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-s9118" xml:space="preserve">
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            per AC vel PM. </s>
            <s xml:id="echoid-s9119" xml:space="preserve">(vel, quòd eodem recidit, dico quòd velocitas
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            puncti deſcendentis in M ad velocitatem quâ fertur recta AZ ſe
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            habet, ut recta TP ad PM.) </s>
            <s xml:id="echoid-s9120" xml:space="preserve">Sumatur enim ubivis in tangente
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            punctum aliquod K, & </s>
            <s xml:id="echoid-s9121" xml:space="preserve">per ipſum ducatur recta KG, curvæ occur-
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            rens in O, parallelis autem AY, & </s>
            <s xml:id="echoid-s9122" xml:space="preserve">PG in D, & </s>
            <s xml:id="echoid-s9123" xml:space="preserve">G. </s>
            <s xml:id="echoid-s9124" xml:space="preserve">Et quia
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            tangens TM duplici concipiatur uniformi motu deſcripta, altero
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            rectæ TZ per AC vel PM parallelωs delatæ, altero puncti deſcen-
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            dentis à T per TZ; </s>
            <s xml:id="echoid-s9125" xml:space="preserve">& </s>
            <s xml:id="echoid-s9126" xml:space="preserve">ſit horum motuum alter per AC, vel
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            PM communis vel idem cum illo quo curva deſcribitnr; </s>
            <s xml:id="echoid-s9127" xml:space="preserve">cùm TZ
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            eſt in ſitu KG, erit AZ in eodem; </s>
            <s xml:id="echoid-s9128" xml:space="preserve">ergò cùm punctum à T deſcendens
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            fuerit in K, erit punctum ab A deſcendens in curvæ cum KG in-
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            terſectione O (nec enim, ut anteà deductum eſt, alibi recta KG
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            curvam ſecat) eſt autem punctum O infra K quia tangens extra cur-
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            vam tota verſatur. </s>
            <s xml:id="echoid-s9129" xml:space="preserve">Jam ſi punctum K ponatur ſupra contactum
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            verſus T, quoniam tum OG minor eſt quàm KG, liquet velo-
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            citatem puncti deſcendentis, quo curva deſcribitur, in curvæ pun-
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            cto O minorem eſſe velocitate motûs uniformis deſcendentis, quâ
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            tangens efficitur; </s>
            <s xml:id="echoid-s9130" xml:space="preserve">quoniam illa ſemper increſcens eodem tempore
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            (per GM repræſentato) minus ſpatium tranſigit, quàm hæc mi-
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            nimè creſcens; </s>
            <s xml:id="echoid-s9131" xml:space="preserve">aſt eadem continuo perſeverans; </s>
            <s xml:id="echoid-s9132" xml:space="preserve">illa ſcilicet rectam
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            OG hæc rectam KG conficit. </s>
            <s xml:id="echoid-s9133" xml:space="preserve">Contra vero ſi punctum K infra
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            contactum ad partes S exiſtat, quoniam OG tum major eſt quàm
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            KG, patet velocitatem puncti deſcendentis, quo curva fit, in pun-
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            cto O majorem eſſe velocitate motûs uniformis itidem deſcenden-
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            tis, quo tangens efficitur; </s>
            <s xml:id="echoid-s9134" xml:space="preserve">quia motus iſte, continuò decreſcens
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            eodem per GM tempore, majus peragit ſpatium OG, quàm hic
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              <note position="right" xlink:label="note-0211-01" xlink:href="note-0211-01a" xml:space="preserve">Fig. 20.</note>
            minimè decreſcens, at in eodem tenore perſiſtens, conficit, ip-
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            ſum nempe ſpatium KG. </s>
            <s xml:id="echoid-s9135" xml:space="preserve">Ergo cùm velocitas curvam deſcribentis
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            puncti quovis in curvæ puncto ſupra contactum verſus A minor ſit
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            velocitate motûs per TP; </s>
            <s xml:id="echoid-s9136" xml:space="preserve">quovis autem in puncto infra contactum
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            eâdem major; </s>
            <s xml:id="echoid-s9137" xml:space="preserve">liquet in ipſo contactu M ei penitus exæquari.
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            </s>
            <s xml:id="echoid-s9138" xml:space="preserve">Q.</s>
            <s xml:id="echoid-s9139" xml:space="preserve">E.</s>
            <s xml:id="echoid-s9140" xml:space="preserve">D.</s>
            <s xml:id="echoid-s9141" xml:space="preserve"/>
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            <s xml:id="echoid-s9142" xml:space="preserve">XII Hujus converſa, conſimili diſcurſu, rem breviùs exponendo,
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            demonſtretur. </s>
            <s xml:id="echoid-s9143" xml:space="preserve">Nempe, ſi velocitas puncti deſcendentis ab A in a-
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            liquo curvæ puncto M æquetur velocitati, quâ punctum T uni-
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            formiter latum, rectam TP deſcriberet tempore PM vel AC
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            (vel ſit velocitas motûs deſcendentis ad M ad velocitatem motûs
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            tranſverſi, ut TP ad PM) recta TMS curvam AMO tan-
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            get ad M.</s>
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