Ἑλληνική Transliterate
English
DEFINITIONS . σημεῖόν ἐστιν , οὗ μέρος οὐθέν . γραμμὴ δὲ μῆκος ἀπλατές . γραμμῆς δὲ πέρατα σημεῖα . εὐθεῖα γραμμή ἐστιν , ἥτις ἐξ ἴσου τοῖς ἐφ᾽ ἑαυτῆς σημείοις κεῖται . ἐπιφάνεια δέ ἐστιν , ὃ μῆκος καὶ πλάτος μόνον ἔχει . ἐπιφανείας δὲ πέρατα γραμμαί . ἐπίπεδος ἐπιφάνειά ἐστιν , ἥτις ἐξ ἴσου ταῖς ἐφ᾽ ἑαυτῆς εὐθείαις κεῖται . ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπ᾽ εὐθείας κειμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις . ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν γραμμαὶ εὐθεῖαι ὦσιν , εὐθύγραμμος καλεῖται ἡ γωνία . ὅταν δὲ εὐθεῖα ἐπ᾽ εὐθεῖαν σταθεῖσα τὰς ἐφεξῆς γωνίας ἴσας ἀλλήλαις ποιῇ , ὀρθὴ ἑκατέρα τῶν ἴσων γωνιῶν ἐστι , καὶ ἡ ἐφεστηκυῖα εὐθεῖα κάθετος καλεῖται , ἐφ᾽ ἣν ἐφέστηκεν . ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς . ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς . ὅρος ἐστίν , ὅ τινός ἐστι πέρας . σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον . κύκλος ἐστὶ σχῆμα ἐπίπεδον ὑπὸ μιᾶς γραμμῆς περιεχόμενον ἣ καλεῖται περιφέρεια , πρὸς ἣν ἀφ᾽ ἑνὸς σημείου τῶν ἐντὸς τοῦ σχήματος κειμένων πᾶσαι αἱ προσπίπτουσαι εὐθεῖαι πρὸς τὴν τοῦ κύκλου περιφέρειαν ἴσαι ἀλλήλαις εἰσίν . κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται . διάμετρος δὲ τοῦ κύκλου ἐστὶν εὐθεῖά τις διὰ τοῦ κέντρου ἠγμένη καὶ περατουμένη ἐφ᾽ ἑκάτερα τὰ μέρη ὑπὸ τῆς τοῦ κύκλου περιφερείας , ἥτις καὶ δίχα τέμνει τὸν κύκλον . ἡμικύκλιον δέ ἐστι τὸ περιεχόμενον σχῆμα ὑπό τε τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπ᾽ αὐτῆς περιφερείας . κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό , ὃ καὶ τοῦ κύκλου ἐστίν . σχήματα εὐθύγραμμά ἐστι τὰ ὑπὸ εὐθειῶν περιεχόμενα , τρίπλευρα μὲν τὰ ὑπὸ τριῶν , τετράπλευρα δὲ τὰ ὑπὸ τεσσάρων , πολύπλευρα δὲ τὰ ὑπὸ πλειόνων ἢ τεσσάρων εὐθειῶν περιεχόμενα . τῶν δὲ τριπλεύρων σχημάτων ἰσόπλευρον μὲν τρίγωνόν ἐστι τὸ τὰς τρεῖς ἴσας ἔχον πλευράς , ἰσοσκελὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλευράς , σκαληνὸν δὲ τὸ τὰς τρεῖς ἀνίσους ἔχον πλευράς . ἔτι δὲ τῶν τριπλεύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν , ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλεῖαν γωνίαν , ὀξυγώνιον δὲ τὸ τὰς τρεῖς ὀξείας ἔχον γωνίας . τῶν δὲ τετραπλεύρων σχημάτων τετράγωνον μέν ἐστιν , ὃ ἰσόπλευρόν τέ ἐστι καὶ ὀρθογώνιον , ἑτερόμηκες δέ , ὃ ὀρθογώνιον μέν , οὐκ ἰσόπλευρον δέ , ῥόμβος δέ , ὃ ἰσόπλευρον μέν , οὐκ ὀρθογώνιον δέ , ῥομβοειδὲς δὲ τὸ τὰς ἀπεναντίον πλευράς τε καὶ γωνίας ἴσας ἀλλήλαις ἔχον , ὃ οὔτε ἰσόπλευρόν ἐστιν οὔτε ὀρθογώνιον : τὰ δὲ παρὰ ταῦτα τετράπλευρα τραπέζια καλείσθω . παράλληλοί εἰσιν εὐθεῖαι , αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ᾽ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις . POSTULATES . Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν . καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖν . καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι . καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι . καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ , ἐκβαλλομένας τὰς δύο εὐθείας ἐπ᾽ ἄπειρον συμπίπτειν , ἐφ᾽ ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες . COMMON NOTIONS . τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα . καὶ ἐὰν ἴσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἴσα . καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀφαιρεθῇ , τὰ καταλειπόμενά ἐστιν ἴσα . καὶ ἐὰν ἀνίσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἄνισα . καὶ τὰ τοῦ αὐτοῦ διπλάσια ἴσα ἀλλήλοις ἐστίν . καὶ τὰ τοῦ αὐτοῦ ἡμίση ἴσα ἀλλήλοις ἐστίν . καὶ τὰ ἐφαρμόζοντα ἐπ᾽ ἄλληλα ἴσα ἀλλήλοις ἐστίν . καὶ τὸ ὅλον τοῦ μέρους μεῖζον ἐστιν . καὶ δύο εὐθεῖαι χωρίον οὐ περιέχουσιν . PROPOSITION 1 ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συστήσασθαι . ἔστω ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ . δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστήσασθαι . κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ὁ ΒΓΔ , καὶ πάλιν κέντρῳ μὲν τῷ Β διαστήματι δὲ τῷ ΒΑ κύκλος γεγράφθω ὁ ΑΓΕ , καὶ ἀπὸ τοῦ Γ σημείου , καθ᾽ ὃ τέμνουσιν ἀλλήλους οἱ κύκλοι , ἐπὶ τὰ Α , Β σημεῖα ἐπεζεύχθωσαν εὐθεῖαι αἱ ΓΑ , ΓΒ . καὶ ἐπεὶ τὸ Α σημεῖον κέντρον ἐστὶ τοῦ ΓΔΒ κύκλου , ἴση ἐστὶν ἡ ΑΓ τῇ ΑΒ : πάλιν , ἐπεὶ τὸ Β σημεῖον κέντρον ἐστὶ τοῦ ΓΑΕ κύκλου , ἴση ἐστὶν ἡ ΒΓ τῇ ΒΑ . ἐδείχθη δὲ καὶ ἡ ΓΑ τῇ ΑΒ ἴση : ἑκατέρα ἄρα τῶν ΓΑ , ΓΒ τῇ ΑΒ ἐστὶν ἴση . τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα : καὶ ἡ ΓΑ ἄρα τῇ ΓΒ ἐστὶν ἴση : αἱ τρεῖς ἄρα αἱ ΓΑ , ΑΒ , ΒΓ ἴσαι ἀλλήλαις εἰσίν . ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον , καὶ συνέσταται ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τῆς ΑΒ . Ἐπὶ τῆς δοθείσης ἄρα εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συνέσταται : ὅπερ ἔδει ποιῆσαι .
DEFINITIONS
.
1
A
point
is
that
which
has
no
part
.
2
A
line
is
breadthless
length
.
3
The
extremities
of
a
line
are
points
.
4
A
straight
line
is
a
line
which
lies
evenly
with
the
points
on
itself
.
5
A
surface
is
that
which
has
length
and
breadth
only
.
6
The
extremities
of
a
surface
are
lines
.
7
A
plane
surface
is
a
surface
which
lies
evenly
with
the
straight
lines
on
itself
.
8
A
plane
angle
is
the
inclination
to
one
another
of
two
lines
in
a
plane
which
meet
one
another
and
do
not
lie
in
a
straight
line
.
9
And
when
the
lines
containing
the
angle
are
straight
,
the
angle
is
called
rectilineal
.
10
When
a
straight
line
set
up
on
a
straight
line
makes
the
adjacent
angles
equal
to
one
another
,
each
of
the
equal
angles
is
right
,
and
the
straight
line
standing
on
the
other
is
called
a
perpendicular
to
that
on
which
it
stands
.
11
An
obtuse
angle
is
an
angle
greater
than
a
right
angle
.
12
An
acute
angle
is
an
angle
less
than
a
right
angle
.
13
A
boundary
is
that
which
is
an
extremity
of
anything
.
14
A
figure
is
that
which
is
contained
by
any
boundary
or
boundaries
.
15
A
circle
is
a
plane
figure
contained
by
one
line
such
that
all
the
straight
lines
falling
upon
it
from
one
point
among
those
lying
within
the
figure
are
equal
to
one
another
;
16
And
the
point
is
called
the
centre
of
the
circle
.
17
A
diameter
of
the
circle
is
any
straight
line
drawn
through
the
centre
and
terminated
in
both
directions
by
the
circumference
of
the
circle
,
and
such
a
straight
line
also
bisects
the
circle
.
18
A
semicircle
is
the
figure
contained
by
the
diameter
and
the
circumference
cut
off
by
it
.
And
the
centre
of
the
semicircle
is
the
same
as
that
of
the
circle
.
19
Rectilineal
figures
are
those
which
are
contained
by
straight
lines
,
trilateral
figures
being
those
contained
by
three
,
quadrilateral
those
contained
by
four
,
and
multilateral
those
contained
by
more
than
four
straight
lines
.
20
Of
trilateral
figures
,
an
equilateral
triangle
is
that
which
has
its
three
sides
equal
,
an
isosceles
triangle
that
which
has
two
of
its
sides
alone
equal
,
and
a
scalene
triangle
that
which
has
its
three
sides
unequal
.
21
Further
,
of
trilateral
figures
,
a
right-angled
triangle
is
that
which
has
a
right
angle
,
an
obtuse-angled
triangle
that
which
has
an
obtuse
angle
,
and
an
acuteangled
triangle
that
which
has
its
three
angles
acute
.
22
Of
quadrilateral
figures
,
a
square
is
that
which
is
both
equilateral
and
right-angled
;
an
oblong
that
which
is
right-angled
but
not
equilateral
;
a
rhombus
that
which
is
equilateral
but
not
right-angled
;
and
a
rhomboid
that
which
has
its
opposite
sides
and
angles
equal
to
one
another
but
is
neither
equilateral
nor
right-angled
.
And
let
quadrilaterals
other
than
these
be
called
trapezia
.
23
Parallel
straight
lines
are
straight
lines
which
,
being
in
the
same
plane
and
being
produced
indefinitely
in
both
directions
,
do
not
meet
one
another
in
either
direction
.
POSTULATES
.
1
Let
the
following
be
postulated
:
To
draw
a
straight
line
from
any
point
to
any
point
.
2
To
produce
a
finite
straight
line
continuously
in
a
straight
line
.
3
To
describe
a
circle
with
any
centre
and
distance
.
4
That
all
right
angles
are
equal
to
one
another
.
5
That
,
if
a
straight
line
falling
on
two
straight
lines
make
the
interior
angles
on
the
same
side
less
than
two
right
angles
,
the
two
straight
lines
,
if
produced
indefinitely
,
meet
on
that
side
on
which
are
the
angles
less
than
the
two
right
angles
.
COMMON
NOTIONS
.
1
Things
which
are
equal
to
the
same
thing
are
also
equal
to
one
another
.
2
If
equals
be
added
to
equals
,
the
wholes
are
equal
.
3
If
equals
be
subtracted
from
equals
,
the
remainders
are
equal
.
7
Things
which
coincide
with
one
another
are
equal
to
one
another
.
8
The
whole
is
greater
than
the
part
.
PROPOSITION
1
On
a
given
finite
straight
line
to
construct
an
equilateral
triangle
.
Let
AB
be
the
given
finite
straight
line
.
Thus
it
is
required
to
construct
an
equilateral
triangle
on
the
straight
line
AB
.
With
centre
A
and
distance
AB
let
the
circle
BCD
be
described
;
[
Post
.
3
]
again
,
with
centre
B
and
distance
BA
let
the
circle
ACE
be
described
;
[
Post
.
3
]
and
from
the
point
C
,
in
which
the
circles
cut
one
another
,
to
the
points
A
,
B
let
the
straight
lines
CA
,
CB
be
joined
.
[
Post
.
1
]
Now
,
since
the
point
A
is
the
centre
of
the
circle
CDB
,
AC
is
equal
to
AB
.
[
Def
.
15
]
Again
,
since
the
point
B
is
the
centre
of
the
circle
CAE
,
BC
is
equal
to
BA
.
[
Def
.
15
]
But
CA
was
also
proved
equal
to
AB
;
therefore
each
of
the
straight
lines
CA
,
CB
is
equal
to
AB
.
And
things
which
are
equal
to
the
same
thing
are
also
equal
to
one
another
;
[
C
.
N
.
1
]
therefore
CA
is
also
equal
to
CB
.
Therefore
the
three
straight
lines
CA
,
AB
,
BC
are
equal
to
one
another
.
Therefore
the
triangle
ABC
is
equilateral
;
and
it
has
been
constructed
on
the
given
finite
straight
line
AB
.
Being
what
it
was
required
to
do
.
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