Table Of ContentEUCLID’S ELEMENTS IN GREEK
The Greek text of J.L. Heiberg (1883–1884)
from Euclidis Elementa, edidit et Latine interpretatus est
I.L. Heiberg, Lipsiae, in aedibus B.G. Teubneri, 1883–1884
with an accompanying English translation by
Richard Fitzpatrick
For Faith
Preface
Euclid’s Elements is by far the most famous mathematical work of classical antiquity, and also
has the distinction of being the world’s oldest continuously used mathematical textbook. Little is
known about the author, beyond the fact that he lived in Alexandria around 300 BCE. The main
subject of this work is Geometry, which was something of an obsession for the Ancient Greeks.
Most of the theorems appearing in Euclid’s Elements were not discovered by Euclid himself,
but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hip-
pocrates of Chios, Theaetetus, and Eudoxus of Cnidos. However, Euclid is generally credited
with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always
with the rigour demanded by modern mathematics) that they necessarily follow from five sim-
ple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of
previously discovered theorems: e.g., Theorem 48 in Book 1.
It is natural that anyone with a knowledge of Ancient Greek, combined with a general interest
in Mathematics, would wish to read the Elements in its original form. It is therefore extremely
surprizing that, whilst translations of this work into modern languages are easily available, the
Greek text has been completely unobtainable (as a book) for many years.
ThispurposeofthispublicationistomakethedefinitiveGreektextofEuclid’sElements—i.e.,
that edited by J.L. Heiberg (1883-1888)—again available to the general public in book form. The
Greek text is accompanied by my own English translation.
The aim of my translation is to be as literal as possible, whilst still (approximately) remain-
ing within the bounds of idiomatic English. Text within square parenthesis (in both Greek and
English) indicates material identified by Heiberg as being later interpolations to the original text
(some particularly obvious or unhelpful interpolations are omitted altogether). Text within round
parenthesis (in English) indicates material which is implied, but but not actually present, in the
Greek text.
My thanks goes to Mariusz Wodzicki for advice regarding the typesetting of this work.
Richard Fitzpatrick; Austin, Texas; December, 2005.
References
Euclidus Opera Ominia, J.L. Heiberg & H. Menge (editors), Teubner (1883-1916).
Euclid in Greek, Book 1, T.L. Heath (translator), Cambridge (1920).
Euclid’s Elements, T.L. Heath (translator), Dover (1956).
History of Greek Mathematics, T.L. Heath, Dover (1981).
ΣΤΟΙΧΕΙΩΝ α΄
ELEMENTS BOOK 1
Fundamentals of plane geometry involving
straight-lines
ΣΤΟΙΧΕΙΩΝ α΄
“Οροι
α΄ Σηµε‹όν ™στιν, οá µέρος οÙθέν.
β΄ Γραµµ¾ δ(cid:157) µÁκος ¢πλατές.
γ΄ ΓραµµÁς δ(cid:157) πέρατα σηµε‹α.
δ΄ ΕÙθε‹α γραµµή ™στιν, ¼τις ™ξ ‡σου το‹ς ™φ' ˜αυτÁς σηµείοις κε‹ται.
ε΄ 'Επιφάνεια δέ ™στιν, Ö µÁκος καˆ πλάτος µόνον œχει.
$΄ 'Επιφανείας δ(cid:157) πέρατα γραµµαί.
ζ΄ 'Επίπεδος ™πιφάνειά ™στιν, ¼τις ™ξ ‡σου τα‹ς ™φ' ˜αυτÁς εÙθείαις κε‹ται.
η΄ 'Επίπεδος δ(cid:157) γωνία ™στˆν ¹ ™ν ™πιπέδJ δύο γραµµîν ¡πτοµένων ¢λλήλων καˆ µ¾ ™π'
εÙθείας κειµένων πρÕς ¢λλήλας τîν γραµµîν κλίσις.
θ΄ “Οταν δ(cid:157) αƒ περιέχουσαι τ¾ν γωνίαν γραµµαˆ εÙθε‹αι ðσιν, εÙθύγραµµος καλε‹ται ¹ γωνία.
ι΄ “Οταν δ(cid:157) εÙθε‹α ™π' εÙθε‹αν σταθε‹σα τ¦ς ™φεξÁς γωνίας ‡σας ¢λλήλαις ποιÍ, Ñρθ¾ ˜κατέρα
τîν ‡σων γωνιîν ™στι, καˆ ¹ ™φεστηκυ‹α εÙθε‹α κάθετος καλε‹ται, ™φ' ¿ν ™φέστηκεν.
ια΄ 'Αµβλε‹α γωνία ™στˆν ¹ µείζων ÑρθÁς.
ιβ΄ 'Οξε‹α δ(cid:157) ¹ ™λάσσων ÑρθÁς.
ιγ΄ “Ορος ™στίν, Ó τινός ™στι πέρας.
ιδ΄ ΣχÁµά ™στι τÕ Øπό τινος ½ τινων Óρων περιεχόµενον.
ιε΄ Κύκλος ™στˆ σχÁµα ™πίπεδον ØπÕ µι©ς γραµµÁς περιεχόµενον [¿ καλε‹ται περιφέρεια],
πρÕς ¿ν ¢φ' ˜νÕς σηµείου τîν ™ντÕς τοà σχήµατος κειµένων π©σαι αƒ προσπίπτουσαι
εÙθε‹αι [πρÕς τ¾ν τοà κύκλου περιφέρειαν] ‡σαι ¢λλήλαις ε„σίν.
ι$΄ Κέντρον δ(cid:157) τοà κύκλου τÕ σηµε‹ον καλε‹ται.
ιζ΄ ∆ιάµετρος δ(cid:157) τοà κύκλου ™στˆν εÙθε‹ά τις δι¦ τοà κέντρου ºγµένη καˆ περατουµένη ™φ'
˜κάτερα τ¦ µέρη ØπÕ τÁς τοà κύκλου περιφερείας, ¼τις καˆ δίχα τέµνει τÕν κύκλον.
ιη΄ `Ηµικύκλιον δέ ™στι τÕ περιεχόµενον σχÁµα Øπό τε τÁς διαµέτρου καˆ τÁς ¢πολαµβα-
νοµένης Øπ' αÙτÁς περιφερείας. κέντρον δ(cid:157) τοà ¹µικυκλίου τÕ αÙτό, Ö καˆ τοà κύκλου
™στίν.
ιθ΄ Σχήµατα εÙθύγραµµά ™στι τ¦ ØπÕ εÙθειîν περιεχόµενα, τρίπλευρα µ(cid:157)ν τ¦ ØπÕ τριîν,
τετράπλευρα δ(cid:157) τ¦ ØπÕ τεσσάρων, πολύπλευρα δ(cid:157) τ¦ ØπÕ πλειόνων À τεσσάρων εÙθειîν
περιεχόµενα.
6
ELEMENTS BOOK 1
Definitions
1 A point is that of which there is no part.
2 And a line is a length without breadth.
3 And the extremities of a line are points.
4 A straight-line is whatever lies evenly with points upon itself.
5 And a surface is that which has length and breadth alone.
6 And the extremities of a surface are lines.
7 A plane surface is whatever lies evenly with straight-lines upon itself.
8 Andaplaneangleistheinclinationofthelines,whentwolinesinaplanemeetoneanother,
and are not laid down straight-on with respect to one another.
9 And when the lines containing the angle are straight then the angle is called rectilinear.
10 And when a straight-line stood upon (another) straight-line makes adjacent angles (which
are) equal to one another, each of the equal angles is a right-angle, and the former straight-
line is called perpendicular to that upon which it stands.
11 An obtuse angle is greater than a right-angle.
12 And an acute angle is less than a right-angle.
13 A boundary is that which is the extremity of something.
14 A figure is that which is contained by some boundary or boundaries.
15 A circle is a plane figure contained by a single line [which is called a circumference], (such
that)allofthestraight-linesradiatingtowards[thecircumference]fromasinglepointlying
inside the figure are equal to one another.
16 And the point is called the center of the circle.
17 And a diameter of the circle is any straight-line, being drawn through the center, which
is brought to an end in each direction by the circumference of the circle. And any such
(straight-line) cuts the circle in half.1
18 And a semi-circle is the figure contained by the diameter and the circumference it cuts off.
And the center of the semi-circle is the same (point) as (the center of) the circle.
19 Rectilinear figures are those figures contained by straight-lines: trilateral figures being con-
tained by three straight-lines, quadrilateral by four, and multilateral by more than four.
1Thisshouldreallybecountedasapostulate,ratherthanaspartofadefinition.
7
ΣΤΟΙΧΕΙΩΝ α΄
κ΄ Τîν δ(cid:157) τριπλεύρων σχηµάτων „σόπλευρον µ(cid:157)ν τρίγωνόν ™στι τÕ τ¦ς τρε‹ς ‡σας œχον
πλευράς, „σοσκελ(cid:157)ς δ(cid:157) τÕ τ¦ς δύο µόνας ‡σας œχον πλευράς, σκαληνÕν δ(cid:157) τÕ τ¦ς τρε‹ς
¢νίσους œχον πλευράς.
κα΄ ”Ετι δ(cid:157) τîν τριπλεύρων σχηµάτων Ñρθογώνιον µ(cid:157)ν τρίγωνόν ™στι τÕ œχον Ñρθ¾ν γωνίαν,
¢µβλυγώνιον δ(cid:157) τÕ œχον ¢µβλε‹αν γωνίαν, Ñξυγώνιον δ(cid:157) τÕ τ¦ς τρε‹ς Ñξείας œχον γωνίας.
κβ΄ Τëν δ(cid:157) τετραπλεύρων σχηµάτων τετράγωνον µέν ™στιν, Ö „σόπλευρόν τέ ™στι καˆ Ñρθο-
γώνιον, ˜τερόµηκες δέ, Ö Ñρθογώνιον µέν, οÙκ „σόπλευρον δέ, ·όµβος δέ, Ö „σόπλευρον
µέν, οÙκ Ñρθογώνιον δέ, ·οµβοειδ(cid:157)ς δ(cid:157) τÕ τ¦ς ¢πεναντίον πλευράς τε καˆ γωνίας ‡σας
¢λλήλαις œχον, Ö οÜτε „σόπλευρόν ™στιν οÜτε Ñρθογώνιον· τ¦ δ(cid:157) παρ¦ ταàτα τετράπλευρα
τραπέζια καλείσθω.
κγ΄ Παράλληλοί ε„σιν εÙθε‹αι, α†τινες ™ν τù αÙτù ™πιπέδJ οâσαι καˆ ™κβαλλόµεναι ε„ς ¥πειρον
™φ' ˜κάτερα τ¦ µέρη ™πˆ µηδέτερα συµπίπτουσιν ¢λλήλαις.
Α„τήµατα
α΄ 'Ηιτήσθω ¢πÕ παντÕς σηµείου ™πˆ π©ν σηµε‹ον εÙθε‹αν γραµµ¾ν ¢γαγε‹ν.
β΄ Καˆ πεπερασµένην εÙθε‹αν κατ¦ τÕ συνεχ(cid:157)ς ™π' εÙθείας ™κβαλε‹ν.
γ΄ Καˆ παντˆ κέντρJ καˆ διαστήµατι κύκλον γράφεσθαι.
δ΄ Καˆ πάσας τ¦ς Ñρθ¦ς γωνίας ‡σας ¢λλήλαις ε(cid:141)ναι.
ε΄ Καˆ ™¦ν ε„ς δύο εÙθείας εÙθε‹α ™µπίπτουσα τ¦ς ™ντÕς καˆ ™πˆ τ¦ αÙτ¦ µέρη γωνίας δύο
Ñρθîν ™λάσσονας ποιÍ, ™κβαλλοµένας τ¦ς δύο εÙθείας ™π' ¥πειρον συµπίπτειν, ™φ' § µέρη
ε„σˆν αƒ τîν δύο Ñρθîν ™λάσσονες.
Κοιναˆ œννοιαι
α΄ Τ¦ τù αÙτù ‡σα καˆ ¢λλήλοις ™στˆν ‡σα.
β΄ Καˆ ™¦ν ‡σοις ‡σα προστεθÍ, τ¦ Óλα ™στˆν ‡σα.
γ΄ Καˆ ™¦ν ¢πÕ ‡σων Šσα ¢φαιρεθÍ, τ¦ καταλειπόµενά ™στιν ‡σα.
δ΄ Καˆ τ¦ ™φαρµόζοντα ™π' ¢λλήλα ‡σα ¢λλήλοις ™στίν.
ε΄ Καˆ τÕ Óλον τοà µέρους µε‹ζόν [™στιν].
8
ELEMENTS BOOK 1
20 And of the trilateral figures: an equilateral triangle is that having three equal sides, an
isosceles (triangle) that having only two equal sides, and a scalene (triangle) that having
three unequal sides.
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 acute-angled (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
another 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 infin-
ity in each direction, meet with one another in neither (of these directions).
Postulates
1 Let it have been postulated to draw a straight-line from any point to any point.
2 And to produce a finite straight-line continuously in a straight-line.
3 And to draw a circle with any center and radius.
4 And that all right-angles are equal to one another.
5 And that if a straight-line falling across two (other) straight-lines makes internal angles
on the same side (of itself) less than two right-angles, being produced to infinity, the two
(other) straight-lines meet on that side (of the original straight-line) that the (internal an-
gles) are less than two right-angles (and do not meet on the other side).2
Common Notions
1 Things equal to the same thing are also equal to one another.
2 And if equal things are added to equal things then the wholes are equal.
3 And if equal things are subtracted from equal things then the remainders are equal.3
4 And things coinciding with one another are equal to one another.
5 And the whole [is] greater than the part.
2Thispostulateeffectivelyspecifiesthatwearedealingwiththegeometryofflat,ratherthancurved,space.
3As an obvious extension of C.N.s 2 & 3—if equal things are added or subtracted from the two sides of an
inequalitythentheinequalityremainsaninequalityofthesametype.
9
ΣΤΟΙΧΕΙΩΝ α΄
α΄
Γ
∆ Α Β Ε
'Επˆ τÁς δοθείσης εÙθείας πεπερασµένης τρίγωνον „σόπλευρον συστήσασθαι.
”Εστω ¹ δοθε‹σα εÙθε‹α πεπερασµένη ¹ ΑΒ.
∆ε‹ δ¾ ™πˆ τÁς ΑΒ εÙθείας τρίγωνον „σόπλευρον συστήσασθαι.
ΚέντρJ µ(cid:157)ν τù Α διαστήµατι δ(cid:157) τù ΑΒ κύκλος γεγράφθω Ð ΒΓ∆, καˆ πάλιν κέντρJ µ(cid:157)ν τù
Β διαστήµατι δ(cid:157) τù ΒΑ κύκλος γεγράφθω Ð ΑΓΕ, καˆ ¢πÕ τοà Γ σηµείου, καθ' Ö τέµνουσιν
¢λλήλους οƒ κύκλοι, ™πί τ¦ Α, Β σηµε‹α ™πεζεύχθωσαν εÙθε‹αι αƒ ΓΑ, ΓΒ.
Καˆ ™πεˆ τÕ Α σηµε‹ον κέντρον ™στˆ τοà Γ∆Β κύκλου, ‡ση ™στˆν ¹ ΑΓ τÍ ΑΒ· πάλιν, ™πεˆ τÕ Β
σηµε‹ον κέντρον ™στˆ τοà ΓΑΕ κύκλου, ‡ση ™στˆν ¹ ΒΓ τÍ ΒΑ. ™δείχθη δ(cid:157) καˆ ¹ ΓΑ τÍ ΑΒ
‡ση· ˜κατέρα ¥ρα τîν ΓΑ, ΓΒ τÍ ΑΒ ™στιν ‡ση. τ¦ δ(cid:157) τù αÙτù ‡σα καˆ ¢λλήλοις ™στˆν ‡σα· καˆ
¹ ΓΑ ¥ρα τÍ ΓΒ ™στιν ‡ση· αƒ τρε‹ς ¥ρα αƒ ΓΑ, ΑΒ, ΒΓ ‡σαι ¢λλήλαις ε„σίν.
'Ισόπλευρον ¤ρα ™στˆ τÕ ΑΒΓ τρίγωνον. καˆ συνέσταται ™πˆ τÁς δοθείσης εÙθείας πεπερασµένης
τÁς ΑΒ· Óπερ œδει ποιÁσαι.
10
