Cursus Philosophicus Thomisticus, Secunda Pars Artis Logicae

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

COORDINATION AND TREE OF THE PREDICAMENT OF QUANTITY.

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

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