Clavius, Christoph
,
Gnomonices libri octo, in quibus non solum horologiorum solariu[m], sed aliarum quo[quam] rerum, quae ex gnomonis umbra cognosci possunt, descriptiones geometricè demonstrantur
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LIBER PRIMVS.
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c
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undem ſemicirculũ E D F, recta erunt, atque idcirco & </
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<
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xml:space
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ctus erit.) </
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<
s
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xml:space
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">Quare D E, arcus erit inclinationis circuli A B C D, ad circulum A F C E, proptereaq́;
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<
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xml:space
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D, ad planum A F C E, demiſſa in E F, communem
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<
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ſectionem planorum A F C E, E D F; </
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<
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xml:space
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in I, cadere. </
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<
s
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xml:space
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">Si enim alio cadat, vt in k, erunt duo
<
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anguli H I G, I G H, trianguli G H I, duobus an-
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gulis D K G, K G D, trianguli G D K, ęquales;
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</
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<
s
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xml:space
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">(Nam I G H, K G D, ęquales ſunt, ob æqualitatem
<
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<
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xlink:label
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xml:space
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">27. tertij.</
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arcuum E H, E D, & </
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<
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xml:space
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">H I G, D K G, recti ſunt, ex
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<
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conſtructione, & </
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<
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<
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">lib. </
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<
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<
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">Euclidis)
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ſuntautem & </
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<
s
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xml:space
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">latera G H, G D, ducta à centro ſphę-
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rę ad eius ſuperficiem, ęquales. </
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<
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xml:space
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">Igitur & </
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<
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xml:space
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">latera G I,
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<
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xlink:label
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note-0111-04
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xlink:href
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xml:space
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">26. primi.</
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G K, æqualia erunt, pars, & </
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<
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<
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xml:space
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">Quod eſt abſur-
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dum. </
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<
s
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xml:space
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">Non ergo perpendicularis à puncto D, de-
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miſſa ad planum A F C E, alio cadit, quàm in I. </
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<
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xml:space
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dem modo reperiemus punctum, in quod cadit
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perpendicularis ex B, demiſſa. </
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<
s
xml:id
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xml:space
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">Cadet autem ſem-
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per in punctum, puta M, quod tantum à centro
<
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G, abeſt, quantum I, ab eodem diſtat. </
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<
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xml:space
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">Quoniam enim in triangulis D G I, B G M, anguli ad I, M,
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<
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ex defin. </
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<
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<
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<
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<
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<
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">recti ſunt, anguliq́; </
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<
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">ad verticem G, æquales; </
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<
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xml:space
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">Item & </
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<
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">latera D G, B G, æqua
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<
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xml:space
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lia, cum ſint ſpheræ ſemidiametri; </
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<
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">erunt & </
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<
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">latera G I, G M, inter ſe ęqualia. </
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<
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xml:space
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">In circunferentia igi-
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<
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tur circuli maximi in ſphæra, &</
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<
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<
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</
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<
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xml:space
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<
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<
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">EX his eadem via inueniemus diametrum minorem ellipſis illius, in quam perpendiculares à circun-
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<
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xlink:label
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xml:space
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">Inuentio min@
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tis diametri El-
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lipſis, quæ fit à
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perpendiculari-
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bus cadenubus
<
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à circunferẽria
<
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circuli inclinati
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ad alium circu-
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lum.</
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>
ferentia circuli inclinati in alium circulum demiſſæ cadunt. </
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<
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xml:space
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">Nam recta I M, inter puncta I, M, in quæ
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dictæ perpendiculares cadunt, minor diameter eſt, per antecedentem propoſitionem.</
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</
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<
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<
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xml:space
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<
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<
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">IN circunferentia circuli maximi in ſphæra ad alium circulum ma-
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ximum inclinati ſumptis quibuslibet punctis, quo loco perpendicula-
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res ab his ductæ in alium circulum cadant, ſi inclinatio fuerit nota,
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inquirere.</
s
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<
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</
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<
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<
s
xml:id
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xml:space
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">SIT in ſphæra circulus maximus A B C D, ad maximum D E B F, inclinatus, & </
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<
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xml:space
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">nota inclina-
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tio, ſitq́; </
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<
s
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xml:space
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">eorum fectio communis diameter D B, per centrum G, tranſiens, ad quam ad angulos
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<
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rectos ducatur in circulo quidem A B C D, diame-
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<
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xlink:href
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number
="
74
">
<
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0111-02
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xlink:href
="
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</
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>
rer A C, in circulo verò D E B F, diameter E F, in
<
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quam cadent perpendiculares ex A, C, in circulum
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D E B F, demiſſæ, vt in propoſitione præcedenti eſt
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oſtenſum. </
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<
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xml:space
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">Cadant ergo in H, I, vt ſit D B, diame-
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ter maior, & </
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<
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xml:space
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">H I, minor eius Ellipſis, quam perpen
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diculares à circunferentia circuli A B C D, in pla-
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num circuli D E B F, demiſſę faciunt, vt demonſtra
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tum eſt. </
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<
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xml:space
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">Sumatur autem quodcunque punctum K,
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in circunferentia A B C D. </
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<
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xml:space
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<
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re, quo loco perpendicularis à K, in planum D E-
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B F, deducta cadat. </
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<
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xml:space
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">Sumatur arcui A K, æqualis ar-
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cus E L, & </
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<
s
xml:id
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xml:space
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">ducatur recta G L, quæ circulum H I,
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circa minorem Ellipſis diametrum H I, deſcriptũ
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<
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ctorum, in quæ
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cadunt perpen-
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diculares à quo
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cunque puncto
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circuli inclinati
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ad alium circu-
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lum.</
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ſecet in M. </
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<
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xml:space
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">Deinde per L, ducatur L N, parallela
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minori diametro H I, quæ ſecet D B, in O; </
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<
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xml:space
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">& </
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<
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M, ducatur P M, parallela maiori Ellipſis diame-
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tro D B, ſecans L N, in Q. </
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>
<
s
xml:id
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xml:space
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">Dico perpendicularem à K, in planum D E B F, demiſſam cadere in
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punctum Q. </
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<
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">Quòd enim cadat in lineam L N, ita oſtendetur. </
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<
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">Ducta recta K O, erit hæc ipſi
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A G, parallela. </
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<
s
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xml:space
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">Ductis enim L S, k T, ad E G, A G, perpendicularibus, cum G O, æqualis ſit ipſi
<
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<
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">34. primi.</
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>
L S, ſinui recto arcus E L; </
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<
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">ſit autem L S, ſinus æqualis ipſi k T, ſinui recto arcus A K, qui </
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