Full text: Viviani, Vincenzio: De maximis et minimis, geometrica divinatio

Ampliùs dico ipſam DEF quò longiùs aberit à vertice E infra EA, eò
magis appropinquare ſectioni B A M. quoniam ducta D M parallela ad
OEB, & MN ad DO, fiet parallelogrammum DN, cuius oppoſita latera
MN, DO æqualia erunt: Itaqueregulæ IG occurrat producta MN in Q, & regulæ LH producta DO in R: cum ſit oſtenſa MN æqualis DO, erit qua-
dratum MN ſiue rectangulum BNQ, æquale quadrato DO ſiue rectangu- lo EOR: ſed in triangulis IBG, LEH ſunt latera IB, LE, & BG, EH inter ſe
æqualia, alterum alteri, quapropter æqualium rectangulorum BNQ, EOR
latera BN, & EO æqualia erunt: quare cum diametri ſegmenta BN, EO ſint æqualia, facta proſtaphereſi, proueniet BE æqualis NO, ſed eſt quoque
MD æqualis NO in parallelogrammo DN, igitur rectæ BE, & MD inter ſe
ſunt æquales, at ſunt etiam parallelæ, ergo coniuncta BM iunctæ ED æqui-
diſtat, ſed BM ſecat NM, quare producta ſecabit quoque OD, ſed extra ſe-
ctionem BMA (cum BM ſit intra ſectionem, producta verò tota cadat extra)
ſitque occurſus in P, & OD occurrat ſectioni BMA in S, PM verò contin-
gentem EA ſecet in T; & in ſecunda figura, in qua punctum A cadit inter
puncta S, & M, iungatur SM, quæ cum tota cadat intra ſectionem, neceſſa-
riò ſecabit applicatam AE: veluti in V.

111.1.

Coroll.
1. huius.
ibidem.
43. h.

Iam, in prima figura, cum in parallelogrammo P E oppoſita latera ET,
DP ſint æqualia, ſitque EA maius ET, erit EA quoque maius ipſo DP, ſed
eſt DP maius intercepto applicatæ ſegmento DS, erit ergo AE, eò maius
ipſo DS. In ſecunda autem figura cum pariter ET, DP ſint æquales, ſitque
dempta TV minor dempta PS, erit reliqua EV maior reliqua DS, & eò ma-
gis EA maior eadem DS. Eodè penitùs modo oſtendetur, quamlibet aliam
interceptam ZY infra SD minorem eſſe ipſa SD: nam ducta YZ æquidiſtan-
ter ad EB, demonſtrabitur item YZ æqualem eſſe eidem BE, ac ideo YZ, & DM eſſe inter ſe ęquales, & parallelas: ex quo ſi iungantur MZ, & DY, ipſę
æquales erunt, & parallelæ; completa igitur conſimili conſtructione, ac ſu-
pra, idem omnino inſequetur, hoc eſt interceptam YX minorem adhuc eſſe
DS: tales ergo interceptæ quò magis à tangente EA remouentur continuè
decreſcunt. Quare ſectiones ABC, DEF ſunt ſemper ſimul accedentes. Quod ſecundò, & c.

Præterea, ſi ad euitandam in hiſce figuris linearum implicationem, con-
cipiatur circumſcriptæ Hyperbolæ ABC centrum eſſe I, aſymptoton IG, & contingens ex vertice BG; at inſcriptæ DEF centrum L, aſymptoton LH,
contingens autem ex vertice ſit EH: cum harum ſectionum latera ſint data
æqualia, erunt quoque ipſorum rectangula inter ſe æqualia, ideoque, & eo-
rum ſubquadrupla hoc eſt quadrata contingentium BG, EH, vnde ipſæ li- neæ BG, EH æquales erunt, ſed eſt etiam BI æqualis EL (nam vtra eſt dimi-
dium æqualium verſorum laterum) quare in triangulis IBG, LEH, cum ſint
latera IB, BG, lateribus LE, EH æqualia, & anguli ad B, E æquales, etiam
anguli ad baſes I, L æquales erunt, vnde aſymptoti IG, LG inter ſe æqui-
diſtant; & cum ſit à puncto L, quod eſt intra angulum ab aſymptotis cir-
cumſcriptæ ſectionis factũ, ducta LH alteri aſymptoto IK æquidiſtans, pro-
ducta ſecabit omnino Hyperbolen ABC: quare LH aſymptotos inſcriptæ ſecat Hyperbolen circumſcriptam; ſecet ergo in 1, per quod applicetur
2 1 3: Dico harum ſectionum interuallum infra applicatam 2 1 3 per in-

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