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342 COMMENTARIA ANTIQUA. za TM, MZ, rcc aga ano AHN x^v ano SHO v7t£Qe%ov6L xolq ^y^q, A' B' yv(6^o6LV. >:al eTtsl l'6ov iati t6 HZ ta ^Sl, t6 ds EK tS ^P, oC ^q, A' B' yvcD^ovsg i'6oi d6l rc3 ts ZB xal ta AQ. t6 8s ^ A0 tui ZA l'aov, ta ds ZA, ZB l'aa iotl ta dlg VTtb ASN, tovts6Tiv vTto AON' ta aga anh tciv AHN, tovts6ti ta AM, MN, tcjv aTto SHO^ tovts6ti t(DV TM, MZ, v7tSQS%SL ta dlg vTto NSA rjtoi totg AZ, ZB. 10 Etg to la, Avvatov i6tL tovto tb dscoQTj^a dsi^at ofiotcog ta TtQO avtov TtOLOvvtag tag dvo svdstag ^Lag to^fjg icpditts^^^aL' aAA' iTtSLdtj Ttdvtfj tavtbv riv tcj inl trjg ^Lccg vTtSQ^oXijg TtQodsdsiy^isvG), avtrj fj dTtodst^tg 15 d7tsXs%d-ri. Elg t6 ky'. "E6tL Tial dXkcog tovto to d-scoQrj^a dst^ai' idv yaQ iTtL^sv^coiisv tdg FA, AZ, icpdilJOvtaL tav to^o5v dLa T« dsdsLy^sva iv ttp ^' tov /3' ^l^Uov. 20 iTtsl ovv ''Aklag tb kd'. "E6to VTtSQPoXrj 71 AB Tcal d6v^7ttcjt0L aC FAE Tcal itpaTtto^svri 7} FBE xal ^taQaXXrjloL ai FAH, ZBH. Xiyo, otL l'6rj r} FA trj AH. 1. AHN] AHM Wp; AH, HN Comm. 2. A' B'] ccB W, or^ p. naC] supra scr. p insl naC p 3. i^tiv W. A''b'] a B W, a§ p. 4. sCglv W. Post rs litt. del. p. 5. ZB] AZB Wp. iGtiv W. rwj corr. ex rd W. d^g] ds Wp, corr. Halley. 6. AZN ^' 7. tovzsgtlvW. ^HO] SH@ Wp; ISH, HO Comm. ^ xovtsgxiv W. 14. Post vnsQ^oXrjg una litt. del. p. 15. dnsXix^i]] Halley, dns- Xsyx^n W, dnriXsyx^^ P- 17. sariv W. 18. FA] scripsi,
20. EUTOCII COMMENTARIA IN CONICA. 343 A/1 = ZN = AT = 0B. quoniam igitur AH^ + HN^ = AM+ MN €t ;s;h^ + HO^ = TM + MZ, erit AW + HN^ = ^H^ + HO^ + ^q + A'B\ et quoniam est HZ = OSl^ UK = OPj erunt gnomones ^q +A'B'=ZB + A0. est autem A^=ZA, et ZB + ZA = 2A^ X SN = 2A0 X ON ergo AH^ + HN^ (siue AM + MN) = SW + HO^ (siue TM + MZ) + 2NS X SA (siue AZ + ZB). Ad prop. XXXI. Fieri -potest, ut haec propositio similiter demonstretur ac praecedens, si utramque rectam eandem sectionem contingentem fecerimus; sed quoniam prorsus idem erat, ac quod in una hyperbola antea demonstratum est [III^ 30], hanc demonstrationem elegimus. Ad prop. XXXIII. Haec propositio etiam aliter demonstrari potest: si enim FA, AZ duxerimus, sectiones contingent propter ea, quae in prop. XL libri II demonstrata sunt. quoniam igitur. . . . Aliter prop. XXXIV. Sit hyperbola AB, asymptotae FA, zJE, contingens FBE, parallelae FAH, ZBH dico, esse FA = AH. TA Wp. . Post ovv magnam lacunam Wp. 23. FBE] IIBE Wp, corr. Comm. rAH] A corr. ex z^ m. 1 W; rJH, H e corr., p. 24. ZBH] ZHB Wp, corr. Comm.
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20.<br />
EUTOCII COMMENTARIA IN CONICA. 343<br />
A/1 = ZN = AT = 0B. quoniam igitur<br />
AH^ + HN^ = AM+ MN<br />
€t ;s;h^ + HO^ = TM + MZ, erit<br />
AW + HN^ = ^H^ + HO^ + ^q + A'B\<br />
et quoniam est HZ = OSl^ UK = OPj erunt gnomones<br />
^q +A'B'=ZB + A0. est autem A^=ZA,<br />
et ZB + ZA = 2A^ X SN = 2A0 X ON ergo<br />
AH^ + HN^ (siue AM + MN) = SW + HO^ (siue<br />
TM + MZ) + 2NS X SA (siue AZ + ZB).<br />
Ad prop. XXXI.<br />
Fieri -potest, ut haec propositio similiter demonstretur<br />
ac praecedens, si utramque rectam eandem<br />
sectionem contingentem fecerimus; sed quoniam prorsus<br />
idem erat, ac quod in una hyperbola antea demonstratum<br />
est [III^ 30], hanc demonstrationem elegimus.<br />
Ad prop. XXXIII.<br />
Haec propositio etiam aliter demonstrari potest:<br />
si enim FA, AZ duxerimus, sectiones contingent<br />
propter ea, <strong>quae</strong> in prop. XL libri II demonstrata<br />
sunt. quoniam igitur. . . .<br />
Aliter prop. XXXIV.<br />
Sit hyperbola AB, asymptotae FA, zJE, contingens<br />
FBE, parallelae FAH, ZBH dico, esse FA = AH.<br />
TA Wp. . Post ovv magnam lacunam Wp. 23. FBE]<br />
IIBE Wp, corr. Comm. rAH] A corr. ex z^ m. 1 W;<br />
rJH, H e corr., p. 24. ZBH] ZHB Wp, corr. Comm.
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