EUCLID S ELEMENTS OF GEOMETRY

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1 EUCLID S ELEMENTS OF GEOMETRY The Greek text of J.L. Heiberg ( ) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, edited, and provided with a modern English translation, by Richard Fitzpatrick NB: This excerpt from Euclid s Element contains the first two propositions (or theorems). Before proving these propositions, however, Euclid formulates definitions, postulates and common notions (or axioms, self-evident facts). In reading the proofs of the two propositions, make sure you identify the definitions, postulates and common notions that are needed in the proof of the first two propositions (or theorems).

2 Οροι. Definitions α. Σηµε όν στιν, ο µέρος ο θέν. 1. A point is that of which there is no part. β. Γραµµ δ µ κος πλατές. 2. And a line is a length without breadth. γ. Γραµµ ς δ πέρατα σηµε α. 3. And the extremities of a line are points. δ. Ε θε α γραµµή στιν, τις ξ σου το ς φ αυτ ς 4. A straight-line is whatever lies evenly with points σηµείοις κε ται. upon itself. ε. Επιφάνεια δέ στιν, µ κος κα πλάτος µόνον 5. And a surface is that which has length and breadth χει. alone.. Επιφανείας δ πέρατα γραµµαί. 6. And the extremities of a surface are lines. ζ. Επίπεδος πιφάνειά στιν, τις ξ σου τα ς φ 7. A plane surface is whatever lies evenly with αυτ ς ε θείαις κε ται. straight-lines upon itself. η. Επίπεδος δ γωνία στ ν ν πιπέδ δύο 8. And a plane angle is the inclination of the lines, γραµµ ν πτοµένων λλήλων κα µ π ε θείας when two lines in a plane meet one another, and are not κειµένων πρ ς λλήλας τ ν γραµµ ν κλίσις. laid down straight-on with respect to one another. θ. Οταν δ α περιέχουσαι τ ν γωνίαν γραµµα 9. And when the lines containing the angle are ε θε αι σιν, ε θύγραµµος καλε ται γωνία. straight then the angle is called rectilinear. ι. Οταν δ ε θε α π ε θε αν σταθε σα τ ς φεξ ς 10. And when a straight-line stood upon (another) γωνίας σας λλήλαις ποι, ρθ κατέρα τ ν σων straight-line makes adjacent angles (which are) equal to γωνι ν στι, κα φεστηκυ α ε θε α κάθετος καλε ται, one another, each of the equal angles is a right-angle, and φ ν φέστηκεν. the former straight-line is called perpendicular to that ια. Αµβλε α γωνία στ ν µείζων ρθ ς. upon which it stands. ιβ. Οξε α δ λάσσων ρθ ς. 11. An obtuse angle is greater than a right-angle. ιγ. Ορος στίν, τινός στι πέρας. 12. And an acute angle is less than a right-angle. ιδ. Σχ µά στι τ πό τινος τινων ρων πε- 13. A boundary is that which is the extremity of someριεχόµενον. thing. ιε. Κύκλος στ σχ µα πίπεδον π µι ς γραµµ ς 14. A figure is that which is contained by some boundπεριεχόµενον [ καλε ται περιφέρεια], πρ ς ν φ ary or boundaries. ν ς σηµείου τ ν ντ ς το σχήµατος κειµένων π σαι 15. A circle is a plane figure contained by a single α προσπίπτουσαι ε θε αι [πρ ς τ ν το κύκλου πε- line [which is called a circumference], (such that) all of ριφέρειαν] σαι λλήλαις ε σίν. the straight-lines radiating towards [the circumference] ι. Κέντρον δ το κύκλου τ σηµε ον καλε ται. from a single point lying inside the figure are equal to ιζ. ιάµετρος δ το κύκλου στ ν ε θε ά τις δι το one another. κέντρου γµένη κα περατουµένη φ κάτερα τ µέρη 16. And the point is called the center of the circle. π τ ς το κύκλου περιφερείας, τις κα δίχα τέµνει 17. And a diameter of the circle is any straight-line, τ ν κύκλον. being drawn through the center, which is brought to an ιη. Ηµικύκλιον δέ στι τ περιεχόµενον σχ µα πό end in each direction by the circumference of the circle. τε τ ς διαµέτρου κα τ ς πολαµβανοµένης π α τ ς And any such (straight-line) cuts the circle in half. περιφερείας. κέντρον δ το µικυκλίου τ α τό, κα 18. And a semi-circle is the figure contained by the το κύκλου στίν. diameter and the circumference it cuts off. And the center ιθ. Σχήµατα ε θύγραµµά στι τ π ε θει ν πε- of the semi-circle is the same (point) as (the center of) the ριεχόµενα, τρίπλευρα µ ν τ π τρι ν, τετράπλευρα circle. δ τ π τεσσάρων, πολύπλευρα δ τ π πλειόνων 19. Rectilinear figures are those figures contained by τεσσάρων ε θει ν περιεχόµενα. straight-lines: trilateral figures being contained by three κ. Τ ν δ τριπλεύρων σχηµάτων σόπλευρον µ ν straight-lines, quadrilateral by four, and multilateral by τρίγωνόν στι τ τ ς τρε ς σας χον πλευράς, σοσκελ ς more than four. δ τ τ ς δύο µόνας σας χον πλευράς, σκαλην ν δ τ 20. And of the trilateral figures: an equilateral trianτ ς τρε ς νίσους χον πλευράς. gle is that having three equal sides, an isosceles (triangle) κα Ετι δ τ ν τριπλεύρων σχηµάτων ρθογώνιον that having only two equal sides, and a scalene (triangle) µ ν τρίγωνόν στι τ χον ρθ ν γωνίαν, µβλυγώνιον that having three unequal sides. 6

3 δ τ χον µβλε αν γωνίαν, ξυγώνιον δ τ τ ς τρε ς ξείας χον γωνίας. κβ. Τ ν δ τετραπλεύρων σχηµάτων τετράγωνον µέν στιν, σόπλευρόν τέ στι κα ρθογώνιον, τερόµηκες δέ, ρθογώνιον µέν, ο κ σόπλευρον δέ, όµβος δέ, σόπλευρον µέν, ο κ ρθογώνιον δέ, οµβοειδ ς δ τ τ ς πεναντίον πλευράς τε κα γωνίας σας λλήλαις χον, ο τε σόπλευρόν στιν ο τε ρθογώνιον τ δ παρ τα τα τετράπλευρα τραπέζια καλείσθω. κγ. Παράλληλοί ε σιν ε θε αι, α τινες ν τ α τ πιπέδ ο σαι κα κβαλλόµεναι ε ς πειρον φ κάτερα τ µέρη π µηδέτερα συµπίπτουσιν λλήλαις. 21. And further of the trilateral figures: a right-angled triangle is that having a right-angle, an obtuse-angled (triangle) that having an obtuse angle, and an acuteangled (triangle) that having three acute angles. 22. And of the quadrilateral figures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled, and a rhomboid that having opposite sides and angles equal to one an- other which is neither right-angled nor equilateral. And let quadrilateral figures besides these be called trapezia. 23. Parallel lines are straight-lines which, being in the same plane, and being produced to infinity in each direction, meet with one another in neither (of these directions). This should really be counted as a postulate, rather than as part of a definition. Α τήµατα. Postulates α. Ηιτήσθω π παντ ς σηµείου π π ν σηµε ον 1. Let it have been postulated to draw a straight-line ε θε αν γραµµ ν γαγε ν. from any point to any point. β. Κα πεπερασµένην ε θε αν κατ τ συνεχ ς π 2. And to produce a finite straight-line continuously ε θείας κβαλε ν. in a straight-line. γ. Κα παντ κέντρ κα διαστήµατι κύκλον γράφεσ- 3. And to draw a circle with any center and radius. θαι. 4. And that all right-angles are equal to one another. δ. Κα πάσας τ ς ρθ ς γωνίας σας λλήλαις ε ναι. 5. And that if a straight-line falling across two (other) ε. Κα ν ε ς δύο ε θείας ε θε α µπίπτουσα straight-lines makes internal angles on the same side (of τ ς ντ ς κα π τ α τ µέρη γωνίας δύο ρθ ν itself whose sum is) less than two right-angles, then, be- λάσσονας ποι, κβαλλοµένας τ ς δύο ε θείας π πει- ing produced to infinity, the two (other) straight-lines ρον συµπίπτειν, φ µέρη ε σ ν α τ ν δύο ρθ ν meet on that side (of the original straight-line) that the λάσσονες. (sum of the internal angles) is less than two right-angles (and do not meet on the other side). This postulate effectively specifies that we are dealing with thegeometryofflat, ratherthancurved,space. Κοινα ννοιαι. Common Notions α. Τ τ α τ σα κα λλήλοις στ ν σα. 1. Things equal to the same thing are also equal to β. Κα ν σοις σα προστεθ, τ λα στ ν σα. one another. γ. Κα ν π σων σα φαιρεθ, τ καταλειπόµενά 2. And if equal things are added to equal things then στιν σα. the wholes are equal. δ. Κα τ φαρµόζοντα π λλήλα σα λλήλοις 3. And if equal things are subtracted from equal things στίν. then the remainders are equal. ε. Κα τ λον το µέρους µε ζόν [ στιν]. 4. And things coinciding with one another are equal to one another. 5. And the whole [is] greater than the part. As an obvious extension of C.N.s 2 & 3 if equal things are addedorsubtractedfromthetwosidesofaninequalitythentheinequality remains an inequality of the same type. 7

4 α. Proposition 1 Επ τ ς δοθείσης ε θείας πεπερασµένης τρίγωνον σόπλευρον συστήσασθαι. Γ To construct an equilateral triangle on a given finite straight-line. C Α Β Ε D A B E Εστω δοθε σα ε θε α πεπερασµένη ΑΒ. Let AB be the given finite straight-line. ε δ π τ ς ΑΒ ε θείας τρίγωνον σόπλευρον So it is required to construct an equilateral triangle on συστήσασθαι. the straight-line AB. Κέντρ µ ν τ Α διαστήµατι δ τ ΑΒ κύκλος Let the circle BCD with center A and radius AB have γεγράφθω ΒΓ, κα πάλιν κέντρ µ ν τ Β διαστήµατι been drawn [Post. 3], and again let the circle ACE with δ τ ΒΑ κύκλος γεγράφθω ΑΓΕ, κα π το Γ center B and radius BA have been drawn [Post. 3]. And σηµείου, καθ τέµνουσιν λλήλους ο κύκλοι, πί τ Α, let the straight-lines CA and CB have been joined from Βσηµε α πεζεύχθωσαν ε θε αι α ΓΑ, ΓΒ. the point C, wherethecirclescutoneanother, to the Κα πε τ Α σηµε ον κέντρον στ το Γ Β κύκλου, points A and B (respectively) [Post. 1]. ση στ ν ΑΓ τ ΑΒ πάλιν, πε τ Β σηµε ον κέντρον And since the point A is the center of the circle CDB, στ το ΓΑΕ κύκλου, ση στ ν ΒΓ τ ΒΑ. δείχθη AC is equal to AB [Def. 1.15]. Again, since the point δ κα ΓΑ τ ΑΒ ση κατέρα ρα τ ν ΓΑ, ΓΒ τ B is the center of the circle CAE, BC is equal to BA ΑΒ στιν ση. τ δ τ α τ σα κα λλήλοις στ ν σα [Def. 1.15]. But CA was also shown (to be) equal to AB. κα ΓΑ ρα τ ΓΒ στιν ση α τρε ς ρα α ΓΑ, ΑΒ, Thus, CA and CB are each equal to AB. Butthingsequal ΒΓ σαι λλήλαις ε σίν. to the same thing are also equal to one another [C.N. 1]. Ισόπλευρον ρα στ τ ΑΒΓ τρίγωνον. κα Thus, CA is also equal to CB. Thus,thethree(straightσυνέσταται π τ ς δοθείσης ε θείας πεπερασµένης τ ς lines) CA, AB, andbc are equal to one another. ΑΒ περ δει ποι σαι. Thus, the triangle ABC is equilateral, and has been constructed on the given finite straight-line AB. (Which is) the very thing it was required to do. The assumption that the circles do indeed cut one another should be counted as an additional postulate. There is also an implicit assumption that two straight-lines cannot share a common segment. β. Proposition 2 Πρ ς τ δοθέντι σηµεί τ δοθείσ ε θεί σην To place a straight-line equal to a given straight-line ε θε αν θέσθαι. at a given point. Εστω τ µ ν δοθ ν σηµε ον τ Α, δ δοθε σα Let A be the given point, and BC the given straightε θε α ΒΓ δε δ πρ ς τ Α σηµεί τ δοθείσ line. So it is required to place a straight-line at point A ε θεί τ ΒΓ σην ε θε αν θέσθαι. equal to the given straight-line BC. Επεζεύχθω γ ρ π το Α σηµείου πί τ Β For let the straight-line AB have been joined from σηµε ον ε θε α ΑΒ, κα συνεστάτω π α τ ς τρίγωνον point A to point B [Post. 1], and let the equilateral trian- σόπλευρον τ ΑΒ, κα κβεβλήσθωσαν π ε θείας gle DAB have been been constructed upon it [Prop. 1.1]. τα ς Α, Β ε θε αι α ΑΕ, ΒΖ, κα κέντρ µ ν τ And let the straight-lines AE and BF have been pro- Β διαστήµατι δ τ ΒΓ κύκλος γεγράφθω ΓΗΘ, duced in a straight-line with DA and DB (respectively) κα πάλιν κέντρ τ κα διαστήµατι τ Η κύκλος [Post. 2]. And let the circle CGH with center B and ra- 8

5 γεγράφθω ΗΚΛ. dius BC have been drawn [Post. 3], and again let the circle GKL with center D and radius DG have been drawn [Post. 3]. Γ C Κ Θ Β K H D B Α Η A G Ζ F Λ L Ε E Επε ο ν τ Β σηµε ον κέντρον στ το ΓΗΘ, ση Therefore, since the point B is the center of (the cir- στ ν ΒΓ τ ΒΗ. πάλιν, πε τ σηµε ον κέντρον cle) CGH, BC is equal to BG [Def. 1.15]. Again, since στ το ΗΚΛ κύκλου, ση στ ν Λ τ Η, ν the point D is the center of the circle GKL, DL is equal Α τ Β ση στίν. λοιπ ρα ΑΛ λοιπ τ ΒΗ to DG [Def. 1.15]. And within these, DA is equal to DB. στιν ση. δείχθη δ κα ΒΓ τ ΒΗ ση κατέρα ρα Thus, the remainder AL is equal to the remainder BG τ ν ΑΛ, ΒΓ τ ΒΗ στιν ση. τ δ τ α τ σα κα [C.N. 3]. But BC was also shown (to be) equal to BG. λλήλοις στ ν σα κα ΑΛ ρα τ ΒΓ στιν ση. Thus, AL and BC are each equal to BG. Butthingsequal Πρ ς ρα τ δοθέντι σηµεί τ Α τ δοθείσ to the same thing are also equal to one another [C.N. 1]. ε θεί τ ΒΓ ση ε θε α κε ται ΑΛ περ δει ποι σαι. Thus, AL is also equal to BC. Thus, the straight-line AL,equaltothegivenstraightline BC, hasbeenplacedatthegivenpointa. (Which is) the very thing it was required to do. This proposition admits of a number of different cases, depending on the relative positions of the point A and the line BC. Insuchsituations, Euclid invariably only considers one particular case usually, the most difficult and leaves the remaining cases as exercises for the reader. γ. Proposition 3 ύο δοθεισ ν ε θει ν νίσων π τ ς µείζονος τ For two given unequal straight-lines, to cut off from λάσσονι σην ε θε αν φελε ν. the greater a straight-line equal to the lesser. Εστωσαν α δοθε σαι δύο ε θε αι νισοι α ΑΒ, Γ, Let AB and C be the two given unequal straight-lines, ν µείζων στω ΑΒ δε δ π τ ς µείζονος τ ς ΑΒ of which let the greater be AB. Soitisrequiredtocutoff τ λάσσονι τ Γ σην ε θε αν φελε ν. astraight-lineequaltothelesserc from the greater AB. Κείσθω πρ ς τ Α σηµεί τ Γ ε θεί ση Α Let the line AD, equaltothestraight-linec, have κα κέντρ µ ν τ Α διαστήµατι δ τ Α κύκλος been placed at point A [Prop. 1.2]. And let the circle γεγράφθω ΕΖ. DEF have been drawn with center A and radius AD Κα πε τ Α σηµε ον κέντρον στ το ΕΖ [Post. 3]. κύκλου, ση στ ν ΑΕ τ Α λλ κα Γ τ Α And since point A is the center of circle DEF, AE στιν ση. κατέρα ρα τ ν ΑΕ, Γ τ Α στιν ση is equal to AD [Def. 1.15]. But, C is also equal to AD. στε κα ΑΕ τ Γ στιν ση. Thus, AE and C are each equal to AD. So AE is also 9