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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            rectæ AB partibus delineabunt, pariter ut antehac in ſiguræ planæ
              <lb/>
            exemplo commonſtratum eſt; </s>
            <s xml:id="echoid-s8660" xml:space="preserve">unde ſi ſuperſicies hoc modo procreatæ
              <lb/>
            à plano quolibet ad rectam ſeu genetricem, ſeu directricem (quam
              <lb/>
            ubique ſitam Superficiei productæ latus appellare licet) parallelo
              <lb/>
            ſecetur, ſectio communis duabus rectis parallelis conſtabit æqualibus
              <lb/>
            inter ſe. </s>
            <s xml:id="echoid-s8661" xml:space="preserve">De Superficiebus autem ità progenitis obſervatu dignum eſt
              <lb/>
            (nec enim planè nudas magnitudinum generationes indigitare, ſed & </s>
            <s xml:id="echoid-s8662" xml:space="preserve">
              <lb/>
            generales nonnullas ipſarum affectiones è diverſis reſultantes generandi
              <lb/>
            modis inſinuare propoſitum eſt nobis) quòd ſi linea directrix recta ſit
              <lb/>
            (ut in figura per literam Z diſcriminata) Superficiei productæ partes
              <lb/>
            parallelis lineis genetricibus interjectæ reſpectivis directricis lineæ
              <lb/>
            partibus ſemper proportionales ſunt (ſuperficies nempe BCCB re-
              <lb/>
            ſpectivis rectis BB:) </s>
            <s xml:id="echoid-s8663" xml:space="preserve">At ſi linea curva pro directrice habeatur (ut in
              <lb/>
            figura Y) non ſemper eveniet, ut interceptæ genetricibus rectis Super-
              <lb/>
            ficies interceptis curvæ directricis partibus proportionentur; </s>
            <s xml:id="echoid-s8664" xml:space="preserve">at ſaltem
              <lb/>
            accidet hoc, cùm recta genetrix AB æqualiter ad curvam BC ubique,
              <lb/>
            vel ſecundum omnia ejus puncta inclinatur; </s>
            <s xml:id="echoid-s8665" xml:space="preserve">quomodo fit in cylindri
              <lb/>
            cujuſcunque, laxè vel ſtrictè dicti, recti ſuperficie; </s>
            <s xml:id="echoid-s8666" xml:space="preserve">quia tum recta
              <lb/>
            genetrix omnibus curvæ punctis (hoc eſt omnibus eam ad dicta puncta
              <lb/>
              <note position="left" xlink:label="note-0194-01" xlink:href="note-0194-01a" xml:space="preserve">Fig. 6.</note>
            tangentibus, eive ſubtenſis rectis eſt perpendicularis.) </s>
            <s xml:id="echoid-s8667" xml:space="preserve">Verum ſi, in
              <lb/>
            exemplum, curva BC ponatur arcus circularis, qui dividatur æqualiter
              <lb/>
            ad puncta B, non erunt neceſſariò ſuperficies ABBA peripheriis
              <lb/>
            æqualibus BB inſiſtentes inter ſe pares, quia (præterquam in caſu
              <lb/>
            prædicto cylindri recti) rectæ AB ubique ad puncta B inæqualiter
              <lb/>
            inclinantur (unam quamvis inclinationem cum alia conferendo) an-
              <lb/>
            gulos nempe cum tangentibus ad B aliis ac aliis, & </s>
            <s xml:id="echoid-s8668" xml:space="preserve">cum ſubtenſis BB
              <lb/>
            mæquales efficiunt. </s>
            <s xml:id="echoid-s8669" xml:space="preserve">E qua re pendet _inſuperabilis illa difficultas,_
              <lb/>
            quacum conflictantur, qui _cylindricas obliquas ſuperſicies conantur_
              <lb/>
            _dimetiri, ſen cum Cylinàricis Superficiebus rectis, aliìſve quadantenus_
              <lb/>
            _cognitis Superficiebus quoad proportionem comparare._ </s>
            <s xml:id="echoid-s8670" xml:space="preserve">Supponunt denique
              <lb/>
            conſimili pacto ſuperſiciem quamvis planam directo motu ſibi parallelo
              <lb/>
            progredi, ſcilicet ut prædicto modo, ſingula ipſius puncta lineas
              <lb/>
            rectas deſcribant, inter ſe pares, ac parallelas; </s>
            <s xml:id="echoid-s8671" xml:space="preserve">vel ut ejus ſingulæ
              <lb/>
            rectæ (id quod indè conſectatur) planas Superficies parallelogrammas
              <lb/>
            effingant; </s>
            <s xml:id="echoid-s8672" xml:space="preserve">cujuſmodi motu deſcribuntur priſmatica quæque cylindricá-
              <lb/>
            que corpora; </s>
            <s xml:id="echoid-s8673" xml:space="preserve">illa nimirum ipſa, de quorum Superficiebus mox egimus,
              <lb/>
            quibúſque ſimili jure poſſunt adaptari, quæ Superficiebus iſtis oſtendi-
              <lb/>
            mus convenire. </s>
            <s xml:id="echoid-s8674" xml:space="preserve">Veluti quod parallelis planis interjectæ Superſicies
              <lb/>
            ipſorum, & </s>
            <s xml:id="echoid-s8675" xml:space="preserve">ipſa corpora lateribus ſuis (ſeu directricis rectæ partibus
              <lb/>
            reſpectivis) proportionantur. </s>
            <s xml:id="echoid-s8676" xml:space="preserve">Quòd & </s>
            <s xml:id="echoid-s8677" xml:space="preserve">ſi definita hujuſmodi </s>
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