Aristotle · a new plain-English translation from the original language
1| [section 1] Are there indivisible lines, and in general, in every quantity, is there something without parts, as some people say? For if the many and the large exist in the same way as their opposites, the few and the small, and what has virtually infinite divisions is not few but many, then it is clear that the few and the small will have finite divisions; and if the divisions are finite, there must be some magnitude without parts, so that in everything there will be something without parts, since indeed the few and the small exist. [section 2] Again, if there is a form of a line, and the form is prior among things that share a name, and the parts are prior in nature to the whole, then the line itself would be divisible, and in the same way the square and the triangle and the other figures, and the plane itself in general and the body; for it will follow that certain things are prior to these. [section 3] Again, if there are elements of body, and nothing is prior to the elements, while the parts are prior to the whole, then fire would be indivisible, and in general each of the elements of body, so that not only among the intelligible things but also among the perceptible things there is something without parts.
2| [section 4] Again, by Zeno's argument there must be some magnitude without parts, if indeed it is impossible in a finite time to touch things infinite in number, touching each one in turn, and yet the moving thing must first arrive at the half, and of what is not without parts there is always a half. [section 5] But if what is carried along the line does touch the infinitely many things in a finite time, and what is faster covers more in an equal time, and the motion of thought is fastest of all, then thought too would touch the infinitely many things one by one in a finite time; so that if touching each one in turn is what counting is, it is possible to count the infinite in a finite time. [section 6] But if this is impossible, there would be some indivisible line. Again, from what the mathematicians themselves say, there would be some indivisible line, as they claim, if lines measured by the same measure are commensurable; and all lines that are commensurable are measured by it. For there would be some length by which all are measured. And this must be indivisible. [section 7] For if it is divisible, its parts too will be a measure of something; for they are commensurable with the whole. So that some part would measure a half that is double, since this would be an impossible measure.
3| [section 8] Likewise the lines measured once by it, like all the lines compounded from the measure, are composed of parts without parts. And the same thing will result also in the case of plane figures; for all the squares on rational lines are commensurable with one another, so that the measure of them will be without parts. [section 9] But then, if some measure is to cut a certain ordered and determinate line, it will be neither rational nor irrational, nor any of the other kinds just spoken of, such as an apotome from two names; rather, in their own right they will have no natures at all, but relative to one another they will be rational and irrational. [section 10] But in the first place, it is not necessary that what has infinite divisions should not be small and few; for we call a place, and a magnitude, and in general the continuous, small, and we apply the term "few" to some of these, and yet we still say they have infinite divisions. Again, if lines are in the composite, the term "small" is applied in respect of these indivisibles, and infinitely many points are present within it. [section 11] Insofar as it is a line, division is by a point, and likewise, in whatever way you take it, every line that is not indivisible would have infinite divisions. Some of these ratios run to great length and are infinite. Every line that is not indivisible can be cut in the ratio assigned. Again, if the large is composed of certain small things, either the large will be nothing, or what has finite divisions will not be large.
4| [section 12] For the whole has the divisions of its parts in like manner. It is reasonable that the small should have finite divisions and the large infinite ones — this is what they claim. So it is clear that the large and the small should not be spoken of in these terms, as having finite and infinite divisions respectively. [section 13] And if because the few, even in numbers, has finite divisions, one should also claim the small in lines has finite divisions, this is foolish. For there, generation is from things without parts, and there is something which is the starting principle of numbers, and every number that is not infinite has finite divisions; but with magnitudes it is not the same. [section 14] Those who construct indivisibles among the forms take on, perhaps, the lesser assumption compared to the one previously laid down, namely positing forms of these things; and in a way they destroy these very things by the arguments through which they demonstrate them. For indeed, through these very arguments the forms are destroyed. Again, in the case of the bodily elements, it is foolish to claim they are without parts. [section 15] For even if certain people do in fact declare this, still, with regard to the inquiry now before us, they are assuming the very thing at issue from the start. And the more it might seem that the original point is being taken up again, the more it seems that body and length are divisible, both in their bulk and in their intervals.
5| [section 16] Zeno's argument does not, in this same way, establish that what is carried touches the infinitely many things in a finite time. For time too is spoken of as infinite and as finite, and a length has just as many divisions. [section 17] Nor indeed is thought's touching each of the infinitely many things, one by one, the same as counting them — even supposing one were to conceive of thought as touching the infinitely many things in this way. And this is perhaps impossible; for the motion of thought is not among continuous and underlying things, as is the motion of things carried along. [section 18] But even if it is possible for it to move in this way, this is not counting; for counting is proceeding with a halt at each stage. But it is perhaps absurd, for those unable to resolve the argument, to be enslaved to their own weakness, and to deceive themselves still further with greater deceptions, propping up their incapacity. [section 19] As for the claim about commensurable lines, that all of them are measured by one and the same thing, this is thoroughly sophistical and least of all in keeping with the hypothesis used in mathematics; for mathematicians do not lay it down in this way, nor is it useful to them. And at the same time it is self-contradictory to claim both that every line becomes commensurable, and that there is a common measure of all the commensurable lines.
6| [section 20] So it is ridiculous, when professing to demonstrate a point by appeal to those men's own opinions and from what they themselves say, to veer off instead into an eristic and sophistical argument, and one so weak besides. For it is weak in many ways, and it is possible in every way to escape both the paradoxes and the refutations. [section 21] Again, it would be absurd that some people should have been talked into making lines indivisible on account of Zeno's argument, merely because they had no answer to it, while on account of the straight line's motion into the one-and-a-half motion, which necessarily at once must be cut, since infinitely many arcs and intervals fall between, and again on account of the easily-credited fact about equal circles, that whatever moves must move through the greater semicircle, and however many other such things have been observed concerning lines showing that it is not possible for such a motion to occur so that it does not fall first on each of the intermediate points — for these are far more generally agreed upon than those other claims. [section 22] That, then, it is neither necessary nor plausible that there be indivisible lines, on the basis of the arguments stated, is clear. And it would become still clearer from the following. First, from the things demonstrated and laid down in mathematics, which it is not right to disturb with less trustworthy arguments.
7| [section 23] For neither the definition of a line nor that of a straight line will fit the indivisible, because it is neither between two things nor has a middle. Further, all lines will be commensurable. For all will be measured by the indivisibles, both those commensurable in length and those commensurable in square. [section 24] But the indivisibles are all commensurable in length, since they are equal; so also in square. And if this is so, the square will be divisible. Further, if the area applied to the greater line produces the width, then the area equal to the sum of the squares on the indivisible and on the foot-long line, applied to the two-foot line, will produce a width smaller than the partless one; the width on the indivisible will be smaller. [section 25] Further, if a triangle is constructed from three given straight lines, one will also be constructed from indivisibles. And in every equilateral triangle the perpendicular falls on the midpoint, so it will also fall on the indivisible. Further, if the square is constructed from the partless ones, with a line falling through the middle and a perpendicular drawn, the side of the square is equal in power to the perpendicular and to half the diagonal, so it is not the smallest. [section 26] Nor will the area on the diagonal be double that on the indivisible. For when an equal area is subtracted, the remainder will be less than the partless one. For perhaps the diagonal would in that case describe a quadruple area, and one could draw other such conclusions besides; for the theory is, so to speak, opposed to everything in mathematics.
8| [section 27] Again, contact of the partless is single, but of a line it is twofold: for a line touches another whole with whole, and also at its extremity, from opposite ends. Further, a line added to the whole does not make the whole greater; for partless things put together will not make anything greater. [section 28] Further, nothing continuous comes to be out of two partless things, because everything continuous has more than one division; and every line other than the indivisible is continuous, so an indivisible line would not be continuous. Further, if every line other than the indivisible is divided into both equal and unequal parts, and not only into three indivisibles or odd numbers generally, then the indivisible is undivided. [section 29] And likewise even if it is cut in half; for this holds of every line composed of an odd number. But if not every line is cut in half, but only the one composed of an even number, while the one divided in half can be cut into as many parts as possible, then in this way too the indivisible will be divided, whenever the line composed of an even number is divided into unequal parts. [section 30] Again, if what is moved will, in the time in which it moves the whole distance, move half of it in half the time, and in a lesser time less than half, then if the length is composed of an odd number of indivisibles, the middle division of the indivisibles will be eliminated, provided it traverses the half in half the time; for the time and the line will be divided in the same way.
9| [section 31] So none of the composite lines will be cut into equal and unequal parts, nor will they be divided in the same way as the times. There will be no indivisible lines. And it belongs to the same argument, as has been said, to make all these things out of partless units. Further, every line that is not infinite has two limits; for a line is defined by these. [section 32] But the indivisible is not infinite, so it will have a limit. It is therefore divisible; for the limit is one thing and that of which it is the limit another. Or else there will be some line, besides these, that is neither infinite nor finite. Further, there will not be a point in every line. For it will not be in the indivisible: for if there is only one point in it, then a line will exist, and then a point; but if there are more, the line is divisible. [section 33] So if a point does not exist in the indivisible, it will not exist in a line at all; for the other lines are composed of the indivisibles. Further, either none of the points will have anything between them, or a line will be between them; and if a line is between them, and there are more points in all the lines, that line will not be indivisible. Further, there will not be a square on every line; for it will have length and breadth, so it will be divisible, since it is one thing in one respect and another in another, [section 34] and if the square is divisible, so is the line. Further, the limit of the line will be a point, not a line. For the point is a limit, while the indivisible is only an extremity. For if it is a point, the limit of the indivisible will be a point, and one line will be greater than another line by a point. But if the point exists in the indivisible, because the limit of the connecting lines is the same, there will be some limit of the partless one,
10| [section 35] and in general, how will a point differ from a line? For the indivisible line will have nothing distinctive as against the point, except the name. Further, the same argument holds likewise for the plane, and a body too is indivisible on this view. For if one of these is indivisible, the others too will follow, because the one is divided in respect of the other. [section 36] A body will not be indivisible, because there is depth and breadth in it, nor could a line be indivisible: for a body is divided according to a plane, and a plane according to a line. Since, then, the arguments by which they attempt to persuade us are weak, and all the opinions opposed to those that carry conviction are false, it is clear that there could be no indivisible line. And it is clear from these arguments that a line could not be composed of points either. [section 37] For pretty much the same arguments, in most cases, will apply. For the point must necessarily be divided, whenever equal parts are cut from an odd number, or unequal parts from an even number. And it follows that a part of a line is not a line, nor a part of a plane a plane. [section 38] And a line will be greater than another line by a point; for it will exceed it by the things it is composed of. That this is impossible is clear both from the results in mathematics, and further it will follow that what is carried traverses the point in time, if indeed it traverses the greater line in more time, and the equal line in an equal time, and the excess of time is itself time.
11| [section 39] But perhaps time too is composed of nows, and it belongs to the same argument to speak of both. If, then, the now is the beginning and end of time, and the line of the point, but the beginning and the end are not continuous but have something between them, then neither the nows nor the points would be continuous with one another. [section 40] Further, the line is a certain magnitude, but the composition of points produces no magnitude, because it does not occupy a greater place either. For when a line is placed upon a line and coincides with it, the breadth does not become any greater. And points too exist in the line, yet the points would not occupy a greater place, so they would not produce a magnitude. [section 41] Further, if everything touches everything either whole with whole, or part with part, or whole with part, and the point is partless, it would touch as a whole. And what touches whole with whole must necessarily be one. For if something remains outside, or the other does not exist in the same way, it would not touch whole with whole. [section 42] But if the partless things are together, then several of them occupy the same place that the one alone did before; for of things that are together and have no extension in the same respects, the place of both is the same, and the partless has no dimension, so a continuous magnitude could not be composed of partless things. Therefore the line is not composed of points, nor is time composed of nows,
12| [section 43] Further, if a line is composed of points, point will touch point. So if AB and CD are drawn out from K, the point in KD will also touch K. So it will touch some other point too: for the partless touches the partless, whole to whole. So it will occupy the same place as K, and points that touch will be in the same place as one another. [section 44] And if they are in the same place, they also touch: for things that are primarily in the same place must touch; and then in this way a straight line will touch a straight line at two points. For the point in AK, and the point in KC, each touches a further point. So the line from CD touches at several points. [section 45] The same account holds even if a line touched another not through these points but in some other way whatever. Further, the line of a circle will also touch a straight line at several points. For at the point of contact, the point in the circle and the point in the straight line touch each other too. [section 46] But if this is not possible, then neither is it possible for point to touch point; and if they do not touch, then the line will not be composed of points either — for it is not necessary that they touch. Further, how then will there be a straight line and a curved line at all? For the joining of points will make no difference between the straight line and the curved one.
13| [section 47] For the partless touches the partless whole to whole, and it is not possible to touch in any other way. So if the lines are different but the joining is undifferentiated, then the line will not come from the joining, and so not from points either. Further, the points must either touch one another or not touch one another. [section 48] Now if what is next in succession must touch, the same account will hold; but if it is possible for something to be next in succession without touching, then note that we call nothing continuous except what is composed of things that touch. So on this reasoning too the points must touch one another, if the line is to be continuous. [section 49] Further, if it is absurd for point to be upon point, so that there should be a line even upon a point — since a line would then be a plane — the things stated are impossible. For either the points are next in succession, in which case the line will be cut at neither of the points but in between them, or they touch, in which case there will be a line that is the place of a single point. But this is impossible. [section 50] Further, everything would be divisible and would be resolved into points, and the point would be a part of body — if indeed body is composed of planes, the plane of lines, and lines of points. And if each thing consists of the primary constituents present in it, and these are elements, then points would be the elements of bodies. So neither set of elements would be synonymous in species with what they compose.
14| [section 51] It is clear, then, from what has been said, that a line is not composed of points. But neither is it possible for a point to be taken away from a line. For if it can be taken away, it can also be added; and when something is added, what results from the addition will be greater than what was there at the start, provided that what is added is of such a kind as to make one whole together with it. [section 52] So one line will be greater than another line by a point. But this is impossible. In its own right, then, this is not possible; but incidentally it is possible to take a point away from a line, since it is present in the line that is taken away. For if, when the whole is taken away, both the starting-point and the limit are taken away with it, and the starting-point and limit of a line was a point, then it is possible to take away a line, and a point could be taken away along with it. [section 53] But this taking-away is incidental. And if the limit touches, what it touches is not the limit either of itself or of anything belonging to it. But the point, insofar as it is the limit of a line, does touch. So insofar as it does, a point will be greater than a line, and the point will be composed of points; for of things that touch, nothing lies between them. [section 54] The same account holds also in the case of a section: if the section is of a point, and the section touches something, this holds also for solid and plane; and likewise the solid is composed of planes and lines. But it is not true to speak in terms of a point, nor is it true that the smallest of the things composing a line has thereby been said to be the smallest of the things present in it. For the smallest, of the things of which it is the smallest, is also less than them.
15| [section 55] In a line nothing else is present except point-like things and lines. But a line is not greater than a point in this way; for neither again is a plane greater than a line in this way. So the point will not be the smallest thing in the line. And if the point is comparable with the line, while "the smallest" applies in three respects, the point will not be the smallest of the things in the line. [section 56] And other things too are present in length besides points and lines; for it is not composed of points. But if the point, being in place, were a length or a plane or a solid or something composed of these, then — since the things of which the line is composed are in place, for so too is the line — and since neither body nor plane nor anything composed of these is present in the line, there will be nothing at all in length besides points and lines. [section 57] Further, if, of what is in place, the greater is called length, surface, or solid, while the point is in place, and what is present in length besides points and lines is none of the things aforesaid, then the point will not be the smallest of the things present in it.
16| [section 58] Further, that which is the smallest of the things in a house is not so called by being set against the house itself; and likewise in the other cases too. So neither will the smallest thing in a line be so called as compared with the line. So "the smallest" will not fit, since what is not in the house is not the smallest of the things in the house. And likewise in the other cases too. [section 59] For it is possible for a point to exist by itself, independently. Of such a point it will not be true to say that it is the smallest thing in a line, because the point is not an indivisible joint. For a joint is always the boundary of two things, whereas a point is the boundary even of a single line. Further, the one, the point, is a limit, while the other, the joint, is rather a division. [section 60] Further, on this reasoning the line and the plane would be joints; for they stand in an analogous relation, since a joint is in some way a differentiated thing — which is why Empedocles too composed the verse, wherefore it must rightly — but the point belongs also among the things that are unmoved. Further, no one has infinitely many joints in his body or in his hand, but he does have infinitely many points. Further, a stone has no joint and does not possess one, but it does have points.