Clavius, Christoph, Geometria practica

Table of contents

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[411.] Tabula quadratorum, & Cuborum.
Page: 410 (382)
[412.] Tabula Quadratorum, & cuborum.
Page: 411 (383)
[413.] Tabula Quadratorum, & Cuborum.
Page: 412 (384)
[414.] Tabula Quadratorum, & cuborum.
Page: 413 (385)
[415.] Tabula Quadratorum, & Cuborum.
Page: 414 (386)
[416.] DE DIFFERENTHS QVADRATO-rum & cuborum, & de continuationeta-bulæ eorundem.
Page: 415 (387)
[417.] VSVS PRÆCEDENTIS TABVLÆ quadratorum, & cuborum in radicibus extrahendis.
Page: 419 (391)
[418.] FINIS LIBRI OCTAVI.
Page: 420 (392)
[419.] ALPHABETICVS RERVM. AC VERBORVM.
Page: 421
[420.] ERRATA.
Page: 445
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          <pb o="387" file="415" n="415" rhead="LIBER OCTAVVS."/>
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        <div xml:id="echoid-div1130" type="section" level="1" n="416">
          <head xml:id="echoid-head443" xml:space="preserve">DE DIFFERENTHS QVADRATO-
            <lb/>
          rum & cuborum, & de continuationeta-
            <lb/>
          bulæ eorundem.</head>
          <p>
            <s xml:id="echoid-s17817" xml:space="preserve">
              <emph style="sc">Qvoniam</emph>
            quadrati numeri creantur per continuam additionem nume-
              <lb/>
              <note position="right" xlink:label="note-415-01" xlink:href="note-415-01a" xml:space="preserve">Differentiæ
                <lb/>
              quadratorum</note>
            rorum imparium, vt Arithmetici demonſtrant: </s>
            <s xml:id="echoid-s17818" xml:space="preserve">fit vt differentia inter quemlibet
              <lb/>
            quadratum, & </s>
            <s xml:id="echoid-s17819" xml:space="preserve">proximè inſequen-
              <lb/>
              <note position="right" xlink:label="note-415-02" xlink:href="note-415-02a" xml:space="preserve">
                <lb/>
              Numeri \\ impares. # Quadra- \\ ti. # Radices.
                <lb/>
              1 # 1 # 1
                <lb/>
              3 # 4 # 2
                <lb/>
              5 # 9 # 3
                <lb/>
              7 # 16 # 4
                <lb/>
              9 # 25 # 5
                <lb/>
              11 # 36 # 6
                <lb/>
              13 # 49 # 7
                <lb/>
              15 # 64 # 8
                <lb/>
              17 # 81 # 9
                <lb/>
              19 # 100 # 10
                <lb/>
              21 # 121 # 11
                <lb/>
              23 # 144 # 12
                <lb/>
              </note>
            tem ſit duplaradicis minoris, addi-
              <lb/>
            ta inſuper vnitate. </s>
            <s xml:id="echoid-s17820" xml:space="preserve">Itaque duobus
              <lb/>
            modis tabula quadratorum com-
              <lb/>
            poni poteſt, & </s>
            <s xml:id="echoid-s17821" xml:space="preserve">continuari. </s>
            <s xml:id="echoid-s17822" xml:space="preserve">Vno
              <lb/>
              <note position="right" xlink:label="note-415-03" xlink:href="note-415-03a" xml:space="preserve">Compoſitio ta
                <unsure/>
              -
                <lb/>
              bulæ quadra-
                <lb/>
              torum.</note>
            modo, ſi omnes numeri impares
              <lb/>
            ordine ponantur, initio ſumpto ab
              <lb/>
            1. </s>
            <s xml:id="echoid-s17823" xml:space="preserve">Nam 1. </s>
            <s xml:id="echoid-s17824" xml:space="preserve">dat primum quadratum
              <lb/>
            1. </s>
            <s xml:id="echoid-s17825" xml:space="preserve">Et ex 1. </s>
            <s xml:id="echoid-s17826" xml:space="preserve">& </s>
            <s xml:id="echoid-s17827" xml:space="preserve">3. </s>
            <s xml:id="echoid-s17828" xml:space="preserve">fit ſecundus 4. </s>
            <s xml:id="echoid-s17829" xml:space="preserve">cui ſi
              <lb/>
            addatur ſequens impar 5. </s>
            <s xml:id="echoid-s17830" xml:space="preserve">fit tertius
              <lb/>
            9. </s>
            <s xml:id="echoid-s17831" xml:space="preserve">& </s>
            <s xml:id="echoid-s17832" xml:space="preserve">ſi addatur impar ſequens 7. </s>
            <s xml:id="echoid-s17833" xml:space="preserve">fit
              <lb/>
            quartus quadratus 16. </s>
            <s xml:id="echoid-s17834" xml:space="preserve">at que ita de-
              <lb/>
            inceps. </s>
            <s xml:id="echoid-s17835" xml:space="preserve">Habet autem quilibet qua-
              <lb/>
            dratusra dicem tot vnitatum, quot
              <lb/>
            numeri impares ipſum conficiunt.
              <lb/>
            </s>
            <s xml:id="echoid-s17836" xml:space="preserve">Vt quia ſolus impar 1. </s>
            <s xml:id="echoid-s17837" xml:space="preserve">dat primum
              <lb/>
            quadratum 1. </s>
            <s xml:id="echoid-s17838" xml:space="preserve">propterea eius radix
              <lb/>
            eſt 1. </s>
            <s xml:id="echoid-s17839" xml:space="preserve">Deinde quia duo impares 1. </s>
            <s xml:id="echoid-s17840" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s17841" xml:space="preserve">3. </s>
            <s xml:id="echoid-s17842" xml:space="preserve">conficiunt ſecundum quadra-
              <lb/>
            tum 4. </s>
            <s xml:id="echoid-s17843" xml:space="preserve">erit eius radix 2. </s>
            <s xml:id="echoid-s17844" xml:space="preserve">Sic quia
              <lb/>
            duo decim numeri impares 1. </s>
            <s xml:id="echoid-s17845" xml:space="preserve">3. </s>
            <s xml:id="echoid-s17846" xml:space="preserve">5. </s>
            <s xml:id="echoid-s17847" xml:space="preserve">7. </s>
            <s xml:id="echoid-s17848" xml:space="preserve">
              <lb/>
            9. </s>
            <s xml:id="echoid-s17849" xml:space="preserve">11. </s>
            <s xml:id="echoid-s17850" xml:space="preserve">13. </s>
            <s xml:id="echoid-s17851" xml:space="preserve">15. </s>
            <s xml:id="echoid-s17852" xml:space="preserve">17. </s>
            <s xml:id="echoid-s17853" xml:space="preserve">19. </s>
            <s xml:id="echoid-s17854" xml:space="preserve">21. </s>
            <s xml:id="echoid-s17855" xml:space="preserve">23. </s>
            <s xml:id="echoid-s17856" xml:space="preserve">compo-
              <lb/>
            nunt quadratum 144. </s>
            <s xml:id="echoid-s17857" xml:space="preserve">erit eius ra-
              <lb/>
            dix 12. </s>
            <s xml:id="echoid-s17858" xml:space="preserve">& </s>
            <s xml:id="echoid-s17859" xml:space="preserve">ſic de cæteris. </s>
            <s xml:id="echoid-s17860" xml:space="preserve">Atq; </s>
            <s xml:id="echoid-s17861" xml:space="preserve">in hũc
              <lb/>
            modum ſi ſemper ſequens nume-
              <lb/>
            us impar adiiciatur ad quadratum præcedentem, conflatur ſequens numerus
              <lb/>
            quadratus, continuabiturque tabula in infinitum: </s>
            <s xml:id="echoid-s17862" xml:space="preserve">ſitamen prius ſeries nume-
              <lb/>
            rorum imparium continuetur. </s>
            <s xml:id="echoid-s17863" xml:space="preserve">Radices ſerie numerorum naturali progre-
              <lb/>
            diuntur.</s>
            <s xml:id="echoid-s17864" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s17865" xml:space="preserve">
              <emph style="sc">Alio</emph>
            modo condi poterit tabula quadratorum, & </s>
            <s xml:id="echoid-s17866" xml:space="preserve">in infinitum continua-
              <lb/>
            ri, ſine numerorum imparium ſerie, ſi omnes radices ponantur ordine, vt in tabu-
              <lb/>
            la vides. </s>
            <s xml:id="echoid-s17867" xml:space="preserve">Cum enim primus quadratus ſit 1. </s>
            <s xml:id="echoid-s17868" xml:space="preserve">cuius radix 1. </s>
            <s xml:id="echoid-s17869" xml:space="preserve">ſi hæc radix dupli-
              <lb/>
            cata, addita inſuper 1. </s>
            <s xml:id="echoid-s17870" xml:space="preserve">addatur primo quadrato 1. </s>
            <s xml:id="echoid-s17871" xml:space="preserve">fit ſecundus 4. </s>
            <s xml:id="echoid-s17872" xml:space="preserve">cuius ra-
              <lb/>
            dix 2. </s>
            <s xml:id="echoid-s17873" xml:space="preserve">Hæc duplicata, & </s>
            <s xml:id="echoid-s17874" xml:space="preserve">inſuper addita 1. </s>
            <s xml:id="echoid-s17875" xml:space="preserve">ſi adiiciatur ſecundo quadra-
              <lb/>
            to 4. </s>
            <s xml:id="echoid-s17876" xml:space="preserve">fit tertius 9. </s>
            <s xml:id="echoid-s17877" xml:space="preserve">cuius radix 3. </s>
            <s xml:id="echoid-s17878" xml:space="preserve">quæ duplicata, & </s>
            <s xml:id="echoid-s17879" xml:space="preserve">inſuper addita 1. </s>
            <s xml:id="echoid-s17880" xml:space="preserve">facit 7.
              <lb/>
            </s>
            <s xml:id="echoid-s17881" xml:space="preserve">Si igitur addantur 7. </s>
            <s xml:id="echoid-s17882" xml:space="preserve">ad quadratum 9. </s>
            <s xml:id="echoid-s17883" xml:space="preserve">fit quartus quadratus 16. </s>
            <s xml:id="echoid-s17884" xml:space="preserve">& </s>
            <s xml:id="echoid-s17885" xml:space="preserve">ſic in infi-
              <lb/>
            nitum.</s>
            <s xml:id="echoid-s17886" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s17887" xml:space="preserve">
              <emph style="sc">Nvmeri</emph>
            autem cubi gignuntur quoque ex additione numerorum impa-
              <lb/>
            rium, hoc modo. </s>
            <s xml:id="echoid-s17888" xml:space="preserve">Deſcripta ſerie imparium numerorum ab 1. </s>
            <s xml:id="echoid-s17889" xml:space="preserve">incipientium, </s>
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