Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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            <s xml:id="echoid-s4986" xml:space="preserve">
              <pb o="44" file="0199" n="199" rhead="DE CENTRO GRAVIT. SOLID."/>
            relinquetur p e ipſi n χ æqualis. </s>
            <s xml:id="echoid-s4987" xml:space="preserve">cum autem b e ſit dupla
              <lb/>
            e d, & </s>
            <s xml:id="echoid-s4988" xml:space="preserve">o p dupla p n, hoc eſt ipſius e χ, & </s>
            <s xml:id="echoid-s4989" xml:space="preserve">reliquum, uideli-
              <lb/>
            cet b o unà cum p e ipſius reliqui χ d duplnm erit. </s>
            <s xml:id="echoid-s4990" xml:space="preserve">eſtque
              <lb/>
              <note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve">19. quinti</note>
            b o dupla ζ d. </s>
            <s xml:id="echoid-s4991" xml:space="preserve">ergo p e, hoc eſt n χ ipſius χ ρ dupla. </s>
            <s xml:id="echoid-s4992" xml:space="preserve">ſed d n
              <lb/>
            dupla eſt n ζ. </s>
            <s xml:id="echoid-s4993" xml:space="preserve">reliqua igitur d χ dupla reliquæ χ n. </s>
            <s xml:id="echoid-s4994" xml:space="preserve">ſunt au-
              <lb/>
            tem d χ, p n inter ſe æquales: </s>
            <s xml:id="echoid-s4995" xml:space="preserve">itemq; </s>
            <s xml:id="echoid-s4996" xml:space="preserve">æquales χ n, p e. </s>
            <s xml:id="echoid-s4997" xml:space="preserve">qua-
              <lb/>
            re conſtat n p ipſius p e duplam eſſe. </s>
            <s xml:id="echoid-s4998" xml:space="preserve">& </s>
            <s xml:id="echoid-s4999" xml:space="preserve">idcirco p e ipſi e n
              <lb/>
            æqualem. </s>
            <s xml:id="echoid-s5000" xml:space="preserve">Rurſus cum ſit μ ν dupla o ν, & </s>
            <s xml:id="echoid-s5001" xml:space="preserve">μ σ dupla σ ν; </s>
            <s xml:id="echoid-s5002" xml:space="preserve">erit
              <lb/>
            etiam reliqua ν σ o dupla. </s>
            <s xml:id="echoid-s5003" xml:space="preserve">Eadem quoque ratione
              <lb/>
            cõcludetur π υ dupla υ m. </s>
            <s xml:id="echoid-s5004" xml:space="preserve">ergo ut ν σ ad σ O, ita π υ ad υ m:
              <lb/>
            </s>
            <s xml:id="echoid-s5005" xml:space="preserve">componendoq;</s>
            <s xml:id="echoid-s5006" xml:space="preserve">, & </s>
            <s xml:id="echoid-s5007" xml:space="preserve">permutando, ut υ o ad π m, ita o σ ad
              <lb/>
            m υ & </s>
            <s xml:id="echoid-s5008" xml:space="preserve">ſunt æquales ν o, π m. </s>
            <s xml:id="echoid-s5009" xml:space="preserve">quare & </s>
            <s xml:id="echoid-s5010" xml:space="preserve">o σ, m υ æquales. </s>
            <s xml:id="echoid-s5011" xml:space="preserve">præ
              <lb/>
            terea σ π dupla eſt π τ, & </s>
            <s xml:id="echoid-s5012" xml:space="preserve">ν π ipſius π m. </s>
            <s xml:id="echoid-s5013" xml:space="preserve">reliqua igitur σ ν re
              <lb/>
            liquæ m τ dupla. </s>
            <s xml:id="echoid-s5014" xml:space="preserve">atque erat ν σ dupla σ o. </s>
            <s xml:id="echoid-s5015" xml:space="preserve">ergo m τ, σ o æ-
              <lb/>
            quales ſunt: </s>
            <s xml:id="echoid-s5016" xml:space="preserve">& </s>
            <s xml:id="echoid-s5017" xml:space="preserve">ita æquales m υ, n φ. </s>
            <s xml:id="echoid-s5018" xml:space="preserve">at o σ, eſt æqualis
              <lb/>
            m υ. </s>
            <s xml:id="echoid-s5019" xml:space="preserve">Sequitur igitur, ut omnes o σ, m τ, m υ, n φ in-
              <lb/>
            ter ſe ſint æquales. </s>
            <s xml:id="echoid-s5020" xml:space="preserve">Sed ut ρ π ad π τ, hoc eſt ut 3 ad 2, ita n d
              <lb/>
            ad d χ: </s>
            <s xml:id="echoid-s5021" xml:space="preserve">permutãdoq; </s>
            <s xml:id="echoid-s5022" xml:space="preserve">ut ρ π ad n d, ita π τ ad d χ. </s>
            <s xml:id="echoid-s5023" xml:space="preserve">& </s>
            <s xml:id="echoid-s5024" xml:space="preserve">ſũt æqua
              <lb/>
            les ζ π, n d. </s>
            <s xml:id="echoid-s5025" xml:space="preserve">ergo d χ, hoc eſt n p, & </s>
            <s xml:id="echoid-s5026" xml:space="preserve">π τ æquales. </s>
            <s xml:id="echoid-s5027" xml:space="preserve">Sed etiam æ-
              <lb/>
            quales n π, π m. </s>
            <s xml:id="echoid-s5028" xml:space="preserve">reliqua igitur π p reliquæ m τ, hoc eſt ipſi
              <lb/>
            n φ æqualis erit. </s>
            <s xml:id="echoid-s5029" xml:space="preserve">quare dempta p π ex p e, & </s>
            <s xml:id="echoid-s5030" xml:space="preserve">φ n dempta ex
              <lb/>
            n e, relinquitur p e æqualis e φ. </s>
            <s xml:id="echoid-s5031" xml:space="preserve">Itaque π, ρ centra figurarũ
              <lb/>
            ſecundo loco deſcriptarum a primis centris p n æquali in-
              <lb/>
            teruallo recedunt. </s>
            <s xml:id="echoid-s5032" xml:space="preserve">quòd ſi rurſus aliæ figuræ deſcribantur,
              <lb/>
            eodem modo demonſtrabimus earum centra æqualiter ab
              <lb/>
            his recedere, & </s>
            <s xml:id="echoid-s5033" xml:space="preserve">ad portionis conoidis centrum propius ad
              <lb/>
            moueri. </s>
            <s xml:id="echoid-s5034" xml:space="preserve">Ex quibus conſtat lineam π φ à centro grauitatis
              <lb/>
            portionis diuidi in partes æquales. </s>
            <s xml:id="echoid-s5035" xml:space="preserve">Si enim fieri poteſt, non
              <lb/>
            ſit centrum in puncto e, quod eſt lineæ π φ medium: </s>
            <s xml:id="echoid-s5036" xml:space="preserve">ſed in
              <lb/>
            ψ: </s>
            <s xml:id="echoid-s5037" xml:space="preserve">& </s>
            <s xml:id="echoid-s5038" xml:space="preserve">ipſi π ψ æqualis fiat φ ω. </s>
            <s xml:id="echoid-s5039" xml:space="preserve">Cum igitur in portione ſolida
              <lb/>
            quædam figura inſcribi posſit, ita ut linea, quæ inter cen-
              <lb/>
            trum grauitatis portionis, & </s>
            <s xml:id="echoid-s5040" xml:space="preserve">inſcriptæ figuræ interiicitur,
              <lb/>
            qualibet linea propoſita ſit minor, quod proxime demon-
              <lb/>
            ſtrauimus: </s>
            <s xml:id="echoid-s5041" xml:space="preserve">perueniet tandem φ centrum inſcriptæ </s>
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