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 in a 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 is 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 in a 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 pleased 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 Sacred Scripture in a spiritual sense but even in a literal sense.

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 adverbs; and from such conjunctions: ‘si,’ ‘et,’ ‘vel,’ ‘at,’ ‘quia’; and from others such abstracts may be formed: ‘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 diffusely 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.

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