Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div979" type="section" level="1" n="349">
          <p>
            <s xml:id="echoid-s15903" xml:space="preserve">
              <pb o="342" file="370" n="370" rhead="GEOMETR. PRACT."/>
            numeratum, quicunqueille ſit; </s>
            <s xml:id="echoid-s15904" xml:space="preserve">ac poſtremo ex parte aliquota ſummæ, cuius de-
              <lb/>
            nominator eſt numerus à te electus, auferre ſimilem partem ex productis ſingu-
              <lb/>
            lorum, hoc eſt, ipſos conceptos numeros: </s>
            <s xml:id="echoid-s15905" xml:space="preserve">reliquus numerus cuiuſque erit ſi-
              <lb/>
            milis pars numeri adiecti.</s>
            <s xml:id="echoid-s15906" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15907" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            ſi malueris diuerſos numeros, dic vt ſecundus ſuum reſiduum du-
              <lb/>
            plicet, & </s>
            <s xml:id="echoid-s15908" xml:space="preserve">tertius triplicet, &</s>
            <s xml:id="echoid-s15909" xml:space="preserve">c. </s>
            <s xml:id="echoid-s15910" xml:space="preserve">Ita enim coniicies, primi reſiduum eſſe illam par-
              <lb/>
            tem aliquotam numeriadiecti: </s>
            <s xml:id="echoid-s15911" xml:space="preserve">ſecundum verò habere duplum illius, & </s>
            <s xml:id="echoid-s15912" xml:space="preserve">ter-
              <lb/>
            tium triplum, &</s>
            <s xml:id="echoid-s15913" xml:space="preserve">c. </s>
            <s xml:id="echoid-s15914" xml:space="preserve">Vbivides eos reſiduum illud per quoſcunque numeros poſ-
              <lb/>
            ſe multiplicare, dummodo memor ſis in coniiciendis numeris, per quos nume-
              <lb/>
            ros factæ ſunt multiplicationes.</s>
            <s xml:id="echoid-s15915" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div981" type="section" level="1" n="350">
          <head xml:id="echoid-head377" xml:space="preserve">PROBL. 5. PROPOS. 9.</head>
          <p>
            <s xml:id="echoid-s15916" xml:space="preserve">DATVM numerum quadratum in quotuis quadratos numeros par-
              <lb/>
            tiri.</s>
            <s xml:id="echoid-s15917" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15918" xml:space="preserve">
              <emph style="sc">Qvamvis</emph>
            problema hoc videatur ferè impoſsibile: </s>
            <s xml:id="echoid-s15919" xml:space="preserve">(qui enim fieri po-
              <lb/>
            teſt, dicet aliquis, vt quilibet numerus quadratus diuidi poſsit in quotlibet nu-
              <lb/>
            meros, qui omnes ſint quadrati?) </s>
            <s xml:id="echoid-s15920" xml:space="preserve">ſolutio tamen eius non eſt difficilis. </s>
            <s xml:id="echoid-s15921" xml:space="preserve">Sit igitur
              <lb/>
            quadratus numerus datus 36. </s>
            <s xml:id="echoid-s15922" xml:space="preserve">diuidendus in 5. </s>
            <s xml:id="echoid-s15923" xml:space="preserve">numeros quadratos. </s>
            <s xml:id="echoid-s15924" xml:space="preserve">Per ea, quæ
              <lb/>
            ad propoſ. </s>
            <s xml:id="echoid-s15925" xml:space="preserve">47. </s>
            <s xml:id="echoid-s15926" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s15927" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15928" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s15929" xml:space="preserve">ſcripſimus, reperiantur tres numeri, quorummaio-
              <lb/>
            ris quadratus reliquorum quadratis ſit æqualis, nimirum 5. </s>
            <s xml:id="echoid-s15930" xml:space="preserve">4. </s>
            <s xml:id="echoid-s15931" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15932" xml:space="preserve">Deinde dic: </s>
            <s xml:id="echoid-s15933" xml:space="preserve">ſi
              <lb/>
            5. </s>
            <s xml:id="echoid-s15934" xml:space="preserve">dant 4. </s>
            <s xml:id="echoid-s15935" xml:space="preserve">quid dabunt 6. </s>
            <s xml:id="echoid-s15936" xml:space="preserve">quadrata videlicet radix dati quadrati? </s>
            <s xml:id="echoid-s15937" xml:space="preserve">Item ſi 5. </s>
            <s xml:id="echoid-s15938" xml:space="preserve">dant
              <lb/>
            3. </s>
            <s xml:id="echoid-s15939" xml:space="preserve">quid dabunt 6? </s>
            <s xml:id="echoid-s15940" xml:space="preserve">Inuenieſq; </s>
            <s xml:id="echoid-s15941" xml:space="preserve">{24/5}. </s>
            <s xml:id="echoid-s15942" xml:space="preserve">& </s>
            <s xml:id="echoid-s15943" xml:space="preserve">{18/5}. </s>
            <s xml:id="echoid-s15944" xml:space="preserve">hoc eſt, 4 {4/5}. </s>
            <s xml:id="echoid-s15945" xml:space="preserve">& </s>
            <s xml:id="echoid-s15946" xml:space="preserve">3 {3/5}. </s>
            <s xml:id="echoid-s15947" xml:space="preserve">radices duorum qua-
              <lb/>
            dratorum quadrato 36. </s>
            <s xml:id="echoid-s15948" xml:space="preserve">dato æqualium. </s>
            <s xml:id="echoid-s15949" xml:space="preserve">Nam cum ita ſe habeat radix 6. </s>
            <s xml:id="echoid-s15950" xml:space="preserve">ad in-
              <lb/>
            uentos duos numeros, vt 5. </s>
            <s xml:id="echoid-s15951" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s15952" xml:space="preserve">& </s>
            <s xml:id="echoid-s15953" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15954" xml:space="preserve">ex conſtructione: </s>
            <s xml:id="echoid-s15955" xml:space="preserve">fiet ex lateribus 6. </s>
            <s xml:id="echoid-s15956" xml:space="preserve">4 {4/5}.
              <lb/>
            </s>
            <s xml:id="echoid-s15957" xml:space="preserve">3 {3/5}. </s>
            <s xml:id="echoid-s15958" xml:space="preserve">triangulum rectangulum, ſimile nimirum triangulo rectangulo ex lateribus
              <lb/>
            5. </s>
            <s xml:id="echoid-s15959" xml:space="preserve">4. </s>
            <s xml:id="echoid-s15960" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15961" xml:space="preserve">conſtructo. </s>
            <s xml:id="echoid-s15962" xml:space="preserve"> Igitur quadrati ex 4 {4/5}. </s>
            <s xml:id="echoid-s15963" xml:space="preserve">& </s>
            <s xml:id="echoid-s15964" xml:space="preserve">{3/5}. </s>
            <s xml:id="echoid-s15965" xml:space="preserve">æquales erunt quadrato
              <note symbol="a" position="left" xlink:label="note-370-01" xlink:href="note-370-01a" xml:space="preserve">47. primi.</note>
            cis 6. </s>
            <s xml:id="echoid-s15966" xml:space="preserve">dato. </s>
            <s xml:id="echoid-s15967" xml:space="preserve">Rurſus ſi fiat, vt 5. </s>
            <s xml:id="echoid-s15968" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s15969" xml:space="preserve">& </s>
            <s xml:id="echoid-s15970" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s15971" xml:space="preserve">ita 3 {3/5}. </s>
            <s xml:id="echoid-s15972" xml:space="preserve">ad aliud; </s>
            <s xml:id="echoid-s15973" xml:space="preserve">(ſumendo mino-
              <lb/>
            rem radicem inuentam, ne coincidamus cum aliqua præcedente radiceiam in-
              <lb/>
            uenta) inueniẽtur alij duo numeri, quorũ quadrati æquales ſint quadrato radi-
              <lb/>
            cis 3 {3/5}. </s>
            <s xml:id="echoid-s15974" xml:space="preserve">nimirum 2 {22/25}. </s>
            <s xml:id="echoid-s15975" xml:space="preserve">& </s>
            <s xml:id="echoid-s15976" xml:space="preserve">2 {4/25}. </s>
            <s xml:id="echoid-s15977" xml:space="preserve">Atqueita iam (relicta radice 3 {3/5}.) </s>
            <s xml:id="echoid-s15978" xml:space="preserve">inuentæ erunt
              <lb/>
            tres radices 4 {4/5}. </s>
            <s xml:id="echoid-s15979" xml:space="preserve">2 {22/25}. </s>
            <s xml:id="echoid-s15980" xml:space="preserve">2 {4/25}. </s>
            <s xml:id="echoid-s15981" xml:space="preserve">quarum quadrati æquales erunt quadrato 36. </s>
            <s xml:id="echoid-s15982" xml:space="preserve">pro-
              <lb/>
            poſito. </s>
            <s xml:id="echoid-s15983" xml:space="preserve">Eodemmodo, ſi fiat, vt 5. </s>
            <s xml:id="echoid-s15984" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s15985" xml:space="preserve">& </s>
            <s xml:id="echoid-s15986" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s15987" xml:space="preserve">ita 2 {4/25}. </s>
            <s xml:id="echoid-s15988" xml:space="preserve">ad aliud, reperientur
              <lb/>
            duæ aliæ radices 1 {91/125}. </s>
            <s xml:id="echoid-s15989" xml:space="preserve">& </s>
            <s xml:id="echoid-s15990" xml:space="preserve">1 {37/125}. </s>
            <s xml:id="echoid-s15991" xml:space="preserve">Quare (relicta radice 2 {4/25}. </s>
            <s xml:id="echoid-s15992" xml:space="preserve">cuius loco duas inue-
              <lb/>
            nimus) inuentæ iam erunt quatuor radices 4 {4/5}. </s>
            <s xml:id="echoid-s15993" xml:space="preserve">2 {22/25}. </s>
            <s xml:id="echoid-s15994" xml:space="preserve">1 {91/125}. </s>
            <s xml:id="echoid-s15995" xml:space="preserve">& </s>
            <s xml:id="echoid-s15996" xml:space="preserve">1 {37/125}. </s>
            <s xml:id="echoid-s15997" xml:space="preserve">quarumnu-
              <lb/>
            meri quadrati quadrato 36. </s>
            <s xml:id="echoid-s15998" xml:space="preserve">æquales erunt. </s>
            <s xml:id="echoid-s15999" xml:space="preserve">Denique ſi rurſus fiat vt 5. </s>
            <s xml:id="echoid-s16000" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s16001" xml:space="preserve">& </s>
            <s xml:id="echoid-s16002" xml:space="preserve">
              <lb/>
            ad 3. </s>
            <s xml:id="echoid-s16003" xml:space="preserve">ita 1 {37/125}. </s>
            <s xml:id="echoid-s16004" xml:space="preserve">minor radix inuenta ad aliud, reperientur duæ aliæ radices 1 {23/625}.
              <lb/>
            </s>
            <s xml:id="echoid-s16005" xml:space="preserve">& </s>
            <s xml:id="echoid-s16006" xml:space="preserve">{486/625}. </s>
            <s xml:id="echoid-s16007" xml:space="preserve">Quocirca (relicta radice 1 {37/@25}. </s>
            <s xml:id="echoid-s16008" xml:space="preserve">pro qua duas proximas inuenimus) in-
              <lb/>
            uentæ erunt quinque radices 4 {4/5}. </s>
            <s xml:id="echoid-s16009" xml:space="preserve">2 {22/25}. </s>
            <s xml:id="echoid-s16010" xml:space="preserve">1 {91/125}. </s>
            <s xml:id="echoid-s16011" xml:space="preserve">1 {23/625}. </s>
            <s xml:id="echoid-s16012" xml:space="preserve">& </s>
            <s xml:id="echoid-s16013" xml:space="preserve">{486/625}. </s>
            <s xml:id="echoid-s16014" xml:space="preserve">quarum quadrati nu-
              <lb/>
            meri 23 {1/25}. </s>
            <s xml:id="echoid-s16015" xml:space="preserve">8 {184/625}. </s>
            <s xml:id="echoid-s16016" xml:space="preserve">2 {1@406/35625}. </s>
            <s xml:id="echoid-s16017" xml:space="preserve">1 {29279/190625}. </s>
            <s xml:id="echoid-s16018" xml:space="preserve">& </s>
            <s xml:id="echoid-s16019" xml:space="preserve">{236196/39@625}. </s>
            <s xml:id="echoid-s16020" xml:space="preserve">conficiunt datum quadratum 36. </s>
            <s xml:id="echoid-s16021" xml:space="preserve">At-
              <lb/>
            que in hunc modum plures quadrati inueniri poterunt æquales numero
              <lb/>
            36. </s>
            <s xml:id="echoid-s16022" xml:space="preserve">ſi nimirum fiat;</s>
            <s xml:id="echoid-s16023" xml:space="preserve">
              <unsure/>
            vt 5. </s>
            <s xml:id="echoid-s16024" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s16025" xml:space="preserve">& </s>
            <s xml:id="echoid-s16026" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s16027" xml:space="preserve">ita vltima radix inuenta {486/625}. </s>
            <s xml:id="echoid-s16028" xml:space="preserve">
              <lb/>
            quæ minima eſt, ad aliud, &</s>
            <s xml:id="echoid-s16029" xml:space="preserve">c.</s>
            <s xml:id="echoid-s16030" xml:space="preserve"/>
          </p>
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