that the continuum is found defined by the Philosopher in two ways. In one way by a formal definition, according as it is stated in the Categories that a continuum is that whose parts are joined at one common boundary: for the unity of a continuum is as it were its form. In another way by a material definition, which is taken from the parts, which have the character of matter, as is stated in Physics II; and thus it is defined here that a continuum is that which is divisible into things always further divisible. For no part of a continuum can be indivisible: because no continuum is composed of indivisibles, as is proved in Physics VI. And quite fittingly this definition is set down here, while the other is set down in the Categories: because the consideration of the natural philosopher deals with matter, whereas the consideration of the logician deals with intelligible notion and species.
[lib. 1 l. 2 n. 3] Then when he says, A body, however, etc., he defines body. And first he proposes the definition, saying that a body is a continuum that is divisible every way, that is, toward every part, or according to every dimension. Second, at Now of magnitude, etc., he proves the proposed definition by the following argument. A body is divided according to three dimensions; now what is divided according to three dimensions is divided according to all dimensions; therefore a body is divisible according to all dimensions. First, therefore, he makes plain the minor premise, as it were by division. For among magnitudes there is one that is divided in one direction, and this is called a line; another that is divided in two directions, and this is called a plane, that is, a surface; and another that is divided according to three dimensions—and since such a magnitude is neither a line nor a surface, it follows that it is a body. The major premise he sets forth at And besides these, etc. And first he states it: saying that besides these magnitudes or dimensions there is no other magnitude or dimension, on account of the fact that three possesses the character of being all, because it possesses the character of a certain totality; and what is threefold seems to be every way, or altogether, that is, according to every mode.
[lib. 1 l. 2 n. 4] Second, at For just as, etc., he proves what he had stated in three ways. First, indeed, according to the reasoning of the Pythagoreans, who said that what is called whole and all is determined by the ternary number. For beginning and middle and consummation, that is, end, possess the number that belongs to the whole and the all: for in divisible things the first part does not suffice for the integrity of the whole, which is constituted through an ultimate term, to which one arrives from the beginning through a middle. Now these—namely beginning, middle, and end—have the ternary number: and thus it is clear that the ternary number belongs to the all and the whole.
[lib. 1 l. 2 n. 5] Second, at Wherefore taking from nature, etc., he proves the same thing through what is observed in divine worship. For we employ this number, namely the ternary, for the sanctifications of the gods (whom, namely, the Gentiles worshiped), that is, in their sacrifices and praises, as though having received from nature its own laws and rules: so that, namely, just as nature perfects all things by the ternary number, so those who instituted divine worship, wishing to attribute to God everything that is perfect, attribute to Him the ternary number.
[lib. 1 l. 2 n. 6] Third, at Moreover, we assign, etc., he proves the same thing through the common usage of speech. And he says that we also assign words to things according to the aforesaid manner, namely that perfection belongs to the ternary. For if things are two, we say both, and two men we call both; we do not say all of them, but first use this word concerning three. And all of us in common follow this manner of speaking on account of the fact that nature inclines us to this. For the things that are peculiar to individuals in their manner of speaking seem to arise from the private conceptions of each person, whereas what is observed commonly among all seems to arise from natural inclination.
[lib. 1 l. 2 n. 7] Now it is to be observed that nowhere else is Aristotle found employing Pythagorean arguments to show what is proposed, nor is he elsewhere found concluding anything about things through the properties of numbers; and perhaps he does so here on account of the affinity of numbers to magnitudes, which are treated here. Even so, it seems that this proof is not efficacious: for it seems no more to follow that dimensions are three on account of the ternary being the number of the whole and of all—otherwise it would follow by the same reasoning that there were only three elements, or three fingers on the hand. Yet one must know that, as \ Simplicius states in his commentary, Aristotle does not proceed here demonstratively, but according to probability; and this mode suffices after demonstrations already premised, or presupposed from another science. Now it is manifest that making determinations concerning the dimensions of bodies insofar as they are such pertains per se to the mathematician, whereas the natural philosopher assumes from the mathematician the things he considers concerning dimensions. And therefore to prove demonstratively that there are only three dimensions pertains to the mathematician—just as \ Ptolemy proves through the fact that it is impossible for more than three perpendicular lines to meet together at the same point, while every dimension is measured along some perpendicular line. Presupposing the demonstration of this from the mathematician, therefore, Aristotle employs testimony and signs, just as he is accustomed to do after demonstrations introduced by himself.
[lib. 1 l. 2 n. 8] Then when he says, Therefore, since all, etc., from what has been shown he proceeds to establish his principal point. And he says that these three—all and whole and perfect—do not differ from one another according to species, that is, according to formal notion, because all of them import a certain integrity; rather, if they differ in anything, they differ in matter and subject, insofar as they are predicated of diverse things. For we use the word all in discrete things, as when we say every man, and we also use it in continua that are ready for division, as when we say all the water and all the air. Whole, on the other hand, is spoken of both in these and in solid continua: for we say the whole people and the whole piece of wood. And perfect we speak of both in these and in forms: for we say perfect whiteness and perfect virtue. Because, therefore, all and perfect are the same, it follows that body is perfect among magnitudes: because body alone is determined by three dimensions, and this possesses the character of all, as was shown above; for since it is divisible in three ways, it follows that it is divisible every way, that is, according to every dimension. Among the other magnitudes, however, one is divisible according to two dimensions, namely surface, and another according to one, namely line. For as they have obtained number—that is, according as magnitudes possess the number of dimensions—so do they possess division and continuity: in such a way, namely, that one magnitude is continuous according to one mode, namely line; another is continuous in two modes, namely surface; whereas body is continuous according to every mode. Hence it is clear that body is the perfect magnitude, as possessing every mode of continuity.
[lib. 1 l. 2 n. 9] Then when he says, Whichever magnitudes, therefore, indeed, etc., he shows what is or is not manifest from the premises, and sets down three points. The first of these is manifest of itself, namely that whichever magnitude is divisible is continuous: for if it were not continuous, it would have the character not of magnitude, but rather of number. The second is the converse of this, namely that every continuum is divisible, just as was placed in its definition. And this indeed is manifest from the things proved in Physics VI, as was stated above; yet it is not manifest from what has just now been said, because here he supposed that a continuum is divisible and did not prove it. The third point is manifest from the premises, namely that there is no transition from body into another genus of magnitude, in the way that there is a transition from length into surface, and from surface into body. And he employs the mode of speaking used by geometers, who imagine that a point moved makes a line, while a line moved makes a surface, and a surface a body. From body, however, there is no transition to another magnitude: because such a going forth, or progression, to another genus of magnitude takes place according to a defect of that from which the transition is made (whence natural motion too is the act of the imperfect). Yet it is not possible that body, which is the perfect magnitude, should fall short according to this character, because it is continuous according to every mode; and therefore no transition can be made from body into another genus of magnitude.
[lib. 1 l. 2 n. 10] Then when he says, Of partial bodies, indeed, etc., he makes plain the proper perfection of the universe by contrast with particular bodies. And first he sets forth how particular bodies stand with respect to perfection. And he says that each of the particular bodies, according to the common notion of body, is such, that is, perfect, insofar as it possesses all dimensions; yet it is bounded against the next body insofar as it is contiguous with it. And thus each of such bodies is in a certain manner many—that is, perfect insofar as it possesses all dimensions, and imperfect insofar as it has another body outside itself against which it is bounded. Or it is many according to its contact with diverse bodies; or it is many because there are several individuals of one species on account of imperfection—which does not happen in the case of the universe.
[lib. 1 l. 2 n. 11] Second, at The whole, however, etc., he shows how the universe stands with respect to perfection. And he says that the whole, that is, the universe, whose parts are the particular bodies, must of necessity be perfect in every way; and just as the very name universe signifies, every way, that is, in all modes, perfect, and not according to one mode while failing in another: because it both possesses all dimensions and comprehends within itself all bodies.
On the Heavens Book II, Lecture 2 — On Positional Directions in Natural Bodies and How They Are Considered by Position in Mathematical Figures¶
Latin Locus: De Caelo II, lect. 2 · Commentary: Notes on De Caelo II, lect. 2
[lib. 2 l. 2 n. 1] After the Philosopher has made determinations concerning the perpetuity of the heaven, here he treats of the diversity of its parts. And first, he treats of the diversity of the parts of the heaven taken according to diversity of position within the same heaven; second, of the diversity of parts taken according to the order of the celestial bodies to one another, at Since, however, there is no contrary motion, etc. Concerning the first he does two things: first, he treats of the diversity of positional parts of the heaven according to the opinion of others; second, according to his own opinion, at For us, however, since it has been determined, etc. Concerning the first he does two things: first, he proposes what he intends; second, he makes plain what is proposed, at Now it has indeed been determined, therefore, etc. He says first, therefore, that because certain thinkers—namely the Pythagoreans, who posited right and left in all things—say that a certain part of the heaven is right and a certain part left, it seems we must consider whether the matter stands in the way they state, or whether these are rather to be attributed to the heaven otherwise than they state, if indeed one ought to apply these principles, namely right and left, to the body of the whole world inasmuch as they belong to the celestial body that contains the whole world. And this seems worthy of consideration because right from the outset it occurs to anyone that, if right and left exist in the heaven, much more and with greater priority must the prior principles—namely above and below, before and behind—be judged to exist in the heaven.
[lib. 2 l. 2 n. 2] Then when he says, Now it has indeed been determined, therefore, etc., he makes plain what is proposed. And first, he shows the condition of these principles according as they are found in other things; second, he shows that they are not found in all bodies, at Wherefore neither in every body, etc. Concerning the first he does two things: first, he shows that not all the aforesaid principles belong to all things; second, he shows in what order they stand to one another, at Now above is indeed, etc. Concerning the first he again does two things: first, he shows that principles of this sort do not all belong to all things, whereas some and not all belong to certain things; second, he shows that it is altogether incongruous to attribute some and not all of them to the heaven, at Yet if one must apply them to the heaven, etc. He says first, therefore, that determinations concerning principles of this sort, which are called differences of position, were made in the book On the Progression of Animals, on account of the fact that they are proper to the nature of those things, namely animate beings. For in animals such differences are manifestly seen to exist according to determinate parts: in such a way indeed that to certain animals—namely perfect ones, which not only sense but are also moved with local motion—all such parts belong, namely right and left, before and behind, above and below; whereas to certain animals, namely imperfect and immobile ones, some of them belong, namely above and below, before and behind; and to plants none belong save above and below.
[lib. 2 l. 2 n. 3] Then when he says, Yet if one must apply them to the heaven, etc., he shows that if any of these be posited in the heaven, it is necessary to posit all things of this sort in it. And he says that if anything of this kind must be attributed to the heaven, namely either right or left, it is reasonable that what is found in perfect animals should exist there preeminently—or it is reasonable that what is first in animals should exist in it, because when the posterior is posited, the prior is posited. Now since there are three oppositions or dimensions, each of these three—namely above, before, and right—is like a certain principle of its own opposition or dimension. And consequently he expounds what those three are that he spoke of: one of them is the opposition or dimension between above and below, in which above is the principle; another is between before and its opposite, which is called behind, wherein what is before is the principle; and the third is between right and left, in which right is the principle. And because the perfect is that which consists of all its parts or principles, it is reasonable that all oppositions or dimensions of this sort be found in perfect bodies, that is, in perfect animals. Hence, since the heaven is preeminently perfect, it is reasonable that if it is capable of these parts, it should possess all of them, and not merely some.
[lib. 2 l. 2 n. 4] Then when he says, Now above is indeed, etc., he shows the order of the aforesaid principles in two ways. First, indeed, on the side of the dimensions themselves. For above is the principle of length: for in man, who is the preeminently perfect animal, length—as his greatest dimension—is reckoned from the head, which is his above, down to the feet, which are his below. Right, however, is the principle of breadth: for the breadth of a man is reckoned according to the distance between right and left. Before, moreover, is the principle of depth: for the depth or thickness of a man is reckoned according to the distance between before and behind. And in other animals the relation holds proportionally. Now length is prior to breadth, and breadth to depth, just as line is prior to surface, and surface to body. Therefore above is prior to right, and right is prior to before.
[lib. 2 l. 2 n. 5] Second, at Moreover, in another way, etc., he proves the same thing on the side of motions. And this is because the things that have been spoken of are certain principles from which motions first begin in animals possessing parts or principles of this sort. For the motion of growth begins from above. And this appears manifestly in human beings: for the head, which is the above of a human being, is also above according to the position of the world; and from the head the motion of growth begins, because through the opening of the mouth, which is in the head, nourishment—the matter of growth—is drawn in. In plants, however, the above is the root, which is proportioned to the head in animals in taking in nourishment; yet what is the above of a plant stands in an opposite position to the above of the world. In other animals, meanwhile, it stands in an intermediate manner. Motion according to place, on the other hand, begins from the right: for naturally animals move the right part before the left, just as in walking they move the right foot first. Yet in the motion of alteration of the senses themselves, the principle is what is before: for that part of the animal in which the senses exist is called the front. Because, therefore, the motion of growth is prior to sensitive motion, which in turn is prior to local motion in animals, it follows that above is prior to before, and before is prior to right.
[lib. 2 l. 2 n. 6] Then when he says, Wherefore neither in every body, etc., he shows that principles of this sort do not exist in all bodies. And first, he concludes from the premises that speaking properly and per se principles of this sort do not exist in inanimate bodies; second, he shows in what manner they are spoken of as existing there, at Yet in these indeed, etc. He says first, therefore, that because the aforesaid differences are principles of certain motions, it follows that above and below, right and left, before and behind are not to be sought in all bodies, save only in animate bodies, whichever possess within themselves a principle of motion; whereas in none of the inanimate bodies do we see any principle whence motion begins. Now this can be understood in two ways. In one way, because in animate bodies there is an active principle of motion, which is the soul; whereas in inanimate bodies there is no active principle of motion—namely one that moves—rather, they are moved by an exterior mover, which is either what generates them or what removes an obstacle, while inwardly they possess a passive principle of motion whereby they are naturally suited to be moved, such as heaviness or lightness, as is clear in Physics VIII. In another way it can be understood because in animate bodies a determinate part of the body is found from which motion begins, as has been said, whereas in inanimate bodies this is not found. For, as he adds, among inanimate bodies some are not moved at all, such as those that rest in their proper places (or rather he says this on account of artificial bodies, which possess no motion of themselves); whereas others are moved, such as natural bodies existing outside their own proper place, yet each of them is moved toward its own place uniformly on every side: just as fire is moved solely upward and earth is moved solely toward the center of the world, with no other difference of position taken into account either on the part of the body moved—so that one part of it should begin to be moved earlier than another—or with respect to place, so that a natural body should be moved toward its own place from one local position and not from another.
[lib. 2 l. 2 n. 7] Then when he says, Yet in these indeed, etc., he shows how the aforesaid positions are sometimes spoken of in inanimate bodies. And he says that in bodies of this sort we speak of above and below, and right and left, and similarly before and behind, solely by comparison to ourselves. And this happens in three ways: in one way, according as we call right that which stands opposite us on our right side, just as diviners, for instance augurs, call a bird on the right that which lies to our right, and on the left that which lies to our left; in a second way, by likeness to our own parts, just as in a statue we call right what resembles the right side of a human being, and left what resembles the left; in a third way, by contrary position, calling left what stands opposite our right, and right what stands opposite our left, as is clear in an image reflected in a mirror. And the same account holds in the other positions. Yet in inanimate things themselves considered according to themselves, no diversity of such parts is found. And this is clear because, if they are turned around relative to us, they will stand in the contrary way from before: for what was right will be called left, and conversely; and it is similar in the other positions. In animate things, on the other hand, however they may be turned, parts of this sort always stand in the same way.
[lib. 2 l. 2 n. 8] Then when he says, Wherefore one might also wonder at the Pythagoreans, etc., he shows that the Pythagoreans wrongly attributed differences of this sort to the heaven: and this in three ways, which can be gathered from what precedes, and therefore he introduces them here by way of conclusion. Now the first way is that, since there are six positions, it seems astonishing why they attributed only two of these to the heaven—namely right and left—and left aside the other four, even though it is reasonable that all of them should belong to the heaven, as was said above.
[lib. 2 l. 2 n. 9] The second way he sets forth at No less, etc.: because, namely, if some had to be passed over so as not to be attributed to the heaven, those ought to have been passed over which are less principal. Now that the four they passed over are not less principal than the two they posited, he shows by four arguments. The first of these he sets forth at For in no way a lesser, etc. For we do not see in any animals whatever that the part which is above has a lesser difference from that which is below, or that which is before from that which is behind, than the right has from the left—nay rather, a greater one. For the right and left parts differ solely in power, while agreeing in figure (for the right hand is stronger than the left, even though it is of the same figure; and similarly the right shoulder is stronger than the left for initiating motion, although the left is stronger for bearing a burden; and similarly the right foot is stronger for motion, whereas the left foot is stronger for firm support); yet it is manifest that the front and back parts of an animal, and its upper and lower parts, differ not only in power, but also in figure; and things that differ more seem to possess a more principal distinction.
[lib. 2 l. 2 n. 10] The second argument he sets forth at And above and below, etc., which is as follows. Above and below are found in all animate bodies, both animals and plants; whereas right and left do not exist in plants, save only in perfect animals; and thus above and below are prior, according as that is called prior from which the consequence of being is not convertible.
[lib. 2 l. 2 n. 11] The third argument he sets forth at Moreover, as length, etc., which is as follows. Length is prior to breadth, and this in the way of generation, because according to geometers a line moved makes a surface; now above is the principle of length, while right is the principle of breadth, as was shown above. Since, therefore, the principle of what is prior is itself prior, it follows that above is prior to right, namely according as something is prior in generation (inasmuch as something is called prior in many ways, as is clear in the Categories and in Metaphysics V).
[lib. 2 l. 2 n. 12] The fourth argument he sets forth at Moreover, if above indeed, etc. And he says that above is that whence motion proceeds, which can be understood of the motion of growth; right, however, is that from which local motion starts; while before is that toward which the animal advances, as lying opposite its sense organs; and thus it is clear that above possesses a certain primacy with respect to the other species of position, just as the motion of growth is more essential and more intrinsic to an animal than local motion. Yet everything stated here can better be referred to local motion, so as to say that in an animal moved according to place, above is the principle whence motion originates—because, namely, in the head, which is above, sense flourishes, and sense is what moves in animals, as is stated in De Anima III; right, on the other hand, is that from which local motion starts, because the right part is moved first, as has been said; whereas before is that toward which the animal is moved. Now the moving principle holds the highest primacy in the motion of an animal; and according to this it is clear that above holds primacy among the other species of position. Thus, therefore, from these four arguments the Philosopher concludes the second way of refuting the statement of the Pythagoreans, concluding that it is just to rebuke them because they neglected the more principal principles by not attributing them to the heaven.
[lib. 2 l. 2 n. 13] The third way he sets forth at And because they thought these, etc., saying that they are also to be rebuked because they posited that right and left exist alike in all things, whereas they exist only in perfect animals, as was said above. Yet one must know that it belonged to the intention of the Pythagoreans to reduce all things to good and evil as to two principles. And because they believed every number to be comprehended under the decad, they posited ten principles on the side of the good and ten opposites on the side of evil, as is clear in Metaphysics I. Moreover, through each of the items they placed in that table of opposites, they understood all things belonging to its genus. Hence through right and left they understood all the other positions, understanding that just as right is referred to the good, so too are above and before, whereas left and behind and below are referred to evil. And they posited right and left rather than the other positions because right is more manifestly accustomed to be referred to the good and left to evil—for we are accustomed to call good fortune right (dextra) and ill fortune left (sinistra)—and therefore they attributed right and left to all things to which they attributed good and evil. Or else they named only right and left, comprehending the others under these, because they saw that in whatever things right and left are found, the others are also found, though not conversely. Perhaps, too, they attributed right and left specially to the heaven rather than the others because in the heaven there is local motion, to which right and left pertain, whereas there is neither growth, to which above and below pertain, nor alteration of sense, to which before and behind pertain; or because above and below, and before and behind, are diversified according to figure, whereas right and left are not: for the parts of the heaven, being circular, are not diversified according to figure.