Abstract
Spurious mathematical treatise that examines and largely refutes the arguments (including one attributed to Zeno) for the existence of indivisible lines as minimal units of magnitude, defending the infinite divisibility of the continuum.
Traduzione di W. S. Hett · Delphi Classics, 2013 · pubblico dominio
THIS is a most interesting and extremely difficult treatise, written by some author of the Peripatetic School. It refers directly to Euclid’s Elementa, Book X., and is unintelligible without some understanding of Euclid’s definitions. Unfortunately the condition of the manuscripts is most unsatisfactory. By kind permission of Messrs. Teubner, Apelt’s text has been used for this volume. This together with his comments in the Introduction has elucidated a number of difficulties, but, even so the thought as well as the terminology is involved. The treatise is mainly concerned with a refutation of the theory that every line contains a unit which is an indivisible line. Without the modern view of infinity, there is much which is mathematically brilliant, and on his own terms the author seems to prove his case. The main argument is a syllogism:
All lines consist of indivisible lines (Zeno).
All indivisible lines are points.
.-. all lines consist of points.
Aristotle then demonstrates the absurdity of this conclusion, thus demolishing the major premiss.
Are there such things as indivisible lines, and must there be in all magnitudes some unit which has no parts, as some say?
If “much” and “big,” and their opposites “few” and “little,” are similarly constituted, and if that which has almost infinite divisions is not small, but big, it is evident that “few” and “little” will have a limited number of divisions; if, then, the divisions are limited, there must be some magnitude which has no parts, so that in all magnitudes there will be some indivisible unit, since in all of them there is a “few” and a “little.”
Moreover, if there is an idea of a line, and the Idea is the first of quantities so called, and if the parts are logically prior to the whole, this unit line must be indivisible, and the same argument will apply to the square, triangle, and other figures, and generally speaking to a plane figure or to any other body; for there must be some unit prior in their case too.
Again, if there are elements in a body, and there is nothing prior to the elements, and if the parts are prior to the whole, fire and, generally speaking, each of the elements of the body would be indivisible, so that there must be a unit without parts, not only in the world of thought, but also in the world of perception.
Again, according to the argument of Zeno, there must be some magnitude without parts, since it is impossible to touch an infinite number of things in a finite time, when touching each of them, and that which moves must first reach half-way, and half clearly belongs to that which is not without parts. But if anything travelling along a line touches an infinite series in a finite time, secondly if the faster it travels the greater the space it covers in the same time, and lastly if the movement of thought is the quickest movement, then even thought must touch an infinite series one by one in a finite time. If, then, thought touching the series one by one is counting, then it must be possible to count an infinite series in finite time. If this is impossible, then there must exist an indivisible line.
The next argument, we are told, is used by the mathematicians to prove that the indivisible line must exist, if we admit that “commensurate” lines are those which are measured by the same unit, and all the lines measured are “commensurate.” For there must be some length by which they are all measured. And this must be incapable of division. For if it is divisible, then its parts can also be expressed in the terms of some unit. For they are commensurate with the whole. So that the measurement of each part would be double its half; since this is impossible the unit of measurement must itself be indivisible.
Again, just as the lines built up from the unit of measurement are all composed of units without parts, so also must those be which are once measured by it. The same thing will also happen in plane figures; for all the squares on rational lines are commensurable with each other, so that their unit of measurement will also be without parts. Moreover if any one of them is cut (on any unit) by a fixed and finite line, this line will neither be rational nor irrational, nor will belong to any of the categories to which the rational functions belong, such as “apotome” or “of two terms”; but in themselves they have no natural characteristics, though they will be rational or irrational in relation to each other.
Now in the first place it does not follow that what admits of infinite division is not either “small” or “little”; for we can apply the term “small” to space, and size, and generally to anything which is continuous, and in a similar way we apply the term “little” where it is applicable, not but what we admit that they have infinite divisions.
Secondly, if among commensurables there are lines, we can apply the term “small” to these indivisible units, and they themselves contain an infinite number of points. But in so far as it is a line it admits of division at a point, and similarly at any other point; consequently every line which is not indivisible must have an infinite number of divisions.
Now some of these divisions are small; and possible ratios between the divisions are infinite. It is possible for every line which is not indivisible to be cut in accordance with any given ratio.
Moreover, if “great” is compounded of a number of “smalls,” “great” either has no meaning at all, or “great” will be that which has finite divisions. For the whole must be susceptible of the same divisions as its parts. But it is illogical to suppose that the small has finite divisions and the great infinite; yet this is what they claim.
So it is clear that the terms “great” and “small” are not applied because the one has finite, and the other infinite, divisions. Again, if any man claims that because in numbers the “little” has finite divisions, the “small” in lines must do the same, his argument is foolish. For in the case of numbers a the whole is built from units which have no parts, and there is some unit which is the basis of all numbers, and every number which is not infinite has finite divisions; but the same thing is not true of magnitudes.
But those who build up their theory of indivisible lines on Ideas have, I fancy, too slight a basis for the superstructure, the supposition that there are Ideas of these indivisible lines; and in a certain sense they destroy their own argument by their demonstration. For the whole theory of Ideas is destroyed by their arguments.
Again, in the case of bodily elements it is foolish to maintain that they are without parts. For, if any do actually demonstrate this, they are for the purpose of the argument under discussion assuming the major premiss of the argument. And the more this major premiss is assumed, the more does it appear that the body and length are divisible both in two dimensions and in one.
Again, Zeno’s reasoning does not prove that what moves along a line touches an infinite series in finite time on this same plan. For “time” and “length” must be called both infinite and finite, and admit of the same divisions.
Again, the process of the mind touching an infinite series one by one is not the process of counting, if indeed anyone supposes that the mind does in this way touch an infinite series. Perhaps this supposition is in itself impossible; for the movement of the mind does not take place like the movement of travelling bodies in continuous matter.
But to resume — even if its movement can be of this kind, this is not counting. For counting involves a series of pauses. But it is perhaps quite unreasonable that those who have failed to solve the riddle should be subservient to their own weakness, and should cheat themselves still more in an effort to reinforce their incapacity.
As for the argument about commensurate lines, namely that all lines are measured by one and the same unit of measurement, this is merely chopping logic, and does not agree with mathematical assumptions; for the mathematician does not lay this down, and it would be of no use to him if he did. In fact the two statements are actually contradictory — that all lines are commensurable, and that there is a common measure of all commensurable lines.
So their position is absurd; after professing that they are going to demonstrate the mathematicians’ own opinions, and to argue from their statements, they merely relapse into a contentious and casuistical argument, and a weak one at that. For it is weak from many points of view, and in every way fails to escape both contradictoriness and refutation.
Moreover it is unreasonable for them to be led astray on the one hand by the reasoning of Zeno, and presume the existence of indivisible lines merely because they cannot disprove their existence; and on the other to be unimpressed by the arguments both from the movement of a straight line in a semicircle, which must clearly touch all the infinite points of the circumference and its divisions, and again to neglect the convincing fact about a circle that there must be movement of some such kind, if the radius moves in a semicircle,” and all the other theorems demonstrated about lines showing that movement is impossible of such a kind that it does not fall upon all the intervening points in turn; for these theorems are far more universally admitted than the others.
It is, then, clear from the arguments we have adduced that it is not inevitable nor even plausible that indivisible lines should exist. But from what follows it will become still more obvious. First of all from theorems demonstrated and laid down as axiomatic in mathematics, which must either be accepted or removed by more convincing arguments.
For neither the definition of “line” nor of “straight line” will fit in with the “indivisible line,” because it does not lie between points nor has it a middle point.
Secondly all lines will be commensurate on the assumption of indivisible lines. For all lines will be measured by indivisible lines, both those which are commensurable in length and in their squares. But indivisible lines are commensurate in length; for they are all equal; so they must also be commensurate in their squares. If this is true, then every square will be rational.
Again, seeing that the line applied to the longer side determines the breadth of a rectangle, the rectangle which is equal in area to the square on the indivisible line (suppose it to be one foot long) will, when applied to a line twice the length, have a breadth shorter than the indivisible line (which is a priori impossible); for its breadth will be less than that of the square on the indivisible line. (See note a (1).)
Again, since a triangle can be made from three given straight lines, it will also be made from three indivisible lines. Now in every equilateral triangle the perpendicular from any angle bisects the base and so must divide the indivisible line. (See note a (2).)
(2) — ABC is an equilateral triangle, and AD the perpendicular dropped on BC from A. This figure produces exactly the same impossibility as the last.
(3) — ABCD is a square, of which AC is the diagonal. A perpendicular is dropped from D to the diagonal. Here again we have the same impossibility.
Again, if a square can be made of indivisible lines, then when a diagonal is drawn and a perpendicular dropped on it from an angle, the side of the square will equal the perpendicular plus half the diagonal, so that it will not be the smallest line. (See note a (3).)
Nor will the area which is the square on the diagonal be double the square on the indivisible line. For when the equal part is taken away, the remainder will be less than the indivisible line; but if it were equal, then the square on the diagonal would be four times that on the original square; one could of course collect other examples; for they are opposed practically to all mathematical principles.
Again, there is only one way of joining what has no parts to anything else, but two ways in the case of a line; for two lines may be joined lengthways, or on the other hand, end to end.
Again, a line fitted to another side by side will not make the whole any greater; for lines without parts when put together will not make them any longer.
Again, no continuous length can be made out of two lines without parts, for every continuous length can be divided into more than one part, and if every line is continuous in contrast with an indivisible line, then there can be no such thing as an indivisible line.
Again, if in contrast with the indivisible line every line can be divided into equal and unequal parts, even if it is constructed out of three indivisible lines or generally speaking out of any odd number, the indivisible line will be capable of division. Equally so every line can be cut in half; for every line made up of odd numbers will involve bisection of the indivisible line. But if no such lines can be bisected, unless they are composed of an even number of lines, even in this case it must be possible to divide a bisected line any number of times, and thus the indivisible line will be divided, whenever the line composed of an even number of parts is divided into unequal parts.
Again, if the moving object moves over half the line in half the time it takes to move over the whole line, it also moves over less than half in less than half the time, so that if the whole length is composed of an odd number of indivisible lines the bisection of indivisible lines will be seen again, if it covers half the length in half the time; for the time and the line will be divided in proportionate divisions. So that none of the component lines will admit of equal and unequal divisions; if they are divided proportionately to the time, they will not be indivisible lines. And yet, as has been said, constructing all these things from lines without parts belongs to the same argument.
Again, everything which is not unlimited has two limits; for by these the line is defined. But the indivisible line is not unlimited, and so will possess a limit. Therefore it is divisible: for the limit is not the same as that of which it is the limit. Or else there will be a line which is neither unlimited nor limited, beyond these two categories.
Again, there will not be a point in every line; for there will be no point in the indivisible line; for if there were one and one only, a line would be a point; if there are more than one, then the line is divisible.
But if there is no point in the indivisible line, then there is not generally in any line; for the other lines are made up of indivisible lines.
Again, (if such points exist in a line) there will be either nothing between them, or a line; if there is a line between, and more than one point in all lines, then the line will not be indivisible.
Again, it will not be possible to construct a square on every line; for a square will have length and breadth, so that it is divisible, since both its length and its breadth are quantities. But if the square is divisible, so also is the line upon which it is constructed.
Again, the limit of a line will be a line, and not a point. For the limit is the ultimate thing, and the indivisible line is ultimate. For if a point is the limit, the point will be the limit of an indivisible line, and a line will then be greater than another line by a point. But if the limiting point is within the indivisible line, because two connected lines have the same limit, there will be a limit to the line without parts. Generally speaking, then, what will be the difference between a point and a line? For in comparison with the point the indivisible line will have no property peculiar to it except the name.
Again, in the same sense, the plane figure and the solid will be indivisible. For if the one is indivisible, it will follow that the others are so, for the one is divided by means of the other. But the solid is not indivisible because it contains both depth and breadth; then a line cannot be indivisible; for a solid is formed by the addition of a line to a plane surface, and a plane surface by the addition of a line to a line.
But since the arguments by which they attempt to prove their case, are not only feeble but even false, and their opinions are opposed to all those which carry conviction, it is evident that there cannot be an indivisible line. For nearly all the same arguments will apply.
For instance, it must be possible to divide the point, when a line consisting of an odd number of points is divided into equal parts, or one consisting of an even number of points into unequal parts; also, the part of a line would not be a line, nor the part of a plane figure a plane figure.
Also, one line would have to be greater than another by a point; and it will then be greater than the elements out of which it is composed. That this is impossible is obvious from the principles of mathematics, and a further consequence will be that a travelling object will pass over a point in a definite time, since it travels over a greater distance in a longer time, and an equal distance in an equal time, but the excess of one time over another is in itself a time.
But perhaps time consists of a succession of “nows,” and both ideas belong to the same theory.
But if a “now” is the beginning and the limit of time, and a point is in a similar relation to a line, the beginning and the end cannot be in themselves continuous, but there must be something in between, so that neither the “nows” (in time), nor the points (in a line) could by themselves form a continuous whole.
Again, the line is a certain magnitude, but an aggregation of points produces no magnitude, because such an aggregation fills no greater space. For when a line is added to a line and fitted on to it, the width does not increase.” If, then, points constitute lines, the points, however many, would occupy no larger space, so that they could not produce a magnitude.
Again, if they all touched every point, whether the whole was in contact with the whole, or a part with a part, or the whole with a part, and since the point is indivisible, the contact would be the whole with the whole. But the whole in contact with the whole must produce a unit. For if anything belongs to one which does not belong to the other, then the whole is not in contact with the whole. But if the indivisible parts are all in one place, then a number of things occupy the same space which was formerly occupied by a unit; for in the case of two things, which are together and yet have no power of extension, the same space must serve for both. But since what has no parts cannot have dimensions, nothing composed of units without parts can produce a continuous magnitude. Hence it follows that a line cannot be made out of a series of points, nor a time out of a series of “nows.”
Moreover, if a line were composed of points, a point would be in contact with a point. Suppose that from K two lines AB and TA are drawn, both the point which terminates ΔK and the point which terminates KΔ will meet in K, so that the two points will be in contact A with each other; for the indivisible touches the indivisible, as a whole touches a whole. So that it will occupy the same space as K, and the points will be in contact with each other in the same place. Conversely, if they are in the same place, they must be in contact; for in the first place things which are in the same space must touch, and, if this is so, the straight line touches a straight line in two points. For the point in AK touches both the points in KΓ, and also another (i e., the next point in AΓ which occupies the same place as K). So that AK touches TA in more points than one. And the same argument applies not merely to two lines in contact but to any number.
Again, the circumference of a circle would touch the tangent in more points than one. For both the point on the circumference and the point on the tangent are touching the point of contact, and each other. If this is impossible, then a point cannot touch a point; but if it cannot, then a line cannot consist of points; for otherwise it would be in contact.
Again, how will it affect the question of straight lines and curves? For there can be no difference between the contact of points in the straight and in the curved line. For the line without parts touches a similar line over all its length, and cannot touch it in any other way. If, then, there are lines of different kinds and no different kind of contact, a line will not depend on the elements of its construction, and so does not depend on points.
Again, the points must either be in contact with each other or not. If they are in contact in series, the argument is the same; if it is possible for the series to be continuous without contact, still by continuous we mean nothing except something whose component parts are in contact, so that on this supposition also the points must touch each other, or eke the line cannot be described as continuous.
Moreover, if it is absurd to put a point on a point to produce a line, and a line on a point to produce a plane surface, what they say cannot be true. For if either of the points is continuous then the line will not be cut at either of the points, but in between them; if, on the other hand, they touch, the line will be in the place of one point, and this is impossible.
Moreover, all geometrical figures could be divided and resolved into points, and a point would be part of a solid (i e., have three dimensions), since the solid is constructed out of the plane figure, the plane figure from lines, and lines from points. But if each thing consists of its original elements, then points would be the elements of solid bodies. So that elements would have the same name, and be no different in kind.
So it is clear from what we have already said that the line is not composed of points. Nor can the point be detached from the line. For if it can be so detached, it can also be added. But, when anything is added, that to which it is added will be greater than it was at the beginning, if the addition is of such a kind as to make a complete unit. Then one line will be greater than another by a point. But this is impossible. It is impossible, that is to say, in itself, but incidentally it is possible to take a point away from a line, by the fact of its existence in the line taken away. For if the whole is taken away, the beginning and end must be taken away, and the beginning and end of a line is a point. If, then, it is possible to take away a line from a line, it must be possible to take away a point. But this taking away of a point is only incidental. But if the end touches that of which it is the end — that is either touches itself, or any part of it — and the point also touches it in virtue of its being the end of a line, — then one line is greater than another by a point, and a point will consist of points; for there can be nothing in between two things which touch.
The same argument will apply to division, if division is of a point and if division touches something, both in the solid and plane figure; just in the same way the solid is made up of plane figures, and the plane figure of lines.
Nor, again, is it true to say of a point that it is the smallest component of a line.
For if it is the smallest component of a line, the “smallest” must be smaller than those things of which it is the smallest, but in the line there is nothing but points and lines, and a line is not greater than a point (any more than a plane figure can be called greater than a line), so that the smallest component of a line will not be a point.
Even if the point could be compared to the line, the word “smallest” can only be used of three terms, so that the point could not be the smallest component of a line. Also, there must be a third element in length beyond points and lines; for it is not composed of points. But if everything in space is either a point or a length or a plane figure or a solid, or is composed of these, and if the components of a line are in space (for a line is), and if there is neither a solid nor a plane figure nor any such thing in a line, there will be nothing in a given length besides points and lines.
Further, the term greater can only be applied to the following things in space — a length, a surface, or a solid, and a point is in space, but that which is in a length, besides points and lines, is none of the foregoing, so that the point cannot be the smallest component of a line.
Again, since the phrase “the smallest of the things in the house” is used without any reference to the size of the house, so also in other cases, nor will the smallest thing in a line have any reference to the line, so that the phrase smallest does not apply to the line.
Further, if that which is not” in the house cannot be the smallest of the things in the house, just in the same way in other cases (for a point can exist by itself) it will not be true to say of the point that it is the smallest thing in the line.
Again, the point is not an indivisible joint; for the joint is the limit of two things, but the point is the limit of a single line. Again, the point is an end, but the joint is more a division. Again, the line and plane figure are joints; for they have some analogy with it. Again, the joint is in a sense connected with movement, wherefore Empedocles wrote the line “A joint binds two things”; but a point is among the immovable things. Again, no one has an infinite number of joints in the body, or in the hand, but they have an infinite number of points. Again, there can be no joints in a stone, nor has it any, but it has points.