Euclid's Elements

546 Pages·2007·4.172 MB·English

by  Fitzpatrick R. (ed.)

EUCLID’S ELEMENTS OF GEOMETRY
The Greek text of J.L. Heiberg (1883–1885)
from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus
B.G. Teubneri, 1883–1885
edited, and provided with a modern English translation, by
Richard Fitzpatrick
(cid:13)c RichardFitzpatrick,2007. Allrightsreserved.
ISBN
Contents
Introduction 4
Book1 5
Book2 49
Book3 69
Book4 109
Book5 129
Book6 155
Book7 193
Book8 227
Book9 253
Book10 281
Book11 423
Book12 471
Book13 505
Greek-EnglishLexicon 539
Introduction
Euclid’sElementsisbyfarthemostfamousmathematicalworkofclassicalantiquity,andalsohasthedistinction
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 subjects of the work are geometry, proportion, and
numbertheory.
Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of
earlier Greekmathematicianssuch as Pythagoras(and hisschool),Hippocratesof Chios, Theaetetusof Athens,and
EudoxusofCnidos. However,Euclidisgenerallycreditedwitharrangingthesetheoremsinalogicalmanner,soasto
demonstrate(admittedly,notalwayswiththerigourdemandedbymodernmathematics)thattheynecessarilyfollow
fromfivesimpleaxioms. Euclidisalsocreditedwithdevisinganumberofparticularlyingeniousproofsofpreviously
discoveredtheorems: e.g.,Theorem48inBook1.
The geometrical constructions employed in the Elements are restricted to those which can be achieved using a
straight-rule and a compass. Furthermore, empirical proofs by means of measurement are strictly forbidden: i.e.,
anycomparisonoftwomagnitudesisrestrictedtosayingthatthemagnitudesareeitherequal,orthatoneisgreater
thantheother.
TheElementsconsistsofthirteenbooks. Book1outlinesthefundamentalpropositionsofplanegeometry,includ-
ingthethreecasesinwhichtrianglesarecongruent,varioustheoremsinvolvingparallellines,thetheoremregarding
thesumoftheanglesinatriangle,andthePythagoreantheorem. Book2iscommonlysaidtodealwith“geometric
algebra”, since most of the theoremscontained within it have simple algebraic interpretations. Book 3 investigates
circlesand theirproperties,and includes theoremsontangentsand inscribed angles. Book4isconcernedwithreg-
ular polygons inscribed in, and circumscribed around, circles. Book 5 developsthearithmetictheoryof proportion.
Book 6 applies the theoryof proportionto plane geometry, and contains theoremson similar figures. Book 7 deals
withelementarynumbertheory: e.g.,primenumbers,greatestcommondenominators,etc. Book8isconcernedwith
geometric series. Book 9 contains various applications of results in the previous two books, and includes theorems
ontheinfinitudeofprimenumbers,aswellasthesumofageometricseries. Book10attemptstoclassifyincommen-
surable(i.e.,irrational)magnitudesusingtheso-called“methodofexhaustion”,anancientprecursortointegration.
Book 11 deals with the fundamental propositions of three-dimensional geometry. Book 12 calculates the relative
volumesofcones,pyramids,cylinders,andspheresusingthemethodofexhaustion. Finally,Book13investigatesthe
fiveso-calledPlatonicsolids.
This edition of Euclid’s Elements presents the definitive Greek text—i.e., that edited by J.L. Heiberg (1883–
1885)—accompanied by a modern English translation, as well as a Greek-English lexicon. Neither the spurious
books14and15,northeextensivescholiawhichhavebeenaddedtotheElementsoverthecenturies,areincluded.
The aim of thetranslation isto make the mathematicalargument asclear and unambiguous aspossible, whilst still
adhering closely to the meaning of the original Greek. Text within square parenthesis (in both Greek and English)
indicatesmaterialidentifiedbyHeibergasbeinglaterinterpolationstotheoriginaltext(someparticularlyobviousor
unhelpfulinterpolationshavebeenomittedaltogether). Textwithinroundparenthesis(inEnglish)indicatesmaterial
whichisimplied,butnotactuallypresent,intheGreektext.
4
ELEMENTS BOOK 1
Fundamentals of plane geometry involving
straight-lines
5
ΣΤΟΙΧΕΙΩΝ α΄. ELEMENTS BOOK 1
“Οροι. Definitions
α΄.Σηµε‹όν ™στιν, οá µέρος οÙθέν. 1.Apointisthatofwhichthereisnopart.
β΄.Γραµµ¾ δ(cid:157) µÁκος ¢πλατές. 2.Andalineisalengthwithoutbreadth.
γ΄.ΓραµµÁς δ(cid:157) πέρατα σηµε‹α. 3.Andtheextremitiesofalinearepoints.
δ΄.ΕÙθε‹α γραµµή ™στιν, ¼τις ™ξ ‡σου το‹ς ™φ' ˜αυτÁς 4. A straight-line is whatever lies evenly with points
σηµείοις κε‹ται. uponitself.
ε΄. 'Επιφάνεια δέ ™στιν, Ö µÁκος καˆ πλάτος µόνον 5.Andasurfaceisthatwhichhaslengthandbreadth
œχει. alone.
$΄.'Επιφανείας δ(cid:157) πέρατα γραµµαί. 6.Andtheextremitiesofasurfacearelines.
ζ΄. 'Επίπεδος ™πιφάνειά ™στιν, ¼τις ™ξ ‡σου τα‹ς ™φ' 7. A plane surface is whatever lies evenly with
˜αυτÁς εÙθείαις κε‹ται. straight-linesuponitself.
η΄. 'Επίπεδος δ(cid:157) γωνία ™στˆν ¹ ™ν ™πιπέδJ δύο 8. And a plane angle is the inclination of the lines,
γραµµîν ¡πτοµένων ¢λλήλων καˆ µ¾ ™π' εÙθείας whentwolinesinaplanemeetoneanother,andarenot
κειµένων πρÕς ¢λλήλας τîν γραµµîν κλίσις. laiddownstraight-onwithrespecttooneanother.
θ΄. “Οταν δ(cid:157) αƒ περιέχουσαι τ¾ν γωνίαν γραµµαˆ 9. And when the lines containing the angle are
εÙθε‹αι ðσιν, εÙθύγραµµος καλε‹ται ¹ γωνία. straightthentheangleiscalledrectilinear.
ι΄. “Οταν δ(cid:157) εÙθε‹α ™π' εÙθε‹αν σταθε‹σα τ¦ς ™φεξÁς 10. And when a straight-line stood upon (another)
γωνίας ‡σας ¢λλήλαις ποιÍ, Ñρθ¾ ˜κατέρα τîν ‡σων straight-line makes adjacent angles (which are) equal to
γωνιîν ™στι, καˆ ¹ ™φεστηκυ‹α εÙθε‹α κάθετος καλε‹ται, oneanother,eachoftheequalanglesisaright-angle,and
™φ' ¿ν ™φέστηκεν. the former straight-line is called perpendicular to that
ια΄.'Αµβλε‹α γωνία ™στˆν ¹ µείζων ÑρθÁς. uponwhichitstands.
ιβ΄.'Οξε‹α δ(cid:157) ¹ ™λάσσων ÑρθÁς. 11.Anobtuseangleisgreaterthanaright-angle.
ιγ΄.“Ορος ™στίν, Ó τινός ™στι πέρας. 12.Andanacuteangleislessthanaright-angle.
ιδ΄. ΣχÁµά ™στι τÕ Øπό τινος ½ τινων Óρων πε- 13.Aboundaryisthatwhichistheextremityofsome-
ριεχόµενον. thing.
ιε΄. Κύκλος ™στˆ σχÁµα ™πίπεδον ØπÕ µι©ς γραµµÁς 14.Afigureisthatwhichiscontainedbysomebound-
περιεχόµενον [¿ καλε‹ται περιφέρεια], πρÕς ¿ν ¢φ' aryorboundaries.
˜νÕς σηµείου τîν ™ντÕς τοà σχήµατος κειµένων π©σαι 15. A circle is a plane figure contained by a single
αƒ προσπίπτουσαι εÙθε‹αι [πρÕς τ¾ν τοà κύκλου πε- line [which is called a circumference], (such that) all of
ριφέρειαν] ‡σαι ¢λλήλαις ε„σίν. the straight-lines radiating towards [the circumference]
ι$΄.Κέντρον δ(cid:157) τοà κύκλου τÕ σηµε‹ον καλε‹ται. from a single point lying inside the figure are equal to
ιζ΄.∆ιάµετρος δ(cid:157) τοà κύκλου ™στˆν εÙθε‹ά τις δι¦ τοà oneanother.
κέντρου ºγµένη καˆ περατουµένη ™φ' ˜κάτερα τ¦ µέρη 16.Andthepointiscalledthecenterofthecircle.
ØπÕ τÁς τοà κύκλου περιφερείας, ¼τις καˆ δίχα τέµνει 17. And a diameter of the circle is any straight-line,
τÕν κύκλον. being drawn through the center, which is brought to an
ιη΄.`Ηµικύκλιον δέ ™στι τÕ περιεχόµενον σχÁµα Øπό end in each direction by the circumference of the circle.
τε τÁς διαµέτρου καˆ τÁς ¢πολαµβανοµένης Øπ' αÙτÁς Andanysuch(straight-line)cutsthecircleinhalf.†
περιφερείας. κέντρον δ(cid:157) τοà ¹µικυκλίου τÕ αÙτό, Ö καˆ 18. And a semi-circle is the figure contained by the
τοà κύκλου ™στίν. diameterandthecircumferenceitcutsoff. Andthecenter
ιθ΄. Σχήµατα εÙθύγραµµά ™στι τ¦ ØπÕ εÙθειîν πε- ofthesemi-circleisthesame(point)as(thecenterof)the
ριεχόµενα, τρίπλευρα µ(cid:157)ν τ¦ ØπÕ τριîν, τετράπλευρα circle.
δ(cid:157) τ¦ ØπÕ τεσσάρων, πολύπλευρα δ(cid:157) τ¦ ØπÕ πλειόνων À 19. Rectilinear figures are those figures contained by
τεσσάρων εÙθειîν περιεχόµενα. straight-lines: trilateral figures being contained by three
κ΄. Τîν δ(cid:157) τριπλεύρων σχηµάτων „σόπλευρον µ(cid:157)ν straight-lines, quadrilateral by four, and multilateral by
τρίγωνόν ™στι τÕ τ¦ς τρε‹ς ‡σας œχον πλευράς, „σοσκελ(cid:157)ς morethanfour.
δ(cid:157) τÕ τ¦ς δύο µόνας ‡σας œχον πλευράς, σκαληνÕν δ(cid:157) τÕ 20. And of the trilateral figures: an equilateral trian-
τ¦ς τρε‹ς ¢νίσους œχον πλευράς. gleisthathavingthreeequalsides,anisosceles(triangle)
κα΄ ”Ετι δ(cid:157) τîν τριπλεύρων σχηµάτων Ñρθογώνιον thathavingonlytwoequalsides,andascalene(triangle)
µ(cid:157)ν τρίγωνόν ™στι τÕ œχον Ñρθ¾ν γωνίαν, ¢µβλυγώνιον thathavingthreeunequalsides.
6
ΣΤΟΙΧΕΙΩΝ α΄. ELEMENTS BOOK 1
δ(cid:157) τÕ œχον ¢µβλε‹αν γωνίαν, Ñξυγώνιον δ(cid:157) τÕ τ¦ς τρε‹ς 21.Andfurtherofthetrilateralfigures: aright-angled
Ñξείας œχον γωνίας. triangle is that having a right-angle, an obtuse-angled
κβ΄.Τëν δ(cid:157) τετραπλεύρων σχηµάτων τετράγωνον µέν (triangle) that having an obtuse angle, and an acute-
™στιν, Ö „σόπλευρόν τέ ™στι καˆ Ñρθογώνιον, ˜τερόµηκες angled(triangle)thathavingthreeacuteangles.
δέ, Ö Ñρθογώνιον µέν, οÙκ „σόπλευρον δέ, ·όµβος δέ, 22. And of the quadrilateral figures: a square is that
Ö „σόπλευρον µέν, οÙκ Ñρθογώνιον δέ, ·οµβοειδ(cid:157)ς δ(cid:157) which is right-angled and equilateral, a rectangle that
τÕ τ¦ς ¢πεναντίον πλευράς τε καˆ γωνίας ‡σας ¢λλήλαις whichisright-angled butnotequilateral,arhombusthat
œχον, Ö οÜτε „σόπλευρόν ™στιν οÜτε Ñρθογώνιον· τ¦ δ(cid:157) whichisequilateralbutnotright-angled,andarhomboid
παρ¦ ταàτα τετράπλευρα τραπέζια καλείσθω. that having opposite sides and angles equal to one an-
κγ΄. Παράλληλοί ε„σιν εÙθε‹αι, α†τινες ™ν τù αÙτù other which is neither right-angled nor equilateral. And
™πιπέδJ οâσαι καˆ ™κβαλλόµεναι ε„ς ¥πειρον ™φ' ˜κάτερα letquadrilateralfiguresbesidesthesebecalledtrapezia.
τ¦ µέρη ™πˆ µηδέτερα συµπίπτουσιν ¢λλήλαις. 23.Parallellinesarestraight-lineswhich,beinginthe
sameplane,andbeingproducedtoinfinityineachdirec-
tion, meet with one another in neither (of these direc-
tions).
†Thisshouldreallybecountedasapostulate,ratherthanaspartofadefinition.
Α„τήµατα. Postulates
α΄. 'Ηιτήσθω ¢πÕ παντÕς σηµείου ™πˆ π©ν σηµε‹ον 1. Let it have been postulated to draw a straight-line
εÙθε‹αν γραµµ¾ν ¢γαγε‹ν. fromanypointtoanypoint.
β΄. Καˆ πεπερασµένην εÙθε‹αν κατ¦ τÕ συνεχ(cid:157)ς ™π' 2. And to produce a finite straight-line continuously
εÙθείας ™κβαλε‹ν. inastraight-line.
γ΄.Καˆ παντˆ κέντρJ καˆ διαστήµατι κύκλον γράφεσ- 3.Andtodrawacirclewithanycenterandradius.
θαι. 4.Andthatallright-anglesareequaltooneanother.
δ΄.Καˆ πάσας τ¦ς Ñρθ¦ς γωνίας ‡σας ¢λλήλαις ε(cid:141)ναι. 5.Andthatifastraight-linefallingacrosstwo(other)
ε΄. Καˆ ™¦ν ε„ς δύο εÙθείας εÙθε‹α ™µπίπτουσα straight-lines makesinternalangles onthe same side (of
τ¦ς ™ντÕς καˆ ™πˆ τ¦ αÙτ¦ µέρη γωνίας δύο Ñρθîν itself whose sum is) less thantwo right-angles, then,be-
™λάσσονας ποιÍ, ™κβαλλοµένας τ¦ς δύο εÙθείας ™π'¥πει- ing produced to infinity, the two (other) straight-lines
ρον συµπίπτειν, ™φ' § µέρη ε„σˆν αƒ τîν δύο Ñρθîν meet on that side (of the original straight-line) that the
™λάσσονες. (sum oftheinternalangles) islessthantworight-angles
(anddonotmeetontheotherside).†
†Thispostulateeffectivelyspecifiesthatwearedealingwiththegeometryofflat,ratherthancurved,space.
Κοιναˆ œννοιαι. Common Notions
α΄.Τ¦ τù αÙτù ‡σα καˆ ¢λλήλοις ™στˆν ‡σα. 1. Things equal to the same thing are also equal to
β΄.Καˆ ™¦ν ‡σοις ‡σα προστεθÍ, τ¦ Óλα ™στˆν ‡σα. oneanother.
γ΄.Καˆ™¦ν ¢πÕ‡σωνŠσα ¢φαιρεθÍ, τ¦καταλειπόµενά 2.And if equal things are added to equal things then
™στιν ‡σα. thewholesareequal.
δ΄. Καˆ τ¦ ™φαρµόζοντα ™π' ¢λλήλα ‡σα ¢λλήλοις 3.Andifequalthingsaresubtractedfromequalthings
™στίν. thentheremaindersareequal.†
ε΄.Καˆ τÕ Óλον τοà µέρους µε‹ζόν [™στιν]. 4. And things coinciding with one another are equal
tooneanother.
5.Andthewhole[is]greaterthanthepart.
†AsanobviousextensionofC.N.s2&3—ifequalthingsareaddedorsubtractedfromthetwosidesofaninequalitythentheinequalityremains
aninequalityofthesametype.
7
ΣΤΟΙΧΕΙΩΝ α΄. ELEMENTS BOOK 1
α΄. Proposition 1
'Επˆ τÁς δοθείσης εÙθείας πεπερασµένης τρίγωνον To construct an equilateral triangle on a given finite
„σόπλευρον συστήσασθαι. straight-line.
Γ C
∆ Α Β Ε D A B E
”Εστω ¹ δοθε‹σα εÙθε‹α πεπερασµένη ¹ ΑΒ. LetAB bethegivenfinitestraight-line.
∆ε‹ δ¾ ™πˆ τÁς ΑΒ εÙθείας τρίγωνον „σόπλευρον Soitisrequiredtoconstructanequilateraltriangleon
συστήσασθαι. thestraight-lineAB.
ΚέντρJ µ(cid:157)ν τù Α διαστήµατι δ(cid:157) τù ΑΒ κύκλος LetthecircleBCDwithcenterAandradiusABhave
γεγράφθωÐΒΓ∆,καˆπάλινκέντρJµ(cid:157)ντùΒδιαστήµατι beendrawn[Post.3], and again let thecircle ACE with
δ(cid:157) τù ΒΑ κύκλος γεγράφθω Ð ΑΓΕ, καˆ ¢πÕ τοà Γ centerB andradiusBAhavebeendrawn[Post.3]. And
σηµείου, καθ' Ö τέµνουσιν ¢λλήλους οƒ κύκλοι, ™πί τ¦ Α, let the straight-lines CA and CB have been joined from
Β σηµε‹α ™πεζεύχθωσαν εÙθε‹αι αƒ ΓΑ, ΓΒ. the point C, where the circles cut one another,† to the
Καˆ™πεˆτÕΑσηµε‹ονκέντρον ™στˆτοàΓ∆Β κύκλου, pointsAandB (respectively)[Post.1].
‡ση ™στˆν ¹ ΑΓ τÍ ΑΒ· πάλιν, ™πεˆ τÕ Β σηµε‹ον κέντρον AndsincethepointAisthecenterofthecircleCDB,
™στˆ τοà ΓΑΕ κύκλου, ‡ση ™στˆν ¹ ΒΓ τÍ ΒΑ. ™δείχθη AC is equal to AB [Def. 1.15]. Again, since the point
δ(cid:157) καˆ ¹ ΓΑ τÍ ΑΒ ‡ση· ˜κατέρα ¥ρα τîν ΓΑ, ΓΒ τÍ B is the center of the circle CAE, BC is equal to BA
ΑΒ ™στιν ‡ση. τ¦ δ(cid:157) τù αÙτù ‡σα καˆ ¢λλήλοις ™στˆν ‡σα· [Def.1.15]. ButCAwasalsoshown(tobe)equaltoAB.
καˆ ¹ ΓΑ ¥ρα τÍ ΓΒ ™στιν ‡ση· αƒ τρε‹ς ¥ρα αƒ ΓΑ, ΑΒ, Thus,CAandCBareeachequaltoAB. Butthingsequal
ΒΓ ‡σαι ¢λλήλαις ε„σίν. tothesamethingarealsoequaltooneanother[C.N.1].
'Ισόπλευρον ¤ρα ™στˆ τÕ ΑΒΓ τρίγωνον. καˆ Thus, CA is also equal toCB. Thus,the three(straight-
συνέσταται ™πˆ τÁς δοθείσης εÙθείας πεπερασµένης τÁς lines)CA,AB,andBC areequaltooneanother.
ΑΒ· Óπερ œδει ποιÁσαι. Thus, the triangle ABC is equilateral, and has been
constructed on the given finite straight-line AB. (Which
is)theverythingitwasrequiredtodo.
† Theassumptionthatthecirclesdoindeedcutoneanothershouldbecountedasanadditionalpostulate. Thereisalsoanimplicitassumption
thattwostraight-linescannotshareacommonsegment.
β΄. Proposition 2
†
ΠρÕς τù δοθέντι σηµείJ τÍ δοθείσV εÙθείv ‡σην To place a straight-line equal to a given straight-line
εÙθε‹αν θέσθαι. atagivenpoint.
”Εστω τÕ µ(cid:157)ν δοθ(cid:157)ν σηµε‹ον τÕ Α, ¹ δ(cid:157) δοθε‹σα Let A be the given point, and BC the given straight-
εÙθε‹α ¹ ΒΓ· δε‹ δ¾ πρÕς τù Α σηµείJ τÍ δοθείσV line. So it is required to place a straight-line at point A
εÙθείv τÍ ΒΓ ‡σην εÙθε‹αν θέσθαι. equaltothegivenstraight-lineBC.
'Επεζεύχθω γ¦ρ ¢πÕ τοà Α σηµείου ™πί τÕ Β For let the straight-line AB have been joined from
σηµε‹ον εÙθε‹α ¹ ΑΒ, καˆ συνεστάτω ™π' αÙτÁς τρίγωνον pointAtopointB [Post.1],andlettheequilateraltrian-
„σόπλευρον τÕ ∆ΑΒ, καˆ ™κβεβλήσθωσαν ™π' εÙθείας gleDABhavebeenbeenconstructeduponit[Prop.1.1].
τα‹ς ∆Α, ∆Β εÙθε‹αι αƒ ΑΕ, ΒΖ, καˆ κέντρJ µ(cid:157)ν τù And let the straight-lines AE and BF have been pro-
Β διαστήµατι δ(cid:157) τù ΒΓ κύκλος γεγράφθω Ð ΓΗΘ, duced in a straight-line with DA and DB (respectively)
καˆ πάλιν κέντρJ τù ∆ καˆ διαστήµατι τù ∆Η κύκλος [Post. 2]. And let the circle CGH with center B and ra-
8
ΣΤΟΙΧΕΙΩΝ α΄. ELEMENTS BOOK 1
γεγράφθω Ð ΗΚΛ. diusBC havebeendrawn[Post.3],andagainletthecir-
cleGKLwithcenterDandradiusDGhavebeendrawn
[Post.3].
Γ C
Θ H
Κ K
∆ D
Β B
Α A
Η G
Ζ F
Λ L
Ε E
'Επεˆ οâν τÕ Β σηµε‹ον κέντρον ™στˆ τοà ΓΗΘ, ‡ση Therefore,since the point B is the center of (the cir-
™στˆν ¹ ΒΓ τÍ ΒΗ. πάλιν, ™πεˆ τÕ ∆ σηµε‹ον κέντρον cle) CGH, BC is equal to BG [Def. 1.15]. Again, since
™στˆ τοà ΗΚΛ κύκλου, ‡ση ™στˆν ¹ ∆Λ τÍ ∆Η, ïν ¹ the pointD isthe center ofthe circle GKL,DL is equal
∆Α τÍ ∆Β ‡ση ™στίν. λοιπ¾ ¥ρα ¹ ΑΛ λοιπÍ τÍ ΒΗ toDG[Def.1.15]. Andwithinthese,DAisequaltoDB.
™στιν ‡ση. ™δείχθη δ(cid:157) καˆ ¹ ΒΓ τÍ ΒΗ ‡ση· ˜κατέρα ¥ρα Thus, the remainder AL is equal to the remainder BG
τîν ΑΛ, ΒΓ τÍ ΒΗ ™στιν ‡ση. τ¦ δ(cid:157) τù αÙτù ‡σα καˆ [C.N. 3]. But BC was also shown (to be) equal to BG.
¢λλήλοις ™στˆν ‡σα· καˆ ¹ ΑΛ ¥ρα τÍ ΒΓ ™στιν ‡ση. Thus,ALandBC areeachequaltoBG. Butthingsequal
ΠρÕς ¥ρα τù δοθέντι σηµείJ τù Α τÍ δοθείσV tothesamethingarealsoequaltooneanother[C.N.1].
εÙθείv τÍ ΒΓ ‡ση εÙθε‹α κε‹ται ¹ ΑΛ· Óπερ œδει ποιÁσαι. Thus,ALisalsoequaltoBC.
Thus,thestraight-lineAL,equaltothegivenstraight-
line BC, has been placed at the given point A. (Which
is)theverythingitwasrequiredtodo.
†Thispropositionadmitsofanumberofdifferentcases,dependingontherelativepositionsofthepointAandthelineBC. Insuchsituations,
Euclidinvariablyonlyconsidersoneparticularcase—usually,themostdifficult—andleavestheremainingcasesasexercisesforthereader.
γ΄. Proposition 3
∆ύο δοθεισîν εÙθειîν ¢νίσων ¢πÕ τÁς µείζονος τÍ For two given unequal straight-lines, to cut off from
™λάσσονι ‡σην εÙθε‹αν ¢φελε‹ν. thegreaterastraight-lineequaltothelesser.
”Εστωσαν αƒ δοθε‹σαι δύο εÙθε‹αι ¥νισοι αƒ ΑΒ, Γ, LetABandC bethetwogivenunequalstraight-lines,
ïν µείζων œστω ¹ ΑΒ· δε‹ δ¾ ¢πÕ τÁς µείζονος τÁς ΑΒ ofwhichletthegreaterbeAB. Soitisrequiredtocutoff
τÍ ™λάσσονι τÍ Γ ‡σην εÙθε‹αν ¢φελε‹ν. astraight-lineequaltothelesserC fromthegreaterAB.
Κείσθω πρÕς τù Α σηµείJ τÍ Γ εÙθείv ‡ση ¹ Α∆· Let the line AD, equal to the straight-line C, have
καˆ κέντρJ µ(cid:157)ν τù Α διαστήµατι δ(cid:157) τù Α∆ κύκλος been placed at point A [Prop. 1.2]. And let the circle
γεγράφθω Ð ∆ΕΖ. DEF have been drawn with center A and radius AD
Καˆ ™πεˆ τÕ Α σηµε‹ον κέντρον ™στˆ τοà ∆ΕΖ [Post.3].
κύκλου, ‡ση ™στˆν ¹ ΑΕ τÍ Α∆· ¢λλ¦ καˆ ¹ Γ τÍ Α∆ And since point A is the center of circle DEF, AE
™στιν ‡ση. ˜κατέρα ¥ρα τîν ΑΕ, Γ τÍ Α∆ ™στιν ‡ση· is equal to AD [Def. 1.15]. But, C is also equal to AD.
éστε καˆ ¹ ΑΕ τÍ Γ ™στιν ‡ση. Thus, AE and C are each equal to AD. So AE is also
9
ΣΤΟΙΧΕΙΩΝ α΄. ELEMENTS BOOK 1
equaltoC [C.N.1].
Γ C
∆ D
Ε E
Β B
Α A
F
Ζ
∆ύο ¥ρα δοθεισîν εÙθειîν ¢νίσων τîν ΑΒ, Γ ¢πÕ Thus,fortwogivenunequalstraight-lines,ABandC,
τÁς µείζονος τÁς ΑΒ τÍ ™λάσσονι τÍ Γ ‡ση ¢φÇρηται ¹ the(straight-line)AE,equaltothelesserC,hasbeencut
ΑΕ· Óπερ œδει ποιÁσαι. offfromthegreaterAB. (Whichis)theverythingitwas
requiredtodo.
δ΄. Proposition 4
'Ε¦ν δύο τρίγωνα τ¦ς δύο πλευρ¦ς [τα‹ς] δυσˆ If two triangles have two corresponding sides equal,
πλευρα‹ς ‡σας œχV ˜κατέραν ˜κατέρv καˆ τ¾ν γωνίαν τÍ and have the angles enclosed by the equal sides equal,
γωνίv ‡σην œχV τ¾ν ØπÕ τîν ‡σων εÙθειîν περιεχοµένην, then they will also have equal bases, and the two trian-
καˆ τ¾ν βάσιν τÊ βάσει ‡σην ›ξει, καˆ τÕ τρίγωνον τù gles will be equal, and the remaining angles subtended
τριγώνJ ‡σον œσται, καˆ αƒ λοιπαˆ γωνίαι τα‹ς λοιπα‹ς by the equal sides will be equal to the corresponding re-
γωνίαις ‡σαι œσονται ˜κατέρα ˜κατέρv, Øφ' §ς αƒ ‡σαι mainingangles.
πλευραˆ Øποτείνουσιν.
Α ∆ A D
B C E F
Β Γ Ε Ζ
”Εστω δύο τρίγωνα τ¦ ΑΒΓ, ∆ΕΖ τ¦ς δύο πλευρ¦ς Let ABC and DEF be two triangles having the two
τ¦ς ΑΒ, ΑΓ τα‹ς δυσˆ πλευρα‹ς τα‹ς ∆Ε, ∆Ζ ‡σας œχοντα sidesAB andAC equaltothetwosidesDE andDF,re-
˜κατέραν ˜κατέρv τ¾ν µ(cid:157)ν ΑΒ τÍ ∆Ε τ¾ν δ(cid:157) ΑΓ τÍ spectively. (Thatis)ABtoDE,andAC toDF. And(let)
∆Ζ καˆ γωνίαν τ¾ν ØπÕ ΒΑΓ γωνίv τÍ ØπÕ Ε∆Ζ ‡σην. the angle BAC (be) equal to the angle EDF. I say that
λέγω, Óτι καˆ βάσις ¹ ΒΓ βάσει τÍ ΕΖ ‡ση ™στίν, καˆ the base BC is also equal to the base EF, and triangle
τÕ ΑΒΓ τρίγωνον τù ∆ΕΖ τριγώνJ ‡σον œσται, καˆ αƒ ABC will be equal to triangle DEF, and the remaining
λοιπαˆ γωνίαι τα‹ς λοιπα‹ς γωνίαις ‡σαι œσονται ˜κατέρα angles subtended by the equal sides will be equal to the
˜κατέρv, Øφ' §ς αƒ Šσαι πλευραˆ Øποτείνουσιν, ¹ µ(cid:157)ν ØπÕ correspondingremainingangles. (Thatis)ABCtoDEF,
ΑΒΓ τÍ ØπÕ ∆ΕΖ, ¹ δ(cid:157) ØπÕ ΑΓΒ τÍ ØπÕ ∆ΖΕ. andACB toDFE.
'Εφαρµοζοµένου γ¦ρτοàΑΒΓτριγώνου™πˆτÕ∆ΕΖ Let the triangle ABC be applied to the triangle
τρίγωνον καˆ τιθεµένου τοà µ(cid:157)ν Α σηµείου ™πˆ τÕ ∆ DEF,† the point A being placed on the point D, and
σηµε‹ον τÁς δ(cid:157) ΑΒ εÙθείας ™πˆ τ¾ν ∆Ε, ™φαρµόσει καˆ the straight-line AB on DE. The point B will also coin-
τÕ Β σηµε‹ον ™πˆ τÕ Ε δι¦ τÕ ‡σην ε(cid:141)ναι τ¾ν ΑΒ τÍ cide with E, on account of AB being equal to DE. So
∆Ε· ™φαρµοσάσης δ¾ τÁς ΑΒ ™πˆ τ¾ν ∆Ε ™φαρµόσει (because of) AB coinciding with DE, the straight-line
10