Cardano, Girolamo
,
De subtilitate
,
1663
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torum: ideóque ex decimaquarta eiuſdem,
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ſi KC eſt maior AC, vel æqualis, vel minor,
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ita CG maior, æqualis, aut minor CB. </
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<
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igitur KC æqualis ſit AF, diuiſa per medium
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in H, per quintam deſcribetur ſemicircu
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lus ſecundum propoſitam magnitudinem,
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quia AF fuit dupla illi latitudini per ſextam:
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erigo igitur CM perpendicularem, & ducta
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GL ex ſectione circuli, & perpendicularis,
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ducemus BM illi æquidiſtantem per trigeſi
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mam mediam: conſtat igitur CM, eſſe pro
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portione mediam inter AC & CB: nam (vt
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demonſtratum eſt) vt KC ad CA, ita CG ad
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CB, quare, vt KC ad CG, ita AC ad CB: at
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ex quarta ſexti, vt CG ad CB, ita LC ad
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CM: LC, autem ex octaua ſexti Elemento
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rum & trigeſimaquarta harum eſt in media
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proportione K C & CG, igitur & CM, in
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media proportione AC & CB. </
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betur vltima ſecundi Element. </
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geſimaſexta. </
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Quintus li
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ber Euclid.
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totus.
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</
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primæ duo
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decim propo
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ſitiones.</
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13 35
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Vltima ſe
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cundi
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Elemẽ-torum
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torum</
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36.
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Reliquum
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ſexti Ele
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mentorum
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præter vlti
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mam.
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<
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<
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17 37
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<
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">Eadem ratione perficiemus Euclidis de
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monſtrationibus omnes ſexti Elementorum
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propoſitiones, vltima duntaxat excepta. </
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<
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">Inde
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decimamſeptimam tertij Elementorum ag
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grediemur. </
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<
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cta igitur ex puncto præter circulum per
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centrum recta, ſumam mediam per trigeſi
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mamquintam inter totam, quæ ex puncto
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vſque ad circumferentiam interiorem, & il
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lam quæ ei exterius adiacet, inde ſuper ter
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minum inuentæ erecta perpendiculari ſe
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cundum quantitatem ſemidiametri circuli,
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ad quem ex puncto propoſito contingens
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ducenda eſt, concludo triangulum. </
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<
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">Ergo
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huic angulo contenta ex vltimo ducta &
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perpendiculari, ſeu oppoſito conſimili ex
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25. facio angulum in centro æqualem verſus
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punctum propoſitum, ex quo ducta recta ad
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extremum lineæ quæ angulum facit, vbi ſci
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licet circulum tangit, contingens erit: nam
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ex ſexta ſecundi Element. </
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">& 47. primi eiuſ
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dem, linea ex puncto ad centrum æqualis
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erit lineæ vltimò ductæ, quæ recto opponi
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tur: ex prima igitur harum, angulus ille in
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baſi iuxta circumferentiam rectus, & ex 16.
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tertij Element. </
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<
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quæ omnes tertij libri, præter 24. & 33. de
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monſtrantur, ex iam demonſtratis. </
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demonſtrabimus locum centri, vt Euclides:
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perficere circulum non eſt poſſibile, cum re
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pugnet iam promiſſis, illa
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tamẽ
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vtemur, quia
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non eſt opus, niſi in circumſcribendis, aut in
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ſcribendis circulis niſi centri inuentione, vt
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demonſtrabimus. </
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id
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">In 33. etiam tertij abſol
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uemus quotquot voluerimus angulos ſupra
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datam lineam: quia omnes, ſi circulus ſupra
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illam deſcriberetur, in circumferentia illius
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eſſent. </
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">Id prius auxilio trigeſimæquartæ ter
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tij, quæ abſque 33. demonſtratur in propo
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ſito tuo circulo, abſoluemus, inde per 25. ſu
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pra conſtitutam lineam: ergo erunt hæ no
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bis loco trigeſimęoctauæ, & trigeſimęnonæ,
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ſicut vltima ſexti Elementorum pro 40. Pòſt
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demonſtrabimus primam quarti Elemento
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rum, hæc erit nobis quadrageſimaprima: per
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duodecimam ſexti Elementorum conſtitu
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tam, vt ſit latitudinis circini propoſiti ad A,
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lineam, vt ſemidiametri circuli, in quo eſt
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linea inſcribenda ad lineam inſcribendam,
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inde collocata A in circulo mihi permiſſo,
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expleo trigonum duûm æqualium laterum,
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& angulo in centro circuli permiſſi, quem
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ſubtendit linea A æqualem, ex vigeſima
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quinta facio in propoſiti circuli centro.
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<
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trigonos eſſe ſimiles, ſubducta ſemidiame
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tris recta, quare ſemidiametri conceſſi ad A,
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vt ſemidiametri propoſiti ad ſubductam ex
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quartaſexti Elementorum, talis verò fuit
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ſemidiametri propoſiti circuli ad lineam
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propoſitam, igitur ſubducta eſt æqualis pro
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poſitæ. </
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Reliquum
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tertij libri
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excepta 24.
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& 33.
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24 37
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33 39
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Eucl. </
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Vltim. </
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40.1. quar
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ti 41.</
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<
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mentorum: quamuis ad Euclidis finem non
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ſit neceſſaria, ſed propter 22. ab eodem ad
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iecta fuerit ſolum, quæ iam ſuperius eſt de
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monſtrata. </
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<
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ABC, ſub conditione ibidem adiecta, & aſ
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ſumo circulum mihi conceſſum, cuius dime
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tiens DH, & medium eius DE, & ſit A ma
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ior B, & B maior C, & ex 12. ſexti Elemen
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torum iam demonſtrata, fiat DE ad EF, vt
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A ad B, & EF ad FG, vt B ad C: & quia B
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& C, ſupponuntur longiores A, erit tota EG
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longior E D, igitur G punctus cadet extra
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circulum: fiat ex eadem 12. ſexti Elemento
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rum, vt GF ad FH, ita DF ad K, cui K adii
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ciatur L, æqualis GF: igitur vt DF ad K, ita
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L ad F H. </
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<
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">Quia verò H F eſt minima 4.
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quantitatum proportionis vnius, erit D F
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maxima, totáque DH maior tota KL, ex 25.
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quinti Elementorum: igitur per 41. præce
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dentem collocabimus KL, quomodolibet in
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circulo, vt ſit M O, & ex 13. faciam M N
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æqualem L, erítque NO æqualis K: & du
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cam ex centro ENP producendo ex aduerſo
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in Q, & iterum ex eodem centro EM. </
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<
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">Ex 16.
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ſexti Element. </
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<
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in NO, æquale eſt producto DF in FH, quia
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K & L fuerunt proportione mediæ inter
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DF & FH: & ex 35. tertij Element. </
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<
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ctum P N in NQ, æquale eſt producto MN
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in NO, igitur ex PN in NQ, fit quantam ex
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DF in FH: igitur cum PQ, ſit æqualis DH,
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erit NP, æqualis FH, & EN æqualis EF:EM
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autem eſt æqualis E D, & FG æqualis L, &
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L æqualis MN, igitur F G æqualis M N,
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igitur trigonus E M N, conſtat ex tribus </
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