The Politics of Aristotle

by Aristotle

  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.

  **TEXT: M. Timpanaro Cardini, Milan, 1970

  1Reading ὅσαι δ’ εἰσὶ σύμμετρoι, πᾶσαί εἰσι μετρoύμεναι.

  2Reading ἐν τῷ συνθέτῳ ἄτoμoί εἰσι γραμμαί.

  3Reading κατὰ τὰς ἐκείνων δόξας … φάσκoντας.

  4Reading ἀνευρεθήσεται.

  5Reading ᾗ θάτερoν.

  6Retaining ἐπέκτασιν, κατὰ ταῦτα.

  7Reading πρώτῳ.

  8Reading καὶ τῆς KΓ.

  9Reading oὕτω γὰρ.

  10Reading ᾗ εἶναι γραμμἠν συνεχῆ.

  11Reading ἔτι εἰ ἄτoπoν στιγμὴν ἐπὶ στιγμῆς εἶναι ἢ γραμμὴν ἐπὶ στιγμῆς ἢ ἐπὶ γραμμῆς ἐπίπεδoν. But the text is quite uncertain, even by the standards of the rest of the treatise.

  12Reading ᾗ τoμή.

  13Omitting Apelt’s πρὀς τἠν oἰκἱαν συμβἁλλεται μἡτε, and reading μἡ τι τῆς for μἡτε τῆς.

  14Reading ἐνδέχεται δέ.

  15Reading δύω δέει ἄρθρoν: frag. 32 Diels-Kranz.

  16Omitting τό.

  THE SITUATIONS AND NAMES OF WINDS**

  E. S. Forster

  Boreas. At Mallus this wind is called Pagreus; for it blows from the high cliffs [973a1] and two parallel ranges known as the Pagrean Mountains. At Caunus it is called Meses; in Rhodes it is known as Caunias, for it blows from Caunus, causing storms [5] in the harbour of that place. At Olbia, near Magydum in Pamphylia, it is called Idyreus; for it blows from an island called Idyris. Some people identify Boreas and Meses, amongst them the Lyrnatians near Phaselis.

  Caecias. In Lesbos this wind is called Thebanas; for it blows from the plain of [10] Thebe, north of the Elaitic Gulf in Mysia. It causes storms in the harbour of Mitylene and very violent storms in the harbour of Mallus. In some places it is called Caunias, which others identify with Boreas.

  Apeliotes. This wind is called Potameus at Tripolis in Phoenicia; it blows from a plain resembling a great threshing-floor, which lies between the mountains of Libanus and Bapyrus; hence it is called Potameus. It causes storms at Posidonium. [15] In the Gulf of Issus
and the neighbourhood of Rosus it is known as Syriandus; it blows from ‘the Syrian Gates’, the pass between the Taurus and the Rosian Mountains. In the Gulf of Tripolis it is called Marseus, from the village of Marsus. [20] In Proconnesus, Teos, Crete, Euboea, and Cyrene it is known as Hellespontias. It causes storms in particular at Caphereus in Euboea, and in the harbour of Cyrene, which is called Apollonia. It blows from the Hellespont. At Sinope it is called Berecyntias, because it blows from the direction of Phrygia. In Sicily it is known as [973b1] Cataporthmias, because it blows from the Straits. Some people identify it with Gaecias, and also call it Thebanas.

  Eurus. This wind is called Scopeleus at Aegae, on the borders of Syria, after the cliff at Rosus. In Cyrene it is known as Carbas after the Carbanians in [5] Phoenicia; hence some people call this same wind Phoenicias. Some people identify it with Apeliotes.

  Orthonotus. Some call this wind Eurus, others Amneus.

  Notus bears the same name everywhere. It is derived from the fact that this wind is unwholesome, while out of doors it brings showers; thus there are two reasons for its name. [10]

  Leuconotus likewise derives its name from its effect; for it clears the sky.

  Lips. This wind gets its name from Libya, whence it blows.

  Zephyrus. This wind is so named because it blows from the west, and the west. . . .

  Iapyx. At Tarentum it is called Scylletinus from the place Scylletium. At [15] Dorylaeum in Phrygia.. . . Some people call it Pharangites, because it blows from a certain ravine in Mount Pangaeus. Many call it Argestes.

  Thracias is called Strymonias in Thrace, for it blows from the river Strymon; [20] in the Megarid it is known as Sciron, after the Scironian cliffs; in Italy and Sicily it is called Circias, because it blows from Circaeum. In Euboea and Lesbos it goes by the name of Olympias, which is derived from Pierian Olympus; it causes storms at Pyrrha.