Pappus Alexandrinus
,
Mathematical Collection, Book 8
,
1876
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<
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id.000246
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1100
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ἡ ΑΖ, καὶ ἀπειλήφθω αὐτῆς τὸ γ# μέρος, καὶ ἔστω ἡ ΑΓ,
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ἐφ' ἧς τμῆμα κύκλου γεγράφθω τὸ ΑΒΓ δεχόμενον γωνίαν
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διμοίρου ὀρθῆς, καὶ οἵων ἐστὶν ἡ ΑΓ ε#, τοιούτων δ# ἀπει-
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λήφθω ἡ ΓΕ, καὶ ἤχθω ἐφαπτομένη τοῦ τμήματος ἡ ΕΒ,
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καὶ ἐπεζεύχθω ἥ τε ΑΒ καὶ ἡ ΖΒ, καὶ ἔτι ἐπιζευχθεῖσα
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5
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ἡ ΒΓ ἐκβεβλήσθω ἐπὶ τὸ Δ, καὶ κείσθω τῇ ΑΒ ἴση ἡ
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ΒΔ, καὶ ἐπεζεύχθω ἡ ΑΔ. </
s
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<
s
id
="
id.000247
">ἐπεὶ οὖν εἰς κύκλον διήχθησαν
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lb
n
="
7
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ἥ τε ΕΓΑ καὶ ἡ ΕΒ, καὶ ἡ μὲν τέμνει τὸν κύκλον ἡ δὲ
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lb
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="
8
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ἐφάπτεται, τὸ ἄρα ὑπὸ ΑΕΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς ΕΒ·
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lb
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="
9
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ἔστιν ἄρα ὡς ἡ ΑΕ πρὸς ΕΒ, οὕτως ἡ ΒΕ πρὸς ΓΕ·
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lb
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="
10
"/>
ἰσογώνιον ἄρα τὸ ΓΒΕ τρίγωνον τῷ ΑΒΕ τριγώνῳ. </
s
>
<
s
id
="
id.000248
">ἔστιν
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lb
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11
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ἄρα ὡς ἡ ΕΑ πρὸς ΑΒ, ἡ ΕΒ πρὸς ΒΓ· καὶ ὡς ἄρα τὸ
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ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ τῆς ΕΒ, τὸ ἀπὸ τῆς ΑΒ πρὸς
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τὸ ἀπὸ τῆς ΒΓ. </
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>
<
s
id
="
id.000249
">ἀλλ' ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ
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14
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τῆς ΕΒ, οὕτως ἐστὶν ἡ ΑΕ πρὸς ΕΓ διὰ κ# τοῦ ς#. </
s
>
<
s
id
="
id.000250
">καὶ
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ὡς ἄρα ἡ ΑΕ πρὸς ΕΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ, τουτέστιν
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τὸ ἀπὸ τῆς ΒΔ, πρὸς τὸ ἀπὸ τῆς ΒΓ· τὸ ἄρα ἀπὸ τῆς
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="
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ΒΔ πρὸς τὸ ἀπὸ τῆς ΒΓ λόγον ἔχει ὃν τὰ θ# πρὸς δ#·
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lb
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ἡμιολία ἄρα ἡ ΒΔ τῆς ΒΓ· διπλασία ἄρα ἡ ΒΓ τῆς ΓΔ.
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19
"/>
</
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<
s
id
="
id.000251
">ἔστιν δὲ καὶ ἡ ΖΓ τῆς ΓΑ διπλασία· ὡς ἄρα ἡ ΖΓ πρὸς
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lb
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="
20
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ΓΑ, ἡ ΒΓ πρὸς ΓΔ. </
s
>
<
s
id
="
id.000252
">καὶ ἴσαι εἰσὶν αἱ πρὸς τῷ Γ γω-
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νίαι· ἴση ἄρα καὶ ἡ μὲν Δ γωνία τῇ ὑπὸ ΖΒΓ, ἡ δὲ Ζ
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τῇ ὑπὸ ΓΑΔ· ἔστιν ἄρα ὡς ἡ ΖΒ πρὸς ΒΓ, οὕτως ἡ ΑΔ
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πρὸς ΔΓ. </
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>
<
s
id
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id.000253
">ἐναλλὰξ ὡς ἡ ΖΒ πρὸς ΑΔ, οὕτως ἡ ΒΓ πρὸς
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ΓΔ. </
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<
s
id
="
id.000254
">διπλασία δὲ ἡ ΒΓ τῆς ΓΔ· διπλασία ἄρα καὶ ἡ ΖΒ
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lb
n
="
25
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τῆς ΑΔ, τουτέστιν τῆς ΑΒ. </
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>
<
s
id
="
id.000255
">καὶ ἔστιν διμοίρου ἡ Δ·
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lb
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διμοίρου ἄρα ὀρθῆς καὶ ἡ ὑπὸ ΖΒΓ. </
s
>
<
s
id
="
id.000256
">ὅλη δὲ ἡ ὑπὸ ΑΒΖ
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27
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</
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</
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</
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</
text
>
</
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>