Cursus Philosophicus Thomisticus, Secunda Pars Artis Logicae

Aristotle does not give a definition of quantity in this treatise on the Categories, but begins directly with the division of quantity into continuous and discrete. He does define quantity in Metaphysics V, text 18,¹ as follows: “A quantum is that which is divisible into the things present in it, each of which, or each taken singly, is naturally one thing and this particular thing.” St. Thomas explains this definition there, lecture 15.² The whole explanation comes down to this: Aristotle defines a quantum as divisible, not into physical parts, that is, matter and form; nor into potential parts, as the soul is divided into intellective and sensitive powers; nor into subjective parts, as a universal is divided into its inferiors; but into integral and quantitative parts, which are composed in such a way that, when division is made, each remains one thing, as is clear when water is divided into various parts. A quantified thing is divisible into these integral parts.

For the quiddity of a thing cannot be explained unless we first understand its separation from the other genera to be a real accident, distinct from substance and the other accidents, so that we may inquire only into its distinctive and proper account. Some Nominalists denied this. On the one hand, they thought that without quantity there are no parts or divisibility; on the other, they thought that the entity of anything whatever is nothing other than its parts. They therefore said that quantity is not really distinct from the thing having parts, whether it is a substance or an accident.

The most sacred mystery of the Eucharist, however, has shown us the opposite. Quantity remains there—the quantity that was formerly bread, as we see with our eyes—while the substance of bread does not remain, as faith teaches. Quantity, then, is distinguished and separated from the quantified thing. The Nominalists reply that the quantity of the substance does not remain, since the substance itself has vanished, but that the quantity of the qualities and other extended accidents remains. But there are several accidents there. Therefore, either each has its own distinct quantity, or there is some quantity common to all. If there is one quantity common to all, it is distinct from each accident, since it is common to several, and it is also distinct from the substance, which is not there; therefore, quantity is distinct from the quantified thing. If the first alternative is proposed, then those quantities are mutually penetrated, since all those accidents are in the same place. This penetration is either supernatural and miraculous, or natural and according to their nature. The first cannot be said, because those accidents have the same quantity after consecration as before, since they have the same entity, and quantity is not distinct from entity. Before consecration, moreover, they were also in the same place; therefore, they did not need a supernatural miracle in order to be penetrated. If the second alternative is proposed, then the quantity of such accidents naturally does not resist penetration, but admits it; thus it will be a quantity of another kind from the quantity we are discussing here, which experience itself shows us resists penetration. We say that this quantity is distinct from the quantified thing.

This also refutes a certain more recent opinion that quantity is a mode, not a real accident of the quantified thing. For a mode cannot be separated from the thing of which it is a mode. But in the Eucharist quantity is separated from substance. This opinion therefore concedes that the quantity of the substance vanishes with it, but says that another dimensional quantity remains. If this quantity is a mode of some accident that remains there, there will be as many modal quantities as there are accidents, and so the opinion falls into the view it set out to refute. If, however, it is not a mode but a reality, we have what we seek: quantity is a thing, not a mode. This view must then explain whose dimensional quantity it is.

We shall reply to the opposing basis by explaining St. Thomas’s view: quantity is said to provide a substance with integral parts, not by constituting those parts, but by ordering them among themselves. This ordering is accidental.

With these points presupposed, authors hold a varied and many-sided VIEW in assigning the formal constituent account of quantity. Most seek it in the effects introduced by quantity, some of which are separable from it and others not. Since everyone commonly understands the proper effect of quantity to be the extension of parts, from which several properties flow—such as the account of measure, divisibility, impenetrability, and filling a place—some assign quantitative extension, distinguishing it from the substantial extension of parts by its account of measure. For measure is common to continuous and discrete quantity, and St. Thomas teaches that Aristotle explained quantity in the Categories according to this account of measure in Metaphysics V, lecture 15.³

Others constitute the essence of quantity in divisibility itself: some in radical divisibility, others in formal and proximate divisibility.

Others constituted the essence of quantity in an extension that fills space—not as regards actual filling, as some Nominalists thought, but as an aptitude or root, for quantity is a form that demands this filling. To make this account include every quantity, continuous, discrete, and successive, they distinguish three kinds of space: of place, duration, and discreteness. Extension of continuous quantity occurs according to the space of place, and is commensurated with different parts of place. Extension of successive quantity occurs according to the space of duration, corresponding to different parts of time or motion. Extension of discrete quantity occurs according to the space of discreteness, corresponding to different parts of number.

Others assign the essence of quantity to the extension of parts insofar as it is apt to resist penetration and to fill a place by its nature, although actual filling and commensuration can be separated from quantity. On these views see Carmelus, Course, here, disp. 13, qq. 2 and 3; Merinero, disputation on Quantity, qq. 1 and 2; Master Cabero, here, disp. 2, dub. 2; and Suarez, Metaphysics, disp. 40, sect. 4.

Finally, others constitute the essence of quantity in the extension of parts, not in relation to a place outside them, but in relation to the whole itself, so that without quantity a substance or subject would not have ordered parts, or one part outside another, which is what it means for them to be extended. The disciples of St. Thomas commonly follow this view, as may be seen in the cited authors, and in Vazquez, First Part, disp. 196, ch. 3; Toletus, Physics I, q. 7; and others.

Amid such a variety of opinions, I presuppose three points as certain and more generally agreed upon among authors; I find some doubt only in one.

THE FIRST is that the formal account of quantity cannot consist primarily and per se in formal measure or formal divisibility, actual filling of place, actual impenetration, or any actual extension in relation to place. For it is clear from the mysteries of faith that quantity is found without these properties. The body of Christ is in the Eucharist with its quantity as well as its other accidents, as St. Thomas proves in the Third Part, q. 76, a. 4; nevertheless, there it is not divisible in a divisible mode, nor measurable in relation to place by division, nor impenetrable in the ordinary mode. It is also clear that it was penetrated together with the tomb, when it came forth from the Virgin’s womb, and when it entered with the doors closed. Nor does it actually fill a place in the Eucharist, since the whole is in the whole and the whole is in each part. Therefore, the account of quantity does not consist in these things. If St. Thomas sometimes says quantity consists in measure, as in Opusculum 52,⁴ or in divisibility, as in 1 Sent., d. 19, q. 1, a. 1, ad 1,⁵ and Metaphysics V, lecture 15,⁶ he is speaking of measure and divisibility as roots.

THE SECOND is that quantitative extension, as the root and fundamental aptitude for these formalities and properties of extension in relation to place, belongs intrinsically to the very account of quantity. By its nature quantity demands to fill a place, not to be penetrated, to be formally divisible, and the like; for these modes are connatural to quantity. Hence the view that quantity consists in extension having an aptitude to fill a place and not to be penetrated is entirely correct. Quantity also consists in an aptitude for measure and divisibility, but that account is incomplete, because it explains the nature of the thing not by what it formally signifies, but by what it demands radically. The difficulty, however, is in what this root formally consists.

THE THIRD is that extension in relation to the whole presuppositionally signifies the entity of the parts, and then their union and ordering. It is clear that quantity does not give the entity of the parts, for the parts are entities substantial in kind. Union, moreover, is considered substantially or entitatively insofar as the united parts compose a substantial whole. It is also certain that quantity does not give union in this sense, because such composition is not accidental but substantial, since the resulting whole itself is substantial. Such union can also be taken accidentally, insofar as it signifies union not in just any way, but as ordered: it removes confusion by uniting one part to another, not as a whole to a whole, but according to some aspect of it, namely, according to an extremity. For this reason it is said to place one part after another, because some part extends outside another part and is not united to it as a whole. This is called the formality of an extended part, not insofar as it is the entity of a part, but insofar as it is distinct and ordered, or unconfused. This depends on extremities and indivisibles, without which the parts are not united in an orderly and unconfused way. It is doubtful whether this is the formal effect of quantity.

Resolution: LET THERE BE ONE CONCLUSION: In the view of St. Thomas, the proper and formal account of quantity is the extension of parts in relation to the whole, which makes the parts formal integrants. Hence, when quantity is removed, a substance does not have integral parts formally ordered and distinct in the account of part.

Before the conclusion is proved, I observe that the term “distinct parts,” or “distinction of parts,” can be considered in two ways: in one way, as the term “distinction” is opposed to an altogether simple unity; in another way, as it is opposed to confusion. For distinction can remove unity when multiplicity comes about, and it can remove confusion when order comes about. Things are called confused when they are without order, and distinct when they are placed in an orderly and proper way. What we assert, then, is that quantity formally, insofar as it is quantity, does not constitute the parts of a substance with respect to their entity and intrinsic account, but with respect to their ordering, which removes confusion. This account of ordering is an accidental formality of the parts, which quantity constitutes as ordered and distinct.

That this is the EXPLICIT VIEW OF ST. THOMAS is proved. First, in Opusculum 42, ch. 16,⁷ he says “that part and part are in something by means of quantity, which is the first accident of a body, since it is its measure.” Likewise, in Opusculum 48, Treatise on Quantity, ch. 4,⁸ he says “that position is taken in one way as the order of parts in a place, and thus is one of the predicaments, called situs; in another way, position is the order of parts in a whole, and thus position is a difference of quantity.” He teaches the same there when discussing the predicament of situs, ch. 1,⁹ and in Physics IV, lecture 7.¹⁰ St. Thomas therefore holds that the order or extension of parts in a whole is the proper and distinctive account of quantity. He speaks in the same way in Sentences IV, d. 10, q. 1, a. 3, q. 3, ad 2,¹¹ where he says: “Although we cannot understand an order of the parts of Christ’s body according to a comparison with the parts of the host, we can nevertheless understand an order among the parts themselves in Christ’s body according to its proper quantity.” St. Thomas therefore holds that the order of parts in a whole pertains to quantity itself.

That when quantity is removed the substance remains indivisible, and thus quantity orders its parts first and per se, is proved from St. Thomas himself. In Opusculum 70, q. 5, a. 3, ad 3,¹² he says: “Matter cannot be divided except with quantity presupposed; when quantity is removed, all substance remains indivisible, and so the first account of diversifying things of one species is according to quantity. This pertains to quantity insofar as, in its account, it has position, as it were, for a constitutive difference; position is nothing other than the order of parts.” Here St. Thomas does not take position formally, but as a root; otherwise, how could he have been unaware that it is a predicament distinct from quantity and not its constitutive difference? Likewise, in Quodlibet 9, a. 6,¹³ he says “that a diversity of parts cannot be understood in matter without understanding division, nor can division be understood without understanding dimension, because when quantity is removed the substance remains indivisible, as is said in Physics I, text 18.”¹⁴ He says the same in the First Part, q. 50, a. 2, and in Contra Gentiles IV, ch. 65: “when quantity is removed, every substance is indivisible.” He adds that “quantity has this property among the other accidents: it is individuated according to itself. This is because position, which is the order of parts in a whole, is included in its account; for quantity is quantity having position.” See how clearly and consistently St. Thomas teaches both that quantity provides the order and account of parts in a whole, and that when quantity is removed the substance remains indivisible. Indeed, even when quantity is removed in relation to place, as in the body of Christ, the body nevertheless remains divisible and extended in relation to the whole by reason of quantity. St. Thomas therefore holds that when quantity is removed, substance remains indivisible both with respect to place and with respect to the whole; that is, it does not have parts formally as parts.

The BASIS of this view is often established by several arguments intended to destroy the entity-like extension of integral parts prior to quantity. The arguments, however, are most effective when reduced principally to two principles:

We cannot explain the first here; it must be presupposed and treated at length in the books on Generation,¹⁵ namely, that the individuation of corporeal things is taken from matter designated by quantity. It is taken from matter as a substantial principle, which individuates not formally but radically. Hence matter is not individuation itself, but the principle of individuation. Quantity is taken as a required condition, not as the per se principle of substantial individuation. This teaching is found explicitly in St. Thomas, Opusculum 70, q. 4, a. 2,¹⁶ and in many other places. From this principle it necessarily follows that if quantity designates matter with respect to individuation or the distinction of individuals, it will also do so with respect to the distinction of parts, because only by actual division do designated and extended parts become several individuals. Thus the designation of distinct parts and of individuals pertains to the same principle. Hence the Philosopher also calls a quantum “divisible into things, each of which is naturally one thing and this particular thing,” that is, something so designated that only by actual division does it become this or that individual.

The second principle to be explained here is taken from the proper account of integral parts. To be extended, unconfused, and ordered, they require some accident in addition to substance. This is proved because extension of parts in a whole is not just any union among them, but a union only at their extremities, so that one part is not united to another as a whole to a whole, penetrated and absorbed into it, as form is united with matter, which is purely substantial. For if one part is united with another in the whole by penetration and according to every part of itself, it is not united extensively—that is, one part is not outside the other. Therefore, for an extensive union in a whole, the parts must be united at their extremities and not as wholes, since otherwise they would be united by penetration, like matter and form. The mode of union among integral parts is distinct from the union of matter and form.

We therefore inquire: what causes the substance of one part, for example, a hand, to be prevented from uniting with the arm according to every part of itself, and to be united and joined to it only at the extremities? This cannot happen by virtue of the substance itself, since substance does not of itself have parts with extremities. Extremities are had only through indivisibles, but substance does not of itself have indivisibles, such as lines, surfaces, and points, since these are proper species of quantity.

If it is said that substance also has its own substantial indivisibles corresponding to the indivisibles of quantity, the following argument presses against this: these indivisibles are posited so that substantial parts may be united at their extremities, that is, extensively and not by penetration. For if they are not united at their extremities, then precisely by this fact they are united as wholes and by penetration, which means they remain confused and unordered among themselves, rather than one being after another. Therefore, if substantial parts are united before quantity by these indivisibles and extremities, they remain unpenetrated or extended in the very whole prior to quantity. Thus substance in itself has a sufficient principle to resist penetration in relation to place and to fill it. Consequently, it is pointless to posit the formal account of quantity as a principle of impenetration in relation to place, since the entity-like extension of substance in itself has this already.

The consequence is proved because the entity-like extension of substance occurs at extremities and indivisibles before quantity, in such a way that this union prevents one part from being penetrated with another in the whole as regards continuity. Therefore, it will also prevent it from being penetrated with the parts of an adjacent, contiguous body. For it is unintelligible that a principle sufficient to prevent the union and penetration of a continuous part would not prevent the union or penetration of a contiguous part. But authors of the opposing view call the principle that prevents penetration by a contiguous body quantitative extension, distinct from entity-like extension. Therefore, before quantity they posit in substance the formal account of quantity itself, precisely by positing that substantial parts are united in an orderly and unconfused way through substantial extremities and indivisibles. We must therefore hold that this extensive union in a whole is accidental and comes from quantity.

And if you say: In the sacramentally present body of Christ, there is quantitative extension in relation to the whole, such that one part is not penetrated with another in the union of the whole; and yet in relation to place it has no extension, but an indivisible mode. Therefore, this extension in relation to the whole comes from a principle other than the union of parts in relation to place. The reply is that the quantity of Christ’s body, which is in the whole itself, is a sufficient principle of impenetrability and extension in relation to place, unless this effect, which is secondary in quantity and properly pertains to circumscriptive location, is divinely impeded. This is why we place the whole force of the argument here: in substance itself, prior to quantity, there is a principle that repels penetration by a contiguous body, precisely because there is a principle that repels penetration of parts within a continuum. But if there is no principle that repels penetration in place, there is likewise no principle that repels penetration in the whole. Thus both effects come not from substance but from quantity, although one effect can be separated from the other by God’s absolute power, because one is primary and the other secondary.

THE ARGUMENTS ARE RESOLVED.

Nearly the whole difficulty and the arguments raised against this resolution depend on the principle that the parts of substance do not depend on quantity in their entity or distinction. Thus substance is considered without quantity and yet with parts. For example, if God were to remove quantity from a substance while preserving it, then either the substance would remain with distinct and extended parts, or without them and indivisible. If the first, we have what we seek; if the second, it follows that all the parts of substance converge to a point. This would occur either in an instant or over time. It cannot occur in an instant, because a flow or motion cannot take place in an instant. If it occurs over time, then those parts last as long as that time lasts, and yet there is no quantity. Nor is there more reason for that substance to converge to one part than to another; therefore it will remain as before.

It is confirmed: if two partial substances are united and newly acquired, as happens in nutrition, and God removes quantity from them, either those parts will remain one or distinct—that is, one part will not be the other. If the latter, we have what we seek. If the former, then they will lose their individuating principles and individual differences while those same principles remain, which is impossible. The same argument applies to two fires, one of which produced the other, if they are united with one another and God removes their quantity. Either the parts will remain distinct or they will not. If they remain, we have what we seek; if not, then when quantity is removed the producer and produced will be the same and one. Likewise, if God removes quantity from the two outer palms of a three-palm piece of wood while leaving quantity in the middle, then those two ends are distinct parts of the wood and without quantity. Therefore.

All these arguments press the same difficulty, and they are resolved by one principle: before quantity, substance does not have parts formally extended—that is, ordered and unconfused—but it does have parts radically, that is, a capacity to receive that ordering and distinction of parts. Without quantity, the parts remain penetrated and united as wholes, which is not to have one part outside another. This should not seem surprising, because when quantity is removed all indivisibles and extremities are removed, without which the parts are not united extensively—that is, at the extremity of one and the other—but by penetration and as wholes. Therefore, the entire entity of the parts remains when quantity is removed, but confused and in the manner of one thing. There is not, however, one integral part and another formally; there are parts only radically.

THEREFORE, AS TO THE FIRST CASE, we say that when quantity is removed substance would not converge to a point, but would lack all points and consequently every union of parts by way of extension. For, as St. Thomas rightly observes in 2 Sent., d. 3, q. 1, a. 1,¹⁷ substance without quantity is not indivisible because its parts are reduced to a point, which is a principle of quantity, but because it lacks all divisibility. Hence there would be no motion in that substance, just as there would be no physical place; rather, it would exist in the universe as a certain part of it, not as something located in a place. For all these imaginings must be set aside, since they follow quantity as located, as Cajetan rightly taught in the First Part, q. 52, a. 1, near the middle. Therefore, that substance is neither distant nor positively somewhere, but would have its existence without a place, like a thing outside the world or an angel not acting.

In reply to the confirmation, it is said that when quantity is removed from the parts, they do not remain actually distinct, but confused, as one entity with a radical capacity for partial distinction. When it is said that one part is not the other, and that the individual differences cannot be lost while the same things remain, we reply by denying the presupposition. For when quantity is removed, no part remains in act, but only radically and in potency. Hence every individual difference in the genus of part ceases, because no part is given in act. Nor do the same parts remain once their differences are removed, because, as we said, the parts do not remain in act; rather, a certain confused substance remains, capable of parts but not actually composed of parts. It is like matter without form: it would have no formal being in act, but only in capacity and potency, as will be said in Physics I.¹⁸

You will insist: Two individuals, for example, two stones, remain distinct according to their individual differences when quantity is removed, although their individuation depends on quantity. Therefore, substantial parts too will remain actually distinct according to substantial distinction, although that distinction depends on quantity.

The reply is to deny the consequence. The difference is that the distinction of individuals depends on quantity as dividing and separating them. Once division has taken place, if quantity is removed, since the substances themselves are not united again, they remain separate and not one; they have this by their order to quantity, although they are not informed by it. So a soul, individuated by its order to the body, is individuated by its order to the body even when separated from it; and the same is true of a separated accident in its order to its subject. But parts in the same whole are not distinguished by quantity as dividing and separating them—for in that way the parts would not remain; rather, the whole would—but by quantity as uniting them in an ordered way, that is, not as wholes and confusedly, but at their extremities, placing one outside another without dividing it from the other. When quantity is removed, this ordered and extended union is dissolved and another takes its place, by which those parts are united as wholes and confusedly, or rather become one in substance. Thus their distinction in the account of part is prevented, and this is to lose the individual differences of parts. For parts in the very whole do not have individual differences simply, which arise only through division, but in a qualified sense, since they truly compose one thing and are not simply distinct individuals. Hence their actual distinction in the genus of part does not remain in act when quantity is removed, but in potency, because the ordered union by which they were distinguished changes into another, confused union that prevents actual distinction.

As to the producer and product becoming the same in that case, we reply that the producer and product are never the same while production is being exercised. But when production ceases, the thing that produces can indeed pass into the substance of the product and be united to it, as if a son ate his father; and then, when quantity is removed, those parts are distinguished only virtually and radically. By absolute power, through transubstantiation, the producing things can be converted absolutely and wholly into the thing produced.

As to the final case, it is said that when quantity is removed from the two ends while quantity remains in the middle, the quantity itself is consequently divided, and those ends remain separated from the whole and become individuals by their order to the divided quantity; they are no longer parts. But if God keeps them united to the intermediate substance, they will not remain parts formally, but radically, as confused and united as wholes with the intermediate substance.

And if you ask what it means for a substance to have parts radically, and how it differs from an indivisible substance such as an angel, the reply is that it differs from a spiritual substance because a spirit lacks parts negatively and by incapacity, whereas a body deprived of quantity lacks parts privatively, while retaining a capacity for them. It is not said to have those parts actually, because it does not have them distinct and multiplied by extension, that is, one outside another; rather, through confusion they are reduced within it to the manner of one thing, and not merely with respect to place. Thus the entity remains ordered toward an extension of parts, but it does not itself have an ordered extension of parts.

Second it is argued: When substance is under quantity, it has distinct parts not only accidentally but also substantially; therefore, even when quantity is removed, that substantial distinction can remain.

The antecedent is proved: the distinction of a hand from an arm now does not occur through quantity alone. For if they differed through quantity alone, they would not be two parts of a substance; just as Christ’s body placed under the Sacrament and in heaven is not two bodies, although its place accidents vary. Therefore, likewise, if those parts differ substantially, they are not distinguished through quantity alone.

It is further confirmed that unity or division follows the being of a thing. But every substantial part has being as a substance not from quantity but from itself; therefore, it will also have substantial unity and distinction from itself, not from quantity. The major premise is clear, because unity is a property of being, and there are as many unities as beings. Therefore, substantial unity or distinction is a property of substance and thus does not depend on quantity. Hence St. Thomas says in Metaphysics V, lecture 15,¹⁹ that only quantity has division into proper parts after substance. Therefore, St. Thomas presupposes that substance has proper parts.

It is confirmed a second time: the quantity of a hand is not in the foot, or vice versa. Therefore, the quantity of one part is not received in another. Therefore, a distinct account of the parts of hand and foot is presupposed before quantity, since the quantity received in one is not received in the other. Again, when quantity is removed, the entity of the hand, the entity of the foot, and the other parts remains in that substance. But the entity of the hand is not the whole entity of that substance; therefore, it is a part. Nor can it be said to be reduced to an indivisible entity, as a spiritual substance is, since a spiritual entity does not contain the entity of a hand and foot as a corporeal substance does when quantity is removed.

THE REPLY to the principal argument is that substance under quantity has parts distinct in two ways, namely substantially and accidentally. But substantial distinction cannot be reduced to act except dependently on quantity as a condition, without which that distinction of substance does not come into act. Just as substance does not come into act in influencing and acting unless a quality is posited as the instrument by which it acts, and when the quality is removed it remains an agent only radically, so without quantity confusion and indistinction remain. When this is posited, substance cannot proceed to the act of substantial distinction through an extension of parts that removes this confusion; rather, the whole remains radically distinguishable through quantity. Thus from substance comes the account of constituting parts; from quantity comes the account of ordering and removing confusion through extension; and substantial distinction itself depends on this ordering as a required condition.

In reply to the first confirmation, it is said that unity or division follows entity, but not always absolutely; it may depend on something extrinsic, without which distinction does not follow in act but remains in potency. Thus unity follows being without abstracting from a state, because sometimes unity involves a state, for example, the unity of individuation, universality, order, and so on. These states can depend on accidents at least as conditions, without which that state does not exist. For example, the state of individuation depends on a collection of properties belonging to one individual rather than another, and on the designation of matter, which occurs through quantity. Likewise, the distinction of integral parts depends on quantity because it does not occur without an order that removes their confusion. For this reason, St. Thomas calls those parts material and accidental in the cited Opusculum 70.²⁰ As to the passage from Metaphysics V, we reply that substance has proper parts radically from itself, but formally and in an ordered way from quantity. It also has proper physical parts, namely matter and form, but not extended and integral parts.

In reply to the second confirmation, it is said that although the quantity of the hand is not received in the foot, it does not follow that before quantity the hand and foot are formally distinct parts. Rather, there is only something capable of and receptive to the quantity of a hand and the quantity of a foot. This is to precede those parts radically and in capacity, as things that can be ordered, not as already ordered. Thus different quantities are received in different parts, not parts that are presupposed to be different, but parts that quantity makes different. This is like the view that distinguishes existence from essence and posits matter as inseparable from every form: existence is received in an existing subject and form in existing matter, not by an existence that it presupposes, but one that it provides; it presupposes only a capacity for existing. Similarly, regarding what is added—that the entity of the hand is not the whole substance when quantity is removed—we say it is not the whole adequately. It does not follow, however, that it is formally a part, but only radically, because it is not capable of the same quantity and ordering that is in the whole; in itself, however, it does not have another ordering and distinction of a part in act. Nor does it follow that this entity becomes spiritual, because it remains with a capacity for quantity, which a spirit does not have; nevertheless, it has a certain mode of spirituality, like Christ’s body in the Sacrament.

Finally it is argued: Substance acquires parts not all at once but successively, so that when one is produced, it is not the same part that existed before, but a distinct one. What is produced is not merely new quantity but new substance, since it is produced through nutrition, which is an action distinct from increase; increase formally concerns a greater quantity. Therefore, if nutrition or generation of an added part is distinct from increase, it must have a distinct formal terminus in the account of substance. But it does not terminate in the whole substance, since that pertains to generation itself; therefore, it terminates in a part. Thus before quantity there is a part in substance.

It is confirmed that the parts of substance are united substantially; therefore, diversity is given according to substance, not only according to quantity, for a union is between distinct extremes. Likewise, when quantity is removed, if the parts are confused, they must be united in a different way from how they were united before, under quantity. Why, then, could God not prevent this new union, so that they remain unconfused without quantity?

Finally, it is not impossible in the Eucharist for one half of the host to remain unconsecrated when the other half has been consecrated. Then it is not distinguished from the consumed half by quantity, because the quantity of that half remains there. Therefore, it is distinguished without quantity and, consequently, one half could be preserved by God in this way without quantity, as could the other.

THE REPLY is that nutrition and every successive acquisition of parts formally terminate in substance, but do so under the condition of quantity, by reason of which the substance is formally a distinct part. Increase, on the contrary, formally terminates in quantity, but in quantity inhering in substance; this serves as a condition with respect to increase. Hence nutrition and generation of an added part are distinguished from increase on the side of the formal terminus, because the terminus of nutrition is substance, not total substance but partial substance. Yet the formal character of part does not come from the formality of the terminus of nutrition, but from the added condition of quantity. Thus if it terminated in substance stripped of all quantity, it would not terminate in a part formally but radically.

In reply to the confirmation, it is said that this substantial union of parts cannot occur independently of quantity, which is a condition required for the ordering and designation of parts. Once this designation is presupposed, the substance of the parts themselves is also united. In the view of St. Thomas, designated matter is likewise a principle of substantial individuation, with quantity as a required condition; for without the designation of matter, it is a principle of confusion rather than of division and individuation. Likewise, since substance without quantity presupposes no ordering of parts, it cannot exercise a substantial union among them as ordered; rather, it is said to remain confused, and consequently without substantial union between formally distinct extremes, but only radically.

As to what is added—that when quantity is removed the confusion of parts occurs through a new union—we reply that this is not necessary; rather, only the removal of every union is necessary. For when quantity is removed, parts are not replicated, penetrated, or reduced to a point, all of which would require a positive union. Rather, they are left in the negation of all divisibility, as St. Thomas says in 2 Sent., d. 30, q. 2, a. 1.²¹ A positive union is not necessary for this negation; for parts do not remain to be united through a positive confusion, but only an entity capable of parts and partibility remains, and this is a negative confusion.

As to the case of the consecration of half the host, we reply that the half that is consumed is distinguished from this half not as a part coordinated and distinct from it, but as a certain whole, because it has been converted into the body of Christ, which has its own distinct quantity. If God were to separate that half without converting it into some other substance, while its quantity remained with the other half of the host, then it would be distinguished from the other half by separation as a whole, not by an ordered union as a part; and it would have an order to the divided and distinct quantity that it demands by virtue of this separation.

There is no less difficulty in understanding the formal character of quantity which, by its extension, unites parts and is continuous quantity than in understanding that which, by its extension, separates them and is discrete quantity, or number. The familiar distinction between numbering number and numbered number must be ASSUMED. Numbering number is that which is the principle of numbering in the intellect, such as two, three, four, and so on: these are the principles by which we number every sort of matter. Numbered number, by contrast, is the things themselves, or the matter, that is subjected to this numbering. Numbered number may be taken generally for every multitude in any way numerable by the intellect, including one found among spiritual things; or, more particularly, for quantitative multitude, which, by reason of quantity, has a special character of measure in numbering. For this reason, the very discreteness of quantity, relative to the substance subjected to it, is sometimes called numbering number—that is, it supplies the principle by which something is numerically measurable as quantitative. This is taken from St. Thomas, Opusculum 48, treatise on Quantity, ch. 1.¹

THE DIFFICULTY, THEN, lies in explaining two things. First, whether this character of number in discrete quantity is something one per se, sufficient to constitute a true species of quantity. Second, from what form number derives its unity, given that it has parts really separate and divided.

THE OPINION OF MANY, THEN, is that, considering the multitude of which number consists, they have said that number is not a being per se but per accidens, and consequently must be excluded from the per se species of quantity. This view may be found in Fonseca, Metaphysics V, ch. 7, q. 5, sections 1 and 2; Suárez, Disputationes Metaphysicae 41, section 1; Vázquez, Prima Pars 26, ch. 3; and the Coimbra commentators, here, q. 2, art. 1.

Others, on the contrary, affirm that number is one per se by the unity of discrete quantity, although it is not one in the genus of continuous quantity. Both the Thomist and Scotist schools commonly follow this view, but they do not all explain its unity in the same way. Some say that number is one by unity of order, a view grounded in St. Thomas, Metaphysics VII, final lecture, on text 60², where he says that a syllable and a number receive their species from order. But, if taken precisely, this unity seems more relational than quantitative. Others say that the last unit gives number its form and unity, as St. Thomas explicitly says in Metaphysics VIII, lect. 3³; this must be understood of the last unit in order to the others. There is, however, a difficulty: in a multitude the last unit is not determined to be this unit rather than another, so it will never remain determinately one in reality. Others say that discreteness itself is the formal character of number, while the units making up the multitude are the subject and matter of this discreteness. This is commonly the view of Scotus’s disciples, as may be seen in Father Merinero’s commentary here on the chapter on Quantity; St. Thomas, Opusculum 48, treatise on the Predicament of Quantity, ch. 1, seems to support them.⁴ Yet it is unclear how discreteness could be one form simply in the genus of quantity rather than relation, while having as its subject, in which it really inheres, things so diverse and really separate. Indeed, discreteness seems to be nothing other than division. Division is not a form that constitutes number as one species, but only as a multitude and a separation of continuous quantity; and multitude as such is not a species of quantity.

Finally, others say that every number is a plurality or multitude not ordered to compose one principle of operations, yet is a species of quantity insofar as it has the character of measure; otherwise it has unity only from our intellect, and in reality has only measurability, which is the numerical plurality of its parts. This is Torrejon’s view here, disputation 3, q. 5. But this simply excludes number from the predicament of quantity: its unity is not real, and the character of measure common to every spiritual and corporeal quantitative multitude is not real either. Number’s unity is not taken from its order to some operation, just as the unity of continuous quantity is not, since quantity is not operative.

Therefore, to explain what seems more probable in this matter and more in accord with the mind of the Holy Doctor, we answer with two conclusions.

FIRST CONCLUSION: Number composed of quantitative units is a true and proper species of quantity.

This conclusion cannot be denied in Aristotle’s doctrine. He places number among per se quantities not only here in the Categories, but also in Metaphysics V, where he enumerates only the per se species of quantity and excludes the species of quantity per accidens.⁵ Nor can Aristotle be explained, as some say, to have placed number among the species of quantity not because it has true unity, but because it has its own mode of extension or multiplication and is commonly conceived as one thing with a definition and properties of its own—enough, they say, for it to be placed in a predicament, since not everything placed in a predicament has its own unity per se. This solution swallows the entire difficulty and destroys the rationale of predicamental order. For if number lacks true unity and is a being per accidens, it is excluded from the predicament: it has no per se definition, since it has no one per se quiddity; consequently, it has neither one genus nor only one difference, and therefore is not placed in one predicament, since a predicament is an ordering of genera and species through their differences. If Aristotle excluded a being per accidens from definition, as is clear in Metaphysics VII⁶, how could he place number, a being per accidens, per se in a predicament as constituted by genus and difference—that is, by a definition? Hence there can be no doubt about the Philosopher’s intention.

There can be even less difficulty about St. Thomas’s intention. He plainly affirms in several places that quantitative number is a being per se and is placed in the predicament of quantity: see Metaphysics V, lect. 15⁷; Metaphysics IV, lect. 2⁸; Sentences I, dist. 24, q. 1, art. 3⁹; De potentia, q. 9, art. 7¹⁰; Summa theologiae I, q. 11, art. 1, ad 1; and Quodlibet 10, art. 1.¹¹ There is no need to cite his words one by one, since everywhere he says that number is a species of quantity and is one per se.

Thus those who explain St. Thomas’s statement that the last unit constitutes the species of number by saying he speaks metaphorically have either not read him or have refused to explain him in the many other places where he teaches absolutely that number is something one per se and belongs in the predicament of quantity. We understand, then, that when he says the last unit gives number its species, he means a true species, one per se, not a metaphorical one. Thus, near the end of Metaphysics VII, lect. 13,¹² he says: “Duality is not two units, but something composed of two units; otherwise number would not be a being per se and truly, but per accidens, like things heaped together.” Here St. Thomas plainly excludes metaphorical composition and unity per accidens, which are precisely the reasons for excluding number from the predicament of quantity. The same may be seen in Metaphysics VIII, lect. 3¹³ and Metaphysics VII, final lecture, on text 60.¹⁴

THE BASIS for this conclusion is given by St. Thomas in the cited Quodlibet 10 in two ways, one a priori, the other a posteriori. The a priori basis is that number expresses true extension and a character of measure arising from division of the continuous, which is not expressed by multitude alone, taken confusedly in the manner of a heap or pile. The reasoning is as follows. Number founded on divided quantity has true and proper measurement by way of extension and multiplication, by which it can exhaust and equal the very parts and extension of the continuous. It does not have this character of measurement and extension as a quantity per accidens, as do time and motion, which are extended by reason of the continuous. Number’s own discrete measurement does not arise from continuous quantity, since it rather divides and dissolves its continuity. Therefore, the character of measure and extension that number has is not a quantity per accidens, that is, through another, but is distinct from the mode of measuring the continuous. That number’s measure is a measure in the genus of quantity and extension is clear from its measuring according to the properties of quantity, such as equality and inequality, evenness and oddness, increase and augmentation. Therefore this measure is quantitative. Accordingly, St. Thomas says in Sentences I, to Hannibald, dist. 24, q. 1, art. 1, ad 3,¹⁵ that “number simply is that whose numbered things make some aggregation.” This is clear because by counting and adding units we make an aggregation and an increase in an ordered way, that is, with a measure of before and after. Thus there is extension—that is, multiplication joined to an ordering of parts—and the sole difference from continuous quantity is that in the latter the parts are united, whereas here they are separate. This accords with St. Thomas’s common teaching that the unit, as the principle of number, adds a character of measurement to the character of unity that is a property of being, as is clear in Metaphysics IV, lect. 2¹⁶; Metaphysics V, lect. 8¹⁷; and Summa theologiae I, q. 11, art. 1. Since number is constituted from units, if, as they say, the principle of measurement consists in those units, number thereby introduces a special measurement and extension.

YOU WILL OBJECT: All this is found in spiritual multitude: one unit added to another makes more, which is to make an aggregation. Even and odd are found there as well, since three angels are odd and four are even. Therefore spiritual number has the same character as quantitative number.

We answer that what is lacking there is extension and aggregation arising from division of the continuous; there is only affirmation and negation of being, as St. Thomas rightly notes in Sentences I, dist. 24, q. 1, art. 3.¹⁸ Accordingly, no aggregation or division commensurable with continuous quantity, and thus of the same genus as it, is found there. This is what belongs to the character of quantitative number: because it is caused by division of the continuous, by its very character as number it measures and exhausts the parts of continuous quantity. Perhaps this is why the Commentator said in Metaphysics X, text 2, that the character of quantity or measure belongs primarily to discrete quantity: we measure the continuous itself by counting its parts, and no measure is better known than counting. St. Thomas explicitly teaches this in Opusculum 36, ch. 1,¹⁹ saying that “the character of measure is found first in discrete quantity.” Therefore number in the genus of quantity adds this to multitude taken in general: it exhausts the measurement of continuous quantity and makes an aggregation and addition of one to another. Spiritual multitude does not do this, although we number angels just as we do quantities from the standpoint of numbering number.

St. Thomas’s second argument is that quantitative number is the proper object of arithmetic; therefore it is a being per se. The antecedent is evident, since arithmetic has no object other than number, or discrete quantity. The consequence is also evident: an object per se knowable must be a being per se, with its own definition, which is the principle of a science. Therefore number is a being per se, and so there is no impediment to placing it among the species of quantity.

BUT THE ANSWER IS GIVEN first, that number is a being per accidens as a thing, but per se as something knowable; thus number, because it has a numerable unity, is the object of arithmetic. Second, number is formally one by a unity of reason, which apprehends number as though it were something one; this unity suffices for it to be an object of science. Third, arithmetic treats number materially—that is, as continuous quantities, not formally as a plurality of units.

But the first solution fails completely. If number is per accidens in the genus of things but one per se in the genus of knowables, then there can be a science of a being per accidens, since a being per accidens can be per se knowable. This is wholly impossible, since a being per accidens cannot be defined per se: it has no per se quiddity, as has often been proved from the Philosopher. What cannot be defined cannot be an object of science, since definition is the principle of demonstration and scientific procedure, as is clear from Posterior Analytics I.²⁰

The second solution is no better. Either that unity of reason is formal and per se in number, or it is only accidental, as every nature considered universally has a unity of reason insofar as it is universal. The latter is irrelevant: just as universality is a condition of the object of knowledge, not the very character that is the knowable object, so that unity of reason would be only a condition. If the former is said, then number is formally a being of reason, since it is formally one by reason; arithmetic would thus treat a being of reason, which it would be remarkable for these authors to concede, since they deny that logic treats of a being of reason. Then numbered number, which arithmetic treats, would not be distinguished from numbering number, since that unity of reason arises in number only from the act of numbering reason. But the arithmetician does not consider that rational denomination per se.

The third solution is even less satisfactory. When arithmetic is said to treat number materially, taking it for continuous quantities, this means either that it takes continuous quantity formally insofar as it is continuous, or materially under the character of being divided and discrete. It plainly does not treat the things themselves that are subject to quantity. If number is taken for continuous quantity formally, that belongs not to arithmetic but to geometry. If it is taken insofar as divided and discrete, then arithmetic treats number per se, not continuous quantity; it is only per accidens to arithmetic to consider ratios of numbers as they occur in lines, surfaces, or some other continuous quantity.

I SAY, SECOND: That by which number is made one per se is not the multitude of units itself, nor their aggregation or discreteness; all this is material in number and is a being per accidens. Rather, the last unit is that from which number derives its unity per se, insofar as several units are ordered under it.

Therefore, the form of discrete quantity does not inform all the parts as continuous quantity does, but orders them; these discrete parts can be extended only by being ordered under a last unit. The unity of number is like the unity of place. Several surfaces succeed one another at one distance, and with the surfaces several relations to the poles of the world; nevertheless the same place remains formally with respect to the fixed terminus it faces, although materially there are several surfaces, as is said in Physics IV²¹. So in number there is the character of a last unit terminating the others. That this or that designated unit performs the office of last unit is material and per accidens.

This conclusion follows from St. Thomas. First, near the end of Metaphysics VII, lect. 13,²² he says: “Duality is not two units, but something composed of units; otherwise number would not be one per se and truly, but per accidens, like things heaped together.” Here he plainly distinguishes the multitude of units, which belongs to heaping, from the character of number, which is one per se. This also accords with the well-known authority from the Philosopher, Metaphysics V, text 19,²³, lect. 16 in St. Thomas²⁴, that “six is not twice three, but once six.” Number, then, does not consist in the mere division of the continuous; it requires a unity per se that results from such division and multitude.

Moreover, we should note St. Thomas’s resolution in Metaphysics VIII, lect. 3,²⁵ where he speaks of the form that constitutes the specific unity of number: “Number is one per se insofar as the last unit gives number its species and unity, just as in things composed of matter and form something is one through the form and receives its unity and species from it. Those who speak of the unity of number as though number were not one per se cannot say by what it is one, if it is one. For since it is composed of many units, either it is not one simply, but the units are aggregated in it by way of heaping, which does not make one simply and consequently does not constitute anything in a species, so that number is not any species of being; or, if number is one simply and not through itself, one must say what makes it one from many units, and this cannot be assigned.” See how plainly St. Thomas teaches that the last unit gives number its species per se. If it does not, no formed or resulting thing made from many units can be found to give number its species. Therefore, on the side of multitude, discreteness, or division of quantity, nothing can be given that constitutes per se the species of a determinate number except the last unit, insofar as it presupposes the others and subordinates them to itself.

That this suffices for number to be one simply and not a being per accidens is drawn from St. Thomas’s teaching in Metaphysics VII, final lecture on text 60,²⁶ where he explains the difference between composition from many things that makes one simply and that which makes one in a qualified sense. “The reason for this difference,” he says, “is that sometimes a composite receives its species from some one thing, which is either form, as in a mixed body, or composition, as in a house, or order, as in a syllable and a number; then the whole composite must be one simply. Sometimes, however, a composite receives its species from the multitude of collected parts themselves, as in a heap, a people, and other such things; in these cases the whole composite is not one simply, but only in a qualified sense.” This teaching must be carefully observed in distinguishing the various modes of unity simply and per accidens, for all these things are not called one in the same way; they are different degrees. An angel and the heavens are one in a more perfect way, admitting no variety, than animals or living things, which undergo continual increase and decrease; and these are more one than a river, which is not the same in matter or form as the river flowing now and the one that flowed twenty years ago, but is the same only by reason of the place through which it flows. Artifacts, similarly, are one only on the side of their artificial form. Yet all these fall under the character of unity simply by St. Thomas’s rule: things are one simply when they receive their species from something one, whether that one is form, composition, or order; only things that receive their species not from one thing but from a multitude are one in a qualified sense. So we say that number, insofar as it expresses multitude, is a being per accidens; both multitude and discreteness—which discontinues or divides quantity—are material and cannot give unity per se. For if number received its species from them, it would receive it from multitude itself and thus be one only in a qualified sense. Number, then, receives its unity per se and discrete extension from all these units being ordered under one last unit, which terminates them, insofar as they are quantitative units that add one divisible unit to another, can be made equal to some continuum, and are terminated by the last unit.

It also follows that discreteness itself, or any other form intrinsic to the whole number, cannot constitute number. Either that form is really in all the units or only in one. If in several, it is either divided among them or undivided. If divided, it is multiplied and consequently constitutes unity per se no more than multitude itself. If undivided, it cannot be identified with so many separate and divided units. But if that form exists in only one of the units, then our point is established: something one, namely the last unit, gives number its species. Therefore this unit cannot give number its species as an intrinsic form. Hence St. Thomas does not reduce the unity of number to unity from form, but to unity from order, as is clear from the cited words of Metaphysics VII, lect. 13,²⁷ insofar as the other units are ordered to it as that which finally terminates and closes them.

Thus the form of continuous quantity, by informing, extends and orders the parts; discrete quantity, however, cannot extend and order by informing, but coordinates one unit with another, so that from the last unit the determinate quantity of the number is taken. For example, that there are just three, five, or any other determinate number depends on the last unit by which the number is terminated and thereby determined.

BUT A DIFFICULTY ARISES HERE, one that commonly troubles the defenders of this opinion: how can that last unit be certain and designated, if, in a number—for example, four—any unit can be the last, and this depends only on the designation made by the intellect that numbers, which would make it something of reason?

We answer that in number there is in reality a last unit according to formal determination and designation, not material designation: that is, insofar as it terminates and closes the units. Four does not go beyond four units, for otherwise it would not be four; this effect is real. That the effect proceeds determinately from this unit or that is material and accidental, since each unit is suited and able to produce it. Thus it is one thing to be last with respect to designation, another with respect to the effect of termination. The latter exists in reality; the former may depend on the intellect’s act of designation, but is material and per accidens to the constitution of number itself. Many things in reality labor under this kind of difficulty: some formality and its effect are found in reality, while the material designation of the subject or part in which that formality exists depends on the intellect. For example, a circle, being of finite and determinate quantity, must have a beginning, middle, and end; otherwise its quantity would not be finite. Yet the character of beginning, middle, or end can be designated at any part, since no such part is designated by the nature of the thing. Similarly, a point in the middle of a line is the end of one part and the beginning of another; yet whether it is the beginning or end with respect to this determinate part or another depends on the intellect’s designation and is material. The same applies to the unity of a place or river: the designated parts or surfaces are material to the unity of the place or river, provided they remain under some relation to the same terminus.

The reason for these points is that the form giving unity to these things does not have an altogether absolute designation, but a relative one. That this part is a beginning or end, last or first, is said relatively; nevertheless, since quantity is finite, there is in reality a last and a first, an end and a beginning, even though their relative designation is not fixed. Thus it is materially variable but formally fixed, because the formal character of what is last always has the character of form, not matter, while what is not last always has the character of matter and of what can be determined. What is called last denominatively and materially can vary in receiving that denomination. Therefore it is false, speaking formally, that some units are matter and form with respect to one another in number: only the character of being last is form. What is last denominatively can shed that formal denomination, so the thing that is now denominatively form can become denominatively matter, or conversely; the very formality of being last cannot become matter.

BUT YOU WILL ASK whether this formality of the last unit is a physical or metaphysical form of number, and, if it does not compose number intrinsically, what the intrinsic form of number is. Some think that the last unit is the difference of number and its metaphysical form. They are entirely mistaken, since a difference is predicated of the whole, whereas the last unit is not predicated of the whole number: five is not the last unit. We must therefore say that the last unit is a partial and extrinsic form with respect to the other units, although it composes number with them—not in the sense that an accident is composed of matter and form, but in the way that, among parts of one extension, one part determines another, as a line determines a surface and a surface a body, or one degree of intensity another, although all together belong to an integral quality or quantity.

When, however, one asks what the intrinsic form of number is, we answer that there is no intrinsic form on the side of the units, because the unity of number is not unity of form but of order, as we said from St. Thomas, Metaphysics VII, final lecture.²⁸ This order is not a pure predicamental relation, for then it would be a unity of relation; it is an order of discrete extension, formally and principally in the last unit, and participated in and presupposed in the others subordinated to it, as will be explained more fully below in the answer to the second argument. This is like the fact that the whole power of acting is in the final disposition and presupposed in the preceding ones, or that the habit of virtue is principally in one power and diffused or disposed, as it were, presuppositionally in another, as St. Thomas teaches in Summa theologiae I–II, q. 56, art. 2. Therefore, as to its formal character, number is principally and inherently not in several subjects but in one, namely the last unit; it is present in the others by order and participation, and the last unit is determined formally, not materially.

THE ARGUMENTS ARE ANSWERED.

Objections are usually made on two grounds: first, to prove that number is not a being per se but per accidens; second, to prove that it is not specified by the last unit.

As to the first point, the common argument is made to prove that number is not one per se but per accidens; therefore it is not a species of quantity.

The antecedent is proved first as follows: by the division of quantity alone, every other form being removed, quantitative multitude is constituted, and this is nothing other than number. But a character drawn from multitude does not express unity per se, as we showed above from St. Thomas. Second, the antecedent is proved because one per se is not made from several beings in act, as the common axiom has it. White is called a being per accidens because it consists of several beings in act, even though they are united as subject and form—much more closely than the units in number. All the more, then, number is a being per accidens, since it consists of divided units in act. Nor can one say it is one per se in the genus of discrete quantity, though not in the genus of continuous quantity. For if the discrete genus consists of several beings in act, it cannot be a genus of unity per se, but only per accidens. Third, several things cannot coalesce into one except by unity or union. Among the units of number there is no union, since they are separate and discontinuous, much less any unity. Therefore number is not one per se. Fourth: Unity is opposed to multitude and is destroyed by it, whereas number is a kind of multitude. How, then, is number one per se? Finally: Number is one per se either insofar as it is a multitude or insofar as it is discrete. It is not one per se insofar as it is a multitude, for then it would not belong to the genus of quantity or express unity. Nor is it one per se insofar as it is discrete, for then it would consist only in the privation of continuity and connection among parts. Hence the Philosopher says in Physics III, text 68,²⁹ that number is several units, and in Metaphysics VIII, ch. 3, that number is not one, but is like a heap.³⁰

THE ANSWER is to deny the antecedent. To the first proof we say that number is not constituted formally by active division, as is evident, but efficiently. Nor is it constituted by passive division alone, since quantity’s being divided or multiplied is material to the character of number: division precisely dissolves continuity, but does not order parts under a determinate measure of number. Division can of itself proceed infinitely or without limit. That it occurs determinately and under a certain measure belongs to the character of number; thus division is opposed only to unity or the union of the parts of a continuum. Just as mere union is not enough for a continuum, but the parts must be ordered and terminated, so conversely, mere separation of the union of parts and division are not enough for discrete quantity: they must be ordered to one last unit, and thus made determinate and finite. In this way discrete quantity takes on the character of a special quantitative measure. Hence St. Thomas said that duality is not two units but something composed—that is, resulting—from them, in Metaphysics VII, lect. 13,³¹ and that twice three is not formally six, but once six³², and that number has unity not of form or composition but of order.³³

To the second proof we answer that from several beings in act there is not one per se by unity of form, but there can be one by unity of order and measure, as from several beings in act one artifact, one sacrament, or one banquet is made. These are one per se insofar as this is opposed to unity per accidens, which is taken from multitude and not from something one—whether form, composition, or order—as we saw from St. Thomas, Metaphysics VII, final lecture. To the objection about white, we answer that white, taken formally, is one per se and definable by a unity drawn from its form, although connotatively and denominatively it includes a subject as the supposit for whiteness. If it is taken not formally but as the third thing constituted from whiteness and a subject, we say that it is a being per accidens, because these do not coalesce into some third unity. Likewise, number formally expresses the last unit under one order; if, however, it is taken as an aggregate of units and order, it is a being per accidens, like other concrete things taken as the constituted thing or aggregate.

To the third proof we answer that there are several modes of unity and union: unity of form, composition, or order, as St. Thomas enumerates. Number has no union or unity of form, continuity, or composition, but has quantitative unity of order under one last unit. The precise nature of this unity of order will be explained in the following argument.

To the fourth proof we say that unity of continuity and form is opposed to multitude and division, but unity of order and discreteness is not—such as is found in number under one last unit terminating the order.

To the final proof we answer that number is one per se insofar as it is discrete, understanding discreteness not privatively, as opposed to continuity—in that sense it is nothing other than division—but in an ordering and formal sense, insofar as it expresses discrete extension under one last unit. The Philosopher in Physics III³⁴ does not say that number is several units; he says that number is “more than one and a certain amount,” because, as the Holy Doctor explains there, lect. 12,³⁵ “every number is denominated from one; hence Metaphysics V says³⁶ that the substance of six consists in its being six times one, not in its being twice three or three times two, for it would follow that one thing had several definitions and several substances.” Here Aristotle and St. Thomas teach our resolution quite openly. In Metaphysics VIII Aristotle does not say that number is like a heap; he says that if number is one, it is either like a heap, or else one must explain by what it is one. He explains this there by saying that one part is like matter and another like form, namely the last unit, as St. Thomas also teaches in lect. 3. When St. Thomas says in Summa theologiae I, q. 30, art. 1, ad 4, that abstract and absolute number exists only in the intellect, he is speaking of numbering number, which is taken apart from every subject.

Second, an argument is made from the other side to prove that the last unit cannot give number its species, nor can any order of units to it do so.

For this order cannot exist in several subjects except by a relation through which they regard and order themselves to one another. Therefore this unity would be a unity of relation, not quantity, and would belong to the predicament of relation. If one says that it is not a predicamental order but an order secundum dici, and so can belong to quantity, the contrary follows: either this quantitative order is the very plurality of the divided quantities, or it is some different quantity supervening on them. If the first, number will be nothing other than multitude or a heap, and not one per se. If the second, the difficulty returns: how can that quantity be in diverse subjects, or, if it is in one, how does it affect them all?

Next, it is specifically proved that the last unit cannot be the form of number. Either that last unit gives number its species because it is a unit, or because it is last. Not because it is a unit, since then the first unit would also give number its species. Not because it is last, since it owes its being last to the intellect that designates it. By the nature of the thing, this unit is no more determined than that one; without a determinate form, however, there is no determinate species. Second, it is unclear how the last unit is united to the remaining units so as to make one per se with them. As we said, it is not an intrinsic form but is extrinsic to the remaining units. Who would believe that the unit in India is united per se with the unit in Spain to constitute a pair or some other number? Third: Either the unit that becomes last is the form of number through itself or through something added to it. If through itself, number adds nothing to the unit, because any unit by itself suffices to constitute number, provided other units are posited extrinsically. If through something added, one must say what that is: a mode or a thing; whether it is found in all the units as such, or only in one when it becomes last. Yet since this designation comes from the intellect, nothing real can result in the unit from it. Finally, if this unit is taken apart from the other units, either it has that mode of number, or that by which it is the form of number, or it does not. If it does, then number is preserved in a single unit, which is absurd. If it does not, then its being last and the form of number come from the intellect alone, and so are not something real.

THE ANSWER to the first part of the argument is that the form of number, which is the last unit, is not a form inhering in all the units; nevertheless, an order to such a unit is present in them. This is not merely a relative unity of order, but also a quantitative one—not because some quantity inheres in all the units, which would make it continuous quantity and give number unity of form rather than order, something St. Thomas denies, but because a new character of measurement and discrete extension results. The order in number is called quantitative because materially it depends on continuous quantity, as a body depends on a surface and a line, although formally it expresses another measurement of extension and aggregation. Thus number, insofar as it takes on the special character of a measure that discretely exhausts continuous quantity, is found only in the genus of quantity. Hence St. Thomas says in Metaphysics V, lect. 8,³⁷ that the character of the one convertible with being consists solely in indivision, while that of the one that is the principle of number consists in measurement. In this way the unity of order in number is distinguished from the unity of an army or a city, which are beings per accidens, because in them there is only an order of relation, insufficient for unity per se. Relation formally concerns several terms, and so unity of order is taken from multitude when it is merely relational. But the unity of quantitative order or extension in number is not merely unity of relation: it is the distinction of divided quantities under one last unit, under which the others are ordered and enclosed, as taking on a new mode of measuring in extension itself and in discrete quantity.

To the second part of the argument: as to the first proof, we answer that the last unit gives number its species not merely because it is a unit, but because it is the last unit. As to material designation, it has the character of lastness from the intellect, and so does not formally constitute number. But as to the effect and formality of terminating—that is, of closing all the units under a determinate terminus so that the number does not remain indeterminate and undecided—this is real.

When it is objected that without a determinate form there is no determinate species, we answer, from what was said above³⁸, that without a determinate form, at least formally, no determinate effect is produced. But if the formality is determinate while the material designation is indeterminate, a determinate species simply exists in reality. This is like the unity of a river, which exists in reality even though materially the water is not the same but is now this water, now that; nevertheless, because it flows under the same relation and succession with respect to its bed, the river remains the same. The same holds for the unity of a place, as different surfaces succeed one another at the same distance. Thus different units can succeed one another by the intellect’s designation under the same character of being last, and so have unity formally in reality, though not materially.

To the second proof we say that units distant from one another are united for the character of number in the same way as units that are not distant. Since they are assumed to be divided and separate, it makes no difference whether they are far apart or near. They are not united by continuity or connection, but by unity of order and discreteness, in which one stands to another by quantitative addition. There is no need here to inquire into another intrinsic form, as has often been said: number has not unity of form, according to what we said from St. Thomas, but quantitative unity of order in discrete quantity, insofar as it introduces a new character of measure that exists formally in the last unit and presuppositionally in the others.

To the third we answer that a unit considered in itself is inadequate and partial matter of number, but insofar as it is last, it is form. Again, its being last with respect to the effect and formality of terminating and closing all the units under the terminus of a determinate number exists in reality, although it receives its material designation from the intellect. This termination, or character of being last, is truly something in every determinate number, because this function of closing and terminating the units under the character of a number—say, five or six—is truly found in reality. This character of terminus seems to be the mode by which the remaining units are terminated, coordinated and coextended with it under the character of discrete extension, which is a new termination. Such a character or mode cannot be separated from the divided units, since they are divided determinately in act: division is not determinate unless a terminus is posited. Thus the last unit, under the formality of being last, cannot be separated from the others with respect to which it is last without the number being destroyed and changing, for example, from five to four.

To the final proof we say that a unit separated from the other units does not have the character of being last, not through any defect in itself, but through the absence of the others, which must necessarily be presupposed and required for it to be last. When it is inferred that being last is nothing real but comes from the intellect, we distinguish: as to the designation of that material unit taken as last, we grant it; as to the effects and formality of terminating and closing the units under a determinate measure and character of discrete extension, we deny it. When it is said that nothing is required for number except that several units be posited, we say that materially, on the side of matter, nothing but plurality of units is required; formally, however—that is, for number to take on a new character of extension and discrete measurement—something further is required: the ordering of the units under one last unit that terminates them, an order that is quantitative, as has been explained.

The Philosopher explicitly enumerated all these, except motion, among the species of quantity in the Categories,¹ but rejected them in 5. Metaph.,² lect. 15. in St. Thomas.³ He places motion neither in the Metaphysics nor here, in his explanation of this category, among the species of quantity. We will fully gather the reason for this variation from the words of St. Thomas to be cited below.

Some define place, time, and motion here, but this question lies outside the present topic, since they should properly be treated in the Physics. Briefly, it suffices to transfer their definitions from there to here, so that we may at least taste, as it were, what they are. Place is defined in 4. Phys.⁴ as follows: “Place is the first unmoving surface of the containing body,” that is, a surrounding surface together with an immobility of distance. Motion is called “the act of a being in potency, insofar as it is in potency”⁵, that is, an act that passes successively, in which something is in potency and something in act. Time is “the number of motion according to before and after”⁶, that is, the measure of the duration of motion according to its earlier and later parts.

There is not, then, one OPINION AMONG AUTHORS, including Thomists, CONCERNING THE EXCLUSION of these species from quantity. As for place and motion, almost everyone agrees that they must be rejected from quantity, though they do not agree on the reason for rejecting them. But concerning time and speech, some admit time as a continuous successive quantity and speech as belonging to discrete quantity, while others exclude them.

I SAY, THEN, FIRST: According to St. Thomas’s opinion, motion, time, and place are not per se species of quantity but quantities per accidens⁷; and the Angelic Doctor excludes one and the other in the same way.

This is the opinion of many Thomists, as can be seen in Sanchez, lib. 5. q. 14., and Araujo, 5. Metaph. q. 3. art. 5. Yet in this matter the testimony of St. Thomas is clearer than daylight, so it is surprising that some of his disciples have doubted it. In 5. Metaph., lect. 15., § Distinguit modos, etc.,⁸ he says: “He posits two modes of quantity per accidens. In one way, some things are called quantities per accidens simply because they are accidents of something quantitative, as white and musical are accidents of a subject that is quantitative. In another way, some things are called quantities per accidens, not because of the subject in which they are, but because they are divided according to the quantity of some other quantity, as motion and time are said to be certain quantities and continuous because the things of which they are are divisible, and they themselves are divided according to the division of those things. For time is divisible and continuous because of motion, while motion is so because of magnitude—not indeed because of the magnitude of what is moved, but because of the magnitude of that in which it is moved. For motion is quantitative because that magnitude is quantitative. And because motion is quantitative, it follows that time is quantitative. Hence these can be called not only quantities per accidens, but quantities by posteriority, insofar as they derive the division of quantity from something prior.” Thus far St. Thomas. He concludes there that they are not posited as species of quantity because the species of quantity are considered according to quantity’s very being. Therefore, those things that have being as quantities only from another are not posited there as species of quantity, but are called quantities per accidens, as motion and time are.

From this it clearly follows that those who say St. Thomas does not deny that time and motion are quantities per se, but only that they have quantity in the most imperfect way because they are divided according to another quantity, are not glossing him correctly. They say these are not quantities by reason of their subject, as white is, which is required for being a quantity per accidens. But St. Thomas plainly calls both modes quantities per accidens, not one per accidens and the other per se, though imperfectly. Whatever one thinks about the mode of quantity per accidens, St. Thomas certainly holds that motion and time are quantities per accidens, and that they are therefore excluded from the category of quantity and its species. Hence they are quantities per accidens, as our purpose requires.

As for place, everyone easily rejects it, since it has nothing of quantity except either a surface or the character of encompassing and surrounding what is located. The first is not a quantity distinct from the species of quantity that is surface. In its character as an encompassing, it adds no new being or account of quantity, but only containment or application. Therefore, St. Thomas briefly rejected place from this category in the cited passage, saying that place is not placed among the species of quantity because it has another account of measure, but not another being of quantity.

As for motion and time, since they are successive, and all succession is either continuous or discrete, we speak especially here of continuous succession. For discrete succession, which excludes all continuity in its parts, consists only of indivisibles or instants and is thus found in spiritual things. Continuous succession, which has flowing and nonpermanent parts, can obtain quantity or extension from three sources. First, from the term or form, which comes about part by part; that it comes about part by part and not all at once may depend on the agent’s imperfection, which is unable to introduce the whole term at once, or on the resistance of a contrary in the patient, which is gradually overcome. Second, solely from the flow itself or the duration by which the parts pass successively. Third, from the space, whose magnitude and distance are overcome only by extension.

Although some recognize quantity per accidens in motion only from the first source, especially because of the agent’s weakness or the resistance of a contrary, and others only from the second, St. Thomas, leaving these aside, derived its magnitude and quantity solely from the space in which motion occurs. The reason is that motion, considered in relation to its term, is something incomplete, as it were the way and coming-to-be of the term itself; it is reduced to the genus or category of its term. But the term of motion is only where, quality, or quantity, as is proved in 5. Phys.⁹ Therefore motion cannot have a quantity that is a species of quantity by reason of its term, because not every motion has quantity as its term, and a motion that does have it is not constituted in the account of a distinct species of quantity from its term by reason of that term; it is simply called quantitative from it. But the succession of motion, taken according to the flow and duration of the motion itself, constitutes time. This does not provide quantity to motion; rather, it receives quantity from motion. Indeed, if motion were quantitative by reason of time, time and motion would not be two distinct species of quantity, since there would not be two quantities; motion would be quantitative through time. But it is incompatible with the nature of a thing that its duration should be quantity in the first and per se sense, since duration is nothing other than the continued being of a thing. That this continuation is divisible and quantitative, as it is in us, or indivisible, as it is in angels, does not follow from duration alone—which in both cases is continued being—but from the thing that endures. Therefore duration is not quantity of itself, but by reason of its subject. This is also clear from an indication: much motion can occur in little time, and vice versa; therefore the quantity of time does not quantify motion per se and primarily.

Hence St. Thomas rightly derived the quantity of motion from the quantity of that in which motion occurs, namely the space or distance between the terms between which there is motion. In local motion, this is a space consisting of several places; in qualitative motion, of several terminations or degrees in one and the same quality, or of passage from one quality to another in different subjects; in growth, of several parts of quantity. Motion has divisibility and quantity according to the distance to be traversed. For motion of itself says only that transition by which some being is in a state of coming-to-be, partly in potency and partly in act. Therefore this transition has quantity to the extent that there is divisibility and distance between the terms at which the transition occurs. If there is no distance, as between privation or negation and form, change from nonbeing to being occurs in an instant. If, however, change occurs between positive terms, motion is divisible precisely because there is distance between them. Motion therefore takes quantity from distance or space; of itself it does not provide it. It only makes the subject moved, not quantitative.

From this it is also shown that time is not quantity per se, but quantified by another. The quantity of time is the quantity of the duration of motion, and according to the before and after in motion there is a before and after in time, as is said in 4. Phys.¹⁰ Consequently, the divisibility and extension of time—which are nothing other than its before and after—are taken from the before and after in motion. Time therefore does not have divisibility and extension per se, but by reason of motion; hence it is quantitative per accidens, that is, through another, and so is not a per se species of quantity.

NOR IS IT VALID to answer, as some do, that time receives quantity from another, as it were efficiently and originally but not formally, because time has distinct indivisibles of its own, different from motion, namely instants, and consequently also distinct parts. The parts of motion, they say, are only presupposed to time, as a line is to a surface and a surface to a body; yet each of these is a per se species of quantity.

But the contrary is the case. Motion and time are related as duration and the thing that endures, and no thing causes its own duration, just as it does not cause its own being. Therefore neither can the quantity of motion cause the quantity of time. Moreover, for time to have a quantity per se distinct from motion, it would have to add a new extension, or parts to parts of motion, as a surface adds to a line and a body to a surface, and as number, by dividing a continuum, adds a new mode of extending and measuring discretely. But time has no divisibility and extension other than that of motion, since it is the before and after in motion. Otherwise, if it added distinct parts, it would be only the duration of motion, nor a number according to the before and after in motion: it would number something else, namely the parts it added over and above motion. We will say in the reply to the second argument how the indivisibles of time, which are instants, are not distinct from the indivisibles of motion.

I SAY, SECOND: Speech is not essentially and per se a species of quantity, but is so in a certain metaphorical way.

The Philosopher in 5. Metaph.¹¹ does not place speech among the species of quantity. Cajetan also holds this in his chapter on Quantity.

THE REASON IS that speech is called a quantity insofar as it consists of long and short syllables, which all pertains to the pronunciation of a voice in a faster or slower manner. But the voice itself is a quality, namely a sound uttered by an animal; pronunciation is motion; and speed and slowness are modes consequent upon motion. If these constituted quantity in speech, they would constitute it in motion too. Therefore, neither by reason of the voice, which is a quality, nor by reason of motion and its speed or slowness, which are quantities per accidens, are speech or its syllables per se species of quantity.

Others turn to the small durations of syllable pronunciation, which seem to imply a certain extension. But these little intervals are in reality nothing other than time, which is the duration of motion. We have placed time, with St. Thomas, among quantities per accidens. Hence St. Thomas in Opusc. 48, tract. On When, last chapter,¹² also reduces speech to a measure of discrete time, or to the pronunciation of that sound.

Others make the quantity of speech consist in a certain harmonious consonance of its parts. But this consonance or harmony is not quantity, but proportion, which pertains to relation or, at most, to quality. Indeed, if the proportion is metrical, as in verse, besides the fact that this all depends on convention, it is not common to every kind of speech but only to poetry. If the harmony and proportion of speech lie in the expression itself and the liveliness of apt signification, all this comes from human convention in the voice, not from real quantity. Proportion and harmony in the pronunciation of a voice do not make a species of quantity any more than the proportion of parts in any motion does.

Finally, others say that speech is a discrete quantity, but of an unnamed and unknown species. They defend themselves by ignorance, which cannot be refuted. Finally, if speech is not a significant voice uttered according to a proportion of short and long syllables, it does not have the character of a special number. But all this is not something natural; it depends on human convention. Therefore speech is a species of quantity only metaphorically.

BUT YOU WILL ASK: Why, then, do the Philosopher in the Categories and St. Thomas in Opusc. 48¹³ enumerate place, time, and speech among the species of quantity? St. Thomas answers in the cited 5. Metaph., lect. 15.,¹⁴ “that the Philosopher in the Categories posited time as a quantity per se, whereas in the Metaphysics he posits it as a quantity per accidens, because there he distinguished the species of quantity according to different accounts of measure. Time, which is an extrinsic measure, has one account of measure, and magnitude, which is an intrinsic measure, another. Here, however, he considers the species of quantity according to quantity’s very being. Therefore, those things that have being as quantities only from another are not posited here as species of quantity, but as quantities per accidens.” Thus St. Thomas.

And if you say: If time is a quantity per accidens, how does it have the account of a per se measure, since measure is a property of quantity? The answer is that different measures can arise from the same quantity according to its different modification and application, just as a quantity measures differently when it informs intrinsically and when it is applied extrinsically. Thus, because time and magnitude or motion have the same quantity, they do not constitute different species of quantity per se; yet they do constitute different measurements because they are applied in different ways. Number differs in this respect, since it not only has a different mode of measuring but also a different mode of extending from continuous quantity: it divides and dissolves continuity.

THE ARGUMENTS ARE ANSWERED.

First it is argued: Place, although materially the same as a surface, formally has a different account of measuring, because a surface extends by informing, while place extends by encompassing. But a surface too is materially composed of lines, and number of continuous quantities, while each constitutes a different species of quantity. Therefore place does as well.

THE REPLY is that the cases differ. A surface adds new parts and a new account of extension to a line, although it presupposes the line’s extension. Number also dissolves the mode of continuous extension and transfers it to a discrete mode, which differs from the continuous even in its account of extension. Place, however, adds nothing to a surface except the application of it to something extrinsic that it surrounds.

Second it is argued: Motion and time have parts extended in a new way, namely by the distinct durations and successions through which the parts flow. They also have different indivisibles by which the successive parts are joined: in motion, the “having-changed,” and in time, instants. Therefore, nothing is lacking in them for the account of quantity per se. For spatial extension is not required essentially for quantity; extension over a duration suffices. In this space, however, there is a true extension of successive parts, one of which does not penetrate another but succeeds it. That this successive quantity depends on continuous quantity does not prevent motion and time from being true quantities, just as a body depends on a surface and yet is a true quantity.

THE REPLY is that these successive parts of motion do not add a new extension to the parts of magnitude or space, but a new mode of opposition to permanence, insofar as the parts of motion—that is, the terms acquired in space or distance—do not exist all at once but pass away. This mode, however, does not formally express a new kind of extension, but a lack of permanence, as is clear in itself. Since motion does not express an account of extension distinct from magnitude, it follows that time, which is its duration, does not have a quantity distinct from motion. For time is the duration of motion and has no parts other than motion, whose number it is according to before and after, although any motion is time only in a more regular and uniform way. Herein lies the great difference between motion and time, on the one hand, and magnitude and body, on the other: although a body presupposes a surface, it adds new parts to it; time, however, adds no new parts to motion, nor does motion add new parts to the magnitude of the space or distance in which it is exercised.

As for the added point about the indivisibles of motion and time, it is answered that an instant of time is not distinct from the indivisibles of motion, which are the “having-changed” in the motion of which time is the subject. In the same way, the parts of time do not differ from the parts of motion. Likewise, the indivisibles of motion are not different from the indivisibles of space or of the term acquired through motion in space or distance; they merely connote succession and a negation of permanence in these indivisibles. But we will treat this at greater length in the Physics.¹⁵

Third it is argued: In metrical or harmonious speech, there are different parts according to the length and shortness of pronunciation, which have different durations, and a different order of before and after according to the number of utterances, which are measured by syllables just as number is measured by units. Why, then, would it not be a distinct species of number?

THE REPLY is that all this says nothing quantitatively distinct from discrete or continuous motion, or from its duration, which is time. Hence speech is quantified discretely by the discrete motion of vocal utterance; it does not itself quantify in a new way. As for the added harmony, high or low pitch, and fast or slow pronunciation of the voice, these are qualities rather than quantities of speech.

The distinction among these three quantities is taken from Aristotle, Metaphysics V, ch. 13¹, and St Thomas, lect. 15²: a continuous magnitude in one dimension is length or line; one continuous in two dimensions is breadth; and one in three is depth.

From this comes the definition of a line as “a dimension in the direction of length,” of a surface as “extension in the direction of breadth,” and of a body as “extension in the direction of depth.” A body is continuous through surfaces, a surface through lines, and a line through points. A point is “indivisible in every respect.” We imagine a line generated by the motion of a point, a surface by the motion of a line in the direction of breadth, and a body by the motion of a surface in the direction of depth. Continuous quantity is common to all these quantitative things and is defined as that “whose parts are joined by a common terminus”; that is, it has parts joined and touching one another at a single indivisible boundary. If they touch at two indivisibles, the parts are contiguous, not continuous.

Without controversy, body among these three dimensions is accepted as a true species of quantity, since it is divisible in every direction. Body in the genus of quantity signifies depth, whereas body in the genus of substance, divided against spirit, is a substance that requires this depth.

Regarding the other two species, SURFACE AND LINE, some authors exclude them from the predicament of quantity, such as Master Cabero here, disp. 2, dub. 5, and Hurtado, Disputationes metaphysicae, disp. 13, lect. 6. They admit them reductively in quantity, however, as something incomplete and indivisible, just as a point is quantitative only reductively, as terminating, initiating, or continuing extension.

The first basis for this view is that line and surface are intrinsically incomplete things, like a point; therefore they cannot constitute a species per se and directly in this predicament, just as other incomplete beings in other predicaments cannot. The antecedent is proved because they are essentially parts composing a body and so essentially incomplete. Just as a point is not in the predicament of quantity because it is ordered per se to compose a line, so, since a line and surface are ordered per se as parts to compose a body, they are not species per se and directly in this predicament.

Second, this view is based on the claim that the differences of these species are negative. A line differs from a surface only by lacking breadth, since the line itself has length; a surface differs from a body by lacking depth. They therefore have no positive difference by which they could be constituted as a positive species. The argument is strengthened because a line has only length and absence of breadth. It is not constituted as a species of quantity by this absence; through length it is an incomplete thing that composes a surface and joins its parts. Therefore it has nothing by which it could be a complete being in the genus of quantity.

Third, these indivisibles interpenetrate other parts; therefore, strictly speaking, they are not species of quantity, since quantity is impenetrable. The antecedent is evident: a continuum is that whose parts are joined by one indivisible terminus. The joining terminus must therefore be within both parts; otherwise it would join neither. Since a line joins the parts of a surface and a surface joins the parts of a body, it must penetrate into the parts of each and so offer no resistance to penetration, as the nature of quantity requires.

It is confirmed because if two equally flat bodies touch one another, one surface penetrates the other: both because they touch at every part of the surface, which is penetration, and because they are contained under the same indivisible place. For if they were contained by two indivisibles of the surrounding air, since one body is immediately adjacent to the other, the two indivisibles in the air would also be adjacent, which is impossible, since in a continuum a part follows every indivisible.

NEVERTHELESS, the opposite view is Aristotle’s in Metaphysics V, ch. 13³, and St Thomas’s there, lect. 15⁴, where he says there are three species of magnitude, which Aristotle lists there: length, breadth, and depth. The same is found in Opusc. 48, treatise on Quantity⁵, and Sentences IV, d. 40, q. 1, a. 2, ad 3⁶: “Line is sometimes taken properly for the dimension that is the first species of continuous quantity.” This is the common view among the followers of Aristotle. See the Carmelite Course here, disp. 13, q. 4.

THE BASIS IS that both line and surface, independently of body and according to their own natures, are extended and divisible into parts. A line consists of parts of length, and it does not share this extension from the body; rather, the body has length from the line. Therefore, in themselves and essentially they participate in the nature of quantity, namely, extension and divisibility into parts. It is not of the nature of quantity in general to be divisible in every part and in every way; hence nothing prevents lines and surfaces from being quantities per se and essentially.

Again, they differ in species, both because the positions of extension themselves are different—length, breadth, and depth—and because different properties follow from them: a line is short or long, a surface broad or narrow, a body high and deep. Therefore they are distinct species of dimension. They also have distinct means of continuity: a line is continuous through points, a surface through lines, and a body through surfaces. All these are signs of specific distinction.

Nor should it seem remarkable that one of these species composes another. This follows both from the special nature of quantity, since its nature is the extension of parts and any species of quantity that provides a position of parts can serve another so that, by joining the parts, an integral continuum results; and because less perfect species can contribute or be materially presupposed to the composition of a more perfect one, as an element virtually composes a mixture, a wall a house, and three a six. In number or multitude, units compose it not by reason of the negation implied by a unit, but by reason of its positive aspect, even though considered by itself a unit is a species distinct from number. Likewise line and surface compose body by giving parts through divisibility; through indivisibility they do not give parts but continuity.

To the first argument for the opposing view, then, it is answered from what has been said that line and surface are true and complete species of quantity, although they materially compose a body. Their formalities do not consist precisely in being ordered to composition as matter and form are in substance; even separately and apart from a body, they have true and proper extension, just as parts separated from a body retain the true nature of quantity. In this respect the argument about line and surface proves no more than an argument about the parts of a body; a body is composed no less from lines than from other parts. Thus, just as parts taken by themselves are true quantities although they can materially compose another whole, so are line and surface. A point is not similar, since it wholly lacks parts and so has no quantity. As to how one species can compose another that is opposed to it: a species does not formally compose another insofar as it is opposed to it, but materially, because the entity of one is presupposed to the entity of the other. Thus a mixed quality, such as whiteness or flavor, presupposes the primary qualities from which it results, for example cold or heat; a unit composes a number, and one number another, materially.

To the second, a line differs formally from a surface not through the absence of breadth but through its own positive length, since a surface as surface does not formally include this, but includes it by reason of the line materially presupposed to it. Thus length, which is material in a surface, is formal in a line, just as three is material in six.

This makes the answer to the confirmation clear: length is not something incomplete in its own genus, since apart from a body it has true quantity. It can, however, contribute materially to another species of quantity, not only as an indivisible but also by giving the parts it has, just as one proposition composes another, a wall a house, and one number another.

To the third, the indivisible joining parts is not penetrated insofar as it is indivisible, since it has no parts; indeed, it makes the parts it joins impenetrable, because they are joined at their extremities. A line and surface are not penetrated insofar as they have parts, but have them extended; insofar as they are indivisible, they have no parts to be penetrated.

To the confirmation, two planes touch at their surfaces, whether one encloses the other or not, without any penetration, because the surfaces at which they touch have no parts in the direction of depth. Their immediate contact does not make them penetrate one another, since in the respect in which the surfaces touch, they lack parts that could be penetrated. That two contiguous surfaces correspond to the same indivisible of the air that occupies a place is not because one surface enters and penetrates the other, but because several indivisibles can correspond to the same place: one indivisible occupies no more place than a thousand or infinitely many. Indeed, an indivisible is not per se in a place, as St Thomas teaches in Opusc. 48, ch. 2, on Where⁷.

Finally, observe here that line, surface, and body are infimae species, which St Thomas affirms of line in Nicomachean Ethics X, lect. 5, near the end⁸. Whatever diversity occurs among them is according to different figures and arrangements, curved or straight, square or round, and so forth; figure, however, is an accident of quantity pertaining to quality.

COORDINATION AND TREE OF THE PREDICAMENT OF QUANTITY.

The highest genus is quantity. It is divided into two species, continuous and discrete. According to Aristotle’s treatment in Logic, discrete quantity is divided into number and spoken utterance. UTTERANCE is divided into various species and combinations of meter and harmony, according to the length and shortness of utterance; these are more artificial than natural quantities. NUMBER is divided into all the species of numbers, since adding one makes a different species of number, such as two, three, and so on. According to the view proved above, utterance is excluded from this division.

According to Aristotle’s treatment in Logic, continuous quantity is divided into that which measures intrinsically and that which measures extrinsically. INTRINSICALLY MEASURING quantity has position in a continuum and is divided into three species—line, body, and surface—each of which is an infima species. EXTRINSICALLY MEASURING quantity is divided into permanent, such as place, and successive, such as time. Place can be divided into different distances and modes of measurement, such as a league, furlong, or mile, and into measures that measure by containing or, as it were, placing, such as a bushel, amphora, or cask. Time is divided into days, months, years, and so on. Metaphysicians, however, enumerate only the three species of continuous quantity: line, surface, and body.