Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s6651" xml:space="preserve">
              <pb o="165" file="195" n="195" rhead="LIBER QVARTVS."/>
            4 {1755/2416}. </s>
            <s xml:id="echoid-s6652" xml:space="preserve">in totam baſem 12. </s>
            <s xml:id="echoid-s6653" xml:space="preserve">conficietur area trianguli ABD, 56 {866/1208}. </s>
            <s xml:id="echoid-s6654" xml:space="preserve">vel 56 {433/604}.
              <lb/>
            </s>
            <s xml:id="echoid-s6655" xml:space="preserve">Quæ etiam producetur, ſi tota perpendicularis in totam baſem ducatur, & </s>
            <s xml:id="echoid-s6656" xml:space="preserve">
              <lb/>
            producti capiatur ſemiſsis.</s>
            <s xml:id="echoid-s6657" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6658" xml:space="preserve">
              <emph style="sc">Vt</emph>
            autem fractiones, quoad eius fieri poteſt, vitentur, curabis, vt quando
              <lb/>
              <note position="right" xlink:label="note-195-01" xlink:href="note-195-01a" xml:space="preserve">Vt fractiones
                <lb/>
              vitentur quid
                <lb/>
              agendum.</note>
            perpendicularis eſt numerus par, & </s>
            <s xml:id="echoid-s6659" xml:space="preserve">baſis numerus impar, accipias ſemiſſem per-
              <lb/>
            pendicularis, eamquein totam baſem ducas: </s>
            <s xml:id="echoid-s6660" xml:space="preserve">quando vero perpendicularis eſt
              <lb/>
            numerus impar, & </s>
            <s xml:id="echoid-s6661" xml:space="preserve">baſis numerus par, ſumas ſemiſſem baſis, eamque ducas in
              <lb/>
            totam perpendicularem. </s>
            <s xml:id="echoid-s6662" xml:space="preserve">Quod ſi tam perpendicularis, quam baſis fuerit nu-
              <lb/>
            merus par, velimpar, nihil intereſt, vtrius ſemiſſem capias.</s>
            <s xml:id="echoid-s6663" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6664" xml:space="preserve">
              <emph style="sc">Qvando</emph>
            etiam perpendicularis eſt radix ſurda, quæ videlicet numero ex-
              <lb/>
              <note position="right" xlink:label="note-195-02" xlink:href="note-195-02a" xml:space="preserve">Quid agen-
                <lb/>
              dum, quando
                <lb/>
              perpendicula-
                <lb/>
              ris eſt nume-
                <lb/>
              r{us} ſurd{us}.</note>
            primi nequeat, qualis fuit DC, in poſteriori triangulo ABD, rectè feceris, ſi eius
              <lb/>
            quadratum (non extracta radiceilla ſurda) multiplices per quadratum ſemiſsis
              <lb/>
            baſis. </s>
            <s xml:id="echoid-s6665" xml:space="preserve">Numerus enim productus erit quadratus numerus areæ trianguli: </s>
            <s xml:id="echoid-s6666" xml:space="preserve">adeo
              <lb/>
            vt radix eius ſit ipſa trianguli area. </s>
            <s xml:id="echoid-s6667" xml:space="preserve">Hac enimratione minus à vero aberrabimus.
              <lb/>
            </s>
            <s xml:id="echoid-s6668" xml:space="preserve">Vtin dicto poſteriori triangulo ABD, ſi quadratum perpendicularis DC, {5719/64}. </s>
            <s xml:id="echoid-s6669" xml:space="preserve">
              <lb/>
            ducamus in 36. </s>
            <s xml:id="echoid-s6670" xml:space="preserve">quadratum ſemiſsis baſis, producemus {@@@884/64}. </s>
            <s xml:id="echoid-s6671" xml:space="preserve">quadratuma-
              <lb/>
            reæ, cuius radix eſt 56 {2605/3628}. </s>
            <s xml:id="echoid-s6672" xml:space="preserve">area videlicet trianguli ABD, paulo maior, quam
              <lb/>
            priusinuenta. </s>
            <s xml:id="echoid-s6673" xml:space="preserve">Pariratione, ſi in aliquo triangulo quadratum perpendicularis
              <lb/>
            foret 72. </s>
            <s xml:id="echoid-s6674" xml:space="preserve">& </s>
            <s xml:id="echoid-s6675" xml:space="preserve">ſemiſsis baſis 6. </s>
            <s xml:id="echoid-s6676" xml:space="preserve">ſi radicem numeri 72. </s>
            <s xml:id="echoid-s6677" xml:space="preserve">nimirum 8 {8/17}. </s>
            <s xml:id="echoid-s6678" xml:space="preserve">hoc eſt, ipſam
              <lb/>
            perpendicularem, ducamus in 6. </s>
            <s xml:id="echoid-s6679" xml:space="preserve">producemus aream 50 {14/17}. </s>
            <s xml:id="echoid-s6680" xml:space="preserve">At ſi ipſũmet qua-
              <lb/>
            dratum 72. </s>
            <s xml:id="echoid-s6681" xml:space="preserve">multiplicemus per 36. </s>
            <s xml:id="echoid-s6682" xml:space="preserve">quadratum videlicet ſemiſsis baſis, procrea-
              <lb/>
            bimus 2592. </s>
            <s xml:id="echoid-s6683" xml:space="preserve">quadratum areæ, cuius radix paulo maior eſt, quam 50 {92/101}. </s>
            <s xml:id="echoid-s6684" xml:space="preserve">quinu-
              <lb/>
            merus aliquanto maior eſt, quam area prius inuenta 50 {14/17}. </s>
            <s xml:id="echoid-s6685" xml:space="preserve">Ratio huius noſtræ
              <lb/>
            regulæ eſt, quòd, vt paulò ante ad finem Num. </s>
            <s xml:id="echoid-s6686" xml:space="preserve">1. </s>
            <s xml:id="echoid-s6687" xml:space="preserve">demonſtrauimus, duo nume-
              <lb/>
            riſeſe multiplicantes producantradicem numeri ex eorum quadratis producti.</s>
            <s xml:id="echoid-s6688" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6689" xml:space="preserve">3. </s>
            <s xml:id="echoid-s6690" xml:space="preserve">
              <emph style="sc">Expositis</emph>
            duabusregulis generalibus, per quas trianguli cuiuslibet
              <lb/>
            area ex cognitis lateribus inueſtigatur, proponemus nunc particularia quædam
              <lb/>
              <note position="right" xlink:label="note-195-03" xlink:href="note-195-03a" xml:space="preserve">Area triangu
                <lb/>
              li rectanguli:</note>
            præcepta pro particularibus triangulis nonnullis, quæ nõnuquam magno vſui
              <lb/>
            erunt, cumper ea ſæpenumero expeditius in aliquibus triangulis areæ reperi-
              <lb/>
            antur, quam perillas generales regulas. </s>
            <s xml:id="echoid-s6691" xml:space="preserve">Area ergo triangulirectanguli produ-
              <lb/>
            cetur,
              <unsure/>
            ſi duo latera circa rectum angulum inter ſe multiplicentur, & </s>
            <s xml:id="echoid-s6692" xml:space="preserve">numeri pro-
              <lb/>
            ductiſemiſsis capiatur. </s>
            <s xml:id="echoid-s6693" xml:space="preserve">Nam ex multiplicatione illa gignitur parallelogrãmum
              <lb/>
            rectangulum ſub duobus lateribus circa angulum rectum comprehenſum, vt c.
              <lb/>
            </s>
            <s xml:id="echoid-s6694" xml:space="preserve">1. </s>
            <s xml:id="echoid-s6695" xml:space="preserve">dictum eſt, cuius rectanguli triangulum ſemiſsis eſt, Quod perinde eſt, ac
              <note symbol="a" position="right" xlink:label="note-195-04" xlink:href="note-195-04a" xml:space="preserve">41. primi.</note>
            ſemiſsis vtriuſuis lateris in totum alterum, tamquam in baſem, multiplicetur. </s>
            <s xml:id="echoid-s6696" xml:space="preserve">Vt
              <lb/>
            in præcedentitriangulo ABC, diuiſo in duo triangula rectangula ADB, ADC; </s>
            <s xml:id="echoid-s6697" xml:space="preserve">ſi
              <lb/>
            AD, 8.</s>
            <s xml:id="echoid-s6698" xml:space="preserve">
              <unsure/>
            ducaturin BD, 6. </s>
            <s xml:id="echoid-s6699" xml:space="preserve">producetur numerus 48. </s>
            <s xml:id="echoid-s6700" xml:space="preserve">cuius ſemiſsis 24. </s>
            <s xml:id="echoid-s6701" xml:space="preserve">erit area tri-
              <lb/>
            anguli ADB. </s>
            <s xml:id="echoid-s6702" xml:space="preserve">Sic ſi AD, 8, ducatur in DC, 15. </s>
            <s xml:id="echoid-s6703" xml:space="preserve">fiet numerus 120. </s>
            <s xml:id="echoid-s6704" xml:space="preserve">cuus ſemiſsis 60.
              <lb/>
            </s>
            <s xml:id="echoid-s6705" xml:space="preserve">eritarea trianguli ADC: </s>
            <s xml:id="echoid-s6706" xml:space="preserve">vbi vides, duo triangula 24. </s>
            <s xml:id="echoid-s6707" xml:space="preserve">& </s>
            <s xml:id="echoid-s6708" xml:space="preserve">60. </s>
            <s xml:id="echoid-s6709" xml:space="preserve">componere totum
              <lb/>
            triangulum ABC, 84. </s>
            <s xml:id="echoid-s6710" xml:space="preserve">vtſuprainuenimus.</s>
            <s xml:id="echoid-s6711" xml:space="preserve"/>
          </p>
          <figure number="125">
            <image file="195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/195-01"/>
          </figure>
          <p>
            <s xml:id="echoid-s6712" xml:space="preserve">4. </s>
            <s xml:id="echoid-s6713" xml:space="preserve">
              <emph style="sc">Area</emph>
            trianguli Iſoſcelis, vel etiam æquilate-
              <lb/>
              <note position="right" xlink:label="note-195-05" xlink:href="note-195-05a" xml:space="preserve">Areatrian-
                <lb/>
              guli Iſoſcelis.</note>
            ri, procreabitur, ſi quadratum ſemiſsis baſis ex qua-
              <lb/>
            drato lateris auferatur, & </s>
            <s xml:id="echoid-s6714" xml:space="preserve">reliquus numerusinidem
              <lb/>
            quadratum ſemiſsis baſis ducatur, ac denique huius
              <lb/>
            ꝓducti radix quadrata eruatur. </s>
            <s xml:id="echoid-s6715" xml:space="preserve">Vtin Iſoſcele ABC,
              <lb/>
            cuius æqualia latera AB, AC, ſint 32. </s>
            <s xml:id="echoid-s6716" xml:space="preserve">32. </s>
            <s xml:id="echoid-s6717" xml:space="preserve">& </s>
            <s xml:id="echoid-s6718" xml:space="preserve">baſis BC,
              <lb/>
            24. </s>
            <s xml:id="echoid-s6719" xml:space="preserve">ſi qua dratum 144. </s>
            <s xml:id="echoid-s6720" xml:space="preserve">ſemiſsis baſis dematur ex 1024. </s>
            <s xml:id="echoid-s6721" xml:space="preserve">quadrato lateris AC, </s>
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