Clavius, Christoph, Geometria practica

Table of contents

< >
[291.] THEOR. 1. PROPOS. 1.
Page: 322 (292)
[292.] PROBL. 2. PROPOS. 2.
Page: 323 (293)
[293.] THEOR. 3. PROPOS. 3.
Page: 324 (294)
[294.] THEOR. 4. PROPOS. 4.
Page: 324 (294)
[295.] THEOR. 5. PROPOS. 5.
Page: 325 (295)
[296.] THEOR. 6. PROPOS. 6.
Page: 326 (296)
[297.] PROBL. 1. PROPOS. 7.
Page: 327 (297)
[298.] SCHOLIVM.
Page: 327 (297)
[299.] THEOR. 7. PROPOS. 8.
Page: 327 (297)
[300.] THEOR. 8. PROPOS. 9.
Page: 328 (298)
[301.] PROBL. 2. PROPOS. 10.
Page: 329 (299)
[302.] THEOR. 9. PROPOS. 11.
Page: 330 (300)
[303.] THEOR. 10. PROPOS. 12.
Page: 333 (303)
[304.] SCHOLIVM.
Page: 334 (304)
[305.] THEOR. 11. PROPOS. 13.
Page: 336 (306)
[306.] COROLLARIVM.
Page: 336 (306)
[307.] THEOR. 12. PROPOS. 14.
Page: 337 (307)
[308.] THEOR. 13. PROPOS. 15.
Page: 337 (307)
[309.] THEOR. 14. PROPOS. 16.
Page: 338 (308)
[310.] THEOR. 15. PROPOS. 17.
Page: 340 (310)
[311.] COROLLARIVM.
Page: 341 (311)
[312.] THEOR. 16. PROPOS. 18.
Page: 341 (311)
[313.] THEOR. 17. PROPOS. 19.
Page: 343 (313)
[314.] SCHOLIVM.
Page: 343 (313)
[315.] PROBL. 3. PROPOS. 20.
Page: 343 (313)
[316.] PROBL. 4. PROPOS. 21.
Page: 344 (314)
[317.] SCHOLIVM.
Page: 346 (316)
[318.] PROBL. 5. PROPOS. 22.
Page: 346 (316)
[319.] SCHOLIVM.
Page: 347 (317)
[320.] APPENDIX.
Page: 347 (317)
< >
page |< < (275) of 450 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div790" type="section" level="1" n="272">
          <p>
            <s xml:id="echoid-s12674" xml:space="preserve">
              <pb o="275" file="305" n="305" rhead="LIBER SEXTVS."/>
            ſcribemus, qua huiuſmodi medias inuenire poſsimus. </s>
            <s xml:id="echoid-s12675" xml:space="preserve">Propoſitis igitur primum
              <lb/>
            duobus numeris quibuſcunque 9. </s>
            <s xml:id="echoid-s12676" xml:space="preserve">& </s>
            <s xml:id="echoid-s12677" xml:space="preserve">25. </s>
            <s xml:id="echoid-s12678" xml:space="preserve">inter quos reperiendus ſit vnus medius
              <lb/>
            proportionalis; </s>
            <s xml:id="echoid-s12679" xml:space="preserve">ſi multiplicentur inter ſe, & </s>
            <s xml:id="echoid-s12680" xml:space="preserve">producti numeri 225. </s>
            <s xml:id="echoid-s12681" xml:space="preserve">radix qua-
              <lb/>
            drata eruatur 15. </s>
            <s xml:id="echoid-s12682" xml:space="preserve">vt in Arithmetica practica cap. </s>
            <s xml:id="echoid-s12683" xml:space="preserve">26. </s>
            <s xml:id="echoid-s12684" xml:space="preserve">docuimus: </s>
            <s xml:id="echoid-s12685" xml:space="preserve"> erit radix
              <note symbol="a" position="right" xlink:label="note-305-01" xlink:href="note-305-01a" xml:space="preserve">17. ſexti, vel
                <lb/>
              20 ſept.</note>
            quadrata medio loco proportionalis inter datos numeros, vt hic 9. </s>
            <s xml:id="echoid-s12686" xml:space="preserve">15. </s>
            <s xml:id="echoid-s12687" xml:space="preserve">25. </s>
            <s xml:id="echoid-s12688" xml:space="preserve">quip-
              <lb/>
            pe cum quadratum medij numeri æquale ſit rectangulo ſub extremis compre-
              <lb/>
            henſo. </s>
            <s xml:id="echoid-s12689" xml:space="preserve">Sic inter 5. </s>
            <s xml:id="echoid-s12690" xml:space="preserve">& </s>
            <s xml:id="echoid-s12691" xml:space="preserve">13. </s>
            <s xml:id="echoid-s12692" xml:space="preserve">medius proportionalis erit radix quadrata numeri 65.
              <lb/>
            </s>
            <s xml:id="echoid-s12693" xml:space="preserve">qui ex multiplicatione datorum numerorum gignitur, quæ radix paulò maior
              <lb/>
            eſt, quam 8 {1/17}. </s>
            <s xml:id="echoid-s12694" xml:space="preserve">& </s>
            <s xml:id="echoid-s12695" xml:space="preserve">paulò minor, quam 8 {1/16}.</s>
            <s xml:id="echoid-s12696" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12697" xml:space="preserve">
              <emph style="sc">Sint</emph>
            deinde duo numeri 2. </s>
            <s xml:id="echoid-s12698" xml:space="preserve">& </s>
            <s xml:id="echoid-s12699" xml:space="preserve">54. </s>
            <s xml:id="echoid-s12700" xml:space="preserve">inter quos inueniendi ſint duo medij
              <lb/>
            proportionales. </s>
            <s xml:id="echoid-s12701" xml:space="preserve">Multiplicetur quadratus minoris in maiorem. </s>
            <s xml:id="echoid-s12702" xml:space="preserve">Producti nam-
              <lb/>
            que numeri 216. </s>
            <s xml:id="echoid-s12703" xml:space="preserve">radix cubica 6. </s>
            <s xml:id="echoid-s12704" xml:space="preserve">erit primus medius iuxta minorem collocan-
              <lb/>
            dus. </s>
            <s xml:id="echoid-s12705" xml:space="preserve">Et ſi maioris quadratus ducatur in minorem, erit producti numeri 5832.
              <lb/>
            </s>
            <s xml:id="echoid-s12706" xml:space="preserve">radix cubica 18. </s>
            <s xml:id="echoid-s12707" xml:space="preserve">alter medius iuxta maiorem ſtatuendus, vt hic 2. </s>
            <s xml:id="echoid-s12708" xml:space="preserve">6. </s>
            <s xml:id="echoid-s12709" xml:space="preserve">18. </s>
            <s xml:id="echoid-s12710" xml:space="preserve">54. </s>
            <s xml:id="echoid-s12711" xml:space="preserve">Ra-
              <lb/>
            tio huius rei eſt, quod datis quatuor lineis continuè proportionalibus, paralle-
              <lb/>
            lepipedum ſub quadrato alterutrius extremarum, & </s>
            <s xml:id="echoid-s12712" xml:space="preserve">ſub altera extrema com-
              <lb/>
            prehenſum, æquale eſt cubo mediæ proportionalis, quæ priori extremo aſſum-
              <lb/>
            pto propinquior eſt, vt in ſequenti Lemmate demonſtrabimus. </s>
            <s xml:id="echoid-s12713" xml:space="preserve">Quoniam ve-
              <lb/>
            rò, vt in ſcholio propoſ. </s>
            <s xml:id="echoid-s12714" xml:space="preserve">19. </s>
            <s xml:id="echoid-s12715" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s12716" xml:space="preserve">8. </s>
            <s xml:id="echoid-s12717" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s12718" xml:space="preserve">oſtendimus, propoſitis hiſce tribus
              <lb/>
            numeris 2. </s>
            <s xml:id="echoid-s12719" xml:space="preserve">2. </s>
            <s xml:id="echoid-s12720" xml:space="preserve">54. </s>
            <s xml:id="echoid-s12721" xml:space="preserve">idem procreatur numerus, ſiue prius ducantur 2. </s>
            <s xml:id="echoid-s12722" xml:space="preserve">in 2. </s>
            <s xml:id="echoid-s12723" xml:space="preserve">deinde
              <lb/>
            productus 4. </s>
            <s xml:id="echoid-s12724" xml:space="preserve">in 54. </s>
            <s xml:id="echoid-s12725" xml:space="preserve">ſiue prius 2. </s>
            <s xml:id="echoid-s12726" xml:space="preserve">in 54. </s>
            <s xml:id="echoid-s12727" xml:space="preserve">deinde productus 108. </s>
            <s xml:id="echoid-s12728" xml:space="preserve">in 2. </s>
            <s xml:id="echoid-s12729" xml:space="preserve">Item datis hiſ-
              <lb/>
            ce tribus numeris 54. </s>
            <s xml:id="echoid-s12730" xml:space="preserve">54. </s>
            <s xml:id="echoid-s12731" xml:space="preserve">2. </s>
            <s xml:id="echoid-s12732" xml:space="preserve">idem numerus gignitur, ſiue prius ducantur 54. </s>
            <s xml:id="echoid-s12733" xml:space="preserve">in
              <lb/>
            54. </s>
            <s xml:id="echoid-s12734" xml:space="preserve">deinde productus 2916. </s>
            <s xml:id="echoid-s12735" xml:space="preserve">in 2. </s>
            <s xml:id="echoid-s12736" xml:space="preserve">ſiue prius 54. </s>
            <s xml:id="echoid-s12737" xml:space="preserve">in 2. </s>
            <s xml:id="echoid-s12738" xml:space="preserve">deinde productus 108. </s>
            <s xml:id="echoid-s12739" xml:space="preserve">in
              <lb/>
            54. </s>
            <s xml:id="echoid-s12740" xml:space="preserve">manifeſto colligitur, ſi minor 2. </s>
            <s xml:id="echoid-s12741" xml:space="preserve">ducatur in maiorem 54. </s>
            <s xml:id="echoid-s12742" xml:space="preserve">& </s>
            <s xml:id="echoid-s12743" xml:space="preserve">productus 108. </s>
            <s xml:id="echoid-s12744" xml:space="preserve">
              <lb/>
            in minorem 2. </s>
            <s xml:id="echoid-s12745" xml:space="preserve">produci quoque cubum medij proportionalis iuxta minorem
              <lb/>
            conſtituendi: </s>
            <s xml:id="echoid-s12746" xml:space="preserve">Item ſi maior 54. </s>
            <s xml:id="echoid-s12747" xml:space="preserve">ducatur in minorem 2. </s>
            <s xml:id="echoid-s12748" xml:space="preserve">& </s>
            <s xml:id="echoid-s12749" xml:space="preserve">productus 108. </s>
            <s xml:id="echoid-s12750" xml:space="preserve">in ma-
              <lb/>
            iorem 54. </s>
            <s xml:id="echoid-s12751" xml:space="preserve">pro creari cubum medij proportionalis iuxta maiorem ſcribendi. </s>
            <s xml:id="echoid-s12752" xml:space="preserve">Sic
              <lb/>
            inter 4 & </s>
            <s xml:id="echoid-s12753" xml:space="preserve">100. </s>
            <s xml:id="echoid-s12754" xml:space="preserve">erunt duo medij proportionales, Radix cubica numeri 1600. </s>
            <s xml:id="echoid-s12755" xml:space="preserve">& </s>
            <s xml:id="echoid-s12756" xml:space="preserve">
              <lb/>
            Radix cubica numeri 40000. </s>
            <s xml:id="echoid-s12757" xml:space="preserve">Cæterum inuento altero mediorum numero-
              <lb/>
            rum, reperietur alter etiam, ſi inuentus per extremum remotiorem multiplice-
              <lb/>
            tur & </s>
            <s xml:id="echoid-s12758" xml:space="preserve">producti numeri radix quadrata capiatur. </s>
            <s xml:id="echoid-s12759" xml:space="preserve">Vt in dato exemplo 2. </s>
            <s xml:id="echoid-s12760" xml:space="preserve">6. </s>
            <s xml:id="echoid-s12761" xml:space="preserve">18. </s>
            <s xml:id="echoid-s12762" xml:space="preserve">
              <lb/>
            54. </s>
            <s xml:id="echoid-s12763" xml:space="preserve">ſi medius inuentus 6. </s>
            <s xml:id="echoid-s12764" xml:space="preserve">ducatur in 54. </s>
            <s xml:id="echoid-s12765" xml:space="preserve">erit producti numeri 324. </s>
            <s xml:id="echoid-s12766" xml:space="preserve">radix quadra-
              <lb/>
            ta 18. </s>
            <s xml:id="echoid-s12767" xml:space="preserve">alter medius: </s>
            <s xml:id="echoid-s12768" xml:space="preserve">Item inuentus medius 18. </s>
            <s xml:id="echoid-s12769" xml:space="preserve">ſi multiplicetur per 2. </s>
            <s xml:id="echoid-s12770" xml:space="preserve">erit pro ducti
              <lb/>
            numeri 36. </s>
            <s xml:id="echoid-s12771" xml:space="preserve">radix quadrata 6. </s>
            <s xml:id="echoid-s12772" xml:space="preserve">alter medius: </s>
            <s xml:id="echoid-s12773" xml:space="preserve">propterea quod tam 2. </s>
            <s xml:id="echoid-s12774" xml:space="preserve">6. </s>
            <s xml:id="echoid-s12775" xml:space="preserve">18. </s>
            <s xml:id="echoid-s12776" xml:space="preserve">quam 6. </s>
            <s xml:id="echoid-s12777" xml:space="preserve">
              <lb/>
            18. </s>
            <s xml:id="echoid-s12778" xml:space="preserve">54. </s>
            <s xml:id="echoid-s12779" xml:space="preserve">ſunt tres continuè proportionales.</s>
            <s xml:id="echoid-s12780" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div792" type="section" level="1" n="273">
          <head xml:id="echoid-head298" xml:space="preserve">LEMMA.</head>
          <p>
            <s xml:id="echoid-s12781" xml:space="preserve">SI ſint quatuor lineæ continuè proportionales: </s>
            <s xml:id="echoid-s12782" xml:space="preserve">parallelepipedum ſub
              <lb/>
            quadrato alterutrius extremarum, & </s>
            <s xml:id="echoid-s12783" xml:space="preserve">altera extrema comprehenſum,
              <lb/>
            æquale eſt cubo mediæ proportionalis, quæ priori extremæ aſſumptæ
              <lb/>
            propinquior eſt.</s>
            <s xml:id="echoid-s12784" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12785" xml:space="preserve">
              <emph style="sc">Repetatvr</emph>
            figura propoſ. </s>
            <s xml:id="echoid-s12786" xml:space="preserve">17. </s>
            <s xml:id="echoid-s12787" xml:space="preserve">in qua lineæ quatuor continuè propor-
              <lb/>
            tionales ſunt A, E, F, D. </s>
            <s xml:id="echoid-s12788" xml:space="preserve">Dico parallelepipedum ſub quadrato extremæ A, &</s>
            <s xml:id="echoid-s12789" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
Impressum