Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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FED. COMMANDINI
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          <p>
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              <pb file="0168" n="168" rhead="FED. COMMANDINI"/>
            ſunt uertice, eandem proportionem habent, quam ipſarũ
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            baſes. </s>
            <s xml:space="preserve">eadem ratione pyramis a c l k pyramidi b c l k: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">py
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            ramis a d l k ipſi b d l k pyramidi æqualis erit. </s>
            <s xml:space="preserve">Itaque ſi a py
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            ramide a c l d auferantur pyramides a clk, a d l k: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">à pyra
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            mide b c l d auferãtur pyramides b c l k, d b l K: </s>
            <s xml:space="preserve">quæ relin-
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            quuntur erunt æqualia. </s>
            <s xml:space="preserve">æqualis igitur eſt pyramis a c d k
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            pyramidi b c d _K_. </s>
            <s xml:space="preserve">Rurſus ſi per lineas a d, d e ducatur pla-
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            num quod pyramidem ſecet: </s>
            <s xml:space="preserve">ſitq; </s>
            <s xml:space="preserve">eius & </s>
            <s xml:space="preserve">baſis communis
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            ſectio a e m: </s>
            <s xml:space="preserve">ſimiliter oſtendetur pyramis a b d K æqualis
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            pyramidi a c d
              <emph style="sc">K</emph>
            . </s>
            <s xml:space="preserve">ducto denique alio piano per lineas c a,
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            a f: </s>
            <s xml:space="preserve">ut eius, & </s>
            <s xml:space="preserve">trianguli c d b communis ſectio ſit c fn, py-
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            ramis a b c k pyramidi a c d
              <emph style="sc">K</emph>
            æqualis demonſtrabitur. </s>
            <s xml:space="preserve">cũ
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            ergo tres pyramides b c d _k_, a b d k, a b c k uni, & </s>
            <s xml:space="preserve">eidem py
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            ramidia c d k ſint æquales, omnes inter ſe ſe æquales erũt.
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            </s>
            <s xml:space="preserve">Sed ut pyramis a b c d ad pyramidem a b c k, ita d e axis ad
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            axem k e, ex uigeſima propoſitione huius: </s>
            <s xml:space="preserve">ſunt enim hæ
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            pyramides in eadem baſi, & </s>
            <s xml:space="preserve">axes cum baſibus æquales con
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            tinent angulos, quòd in eadem recta linea conſtituantur. </s>
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              <lb/>
            quare diuidendo, ut tres pyramides a c d k, b c d _K_, a b d _K_
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            ad pyramidem a b c _K_, ita d _k_ ad _K_ e. </s>
            <s xml:space="preserve">conſtat igitur lineam
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            d K ipſius _K_ e triplam eſſe. </s>
            <s xml:space="preserve">ſed & </s>
            <s xml:space="preserve">a k tripla eſt K f: </s>
            <s xml:space="preserve">itemque
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            b K ipſius _K_ g: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">c
              <emph style="sc">K</emph>
            ipſius
              <emph style="sc">K</emph>
            l tripla. </s>
            <s xml:space="preserve">quod eodem modo
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            demonſtrabimus.</s>
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          </p>
          <div type="float" level="2" n="1">
            <note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">17. huíus</note>
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            <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a">
              <image file="0167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0167-01"/>
            </figure>
            <note position="right" xlink:label="note-0167-02" xlink:href="note-0167-02a" xml:space="preserve">1. ſexti.</note>
            <note position="right" xlink:label="note-0167-03" xlink:href="note-0167-03a" xml:space="preserve">5. duode-
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            cimi.</note>
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          <p>
            <s xml:space="preserve">Sit pyramis, cuius baſis quadrilaterum a b c d; </s>
            <s xml:space="preserve">axis e f:
              <lb/>
            </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">diuidatur e fin g, ita ut e g ipſius g f ſit tripla. </s>
            <s xml:space="preserve">Dico cen-
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            trum grauitatis pyramidis eſſe punctum g. </s>
            <s xml:space="preserve">ducatur enim
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            linea b d diuidens baſim in duo triangula a b d, b c d: </s>
            <s xml:space="preserve">ex
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            quibus intelligãtur cõſtitui duæ pyramides a b d e, b c d e: </s>
            <s xml:space="preserve">
              <lb/>
            ſitque pyramidis a b d e axis e h; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">pyramidis b c d e axis
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            e K: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">iungatur h _K_, quæ per ftranſibit: </s>
            <s xml:space="preserve">eſt enim in ipſa h K
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            centrum grauitatis magnitudinis compoſitæ ex triangulis
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            a b d, b c d, hoc eſt ipſius quadrilateri. </s>
            <s xml:space="preserve">Itaque centrum gra
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            uitatis pyramidis a b d e ſit punctum l: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">pyramidis b c d e
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            ſit m. </s>
            <s xml:space="preserve">ductaigitur l m ipſi h m lineæ æquidiſtabit: </s>
            <s xml:space="preserve">nam el ad
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              <anchor type="note" xlink:label="note-0168-01a" xlink:href="note-0168-01"/>
            </s>
          </p>
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