Barrow, Isaac
,
Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur
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lata, pèrque motum iſtum in curva deſcribenda conſpirans, percurrit
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rectam PM. </
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<
s
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echoid-s9220
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xml:space
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">Cùm igitur ſint TP, PM ex conſtructione pares,
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adeóque velocitates motuum, quibus ſimul peraguntur, æquales;
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</
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<
s
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xml:space
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">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. </
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<
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Ergo punctum S eſt id ipſum, quod inveniri debuit, & </
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<
s
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">abſolutum eſt
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<
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note
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propoſitum.</
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<
s
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xml:space
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">| Exemplo ſit _parabola_, quæ facta concipitur ex motu
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uniformi horizontali, & </
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<
s
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echoid-s9225
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">deſcenſivo pariter accelerato; </
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<
s
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xml:space
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">tum punctum
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P ità facilè per _Analyſin_ inveſtigatur. </
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<
s
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echoid-s9227
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xml:space
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">Sit recta R _datæ parabo
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læ_
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_rectuns
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latus._ </
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<
s
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xml:space
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">Eſt igitur ex _parabolæ_ natura, R x AP. </
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<
s
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echoid-s9229
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xml:space
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">= PMq
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= TPq (exhypotheſi modi noſtri generalis.) </
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<
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echoid-s9230
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xml:space
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">Item, ex parabolæ
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nota proprietate eſt TPq = 4 APq. </
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<
s
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xml:space
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">Ergo eſt R x AP = 4 APq.
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</
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<
s
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echoid-s9232
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xml:space
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">Adeóque R = 4AP; </
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<
s
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echoid-s9233
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xml:space
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">vel {1/4} R = AP = SA. </
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<
s
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echoid-s9234
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xml:space
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">Nimirum ita _Gali-_
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_læus_ determinavit. </
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<
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xml:space
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">In hoc autem caſu puncta T, S coincidunt. </
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<
s
xml:id
="
echoid-s9236
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xml:space
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">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, & </
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<
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xml:space
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">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. </
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<
s
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echoid-s9238
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xml:space
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">Et ex hujus regulæ generalis præſcripto
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eſt PM = TP, adeóque PM cub. </
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<
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echoid-s9239
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xml:space
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">= TP cub. </
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<
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xml:space
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">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. </
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<
s
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="
echoid-s9241
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xml:space
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">= 27 AP cub. </
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<
s
xml:id
="
echoid-s9242
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xml:space
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">Erit
<
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igitur R q AP = 27 AP cub. </
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<
s
xml:id
="
echoid-s9243
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xml:space
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">Adeóque R q = 27 APq; </
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<
s
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echoid-s9244
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xml:space
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">vel
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{Rq/27} = APq = SAq. </
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<
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xml:space
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">In reliquis ſimili ratione procedentes
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aſſequemur propoſitum. </
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<
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xml:space
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">Poſſent opinor & </
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<
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">hinc nedum pleræque
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_Galilæipoſitiones_ huic affines, & </
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<
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">hanc attingentes materiam utcun-
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<
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que deduci, ſed & </
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<
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xml:space
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">generaliores reddi, vel ad alia curvas omnigenas
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extendi. </
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<
s
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xml:space
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">Verùm parco pluribus, hoc _ſpecimine_ (quoad iſta) con-
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tentus; </
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<
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">huc non niſi per tranſcurſum adducto. </
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<
s
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echoid-s9252
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xml:space
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">Ad alia pergo præ-
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dictis cohærentia.</
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<
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<
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xml:space
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">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.) </
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<
s
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echoid-s9256
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xml:space
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">Hujuſce ſuperficiei ad paral-
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lelogrammum æquealtum, ſuper eadem baſe conſtitutum, proportio
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proportionem indicabit ipſarum AP; </
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<
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">TP, à puncto P vertici, tan-
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gentique interjectarum.</
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