Clavius, Christoph, Geometria practica

Table of Notes

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            <s xml:id="echoid-s9329" xml:space="preserve">
              <pb o="208" file="238" n="238" rhead="GEOMETR. PRACT."/>
            æquali ipſi DE) vt SB, ad DE, ita AD, ad DH. </s>
            <s xml:id="echoid-s9330" xml:space="preserve"> Quia vero plana parallela
              <note symbol="a" position="left" xlink:label="note-238-01" xlink:href="note-238-01a" xml:space="preserve">17. vnde.</note>
            DEF, ſecant rectas AH, GH, proportionaliter in I, G; </s>
            <s xml:id="echoid-s9331" xml:space="preserve">erit quoque vt SB, ad
              <lb/>
            DE, ita GI, ad IH. </s>
            <s xml:id="echoid-s9332" xml:space="preserve">Siigitur fiat, vt SB, differentia inter latera homologa AB,
              <lb/>
            DE, baſium ad DE, ita GI, altitudo Fruſti pyramidis (quæ cognoſcetur per li-
              <lb/>
            neam perpendicularem demiſſam ad baſem ex aliquo puncto plani DEF, et-
              <lb/>
            iam producti, ſi opus eſt,) ad aliud, prodibit recta IH, altitudo nimirum pyra-
              <lb/>
            midis DEFH: </s>
            <s xml:id="echoid-s9333" xml:space="preserve">qua addita ad GI, tota altitudo GH, cognita erit. </s>
            <s xml:id="echoid-s9334" xml:space="preserve">Quocirca ſi per
              <lb/>
              <note position="left" xlink:label="note-238-02" xlink:href="note-238-02a" xml:space="preserve">Area fruſti
                <lb/>
              pyramidis.</note>
            caput præcedens inueniatur area tam integræ pyramidis ABCH, quam abſciſ-
              <lb/>
            ſæ pyramidis D E F H, & </s>
            <s xml:id="echoid-s9335" xml:space="preserve">hęc ab illa dematur, reliquum fiet Fruſtum
              <lb/>
            A B C D E F.</s>
            <s xml:id="echoid-s9336" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9337" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9338" xml:space="preserve">
              <emph style="sc">Non</emph>
            aliter fruſtum Coni ABCD, inueſtigabitur, vt patet, ſi integer Co-
              <lb/>
              <note position="left" xlink:label="note-238-03" xlink:href="note-238-03a" xml:space="preserve">Areafruſti
                <lb/>
              coni.</note>
            nus intelligatur ABH, &</s>
            <s xml:id="echoid-s9339" xml:space="preserve">c.</s>
            <s xml:id="echoid-s9340" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s9341" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9342" xml:space="preserve">
              <emph style="sc">Alio</emph>
            modo idem fruſtum tam Pyramidis, quam Coni cognoſcemus,
              <lb/>
            etiamſi neque pyramis, neque conus integretur. </s>
            <s xml:id="echoid-s9343" xml:space="preserve"> Fiant quadrata K L M N, NOPQ, baſibus ABC, DEF, notis æqualia, inueniaturque inter quadrata KM,
              <lb/>
              <note symbol="b" position="left" xlink:label="note-238-04" xlink:href="note-238-04a" xml:space="preserve">14. ſecundi.</note>
            NP, ſuperficies media proportionalis, qua-
              <lb/>
              <figure xlink:label="fig-238-01" xlink:href="fig-238-01a" number="154">
                <image file="238-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/238-01"/>
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            lis eſt rectangula figura OK, producto la-
              <lb/>
            tere O P, ad R. </s>
            <s xml:id="echoid-s9344" xml:space="preserve"> Quoniam enim eſt,
              <note symbol="c" position="left" xlink:label="note-238-05" xlink:href="note-238-05a" xml:space="preserve">1. ſexti.</note>
            MN, ad NO, ita KM, ad NR. </s>
            <s xml:id="echoid-s9345" xml:space="preserve">Item vt KN,
              <lb/>
            ad NQ, ita NR, ad NP. </s>
            <s xml:id="echoid-s9346" xml:space="preserve"> eſtque M N,
              <note symbol="d" position="left" xlink:label="note-238-06" xlink:href="note-238-06a" xml:space="preserve">7. quinti.</note>
            NO, vt KN, ad N Q: </s>
            <s xml:id="echoid-s9347" xml:space="preserve">erit quadratum K M,
              <lb/>
            ad rectangulum NR, vt rectangulum N R,
              <lb/>
            ad quadratum N P: </s>
            <s xml:id="echoid-s9348" xml:space="preserve">ideoque N R, medio
              <lb/>
            loco proportionale eſt inter quadrata
              <lb/>
            KM, NP. </s>
            <s xml:id="echoid-s9349" xml:space="preserve">Quamobrem, ſi radix quadrata
              <lb/>
            baſis ABC, notæ, id eſt, latus KN, quadrati KM, ducatur in radicem quadratam
              <lb/>
            baſis DEF, notę, hoc eſt, in latus NO, quadrati N P, producetur area rectangu-
              <lb/>
            li N R.</s>
            <s xml:id="echoid-s9350" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s9351" xml:space="preserve">
              <emph style="sc">Iam</emph>
            vero ducatur GI, altitudo fruſti in ſummam ex quadrato KM, hoc eſt,
              <lb/>
            ex baſe ABC, & </s>
            <s xml:id="echoid-s9352" xml:space="preserve">quadrato NP, ſiue baſe DEF, & </s>
            <s xml:id="echoid-s9353" xml:space="preserve">ſuperficie NR, media propor-
              <lb/>
            tionali inter baſes, vel dicta quadrata collectam. </s>
            <s xml:id="echoid-s9354" xml:space="preserve">Productus enim numerus
              <lb/>
            triplus erit fruſti pyramidis A B C D E F; </s>
            <s xml:id="echoid-s9355" xml:space="preserve">ideoque tertia producti pars area erit
              <lb/>
            prædicti fruſti. </s>
            <s xml:id="echoid-s9356" xml:space="preserve"> Quoniam enim priſma, quod fit ex GH, altitudine
              <note symbol="e" position="left" xlink:label="note-238-07" xlink:href="note-238-07a" xml:space="preserve">7. duodec.</note>
            in baſem ABC, ſiue quadratum KM, triplum eſt pyramidis ABCDEFH: </s>
            <s xml:id="echoid-s9357" xml:space="preserve">erit
              <lb/>
            quoque parallelepipedum factum ex GI, in quadratum KM, vna cum paralle-
              <lb/>
            lepipedo facto ex IH, in idem quadratum KM, triplum pyramidis eiuſdem. </s>
            <s xml:id="echoid-s9358" xml:space="preserve">
              <note symbol="f" position="left" xlink:label="note-238-08" xlink:href="note-238-08a" xml:space="preserve">7. duodec.</note>
            aũt & </s>
            <s xml:id="echoid-s9359" xml:space="preserve">ablatum parallelepipedum factum ex IH, in baſem DEF, hoc eſt, in qua-
              <lb/>
            dratum NP, triplum ablatæ pyramidis DEFH, Igitur & </s>
            <s xml:id="echoid-s9360" xml:space="preserve">reliquum, quod fit
              <note symbol="g" position="left" xlink:label="note-238-09" xlink:href="note-238-09a" xml:space="preserve">5. quinti.</note>
            GI, in quadratum KM, vna cum iis, quæfiunt ex IH, in KP, & </s>
            <s xml:id="echoid-s9361" xml:space="preserve">in RM, triplum
              <lb/>
            erit fruſti reliqui ABCDEF.</s>
            <s xml:id="echoid-s9362" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s9363" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9364" xml:space="preserve">
              <emph style="sc">Qvia</emph>
            verò æquales ſuperficies A B C, K M, ad ſuperficies
              <note symbol="h" position="left" xlink:label="note-238-10" xlink:href="note-238-10a" xml:space="preserve">7. quinti.</note>
            DEF, NR, eandem habent proportionem, erit permutando ABC, ad DEF, vt
              <lb/>
            KM, ad NP. </s>
            <s xml:id="echoid-s9365" xml:space="preserve"> ideoque latus AB, ad latus DE, erit, vt latus KN, ad latus NQ, &</s>
            <s xml:id="echoid-s9366" xml:space="preserve">
              <note symbol="i" position="left" xlink:label="note-238-11" xlink:href="note-238-11a" xml:space="preserve">22. ſexti.</note>
            diuidendo (ſubtracta recta AS, æquali ipſi DE, ex AB,) SB, ad DE, vt KQ, ad
              <lb/>
            QN. </s>
            <s xml:id="echoid-s9367" xml:space="preserve">Eſt autem, vt Num. </s>
            <s xml:id="echoid-s9368" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9369" xml:space="preserve">demonſtrauimus, vt SB, ad DE, ita GI, ad IH. </s>
            <s xml:id="echoid-s9370" xml:space="preserve">Igi-
              <lb/>
            tur erit etiam vt KQ, ad QN, ideoque vt MO, ad ON, ita GI, ad IH. </s>
            <s xml:id="echoid-s9371" xml:space="preserve"> Sed vt KQ, ad QN, ita eſt KP, ad PN: </s>
            <s xml:id="echoid-s9372" xml:space="preserve">Et vt MO, ad ON, ita MR, ad RN. </s>
            <s xml:id="echoid-s9373" xml:space="preserve">Igitur erit
              <lb/>
              <note symbol="k" position="left" xlink:label="note-238-12" xlink:href="note-238-12a" xml:space="preserve">1. ſexti.</note>
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