Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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9672 in ipſa DEF, quocunque puncto D, per ipſum ordinatim applicetur ODS,
alteram ſectionem ſecans in S:
erit quadratum SO æquale rectãgulo 111. huius.& quadratum DO rectangulo OEH, ſed rectangulum OBG maius eſt rectã-
gulo OEH, cum latitudo BG æqualis ſit EH, altitudo verò BO maior EO,
quare SO quadratum, maius eſt quadrato DO;
vnde punctum D cadit intra
Parabolen BA, &
ſic de quibuslibet alijs pũctis Parabolæ DEF; ergo huiuſ-
modi ſectiones inter ſe nunquam conueniunt.
Quod primò, & c.
Has tamen dico, licet in infinitum productas, infra contingentem EA ad
ſe propiùs accedere:
Ducta enim DM parallela ad ON, & per M applicata
MN, fiet parallelogrammum DN, cuius oppoſita latera MN, DO æqualia
erunt.
Iam quadratum MN, ſiue rectangulum NBG æquatur 22ibidem. DO, ſiue rectangulo OEH, ſed horum latera BG, EH æqualia ſunt, 33ibidem.& latera BN, EO æqualia erunt: itaque per proſtaphereſim, erit BE æqualis
NO, ſed eſt quoque MD æqualis eidem NO, igitur BE, &
MD inter ſe ſunt
65[Figure 65] æquales, at ſunt quoque parallelæ, igitur coniunctæ BM, &
ED æquales
erunt, &
parallelæ, ſed BM ſecat NM, quare producta ſecabit quoque alte-
ram parallelarum OD, ſed extra ſectionem BMA (cum ſit BM intra ſectio-
nem, producta verò, tota cadat extra) ſit ergo occurſus cum ODS in P, &

cum contingente EA in T;
& in ſecunda figura, in qua contingens EA cadit
inter applicatas MN, OS, iungatur SM ſecans EA in V.
Iam in prima figura, cum in parallelogrammo PE latera ET, DP, ſint æ-
qualia, ſitque EA maius ET, erit EA quoque maius DP, eſtque DP maius
intercepto ſegmento DS, quare AE, eò maius erit ipſo DS.
In ſecunda au-
tem figura, cum pariter ET, DP ſint æquales, ſitque ablata TV minor abla-
ta PS, erit reliqua EV maior reliqua DS, &
eò magis EA maior eadem DS.
Non abſimili modò oſtendetur quamcunque interceptam XY infra SD, mi-
norem eſſe ipſa SD;
ducta enim YZ æquidiſtanter EB, demonſtrabitur item
YZ æqualis eidem BE, ideoque YZ, &
DM inter ſe æquales erunt, &

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