§42Εἰ δ' ἅμα ἐστὶ τὰ ἀμερῆ, τὸν αὐτὸν κατέχει τόπον πλείων ὃν καὶ πρότερον τὸ ἕν·
If indivisible things are together, several will occupy the same space which one occupied before; for those things which are together and do not have extension in the same direction have the same space.
τῶν γὰρ ἅμα ὄντων καὶ μὴ ἐχόντων ἐπέκτασιν κατὰ ταὐτὰ ὁ αὐτὸς ἀμφοῖν τόπος, Τὸ δ’ ἀμερὲς οὐκ ἔχει διάστασιν, ὥστ’ οὐκ ἂν εἴη μέγεθος συνεχὲς ἐξ ἀμερῶν.
But the indivisible has no dimension, so that there would be no continuous magnitude composed of indivisibles.
Οὐκ ἄρα οὔθ’ ἡ γραμμὴ ἐκ στιγμῶν οὔθ’ ὁ χρόνος ἐκ τῶν νῦν,
Therefore, neither is a line composed of points, nor is time composed of 'nows'.
§43Ἔτι εἰ ἔστιν ἐκ στιγμῶν, ἅψεται στιγμὴ στιγμῆς·
Furthermore, if a line is composed of points, a point will touch a point.
ἐὰν οὖν ἐκ τοῦ Κ ἐκβληθῇ ἡ Α Β καὶ Γ Δ, ἅψεται τοῦ Κ καὶ ἡ ἐν τῇ Κ Δ στιγμή.
If, then, from K the lines AB and CD are drawn, the point in KD will also touch K.
Ὤστε καὶ ἄλλῳ τινί· τὸ γὰρ ἀμερὲς τοῦ ἀμεροῦς ὅλον ὅλου ἐφάπτεται.
Therefore it will also touch some other point; for the indivisible touching the indivisible touches as a whole touching a whole.
Ὤστε τὸν αὐτὸν ἐφέξει τόπον τοῦ Κ, καὶ ἁπτόμεναι στιγμαὶ ἐν τῷ αὐτῷ τόπῳ ἀλλήλαις.
Therefore it will occupy the same space as K, and the touching points will be in the same space as one another.
§44Εἰ δ’ ἐν τῷ αὑτῷ, καὶ ἄπτονται· τὰ γὰρ ἐν τῷ αὐτῷ τόπῳ ὄντα πρῶτα ἅπτεσθαι ἀναγκαῖον, εἶθ’ οὕτως εὐθεῖα εὐθείας ἅψεται κατὰ δύο στιγμάς.
And if they are in the same space, they also touch; for those things which are primarily in the same space must touch, so that in this way a straight line will touch a straight line at two points.
Ἡ γὰρ ἐν τῇ Α Κ στιγμὴ καὶ τῇ Κ Γ καὶ ἑτέρας ἅπτεται στιγμῆς.
For the point in AK touches the point in KC and also another point.
῎Ωστε ἡ ἐκ τῆς Γ Δ κατὰ πλείους ἅπτεται στιγμάς.
Thus the line from GD touches at several points.
§45Ὁ αὐτὸς δὲ λόγος καὶ εἰ μὴ δι' ἀλλήλων ἀλλ’ ὁπωσοῦν ἥψατο γραμμῆς.
The same argument holds even if they do not pass through one another but touch a line in any way whatever.
Ἔτι καὶ ἡ τοῦ κύκλου τῆς εὐθείας ἅψεται κατὰ πλείω.
Furthermore, a circle will also touch a straight line at several points.
Τῆς γὰρ συναφῆς καὶ ἡ ἐν τῷ κύκλῳ καὶ ἡ ἐν τῇ εὐθείᾳ ἅπτεται καὶ ἀλλήλων.
For at the point of contact, both the point in the circle and the point in the straight line touch it and one another.
§46Εἰ δὲ τοῦτο μὴ δυνατόν, οὐδὲ τὸ ἅπτεσθαι στιγμὴν στιγμῆς·
But if this is not possible, neither is it possible for a point to touch a point.
εἰ δὲ μὴ ἅπτεσθαι, οὐδ’ εἶναι τὴν γραμμὴν στιγμήν· οὐδὲ γὰρ ἅπτεσθαι ἀναγκαῖον.
And if they do not touch, neither is a line composed of points; for then it is not even necessary for them to touch.
Ἔτι πῶς ποτὲ ἔσται εὐθεῖα γραμμὴ καὶ περιφερής;
Furthermore, how will there ever be a straight line and a curved line?
Οὐδὲν γὰρ διοίσει ἡ σύναψις τῶν στιγμῶν ἐν τῇ εὐθείᾳ καὶ τῇ περιφερεῖ.
For there will be no difference between the contact of points in a straight line and that in a curved line.
§47Τὸ γὰρ ἀμερὲς τοῦ ἀμεροῦς ὅλον ὅλου ἅπτεται, καὶ οὐκ ἔστιν ὅλως ἅπτεσθαι.
For the indivisible touching the indivisible touches as a whole touching a whole, and it is not possible to touch in part at all.
Εἰ οὖν αἱ μὲν γραμμαὶ διάφοροι, ἡ δὲ σύναψις ἀδιάφορος, οὐκ ἔσται δὴ γραμμὴ ἐκ τῆς συνάψεως, ὥστ’ οὐδ’ ἐκ στιγμῶν.
If, then, the lines are different, but the contact is not different, there will indeed be no line from the contact, so that neither from points.
Ἔτι ἀναγκαῖον ἢ ἅπτεσθαι ἢ μὴ ἅπτεσθαι τὰς στιγμὰς ἀλλήλων.
Furthermore, it is necessary for points either to touch or not to touch one another.
§48Εἰ μὲν οὖν τὸ ἐφεξῆς ἅπτεσθαι ἀνάγκη, ὁ αὐτὸς ἔσται λόγος· εἰ δὲ ἐνδέχεται ἐφεξῆς τι εἶναι μὴ ἁπτόμενον, τὸ δὲ συνεχὲς οὐδὲν ἄλλο λέγομεν ἢ τὸ ἐξ ὦν ἐστὶν ἁπτομένων· ὥστε καὶ οὕτως ἀνάγκη τὰς στιγμὰς ἅπτεσθαι ἀλλήλων, ἢ εἶναι γραμμὴν συνεχῆ.
If, then, it is necessary for adjacent things to touch, the same argument will hold; but if it is possible for something to be adjacent without touching, and we define the continuous as nothing other than that which is composed of touching things, then even so it is necessary either for points to touch one another, or for the line to fail to be continuous.
§49Ἔτι εἰ ἄτοπον στιγμὴ ἐπὶ στιγμῆς, ἴν’ ᾖ γραμμὴ καὶ ἐπὶ στιγμῇ, ἐπεὶ ἡ γραμμὴ ἐπίπεδον, ἀδύνατον τὰ εἰρημένα εἶναι.
Furthermore, if it is absurd for a point to be placed upon a point in order that there may be a line, and on a point, since a line is a plane, it is impossible for the aforesaid things to be.
Εἴτε γὰρ ἐφεξῆς αἱ στιγμαί εἰσι, τμηθήσεται ἡ γραμμὴ κατ’ οὐδετέραν τῶν στιγμῶν, ἀλλ’ ἀνὰ μέσον· εἴθ’ ἅπτονται, γραμμὴ ἔσται τῆς μιᾶς στιγμῆς χώρα.
For if the points are adjacent, the line will be cut not at either of the points, but in between them; and if they touch, the space of a single point will be a line.
Τοῦτο δ’ ἀδύνατον.
But this is impossible.
§50Ἔτι διαιροῖτ’ ἂν ἅπαντα καὶ ἀναλύοιτο εἰς στιγμάς, καὶ ἡ στιγμὴ μέρος σώματος, εἴπερ τὸ μὲν σῶμα ἐξ ἐπιπέδων, τὸ δ' ἐπίπεδον ἐκ γραμμῶν, αἱ δὲ γραμμαὶ ἐκ στιγμῶν.
Furthermore, everything would be divided and resolved into points, and the point would be a part of a body, if indeed a body is composed of planes, a plane of lines, and lines of points.
Εἰ δ’ ἐξ ὦν πρώτων ἐνυπαρχόντων ἕκαστά ἐστι, στοιχεῖά ἐστι ταῦτα, αἱ στιγμαὶ ἂν εἴησαν στοιχεῖα σωμάτων.
And if those primary inherent things of which each is composed are elements, points would be elements of bodies.
Ὥστε συνώνυμα στοιχεῖα οὐδέτερα τῷ εἴδει.
So that they would be homonymous elements, neither of them differing in kind.
§51Φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι οὐκ ἔστι γραμμὴ ἐκ στιγμῶν.
It is clear, then, from what has been said, that a line is not composed of points.
Ἀλλ' οὐδ’ ἀφαιρεθῆναι οἶόν τε στιγμὴν ἀπὸ γραμμῆς.
But neither is it possible for a point to be subtracted from a line.
Εἰ γὰρ ἐνδέχεται ἀφαιρεθῆναι, καὶ προστεθῆναι δυνατόν· προστεθέντος δέ τινος τὸ προστεθὲν μεῖζον ἔσται τοῦ ἐξ ἀρχῆς, ἐὰν τοιοῦτον ᾖ τὸ προστιθέμενον ὥστε ἓν ὅλον ποιεῖν.
For if it is possible to be subtracted, it is also possible to be added; and when something is added, that to which it is added will be larger than the original, if that which is added is such as to make one whole.