Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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DE CENTRO GRAVIT. SOLID.
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              <pb o="43" file="0197" n="197" rhead="DE CENTRO GRAVIT. SOLID."/>
            b m. </s>
            <s xml:space="preserve">ergo circulus a c circuli _k_ g: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">idcirco cylindrus
              <lb/>
            a h cylindri _k_ l duplus erit. </s>
            <s xml:space="preserve">quare & </s>
            <s xml:space="preserve">linea o p dupla
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            ipſius p n. </s>
            <s xml:space="preserve">Deinde inſcripta & </s>
            <s xml:space="preserve">circumſcripta portioni
              <lb/>
            alia figura, ita ut inſcripta conſtituatur ex tribus cylin-
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            dris q r, s g, tu: </s>
            <s xml:space="preserve">circumſcripta uero ex quatuor a x, y z,
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            _K_ ν, θ λ: </s>
            <s xml:space="preserve">diuidantur b o, o m, m n, n d bifariam in punctis
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            μ ν π ρ. </s>
            <s xml:space="preserve">Itaque cylindri θ λ centrum grauitætis eſt punctum
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            μ: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">cylindri
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            ν centrum ν. </s>
            <s xml:space="preserve">ergo ſi linea μ ν diuidatur in σ,
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            ita ut μ σ ad σ ν proportionẽ eã habeat, quam cylindrus K ν
              <lb/>
            ad cylindrum θ λ, uidelicet quam quadratum
              <emph style="sc">K</emph>
            m ad qua-
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            dratum θ o, hoc eſt, quam linea m b ad b o: </s>
            <s xml:space="preserve">erit σ centrum
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              <anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/>
            magnitudinis compoſitæ ex cylindris
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            ν, θ λ. </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">cum linea
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            m b ſit dupla b o, erit & </s>
            <s xml:space="preserve">μ σ ipſius σ ν dupla. </s>
            <s xml:space="preserve">præterea quo-
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            niam cylindri y z centrum grauitatis eſt π, linea σ π ita diui
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            ſa in τ, ut σ τ ad τ π eam habeat proportionem, quam cylin
              <lb/>
            drus y z ad duos cylindros K ν, θ λ: </s>
            <s xml:space="preserve">erit τ centrum magnitu
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            dinis, quæ ex dictis tribus cylindris conſtat. </s>
            <s xml:space="preserve">cylindrus au-
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            tẽ y z ad cylindrum θ λ eſt, ut linea n b ad b o, hoc eſt ut 3
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            ad 1: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ad cylindrum k ν, ut n b ad b m, uidelicet ut 3 ad 2.
              <lb/>
            </s>
            <s xml:space="preserve">quare y z cylĩdrus duobus cylindris k ν, θ λ æqualis erit. </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">
              <lb/>
            propterea linea σ τ æqualis ipſi τ π. </s>
            <s xml:space="preserve">denique cylindri a x
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            centrum grauitatis eſt punctum ρ. </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">cum τ ζ diuiſa fuerit
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            in eã proportionem, quam habet cylindrus a x ad tres cy-
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            lindros y z, _k_ ν, θ λ: </s>
            <s xml:space="preserve">erit in eo puncto centrum grauitatis
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            totius figuræ circũſcriptæ. </s>
            <s xml:space="preserve">Sed cylindrus a x ad ipſum y z
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            eſt ut linea d b ad b n: </s>
            <s xml:space="preserve">hoc eſt ut 4 ad 3: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">duo cylindri _k_ ν
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            θ λ cylindro y z ſunt æquales. </s>
            <s xml:space="preserve">cylindrns igitur a x ad tres
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            iam dictos cylindros eſt ut 2 ad 3. </s>
            <s xml:space="preserve">Sed quoniã μ σ eſt dua-
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            rum partium, & </s>
            <s xml:space="preserve">σ ν unius, qualium μ π eſt ſex; </s>
            <s xml:space="preserve">erit σ π par-
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            tium quatuor: </s>
            <s xml:space="preserve">proptereaq; </s>
            <s xml:space="preserve">τ π duarum, & </s>
            <s xml:space="preserve">ν π, hoc eſt π ρ
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            trium. </s>
            <s xml:space="preserve">quare ſequitur ut punctum π totius figuræ circum
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            ſcriptæ ſit centrum. </s>
            <s xml:space="preserve">Itaque fiat ν υ ad υ π, ut μ σ ad σ ν. </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">υ ρ
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            bifariam diuidatur in φ. </s>
            <s xml:space="preserve">Similiter ut in circumſcripta figu
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            ra oſtendetur centrum magnitudinis compoſitæ ex cylin-</s>
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