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

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            <s xml:id="echoid-s9259" xml:space="preserve">Ut ſi ad rectam α δ applicetur plana ſuperficies α δ μ, & </s>
            <s xml:id="echoid-s9260" xml:space="preserve">utcun-
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              <note position="right" xlink:label="note-0215-01" xlink:href="note-0215-01a" xml:space="preserve">Fig. 23, 24.</note>
            que divisâ AD punctis B, C, ſimilitérque dicisâ rectâ α δ punctis
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              <note style="it" position="right" xlink:label="note-0215-02" xlink:href="note-0215-02a" xml:space="preserve">Hæc poſthac
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              γτωμετριηίο
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              τερον demo@-
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              ſtrata haben-
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              tur.</note>
            que divisâ AD punctis B, C, ſimilitérque dicisâ rectâ α δ punctis
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            β, γ, fuerit ut BM ad CM ità ſuperficies β α μ, ad ſuperficiem
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            γ α μ, & </s>
            <s xml:id="echoid-s9261" xml:space="preserve">hoc in comparationibus univerſis taliter inſtitutis contingat;
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            </s>
            <s xml:id="echoid-s9262" xml:space="preserve">_completo parallelogrammo α δ μ φ, ſe habebit recta_ AP _adrectam_ TP
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            _ut ſuperficies αδ μ adl
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            parallelogrammum_ α δ μ φ. </s>
            <s xml:id="echoid-s9263" xml:space="preserve">Et enim ſi recta
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            α δ commune tempus defignare concipiatur, quo recta AD motu
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            æquabili, rectáque DM motu continuè accelerato tranſiguntur,
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            recta δ μ bene deſignabit velocitatem hujus definiti temporis maxi-
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            mam, quam habet punctum deſcendens in curvæ puncto M infimo; </s>
            <s xml:id="echoid-s9264" xml:space="preserve">
              <lb/>
            hoc eſt velocitatem quâ recta TP uniformiter decurritur eodem tem-
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            pore; </s>
            <s xml:id="echoid-s9265" xml:space="preserve">quapropter (ut antehac commonſtratum eſt.) </s>
            <s xml:id="echoid-s9266" xml:space="preserve">_Parallelogram-_
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            _mum_ α δ μ φ optimè _Spatium_ repræſentabit, quod hâc eâdem per-
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            manente velocitate per totum tempus α δ uniformiter deſcribitur,
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            hoc eſt ipſam rectam TP. </s>
            <s xml:id="echoid-s9267" xml:space="preserve">Cu
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            m igitur, ex hypotheſis præſtratæ con
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            ditione, figura δ α μ rectam DM, vel AP, repræſentet, erit ut figura
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            δ αμ ad parallelogrammum α δ μ φ, ità AP ad TP; </s>
            <s xml:id="echoid-s9268" xml:space="preserve">cognitáque
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            proinde modo quovis iſtâ proportione, ſimul hæc innoteſcet; </s>
            <s xml:id="echoid-s9269" xml:space="preserve">& </s>
            <s xml:id="echoid-s9270" xml:space="preserve">re-
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            ciprocè. </s>
            <s xml:id="echoid-s9271" xml:space="preserve">Exemplo res manifeſtior evadet uno, vel altero. </s>
            <s xml:id="echoid-s9272" xml:space="preserve">Propoſita
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            curva ſit _parabola quadratica_, ſeu in qua rectæ BM, CM ſe
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            habent, ut quadrata ex AB, AC, hoc eſt ut quadrata ex α β, α γ. </s>
            <s xml:id="echoid-s9273" xml:space="preserve">
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            Ergò ſi figura α δ μ ſit triangulum, id optimè quadrabit huic negotio. </s>
            <s xml:id="echoid-s9274" xml:space="preserve">
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            Nam eo ſuppoſito ſemper triangula βαμ, γαμ proportionalia erunt
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            quadratis ex α β, αγ, hoc eſt rectis BM; </s>
            <s xml:id="echoid-s9275" xml:space="preserve">CM. </s>
            <s xml:id="echoid-s9276" xml:space="preserve">Quoniam verò
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            triangulum δ α μ parallelogrammi δ α φ μ eſt ſubduplum, erit
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            recta AP quoque rectæ TP ſubdupla; </s>
            <s xml:id="echoid-s9277" xml:space="preserve">quod ità ſe habere demon-
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            ſtratum habetur in _conicis elementis_, & </s>
            <s xml:id="echoid-s9278" xml:space="preserve">paſſim agnoſcitur. </s>
            <s xml:id="echoid-s9279" xml:space="preserve">Sit rurſus
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            curva AMM _parabola cubica_; </s>
            <s xml:id="echoid-s9280" xml:space="preserve">& </s>
            <s xml:id="echoid-s9281" xml:space="preserve">quoniam in ea rectæ BM, CM
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            ſe habent ut cubi rectarum AB, AC, hoc eſt ut cubi rectarum α β,
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            α γ; </s>
            <s xml:id="echoid-s9282" xml:space="preserve">& </s>
            <s xml:id="echoid-s9283" xml:space="preserve">ſi _ſuperficies α δ μ fuerit complementum ſemiparabolicæ qua-_
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            _draticæ portionis, trilinea α β μ, α γ μ cubis ex α β, α γ proportionalia_
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            _erunt_ (ut à _Pappo_, ac aliis oſtenditur, & </s>
            <s xml:id="echoid-s9284" xml:space="preserve">ex _Archimidea parabolæ_
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            _dimenſione_ quàm facillimè deducitur) itaque negotio propoſito quàm
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            rectiſſimè adaptetur _parabola quadratica_; </s>
            <s xml:id="echoid-s9285" xml:space="preserve">cúmque conſtiterit ali-
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            undè tum figuram α δ μ ſubtriplam fore parallelogrammi α δ μ φ; </s>
            <s xml:id="echoid-s9286" xml:space="preserve">
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            erit etiam juxta regulæ jam aſſignatæ præſcriptum recta AP quoque
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            ſubtripla rectæ TP. </s>
            <s xml:id="echoid-s9287" xml:space="preserve">De qua concluſione ſatis convenit inter _Geo-_
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            _metras_.</s>
            <s xml:id="echoid-s9288" xml:space="preserve"/>
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