I aim to make these inadequacies clear , and then to add some brief speculations on why they are present .sx There is one significant feature of Aristotle's whole discussion in Book VI , which I would not regard as an 'inadequacy' , but which deserves a mention here at the outset .sx That is the tension between Aristotle's reasoning in this book and his discussion of infinity in Book III .sx To put it very roughly , his position in Book III is that there is no 'actual' or 'completed' infinite , and all infinity is merely 'potential' .sx But in Book VI he shows no such tendency to be suspicious of the infinite , and apparently accepts without qualms that a line does ( 'actually' ) contain infinitely many points , that a stretch of time does ( 'actually' ) contain infinitely many instants , and so on .sx Thus his answer in this book to Zeno's most famous paradox on motion apparently accepts that one who traverses a finite distance has thereby completed a series of infinitely many distinct tasks , in traversing the infinitely many 'half - distances' contained within the original distance .sx This is possible , he here says , because the finite time available may equally be divided into a corresponding and infinite series of 'half-times' ( 233 a 13-31 , 239 b 9-29) .sx Later in Book VIII he will reconsider this paradox , and with the doctrine of Book III in mind he will deny that a moving body ever does pass over infinitely many 'actual' points ( 263 a 4- b 9) .sx But here in Book VI we find no such complications .sx A simple hypothesis evidently suggests itself :sx Book VI was written before the doctrine of Book III was worked out .sx ( In my concluding section I shall note some other pointers to the early date of Book VI .sx ) .sx I The Definition of a Continuum .sx In chapter 3 of Book V Aristotle defines what it is for one thing to be continuous with another .sx He begins by defining 'succession' ( t o-grave veFexes ) :sx one thing succeeds another when it comes after it , in position or in some other way , and there is nothing between them that is of the same kind ( 226 b 34-227 a 6) .sx A special case of succession is 'being next to' ( t o-grave vech o menon ) :sx one thing is next to another when it succeeds it and is also in contact with it ( 227 a 6 ) , contact being defined as occurring when ( some part of ) the 'limits' of the two things are in exactly the same place ( 226 b 21-3) .sx A special case of being next to is 'continuity' ( t o-grave sunech e s ) :sx one thing is continuous with another when they are next to one another and in addition the limits at which they touch are the same thing , or 'have become one' , being indeed 'held together' as the word sun-ech e s implies .sx Aristotle's thought evidently is that things which are merely next to one another need not hold together - one may move one of them leaving the other where it is - whereas if they are continuous then they move together ( 227 a 10-17) .sx At the opening of Book VI Aristotle recalls this definition , and appears to suppose that it has also explained to us what it is for a single thing to be a continuous thing - or , as I shall say , a continuum - though clearly it has not explained this at all ( 231 a 21-6) .sx But if we follow through the argument that he at once plunges into , we can I think reconstruct the definition that he has failed to state but must have had in mind .sx For he claims that it follows from the definition that a continuum ( such as a line ) cannot be made up from indivisible things ( such as points) .sx The first argument is simply that an indivisible thing has no limits , for a limit ( v e-grave schaton ) must be different from what it limits , and thus what has limits must have parts , and must therefore be divisible ( 231 a 25-9) .sx As a second argument Aristotle adds that indivisible things such as points cannot form a continuum by being in contact with one another , even if we waive the point about their lack of limits .sx For , he says , if a point could be in contact with a point then they would have to be in contact 'as wholes' , and this ( I think ) he takes to imply that they would have to be in exactly the same place .sx At any rate he objects that contact of this kind will not produce anything continuous , because a continuous thing will be divisible into parts that occupy different places ( 231 a 29- b 6) .sx ( The thought here seems to be this .sx When one ( perfect ) sphere touches another there will be a point on the surface of the one that touches a point on the surface of the other , and these are different points , just as the face of one cube in contact with the face of another are different faces ( see n. 5 above) .sx Hence one point may be said to be in contact with another , but such contact yields spatial coincidence , so no matter how many points we may put together that are in contact with one another in this way we shall still not form anything bigger than a single point .sx But this subtlety is generally ignored in what follows , where it is simply claimed that indivisible things cannot be in contact ( e.g. 231 b 17) .sx ) Finally he adds as a third argument that points cannot even be successive , since between any two points there will be a line ( and hence , he presumably means to add , a further point ) ( 231 b 6-10) .sx Summing up this reasoning , Aristotle concludes that what a thing is made up from it may also be divided into , and so we have shown that a continuum cannot be divided into indivisibles .sx Anything whatever must be either divisible or indivisible , and if divisible then either divisible into indivisibles or divisible only into what may always be divided further ; this last is a continuum .sx And conversely every continuum is divisible into what may always be divided further , for if it could be divided into indivisibles then one indivisible could touch another , " for the limits of ( two ) continuous things touch , and are one " ( 231 b 10-18) .sx Now , if this argument is to be intelligible , what must we take the definition of a continuum to be ?sx First , it must evidently be given as a premiss that a continuum has at least two parts which do not coincide ( 213 b 4-6) .sx For Aristotle clearly does not count a point as itself a continuum , and this seems to be the minimal premiss needed to rule out that suggestion .sx To avoid some complications I should like to strengthen this premiss a little to the following :sx a continuum may always be divided ( without remainder ) into two parts that do not coincide .sx It is true that , given a 'logic of parts' that is nowadays orthodox , the supposedly stronger version would follow at once from the original .sx For the now orthodox logic supposes that , given any one proper part of a thing , there will always be some one further part of the thing , which comprises all the rest of it excluding the given part .sx Thus the whole may be said to be divided into these two parts in the sense that the two do not overlap one another - i.e. they have no common part - and together they exhaust the original whole , in so far as every part of it must overlap at least one of these two .sx But it seems to me doubtful that Aristotle would accept the assumption on which this reasoning is based .sx For example , it implies that if we begin with a line three inches long , and subtract a part one inch long from the middle of it , then the two separated line-segments remaining may be counted as together forming one part of the original .sx But this is not a very natural use of the notion of one part , and equally it is not very natural to say that the operation just envisaged would divide our three-inch line into just two parts .sx I shall come back to this problem shortly , but for the moment let us simply side-step it , by adopting the stronger premiss :sx any continuum may always be divided into two .sx Then it is perfectly clear from the whole run of Aristotle's discussion that we must add a further premiss :sx any division of a continuum into two ( non-coincident ) parts must divide it into parts that are continuous with one another in the sense defined in Book V , i.e. the two parts must touch , and the limits where they touch must 'be one' .sx These two premisses are , I imagine , the premisses that Aristotle regards his argument in this passage as depending upon , and as given by the ( unstated ) definition of what a continuum is .sx We may therefore re - construct the definition by designing it to yield just these premisses :sx a continuum is anything which ( i ) can be divided into two parts , and ( ii ) is such that any two parts into which it is divided must share a limit .sx But if this is indeed the definition Aristotle has in mind , then either it is incorrect or the conclusion that he attempts to deduce from it will not follow .sx That is , it will not follow that the parts into which a continuum may be divided must themselves be further divisible .sx Suppose we take a finite line , and suggest 'dividing' it into these two parts :sx one part is to be an end-point of the line , and the other part is to be all the rest of the line , excluding this end-point .sx ( Thus the second part is what we call a 'half-open' interval , containing all the points of the line except this one end-point .sx ) Now we may surely assume that a finite line is a continuum , if anything is , so if Aristotle's definition is correct then the two parts into which we have divided it must share a limit .sx According to one of Aristotle's lines of thought , they do not , since he claims that a point has no limit .sx But in that case we must simply reject the definition as incorrect .sx However , if my interpretation is right , then he also has another line of thought which allows points to be in contact with one another ( 'as wholes' ) despite their alleged lack of limits .sx This is most easily harmonized with his general position by allowing that a point may be said to be its own limit .sx If we do allow this , then the suggested definition of a continuum may be retained , even in the face of this example .sx For the end-point of the line , and the rest of the line , now do share a limit , namely the end-point itself .sx It is its own limit , and it must also be reckoned to be one of the limits of the rest of the line .sx For we cannot say that the rest of the line has no limit in this direction without once more subverting the suggested definition , and if it is to have a limit then clearly there is no other candidate than this same end-point .sx Thus , if the definition is to be retained , while admitting this example as a 'division' , then we must conclude that it is possible to divide a line into two parts , where one of these parts is a mere point , and hence not further divisible .sx ( And we may add , incidentally , that any point of the line may thus be exhibited as one of the parts into which it may be divided , by considering a division into three parts , one of which is the given point , one the rest of the line ( if any ) that is to the left of it , and one the rest of the line ( if any ) that is to the right of it .sx ) .sx There are two possible avenues one might explore , in seeking to rescue Aristotle from this objection .sx One would be to note that I have merely conjectured his definition of a continuum ( since he fails to state any definition himself ) , and to seek for an alternative conjecture .sx Here one may remark that he quite often appears to take , as definitive of a continuum , the thesis that in this passage he is attempting to prove .sx