Euclid's Elements of Geometry

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ELEMENTS BOOK 7



αὑτὸ μέρος ἐστὶ καὶ ὁ ΔΘ τοῦ Ζ, ὃ ἄρα μέρος ἐστὶν ὁ ΑΗ HE. And since which(ever) part AG is of C, DH is also
τοῦ Γ, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ὁ ΑΗ, ΔΘ the same part of F, thus which(ever) part AG is of C,
συναμφοτέρου τοῦ Γ, Ζ. διὰ τὰ αὐτὰ δὴ καὶ ὃ μέρος ἐστὶν ὁ the sum AG, DH is also the same part of the sum C, F
zþ (Which is) the very thing it was required to show.
ΗΒ τοῦ Γ, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ὁ ΗΒ, ΘΕ [Prop. 7.5]. And so, for the same (reasons), which(ever)
συναμφοτέρου τοῦ Γ, Ζ. ἃ ἄρα μέρη ἐστὶν ὁ ΑΒ τοῦ Γ, τὰ part GB is of C, the sum GB, HE is also the same part
αὐτὰ μέρη ἐστὶ καὶ συναμφότερος ὁ ΑΒ, ΔΕ συναμφοτέρου of the sum C, F. Thus, which(ever) parts AB is of C,
τοῦ Γ, Ζ· ὅπερ ἔδει δεῖξαι. the sum AB, DE is also the same parts of the sum C, F.



† In modern notation, this proposition states that if a = (m/n) b and c = (m/n) d then (a + c) = (m/n) (b + d), where all symbols denote
numbers.

.
†
Proposition 7
᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρος ᾖ, ὅπερ ἀφαιρεθεὶς ἀφαι- If a number is that part of a number that a (part)
ρεθέντος, καὶ ὁ λοιπὸς τοῦ λοιποῦ τὸ αὐτὸ μέρος ἔσται, taken away (is) of a (part) taken away then the remain-
ὅπερ ὁ ὅλος τοῦ ὅλου. der will also be the same part of the remainder that the
whole (is) of the whole.
Α Ε Β A E B



Η Γ Ζ ∆ G C F D

᾿Αριθμὸς γὰρ ὁ ΑΒ ἀριθμοῦ τοῦ ΓΔ μέρος ἔστω, ὅπερ For let a number AB be that part of a number CD
ἀφαιρεθεὶς ὁ ΑΕ ἀφαιρεθέντος τοῦ ΓΖ· λέγω, ὅτι καὶ λοιπὸς that a (part) taken away AE (is) of a part taken away
ὁ ΕΒ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὅλος ὁ ΑΒ CF. I say that the remainder EB is also the same part of
ὅλου τοῦ ΓΔ. the remainder FD that the whole AB (is) of the whole
῝Ο γὰρ μέρος ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος ἔστω CD.
καὶ ὁ ΕΒ τοῦ ΓΗ. καὶ ἐπεί, ὃ μέρος ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ For which(ever) part AE is of CF, let EB also be the
αὐτὸ μέρος ἐστὶ καὶ ὁ ΕΒ τοῦ ΓΗ, ὃ ἄρα μέρος ἐστὶν ὁ ΑΕ same part of CG. And since which(ever) part AE is of
τοῦ ΓΖ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΑΒ τοῦ ΗΖ. ὃ δὲ μέρος CF, EB is also the same part of CG, thus which(ever)
ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος ὑπόκειται καὶ ὁ ΑΒ τοῦ part AE is of CF, AB is also the same part of GF
ΓΔ· ὃ ἄρα μέρος ἐστὶ καὶ ὁ ΑΒ τοῦ ΗΖ, τὸ αὐτὸ μέρος ἐστὶ [Prop. 7.5]. And which(ever) part AE is of CF, AB is
καὶ τοῦ ΓΔ· ἴσος ἄρα ἐστὶν ὁ ΗΖ τῷ ΓΔ. κοινὸς ἀφῃρήσθω also assumed (to be) the same part of CD. Thus, also,
ὁ ΓΖ· λοιπὸς ἄρα ὁ ΗΓ λοιπῷ τῷ ΖΔ ἐστιν ἴσος. καὶ ἐπεί, which(ever) part AB is of GF, (AB) is also the same
ὃ μέρος ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος [ἐστὶ] καὶ ὁ ΕΒ part of CD. Thus, GF is equal to CD. Let CF have been
τοῦ ΗΓ, ἴσος δὲ ὁ ΗΓ τῷ ΖΔ, ὃ ἄρα μέρος ἐστὶν ὁ ΑΕ τοῦ subtracted from both. Thus, the remainder GC is equal
ΓΖ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΕΒ τοῦ ΖΔ. ἀλλὰ ὃ μέρος to the remainder FD. And since which(ever) part AE is
ἐστὶν ὁ ΑΕ τοῦ ΓΖ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΑΒ τοῦ ΓΔ· of CF, EB [is] also the same part of GC, and GC (is)
hþ der EB is also the same part of the remainder FD that
καὶ λοιπὸς ἄρα ὁ ΕΒ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστίν, equal to FD, thus which(ever) part AE is of CF, EB is
ὅπερ ὅλος ὁ ΑΒ ὅλου τοῦ ΓΔ· ὅπερ ἔδει δεῖξαι. also the same part of FD. But, which(ever) part AE is of
CF, AB is also the same part of CD. Thus, the remain-
the whole AB (is) of the whole CD. (Which is) the very
thing it was required to show.

† In modern notation, this proposition states that if a = (1/n) b and c = (1/n) d then (a− c) = (1/n) (b− d), where all symbols denote numbers.

Proposition 8
†
.
᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, ἅπερ ἀφαιρεθεὶς ἀφαι- If a number is those parts of a number that a (part)
ρεθέντος, καὶ ὁ λοιπὸς τοῦ λοιποῦ τὰ αὐτὰ μέρη ἔσται, taken away (is) of a (part) taken away then the remain-
ἅπερ ὁ ὅλος τοῦ ὅλου. der will also be the same parts of the remainder that the

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whole (is) of the whole. ELEMENTS BOOK 7
Γ Ζ ∆ C F D



Η Μ Κ Ν Θ G M K N H



Α Λ Ε Β A L E B

᾿Αριθμὸς γὰρ ὁ ΑΒ ἀριθμοῦ τοῦ ΓΔ μέρη ἔστω, ἅπερ For let a number AB be those parts of a number CD
ἀφαιρεθεὶς ὁ ΑΕ ἀφαιρεθέντος τοῦ ΓΖ· λέγω, ὅτι καὶ λοιπὸς that a (part) taken away AE (is) of a (part) taken away
ὁ ΕΒ λοιποῦ τοῦ ΖΔ τὰ αὐτὰ μέρη ἐστίν, ἅπερ ὅλος ὁ ΑΒ CF. I say that the remainder EB is also the same parts
ὅλου τοῦ ΓΔ. of the remainder FD that the whole AB (is) of the whole
Κείσθω γὰρ τῷ ΑΒ ἴσος ὁ ΗΘ, ἃ ἄρα μέρη ἐστὶν ὁ ΗΘ CD.
τοῦ ΓΔ, τὰ αὐτὰ μέρη ἐστὶ καὶ ὁ ΑΕ τοῦ ΓΖ. διῃρήσθω ὁ For let GH be laid down equal to AB. Thus,
μὲν ΗΘ εἰς τὰ τοῦ ΓΔ μέρη τὰ ΗΚ, ΚΘ, ὁ δὲ ΑΕ εἰς τὰ τοῦ which(ever) parts GH is of CD, AE is also the same
ΓΖ μέρη τὰ ΑΛ, ΛΕ· ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΗΚ, ΚΘ parts of CF. Let GH have been divided into the parts
τῷ πλήθει τῶν ΑΛ, ΛΕ. καὶ ἐπεί, ὃ μέρος ἐστὶν ὁ ΗΚ τοῦ of CD, GK and KH, and AE into the part of CF, AL
ΓΔ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΑΛ τοῦ ΓΖ, μείζων δὲ ὁ ΓΔ and LE. So the multitude of (divisions) GK, KH will be
τοῦ ΓΖ, μείζων ἄρα καὶ ὁ ΗΚ τοῦ ΑΛ. κείσθω τῷ ΑΛ ἴσος equal to the multitude of (divisions) AL, LE. And since
ὁ ΗΜ. ὃ ἄρα μέρος ἐστὶν ὁ ΗΚ τοῦ ΓΔ, τὸ αὐτὸ μέρος ἐστὶ which(ever) part GK is of CD, AL is also the same part
καὶ ὁ ΗΜ τοῦ ΓΖ· καὶ λοιπὸς ἄρα ὁ ΜΚ λοιποῦ τοῦ ΖΔ of CF, and CD (is) greater than CF, GK (is) thus also
τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὅλος ὁ ΗΚ ὅλου τοῦ ΓΔ. πάλιν greater than AL. Let GM be made equal to AL. Thus,
ἐπεί, ὃ μέρος ἐστὶν ὁ ΚΘ τοῦ ΓΔ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ which(ever) part GK is of CD, GM is also the same part
ΕΛ τοῦ ΓΖ, μείζων δὲ ὁ ΓΔ τοῦ ΓΖ, μείζων ἄρα καὶ ὁ ΘΚ of CF. Thus, the remainder MK is also the same part of
τοῦ ΕΛ. κείσθω τῷ ΕΛ ἴσος ὁ ΚΝ. ὃ ἄρα μέρος ἐστὶν ὁ ΚΘ the remainder FD that the whole GK (is) of the whole
τοῦ ΓΔ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΚΝ τοῦ ΓΖ· καὶ λοιπὸς CD [Prop. 7.5]. Again, since which(ever) part KH is of
ἄρα ὁ ΝΘ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστίν, ὅπερ ὅλος ὁ CD, EL is also the same part of CF, and CD (is) greater
ΚΘ ὅλου τοῦ ΓΔ. ἐδείχθη δὲ καὶ λοιπὸς ὁ ΜΚ λοιποῦ τοῦ than CF, HK (is) thus also greater than EL. Let KN be
ΖΔ τὸ αὐτὸ μέρος ὤν, ὅπερ ὅλος ὁ ΗΚ ὅλου τοῦ ΓΔ· καὶ made equal to EL. Thus, which(ever) part KH (is) of
συναμφότερος ἄρα ὁ ΜΚ, ΝΘ τοῦ ΔΖ τὰ αὐτὰ μέρη ἐστίν, CD, KN is also the same part of CF. Thus, the remain-
ἅπερ ὅλος ὁ ΘΗ ὅλου τοῦ ΓΔ. ἴσος δὲ συναμφότερος μὲν der NH is also the same part of the remainder FD that
ὁ ΜΚ, ΝΘ τῷ ΕΒ, ὁ δὲ ΘΗ τῷ ΒΑ· καὶ λοιπὸς ἄρα ὁ ΕΒ the whole KH (is) of the whole CD [Prop. 7.5]. And the
λοιποῦ τοῦ ΖΔ τὰ αὐτὰ μέρη ἐστίν, ἅπερ ὅλος ὁ ΑΒ ὅλου remainder MK was also shown to be the same part of
τοῦ ΓΔ· ὅπερ ἔδει δεῖξαι. the remainder FD that the whole GK (is) of the whole
CD. Thus, the sum MK, NH is the same parts of DF
jþ remainder EB is also the same parts of the remainder
that the whole HG (is) of the whole CD. And the sum
MK, NH (is) equal to EB, and HG to BA. Thus, the

FD that the whole AB (is) of the whole CD. (Which is)
the very thing it was required to show.

† In modern notation, this proposition states that if a = (m/n) b and c = (m/n) d then (a − c) = (m/n) (b − d), where all symbols denote
numbers. †

.
Proposition 9
᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρος ᾖ, καὶ ἕτερος ἑτέρου τὸ If a number is part of a number, and another (num-
αὐτὸ μέρος ᾖ, καὶ ἐναλλάξ, ὃ μέρος ἐστὶν ἢ μέρη ὁ πρῶτος ber) is the same part of another, also, alternately,
τοῦ τρίτου, τὸ αὐτὸ μέρος ἔσται ἢ τὰ αὐτὰ μέρη καὶ ὁ which(ever) part, or parts, the first (number) is of the
δεύτερος τοῦ τετάρτου. third, the second (number) will also be the same part, or

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the same parts, of the fourth. ELEMENTS BOOK 7
Ε E

Β B



Θ H
Η G




Γ Ζ C F
Α ∆ A D
᾿Αριθμὸς γὰρ ὁ Α ἀριθμοῦ τοῦ ΒΓ μέρος ἔστω, καὶ ἕτε- For let a number A be part of a number BC, and an-
ρος ὁ Δ ἑτέρου τοῦ ΕΖ τὸ αὐτὸ μέρος, ὅπερ ὁ Α τοῦ ΒΓ· other (number) D (be) the same part of another EF that
λέγω, ὅτι καὶ ἐναλλάξ, ὃ μέρος ἐστὶν ὁ Α τοῦ Δ ἢ μέρη, τὸ A (is) of BC. I say that, also, alternately, which(ever)
αὐτὸ μέρος ἐστὶ καὶ ὁ ΒΓ τοῦ ΕΖ ἢ μέρη. part, or parts, A is of D, BC is also the same part, or
᾿Επεὶ γὰρ ὃ μέρος ἐστὶν ὁ Α τοῦ ΒΓ, τὸ αὐτὸ μέρος ἐστὶ parts, of EF.
καὶ ὁ Δ τοῦ ΕΖ, ὅσοι ἄρα εἰσὶν ἐν τῷ ΒΓ ἀριθμοὶ ἴσοι τῷ For since which(ever) part A is of BC, D is also the
Α, τοσοῦτοί εἰσι καὶ ἐν τῷ ΕΖ ἴσοι τῷ Δ. διῃρήσθω ὁ μὲν same part of EF, thus as many numbers as are in BC
ΒΓ εἰς τοὺς τῷ Α ἴσους τοὺς ΒΗ, ΗΓ, ὁ δὲ ΕΖ εἰς τοὺς τῷ equal to A, so many are also in EF equal to D. Let BC
Δ ἴσους τοὺς ΕΘ, ΘΖ· ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, have been divided into BG and GC, equal to A, and EF
ΗΓ τῷ πλήθει τῶν ΕΘ, ΘΖ. into EH and HF, equal to D. So the multitude of (di-
Καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΒΗ, ΗΓ ἀριθμοὶ ἀλλήλοις, εἰσὶ visions) BG, GC will be equal to the multitude of (divi-
δὲ καὶ οἱ ΕΘ, ΘΖ ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν ἴσον τὸ sions) EH, HF.
πλῆθος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, ΘΖ, ὃ ἄρα μέρος And since the numbers BG and GC are equal to one
ἐστὶν ὁ ΒΗ τοῦ ΕΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΗΓ another, and the numbers EH and HF are also equal to
τοῦ ΘΖ ἢ τὰ αὐτὰ μέρη· ὥστε καὶ ὃ μέρος ἐστὶν ὁ ΒΗ τοῦ one another, and the multitude of (divisions) BG, GC
ΕΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ὁ ΒΓ is equal to the multitude of (divisions) EH, HC, thus
συναμφοτέρου τοῦ ΕΖ ἢ τὰ αὐτὰ μέρη. ἴσος δὲ ὁ μὲν ΒΗ which(ever) part, or parts, BG is of EH, GC is also
τῷ Α, ὁ δὲ ΕΘ τῷ Δ· ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ Δ ἢ μέρη, the same part, or the same parts, of HF. And hence,
τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ ΒΓ τοῦ ΕΖ ἢ τὰ αὐτὰ μέρη· ὅπερ which(ever) part, or parts, BG is of EH, the sum BC
iþ the same part, or the same parts, of EF. (Which is) the
ἔδει δεῖξαι. is also the same part, or the same parts, of the sum EF
[Props. 7.5, 7.6]. And BG (is) equal to A, and EH to D.
Thus, which(ever) part, or parts, A is of D, BC is also

very thing it was required to show.

† In modern notation, this proposition states that if a = (1/n) b and c = (1/n) d then if a = (k/l) c then b = (k/l) d, where all symbols denote
numbers. †

.
Proposition 10
᾿Εὰν ἀριθμὸς ἀριθμοῦ μέρη ᾖ, καὶ ἕτερος ἑτέρου τὰ αὐτὰ If a number is parts of a number, and another (num-
μέρη ᾖ, καὶ ἐναλλάξ, ἃ μέρη ἐστὶν ὁ πρῶτος τοῦ τρίτου ἢ ber) is the same parts of another, also, alternately,
μέρος, τὰ αὐτὰ μέρη ἔσται καὶ ὁ δεύτερος τοῦ τετάρτου ἢ which(ever) parts, or part, the first (number) is of the
τὸ αὐτὸ μέρος. third, the second will also be the same parts, or the same
᾿Αριθμὸς γὰρ ὁ ΑΒ ἀριθμοῦ τοῦ Γ μέρη ἔστω, καὶ ἕτερος part, of the fourth.
ὁ ΔΕ ἑτέρου τοῦ Ζ τὰ αὐτὰ μέρη· λέγω, ὅτι καὶ ἐναλλάξ, For let a number AB be parts of a number C, and
ἃ μέρη ἐστὶν ὁ ΑΒ τοῦ ΔΕ ἢ μέρος, τὰ αὐτὰ μέρη ἐστὶ καὶ another (number) DE (be) the same parts of another F.
ὁ Γ τοῦ Ζ ἢ τὸ αὐτὸ μέρος. I say that, also, alternately, which(ever) parts, or part,


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ELEMENTS BOOK 7



AB is of DE, C is also the same parts, or the same part,
of F.




∆ D

Α A

Θ H
Η G


Β Ε B E
Γ Ζ C F
᾿Επεὶ γάρ, ἃ μέρη ἐστὶν ὁ ΑΒ τοῦ Γ, τὰ αὐτὰ μέρη ἐστὶ For since which(ever) parts AB is of C, DE is also
καὶ ὁ ΔΕ τοῦ Ζ, ὅσα ἄρα ἐστὶν ἐν τῷ ΑΒ μέρη τοῦ Γ, the same parts of F, thus as many parts of C as are in
τοσαῦτα καὶ ἐν τῷ ΔΕ μέρη τοῦ Ζ. διῃρήσθω ὁ μὲν ΑΒ εἰς AB, so many parts of F (are) also in DE. Let AB have
τὰ τοῦ Γ μέρη τὰ ΑΗ, ΗΒ, ὁ δὲ ΔΕ εἰς τὰ τοῦ Ζ μέρη τὰ been divided into the parts of C, AG and GB, and DE
ΔΘ, ΘΕ· ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΑΗ, ΗΒ τῷ πλήθει into the parts of F, DH and HE. So the multitude of
τῶν ΔΘ, ΘΕ. καὶ ἐπεί, ὃ μέρος ἐστὶν ὁ ΑΗ τοῦ Γ, τὸ αὐτὸ (divisions) AG, GB will be equal to the multitude of (di-
μέρος ἐστὶ καὶ ὁ ΔΘ τοῦ Ζ, καὶ ἐναλλάξ, ὃ μέρος ἐστὶν ὁ visions) DH, HE. And since which(ever) part AG is
ΑΗ τοῦ ΔΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Ζ ἢ of C, DH is also the same part of F, also, alternately,
τὰ αὐτὰ μέρη. διὰ τὰ αὐτὰ δὴ καί, ὃ μέρος ἐστὶν ὁ ΗΒ τοῦ which(ever) part, or parts, AG is of DH, C is also the
ΘΕ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Ζ ἢ τὰ αὐτὰ same part, or the same parts, of F [Prop. 7.9]. And so,
μέρη· ὥστε καί [ὃ μέρος ἐστὶν ὁ ΑΗ τοῦ ΔΘ ἢ μέρη, τὸ for the same (reasons), which(ever) part, or parts, GB is
αὐτὸ μέρος ἐστὶ καὶ ὁ ΗΒ τοῦ ΘΕ ἢ τὰ αὐτὰ μέρη· καὶ ὃ of HE, C is also the same part, or the same parts, of F
ἄρα μέρος ἐστὶν ὁ ΑΗ τοῦ ΔΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ [Prop. 7.9]. And so [which(ever) part, or parts, AG is of
καὶ ὁ ΑΒ τοῦ ΔΕ ἢ τὰ αὐτὰ μέρη· ἀλλ᾿ ὃ μέρος ἐστὶν ὁ ΑΗ DH, GB is also the same part, or the same parts, of HE.
τοῦ ΔΘ ἢ μέρη, τὸ αὐτὸ μέρος ἐδείχθη καὶ ὁ Γ τοῦ Ζ ἢ τὰ And thus, which(ever) part, or parts, AG is of DH, AB is
αὐτὰ μέρη, καὶ] ἃ [ἄρα] μέρη ἐστὶν ὁ ΑΒ τοῦ ΔΕ ἢ μέρος, also the same part, or the same parts, of DE [Props. 7.5,
iaþ C is also the same parts, or the same part, of F. (Which
τὰ αὐτὰ μέρη ἐστὶ καὶ ὁ Γ τοῦ Ζ ἢ τὸ αὐτὸ μέρος· ὅπερ ἔδει 7.6]. But, which(ever) part, or parts, AG is of DH, C
δεῖξαι. was also shown (to be) the same part, or the same parts,
of F. And, thus] which(ever) parts, or part, AB is of DE,

is) the very thing it was required to show.

† In modern notation, this proposition states that if a = (m/n) b and c = (m/n) d then if a = (k/l) c then b = (k/l) d, where all symbols denote
numbers.

.
Proposition 11
᾿Εαν ᾖ ὡς ὅλος πρὸς ὅλον, οὕτως ἀφαιρεθεὶς πρὸς ἀφαι- If as the whole (of a number) is to the whole (of an-
ρεθέντα, καὶ ὁ λοιπὸς πρὸς τὸν λοιπὸν ἔσται, ὡς ὅλος πρὸς other), so a (part) taken away (is) to a (part) taken away,
ὅλον. then the remainder will also be to the remainder as the
῎Εστω ὡς ὅλος ὁ ΑΒ πρὸς ὅλον τὸν ΓΔ, οὕτως ἀφαι- whole (is) to the whole.
ρεθεὶς ὁ ΑΕ πρὸς ἀφαιρεθέντα τὸν ΓΖ· λέγω, ὅτι καὶ λοιπὸς Let the whole AB be to the whole CD as the (part)
ὁ ΕΒ πρὸς λοιπὸν τὸν ΖΔ ἐστιν, ὡς ὅλος ὁ ΑΒ πρὸς ὅλον taken away AE (is) to the (part) taken away CF. I say
τὸν ΓΔ. that the remainder EB is to the remainder FD as the
whole AB (is) to the whole CD.





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Γ C ELEMENTS BOOK 7




Ζ F
Α A


Ε E




Β ∆ B D
᾿Επεί ἐστιν ὡς ὁ ΑΒ πρὸς τὸν ΓΔ, οὕτως ὁ ΑΕ πρὸς (For) since as AB is to CD, so AE (is) to CF, thus
ibþ (Which is) the very thing it was required to show.
τὸν ΓΖ, ὃ ἄρα μέρος ἐστὶν ὁ ΑΒ τοῦ ΓΔ ἢ μέρη, τὸ αὐτὸ which(ever) part, or parts, AB is of CD, AE is also the
μέρος ἐστὶ καὶ ὁ ΑΕ τοῦ ΓΖ ἢ τὰ αὐτὰ μέρη. καὶ λοιπὸς same part, or the same parts, of CF [Def. 7.20]. Thus,
ἄρα ὁ ΕΒ λοιποῦ τοῦ ΖΔ τὸ αὐτὸ μέρος ἐστὶν ἢ μέρη, ἅπερ the remainder EB is also the same part, or parts, of the
ὁ ΑΒ τοῦ ΓΔ. ἔστιν ἄρα ὡς ὁ ΕΒ πρὸς τὸν ΖΔ, οὕτως ὁ remainder FD that AB (is) of CD [Props. 7.7, 7.8].
ΑΒ πρὸς τὸν ΓΔ· ὅπερ ἔδει δεῖξαι. Thus, as EB is to FD, so AB (is) to CD [Def. 7.20].



† In modern notation, this proposition states that if a : b :: c : d then a : b :: a − c : b − d, where all symbols denote numbers.
.
†
Proposition 12
᾿Εὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἀνάλογον, ἔσται ὡς εἷς If any multitude whatsoever of numbers are propor-
τῶν ἡγουμένων πρὸς ἕνα τῶν ἑπομένων, οὕτως ἅπαντες οἱ tional then as one of the leading (numbers is) to one of
ἡγούμενοι πρὸς ἅπαντας τοὺς ἑπομένους. the following so (the sum of) all of the leading (numbers)
will be to (the sum of) all of the following.
















Α Β Γ ∆ A B C D
῎Εστωσαν ὁποσοιοῦν ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ, Let any multitude whatsoever of numbers, A, B, C,
ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ· λέγω, ὅτι ἐστὶν D, be proportional, (such that) as A (is) to B, so C (is)
ὡς ὁ Α πρὸς τὸν Β, οὕτως οἱ Α, Γ πρὸς τοὺς Β, Δ. to D. I say that as A is to B, so A, C (is) to B, D.
᾿Επεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς For since as A is to B, so C (is) to D, thus which(ever)
τὸν Δ, ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ Β ἢ μέρη, τὸ αὐτὸ μέρος part, or parts, A is of B, C is also the same part, or parts,
ἐστὶ καὶ ὁ Γ τοῦ Δ ἢ μέρη. καὶ συναμφότερος ἄρα ὁ Α, of D [Def. 7.20]. Thus, the sum A, C is also the same
Γ συναμφοτέρου τοῦ Β, Δ τὸ αὐτὸ μέρος ἐστὶν ἢ τὰ αὐτὰ part, or the same parts, of the sum B, D that A (is) of B
μέρη, ἅπερ ὁ Α τοῦ Β. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως [Props. 7.5, 7.6]. Thus, as A is to B, so A, C (is) to B, D
οἱ Α, Γ πρὸς τοὺς Β, Δ· ὅπερ ἔδει δεῖξαι. [Def. 7.20]. (Which is) the very thing it was required to
show.



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† In modern notation, this proposition states that if a : b :: c : d then a : b :: a + c : b + d, where all symbols denote numbers.
Proposition 13
†
.
᾿Εὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, καὶ ἐναλλὰξ If four numbers are proportional then they will also
ἀνάλογον ἔσονται. be proportional alternately.
















Α Β Γ ∆ A B C D
῎Εστωσαν τέσσαρες ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ, Let the four numbers A, B, C, and D be proportional,
ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ· λέγω, ὅτι καὶ (such that) as A (is) to B, so C (is) to D. I say that they
ἐναλλὰξ ἀνάλογον ἔσονται, ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Β will also be proportional alternately, (such that) as A (is)
πρὸς τὸν Δ. to C, so B (is) to D.
᾿Επεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν For since as A is to B, so C (is) to D, thus which(ever)
idþ thing it was required to show.
Δ, ὃ ἄρα μέρος ἐστὶν ὁ Α τοῦ Β ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ part, or parts, A is of B, C is also the same part,
καὶ ὁ Γ τοῦ Δ ἢ τὰ αὐτὰ μέρη. ἐναλλὰξ ἄρα, ὃ μέρος ἐστὶν or the same parts, of D [Def. 7.20]. Thus, alterately,
ὁ Α τοῦ Γ ἢ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Β τοῦ Δ ἢ τὰ which(ever) part, or parts, A is of C, B is also the same
αὐτὰ μέρη. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Β πρὸς part, or the same parts, of D [Props. 7.9, 7.10]. Thus, as
τὸν Δ· ὅπερ ἔδει δεῖξαι. A is to C, so B (is) to D [Def. 7.20]. (Which is) the very


† In modern notation, this proposition states that if a : b :: c : d then a : c :: b : d, where all symbols denote numbers. †

Proposition 14
.
᾿Εὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ καὶ ἄλλοι αὐτοῖς ἴσοι τὸ If there are any multitude of numbers whatsoever,
πλῆθος σύνδυο λαμβανόμενοι καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ δι᾿ and (some) other (numbers) of equal multitude to them,
ἴσου ἐν τῷ αὐτῷ λόγῷ ἔσονται. (which are) also in the same ratio taken two by two, then
they will also be in the same ratio via equality.
Α ∆ A D
Β Ε B E
Γ Ζ C F
῎Εστωσαν ὁποσοιοῦν ἀριθμοὶ οἱ Α, Β, Γ καὶ ἄλλοι αὐτοῖς Let there be any multitude of numbers whatsoever, A,
ἴσοι τὸ πλῆθος σύνδυο λαμβανόμενοι ἐν τῷ αὐτῷ λόγῳ οἱ B, C, and (some) other (numbers), D, E, F, of equal
Δ, Ε, Ζ, ὡς μὲν ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Ε, multitude to them, (which are) in the same ratio taken
ὡς δὲ ὁ Β πρὸς τὸν Γ, οὕτως ὁ Ε πρὸς τὸν Ζ· λέγω, ὅτι two by two, (such that) as A (is) to B, so D (is) to E,
καὶ δι᾿ ἴσου ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς τὸν and as B (is) to C, so E (is) to F. I say that also, via
Ζ. equality, as A is to C, so D (is) to F.
᾿Επεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς For since as A is to B, so D (is) to E, thus, alternately,
τὸν Ε, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, οὕτως ὁ Β as A is to D, so B (is) to E [Prop. 7.13]. Again, since
πρὸς τὸν Ε. πάλιν, ἐπεί ἐστιν ὡς ὁ Β πρὸς τὸν Γ, οὕτως ὁ as B is to C, so E (is) to F, thus, alternately, as B is


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Ε πρὸς τὸν Ζ, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Β πρὸς τὸν Ε, οὕτως to E, so C (is) to F [Prop. 7.13]. And as B (is) to E,
ὁ Γ πρὸς τὸν Ζ. ὡς δὲ ὁ Β πρὸς τὸν Ε, οὕτως ὁ Α πρὸς so A (is) to D. Thus, also, as A (is) to D, so C (is) to F.
τὸν Δ· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Δ, οὕτως ὁ Γ πρὸς τὸν Thus, alternately, as A is to C, so D (is) to F [Prop. 7.13].
Ζ· ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Δ πρὸς (Which is) the very thing it was required to show.
τὸν Ζ· ὅπερ ἔδει δεῖξαι.
† In modern notation, this proposition states that if a : b :: d : e and b : c :: e : f then a : c :: d : f, where all symbols denote numbers.

Proposition 15
.
᾿Εὰν μονὰς ἀριθμόν τινα μετρῇ, ἰσακις δὲ ἕτερος ἀριθμὸς If a unit measures some number, and another num-
ἄλλον τινὰ ἀριθμὸν μετρῇ, καὶ ἐναλλὰξ ἰσάκις ἡ μονὰς τὸν ber measures some other number as many times, then,
τρίτον ἀριθμὸν μετρήσει καὶ ὁ δεύτερος τὸν τέταρτον. also, alternately, the unit will measure the third num-
ber as many times as the second (number measures) the
fourth.
Β Η Θ Γ B G H C
Α A

Ε Κ Λ Ζ E K L F
∆ D
Μονὰς γὰρ ἡ Α ἀριθμόν τινα τὸν ΒΓ μετρείτω, ἰσάκις δὲ For let a unit A measure some number BC, and let
ἕτερος ἀριθμὸς ὁ Δ ἄλλον τινὰ ἀριθμὸν τὸν ΕΖ μετρείτω· another number D measure some other number EF as
λέγω, ὅτι καὶ ἐναλλὰξ ἰσάκις ἡ Α μονὰς τὸν Δ ἀριθμὸν many times. I say that, also, alternately, the unit A also
μετρεῖ καὶ ὁ ΒΓ τὸν ΕΖ. measures the number D as many times as BC (measures)
᾿Επεὶ γὰρ ἰσάκις ἡ Α μονὰς τὸν ΒΓ ἀριθμὸν μετρεῖ καὶ ὁ EF.
Δ τὸν ΕΖ, ὅσαι ἄρα εἰσὶν ἐν τῷ ΒΓ μονάδες, τοσοῦτοί εἰσι For since the unit A measures the number BC as
καὶ ἐν τῷ ΕΖ ἀριθμοὶ ἴσοι τῷ Δ. διῃρήσθω ὁ μὲν ΒΓ εἰς τὰς many times as D (measures) EF, thus as many units as
ἐν ἑαυτῷ μονάδας τὰς ΒΗ, ΗΘ, ΘΓ, ὁ δὲ ΕΖ εἰς τοὺς τῷ Δ are in BC, so many numbers are also in EF equal to
ἴσους τοὺς ΕΚ, ΚΛ, ΛΖ. ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, D. Let BC have been divided into its constituent units,
ΗΘ, ΘΓ τῷ πλήθει τῶν ΕΚ, ΚΛ, ΛΖ. καὶ ἐπεὶ ἴσαι εἰσὶν αἱ BG, GH, and HC, and EF into the (divisions) EK, KL,
ΒΗ, ΗΘ, ΘΓ μονάδες ἀλλήλαις, εἰσὶ δὲ καὶ οἱ ΕΚ, ΚΛ, ΛΖ and LF, equal to D. So the multitude of (units) BG,
ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν ἴσον τὸ πλῆθος τῶν ΒΗ, GH, HC will be equal to the multitude of (divisions)
ΗΘ, ΘΓ μονάδων τῷ πλήθει τῶν ΕΚ, ΚΛ, ΛΖ ἀριθμῶν, EK, KL, LF. And since the units BG, GH, and HC
ἔσται ἄρα ὡς ἡ ΒΗ μονὰς πρὸς τὸν ΕΚ ἀριθμόν, οὕτως ἡ are equal to one another, and the numbers EK, KL, and
ΗΘ μονὰς πρὸς τὸν ΚΛ ἀριθμὸν καὶ ἡ ΘΓ μονὰς πρὸς τὸν LF are also equal to one another, and the multitude of
ΛΖ ἀριθμόν. ἔσται ἄρα καὶ ὡς εἷς τῶν ἡγουμένων πρὸς ἕνα the (units) BG, GH, HC is equal to the multitude of the
τῶν ἑπομένων, οὕτως ἅπαντες οἱ ἡγούμενοι πρὸς ἅπαντας numbers EK, KL, LF, thus as the unit BG (is) to the
τοὺς ἑπομένους· ἔστιν ἄρα ὡς ἡ ΒΗ μονὰς πρὸς τὸν ΕΚ number EK, so the unit GH will be to the number KL,
ἀριθμόν, οὕτως ὁ ΒΓ πρὸς τὸν ΕΖ. ἴση δὲ ἡ ΒΗ μονὰς τῇ and the unit HC to the number LF. And thus, as one of
Α μονάδι, ὁ δὲ ΕΚ ἀριθμὸς τῷ Δ ἀριθμῷ. ἔστιν ἄρα ὡς ἡ the leading (numbers is) to one of the following, so (the
Α μονὰς πρὸς τὸν Δ ἀριθμόν, οὕτως ὁ ΒΓ πρὸς τὸν ΕΖ. sum of) all of the leading will be to (the sum of) all of
ἰσάκις ἄρα ἡ Α μονὰς τὸν Δ ἀριθμὸν μετρεῖ καὶ ὁ ΒΓ τὸν the following [Prop. 7.12]. Thus, as the unit BG (is) to
ΕΖ· ὅπερ ἔδει δεῖξαι. the number EK, so BC (is) to EF. And the unit BG (is)
equal to the unit A, and the number EK to the number
D. Thus, as the unit A is to the number D, so BC (is) to
EF. Thus, the unit A measures the number D as many
times as BC (measures) EF [Def. 7.20]. (Which is) the
very thing it was required to show.

† This proposition is a special case of Prop. 7.9.





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†
Proposition 16
.
Εὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί If two numbers multiplying one another make some
τινας, οἱ γενόμενοι ἐξ αὐτῶν ἴσοι ἀλλήλοις ἔσονται. (numbers) then the (numbers) generated from them will
be equal to one another.
Α A


Β B

Γ C

∆ D

Ε E
῎Εστωσαν δύο ἀριθμοὶ οἱ Α, Β, καὶ ὁ μὲν Α τὸν Β πολ- Let A and B be two numbers. And let A make C (by)
λαπλασιάσας τὸν Γ ποιείτω, ὁ δὲ Β τὸν Α πολλαπλασιάσας multiplying B, and let B make D (by) multiplying A. I
τὸν Δ ποιείτω· λέγω, ὅτι ἴσος ἐστὶν ὁ Γ τῷ Δ. say that C is equal to D.
᾿Επεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, For since A has made C (by) multiplying B, B thus
ὁ Β ἄρα τὸν Γ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας. μετρεῖ δὲ measures C according to the units in A [Def. 7.15]. And
καὶ ἡ Ε μονὰς τὸν Α ἀριθμὸν κατὰ τὰς ἐν αὐτῷ μονάδας· the unit E also measures the number A according to the
ἰσάκις ἄρα ἡ Ε μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ ὁ Β τὸν units in it. Thus, the unit E measures the number A as
Γ. ἐναλλὰξ ἄρα ἰσάκις ἡ Ε μονὰς τὸν Β ἀριθμὸν μετρεῖ καὶ many times as B (measures) C. Thus, alternately, the
ὁ Α τὸν Γ. πάλιν, ἐπεὶ ὁ Β τὸν Α πολλαπλασιάσας τὸν Δ unit E measures the number B as many times as A (mea-
πεποίηκεν, ὁ Α ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Β μονάδας. sures) C [Prop. 7.15]. Again, since B has made D (by)
μετρεῖ δὲ καὶ ἡ Ε μονὰς τὸν Β κατὰ τὰς ἐν αὐτῷ μονάδας· multiplying A, A thus measures D according to the units
ἰσάκις ἄρα ἡ Ε μονὰς τὸν Β ἀριθμὸν μετρεῖ καὶ ὁ Α τὸν Δ. in B [Def. 7.15]. And the unit E also measures B ac-
izþ equal number of times. Thus, C is equal to D. (Which is)
ἰσάκις δὲ ἡ Ε μονὰς τὸν Β ἀριθμὸν ἐμέτρει καὶ ὁ Α τὸν Γ· cording to the units in it. Thus, the unit E measures the
ἰσάκις ἄρα ὁ Α ἑκάτερον τῶν Γ, Δ μετρεῖ. ἴσος ἄρα ἐστὶν number B as many times as A (measures) D. And the
ὁ Γ τῷ Δ· ὅπερ ἔδει δεῖξαι. unit E was measuring the number B as many times as
A (measures) C. Thus, A measures each of C and D an

the very thing it was required to show.

† In modern notation, this proposition states that a b = b a, where all symbols denote numbers.
Proposition 17
†
.
᾿Εὰν ἀριθμὸς δύο ἀριθμοὺς πολλαπλασιάσας ποιῇ τινας, If a number multiplying two numbers makes some
οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον τοῖς πολλα- (numbers) then the (numbers) generated from them will
πλασιασθεῖσιν. have the same ratio as the multiplied (numbers).
Α A

Β Γ B C
∆ Ε D E
Ζ F
᾿Αριθμὸς γὰρ ὁ Α δύο ἀριθμοὺς τοὺς Β, Γ πολλα- For let the number A make (the numbers) D and
πλασιάσας τοὺς Δ, Ε ποιείτω· λέγω, ὅτι ἐστὶν ὡς ὁ Β πρὸς E (by) multiplying the two numbers B and C (respec-
τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ε. tively). I say that as B is to C, so D (is) to E.
᾿Επεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Δ πεποίηκεν, For since A has made D (by) multiplying B, B thus
ὁ Β ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Α μονάδας. μετρεῖ measures D according to the units in A [Def. 7.15]. And


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δὲ καὶ ἡ Ζ μονὰς τὸν Α ἀριθμὸν κατὰ τὰς ἐν αὐτῷ μονάδας· the unit F also measures the number A according to the
ἰσάκις ἄρα ἡ Ζ μονὰς τὸν Α ἀριθμὸν μετρεῖ καὶ ὁ Β τὸν Δ. units in it. Thus, the unit F measures the number A as
ihþ (Which is) the very thing it was required to show.
ἔστιν ἄρα ὡς ἡ Ζ μονὰς πρὸς τὸν Α ἀριθμόν, οὕτως ὁ Β many times as B (measures) D. Thus, as the unit F is
πρὸς τὸν Δ. διὰ τὰ αὐτὰ δὴ καὶ ὡς ἡ Ζ μονὰς πρὸς τὸν Α to the number A, so B (is) to D [Def. 7.20]. And so, for
ἀριθμόν, οὕτως ὁ Γ πρὸς τὸν Ε· καὶ ὡς ἄρα ὁ Β πρὸς τὸν the same (reasons), as the unit F (is) to the number A,
Δ, οὕτως ὁ Γ πρὸς τὸν Ε. ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Β πρὸς so C (is) to E. And thus, as B (is) to D, so C (is) to E.
τὸν Γ, οὕτως ὁ Δ πρὸς τὸν Ε· ὅπερ ἔδει δεῖξαι. Thus, alternately, as B is to C, so D (is) to E [Prop. 7.13].



† In modern notation, this proposition states that if d = a b and e = a c then d : e :: b : c, where all symbols denote numbers.

†
.
Proposition 18
᾿Εὰν δύο ἀριθμοὶ ἀριθμόν τινα πολλαπλασιάσαντες If two numbers multiplying some number make some
ποιῶσί τινας, οἱ γενόμενοι ἐξ αὐτῶν τὸν αὐτὸν ἕξουσι λόγον (other numbers) then the (numbers) generated from
τοῖς πολλαπλασιάσασιν. them will have the same ratio as the multiplying (num-
bers).
Α A

Β B

Γ C

∆ D

Ε E
Δύο γὰρ ἀριθμοὶ οἱ Α, Β ἀριθμόν τινα τὸν Γ πολλα- For let the two numbers A and B make (the numbers)
πλασιάσαντες τοὺς Δ, Ε ποιείτωσαν· λέγω, ὅτι ἐστὶν ὡς ὁ D and E (respectively, by) multiplying some number C.
Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς τὸν Ε. I say that as A is to B, so D (is) to E.
᾿Επεὶ γὰρ ὁ Α τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν, For since A has made D (by) multiplying C, C has
ijþ it was required to show.
καὶ ὁ Γ ἄρα τὸν Α πολλαπλασιάσας τὸν Δ πεποίηκεν. διὰ thus also made D (by) multiplying A [Prop. 7.16]. So, for
τὰ αὐτὰ δὴ καὶ ὁ Γ τὸν Β πολλαπλασιάσας τὸν Ε πεποίηκεν. the same (reasons), C has also made E (by) multiplying
ἀριθμὸς δὴ ὁ Γ δύο ἀριθμοὺς τοὺς Α, Β πολλαπλασιάσας B. So the number C has made D and E (by) multiplying
τοὺς Δ, Ε πεποίηκεν. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως the two numbers A and B (respectively). Thus, as A is to
ὁ Δ πρὸς τὸν Ε· ὅπερ ἔδει δεῖξαι. B, so D (is) to E [Prop. 7.17]. (Which is) the very thing


† In modern notation, this propositions states that if a c = d and b c = e then a : b :: d : e, where all symbols denote numbers.

.
Proposition 19
†
᾿Εὰν τέσσαρες ἀριθμοὶ ἀνάλογον ὦσιν, ὁ ἐκ πρώτου καὶ If four number are proportional then the number cre-
τετάρτου γενόμενος ἀριθμὸς ἴσος ἔσται τῷ ἐκ δευτέρου καὶ ated from (multiplying) the first and fourth will be equal
τρίτου γενομένῳ ἀριθμῷ· καὶ ἐὰν ὁ ἐκ πρώτου καὶ τετάρτου to the number created from (multiplying) the second and
γενόμενος ἀριθμὸς ἴσος ᾖ τῷ ἐκ δευτέρου καὶ τρίτου, οἱ third. And if the number created from (multiplying) the
τέσσασρες ἀριθμοὶ ἀνάλογον ἔσονται. first and fourth is equal to the (number created) from
῎Εστωσαν τέσσαρες ἀριθμοὶ ἀνάλογον οἱ Α, Β, Γ, Δ, (multiplying) the second and third then the four num-
ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ, καὶ ὁ μὲν Α bers will be proportional.
τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Β τὸν Γ πολ- Let A, B, C, and D be four proportional numbers,
λαπλασιάσας τὸν Ζ ποιείτω· λέγω, ὅτι ἴσος ἐστὶν ὁ Ε τῷ Ζ. (such that) as A (is) to B, so C (is) to D. And let A make
E (by) multiplying D, and let B make F (by) multiplying
C. I say that E is equal to F.

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G
F














῾Ο γὰρ Α τὸν Γ πολλαπλασιάσας τὸν Η ποιείτω. ἐπεὶ For let A make G (by) multiplying C. Therefore, since
οὖν ὁ Α τὸν Γ πολλαπλασιάσας τὸν Η πεποίηκεν, τὸν δὲ A has made G (by) multiplying C, and has made E (by)
Δ πολλαπλασιάσας τὸν Ε πεποίηκεν, ἀριθμὸς δὴ ὁ Α δύο multiplying D, the number A has made G and E by mul-
ἀριθμοὺς τοὺς Γ, Δ πολλαπλασιάσας τούς Η, Ε πεποίηκεν. tiplying the two numbers C and D (respectively). Thus,
ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Η πρὸς τὸν Ε. ἀλλ᾿ as C is to D, so G (is) to E [Prop. 7.17]. But, as C (is) to
ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Β· καὶ ὡς ἄρα D, so A (is) to B. Thus, also, as A (is) to B, so G (is) to
ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν Ε. πάλιν, ἐπεὶ ὁ Α E. Again, since A has made G (by) multiplying C, but,
τὸν Γ πολλαπλασιάσας τὸν Η πεποίηκεν, ἀλλὰ μὴν καὶ ὁ in fact, B has also made F (by) multiplying C, the two
Β τὸν Γ πολλαπλασιάσας τὸν Ζ πεποίηκεν, δύο δὴ ἀριθμοὶ numbers A and B have made G and F (respectively, by)
οἱ Α, Β ἀριθμόν τινα τὸν Γ πολλαπλασιάσαντες τοὺς Η, Ζ multiplying some number C. Thus, as A is to B, so G (is)
πεποιήκασιν. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς to F [Prop. 7.18]. But, also, as A (is) to B, so G (is) to
τὸν Ζ. ἀλλὰ μὴν καὶ ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς E. And thus, as G (is) to E, so G (is) to F. Thus, G has
τὸν Ε· καὶ ὡς ἄρα ὁ Η πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν the same ratio to each of E and F. Thus, E is equal to F
Ζ. ὁ Η ἄρα πρὸς ἑκάτερον τῶν Ε, Ζ τὸν αὐτὸν ἔχει λόγον· [Prop. 5.9].
ἴσος ἄρα ἐστὶν ὁ Ε τῷ Ζ. So, again, let E be equal to F. I say that as A is to B,
῎Εστω δὴ πάλιν ἴσος ὁ Ε τῷ Ζ· λέγω, ὅτι ἐστὶν ὡς ὁ Α so C (is) to D.
kþ
πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ. For, with the same construction, since E is equal to F,
Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ ἴσος ἐστὶν ὁ thus as G is to E, so G (is) to F [Prop. 5.7]. But, as G
Ε τῷ Ζ, ἔστιν ἄρα ὡς ὁ Η πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν (is) to E, so C (is) to D [Prop. 7.17]. And as G (is) to F,
Ζ. ἀλλ᾿ ὡς μὲν ὁ Η πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς τὸν Δ, ὡς so A (is) to B [Prop. 7.18]. And, thus, as A (is) to B, so
δὲ ὁ Η πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Β. καὶ ὡς ἄρα ὁ Α C (is) to D. (Which is) the very thing it was required to
πρὸς τὸν Β, οὕτως ὁ Γ πρὸς τὸν Δ· ὅπερ ἔδει δεῖξαι. show.

† In modern notation, this proposition reads that if a : b :: c : d then a d = b c, and vice versa, where all symbols denote numbers.

.
Proposition 20
Οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων The least numbers of those (numbers) having the
αὐτοῖς μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ same ratio measure those (numbers) having the same ra-
τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα. tio as them an equal number of times, the greater (mea-
῎Εστωσαν γὰρ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον suring) the greater, and the lesser the lesser.
ἐχόντων τοῖς Α, Β οἱ ΓΔ, ΕΖ· λέγω, ὅτι ἰσάκις ὁ ΓΔ τὸν For let CD and EF be the least numbers having the
Α μετρεῖ καὶ ὁ ΕΖ τὸν Β. same ratio as A and B (respectively). I say that CD mea-
sures A the same number of times as EF (measures) B.










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Θ H
Η G

Ζ F

∆ D




῾Ο ΓΔ γὰρ τοῦ Α οὔκ ἐστι μέρη. εἰ γὰρ δυνατόν, ἔστω· For CD is not parts of A. For, if possible, let it be
καὶ ὁ ΕΖ ἄρα τοῦ Β τὰ αὐτὰ μέρη ἐστίν, ἅπερ ὁ ΓΔ τοῦ (parts of A). Thus, EF is also the same parts of B that
Α. ὅσα ἄρα ἐστὶν ἐν τῷ ΓΔ μέρη τοῦ Α, τοσαῦτά ἐστι καὶ CD (is) of A [Def. 7.20, Prop. 7.13]. Thus, as many parts
ἐν τῷ ΕΖ μέρη τοῦ Β. διῃρήσθω ὁ μὲν ΓΔ εἰς τὰ τοῦ Α of A as are in CD, so many parts of B are also in EF. Let
μέρη τὰ ΓΗ, ΗΔ, ὁ δὲ ΕΖ εἰς τὰ τοῦ Β μέρη τὰ ΕΘ, ΘΖ· CD have been divided into the parts of A, CG and GD,
ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΓΗ, ΗΔ τῷ πλήθει τῶν ΕΘ, and EF into the parts of B, EH and HF. So the multi-
ΘΖ. καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΓΗ, ΗΔ ἀριθμοὶ ἀλλήλοις, εἰσὶ tude of (divisions) CG, GD will be equal to the multitude
δὲ καὶ οἱ ΕΘ, ΘΖ ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν ἴσον τὸ of (divisions) EH, HF. And since the numbers CG and
πλῆθος τῶν ΓΗ, ΗΔ τῷ πλήθει τῶν ΕΘ, ΘΖ, ἔστιν ἄρα ὡς GD are equal to one another, and the numbers EH and
ὁ ΓΗ πρὸς τὸν ΕΘ, οὕτως ὁ ΗΔ πρὸς τὸν ΘΖ. ἔσται ἄρα HF are also equal to one another, and the multitude of
καὶ ὡς εἷς τῶν ἡγουμένων πρὸς ἕνα τῶν ἑπομένων, οὕτως (divisions) CG, GD is equal to the multitude of (divi-
ἅπαντες οἱ ἡγούμενοι πρὸς ἅπαντας τοὺς ἑπομένους. ἔστιν sions) EH, HF, thus as CG is to EH, so GD (is) to HF.
ἄρα ὡς ὁ ΓΗ πρὸς τὸν ΕΘ, οὕτως ὁ ΓΔ πρὸς τὸν ΕΖ· οἱ Thus, as one of the leading (numbers is) to one of the
ΓΗ, ΕΘ ἄρα τοῖς ΓΔ, ΕΖ ἐν τῷ αὐτῷ λόγῳ εἰσὶν ἐλάσσονες following, so will (the sum of) all of the leading (num-
ὄντες αὐτῶν· ὅπερ ἐστὶν ἀδύνατον· ὑπόκεινται γὰρ οἱ ΓΔ, bers) be to (the sum of) all of the following [Prop. 7.12].
ΕΖ ἐλάχιστοι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς. οὐκ Thus, as CG is to EH, so CD (is) to EF. Thus, CG
ἄρα μέρη ἐστὶν ὁ ΓΔ τοῦ Α· μέρος ἄρα. καὶ ὁ ΕΖ τοῦ Β τὸ and EH are in the same ratio as CD and EF, being less
kaþ A. Thus, (it is) a part (of A) [Prop. 7.4]. And EF is the
αὐτὸ μέρος ἐστίν, ὅπερ ὁ ΓΔ τοῦ Α· ἰσάκις ἄρα ὁ ΓΔ τὸν than them. The very thing is impossible. For CD and
Α μετρεῖ καὶ ὁ ΕΖ τὸν Β· ὅπερ ἔδει δεῖξαι. EF were assumed (to be) the least of those (numbers)
having the same ratio as them. Thus, CD is not parts of

same part of B that CD (is) of A [Def. 7.20, Prop 7.13].
Thus, CD measures A the same number of times that EF
(measures) B. (Which is) the very thing it was required
to show.

.
Proposition 21
Οἱ πρῶτοι πρὸς ἀλλήλους ἀριθμοὶ ἐλάχιστοί εἰσι τῶν τὸν Numbers prime to one another are the least of those
αὐτὸν λόγον ἐχόντων αὐτοῖς. (numbers) having the same ratio as them.
῎Εστωσαν πρῶτοι πρὸς ἀλλήλους ἀριθμοὶ οἱ Α, Β· λέγω, Let A and B be numbers prime to one another. I say
ὅτι οἱ Α, Β ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων that A and B are the least of those (numbers) having the
αὐτοῖς. same ratio as them.
Εἰ γὰρ μή, ἔσονταί τινες τῶν Α, Β ἐλάσσονες ἀριθμοὶ For if not then there will be some numbers less than A
ἐν τῷ αὐτῷ λόγῳ ὄντες τοῖς Α, Β. ἔστωσαν οἱ Γ, Δ. and B which are in the same ratio as A and B. Let them
be C and D.






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D













᾿Επεὶ οὖν οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον Therefore, since the least numbers of those (num-
ἐχόντων μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις bers) having the same ratio measure those (numbers)
ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάττων τὸν ἐλάττονα, having the same ratio (as them) an equal number of
τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος times, the greater (measuring) the greater, and the lesser
τὸν ἑπόμενον, ἰσάκις ἄρα ὁ Γ τὸν Α μετρεῖ καὶ ὁ Δ τὸν Β. the lesser—that is to say, the leading (measuring) the
ὁσάκις δὴ ὁ Γ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν leading, and the following the following—C thus mea-
τῷ Ε. καὶ ὁ Δ ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας. sures A the same number of times that D (measures) B
καὶ ἐπεὶ ὁ Γ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας, καί ὁ [Prop. 7.20]. So as many times as C measures A, so many
Ε ἄρα τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Γ μονάδας. διὰ τὰ αὐτὰ units let there be in E. Thus, D also measures B accord-
δὴ ὁ Ε καὶ τὸν Β μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας. ὁ Ε ing to the units in E. And since C measures A according
ἄρα τοὺς Α, Β μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ to the units in E, E thus also measures A according to
kbþ be any numbers less than A and B which are in the same
ἐστὶν ἀδύνατον. οὐκ ἄρα ἔσονταί τινες τῶν Α, Β ἐλάσσονες the units in C [Prop. 7.16]. So, for the same (reasons), E
ἀριθμοὶ ἐν τῷ αὐτῷ λόγῳ ὄντες τοῖς Α, Β. οἱ Α, Β ἄρα also measures B according to the units in D [Prop. 7.16].
ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς· ὅπερ Thus, E measures A and B, which are prime to one an-
ἔδει δεῖξαι. other. The very thing is impossible. Thus, there cannot

ratio as A and B. Thus, A and B are the least of those
(numbers) having the same ratio as them. (Which is) the
very thing it was required to show.

Proposition 22
.
Οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων The least numbers of those (numbers) having the
αὐτοῖς πρῶτοι πρὸς ἀλλήλους εἰσίν. same ratio as them are prime to one another.
Α A

Β B


Γ C

∆ D

Ε E
῎Εστωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον Let A and B be the least numbers of those (numbers)
ἐχόντων αὐτοῖς οἱ Α, Β· λέγω, ὅτι οἱ Α, Β πρῶτοι πρὸς having the same ratio as them. I say that A and B are
ἀλλήλους εἰσίν. prime to one another.
Εἰ γὰρ μή εἰσι πρῶτοι πρὸς ἀλλήλους, μετρήσει τις For if they are not prime to one another then some
αὐτοὺς ἀριθμός. μετρείτω, καὶ ἔστω ὁ Γ. καὶ ὁσάκις μὲν number will measure them. Let it (so measure them),
ὁ Γ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Δ, and let it be C. And as many times as C measures A, so


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ὁσάκις δὲ ὁ Γ τὸν Β μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν many units let there be in D. And as many times as C
τῷ Ε. measures B, so many units let there be in E.
᾿Επεὶ ὁ Γ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας, ὁ Since C measures A according to the units in D, C
Γ ἄρα τὸν Δ πολλαπλασιάσας τὸν Α πεποίηκεν. διὰ τὰ has thus made A (by) multiplying D [Def. 7.15]. So, for
αὐτὰ δὴ καὶ ὁ Γ τὸν Ε πολλαπλασιάσας τὸν Β πεποίηκεν. the same (reasons), C has also made B (by) multiplying
kgþ
ἀριθμὸς δὴ ὁ Γ δύο ἀριθμοὺς τοὺς Δ, Ε πολλαπλασιάσας E. So the number C has made A and B (by) multiplying
τοὺς Α, Β πεποίηκεν· ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Ε, οὕτως the two numbers D and E (respectively). Thus, as D is
ὁ Α πρὸς τὸν Β· οἱ Δ, Ε ἄρα τοῖς Α, Β ἐν τῷ αὐτῷ λόγῳ to E, so A (is) to B [Prop. 7.17]. Thus, D and E are in
εἰσὶν ἐλάσσονες ὄντες αὐτῶν· ὅπερ ἐστὶν ἀδύνατον. οὐκ the same ratio as A and B, being less than them. The
ἄρα τοὺς Α, Β ἀριθμοὺς ἀριθμός τις μετρήσει. οἱ Α, Β ἄρα very thing is impossible. Thus, some number does not
πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι. measure the numbers A and B. Thus, A and B are prime
to one another. (Which is) the very thing it was required
to show.
Proposition 23
.
᾿Εὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, ὁ τὸν ἕνα If two numbers are prime to one another then a num-
αὐτῶν μετρῶν ἀριθμὸς πρὸς τὸν λοιπὸν πρῶτος ἔσται. ber measuring one of them will be prime to the remaining
(one).
















Α Β Γ ∆ A B C D
.
῎Εστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους οἱ Α, Β, Let A and B be two numbers (which are) prime to
τὸν δὲ Α μετρείτω τις ἀριθμὸς ὁ Γ· λέγω, ὅτι καὶ οἱ Γ, Β one another, and let some number C measure A. I say
πρῶτοι πρὸς ἀλλήλους εἰσίν. that C and B are also prime to one another.
Εἰ γὰρ μή εἰσιν οἱ Γ, Β πρῶτοι πρὸς ἀλλήλους, μετρήσει For if C and B are not prime to one another then
kdþ
[τις] τοὺς Γ, Β ἀριθμός. μετρείτω, καὶ ἔστω ὁ Δ. ἐπεὶ ὁ [some] number will measure C and B. Let it (so) mea-
Δ τὸν Γ μετρεῖ, ὁ δὲ Γ τὸν Α μετρεῖ, καὶ ὁ Δ ἄρα τὸν Α sure (them), and let it be D. Since D measures C, and C
μετρεῖ. μετρεῖ δὲ καὶ τὸν Β· ὁ Δ ἄρα τοὺς Α, Β μετρεῖ measures A, D thus also measures A. And (D) also mea-
πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον. οὐκ sures B. Thus, D measures A and B, which are prime
ἄρα τοὺς Γ, Β ἀριθμοὺς ἀριθμός τις μετρήσει. οἱ Γ, Β ἄρα to one another. The very thing is impossible. Thus, some
πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι. number does not measure the numbers C and B. Thus,
C and B are prime to one another. (Which is) the very
thing it was required to show.

Proposition 24
.
᾿Εὰν δύο ἀριθμοὶ πρός τινα ἀριθμὸν πρῶτοι ὦσιν, καὶ ὁ If two numbers are prime to some number then the
ἐξ αὐτῶν γενόμενος πρὸς τὸν αὐτὸν πρῶτος ἔσται. number created from (multiplying) the former (two num-
bers) will also be prime to the latter (number).





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Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρός τινα ἀριθμὸν τὸν Γ πρῶτοι For let A and B be two numbers (which are both)
ἔστωσαν, καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Δ ποιείτω· prime to some number C. And let A make D (by) multi-
λέγω, ὅτι οἱ Γ, Δ πρῶτοι πρὸς ἀλλήλους εἰσίν. plying B. I say that C and D are prime to one another.
Εἰ γὰρ μή εἰσιν οἱ Γ, Δ πρῶτοι πρὸς ἀλλήλους, μετρήσει For if C and D are not prime to one another then
[τις] τοὺς Γ, Δ ἀριθμός. μετρείτω, καὶ ἔστω ὁ Ε. καὶ ἐπεὶ [some] number will measure C and D. Let it (so) mea-
οἱ Γ, Δ πρῶτοι πρὸς ἀλλήλους εἰσίν, τὸν δὲ Γ μετρεῖ τις sure them, and let it be E. And since C and A are prime
ἀριθμὸς ὁ Ε, οἱ Α, Ε ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν. to one another, and some number E measures C, A and
ὁσάκις δὴ ὁ Ε τὸν Δ μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν E are thus prime to one another [Prop. 7.23]. So as
τῷ Ζ· καὶ ὁ Ζ ἄρα τὸν Δ μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας. many times as E measures D, so many units let there
ὁ Ε ἄρα τὸν Ζ πολλαπλασιάσας τὸν Δ πεποίηκεν. ἀλλὰ be in F. Thus, F also measures D according to the units
μὴν καὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Δ πεποίηκεν· ἴσος in E [Prop. 7.16]. Thus, E has made D (by) multiply-
ἄρα ἐστὶν ὁ ἑκ τῶν Ε, Ζ τῷ ἐκ τῶν Α, Β. ἐὰν δὲ ὁ ὑπὸ ing F [Def. 7.15]. But, in fact, A has also made D (by)
τῶν ἄκρων ἴσος ᾖ τῷ ὑπὸ τῶν μέσων, οἱ τέσσαρες ἀριθμοὶ multiplying B. Thus, the (number created) from (multi-
ἀνάλογόν εἰσιν· ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Α, οὕτως ὁ Β plying) E and F is equal to the (number created) from
πρὸς τὸν Ζ. οἱ δὲ Α, Ε πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ (multiplying) A and B. And if the (rectangle contained)
δὲ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς by the (two) outermost is equal to the (rectangle con-
μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων tained) by the middle (two) then the four numbers are
τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, τουτέστιν ὅ τε proportional [Prop. 6.15]. Thus, as E is to A, so B (is)
ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· ὁ to F. And A and E (are) prime (to one another). And
Ε ἄρα τὸν Β μετρεῖ. μετρεῖ δὲ καὶ τὸν Γ· ὁ Ε ἄρα τοὺς Β, Γ (numbers) prime (to one another) are also the least (of
μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον. those numbers having the same ratio) [Prop. 7.21]. And
οὐκ ἄρα τοὺς Γ, Δ ἀριθμοὺς ἀριθμός τις μετρήσει. οἱ Γ, Δ the least numbers of those (numbers) having the same
ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι. ratio measure those (numbers) having the same ratio as
them an equal number of times, the greater (measuring)
keþ following [Prop. 7.20]. Thus, E measures B. And it also
the greater, and the lesser the lesser—that is to say, the
leading (measuring) the leading, and the following the

measures C. Thus, E measures B and C, which are prime
to one another. The very thing is impossible. Thus, some
number cannot measure the numbers C and D. Thus,
C and D are prime to one another. (Which is) the very
thing it was required to show.

Proposition 25
.
᾿Εὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, ὁ ἐκ τοῦ If two numbers are prime to one another then the
ἑνὸς αὐτῶν γενόμενος πρὸς τὸν λοιπὸν πρῶτος ἔσται. number created from (squaring) one of them will be
῎Εστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους οἱ Α, Β, prime to the remaining (number).
καὶ ὁ Α ἑαυτὸν πολλαπλασιάσας τὸν Γ ποιείτω· λέγω, ὅτι Let A and B be two numbers (which are) prime to


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οἱ Β, Γ πρῶτοι πρὸς ἀλλὴλους εἰσίν. one another. And let A make C (by) multiplying itself. I
say that B and C are prime to one another.
Α Β Γ ∆ A B C D
















kþ
Κείσθω γὰρ τῷ Α ἴσος ὁ Δ. ἐπεὶ οἱ Α, Β πρῶτοι πρὸς For let D be made equal to A. Since A and B are
ἀλλήλους εἰσίν, ἴσος δὲ ὁ Α τῷ Δ, καί οἱ Δ, Β ἄρα πρῶτοι prime to one another, and A (is) equal to D, D and B are
πρὸς ἀλλήλους εἰσίν· ἑκάτερος ἄρα τῶν Δ, Α πρὸς τὸν Β thus also prime to one another. Thus, D and A are each
πρῶτός ἐστιν· καὶ ὁ ἐκ τῶν Δ, Α ἄρα γενόμενος πρὸς τὸν Β prime to B. Thus, the (number) created from (multily-
πρῶτος ἔσται. ὁ δὲ ἐκ τῶν Δ, Α γενόμενος ἀριθμός ἐστιν ὁ ing) D and A will also be prime to B [Prop. 7.24]. And C
Γ. οἱ Γ, Β ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι. is the number created from (multiplying) D and A. Thus,
C and B are prime to one another. (Which is) the very
thing it was required to show.
.
Proposition 26
᾿Εὰν δύο ἀριθμοὶ πρὸς δύο ἀριθμοὺς ἀμφότεροι πρὸς If two numbers are both prime to each of two numbers
ἑκάτερον πρῶτοι ὦσιν, καὶ οἱ ἐξ αὐτῶν γενόμενοι πρῶτοι then the (numbers) created from (multiplying) them will
πρὸς ἀλλήλους ἔσονται. also be prime to one another.
Α Γ A C
Β ∆ B D

Ε E
Ζ F
Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς δύο ἀριθμοὺς τοὺς Γ, For let two numbers, A and B, both be prime to each
Δ ἀμφότεροι πρὸς ἑκάτερον πρῶτοι ἔστωσαν, καὶ ὁ μὲν of two numbers, C and D. And let A make E (by) mul-
Α τὸν Β πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Γ τὸν Δ tiplying B, and let C make F (by) multiplying D. I say
πολλαπλασιάσας τὸν Ζ ποιείτω· λέγω, ὅτι οἱ Ε, Ζ πρῶτοι that E and F are prime to one another.
πρὸς ἀλλήλους εἰσίν. For since A and B are each prime to C, the (num-
᾿Επεὶ γὰρ ἑκάτερος τῶν Α, Β πρὸς τὸν Γ πρῶτός ἐστιν, ber) created from (multiplying) A and B will thus also
καὶ ὁ ἐκ τῶν Α, Β ἄρα γενόμενος πρὸς τὸν Γ πρῶτος ἔσται. be prime to C [Prop. 7.24]. And E is the (number) cre-
ὁ δὲ ἐκ τῶν Α, Β γενόμενός ἐστιν ὁ Ε· οἱ Ε, Γ ἄρα πρῶτοι ated from (multiplying) A and B. Thus, E and C are
πρὸς ἀλλήλους εἰσίν. διὰ τὰ αὐτὰ δὴ καὶ οἱ Ε, Δ πρῶτοι prime to one another. So, for the same (reasons), E and
πρὸς ἀλλήλους εἰσίν. ἑκάτερος ἄρα τῶν Γ, Δ πρὸς τὸν Ε D are also prime to one another. Thus, C and D are each
πρῶτός ἐστιν. καὶ ὁ ἐκ τῶν Γ, Δ ἄρα γενόμενος πρὸς τὸν prime to E. Thus, the (number) created from (multiply-
Ε πρῶτος ἔσται. ὁ δὲ ἐκ τῶν Γ, Δ γενόμενός ἐστιν ὁ Ζ. οἱ ing) C and D will also be prime to E [Prop. 7.24]. And
Ε, Ζ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι. F is the (number) created from (multiplying) C and D.
Thus, E and F are prime to one another. (Which is) the
very thing it was required to show.




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ELEMENTS BOOK 7


Proposition 27
†
.
᾿Εὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, καὶ πολ- If two numbers are prime to one another and each
λαπλασιάσας ἑκάτερος ἑαυτὸν ποιῇ τινα, οἱ γενόμενοι ἐξ makes some (number by) multiplying itself then the num-
αὐτῶν πρῶτοι πρὸς ἀλλήλους ἔσονται, κἂν οἱ ἐξ ἀρχῆς bers created from them will be prime to one another, and
τοὺς γενομένους πολλαπλασιάσαντες ποιῶσί τινας, κἀκεῖνοι if the original (numbers) make some (more numbers by)
πρῶτοι πρὸς ἀλλήλους ἔσονται [καὶ ἀεὶ περὶ τοὺς ἄκρους multiplying the created (numbers) then these will also be
τοῦτο συμβαίνει]. prime to one another [and this always happens with the
extremes].
Α Β Γ ∆ Ε Ζ A B C D E F

















῎Εστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους οἱ Α, Β, Let A and B be two numbers prime to one another,
καὶ ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Γ ποιείτω, τὸν and let A make C (by) multiplying itself, and let it make
δὲ Γ πολλαπλασιάσας τὸν Δ ποιείτω, ὁ δὲ Β ἑαυτὸν μὲν D (by) multiplying C. And let B make E (by) multiplying
πολλαπλασιάσας τὸν Ε ποιείτω, τὸν δὲ Ε πολλαπλασιάσας itself, and let it make F by multiplying E. I say that C
τὸν Ζ ποιείτω· λέγω, ὅτι οἵ τε Γ, Ε καὶ οἱ Δ, Ζ πρῶτοι πρὸς and E, and D and F, are prime to one another.
ἀλλήλους εἰσίν. For since A and B are prime to one another, and A has
᾿Επεὶ γὰρ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους εἰσίν, καὶ ὁ made C (by) multiplying itself, C and B are thus prime
Α ἑαυτὸν πολλαπλασιάσας τὸν Γ πεποίηκεν, οἱ Γ, Β ἄρα to one another [Prop. 7.25]. Therefore, since C and B
πρῶτοι πρὸς ἀλλήλους εἰσίν. ἐπεὶ οὖν οἱ Γ, Β πρῶτοι πρὸς are prime to one another, and B has made E (by) mul-
ἀλλήλους εἰσίν, καὶ ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Ε tiplying itself, C and E are thus prime to one another
πεποίηκεν, οἱ Γ, Ε ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν. πάλιν, [Prop. 7.25]. Again, since A and B are prime to one an-
ἐπεὶ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους εἰσίν, καὶ ὁ Β ἑαυτὸν other, and B has made E (by) multiplying itself, A and
πολλαπλασιάσας τὸν Ε πεποίηκεν, οἱ Α, Ε ἄρα πρῶτοι πρὸς E are thus prime to one another [Prop. 7.25]. Therefore,
ἀλλήλους εἰσίν. ἐπεὶ οὖν δύο ἀριθμοὶ οἱ Α, Γ πρὸς δύο since the two numbers A and C are both prime to each
ἀριθμοὺς τοὺς Β, Ε ἀμφότεροι πρὸς ἑκάτερον πρῶτοί εἰσιν, of the two numbers B and E, the (number) created from
καὶ ὁ ἐκ τῶν Α, Γ ἄρα γενόμενος πρὸς τὸν ἐκ τῶν Β, Ε (multiplying) A and C is thus prime to the (number cre-
khþ thing it was required to show.
πρῶτός ἐστιν. καί ἐστιν ὁ μὲν ἐκ τῶν Α, Γ ὁ Δ, ὁ δὲ ἐκ ated) from (multiplying) B and E [Prop. 7.26]. And D is
τῶν Β, Ε ὁ Ζ. οἱ Δ, Ζ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· the (number created) from (multiplying) A and C, and F
ὅπερ ἔδει δεῖξαι. the (number created) from (multiplying) B and E. Thus,
D and F are prime to one another. (Which is) the very


3
2
2
3
† In modern notation, this proposition states that if a is prime to b, then a is also prime to b , as well as a to b , etc., where all symbols denote
numbers.
Proposition 28
.
᾿Εὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, καὶ συ- If two numbers are prime to one another then their
ναμφότερος πρὸς ἑκάτερον αὐτῶν πρῶτος ἔσται· καὶ ἐὰν sum will also be prime to each of them. And if the sum
συναμφότερος πρὸς ἕνα τινὰ αὐτῶν πρῶτος ᾖ, καὶ οἱ ἐξ (of two numbers) is prime to any one of them then the
ἀρχῆς ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ἔσονται. original numbers will also be prime to one another.


216

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Α Β Γ A ELEMENTS BOOK 7
B
C

∆ D
Συγκείσθωσαν γὰρ δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλ- For let the two numbers, AB and BC, (which are)
ους οἱ ΑΒ, ΒΓ· λέγω, ὅτι καὶ συναμφότερος ὁ ΑΓ πρὸς prime to one another, be laid down together. I say that
ἑκάτερον τῶν ΑΒ, ΒΓ πρῶτός ἐστιν. their sum AC is also prime to each of AB and BC.
Εἰ γὰρ μή εἰσιν οἱ ΓΑ, ΑΒ πρῶτοι πρὸς ἀλλήλους, For if CA and AB are not prime to one another then
μετρήσει τις τοὺς ΓΑ, ΑΒ ἀριθμός. μετρείτω, καὶ ἔστω ὁ Δ. some number will measure CA and AB. Let it (so) mea-
ἐπεὶ οὖν ὁ Δ τοὺς ΓΑ, ΑΒ μετρεῖ, καὶ λοιπὸν ἄρα τὸν ΒΓ sure (them), and let it be D. Therefore, since D measures
μετρήσει. μετρεῖ δὲ καὶ τὸν ΒΑ· ὁ Δ ἄρα τοὺς ΑΒ, ΒΓ με- CA and AB, it will thus also measure the remainder BC.
τρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον. And it also measures BA. Thus, D measures AB and
οὐκ ἄρα τοὺς ΓΑ, ΑΒ ἀριθμοὺς ἀριθμός τις μετρήσει· οἱ BC, which are prime to one another. The very thing is
ΓΑ, ΑΒ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν. διὰ τὰ αὐτὰ δὴ impossible. Thus, some number cannot measure (both)
καὶ οἱ ΑΓ, ΓΒ πρῶτοι πρὸς ἀλλήλους εἰσίν. ὁ ΓΑ ἄρα πρὸς the numbers CA and AB. Thus, CA and AB are prime
ἑκάτερον τῶν ΑΒ, ΒΓ πρῶτός ἐστιν. to one another. So, for the same (reasons), AC and CB
῎Εστωσαν δὴ πάλιν οἱ ΓΑ, ΑΒ πρῶτοι πρὸς ἀλλήλους· are also prime to one another. Thus, CA is prime to each
λέγω, ὅτι καὶ οἱ ΑΒ, ΒΓ πρῶτοι πρὸς ἀλλήλους εἰσίν. of AB and BC.
Εἰ γὰρ μή εἰσιν οἱ ΑΒ, ΒΓ πρῶτοι πρὸς ἀλλήλους, So, again, let CA and AB be prime to one another. I
μετρήσει τις τοὺς ΑΒ, ΒΓ ἀριθμός. μετρείτω, καὶ ἔστω ὁ Δ. say that AB and BC are also prime to one another.
καὶ ἐπεὶ ὁ Δ ἑκάτερον τῶν ΑΒ, ΒΓ μετρεῖ, καὶ ὅλον ἄρα τὸν For if AB and BC are not prime to one another then
ΓΑ μετρήσει. μετρεῖ δὲ καὶ τὸν ΑΒ· ὁ Δ ἄρα τοὺς ΓΑ, ΑΒ some number will measure AB and BC. Let it (so) mea-
kjþ is impossible. Thus, some number cannot measure (both)
μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον. sure (them), and let it be D. And since D measures each
οὐκ ἄρα τοὺς ΑΒ, ΒΓ ἀριθμοὺς ἀριθμός τις μετρήσει. οἱ of AB and BC, it will thus also measure the whole of
ΑΒ, ΒΓ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι. CA. And it also measures AB. Thus, D measures CA
and AB, which are prime to one another. The very thing
the numbers AB and BC. Thus, AB and BC are prime
to one another. (Which is) the very thing it was required
to show.
.
Proposition 29
῞Απας πρῶτος ἀριθμὸς πρὸς ἅπαντα ἀριθμόν, ὃν μὴ με- Every prime number is prime to every number which
τρεῖ, πρῶτός ἐστιν. it does not measure.
Α A

Β B

Γ C
῎Εστω πρῶτος ἀριθμὸς ὁ Α καὶ τὸν Β μὴ μετρείτω· λέγω, Let A be a prime number, and let it not measure B. I
ὅτι οἱ Β, Α πρῶτοι πρὸς ἀλλήλους εἰσίν. say that B and A are prime to one another. For if B and
Εἰ γὰρ μή εἰσιν οἱ Β, Α πρῶτοι πρὸς ἀλλήλους, μετρήσει A are not prime to one another then some number will
τις αὐτοὺς ἀριθμός. μετρείτω ὁ Γ. ἐπεὶ ὁ Γ τὸν Β μετρεῖ, measure them. Let C measure (them). Since C measures
ὁ δὲ Α τὸν Β οὐ μετρεῖ, ὁ Γ ἄρα τῷ Α οὔκ ἐστιν ὁ αὐτός. B, and A does not measure B, C is thus not the same as
καὶ ἐπεὶ ὁ Γ τοὺς Β, Α μετρεῖ, καὶ τὸν Α ἄρα μετρεῖ πρῶτον A. And since C measures B and A, it thus also measures
ὄντα μὴ ὢν αὐτῷ ὁ αὐτός· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα A, which is prime, (despite) not being the same as it.
τοὺς Β, Α μετρήσει τις ἀριθμός. οἱ Α, Β ἄρα πρῶτοι πρὸς The very thing is impossible. Thus, some number cannot
ἀλλήλους εἰσίν· ὅπερ ἔδει δεῖξαι. measure (both) B and A. Thus, A and B are prime to
one another. (Which is) the very thing it was required to



217

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.

᾿Εὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί show. Proposition 30
If two numbers make some (number by) multiplying
τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, one another, and some prime number measures the num-
καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει. ber (so) created from them, then it will also measure one
of the original (numbers).
Α A

Β B


Γ C

∆ D

Ε E
Δύο γὰρ ἀριθμοὶ οἱ Α, Β πολλαπλασιάσαντες ἀλλήλους For let two numbers A and B make C (by) multiplying
τὸν Γ ποιείτωσαν, τὸν δὲ Γ μετρείτω τις πρῶτος ἀριθμὸς ὁ one another, and let some prime number D measure C. I
Δ· λέγω, ὅτι ὁ Δ ἕνα τῶν Α, Β μετρεῖ. say that D measures one of A and B.
Τὸν γὰρ Α μὴ μετρείτω· καί ἐστι πρῶτος ὁ Δ· οἱ Α, For let it not measure A. And since D is prime, A
Δ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν. καὶ ὁσάκις ὁ Δ τὸν Γ and D are thus prime to one another [Prop. 7.29]. And
μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ε. ἐπεὶ οὖν ὁ Δ as many times as D measures C, so many units let there
τὸν Γ μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας, ὁ Δ ἄρα τὸν Ε be in E. Therefore, since D measures C according to
πολλαπλασιάσας τὸν Γ πεποίηκεν. ἀλλὰ μὴν καὶ ὁ Α τὸν the units E, D has thus made C (by) multiplying E
Β πολλαπλασιάσας τὸν Γ πεποίηκεν· ἴσος ἄρα ἐστὶν ὁ ἐκ [Def. 7.15]. But, in fact, A has also made C (by) multi-
τῶν Δ, Ε τῷ ἐκ τῶν Α, Β. ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Α, plying B. Thus, the (number created) from (multiplying)
οὕτως ὁ Β πρὸς τὸν Ε. οἱ δὲ Δ, Α πρῶτοι, οἱ δὲ πρῶτοι καὶ D and E is equal to the (number created) from (mul-
ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον tiplying) A and B. Thus, as D is to A, so B (is) to E
ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν [Prop. 7.19]. And D and A (are) prime (to one another),
ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ and (numbers) prime (to one another are) also the least
ἐπόμενος τὸν ἑπόμενον· ὁ Δ ἄρα τὸν Β μετρεῖ. ὁμοίως δὴ (of those numbers having the same ratio) [Prop. 7.21],
δείξομεν, ὅτι καὶ ἐὰν τὸν Β μὴ μετρῇ, τὸν Α μετρήσει. ὁ Δ and the least (numbers) measure those (numbers) hav-
laþ ing, and the following the following [Prop. 7.20]. Thus,
ἄρα ἕνα τῶν Α, Β μετρεῖ· ὅπερ ἔδει δεῖξαι. ing the same ratio (as them) an equal number of times,
the greater (measuring) the greater, and the lesser the
lesser—that is to say, the leading (measuring) the lead-

D measures B. So, similarly, we can also show that if
(D) does not measure B then it will measure A. Thus,
D measures one of A and B. (Which is) the very thing it
was required to show.

.
Proposition 31
῞Απας σύνθεντος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ με- Every composite number is measured by some prime
τρεῖται. number.
Let A be a composite number. I say that A is measured
῎Εστω σύνθεντος ἀριθμὸς ὁ Α· λέγω, ὅτι ὁ Α ὑπὸ πρώτου by some prime number.
τινὸς ἀριθμοῦ μετρεῖται. For since A is composite, some number will measure
᾿Επεὶ γὰρ σύνθετός ἐστιν ὁ Α, μετρήσει τις αὐτὸν it. Let it (so) measure (A), and let it be B. And if B


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ELEMENTS BOOK 7



ἀριθμός. μετρείτω, καὶ ἔστω ὁ Β. καὶ εἰ μὲν πρῶτός ἐστιν ὁ is prime then that which was prescribed has happened.
Β, γεγονὸς ἂν εἴη τὸ ἐπιταχθέν. εἰ δὲ σύνθετος, μετρήσει And if (B is) composite then some number will measure
τις αὐτὸν ἀριθμός. μετρείτω, καὶ ἔστω ὁ Γ. καὶ ἐπεὶ ὁ Γ it. Let it (so) measure (B), and let it be C. And since
τὸν Β μετρεῖ, ὁ δὲ Β τὸν Α μετρεῖ, καὶ ὁ Γ ἄρα τὸν Α C measures B, and B measures A, C thus also measures
μετρεῖ. καὶ εἰ μὲν πρῶτός ἐστιν ὁ Γ, γεγονὸς ἂν εἴη τὸ A. And if C is prime then that which was prescribed has
ἐπιταχθέν. εἰ δὲ σύνθετος, μετρήσει τις αὐτὸν ἀριθμός. happened. And if (C is) composite then some number
τοιαύτης δὴ γινομένης ἐπισκέψεως ληφθήσεταί τις πρῶτος will measure it. So, in this manner of continued inves-
ἀριθμός, ὃς μετρήσει. εἰ γὰρ οὐ ληφθήσεται, μετρήσουσι tigation, some prime number will be found which will
τὸν Α ἀριθμὸν ἄπειροι ἀριθμοί, ὧν ἕτερος ἑτέρου ἐλάσσων measure (the number preceding it, which will also mea-
ἐστίν· ὅπερ ἐστὶν ἀδύνατον ἐν ἀριθμοῖς. ληφθήσεταί τις ἄρα sure A). And if (such a number) cannot be found then an
πρῶτος ἀριθμός, ὃς μετρήσει τὸν πρὸ ἑαυτοῦ, ὃς καὶ τὸν Α infinite (series of) numbers, each of which is less than the
μετρήσει. preceding, will measure the number A. The very thing is
impossible for numbers. Thus, some prime number will
(eventually) be found which will measure the (number)
preceding it, which will also measure A.
Α A
lbþ
Β B

Γ C

῞Απας ἄρα σύνθεντος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ Thus, every composite number is measured by some
μετρεῖται· ὅπερ ἔδει δεῖξαι. prime number. (Which is) the very thing it was required
to show.

.
Proposition 32
῞Απας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς Every number is either prime or is measured by some
ἀριθμοῦ μετρεῖται. prime number.
Α A
lgþ
῎Εστω ἀριθμὸς ὁ Α· λέγω, ὅτι ὁ Α ἤτοι πρῶτός ἐστιν ἢ Let A be a number. I say that A is either prime or is
ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται. measured by some prime number.
Εἰ μὲν οὖν πρῶτός ἐστιν ὁ Α, γεγονὸς ἂν εἴη τό In fact, if A is prime then that which was prescribed
ἐπιταχθέν. εἰ δὲ σύνθετος, μετρήσει τις αὐτὸν πρῶτος has happened. And if (it is) composite then some prime
ἀριθμός. number will measure it [Prop. 7.31].
῞Απας ἄρα ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου Thus, every number is either prime or is measured
τινὸς ἀριθμοῦ μετρεῖται· ὅπερ ἔδει δεῖξαι. by some prime number. (Which is) the very thing it was
required to show.

Proposition 33
.
᾿Αριθμῶν δοθέντων ὁποσωνοῦν εὑρεῖν τοὺς ἐλαχίστους To find the least of those (numbers) having the same
τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς. ratio as any given multitude of numbers.
῎Εστωσαν οἱ δοθέντες ὁποσοιοῦν ἀριθμοὶ οἱ Α, Β, Γ· δεῖ Let A, B, and C be any given multitude of numbers.
δὴ εὑρεῖν τοὺς ἐλαχίστους τῶν τὸν αὐτὸν λόγον ἐχόντων So it is required to find the least of those (numbers) hav-
τοῖς Α, Β, Γ. ing the same ratio as A, B, and C.
Οἱ Α, Β, Γ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ. For A, B, and C are either prime to one another, or
εἰ μὲν οὖν οἱ Α, Β, Γ πρῶτοι πρὸς ἀλλήλους εἰσίν, ἐλάχιστοί not. In fact, if A, B, and C are prime to one another then
εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς. they are the least of those (numbers) having the same
ratio as them [Prop. 7.22].




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ELEMENTS BOOK 7


Α Β Γ ∆ Ε Ζ Η Θ Κ Λ Μ A B C D E F G H K L M











Εἰ δὲ οὔ, εἰλήφθω τῶν Α, Β, Γ τὸ μέγιστον κοινὸν And if not, let the greatest common measure, D, of
μέτρον ὁ Δ, καὶ ὁσάκις ὁ Δ ἕκαστον τῶν Α, Β, Γ μετρεῖ, A, B, and C have be taken [Prop. 7.3]. And as many
τοσαῦται μονάδες ἔστωσαν ἐν ἑκάστῳ τῶν Ε, Ζ, Η. καὶ times as D measures A, B, C, so many units let there
ἕκαστος ἄρα τῶν Ε, Ζ, Η ἕκαστον τῶν Α, Β, Γ μετρεῖ κατὰ be in E, F, G, respectively. And thus E, F, G mea-
τὰς ἐν τῷ Δ μονάδας. οἱ Ε, Ζ, Η ἄρα τοὺς Α, Β, Γ ἰσάκις sure A, B, C, respectively, according to the units in D
μετροῦσιν· οἱ Ε, Ζ, Η ἄρα τοῖς Α, Β, Γ ἐν τῷ αὐτῷ λόγῳ [Prop. 7.15]. Thus, E, F, G measure A, B, C (respec-
εἰσίν. λέγω δή, ὅτι καὶ ἐλάχιστοι. εἰ γὰρ μή εἰσιν οἱ Ε, Ζ, tively) an equal number of times. Thus, E, F, G are in
Η ἐλάχιστοι τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ, the same ratio as A, B, C (respectively) [Def. 7.20]. So I
ἔσονται [τινες] τῶν Ε, Ζ, Η ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ say that (they are) also the least (of those numbers hav-
λόγῳ ὄντες τοῖς Α, Β, Γ. ἔστωσαν οἱ Θ, Κ, Λ· ἰσάκις ἄρα ὁ ing the same ratio as A, B, C). For if E, F, G are not
Θ τὸν Α μετρεῖ καὶ ἑκάτερος τῶν Κ, Λ ἑκάτερον τῶν Β, Γ. the least of those (numbers) having the same ratio as A,
ὁσάκις δὲ ὁ Θ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν B, C (respectively), then there will be [some] numbers
τῷ Μ· καὶ ἑκάτερος ἄρα τῶν Κ, Λ ἑκάτερον τῶν Β, Γ μετρεῖ less than E, F, G which are in the same ratio as A, B, C
κατὰ τὰς ἐν τῷ Μ μονάδας. καὶ ἐπεὶ ὁ Θ τὸν Α μετρεῖ κατὰ (respectively). Let them be H, K, L. Thus, H measures
τὰς ἐν τῷ Μ μονάδας, καὶ ὁ Μ ἄρα τὸν Α μετρεῖ κατὰ τὰς A the same number of times that K, L also measure B,
ἐν τῷ Θ μονάδας. διὰ τὰ αὐτὰ δὴ ὁ Μ καὶ ἑκάτερον τῶν Β, C, respectively. And as many times as H measures A, so
Γ μετρεῖ κατὰ τὰς ἐν ἑκατέρῳ τῶν Κ, Λ μονάδας· ὁ Μ ἄρα many units let there be in M. Thus, K, L measure B,
τοὺς Α, Β, Γ μετρεῖ. καὶ ἐπεὶ ὁ Θ τὸν Α μετρεῖ κατὰ τὰς C, respectively, according to the units in M. And since
ἐν τῷ Μ μονάδας, ὁ Θ ἄρα τὸν Μ πολλαπλασιάσας τὸν Α H measures A according to the units in M, M thus also
πεποίηκεν. διὰ τὰ αὐτὰ δὴ καὶ ὁ Ε τὸν Δ πολλαπλασιάσας measures A according to the units in H [Prop. 7.15]. So,
τὸν Α πεποίηκεν. ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Ε, Δ τῷ ἐκ for the same (reasons), M also measures B, C accord-
τῶν Θ, Μ. ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Μ πρὸς ing to the units in K, L, respectively. Thus, M measures
τὸν Δ. μείζων δὲ ὁ Ε τοῦ Θ· μείζων ἄρα καὶ ὁ Μ τοῦ Δ. A, B, and C. And since H measures A according to the
καὶ μετρεῖ τοὺς Α, Β, Γ· ὅπερ ἐστὶν ἀδύνατον· ὑπόκειται units in M, H has thus made A (by) multiplying M. So,
γὰρ ὁ Δ τῶν Α, Β, Γ τὸ μέγιστον κοινὸν μέτρον. οὐκ ἄρα for the same (reasons), E has also made A (by) multiply-
ἔσονταί τινες τῶν Ε, Ζ, Η ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ ing D. Thus, the (number created) from (multiplying)
λόγῳ ὄντες τοῖς Α, Β, Γ. οἱ Ε, Ζ, Η ἄρα ἐλάχιστοί εἰσι τῶν E and D is equal to the (number created) from (multi-
τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ· ὅπερ ἔδει δεῖξαι. plying) H and M. Thus, as E (is) to H, so M (is) to
D [Prop. 7.19]. And E (is) greater than H. Thus, M
ldþ assumed (to be) the greatest common measure of A, B,
(is) also greater than D [Prop. 5.13]. And (M) measures
A, B, and C. The very thing is impossible. For D was

and C. Thus, there cannot be any numbers less than E,
F, G which are in the same ratio as A, B, C (respec-
tively). Thus, E, F, G are the least of (those numbers)
having the same ratio as A, B, C (respectively). (Which
is) the very thing it was required to show.

Proposition 34
.
Δύο ἀριθμῶν δοθέντων εὑρεῖν, ὃν ἐλάχιστον μετροῦσιν To find the least number which two given numbers
ἀριθμόν. (both) measure.
῎Εστωσαν οἱ δοθέντες δύο ἀριθμοὶ οἱ Α, Β· δεῖ δὴ εὑρεῖν, Let A and B be the two given numbers. So it is re-


220

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ὃν ἐλάχιστον μετροῦσιν ἀριθμόν. quired to find the least number which they (both) mea-
sure.
Α Β A B

Γ C

∆ D

Ε Ζ E F
Οἱ Α, Β γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ. For A and B are either prime to one another, or not.
ἔστωσαν πρότερον οἱ Α, Β πρῶτοι πρὸς ἀλλήλους, καὶ ὁ Α Let them, first of all, be prime to one another. And let A
τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω· καὶ ὁ Β ἄρα τὸν Α make C (by) multiplying B. Thus, B has also made C
πολλαπλασιάσας τὸν Γ πεποίηκεν. οἱ Α, Β ἄρα τὸν Γ με- (by) multiplying A [Prop. 7.16]. Thus, A and B (both)
τροῦσιν. λέγω δή, ὅτι καὶ ἐλάχιστον. εἰ γὰρ μή, μετρήσουσί measure C. So I say that (C) is also the least (num-
τινα ἀριθμὸν οἱ Α, Β ἐλάσσονα ὄντα τοῦ Γ. μετρείτωσαν ber which they both measure). For if not, A and B will
τὸν Δ. καὶ ὁσάκις ὁ Α τὸν Δ μετρεῖ, τοσαῦται μονάδες (both) measure some (other) number which is less than
ἔστωσαν ἐν τῷ Ε, ὁσάκις δὲ ὁ Β τὸν Δ μετρεῖ, τοσαῦται C. Let them (both) measure D (which is less than C).
μονάδες ἔστωσαν ἐν τῷ Ζ. ὁ μὲν Α ἄρα τὸν Ε πολλα- And as many times as A measures D, so many units let
πλασιάσας τὸν Δ πεποίηκεν, ὁ δὲ Β τὸν Ζ πολλαπλασιάσας there be in E. And as many times as B measures D,
τὸν Δ πεποίηκεν· ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Α, Ε τῷ ἐκ τῶν so many units let there be in F. Thus, A has made D
Β, Ζ. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Ζ πρὸς τὸν (by) multiplying E, and B has made D (by) multiply-
Ε. οἱ δὲ Α, Β πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ing F. Thus, the (number created) from (multiplying)
ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ A and E is equal to the (number created) from (multi-
τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα· ὁ Β plying) B and F. Thus, as A (is) to B, so F (is) to E
ἄρα τὸν Ε μετρεῖ, ὡς ἑπόμενος ἑπόμενον. καὶ ἐπεὶ ὁ Α τοὺς [Prop. 7.19]. And A and B are prime (to one another),
Β, Ε πολλαπλασιάσας τοὺς Γ, Δ πεποίηκεν, ἔστιν ἄρα ὡς and prime (numbers) are the least (of those numbers
ὁ Β πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς τὸν Δ. μετρεῖ δὲ ὁ Β τὸν having the same ratio) [Prop. 7.21], and the least (num-
Ε· μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ ὁ μείζων τὸν ἐλάσσονα· ὅπερ bers) measure those (numbers) having the same ratio (as
ἐστὶν ἀδύνατον. οὐκ ἄρα οἱ Α, Β μετροῦσί τινα ἀριθμὸν them) an equal number of times, the greater (measuring)
ἐλάσσονα ὄντα τοῦ Γ. ὁ Γ ἄρα ἐλάχιστος ὢν ὑπὸ τῶν Α, Β the greater, and the lesser the lesser [Prop. 7.20]. Thus,
μετρεῖται. B measures E, as the following (number measuring) the
following. And since A has made C and D (by) multi-
plying B and E (respectively), thus as B is to E, so C
(is) to D [Prop. 7.17]. And B measures E. Thus, C also
measures D, the greater (measuring) the lesser. The very
thing is impossible. Thus, A and B do not (both) mea-
sure some number which is less than C. Thus, C is the
least (number) which is measured by (both) A and B.
Α Β A B

Ζ Ε F E

Γ C

∆ D

Η Θ G H
Μὴ ἔστωσαν δὴ οἱ Α, Β πρῶτοι πρὸς ἀλλήλους, So let A and B be not prime to one another. And
καὶ εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον let the least numbers, F and E, have been taken having
ἐχόντων τοῖς Α, Β οἱ Ζ, Ε· ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν Α, Ε τῷ the same ratio as A and B (respectively) [Prop. 7.33].


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ἐκ τῶν Β, Ζ. καὶ ὁ Α τὸν Ε πολλαπλασιάσας τὸν Γ ποιείτω· Thus, the (number created) from (multiplying) A and E
καὶ ὁ Β ἄρα τὸν Ζ πολλαπλασιάσας τὸν Γ πεποίηκεν· οἱ Α, is equal to the (number created) from (multiplying) B
Β ἄρα τὸν Γ μετροῦσιν. λέγω δή, ὅτι καὶ ἐλάχιστον. εἰ γὰρ and F [Prop. 7.19]. And let A make C (by) multiplying
μή, μετρήσουσί τινα ἀριθμὸν οἱ Α, Β ἐλάσσονα ὄντα τοῦ E. Thus, B has also made C (by) multiplying F. Thus,
Γ. μετρείτωσαν τὸν Δ. καὶ ὁσάκις μὲν ὁ Α τὸν Δ μετρεῖ, A and B (both) measure C. So I say that (C) is also the
τοσαῦται μονάδες ἔστωσαν ἐν τῷ Η, ὁσάκις δὲ ὁ Β τὸν Δ least (number which they both measure). For if not, A
μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Θ. ὁ μὲν Α ἄρα and B will (both) measure some number which is less
τὸν Η πολλαπλασιάσας τὸν Δ πεποίηκεν, ὁ δὲ Β τὸν Θ than C. Let them (both) measure D (which is less than
πολλαπλασιάσας τὸν Δ πεποίηκεν. ἴσος ἄρα ἐστὶν ὁ ἐκ τῶν C). And as many times as A measures D, so many units
Α, Η τῷ ἐκ τῶν Β, Θ· ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως let there be in G. And as many times as B measures D,
ὁ Θ πρὸς τὸν Η. ὡς δὲ ὁ Α πρὸς τὸν Β, οὕτως ὁ Ζ πρὸς so many units let there be in H. Thus, A has made D
τὸν Ε· καὶ ὡς ἄρα ὁ Ζ πρὸς τὸν Ε, οὕτως ὁ Θ πρὸς τὸν (by) multiplying G, and B has made D (by) multiplying
Η. οἱ δὲ Ζ, Ε ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν H. Thus, the (number created) from (multiplying) A and
αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ G is equal to the (number created) from (multiplying) B
ἐλάσσων τὸν ἐλάσσονα· ὁ Ε ἄρα τὸν Η μετρεῖ. καὶ ἐπεὶ ὁ and H. Thus, as A is to B, so H (is) to G [Prop. 7.19].
Α τοὺς Ε, Η πολλαπλασιάσας τοὺς Γ, Δ πεποίηκεν, ἔστιν And as A (is) to B, so F (is) to E. Thus, also, as F (is)
ἄρα ὡς ὁ Ε πρὸς τὸν Η, οὕτως ὁ Γ πρὸς τὸν Δ. ὁ δὲ Ε τὸν to E, so H (is) to G. And F and E are the least (num-
Η μετρεῖ· καὶ ὁ Γ ἄρα τὸν Δ μετρεῖ ὁ μείζων τὸν ἐλάσσονα· bers having the same ratio as A and B), and the least
ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα οἱ Α, Β μετρήσουσί τινα (numbers) measure those (numbers) having the same ra-
ἀριθμὸν ἐλάσσονα ὄντα τοῦ Γ. ὁ Γ ἄρα ἐλάχιστος ὢν ὑπὸ tio an equal number of times, the greater (measuring)
τῶν Α, Β μετρεῖται· ὅπερ ἔπει δεῖξαι. the greater, and the lesser the lesser [Prop. 7.20]. Thus,
leþ (is) to D [Prop. 7.17]. And E measures G. Thus, C also
E measures G. And since A has made C and D (by) mul-
tiplying E and G (respectively), thus as E is to G, so C

measures D, the greater (measuring) the lesser. The very
thing is impossible. Thus, A and B do not (both) mea-
sure some (number) which is less than C. Thus, C (is)
the least (number) which is measured by (both) A and
B. (Which is) the very thing it was required to show.

.
Proposition 35
᾿Εὰν δύο ἀριθμοὶ ἀριθμόν τινα μετρῶσιν, καὶ ὁ ἐλάχιστος If two numbers (both) measure some number then the
ὑπ᾿ αὐτῶν μετρούμενος τὸν αὐτὸν μετρήσει. least (number) measured by them will also measure the
same (number).
Α Β A B
Γ Ζ ∆ C F D



Ε E
Δύο γὰρ ἀριθμοὶ οἱ Α, Β ἀριθμόν τινα τὸν ΓΔ με- For let two numbers, A and B, (both) measure some
τρείτωσαν, ἐλάχιστον δὲ τὸν Ε· λέγω, ὅτι καὶ ὁ Ε τὸν ΓΔ number CD, and (let) E (be the) least (number mea-
μετρεῖ. sured by both A and B). I say that E also measures CD.
Εἰ γὰρ οὐ μετρεῖ ὁ Ε τὸν ΓΔ, ὁ Ε τὸν ΔΖ μετρῶν For if E does not measure CD then let E leave CF
λειπέτω ἑαυτοῦ ἐλάσσονα τὸν ΓΖ. καὶ ἐπεὶ οἱ Α, Β τὸν Ε less than itself (in) measuring DF. And since A and B
μετροῦσιν, ὁ δὲ Ε τὸν ΔΖ μετρεῖ, καὶ οἱ Α, Β ἄρα τὸν (both) measure E, and E measures DF, A and B will
ΔΖ μετρήσουσιν. μετροῦσι δὲ καὶ ὅλον τὸν ΓΔ· καὶ λοιπὸν thus also measure DF. And (A and B) also measure the
ἄρα τὸν ΓΖ μετρήσουσιν ἐλάσσονα ὄντα τοῦ Ε· ὅπερ ἐστὶν whole of CD. Thus, they will also measure the remainder
ἀδύνατον. οὐκ ἄρα οὐ μετρεῖ ὁ Ε τὸν ΓΔ· μετρεῖ ἄρα· ὅπερ CF, which is less than E. The very thing is impossible.
ἔδει δεῖξαι. Thus, E cannot not measure CD. Thus, (E) measures


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Proposition 36
.
Τριῶν ἀριθμῶν δοθέντων εὑρεῖν, ὃν ἐλάχιστον με- (CD). (Which is) the very thing it was required to show.
To find the least number which three given numbers
τροῦσιν ἀριθμόν. (all) measure.
῎Εστωσαν οἱ δοθέντες τρεῖς ἀριθμοὶ οἱ Α, Β, Γ· δεῖ δὴ Let A, B, and C be the three given numbers. So it is
εὑρεῖν, ὃν ἐλάχιστον μετροῦσιν ἀριθμόν. required to find the least number which they (all) mea-
sure.
Α A

Β B
Γ C

∆ D

Ε E

Ζ F
Εἰλήφθω γὰρ ὑπὸ δύο τῶν Α, Β ἐλάχιστος μετρούμενος For let the least (number), D, measured by the two
ὁ Δ. ὁ δὴ Γ τὸν Δ ἤτοι μετρεῖ ἢ οὐ μετρεῖ. μετρείτω (numbers) A and B have been taken [Prop. 7.34]. So C
πρότερον. μετροῦσι δὲ καὶ οἱ Α, Β τὸν Δ· οἱ Α, Β, Γ either measures, or does not measure, D. Let it, first of
ἄρα τὸν Δ μετροῦσιν. λέγω δή, ὅτι καὶ ἐλάχιστον. εἰ γὰρ all, measure (D). And A and B also measure D. Thus,
μή, μετρήσουσιν [τινα] ἀριθμὸν οἱ Α, Β, Γ ἐλάσσονα ὄντα A, B, and C (all) measure D. So I say that (D is) also
τοῦ Δ. μετρείτωσαν τὸν Ε. ἐπεὶ οἱ Α, Β, Γ τὸν Ε μετροῦσιν, the least (number measured by A, B, and C). For if not,
καὶ οἱ Α, Β ἄρα τὸν Ε μετροῦσιν. καὶ ὁ ἐλάχιστος ἄρα ὑπὸ A, B, and C will (all) measure [some] number which
τῶν Α, Β μετρούμενος [τὸν Ε] μετρήσει. ἐλάχιστος δὲ ὑπὸ is less than D. Let them measure E (which is less than
τῶν Α, Β μετρούμενός ἐστιν ὁ Δ· ὁ Δ ἄρα τὸν Ε μετρήσει D). Since A, B, and C (all) measure E then A and B
ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα οἱ thus also measure E. Thus, the least (number) measured
Α, Β, Γ μετρήσουσί τινα ἀριθμὸν ἐλάσσονα ὄντα τοῦ Δ· οἱ by A and B will also measure [E] [Prop. 7.35]. And D
Α, Β, Γ ἄρα ἐλάχιστον τὸν Δ μετροῦσιν. is the least (number) measured by A and B. Thus, D
Μὴ μετρείτω δὴ πάλιν ὁ Γ τὸν Δ, καὶ εἰλήφθω ὑπὸ τῶν will measure E, the greater (measuring) the lesser. The
Γ, Δ ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Ε. ἐπεὶ οἱ Α, Β very thing is impossible. Thus, A, B, and C cannot (all)
τὸν Δ μετροῦσιν, ὁ δὲ Δ τὸν Ε μετρεῖ, καὶ οἱ Α, Β ἄρα measure some number which is less than D. Thus, A, B,
τὸν Ε μετροῦσιν. μετρεῖ δὲ καὶ ὁ Γ [τὸν Ε· καὶ] οἱ Α, Β, and C (all) measure the least (number) D.
Γ ἄρα τὸν Ε μετροῦσιν. λέγω δή, ὅτι καὶ ἐλάχιστον. εἰ So, again, let C not measure D. And let the least
γὰρ μή, μετρήσουσί τινα οἱ Α, Β, Γ ἐλάσσονα ὄντα τοῦ Ε. number, E, measured by C and D have been taken
μετρείτωσαν τὸν Ζ. ἐπεὶ οἱ Α, Β, Γ τὸν Ζ μετροῦσιν, καὶ οἱ [Prop. 7.34]. Since A and B measure D, and D measures
Α, Β ἄρα τὸν Ζ μετροῦσιν· καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν E, A and B thus also measure E. And C also measures
Α, Β μετρούμενος τὸν Ζ μετρήσει. ἐλάχιστος δὲ ὑπὸ τῶν [E]. Thus, A, B, and C [also] measure E. So I say that
Α, Β μετρούμενός ἐστιν ὁ Δ· ὁ Δ ἄρα τὸν Ζ μετρεῖ. μετρεῖ (E is) also the least (number measured by A, B, and C).
δὲ καὶ ὁ Γ τὸν Ζ· οἱ Δ, Γ ἄρα τὸν Ζ μετροῦσιν· ὥστε καὶ ὁ For if not, A, B, and C will (all) measure some (number)
ἐλάχιστος ὑπὸ τῶν Δ, Γ μετρούμενος τὸν Ζ μετρήσει. ὁ δὲ which is less than E. Let them measure F (which is less
ἐλάχιστος ὑπὸ τῶν Γ, Δ μετρούμενός ἐστιν ὁ Ε· ὁ Ε ἄρα than E). Since A, B, and C (all) measure F, A and B
τὸν Ζ μετρεῖ ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον. thus also measure F. Thus, the least (number) measured
οὐκ ἄρα οἱ Α, Β, Γ μετρήσουσί τινα ἀριθμὸν ἐλάσσονα ὄντα by A and B will also measure F [Prop. 7.35]. And D
τοῦ Ε. ὁ Ε ἄρα ἐλάχιστος ὢν ὑπὸ τῶν Α, Β, Γ μετρεῖται· is the least (number) measured by A and B. Thus, D
ὅπερ ἔδει δεῖξαι. measures F. And C also measures F. Thus, D and C
(both) measure F. Hence, the least (number) measured
by D and C will also measure F [Prop. 7.35]. And E


223

ST EW zþ. ELEMENTS BOOK 7







lzþ is the least (number) measured by C and D. Thus, E



measures F, the greater (measuring) the lesser. The very
thing is impossible. Thus, A, B, and C cannot measure
some number which is less than E. Thus, E (is) the least
(number) which is measured by A, B, and C. (Which is)
the very thing it was required to show.

.
Proposition 37
᾿Εὰν ἀριθμὸς ὑπό τινος ἀριθμοῦ μετρῆται, ὁ μετρούμενος If a number is measured by some number then the
ὁμώνυμον μέρος ἕξει τῷ μετροῦντι. (number) measured will have a part called the same as
the measuring (number).
Α A

Β B

Γ C

∆ D
᾿Αριθμὸς γάρ ὁ Α ὑπό τινος ἀριθμοῦ τοῦ Β μετρείσθω· For let the number A be measured by some number
λέγω, ὅτι ὁ Α ὁμώνυμον μέρος ἔχει τῷ Β. B. I say that A has a part called the same as B.
῾Οσάκις γὰρ ὁ Β τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστω- For as many times as B measures A, so many units
σαν ἐν τῷ Γ. ἐπεὶ ὁ Β τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Γ let there be in C. Since B measures A according to the
μονάδας, μετρεῖ δὲ καὶ ἡ Δ μονὰς τὸν Γ ἀριθμὸν κατὰ τὰς units in C, and the unit D also measures C according
ἐν αὐτῷ μονάδας, ἰσάκις ἄρα ἡ Δ μονὰς τὸν Γ ἀριθμὸν με- to the units in it, the unit D thus measures the number
τρεῖ καὶ ὁ Β τὸν Α. ἐναλλὰξ ἄρα ἰσάκις ἡ Δ μονὰς τὸν Β C as many times as B (measures) A. Thus, alternately,
lhþ the same as B (i.e., C is the Bth part of A). Hence, A has
ἀριθμὸν μετρεῖ καὶ ὁ Γ τὸν Α· ὃ ἄρα μέρος ἐστὶν ἡ Δ μονὰς the unit D measures the number B as many times as C
τοῦ Β ἀριθμοῦ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Γ τοῦ Α. ἡ δὲ Δ (measures) A [Prop. 7.15]. Thus, which(ever) part the
μονὰς τοῦ Β ἀριθμοῦ μέρος ἐστὶν ὁμώνυμον αὐτῷ· καὶ ὁ Γ unit D is of the number B, C is also the same part of A.
ἄρα τοῦ Α μέρος ἐστὶν ὁμώνυμον τῷ Β. ὥστε ὁ Α μέρος And the unit D is a part of the number B called the same
ἔχει τὸν Γ ὁμώνυμον ὄντα τῷ Β· ὅπερ ἔδει δεῖξαι. as it (i.e., a Bth part). Thus, C is also a part of A called


a part C which is called the same as B (i.e., A has a Bth
part). (Which is) the very thing it was required to show.

.
Proposition 38
᾿Εὰν ἀριθμος μέρος ἔχῃ ὁτιοῦν, ὑπὸ ὁμωνύμου ἀριθμοῦ If a number has any part whatever then it will be mea-
μετρηθήσεται τῷ μέρει. sured by a number called the same as the part.
Α A

Β B

Γ C

∆ D
᾿Αριθμὸς γὰρ ὁ Α μέρος ἐχέτω ὁτιοῦν τὸν Β, καὶ τῷ Β For let the number A have any part whatever, B. And
μέρει ὁμώνυμος ἔστω [ἀριθμὸς] ὁ Γ· λέγω, ὅτι ὁ Γ τὸν Α let the [number] C be called the same as the part B (i.e.,
μετρεῖ. B is the Cth part of A). I say that C measures A.
᾿Επεὶ γὰρ ὁ Β τοῦ Α μέρος ἐστὶν ὁμώνυμον τῷ Γ, ἔστι For since B is a part of A called the same as C, and
δὲ καὶ ἡ Δ μονὰς τοῦ Γ μέρος ὁμώνυμον αὐτῷ, ὃ ἄρα μέρος the unit D is also a part of C called the same as it (i.e.,


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ELEMENTS BOOK 7

ljþ number B as many times as C (measures) A [Prop. 7.15].
ἐστὶν ἡ Δ μονὰς τοῦ Γ ἀριθμοῦ, τὸ αὐτὸ μέρος ἐστὶ καὶ ὁ Β D is the Cth part of C), thus which(ever) part the unit D
τοῦ Α· ἰσάκις ἄρα ἡ Δ μονὰς τὸν Γ ἀριθμὸν μετρεῖ καὶ ὁ Β is of the number C, B is also the same part of A. Thus,
τὸν Α. ἐναλλὰξ ἄρα ἰσάκις ἡ Δ μονὰς τὸν Β ἀριθμὸν μετρεῖ the unit D measures the number C as many times as B
καὶ ὁ Γ τὸν Α. ὁ Γ ἄρα τὸν Α μετρεῖ· ὅπερ ἔδει δεὶξαι. (measures) A. Thus, alternately, the unit D measures the

Thus, C measures A. (Which is) the very thing it was
required to show.


.
Proposition 39
᾿Αριθμὸν εὐρεῖν, ὃς ἐλάχιστος ὢν ἕξει τὰ δοθέντα μέρη. To find the least number that will have given parts.

Α Β Γ A B C


∆ Ε D E



Ζ F


Η G


Θ H

῎Εστω τὰ δοθέντα μέρη τὰ Α, Β, Γ· δεῖ δὴ ἀριθμὸν Let A, B, and C be the given parts. So it is required
εὑρεῖν, ὃς ἐλάχιστος ὢν ἕξει τὰ Α, Β, Γ μέρη. to find the least number which will have the parts A, B,
῎Εστωσαν γὰρ τοῖς Α, Β, Γ μέρεσιν ὁμώνυμοι ἀριθμοὶ and C (i.e., an Ath part, a Bth part, and a Cth part).
οἱ Δ, Ε, Ζ, καὶ εἰλήφθω ὑπὸ τῶν Δ, Ε, Ζ ἐλάχιστος με- For let D, E, and F be numbers having the same
τρούμενος ἀριθμὸς ὁ Η. names as the parts A, B, and C (respectively). And let
῾Ο Η ἄρα ὁμώνυμα μέρη ἔχει τοῖς Δ, Ε, Ζ. τοῖς δὲ Δ, the least number, G, measured by D, E, and F, have
Ε, Ζ ὁμώνυμα μέρη ἐστὶ τὰ Α, Β, Γ· ὁ Η ἄρα ἔχει τὰ Α, Β, been taken [Prop. 7.36].
Γ μέρη. λέγω δή, ὅτι καὶ ἐλάχιστος ὤν, εἰ γὰρ μή, ἔσται τις Thus, G has parts called the same as D, E, and F
τοῦ Η ἐλάσσων ἀριθμός, ὃς ἕξει τὰ Α, Β, Γ μέρη. ἔστω ὁ [Prop. 7.37]. And A, B, and C are parts called the same
Θ. ἐπεὶ ὁ Θ ἔχει τὰ Α, Β, Γ μέρη, ὁ Θ ἄρα ὑπὸ ὁμωνύμων as D, E, and F (respectively). Thus, G has the parts A,
ἀριθμῶν μετρηθήσεται τοῖς Α, Β, Γ μέρεσιν. τοῖς δὲ Α, Β, B, and C. So I say that (G) is also the least (number
Γ μέρεσιν ὁμώνυμοι ἀριθμοί εἰσιν οἱ Δ, Ε, Ζ· ὁ Θ ἄρα ὑπὸ having the parts A, B, and C). For if not, there will be
τῶν Δ, Ε, Ζ μετρεῖται. καί ἐστιν ἐλάσσων τοῦ Η· ὅπερ some number less than G which will have the parts A,
ἐστὶν ἀδύνατον. οὐκ ἄρα ἔσται τις τοῦ Η ἐλάσσων ἀριθμός, B, and C. Let it be H. Since H has the parts A, B, and
ὃς ἕξει τὰ Α, Β, Γ μέρη· ὅπερ ἔδει δεῖξαι. C, H will thus be measured by numbers called the same
as the parts A, B, and C [Prop. 7.38]. And D, E, and
F are numbers called the same as the parts A, B, and C
(respectively). Thus, H is measured by D, E, and F. And
(H) is less than G. The very thing is impossible. Thus,
there cannot be some number less than G which will have
the parts A, B, and C. (Which is) the very thing it was
required to show.







225

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Continued Proportion †



































































† The propositions contained in Books 7–9 are generally attributed to the school of Pythagoras.

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Proposition 1
.
᾿Εὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ δὲ If there are any multitude whatsoever of continuously
ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους ὦσιν, ἐλάχιστοί εἰσι proportional numbers, and the outermost of them are
τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς. prime to one another, then the (numbers) are the least
of those (numbers) having the same ratio as them.
Α Ε A E
Β Ζ B F
Γ Η C G
∆ Θ D H
῎Εστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Let A, B, C, D be any multitude whatsoever of con-
Γ, Δ, οἱ δὲ ἄκροι αὐτῶν οἱ Α, Δ, πρῶτοι πρὸς ἀλλήλους tinuously proportional numbers. And let the outermost
ἔστωσαν· λέγω, ὅτι οἱ Α, Β, Γ, Δ ἐλάχιστοί εἰσι τῶν τὸν of them, A and D, be prime to one another. I say that
αὐτὸν λόγον ἐχόντων αὐτοῖς. A, B, C, D are the least of those (numbers) having the
Εἰ γὰρ μή, ἔστωσαν ἐλάττονες τῶν Α, Β, Γ, Δ οἱ Ε, same ratio as them.
Ζ, Η, Θ ἐν τῷ αὐτῷ λόγῳ ὄντες αὐτοῖς. καὶ ἐπεὶ οἱ Α, For if not, let E, F, G, H be less than A, B, C, D
Β, Γ, Δ ἐν τῷ αὐτῷ λόγῳ εἰσὶ τοῖς Ε, Ζ, Η, Θ, καί ἐστιν (respectively), being in the same ratio as them. And since
ἴσον τὸ πλῆθος [τῶν Α, Β, Γ, Δ] τῷ πλήθει [τῶν Ε, Ζ, Η, A, B, C, D are in the same ratio as E, F, G, H, and the
Θ], δι᾿ ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Δ, ὁ Ε πρὸς τὸν multitude [of A, B, C, D] is equal to the multitude [of E,
Θ. οἱ δὲ Α, Δ πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ F, G, H], thus, via equality, as A is to D, (so) E (is) to H
ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας [Prop. 7.14]. And A and D (are) prime (to one another).
ἰσάκις ὅ τε μείζων τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, And prime (numbers are) also the least of those (numbers
τουτέστιν ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος having the same ratio as them) [Prop. 7.21]. And the
τὸν ἑπόμενον. μετρεῖ ἄρα ὁ Α τὸν Ε ὁ μείζων τὸν ἐλάσσονα· least numbers measure those (numbers) having the same
ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα οἱ Ε, Ζ, Η, Θ ἐλάσσονες ratio (as them) an equal number of times, the greater
bþ impossible. Thus, E, F, G, H, being less than A, B, C,
ὄντες τῶν Α, Β, Γ, Δ ἐν τῷ αὐτῷ λόγῳ εἰσὶν αὐτοῖς. οἱ Α, (measuring) the greater, and the lesser the lesser—that
Β, Γ, Δ ἄρα ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων is to say, the leading (measuring) the leading, and the
αὐτοῖς· ὅπερ ἔδει δεῖξαι. following the following [Prop. 7.20]. Thus, A measures
E, the greater (measuring) the lesser. The very thing is

D, are not in the same ratio as them. Thus, A, B, C, D
are the least of those (numbers) having the same ratio as
them. (Which is) the very thing it was required to show.

.
Proposition 2
Αριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον ἐλαχίστους, ὅσους ἂν To find the least numbers, as many as may be pre-
ἐπιτάξῃ τις, ἐν τῷ δοθέντι λόγῳ. scribed, (which are) continuously proportional in a given
῎Εστω ὁ δοθεὶς λόγος ἐν ἐλάχίστοις ἀριθμοῖς ὁ τοῦ ratio.
Α πρὸς τὸν Β· δεῖ δὴ ἀριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον Let the given ratio, (expressed) in the least numbers,
ἐλαχίστους, ὅσους ἄν τις ἐπιτάξῃ, ἐν τῷ τοῦ Α πρὸς τὸν Β be that of A to B. So it is required to find the least num-
λόγῳ. bers, as many as may be prescribed, (which are) in the
᾿Επιτετάχθωσαν δὴ τέσσαρες, καὶ ὁ Α ἑαυτὸν πολλα- ratio of A to B.
πλασιάσας τὸν Γ ποιείτω, τὸν δὲ Β πολλαπλασιάσας τὸν Δ Let four (numbers) have been prescribed. And let A
ποιείτω, καὶ ἔτι ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Ε ποιείτω, make C (by) multiplying itself, and let it make D (by)
καὶ ἔτι ὁ Α τοὺς Γ, Δ, Ε πολλαπλασιάσας τοὺς Ζ, Η, Θ multiplying B. And, further, let B make E (by) multiply-
ποιείτω, ὁ δὲ Β τὸν Ε πολλαπλασιάσας τὸν Κ ποιείτω. ing itself. And, further, let A make F, G, H (by) mul-
tiplying C, D, E. And let B make K (by) multiplying
E.


228

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Α Γ A C ELEMENTS BOOK 8

Β ∆ B D

Ε E

Ζ F
Η G

Θ H

Κ K
Καὶ ἐπεὶ ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Γ And since A has made C (by) multiplying itself, and
πεποίηκεν, τὸν δὲ Β πολλαπλασιάσας τὸν Δ πεποίηκεν, has made D (by) multiplying B, thus as A is to B, [so] C
ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, [οὕτως] ὁ Γ πρὸς τὸν Δ. (is) to D [Prop. 7.17]. Again, since A has made D (by)
πάλιν, ἐπεὶ ὁ μὲν Α τὸν Β πολλαπλασιάσας τὸν Δ πεποίηκεν, multiplying B, and B has made E (by) multiplying itself,
ὁ δὲ Β ἑαυτὸν πολλαπλασιάσας τὸν Ε πεποίηκεν, ἑκάτερος A, B have thus made D, E, respectively, (by) multiplying
ἄρα τῶν Α, Β τὸν Β πολλαπλασιάσας ἑκάτερον τῶν Δ, Ε B. Thus, as A is to B, so D (is) to E [Prop. 7.18]. But, as
πεποίηκεν. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Δ πρὸς A (is) to B, (so) C (is) to D. And thus as C (is) to D, (so)
τὸν Ε. ἀλλ᾿ ὡς ὁ Α πρὸς τὸν Β, ὁ Γ πρὸς τὸν Δ· καὶ ὡς D (is) to E. And since A has made F, G (by) multiplying
ἄρα ὁ Γ πρὸς τὸν Δ, ὁ Δ πρὸς τὸν Ε. καὶ ἐπεὶ ὁ Α τοὺς Γ, C, D, thus as C is to D, [so] F (is) to G [Prop. 7.17].
Δ πολλαπλασιάσας τοὺς Ζ, Η πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ And as C (is) to D, so A was to B. And thus as A (is)
πρὸς τὸν Δ, [οὕτως] ὁ Ζ πρὸς τὸν Η. ὡς δὲ ὁ Γ πρὸς τὸν to B, (so) F (is) to G. Again, since A has made G, H
Δ, οὕτως ἦν ὁ Α πρὸς τὸν Β· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, ὁ (by) multiplying D, E, thus as D is to E, (so) G (is) to
Ζ πρὸς τὸν Η. πάλιν, ἐπεὶ ὁ Α τοὺς Δ, Ε πολλαπλασιάσας H [Prop. 7.17]. But, as D (is) to E, (so) A (is) to B.
τοὺς Η, Θ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Ε, ὁ Η And thus as A (is) to B, so G (is) to H. And since A, B
πρὸς τὸν Θ. ἀλλ᾿ ὡς ὁ Δ πρὸς τὸν Ε, ὁ Α πρὸς τὸν Β. καὶ have made H, K (by) multiplying E, thus as A is to B,
ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς τὸν Θ. καὶ ἐπεὶ so H (is) to K. But, as A (is) to B, so F (is) to G, and
οἱ Α, Β τὸν Ε πολλαπλασιάσαντες τοὺς Θ, Κ πεποιήκασιν, G to H. And thus as F (is) to G, so G (is) to H, and H
ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Θ πρὸς τὸν Κ. ἀλλ᾿ to K. Thus, C, D, E and F, G, H, K are (both continu-
ὡς ὁ Α πρὸς τὸν Β, οὕτως ὅ τε Ζ πρὸς τὸν Η καὶ ὁ Η πρὸς ously) proportional in the ratio of A to B. So I say that
τὸν Θ. καὶ ὡς ἄρα ὁ Ζ πρὸς τὸν Η, οὕτως ὅ τε Η πρὸς (they are) also the least (sets of numbers continuously
τὸν Θ καὶ ὁ Θ πρὸς τὸν Κ· οἱ Γ, Δ, Ε ἄρα καὶ οἱ Ζ, Η, proportional in that ratio). For since A and B are the
Θ, Κ ἀνάλογόν εἰσιν ἐν τῷ τοῦ Α πρὸς τὸν Β λόγῳ. λέγω least of those (numbers) having the same ratio as them,
δή, ὅτι καὶ ἐλάχιστοι. ἐπεὶ γὰρ οἱ Α, Β ἐλάχιστοί εἰσι τῶν and the least of those (numbers) having the same ratio
τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς, οἱ δὲ ἐλάχιστοι τῶν τὸν are prime to one another [Prop. 7.22], A and B are thus
αὐτὸν λόγον ἐχόντων πρῶτοι πρὸς ἀλλήλους εἰσίν, οἱ Α, Β prime to one another. And A, B have made C, E, respec-
ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν. καὶ ἑκάτερος μὲν τῶν Α, tively, (by) multiplying themselves, and have made F, K
Β ἑαυτὸν πολλαπλασιάσας ἑκάτερον τῶν Γ, Ε πεποίηκεν, by multiplying C, E, respectively. Thus, C, E and F, K
ἑκάτερον δὲ τῶν Γ, Ε πολλαπλασιάσας ἑκάτερον τῶν Ζ, Κ are prime to one another [Prop. 7.27]. And if there are
πεποίηκεν· οἱ Γ, Ε ἄρα καὶ οἱ Ζ, Κ πρῶτοι πρὸς ἀλλήλους any multitude whatsoever of continuously proportional
εἰσίν. ἐὰν δὲ ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ numbers, and the outermost of them are prime to one
δὲ ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους ὦσιν, ἐλάχιστοί εἰσι another, then the (numbers) are the least of those (num-
τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς. οἱ Γ, Δ, Ε ἄρα καὶ bers) having the same ratio as them [Prop. 8.1]. Thus, C,
οἱ Ζ, Η, Θ, Κ ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων D, E and F, G, H, K are the least of those (continuously
τοῖς Α, Β· ὅπερ ἔδει δεῖξαι. proportional sets of numbers) having the same ratio as A
and B. (Which is) the very thing it was required to show.







229

ST EW hþ. ìri sma






ELEMENTS BOOK 8
gþ

Corollary
.
᾿Εκ δὴ τούτου φανερόν, ὅτι ἐὰν τρεῖς ἀριθμοὶ ἑξῆς So it is clear, from this, that if three continuously pro-
ἀνάλογον ἐλάχιστοι ὦσι τῶν τὸν αὐτὸν λόγον ἐχόντων portional numbers are the least of those (numbers) hav-
αὐτοῖς, οἱ ἄκρον αὐτῶν τετράγωνοί εἰσιν, ἐὰν δὲ τέσσαρες, ing the same ratio as them then the outermost of them
κύβοι. are square, and, if four (numbers), cube.

.
Proposition 3
᾿Εὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον ἐλάχιστ- If there are any multitude whatsoever of continu-
οι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς, οἱ ἄκροι αὐτῶν ously proportional numbers (which are) the least of those
πρῶτοι πρὸς ἀλλήλους εἰσίν. (numbers) having the same ratio as them then the outer-
most of them are prime to one another.
Α Ε Η A E G

Β Ζ Θ B F H
Γ Κ C K
∆ D

Λ L

Μ M
Ν N
Ξ O
῎Εστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον ἐλάχιστοι Let A, B, C, D be any multitude whatsoever of con-
τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς οἱ Α, Β, Γ, Δ· λέγω, tinuously proportional numbers (which are) the least of
ὅτι οἱ ἄκροι αὐτῶν οἱ Α, Δ πρῶτοι πρὸς ἀλλήλους εἰσίν. those (numbers) having the same ratio as them. I say
Εἰλήφθωσαν γὰρ δύο μὲν ἀριθμοὶ ἐλάχιστοι ἐν τῷ τῶν that the outermost of them, A and D, are prime to one
Α, Β, Γ, Δ λόγῳ οἱ Ε, Ζ, τρεῖς δὲ οἱ Η, Θ, Κ, καὶ ἑξῆς another.
ἑνὶ πλείους, ἕως τὸ λαμβανόμενον πλῆθος ἴσον γένηται τῷ For let the two least (numbers) E, F (which are)
πλήθει τῶν Α, Β, Γ, Δ. εἰλήφθωσαν καὶ ἔστωσαν οἱ Λ, Μ, in the same ratio as A, B, C, D have been taken
Ν, Ξ. [Prop. 7.33]. And the three (least numbers) G, H, K
Καὶ ἐπεὶ οἱ Ε, Ζ ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον [Prop. 8.2]. And (so on), successively increasing by one,
ἐχόντων αὐτοῖς, πρῶτοι πρὸς ἀλλήλους εἰσίν. καὶ ἐπεὶ until the multitude of (numbers) taken is made equal to
ἑκάτερος τῶν Ε, Ζ ἑαυτὸν μὲν πολλαπλασιάσας ἑκάτερον the multitude of A, B, C, D. Let them have been taken,
τῶν Η, Κ πεποίηκεν, ἑκάτερον δὲ τῶν Η, Κ πολλα- and let them be L, M, N, O.
πλασιάσας ἑκάτερον τῶν Λ, Ξ πεποίηκεν, καὶ οἱ Η, Κ ἄρα And since E and F are the least of those (numbers)
καὶ οἱ Λ, Ξ πρῶτοι πρὸς ἀλλήλους εἰσίν. καὶ ἐπεὶ οἱ Α, Β, having the same ratio as them they are prime to one an-
Γ, Δ ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς, other [Prop. 7.22]. And since E, F have made G, K, re-
εἰσὶ δὲ καὶ οἱ Λ, Μ, Ν, Ξ ἐλάχιστοι ἐν τῷ αὐτῷ λόγῳ ὄντες spectively, (by) multiplying themselves [Prop. 8.2 corr.],
τοῖς Α, Β, Γ, Δ, καί ἐστιν ἴσον τὸ πλῆθος τῶν Α, Β, Γ, and have made L, O (by) multiplying G, K, respec-
Δ τῷ πλήθει τῶν Λ, Μ, Ν, Ξ, ἕκαστος ἄρα τῶν Α, Β, Γ, tively, G, K and L, O are thus also prime to one another
Δ ἑκάστῳ τῶν Λ, Μ, Ν, Ξ ἴσος ἐστίν· ἴσος ἄρα ἐστὶν ὁ [Prop. 7.27]. And since A, B, C, D are the least of those
μὲν Α τῷ Λ, ὁ δὲ Δ τῷ Ξ. καί εἰσιν οἱ Λ, Ξ πρῶτοι πρὸς (numbers) having the same ratio as them, and L, M, N,
ἀλλήλους. καὶ οἱ Α, Δ ἄρα πρῶτοι πρὸς ἀλλήλους εἰσίν· O are also the least (of those numbers having the same
ὅπερ ἔδει δεῖξαι. ratio as them), being in the same ratio as A, B, C, D, and
the multitude of A, B, C, D is equal to the multitude of
L, M, N, O, thus A, B, C, D are equal to L, M, N, O,
respectively. Thus, A is equal to L, and D to O. And L
and O are prime to one another. Thus, A and D are also
prime to one another. (Which is) the very thing it was

230

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.

For any multitude whatsoever of given ratios, (ex-
Λόγων δοθέντων ὁποσωνοῦν ἐν ἐλαχίστοις ἀριθμοῖς required to show. Proposition 4
ἀριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον ἐλαχίστους ἐν τοῖς δοθεῖσι pressed) in the least numbers, to find the least numbers
λόγοις. continuously proportional in these given ratios.
Α Β A B
Γ ∆ C D
Ε Ζ E F

Ν Θ N H
Ξ Η O G

Μ Κ M K
Ο Λ P L
῎Εστωσαν οἱ δοθέντες λόγοι ἐν ἐλαχίστοις ἀριθμοῖς ὅ Let the given ratios, (expressed) in the least numbers,
τε τοῦ Α πρὸς τὸν Β καὶ ὁ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι ὁ be the (ratios) of A to B, and of C to D, and, further,
τοῦ Ε πρὸς τὸν Ζ· δεῖ δὴ ἀριθμοὺς εὑρεῖν ἑξῆς ἀνάλογον of E to F. So it is required to find the least numbers
ἐλαχίστους ἔν τε τῷ τοῦ Α πρὸς τὸν Β λόγῳ καὶ ἐν τῷ τοῦ continuously proportional in the ratio of A to B, and of
Γ πρὸς τὸν Δ καὶ ἔτι τῷ τοῦ Ε πρὸς τὸν Ζ. C to B, and, further, of E to F.
Εἰλήφθω γὰρ ὁ ὑπὸ τῶν Β, Γ ἐλάχιστος μετρούμενος For let the least number, G, measured by (both) B and
ἀριθμὸς ὁ Η. καὶ ὁσάκις μὲν ὁ Β τὸν Η μετρεῖ, τοσαυτάκις C have be taken [Prop. 7.34]. And as many times as B
καὶ ὁ Α τὸν Θ μετρείτω, ὁσάκις δὲ ὁ Γ τὸν Η μετρεῖ, το- measures G, so many times let A also measure H. And as
σαυτάκις καὶ ὁ Δ τὸν Κ μετρείτω. ὁ δὲ Ε τὸν Κ ἤτοι μετρεῖ many times as C measures G, so many times let D also
ἢ οὐ μετρεῖ. μετρείτω πρότερον. καὶ ὁσάκις ὁ Ε τὸν Κ με- measure K. And E either measures, or does not measure,
τρεῖ, τοσαυτάκις καὶ ὁ Ζ τὸν Λ μετρείτω. καὶ ἐπεὶ ἰσάκις ὁ K. Let it, first of all, measure (K). And as many times as
Α τὸν Θ μετρεῖ καὶ ὁ Β τὸν Η, ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν E measures K, so many times let F also measure L. And
Β, οὕτως ὁ Θ πρὸς τὸν Η. διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Γ πρὸς since A measures H the same number of times that B also
τὸν Δ, οὕτως ὁ Η πρὸς τὸν Κ, καὶ ἔτι ὡς ὁ Ε πρὸς τὸν Ζ, (measures) G, thus as A is to B, so H (is) to G [Def. 7.20,
οὕτως ὁ Κ πρὸς τὸν Λ· οἱ Θ, Η, Κ, Λ ἄρα ἑξῆς ἀνάλογόν Prop. 7.13]. And so, for the same (reasons), as C (is) to
εἰσιν ἔν τε τῷ τοῦ Α πρὸς τὸν Β καὶ ἐν τῷ τοῦ Γ πρὸς τὸν D, so G (is) to K, and, further, as E (is) to F, so K (is)
Δ καὶ ἔτι ἐν τῷ τοῦ Ε πρὸς τὸν Ζ λόγῳ. λέγω δή, ὅτι καὶ to L. Thus, H, G, K, L are continuously proportional in
ἐλάχιστοι. εἰ γὰρ μή εἰσιν οἱ Θ, Η, Κ, Λ ἑξῆς ἀνάλογον the ratio of A to B, and of C to D, and, further, of E to
ἐλάχιστοι ἔν τε τοῖς τοῦ Α πρὸς τὸν Β καὶ τοῦ Γ πρὸς τὸν F. So I say that (they are) also the least (numbers con-
Δ καὶ ἐν τῷ τοῦ Ε πρὸς τὸν Ζ λόγοις, ἔστωσαν οἱ Ν, Ξ, tinuously proportional in these ratios). For if H, G, K,
Μ, Ο. καὶ ἐπεί ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Ν πρὸς L are not the least numbers continuously proportional in
τὸν Ξ, οἱ δὲ Α, Β ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς the ratios of A to B, and of C to D, and of E to F, let N,
τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων τὸν μείζονα O, M, P be (the least such numbers). And since as A is
καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος to B, so N (is) to O, and A and B are the least (numbers
τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον, ὁ Β ἄρα τὸν which have the same ratio as them), and the least (num-
Ξ μετρεῖ. διὰ τὰ αὐτὰ δὴ καὶ ὁ Γ τὸν Ξ μετρεῖ· οἱ Β, Γ bers) measure those (numbers) having the same ratio (as
ἄρα τὸν Ξ μετροῦσιν· καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν Β, Γ them) an equal number of times, the greater (measur-
μετρούμενος τὸν Ξ μετρήσει. ἐλάχιστος δὲ ὑπὸ τῶν Β, Γ ing) the greater, and the lesser the lesser—that is to say,
μετρεῖται ὁ Η· ὁ Η ἄρα τὸν Ξ μετρεῖ ὁ μείζων τὸν ἐλάσσονα· the leading (measuring) the leading, and the following
ὅπερ ἐστὶν ἀδύντατον. οὐκ ἄρα ἔσονταί τινες τῶν Θ, Η, Κ, the following [Prop. 7.20], B thus measures O. So, for
Λ ἐλάσσονες ἀριθμοὶ ἑξῆς ἔν τε τῷ τοῦ Α πρὸς τὸν Β καὶ the same (reasons), C also measures O. Thus, B and C
τῷ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι τῷ τοῦ Ε πρὸς τὸν Ζ λόγῷ. (both) measure O. Thus, the least number measured by
(both) B and C will also measure O [Prop. 7.35]. And
G (is) the least number measured by (both) B and C.

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Thus, G measures O, the greater (measuring) the lesser.
The very thing is impossible. Thus, there cannot be any
numbers less than H, G, K, L (which are) continuously
(proportional) in the ratio of A to B, and of C to D, and,
further, of E to F.
Α Β A B
Γ ∆ C D
Ε Ζ E F

Ν Θ N H
Ξ Η O G
Μ Κ M K
Ο P
Π Q
Ρ R
Σ S
Τ T
Μὴ μετρείτω δὴ ὁ Ε τὸν Κ, καὶ εἰλήφθω ὑπὸ τῶν Ε, So let E not measure K. And let the least num-
Κ ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Μ. καὶ ὁσάκις μὲν ber, M, measured by (both) E and K have been taken
ὁ Κ τὸν Μ μετρεῖ, τοσαυτάκις καὶ ἑκάτερος τῶν Θ, Η [Prop. 7.34]. And as many times as K measures M, so
ἑκάτερον τῶν Ν, Ξ μετρείτω, ὁσάακις δὲ ὁ Ε τὸν Μ με- many times let H, G also measure N, O, respectively.
τρεῖ, τοσαυτάκις καὶ ὁ Ζ τὸν Ο μετρείτω. ἐπεὶ ἰσάκις ὁ Θ And as many times as E measures M, so many times let
τὸν Ν μετρεῖ καὶ ὁ Η τὸν Ξ, ἔστιν ἄρα ὡς ὁ Θ πρὸς τὸν F also measure P. Since H measures N the same num-
Η, οὕτως ὁ Ν πρὸς τὸν Ξ. ὡς δὲ ὁ Θ πρὸς τὸν Η, οὕτως ber of times as G (measures) O, thus as H is to G, so
ὁ Α πρὸς τὸν Β· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Ν N (is) to O [Def. 7.20, Prop. 7.13]. And as H (is) to G,
πρὸς τὸν Ξ. διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως so A (is) to B. And thus as A (is) to B, so N (is) to
ὁ Ξ πρὸς τὸν Μ. πάλιν, ἐπεὶ ἰσάκις ὁ Ε τὸν Μ μετρεῖ καὶ O. And so, for the same (reasons), as C (is) to D, so O
ὁ Ζ τὸν Ο, ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Ζ, οὕτως ὁ Μ πρὸς (is) to M. Again, since E measures M the same num-
τὸν Ο· οἱ Ν, Ξ, Μ, Ο ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τοῖς τοῦ ber of times as F (measures) P, thus as E is to F, so
τε Α πρὸς τὸν Β καὶ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι τοῦ Ε πρὸς M (is) to P [Def. 7.20, Prop. 7.13]. Thus, N, O, M, P
τὸν Ζ λόγοις. λέγω δή, ὅτι καὶ ἐλάχιστοι ἐν τοῖς Α Β, Γ are continuously proportional in the ratios of A to B, and
Δ, Ε Ζ λόγοις. εἰ γὰρ μή, ἔσονταί τινες τῶν Ν, Ξ, Μ, Ο of C to D, and, further, of E to F. So I say that (they
ἐλάσσονες ἀριθμοὶ ἑξῆς ἀνάλογον ἐν τοῖς Α Β, Γ Δ, Ε Ζ are) also the least (numbers) in the ratios of A B, C D,
λόγοις. ἔστωσαν οἱ Π, Ρ, Σ, Τ. καὶ ἐπεί ἐστιν ὡς ὁ Π πρὸς E F. For if not, then there will be some numbers less
τὸν Ρ, οὕτως ὁ Α πρὸς τὸν Β, οἱ δὲ Α, Β ἐλάχιστοι, οἱ δὲ than N, O, M, P (which are) continuously proportional
ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς in the ratios of A B, C D, E F. Let them be Q, R, S,
ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν T . And since as Q is to R, so A (is) to B, and A and B
ἑπόμενον, ὁ Β ἄρα τὸν Ρ μετρεῖ. διὰ τὰ αὐτὰ δὴ καὶ ὁ Γ (are) the least (numbers having the same ratio as them),
τὸν Ρ μετρεῖ· οἱ Β, Γ ἄρα τὸν Ρ μετροῦσιν. καὶ ὁ ἐλάχιστος and the least (numbers) measure those (numbers) hav-
ἄρα ὑπὸ τῶν Β, Γ μετούμενος τὸν Ρ μετρήσει. ἐλάχιστος ing the same ratio as them an equal number of times,
δὲ ὑπὸ τῶν Β, Γ μετρούμενος ἐστιν ὁ Η· ὁ Η ἄρα τὸν Ρ the leading (measuring) the leading, and the following
μετρεῖ. καί ἐστιν ὡς ὁ Η πρὸς τὸν Ρ, οὕτως ὁ Κ πρὸς τὸν the following [Prop. 7.20], B thus measures R. So, for
Σ· καὶ ὁ Κ ἄρα τὸν Σ μετρεῖ. μετρεῖ δὲ καὶ ὁ Ε τὸν Σ· οἱ Ε, the same (reasons), C also measures R. Thus, B and C
Κ ἄρα τὸν Σ μετροῦσιν. καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν Ε, Κ (both) measure R. Thus, the least (number) measured by
μετρούμενος τὸν Σ μετρήσει. ἐλάχιστος δὲ ὑπὸ τῶν Ε, Κ (both) B and C will also measure R [Prop. 7.35]. And G
μετρούμενός ἐστιν ὁ Μ· ὁ Μ ἄρα τὸν Σ μετρεῖ ὁ μείζων τὸν is the least number measured by (both) B and C. Thus,
ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα ἔσονταί τινες τῶν G measures R. And as G is to R, so K (is) to S. Thus,



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Ν, Ξ, Μ, Ο ἐλάσσονες ἀριθμοὶ ἑξῆς ἀνάλογον ἔν τε τοῖς K also measures S [Def. 7.20]. And E also measures
τοῦ Α πρὸς τὸν Β καὶ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι τοῦ Ε πρὸς S [Prop. 7.20]. Thus, E and K (both) measure S. Thus,
τὸν Ζ λόγοις· οἱ Ν, Ξ, Μ, Ο ἄρα ἑξῆς ἀνάλογον ἐλάχιστοί the least (number) measured by (both) E and K will also
εἰσιν ἐν τοῖς Α Β, Γ Δ, Ε Ζ λόγοις· ὅπερ ἔδει δεῖξαι. measure S [Prop. 7.35]. And M is the least (number)
eþ sible. Thus there cannot be any numbers less than N, O,
measured by (both) E and K. Thus, M measures S, the
greater (measuring) the lesser. The very thing is impos-

M, P (which are) continuously proportional in the ratios
of A to B, and of C to D, and, further, of E to F. Thus,
N, O, M, P are the least (numbers) continuously propor-
tional in the ratios of A B, C D, E F. (Which is) the very
thing it was required to show.

Proposition 5
.
Οἱ ἐπίπεδοι ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσι τὸν Plane numbers have to one another the ratio compoun-
†
συγκείμενον ἐκ τῶν πλευρῶν. ded out of (the ratios of) their sides.
Α A
Β B

Γ ∆ C D
Ε Ζ E F

Η G
Θ H
Κ K

Λ L
῎Εστωσαν ἐπίπεδοι ἀριθμοὶ οἱ Α, Β, καὶ τοῦ μὲν Α Let A and B be plane numbers, and let the numbers
πλευραὶ ἔστωσαν οἱ Γ, Δ ἀριθμοί, τοῦ δὲ Β οἱ Ε, Ζ· λέγω, C, D be the sides of A, and (the numbers) E, F (the
ὅτι ὁ Α πρὸς τὸν Β λόγον ἔχει τὸν συγκείμενον ἐκ τῶν sides) of B. I say that A has to B the ratio compounded
πλευρῶν. out of (the ratios of) their sides.
Λόγων γὰρ δοθέντων τοῦ τε ὃν ἔχει ὁ Γ πρὸς τὸν Ε καὶ For given the ratios which C has to E, and D (has) to
ὁ Δ πρὸς τὸν Ζ εἰλήφθωσαν ἀριθμοὶ ἑξῆς ἐλάχιστοι ἐν τοῖς F, let the least numbers, G, H, K, continuously propor-
Γ Ε, Δ Ζ λόγοις, οἱ Η, Θ, Κ, ὥστε εἶναι ὡς μὲν τὸν Γ πρὸς tional in the ratios C E, D F have been taken [Prop. 8.4],
τὸν Ε, οὕτως τὸν Η πρὸς τὸν Θ, ὡς δὲ τὸν Δ πρὸς τὸν Ζ, so that as C is to E, so G (is) to H, and as D (is) to F, so
οὕτως τὸν Θ πρὸς τὸν Κ. καὶ ὁ Δ τὸν Ε πολλαπλασιάσας H (is) to K. And let D make L (by) multiplying E.
τὸν Λ ποιείτω. And since D has made A (by) multiplying C, and has
Καὶ ἐπεὶ ὁ Δ τὸν μὲν Γ πολλαπλασιάσας τὸν Α made L (by) multiplying E, thus as C is to E, so A (is) to
πεποίηκεν, τὸν δὲ Ε πολλαπλασιάσας τὸν Λ πεποίηκεν, L [Prop. 7.17]. And as C (is) to E, so G (is) to H. And
ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν Λ. ὡς thus as G (is) to H, so A (is) to L. Again, since E has
δὲ ὁ Γ πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Θ· καὶ ὡς ἄρα ὁ made L (by) multiplying D [Prop. 7.16], but, in fact, has
Η πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Λ. πάλιν, ἐπεὶ ὁ Ε τὸν also made B (by) multiplying F, thus as D is to F, so L
Δ πολλαπλασιάσας τὸν Λ πεποίηκεν, ἀλλὰ μὴν καὶ τὸν Ζ (is) to B [Prop. 7.17]. But, as D (is) to F, so H (is) to
πολλαπλασιάσας τὸν Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Δ πρὸς K. And thus as H (is) to K, so L (is) to B. And it was
τὸν Ζ, οὕτως ὁ Λ πρὸς τὸν Β. ἀλλ᾿ ὡς ὁ Δ πρὸς τὸν Ζ, also shown that as G (is) to H, so A (is) to L. Thus, via
οὕτως ὁ Θ πρὸς τὸν Κ· καὶ ὡς ἄρα ὁ Θ πρὸς τὸν Κ, οὕτως equality, as G is to K, [so] A (is) to B [Prop. 7.14]. And
ὁ Λ πρὸς τὸν Β. ἐδείχθη δὲ καὶ ὡς ὁ Η πρὸς τὸν Θ, οὕτως G has to K the ratio compounded out of (the ratios of)
ὁ Α πρὸς τὸν Λ· δι᾿ ἴσου ἄρα ἐστὶν ὡς ὁ Η πρὸς τὸν Κ, the sides (of A and B). Thus, A also has to B the ratio
[οὕτως] ὁ Α πρὸς τὸν Β. ὁ δὲ Η πρὸς τὸν Κ λόγον ἔχει compounded out of (the ratios of) the sides (of A and B).

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ELEMENTS BOOK 8



τὸν συγκείμενον ἐκ τῶν πλευρῶν· καὶ ὁ Α ἄρα πρὸς τὸν (Which is) the very thing it was required to show.
Β λόγον ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν· ὅπερ ἔδει
δεῖξαι.

† i.e., multiplied.

Proposition 6
.
᾿Εὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, ὁ δὲ If there are any multitude whatsoever of continuously
πρῶτος τὸν δεύτερον μὴ μετρῇ, οὐδὲ ἄλλος οὐδεὶς οὐδένα proportional numbers, and the first does not measure the
μετρήσει. second, then no other (number) will measure any other
(number) either.
Α A
Β B
Γ C

∆ D
Ε E

Ζ F
Η G

Θ H
῎Εστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Let A, B, C, D, E be any multitude whatsoever of
Γ, Δ, Ε, ὁ δὲ Α τὸν Β μὴ μετρείτω· λέγω, ὅτι οὐδὲ ἄλλος continuously proportional numbers, and let A not mea-
οὐδεὶς οὐδένα μετρήσει. sure B. I say that no other (number) will measure any
῞Οτι μὲν οὖν οἱ Α, Β, Γ, Δ, Ε ἑξῆς ἀλλήλους οὐ με- other (number) either.
τροῦσιν, φανερόν· οὐδὲ γὰρ ὁ Α τὸν Β μετρεῖ. λέγω Now, (it is) clear that A, B, C, D, E do not succes-
δή, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει. εἰ γὰρ δυ- sively measure one another. For A does not even mea-
νατόν, μετρείτω ὁ Α τὸν Γ. καὶ ὅσοι εἰσὶν οἱ Α, Β, Γ, sure B. So I say that no other (number) will measure
τοσοῦτοι εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν any other (number) either. For, if possible, let A measure
λόγον ἐχόντων τοῖς Α, Β, Γ οἱ Ζ, Η, Θ. καὶ ἐπεὶ οἱ Ζ, C. And as many (numbers) as are A, B, C, let so many
Η, Θ ἐν τῷ αὐτῷ λόγῳ εἰσὶ τοῖς Α, Β, Γ, καί ἐστιν ἴσον τὸ of the least numbers, F, G, H, have been taken of those
πλῆθος τῶν Α, Β, Γ τῷ πλήθει τῶν Ζ, Η, Θ, δι᾿ ἴσου ἄρα (numbers) having the same ratio as A, B, C [Prop. 7.33].
ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Ζ πρὸς τὸν Θ. καὶ ἐπεί And since F, G, H are in the same ratio as A, B, C, and
ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Ζ πρὸς τὸν Η, οὐ μετρεῖ the multitude of A, B, C is equal to the multitude of F,
δὲ ὁ Α τὸν Β, οὐ μετρεῖ ἄρα οὐδὲ ὁ Ζ τὸν Η· οὐκ ἄρα μονάς G, H, thus, via equality, as A is to C, so F (is) to H
ἐστιν ὁ Ζ· ἡ γὰρ μονὰς πάντα ἀριθμὸν μετρεῖ. καί εἰσιν οἱ [Prop. 7.14]. And since as A is to B, so F (is) to G,
zþ F is to H, so A (is) to C. And thus A does not measure
Ζ, Θ πρῶτοι πρὸς ἀλλήλους [οὐδὲ ὁ Ζ ἄρα τὸν Θ μετρεῖ]. and A does not measure B, F does not measure G either
καί ἐστιν ὡς ὁ Ζ πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Γ· οὐδὲ [Def. 7.20]. Thus, F is not a unit. For a unit measures
ὁ Α ἄρα τὸν Γ μετρεῖ. ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἄλλος all numbers. And F and H are prime to one another
οὐδεὶς οὐδένα μετρήσει· ὅπερ ἔδει δεῖξαι. [Prop. 8.3] [and thus F does not measure H]. And as

C either [Def. 7.20]. So, similarly, we can show that no
other (number) can measure any other (number) either.
(Which is) the very thing it was required to show.

.
Proposition 7
᾿Εὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ [ἑξῆς] ἀνάλογον, ὁ δὲ If there are any multitude whatsoever of [continu-
πρῶτος τὸν ἔσχατον μετρῇ, καὶ τὸν δεύτερον μετρήσει. ously] proportional numbers, and the first measures the

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last, then (the first) will also measure the second.
Α A

Β B

Γ C

∆ D
hþ
῎Εστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, Let A, B, C, D be any number whatsoever of continu-
Δ, ὁ δὲ Α τὸν Δ μετρείτω· λέγω, ὅτι καὶ ὁ Α τὸν Β μετρεῖ. ously proportional numbers. And let A measure D. I say
Εἰ γὰρ οὐ μετρεῖ ὁ Α τὸν Β, οὐδὲ ἄλλος οὐδεὶς οὐδένα that A also measures B.
μετρήσει· μετρεῖ δὲ ὁ Α τὸν Δ. μετρεῖ ἄρα καὶ ὁ Α τὸν Β· For if A does not measure B then no other (number)
ὅπερ ἔδει δεῖξαι. will measure any other (number) either [Prop. 8.6]. But
A measures D. Thus, A also measures B. (Which is) the
very thing it was required to show.

.
Proposition 8
᾿Εὰν δύο ἀριθμῶν μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον If between two numbers there fall (some) numbers in
ἐμπίπτωσιν ἀριθμοί, ὅσοι εἰς αὐτοὺς μεταξὺ κατὰ τὸ συ- continued proportion then, as many numbers as fall in
νεχὲς ἀνόλογον ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ εἰς τοὺς between them in continued proportion, so many (num-
τὸν αὐτὸν λόγον ἔχοντας [αὐτοῖς] μεταξὺ κατὰ τὸ συνὲχες bers) will also fall in between (any two numbers) having
ἀνάλογον ἐμπεσοῦνται the same ratio [as them] in continued proportion.
Α Ε A E
Γ Μ C M

∆ Ν D N

Β Ζ B F

Η G

Θ H
Κ K
Λ L
Δύο γὰρ ἀριθμῶν τῶν Α, Β μεταξὺ κατὰ τὸ συνεχὲς For let the numbers, C and D, fall between two num-
ἀνάλογον ἐμπιπτέτωσαν ἀριθμοὶ οἱ Γ, Δ, καὶ πεποιήσθω ὡς bers, A and B, in continued proportion, and let it have
ὁ Α πρὸς τὸν Β, οὕτως ὁ Ε πρὸς τὸν Ζ· λέγω, ὅτι ὅσοι εἰς been contrived (that) as A (is) to B, so E (is) to F. I say
τοὺς Α, Β μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεπτώκασιν that, as many numbers as have fallen in between A and
ἀριθμοί, τοσοῦτοι καὶ εἰς τοὺς Ε, Ζ μεταξὺ κατὰ τὸ συνεχὲς B in continued proportion, so many (numbers) will also
ἀνάλογον ἐμπεσοῦνται. fall in between E and F in continued proportion.
῞Οσοι γάρ εἰσι τῷ πλήθει οἱ Α, Β, Γ, Δ, τοσοῦτοι For as many as A, B, C, D are in multitude, let so
εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον many of the least numbers, G, H, K, L, having the same
ἐχόντων τοῖς Α, Γ, Δ, Β οἱ Η, Θ, Κ, Λ· οἱ ἄρα ἄκροι ratio as A, B, C, D, have been taken [Prop. 7.33]. Thus,
αὐτῶν οἱ Η, Λ πρῶτοι πρὸς ἀλλήλους εἰσίν. καὶ ἐπεὶ οἱ Α, the outermost of them, G and L, are prime to one another
Γ, Δ, Β τοῖς Η, Θ, Κ, Λ ἐν τῷ αὐτῷ λόγῳ εἰσίν, καί ἐστιν [Prop. 8.3]. And since A, B, C, D are in the same ratio
ἴσον τὸ πλῆθος τῶν Α, Γ, Δ, Β τῷ πλήθει τῶν Η, Θ, Κ, as G, H, K, L, and the multitude of A, B, C, D is equal
Λ, δι᾿ ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Η πρὸς to the multitude of G, H, K, L, thus, via equality, as A is
τὸν Λ. ὡς δὲ ὁ Α πρὸς τὸν Β, οὕτως ὁ Ε πρὸς τὸν Ζ· καὶ to B, so G (is) to L [Prop. 7.14]. And as A (is) to B, so


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ὡς ἄρα ὁ Η πρὸς τὸν Λ, οὕτως ὁ Ε πρὸς τὸν Ζ. οἱ δὲ Η, Λ E (is) to F. And thus as G (is) to L, so E (is) to F. And
πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι ἀριθμοὶ G and L (are) prime (to one another). And (numbers)
μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις ὅ τε μείζων prime (to one another are) also the least (numbers hav-
τὸν μείζονα καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, τουτέστιν ὅ τε ing the same ratio as them) [Prop. 7.21]. And the least
ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον. numbers measure those (numbers) having the same ratio
ἰσάκις ἄρα ὁ Η τὸν Ε μετρεῖ καὶ ὁ Λ τὸν Ζ. ὁσάκις δὴ ὁ Η (as them) an equal number of times, the greater (measur-
τὸν Ε μετρεῖ, τοσαυτάκις καὶ ἑκάτερος τῶν Θ, Κ ἑκάτερον ing) the greater, and the lesser the lesser—that is to say,
τῶν Μ, Ν μετρείτω· οἱ Η, Θ, Κ, Λ ἄρα τοὺς Ε, Μ, Ν, Ζ the leading (measuring) the leading, and the following
ἰσάκις μετροῦσιν. οἱ Η, Θ, Κ, Λ ἄρα τοῖς Ε, Μ, Ν, Ζ ἐν τῷ the following [Prop. 7.20]. Thus, G measures E the same
αὐτῷ λόγῳ εἰσίν. ἀλλὰ οἱ Η, Θ, Κ, Λ τοῖς Α, Γ, Δ, Β ἐν τῷ number of times as L (measures) F. So as many times as
αὐτῷ λόγῳ εἰσίν· καὶ οἱ Α, Γ, Δ, Β ἄρα τοῖς Ε, Μ, Ν, Ζ ἐν G measures E, so many times let H, K also measure M,
τῷ αὐτῷ λόγῳ εἰσίν. οἱ δὲ Α, Γ, Δ, Β ἑξῆς ἀνάλογόν εἰσιν· N, respectively. Thus, G, H, K, L measure E, M, N,
καὶ οἱ Ε, Μ, Ν, Ζ ἄρα ἑξῆς ἀνάλογόν εἰσιν. ὅσοι ἄρα εἰς F (respectively) an equal number of times. Thus, G, H,
τοὺς Α, Β μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεπτώκασιν K, L are in the same ratio as E, M, N, F [Def. 7.20].
jþ E, M, N, F are also continuously proportional. Thus,
ἀριθμοί, τοσοῦτοι καὶ εἰς τοὺς Ε, Ζ μεταξὺ κατὰ τὸ συνεχὲς But, G, H, K, L are in the same ratio as A, C, D, B.
ἀνάλογον ἐμπεπτώκασιν ἀριθμοί· ὅπερ ἔδει δεῖξαι. Thus, A, C, D, B are also in the same ratio as E, M, N,
F. And A, C, D, B are continuously proportional. Thus,
as many numbers as have fallen in between A and B in
continued proportion, so many numbers have also fallen
in between E and F in continued proportion. (Which is)
the very thing it was required to show.

Proposition 9
.
᾿Εὰν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους ὦσιν, καὶ If two numbers are prime to one another and there
εἰς αὐτοὺς μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπίπτωσιν fall in between them (some) numbers in continued pro-
ἀριθμοί, ὅσοι εἰς αὐτοὺς μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον portion then, as many numbers as fall in between them
ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ ἑκατέρου αὐτῶν καὶ in continued proportion, so many (numbers) will also fall
μονάδος μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεσοῦνται. between each of them and a unit in continued proportion.
Α Θ A H
Γ Κ C K

∆ Λ D L
Β B

Μ M
Ε E
Ν N
Ζ Ξ F O

Η Ο G P
῎Εστωσαν δύο ἀριθμοὶ πρῶτοι πρὸς ἀλλήλους οἱ Α, Let A and B be two numbers (which are) prime to
Β, καὶ εἰς αὐτοὺς μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον one another, and let the (numbers) C and D fall in be-
ἐμπιπτέτωσαν οἱ Γ, Δ, καὶ ἐκκείσθω ἡ Ε μονάς· λέγω, tween them in continued proportion. And let the unit E
ὅτι ὅσοι εἰς τοὺς Α, Β μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον be set out. I say that, as many numbers as have fallen
ἐμπεπτώκασιν ἀριθμοί, τοσοῦτοι καὶ ἑκατέρου τῶν Α, in between A and B in continued proportion, so many
Β καὶ τῆς μονάδος μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον (numbers) will also fall between each of A and B and
ἐμπεσοῦνται. the unit in continued proportion.
Εἰλήφθωσαν γὰρ δύο μὲν ἀριθμοὶ ἐλάχιστοι ἐν τῷ τῶν For let the least two numbers, F and G, which are in
Α, Γ, Δ, Β λόγῳ ὄντες οἱ Ζ, Η, τρεῖς δὲ οἱ Θ, Κ, Λ, καὶ ἀεὶ the ratio of A, C, D, B, have been taken [Prop. 7.33].


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ELEMENTS BOOK 8



ἑξῆς ἑνὶ πλείους, ἕως ἂν ἴσον γένηται τὸ πλῆθος αὐτῶν τῷ And the (least) three (numbers), H, K, L. And so on,
πλήθει τῶν Α, Γ, Δ, Β. εἰλήφθωσαν, καὶ ἔστωσαν οἱ Μ, Ν, successively increasing by one, until the multitude of the
Ξ, Ο. φανερὸν δή, ὅτι ὁ μὲν Ζ ἑαυτὸν πολλαπλασιάσας τὸν (least numbers taken) is made equal to the multitude of
Θ πεποίηκεν, τὸν δὲ Θ πολλαπλασιάσας τὸν Μ πεποίηκεν, A, C, D, B [Prop. 8.2]. Let them have been taken, and
καὶ ὁ Η ἑαυτὸν μὲν πολλαπλασιάσας τὸν Λ πεποίηκεν, τὸν let them be M, N, O, P. So (it is) clear that F has made
δὲ Λ πολλαπλασιάσας τὸν Ο πεποίηκεν. καὶ ἐπεὶ οἱ Μ, Ν, H (by) multiplying itself, and has made M (by) multi-
Ξ, Ο ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Ζ, plying H. And G has made L (by) multiplying itself, and
Η, εἰσὶ δὲ καὶ οἱ Α, Γ, Δ, Β ἐλάχιστοι τῶν τὸν αὐτὸν λόγον has made P (by) multiplying L [Prop. 8.2 corr.]. And
ἐχόντων τοῖς Ζ, Η, καί ἐστιν ἴσον τὸ πλῆθος τῶν Μ, Ν, Ξ, since M, N, O, P are the least of those (numbers) hav-
Ο τῷ πλήθει τῶν Α, Γ, Δ, Β, ἕκαστος ἄρα τῶν Μ, Ν, Ξ, Ο ing the same ratio as F, G, and A, C, D, B are also the
ἑκάστῳ τῶν Α, Γ, Δ, Β ἴσος ἐστίν· ἴσος ἄρα ἐστὶν ὁ μὲν Μ least of those (numbers) having the same ratio as F, G
τῷ Α, ὁ δὲ Ο τῷ Β. καὶ ἐπεὶ ὁ Ζ ἑαυτὸν πολλαπλασιάσας [Prop. 8.2], and the multitude of M, N, O, P is equal
τὸν Θ πεποίηκεν, ὁ Ζ ἄρα τὸν Θ μετρεῖ κατὰ τὰς ἐν τῷ Ζ to the multitude of A, C, D, B, thus M, N, O, P are
μονάδας. μετρεῖ δὲ καὶ ἡ Ε μονὰς τὸν Ζ κατὰ τὰς ἐν αὐτῷ equal to A, C, D, B, respectively. Thus, M is equal to
μονάδας· ἰσάκις ἄρα ἡ Ε μονὰς τὸν Ζ ἀριθμὸν μετρεῖ καὶ ὁ Ζ A, and P to B. And since F has made H (by) multiply-
τὸν Θ. ἔστιν ἄρα ὡς ἡ Ε μονὰς πρὸς τὸν Ζ ἀριθμόν, οὕτως ing itself, F thus measures H according to the units in F
ὁ Ζ πρὸς τὸν Θ. πάλιν, ἐπεὶ ὁ Ζ τὸν Θ πολλαπλασιάσας [Def. 7.15]. And the unit E also measures F according to
τὸν Μ πεποίηκεν, ὁ Θ ἄρα τὸν Μ μετρεῖ κατὰ τὰς ἐν τῷ Ζ the units in it. Thus, the unit E measures the number F
μονάδας. μετρεῖ δὲ καὶ ἡ Ε μονὰς τὸν Ζ ἀριθμὸν κατὰ τὰς ἐν as many times as F (measures) H. Thus, as the unit E is
αὐτῷ μονάδας· ἰσάκις ἄρα ἡ Ε μονὰς τὸν Ζ ἀριθμὸν μετρεῖ to the number F, so F (is) to H [Def. 7.20]. Again, since
καὶ ὁ Θ τὸν Μ. ἔστιν ἄρα ὡς ἡ Ε μονὰς πρὸς τὸν Ζ ἀριθμόν, F has made M (by) multiplying H, H thus measures M
οὕτως ὁ Θ πρὸς τὸν Μ. ἐδείχθη δὲ καὶ ὡς ἡ Ε μονὰς πρὸς according to the units in F [Def. 7.15]. And the unit E
τὸν Ζ ἀριθμόν, οὕτως ὁ Ζ πρὸς τὸν Θ· καὶ ὡς ἄρα ἡ Ε μονὰς also measures the number F according to the units in it.
πρὸς τὸν Ζ ἀριθμόν, οὕτως ὁ Ζ πρὸς τὸν Θ καὶ ὁ Θ πρὸς Thus, the unit E measures the number F as many times
τὸν Μ. ἴσος δὲ ὁ Μ τῷ Α· ἔστιν ἄρα ὡς ἡ Ε μονὰς πρὸς τὸν as H (measures) M. Thus, as the unit E is to the number
Ζ ἀριθμόν, οὕτως ὁ Ζ πρὸς τὸν Θ καὶ ὁ Θ πρὸς τὸν Α. διὰ F, so H (is) to M [Prop. 7.20]. And it was shown that as
τὰ αὐτὰ δὴ καὶ ὡς ἡ Ε μονὰς πρὸς τὸν Η ἀριθμόν, οὕτως ὁ the unit E (is) to the number F, so F (is) to H. And thus
Η πρὸς τὸν Λ καὶ ὁ Λ πρὸς τὸν Β. ὅσοι ἄρα εἰς τοὺς Α, Β as the unit E (is) to the number F, so F (is) to H, and H
iþ have fallen in between A and B in continued proportion,
μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεπτώκασιν ἀριθμοί, (is) to M. And M (is) equal to A. Thus, as the unit E is
τοσοῦτοι καὶ ἑκατέρου τῶν Α, Β καὶ μονάδος τῆς Ε μεταξὺ to the number F, so F (is) to H, and H to A. And so, for
κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεπτώκασιν ἀριθμοί· ὅπερ ἔδει the same (reasons), as the unit E (is) to the number G,
δεῖξαι. so G (is) to L, and L to B. Thus, as many (numbers) as

so many numbers have also fallen between each of A and
B and the unit E in continued proportion. (Which is) the
very thing it was required to show.

Proposition 10
.
᾿Εάν δύο ἀριθμῶν ἑκατέρου καὶ μονάδος μεταξὺ κατὰ If (some) numbers fall between each of two numbers
τὸ συνεχὲς ἀνάλογον ἐμπίπτωσιν ἀριθμοί, ὅσοι ἑκατέρου and a unit in continued proportion then, as many (num-
αὐτῶν καὶ μονάδος μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον bers) as fall between each of the (two numbers) and the
ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ εἰς αὐτοὺς μεταξὺ κατὰ unit in continued proportion, so many (numbers) will
τὸ συνεχὲς ἀνάλογον ἐμπεσοῦνται. also fall in between the (two numbers) themselves in con-
Δύο γὰρ ἀριθμῶν τῶν Α, Β καὶ μονάδος τῆς Γ με- tinued proportion.
ταξύ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπιπτέτωσαν ἀριθμοὶ οἵ For let the numbers D, E and F, G fall between the
τε Δ, Ε καὶ οἱ Ζ, Η· λέγω, ὅτι ὅσοι ἑκατέρου τῶν Α, numbers A and B (respectively) and the unit C in con-
Β καὶ μονάδος τῆς Γ μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον tinued proportion. I say that, as many numbers as have
ἐμπεπτώκασιν ἀριθμοί, τοσοῦτοι καὶ εἰς τοὺς Α, Β μεταξὺ fallen between each of A and B and the unit C in contin-
κατὰ τὸ συνεχὲς ἀνάλογον ἐμπεσοῦνται. ued proportion, so many will also fall in between A and
B in continued proportion.


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Γ Γ C C ELEMENTS BOOK 8

∆ Ζ D F
Ε Η E G
Α Β A B


Θ H
Κ K
Λ L
῾Ο Δ γὰρ τὸν Ζ πολλαπλασιάσας τὸν Θ ποιείτω, For let D make H (by) multiplying F. And let D, F
ἑκάτερος δὲ τῶν Δ, Ζ τὸν Θ πολλαπλασιάσας ἑκάτερον make K, L, respectively, by multiplying H.
τῶν Κ, Λ ποιείτω. As since as the unit C is to the number D, so D (is) to
Καὶ ἐπεί ἐστιν ὡς ἡ Γ μονὰς πρὸς τὸν Δ ἀριθμόν, οὕτως E, the unit C thus measures the number D as many times
ὁ Δ πρὸς τὸν Ε, ἰσάκις ἄρα ἡ Γ μονὰς τὸν Δ ἀριθμὸν μετρεῖ as D (measures) E [Def. 7.20]. And the unit C measures
καὶ ὁ Δ τὸν Ε. ἡ δὲ Γ μονὰς τὸν Δ ἀριθμὸν μετρεῖ κατὰ τὰς the number D according to the units in D. Thus, the
ἐν τῷ Δ μονάδας· καὶ ὁ Δ ἄρα ἀριθμὸς τὸν Ε μετρεῖ κατὰ number D also measures E according to the units in D.
τὰς ἐν τῷ Δ μονάδας· ὁ Δ ἄρα ἑαυτὸν πολλαπλασιάσας τὸν Thus, D has made E (by) multiplying itself. Again, since
Ε πεποίηκεν. πάλιν, ἐπεί ἐστιν ὡς ἡ Γ [μονὰς] πρὸς τὸν as the [unit] C is to the number D, so E (is) to A, the
Δ ἀριθμὸν, οὕτως ὁ Ε πρὸς τὸν Α, ἰσάκις ἄρα ἡ Γ μονὰς unit C thus measures the number D as many times as E
τὸν Δ ἀριθμὸν μετρεῖ καὶ ὁ Ε τὸν Α. ἡ δὲ Γ μονὰς τὸν (measures) A [Def. 7.20]. And the unit C measures the
Δ ἀριθμὸν μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας· καὶ ὁ Ε ἄρα number D according to the units in D. Thus, E also mea-
τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Δ μονάδας· ὁ Δ ἄρα τὸν Ε sures A according to the units in D. Thus, D has made
πολλαπλασιάσας τὸν Α πεποίηκεν. διὰ τὰ αὐτὰ δὴ καὶ ὁ A (by) multiplying E. And so, for the same (reasons), F
μὲν Ζ ἑαυτὸν πολλαπλασιάσας τὸν Η πεποίηκεν, τὸν δὲ Η has made G (by) multiplying itself, and has made B (by)
πολλαπλασιάσας τὸν Β πεποίηκεν. καὶ ἐπεὶ ὁ Δ ἑαυτὸν μὲν multiplying G. And since D has made E (by) multiplying
πολλαπλασιάσας τὸν Ε πεποίηκεν, τὸν δὲ Ζ πολλαπλασιάσας itself, and has made H (by) multiplying F, thus as D is to
τὸν Θ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Ε F, so E (is) to H [Prop 7.17]. And so, for the same rea-
πρὸς τὸν Θ. διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ sons, as D (is) to F, so H (is) to G [Prop. 7.18]. And thus
Θ πρὸς τὸν Η. καὶ ὡς ἄρα ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Θ πρὸς as E (is) to H, so H (is) to G. Again, since D has made
τὸν Η. πάλιν, ἐπεὶ ὁ Δ ἑκάτερον τῶν Ε, Θ πολλαπλασιάσας A, K (by) multiplying E, H, respectively, thus as E is to
ἑκάτερον τῶν Α, Κ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Θ, H, so A (is) to K [Prop 7.17]. But, as E (is) to H, so D
οὕτως ὁ Α πρὸς τὸν Κ. ἀλλ᾿ ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Δ (is) to F. And thus as D (is) to F, so A (is) to K. Again,
πρὸς τὸν Ζ· καὶ ὡς ἄρα ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν since D, F have made K, L, respectively, (by) multiply-
Κ. πάλιν, ἐπεὶ ἑκάτερος τῶν Δ, Ζ τὸν Θ πολλαπλασιάσας ing H, thus as D is to F, so K (is) to L [Prop. 7.18]. But,
ἑκάτερον τῶν Κ, Λ πεποίηκεν, ἕστιν ἄρα ὡς ὁ Δ πρὸς τὸν as D (is) to F, so A (is) to K. And thus as A (is) to K,
Ζ, οὕτως ὁ Κ πρὸς τὸν Λ. ἀλλ᾿ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ so K (is) to L. Further, since F has made L, B (by) mul-
Α πρὸς τὸν Κ· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Κ, οὕτως ὁ Κ πρὸς tiplying H, G, respectively, thus as H is to G, so L (is) to
τὸν Λ. ἔτι ἐπεὶ ὁ Ζ ἑκάτερον τῶν Θ, Η πολλαπλασιάσας B [Prop 7.17]. And as H (is) to G, so D (is) to F. And
ἑκάτερον τῶν Λ, Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Θ πρὸς τὸν thus as D (is) to F, so L (is) to B. And it was also shown
Η, οὕτως ὁ Λ πρὸς τὸν Β. ὡς δὲ ὁ Θ πρὸς τὸν Η, οὕτως that as D (is) to F, so A (is) to K, and K to L. And thus
ὁ Δ πρὸς τὸν Ζ· καὶ ὡς ἄρα ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Λ as A (is) to K, so K (is) to L, and L to B. Thus, A, K,
πρὸς τὸν Β. ἐδείχθη δὲ καὶ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὅ τε L, B are successively in continued proportion. Thus, as
Α πρὸς τὸν Κ καὶ ὁ Κ πρὸς τὸν Λ· καὶ ὡς ἄρα ὁ Α πρὸς many numbers as fall between each of A and B and the
τὸν Κ, οὕτως ὁ Κ πρὸς τὸν Λ καὶ ὁ Λ πρὸς τὸν Β. οἱ Α, unit C in continued proportion, so many will also fall in
Κ, Λ, Β ἄρα κατὰ τὸ συνεχὲς ἑξῆς εἰσιν ἀνάλογον. ὅσοι between A and B in continued proportion. (Which is)
ἄρα ἑκατέρου τῶν Α, Β καὶ τῆς Γ μονάδος μεταξὺ κατὰ the very thing it was required to show.
τὸ συνεχὲς ἀνάλογον ἐμπίπτουσιν ἀριθμοί, τοσοῦτοι καὶ εἰς
τοὺς Α, Β μεταξὺ κατὰ τὸ συνεχὲς ἐμπεσοῦνται· ὅπερ ἔδει



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δεῖξαι. iaþ ELEMENTS BOOK 8
.
Proposition 11
Δύο τετραγώνων ἀριθμῶν εἷς μέσος ἀνάλογόν ἐστιν There exists one number in mean proportion to two
†
ἀριθμός, καὶ ὁ τετράγωνος πρὸς τὸν τετράγωνον δι- (given) square numbers. And (one) square (number)
‡
πλασίονα λόγον ἔχει ἤπερ ἡ πλευρὰ πρὸς τὴν πλευράν. has to the (other) square (number) a squared ratio with
respect to (that) the side (of the former has) to the side
(of the latter).
Α A

Β B

Γ ∆ C D

Ε E
῎Εστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, Β, καὶ τοῦ μὲν Α Let A and B be square numbers, and let C be the side
πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· λέγω, ὅτι τῶν Α, Β εἷς of A, and D (the side) of B. I say that there exists one
μέσος ἀνάλογόν ἐστιν ἀριθμός, καὶ ὁ Α πρὸς τὸν Β δι- number in mean proportion to A and B, and that A has
πλασίονα λόγον ἔχει ἤπερ ὁ Γ πρὸς τὸν Δ. to B a squared ratio with respect to (that) C (has) to D.
῾Ο Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω. καὶ For let C make E (by) multiplying D. And since A is
ἐπεὶ τετράγωνός ἐστιν ὁ Α, πλευρὰ δὲ αὐτοῦ ἐστιν ὁ Γ, ὁ Γ square, and C is its side, C has thus made A (by) multi-
ἄρα ἑαυτὸν πολλαπλασιάσας τὸν Α πεποίηκεν. διὰ τὰ αὐτὰ plying itself. And so, for the same (reasons), D has made
δὴ καὶ ὁ Δ ἑαυτὸν πολλαπλασιάσας τὸν Β πεποίηκεν. ἐπεὶ B (by) multiplying itself. Therefore, since C has made A,
οὖν ὁ Γ ἑκάτερον τῶν Γ, Δ πολλαπλασιάσας ἑκάτερον τῶν E (by) multiplying C, D, respectively, thus as C is to D,
Α, Ε πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α so A (is) to E [Prop. 7.17]. And so, for the same (rea-
πρὸς τὸν Ε. διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ sons), as C (is) to D, so E (is) to B [Prop. 7.18]. And
Ε πρὸς τὸν Β. καὶ ὡς ἄρα ὁ Α πρὸς τὸν Ε, οὕτως ὁ Ε πρὸς thus as A (is) to E, so E (is) to B. Thus, one number
τὸν Β. τῶν Α, Β ἄρα εἷς μέσος ἀνάλογόν ἐστιν ἀριθμός. (namely, E) is in mean proportion to A and B.
Λέγω δή, ὅτι καὶ ὁ Α πρὸς τὸν Β διπλασίονα λόγον ἔχει So I say that A also has to B a squared ratio with
ἤπερ ὁ Γ πρὸς τὸν Δ. ἐπεὶ γὰρ τρεῖς ἀριθμοὶ ἀνάλογόν εἰσιν respect to (that) C (has) to D. For since A, E, B are
οἱ Α, Ε, Β, ὁ Α ἄρα πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ three (continuously) proportional numbers, A thus has
ibþ show.
ὁ Α πρὸς τὸν Ε. ὡς δὲ ὁ Α πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς to B a squared ratio with respect to (that) A (has) to E
τὸν Δ. ὁ Α ἄρα πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ ἡ [Def. 5.9]. And as A (is) to E, so C (is) to D. Thus, A has
Γ πλευρὰ πρὸς τὴν Δ· ὅπερ ἔδει δεῖξαι. to B a squared ratio with respect to (that) side C (has)
to (side) D. (Which is) the very thing it was required to


† In other words, between two given square numbers there exists a number in continued proportion.
‡ Literally, “double”.

Proposition 12
.
Δύο κύβων ἀριθμῶν δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοί, There exist two numbers in mean proportion to two
†
καὶ ὁ κύβος πρὸς τὸν κύβον τριπλασίονα λόγον ἔχει ἤπερ ἡ (given) cube numbers. And (one) cube (number) has to
‡
πλευρὰ πρὸς τὴν πλευράν. the (other) cube (number) a cubed ratio with respect
῎Εστωσαν κύβοι ἀριθμοὶ οἱ Α, Β καὶ τοῦ μὲν Α πλευρὰ to (that) the side (of the former has) to the side (of the
ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· λέγω, ὅτι τῶν Α, Β δύο μέσοι latter).
ἀνάλογόν εἰσιν ἀριθμοί, καὶ ὁ Α πρὸς τὸν Β τριπλασίονα Let A and B be cube numbers, and let C be the side
λόγον ἔχει ἤπερ ὁ Γ πρὸς τὸν Δ. of A, and D (the side) of B. I say that there exist two
numbers in mean proportion to A and B, and that A has


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to B a cubed ratio with respect to (that) C (has) to D.
Α Ε A E
Β Ζ B F
Γ Η C G
∆ Θ D H
Κ K
῾Ο γὰρ Γ ἑαυτὸν μὲν πολλαπλασιάσας τὸν Ε ποιείτω, For let C make E (by) multiplying itself, and let it
τὸν δὲ Δ πολλαπλασιάσας τὸν Ζ ποιείτω, ὁ δὲ Δ ἑαυτὸν make F (by) multiplying D. And let D make G (by) mul-
πολλαπλασιάσας τὸν Η ποιείτω, ἑκάτερος δὲ τῶν Γ, Δ τὸν tiplying itself, and let C, D make H, K, respectively, (by)
Ζ πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ ποιείτω. multiplying F.
Καὶ ἐπεὶ κύβος ἐστὶν ὁ Α, πλευρὰ δὲ αὐτοῦ ὁ Γ, καὶ ὁ And since A is cube, and C (is) its side, and C has
Γ ἑαυτὸν μὲν πολλαπλασιάσας τὸν Ε πεποίηκεν, ὁ Γ ἄρα made E (by) multiplying itself, C has thus made E (by)
ἑαυτὸν μὲν πολλαπλασιάσας τὸν Ε πεποίηκεν, τὸν δὲ Ε multiplying itself, and has made A (by) multiplying E.
πολλαπλασιάσας τὸν Α πεποίηκεν. διὰ τὰ αὐτὰ δὴ καὶ ὁ And so, for the same (reasons), D has made G (by) mul-
Δ ἑαυτὸν μὲν πολλαπλασιάσας τὸν Η πεποίηκεν, τὸν δὲ Η tiplying itself, and has made B (by) multiplying G. And
πολλαπλασιάσας τὸν Β πεποίηκεν. καὶ ἐπεὶ ὁ Γ ἑκάτερον since C has made E, F (by) multiplying C, D, respec-
τῶν Γ, Δ πολλαπλασιάσας ἑκάτερον τῶν Ε, Ζ πεποίηκεν, tively, thus as C is to D, so E (is) to F [Prop. 7.17]. And
ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ε πρὸς τὸν Ζ. so, for the same (reasons), as C (is) to D, so F (is) to G
διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ζ πρὸς [Prop. 7.18]. Again, since C has made A, H (by) multi-
τὸν Η. πάλιν, ἐπεὶ ὁ Γ ἑκάτερον τῶν Ε, Ζ πολλαπλασιάσας plying E, F, respectively, thus as E is to F, so A (is) to
ἑκάτερον τῶν Α, Θ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν H [Prop. 7.17]. And as E (is) to F, so C (is) to D. And
Ζ, οὕτως ὁ Α πρὸς τὸν Θ. ὡς δὲ ὁ Ε πρὸς τὸν Ζ, οὕτως ὁ Γ thus as C (is) to D, so A (is) to H. Again, since C, D
πρὸς τὸν Δ· καὶ ὡς ἄρα ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν have made H, K, respectively, (by) multiplying F, thus
Θ. πάλιν, ἐπεὶ ἑκάτερος τῶν Γ, Δ τὸν Ζ πολλαπλασιάσας as C is to D, so H (is) to K [Prop. 7.18]. Again, since D
ἑκάτερον τῶν Θ, Κ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν has made K, B (by) multiplying F, G, respectively, thus
Δ, οὕτως ὁ Θ πρὸς τὸν Κ. πάλιν, ἐπεὶ ὁ Δ ἑκάτερον τῶν as F is to G, so K (is) to B [Prop. 7.17]. And as F (is)
Ζ, Η πολλαπλασιάσας ἑκάτερον τῶν Κ, Β πεποίηκεν, ἔστιν to G, so C (is) to D. And thus as C (is) to D, so A (is)
ἄρα ὡς ὁ Ζ πρὸς τὸν Η, οὕτως ὁ Κ πρὸς τὸν Β. ὡς δὲ ὁ Ζ to H, and H to K, and K to B. Thus, H and K are two
πρὸς τὸν Η, οὕτως ὁ Γ πρὸς τὸν Δ· καὶ ὡς ἄρα ὁ Γ πρὸς (numbers) in mean proportion to A and B.
τὸν Δ, οὕτως ὅ τε Α πρὸς τὸν Θ καὶ ὁ Θ πρὸς τὸν Κ καὶ So I say that A also has to B a cubed ratio with re-
ὁ Κ πρὸς τὸν Β. τῶν Α, Β ἄρα δύο μέσοι ἀνάλογόν εἰσιν spect to (that) C (has) to D. For since A, H, K, B are
οἱ Θ, Κ. four (continuously) proportional numbers, A thus has
Λέγω δή, ὅτι καὶ ὁ Α πρὸς τὸν Β τριπλασίονα λόγον ἔχει to B a cubed ratio with respect to (that) A (has) to H
igþ
ἤπερ ὁ Γ πρὸς τὸν Δ. ἐπεὶ γὰρ τέσσαρες ἀριθμοὶ ἀνάλογόν [Def. 5.10]. And as A (is) to H, so C (is) to D. And
εἰσιν οἱ Α, Θ, Κ, Β, ὁ Α ἄρα πρὸς τὸν Β τριπλασίονα λόγον [thus] A has to B a cubed ratio with respect to (that) C
ἔχει ἤπερ ὁ Α πρὸς τὸν Θ. ὡς δὲ ὁ Α πρὸς τὸν Θ, οὕτως ὁ (has) to D. (Which is) the very thing it was required to
Γ πρὸς τὸν Δ· καὶ ὁ Α [ἄρα] πρὸς τὸν Β τριπλασίονα λόγον show.
ἔχει ἤπερ ὁ Γ πρὸς τὸν Δ· ὅπερ ἔδει δεῖξαι.

† In other words, between two given cube numbers there exist two numbers in continued proportion.
‡ Literally, “triple”.

Proposition 13
.
᾿Εὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, καὶ If there are any multitude whatsoever of continuously
πολλαπλασιάσας ἕκαστος ἑαυτὸν ποιῇ τινα, οἱ γενόμενοι proportional numbers, and each makes some (number
ἐξ αὐτῶν ἀνάλογον ἔσονται· καὶ ἐὰν οἱ ἐξ ἀρχῆς τοὺς by) multiplying itself, then the (numbers) created from
γενομένους πολλαπλασιάσαντες ποιῶσί τινας, καὶ αὐτοὶ them will (also) be (continuously) proportional. And if
ἀνάλογον ἔσονται [καὶ ἀεὶ περὶ τοὺς ἄκρους τοῦτο συμβαίνει]. the original (numbers) make some (more numbers by)
῎Εστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ Α, Β, multiplying the created (numbers) then these will also


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Γ, ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Β πρὸς τὸν Γ, καὶ οἱ be (continuously) proportional [and this always happens
Α, Β, Γ ἑαυτοὺς μὲν πολλαπλασιάσαντες τοὺς Δ, Ε, Ζ with the extremes].
ποιείτωσαν, τοὺς δὲ Δ, Ε, Ζ πολλαπλασιάσαντες τοὺς Η, Let A, B, C be any multitude whatsoever of contin-
Θ, Κ ποιείτωσαν· λέγω, ὅτι οἵ τε Δ, Ε, Ζ καὶ οἱ Η, Θ, Κ uously proportional numbers, (such that) as A (is) to B,
ἑξῆς ἀνάλογον εἰσιν. so B (is) to C. And let A, B, C make D, E, F (by)
multiplying themselves, and let them make G, H, K (by)
multiplying D, E, F. I say that D, E, F and G, H, K are
continuously proportional.
Α Λ A L
Β Ξ B O
Γ Μ C M
Ν N
∆ D
Ο P
Ε E
Π Q
Ζ F

Η G
Θ H
Κ K
῾Ο μὲν γὰρ Α τὸν Β πολλαπλασιάσας τὸν Λ ποιείτω, For let A make L (by) multiplying B. And let A, B
ἑκάτερος δὲ τῶν Α, Β τὸν Λ πολλαπλασιάσας ἑκάτερον τῶν make M, N, respectively, (by) multiplying L. And, again,
Μ, Ν ποιείτω. καὶ πάλιν ὁ μὲν Β τὸν Γ πολλαπλασιάσας τὸν let B make O (by) multiplying C. And let B, C make P,
Ξ ποιείτω, ἑκάτερος δὲ τῶν Β, Γ τὸν Ξ πολλαπλασιάσας Q, respectively, (by) multplying O.
ἑκάτερον τῶν Ο, Π ποιείτω. So, similarly to the above, we can show that D, L,
῾Ομοίως δὴ τοῖς ἐπάνω δεῖξομεν, ὅτι οἱ Δ, Λ, Ε καὶ οἱ E and G, M, N, H are continuously proportional in the
Η, Μ, Ν, Θ ἑξῆς εἰσιν ἀνάλογον ἐν τῷ τοῦ Α πρὸς τὸν ratio of A to B, and, further, (that) E, O, F and H, P, Q,
Β λόγῳ, καὶ ἔτι οἱ Ε, Ξ, Ζ καὶ οἱ Θ, Ο, Π, Κ ἑξῆς εἰσιν K are continuously proportional in the ratio of B to C.
idþ
ἀνάλογον ἐν τῷ τοῦ Β πρὸς τὸν Γ λόγῳ. καί ἐστιν ὡς ὁ Α And as A is to B, so B (is) to C. And thus D, L, E are in
πρὸς τὸν Β, οὕτως ὁ Β πρὸς τὸν Γ· καὶ οἱ Δ, Λ, Ε ἄρα τοῖς the same ratio as E, O, F, and, further, G, M, N, H (are
Ε, Ξ, Ζ ἐν τῷ αὐτῷ λόγῳ εἰσὶ καὶ ἔτι οἱ Η, Μ, Ν, Θ τοῖς in the same ratio) as H, P, Q, K. And the multitude of
Θ, Ο, Π, Κ. καί ἐστιν ἴσον τὸ μὲν τῶν Δ, Λ, Ε πλῆθος τῷ D, L, E is equal to the multitude of E, O, F, and that of
τῶν Ε, Ξ, Ζ πλήθει, τὸ δὲ τῶν Η, Μ, Ν, Θ τῷ τῶν Θ, Ο, G, M, N, H to that of H, P, Q, K. Thus, via equality, as
Π, Κ· δι᾿ ἴσου ἄρα ἐστὶν ὡς μὲν ὁ Δ πρὸς τὸν Ε, οὕτως ὁ D is to E, so E (is) to F, and as G (is) to H, so H (is) to
Ε πρὸς τὸν Ζ, ὡς δὲ ὁ Η πρὸς τὸν Θ, οὕτως ὁ Θ πρὸς τὸν K [Prop. 7.14]. (Which is) the very thing it was required
Κ· ὅπερ ἔδει δεῖξαι. to show.

Proposition 14
.
᾿Εὰν τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ τὴν If a square (number) measures a(nother) square
πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, καὶ (number) then the side (of the former) will also mea-
ὁ τετράγωνος τὸν τετράγωνον μετρήσει. sure the side (of the latter). And if the side (of a square
῎Εστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, Β, πλευραὶ δὲ αὐτῶν number) measures the side (of another square number)
ἔστωσαν οἱ Γ, Δ, ὁ δὲ Α τὸν Β μετρείτω· λέγω, ὅτι καὶ ὁ then the (former) square (number) will also measure the
Γ τὸν Δ μετρεῖ. (latter) square (number).
Let A and B be square numbers, and let C and D be
their sides (respectively). And let A measure B. I say that
C also measures D.




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Α Γ A ELEMENTS BOOK 8
C
Β ∆ B D



Ε E
῾Ο Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω· οἱ Α, Ε, For let C make E (by) multiplying D. Thus, A, E,
Β ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ. B are continuously proportional in the ratio of C to D
καὶ ἐπεὶ οἱ Α, Ε, Β ἐξῆς ἀνάλογόν εἰσιν, καὶ μετρεῖ ὁ Α τὸν [Prop. 8.11]. And since A, E, B are continuously pro-
Β, μετρεῖ ἄρα καὶ ὁ Α τὸν Ε. καί ἐστιν ὡς ὁ Α πρὸς τὸν Ε, portional, and A measures B, A thus also measures E
οὕτως ὁ Γ πρὸς τὸν Δ· μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ. [Prop. 8.7]. And as A is to E, so C (is) to D. Thus, C
Πάλιν δὴ ὁ Γ τὸν Δ μετρείτω· λέγω, ὅτι καὶ ὁ Α τὸν Β also measures D [Def. 7.20].
μετρεῖ. So, again, let C measure D. I say that A also measures
Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, B.
ὅτι οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ For similarly, with the same construction, we can
λόγῳ. καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς show that A, E, B are continuously proportional in the
τὸν Ε, μετρεῖ δὲ ὁ Γ τὸν Δ, μετρεῖ ἄρα καὶ ὁ Α τὸν Ε. καί ratio of C to D. And since as C is to D, so A (is) to E,
εἰσιν οἱ Α, Ε, Β ἑξῆς ἀνάλογον· μετρεῖ ἄρα καὶ ὁ Α τὸν Β. and C measures D, A thus also measures E [Def. 7.20].
᾿Εὰν ἄρα τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ And A, E, B are continuously proportional. Thus, A also
ieþ ber) measures the side (of another square number) then
τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, measures B.
καὶ ὁ τετράγωνος τὸν τετράγωνον μετρήσει· ὅπερ ἔδει Thus, if a square (number) measures a(nother) square
δεῖξαι. (number) then the side (of the former) will also measure
the side (of the latter). And if the side (of a square num-

the (former) square (number) will also measure the (lat-
ter) square (number). (Which is) the very thing it was
required to show.

Proposition 15
.
᾿Εὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μετρῇ, καὶ ἡ πλευρὰ If a cube number measures a(nother) cube number
τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, then the side (of the former) will also measure the side
καὶ ὁ κύβος τὸν κύβον μετρήσει. (of the latter). And if the side (of a cube number) mea-
Κύβος γὰρ ἀριθμὸς ὁ Α κύβον τὸν Β μετρείτω, καὶ τοῦ sures the side (of another cube number) then the (for-
μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· λέγω, ὅτι ὁ Γ τὸν mer) cube (number) will also measure the (latter) cube
Δ μετρεῖ. (number).
For let the cube number A measure the cube (num-
ber) B, and let C be the side of A, and D (the side) of B.
I say that C measures D.
Α Γ A C
Β ∆ B D



Ε Θ E H
Η Κ G K



Ζ F
῾Ο Γ γὰρ ἑαυτὸν πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Δ For let C make E (by) multiplying itself. And let


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ἑαυτὸν πολλαπλασιάσας τὸν Η ποιείτω, καὶ ἔτι ὁ Γ τὸν Δ D make G (by) multiplying itself. And, further, [let] C
πολλαπλασιάσας τὸν Ζ [ποιείτω], ἑκάτερος δὲ τῶν Γ, Δ τὸν [make] F (by) multiplying D, and let C, D make H, K,
Ζ πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ ποιείτω. φανερὸν respectively, (by) multiplying F. So it is clear that E, F,
δή, ὅτι οἱ Ε, Ζ, Η καὶ οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν G and A, H, K, B are continuously proportional in the
ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ. καὶ ἐπεὶ οἱ Α, Θ, Κ, Β ἑξῆς ratio of C to D [Prop. 8.12]. And since A, H, K, B are
ἀνάλογόν εἰσιν, καὶ μετρεῖ ὁ Α τὸν Β, μετρεῖ ἄρα καὶ τὸν continuously proportional, and A measures B, (A) thus
Θ. καί ἐστιν ὡς ὁ Α πρὸς τὸν Θ, οὕτως ὁ Γ πρὸς τὸν Δ· also measures H [Prop. 8.7]. And as A is to H, so C (is)
μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ. to D. Thus, C also measures D [Def. 7.20].
iþ
᾿Αλλὰ δὴ μετρείτω ὁ Γ τὸν Δ· λέγω, ὅτι καὶ ὁ Α τὸν Β And so let C measure D. I say that A will also mea-
μετρήσει. sure B.
Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δὴ δείξομεν, For similarly, with the same construction, we can
ὅτι οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς show that A, H, K, B are continuously proportional in
τὸν Δ λόγῳ. καὶ ἐπεὶ ὁ Γ τὸν Δ μετρεῖ, καί ἐστιν ὡς ὁ Γ the ratio of C to D. And since C measures D, and as C is
πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Θ, καὶ ὁ Α ἄρα τὸν Θ to D, so A (is) to H, A thus also measures H [Def. 7.20].
μετρεῖ· ὥστε καὶ τὸν Β μετρεῖ ὁ Α· ὅπερ ἔδει δεῖξαι. Hence, A also measures B. (Which is) the very thing it
was required to show.

Proposition 16
.
᾿Εὰν τετράγωνος ἀριθμὸς τετράγωνον ἀριθμὸν μὴ If a square number does not measure a(nother)
μετρῇ, οὐδὲ ἡ πλευρὰ τὴν πλευρὰν μετρήσει· κἂν ἡ πλευρὰ square number then the side (of the former) will not
τὴν πλευρὰν μὴ μετρῇ, οὐδὲ ὁ τετράγωνος τὸν τετράγωνον measure the side (of the latter) either. And if the side (of
μετρήσει. a square number) does not measure the side (of another
square number) then the (former) square (number) will
not measure the (latter) square (number) either.
Α Γ . A C

Β ∆ B D
῎Εστωσαν τετρὰγωνοι ἀριθμοὶ οἱ Α, Β, πλευραὶ δὲ αὐτῶν Let A and B be square numbers, and let C and D be
ἔστωσαν οἱ Γ, Δ, καὶ μὴ μετρείτω ὁ Α τὸν Β· λὲγω, ὅτι οὐδὲ their sides (respectively). And let A not measure B. I say
ὁ Γ τὸν Δ μετρεῖ. that C does not measure D either.
Εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, μετρήσει καὶ ὁ Α τὸν Β. οὐ For if C measures D then A will also measure B
izþ
μετρεῖ δὲ ὁ Α τὸν Β· οὐδὲ ἄρα ὁ Γ τὸν Δ μετρήσει. [Prop. 8.14]. And A does not measure B. Thus, C will
Μὴ μετρείτω [δὴ] πάλιν ὁ Γ τὸν Δ· λέγω, ὅτι οὐδὲ ὁ Α not measure D either.
τὸν Β μετρήσει. [So], again, let C not measure D. I say that A will not
Εἰ γὰρ μετρεῖ ὁ Α τὸν Β, μετρήσει καὶ ὁ Γ τὸν Δ. οὐ measure B either.
μετρεῖ δὲ ὁ Γ τὸν Δ· οὐδ᾿ ἄρα ὁ Α τὸν Β μετρήσει· ὅπερ For if A measures B then C will also measure D
ἔδει δεῖξαι. [Prop. 8.14]. And C does not measure D. Thus, A will
not measure B either. (Which is) the very thing it was
required to show.

Proposition 17
.
᾿Εὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μὴ μετρῇ, οὐδὲ ἡ If a cube number does not measure a(nother) cube
πλευρὰ τὴν πλευρὰν μετρήσει· κἂν ἡ πλευρὰ τὴν πλευρὰν number then the side (of the former) will not measure the
μὴ μετρῇ, οὐδὲ ὁ κύβος τὸν κύβον μετρήσει. side (of the latter) either. And if the side (of a cube num-
ber) does not measure the side (of another cube number)
then the (former) cube (number) will not measure the
(latter) cube (number) either.





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C
Β ∆ B D
Κύβος γὰρ ἀριθμὸς ὁ Α κύβον ἀριθμὸν τὸν Β μὴ με- For let the cube number A not measure the cube num-
τρείτω, καὶ τοῦ μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ· ber B. And let C be the side of A, and D (the side) of B.
λέγω, ὅτι ὁ Γ τὸν Δ οὐ μετρήσει. I say that C will not measure D.
Εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, καὶ ὁ Α τὸν Β μετρήσει. οὐ For if C measures D then A will also measure B
ihþ
μετρεῖ δὲ ὁ Α τὸν Β· οὐδ᾿ ἄρα ὁ Γ τὸν Δ μετρεῖ. [Prop. 8.15]. And A does not measure B. Thus, C does
᾿Αλλὰ δὴ μὴ μετρείτω ὁ Γ τὸν Δ· λέγω, ὅτι οὐδὲ ὁ Α not measure D either.
τὸν Β μετρήσει. And so let C not measure D. I say that A will not
Εἰ γὰρ ὁ Α τὸν Β μετρεῖ, καὶ ὁ Γ τὸν Δ μετρήσει. οὐ measure B either.
μετρεῖ δὲ ὁ Γ τὸν Δ· οὐδ᾿ ἄρα ὁ Α τὸν Β μετρήσει· ὅπερ For if A measures B then C will also measure D
ἔδει δεῖξαι. [Prop. 8.15]. And C does not measure D. Thus, A will
not measure B either. (Which is) the very thing it was
required to show.

.
Proposition 18
Δύο ὁμοίων ἐπιπέδων ἀριθμῶν εἷς μέσος ἀνάλογόν ἐστιν There exists one number in mean proportion to two
ἀριθμός· καὶ ὁ ἐπίπεδος πρὸς τὸν ἐπίπεδον διπλασίονα λόγον similar plane numbers. And (one) plane (number) has to
†
ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν. the (other) plane (number) a squared ratio with respect
to (that) a corresponding side (of the former has) to a
corresponding side (of the latter).
Α Β A B
Γ Ε C E
∆ Ζ D F

Η G
῎Εστωσαν δύο ὅμοιοι ἐπίπεδοι ἀριθμοὶ οἱ Α, Β, καὶ τοῦ Let A and B be two similar plane numbers. And let
μὲν Α πλευραὶ ἔστωσαν οἱ Γ, Δ ἀριθμοί, τοῦ δὲ Β οἱ Ε, the numbers C, D be the sides of A, and E, F (the sides)
Ζ. καὶ ἐπεὶ ὅμοιοι ἐπίπεδοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς of B. And since similar numbers are those having pro-
πλευράς, ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ε πρὸς portional sides [Def. 7.21], thus as C is to D, so E (is) to
τὸν Ζ. λέγω οὖν, ὅτι τῶν Α, Β εἷς μέσος ἀνάλογόν ἐστιν F. Therefore, I say that there exists one number in mean
ἀριθμός, καὶ ὁ Α πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ ὁ proportion to A and B, and that A has to B a squared
Γ πρὸς τὸν Ε ἢ ὁ Δ πρὸς τὸν Ζ, τουτέστιν ἤπερ ἡ ὁμόλογος ratio with respect to that C (has) to E, or D to F—that is
πλευρὰ πρὸς τὴν ὁμόλογον [πλευράν]. to say, with respect to (that) a corresponding side (has)
Καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ε πρὸς τὸν to a corresponding [side].
Ζ, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Γ πρὸς τὸν Ε, ὁ Δ πρὸς τὸν Ζ. For since as C is to D, so E (is) to F, thus, alternately,
καὶ ἐπεὶ ἐπίπεδός ἐστιν ὁ Α, πλευραὶ δὲ αὐτοῦ οἱ Γ, Δ, ὁ Δ as C is to E, so D (is) to F [Prop. 7.13]. And since A is
ἄρα τὸν Γ πολλαπλασιάσας τὸν Α πεποίηκεν. διὰ τὰ αὐτὰ plane, and C, D its sides, D has thus made A (by) mul-
δὴ καὶ ὁ Ε τὸν Ζ πολλαπλασιάσας τὸν Β πεποίηκεν. ὁ Δ tiplying C. And so, for the same (reasons), E has made
δὴ τὸν Ε πολλαπλασιάσας τὸν Η ποιείτω. καὶ ἐπεὶ ὁ Δ τὸν B (by) multiplying F. So let D make G (by) multiplying
μὲν Γ πολλαπλασιάσας τὸν Α πεποίηκεν, τὸν δὲ Ε πολλα- E. And since D has made A (by) multiplying C, and has
πλασιάσας τὸν Η πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Ε, made G (by) multiplying E, thus as C is to E, so A (is) to
οὕτως ὁ Α πρὸς τὸν Η. ἀλλ᾿ ὡς ὁ Γ πρὸς τὸν Ε, [οὕτως] G [Prop. 7.17]. But as C (is) to E, [so] D (is) to F. And
ὁ Δ πρὸς τὸν Ζ· καὶ ὡς ἄρα ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Α thus as D (is) to F, so A (is) to G. Again, since E has
πρὸς τὸν Η. πάλιν, ἐπεὶ ὁ Ε τὸν μὲν Δ πολλαπλασιάσας τὸν made G (by) multiplying D, and has made B (by) multi-
Η πεποίηκεν, τὸν δὲ Ζ πολλαπλασιάσας τὸν Β πεποίηκεν, plying F, thus as D is to F, so G (is) to B [Prop. 7.17].
ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Η πρὸς τὸν Β. And it was also shown that as D (is) to F, so A (is) to G.
ἐδείχθη δὲ καὶ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν And thus as A (is) to G, so G (is) to B. Thus, A, G, B are


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ELEMENTS BOOK 8



Η· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Η, οὕτως ὁ Η πρὸς τὸν Β. οἱ continously proportional. Thus, there exists one number
Α, Η, Β ἄρα ἑξῆς ἀνάλογόν εἰσιν. τῶν Α, Β ἄρα εἷς μέσος (namely, G) in mean proportion to A and B.
ἀνάλογόν ἐστιν ἀριθμός. So I say that A also has to B a squared ratio with
Λέγω δή, ὅτι καὶ ὁ Α πρὸς τὸν Β διπλασίονα λόγον respect to (that) a corresponding side (has) to a corre-
ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν, sponding side—that is to say, with respect to (that) C
ijþ
τουτέστιν ἤπερ ὁ Γ πρὸς τὸν Ε ἢ ὁ Δ πρὸς τὸν Ζ. ἐπεὶ γὰρ (has) to E, or D to F. For since A, G, B are continuously
οἱ Α, Η, Β ἑξῆς ἀνάλογόν εἰσιν, ὁ Α πρὸς τὸν Β διπλασίονα proportional, A has to B a squared ratio with respect to
λόγον ἔχει ἤπερ πρὸς τὸν Η. καί ἐστιν ὡς ὁ Α πρὸς τὸν Η, (that A has) to G [Prop. 5.9]. And as A is to G, so C (is)
οὕτως ὅ τε Γ πρὸς τὸν Ε καὶ ὁ Δ πρὸς τὸν Ζ. καὶ ὁ Α ἄρα to E, and D to F. And thus A has to B a squared ratio
πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ ὁ Γ πρὸς τὸν Ε ἢ with respect to (that) C (has) to E, or D to F. (Which
ὁ Δ πρὸς τὸν Ζ· ὅπερ ἔδει δεῖξαι. is) the very thing it was required to show.

† Literally, “double”.

Proposition 19
.
Δύο ὁμοίων στερεῶν ἀριθμῶν δύο μέσοι ἀνάλογον Two numbers fall (between) two similar solid num-
ἐμπίπτουσιν ἀριθμοί· καὶ ὁ στερεὸς πρὸς τὸν ὅμοιον στερεὸν bers in mean proportion. And a solid (number) has to
†
τριπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν a similar solid (number) a cubed ratio with respect to
ὁμόλογον πλευράν. (that) a corresponding side (has) to a corresponding side.
Α Γ A C
∆ D
Ε E

Β Ζ B F
Η G
Θ H
Κ K
Μ Ν M N
Λ Ξ L O
῎Εστωσαν δύο ὅμοιοι στερεοὶ οἱ Α, Β, καὶ τοῦ μὲν Α Let A and B be two similar solid numbers, and let
πλευραὶ ἔστωσαν οἱ Γ, Δ, Ε, τοῦ δὲ Β οἱ Ζ, Η, Θ. καὶ ἐπεὶ C, D, E be the sides of A, and F, G, H (the sides) of
ὅμοιοι στερεοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς, ἔστιν B. And since similar solid (numbers) are those having
ἄρα ὡς μὲν ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ζ πρὸς τὸν Η, ὡς δὲ proportional sides [Def. 7.21], thus as C is to D, so F
ὁ Δ πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Θ. λέγω, ὅτι τῶν Α, (is) to G, and as D (is) to E, so G (is) to H. I say that
Β δύο μέσοι ἀνάλογόν ἐμπίπτουσιν ἀριθμοί, καὶ ὁ Α πρὸς two numbers fall (between) A and B in mean proportion,
τὸν Β τριπλασίονα λόγον ἔχει ἤπερ ὁ Γ πρὸς τὸν Ζ καὶ ὁ and (that) A has to B a cubed ratio with respect to (that)
Δ πρὸς τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ. C (has) to F, and D to G, and, further, E to H.
῾Ο Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Κ ποιείτω, ὁ δὲ For let C make K (by) multiplying D, and let F make
Ζ τὸν Η πολλαπλασιάσας τὸν Λ ποιείτω. καὶ ἐπεὶ οἱ Γ, L (by) multiplying G. And since C, D are in the same
Δ τοὶς Ζ, Η ἐν τῷ αὐτῷ λόγῳ εἰσίν, καὶ ἐκ μὲν τῶν Γ, ratio as F, G, and K is the (number created) from (mul-
Δ ἐστιν ὁ Κ, ἐκ δὲ τῶν Ζ, Η ὁ Λ, οἱ Κ, Λ [ἄρα] ὅμοιοι tiplying) C, D, and L the (number created) from (multi-
ἐπίπεδοί εἰσιν ἀριθμοί· τῶν Κ, Λ ἄρα εἷς μέσος ἀνάλογόν plying) F, G, [thus] K and L are similar plane numbers
ἐστιν ἀριθμός. ἔστω ὁ Μ. ὁ Μ ἄρα ἐστὶν ὁ ἐκ τῶν Δ, [Def. 7.21]. Thus, there exits one number in mean pro-
Ζ, ὡς ἐν τῷ πρὸ τούτου θεωρήματι ἐδείχθη. καὶ ἐπεὶ ὁ portion to K and L [Prop. 8.18]. Let it be M. Thus, M is
Δ τὸν μὲν Γ πολλαπλασιάσας τὸν Κ πεποίηκεν, τὸν δὲ Ζ the (number created) from (multiplying) D, F, as shown
πολλαπλασιάσας τὸν Μ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ πρὸς in the theorem before this (one). And since D has made
τὸν Ζ, οὕτως ὁ Κ πρὸς τὸν Μ. ἀλλ᾿ ὡς ὁ Κ πρὸς τὸν Μ, K (by) multiplying C, and has made M (by) multiplying
ὁ Μ πρὸς τὸν Λ. οἱ Κ, Μ, Λ ἄρα ἑξῆς εἰσιν ἀνάλογον ἐν F, thus as C is to F, so K (is) to M [Prop. 7.17]. But, as


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τῷ τοῦ Γ πρὸς τὸν Ζ λόγῷ. καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς τὸν K (is) to M, (so) M (is) to L. Thus, K, M, L are contin-
Δ, οὕτως ὁ Ζ πρὸς τὸν Η, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Γ πρὸς uously proportional in the ratio of C to F. And since as
τὸν Ζ, οὕτως ὁ Δ πρὸς τὸν Η. διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Δ C is to D, so F (is) to G, thus, alternately, as C is to F, so
πρὸς τὸν Η, οὕτως ὁ Ε πρὸς τὸν Θ. οἱ Κ, Μ, Λ ἄρα ἑξῆς D (is) to G [Prop. 7.13]. And so, for the same (reasons),
εἰσιν ἀνάλογον ἔν τε τῷ τοῦ Γ πρὸς τὸν Ζ λόγῳ καὶ τῷ as D (is) to G, so E (is) to H. Thus, K, M, L are contin-
τοῦ Δ πρὸς τὸν Η καὶ ἔτι τῷ τοῦ Ε πρὸς τὸν Θ. ἑκατερος uously proportional in the ratio of C to F, and of D to G,
δὴ τῶν Ε, Θ τὸν Μ πολλαπλασιάσας ἑκάτερον τῶν Ν, Ξ and, further, of E to H. So let E, H make N, O, respec-
ποιείτω. καὶ ἐπεὶ στερεός ἐστιν ὁ Α, πλευραὶ δὲ αὐτοῦ εἰσιν tively, (by) multiplying M. And since A is solid, and C,
οἱ Γ, Δ, Ε, ὁ Ε ἄρα τὸν ἐκ τῶν Γ, Δ πολλαπλασιάσας τὸν D, E are its sides, E has thus made A (by) multiplying
Α πεποίηκεν. ὁ δὲ ἐκ τῶν Γ, Δ ἐστιν ὁ Κ· ὁ Ε ἄρα τὸν Κ the (number created) from (multiplying) C, D. And K
πολλαπλασιάσας τὸν Α πεποίηκεν. διὰ τὰ αὐτὰ δὴ καὶ ὁ Θ is the (number created) from (multiplying) C, D. Thus,
τὸν Λ πολλαπλασιάσας τὸν Β πεποίηκεν. καὶ ἐπεὶ ὁ Ε τὸν E has made A (by) multiplying K. And so, for the same
Κ πολλαπλασιάσας τὸν Α πεποίηκεν, ἀλλὰ μὴν καὶ τὸν Μ (reasons), H has made B (by) multiplying L. And since
πολλαπλασιάσας τὸν Ν πεποίηκεν, ἔστιν ἄρα ὡς ὁ Κ πρὸς E has made A (by) multiplying K, but has, in fact, also
τὸν Μ, οὕτως ὁ Α πρὸς τὸν Ν. ὡς δὲ ὁ Κ πρὸς τὸν Μ, made N (by) multiplying M, thus as K is to M, so A (is)
οὕτως ὅ τε Γ πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ἔτι ὁ Ε to N [Prop. 7.17]. And as K (is) to M, so C (is) to F,
πρὸς τὸν Θ· καὶ ὡς ἄρα ὁ Γ πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν and D to G, and, further, E to H. And thus as C (is) to
Η καὶ ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Ν. πάλιν, ἐπεὶ F, and D to G, and E to H, so A (is) to N. Again, since
ἑκάτερος τῶν Ε, Θ τὸν Μ πολλαπλασιάσας ἑκάτερον τῶν E, H have made N, O, respectively, (by) multiplying M,
Ν, Ξ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Ν thus as E is to H, so N (is) to O [Prop. 7.18]. But, as
πρὸς τὸν Ξ. ἀλλ᾿ ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὅ τε Γ πρὸς τὸν E (is) to H, so C (is) to F, and D to G. And thus as C
Ζ καὶ ὁ Δ πρὸς τὸν Η· καὶ ὡς ἄρα ὁ Γ πρὸς τὸν Ζ καὶ ὁ Δ (is) to F, and D to G, and E to H, so (is) A to N, and
πρὸς τὸν Η καὶ ὁ Ε πρὸς τὸν Θ, οὕτως ὅ τε Α πρὸς τὸν Ν N to O. Again, since H has made O (by) multiplying M,
καὶ ὁ Ν πρὸς τὸν Ξ. πάλιν, ἐπεὶ ὁ Θ τὸν Μ πολλαπλασιάσας but has, in fact, also made B (by) multiplying L, thus as
τὸν Ξ πεποίηκεν, ἀλλὰ μὴν καὶ τὸν Λ πολλαπλασιάσας τὸν M (is) to L, so O (is) to B [Prop. 7.17]. But, as M (is)
Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Μ πρὸς τὸν Λ, οὕτως ὁ Ξ πρὸς to L, so C (is) to F, and D to G, and E to H. And thus
τὸν Β. ἀλλ᾿ ὡς ὁ Μ πρὸς τὸν Λ, οὕτως ὅ τε Γ πρὸς τὸν Ζ as C (is) to F, and D to G, and E to H, so not only (is)
καὶ ὁ Δ πρὸς τὸν Η καὶ ὁ Ε πρὸς τὸν Θ. καὶ ὡς ἄρα ὁ Γ O to B, but also A to N, and N to O. Thus, A, N, O,
πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ὁ Ε πρὸς τὸν Θ, οὕτως B are continuously proportional in the aforementioned
οὐ μόνον ὁ Ξ πρὸς τὸν Β, ἀλλὰ καὶ ὁ Α πρὸς τὸν Ν καὶ ὁ ratios of the sides.
Ν πρὸς τὸν Ξ. οἱ Α, Ν, Ξ, Β ἄρα ἑξῆς εἰσιν ἀνάλογον ἐν So I say that A also has to B a cubed ratio with respect
τοῖς εἰρημένοις τῶν πλευρῶν λόγοις. to (that) a corresponding side (has) to a corresponding
Λέγω, ὅτι καὶ ὁ Α πρὸς τὸν Β τριπλασίονα λόγον side—that is to say, with respect to (that) the number C
ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν, (has) to F, or D to G, and, further, E to H. For since A,
τουτέστιν ἤπερ ὁ Γ ἀριθμὸς πρὸς τὸν Ζ ἢ ὁ Δ πρὸς τὸν N, O, B are four continuously proportional numbers, A
Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ. ἐπεὶ γὰρ τέσσαρες ἀριθμοὶ ἑξῆς thus has to B a cubed ratio with respect to (that) A (has)
ἀνάλογόν εἰσιν οἱ Α, Ν, Ξ, Β, ὁ Α ἄρα πρὸς τὸν Β τρι- to N [Def. 5.10]. But, as A (is) to N, so it was shown (is)
πλασίονα λόγον ἔχει ἤπερ ὁ Α πρὸς τὸν Ν. ἀλλ᾿ ὡς ὁ Α C to F, and D to G, and, further, E to H. And thus A has
kþ
πρὸς τὸν Ν, οὕτως ἐδείχθη ὅ τε Γ πρὸς τὸν Ζ καὶ ὁ Δ πρὸς to B a cubed ratio with respect to (that) a corresponding
τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ. καὶ ὁ Α ἄρα πρὸς τὸν Β side (has) to a corresponding side—that is to say, with
τριπλασίονα λόγον ἔχει ἤπερ ἡ ομόλογος πλευρὰ πρὸς τὴν respect to (that) the number C (has) to F, and D to G,
ὁμόλογον πλευράν, τουτέστιν ἤπερ ὁ Γ ἀριθμὸς πρὸς τὸν and, further, E to H. (Which is) the very thing it was
Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ· ὅπερ ἔδει required to show.
δεῖξαι.
† Literally, “triple”.


.
Proposition 20
᾿Εὰν δύο ἀριθμῶν εἷς μέσος ἀνάλογον ἐμπίπτῇ ἀριθμός, If one number falls between two numbers in mean
ὅμοιοι ἐπίπεδοι ἔσονται οἱ ἀριθμοί. proportion then the numbers will be similar plane (num-


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Δύο γὰρ ἀριθμῶν τῶν Α, Β εἷς μέσος ἀνάλογον bers). ELEMENTS BOOK 8
ἐμπιπτέτω ἀριθμὸς ὁ Γ· λέγω, ὅτι οἱ Α, Β ὅμοιοι ἐπίπεδοί For let one number C fall between the two numbers A
εἰσιν ἀριθμοί. and B in mean proportion. I say that A and B are similar
plane numbers.
Α ∆ A D
Γ Ζ C F
Β B
Ε E
Η G
Εἰλήφθωσαν [γὰρ] ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν [For] let the least numbers, D and E, having the
λόγον ἐχόντων τοῖς Α, Γ οἱ Δ, Ε· ἰσάκις ἄρα ὁ Δ τὸν Α same ratio as A and C have been taken [Prop. 7.33].
μετρεῖ καὶ ὁ Ε τὸν Γ. ὁσάκις δὴ ὁ Δ τὸν Α μετρεῖ, τοσαῦται Thus, D measures A as many times as E (measures) C
μονάδες ἔστωσαν ἐν τῷ Ζ· ὁ Ζ ἄρα τὸν Δ πολλαπλασιάσας [Prop. 7.20]. So as many times as D measures A, so
τὸν Α πεποίηκεν. ὥστε ὁ Α ἐπίπεδός ἐστιν, πλευραὶ δὲ many units let there be in F. Thus, F has made A (by)
αὐτοῦ οἱ Δ, Ζ. πάλιν, ἐπεὶ οἱ Δ, Ε ἐλάχιστοί εἰσι τῶν τὸν multiplying D [Def. 7.15]. Hence, A is plane, and D,
αὐτὸν λόγον ἐχόντων τοῖς Γ, Β, ἰσάκις ἄρα ὁ Δ τὸν Γ με- F (are) its sides. Again, since D and E are the least
τρεῖ καὶ ὁ Ε τὸν Β. ὁσάκις δὴ ὁ Ε τὸν Β μετρεῖ, τοσαῦται of those (numbers) having the same ratio as C and B,
μονάδες ἔστωσαν ἐν τῷ Η. ὁ Ε ἄρα τὸν Β μετρεῖ κατὰ τὰς D thus measures C as many times as E (measures) B
ἐν τῷ Η μονάδας· ὁ Η ἄρα τὸν Ε πολλαπλασιάσας τὸν Β [Prop. 7.20]. So as many times as E measures B, so
πεποίηκεν. ὁ Β ἄρα ἐπίπεδος ἐστι, πλευραὶ δὲ αὐτοῦ εἰσιν many units let there be in G. Thus, E measures B ac-
οἱ Ε, Η. οἱ Α, Β ἄρα ἐπίπεδοί εἰσιν ἀριθμοί. λέγω δή, ὅτι cording to the units in G. Thus, G has made B (by) mul-
καὶ ὅμοιοι. ἐπεὶ γὰρ ὁ Ζ τὸν μὲν Δ πολλαπλασιάσας τὸν tiplying E [Def. 7.15]. Thus, B is plane, and E, G are
Α πεποίηκεν, τὸν δὲ Ε πολλαπλασιάσας τὸν Γ πεποίηκεν, its sides. Thus, A and B are (both) plane numbers. So I
ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν Γ, say that (they are) also similar. For since F has made A
τουτέστιν ὁ Γ πρὸς τὸν Β. πάλιν, ἐπεὶ ὁ Ε ἑκάτερον τῶν Ζ, (by) multiplying D, and has made C (by) multiplying E,
Η πολλαπλασιάσας τοὺς Γ, Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Ζ thus as D is to E, so A (is) to C—that is to say, C to B
†
πρὸς τὸν Η, οὕτως ὁ Γ πρὸς τὸν Β. ὡς δὲ ὁ Γ πρὸς τὸν Β, [Prop. 7.17]. Again, since E has made C, B (by) multi-
οὕτως ὁ Δ πρὸς τὸν Ε· καὶ ὡς ἄρα ὁ Δ πρὸς τὸν Ε, οὕτως plying F, G, respectively, thus as F is to G, so C (is) to
ὁ Ζ πρὸς τὸν Η· καὶ ἐναλλὰξ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ B [Prop. 7.17]. And as C (is) to B, so D (is) to E. And
kaþ [Def. 7.21]. (Which is) the very thing it was required to
Ε πρὸς τὸν Η. οἱ Α, Β ἄρα ὅμοιοι ἐπίπεδοι ἀριθμοί εἰσιν· αἱ thus as D (is) to E, so F (is) to G. And, alternately, as D
γὰρ πλευραὶ αὐτῶν ἀνάλογόν εἰσιν· ὅπερ ἔδει δεῖξαι. (is) to F, so E (is) to G [Prop. 7.13]. Thus, A and B are
similar plane numbers. For their sides are proportional

show.

† This part of the proof is defective, since it is not demonstrated that F × E = C. Furthermore, it is not necessary to show that D : E :: A : C,
because this is true by hypothesis.
Proposition 21
.
᾿Εὰν δύο ἀριθμῶν δύο μέσοι ἀνάλογον ἐμπίπτωσιν If two numbers fall between two numbers in mean
ἀριθμοί, ὅμοιοι στερεοί εἰσιν οἱ ἀριθμοί. proportion then the (latter) are similar solid (numbers).
Δύο γὰρ ἀριθμῶν τῶν Α, Β δύο μέσοι ἀνάλογον For let the two numbers C and D fall between the two
ἐμπιπτέτωσαν ἀριθμοὶ οἱ Γ, Δ· λέγω, ὅτι οἱ Α, Β ὅμοιοι numbers A and B in mean proportion. I say that A and
στερεοί εἰσιν. B are similar solid (numbers).
Εἰλήφθωσαν γὰρ ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν For let the three least numbers E, F, G having the
λόγον ἐχόντων τοῖς Α, Γ, Δ τρεῖς οἱ Ε, Ζ, Η· οἱ ἄρα ἄκροι same ratio as A, C, D have been taken [Prop. 8.2]. Thus,
αὐτῶν οἱ Ε, Η πρῶτοι πρὸς ἀλλήλους εἰσίν. καὶ ἐπεὶ τῶν the outermost of them, E and G, are prime to one an-
Ε, Η εἷς μέσος ἀνάλογον ἐμπέπτωκεν ἀριθμὸς ὁ Ζ, οἱ Ε, other [Prop. 8.3]. And since one number, F, has fallen
Η ἄρα ἀριθμοὶ ὅμοιοι ἐπίπεδοί εἰσιν. ἔστωσαν οὖν τοῦ μὲν (between) E and G in mean proportion, E and G are


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Ε πλευραὶ οἱ Θ, Κ, τοῦ δὲ Η οἱ Λ, Μ. φανερὸν ἄρα ἐστὶν thus similar plane numbers [Prop. 8.20]. Therefore, let
ἐκ τοῦ πρὸ τούτου, ὅτι οἱ Ε, Ζ, Η ἑξῆς εἰσιν ἀνάλογον ἔν H, K be the sides of E, and L, M (the sides) of G. Thus,
τε τῷ τοῦ Θ πρὸς τὸν Λ λόγῳ καὶ τῷ τοῦ Κ πρὸς τὸν Μ. it is clear from the (proposition) before this (one) that E,
καὶ ἐπεὶ οἱ Ε, Ζ, Η ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον F, G are continuously proportional in the ratio of H to
ἐχόντων τοῖς Α, Γ, Δ, καί ἐστιν ἴσον τὸ πλῆθος τῶν Ε, L, and of K to M. And since E, F, G are the least (num-
Ζ, Η τῷ πλήθει τῶν Α, Γ, Δ, δι᾿ ἴσου ἄρα ἐστὶν ὡς ὁ Ε bers) having the same ratio as A, C, D, and the multitude
πρὸς τὸν Η, οὕτως ὁ Α πρὸς τὸν Δ. οἱ δὲ Ε, Η πρῶτοι, οἱ of E, F, G is equal to the multitude of A, C, D, thus,
δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν via equality, as E is to G, so A (is) to D [Prop. 7.14].
αὐτὸν λόγον ἔχοντας αὐτοῖς ἰσάκις ὅ τε μείζων τὸν μείζονα And E and G (are) prime (to one another), and prime
καὶ ὁ ἐλάσσων τὸν ἐλάσσονα, τουτέστιν ὅ τε ἡγούμενος (numbers) are also the least (of those numbers having
τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον· ἰσάκις ἄρα the same ratio as them) [Prop. 7.21], and the least (num-
ὁ Ε τὸν Α μετρεῖ καὶ ὁ Η τὸν Δ. ὁσάκις δὴ ὁ Ε τὸν Α bers) measure those (numbers) having the same ratio as
μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ν. ὁ Ν ἄρα τὸν Ε them an equal number of times, the greater (measuring)
πολλαπλασιάσας τὸν Α πεποίηκεν. ὁ δὲ Ε ἐστιν ὁ ἐκ τῶν the greater, and the lesser the lesser—that is to say, the
Θ, Κ· ὁ Ν ἄρα τὸν ἐκ τῶν Θ, Κ πολλαπλασιάσας τὸν Α leading (measuring) the leading, and the following the
πεποίηκεν. στερεὸς ἄρα ἐστὶν ὁ Α, πλευραὶ δὲ αὐτοῦ εἰσιν following [Prop. 7.20]. Thus, E measures A the same
οἱ Θ, Κ, Ν. πάλιν, ἐπεὶ οἱ Ε, Ζ, Η ἐλάχιστοί εἰσι τῶν τὸν number of times as G (measures) D. So as many times as
αὐτὸν λόγον ἐχόντων τοῖς Γ, Δ, Β, ἰσάκις ἄρα ὁ Ε τὸν Γ E measures A, so many units let there be in N. Thus, N
μετρεῖ καὶ ὁ Η τὸν Β. ὁσάκις δὴ ὁ Ε τὸν Γ μετρεῖ, τοσαῦται has made A (by) multiplying E [Def. 7.15]. And E is the
μονάδες ἔστωσαν ἐν τῷ Ξ. ὁ Η ἄρα τὸν Β μετρεῖ κατὰ τὰς (number created) from (multiplying) H and K. Thus, N
ἐν τῷ Ξ μονάδας· ὁ Ξ ἄρα τὸν Η πολλαπλασιάσας τὸν Β has made A (by) multiplying the (number created) from
πεποίηκεν. ὁ δὲ Η ἐστιν ὁ ἐκ τῶν Λ, Μ· ὁ Ξ ἄρα τὸν ἐκ (multiplying) H and K. Thus, A is solid, and its sides are
τῶν Λ, Μ πολλαπλασιάσας τὸν Β πεποίηκεν. στερεὸς ἄρα H, K, N. Again, since E, F, G are the least (numbers)
ἐστὶν ὁ Β, πλευραὶ δὲ αὐτοῦ εἰσιν οἱ Λ, Μ, Ξ· οἱ Α, Β ἄρα having the same ratio as C, D, B, thus E measures C the
στερεοί εἰσιν. same number of times as G (measures) B [Prop. 7.20].
So as many times as E measures C, so many units let
there be in O. Thus, G measures B according to the units
in O. Thus, O has made B (by) multiplying G. And
G is the (number created) from (multiplying) L and M.
Thus, O has made B (by) multiplying the (number cre-
ated) from (multiplying) L and M. Thus, B is solid, and
its sides are L, M, O. Thus, A and B are (both) solid.
Α Θ A H
Γ Κ C K
∆ Ν D N

Β B


Ε Λ E L

Ζ Μ F M
Η Ξ G O
Λέγω [δή], ὅτι καὶ ὅμοιοι. ἐπεὶ γὰρ οἱ Ν, Ξ τὸν Ε πολ- [So] I say that (they are) also similar. For since N, O
λαπλασιάσαντες τοὺς Α, Γ πεποιήκασιν, ἔστιν ἄρα ὡς ὁ Ν have made A, C (by) multiplying E, thus as N is to O, so
πρὸς τὸν Ξ, ὁ Α πρὸς τὸν Γ, τουτέστιν ὁ Ε πρὸς τὸν Ζ. A (is) to C—that is to say, E to F [Prop. 7.18]. But, as
ἀλλ᾿ ὡς ὁ Ε πρὸς τὸν Ζ, ὁ Θ πρὸς τὸν Λ καὶ ὁ Κ πρὸς τὸν E (is) to F, so H (is) to L, and K to M. And thus as H
Μ· καὶ ὡς ἄρα ὁ Θ πρὸς τὸν Λ, οὕτως ὁ Κ πρὸς τὸν Μ καὶ (is) to L, so K (is) to M, and N to O. And H, K, N are
ὁ Ν πρὸς τὸν Ξ. καί εἰσιν οἱ μὲν Θ, Κ, Ν πλευραὶ τοῦ Α, the sides of A, and L, M, O the sides of B. Thus, A and



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ELEMENTS BOOK 8



οἱ δὲ Ξ, Λ, Μ πλευραὶ τοῦ Β. οἱ Α, Β ἄρα ἀριθμοὶ ὅμοιοι B are similar solid numbers [Def. 7.21]. (Which is) the
στερεοί εἰσιν· ὅπερ ἔδει δεῖξαι. very thing it was required to show.

† The Greek text has “O, L, M”, which is obviously a mistake.

.
Proposition 22
᾿Εὰν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ δὲ πρῶτος If three numbers are continuously proportional, and
τετράγωνος ᾖ, καὶ ὁ τρίτος τετράγωνος ἔσται. the first is square, then the third will also be square.
Α A

Β B

Γ C
kgþ
῎Εστωσαν τρεῖς ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, ὁ δὲ Let A, B, C be three continuously proportional num-
πρῶτος ὁ Α τετράγωνος ἔστω· λέγω, ὅτι καὶ ὁ τρίτος ὁ Γ bers, and let the first A be square. I say that the third C
τετράγωνός ἐστιν. is also square.
᾿Επεὶ γὰρ τῶν Α, Γ εἷς μέσος ἀνάλογόν ἐστιν ἀριθμὸς For since one number, B, is in mean proportion to
ὁ Β, οἱ Α, Γ ἄρα ὅμοιοι ἐπίπεδοί εἰσιν. τετράγωνος δὲ ὁ Α· A and C, A and C are thus similar plane (numbers)
τετράγωνος ἄρα καὶ ὁ Γ· ὅπερ ἔδει δεῖξαι. [Prop. 8.20]. And A is square. Thus, C is also square
[Def. 7.21]. (Which is) the very thing it was required to
show.

Proposition 23
.
᾿Εὰν τέσσαρες ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ δὲ πρῶτος If four numbers are continuously proportional, and
κύβος ᾖ, καὶ ὁ τέταρτος κύβος ἔσται. the first is cube, then the fourth will also be cube.
Α . A

Β B

Γ C


∆ D
kdþ
῎Εστωσαν τέσσαρες ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, Let A, B, C, D be four continuously proportional
Δ, ὁ δὲ Α κύβος ἔστω· λέγω, ὅτι καὶ ὁ Δ κύβος ἐστίν. numbers, and let A be cube. I say that D is also cube.
᾿Επεὶ γὰρ τῶν Α, Δ δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοὶ For since two numbers, B and C, are in mean propor-
οἱ Β, Γ, οἱ Α, Δ ἄρα ὅμοιοί εἰσι στερεοὶ ἀριθμοί. κύβος δὲ tion to A and D, A and D are thus similar solid numbers
ὁ Α· κύβος ἄρα καὶ ὁ Δ· ὅπερ ἔδει δεῖξαι. [Prop. 8.21]. And A (is) cube. Thus, D (is) also cube
[Def. 7.21]. (Which is) the very thing it was required to
show.

Proposition 24
.
᾿Εὰν δύο ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν If two numbers have to one another the ratio which
τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ὁ δὲ a square number (has) to a(nother) square number, and
πρῶτος τετράγωνος ᾖ, καὶ ὁ δεύτερος τετράγωνος ἔσται. the first is square, then the second will also be square.
Α Γ A C

Β ∆ B D
Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς ἀλλήλους λόγον For let two numbers, A and B, have to one another


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ἐχέτωσαν, ὃν τετράγωνος ἀριθμὸς ὁ Γ πρὸς τετράγωνον the ratio which the square number C (has) to the square
ἀριθμὸν τὸν Δ, ὁ δὲ Α τετράγωνος ἔστω· λέγω, ὅτι καὶ ὁ number D. And let A be square. I say that B is also
keþ
Β τετράγωνός ἐστιν. square.
᾿Επεὶ γὰρ οἱ Γ, Δ τετράγωνοί εἰσιν, οἱ Γ, Δ ἄρα ὅμοιοι For since C and D are square, C and D are thus sim-
ἐπίπεδοί εἰσιν. τῶν Γ, Δ ἄρα εἷς μέσος ἀνάλογον ἐμπίπτει ilar plane (numbers). Thus, one number falls (between)
ἀριθμός. καί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, ὁ Α πρὸς τὸν Β· C and D in mean proportion [Prop. 8.18]. And as C is
καὶ τῶν Α, Β ἄρα εἷς μέσος ἀνάλογον ἐμπίπτει ἀριθμός. καί to D, (so) A (is) to B. Thus, one number also falls (be-
ἐστιν ὁ Α τετράγωνος· καὶ ὁ Β ἄρα τετράγωνός ἐστιν· ὅπερ tween) A and B in mean proportion [Prop. 8.8]. And A
ἔδει δεῖξαι. is square. Thus, B is also square [Prop. 8.22]. (Which is)
the very thing it was required to show.

.
Proposition 25
᾿Εὰν δύο ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν If two numbers have to one another the ratio which
κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν, ὁ δὲ πρῶτος κύβος ᾖ, a cube number (has) to a(nother) cube number, and the
καὶ ὁ δεύτερος κύβος ἔσται. first is cube, then the second will also be cube.
Α Γ A C
Ε E

Ζ F
Β ∆ B D
Δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς ἀλλήλους λόγον For let two numbers, A and B, have to one another
ἐχέτωσαν, ὃν κύβος ἀριθμὸς ὁ Γ πρὸς κύβον ἀριθμὸν τὸν the ratio which the cube number C (has) to the cube
Δ, κύβος δὲ ἔστω ὁ Α· λέγω [δή], ὅτι καὶ ὁ Β κύβος ἐστίν. number D. And let A be cube. [So] I say that B is also
᾿Επεὶ γὰρ οἱ Γ, Δ κύβοι εἰσίν, οἱ Γ, Δ ὅμοιοι στε- cube.
ρεοί εἰσιν· τῶν Γ, Δ ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν For since C and D are cube (numbers), C and D are
ἀριθμοί. ὅσοι δὲ εἰς τοὺς Γ, Δ μεταξὺ κατὰ τὸ συνεχὲς (thus) similar solid (numbers). Thus, two numbers fall
ἀνάλογον ἐμπίπτουσιν, τοσοῦτοι καὶ εἰς τοὺς τὸν αὐτὸν (between) C and D in mean proportion [Prop. 8.19].
λόγον ἔχοντας αὐτοῖς· ὥστε καὶ τῶν Α, Β δύο μέσοι And as many (numbers) as fall in between C and D in
kþ (so) fall. Therefore, since the four numbers A, E, F, B
ἀνάλογον ἐμπίπτουσιν ἀριθμοί. ἐμπιπτέτωσαν οἱ Ε, Ζ. ἐπεὶ continued proportion, so many also (fall) in (between)
οὖν τέσσαρες ἀριθμοὶ οἱ Α, Ε, Ζ, Β ἑξῆς ἀνάλογόν εἰσιν, those (numbers) having the same ratio as them (in con-
καί ἐστι κύβος ὁ Α, κύβος ἄρα καὶ ὁ Β· ὅπερ ἔδει δεῖξαι. tinued proportion) [Prop. 8.8]. And hence two numbers
fall (between) A and B in mean proportion. Let E and F

are continuously proportional, and A is cube, B (is) thus
also cube [Prop. 8.23]. (Which is) the very thing it was
required to show.

.
Proposition 26
Οἱ ὅμοιοι ἐπίπεδοι ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχου- Similar plane numbers have to one another the ratio
σιν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν. which (some) square number (has) to a(nother) square
number.
Α ∆ A D

Γ Ε C E

Β Ζ B F
῎Εστωσαν ὅμοιοι ἐπίπεδοι ἀριθμοὶ οἱ Α, Β· λέγω, ὅτι Let A and B be similar plane numbers. I say that A
ὁ Α πρὸς τὸν Β λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς has to B the ratio which (some) square number (has) to


250