Euclid, Elements, Proposition V.5

Euclid, Elements, Proposition V.5

Two segments, ΑΒ and ΓΔ. ΑΒ is divided at Ε. ΓΔ is divided at Ζ and extended beyond Γ to Η. ΑΕ is as many multiples of ΓΖ as ΕΒ is of ΗΓ

We give

The Greek text and the diagram above are from the handy Euclid homepage of Dimitrios Mourmouras in the physics department of the National Technical University of Athens. One can download the page, or rather site, as I have done; and it turns out the webmaster also has a text of the Nicomachean Ethics, and much else.

The present page is interconvertible between txt and html formats by the pandoc program.

Translation of the Greek is made with the understanding that the following are cognate:

  • “-ple” in “multiple,”
  • “-fold” in “manifold,”
  • -πλάσιον in
    • πολλαπλάσιον “multiple, manifold,”
    • ὁσαπλάσιον “as/however many times, how manifold, as manifold as,” and
    • τοσαυταπλάσιον “so many times, so manifold.”

One can infer something of the relations of the three Greek words from the table in Smyth’s Greek Grammar as posted at Project Perseus:

Correlative Pronouns.—Many pronominal adjectives correspond to each other in form and meaning. In the following list poetic or rare forms are placed in ().

Interrogative: Direct or Indirect Indefinite (Enclitic) Demonstrative Relative (Specific) or Exclamatory Indefinite Relative or Indirect Interrogative
τίς who? which? what? qui? τὶς some one, any one, aliquis, quidam (ὁ, ὅς) ὅδε this (here), hic
οὗτος this, that is, ille
ἐκεῖνος ille		 ὅς who, which, qui ὅστις whoever, any one who, quisquis, quicunque
πότερος which of two? uter? πότερος or ποτερός one of two (rare) ἕτερος the one or the other of two, alter ὁπότερος whichever of the two ὁπότερος whichever of the two, utercumque
πόσος how much? how many? quantus? quot? ποσός of some quantity or number (τόσος,) τοσόσδε, τοσοῦτος so much, so many, tantus, tot ὅσος as much as, as many as, quantus, quot ὁπόσος of whatever size, number, quantuscumque, quotquot
ποῖος of what sort? qualis? ποιός of some sort (τοῖος,) τοιόσδε, τοιοῦτος such, talis οἷος of which sort, (such) as, qualis ποῖος of whatever sort, qua lis­cumque
πηλίκος how old? how large? πηλίκος of some age, size (τηλίκος,) τηλικόσδε, τηλικοῦτος so old, young, so large, so great ἡλίκος of which age, size, (as old, as large) as ὁπηλίκος of whatever age or size

Now Euclid.

Πρότασις / Enunciation

Ἐὰν μέγεθος μεγέθους ἰσάκις ᾖ πολλαπλάσιον,
If magnitude of magnitude equally be manifold,
ὅπερ ἀφαιρεθὲν ἀφαιρεθέντος,
that subtrahend of subtrahend,
καὶ τὸ λοιπὸν τοῦ λοιποῦ
also the remainder of the remainder
ἰσάκις ἔσται πολλαπλάσιον,
equally will be manifold,
ὁσαπλάσιόν ἐστι τὸ ὅλον τοῦ ὅλου.
how-manifold is the whole of the whole.

Algebraically, if

A = B + C,
D = E + F,
A = nD,
B = nE,

then

C = nF.

For, suppose

C = nG.

By Proposition V.1 (whose enunciation is below),

A = B + C = nE + nG = n(E + G).

However,

A = nD = n(E + F).

Thus

n(E + G) = n(E + F),
E + G = E + F,
G = F.

If you don’t like the assumption

C = nG,

you can let

H = nF.

Then

B + H = nE + nF = n(E + F) = nD,

but also

B + C = A = nD,

so

B + H = B + C,
H = C.

Ἔκθεσις / Exposition

 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


                     Γ           Ζ     Δ
                     |--|--|--|--|--|--|
                     G           Z     D
 
Μέγεθος γὰρ τὸ ΑΒ μεγέθους τοῦ ΓΔ
For, magnitude the AB of magnitude the GD
ἰσάκις ἔστω πολλαπλάσιον,
equally be manifold,
ὅπερ ἀφαιρεθὲν τὸ ΑΕ ἀφαιρεθέντος τοῦ ΓΖ.
that subtrahend the AE of subtrahend the GZ.

Note. ἔστω “[let] be” is a third-person imperative, as in “Anger be now your song, immortal one” (Fitzgerald’s translation of the first three words of the Iliad, Μῆνιν ἄειδε θεά).

Διορισμός / Definition

 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


                     Γ           Ζ     Δ
                     |--|--|--|--|--|--|
                     G           Z     D
 
λέγω, ὅτι καὶ λοιπὸν τὸ ΕΒ λοιποῦ τοῦ ΖΔ
I say that also remainder the AB of remainder the ZD
ἰσάκις ἔσται πολλαπλάσιον,
equally will be manifold,
ὁσαπλάσιόν ἐστιν ὅλον τὸ ΑΒ ὅλου τοῦ ΓΔ.
how-manifold is whole the AB of whole the GD.

Κατασκευή / Construction

 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


               Η     Γ           Ζ     Δ
               |--|--|--|--|--|--|--|--|
               H     G           Z     D
 
Ὁσαπλάσιον γάρ ἐστι τὸ ΑΕ τοῦ ΓΖ,
For, how-manifold is the AE of the GZ,
τοσαυταπλάσιον γεγονέτω καὶ τὸ ΕΒ τοῦ ΓΗ.
so-manifold become also the EB of the GH.

Note. Here is another third-person imperative: γεγονέτω “[let] become.” Why we can make this command is a good question. An alternative is, however many AE is of GZ, so many to be AK of ZD.

 
Κ                 Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
K                 A                                   E                 B


                           Γ           Ζ     Δ
                           |--|--|--|--|--|--|
                           G           Z     D
 

Ἀπόδειξις / Demonstration

 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


               Η     Γ           Ζ     Δ
               |--|--|--|--|--|--|--|--|
               H     G           Z     D
 
Καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον
And since equally is manifold
τὸ ΑΕ τοῦ ΓΖ καὶ τὸ ΕΒ τοῦ ΗΓ,
the AE of the GZ and the EB of the HG,
ἰσάκις ἄρα ἐστὶ πολλαπλάσιον
equally then is manifold
τὸ ΑΕ τοῦ ΓΖ καὶ τὸ ΑΒ τοῦ ΗΖ.
the AE of the GZ and the AB of the HZ.

Note. This is by Proposition V.1.

 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


               Η     Γ           Ζ     Δ
               |--|--|--|--|--|--|--|--|
               H     G           Z     D
 
κεῖται δὲ ἰσάκις πολλαπλάσιον
And, lie equally manifold
τὸ ΑΕ τοῦ ΓΖ καὶ τὸ ΑΒ τοῦ ΓΔ.
the AE of the GZ and the AB of the GD.
ἰσάκις ἄρα ἐστὶ πολλαπλάσιον
Equally then is manifold
τὸ ΑΒ ἑκατέρου τῶν ΗΖ, ΓΔ·
the AB of either of HZ, GD;
ἴσον ἄρα τὸ ΗΖ τῷ ΓΔ.
equal then the HZ to the GD. 
 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


               Η     Γ           Ζ     Δ
               |--|--|--|--|--|--|--|--|
               H     G           Z     D
 
κοινὸν ἀφῃρήσθω τὸ ΓΖ·
Common be subtracted the GZ.
λοιπὸν ἄρα τὸ ΗΓ λοιπῷ τῷ ΖΔ ἴσον ἐστίν.
Remainder then the HG to remainder the ZD equal is. 
 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


               Η     Γ           Ζ     Δ
               |--|--|--|--|--|--|--|--|
               H     G           Z     D
 
καὶ ἐπεὶ ἰσάκις ἐστὶ πολλαπλάσιον
And since equally is manifold
τὸ ΑΕ τοῦ ΓΖ καὶ τὸ ΕΒ τοῦ ΗΓ,
the AE of the GZ and the EB of the HG,
ἴσον δὲ τὸ ΗΓ τῷ ΔΖ,
and equal the HG to the DZ,
ἰσάκις ἄρα ἐστὶ πολλαπλάσιον
equally then is manifold
τὸ ΑΕ τοῦ ΓΖ καὶ τὸ ΕΒ τοῦ ΖΔ.
the AE of the GZ and the EB of the ZD. 
 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


               Η     Γ           Ζ     Δ
               |--|--|--|--|--|--|--|--|
               H     G           Z     D
 
ἰσάκις δὲ ὑπόκειται πολλαπλάσιον
And equally lie manifold
τὸ ΑΕ τοῦ ΓΖ καὶ τὸ ΑΒ τοῦ ΓΔ·
the AE of the GZ and the AB of the GD;
ἰσάκις ἄρα ἐστὶ πολλαπλάσιον
equally then is manifold
τὸ ΕΒ τοῦ ΖΔ καὶ τὸ ΑΒ τοῦ ΓΔ.
the EB to the ZD and the AB of the GD. 
 
Α                                   Ε                 Β
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
A                                   E                 B


               Η     Γ           Ζ     Δ
               |--|--|--|--|--|--|--|--|
               H     G           Z     D
 
καὶ λοιπὸν ἄρα τὸ ΕΒ λοιποῦ τοῦ ΖΔ
Also remainder then the EB to remainder the ZD
ἰσάκις ἔσται πολλαπλάσιον,
equally will be manifold,
ὁσαπλάσιόν ἐστιν ὅλον τὸ ΑΒ ὅλου τοῦ ΓΔ.
so-manifold is whole the AB of whole the GD.

Συμπέρασμα / Conclusion

Ἐὰν ἄρα μέγεθος μεγέθους ἰσάκις ᾖ πολλαπλάσιον,
If then magnitude of magnitude equally be manifold,
ὅπερ ἀφαιρεθὲν ἀφαιρεθέντος,
that subtrahend of subtrahend
καὶ τὸ λοιπὸν τοῦ λοιποῦ
also the remainder of the remainder
ἰσάκις ἔσται πολλαπλάσιον,
equally will be manifold,
ὁσαπλάσιόν ἐστι καὶ τὸ ὅλον τοῦ ὅλου.
how-manifold is also the whole of the whole.
ὅπερ ἔδει δεῖξαι.
Q. E. D.

Proposition V.1

Ἐὰν ᾖ ὁποσαοῦν μεγέθη
If be however many magnitudes
ὁποσωνοῦν μεγεθῶν ἴσων τὸ πλῆθος
of however many magnitudes equal the multitude
ἕκαστον ἑκάστου ἰσάκις πολλαπλάσιον,
each of each equally manifold,
ὁσαπλάσιόν ἐστιν ἓν τῶν μεγεθῶν ἑνός,
how-manifold is one of the magnitudes of one,
τοσαυταπλάσια ἔσται καὶ τὰ πάντα τῶν πάντων.
so-manifold will be also the all of the all.