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-s9219" xml:space="preserve">
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            lata, pèrque motum iſtum in curva deſcribenda conſpirans, percurrit
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            rectam PM. </s>
            <s xml:id="echoid-s9220" xml:space="preserve">Cùm igitur ſint TP, PM ex conſtructione pares,
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            adeóque velocitates motuum, quibus ſimul peraguntur, æquales;
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            </s>
            <s xml:id="echoid-s9221" xml:space="preserve">etiam motus deſcenſivus in P, vel M æquabitur motui tranſverſo, cur-
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            vam deſcribenti, hoc eſt motûs ab S ad A velocitas in A eidemæquatur. </s>
            <s xml:id="echoid-s9222" xml:space="preserve">
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            Ergo punctum S eſt id ipſum, quod inveniri debuit, & </s>
            <s xml:id="echoid-s9223" xml:space="preserve">abſolutum eſt
              <lb/>
              <note position="left" xlink:label="note-0214-01" xlink:href="note-0214-01a" xml:space="preserve">Fig. 22.</note>
            propoſitum.</s>
            <s xml:id="echoid-s9224" xml:space="preserve">| Exemplo ſit _parabola_, quæ facta concipitur ex motu
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            uniformi horizontali, & </s>
            <s xml:id="echoid-s9225" xml:space="preserve">deſcenſivo pariter accelerato; </s>
            <s xml:id="echoid-s9226" xml:space="preserve">tum punctum
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            P ità facilè per _Analyſin_ inveſtigatur. </s>
            <s xml:id="echoid-s9227" xml:space="preserve">Sit recta R _datæ parabo
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            læ_
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            _rectuns
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            latus._ </s>
            <s xml:id="echoid-s9228" xml:space="preserve">Eſt igitur ex _parabolæ_ natura, R x AP. </s>
            <s xml:id="echoid-s9229" xml:space="preserve">= PMq
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            = TPq (exhypotheſi modi noſtri generalis.) </s>
            <s xml:id="echoid-s9230" xml:space="preserve">Item, ex parabolæ
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            nota proprietate eſt TPq = 4 APq. </s>
            <s xml:id="echoid-s9231" xml:space="preserve">Ergo eſt R x AP = 4 APq.
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            </s>
            <s xml:id="echoid-s9232" xml:space="preserve">Adeóque R = 4AP; </s>
            <s xml:id="echoid-s9233" xml:space="preserve">vel {1/4} R = AP = SA. </s>
            <s xml:id="echoid-s9234" xml:space="preserve">Nimirum ita _Gali-_
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            _læus_ determinavit. </s>
            <s xml:id="echoid-s9235" xml:space="preserve">In hoc autem caſu puncta T, S coincidunt. </s>
            <s xml:id="echoid-s9236" xml:space="preserve">Quòd
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            ſi rurſus gravia juxta _triplicatam temporum rationem_ velocitate creſcen-
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            do deſcendant, adeóque motus ipſorum talis cum uniformi tranſverſo
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            compoſitus _parabolam cubicam_ deſcribat, & </s>
            <s xml:id="echoid-s9237" xml:space="preserve">ſit R iſtius curvæ _para-_
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            _meter_, erit eo in caſù SA = √ {R q/27} nam ex hujuſce curvæ proprie-
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            tate eſt R q AP = PM cub. </s>
            <s xml:id="echoid-s9238" xml:space="preserve">Et ex hujus regulæ generalis præſcripto
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            eſt PM = TP, adeóque PM cub. </s>
            <s xml:id="echoid-s9239" xml:space="preserve">= TP cub. </s>
            <s xml:id="echoid-s9240" xml:space="preserve">Denique quoniam
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            in hujuſmodi _parabola_ tangentis intercepta ſemper triſecatur à vertice
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            (nimirum ut ſit AP = {1/3} TP) eſt TP cub. </s>
            <s xml:id="echoid-s9241" xml:space="preserve">= 27 AP cub. </s>
            <s xml:id="echoid-s9242" xml:space="preserve">Erit
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            igitur R q AP = 27 AP cub. </s>
            <s xml:id="echoid-s9243" xml:space="preserve">Adeóque R q = 27 APq; </s>
            <s xml:id="echoid-s9244" xml:space="preserve">vel
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            {Rq/27} = APq = SAq. </s>
            <s xml:id="echoid-s9245" xml:space="preserve">In reliquis ſimili ratione procedentes
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            aſſequemur propoſitum. </s>
            <s xml:id="echoid-s9246" xml:space="preserve">Poſſent opinor & </s>
            <s xml:id="echoid-s9247" xml:space="preserve">hinc nedum pleræque
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            _Galilæipoſitiones_ huic affines, & </s>
            <s xml:id="echoid-s9248" xml:space="preserve">hanc attingentes materiam utcun-
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              <handwritten xlink:label="hd-0214-01" xlink:href="hd-0214-01a" number="5"/>
            que deduci, ſed & </s>
            <s xml:id="echoid-s9249" xml:space="preserve">generaliores reddi, vel ad alia curvas omnigenas
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            extendi. </s>
            <s xml:id="echoid-s9250" xml:space="preserve">Verùm parco pluribus, hoc _ſpecimine_ (quoad iſta) con-
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            tentus; </s>
            <s xml:id="echoid-s9251" xml:space="preserve">huc non niſi per tranſcurſum adducto. </s>
            <s xml:id="echoid-s9252" xml:space="preserve">Ad alia pergo præ-
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            dictis cohærentia.</s>
            <s xml:id="echoid-s9253" xml:space="preserve"/>
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            <s xml:id="echoid-s9254" xml:space="preserve">XVI. </s>
            <s xml:id="echoid-s9255" xml:space="preserve">Si ad rectam lineam applicetur _planæ ſuperficies_, cujus
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            ſingulæ quæque partes applicatis ad iſtam rectam parallelis inter-
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            ceptæ proportionales ſint rectis ad rectam AY ſimpliciter diviſam
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            applicatis (ad AZ nempe parallelis.) </s>
            <s xml:id="echoid-s9256" xml:space="preserve">Hujuſce ſuperficiei ad paral-
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            lelogrammum æquealtum, ſuper eadem baſe conſtitutum, proportio
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            proportionem indicabit ipſarum AP; </s>
            <s xml:id="echoid-s9257" xml:space="preserve">TP, à puncto P vertici, tan-
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            gentique interjectarum.</s>
            <s xml:id="echoid-s9258" xml:space="preserve"/>
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