§16.5Διὰ τί ὁ μὲν κύλινδρος ὠσθεὶς εἰς εὐθύ τε φέρεται καὶ γράφει εὐθείας τοῖς ὁρίζουσιν αὐτὸν κύκλοις, ὁ δὲ κῶνος κύκλῳ περιφέρεται, τῆς κορυφῆς μενούσης, καὶ γράφει τὸν κύκλον τῷ ὁρίζοντι κύκλῳ;
Why does a cylinder, when pushed, travel in a straight line and describe straight lines with its limiting circles, while a cone revolves in a circle, with its vertex remaining stationary, and describes a circle with its limiting circle?
κύκλῳ μὲν ἀμφότερα φέρεται, γράφει δ’ ἐν τῷ ἐπιπέδῳ ὁ μὲν κύλινδρος εὐθείας, ὁ δὲ κῶνος κύκλους, διὰ τὸ τοὺς μὲν ἐν τῷ κώνῳ ἀνίσους εἶναι κύκλους, φέρεσθαι δὲ ἀεὶ θᾶττον τὸν μείζονα τῶν περὶ τὸ αὐτὸ κέντρον.
Both are carried in a circle, but on the plane the cylinder describes straight lines, while the cone describes circles, because the circles in the cone are unequal, and the larger of those around the same center always travels faster.
φερομένων δὲ ἀνίσως πάντων ἅμα τῶν ἐν τῷ κώνῳ κύκλων, συμβαίνει τοὺς ἐξωτάτω πλεῖστον ἐν ταὐτῷ χρόνῳ τόπον καὶ γραμμὴν φέρεσθαι· διὸ καὶ κύκλῳ φέρονται.
Since all the circles in the cone are carried simultaneously at unequal speeds, it happens that the outermost circle traverses the greatest distance and line in the same time; which is why they are carried in a circle.
γράφονταί τε γὰρ πάντες τῇ αὐτῇ εὐθείᾳ, καὶ τῆς εὐθείας κύκλῳ μὲν φερομένης οὐ πάντα τὰ ἐν αὐτῇ σημεῖα ἴσην ἐν ταὐτῷ χρόνῳ γράφει γραμμήν, εἰς εὐθὺ δὲ φέρει τὴν ἴσην.
For they are all described by the same straight line, and when this straight line is carried in a circle, not all the points on it describe an equal line in the same time, but when it is carried in a straight line, it describes an equal one.
τοῦ δὲ κυλίνδρου πάντων ἴσων ὄντων τῶν κύκλων καὶ περὶ ταὐτὸ κέντρον, συμβαίνει τὰ ἅμα τοῦ ἐπιπέδου ἐν αὐτοῖς πάνθ’ ἁπτομένοις σημεῖα, φέρεσθαί τε ἰσοταχεῖς κυλιομένους διὰ τὸ τοὺς κυλίνδρους ἴσους εἶναι, καὶ ἥκειν ἐπὶ τὸ ἐπίπεδον πάλιν ἅμα ἐκκυλισθέντα ἕκαστον τὸν αὑτοῦ κύκλον, ὥστε καὶ τὰς ἐν τῷ ἐπιπέδῳ εὐθείας ἴσας γίνεσθαι·
But in the case of the cylinder, since all the circles are equal and around the same center, it happens that all their points which touch the plane at the same time are carried at equal speeds as they roll, because the cylinders are equal, and each returns to the plane again at the same time after having completed its own circle; so that the straight lines on the plane also become equal.
τῇ γὰρ αὑτῶν ἀφῇ αὐτὰς ἔγραψαν, ὄντες ἴσοι τε καὶ ἰσοταχεῖς.
For they described them by their own contact, being equal and of equal speed.
ἐγίνοντο δὲ εὐθεῖαι αἱ ὑπὸ τῆς αὐτῆς γραφεῖσαι γραμμῆς εἰς εὐθὺ φερομένης, ὥστε διὰ ταύτας εἰς εὐθὺ ἀναφέροιτο ὁ κύλινδρος.
And the lines described by the same line when carried in a straight line became straight, so that because of these the cylinder would be carried in a straight line.
διαφέρει γὰρ οὐθέν, ᾗ ἡ πρώτη ἥψατο ὁ κύλινδρος τοῦ ἐπιπέδου γραμμῇ, ταύτῃ ἕλκειν ἐν τῷ ἐπιπέδῳ, ἢ ἐγκυλίειν αὐτό· ἀεὶ γὰρ ἴσην καὶ ὁμοίαν γραμμὴν τῶν ἐν τῷ κυλίνδρῳ συμβήσεται ἅπτεσθαι τοῦ ἐπιπέδου, ἑλκομένου τε καὶ κυλιομένου τοῦ κυλίνδρου.
For it makes no difference whether one drags the cylinder along the plane by the line with which it first touched the plane, or rolls it; for always an equal and similar line of those on the cylinder will happen to touch the plane, whether the cylinder is dragged or rolled.
§16.6Διὰ τί τῶν βιβλίων ἡ τομὴ οὖσα ἐπίπεδος καὶ εὐθεῖα, ἐὰν μέν τις τέμῃ παρὰ τὴν βάσιν, γίνεται εὐθεῖα ἀνελιττομένη, ἐὰν δὲ ἐγκλίνας, σκολιά;
Why is it that, although the section of scrolls is flat and straight, if one cuts parallel to the base, it becomes a straight line when unrolled, but if one cuts obliquely, it becomes crooked?
ἢ ὅτι συμβαίνει τῶν ἐν τῇ ἑτέρᾳ τομῇ κύκλων ἐν ταὐτῷ ἐπιπέδῳ ὄντων τὴν ἐγκεκλιμένην τομὴν μὴ παρακειμένην εἶναι, ἀλλ’ ἐν τῇ μὲν πλεῖον, τῇ δὲ ἔλαττον αὐτῆς ἀπέχειν, ὥστε ἐξελιττομένου οἱ μὲν ἐν ταὐτῷ ἐπιπέδῳ ὄντες κύκλοι, καὶ τὴν ἀρχὴν ἔχοντες ἐν ταὐτῷ ἐπιπέδῳ.
Is it because it happens that, while the circles in the other section are in the same plane, the oblique section is not parallel to it, but is at a greater distance from it in one part and less in another? Consequently, when unrolled, the circles which are in the same plane, and have their origin in the same plane, will produce the line consisting of themselves when unrolled.
τὴν ἐξ αὑτῶν ποιήσουσι γραμμὴν ἐξελιττόμενοι. ἔστι γὰρ ἡ γινομένη γραμμὴ ἐκ τῶν κύκλων οἵ εἰσιν ἐν ταὐτῷ ἐπιπέδῳ· ὥστε καὶ εὐθεῖα οὖσα ἐν ἐπιπέδῳ.
For the resulting line is composed of the circles which are in the same plane; so that it is also a straight line in a plane.
ἥ τε τῆς λοξῆς τομῆς ἐξελιττομένη γραμμὴ οὐκ οὖσα παρὰ τὴν πρώτην, ἀλλὰ τῇ μὲν πλέον, τῇ δὲ ἔλαττον αὐτῆς διεστηκυῖα διὰ τὸ καὶ τὴν τομὴν οὕτως ἔχειν πρὸς αὐτήν, οὐκ ἐν ἐπιπέδῳ ἔσται, ὥστε οὐδ’ εὐθεῖα· τῆς γὰρ εὐθείας οὐκ ἔστι τὸ μὲν ἐν ἄλλῳ, τὸ δὲ ἐν ἄλλῳ ἐπιπέδῳ.
But the unrolled line of the oblique section, not being parallel to the first, but standing apart from it more in one place and less in another because the section also stands in this relation to it, will not be in one plane, so that it is not a straight line either; for of a straight line, it is impossible for one part to be in one plane and another in another.
§16.7Διὰ τί διαιρούμενα τὰ μεγέθη ἐλάττω φαίνεται πάντα τοῦ ὅλου;
Why do magnitudes when divided always appear smaller than the whole?
ἢ ὅτι διαιρούμενα μὲν ἀριθμὸν ἔχει πάντα, μεγέθει δὲ ἐλάττω ἐστὶ τοῦ ἑνός;
Is it because when divided they all have number, but in magnitude they are smaller than the one?
τὸ μὲν γὰρ μέγα τῷ κατὰ συνέχειαν εἶναι καὶ ποσόν τι μέγα λέγεται, ὁ δὲ ἀριθμός τε πᾶς παντὸς μεγέθους ἀριθμοῦ μείζων.
For that which is large is called a certain large quantity by virtue of being continuous, but every number is greater than any magnitude of number.
διόπερ εἰκὸς τὸ ὅλον διαιρεθέντων τῶν μερῶν μεῖζον φαίνεσθαι· τῶν αὐτῶν γὰρ ὄντων αὐτῶν τὸ μὲν ὅλον τὴν τοῦ μεγέθους ἔχει μᾶλλον φύσιν, συνεχὲς ὄν, τὰ δὲ μέρη τὴν τοῦ ἀριθμοῦ.
Therefore, it is natural that the whole should appear larger when the parts are divided; for though they are the same, the whole has more of the nature of magnitude, being continuous, while the parts have that of number.