Archimedes
,
Archimedis De iis qvae vehvntvr in aqva libri dvo
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[Figure 151]
Page: 208
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DE CENTRO GRAVIT. SOLID.
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grauitatis magnitudinis, quæ ex utriſque pyramidibus cõ
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ſtat; </
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<
s
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<
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xml:space
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">Sed fruſti centrum eſt etiam in a-
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xe g h. </
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<
s
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xml:space
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">ergo in puncto φ, in quo lineæ z u, g h conueniunt.
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</
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<
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xml:space
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">Itaque u φ ad φ z eam proportionem habet, quam pyramis
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xml:space
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">8. prim I
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libri Ar-
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chimedis
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de cẽtro
<
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grauita-
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tis plano
<
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runi</
note
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b c f e d ad pyramidem a b c d. </
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<
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xml:space
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">& </
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<
s
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xml:space
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">componendo u z ad z φ
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eam habet, quam fruſtum ad pyramidem a b c d. </
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<
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xml:space
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">Vtuero
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u z ad z φ, ita o p ad p φ ob ſimilitudinem triangulorum,
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u o φ, z p φ. </
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<
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xml:space
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">quare o p ad p φ eſt ut fruſtum ad pyramidem
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a b c d. </
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<
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xml:space
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<
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xml:space
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">æquales igitur ſunt p φ, p q: </
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xml:space
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<
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<
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xml:space
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">7. quinti.</
note
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q φ unum atque idem punctum. </
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<
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xml:space
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">ex quibus ſequitur lineam
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z u ſecare o p in q: </
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xml:space
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">& </
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<
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xml:space
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">propterea pũctum q ipſius fruſti gra-
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uitatis centrum eſſe.</
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<
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xml:space
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</
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<
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<
s
xml:id
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xml:space
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">Sit fruſtum a g à pyramide, quæ quadrangularem baſim
<
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habeat abſciſſum, cuius maior baſis a b c d, minor e f g h,
<
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& </
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<
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xml:id
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<
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xml:space
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">diuidatur autem primũ _k_ l, ita ut quam propor-
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tionem habet duplum lateris a b unà cum latere e f ad du
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plum lateris e f unà cum a b; </
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<
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xml:id
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xml:space
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<
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xml:space
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">deinde à
<
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púcto m ad k ſumatur quarta pars ipſius m k, quæ ſit m n.
<
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</
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<
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xml:space
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<
s
xml:id
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xml:space
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">rurſus ab l ſumatur quarta pars totius axis l k, quæ ſit
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l o. </
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<
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xml:space
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">poſtremo fiat o n ad n p, ut fruſtum a g ad pyramidẽ,
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cuius baſis ſit eadem, quæ fruſti, & </
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<
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xml:id
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">altitudo æqualis. </
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<
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xml:id
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xml:space
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">Dico
<
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punctum p fruſti a g grauitatis centrum eſſe. </
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<
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enim a c, e g: </
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<
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xml:space
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">intelligantur duo fruſta triangulares ba-
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ſes habentia, quorum alterum l f ex baſibus a b c, e f g cõ-
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ſtet; </
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<
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xml:id
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">alterum l h ex baſibus a c d, e g h. </
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<
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xml:space
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q r; </
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<
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<
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<
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x grauitatis centrum: </
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<
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xml:space
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">deinde iungantur u r, t q, x s. </
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xml:space
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bit u r per l: </
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<
s
xml:id
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xml:space
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">quoniam l eſt centrum grauitatis quadran-
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guli a b c d: </
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<
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xml:space
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">& </
s
>
<
s
xml:id
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xml:space
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">puncta r u grauitatis centra triangulorum
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/>
a b c, a c d; </
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<
s
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">in quæ quadrangulum ipſum diuiditur. </
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<
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xml:id
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xml:space
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">eadem
<
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quoque ratione t q per punctum _k_ tranſibit. </
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<
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xml:space
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">At uero pro
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portiones, ex quibus fruſtorum grauitatis centra inquiri-
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mus, eædem ſunt in toto ſruſto a g, & </
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<
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xml:id
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">in fruſtis l f, l h. </
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<
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xml:id
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xml:space
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<
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enim per octauam huius quadrilatera a b c d, e f g h ſimilia:</
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