Barrow, Isaac, Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

Table of Notes

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            ſionum puncta lineam conſtituent, ſaltem ad lineam conſiſtent, ipſi
              <lb/>
            BC ſimilem. </s>
            <s xml:id="echoid-s8733" xml:space="preserve">Ductis enim quotlibet lateribus VB, VD, VE, VC,
              <lb/>
            & </s>
            <s xml:id="echoid-s8734" xml:space="preserve">ducto plano GKLH ad planum BDEC parallelo, ſint com-
              <lb/>
              <note position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">16. XI.Elem.</note>
            munes plani VBD cum planis BC, GH ſectiones rectæ BD, CH;
              <lb/>
            </s>
            <s xml:id="echoid-s8735" xml:space="preserve">hæ parallelæ erunt. </s>
            <s xml:id="echoid-s8736" xml:space="preserve">Item communes plani VDE cum iiſdem planis
              <lb/>
            BC, GH Sectiones DE, KL parallelæ erunt. </s>
            <s xml:id="echoid-s8737" xml:space="preserve">Ergò anguli BDE,
              <lb/>
            GKL ſunt æquales. </s>
            <s xml:id="echoid-s8738" xml:space="preserve">Item ſe habet recta BD ad GK, ut DE ad
              <lb/>
              <note position="right" xlink:label="note-0197-02" xlink:href="note-0197-02a" xml:space="preserve">10. XI. elen
                <unsure/>
              .</note>
            KL, quia utraque hæc proportio æqualis eſt illi, quam habet VD
              <lb/>
            ad VK (ſimilia quippe ſunt triangula VDB, VKG, & </s>
            <s xml:id="echoid-s8739" xml:space="preserve">triangula
              <lb/>
            VDE, VKL) permutandóque BD. </s>
            <s xml:id="echoid-s8740" xml:space="preserve">DE:</s>
            <s xml:id="echoid-s8741" xml:space="preserve">: GK. </s>
            <s xml:id="echoid-s8742" xml:space="preserve">KL. </s>
            <s xml:id="echoid-s8743" xml:space="preserve">ergò omnes
              <lb/>
            ſubtenſæ in GH proportionales ſunt ſubtenſis omnibus in BC, eas
              <lb/>
            nimirum in utraque linea ordinatim & </s>
            <s xml:id="echoid-s8744" xml:space="preserve">deinceps accipiendo; </s>
            <s xml:id="echoid-s8745" xml:space="preserve">& </s>
            <s xml:id="echoid-s8746" xml:space="preserve">quæ
              <lb/>
            ſibimet adjacent in una pariter inflectuntur cum iis, quæ ſibi adjacent
              <lb/>
            in altera. </s>
            <s xml:id="echoid-s8747" xml:space="preserve">Ergò ſecundum ſuperiùs inſinuata lineas BC, GH ſimiles
              <lb/>
            eſſe conſtat.</s>
            <s xml:id="echoid-s8748" xml:space="preserve">‖ Hinc etiam patet lineas curvas ſimiles BC, GH ean-
              <lb/>
            dem ad ſe proportionem habere, quam Superficierum, in eadem
              <lb/>
            qualibet recta ſita, latera VB, VG. </s>
            <s xml:id="echoid-s8749" xml:space="preserve">Quum enim ſubtenſarum
              <lb/>
            iiſdem angulis incluſarum (ut BD, GK, vel DE, KL) ſingulæ
              <lb/>
            rationes æquales ſint rationi laterum VB, VG; </s>
            <s xml:id="echoid-s8750" xml:space="preserve">etiam omnes ante-
              <lb/>
              <note position="right" xlink:label="note-0197-03" xlink:href="note-0197-03a" xml:space="preserve">12. V. Elem.</note>
            cedentes conjunctæ (hoc eſt tota BC) ad omnes conſequentes con-
              <lb/>
            junctas (hoc eſt totam GH) ſe habebunt ut VB ad VG. </s>
            <s xml:id="echoid-s8751" xml:space="preserve">Hinc etiam
              <lb/>
            tali motu productarum ſuperficierum emergit hæc proprietas; </s>
            <s xml:id="echoid-s8752" xml:space="preserve">quòd
              <lb/>
            interceptæ ſcilicet à parallelis ad BC planis, à vertice deſumptæ,
              <lb/>
            quibuſcunque lateribus iiſdem incluſæ partes ipſarum ſint inter ſe ſi-
              <lb/>
            miles; </s>
            <s xml:id="echoid-s8753" xml:space="preserve">ut puta Superficies BVC, GVH; </s>
            <s xml:id="echoid-s8754" xml:space="preserve">& </s>
            <s xml:id="echoid-s8755" xml:space="preserve">BVD, GVK.
              <lb/>
            </s>
            <s xml:id="echoid-s8756" xml:space="preserve">(Quod ex generali ſimilitudinis doctrina poſthac explicanda luculen-
              <lb/>
            tiùs apparere poterit; </s>
            <s xml:id="echoid-s8757" xml:space="preserve">interim ex ſimilitudine linearum curvarum, & </s>
            <s xml:id="echoid-s8758" xml:space="preserve">
              <lb/>
            earum cum Superficiei lateribus analogia, penitúſque conſimili Superſi-
              <lb/>
            cierum generatione ſatìs eluceſcit; </s>
            <s xml:id="echoid-s8759" xml:space="preserve">ſaltem ex triangulorum VBD,
              <lb/>
            VGK; </s>
            <s xml:id="echoid-s8760" xml:space="preserve">& </s>
            <s xml:id="echoid-s8761" xml:space="preserve">VDE, VKL, & </s>
            <s xml:id="echoid-s8762" xml:space="preserve">talium omnium ſimilitudine ſatìs con-
              <lb/>
            ſtat; </s>
            <s xml:id="echoid-s8763" xml:space="preserve">ſiquidem ex talibus infinitis triangulis utraque Superficies com-
              <lb/>
            poſita cenſeatur.) </s>
            <s xml:id="echoid-s8764" xml:space="preserve">Unde ſimilium Superficierum proprietates iis con-
              <lb/>
            venient. </s>
            <s xml:id="echoid-s8765" xml:space="preserve">Verùm quòd interceptas attinet à diverſis lateribus Super-
              <lb/>
            ficies, eas inter ſe comparando, notandum eſt quòd baſibus ſuis, ſeu
              <lb/>
            directricis lineæ reſpectivis partibus non ſemper proportionales ſunt; </s>
            <s xml:id="echoid-s8766" xml:space="preserve">
              <lb/>
            at ſaltem hoc tum evenit, cùm omnia dictæ Superficiei latera ſunt
              <lb/>
            æqualia inter ſe, adeóque cùm linea directrix eſt peripheria circuli; </s>
            <s xml:id="echoid-s8767" xml:space="preserve">
              <lb/>
            quo caſu producta Superficies erit conica Superficies ſtrictè dicta,
              <lb/>
            rectúmque quidem ad conum pertinens. </s>
            <s xml:id="echoid-s8768" xml:space="preserve">Quod ſi directrix BC ſup-
              <lb/>
            ponatur e. </s>
            <s xml:id="echoid-s8769" xml:space="preserve">g. </s>
            <s xml:id="echoid-s8770" xml:space="preserve">peripheria circularis, lateráque ſibimer inæqualia, </s>
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