Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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            <s xml:id="echoid-s4165" xml:space="preserve">
              <pb file="0168" n="168" rhead="FED. COMMANDINI"/>
            ſunt uertice, eandem proportionem habent, quam ipſarũ
              <lb/>
            baſes. </s>
            <s xml:id="echoid-s4166" xml:space="preserve">eadem ratione pyramis a c l k pyramidi b c l k: </s>
            <s xml:id="echoid-s4167" xml:space="preserve">& </s>
            <s xml:id="echoid-s4168" xml:space="preserve">py
              <lb/>
            ramis a d l k ipſi b d l k pyramidi æqualis erit. </s>
            <s xml:id="echoid-s4169" xml:space="preserve">Itaque ſi a py
              <lb/>
            ramide a c l d auferantur pyramides a clk, a d l k: </s>
            <s xml:id="echoid-s4170" xml:space="preserve">& </s>
            <s xml:id="echoid-s4171" xml:space="preserve">à pyra
              <lb/>
            mide b c l d auferãtur pyramides b c l k, d b l K: </s>
            <s xml:id="echoid-s4172" xml:space="preserve">quæ relin-
              <lb/>
            quuntur erunt æqualia. </s>
            <s xml:id="echoid-s4173" xml:space="preserve">æqualis igitur eſt pyramis a c d k
              <lb/>
            pyramidi b c d _K_. </s>
            <s xml:id="echoid-s4174" xml:space="preserve">Rurſus ſi per lineas a d, d e ducatur pla-
              <lb/>
            num quod pyramidem ſecet: </s>
            <s xml:id="echoid-s4175" xml:space="preserve">ſitq; </s>
            <s xml:id="echoid-s4176" xml:space="preserve">eius & </s>
            <s xml:id="echoid-s4177" xml:space="preserve">baſis communis
              <lb/>
            ſectio a e m: </s>
            <s xml:id="echoid-s4178" xml:space="preserve">ſimiliter oſtendetur pyramis a b d K æqualis
              <lb/>
            pyramidi a c d
              <emph style="sc">K</emph>
            . </s>
            <s xml:id="echoid-s4179" xml:space="preserve">ducto denique alio piano per lineas c a,
              <lb/>
            a f: </s>
            <s xml:id="echoid-s4180" xml:space="preserve">ut eius, & </s>
            <s xml:id="echoid-s4181" xml:space="preserve">trianguli c d b communis ſectio ſit c fn, py-
              <lb/>
            ramis a b c k pyramidi a c d
              <emph style="sc">K</emph>
            æqualis demonſtrabitur. </s>
            <s xml:id="echoid-s4182" xml:space="preserve">cũ
              <lb/>
            ergo tres pyramides b c d _k_, a b d k, a b c k uni, & </s>
            <s xml:id="echoid-s4183" xml:space="preserve">eidem py
              <lb/>
            ramidia c d k ſint æquales, omnes inter ſe ſe æquales erũt.
              <lb/>
            </s>
            <s xml:id="echoid-s4184" xml:space="preserve">Sed ut pyramis a b c d ad pyramidem a b c k, ita d e axis ad
              <lb/>
            axem k e, ex uigeſima propoſitione huius: </s>
            <s xml:id="echoid-s4185" xml:space="preserve">ſunt enim hæ
              <lb/>
            pyramides in eadem baſi, & </s>
            <s xml:id="echoid-s4186" xml:space="preserve">axes cum baſibus æquales con
              <lb/>
            tinent angulos, quòd in eadem recta linea conſtituantur. </s>
            <s xml:id="echoid-s4187" xml:space="preserve">
              <lb/>
            quare diuidendo, ut tres pyramides a c d k, b c d _K_, a b d _K_
              <lb/>
            ad pyramidem a b c _K_, ita d _k_ ad _K_ e. </s>
            <s xml:id="echoid-s4188" xml:space="preserve">conſtat igitur lineam
              <lb/>
            d K ipſius _K_ e triplam eſſe. </s>
            <s xml:id="echoid-s4189" xml:space="preserve">ſed & </s>
            <s xml:id="echoid-s4190" xml:space="preserve">a k tripla eſt K f: </s>
            <s xml:id="echoid-s4191" xml:space="preserve">itemque
              <lb/>
            b K ipſius _K_ g: </s>
            <s xml:id="echoid-s4192" xml:space="preserve">& </s>
            <s xml:id="echoid-s4193" xml:space="preserve">c
              <emph style="sc">K</emph>
            ipſius
              <emph style="sc">K</emph>
            l tripla. </s>
            <s xml:id="echoid-s4194" xml:space="preserve">quod eodem modo
              <lb/>
            demonſtrabimus.</s>
            <s xml:id="echoid-s4195" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4196" xml:space="preserve">Sit pyramis, cuius baſis quadrilaterum a b c d; </s>
            <s xml:id="echoid-s4197" xml:space="preserve">axis e f:
              <lb/>
            </s>
            <s xml:id="echoid-s4198" xml:space="preserve">& </s>
            <s xml:id="echoid-s4199" xml:space="preserve">diuidatur e fin g, ita ut e g ipſius g f ſit tripla. </s>
            <s xml:id="echoid-s4200" xml:space="preserve">Dico cen-
              <lb/>
            trum grauitatis pyramidis eſſe punctum g. </s>
            <s xml:id="echoid-s4201" xml:space="preserve">ducatur enim
              <lb/>
            linea b d diuidens baſim in duo triangula a b d, b c d: </s>
            <s xml:id="echoid-s4202" xml:space="preserve">ex
              <lb/>
            quibus intelligãtur cõſtitui duæ pyramides a b d e, b c d e: </s>
            <s xml:id="echoid-s4203" xml:space="preserve">
              <lb/>
            ſitque pyramidis a b d e axis e h; </s>
            <s xml:id="echoid-s4204" xml:space="preserve">& </s>
            <s xml:id="echoid-s4205" xml:space="preserve">pyramidis b c d e axis
              <lb/>
            e K: </s>
            <s xml:id="echoid-s4206" xml:space="preserve">& </s>
            <s xml:id="echoid-s4207" xml:space="preserve">iungatur h _K_, quæ per ftranſibit: </s>
            <s xml:id="echoid-s4208" xml:space="preserve">eſt enim in ipſa h K
              <lb/>
            centrum grauitatis magnitudinis compoſitæ ex triangulis
              <lb/>
            a b d, b c d, hoc eſt ipſius quadrilateri. </s>
            <s xml:id="echoid-s4209" xml:space="preserve">Itaque centrum gra
              <lb/>
            uitatis pyramidis a b d e ſit punctum l: </s>
            <s xml:id="echoid-s4210" xml:space="preserve">& </s>
            <s xml:id="echoid-s4211" xml:space="preserve">pyramidis b c d e
              <lb/>
            ſit m. </s>
            <s xml:id="echoid-s4212" xml:space="preserve">ductaigitur l m ipſi h m lineæ æquidiſtabit: </s>
            <s xml:id="echoid-s4213" xml:space="preserve">nam el ad
              <lb/>
              <note position="right" xlink:label="note-0168-01" xlink:href="note-0168-01a" xml:space="preserve">2. ſexti.</note>
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