Treatise on Quantity

THE TRACT OF THE VENERABLE INCEPTOR, WILLIAM OF OCKHAM, CONCERNING THE SACRAMENT OF THE ALTAR

The very renowned tract of the Venerable Inceptor, William of Ockham, of England, a very famous teacher of sacred theology, a most discriminating investigator of realities, of the order of Friars Minor, concerning the sacrament of the altar and with special reference to the distinction of point, line, surface, body, quantity, quality, and substance. It begins aus piciously.

Concerning the conversion of the bread into the body of Christ, as says a certain gloss on the De Consecratione, Distinction IL, Chapter, “In sacramentorum.’ “There are various opinions. One opinion asserts that that substance, which was bread at first, afterwards is the flesh and the blood of Christ. It is argued later in the Chapter, ‘Panis est in altari,’ and in the Chapter, ‘Quia corpus.’ The second opinion holds that the substance of the bread and of the wine ceases to be there, and only the accidents remain: namely, taste, color, and weight, and similar accidents; and under those accidents the body of Christ begins to be there. The third opinion holds that the substance of the bread and of the wine remains there, and that the body of Christ is in the same place and under the same species. It is argued later in the Chapter, ‘Ego.’ Each opinion, however, acknowledges that the body of Christ is there. The second opinion is more true, as in the margin of the De Summa Trinitate, Chapter, ‘Firmiter,’ Una.’’ These are the words of the gloss.

From this it appears that the substance of the bread is converted into the body of Christ, so that the substance of the bread does not remain but only the accidents remain which were before in the substance of the bread. And in confirmation of this truth it is said in the margin of the De Summa Trinitate, ‘Firmiter,’ as the previous gloss asserts: “There is one universal church of all the faithful, outside of which no one at all is saved; in which the priest and the sacrifice are the very same, Jesus Christ, whose body and blood are truly contained in the sacrament of the altar under the species of bread and of wine, since the bread is transubstantiated into body and the wine into blood by divine power.”

With a view to the exposition of this truth, some questions must be asked about two points principally, as is obvious; first, about the entity and the distinction of those accidents; secondly, concerning the separation of them from a subject.

Moreover, since it is commonly conceded by all that color, taste, and sensible qualities of such a kind, are really distinguished from each other and also from substance; therefore, since this is conceded, some questions must be raised about the distinction of these qualities from quantity, figure, rectitude, and the like; concerning which there is great doubt as to how they may be distinguished.

And because this consideration pertains not only to theology but also to philosophy, since it may be the province of philosophy to come to know the natures of things and their distinctions; therefore, it is necessary to investigate what the philosophers have thought of this matter; and especially the prince of philosophers, Aristotle, and also what the theologians may say. But that I may proceed with more regard for logical order, I shall inquire first concerning the distinction of point, line, surface, and body; and afterwards of other matters,

Question I

Therefore, I ask, first, whether a point is an absolute thing really distinct from quantity. Which is answered affirmatively; for the terminus of every single thing is distinguished from that of which it is a terminus. But a point is a terminus of a line; therefore, it is really distinguished from a line, and consequently it is for the same reason really distinguished from any quantity.

On the contrary, every positive thing is substance or quantity or quality or relation, and so of the other categories. But a point is not a substance distinct from quantity, nor a quality distinct from quantity, nor a relation distinct from quantity, and so of other respective genera; therefore, a point is not a thing other than quantity.

Before answering this question and all following concerning this matter, I make in advance one statement; namely, that whatever I shall say under whatever form of words, which can in any mode be regarded as contrary to whatever has been said in Sacred Scripture or in the writings of the Saints, or contrary to the determination and the doctrine of the Roman Church, or contrary to the opinion of the Doctors approved by the church, I shall not say by affirming but by quoting precisely in the person of those who also hold the opinion to be discussed; whether that opinion be true or false, or Catholic, or heretical, or erroneous. Whence, if I shall use such words as “I say,’ “It must be said,” and the like; I desire those words to be understood not as used by me personally but as used by those who hold such opinions.

With this understood, I shall proceed thus with respect to the question proposed. First, I shall show that a point is not some positive and absolute thing really distinct from any quantity and especially from a line. In the second place, I shall posit some reasons and passages to prove that a point is another thing. In the third place, I shall reply to these.

First, therefore, I prove that a point is not a thing other than a line, or any quantity; and in the first place, through philosophical reasons; in the second place, through passages of the philosophers; in the third place, through theological reasons.

According to the first way; I, first, argue ostensively thus: if a point be a thing other than quantity, it is either substance or accident; but a point is not substance distinct from quantity nor accident distinct from quantity; therefore, a point is not a thing other than quantity.

The major premise is known; the minor premise I prove. And, first, I prove the first part; namely, that a point is not a substance distinct from quantity; for every substance is either matter or form or a composite or substance abstracted from matter. But it is manifest that a point is not matter or form or a composite; nor is it form abstracted from matter, if a point be indivisible and in no way a quantum, as is posited by the opponents; therefore, et cetera.

I prove the second part thus; namely, that a point may not be an indivisible accident distinct from quantity. In the first place thus: every absolute accident is placed in any subject adequate to it, so that that accident is either a whole in a whole subject and in each part, or a whole in a whole and a part in a part or is an indivisible accident having an indivisible subject, as a spiritual accident is in the rational soul as if in its primary and adequate subject. But a point in none of these three modes has or can have any primary and adequate subject; therefore, the major premise of that syllogism is obvious through induction as regards any absolute accident.

The first part of the minor premise is obvious: namely, that a point does not have any adequate subject so that there may be a whole in a whole and in each part. For I take a point terminating the line A, and let that be called B; and I take another corresponding point, and let it be called C; then I ask whether B is in A, as if in its adequate subject or in the subject of itself; and it follows in either way, that C will be in the same adequate subject for the same reason.

Then I argue thus: whatever things are in the same adequate subject, they do not differ either in situation or in place; but B and C are distinguished or separate in place and in situation; therefore, they are not in the same adequate subject. If it be said that B is not in the whole line A, nor in the subject of the same line A, but in a part of the same line or in part of the subject (of the line) ; on the contrary, if B be not in the whole line as if in an adequate subject nor in the whole subject of the same line; then it is in a part as if in an adequate subject. Let that part be D; then I argue, B is in D as if in an adequate subject; therefore, for the same reason another point corresponding to the same B, at which D is terminated out of another part, will be in the same D as if in an adequate subject, and let that point be called E. From which it follows that B and E are in the same adequate subject; and consequently they are not separate in place and in situation, which is most manifestly false. So, therefore, there can not be any point in any divisible quantum, and as if in its adequate subject, so that a whole may be in a whole and a whole in each part; not only because two points of that divisible would not then be separate in situation, which is false; but also because a point the same numerically would then be in two situations, which is false.

If it be said that a point is in some divisible part but another point does not correspond actually to that part, it can not correspond to it, because that part can not be existent per se, but only non-existent to another.

On the contrary; whatever part is designated, God can of His absolute power separate it from another part and can preserve it existent per se; therefore, this part, which is the primary and adequate subject of a point, God can preserve without another. Therefore, let it be posited in being, then it will actually have another point, which will be for the same reason in that subject as if in its primary subject; for there is no greater reason that one point terminating a line may be in that whole line or in that whole subject of that line than for another point terminating the same line; and consequently these two points will not be separate in situation; for those accidents, which are in the same adequate subject, are not separate in situation. Moreover, a corresponding point is not merely potentially in whatever way it is about to be potentially ; therefore, it is in the nature of things and consequently has another primary subject, and this can not be granted unless a line or a subject of a line terminating at another point; therefore, these two points are in the same primary subject and consequently are not separate in situation, which is false. Moreover, if it were in some divisible subject, as if in a primary subject, then they would not be separate in situation from another part, nor could it be more separate from one part than from another.

The second principal part of the minor premise is thus proved; namely, that a point is not a whole in a whole subject and a part in a part; for a point would then be a quantum, and the thesis is established.

The third part of that minor premise; namely, that it is not in some indivisible as if in its primary and adequate subject, is obvious; because that indivisible would be either substance or accident. It is not accident; for I ask in what genus? And it is obvious that it can not be conceded to be in a genus other than the genus ‘substance’.

Similarly, inquiry must be made concerning the subject of that accident, as about the subject of the point. Nor is that indivisible a substance; for, if it be a substance, it is either matter or form or a composite or a substance abstracted from matter; but it is not matter, since all matter is divisible; nor is it form; for, if it be form, it is either form extended and consequently is divisible or it is form non-extended and consequently is a rational soul; for the rational soul alone is non-extended; but it is manifest that it is not a rational soul; therefore, it is manifest that it is not form which is one part of a composite, nor is it a composite; and it is manifest that it is not form abstracted from matter; therefore, it is not some indivisible substance, which may be able to be the primary and adequate subject of a point.

Moreover, if there be in a stone some indivisible substance which may be able to be the primary and adequate subject of a point; then I take this substance, and I ask whether or not it makes one per se with the matter and the form of the stone. If so, there would, therefore, be some indivisible substance in a stone which would be matter or form with respect to the matter of the stone; for whenever some making one per se are of diverse grounds in the genus ‘substance, one of these is act, the rest indeed power; and consequently one is matter and the rest form.

But it is impossible that that indivisible is primary matter, for primary matter is extended. But if this indivisible substance does not make one per se with the matter and the form of the stone, it is surely not part of the stone and consequently can not be part of the stone; and consequently can not be part of any substance.

But every substance, which can not be part of some substance, is subsistent per se and per se in a genus and in a species; and consequently this substance would be subsistent per se and per se in a genus and in a species, which is false; therefore, it remains that there is no substance in a stone except matter or form or a composite; but every such is divisible and possessed of parts really distinct; therefore, none of them is indivisible. So, therefore, it is obvious that neither any indivisible nor any divisible can be a primary and adequate subject of the same point, if a point be a certain absolute thing really distinct from every divisible.

If it be said as regards the principle reason, that the major premise is false; now there are many accidents which are not wholes in a whole and in each part, nor wholes in wholes and parts in parts, nor are they indivisible accidents having indivisible subjects but indivisible accidents having divisible subjects; and yet they are not in any part but a whole in a whole, as time is in the whole movement of the whole heaven, and yet it is not any part of the movement of any part of heaven. Similarly, doubleness is a whole in a whole double, and neither doubleness nor any part of it is ina part of a double with respect to the same half. And it seems so of many other accidents, that they are in a whole; and yet neither they themselves nor a part of them are in any part of the whole.

But this reply does not suffice. First, because it accepts the fallacy that there are some indivisible accidents existing at first in some divisible and yet in no part of it; whence, time is not some absolute accident really distinct from motion and from permanent things, existing subjectively in a whole motion and in no part.

In the second place, this reply does not suffice; for, if this is granted, the argument still obtains; for a point is not such an accident. First; because it is obvious, that no accident existing in any adequate subject, whether it be a whole in a whole and a whole in each part or a whole in a whole and yet in no part, is separate in place and in situation from any accident existing in the same subject adequate to it; for color is not separate from the color of the same subject. Similarly, time, if it be another accident, is not separate in situation from some accident in the same subject adequate to it. Similarly, doubleness is not separate in situation from some accident existing in the same subject, as if adequate to it; and so in general of every accident. It is obvious, that it is not separate in situation from some accident existing in the same subject adequate to it. But a point is separate in situation from each point of that quantum, of which it is a point; therefore, two points can not be in the same primary and adequate subject, which however would be necessary if a point were an indivisible accident having some divisible for an adequate subject.

In the third place, no accident is separate in situation from each part of its primary and adequate subject; as the whiteness of that part A is not separate in situation from each part of the same A, although it is separate in situation from another part joining with A.

Similarly, if time or doubleness be an accident really distinct from a double or motion, it is manifest that time is not separate in situation from some part of motion, nor is doubleness separate in situation from some part of a double; and so of every other accident. But if each divisible is designated on the ground that it ought to be considered the primary subject of that point, it is manifest that this is separate in situation from another part of this divisible; therefore, that divisible is not the primary and adequate subject of the same point.

This reason is confirmed; for, when any accident is in some primary and adequate subject, it 1s obvious by induction that that accident is not more separate in situation from one part of this subject than from another. But it is manifest that that point is more separate in situation from one part of that divisible, whatever divisible is designated, than from another; therefore, that divisible is not the primary subject of this point.

From which it is obvious, that a point is not some absolute accident really distinct from every divisible, having some divisible for a primary and adequate subject; namely, that which is a subject in itself or per se and not through its own part.

Moreover, it can not be conceded that a point connecting two halves, in which it is subjectively, is in one half more than in the other, or, therefore, in both; and then an accident the same numerically would be in subjects several numerically, which is impossible, or in neither ; and consequently in none subjectively.

If it be said that a point may be in some indivisible subject, as was mentioned before, this does not obtain, as was proved before.

And I shall prove it again; since some indivisible would be in a divisible subject, say, in a stone. Let that stone be divided in correspondence with that indivisible, and I ask, whether that indivisible remains or does not remain. If it remains; I ask, where it remains, whether with each part, or in one and not in the other, or separate from both. The first can not be conceded; for then some substance one numerically would be in distinct places which do not make one, which is impossible naturally. The second can not be conceded; for there is no greater reason that it be in one part than in another. Similarly, the third can not be conceded; since then nothing existing per se or in some other would make one per se; each of which is obviously false and impossible.

In the second place, I argue principally thus. Every positive and absolute thing really distinct from every other thing not making one per se, as if a part with another thing, is per se in a genus. But a point, if it be such an indivisible thing, is not a part of any divisible, nor does it make one per se with any other; therefore, it will be per se in some category ; but this is false, for then it would be either substance or quantity or quality or respect, any one of which is false according to those positing such indivisibles.

The major premise of that syllogism is manifest: for when there are some things distinct as regards themselves as wholes, of which neither is part of the other, there is no greater reason that one of them be per se in some category than for the other, as is obvious by induction.

But it is said in reply to this, that a point is not per se in a genus, but only through reduction; for it is not, unless a terminus of quantity.

But that reply does not obtain; first, because a line could not then be per se in the genus ‘quantity,’ since it may not be unless a terminus of quantity. If it be said that a line is not only a terminus of quantity but even a quantity; it is, therefore, per se in a genus. But a point is not per se a quantity; therefore, it is not per se in the genus ‘quantity’ but only through reduction. This does not obtain; for, if a point be such an indivisible thing, not only will it be a terminus of quantity, but it will be one absolute thing really distinct as regards itself as a whole from every other thing, and consequently will be per se in some category.

This is confirmed; for every thing absolute and permanent and one per se, not merely one through aggregation or order, is per se in some category or is a part of something existing per se in a category. This is obvious by induction of every thing, unless you insist upon the proposition. It is also obvious; for, if that major premise be denied, every way of proving that some thing is per se in a genus is destroyed; for thus of other things I may say, that something is not per se in a genus because it is the terminus of any substance; as you say, that a point is not per se in a genus, for it is the terminus of something existing per se ina genus: therefore, this is obvious, that everything one per se is per se in some category or is part of something existing in a category; but it is obvious by induction that a point is not a part of something existing per se in a category; therefore, a point is per se in a category; which was proved to be false. From which it follows that a point is not ‘such an indivisible thing.

In the third place, I argue thus. It is impossible that there may be things infinite actually, distinct as regards themselves as wholes, and of which none may be part of another, and which do not make some one. But if a point were such an indivisible thing, such would be infinite actually. For example, I take wood. It is manifest according to those writers that there are here infinite lines; for there are not just so many so that there could not be more, and each of them is terminated actually at some point; for otherwise no line could be terminated actually at some point. And it is manifest that they are distinguished as regards themselves as wholes, if they be indivisibles; nor do they make one per se, since they may not be parts of anything; therefore, such will be infinite actually, which is impossible and contrary to all philosophy. Therefore, it remains that a point is not any such indivisible thing really distinct from every divisible.

In the fourth place thus: some thing positive and absolute is not caused through division only of a continuum. Let, therefore, wood be divided; and I ask about the terminating points, whether they were before or not. If they were, it is certain that they will not be separate in situation; therefore, a point was immediate to a point; which is false even according to those writers. If they were not before and now are; they are, therefore, generated de novo, and consequently some absolute thing necessarily would be generated through division alone, which is false.

In the fifth place; since they can become continuous through the union only of parts of water or of another, whose parts were separate before, some new absolute thing is not generated nor is an absolute corrupted. But if a point were such an indivisible thing, whenever such parts would become continuous, of necessity some absolute thing would be corrupted; say, these two points which before terminated two lines, and a new thing would be generated; namely, one point connecting those two parts. Indeed, it would follow that infinite, absolute things distinct as regards themselves as wholes would be corrupted, and infinites would be generated; as can be deduced through the third principal reason previously stated; all of which are impossible, and consequently a point is not such an indivisible thing.

I prove this same conclusion through theological reasons. First thus: every absolute thing distinct as regards itself as a whole from another absolute thing, of which it is not subject nor effect, nor conversely, God can create without the second thing.

But if a point be such an indivisible thing, it is manifest that a line as regards itself as a whole is distinguished from a point; for it is manifest that a line is not a part of a point, nor is a point a part of a line, nor again is a line the subject of a point, as was proved before; and conversely a point is not the subject of a line. Moreover, a line is not the cause of a point, nor conversely; as is obvious by a survey of all classes of causes; therefore, God can of His absolute power make a line without any point.

Then I ask, whether that line is finite or infinite. If it be finite and without any point; it is consequently needless to posit a point to terminate a line, and yet it is not posited by virtue of another by those who thus posit it. If, however, that line be not terminated nor finite, and is in the nature of things per casum; it, therefore, is infinite, which obviously is manifestly false ; for that line will not by virtue of this be greater, or longer because that point is separated or destroyed; therefore, in no mode will there be that infinite, no matter to what extent all the points may be destroyed. Moreover, if all the points be destroyed by God, and the line is saved; I ask whether or not that line is continuous. If it is, and not through a point, for it is not per casum; there is, therefore, a continuous line without any point, and consequently a point is needlessly posited there. If it be not continuous; I ask whether any part of it be continuous, in which case the preceding argument applies, or else no part is continuous; from which it follows that there is some line which is not composed of continua, which is impossible.

Similarly, that line is continuous or discrete; if continuous, the thesis is established; if discrete, then each part is divided into a quantum infinite as regards all its parts, which is impossible. But this reason proves that a line is sufficiently continuous and finite through its proper nature without any other thing added to it; and consequently, since a point ought not to be posited by virtue of another, a point is needlessly posited to be such an indivisible thing.

If it be said that God can not of His absolute power make a line without a point; on the contrary, when any absolute thing can through nature be made without many things separately so that it may be without any of them singly, that thing can be made through the divine power without all of them joined together so that none of them may be with it; which is especially true, when these things as regards themselves as wholes are really distinct and neither of them is an effect or an accident of the other; as is the case of the point and of the line. If the point be such an indivisible thing, (but this line can through nature be without this point and that point and so of others and so of any,) God can, therefore, make this without any point; and so the thesis is established.

Moreover, if that, which essentially depends on something as an effect depends on a cause and an accident on its subject, can be separated from that through divine power; much more firmly can God make that, which does not depend on another either as a whole on a part or as an accident on its subject or as an effect on another, without that.

But an accident in the sacrament of the altar depends before the separation on a substance as if on a cause and on its subject; and yet through divine power it becomes an accident without a subject; therefore, much more firmly can the line, which does not depend on the point either as if on its part or as if on a subject or as if on its cause, be made without it; and so the first argument obtains.

That reason is confirmed; for, according to the Philosopher in the second book of the Physics, “Quantity is per se finite or infinite.” But if a quantity were finite through another thing, as a line is finite through points as if through another thing, it itself would not be finite per se but through another. In the sacrament of the altar a quantity is separated from every substance; therefore, a line is separated from every subject. Then I ask about the point terminating that line, whether or not it is separated from every subject; if so, then that point and the line do not in any mode make one, for they are neither one per se nor one through an accident, since neither may be receptive of the other; and consequently the line can not otherwise be terminated through a point, unless as water is terminated by a solid body; namely, earth or something similar, which can not correspond to a point.

But if a point may not be without a subject, and it is manifest that it is not subjectively in a substance; since it may not be there unless the body of Christ, in which a point can not be posited; that, therefore, is subjectively in a line, but this is impossible as was proved before; for if it were subjectively in a line or in some part of a line, as if in a primary and adequate subject and not only through a part, as this whiteness is said to be in Sortes; for it is in one part of Sortes, a point might be able to be separate in place and in situation from no part of a subject, say, a line or a part of a line, as was affirmed before; which is manifestly false, as was proved before.

By virtue of the previous statements and many others, which can be adduced, those positing might say that nothing is simply indivisible in things through the creation of all parts really distinct except substances separate from matter and principally the rational soul and its accidents; and that a point is not a thing indivisible, positive and absolute, really distinct from a line and every quantity. But they might say that any thing divisible in itself is formally finite, terminated and continuous; if it be continuous without any other thing added to it, it is causally terminated, finite, and continuous by God and other causes per se placing it in being, whatever may be those causes; whence this line without any other thing added to it. Indeed, if any other thing were destroyed and this line were preserved by God, it would truly be finite, terminated and continuous; and, indeed, since such an indivisible thing may not be posited by virtue of others, it seems as impossible as superfluous to posit that a point is such an indivisible thing.

And if it is asked: What is a point? Whether it is a thing divisible or an indivisible? Which must be answered thus by saying, ‘a point is something’ or ‘a point is a thing’ or anything else of the kind. If a point stand for something so that it may have precisely the force of a name and may not embrace as an equivalent one compounded of a noun and a verb, or something very similar which can represent a supposition for a verb according to a property of speech; it ought to be conceded that a point is something and that a point is a thing, and this because it ought to be conceded that a point is a line or a point is a quantity; for then this name ‘point’ will be equivalent to this ‘a line of such or such length,’ or ‘a line not farther lengthened or extended,’ or to some whole compounded of a noun and a verb with an intervening conjunction or adverb or this pronoun ‘quae’; according to which it is pleasing to give diverse definitions expressing some characteristic of that name ‘point.’ And indeed, as this predicate ‘a divisible thing’ is predicated of such a subject and consequently of a composite of an adjective and a substantive, so it is predicated of a point.

Whence, as some say that this name ‘privation’ is equivalent in significance to this whole ‘subjectum privatum’; and by virtue of this, as they concede this, ‘subjectum privatum’ is matter; so they concede this, ‘privation’ is matter. So if this name ‘point’ is equivalent in signification to this whole, ‘a line not farther lengthened’ or to something similar; of which one part may be a line, and other parts may be some adjectives or a verb connected with line, this pronoun ‘quae’ intervening, or another similar. As this is conceded, ‘a line is divisible,’ so this is conceded according to the virtue of signification, ‘a point is a line,’ ‘a point is divisible.’ For whatever is true of this name ‘line,’ when used significatively, is true of this name ‘point’ when used significatively, and this by speaking according to the virtue of signification; yet this, notwithstanding that one proposition, in which this name ‘point’ is substituted, could be per se; and another, in which this name ‘line’ is posited, will not be per se, and conversely.

Just as whatever is predicated of one of the convertibles, when used significatively, is true of the rest when used significatively, and yet one of them per se and not the rest. Whence, this is per se in the first mode, ‘every man is an animal,’ and not this, ‘every risible is an animal.’ But if this name ‘point’ will be equivalent in signification to some composite of a noun or a verb or a conjunction or an adverb and this pronoun ‘qui,’ or of a composite of an oblique case and a verb, which according to a property of common speech can not express, speaking grammatically, a supposition for a verb; consequently, too, any language, in which it is posited, is not according to a property, but is figurative speech and can be reduced to a figure which is called by the grammarians ‘hypallage,’ of which they say that hypallage is the conversion of cases or of a construction or sometimes of a whole sentence.

So it ought to be conceded that ‘a point is an indivisible thing’ and that ‘a point does not have parts,’ and anything else of the kind. But those propositions and similar ones must not be conceded according to a property of common speech, as proper propositions must not be disposed of through any grammatical figure; and consequently in the investigation of reality it must not be argued from these, unless they are taken according to the sense which they have according to the intention of the authors; but they must be conceded under that sense, which they have from this, that they are disposed of through the aforesaid figure; for this reason, namely, that one expression is posited in place of one form of language or of a composite of many expressions, and at the same time with this it is necessary to convert the language by transposing and even by changing sometimes one part of the language into another, and sometimes one expression into another, and a case into a case, or one grammatical accident into another, or in a similar mode by conversion; that the sense of this proposition, ‘a point is indivisible, may be this, ‘a line is not farther lengthened,’ of which, however, each part is divisible; or another similar sense according as one is neeed to explain the previous proposition. But the sense of this proposition, ‘a point is divisible,’ may be that or some similar sense; for whatever part of the line is taken, it is still proper to take the more remote.

And as it is said of these propositions, so it must be said of similar propositions; and not only is this true of this name ‘point’; but it is also true of all verbal names and all names formed from adverbs, conjunctions, pronouns, prepositions, and in general of all syncategorematics, and in general of all, which do not precisely have the virtue of a name; so that they can not be the extremes of a proposition separated by a copula without any figure and a property of common speech. And according to this, all such: ‘action is in passion,’ ‘motion is in the mobile,’ ‘likeness is in the similar,’ ‘whiteness is the terminus of motion,’ ‘every thing is produced through production,’ ‘fire makes warm by heating,’ and innumerable propositions of a similar kind, which are found in the statements of the philosophers and of the Saints and even in Sacred Scriptures, must be figuratively disposed of through some grammatical figure. Nor is it inconsistent to concede that the philosophers and the Saints spoke thus figuratively, since many such may be found in Sacred Scripture; not however by accepting the statements of

Indeed, even the popular and common mode of speaking frequently employs such figurative expressions ; as can be proved through the statements of the grammarians treating of such. Whence, not only the expression can be transferred from a proper to an improper signification, but even sometimes terms are given which have no proper signification, such as they have when first instituted. And as a translation is made according to the grammarians for a threefold cause : sometimes for the sake of metre as in the poets; sometimes for the sake of ornament as in rhetorical speech; sometimes for the sake of necessity, either brevity or utility, as in philosophy; and in all of these ways a translation is made in theology. So the application of such names, whether they be abstract or concrete, is made for a three-fold cause; namely, for the sake of metre, ornament, and utility or brevity. Although from the previous causes many errors occasionally originate among the simple minded, who wish to accept all the statements of the philosophers and of the Saints according to a property of common speech, when they must be taken figuratively.

The second principal division. But a certain Doctor argues first thus against the previous statements. It follows from that opinion that sudden generation, which is not per se the terminus of a continuum, will be nothing or at least in nothing; for it does not have any other positive measure, and so is it of illumination and of all sudden changes, which are not per se terminations of motion. And although they can escape from the changes, which are the terminations of motions and happen in an instant, as in nothing, or the privation of continuity in the privation of continuity ; yet concerning these it seems absurd; for they are not the terminations of the continuity of some continuum ; for they are by no means characteristic of any continuum, either positively or privatively.

Moreover, in the first book of the Posteriores, the ground of the line is from points; that is, a point falls within the essential ground of the line, which is called the ground according to line in the first mode of speaking per se; but no privation pertains per se to the essential ground of a positive.

In the third place; it follows from the same, that, if a point is only privation, a line will be only privation and surface and body; for the terminate is always limited through a terminant, and a positive does not essentially include privation.

In the fourth place; Aristotle in the Categories seems to be in opposition to that opinion, where he says that parts of a line are terminated at a point as if at a termination; but the terminus is always other than that which is terminated at itself; therefore, a point is such an indivisible thing.

In the fifth place; the philosopher in the first book of the Physics says that ‘an indivisible terminus is not infinite’; according to this he affirms that a point, because it is indivisible, is neither finite nor infinite.

In the sixth place; the philosopher in the fourth book of the Physics, where he proves that place is not body, says that we do not have one difference of point and of place; therefore, according to him a point is something distinct from divisible things.

In the seventh place; according to the philosopher in the second book of the Physics, he says that “they touch each other whose extremes are together” ; therefore, the two extremes of two lines touching each other are at the same time; but no two parts of these lines are at the same time, nor are the lines themselves at the same time; therefore, besides the lines and their parts, there are others, which are at the same time, which can not be unless indivisible points.

In the eighth place; in the fifth and the sixth book of the Physics he says that those are continuous of which the extremes are one; therefore, the two extremes of two halves in one line are one; but it is manifest that neither those halves nor any parts of them are one; therefore, besides those there are some which are one.

In the ninth place; the philosopher in the same place posits, that a line is divisible and a point invisible, and this is proved through this, that a line is not composed of points, nor is any continuum composed of indivisibles; which would be futile unless there could be such indivisibles other than continua.

In the tenth place; the philosopher says in the same place that, since however an impartible is indivisible, it is necessary for a whole to touch, et cetera; from which it is obvious that he posits some such indivisible.

Innumerable statements of such a character might be adduced by virtue of which it seems that it may be in accordance with the intention of the philosopher to posit such indivisible things really distinct from divisible things.

The third principal division responds to the reasons; but because another way of solving could appear from the solution of the previous statements; I, therefore, omit those arguments for the sake of brevity, and I reply to these.

As regards the first; I say that an instant is not such an absolute thing really distinct from every divisible thing, which can be proved in various ways.

First ; because it does not have any primary subject in which it may be able to be in a primary and adequate subject as if an accident in its subject, as is obvious inductively.

In the second place; because then God might not be able to make time to exist, unless he could necessarily produce infinite things as regards themselves distinct wholes and corrupt the same.

In the third place; because such things, when produced, would not be able naturally to be destroyed; for they could not be destroyed either through the corruption of their subject or through the absence of their preserving cause, or through the induction of their contrary.

In the fourth place; because, if there were such a thing, it could be in a genus per se, which is manifestly false.

In the fifth place; because God can preserve at the same time in the nature of things all things of the same ground, since they may not repel one another; therefore, God could bring it to pass that all past instants might be now, which however are infinite.

In the sixth place, because God may be able to make an instant apart from any subject of its own, which is impossible.

For the sake of brevity I pass by the presentation of those reasons; but I say that neither change nor generation is such an indivisible thing. And consequently, therefore, as regards the previous statements that all such expressions are figurative and metaphorical: ‘an instant is indivisible,’ ‘generation is in an instant,’ “sudden change is in an instant,’ ‘generation is sudden,’ ‘substantial form is a terminus of generation,’ ‘to have been changed is a terminus of motion,’ and any other of the kind; because in all such formal expressions in place of the other extreme of the proposition, one name is posited which has not only the force of a name in signification, but has the force of a noun and of a verb, or has the force of something composed of diverse terms, which compounded according to a property of common speech can not express a supposition for a verb, nor be the extreme of a proposition distinct from the other extreme and a copula.

Whence, so far as this is concerned; it is similar of such expressions, as it would be if there were one name put in to signify that which signifies ‘tunc bene,’ say, A; according to a property of common speech it would not be well said, ‘tunc bene est.’ So it would not be well said, ‘A is something’; yet there might be such a saying, ‘A is something’ to be accepted, as if a metaphorical and figurative expression that might be equivalent to this, ‘that, which is tunc bene, is something,’ or any other of a like kind. So I say of such, ‘generation is sudden,’ and of any other of a like kind, that they are not proper expressions but figurative; and it is from this, that this name ‘instant’ and this name “generation,’ and any other of a like kind, are not finite names having finite signification, as if they signify precisely certain things for which they may have to substitute, according to the mode in which such names as ‘angel,’ ‘whiteness,’ ‘blackness,’ ‘heat,’ ‘cold,’ have finite significations; but there are certain derivatives from verbs or other parts of formal speech or similar to such, which are not instituted, unless for the sake of metre, or ornament, or for the sake of brevity of speech.

And, therefore, as many statements in poetry and in an ornate mode of speaking must not be quoted unless they be explained and interpreted; so many statements must not be accepted for the sake of brevity according to a property of common speech, but according to the intention of the authors saying these things, who spoke figuratively by virtue of the cause stated; and this, because such were sufficiently well known in their times, that they were not true according to a property of common speech, although some later writers have ignored this.

Whence, I say that this proposition, ‘generation is in an instant,’ must not be accepted under the meaning which it signifies, as if one thing may be in another; as through this proposition, ‘water is in a vase,’ it is denoted that one thing is in another in a distinct thing ; whence, in distributing through all the modes in which this proposition ‘in’ is taken equivocally, of which the philosopher speaks in book four of the Physics, it is obvious that according to none of those can it be said that generation is in an instant; but this proposition, “generation is in an instant,’ ought to be accepted under this meaning, ‘when something is generated, a part is not generated before a part, but the whole is generated at the same time’ ; that this brief proposition, ‘generation is in an instant,’ may be posited for the sake of brevity in place of that long formal expression, ‘when something is generated, a part is not generated before a part but the whole at the same time.’ Similarly this proposition, ‘motion is in time,’ ought not to be accepted under that meaning which it signifies according to the form of the proposition; namely, that one thing is in another thing really distinct from it, but it ought to be accepted under this meaning, ‘when something is moved, it acquires one part before another, or it acquires one before another without interruption,’ or under such a meaning, ‘when something is moved, it acquires or loses one thing after another, co-existing at the same time with a movable first acquiring one after another,’ or in some such meaning as that brief formal expression, ‘motion is in time,’ is posited in place of such a long formal expression or of some very similar expression. And when it is said that according to the preceding statements, ‘generation would be nothing,’ I say that ‘generation is something’ under this meaning, ‘when something is generated, one whole form is acquired at the same time and not a part before a part is added to matter.’ And under a meaning opposed to this, this is false, ‘generation is nothing.’ And if it be said, ‘generation is not per se the terminus of a continuum, nor is it something continuous’; then it is nothing. For everything, which is in these inferiors, is a continuum or part of a continuum. It must be said that, according to the virtue of common speech and according to every property of common speech, that that proposition is true, ‘everything, which is in these inferiors, is per se one and a continuum or the terminus of a continuum’; for the other part is true; namely, ‘everything one per se in these inferiors is a continuum’; for there is nothing in these inferiors except a continuum, unless perchance the rational soul may be included with these inferiors. And, therefore, if this proposition, ‘generation is something,’ were true according to a property of common speech, this would be true according to a property of common speech, ‘generation is continuous’ : but neither is true according to a property of common speech; and there is a reason why this is not true according to a property of common speech, ‘generation is continuous or the terminus of a continuum.’ Nor is it true according to the meaning of the authors; for they understand this, ‘when something is generated, one part of the form is acquired mediately before another or before something of the same ground was acquired.’ And, therefore, it must not be imagined that generation may be something positive or privative really distinct from every continuum and from every permanent thing.

On the contrary, according to a property of common speech, neither of these is true, ‘generation is something,’ ‘generation is nothing,’ whether generation is negation or privation; as any one of these is not true according to a property of common speech, ‘man is something,’ ‘man is nothing,’ whether negation or privation; and this by accepting significatively all the terms; for according to a property of common speech such an abstract ‘generatio’ ought not any more to be formed from this verb ‘generari,’ than such an abstract ‘hominitas’ from this accusative case ‘hominem.’ And, therefore, through a similar mode through which you might reply to such questions raised about an abstract corresponding precisely to this accusative case ‘hominem.’ What may such humanity be? What is humanity? Is humanity either something or nothing, or is it a thing? I reply to such questions: generation is either something or nothing, and so of others: for thus even such questions are not proper according to a property of common speech; but if they ought to be accepted, they ought to be accepted according to the good sense of those using. So I say of such, whether ‘generation may be something,’ and of similar questions.

And in general, I say proportionately of all abstracts, which are formed from adverbs, prepositions, verbs, conjunctions, and in general of syncategorematics and of other abstracts for which their concretes stand ; not for those for which their concretes stand, as if they might be able to stand for those for which their own concretes stand, unless they may equally include in signification something syncategorematic. Whatever is conceded of a concrete, ought to be conceded according to a property of common speech of an abstract.

And, therefore, as Aristotle concedes this, ‘man runs’; so he would concede this, ‘humanity runs,’ unless humanity may precisely signify form. And so it may be said that whatever is predicated of any abstract, if it be truly predicated, is predicated of that per se in the first mode. It must be said that that proposition is simply false. For if that proposition thus accepted were true, all such would be false: ‘deity is seen by the blessed,’ ‘deity is really distinguished from creatures,’ ‘deity is omnipotent,’ ‘humanity is in the nature of things,’ ‘humanity is unible with a supposition for a verb,’ ‘humanity is distinguished from angel,’ and many such; since they may not be per se in the first mode; and yet it is manifest that they may be true and that many of them fall under faith, so that the opposite are heretical.

Whence, neither among the philosophers nor among the ancient Saints nor among others, is it found that they have posited such a distinction between an abstract and a concrete, as the modernist make. But among the ancient writers there have been many synonyms with their concretes, and many abstracts were not necessarily considered according to a property of common speech; but they could become formed only by virtue of any of the three causes previously mentioned. And, therefore, I do not censure the ancients, because they formed such abstracts for the sake of utility and metre, or the embellishment of common speech, or brevity. And this was sufficiently convenient for them, for it was about a thing sufficiently well known to them; at least to the wise, especially to Aristotle. And indeed Aristotle cared less about a mode of speaking; from which the thing was well known, as is obvious in the first book of the Physics, that he had moderate solicitude about names. And indeed many fall into various errors by virtue of this, that they accept the passages of the ancients as they sound according to the letter and according to a property of common speech, which intent, however, the ancients have not held.

Consequently, it ought to be said according to the previous statements that all such propositions are improper ; ‘aditas is something,’ ‘abietas is relation,’ ‘aquietas is respect,’ ‘haectitas is distinct from nature,’ and so of many such, which are accepted by the modernists. Nor are they to be accepted according to a property of common speech any more than if from these adverbs: “dum,’ ‘cum,’ ‘tunc,’ ‘iam,’ and from other args and from such conjunctions: ‘si,’ ‘et,’ ‘vel,’ ‘at,’ ‘quia’; and from others such abstracts may be patel : ‘dummitas,’ ‘cummitas,’ ‘tunctitas,’ ‘iammitas,’ ‘siitas,’ ‘attitas,’ ‘etitas,’ ‘quiitas.’ And by virtue of this, such propositions may be formed: ‘dummitas is something,’ ‘siitas is something,’ ‘velitas is God,’ and so of others.

As regards the second point, I say that the philosopher does not intend that a point falls within the definition of a line, as if something of the essence of a line, or as if signifying something of the essence of a line and distinct from a line. But he intends that in a definition expressing something characteristic of the same name ‘line,’ this name ‘point’ is posited, and according to the famous definition of a line, whether or not that be a true definition of a line; which however is not a true definition except according to the intention of the one judging correctly of a point and a line. Nor is it, however, to be understood that a point may be any privation totally distinct from a line, as men commonly imagine; which it is impossible to imagine ; namely, that a point may be something, either a privative or a positive, distinct as regards itself as a whole, or not the same as a line. On the contrary, according to a property of common speech or of signification, it must not be conceded that it is a positive, nor that it is a privative.

And if it be said that these are contradictories: ‘a point is a positive,’ ‘a point is not a positive’ ; then, according to a property of common speech, one is true and the other is false. I say that, as according to property, these are not contradictories: ‘to see a man is something,’ ‘to see a man is not something’ ; nor these: ‘it does not have farther extension is something,’ ‘it does not have farther lengthening is not something,’ ‘a point is something,’ ‘a point is not something’; and this by accepting all the terms significatively. And by virtue of these neither of them is to be conceded according to the virtue of common speech. And this with the preceding statements may be said according to that mode of positing of those who say that this name ‘point’ is equivalent in signification to one aggregate of many expressions, which can not express a supposition for a verb according to grammatical science; and, therefore, no language is proper, in which it is posited, that in speaking grammatically it may be substituted with respect to a verb. But if another mode of positing concerning that, which was stated before, be held; it must be so conceded that a point is something positive, and a point is a quantum, and a line is divisible; for it is equivalent to this: ‘so much quantum,’ or ‘something is of so much extension,’ and ‘a quantum is divisible,’ and so of others.

As regards the third point, I concede proportionately of a line and of a surface as of a point; whence, as according to one mode of speaking this must be conceded, “a point is divisible,’ so this, ‘a line is a body.’ But according to another mode of speaking, as this is not proper, ‘a point is something,’ so neither this, ‘a line is something,’ nor this, ‘a surface is something’ ; but each of them is stated figuratively and must be accepted.

And when it is said that a terminate is defined through a terminant, it must be said that that proposition, as it is formed, is false. Whence, it must not be imagined that there may be some terminate there, and by virtue of that terminate there may be another terminant distinct from a terminate, whether a privative or a positive terminating that terminate; but there is there only a mode of expression of names signifying the same things in different grammatical modes, as these two in oblique cases ‘hominis’ and ‘homini’ Signify the same things, yet in entirely diverse grammatical modes.

And there is no inconsistency concerning such names, if one falls within a definition expressing something characteristic of another name. For thus this name ‘entity’ falls within a definition expressing some characteristic of that name ‘unum,’ and of that name ‘verum,’ and of that name ‘bonum’; and yet nothing is signified through one name that may not be signified through the rest, although it is understood through one name in an oblique case as well as in a direct case and through another name not unless in a direct case; whence, we know through this name ‘verum,’ that there is as much signified in an oblique case as in a direct case; for this that we understand, ‘entity is knowable and cognizable by the intellect.’ But through this name ‘entity’ it is not signified except in a direct case only, and indeed in the proposition it might be able to be conceded either properly or figuratively according to a diverse mode of positing that those names ‘terminant’ and ‘terminate’ are significative; that ‘the same’ is ‘terminant’ and ‘the same’ is ‘terminate,’ and ‘the same’ terminates itself; as ‘the same’ understands itself, or ‘the same’ is that which is ‘the same’ to itself. And when it is said that a positive never includes essentially a privation, it is obvious; for this must not be conceded, ‘a point is privation,’ ‘a surface is privation,’ and this according to a property of common speech and according to the fancy of those who imagine that privation may be something apart from a thing in any mode whatever distinct from positive things.

Whence, those arguments are cited or made contrary to that fancy which imagines, that a point may be apart from a thing, a certain privation distinct from a line as regards itself as a whole in every mode imaginable. According to which intent that Doctor seems to reject that opinion in opposition to which he offers the arguments touched upon before.

But that opinion does not thus posit, nor does it posit that a point is privation any more than that it is a positive; although in a definition expressing some characteristic of the same name, one name ‘negative’ or a concept ‘negative’ may necessarily be posited. And, therefore, according to the virtue of common speech either it must be conceded that a point may be an absolute thing, a quantum and divisible, or that every formal expression is improper in which this name ‘point’ is posited for one extreme distinct from the other extreme and from a copula; and no such expression ought to be accepted according to property but as if stated figuratively, and the arguments of the Doctors in no mode proceed contrary to that mode of positing.

As regards the fourth point, which was touched upon after the arguments of the Doctor, it can be said in various modes that the philosopher there as well as in the book of the Physics as also in other places where he speaks of a point, was speaking according to the famous opinion according to his custom; because he adapted himself to the mode of speaking of others, when he was treating a matter not according to intention. Whence, concerning his passages touching upon any matter which he treated not according to intention, it does not seem that he exercised much care; for he then frequently spoke as many do; and, therefore, where he treated such matters according to intention, he frequently said the opposite; as he says in the book of the Categories that “genera and species are secondary substances.” But in the seventh book of the Metaphysics, where he inquires according to intention whether ‘genus’ and ‘species,’ generally universals, may be substances, he decides according to intention that no universal is a substance; so is it in the proposition.

Nowhere is it found that Aristotle according to intention thoroughly treated that question, whether a point may be an indivisible thing really distinct from every divisible thing. Therefore, all his passages, which affirm that a point is such another thing, must be explained and interpreted, because either he spoke as many do or according to that famous opinion or for the sake of an example or figuratively; or diverse passages must be explained in different ways, as will be obvious further on of diverse passages.

Whence, the philosopher in the book of the Categories does not intend that one absolute thing really distinct from parts of a line may mutually connect parts of a line, and by virtue of this a line may be called continuous; for then a line would not be continuous through itself or through some of its parts, but through one thing totally distinct from a line; and it might be said with the same facility that two pieces of wood are continuous through one stone. Nor can it any more be said that a point, if there be such a thing, may be the terminus of a line than that a stone is the terminus of wood, since the distinction between a line and a point may be as much or greater than the distinction between stone and wood.

Nor can it any more be said that a point may be an accident of a substance existing in a line than that a stone is in wood. And, therefore, the philosopher understands through this proposition, ‘parts of a line are joined at one common terminus,’ this proposition, ‘nothing is intermediate between parts of a line and they make one per se’; so that two are required for this, that ‘a line may be continuous’; namely, that ‘nothing may be in situation intermediate between parts of a line.’ And by virtue of this when one body is in one situation and another body is in another situation and between those bodies there is in situation another intermediate, those two bodies are not continuous.

A second statement is required; namely, that they make one per se, and by virtue of this the contiguous are not continuous; and so through this proposition, ‘parts of a line are joined at a point as if at a common terminus,’ and through similar propositions the philosopher understands that copulative, ‘nothing is intermediate between parts of a line and they themselves make one line per se.’ And when it is said that a terminus is always distinguished from a terminate; it must be said that, if this name ‘terminus’ is precisely substituted for another thing so that it may have precisely the force of the name ‘category’ in signification, that it may not be equivalent to any other part of language nor to any composite of such a part; and similarly this name ‘terminate’; then it might be able to be conceded that ‘terminus’ is really distinguished from ‘terminate, if one be a ‘terminus’ and another a ‘terminate.’ But in taking them thus; this is false, ‘a point is a terminus,’ and ‘a line is terminated through a point.’ If those names ‘terminant’ and ‘terminate’ signify differently; then that proposition can be denied or conceded according as one is pleased to express in various modes the significations of terms.

Yet I say this, that it must not be imagined that a line may be a certain thing distinct from a thing which is a terminus or which may be a terminus; but from this itself, that ‘a line is such a thing and is not farther lengthened,’ it is said to be terminated there. As when we say that whiteness is the terminus of whitening and substantial form is the terminus of generation, it must not there be imagined that where there is one thing which may be whitening and generation, and afterwards or at the same time one other thing which is whiteness and form; for no contradiction appears but that God might be able to make form without any other thing even in a subject; nor does it appear why that thing ought to be destroyed if it be posited. But through this proposition, ‘form is the terminus of generation,’ this ought to be understood, ‘when something is generated, form is added de novo to pre-existing matter ; and as it is said of this, so it'must be said proportionately of many other similar propositions.

As regards the fifth and the following point, it can be said that the philosopher speaks according to the famous opinion which posits that a point is such another thing, not according to a proper opinion.

As regards the seventh point, I say that this term ‘simul’ is accepted in various ways. Sometimes, ‘simul’ is equivalent to this whole, ‘nothing is intermediate,’ so that it may be the sense of such a proposition, ‘these are at the same time’; that is, ‘nothing is intermediate between them.’ And so this must be conceded, ‘those touch each other whose extremes are at the same time’ under this sense, ‘those touch each other between which nothing’ is intermediate.’ Sometimes ‘simul’ is equivalent in signification to this, ‘not to be separate in situation’; and so no two things, which may be touching each other in any mode whatever, are at the same time; nor does the philosopher wish this. But through this proposition, ‘those touch each other between which nothing is intermediate,’ he understands this, as was said, ‘those touch each other between which nothing is intermediate,’ and if there could be such indivisibles in these touching, these would be at the same time; and this by taking this term ‘simul’ in the second mode; for they would not be separate in place nor in situation. And when it is said that no parts of two lines are at the same time; it must be said that in accepting ‘simul’ in the second mode it is true, and this is not required for this, that some touch each other; yet in accepting ‘simul’ in the first mode parts of a line are at the same time, for nothing is intermediate between them.

As regards the eighth point; when it is said that those are continuous whose extremes are one, it is manifest that that proposition ought to be explained; for it is false according to the virtue of common speech since it involves a contradiction, as that involves a contradiction, ‘more men are one man’; for, if they are extremes, they are more, and consequently are not one. Therefore, I say that the understanding of the philosopher is this, ‘those are continuous which make one per se,’ that is, ‘they are not separate,’ or ‘between which nothing is intermediate.’

As regards the ninth point; when it is said that a point is indivisible, it must be said that the philosopher either was speaking according to the famous opinion or the proposition must be explained, as was stated before.

When it is said for confirmation that the philosopher proves that no quantum is composed of indivisibles, it must be said that it is true; and that, since it may be negative, does not infer any such affirmative, ‘something is indivisible.’

And when it is said that it would have been useless to prove so diffusedly that a continuum is not composed of indivisibles unless there were some such indivisibles, it must be said that it is not useless to prove one conclusion through diverse intermediates, although it might be able to be proved sufficiently through one intermediate. Whence, an intermediate was sufficient to prove that there is no indivisible, but however others would not be superfluous, and this especially in that case; for to prove that nothing is indivisible in these inferiors can not be accomplished except through subtle reasons, which mathematicians and others less skilled in metaphysics and logic can not comprehend. But to prove that a continuum is not composed of indivisibles through other intermediates, even if there were such indivisibles, can be accomplished through reasons more apparent to mathematicians and to all others, as will be shown elsewhere; and, therefore, the reasons are not superfluous.

As regards the tenth point; it can be said just as about the preceding; for he was speaking according to the famous opinion.

As regards the principal argument it is obvious, that this, ‘a point is the terminus of a line,’ is false according to the virtue of common speech, yet it is true according to the sense that it alleges, as has been said.

Question II

In the second place it is asked, whether a line and a surface may really be distinguished from each other and from a body. And it seems that they are; for they are distinct species of quantity according to the philosopher in the Categories; therefore, they are really distinguished from each other and from a body.

As regards the opposite; as is the relation of a point to a line so is the relation of a line to a surface, and a surface to a body; but a point is not really distinct from a line, as was proved in the Categories; therefore, a line is not distinct from a surface nor a surface from a body, and consequently a line and a surface are not distinguished from each other or from a body.

With respect to that question, I say that a line and a surface are not things distinct from each other and froma body. For that a line may not be distinct from a surface can be proved through the first, third, and fourth reasons, and through other theological reasons through which it is proved that a point is not an indivisible thing really distinct from a line; for if a line were such another thing, it would be in some primary and adequate subject. Similarly, there would be infinite lines, none of which would be a part of another.

Similarly, God might not be able to divide one continuum unless He could create more absolute things, and could destroy some absolute things; nor would He be able to unite, unless He could very similarly corrupt and produce new things. Similarly, then it does not seem contradictory that there could be a surface without a line, and consequently there could be a surface not terminated, and yet it would be of finite length.

But that a surface could be distinct from a body can be proved through the first theological reason; for another primary and adequate subject can not be assigned to itself, as is evidently clear. Similarly, this can be proved through the third reason, for there would be surfaces infinite actually but potentially only finite; for the extreme surfaces alone of a body are actually ; but the others are not except only potentially.

In opposition to those positing thus; either the parts of the body are joined to a surface, which is in the nature of things and is existent, or to a surface which is not in the nature of things but can be. If the first be granted; then that surface is actually, and consequently there are surfaces infinite actually. If the second be conceded, then those parts of a body are not actually continuous but only potentially; for no parts are actually continuous through that which is not in the nature of things.

And then I ask, how that potentiality may be reduced to actuality; whether when those parts are continuous or when they are not continuous. Not in the first mode, for then according to those writers a continuant surface always is potentially, not actually; nor in the second mode, for then they are not continuous. But it may be said that that surface is not indeed said to be potentially; for it is not in the nature of things; but because it is not separate from a body; as parts are said to be potentially, not because they are not in the nature of things, but because they are not separate from a whole.

On the contrary, this does not suffice. First, because according to this such an inconsistency follows; namely, that infinites are in the nature of things of the same quantity. In the second place, because everything, which is in the nature of things and is not part of another, is simply actually suitable to itself, but that surface is in the nature of things and is not a part of another; therefore, it simply is actually.

In the third place thus; for, as that intermediate surface is not separate from a body, so neither is the extreme separate from a body; therefore, as the extreme is actually, so that is actually. Thus, therefore, it is obvious that, if a surface be a thing other than a body; this inconsistency follows, that infinites of the same quantity, of which none is part of another, are actually.

However, the principal conclusion can be proved through four and five reasons and through other reasons which follow. For if a surface be a thing as regards itself as a whole distinct from a body, there does not appear any contradiction that God may make it without a body, or conversely; and consequently a body can be without any such surface, and yet it will be finite and continuous; therefore, a surface is needlessly posited.

Similarly, I ask how a surface and a body make one in the sacrament of the altar; whether they make one per se or one through accident or one only by aggregation. The first can not be granted; for then a surface would be part of a body or both would be essential parts of another, which can not be conceded. And if it be said that a surface, although it may not be a quantitative part of a body, is however an essential part of a body; it is not refuted by the philosopher, although it may be refuted, that a surface may be a quidditative part of a body.

On the contrary, whenever something is part of another, either it is of the same ground as another part or it 1s of another ground. Therefore, if a surface be a part of a body which is per se one, either it is of the same ground as another part of a body or it is of another ground, which also can not be conceded; for, whenever some parts of diverse grounds make one per se, one of them is matter or potentiality, and the other is actuality or form; but potentiality and actuality making One per se are never distinguished in place or situation, as is manifestly obvious; therefore, in no mode are body and surface potentiality and actuality, and consequently they do not make one per se.

The second also can not be granted; namely, that a surface and a body make one by accident; for then one would be the actuality of another, and consequently a surface would be subjectively in a body, or conversely ; each of which is false according to the arguments set forth in the preceding question.

Nor can the third be conceded; namely, that they make one by aggregation, since it could not then be affirmed that a surface would otherwise be a terminus of a body than that a vase is a terminus of water. Then also no contradiction appears in the statement that God might be able to make a body in the same place and situation without a surface, as he can make water in the same place and situation without a vase; and yet a body would then be an infinite continuum without any surface; if this is conceded, it is obvious that such a surface is posited entirely in vain.

Similarly, then before transubstantiation a surface either would be without a substance or would be in another substance, as if in a primary and adequate subject, and not in the same substance in which a body is in a primary and adequate subject; as can be proved through the arguments set forth in the preceding question, and consequently there would be another substance there besides matter and form in which a body is posited, which is a quantity subjectively.

From the preceding statements it is obvious, that a line and a surface are not really distinguished from each other; and because it was proved that a surface is not distinguished from a body; therefore, neither is a line, and consequently they are not really distinguished from each other or from a body.

As regards the principal argument, I say that these concepts ‘line’ and ‘surface’ are distinct, and this suffices for a distinction of species; yet notwithstanding this, those, which are signified, are not distinct; for ‘place’ and ‘surface’ are thus distinct species, and yet they are not really distinct; but it suffices that they signify the same in diverse modes, as the commentator says in Book IV. of the Metaphysics concerning ‘entity’ and ‘one,’ where he states that ‘entity’ and ‘one’ signify the same, but in diverse modes.

Whence, as such diversity of modes of signifying the same thing vitiates the predication of one of another, although there may be only two such modes of signification, namely, ‘case’ and ‘number’; so a diversity of modes of signification suffices for the distinction of species, no matter how frequently one be truly predicated of another.

There are many arguments contrary to these which I pass over for the present; for I have spoken of them elsewhere in diverse places.

Question III

It is asked in the third place whether a body, which is a quantity, may be an absolute thing really distinct from substance. Which is thus proved; for according to the philosopher in the first book of the Posteriores, a negative is immediate in which one category is denied of another, as this is immediate, ‘no substance is a quantity,’ and consequently a substance is a thing other than a quantity.

As regards the opposite; a substance has of itself parts; therefore, it is of itself a quantum; therefore, it is not a quantum through another thing; and consequently it is not another thing.

Regarding this question; in the first place, I shall declare one opinion through the mode of narration; in the second place, through the mode of demonstration; in the third place, I shall posit the objections against this opinion and I shall refute them.

Regarding the first, I say that quantity is not another absolute thing really distinct from substance and quality. But if this name ‘quantity’ be substituted for another thing and have precisely the force of the name ‘categorematic’ in signification, it ought to be conceded that any quantity is not substance but quality, so that quantity is predicated particularly of substance as well as of quality; for any substance is a quantity, and any quality is a quantity.

But the mode of positing is this: some substance is simply not divisible into parts of the same ground, which substance is an angel and a rational soul; but every other substance is composed of some parts of the same ground among themselves, although not all parts may be of the same ground with the whole among themselves; as also in man some parts are of the same ground among themselves, although not all the parts are of the same ground with man; but such as are so constituted out of their own natures without any thing added, that they are more, although they are not more causally out of themselves but through their essential causes, say, efficient and final; so out of their natures are they constituted that they are not in the same place and situation naturally and through created agency, whatever may be of the absolute divine power. And indeed from this, that they are produced by whatever created agency they are produced, they are in a distinct situation so that one part is separate from another in situation and is outside another, no matter how much no other thing may be produced with it; and if there be another place surrounding the same product, that product, no matter how much no other thing might inhere in it, would be circumscriptively in that place, so that the whole would be in a whole place and a part in a part; and consequently by this itself it would be a quantum; since nothing other could be quantity, concerning which there is a question, than a continuous body per se one or a thing having parts or a part outside of a part existing circumscriptively in a place, if there be a place surrounding it. And so the substance itself through this alone, that it has part outside of part without any thing advening to it, is a quantum; and consequently for this, that it may be a quantum, such another thing is not required. And as it 1s of substance that something has such parts, so is it of any quality, which would be a quantum without any thing added to it; even if it could be separated from every other thing and could have part outside of part, and consequently this quality would not really differ from this quantity. So, therefore, any quantity is substance, and any quantity is quality; as any substance has part outside of part, and any quality has part outside of part.

In the second place, I prove this conclusion; namely, that any substance is quantity, and in the same mode it can be proved that any quality is quantity; so, namely, that it may not be another distinct thing.

First in this way: God can of His own absolute power preserve every absolute thing prior to another absolute thing, and can destroy a posterior not by changing that prior thing locally or in situation. Therefore, if this substance be quantum, say, that stone, God can of His absolute power preserve that substance and destroy that posterior thing, which is posited as quantity, not by changing the substance of that stone locally or in situation, but with the repose of that substance; for it does not seem necessary, so that the opposite of this may involve a contradiction, if God wills to preserve that absolute thing prior, that He be compelled to change it from place to place.

Therefore, let it be posited that that prior thing may be preserved without local change and that a posterior may be destroyed. Then I ask whether that substance is circumscriptively in a place or is not circumscriptively ina place. If the first is granted, then that substance is quantum; therefore, this is some quantity and this is not another posterior thing; therefore, this, ‘some quantity is without quantity,’ which is false; if it be not circumscriptively in a place.

On the contrary, I take one part of that substance and I ask, whether or not it is either definitively or circumscriptively in the same place in which there is another part. If so, then it has been changed locally, because it was not before in the same place. If it be not in the same place, then it is in another and consequently a whole is in a whole place and a part in a part; and consequently the whole is circumscriptively in a place, and consequently the whole is a quantum without any other thing added to it.

Whence, it much more restricts the divine power by saying that God can not preserve a thing prior without local change of it and destroy a thing posterior, which however follows according to the contrary opinion, than to posit that some quantity is substance; if it be said that that substance does not have parts except through quantity; and, therefore, when quantity is destroyed, it does not have part and consequently will not be in a place circumscriptively but only definitively.

On the contrary, this proposition does not vitiate a response; for there always follows that inconsistency, that God of His absolute power can not preserve a thing prior without local change by destroying the thing posterior. Moreover, the response embraces a falsehood; namely, that substance does not have parts except through quantity, if quantity be another thing.

First; because, if a whole substance is prior to a whole quantity and the subject of it, so a part of a substance is prior to a part of a quantity and the subject of it; therefore, parts of a substance are prior to parts of a quantity, if quantity is a thing existing subjectively in substance; therefore, substance does not have parts precisely through another thing advening to it. Moreover, that, which remains without another, is before this, but this part of substance is without that quantity; therefore, it is prior to it, and for the same reason is prior to another quantity.

If it be said that substance does not have other parts except parts which are parts of a quantity, so that these are not substances but are quantities, (on the contrary that part of a quantity is in some primary subject; therefore, either in the same primary subject in which there is another part of quantity, or in another ; therefore, if in the same, that subject is not extended through quantity more than the rational soul, which is a whole in a whole body and a whole in each part; if it be in another subject, then there are distinct parts of quantity there say, wood may be divided into two parts) it is manifest; that each quantity is in some subject, and not in the same; therefore in distinct, and consequently either a new substance is caused through division or these substances, which are of these quantities, were before division; and the proposition is maintained.

In the second place, I argue thus: it is not more impossible for God to make ‘a thing having parts really distinct’ to coexist with some whole not informing so that a whole coexists with a whole and a part with a part, than to make this same thing to coexist with such a whole informing. But according to those holding this opinion in common, ‘a substance having parts really distinct coexists with a whole quantity informing so that a whole substance is under a whole quantity,’ and ‘a part of a substance is under part of a quantity’ ; therefore, it is possible for God to bring it to pass that a whole substance may be coexisting with the same quantity not informing and part with the whole. This having been posited, this substance is truly a quantum; for it truly has part distinct in situation from part, and a substance is not a quantum through some other quantity informing; for it is posited that no quantity of a substance may inform it, but it may only coexist with it; therefore, it is a quantum through itself; therefore, the substance itself is a certain quantity, which is the thesis.

I posit the confirmation of this ground through the statements of those holding opposite views; for according to them, whatever may be of a respect intrinsically advening, is said of a respect extrinsically advening, yet it is true; but God can make extremes without such a respect; for the respect of inherence, by which quantity inheres in a subject, is a respect extrinsically advening according to those writers; therefore, God can make these extremes without such a respect. Therefore, God may preserve this substance of stone and that quantity by destroying precisely that respect of inherence. ‘This having been posited, still this substance will have the respect of presentiality as regards this quantity, so that a whole will be present with a whole and a part with a part; therefore, that substance has part outside of part, and consequently would be a quantum and yet without any quantity informing; therefore, the substance itself is then quantity. Those two reasons prove that substance is truly quantum through its substantial parts without any other thing advening to itself, and consequently that it itself is quantity.

From which I argue farther that there is not any other quantity except substance or quality; for that is needlessly accomplished through more which can be accomplished through fewer; but all, which can be saved through such a quantity distinct from substance and quality, can be saved without it, as is obvious inductively; therefore, such a quantity distinct from substance and quality must not be posited. It can also be proved through these two reasons, that quality is quantity; since this major premise is assumed, that God for this reason can make quality without quantity, if it be another thing, since it may no more depend on quantity than on substance. Similarly, if God can make such quantity, which they posit, without substance; for the same reason he could make quality without quantity, since quality no more depends on quantity than quality on substance. I confirm the previous arguments.

It is not more inconsistent for a substance having parts through its own self without every other thing informing it to be in a divisible place so that a whole thing may be in a whole place and parts in parts of a place, than it may be inconsistent for an indivisible thing without every thing informing it to be in a whole place and in each part; but an indivisible thing, as an angel and a rational soul, is a whole in a whole place and in each part; therefore, it is not inconsistent for a thing having parts really distinct, of such a kind is material substance and also many qualities, to be a whole in a whole place and a part in a part without every other thing advening. But if this is posited, such a thing will truly have part outside of part, and consequently will be quantum without any other thing advening to it; whence they seem to detract much from the divine power who say that God of His own absolute power can not bring it to pass that one thing may be separate from another in situation, unless He create one other thing making those things to be separate ; and they seem to reject the divine omnipotence more than those who say that a substance having parts is a quantum without any other thing advening to itself and is circumscriptively in a place.

Again, the preceding reasons are confirmed. It is not more inconsistent for a substance composed of some things really distinct and created to be separate in situation to be through itself without a thing added to itself in a place in a mode suitable to itself, than it is inconsistent for an indivisible substance not having parts to be in a place in a mode suitable to itself without a thing added to itself. But an angel without any thing added to itself is in a place definitively; therefore, it is not inconsistent for a substance having parts really distinct to be in a place circumscriptively without any thing added to itself. But nothing is in a place circumscriptively except a quantum; therefore, a substance can be a quantum without any thing really distinct added to it; therefore, such another quantity is needlessly posited.

Again, it is confirmed; for every thing, which is present to another thing locally, is present to itself through itself or through another intrinsic; for it has not been said that one thing is locally present to another, unless because its end is present there locally. If, therefore, the substance of a stone be present in a place, it is present in a place through itself, and consequently for the same reason parts of a substance are present in a place through themselves, and consequently the whole substance is through itself fundamentally and circumscriptively in a place; but every thing, which in itself is circumscriptively in a place, is in itself a quantum; therefore, substance in itself is quantum and consequently is quantity. The same reason can be applied concerning quality.

Again, the previous reasons are confirmed; for when some are created to make one which however can not make one, of such kind are parts of water and of air and of many others; for sometimes the same parts make one per se, sometimes not. If God can make these to be distinct in place and in situation, when they do not make one per se without any thing added to them, God can make these same to be distinct in place and in situation without any thing added to them, although they make one per se; for no reason appears why one more than the rest ought to be denied of the divine omnipotence. But God can make this air, which is here, and that air which is at Rome, to be distinct in place and in situation by destroying all the absolute accidents in them and without any quantity inhering in them; therefore, God of His absolute power can bring it to pass that those two bodies of air may be united and make one body of air, and yet that they may be distinct in place and situation without any such thing advening to them; therefore, that air will have part outside of part without a thing however added, and consequently will be a quantum without such a thing added; therefore, such an added thing is needlessly posited for this, that air may be a quantum.

The previous reasons can be confirmed in many other ways, which I pass by for the sake of brevity.

In the third place, it is principally argued thus according to the Master of the Sentences in Book IV., Distinction XII. Accidents of such a kind, taste, color, weight, and any other of the kind, are without a subject in the sacrament of the altar. But if quantity were another thing bearing those qualities, those qualities would not be without a subject, but would be in a subject; on the contrary the common dictum of the Doctors could in no mode be able to be verified, and very incorrectly, as it seems, the determination of the church that the accidents remain in the sacrament of the altar without a subject; for thus only one accident could remain without a subject, and all others would be ina subject; therefore, a quantity would not be such a distinct thing, et cetera. And so in carefully considering the Master and the text with the gloss on the De Consecratione and the statements of the Saints, there is not found a passage so apparent that by virtue of the sacrament of the altar it may be necessary to posit such another thing intermediate between substance and qualities, as is this against such an intermediate thing bearing qualities. No mention is made in all those places of an intermediate quantity nor is this term even once mentioned, nor does any passage speak of quantity, whether or not it be, except that many seem to posit that all the qualities in the sacrament of the altar are not in any subject; whence, the Master says thus in Book IV., Distinction XII., A:

“Those accidents, therefore, remain,’ which he enumerated before; namely, taste, weight, and any other of the kind, “subsisting per se at the celebration of the mystery,” et cetera. However, I do not intend to say that those who posit such a thing bearing such accidents, of which the Saints speak and which they seem to posit without any subject, may be heretical or erring in the faith or about the sacrament of the Eucharist; for they were permitted to explain such passages; nor are all who have uttered a falsehood to be considered in error. And as they were permitted to explain such passages of the Doctors approved by the church, so I shall be permitted not only to explain but even to refute the statements of the Saints who mutually condemn one another, and in no way were approved as if authorized by the church.

In the fourth place, I argue thus: no absolute accident is more simple than its primary and adequate subject; but, if a quantity were any thing other than substance and quality, it would be an accident. Therefore, I ask about its primary and adequate subject, whether it is matter or form or a composite. It is not a composite; for a subject would be more composite than its absolute accident, which seems false; nor is matter its own subject, because then some absolute accident would precede form in matter; nor is form its own primary subject, because then matter, if it were separated, could not be a quantum, which seems to be false.

Similarly, dimensions precede form without limits in matter according to the commentator; therefore, for the same reason dimensions are limited at first in matter. That reason is valid, as I think, if this opinion hold which rejects this entity distinct from matter and form, which some posit.

In the fifth place, I argue thus: according to the philosopher in the Categories, substance alone is susceptive of contraries; but quantity is not a substance according to those writers, and yet it is susceptive of contraries, which is contrary to Aristotle.

In the sixth place, it is argued thus: when a thing becomes rare from dense, I ask whether or not there is there another new quantity; but it is precisely the same quantity numerically; if it is new there, then any preceding quantity is corrupted, which seems false. If a part be new; I ask about the subject of that quantity, for it is necessary that there be a whole rarefied substance or a part of it; not the whole because then the two quantities would be at the same time, which they deny who think thus. If a part, the same argument follows, or that an accident might travel from subject into subject, as would be able to be clearly shown; but it is omitted for the sake of brevity. If it be the same quantity, then it coexists in a larger place now than before without the variation of an absolute; but this inconsistency is posited of substance; therefore, such a quantity distinct from quality and substance is needlessly posited.

Therefore, I say by virtue of those reasons and many other reasons, physical as well as theological, that quantity is not a thing really distinct from substance and quality, but any quantity is really the same as substance, and any quantity is really the same as quality; whence, quantity is not, unless ‘a thing having part outside of part and having part separate from another in situation,’ or ‘a thing existing circumscriptively in a place,’ 1f it be something surrounding itself. And because some substance in itself fundamentally, although causally by God or other extrinsic causes, has part outside of part; it is not necessary to posit another thing bearing one part outside of another part. But the extrinsic causes of a thing, say, efficient and final, are sufficient to produce things diverse, one in one situation and another in another, without any intermediate thing between them; therefore, the substance itself is a quantum without any other thing connected with itself or made at the same time with it.

And as I say of substance, so I say proportionately of any corporeal thing. Nor does that opinion in any way detract from the sacrament of the altar; but it avoids many difficulties respecting the sacrament of the altar, as will be obvious in replying and in objecting.

And, therefore, although some modern writers, stimulated perhaps by envy, would censure that opinion as if erroneous not by arguments but by detractions; yet the ancient Doctors, although they could not understand it, did not, however, repudiate it as heretical respecting faith or character, whose reasons I intend to cite and to explain. Whence, a certain Doctor arguing against this opinion says thus. This opinion is contrary to the philosopher, who in the Categories distinguishes quantity from substance and quality. It is also contrary to the common opinion of the Doctors. It is also contrary to experience; and when a corporeal substance is condensed without conversion of anything into itself, it is effected of a less quantity without the loss of any part. It is also contrary to reason, for in diverse subjects differing in species a quantity the same in species seems to be able to be; for the length, or breadth, or depth in a handful of air is not other than that ina handful of water. That they, however, say that the quantity of a quality really differs from the quantity of a substance and that it itself is really the same as quality, is contrary to authority and reason. For the philosopher in the sixth book of the Physics says that “quality is not quantity except through accident,’ which would not be true if it were properly called quantity. It is argued through the reasons thus: it is naturally impossible for a body, which is a quantity, to be at the same time with another body which is a quantity; but quality is at the same time with substance; therefore, from the previous Opinion it follows that this is impossible, that two thicknesses or even many more are naturally at the same time; because for whatever reason whiteness would have its proper quantity, for the same reason taste would have its own, and coldness or frigidity its own, and humidity and dryness, and so of others; and so not only will the quantity of substance and the quantity of quality be at the same time, but there would also be at the same time the thicknesses of several qualities; therefore, it is obvious that the opinion cited above implies many inconsistencies. These are the reasons of that Doctor, which are more apparent than others. But besides these I add others; first, because the substance of the bread is not transubstantiated into the quantity of the body of Christ but only into the substance of the body of Christ. But if substance could be quantity, it would have been truly transubstantiated into quantity. It is confirmed; for the quantity of the body of Christ is not in the sacrament of the altar by the power of conversion but only concomitantly. But if the substance of the body of Christ were a quantity, the quantity of the body of Christ would truly be there by the power of conversion. Moreover, some argue: if substance is quantity, then since the substance of the bread may not remain in the sacrament of the altar; therefore, a quantity is not there, which is contrary to sense and contrary to the reality of the sacrament. Other reasons can be adduced; but, because the solution of these could be made clear from the explanations of those already offered, I, therefore, pass them by.

As regards the first argument of that Doctor, I say that that opinion is in no mode contrary to the philosopher either in the Categories or elsewhere. And when he says that in the Categories he distinguishes quantity from substance and quality, I reply that the philosopher does not intend to distinguish between substance, quantity and quality; as if they be absolute things really distinct from one another. But he intends to distinguish between those categories which signify things, not indeed distinct things but the same things although in diverse modes, which are not modes, unless grammatical or logical modes; and indeed although this category ‘quantity’ may be really distinct, if it be a thing in mind different from other categories, of which distinction a statement was made elsewhere; yet it does not signify another thing one per se, unless that thing be a substance or a quality; and so the distinction of categories, of which the philosopher speaks, obtains at the same time with this, that quantity may not be a thing other than substance and quality; as according to him ‘place’ and ‘surface’ are not really distinguished, and yet they are distinct species of quantity. For that there may be distinct species of quantity is obvious through the philosopher in the Categories. That these species do not imply quantities distinct from a part of a thing, is obvious; since there would then be two continuous quantities at the same time in situation and in place, neither of which would be created to inform another.

Similarly, according to the philosopher in the Categories in the chapter on ‘quality’; the same is in diverse categories; therefore, categories sometimes signify the same thing; therefore, notwithstanding that quantity and substance and quality may be distinct categories, yet some of them have been able to signify the same thing, and so it is possible to agree with the philosopher that quantity is not an absolute thing other than substance and quality, as resemblance and white are in diverse categories, whether per se or through reduction; for the present I do not care, and yet they signify some thing the same.

Quite similarly I speak of ‘number’ which signifies no thing one per se, unless that which is under some species other than that under this common (species) ‘number’; as was proved elsewhere; and yet they are different species. Similarly, ‘ternary’ and ‘quaternary are distinct species, and yet no thing one per se is signified through ‘ternary,’ but that it is or can be of the same species with another thing signified through ‘quaternary,’ or conversely. And, indeed, as was stated before, only a diversity of modes of signification suffices to distinguish one concept from another; not that no thing may be signified absolutely through one, but that it may be signified by another. So also it is obvious from these two ‘homo’ and ‘homines’; for it is impossible to grant any thing signified through one that may not be signified through the rest; yet this is false, ‘man is species.’ So the identity of significators through some species or genera obtains with the distinction of species and of genera; and, as was frequently proved, genera or species are not, unless concepts or names.

As regards the second point; since it is said that that opinion is contrary to the common opinion of the Doctors, I reply that, if it be understood through the Doctors approved by the Roman Church, it is not true. On the contrary, this can be proved through no passage of any Doctor approved by the Roman Church. And if there be found any high-sounding passage of such a Doctor, this must be sufficiently explained. Nor is it inconsistent to explain their statements, since many statements even of the Sacred Page may also be in need of exposition; for many are not true according to a property of common speech, but are most true according to the sense which we ought to have.

Whence, such propositions which the Saviour uttered: “I am the true vine and my Father is the husbandman,” “My doctrine is not mine,” and many of a like kind, are not true according to the propriety of common speech; as the blessed Augustine says in a certain Homily on John: ‘Although in the sense in which they were spoken and in which they ought to be accepted, they may be most true.” And as many passages of Sacred Scripture must be explained, so the passages of the Doctors approved by the Roman Church declaring that quantity may be a thing other than substance and quality must also be explained. But if through the Doctors they may understand the modern Doctors who mutually censure each other publicly and privately and even in their writings, I concede; but it is not inconsistent to contradict them; for nothing, which they say, must be accepted, except that which they can prove through clear reason or through the authority of Sacred Scripture or through a decree of the church or through the Doctors approved by the church.

And, therefore, since they can prove in none of the preceding modes that quantity is a thing other than substance and quality, although some sophistications may be able to be adduced for this; I, therefore, do not hold it as an inconsistency to refute them in this part; for by what reason they themselves mutually censure one another, and the latter reprove the former, I can refute them, when they have for their support nothing but sophisms or passages wrongly understood ; nor do I think that I ought to be limited to those contrary to the dictate of reason.

On the contrary, I consider it dangerous and rash to wish to limit any one to the holding captive of his ability and to the believing of something which reason dictates to him to be false, unless it may be able to be drawn from Sacred Scripture or from a decree of the Roman Church or from the statements of the approved Doctors. Such is not this, ‘quantity is an absolute thing other than substance and quality’; whence, I assert this, that no passage of any philosopher, nor any natural reason, which I have ever heard or read, moves me to believe that quantity is an absolute thing other than substance and quality more than to believe that flesh is a thing other than God and creature. Yet if it may be able to be proved that it may be from the mind of some Saint or Doctor approved by the church, whom it is not permitted to refute; for his sake I wish to hold in check my ability and to concede that there may be a thing other than substance and quality.

As regards the third, I say that it is not contrary to experience, but rather consonant with experience; whence, if he should not have another reason, except that experience which he adduces, I would concede that quantity is not such a thing as they themselves imagine.

Whence, when a corporeal substance is condensed, I ask whether any part or no part of a quantity is lost. If no part, now then condensation can be saved; then for the same reason, although substance may be quantity and no part of substance may be lost, condensation will have been saved. But if some part of quantity is lost, I ask about the primary and adequate subject of the quantity itself, whether it remains without any quantity, which is contrary to experience, or whether it has any quantity, and then whether new or preexisting. It does not have a new quantity, because for the same reason another part would have a new quantity, since there may be no greater reason for one than for another; and consequently the whole substance would utterly lose the whole pre-existing quantity and would acquire a new quantity, which is contrary to experience. Nor does it have a pre-existing (quantity), since then an accident would change from subject into subject, which is not consonant with experience.

And so it is obvious that in no way is it contrary to experience to say, that quantity is not a thing other than substance and quality.

Therefore, a reply is made to that experience and by conceding that, when corporeal substance is condensed, it itself is effected of a smaller quantity without the loss of any of its part; but not without local motion of all of its parts.

Whence, through this alone, that all the parts are so moved locally that they may occupy a smaller place now than before and one part may be less separate from another now than before, the whole substance is condensed, and otherwise a substance is not to be condensed. And this is that which the philosopher says in the Categories, that “that is dense whose parts lie closer together,”’ which is not to be understood of the parts between which there is not an intermediate; for those can not as regards themselves as wholes lie closer together; but it must be understood of the parts between which there are intermediate parts, for those parts lie closer together in a dense body than they did in the same body when it was rare. And from this itself, that all such parts lie closer together whether they may be under another thing or not; they in themselves lie closer together, and that body is more dense now than before.

Whence, I do not see that condensation and rarefication may be able conveniently to be saved, except by positing that quantity is not a thing other than substance and quality, and through that mode is easily saved. And if it be said that the whole substance remains after condensation (but the whole quantity does not remain; for, if the whole could remain, it follows that it has as much quantity as there was before) ; then, that preceding quantity was not itself substance. It must be said that those who perfectly understand logic clearly see that all such are sophistic, for such consequences inferring one of the past from two of the present are not valid; as frequently two of the past or of the future do not infer one of the present, and frequently even one of the past and one of the present do not infer one of the present. But it would take too long to tell how it must be argued from such; therefore, I pass on from this. And I say that this consequence is not valid: ‘the whole substance remains’ and ‘so much quantity does not remain’; therefore, ‘so much quantity was not in that whole substance.’ And quite similarly such consequences are not valid: ‘that foot of quantity does not remain’ and ‘this substance remains ; therefore, ‘that substance will not be of the quantity of a foot’; nor does it infer more; therefore, that substance was other than a quantity of a foot. And as it is said of these, so proportionately must it be said of all of like kind, which are considered demonstrations by many.

Whence, as it does not follow: ‘making happy is not.’ It may be posited that ‘God may make no one happy,’ “God is’; therefore, ‘God was not making happy.’ Similarly, it does not follow: ‘the dead is not,’ “Christ is’; therefore, ‘Christ is not dead.’ So it does not follow: ‘so much quantity is not,’ ‘substance is’; therefore, ‘substance was not so much quantity.’ And so it evidently is obvious that, although such premises may not infer that of the present, yet they do not infer that of the past. Nor does it refer quantum to a mode, whether the terms ‘abstracts’ or ‘concretes’ be accepted.

And as it is said of these, so proportionately of those of a ‘possible’; for frequently two of a ‘possible,’ and even one of an ‘actual’ and another of a ‘possible,’ do not infer one negative of an ‘actual,’ although they may infer one of a ‘possible’; and, therefore, such modes of arguing are not valid. This is possible, ‘this likeness is not,’ while this appears true, ‘this whiteness is’; therefore, ‘this whiteness is not this likeness’; although that follows from a ‘possible’; therefore, it is possible for ‘this whiteness not to be this likeness.’ It does not follow: it is possible for ‘this whiteness not to be this likeness’; therefore, ‘this whiteness’ and ‘this likeness’ are not the same thing. So it does not follow: ‘it is possible for God not to be creating’ ; therefore, ‘God’ and ‘creating’ are not the same thing. But it certainly correctly follows that this is possible, ‘God and creating are not the same’; and this is true.

Quite similarly is it in the proposition, that because that substance can remain, although this may not be true, ‘the whole quantity is’; yet from this it does not follow that this may be true, ‘this substance’ and ‘so much quantity’ are not the same thing; but it follows that it may be possible; and this is true.

And if it be said that this is possible, ‘this substance’ and ‘this quantity’ are not the same thing, if the substance remains; by virtue of the same reason I say that this is fully possible, ‘this substance’ and ‘this quantity’ are not the same thing, so that it can be true, while this appears true, ‘this substance is’; but it can not be true, while this appears true, ‘this quantity is.’ Whence, this will then be true by virtue of that cause of reality, ‘this quantity is not,’ which follows from this, ‘this substance is not quantum,’ which is possible.

Another argument will be set forth later. And if it be said further that it is impossible to pass from a contradictory to a contradictory without entire newness, (if, therefore, this be at first true, ‘that substance is of so much quantity’ and afterwards it be false;) there is need that there be some newness or at least a destruction of something old. It must be said that it is not true; but sometimes local motion alone suffices, through which it is not necessary for something to be acquired or destroyed nor yet something old, say, a place always remaining.

On the contrary; as is shown elsewhere, a place surrounding a movable body can remain and this either in a whole or ina part. And, therefore, in the proposition regarding this, that the substance itself may be of a larger quantity or a larger quantity now than before as in rarefaction, or a smaller quantity now than before as in condensation, only local motion of all the parts of this substance suffices without entire newness of any imaginable thing whatever and without entire destruction of any thing old whatever.

As regards the fourth point; when it is said that that opinion is contrary to reason, it must be said that it is not, but is consonant with reason, as was shown before. And when it is said that in diverse substances differing in species it seems to be a quantity the same in species, it must be said that any quantity, which is not quality which is in substances diverse in species, differs in species; whence every quantity of fire, which is not a quality, differs in species from every quantity of air which is not a quality; and indeed I concede that the length of fire, and similarly the depth and the width, differ in species. And yet notwithstanding this, that the same most particular species could be predicated of them, although not ‘in quid’ and ‘per se’ in the first mode. Whence, two men and two horses differ in species, and yet ‘binarius,’ which is a most particular species, is applied to them. Whence, as it is not inconsistent that the same may be in diverse categories, so it is not inconsistent that the same may be in diverse species of one category, and yet they are in one most particular species of another category; and, therefore, it is not inconsistent that the same may be simply in diverse species of substance, and yet that they may be in the same most particular species of quantity. And whenever it is so, then that most particular species is not predicated of them ‘in quid’ and ‘per se’ in the first mode, and many passages concerning “genera’ as well as concerning ‘species’ speak of them with respect to those of which they are predicated ‘in quid’ and ‘per se’ in the first mode; yet there are many of no proper noun of any thing nor of a pronoun signifying some thing which are predicated ‘in quid’ and ‘per se’ in the first mode. Indeed, according to the intention of the philosopher relative to that which is replied in diverse modes to the question raised concerning an individual of a substance, as regards this diverse categories are accepted, as was said elsewhere; and indeed that is false which some say, that each category has a proper thing of which it is predicated ‘in quid.’

As regards the fifth point; I say that quantity, which is quality, is really distinguished from quantity, which is at least some substance; say, any such, as color, heat, cold, and any other of the kind; although this may not be true of figure and of any such. And when it is said that quality according to the philosopher is not a quantum except through accident, I say that through all such propositions in the Categories and in the book of the Physics and elsewhere, “‘substance is not a quantum except through accident,” and so of quality. Finite and infinite do not combine with a substance or a quality or through accident and anything else of the kind, It means nothing other than that that predicable ‘quantum’ is not predicated of substance and quality except through accident and not per se in the first mode. Whence, such is not per se in the first mode; neither by strictly nor freely accepting per se in the first mode: ‘substance is quantum,’ ‘quality is quantum,’ “man is quantum,’ ‘man is long,’ ‘whiteness is wide,’ and any other of the kind; since this however holds that quantity may not be a thing different from quality and substance. So ‘God’ and ‘creating’ are not distinct things, and yet this is not per se, ‘creating is immortal’; as this, ‘God is immortal.’ Similarly, finite or infinite per se in the second or the first mode is applied to quantum and not to substance or quality. And so it appears, since those substantives and predicables, ‘quantity,’ ‘substance,’ ‘quality,’ are distinct; since this however holds that they do not signify distinct things, and through that mode the reply must be made to many passages.

As regards the last point of that Doctor, I say that in speaking of bodies of diverse grounds of which one was created to be in the form of another, or of which each was created to be in the form of a third, it is not inconsistent for two bodies to exist naturally in the same place; and accident and subject are so constituted; so also are constituted diverse accidents which are created to inform the same subject; so also whiteness and sweetness in milk are constituted; and, therefore, it ought to be conceded according to the virtue of common speech that several bodies can be at the same time, that is, that several things existing circumscriptively in this place, and each one of which has part outside of part, are at the same time; yet it is inconsistent that two bodies of the same ground (each of which may be completely in a genus or in some species of a substance, or that two bodies of diverse grounds, neither of which was created to be in the form of another nor both (in the form) of a third), may be at the same time; and of such bodies Aristotle speaks and other authors who say that two bodies can not be at the same time.

As regards other arguments. As regards the first; when it is said that the substance of the bread is not transubstantiated into the quantity of the body of Christ but only into the substance of the body of Christ, it must be said that according to the virtue of common speech this must be conceded, ‘the substance of the bread is transubstantiated into the quantity of the body of Christ’; nor is the opposite of that found in the Bible nor in the statements of the Saints nor in the canonical law, nor in any authorized book; although it may be found in the writings of some who are censured by many Catholics and religious writers. I do not say that they censure them as heretics, but as those holding false opinions; whence, the subtle Doctor censures very much the profane and sacred writer and all, in whose writings is found the opposite of that, as regards many opinions. Many Catholic Doctors also censure the subtle Doctor in many conclusions; and, indeed, although a conclusion the opposite of these may be found, it need not make much difference; for no one is bound to accept their authority, although thanks must be rendered to them, for this reason that, when they have made false statements, they have stimulated our abilities, and they have rather given to us occasions for finding the truth. I say, therefore, that this must really be conceded, ‘the substance of the bread is transubstantiated into the body of Christ which is in heaven.’ Similarly, this must be conceded according to the virtue of common speech, ‘the substance of the bread is transubstantiated into the body of Christ which is in heaven locally.’ In the same mode I say in confirmation that this must be conceded according to the virtue of common speech, ‘the quantity of the body of Christ is in the sacrament of the altar by the power of conversion’; as according to the virtue of common speech this must be conceded, ‘the body of Christ existing in heaven is in the sacrament of the altar by the power of conversion’; and precisely the opposite is not found in any authorized writer. But yet no matter how much each of those may be true according to the virtue of common speech and according to fact, yet according to the virtue of common speech both can be false; while this appears true, ‘the substance of the body of Christ is in the sacrament of the altar by the power of conversion’; but not except in one case, say, if the substance of the body of Christ should exist nowhere circumscriptively. Whence, it does not seem to involve a contradiction, that, when the transubstantiation into the body of Christ has been accomplished and the body of Christ remains there under the species of bread, God could make the body of Christ to be nowhere locally or circumscriptively. When this has been done, the priest might be able to celebrate, and then this would be true according to the virtue of common speech, ‘the substance of the body of Christ is under the species of bread by the power of transubstantiation’ ; and then this would be false, ‘the quantity of the body of Christ ts under the species of that bread,’ and this by virtue of a false implication; for it would be implied that the substance of the body of Christ would be a quantum, which would not be true; for nothing is a quantum except that whose part is outside of part and whose part is separate from part. But, since the body of Christ could be nowhere locally, never could it be able in that mode to be assigned that one part could be separate from another part; but wherever it would be, one part would be in the same situation as another; and then this would be true, ‘the body of Christ is nowhere circumscriptively’; and for the same reason this would be true, ‘the body of Christ is nowhere a quantum’; and consequently this would be false, ‘the body of Christ is a quantity’; and so by virtue of a false implication this would be false, ‘the quantity of the body of Christ is in the sacrament of the altar.’ And so he who would desire to excuse those who say that the quantity of the body of Christ is not in the sacrament of the altar by the power of conversion, could say that they say this, because that consequence is not valid; namely, ‘the substance of the body of Christ is in the sacrament of the altar by the power of conversion’ ; therefore, the quantity of the body of Christ is in the sacrament of the altar by the power of conversion; for the antecedent can be true, while the consequent appears false. But this can not happen unless the substance of the body of Christ nowhere have part separate from part.

And if it be said contrary to this that, wherever the quantity of the body of Christ is by the power of conversion, there the body of Christ is a quantity or a quantum by the power of conversion. If, therefore, the quantity of the body of Christ be in the sacrament of the altar by the power of conversion, the body of Christ is a quantum or quantity in the sacrament of the altar. But every quantum is circumscriptively in a place; therefore, the body of Christ is circumscriptively in a place in the sacrament of the altar, which is heretical; for it could not then be in the whole host and in each part of the host. It must be said that that argument alone would be able to move a Catholic to hold that quantity is not a thing other than substance and quality, and would be able to be formulated thus: wherever any thing has another thing created to designate it, it is true to say that the thing there is such; as when a body has whiteness, it is true to say that the body there is white; and so is it of all. If, therefore, the quantity of the body of Christ be a thing other than the substance of the body of Christ, it is not separate from that quantity; for then those holding this opinion thus say, that the quantity of the body of Christ is there, although concomitantly; but whenever there is some quantum there, it truly has part outside of part and part separate from part; for whenever a definition corresponds to something, the defined corresponds there to the same. Therefore, since this may be a definition of quantity or of quantum, ‘to have part outside of part’; if the body of Christ in the sacrament of the altar be a quantum either concomitantly or in another mode, it is necessary that the body of Christ there be ‘having part outside of part, separate from part,’ which is manifestly false and perilous, and it does not seem at all safe to concede that the body of Christ may be a quantum here. And indeed I say that it must be conceded according to the virtue of common speech that the body of Christ is not a quantum in the sacrament of the altar. As according to the virtue of common speech it must be conceded that the body of Christ in the sacrament does not have part outside of part, although the body of Christ may have part outside of part in heaven, and consequently may be a quantum in heaven. And if it be said that this is contradictory to a prior statement, (for it was said that this is true, ‘the quantity of the body of Christ is in the sacrament of the altar’; from which it follows that the body of Christ is a quantum or a quantity in the sacrament of the altar, which was just denied;) it must be stated that this is simply true, ‘the quantity of the body of Christ is in the sacrament of the altar.’ But the former does not infer that; therefore, the body of Christ is a quantum in the sacrament of the altar; as this is true according to the virtue of speech, ‘the body of Christ existing locally in heaven is in the sacrament of the altar’; and yet this is false, ‘the body of Christ is locally existent in the sacrament of the altar.’

Similarly, this is true, ‘the body of Christ existing in a place is in the sacrament of the altar’; for nothing else is designated through that except that for which it substitutes this, ‘the whole body of Christ existing circumscriptively in a place,’ which is not unless the body of Christ is in the sacrament of the altar, which is true; and yet this is false, ‘the body of Christ is circumscriptively existent in a place in the sacrament . of the altar.’

Similarly, this is true, ‘the body of Christ having part outside of part is in the sacrament of the altar.’ In the same way, notwithstanding that this may be true, ‘the quantity of the body of Christ is in the sacrament of the altar’; yet this is false according to a property of common speech, ‘the body of Christ is a quantum in the sacrament of the altar’; and so, although the quantity of the body of Christ be truly there, yet the body of Christ is not a quantity there; as, although the body of Christ having part separate in situation from part may be there, yet it does not there have part separate from part.

And that reply is confirmed; for, whenever some are convertibles, of whatever one term is predicated the rest is also assumed significatively; and similarly of whatever one term is predicated, the rest also. But those are convertibles: ‘a continuous and permanent quantum’ and ‘having part separate in situation from part,’ and conversely; therefore, of whatever one is said, the rest also. If therefore, this be true, ‘the body of Christ is in the sacrament of the altar having part separate in situation from part,’ which is manifestly false; for no part is separate in situation from another in the sacrament of the altar; therefore, this is false according to a property of common speech, ‘the substance of the body of Christ is a continuous and permanent quantum in the sacrament of the altar.’

Therefore, I say that the body of Christ is not a quantum there; for it does not have there part separate from part; but in heaven, as elsewhere, it has part separate in situation from part, so in heaven it is a quantum. If only the quantity of the body of Christ were another thing, there would truly be a quantity there; namely, in the sacrament of the altar ; and consequently the body of Christ would truly be a quantum there, and consequently would truly have there part separate from part, and consequently would not be a whole under a whole host and a whole under each part.

As to the other; when it is said that, if the substance of the body of Christ were a quantity, then the quantity could not remain in the sacrament of the altar; so the substance of the bread does not remain there.

It must be said that that argument ought not to be advanced here, unless because some maliciously provoke some frank statements contrary to the opinion; as if that opinion could posit, that every quantity would be substance; and they indeed maliciously propose this proposition, ‘substance is quantity’ and withdraw this, ‘quality is quantity,’ that they may grant through this an opportunity for the unlearned who pay no attention to the preceding opinion to understand and to posit this, “every quantity is a substance’; and by virtue of a defect in the arguments they carefully strive to enlist some simple minded folk against the opinion.

Therefore, I say as regards the argument, that it is too stupid; for it does not follow, ‘substance is quantity,’ and substance ceases to be there; then some quantity does not remain there, as substance does not remain there; as it does not follow, ‘substance is an entity’; therefore, no entity remains there; as a preexisting substance does not remain there.

Whence, I say that one pre-existing quantity remains in the sacrament of the altar; but quantity does not remain there, which is substance or which was substance; but one quantity remains there, which is quality; for quantity remains there, which is not a thing other than whiteness.

Similarly, a quantity remains there which is not a thing other than taste, and so of other quantities. And I say with the Doctors approved by the church, that color, taste, weight, that is, gravity, and some qualities of a like kind, remain there; none of which is a thing other than a quantity subsisting per se in situation outside of every subject by divine power; none of which is subjectively in another; but each one exists outside of a subject through the divine power, and they are preserved at the same time in the same place and situation through the divine power; and indeed there remains there one quantity having part separate from part, which never was nor ever will be substance, but is simply a thing other than substance. And these arguments concerning that matter may suffice for the present, for I shall elsewhere treat at length of it.

The introduction to the tract concerning the body of Christ begins.

Wondrous gifts from the abundance on high, expended by Him for the redemption of the human race, place it under an infinite obligation to its glorious Redeemer; for the Only-begotten Son of God, after He had assumed the substance of our frailty in order that He might redeem us from the servitude of the devil and restore us to our original liberty and also bring us at last to our heavenly inheritance, offered Himself on the altar of the cross as the price for the redemption of the human race.

Moreover, that a perpetual memorial of so great a gift might remain among us and also that Christ might daily be mystically offered as a sacrifice for us who err daily, He left behind His body in the food and His blood in the drink to be received by the faithful in the sacrament of the Eucharist, the unspeakable loftiness of which is deemed an improper subject for human investigation; by virtue of which there is need to speak of this sublime and most excellent sacrament with all fear and modesty; for with respect to no sacrament is error more perilous, nor inquiry more laborious, nor invention more fruitful.

Accordingly, since I am about to examine carefully some views of this most worthy sacrament, I protest that I will not assert anything except that which the Roman Church holds and teaches. Moreover, subjecting and exposing myself to the correction of all the skilled, orthodox Catholics who are interested, I shall explain the statements of physics which are to be inserted and whatever statements are in general to be quoted, which are not authorized by the Roman Church, with all humility and modesty, that is, not by rashly approving them but by citing them only for the sake of practice and of inquiring into the truth.

However, let all men understand that I do not undertake this present little work that I may inflict some profane novelties of speech upon the ears of the faithful, but that I may show how blamelessly and temperately I have spoken on this most worthy sacrament when I expressed my opinions.