[968a1] Are there indivisible lines? And, generally, is there something partless in every class of quanta, as some say?
For if, where ‘many’ and ‘large’ apply, so do their opposites, ‘few’ and ‘small’; [5] and if that which admits practically an infinite number of divisions is many not few, then what is few and what is small will clearly admit only a finite number of divisions. But if the divisions are finite in number, there must be a partless magnitude. Hence in all classes of quanta there will be found something partless, since in all of them ‘few’ and ‘small’ apply.
Again, if there is an Idea of line, and if the Idea is first of the things called by [10] its name, then, since the parts are by nature prior to their whole, the Ideal Line must be indivisible. And, on the same principle, the Ideal Square, the Ideal Triangle, and all the other Ideal Figures—and, generalizing, the Ideal Plane and the Ideal Solid—must be without parts; for otherwise it will result that there are things prior [15] to each of them.
Again, if body consists of elements, and if there is nothing prior to the elements, and if parts are prior to their whole, then fire and, generally, each of the elements which are the constituents of body must be indivisible. Hence there must be something partless in the objects of sense as well as in the objects of thought.
Again, Zeno’s argument proves that there must be partless magnitudes. For it [20] is impossible to touch an infinite number of things in a finite time, touching them one by one; and the moving body must reach the half-way point before it reaches the end; and there always is a half-way point in any non-partless thing.
But even if the body, which is moving along the line, does touch the infinity of points in a finite time; and if the quicker the movement of the moving body, the greater the stretch which it traverses in an equal time; and if the movement of [968b1] thought is quickest of all movements:—it follows that thought too will come successively into contact with an infinity of objects in a finite time. And since thought’s coming into contact with objects one-by-one is counting, it is possible to count infinitely many objects in a finite time. But since this is impossible, there must be such a thing as an indivisible line.
[5] Again, the being of indivisible lines (it is maintained) follows from the mathematicians’ own statements. For if commensurate lines are those which are measured by the same unit of measurement, and if all commensurate lines are being measured,1 there will be some length by which all of them will be measured. And this length must be indivisible. For if it is divisible, its parts—since they are commensurate with the whole—will involve some unit of measurement. Thus half [10] of a certain part will be double it. But since this is impossible, there must be an indivisible unit of measurement. And just as all the lines which are compounded of the unit are composed of partless elements, so also are the lines which the unit measures once.
And the same can be shown to follow in the plane figures too. For all which are [15] drawn on the rational lines are commensurate with one another; and therefore their unit of measurement will be partless.
But if any such plane be cut along any prescribed and determinate line, that line will be neither rational nor irrational, nor will any of the other kinds of lines which produce rational squares, such as the ‘apotome’ or the ‘line ex duobus nominibus’. Such lines will have no nature of their own at all; though, relatively to [20] one another, they will be rational or irrational.
Now in the first place, it does not follow that that which admits an infinite number of divisions is not small or few. For we apply the predicate ‘small’ to place and magnitude, and generally to the continuous (and we apply ‘few’ where that is [25] applicable); and nevertheless we affirm that these quanta admit an infinite number of divisions.
Moreover, if in the composite magnitude there are contained indivisible lines,2 the predicate ‘small’ is applied to these indivisible lines, and each of them contains an infinite number of points. But each of them, quâ line, admits of division at a [969a1] point, and equally at any and every point: hence each of these non-indivisible lines would admit an infinite number of divisions. Moreover, some amongst the non-indivisible lines are small. The ratios are infinite in number; and every non-indivisible line admits of division in accordance with any prescribed ratio. [5]
Again, since the great is compounded of certain small things, the great will either be nothing, or it will be identical with that which admits a finite number of divisions. For the whole admits the divisions admitted by its parts. It is unreasonable that, whilst the small admits a finite number of divisions only, the great should admit an infinite number; and yet this is what the advocates of the theory postulate. [10]
It is clear, therefore, that it is not quâ admitting a finite and an infinite number of divisions that quanta are called small and great respectively. And to argue that, because in numbers what is few admits a finite number of divisions, therefore in lines the small line must admit only a finite number of divisions, is childish. For in numbers the development is from partless objects, and there is a determinate something from which the whole series of the numbers starts, and every number [15] which is not infinite admits a finite number of divisions; but in magnitudes the case is not parallel.
As to those who try to establish indivisible lines by arguments drawn from the Ideal Lines, we may perhaps say that, in positing Ideas of these quanta, they are assuming a premiss too narrow to carry their conclusion; and, by arguing thus, they [20] in a sense destroy the premisses which they use to prove their conclusion. For their arguments destroy the Ideas.
Again, as to the corporeal elements, it is childish to postulate them as partless. For even though some do as a matter of fact make this statement about them, yet to assume this for the present inquiry is to assume the point at issue. Or rather, the [25] more obviously the argument would appear to assume the point at issue, the more the opinion is confirmed that solids and lengths are divisible in bulk and distance.
The argument of Zeno does not establish that the moving body comes into contact with the infinite number of points in a finite time in the same way. For the [30] time and the length are called infinite and finite and admit of the same divisions.
Nor is thought’s coming into contact with the members of an infinite series one-by-one counting, even if it were supposed that thought does come into contact in this way with the members of an infinite series. Such a supposition perhaps assumes what is impossible: for the movement of thought does not, like the movement of [969b1] moving bodies, essentially involve continua and substrata.
If, however, the possibility of thought moving in this fashion be admitted, still this moving is not counting; for counting is movement combined with pausing.
It is surely absurd that, because you are unable to solve Zeno’s argument, you [5] should make yourselves slaves of your inability, and should commit yourselves to still greater errors, in the endeavour to support your incompetence.
As to what they say about commensurate lines—that all lines are measured by one and the same unit of measurement—this is sheer sophistry; nor is it in the least in accordance with the mathematical assumption as to commensurability. For the mathematicians do not make the assumption in this form, nor is it of any use to them.
[10] Moreover, it is actually inconsistent to postulate both that every line becomes commensurate, and that there is a common measure of all commensurate lines.
Hence their procedure is ridiculous, since, whilst professing that they are going to demonstrate their thesis in accordance with3 the opinions of the mathematicians, and by premisses drawn from the mathematicians’ own statements, they lapse into an argument which is a mere piece of contentious and sophistical dialectic—and [15] such a feeble piece of sophistry too! For it is feeble in many respects, and totally unable to escape paradox and refutation.
Moreover, it would be absurd for people to be led astray by Zeno’s arguments, and to be persuaded—because they cannot refute it—to invent indivisible lines; and yet because of the movement of a straight line to make a semicircle, which must [20] touch infinitely many arcs and distances in between, and because of its movement to form a circle, which readily shows that it must move at every point if it moves to make a semicircle, and because of other similar considerations about lines—to refuse to accept that a movement can be generated such that in it the moving thing does not fall successfully on each of the intervening points before reaching the [25] end-point. For the theorems in question are more generally admitted, than the arguments of Zeno.
It is clear, then, that the being of indivisible lines is neither demonstrated nor rendered plausible—at any rate by the arguments which we have quoted. And this conclusion will grow clearer in the light of the following considerations.
In the first place, our result will be confirmed by reflection on the conclusions proved in mathematics, and on the assumptions there laid down—conclusions and [30] assumptions which must either stand or be overthrown by more convincing arguments.
For neither the definition of line, nor that of straight line, will apply to the indivisible line, since the latter is not between any terminal points, and does not possess a middle.
Secondly, all lines will be commensurate. For all lines—both those which are [970a1] commensurate in length, and those which produce commensurate squares—will be measured by the indivisible lines.
And the indivisible lines are all of them commensurate in length (for they are all equal to one another), and therefore also they all produce commensurate squares. But if so, then the square on any line will always be rational. [5]
Again, since the line applied to the longer side determines the breadth of the figure, the rectangle, which is equal in area to the square on the indivisible line (e.g. on the line one foot long), will, if applied to a line double the indivisible line, have a breadth determined by a line shorter than the indivisible line: for its breadth will be less than the breadth of the square on the indivisible line.
Again, since any three given straight lines can be combined to form a triangle, [10] a triangle can also be formed by combining three given indivisible lines. But in every equilateral triangle the perpendicular dropped from the apex bisects the base. Hence, it will bisect the indivisible base too.
Again, if the square can be constructed of partless lines, then let its diagonal be drawn, and a perpendicular be dropped. The square on the side will be equal to the square on the perpendicular together with the square on half the diagonal. Hence it will not be the smallest line. [15]
Nor will the area which is the square on the diagonal be double the square on the indivisible line. For if from the diagonal a length equal to the side of the original square be subtracted, the remaining portion of the diagonal will be less than the partless line. For if it were equal the square on the diagonal would have been four times the original square.
And one might collect other similar absurdities to which the doctrine leads; for indeed it conflicts with practically everything in mathematics. [20]
Again, what is partless admits of only one mode of conjunction, but a line admits of two: for one line may be conjoined to another either along the whole length of both lines, or by contact at either of its opposite terminal points.
Further, the addition of a line will not make the whole line any longer; for partless items will not, by being added together, produce an increased total magnitude.
[25] Further, every continuous quantum admits more divisions than one, and therefore no continuous quantum can be formed out of two partless items. And since every line (other than the indivisible line) is continuous, there can be no indivisible line.
Further, if every line (other than the indivisible line) can be divided both into equal and into unequal parts—every line, even if it consist of three or any odd number of indivisible lines—it will follow that the indivisible line is divisible.
And the same will result if every line admits of bisection; for then every line [30] consisting of an odd number of indivisible lines will admit of bisection.
And if not every line, but only lines consisting of an even number of units admit of bisection, and if it is possible to cut the line being bisected any number of times, still, even so, the ‘indivisible’ line will be divided, when the line consisting of an even number of units is divided into unequal parts.
[970b1] Again, if a body has been set in motion and takes a certain time to traverse a certain stretch, and half that time to traverse half that stretch, it will traverse less than half the stretch in less than half the time. Hence if the stretch be a length consisting of an odd number of indivisible unit-lines, we shall here again find4 the [5] bisection of the indivisible lines, since the body will traverse half the stretch in half the time: for the time and the line will be correspondingly divided.
So that none of the composite lines will admit of division both into equal and into unequal parts; and if they are divided in a way corresponding to the division of the times, there will not be indivisible lines. And yet (as we said) the truth is, that [10] the same argument implies that all these things consist of partless items.
Further, every line which is not infinite has two terminal points; for line is defined by these. Now, the indivisible line is not infinite, and will therefore have a terminal point. Hence it is divisible: for the terminal point and that which it terminates are different from one another. Otherwise there will be a third kind of line, which is neither finite nor infinite.
[15] Further, there will not be a point contained in every line. For there will be no point contained in the indivisible line; since, if it contains one point only, a line will be a point, whilst if it contains more than one point it will be divisible. And if there is no point in the indivisible line, neither will there be a point in any line at all: for all the other lines are made up of the indivisible lines.
Moreover, there will either be nothing between the points, or a line. But if [20] there is a line between them, and if all lines contain more points than one, the line will not be indivisible.
Again, it will not be possible to construct a square on every line. For a square will always possess length and breadth, and will therefore be divisible, since each of its dimensions is a determinate something. But if the square is divisible, then so will be the line on which it is constructed.
Again, the limit of the line will be a line and not a point. For it is the ultimate thing which is a limit, and it is the indivisible line which is ultimate. For if the [25] ultimate thing be a point, then the limit to the indivisible line will be a point, and one line will be longer than another by a point. But if the point is contained within the indivisible line, because two lines united so as to form a continuous line have one and the same limit at their juncture, then the partless line will after all have a limit belonging to it.
And, indeed, how will a point differ at all from a line on their theory? For the indivisible line will posses nothing characteristic to distinguish it from the point, [30] except the name.
Again, there must, by parity of reasoning, be indivisible planes and solids too. For if one is indivisible, the others will follow suit; for each divides at one of the others. But there is no indivisible solid; for a solid contains depth and breadth. [971a1] Hence neither can there be an indivisible line. For a solid is divisible at a plane, and a plane is divisible at a line.
But since the arguments by which they endeavour to convince us are weak and false, and since their opinions conflict with all the most convincing arguments, it is [5] clear that there can be no indivisible line.
And it is further clear from the above considerations that a line cannot be composed of points. For the same arguments, or most of them, will apply.
For it will necessarily follow that the point is divided, when the line composed of an odd number of points is divided into equal parts, or when the line composed of an even number of points is divided into unequal parts. [10]
And it will follow that the part of a line is not a line, nor the part of a plane a plane.
Further it will follow that one line is longer than another by a point; for it is by its constituent elements that one line will exceed another. But that this is impossible is clear both from what is proved in mathematics and from the following argument. For it would result that a moving body would take a time to traverse a point. For, as [15] it traverses an equal line in an equal time, it will traverse a longer line in a greater time: and that by which the greater time exceeds the equal time is itself a time.
Perhaps, however, time consists of ‘nows’, and both theses belong to the same way of thinking.
Since, then, the now is a beginning and end of a time, and the point a beginning and end of a line; and since the beginning of anything is not continuous with its end, but they have an interval between them; it follows that neither nows nor points can [20] be continuous with one another.
Again, a line is a magnitude; but the putting together of points constitutes no magnitude, because several points put together occupy no more space than one. For when one line is superimposed on another and coincides with it, the breadth is in no way increased. And if points too are contained in the line, neither would points [25] occupy more space. Hence points would not constitute a magnitude.
Again, whenever one thing is contiguous with another, the contact is either whole-with-whole, or part-with-part, or whole-with-part. But the point is without parts. Hence the contact of point with point must be a contact whole-with-whole.
But if one thing is in contact with another whole-with-whole, the two things must be one. For if either of them is anything in any respect in which the other5 is [30] not, they would not be in contact whole-with-whole.
But if the partless items are together, then a plurality occupies the same place [971b1] which was formerly occupied by one; for if two things are together and neither admits of being extended, just so far6 the place occupied by both is the same. And since the partless has no dimension, it follows that a continuous magnitude cannot be composed of partless items. Hence neither can a line consist of points nor a time of nows.
[5] Further, if a line consists of points, point will be in contact with point. If, then, from K there be drawn the lines AB and CD, the point in the line AK and the point in the line KD will both be in contact with K. So that they will also be in contact with one another; for what is partless when in contact with what is partless is in contact whole-with-whole. So that the points will occupy the same place as K, and, being in [10] contact, will be in the same place with one another. But if they are in the same place with one another, they must also be in contact with one another; for things which are in the same primary7 place must be in contact. But, if this is so, one straight line will touch another straight line at two points. For the point in the line AK touches both the point KC8 and another. Hence the line AK touches the line CD at more points than one.
[15] And the same argument would apply not only where two lines were in contact but also if there had been any number of lines touching one another.
Further, the circumference of a circle will touch the tangent at more points than one. For both the point on the circumference and the point in the tangent touch the point of junction and also touch one another. But since this is not possible, neither is it possible for point to touch point. And if point cannot touch point, [20] neither can a line consist of points; for if it did,9 they would necessarily be in contact.
Moreover, how will there any longer be straight and curved lines? For the conjunction of the points in the straight line will not differ in any way from their conjunction in the curved line. For the contact of what is partless with what is partless is contact whole-with-whole, and no other mode of contact is possible. Since, then, the lines are different, but the conjunction of points is the same, clearly [25] a line will not depend on the conjunction: hence neither will a line consist of points.
Further, the points must either touch or not touch one another. Now if the next in a series must touch the preceding term, the same arguments will apply; but if there can be a next without its being in contact yet by the continuous we mean [30] nothing but a composite whose constituents are in contact. So that even so the points must be in contact, in so far as the line must be continuous.10
Again, if it is absurd for a point to be by a point, or a line by a point, or a plane [972a1] by a line, what they say is impossible.11 For if the points form a series, the line will be divided not at either of the points, but between them; whilst if they are in contact, a line will be the place of the single point. And this is impossible. [5]
Further, all things would be divided, i.e. be dissolved, into points; and the point would be a part of a solid, since a solid consists of planes, a plane of lines, and lines of points. And since those constituents, of which (as primary immanent factors) the various groups of things are composed, are elements, points would be elements of [10] bodies. Hence elements would be synonymous, and not specifically different.
It is clear, then, from the above arguments that a line does not consist of points.
But neither is it possible to subtract a point from a line. For, if a point can be subtracted, it can also be added. But if anything is added, that to which it was added [15] will be bigger than it was at first, if that which is added be such as to form one whole with it. Hence a line will be bigger than another line by a point. And this is impossible.
But though it is not possible to subtract a point as such from a line, one may subtract it incidentally, viz. in so far as a point is contained in the line which one is subtracting from another line. For since, if the whole be subtracted, its beginning [20] and its end are subtracted too; and since the beginning and the end of a line are points: then, if it be possible to subtract a line from a line, it will be possible also thereby to subtract a point. But such a subtraction of a point is accidental.
But if the limit touches that of which it is the limit (touches either it or some [25] one of its parts), and if the point quâ limit of the line, touches the line, then the line will be greater than another line by a point, and the point will consist of points. For there is nothing between two things in contact.
The same argument applies in the case of division, since the division is a point and, quâ dividing-point,12 is in contact with something. It applies also in the case of a solid and a plane. And the solid must consist of planes, the plane of lines. [30]
Neither is it true to say of a point that it is the smallest constituent of a line.
For if it be called the smallest of the things contained in the line, what is smallest is also smaller than those things of which it is the smallest. But in a line [972b1] there is contained nothing but points and lines: and a line is not bigger than a point, for neither is a plane bigger than a line. Hence a point will not be the smallest of the constituents in a line.
And if a point is commensurate with a line, yet, since the smallest involves [5] three degrees of comparison, the point will not be the smallest of the constituents of the line; and there are other things in the length besides points and lines; for it will not consist of points. But, since that which is in place is either a point or a length or a plane or a solid, or some compound of these; and since the constituents of a line are [10] in place (for the line is in place); and since neither a solid nor a plane, nor anything compounded of these, is contained in the line:—there can be absolutely nothing in the length except points and lines.
Further, since that which is called greater than that which is in place is a [15] length or a surface or a solid; then, since the point is in place, and since that which is contained in the length besides points and lines is none of the aforementioned:—the point cannot be the smallest of the constituents of a length.
Further, since the smallest of the things contained in a house is so called, without in the least comparing the house with it,13 and so in all other cases:—neither [20] will the smallest of the constituents in the line be determined by comparison with the line. Hence the term ‘smallest’ applied to the point will not be suitable.
Further, that which is not in the house is not the smallest of the constituents of the house, and so in all other cases. Hence, since14 a point can exist by itself, it will [25] not be true to say of it that it is the smallest thing in the line.
Again, a point is not an indivisible joint.
For a joint is always a limit of two things, but a point is a limit of one line. Moreover a point is a limit, but a joint is more of the nature of a division.
Again, a line and a plane will be joints; for they are analogous to the point. Again a joint is in a sense on account of movement (which explains the verse of Empedocles ‘a joint binds two’15); but a point is found also in immovable [30] things.16
Again, nobody has an infinity of joints in his body or his hand, but he has an infinity of points. Moreover, there is no joint of a stone, nor has it any; but it has points.