Are there indivisible lines? And, generally, is there a simple unit in every class of quanta?
§ 1. Some people maintain this thesis on the following grounds:—
(i) If we recognize the validity of the predicates ‘big’ and ‘great’, we must equally recognize the validity of their opposites, ‘little’ and ‘small’. Now that which admits practically an infinite number of divisions, is ‘big’ not ‘little’ ⟨or ‘great’ not ‘small’⟩. Hence, the ‘little’ quantum and the ‘small’ quantum will clearly admit only a finite number of divisions. But if the divisions are finite in number, there must be a simple magnitude. Hence in all classes of quanta there will be found a simple unit, since in all of them the predicates ‘little’ and ‘small’ apply.
(ii) Again, if there is an Idea of line, and if the Idea is first of the things called by 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 elements prior to each of them.
(iii) Again, if Body consists of elements, and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indivisible: for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought.
(iv) Again, Zeno’s argument proves that there must be simple magnitudes. For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any ‘stretch’ which is not simple, motion—unless there be simple magnitudes—involves that the moving body touches successively one-by-one an infinite number of points in a finite time: which is impossible.
But even if the body, which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the ‘stretch’ which it traverses in an equal time: and since the movement of 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, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible, there must be such a thing as an ‘indivisible line’.
(v) Again, the being of ‘indivisible lines’ (it is maintained) follows from the Mathematicians’ own statements. For if we accept their definition of ‘commensurate’ lines as those which are measured by the same unit of measurement, and if we suppose that all commensurate lines actually are being measured, there will be some actual 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 measuring both them and their whole. And thus the original unit of measurement would turn out to be twice one of its parts, viz. twice its half. 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 ‘simples’, so also the lines, which the unit measures once, consist of ‘simples’.
And the same can be shown to follow in the plane figures too. For all the squares, which are drawn on the rational lines, are commensurate with one another; and therefore ⟨by the preceding argument⟩ their unit of measurement will be simple.
But if ⟨per impossibile⟩ any such unit-square be cut along any prescribed and determinate line, that line will be neither ‘rational’ nor ‘irrational’, nor any of the recognized kinds of ⟨irrational⟩ lines which produce rational squares, such as the ‘apotome’ or the ‘line ex duobus nominibus’. Such lines, at which the unit-square might be divided, will have no nature of their own at all; though, relatively to one another, they will be rational or irrational.
§ 2. To these arguments we must make the following answers:—
(i) (a) In the first place, it does not follow that the quantum, which admits an infinite number of divisions, is not ‘small’ or ‘little’. For we apply the predicate ‘small’ to place and magnitude, and generally to the continuous (and in some quanta the predicate ‘little’ is suitably applied); and nevertheless we affirm that these quanta admit an infinite number of divisions.
(i) (b) Moreover, if in the composite magnitude there are contained ⟨indivisible⟩ lines, 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 point, and equally at any and every point: hence each of these indivisible lines would admit an infinite number of divisions just like the non-indivisible lines. Moreover, some amongst the non-indivisible lines are ‘small’. But every non-indivisible line admits of division in accordance with any prescribed ratio: and the ratios, ⟨in accordance with which any such line may be divided⟩, are infinite in number.
(i) (c) Again, since the ‘great’ is compounded of certain ‘smalls’, 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: i.e. its divisions are finite or infinite, according as their divisions are finite or infinite. 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.
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 the ‘little’ number 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 more complex are developed out of ‘simples’, and there is a determinate something from which the whole series of the numbers starts, and every number which is not infinite admits a finite number of divisions; but in magnitudes the case is not parallel.
(ii) As to those who try to establish the being of the 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 in a sense destroy the premisses which they use to prove their conclusion. For their arguments destroy the Ideas.
(iii) Again, as to the corporeal elements, it is childish to postulate them as ‘simple’. For even though some physicists do as a matter of fact make this statement about them, yet to assume this for the present inquiry is a petitio principii. Or rather, the more obviously the argument would appear to involve a petitio principii, the more the opinion is confirmed that Solids and Lengths are divisible in bulk and distance.
(iv) The argument of Zeno does not establish that the moving body comes into contact with the infinite number of points in a finite time, if the period and the path of the motion are considered on the same principle. For the time and the length are called ⟨both⟩ infinite and finite ⟨from different points of view⟩, and admit of the same divisions ⟨if considered both on the same principle⟩.
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 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 absurd—we may perhaps suggest to our opponents—that, because you are unable to solve Zeno’s argument, you should make yourselves slaves of your inability, and should commit yourselves to still greater errors, in the endeavour to support your incompetence.
(v) As to what they say about ‘commensurate lines’—that all lines, because commensurate, are measured by one and the same actual 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.
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 with 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 such a feeble piece of sophistry too! For it is feeble in many respects, and totally ⟨unable⟩ to escape paradox on the one side, and destructive scientific criticism on the other.
Moreover, it would be absurd for people to be led astray by Zeno’s argument, and to be persuaded—because they cannot refute it—to invent indivisible lines: and yet to pay no attention to all those theorems concerning lines, in which it is proved that it is impossible for a movement to be generated such that in it the moving thing does not fall successively on each of the intervening points before reaching the end-point. For the theorems in question are far better established, and more generally admitted, than the arguments of Zeno.
§ 3. 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:—
(A) 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 assumptions which we have no right to reject except on more convincing arguments than those adduced by the advocates of indivisible lines.
For (i) 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.
(ii) Secondly, all lines will be commensurate. For all lines—both those which are 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.
(iii) Again, since, in a rectangle, the line applied at right angles to the longer side determines the breadth of the figure: the rectangle, which is equal in area to the square on the indivisible line (v.g. on the line one foot long), will, if applied to a line double the indivisible line (v.g. to a line two feet long), 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.
(iv) Again, since any three given straight lines can be combined to form a triangle, a triangle can also be formed by combining three given indivisible lines. Such a triangle will be equilateral: but in every equilateral triangle the perpendicular dropped from the apex bisects the base. Hence, in the equilateral triangle whose sides are the indivisible lines, the ‘indivisible’ base will be bisected by the perpendicular dropped from its apex.
(v) Again, if the square can be constructed of Simples, then let its diagonal be drawn, and a perpendicular dropped from one angle on to the diagonal. The square on the side will be equal to the square on the perpendicular together with the square on half the diagonal. Hence the side of the square—i.e. the ‘indivisible’ line—will not be the smallest line.
Nor will the area, which is the square on the diagonal, be double the square on the indivisible line. For ⟨suppose it to be so: then,⟩ 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 ‘simple’ line. For if the remaining portion of the diagonal were ⟨not less than, but⟩ equal to the ‘simple’ line, 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.
(B) Then again ⟨the following arguments support our criticism of the doctrine⟩:—
(i) The Simple admits of only one mode of conjunction, but a line admits of two: for one line may be conjoined to another either by contact along the whole length of both lines, or by contact at either of its opposite terminal points.
(ii) Further, the addition of a line will not (on the theory) make the whole line any longer than the original line to which the addition was made: for Simples will not, by being added together, produce an increased total magnitude.
(iii) Further, every continuous quantum admits more divisions than one, and therefore no continuous quantum can be formed out of two Simples. And since every line (other than the indivisible line) is admittedly continuous, there can be no indivisible line: ⟨for if there were, a continuous quantum—viz. the line formed by the conjunction of two indivisible lines—would be formed out of two Simples.⟩
(iv) 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 consisting of an odd number of indivisible lines will admit of bisection, and this will involve the division of the ‘indivisible’ line.
And if not every line, but only lines consisting of an even number of units admit of bisection: still, even so, the ‘indivisible’ line will be divided, when the line consisting of an even number of units is divided into unequal parts (by progressive bisection).
(C) Again, ⟨the following arguments must be considered against the doctrine⟩:—
(i) 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 find the bisection of the ‘indivisible’ lines, since the body will traverse half the stretch in the half 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, nor will they admit of division corresponding to the division of the times, if there are to be ‘indivisible’ lines. And yet (as we said) the truth is, that the same argument, which leads to the view that lines consist of Simples, leads by logical necessity to the view that all these things (composite times, e.g., as well as composite lines) consist of Simples.
(ii) 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.
(iii) 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 out of the indivisible lines.
Moreover, if there are points in the indivisible line, there will either be nothing between the points, or a line. But if there is a line between them, and if all lines contain more points than one, the unit-line will not be indivisible.
(iv) 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—its length and its breadth—is a determinate something. But if the square is divisible, then so will be the line on which it is constructed.
(v) 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 ultimate thing be ‘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 it be urged that the limiting point is contained within the indivisible line, on the ground that two lines united so as to form a continuous line have one and the same limit at their juncture, then the simple 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 possess nothing characteristic to distinguish it from the point, except the name.
(vi) Again, if there be indivisible lines, there must, by parity of reasoning, be indivisible planes and solids too. For the being of an indivisible unit in one dimension will carry with it the being of indivisibles in the remaining dimensions too, since it is at a plane that a solid is divided, and at a line that a plane is divided. But there is no indivisible solid: for a solid contains depth and breadth. 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 the opinions ⟨which they are trying to establish⟩ conflict with all the most convincing arguments, it is clear that there can be no indivisible line.
§ 4. And it is further clear from the above considerations that a line can no more be composed of points than of indivisible lines. For the same arguments, or most of them, will apply equally against both views.
For (i) 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.
And (ii) it will follow that the part of a line is not a line, nor the part of a plane a plane.
Further (iii) 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 it is impossible for one line to be longer than another by a point, is clear both from what is proved in mathematics and from the following argument. For, if it were possible, the absurd consequence would result that the moving body would take a time to traverse the point. For, as it traverses the equal line in an equal time, it will traverse the longer line in a greater time: and that by which the greater time exceeds the equal time is itself a time.
Perhaps, however, we are to suppose that just as a line consists of points, so also time consists of ‘nows’, and both theses belong to the same way of thinking. ⟨Let us then examine the doctrine that a line, or generally continua, like times and lengths, consist of discrete elements.⟩
(a) 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 be continuous with one another.
(b) Again, a line is a magnitude: but the ‘composition’ 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 wise increased. And since points too are contained in the line thus superimposed, it follows that neither would points, by being superimposed on points, occupy more space. Hence points would not constitute a magnitude by composition.
(c) 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 other is not, they would not be in contact whole-with-whole.
But if the Simples ⟨when in contact⟩ are ⟨not ‘one’, but⟩ ‘coincident’, then a plurality occupies the same place which was formerly occupied by one: for if two things are coincident and neither admits of being extended beyond the coincidence, just so far the place occupied by both is the same. And since the Simple has no dimension, it follows that a continuous magnitude cannot be composed of Simples. Hence neither can a line consist of Points nor a time of Nows.
(d) Further, if the 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 B in the line A(B)K and the point C in the line K(C)D will both be in contact with K. So that the points B and C will also be in contact with one another: for the Simple, when in contact with the Simple, is in contact whole-with-whole. So that the points will occupy the same place as K, and, quâ in contact with K, 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 ‘continent’ place must be in contact. But, if this is so, one straight line will touch another straight line in two points. For the point (B) in the line AK touches both the point KC and another ⟨viz. the point contiguous to C in the line K(C)D⟩. Hence the line AK touches the line CD in more points than one.
And the same argument would apply not only in the case supposed, where two lines were in contact with one another at the point K, but also if there had been any number of lines touching one another at K.
(e) Further, if a line consist of points in contact with one another, 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, neither can the line consist of points: for if it did, they would necessarily be in contact.
(f) Moreover, how—on the supposition that the line consists of points—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 Simple with Simple is contact whole-with-whole, and Simples admit no other mode of contact. Since, then, the straight and curved lines are different, but the conjunction of points is invariably the same, clearly a line will not be curved or straight because of the conjunction: hence neither will a line consist of points.
(g) Further, the points (of which the line consists) must either touch or not touch one another. Now if ‘the next’ in a series must touch the preceding term, the same arguments, which were advanced above, will apply: but if there can be ‘a next’ without its being in contact ⟨with its predecessor or successor⟩, yet by ‘the continuous’ we mean nothing but a composite whose constituents are in contact. So that the points forming the line must be in contact, in so far as the line must be continuous, even though we suppose the points to be a ‘series’.
(h) † ἔτι εἰ ἄτοπον στιγμὴ ἐπὶ στιγμῆς, ἵν᾽ ᾖ γραμμὴ καὶ ἐπὶ στιγμῇ,, ἐπεὶ ἡ γραμμὴ ἐπίπεδον, ἀδύνατον τὰ εἰρημένα εἶναι. † For if the points form a series without contact, 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.
(j) Further, all things would be divided, i.e. be dissolved, into points; and the point would be a part of a solid, since the solid—on the theory—consists of planes, the plane of lines, and the lines of points. And since those constituents, of which (as their primary immanent factors) the various groups of things are composed, are ‘elements’, points would be ‘elements’ of bodies. Hence ‘elements’ would be identical in nature as well as in name, and not even specifically different.
§ 5. It is clear, then, from the above arguments that a line does not consist of points.
(a) 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 will be bigger than it was at first, if that which is added be such as to coalesce and 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 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 incidental or per accidens.
(b) But if the limit touches that of which it is the limit (touches either it or some 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, 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, just as (on the theory) the line consists of points.
(c) Neither is it true to say of a point that it is ‘the smallest constituent of a line’.
(i) 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 the line there is contained nothing but points and lines: and the line is not bigger than the point, for neither is the plane bigger than the line. Hence the point will not be the smallest of the constituents in the line.
(ii) And if the point is comparable in magnitude with the line, yet, since ‘the smallest’ involves three degrees of comparison, the point will not be the smallest of the constituents of the line: or there will be other things in the length besides the points and lines, so that 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 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.
(iii) Further, since that which is called ‘greater’ than that which is in place is a 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.
(iv) Further, since ‘the smallest of the things contained in a house’ is so called, without in the least comparing the house with it, and so in all other cases:—neither 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.
(v) 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, since the point can exist per se, it will not be true to say of it that it is ‘the smallest thing in the line’.
(d) Lastly, the point is not an ‘indivisible joint’.
For (i) the joint is always a limit of two things, but the point is a limit of one line as well as of two. Moreover (ii) the point is an end, but the joint is more of the nature of a division.
Again (iii) the line and the plane will be ‘joints’ ⟨too⟩: for they are analogous to the point. Again (iv) the joint is in a sense on account of movement (which explains the verse of Empedocles): but the point is found also in the immovable things.
(v) Again, nobody has an infinity of joints in his body or his hand, but he has an infinity of points. (vi) Moreover, there is no joint of a stone, nor has it any: but it has points.