End of Chapter II and Chapter III. How line, surface, and body are continuous quantities

which the aggregation stands, as in that completing unity, so that as end and complement it is the form of that number, as in the quinary the fifth, and in the denary the tenth, and so concerning the others. And therefore Aristotle says ¹ that ten is not three and seven, or twice five, or eight and two; but one must receive the form in the last unity insofar as it is the end of the ordered aggregation toward that unity which is the complement of such an aggregation.

Similarly also in speech insofar as it is in lettered pronunciation: for that is an aggregation of sounds, elements, syllables, and dictions, completing an ordered aggregation toward one whole, just as the aggregation of letters in a syllable is one completing thing, and the aggregation of syllables in a diction is one completing thing, from which the diction is called one in the whole aggregation of syllables. Similarly also the aggregation of dictions in speech is one completing thing, from which that whole aggregation of distinct and ordered sounds is called one speech, since the sound of each element or syllable remains distinct in itself and neither continuous nor mixed, so that one is short and another long, and one is shorter than the short, but another is longer than the long; and in such things nothing makes one except order toward one. And therefore that extended aggregation according to itself is discrete quantity like number; on account of which speech is quantity, whose indivisible is not one thing, whose repetition makes that aggregation, but the indivisibles are many, which are the sounds of elements in letters, and sounds of syllables in dictions, and sounds of dictions in speech; but among them the only indivisible is a letter or the element of a letter, from which, ordered with another, a syllable is made, and from a syllable ordered with another a diction is made, and from a diction ordered with another speech is made.

But among the elements of letters there is not one indivisible which, repeated through substance, completes that whole aggregation, as unity is in number. Unless someone should say that there is one element of some letter which is first, which is repeated syllabically in all the elements of other letters, and that the elements of other letters consist from the repetition of that one. But although some have said this, nevertheless it is false and cannot be proved. Therefore we say in this way that speech is composed by the order of many indivisibles according to the sound of elements, and in this speech differs from number, and is diverse from number itself according to the species of quantity.

But if it is objected that every quantity proceeds from an indivisible, one must say that this is true: that just as number proceeds from unity, and the continuous from a point, and time from an instant, so speech proceeds from an indivisible of lettered sound. But this indivisible is not the element of some letter, whose repeated element would make such a quantity of speech, but an indivisible whose account is preserved in any sound whatever of any letter whatever aggregated with another in such pronunciation.

For hence Democritus and Leucippus, and certain others positing atoms, did not say that the principle was one atom, some one principle, but many atoms differing in figure and order and thus constituting all things. And they gave an example of their account in letters: for in lettered sound or pronunciation there is not some one indivisible as the cause of aggregation, but many things differing in figure, sound, and order. For thus speech is a quantity measured by syllable, or by a long or short letter of pronunciation, and distinct in the whole mass of such aggregation, and it is quantity in itself, so that its substance and quiddity is quantity differing from number in the aforesaid mode. Therefore these are the two first species of discrete quantity.

CHAPTER III.

How line, surface, and body are continuous quantities.

Now it remains to say how continuous

quantities are line, surface, and body. For it was fitting to speak first of discrete quantity, on account of this, that the indivisible from which number is perfected, and the indivisible from which speech is perfected, are simpler than the indivisible from which the continuous is perfected. For unity is an indivisible, neither having position in the continuous nor continuing those things of which it is the term. Lettered sound also is indivisible, neither an indivisible placed in a continuum having position, nor continuing those things of which it is the term or principle; because if it is a principle, it is not that from which there is continuation; and if it is a middle, it is not one and the same end of one thing and principle of another, as a point is continuing; and if it is an end, it is not the term to which the continuum is joined continuously in the form of continuation. And therefore also the indivisible itself of lettered sound is simpler than a point; on account of which the natural order requires that first one speak of number, and afterward of speech, and thirdly of the species of continuous quantity.

But what certain people say, that for this reason one should first have spoken of number, because number is found in body and in incorporeal things, and not the continuous, and because things from the institution of the world were made by proportions of numbers: those arguments are not true, nor do they make for the proposed point, and they are founded upon what is false; but concerning those things let those who assert such arguments see to it.

Therefore by this necessity which we state, after number and speech we say that line is of the number of continuous things: for from any point a continuous term can be taken, to which all the parts of a line which are under one form of continuity are joined. But this common term is point, whose flux into a continuum, or drawing-out, makes a line; and therefore, although line is not divided in a point, nevertheless it cannot be divided except in a point, because line is everywhere substantially point. Substantially, I say, from the substance which is as matter: for point extended or stretched out into a continuum is line, whose very stretching-out is quantity, and whose stretched-out substance is point. And thus line is from one substantial thing as from a subject, which is point, and from one formal thing, which is stretching-out extended into a continuum, which is continuation. And this indeed is line; and therefore because everywhere the material substance of line is point, wherever a point not detached from extension into a continuum is taken, there the point is in the account of end and principle: end indeed toward the prior thing, and principle toward the posterior thing. Therefore it is the common term of this joining of the prior part with the following part through the whole continuum; and because point is a placed indivisible, therefore its whole intellectual and real extension in the whole length of substance preserves indivisibility. On account of this line according to breadth and depth is indivisible, divisible only by length, which is caused not by point, but by the extension of point and flux into a continuum, because that extension gives the form of continuity, which according to itself is always divisible.

Similarly surface is continuous quantity: for it is generated by the expansion of line broadwise into a continuum, or from the flux of line or from the drawing-out of line, as Euclid says; and thus it will be breadth without depth, everywhere measured from two lines cutting one another at a right angle. Because this flows from an indivisible according to depth, therefore it does not have depth; but because it flows from a divisible according to length, therefore it has divisibility according to length. And because the flux of that line is to the side and not to a point which is the term of line, and the flux is continuous, therefore surface has continuity in breadth, and from that it is surface. For the continuity which is lengthwise in surface is from line; and therefore the substance of surface is line, just as point is the substance of line. On account of this, from whatever direction surface is divided, there will be a line dividing it just as point divides line; and wherever two parts are taken in a surface, there will be a line

joining those parts, and it is the common term of their joining; for it is the principle of one, from which its continuity is, and it is the end of the other, to which its continuation is. For a plane is that whose particles of surface are joined to a certain common term in the said mode. Surface is called plane because, by the flux of line, it is understood to be spread out as into a plane, with no part of it standing out above another. Therefore such is the understanding of surface according as it is a species of the continuous.

Similarly also in body one must take a common term joining the particles of body. For body is threefold dimension itself, which is by three diameters in every part of the body touching one another and cutting one another at a right angle, one in length, another in breadth from the point of the preceding one, and the third drawn out in depth from the angle of the aforesaid ones: for it is generated from the flux or stretching-out of surface, with the surface moved or stretched out not in length or in breadth, but in depth. And therefore Plato used to say that the substance of body according to what it is, is surface, just as point is the substance of line and line the substance of surface; and therefore body, as it is body, is divided by surface, as surface by line, and line by point, and wherever you take two parts of body, they have a surface to which the continuity of the preceding part belongs, and from which the continuity of the following part begins. Therefore that surface in body is a common term, which is the end of one and the principle of the other; and therefore it is something of one and something of the other; on account of which it is called the common term of continuous things. But if you wish to take the common term in body according to breadth, then, as was said before, you will take line as the term of joining. But if you wish to take the common term of joining in length, it is necessary that you take a common point, which joins the parts in length.

Therefore although a point is no part of line, nevertheless it is something of line, because it is the term of line and joins its parts; and according to this, that point has position in the continuous, thus it is the substance of line, but the formal being is continuity in length, and that is from the intellectual flux of point into the continuous. And in every same mode line is related to surface, so that it is not some part of surface, and yet it is something of it, both because it is the term of its joining, and also in this, that it is the substance of surface, and the form of continuity which it places around the substance is from the intellectual flux of line to the breadth of line, and not to a point. According to the same mode also surface is related to body, both in this, that it is the common term of joining, and in this, that it is the material substance of body, around which the form of the continuity of body is placed from the intellectual flux of surface into depth.

From this it is clear that the continuous is divided to infinity insofar as it is continuous: for in whatever term the indivisible is taken, it is the term of joining; therefore consequently it will be the cause of continuity, because it is understood in the account of end and principle, which cannot be understood except through this, that it flows from one thing into another. Therefore the account of division does not remove the form of continuity: while the form of continuity remains, the divisible is posited. Therefore whatever things are understood to be divided, this indivisible is always posited for this.

But let it move no one that we say this to happen according to understanding, since in this book we are speaking of the ten genera, the principles of things, to which all predicables are reduced. For all abstract and mathematical things are taken according to understanding; for according to being in nature this or that is not found except in light alone, as certain people say, although their saying does not agree with the Philosophers. Therefore although those things are taken according to understanding, nevertheless that understanding is caused by special things and is referred to being, because it says the nature and being of the thing as it is this which it is, as it is signified