Clavius, Christoph, Geometria practica

Table of Notes

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            <s xml:id="echoid-s15537" xml:space="preserve">
              <emph style="sc">Hæc</emph>
            ergo ſunt tria lemmata, quæ Cardanus præmittit: </s>
            <s xml:id="echoid-s15538" xml:space="preserve">quibus adiungit
              <lb/>
              <note position="right" xlink:label="note-363-01" xlink:href="note-363-01a" xml:space="preserve">Principium
                <lb/>
              Cardani.</note>
            hoc poſtulatum, ſine principium. </s>
            <s xml:id="echoid-s15539" xml:space="preserve">Cum arcus quilibet maior ſit quotcunque
              <lb/>
            rectis in eo ſubtenſis ſimul ſumptis, & </s>
            <s xml:id="echoid-s15540" xml:space="preserve">quo plures ſubtenſæ fuerint, eo minori
              <lb/>
            exceſſu arcus illas ſuperet: </s>
            <s xml:id="echoid-s15541" xml:space="preserve">fieri poteſt, vt tot ſubtenſæ duci poſsint, ita vt ex-
              <lb/>
            ceſſus, quo arcus illas ſuperet, minor ſit quauis recta propoſita. </s>
            <s xml:id="echoid-s15542" xml:space="preserve">Hoc princi-
              <lb/>
            pium videtur eſſe manifeſtum, cumtam paruus arcus poſsit accipi, vt eius chor-
              <lb/>
            da illi ferè æqualis ſit; </s>
            <s xml:id="echoid-s15543" xml:space="preserve">adeò vt ſenſus nullam percipere poſsit inter arcum, & </s>
            <s xml:id="echoid-s15544" xml:space="preserve">
              <lb/>
            chordam differentiam. </s>
            <s xml:id="echoid-s15545" xml:space="preserve">A poſteriori tamen illud confirmari poteſt per nume-
              <lb/>
            ros. </s>
            <s xml:id="echoid-s15546" xml:space="preserve">Poſita enim proportione circumferentiæ ad diametrum fermè 31415926.
              <lb/>
            </s>
            <s xml:id="echoid-s15547" xml:space="preserve">ad 10000000. </s>
            <s xml:id="echoid-s15548" xml:space="preserve">vt lib. </s>
            <s xml:id="echoid-s15549" xml:space="preserve">4. </s>
            <s xml:id="echoid-s15550" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s15551" xml:space="preserve">7. </s>
            <s xml:id="echoid-s15552" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s15553" xml:space="preserve">5. </s>
            <s xml:id="echoid-s15554" xml:space="preserve">ex probatis auctoribus retulimus, depre-
              <lb/>
            hendemus, ſi arcus propoſitus continuè ſecetur bifariam per rectas ſubtenſas,
              <lb/>
            ſemper à præ cedenti exceſſu plus dimidio auferri; </s>
            <s xml:id="echoid-s15555" xml:space="preserve"> ac proinde tandem
              <note symbol="a" position="right" xlink:label="note-363-02" xlink:href="note-363-02a" xml:space="preserve">1. decimi.</note>
            qui exceſſum omni quantitate minorem. </s>
            <s xml:id="echoid-s15556" xml:space="preserve">Nam arcus verbi gratia graduum 4.
              <lb/>
            </s>
            <s xml:id="echoid-s15557" xml:space="preserve">erit 698131. </s>
            <s xml:id="echoid-s15558" xml:space="preserve">& </s>
            <s xml:id="echoid-s15559" xml:space="preserve">eius chorda ex tabula ſinuum eruta 697990. </s>
            <s xml:id="echoid-s15560" xml:space="preserve">ita vt arcus chor-
              <lb/>
            dam ſuperet hoc numero 141. </s>
            <s xml:id="echoid-s15561" xml:space="preserve">Summa deinde duarum chordarum graduum 2. </s>
            <s xml:id="echoid-s15562" xml:space="preserve">
              <lb/>
            erit 698096. </s>
            <s xml:id="echoid-s15563" xml:space="preserve">quæ ſuperatur ab eodem arcu grad. </s>
            <s xml:id="echoid-s15564" xml:space="preserve">4. </s>
            <s xml:id="echoid-s15565" xml:space="preserve">numero hoc 35. </s>
            <s xml:id="echoid-s15566" xml:space="preserve">qui minor
              <lb/>
            eſt ſemiſſe præcedentis exceſſus 141. </s>
            <s xml:id="echoid-s15567" xml:space="preserve">ac proinde plus dimidio ab eo ablatum
              <lb/>
            erit. </s>
            <s xml:id="echoid-s15568" xml:space="preserve">Rurſus ſumma quatuor chordarum gradus 1. </s>
            <s xml:id="echoid-s15569" xml:space="preserve">erit 698120. </s>
            <s xml:id="echoid-s15570" xml:space="preserve">quam idem ar-
              <lb/>
            cus 4. </s>
            <s xml:id="echoid-s15571" xml:space="preserve">graduum ſuperat hoc numero 11. </s>
            <s xml:id="echoid-s15572" xml:space="preserve">qui etiam minor eſt ſemiſſe proximi ex-
              <lb/>
            ceſſus 35. </s>
            <s xml:id="echoid-s15573" xml:space="preserve">Item ſumma 8. </s>
            <s xml:id="echoid-s15574" xml:space="preserve">chordarum, quarum quælibet 30. </s>
            <s xml:id="echoid-s15575" xml:space="preserve">minutis debetur, erit
              <lb/>
            698128. </s>
            <s xml:id="echoid-s15576" xml:space="preserve">exceſſus autem inter eam, & </s>
            <s xml:id="echoid-s15577" xml:space="preserve">eundem arcum 4. </s>
            <s xml:id="echoid-s15578" xml:space="preserve">graduum, numerus 3. </s>
            <s xml:id="echoid-s15579" xml:space="preserve">
              <lb/>
            qui minor quoque eſt, quam ſemiſsis proximi exceſſus 11. </s>
            <s xml:id="echoid-s15580" xml:space="preserve">& </s>
            <s xml:id="echoid-s15581" xml:space="preserve">ſic deinceps. </s>
            <s xml:id="echoid-s15582" xml:space="preserve">
              <lb/>
            Scio confirmationem hanc propoſiti principii non eſſe demonſtratiuam, cum
              <lb/>
            prop ortio circumferentiæ ad diametrum colligatur ex eo, quod demonſtrare
              <lb/>
            conamur, nimirum figuram circulo circumſcriptam habere maiorem ambitum
              <lb/>
            ambitu circuli: </s>
            <s xml:id="echoid-s15583" xml:space="preserve">eam tamen probabilem eſſe, nemo dubitabit, cum vix credi@-
              <lb/>
            le videatur, (ſi illa proportio longè à vero abeſſet) exceſſus illos paulatim ita
              <lb/>
            minui, vt ſemper minor numerus ſemiſſe præcedentis exceſſus relin quatur; </s>
            <s xml:id="echoid-s15584" xml:space="preserve">adeò
              <lb/>
            vt tandem nulla ferè differentia inter arcum, & </s>
            <s xml:id="echoid-s15585" xml:space="preserve">ſummam chordarum ſubtenſa-
              <lb/>
            rum rep eriatur.</s>
            <s xml:id="echoid-s15586" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s15587" xml:space="preserve">
              <emph style="sc">His</emph>
            præmiſsis, tangant duæ rectæ AB, AL, arcum BCL. </s>
            <s xml:id="echoid-s15588" xml:space="preserve">Dico eas eſſe ma-
              <lb/>
              <note position="right" xlink:label="note-363-03" xlink:href="note-363-03a" xml:space="preserve">Demonſtra-
                <lb/>
              tio Cardani</note>
            iores arcu. </s>
            <s xml:id="echoid-s15589" xml:space="preserve">Sint enim, ſi fieri poteſt, non maiores, ac proinde arcus BCL, ſit vel
              <lb/>
              <figure xlink:label="fig-363-01" xlink:href="fig-363-01a" number="254">
                <image file="363-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/363-01"/>
              </figure>
            æqualis rectis AB, AL, vel maior. </s>
            <s xml:id="echoid-s15590" xml:space="preserve">Secto ergo arcu bifariam
              <lb/>
            in C, ducatur DCE, tangens arcum in C. </s>
            <s xml:id="echoid-s15591" xml:space="preserve">Diuiſis quo que ar-
              <lb/>
            cubus CB, CL, bifariam in G, F, iungantur rectæ B G, G C,
              <lb/>
            CF, FL. </s>
            <s xml:id="echoid-s15592" xml:space="preserve"> Et quia AD, AE, maiores ſunt quam DE; </s>
            <s xml:id="echoid-s15593" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-363-04" xlink:href="note-363-04a" xml:space="preserve">20. primi.</note>
            tis DL, EB, communibus, quæ æquales ſunt; </s>
            <s xml:id="echoid-s15594" xml:space="preserve">(Namiunctis
              <lb/>
            rectis NA, NB, NL, ex centro N; </s>
            <s xml:id="echoid-s15595" xml:space="preserve">quoniã tria latera trianguli
              <lb/>
            ABN, tribus lateribus triãguli ALN, æqualia ſunt; </s>
            <s xml:id="echoid-s15596" xml:space="preserve"> erunt
              <note symbol="c" position="right" xlink:label="note-363-05" xlink:href="note-363-05a" xml:space="preserve">8. primi.</note>
            anguli ad N, quam ad A, æquales; </s>
            <s xml:id="echoid-s15597" xml:space="preserve"> ideoq; </s>
            <s xml:id="echoid-s15598" xml:space="preserve">arcus CB,
              <note symbol="d" position="right" xlink:label="note-363-06" xlink:href="note-363-06a" xml:space="preserve">26. tertii.</note>
            æquales erunt; </s>
            <s xml:id="echoid-s15599" xml:space="preserve">ac proinde recta N A, per contactum C,
              <lb/>
              <note symbol="e" position="right" xlink:label="note-363-07" xlink:href="note-363-07a" xml:space="preserve">18. tertii.</note>
            tranſibit, eritque ad DE, perpendicularis. </s>
            <s xml:id="echoid-s15600" xml:space="preserve">Cum igitur duo anguli DAC,
              <note symbol="f" position="right" xlink:label="note-363-08" xlink:href="note-363-08a" xml:space="preserve">26. primi.</note>
            duobus angulis E A C, E C A, æquales ſint, & </s>
            <s xml:id="echoid-s15601" xml:space="preserve">latus adiacens A C, commune;
              <lb/>
            </s>
            <s xml:id="echoid-s15602" xml:space="preserve"> erunt latera AD, AE, æqualia: </s>
            <s xml:id="echoid-s15603" xml:space="preserve">proptereaque & </s>
            <s xml:id="echoid-s15604" xml:space="preserve">reliquæ DL, EB, æquales erunt, cum tangentes AL, AB, æquales ſint) erunt A L, A B, maiores tribus L D, DE,
              <lb/>
            EB. </s>
            <s xml:id="echoid-s15605" xml:space="preserve">Sit ergo exceſſus H. </s>
            <s xml:id="echoid-s15606" xml:space="preserve">Rurſus quia arcus BL, maior eſt rectis BG, GC, CF, FL,
              <lb/>
            ſit exceſſus I, qui minor ſit exceſſu H. </s>
            <s xml:id="echoid-s15607" xml:space="preserve">Si nam que minor non eſt, diuidemus </s>
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